Thermal Convection: Patterns, Evolution and Stability

Thermal Convection Thermal Convection: Patterns, Evolution and Stability MARCELLO LAPPA Naples, Italy A John Wiley ...

Author: Marcello Lappa

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Thermal Convection

Thermal Convection: Patterns, Evolution and Stability

MARCELLO LAPPA Naples, Italy

A John Wiley and Sons, Ltd., Publication

This edition ﬁrst published 2010  2010 John Wiley & Sons Registered ofﬁce John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial ofﬁces, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identiﬁed as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. The publisher and the author make no representations or warranties with respect to the accuracy or completeness of the contents of this work and speciﬁcally disclaim all warranties, including without limitation any implied warranties of ﬁtness for a particular purpose. This work is sold with the understanding that the publisher is not engaged in rendering professional services. The advice and strategies contained herein may not be suitable for every situation. In view of ongoing research, equipment modiﬁcations, changes in governmental regulations, and the constant ﬂow of information relating to the use of experimental reagents, equipment, and devices, the reader is urged to review and evaluate the information provided in the package insert or instructions for each chemical, piece of equipment, reagent, or device for, among other things, any changes in the instructions or indication of usage and for added warnings and precautions. The fact that an organization or Website is referred to in this work as a citation and/or a potential source of further information does not mean that the author or the publisher endorses the information the organization or Website may provide or recommendations it may make. Further, readers should be aware that Internet Websites listed in this work may have changed or disappeared between when this work was written and when it is read. No warranty may be created or extended by any promotional statements for this work. Neither the publisher nor the author shall be liable for any damages arising herefrom.

Library of Congress Cataloging-in-Publication Data Lappa, Marcello. Thermal convection : patterns, evolution and stability (historical background and current status) / Marcello Lappa. p. cm. Includes bibliographical references and index. ISBN 978-0-470-69994-2 (cloth) 1. Thermal conductivity. 2. Density currents. 3. Viscous ﬂow. 4. Fluid dynamics. I. Title. TA418.54.L37 2009 620.1 1296–dc22 2009025407 A catalogue record for this book is available from the British Library. 978-0-470-69994-2 Typeset in 9/11pt Times-Roman by Laserwords Private Limited, Chennai, India. Printed and bound in Singapore by Fabulous Printers Private Ltd

To a red rose . . . to my sons

Contents

Preface Acknowledgements 1

Equations, General Concepts and Methods of Analysis 1.1

1.2

1.3 1.4 1.5

1.6

1.7

1.8

Pattern Formation and Nonlinear Dynamics 1.1.1 Some Fundamental Concepts: Pattern, Interrelation and Scale 1.1.2 PDEs, Symmetry and Nonequilibrium Phenomena The Navier–Stokes Equations 1.2.1 A Satisfying Microscopic Derivation of the Balance Equations 1.2.2 A Statistical Mechanical Theory of Transport Processes 1.2.3 The Continuity Equation 1.2.4 The Momentum Equation 1.2.5 The Total Energy Equation 1.2.6 The Budget of Internal Energy 1.2.7 Newtonian Fluids 1.2.8 Some Considerations About the Dynamics of Vorticity 1.2.9 Incompressible Formulation of the Balance Equations 1.2.10 Nondimensional Form of the Equations for Thermal Problems Energy Equality and Dissipative Structures Flow Stability, Bifurcations and Transition to Chaos Linear Stability Analysis: Principles and Methods 1.5.1 Conditional Stability and Inﬁnitesimal Disturbances 1.5.2 The Exponential Matrix and the Eigenvalue Problem 1.5.3 Linearization of the Navier–Stokes Equations 1.5.4 A Simple Example: The Stability of a Parallel Flow with an Inﬂectional Velocity Proﬁle 1.5.5 Weaknesses and Limits of the Linear Stability Approach Energy Stability Theory 1.6.1 A Global Budget for the Generalized Disturbance Energy 1.6.2 The Extremum Problem Numerical Integration of the Navier–Stokes Equations 1.7.1 Vorticity Methods 1.7.2 Primitive Variables Methods Some Universal Properties of Chaotic States 1.8.1 Feigenbaum, Ruelle–Takens and Manneville–Pomeau Scenarios

xv xix 1 1 2 4 6 6 7 9 10 11 13 13 15 18 19 21 25 27 27 28 30 32 35 36 36 39 40 41 42 46 46

viii

Contents

1.8.2 1.8.3 1.8.4 1.8.5 1.9

Phase Trajectories, Attractors and Strange Attractors The Lorenz Model and the Butterﬂy Effect A Possible Quantiﬁcation of SIC: The Lyapunov Spectrum The Mandelbrot Set: The Ubiquitous Connection Between Chaos and Fractals The Maxwell Equations

2 Classical Models, Characteristic Numbers and Scaling Arguments 2.1 2.2

2.3

2.4

2.5

Buoyancy Convection and the Boussinesq Model Convection in Space 2.2.1 A Deﬁnition of Microgravity 2.2.2 Experiments in Space 2.2.3 Surface Tension-driven Flows 2.2.4 Acceleration Disturbances on Orbiting Platforms and Vibrational Flows Marangoni Flow 2.3.1 The Genesis and Relevant Nondimensional Numbers 2.3.2 Microzone Facilities and Microscale Experimentation 2.3.3 A Paradigm Model: The Liquid Bridge Exact Solutions of the Navier–Stokes Equations for Thermal Problems 2.4.1 Thermogravitational Convection: The Hadley Flow 2.4.2 Marangoni Flow 2.4.3 Hybrid States 2.4.4 General Properties 2.4.5 The Inﬁnitely Long Liquid Bridge 2.4.6 Inclined Systems Conductive, Transition and Boundary-layer Regimes

3 Examples of Thermal Fluid Convection and Pattern Formation in Nature and Technology 3.1

3.2 3.3

3.4

Technological Processes: Small-scale Laboratory and Industrial Setups 3.1.1 Crystal Growth from the Melt: Typical Techniques 3.1.2 Detrimental Effects Induced by Convective Phenomena Examples of Thermal Fluid Convection and Pattern Formation at the Mesoscale Planetary Structure and Dynamics: Convective Phenomena 3.3.1 Earth’s ‘Layered’ Structure 3.3.2 Earth’s Mantle Convection 3.3.3 Plate Tectonics Theory 3.3.4 Earth’s Core Convection 3.3.5 The Icy Galilean Satellites Atmospheric and Oceanic Phenomena 3.4.1 A Fundamental Model: The Hadley Circulation 3.4.2 Mesoscale Shallow Cellular Convection: Collection of Clouds and Related Patterns

47 48 51 53 58 63 64 66 66 67 68 68 70 71 75 75 78 80 80 83 83 85 86 89

95 95 96 101 103 103 103 104 104 106 107 108 108 110

Contents

3.4.3 3.4.4 4

112 116

Thermogravitational Convection: The Rayleigh–B´enard Problem

119

4.1

119 119 122 124 125 127 133 135 138 142 149 151 151 155

4.2

4.3 4.4 4.5 4.6 4.7 4.8

4.9

4.10

4.11

4.12 4.13

5

The Planetary Boundary Layer Atmospheric Convection in Other Solar System Bodies

ix

Nonconﬁned Fluid Layers and Ideal Straight Rolls 4.1.1 The Linearized Problem: Primary Convective Modes 4.1.2 Systems Heated from Above: Internal Gravity Waves The Busse Balloon 4.2.1 Toroidal–Poloidal Decomposition 4.2.2 The Zoo of Secondary Modes Some Considerations About the Role of Dislocation Dynamics Tertiary and Quaternary Modes of Convection Spoke Pattern Convection Spiral Defect Chaos, Hexagons and Squares Convection with Lateral Walls Two-dimensional Models 4.8.1 Distinct Modes of Convection and Possible Symmetries 4.8.2 Higher Modes of Convection and Oscillatory Regimes Three-dimensional Parallelepipedic Enclosures: Classiﬁcation of Solutions and Possible Symmetries 4.9.1 The Cubical Box 4.9.2 The Onset of Time Dependence The Circular Cylindrical Problem 4.10.1 Moderate Aspect Ratios: Azimuthal Structure and Effect of Lateral Boundary Conditions 4.10.2 Small Aspect Ratios: Targets and PanAm Textures Spirals: Genesis, Properties and Dynamics 4.11.1 The Archimedean Spiral 4.11.2 Spiral Wavenumber 4.11.3 Multi-armed Spirals and Spiral Core Instability From Spirals to SDC: The Extensive Chaos Problem Three-dimensional Convection in a Spherical Shell 4.13.1 Possible Patterns of Convection and Related Symmetries 4.13.2 The Heteroclinic Cycles 4.13.3 The Highly Viscous Case 4.13.4 The Geodynamo Problem

157 160 161 165 165 170 173 175 175 176 179 182 183 183 185 188

The Dynamics of Thermal Plumes and Related Regimes of Motion

195

5.1 5.2

195 196 197 198 198 200 200

Introduction Free Plume Regimes 5.2.1 The Diffusive–Viscous Regime 5.2.2 The Viscous–Nondiffusive Regime 5.2.3 The Inviscid–Diffusive Regime 5.2.4 The Inviscid–Nondiffusive Regime 5.2.5 Sinuous Instabilities Created by Horizontal Shear

x

Contents

5.3

5.4

5.2.6 Geometric Constraints The Flywheel Mechanism: The ‘Wind’ of Turbulence 5.3.1 Upwelling and Downward Jets and Alternating Eruption of Thermal Plumes 5.3.2 Geometric Effects 5.3.3 The Origin of the Large-scale Circulation: The Childress and Villermaux Theories 5.3.4 The Role of Thermal Diffusion in Turbulent Rayleigh–B´enard Convection Multiplume Conﬁgurations Originated from Discrete Sources of Buoyancy

6 Systems Heated from the Side: The Hadley Flow 6.1

6.2

6.3 6.4 6.5

The Inﬁnite Horizontal Layer 6.1.1 The Hadley Flow and its General Perturbing Mechanisms 6.1.2 Hydrodynamic Modes and Oscillatory Longitudinal Rolls 6.1.3 The Rayleigh Mode 6.1.4 Competition of Disturbances and Tertiary Modes of Convection Two-dimensional Horizontal Enclosures 6.2.1 Geometric Constraints and Multiplicity of Solutions 6.2.2 Instabilities Originating from Boundary Layers and Patterns with Internal Waves The Inﬁnite Vertical Layer: Cats-eye Patterns and Temperature Waves Three-dimensional Parallelepipedic Enclosures Cylindrical Geometries under Various Heating Conditions

7 Thermogravitational Convection in Inclined Systems 7.1

7.2

Inclined Layer Convection 7.1.1 The Codimension-two Point 7.1.2 Tertiary and High-order Modes of Convection Inclined Side-heated Slots 7.2.1 Stationary Longitudinal Long-wavelength Instability 7.2.2 Stationary Transversal Instability 7.2.3 Oscillatory Transversal Long-wavelength Instability 7.2.4 Stationary Longitudinal Short-wavelength Instability 7.2.5 Oscillatory Longitudinal Instability 7.2.6 Interacting Longitudinal and Transversal Multicellular Modes

8 Thermovibrational Convection 8.1 8.2 8.3 8.4 8.5

Equations and Relevant Parameters Fields Decomposition The TFD Distortions High Frequencies and the Thermovibrational Theory States of Quasi-equilibrium and Related Stability 8.5.1 The Vibrational Hydrostatic Conditions

201 202 203 204 205 208 208 215 215 216 219 223 225 228 228 235 247 253 262 271 272 273 275 279 281 282 284 284 284 286 289 289 290 291 293 294 294

Contents

8.6 8.7

9

8.5.2 The Linear Stability Problem 8.5.3 Solutions for the Inﬁnite Layer Primary and Secondary Patterns of Symmetry Medium and Low Frequencies: Possible Regimes and Flow Patterns 8.7.1 Synchronous, Subharmonic and Nonperiodic Response 8.7.2 Reduced Equations and Related Ranges of Validity

xi

295 297 299 303 303 305

Marangoni–B´enard Convection

317

9.1 9.2 9.3 9.4

317 320 325 326 328 331 334

9.5

Introduction High Prandtl Number Liquids: Patterns with Hexagons, Squares and Triangles Liquid Metals: Inverted Hexagons and High-order Solutions Effects of Lateral Conﬁnement 9.4.1 Circular Containers 9.4.2 Rectangular Containers Temperature Gradient Inclination

10 Thermocapillary Convection 10.1 10.2

10.3 10.4

Basic Features of Steady Marangoni Convection Stationary Multicellular Flow and Hydrothermal Waves 10.2.1 Basic Velocity Proﬁles: The Linear and Return Flows 10.2.2 Linear Stability Analysis 10.2.3 Weakly Nonlinear Analysis 10.2.4 Boundary Effects: 2D and 3D Numerical Studies Annular Conﬁgurations The Liquid Bridge 10.4.1 Historical Perspective 10.4.2 Liquid Metals and Semiconductor Melts 10.4.3 The First Bifurcation: Structure of the Secondary 3D Steady Flow 10.4.4 Effect of Geometric Parameters 10.4.5 A Generalized Theory for the Azimuthal Wavenumber 10.4.6 The Second Bifurcation: Tertiary Modes of Convection 10.4.7 High Prandtl Number Liquids 10.4.8 Standing Waves and Travelling Waves 10.4.9 Symmetric and Asymmetric Oscillatory Modes of Convection 10.4.10 System Dynamic Evolution 10.4.11 The Hydrothermal Mechanism in Liquid Bridges 10.4.12 Noncylindrical Liquid Bridges 10.4.13 The Intermediate Range of Prandtl Numbers

11 Mixed Buoyancy–Marangoni Convection 11.1 11.2 11.3 11.4

The Canonical Problem: The Inﬁnite Horizontal Layer Finite-sized Systems Filled with Liquid Metals Typical Terrestrial Laboratory Experiments with Transparent Liquids The Rectangular Liquid Layer

341 342 345 346 346 354 359 368 375 375 378 379 381 389 390 393 399 407 412 417 421 423 427 429 436 449 450

xii

Contents

11.4.1 11.4.2

11.5

11.6

11.7

11.8

Waves and Multicellular Patterns Tertiary Modes of Convection: OMC and HTW with Spatiotemporal Dislocations Effects Originating from the Walls 11.5.1 Lateral Boundaries as a Permanent Stationary Disturbance 11.5.2 Collision Phenomena of HTW and Wall-generated Steady Patterns 11.5.3 Streaks Generated by a Lift-up Process and Instabilities of a Mechanical Nature The Open Vertical Cavity 11.6.1 Volume Driving Actions and Rising Thermal Plumes 11.6.2 Aiding Marangoni and Buoyant Flows 11.6.3 Counteracting Driving Forces and Separation Phenomena 11.6.4 Surface Driving Actions and Vertical Temperature Gradients The Annular Pool 11.7.1 Target-like Wave Patterns (HW2 ) 11.7.2 Waves with Spiral Pattern (HW1 ) 11.7.3 Stationary Radial Rolls 11.7.4 Progression Towards Chaos and Fractal Behaviour 11.7.5 The Reverse Annular Conﬁguration: Incoherent Spatial Dynamics 11.7.6 Some Considerations About the Role of Curvature, Heating Direction and Gravity The Liquid Bridge on the Ground 11.8.1 Microscale Experiments 11.8.2 Heating from Above or from Below 11.8.3 The Route to Aperiodicity

12 Hybrid Regimes with Vibrations 12.1 12.2

12.3 12.4

12.5

12.6

RB Convection with Vertical Shaking Complex Order, Quasi-periodic Crystals and Superlattices 12.2.1 Purely Harmonic Patterns 12.2.2 Purely Subharmonic Patterns 12.2.3 Coexistence and Complex Order RB Convection with Horizontal or Oblique Shaking Laterally Heated Systems and Parametric Resonances 12.4.1 The Inﬁnite Horizontal Layer 12.4.2 Domains with Vertical Walls 12.4.3 The Inﬁnite Vertical Layer 12.4.4 Inclined Systems Control of Thermogravitational Convection 12.5.1 Cell Orientation as a Means to Mitigate Convective Disturbances on Orbiting Platforms 12.5.2 Control of Convection Patterning and Intensity in Shallow Enclosures 12.5.3 Modulation of Thermal Boundary Conditions Mixed Marangoni–Thermovibrational Convection 12.6.1 Basic Solutions

450 456 458 459 460 464 468 470 470 472 474 475 476 478 480 483 487 488 491 492 499 510 517 519 525 527 529 529 533 538 538 544 548 550 550 551 553 559 561 561

Contents

12.7

12.6.2 Control of Convection Patterning and Intensity in Shallow Enclosures 12.6.3 Control of Hydrothermal Waves Modulation of Marangoni–B´enard Convection

13 Flow Control by Magnetic Fields 13.1

13.2

13.3 13.4

Static and Uniform Magnetic Fields 13.1.1 Physical Principles and Governing Equations 13.1.2 Hartmann Boundary Layers Historical Developments and Current Status 13.2.1 Stabilization of Thermogravitational Flows 13.2.2 Stabilization of Surface Tension-driven Flows Rotating Magnetic Fields Gradients of Magnetic Fields and Virtual Microgravity

xiii

566 567 575 581 582 582 584 584 584 597 604 607

References

609

Index

659

Preface

Most of the ﬂuid motion we are accustomed to on Earth is driven by gravity. The presence of Earth creates a gravitational ﬁeld that acts to attract objects with a force that is inversely proportional to the square of the distance between the mass centre of the object and the centre of Earth. A very common example of gravity’s impact on ﬂuids is the creation of ﬂows around our bodies, around the ﬂame of a candle, in a container of water heated from below or from the side and in atmospheric and oceanic circulation at every scale. The presence of ﬂows of gravitational origin is not limited to ﬂuids that affect our lives every day. They are also found inside planetary bodies. This is the reason why, for instance, continents ‘move’ (the ‘solid’ Earth itself undergoes a ﬂuid-like internal circulation on time-scales of millions of years, the surface expression of which is continental drift) and a magnetic ﬁeld is present around our planet (as a consequence of liquid metal motion in the Earth ‘core’). Gravitational attraction is a fundamental property of matter that exists throughout the known universe; hence ﬂuid motion of a gravitational origin also occurs in and around other celestial bodies and is presumed to play an important role in the dynamics of stars like the Sun. Instability of such ﬂows and their transition to turbulence are widespread phenomena in the natural environment at several scales and are at the root of typical problems in meteorology, oceanography, geophysics and astrophysics. The possible origin of natural ﬂows, however, is not limited to the action of the gravitational force. Other volume or ‘surface’ forces may be involved in the process related to the generation of ﬂuid motion and ensuing evolutionary progress. In particular, in the presence of a free interface (e.g. a surface separating two immiscible liquids or a liquid and a gas), surface tension-driven convection (also referred to as ‘Marangoni’ ﬂow) may arise as a consequence of temperature or concentration gradients. In such a context, it should be stressed that the universal nature of all these ﬂuid phenomena makes their study fundamental not only to science, but also to engineering and industrial practical applications (e.g. the processing of metal alloys and inorganic or organic emulsions, cooling systems, the production of semiconductor crystals and various biological and biotechnological processes). The study of these topics has extensive background application in many ﬁelds. As a relevant and important example, most widely used technologies for single-crystalline materials (e.g. horizontal and vertical Bridgman growth, Czochralski method, ﬂoating-zone technique) are affected by the presence of ﬂuid convection. All conventional melt growth conﬁgurations require, in fact, the application of thermal gradients across the phase boundary: the axial and/or radial components of these gradients are destabilizing and provide driving forces for free convection in all ﬂuid phases involved. Melt growth processes are, therefore, subject to varying heatand mass-transfer conditions, which in recent years have been found to be directly or indirectly responsible for most bulk deﬁciencies in many materials. In particular, instabilities of the melt ﬂow usually lead to three-dimensional oscillatory effects which strongly affect the quality of the growing crystals at microscopic scale length and therefore are very undesirable.

xvi

Preface

Some of these effects are known to be independent of gravity, that is, they are related to the other types of forces mentioned before. Along these lines, it is worth mentioning that (because in many circumstances the inﬂuence of gravity on ﬂuids is strong and masks or overshadows these important factors), a number of experiments have been carried out in recent years on orbiting platforms (the so-called ‘microgravity’ conditions). The peculiar behaviour of physical systems in space, and ultimately the interest in this ‘new’ environment, have come from the virtual disappearance of the gravity forces and related effects mentioned above and the appearance of phenomena unobservable on Earth, especially those driven by surface forces (that become largely predominant when terrestrial gravity is removed). The use of such an environment has also led, however, to the identiﬁcation of a new type of ﬂuid motion induced by the presence of ‘vibrations’ of the considered orbiting platform (usually referred to as g-jitters). This kind of convection, initially studied due to its perturbing and undesired inﬂuence on microgravity experiments, has recently witnessed renewed interest due to its possible application in terrestrial conditions as a means to ‘control’ ﬂow intensity and patterning in other types of convection (as a possible variant to the use of magnetic ﬁelds traditionally employed for such a purpose).

Aims and Scope As a natural consequence of all the arguments illustrated above, the present book is devoted to a critical, focused and ‘comparative’ study of all these different types of thermal convection. Gravitational (also referred to in the literature as ‘natural’ or buoyancy), surface tension-driven, vibrational and magnetic ﬂows are considered in various geometric models (inﬁnite horizontal and vertical layers, open and closed geometries, shallow and tall cavities, cubic and parallelepiped slots, annular and spherical conﬁgurations, cylindrical enclosures, ﬂoating zones, liquid bridges, etc., many of which have enjoyed widespread use over recent years as ‘paradigm‘ models for the study of these topics), under various heating conditions (from below, from above or from the side), for different ﬂuids (liquid metals, molten salts and semiconductors, gases, water, oils, many organic and inorganic transparent liquids, etc.) and possible combinations of all these variants. A signiﬁcant effort is provided to illustrate the genesis of these kinds of ﬂows, the governing nondimensional parameters, the scaling properties, their structure and, in particular, the stability behaviour and the possible bifurcations to different patterns of symmetry and/or spatiotemporal regimes. The book presents, in fact, a discussion of the main modes of two- and three-dimensional ﬂows, pattern defects and the scenarios of convection-regime changes (together with the related transitional stages of evolution). To name some examples: striped patterns, various types of planforms (related to Rayleigh–B´enard or Marangoni–B´enard convection), textures (hexagons, squares, triangles, diamonds, spirals, panam structures, targets, spoke pattern), rhombic, square and star-like ‘lattices’ or ‘super-lattices’ (in vibrational convection), multiplume and multicellular conﬁgurations, cats-eye structures, patterns exhibiting the shape of a ‘ﬂower’, a variety of symmetry-breaking effects, and so on. A categorization and description of many kinds (both canonical and ‘exotic’) of instability are provided; to name just a few: Eckhaus, oscillatory skewed varicose, cross-roll, bimodal, the Busse oscillatory instability, zig-zag, knot, oscillatory blob, spiral-defect chaos, transverse hydrodynamic modes, oscillatory longitudinal rolls, transverse, longitudinal and oblique hydrothermal waves, steady and oscillatory multicellular ﬂows, pulsating and rotating regimes, and so on, with the related discussion not limited to the ﬁrst bifurcation of the ﬂow, but also considering secondary, tertiary and high-order states. Some emphasis is also given to the transition to chaos, related theories and possible means of ﬂow control.

Preface

xvii

The analysis, moreover, does not cover only the cases in which all these types of convection (thermogravitational, thermocapillary, thermovibrational) act separately. Signiﬁcant space is also devoted to elucidate the possible ‘interplay’ of several effects in situations where driving forces of different nature are simultaneously responsible for the generation of ﬂuid motion. This subject (hybrid or mixed convection) is of particular importance as the identiﬁcation of the most dominant mechanism and/or the mutual interference of different mechanisms involved with a comparable intensity may help researchers in elaborating rational guidelines relating to physical factors that can increase the probability of success in practical technological processes. A number of existing analyses are reviewed and discussed through a focused and critical comparison of experimental and numerical results and theoretical arguments introduced over the years by investigators to explain the observed phenomena. The text has elicited information from about 100 of the author’s relevant and recent papers and about 1000 analyses available in the literature to illustrate possible approaches to the considered problems, practical applications and the ensuing insights into the physics. A deductive approach is followed with systems of growing complexity being treated as the discussion progresses. The book, however, is not limited to a systematic survey of landmark and recent results in the literature. Speciﬁc experimental and numerical examples are conceived and presented to provide inputs for an increased understanding of the underlying ﬂuid ﬂow mechanisms. Of course, an important part of these examples is based on numerical simulations (CFD). This branch of ﬂuid dynamics complements experimental and theoretical ﬂuid dynamics by providing an alternative cost-effective means of simulating real processes. It offers the means of testing theoretical advances for conditions often unavailable experimentally or having a prohibitive cost. To summarize, the book progresses with the aid and support of both experimental results and numerical simulations for a better representation of the structure of convection and moves through very focused examples and situations, many of which are of a prototypical nature (some unpublished and heretofore unseen material is used to support the discussions). The declared objectives are: (a) to provide the reader with an ensemble picture of the subject (illustrating the state-of-the-art and providing researchers from universities and industry with a basis on which they are able to estimate the possible impact of a variety of parameters); (b) to clarify some still unresolved controversies pertaining to the physical nature of the dominant driving force responsible for asymmetric/oscillatory convection in various natural phenomena and/or technologically important processes; (c) to elucidate some unexpected theoretical kinships existing among ﬂuid-mechanical behaviours arising in different contexts (such a philosophy, in particular, being used in the attempt to build a common theoretical source for the community of ﬂuid physicists under the optimistic idea that an ongoing, mutually beneﬁcial dialogue is established among different branches of research in these ﬁelds). Each chapter of the book deals with a different aspect of the aforementioned topics, providing the necessary background information (i.e. literature, fundamental concepts, equations and mathematical models, information on the experimental and numerical techniques, etc.), focusing on the latest advances, describing in detail the insights into the physics provided by the experiments and/or numerical simulations and introducing (where necessary) theoretical and critical links with the other book chapters and related topics. As anticipated, the ﬁnal goal of such a treatise is to help the scientiﬁc community signiﬁcantly in elaborating and validating new, more complex models, in accelerating the current trend towards predictable and reproducible natural phenomena and ﬁnally in establishing an adequate scientiﬁc foundation to industrial processes which are still conducted on a largely empirical basis.

xviii

Preface

In practice, the text is conceived in order to be a useful reference guide for other specialists in these disciplines (including professionals in the metallurgy and foundry ﬁeld; researchers and scientists who are now coordinating their efforts to improve the quality of semiconductor or macromolecular crystals; organic chemists and materials scientists; and atmosphere and planetary physicists) and also an advanced-level text for students taking part in courses on the physics of ﬂuids, ﬂuid mechanics, the behaviour and evolution of nonlinear systems, environmental phenomena and materials engineering. It is directed at readers already engaged or starting to be engaged in these topics. Physicists, engineers, designers and students will ﬁnd the necessary information and revealing insights into the behaviour of many phenomena (including, as outlined before, both historical developments and very recent contributions). Finally, it is also worth pointing out that the study of pattern formation (convective ﬂows can form more or less ordered spatial structures) also falls under the broader heading of nonequilibrium phenomena. Beyond practical applications, it is therefore clear that these problems also exert an appeal to researchers and scientists as a consequence of the complexity of the possible stages of evolution, of the nonlinear behaviour and because these organized structures are aesthetically and philosophically pleasing as well as irresistible to theoretical physicists. This complexity is shared with other systems in Nature and constitutes a remarkable challenge for any theoretical model. Indeed, convection problems are a rich source of material propaedeutical to the development of new ideas concerning the relationship between order and chaos in ﬂuid dynamics and, in general, between simplicity and complexity in the structure and behaviour of systems governed by nonlinear equations. In view of the foregoing discussion, there is no doubt that elucidating the mechanisms for the formation and evolution of hydrodynamic structures can be regarded as a subject of paramount importance not only for the aforementioned meteorology, oceanography, astrophysics, geophysics and (on a smaller scale) crystal growth, the processing of metal alloys and a variety of other technological processes, but also from an ‘ideological’ synergetic point of view for further progress in the understanding of pattern-forming systems of different nature. Unlike earlier books on the subject, here, even if partial differential equations and related methods of solution are widely used in the text and CFD is actually at the root of many of the proposed examples, the heavy mathematical background underlying and governing the behaviours illustrated is kept to the minimum. Much of the available space is devoted to the description (both qualitative and quantitative) of the spatial and temporal convection structures, related thresholds in terms of characteristic numbers and to the ‘physics’. This is done under the optimistic hope that such a philosophy may signiﬁcantly increase the readability of the book and, in particular, make it understandable also to those individuals who are not ‘pure’ ﬂuid physicists or mathematicians. In the same spirit, the use of jargon is limited as much as possible and most of the mathematical arguments are concentrated in the ﬁrst chapter (this chapter is devoted to the description of the numerical algorithms used to perform the time integration, to compute directly the steady or oscillatory states and to investigate their stability), allowing readers who are not interested in these aspects to skip them and jump directly into the results. Marcello Lappa

Acknowledgements

This book is a composite of many ideas. It was authored between 2006 and 2009 in the pleasant atmosphere provided by my writing desk and warm lamp at home, especially in the evening and at night. It was originally conceived (in 2005) as an enriched version of Chapter 2 of my earlier monograph Fluids, Materials and Microgravity, published in 2004 by Elsevier Science, for which I was preparing a second edition. After writing about 100 pages, I realized, however, that the subject of thermal convection would deserve its own separate and exhaustive treatment. I gratefully acknowledge the many unknown reviewers selected by John Wiley & Sons, who initially examined the new book project, for their critical reading and valuable comments on the work. I wish also to express my special thanks to many colleagues around the world for generously sharing with me their precious recent experimental and numerical data (in alphabetical order): Prof. J. Iwan D. Alexander, Dr. I. Aranson, Prof. E. Bodenschatz, Prof. F. H. Busse, Dr. R. Delgado-Buscalioni, Dr. N. Garnier, Prof. A. Yu. Gelfgat, Prof. S. Hoyas, Prof. W. R. Hu, Prof. Y. Kamotani, Prof. H. Kuhlmann, Dr. P. Laure, Prof. G. Lebon, Dr. G. D. McBain, Prof. J. Mizushima, Prof. M. Paul, Dr. C. Piccolo, Dr. B. Plapp, Dr. J. Priede, Prof. D. Schwabe, Prof. V. Shevtsova, Dr. J. Stiller, Dr. Z. M. Tang, Dr. Lev S. Tsimring, Prof. I. Ueno and Prof. A. M. G. Zebib. I also acknowledge the Italian Aerospace Centre (CIRA), the Italian Inter-University Centre for Supercomputing (CINECA) for their kind help in efﬁciently using their parallel machines, the Microgravity Advanced Research and Support Center (MARS, Italy) whose laboratories were used for conducting some of the experiments, and the NASA, ESA and JAXA space agencies for making available some interesting data and pictures. In particular, I would like to express my deepest appreciation to Prof. F. H. Busse and Prof. W. Pesch (Institute of Physics, University of Bayreuth, Germany) for their valuable suggestions for Chapters 1 and 4 and Prof. E. J. Villamizar Roa (Escuela de Matem´aticas, Universidad Industrial de Santander-UIS, Colombia) for some minor but helpful comments about Chapter 1. Most of all, I am indebted to Prof. N. Imaishi (Kyushu University, Institute for Materials, Chemistry and Engineering, Division of Advanced Device Materials, formerly Department of Advanced Material Study, Fukuoka, Japan), who supported me in numerous stages of evolution of this book by providing groundbreaking articles, data, ﬁgures and also useful comments on Chapters 10 and 11. His help both with this work and in my past scientiﬁc career, especially during the period when I spent some months at Kyushu University in Japan, was invaluable and will be an ever-sweet souvenir in my life. Finally, I would like to mention that a signiﬁcant amount of the insights that I have tried to convey in this book resulted from the last 5 years of work I have done in the position of Editor-in-Chief of the journal Fluid Dynamics and Materials Processing (FDMP), which obliged me to keep myself informed on the latest advancements in the ﬁeld, to interact almost daily with

xx

Acknowledgements

article authors, reviewers, experts in various ﬁelds and other Editorial Board members, to whom collectively I also express my appreciation. As a concluding remark, let me also point out that the overt intention of including so many references (there are more than 1000) is to encourage readers and students to follow up on various details and, most importantly, not to limit their readings to the relatively synthetic and didactic account I have provided here. It is obvious that if one tries to survey the developments of the last 200 years, one cannot follow carefully all the twigs of the tree. It is also evident that one will possibly emphasize some results due to personal taste, interests and experiences. Let me apologize for this right at the beginning.

To contact the author: Marcello Lappa Via Salvator Rosa 53 San Giorgio a Cremano (Na) 80046 – Italy Email: [email protected] [email protected] [email protected] Websites: www.thermalconvection.net www.ﬂuidsandmaterials.com www.techscience.com/FDMP

1 Equations, General Concepts and Methods of Analysis 1.1

Pattern Formation and Nonlinear Dynamics

Regular structures arise everywhere in Nature and virtually every technological process involves their formation at some stage. By injecting energy into a dynamic system, typically an initial equilibrium state becomes unstable above a certain threshold and, as a result of this instability, well-deﬁned space–time structures emerge. Beyond the speciﬁc situation or system considered, these structures are characterized by a recognizable level of self-organization (i.e. a precise morphology and/or topology in space and/or lines of evolution in time) and under certain idealizations it is natural to consider the process leading to their formation as the life of the considered dynamic system. The features of this life as t → ∞ then determine the characteristic aspects of these structures, be they perfect or irregular. In general, there exist, between the limiting purely regular and irregular ﬁeld distributions in space (and/or time), numerous intermediate situations. One of the most remarkable achievements obtained in recent years is the discovery that these dynamics and the related transitional stages are largely determined by a sort of obscure dialectics between the tendency that every natural system exhibits towards order or disorder, self-organization or chaos. This seems to be an intrinsic feature of the way in which our Universe (and all the dynamic systems which are contained within it) works. Among other things, it also constitutes one of the most fascinating philosophical questions to which humankind is trying to ﬁnd a decisive answer. The fact that strikingly well-ordered and similar phenomena are found across disciplines is indeed an important impetus for research in this theoretical ﬁeld. It stands at the intersection of many scientiﬁc branches, which make it a multi-domain ﬁeld of investigations and a truly interdisciplinary science. The problem has always been widely open and has been approached from different directions and by different research groups with various backgrounds and perspectives. In particular, the similarity in fundamental mechanisms and the accompanying mathematics has brought together scientists from many ﬁelds, such as fundamental ﬂuid dynamics (e.g. Cross and Hohenberg, Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

2

Thermal Convection: Patterns, Evolution and Stability

1993), meteorology, oceanography, astrophysics, geophysics, material science (e.g. Langer, 1980), chemistry (e.g. Weaire and Rivier, 1984; Henisch, 1991), surface science (e.g. Zander et al., 1990), biology (e.g. Gierer and Meinhardt, 1972), medicine and so on (the reader is also referred to the discussions in the preface of this book). This synergy has led over the years to the establishment of a common, elegant theoretical framework that is now generally referred to as the ﬁeld of pattern formation or, in other acceptations, the study of the related stability and possible evolution. The above-mentioned commonality, whose most evident articulation has been over recent years the deﬁnition of general objectives and a general modus operandi (as discussed below), can be regarded as the spark at the root of the present work. The principal objectives of such research are (i) the analysis of the hierarchy of instabilities and the birth of various structures in the course of evolution from an initial state, (ii) the investigation of the mutual transformation of these structures as some control conditions are varied and (iii) understanding the cause-and-effect relationships at the root of the observed behaviours. The common modus operandi consists of a general way of thinking, which, from a more precise mathematical point of view, means the adoption of speciﬁc tools of analysis and techniques to be used when the dynamics of interest do not follow linear laws (i.e. are not characterized by a direct proportionality between cause and effect). Although the above arguments are often used narrowly to describe this ﬁeld, in general they may be applied to describe more or less everything that happens in the Universe. Hence these statements can hardly be used as rigorous deﬁnitions. In practice, this topic must be placed in a more precise theoretical context by introducing some necessary concepts and notions. Such a theoretical melange is propaedeutical to a better recognition, deﬁnition and characterization of the aforementioned phenomena. Also, these general considerations facilitate the subsequent introduction of more complex notions and will signiﬁcantly help the reader in the understanding of the theoretical explanations and arguments given throughout this chapter. Some of them, such as stability, instability and evolution, have been already used above without providing, however, an adequate basis (we shall come back to these later in this chapter). Other fundamental and propaedeutical ingredients are illustrated and elaborated in Section 1.1.1. Many of them are not independent of one another and the related relationships are difﬁcult to discern, which requires careful treatment.

1.1.1 Some Fundamental Concepts: Pattern, Interrelation and Scale Given the complexity of the considered topic, following the elegant approach of Bar-Yam (2001), it is convenient to start the discussion from the introduction of three simple (but illuminating) ideas only. The ﬁrst is the concept of pattern per se. The other ones are the deﬁnitions of interrelation and scale. To some extent, these concepts simplify the problem by abstracting from speciﬁc cases the features which are essential in the description of pattern formation. In the process of abstraction we get a more general problem in which thermal convection (the subject of this book) is just one effective realization. A pattern is

• a set of relationships that can be identiﬁed by observations of a system, or an ensemble of sub-systems

• a simple type of emergent property of a system, where a pattern is a feature of the system as a whole but does not apply to constituent sub-parts of the system.

• a property of a system by which the description of the system becomes relatively simple and short with respect to detailing the characteristics of its components.

Equations, General Concepts and Methods of Analysis

3

A simple type of pattern is a repetitive structure in space. Shifting the view by one repeat length leads to seeing the same thing (this may occur along a single direction or along more than one direction). Similarly to repeating patterns in space, we can also have a repeating pattern in time (this may occur in the form of a simple harmonic process or as the superposition of many of these behaviours with different amplitude and frequency). Generally, a pattern can have both features. We also think of patterns as prototypes or exemplars. This is the sense in which we use it to describe a given structure (in space or in time) with well-deﬁned features. In this case the pattern is not about the relationships within the structure, but about the possibility of repeating such a scenario many times in certain (well-deﬁned and reproducible) circumstances. The connection between the pattern as repetition and the pattern as prototype is just like the relationship between two types of properties: properties that appear as a consequence of the mutual interference among components of a system and properties that arise from interaction of a system with its environment (the larger system of which it is a part). Interrelation is

• what parts of a system do as a consequence of mutual interplay that they would not do by themselves: collective behaviour

• what a system does by virtue of interaction with its environment that it would not do by itself, for example its function. According to the ﬁrst statement, interrelation refers to understanding how ensemble properties arise from the cooperative behaviour of parts. More generally, it refers to how behaviour at a larger scale of the system arises from detailed structures and interdependencies on a ﬁner scale. In practice, it is about how a macroscopic scenario arises from microscopic behaviours (for interesting effective examples in various ﬁelds not covered by the present book, the reader is referred to, for example, Piccolo et al., 2002; Carotenuto et al., 2002; Lappa et al., 2002, 2003b, Lappa, 2002b, 2003c,d, 2005c, 2006c; Lappa and Castagnolo, 2003; and references cited therein). According to the second statement, interrelation refers to all the properties that we assign to a system due to interaction between it and its environment. In practice, the second aspect of interrelation may be linked theoretically to the ﬁrst aspect because the system can be viewed along with parts of its environment as together forming a larger system. The collective behaviours due to the relationships of the larger system’s parts reﬂect the relationships of the original system and its environment. In general, however, there is a tendency to separate expressly the interdependence between the components of a system that creates its recognizable identiﬁcation (i.e. the pattern) from its environment: The point of transition from the system to its environment is generally referred to as the ‘boundary’ of the system; such a boundary, together with its functions, that is, the related protocols of interaction with the external environment, are typically regarded as an additional property of the considered system. Scale is

• the size of a system or an appropriate reference quantity for a property that one is describing • the required precision of observation or description. A somewhat related concept is that of scaling or scalings that refer to some general analytical relationships which can be established between certain properties of the considered system and fundamental reference quantities (e.g. a length scale). Additional useful and important ideas such as dissipative structures, stability, bifurcation, uniqueness, multiplicity of solutions and attractor will be introduced in the following sections as required and with an increasing degree of complexity as the discussion progresses. It will be illustrated how these interwoven deﬁnitions can be applied to systems that are vastly different in their meaning, shape, scales and physics.

4

Thermal Convection: Patterns, Evolution and Stability

In particular, starting from the derivation of the governing (balance) equations of a dynamic system from the microscopic collective behaviour of its molecules, it will be shown how all these concepts have extensive background application at large scales (when discussing the properties of the natural patterns provided by laboratory or numerical experiments) and again at relatively small scales (both in space and time) when the considered system approaches a special condition known as ‘spatiotemporal chaos’ in which it exhibits an increasing degree of complexity and ﬁner (sub-)structures.

1.1.2 PDEs, Symmetry and Nonequilibrium Phenomena From a purely mathematical point of view, in typical pattern-formation phenomena organized structures are formed due to intrinsic nonlinearities of the considered system. It is a well-known and universally recognized concept that Nature does not follow a linear pattern, and linearity, if it exists in Nature, is a special case of nonlinearity. Trying to provide a deﬁnition for nonlinearity (or of nonlinear science) makes almost no sense given the excessive level of abstraction that would be required by such an attempt. From an intuitive standpoint, however, nonlinearity can be regarded as ‘a feedback loop’ acting as an intrinsic property of the system that feeds information back into the system where it is iterated or used multiplicatively. This feedback loop is created when, as explained before, the system parts are connected in a network of specialized functions. This leads to collective behaviours more complex than those of the individual constituent components; the related feedback loop and iterative process make the system extremely sensitive to its (even though very small) internal variations. It is by virtue of these mechanisms that these systems contain their own capacity for transformation (requiring only the right conditions for activation) and that we speak about nonlinear behaviour. In general, studies of pattern formation use a common set of fundamental concepts to describe how non-equilibrium processes cause structures to appear. The theoretical starting point is usually a set of deterministic equations governing the possible evolutionary progress of the considered system. Obviously, these equations, typically in the form of partial differential equations (PDEs), are nonlinear. As explained above, nonlinearity of these equations reﬂects how the system parts interfere with one another exchanging some kind of information. This, however, is not the only factor playing a signiﬁcant role. In canonical studies on these subjects the nonlinear model equations are often considered on ﬁnite spatial domains and (according to the earlier discussion on the concept of interrelation, Section 1.1.1) need a speciﬁcation for the system interaction with its environment. These interactions are generally modelled as additional mathematical constraints known as ‘boundary conditions’. Obviously, these conditions are also vital in determining the pattern-selection processes and underlying mechanisms. It is also worth noting that, beyond the mathematical form (i.e. the functional dependences relating the system properties to those of the environment) of the these protocols of interaction, the spatial shape of the system boundary per se can signiﬁcantly enter the dynamics (e.g. Lappa et al., 2002; Lappa, 2005a,b, 2006a). Along these lines, over the years simpliﬁed (easy to handle) conﬁgurations (where well-established parameters can be ﬁxed and the behaviour of the system in response to changes of these parameters can be investigated) have been conceived by researchers. In general, well-deﬁned geometric shapes make mathematical analysis and computations simpler and some boundary conditions are more attractive than others. For these reasons they have enjoyed widespread use in the deﬁnition and ensuing analysis of this subject.

Equations, General Concepts and Methods of Analysis

5

These simple geometric domains and mathematically friendly boundary conditions usually imply symmetry. Such symmetries may be present because of the domain geometry or as a result of some modelling assumptions (large systems are often discussed using periodic boundary conditions). Symmetry is a very important ingredient (together with nonlinearity) in pattern formation phenomena. The aforementioned partial differential equations are often invariant under some groups (G) of Euclidean transformations (translations, rotations and reﬂections of the physical space). Any PDE that is posed on a domain and is invariant under a group G will inherit those symmetries in G that preserve the domain and the boundary conditions (symmetries enter into problems of this type from the invariance properties of the governing equations and the shape of the boundary of the considered system, the container in the case of a ﬂuid). Remarkably, the existence of these symmetries implies the possibility of symmetry breaking, which is one of the fundamental concepts at the root of pattern formation phenomena (it is strictly associated, in particular, with some fundamental companion notions such as stability and bifurcation that will be treated later in this chapter together with related tools of analysis). Although much progress towards possible techniques for the integration of the governing equations and the study of stability and bifurcation of related solutions has been achieved in recent years, fundamental challenges remain, many of which are of a ‘philosophical’ or ‘archetypical’ nature. Systems that are driven out of equilibrium often show similar patterns, although the underlying processes can be quite different. One challenge is to ﬁnd measures that can quantitatively assess the similarity of different patterns. The deep question of whether universality classes exist for patterning behaviour, however, is still unanswered. The characterization of dynamics that are complex in both space and time (the aforementioned spatiotemporal chaos, Section 1.1.1) is far from complete. It is known that sudden changes from ‘normal’ to alternate realities are common. A minute change in one variable can yield a vastly disproportionate change in the system at a later time (Section 1.8.3). In some situations, these systems are deterministic (i.e. there is a unique, well-deﬁned consequent to every state), but in other circumstances they exhibit a stochastic or random behaviour (there is more than one consequent chosen from some probability distribution, for example the ‘perfect’ coin toss has two consequents with equal probability for each initial state). Some researchers have modelled some of these behaviours as the patterns formed by nonlinear systems were controlled by one or more ‘attractors’ (and it is known that more complex patterns, such as fractals, are formed by strange attractors), but the underlying mechanisms are still obscure. In such a context, ﬂuid convection induced by body and/or surface forces can certainly be regarded as one of the most distinguished physical phenomena to test existing theories and concepts and probe new ideas about dynamic systems. First studies of these subjects can be tracked back to almost 2000 years ago. Indeed, the origin of the word convection should be ascribed to the Latin word convectio–convectionis (which means ‘transport’), and the word thermal has its root in the Greek preﬁx thermo- (θ ερµ´oς, meaning heat, hot, warm) and/or in the derived Latin word thermanticus, (meaning ‘which transports heat’). Over the last century, ﬂuid dynamics has motivated much of the basic research on pattern formation and books are still being published on this subject. As an example, every year the international journal The Physics of Fluids devotes one issue to illustrate the variety and beauty of natural ﬂuid ﬂows (under the heading ‘a gallery of ﬂuid motion’); the formation of patterns in ﬂuids is also the primary focus of journals focused expressly on technological and industrial applications (the international journal Fluid Dynamics and Materials Processing, above all).

6

Thermal Convection: Patterns, Evolution and Stability

Indeed, convection problems of the type considered in the present book (thermogravitational, thermal Marangoni and thermovibrational ﬂows) can provide fundamental information on the relationship between determinism and chaos in ﬂuid dynamics and, in general, between simplicity and complexity in the structure and behaviour of systems governed by nonlinear equations. In the case of ﬂuids, the governing equations correspond to the Navier–Stokes equations, one of the most intensively studied set of PDEs.

1.2

The Navier–Stokes Equations

1.2.1 A Satisfying Microscopic Derivation of the Balance Equations The Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of ﬂuid substances such as gases, liquids and even solids of geological sizes and time-scales. These equations establish that changes in momentum in inﬁnitesimal volumes of ﬂuid are simply the sum of dissipative viscous forces, changes in pressure, gravity, surface tension (in the presence of a free surface) and other forces acting on the ﬂuid: an application of Newton’s second law (Navier, 1822; Stokes, 1845). They are one of the most useful sets of equations because they describe the physics of a large number of phenomena of academic and practical interest. They may be used to model weather, ocean currents, ﬂow around an airfoil (wing), ﬂuid motion inside a crucible used for crystal growth or for the treatment of metal alloys, blood ﬂow in an artery and even motion of stars inside a galaxy. Although ﬂuid dynamics is a well-established discipline, its focus has shifted over the years and the range of applications has diversiﬁed. As such, these equations, in both full and simpliﬁed forms, are used in the design of aircraft and cars, the study of natural convection, the design of power stations, the analysis of the effects of pollution, the study of biokinetics of protein crystals (e.g. Lappa, 2003b, 2004d, 2005c) and the biomechanics of biological tissues (e.g. Lappa, 2003e, 2004e, 2006b) and so on. Coupled with Maxwell’s equations, they can be used to model and study magnetohydrodynamics in typical problems of crystal growth or Earth-core dynamics. The are many ways to derive these equations. They can be introduced starting directly from the conservation of mass and momentum being written for an arbitrary macroscopic control volume (this is the usual ‘point of view’ taken by engineers), as an application of Newton’s second law to a continuum (this is the usual ‘point of view’ taken by physicists). In these treatments, the (geometric) continuum hypothesis is invoked from the start (the underlying idea is that ‘matter’ occupies all points of the space of interest and that properties of the ﬂuid can be represented by piecewise continuous functions of space and time, as long as length and time scales are not too small). The Navier–Stokes equations can be also derived from microscopic models, i.e. by obtaining these classical partial differential equations as the scaling limits of large microscopic systems (the mathematician’s point of view). The latter strategy provides a wealth of additional aspects that are overlooked or somehow ‘hidden’ when using other approaches (e.g. the meaning of the ‘mass velocity’, the relationship between the stress tensor and the exchange of momentum at a molecular level) and which otherwise have to be introduced on empirical bases. Here the goal is to stake out some common ground by providing a synergetic synthesis of the distinct approaches/points of view. Towards this end, it is worth noting that the microscopic approach for deriving the Navier–Stokes equations can be elaborated in its simplest form by reinterpreting (at a different length scale) the descriptive models originally introduced for Maxwell’s equations, that is, explaining the macrophysical properties of ﬂuids based on microscopic models of their constituent particles (i.e. deriving thermodynamic and hydrodynamic limits for stochastic particle systems, i.e. ‘building’ a macroscopic state from microscopic statistics).

Equations, General Concepts and Methods of Analysis

7

Lorentz (1902) was the ﬁrst to give a derivation of Maxwell’s equations in material bodies from the fundamental equations of his electron theory by averaging the microscopic ﬁeld quantities over physically inﬁnitesimal space and time regions (a century ago, Lorentz deduced the macroscopic Maxwell equations by spatially averaging a set of postulated equations for the microscopic electromagnetic ﬁeld). This procedure has with only slight modiﬁcations been taken over by various authors. A straightforward derivation of Maxwell’s equations from electron theory was given by Mazur and Nijboer (1953) on the basis of ensemble averaging. The formalism used was analogous to that applied by Kirkwood (1946) and Irving and Kirkwood (1950). This philosophy gives justice to the fundamental discontinuous nature of matter. Moreover, it allows the establishment of a fruitful theoretical link to the concepts elaborated in Section 1.1. At ﬁrst glance, recalling the ideas illustrated in Section 1.1.1, the inﬁnite variety of ﬂow patterns and convective spatiotemporal structures displayed by ﬂuids could be simply regarded as the ensemble behaviour created by the collective motion of the molecules of which the considered ﬂuid consists. It will be shown in the following sections how, at a deeper level of analysis, an intimate correspondence can be established between the theoretical concept of the set of system parts mentioned in Section 1.1.1 (connected in a network of specialized functions responsible for nonlinear dynamics) and a set of elementary volumes (or parcels) of ﬂuid exchanging at any instant mass, momentum and energy such as the biological cells of a living organism (e.g. Lappa, 2008). Remarkably, such exchange is basically responsible for the nonlinear nature of the Navier–Stokes equations. The macroscopic protocols of interaction among the various subparts (the so-called convective ﬂuxes introduced in Section 1.2.3), in fact, admit mathematical representation in which some of the system variables are involved multiplicatively. Not to be too cryptic and to prevent the ﬂavour being too philosophical, hereafter the discussion progresses with the support of precise mathematical arguments.

1.2.2 A Statistical Mechanical Theory of Transport Processes As outlined before, in their ﬁnal form the Navier–Stokes equations assume a ﬂuid to be a continuum, whereas in reality a ﬂuid is a collection of discrete molecules. To model the underlying microscopic physics, statistical mechanics begins considering the characterization of a generic ensemble of N particles as deﬁned in classical mechanics, that is, in terms of the complete speciﬁcation (at a given instant) of the individual particle position r i , particle mass mi and velocity ci ; moreover, the following obvious relationships are used (a microscopic application of Newton’s laws): N

mi = constant

(1.1a)

i=1

dci (1.1b) dt The considered system consisting of N particles is assumed to be hosted in a generic volume DREV (further necessary information on the nature of this volume will be provided later in this section). In the following, the generic quantity associated with a single particle is denoted by i , and its • time derivative byi . Moreover, given a generic function a, the symbol < a > is used to indicate its ‘average’ value from a stochastic standpoint. In such a context, the density of the generic quantity i is introduced as a function P (r, t) deﬁned in such a way that its integral computed over the domain DREV gives the stochastic average value of the sum of all the quantities i related to the particles hosted in DREV . From a mathematical point of view, the deﬁnition of the density function P (r, t) can be based on the well-known Dirac function δ(r) [whose properties are: δ(s) = 0 for s = 0, δ = ∞ for s = 0 f i = mi

8

Thermal Convection: Patterns, Evolution and Stability

and

D

δdD = 1 if the generic domain D contains the origin s = 0]. It reads N P (r, t) = i δ(r − r i )

(1.2)

i=1

In fact, taking into account the properties of the Dirac function: N N P (r, t)dD = i δ(r − r i )dD = i DREV

i=1

DREV

(1.3)

i=1

which satisﬁes the deﬁnition given above for P (r, t). By taking the derivative with respect to time of Eq. (1.2): N • ∂ ∂ i δ(r − r i ) + i δ(r − r i ) P (r, t) = ∂t ∂t

(1.4)

i=1

and since from a mathematical point of view ∂ ∂ ∂ δ[r − r i (t)] [r − r i (t)] = −ci · ∇[δ(r − r i )] = −∇ · [ci δ(r − r i )] (1.5) δ[r − r i (t)] = ∂t ∂r ∂t Eq. (1.4) becomes N • ∂ i δ(r − r i ) − i ∇ · [ci δ(r − r i )] P (r, t) = ∂t

(1.6)

i=1

Introducing P =

N

i ci δ(r − r i )

(1.7a)

i=1

N

• P = i δ(r − r i ) ∗

(1.7b)

i=1

Eq. (1.4) can be rewritten as ∂ P (r, t) + ∇ · P = P ∗ ∂t

(1.8)

which is known as the general balance equation, where P and P ∗ are known as ﬂux density and production density, respectively. In such an equation P (r, t) and P ∗ have the same tensorial order (e.g. both are scalars or vectors), whereas P has a larger order [e.g. it is a vector if P (r, t) is a scalar and becomes a tensor with order two when P (r, t) has order one]. At this stage, some additional insights can be provided about the ‘nature’ of the volume DREV used for determining Eq. (1.8). In classical thermodynamics, the problem of deriving governing laws is typically investigated in the limit as N → ∞ and D → ∞, while the density n = N/D remains constant (this is called the thermodynamic limit). The various properties of the system are separated into extensive and intensive quantities. Extensive quantities are proportional to the effective size of the system, whereas intensive quantities are independent of the size of the system (this reﬂects the intuition that local properties of a macroscopic object do not depend on the size of the system). As a relevant example, all the quantities appearing in Eq. (1.8) are intensive macroscopic quantities.

Equations, General Concepts and Methods of Analysis

9

For the case treated here, namely the derivation of thermoﬂuid-dynamic balance laws as partial differential equations from averaging the mechanics of molecular motion, D is not assumed to be inﬁnite or very extended. Rather, it must be regarded as a ‘representative elementary volume’ (REV) deﬁned in a way such that it is sufﬁciently small to be considered elementary, but sufﬁciently large to contain a very high number of molecules (and to be more extended than their mean free-path). Remarkably, Eq. (1.8) can be applied to every point in a ﬂuid and even integrated over a macroscopic control volume since an elementary representative volume DREV with the properties discussed earlier can be associated with each point of the physical domain hosting the ﬂuid. This means that by averaging, smooth ﬁelds for the various macroscopic quantities of interest P (r, t) can be deﬁned over the whole region occupied by the considered ﬂuid. In practice, however, it should be pointed out that a REV exists at a given point provided that the ﬂuxes P are sufﬁciently small. This assumption is often called ‘local equilibrium’ or ‘local thermodynamic equilibrium’ (LTE) in textbooks. In general, it is useful to distinguish between global and local equilibrium. In thermodynamics, exchanges within a system and between the system and the outside are controlled by intensive parameters. Global thermodynamic equilibrium (GTE) means that those intensive parameters are homogeneous throughout the whole system, whereas local thermodynamic equilibrium (LTE) means that those intensive parameters are varying in space and time, but are varying so slowly that for any point, one can assume equilibrium in some neighbourhood (the aforementioned REV) about that point. Throughout this book, we will assume that this condition is satisﬁed over the relevant physical domain. The speciﬁc central macroscopic intensive variables that are to be deﬁned and related in the following sections under such assumptions are the mass density, mass velocity, pressure, stress tensor, internal energy and temperature. In the presence of magnetic ﬁelds, the quantities should also include the magnetic ﬂux density B (this aspect will be treated in Chapter 13). Other macroscopic quantities that are relevant will be added as necessary within the framework developed. Obviously, like Newtonian mechanics, a key aspect of this approach is also to understand how systems can be acted upon or can act upon each other. Along these lines, in addition to the quantities mentioned above, there are two quantities that describe actions that may be made on a system to change its state and that, hence, must be taken into account: work and heat transfer (also known as ﬂux of kinetic energy and internal energy, respectively). The determination of an expression for these quantities in terms of molecular variables is also of fundamental importance. Hence it is clear that the major outcome of all such effort will be a statistical mechanical theory of transport processes whose most interesting articulation is a well-founded formulation of the balance equations for mass, momentum and energy (and some related quantities such as vorticity). The elaboration of this theory will be illustrated in Sections 1.2.3, 1.2.4, 1.2.5 and 1.2.8.

1.2.3 The Continuity Equation The mass balance equation (generally referred to in the literature as the continuity equation) can be obtained formally by simply setting i = mi in Eqs (1.2), (1.7) and (1.8). Equation (1.7b) gives simply P ∗ = 0 as a consequence of Eq. (1.1a). In general, Pm is denoted by ρ; moreover, by deﬁnition, the mass velocity V can be introduced as N m = [mi ci δ(r − r i )] = ρV (1.9) i=1

that is, by expressing the ﬂux of mass as product of ρ and V .

10

Thermal Convection: Patterns, Evolution and Stability

Accordingly, the continuity equation reads ∂ρ + ∇ · (ρV ) = 0 ∂t

(1.10a)

which, in terms of the substantial derivative D/Dt = ∂/∂t + V · ∇ (also known as the ‘material’ or ‘total’ derivative), can be rewritten as Dρ + ρ∇ · V = 0 Dt

(1.10b)

The introduction of the mass velocity V has some important consequences. It is worth underlying that such a velocity accounts for the macroscopic motion of the considered ﬂuid. Accordingly, the generic velocity ci of the particle with mass mi can be split into two contributions: ci = V + C i

(1.11)

where the latter represents the well-known random motion of molecules. By comparison of Eqs (1.9) and (1.11), it follows that N [mi C i δ(r − r i )] = 0

(1.12)

i=1

which has some useful and important implications for the derivation of the balance equations for other typical quantities. For the generic scalar quantity i , moreover, on the basis of Eq. (1.11) the ﬂux density can be split into two distinct contributions: N N N [i ci δ(r − r i )] = [i V δ(r − r i )] + [i C i δ(r − r i )] = P V + J P P = i=1

i=1

i=1

known as the convective and diffusive ﬂuxes, respectively. They take into account transport of the considered quantity i as due to bulk ﬂuid motion (macroscopically represented by the mass velocity) and transport at microscopic scale (due to the average effect of random molecular motion), respectively. Remarkably, in the expression of the convective contribution the density of the generic quantity (P ) is multiplied by the mass velocity, which acts as a source of nonlinearity for the overall Navier–Stokes equations.

1.2.4 The Momentum Equation The momentum balance equation can be obtained formally by simply setting i = mi ci in Eqs (1.2), (1.7) and (1.8). Taking into account the mass ﬂux introduced with Eq. (1.9), Eq. (1.2) gives N mi ci δ(r − r i ) = ρV (1.13) P mt = i=1

Moreover,

mt ∗

=

P =

N

[mi ci ci δ(r − r i )]

i=1 N i=1

dc mi i δ(r − r i ) dt

(1.14)

=

N i=1

[f i δ(r − r i )] = F b

(1.15)

Equations, General Concepts and Methods of Analysis

11

where F b is the generic body force acting on the ﬂuid, for instance steady terrestrial gravity (see Chapter 2), oscillatory g-jitters on a space platform (see Chapter 8) or an artiﬁcial force produced by a different effect, for example, ﬂuid interaction with a magnetic ﬁeld (see Section 1.9 and Chapter 13) or dielectrophoresis (e.g. Pohl, 1978). The momentum equation reads ∂ ρV + ∇ · mt = F b ∂t

(1.16)

Substituting Eq. (1.11) into Eq. (1.14): N N [mi V V δ(r − r i )] + [mi V C i δ(r − r i )] mt = i=1

+

N

[mi C i V δ(r − r i )] +

i=1

=

N

+

[mi C i C i δ(r − r i )]

i=1

[mi δ(r − r i )] V V + V

i=1

N

i=1

N

[mi C i δ(r − r i )] V +

i=1

N

[mi C i δ(r − r i )]

i=1

N

[mi C i C i δ(r − r i )]

(1.17)

i=1

and taking into account Eqs (1.9) and (1.12), the density of ﬂux for momentum can be written as mt = ρV V − τ

(1.18)

where τ is known as the stress tensor: τ =−

N

[mi C i C i δ(r − r i )]

(1.19)

i=1

Such a tensor can be regarded as a stochastic measure of the exchange of microscopic momentum mi C i (related to the random component C i of the particle total velocity) induced at molecular level by particle random motion [Eq. (1.19) provides clear evidence of the fact that viscous forces originate in molecular interactions; we shall come back to this concept later]. Substituting Eq. (1.18) into Eq. (1.16), we obtain ∂ ρV + ∇ · (ρV V − τ ) = F b ∂t DV − ∇ · τ = Fb ρ Dt

(1.20a) (1.20b)

1.2.5 The Total Energy Equation The genesis of the total energy balance equation is analyzed in the following by formally setting i = ei = 12 mi ci2 (kinetic energy associated with the generic particle at microscopic level) in Eqs (1.2), (1.7) and (1.8). The density of total energy reads N 1 (1.21) mi ci2 δ(r − r i ) Pe = E = 2 i=1

12

Thermal Convection: Patterns, Evolution and Stability

Using Eq. (1.11), formally, ci2 = ci · ci = (V + C i ) · (V + C i ) = V 2 + Ci2 + 2V · C i , hence N N N 1 1 1 2 2 E= mi V δ(r − r i ) + mi Ci δ(r − r i ) + mi 2V · C i δ(r − r i ) 2 2 2 i=1 i=1 i=1 N N N 1 1 2 2 = mi δ(r − r i ) V + mi C i δ(r − r i ) (1.22a) mi Ci δ(r − r i ) + V · 2 2 i=1

i=1

i=1

which, taking into account Eq. (1.12) and the deﬁnition of ρ, becomes E= where by deﬁnition

ρuint =

1 ρV 2 + ρuint 2

N 1 i=1

2

(1.22b)

mi Ci2 δ(r

− ri)

uint being the so-called internal energy. According to Eq. (1.7a), the density of energy ﬂux is N 1 mi ci2 ci δ(r − r i ) e = 2

(1.23)

(1.24a)

i=1

Substituting Eq. (1.11) into Eq. (1.24a): N 1 2 2 e = mi (V + Ci + 2V · C i )(V + C i )δ(r − r i ) 2 i=1 N N 1 1 2 2 mi V V δ(r − r i ) + mi V C i δ(r − r i ) = 2 2 i=1 i=1 N N 1 1 2 2 + mi Ci V δ(r − r i ) + mi Ci C i δ(r − r i ) 2 2 i=1 i=1 N N 1 1 + mi (2V · C i )V δ(r − r i ) + mi (2V · C i )C i δ(r − r i ) 2 2 i=1

(1.24b)

i=1

which can be rewritten as N N 1 1 2 [mi δ(r − r i )] V V + [mi C i δ(r − r i )] V 2 e = 2 2 i=1 i=1 N N 1 1 2 2 + mi Ci δ(r − r i ) V + mi Ci C i δ(r − r i ) 2 2 i=1 i=1 N N [mi C i δ(r − r i )] V + V · [mi C i C i δ(r − r i )] +V · i=1

(1.24c)

i=1

Taking into account the deﬁnition of ρ, Eqs (1.12) and (1.19) and the deﬁnition of the internal energy, Eq. (1.23), then Eq. (1.24c) becomes 1 (1.25) ρV 2 + ρuint V + J u − V · τ e = 2

Equations, General Concepts and Methods of Analysis

where by deﬁnition

Ju =

N 1 i=1

2

mi Ci2 C i δ(r

− ri)

13

(1.26)

is the (density of) diffusive ﬂux of internal energy (it measures transport at the microscopic level of molecular kinetic energy 12 mi Ci2 due to molecular random motion). The density of production of total energy simply corresponds to the work done per unit time by the body forces, that is, it can simply be written as P∗ = Fb · V

(1.27)

In the light of the above considerations, the total energy balance equation can be cast in condensed form as 1 2 1 2 ∂ ρ (1.28a) V + uint + ∇ · ρ V + uint V + J u − V · τ = F b · V ∂t 2 2 or in terms of the substantial derivative as D 1 2 ρ V + uint + ∇ · [J u − V · τ ] = F b · V Dt 2

(1.28b)

1.2.6 The Budget of Internal Energy A speciﬁc balance equation for the single internal energy can be obtained from subtracting the kinetic energy balance equation from the total energy balance equation introduced before (Eq. 1.28). Obviously, a balance equation for the pure kinetic energy can be introduced by taking the product of the momentum balance equation with V : D V2 (1.29a) − (∇ · τ ) · V = F b · V ρ Dt 2 This equation, using the well-known vector identity ∇ · (V · τ ) = (∇ · τ ) · V + τ : ∇V , can be rewritten as D V2 − ∇ · (V · τ ) = F b · V − τ : ∇V (1.29b) ρ Dt 2 from which, among other things, it is evident that the diffusive ﬂux of kinetic energy can be simply expressed as the scalar product between V and the stress tensor. Subtracting, as explained before, Eq. (1.29b) from Eq. (1.28b), one obtains Duint + ∇ · J u = τ : ∇V Dt ∂ρuint + ∇ · [ρuint V + J u ] = τ : ∇V ∂t which is the aforementioned balance equation for the internal energy. ρ

(1.30a) (1.30b)

1.2.7 Newtonian Fluids In the preceding sections, it has been shown how the statistical treatment of the many particles of a ﬂuid, with a key set of assumptions, simply reveals that the governing laws of continuum can be derived as a natural consequence of the ensemble behaviour of many microscopic particles interacting with each other. In general, however, the ‘closure‘ of the thermoﬂuid-dynamic balance

14

Thermal Convection: Patterns, Evolution and Stability

equations provided by such approach, that is, the determination of a precise mathematical formalism relating the diffusive ﬂuxes [stress tensor and the diffusive ﬂux of internal energy deﬁned by Eqs (1.19) and (1.26), respectively] to the macroscopic variables involved in the process, is not as straightforward as many would assume. For a particular but fundamental category of ﬂuids, known as ‘Newtonian’ ﬂuids, the treatment of this problem, however, is relatively simple. For the case considered in the present book (nonpolar ﬂuids and absence of torque forces), the stress tensor can be taken symmetric, that is, τij = τj i (conversely, a typical example of ﬂuids for which the stress tensor is not symmetric is given by ‘micropolar’ ﬂuids, which represent ﬂuids consisting of rigid, randomly oriented particles suspended in a viscous medium; see, e.g. Ferreira and Villamizar Roa, 2007, and references therein). If the considered ﬂuid is in quiescent conditions (i.e. there is no macroscopic motion), the stress tensor is diagonal and simply reads τ = −pI

(1.31a)

where I is the unity tensor and p is the pressure. In the presence of bulk convection, the above expression must be enriched with the contributions induced by macroscopic ﬂuid motion. According to Isaac Newton’s observation, this contribution is simply provided by gradients of mass velocity through a proportionality constant that does not depend on such gradients: τ = −pI + 2µ(∇V )so

(1.31b)

where the constant of proportionality µ is known as the dynamic viscosity (it may be regarded as a macroscopic measure of the intermolecular attraction forces) and the tensor (∇V )so (known as viscous stress tensor or dissipative part of the stress tensor) comes from the following decomposition of ∇V : 1 ∇V = (∇ · V )I + (∇V )so + (∇V )a (1.32) 3 where 1 ∇V + ∇V T (∇V )so = (∇V )s − (∇ · V )I (∇V )s = (1.33a) 3 2 T ∇V − ∇V (∇V )a = (1.33b) 2 The three contributions in Eq. (1.32) are known to be responsible for volume changes, deformation and rotation, respectively, of a generic (inﬁnitesimal) parcel of ﬂuid (moving under the effect of the velocity ﬁeld V ; see Section 1.2.8 for additional details about the meaning of (∇V )a and its kinship with the concept of vorticity). Moreover, in general, the diffusive ﬂux of internal energy can be written as (Fourier law) J u = −λ∇T

(1.34)

where λ is the thermal conductivity and T the ﬂuid temperature. Accordingly, and taking into account the following vector and tensor identities: ∇ · (pI ) = ∇p ∇ · (pV ) = p∇ · V + V · ∇p pI : ∇V = p∇ · V 1 (∇ · V )I + (∇V )so + (∇V )a = (∇V )so : (∇V )so (∇V )so : ∇V = (∇V )so : 3 the affected balance equations can be rewritten in compact form as follows:

(1.35) (1.36) (1.37) (1.38)

Equations, General Concepts and Methods of Analysis

15

Momentum: ∂ ρV + ∇ · (ρV V ) + ∇p = ∇ · [2µ(∇V )so ] + F b ∂t DV + ∇p = ∇ · [2µ(∇V )so ] + F b ρ Dt

(1.39a) (1.39b)

Kinetic energy: V2 V2 ∂ ρ +∇· ρ V + V · ∇p = ∇ · [2µV · (∇V )so ] − 2µ(∇V )so : (∇V )so + F b · V ∂t 2 2 (1.40a) D V2 + V · ∇p = ∇ · [2µV · (∇V )so ] − 2µ(∇V )so : (∇V )so + F b · V ρ (1.40b) Dt 2 Internal energy: ∂ρuint (1.41a) + ∇ · [ρuint V ] = ∇ · (λ∇T ) − p∇ · V + 2µ(∇V )so : (∇V )so ∂t Duint (1.41b) ρ = ∇ · (λ∇T ) − p∇ · V + 2µ(∇V )so : (∇V )so Dt Total (internal + kinetic) energy: 1 2 ∂ 1 2 ρ V + uint + ∇ · ρ V + uint V = ∇ · [λ∇T − pV + 2µV · (∇V )so ] + F b · V ∂t 2 2 (1.42a) D 1 2 (1.42b) V + uint = ∇ · [λ∇T − pV + 2µV · (∇V )so ] + F b · V ρ Dt 2

1.2.8 Some Considerations About the Dynamics of Vorticity Vorticity is an additional useful mathematical concept widely used in ﬂuid dynamics for a better description or characterization of some ﬂows (vorticity plays a fundamental role in the physics of vortex-dominated ﬂows, its dynamics being the primary tool to understand the time evolution of dissipative vortical structures) and for the introduction of alternative mathematical models and numerical methods [as possible variants of those based on the traditional formulation of the momentum balance equation Eq. (1.39), as will be illustrated in Section 1.7.1]. It also plays a crucial role in the development of turbulence and related mathematical models. In general, it can be related to the amount of ‘circulation’ or ‘rotation’ (or, more strictly, the local angular rate of rotation) in a ﬂuid (it is intimately linked to the moment of momentum of a generic small ﬂuid particle about its own centre of mass). The average vorticity in a small region of ﬂuid ﬂow, in fact, can be deﬁned as the circulation around the boundary of the small region, divided by the area A of the small region: (1.43a) A where the ﬂuid circulation is deﬁned as the line integral of the velocity V around the closed curve in Figure 1.1: (1.43b) = V · tˆd ζ =

tˆ being the unit vector tangent to .

16

Thermal Convection: Patterns, Evolution and Stability

Figure 1.1 Vorticity as a measure of the rate of rotational spin in a ﬂuid

In practice, the vorticity at a point in a ﬂuid can be regarded as the limit of Eq. (1.43a) as the area of the small region of ﬂuid approaches zero at the point: d (1.43c) dA In addition to the previous modelling, using the Stokes theorem (purely geometric in nature), which equates the circulation around to the ﬂux of the curl of V through any surface area bounded by : (∇ ∧ V ) · ndS ˆ (1.44) = V · tˆd = ζ =

A

where nˆ is the unit vector perpendicular to the surface A bounded by the closed curve (it is implicitly assumed that is smooth enough, i.e. that it is locally Lipschitzian; this implies that the existence of the unit vector perpendicular to the surface is guaranteed), it becomes evident that from a mathematical point of view the vorticity at a point can be deﬁned as the curl of the velocity: ζ =∇ ∧V

(1.45)

It is a vector quantity, whose direction is along the axis of the ﬂuid’s rotation. Notably, ζ has the same components of the antisymmetric part of ∇V , that is in line with the explanation given in Section 1.2.7 about the physical meaning of (∇V )a . Related concepts are the vortex line, which is a line which is everywhere tangent to the local vorticity, and a vortex tube, which is the surface in the ﬂuid formed by all vortex lines passing through a given (reducible) closed curve in the ﬂuid. The ‘strength’ of a vortex tube is the integral of the vorticity across a cross-section of the tube and is the same everywhere along the tube (because vorticity has zero divergence). In general, it is possible to associate a vector vorticity with each point in the ﬂuid; hence the whole ﬂuid space may be thought of as being threaded by vortex lines which are everywhere tangential to the local vorticity vector. These vortex lines represent the local axis of spin of the ﬂuid particle at each point. The related scalar quantity: ζ2 (1.46) 2 is generally referred to in the literature as ‘density of enstrophy’. It plays a signiﬁcant role in some theories and models for the characterization of turbulence and in some problems related to the uniqueness of solutions of the Navier–Stokes equations (as will be outlined in Section 1.3). By simple mathematical manipulations it can also appear in global budgets of kinetic energy (again, see Section 1.3). (∇V )a : (∇V )a =

Equations, General Concepts and Methods of Analysis

17

Figure 1.2 Vorticity as the sum of the angular velocity of two short ﬂuid line elements that happen, at that instant, to be mutually perpendicular (Shapiro, 1969)

For two-dimensional ﬂows, the vorticity vector is perpendicular to the plane. Its intensity at any instant is equal to the sum of the angular velocities of any pair of mutually-perpendicular, inﬁnitesimal ﬂuid lines passing through the considered point (see Figure 1.2); thereby, ζ /2 can be regarded as the average angular velocity of the considered ﬂuid element. It is in this precise sense the vorticity acts as a measure of the local rotation or spin, of ﬂuid elements as mentioned before (for a ﬂuid having locally a ‘rigid rotation’ around an axis, i.e. moving like a rotating cylinder, vorticity would be simply twice the system angular velocity). From a mathematical point of view, two-dimensional vorticity can be related to the well-known stream function ψ (that gives the velocity components u and v along two perpendicular directions x and y as u = ∂ψ/∂y and v = −∂ψ/∂x, respectively) through a simple Poisson equation (∇ 2 ψ = −ζ ). In general, for any ﬂow (2D or 3D), the governing equations can be written in terms of vorticity rather than velocity by simply taking the curl of the momentum equations and taking into account the following identities: ∇ · ζ = ∇ · (∇ ∧ V ) = 0 2 V +ζ ∧V V · ∇V = ∇ 2 2 V =0 ∇∧∇ 2 ∇ ∧ (V ∧ ζ ) = V (∇ · ζ ) − ζ (∇ · V ) + ζ · ∇V − V · ∇ζ 1 1 1 1 ∇∧ ∇p = ∇ ∧ ∇p − 2 ∇ρ ∧ ∇p = − 2 ∇ρ ∧ ∇p ρ ρ ρ ρ

(1.47) (1.48) (1.49) (1.50) (1.51)

This leads to Dζ Dt

=

∂ζ ∂t

+ V · ∇ζ = ζ · ∇V − ζ (∇ · V )

+

1 ∇ · 2µ(∇V )so 1 + ∇ ∇ρ ∧ ∇p + ∇ ∧ ∧ F b ρ2 ρ ρ

(1.52)

The ﬁrst term on the right-most side of this equation, ζ · ∇V , is known to be responsible for possible stretching of vortex ﬁlaments along their axial direction; this leads to contraction of cross-sectional area of ﬁlaments and, as a consequence of the conservation of angular momentum, to an increase in vorticity (this term is absent in the case of two-dimensional ﬂows). The second term, ζ (∇ · V ),

18

Thermal Convection: Patterns, Evolution and Stability

described possible stretching of vorticity due to ﬂow compressibility. The third term is generally known as the baroclinic term (it accounts for changes in vorticity due to interaction of density and pressure gradients acting inside the ﬂuid). The fourth term shows that vorticity can be produced or damped by the action of viscous stresses. The last term accounts for possible production of vorticity due to the effect of body forces.

1.2.9 Incompressible Formulation of the Balance Equations

N Remarkably, in the so-called incompressible form, ρ = i=1 mi δ(r − r i ) = constant = ρ0 , traditionally assumed in studies of internal thermal convection, all the governing equations derived in the preceding subsection can be rewritten in a simpler form. In such a context, it is also worth noting that, in general, the approximation of constant density is considered together with that of constant transport coefﬁcients (µ and λ), which leads to additional useful simpliﬁcations. The continuity equations reads ∇ ·V =0

(1.53)

∇ · [2µ(∇V )so ] = µ∇ · [∇V + ∇V T ] = µ[∇ 2 V + ∇(∇ · V )] = µ∇ 2 V

(1.54)

and, as a consequence, in Eq. (1.39a)

and the momentum equation reads ∂V + ρ0 ∇ · [V V ] + ∇p = µ∇ 2 V + F b ∂t The internal energy equation becomes ρ0

ρ0

∂uint + ρ0 ∇ · [uint V ] = λ∇ 2 T + 2µ(∇V )so : (∇V )so ∂t

(1.55)

(1.56)

where the last term, 2µ(∇V )so : (∇V )so , represents the production of internal energy due to viscous stresses (also referred to in the literature as density of viscous heating or kinetic energy degradation: the rate at which the work done against viscous forces is irreversibly converted into internal energy). In general, the order of magnitude of this term is negligible with respect to the other terms and for that reason it can be ignored (as shown by Gebhart, 1962, the effect of viscous dissipation in natural convection becomes appreciable only when the induced kinetic energy is appreciable compared with the amount of heat transferred; this occurs when either the equivalent body force is large or when the convection region is extensive). It is also worth noting that using thermodynamic relationships, the internal energy can be written as a function of the temperature T . In fact: duint = Cv dT where Cv is the speciﬁc heat at constant volume: ∂uint Cv = ∂T v=constant

(1.57a)

(1.57b)

v being the speciﬁc volume, v = 1/ρ. Interestingly, by comparison of Eqs (1.23) and (1.57a), among other things, the well-known intimate relationship between the temperature of a ﬂuid and the kinetic energy of motion of its molecules becomes evident. Taking into account that, in particular, for liquids Cv ∼ = Cp , where Cp is speciﬁc heat at constant pressure and introducing the thermal diffusivity α deﬁned as α = λ/ρCp , the energy equation can

Equations, General Concepts and Methods of Analysis

19

be cast in compact form as ∂T (1.58) + ∇ · [V T ] = α∇ 2 T ∂t Hereafter and throughout this book, the ﬂuid will be assumed to be incompressible and the coefﬁcients µ, α, λ and Cp to be constant . This cardinal simpliﬁcation is extremely accurate for many ﬂows and makes the related mathematics simpler. Remarkably, Eqs (1.53), (1.55) and (1.58) represent a set of three coupled equations whose solution is sufﬁcient for the determination of the problem unknowns, namely V , p and T .

1.2.10 Nondimensional Form of the Equations for Thermal Problems Finding solutions to the Navier–Stokes equations is extremely challenging. In fact, only a handful of exact solutions are known (see, e.g., Section 2.4) and these solutions are generally available only for very simple systems. An advantageous alternative for obtaining useful information is through the approximations which assess the relative magnitude of driving forces by appropriate scaling of the equations. The basic idea behind such analyses is to choose characteristic scales for length, time, velocity and so on so that the equations are made dimensionless. The resulting grouping of physical properties and characteristic scales form dimensionless numbers which represent ratios of various forces or quantities. Theoreticians often communicate through this mysterious melange of dimensionless parameters (many of these parameters will be deﬁned in Chapter 2). Here the attention is limited to the typical (most general) choice of characteristic reference quantities for thermal convection. In such a case, the nondimensional form of these equations usually results from scaling the lengths by a reference distance (L) and the velocity by the energy diffusion velocity Vα = α/L; the scales for time and pressure are, respectively, L2 /α and ρα 2 /L2 . The temperature, measured with respect to a reference value T0 , is scaled by a reference temperature gradient T . This approach leads to (in the following, for the sake of simplicity, the same symbols as used for the equations in dimensional form are also used for the nondimensional formulation): ∇ ·V =0 ∂V + ∇ · [V V ] + ∇p = Pr∇ 2 V + F b ∂t

(1.59) (1.60a)

∂T (1.61a) + ∇ · [V T ] = ∇ 2 T ∂t where Pr is the Prandtl number (Pr = ν/α) and ν is the constant kinematic viscosity (ν = µ/ρ). In the light of the arguments elaborated in Sections 1.2.4–1.2.7, this ﬁrst nondimensional parameter measures the relative importance of transport at a molecular level of momentum (via ν) and kinetic energy, (via α). It is often regarded as a clear ‘signature’ of the ﬂuid considered. This is the reason why researchers often identify considered ﬂuids with the related values of Pr instead of providing details (nomenclature, composition, etc.) about the chemical structure. Table 1.1 gives some typical values of the Prandtl number for different categories of substances. According to Eq. (1.59), Eqs (1.60a) and (1.61a) can be also written as ∂V + V · ∇V + ∇p = Pr∇ 2 V + F b ∂t ∂T + V · ∇T = ∇ 2 T ∂t known as the ‘nonconservative’ form.

(1.60b) (1.61b)

20

Thermal Convection: Patterns, Evolution and Stability

Table 1.1 Typical values of the Prandtl number for various substances (where not speciﬁed, ﬂuids are assumed to be at room temperature, ∼25◦ C; Tm is the melting temperature of the considered substance) Type

Fluid

T (K)

Pr

Liquid metals [Pr < O (1)]

Al

934 1100 544 800 303 300 400 400 800 500 1000 400 1000 607 800 1685 505 693 1000 400 800 Tm

0.053 0.011 0.021 0.011 0.0207 0.025 0.015 0.0066 0.0036 0.059 0.018 0.0099 0.0040 0.026 0.018 0.01 0.009 0.028 0.015 0.045 0.013 0.054

Tm Tm Tm Tm Tm – – – – 1043 1073 620 800 590 633 750 883 1121 823 1243 996

0.068 0.037 0.04 0.015 0.039 0.71 0.68 0.77 0.72 1.16 1.02 8.13 3.27 9.28 7.00 4.31 1.52 4.69 3.47 7.03 9.5

Bi Ga Hg K Li Na Pb Si Sn Zn Pb–Bi Compound semiconductors and metal alloys [Pr < O (1)]

Gases [Pr = O (1)]

Molten salts [Pr ≥ O (1)]

AlSb (aluminium antimonide) GaAs (gallium arsenide) GaSb (gallium antimonide) InAs (indium arsenide) InP (indium phosphide) InSb (indium antimonide) Air Helium Carbon dioxide Oxygen KCl NaCl KNO3 NaNO3

LiCl LiF LiBr BaCl2 Li2 C03

Equations, General Concepts and Methods of Analysis

21

Table 1.1 (Continued) Type

Fluid

Transparent organic or inorganic liquids [Pr > O (1)]

Silicone oil

Fluids of geophysical interest [Pr O (1)]

1 cSt 10 cSt 100 cSt 1000 cSt Parafﬁn Molten hexatriacontane (C36 H74 ) Molten tetracosane (C24 H50 ) Hexadecane Tetradecamethylhexasiloxane Cyclohexane Octane Ethylene glycol Molten glass Molten succinonitrile FC-70 FC71 FC-75 FC-77 Glycerin Acetone Benzene Butyl alcohol Ethyl alcohol Methanol Water Solid–liquid mixture of the inner–outer Earth core boundary Magma Earth rocky mantle

T (K)

Pr

– – – – (330–340) 370 360 – – – – – 1873 332 – – – – – – – – – – –

17.7 134.5 900 9000 O (102 ) 65 49 50 35 14.6 6.92 159 6.75 × 103 22.9 389 2060 23.3 16.4 1.17 × 104 4.3 6.1 40.3 11.6 10 6.9 1013 104 − 108 1023

1.3 Energy Equality and Dissipative Structures In the preceding sections, attention has mainly been devoted to deriving (and commenting on) the balance equations in their local formulation (i.e. applicable to representative elementary volumes, in the sense that has been given to such a deﬁnition in Section 1.2.2) for the various quantities of interest. A fruitful alternative for obtaining insights into thermal convection and, in particular, about the so-called dissipative structures, those of thermal convection being the typical manifestation, is to consider some global budgets obtained by integrating the local form of the balance equations over the total physical domain occupied by the ﬂuid. A typical example of such an approach is the introduction of the so-called energy equality (that also leads to the somewhat connected problem of the existence and unicity of solutions of the Navier–Stokes equations). Projecting the momentum equation Eq. (1.60b) on a divergence-free vector N with homogeneous component on the frontier ∂D (i.e. N = 0), assuming that V = 0 on ∂D and integrating over the

22

Thermal Convection: Patterns, Evolution and Stability

whole volume D occupied by the ﬂuid, the global budget reads ∂V · N + (V · ∇V ) · N + ∇p · N − Pr∇ 2 V · N − F b · N dD = 0 ∂t D

(1.62)

The domain D denotes the relevant geometric region where the spatial variables are ranging; therefore, it will coincide with the region of ﬂow for three-dimensional motions, that is, D ⊂ 3 , whereas it will coincide with a two-dimensional region in the case of plane ﬂows, that is, D ⊂ 2 . Since mathematically ∇p · N = ∇ · (pN) − p∇ · (N) = ∇ · (pN), applying the divergence theorem, ∇ · (pN)dD = (pN · n)dS ˆ =0 (1.63) D

∂D

where nˆ is the unit vector perpendicular to ∂D, Eq. (1.62) becomes ∂V · N + (V · ∇V ) · N − Pr∇ 2 V · N − F b · N dD = 0 ∂t D

(1.64)

and since ∇ 2 V · N = ∇ · (∇V · N ) − ∇V : ∇N

(1.65)

and taking into account the divergence theorem and the considered boundary conditions for N, it follows that ∂V · N + (V · ∇V ) · N + Pr(∇V : ∇N) − F b · N dD = 0 (1.66) ∂t D Integrating by parts the convective term, the momentum equation in integral form can be ﬁnally written as ∂V · N − (∇N · V ) · V + Pr(∇V : ∇N) − F b · N dD = 0 (1.67) ∂t D known as a ‘weak formulation’. The term ‘weak’ refers to the fact that since the velocity in Eq. (1.67) appears with a lower order of derivation with respect to the corresponding terms in Eq. (1.62), the new equation, Eq. (1.67), allows a larger class of solutions with respect to the initial equation [it is clear that if V is a solenoidal vector ﬁeld that satisﬁes Eq. (1.67) but is not sufﬁciently differentiable, we cannot go from Eq. (1.67) to Eq. (1.62) and it is precisely in this sense that Eq. (1.67) has to be considered as the weak formulation of the original equations]. Since the velocity V is a divergence-free ﬁeld and V must be equal to zero on solid walls (the well-known no-slip condition), in particular, V can be used in place of the generic vector N. Taking into account that V2 (1.68) (V · ∇V ) · V = ∇ · V 2 2 V V2 ∇· V (V · n) ˆ dS = 0 (1.69) dD = 2 2 D ∂D Eq. (1.66) becomes

D

∂V · V + Pr∇V : ∇V − F b · V dD = 0 ∂t

Integrating this in time ﬁnally gives the aforementioned energy equality: t t ε(τ )dτ = K(0) + ℘ (τ )dτ K(t) + 0

0

(1.70)

(1.71)

Equations, General Concepts and Methods of Analysis

where

1 V 2 dD 2 D ε(t) = Pr (∇V : ∇V )dD D ℘ (t) = (F b · V )dD

K(t) =

23

(1.72a) (1.72b) (1.72c)

D

K(t) being the total kinetic energy at time t, ε(t) the energy dissipation rate and ℘(t) the power input due to the volume force/ﬂow interaction. By simple mathematical manipulations, the energy dissipation rate, in turn, can be written as the sum of two contributions: ζ2 (∇V )so : (∇V )so dD + Pr dD (1.73) ε(t) = Pr D D 2 where the ﬁrst integral is the integral of the density of viscous heating deﬁned in Section 1.2.9 and the second is the integral of the enstrophy density deﬁned in Section 1.2.8. At this stage, it should also be pointed out that when the boundary is not formed solely by solid walls ∂DW (where V = 0 due to no-slip conditions as considered until here), but also includes a free surface1 ∂DS (i.e. ∂D = ∂DW ∪ ∂DS ), the power input ℘ (t) will exhibit a second contribution related to the work done per unit time by tangential surface tension forces F Tσ , that is, (F b · V )dD + (F Tσ · V )dS (1.74) ℘ (t) = D

∂DS

Equation (1.71) with the related Eqs (1.72)–(1.74) provides simple but interesting insights into the physics of the problem, as it shows that energy dissipated by viscous forces can be replaced in time by new kinetic energy produced by the action of body and/or surface forces. This is the reason why the related ﬂow patterns are also known as ‘dissipative structures’. As already discussed, Eq. (1.67) is known as the ‘weak formulation’ of the Navier–Stokes equations. The related solutions are referred to as weak solutions to separate them from the ‘classical’ or ‘strong’ solutions of Eq. (1.60): a clear distinction is given by application of Eq. (1.71), which for such solutions holds in a weak sense (it holds if the symbol ‘=’ is replaced by ‘≤’, which means that weak solutions obey only an energy inequality rather than the energy equality). Interestingly, as mentioned at the beginning of this section, many of these considerations are somewhat linked to the subject of the existence and uniqueness of solutions of the 3D Navier–Stokes equations, which surprisingly is still an open mathematical problem. Several mathematical properties for the system Eqs (1.59)–(1.60) have been widely investigated over the years and are still the subject of fundamental research. However, more than 170 years after their formulation, the Fundamental Problem (FP) related to them (often referred to as the sixth problem of the millennium; Wiegner, 1999; Ladyzhenskaya 2003) still remains unsolved; in its general form, this problem can be stated as follows: Given the body force F b and the initial distribution of velocity V 0 = V (r, 0) (no matter how smooth), to determine a corresponding unique regular solution V (r, t), p(r, t) to Eqs (1.59) and (1.60) for all times t > 0. Notably, following Leray (1934a,b) and Hopf (1950–1951), the aforementioned weak solutions play a major role in the mathematical theory of Navier–Stokes equations, in that they are the only solutions, known so far, which exist for all times and without restrictions on the size of the data. 1 Throughout the book, free surfaces will be assumed to be undeformable and with a ﬁxed location in space.

24

Thermal Convection: Patterns, Evolution and Stability

In 2D, existence and uniqueness of weak solutions for all times were shown by Leray (1933, 1934a). Later, Ladyzhenskaya (1969) gave a complete proof of the existence and uniqueness theorem for strong solutions. Leray (1934a) also elaborated the theory for the existence of weak solutions in the 3D case whereas uniqueness is still a matter of discussion. Additional useful information about the main mathematical difﬁculties relating to Eqs (1.59) and (1.60), which oppose the formulation of a general existence and uniqueness theorem in 3D, can be summarized as follows. First, it should be highlighted that the unknowns V and p do not appear in a ‘symmetric way’. In other words, the equation of conservation of mass is not of the form ∂p/∂t = G(p, V ). This is due to the fact that, from the mechanical point of view, the pressure plays the role of a reaction force (Lagrange multiplier) associated with the zero-divergence constraint div V = 0. In these regards, in a perfect analogy with problems of motion of constrained rigid bodies, the pressure ﬁeld must generally be deduced in terms of the velocity ﬁeld; once the latter has been determined [in particular, the ﬁeld p(r, t) can be formally obtained by operating with the divergence operator on both sides of the momentum equation and projecting such equation on the direction perpendicular to ∂D; see Section 1.7.2 for additional details on this aspect]. Because of the mentioned lack of ‘symmetry’ in V and p, the system Eqs (1.59)–(1.60) does not fall into any of the known classical categories of equations, even though, in a sense, it could be considered close to a quasi-linear parabolic system. In practice, however, the basic difﬁculty related to the problem with the system Eqs (1.59)–(1.60) does not arise from the lack of such a symmetry but, rather, from the coupled effect of the lack of symmetry and of the presence of the nonlinear term (the convective contribution in the momentum equation). In fact, the FP formulated for any of the systems obtained from the original one, Eqs (1.59) and (1.60), by disregarding either the nonlinear term or the zero-divergence condition can be completely solved (Galdi, 2000). Important extensions to the results of Leray have been provided over subsequent years by many respected authors (the courageous reader may consider, e.g., Hopf, 1950, 1951; Prodi, 1959; Lions and Prodi, 1959; Lions, 1960; Foias¸, 1961; Serrin, 1963; Fujita and Kato, 1964; Ladyzhenskaya, 1967, 1969; Rionero and Galdi, 1976; Scheffer, 1978, 1980; Ma, 1981; Caffarelli et al., 1982; Giga, 1983, 1986; Kato, 1984; Masuda, 1984; Sohr and von Wahl, 1984, 1985, 1987; Giga and Miyakawa, 1985; von Wahl, 1986; Kajikiya and Miyakawa, 1986; Beir˜ao da Veiga, 1987; Wiegner, 1987, 1990; Struwe, 1988; Kozono, 1988, 1989; Calderon, 1990; Kozono and Ogawa, 1994; Frehse and R˚uzˇ iˇcka, 1995, 1996; Carpio, 1996; Kozono and Sohr, 1996, 1997; Taniuchi, 1997; Kozono, 1998; Berselli, 1999; Chemin, 1999; Mattingly and Sinai, 1999; Ladyzhenskaya and Seregin, 1999; Galdi, 2000; He and Hsiao, 2002; Sani et al., 2006; Kukavica, 2006; and, in particular, for the thermal problems of interest in this book, Rabinowitz, 1968; Joseph, 1969; Fife and Joseph, 1969; Cannon and DiBenedetto, 1980; Yingcai and Yuting, 1989; Oeda, 1989; Morimoto, 1989, 1991; Hishida, 1991; Fang and Grillakis, 1996; Skal´ak and Kucera, 2000; and most recently da Rocha et al., 2003; Danchin, 2003; Li, 2004; Shilkin 2004; Gonz´alez et al., 2005; Villamizar-Roa et al., 2006; Naumann, 2006; Kaizu and Saito, 2007; Hmidi and Keraani, 2007; Abidi and Hmidi, 2007; Attaoui, 2009). An intriguing outcome of all such theoretical effort is that the uniqueness problem may be somewhat connected to the spontaneous tendency that all dissipative structures exhibit (when the intensity of the driving forces becomes sufﬁciently high with respect to the opposing viscous ones) to evolve towards a particular condition in which they appear as they were evolving under random forces rather than deterministic laws (usually referred to as chaos or turbulence). The theoretical link between the uniqueness open problem and the existence of the regimes mentioned above (in which the connection between the data of the problem, i.e. the initial conditions and applied forces and the system response becomes very elusive; see Section 1.4 for additional details on the behaviour and properties of these regimes), was already postulated by

Equations, General Concepts and Methods of Analysis

25

Leray in his landmark initial studies. He, in fact, was the ﬁrst to speculate that uniqueness may be connected to the time regularity of the energy dissipation rate ε(t) and that turbulence might be related to a ‘breakdown of uniqueness in weak solutions to the Navier–Stokes equations’. In particular, he conjectured that ε(t) might have singularities which are integrable but not square integrable (see, e.g., Galdi, 2000 for a complete elaboration of this theory, which is beyond the scope of the present book).

1.4

Flow Stability, Bifurcations and Transition to Chaos

The main goal of ﬂuid dynamic research is to describe and predict the motions of ﬂuids in response to applied surface and/or body forces. In many circumstances, these forces scale with a nondimensional (properly deﬁned) number. When this number (hereafter simply referred to as ‘characteristic’ number or ‘control’ parameter and generically denoted by R) is small, the problem is not so difﬁcult because there is a unique correspondence between the given boundary and internal forcing data and the predicted motions. But when the characteristic number R is larger, the dynamics of the system can be complicated; there are many solutions; nonuniqueness is the rule; sets of solutions must be described and stable and observable subsets must be separated from the others. A mathematical basis for the study of these problems is the theory of stability and the theory of bifurcation, even if, as will be pointed out in subsequent sections of this chapter, also direct numerical discretization and solution of the system model equations (the Navier–Stokes equations in their complete form together with the energy equation, also referred to as thermal convection equations) can provide signiﬁcant information about these topics. Hereafter, following Joseph (1976), the boundary conditions (for instance, surface tension forces acting on the interface separating two liquids or a liquid and a gas) and the body forces (acceleration ﬁelds) that drive the motion are referred to as ‘data’ of the problem. When the characteristic number R is small, all solutions of the Navier–Stokes equations tend to a single solution, determined by the data after the initial conditions have died away. So if the data are steady (for instance, a steady body force), the solution is steady; if the data are time periodic (g-jitters, i.e. a periodic acceleration ﬁeld induced by a periodic displacement of the considered geometric conﬁguration), so is the solution. In other words, the ﬂow that evolves after a time is uniquely determined by the data, independent of the initial conditions. This unique ﬂow has the maximum symmetry consistent with the data. At higher values of the aforementioned number, however, uniqueness is lost. Other ﬂows with different, usually more complicated patterns of symmetry are then observed after transients have decayed away. For instance, as anticipated in Section 1.1, convection that is spatially uniform in certain directions can be replaced by motions that are spatially periodic, quasi-periodic or aperiodic. Also, motions that are steady can be taken over by ﬂows that are time periodic, quasi-periodic or aperiodic. Hence at larger values of the characteristic number, ﬂows that do not follow the symmetry of the boundary conditions can appear. The loss of symmetry implies the existence of a new solution that bifurcates from the pre-existing one due to the selection and ensuing ampliﬁcation of disturbances. These disturbances, whose possible nature, structure and mathematical representation will be clariﬁed in subsequent sections of this chapter, determine a departure of the system from its initial state. If the characteristic number is increased many times, the bifurcations become faster (bifurcations sequence) and faster until the system becomes chaotic and turbulent. This breakdown in the symmetry of solutions is especially dramatic in the turbulent condition. In such ﬂows, the connection between the data and the ﬂow is very elusive; even with steady data, ﬂows arise whose behaviour after a long time is aperiodic with no regularity. The spatial

26

Thermal Convection: Patterns, Evolution and Stability

structure of turbulent ﬂow is also very complicated, with many small eddies and ﬂuctuations that are sometimes called random because nobody knows how to characterize them in a precise way (a peculiar feature of turbulence is that it contains reasonably well-deﬁned structures, appearing, however, in a mosaic of different combinations and ever-changing patterns). These ﬂows are extremely sensitive to the initial conditions (Section 1.8). A chaotic system may be deﬁned, in fact, as one that shows ‘sensitivity to initial conditions’; that is, any uncertainty in the initial state of the given system, no matter how small, would lead to rapidly growing errors in any effort to predict the future behaviour (its behaviour could be predicted only if the initial conditions were known to an inﬁnite degree of accuracy, which is impossible). Two ﬂows with the same data but slightly different initial conditions evolve into two very different ﬂows. It may be useful to think of turbulence as the least symmetric state of motion consistent with the given data. Instability of ﬂows and their transition to turbulence are widespread phenomena in Nature (the most relevant example is the dynamics of the atmosphere; see Section 1.8.3) and especially in engineering and are also important in many applied sciences. Even if limited to the ﬁrst or second transition of the ﬂow along the sequence that leads to complete turbulence (hereafter referred to as the ‘route to chaos’), the loss of spatial and/or temporal symmetry can have, in fact, detrimental effects in a number of technological processes where the spatial symmetry of the thermoﬂuid-dynamic ﬁeld and/or steady conditions are crucial factors to guarantee a good quality of the ﬁnal product (the growth process of many inorganic and organic crystals and materials). An obvious justiﬁcation for the long-lasting (and continuing) efforts in this ﬁeld can be found, as mentioned above, in the relevance that these dynamics have in several natural phenomena of interest (meteorology, oceanography, geophysics, etc.) and applications in the industrial realm. However, it is fairly clear that these problems have also generated an appeal to scientists and engineers as a consequence of the complexity of the possible stages of evolution and of the nonlinear behaviour. As anticipated in Section 1.1, this complexity is shared with other systems in Nature and constitutes a remarkable challenge for any theoretical model. Along these lines, over the years a number of possible strategies of attack on this complex subject have been elaborated and in the future, undoubtedly, more exciting activity in this area will be stimulated. The next three sections of this chapter provide a synthetic account of the current state of stability theory for ﬂuid motions. Basically, they are focused on a critical comparison of existing methods and possible strategies of analysis, illustrating advantages and disadvantages, strengths and weaknesses. Of course, such sections are necessarily limited in scope and depth, both because of the page limit and because of the amount of published literature on these topics, which is now enormous. Consequently, the aspects emphasized have been selected according to the opinion and experience of the present author. Owing to the elaborate nature of some of these studies, it has not always been possible to ﬁt an adequate account of them into the framework of the present chapter, but attempts are made to give some indication of the most important results and of the methods employed. The reader will quickly become aware that, although the transition problem is not yet solved, great progress has been made during the past decades and the possibility of decisive further advances has been uncovered. An exception is given solely by the treatment of the so-called ‘amplitude equations’ coming from weakly nonlinear theories (and similar approaches). They are expressly ignored here. These models are usually developed by resorting to simplifying assumptions which are believed to retain the essence of the system. Indeed, these simpliﬁed formulations have provided a great amount of valuable information over the years; however, they often preclude any detailed and quantitative comparison with real situations.

Equations, General Concepts and Methods of Analysis

1.5

27

Linear Stability Analysis: Principles and Methods

For a generic system of coupled nonlinear partial differential equations (PDEs), introducing the ‘state vector’ x(t), deﬁned as the column vector containing the unknowns of the problem (more precisely, the state variables are typically a set of time-dependent variables that, together with knowledge of any external inputs to the system, are sufﬁcient to describe the future time evolution of the system), the PDEs and related boundary conditions can be formally rewritten as dx (1.75) = (x) dt where is a nonlinear vector function of the state vector x. The equations are then said to be in state-space form (for a PDE system, the full state is inﬁnite-dimensional; for a discretized ﬂuid system, the state variables are the relevant ﬂuid variables at each spatial point). It is worth noting that unlike linear systems for which the system response upon the application of an initial disturbance δx on a given solution does not depend on the amplitude of the considered perturbation, for systems governed by nonlinear equations such as those of thermal convection, the possible stages of evolution (which the solution can undergo transition to) are highly sensitive to the initial intensity of the applied disturbances.

1.5.1 Conditional Stability and Inﬁnitesimal Disturbances The so-called linear stability analysis deals with the ‘conditional’ (or local ) stability of such systems (Joseph, 1976): a solution of a nonlinear system is said to be ‘conditionally’ stable (or unstable) when it is (un)stable to small inﬁnitesimal disturbances. Such a deﬁnition can be formalized according to the well-known notion of Lyapunov stability for ˜ ˜ > 0 such that, if ||δx(0)|| < δ, which a solution is stable if for every ε > 0, there exists a δ˜ = δ(ε) then ||δx(t)|| < ε for every t > 0, (where, in general, the norm || . . . || might be chosen in different ways). The considered solution, in particular, is said to be asymptotically stable if it is Lyapunov stable and additionally ||δx(t)|| → 0 in the limit as t → ∞. The linear stability analysis technique relies on the assumption that the behaviour of the considered system can be studied ‘locally’ and around (i.e. in a certain neighbourhood of) an equilibrium solution x 0 (for which dx 0 /dt = 0) simply by replacing the right-hand side of Eq. (1.75) with a linearized form of it (from which the denotation of linear analysis). From a formal point of view this can be obtained by considering the related multivariate Taylor expansion in series: • ∂ 1 ∂ 2 1 ∂ n 2 x = (x 0 ) + (x − x 0 ) + (x − x 0 ) + . . . (x − x o )n + . . . (1.76) ∂x 2! ∂x 2 n! ∂x n x0

x0

xo

and neglecting all the terms of order higher than one (this under the assumption given above that the considered displacement (x − x 0 ) with respect to the equilibrium solution is ‘sufﬁciently small’): • ∂ x= (x − x 0 ) (1.77) ∂x x0

where the partial derivative in the above equation is a square matrix that has to be interpreted as the Jacobian of the initial system of equations: ∂ (1.78) J = ∂x xo

Equation (1.77) provides the necessary mathematical basis for the application of linear stability analysis techniques.

28

Thermal Convection: Patterns, Evolution and Stability

The related mathematical treatment of the problem generally proceeds along the following lines. The initial ﬂow x 0 that represents a stationary state of the system is usually referred to as the ‘basic state’. By supposing that the various physical variables describing the ﬂow suffer small (inﬁnitesimal) increments δx, the equations governing these increments are obtained by simply taking into account that • • δx = (x − x 0 ) → δ x =x → and substituting into Eq. (1.77): •

δ x = J · δx

(1.79)

which gives the equation of evolution for the disturbances.

1.5.2 The Exponential Matrix and the Eigenvalue Problem A Taylor expansion in series of the increment δx with respect to time reads ∂(δx) 1 ∂ 2 (δx) 1 ∂ n (δx) 2 t + t + . . . tn + . . . δx = δx(0) + ∂t t=0 2! ∂t 2 t=0 n! ∂t n t=0

(1.80)

with t = time being measured from the initial time at which the inﬁnitesimal increments are supposed to be superimposed on the basic state. The ﬁrst-order and higher-order derivatives in such an expansion can be rewritten according to Eq. (1.79) as follows: ∂(δx) = J · δx(0) (1.81a) ∂t t=0

∂(J · δx) ∂ 2 (δx) = = J 2 · δx(0) ∂t 2 t=0 ∂t t=0 ∂ n (δx) = J n · δx(0) ∂t n t=0

(1.81b)

(1.81c)

and substituting Eqs (1.81) into Eq. (1.80) gives 1 2 J · δx(0)t 2 2! ∞ tk 1 + . . . J n · δx(0)t n + . . . = J k · δx(0) n! k!

δx(t) = δx(0) + J · δx(0)t +

(1.82)

k=1

In mathematics, the ﬁnal sum in Eq. (1.82) is known as the ‘exponential matrix’ with argument J t (it can be formally obtained by replacing the generic variable x with J t in the series expansion of the ordinary exponential function ex ) and is denoted eJ t =

∞ k t k J k!

(1.83)

k=1

Through the introduction of the matrix exponential, Eq. (1.82) can be rewritten in a relatively compact form as δx(t) = eJ t · δx(0)

(1.84)

At this stage, before going further into the mathematical bases and principles of the linear stability analyses it is worth recalling some fundamental deﬁnitions and notions (to be used for additional expansion and manipulation of the above relationships). For the sake of brevity, it is assumed,

Equations, General Concepts and Methods of Analysis

29

however, that readers of this book are familiar with the basic elements of linear algebra. For this reason, many well-known concepts will be provided hereafter without a thorough demonstration or extensive explanation. The so-called eigenvalues (λ) of the Jacobian matrix can be computed according to the following equation: det(J − λI ) = 0

(1.85)

The vectors deﬁned by the following equations: ξ k · J = λk ξ k

(1.86a)

J · χ k = λk χ k

(1.86b)

are the left and right eigenvectors of J , respectively. They satisfy the following relationships: 1 for i = j (1.87) ξi · χj = 0 for i = j known as the orthogonality condition, and J =

N

χ k λk ξ k

(1.88a)

k=1

known as the eigenvector representation (spectral decomposition) of J (under the assumption that J admits N distinct eigenvalues). By introducing the matrix M, collecting the right eigenvectors χ k of J in the form of matrix columns and observing that, as a consequence, its inverse M −1 will be a matrix whose rows correspond to the left eigenvectors ξ k of J , Eq. (1.88a) can be also formally written as J = M · · M −1

(1.88b)

where is a diagonal matrix whose elements are the aforementioned distinct eigenvalues of J ; the matrix M is known as the modal matrix (it is always invertible because the eigenvectors must be linearly independent given the assumption of distinct eigenvalues). According to Eq. (1.87), the following relationships also hold: J2 = Jn =

N k=1 N

χ k (λk )2 ξ k = M · 2 · M −1

(1.89a)

χ k (λk )n ξ k = M · n · M −1

(1.89b)

k=1

and as a consequence eJ =

N

χ k eλk ξ k = M · e · M −1

(1.90)

k=1

Accordingly, Eq. (1.84) can be represented as N χ k eλk t ξ k · δx(0) δx(t) =

(1.91a)

k=1

or δx(t) = (M · et · M −1 ) · δx(0)

(1.91b)

30

Thermal Convection: Patterns, Evolution and Stability

hence the most interesting implication of this equation is that the problem of stability can be reduced to an eigenvalue problem. The eigenvalues of the Jacobian are, in general, complex numbers: λk = uk + ivk

(1.92)

where uk and vk are, respectively, the real and imaginary parts of the eigenvalue. Each of the exponential terms in the expansion Eq. (1.91) can, therefore, be written as eλk t = euk t eivk t → eλk t = euk t [cos(vk t) + i sin(vk t)]

(1.93)

Equations (1.91) and (1.93) ﬁnally provide a simple theoretical basis on which the general principles (and related theorems) of the linear stability analysis can be illustrated. At this stage, it is evident, in fact, that while the complex part of the generic eigenvalue of the Jacobian may contribute an oscillatory component to the solution δx(t), its real part if positive can cause the amplitude of the disturbances to grow indeﬁnitely (as time increases), thereby moving the considered system away from the initial equilibrium condition x 0 . These arguments provide the necessary theoretical background for the following precise statements:

• The considered initial equilibrium point is asymptotically stable if and only if the Jacobian has eigenvalues only with strictly negative real parts, that is, they are located in the left half of the complex plane. • The considered initial equilibrium point is marginally stable (i.e. only Lyapunov stable) if and only if the real part of every eigenvalue of the Jacobian is non-positive. • The considered initial equilibrium point is unstable if at least one of the eigenvalues has a positive real part . These important extensions of the well-known principles of linearized stability for ordinary differential equations were obtained by Prodi (1962), Kirchg¨assner and Sorger (1968) and Sattinger (1970). In general, the method requires the determination of the exponents λk as a function of the characteristic number R (Section 1.4) and especially the critical value of R at which the real part of an exponent ﬁrst changes from negative to positive as R increases. As explained before, the loss of stability of x implies the existence of a new solution which bifurcates from x. The transition from steady to periodic behaviour as the characteristic parameter is varied (and as a consequence the initial solution x 0 is varied) corresponds to a Hopf bifurcation (Hopf, 1942), where an eigenvalue of the Jacobian matrix (in general a complex conjugate pair of eigenvalues) of the governing equations crosses the imaginary axis (i.e. the real part of the eigenvalue becomes positive with a corresponding value of imaginary part = 0). If a single real eigenvalue crosses through zero to the positive half plane (i.e. the real part of the eigenvalue becomes positive but the corresponding imaginary part is zero), the emerging new solution is steady, that is, some breakdown in the initial symmetry of the ﬂow occurs, but such a phenomenon is not associated with the onset of oscillatory convection (stationary bifurcation).

1.5.3 Linearization of the Navier–Stokes Equations For the speciﬁc case of interest in the present book, that is, the thermal-convection equations [Eqs (1.59)–(1.61)], the state vector can be deﬁned as   V (1.94) x= p  T

Equations, General Concepts and Methods of Analysis

and as a consequence

31



 δV δx =  δp  δT

(1.95)

Substituting x 0 + δx into Eqs (1.59)–(1.61) and linearizing with respect to the perturbation quantities (i.e. neglecting all products and powers of the increments higher than the ﬁrst while retaining only terms that are linear in them) yields ∇ · δV = 0 ∂ δV + V 0 · ∇(δV ) + δV · ∇(V 0 ) + ∇(δp) = Pr∇ 2 (δV ) + δF b ∂t ∂ δT + V 0 · ∇(δT ) + δV · ∇(T0 ) = ∇ 2 (δT ) ∂t

(1.96) (1.97) (1.98)

where δF b is the change eventually induced in the body forces acting on the system by the disturbance δx. The basic solution {V 0 , p0 , T0 } may be known explicitly in analytical terms or known only numerically. Since stability means stability with respect to all possible (inﬁnitesimal) disturbances, for an investigation on stability to be complete it is necessary that the reaction of the system to all possible disturbances be examined. This is accomplished by expressing an arbitrary disturbance as a superposition of certain basic possible modes and examining the stability of the system with respect to each of these modes. In general, the linearized stability problem admits solutions of the form δf eλt eib , where b is a linear function of the spatial coordinates. Let us consider, for instance, the stability of an initial steady two-dimensional solution in the (x, y) plane for a layer having depth d along the y-axis and inﬁnite extension along the x- and z-axes. The perturbations can be expressed with a Fourier expansion along the x and z directions:   δV   λt i(q x+q z) (1.99)  δp  = δf (y)e e x z δT where qx and qz are the disturbance wavenumbers along x and z, respectively. In a similar way, when studying the stability of an initial axisymmetric solution in a cylindrical or annular geometry, the general solution of the linear system of Eqs (1.96–1.98) can be expressed as the superposition of normal modes (r and ϕ being the radial and azimuthal angular coordinates, respectively):   δV   λt i(mϕ) (1.100)  δp  = δf (r, z)e e δT where m is the (integer) azimuthal wavenumber. As an example of realization of the general concepts outlined in Section 1.1.2, it is worth noting that for such a case the possible solutions represented by Eq. (1.100) can be directly related to the number of broken symmetries; the general possible symmetries of the problem are the rotations around the z-axis, the reﬂections with respect to the vertical planes containing this axis and, in some cases, the reﬂection with respect to the horizontal midplane (Bajaj, 1982; Golubitsky and Stewart, 1985; Crawford and Knobloch, 1991).

32

Thermal Convection: Patterns, Evolution and Stability

In both cases corresponding to Eqs (1.99) and (1.100), the insertion of the disturbances into the linearized equations leads to a generalized eigenvalue problem of the type A · χ = λB · χ

(1.101)

where A and B are the matrix representation of the set of linear volume equations and considered boundary conditions, λ = σ + iω, as explained earlier, is the eigenvalue (and χ denotes the eigenvector of the ﬁeld entities). Its real part σ is, in general, referred to as the disturbance growth rate and ω as its angular frequency. As illustrated before, the ﬂow is stable if, for given values of the considered characteristic numbers, σ is negative for all the possible values of the wavenumber (or wavenumbers if ampliﬁcation of the disturbance is possible along different directions); conversely, the ﬂow is unstable if σ is positive for some wavenumbers. The states for which σ is zero are called states of neutral stability. Finally, the neutral state for which the characteristic number attains a minimum (as a function of the wavenumbers) is usually referred to as the critical condition. This means that a system must be considered as unstable even if there is only one special mode of disturbance with respect to which it is unstable; and a system cannot be considered stable unless it is stable with respect to every possible disturbance to which it can be subjected. In other words, stability must imply that there exists no mode of disturbance for which it is unstable. If all initial states are classiﬁed as stable or unstable according to the criteria stated, then the locus in the space of parameters which separates these two classes of states deﬁnes the aforementioned conditions of neutral stability. The reader may be interested in knowing that the validity of these principles for the speciﬁc cases of interest in the present book has been proven by Pellew and Southwell (1940) (for pure buoyancy instabilities), Vidal and Acrivos (1966) (for pure thermocapillary ﬂow) and Takashima (1970) (for coupled buoyancy and thermocapillary instabilities).

1.5.4 A Simple Example: The Stability of a Parallel Flow with an Inﬂectional Velocity Proﬁle As a relevant example of the theory illustrated earlier and its potential, the reader may consider the canonical problem concerning the stability properties of a plane parallel ﬂow . In the present section, in particular, the limit of this problem for Pr → 0 is considered. Before embarking on the related mathematical treatment, it is worth noting that such a subject can be regarded as more than a simple test bed for the application of the principles of the linear stability analysis discussed before; in fact, it goes beyond such a simple purpose by providing relevant and illuminating insights for a better characterization of the stability of ﬂows of gravitational origin and/or induced by interfacial tension forces (if a free surface is present) that will be treated extensively in this book for low-Pr ﬂuids (Pr 1, e.g. liquid metals and semiconductor melts). Under the considered assumption of vanishing Prandtl number, the governing equations can be written as ∇ ·V = 0

(1.102)

∂V (1.103) + V · ∇V + ∇p = F b ∂t The consideration of the energy equation is no longer necessary as for Pr → 0 it becomes uncoupled from the mass and momentum conservation equations and, as a consequence, the distribution of temperature does not depend on convection (i.e. it is ﬁxed and satisﬁes ∇ 2 T = 0); accordingly, if F b were dependent on the temperature ﬁeld, it would behave merely as a function of the space coordinates, that is, it would not depend on the solution of Eqs (1.102) and (1.103) and related evolution in time.

Equations, General Concepts and Methods of Analysis

33

Figure 1.3 Sketch of a plane-parallel ﬂow

Let us consider a velocity ﬁeld representing a parallel ﬂow [i.e. u0 = u(y) and v = 0] in the x, y plane (x being the main inﬁnite direction of ﬂow and u and v the velocity components along x and y, respectively; see Figure 1.3). It can easily be veriﬁed that such a ﬂow satisﬁes the governing equations regardless of the shape of the function u = u(y) (all the equation terms, in fact, are zero with the exception of ∇p and F b , which balance each other); obviously, u(y) can be also ‘built’ to take into account the possible presence of tangential surface forces at the boundaries (∂u/∂y ∝ surface stress). It was proved by Squire (1933) that for this problem the most critical disturbances are two-dimensional (Squire’s theorem). Adding a two-dimensional inﬁnitesimal disturbance to the basic solution described earlier gives   u0 (y) + δu  δv (1.104) x 0 + δx =  p0 + δp The governing equations, Eqs (1.102) and (1.103), become ∂δu ∂δv + =0 ∂x ∂y ∂δu ∂δu ∂(u0 + δu) ∂p0 ∂δp + (u0 + δu) + δv + + = F b · ix ∂t ∂x ∂y ∂x ∂x ∂δv ∂δv ∂δv ∂p0 ∂δp + (u0 + δu) + δv + + = F b · iy ∂t ∂x ∂y ∂y ∂y

(1.105a) (1.106a) (1.106b)

where i x and i y are the unit vectors along the x and y directions, respectively. Taking into account the governing equations written for the basic state and linearizing the momentum equations, Eq. (1.106), by omitting quadratic terms, it follows that ∂u0 ∂δp ∂δu ∂δu + u0 + δv =− ∂t ∂x ∂y ∂x ∂δp ∂δv ∂δv + uo =− ∂t ∂x ∂y Eliminating pressure by cross-differentiation: du0 ∂ ∂δv ∂δu ∂δv ∂ ∂δu + u0 + δv = + u0 ∂y ∂t ∂x dy ∂x ∂t ∂x

(1.107a) (1.107b)

(1.108)

Introducing the stream function by ∂ψ ∂y ∂ψ δv = − ∂x

δu =

(1.109a) (1.109b)

34

Thermal Convection: Patterns, Evolution and Stability

then Eq. (1.108) reads

ψyt + u0 ψxy −

du0 ψx dy

= −(ψxt + u0 ψxx )x

(1.110)

y

Introducing a two-dimensional wave-like disturbance: ψ = Q(y)eλt ei(qx) = Q(y)eσ t+i(ωt+qx) = Q(y)eiq(x−Ct) where C = Cr + iCi = −

(1.111)

ω iσ + , then Eq. (1.110) reads q q

(−iqCQ + u0 iqQ − u0 Qiq)y eiq(x−Ct) = [i 2 q 2 CQeiq(x−Ct) − u0 i 2 q 2 Qeiq(x−Ct) ]x → (u0 − C)(Q − q 2 Q) − u0 Q = 0 (1.112) which is known as the Rayleigh’s equation in hydrodynamic instability. This equation can be used to obtain an important theorem about necessary conditions for instability of parallel ﬂows in the limit as Pr → 0. Theorem: In a shear ﬂow, a necessary condition for instability is that there must be a point of inﬂection in the velocity proﬁle u = u(y), that is, a value of y where: d2 u0 =0 dy 2

(1.113)

A proof of such a theorem is due to Rayleigh (1880) and runs as follows. Equation (1.112) can be rewritten as (Q − q 2 Q) −

u0 Q=0 (u0 − C)

(1.114)

Then it is supposed that the ﬂow is unstable to the considered disturbance δx (i.e. the disturbance growth rate σ is positive → Ci > 0). Multiplying Eq. (1.114) by the complex conjugate Q∗ of Q, we obtain (Q Q∗ − q 2 QQ∗ ) −

uo QQ∗ = 0 (uo − C)

(1.115a)

which can be rewritten by partial integration as u0 d (Q∗ Q ) − |Q |2 − q 2 |Q|2 − |Q|2 = 0 dy (u0 − C)

(1.115b)

Integrating between the boundaries y = − 1/2 and y = 1/2 and using the boundary condition Q(− 1/2) = Q( 1/2) = 0, it follows that

1 2 1 −2

(|Q |2 + q 2 |Q|2 )dy+

1 2 1 −2

u0 |Q|2 dy = 0 u0 − C

(1.116)

Moreover, since u0 u (u0 − Cr + iCi )|Q|2 |Q|2 = 0 u0 − C |u0 − C|2

(1.117)

then Eq. (1.116) becomes

1 2 1 −2

2

(|Q | + q |Q| )dy+ 2

2

1 2 1 −2

u0 (u0 − Cr + iCi )|Q|2 dy = 0 |u0 − C|2

(1.118)

Equations, General Concepts and Methods of Analysis

35

and its imaginary part reads Ci

1 2 1 −2

u0 |Q|2 dy = 0 |u0 − C|2

(1.119)

Thus, according to Eq. (1.119), for instability it is necessary that u0 is positive for some y and negative elsewhere, i.e. it must vanish somewhere in the ﬂow (the velocity proﬁle must have an inﬂection point inside the ﬂow). This is the Rayleigh necessary condition for instability. It should be emphasized that even if it was derived as a condition necessary for the instability, in practice, as illustrated by Tollmien (1936), it can be regarded also as a sufﬁcient condition in many situations. For more comprehensive discussions, the interested readers may consider Lin (1944), Rosenbluth and Simon (1964), Drazin and Howard (1966) (who reviewed the theory of the linear instability of parallel ﬂows of inviscid ﬂuids, giving emphasis to its possible applications in the ﬁelds of astrophysics, engineering, meteorology, oceanography and some cases of technological interest) and more recent work by Balmforth and Morrison (1999). Let us also recall that Squire’s theorem applies strictly to non-viscous systems and that the fastest-growing unstable mode of a dissipative (Pr > 0) parallel shear ﬂow is generally three-dimensional (see, e.g., Smyth and Peltier, 1990). As anticipated at the beginning of this section, the arguments and theorems elaborated here ﬁnd important direct or indirect application in explaining the dynamics of various types of ﬂows that will be treated in subsequent chapters.

1.5.5 Weaknesses and Limits of the Linear Stability Approach Direct application of the stability analysis as illustrated in Sections 1.5.1 and 1.5.2 is possible only in the simplest cases when initial stationary ﬂow can be obtained analytically (as in the example treated in Section 1.5.4). In general, when the initial stationary ﬂow must be obtained numerically and a set of all possible perturbations has to be considered, application of the stability theory (as illustrated in Section 1.5.3) leads to a generalized eigenvalue problem of very high order, namely the order of the eigenvalue problem is equal to the number of unknown scalar variables used by a numerical method for approximation of the solution: number of unknown functions multiplied by number of discretization elements (nodes of grid, ﬁnite elements or collocation points). This means that the possibilities of applying stability analysis for investigation of convective ﬂows remain strongly restricted because of the very large order of the eigenvalue problems to be solved. As the dimension of the Jacobian, which varies as the number of discretization elements times number of unknowns, increases very rapidly, direct methods using explicit building of the Jacobian are limited to conﬁgurations in which the bifurcation takes place at relatively small values of the characteristic number so that the solution can be adequately approximated by relatively low spatial resolution. Superimposed on this bottleneck, there are some drawbacks of conceptual nature. The unravelling of the linearized stability problem can certainly be regarded as a necessary ﬁrst step in the elucidation of the transition process, but it is clear that with such an approach many important questions remain unanswered. As a supposed explanation of transition, linearized stability theory has some serious deﬁciencies, of which the following are perhaps the most crucial: As Eq. (1.91) shows, the linear theory predicts that an unstable disturbance grows exponentially without limit. Exponential growth may be an adequate description of the behaviour of a disturbance in its incipient stage, but it cannot be acceptable as a description of the evolution of the disturbance over a period of time. Moreover, the stability should consider the fate of all disturbances which are physically possible (not the inﬁnitesimal ones only).

36

Thermal Convection: Patterns, Evolution and Stability

Over the past four decades, research on the transition problem has been concentrated on attempting to resolve the questions listed above. From a mathematical point of view, this has meant a return to the study of the nonlinear problem deﬁned by Eqs (1.59)–(1.61), and the formidable nature of this system has necessitated special modes of attack on particular aspects (see Sections 1.6 and 1.7).

1.6

Energy Stability Theory

One approach that has enjoyed considerable success has been concerned with the determination of a sufﬁcient condition for stability. In other words, this may be regarded as the following question: is there a value RE such that the basic ﬂow is globally (unconditionally) stable if R < RE ? In the afﬁrmative case, such a question can be formalized as follows: For every δx(0), ||δx(t)|| → 0 in the limit as t → ∞. This condition means that the disturbance tends to zero for all possible choices of it, i.e. for disturbances of arbitrary amplitude. A ﬁrst attempt to answer the above question was made by Orr (1907) and subsequent contributions were provided by several authors (e.g. Hopf, 1941). The modern approach, however, is due to Serrin (1959), whose pioneering work stimulated and laid the basis for important results by others, especially Joseph (Joseph, 1966). These problems have a rich mathematical and physical structure and are among the most important problems in applied mathematics. They come under the common heading of energy theory. The analysis begins in the usual way by assuming a solution to the governing equations which consists of the basic state plus a disturbance: ˜ V = V 0 + v,

p = p0 + p, ˜

T = T0 + T˜

(1.120)

with ∇ · v˜ = 0 over D v| ˜ ∂DW = 0 ˆ ∂DS = 0 v˜ · n|

(1.121) (1.122a) (1.122b)

where, as already explained in Section 1.3, ∂D = ∂DW ∪ ∂DS is the frontier of D and nˆ the unit vector perpendicular to it.

1.6.1 A Global Budget for the Generalized Disturbance Energy Energy theory proceeds from this point by taking the inner product of the disturbance momentum equation with v, ˜ multiplying the disturbance energy equation by cT˜ (the meaning of the parameter c will be explained later in this section), adding these equations together and integrating the result over the volume D occupied by the ﬂuid. Assuming that F b depends on T (the reader is referred to Chapter 2 for further elaboration of this aspect) and considering an expansion in series with respect to T˜ : ∞ 1 ∂ k F b ˜ k ˜ T (1.123) F b (T0 + T ) = F b (T0 ) + k! ∂T k T0 k=1

the disturbance momentum equation reads

∞ ∂ v˜ 1 ∂ k F b ˜ k T + V 0 · ∇ v˜ + v˜ · ∇V 0 + v˜ · ∇ v˜ + ∇ p˜ = Pr∇ 2 v˜ + ∂t k! ∂T k T0 k=1

(1.124)

Equations, General Concepts and Methods of Analysis

Multiplying by v˜ and taking into account the following identities: v˜ 2 v˜ 2 (V 0 · ∇ v˜ + v˜ · ∇ v) ˜ · v˜ = ∇ · V 0 + v˜ 2 2 ˜ ∇ p˜ · v˜ = ∇ · (p˜ v) 2 v˜ − ∇ v˜ : ∇ v˜ ˜ · v˜ = ∇ · (∇ v˜ · v) ˜ − ∇ v˜ : ∇ v˜ = ∇ · ∇ (∇ 2 v) 2

37

(1.125a) (1.126a) (1.127a)

assuming that the velocity is equal to zero on the overall frontier ∂D, i.e. ∂D = ∂DW , integrating over D and applying the divergence theorem: 2 v˜ 2 v˜ 2 v˜ dD = ∇ · V 0 + v˜ ˆ dS = 0 (1.125b) (V 0 · nˆ + v˜ · n) 2 2 2 D ∂D ∇ · (p˜ v)dD ˜ = p( ˜ v˜ · n)dS ˆ =0 (1.126b)

D

∂D

(∇ v) ˜ · vdD ˜ =−

∇ v˜ : ∇ vdD ˜ =0

2

D

it follows that 1 d 2 dt

(1.127b)

D

v˜ 2 dD = − D

D

+

(v˜ · ∇V 0 ) · vdD ˜ − Pr

(∇ v˜ : ∇ v)dD ˜ D

∞ 1 ∂ k F b ˜ k T ·vdD ˜ k! ∂T k T0 D

(1.128)

k=1

known as the Reynolds–Orr equation (Reynolds, 1895; Orr, 1907). If the boundary is not limited to solid walls ∂DW (where V = v˜ = 0 due to no-slip conditions), but a free surface ∂DS is present (where v˜ · nˆ = 0 and the surface stress is balanced by surface tension forces acting in tangential direction), the second member of the above equation also includes (Mittelman, 1994) a contribution related to the work done per unit time by tangential surface tension forces F Tσ , that is 1 d v˜ 2 dD = − (v˜ · ∇V 0 ) · vdD ˜ − Pr (∇ v˜ : ∇ v)dD ˜ 2 dt D D D ∞ 1 ∂ k F b ˜ k T ·vdD + + (F Tσ |T =T˜ · v)dS ˜ (1.129) ˜ k! ∂T k T0 D ∂DS k=1

The additional contribution represents the power input to energy given by disturbances acting on the free surface ( F Tσ |T =T˜ is the tangential component of the surface tension forces related to T˜ ; in practice, it depends on ∇ T˜ , as will be illustrated in Chapter 2). The (internal) energy disturbance equation reads ∂ T˜ + V 0 · ∇ T˜ + v˜ · ∇T0 + v˜ · ∇ T˜ = ∇ 2 T˜ ∂t

(1.130)

Multiplying by T˜ and taking into account that

T˜ 2 T˜ 2 + v˜ (V 0 · ∇ T˜ + v˜ · ∇ T˜ )T˜ = ∇ · V 0 2 2 (∇ 2 T˜ )T˜ = ∇ · (T˜ ∇ T˜ ) − ∇ T˜ · ∇ T˜

(1.131a) (1.132a)

38

Thermal Convection: Patterns, Evolution and Stability

and integrating over D and using the divergence theorem: ˜2 T˜ 2 T˜ 2 T dD = ∇ · V0 ˆ dS = 0 + v˜ (V 0 · nˆ + v˜ · n) 2 2 2 D ∂D T˜ (∇ T˜ · n)dS ˆ (∇ 2 T˜ )T˜ dD = − (∇ T˜ · ∇ T˜ )dD + D

D

(1.131b) (1.132b)

∂D

then the disturbance energy equation can be ﬁnally written in integral form as 1 d T˜ 2 dD = − T˜ (∇ T˜ · n)dS ˆ (v˜ · ∇T0 )T˜ dD − (∇ T˜ · ∇ T˜ )dD + 2 dt D D D ∂D

(1.133)

At this stage, as explained before, a global budget for the generalized disturbance energy can be simply obtained by combining Eqs (1.129) and (1.133) through the parameter c. The result of this process is an exact equation for the volume integrated disturbance energy: dE˜ = −Iε − Ii + I℘ + IS dt

(1.134)

and this energetic identity can be regarded as a functional that governs the time rate of change of the global measure of system energy. In such an equation: (v˜ 2 + cT˜ 2 )dD (1.135a) E˜ = D

is the generalized disturbance energy; Iε = [Pr(∇ v˜ : ∇ v) ˜ + c(∇ T˜ · ∇ T˜ )]dD

(1.135b)

D

represents the viscous dissipation and thermal dissipation rates; Ii = [(v˜ · ∇V 0 ) · v˜ + c(v˜ · ∇T0 )T˜ ]dD

(1.135c)

D

represents the viscous and thermal injection rates (energy production terms due to interaction of the disturbances with the basic state; in other words, these terms measure the power exchange between the basic ﬂow and the disturbance); ∞ 1 ∂ k F b ˜ k T · vdD ˜ + (F Tσ |T =T˜ · v)dS ˜ (1.135d) I℘ = k! ∂T k T0 D ∂DS k=1

is the power input to E˜ due to body and surface forces; ˆ ˆ cT˜ (∇ T˜ · n)dS = cT˜ (∇ T˜ · n)dS + IS = ∂D

∂DW

ˆ cT˜ (∇ T˜ · n)dS

(1.135e)

∂DS

Is is the sum of two contributions: the ﬁrst, related to ∂DW , is always zero (as, in general, solid walls are assumed to have a ﬁxed imposed temperature or to be adiabatic → T˜ = 0 or ∇ T˜ · nˆ = 0, respectively); the second takes into account possible heat exchange at ∂DS and becomes zero if the considered surface is thermally insulated. At this stage, the meaning of the parameter c can be clariﬁed. The constant c is a coupling parameter used to combine the kinetic energy with T˜ 2 . Since it is a free parameter and since energy theory is intended to provide a sufﬁcient condition for stability, in general, c is selected to maximize the value of the stability limit (it is restricted, however, to be positive so that E˜ remains a positive deﬁnite quantity).

Equations, General Concepts and Methods of Analysis

39

1.6.2 The Extremum Problem The energy identity Eq. (1.134) can be used to formulate a maximum problem that would ensure that E˜ approaches zero as time approaches inﬁnity and provide a condition for global stability. A reformulation of energy theory described by Davis and von Kerczek (1973) deﬁnes this maximum ˜ resulting in problem. The theory begins by dividing both sides of the energy identity by E, 1 1 dE˜ = (−Iε − Ii + I℘ + IS ) E˜ dt E˜

(1.136)

Since E˜ is a deﬁnite-positive functional, determining the maximum of this new functional over a set of kinematically admissible disturbances H provides the following inequality: (−Iε − Ii + I℘ + IS ) 1 dE˜ = σE ≤ max H E˜ dt E˜

(1.137)

where H is deﬁned by Eqs (1.121) and (1.122); σE is a scalar and independent of time since the basic state is assumed to be steady. Integrating this expression over time from 0 to τ results in σE τ ˜ ) ≤ E(0)e ˜ E(τ

(1.138)

Hence the basic state is asymptotically stable from a global point of view according to this inequality when σE < 0. In general, the problem is formulated in a way such that the characteristic number R is the stability parameter; then R ∗ is deﬁned as the smallest value of R for which σE = 0 (in practice, the stability problem is one of variational calculus to determine the R that maximizes the functional at a value of zero). The limit RE that guarantees sufﬁcient conditions for stability is ﬁnally computed as RE = max R ∗ c>0

(1.139)

This ensures that all kinematically admissible disturbances decay monotonically in time when R < RE . It is worth highlighting that results obtained in the framework of this theory are not limited to the mere evaluation of the aforementioned stability boundary. The value of the parameter c that maximizes the energy limit, in fact, also provides useful information on the nature of the critical disturbance (as being primarily hydrodynamic or thermal). It is also worth comparing this theory with the linear approach treated earlier, which provides additional illuminating insights into both techniques. As widely explained, linear and energy stability theories provide two extremes which prove instability or stability, respectively. Linear theory is a local analysis assuming that disturbances are small so that the nonlinear terms are negligible. Then a limit is determined to guarantee that inﬁnitesimal disturbances will grow exponentially in time ensuring unstable ﬂow. Energy theory integrates a generalized ‘energy’ of the disturbances over the considered spatial domain and provides a global stability limit by ensuring that a class of functions, which includes the energy of all disturbances, decays in time. If RE and the value provided by the linear stability analysis (RL ) should coincide, a rigorous stability bound would be obtained. However, this is usually not the case (in general RE < RL ) and the proximity of RE to RL depends on the physical mechanism that gives rise to the instability (see, e.g., Sagalakov and Shtern, 1971 for the case of ﬂows having velocity proﬁle with an inﬂection point, already treated in Section 1.5.4 in the framework of the linear theory). Indeed, as highlighted by Mittelman (1994), two such mechanisms for which RE and RL may be expected to be relatively close to each other are two of those treated in the present book, namely buoyancy and thermocapillarity.

40

1.7

Thermal Convection: Patterns, Evolution and Stability

Numerical Integration of the Navier–Stokes Equations

As widely illustrated in the preceding sections, mathematical modelling is the art and craft of building a system of equations that is both sufﬁciently complex to do justice to physical reality and sufﬁciently simple to focus on the most signiﬁcant aspects of the given situation. Stability analysis is the art of deducing fundamental information about the behaviour (in terms of possible evolution) of certain solutions (or categories of solutions) of these equations working on an alternative set of equations derived from the original ones through mathematical manipulation (e.g. linearization or integration in space). Despite the aforementioned important information offered to investigators by linear stability analyses and theories based on energy budgets and variational principles, it is the opinion of the author (and the present book provides a solid basis for such a conviction) that the recent development of numerical methods and high-performance computing is enabling the possibility to approach these problems in terms of direct numerical solution of the thermal convection equations in their nonstationary, nonlinear and complete form. Along these lines, it should be stressed how in recent years improvements in computer hardware performance have occurred hand-in-hand with decreasing hardware costs. Consequently, for a given numerical algorithm and problem, the relative cost of a computational simulation has decreased signiﬁcantly. Moreover, paralleling the improvement in computer hardware has been an improvement in the efﬁciency of computational algorithms (and current improvements in hardware costs and computational algorithm efﬁciency seem to show no obvious sign of reaching a limit). All these factors combine to make numerical computations increasingly cost effective in comparison with other strategies of analysis. Moreover, this philosophy provides a wealth of additional aspects that cannot be obtained on the basis of linear stability computations and energy stability theories, and in particular:

• the effective amplitude of the disturbances when they are saturated • the nonlinear behaviour of the system far from the threshold point • a quite exhaustive picture of the subsequent stages of temporal evolution. Linearized theories do not provide quantitative data about the amplitude of the disturbances after a new state has emerged (i.e. after the bifurcation has occurred) and give only qualitative information about their spatial structure. Of course, the straightforward direct integration of the unsteady nonlinear equations also has some bottlenecks. In particular, this approach is not ideally suited for determining the structure of the unstable eigenmodes or for the accurate determination of the critical characteristic number corresponding to a bifurcation (in general, in fact, linear stability analysis provides a better understanding of the instability phenomena and also more precise critical values of the governing parameters). A synthesis of the advantages and bottlenecks of linear stability analysis compared with straightforward direct integration has recently been provided by Gelfgat (2007a,b) for thermal convection in some canonical geometries. There is no doubt, however, that most of the progress in recent years in the understanding of the structure and dynamics of thermal convection (especially concerning the nonlinear behaviour) has been obtained through the latter approach. For this reason, this section is devoted to illustrating various possible methods, their genesis, theoretical basis and directions of improvement, while providing (in line with the spirit of this chapter) a critical comparison of them (Lappa, 2002a). Prior to expanding on such a theoretical framework, it is opportune, however, to open a short discussion about the way in which these methods can capture ﬂow instabilities, that is, the relationships between the application of these techniques and the theory of stability illustrated in the previous sections. Such a relationship is not obvious and requires some clariﬁcation.

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As described in Section 1.5, linear stability analysis relies on a spectral problem where the stability properties of the basic state are gathered from the behaviour of the eigenvalues of the Jacobian of the linearized equations obtained by adding inﬁnitesimal disturbances to the basic state (the stability is strictly related to the sign of the real parts of the eigenvalues). In general, the addition of such disturbances is not necessary when the problem is approached in terms of straightforward time integration of the governing nonlinear equations in discretized form. No a priori -known small disturbances have to be superimposed on the solution, since such disturbances, in fact, may be produced freely (i.e. in a spontaneous way) by the computations. In practice, disturbances perturbing the system inﬁnitesimally follow directly from the truncation error originating from the ﬁnite length of the computer registers storing the ﬂow and temperature values (see also Section 1.8.3 for additional insights into this process). In other words, this means that the role of the inﬁnitesimal disturbances is played by the numerical noise introduced in the computation by the round-off errors of the memory registers of the computer used for the simulation [for instance, if double precision is used, the initial disturbances are of the order O(10−16 )]. Remarkably, no initial guesses are necessary about the nature and shape of the disturbances. Interestingly, according to some numerical experiments available in the literature, if arbitrary very small disturbances were added to the initial solution, the computations would show noncritical inﬁnitesimal disturbances to decay, whereas the dangerous part (i.e. with the most dangerous value of the wavenumber as predicted by the linear stability analysis) has to be ampliﬁed. This simple observation has led some investigators to add small disturbances in their computations with the most dangerous value of the wavenumber predicted by the linear stability results for the problem under consideration. This has been done in an attempt to shorten the time required for the free selection and ampliﬁcation of perturbations (that otherwise might be rather lengthy and time consuming, especially when performing computations with a very high precision). It is obvious that such a strategy, however, can be adopted only when linear stability results for the considered problem are already known. In the opposite case, the equations must be integrated long enough to let the solution converge to the least damped or most ampliﬁed disturbance and the instability to manifest itself.

1.7.1 Vorticity Methods In some cases, equivalent forms of the momentum equation, Eq. (1.60) (equivalent in the continuum, but generally not equivalent when spatially discretized), can be used to advantage. These alternative equations are all derived from the ‘primitive’ equation from differentiation (in part) and often after additional insights regarding incompressible ﬂow. They are also often used as the starting point for generating alternative numerical solution methods (Gresho, 1991). For instance, for two-dimensional cases, streamfunction–vorticity (ψ − ζ ) methods have enjoyed widespread use during recent years owing to computational efﬁciency and simplicity. The major advantage of these method relies on the possibility of solving the equations leaving aside problems related to the pressure gradient that is usually regarded as a ‘strange’ object by numerical investigators (given its subtle connection to the other problem unknowns; see Section 1.7.2). For the same reason, methods based on velocity and vorticity as the problem variables have witnessed an increase in interest in application to three-dimensional problems. In such methods, the presence of the pressure gradient is removed through application of the curl operator to the momentum equation (the curl of the gradient of a scalar quantity, in fact, is zero). Applying the curl operator as illustrated in Section 1.2.8 [see Eq. (1.52)], taking into account the following vector identities (and using the fact that both V and ζ = ∇ ∧ V are div-free): ∇ · [2(∇V )so ] = ∇ · [∇V + ∇V T ] = [∇ 2 V + ∇(∇ · V )] = ∇ 2 V

(1.140)

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Thermal Convection: Patterns, Evolution and Stability

∇ ∧ (∇ ∧ V ) = ∇(∇ · V ) − ∇ 2 V → ∇ 2 V = −∇ ∧ (∇ ∧ V ) = −∇ ∧ ζ

(1.141)

∇ ∧ (∇ ∧ ζ ) = ∇(∇ · ζ ) − ∇ ζ → 2

−∇ ∧ (∇ ∧ ζ ) = ∇ 2 ζ

(1.142)

Equation (1.52) in nondimensional form and for incompressible ﬂow can be rewritten as ∂ζ

(1.143) + V · ∇ζ = ζ · ∇V + Pr∇ 2 ζ + ∇ ∧ F b ∂t This equation is not sufﬁcient for establishing a ‘closed’ problem as it contains both the unknowns ζ and V . Towards such an end, another higher order equation can be introduced taking into account Eq. (1.141), which provides a Poisson equation for the velocity: ∇ 2 V = −∇ ∧ ζ

(1.144)

which is still a kinematic equation (corresponding to three scalar elliptic equations in the three-dimensional case). In practice, Eqs (1.143) and (1.144) are at the basis of the so-called velocity–vorticity formulations (e.g. Farouk and Fusegi, 1985; Orlandi, 1987; Speziale, 1987; Guj and Stella, 1988, 1993; Stella and Guj, 1989; Dacles and Hafez, 1990; Napolitano and Pascazio, 1991; Pascazio and Napolitano, 1996; Lo et al., 2005). Some numerical investigators (e.g. Aziz and Hellums, 1967) have also tried another variant based on the introduction of a 3D streamfunction. deﬁned as ∇ ∧ =V

(1.145)

known as the 3D vorticity/vector–potential formulation. Accordingly, the dimensionless equations governing the evolution of ζ and are Eqs (1.143) and (1.146), respectively: ∇∧∇∧ =ζ

(1.146)

These methods on the one hand have enjoyed reasonable success for the aforementioned reasons, but on the other hand have been characterized by some conceptual difﬁculties, mainly originating from an incomplete knowledge of the boundary conditions making the problem well posed from a computational/theoretical point of view (Gresho, 1991).

1.7.2 Primitive Variables Methods A different class of methods not needing mathematical manipulation and transformation (i.e. the introduction of the vorticity through application of the curl operator), however, has been simultaneously developed. In such a class the pressure gradient is not eliminated. Rather, it is used for introducing (with the aid of the continuity equations) an additional equation that leads to a closed set of equations. The major advantage offered by this category of methods relies on the possibility of solving the problem directly in terms of the primitive variables (i.e. velocity and pressure) of the ﬂuid. Superimposed on this, the problem related to the boundary conditions that make the problem well posed has been investigated to a certain extent, leading to widely recognized theoretical results (Gresho and Sani, 1987). These techniques are known under several names: projection method, fractional-step method or pressure-correction method (hereafter simply referred to as primitive-variables methods). The approach was originally introduced by Harlow and Welch (1965) and Welch et al. (1965) as the MAC method and successively modiﬁed in the projection method developed independently by Chorin (1968) and Temam (1968, 1969). These techniques have been widely studied since then

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(relevant and important studies include Moin and Kim, 1980; Kim and Moin, 1985; Van Kan, 1986; Orszag et al., 1986; Bell et al., 1989; Gresho, 1991; Karniadakis et al., 1991; Temam, 1991; Shen, 1992, 1996; Rannacher, 1992; Quartapelle, 1993; Perot, 1993; E and Liu, 1995; Guermond, 1996; Lappa, 1997; Guermond and Quartapelle, 1998; Strikwerda and Lee, 1999; Armﬁeld and Street, 1999; Lappa and Savino, 1999; Brown et al., 2001; Lee et al., 2001; Petersson, 2001; Armﬁeld and Street, 2002; Guermond et al., 2006). Despite some minor differences, basically, a common feature of all these methods is that they were conceived to ‘turn around’ the coupling between the pressure and the velocity that is implied by the incompressibility constraint: div(V ) = 0. Due to page limits, the discussion here is limited to illustrating some fundamental concepts and guiding principles (the reader is referred to Guermond et al., 2006, for a more comprehensive treatment of the subject). The basic idea consists of devising time-marching procedures that uncouple viscous and incompressibility effects. In practice, the theoretical foundation of these algorithms is given by the ‘inverse theorem of calculus’. By this theorem, in fact, a vector a of which the divergence, the curl and the normal component at the boundary are given, ∇ · a = s in D ∇ ∧ a = q in D

(1.147) (1.148)

a · n = fn on ∂D with

sdD = D

fn dS

(1.149)

∂D

is uniquely determined (D being a simple connected domain). This is the reason why all the variants of primitive-variables techniques have been conceived in order to ‘preserve’ the divergence and the curl of the vector ∂V /∂t related to the original set of equations. The philosophy leading to the introduction of this kind of method is illustrated to a certain extent in the following. It requires that the momentum equation is discretized with respect to time. For instance, forward differences in time can be used for such a purpose (of course, this is just an example used here for the convenience of the reader; high-order temporal schemes should be used to obtain reliable accuracy in effective applications), obtaining V n+1 = Z(V n , F b ) − t∇pn

(1.150)

where the operator Z reads Z(V , F b ) = V + t[−∇ · (V V ) + + tF b and the superscript n denotes the time step. The pressure ﬁeld can be computed by solving a Poisson equation resulting from the divergence of the momentum equation, Eq. (1.150), with the help of the continuity equation ∇ · V n+1 = 0: Pr∇ 2 V ]

1 (1.151) (∇ · Z n ) t This means that in this type of method the incompressibility constraint (continuity equation) does not appear directly, but is accounted for through the above Poisson equation solved for the pressure. Remarkably, unlike the vorticity-based methods that require the solution of three scalar time-consuming elliptical equations [Eq. (1.144) or (1.146)], this approach leads to a single equation of such a type [Eq. (1.151)]. Some difﬁculties arise, however, due to the necessity to introduce proper boundary conditions for this equation. In general, a boundary condition is referred to as a physical boundary condition (PBC) when it speciﬁes the known physical behaviour of one or more of the dependent variables at the boundaries (e.g. no-slip conditions on solid walls or the surface tension-driven stress condition on a ﬂuid/ﬂuid interface). These conditions are independent of the numerical method used to solve the relevant equations. ∇ 2 pn =

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Thermal Convection: Patterns, Evolution and Stability

Knowing which PBCs to impose is not sufﬁcient to solve the problem numerically. Other conditions have to be added to the original set of PBCs. A boundary condition is referred to as a ‘numerical’ (NBC) when no explicit boundary condition ﬁxes one of the dependent variables, but the numerical implementation requires that something is speciﬁed about this variable. It is called ‘numerical’ because it appears to be needed for the numerical method while not being explicitly given by the physics of the problem. This is the case with Eq. (1.151). A rigorous method to introduce NBCs (when needed) is to use the conservation equations themselves on the boundary to complement the set of PBCs. Variables which are not imposed by PBCs are computed on the boundaries by solving the same conservation equations as in the domain. This leads to the introduction of NBCs in the form of ‘compatibility relations’. It is worth noting that in the case of the Poisson pressure equation, the introduction of compatibility relations, coming from evaluation of the pressure gradient on the boundary, is not straightforward as one would assume, since it involves three different components of this gradient along x, y and z. It has been shown, however, that, for retaining coherence with respect to Eq. (1.151), i.e. for obtaining a well-posed problem, at the physical boundaries of the computational domain the pressure must satisfy conditions obtained by simply projecting the operator Z along the direction perpendicular to the boundary itself, i.e. 1 ∂p = Z · nˆ ∂n t

(1.152)

where nˆ is the unit vector pointing into the liquid domain. Gresho and Sani (1987), in particular, were the ﬁrst to demonstrate that the most reliable boundary condition (BC) is simply the Neumann BC obtained by applying the normal component of the momentum equation on the frontier of the domain [i.e. Eq. (1.152)] because this is the only BC that can always assure that ∇ · V = 0 in the domain. Subtracting the Poisson equation from the divergence of the momentum equation yields ∂(∇ · V ) =0 ∂t

(1.153)

This equation is also important since it shows that ∇ · V = 0 remains zero in the computational domain if it is zero at the initial instant, i.e. ∇ · V 0 = 0. Equations (1.150) and (1.151) can be used for a time-marching procedure where pn is computed as a function of V n [Eq. (1.151)] and V n+1 is computed as a function of V n and pn [Eq. (1.150). This approach was referred to as the consistent pressure Poisson equation (CPPE) by Gresho and Sani (1987). If this CPPE formulation is employed and the initial velocity ﬁeld is div-free, according to Eq. (1.153) the solution will be div-free for all time. In contrast, the initial conditions for vorticity formulations are poorly understood: there are no constraints and therefore the problem is often ill posed (Gresho, 1991). There are no BCs on the vorticity, but the value (which is often poorly behaved) of the vorticity on the boundary does play a major role in most vorticity-based methods. Although vortex sheets are little more than a minor inconvenience in primitive variable formulations, they are often the principal focus of attention in vorticity-based methods, partly because they are computationally cumbersome. Vorticity is a challenging variable to compute. Usually more grid points are needed in a momentum boundary layer (see Section 2.5) with a vorticity method with respect to a primitive variables method to obtain the same accuracy and the prescription of an effective solid-surface boundary condition is often a weak point. Some difﬁculties arise, however, also with the primitive variable formulation due to the complexity of the nonhomogeneous boundary condition in Eq. (1.152). For this reason, simpliﬁed

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variants have been proposed (Amsden and Harlow, 1970) that allow one to impose homogeneous boundary conditions for the pressure. The introduction of such variants requires repeating in a different perspective some of the concepts just illustrated. It is necessary, in fact, to make the concept of provisional or intermediate velocity (that is typical of these methods) more explicit. Many variants of the primitive-variable approach generally proceed as a type of fractional step method by ﬁrst writing a modiﬁed momentum equation and then updating the velocity ﬁeld using the computed pressure to account for the continuity equation. More precisely, at each time step, an intermediate velocity ﬁeld is determined without the knowledge of the correct pressure ﬁeld and therefore no incompressibility condition is enforced. The intermediate velocity ﬁeld is then corrected by a second step in which a pressure equation is solved and then the computed pressure is used to produce a divergence-free velocity ﬁeld. This means that the computation of the velocity ﬁeld at each time step is split into two substeps. In the ﬁrst, an approximate velocity ﬁeld V ∗ corresponding to the correct vorticity of the ﬁeld, but with ∇ · V ∗ = 0, is computed at time (n + 1) neglecting the pressure gradient in the momentum equation: V ∗ = V n + t[−∇ · (V V ) + Pr∇ 2 V ]n + tF nb

(1.154)

In the second substep, the pressure ﬁeld is computed by solving a Poisson equation obtained, imposing that ∇ · V n+1 = 0: ∇ 2 pn =

1 ∇ ·V∗ t

(1.155)

Finally, the velocity ﬁeld is updated using the computed pressure ﬁeld to account for continuity: V n+1 = V ∗ − t∇pn

(1.156)

If the effective velocity boundary conditions are used for the solution of Eq. (1.154), then V ∗ need not be corrected on the boundary (i.e. V n+1 = V ∗ ); this leads to homogeneous boundary conditions for the pressure equation: ∂p =0 ∂n

(1.157)

It is worth concluding this section by observing that, since according to Eqs (1.154), (1.155) and (1.156), an initial approximation to the momentum equation is initially advanced to determine the provisional velocity V ∗ and then an elliptical equation is solved that enforces the divergence constraint, this process basically relies on the Ladyzhenskaya decomposition theorem (Ladyzhenskaya, 1969), which states that any vector function V ∗ can be decomposed into a divergence-free part V plus the gradient of a scalar potential −P (a curl-free part), that is, V = V ∗ + ∇P and ∇P = −∇p/t (the conservation of mass is ensured in this method since V n+1 plays the role of the solenoidal part in the Ladyzhenskaya theorem). These techniques are very efﬁcient and were probably the ﬁrst numerical schemes allowing a cost-effective solution of three-dimensional time-dependent problems. Their simplicity and sometimes surprising efﬁciency render them particularly attractive to the CFD community. Although these algorithms have long been used for calculating steady-state solutions to Navier–Stokes equations, they are now regaining their status as true time-marching procedures for calculating time-dependent incompressible viscous ﬂows. As anticipated at the beginning of Section 1.7 this renewed interest for time-dependent solutions to Navier–Stokes equations is prompted by the increasing capacities of computers and the increasing success of direct numerical simulation in capturing bifurcations and instability phenomena.

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1.8

Thermal Convection: Patterns, Evolution and Stability

Some Universal Properties of Chaotic States

This section is devoted to illustrating some fundamental concepts related to chaos and the theories introduced over the years in the attempt to model the related underlying mechanisms. Chaos theory, a modern development in mathematics and science, provides a framework for understanding irregular or erratic ﬂuctuations in Nature. As already outlined in Section 1.4, certain seemingly simple natural nonlinear processes, for which the laws of motion are known and completely deterministic, can exhibit enormously complex behaviour, often appearing as if they were evolving under random forces rather than deterministic laws. One consequence is the remarkable result that these processes, although completely deterministic, are essentially unpredictable for long times. From a theoretical point of view, chaos is a subject that has always attracted the interest of scientists. The possibility of chaos in a natural, or deterministic, system was ﬁrst envisaged by the French mathematician Henri Poincar´e in the late 19th century, in his work on planetary orbits. In 1944, Landau (see Landau and Lifshits, 1971, and references therein) developed a theory that asserts that only a system characterized by a high number of degrees of freedom can become chaotic. Landau (1944) was the ﬁrst to publish his ideas of a transition to chaos due to a gradual excitation of new oscillators while successively increasing the control parameter. The modern study of chaotic dynamics, however, may be said to have begun in 1963, when American meteorologist Edward Lorenz demonstrated that a simple, deterministic model of thermal convection in the Earth’s atmosphere showed sensitivity to initial conditions or, in current terms, that it was a chaotic system (the reader is referred to Section 1.8.3 for additional historical details).

1.8.1 Feigenbaum, Ruelle–Takens and Manneville–Pomeau Scenarios In the last 30 years, certain universal routes which systems will take in transitioning from regular to irregular motion have been identiﬁed. In particular, three main routes that lead to chaos have been classiﬁed and investigated: the Ruelle–Takens scenario, the Feigenbaum scenario and the Manneville–Pomeau scenario. In 1971, a Belgian physicist, David Ruelle, and a Dutch mathematician, Floris Takens, together predicted that the transition to chaotic turbulence in a moving ﬂuid would take place at a well-deﬁned critical value of the ﬂuid’s velocity (or some other important factor controlling the ﬂuid’s behaviour). They predicted that this transition to turbulence would occur after the system had developed oscillations with at least three distinct frequencies. This route involves, therefore, at least two Hopf bifurcations, the ﬁrst leading from a periodic regime to a quasi-periodic regime with two incommensurate frequencies and the second leading to a quasi-periodic regime with three incommensurate frequencies. As the third frequency is about to occur, some broadband components appear simultaneously. Another American physicist, Mitchell Feigenbaum, then predicted that at the critical point when an ordered system begins to break down into chaos, a consistent sequence of period-doubling bifurcations would be observed (that means: as the characteristic parameter is increased, a periodic regime with period τ is replaced by one with a period 2τ , which will be replaced by another with a period 4τ and so on). This inverse cascade (also referred to as ‘subharmonic cascade’) gives in the Fourier space a broadening of the spectrum and so a transition to chaos. The third route involves ‘intermittency’ (Pomeau and Manneville, 1980): a periodic regime is replaced by another that is characterized by phases of regular periodic behaviour, interrupted from time to time by phases of apparently anarchical behaviour (turbulent bursts). As illustrated above, ‘typical’ appearances (they are ‘universal’, or broadly applicable, in the sense of obeying laws that do not depend on details of the physical system considered) that might

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announce chaos are: quasi-periodic ﬂow states (with two or more incommensurate frequencies in the Fourier spectrum), period doubling phenomena and spatiotemporal intermittency. However, the route to chaos is not limited to these ‘typical’ phenomena. Very ‘special’ transitional events may be also observed. Examples of very special things, on the other hand, are: the splitting of subharmonics in the Fourier spectrum (i.e. the frequency peak with the largest amplitude in the spectrum can be separated into two peaks having approximately equivalent amplitudes), ‘instantaneous’ changes of the dominant oscillatory frequency caused by moderately increasing the characteristic number (this phenomenon is often referred to as ‘frequency skip’; Swinney, 1983) and frequency locking (Gollub and Benson, 1980; Swinney, 1983) as an exceptional case of quasi-periodic behaviour (also referred to as ‘phase locking’: the two main frequencies of a biperiodic ﬂow modify themselves in such a way that their ratio is a rational number; sudden transition from this regime to chaos is observed for a further increase of the characteristic parameter). In practice, there is no necessity that only one of the phenomena just discussed can be observed at a certain time. Any combination of these effects possibly occurs and accompanies the system’s way to a chaotic behaviour. Once a chaotic regime has been established, although there is no scientiﬁc deﬁnition of it, common experience is related to some of its typical properties, the most representative of which, as explained before, is the difﬁculty in predictability of results. A chaotic system loses the memory of itself or, in other words, the knowledge of the status of the system for a ﬁnite time interval does not allow investigators to predict its further evolution.

1.8.2 Phase Trajectories, Attractors and Strange Attractors As discussed by Berg´e et al. (1984), the evolution of a dynamic system can be described by means of a phase trajectory, which is a curve traced in the phase space having as many dimensions as the number of degrees of freedom of the system. One of the many characteristics of all dissipative systems (systems for which, as highlighted in Section 1.3, evolution is driven by competition between a driving force and dissipation of energy) is that their phase trajectories are attracted by a geometric object called ‘attractor’. This means that different trajectories, arising from different points of the attractor, end on the attractor anyway. For the simple case of only two incommensurate frequencies mentioned in Section 1.8.1, the phase trajectory lies on a toroidal surface T 2 in the three-dimensional phase space. In the classical scenario foreseen by Ruelle and Takens, a third independent frequency appears, that is, the attractor becomes a hypertorus T 3 . If this attractor is unstable against perturbations, the system dynamics become chaotic. Curry and Yorke (1977), also constructed a numerical model that permits a transition to chaos from a torus T 2 , that is, two essential frequencies should be sufﬁcient to produce chaotic behaviour. With their theory a transition from a torus T 2 to chaos should simply be featured by a gradual deformation of an initially closed curve in the phase space by a stretching and folding process. In 1971, Ruelle and Takens (1971), by introducing the concept of a ‘strange attractor’, demonstrated that three degrees of freedom (i.e. a hypertorus T 3 ) are necessary to give rise to a chaotic regime. This theory was successively formalized by Newhouse et al. (1978). In this context, it is worth opening a short discussion about the Newhouse–Ruelle–Takens theorem (Newhouse et al., 1978). It asserts that a torus T 3 , under the actions of some perturbations, degenerates to a ‘strange attractor’ and therefore the existence of three frequencies (i.e. three degrees of freedom) is a necessary and sufﬁcient condition for the onset of a chaotic regime. The introduction of the concept of a strange attractor revolutionized the study of chaos and turbulence. It is well known (as explained before) that a chaotic regime loses the memory of itself and therefore two trajectories, initially adjacent, tend to diverge rapidly (sensitivity to initial conditions, hereafter referred to as SIC). An attractor that exhibits SIC is representative of a chaotic

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Thermal Convection: Patterns, Evolution and Stability

regime: it is called ‘strange attractor’ (see Grassberger and Procaccia, 1983a, for further insights into this deﬁnition; see also Section 1.8.3 for some historical background about the discovery of these objects and the related interesting concept of the ‘butterﬂy effect ’). It seems to be a contradiction between attraction, which implies the convergence of trajectories, and SIC, which implies their divergence. However, the divergence merely sets a lower bound on the attractor dimension. Ruelle and Takens (1971) demonstrated that a two-dimensional attractor cannot show SIC for topological reasons and, therefore, the dimension of a strange attractor must be strictly greater than two. As demonstrated by Berg´e et al. (1984), this value is not necessarily an integer. The strange attractor may be, therefore, an object characterized by a fractal dimension: a dimension that corresponds in a unique fashion to the geometric shape under study and, as outlined above, often is not an integer (this ‘scale symmetry’ has the implication that objects look the same on many different scales of observation; further and more detailed elaboration along these lines will be provided in Section 1.8.5). Finally, it is worth pointing out that even if in the light of these arguments, the existence of stable attractors with three incommensurate frequencies appears unlikely (according to the aforementioned Newhouse–Ruelle–Takens theorem, the coexistence of three frequencies should always lead to corrugation of the T 3 torus and ultimately to a strange attractor); however, both experiments (Gollub and Benson, 1980; Linsay and Cummings, 1989) and numerical studies (Grebogi et al., 1983; Paz´o et al., 2001; Bratsun et al., 2003; Mercader et al., 2005) seem to give support to the existence of these attractors, insisting that in some cases they can be structurally stable (although the apparent departure from the Newhouse–Ruelle–Takens theorem is unclear at this level). Some further light has been shed on this topic after the work of Feudel et al. (1996), who presented convincing arguments on the stability of these 3D tori on systems with certain types of symmetries.

1.8.3 The Lorenz Model and the Butterﬂy Effect From a historical point of view, one of the most important characteristics of strange attractors, namely their great sensitivity to initial conditions (SIC), was discovered by accident by Edward Lorenz, a mathematician and research meteorologist. The detailed sequence of events which led this mathematician to his important discovery deserves some attention and is discussed in depth in the following. Apart from the general interest from the point of view of applied mathematics (which is attached to this fact), these historical facts can be regarded as a paradigm landmark example in which an unexpected and fruitful synergy was established between the science of computer simulation and the theory of chaos. It really stands at the basis of all the efforts that have been provided over subsequent years to capture the features of chaos and understand the underlying cause–effect relationships through direct numerical solutions of the governing nonlinear equations. In the attempt to construct a mathematical model of the weather, namely a set of differential equations that represented changes in temperature, pressure, wind velocity and so on, Lorenz initially elaborated a crude model containing a set of 12 differential equations. On a particular day in the winter of 1961, he re-examined a sequence of data coming from this model, but, instead of restarting the entire run, the simulation was continued from somewhere in the middle. Using data printouts, Lorenz entered the conditions at some point near the middle of the previous run and restarted the model computations. What he found was very unusual and unexpected. The data from the second run should have exactly matched the data from the ﬁrst run. Although they matched at ﬁrst, the runs eventually began to diverge dramatically, the second run losing all resemblance to the ﬁrst within a few model months. After discovering that there was no malfunction in the computer used for the calculations, Lorenz ﬁnally realized the source of the problem. To save space, his printouts only showed three digits whereas the data in the computer’s memory contained six digits. Lorenz had entered the

Equations, General Concepts and Methods of Analysis

49

rounded-off data from the printouts assuming that the difference was inconsequential (but this was not the case). To explore this strange effect better, he decided to look for complex behaviour in an even simpler set of equations and was led to the phenomenon of rolling ﬂuid convection. In particular, Lorenz considered a rectangular slice of air heated from below and cooled from above by edges kept at constant temperatures (this is our atmosphere in its simplest description: the bottom is heated by the Earth and the top is cooled by the void of outer space; within this slice, warm air rises and cool air sinks; in the model as in the atmosphere, convection cells develop, transferring heat from bottom to top). In the framework of such a model, the Navier–Stokes equations were simpliﬁed to obtain a set of three nonlinear ordinary differential equations only (Lorenz, 1963). The simpliﬁed equations were written as follows: dx = Pr(y − x) (1.158a) dt dy = Rx − y − xz (1.158b) dt dz = xy − Az (1.158c) dt where Pr is the Prandtl number, R represents the difference in temperature between the top and bottom of the system (it is the ratio between the characteristic number of this type of convection and the related critical value; see Section 2.1 for additional information on the meaning of such parameters) and A is the ratio of the width to height of the box used to hold the system. Moreover, x(t) is the amplitude of the convective motion, y(t) the temperature difference between rising and descending currents and z(t) the temperature deviation with respect to a linear temperature proﬁle (the so-called TFD distortion, as will be illustrated in Chapter 2). The values used by Lorenz were Pr = 10 and A = 8/3. The behaviour of the solution was found to change according to the parameter R as discussed in the following. If R < 1 the system admits a single (trivial) stable solution x = 0, y = 0, z = 0, corresponding to a uniform regime where convection is absent (a linear distribution of temperature). If 1 < R < 24, two stable stationary solutions exist [they are represented in the phase space (x, y, z) by two points that behave as attractors in the sense given before to such a deﬁnition]; from a physical point of view, these two attractors correspond to the appearance of convective rolls of motion (clockwise or anticlockwise oriented according to which of the two aforementioned solutions is considered). For R > 24, stable stationary solutions no longer exist (Lorenz used R = 28). All the trajectories in the phase space are ‘attracted’ towards a well-deﬁned region where they tend to be conﬁned. If one plots the results in three dimensions a ﬁgure called the Lorenz attractor is obtained (Figure 1.4). As one of his list of challenging problems for mathematics (Smale’s problems), Smale (1998) posed the open question of whether the Lorenz attractor is a strange attractor (see also Guckenheimer and Williams, 1979). This question was answered in the afﬁrmative by Tucker (2002). The Lorenz attractor is a classical example of a strange attractor. These attractors are unique from other phase space attractors in that one does not know exactly where on the attractor the system will be. Two points on the attractor that are near each other at one time will be arbitrarily far apart at later times. The only restriction is that the state of the system remains on the attractor. Strange attractors are also unique in that, as shown in Figure 1.4 (see also Figure 1.5), they never close on themselves, that is, the motion of the system never repeats (nonperiodic behaviour). In practice, with his primitive computer at MIT, Lorenz discovered that complex open systems such as the weather may have an enormous sensitivity to initial conditions in some circumstances. The sensitivity is so great that prediction is inherently impossible.

50

Thermal Convection: Patterns, Evolution and Stability

40

30 z 20 20

0 −10

y

10

10 −10 0 x

10

−20 20

Figure 1.4 3D plot of the Lorenz attractor: x(t) represents amplitude of the convective motion, y(t) the temperature difference between rising and descending currents and z(t) the temperature deviation with respect to a linear temperature proﬁle. The attractor appears as two warped disks (M. Lappa)

(a)

(b)

Figure 1.5 Projection of the Lorenz attractor in different planes (M. Lappa)

Equations, General Concepts and Methods of Analysis

51

As described in the ﬁrst part of this section, the slightest rounding off of the initial numbers in the weather formula would lead to entirely different results. In other words, extremely tiny differences in the initial numbers would quickly lead to huge variations in the calculations. In time, the slightest thing could end up making a huge difference in the end result (small variations of the initial condition produce large variations in the long-term behaviour of the system). Edward Lorenz’s important discovery is now exempliﬁed by the famous example which he gave to explain his discovery: ‘the wing movements of a butterﬂy in Peru may later through an extremely complex series of unpredictably linked events magnify air movements and ultimately cause a hurricane in Texas’. This law is now nicknamed the ‘butterﬂy effect’. The phrase refers to the idea that a butterﬂy’s wings might create tiny changes in the atmosphere that ultimately cause a hurricane to appear (or prevent a hurricane from appearing). The ﬂapping wing represents a small change in the initial condition of the system, which causes a chain of events leading to large-scale phenomena. Had the butterﬂy not ﬂapped its wings, the trajectory of the system might have been vastly different. This is now known to apply to all chaotic systems, not just the weather. It is an inherent characteristic of all strange attractors. The butterﬂy effect applies in all complex open systems which change over time. It applies to all dynamic systems (such as the weather) so that the smallest of changes triggers a chain reaction of unexpected exponential consequences. In view of these arguments, the butterﬂy effect is currently used as a general concept that encapsulates the more technical notion of sensitive dependence on initial conditions (SIC) in chaos theory discussed in Section 1.8.2. The next two sections give an account of the main mathematical ideas (and their concrete implementation) behind these aspects. The main subjects are the theory of dimensions (number of excited degrees of freedom), entropy (production of information) and characteristic exponents (describing sensitivity to initial conditions). The relations between these quantities are discussed. It is worth stressing that the systematic introduction of these quantities is mandatory for a reasonable characterization and understanding of dynamic systems, excited well beyond quasi-periodic regimes.

1.8.4 A Possible Quantiﬁcation of SIC: The Lyapunov Spectrum Prior to expanding on how a possible quantiﬁcation of SIC can be obtained, a rigorous deﬁnition of it must be introduced. In the light of all the considerations provided in Sections 1.4, 1.8.2 and 1.8.3, this can be done in a relatively simple way. A dynamic system with evolution function t in a m-dimensional phase space (whose dimension corresponds to the number of degrees of freedom of the dynamic system considered, hereafter simply referred to as M space), in fact, will display a sensitive dependence on the initial conditions if certain points in such a space, no matter how close, become separate with increasing t. From a purely mathematical point of view, this means that there is a δ > 0 such that for any neighbourhood dM containing an initial point z0 there exists a point z0 + z0 from that neighbourhood M and a time t such that ||t (z0 , t) − t (z0 + z0 , t)|| > δ

(1.159)

This deﬁnition does not imply that all points from dM separate from z0 . Moreover, it is susceptible to an interesting geometric interpretation (see Figure 1.6). Each of the two points z0 and z0 + z0 in Figure 1.6 will generate an orbit in the M space according to the evolution function t . These orbits can be thought of as parametric functions of time (for instance, see Figure 1.4). By using one of the orbits a reference orbit, then the separation z between the two orbits will also be a function of time. Because sensitive dependence can arise only in some portions of M (on the strange attractor), this separation will be also a function of the location of the initial value [the separation or distance has the form given by the left member in Eq. (1.159), which means z = z(z0 , t)].

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Thermal Convection: Patterns, Evolution and Stability

Figure 1.6 Two trajectories originating in the phase space from two close points initially at a distance z0

In a system with attracting ﬁxed points or attracting periodic points, z(z0 , t) will diminish asymptotically with time. If z0 belongs to a strange attractor, then the orbits will diverge exponentially, at least for a while. These simple arguments lead to the natural introduction of the concept of the Lyapunov exponent . The latter behaviour, in fact, can be roughly represented as ||z(z0 , t)|| ∝ eλt ||z0 ||

(1.160)

where λ can be regarded as a possible ‘measure’ of the exponential rate of separation of the two arbitrarily close trajectories. In a more rigorous way, this exponent can be deﬁned using the following relationship: 1 ||z(z0 , t)|| λ = lim ln (1.161) t→∞ t ||z0 || ||z ||→0 0

This quantity, normally represented by the lambda symbol, is the Lyapunov exponent mentioned above. According to Eq. (1.161), it simply captures the average rate of divergence of all inﬁnitesimally close trajectories originating from z0 . Another way to think about the Lyapunov exponent is as the rate at which information about initial conditions is lost: As explained before, trajectories emanating from initially close points become exponentially further apart with increasing time, leading to the ampliﬁcation of very small perturbations into global uncertainties; such a loss of positional information with time is often regarded as a tendency of the system to increase its entropy. It is this increase of entropy (loss of memory of initial conditions in any numerical approximation over time) that prevents accurate long-term numerical approximation of the system; also, it is in this speciﬁc sense that the concept of ‘loss of memory’ anticipated in Section 1.8.2, and repeatedly invoked in subsequent sections, should be considered (further clariﬁcation along these lines will be given later with the more precise notion of Kolmogorov entropy). In reality, things tend to be even more complex than as described above (Young, 1982). In its more general and ambitious form, the theory predicts that there are as many Lyapunov dimensions as the dimensions of M, (that is, as the system degrees of freedom). This means that the exponential rate of divergence of trajectories of solutions originating from nearly identical initial conditions is quantiﬁed by a set of Lyapunov exponents λi where i = 1, 2, . . ., m, with λi arranged in descending order (i.e. λ1 ≥ λ2 ≥ . . . ≥ λp ), usually referred to as the Lyapunov spectrum. This spectrum can be used for a possible effective quantitative characterization of SIC. Recalling the geometric analogy introduced earlier, the leading order exponent λ1 [often deﬁned in the literature according to Eq. (1.161)] describes the growth of the line separating two trajectories in the phase space; in combination with the other ones, it leads to even more interesting geometric analogies: λ1 + λ2 will describe the growth of a two-dimensional area of initial perturbations; the summation of all the exponents m λi (1.162) i=1

Equations, General Concepts and Methods of Analysis

53

will describe the dynamics of an m-dimensional ball of initial perturbations. As time progresses, this sphere will evolve into an ellipsoid whose principal axes expand (or contract) at rates given by the exponents. In practice, each exponent corresponds to a Lyapunov vector that is simply a particular perturbation (out of all the possible choices of perturbations) away from the reference state z0 and each Lyapunov vector is orthogonal to all the others (each Lyapunov exponent quantiﬁes the average exponential rate of growth of the perturbation represented by the corresponding Lyapunov vector). In general, the presence of a positive exponent is sufﬁcient for diagnosing chaos and represents local instability along a particular direction (a system with more than one positive exponent is referred to as hyperchaotic, R¨ossler 1979). It is worth highlighting at this stage, however, that since for the existence of an attractor the overall dynamics must be globally stable (as illustrated in Section 1.8.2, different trajectories, arising from different points of the attractor, must end on the attractor anyway), the total rate of contraction must outweigh the total rate of expansion. As a consequence, even when there are several positive Lyapunov exponents, the sum across the entire spectrum, namely Eq. (1.162), must be negative (this is a typical prerogative of dissipative systems; if the system is conservative, that is, there is no dissipation, in fact, the sum of the Lyapunov exponents is zero). The sum of the positive exponents equals the so-called Kolmogorov entropy (Eckman and Ruelle, 1985). In fact, only the positive (expanding) Liapunov exponents contribute to the spreading process of a ball of initial perturbations discussed earlier and so the entropy associated to the loss of spatial (positional) information with time is generally identiﬁed with the sum of such exponents. In general, the number of exponents that yield a positive result when added together is ﬁnite. The exact number of exponents required for the sum to vanish corresponds to the dimension of the ball of initial conditions that will neither grow nor shrink under the dynamics (often called the Lyapunov dimension Dλ ). Given only the Lyapunov exponents, D λ can be determined from the Kaplan–Yorke equation (see, e.g., Frederickson et al., 1983): Dλ = k +

k i=1

λi |λk+1 |

(1.163)

j where k is the largest j for which i=1 λi > 0. The value of Dλ is the minimum number of active degrees of freedom that contribute to the chaotic dynamics (Farmer et al., 1983; Goldhirsch et al., 1987). It may be regarded as a global measure of the complexity of the dynamics (i.e. how chaotic the system is). To conclude this section, it should be noted that, in general, the Lyapunov exponents are extremely difﬁcult to measure experimentally, since it is usually not possible to begin experiments from slightly different initial conditions; moreover, the computation of these exponents is also very intensive (see, e.g., Wolf et al., 1985). The reader may be interested in knowing that the values of the Lyapunov exponents for the Lorenz attractor described in Section 1.8.3 are (0.906, 0, −14.572). From these exponents, the Kaplan–Yorke dimension can be calculated as Dλ = 2 + λ1 /|λ3 | = 2.062. Hence the Lyapunov dimension of the Lorenz attractor is 2.06 (see, e.g., Frøyland and Alfsen, 1984; Viswanath, 2004).

1.8.5 The Mandelbrot Set: The Ubiquitous Connection Between Chaos and Fractals Last but not least, one of the most signiﬁcant and ‘typical’ features of a turbulent regime (that makes the study of such regime really difﬁcult) is its ‘non analytical dependence’ on the system parameters.

54

Thermal Convection: Patterns, Evolution and Stability

Figure 1.7 Bifurcation diagram related to the simple equation xn+1 = rxn (1 − xn ), also known as the ‘logistic map’ (the logistic map is an object often cited as an archetypal example of the Feigenbaum scenario and, in general, of how complex chaotic behaviour can arise from very simple nonlinear dynamic equations; it is a fractal as zooming in on part of the diagram it will look similar or exactly like the overall shape)

Interestingly, as a consequence of such a property, the loci in the space of parameters which separate stable and unstable states of the considered system deﬁned in Section 1.5.3 (neutral stability curves) tend to display really complex geometric conﬁgurations. A relevant mathematical formalism for such geometric conﬁgurations has been provided only in very recent years (in the framework of the theories and discoveries of Professor Benoit B. Mandelbrot). The widespread study of the so-called Mandelbrot set, a set of points in the complex plane that forms an object known as ‘fractal’, that is, a geometric object that is similar to itself on all scales, a property known as self-similarity (the structure on the coarsest scale is repeated on ﬁner length scales), has revealed on the one hand a hidden fractal order underlying all seemingly chaotic events in Nature (from the behaviour of ﬂuids in closed containers to the evolution of the universe) and, on the other, new insights into the chaotic behaviour of nonlinear systems. In such a context, as mentioned before, a fractal can be used to represent the complex interplay between the regions of stability and instability, that is, the ensemble of the transitional stages taken by a nonlinear system in the parameter space during its evolution towards fully developed chaos (see, for example, the bifurcation diagram plotted in Figure 1.7). These objects, that in the framework of a certain deﬁnition (as will be shown later) are characterized by a number of dimensions that is not an integer, also correspond (according to some theories) to possible geometric representations of strange attractors (as already outlined in Section 1.8.2). A classical example of such behaviour is given again by the Lorenz attractor discussed in Section 1.8.3 and shown in Figure 1.4. Recall how the numbers that Lorenz generated with his computer were ergodic (approaching every possible value) and aperiodic (never repeating), this feature being reﬂected in the fact that the curve never meets (i.e. the volutes are plotted ‘at random’ but without crossing). A geometric ﬁgure of this sort (with an inﬁnite level of detail) is clearly a fractal. Such a property was already

Equations, General Concepts and Methods of Analysis

55

Figure 1.8 Example of fractal: the Mandelbrot set. Image generated using the Mandelbrot-set calculation/drawing program v3.3, free software by SADA with the following input data: x = −1.56947810295, y = 2 × 10−8 , w = 1.5 × 10−6

noted by Lorenz in his landmark initial studies (Lorenz, 1963); he described the attractor, in fact, as an inﬁnite complex of surfaces. Further elaboration along these lines requires the introduction of many propaedeutical concepts, in particular, the notion of ‘dimension’ (it will be illustrated how this word may be somewhat ambiguous and exhibit several possible meanings) as well as a more rigorous deﬁnition of the concept of fractal. A fractal can be deﬁned as ‘a fragmented geometric shape that can be subdivided into parts, each of which is (at least approximately) a reduced-size copy of the whole’ (Figure 1.8). The term was coined by Mandelbrot in 1975 and was derived from the Latin fractus, meaning ‘broken’ or ‘fractured’. A fractal has the following features:

• • • • •

It It It It It

has a ﬁne structure at arbitrarily small scales. is too irregular to be easily described in traditional Euclidean geometric language. has a simple and recursive deﬁnition. is self-similar. has a Hausdorff dimension which is greater than its topological dimension.

In mathematics, the Hausdorff dimension (also known as the Hausdorff–Besicovitch dimension) is an extended non-negative real number associated with any metric space. It was originally introduced by the mathematician Felix Hausdorff (Hausdorff, 1919), while many of the subsequent technical developments used to compute its effective value for highly irregular sets were obtained by Abram Samoilovitch Besicovitch (Besicovitch, 1929). Less frequently, it is also called the capacity dimension or fractal dimension (the latter is somewhat misleading as there are many other choices of deﬁnition, as will be shown later). Intuitively, the dimension of a set (for example, a subset of Euclidean space) is the number of independent parameters needed to describe a point in the set. One mathematical concept which closely models this naive idea is that of topological dimension of a set. For example a point in the plane is described by two independent parameters (the Cartesian coordinates of the point), so in this sense, the plane is two-dimensional. As one would expect, topological dimension is always a natural number.

56

Thermal Convection: Patterns, Evolution and Stability

The most natural way to introduce (before providing a rigorous deﬁnition) the idea of Hausdorff dimension for a generic set ( being a bounded subset of R n ) is to consider a number N of ‘open sets’ of size less than or equal to a given distance r, forming a ﬁnite cover of . For the sake of simplicity, the aforementioned generic open set may be imagined as a ball, that is, a sphere of radius at most r or as a cube of edge length equal to r. In the ﬁrst case N will be the minimum number of balls required to cover completely, and in the latter it will correspond to the minimum number of elements of the rectangular grid required to approximate . In both cases (in practice, these arguments do not depend on the shape of the element selected to form a ﬁnite cover of ), N will grow as 1/r D0 [ → r ∝ 1/N 1/D0 ] when r is reduced (where D0 is an exponent depending on the initial set considered). If the initial set were simply a one-dimensional line segment, dividing it into N identical parts, each part would be scaled down by the ratio r = 1/N (e.g. cutting a line into two equal pieces, two lines each of half the original length are obtained) and an exponent D0 = 1 would be obtained; similarly, a two-dimensional object, such as a square, could be divided into N self-similar parts, each part being scaled down by the factor r = 1/N 1/2 [cutting a square into four equally sized squares, each new square is half the size (side length) of the original square], which means D0 = 2. These simple considerations illustrate that the exponent deﬁned in the framework of the above discussion merely corresponds to the topological dimension of the considered set when this set is a classical Euclidean geometric object. For fractals, however, the exponent is not an integer. By generalization of the relationships described earlier, it can be computed starting from the following identity: r = 1/N 1/D0 → D0 =

ln N ln(r −1 )

(1.164)

which leads to a more rigorous deﬁnition of the Hausdorff dimension: D0 = lim

r→0

ln N ln(r −1 )

(1.165)

It is evident at this stage that the Hausdorff dimension can be regarded as an extension of the concept of topological dimension. It is worth remarking, however, that the nature of strange attractors can be complex and difﬁcult to describe. It is useful, therefore, to have several possible quantitative characterizations (Russell et al., 1980; Farmer et al., 1983; Grassberger, 1985). Obviously, the most basic characterization is given by the Hausdorff dimension of the attractor deﬁned by Eq. (1.165). However, the notion of dimension can be extended to incorporate other features (not solely the geometry of the attractor) and, in particular, singular properties of the density with which a typical orbit ‘visits’ different parts of the attractor. Such a feature is naturally embodied into the deﬁnition of the natural measure. The natural measure associated with a strange attractor gives the fraction of the time that a long orbit on the attractor spends in any given region of the state space. In particular, by indicating with C a generic cube in the phase space of the system and with z(z0 , t) the orbit originating from a generic initial condition z0 and with τ (C, z0 , t) the fraction of time that z(z0 , t) spends in C in a time interval t and assuming the limit τ (C, z0 ) = lim τ (C, z0 , t) t→∞

(1.166)

exists, if τ (C, z0 ) takes on the same value for almost all z0 in the basin of the attractor, the natural measure simply corresponds to such a common value and is usually denoted by τ (C). The concept of natural measure leads to the ensuing more complex notion of Renyi dimension (also called the generalized dimension; Balatoni and Renyi, 1956). This dimension (sometimes also referred to as the Renyi exponent) takes into account the frequency with which cubes are visited via weighting them according to their natural measure. The

Equations, General Concepts and Methods of Analysis

57

strength of this weighting is given by an index q. For q > 0, the larger is q, the stronger is the relative weighting of the higher measure boxes. Considering a partition of the phase space by an r-grid [similar to that considered before for the introduction of Eq. (1.165)] and evaluating τ (Cj ), i.e. the natural measure of the j th r-cube Cj needed to cover the attractor, the order q Renyi dimension of the attractor can be deﬁned as (Balatoni and Renyi, 1956) q [τ (C )] ln j j 1 (1.167) Dq = lim r→0 1 − q ln(r −1 ) For q = 0, Eq. (1.167) yields Eq. (1.165), that is, the Hausdorff dimension (Grassberger, 1981). For q = 1 and 2, it gives the so-called information dimension and correlation dimension, respectively, which have been the subject of signiﬁcant interest over recent years (Grassberger and Procaccia, 1983a,b; Hentschel and Procaccia, 1983). In particular, the information dimension can be obtained from Eq. (1.167) by taking the limit q → 1 and using l’Hospitals’s rule: j τ (Cj ) log[τ (Cj )] (1.168) D1 = lim r→0 ln r It is worth emphasizing that according to the so-called Kaplan–Yorke conjecture D1 = Dλ

(1.169)

that is, the information dimension is equal to the Lyapunov dimension. This relationship is remarkable in that it relates dynamics (Lyapunov exponents) to attractor geometry and natural measure. Young (1982) showed that this conjecture is true for certain systems. It is now widely believed that almost every attractor has its Lyapunov dimension equal to its information dimension. An important property of the family of dimensions Dq is that it is a non-increasing function of q D2 ≤ D1 ≤ D0

(1.170a)

which according to Eq. (1.169), also implies D2 ≤ Dλ ≤ D0

(1.170b)

that is, the Hausdorff and the correlation dimensions behave as upper and lower bounds, respectively, for the Lyapunov dimension. For the Lorenz attractor, as an example, the Hausdorff dimension appears to be D0 = 2.0627, whereas the correlation dimension (D2 ) is about 2.05. Recalling that the Lyapunov dimension Dλ = 2.062 (Section 1.8.4), this gives (Grassberger and Procaccia, 1983b; Viswanath, 2004) D0 ∼ = Dλ = D1 > D2

(1.171)

In all cases the dimension is greater than two in agreement with the concepts provided in Section 1.8.2. For relevant examples in which the fractal dimension has been determined in practical situations, that is, realistic conﬁgurations, the reader may consider the work of Paul et al. (2007), Ueno et al. (2003) and Liz´ee and Alexander (1997) for the cases of thermogravitational (Rayleigh–B´enard), Marangoni and thermovibrational convection, respectively, and related discussions in Sections 4.12, 11.8.3 and 12.4, respectively. It is worth concluding this section by noting that an additional link between fractal objects and chaos in ﬂuid dynamics (and related properties) is given by the previously mentioned intrinsic feature of fractals by which they are self-similar geometric items embodying the concept of progressively increasing structure on ﬁner and ﬁner length scales typical of turbulence. Further elaboration of these theories is beyond the scope of the present book. It should be pointed out at least, however, that in the light of the concepts introduced in Section 1.1.1 (pattern,

58

Thermal Convection: Patterns, Evolution and Stability

interrelation and scale) also these special objects can be regarded as ‘patterns’ partly associated with the behaviour of ﬂuid systems (they can be seen as patterns created by the considered system in the space of parameters or in the phase space rather than in the ‘physical’ space). Despite these theoretical achievements on the connection between strange attractors and fractals, between patterns in the physical space and in the phase space and related possible ‘tools’ of measure, many signiﬁcant open questions of a scientiﬁc nature remain, including, in particular, how many of the inﬁnite degrees of freedom in a continuous system are excited, the origin, structure and dynamics of these chaotic degrees of freedom, how they enter the dynamics as the system size is increased and so on. Some of these questions will be answered (or at least an attempt will be given along such lines) in the various chapters of this book (see, in particular, the discussions about spiral defect chaos in Chapter 4 and the related ‘bistability’ problem).

1.9

The Maxwell Equations

Since the topics treated in the present book also include possible interplay between thermal convection and magnetic ﬁelds, created by ﬂuid motion itself (e.g. the geodynamo model discussed in Chapter 4) or externally imposed for purposes of ﬂow control (Chapter 13), this (ﬁnal) section is devoted to the introduction of the related popular Maxwell equations in their general and most complete formulation (already invoked in Section 1.2.1 as necessary equations supplementing the Navier–Stokes equations for the study of typical magnetohydrodynamics problems). For consistency with the general philosophy for the treatment of continua undertaken at the beginning of this chapter, Maxwell’s equations for ﬂuids are derived starting from fundamental models valid at a microscopic scale (coming from Lorentz’s electronic theory) and applying again the same kind of procedure based on the ensemble averaging of microscopic ﬁeld quantities over physically inﬁnitesimal space and time regions (Mazur and Nijboer, 1953). This way of proceeding is always relevant for obtaining the macroscopic behaviour of a medium from the interaction between the microscopic ﬁelds and the particles of the medium (in particular, as for the thermal convection equations, crucial in this process is the identiﬁcation of the average of the microscopic quantities in terms of macroscopic quantities). As mentioned above, the starting point for such a theoretical development is given by the Lorentz postulated Maxwell equations for the microscopic electromagnetic ﬁeld (the essence of these equations will be clariﬁed later; they play, for the present case, the same role as played by Newton’s laws in Section 1.2). Resorting to the same concepts as introduced in Section 1.2.2, it is convenient to consider a system of N atoms numbered by the index i (i = 1, 2, . . . , N) with the generic atom consisting of a point-nucleus with charge qi and of p point-electrons with charges qih (h = 1, 2, . . . , p). Let us denote the position of the nucleus of the ith atom by the vector r i while the relative positions of the electrons to the nucleus are denoted by r ih . Note that even if the formalism developed in the following is based on such assumptions, the resulting equations, however, are not restricted to identical and (or) neutral atoms. In practice, mixtures of atoms and (or) ions are also included in this treatment (the formulation is valid, in general, for any stable association of charges). It should also be pointed out that in the above notation electrons with relative coordinates r ih are supposed to be bound to a nucleus. If free electrons do occur they should be included by treating them formally on the same footing as the nuclei. Within such a general theoretical framework, the average number of atoms per unit volume at position r at time t (hereafter referred to as the number density n) can be deﬁned simply as N n(r, t) = δ(r − r i ) (1.172) i=1

where is used to indicate an ‘average’ value from a stochastic standpoint.

Equations, General Concepts and Methods of Analysis

59

The charge of the generic atom is given by qiTOT =

p

(qi + qih )

(1.173)

h=1

and, as a natural consequence, the average charge density ρq at r can be written as ρq (r, t) = (qi + qih )δ(r − r i )

(1.174)

i,h

Along the same lines, the electric current due to the motion (with velocity ci = dr i /dt) of the ith atom reads p (qi + qih )ci (1.175) h=1

Therefore, the mean electric current at r and time t can be written as (qi + qih )ci δ(r − r i ) i,h

N

(1.176)

δ(r − r i )

i=1

From this expression, the average current density J f can be obtained by multiplication with the number density n, which leads to (qi + qih )ci δ(r − r i ) (1.177) J f (r, t) = i,h

Taking into account Eq. (1.5), moreover, it is possible to combine Eqs (1.174) and (1.177) into the well-known law of conservation of charge: ∂ρq (1.178) + ∇ · (J f ) = 0 ∂t which represents in the context of electromagnetism the analogue of the continuity equation, Eq. (1.10a), for the conservation of mass in ﬂuids. At this stage, the aforementioned Lorentz postulated Maxwell equations for the microscopic electromagnetic ﬁeld can be introduced as follows: ∇·b = 0

(1.179)

∂b ∂t qi δ(r − r i ) + qih δ(r − r i − r ih ) ∇ · (ε0 e) = ∇ ∧e = −

i

µ−1 0 ∇ ∧ b = ε0

(1.180) (1.181)

i,h

∂e qi ci δ(r − r i ) + qih (ci + cih )δ(r − r i − r ih ) + ∂t i

(1.182)

i,h

where e and b are the microscopic electric ﬁeld strength and the microscopic magnetic ﬂux density, respectively, and ε0 and µ0 are the so-called electric and magnetic constants (permittivity and permeability of vacuum, respectively).

60

Thermal Convection: Patterns, Evolution and Stability

The essence of the last two equations, in particular, reﬂects Lorentz’s original physical intuition that the spatial structure (let us recall the divergence operator ∇· appearing in the left member is a spatial operator) of the microscopic electric ﬁeld simply depends on the related spatial distribution of charges [Eq. (1.181)], whereas the spatial structure (via the curl operator ∇∧) of the microscopic magnetic ﬁeld depends on changes in time of the microscopic electric ﬁeld and on any microcurrents that may be present in the considered medium [Eq. (1.182)]. Equation (1.179) gives an additional geometric constraint on the microscopic magnetic ﬁeld (whose ﬁeld lines must be closed curves), whereas Eq. (1.180) provides further coupling between the spatial structure of e and variations in time of b (these two equations, among other things, also make the ﬁelds e and b unique according to the ‘inverse theorem of calculus’ mentioned in Section 1.7.2, by which a vector ﬁeld with given divergence and curl is uniquely determined). Additional related physical microscopic quantities are di =

p

qih r ih

(1.183)

h=1

1 qih r ih ∧ cih 2 p

hi =

(1.184)

h=1

known as electric dipole and magnetic dipole moments, respectively (these moments of the ith atom are simply deﬁned with respect to the position r i of its nucleus as origin). They are connected to the response the medium may exhibit to the global electromagnetic ﬁeld. If an electric ﬁeld is applied to a dielectric material (an electrically nonconducting substance), each of the molecules responds by forming a microscopic dipole (its atomic nucleus will move a relatively small distance in the direction of the ﬁeld, while its electrons will move a relatively small distance in the opposite direction). This effect is generally known as polarization of the material. If a magnetic ﬁeld is applied to a magnetic material (diamagnetic, paramagnetic or ferromagnetic; we shall come back to these deﬁnitions later), then a magnetic dipole moment arises. This effect is generally known as magnetization of the material. Magnetic dipole moments may also be an intrinsic feature of the material itself (i.e. not depending on the application of an external ﬁeld, for example, in ferromagnets). These dipole moments appear explicitly in the generic expression of the electromagnetic force acting on any generic particle with charge qi moving with velocity ci under the effect of e and b: f qi = qiTOT (e + ci ∧ b) + ∇e · d i + ∇b · hi

(1.185)

A macroscopic formulation of the Maxwell equations can be obtained from Eqs (1.179–1.182) applying the ensemble averaging process mentioned earlier. For Eqs (1.179) and (1.180), this method yields simply ∇ ·B = 0 ∂B ∇∧E = − ∂t

(1.186) (1.187)

where E = e and B = b are the macroscopic electric ﬁeld and magnetic ﬁeld (magnetic induction or magnetic ﬂux density), respectively. For Eqs (1.181) and (1.182), the derivation is more complicated (the relevant mathematical development is not described in detail due to page limits). For Eq. (1.181), in particular, the ensemble averaging process gives ∇ · (ε0 E) = ρq − ∇ · P

(1.188)

Equations, General Concepts and Methods of Analysis

where P is the density of the electric dipole moment [deﬁned by Eq. (1.183)]: N d i δ(r − r i ) P (r, t) =

61

(1.189)

i=1

generally referred to as density of polarization of the medium. As explained earlier, electric polarization corresponds to a rearrangement of the bound electrons in the medium, which creates an additional charge density, known as the bound charge density ρb . This charge density can be related to the density of polarization through application of the divergence operator: −∇ · P = ρb Its derivative with respect to time, on the basis of Eq. (1.5), can be written as ∂P qih cih δ(r − r i ) − qih r ih {ci · ∇[δ(r − r i )]} = ∂t i,h

(1.190)

(1.191)

i,h

Applying the ensemble averaging process to Eq. (1.182), taking into account Eqs (1.191) and (1.177), it follows: that ∂P ∂E + + Jf + ∇ ∧ M ∂t ∂t where M is the density of magnetic dipole moment [deﬁned by Eq. (1.184)]: N hi δ(r − r i ) M(r, t) = µ−1 0 ∇ ∧ B = ε0

(1.192)

(1.193)

i=1

generally referred to as density of magnetization of the medium. The macroscopic Maxwell equations can therefore be cast in compact form as ∇·B = 0

(1.194) ∂B ∇∧E = − (1.195) ∂t (1.196) ∇ · (ε0 E) = ρq − ∇ · P ∂E µ−1 (1.197) + J TOT 0 ∇ ∧ B = ε0 ∂t where J TOT = J f + J p + J M is the total electric current density due to the motion of both the free and the bound electric charges. It includes three contributions: the ﬁrst, J f , is due to the motion of free, electrically charged particles and represents the current density deﬁned by Eq. (1.177); the second, J p , is caused by the motion of the bound electric charges of the electric dipoles (it corresponds to the polarization current density and depends on the polarization P of the medium as J p = ∂P /∂t); the third, J M , is due to the motion of the bound electric charges of the magnetic dipoles (it is equal to the magnetization current density and depends on the magnetization M of the medium as J M = ∇ ∧ M). As already discussed, net magnetization, in general, results from the response of a material to an external magnetic ﬁeld, together with any unbalanced magnetic dipole moments that may be inherent in the material itself, for example, in ferromagnets. In Nature, however, distinct types of magnetism exist. Diamagnetic magnetization is the most common magnetic behaviour; it is proportional and opposed to the applied magnetic ﬁeld. Many materials present a diamagnetic response (see Section 13.4). This is a weak form of magnetism that is nonpermanent and persists only while an external ﬁeld is applied.

62

Thermal Convection: Patterns, Evolution and Stability

Paramagnetic materials exhibit a magnetization that is proportional to the applied ﬁeld and reinforces it. Paramagnetism varies inversely with temperature. When placed between the poles of a strong electromagnet, paramagnetic materials are pulled towards the region where the ﬁeld is stronger. Ferromagnetic materials present a magnetization much larger than other materials. This behaviour arises from the strong coupling between the neighbouring magnetic dipoles in the material. As anticipated, ferromagnetic materials can display spontaneous magnetization. They are known to change their behaviour to paramagnetic when a threshold temperature is exceeded (the Curie temperature). Introducing the magnetic ﬁeld intensity vector as H = B/µ0 − M

(1.198)

and taking into account that, in general, for magnetic materials M = ρχg H

(1.199)

where χg is the so-called mass magnetic susceptibility, the magnetic ﬂux density (also referred to as magnetic induction) can be ﬁnally written as B = µp H

(1.200)

µp = µ0 (1 + ρχg )

(1.201)

where

is the magnetic permeability of the material. Accordingly, Eq. (1.197) can be also cast in condensed form as ∂D + Jf ∂t where D is the so-called electric displacement ﬁeld : ∇ ∧H =

D = ε0 E + P

(1.202)

(1.203)

2 Classical Models, Characteristic Numbers and Scaling Arguments In Chapter 1, to truly understand the rich and wide-ranging phenomena treated in the present book, a general framework was built by ideally transcending speciﬁc cases and introducing the fundamental equations and related methods of analysis (which can provide vital insights into the solutions of these equations) essential in the treatment of continuum transport processes. However, one must be well acquainted with the nature of the driving forces responsible for the genesis of such phenomena and also with the related fundamental models introduced over the years by investigators. In this context, it is worth noting that physicists have often looked to applied mathematicians and engineers of various sorts for turning such effects into precise mathematical relationships. This strategy has been largely beneﬁcial to advancement of understanding. Remarkably, it has been fed by a fruitful interaction between theoreticians on one side and experimenters on the other. In particular, in such a process theoreticians have brought forward their own peculiar way of thinking about ﬂows and their effects, such as the pervasive use of scaling analysis and dimensionless numbers (Lappa, 2002a). Direct experimental analysis has permitted us to assess the validity of such a way of thinking, feeding back, in turn, vital information for further reﬁnement and/or theoretical elaboration. It is by virtue of such dialectics that theoretical approximations have moved beyond intuition towards a ﬁrm quantitative foundation. This chapter provides some simple and fundamental information along these lines. It is worth recalling, anyhow, that a number of applications and effective distinct cases will be presented in the following chapters. This means that the reader will be taken beyond the theoretical to demonstrate how the governing ﬂow equations and related models can be solved to provide applied, practical, predictive solutions to a variety of natural or technological phenomena or processes both on Earth and in space. Along these lines, a signiﬁcant effort is provided in many parts of the book to show how the numerical or theoretical results ﬁt the corresponding experimental counterparts. The consistency of numerical predictions with experimental data suggests, in fact, that rate-controlling steps are taken into account, that the unavoidable simpliﬁcations do not distort actual behaviour and ﬁnally provides validation for the theoretical models discussed here. The present chapter runs as follows: Sections 2.1–2.3 deal with landmark models for representing body and surface forces; Section 2.4 illustrates some exact solutions of the Navier–Stokes equations when ﬂuid motion is driven by these forces; and ﬁnally, Section 2.5 introduces the concept of the ‘boundary layer’. This concept is essential in the treatment of many ﬂows and also provides some necessary information Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

64

Thermal Convection: Patterns, Evolution and Stability

complementary from a theoretical point of view to the notion of scale discussed in Section 1.1.1 in a more abstract context.

2.1

Buoyancy Convection and the Boussinesq Model

Gravitational attraction is a fundamental property of matter that exists throughout the known universe. The presence of Earth creates a gravitational ﬁeld that acts to attract objects with a force inversely proportional to the square of the distance between the centre of the object and the centre of Earth. A remarkable impact of this body force on ﬂuids is the creation of ﬂows due to density differences (buoyancy-induced convection). Consider, for instance, what happens when a container of water is heated from below. As the water in the bottom is heated by conduction through the container, it becomes less dense than the unheated, cooler water. Because of gravity, the cooler, denser water sinks to the bottom of the container and the heated water rises to the top due to buoyancy; thus a circulation pattern is produced that mixes the hot water with the colder water. This is an example of buoyancy-driven (or gravity-driven) convection. The convection causes the water to be heated more quickly and uniformly than if it was heated by conduction (thermal diffusion) alone. This is the same density-driven convection process to which we refer when we state matter-of-factly that ‘hot air rises’. From a mathematical point of view, the buoyancy force can be simply obtained by multiplying the density of the considered ﬂuid by the acceleration due to gravity, g. This means that the body force term in the momentum equation Eq. (1.55) will simply read F b = ρg

(2.1)

Following the usual Boussinesq approximation for incompressible ﬂuids (Boussinesq, 1903), the physical properties can be assumed to be constant, except for the density ρ in such a generation term, which is assumed to be a linear function of temperature: ∞ 1 dk ρ (T − TREF )k (2.2) ρ = ρ(T ) = ρ(TREF ) + k! dT k k=1 TREF dk ρ dρ ∼ 0 for k ≥ 2 = −βT = 0 for k = 1 and with = dT T dT k REF

TREF

→ρ∼ = ρ0 [1 − βT (T − TREF )]

(2.3)

where βT is simply known as the thermal expansion coefﬁcient and TREF is a reference value for temperature. In his attempts to explain the motion of the light in the ether, Boussinesq (1903) opened a wide perspective of mechanics and thermodynamics. With a theory of heat convection in ﬂuids and of propagation of heat in deforming or vibrating solids, he showed that density ﬂuctuations are of minor importance in the conservation of mass. The motion of a ﬂuid initiated by heat results mostly in an excess of buoyancy and is not due to internal waves excited by density variations. In practice, this approximation states that (the reader is referred to Chapter 1 for the meaning of symbols in the second member of such expressions): N mi δ(r − r ) ∼ (2.4a) ∇ρ = ∇ =0 i

∂ ∂ ρ= ∂t ∂t

i=1 N i=1

mi δ(r − r i ) ∼ =0

(2.4b)

Classical Models, Characteristic Numbers and Scaling Arguments

65

that is, density differences are small enough to be neglected, except where they appear in terms multiplied by g, the acceleration due to gravity. The essence of this approximation is that the difference in inertia is negligible but gravity is strong enough to make the speciﬁc weight appreciably different between two ﬂuids particles of different temperatures. As a consequence, the continuity equation may be reduced to the vanishing of the divergence of the velocity ﬁeld (that is typical of incompressible ﬂows as shown in Section 1.2.9) and variations of the density can be ignored in the momentum equation also, except insofar as they give rise to a gravitational force. The derivation of conditions for the validity of the Boussinesq approximations is not as straightforward as many would assume. In the literature, a variety of sets of conditions have been assumed which, if satisﬁed, allow application of this approximation (e.g. Mihaljan, 1962; Mahrt, 1986). Basically, as illustrated by Gray and Giorgini (1976), such an assumption leads to reliable results if both the product (βT T ) and the ratio T /Tbulk are well below the value of 0.1, taken as the limit for the applicability of this model. Although used earlier by Oberbeck (1879), Boussinesq’s theoretical approach established a cardinal simpliﬁcation that is extremely accurate for many ﬂows and makes the mathematics and physics simpler (see the discussions in Chandrasekhar, 1961). In the framework of such an approximation, the momentum equation in dimensional form can be written as 1 ∂V (2.5) + ∇ · [V V ] + ∇p = ν∇ 2 V − [βT (T − TREF )]g ∂t ρ0 The identiﬁcation of the signiﬁcant parameters in the momentum equation requires this equation to be posed in nondimensional form through the choice of relevant reference quantities. Following the approach deﬁned in Section 1.2.10, the nondimensional form reads ∂V = −∇p − ∇ · [V V ] + Pr∇ 2 V − PrRaT i g ∂t

(2.6)

where Ra =

gβT T L3 να

(2.7)

is the Rayleigh number and i g the unit vector along the direction of gravity [throughout this book, unless expressly indicated otherwise, it is assumed i g = −i y where i y is the unit vector along the direction of the y axis (see Figure 2.1)]. This nondimensional parameter measures the magnitude of the buoyancy velocity Vg (it is traditionally employed as a reference quantity in this kind of

Figure 2.1 Gravity vector and coordinate reference system

66

Thermal Convection: Patterns, Evolution and Stability

problem) to the thermal diffusive velocity, where Vg reads Vg=

gβT T (L)2 ν

(2.8)

Moreover, gβT T L3 (2.9) ν2 is the Grashof number, which represents the ratio of buoyant to molecular viscous transport (obviously, only two of the Gr, Ra and Pr numbers are independent). When gravity is opposite to ∇T , g has no effect (it does not induce convective ﬂows) for any value of its magnitude (this effect is well known: for instance, ﬂuids uniformly heated from above do not exhibit convective motion); when gravity is concurrent with ∇T (ﬂuid heated from below), then distortions of the thermoﬂuid–dynamic ﬁeld (hereafter referred to as TFD and deﬁned as the ‘difference’ between the values of the ﬁeld variables in the presence of convection and the related values in quiescent diffusive conditions) arise only if the critical conditions for the onset of convection are exceeded (i.e. if the Rayleigh number is larger than the critical value Ra = Racr , so that instability sets in). Finally, when gravity is orthogonal to ∇T (ﬂuid heated from the side), then TFD distortions arise for any value of T . Their intensity depends on the Rayleigh number. Onset of gravity-induced convection for different heating conditions (from the side or from below) and for various geometric conﬁgurations is a basic problem in many heat-transfer systems in technical applications and has been widely investigated in the literature. The behaviour of ﬂuids uniformly heated from below falls into the category of phenomena known as the ‘Rayleigh–B´enard’ problem (Lord Rayleigh, 1916), which is one of the most intensively studied hydrodynamic systems (see Chapters 4 and 5). Also, the case of conﬁgurations heated from the side has instigated much research (Chapters 6 and 7). Some interesting examples in Nature and technology will be discussed in Chapter 3. Gr = Ra/ Pr =

2.2

Convection in Space

2.2.1 A Deﬁnition of Microgravity When the acceleration of an object acted upon only by Earth’s gravity at the Earth’s surface is measured, it is commonly referred to as one-g (1g) or one Earth gravity. This acceleration is approximately 9.8 metres per second squared (m s−2 ). The weight of an object is the gravitational force exerted on it by Earth. While the mass of an object is constant and the weight of an object is constant (ignoring differences in g at different locations on the Earth’s surface), the environment of an object may be changed in such a way that its apparent weight changes. Although gravity is a universal force, there are times, in fact, when it is not desirable to conduct scientiﬁc research under its full inﬂuence. In these cases, scientists perform their experiments in microgravity, a condition in which the effects of gravity are greatly reduced. This description may bring to mind images of astronauts and objects ﬂoating around inside an orbiting spacecraft, seemingly free of Earth’s gravitational ﬁeld, but these images are misleading. The pull of Earth’s gravity actually extends far into space. To reach a point where Earth’s gravity is reduced to one-millionth of that on the Earth’s surface, one would have to be 6.37 × 106 km away from Earth (almost 17 times further away than the Moon). Since spacecraft usually orbit only 200–450 km above the Earth’s surface, there must be another explanation for the microgravity environment found aboard these vehicles (Rogers et al., 1997). Astronauts ﬂoating in the Shuttle

Classical Models, Characteristic Numbers and Scaling Arguments

67

appear weightless not because they have escaped Earth’s gravity, but because they are in a state of free-fall . In practice, gravity cannot simply be switched off, but its effects can be compensated with the help of an appropriate acceleration force. This acceleration force must have exactly the same absolute value as the gravity force and it must point into the opposite direction of the local gravity vector. The resulting equilibrium of forces is called in normal language ‘weightlessness’. As an example, the propulsion-less ﬂight of a space vehicle or a space station around the Earth is a special form of a free-fall trajectory. The attraction force of the Earth’s gravity is permanently compensated by the centrifugal force resulting from the curved shape of the orbit. In general, however, an exact equilibrium state is difﬁcult to obtain and a very small gravity force always remains. This is the reason why specialists speak of ‘microgravity’ rather than ‘weightlessness’.

2.2.2 Experiments in Space Microgravity is instrumental in unravelling processes that are interwoven or overshadowed in normal gravity. Because gravity’s effect on ﬂuids is strong, scientists cannot determine what effect other forces are having on ﬂuid behaviour. Many of these forces become dominant in microgravity, allowing scientists to observe them without other competing inﬂuences. Along these lines, research conducted in microgravity is increasing our understanding of ﬂuid physics (otherwise hidden in normal-gravity conditions) to provide a foundation for predicting, controlling and improving a vast range of processes. Over recent years, both through the results of space experiments and through related groundbased research (normally, the effect of a microgravity environment is judged on the basis of comparison of experiments under identical conditions and in identical set-ups in ground conditions and in microgravity), an immense and signiﬁcant amount has been learned about gravitational and nongravitational contributions in a variety of natural phenomena and technological activities. Gravity has been found to inﬂuence processes that were thought to be independent of gravity. On the contrary: by means of microgravity experimentation, researchers have focused on many effects that have also been proved to inﬂuence the processes carried out on the ground. It should be stressed that at the present stage, the results obtained in microgravity are mostly of a fundamental nature, quantifying, as explained earlier, theoretical models of gravity inﬂuences on ﬂuid phenomena or leading to better insights into the signiﬁcance of forces and interactions which, during experiments on Earth, are masked by gravity-induced ﬂows; however, it has been demonstrated that such effects can be relevant in a number of phenomena of scientiﬁc and technological interest (Minster et al., 1999). For an account of all the experiments performed in microgravity, the interested reader may consult the ESA Microgravity Sciences Data Base (MSDB), the NASA Microgravity Research Experiments (MICREX) Database and JAXA’s International Space Environment Utilization Research Database (ISRDB) archives. These archives have been merged in the Microgravity International Distributed Experiment Archives (IDEA) Database, now included in the ESA Erasmus Experiment Archive (EEA) (http://spaceﬂight.esa.int/eea/). The ESA MSDB is a database of ESA-funded or co-funded experiments covering a wide range of scientiﬁc areas, which were performed during missions and campaigns on/in various space platforms and microgravity ground-based facilities over the past 30 years. It contains information about research areas, mission names and dates, team members, processing facilities, experimental objectives, experimental procedures and results. MICREX is a database created at NASA Marshall Space Flight Center, identifying over 800 ﬂuids and materials processing experiments performed in a low-gravity environment. It was designed

68

Thermal Convection: Patterns, Evolution and Stability

to document all such experimental efforts performed on United States manned space vehicles, on payloads deployed from United States manned space vehicles and on all domestic and international sounding rockets (excluding those of China and the former USSR). ISRDB is a similar database created by JAXA, collecting data on Japanese microgravity experiments. There also exist some books on the subject (e.g. Lappa, 2004a). Moreover, the list of acronyms and abbreviations below provides disjointed glimpses of the rich variety of multi-user laboratories and facilities currently accommodated in different science modules of the International Space Station (ISS) for future studies in the ﬁeld of ﬂuids [e.g. the American Fluids and Combustion Facility, FCF; the European Fluid Science Laboratory, FSL (the reader is referred to Albanese et al., 2007, and Trinchero et al., 2007, for further details about this facility expressly devoted to the study of thermal convection); the Japanese Fluid Physics Experimental facility, FPEF (see, e.g., Shevtsova et al., 2008)], inorganic material science (e.g. the Materials Sciences Laboratory, MSL, with the Low Gradient Furnace, LGF, the Solidiﬁcation and Quenching Furnace, SQF, and the Float Zone Furnace, FMF; the High Gradient Directional Solidiﬁcation Furnace Experiment Module, HGDS; the Directional Solidiﬁcation and Vapor Transport Experiment Module, DSVT; the Advanced Tubular Furnace With Integrated Thermal Analysis Under Space Conditions, TITUS; the Quench Module Insert, QMI; the Diffusion Module Insert, DMI; the Advanced Thermal Environment Furnace, ATEN; the Advanced Furnace for Microgravity Experiment with X-ray Radiography, AFEX; the Gradient Heating Furnace, GHF, etc.), organic material science (the Protein Crystallization Diagnostics Facility, PCDF; the Solution Protein Crystal Growth Facility, SPCF; etc.) and biotechnology (the BIOLAB; etc.).

2.2.3 Surface Tension-driven Flows The most intensively studied type of convection in space is ﬂuid ﬂow induced by surface tension-driven forces; as outlined earlier, in fact, microgravity gives the possibility of avoiding some limitations related to the ground environment that adversely affect the experimental study of this problem (in particular, the aforementioned buoyancy-driven convection that in many circumstances overshadows this kind of convection). Moreover, it is worth stressing that in zero-g conditions it is possible to form very large ﬂoating liquid volumes with extended liquid/gas interfaces that facilitate signiﬁcantly the development and ensuing study of these ﬂows; indeed, during recent years, the availability of sounding rockets, orbiting laboratories such as the Spacelab and especially the ISS, has made possible microgravity experiments with large free surfaces, which could not be performed on Earth under normal-gravity conditions. Prior to the space program, this phenomenon had been ignored in investigations of materials processing on Earth. In microgravity, the reduced level of buoyancy-driven convection allowed convection driven by gradients of surface tension to become obvious. Once it became recognized, it was found to be signiﬁcant in some Earth-based processes also (especially semiconductor crystal growth, but also other important technological processes and instances in Nature; see, e.g., DebRoy and David, 1995; Mills et al., 1998; Monti et al., 1998c; Limmaneevichitr and Kou, 2000; Monti et al., 2000c; Savino et al., 2003; Lappa, 2004a, 2005f–i, 2006b, 2007c; Amberg and Shiomi, 2005, etc.). It was found that this surface tension-driven convection could not only be vigorous, but also become asymmetric, oscillatory and even turbulent. This phenomenon is usually referred to in the literature as ‘Marangoni convection’ (named after the Italian physicist Carlo Giuseppe Matteo Marangoni (Marangoni, 1871).

2.2.4 Acceleration Disturbances on Orbiting Platforms and Vibrational Flows The preﬁx micro- (µ) traditionally used for the word microgravity (µg) derives from the Greek mikros, meaning small.

Classical Models, Characteristic Numbers and Scaling Arguments

69

Quantitative systems of measurement, such as the metric system, commonly use micro- to mean one part in a million. Using that deﬁnition, the acceleration experienced by an object in a microgravity environment would be one-millionth (10−6 ) of that experienced at the Earth’s surface (g0 ). In practice, the microgravity environments used by scientiﬁc researchers range from about 1% of the Earth’s gravitational acceleration (aboard aircraft in parabolic ﬂight) to better than one part in a million (for example, onboard Earth-orbiting research satellites). True accelerationless environments do not exist in the real world; also, the facilities for microgravity experimentation suffer from some degree of residual acceleration. A free-fall trajectory around the Earth can, in principle, last for eternity. In practice, however, at the low altitude of a space station (typically about 400 km above the Earth’s surface) there are small disturbances resulting from the fact that the Earth’s atmosphere does not stop abruptly, but only becomes thinner and thinner. At an altitude of 400 km, there are still some oxygen atoms which can decelerate an object. This disturbs the microgravity level onboard the object. It also leads to a slow, but constant, loss of altitude which needs to be compensated from time to time by a manoeuvre which is called ‘reboost’, where the platform uses small rocket engines to accelerate it and bring it back to a higher orbit. During these reboost manoeuvres, the microgravity level onboard is also disturbed. Zero residual gravity would be obtained only in the platform centre of mass, where gravity and inertia perfectly balance themselves in the ideal case of a circular orbit, constant angular speed and gravity as the only acting external force. In real circumstances, the ideal zero gravity intensity is perturbed by various factors, some of which have been mentioned above, that contribute to create steady levels of residual gravity and also a time-dependent residual gravity ﬁeld. In general, the real microgravity environment for medium/long-duration missions consists of a spectrum of accelerations at different frequencies ranging from zero (steady residual acceleration) to hundreds of hertz. The unsteady contributions to the residual gravity ﬁeld, in turn, can be separated into ‘quasi-steady’, having a low frequencies content, such as those due to atmospheric drag, solar radiation pressure and higher harmonics of the Earth gravitational ﬁeld, and the so-called ‘g-jitters’ having a higher frequencies content, such as those coming from pulsating or impulsive external loads, like motor ﬁrings during manoeuvres, crew activity, station motorized equipment, station moving elements, space debris impacts and the station structural elastic response to these loads. In the following, as a relevant and clarifying example, these disturbances and the related magnitudes are listed for the International Space Station (ISS), the largest orbiting object ever built by the humankind: 1. Steady (or quasi-steady) residual-g. These include aerodynamic drag (1–3 × 10−7 g0 ), radiation pressure (10−8 g0 ), micrometeorite impacts (10−9 g0 ) and, for points distant from the centre of mass, gravity gradient and rotation periodic with the orbit [O(10−7 )g0 m−1 ]. 2. Periodic, high frequency, due to onboard machineries and natural frequencies excited by external forces (10−6 < g/g0 < 10−2 , 0.1 Hz < f < 300 Hz). The global effect of steady (or quasi-steady) residual-g is of the order O(10−6 g0 ). The resultant combination of atmospheric drag, gravity gradient and other secondary effects produce a set of gravity contours (i.e. locations of equal gravity level) which deﬁne the quasi-steady-state microgravity environment of the ISS. These gravity contours are in the form of coaxial elliptical cylinders aligned parallel to the ISS velocity vector (Figure 2.2). In general, g-jitter varies randomly in magnitude and direction. With the exception of attitude adjustments, these disturbances are usually transmitted as structural vibrations, eventually to the support structure or container walls of the ﬂuid system in question, namely laboratory experiments. Therefore, the actual excitation experienced by the ﬂuid may sometimes be approximated by a harmonic forcing with the frequency of the resonant structural mode.

70

Thermal Convection: Patterns, Evolution and Stability

1 µg 2 µg

Destiny (US-Lab)

Kibo (Jaxa-module)

Columbus (ESA-COF)

Figure 2.2 Gravity contours which deﬁne the quasi-steady-state microgravity environment of the ISS. Courtesy of ESA and NASA

The predicted high-frequency g-jitter (e.g. NIRA 99 predictions for the US-Lab and for the ESA-COF) are usually reported as a plot of acceleration amplitude versus frequency and compared with the System Allowable (so-called ‘ISS requirements curve’; see Figure 2.3). Similarly to the case of Marangoni convection, the nonlinear response of ﬂuid systems to high-frequency disturbances has become a topic of great interest in the last two decades. Also in this case, in fact, the system can exhibit transition to subsequent complex patterns of ﬂow (thermovibrational convection) according to the amplitude, oscillation frequency and orientation of the external forces and deviate in structure and magnitude from the diffusive (motionless) conditions that should be ideally maintained by the microgravity condition (in space, these ﬂows usually represent an undesired effect when carrying out experimentation that would beneﬁt from the establishment of quiescent regimes (see, e.g., Monti et al., 2001; Lappa and Carotenuto, 2003). However, there is also another reason why the study of the effect of vibrations on ﬂuid behaviour has witnessed increased interest over recent years. It has been understood, in fact, that owing to the peculiar properties of this type of convection (discussed in Chapter 8), it can be used as a useful means to ‘mitigate’ buoyancy or Marangoni convection present under terrestrial conditions and/or somehow control their instabilities (this subject will be extensively treated in Chapter 12).

2.3 Marangoni Flow The conditions for which a system of two immiscible ﬂuid phases with their ‘interphase layer’ can be modelled at the microscopic level as two volume phases separated by an ‘interface’ were examined in a unitary, exhaustive and consistent way by Napolitano (1979). In such a theoretical study thermodynamic and dynamic theories of the surface phase were developed for increasing levels of sophistication according to the nature and relevance of the interactions between the considered volume phases and their interphase layer. Here we limit to consider the canonical case in which the interface separating a liquid and a gas can be modelled as

Classical Models, Characteristic Numbers and Scaling Arguments

71

1E + 5 Lab Ergometer

Acceleration (micro-g)

1E + 4

1E + 3

1E + 2

nt

me

e uir

q

ISS

1E + 1

Re

1E + 0

1E − 1 1E − 2

1E − 1

1E + 0

1E + 1

1E + 2

Frequency (Hz)

Figure 2.3 Predicted high-frequency g -jitter (Theoretical Non-Isolated Rack Assessment) reported in a plot of acceleration amplitudes versus frequencies and compared with the System Allowable, the so-called ‘ISS requirements curve’. Courtesy of NASA

a mathematical boundary with no mass and zero thickness, assumed (as also mentioned in Section 1.7) to be undeformable and with a ﬁxed location in space. The set of equations constituting the appropriate boundary conditions for the corresponding closed set of ﬁeld equations in the volume phases is provided under the assumption (already invoked in Section 1.2.2) that local formulation of equilibrium thermodynamics applies (reference being made to Defay et al ., 1977 for a discussion of the related implications in the case of surface dynamics).

2.3.1 The Genesis and Relevant Nondimensional Numbers The surface tension σ = σ (T ) for many cases of practical interest exhibits a linear dependence on temperature: ∞ 1 dk σ (T − TREF )k (2.10) σ = σ (T ) = σ (TREF ) + k! dT k k=1

dk σ dσ = −σT = 0 for k = 1 and with dT TREF dT k

TREF

∼ = 0 for k ≥ 2

TREF

→σ ∼ = σ0 [1 − σT (T − TREF )]

(2.11)

where σ0 is the surface tension for T = TREF (TREF is a reference value), σT =−dσ /dT > 0 (σ is a decreasing function of T ).

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Thermal Convection: Patterns, Evolution and Stability

If a nonisothermal free surface is involved in the process considered, then surface tension forces F Tσ = ∇ S σ (∇ S derivative tangential to the interface) arise that must be balanced by viscous stresses in the liquid (throughout this book, the dynamic viscosity of the gas surrounding the free liquid surface will be assumed to be negligible with respect to the viscosity of the considered liquid); from a mathematical point of view, this condition can be written as ˆ · ∇T τ · nˆ = −σT (I − nˆ n) d

(2.12a)

where τ = is the dissipative part of the stress tensor [see Eq. (1.31b)], nˆ is the unit d vector perpendicular to the liquid/gas interface (directed from liquid to gas) and I is the unity matrix. For a planar surface, the balance above simply yields ∂V (2.12b) µ S = −σT ∇ S T ∂n where n is the direction locally perpendicular to the considered elementary portion of the free interface and V s the surface velocity vector. In nondimensional form (scaling the lengths by a reference distance L, the velocities by the energy diffusion velocity and the temperature by a reference T ), Eq. (2.12b) reads ∂V S (2.13) = −Ma∇S T ∂n where Ma = σT T L/µα (µ being the dynamic viscosity) is the so-called Marangoni number. This condition enforces a ﬂow by tangential variation of the surface tension. The motion (thermocapillary convection) immediately results whenever a temperature gradient exists along the considered interface, no matter how small. The surface moves from the region with a low surface tension (relatively hot) to that with a high surface tension (relatively cold). The viscosity transfers this motion to the underlying ﬂuid, that is, the ﬂow penetrates into the bulk through viscous coupling to the motion at the interface (Scriven and Sternling, 1960). The following nondimensional numbers are of particular importance in such a context (see, e.g., Kuhlmann, 1995): σT T L (2.14) Re = ρν 2 σT T (2.15) Ca = σ0 σT T L Ma = RePr = (2.16) µα The Reynolds number (Re) measures the magnitude of the tangential stress σT T /L to the viscous stress ρν 2 /L2 ; the Prandtl number (Pr, already introduced in Chapter 1) measures the rate of momentum diffusion ν to that of heat diffusion α and the Marangoni number is Ma = RePr (only two of these three numbers are independent). The capillary number (Ca = ReOh, Oh = ρν 2 /σ0 L being the Ohnesorge number) gives the relative change of the surface tension due to temperature variations. It involves the mean surface tension σ0 and it is an important measure of the dynamic deformability of the free surface (deformability induced by ﬂuid motion). The interface location is determined, in fact, by the hydrodynamic and the Laplace pressure. The relative importance of both is given by the ratio of the relative magnitudes σT T /L and σ0 /L. This ratio is the above-mentioned capillary number, Ca = σT T /σ0 . If Ca → 0 (i.e. Ca 1), the dynamic surface deformation can be neglected. In such a case, the liquid/gas interface of liquid layers and open cavities is usually assumed to be ﬁxed, ﬂat and plane in the xz plane (Figure 2.4b); thus Eq. (2.13) reads: ∂T ∂u = −Ma (2.17a) ∂y ∂x 2µ(∇V )so

Classical Models, Characteristic Numbers and Scaling Arguments

(a)

73

(b)

(c)

Figure 2.4 Fundamental models for the study of Marangoni convection: (a) Liquid column; (b) rectangular layer or slot; (c) annular pool

∂T ∂w = −Ma ∂y ∂z

(2.17b)

u and w being the nondimensional velocity components along x and z, respectively. For a cylindrical free interface (Figure 2.4a), the same condition reads ∂T ∂u = −Ma ∂r ∂z ∂T ∂Vϕ − Vϕ = −Ma r ∂r ∂ϕ

(2.18a) (2.18b)

u and Vϕ being the velocity components along z and the angular coordinate, respectively (z is the axis of the cylinder and r the radial coordinate, Figure 2.4a). For the annular pool shown in Figure 2.4c: ∂T ∂v = −Ma ∂y ∂r ∂T ∂Vϕ = −Ma r ∂y ∂ϕ

(2.19a) (2.19b)

v and Vϕ being the velocity components along r and the angular coordinate, respectively. Features of interest are the fundamental scales of velocity and temperature that determine the strength of the ﬂow.

74

Thermal Convection: Patterns, Evolution and Stability

It was noted by Rybicki and Floryan (1987) that the appropriate scaling of the dimensional velocity is the Marangoni velocity: T = VMa

σT T µ

(2.20)

Then the Marangoni number can also be seen as the measure of the relative importance of the Marangoni and of the thermal diffusion velocities. Moreover, for small Reynolds and Prandtl numbers, the dimensional temperature ﬁeld scales as RePrT . In many situations, the capillary number is small and the ﬂow-induced surface deformations can be expanded in a power series of Ca to yield the leading-order surface deﬂection δ0 , which turns out to be of O(CaL). To summarize, in dimensional form the scaling for small Re, Pr and Ca reads σT T (2.21) V =O µ TFD = max(T − Tdiff ) = O(RePrT ) = O(MaT ) (2.22) δ0 = O(CaL) (2.23) where Tdiff is the reference diffusive temperature ﬁeld (i.e. the temperature distributions that would be established in the absence of convection) and TFD is the corresponding temperature distortion, i.e. the departure from the diffusive temperature proﬁle. An additional relevant parameter with which researchers often have to deal with is represented by the Biot number, which is deﬁned as hL (2.24) λ where h is the so-called convective heat transfer coefﬁcient at the free surface. This nondimensional number is used to take into account possible thermal coupling of liquid with ambient. On the free surface, it is generally assumed, in fact, that the heat transfer between the liquid and the surrounding gas can be conveniently approximated by the following nondimensional relation: Bi =

∂T (2.25) = −Bi(T − Ta ) ∂n where Ta is the gas (ambient) temperature: T < Ta means liquid is heated from the surrounding gas and, vice versa, T > Ta means there exists a ﬂux of heat from the liquid to the ambient; an adiabatic surface can be seen as a special case of Eq. (2.25) with Bi = 0. Marangoni convection has attracted increasing interest in recent years with regard to many different geometric conﬁgurations and heating conditions. Indeed, the mechanics of response of these ﬂuid dynamic systems depends on the type of heating applied to the interface. In the geometrically simplest case of liquid contained in a rectangular cavity open from above, the heating can be applied either through the bottom (or from above) or through the sidewalls. The response of the system is markedly different in each of these cases. As in the case of thermogravitational convection described in Section 2.1, the direction of the imposed T plays a crucial role: if the externally imposed T yields imposed temperature gradients that are primarily perpendicular to the interface, the basic state is static with a diffusive temperature distribution and motion (Marangoni–B´enard convection) ensues with the onset of instability when T exceeds some threshold (this subject is treated in Chapter 9); if the externally imposed T yields imposed temperature gradients that are primarily parallel to the interface, as anticipated, in these cases motion occurs for any value of T (see Chapter 10).

Classical Models, Characteristic Numbers and Scaling Arguments

75

2.3.2 Microzone Facilities and Microscale Experimentation Despite the presence of buoyancy convection that in many cases obscures Marangoni ﬂow, it must be pointed out how some insights into this type of convection can be obtained also by experimental investigation on Earth if the typical scale of the liquid volume is reduced. Usually this kind of study and the related facility are referred to as microscale experimentation and microzone apparatus, respectively. In practice, simulated microgravity can be obtained on the ground by reducing the characteristic length of the system under investigation. This condition, of course, does not really cancel gravity, but it is possible to emphasize surface tension effects with respect to buoyancy-induced effects. In fact, the ratio of Rayleigh number (see Section 2.1 for the related deﬁnition) to Marangoni number (which measures the driving actions of the gravity and of the surface tension imbalance) grows quadratically with the linear dimension of the liquid zone, making thermal buoyancy convection less important for small liquid volumes: 1 gρβT 2 Ra L = = Ma W σT

(2.26)

If this ratio is kept small (W is kept large) by reducing L (a few millimetres), the Marangoni effect is emphasized in comparison with the buoyancy effect. This approach provides a relevant alternative to the use of expensive microgravity or other expensive means (e.g. virtual microgravity that can be obtained on Earth using intense gradients of magnetic ﬁelds, as will be explained in Chapter 13). From a historical point of view, it has been largely used over the last three decades especially by scientists not having the opportunity to carry out research in true microgravity conditions, that is, on orbiting platforms. Remarkably, the reduction of the scale on the ground also has another purpose: it can be used to avoid large curvature of the liquid/gas interface. Such deformation is usually approximately proportional to the so-called Bond number: Bo =

ρgL2 σ0

(2.27)

where ρ is the density jump between the liquid and the surrounding gas (ρ ∼ = ρliquid ). The parameter Bo represents the ratio of internal hydrodynamic pressure to surface-tension force. If this number is sufﬁciently small, the ﬂuid/ﬂuid interfaces behave (approximately) not much differently with respect to the case of zero-g.

2.3.3 A Paradigm Model: The Liquid Bridge For Marangoni convection, a special conﬁguration has been conceived for the microgravity environment and has become, over the years, a paradigm model for the study of these ﬂows, their stability and their bifurcations. It was introduced in the mid-1970s as a vehicle for performing experiments in well-controlled conditions and is usually referred to as the liquid bridge, a drop of liquid with cylindrical or quasi-cylindrical free liquid/gas interface held between two disks of radius R placed L apart (Figure 2.5). This model has distinct advantages for both the experimentalist and the numerical investigator. In particular, the ends of the domain are isothermal, the interface is generally considered adiabatic and the applied temperature difference (difference of the temperatures of the supporting disks) driving the Marangoni surface ﬂow can be ﬁxed a priori in the analyses. Moreover, unlike other traditional geometric models (open cavities and annular pools), it is the only conﬁguration for

76

Thermal Convection: Patterns, Evolution and Stability

z r = c(z)

S>1 S=1 S <1

r j

(a)

z

r ϕ (b)

Figure 2.5 Non-cylindrical liquid bridges: (a) microgravity conditions (concave and convex surfaces); (b) on the ground shape

which the ratio between the areas of the free surface (2π RL, where surface stress supporting the development of convection arise) and of the solid walls (2π R 2 ) can be larger than unity. Such a ratio is given by L 2π RL = (2.28) 2π R 2 R Hence to make it >1 it is sufﬁcient that L > R. Notably, the comparatively high surface to solid wall ratio for this model leads to a pronounced surface tension-driven convection. Possible static conﬁgurations of the liquid bridge, however, are not limited to a cylindrical shape. Unlike the other conﬁgurations in Figure 2.4 for which the interface is ﬂat and planar, if the effective volume of liquid held between the disks is larger or smaller than the volume of the corresponding cylinder with the same height and basis, then the shape can be convex or concave, respectively (see Figure 2.5a). This shape can undergo deformation in normal gravity conditions as shown in Figure 2.5b (more or less pronounced according to the surface tension of the liquid and the height of the column;

Classical Models, Characteristic Numbers and Scaling Arguments

(a)

77

(b)

Figure 2.6 Snapshots of: (a) liquid bridge of opaque molten metal (tin) held between two stainless steel rods of 4 mm diameter (after Yang and Kou, 2001; Copyright Elsevier, 2001); (b) liquid bridge of silicone oil sustained between two disks of 4 mm diameter (after Monti et al., 2000a; the liquid motion in the meridian plane is visualized using tracers scattering the light generated by an He–Ne laser diode with a wavelength of 635 nm, forming a light sheet; the laser beam is oriented orthogonal to the main optical path of a CCD camera)

see Figure 2.6). In general, the gravitational acceleration results in a surface shape that bulges out below the equatorial plane and necks in above it. The geometric aspect ratio of the liquid bridge is usually deﬁned as AH = L/D (where D is the diameter of the supporting disks). The effect of the shape (cylindrical, convex or concave under microgravity conditions, amphora-like on the ground), as expected, must be also regarded as an important aspect of the problem. Usually these aspects are studied in terms of the nondimensional parameter S (the so-called volume or shape factor), the ratio of the volume held between the supporting disks and the volume of the corresponding straight conﬁguration (for a ﬁxed aspect ratio, the effective volume of liquid suspended between the supporting disks, in fact, determines the shape of the melt/gas interface). As explained in Section 2.3.1, if Ca 1, the interface can be assumed to be rigid and axisymmetric around the z-axis, its radial coordinate being a function solely of the z variable [r = c(z)]. Under such assumptions, the hydrostatic shape can be obtained from the Young–Laplace equation, relating the local curvature of the surface to the pressure jump along the liquid/gas interface: 1 1 (2.29a) + p + ρgz = σ0 R1 R2 where R1 and R2 are the principal radii of curvature at each point of the surface. Equation (2.29a) may be reformulated in dimensionless form (scaling the radial and axial coordinates by L) in cylindrical coordinates by substituting the analytical expression of the principal radii of curvature in the axisymmetric geometry in terms of the function r = c(z) representing the location in space of the free surface: 1 − c(1 + c2 )

c 3

(1 + c2 ) 2

=

Lp ρgL2 + z = k1 + k2 z σ0 σ0

(2.29b)

The parameter k2 = ρgL2 /σ0 , in practice, is the Bond number already introduced in Section 2.3.2. It takes into account the deformation of the shape under gravity conditions and depends on the

78

Thermal Convection: Patterns, Evolution and Stability

1.00

Axial coordinate

0.80

0.60 Bo

0

0.40

0.20

0.00 0.40

0.44

0.48 0.52 Radial coordinate

0.56

0.60

Figure 2.7 Shape of the liquid bridge surface as a function of the Bond number (silicone oil 1 cSt, normal-gravity conditions): Bond numbers in order from the most arched proﬁle to the ﬂattest one are Bo = 11.5 (L = 0.5 cm), 7.36 (L = 0.4 cm), 4.14 (L = 0.3 cm), 1.84 (L = 0.2 cm), 0.46 (L = 0.1 cm). Bo = 0 would correspond to the cylindrical shape (zero-g conditions)

value of the g-level, on the height of the ﬂoating zone and on the properties of the liquid under investigation (ρ and σ ). Each value of the parameter k1 = Lp/σ0 corresponds to a ﬁxed volume of the liquid zone. The value of k1 can be changed in Eq. (2.29b) in order to obtain the desired volume and the equation can be integrated by a shooting method with the condition that the liquid is attached to the solid supports: c(0) = c(1) = R/L

(2.30)

As an example, Figure 2.7 shows the shape of a silicone oil liquid bridge with a ﬁxed aspect ratio (AH = 1) for different values of Bo (corresponding to different heights L of the liquid column). It is shown that, as expected, the shape tends to the cylindrical interface (microgravity conditions) when the Bond number tends to zero.

2.4

Exact Solutions of the Navier–Stokes Equations for Thermal Problems

As discussed in Chapter 1, in general, the thermal convection equations are nonlinear partial differential equations that in most cases require the use of complex algorithms (see Section 1.7) in combination with opportune discretization techniques for obtaining reliable numerical solutions.

Classical Models, Characteristic Numbers and Scaling Arguments

79

Figure 2.8 Sketch of layer of inﬁnite extent subject to horizontal heating

There are some cases, however, in which such equations admit analytical solutions. Such exact solutions have enjoyed fairly wide use in the literature as basic states for determining the linear stability limits in some idealized situations. For the cases of interest in the present book (buoyancy and Marangoni convection), in particular, these exact solutions exist when the considered system is inﬁnitely extended along the direction of the imposed temperature gradient or the orthogonal direction; in general, these solutions are regarded as reasonable approximations in the steady state of the ﬂow occurring in the core of real conﬁgurations which are sufﬁciently elongated (the core is the region sufﬁciently far away from the end regions, where the ﬂuid turns around, to be considered not to be inﬂuenced by such edge effects). Even though limited to cases of great simplicity, these solutions have proved able to yield directly or indirectly insights and understanding that would have been difﬁcult to obtain otherwise. Hereafter, ﬁrst the attention is focused on the case of systems subject to horizontal heating (or to a temperature gradient along the direction in which they are assumed to be inﬁnitely extended), then other possible variants and conﬁgurations are considered in Section 2.4.6. In particular, in Sections 2.4.1, 2.4.2 and 2.4.3 the horizontal boundaries are assumed to be located at y = − 1/2 and 1/2, respectively (see Figure 2.8). The velocity components along y and z are zero (v = w = 0) and the component along x is solely a function of y [u = u(y)]. Temperature depends on x and y (i.e. these solutions are essentially two-dimensional). Rayleigh and Marangoni numbers are deﬁned as Ra = GrPr = gβT γ d 4 /να and Ma = RePr = σT γ d 2 /µα, respectively (where γ is a rate of uniform temperature increase along the x axis and d is the distance between the boundaries). Velocity and temperature are referred to the scales α/d and γ d, respectively (moreover, all distances are scaled on d). In the most general case, such analytical solutions can be written as (with Ma = 0 or Ra = 0 for pure buoyancy or Marangoni ﬂows, respectively)     Rag1 (y) + Mag2 (y) u  0 (2.31) V = v = w 0 T = x + Raf1 (y) + Maf2 (y)

(2.32)

(the temperature ﬁeld is linear in x plus a distribution in y obtained from a balance between vertical thermal diffusion and horizontal convection; the latter contribution is the ﬂow-induced temperature ﬁeld, i.e. the TFD distortion), f1 , f2 , g1 and g2 being polynomial expressions with constant coefﬁcients satisfying speciﬁc equations obtained by substituting Eqs (2.31) and (2.32) into the energy equation and into the equation resulting from cross-differentiation of the two projections of the momentum equation in the x and y directions. Such a system of equations reads g1 =

d2 f1 dy 2

(2.33a)

80

Thermal Convection: Patterns, Evolution and Stability

d2 f2 g2 = dy 2 3 d3 g 2 d g1 − 1 + Ma 3 = 0 Ra 3 dy dy

(2.33b) (2.33c)

to be supplemented with the proper boundary conditions as follows: Kinematic conditions: u = 0 → g1 (y) = g2 (y) = 0 dg2 dg1 = 0 and = −1 Eq.(2.13) → dy dy

solid boundary : free surface :

(2.33d) (2.33e)

Thermal conditions: adiabatic boundary : conducting boundary :

df1 df2 ∂T =0→ = =0 ∂y dy dy T = x → f1 = f2 = 0

(2.33f) (2.33g)

Before going further with the description of solutions satisfying such a system of equations, Eqs (2.33a–g), it is worth noting that some insights into their expected polynomial order can be given immediately on the basis of Eq. (2.33c). According to such an equation, in fact, d3 g1 /dy 3 = 1 for Ma = 0, while d3 g2 /dy 3 = 0 for Ra = 0, which leads to the general conclusion that the polynomial expression for u will be of third order for pure buoyancy ﬂow and of second order for pure Marangoni ﬂow, whereas [on the basis of Eqs (2.33a) and (2.33b)] the respective temperature proﬁles are of ﬁfth and fourth order in y.

2.4.1 Thermogravitational Convection: The Hadley Flow In the case of liquid conﬁned between two horizontal inﬁnite walls with perpendicular gravity, the Navier–Stokes equations admit as an exact solution the following velocity proﬁle: Ra " 3 y # y − (2.34) u= 6 4 The corresponding temperature proﬁle changes according to the type of boundary conditions considered. For adiabatic walls it reads 5 5 Ra (2.35a) T =x+ y y4 − y2 + 120 6 16 whereas for conducting boundaries it becomes

Ra 5 2 7 4 T =x+ y y − y + 120 6 48

(2.35b)

The polynomial expressions f and g for such solutions are plotted in Figure 2.9.

2.4.2 Marangoni Flow In the absence of gravity and replacing the upper solid wall with a liquid/gas interface supporting the development of surface tension-driven (Marangoni) convection, the exact solution reads (Birikh, 1966b) 1 Ma 3y 2 + y − (2.36) u=− 4 4

Classical Models, Characteristic Numbers and Scaling Arguments

81

(a)

(b)

Figure 2.9 Exact solution for the case of thermogravitational convection in an inﬁnite layer with top and bottom solid walls: (a) velocity proﬁle g(y); (b) temperature proﬁle f (y) for adiabatic (solid) and conducting (dashed) boundaries

For an adiabatic interface and insulated bottom wall, the temperature distribution can be expressed as 3 3 5 Ma 3y 4 + 2y 3 − y 2 − y − (2.37a) T =x− 48 2 2 16

82

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 2.10 Exact solution for the case of thermocapillary convection in an inﬁnite layer with bottom solid wall and upper free surface: (a) velocity proﬁle g(y); (b) temperature proﬁle f (y) for adiabatic (solid) and conducting (dashed) boundaries

which for conducting boundaries must be replaced with T =x−

3 1 3 Ma 3y 4 + 2y 3 − y 2 − y + 48 2 2 16

The polynomial expressions f and g for such solutions are plotted in Figure 2.10.

(2.37b)

Classical Models, Characteristic Numbers and Scaling Arguments

83

2.4.3 Hybrid States Under the considered conditions (u = 0, v = w = 0), the nonlinear convective term of the momentum equation becomes zero and the energy equation reduces to u = ∂ 2 T /∂y 2 , that is, the governing equations are linear; as a consequence, more complex solutions can be built as a superposition (addition) of other simple existing solutions. Along these lines, in the presence of vertical gravity and a liquid/gas interface supporting the development of surface tension-driven (Marangoni) convection, the velocity proﬁle can be obtained as the sum of two terms, the ﬁrst term corresponding to the pure buoyancy-driven ﬂow and the second representing the contribution of a pure thermocapillary-driven ﬂow, which provides a simple theoretical explanation for the general form given by Eqs (2.31) and (2.32). The resulting shear ﬂow is set up by a combined effect of buoyancy and viscous surface stress due to the temperature dependence of surface tension. The ﬁrst contribution related to pure buoyancy ﬂow, however, exhibits some differences with respect to Eq. (2.34) as the upper solid wall considered in Section 2.4.1 must be replaced with a stress-free boundary (see Figure 2.11); the velocity proﬁle reads 1 Ra 3 2 8y − 3y − 3y + (2.38) u= 48 4 Hence the resulting proﬁle for mixed gravitational–Marangoni convection has the form Ra 1 1 Ma u= 8y 3 − 3y 2 − 3y + − 3y 2 + y − (2.39) 48 4 4 4 The associated temperature distribution is 5 19 1 Ra 1 8y 5 − 5y 4 − 10y 3 + y 2 + 5y + T =x+ − 3η y + 48 20 2 16 2 $ 3 3 5 1 (2.40) −Ma 3y 4 + 2y 3 − y 2 − y − +η y+ 2 2 16 2 The cases of adiabatic and conducting horizontal surfaces can be obtained by setting η = 0 and η = 1, respectively, in this relation. Equation (2.40) can be extended to the more general case in which the bottom is conducting and a Biot number is introduced to describe heat transfer at the top free surface by assuming η = Bi/(1 + Bi). The solutions for symmetrical (top and bottom) conditions can again be recovered as limiting cases when Bi → 0 and Bi → ∞. When Bi → ∞, in fact, η → 1 and Eq. (2.40) reduces to the conducting case; when Bi → 0 (η → 0), it reduces to the insulating boundary conditions for the temperature proﬁle. Other ‘hybrid’ exact solutions for which the fundamental mechanism (buoyancy or Marangoni effect) interact with other forces will be reported in the other chapters of this book as necessary (e.g. Sections 12.4 and 12.6 for gravitational and Marangoni ﬂow interacting with vibrations, respectively, Section 8.5 for the case of pure thermovibrational convection and Section 13.2 for the case in which a new body force is introduced due to the presence of magnetic ﬁelds).

2.4.4 General Properties It is worth noting that all the solutions described in Sections 2.4.1–2.4.3 are of the parallel-ﬂow type and maintain zero mass ﬂux through any vertical plane (since the ﬂow is a parallel ﬂow intended to model a slot with distant end walls, continuity requires that the net ﬂux of ﬂuid at any cross section of the slot be zero), that is 1 2 udy = 0 (2.41) 1 −2

84

Thermal Convection: Patterns, Evolution and Stability

(a)

(b) Figure 2.11 Exact solution for the case of pure thermogravitational convection in an inﬁnite layer with bottom solid wall and upper stress-free surface: (a) velocity proﬁle g(y); (b) temperature proﬁle f (y) for adiabatic (solid) and conducting (dashed) boundaries

However, they exhibit different behaviours with respect to the Rayleigh necessary condition (for the instability of parallel ﬂows in the limit as Pr → 0 illustrated in Section 1.5.4) that reads d2 u =0 dy 2

(2.42)

Classical Models, Characteristic Numbers and Scaling Arguments

85

The second derivative of the solution Eq. (2.34) gives d 2u = Ray (2.43) dy 2 which means that the proﬁle has an inﬂection point at mid-height (y = 0) and satisﬁes the Rayleigh necessary condition. For pure buoyancy ﬂow with upper free surface, i.e. the solution Eq. (2.38), the second derivative reads Ra d2 u = (48y − 6) (2.44) dy 2 48 which gives the inﬂection point at a slightly different position (y = 1/8). The inﬂection point, however, is no longer present for the case of pure Marangoni ﬂow, since 3 d2 u = − Ma = 0 (2.45) dy 2 2 which means that Marangoni ﬂow solutions of the type given by Eq. (2.36) do not satisfy the Rayleigh necessary condition. The most interesting case in this regard is, perhaps, given by the mixed state represented by Eq. (2.39). In such a case Ra d2 u = (48y − 6 − 72W ) (2.46) dy 2 48 which makes the location of the inﬂection point a linear function of the nondimensional parameter W = Ma/Ra: 12W + 1 (2.47) y= 8 The inﬂection point disappears when y > 1/2, that is, for W > 1/4. Accordingly, these solutions satisfy the Rayleigh necessary condition only if 0 < W < 1/4 (see Figure 2.12).

2.4.5 The Inﬁnitely Long Liquid Bridge For pure Marangoni ﬂow, the equations of thermal convection admit analytic solutions also for the case of a liquid bridge assumed to be inﬁnitely extended along the axial direction z (Xu and Davis, 1983). For such a case, the solution is axisymmetric and reads Ma 2 1 r − (2.48) w= 2 2 Ma (2.49) (1 − r 2 )2 T = −z − 32 The interface (adiabatic) is assumed to be located at r = 1. The Marangoni number is deﬁned as Ma = RePr = σT γ R 2 /µα (where γ is the rate of constant temperature increase along the z axis and R is the radius of the liquid bridge). Velocity and temperature are referred to the scales α/R and γ R, respectively (moreover, all distances are scaled on R). According to Eq. (2.48), the velocity component along the axis of the liquid bridge is solely a function of z [w = w(z)]. Temperature is a function of both axial and radial coordinates (z and r). In such a context, it is worth pointing out that similar solutions also exist for other geometries with rotational symmetry, for example the case shown in Figure 2.4c, that is, the so-called annular conﬁguration (pool subjected to a radial temperature gradient); the reader is referred, for instance, to the study of Li et al. (2008c) for relevant analytical expressions, which these authors obtained for the case with heating from the outer cylindrical wall, cooling at the inner wall, adiabatic bottom wall and top free surface in the limit as the ratio of the pool height to the gap width goes to zero (asymptotic core-ﬂow solution).

86

Thermal Convection: Patterns, Evolution and Stability

Figure 2.12 Velocity proﬁle g1 (y)+ Wg2 (y) for the case of mixed thermogravitational–thermocapillary convection in an inﬁnite layer with bottom solid wall and upper free surface: the inﬂection point disappears for W = 0.25

Figure 2.13 Sketch of a layer of inﬁnite extent inclined with respect to the horizontal direction (temperature gradient along the x direction; θ - angle between ∇T and g )

2.4.6 Inclined Systems The solutions given in Section 2.4.1 for the case of a horizontal layer of ﬂuid with no-slip walls subjected to a uniform temperature gradient along x can be extended to the more general case in which the layer is inclined with respect to the horizontal direction (see Figure 2.13). In particular, the analytical solution for such a case has distinct expressions depending on the projection of the temperature gradient to the gravity vector, that is, according to whether from a global point of view the layer tends to behave as a system heated from below or from above (depending on the sign of 90◦ – θ in Figure 2.13) and depending on the type of thermal conditions along the walls. In the following, these expressions are given ﬁrst for the case of adiabatic walls [Eqs (2.50)–(2.53)], then the conﬁguration with conducting boundaries is considered [Eqs (2.56)–(2.59)].

Classical Models, Characteristic Numbers and Scaling Arguments

Heating from below (0◦ < θ < 90◦ ): Ra sin(θ ) sinh(2ξy) sin(ξ ) − sinh(ξ ) sin(2ξy) u= 2 2 16 ξ [sinh(ξ ) cos(ξ ) + cosh(ξ ) sin(ξ )] T =x+

1 sinh(2ξy) sin(ξ ) + sinh(ξ ) sin(2ξy) tan(θ ) 2y − 2 sinh(ξ ) cos(ξ ) + cosh(ξ ) sin(ξ )

87

(2.50)

(2.51)

Heating from above (90◦ < θ < 180◦ ): Ra sin(θ ) cosh(2ξy) sin(2ξy) sinh(ξ ) cos(ξ ) − sinh(2ξy) cos(2ξy) cosh(ξ ) sin(ξ ) u(y) = 16 ξ 3 [sinh(ξ ) cosh(ξ ) + sin(ξ ) cos(ξ )] 

 sinh(2ξy) cos(2ξy) sinh(ξ ) cos(ξ ) 1 + cosh(2ξy) sin(2ξy) cosh(ξ ) sin(ξ )   T = x + tan(θ ) 2y −  2 ξ [sinh(ξ ) cosh(ξ ) + sin(ξ ) cos(ξ )]

(2.52)

(2.53) where 1 1 (2.54a) [Ra cos(θ )] 4 for θ < 90◦ 2 1 1 (2.54b) ξ = √ [−Ra cos(θ )] 4 for θ > 90◦ 2 2 Interestingly, some fundamental insights into this kind of solution can be obtained (as noticed by Delgado-Buscalioni and Crespo del Arco, 2001) considering the vorticity-production term of the vorticity balance equation [Eq. (1.143)]. For the present case, such a term reads ∂T ∂T sin(θ ) + cos(θ ) (2.55) Ra ∂x ∂y

ξ=

The evolution of these ﬂow proﬁles with Ra is ruled by the balance of dissipation and production of vorticity by buoyant forces. According to Eq. (2.55), the production of vorticity due to the y and x components of buoyancy are proportional to Ra[sin(θ )∂T /∂x] and Ra[cos(θ )∂T /∂y], respectively. The temperature y gradient is created by the ﬂow advection and at low enough values of Ra it is negligibly small; therefore, at small Ra and for any (not vertical) inclination, the ﬂow is generated solely by the y component of buoyancy [at low values of Ra, in the conducting regime the cross-stream temperature gradient is vanishingly small and the vorticity is generated by the cross-stream component of gravity, at a rate given by sin(θ )∂T /∂x; this induces a cellular ﬂow whose y-dependence coincides for θ = 90◦ with the proﬁle given by Eq. (2.34)]. As Ra increases, the streamwise advection creates an increasing (positive) temperature gradient along the y-axis, which acts as another source of motion owing to the presence of the streamwise component of buoyancy; as explained earlier, this term produces vorticity at a rate given by cos(θ )∂T /∂y and hence its effect depends greatly on the range of the inclination angle. When heating from above (θ > 90◦ ) the effect of the axial buoyancy is to suppress the convection in the central part of the layer, provided that cos(θ )∂T /∂y > 0 while sin(θ )∂T /∂x < 0. For large enough Ra and θ > 90◦ , the ﬂow is conﬁned to small regions near the walls where ∂T /∂y ∼ = 0. In contrast, if the cavity is heated from below (θ < 90◦ ), both sources of vorticity have the same sign and as Ra increases a positive feedback loop between u(y) and T (y) occurs: any increment of the ﬂow intensity increases the cross-stream temperature gradient, which in turn enhances the intensity of the ﬂow. For conducting boundaries, the expressions for heating from below and from above read as follows.

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Heating from below (0◦ < θ < 90◦ ): Ra sin(θ ) sinh(2ξy) sin(ξ ) − sinh(ξ ) sin(2ξy) u= 16 ξ 2 [sinh(ξ ) sin(ξ )] 1 1 sinh(2ξy) sin(ξ ) + sinh(ξ ) sin(2ξy) T = x + tan(θ ) 2y − 2 2 sinh(ξ ) sin(ξ ) Heating from above (90◦ < θ < 180◦ ):   cosh(2ξy) sin(2ξy) sinh(ξ ) cos(ξ ) Ra sin(θ )  − sinh(2ξy) cos(2ξy) cosh(ξ ) sin(ξ )  u(y) =   2 16 ξ [sinh2 (ξ ) cos2 (ξ ) + cosh2 (ξ ) sin2 (ξ )]   sinh(2ξy) cos(2ξy) sinh(ξ ) cos(ξ ) 1 + cosh(2ξy) sin(2ξy) cosh(ξ ) sin(ξ )   T = x + tan(θ ) 2y −  2 sinh2 (ξ ) cos2 (ξ ) + cosh2 (ξ ) sin2 (ξ )

(2.56) (2.57)

(2.58)

(2.59)

with ξ given by Eq. (2.54). Equations (2.50)–(2.53) and (2.56)–(2.59) provide analytical solutions if the imposed T is parallel to the walls as shown in Figure 2.13; thermogravitational convection in an inclined layer, however, also admits exact solutions if the imposed T is primarily perpendicular to the walls, that is, if such a temperature gradient acts across the thickness of the layer (i.e. a layer with upper and lower walls kept at uniform different temperatures as shown in Figure 2.14). Such a solution reads " y# Ra sin(θ ) y 3 − (2.60) u= 6 4 with Ra = gβT T d 3 /να The corresponding distribution of temperature along y is governed by diffusion only and is approximately uniform throughout the plane of the layer. Velocity (u) tends to zero (as expected) in the limit as θ → 0 (in such a case, convection arises only if a given threshold of the Rayleigh number is exceeded, as already outlined in Section 2.1; see also Chapter 4 for further details). It is also worth noting that for θ = 90◦ one recovers the idealized case of a transversely heated vertical cavity (in the limit as the vertical scale of motion tends to inﬁnity; see Figure 2.15): Ra " 3 x # x − (2.61) v= 6 4 This proﬁle is generally referred to as ‘conduction regime’ solution (owing to the associated temperature proﬁle along x, which as mentioned earlier is linear).

Figure 2.14 Sketch of a layer of inﬁnite extent inclined with respect to the horizontal direction with heating applied through the bottom wall (temperature gradient imposed along the y direction)

Classical Models, Characteristic Numbers and Scaling Arguments

89

Figure 2.15 Sketch of a transversely heated vertical cavity in the limit as the vertical scale of motion tends to inﬁnity

2.5

Conductive, Transition and Boundary-layer Regimes

The solutions described in Section 2.4 assume that the system is inﬁnitely extended along a speciﬁc direction. When this is not the case, generally, things are more complex and general exact solutions are no longer available. There exist, however, some universal and fundamental behaviours of thermal convection in geometric models of ﬁnite size that are common to the distinct kinds of convective ﬂows deﬁned in the earlier sections. In general, when the characteristic number of the considered type of ﬂow increases, in fact, diffusion (which dominates at low values of this parameter) gradually diminishes and convection begins to control heat transfer. In the literature, this trend is usually described by classifying the ﬂow into three possible regimes ordered in increasing values of the control parameter (Gr or Ra for buoyancy ﬂow, Re or Ma for thermocapillary convection): conduction, intermediate and boundary-layer regimes. The last is of particular interest due to the existence of some regions (the boundary layers) where extremely steep gradients of velocity and/or temperature are present. A ﬁrst step towards the characterization of these phenomena and the deﬁnition of the frontiers separating the related ﬂow regimes is represented by Order of Magnitude Analysis (OMA), whose general formulation was derived by Napolitano (see, e.g., Napolitano, 1982; Russo and Napolitano, 1984) and applied to Marangoni, buoyancy and combined free convection (here, due to page limits, the discussion is limited to providing the reader with the fundamental results of this approach without entering into the process leading to them). It allows, in particular, the individuation of the ranges of the nondimensional characteristic numbers and of the conditions for which boundary layers develop. The ability to predict the emergence and properties of these regimes is a critical factor for the study of problems concerning Marangoni and buoyancy ﬂows (pure or mixed) and of their steady states, stability properties and possible bifurcations. In particular, the knowledge of the nature (momentum, thermal) and of the scale factors governing the evolution of the aforementioned layers is propaedeutical to the appropriate choice of the regions and directions where mesh stretching for numerical solution is necessary (this plays a crucial role for the correct application of both linear stability tools and nonlinear numerical solution methods described in Chapter 1). The results of the Napolitano’s OMA for convection in geometries heated from the side are summarized in Tables 2.1 and 2.2. Therein, is a length scale factor at most of order one: if

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Table 2.1 Governing equations and Order of Magnitude Analysis Pr

χ

Governing equation

≤ O (1) > O (1)

1 Pr

Momentum Internal energy

Table 2.2 Length scale factor and velocity scaling Test on χGr ≤ O (1) χGr > O (1) χRe ≤ O (1) χRe > O (1)

1 (χGr)−1/4 1 (χRe)−1/3

Velocity scaling

Vg Vg (χGr)−1/2 VMa VMa (χRe)−1/3

= 1, no boundary layers are present for the considered phenomenon; if = 1, boundary layers exist and their thickness scales according to . Consider, for instance, the case of pure thermal buoyancy ﬂow and a very small value of the Prandtl number (Pr 1, e.g. a liquid metal or a semiconductor melt). For this case, Table 2.1 shows that the existence of boundary layers is possible with respect to momentum and gives χ = 1. Correspondingly, in Table 2.2, two cases are possible, Gr ≤ O(1) with = 1 and Gr > O(1) with

= Gr−1/4 . In the second case, the existence of a boundary layer is predicted with a thickness changing according to Gr−1/4 (as Gr increases it becomes thinner). For the opposite case, Pr > O(1), boundary layers appear primarily with respect to energy (χ = Pr) and their thickness would scale according to Ra−1/4 . There have been a variety of theoretical, numerical and experimental studies focused on these subjects. One of the earliest workers to investigate ﬂow conﬁned within a cavity heated from the side was Batchelor (1954a), who analytically showed that, for Ra < 103 and large aspect ratios (length/height), the heat transfer mode is dominated by diffusion. Gill (1966), however, illustrated that, for a ﬁxed large aspect ratio and high Rayleigh numbers, the ﬂow is driven by the boundary layer regions of the ﬂow, where convection dominates the heat transfer mode while the interior region tends to remain stagnant and vertically stratiﬁed. As shown in Figure 2.16, for Pr > O(1), hot/cold thermal boundary layers develop at the left/right walls as the Rayleigh number is increased. These hot/cold thermal boundary layers also turn around the upper-left/lower-right corners and develop into intrusion layers that extend across the top and bottom walls (the vertical boundary layers tend to discharge into the core and these horizontal counterparts are also referred to as ‘boundary layer exit jets’). Accordingly the vorticity is mostly distributed on the sidewalls, whereas the core remains stagnant , which allows for the development of a vertical stratiﬁcation. These behaviours are usually described in the literature in terms of the nondimensional parameter K; this parameter is deﬁned as the ratio of the horizontal temperature gradient at the core region and the overall imposed temperature gradient and is used to distinguish between three unicellular ﬂow regimes ordered in increasing Ra: the diffusive regime (K ∼ = 1), the transition regime (0 < K < 1) and the boundary layer (or convective) regime (K ∼ = 0). The pioneering theoretical study of Gill (1966) mentioned above gave the basis for understanding this problem. More recently, Boehrer (1997) obtained the frontiers of these ﬂow regimes in long horizontal cavities and Pr ≥ O(1) as RaA−2 ∼ = 102 and RaA−2 ∼ = 104 , respectively (A = length/height).

Classical Models, Characteristic Numbers and Scaling Arguments

91

15

11 4 7 9

2

6

0.25

0

1

13

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0.75

1

10

14 11 9

7 6

4

8

2 0

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1 Level 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

14

4 11

10 9

7

6 5 3 0

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0.75

T 0.94 0.88 0.81 0.75 0.69 0.63 0.56 0.50 0.44 0.38 0.31 0.25 0.19 0.13 0.06

1

Figure 2.16 Temperature ﬁeld for buoyancy convection in a closed cavity (Pr = 15, A = 2): (a) Ra = 103 ; (b) Ra = 104 ; (c) Ra = 105 (Ra based on the depth; cold wall on the left side, hot wall on the right side, upper and lower boundaries with adiabatic conditions; when the Rayleigh number is increased well-deﬁned boundary layers appear close to the lateral solid walls together with a core region showing a stratiﬁed temperature distribution; numerical simulations) (M. Lappa)

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When the boundary layer regime is attained, mesh stretching is necessary near the vertical walls according to the information provided in Table 2.2 (the related thickness scales as Ra−1/4 ). Some care may be also necessary to obtain a sufﬁcient numerical resolution in the aforementioned horizontal intrusion layers along the top and bottom walls. For spatial resolution requirements concerning pure buoyancy and enclosures heated from below (Rayleigh–B´enard problem), things become more complex. A good reference is a paper by Gr¨otzbach (1983), who obtained different criteria for the prediction of grids, which allows for accurate direct numerical simulations of high-Ra ﬂows, based on wavelength considerations and boundary layer thickness estimates. Additional relevant information has been provided by recent experimental works. As an example, for the case of water (Pr = 7; Xin and Xia, 1997), the thermal and velocity boundary layers were found to scale with Ra−1/4 and Ra−4/25 , respectively [Ra = O(107 –1011 )]. For the latter, dependence upon the Prandtl number was investigated in a later study by Lam et al. (2002) for 6 ≤ Pr ≤ 1027. They found the viscous layer thickness to scale as Pr6/25 Ra−4/25 . In such cases, the problem related to the choice of the mesh is basically made more complex by the possible presence of plumes detaching from the top and bottom thermal boundary layers (these phenomena will be treated in Chapter 5). These structures have horizontal scales comparable to the thicknesses of the boundary layers in which they arise (see, e.g., Parodi et al., 2004). Moreover, as illustrated by Hier Majumder et al. (2004), additional thermal and/or velocity boundary layers develop (vertically) along the plume body and different regimes of plume growth (categorized in terms of relative thickness of these thermal and velocity boundary layers) are possible in the Prandtl–Rayleigh space (see Section 5.2). Since the positions where the plumes arise and detach from the thermal boundary layers cannot be predicted a priori and change as a function of time, in these cases grids uniform and dense throughout the physical domain are required to capture the dynamics of the system correctly. Let us now switch to the case of surface tension-driven ﬂows in transversely heated cavities. For the case of pure thermal Marangoni ﬂow and a very small value of the Prandtl number (Pr 1, e.g., a liquid metal or a semiconductor melt), Table 2.1 shows that boundary layers occur primarily with respect to momentum and gives χ = 1. Correspondingly, in Table 2.2, two cases are possible, Re ≤ O(1) with = 1 and Re > O(1) with = Re−1/3 . In the second case (which is the usual situation arising in material processing), the existence of a boundary layer is predicted with a thickness changing according to Re−1/3 (as Re increases it becomes thinner; Rivas, 1991). This means that for the case of liquid metals and semiconductor melts, nonuniform grids stretched close to the free surface should be used. This zone plays a critical role in the computations since therein a Marangoni boundary layer is present with extremely steep radial gradients of velocity induced by the driving force acting on the free surface (in such a context it is worth stressing that, as noted by Carpenter and Homsy, 1990, a complete structural analogy could be established between ﬂow in a cavity driven by a moving lid and thermocapillary ﬂow in the boundary layer limit). Direct numerical experimentation shows that this is particularly true for very large values of Re [O(Ma) > 102 ]. The results provided by the OMA in this case become even more important if one considers that, on the basis of Table 2.1, no thermal boundary layers should be present (or, if present, they should be weak): ‘Momentum’ in the last column means, in fact, that the prominent features of the ﬁeld should be related to this quantity and that other (thermal) effects should be negligible in terms of steepness of the related gradients (hereafter the word ‘prominent’ is used to denote only the features of the thermoﬂuid-dynamic ﬁeld that exhibit steep gradients). From a physical point of view, the absence of thermal boundary layers for Pr < O(1) follows from the large thermal diffusivity of liquid metals that spreads thermal gradients over wide regions. These are the reasons why, for relatively small values of the Marangoni number [Ma = O(10)] grid-reﬁnement studies simply show that for the case of liquid metals (Pr 1), the numerical

Classical Models, Characteristic Numbers and Scaling Arguments

9

10

6

13

7

14

8

5

4

3

2

11

93

12 15 1

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0.7

0.8

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9 8

10 15

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(b)

10 10

9 8 7

9

2 5

8

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0.6

0.7

0.8

9

15

0.9

1

Level 15 14 13 12 11 10 9 8 7 6 5 4 3 2 1

T 0.94 0.88 0.81 0.75 0.69 0.63 0.56 0.50 0.44 0.37 0.31 0.25 0.19 0.12 0.06

Figure 2.17 Temperature ﬁeld for Marangoni convection in an open cavity (Pr = 15, A = 2): (a) Ma = 103 ; (b) Ma = 104 ; (c) Ma = 105 (Ma based on the extension of the free interface; cold wall on the left side, hot wall on the right side, upper and lower boundaries with adiabatic conditions; when the Marangoni number is increased well-deﬁned boundary layers appear close to the lateral solid walls together with a core region displaying an almost uniform temperature distribution; numerical simulations) (M. Lappa)

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simulation of the problem is not adversely affected by the use of uniform grids. For this case, in fact, in the light of the aforementioned arguments, both momentum and thermal boundary layers are absent or weak and grid convergence is achieved fairly easily. As expected, when Ma becomes large, stretching functions clustering the computational points close to the free surface should be chosen. Since the isotherms tend to be compressed by the ﬂow towards the cold wall, some points should also be accumulated close to this boundary. Strong thermal gradients, however, are expected for the case of high Prandtl number (transparent) liquids; extensive numerical experimentation shows that for Pr 1, in fact, very thin thermal boundary layers appear close to the heated (or cooled) sides. This situation is also predicted in principle by the OMA results in Tables 2.1 and 2.2. In fact, for Pr > O(1) the prominent features of the ﬁeld are related to its thermal structure (‘Energy’ appears in the last column of Table 2.1); in this case χ = Pr and the thickness of the thermal boundary layers scales according to = (PrRe)−1/3 = Ma−1/3 . Such behaviour is qualitatively conﬁrmed by the numerical results shown in Figure 2.17. The temperature ﬁelds clearly exhibit bands close to the heated (or cooled) walls where the isotherms tend to be compressed (see also Cowley and Davis, 1983; Zebib et al., 1985a; Canright, 1994). The width of these layers is reduced as Ma increases, leading to an increasing demand for points in these regions. It is also worth stressing that unlike buoyancy convection, for which, when Ra is sufﬁciently increased, vorticity tends to be mostly distributed on the sidewalls whereas the core remains stagnant (leading to vertical thermal stratiﬁcation), for Marangoni ﬂow the region between the lateral boundary layers is characterized by a strong vortical structure that leads to an almost uniform temperature over it (Figure 2.17c). When both buoyancy and thermocapillarity are present for Pr > O(1), since the thermal gravitational and Marangoni boundary layers scale as Ra−1/4 and Ma−1/3 , respectively, in general, thermocapillarity tends to ultimately dominate such ﬂows for Ra/Ma ≤ O(10). The case of combined buoyancy–thermocapillary convection in a cavity with a free surface heated differentially in the horizontal direction was initially treated by Carpenter and Homsy (1989) for Pr > 1. They found that for Ra/Ma = O(1), the ﬂow evolves toward its boundary-layer limit in a fashion identical to that for Ra = 0 (pure Marangoni ﬂow). For Ra/Ma = O(10), the evolution is from a buoyancy-dominated structure, through a transition, to a thermocapillary-dominated structure. Similar concepts for the case of liquid metals were derived by Camel et al. (1986).

3 Examples of Thermal Fluid Convection and Pattern Formation in Nature and Technology This chapter deals with some typical prototypes or exemplars of thermal convection. The focus is on how these phenomena can be responsible for pattern formation at a multitude of spatial length scales and in several distinct contexts. Some effort is provided to illustrate possible kinships, transdisciplinary commonalities, similarities, analogies and so on.

3.1

Technological Processes: Small-scale Laboratory and Industrial Setups

In small-scale laboratory and industrial setups, both buoyancy- and surface tension-driven convective instabilities can signiﬁcantly affect the heat/mass transfer characteristics and create interesting patterns. There is a plethora of examples that could be considered in such a context, for instance, heat exchangers, power plants, solar collectors, nuclear reactors, drying processes, thermal control of electronic components, heat pipes and lab-on-a-microchip applications. Many other relevant examples lie in the speciﬁc realm of materials science (see, e.g., the reviews by Amberg and Shiomi, 2005; Saeedi et al., 2006; Hong et al., 2006; Tang et al., 2007; Prud’homme and El Ganaoui, 2007; Hirata, 2007); to name just a few: crystal growth from the melt (Lappa, 2005b, and references therein), vapour crystal growth, casting processes, soldering and welding processes (in practice, all the technological methods in which the material is initially liquid and then undergoes solidiﬁcation or vice versa), the production of inorganic (e.g. metal), organic alloys, emulsions and polymers and phase separation techniques (in general, all the practical and engineering applications in which a ﬂuid undergoes a thermal quench or increase). For the sake of brevity, the discussion in the following will be limited to some typical cases concerning the production of high-quality single crystals from the melt (of initially polycrystalline materials; the reader is referred to Table 1.1 for typical values of the involved Prandtl number). Such a topic exhibits a wealth of details and variants, which make it a good paradigm for the illustration of some fundamental concepts. The various techniques traditionally used for growing such materials involve almost all of the basic geometric models (layer, rectangular or parallelepipedic cavities, annular or cylindrical Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Thermal Convection: Patterns, Evolution and Stability

conﬁguration) deﬁned in Chapter 2 as reference models for the study of both gravitational and surface tension-driven ﬂows (Lappa, 2005a, 2006a, and references therein). Moreover, there is an additional important reason for considering these cases as important exemplars: these techniques and related geometric models have been the subject of intense study in the microgravity environment. At the beginning, research in such an environment on these processes was mainly pursued to develop materials with unique properties, but later the objective was broadened to use microgravity to seek and understand quantitative cause-and-effect relationships between the materials processing, properties and speciﬁc convective phenomena or to gain fundamental information on such convection phenomena (that become evident or dominant when gravity is removed), regardless of any potential application.

3.1.1 Crystal Growth from the Melt: Typical Techniques The properties that an electronic material has (such as how strong it is or whether it conducts electricity) are determined by its structure. Hence establishing quantitative and predictive relationships between the way in which it is produced (processing), its structure (how atoms or larger inclusions are arranged) and its properties is of paramount importance. A category of materials of particular interest in such a context is represented by doped semiconductors and some special alloys. Most semiconductor devices are currently based on wafers cut from single crystals of either elemental or compound semiconductors. In the following, a brief description of the different growth techniques currently used to obtain high-quality single crystals from the melt (of initially polycrystalline materials) is provided. The prominent features of each method are discussed, placing emphasis on the types of thermal convection involved and on related potential effects. The existing terrestrial technology is brieﬂy reviewed. Attention is also given to the potential of microgravity and to a critical and focused comparison of classical terrestrial methods with the so-called containerless processing (the ﬂoating-zone technique) that, as anticipated in Chapter 2, has become over the years a paradigm for the study of Marangoni ﬂows, their structure and bifurcations.

3.1.1.1 Bridgman Method (Directional Solidiﬁcation) In Bridgman growth, directional solidiﬁcation occurs within a vertical ampoule (VB) or a horizontal open boat (HB). The ampoule or boat is inside a furnace with a temperature that varies from above the melting point of the considered material at one end to below the melting point at the other end. Freezing is induced by moving the ampoule through the furnace or vice versa or by slowly lowering the furnace temperature with both ampoule and furnace immobile. The latter method is often called the ‘gradient-freeze technique’ or ‘power-down method’. Obviously, in VB the ampoule, which is, for example, a quartz glass or a graphite container, is mandatory to support the melt, because the whole feed material (above the crystal) is molten (Figure 3.1). The HB can be basically sketched as a rectangular zone with a free upper surface exposed to an ambient inert gas (Figure 3.2). For semiconductors such as GaAs, HB is preferred to VB because of the observability of the growth process and the possibility of volume expansion. Since in the HB case the liquid is put in contact with a cold lateral surface on one side (the solidifying interface) while most of heat is provided through the other side, convection of the Hadley type develops typically in such a conﬁguration (see Sections 2.4.1 and 2.4.3; it is worth anticipating here, however, that this ﬂow is also a fundamental model of convection occurring at

Examples of Thermal Fluid Convection and Pattern Formation in Nature and Technology

97

Melt

Single Crystal

Ampoule

Figure 3.1 Sketch of the vertical Bridgman (VB) technique Free surface Single Crystal

Inert gas

Melt

Ampoule

Figure 3.2 Sketch of the horizontal Bridgman (HB) technique (often also referred to as the ‘open boat’)

the mesoscale in the Earth’s atmosphere as a consequence of the temperature difference between the warm equator and the cold poles, as will be shown in Section 3.4). In general, buoyancy convection tends to be stronger in HB than in VB (in the ﬁrst case, as explained above, ﬂow of the Hadley type occurs since a temperature gradient acting primarily along the horizontal direction is established, whereas in the latter case it is essentially Rayleigh–B´enard convection). It is also worth noting that in the HB conﬁguration, in addition to ﬂow of the Hadley type, Marangoni ﬂow can arise on the free melt surface as a consequence of the temperature gradient established along the horizontal direction (see, e.g., Roux et al., 1988, and references therein). Such a ﬂow becomes dominant in microgravity conditions where thermogravitational convection is absent or signiﬁcantly weakened.

3.1.1.2 Floating-zone (FZ) Method During the ﬂoating-zone (FZ) process, a melt zone is established between a lower seed material and upper feed material by applying localized heating (see Figure 3.3). This ﬂoating zone is moved along the rod (by means of relative motion of the heating device) in such a way that the crystal grows on the seed (which is below the melt) and simultaneously melting the feed material above the ﬂoating zone. Both the seed material and the feed rod are supported but (in contrast to the VB

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Thermal Convection: Patterns, Evolution and Stability

Feed rod Melt Single Crystal

Free surface

Ring Heater

Figure 3.3 Sketch of the ﬂoating-zone (FZ) method

technique) no container is in contact with the growing crystal or the melt, which is held in place only by surface tension. Thereby, it becomes evident that the key characteristic of this method is that the molten zone does not need to be in contact with a foreign solid (crucible), which, in addition to being awkward to realize in practice (the working temperature of the crucible must be well above the 1690 K melting temperature of silicon, for example), would introduce impurities unacceptable for the applications envisaged (e.g. molten silicon is a very reactive material). In general, the FZ can be regarded as a partially containerless system (the circumferential liquid surface is almost entirely in contact with the gas surrounding it except at the solidus walls). Such a containerless processing eliminates effectively wall effects such as contamination and nucleation at the lateral boundary and allows the formation of more pure and perfect crystals. In addition, the absence of the ampoule wall allows unconstrained material expansion during freezing, preventing sample breakage and other constraining effects (defects incurred by differential contraction). Of course, containerless processing on massive samples can only be done in the microgravity environment of space, where the forces used for suspending and manipulating the specimens are not overwhelmed by gravity. Microgravity requires much smaller forces to control the position of containerless samples, so the materials being studied are not disturbed as much as they would be if they were levitated on Earth. In practice, under Earth conditions the zone height is limited because the liquid will run down if the molten zone becomes too large; this effect limits the possible diameter of crystals that are grown in terrestrial conditions. In space, the maximum zone height is given by the circumference of the crystal (a reduction in gravity, while tending to minimize buoyancy-driven convection, also results in the reduction of the hydrostatic pressure; such a reduction in pressure prevents liquid in a ﬂoating zone conﬁguration from deforming under its own weight and allows longer, more stable zones to be formed); as a natural consequence, ﬂoating-zone experiments with greater zone heights and larger diameters are feasible under microgravity (see Figure 3.4). For both cases (experiments in space with large ﬂoating liquid columns or on the ground with microzones) the ﬂow in the FZ melt is governed essentially by Marangoni ﬂow (driven by the temperature gradient acting along the axial direction; see, e.g., Lappa, 2003a, 2004b, c, 2005d, and references therein), which makes the study of this type convection and the understanding of the related instabilities of crucial importance for this kind of problem. The Liquid Bridge. The liquid bridge introduced in Section 2.3.3 has to be regarded as a simpliﬁed model of the effective FZ process discussed above. One of the major difﬁculties in the experimental analysis of Marangoni ﬂow in real ﬂoating zones is that, due to the aforementioned phase change related to melting and solidiﬁcation of the material, the geometry of the boundary of the liquid volume is not known a priori . For these reasons, the liquid bridge was introduced in the mid-1970s as a vehicle for performing

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16mm

mg: FR15

mg: FR9

1g

Figure 3.4 GaSb FZ single crystals grown during the Spacehab-4 mission (left and centre). For comparison, a 1g crystal, of the maximum diameter that can be grown by the FZ method, is shown on the right. The crystals are single crystalline, exempliﬁed by the appearance of facets at the crystal periphery. After Croll ¨ et al. (1998); Copyright Elsevier, 1998

experiments in well-controlled conditions: In practice, it simulates half of a real ﬂoating zone (the liquid between one of the ends of the domain and the equatorial plane where local heating is applied). Both the real ﬂoating zone and the liquid bridge are held by surface tension forces, spanning between two sharp-edged coaxial solids against the natural tendency of liquids to adopt a spherical shape in the absence of other forces and the tendency to creep down the rod in a gravity ﬁeld. For the liquid bridge, the presence of a ring heater around the equatorial plane is simulated by heating one of the two supporting disks with respect to the other, whereas heat ﬂow is often neglected through the free surface. As already outlined, in fact, it is assumed to approximate half of the actual ﬂoating-zone process, with the hotter disk representing the plane of the hottest circumference and the colder disk representing either the crystal/melt or the feed rod/melt interface (axial heat ﬂux is modelled in the half-zone by simply making the bottom boundary hot, the top the melting temperature and the free surface insulating, producing a temperature gradient along the free surface). Although it is a very crude simpliﬁcation (the real FZ is not a static conﬁguration but a dynamic process governed by temperature gradients that force the tip of the feeding rod to melt and the tip of the grown material to freeze), the model of a quasi-steady series of liquid bridges has already shown to be relevant to some key aspects of the problem (Lappa, 2007d), in spite of the fact that real ﬂoating zone experiments are much more awkward to analyse in real time than liquid bridges.

3.1.1.3 Czochralski (CZ) Method Like the FZ process, the Czochralski (CZ) method is relatively simple in concept but rather difﬁcult to control. The method starts with a base material from which the desired crystalline structure can be grown (chunks of polysilicon for silicon single-crystal growth).

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Free surface

Feed rod Melt Single Crystal

Crucible

Figure 3.5 Sketch of the Czochralski (CZ) method

It is also often used for the production of oxide-single crystals such as yttrium aluminium garnet, gadolinium gallium garnet and lithium niobate, utilized as solid-state laser hosts and materials for acoust-opt-electronic devices. The melting of the base material takes place in a circular crucible, which can be heated arbitrarily at its bottom and sidewall. The crucible can rotate around its axis. When the base material has melted and is at the proper temperature, a seed is brought into contact with the free melt surface to begin the growth process. The melt will freeze at the seed. At this point, the crystal growth equipment begins to pull the seed up slowly from the melt as the material solidiﬁes forming the high-quality crystals (Figure 3.5). Solid crystals are subsequently cut to form thin semiconductor wafers from which integrated circuits are produced. Cooling from above (the seed is at a lower temperature with respect to the melt) can lead to the onset of convection of the Rayleigh–B´enard type in such a system (e.g. descending cold plumes), whereas heating from the side yields a ﬂow of the Hadley type (these mechanisms can also be simultaneously present due to the combined action of vertical and radial temperature gradients). Moreover, owing to the presence of a relatively large free surface, Marangoni ﬂow can play a signiﬁcant role in such conﬁgurations even under normal gravity conditions. This makes the CZ conﬁguration a good example of situations in which ﬂuid motion is brought about by different coexisting mechanisms: Marangoni convection, generated by the interfacial stresses due to horizontal temperature gradients along the free surface (see, e.g., Kumar, 2003) and gravitational convection driven by the volumetric buoyancy forces caused by thermally and/or solutally generated density variations in the bulk of the ﬂuid (see, e.g., Jones, 1984). The delicate interplay among these effects makes the problem very complex (Kakimoto, 1995). Superimposed on this, is the fact that, given a melting temperature of the order of 103 K, the surface will be cooled very effectively by radiation. This can induce Marangoni–B´enard cells which tend to increase the heat transfer from the bulk to the free surface and through the free surface; hence the heat ﬂux through the free surface is enhanced, which in turn, generally, increases the thermocapillary effect. The CZ technique has attracted much attention over recent years by virtue of its connection to the so-called spoke-pattern problem (well known to the community of crystal growers): when semiconductor materials [Pr < O(1)] or oxide melts [Pr ≥ O(1)] are processed with this method, more or less curved radial spokes (Figure 3.6) can appear on the melt free surface (Takagi et al., 1976; Jones, 1983; Yamagishi and Fusegawa, 1990; Azami et al., 2001a). In some circumstances, these patterns may be ascribed to surface tension-driven ﬂows, in particular to surface tension-driven convection of the Marangoni–B´enard type or of the type induced by lateral heating, depending on whether temperature gradients perpendicular (e.g. Yi et al., 1994; Jing et al., 1999; Tsukada et al., 2005) or parallel (Azami et al., 2001a; Li et al., 2004a,b) to the

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Figure 3.6 Sketch of patterns on oxide melt surface in a CZ furnace

free surface are dominant (these phenomena are treated in Chapters 9 and 10, respectively) and according to the considered category of materials [Pr ≥ O(1) or Pr < O(1)]. To ‘counteract’ ﬂow arising in the melt owing to buoyancy and surface tension forces, the crucible where the crystal is being grown is generally rotated while the mechanism pulling it out of the melt is rotated in the opposite direction (in practice, a relative rotatory motion is established between the crystal and the crucible with the intention of controlling the melt ﬂow pattern and heat/mass transport in the melt phase and of producing a cylindrical crystal). Notably, in such rotating systems, the Coriolis force combined with the buoyancy force can cause additional ﬂuid-dynamic phenomena (e.g. Enger et al., 2000; Jing et al., 2004; Tsukada et al., 2005) known as baroclinic instabilities (which also represents a relevant aspect of the problem). In the absence of Marangoni forces (surface tension effects neglected), the melt below the crystal would tend to be sucked upwards and expelled out towards the crucible by the centrifugal force induced by the rotating crystal; on the other hand, the melt near the crucible would tend to rise along the crucible sidewall and ﬂow towards the crystal along the free surface owing to the buoyancy effect. For oxide melts, it is known that when the buoyancy-driven ﬂow and crystal rotation-driven ﬂow are of a comparable magnitude, a wavy pattern appears on the melt surface (induced by the aforementioned baroclinic mechanism). A detailed and exhaustive picture of the mechanisms leading to similar wavy patterns on the free surface of semiconductor melts and of the role played in the related context by Marangoni effects has not yet emerged. Anyhow, all these convective-transport mechanisms affect the melt ﬂow, the crystal/melt interface and the crystal itself.

3.1.2 Detrimental Effects Induced by Convective Phenomena One aspect of the solidiﬁcation of semiconductors that inﬂuences their microstructures is the shape of the boundary or interface that exists between the melt and the solidifying material. It is known that during the solidiﬁcation process, the shape of the solidifying interface can go through a series of transitions and morphological instabilities (Delves, 1971; Coriell and Sekerka, 1981; Glicksman et al., 1986; Schulze and Davis, 1994; Davis and Schulze, 1996; Murray et al., 2000, and references therein). These instabilities are inﬂuenced by many factors. Gravity plays an important role in a number of them (there is a plethora of publications on the subject; see, in particular, Glicksman et al., 1986; Muller, 1988). A similar effect can be induced by Marangoni convection (see, in particular, Benz, 1990; Lan and Kou, 1990; Chen et al., 1994; Lan and Chian, 2001; Sumiji et al., 2001; Lappa and Savino, 2002; Matsunaga and Kawamura, 2006). Moreover, these convective phenomena can lead to differences between the locally averaged crystal’s dopant concentrations in different parts of the crystal, called macrosegregation (axial and

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Figure 3.7 Striations in crystallized samples of doped silicon. After Dold et al. (2001); Copyright Elsevier, 2001

radial segregation acting on the scale of the crystal dimensions) and to small-scale spatial concentration oscillations (micrometres to millimetres), called microsegregation or striations, respectively. When convection in the melt is oscillatory, the associated oscillations in the heat ﬂux from the melt to the crystal produce ﬂuctuations in the local growth rate, with alternating periods of growth and remelting in severe situations. The elements or dopants which give the crystal the desired properties are generally rejected during crystallization, so that there is a large dopant concentration in the melt adjacent to the growth interface. When the local growth rate decreases, dopant can diffuse away from the growth interface, leading to a lower local concentration in the crystal, and when the local growth rate increases, the crystal overtakes the rejected dopant, leading to a higher local concentration in the crystal (the spatial oscillations of the dopant concentration in the crystal are the aforementioned striations; see, e.g., Figure 3.7). These effects are highly undesired and affect the performances of these materials in a very detrimental way. Microscopic inhomogeneity with respect to the distribution of dopants is very common in mixed crystals. In normal gravity it is gravity-driven unsteady convection and as a consequence unsteady heat transport (Lappa, 2007a,b) that cause temperature ﬂuctuations on the solid/liquid interface and create striations via the growth rate dependence of the segregation. These imperfections along with the other gravity-caused ones, such as dislocations, strain ﬁelds, voids and grain boundaries, can be eliminated by a decrease in gravity. Buoyancy, however, is not the only mechanism that can be responsible for oscillatory instabilities. Marangoni convection also may be a reason for the onset of oscillations. Strain ﬁelds, dislocations, voids and striations can also occur in the absence of gravity. Each inhomogeneity has its own mechanism but it is always the Marangoni convection that remains behind all them. In general, it does not matter if the thermocapillary convective ﬂow is steady or time dependent – it is always undesirable. For example, microscopic striations are caused by time-dependent Marangoni convection whereas steady convection causes an increase in macrosegregation. It is necessary, therefore, to predict the appearance of all these instabilities of both gravitational and non-gravitational origin, to understand the related physical mechanisms and to ﬁnd ﬂow control means capable of stabilizing the ﬂow. In many cases, stabilizing the primary instability (the ﬁrst bifurcation of the ﬂow) would mean stabilizing the process as a whole, which is extremely desirable for the various crystal growth technologies illustrated earlier.

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Beyond examples in the industrial realm, thermal convection can explain a variety of phenomena occurring in Nature (on Earth and/or other objects in our Solar System). It occurs within the major planetary subsystems, that is

• the geosphere • the atmosphere • the hydrosphere on a large range of time and length scales. The range of possible values of the Prandtl number involved in these phenomena is also very wide (the atmosphere, Pr ∼ = 1; lakes and oceans, Pr ∼ = 7; the Earth’s outer core, whose Prandtl number can range from 0.01 in completely ﬂuid regions to 1013 in the solid–liquid mixture of the inner–outer core boundary). Thermal convection is also an important process in magma chambers, where the Prandtl numbers of magmas range from 104 to 108 . The rocky mantles of terrestrial planets like Earth convect with Pr ∼ = 1023 . Many of the features that we see on the surface of Jupiter (Pr 1) and the Sun (Pr ∼ = 0) are the result of thermal convection. The seas of Saturn’s moon Titan are composed of a hydrocarbon liquid (Pr ∼ = 2). Europa and Ganymede both have global oceans of water with icy outer lids (Spohn and Schubert, 2003). It is believed convective phenomena also occur in these oceans.

3.3

Planetary Structure and Dynamics: Convective Phenomena

Fluid motion in the mantle and in the Earth’s liquid core are typical examples of Rayleigh–B´enard convection. The outcome of ﬂuid motion in the mantle is perhaps the most evident example of how convective phenomena at the mesoscale and pattern formation on the Earth’s surface can be intimately interrelated. Energy sustaining this phenomenon (i.e. heat) is provided by the decay of radioactive elements and heat left over (cooling down) from planetary formation. Further elaboration along these lines requires providing the reader with some synthetic but necessary information about the internal structure of our planet.

3.3.1 Earth’s ‘Layered’ Structure The Earth consists of three concentric layers: the core, the mantle and the crust . This orderly division results from density differences between the layers as a function of variations in composition, temperature and pressure (Figure 3.8). The core has a calculated density of 10–13 g cm−3 and occupies about 16% of the Earth’s total volume. Seismic (earthquake) data indicate that it consists of a small, solid inner region and a larger liquid outer portion. Both are thought to include large amounts of iron and a smaller amount of nickel. The mantle surrounds the core and comprises about 83% of the Earth’s volume. It lies between the crust and the molten core (extension 3 × 103 km) and consists of rocks at temperatures so high that they behave as a highly viscous liquid. This region, however, is less dense than the core (3.3–5.7 g cm−3 ) and is thought to be composed largely of peridotite, a dark, dense, igneous rock containing abundant iron and magnesium.

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Figure 3.8 Sketch of the Earth’s ‘layered’ structure

3.3.2 Earth’s Mantle Convection The Earth’s mantle, the 3000 km thick shell that extends from the liquid outer core to the Earth’s surface, has been deforming slowly over geological times, resulting in large interior mass displacements (and a gradual resurfacing of the Earth as will be illustrated in Section 3.3.3). As outlined earlier, the mantle is solid, but over very long periods of time most of it behaves like a very viscous liquid and ﬂows (it is kept soft by the aforementioned heat generated by radioactive decay). As shown in Figure 3.8, the temperature difference between the top and bottom of the mantle causes recirculating ﬂows. The uppermost mantle and crust form an outer shell of rigid plates. The plates and the continents that they contain move across the Earth’s surface on the convection currents. Indeed, the Rayleigh number of the mantle, which quantiﬁes this convective instability is estimated to range between 107 and 109 (104 − 106 times the value at which convection begins), yielding ﬂow velocities of 1–10 cm yr−1 (Lithgow-Bertelloni et al., 2001, and references therein). Plate tectonics are the prime manifestation of these slow deformational processes, but ultimately all large-scale geological activity and dynamics of our planet, such as earthquakes, mountain building and the opening and closure of major ocean basins, is the result of sub solidus convection within the mantle. The increasing evidence for solid-state convection in the Earth and in other terrestrial planets, at least throughout parts of their thermal history, has stimulated a number of nonlinear analyses of the problem of convection in spherical shells, as will be extensively illustrated in Chapter 4.

3.3.3 Plate Tectonics Theory The aesthenosphere (the region of the Earth between 100 and 200 km below the surface, but perhaps extending as deep as 400 km), is part of the upper mantle that is the weak or ‘soft’ zone in the mantle. It has the same composition as the lower mantle but behaves more plastically; moreover, partial melting within the aesthenosphere generates magma, some of which rises to the surface (because it less dense than the rock from which it was derived).

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The upper part of the aesthenosphere is believed to be the zone upon which the rigid plates of the Earth’s crust move. Due to the temperature and pressure conditions in the aesthenosphere, rock becomes ductile, moving at rates of deformation measured in centimetres per year over lineal distances eventually measuring thousands of kilometres. In this way, it ﬂows like a convection current. Above the aesthenosphere, at the same rate of deformation, rock behaves elastically and, being brittle, can break, causing faults. The solid portion of the upper mantle and the overlying crust underlain by the aesthenosphere constitute the more rigid lithosphere (0–70 km), which, as explained earlier, is broken into numerous individual pieces. The rigid lithosphere is thought to ‘ﬂoat’ or move about on the slowly ﬂowing aesthenosphere, creating the movement of crustal plates described by plate tectonics theory (Dietz, 1961; Hess, 1962; Morgan, 1971, 1972). According to these arguments, the boundary between the lithosphere and the underlying aesthenosphere can be simply deﬁned by a difference in response to stress: the lithosphere remains rigid for long periods of geological time, whereas the aesthenosphere ﬂows much more readily (in practice, the lithosphere responds essentially as a rigid shell and thus deforms primarily through brittle failure, whereas the aesthenosphere is heat softened and accommodates strain through plastic deformation). Convective effects, however, are not limited to the mantle: they also occur in the lithosphere where plumes of less dense magma transfer heat to the surface, breaking apart the plates at the spreading centres and creating divergent plate boundaries. Movement in the mantle cracking the overlying crust allows the magma to escape and build volcanoes on the sea ﬂoor. The crust may be regarded as just a thin layer of buoyant rock embedded in lithospheric plates; it consists of two types. Continental crust is thick (20–90 km), has an average density of 2.7 g cm−3 and contains considerable silicon and aluminium. Oceanic crust is thin (5–10 km), denser than continental crust (3.0 g cm−3 ) and is composed of basaltic rock. Each time the crust cracks, the rocks on either side of the ridge are moved a small distance sideways to make room for the new volcanic rock. Repeated cracking gradually moves old volcanoes away from the hot, active ridge area (Figures 3.9 and 3.10).

Figure 3.9 Theory of ‘plate tectonics and continental drift’: the Earth’s plates move as a result of underlying mantle convection cells in which warm material from deep within Earth rises toward the surface, cools and then, upon losing heat, descends back into the interior

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Figure 3.10 Surface expression of mantle convection in the Earth. It consists of linear arcuate subduction zone delineating sheet-like downﬂows or descending slabs, hot spots indicative of quasi-cylindrical upﬂow plumes and linear ridges representing sheet-like upﬂow

When plate edges override one another, one of the plates is forced down into the hot mantle and melts. This process is called subduction. In general, when a continental plate collides with an oceanic plate, the dense oceanic plate is usually forced underneath the lighter continental plate. Continents therefore endure by ﬂoating on the surface of the mantle, but ocean crust is consumed by subduction. Ocean crust is easily subducted because, as outlined in the preceding section, it is made of heavy basalt, gabbro and ultramaﬁc rock, which ‘sink’ into the mantle, whereas continent crust, being made of lighter rocks such as granite, ‘ﬂoats’ on the mantle, instead of being subducted. When a plate descends into the mantle at a subduction zone, part of it melts in the hot interior of the Earth; the melted rock then erupts on the surface as a line of volcanoes (since molten crust material is lighter than mantle, it tends to rise, melting its way through the overlying solid rock and erupting as volcanic lava). It is also worth mentioning that volcanoes can also be induced by a second mode of mantle convection present in the form of hot upwelling plumes that are thought to be the underlying cause of typical ‘hotspots’ on the Earth’s surface (Morgan, 1972). These plumes probably result from a thermal boundary layer at the bottom of the mantle that forms in response to the ﬂow of heat out of the core.

3.3.4 Earth’s Core Convection The detailed description of the generation mechanism for the Earth’s magnetic ﬁeld is beyond the scope of the present chapter (the interested reader is referred to Section 4.13.4 for solid mathematical arguments). Here the discussion will be limited to providing just one fundamental concept and some related insights. The strength of the ﬁeld at the Earth’s surface ranges from less than 30 µT (0.3 G) in an area including most of South America and South Africa to over 60 µT (0.6 G) around the magnetic poles in northern Canada and south of Australia and in part of Siberia. The ﬁeld is similar to that of a bar magnet, but this similarity is superﬁcial. The magnetic ﬁeld of a bar magnet, or any other type of permanent magnet, is created by the coordinated spins of electrons and nuclei within iron atoms. The Earth’s core, however, is hotter

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than 1043 K, the Curie point temperature at which the orientations of spins within iron become randomized. Such randomization causes the substance to lose its magnetic ﬁeld. Hence the Earth’s magnetic ﬁeld is caused not by magnetized iron deposits, and there must be another explanation. It is now fairly universally accepted, in fact, that its generation mechanism is a self-exciting dynamo action operative in the liquid outer core (Jackson, 1996; Stevenson, 2003, and references therein), which means that ﬂuid currents in the liquid core are responsible for such a phenomenon. It is thought that convective motions driven by thermal buoyancy have been capable of sustaining the ﬁeld over much of Earth’s history (at least 3.5 billion years). The genesis of this phenomenon is similar to that of ﬂuid motion in the mantle (thermal convection), but the dynamics are completely different, as witnessed by the opposite assumptions often introduced by investigators to model these phenomena (Pr → 0 for the core, Pr → ∞ for the mantle).

3.3.5 The Icy Galilean Satellites Geological activity is not an exclusive prerogative of the Earth: other bodies in the Solar System have been found to exhibit similar phenomena. The most interesting example along these lines is perhaps represented by the Jupiter’s moon Io. It is so interesting because its heating does not come from decay of radioactive isotopes or heat left over from planetary formation. The extreme geological activity of this moon is the result of tidal heating from friction generated within Io’s interior by Jupiter’s varying pull. Several volcanoes produce plumes of sulfur and sulfur dioxide that climb as high as 500 km. Io’s surface is also dotted with more than 100 mountains that have been uplifted by extensive compression at the base of the moon’s silicate crust. Recently, it has been proposed (Spohn and Schubert, 2003, and references therein) that tidal heating of the type described above may be also responsible for the presence of internal (water) oceans underneath ice shells in the three satellites of Jupiter, Europa, Ganymede and Callisto. Observations of electromagnetic induction signatures at Europa and Callisto have been interpreted directly as indicative of sublithosphere liquid water oceans in these satellites. The magnetic ﬁeld data gathered during several close ﬂy-bys of Ganymede have also been proposed to be in part due to induction in an ocean in this satellite.

Figure 3.11 Sketch of convective currents inside the oceans of Ganymede and Callisto

For Europa, the strength of the induced signal suggests an ocean underneath a thin ice shell a few tens of kilometres thick. For Ganymede and Callisto, the oceans are at greater depths of 100 km or more.

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For Europa, the existence of an ocean underneath tens of kilometres of ice is also supported by geological evidence (Pappalardo et al., 1999), and some authors favour even thinner ice layers. Similar geological evidence is not available for Ganymede and Callisto, but equilibrium models of heat transfer by heat conduction and thermal convection seem to conﬁrm the existence of such oceans (Figure 3.11). For the ﬂuid dynamic analysis of these cases, a Prandtl number of O(10) is generally assumed, which is intermediate between the cases represented by the Earth’s core and mantle, respectively.

3.4

Atmospheric and Oceanic Phenomena

3.4.1 A Fundamental Model: The Hadley Circulation Beyond systems in which the ﬂuid is heated from below (such as those considered in the preceding section), ‘the heating from the side condition’ can also be used as a relevant model of phenomena occurring at the mesoscale. Since the pioneering work of Hadley (1735), who proposed a single cell, thermally driven, zonally symmetrical model of the Earth’s general circulation, such regimes of motion are generally referred to as the Hadley circulations (Figure 3.12). The Earth’s atmosphere is put into motion because of the differential heating of the Earth’s surface (uneven heating of its surface by the Sun’s rays). Daytime solar heating is greatest near the equator, where incoming sunlight is nearly vertical to the ground, and smallest near both poles, where sunlight arrives nearly horizontal to the ground. In particular, near the poles, heat lost to space by radiation exceeds the heat gained from sunlight, so air near the poles is losing heat. Conversely, heat gained from sunlight near the equator exceeds heat losses, so air near the equator is gaining heat. The heated air near the equator expands and rises, whereas the cooled air near the poles contracts and sinks. The combination of these two processes was initially thought to lead to a general circulation pattern with air rising near the equator, ﬂowing north and south away from the equator at high altitudes, sinking near the poles and ﬂowing back along the surface from both poles to the equator (Figure 3.13). This type of ﬂow corresponds to the aforementioned model of circulation after Hadley (1735) who, as mentioned above, ﬁrst theorized such a process. In real life, this ‘single cell’ circulation, however, cannot be sustained over the long distances between the equator and the poles, within a relatively shallow atmosphere. As far as Earth is concerned, it is known observationally, in fact, that a steady elongated circulation does not exist. The circulation breaks up into three cells in each hemisphere. Rising air at the equator descends around latitude 30◦ , continues towards the pole near the surface until about latitude 60◦ , then rises before continuing towards the pole in the upper atmosphere. The return ﬂow reverses this and so forms a double ﬁgure-of-eight circulation (Figure 3.14). This is partly attributed to the fact that, at mid-latitudes, ﬂuid motion is unstable to a class of asymmetric disturbances called baroclinic waves (Eady, 1949). In this instability the effect of the Earth’s rotation plays a fundamental role.

Figure 3.12 A ﬂuid system heated and cooled from the sides: the Hadley circulation

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Figure 3.13 ‘Hadley cells’ as a primitive concept of atmospheric wind systems

Figure 3.14 Atmospheric circulation: three major convective cells between the equator and the pole

As discussed by Hart (1972), however, the Hadley model can be relevant to a number of phenomena at the mesoscale covering a wide range of topics, from motions in storm windows (Gill, 1966) and the narrow sinking regions of the ocean (Stommel, 1951), to the general circulation of atmospheres of other planets in the Solar System (Stone, 1968; Malkus, 1970). In the industrial realm, the study of such ﬂow and the related instabilities has important extensive background applications in the ﬁeld of crystal growth. These aspects will be treated in Chapter 6.

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Perhaps it is worth anticipating here, however, that also in the HB conﬁguration (Section 3.1) the ideal horizontally elongated ﬂow corresponding to the Hadley model is unstable and tends to be replaced by multicellular patterns with transverse or longitudinal rolls (with their axis perpendicular or parallel to the temperature gradient, respectively). Despite the apparent similarity, the underlying mechanisms responsible for the instability of the Hadley ﬂow in the HB conﬁguration, however, exhibit notable differences with respect to those affecting the atmospheric circulation. This is due to the different value of the Prandtl number related to the ﬂuids involved in these two cases (Pr ∼ = 0.7 for the atmosphere, Pr 1 in the case of liquid metals and semiconductor melts).

3.4.2 Mesoscale Shallow Cellular Convection: Collection of Clouds and Related Patterns In addition to the existence of the giant convective cells between the equator and the poles driven by solar differential heating, one of the most interesting features of our atmosphere is the possibility of observing directly multicellular convective structures made visible by the presence of clouds (in this regard, clouds serve basically as convenient indicators of convective processes as they act as tracers of vertical motion). Prior to expanding on this topic, the reader should note that depending on the strength of such vertical motion, this kind of convection can be deep or shallow . Deep convection extends through the depth of the troposphere and includes mesoscale convective systems and hurricanes. Shallow convection is much weaker and typically occurs within the lowest few kilometres of the atmosphere. Although deep convective systems have been extensively studied due to their impact on human life and property, shallow convective systems are also of interest because they reveal complex processes that affect cloud appearance and behaviour in regions that play a signiﬁcant role in the global climate system, particularly the tropical oceans. Cloud systems take their names (cirrus, cumulus and stratus) from Latin terms that describe the way clouds appear to an observer standing on the ground. Cirrus clouds resemble curls of hair, cumulus clouds look like piled up heaps and stratus clouds are spread out like blankets. While individual clouds may take innumerable forms, these systems work because the shape of clouds is not completely random, but is instead governed by organizational principles driven by convection phenomena. Satellite photography over the Earth’s oceans suggests a similarity between organized mesoscale shallow convection systems of clouds and the patterns that can be produced by convection of the Rayleigh–B´enard type already invoked in Section 3.3 to explain the Earth’s geophysical activity in its mantle and core (e.g. Agee and Dowell, 1974) and in Section 3.1 with regard to several technological processes of interest. The observed mesoscale systems are organized into patterns of either open or closed polygonal cells (Figure 3.15). These atmospheric manifestations of Rayleigh–B´enard convection are due to air–sea interaction processes and are commonly referred to as mesoscale cellular convection (MCC) to distinguish them from the (Hadley) large-scale ﬂows induced by lateral heating of huge portions of atmosphere at the equator or cooling at the poles as discussed in Section 3.4.1. Open cells with broad downwelling regions at their centres and thin regions of ascent at their boundaries are generally observed when the sea surface temperature exceeds the temperature of the surface air, whereas closed cells with broad upwelling regions at their centres and thin regions of descent at their boundaries generally occur where the temperature of the sea surface is slightly cooler than that of the adjacent air. Open cells usually occur in regions of cold air advection, whereas closed cells favour regions of warm air advection.

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(a)

(b)

Figure 3.15 Examples of (a) open and (b) closed cells (satellite perspective). Courtesy of NASA: image by Jesse Allen, Earth Observatory, using data obtained from the MODIS Rapid Response Team at NASA GSFC; http://earthobservatory.nasa.gov. Organized intersecting lines of cumulus convection clouds form the open cells (the centre of the cell is clear), whereas the closed cells consist of stratus cloud decks that have broken up into organized stratocumulus cloud layers (the centre of the cell is cloudy)

Although Rayleigh–B´enard convection is extensively treated in Chapter 4, for the convenience of the reader it is worth noting here some of its fundamental properties (at least in shallow layers). In general, the patterns formed by this type of convection depend strongly on the presence or absence of the so-called ‘inversion symmetry’ (spatial invariance under vertical reﬂection about the midplane of the ﬂuid layer). If this symmetry is present (which, from a practical point of view, means that the assumptions discussed in Section 2.1 hold, i.e. the system thermodynamic coefﬁcients are uniform throughout the layer), striped patterns (i.e. convection rolls) are typically observed (these are the usual phenomena observed in small-scale laboratory experiments or in technological applications). In the absence of this symmetry (likely to occur in the atmosphere due to the large temperature variations with the height and related changes in density and viscosity), polygonal structures

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Figure 3.16 Example of actinoform clouds with radial arms (spokes) of convective structure clearly visible (satellite perspective). Courtesy of NASA: image by Jacques Descloitres, MODIS Rapid Response Team, NASA/GSFC; http://visibleearth.nasa.gov

should form such as those observed in Figure 3.15. These structures can also appear when vertical reﬂectional symmetry is preserved provided that the Rayleigh number lies in a given range (Section 4.6). An additional class of MCC, which does not ﬁt into either of the other categories discussed earlier, also exists (see Figure 3.16). These cloud systems are referred to as ‘actinoform’ clouds, where, as with other types of MCC, the term refers to the collection of clouds and not to any individual cloud. The radial arms of convective cloudiness in these systems are known as ‘actiniae’ (Agee, 1984). The root word for naming actiniae clouds is of Greek origin, anemone, which refers to plants/animals with radial extensions such as tentacles and arms [the sea anemone is appropriately named (the home of Nemo); actinia is singular, the word actiniae is plural and actiniae clouds are clouds with radial arms of convective structure]. Investigators have focused on the possible mechanisms leading to the formation of such structures only recently (e.g., Garay et al., 2004; Agee, 2006). Even if these bizarre cloud systems look similar to the curved radial arms described in Section 3.1 with regard to typical ﬂow in CZ systems, it has been understood that the underlying mechanism is still given by Rayleigh–B´enard convection. In this regard, the mechanism is somewhat linked to the phenomena that occur in laboratory experiments known as spoke pattern convection (see Chapter 4). For the atmosphere, however, this phenomenon operates on a variety of spatial scales (this is supported by the evidence that individual convective clouds organize themselves into radial structures at intermediate scales and lines of these radial structures are organized at even larger scales, thus leading to a pattern that could be classiﬁed as a ‘fractal’ according to the deﬁnition given in Chapter 1, Section 1.8.5).

3.4.3 The Planetary Boundary Layer Other interesting phenomena can form on Earth within the so-called planetary boundary layer, that is, the lower layer (1–3 km) of the atmosphere (which is the region of the atmosphere in which the motion of the air is strongly inﬂuenced by interaction with the surface of the Earth).

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(a)

(b)

Figure 3.17 Examples of striped cloud patterns: (a) cloud streets; (b) atmospheric gravity waves revealing themselves in double, overlapping arcs of clouds (satellite perspectives). Courtesy of NASA: images by Jesse Allen and Jacques Descloitres, respectively, using data obtained from the MODIS Rapid Response Team at NASA GSFC; http://earthobservatory.nasa.gov

As an example, Figure 3.17a shows linear bands of clouds oriented nearly parallel to the mean ambient ﬂow direction, a phenomenon usually referred to as cloud streets. Cloud streets are essentially horizontal helices of air ﬂow, that is, longitudinal rolls with their axis almost parallel to the direction of the horizontal wind. These roll vortices are driven by a warming of the lower part of the boundary layer by the warm ocean surface, i.e. by a thermal mechanism of the Rayleigh–B´enard type. This mechanism alone, however, is not sufﬁcient to explain the dynamics of such cloud systems. As mentioned above, regularly spaced cloud streets indicate the alignment of convection rolls generated by thermal buoyancy with the shear of the mean wind, which means that also the horizontal mean wind (of the Hadley type) plays some role in these phenomena. Indeed, it is

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believed (see, e.g., the review by Etling and Brown, 1993) that the formation of cloud streets is mainly caused by the interplay between two types of instability: the thermal one and a dynamic (inﬂection) instability. As illustrated in Section 1.5.4, the latter phenomenon is generally caused by an inﬂection point in the wind velocity component perpendicular to the roll system. This kind of instability can develop in neutral or even slightly stable stratiﬁcation of the lower troposphere (see related discussions later). However, cloud streets usually occur under unstable stratiﬁcation, bounded by an inversion layer above (which implies thermal Rayleigh–B´enard instability). As explained earlier many times, this instability is created when cold air ﬂows over a relatively warm surface. Therefore, it is most likely that the two types of instabilities can act together, leading to vortex roll development and the formation of cloud streets. Along these lines, in 1972 Brown was the ﬁrst to point out that under unstable stratiﬁcation, the inﬂection instability might be enhanced by the thermal instability (that tends to shift the related disturbances, otherwise transverse, towards a longitudinal orientation) (Brown, 1972). Whether inﬂection instability or thermal instability is more dominant for cloud street formation is still a matter of discussion. It is worth pointing out, however, that this phenomenon exhibits a notable analogy with the thermal ﬂow instabilities that occur in containers heated from below and slightly ‘inclined’ with respect to the horizontal direction (see Chapter 7). In such systems, the horizontal component of the imposed temperature gradient (due to the inclination) leads to a symmetry-breaking shear ﬂow (of the Hadley type) and to the formation of steady longitudinal rolls (with their axes aligned with the shear ﬂow) representing the preferred mode of convection for a wide range of parameters. When the thermal stratiﬁcation is stable (i.e. temperature increasing with the height), the inﬂectional instability can still be effective in determining speciﬁc patterns of convection in the troposphere (in this regard, note that since the wind velocity must vanish at the Earth’s surface due to the well-known no-slip condition determined by viscous effects, the planetary boundary layer is always characterized by the presence of shear). In such a case, it generally leads to the onset of rolls with their axis perpendicular to the direction of the mean horizontal wind (transverse rolls). The related mechanism, generally known as Kelvin–Helmholtz instability, is basically a speciﬁc case of inﬂectional instability occurring in conditions for which buoyancy acts as a restoring force (i.e. as a force counteracting the effects of the shear-driven disturbances, which tends to break the initial parallel structure of the ﬂow). This phenomenon is related quantitatively to the differences in velocity and density between two adjacent horizontal layers of ﬂuid. The greater the velocity difference (the shear), the stronger is the instability. The density difference plays a role in the instability mechanism as the more the density of the lower layer exceeds that of the upper layer, the stronger is the buoyant stability and hence the more the deformation of the boundary between the two layers will be suppressed by the buoyancy restoring force. The classical mathematical expression for the stability condition is generally obtained by considering the idealized case in which the effect of viscosity is neglected (Miles, 1961; Howard, 1961; Chandrasekhar, 1961):

Riloc

g ∂ρ − 1 ρ0 ∂y = 2 > 4 ∂u ∂y

(3.1)

where ρ is the density considered as a function of the vertical (upward) coordinate y (ρ0 is a reference value) and ∂u/∂y is the gradient of horizontal velocity. Equation (3.1), if satisﬁed everywhere along the velocity proﬁle u = u(y), gives a necessary condition for stability (typically instabilities develop for Ri < 1/4; in practice, the gradient Richardson number deﬁned by Eq. (3.1)

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represents the capacity of the shear ﬂow vorticity to mix ﬂuid against the restoring action exerted by buoyancy, such that large values of Ri are associated with a low mixing capacity of the ﬂow; in another acceptation it can also be regarded as the ratio of the rate of removal of energy by buoyancy forces to its production by shear). Recalling Squire’s theorem introduced in Section 1.5.4 for the speciﬁc situation of inviscid ﬂows, the most critical disturbances for such a case are two-dimensional, which, among other things, provides a justiﬁcation for the aforementioned transverse orientation of the rolls (as mentioned before, they have their axis perpendicular to the direction of the mean horizontal wind; see, e.g., Hardy et al., 1973). Other interesting cloud patterns in the presence of horizontal wind and a stable thermal stratiﬁcation can be originated from another type of mechanism (known as internal waves) whose genesis will be discussed in Chapter 4 for the ideal case of initial conditions corresponding to stagnant ﬂuid (for some theoretical background related to the case in which such waves arise on a basic shear ﬂow, the reader is referred to, e.g., Thorpe, 1978, and the review by Staquet and Sommeria, 2002). Here we limit the discussion to observe that it can occur when a uniform layer of stably stratiﬁed ﬂuid blows over a large obstacle, such as a mountain or island (see, e.g., Clark and Peltier, 1977, and references therein; in particular, landmark work on the subject included the contributions of Queney, 1941; Lyra, 1943; Scorer, 1949). When air hits the obstacle, horizontal ribbons of uniform air are disturbed, which forms a wave pattern (wave clouds can also form, as shown in Figure 3.17b). Such motions are commonly known as ‘mountain waves’. Induced internal waves are also a prevalent feature of the oceanic interior (Olbers, 1983). Internal waves arise in much the same way as do atmospheric gravity waves, the main difference being that the waves occur between layers of water with different densities instead of layers of the atmosphere. In particular, wind-generated waves on the water surface are a speciﬁc example of gravity waves induced in the ocean by coupling with atmospheric phenomena (small wind ﬂuctuations). For both the oceans and the atmosphere, it is also known that these waves can ‘break’, leading to the release of momentum and energy in the form of a mean ﬂow into the ambient (although it transports momentum from its source to the dissipation region, in general, before breaking a single wave has no mean ﬂuctuating velocity) and to the ensuing production of small-scale turbulence. This is due to the growth of shear instabilities similar to the Kelvin-Helmholtz type discussed above originating in the high-shear wave crest (see, e.g., Troy and Koseff, 2005 and references list in Staquet and Sommeria, 2002). As a concluding remark for this section, let us observe that in addition to the phenomena with cloud streets discussed earlier, there are other examples in which the horizontal mean wind interacts with phenomena originating from localized heating at the bottom. Thermal plumes are one of the many examples in the natural, industrial or urban environment where localized sources of buoyancy cause materials to rise into an overlying ﬂuid and forced circulation to develop (Chapter 5). Plumes from terrestrial volcanoes, forest ﬁres (Figure 3.18) and industrial stacks provide instances of buoyant plumes or buoyant jets occurring in a cross ﬂow. Fire scenarios, in particular, commonly involve multiple combustion sources interacting with a nonstationary atmosphere. Examples include the burning of oil spills on water, urban mass ﬁres and a variety of industrial accidents. The interest in these ﬂows stems from the fact that mixing and the consequent dilution of the efﬂuent with downstream spread must be known to determine their impact on their surroundings. Depending on the circumstances, the near- or the far-ﬁeld of these ﬂows may be of interest. For example, the near-ﬁeld entrainment characteristics of ﬁre plumes plays an important role in determining the production of combustion products and their accumulation in inhabitable space. In forest ﬁres or large-scale accidental gas spills or ﬁres, knowledge of the far-ﬁeld plume behaviour is valuable in determining their environmental impact.

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Figure 3.18 Forest ﬁres. Courtesy of P. Baldino

3.4.4 Atmospheric Convection in Other Solar System Bodies Interesting patterns due to thermal convection can be observed also in other bodies pertaining to the Solar System. As an example, Figure 3.19 shows the polygon-shaped feature circling the entire north pole of Saturn observed recently (Cassini–Huygens mission). Rather than the normally sinuous cloud

Figure 3.19 Odd, six-sided, honeycomb-shaped feature circling the entire north pole of Saturn (infrared mapping spectrometer picture). Courtesy of NASA, the Cassini–Huygens mission is a cooperative project of NASA, the European Space Agency and the Italian Space Agency; http://www.jpl.nasa.gov/images/cassini/

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Figure 3.20 Image from the Solar Optical Telescope showing a greatly magniﬁed portion of the solar surface. Energy from below the surface of the Sun is transported by convection and results in the convection cells or granulation, seen in this image. The lighter areas reveal where gases are rising from below, whereas the darker ‘intergranular lanes’ reveal where cooler gases are sinking back down. Courtesy of NASA and JAXA: Hinode JAXA/NASA/PPARC; http://www.nasa.gov/mission_pages/solar-b

structures seen on all planets that have atmospheres, this structure is a hexagon. The hexagon is similar to Earth’s polar vortex, which has winds blowing in a circular pattern around the polar region. On Saturn, however, the vortex has a hexagonal rather than circular shape. It has an extension of about 25 000 km. The images taken in thermal infrared light have also shown that the hexagon extends into the atmosphere about 100 km. A system of clouds lies within the hexagon. The nature of this structure is still a matter of discussion in the scientiﬁc community at the time of publication of the present book. A similar behaviour, however, can be observed in some typical terrestrial laboratory experiments in the realms of both thermogravitational (see Figure 4.49) and thermocapillary ﬂows (see Figure 11.45). Convection also occurs on the Sun. A high-resolution white light image of the Sun (Figure 3.20) shows a pattern that looks something like rice grains (Stein and Nordlund, 1989). Very large convection cells cause this granulation. The bright centre of each cell is the top of a rising column of hot gas. The dark edges of each grain are the cooled gas beginning its descent to be reheated. These granules are the size of the Earth and larger. They constantly evolve and change. Remarkably, these granules can be regarded as examples of the ‘eddies’ which are typically observed in turbulent convection as discussed in Section 1.4.

4 Thermogravitational Convection: The Rayleigh–B´enard Problem For the past century, Rayleigh–B´enard (RB) convection has been the subject of very intensive theoretical, experimental and numerical studies. As illustrated in Chapter 3, analysis of the RB problem is of practical importance for many engineering applications and natural phenomena. However, it is worth emphasizing that the main interest in this problem to researchers is of a theoretical nature, since RB convection presents, during the evolution from the stationary state to the fully developed turbulent regime, such a rich scenario of different structures and bifurcations that it is widely regarded as a reference problem for the study of different transition mechanisms in ﬂuid dynamics (Clever and Busse, 1974; Busse, 1978a; Gollub and Benson, 1980; Curry et al., 1984; Daniels and Ong, 1990).

4.1

Nonconﬁned Fluid Layers and Ideal Straight Rolls

4.1.1 The Linearized Problem: Primary Convective Modes The ‘paradigm’ example traditionally used in the literature for classical studies on this type of convection is represented by a layer of inﬁnite extent. Most studies have focused, in fact, on the stability of the quiescent state of nonconﬁned ﬂuid layers heated from below. Even if such a conﬁguration may be regarded as an idealized system (due to the nonphysical absence of lateral boundaries), experimental conditions effectively approaching this theoretical model can be yielded in practice by considering a ﬂuid layer with a horizontal extent that is large compared with the vertical thickness d (such that the physical conditions are approximately isotropic and homogeneous with respect to the horizontal dimension). It is known from experiments that as the temperature on the bottom of the ﬂuid layer is increased, the inherently unstable arrangement of heavy ﬂuid over light ﬂuid starts to break down and an overturning begins. The overturning is not a random, featureless motion in the ﬂuid. Rather, what appears is a highly structured state, a well-ordered spatially periodic pattern of alternatively upwelling and downwelling ﬂuid. The appearance of such a pattern is due to competition between a driving force and a dissipation of energy, hence these patterns are also known as ‘dissipative structures’ [see Eq. (1.71) and the discussion in Section 1.3]. Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Buoyancy forces of the lower warmer ﬂuid drive the instability and the viscous forces and thermal diffusion of the ﬂuid serve to counteract it. Hence the viscosity of the ﬂuid has a stabilizing effect and for small temperature gradients the ﬂuid remains at rest and there is only heat diffusion in the system (under such conditions small perturbations of the stationary conducting state decay, so the conducting state is linearly stable). Three cases of the boundary conditions are possible, in principle, for this conﬁguration and have also been investigated theoretically in the literature: (1) stress-free isothermal horizontal boundaries, (2) no-slip isothermal horizontal boundaries and (3) stress-free thermally insulated upper and no-slip isothermal lower boundary. Cases (1) and (2) are classical and the critical Rayleigh numbers based on the height of the ﬂuid layer d for the onset of convection are Racr = 657 and 1707, respectively. In the third case, Racr = 669 (it becomes 1101 if the upper free surface is assumed to be isothermal). These threshold values of the Rayleigh number do not depend on the Prandtl number, i.e. on the model liquid used for the experiments. Moreover, in all three problems the aforementioned isotropy of the layer is responsible for the onset of rotationally invariant (‘isotropic’) convection. In practice, the instability is driven by two-dimensional perturbations of the diffusive state (see, e.g., the discussions in Chandrasekhar, 1961) and the ensuing ﬂuid motion is regular and organized as a set of horizontal parallel rolls all aligned along an arbitrary direction (perpendicular to the planes containing the original two-dimensional disturbances mentioned earlier). These rolls are often referred to in the literature as ideal straight rolls (ISRs). Following the concepts illustrated in Section 1.5 and taking into account that the system initial state corresponds to the absence of convection (V 0 = 0) and a linear temperature proﬁle along the y direction (∇T0 = −i y ), the linear stability equations Eqs (1.96)–(1.98) simply reduce to ∇ · (δV ) = 0 ∂ δV + ∇(δp) = Pr ∇ 2 (δV ) + Pr RaδT i y ∂t ∂ δT − δv = ∇ 2 (δT ) ∂t δv being the vertical velocity component. The curl of the momentum equation gives

(4.1) (4.2) (4.3)

∂ζ

(4.4) = Pr ∇ 2 ζ + Pr Ra∇(δT ) ∧ i y ∂t where ζ = ∇ ∧ (δV ) is the disturbance vorticity. Taking the curl of this equation again leads to ∂ 2 ∂(δT ) 2 2 2 (4.5) ∇ (δV ) = Pr ∇ [∇ (δV )] + Pr Ra ∇ (δT )i y − ∇ ∂t ∂y Projecting this equation on the y-axis yields ∂ 2 ∇ (δv) = Pr ∇ 2 [∇ 2 (δv)] + Pr Ra[2 (δT )] ∂t where 2 denotes the horizontal Laplacian, 2 = ∇ 2 − ∂ 2 /∂y 2 . In compact form, this equation can be also written as 1 ∂ 2 − ∇ ∇ 2 (δv) = Ra[2 (δT )] Pr ∂t Similarly, the heat equation Eq. (4.3) can be cast in condensed form as ∂ − ∇ 2 δT = δv ∂t

(4.6a)

(4.6b)

(4.7)

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Eliminating δT , Eqs (4.6) and (4.7), in turn, can be used to obtain a compact high-order equation equivalent to the initial system Eqs (4.6)–(4.7): 1 ∂ ∂ 2 2 (4.8) −∇ − ∇ ∇ 2 (δv) = Ra[2 (δv)] ∂t Pr ∂t that can be used as a relevant alternative for introducing the eigenvalue problem typical of linear stability analysis. As illustrated (in the most general case) in Section 1.5, disturbances can be simply represented as plane waves of the form δv = vd (y)eλt ei(qx x+qz z)

(4.9a)

δT = Td (y)eλt ei(qx x+qz z)

(4.10a)

where qx and qz are the disturbance wavenumbers along x and z, respectively. Since, in the RB problem, as mentioned earlier, such disturbances are known to be two-dimensional, the above expressions can be simpliﬁed as follows: δv = vd (y)eλt ei(qx)

(4.9b)

δT = Td (y)eλt ei(qx)

(4.10b)

where it has been assumed (without sacriﬁcing generality given the aforementioned system isotropy) that the two-dimensional disturbances act in the generic xy plane (that is, the rolls are oriented along the z direction). Substituting them into Eqs (4.6) and (4.7) leads to a system of two ordinary differential equations: Pr(vdI V − 2q 2 vd + q 4 vd ) − λ(vd − q 2 vd ) = Pr Raq 2 Td Td − (λ + q 2 )Td = −vd

(4.11a) (4.11b)

(where the prime denotes derivative with respect to y), which must be solved together with the following boundary conditions (obtained via simple mathematical manipulations and using the continuity equation): vd = vd = Td = 0 vd = vd = Td = 0

(rigid boundaries) (stress-free boundaries)

Similarly, Eq. (4.8) becomes a sixth-order ordinary differential equation: 1 1 λ2 vdI V + 3q 4 + 2λ 1 + q2 + v vdV I − 3q 2 + λ 1 + Pr Pr Pr d 1 λ2 q 4 + q 2 vd = −q 2 Ravd − q6 + λ 1 + Pr Pr to be solved [in place of Eqs (4.11)] with the conditions λ v = 0 vd = vd = vdI V − 2q 2 + Pr d vd = vd = vdI V = 0

(4.12a) (4.12b)

(4.13)

(rigid boundaries)

(4.14a)

(stress-free boundaries)

(4.14b)

where Eqs (4.14) follow from Eqs (4.11a) and (4.12). Results obtained via this approach are shown in Figure 4.1 [remarkably, for stress-free boundaries the problem admits analytical solution of the form Racr = (q 2 + π 2 )3 /q 2 ]. The governing instability equations have a particular symmetry which determines that all eigenvalues of the linearized problem are real (Pellew and Southwell, 1940). This means that the emerging pattern is always stationary.

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Figure 4.1 Marginal stability curves for Rayleigh–B´enard convection in an inﬁnite layer for stress-free and no-slip boundary conditions

Historically, Jeffreys (1926, 1928) was the ﬁrst (after some numerical problems) to yield the value Racr = 1707 relevant to experiments using ﬂuids conﬁned between well-conducting solid parallel plates (the reader is also referred to the accurate analysis by Reid and Harris, 1958). The theory for stress-free conditions was developed by Malkus and Veronis (1958), who expanded the nonlinear equations describing the ﬁelds of motion and temperature in a sequence of inhomogeneous linear equations dependent upon the solutions of the linear stability problem. They found that there are an inﬁnite number of steady-state ﬁnite amplitude solutions (having different horizontal planforms) that formally satisfy these equations. The foundation for much of the ‘modern’ work on RB convection, however, was laid during the 1960s by the weakly nonlinear analysis of Schl¨uter et al. (1965) for rigid boundaries, which predicted stable, straight rolls above onset for this realistic case (see also the landmark studies of Davis, 1967, and Busse, 1967a–c). The expected nondimensional diameter of the convection cells (ratio of the horizontal cell extension to the depth of the liquid layer) is half the critical wavelength, λc = 2π/qc , where qc is the horizontal wavenumber at onset. In the no-slip case qc ∼ = 3.12, which makes the predicted horizontal size very close to the height of the layer (π/3.12 ∼ = 1.007). In the stress-free case 1 qc = π/(2) /2 , which gives a somewhat larger diameter (Figure 4.1). These theoretical ﬁndings are reﬂected in real experiments where, apart from the usually present irregularities or pattern defects, the velocity ﬁeld of roll convection is nearly two-dimensional (see Figure 4.2).

4.1.2 Systems Heated from Above: Internal Gravity Waves It is obvious that in the opposite situation in which the liquid is uniformly heated from above and cooled from below, no convection arises and the system will maintain a stable temperature stratiﬁcation. Anyhow, for the sake of completeness, it should be reported that in the limit as the dissipative (Laplacian) terms disappear from the thermal-convection equations and in the framework of the so-called approximate parcel theory for the atmosphere (by which pressure perturbations are

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Figure 4.2 Shadowgraph image [plane (x, z)] of nearly two-dimensional convection (Pr ∼ = 1 and Ra ∼ = 2 × Racr ; the parallel strips correspond to straight convection rolls. Courtesy of B. Plapp and E. Bodenschatz

neglected), the linearized equation for the disturbance velocity component in the y direction, δv, reduces to the undamped harmonic oscillator equation: ∂2 2 (δv) + ωBV (δv) = 0 (4.15a) ∂t 2 2 with ωBV = Pr Ra (the derivation of this equation is relatively simple; it is not reported here for the sake of brevity), which means that a thermally stratiﬁed system would admit under such assumptions oscillatory disturbances in the form of simple harmonic motion-like waves. From a physical point of view, the oscillatory motion represented by such waves should be regarded as the outcome of the spontaneous tendency displayed by any generic parcel of ﬂuid to oscillate (if somehow displaced to a different location with respect to the initial equilibrium one) around the equilibrium position under the actions of inertia and the restoring effect of buoyancy (consider, for instance, a parcel of ﬂuid at initial temperature T and position y that is displaced vertically and upwards due to some ﬂuctuation of the vertical component of ﬂuid velocity: due to the stable stratiﬁcation, the ﬂuid parcel will have in its new location, y + dy, a lower temperature than that of the ambient ﬂuid; owing to the related excess weight, the ﬂuid parcel will tend to sink, due to inertia, to a location that is deeper than its initial position, where the ambient temperature is lower than that of the ﬂuid parcel; as a consequence of buoyancy, the particle will move upwards again, thus deﬁning an oscillatory harmonic motion). These modes are generally known in the literature as internal waves, while the angular frequency 1 ωBV = (PrRa) /2 is referred to as the Brunt–V¨ais¨al¨a frequency. In reality, however, such disturbances tend to be damped by the effect of viscous and thermal dissipation (the waves propagate until they dissipate), which makes their permanence possible only if there is an external source continuously exciting them (see, e.g., Thorpe, 1968, 1975; Voisin, 1994) (it is also worth mentioning that in the framework of the aforementioned analogy with the dynamics of an oscillator, the development of these modes could be successfully compared with the behaviour of the so-called damped driven harmonic oscillator for which an external forcing is required for balancing the damping action exerted by a dissipative force). In some

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circumstances, a mean ﬂow can be directly produced by nonlinear effects (M¨uller et al., 1986; Staquet and Sommeria, 2002), but the resulting changes are weak and reversible and such mean ﬂow disappears after the wave emission has stopped. A more rigorous derivation of the properties of these waves can be obtained directly from Eq. (4.8) that, neglecting dissipative terms and in the case of heating from above, reduces to ∂2 2 2 ∇ (δv) + ωBV [2 (δv)] = 0 (4.15b) ∂t 2 Notably, assuming a plane wave solution of the form δv = vd (y)ei(qx−ωt) and substituting into Eq. (4.15b) gives for the amplitude 2 ωBV − 1 q 2 vd = 0 (4.15c) vd + ω2 which leads to the conclusion that effective solutions of this equation only have a wave-like character when ω < ωBV (whose most interesting articulation is that Brunt–V¨ais¨al¨a frequency should be regarded as the upper limit of frequency for which wave motion can exist in a stably stratiﬁed ﬂuid). In general, these waves are ubiquitous dynamic features in the stratosphere and in the interior of the ocean. Some examples have been discussed in Chapter 3. An effective circumstance in which these otherwise damped internal waves are sustained by adequate stimuli will be examined in detail in Chapter 6 for a case of technological interest. Hereafter, internal gravity waves are not considered further (we will come back to them in Section 6.2.2) and, unless explicitly mentioned, attention is devoted to the case with solid walls heated from below for which the expected primary modes, as discussed earlier, are stationary convective rolls.

4.2

The Busse Balloon

As illustrated in the preceding section, the simplest pattern which can occur in RB convection is that of straight, parallel convection rolls with a dimensional horizontal wavelength λ ∼ = 2d. Such rolls can be found near the onset; however, as Ra increases, the pattern becomes progressively more complicated. Although at Ra = Racr there is only one possible roll wavenumber qc , for every Ra > Racr there is a range of wavenumbers for which convective rolls can exist. However, not all of the roll wavenumbers are stable states. A variety of secondary instabilities that further restrict the domain of stable convection exist. In a series of papers, Busse and co-workers (Schl¨uter et al., 1965; Clever and Busse, 1974; Busse, 1978a, Busse and Clever, 1979) gave a complete classiﬁcation of the secondary instabilities which occur in RB convection in inﬁnite layers bounded from above and below by isothermal solid walls. In these studies, ideal straight rolls were represented (using a Galerkin expansion) as (i) amn exp(imqx)G(i) (4.16) n (y) m,n

G(i) n (y)

(i) where are the Chandrasekhar functions (Chandrasekhar, 1961) and amn are the coefﬁcients related to the expansion (such a representation corresponds to a straight roll pattern with roll axes along the z direction and wavenumber qx = q, as shown in Figure 4.3). Possible disturbances potentially affecting the above conﬁguration were represented as (i) a˜ mn exp(imqx)G(i) (4.17) exp(iSx x + iSz z + σ t) n (y) m,n

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Figure 4.3 Schematic diagram of rolls in Rayleigh–B´enard convection: for a laterally unlimited domain, the ﬂuid motion is regular and organized as a set of horizontal parallel rolls (arrows indicate the direction of ﬂuid ﬂow; the wavelength of the pattern is approximately equal to twice the layer height d )

Sx and Sz being the so-called Floquet (disturbance) wavenumbers and σ the disturbance growth rate. Accordingly, secondary modes of convection were expressed as (i) almn exp(imSx x + ilSz z)G(i) (4.18) n (y) l,m,n

The authors revealed the existence of a whole zoology of such secondary modes, with exotic names such as ‘Eckhaus’, ‘cross-roll’, ‘knot’, ‘zig-zag’, ‘skewed-varicose’, ‘oscillatory blob’ and others.

4.2.1 Toroidal–Poloidal Decomposition Before going further into the description of these results, it is worth providing some details on the method that was used by Busse and co-workers mentioned above. The mathematical model that they developed, in fact, exhibits outstanding elements of originality and can be regarded as a valuable variant (to be used in the context of studies devoted to the RB problem) of the models and methods illustrated in Section 1.7. Since the velocity ﬁeld is solenoidal (divergence free), they used the following vectorial representation: " # " # (4.19) V = ∇ ∧ ∇ ∧ φi y + ∇ ∧ ϕi y (where i y is the unit vector along the y direction; see Figure 2.1) known as the poloidal–toroidal decomposition of the velocity ﬁeld V (for the rectangular cell geometry the decomposition has been proven to be unique and complete; Schmitt and von Wahl, 1992). They also applied the following decomposition for the temperature ﬁeld: T = Tdiff + Tv

(4.20)

where Tv is the dimensionless ﬁeld measuring the deviation of T from the solution in the absence of convection (Tdiff coming from ∇ 2 Tdiff = 0; the reader is referred to Section 2.1 for some additional information about the deﬁnition of TFD distortions and their meaning). Accordingly, the energy equation Eq. (1.61) was rewritten as ∂Tv + V · ∇Tdiff + V · ∇Tv = ∇ 2 Tv ∂t

(4.21a)

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where (obviously) ∇Tdiff = −i y → ∂Tv (4.21b) + V · ∇Tv = V · i y + ∇ 2 Tv ∂t Rescaling Tv by the Rayleigh number, that is, introducing = RaTv and letting ∇π = ∇p + Pr Ray, the momentum and energy balance equations [Eqs (1.60) and (1.61), respectively], were rewritten as ∂V (4.22) + V · ∇V + ∇π = Pr ∇ 2 V + Pr i y ∂t ∂ + V · ∇ = RaV · i y + ∇ 2 ∂t Then the poloidal–toroidal decomposition was applied as follows: ix iy i z ∂φ ∂φ ∂ ∂ ∂ ix + i = ξφ ∇ ∧ (φi y ) = = − ∂x ∂y ∂z ∂z ∂x z 0 φ 0

(4.23)

(4.24)

where the ξ operator reads

∂ ∂ ix + i ξ = − ∂z ∂x z ix iy iz ∂ ∂ ∂ ∇ ∧ [∇ ∧ (φi y )] = ∂x ∂y ∂z ∂φ ∂φ − 0 ∂z ∂x 2 2 2 ∂ 2φ ∂ φ ∂ φ ∂ φ i − i = ηφ + 2 iy + = ∂x∂y x ∂z2 ∂x ∂y∂z z

where the η operator reads 2 2 ∂2 ∂ ∂ i x − 2 i y + i z with 2 = ∇ 2 − 2 η= ∂x∂y ∂y∂z ∂y

(4.25)

(4.26)

(4.27)

The most useful property of these operators is that they are orthogonal: ξ ·η =0

(4.28)

ξ ·∇ = η·∇ =0 ξ · ξ = 2

(4.29) (4.30)

Other useful relations are

η · η = 2 ∇ 2

(4.31) (curl)2

and the curl of the momentum Following this approach and taking the y component of the equation [the continuity equation is satisﬁed implicitly as ∇ · (∇ ∧ V ) = 0], the following equations were obtained for ϕ and φ: ∂ 2 ϕ ∂t ∂ (∇ 4 2 φ − 2 ) Pr = η · [(ηφ + ξ ϕ) · ∇(ηφ + ξ ϕ)] + ∇ 2 2 φ ∂t (∇ 2 2 ϕ) Pr = ξ · [(ηφ + ξ ϕ) · ∇(ηφ + ξ ϕ)] +

(4.32) (4.33)

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and the internal energy (heat) equation becomes ∂ (4.34) ∂t with the boundary conditions φ = ∂φ/∂y = ϕ = = 0 at y = − 1/2 and y = 1/2. Additional related quantities are the densities of poloidal and toroidal kinetic energies, deﬁned by (∇ 2 − Ra2 φ) = (ηφ + ξ ϕ) · ∇ +

1 (4.35) |∇ ∧ [∇ ∧ (φi y )]|2 2 1 (4.36) kϕ = |∇ ∧ (ϕi y )|2 2 Busse and co-workers used the system of equations Eqs (4.32)–(4.34) to deﬁne mathematically and solve the stability problem. Additional insights were obtained through evaluation and study of the behaviour of the kinetic energies deﬁned by Eqs (4.35) and (4.36). Further details are given in Section 4.2.2. kφ =

4.2.2 The Zoo of Secondary Modes Hereafter, following the work of Busse and co-workers, the conﬁguration in Figure 4.3 is considered as the possible initial state of RB convection and used to provide the reader with a clear and exhaustive picture of the possible scenarios related to the secondary instabilities which can affect such initial state. As illustrated earlier [(Eq. (4.16)], such a considered initial state is simply proportional to exp(iqx), where q is the wavenumber of the pattern. Sketching the boundaries of the region in the space of wavenumber and control parameter where these parallel rolls are stable, one obtains the so-called ‘Busse balloon’, whose detailed form depends only on the Prandtl number (whereas, as explained in Section 4.1, for different Prandtl numbers, the primary onset of convection is unchanged, and the secondary instability boundaries take different shapes according to Pr; see Figure 4.4). Inside this balloon, the periodic roll structure is linearly stable, but on crossing one of its boundaries, the rolls are destabilized by a secondary instability. Some secondary modes are universal, that is, they do not depend on the fact that the pattern is generated by the RB mechanism but on the symmetries of the rolls (invariance through translation, e.g. the Eckhaus instability, and rotation, e.g. the zig-zag instability) or on the fact that the intensity of the convection is weak close to the marginal curve (cross-roll instability). Other secondary modes are much more speciﬁc to RB convection, with structures that depend strongly on the value of Pr, for example the bimodal instability with secondary rolls localized in thermal boundary layers at right-angles to the primary pattern when Pr is large. Some of the properties of the secondary modes are summarized in Tables 4.1 and 4.2. The Eckhaus instability is the simplest secondary instability. It is a long-wavelength modulation of the form exp(iqx )exp(iS x x), where Sx q. If a roll pattern is at some Ra and q near (but above) the Eckhaus boundary (this boundary limits the Busse balloon from below) and Ra is dropped, the evolution of the Eckhaus instability leads to the addition of a roll pair and an increase in the wavenumber. The Eckhaus instability is a slow-growing instability and for this reason in many cases the cross-roll instability, which is a short-wavelength, fast-growing instability, will occur before the Eckhaus instability can. The cross-roll instability is related to a modulation of the pattern parallel to the axes of the initial rolls. It causes a perturbation of the form exp(iS z z), where Sz is of the order of q. The appearance will be that of rolls growing perpendicularly to the original pattern of rolls (Busse and Whitehead, 1971). In some circumstances it can produce ‘totem’ structures like that shown in Figure 4.5.

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Figure 4.4 Sketch of the Busse balloon in three dimensions: Rayleigh number, Prandtl number and wavenumber. The various solid curves mark the boundaries outside of which two-dimensional convecting rolls of wavenumber q are unstable. The different boundaries are denoted as follows: the Eckhaus instability (ECK), the oscillatory skewed varicose instability (SV), the cross-roll instability (CR), the oscillatory instability with travelling waves (OS), the zig-zag instability (ZZ), the knot instability (KN) and the oscillatory blob instability (OB). Courtesy of F.H. Busse

At Rayleigh numbers not too far above the critical value, the growth of the perpendicular rolls and the concurrent decay of the original rolls proceed until the original rolls with their too short wavelength are taken over by the perpendicular or cross-rolls with a wavenumber close to the critical value (Busse and Whitehead, 1971). However, at high Ra (of the order 2 × 104 ), the cross-roll instability is characterized by a value of the wavenumber Sz that is much larger that the critical value qc and hence cannot just lead to replacement of the given roll pattern by a more stable one. Instead, a steady boundary layer-type structure appears (see Figure 4.6) and the evolution of the cross-roll instability thus leads to a new form convection known as ‘bimodal’ [Clever and Busse (1994) investigated such behaviours for Prandtl numbers in the range 10 ≤ Pr ≤ 100]. Since the secondary set of rolls has a smaller wavelength, it is especially suited to increase the heat transport in the thermal boundary layers. In this way, bimodal convection is more efﬁcient in transporting heat than two-dimensional rolls. A remarkable feature of this type of convection is that, as noted by Busse and Whitehead (1974), a striking similarity between its appearance (Figure 4.7) and a two-dimensional crystal lattice could be established (this similarity includes various kinds of irregularities found in the lattice such as edge dislocations). This phenomenological analogy is particularly interesting as it suggests that

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Table 4.1 Symmetry properties of ideal straight rolls (υ is the velocity component in the vertical direction y) A

Translation in time

B

Translation along the roll axis

C

Transverse periodicity

D

Transverse reﬂection

E

Inversion about roll axis

∂υ =0 ∂t ∂υ =0 ∂ z

2π , y = υ(x , y ) q υ(−x , y ) = υ(x , y ) π υ x + , y = −υ(x , y ) q υ x+

Table 4.2 Symmetries broken by bifurcation from ISR states Disturbance properties

Eckhaus instability Oscillatory instability Skewed varicose inst. Oscillatory skewed varicose instability Knot instability Zig-zag instability Cross-roll instability Even blob instability Odd blob instability

Broken symmetries

(ECK) (OS) (SV) (SV) (KN) (ZZ) (CR) (OBe) (OBo)

A σ = 0

B Sz = 0

X

X X X X X X X X

X

X X

C Sx = 0

D

X

X X X X

X X

E

X X X X

the transitions from cross-roll convection to the bimodal structure could be regarded as a phase transitions from one kind of lattice structure to another. Technically speaking (Busse, 1967c), the transition from rolls to bimodal convection arises from an instability of the thermal boundary layers at the rigid upper and lower boundaries of the ﬂuid layer; as explained above; in fact, the additional small-wavelength convection rolls that develop at right-angles to the basic roll are particularly suited to take advantage of the buoyancy stored in the thermal boundary layers and their presence makes the system more stable. Another mechanism by which instabilities may lead to a more efﬁcient convective heat transport is presented by the (stationary) knot instability. This instability breaks the same symmetries as the cross-roll instability, but with a much smaller value of the wavenumber Sz along the axis of the rolls. For Ra > 3 × 104 , strong plumes evolve along the currents of rising and descending liquid. Concurrently, ‘streamers’ evolve in the thermal boundary layers feeding the plumes. These features look like knots in the shadowgraph observations of convection and have given rise to the name for the instability (Busse and Clever, 1979; Clever and Busse, 1989). These nearly axisymmetric plumes are more efﬁcient in transporting heat than rolls at not too high Prandtl numbers (for Pr ≤ 7) because of the concentrated upward (downward) momentum with which these plumes impinge on the cold (hot) boundary of the layer (thereby creating strong temperature gradients). The knot instability disappears in the limits of both small and large Prandtl numbers (Busse and Clever, 1979).

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Figure 4.5 Snapshot [shadowgraph image, plane (x, z)], of a totem structure (cross-roll instability, Pr = 1.09, Ra/Racr = 1.7). Courtesy of B. Plapp and E. Bodenschatz

As an additional variant, the zig-zag perturbation has the form exp(iqx )exp(±iSz z) with Sz q (Busse and Whitehead, 1971). In practice, as shown by the Busse balloon, the predominant modes at high Ra are bimodal convection at large Prandtl numbers [Pr ≥ O(10)], knot convection at moderate Prandtl numbers [O(1) < Pr < O(10)] and oscillatory convection of the OS type in low-Pr ﬂuids [Pr ≤ O(1); further details on this mode are given later]. Bimodal convection is the preferred state of convection at high Prandtl numbers and knot convection assumes this role in the range 2 ≤ Pr ≤ Pr∗ , where Pr∗ increases from about 10 at a Rayleigh number of the order 3 × 104 to much higher values as Ra increases. The stability boundary at Pr∗ corresponds to the transition from knot convection to bimodal cells and vice versa. In the range 2 ≤ Pr ≤ 12, oscillatory blob (OB) convection can also occur as an instability of two-dimensional convection rolls. This instability competes with knot convection in the parameter space of the problem (Bolton et al., 1986; Clever and Busse, 1995a). The so-called thermal ‘blobs’ represent a main feature in high-Ra convection as shown in modern experiments by Zocchi et al. (1990) and Solomon and Gollub (1990). The ﬂow is featured by the presence of hot and cold blobs which circulate around the generic convection roll. Since, for the relatively high values of Ra associated with this mode of convection, heat transport provided

y

x z

Figure 4.6 Sketch of bimodal convection

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Figure 4.7 Experimental snapshot [plane (x, z)] of bimodal convection (Pr = 63, Ra = 2.1 × 105 ). After Busse and Whitehead (1974); Reproduced by permission of Cambridge University Press

by the rolls is not sufﬁciently high to keep the thermal boundary layer stable (in terms of the Rayleigh criterion for the stability of a ﬂuid layer heated from below), periodic eruptions from the thickening thermal boundary layers occur; along these lines, the OB mode of convection may be regarded as a kind of resonance phenomenon between these periodic eruptions of ﬂuid from the thermal boundary layers and a simple fraction of the circulation period of the basic roll velocity. In general, the motion of the blobs, however, is not limited to a simple circulation in a plane perpendicular to the roll axis, but it also involves propagation along its extension (Figure 4.8); thus, thermal blob convection occurs primarily as a wave travelling along the basic convection roll. Standing states are also possible, but they should be regarded as the superposition of two waves travelling in opposite directions (Clever and Busse, 1995a). The parameter range where travelling blob convection can be expected is not very large and other instabilities leading from two-dimensional rolls to three-dimensional convection are more

Figure 4.8 Sketch of travelling blob convection at a given instant in time (the pattern propagates in the positive z direction; regions where the ﬂuid near the boundary of a roll is hotter that its time-averaged temperature are indicated by the bands on the roll boundary). After Clever and Busse (1995); Reproduced by permission of Cambridge University Press

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Figure 4.9 Snapshot [shadowgraph image, plane (x, z)] of skewed varicose instability (Pr = 1.07, Ra = 3.26 × Racr ). Courtesy of B. Plapp and E. Bodenschatz

commonly encountered in the relevant Prandtl number range (2 ≤ Pr ≤ 10), i.e. the aforementioned steady knot and the skewed varicose (SV) modes. The latter, that is, the SV oscillatory instability, is a modulation of the form exp(iqx )exp(iSx x + iSz z), with Sx , Sz q and Sz /Sx ∼ = O(1). The angle of the SV instability relative to the original rolls tan−1 (Sz /Sx ) is near 45◦ (see, e.g., Figure 4.9). This instability causes a shearing to occur across the rolls and a roll pair is torn, creating two defects in the roll pattern. These defects migrate in such a way as to remove a roll pair from the pattern and thereby move the wavenumber to a lower value (Busse and Clever, 1979). It is worth noting that this type of mechanism (disclosed in the theoretical study of Clever and Busse, 1978, and observed in the experiments of Busse and Clever, 1979, for Pr ≤ 7, of Gollub et al., 1982, for Pr = 2.5, of Motsay et al., 1988, for 4 He with Pr = 0.52 and 0.70 and in numerical simulations by Gr¨otzbach, 1983, for Pr = 0.71) is a relevant example of situations in which dislocations, that is, the topological point defects where stripes (i.e. the rolls) terminate in the interior of the system (i.e. localized defects in the pattern), play an important role (see Section 4.3 for further elaboration of this concept). For this instability, the time dependence is directly related to the repetitive nucleation and migration of pairs of dislocations in the convective structure (Croquette et al., 1986). Like the knot mode, the SV instability disappears in the limits of high and small Pr. For Pr ≤ O(1), in particular, the oscillatory instability becomes the dominant secondary mode of convection (even though for Pr ∼ = 1 the SV mode is still possible). Some initial studies related to the range 10−3 ≤ Pr ≤ 0.71 are due to Busse and Clever (1981) and Clever and Busse (1981). Clever and Busse (1987) showed that the OS mode appears as a short-wavelength wave propagating along the rolls with the form exp(iSz z) exp(iωt), where Sz ≈ q and ω is the angular frequency of the wave. This instability does not destroy the roll structure and does not create defects; it simply adds a wave that propagates along the rolls. The oscillatory instabilities of long, straight, parallel rolls (in shallow layers) calculated by Clever and Busse (1990) for Pr 1 (e.g. Pr = 0.01) appear, for example, as simple travelling waves. They found that the transition from thermal convection

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Figure 4.10 Snapshot [shadowgraph image, plane (x, z)] of travelling wave related to the OS instability (Pr ∼ = 1 and Ra/Racr ∼ = 6). Courtesy of B. Plapp and E. Bodenschatz

in the form of rolls in a ﬂuid layer heated from below to travelling wave convection occurs at Ra = 1854 in the limit of low Prandtl numbers and in the presence of no-slip boundaries. They also revealed that for Pr < 0.02 the travelling wave convection remains stationary with respect to a moving frame of reference. This type of convection exhibits similar properties at very low and moderately low Pr. As an example, for a ﬂuid with Pr ∼ = 1, Croquette (1989b) found by experimental investigation that the oscillations along the rolls could be described as simply ‘decorating’ the rolls (similar results were obtained by Plapp, 1997; see Figure 4.10). These waves were also obtained numerically by Meneguzzi et al. (1987) for Pr = 0.025 and Thual (1992) in the limit as Pr → 0 and by Lipps (1976) for Pr = 0.71 and McLaughlin and Orszag (1982). For the special case of stress-free boundaries (for a variety of astrophysical and geophysical applications the convective layers in stars such as the Sun or convection in the Earth’s mantle, stress-free conditions are probably a better choice than no-slip conditions, which are realized in most laboratory experiments) and related evolution of the ﬂow pattern when Ra is increased, the reader may consult Busse (1972), Busse and Bolton (1984), Bolton and Busse (1985), Schnaubelt and Busse (1989) and Busse et al. (1992). Basically, these analyses have demonstrated that for such a case the convection rolls are unstable in almost the entire parameter space of the problem, primarily through the action of the SV instability.

4.3

Some Considerations About the Role of Dislocation Dynamics

Given their crucial importance in explaining many of the features of RB convection (e.g. the transitional stages of evolution of the SV instability), the dynamics of dislocations outlined in the preceding section deserve some additional discussion.

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Dislocations present a simple realization of topological singularities in a ﬁeld description of continuous extended systems. In general, travel of dislocations through an otherwise uniformly periodic structure can take two forms: ‘climbing’ or travel of the dislocations parallel to the axes of the rolls, and ‘gliding’ or travel of the dislocations perpendicularly to the axes of the rolls (Siggia and Zippelius, 1981; Pomeau et al., 1983; Tesauro and Cross, 1986). Most existing studies on dislocation propagation in RB convection have dealt with climbing. In climbing, a dislocation either removes or creates a roll pair (depending on the direction of propagation). In gliding, the dislocation does not create such modiﬁcations even though it tends to modify slightly the orientation of existing rolls. The reader may consider the theoretical analysis by Walter et al. (2004) for additional interesting details about various possible aspects of the dynamics of dislocations in RB convection and to understand how various types of defects can sustain a speciﬁc persistent dynamics; as shown by these authors, the nucleation, motion and annihilation of dislocations are essential for many pattern-selection processes, which are initiated by modulational instabilities. In such work, these authors approached the problem numerically [for Pr = O(1)] by using both the full hydrodynamic equations and standard, well-accepted model equations. Interestingly, by such a strategy they proved that dislocations are driven by a superposition of two independent forces, as follows. One is the well-known Peach–Koehler force, which describes the tendency of the system to develop towards a striped pattern with an optimal average wavenumber (the reader may be interested in knowing that this concept, which can be regarded as an energy-minimization principle, was ﬁrst introduced in solid-state physics to describe the dislocation dynamics under the inﬂuence of an external stress, which is crucial to understanding the strength of crystals). The second force, an advection force, is due to a long-range pressure ﬁeld caused by strong roll curvature gradients in the vicinity of a dislocation, which excite a ﬂow ﬁeld with a ﬁnite vertical average (mean) ﬂow. This advection force acts in general to remove dislocations from the system so as to reduce the wavenumber of the pattern. For a certain wavenumber qD the two forces balance and a single dislocation is stationary. In the vicinity of qD , two approaching dislocations of opposite topological charge can form bound states which are also typically observed in experiments. In their numerical analysis, Walter et al. (2004) focused, in particular, on dislocation climbing, that is, the aforementioned motion parallel to the roll axis, in a rectangular domain −Lx,z /2 < x, z < Lx,z /2 with width Lx and length Lz = 2Lx . Dislocations were seeded at time t = 0 into initial striped patterns with different q: a wavenumber qD (Ra) was identiﬁed such that for q > qD the dislocation climbs with vD < 0 whereas it moves in the opposite direction for q < qD and becomes stationary for q = qD . According to such ﬁndings, for instance, the totem structure shown in Figure 4.5 could be explained according to a local cross-roll instability triggered by a ﬁnite perturbation due to a dislocation. A dislocation moving along the rolls into a pattern with q < qD , in fact, would split into a ‘chain of bubbles’. Such bubbles would develop from localized transverse bridging of adjacent rolls and form the totem structure. The possible crucial role played by dislocation dynamics in determining the formation of pattern is not limited to the SV or CR modes. In addition to the short-wavelength CR instability of a roll pattern, as widely illustrated in the preceding section, the Busse balloon is also organized by several modulation instabilities with general wavevector S = (Sx , Sz ). The case Sx = 0 denotes the zig-zag (ZZ) instability, the case Sz = 0 the Eckhaus instability and the general case with both Sx and Sz = 0 corresponds to the SV instability (in the case of an SV instability, in particular, the dislocation moves in a direction perpendicular to S). So far, an appropriate and exhaustive theoretical modelling of combined climb and glide dynamics is still lacking. Along these lines, however, it is worth noting that to obtain additional insights,

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the problem of defect dynamics could somehow be connected to the more general one related to the study of stationary and oscillatory spatially localized structures (the term ‘spatially localized structure’ is generally used to refer to the presence of one state ‘embedded’ in a background consisting of a different state). These structures, referred to as dissipative solitons and oscillons, respectively, are known to arise (beyond RB convection) in many interesting and important applications. Any defect that is stationary in time may be regarded as a soliton. The recent resurgence in interest in these structures has led to signiﬁcant advances in our understanding of the origin and properties of these states (Knobloch, 2008) and these in turn suggest new questions, both general and system speciﬁc. Further elaboration of these concepts is beyond the scope of the present book. Some examples of oscillons, however, will be presented in Section 4.11 when discussing the peculiar behaviour displayed by the core of some structures known as ‘spirals’ (these localized oscillations leave the rest of the system undisturbed and may form ‘bound’ states).

4.4 Tertiary and Quaternary Modes of Convection In general, not too far from threshold, the theoretical results for no-slip conditions and the related Busse balloon have been found to agree well with real experiments for large Pr (water and silicone oils in the experiments of Busse and Whitehead, 1971) and reasonably well for gases with Pr ∼ =1 (Croquette, 1989a). Secondary instabilities, however, are just a step towards more complex behaviour as Ra is increased. Different scenarios have indeed been observed, depending on the value of Pr. While the solution after the ﬁrst bifurcation leading to the onset of convection is rather similar in all cases due to its intrinsic two-dimensional nature, the higher order (tertiary, quaternary, etc.) states (introduced by the third and higher bifurcations) are speciﬁc to the respective problems. Within this context, it is worth noting that beyond the primary instability responsible for the formation of time-independent two-dimensional rolls, according to the Busse balloon, time dependence introduces itself ﬁrst and at relatively low Ra when Pr is small, but only after the secondary instabilities adding space dependence along the rolls (i.e. leading to three-dimensional time-independent states) at large Pr at higher Ra. For example, an oscillatory secondary instability can occur at medium and small Prandtl numbers (OS or SV modes), in contrast to stationary bimodal or knot bifurcations at large Pr. This is the reason why, for instance, oscillations in air occur as a secondary instability and appears as waves propagating along the axes of the convection rolls (Willis and Deardorff, 1970), while oscillations in water are preceded by the transition to bimodal convection (Busse and Whitehead, 1974). The subject of transition to time dependence in thermal RB convection of high-Pr ﬂuids has long been a controversial one. Because the inertial terms in the equations of motion become unimportant for sufﬁciently high Prandtl numbers, the time derivative and convective transport terms in the energy equation are the only source of time dependence and nonlinearity, respectively, in the basic equations of the problem. In practice, as already discussed, the oscillatory instability occurs in low-Pr ﬂuids at smaller Rayleigh number and via lower-order bifurcations than in high-Pr ﬂuids. Interesting experimental results along these lines were obtained by Krishnamurti (1973), who investigated the time-dependent transition in a ﬂuid layer for a wide range of Prandtl numbers (0.025 < Pr < 8500) and found a systematic increase in the related critical Rayleigh numbers (except for the ﬁrst, Racr ) when the Prandtl number was increased. Experiments on bimodal convection and related transition to time-dependent states (via a tertiary bifurcation), in particular, were carried out by Krishnamurti (1970a) and Busse and Whitehead

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(1974). Interestingly, in her experiments with ﬂuids of different Prandtl numbers, Krishnamurti (1970b) found that the Rayleigh number for the onset of time dependence in this type of convection becomes essentially independent of Pr once Pr exceeds a value of about 50. Busse and Whitehead (1974) observed that since (see Section 4.2) bimodal convection can be visualized roughly as the superposition of two roll patterns with different wavelengths at right-angles and since the homogeneity along the axis of the basic rolls is lost, travelling waves for such a case can no longer be expected. Indeed, standing waves are, in fact, the typical outcome of experiments (oscillating bimodal convection can be regarded as a standing wave on the boundary of the second mode with a wavelength given by the ﬁrst mode for which the theory of oscillating convection rolls applies qualitatively). According to Busse and Whitehead (1974), more speciﬁcally, it is since the onset of oscillations is seen in the form of standing waves on the boundary of the short-wavelength component of the bimodal convection, and since the dissipation rises rapidly with the wavenumber of the oscillation, that waves on the long-wavelength or basic component of bimodal convection cannot occur. The critical Rayleigh number for the onset of oscillations was found to behave approximately as a linear function of the Prandtl number (Ra = 2.5 × 103 Pr) and the period of oscillations as τ ∝ Ra−2/3 . The dependence of the critical Ra on the wavenumber q2 (wavenumber of the short-wavelength component) is shown in Figure 4.11. This problem has been investigated from a theoretical point of view more recently by Clever and Busse (1994) for 10 ≤ Pr ≤ 100, who also considered transition to oscillatory regimes for Knot convection (extending the earlier study by Clever and Busse, 1989). Knot convection and bimodal convection exhibit, in fact, notable similarities in their oscillatory form. From a physical point of view, they are rather similar in that the emergence of hot and cold blobs from the hot and cold boundary layers, respectively, is the dominant mechanism for the transition from steady to time-dependent convection.

Figure 4.11 Rayleigh number for the onset of regular oscillations as a function of the wavenumbers of the short- and long-wavelength components of bimodal convection (Pr = 64; the upper and lower curves correspond to values of the long-wavelength component of 2.40 and 2.04, respectively). After Busse and Whitehead (1974); Reproduced by permission of Cambridge University Press

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It is by virtue of such a similarity that Clever and Busse (1989, 1994) used a common theoretical and mathematical framework to examine the transition to oscillatory behaviour for these two types of convection (for which it is worth opening a short discussion; among other things, examining these ‘similar’ features will also help the reader in understanding some considerations given in Section 4.5). By expanding the dependent variables into complete systems of functions, Clever and Busse (1989, 1994) obtained the following representation for steady knot or bimodal convection: (i) almn cos(mSx x) cos(lSz z)G(i) (4.37a) φ= n (y) l,m,n

1 (i) clmn sin(mSx x) sin(lSz z) sin nπ y + 2 l,m,n (i) 1 = blmn cos(mSx x) cos(lSz z) sin nπ y + 2 ϕ=

(4.37b) (4.37c)

l,m,n

To study the stability of the steady solutions of the form given by Eqs (4.37), these authors superimposed inﬁnitesimal 3D disturbances expressed as (i) φ˜ = a˜ lmn exp[i(mSx + d)x + i(lSz + b)z + σ t]G(i) (4.38a) n (y) l,m,n

1 (i) c˜lmn exp[i(mSx + d)x + i(lSz + b)z + σ t] sin nπ y + 2 l,m,n (i) 1 ˜ = b˜lmn exp[i(mSx + d)x + i(lSz + b)z + σ t] sin nπ y + 2 ϕ˜ =

(4.38b) (4.38c)

l,m,n

(i) ˜ (i) (i) where b and d are the disturbance wavenumbers, a˜ lmn , blmn and c˜lmn are complex coefﬁcients and the summation indices l and m run through integers. The analysis was restricted to the case in which the Floquet wavenumbers b and d are equal to zero (according to some experimental evidence suggesting that the horizontal periodicity interval of the steady knot and bimodal solutions is not changed by ensuing oscillatory instabilities). By virtue of this restriction, the disturbances were separated into several subclasses (because of the initial symmetry of the steady solutions). In particular, three different symmetry properties were identiﬁed according to which the general manifold of disturbances [Eqs (4.38)] was partitioned into eight classes, as explained in detail below. (i) only for even l + m + n, Because the steady solutions exhibit non-vanishing coefﬁcients almn the disturbances can be separated into a class with even l + m + n and into a class with odd l + m + n. These two classes were referred to by Clever and Busse as ‘E’ and ‘O’, respectively. The classiﬁcation for two other symmetries follows from the symmetry of the function φ in z and x: the class of disturbances that is symmetric in z can be separated from the class that is antisymmetric provided the Floquet wavenumber b is equal to zero; the same property holds with respect to the x dependence. This led Clever and Busse (1989, 1994) to categorize disturbances as ECC, ECS, ESC, ESS, OCC, OCS, OSC and OSS, where the second letter indicates that the z dependence of φ˜ is described by cos(lSz z) (C) or by sin(lSz z) (S) and the third letter indicates the ˜ has corresponding property for the x dependence; moreover, as in the case of steady solutions, the same symmetries as φ˜ whereas ϕ˜ always has the opposite symmetries. For a given steady solution characterized by the parameters Pr, Ra, Sx and Sz , the equations for ˜ ϕ˜ and ˜ were solved by Busse and co-workers as an eigenvalue problem with the disturbances φ, the growth rate σ as eigenvalue. According to such results, there are two possible oscillatory instabilities of steady bimodal cells, one that does not change the spatial symmetry and is therefore referred to as ‘symmetric

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Figure 4.12 Stability boundaries as a function of Ra and Pr for steady bimodal convection with Sx = 2.5 and Sz = 5.4 (solid lines) or Sz = 4.5 (dashed lines) with respect to the transition to steady knot convection and with respect to the transition to oscillatory knot convection. Data after Clever and Busse (1994); Reproduced by permission of Cambridge University Press

oscillatory instability’, and the ‘wavy oscillatory instability’ that breaks the reﬂection symmetry of the superimposed small-wavelength rolls. The former is preferred at somewhat higher Pr than the latter that occurs primarily in the range 10 ≤ Pr ≤ 50. In the oscillatory symmetric regime, the secondary cross-rolls periodically contract towards and expand from the points of intersection with the primary rolls. This ‘breathing’ is associated with alternating hotter and colder (than average) liquid circulating around the convection rolls. The wavy oscillatory instability exhibits the additional feature that the up- and down-moving sections of the secondary cross-rolls shift their position back and forth in the direction along the axis of the primary rolls. This shift occurs in opposite directions in the upper and lower halves of the layer. With further increase in Ra, a subharmonic instability (Clever and Busse, 1994) leads to a destruction of the oscillatory bimodal cells and introduces a larger scale pattern. This type of convection is usually referred to as ‘spoke pattern convection’. Whereas the large-scale structure remains nearly steady, the hot and cold streamers in the respective thermal boundary layers are strongly time dependent (Busse, 1994). For moderate values of the Prandtl number, spoke convection appears in the form of oscillatory knot convection (Clever and Busse, 1989). Transitions from knot convection to bimodal convection and vice versa are also possible (see Figure 4.12) when the Rayleigh number is varied. Further information about spoke pattern convection is given in the following section.

4.5

Spoke Pattern Convection

The term ‘spoke’ pattern convection mentioned in the preceding section was originally introduced by Busse and Whitehead (1974) to describe the time-dependent form of convection that is observed

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Figure 4.13 Shadowgraph image [plane (x, z)] of spoke pattern convection in a layer of silicone oil (Pr = 63). Courtesy of F.H. Busse

in ﬂuid layers heated from below when the Rayleigh number exceeds a value of the order of 3 × 104 for Prandtl numbers above the order of unity. For such conditions, the thermal boundary layers at the rigid upper and lower boundaries of the ﬂuid layer are typically unstable and erupting sheets of hot and cold ﬂuid tend to move radially inwards towards central plumes which carry ﬂuid to the opposite boundary. In the visualization of convection with the shadowgraph method, radial ‘spokes’ together with the aforementioned plumes represent a characteristic feature of this type of convection (see Figure 4.13) from which its name is derived. The mechanism underlying this instability is of particular interest since, as mentioned at the end of Section 4.4, it represents a subharmonic response of oscillatory bimodal convection. A new wavenumber several times smaller than the basic wavenumber q2 in the same direction appears (most interestingly, this contradicts the intuitive notion that increasingly higher wavenumbers are introduced in the transition to turbulence). When the amplitude of oscillation of a particular cell is sufﬁciently large, it combines with the neighbouring cells on the same basic roll. In particular, a spoke structure is formed when the knots corresponding to the intersections of the bimodal cells with the basic rolls gravitate towards a common centre on the boundary of the basic rolls. The boundaries of the short-wavelength mode form the spokes while continuing to oscillate at roughly the same frequency. The process leading from bimodal to spoke pattern convection is shown in Figure 4.14. Interestingly, the organization of spokes in this type of convection leads to structures which could be categorized as fractals according to the deﬁnition provided for such geometric objects in Chapter 1 (see, in particular, Figure 4.14f). The appearance of spoke pattern convection changes, however, with varying Ra and Pr. The spokes tend to be nearly stationary at low Ra close to the onset of this type of ﬂow, but at Ra = O(106 ) they ﬂuctuate strongly in time and display a chaotic appearance. For Ra ≤ O(105 ), however, is spite of the small-scale turbulence that increases strongly with Ra, the convection pattern retains a large-scale nearly steady structure in which rising and descending plumes are arranged at well-deﬁned distances (the spoke pattern cells often exhibit the approximate symmetry of a square lattice in which upward and downward plumes alternate; assuming a square lattice, the wavelength of the pattern can be deﬁned as the mean distance between spoke centres of the same sign). In such regimes, the spacing of the spoke pattern is surprisingly regular even though individual spoke structures vary considerably and the oscillations of spokes tend to have a random phase. The form of the oscillations sometimes looks more like a ‘breathing blob’ and at other times more like a ‘waving sheet’.

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(a)

(b)

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Figure 4.14 Snapshots [plane (x, z)] showing irregular transition from bimodal convection to spoke pattern convection [Pr = 63, Ra = 2.4 × 105 , except for (a) which was taken at Ra = 2.1 × 105 ]. After Busse and Whitehead (1974); Reproduced by permission of Cambridge University Press

Although, as explained above, the large-scale wavelength of spoke pattern convection changes relatively little with Ra, however, as shown in Figure 4.15a and b for Pr = 7 and 15, respectively, there is a strong inﬂuence of Pr on such a wavelength (the wavelength decreases as Pr increases; Figure 4.16). Continuing with the description of these behaviours, it is worth noting that some of the characteristic features of spoke pattern convection, namely the eruption of thermal blobs from the boundary

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(a)

141

(b)

Figure 4.15 Temperature ﬁeld related to RB spoke pattern convection in the midplane [plane (x, z)] of a 20 × 1 × 20 layer for Ra = 5 × 104 and two different values of the Prandtl number: (a) Pr = 7; (b) Pr = 15 (numerical simulation, M. Lappa)

Figure 4.16 Wavelength of spoke pattern convection as a function of the Prandtl number. Courtesy of F.H. Busse

layer and their transport in the form of plumes to the opposite boundary, can already be seen in the aforementioned steady knot convection which arises as an instability of two-dimensional convection rolls (Section 4.2); hence the oscillatory knot convection (treated in Section 4.4 together with oscillatory bimodal convection) should be regarded as the initial stage of spoke-pattern convection at moderate Prandtl number in contradistinction to the high-Pr case where it arises via a subharmonic instability of oscillatory bimodal structures, as illustrated in the foregoing.

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4.6 Spiral Defect Chaos, Hexagons and Squares As an additional example of the astonishing complexity displayed by RB convection in layers, in 1993 a new chaotic state referred to as spiral-defect chaos (SDC) was discovered by Morris et al. (1993) for Pr ∼ = 1 in a parameter regime where on the basis of the theory for an inﬁnitely extended system (Section 4.2) parallel straight rolls [ideal straight rolls (ISRs)] should be stable (compare, e.g., Figure 4.17 with Figure 4.2; these were both obtained for Pr ∼ = 2, but show = 1 and Ra/Racr ∼ completely different patterns). Later, Morris et al. (1996) proved that SDC can be effectively made to exist in the same experimental conditions and setup as those displaying straight-roll states; as a consequence, this problem is now generally known as the ‘bistability’ problem [as it is featured by competition between spatiotemporal chaotic (SDC) and ﬁxed-point (ISR) attractors; Cakmur et al., 1997]. Remarkably, this type of chaos can be reproduced in amplitude equation model calculations (Cross, 1996, and references therein) and in numerical simulations (Pesch, 1996, and references therein; see, e.g., Figure 4.18). It is known that it generally occurs in ﬂuids with Pr = O(1) [see, e.g., the experiments of Liu and Ahlers (1996) for 0.3 ≤ Pr ≤ 1] and that the related onset moves to smaller values of Ra as Pr is decreased. Additional and more detailed information about this form of chaos occurring for slightly supercritical values of Ra will be provided in Section 4.12 after introducing the concept of ‘spiral’ in RB convection (cylindrical geometry) and discussing its possible dynamics (Section 4.11). Over recent years, other modes of convection representing a ‘deviation’ with respect to the standard occurrence of stable straight rolls have been detected and studied. Assenheimer and Steinberg (1996) were the ﬁrst to observe experimentally patterns consisting of domains of upﬂow hexagons coexisting with domains of downﬂow hexagons in RB convection. Hexagons were found to occur for 2.8 ≤ Pr ≤ 28 and surprisingly in a range where only rolls were known to be stable.

(a)

(b)

(c)

Figure 4.17 Snaphots [shadowgraph images, plane (x, z)] for SDC convection for Pr ∼ = 1 and three values of the Rayleigh number: (a) Ra/Racr ∼ = 2; (b) Ra/Racr ∼ = 4; (c) Ra/Racr ∼ = 6. Courtesy of B. Plapp and E. Bodenschatz

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Figure 4.18 Snapshot of spiral-defect chaos [temperature distribution in the midplane (x, z)] (Pr = 1, Ra = 4 × 103 , 20 × 1 × 20 domain with periodic boundary conditions along the frontier) (numerical simulation, M. Lappa)

The main feature of the observed hexagonal patterns was a wavelength signiﬁcantly larger than that of rolls at the same Rayleigh number (a wavenumber ratio of 1.2–1.3 between rolls and hexagons). Clever and Busse (1996) proved by numerical simulation that such hexagonal convection (see, e.g., Figure 4.19) can effectively become stable at Rayleigh numbers of about two times Racr if the Prandtl number exceeds a value of order unity (Pr > 1.2). Along these lines, Figures 4.20–4.22 show the regions in the (Ra,q) plane of stable hexagonal convection obtained by these researchers for Pr = 2.5, 7 and 16, respectively. Interestingly, according to such ﬁgures, some common features can be identiﬁed and discussed, as follows. Stable hexagons always have wavenumbers much lower than the critical value qc of ISR (∼3.12; Section 4.1). For Prandtl numbers less than about 10 (Figures 4.20 and 4.21), the region of stable

Figure 4.19 Lines of constant vertical velocity with solid (dashed) lines for positive (negative) values in the plane y = −0.4 for hexagonal convection (Pr = 4, q = 2 and Ra = 5 × 103 ). Courtesy of F.H. Busse

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Figure 4.20 Region of stable hexagons in the (Ra,q ) plane for Pr = 2.5; at low values of the Rayleigh number the stability region is bounded from below by the onset of two types of disturbances (solid and dashed lines). Courtesy of F.H. Busse

Figure 4.21 Region of stable hexagons in the (Ra,q ) plane for Pr = 7. Courtesy of F.H. Busse

hexagons corresponds to a strip in the (Ra,q) plane, the average wavenumber of which decreases with increase in the Rayleigh number. In all cases, stable hexagons do not exist for Ra < 3 × 103 . Furthermore, all instabilities involved in the stability boundaries are characterized by a vanishing imaginary part of the growth rate (which means that the emerging pattern is stationary). These results agree with the aforementioned experimental observations of Assenheimer and Steinberg (1996).

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Figure 4.22 Region of stable hexagons in the (Ra,q ) plane for Pr = 16. Courtesy of F.H. Busse

Interestingly, Busse and Clever (1998) showed that similar properties hold for convection ﬂows in the form of squares: they observed that asymmetric squares with rising or with descending motion in the centre (and descending or rising motion near the boundary) become stable at elevated Rayleigh numbers (Ra in excess of 3–4 times the critical value). It was found that such asymmetric squares differ from the dual hexagons essentially only in that the hexagonal boundary is replaced by a square boundary. In order to study the stability of this kind of convection, Busse and Clever (1998) superimposed on to steady solutions of the form given by Eqs (4.39) inﬁnitesimal disturbances of the general form given by Eqs (4.40): (i) almn cos(mqx) cos(lqz)G(i) (4.39a) φ= n (y) l,m,n

1 (i) clmn sin(mqx) sin(lqz) sin nπ y + 2 l,m,n (i) 1 = blmn cos(mqx) cos(lqz) sin nπ y + 2 ϕ=

(4.39b)

(4.39c)

l,m,n

φ˜ =

(i) a˜ lmn exp[i(mq + d)x + i(lq + b)z + σ t]G(i) n (y)

(4.40a)

l,m,n

1 (i) c˜lmn exp[i(mq + d)x + i(lq + b)z + σ t] sin nπ y + 2 l,m,n (i) 1 ˜ = b˜lmn exp[i(mq + d)x + i(lq + b)z + σ t] sin nπ y + 2

ϕ˜ =

(4.40b)

(4.40c)

l,m,n

Symmetric solutions were separated from asymmetric ones taking into account that among the solutions described by the representation in Eq. (4.39), those with a square symmetry are characterized

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Figure 4.23 Region of stable asymmetric squares in the (Ra,q ) space for Pr = 7; such a region is bounded by a singly subharmonic instability from below and by a doubly subharmonic instability from above (solid lines); the boundary towards high q (solid line) indicates the transition to rolls; also shown are the boundaries (dashed lines) beyond which disturbances are growing that do not change the horizontal periodicity of asymmetric square convection. Courtesy of F.H. Busse

by the property that the coefﬁcients are symmetric in the subscripts l and m: almn = blmn = 0 for l = m = n = odd clmn = 0 for l = m = n = even

(4.41a) (4.41b)

This type of convection was found to be generally unstable. In contrast, convection in the form of asymmetric squares was observed to exist as up squares or as down squares (corresponding to opposite signs of the coefﬁcients almn with odd l + m + n, whereas those with even l + m + n have the same values). The related stability results in the (Ra,q) plane are shown in Figures 4.23 and 4.24, where two kinds of stability boundaries are displayed. The outer stability boundaries (dashed lines) correspond to disturbances with d = b = 0 (disturbances that do not change the periodicity interval of the steady solution). Figure 4.23 clearly demonstrates that there exists a region of Rayleigh numbers extending from above 6 × 103 to about 2.7 × 104 where steady asymmetric square convection is stable with respect to general inﬁnitesimal disturbances. Like hexagonal convection discussed earlier, the wavenumber of the stable square cells is considerably smaller than the critical value qc and tends to decrease with increase in Rayleigh number. It is also worth noting that the most strongly growing disturbances outside the stability region are subharmonic disturbances. Towards lower values of Ra, the strongest growing disturbances are usually subharmonic in one dimension, that is, they correspond to d = q/2, b = 0 or b = q/2, d = 0 for reasons of symmetry. Towards higher values of Ra, doubly subharmonic disturbances with d = b = q/2 are the most critical ones. The boundary towards high wavenumbers corresponds to disturbances with d = b = 0. These disturbances tend to provide the transition to rolls in one of the two directions deﬁned by the squares.

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Figure 4.24 Region of stable asymmetric squares in the (Ra,q ) space for Pr = 16; such a region is bounded by a singly subharmonic instability from below and by a doubly subharmonic instability from above (solid lines); the boundary towards high q (solid line) indicates the transition to rolls; also shown are the boundaries (dashed lines) beyond which disturbances are growing that do not change the horizontal periodicity of asymmetric square convection. Courtesy of F.H. Busse

Results analogous to those found in the case Pr = 7 apply to Pr = 16 (as shown in Figure 4.24). Again, a range of stable asymmetric square convection exists for Rayleigh numbers above 6 × 103 (Busse and Clever, 1998). According to their results, nevertheless, when the Prandtl number is decreased to 2.5 a range of stable asymmetric square convection can be obtained only when disturbances are restricted to those with d = b = 0. With respect to the latter type of disturbances, the stability boundaries resemble the corresponding ones in Figures 4.23 and 4.24, but without the restriction d = b = 0 growing disturbances occur in all parts of the relevant domain in the (Ra,q) plane. Most of these theoretical results have been conﬁrmed by effective laboratory experiments. As an example, an experimental observation indicating a tendency towards asymmetric square cells (in addition to hexagonal cells) is shown in Figure 4.25. Interestingly, since cells with rising motion in the centre and those with descending motion have the same stability properties, domains with either types of cells tend to be realized in effective experiments (as also demonstrated by Assenheimer and Steinberg, 1996). At this stage, it is worth stressing that in these studies (Clever and Busse, 1996; Busse and Clever, 1998), the ﬂuid properties and the boundary conditions were assumed to be symmetric about the midplane of the convective layer. Usually, in fact, hexagons and asymmetric squares are not observed in such conditions. Rather, they frequently occur in convection lacking up–down reﬂection symmetry (Palm, 1960; Golubitsky et al., 1984), namely in ﬂuids with strongly temperature-dependent viscosity or in atmospheric convection (see, e.g., the discussion about the mesoscale shallow cellular convection given in Section 3.4.2). Interestingly, under the same assumption of reﬂectional symmetry considered by Busse and Clever (1998), Demircan and Seehafer (2001) observed (numerical simulations for Pr = 6.8) that when stress-free conditions are considered at the horizontal boundaries in place of no-slip ones the square patterns appear in RB convection via the SV instability of rolls. In particular, they

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Figure 4.25 Snaphot of convection [shadowgraph image, plane (x, z)] in a layer of methyl alcohol [Pr = 7 at about 23◦ C, d = 5.6 mm, Ra ∼ = 4.5 × 104 , dark (white) lines indicate rising (descending) motions]. Courtesy of F.H. Busse

are determined by modes with two different wavenumbers that are simultaneously excited and the nonlinear interaction between them. As the buoyancy forces increase, these interacting modes give rise to bifurcations leading to periodic alternation between a nonequilateral hexagonal pattern and a square pattern or to different kinds of standing oscillations. One of them, shown in Figure 4.26, appears like the superposition of two standing waves, one with oscillations in the x direction and the other with oscillations in the z direction, the vertex points of the square pattern being located at the nodes of the standing waves. A further kind of oscillation can result from a single Hopf bifurcation. It consists of a periodic alternation between two nonequilateral hexagonal patterns, seen in Figure 4.27 (Demircan and Seehafer, 2001). These are at right-angles to each other and during the transition between them the square pattern is passed through. It is worth stressing that this oscillatory behaviour of dynamic side swapping in square convection was also observed in experiments done for Marangoni–B´enard convection. Ondarcuhu et al. (1993), for example, reported for a small square vessel (L/d = 4.46) a sequence of qualitative changes in a convection pattern consisting of four square cells determined by the dynamics of one vertex. Krmpotic et al. (1996) described such behaviour as the nonlinear interaction between different critical modes whose linear superposition generates the experimentally observed pattern. Transformation from a hexagonal pattern into a square pattern via a merging of cell knots was also observed by Eckert et al. (1998).

Thermogravitational Convection: The Rayleigh–B´enard Problem

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Figure 4.26 Convection resulting from the superposition of two standing waves, one with oscillations in the x direction and the other with oscillations in the z direction, the vertex points of the square pattern being located at the nodes of the standing waves (Pr = 6.8, layer with nondimensional periodic length L/d = 4.1, Ra = 8 × 103 ). Snapshots of the vertical velocity component in the horizontal midplane (x, z) are taken at t = 0 (a), t = τ/4 (b), t = τ/2 (c) and t = 3τ/4 (d), where τ is the time period. After Demircan and Seehafer (2001); Reproduced by permission of IOP

Further details about the emergence of square and hexagonal patterns in Marangoni–B´enard convection are provided in Chapter 9, which is entirely devoted to these aspects.

4.7

Convection with Lateral Walls

Although ideal periodic patterns can be created in the experiments when the horizontal extent is very large compared with the vertical thickness, convective patterns, particularly when they form in the presence of lateral walls, typically are characterized by local roll patches with grain boundaries and point defects. Furthermore, when the ratio of the vertical to the horizontal extension is of O(1) or even O(10−1 ), in general, the lateral walls tend to have a signiﬁcant inﬂuence on the overall structure of the ﬂow pattern that develops when the Rayleigh number exceeds its critical value (Greenside and Coughran, 1984; Chana and Daniels, 1989).

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Figure 4.27 Convection resulting from the periodic alternation between two nonequilateral hexagonal patterns (Pr = 6.8, layer with nondimensional periodic length L/d = 4.5, Ra = 5.8 × 103 ). Snapshots of the vertical velocity component in the horizontal midplane (x, z) are taken at t = 0 (a), t = τ/4 (b), t = τ/2 (c) and t = 3τ/4 (d), where τ is the time period. After Demircan and Seehafer (2001); Reproduced by permission of IOP

In the theory of pattern formation in continuous systems, it is known that even distant boundaries can have surprisingly important effects. This is because the boundaries break continuous symmetries (translations/rotations) present in the unbounded system. Thus results for the corresponding inﬁnite layer cannot, in general, be used to make predictions about either the detailed structure or the stability of the roll pattern in these situations (Catton, 1972). The effective properties of convection in a container of ﬁnite size depend on the relative ratios of the related extensions along the different spatial directions (length, height and width). In the following, the vital role played by the presence of limiting lateral walls and effective system geometry is ﬁrst discussed for the idealized case of RB convection under the constraint of two-dimensional ﬂow, that is, no velocity component along the third direction z, which is assumed to be inﬁnite (this hypothesis might be regarded as a reliable assumption for Ra not too far from the critical threshold since, as illustrated in Section 4.1, the disturbances responsible for onset of convection from the quiescent state are two-dimensional in nature) and then for the more realistic case of three-dimensional geometric conﬁgurations.

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4.8

151

Two-dimensional Models

4.8.1 Distinct Modes of Convection and Possible Symmetries For the two-dimensional case, it has been clearly illustrated by many investigators that there are several modes of the most dangerous perturbation that replace each other when the aspect ratio A (ratio of the horizontal extension and of the height) is varied. Moreover, a new variable enters the dynamics with respect to the idealized case of inﬁnite horizontal extension: the type of thermal boundary condition along the lateral walls (conducting or adiabatic sidewalls). Velte (1964) was the ﬁrst to consider perfectly conducting side walls and to determine the critical Rayleigh number in the range 0.7 ≤ A ≤ 7 (Racr = 13 380, 5030, 3510, 3090 and 2350 for A = 2/3, 1, 1.3, 1.5 and 2, respectively). The case with adiabatic lateral walls was considered a year later by Kurzweg (1965). For such a case, in particular, Luijkx and Platten (1981) found A = 1, Racr ∼ = 2585, A = 2, Racr ∼ = 2013, A = 3, Racr ∼ = 1870, A = 4, Racr ∼ = 1810, A = 5, Racr ∼ = 1779 and A → ∞, Ra → 1707 (for no-slip horizontal boundaries). A rigorous categorization of solutions in terms of the related symmetries can be found in Mizushima (1995), who considered a 2D box with aspect ratio in the range 0.1 ≤ A ≤ 10 and investigated both types of lateral boundary conditions mentioned before (see also Chana and Daniels, 1989). In general, the distinct modes of convection can be delineated by considering various combinations of the possible symmetries along the horizontal and vertical directions. This leads to partition the set of possible modes into four cases, as illustrated in Figure 4.28:

• (aa): The antisymmetric–antisymmetric mode. This mode has an odd number of vortex cells along both the horizontal and the vertical directions.

• (sa): The symmetric–antisymmetric mode. This mode is characterized by an even number of rolls along the horizontal direction and an odd number of vortices along the y direction.

• (as): The antisymmetric–symmetric mode. This mode exhibits an odd number of rolls along x and an even number of cells in the perpendicular direction.

• (ss): The symmetric–symmetric mode. This mode has an even number of vortex cells along both the horizontal and the vertical directions. The corresponding neutral curves for the case of adiabatic and conducting lateral walls are shown in Figures 4.29 and 4.30, respectively. According to these ﬁgures, the critical Rayleigh number is given alternatively by the (aa) and the (sa) modes (the related neutral curves, in fact, intersect many times), whereas the curves for the (as) and (ss) modes are located signiﬁcantly above. Within this context, it should also be mentioned that the decrease in the aspect ratio A generally results in an increase in the critical threshold for the onset of convection (Mizushima, 1995; Gelfgat, 1999a).

Figure 4.28 Categorization of possible solutions of RB convection in 2D ﬁnite enclosures in terms of related symmetries

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(a)

(b)

Figure 4.29 Neutral Rayleigh number as a function of the enclosure aspect ratio in the case of adiabatic sidewalls (the numbers adjacent to the curves indicate the number of vortices): (a) 1/10 < A < 10; (b) 5 < A < 10. Data after Mizushima (1995); Reproduced by permission of the Physical Society of Japan

It is also evident that the stability characteristics for the two cases of adiabatic or conducting sidewalls are rather similar, the differences being limited to a more evident gap between the (aa) and (sa) neutral curves in the adiabatic case. The most interesting outcome of the neutral curves, perhaps, is that the number of rolls in the x direction increases by one as A increases by one; this means that the scale of the vortices in horizontal direction is approximately one (see Figure 4.31), as in the case of the inﬁnitely extended ﬂuid layer discussed in Section 4.1.

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(a)

(b)

Figure 4.30 Neutral Rayleigh number as a function of the enclosure aspect ratio in the case of conducting sidewalls (the numbers adjacent to the curves indicate the number of vortices): (a) 1/10 < A < 10; (b) 5 < A < 10. Data after Mizushima (1995); Reproduced by permission of the Physical Society of Japan

Another interesting feature highlighted by Figures 4.29 and 4.30 is that the critical Rayleigh number increases by almost two order of magnitudes when A is reduced from 1 to 0.1. When these tall enclosures are considered, in general, a single vertically elongated cell appears at the onset. It is known, however, that the number of cells increases along the vertical direction for higher values of Ra, as shown in Figure 4.32. Mizushima (1995) also determined the asymptotic form of the critical Rayleigh number in both limits A → 0 and A → ∞ (he evaluated the coefﬁcients and the exponents for such relationships

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(a)

(b)

Figure 4.31 Two-dimensional Rayleigh–B´enard convection [plane (x, y )] in rectangular containers (Pr = 0.01, Ra = 3 × 103 ; container heated from below and cooled from above with adiabatic sidewalls) for two values of the aspect ratio: (a) A = 3; (b) A = 4 (Ra based on the depth of the box) (numerical simulations, M.Lappa)

(a)

(b)

(c)

Figure 4.32 Two-dimensional steady Rayleigh–B´enard convection [plane (x, y )] in a box with A = 1/4 (Pr = 0.01; box heated from below and cooled from above with adiabatic side walls) for three values of the Rayleigh number: (a) Ra = 2 × 105 ; (b) Ra = 3 × 105 ; (c) Ra = 4 × 105 (Ra based on the box height) (numerical simulations, M. Lappa)

from the data obtained for the ranges 0.01 ≤ A ≤ 0.05 and 7 ≤ A ≤ 10, respectively). They are as follows: Adiabatic case: Racr = 506 A−4 Racr − Racr∞ = 1781 A−1.8902

as A → 0 as A → ∞

(4.42a) (4.42b)

Racr = 1567 A−4

as A → 0

(4.43a)

Conducting case:

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Racr − Racr∞ = 2778 A−2.0367

as A → ∞

155

(4.43b)

where Racr∞ = 1707 is the critical Rayleigh number for the corresponding inﬁnite layer (Section 4.1).

4.8.2 Higher Modes of Convection and Oscillatory Regimes Like idealized systems of inﬁnite horizontal extension, also for laterally bounded conﬁgurations the thermoﬂuid–dynamic ﬁeld tends to display an increasing degree of complexity as the Rayleigh number is increased. As an example, in the speciﬁc case of relatively extended conﬁgurations (where the horizontal extent is much larger than the depth), several studies (Cross et al., 1983; Daniels, 1984) elucidated that for ﬁxed values of A and Rayleigh numbers in the range Ra = Racr∞ + O(2/A), there exists a class of ﬁnite-amplitude, steady-state, two-dimensional ‘phase-winding’ solutions that correspond physically to the possibility of an adjustment in the number of rolls in the container as the local value of Rayleigh number is varied; the rolls simply undergo a wavenumber adjustment without breaking the two-dimensionality by a simple expansion process (sometimes referred to as roll relaxation, i.e. a 2D process by which the convection rolls increase in wavelength to values larger than the value at threshold; Paul and Catton, 2004). Interesting and rich behaviours are also possible in enclosures with A = O(1). When the Rayleigh number is increased beyond a certain critical threshold, it is known that even under the nonphysical constraint of two-dimensional ﬂow, RB convection can undergo transition to relatively complex and/or time-dependent regimes. Mizushima and Adachi (1997) proved clearly (in the case of conducting boundaries and Pr = 7) that the initial modes with the (aa) or (sa) symmetries can produce modes with different symmetries via a nonlinear interaction mechanism when the Rayleigh number is sufﬁciently increased. If the initial mode has (aa) symmetry, in particular, the (ss) mode is excited; the thermal convection arising in such a case after the ﬂow bifurcation can be expressed as a linear combination of these two modes. If the initial mode has (sa) symmetry, the aforementioned nonlinear interaction can produce all four possible modes [(aa), (sa), (as) and (ss)]; in such a case, the ﬂow exhibits no symmetry. For instance, Figures 4.33 and 4.34 show some possible steady and oscillatory 2D regimes for Pr = 15 and adiabatic lateral walls which occur in the simplest case of a square cavity.

(a)

(b)

Figure 4.33 Steady patterns [temperature distribution and velocity ﬁeld in the plane (x, y )] of 2D Rayleigh–B´enard convection (Pr = 15, A = 1; adiabatic lateral walls) for two values of the Rayleigh number: (a) Ra = 1 × 104 (single-cell ﬂow); (b) Ra = 1 × 105 (bicellular ﬂow) (numerical simulations, M. Lappa)

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(a)

(b)

(c)

(d)

Figure 4.34 Snapshots of temperature distribution and related convective cells [plane (x, y )] for 2D oscillatory Rayleigh–B´enard convection (Pr = 15, A = 1, Ra = 5 × 105 ): the pattern is shown at t = 0 (a), t = τ/3 (b), t = τ/2 (c) and t = 2τ/3 (d), where τ is the time period of the oscillatory phenomenon. The ﬁeld is featured by a recurrent appearance of one- and four-cell ﬂows (numerical simulation, M. Lappa)

For Ra = 5 × 105 (Figure 4.34), in particular, the ﬂow is oscillatory. One- and four-cell ﬂows appear cyclically. In Figure 4.34a, the main structure has one diagonal clockwise-oriented cell, in Figure 4.34b it has four cells, then breaks symmetry again to form one main diagonal counterclockwise-oriented cell. In practice, due to the symmetry/antisymmetry of the governing equations and of the boundary conditions, different solutions appear which can be obtained by reﬂection about the vertical cavity centreline (parallel to the applied temperature gradient), about the horizontal cavity centreline (perpendicular to the gradient) and about both of them. A number of other 2D numerical results are available in the literature for time-dependent RB convection. As an additional example for the case of the square cavity, Goldhirsch et al. (1989) investigated the problems of dynamic onset of convection, textural transitions and chaotic dynamics using well-resolved, pseudo-spectral simulations. Complicated ﬂow patterns and textural transitions were observed in both nonchaotic and chaotic ﬂow regimes together with multistability. According to this study, in particular, intermediate Prandtl number ﬂuids (e.g. Pr = 0.71) display a quasi-periodic time dependence up to Rayleigh numbers of order 106 , whereas when the Prandtl number is raised to 6.8, aperiodic (chaotic) ﬂows of non-integer dimension may occur (see Section 1.8 for some necessary concepts about the meaning of ﬂows with a non-integer dimension). In this case, roll merging and separation are an important feature of the dynamics; in some cases, corner rolls

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migrate into the interior of the cell and grow into regular rolls; moreover, the large rolls may shrink and retreat into corners. These results basically proved that there are many and complicated routes for two-dimensional convection to attain its equilibrium or chaotic states. Fluids with Pr = O(1) were also considered by Yahata (1986), who studied the possible instability mechanisms for Pr = 0.2 and 0.5, and Kenjereˇs and Hanjali´c (2000), who investigated shallow cavities with 4 ≤ A ≤ 32 up to the onset of chaos. Wilson and Rydin (1990) used a nodal integral method to investigate bifurcation phenomena for aspect ratios in the range 1 ≤ A ≤ 9. For the case of liquid metals, it is worth mentioning Ozoe and Hara (1995), who illustrated the possible oscillatory behaviours for 10−3 ≤ Pr ≤ 10−1 and a ﬁxed aspect ratio (A = 4), and Crunkleton et al. (2006), who analysed the transition from steady to oscillatory ﬂow for a very low Prandtl number ﬂuid (Pr = 0.008) and aspect ratios of 0.25, 0.4, 1.0 and 2.0 (transitions were found to occur at Rayleigh numbers of 2.5 × 105 , 1.3 × 105 , 8.35 × 104 and 3 × 104 , respectively); the structures of the oscillations were graphically depicted as corner cells which dissipate into centred cells and then into opposite corner cells (these authors also detected a secondary ﬂow transition for a geometry with an aspect ratio of 1.0 at Ra/Racr2 = 1.2). Finally, for the case of tall enclosures (A < 1), in a numerical study Cappelli d’Orazio et al. (2004) treated rectangular enclosures (ﬁlled with air) with aspect ratio 1/6 ≤ A ≤ 1/2; interestingly, it was found that for Pr = O(1), after the departure from the motionless conduction state, the following ﬂow-pattern evolution with a columnar arrangement of cells takes place: one-cell steady → two-cell steady → two-cell periodic → one-to-three-cell periodic → three-cell periodic.

4.9

Three-dimensional Parallelepipedic Enclosures: Classiﬁcation of Solutions and Possible Symmetries

For real three-dimensional shallow enclosures the ﬂow can be quasi-two-dimensional (near the onset), but it is also known that an increase in Ra can cause a loss of stability of initial two-dimensional ﬂow patterns, which are replaced by fully-three-dimensional ﬂow (that develops peculiar spatiotemporal dynamics as Ra increases). The process by which this occurs is complicated (possibly involving the effects of multiple incommensurate selection mechanisms). As shown by many authors (e.g. experiments by Krishnamurti, 1970a; Busse and Whitehead, 1971; Kolodner et al., 1986; numerical simulations by Stella et al., 1993), there are, in fact, several possible instability mechanisms, such as the (aforementioned) cross-roll ﬂow, bimodal convection and ‘soft-roll’, according to the Prandtl number and the wavenumber. Under the same conditions, the thermal-convection equations may have more than one solution. According to relevant information available in the literature, the several solutions can be categorized as follows: 1. Quasi two-dimensional patterns in the form of rectilinear rolls consisting of (a) n transverse rolls (nT regime, n being an integer), which are orthogonal to the longest side of the considered parallelepipedic box (see, e.g., the landmark linear stability analysis of Davis, 1967, the theoretical study of Segel, 1969, the experiments of Gollub and Benson, 1980, and Mukutmoni and Yang, 1993a, and the numerical investigation of Stella and Bucchignani, 1999); (b) n longitudinal rolls (nL regime), which are parallel to the longest side wall of the box. For case (a), in particular, it is worth discussing brieﬂy the interesting insights that were initially provided by Davis (1967). He treated the stability problem for the onset of three-dimensional convection in a rectangular

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parallelepiped cavity heated from below for the case of perfectly conducting sidewalls and the assumption of two-dimensional ﬁnite rolls (that means only two non-zero velocity components which are dependent on all three spatial variables). He evaluated the critical Rayleigh number for the onset of convection for 1/4 ≤ A ≤ 6 and concluded that ﬁnite rolls with axes parallel to the shorter side are the preferred mode of convection, which leads to the interesting conclusion that the assumption of two-dimensional ﬂow should be regarded as a good approximation of very short enclosures rather than of long ones. 2. Fully three-dimensional ﬂow caused by (a) bimodal convection, consisting (as explained earlier for the inﬁnite layer, in Section 4.2) of a base ﬂow superimposed with cross-rolls of approximately the same strength as the base ﬂow (these rolls emerge at right-angles to the original rolls; see, for example, the numerical investigation of Edwards, 1988); (b) distortion of the original rolls into an L shape (this conﬁguration, called soft-roll as it is given by the superposition of two rolls whose axes are not parallel, allows a continuous transition between different wavenumber ﬂow patterns; it has been observed experimentally by Kolodner et al., 1986, and numerically by Stella et al., 1993, and Lappa, 2007a, in relatively shallow enclosures). It is also worth citing the numerical investigation (ﬁnite element method) of Tang and Tsang (1997), who found several physical phenomena, such as multicellular ﬂow pattern, oscillatory transient solution, ‘T-shaped’ rolls at the ends of a rectangular box (see also Mukutmoni and Yang, 1995a) and roll alignment. In general, the pattern selection process depends strongly on the effective system extension in the 3D space. As a relevant and interesting example of the sensitivity that the ﬁnal state of the system can exhibit to the presence of limiting sidewalls along the third direction (z), the reader may compare the ﬁelds illustrated in Figure 4.31 for the case of idealized 2D computations with corresponding 3D results shown in Figure 4.35 for a 3 × 1 × 3 enclosure (Lappa, 2005a).

(a)

(b)

(c)

Figure 4.35 Steady state of Rayleigh–B´enard convection in a 3 × 1 × 3 enclosure (Pr = 0.01, Ra = 3.5 × 103 ): (a) velocity ﬁeld in the (x, y ) midplane; (b) vertical velocity contours in the horizontal midplane (x, z); (c) temperature ﬁeld in the horizontal midplane (x, z). After an initial stage, the ensuing steady 3D pattern is featured by the presence of parallel convection rolls with a diagonal prevailing direction. After Lappa, 2005a

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Figures 4.35 clearly proves that even if the z extension comparable to that in the x direction, the ﬂow may exhibit a fully 3D structure. In general, for the 3D geometry, similarly to the 2D idealized case, there are several modes of the most dangerous perturbation that replace each other when the aspect ratio is varied (see, e.g., the numerical simulations of Yahata, 1989; Mukutmoni and Yang, 1995a; Mizushima and Nakamura, 2003). However, the spectrum of possible perturbations is more complicated in the 3D case. The larger variety of perturbations is obviously caused by the 3D geometry; see, for example, the interesting parametric study of the RB instability in 3D boxes carried out by Gelfgat (1999a) and the linear stability analysis of Mizushima and Nakamura (2003). The latter authors, in particular, evaluated the critical Rayleigh numbers for parallelepiped cavities with a square vertical cross-section (Lz /height = 1) and several possible aspect ratios Ax (deﬁned as Lx /height) showing the possible alternance of disturbances with different symmetries. The divergence-free velocity ﬁeld was expressed as V = ∇ ∧ (ψ (1) i x ) + ∇ ∧ (ψ (2) i z )

(4.44)

where i x and i z are the unit vectors along the x and z directions, respectively. Using the same rescaling deﬁned in Section 4.2.1, the linear disturbance equations for ψ (1) , ψ (2) and can be written in nondimensional form as 2 ∂2 ∂2 ∂ ∂ 2 (1) ∇ + ψ − ∇ 2 ψ (2) − =0 (4.45a) ∂y 2 ∂z2 ∂x∂z ∂z ∂2 ∂ ∂2 ∂2 + 2 ∇ 2 ψ (2) − ∇ 2 ψ (1) − =0 (4.45b) 2 ∂x∂z ∂x ∂y ∂x ∂ψ (1) ∂ψ (2) =0 (4.45c) ∇ 2 + Ra − + ∂z ∂x The following four disturbance modes were considered by Mizushima and Nakamura (2003): Mode Mode Mode Mode

1: 2: 3: 4:

[ψ (1) (o,e,o), ψ (2) (e,e,e), (o,e,e)] [ψ (1) (e,e,o), ψ (2) (o,e,e), (e,e,e)] [ψ (1) (o,e,e), ψ (2) (e,e,o), (o,e,o)] [ψ (1) (e,e,e), ψ (2) (o,e,o), (e,e,o)]

(4.46a) (4.46b) (4.46c) (4.46d)

where ψ (1) (o,e,o) indicates that ψ (1) is and odd function of x and z and an even function of y. For the convenience of the reader, a description of the ﬂow structure corresponding to these modes is given below. The ﬂow ﬁeld of mode 1 (mode 4) appears like a roll with the axis parallel to the z direction (x direction), which seems to be an almost two-dimensional ﬂow in the xy plane (yz plane) although there is a small velocity component in the z direction (x direction). The ﬂuid particles make outward spiral motions in the xy plane with their centres moving outward in the z direction towards each sidewall. The ﬂuid particles in the ﬂow ﬁeld pertaining to mode 2 rise up at the central region and sink down near the walls forming toroidal surfaces. The ﬂow ﬁeld related to mode 2 is almost axisymmetric around the y-axis. Hence one can observe a pair of vortices at any vertical cross-section through the axis of symmetry. The ﬂow ﬁeld corresponding to mode 3 is symmetric along the diagonal cross-sections x = z and x = −z and looks like two pairs of circular cones connected at their apexes at the centre of the cavity, one pair of which has an axis in the x direction and the other in the z direction. The circular ﬂuid motions on the two cones with the same axis have opposite rotation directions. Each ﬂuid particle initially placed near the central region makes an outward spiral motion in the xy or yz plane with its centre moving in the z or x direction towards each sidewall. In other words, the

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Figure 4.36 Critical Rayleigh number for the onset of convection in a rectangular parallelepiped cavity with a square vertical cross-section and adiabatic vertical walls (Ra based on the cavity height). Data after Mizushima and Nakamura (2003); Reproduced by permission of the Physical Society of Japan

ﬂuid particles take almost circular paths near the sidewalls and the directions of rotation of the circular motion near two adjacent sidewalls are opposite to each other. The magnitude of the ﬂuid velocity is very small near the central region of the cavity and large near the sidewalls. The corresponding values of the neutral Rayleigh number are plotted against the aspect ratio Ax in Figure 4.36. The smallest neutral Rayleigh number of the four modes gives the critical Rayleigh number for each aspect ratio Ax . It can be seen that mode 4 gives the critical Rayleigh number for Ax < 1 and 1.3 < Ax < 1.8. For Ax > 1.8, modes 1 and 2 give the critical Rayleigh number alternately, which means that an odd or even number of rolls should be observed alternately on the xy cross-section at onset as the aspect ratio Ax is increased above 1.8. In practice, the curve of the critical value of the Rayleigh number has a wavy pattern due to the switching of the critical modes, which is similar to the behaviour already discussed in Section 4.8.1 for the case of two-dimensional convection in a rectangular cavity. When the Rayleigh number is increased to higher values, more complex behaviours arise, as already illustrated in Section 4.8.2 for two-dimensional models.

4.9.1 The Cubical Box As a very relevant (paradigm) example of the problem complexity, the reader may consider, in particular, the case of a cubical box. In fact, in such a conﬁguration, a variety of symmetries are possible. Remarkably, seven distinct branches of possible supercritical states for the RB convection in a cube were reported by Pallar`es et al. (1999). Because of the intrinsic symmetry of the cubical geometry, this natural convection ﬂow tends to produce state solutions with manifold symmetries. As the Rayleigh number is increased, new possible patterns of symmetry are prone to occur. Furthermore, the delicate evolutionary route to

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a ﬁnal state can be coupled to signiﬁcant and intriguing adjustments in the roll pattern inside the domain; this aspect, (often the subject of controversy in the literature) also forms a relevant and important part of the problem. According to the studies of Mizushima and Nakamura (2003) and Puigjaner et al. (2004), for a cubical box with adiabatic vertical walls, the critical Rayleigh number is Racr = 3389; such a large value compared with the corresponding two-dimensional case Racr ∼ = 2585 obviously follows from the presence of solid walls along the third dimension and related stabilization inﬂuence, such an effect being absent in 2D models. Similarly, for vertical conducting walls, Mizushima and Matsuda (1997) found Racr = 6796, whereas the corresponding 2D value is 5030 (the critical Marangoni numbers for conducting side walls are always larger than for adiabatic ones; this is due to the fact that a temperature perturbation arriving at an insulating boundary is reﬂected towards the bulk of the ﬂuid, whereas it is dissipated in the walls when these are conducting; for this reason, adiabatic side walls give rise to less stable systems). These critical values correspond to a double eigenvalue. In practice, this indicates that any linear combination of the two eigenvectors, that is, of one x roll and one z roll, is possible in principle (the curves related to the modes 1 and 4 in Figure 4.36 intersect for Ax = 1, which means that both modes are equally likely to occur in a cubical box). Nevertheless, among all the combinations, Puigjaner et al. (2004) found that only a stable single x or z roll is possible whereas, unlike the case already discussed for the 3 × 1 × 3 container (Figure 4.35), the diagonal x ± z rolls tend to be unstable, that is, they may only appear as a transitional stage of evolution. Some examples of the possible patterns of symmetry and their ranges of existence in the case of silicon melt in a cubical box with adiabatic lateral walls are shown in Figure 4.37 (obtained as bifurcations from an initial diffusive state; Lappa, 2005a). Following Puigjaner et al. (2004), they can be categorized as Si states with the subscript i = 1, 2, . . . , n. A possible ﬂow pattern (S1 shown in Figure 4.37a), that occurs in a relatively wide range of the Rayleigh number (Racr < Ra < 6 × 104 ), is formed by one x roll. The S2 state shown in Figure 4.37b appears as a transitional solution along the evolutionary process that leads to S1 . It occurs in a limited range of the Rayleigh number. This S2 solution can be essentially obtained by subtracting one x roll from one z roll. In a similar way, a symmetric solution can be found by adding them. For large values of Ra, the transitional S2 state is no longer possible and is replaced by a solution with a different symmetry pattern (S4 ). The S4 solution has a toroidal conﬁguration and was also observed in the numerical simulations of Hern`andez and Frederick (1994) for Pr = 0.71 and in the experiments of Pallar`es et al. (2001) as a transitional convective state for Pr = 130. It can be essentially regarded as the sum of two x rolls and two z rolls with no change of rotation along their axes (see Figure 4.37c). At very large values of the Rayleigh number (Ra > 7 × 104 ) a new pattern of symmetry can arise. This S8 structure (Figure 4.37d) consists of two asymmetric counter-rotating rolls aligned along one of the x = ±z diagonals (the S8 solution bears a certain similarity to the S2 pattern illustrated in Figure 4.37b).

4.9.2 The Onset of Time Dependence Convection in the cubical geometry becomes oscillatory for sufﬁciently high values of Ra (this occurs for Ra > 104 and Ra > 1.2 × 105 for the Pr = 0.01 and Pr = 0.71 cases, respectively; see Lappa, 2005a, and Puigjaner et al., 2004, respectively). The ﬁnal state of the system is represented by an oscillating S1 pattern.

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(a)

(b)

(c)

(d)

Figure 4.37 Contours of vertical velocity in the horizontal midplane (x, z) of a cubical enclosure heated from below with adiabatic vertical walls for Pr = 0.01: (a) pattern with symmetry S1 (Ra = 5 × 103 , steady state); (b) pattern with symmetry S2 (Ra = 9 × 103 , transitional state during the evolution that leads to S1 ); (c) pattern with symmetry S4 (Ra = 7 × 104 , transitional state); (d) pattern with symmetry S8 (Ra = 8 × 104 ) (numerical simulation, M. Lappa)

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Similar results for higher values of the Prandtl number were reported by Puigjaner et al. (2006). They determined the bifurcation diagram of steady convective ﬂow patterns inside a cubical cavity with adiabatic lateral walls heated from below and ﬁlled with Pr = 130 silicone oil for values of the Rayleigh number up to 1.5 × 105 . In particular, a continuation procedure based on the Galerkin spectral method was used to determine the steady convective solutions as a function of Ra. Bifurcations leading to either new steady or time-dependent solutions were identiﬁed and new steady solution branches were also continued. Interestingly, a total of 15 steady solutions were tracked and the stability analysis predicted that six ﬂow patterns are stable and that two, three or even four of these patterns can coexist over certain ranges of Ra in the studied domain. In general, the Rayleigh number for the onset of time-dependent ﬂow in parallelepipedic boxes depends strongly on both the aspect ratio of the enclosure and the Prandtl number (see the experiments by Gollub and Benson, 1980, for 2.5 ≤ Pr ≤ 5 and Mukutmoni and Yang, 1993a,b, 1995b, for Pr = 5). Whereas, as illustrated earlier, the ﬁrst critical Rayleigh number (Racr ) depends only on the box geometry, the critical value for the appearance of the time-dependent behaviour is also strongly dependent on the Prandtl number (as pointed out by Busse and co-workers for the inﬁnite-layer case; see, e.g., Clever and Busse, 1974, also in ﬁnite-sized containers, although the onset of convection from the diffusive state is independent of Pr, signiﬁcant Prandtl number speciﬁc peculiarities develop in the supercritical regime). In particular, in qualitative agreement with the studies for the idealized case of an inﬁnite layer (Section 4.2), ﬂow in low-Pr liquid metals is distinctly different to that in high-Pr ﬂuids. For more recent experiments with Pr 1, the reader may consider Takeshita et al. (1996), who studied the onset of thermal turbulence in mercury, and Koster (1997) and Yamanaka et al. (1998), who studied liquid gallium. Given the obvious difﬁculties in directly observing ﬂuid ﬂow in liquid metals (they are opaque), many numerical studies have been also devoted to this subject. As an example, Figure 4.38 shows that for Pr = 0.01 and a 4 × 1 × 4 enclosure with adiabatic lateral solid walls, the oscillatory ﬂow at Ra = 4 × 103 is characterized by a sinusoidal wave travelling the axes of the rolls, moving them alternately left and right as predicted by Busse and co-workers for the OS instability (this oscillatory instability give rise to wavy distortions of the convection rolls which travel along the axis of the rolls). Although standing waves are possible in principle, they always appear to be unstable. It is worth noting that the appearance of these travelling waves for Pr 1 does not differ signiﬁcantly from those at higher Prandtl number (Pr ∼ = 1) shown in Figure 4.10. In general, the onset of oscillatory ﬂow in three-dimensional parallelepipedic enclosures has been simulated for a variety of ﬂuids and aspect ratios using the methods described in Section 1.7. Nakano et al. (1998), Tomita and Abe (1999) and Stella and Bucchignani (1999) considered shallow conﬁgurations for the cases of silicon (Pr = 0.01), air and water, respectively (see also the earlier investigation of Kessler, 1987, for Pr = 0.71 and 7, of Yahata, 1989, for Pr = 7 and of Stella et al., 1993, for Pr = 0.71 and 15). Moreover, Yahata (2005) investigated the case when Pr = 2.5. Xia and Murthy (2002) considered deep (tall) conﬁgurations. Bucchignani and Stella (1999) and Xia and Murthy (2002), in particular, focused on the route to chaos. Along these lines, it should be pointed out that in Bucchignani and Stella (1999) two individual mechanisms for the transition to nonperiodic motion were captured (by means of numerical simulation), in qualitative agreement with the experiments of Gollub and Benson (1980): the subharmonic cascade and the quasi-periodicity with three incommensurate frequencies. A quantitative characterization of these chaotic regimes was also attained by evaluating the correlation dimension D2 [see Eq. (1.167)] of the attractor (which was found to be equal to 3.5 for the former behaviour and 3.4 for the latter). Bucchignani and Stella (1999) also observed phase-locking phenomena.

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(a)

(b)

(c)

(d)

Figure 4.38 Travelling wave for the OS instability [Pr = 0.01, cavity 4 × 1 × 4 with adiabatic solid lateral walls, Ra = 4 × 103 , Ra based on the depth; the computed temperature ﬁeld in the midplane (x, z) is shown in four snapshots evenly distributed within one oscillation period; the points P1 and P2 highlight the propagation of two consecutive wave crests along the z direction] (numerical simulation, M. Lappa)

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For phenomena related to spatiotemporal intermittency, the reader may consider the landmark study by Pomeau and Manneville (1980), whereas for exotic scenarios representing a deviation with respect to the Ruelle–Takens theorem enunciated in Section 1.8.2 (which asserts that stable coexistence of three incommensurate frequencies should not be possible) a famous example is the experimental study by Walden et al. (1984), who observed long-lasting states with three, four and even ﬁve distinct frequencies. As already outlined in Section 4.1, RB convection has concentrated a large part of the efforts in the study of nontrivial features of nonlinear dynamics as applied to physical problems, namely chaos, transition scenarios, strange attractors and the empirical reconstruction of experimental nonlinear dynamics (see Section 1.8 for additional explanations about the possible universal routes that systems will take in transitioning from regular to irregular motion). Because of the light shed on the problem by these and previous studies, most of the scientiﬁc community is currently aware of the importance that the control of the boundary conditions and the geometry of the container may have on the threshold values and observed ﬂow patterns. Nonlinear effects, geometric constraints and the multiplicity of solutions are all essential ingredients of this type of ﬂow.

4.10

The Circular Cylindrical Problem

Compared with the buoyant ﬂow between parallel planes of inﬁnite extent and the case of rectangular two-dimensional or parallelepipedic three-dimensional enclosures, the circular cylindrical problem has received less attention.

4.10.1 Moderate Aspect Ratios: Azimuthal Structure and Effect of Lateral Boundary Conditions Some initial studies dealing with the primary instability of the motionless thermal diffusive state in bounded cylindrical ﬂuid layers for the aspect ratio A (height/diameter) in the range O(10−1 ) ≤ A ≤ O(1) are due to Catton and Edwards (1970), Charlson and Sani (1970, 1971), Stork and Muller (1975), Rosenblat (1982), Buell and Catton (1983) and Yamaguchi et al. (1984). All these studies showed that the structure of the emerging buoyancy ﬂow, which is given by the ﬁrst primary threshold, depends on A. Charlson and Sani (1970) investigated by a numerical variational technique the onset of axisymmetric convection in cylinders of aspect ratios between 1/16 and 1, with insulating and conducting sidewalls. They then extended this analysis (Charlson and Sani, 1971) to include non-axisymmetric modes and predicted the critical Rayleigh number and the corresponding azimuthal structure. Stork and M¨uller (1975) observed experimentally convective patterns in annuli and cylinders of aspect ratio 0.15 ≤ A ≤ 0.7, varying the sidewall insulation. Rosenblat (1982) analysed the onset of convective motion by means of numerical simulation and described non-axisymmetric motions existing just above the onset for aspect ratios between 0.25 and 1.0. They also considered the interaction between different critical modes, deriving coupled amplitude equations and obtaining bifurcation diagrams involving a transition from a pure mode to a mixed mode. Finally, Buell and Catton (1983) performed a linear analysis for A ≥ 0.125 providing the critical Rayleigh number and azimuthal wavenumber m (this gives the number of disturbance nodes in the azimuthal direction) as a function of both aspect ratio and sidewall conductivity, thus completing the results of the earlier investigations, which considered either perfectly insulating or perfectly conducting walls. Muller et al. (1984) investigated two different types of liquids, water (Pr = 6.7) and Ga and GaSb melts (Pr ∼ = 2 × 10−2 ) for various aspect ratios (0.5 ≤ A ≤ 5) and Rayleigh numbers up to 108 . Both experimental results and numerical analysis for the onset of convection and the state

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Figure 4.39 Curves of neutral stability for the onset of Rayleigh–B´enard convection in cylindrical enclosures with adiabatic lateral wall (0.6 ≤ radius/height ≤ 3 corresponding to 0.16 ≤ A ≤ 0.83; Racr based on the cylinder height). After Wanschura et al. (1996); Reproduced by permission of Cambridge University Press

of convective ﬂow were presented. The values of Racr for 0.5 ≤ A ≤ 5 and the symmetries of the basic ﬂow (axial symmetry for A = 0.5 and nonaxial symmetry for 1 ≤ A ≤ 5) were in good agreement with the earlier theoretical predictions of Charlson and Sani (1971). According to all these studies, the ﬂow structure that appears after the ﬁrst bifurcation from the diffusive state depends strongly on both the aspect ratio and type of lateral boundary conditions. For the case of adiabatic sidewalls (see Figure 4.39), in particular, the ﬂow is axisymmetric for A < 0.55 (m = 0) and asymmetric for larger values of A (m = 1). The transition between axisymmetric and asymmetric modes occurs around A = 0.72 if conductive lateral boundaries instead of adiabatic walls are considered. In agreement with the results for other geometric models, the thresholds for this primary (stationary) instability are independent of the Prandtl number; for the adiabatic case and 0.32 < A < 0.55, the primary bifurcation to convection occurs at Racr ∼ = 2000 (Ra based on the height of the cylinder). The critical Rayleigh number behaves merely as an increasing function of the aspect ratio, for example Racr ∼ = 3700 for A = 1 and Racr ∼ = 2250 for A = 0.5, with Racr decreasing asymptotically towards 1707 for A → 0. Like the other geometric conﬁgurations considered in the preceding sections, if the Rayleigh number is further increased the ﬂow undergoes a subsequent transition to more complicated ﬂow patterns. This aspect has been the subject of many numerical studies (see, e.g., Muller et al., 1984; Figliola, 1986; Crespo Del Arco and Bontoux, 1989; Neumann, 1990; Hardin and Sani, 1993; Wagner et al., 1994). Some authors have clearly observed the existence of a secondary stationary bifurcation. Wanschura et al. (1996) made a complete linear stability analysis of this secondary instability, but the study was limited to a range of aspect ratios for which the primary threshold corresponds

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(a)

(b)

Figure 4.40 Curves of neutral stability for the three-dimensional instability of the axisymmetric basic ﬂow in cylindrical enclosures with adiabatic lateral wall for two values of Pr (Racr based on the cylinder height; 0.9 ≤ radius/height ≤ 1.57 corresponding to 0.318 ≤ A ≤ 0.55; the lower dotted line marks the onset of 2D convection): (a) Pr = 0.02; (b) Pr = 1 (the instabilities for m = 3 and m = 4 are oscillatory). After Wanschura et al. (1996); Reproduced by permission of Cambridge University Press

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to steady axisymmetric ﬂow. For this case, the unstable mode for the secondary transition is generally a steady mode with m = 2, but it can also be a steady mode with m = 1, 3 or 4 according to the aspect ratio and the Prandtl number (m = 1 and 3 for Pr = 0.02 whereas m = 4 for Pr = 1; see Figure 4.40). According to their analysis, the secondary threshold increases with the Prandtl number and also depends strongly on the aspect ratio. They also elucidated how, for large Prandtl numbers, the axisymmetric ﬂow becomes unstable due to the classical thermal (RB) instability mechanism, whereas for small Prandtl numbers (liquid metals) the secondary instability is inertial (hydrodynamic) in nature. Three years later, Touihri et al. (1999a) re-examined the possible sequence of transitions showing that for A < 0.55, convection sets in with the mode m = 0 followed by m = 2 and m = 1, whereas for 0.55 < A < 0.63, the ﬁrst mode is m = 1 and the next are m = 0 and m = 2, and ﬁnally for A > 0.63 the order of appearance is m = 1, = 2, and = 0. In such an analysis, the critical Rayleigh number for the secondary bifurcation was found to increase with the Prandtl number quadratically at low Pr and linearly at larger Pr (the considered values of the Prandtl number were 0.02, 1 and 6.7; see Figure 4.41). Similar results were obtained in the earlier investigation of Crespo del Arco and Bontoux (1989), who found that for A = 2 at low Rayleigh number the core ﬂow exhibits characteristic features of the m = 1 dominant mode, whereas at elevated Rayleigh numbers secondary vortices corresponding to the m = 0 mode appear and develop differently in both size and magnitude according to the considered Prandtl number (Pr = 0.02 or 6.7 in their analysis). 3000 2492

Crititcal Rayleigh Number

2900 2488

2800

2700

2484

2480 0.000

0.002

0.004

0.006

0.008

2600

2500

0.00

0.20

0.40

0.60

0.80

1.00

Pr

Figure 4.41 Inﬂuence of the Prandtl number on the secondary bifurcation of Rayleigh–B´enard convection in a cylindrical enclosure with A = 0.5 (Racr based on the cylinder height, adiabatic sidewall). After Touihri et al. (1999a); Reproduced by permission of the American Institute of Physics

Subsequent analyses (both experimental and numerical) have proven that if shallow domains are considered, an even richer variety of patterns is possible; for example, varying the Rayleigh number through different sequences of values, for a ﬁxed A = 0.25 and Pr = 6.7, Hof et al. (1999) obtained experimentally several different steady stable patterns for the same ﬁnal Rayleigh number

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Ra = 14 200 (which they categorized in terms of their symmetry properties as ‘rolls with hot ﬂuid rising along the centre’, ‘rolls with cold ﬂuid falling along the centre’, ‘spoke patterns with cold ﬂuid falling along the spokes’, ‘spoke patterns with hot ﬂuid rising along the spokes’ and ‘axisymmetric pattern with hot ﬂuid rising in the centre’; see Figure 4.42).

Figure 4.42 Shadowgraph images of convection patterns (water, diameter = 12.8 mm, height = 3.2 mm, A = 0.25). In (a)–(h), Ra = 1.42 × 104 : (a) three rolls; (b) two rolls with hot ﬂuid rising along the centre; (c) two rolls with cold ﬂuid falling along the centre; (d) four rolls with cold ﬂuid falling along the centre; (e) four rolls with hot ﬂuid rising along the centre; (f) three spoke patterns with cold ﬂuid falling along the spokes; (g) three spoke patterns with hot ﬂuid rising along the spokes; (h) axisymmetric pattern with hot ﬂuid rising in the centre; (i) rotating pattern at Ra = 2.6 × 104 ; (j) pulsating spoked pattern at Ra = 3.3 × 104 . After Hof et al. (1999); Reproduced by permission of the American Institute of Physics

Similarly, Leong (2002) computed several steady convective solutions (four main types of ﬂow structure: concentric, radial, parallel and cross rolls) for Ra > Racr , all of which were stable in the range 6250 ≤ Ra ≤ 37 500 (for aspect ratios A = 0.125 and 0.25 with Pr = 7). Some studies have also appeared where attention was specially focused on the transition to oscillatory behaviours. Landmark experiments were initially carried out by Abernathey and Rosenberger (1985). Wanschura et al. (1996) predicted the secondary ﬂow to be steady except over a narrow aspect ratio range for relatively shallow cylinders (0.32 ≤ A ≤ 0.34) at Pr = 1, where they found oscillatory instabilities at Ra = 2.5 × 104 . Hof et al. (1999) observed (at A = 0.25, Pr = 6.7) the ﬂow to undergo transition to a time-dependent motion in the form of three loops rotating either clockwise or anticlockwise depending on the initial conditions on further increasing Ra beyond 2.3 × 104 . Another interesting oscillatory state was reported for Ra = 3.3 × 104 (consisting of 13 spokes pulsating such that the central dark area of hot ﬂuid expands and contracts as shown in Figure 4.42j). The effect of lateral boundary conditions (adiabatic or conducting wall) on the onset of time-dependent convection was evaluated via three-dimensional unsteady numerical computation by Cheng et al. (2000) in the case of air (Pr = 0.71). More recently, Boronska and Tuckermann (2006) used both nonlinear simulations and linear stability analysis to elucidate the behaviour of RB convection in the parameter region 0.318 ≤ A ≤ 0.345, Pr = 1 and adiabatic lateral wall, originally considered by Wanschura et al. (1996). According to their analysis (which, given the amount of new observations, deserves some additional discussion), in this regime, the primary axisymmetric convective state loses stability to an m = 3 perturbation via a Hopf bifurcation; the related bifurcation scenario guarantees that both branches of standing waves and of travelling waves are created at the bifurcation, but that at most one of these branches is stable. The nonlinear simulations showed, in fact, a supercritical bifurcation leading to long-lived standing waves which were eventually succeeded by travelling waves, both as time progressed and as the Rayleigh number was increased. Most interestingly in this regard, the ﬂow was found to exhibit a notable similarity to oscillatory ﬂows typically observed in the context of studies devoted to Marangoni ﬂow in liquid bridges (see Chapter 10 for further information).

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Rotating and pulsating regimes (nevertheless with features different with respect to those described by Boronska and Tuckermann, 2006) were also observed, as outlined earlier, in the experiments of Hof et al. (1999) in the case of Pr = 6.7 and A = 0.25. Transition to more complex (chaotic) states has been also considered over the years. Walden (1983) carried out experiments using ﬂuids with Prandtl number between 0.49 and 0.78 and reported several types of possible transitions from steady to time-dependent ﬂow. The subsequent evolution towards fully developed turbulence for Ra ranging from 106 to 1011 (Pr = 0.7, corresponding to gaseous helium) was considered by Heslot et al. (1987) and Castaing et al. (1989) via experimental analysis and by Verzicco and Camussi (1999, 2003) through numerical simulations. For the case of water, the reader may consider the experimental analysis of Sun et al. (2005) and references therein. For the case of liquid metals, Verhoeven (1969) was the ﬁrst to provide very precise measurements of temperature oscillations in vertical columns ﬁlled with mercury or a mercury–zinc alloy. Crespo del Arco et al. (1988) simulated complex (steady and time-dependent) supercritical regimes in cylinders of aspect ratios A = 1 and 2 for Pr = 0.02 and in ranges of Ra up to 8Rac and 6Rac , respectively. Transition to fully developed turbulence for a similar conﬁguration (A = 1 and Pr = 0.022, mercury) was considered through three-dimensional numerical simulations by Verzicco and Camussi (1997) for Ra < 106 and via experimental analysis by Takeshita et al. (1996) for 106 < Ra < 108 and Cioni et al. (1997) for 5 × 106 < Ra < 5 × 109 , respectively. Verzicco and Camussi (1997), in particular, showed that the increase in the Rayleigh number leads to the appearance of higher order harmonics and subharmonics of the fundamental oscillation frequency, indicating the transition towards a chaotic state following the classical period-doubling scenario described in Section 1.8. Marked oscillatory phenomena were also detected in the experiments of Kamotani et al. (1994) dealing with tall cylinders (A = 3) and gallium.

4.10.2 Small Aspect Ratios: Targets and PanAm Textures The discussion in the foregoing text provides a fairly exhaustive picture of the state-of-the-art for geometries with O(10−1 ) ≤ A ≤ O(1). It does not consider, however, the case with A = O(10−2 ), for which a rich literature has been developed especially over recent years (thanks to the introduction of the shadowgraph technique for compressed gases). The ﬁnal part of this section is therefore devoted to a survey of such a problem from the primary instability of the diffusive state up to the development of chaos. We will content ourselves, however, with giving only some ﬂavour of the most important ﬁndings by showing some examples. The reader interested in a more thorough discussion should consult the relevant references cited hereafter. Let us begin by pointing out that, basically, the behaviours for A = O(10−2 ) can be divided into two categories: those with rolls parallel (axisymmetric states) or perpendicular to the sidewalls (non-axisymmetric states). To some extent this is an artiﬁcial division and there is much overlap between the two sets of topics, but, perhaps, it is the best way to introduce the subject. The ﬁrst category includes the so-called targets with axisymmetric (circular) concentric rolls (Hoard et al., 1970; Koschmieder and Pallas, 1974; Croquette et al., 1983; Plapp et al., 1998) and the spiral patterns (see Figure 4.43; spirals are not perfectly axisymmetric, but their relationship to targets indicated that including them in that category is appropriate; this type of pattern was studied in depth by Plapp, 1997). The second category consists of patterns including the aforementioned ISRs (ideal straight rolls; see the experiments of Croquette et al., 1986, Croquette, 1989a,b, and Hu et al., 1995a), the PanAm structures shown in Figure 4.44 (see, e.g., Steinberg et al., 1985;

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Figure 4.43 Shadowgraph image of a two-armed, clockwise rotating spiral for Pr ∼ = 1. Courtesy of B. Plapp and E. Bodenschatz

Figure 4.44 Shadowgraph image of a PanAm texture for Pr ∼ = 1. The basic PanAm structure is that of some foci afﬁxed to opposite sides of the walls of the cylindrical container (arches with several centres of curvature). Courtesy of B. Plapp and E. Bodenschatz

the origin of this term is explained later) and the spiral-defect chaos discovered by Morris et al. (1993) and partially discussed in Section 4.6. Close to the onset of convection (Sections 4.1 and 4.2), ISRs are stable in both circular and rectangular geometries. However, for circular containers there is a tendency to form short cross-rolls near the part of the sidewall where the rolls would otherwise be parallel to the wall.

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Thermal Convection: Patterns, Evolution and Stability

This behaviour is a manifestation of the already mentioned (Section 4.9) tendency for rolls to terminate with their axes perpendicular to the sidewalls (as proved by Segel, 1969, this conﬁguration minimizes the friction experienced by the rolls at the sidewalls). With increasing Ra this effect becomes more pronounced. In circular convection cells it leads to enhanced rolls curvature in the pattern interior. As the curvature increases, focus singularities are formed at the wall (wall foci). Typically in the existing experiments, two wall foci have been observed and the resulting structures (arches with several centres of curvature) have been often referred to as ‘PanAm’ (by similarity with the logo of the well-known American airline company). Target textures have been observed to be stable in the same Ra range as the initial straight rolls or the curved rolls with wall foci; however, according to many experimental analyses, it seems that the occurrence of these textures must be somewhat supported by an initial sidewall forcing, that is, horizontal temperature gradients perpendicular to the sidewalls forcing the rolls to align with the circular boundary. There are occasional mixtures of the two categories; for example, Hu et al. (1993) generated a stable target pattern with an outside ring of rolls perpendicular to the boundary. In addition complicated, non-axisymmetric patterns, in which at some positions along the sidewalls the rolls are parallel to the boundary and at other positions perpendicular, have been also reported. When the Rayleigh number is increased, these ﬂow patterns may undergo transition to more complex spatiotemporal states and/or transitions from a type of pattern to another may occur. With regard to PanAm structures, for instance, Pocheau et al. (1985) made observations of skewed varicose instabilities (Section 4.2) in the middle of the patterns. Particularly interesting was the fact that the focus singularities at the sidewalls were seen to emit rolls, which then caused the skewed varicose instability to occur in the region in the centre. The mean ﬂow (mean ﬂow in the azimuthal direction) shown in Figure 4.45 was apparently responsible for this time-dependent behaviour. For detailed explorations of the stationary states and time dependence of the wavevector ﬁeld in PanAm patterns, the reader may consider the experimental studies of Gollub and co-workers (e.g. Heutmaker et al., 1985; Heutmaker and Gollub, 1986, 1987).

Figure 4.45 Sketch of PanAm pattern and related mean ﬂow

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173

Concerning targets, it is known that when the Rayleigh number is changed, a target roll pattern can simply expand or contract, or, when the required shift is too great for that process, the pattern may undergo adjustments in wavenumber by losing or gaining rolls at the core of the pattern. Systematic experiments along these lines are due to Koschmieder and Pallas (1974) and Steinberg et al. (1985) in silicone oils with Pr = 511 − 1673 and in water with Pr = 6.1, respectively. It is also worth citing Croquette et al. (1983) and Tuckerman and Barkley (1988), who clearly observed the existence of radially propagating patterns of concentric rolls. Target instabilities in which the targets are seen to move off-centre (targets with dislocated centre; see Figure 4.46) have been also reported by some authors (Steinberg et al., 1985; Croquette, 1989a,b; Hu et al., 1993). The pattern ‘umbilicus’ (the origin of the concentric pattern) drifts away from the centre and toward the wall, thus destroying the concentric pattern. In reporting this observation, Croquette conjectured that the off-centre motion of the target was due to a dipolar mean ﬂow driving the core to one side of the cell. The presence of such ﬂow was effectively conﬁrmed in a later work. Figure 4.46 shows a sketch of the mean ﬂow (loops with arrows) superimposed on the off-centre target pattern. By symmetry such a ﬂow would not be sustained in an on-centre target pattern. A further possible mechanism involves an off-centre displacement of the umbilicus which is followed by radial oscillations of its position (Plapp, 1997) or emission of radially travelling waves (Hu et al., 1993).

4.11

Spirals: Genesis, Properties and Dynamics

As shown by Plapp (1997) and Plapp et al. (1998), target patterns can also evolve towards spirals. In these studies, the primary motivation of which was the detailed investigation of the steady state and transient behaviour of isolated or global spirals (that is, spirals that ﬁll the entire ﬂuid container), it was clearly observed for Pr ∼ = 1 (compressed CO2 ) that target patterns can give way to single-armed rigidly rotating spirals (the skewed varicose instability being the primary means of making such a transition).

Figure 4.46 Sketch of off-centre target and related mean ﬂow

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The process leading to the formation of spirals (as typically observed in the experiments) can be summarized as follows. Upon crossing the onset of convection, target patterns initially develop, which with increasing Ra change their wavenumbers by annihilating rolls in the centre. With a further increase in Ra, the target moves off-centre and the concentric pattern is compressed on one side and dilated on the other until the wavenumber in the compressed region increases beyond the skewed varicose instability. At this stage, a defect pair nucleate to decrease the wavenumber. One of the defects then moves to the centre while the other glides radially outward before coming to rest at a distance rd from the geometric centre (while continuously revolving around the centre). The pattern ﬁnally relaxes to an on-centre, one-armed rotating spiral of radius rd . The value of Ra at which the target instability occurs has been found to increase with increasing aspect ratio A (i.e. increasing the relative depth of the ﬂuid). It is worth remarking that, in the light of the explanation given above, the rigid rotation of a ﬁnite spiral of radius rd necessitates that the spiral waves which propagate from the spiral’s core are annihilated at r = rd by a circular motion of the outer defect. Thus, the pattern simply consists of stationary, concentric rolls for r > rd and rd can be regarded as the distance from the inner to the outer defect (see Figure 4.47). This balancing mechanism has been placed on a more precise theoretical framework by Cross and Tu (1995) and Cross (1996). These authors elucidated that, in general, the rotation of a spiral requires the reconciliation of two competing selection principles acting far away from the spiral’s core: (1) wavelength selection by climbing of the outer defect and (2) the emission of radially travelling waves due to target selection (another interesting way to explain the mechanism just stated is that a rigidly rotating spiral is driven by two wavenumber selection mechanisms, one that determines the velocity of the outer defect around the spiral centre and the other that determines the velocity of outward travelling rolls). The interpretation given by Cross and Tu (1995) and Cross (1996) is particularly interesting as it has been proved that spiral rotation is driven by wavevector frustration and not by mean ﬂow effects (as was previously speculated by other investigators). In the remainder of this section (and also in Section 4.12), a number of outstanding aspects in the theory of spiral formation that are of particular interest are mentioned. The list is not intended to be exhaustive, but rather to stimulate the interest of the reader in certain topics, some longstanding, some new, where progress is needed and to summarize the state of the ﬁeld and formulate the right questions.

Figure 4.47 Sketch of ﬁnite single-armed spiral indicating the defect radius (rd ), the outward velocity of the rolls (vR ) and the velocity of the revolving defect (vD )

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Before embarking on a more precise theoretical treatment of the concepts illustrated above, however, a short discussion of the mathematical properties of a generic Archimedean spiral is required in order to provide the reader with some necessary, helpful and fundamental information (some of these notions will be also used in Chapter 11 when discussing the properties of hydrothermal waves in Marangoni ﬂows).

4.11.1 The Archimedean Spiral Towards this end, as a ﬁrst step, let us consider the equation for a target pattern: f (r) = F cos(qr)

(4.47)

where q is the wavenumber, F is an arbitrary amplitude and % r = (x − x0 )2 + (z − z0 )2

(4.48)

with (x0 , z0 ) being the centre of the target. The equation for an Archimedean spiral is a target pattern with an extra phase term added to the argument of the cosine: f (r) = F cos(qr + φ) where φ = − tan−1

z − z0 x − x0

(4.49)

(4.50)

is the polar angle around (x0 , z0 ). It should be pointed out, however, that since Eq. (4.49) gives a spiral that winds out ad inﬁnitum, it does not describe effective spirals in RB convection that terminate at a radius rd . To describe these ﬁnite spirals, an additional phase term φD must be added inside the cosine; this leads to f (r) = F cos(qr + ϑ) where ϑ = φ + φD and φD = tan

−1

z − z1 x − x1

(4.51)

(4.52)

is an additional polar angle around the defect position (x1 , z1 ).

4.11.2 Spiral Wavenumber Following the theoretical explanations provided by Plapp (1997), let us now observe from a geometric point of view how the two motions of roll travel and defect revolution mentioned earlier are related. Consider a rigidly rotating spiral with a period of rotation τ and a roll wavenumber q = 2π/λ, where λ is the wavelength. Figure 4.47 shows a sketch of such a ﬁnite spiral. The outer defect revolves around the centre at a radius rd with a velocity vD and the motion of rolls outwards has a velocity vR . Outside the defect there are concentric rolls. The period of the outer defect revolution can be expressed as τD =

2π rd vD

(4.53)

The period of the travelling rolls (as seen from a stationary point in the spiral) will read τR =

λ 2π = vR qvR

(4.54)

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As discussed above, for the rigidly rotating spiral, these periods must be equal, giving vD = qrd vR

(4.55)

The spiral wavenumber can be derived from this expression. This, however, requires approximate expressions for vR and vD . Towards this end, following Cross and Tu (1995) and Cross (1996), the velocity of a climbing defect can be written as vD = b(q − qD )

(4.56)

where b is a proportionality constant, q is the background wavenumber of the defect and qD is the ‘optimal’ wavenumber, at which the defect is stationary (see Section 4.3 from some additional theoretical background). Moreover, according to Cross and Tu (1995) and Cross (1996), the velocity with which a curved roll pattern (e.g. a target) will adjust its wavenumber until the selected wavenumber is reached can be written in the form ∼ a(qT − q) (4.57) vR = qrd where a is a proportionality constant, qT is the selected target wavenumber and rd is the radius of the outer dislocation. Using the wavenumber relations for the dislocation and outward roll motions [Eqs (4.56) and (4.57)] and combining them with the relationship between the two velocities in a rigidly rotating spiral [Eq. (4.55)], it follows that q=

aqT + bqD a+b

(4.58)

This leads to the notable result that the wavenumber of the spiral at a particular Ra can be seen as the weighted average of the selected target wavenumber qT and the optimal defect wavenumber qD . In general, as Ra is increased to relatively high values, however, the spiral structure becomes unstable. Typically the core moves off-centre. The patterns develop many dislocations. In particular, as shown by Plapp and Bodenschatz (1996), the single-armed spirals move off-centre in a manner very similar to the target instability described in Section 4.10.2; however, the occurrence of a skewed varicose event in the compression region leads to the migration of the dislocations to the dilated region, further pushing the core to one side and leading to more skewed varicose instabilities with the nucleation of new defect pairs. Depending on parameters, the structure then evolves into a multi-armed spiral , a texture with wall foci or spiral-defect chaos.

4.11.3 Multi-armed Spirals and Spiral Core Instability Interestingly, the analysis elaborated earlier for a single-armed spiral [Eqs (4.53)–(4.58)] can be applied to m-armed spirals (Li et al., 1996; Plapp, 1997), with the recognition, however, that one period of revolution for the m outer defects will be equal to m periods of the rolls travelling outwards as seen by an observer in the spiral. The roll velocity vR as given in Eq. (4.57) and the defect velocity from Eq. (4.56) are still valid, but instead of τD being equal to τR , one has τD = mτR

(4.59)

which means that vD = qrd

vR m

(4.60)

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As a consequence, the resultant wavenumber of the m-armed spiral at the location of the defects becomes aqT + mbqD (4.61) qm = a + mb In general, multi-armed (m) spirals display two types of periodic behaviour (Plapp and Bodenschatz, 1996; Assenheimer and Steinberg, 1996). The ﬁrst, obviously, is the slow, large-scale rotation that can be described by outward travelling waves, which, as explained previously, are annihilated by (m) defects revolving on circular trajectories. Second, they exhibit a much faster spatiotemporal periodic behaviour in a radial region of approximate size r ∼ = mλ/2 (this region usually referred to in the literature as the spiral ‘core’). The core oscillations can be illustrated by comparing the temporal behaviour at points near the core and at the rim. At the edge, the core oscillations are negligible so that mainly the background rotation is sensed. Cyclic oscillations are evident in the core and visually it appears as that such a region rotates against the overall spiral rotation (Figure 4.48). The striking feature of this instability is that, as outlined above, spiral cores oscillate periodically with a frequency considerably higher than the frequency of the overall spiral rotation. The core exhibits a cyclic motion with a period of several vertical diffusion times as compared with a period of overall spiral rotation of a few hundred vertical diffusion times (in general, the related dynamics can be regarded as a ‘dance’ of the inner m defects with alternating connecting and disconnecting of the tips). In this regard, such a localized behaviour could be categorized as an ‘oscillon’ (on the basis of the meaning that has been given to such a deﬁnition in Section 4.3). Some interesting insights into this mechanism have been provided by Aranson et al. (1997). They showed that this instability is driven by self-generated vorticity, namely that the vorticity ﬁeld generated at the spiral core plays a major role in the origin of the oscillations. Such theoretical arguments have been elaborated within the framework of an interesting model in which a one-armed spiral of the form given by Eq. (4.51), namely F cos() (with = qr + ϑ) has been coupled to an external velocity ﬁeld generated by a ﬁxed-point vortex with circulation , placed at the centre of rotation (see Section 1.2.8 for fundamental information about the meaning of and its relationship with the concept of vorticity).

Figure 4.48 Time evolution of the spiral’s core of a counterclockwise rotating four-armed spiral [Pr = 1.38, Ra/Racr = 1.78; the sequence of evenly distributed snapshots (shadowgraph images) shows a quarter period; pictures are taken 1 s apart]. Four defects are bound in the core and four defects with opposite topological charge revolve around the core so as to eliminate the outward travelling spiral waves. Courtesy of B. Plapp and E. Bodenschatz

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In practice, such a model has been based on the earlier observations of Bestehorn et al. (1993), according to which the spiral tip generates a highly localized vorticity peak at the core and for m-armed spirals, m identical vortices are created at the core. With this model, Aranson et al. (1997) found that ∂ ∂ =− 2 + ... (4.62) ∂t r ∂ϑ ∂q → (4.63) ∝− 3 ∂t r This means that the local wavenumber decreases linearly in time (in other words, the external velocity ﬁeld winds the spiral up near the core); as a consequence of such behaviour, eventually the local wavenumber is carried away from the region of stability and the Eckhaus instability is initiated; phase slips then occur, locally winding the spiral up and returning the wavenumber into the stable domain. This process then recurs, leading to quasi-periodic oscillations. Aranson et al. (1997) simulated the spiral dynamics using the simpliﬁed model described above with a velocity proﬁle of the form u = /r (where u is the circumferential velocity component) and observed the above-mentioned scenario. As the magnitude of the circulation was increased, a bifurcation similar to that described above was observed (at small a steady rotation persists, whereas for > c the core starts to oscillate). The authors also reported that when the wavenumber near the core reaches zero due to local spiral unwinding, a zero mode is generated that can lead to proliferation of up- and downﬂow hexagons (see Figure 4.49). Remarkably, this means the model introduced by Aranson et al. (1997) can also be used to introduce a theoretical justiﬁcation for the transition from spirals to hexagons that was reported previously by Assenheimer and Steinberg (1996). In the experiments of these researchers the hexagons were found to develop and invade the system primarily from spiral and target cores and other defects. According to the analysis of Aranson et al. (1997), such behaviour must be regarded simply as a consequence of the spiral core instability; in their numerics and also in the experiments of Assenheimer and Steinberg (1996), in fact, the core oscillations always precede the transition to the hexagonal state. To conclude this discussion, it is worth repeating that while interaction between the spiral and the mean ﬂow generated by the curved rolls near its core (i.e. the aforementioned vortex) is required for both the spiral core instability and subsequent generation of up- and downﬂow hexagons, mean

Figure 4.49 Hexagon nucleation at a spiral core obtained for Pr = 1 and Ra = 2.9 × Racr (snapshots taken at nondimensional times t = 10, 110, 470, 650, 340, 2350). Courtesy of I. Tsimring and L.S. Aranson

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ﬂow effects play a minor role in the overall spiral rotation that, as explained many times before, arises as a consequence of the delicate balancing mechanism theorized by Cross and Tu (1995) (for which rotation of a spiral requires the reconciliation of two competing processes: wavelength selection by climbing of an outer defect and the emission of radially travelling waves).

4.12

From Spirals to SDC: The Extensive Chaos Problem

In another scenario, as outlined in Section 4.11.2, spirals evolve directly into spiral-defect chaos (SDC). This state is comprised of a large number of rotating spirals of various sizes. Spirals nucleate, interact and annihilate, yielding a macroscopically disordered pattern. Essential features of SDC include spontaneous spiral creation, quasi-stationary spiral rotation, spiral-core instability and eventual spiral destruction by other spirals. In practice, this regime is characterized by the spontaneous and perpetual emergence and disintegration of large extended spiral and target patterns and other defects in the roll structure. Defects present in the pattern have a modest lifetime and drift about irregularly, and new ones are constantly created as old ones disappear. Moreover, the spirals may coexist with regions of more or less straight rolls (as an example, for Pr = 0.96 and Ra/Racr = 1.72 these regions have a width of only a few wavelengths, but near the onset of SDC and particularly for very horizontally extended cells, the straight-roll regions can become fairly large). Due to such complexity, theoretical and experimental investigations of SDC in RB have played an important role in the fundamental study of pattern formation and related mechanisms. A quantitative understanding of SDC, however, has not yet been achieved (the problem is very difﬁcult because, as illustrated in Section 4.11, the chaotic state evolves from a ground state which is already extremely complex). Nevertheless, it should be stressed that despite the aforementioned difﬁculties, some important insights into the dynamics of this state have been gained (Bodenschatz et al., 2000). As an example, most surprisingly, the average spatial periodicity (wavenumber) of SDC seems to be located in the middle of the Busse balloon (Section 4.2). Moreover, a central feature of the dynamics seems to be the competition between two wavenumber selection processes as in the case of a single spiral treated in the preceding section. The tip of the generic spiral selects one wavenumber and the far ﬁeld which is dominated by a number of different defect types selects another; the resulting wavenumber gradient orthogonal to the spiral arms leads to outward-travelling waves surrounding the spiral tips which are equivalent to spiral rotation. Another remarkable feature of SDC is the aforementioned competition between it and ISR states. As already mentioned in Section 4.6, SDC is not caused by a bulk instability of the straight-roll patterns; rather, there is bistability of SDC and the usual roll state [that is, over a wide parameter range both straight rolls (a ﬁxed point) and SDC (a chaotic attractor) are stable solutions of the equations of thermal convection]. According to typical experiments for Prandtl numbers close to or less than one, the initial and boundary conditions fall within the attractor basin of SDC and rolls without spirals are rarely observed for Ra greater than some onset value RaSDC (Cakmur et al., 1997). Additional conﬁrmation of the bistability comes from the integration of the thermal-convection equations with periodic boundary conditions which always yield SDC from random initial conditions but give stable ISRs when ISR-like initial conditions are used (Decker et al., 1994; Rudiger and Feudel, 2000). In the experiments, SDC is the generically selected state above RaSDC when Ra is quasi-statically increased from onset (Liu and Ahlers 1996), as shown in Figures 4.50 and 4.51. The patterns of SDC can be characterized in several ways: by global Fourier transform methods (Hu et al. 1995a,b; Morris et al., 1993, 1996), with local roll properties such as curvature and

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Figure 4.50 Onset of spiral defect chaos as a function of the radius to height ratio of the container (Pr ∼ = 1; curve obtained via interpolation of several experimental data available in the literature)

Figure 4.51 Onset of spiral defect chaos as a function of the Prandtl number and two values of the aspect ratio A = L/D . Curves obtained via interpolation of experimental data reported by Liu and Ahlers (1996)

wavenumber variations (Hu et al. 1995a,b; Egolf et al., 1998) and by the statistics of spiral-defect populations (Ecke et al., 1995; Egolf et al., 1998). Perhaps the most interesting insights into this type of ﬂow have been yielded via analysis of the related Lyapunov spectrum (Egolf et al., 2000; Paul et al., 2007). Some fundamental information about possible means to characterize strange attractors and chaotic states has been provided in Section 1.8 with the deﬁnition and ensuing illustration of the concepts of Lyapunov exponents, Hausdorff, Lyapunov and correlation dimensions (and related

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∫λ(i/L2s)d(i/L2s)

interplay) for some canonical examples of interest in the ﬁeld of thermal convection (e.g. the Lorenz model). It should be pointed out, however, that although insights into the crucial dynamic degrees of freedom in low-dimensional chaos have been routinely obtained through the analysis of these dynamic quantities, high-dimensional spatiotemporal chaos like SDC has proven difﬁcult to understand in the same theoretical framework. Initial attempts to extend the dynamic approach based on the evaluation of the Lyapunov exponents (and related quantities) to higher-dimensional systems demonstrated numerically that the spatiotemporal chaos in several simple models is ‘extensive’ in that the number of dynamic degrees of freedom (the Lyapunov dimension) scales with the system volume (Egolf and Greenside, 1994). Ruelle was the ﬁrst to conjecture that for very large systems the Lyapunov dimension [Eq. (1.163)] should scale extensively with the size of the system, that is, Dλ ∝ (Ls )n , where Ls is the nondimensional system size (for the present case the ratio between the radius and the height of the shallow cylindrical ﬂuid container) and n is the number of spatially extended dimensions (n = 2 for SDC). Very interesting analyses along these lines were performed by Egolf et al. (2000) and later by Paul et al. (2007). Egolf et al. (2000) were the ﬁrst to provide a direct demonstration of the principle of extensivity of the dynamics of an experimentally relevant spatiotemporal chaotic system. Similar results were yielded by Paul et al. (2007). They considered six aspect ratios in the range 3.3 × 10−2 ≤ A ≤ 0.1(4.72 ≤ Ls ≤ 15) with a ﬁxed thermal driving Ra/Racr = 3.5 and Pr = 1. By means of numerical simulations and ensuing computation of the Lyapunov exponents, they showed that the Lyapunov spectra collapse into a single curve for Ls > 10 (see Figure 4.52), that is, when the system becomes sufﬁciently extended along the horizontal dimension, demonstrating unambiguously that SDC is extensive.

Figure 4.52 Lyapunov spectrum λ(i/L2s ) as a function of i/L2s where i = 1, . . . , N and N is the number of Lyapunov exponents (Pr = 1, Ls > 10, Ra/Racr = 3.5). Courtesy of M. Paul; data originally published in M.R. Paul, M.I. Einarsson, P.F. Fischer and M.C. Cross, Phys. Rev. E, 2007, 75, 045203

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Figure 4.53 Extensive chaos in RB convection as illustrated by the linear relationship between the Lyapunov dimension Dλ and system size L2s . Courtesy of M. Paul; data originally published in M.R. Paul, M.I. Einarsson, P.F. Fischer and M.C. Cross, Phys. Rev. E, 2007, 75, 045203

The major outcome of the more recent analysis of Paul et al. (2007), however, is that a clear linear relationship between the Lyapunov dimension and L2s becomes evident when Dλ is plotted as a function of L2s (Figure 4.53). The slope of the curve corresponds to the so-called dimension density δλ . It is δλ ∼ = 0.25 that leads to the interesting theoretical consequence that for Ls = O(102 ) [i.e. A = O(10−2 ), a common size used in experiment] the Lyapunov dimension would be Dλ = O(103 ), indicating the impressive presence of a high number of chaotic degrees freedom. An interesting related notion is the concept of ‘natural length scale for an individual degree of freedom’ (also known as chaotic length scale ξ ). A volume Lns contains Dλ degrees of freedom, which suggests that such a quantity can be deﬁned as ξ=

Dλ Lns

− 1

n

(4.64)

For extensive chaos ξ is independent of system size and for SDC ξ ∼ = 2, which suggests that an individual degree of freedom would occupy on average an area ξ 2 ∼ = 4 (Paul et al., 2007).

4.13

Three-dimensional Convection in a Spherical Shell

In addition to the well-known RB problem of convection in a layer heated from below or in other geometric conﬁgurations of great interest in the ﬁeld of crystal growth (and, in general, in materials science), which have been extensively treated in the preceding sections, the problem of convection in a spherical shell heated from within has attracted much attention. The most widespread attention has come from, in particular, geophysicists and astrophysicists interested in convective processes (see the discussions in Chapter 3). Notably, such problem has outstanding background application in the cases of both very small and very large values of the Prandtl number.

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In the former case (liquid metals), RB convection in spherical geometries, as will be discussed here, is known to undergo transition to particular solutions (known as ‘heteroclinic cycles’). Moreover, taking into account magnetohydrodynamic effects, it can be used to explain the origin of the Earth’s magnetic ﬁeld and many of its typical features (Section 4.13.4). The latter case, the study of thermal convection in highly viscous spherical ﬂuid shells, is important for its application to the structure and evolution of terrestrial planets. In general, highly viscous thermal convection in the silicate mantles of solid planets is the primary process governing their thermal and mechanical evolution over long time-scales. This process is driven primarily by the transfer of heat from the interior to the surface.

4.13.1 Possible Patterns of Convection and Related Symmetries This subject was initially approached by Busse and co-workers (Busse, 1975; Busse and Riahi, 1982), who theoretically depicted the possible patterns of symmetry in the most general case by considering different orders of the spherical harmonics that describe the solution of the linear stability problem (their analyses were based on an expansion in spherical harmonics similar to that already used by Chandrasekhar, 1961). These authors obtained solutions for = 2, 4 and 6 (Busse, 1975) and = 3 (Busse and Riahi, 1982). The solution = 2 was found to be axisymmetric, whereas = 4 gives a pattern with cubic symmetry (Figure 4.54a) (also referred to as six-cell solutions, as there are six regions with either ascending or descending convective motions) and = 6 corresponds to a pattern of convection with the symmetry of a dodecahedron (Figure 4.54c). Busse and Riahi (1982) showed the solution for = 3 to correspond to a tetrahedron (Figure 4.54b) and found this solution to be generally preferred with respect to the others. The major outcome of these studies was the qualitative difference between convective patterns of odd and even order . This difference does not have a direct analogue in the case of a plane layer (Section 4.1), where solutions with different wavenumber differ only quantitatively.

4.13.2 The Heteroclinic Cycles Busse and Riahi (1988) derived a number of additional patterns which are likely to occur in bifurcations from spherically symmetric basic states when two neighbouring degrees and ∗ of spherical harmonics yield nearly the same lowest value of the control parameter (i.e. when and

∗ compete in determining the instability threshold). As explained before on the basis of the results of Busse (1975) and Busse and Riahi (1982), cellular-convection ﬂows with two and six cells are attained for = 2 and 4, respectively, whereas a cellular ﬂow with an asymmetry between rising and descending motion cannot be described by solutions with = 3. Busse and Riahi (1988), however, illustrated that the set of possible solutions can be enriched with regular patterns of one, three, four or seven cells by considering bifurcations in which modes with and ∗ participate. The effect of the combination of two wavelengths was shown to be most pronounced in the case ∗ = 1, = 2 where the wavelengths differ by a factor of two. The resulting solution is axisymmetric. In looking at the case = 2, ∗ = 3 they found a three-cell pattern. The four-cell pattern was yielded for ∗ = 3, = 4 (this pattern exhibits the symmetry of a tetrahedron just as the preferred solution for = 3 discussed by Busse and Riahi, 1982). Since (as shown by Busse, 1975) the conﬁguration for = 4 is featured by six cells, there is no combination of and ∗ available which could yield a pattern with ﬁve cells. However, a seven-cell pattern was reported for = 4, ∗ = 5.

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(a)

(b)

(c)

Figure 4.54 Typical patterns of convection in spherical shells: (a) cubic symmetry for = 4; (b) symmetry of a tetrahedron for = 3; (c) symmetry of a dodecahedron for = 6. Courtesy of F.H. Busse

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Interestingly, Busse and Riahi (1988) also conjectured that secondary and tertiary bifurcations may introduce patterns of lower symmetry and new phenomena such as time-periodic and aperiodic solutions. Interesting results in this direction of research were obtained by Friedrich and Haken (1986). They proved the effective occurrence of motion reversal (the aforementioned heteroclinic cycles) for ∗ = 1, = 2. Some relevant theoretical background for the existence of these time-dependent solutions was provided over subsequent years by other authors (Iooss and Rossi, 1989; Chossat et al., 1990; Chossat and Guyard, 1996). Perhaps the most interesting outcome of the above-mentioned study of Friedrich and Haken (1986) is that the effective condition for which two groups of modes can simultaneously become unstable and determine time-dependent behaviour depends on the system aspect ratio (the inner to outer radius ratio of the considered spherical shell). By varying the Rayleigh number and the aspect ratio, they observed, in fact, a variety of interesting spatiotemporal behaviours (static, time periodic and quasi-periodic ﬂuid motions and also various types of chaotic ﬂuid ﬂow were analysed in detail). The problem was re-examined (in a slightly different context) by Beltrame et al. (2006). For a central ﬁeld exhibiting a dependence on the radius of the type r −5 (instead of r −2 as in the classical case of gravity), these authors reported that a rich variety of heteroclinic cycles is possible (of the type ∗ = 2, = 3) provided that the Prandtl number is less than Pr ∼ = 0.24 (i.e. for the case of liquid metals).

4.13.3 The Highly Viscous Case Many analyses have appeared in the literature considering inﬁnite-Pr convection in basally heated spherical shells with a central gravity ﬁeld (a spherically symmetric distribution of gravity) and stress-free boundary conditions (a reasonable approximation of the Earth’s mantle conditions). The majority of these numerical analyses assumed axisymmetry (Schubert and Zebib, 1980; Zebib et al., 1980, 1983, 1985b; Machetel and Yuen, 1986, 1987). However, analytical studies (Busse, 1975; Busse and Riahi, 1982) and stability analyses (Schubert and Zebib, 1980; Zebib et al., 1980, 1983, 1985b) indicated that there are only a limited number of axisymmetric solutions that are stable to three-dimensional (i.e. azimuthal) perturbations. Only after 1985 were fully 3D solutions generated (see, e.g., Baumgardner, 1985; Machetel et al., 1986; Bercovici et al., 1989; Ratcliff et al., 1995; Yanagisawa and Yamagishi, 2005). In particular, Machetel et al. (1986) studied both axisymmetric and three-dimensional convective solutions for a shell with constant gravity and an inner to outer radius ratio of 0.62. They examined the multiplicity of nonlinear solutions that exist for a variety of initial conditions at a slightly supercritical Ra. For Ra as high as 13Racr , they found that only ‘polygonal’-type solutions are stable. In the numerical study of Bercovici et al. (1989), the inner to outer radius ratio was reduced to 0.55 and a wider range of Ra was considered (from the onset of convection at Ra = 712 up to Ra = 7 × 104 ). They reported on the existence of two dominant planforms of convection occurring in the set of three-dimensional solutions: odd (i.e. non-equatorially symmetric) and even (equatorially symmetric) solutions. Related examples are shown in Figure 4.55, where the protrusions represent upwelling regions or plumes and the apparent canyons are downwelling areas. If the upwelling regions are assumed to mark the apexes of a polyhedron, then the odd and even solutions form a tetrahedron and an octahedron, respectively. It is worth noting that following the earlier studies by Busse and co-workers, Bercovici et al. (1989) denoted the ﬁrst pattern tetrahedral and the latter cubic to denote a certain family of

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Figure 4.55 Three-dimensional solutions showing dominant planforms of convection in a spherical shell in the limit as Pr → ∞ (Ra = 7 × 103 , constant-viscosity approach): (a) cubic pattern with six upwelling cylindrical plumes; (b) tetrahedral pattern with four upwelling plumes. After Hernlund and Tackley (2003); Copyright Elsevier, 2003

polyhedra with cubic symmetry to which the octahedron belongs (this nomenclature has been maintained in subsequent analyses also by other authors). All these solutions have even symmetry about at least two planes of constant longitude (the cubic pattern has four of such planes). At the onset of convection, both patterns have a nearly equal likelihood of occurring and one pattern is not preferred over the other. For the convenience of the reader, the structure of these patterns and the related dependence on the Rayleigh number, as reported originally in the landmark analysis of Bercovici et al. (1989), are described in the following. For both the tetrahedral and cubic patterns, the upwelling areas are cylindrical and separated by downwelling ﬂuid. Although downwelling regions have local extrema, they are usually connected in a network of linear features (this structure is characteristic of the aforementioned polygonal solutions observed by Machetel et al., 1986). As Ra increases, the up- and downwelling regions become more conﬁned to narrow areas and the virtually stagnant region between them grows in width. At high Rayleigh numbers, the downwelling regions are manifested as a network of narrow linear sheets. At all Ra and for both patterns, the maximum velocity and temperature deviation with respect to the diffusive solution (the TFD distortions deﬁned in Section 2.1) always occur in the upwelling regions. Midway through the shell, the maximum upwelling velocities are three to four times the maximum velocities of the downwelling regions and the magnitude of the hot temperature distortions is three to ﬁve times the magnitude of the cold ones. Interestingly, the velocity and temperature maxima of upwelling are greater for the tetrahedral pattern than for the cubic pattern, whereas the reverse occurs for the downwelling velocity and temperature. Anyhow, for both patterns, as Ra increases the maximum temperature distortion of the upwelling region increases, whereas that of the downwelling region decreases slightly. According to Bercovici et al. (1989), this probably reﬂects the narrowing and subsequent intensiﬁcation of upwelling plumes and the spreading of downwelling ﬂuid more uniformly into sheets. In such a context, it is also worth recalling that the temperature TFD distortions of these regions will exhibit a tendency to be more detailed than the velocity features because in high-Pr systems internal (thermal) energy undergoes much less diffusion than momentum. Interestingly, for both patterns, the areas of negative radial velocity in the downwelling zones are broader than the corresponding areas of negative (cold) temperature TFD distortions; this means that areas of downwelling overlap areas of relatively warm material. Despite the relative buoyancy of this ﬂuid, it is entrained into the downwelling currents. Furthermore, the up- and downwelling regions of both patterns do not maintain the same horizontal structure throughout the depth of the shell. At all Rayleigh numbers considered by Bercovici et al. (1989), the upwelling regions of the cubic pattern are nearly circular at the top of the shell;

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near the middle of the shell, the upwelling regions are fairly complex, as though undergoing a sharp transition with depth, and at the bottom of the shell they are distinctly in the shape of diamonds. At low Ra, the upwelling regions of the tetrahedral pattern do not change shape with depth. However, for Ra > 3.5 × 104 , the upwelling regions are approximately triangular at the top and almost clover-shaped at the bottom. Bercovici et al. (1989) explained this effect as due to the breakup of the downwelling regions from a network of linear sheets at the top to a pattern of connected plumes at the bottom. This effect was found to be most striking at Ra = 7 × 104 , where the breakup begins midway through the shell. Since the downwelling sheets do not have uniform intensity, when they impinge on the bottom boundary they do not spread out into the boundary layers uniformly, hence the mass ﬂux which the boundary layers feed into the upwelling plumes is not isotropic with respect to the axes of the plumes. In contrast, the upwelling regions impinge on the upper boundary as narrow cylinders (especially at Ra = 7 × 104 ) and are more likely to spread uniformly. It is worth noting that the large-scale structure of these patterns is closely related to the geometric planforms predicted previously within the framework of the analytical theories of slightly supercritical spherical convection introduced by Busse (1975) and Busse and Riahi (1982). In practice, polygonal patterns are the spherical analogues of hexagonal patterns in plane layer convection (along the same lines, axisymmetric patterns can be regarded as the analogues of ideal straight rolls). Since, as explained in Chapter 3 and repeated in Section 4.6, convection in a ﬂuid layer without midplane symmetry can only have a hexagonal planform, Busse (1975) predicted that the polygonal convective patterns in a spherical shell (that obviously destroys the midplane symmetry as its is homogeneous with respect to two dimensions but inhomogeneous with respect to the radial coordinate) would be the only stable patterns and would exist for Rayleigh numbers much greater than the slightly supercritical Ra used in small-amplitude theory. Additional insights can be found again in the numerical study by Bercovici et al. (1989). Since upwelling regions have both a small cross-sectional area and a high velocity, they are more sheared than the downwelling regions. Hence the net shear on the up- and downwelling regions is minimized by allowing the region with maximum shear (the upwelling region) to assume the shape with minimum effective surface area (a cylinder), while the region with less shear (the downwelling region) assumes the shape with the larger surface area (a two-dimensional sheet). The predominance of upwelling cylinders and downwelling sheets may also be largely determined by the spherical geometry of the ﬂuid layer. When two planar horizontal boundary-layer ﬂows converge, they are eventually forced to bend and move away from the boundary surface in a sheet-like ﬂow. If the boundary-layer ﬂows are constrained to move on a spherical surface, the sheet-like ﬂow away from the surface will be stretched (compressed) along the plane of the sheet as it moves radially outwards (inwards) from the spherical surface; radial outward (inward) motion corresponds to upwelling (downwelling) in the spherical shell. Hence a downwelling sheet tends to thicken and become concentrated as it descends. An upwelling sheet would become stretched and dispersed. Downwelling motion in a spherical geometry, therefore, tends to preserve sheet-like structures, whereas upwelling motion tends to disrupt them. The cylindrical shape of upwelling plumes may then occur because once the sheet-like structure disintegrates, the ﬂow coalesces into shapes with minimum surface area to reduce the net viscous shear opposing their motion. Bercovici et al. (1989) found the steady cubic and tetrahedral patterns to persist for Ra at least up to 7 × 104 . The study of this problem was extended to the case of high Ra values (up to 108 ) by Yanagisawa and Yamagishi (2005). They basically conﬁrmed the earlier results of Zebib and co-workers for Ra = O(104 ) and revealed that for Ra ≥ O(106 ) the velocity and temperature ﬁelds become time dependent (see, e.g., Figure 4.56). For the highest Ra considered (108 ), in particular, the pattern exhibits a spoke pattern-like structure very similar to those already described in Section 4.5, with unsteady release of thermal

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Figure 4.56 Snapshot of time-dependent convection in a spherical shell in the limit as Pr → ∞ (Ra = 5 × 105 , constant-viscosity approach). After Hernlund and Tackley (2003); Copyright Elsevier, 2003

plumes from both top and bottom boundary layers. Further details on the peculiar dynamics of these plumes are provided in Chapter 5.

4.13.4 The Geodynamo Problem Due to the importance of the magnetic ﬁeld for terrestrial life, a proper understanding of the dynamo processes in the Earth’s metallic core is one of the great challenges in modern geophysics. By its very nature, the problem is interdisciplinary and lies at the interface of theoretical physics, geophysics, ﬂuid dynamics, electromagnetism and applied mathematics. The subject of the Earth’s dipolar magnetic ﬁeld and its fundamental nature is a long-standing issue, which has captured the attention of many renowned scientists: William Gilbert, Andr´e-Marie Amp`ere, Ren´e Descartes, Edmond Halley, Karl-Friedrich Gauss, Lord Blackett and many others who contributed to the development of science have worked on this problem. Every well-educated student knows that the dipolar magnetic ﬁeld of a bar magnet or any other type of permanent magnet is created by the coordinated spins of electrons and nuclei within iron atoms (the reader is referred to the fundamental concepts illustrated in Section 1.9). As anticipated in Section 3.3.3, however, the Earth’s core is hotter than 1043 K, the Curie point temperature at which the orientations of spins within iron become randomized (Tcore ∼ = 4200 K). Such randomization causes the substance to lose its magnetic ﬁeld, which leads to the conclusion that the origin of the Earth’s magnetic ﬁeld cannot be related to an effect of permanent magnetization of the medium, as expressed by Eq. (1.193) in Chapter 1. The current consensus is that ﬂow of the liquid iron alloy within the Earth’s outer core, driven by buoyancy forces and inﬂuenced by the Earth’s rotation, generates large electric currents that induce a magnetic ﬁeld. This process is universally referred to as a convective dynamo. More speciﬁcally, by deﬁnition, a convective dynamo is the mechanism by which a magnetic ﬁeld is generated and self-sustained by the motion of an electrically conducting ﬂuid in a simply connected domain (Busse, 1978). The motion can itself be produced by thermal convection. The electromotoric force produced by the ﬂuid motion tends to generate a magnetic ﬁeld. This effect is counterbalanced, however, by ohmic dissipation (there is a natural decay of the ﬁeld; the dynamo process converts mechanical energy into magnetic energy and dissipates it in the form of ohmic heat). On the other hand, the magnetic ﬁeld itself reacts on the motion through the Lorentz force. The ﬁeld is self-sustaining if, on average, the generation of ﬁeld is balanced by its decay. From a mathematical point of view, such a complex mutual interference generally leads to a bifurcation problem for a magnetohydrodynamic system (in practice, the onset of magnetic ﬁeld generation in a moving ﬂuid of sufﬁciently high electrical conductivity differs from the usual hydrodynamic instabilities only in that a new physical quantity, the magnetic ﬁeld is introduced).

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Historically, the self-excited dynamo mechanism was ﬁrst proposed by Sir Joseph Larmor in 1919. The guiding principle at the basis of this theory was that the non-magnetic solution (B = 0) in a magnetohydrodynamic ﬂow can become unstable if the ﬂow is vigorous enough. It was soon realized, however, that this sort of instability, which can easily be introduced using a simple mechanical disk device (see, e.g., Busse, 1978), is not as straightforward when it concerns a uniform volume of conducting ﬂuid. A major cause of concern was ﬁrst identiﬁed by Thomas Cowling, as early as 1934. Cowling found that an axially symmetric ﬁeld cannot be maintained by such a dynamo (Cowling’s theorem). This result suppressed the hope for any simple axially symmetric description of the dynamo process (a tempting approach considering the axisymmetric structure of the Earth’s ﬁeld to a ﬁrst approximation, at least as seen from the outside). An even more general result came afterwards and is known as Zeldovich’s theorem: no two-dimensional solution can be sought. These initial efforts led to the conclusion that the problem has to be envisaged directly in the three dimensions of space, without the hope of a ﬁrst simpliﬁed spatial approach. In general, the stability of a given velocity three-dimensional ﬁeld V with respect to growing magnetic disturbances B is governed by the so-called magnetic induction equation. The question of whether growing solutions B of the magnetic induction equation (which is linear in B for assigned ﬁxed V ) exist is called the kinematic dynamo problem. The fully nonlinear problem in which the reaction of the magnetic ﬁeld on the velocity ﬁeld through the Lorentz force is taken into account is generally referred to as the magnetohydrodynamic dynamo problem. Obviously (see, e.g., Braginsky and Roberts, 1995), the model equations are the Navier–Stokes equations coupled with the energy equation (the thermal-convection equations) and the Maxwell equations. For an electrically conducting ﬂuid and above the Curie point, the Maxwell equations in their magnetohydrodynamic approximation in which the displacement current is neglected (the reader is referred again to the general theoretical background provided in Section 1.9) reduce to ∇·B = 0

(4.65)

∂B ∇∧E = − ∂t ∇ ∧ B = µ0 J f

(4.67)

J f = σe (E + V ∧ B)

(4.68)

(4.66)

and Ohm’s law reads

where B, E, J f , µ0 and σe are the magnetic ﬂux density, electric ﬁeld, electric current density, magnetic permeability and electrical conductivity, respectively. Simple manipulations of such equations lead to a speciﬁc equation for the magnetic induction, as shown in the following. Combining Eq. (4.66) with Eq. (4.68) yields Jf ∂B (4.69) +V ∧B =∇∧ − ∂t σe which, taking into account Eq. (4.67), can be rewritten as 1 ∂B ∇ ∧B +V ∧B =∇∧ − ∂t µ 0 σe

(4.70)

and since mathematically ∇ ∧ (∇ ∧ B) = ∇(∇ · B) − ∇ 2 B and according to Eq. (4.65) the divergence of B is equal to zero, this ﬁnally leads to ∂B = ∇ ∧ (V ∧ B) + η∇ 2 B ∂t

(4.71)

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which represents the aforementioned magnetic induction equation [η = (µ0 σe )−1 is known as the magnetic diffusivity] in which the ﬁrst term in the second member accounts for the buildup or breakdown of the magnetic ﬁeld and the last term is the rate of decay of the magnetic ﬁeld due to Ohmic dissipation. In the framework of such an approximation, the momentum equation in dimensional form [Eq. (2.5)] can be written as 1 1 ∂V + ∇ · [V V ] + ∇p = ν∇ 2 V − [βT (T − TREF )]g + F L ∂t ρ0 ρ0

(4.72)

where F L is the dimensional Lorentz force: FL = Jf ∧ B

(4.73a)

which according to Eq. (4.67) can also be expressed as FL =

1 (∇ ∧ B) ∧ B µ0

(4.73b)

As a consequence, Eq. (4.72) ﬁnally reads 1 1 ∂V (∇ ∧ B) ∧ B + ∇ · [V V ] + ∇p = ν∇ 2 V − [βT (T − TREF )]g + ∂t ρ0 ρ0 µ0

(4.74)

1

Introducing nondimensional quantities by using d 2 /α, α/d, ρ0 (α/d)2 and ν(ρ0 µ0 ) /2 /d as units of time, velocity, pressure and magnetic ﬂux density, respectively (where d is the thickness of the spherical shell), the magnetic induction and momentum equations in nondimensional form read (as usual in this book, for simplicity the nondimensional quantities are denoted by the same symbols as used before scaling them by the reference units) Pr 2 ∂B ∇ B (4.75) = ∇ ∧ (V ∧ B) + ∂t Prm ∂V (4.76) = −∇p − ∇ · [V V ] = Pr ∇ 2 V − Pr RaTf (r)i g + Pr2 (∇ ∧ B) ∧ B ∂t where Ra is the Rayleigh number based on a reference gravity value [the function f (r) takes into account the effective dependence of gravity on the radius; the temperature, measured with respect to TREF , is scaled by a reference T ] and Prm is the magnetic Prandtl number deﬁned as the ratio of the kinematic viscosity to the magnetic diffusivity (ν/η). These equations, obviously, must be supplemented with the continuity and energy equations [Eqs (1.59) and (1.61), respectively]. The problem is made even more complex by the presence of other effects, which have to be properly modelled (inﬂuence of the Earth’s rotation, Coriolis force, etc., that lead to additional source terms in the momentum equation) and by the variety of possible boundary conditions which can be used to simulate in a more or less realistic way the exchange of heat, mass and momentum at the outer core/mantle and outer core/inner core boundaries. The detailed treatment of these aspects is beyond the scope of the present section, which is limited to a survey of the salient concepts, equations, methods of analysis and most important results that have appeared in the literature. As mentioned before, over recent years there have so far been two basic approaches for the treatment of the three-dimensional dynamo instability problem. In the ﬁrst approach, one starts from a given, known ﬂuid ﬂow and studies (numerically) its linear stability under magnetic disturbances (the aforementioned kinematic dynamo problem). There are four limitations to this modus operandi : ﬁrst, it does not fully solve the bifurcation problem since it only deals with linear stability (see, e.g., the arguments provided in Section 1.5.5). Second the convective ﬂows used as input for the induction equation are usually fairly simple models of ﬂows rather than solutions of the actual magnetohydrodynamic equations. Investigators

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resorting to this method, in general, use as initial (known) ﬂuid ﬂow the leading part of the primary branches of pure convective solution to the classical RB problem in a spherical shell (the pattern selection rules and stability of these ﬂows have been discussed in Section 4.13.1 according to the studies of Busse and co-workers). After determining the critical parameter values and the critical modes by the linear stability analysis of the induction equations, the symmetry-breaking type of the bifurcation is evaluated by examination of the action of the symmetry group of the pure convective ﬂow on the critical eigenvectors. Then standard bifurcation theory is applied to list the bifurcated solutions with their symmetry and possible stability properties (see, e.g., Gubbins, 1973, and Vivancos et al., 1999). Third, with such an approach the resulting structure of the magnetic ﬁeld depends essentially on the basic thermal convection considered. Fourth, the resulting magnetic ﬁeld cannot exert any feedback on convection, which is, perhaps, the most important limitation of this method. The second approach is a direct simulation of the full magnetohydrodynamic (MHD) system (Glatzmaier, 1984). We will discuss the various bottlenecks of this strategy later. Here it is worth starting the discussion by stressing its major advantages. Remarkably, when such a model generates a magnetic ﬁeld that, at the model’s surface, looks qualitatively similar to the Earth’s surface ﬁeld in terms of structure, intensity and time dependence, then it is plausible that the 3D ﬂows and ﬁelds inside the model core are qualitatively similar to those in the Earth’s core. Analysing these detailed simulated data provides a physical description and explanation of the model’s dynamo mechanism and, by assumption, of the geodynamo. With this method, the set of coupled nonlinear differential equations described earlier, with a set of prescribed boundary conditions, is solved for each numerical time step to obtain the evolution in 3D of the ﬂuid ﬂow, magnetic ﬁeld and thermodynamic perturbations. Most geodynamo models have employed spherical harmonic expansions in the horizontal directions and either Chebyshev polynomial expansions or ﬁnite differences in radius. Historically, the ﬁrst MHD models of the Earth’s dynamo that successfully produced a time-dependent and dominantly dipolar ﬁeld at the model’s surface were not published until 1995 (Glatzmaier and Roberts, 1995; Jones et al., 1995; Kageyama and Sato, 1995, 1997). Since then, several groups around the world have developed dynamo models and several others are currently being designed. Some features of the various simulated ﬁelds are robust, such as the dominance of the dipolar part of the ﬁeld outside the core. These simulations have revealed that within the ﬂuid outer core, where the ﬁeld is generated, ﬁeld lines are twisted and sheared by the ﬂow, leading to a very intricate pattern. The ﬁeld that extends beyond the core, however, is signiﬁcantly weaker and dominantly dipolar at the model’s surface, not unlike the Earth’s geomagnetic ﬁeld. Simulations are available over a wide range of the magnetic Prandtl number [O(10−1 ) ≤ Prm ≤ O(102 )]. Dipolar dynamos have been found for low amplitudes of convection and high magnetic Prandtl numbers. At the opposite side, quadrupolar dynamos set in ﬁrst, but as the amplitude of convection increases, an evolution towards magnetic ﬁelds with prevalent dipolar symmetry occurs. According to such simulations, the threshold Rayleigh number able to maintain a stable dipolar ﬁeld generally increases with decreasing Prm . This behaviour simply reﬂects the need for a higher velocity to sustain the geodynamo as the magnetic diffusivity of the liquid increases (recall the magnetic Prandtl number is inversely proportional to the magnetic diffusivity); in other words, the dynamo action requires motions to be vigorous enough to counteract magnetic diffusion and ﬁeld decay [in practice, as discussed by Busse, 2002, the magnetic Reynolds number, which is deﬁned as product of Prm and the mean velocity of convection, must exceed a value of O(102 ) to sustain the dynamo action]. These studies have also provided interesting information about the pattern of convection established inside the outer core. Convection in such a region occurs in columns parallel to the rotation

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Figure 4.57 Columnar structure of convective ﬂow in the Earth’s outer core (the Coriolis effect induced by the rotation of the Earth’s forces liquid metal to follow helical paths)

axis (Figure 4.57). These columns drift slowly around the rotation axis in time leading to appreciable variations of the magnetic ﬁeld (which might explain the so-called secular variations of the terrestrial magnetic ﬁeld). These spiralling convective ﬂows that align with the rotation axis have been proved to be caused by the effect of planetary rotation. The magnetic ﬁeld is ampliﬁed by these helical motions through a mechanism known as the α effect (see, e.g., Volk et al., 2008, and references therein). The strength of the shear ﬂow on the ‘tangent cylinder’ (the imaginary cylinder tangential to the inner core equator), which depends on the relative dominance of the Coriolis forces, is not the same for all simulations. Likewise, the vigour of the convection and the resulting magnetic ﬁeld generation tends to be greater outside this tangential cylinder for some models and inside for others. However, all the solutions have a dominantly dipolar magnetic ﬁeld outside the core. Some of these simulations also displayed the capability to capture some of the features of the Earth’s magnetic ﬁeld on very large time-scales. From measurements of changes in fossil magnetism over time (the direction and intensity of the magnetic ﬁeld in the past can be inferred from the magnetization acquired by rocks at the time of their formation), it is known that the Earth’s magnetic ﬁeld has reversed polarity at irregular intervals in the past. The Earth’s dipolar magnetic ﬁeld from time to time ‘starts to oscillate’ and changes its polarity from one sign to the other. This phenomenon has been called reversals and typically lasts 103 − 104 years. The duration of such an ‘oscillating’ phase is extremely short compared with the time between consecutive reversals. In fact, the average time between two reversals is about 50 times longer than the duration of the reversal itself. The appearance of the reversals seems to be chaotic rather than periodic. Remarkably, several dynamo simulations produced spontaneous nonperiodic magnetic dipole reversals. Simulations using the Glatzmaier–Roberts dynamo, in particular, demonstrated that

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changes in the pattern of heat ﬂow at the core–mantle boundary can alter the frequency of reversals, depending on the imposed pattern of heterogeneity. Many review articles have been written that describe and compare the various mathematical models elaborated by researchers in the framework of the magnetohydrodynamic dynamo problem (differing, as mentioned before, especially in the choice of the boundary conditions and of the characteristic numbers) and the related numerical results (e.g. Hollerbach, 1996; Glatzmaier and Roberts, 1997; Fearn, 1998; Busse, 2000, 2002; Dormy et al., 2000; Roberts and Glatzmaier, 2000; Buffett, 2000; Glatzmaier, 2002; Dormy, 2006). There is no doubt that geodynamo models will continue to open new opportunities for studying interactions between the core and the rest of the Earth. Thermal interactions with the mantle may prove to be the most important. Estimates of the heat ﬂow across the core–mantle boundary inﬂuence the vigour of thermal convection in the core. These inﬂuences should be reﬂected in the magnetic ﬁeld. A better understanding of these inﬂuences will allow their signature in paleomagnetic observations to be interpreted. Current limitations related to this strategy are essentially of a computational nature. When assuming Earth values for the radius and rotation rate of the core, all models of the geodynamo have been forced (due to computational limitations) to use a viscous diffusivity that is at least three to four orders of magnitude larger than estimates of what a turbulent (or eddy) viscosity should be for the spatial resolutions that have been employed. Because of the large turbulent diffusion coefﬁcients, all geodynamo simulations have produced large-scale laminar convection, that is, convective cells and plumes of the simulated ﬂow typically span the entire depth of the ﬂuid outer core, unlike the small-scale turbulence that likely exists in the Earth’s core. The magnetic Prandtl number Prm = O(10−6 ) for liquid metals and the iron core, but for computational efﬁciency it is usually assumed to be in the range O(10−1 ) ≤ Prm ≤ O(102 ). Turbulence in the outer core is a natural consequence of the properties of the dynamo mechanism. Like all liquid metals, liquid iron dissipates electrical currents much more efﬁciently than heat and momentum. In other words, the ratios of the kinematic viscosity or thermal diffusivity to the magnetic diffusivity (these ratios are the aforementioned magnetic Prandtl number and the Roberts number Rb = α/η, respectively) are exceedingly small [of the order of O(10−6 ), as mentioned earlier]. Since, as discussed previously, dynamo action requires motions to be vigorous enough to counteract magnetic diffusion, it means that they are extremely vigorous in comparison with, say, viscous effects. In practice, the ratio of these two effects is the aforementioned Reynolds number. A large Reynolds number leads naturally to a strong connection with the difﬁcult problem of hydrodynamic turbulence. In the case of Earth core dynamics, this issue is made even more challenging by the rapid rotation of our planet. The Coriolis force, in addition to the magnetic Lorentz force, strongly affects the turbulent ﬂow, reducing turbulence, but also making it very anisotropic. The properties of this rapidly rotating magnetohydrodynamic turbulence are, as yet, far from understood. Greater spatial resolution and longer integrations will reduce some of the difﬁculties that are currently encountered. However, future progress will not rely solely on incremental advances in computing capabilities. A better understanding of turbulence in the core and its inﬂuence on the resolvable part of the ﬂow will be essential for improving the reliability of geodynamo simulations. Present schemes that account for turbulence with isotropic diffusivities may not be adequate because the effect of rotation and magnetic ﬁeld can make turbulence strongly anisotropic. A potentially more serious limitation is that turbulent diffusivities only account for the transfer of energy from large scales to small scales. More effort will be required to describe reliably the turbulent interactions that transfer magnetic energy back to the large scales.

5 The Dynamics of Thermal Plumes and Related Regimes of Motion

5.1

Introduction

Thermal plumes are widespread phenomena in Nature and technology. They are one of the many examples in the natural, industrial or urban environment where lighter materials rise into an overlying ﬂuid (plumes from urban mass ﬁres, industrial stacks and terrestrial volcanoes provide typical instances of such behaviours; see also the discussions in Chapter 3). A feature common to all these circumstances is the existence of localized regions where the temperature is higher than that of the surroundings and/or heat is injected directly into the ﬂuid. Plumes, however, are not an exclusive prerogative of systems for which energy is provided to the ﬂuid via a single source or a discrete distribution of sources of buoyancy. Remarkably, they can become a persistent feature of ﬂow dynamics also when localized heating is replaced by homogeneous thermal boundary conditions and the system is limited by solid walls [which leads again to the classical subject of Rayleigh–B´enard (RB) convection]. The role of thermal plumes in determining some spatial and temporal features of RB ﬂows has been already discussed to a certain extent in Sections 4.2 and 4.4 with regard to some secondary and tertiary modes of convection in an inﬁnite layer (e.g. oscillatory knot, oscillatory bimodal and ensuing spoke-pattern convection). When the Rayleigh number is increased to relatively large values, it is known, however, that in all the ﬁnite-sized systems discussed in Chapter 4 (rectangular, parallelepipedic or cylindrical enclosures, etc.), well-deﬁned boundary layers develop along the solid walls (see Section 2.5 for additional details about the existence and properties of these structures) and that, in general, ensuing convection is characterized by the time-dependent (often chaotic) release of unsteady thermal plumes, which detach from the aforementioned boundary layers (Lappa, 2007a). These phenomena are intimately related to the development of the turbulent regime and its features in many cases of practical interest [for which the geometric aspect ratio, i.e. the ratio of the vertical extension to the horizontal extension is of O(1) or even O(10−1 )].

Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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In particular, recalling the fundamental concepts introduced at the beginning of this book (Section 1.1), these thermal plumes can be regarded as the atomic (sub-)structures whose interrelations (or interactions) are the primary mechanisms leading to the formation of the pattern. In general, such turbulent patterns display a complex spatial structure on distinct length scales. Typically, they are given by the superposition of two well-deﬁned coherent structures: (1) a large-scale circulation spanning the height of the considered domain (e.g. the ﬂuid container) and (2) intermittent bursts of thermal plumes from the upper and lower thermal boundary layers. It is known that the resulting complex dynamics are the consequence of many competing factors including the strong external driving, many interacting components and the relatively large extension of the system in the vertical direction (comparable to the horizontal dimension). In this chapter, we explore the development of spatiotemporal chaos in such systems. For simplicity, the discussion is divided into three sections. The ﬁrst (Section 5.2) is devoted to the general description (i.e. with a fairly strong degree of abstraction) of the possible regimes of plume growth for phenomena originated from single and localized (points or segments) sources of heat and the typical instabilities to which they can be subjected. The second (Section 5.3) focuses on the efforts made over the years to elaborate effective constitutive relations for a macroscopic theory able to explain how large-scale behaviours can arise from interactions (at a smaller scale) among plumes. The last section (Section 5.4) comes back to the original problem in which convection is originated from a discrete proﬁle of sources. These sources are supposed to be located at the bottom of a closed container and various insights are provided (allowing the number of sources to change while keeping the geometry ﬁxed) by means of critical comparison with the analogous case in which the container is uniformly heated from below.

5.2

Free Plume Regimes

In this section, attention is focused on the development of single plumes. A single laminar plume, typically referred to in the literature as a free or unbounded plume, develops as the ﬂuid near a point or horizontal line heat source begins to rise (other shapes of the source are also possible in principle, e.g. square, cylindrical, spherical; see, e.g., Lappa et al., 2004a). As the heated ﬂuid rises, it pushes aside the material above it and is, itself, in turn deﬂected. The rising material produces a stalk, while the deﬂected ﬂuid produces a cap on top. As the pushing and deﬂection continue, the edge of the cap may further fold over. The result is something that looks like a mushroom. As outlined above, there are two types of commonly studied free plumes: a 2D line source plume that is produced by placing a horizontal heated wire in a box of cold ﬂuid (see, e.g., the experiments of Rouse et al., 1952) and an axisymmetric plume created by a point heat source. The theory for steady laminar plumes is well established. Scaling arguments indicate that the vertical velocity is constant (Batchelor, 1954a). Numerical results are available up to a Prandtl number of 10 (e.g. Fujii, 1963; Brand and Lahey, 1967; Worster, 1986) and asymptotic analyses for larger Prandtl numbers may be found in Gebhart et al. (1970), Worster (1986) and Li˜na´ n and Kurdyumov (1998). These analyses are valid for the plume stem far from the leading edge (the cap) and do not specify the cap behaviour. Studies on the starting (cap) behaviour of thermal plumes have been reported by Moses et al. (1993) and Kaminski and Jaupart (2003). It has been well known since 1954, however, that as a plume rises, it widens due to the diffusion of both heat and momentum in the lateral plane. A line source plume forms a wedge, with the angle controlled by the rate of entrainment (Batchelor, 1954a). Hier Majumder et al. (2004) extended these investigations to Prandtl numbers ranging from 10−2 up to 104 , illustrating the possible existence of different regimes of plume growth (categorized in

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Figure 5.1 Sketch of development of thermal and velocity boundary layers along the plume body

terms of relative thickness of thermal and velocity boundary layers along the plume body, see Figure 5.1) in the Prandtl-Rayleigh space. According to these studies, the thickness of thermal and vertical-velocity boundary layers grows typically as δT ∝ s a

(5.1)

δV ∝ s b

(5.2)

and

where s is the coordinate along the plume axis and a and b are exponents depending on the considered regime (in regimes with high thermal diffusivity, the thermal boundary layers are characterized by a < 1/4; plumes in nondiffusive regimes have a > 1/4; in regimes with low viscosity, vertical velocity boundary layers are featured by b < 1/4; and plumes in high-viscosity regimes have thick boundary layers, i.e. b > 1/4). The distinct regimes of plume growth for ﬁnite Prandtl number convection were delineated precisely by varying systematically the Rayleigh and Prandtl numbers. The variation in the thickness of these boundary layers led, in particular, to partition the Ra–Pr number space into four regimes of plume growth as shown in Table 5.1 and Figure 5.2.

5.2.1 The Diffusive–Viscous Regime This regime (DV) is deﬁned by both wide thermal and velocity boundary layers. Plumes in this regime have thick plume stems with no distinct plume heads and no strong vortices. Table 5.1 Distinct regimes of plume growth classiﬁed according to the exponents a and b in Eqs. (5.1) and (5.2) Regime DV IVD VND IVND

a

b

>1/4 >1/4 <1/4 <1/4

>1/4 <1/4 >1/4 <1/4

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Figure 5.2 Plume convective regimes in the (Pr,Ra) plane deﬁned on the basis of the rate of growth of thermal and velocity boundary layers with the plume height: DV = diffusive–viscous regime; IVD = inviscid–diffusive regime; VND = viscous–nondiffusive regime; IVND = inviscid–nondiffusive regime. After Hier Majumder et al., 2004; reproduced by permission of the American Institute of Physics

The strength of the diffusive component for such case is generally evident in the temperature map. It is responsible for the homogenization of the temperature ﬁeld around the plume, which leads, as mentioned above, to a thick plume stem without a separate plume cap. The velocity ﬁeld for a plume with Pr = 1 and Ra = 105 is shown in Figure 5.3a. There are cells of opposite vorticity on the right and left halves of the plume. The cells are relatively diffuse and no strong spiralling vortices can be seen. As expected from the relatively strong viscosity, the velocity is slowed signiﬁcantly by viscous friction. This regime is characterized by relatively low Rayleigh numbers [O(105 )] and not too high Prandtl numbers. The development of turbulence is generally impeded by both the high viscosity and thermal diffusivity. Due to the lack of ﬁne-scale turbulent structures, numerical simulations can be conducted with relatively coarse grids.

5.2.2 The Viscous–Nondiffusive Regime This regime (VND) is deﬁned by thin temperature boundary layers and thick velocity boundary layers. It has thin plumes with sharp features and no strong vortices. Figure 5.3b conﬁrms that the plume shape is clear and distinct with a thin stem. It has a well-deﬁned head with side lobes. With time, the plume lobes remain relatively distinct and are not signiﬁcantly deformed by vortex structures (the vorticity ﬁeld is featured by two broad regions of opposing vorticity on each side of the plume without sharp vortices). The thick velocity boundary layers seen in this ﬂow result in the velocity being dissipated by viscous friction rather than by vortices. This regime occurs at moderate Rayleigh numbers [O(105 ) ≤ Ra ≤ O(106 )] and relatively high Prandtl number [Pr ≥ O(10)]. Since it is characterized by small, thin plume structures due to the low thermal diffusivity, ﬁne grids are required to capture the related dynamics properly.

5.2.3 The Inviscid–Diffusive Regime This regime (IVD) is deﬁned by thin velocity and thick temperature boundary layers. The plume has a thick, ﬂame-shaped stem (Figure 5.3c).

The Dynamics of Thermal Plumes and Related Regimes of Motion

(a)

(b)

(c)

(d)

199

Figure 5.3 Example of plumes (temperature contour map and streamlines) for various possible regimes (the plume is obtained by placing a line heat source at the centre of the bottom of a box of colder ﬂuid). (a) Viscous–diffusive regime (DV) for Pr = 1 and Ra = 105 ; (b) viscous–nondiffusive regime (VND) for Pr = 15 and Ra = 106 ; (c) inviscid–diffusive regime (IVD) for Pr = 0.01 and Ra = 107 ; (d) inviscid– nondiffusive regime (IVND) for Pr = 1 and Ra = 108 (numerical simulation, M. Lappa)

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In the temperature map, it has a weakly developed head area with no lobes. Moreover, the head tends to merge with the plume stem over time due to the high thermal diffusion. The head region, however, is well deﬁned in the vorticity ﬁeld by strong vortices. Vortices develop in the head region of the plume because the low viscosity does not allow viscous dissipation by diffusion. This occurs at small Prandtl numbers [Pr < O(1)] and moderate Rayleigh numbers [Ra = O(106 )].

5.2.4 The Inviscid–Nondiffusive Regime This fourth regime (IVND) is deﬁned by both thin temperature and velocity boundary layers. The nondiffusive nature of the regime can be clearly observed in both the temperature and velocity ﬁelds (Figure 5.3d). The plume has a thin, sharp stem with a well-deﬁned cap and lobes that are signiﬁcantly deformed by vortex structures. The ranges of existence of this regime are Ra ≥ O(108 ) for Pr = 10−3 , Ra ≥ O(106 ) for Pr ∼ =1 and Ra ≥ O(107 ) for Pr > O(104 ). With time, horizontal shear develops and the plume stem becomes sinuous. Alternating vortices are shed from either side of the plume lobes. The existence of these vortices further conﬁrms the inviscid nature of the ﬂow and distinguishes it from the VND regime. The thin velocity boundary layers, in particular, indicate that the velocity cannot be dissipated by viscous friction and that it is dissipated in the vortices. The tendency to develop such vortices makes this regime very susceptible to the development of turbulent structures (it is the most turbulent of the four regimes, being characterized by ﬁne, turbulent structures that cover a wide range of small scales). This makes the regime the most difﬁcult one from a computational point of view.

5.2.5 Sinuous Instabilities Created by Horizontal Shear As explained earlier, in the DV and the IVD regimes, the diffusion of the heat away from the plume into the surrounding cold ﬂuid due to the relatively high thermal diffusivity tends to result in broader, more large-scale structures, whereas the VND and IVND regimes are generally characterized by a wide range of small-scale structures. The plumes in all these situations, however, are susceptible to the sinuous instabilities created by horizontal shear (see Section 1.5.4 for some theoretical background about this type of instability in inviscid ﬂows by simply replacing there the x coordinate with the coordinate s along the plume axis). With time, the plumes develop a sinuous, snake-like appearance due to horizontal shear. This shear can cause the plume stem to become so distorted that the local Rayleigh number becomes supercritical. This causes the original plume to collapse and new plumes to develop from the original plume stem (see, e.g., Cortese and Balachandar, 1993). An investigation of the hydrodynamic stability of a laminar plume arising from a horizontal line source of heat in the framework of the Tollmien–Schlichting theory of small disturbances was originally carried out by Pera and Gebhart (1971). Inviscid solutions of the Orr–Sommerfeld equation were obtained for both symmetric and asymmetric disturbances and the effect of the Prandtl number on the inviscid stability was calculated for asymmetric disturbances. The base ﬂow was found to be less stable for the asymmetric mode. In addition, the full disturbance momentum equations, coupled and uncoupled from the energy equation, were numerically integrated with the boundary conditions appropriate for asymmetric disturbances superimposed on the symmetric plume base ﬂow. Neutral stability curves were obtained in terms of the Grashof number. As discussed by Hier Majumder et al. (2004), phenomena of this kind are of great importance in many atmospheric and geophysical ﬂows. For example, thunderstorms and dust devils are associated with spiralling plumes of hot updrafts and cold downdrafts. The large component of vertical vorticity responsible for the spiralling behaviour is created by the interaction between the

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buoyancy-induced vertical ﬂow and shear associated with a mean horizontal ﬂow (wind in the atmospheric boundary layer). As discussed by Farnetani et al. (2002), plume deformation also plays an important role in the dynamics of the Earth’s mantle (Pr → ∞). Sinuous instabilities have also been observed in numerical experiments of solar convection (Pr ∼ = 0). They develop from shear instabilities immediately behind the plume head and propagate upwards along the stem. It is generally assumed that such astrophysical phenomena correspond to behaviour of the IVND regime. As outlined earlier, in fact, beyond its signiﬁcant tendency to evolve towards turbulence, this regime exhibits features almost independent of both viscosity and thermal diffusivity (which are similar to those typical of stellar convection). In particular, as proved by Hier Majumder et al. (2004), the vortices in the lobes of the IVND plume can form Kelvin–Helmholtz waves. The susceptibility to form such waves is measured by the local gradient Richardson number: −gβT ∂T Riloc = 2 ∂s ∂v ∂s

(5.3)

where the dimensional quantities have been already deﬁned in Section 2.1 and ∂v/∂s is the gradient of the radial velocity component (the shear). When the local Richardson number drops below 0.25, Kelvin–Helmholtz waves can form (the reader is also referred to the propaedeutical arguments provided in Section 3.4.3). Another potential instability in single plume turbulence occurs when a heavy ﬂuid is found on top of a lighter ﬂuid. This is known as a local Richardson shear instability. As shown by Vincent and Yuen (2000), this instability is generally associated with a large ‘local’ Rayleigh number. For instance, these authors conducted high-resolution two-dimensional calculations for a Boussinesq convection model with a Prandtl number of unity and Rayleigh numbers between 108 and 1014 and found evidence for a transition involving the branching of plumes at a Rayleigh number of O(1010 ). Inside the core of these ‘superplumes’ the structure was found to be extremely complex. Moreover, they conjectured about the existence of a second transition at Ra of O(1012 ), where a secondary instability may develop in regions of the local Rayleigh number which becomes supercritical inside the core of the superplumes. For additional information about the typical oscillatory instabilities of rising buoyant jets for a number of different cases and ﬂuid–ﬂuid systems of typical interest in the technological and industrial context, the reader may consider the interesting work of Cetegen and Ahmed (1993), Cetegen and Kasper (1996), Cetegen (1997) and Cetegen et al. (1998) (candle ﬂames and plumes of gas mixtures) and Lappa and Piccolo (2004), Lappa et al. (2004a, b), Savino and Lappa (2003b) and Savino et al. (2005) (solutal plumes originating in non-equilibrium conditions in organic systems with a miscibility gap), and references cited therein.

5.2.6 Geometric Constraints For the cases in which the environment where the plumes develop is not unbounded, but relatively limited, the ﬁnite size of the considered geometry also plays a role in determining plume behaviour and evolution. A relevant example is the work of Desrayaud and Lauriat (1993). They studied numerically two-dimensional time-dependent buoyancy-induced ﬂows above a horizontal line heat source inside rectangular vessels, with adiabatic sidewalls and top and bottom walls maintained at uniform temperature for Pr = 0.71 (air). Transitions to unsteady ﬂows were obtained by direct simulations for various depths of immersion of the source in the central vertical plane of the considered vessels.

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According to such simulations, for a square vessel (A = length/depth = 1) and a line source near the bottom wall, the thermoﬂuid-dynamic ﬁeld exhibits a sequence of instabilities beginning with a periodic regime having a high fundamental frequency followed by a two-frequency locked regime. Then, broadband components appearing in the spectra indicate chaotic behaviour and a weakly turbulent motion arises via an intermittent route to chaos. This behaviour is generally referred to (see, e.g., the experimental work of Noto, 1989) as natural swaying motion of conﬁned plumes. For rectangular vessels of aspect ratio lower than 1/2 (tall enclosures), however, the dynamics become very different. If the depths of immersion are greater than the width, the ﬂow undergoes a stationary bifurcation. This symmetry breaking is driven by the destabilization of an upper unstable layer of stagnant ﬂuid above the plume. Then a subcritical Hopf bifurcation occurs. If the depth of immersion is lower than the width of the vessel, a stable layer of ﬂuid is at rest below the line source. Then penetrative convection sets the whole air-ﬁlled vessel in motion and an oscillatory motion of very low frequency arises through supercritical Hopf bifurcation followed by a two-frequency locked state (Desrayaud and Lauriat, 1993).

Figure 5.4 Rayleigh–B´enard convection (Pr = 15, A = 1, Ra = 108 ; adiabatic lateral walls). Early stage of convection with multiple plume-detachment phenomena along the uniformly heated (cooled) bottom (top) wall (numerical simulation, M. Lappa)

5.3

The Flywheel Mechanism: The ‘Wind’ of Turbulence

The simplest model to study the nonlinear dynamics of ﬂuid systems heated from below and cooled from above is that derived by Lorenz (1963), which, as illustrated in Section 1.8.3, consisted of only three ordinary differential equations, resulting from a hard truncation of a Fourier expansion of the equations of two-dimensional thermal convection. The most important property of this

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model was the existence of a single clockwise or anticlockwise oriented convective cell and the appearance of chaos in certain intervals of the system parameters. In reality, however, things are more complex. As mentioned in Section 5.1, real ﬂow structures exhibit really intricate features when turbulent states are reached and thermal plumes play a crucial role in a number of them (Taylor et al., 1956; Turner, 1969; Chu and Goldstein, 1973; Castaing et al., 1989; Solomon and Gollub, 1990; Siggia, 1994). Turbulent Rayleigh–B´enard convection in enclosures with comparable horizontal and vertical sizes produces ﬁelds of intense updrafts and downdrafts (rising and descending plumes) that are responsible for much of the vertical heat transport. These structures (sometimes also referred to in the literature as ‘thermals’) have horizontal scales comparable to the thicknesses of the boundary layers in which they arise (see, e.g., Parodi et al., 2004). Figure 5.4 shows an example of such structures for the same conﬁgurations (the square cavity with Pr = 15 and adiabatic sidewalls) already shown in Figures 4.33 and 4.34 for lower values of the Rayleigh number. As time passes, they organize themselves into clusters with horizontal scales that grow with time and reach the width of the physical domain. According to experimental analyses (mostly available for transparent liquids with Pr > 1), it is known that the ﬂow tends to develop into time-dependent convection with a strong asymmetry and highly convoluted thermal plumes delineating a large-scale circulation. In such conditions, smaller thermal plumes detach from thermal boundary layers and extend over the entire cell, coalescing and creating local inversions of the temperature gradient adjacent to the boundary layers (Werne, 1993; Vincent and Yuen, 1999).

5.3.1 Upwelling and Downward Jets and Alternating Eruption of Thermal Plumes Figure 5.5 illustrates the typical ﬂow structure. Hot plumes congregate in an upwelling jet of ﬂuid near the right-hand wall of the container. A similar, downward jet formed from cold plumes

Figure 5.5 Sketch of the so-called ﬂywheel mechanism: the spatial organization of the thermal plumes produces a unique ﬂow structure, which undergoes a coherent rotation; plumes evolve from the boundary layers above (below) the bottom (top) plate; ﬁrst they move laterally toward the sidewall, driven by the prevailing circulation (the horizontal ‘wind’); they then travel vertically in the region near the sidewall, leaving the central vertical section of the cell relatively free

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occurs on the left-hand wall. Large numbers of hot plumes are also found in left-to-right motion in a mixing zone or viscous boundary layer, near the bottom of the container. A similar layer on the top contains cold plumes, moving from right to left. The central region contains a few plumes, hot and cold, in a partially random motion. These plumes wander chaotically, but also participate in an overall large-scale rotatory ﬂow (ﬂuid circulates as a loop with the plumes rising and falling on opposite sides of the loop). Since, as explained before, the cold and warm plumes are separated laterally in two opposing sidewall regions and exert buoyancy forces to the bulk ﬂuid, an alternating eruption of the thermal plumes, therefore, gives rise to a periodic impulsive torque, which drives a large-scale circulation continuously (see also Section 5.3.3 for additional and relevant information on the origin of such behaviour). This complex mechanism may be regarded as a ‘machine’ (Kadanoff, 2001) containing many different working parts: boundary layers, mixing zones, jets and a relatively free central region. In the light of the notions and explanations given at the beginning of this book in Section 1.1.1, these parts may be seen as the constitutive ‘ingredients’ whose interplay leads to the emergence of a macroscopic pattern with well-deﬁned properties. As outlined above and as illustrated experimentally by Qiu et al. (2000) and Qiu and Tong (2001), despite the large velocity ﬂuctuations, the time-averaged ﬂow ﬁeld maintains, in fact, a large-scale quasi-two-dimensional structure, which rotates in a coherent manner. As in the Lorenz model (but with the due differences), it has a prevailing two-dimensional nature and can be oriented clockwise or anticlockwise (both conﬁgurations are equally likely to occur). It is usually referred to in the literature as ‘wind of turbulence’ (Grossman and Lohse, 2000) or simply ‘wind’, ‘mean wind’ (Niemela and Sreenivasan, 2003) or ‘ﬂywheel’ (Kadanoff, 2001); accordingly, hereafter these terms will be used as synonyms together with ‘large-scale circulation’ or motion.

5.3.2 Geometric Effects This coherent single-roll structure scales with the aspect ratio (see Grossman and Lohse, 2003; for aspect ratio unity, the mean wind is comparable in scale to the container size) and with Ra (see, e.g., Chill`a et al., 1993; Grossman and Lohse, 2003; Qiu and Tong, 2001); as shown by Niemela and Sreenivasan (2003) the self-organized and coherent large-scale motion is a standard feature of turbulent convection at least up to Rayleigh numbers of the order 1013 , its shape, however, being a function of the effective Ra (from a tilted and nearly elliptical shape at moderately high Rayleigh numbers to a squarish shape at very high Rayleigh numbers). Interestingly, several investigators (see the experiments of Niemela et al., 2001, and Sreenivasan et al., 2002, and the recent theoretical analysis of Araujo et al., 2005) also revealed that even if the wind survives even when the dynamic parameter, namely the Rayleigh number, is very large, however, over a wide range of time-scales greater than its characteristic turnover time the wind velocity can exhibit occasional and irregular ‘reversals’ without a change in magnitude (this aspect is still a subject of intense investigation; see also Lappa, 2006b, and references therein). The existence of large-scale ﬂow of the type illustrated before for aspect ratio unity and Pr > 1 has been reported even for the case of relatively shallow conﬁgurations. Relevant experiments along these lines (for which it is worth opening a short discussion), for instance, were carried out by Krishnamurti and Howard (1981), who considered containers with width L an order of magnitude larger than the depth d [O(10) ≤ A ≤ O(102 ), L = 48 cm, Pr = 7]. They reported that for small enough values of the Rayleigh number (Ra < 106 ), multicellular convection with horizontal length scale comparable to the depth occurs. As the Rayleigh number was increased, however, this cellular ﬂow was found to disappear and to be replaced by a state with no permanent cell boundaries and a random array of transient

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plumes. Upon further increase (Ra ∼ = 2 × 106 ), these plumes displayed a drift in one direction near the bottom and in the opposite direction near the top of the container with the axes of plumes tilted in such a way as to allow upward transport of horizontal momentum via the Reynolds stress. With the onset of this large-scale ﬂow, the largest scale of motion was observed clearly to increase from that comparable to the depth to a scale comparable to the width. Thereby, it was concluded that there is not only a (time and/or horizontal) mean Eulerian velocity u(y) but also a net horizontal Lagrangian transport extending the entire width of the tank. On repetitions of the experiment, the large-scale ﬂow was seen to be sometimes from left to right along x at the bottom and from right to left along x at the top (counterclockwise). A model for such phenomena was proposed later by Howard and Krishnamurti (1986). According to this model, the ﬂow has primarily two scales of motion. The smaller scale ﬂow, with horizontal scale comparable to the layer depth d, is best described as transient bubbles or plumes that have an organized tilt away from the vertical. The larger scale ﬂow, having scale L d, is a horizontal ﬂow with vertical shear such that the ﬂow is oppositely directed near the bottom and the top. It is also worth mentioning that in the original experiments of Krishnamurti and Howard (1981), some attempts were made to control the emerging orientation of such large-scale ﬂow by sidewall forcing (i.e. lateral heating), which led to interesting results as discussed in the following: When the ﬂuid layer was heated from the side, the direction of the large-scale ﬂow could be determined only when the heating rate was large enough. Several different heating rates were tried. For example, when the power to the side heater was 3 W, this heating rate did not reverse an already established large-scale ﬂow. The rising motion induced at the heated side wall was accompanied by sinking motion a short distance (
5.3.3 The Origin of the Large-scale Circulation: The Childress and Villermaux Theories In the general case for which A = O(1), as outlined earlier, the self-organized and coherent large-scale structure can be divided into three regions in the rotation plane: (1) a thin viscous

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boundary layer, (2) a fully mixed central core region with a constant mean velocity gradient and (3) an intermediate plume-dominated buffer region. In such a context (all the plume activity is carried along in the overall rotatory motion), it is worth remarking that the plumes play a very important role as they act as heat pumps; in fact, they shuttle heat ﬂux directly between the top and bottom boundaries (the hot plumes rising with their caps on top, cold plumes falling cap-down). Because of contributions of a number of investigators, a lot of progress has been made in the understanding of such dynamics, and, in particular, of the large-scale circulation, its periodicity, the boundary layers and the plume interaction with the large-scale wind (i.e. all those factors which come under the heading of interrelation in the sense that has been given to this deﬁnition in Section 1.1.1). The large-scale coherent circulation is responsible for the aforementioned horizontal motion of the plumes along the top and bottom walls (the horizontal motion of the plumes is controlled by the wind velocity, since the only independent force acting on the plumes is the buoyancy force in the vertical direction). The large-scale motion, in turn, is a result of plume interaction phenomena, as proved theoretically by Childress (2000) and observed experimentally by Xi et al. (2004). Childress (2000) considered the dynamic origin of large-scale ﬂows in systems driven by concentrated Archimedean forces. He introduced a two-dimensional model of plumes, such as those observed in thermal convection at large Rayleigh and Prandtl numbers. According to this model, the onset of mean ﬂow should be regarded as an instability of a convective state consisting of parallel vertical ﬂow supported by buoyancy forces. In practice, the essence of this model can be synthesized as follows: as illustrated in Section 5.2, the vertical parallel ﬂow associated to the motion of a plume can undergo sinuous instabilities induced by horizontal shear. These instabilities lead to the appearance of localized vortices. The large-scale ﬂow ﬁnally results from the spontaneous tendency that the distinct vortices generated by the instabilities of several plumes exhibit to coalesce in vortices of larger scale. In agreement with this model, Xi et al. (2004) illustrated by direct experimental analysis that the interaction and merging of the vortices from neighbouring plumes effectively lead to groupings and/or merging of plumes, which in turn generate vortices of even larger scale; as a result of these interactions, the convective ﬂow eventually evolves into the coherent rotatory motion (the ﬂywheel) consisting mainly of the plumes themselves and spanning the whole convection box. These studies clearly demonstrated that it is the thermal plumes that initiate the horizontal large-scale ﬂow across the top and bottom conducting walls. Some authors have also shown that the phenomenon related to the transport of plumes along the top and bottom boundary layers (due to the action of the horizontal wind) may be somewhat theoretically linked (and/or ascribed) to the propagation of horizontal waves (Gluckmann et al., 1993; Villermaux, 1995; Kadanoff, 2001). Shelley and Vinson (1992) conjectured initially that the observed waves were normal modes associated with the buoyant shear layer at the wall. They introduced a simple model of such an unstably stratiﬁed boundary layer. The model consisted of a two-dimensional layer of constant vorticity against a wall and was aimed at modelling the effect of viscosity and the large-scale rolling motion in creating shear near the horizontal plates. The effect of buoyancy was included as a sharp density jump across the boundary of the shear layer. They studied both the linear analysis of the model and its nonlinear behaviour through numerical simulation. The model was found to reproduce much of the experimental behaviour. They observed, in fact, the formation of both plumes and swirls (propagating regions of rotational ﬂuid), similar in form and dynamics to those in the experiments. Also, they found that plumes and swirls arise from the same instability mechanism, differing only in the amount of shear acting upon them. Some years later (Villermaux’s theory), it was proposed that the temperature oscillation and related horizontally propagating waves are caused by a thermal boundary layer instability triggered by the incoming thermal plumes from the opposite conducting surface.

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This theory has proved to be in very good agreement with both experimental and numerical results. The related details coming from synergetic elaboration of experimental observations and theoretical arguments, are illustrated in the following for the convenience of the reader. The waves form on the thermal boundary layers in regions of high horizontal velocity gradient. The plumes, in turn, form on top of the waves, ride with the wind and are squeezed into the side walls. The waves appear as bulges in the height of the thin boundary layer at the bottom of the cell. As they move, the waves throw up a hot spray that tends to rise, forming sheets of hot, upward motion. These sheets break up into columnar structures that form the plumes shown in Figure 5.5. As they move across the bottom of the cell, these structures grow larger, as shown in Figure 5.6. A few plumes come loose and move into the central region. Most hit the right-hand wall and move upwards as a jet aimed toward the top of the cell. As a plume hits the upper wall, it makes a splash, and the splash makes a wave. Each wave moves along the top, producing a cold spray and, hence, cold plumes. The plumes form into a downward jet at the left-hand side, splash on the bottom and produce hot waves, thus keeping the motion going indeﬁnitely. These mechanisms lead to the alternating emission of cold and warm plumes between the upper and lower thermal boundary layers and the side-jets resulting from plume-clustering phenomena pull the ﬂuid around the container, forming the typical ﬂywheel pattern. Concerning the alternating emission of plumes, some interesting quantitative information can be also deduced from Villermaux’s theory. Assuming that the thermal plumes are transported by the large-scale circulation with a mean velocity U , Villermaux’s model assumes that the unstable modes (i.e. the thermal plumes) in the upper and lower thermal boundary layers interact through a delayed nonlinear coupling with a time constant equal to the cell crossing time τ = L/U , where L is the cell height. According to this model, because of this delayed coupling, the thermal plumes are excited alternately between the upper and lower boundary layers with a local frequency f = 1/2τ = U/2L (i.e. thermal plumes synchronize their emissions and organize their motions spatially between the

(a)

(b)

Figure 5.6 Turbulent Rayleigh–B´enard convection (Pr = 15, A = 1, Ra = 108 ; adiabatic lateral walls). Frames (a) and (b) show the temperature and velocity ﬁelds at two close instants during the fully developed regime in which a large-scale coherent circulation (the wind) has been established (numerical simulation, M. Lappa)

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top and bottom plates, leading to an oscillatory motion in the bulk region of the ﬂuid with a period equal to twice the plume’s cell-crossing time; similar conclusions seem to be also supported by the experiments of Sun et al., 2005).

5.3.4 The Role of Thermal Diffusion in Turbulent Rayleigh–B´enard Convection Grossman and Lohse (2000) identiﬁed several possible regimes for the phenomena discussed above in the Rayleigh number Ra versus Prandtl number Pr phase space (on the basis of available experimental results), according to whether the boundary layer or the bulk dominates the global kinetic and thermal dissipation, respectively, and by whether the thermal or the momentum boundary layer is thicker. Some relevant information in this direction was also provided by the numerical investigations of Verzicco and Camussi (1999). Their results (obtained for Pr in the range 0.022 ≤ Pr ≤ 15) revealed that, depending on the Prandtl number, in practice, there are two distinct ﬂow regimes: in the ﬁrst (Pr > 0.35), the large-scale ﬂow plays a negligible role in the heat transfer, which is mainly transported by the thermal plumes; in the second regime (Pr ≤ 0.35), the ﬂow is dominated by the large-scale recirculation cell (the wind) that is the most important ‘engine’ for heat transfer. These authors observed that whereas, as illustrated earlier (Figure 5.6), the thermal boundary layers for high-Pr ﬂuids are dominated by plumes shed at random positions from the horizontal plates, the low-Pr temperature ﬁeld is substantially different. In particular, the large thermal diffusivity prevents thermal plumes from being generated and the persistent recirculation cell induces a rising hot current on the right and a sinking cold current on the left (or vice versa) that dominate the ﬁeld. To summarize, from the integrated analysis of the data from experiments and simulations, for relatively high Prandtl numbers the heat transport is essentially due to the thermal plumes (mostly pertaining to the IVND regime discussed in Section 5.2), consistent with the scenario proposed by Castaing et al. (1989) and the observations of Ciliberto et al. (1996). In contrast, below Pr = 0.35, the thermal plumes are not generated or are weak (see the discussion about the inviscid–diffusive regime IVD in Section 5.2) and the heat is efﬁciently transported by the large-scale ﬂow. Also in this case, the large-scale motion induces viscous and thermal boundary layers; however, given the low value of Pr, the thermal boundary layers are spread over a relatively large vertical extension (the thickness is considerably larger than the thickness of the momentum boundary layer) and accordingly the mechanism related to the onset and propagation of waves along their external boundary elucidated in Section 5.3.3 is suppressed [vice versa for Pr > O(1), thermal boundary layers are relatively thin as illustrated in Figures 5.5 and 5.6; for further details on the thickness of these layers and related scaling laws, see Section 2.5].

5.4

Multiplume Conﬁgurations Originated from Discrete Sources of Buoyancy

As an example of the complexity of ﬂows that can be found also in the case in which the pattern is originated from discrete distributions of sources of buoyancy, this section deals with the case in which a relatively large number (N) of droplets (at a different temperature with respect to the surrounding immiscible liquid) are uniformly distributed on the bottom of a container (i.e. a liquid matrix conﬁned between two parallel and vertical walls with a periodic array of evenly spaced spherical liquid bodies in the interior). This example has important background applications in the technological processes in which two immiscible phases (one dispersed within the other) at different temperatures are involved.

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As a relevant reference case for illustrating these behaviours, in particular, as in Section 5.3 a square container is considered (A = 1). Other relevant non-dimensional geometric parameters, however, enter the problem: they are the non-dimensional diameter of the droplets d/L = 0.067 (L being the characteristic length of the container and d the droplet diameter) and the parameter κ = d/ (ratio of drop diameter to the distance between the centres of two consecutive droplets; see Figure 5.7). All the walls are assumed to be adiabatic, the Prandtl number of the matrix ﬂuid is Pr ∼ = 1.5 × 108 ) as in the = 630 and the corresponding Rayleigh number is of O(108 ) (Ra ∼ example considered in the preceding section (Figure 5.6). The number of droplets is changed in order to analyze the sensitivity of the overall system to this parameter (N = 4 → κ = 0.33 and N = 5 → κ = 0.4). The general aim of the present section is to show that even if, as illustrated in Section 5.2, a single source of buoyancy can behave as an intriguing pattern-forming dynamic system (leading to a wealth of different spatiotemporal modes of convection when the imposed temperature gradient is increased), the case of a multi-source conﬁguration can be even more complex. Interestingly, according to numerical simulations, the major outcomes of which are illustrated in the following pages, the emerging distribution of temperature depends on the multicellular structure of the convective ﬁeld and on associated pluming phenomena. Signiﬁcant adjustments to such patterns take place as time passes. The structure of the velocity ﬁeld and the number of rising plumes exhibit strong sensitivity to the number of sources of buoyancy as shown in Figures 5.8 and 5.9 and discussed below. For N = 4, at the beginning each droplet is characterized by its own rising plume. In this early phase of the process, the dynamics of the different droplets can be regarded as independent. Two convective cells are generated around each droplet and then rise attached to the mushroom-shaped head (the cap) of the related jet (hereafter the droplets will be referred to as droplet i with i = 1 → N, starting from the left side of the container). However, after a short initial transient behaviour, the two central rising jets i = 2 and i = 3 tend to be featured by a reduced rising velocity with respect to those at the edges (i = 1 and i = 4; see Figure 5.8a). In practice, they experience the return ﬂow of the lateral jets and for this reason tend to be retarded. After 46 s (see Figure 5.8c), the lateral plumes (1 and 2) merge, giving rise to a single rising current. The same behaviour holds for the plumes 3 and 4 since the system exhibits mirror symmetry with respect to the mid-section x = L/2. For this reason, after an initial stage, the lighter ﬂuid is transported upwards by two rising jets only. They are located approximately at x = L/5

g y

L

d l L

Figure 5.7 Sketch of the multidroplet conﬁguration and related geometric parameters

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(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

Figure 5.8 Pattern originated from multiple sources of buoyancy (Pr = 630, L = 3 cm, N = 4, Ra = 1.5 × 108 ): subsequent snapshots corresponding to (a) t = 35.6 s, (b) t = 39.2 s, (c) t = 46.3 s, (d) t = 57 s, (e) t = 71.3 s, (f) t = 82 s, (g) t = 89.1 s, (h) t = 114.2 s (numerical simulation, M. Lappa)

and x = 4L/5, respectively (instead of the four plumes that characterize the system behaviour in the early phase located at x = iL/5, i = 1 → 4). Evolution of the surviving plumes results in a shape that bulges out below y = L/2 and necks in above it. This leads to an amphora-like pattern (Figure 5.8d). With a further increase in time, two new plumes are originated from the central droplets 2 and 3. As they rise, the related combined thermal paths result in a shape that necks below y = L/2 and bulges out above it (Figure 5.8f). This phenomenon is then followed in time by the generation of two new plumes above the lateral droplets 1 and 4 (Figure 5.8g).

The Dynamics of Thermal Plumes and Related Regimes of Motion

(a)

(b)

(c)

(d)

(e)

(f)

(g)

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(h)

Figure 5.9 Pattern originated from multiple sources of buoyancy (Pr = 630, L = 3 cm, N = 5, Ra = 1.5 × 108 ): subsequent snapshots corresponding to (a) t = 35.6 s, (b) t = 39.2 s, (c) t = 42.75 s, (d) t = 53.4 s, (e) t = 64.1 s, (f) t = 71.2 s, (g) t = 81.9 s, (h) t = 99.75 s (numerical simulation, M. Lappa)

During these stages of evolution, the droplets seem to ‘work’ in mirror groups of two. Rising jets are initially created above the couple (1,4), then above the couple (2,3). Then the couple (1,4) is again involved in the generation of rising jets and so on. Therefore, a certain degree of periodicity in time can be highlighted with mechanisms exhibiting a somewhat repetitive behaviour. As time passes, new regimes evolve which exhibit a periodic alternation of central or lateral rising plumes. These phenomena are also susceptible to an interesting interpretation in terms of relative temporal phase shifts. The phase shift is deﬁned here as π L/L, L being the difference between the height (along y) of two plumes at a given instant (this simply means that two plumes having the same

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height are featured by a zero phase shift, whereas the phase shift is approximately π if two plumes, one extended throughout the container and the other conﬁned to the bottom, are considered). Periodic convective structures originated from droplets 1 and 4 do not exhibit a phase shift (φ14 = 0). Convective structures originated from droplets 2 and 3 also do not exhibit a phase shift (φ23 = 0). A phase shift φ = π occurs between droplets 1 and 2 or 3 and 4 (e.g. a new jet is generated above the second drop when the plume above the ﬁrst drop dies at the top of the cell). This also means that only two values of φij are possible (0 or π ). With a further increase in time however, the spectrum of possible phase shifts becomes more complex. The initial regular pulsating mechanism described above, with a rhythmic alternation of central and lateral couples of rising jets (featured by mirror symmetry with respect to the midsection x = L/2) is taken over by a new disordered regime in which the release of plumes seems to be erratic or chaotic (see Figure 5.8h). In such a regime, the pattern becomes asymmetric and such a symmetry breaking is associated with the presence of a variety of possible phase shifts φij . These trends are partially conﬁrmed by the simulations carried out for N = 5 (Figure 5.9). In such a case, the major stages of the system evolution are similar to those discussed before for the case N = 4: at an early stage, the ﬂow undergoes transition to an oscillatory state featured by a mirror symmetric regular alternation of plumes released by lateral and internal droplets [(1,5) and (2,4), respectively]. In terms of possible phase shift, however, the central drop i = 3 seems to work independently. The related plume, in fact, is extended along half of the height of the domain when the others have already reached the top of the domain and/or are still conﬁned to a region close to the surface of the droplets (see Figure 5.9d, where plumes 1 and 5 are extended throughout the domain, plumes 2 and 4 are still in very embryonic conditions and plume 3 affects the lower half of the container). Accordingly, it exhibits a phase shift φ3j ∼ = π /2 with respect to the pulsating phenomena pertaining to the other companion droplets. As time passes, however, as for the case already discussed for the array of four evenly spaced droplets, these regular behaviours are replaced by a fairly erratic generation/release of plumes and related possible time shifts (e.g. Figure 5.9h); this can be regarded as a signiﬁcant result as it highlights the spontaneous tendency of these conﬁgurations to evolve towards ﬁnal erratic pulsations regardless of the even or odd number of droplets. These simulations basically prove that fascinating time-dependent ‘pulsating’ phenomena can occur if evenly spaced distributions of sources of buoyancy are considered for Pr > O(1) at values of the Rayleigh number for which the corresponding case with uniform distribution of temperature at the horizontal walls (i.e. the canonical RB problem) would exhibit the ﬂywheel mechanism discussed in Section 5.3. When there are many independent sources of buoyancy, the generation of jets becomes modulated in time with an intriguing temporal alternation of buoyant plumes originated from different locations within the container. After a short initial transient time in which the system exhibits mirror symmetry with respect to the midsection, progression from order to disorder occurs and the system undergoes transition to a condition in which the time-dependent release of thermal plumes becomes erratic from both spatial and temporal points of view, without exhibiting, however, the coherent rotatory regime that occurs in classical RB case at comparable values of the control parameters. This difference may perhaps be explained by the fact that in the present case (all walls are assumed to be adiabatic), the thermal boundary layers which occur on both top and bottom walls in the case of RB convection are not formed. Hence the system lacks the necessary feedback mechanism depicted in detail in Section 5.3, by which horizontal thermal waves lead to the alternating emission of cold and warm plumes between the upper and lower thermal boundary layers and the side-jets resulting from plume-clustering phenomena pull the ﬂuid around the container. As a ﬁnal remark, it should be noted that the cases treated in this section are not intended to be an exhaustive treatment of the problem of convection with multiple sources of buoyancy, but

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213

rather to stimulate interest in this subject. The reader will ﬁnd additional information in articles by, for example, Pera and Gebhart (1975), Sparrow and Azevedo (1985), Moses et al. (1991), Ho and Chang (1994), Kwak and Song (1998), Barozzi and Corticelli (2000), Liaqat and Baytas (2001), Wang and Jaluria (2002), Najam et al. (2003), Lee et al. (2004), Icoz and Jaluria (2005), Abourida and Hasnaoui (2005), Bazylak et al. (2006) and Bakkas et al. (2006) (see also Lappa, 2005c, for the case in which the plumes are of a solutal origin). Many studies have also appeared in which the interaction of localized thermal plumes with an external cross-ﬂow was investigated (e.g. Lavelle, 1997; Trelles et al., 1999; Davies et al., 2000; Bornoff and Mokhtarzadeh-Dehghan, 2001; K¨onig and Mokhtarzadeh-Dehghan, 2002). Most of these studies concerned large-scale scenarios, that is, geological ﬂows or environmental problems (interaction of localized plumes with wind of the Hadley type, the planetary boundary layer, see, e.g., Figure 3.18, or oceanic currents). As an example, studies on smoke dispersion from multiple ﬁre plumes have shown that multiple-plume interactions can push particulate up to altitudes exceeding that of an equivalent single plume (Trelles et al., 1999). These interactions can have unexpected results for the containment of airborne material. The mixing promoted by the large-scale plume vortex structures can transport combustion products to areas that would not have been covered by a single plume produced by a single ﬁre. The investigation of K¨onig and Mokhtarzadeh-Dehghan (2002) was aimed at verifying the question of how closely the overall characteristics of merged plumes from a multi-ﬂue chimney match those of an equivalent single plume. The results for multi-ﬂue plumes were compared with those for a single plume under the same release conditions for volume ﬂow rate, momentum and temperature. The differences in the velocity, temperature and turbulence energy ﬁelds of a single plume and multiple plumes were found to be signiﬁcant mainly in the early stages of rise and spreading. Bornoff and Mokhtarzadeh-Dehghan (2001) presented the results of a numerical investigation into the interaction of two adjacent plumes in a cross-ﬂow. The computations were performed for three-dimensional, turbulent, buoyant and interacting plumes and for a single plume for comparison. Two double-source arrangements, namely tandem and side-by-side with respect to the oncoming atmospheric boundary layer, were considered. The results showed that the interaction of side-by-side plumes is dominated by the interaction of the rotating vortex pairs within the plumes. A tandem source arrangement leads to early merging and efﬁcient rise enhancement. Davies et al. (2000) carried out experiments to measure concentration ﬂuctuations downwind of two tracer sources in the atmospheric surface layer. They showed that the correlation between the concentrations from the two sources varies with downwind distance and source separation. In smaller scale contexts, combined with other driving forces (e.g. Marangoni effects), patterns created by multiple sources of buoyancy can become even more complex. The reader is referred to, for example, Lappa (2006b, 2007c) for liquid systems (inorganic and organic alloys) with a miscibility gap.

6 Systems Heated from the Side: The Hadley Flow Like the Rayleigh–B´enard (RB) convection treated in Chapters 4 and 5, the heated from the side problem has also received much attention over the years. As illustrated in Chapter 3, in terms of potential applications in both the natural environment and the industrial domain, it is in no way inferior to the companion subject in which gravity and temperature gradients act primarily in the same direction. Since the pioneering work of Hadley (1735), who proposed a single-cell, thermally driven model of the Earth’s general circulation (induced by differential solar heating at the poles and the equator), such regimes of motion have been largely studied as relevant models of the dynamics of atmosphere (Section 3.4). From a technological point of view, they are strongly relevant to the manufacture of bulk semiconductor crystals (see, e.g., Carruthers, 1977; Thevenard et al., 1991). In many industrial settings, the crucible containing the molten crystal is withdrawn horizontally from a furnace, resulting in a horizontal temperature difference (Dupret and Van der Bogaert, 1994; Monberg, 1994; Le Qu´er´e and Gobin, 1999). Hurle (1966) was the ﬁrst to show that oscillatory buoyancy convection of this type could be responsible for the presence of undesired defects in crystals growing from melts (Section 3.1). Before expanding on such details, it is worth highlighting, however, that even if the driving force and the fundamental models used by investigators over the years for the study of these problems are basically the same as already invoked in Chapter 4, simply rotating the direction of the applied temperature gradient by 90◦ induces dramatic changes in the dynamics. It is by virtue of such differences that despite its undeniable theoretical kinship with the RB case, this topic is now regarded as a selfstanding and independent important sub-ﬁeld of thermogravitational convection. Along these lines, it is worth beginning the discussion on the subject by observing that, as opposed to the classical RB problem, the quiescent state for these cases is not a solution; consequently, ﬂuid motion is present without the need to overcome a threshold value of any parameter.

6.1

The Inﬁnite Horizontal Layer

The ﬁrst and simplest model for this kind of ﬂows has been treated in Section 2.4 when discussing the existence of exact solutions to the thermal convection equations and related properties. It is Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Figure 6.1 Basic-state velocity and temperature proﬁles for an inﬁnite horizontal liquid layer conﬁned by two rigid adiabatic walls

generally referred to as the Hadley model and consists of a parallel ﬂow in a layer of inﬁnite extent with adiabatic or conducting horizontal boundaries (the ﬂow is featured by a single velocity component along the direction of the imposed temperature gradient as shown in Figure 6.1). Some fundamental information on and insights into the treatment of the stability properties of plane parallel ﬂows have been already given in Section 1.5.4 for the limiting case Pr → 0, mentioning Squire’s theorem (about the two-dimensional nature of disturbances) and providing a general necessary condition for instability [Eq. (1.113)]. It is known, however, that when momentum and energy equations cannot be uncoupled (i.e. Pr = 0), Squire’s theorem is no longer valid. For this reason, in order to capture the system evolution, equations cannot be reduced to the corresponding plane problem, and, in general, three-dimensional disturbances have to be considered. As shown by many authors, in fact, several perturbing mechanisms become possible in the most general case (i.e. when there are no assumptions about the Prandtl number). Two of them occur for both adiabatic and conducting horizontal boundaries; other modes are more speciﬁc to the type of boundary conditions considered. In the following, ﬁrst attention is focused on such general modes of instability and some general useful generalizations are made (present section and Section 6.1.2); then other speciﬁc cases are considered in Sections 6.1.3 and 6.1.4. Following the same approach as in Chapter 4, relevant information is given about historical developments while providing all the necessary mathematical details about the methods of analysis used. The two general modes mentioned above are related to transverse and longitudinal perturbations, respectively (see Figures 6.2 and 6.3).

6.1.1 The Hadley Flow and its General Perturbing Mechanisms Recalling the Hadley ﬂow model described in Section 2.4, the basic solution can be represented as 

 u0 (y) V 0 = Ra  0  0

(6.1)

T0 = x + Raf (y)

(6.2)

p0 = p0 (x, y)

(6.3)

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217

Figure 6.2 Sketch of convection with transverse rolls

Figure 6.3 Sketch of convection with longitudinal rolls

where u0 (y) and f (y) are known polynomial expressions (with constant coefﬁcients) obtained via analytical solution of the Navier–Stokes equations coupled with the energy equation. Following the concepts illustrated in Chapter 1 about the linear stability theories and related tools of analysis, generic 3D disturbances potentially arising on such a basic state can be represented as 

 ud (y) δV =  vd (y)  eλt ei(qx x+qz z) wd (y)

(6.4)

δT = Td (y)eλt ei(qx x+qz z) δp = pd (y)eλt ei(qx x+qz z)

(6.5) (6.6)

where qx and qz are the wavenumbers for the x and z directions, respectively (see also Section 1.5.3 for additional theoretical background). Then, the eigenvalue problem can be formulated as a system of ordinary differential equations for the amplitudes ud , vd , wd , pd and Td : Continuity equation: i(qx ud + qz wd ) + vd = 0

(6.7)

Pr(ud − q 2 ud ) − Ra(iqx u0 ud + u0 vd ) − iqx pd = λud Pr(vd − q 2 vd ) − Ra(iqx u0 vd ) − pd + Pr RaTd = λvd Pr(wd − q 2 wd ) − Ra(iqx u0 wd ) − iqz pd = λwd

(6.8a) (6.8b) (6.8c)

Momentum:

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Energy: Td − q 2 Td − (iqx Ra u0 Td + ud + vd Raf ) = λTd

(6.9)

and the primes denotes derivatives with respect to the vertical coordinate. where q = + The amplitude equations Eqs (6.7)–(6.9) with the prescribed boundary conditions deﬁne the characteristic λ = σ + iω and the associated eigenfunctions. As illustrated in Section 1.5.3 for the general case, the condition σ = 0 gives the critical (neutral) value of the Grashof number (Gr) as a function of qx , qz and Pr, and ω is the frequency of the critical disturbance. For the two speciﬁc cases shown in Figures 6.2 and 6.3, namely two-dimensional transverse and longitudinal disturbances, respectively, the problem can also be treated in the framework of simpler approaches derived from the original one by considering the peculiar features of the considered disturbance (qz = 0 in the former case and qx = 0 in the latter); such a strategy has enjoyed fairly widespread use in the literature and is also brieﬂy discussed in the following. 2

qx2

qz2

6.1.1.1 Plane Disturbances In the case of plane disturbances (wd = 0, qz = 0), a stream function (see Section 1.2.8) can be introduced in such a way that ψ = ψd (y)eλt+iqx x

(6.10)

∂ψ (6.11a) = ψd eλt+iqx x ∂y ∂ψ δv = − (6.11b) = −iqx ψd eλt+iqx x ∂x Eliminating the pressure in Eqs (6.7)–(6.9) leads to two amplitude equations for ψd and Td : δu =

Pr(ψdI V − 2qx2 ψd + qx4 ψd ) + iqx [u0 ψd + Ra u0 (qx2 ψd − ψd )] − iqx Pr RaTd = λ(ψd − qx2 ψd ) Td − qx2 Td + iqx Ra(ψd f − u0 Td ) − ψd = λTd

(6.12) (6.13)

with the following boundary conditions for velocity: ψd = ψd = 0 for y = − 1/2 and y = 1/2

(6.14)

for rigid top and bottom walls (R–R case); ψd = ψd = 0 for y = − 1/2 and ψd = ψd = 0 for y = 1/2

(6.15)

for rigid bottom wall and upper (stress-free) free surface (R–F case), and with the following boundary conditions for temperature: Td = 0 for y = − 1/2 and y = 1/2 in the case of adiabatic horizontal walls Td = 0 for y = − 1/2 and y = 1/2 in the case of conducting horizontal walls

(6.16) (6.17)

6.1.1.2 Longitudinal Rolls In the speciﬁc case of 3D spatial disturbances with qx = 0 and qz = 0, after eliminating wd and pressure, the problem for ud , vd and Td has the form Pr(ud − qz2 ud ) − Ra u0 vd = λud

Pr(vdI V

− 2qz2 vd + qz4 vd ) − qz2 Pr RaTd = λ(vd − qz2 vd ) Td − qz2 Td − (ud + vd Raf ) = λTd

(6.18) (6.19) (6.20)

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219

with boundary conditions for velocity ud = 0, vd = vd = 0 for y = − 1/2 and y = 1/2

(6.21)

ud = 0 for y = − 1/2 and ud = 0 for y = 1/2 vd = vd = 0 for y = − 1/2 vd = vd = 0 for y = 1/2

(6.22a) (6.22b) (6.22c)

in the R–R case;

in the R–F case, and with the following boundary conditions for temperature: Td = 0 for y = − 1/2 and y = 1/2 in the case of adiabatic horizontal walls Td = 0 for y = − 1/2 and y = 1/2 in the case of conducting horizontal walls

(6.23) (6.24)

6.1.2 Hydrodynamic Modes and Oscillatory Longitudinal Rolls Hart (1972, 1983) determined the sensitivity the Hadley ﬂow (i.e. the property of the ﬁrst ﬂow bifurcation) to both these transverse and longitudinal disturbances (as shown in Figures 6.2 and 6.3), whereas Gill (1974) focused speciﬁcally on the longitudinal disturbances. These authors also revealed that the transversal instability is driven by the mean shear stress (this is the reason why it is often referred to as ‘shear instability’ and the related disturbances as hydrodynamic ones; accordingly, throughout this section, the terms ‘transverse’, ‘shear’ and ‘hydrodynamic’ will be used as synonyms) whereas the longitudinal instability arises as a consequence of a dynamic coupling between the mean shear stress and the buoyancy force (a dynamic balance between the inertial and gravitational forces that makes the role played the thermal effects signiﬁcant in the instability mechanism). In the ﬁrst case, two-dimensional vortices appear on the frontier (i.e. across the midsection of the layer) of the two opposing ﬂows characterizing the basic state (i.e. close to the inﬂection point of the basic velocity proﬁle; see Figure 6.2). These perturbation rolls have axes perpendicular to the imposed temperature gradient , which means that the general outcome of this instability is the replacement of the initial unicellular Hadley ﬂow with a two-dimensional multicellular convective structure. The latter mode of convection is basically due to the onset of a pair of gravitational waves travelling in the spanwise direction, that is, perpendicularly to the basic ﬂow, namely along the z-axis in Figure 6.1 (it is therefore, oscillatory and three-dimensional at the same time). The perturbation rolls have their axes parallel to the applied temperature gradient (Figure 6.3). In practice, these longitudinal rolls combine with the basic parallel Hadley ﬂow to produce helical trajectories of the ﬂuid particles, which explains why this mode of convection is generally referred to as the ‘helical wave’ mode or OLR (oscillatory longitudinal rolls). In the case of adiabatic top and bottom solid boundaries, Hart (1983) found transverse stationary modes to be the most unstable if Pr < 0.015, the longitudinal oscillating modes taking over in the range 0.015 < Pr < 0.27 and longitudinal stationary modes dominating for ﬂuid of larger Pr (this third possible mode with 3D steady longitudinal rolls, hereafter is referred to as SLR). These ranges were reﬁned by Kuo and Korpela (1988). They found that for Prandtl numbers less than 0.033, the aforementioned shear instability causes stationary transverse cells to be formed in the ﬂow, whereas for larger values of Pr, the instability sets in as oscillating longitudinal rolls in the range 0.033 < Pr < 0.2 and as stationary longitudinal rolls (SLR) for larger values of Pr (Figure 6.4). Although the results of these authors were similar to those published by Hart, they differ in two respects. First, the curve for the transverse cells becomes very steep just beyond Pr = 0.1 so

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(a)

(b)

Figure 6.4 Results of linear stability analysis for the Hadley ﬂow with top and bottom adiabatic walls: Critical states (a) and wavenumbers (b) as a function of the Prandtl number for various instability modes (Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the rate of uniform temperature increase along the x -axis). After Kuo and Korpela (1988); Reproduced by permission of the American Institute of Physics

that these modes are strongly stabilized for large values of Pr. Second, the neutral states for the longitudinal oscillatory modes occur at larger values of Gr than those found by Hart. Furthermore, the stationary longitudinal rolls excited at larger values of Gr were found to be the preferred ﬁrst mode of instability solely for 0.2 < Pr < 2 and to be replaced for Pr > 2 by three transverse travelling modes close to one another (appearing at values of Gr larger than 106 ). Subsequently, Laure and Roux (1989) extended these studies by considering the presence of an upper free surface (R–F conﬁguration) and the case of conducting boundaries [the analysis of Kuo and Korpela (1988) was limited to adiabatic solid walls].

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221

For rigid and adiabatic boundaries, they basically conﬁrmed earlier ﬁndings, whereas the new boundary conditions were shown to be the source of a signiﬁcant departure from already known behaviours (especially on replacing the upper solid wall with a free surface). According to such results shown in Figures 6.5 and 6.6 and summarized in Tables 6.1 and 6.2, some general and interesting conclusions can be drawn about the properties and trends of these types of ﬂow.

Figure 6.5 Stability limits for the Hadley ﬂow: R–R case, adiabatic (solid line) and conducting (dashed line) boundaries (Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the uniform rate of temperature increase along the x -axis). Courtesy of P. Laure

Figure 6.6 Stability limits for the Hadley ﬂow: R–F case, adiabatic (solid line) and conducting (dashed line) boundaries (Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the uniform rate of temperature increase along the x -axis). Courtesy of P. Laure

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Table 6.1 Stability results and characteristics of the bifurcated solutions in the rigid–rigid case (courtesy of P. Laure) Thermal condition Adiabatic Conducting

Prandtl number 0.001 Pr 0.034 0.034 Pr 0.2 0.001 Pr 0.114 0.114 Pr 0.45

Type of bifurcation 2D stable stationary ﬂow 3D stable oscillatory ﬂow (travelling wave) 2D stationary ﬂow 3D oscillatory ﬂow

Table 6.2 Stability results and characteristics of the bifurcated solutions in the rigid–free case (courtesy of P. Laure) Thermal condition Adiabatic

Conducting

Prandtl number 0.001 Pr 0.0045 0.0045 Pr 0.38 0.38 Pr 0.41 0.001 Pr 0.077 0.077 Pr 1

Type of bifurcation 2D oscillatory ﬂow 3D stable oscillatory ﬂow (travelling wave) 3D stable oscillatory ﬂow (standing wave) 2D stable oscillatory ﬂow(travelling wave) 3D oscillatory ﬂow

In the R–R case with adiabatic walls, as shown on Figure 6.5, the intersection between the 3D oscillatory mode (OLR) and the 2D stationary (hydrodynamic) mode is located at Pr = 0.034 (in good agreement with Kuo and Korpela, 1988); such a point, however, shifts to Pr = 0.114 for conducting walls. Interestingly, both the two-dimensional branches (in the adiabatic and conducting cases) tend asymptotically to a unique limit (Gr = 7942) as Pr tends to zero, thus conﬁrming that the origin of these disturbances is of a hydrodynamic nature (they survive when the velocity and the thermal ﬁelds are no longer coupled since they take their energy from inertial forces of the basic ﬂow). Nevertheless, for Pr > 0 the critical Grashof number increases smoothly with Pr, which can be regarded as clear evidence of an increasing effect of the thermal ﬁeld (it should be pointed out, however, that the inﬂuence of the temperature ﬁeld does not change the intrinsic nature of the instability mode that remains intimately related to shear stress, the thermal effect being limited to a weak stabilizing action induced by some positive thermal stratiﬁcation established inside the layer as Pr increases). The hydrodynamic instability branch, however, changes from smooth to become fairly steep when the Prandtl number approaches a certain value (PrL ). This value [found to be 0.1 by Kuo and Korpela (1988) for the adiabatic case], shifts to PrL = 0.2 for the conducting case, as is evident in Figure 6.5 (the curve of the hydrodynamic mode for conducting walls in Figure 6.5 admits a vertical asymptote for such a value of the Prandtl number). This means that for both cases, PrL represents a limiting value above which hydrodynamic disturbances are suppressed. At this stage, it is worth recalling that for all cases considered both the results of Laure and Roux and the earlier ones of Kuo and Korpela are in agreement with Squire’s theorem (Section 1.5.4) as in the limit as Pr → 0 the most dangerous disturbances are always two-dimensional (see both Figures 6.5 and 6.6). Remarkably, such a behaviour persists for Pr = 0 until the hydrodynamic disturbances are replaced by OLR modes (as the most dangerous ones) when Pr exceeds a value Pr∗ < PrL , which depends on the speciﬁc boundary conditions considered for velocity and temperature (as shown in Tables 6.1 and 6.2, Pr∗ = 0.034 for the adiabatic R–R case, Pr∗ = 0.114 for the conducting R–R case, Pr∗ = 0.0045 for the adiabatic R–F case and Pr∗ = 0.077 for the conducting R–F case).

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223

The branches for the oscillatory longitudinal modes exhibit completely different behaviour. The neutral curves have a minimum with respect to Pr (at a larger Pr in the conducting case). Unlike the 2D modes, the critical Grashof number increases rapidly when Pr tends to zero, indicating that the origin of oscillations for these modes is mainly thermal. Also for these disturbances, however, there exists a value of Pr where the curves seem to admit vertical asymptotes (PrL2 = 0.2 for the adiabatic R–R case, PrL2 = 0.45 for the conducting R–R case, PrL2 = 0.41 for the adiabatic R–F case and PrL2 = 1 for the conducting R–F case), which means that they are strongly stabilized or suppressed when such a value is exceeded. Beyond the differences just highlighted in terms of values of Pr∗ , PrL and PrL2 , it should be emphasized that the most important and evident difference between the R–R and R–F cases lies in the fact that in the latter case the 2D stationary mode is replaced by an oscillatory branch. This branch still has a hydrodynamic origin and appears for both conducting and adiabatic upper surfaces (the curves are plotted in Figure 6.6), in the range 0 Pr 0.0045 for adiabatic conditions and in the range 0 Pr 0.077 in the conducting case. For the R–F case with adiabatic boundaries, moreover, the OLR mode displays different spatiotemporal behaviours according to the considered range of Pr. More precisely, travelling waves are the preferred mode for 0.0045 Pr 0.38, whereas standing waves (not reported for the other cases) prevail for 0.38 Pr 0.41. Along these lines, it is also worth mentioning that in 1993 additional results for the R–F conﬁguration and adiabatic boundaries were provided by the linear stability analysis of Parmentier et al. (1993). Two distinct curves which intersect at Pr = 0.4 were determined. The ﬁrst one, in the range 4 × 10−3 Pr 0.4, reproduced the results found by Laure and Roux (1989) for the OLR mode shown in Figure 6.6; whereas the second one, for Pr > 0.4 (not mentioned in earlier analyses), was found to correspond to an ‘oblique’ type of disturbance. The angle of propagation for this new oscillatory mode of instability (angle between the disturbance travelling direction and the x-axis) was observed, in fact, to vary in the range 0 63◦ with ∼ = 0◦ (corresponding ∼ to qz ∼ 0, i.e. to transverse rolls) for Pr 2.6 only. = =

6.1.3 The Rayleigh Mode The linear stability analyses considered in the preceding subsections were limited to Pr O(1). As has been widely discussed, despite some differences for Pr around O(1), a common feature of all these studies is the existence of two major branches of instability with 2D (transverse) and 3D (longitudinal) perturbing mechanisms arising in well-deﬁned ranges of the Prandtl number depending on the type of boundary conditions considered. A linear stability analysis also including the missing range of high Prandtl numbers [Pr O(1)] is due to Gershuni et al. (1992). Interestingly, these authors identiﬁed a third fundamental instability for relatively high-Pr ﬂuids and conducting boundaries, referred to as the ‘Rayleigh’ mode, following from the presence of zones of potentially unstable stratiﬁcation near the upper and lower horizontal boundaries (induced by the basic ﬂow, as sketched in Figure 6.7). It was proved that these regions could make possible the onset of instability of the Rayleigh–B´enard type therein (which explains the denomination of this mode) in the form of steady rolls with their axes aligned with the shear ﬂow (i.e. longitudinal rolls) or (at slightly larger values of Ra) as oscillatory rolls with their axes perpendicular to the shear ﬂow (2D oscillatory Rayleigh mode). The latter mode was found to be characterized by disturbances travelling in the form of cells localized in the upper and lower parts of the layer, leading to the emergence of opposing waves as a result of the superposition of the cells with the basic ﬂow (in practice, at the critical point a pair of growing disturbances appears as travelling waves extending in the two opposing basic ﬂows: one of the waves, possessing negative phase velocity, spreads along the upper, warmer ﬂow

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Figure 6.7 Basic-state velocity and temperature proﬁles for an inﬁnite horizontal liquid layer conﬁned by two rigid conducting walls

appearing as a drift of the convective Rayleigh cells originated in the upper, unstable stratiﬁed zone; another wave, with positive phase velocity, drifts to the right in the lower, colder, unstably stratiﬁed region). This planar mode was found to be less dangerous than the stationary (longitudinal) Rayleigh disturbance. The results of this landmark study are summarized in Figures 6.8 and 6.9. Figure 6.8 provides a general and exhaustive picture of all the possible disturbances responsible for ﬂow instability in the speciﬁc case of solid conducting walls. Three well-deﬁned mechanisms occur in delimited intervals

Figure 6.8 Minimal Grashof number (Grm = inf qx ,qz {Grcr }) as a function of the Prandtl number for various instability modes (Hadley ﬂow with top and bottom conducting walls; Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the uniform rate of temperature increase along the x -axis). Courtesy of P. Laure

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Figure 6.9 Critical Rayleigh number as a function of qz and different values of the Prandtl number for the OLR instability (Rayleigh number deﬁned as Ra = gβT γ d 4 /να , where γ is the rate of uniform temperature increase along the x -axis; conducting horizontal boundaries). Courtesy of P. Laure

of Prandtl number. For small Prandtl numbers (0 < Pr < 0.14), the most dangerous is the plane (2D) hydrodynamic mode; in a narrow range of Prandtl numbers (0.14 < Pr < 0.44), instability is caused by the OLR mode (see Figure 6.9); for Pr > 0.44, the instability source is transferred to the Rayleigh mode, which remains the most dangerous up to extremely large Pr. Naturally, this mode does not exist in the case of an absence of unstable stratiﬁcation zones, such as the case of layers with horizontal insulated boundaries (Kuo and Korpela, 1988) where, as illustrated in Section 6.1.2, only hydrodynamic and OLR modes occur in the range of small Pr and the ﬂow tends to become rather stable when the Prandtl number is increased towards unit (or larger) order.

6.1.4 Competition of Disturbances and Tertiary Modes of Convection The (secondary) regimes of convection with transverse or longitudinal rolls described in the previous pages are just a step towards more complex behaviours as the control parameter is increased. A ﬁrst sign of such growing complexity is given by the relative proximity of the curves pertaining to distinct disturbances in some ranges of the Prandtl number. As illustrated in Figures 6.5 and 6.6, for Pr > Pr∗ the curve of critical states of the oscillatory longitudinal modes lies below that of the transverse states; for this reason, in laboratory experiments with Pr > Pr∗ , longitudinal rolls should be the expected outcome. In practice, however, as emphasized by Wang and Korpela (1989), the critical values of Grashof number for longitudinal and transverse modes are sufﬁciently close to one another that when the nonlinear terms become sizable, a competition for dominance between these modes should be expected. Such a competition should take place also for Pr < Pr∗ ; even if in such a case transverse cells appear ﬁrst, since the transition curve for longitudinal modes is not far removed from that for transverse modes, on increasing the temperature gradient in liquid metal ﬂows the actual outcome from the instability should be determined again by the interplay between longitudinal and transverse disturbances.

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Given the crucial potential impact of such dynamics in practical technological applications (especially in the ﬁeld of crystal growth), some studies have also appeared which were expressly devoted to studying the mutual interference of these modes and/or the related subsequent bifurcations for Pr < O(1). As a relevant example, Wang and Korpela (1992) focused on the case of mercury (Pr = 0.027) for which the transverse cells appear at Gr ∼ = 9.16 × 103 (qx = 2.7) and the onset of OLR is located at Gr ∼ = 1.06 × 104 (qz = 0.7). Assuming an initial state consisting of transverse rolls, they identiﬁed several secondary stationary instabilities (referred to as ‘resonances’ by virtue of the precise relationship existing between their wavenumber and that of the initial state). The periodicity of the secondary ﬂow, resulting from the ﬁrst ﬂow bifurcation induced by two-dimensional hydrodynamic disturbances and consisting of a set of transverse cells, was characterized using the following form for the ﬂow variables: F m (y) exp(imqx x) (6.25) fS (x, y) = m

The perturbation variables were written as Fn (y) exp(inqx x + iSx x + iSz z + σ t) fp (x, y, z, t) =

(6.26)

n

where Sx is the wavenumber characterizing the x dependence of the three-dimensional ﬂow potentially emerging as a result of the secondary transition (according to the present symbolism, the corresponding wavenumber in the primary instability analysis is indicated by qx , the different symbols q and S distinguishing whether primary or secondary instabilities are considered, respectively). The reader is referred to the same concepts already illustrated in Section 4.2 [Eqs (4.16)–(4.18)] for the study of the stability of RB convection for further details on this kind of approach. In analogy with results obtained by Busse and co-workers for the analogous case concerning the stability of two-dimensional RB rolls, Wang and Korpela (1992) found (see Figure 6.9) the regions of stability of transverse cells to be restricted by a variety of secondary instabilities. As anticipated, these instabilities were classiﬁed according to the related spatial wavelength and the ratio of the wavenumber of the emerging tertiary ﬂow to that of the pre-existing secondary ﬂow (as ‘subharmonic resonance’, ‘combination resonance’ and Eckhaus instability). It was illustrated, in particular, that transverse cells with wavelengths shorter than the critical value become unstable by a subharmonic resonance, whereas the instability for longer cells sets in by a combination resonance. The Eckhaus instability [branch (b) in Figure 6.10] was ascribed to a three-mode interaction mechanism (the fundamental one and two so-called sideband disturbances, with wavelengths such that one is larger and the other smaller than the fundamental one and with an average wavelength equal to the wavelength of the fundamental one). The speciﬁc spatial properties of all these modes can be summarized as follows. Like the RB case treated in Section 4.2, the Eckhaus instability is stationary and two-dimensional (Sz = 0). The distinguishing feature of the subharmonic resonance is that a spatial disturbance with a wavenumber Sx = qx /2 is ampliﬁed the most. It is stationary but, unlike the Eckhaus instability, it is a 3D phenomenon (interestingly, in fact, the maximum ampliﬁcation rate was obtained for a wavenumber Sz = 1.35). The combination resonance is also a 3D stationary phenomenon (setting in as a system of oblique disturbances). Both Sx and Sz differ from zero (from which the name ‘oblique modes’ derives). For qx = 2.6 and Gr = 1.15 × 104 , in particular, the maximum growth was observed near Sx = 0.59 and Sz = 0.88 and in symmetrical location around Sx = qx /2. Various wave interactions were also identiﬁed. Of these, the principal ones involve the waves with wavenumbers (2qx − Sx , −Sz ), (Sx , Sz ) and the ﬁrst harmonic with wavenumber (2qx , 0). The other similar set with the sign of Sz changed differs from this only by having the wavevectors of the corresponding waves inclined

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Figure 6.10 Stability map for the two-dimensional multicellular ﬂow for Pr = 0.027. Branch (a) indicates the results of the primary stability analysis. The region of stable ﬂow is bounded by (b) the Eckhaus instability boundary from below, (c) the subharmonic resonance from the right and (d) the combination resonance from the left; the boundary of the onset of the oscillatory instability (e) is also shown (Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the rate of uniform temperature increase along the x -axis). After Wang and Korpela (1992); Reproduced by permission of Cambridge University Press

in the opposite direction of the x-axis than in the ﬁrst set. These stationary waves, in general, do not form a periodic pattern. Oscillatory behaviours were predicted for high values of the control parameter. In Figure 6.9, the stability boundary above which oscillatory states are possible has been plotted as curve (e). The maximum ampliﬁcation rate for these states occurs for Sx = 0 and Sx = qx . As a consequence, these states are basically characterized by the wavenumber pair (qx , Sz ). Notably, for a small amplitude of the transverse cells, this oscillatory mode has the same characteristics of the aforementioned pure OLR. The wavenumber falls into the same range of classical longitudinal modes as shown in Figure 6.4b (0.7 < Sz < 0.8). The oscillation frequency is also practically the same as given by the primary instability analysis. This observation has important consequences. It means, in fact, that the inﬂuence of pre-existing transverse cells is relatively weak. It is limited to a slight stabilization of the ﬂow; such a stabilization, in particular, is evident from the neutral stability curve of the oscillatory mode having being bent upwards at the centre (in the absence of initial transverse cells, this mode is longitudinal and the theory for the ﬁrst ﬂow bifurcation discussed in Sections 6.1.1 and 6.1.2 requires that its neutral states are independent of qx ; hence if the transverse cells were to have no inﬂuence on the oscillatory modes at all, their neutral curve would appear in Figure 6.10 as a straight horizontal line). In practice, even if the curve for the onset of oscillatory behaviours lies above both the subharmonic curve and the one characterizing the combination resonance, it does not lie far above them, i.e. the OLR is only slightly delayed by the presence of the transverse cells. Remarkably, this provides a solid theoretical basis to the aforementioned possibility that transverse and longitudinal disturbances can effectively compete in practical applications. We shall come back to this important subject in Section 6.4.

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6.2

Two-dimensional Horizontal Enclosures

Since the Hadley ﬂow is an inﬁnite-layer model, some analyses were devoted to assessing the range of validity of this approximation in bounded containers (two-dimensional slender cavities with imposed end-to-end temperature differences). Many of these studies were based on the seminal work of Batchelor (1954b). In particular, the effect of endwalls in a long cavity was considered by Cormack et al. (1974) in the framework of an asymptotic approach; they divided the cavity into three regions: a central (core) region consisting of a parallel ﬂow and two approximately square end regions in which the ﬂow turns round (a simple unicellular circulation, where motion is up the hot wall, across the top, down the cold wall and returning along the bottom), showing that the aforementioned Hadley model gives a reasonable solution to the ﬂow in the centre of the cavity provided that Gr2 Pr2 /A 3 (A being the aspect ratio, deﬁned as length/height). Bejan and Tien (1978b) described the changes that occur when Ra is increased by classifying the ﬂow into three possible regimes: conduction, intermediate and boundary-layer regimes (see also Section 2.5). As the Rayleigh number increases, diffusion that dominates at low Ra gradually diminishes and convection in the boundary layers at the vertical walls begins to control heat transfer. The intermediate regime is characterized by increasing vertical stratiﬁcation of the temperature in the core as Ra increases (see, e.g., the landmark study by Imberger, 1974, for Ra between 104 and 106 ). Over subsequent years, the model with lateral walls was intensively studied from a theoretical point of view (Laure et al., 1990) and also both experimentally and numerically (see the very rich reference list in Bontoux et al., 1986, and Lappa, 2007b) and used as a convenient benchmark problem for particular values of the aspect ratio and the Prandtl number (e.g. the GAMM Workshop: Roux et al., 1989a; Roux, 1990).

6.2.1 Geometric Constraints and Multiplicity of Solutions The results for the different ﬂow instabilities for inﬁnite layer discussed in Sections 6.1 and 6.2 were reﬁned for the case of ﬁnite-sized enclosures with adiabatic horizontal solid boundaries by Laure (1987), Kuo and Korpela (1988) and many other investigators (hereafter, unless explicitly indicated, cavities with both top and bottom solid walls will be considered). According to such studies, in general, the spacing of the cells pertaining to the 2D hydrodynamic mode depends on a number of factors. First is the natural spacing predicted by the stability theory for the inﬁnite layer. Second, an integral number of cells must ﬁt into a cavity of ﬁnite aspect ratio and the cell spacing must accommodate this constraint. The third factor is the magnitude of the departure from the onset of instability. Fourth is the extent of the end regions and how this changes with the Grashof and Prandtl numbers. The problem still continues to attract much attention. Further theoretical exploration along these lines of research has been motivated over recent years, in particular, by the remarkable features that this two-dimensional mode exhibits when it is considered in systems of ﬁnite lateral extent. The most interesting of such features is, perhaps, the ‘multiplicity’ of possible solutions, that is, the existence of distinct branches of ﬂow for speciﬁc values of the aspect ratio (unlike the bifurcation to transverse-roll ﬂow in the inﬁnite ﬂuid layer, the roll-type structure of convective ﬂow in ﬁnite cavities can develop a continuous change of the ﬂow pattern). Moreover, the dependence of the critical Grashof number on the aspect ratio and Prandtl number is very complex. Motivated by potential applications in the ﬁeld of semiconductor crystal growth, Gelfgat and co-workers (Gelfgat et al., 1999b, 1999c), in particular, carried out rich and exhaustive parametric numerical studies in this area by examining multiple steady states, their stability, onset of oscillatory instability and some supercritical unsteady regimes of convective ﬂow in two-dimensional

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Figure 6.11 Critical Grashof number as a function of the aspect ratio for Pr = 0.015 (cavities with adiabatic horizontal walls; Grashof number based on the cavity height). ◦, Flows with a single roll; ♦, ﬂows with two rolls; , ﬂows with three rolls; , ﬂows with four rolls. After Gelfgat et al. (1999b); Reproduced by permission of Cambridge University Press

laterally heated rectangular containers of ﬁnite extent with upper and lower adiabatic solid (no-slip) boundaries for Pr = 0 and 0.015. A complete study of stability of each branch of steady-state ﬂow was performed for the aspect ratio of the rectangular container varying continuously from 1 to 11. The results obtained by these authors for Pr = 0.015 are summarized in Figure 6.11 as stability diagrams showing the critical Grashof number corresponding to transitions from steady to oscillatory states, appearance of multi-roll states, merging of multiple states and backward transitions from multi-roll to single-roll states. In particular, the upper parts of the neutral curves in such a ﬁgure correspond to transition from steady to oscillatory ﬂow, while the lower branches represent stationary bifurcations from multi-roll to single-roll ﬂows. As an example, Figures 6.12 show the multiple steady states of convection for Pr = 0.01 in a laterally heated closed cavity with A = 4 [this particular value of the aspect ratio is characteristic for the industrial HB method and was used in the landmark experiments of Hurle et al. (1974); the reader is also referred to the numerical studies of Winters (1988) and Skeldon et al. (1996)]. According to the simulations for a relatively low Rayleigh number Ra = 103 , the ﬂow is simply given by a ‘twisted’ elongated recirculation embracing three corotating (anticlockwise in the ﬁgures) vortices distributed along the horizontal extension of the enclosure; when the Rayleigh number is increased, transition to a two-roll conﬁguration occurs; these steady ﬂows are centrally symmetric (namely, with respect to rotation through 180◦ about the centre of the cavity). Figure 6.13 displays the stability diagram for the single-roll state in such a cavity for 0.015 Pr 0.03. It can be seen there that the dependence Grcr (Pr) becomes complicated when Pr exceeds 0.023; there exist, in fact, three hysteresis loops of steady/oscillatory/steady transitions within the narrow interval 0.023 < Pr < 0.027. Some analyses have also appeared where attention was expressly focused on the case of horizontal conducting boundaries. For instance, interesting information was provided by Drummond and Korpela (1987) and Wang and Daniels (1994), who considered for Pr < O(1) both conducting and insulated top and bottom walls for several values of the aspect ratio (e.g. 8 A 20 in Drummond and Korpela, 1987).

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(a)

(b)

Figure 6.12 Multiple steady states of two-dimensional convection [plane (x, y)] in a laterally heated enclosure (Pr = 0.01, A = 4; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions). Single- and two-roll steady-state ﬂows for Ra = 1 × 103 (a) and 2 × 103 (b), respectively (Ra based on the depth of the container; two branches of possible steady state exist) (numerical simulation, M. Lappa)

Figure 6.13 Critical Grashof number as a function of the Prandtl number for A = 4 (cavity with adiabatic horizontal walls; Grashof number based on the cavity height). After Gelfgat et al. (1999b); Reproduced by permission of Cambridge University Press

According to such studies, for ﬂuids of small Prandtl number the differences in the ﬂow patterns in these two cases are slight, the strength of the circulation in the cells being somewhat weaker when the boundaries are insulated (notably, this is a result of a more stable ﬂow in this case, caused by the kinetic energy being more vigorously expended in the work against the buoyant forces; adiabatic boundaries allow the temperature ﬁeld to adjust more freely in the end regions leading to crowding of the isotherms there and consequently to larger heat transfer than when the boundaries are conducting).

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Similarly, the critical Grashof number does not depend signiﬁcantly on whether the horizontal boundaries are conducting or adiabatic. This can be ascribed to the fact that, as illustrated in Section 6.1, this type of instability arises from the shear and not from the buoyancy ﬁeld. For the speciﬁc case A = 4, the two-dimensional oscillatory instabilities following the onset of stationary shear rolls have been studied by many investigators (the reader is referred to, e.g., Pulicani et al., 1990; Skeldon et al., 1996; Gelfgat et al., 1999b,c, and, in particular, Mercader et al., 2004, for fully developed turbulent regimes). It is known that the ﬂow gains in complexity by several possible paths. For the sake of brevity and clarity, in the present section the survey is limited to the simulations of Mercader et al. (2005). They performed an extensive numerical study of the instabilities of natural convection in a differentially heated rectangular cavity for a ﬂuid of low Prandtl number (Pr ∼ = 0.01). The aspect ratio was speciﬁcally selected in order to maintain a basic single-roll ﬂow conﬁguration (Figure 6.14) and to obtain instabilities leading to time-dependent behaviour without breaking this basic structure (as illustrated earlier, more elongated cells typically introduce new instabilities, which break the basic roll into more than one, making the problem very complex). It is also worth noting that they considered A = 2 for which the threshold value of the Rayleigh number for the Hopf bifurcation of the single-roll stationary solution presents a minimum and hence the single roll is most unstable to time-dependent solutions. As explained previously, this allowed these researchers to focus expressly on the onset of time dependence and ensuing route to chaos while keeping a very simple basic spatial ﬂow structure. The main focus of such work was the elucidation of the dynamic scenario in terms of Rayleigh number as the main control parameter and in the effects of the type of boundary conditions, either perfectly conductive or adiabatic, which ﬁts perfectly the scope of the present discussion. Varying the Rayleigh number over a broad range, these authors unveiled a very rich scenario of dynamic behaviours for both boundary conditions, the most interesting aspects of which are discussed in the following. According to their results, for relatively small Ra values, in agreement with earlier ﬁndings of other authors, the behaviours for both conﬁgurations, including the ﬁrst bifurcations, are similar, with their steady solutions losing stability in supercritical Hopf bifurcations of hydrodynamic nature and giving rise to periodic solutions. The effects of the two different thermal boundary conditions, however, start to deviate from each other as Ra is increased further. Bifurcations of a different type in the conducting case and in the adiabatic case give rise to quasi-periodic solutions.

Figure 6.14 Steady convection [plane (x, y)] for A = 2, Pr = 0.01, Ra = 2 × 103 (adiabatic horizontal walls; hot side on the right, cold side on the left) (numerical simulation, M. Lappa)

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A complex sequence of further bifurcations occurs next. In the conductive case, in particular, the transition to chaos occurs due to the classical torus breakdown (see Section 1.8 for the necessary theoretical background). Before the destruction of the torus, the system passes through various resonance horns (frequency locking), in which the stable limit cycles lose and gain stability following several typical scenarios. The torus breakdown occurs via a soft transition due to loss of torus smoothness. In the adiabatic case, the dynamics are dominated by the interaction of two bifurcations of the basic periodic solutions, leading to the stable coexistence of three incommensurate frequencies and ﬁnally to chaos. Thereby, for the considered parameters, the adiabatic case was found to be more unstable, displaying a richer bifurcation structure. Interestingly, to gain further insight into the physical nature of these secondary instabilities (when a different frequency arises), the authors undertook stroboscopic ﬂow decompositions of the bifurcated solutions, as shown in Figures 6.15 and 6.16 (in these ﬁgures the evolution given by the new, smaller frequency can be clearly visualized whereas the rapid variations of the base cycle are hidden). In the stroboscopic decomposition of the conductive case (Figure 6.16), the perturbation has the form of two central rolls which exchange the sense of rotation periodically, implying vertical, oscillatory displacement of the ﬂuid in the central region. The centre of the cell is also periodically overheated and underheated with the same frequency. In the case of adiabatic conditions, shown in Figure 6.15, the perturbed ﬂows look very different, reﬂecting the importance of the thermal boundary conditions for the instability. The occurrence of two-roll structures is now located at the corners, producing a similar overheating and underheating but now at those same corners. Mercader et al. (2005) basically ascribed these differences to the different capability of heat exchange through the horizontal boundaries in the two cases (whereas in the conductive case heat can be exchanged in the vertical direction through the horizontal walls, the heat exchanged in the adiabatic case is restricted to the vertical sidewalls). On the basis of such observations, they conjectured that the different behaviour displayed by the cell with adiabatic and conducting horizontal walls could be somewhat related to the different degree of stratiﬁcation established in the interior of the domain. Indeed, the apparent saturation of the frequency with Ra that they observed in the adiabatic case could be associated with the fact that the stratiﬁcation of the central region was already built and became roughly insensitive to Ra. In contrast, since the conductive boundary conditions forced the isotherms to be more contorted in order to produce a stable stratiﬁcation at the centre and simultaneously fulﬁl the thermal boundary condition at the horizontal walls, the degree of stratiﬁcation was less pronounced in such a case, making the system more sensitive to Ra. Despite these differences, some commonalities were identiﬁed, for example, that chaos emerges at Rayleigh numbers an order of magnitude larger than the ﬁrst instability, with windows of periodic behaviour (intermittency). For all of these cases, the computed value of the maximum Lyapunov exponent (see Section 1.8.4 for the meaning and deﬁnition) was positive, consistent with the chaotic character of the temporal series. To estimate the dimension of the attractor, the standard method of time delays was used (they constructed an n-dimensional vector out of the time series by taking n values of the continuous time evolution, separated by a ﬁxed delay time τ ). At this stage, it is worth underlining that both of the studies devoted to the ﬁrst ﬂow bifurcation cited earlier and those focused on the ensuing route to chaos (for which a relevant and signiﬁcant example is given by the study by Mercader et al., 2005, described above) considered canonical cavities with straight walls. As a concluding remark for this section, it should be mentioned, however, that some numerical analyses have appeared in which one of the lateral straight walls was replaced by an effective

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(a)

233

(b)

Figure 6.15 Perturbation associated with the secondary instability for the case with adiabatic horizontal walls (Pr = 0.00715, A = 2, Ra = 2.67 × 104 ). Streamlines (a) and temperature ﬁelds (b) are represented [plane (x, y)] at times chosen by approximating the quasiperiodic solution by an approximate frequency locking. After Mercader et al. (2005); Reproduced by permission of the American Institute of Physics

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(a)

(b)

Figure 6.16 Perturbation associated with the secondary instability for the case with conductive horizontal walls (Pr = 0.00715, A = 2, Ra = 2.29 × 104 ). Streamlines (a) and temperature ﬁelds (b) are represented [plane (x, y)] at equispaced times, multiples of the periodicity of the fast frequency. After Mercader et al. (2005); Reproduced by permission of the American Institute of Physics

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solidifying interface. It has been found that with such a free solid/liquid interface, the bifurcation diagrams are slightly changed due to interface deformation, but their qualitative characteristics remain the same (Lan et al., 1998; El Ganaoui et al., 2002; Semma et al., 2003). All these analyses were limited to two-dimensional disturbances. As shown by Daniels et al. (1987), nevertheless, when the Prandtl number is increased the work done by the ﬂuid particles against the force of buoyancy increases, causing the amplitude of the hydrodynamic mode to decrease and at Pr > 0.12 the multicellular ﬂow to disappear completely. As already illustrated for the case of inﬁnite layer in Section 6.1, in fact, for Pr∗ = O(10−1 ) the mechanism of instability is transferred to the OLR mode. This means that the effect of additional instabilities related to three-dimensional disturbances could make the bifurcation scenario even more complex, presumably leading to an earlier occurrence of the time-dependent behaviour. Finally, both of these kinds of instabilities (hydrodynamic and OLR) typical of liquid metals and relevant to semiconductor crystal growth disappear when the Prandtl number is further increased [Pr O(1)].

6.2.2 Instabilities Originating from Boundary Layers and Patterns with Internal Waves For Pr O(1), the scenario changes dramatically. Oscillatory instabilities are still possible, but the related dynamics and genesis exhibit a signiﬁcant departure from known behaviours for liquid metals and semiconductor melts. It has been known since 1980 that for Pr > 1 and cavities with insulated horizontal walls, oscillations (affecting the time history of the Nusselt number) can be generated during the transient phase approaching the steady state; this behaviour was investigated by Patterson and Imberger (1980), Patterson and Armﬁeld (1990) and Schladow (1990); for relatively large Rayleigh values, the transients are dominated by internal waves and the approach to the steady state is achieved in an oscillatory manner by decay of internal wave motion (Sch¨opf and Patterson, 1996). It is also known that for sufﬁciently high Ra the ﬂow can undergo transition to permanent 2D oscillatory states. There is a long tradition of numerical studies on such a subject (e.g. the numerical simulations for the case of cavities with horizontal adiabatic walls ﬁlled with air of Le Qu´er´e and Penot, 1987; Henkes and Hoogendoorn, 1990; Bucchignani, 2009, for a square cavity; Yahata, 1999, over the range of the cavity aspect ratio A from 1/10 to 1 and Auteri and Parolini, 2002, for A = 1/8). The ﬂow can even become turbulent (Paolucci and Chenoweth, 1989; Paolucci, 1990; Janssen and Henkes, 1995; Xin and Le Qu´er´e, 1995; Farhangnia et al., 1996; Yahata, 1997; Le Qu´er´e and Behnia, 1998; Mayne et al., 2000). As already outlined above, these instabilities, however, have nothing to do with the typical perturbing mechanisms elucidated in the preceding sections for Pr < O(1). Some illuminating information along these lines can be found in the work of Delgado-Buscalioni (2001a). Notably, in that study the problem was placed in a precise and more comprehensive theoretical framework focusing expressly on the characterization of the convection patterns arising at the core of the basic steady Hadley unicellular ﬂow [over the whole range of Prandtl numbers (0 Pr ∞) and adiabatic horizontal boundaries]. It was shown that the onset of the ﬂow instabilities basically depends on the core Rayleigh number, deﬁned in terms of the local streamwise temperature gradient, and that the effect of conﬁnement can decisively change the stability properties of the core. As already explained in Section 2.5, as the Rayleigh number increases the ﬂow evolves from a diffusive regime to a boundary-layer regime in which almost all the temperature drop and the vorticity production are localized in thin layers adjacent to the end lateral walls. As a consequence, a stagnant region is formed at the centre part of the enclosure (Figure 6.17) and the ﬂow at the core occurs only as a result of the entrainment of mass from the vertical boundary layers being

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(a)

(b)

Figure 6.17 Regions with stable thermal stratiﬁcation, horizontal intrusion thermal layers and vertical thermal boundary layers in a cavity with A = 1 (Pr = 0.71 and Ra = 1 × 107 ; adiabatic horizontal walls): (a) temperature distribution; (b) velocity ﬁeld (numerical simulation, M. Lappa)

conﬁned in thin intruding ﬂowing layers near the horizontal walls (see also Figure 2.16 and related description in Chapter 2). If the steady unicell reaches such a boundary-layer regime, the local temperature gradient vanishes at the core, leaving a completely stable core region with a cross-stream thermal stratiﬁcation. As conﬁrmed by numerical calculations and in theoretical agreement with the studies previously carried out for the inﬁnite Hadley parallel ﬂow in the case of adiabatic horizontal walls, Delgado-Buscalioni (2001a) proved, basically, that core-ﬂow instabilities can only develop for Pr < O(1), whereas at larger Pr the core region remains stable and the instabilities may only develop at the boundary layers arising close to the sidewalls. According to such important arguments, most of the oscillatory instabilities reported over the years by investigators in cavities of ﬁnite extent for Pr O(1) (e.g. air, water and silicone oils) must be ascribed to the presence of such boundary layers. In geometries of ﬁnite lateral extent, in fact, the mechanisms responsible for such a kind of instability tend to replace as the most dangerous one those typical of OLR modes as Pr is increased. In practice, there is a sudden increase in stability already for Pr > O(10−1 ), owing to the presence of the aforementioned completely stable cross-stream stratiﬁcation (as shown, for instance, in Figure 6.17a) and for Pr > 0.2 where the OLR instability is no longer possible as the most unstable mode (as shown in Figures 6.4 and 6.5 and already discussed in Section 6.1.2, the curve for the

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OLR tends to a vertical asymptote for Pr ∼ = 0.2), the breakdown of the steady unicell takes place inside the vertical boundary layers or in particular structures formed near the corners of the cavity (whereas, as mentioned earlier, the core ﬂow tends to be stable). Among other things, this also explains why the existing experiments in horizontal cavities with thermally insulated horizontal walls concerned with the investigation of core-ﬂow instabilities (hydrodynamic mode and/or OLR) worked with liquid metals and A 1, whereas those interested in boundary-layer instabilities used ﬂuids with Pr O(1), that is, gases (see the studies cited above) or transparent liquids (see, for instance, Le Qu´er´e, 1990, for the case of water and A = 1/10) and cavities with A 1 (tall geometries). Some fundamental information about the instabilities of boundary layers of natural convection of inﬁnite extent can be found in the theoretical analyses of Gill and Davey (1969) and Daniels and Patterson (1997, 2001). For ﬁnite cavities, as outlined above, there exists a rich body of numerical analyses, which is reviewed in detail in the following. The features of high-Ra convection in the simple case of a square cavity and Pr = 0.71 (air) were initially investigated numerically by Ravi et al. (1994). The study was limited to steady conditions, but the authors illustrated that at high values of Ra, a recirculating pocket appears near the corners downstream of the vertical walls and the ﬂow separates and reattaches at the horizontal walls in the vicinity of this recirculation (together with a considerable thickening of the horizontal layer; see Figure 6.17b). It was elucidated that the corner structure is caused by thermal effects (in practice, owing to the temperature undershoots in vertical boundary layer, which are known to be caused by the stable thermal stratiﬁcation of the core, relatively cold ﬂuid reaches the upper corner; this cold ﬂuid detaches from the ceiling like a plume at high Rayleigh numbers and causes the separation and recirculation). Paolucci and Chenoweth (1989) and Le Qu´er´e and Behnia (1998) determined numerically that for such a conﬁguration (A = 1 and Pr = 0.71), unsteadiness sets in at a Rayleigh number value of O(108 ) (some experimental results along the same lines are due to Tian and Karayiannis, 2000, who for the ﬁrst time reported the experimental vector plot of the air ﬂow for Ra = 1.58 × 109 in a cavity 0.75 × 0.75 × 1.5 m). This behaviour is known to be an instability that takes place near the aforementioned ‘departing’ corners in the detached ﬂow region along the horizontal boundaries. The boundary layers themselves become unstable at a slightly higher Rayleigh number (this must be regarded as a second instability) and, thereby, the route to chaos proceeds through quasi-periodicity. Le Qu´er´e and Behnia (1998) revealed that the ﬁrst instability mode breaks the usual centrosymmetry of the solution. They also provided interesting insights into the underlying instability mechanisms. In particular, internal gravity waves were shown (see the description given later) to play an important role in the time-dependent dynamics of the solutions, both at the onset of unsteadiness and in the fully chaotic regime. Let us recall that, as illustrated in Section 4.1, such phenomena are generally damped by the effect of thermal and viscous dissipation, but can be maintained in the presence of an adequate excitation source. In their analysis, in the ﬁrst instability mechanism the ﬂuctuating temperature ﬁeld was observed to reach its maximum amplitude in a region that corresponds to the base of the above-mentioned detached ﬂow region which appears along the top horizontal wall in the structure of the steady ﬂow at large enough Rayleigh numbers. Away from this region, the contour lines were found to be inclined at an angle of approximately 20◦ with respect to the horizontal and to propagate (in time) orthogonally to their direction (Figure 6.18). These lines were thought to be the wavefronts of internal waves shed from the region where the instability mechanism takes place (this conjecture being supported by the earlier ﬁndings of Patterson and Imberger, 1980, that the stratiﬁed core region in a differentially heated cavity is

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(a)

(b)

(c)

(d)

Figure 6.18 Fluctuating temperature ﬁeld [plane (x, y)] for buoyancy convection in a square cavity ﬁlled with air (Pr = 0.71, A = 1, Ra = 1.84 × 108 ; hot wall on the right side, cold wall on the left side; adiabatic horizontal walls; evenly distributed snapshots show contour lines inclined at an angle of approximately 20◦ with respect to the horizontal and propagating orthogonally to their direction). After Le Qu´er´e and Behnia (1998); Reproduced by permission of Cambridge University Press

capable of sustaining internal gravity waves). These waves, in fact, were found to be characterized by a frequency less than the Brunt–V¨ais¨al¨a (BV) frequency. Such a characteristic frequency, which in nondimensional form (based on the thermal diffusive time) reads 1

ωBV = (κRa Pr) 2

(6.27)

(where κ is the nondimensional stratiﬁcation parameter in the core of the cavity, i.e. ∂T /∂y), is known to be the upper possible limit for wave patterns arising in stratiﬁed regions (it represents the highest frequency that any stratiﬁed medium can support, hence no internal waves of higher frequency can be excited). Some theoretical background about these waves and the related genesis has been provided in Section 4.1, where it was illustrated via precise mathematical arguments that when a stratiﬁed ﬂuid is excited with a particular frequency ω, internal waves can be generated only if ω ωBV .

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In general, the frequency of these waves is related to the wavenumbers in horizontal and vertical directions by the relation ω = ωBV

qx2 qx2 + qy2

1 2

(6.28a)

which means that they propagate along a wavevector inclined (with respect to the vertical) at an angle θ deﬁned by ω = ωBV sin(θ )

(6.28b)

The particle velocities are all in lines perpendicular to the wavevector and parallel to the direction of the wavefronts (interestingly, in the speciﬁc case in which qy = 0 one recovers the situation with vertical wavefronts and particle motion represented by Eq. (4.15a) in agreement with the original deﬁnition of ωBV as the natural frequency of a vertically displaced particle given in Section 4.1). Application of Eq. (6.28b) to the conditions corresponding to Figure 6.18 gives an angle θ = 20◦ , in excellent agreement with the inclination of the contour lines in such a ﬁgure, hence providing a solid basis to the arguments of Le Qu´er´e and Behnia (1998) by which the oscillatory phenomena in the cavity core were ascribed to internal waves excited by a basic instability mechanism characterized by a frequency below the cutoff frequency (ωBV ). At higher values of Ra, it was found that the wave pattern in the core becomes even more complex than that described above. On the other hand, such a feature is also in agreement with the general properties of patterns determined by internal waves, which, as illustrated by several authors (e.g., Thorpe, 1968; Armﬁeld and Patterson, 1991; Merzkirch and Peters, 1992) can take many forms as a consequence of the emergence of various modes characterized by several wavenumbers in the vertical and horizontal directions (a variety of modal structures are possible) and related interaction. In closed cavities, in particular, internal waves can build standing patterns determined by the superposition of two identical wavetrains travelling in opposite directions (the second wavetrain is generally formed by reﬂection of a travelling wavetrain from a solid wall). The related frequency obeys the law ω = ωBV

A2core p2 2 Acore p2 + n2

12 (6.29)

where p and n are the half-wavelengths in the horizontal and vertical directions, respectively, and Acore is the aspect ratio of the linearly stratiﬁed region, which was used by Le Qu´er´e and Behnia (1998) to justify some of the behaviours obtained numerically for 109 Ra 1010 . Concerning the physics of the aforementioned basic (oscillatory) forcing mechanisms responsible for the excitation of internal waves, so far there has not been any deﬁnitive explanation of their origin. However, as mentioned earlier, Le Qu´er´e and Behnia (1998) noted that the primary instability takes place at the base of the detached ﬂow region and it is therefore believed that the instability mechanism is related to this particular ﬂow structure. In some previous numerical studies (Paolucci and Chenoweth, 1989), the corner ﬂow was considered to be caused by an internal hydraulic jump and the jump theory was used to predict bifurcation of the steady ﬂow into periodic ﬂow. The work of Ravi et al. (1994), however, examined the corner phenomenon more closely and proved that there was no connection between it and a hydraulic jump (with the natural consequence that the instability mechanism should not be related to hydraulic jumps). Janssen and Henkes (1995) suggested that this instability could be of a Kelvin–Helmholtz type (the reader is referred to the fundamental information about this instability provided in Section 3.4.3) of the jet-like structure emerging from the vertical boundary layers (exiting from those

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corners of the cavity where the vertical boundary layers are turned horizontal). This was supported by an examination of the spatial structure of the ﬂuctuating ﬁeld obtained (numerically) for various aspect ratios and Prandtl numbers. The transition from a laminar to a turbulent state for the speciﬁc case of a square cavity was investigated by these authors for Prandtl numbers between 0.25 and 2.0. They reported that such a transition occurs through periodic and quasi-periodic ﬂow regimes (in agreement with the subsequent ﬁndings of Le Qu´er´e and Behnia, 1998). The ﬁrst bifurcation was found to be related to an instability occurring in the jet-like ﬂuid layer (this instability being mainly shear driven), whereas (as also observed by Paolucci and Chenoweth, 1989, for Pr = 0.71) the other bifurcation occurs in the boundary layers along the vertical walls. For Prandtl numbers between 2.5 and 7.0, interestingly, an immediate transition was detected from the steady to the chaotic ﬂow regime without intermediate regimes (this transition is also caused by instabilities originating and concentrated in the vertical boundary layers). In the same year, Xin and Le Qu´er´e (1995) focused more speciﬁcally on the instabilities originating from waves propagating along the vertical boundary layers. They considered the case of a vertical tall cavity (A = 1/4) for Pr = 0.71 (air) and performed numerical simulations for three different values of the Rayleigh number (Ra = 6.4 × 108 , 2 × 109 and 1010 ). It was clearly illustrated for Ra = 6.4 × 108 that the time-dependent solution for the considered cases is made up of travelling waves which run downstream in the boundary layers (in the instantaneous ﬂow they noted periodic oscillations of the isotherms in the downstream parts of the boundary layers, which correspond to the primary instability, in the form of Tollmien–Schlichting waves). According to such simulations, the amplitude of these waves grows as they travel downstream and hook-like temperature patterns form at the outer edge of the thermal boundary layer. At Ra = 2 × 109 , the oscillations of isotherms in the downstream part of the boundary layer, however, become so irregular that eddies are formed there and ejected into the cavity core. The stratiﬁed core is no longer motionless and the isotherms oscillate periodically around their mean horizontal position. Since the cavity core is well stratiﬁed, this phenomenon was ascribed to the propagation of internal waves (as in the case discussed earlier for A = 1). Xin and Le Qu´er´e (1995), in fact, suggested that large-amplitude ﬂuctuations due to the sidewall boundary layer instability are capable of permanently exciting internal waves with a characteristic frequency less than the Brunt–V¨ais¨al¨a (BV) frequency. At the largest Rayleigh number investigated (1010 ), all these phenomena were found to become more marked. For such conditions, the sidewall waves grow to such an extent that they result in the formation of large, unsteady eddies that totally disrupt the boundary layers. These eddies throw hot and cold ﬂuid into the upper and lower parts of the core region, resulting in thermally more homogeneous top and bottom regions that squeeze a region of increased stratiﬁcation near the mid-cavity height. Also in this case internal waves are excited and propagate vigorously through the stratiﬁed region. The mechanism by which this excitation becomes possible was explained assuming that when the Rayleigh number is increased to a sufﬁciently high value (2 × 109 in their simulations), successive bifurcations and nonlinear interactions produce frequencies in the time spectra smaller than the BV frequency. In practice, this corresponds to the appearance of eddies of sufﬁciently large scale as to be characterized by frequencies smaller that the BV frequency (it is hence clear that the permanent excitation of the internal waves is not directly related to the sidewall waves travelling downstream, but is due to the eddies resulting from the nonlinear interactions of the travelling waves in the boundary layers). About 10 years later, the same authors revisited the problem (Pr = 0.71) by using stability analysis algorithms such as Newton’s iteration (steady-state solving), Arnoldi’s method and the

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Figure 6.19 Critical Rayleigh number as a function of the aspect ratio A = length/height for Pr = 0.71 (differentially heated cavity with adiabatic horizontal walls). With increasing aspect ratio A, Racr ﬁrst decreases, reaches a minimum at A = 0.2 and then increases again; for A = 1, 1/2 and 1/3, the critical modes are anticentrosymmetric (ACS), oscillate at a low frequency and completely ﬁll the cavities; at lower aspect ratios, the critical modes are travelling waves in the vertical boundary layers and oscillate at higher frequencies (for A = 1/4, 1/5 and 1/6, the most unstable modes are centrosymmetric and for A = 1/7 and 1/8 they are anticentrosymmetric). Data • after Xin and Le Qu´er´e (2006); Reproduced by permission of Taylor & Francis; http://dx.doi.org/10.1080/10407780600605039

continuation method (Xin and Le Qu´er´e, 2006). They were particularly interested in computing Hopf bifurcation points characterizing the onset of time-dependent ﬂows in cavities with A < 1. Aspect ratios of 1/8 A 1, in fact, were investigated and accurate critical points of several unstable modes which were not fully available in the literature provided for each cavity (Figure 6.19). Results for 1/3 A 1 revealed that for such a range of aspect ratios the mechanism leading to the ﬁrst instability is basically the same as already identiﬁed by Le Qu´er´e and Behnia (1998) for the square cavity. In the separated boundary-layer regime at Rayleigh numbers of the order of 107 , most of the ﬂuid ﬂows up along the hot vertical wall, turns smoothly around the exiting corner, bounces slightly downward and follows the top wall before joining the cold wall. With increasing Rayleigh number, the rebounce of the main stream at the exiting corner becomes more and more important so as to create a detached ﬂow region: after impinging on the top wall, the main stream ﬁrst bounces back strongly downwards and then moves up again to join the top wall, which it follows until it meets the cold wall, spreading gently to ensure the entrainment of the downward boundary layer. The detached ﬂow, which is very weak, takes place for distances from the wall approximately equal to one-quarter to one-third of the cavity height. It is such a detached ﬂow near the exiting corners of the vertical boundary layers that becomes unstable and the corresponding time-dependent ﬂow has a relatively low frequency (less than the BV frequency). No matter what value of A is considered, the most unstable mode is anticentrosymmetric, that is, the centrosymmetry of the base solution is broken.

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Figure 6.20 Unstable base solution [plane (x, y)] and the ﬁrst six unstable modes (eigentemperature) for a tall cavity ﬁlled with air (Pr = 0.71, A = 1/4, Ra = 1.1 × 108 ; hot wall on the right side, cold wall on the left side; adiabatic horizontal walls). From left to right: temperature, stream function and modes 1–6. After Xin and Le Qu´er´e (2006); Reproduced by permission of Taylor & Francis; http://dx.doi.org/10.1080/10407780600605039

Unlike in cavities of 1/3 A 1, the unstable modes responsible for the onset of time-dependent ﬂows for A = 1/4, 1/5, 1/6 and 1/7 were found to display travelling waves around the cavity, in agreement with the earlier study by Xin and Le Qu´er´e (1995). For these cases, the base solutions that become unstable do not exhibit any detached ﬂow region. The leading eigenmodes are made up of waves consisting of alternative positive and negative patches which travel along the cavity walls in an anticlockwise sense in Figure 6.20. Interestingly, if one positive and one neighbouring negative patch are considered as one wave structure (one spatial period), each unstable mode is made up of a well-deﬁned number of wave structures and the symmetry of the unstable modes depends on this number: even numbers mean anticentrosymmetry of the unstable modes and odd numbers imply centrosymmetry. The angular frequency ωc of the unstable modes was found to be an increasing function of the number of such wave structures. These aspects need some additional discussion in order to be explicitly analysed and understood. As an example, for the cavity with aspect ratio 1/4, Xin and Le Qu´er´e calculated six unstable modes and the corresponding critical points. These unstable modes were numbered with increasing order of Racr , which is also the order of increasing ωc and the number of wave structures (Figure 6.20). In Figure 6.20, mode 1 has Racr = 1.031917 × 108 , its number of wave structures is equal to 7 and mode 1 is thus centrosymmetric. Modes 2–6 have 8, 9, 10, 11 and 12 wave structures, respectively, and are alternately anticentrosymmetric (ACS) and centrosymmetric (CS). Qualitatively similar behaviours hold for other values of the aspect ratio. For A = 1/5, four unstable modes emerge with increasing Racr ; they are made of 7 (CS), 8 (ACS), 9 (CS) and 10 (ACS) wave structures, respectively. In a cavity of A = 1/6, the two most unstable modes possess 9 (CS) and 10 (ACS) wave structures, respectively, and the critical values are equal to Racr = 1.109131 × 108 . For A = 1/7, the two most unstable modes are made of 10 (ACS) and 11 (CS) wave structures, respectively, and the same is true for A = 1/8. For 1/8 A 1/4, these unstable modes consisting of travelling waves were found to oscillate at higher frequencies (approximately 10 times those observed in cavities of larger A). As illustrated above, depending on the numbers of wave structures contained, they are either centrosymmetric (odd numbers) or anticentrosymmetric (even numbers). For A = 1/4, 1/5 and 1/6 the most unstable modes are centrosymmetric and for A = 1/7 and 1/8 they are anticentrosymmetric.

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As already pointed out by Xin and Le Qu´er´e (1995), their spatial distribution is limited to the region near the cavity walls. The excitation of internal waves propagating in the internal stratiﬁed region, typical of 1/3 A 1, does not occur (or occurs at higher Ra as discussed earlier for the case A = 1/4) in taller cavities where the onset of unsteadiness is due to a travelling wave instability characterized by typical frequencies larger than the BV frequency (known to be the upper bound for possible wave patterns developed by the linearly stratiﬁed and almost quiescent core). The main outcome of all such numerical effort provided by Le Qu´er´e and co-workers over the years can, therefore, be summarized as follows. For aspect ratio A between 1/10 and 1, there are two distinguishable fundamental mechanisms for the onset of time-dependent ﬂows: for 1/3 A 1, it is the ﬂow structure at the exiting corners of the vertical boundary layers in the form of detached ﬂow that becomes unstable and the corresponding time-dependent ﬂows have a low frequency capable of exciting internal waves propagating in the internal stratiﬁed region; for 1/10 A 1/4, it is the vertical boundary layers that become unstable to travelling waves, which oscillate at a frequency higher than the BV frequency (the reader is referred, in particular, to the theoretical studies by Tao et al., 2004a,b for detailed models and explanations about the delicate mechanisms leading to instability of these boundary layers). In the latter case, no internal waves are excited unless nonlinear interactions of vertically travelling waves produce frequencies in the time spectra smaller than ωBV and/or relatively large eddies are formed and injected into the cavity core (the reader is referred, e.g., to the work of Voisin, 1994, for precise mathematical arguments about the mechanisms of internal wave generation in stratiﬁed ﬂuids as induced by moving point sources). Interestingly, Xin and Le Qu´er´e (2006) also discovered multiple steady-state solutions at A ∼ = 1/3 never reported before. For 1/8 A 1/4, they conﬁrmed for slightly supercritical Rayleigh numbers the possible existence of multiple branches of time-dependent ﬂows in agreement with their earlier results (Xin and Le Qu´er´e, 1995), which indicated the existence of six distinct branches (each characterized by a constant angular frequency) for A = 1/4, and those of Gelfgat (2004), who studied a similar behaviour for A = 1/8 (in practice, the multiple unstable modes, whose properties have been discussed in the text above, and the corresponding bifurcation points are approximately the lower bounds of such solution branches). Air-ﬁlled cavities with even lower values of the aspect ratio (A < 1/12) were studied numerically by Chenoweth and Paolucci (1986), Wakitani (1997, 1998) and Mizushima and Tanaka (2002). The most interesting outcome of these studies with very slender cavities (A 1) is that for such cases the ﬂow, in general, undergoes a ﬁrst transition from an initial single vertically elongated cell to a steady multicellular ﬂow when the Rayleigh number is sufﬁciently increased (e.g. four-cell convection for 6 × 103 Ra 7 × 103 ). In a study by Wakitani (1998), for aspect ratios from A = 1/24 to 1/10 the various cellular structures were clariﬁed by gradually increasing Ra to 106 ; unsteady multicellular solutions were found in some ranges of the Rayleigh number. All these results and related insights into the physics discussed so far were obtained for cavities with adiabatic horizontal walls. The case with conducting boundaries exhibits notable differences with respect to those with adiabatic walls. From a theoretical point of view, the problem is formally equivalent to that treated earlier, but in contradistinction to ﬂuids with Pr < O(1) (where the type of horizontal thermal boundary conditions enters signiﬁcantly the dynamics only if the system is far from the threshold of the primary instability), here the scenario changes dramatically already for the ﬁrst ﬂow bifurcation. This result is not unexpected if one considers that, in the light of the analysis of Gershuni et al. (1992) for inﬁnite layers (Section 6.1.3) and Pr O(1), the presence of zones of potentially unstable stratiﬁcation near the upper and lower horizontal boundaries (induced by the basic ﬂow) may be responsible for the onset of mechanisms of the Rayleigh–B´enard type (and, thereby, signiﬁcantly alter the nature of the ﬂow with respect to the case of insulating boundaries).

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Thermal Convection: Patterns, Evolution and Stability

As an example, the onset of unsteadiness that for a square cavity (Pr = 0.71) with adiabatic walls takes place at Ra = O(108 ), occurs at Ra = O(106 ) when conducting boundaries are considered (Jones and Briggs, 1989). Some additional interesting numerical results for such a case (A = 1 with conducting horizontal walls) are due to Winters (1987), who performed two-dimensional numerical simulations for Pr = 0.71, and Gelfgat and Tanasawa (1994), who also examined Pr = 0.015 and Pr = 7. They identiﬁed several possible modes in the form of waves travelling along the cavity walls breaking or preserving the symmetry of the basic ﬂow and slightly differing in the related critical value [as an example, for Pr = 7 three possible modes were observed, the ﬁrst and the third anticentrosymmetric and the second centrosymmetric, i.e. possessing the same symmetry of the base solution, namely V (1 − x, 1 − y) = −V (x, y) and T (1 − x, 1 − y) = −T (x, y)]. This problem was re-examined later by Xin and Le Qu´er´e (2001). They found the unstable modes to be all oscillatory and to consist of waves travelling in the boundary layers. Moreover, some general trends were obtained as a function of the Prandtl number (Pr = 0.1, 1, 3 and 20), as discussed in detail in the remainder of this section. Interestingly, it was observed that for Pr 3 the perturbations in the two vertical boundary layers are almost completely disconnected [because the amplitude of the eigenmode along the horizontal walls is of O(10−3 ) of the maximum amplitude]. Starting from midheight, the ﬂuctuations are ﬁrst ampliﬁed and then strongly damped after turning around the cavity corners, and what happens in one boundary layer is more or less disconnected from what happens in the other. This feature was also observed for smaller Prandtl numbers (but increasingly pronounced with increasing Prandtl number, as evidenced by their results for Pr = 3 and 7). The global trend in terms of instability threshold is that with increasing Prandtl number the critical Rayleigh number increases, although for Pr = 1 the critical Rayleigh number is slightly smaller than that obtained for Pr = 0.71. Other notable properties can be summarized as follows (Xin and Le Qu´er´e, 2001). As the boundary layers become thinner (when Rayleigh number is higher), the wave structures in the critical mode become smaller as the Prandtl number increases and the corresponding angular frequency increases. Moreover, depending on the thickness of the boundary layers at the critical Rayleigh number, the unstable mode can be either centrosymmetric or anticentrosymmetric. Although these authors showed that for small Prandtl number (Pr 1) the most unstable mode is centrosymmetric and for larger Prandtl number it is anticentrosymmetric, this should not be considered as a general conclusion and may not hold for other Prandtl numbers which were not treated in their work. Even though limited to the case A = 1, this analysis also considered the stability of two-dimensional base solutions with respect to 3D disturbances (which is perhaps the most interesting and original aspect of such work). Notably, it was found that for a Prandtl number of 1 the base solutions are more unstable to three-dimensional perturbations and the most unstable three-dimensional mode is oscillatory. For larger Prandtl numbers (3, 7 and 20), the two-dimensional base ﬂows, however, become more stable to three-dimensional perturbations than to the two-dimensional ones and the most unstable modes are thus two-dimensional. For smaller Prandtl numbers (0.71, 0.1 and 0.015), the base solutions are again more unstable to three-dimensional perturbations. Nevertheless, these three-dimensional modes are no longer connected to the most unstable two-dimensional oscillatory modes and correspond to stationary perturbations. Furthermore, for Pr = 0.015 and 0.1, the critical Rayleigh number is approximately 40 times smaller than that found for two-dimensional perturbations, indicating, as expected, a dramatic change in the nature of the instability. The stability results for Pr = 0.71 are shown in detail in Figure 6.21. The ﬁrst three modes are oscillatory. The structure of these three-dimensional perturbations, if compared with two-dimensional perturbations, reveals that the most signiﬁcant difference occurs in the vertical

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Figure 6.21 Neutral curves of the three most unstable oscillatory modes and of the stationary mode for a square cavity with horizontal conducting boundaries and Pr = 0.71 (three-dimensional perturbations). After Xin and Le Qu´er´e (2001); Reproduced by permission of the American Institute of Physics

boundary layers and at the exiting corners: structures are smaller there. For modes 1 and 2, perturbations are damped before turning the corners and are subsequently reampliﬁed slightly, whereas for mode 3 there is one more wave period of very small size at the corners. The most important consequence of Figure 6.21, however, is that it clearly shows the existence of a stationary mode (mode 4) as the most critical one (with wavenumber qc = 5.85). Interestingly, this could be the reason why Henkes and Le Qu´er´e (1996) obtained a steady numerical solution in their 3D numerical simulations at Ra = 1.8 × 106 . As the neutral curve indicates, this mode does not seem to be connected to a two-dimensional disturbance. Figure 6.22 collects the results for Pr = 0.1. Modes 1 and 3 are more stable in the three-dimensional case and mode 2 is less stable. Modes 4–6 are more unstable. The corresponding critical values are Racr = 66 764 and qc = 20.25 for mode 4, Racr = 44 231.39 and qc = 17.78 for mode 5 and Racr = 39 177.84 and qc = 17.36 for mode 6. Modes 4 and 5 correspond to travelling waves whereas mode 6 is stationary. Moreover, modes 4 and 6 are centrosymmetric and mode 5 is anticentrosymmetric. For Pr = 0.015, the neutral curve for mode 1 (Figure 6.23) shows critical values Racr = 7974.46 and qc = 21.50. The critical values for the other modes are Racr = 2228.406 and qc = 14.5 for mode 2, Racr = 1348.121 and qc = 13.58 for mode 3, Racr = 1060.562 and qc = 11.68 for mode 4 and Racr = 976.260 and qc = 10.84 for mode 5. Modes 2, 3 and 4 are travelling waves whereas mode 5, the most unstable mode, is stationary. Modes 3 and 5 are centrosymmetric whereas mode 4 is anticentrosymmetric. It is clear from Figures 6.22 and 6.23 that for Pr = 0.1 and 0.015, the most unstable modes possess the same symmetry and have similar structure. As outlined earlier, this suggests that the nature of the corresponding instability mechanism is the same. To conclude this long discussion, it is worth noting that the two fundamental models considered in the present section as thermal horizontal boundary conditions [i.e. adiabatic or conducting limits,

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(a)

(b) Figure 6.22 Neutral curves of the ﬁve most unstable oscillatory modes and of the stationary mode for a square cavity with horizontal conducting boundaries and Pr = 0.1, (three-dimensional perturbations). After Xin and Le Qu´er´e (2001); Reproduced by permission of the American Institute of Physics

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Figure 6.23 Neutral curves of the four most unstable oscillatory modes and of the stationary mode for a square cavity with horizontal conducting boundaries and Pr = 0.015 (three-dimensional perturbations). After Xin and Le Qu´er´e (2001); Reproduced by permission of the American Institute of Physics

which led to the discussion being split for ﬂuids with Pr > O(1) into two distinct parts] must be regarded to a certain extent as an artiﬁcial division. In many practical cases with Pr O(1), the boundary condition will fall between these two extremes: given the high sensitivity that these systems have proved to exhibit with respect to them, an astonishing variety of possible ﬂow solutions (whose structure and stability depend sensitively on the assumptions made) must be expected.

6.3

The Inﬁnite Vertical Layer: Cats-eye Patterns and Temperature Waves

The scenarios and general trends described in the previous section were mainly concerned with enclosures with O(10−1 ) A O(10) (e.g. square cavities and rectangular cavities with moderate aspect ratio). Some analyses have also appeared where the problem was considered in the limit in which the aspect ratio tends to zero (A → 0), that is, for vertically elongated geometries. In such a case, the core ﬂow in the horizontal direction is no longer a feature of the system, but a similar behaviour (a parallel ﬂow) can be observed along the vertical direction. In 1954, Batchelor (1954b) investigated the ﬂow in a ﬂuid-ﬁlled container with one opposing pair of vertical walls at different temperatures. His attention was restricted to the limiting case of inﬁnite spanwise aspect ratio. He realized that if the cavity was narrow enough, that is, with a small enough A, a fully developed region might exist in which a one-dimensional solution would apply (that is, the temperature would vary linearly between the hot and cold walls and the purely vertical velocity would have an odd-symmetric cubic proﬁle; see Figure 6.24). Eckert and

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Thermal Convection: Patterns, Evolution and Stability

Figure 6.24 Sketch of the velocity and temperature proﬁles in the ‘conduction regime’ for an inﬁnite vertical ﬂuid layer

Carlson (1961) named such a solution the ‘conduction regime’ and observed its effective existence in a cavity with A 1. The general properties of this exact solution to the thermal-convection equations have been described in Section 2.4 [see Eq. (2.61); the reader is also referred to Daniels and Wang, 1994, for the ranges of Prandtl number and cavity aspect ratio allowing the effective existence of this solution]. A lot of work by different authors has been devoted to the study of the instabilities associated with such inﬁnite ﬂow. It has been shown that, even if the most dangerous disturbances are planar (two-dimensional) over the whole range of Pr (0 Pr ∞), however, the type of instability is determined by the value of the Prandtl number (Korpela, 1974). In particular, the critical disturbance modes are hydrodynamically driven and stationary when Pr < 12.45 (Birikh, 1966a; Ruth, 1979), but they are thermally driven and oscillatory when Pr > 12.45 (Birikh et al., 1972; Korpela et al., 1973). Nonlinear analyses revealed, in particular, that former disturbances evolve into a multicellular pattern of steady transverse rolls (a regular cellular pattern becomes superimposed on the basic ﬂow to produce a ‘cats-eye’ pattern of streamlines, Gershuni et al., 1968) (see, e.g., Figure 6.25) and the latter cause the convection in the form of two counter-propagating waves (Fujimura and Mizushima, 1991). In such a case, the stream pattern replacing the initial unicellular (vertical) base state is a system of rotating vortices, periodic in the vertical direction, which stand between ascending and descending ﬂows (Figure 6.26a); intensities of neighbouring vortices are changed periodically in anti-phase; in contrast with the standing pattern shown by streamlines, the temperature ﬁeld is a pair of travelling waves, one of which propagates along the hot wall upwards and the other propagates along the cold wall downwards (that is why this regime is sometimes also referred to as ‘temperature waves’) (Figure 6.26b).

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(a)

249

(b)

Figure 6.25 Critical mode (stationary) in the plane (x, y) for Pr = 0.7: (a) streamlines; (b) temperature ﬁeld. Courtesy of. G.D. McBain

Hart (1971a) was the ﬁrst to discern that at low Prandtl numbers the disturbances causing the secondary cellular ﬂows draw their energy from the base ﬂow and the instability, therefore, is hydrodynamic in nature (caused by the shear between the upward- and downward-ﬂowing ﬂuid streams); Korpela et al. (1973) found that as the Prandtl number is increased, more of the energy comes from the buoyancy ﬁeld and, as explained above, the mode of instability changes from steady multicellular ﬂow to travelling waves. A linear stability analysis providing the critical Grashof number for this problem in the whole range 0 Pr ∞ (ﬁrst bifurcation) is due to McBain and Armﬁeld (2004). They extended the earlier results of Korpela et al. (1973) and Ruth (1979), illustrating that for Pr < 12.45 the critical Grashof number is nearly independent of Pr [as an example, Figure 6.27 shows that the stability margins for Pr = 0.71 (air) and water (Pr = 7) differ little from the zero-Pr asymptote], whereas for Pr > 12.45 the threshold is markedly inﬂuenced by the Prandtl number.

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(a)

(b)

Figure 6.26 Critical oscillatory mode (temperature wave) in the plane (x, y) for Pr = 12.454: (a) streamlines; (b) temperature ﬁeld. Courtesy of. G.D. McBain

As shown in Figure 6.28, as Pr increases, a second lobe representing oscillatory behaviour appears for a narrow range of qy . As Pr increases further, the oscillatory lobe widens and for Pr 12.454 becomes dominant. Although the dependence of the critical Grashof number on Pr for low Prandtl numbers is weak, it is rather complicated with a minimum near Pr = 0.1 and a maximum near Pr = 0.5 (Rudakov, 1967). In contrast, as already mentioned, for Pr > 12.454 the critical Grashof number exhibits a 1 pronounced dependence on the Prandtl number and behaves approximately as Grcr ∝ Pr− /2 (Gill and Kirkham, 1970). These behaviours are summarized in Figure 6.29.

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Figure 6.27 Linear stability margins for the inﬁnite vertical differentially heated layer at Pr = 0.7 and 7 also showing the Pr → 0 asymptote (Grashof number deﬁned as Gr = gβT T d 3 /ν 2 . Courtesy of. G.D. McBain

Figure 6.28 Linear stability margins for the inﬁnite vertical differentially-heated layer at Pr = 11.7, 12.454, 80 and 103 (Grashof number deﬁned as Gr = gβT T d 3 /ν 2 . Courtesy of. G.D. McBain

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Thermal Convection: Patterns, Evolution and Stability

Figure 6.29 Critical Grashof number (Gr = gβT T d 3 /ν 2 as a function of Pr. Courtesy of. G.D. McBain

When the Grashof number is further increased, as usual, secondary, tertiary and other bifurcations take place in the evolutionary process from laminar to turbulent ﬂuid ﬂow. As an example, Lee and Korpela (1983), Chait and Korpela (1989) and Clever and Busse (1995b) investigated the subsequent oscillatory modes of convection, to which the steady transverse rolls undergo transition (when the temperature gradient is further increased). The evolution of the travelling-wave structures (which occur at Pr > 12.45) was considered, in particular, by Bratsun et al. (2003). Their mixed analysis (both experimental and numerical for Pr = 26) yielded some interesting and original results, which deserve some additional discussion here. According to their study, the evolution of ﬂow pattern to space irregularity includes an interesting sequence of events. At the beginning, as explained earlier, the basic plane-parallel ﬂow becomes unstable to pulsating transverse rolls located between ascending and descending ﬂows. As the Grashof number increases, these rolls become wavy and eventually are pulled apart, giving rise to a cellular-like pattern. Finally, the system evolves to rolls chaotically modulated along their axes, which makes the consideration of three-dimensional ﬂows necessary. In particular, the nonlinear dynamics of ﬂow in time was found to become irregular already for the wavy rolls and to follow generally the Ruelle–Takens route to chaos (Section 1.8). Interestingly, these authors also pointed out (see also Eckert and Carlson, 1961) that, in many effective circumstances [a moderately high cavity and Pr > O(1)], when Gr is increased, the ﬂow may undergo transition to a boundary-layer regime before the conduction state becomes unstable with respect to oscillatory disturbances of the travelling-wave type. In practice, in many real circumstances, as the Prandtl number is increased and the heat advection in such ﬂuids becomes more effective in comparison with thermal conduction, the probability of

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a boundary-layer regime taking place becomes higher; as a natural consequence, it is expected that in such circumstances the ﬁrst ﬂow bifurcation will again be related to an instability of the boundary layers as discussed earlier for Pr O(1) (Section 6.2.2). When such a boundary-layer regime is attained, the basic state is no longer represented by the conduction regime discussed earlier (with the temperature varying linearly between the hot and cold walls); rather, as already illustrated in Figure 6.17 for the square enclosure, the steady base natural convection will be featured by a stably stratiﬁed ﬂuid between the vertical walls maintained at different uniform temperatures. Thin boundary layers will exist along the lateral walls whereas in the central core the temperature is uniform in horizontal planes and increases in the vertical direction (in a different perspective, the concepts just illustrated can be repeated observing that the buoyancy-driven boundary-layer ﬂow between inﬁnite vertical planes generally involves the presence of both a constant vertical temperature gradient and a constant horizontal temperature difference; Daniels, 1985). Interesting stability analyses based on the assumption of a basic temperature ﬁeld of such a kind were carried out by Vest and Arpaci (1969), Mizushima and Gotoh (1976) and Bergholz (1978). The last author, in particular, considered widely varying levels of stable background stratiﬁcation for Pr ranging from 0.73 to 103 . Mizushima and Gotoh (1976) found, as an example, that the convection of water (Pr = 7.5) is unstable to travelling disturbances for A > 1.42 × 102 , whereas it is unstable to stationary disturbances for A < 1.42 × 102 , hence with a signiﬁcant departure from the typical scenario discussed for the conduction regime. Most of the available experimental results dealing with boundary-layer instabilities in tall slots are related to transparent liquids: Elder (1965), Pr ∼ = 480; Chen = 103 ; Seki et al. (1978), Pr ∼ ∼ and Thangam (1985), Pr = 158–720; Chen and Wu (1993), Pr ∼ = 720; Wakitani (1994), Pr ∼ = 900. Some numerical results are shown in Figures 6.30 and 6.31, which show typical instabilities with disturbances travelling downstream in the boundary layers. In many circumstances, stationary secondary ﬂows were observed in place of the temperature waves, which should occur for a basic ﬂow corresponding to the conduction regime (Elder, 1965; Seki et al., 1978; Linthorst et al., 1981).

6.4

Three-dimensional Parallelepipedic Enclosures

Since two-dimensional ﬂows cannot be rigorously realized as sidewalls along the third direction (z) play an important role in ﬁnite ﬂow conﬁgurations, that is, the end wall bias or ‘contamination’ is unavoidable (Crochet et al., 1987, Dupont et al., 1987) and/or the ﬂuid motion displays a spontaneous tendency to break two-dimensional symmetry due to an intrinsic 3D nature of the most dangerous disturbances, many studies (experimental and numerical) have been devoted to effective three-dimensional boxes. With the spectacular rise in its industrial importance, the initially largely theoretical development of the subject (mostly based on the application of linear stability analysis to ideally inﬁnite systems) received a continuation with new studies based on well-controlled laboratory experiments (conceived ad hoc to enlighten some speciﬁc properties of the ﬂow) or based on the solution of the nonlinear and unsteady thermal-convection equations. This section provides a synthetic account of such efforts. As usual, in line with the spirit of the present book, the goal is to stake out some common ground by providing a survey of the distinct

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(a)

(b)

(c)

(d)

T: 0.06 0.13 0.19 0.25 0.31 0.38 0.44 0.50 0.56 0.63 0.69 0.75 0.81 0.88 0.94

Figure 6.30 Subsequent snapshots of convection in an elongated vertical cavity ﬁlled with a high-Pr ﬂuid (Pr = 15, A = 1/20, Ra = 1 × 1010 ; Ra based on the height; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions). The snapshots clearly show travelling waves running downstream in the boundary layers (numerical simulation, M. Lappa)

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Figure 6.31 Detail of ﬂuid motion near the boundary layer adjacent to the hot wall (Pr = 15, A = 1/20, Ra = 1 × 1010 )

approaches and points of view. Much room is also devoted to highlight hidden or overlooked aspects, to give some indications of the most important results, open issues and advances in the ﬁeld. Anyhow, due to page limits, discussions are limited to enclosures differentially heated over two opposing vertical walls with the other horizontal and lateral walls thermally insulated. It is convenient to start to deal with such a topic by considering the pioneering numerical study of Millinson and de Vahl Davis (1977), who showed three-dimensional effects for Ra = 104 to be the result of the inertial interaction of the ﬂow with the stationary walls together with a contribution arising from buoyancy forces generated by longitudinal temperature gradients. The inertial effect was found to be inversely dependent on the Prandtl number, whereas the thermal effect is nearly constant. For higher values of Ra, multiple longitudinal ﬂows were observed to develop as a delicate function of Ra, Pr and the cavity aspect ratios. For the sake of clarity, following the same approach as used in Section 6.2, prior to expanding on the cases with Pr O(1) the nature and structure of these ﬂows are illustrated here using examples with relevance to crystal growth, namely Pr < O(1).

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As a ﬁrst example, Figure 6.32 shows the three-dimensional solution corresponding to the same conditions as in Figure 6.12b (Pr = 0.01), but with a ﬁnite width along z equal to the extension along x (4 × 1 × 4 enclosure). It is evident that, unlike the multicellular ﬂow predicted by two-dimensional studies and shown in Figure 6.12b, the ﬂow in the generic (x, y) plane is represented by a single cell. Such a simple comparison provides some initial evidence that the ﬂow in the presence of constraints along the third direction can exhibit signiﬁcant differences with respect to idealized models with inﬁnite spanwise aspect ratio. In particular, the multiplicity of solutions described in Section 6.2.1 for the two-dimensional hydrodynamic mode seems to be suppressed by the ﬁnite size. Notably, differences are not limited to the spatial structure of convection, but also apply to its stationary or oscillatory behaviour; as for the same value of Ra as considered in Figure 6.12b, the 3D ﬂow shown in Figure 6.32 for a cavity with equal lengthwise and spanwise aspect ratios (Ax = Lx /d and Az = Lz /d, respectively) is no longer steady (the transition to time dependence occurs for higher values of Ra for the two-dimensional conﬁguration of Figure 6.12b with Az = ∞). Qualitatively similar numerical results were originally reported by Afrid and Zebib (1990) for a zero Prandtl number ﬂuid in 4 × 1 × 2 (length by height by width) and 4 × 1 × 1 rectangular boxes. They found the ﬂow ﬁeld to be represented by one cell, unlike the multicellular ﬂow predicted by two-dimensional studies for Ax = 4. They also proved the extension along z of the enclosure to have an important effect on transition to oscillatory convection, as it was shown that reducing this extension from two to one led to a much higher Racr , making the results of two-dimensional numerical simulations somewhat inadequate for many circumstances of practical interest. A similar stabilizing effect was also observed in the experiments of Hung and Andereck (1988) and Pratte and Hart (1990) for Pr = 0.026 (Hung and Andereck, 1988, studied the transition to oscillatory convection in mercury for cavities with aspect ratios 4 × 1 × 1 and 4 × 1 × 2; Pratte and Hart, 1990, considered closed cavities of aspect ratios 4 × 1 × 1, 4 × 1 × 2 and 8 × 1 × 8). Interestingly, the latter authors found the ﬁrst oscillatory instabilities to appear in the form of standing longitudinal waves [with the axis of the roll(s) perpendicular to the heated endwalls]. Continuing with the description of available results, it should be pointed out that, in practice, the nature of the ﬁrst oscillatory instability that appears in 3D boxes for a given Pr is still an open question. The ranges of existence of different types of ﬂow determined in the case of inﬁnite systems, in fact, might not hold when turning to system of ﬁnite extent, that is, the location of the points in the parameter space (i.e. Pr∗ in Section 6.1.2) where the branches pertaining to the different instability mechanisms, transverse or longitudinal, meet might be signiﬁcantly affected by the ﬁnite size of the system (Lappa, 2007b). Even if, as illustrated in Section 6.2.1, the 2D hydrodynamic mode can become oscillatory when Ra is increased (the reader is referred to Figure 6.11 in particular and related discussion in the text), however, there is some experimental evidence that the oscillatory instabilities found in ﬁnite-sized boxes ﬁlled with liquid metals (molten gallium and mercury) with Pr < Pr∗ (Pr∗ = 0.034 for the adiabatic R–R case, while Pr = 0.02 and 0.027 for molten gallium and mercury, respectively) could be due to a longitudinal mode (see the discussion in Hung and Andereck, 1988) and/or to the superposition of a secondary transversal disturbance on a primary longitudinal one as sketched in Figure 6.33 (Pratte and Hart, 1990; Delgado-Buscalioni et al., 2001b). The possible competition of transverse and longitudinal disturbances has already been discussed to a certain extent in Section 6.1.4 for the speciﬁc case Pr = 0.027, where some theoretical information was provided on the basis of the analysis of Wang and Korpela (1992) for the inﬁnite layer.

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(a)

(b)

Figure 6.32 Snapshot of 3D oscillatory buoyancy convection in a 4 × 1 × 4 enclosure differentially heated from sides (the other solid boundaries thermally insulated) for Pr = 0.01 and Ra = 2 × 103 ; (a) and (b) show cuts of the vector ﬁeld at respectively z = 1, 2, 3 and y = 0.5 planes. After Lappa (2007b)

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Figure 6.33 Sketch of convection given by the superposition of transverse and longitudinal rolls

Along these same lines, however, there have also been some experimental studies focusing expressly on regions of the space of parameters where these two different types of bifurcation may intersect; as an example, Braunsfurth and Mullin (1996) reported new types Hopf–Hopf interaction (in particular, they found four different modes of oscillation at onset in the parameter range between Pr = 0.016 and 0.022). In the attempt to shed some light on these behaviours, Wakitani (2001) investigated numerically the onset of oscillations as a function of the system dimensions (a variety of cavities with lengthwise aspect ratios between 2 and 4 and spanwise aspect ratios between 0.5 and 4.2) and Pr in the range 0 Pr 0.027. He found the thresholds to increase globally both with Pr and with a reduction in the spanwise aspect ratio. However, he conﬁrmed that these variations are not regular, indicating possible changes in the oscillatory modes at onset. The effect exerted on these dynamics by a small inclination of the cavity with respect to the horizontal was investigated by Delgado-Buscalioni et al. (2001b) (see Section 7.2.6 for further elaboration of this aspect). In view of all these arguments, it is evident that a coherent picture of the role played by different types of disturbances in determining the ﬁrst transition to time-dependent convection for three-dimensional enclosures and liquid metals has not yet emerged. Several authors have also investigated the subsequent states which the system undergoes transition to when the control parameter is further increased. ‘Classical’ behaviours were reported (see Section 1.8 for some additional and necessary theoretical background): period doubling transitions (McKell et al., 1990), quasi-periodic ﬂow (see, e.g., Hung and Andereck, 1988; Braunsfurth and Mullin, 1996), followed in some cases by frequency-locked states (Pratte and Hart, 1990). In Section 6.2.1, it was discussed how also in 2D enclosures at a low Prandtl number the ﬂow becomes oscillatory after the onset of stationary shear rolls and it gains in complexity by several possible paths. Nevertheless, these patterns do not reproduce quantitatively (or qualitatively) the route to chaos found in the experiments and 3D numerical simulations. Remarkably, the ﬁrst transition to time-dependent ﬂow seems to be of different natures in the 2D and 3D situations and the same applies to the route to chaos. As an other aspect enriching the subject with additional complexity, some authors provided some lines of evidence supporting the idea that the occurrence of possible unsteady oscillations in ﬁnite-sized boxes may be preceded in some circumstances by a steady 3D instability (see, e.g., Figure 6.24 for the case of the cubical enclosure), which originates in an internal, stratiﬁed shear layer that separates from the adiabatic walls of the cavity.

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Despite the appreciable number of studies mentioned earlier (Hung and Andereck, 1988; Kamotani and Sahraoui, 1990; Braunsfurth and Mullin, 1996; Braunsfurth et al., 1997; Henry and Buffat, 1998; Wakitani, 2001) on the oscillatory instabilities that can occur in these containers for Pr 1 (motivated by the theoretical kinship with the horizontal Bridgman method for the production of semiconductor crystals), there seems to be an outstanding lack of information dealing with this primary stationary bifurcation. Juel et al. (2001) and Hof et al. (2004) were the ﬁrst to focus on the structure of this steady three-dimensional ﬂow by means of a combined experimental–numerical investigation in the case of a small value of the Prandtl number (molten gallium) and shallow containers. The 3D nature of the steady ﬂow was clearly demonstrated by quantitative experimental temperature measurements, which gave an indication of the strength of the convective ﬂow. This convection is inherently three-dimensional and characterized by cross-ﬂows which are an order of magnitude smaller than the main circulation and spread from the endwall regions to the entire bulk when the Grashof number is increased beyond a certain threshold (Gr ∼ = 5 × 103 in the case of silicon melt and a cubic box; Figure 6.34b). A couple of vortices located in the upper half of the box are also clearly visible in the (y, z) midplane (Figure 6.34c).

(a)

(b)

(c)

Figure 6.34 Velocity ﬁeld (steady 3D state) in the three orthogonal coordinate planes of a cubic enclosure differentially heated from sides with the other solid boundaries thermally insulated (Pr = 0.01, Ra = 102 ). After Lappa (2005a)

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For the sake of completeness, it should also be mentioned that similar steady three-dimensional convective structures were reported by means of numerical computations in the case of cubical cavities ﬁlled with air (Pr = 0.71). Fusegi et al. (1991a) found such a steady ﬂow in a relatively wide range of the Rayleigh number (103 Ra 106 ); Janssen and Henkens (1996) highlighted that the subsequent unsteady instability is strongly inﬂuenced by this steady instability and as a result its frequency differs strongly from its counterpart in the two-dimensional square cavity (see Section 6.2.2 for two-dimensional studies on the subject); some 3D numerical studies are also due to Henkes and Le Qu´er´e (1996). Owing to the value of the Prandtl number corresponding to air (Pr = 0.71), this subject would deserve further investigation to discern the intrinsic nature of this 3D instability for values of the Prandtl number of O(1) and the reason why it was not reported in other numerical studies (e.g. Janssen et al., 1993). Perhaps it could be ascribed to the stationary longitudinal instability (SLR) predicted in the linear stability analyses of Kuo and Korpela (1988) for 0.2 < Pr < 2, see Figure 6.4b, and Delgado-Buscalioni (2001a), but never explicitly identiﬁed in numerical calculations or experiments. The problem might be even more complex than as discussed above if one considers that de Gassowski et al. (2003) observed by 3D direct simulation the cubic cavity ﬁlled with air, to return after onset of time dependence to a steady state for higher values of the Rayleigh number (7 × 107 and 108 , for example). Most surprisingly, these authors also reported multiple steady ﬂows differing by their symmetry properties for Ra = 108 . Thereby, they opened up a new perspective on the subject with the discovery of a sort of intermittent behaviour given by the alternance in the space of parameter of oscillatory and stationary (eventually multiple) states for increasing values of the characteristic number. The ﬂow was found to revert again to unsteadiness for Ra = 3 × 108 . In this latter case, however, the instability was proved to be due to the classical mechanism related to vertical boundary layers (Section 6.2.2). Boundary-layer instabilities were also clearly detected in experimental studies on the subject. As an example, Belmonte et al. (1995) clearly observed for a cubic air-ﬁlled cavity a turbulent large-scale circulation (Ra > 3 × 107 ) around the periphery of the cell, with side eddies along each plate. The turbulent ﬂuctuations were found to be conﬁned to the regions near the hot and cold plates, whereas the bulk of the cell was stably stratiﬁed. In the central part of the cell they also detected internal waves with a frequency corresponding to the Brunt–V¨ais¨al¨a frequency of the mean vertical temperature gradient (in this regard this instability exhibits notable analogies to that occurring in two-dimensional cavities discussed in Section 6.2.2). The emergence of well-deﬁned boundary layers was also pointed out by Fusegi et al. (1991b), who reported on the features of three-dimensional natural convection for Pr = 0.71 over a range of Rayleigh number from 103 to 1010 , pointing out similarities and discrepancies between the three- and two-dimensional computations. Differences between 2D and 3D solutions were also evaluated by Labrosse et al. (1997), Soria et al. (2004) and Trias et al. (2007), as discussed in detail in the following. Labrosse et al. (1997), in particular, identiﬁed a hysteretic behaviour, characterized by two critical Rayleigh number values, Racr1 and Racr2 , found to lie in the ranges [3.3, 3.57] × 107 and [3.1, 3.2] × 107 , respectively (these values are about six times smaller than the corresponding values for the two-dimensional square cavity). Trias et al. (2007) presented a set of complete two- and three-dimensional direct numerical simulations in a differentially heated vertical air-ﬁlled cavity of aspect ratio 1/4 with adiabatic horizontal walls. In practice, the conﬁgurations selected by these authors (Rayleigh number based on the cavity height 6.4 × 108 , 2 × 109 and 1010 , Pr = 0.71) should be regarded as an extension to three dimensions of the earlier two-dimensional problem treated by Xin and Le Qu´er´e (1995), whose results were discussed in Section 6.2.2.

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The signiﬁcant differences that they observed between two- and three-dimensional results can be summarized as follows. For two-dimensional simulations, the oscillations at the downstream part of the vertical boundary layer are clearly stronger, ejecting large eddies to the cavity core. In the three-dimensional simulations, these large eddies do not persist and their energy is rapidly passed down to smaller scales of motion. This leads to a reduction in the large-scale mixing effect at the hot upper and cold lower regions and consequently the cavity core still remains almost motionless even for the highest Rayleigh number. The boundary layers remain laminar in their upstream parts up to the point where these eddies are ejected. The point where this phenomenon occurs clearly moves upstream for the three-dimensional simulations. Even for the three-dimensional simulations, these eddies are large enough to excite permanently an internal wave motion in the stratiﬁed core region. All these differences become more marked for the highest Rayleigh number. Further investigations are obviously required along these lines. The problem is very complex, far from being well understood, and this applies to both categories of ﬂuids with Pr < O(1) and Pr ∼ =1

1.0

0.8

0.6 y 0.4

0.2

0.0 1.0 0.8 0.6 z

1.0 0.8

0.4

0.6 0.2

0.4 0.0 0.0

0.2

x

Figure 6.35 Particle spiralling motion in a cubic enclosure ﬁlled with a high-Pr ﬂuid (Pr = 7 and Ra ∼ = 103 ). Courtesy of. D. Melnikov and V. Shevtsova

For high-Pr liquids (Pr > 1), the behaviour is completely different and basically less intricate. The reader may consider the studies (dealing with water-ﬁlled cavities) of Schladow et al. (1989), Hiller et al. (1989) and Kowalewski (1998) for some relevant information. In such a case, the differences between the experimental ﬂow and the idealized two-dimensional convection with inﬁnite spanwise aspect ratio (see, e.g., the discussion in Hiller et al., 1989) are almost negligible

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and mainly occur as a consequence of the failure to achieve strictly adiabatic conditions on the insulating sidewalls. When idealized adiabatic boundaries are considered in the numerical simulations (Schladow et al. 1989, Kowalewski, 1998; Melnikov and Shevtsova, 2005), the 3D effects are somewhat limited to weak spiralling motions responsible, besides the main recirculation, for cross-ﬂow along z (from the sidewalls towards the midplane z = Lz /2). According to such numerical results, as Ra is increased, the streamlines develop from a single spiral, starting at both the nonheated sidewalls and moving towards the midplane (see Figure 6.35), into a double spiral conﬁguration, where two adjacent spirals start at the nonheated sidewalls and move towards the midplane; however, the majority of the streamlines are located within the centre plane and if a particle were released into this spiral, it would spend more time in the centre plane than it would in the rest of the ﬂow.

6.5

Cylindrical Geometries under Various Heating Conditions

As in the case of Rayleigh–B´enard convection (Section 4.10), the study of natural convection induced by lateral heating has also been considered for geometries with circular symmetry (Figure 6.36). Of course, the present section is necessarily limited in scope and depth because of the page limitation. Owing to the elaborate nature of some of these studies, it is not possible to ﬁt an adequate account of them into the framework of the present chapter; however, an attempt is made to give some glimpses into the most important results. Three possible conﬁgurations have attracted some attention in the literature: (1) horizontal cylinder with differentially heated end walls, (2) horizontal cylinder subjected to a temperature gradient that is normal to the cylinder axis and to gravity and (3) laterally heated vertical cylinders with isothermal bases at the same temperature. For the ﬁrst case, in the past, ﬂow predictions were mostly obtained via asymptotic analytical approximations in the core (Bejan and Tien, 1978a) or by assuming a two-dimensional solution for the plane of symmetry. Three-dimensional computations of the steady state by Smutek et al.

(a)

(b)

(c)

Figure 6.36 Possible conﬁgurations for the orientation of a circular cylinder with respect to gravity and imposed temperature gradients: (a) horizontal cylinder heated from the end-walls; (b) horizontal cylinder subjected to a temperature gradient that is normal to the cylinder axis and to gravity; (c) laterally heated vertical cylinder

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(1985), Bontoux et al. (1986a, b) and Viviani et al. (1996) conﬁrmed, however, that in reality the steady buoyancy ﬂow in this conﬁguration is highly three-dimensional. Some subsequent studies were concerned with the onset of time dependence (given the important technological relevance to the ﬁeld of crystal growth). As an example, Vaux et al. (2006) investigated numerically the origin of these instabilities for Pr = 0.026 and for aspect ratios varying from A = 1.5 to 10. In order to understand the mechanisms of ﬂow transition, they also performed ﬂuctuating energy analyses close to the threshold; interestingly, they identiﬁed shear as the main instability factor but with the way in which it acts differing according to the aspect ratio. For case (2) (it is generally treated assuming a sinusoidal distribution of temperature along the cylinder circumference), a relatively large number of analytical and experimental investigations are available in the literature. Most of them were realized by Ostrach and co-workers (see, e.g., Brooks and Ostrach, 1970) and for a complete review the reader may consider, in particular, Ostrach and Hantman (1981). Numerical simulations deserving some speciﬁc discussions here are due to Xin et al. (1997) (they considered both steady and oscillatory regimes). They obtained steady solutions as time-asymptotic solutions for Rayleigh numbers (based on the cylinder radius) 103 Ra 106 and Prandtl number in the range from Pr = 0.71 to 20. Some interesting general properties and trends were identiﬁed, as described below. The velocity ﬁeld was found to be symmetrical with respect to the cylinder centre, whereas the temperature was antisymmetric. According to such simulations, moreover, the core is in pure rotation (i.e. zero radial velocity) only for very small Rayleigh number. With a behaviour very similar to that described for the rectangular cavity in Section 6.2.2, with increasing Rayleigh number, a region of uniform thermal stratiﬁcation develops in the core of the circular cross-section whereas the mass ﬂow rate concentrates in boundary layers near the wall and the ﬂuid ﬂow accordingly weakens in the core (as also shown by Weinbaum, 1964, as the Rayleigh number increases the core motion is progressively arrested, leaving a narrow circulating band of ﬂuid adjacent to the wall). As already mentioned, Xin et al. (1997) also investigated the subsequent onset of unsteadiness (for Pr = 0.71, 3, 6, 9, 20 and 50). By means of both direct numerical integration and linear stability analysis, they illustrated that the steady solutions can undergo a Hopf bifurcation and that, depending on the Prandtl number, the most unstable eigenvector may break or keep the symmetry of the base ﬂow. In particular, the critical Rayleigh number was found to achieve an asymptotic value for large enough Prandtl numbers as displayed in Figure 6.37. As shown in Figure 6.38, moreover, the perturbations responsible for the onset of such unsteadiness take the form of travelling waves conﬁned in a region close to the cylinder wall, which leads to the important conclusion that for the range of Prandtl numbers considered by these authors [Pr O(1)], the onset of unsteadiness can be ascribed to the same mechanism already elucidated in Section 6.2.2 for tall vertical rectangular cavities, that is, to an instability of the boundary layers. As for cavities with straight walls, the major outcome of the instability is the emergence of convective structures propagating along the system outer boundary with number depending on the Prandtl number, as described in detail below: For Pr = 0.71, the temperature eigenfunction is symmetric about the cylinder axis, that is, it is part of an antisymmetric mode; the symmetry of the base ﬂow is thus broken at the onset of unsteadiness for this Pr value. The eigenmode is made of six travelling structures which are in turn ampliﬁed and damped as they travel around the cavity (it was also observed that the wavelength varies as the wave circulates; it is shorter when the amplitude is larger and longer when the amplitude is smaller; the structures are thus compressed and stretched alternately). For Pr = 3, the temperature eigenmode is also antisymmetric and comprises eight travelling structures. For Pr = 6, the eigenmode is made of nine travelling structures. The odd number of travelling structures corresponds to a symmetric ﬁeld and the symmetry of the base ﬂow is thus preserved.

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Figure 6.37 Critical Rayleigh number for the onset of oscillatory ﬂow in a horizontal cylinder (with a sinusoidal distribution of temperature along its circumference) as a function of the Prandtl number (Rayleigh number based on the cylinder radius). Data after Xin et al. (1997); Reproduced by permission of the American Institute of Physics

For Pr = 9, the eigenmode is again antisymmetric since it consists of eight travelling structures. The symmetry of the base ﬂow is thus broken. Finally, for Pr = 12, 20 and 50, the same unstable mode composed of seven travelling structures occurs (it is thus symmetrical and the symmetry of the base ﬂow is kept). Let us now switch to the third case shown in Figure 6.36, namely the laterally heated vertical conﬁguration. As in the case of rectangular containers treated in Section 6.2.1, it has extensive background application in the ﬁeld of bulk crystal growth (as highlighted in Chapter 3, in fact, many bulk crystal growth processes are carried out in axisymmetric geometric conﬁgurations; see, e.g., Crochet et al., 1989; Favier, 1990; Lopez et al., 1999). For these reasons, this model has enjoyed a long tradition of numerical studies and deserves some special attention here also. It is known that if the lateral boundary conditions are perfectly axisymmetric, stratiﬁed ﬂuid layers develop near the top and the bottom. The stratiﬁcation near the bottom (the cold ﬂuid below the hot ﬂuid) is stable, whereas stratiﬁcation near the top (the cold ﬂuid above the hot ﬂuid) is unstable with respect to the action of buoyancy forces (i.e. it tends to be unstable with respect to a mechanism of the Rayleigh–B´enard type). Deviation from axisymmetric wall temperature conditions in vertical cylinders was initially investigated in 1989 by Baumgartl et al. (1989) and later by Pulicani et al. (1992) as a possible source of non-axisymmetric convection observed during vertical directional solidiﬁcation (VB) experiments (see Section 3.1.1 for further details on this technique). Subsequent studies demonstrated that even under axisymmetric external conditions, the axisymmetric melt ﬂows frequently

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Pr = 3 and Ra = 1.5 × 106

Pr = 6 and Ra = 3.6 × 106

Pr = 9 and Ra = 6.8 × 106

Pr = 12 and Ra = 7.8 × 106

Pr = 20 and Ra = 8.0 × 106

Pr = 50 and Ra = 8.0 × 106

265

Figure 6.38 Eigenfunctions of unstable modes for Pr = 3, 6, 9, 12, 20 and 50 (Ra based on the radius). After Xin et al. (1997); Reproduced by permission of the American Institute of Physics

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Figure 6.39 Isosurfaces of temperature perturbation: (a) A = 0.5, Pr = 0.015, Grcr = 3.59 × 105 , m = 4, Hopf bifurcation; (b) A = 1, Pr = 0.03, Grcr = 9.6 × 104 , Hopf bifurcation, oscillatory disturbance; (c) A = 2, Pr = 0.03, Grcr = 2.55 × 104 , m = 1, stationary disturbance (A = height/diameter). After Gelfgat et al. (2000); Copyright Elsevier, 2000

become unstable and bifurcate to asymmetric steady or oscillatory states. Such instabilities lead to temperature oscillations and asymmetric ﬂow patterns that, in turn, have been shown to be directly or indirectly responsible for undesired inhomogeneities in the structure of growing crystals (see Section 3.1.2). Some interesting and exhaustive numerical results concerning the stability of buoyant axisymmetric convection in vertical cylinders with a parabolic temperature proﬁle at the sidewall are due to Gelfgat et al. (2000) for 0 Pr 0.05 and 0.5 A 2 (A = height/diameter). The critical parameters corresponding to a transition from the steady axisymmetric (basic state) to a three-dimensional asymmetric (steady or oscillatory) ﬂow pattern were determined and it was elucidated that the instability of the ﬂow is three-dimensional for the whole range of governing parameters studied (see Figure 6.39). According to this study, in particular, the axisymmetric ﬂow in relatively shallow cylinders tends to be oscillatorily unstable via a hydrodynamic Hopf bifurcation of the circulating ﬂow (Figure 6.39a and 6.39b), whereas in tall cylinders the instability sets in due to a steady bifurcation of the aforementioned unstably stratiﬁed ﬂuid layer caused by the Rayleigh–B´enard mechanism (Figure 6.39c). In the ﬁrst case, the three-dimensional perturbation has the form of a travelling wave; accordingly, the oscillatory perturbation patterns (and also the oscillatory component of the ﬂow) rotate around the axis of the cylinder with angular velocity 2πf /m (where f is the frequency of the oscillations and m the azimuthal wavenumber). In the latter case, the ﬂow is initially three-dimensional and steady, but it can undergo a subsequent transition to time-dependent states if the Rayleigh number is further increased. A qualitatively similar behaviour was also observed in the earlier experiments of Selver et al. (1998) that they conducted with liquid gallium, localized heating from the circumference and aspect ratio ranging from 2 to 10. There are both experimental and numerical lines of evidence (in qualitative agreement), hence proving that for liquid metals the toroidal ﬂow in such geometric conﬁgurations becomes non-axisymmetric and oscillatory at the same time beyond a certain Ra when the aspect ratio of the cylinder is smaller than a certain value, whereas for larger aspect ratios the azimuthal symmetry of the ﬂow is broken with the emergence of a new non-axisymmetric steady state (that is taken over by a time-dependent 3D regime at larger Ra).

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Figure 6.40 Critical Grashof number as a function of the aspect ratio for Pr = 0.015 (Grashof number based on the cylinder height; m is the azimuthal wavenumber). After Gelfgat et al. (2000); Copyright Elsevier, 2000

Both the consecutive transitions for the latter case are the result of pure thermal instabilities, as also demonstrated by the experimental evidence of oscillations affecting mainly the upper half of the domain (the zone where, as explained earlier, thermal unstable stratiﬁcation occurs). As already mentioned, these mechanisms were placed in a more precise theoretical framework by Gelfgat et al. (2000). In general, the azimuthal wavenumber (number of disturbance nodes in the azimuthal direction) depends on the geometric aspect ratio. Gelfgat et al. (2000), however, also predicted the possible existence of very complex three-dimensional scenarios for ﬁxed values of this geometric parameter with the possibility of an adjustment in the number of azimuthal disturbance nodes as the value of the Rayleigh number is varied in a restricted range: For instance, as shown in Figure 6.40, for A = 0.5 the critical Rayleigh numbers corresponding to the azimuthal modes with m = 2 and m in the range 5–10 are relatively close and grow rapidly with increase in Pr. For this reason, for Pr < 0.02 a complex interaction of several azimuthal modes at relatively large supercriticalities is possible. In particular, for Pr = 0.01 the most dangerous mode with m = 4 is excited at Ra ∼ = 3.7 × 103 and = 3 × 103 , then the m = 2 mode would be excited at Ra ∼ 3 ∼ at Ra = 4.5 × 10 many additional modes would be excited (see Figures 6.40 and 6.41). A range of Prandtl numbers similar to that considered by Gelfgat et al. (2000) was investigated by Gemeny et al. (2007). Results by linear stability analysis were presented for 0 Pr 0.1 and for two different thermal boundary conditions at the vertical wall: a prescribed parabolic temperature variation and a prescribed parabolic radial heat ﬂux variation. Interestingly, the results were shown to be radically different for the two thermal boundary conditions. It is also worth mentioning that, as a variant to this problem, other authors (Rubinov et al., 2004; Ma et al., 2005) simulated three-dimensional steady and oscillatory ﬂows in vertical cylinders partially heated from the side (vertical wall heated in a zone at mid-height and insulated above and below this middle zone, with both ends of the cylinder cooled); they considered a ﬁxed length of the heated zone, equal to the cylinder radius. For Pr = 0.021, in particular, Rubinov et al.

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Figure 6.41 Critical Grashof number as a function of the Prandtl number for A = 0.5 (Grashof number based on the cylinder height). After Gelfgat et al. (2000); Copyright Elsevier, 2000

(2004) found three different modes of the most dangerous three-dimensional perturbations, which replace each other with variation of the aspect ratio. Comparison with experiment showed good agreement at aspect ratios A = 4 and 6, whereas at A = 2 signiﬁcant disagreement was observed. For such a speciﬁc value of the aspect ratio, the authors investigated the dependence of the critical Grashof number on the Prandtl number in the range 0 < Pr < 0.05, to rule out the possibility that the disagreement was due to uncertainty in the values of ﬂuid properties. The computations were carried out using two independent numerical approaches, which cross-validate each other. A year later, Ma et al. (2005) considered the same conﬁguration and Prandtl number (0.021) with the aspect ratio ranging from A = 1 to 4. Three-dimensional steady and unsteady simulations and also mode decomposition techniques and energy transfer analyses were used to characterize the ﬂows and their transitions. In agreement with the earlier studies, they found that the ﬂows that develop from the steady toroidal pattern beyond the ﬁrst instability threshold break the axisymmetry. At small A (1 A 1.25), the ﬂow corresponds to a two-roll rotating pattern, which is triggered by an m = 2 azimuthal mode as a result of a hydrodynamic (shear) instability. At large A (1.5 A 4), the ﬂow is steady and corresponds to a main one-roll pattern in the upper part of the cylinder; this ﬂow is triggered by an m = 1 mode as a result of buoyancy effects affecting the unstably stratiﬁed upper part (Rayleigh–B´enard instability); these steady ﬂows then transit at a higher threshold to a standing-wave oscillatory one-roll pattern. For intermediate values of A (1.35 A 1.45), the ﬁrst transition is towards an oscillatory pattern, but hysteresis phenomena with a multiplicity of steady and oscillatory states can also take place, further complicating the subject. As a concluding remark for this section, it should be pointed out that the majority of these studies was motivated by potential application to the vertical Bridgman technique, which, as illustrated in Section 3.1, generally involves a cylindrical ampoule containing a molten substance (with Pr 1) subjected to localized heating around the circumference. Buoyancy effects in cylindrical geometries

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heated circumferentially, however, are also relevant to other techniques traditionally employed for the production of semiconductor materials. Another example of interest in the context of the present book (the reader is referred again to Section 3.1) is given by the FZ technique. Related models differ from the vertical Bridgman technique in that the rigid lateral wall typical of VB is replaced by a free surface subjected to an (approximately) parabolic ﬂux. Such (both kinematic and thermal) differences in terms of lateral boundary lead to profound differences in the related dynamics even if one does not consider surface tension-driven effects. This subject (pure thermogravitational ﬂow in the FZ) is still open. The interested reader may ﬁnd some limited results in Lappa (2007d).

7 Thermogravitational Convection in Inclined Systems As is evident after comparative readings of Chapters 4 and 6, the relative direction between the acceleration due to gravity g and ∇T plays a crucial role in the dynamics of thermogravitational convection. Apart from the effect of the speciﬁc geometry considered (rectangular, circular, etc.), in fact, the emerging pattern or convective structure and its properties (in terms of stability thresholds and possible spatiotemporal stages of evolution) change dramatically according to whether gravity and the applied temperature gradient act along the same direction or are perpendicular. If the externally imposed T yields imposed temperature gradients that are primarily vertical, the basic state is static with a diffusive temperature distribution and motion ensues with the onset of instability when T exceeds some threshold (Chapter 4); if the externally imposed T yields imposed temperature gradients that are primarily horizontal, in these cases motion occurs for any value of T (Chapter 6). Remarkably, the secondary, tertiary and high-order modes of convection also depend signiﬁcantly on the situation considered (Nield, 1994). In practice, as the angle θ between g and ∇T is varied in the range 0◦ ≤ θ ≤ 180◦ , a variety of new possible regimes of convection become possible. The related ﬁeld of analysis (discussed to some extent in the present chapter) in its broadest sense attempts to classify and characterize the properties of all these solutions, which are relevant to a number of both industrial and natural contexts (some of which are mentioned in the following). A relevant example is the process of crystal growth from melts in tilted ampoules with axial heating (see Section 3.1 for relevant background information on these processes). It is known that experimentally unavoidable tilt angles as small as θ ∼ = 0.5◦ can cause non-axisymmetric growth conditions with the Bridgman technique. Also, by slightly tilting the horizontal Bridgman conﬁguration larger mass transport rates are typically obtained. In the same way, the heat transfer in heat exchangers and thermosiphons can be enhanced by selecting an optimum inclination (indeed, their efﬁciency can be improved by using an inclined setup). The relation between the inclination and the Nusselt number also has a direct interest for reducing the loss of energy in honeycomb solar collector plates. Similar problems of convective transport also appear in many geophysical situations occurring, for example, in mining and geological processes. A particularly important problem in this ﬁeld is the transport rate of the spread of passive contaminants such as radioactive materials in long rock fractures arbitrarily inclined with respect to the gravity vector. Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Another case, of particular meteorological and oceanographic interest, is convection in a ﬂuid layer inclined with respect to gravity (see Section 3.4 for some important examples in which ﬂuid convection plays a crucial role in phenomena at the mesoscale). This system not only is well suited for the study of buoyancy- and shear ﬂow-driven instabilities in the Earth’s atmosphere and hydrosphere, but may also serve, along with liquid crystal convection, as a paradigm for anisotropic pattern forming systems. In general, cases with θ = 0◦ , 90◦ or 180◦ are idealized situations. To understand truly the rich and wide-ranging phenomena displayed by many effective natural and technological processes, one must be well acquainted with the intricacies of ﬂuid dynamics, heat transfer and mass transfer in a more general and realistic context. Beyond practical applications, as one may imagine, the work of review set forth in this chapter should be also regarded as a speciﬁc attempt to create a theoretical bridge between the fundamental mechanisms of instability described in Chapters 4 and 6. As anticipated, in fact, from a fundamental research standpoint the inclined conﬁguration can effectively be used as a paradigm system for the study of several types of instabilities and their corresponding interactions. It will be shown here how, in transitioning from the canonical RB problem (θ = 0◦ ) to the canonical ﬂow in heated-from-the-side systems (θ = 90◦ ), new types of instabilities and patterns appear which are not possible in such limit cases. Owing to page limits, attention will be limited to some exemplars (layer of inﬁnite extent and rectangular cavities). Two fundamental situations, in particular, are considered: (1) an inﬁnite layer with top and bottom isothermal walls at different temperatures (Section 7.1), which reduces in the limit as θ → 0◦ to the traditional layer heated from below extensively treated in Chapter 4, and, in the limit as θ → 90◦ , to the transversely heated vertical layer considered in Section 6.3, respectively; and (2) inﬁnite layers or slender (but ﬁnite) rectangular cavities subjected to a temperature gradient directed primarily along the direction for which these systems are assumed to be more extended (Section 7.2), which reduce in the limit as θ → 90◦ to the ﬂows of the Hadley type analysed in Sections 6.1, 6.2 and 6.4.

7.1

Inclined Layer Convection

In this section, convection in a ﬂuid layer conﬁned between two conducting walls that is inclined relative to gravity is examined [this case is usually referred to in the literature as inclined layer convection (ILC)]. It is worth starting the analysis with the major point that, unlike RB in ILC, the patternless base state is characterized not only by a linear temperature gradient but also by a symmetry-breaking shear ﬂow . As shown in Figure 7.1, in fact, the component of gravity tangential to the ﬂuid layer causes buoyant ﬂuid to ﬂow up along the warm wall and down along the cold wall. This shear ﬂow breaks the in-plane isotropy of the usual horizontal layer heated from below (discussed in Section 4.1), causing a departure of the dynamics from well-known RB behaviours. A simple model for this kind of ﬂow was treated in Section 2.4 when discussing the existence of exact solutions to the thermal-convection equations and related properties [Eq. (2.60)]. Depending on the inclination angle θ with respect to the horizontal direction and on Pr, longitudinal, oblique, transverse and travelling transverse rolls are the possible ﬂow structures emerging on such initial state. As observed by Busse and Clever (1992), in particular, steady longitudinal rolls (LR) with their axes aligned with the shear ﬂow (i.e. aligned with the component of gravity parallel to the layer) represent the preferred mode of convection at onset for a wide range of parameters. Notably, like the classical RB convection (θ = 0◦ ), the Rayleigh number for the onset of these rolls is independent of Pr. As proved by Kurzweg (1970) for the linearized equations and later

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(a)

(b)

Figure 7.1 Sketch of the base ﬂow in an inclined layer for different heating conditions: (a) heating from below (θ < 90◦ ); (b) heating from above (θ > 90◦ )

by Clever (1973) in the nonlinear case, in fact, longitudinal rolls can be described by a suitable rescaling of any two-dimensional solution of the classical horizontal RB problem; it is known, in particular (see Chen and Pearlstein, 1989), that the bifurcation to longitudinal rolls occurs at Racr (θ ) = Racr(θ =0) / cos θ , where Racr(θ =0) = 1707 as illustrated in Section 4.1. The effective occurrence of this type of rolls as the emerging pattern, however, depends on Pr and the angle of inclination, as discussed in detail in the following.

7.1.1 The Codimension-two Point For small angles of inclination, buoyancy dominates over shear ﬂow and the primary instability is to LR. With increasing angle θ , buoyancy effects decrease and shear effects tend to become more important. Above a critical angle θc ≤ 90◦ the shear ﬂow causes a primary instability to transverse rolls (TR) with roll axes perpendicular to the shear ﬂow (Chen and Pearlstein, 1989). The angle θc where transverse rolls and longitudinal rolls bifurcate at the same threshold is generally referred to as the ‘codimension-two point’. It corresponds to a competition of two different physical instability mechanisms: the thermal, leading to longitudinal rolls and the hydrodynamic (shear-ﬂow), leading to transverse rolls. Figure 7.2 shows, in particular, that with increasing Pr the codimension-two point moves quickly to angles close to 90◦ (the dashed-dotted line). Above 90◦ , the ﬂuid layer is heated from above and the instabilities are basically shear ﬂow driven (in such a condition, the effects of buoyancy are obviously limited to a stabilizing inﬂuence on the ﬂow). According to the linear stability analysis of Fujimura and Kelly (1993), the ranges of existence of the different modes in terms of Prandtl number and inclination angle (see Figures 7.2 and 7.3) can be described as follows. For Pr < 0.26, stationary transverse rolls occur for all angles of inclination as inertial effects tend to be dominant. For Pr > 0.26 and 0 < θ < θc , stationary longitudinal rolls are the linear perturbation that becomes unstable ﬁrst. For 0.26 < Pr < 12.42 and θ > θc , stationary

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Figure 7.2 Stability diagram as a function of the inclination angle and the Prandtl number for longitudinal rolls and transverse rolls (Rayleigh number deﬁned as Ra = gβT T d 3 /να ). Courtesy of B. Plapp and E. Bodenschatz

Figure 7.3 Cross-over points among different modes in inclined layer convection. Three curves intersect at (Pr, δ , Ra) = (12.42, 1.00, 97216) that is the theoretical condition for a three-mode interaction. After Fujimura and Kelly (1993); Reproduced by permission of Cambridge University Press

transverse rolls have a lower threshold. For Pr > 12.42, travelling waves in the form of travelling transverse rolls (TW) are realized at the onset for angles close to 90◦ . The angle θc depends on Pr. It was found to vary from 1◦ to 90◦ as the Prandtl number increases from 0.24 to 12.7 in an earlier study by Korpela (1974). The distribution of the cross-over points between stationary longitudinal rolls (LR) and stationary transverse rolls (TR), between stationary longitudinal rolls (LR) and transverse travelling

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waves (TW) and between stationary transverse rolls (TR) and transverse travelling waves (TW) is illustrated in detail in Figure 7.3 in the (Pr, δ)-plane (where δ = 90◦ − θ ). For Pr 1, the diagram for LR–TR has the asymptotic behaviour δ ∼ = 12.533 × Pr−1.0019 . The −0.54299 ∼ diagram for LR–TW is featured by δ = 17.156 × Pr for Pr 1. The curve for TW–TR is almost vertical but slightly inclined with a very large slope. The Prandtl number on this curve tends to Pr = 12.45 as δ → 0◦ (θ → 90◦ ), hence one recovers the same behaviours already elucidated in Chapter 6 for the laterally heated inﬁnite vertical layer (θ = 90◦ ): the critical disturbance modes are hydrodynamically driven and stationary when Pr < 12.45, but they are thermally driven and oscillatory when Pr > 12.45 (the so-called ‘temperature waves’; see Section 6.3 for additional information on this speciﬁc case). Earlier experimental investigations on the problem were summarized by Shadid and Goldstein (1990) (it is also worth citing Hart, 1971a,b, Ruth, 1980, Ruth et al., 1980a,b and Kirchartz and Oertel, 1988, who used transparent liquids). It is also worth mentioning that Fujimura and Kelly (1993) did not limit their study to a linear stability analysis; interestingly, they also performed a weakly nonlinear analysis of the possible nonlinear interaction between transverse disturbances and longitudinal rolls (in practice, they examined the conditions in which both modes of instability occur at nearly the same value of the control parameter). The salient results of such theoretical study can be summarized as follows. For slightly supercritical values of the control parameter when the critical Rayleigh number for stationary longitudinal TR rolls (RaLR cr ) is somewhat less than that for stationary transverse rolls (Racr ) and for transverse TW travelling waves (Racr ), longitudinal rolls occur ﬁrst and then remain stable as Ra is increased TW TR TW beyond RaTR cr or Racr ; no mixed-mode state occurs. In contrast, if Racr or Racr is slightly below RaLR , pure transverse modes exist for only a relatively small range of Ra beyond RaTR cr cr TW LR or Racr . Thereafter, a three-dimensional mixed-mode state occurs well before Racr is reached, that is, three-dimensionality sets in on a subcritical basis, but ﬁnally the pure longitudinal mode becomes stable as Ra is increased further.

7.1.2 Tertiary and High-order Modes of Convection The secondary instabilities of steady LR modes were analysed via a fully non-linear approach (numerical simulation) by Busse and Clever (1992). They revealed that as the Rayleigh number is increased beyond a critical value, the longitudinal rolls become unstable with respect to a three-dimensional ‘wavy’ instability (already predicted in an earlier theoretical study by Clever and Busse, 1977) leading to the emergence of undulations patterns. Many such theoretical ﬁndings have been conﬁrmed by relevant experimental investigations for Pr ≥ O(10). Along these lines, for instance, in agreement with theory, Shadid and Goldstein (1990) observed a region of unsteady wavy longitudinal rolls at sufﬁciently high Rayleigh numbers for low to moderate angles of inclination. In general, the wavenumber of the longitudinal rolls was found to increase with the angle of inclination from the horizontal. Two distinct types of instability mechanisms were discerned, in particular, which could modify the wavenumber of the longitudinal rolls: an instability of the cross-roll type, which is a disturbance perpendicular to the original roll axis (see Section 4.2 for a somewhat qualitatively related theoretical background to this kind of ﬂow instability); and a pinching mechanism combining two neighbouring longitudinal roll pairs into a longer wavelength roll pair. For the opposite case of small Prandtl number, no experimental results concerning the subsequent system transition are available (due to difﬁculties typically encountered with the use of such liquids). The limit of small Prandtl number was assumed, however, in the nonlinear analysis of Nagata and Busse (1983). According to their study, buoyancy effects caused by temperature perturbations

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are negligible and, after the ﬁrst transition to transverse rolls, the ﬂow becomes unstable at slightly supercritical Grashof numbers to a vortex-pairing instability with alternating pairing in the spanwise direction, which make the resulting ﬂow highly three-dimensional. The intermediate case with Pr ∼ = 1 (for which, as illustrated in Section 4.6, spiral-defect chaos is known to exist in the classical case with θ = 0◦ ) was considered by Daniels et al. (2000). Exploring experimentally large aspect ratio systems over a relatively wide range of inclination angles 0 ≤ θ ≤ 120◦ , that is, from horizontal (heated from below) to past vertical (heated from above), a rich phase diagram with many unpredicted states on increasing Ra above the critical temperature difference was revealed (see Figure 7.4 and additional descriptions in the text below). In qualitative agreement with the aforementioned theoretical ﬁndings of Clever and Busse (1977), Daniels et al. (2000) found transverse modes to trigger a secondary bifurcation of LR to three-dimensional undulation patterns [slightly above onset, i.e. (Ra − Racr )/Racr ∼ = 0.015, see Figure 7.4] over a range of intermediate angles (15◦ ≤ θ ≤ 70◦ ), with a tertiary instability to a state of crawling rolls [(Ra − Racr )/Racr ∼ = 0.3] limiting the existence region of undulations from above. Such undulation patterns were observed to be rather chaotic (UC – undulation chaos) in the region between the lower and the upper thresholds mentioned above (undulating convection rolls perpetually breaking and reconnecting via moving point defects). As already outlined, a variety of other possible states, however, were reported, as follows: for 0 ≤ θ ≤ 77.5◦ , longitudinal rolls, subharmonic oscillations (these oscillations occurring for θ < 13◦ and emerging as a pearl necklace-like pattern of cold spots that travel along a standing wave pattern of wavy rolls), Busse oscillations, undulation chaos (Figure 7.5a) and crawling rolls (Figure 7.5b); in the neighbourhood of the aforementioned codimension-two point for thermal and shear-driven instability (77.5◦ ≤ θ ≤ 84◦ for Pr ∼ = 1 as also shown in Figure 7.2), drifting bimodals, drifting transverse rolls and localized longitudinal and transverse bursts; and for inclinations θ > 84◦ , drifting transverse rolls, switching ‘diamond-panes’ and longitudinal bursts (see, e.g., Figure 7.6).

Figure 7.4 Boundaries between different possible nonlinear states of inclined layer convection in the (Ra, θ ) phase space (Grashof number deﬁned as Gr = gβT T d 3 /ν 2 ): LR, longitudinal rolls; BO, Busse oscillations; SO, subharmonic oscillations; UC, undulation chaos; CR, crawling rolls; DTR, drifting transverse rolls; DB, drifting bimodals; LB, longitudinal bursts; TB, transverse bursts; SDP, switching diamond panes. The dotted line is the predicted onset of Busse oscillations for Pr = 0.7 (Clever and Busse, 1977). Courtesy of B. Plapp and E. Bodenschatz

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(a)

(b) ∼ 1 and θ = 40◦ (warm Figure 7.5 Snapshots (shadograph images) of possible states of ILC for Pr = up-ﬂow to the right and cold down-ﬂow to the left): (a) undulation chaos at (Ra − Racr )/Racr = 0.07 (the characteristic feature of a UC pattern is undulating stripes containing topological point defects; defects locally change the spacing of rolls and their orientation); (b) crawling rolls at (Ra − Racr )/Racr = 0.88. Courtesy of B. Plapp and E. Bodenschatz

Interestingly, Daniels et al. (2000) found most of these novel (spatiotemporally chaotic) states very close to onset; in such a context, it is also worth mentioning the related theoretical analysis of Busse and Clever (2000). The subject was re-examined 8 years later by Daniels et al. (2008), who approached the problem from both experimental and numerical points of view. Particular emphasis was given to the possible competition (bistability) between the aforementioned spatiotemporal chaotic state of undulation chaos (UC) and stationary patterns of ordered undulations (OU) predicted by theory but rarely observed in experiments for Pr ∼ = 1. The analysis of the bistability problem was concentrated on a detailed experimental study for a speciﬁc ﬁxed inclination θ = 30◦ . As mentioned above, experiments were also supported by numerical simulations of the full equations, which allowed the properties of the distinct OU and UC attractors to be explored in a controlled manner. Thereby, these two states were characterized in terms of spectral patterns of entropy, spatial correlation length and defect density. Interestingly, in the simulations the emergence of ordered undulations or undulation chaos was found to be dependent on the initial conditions. Although numerical simulations demonstrated

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(a)

(b) ∼ 1 and θ = 100◦ (warm Figure 7.6 Snapshots (shadograph images) of possible states of ILC for Pr = up-ﬂow to the right and cold down-ﬂow to the left): (a) switching diamond panes at (Ra – Racr )/Racr = 0.1; (b) longitudinal bursts within diamond panes at (Ra – Racr )/Racr = 0.19. Courtesy of B. Plapp and E. Bodenschatz

stable ordered undulations at all Ra above the secondary instability, such a state, however, was observed to be only intermittently accessible in experiments and only at relatively high driving (where the stability region for the undulations is largest). In particular, at relatively large Ra an intermittent switching between states with very few defects and others with many was found. Such a scenario was interpreted as clearly reﬂecting the expected competition between OU and UC states. To conclude this section, it is worth mentioning that some numerical studies have also appeared in which the problem was treated for the cases of two- or three-dimensional enclosures with an aspect ratio of O(1) (i.e. square or cubical cavities). Like pure RB convection treated in Sections 4.8 and 4.9, also for inclined systems the type of lateral thermal boundary conditions becomes an important dynamics-determining parameter when geometries with comparable horizontal and vertical dimensions are considered. Cliffe and Winters (1984) were the ﬁrst to show for the case of a square cavity with adiabatic vertical sidewalls that the perfect stationary bifurcation typical of RB convection becomes

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279

structurally unstable to the tilt and imperfect . The case with conducting sidewalls (2D) was considered by Mizushima and Adachi (1995). They proved that the bifurcation becomes imperfect even if the tilt angle is very small [e.g. O(10−3 )]. More recently, Crunkleton and Anderson (2006) carried out a numerical study of the classical Rayleigh–B´enard problem with the addition of various tilt angles in cubical enclosures of liquid tin (Pr = 0.008), providing interesting information about the ﬂow structure in steady conditions. This subject is not discussed further here due to page limits.

7.2

Inclined Side-heated Slots

As a variant of the classical problem related to the stability of systems heated from the side treated in Chapter 6 (generally falling in the general theme of the Hadley circulation and related ﬂows), some studies have also been devoted to the case in which such systems [horizontally elongated and differentially heated rectangular boxes (see Figure 7.7), which tend asymptotically to the classical layer of inﬁnite extent as the aspect ratio A → ∞] are inclined . The structure of natural convection in these geometries was initially studied analytically and numerically by many authors with the intention of determining in the space of parameters the regions of existence of possible ‘basic’ ﬂow solutions (categorized according to the dominant mechanism of heat transfer as conductive, transition and boundary-layer regimes; see Delgado-Buscalioni and Crespo del Arco, 2001, and references therein; see also Section 2.5 for additional theoretical background). The major outcome of such studies can be summarized as follows. If the cavity is heated from above, the ﬂow always remains in the conductive regime if θ is made large enough (typically θ > 100◦ ), whereas for θ < 90◦ , in contrast to the canonical case of a temperature gradient perpendicular to gravity treated in Chapter 6 (Section 6.2), for increasing values of Ra no stagnant region is formed at the centre of the core, but instead a region with approximately constant shear (which can be regarded as a clear distinguishing mark of these inclined systems with respect to the horizontal counterparts). Numerical simulations of possible three-dimensional steady states in a cubical cavity (ﬁlled with air) depending on the Rayleigh number and inclination angle were carried out by Lee and Lin (1995). Other studies focused expressly on possible ﬂow instabilities, for example, Kuyper et al. (1993) and Adachi and Mizushima (1996) for the square cavity considering two-dimensional disturbances only and Delgado-Buscalioni et al. (1998), Delgado-Buscalioni and Crespo del Arco (1999) and Delgado-Buscalioni (2001a) for rectangular long enclosures with convection driven by end-to-end temperature differences subjected to the full range of possible perturbations. In a tilted conﬁguration, the ﬂuid is stably stratiﬁed along the cross-stream direction, whereas it is unstably stratiﬁed in the perpendicular (streamwise) direction; the available analyses of the related instability mechanisms (mentioned above) revealed several stabilizing or destabilizing couplings between the momentum and temperature ﬁelds that are not possible in the horizontal (θ = 90◦ )

Figure 7.7 Inclined (horizontally elongated) rectangular box heated from the side

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or vertical (θ = 0◦ ) limits (treated in Chapters 6 and 4, respectively), which need to be explicitly discussed and understood. It is worth mentioning that most of these studies considered as basic state (for application of the typical protocols of linear stability analysis), the exact solution for the inclined conﬁguration (inﬁnitely extended along the direction of the imposed temperature gradient) whose general properties have been discussed in Section 2.4.6 [Eqs (2.50)–(2.54)]. Delgado-Buscalioni (2001a), in particular, elucidated that when the system is inclined, a rich dynamic behaviour arises as a consequence of the competition between several instabilities and new types of instabilities occur [as illustrated in Figure 7.8 in the space (θ , Pr) and summarized in Table 7.1, where some general trends are reported for each type of instability in terms of critical Rayleigh number and wavenumber]. The ‘stationary longitudinal long-wavelength instability’ (SLL) appears for any θ < 90◦ and has essentially the same origin as the critical mode of an unstable vertical conﬁguration (i.e. a classical Rayleigh–B´enard mode). The ‘oscillatory transversal long-wavelength instability’ (OTL) is a standing wave with a rather long wavelength (typically nine times the height d); it only comes up if the cavity is inclined and heated from below (0◦ < θ < 90◦ ). The other well-known classical instabilities for the horizontal conﬁguration (θ = 90◦ ) discussed in Sections 6.1 and 6.2 for the Hadley ﬂow are also affected by the inclination. In particular, for θ < 90◦ the stationary transverse rolls (the 2D hydrodynamic instability), which for θ = 90◦ (as widely discussed in Section 6.1.2) are generally observed for ﬂuids with Pr < 0.034 and are suppressed for Pr > PrL ∼ = 0.1 (Figure 6.4a), can become unstable even in gases (Pr ∼ = 1).

Figure 7.8 Various instability modes as a function of the inclination angle and the Prandtl number. The instability with the lowest critical Rayleigh number is indicated with bold letters and that with the second lowest critical Ra is labeled in italics and between parentheses: ST, stationary transversal; OL, oscillatory longitudinal; SLL, stationary longitudinal long-wavelength; OTL, oscillatory transversal long-wavelength; SLS, stationary longitudinal short-wavelength. Courtesy of R. Delgado-Buscalioni

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Table 7.1 ‘Trends for the critical parameters of typical modes of instability in inclined conﬁgurations (Rayleigh number deﬁned as Ra = g βT γ d 4 /να , where γ is the rate of temperature increase along the x-axis; the conﬁguration is assumed to have the longest sidewalls insulated; the constants R0 , c1 , c2 and c3 are given in Delgado-Buscalioni, 2001a). Type of instability

Critical Ra

Critical wavenumber

ST

Pr Pr 1, Ra ∝ sin θ

qx = 1.35

OTL

R0 Pr ∼ = 1 (θ < 90◦ ), Ra ∼ = cos θ ∼ R0 Ra = cos θ Racr (θ =0) ∼ Ra = cos θ

SLL

qx = 1.6 qx ∼ = 0.3 qz = 0

1

SLS

Ra ∝

OL

Ra ∝

Pr 2 (c1 − Pr)2

qz ∼ = 2.9

1

2

Pr 2 1

(c2 − Pr) 2

qz ∝

(c3 − Pr) 5 1

Pr− 2

Moreover, by tilting the cavity with respect to θ = 0◦ , the (Rayleigh–B´ernard) stationary thermal mode is suppressed in cavities whose depth is smaller than a theoretically predicted cutoff wavelength. The inclination also alters the properties of the oscillatory longitudinal instability (OLR) (this instability is damped at a certain cutoff value of Pr, which decreases with the inclination angle; for example, for θ = 90◦ and 80◦ , OLR perturbations are damped for Pr ≥ 0.21 and Pr ≥ 0.26, respectively). Following the interesting studies by Delgado-Buscalioni and co-workers, further and more detailed information is given in the following subsections.

7.2.1 Stationary Longitudinal Long-wavelength Instability The SLL instability leads to the emergence of stationary rolls oriented longitudinally. As mentioned earlier, it has essentially the same origin as the critical mode of an unstable vertical conﬁguration. In fact, a general relationship that is valid for it (Delgado-Buscalioni, 2001a) is RaSLL (θ ) = RaSLL(θ =0) / cos(θ ), meaning that the onset of the SLL perturbation occurs once the vertical projection of the temperature stratiﬁcation reaches the same critical value for the vertical conﬁguration (in this regard, it exhibits notable analogies and a kinship with the LR mode of convection discussed in Section 7.1). As shown in the neutral curves in Figure 7.9, however, there is an important difference between the inclined and the vertical conﬁgurations. In a vertical cavity, any SLL modes with arbitrary wavenumber can become unstable provided that Ra has a large enough value. However, if the cavity is inclined, the SLL instability presents a cutoff wavenumber such that perturbations with q > qcut are damped. The value of qcut depends on θ and Pr. On increasing θ or decreasing Pr, the ﬂow becomes stable to SLL perturbations with larger wavelengths (this result suggests the possibility of ﬁltering the SLL instability in closed tilted 3D cavities by choosing a depth smaller than the cutoff wavelength).

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Figure 7.9 Neutral curve for SLL perturbations for θ = 0◦ and 50◦ (Ra based on d ; Pr → ∞). Courtesy of R. Delgado-Buscalioni

7.2.2 Stationary Transversal Instability The ST instability leads to the emergence of stationary transverse rolls. It takes most of its kinetic energy from the mean shear stress and, therefore, in previous studies for the purely horizontal case (see Section 6.1.2), it has been usually called ‘shear instability’ or hydrodynamic mode. Nevertheless, if two components of buoyancy exist, the ST perturbations can obtain a large amount of kinetic energy from the thermal ﬁeld. A deeper understanding of such a feature can be obtained by ﬁrst discussing the hydrodynamic limit (Pr → 0 and Ra → 0) for arbitrary inclination. In this limit, the governing parameter is the Grashof number Gr = Ra/Pr, which controls the ratio between the inertial and viscous forces. The temperature disturbances are homogenized instantaneously and, therefore, with regard to the perturbative ﬂow, the buoyancy forces are absent. The equation for the amplitude of transversal perturbations for such a case is the well-known Orr–Sommerfeld equation whose solution gives GrST = 495/ sin(θ ) and qST = 1.345 (the asymptotic trends for Pr → 0 of GrST and q are shown with dashed lines in Figure 7.10). It should be pointed out that such an asymptotic limit fails for small enough inclinations and for Pr > 0.05 (Figure 7.10b). This means that as the Prandtl number increases, the thermal effects become increasingly important. The time needed to homogenize a temperature excess in a ﬂuid parcel, in fact, is proportional to O(Prd 2 /ν) and, therefore, at larger Pr, the buoyant force acts on the differentially heated particles during a larger interval of time. An inspection in the trend of qST versus Pr in Figure 7.10b, in particular, indicates that the effect of buoyancy in the perturbative ﬂow that, as explained above, becomes relevant for Pr > 0.05, depends on the inclination angle. At θ = 90◦ , the mean cross-stream stratiﬁcation is stable and buoyancy acts as a restoring force. As a consequence, as Pr increases, the critical ST rolls reduce the relative amount of cross-stream ﬂow (qST decreases) and the ST perturbation is ﬁnally damped for Pr > 0.12 (as already elucidated in Section 6.1.2 and clearly shown in Figure 6.4a). For θ > 90◦ , the ﬂuid is also stably stratiﬁed along the x direction and the ST rolls are damped at even lower values of Pr.

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(a)

(b)

Figure 7.10 Critical Grashof number (based on d) as a function of θ for Pr → 0 (dashed curve), Pr = 0.01, Pr = 0.05 and Pr = 0.7 (a) and critical wavenumber qx (b) versus Pr for three inclinations: θ = 30◦ , 70◦ and 90◦ . Courtesy of R. Delgado-Buscalioni

For θ < 90◦ the ST rolls (which, as repeatedly mentioned, for θ = 90◦ can emerge solely for ﬂuids with Pr < PrL ∼ =1) and with = 0.1) can be observed at relatively large values of Pr (∼ larger wavenumbers (see Figure 7.10b). This is a consequence of the following thermal mechanism: owing to the mean cross-stream temperature difference, ﬂuid particles moving along the y direction carry their local temperature to a new thermal surrounding where they are accelerated by the streamwise buoyancy force, which draws within an unstable stratiﬁcation; in other words, transversal perturbations can take kinetic energy out from the buoyant excess generated by the cross-stream perturbative advection (Delgado-Busaclioni and Crespo del Arco, 1999; Delgado-Buscalioni, 2001a). It is also worth remarking that the increased range of existence of transverse modes in terms of the Prandtl number when the system is inclined and basically heated from below is in qualitative

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agreement with the trends illustrated in Section 7.1 for θ > θc [it was explained there that for θ larger than the codimension-two point, stationary transverse rolls (TR) become the expected emerging pattern in a relatively wide range of Pr, namely O(10−1 ) ≤ Pr ≤ O(10)].

7.2.3 Oscillatory Transversal Long-wavelength Instability The OTL instability is a standing wave with a rather long wavelength (typically nine times the depth). It exists if the cavity is inclined and heated from below (0◦ < θ < 90◦ ). The perturbation gains most of its kinetic energy from the streamwise buoyant force (Delgado-Buscalioni and Crespo del Arco, 1999; Delgado-Buscalioni, 2001a). For θ → 0◦ , the OTL critical perturbation recovers the critical transversal mode in the vertical cavity. The critical wavenumber reaches its maximum value (q = 0.36) at θ ∼ = 20◦ and Pr ∼ = 1 and ◦ decreases to zero for θ = 90 or Pr → 0 or Pr → ∞. The cross-stream motion takes energy out from the mean shear stress and also from the coupling between the cross-stream advection and the streamwise buoyancy force. This latter mechanism is essentially the same as that explained in Section 7.2.2 for the ST instability at Pr ∼ = 1.

7.2.4 Stationary Longitudinal Short-wavelength Instability As seen in the stability diagram in Figure 7.8, the SLS instability only takes place if the inclination is near or equal to 90◦ . Like the SLL instability, it is responsible for the appearance of stationary longitudinal rolls; but unlike the SLL mode (which, as explained in Section 7.2.1, has basically the same origin of a pure Rayleigh–B´enard mode), the SLL corresponds essentially to the SLR mode of the Hadley ﬂow originally identiﬁed by Kuo and Korpela (1988) and whose general properties were discussed in Section 6.1.2 (see Figure 6.4). 1 For Pr < 0.1, the critical Rayleigh ﬁts to RaSLS ∼ = 1.7 × 103 Pr /2 and qSLS ∼ = 2.9. For θ ≤ 90◦ , the ﬂow becomes stable to SLS perturbations at Pr ∼ = 1, whereas for θ > 90◦ , the stabilization occurs at much lower values of Pr (for instance, if θ = 93◦ for Pr > 0.1). The SLS instability is generated by a mechanism that couples the mean shear stress and the streamwise temperature gradient. Owing to the mean streamwise temperature gradient, the perturbative advection in the x direction generates a temperature pattern in the (yz )-plane that activates lift forces along the cross-stream direction and a perturbative ﬂow in the (yz )-plane, which in turn feeds the shear force. For Pr > 1, the SLS instability is damped because of the effect of the cross-stream stable stratiﬁcation and the decrease of the inertial forces. The above-described mechanism makes possible, in principle, the onset of SLS rolls in the transition regime of a ﬂow within a completely stable cross-stream stratiﬁcation, i.e. with adiabatic walls. Nevertheless, consideration of the ﬁnite size of the cavity leads to the conclusion that for a broad range of values of the aspect ratio, the boundary layer regime appears before the onset of the SLS instability, which could explain why this type of instability has not been explicitly reported in experiments.

7.2.5 Oscillatory Longitudinal Instability The outcome of the OL instability is represented by oscillatory longitudinal rolls. For θ = 90◦ , it reduces to the standard OLR mode of the Hadley ﬂow (see Section 6.1.2), whereas in inclined systems it arises at low enough Pr and typically for θ < 115◦ (see Figure 7.8). Figure 7.11 plots the values of RaOL and qOL as a function of θ for different values of Pr.

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285

(a)

(b) 1

Figure 7.11 Critical Rayleigh number (a) and wavenumber scaled with Pr /2 (b) of the OL instability for three values of the Prandtl number (Pr = 0.025, 0.05 and 0.1). Courtesy of R. Delgado-Buscalioni

Any tilt with respect to the horizontal position tends to increase the critical Rayleigh number and thus the critical frequency. However, the effect of inclination is also strongly dependent on the other component of buoyancy which acts along the streamwise direction, as explained in the following. For θ < 90◦ , the unstable stratiﬁcation along the streamwise direction favours perturbations with larger wavelengths. The critical Rayleigh number decreases for inclinations slightly smaller than 90◦ as a consequence of the larger mean ﬂow velocities. However, as seen in Figure 7.11a, at

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a certain tilt this trend is reverted and Racr starts to increase strongly. This means that for any arbitrarily small value of Pr this instability can be suppressed by heating the cavity above a theoretically predicted Pr-dependent angle (which decreases when Pr is increased; Delgado-Buscalioni, 2002). The frequency of this instability is also inﬂuenced by the inclination. For θ > 90◦ , the streamwise buoyant force has no direct inﬂuence on the critical frequency, which grows as [Ra2 sin(θ )]1/3 . The reason for this fact is that the critical frequency is much larger than the Br¨unt–V¨ais¨al¨a cutoff frequency for excitation of internal gravity waves (the reader is referred to Delgado-Buscalioni, 2001a, for further elaboration of this theory). In contrast, for θ < 90◦ , the streamwise buoyant force tends to reduce the critical frequency as 1 θ decreases. However, its relative contribution to the frequency varies as Pr /2 cot(θ )/Raq 2 .

7.2.6 Interacting Longitudinal and Transversal Multicellular Modes As outlined earlier, the OL instability is damped at a certain value of Pr (PrL2 according to the symbolism introduced in Section 6.2.1), which decreases with the inclination angle (e.g. PrL2 ∼ = 0.21 for θ = 90◦ , PrL2 ∼ = 0.26 for θ = 80◦ ). By contrast, for lower values of Pr [Pr ≤ O(10−2 )], the inclination can be expressly chosen to promote interaction between longitudinal and transversal multicellular modes (Delgado-Buscalioni et al., 2001b). Some studies have been devoted expressly to this topic as it has important background applications in the ﬁeld of crystal growth where transverse and longitudinal modes are expected to compete for typical cases of interest (as already emphasized and discussed to a certain extent in Sections 6.1.4 and 6.4). As an example, Delgado-Buscalioni et al. (2001b) performed 3D numerical calculations of natural convection for Pr = 0.025 in an enclosure with dimensions Lx = 4, Ly = 1 and Lz = 6 inclined at 80◦ with respect to the vertical position. The case was selected according to the earlier stability analyses by Delgado-Buscalioni et al. (1998) and Delgado-Buscalioni and Crespo del Arco (1999), which indicated that for Pr = 0.025 the codimension-two line for the transversal and longitudinal instabilities passes through θ = 80◦ . A width Lz = 6 was expressly considered to enable the development of a longitudinal standing wave with a multicellular structure (three counter-rotating rolls) along the (y, z)-plane (further details are given in the following). For the sake of comparison between 3D and 2D results, a 2D enclosure was also examined for the same set of parameters (i.e. θ = 80◦ and A = 4) and studied focusing on the oscillatory (purely two-dimensional) hydrodynamic mode. A stationary shear roll gradually detaching from the walls (and assuming the classical twisted shape shown in Figure 6.12.a for increasing Ra) was found to begin to be formed for Ra > 960. Oscillatory ﬂow (Hopf bifurcation) was detected for 3520 < Ra ≤ 3840. Figure 7.12 shows snapshots of such oscillatory ﬂow along a period of oscillation for Ra = 5760. These snapshots are very similar to those presented by Pulicani et al. (1990) for Pr = 0.015 and θ = 90◦ , who found the Hopf bifurcation at Gr = 1.52 × 105 . For the same parameters (A = 4 and Pr = 0.015) Skeldon et al. (1996) calculated the dependence of the critical Grashof number for the onset of oscillations with the inclination angle showing that it has a minimum value around θ ∼ = 50◦ 4 ) whereas for θ = 80◦ it is around Gr ∼ 1.25 × 105 . (Gr ∼ 8.64 × 10 = = For the 3D case with Lz = 6Ly inclined at θ = 80◦ , the ﬂow was found to remain stable to the longitudinal instability for Ra ≤ 3 × 103 and to be essentially two-dimensional. In particular, for Ra ≥ 1280, a transversal shear roll was observed at the centre of the xy plane with its amplitude gradually increasing with Ra. The two-dimensional stationary ﬂow, however, was found to undergo a Hopf bifurcation at values of Ra lower than the corresponding ones for the 2D case. The ﬂow, in fact, was observed

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287

Figure 7.12 Oscillatory convection (two-dimensional computations) in rectangular cavity (Pr = 0.025, A = 4, θ = 80◦ and Ra = 5760; snapshots of the ﬂow evenly distributed within a period of motion). After Delgado-Buscalioni et al. (2001b); Copyright Elsevier, 2001

to break already at 2560 < Ra < 3008 due to the onset of an oscillatory ﬂow with three longitudinal (counter-rotating) rolls along the z direction. The results in Figure 7.13 show that when such oscillatory ﬂow is established, it is highly three-dimensional as the transversal cell originally created in the centre of the xy plane for Ra > 960 is convected by the longitudinal standing wave. Along a period of the oscillation the centre of the shear roll moves in a ellipse at each xy section; the amplitude of the ellipse is maximum at those values of z where the standing wave has a valley and minimum where it has a node (for instance, at the centre section). If one observes the sections of the velocity ﬁeld shown Figure 7.13, it can be seen, in fact, that the position of the centre of the shear roll describes a sinusoidal path along the z-axis. It should be pointed out that, essentially, the motion of the centre of the transversal

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3

5 1 (d)

2 3 4 5 1

(a)

2 3 4 1

5 2

(c)

3 4 5 (b)

Figure 7.13 Snapshots of the ﬂow for Pr = 0.025 and Ra ∼ = 1.25 × 104 in an inclined (θ = 80◦ )3D cavity (4 × 1 × 6) at an instant of maximum intensity of the perturbative ﬂow in the (y, z)-plane: (a), (b) and (c) show cuts of the vector ﬁeld along the z, x and y directions, respectively; (d) magniﬁcation of the vector ﬁeld in the planes along z labelled in (a) by 3 and 5. After Delgado-Buscalioni et al. (2001b); Copyright Elsevier, 2001

cell coincides with the description that was also given by Henry and Buffat (1998) in an enclosure with the same depth-to-length ratio, Lx /Ly = 4 and a shorter z direction (Lz = 2Ly ) occupied by one longitudinal roll. Interestingly, Delgado-Buscalioni et al. (2001b) also investigated the subsequent Hopf bifurcation (i.e. leading to the emergence of a second frequency) for both 2D and 3D cases. They highlighted that, whereas for the 2D case the ﬂow increases in complexity with a period-doubling bifurcation for 5760 < Ra ≤ 6400, the secondary frequency for the 3D ﬂow appears at Ra = 12 352 and is basically related to the formation of a bicellular transversal pattern coexisting with the pre-existing longitudinal oscillatory disturbance. Thereby, these authors provided a convincing theoretical basis to the widespread opinion that quasi-periodic ﬂows observed in experiments (Hung and Andereck, 1988; Pratte and Hart, 1990; Braunsfurth and Mullin, 1996) with liquid metals may be due to the coexistence of two distinct disturbances (see also the discussions in Section 6.4).

8 Thermovibrational Convection Thermovibrational convection can be regarded as a ‘variant’ of the standard thermogravitational convection treated in Chapters 4–7 for which the steady Earth gravity acceleration is replaced by an acceleration oscillating in time with a given frequency. This kind of ﬂuid motion typically occurs on orbiting platforms [e.g. the International Space Station (ISS)] as a result of acceleration disturbances arising as undesirable and unavoidable deviations from true weightlessness. The reader is referred to Section 2.2.4 for some additional theoretical background on these aspects. It is convenient to recall here, however, that the most common sources of such accelerations on the ISS are structural vibrations (e.g. at the fundamental natural frequencies), equipment operations and crew activity (e.g. repetitive exercises that induce cyclic displacement of the position of all objects hosted on board).

8.1

Equations and Relevant Parameters

Disturbances induced in a ﬂuid by a sinusoidal displacement of the related container along a given direction (nˆ is the related unit vector) s(t) = bsin(ωt)nˆ

(8.1)

where b is the amplitude and ω = 2πf (f is the frequency) induce an acceleration: g(t) = g ω sin(ωt)

(8.2)

ˆ where g ω = bω2 n. which means vibrating a system with frequency f and displacement amplitude b corresponds to a sinusoidal gravity modulation with the same frequency and acceleration amplitude bω2 and vice versa (accordingly, hereafter the terms ‘gravity modulation’, ‘periodic acceleration’, ‘container vibration’ and g-jitter will be used as synonyms). Equation (8.2) represents the very idealized situation of a single frequency (generally referred to as ‘monochromatic’ disturbance). In reality, however, an effective microgravity environment is given by the superposition of disturbances with different amplitudes and frequencies (multifrequency or multicomponent spectrum) and, eventually, distinct directions, whose presence can be modelled by replacing Eq. (8.2) with a Fourier series expansion in terms of ω. Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Nevertheless, it is worth mentioning that given the complexity of the problem even in the simplest case of monochromatic acceleration, most of studies reported in the literature have dealt with the fundamental situation in which a single disturbance is considered per time. Using the Boussinesq approximation deﬁned in Section 2.1, in nondimensional form the continuity equation Eq. (1.59) and momentum equation Eq. (1.60) for such a condition will simply read ∇·V = 0 2

bω2 βT TL3 ∂V L ω 2 = −∇p − ∇ · V V + Pr ∇ V + Pr T sin t nˆ ∂t να α

(8.3) (8.4)

where bω2 βT TL3 /να = Raω can be regarded as a variant of the classical Rayleigh number with the steady acceleration being replaced by the amplitude of the considered monochromatic periodic acceleration. Remarkably, in this form the equation allows the treatment of the problem in terms of three independent nondimensional parameters only, where the ﬁrst is the well-known Prandtl number (Pr) and the others are the nondimensional frequency (#) and displacement (), deﬁned as ωL2 α βT T =b L which lead to the following compact nondimensional form of the momentum equation:

∂V = −∇p − ∇ · V V + Pr ∇ 2 V + #2 T sin(#t)nˆ ∂t #=

(8.5) (8.6)

(8.7)

where, obviously #2 = PrRaω . This equation, obviously, must be considered together with the energy equation Eq. (1.61). This adds a further degree of freedom to the problem due to the strong sensitivity that it exhibits to the angle between the direction nˆ of the acceleration and the direction of the applied temperature gradient (we shall come back to this aspect later).

8.2

Fields Decomposition

Many theoretical (order of magnitude analysis) and numerical studies of Eqs (8.3) and (8.4) together with the additional one for energy have shown that when soliciting a ﬂuid container by a periodic acceleration, an initial diffusive temperature distribution Tdiff (r,t) is distorted and the difference (suitably deﬁned) between the temperature distribution and the diffusive (ideal) one can be used to deﬁne a related TFD distortion (see, e.g., Monti et al., 1987; Schneider and Straub 1989; Alexander, 1990; Monti and Savino, 1994a,b, 1995, 1996a,b; Feonychev and Dolgikh, 1994; Savino, 1997; Savino et al., 1998; Savino and Monti, 1999a,b; Naumann, 2000). It is also known (see, e.g., Monti et al., 1998a, 2001; Savino and Lappa, 2003a) that the velocity ﬁeld V , induced by a periodic acceleration, in general, is made up by an average value V plus a periodic oscillation of amplitude V (V = V − V ) at the acceleration frequency f or at frequencies that are multiple of f . As a result of such a convective ﬁeld, the scalar quantities (temperature) are also distorted. These distortions in turn are also made up by a steady plus an oscillatory contribution (T = T + T ). This leads naturally to the need for different deﬁnitions of the TFD distortion parameters (one for the steady and the other for the oscillatory contribution, i.e. ε¯ and ε , respectively). With regard to these aspects, it is worth emphasizing how different situations may occur, depending on the oscillation frequency. For instance, it is known that on increasing the frequency there

Thermovibrational Convection

291

non-dimensional temperature disturbances

0.25 Ω = 104

Rav = 33 000

0.20 Ω = 103 0.15 T − Tdiff 0.10 T − Tdiff Ω = 102

0.05

0.00 0.00

0.05

0.10 0.15 non-dimensional time

0.20

Figure 8.1 Temperature signals as a function of time in a square test cell ﬁlled with a Pr = 15 liquid (subjected to vibrations perpendicular to the applied temperature gradient) versus the nondimensional frequency # for a ﬁxed value 2 #2 = 106 (at relatively low frequency, there is a regime characterized by relatively large oscillatory temperature distortions and relatively small time-average steady component; conversely, at relatively high frequencies, the oscillatory thermal distortions are very small with respect to the steady ones induced by the time-average part of the velocity ﬁeld) (M. Lappa)

is a ﬁrst regime characterized by a relatively large oscillatory velocity and oscillatory temperature contributions and relatively small time-average steady contributions. In contrast, at high frequencies the oscillatory thermal contributions are very small with respect to the steady ones induced by the time-average part of the velocity ﬁeld (as an example, these behaviours are very evident in Figure 8.1).

8.3 The TFD Distortions In the following, the subscript ‘diff’ denotes the parameters corresponding to the purely diffusive conditions [e.g. the temperature Tdiff (x, y, z, t)]. The thermal deviations induced by any acceleration will, therefore, be deﬁned at each point (x, y, z) and at any time (t) as δT = T (x, y, z, t) − Tdiff (x, y, z, t)

(8.8)

In the case in which the problem is discretized on a mesh, Eq. (8.8) at the generic grid point j will read δTj (t) = Tj (t) − Tj diff (t)

(8.9)

Taking into account that, according to the previous discussion (Section 8.2), the local temperature can be split into a time-averaged steady component plus an oscillating part (Tj = T j + Tj ), the local temperature distortion can be rewritten as δTj = Tj − Tj diff = (T j − Tj diff ) + Tj

(8.10)

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where T j is the nondimensional local temperature (scaled by the characteristic temperature difference T imposed across the container) averaged over the period 2π/ω and Tj is the oscillatory part expressed in terms of the local amplitude of the temperature oscillation (Tj ): Tj = Tj fj (ω)

(8.11)

where fj (ω) is, obviously, a periodic, zero time-average function [e.g. fj (ω) = sin(ωt + φj )]. As a natural consequence of these simple considerations, the two aforementioned different parameters accounting for the overall TFD distortions (over all the N grid points) can be deﬁned as follows: ε¯ T =

N 1 |(Tj − Tj diff )| N

(8.12a)

N 1 |(Tj )| N

(8.12b)

j =1

εT =

j =1

If both ε¯ T and εT are 1, one can assume that the purely diffusive regime is not disturbed by the presence of the periodic acceleration g(t). It is worth highlighting, however, that in many circumstances no comparison can be made between the two types of distortions; the order of magnitude of ε¯ T and εT for the negligibility of the thermal distortions usually is different. In fact, the results that the distortions depend on acceleration and frequency as (gω )2 /ω2 (Monti and Savino, 1996b) or as gω /ω2 (Monti et al., 1987) are both correct but refer to the two types of distortions: the ﬁrst refers to ε¯ T and the second to εT : ε¯ T = O(gω 2 /ω2 ) εT = O(gω /ω2 )

(8.13a) (8.13b)

At low frequencies, the main contribution is given by ε (which for # 1 refers to quasi-steady regimes); at relatively high frequencies, that is, for g-jitters (# 1), ε¯ ε (see Figure 8.2).

Figure 8.2 TFD distortions (for 2 #2 = 106 ) as a function of # for a square test cell ﬁlled with a Pr = 15 liquid (subjected to vibrations perpendicular to the applied temperature gradient) (M. Lappa)

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8.4

293

High Frequencies and the Thermovibrational Theory

As mentioned earlier, for given experimental conditions there is a strong dependence of both types of local TFD distortions (average and periodic) on the acceleration frequency. Amplitudes of the periodic temperatures tend to decrease quickly with frequency; conversely, the average quantities are less dependent on the frequency so that one expects the steady distortions to prevail over the unsteady distortions at high frequencies (and vice versa). Historically, this suggested that investigators (Simonenko and Zen’kovskaja, 1966, were the ﬁrst) should introduce strong simpliﬁcations in the treatment of the problem for the speciﬁc case of high-frequency vibrations (the so-called averaged formulation or thermovibrational theory). According to this formulation (Simonenko 1972; Gershuni and Zhukhovitskii, 1979, 1986; Gershuni et al., 1982), the time-averaged contributions can be simply computed (i.e. with much less computation time) by a simpliﬁed set of equations in terms of quantities averaged over the oscillation period. Accordingly, so far, two methods for g-jitter analysis have been considered in the literature: (a) numerical solution of the full nonlinear and time-dependent thermal convection equations with a time-dependent body force that gives rise to a time-dependent ﬂow [i.e. Eqs (8.3), (8.7) and (1.61)]; and (b) numerical solutions of the time-averaged ﬁeld equations (Gershuni formulation) for the thermovibrational convection problem. The ﬁrst case requires very lengthy computations since the time step for the integration of the equations must (of course) be smaller than the period τ = 1/f of the oscillations. The second case is very cheap from a computational point of view but is restricted to the case of sufﬁciently small amplitudes ( 1) and sufﬁciently large frequencies (# 1) of the vibrations (e.g. ISS g-jitters). Remarkably, under these conditions Gershuni and co-workers showed that, for given Prandtl number, the steady convection depends only on one relevant dimensionless parameter, the vibrational Rayleigh number: Rav =

(bωβT TL)2 #2 2 (βT TL)2 " gω #2 = = 2να 2να ω 2 Pr

(8.14)

Under the assumptions of small amplitudes ( 1) and large frequencies of the vibrations (# 1), this formulation leads to a closed set of equations for the time-averaged quantities. The time-averaged continuity and energy equations remain formally unchanged [i.e. they correspond to Eqs (8.3) and (1.61), respectively, with T = T and V = V ]; the time-averaged momentum equation must be rewritten as

∂V = −∇p − ∇ · V V + Pr ∇ 2 V + Pr Rav (w · ∇T )nˆ − w · ∇w ∂t

(8.15)

where w appearing in the production term (the mathematical details related to the derivation of this term will be provided later; see Section 8.7.2) is an auxiliary potential function satisfying the equations ∇ ·w = 0 ∇ ∧ w = ∇T ∧ nˆ → ∇ 2 w = −∇ ∧ (∇T ∧ n) ˆ

(8.16a) (8.16b)

Although there is a plethora of numerical results that could be used to demonstrate the validity of this approach, for the sake of brevity and due to page limits, however, they are not reviewed here.

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8.5

States of Quasi-equilibrium and Related Stability

As anticipated in Section 8.1 (and as expected by analogy with the canonical cases of buoyancy convection induced by static gravity treated in Chapters 4, 6 and 7), the relative orientation of the imposed temperature gradient and of the direction of the periodic acceleration is a very sensitive parameter for the onset of vibrational time-average convection and its strength. This is the reason why, for instance, for typical geometric containers the condition of highfrequency vibrations perpendicular to the temperature gradient is usually referred to as ‘the worst case’ (which means that the TFD distortion is relatively large; this jargon derives essentially from microgravity experimentation for which, in general, convection induced by vibrations is an undesired effect), whereas in the case of high-frequency vibrations parallel to the temperature gradient, thermal diffusive conditions are expected (Kamotani et al., 1981). The latter behaviour can be placed in a precise theoretical framework on the basis of the Eqs (8.15) and (8.16) related to the Gershuni formulation. If g-jitter parallel to the temperature gradient is considered, in fact, the driving force appearing in the momentum equation vanishes: ∇ · w = 0, ∇T ∧ nˆ = 0 → ∇ ∧ w = 0 → ∇ 2 w = 0, and since w = 0 on the boundary of the domain, the vector Poisson equation for w gives w = 0 → V = 0 (i.e. time-averaged diffusive conditions for the temperature ﬁeld).

8.5.1 The Vibrational Hydrostatic Conditions Notably, even if vibrations and temperature gradient are not parallel, other states of mechanical quasi-equilibrium (no time-average ﬂow) are possible in principle. As illustrated by many authors (e.g. Braverman and Oron, 1994; Gershuni and Demin, 1998, and references therein), the related mathematical conditions of existence can be obtained, in fact, by simply setting V = 0 in Eq. (8.15), assuming that the equilibrium ﬁelds T 0 , w0 and p0 do not depend on time and taking the curl of the resulting equation. This leads to ∇(w 0 · n) ˆ ∧ ∇T 0 = 0 ∇2T 0 = 0

(8.17a) (8.17b)

which, supplemented with the equations for the auxiliary function [Eqs (8.16)], are generally known as vibrational hydrostatic conditions (they guarantee, in fact, the time-averaged body force is balanced by the pressure gradient). Technically speaking, from a mathematical standpoint these equations must be regarded as conditions necessary, but not sufﬁcient, which means when they are satisﬁed → V 0 = 0 and T 0 = Tdiff , but this holds only if the vibrational Rayleigh number is lower than a given threshold Ravcr , whereas (as will be shown later) for Rav > Ravcr time-average convection arises. In practice, beyond considerations on the necessary but non-sufﬁcient nature of the mathematical requisites expressed by Eqs (8.17), these equilibrium states can be effectively attained only for special cases of (uniform) heating, shape and degrees of freedom of the system and direction of the vibrations axis. The simplest example along these lines is given, perhaps, by the layer of inﬁnite extent (Figure 8.3), which has attracted much attention in the literature (and also for other types of convection in the present book) and for which it is worth providing some speciﬁc information. In the following examples, in particular, not to expand excessively the discussion with a lot of variants, both the temperature gradient and vibrations are assumed to lie in the (x,y)-plane (i.e. no spanwise components). Denoting with φT and φv the angles between the direction of the solid boundaries of the layer (x-axis) and the imposed temperature gradient and vibrations, respectively (their difference θ = φT − φv representing as a natural consequence the angle between the imposed temperature gradient and vibrational axis; see Figure 8.3), the two vectors ∇T0 = i T (referred to the scale γ d,

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295

Figure 8.3 Sketch of layer of inﬁnite extent subjected to temperature gradient and vibrations inclined with respect to the walls

where γ is the uniform rate of temperature increase along the direction of the related unit vector i T ) and nˆ will read ∇T 0 = cos(φT )i x + sin(φT )i y nˆ = cos(φv )i x + sin(φv )i y

(8.18) (8.19)

Accordingly, equations (8.17) will be satisﬁed (Gershuni and Demin, 1998) for φT and φv such that [sin(φT ) cos(φv ) − cos(φT ) sin(φv )] cos(φT ) cos(φv ) = 0

(8.20)

with the component of w 0 along the x-axis given by w0 (y) = [sin(φT ) cos(φv ) − cos(φT ) sin(φv )] y

(8.21)

(the other two components along y and z being equal to zero). Equation (8.20) immediately reveals that for a temperature gradient perpendicular to the walls (φT = 90◦ ), quasi-equilibrium is possible for arbitrary directions φv of the vibration axis [cos (φT ) = 0 in Eq. (8.20), which makes this equation always satisﬁed], whereas for ∇T 0 parallel to the walls (φT = 0◦ ), quasi-equilibrium can be obtained solely for vibrations parallel or perpendicular to the boundaries [φv = 0◦ or 90◦ , which give sin(φv ) = 0 and cos(φv ) = 0, respectively, in Eq. (8.20)].

8.5.2 The Linear Stability Problem As outlined earlier, these states of quasi-equilibrium undergo transition to average convective ﬂow (V = 0) when the control parameter (the vibrational Rayleigh number) exceeds a certain threshold Ravcr . The problem is similar in concept to the onset of classical Rayleigh–B´enard (RB) convection treated in Chapter 4 (Section 4.1), the difference being given by a transposition of it on a time-averaged space. Let us recall, in fact, that while RB convection emerges from a true quiescent state V 0 = 0, here this property of the basic state holds from a time-averaged point of view only (V 0 = 0 but the oscillatory component V , although small, is not equal to zero). The determination of the related threshold requires the formulation of a stability problem of the type given by Eqs (1.96)–(1.98) in Section 1.5.3 by replacing physical quantities appearing there with the corresponding time-averaged ones. Assuming a basic state corresponding to quasiequilibrium (V 0 = 0, ∇T 0 and w0 given by Eqs (8.18) and (8.21), respectively) and introducing inﬁnitesimal disturbances for velocity, temperature and auxiliary function as δV , δT and δw, respectively, such a linear stability problem reduces to ∇ · (δV ) = 0

(8.22)

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Thermal Convection: Patterns, Evolution and Stability

∂ δV + ∇(δp) = Pr ∇ 2 (δV ) + δF b ∂t ∂ δT + δV · ∇(T 0 ) = ∇ 2 (δT ) ∂t ∇ 2 (δw) = −∇ ∧ (∇(δT ) ∧ n) ˆ

(8.23) (8.24) (8.25)

where δF b (the change induced in the body forces acting on the system by the disturbance) can be determined taking into account that for the present case the body force reads F b = Pr Rav (w · ∇T )nˆ − w · ∇w ; following the general concepts explained in Section 1.5.3, the corresponding inﬁnitesimal variation can be written as

&

' (8.26) δF b = Pr Rav w 0 · ∇(δT ) + δw · ∇T 0 nˆ − w 0 · ∇(δw) + δw · ∇w0 Interestingly, the problem is qualitatively similar to canonical RB convection also because the related instability is driven by two-dimensional perturbations of the (time-averaged) diffusive state (Gershuni and Demin, 1998) and does not depend on the speciﬁc value of Pr considered. These disturbances emerge and saturate their amplitude in the plane containing ∇T 0 and nˆ (xy plane for the present case). Taking into account that both the velocity and the auxiliary function are solenoidal (their divergence is zero), this makes possible the introduction in such a plane of stream functions for both δV and δw as follows: (8.27) ψ = ψd (y)eλt+iqx ∂ψ δu = (8.28a) = ψd eλt+iqx ∂y ∂ψ (8.28b) δv = − = −iqψd eλt+iqx ∂x λt+iqx = d (y)e (8.29) ∂ λt+iqx δwx = (8.30a) = d e ∂y ∂ δwy = − (8.30b) = −iqd eλt+iqx ∂x where the prime denotes differentiation with respect to y. Similarly, for the other disturbances, one can assume δT = Td (y)eλt+iqx δp = pd (y)eλt+iqx

(8.31) (8.32)

Substituting such disturbances into Eqs (8.23)–(8.25), eliminating the pressure in Eq. (8.23) leads to a closed set of three coupled amplitude equations for ψd , d and Td : Pr(ψdIV − 2q 2 ψd + q 4 ψd ) + &

Pr Rav cos(φT ) cos(φv )d − iq(cos(φT ) sin(φv ) + sin(φT ) cos(φv ))d − q 2 sin(φT ) sin(φv )d

' (8.33) −iqw0 (ψd − q 2 ψd ) + iq cos(φv )(w0 Td ) − iq sin(φv )w0 Td = λ(ψd − q 2 ψd )

(8.34) Td − q 2 Td − cos(φT )ψd − iq sin(φT )ψd = λTd

2 d − q d = cos(φv )Td − iq sin(φv )Td (8.35) to be considered together with the boundary conditions at y = ± 1/2: ψd d Td Td

= = = =

ψd = 0 0 0 (for adiabatic walls) 0 (for conducting walls)

(8.36a) (8.36b) (8.36c) (8.36d)

Thermovibrational Convection

297

which represent the aforementioned problem to be solved for the determination of the critical threshold related to the onset of time-average convection from initial (basic) time-averaged diffusive conditions satisfying Eqs (8.17).

8.5.3 Solutions for the Inﬁnite Layer Let us ﬁrst focus on the layer of inﬁnite extent with walls maintained at different constant temperatures (φT = 90◦ ; see Figure 8.4a) that was the main subject of the studies (linear stability analysis) of Gershuni and Zhukhovitskii (1979, 1981, 1986). As mentioned earlier, for such a case quasi-equilibrium is possible for arbitrary directions of the vibration axis (i.e. for arbitrary angle θ between vibrations and temperature gradient). The corresponding critical vibrational Rayleigh number for the onset of mean convection [coming from the solution of the linear stability problem represented by Eqs (8.33)–(8.36)] is shown in Figure 8.5. The threshold depends on θ with the stability being minimal for θ = 90◦ (Ravcr = 2129) and Ravcr increasing monotonically with reducing the angle of inclination (Figure 8.5a). Conversely, the critical wavenumber displays an increasing trend with θ , reaching qc = 3.23 for θ = 90◦ (Figure 8.5b). Accordingly, what appears at the onset is a highly structured state, a well-ordered spatially periodic pattern of alternatively upwelling and downwelling ﬂuid with convective cells having characteristic size determined by q (since, as already discussed, perturbations leading to the onset of mean convection are planar, the ensuing ﬂow is featured by the emergence of two-dimensional rolls). Gershuni and Zhukhovitskii (1981) also showed that this ﬂow may undergo a second bifurcation at larger values of Rav ; as an example, they found the multicellular ﬂow emerging for θ = 90◦ at Ravcr = 2129 with qc = 3.23 (extension of the generic cell corresponding approximately to the thickness of the layer) to turn into a new ﬂow with qc = 6.5 (two-level cells) at Rav = 2.95 × 104 . Obviously, for the speciﬁc condition θ = 0◦ (vibrations and temperature gradient parallel) the

(a)

(b)

Figure 8.4 Sketch of layer of inﬁnite extent subjected to vibrations: (a) temperature gradient perpendicular to the walls; (b) temperature gradient parallel to the walls

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Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 8.5 Critical vibrational Rayleigh number (a) and wavenumber (b) as a function of the angle θ between the direction of vibrations and the imposed temperature gradient [vibrational Rayleigh deﬁned as Rav = (bωβT Td)2 /2να , where T = γ d and d is the thickness of the layer]. Data after Gershuni and Zhukhovitskii (1981); Reproduced with kind permission of Springer Science and Business Media

value of Ravcr is inﬁnitely large (correspondingly qc = 0) in agreement with the mathematical explanation given before by which for such a case the production term in the momentum equation vanishes (hence this equilibrium state is a stable condition regardless of the value of Rav ). The other canonical conﬁguration in which the inﬁnite layer has a uniform temperature gradient parallel to the walls (φT = 0◦ as shown in Figure 8.4b) and is subjected to inclined vibrations was originally considered by Braverman (1984, 1987) and Birikh (1990). The latter author, in particular, clearly reported that for such a case vibrations lead to the development of a plane-parallel convective ﬂow (similar to those described in Section 2.4 for other types of driving forces) instead of the multicellular convection with wavenumber q discussed earlier for the layer with temperature gradient perpendicular to the boundaries. The related mathematical problem can be deﬁned as follows. Assuming a generic plane-parallel ﬂow solution (see Section 2.4 for additional theoretical background about exact solutions of the thermal-convection equations) in the form   u(y) (8.37) V = 0  0 T = x + f (y)   w(y) w= 0  0

(8.38) (8.39)

the original Gershuni’s partial differential equations for momentum, energy and the auxiliary potential function [Eqs (8.15), (1.61) and (8.16), respectively] can be reduced to a system of ordinary differential equations: u + Rav w cos(θ ) = 0 f + u = 0 w = f cos(θ ) − sin(θ )

(8.40) (8.41) (8.42)

which, combining Eq. (8.40) with Eq. (8.42), can be cast in compact form as u + Rav f cos2 (θ ) = Rav cos(θ ) sin(θ ) f + u = 0

(8.43) (8.44)

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( with Rav = (bωβT γ d 2 )2 2να, where γ is the rate of uniform temperature increase along the x-axis and boundary conditions at y = ± 1/2 : u = 0 (no-slip) and f = 0 or f = 0 for adiabatic or conducting walls, respectively. Notably, the dimensionless characteristic number of the problem together with the components of the unit vector nˆ appearing in these equations can be grouped in two parameters only: R1 = Rav cos(θ ) sin(θ ) R2 = Rav cos2 (θ )

(8.45) (8.46)

of which the ﬁrst can be used to reobtain conditions corresponding to the existence of the states of quasi-equilibrium by simply setting it equal to zero [hence obtaining again the two cases already discussed before for such a conﬁguration, i.e. the usual condition of vibrations parallel to the temperature gradient (θ = 0◦ → sin(θ ) = 0 → R1 = 0) and vibrations perpendicular to the temperature gradient (θ = 90◦ → cos(θ ) = 0 → R1 = 0)]. In practice, the analytical form of the plane-parallel ﬂow established in the layer for other values of θ not corresponding to equilibrium (θ = 0◦ , θ = 90◦ ) can be determined by solution of the system Eqs. (8.40)–(8.42) with the additional constraints

1/2

−1/2

u(y)dy = 0 and

1 2 1 −2

w(y)dy = 0

(8.47)

As illustrated in detail by Birikh (1990), this approach leads to the following mathematical expressions: R1 cosh(2ξy) sin(2ξy) sinh(ξ ) cos(ξ ) − sinh(2ξy) cos(2ξy) cosh(ξ ) sin(ξ ) (8.48) u(y) = − 16 ξ 3 (sinh(ξ ) cosh(ξ ) + sin(ξ ) cos(ξ )) 1 sinh(2ξy) cos(2ξy) sinh(ξ ) cos(ξ ) + cosh(2ξy) sin(2ξy) cosh(ξ ) sin(ξ ) f (y) = tan(θ ) 2y − 2 ξ(sinh(ξ ) cosh(ξ ) + sin(ξ ) cos(ξ )) (8.49) for the case with adiabatic walls and R1 cosh(2ξy) sin(2ξy) sinh(ξ ) cos(ξ ) − sinh(2ξy) cos(2ξy) cosh(ξ ) sin(ξ ) u(y) = − (8.50) 16 ξ 2 (sinh2 (ξ ) cos2 (ξ ) + cosh2 (ξ ) sin2 (ξ )) 1 sinh(2ξy) cos(2ξy) sinh(ξ ) cos(ξ ) + cosh(2ξy) sin(2ξy) cosh(ξ ) sin(ξ ) f (y) = tan(θ ) 2y − 2 (sinh2 (ξ ) cos2 (ξ ) + cosh2 (ξ ) sin2 (ξ )) (8.51) for the case with conducting walls, where 1 1

ξ = √ Rav cos2 (θ ) 4 2 2

8.6

(8.52)

Primary and Secondary Patterns of Symmetry

For enclosures of ﬁnite extent, the ﬂow structure is more complex. In general, these cases do not admit quasi-equilibrium states (with the exception solely of the case in which θ = 0◦ ) and the emerging pattern and ensuing hierarchy of bifurcations depend on many factors. For the sake of simplicity, here such analysis is limited to the case of two-dimensional square or rectangular cavities shown in Figure 8.6.

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Figure 8.6 Sketch of the test cell, its boundary conditions and orientation of g -jitter

For vibrations perpendicular to the temperature gradient (θ = 90◦ ), it is known that if the Rav value is not too high, the time-averaged ﬂow is characterized by a four-vortex structure (Figure 8.7). It is symmetrical by reﬂection with respect to both x = 1/2 and y = 1/2 planes (often referred to as quadrupolar ﬁeld ). This structure is replaced by a different pattern of symmetry if the orientation of the g-jitter is changed (Figure 8.8 shows that for θ = 90◦ a single vortex pervasive throughout the test cell appears). Figure 8.9 displays the intensity of the ﬂow (in terms of maximum of the streamfunction)

Figure 8.7 Primary pattern of symmetry [streamlines in the plane (x, y) showing the socalled quadrupolar ﬁeld] of thermovibrational convection induced by g -jitter perpendicular to the imposed temperature gradient in a square cavity (Pr = 15 and Rav = 3.3 × 102 ; cold and hot walls on the left and right sides, respectively; horizontal boundaries with adiabatic conditions) (numerical simulation, M. Lappa)

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301

(a)

(b)

θ = 90˚

θ = 45˚ (c)

θ = 30˚

Figure 8.8 Velocity ﬁeld [streamlines in the plane (x, y)] as a function of the geometric aspect ratio and the angle θ (silicone oil, Pr = 15, Rav = 3.3 × 102 ; cold and hot walls on the left and right sides, respectively; horizontal boundaries with adiabatic conditions; vibrational Rayleigh number based on the distance between the hot and cold walls): (a) A = 1; (b) A = 2; (c) A = 4 (numerical simulation, M. Lappa)

Figure 8.9 Intensity of the ﬂow (in terms of maximum of the stream function) versus the angle θ (Pr = 15, A = 1, Rav = 3.3 × 102 ; vibrational Rayleigh number based on the distance between the hot and cold walls) (numerical simulation, M. Lappa)

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versus the angle θ . Interestingly, the maximum intensity occurs for θ ∼ = 45◦ . For θ = 0◦ as previously elucidated, no ﬂow arises and consequently the maximum value of the streamfunction is zero. When the aspect ratio (A = length/depth) is increased, for θ = 90◦ the pattern with four corner rolls is maintained (Figure 8.8b and c). The intensity of the related convection, however, decreases as A increases; in fact, as A → ∞ the system tends ideally to the state of equilibrium (no average convection) predicted by Birikh (1990) for the limiting case of inﬁnite layer, as explained earlier (according to Gershuni et al., 1982, in particular, the character of the bifurcation becomes practically the same as for the inﬁnitely long layer when the aspect ratio exceeds the value A = 8). Similarly, for θ = 90◦ when A is increased the ﬂow tends to develop in the central part (the core) the plane-parallel convective ﬂow represented by Eqs (8.48) and (8.49). Like classical buoyancy ﬂows, thermovibrational convection can also undergo transition to new patterns of symmetry with increasing degree of complexity when the characteristic number is increased. Along these lines, Figure 8.10 shows that for θ = 90◦ ﬁxed, if Rav is increased sufﬁciently a secondary stable regime appears possessing another symmetry and bifurcating from the quadrupolar ﬁeld. This pattern forms through fusion of two diagonal vortices with opposite direction of circulation (Figure 8.10 shows, in fact, a diagonal extended roll). Such a secondary ﬂow is usually referred to as a one-vortex regime, because it consists of one main circulation and two small counter-rotative cells with a weaker intensity conﬁned to the corners of the enclosure. In general, as illustrated by Khallouf et al. (1995), the transition from the four-vortex regime to inversional symmetry takes place at a critical value of Rav , which increases monotonically with the aspect ratio (their study was conducted for Pr = 1 and vibrational Rayleigh number Rav = 105 ). A clear experimental conﬁrmation of such a transition has been provided recently by Mialdun et al. (2008) in the context of parabolic-ﬂight experiments or the case of a cubical cell (L = 5 cm) ﬁlled with 2-propanol.

Figure 8.10 Secondary patterns of symmetry [plane (x, y)] induced by g -jitter perpendicular to the imposed temperature gradient (Pr = 15, square cell and Rav = 3.3 × 104 ; cold and hot walls on the left and right sides, respectively; horizontal boundaries with adiabatic conditions) (numerical simulation, M. Lappa)

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8.7

303

Medium and Low Frequencies: Possible Regimes and Flow Patterns

The prevention of signiﬁcant convection (i.e. the attainment of a velocity ﬁeld with zero time-average component and almost negligible oscillatory part) in the case of vibrations parallel to the imposed temperature gradient discussed earlier and shown in Figure 8.9 (θ = 0◦ ) is no longer possible when the Gershuni formulation is not valid (e.g. relatively low frequencies). In such a case, according to available computations (Biringen and Danabasoglu, 1990; Liz´ee and Alexander, 1997), it is known that different regimes of time-dependent convection occur depending on the ‘response’ of the system to the imposed vibration (or gravity modulation).

8.7.1 Synchronous, Subharmonic and Nonperiodic Response A fairly coherent and exhaustive picture of the richness of possible scenarios for the canonical reference case of a 2D square cell has been provided by Hirata et al. (2001). They investigated numerically vibrational convection in a two-dimensional square-section enclosure ﬁlled with a liquid with Pr = 7 vibrating sinusoidally parallel to the applied temperature gradient (with the other two walls assumed to be perfectly conducting). Although for such a conﬁguration Gershuni’s formulation (Sections 8.4 and 8.6) would predict no time-averaged ﬂow and time-averaged thermal diffusive conditions, they found well-deﬁned convective patterns characterized mainly by one- or two-cell structures with transition from one to the other as a function of Raω = #2 /Pr (the Rayleigh number based on the acceleration amplitude deﬁned in Section 8.1). This parameter was varied from 104 to 105 (Figure 8.11). The observations were not limited to the spatial description of the time-dependent convective structures; remarkably, it was also found that there exist regimes where the behaviour of the velocity ﬁeld can be non-synchronous with the frequency of the imposed acceleration and even

Figure 8.11 Time-dependent convective structures arising in a two-dimensional square-section enclosure ﬁlled with a liquid with Pr = 7 vibrating sinusoidally parallel to the applied temperature gradient: 1, one cell; 2, two cells; M, starting with two cells and followed by one and/or four cells; A, alternation of one and four cells; F, random ﬂuctuations between one and two cells; X, stable. After Hirata et al. (2001); Reproduced by permission of Cambridge University Press

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Figure 8.12 Response of the velocity ﬁeld to the imposed periodic acceleration for # in the range 1 ≤ # ≤ 103 (two-dimensional square-section enclosure ﬁlled with a liquid with Pr = 7 vibrating sinusoidally parallel to the applied temperature gradient): synchronous case (SY); half-subharmonic case (SU); non-periodic case (NP); stable case (ST). The shaded area shows where the ﬂuid is stationary everywhere over some interval of time during each period of imposed oscillation. After Hirata et al. (2001); Reproduced by permission of Cambridge University Press

non-periodic; Hirata et al. (2001), in fact, reported on synchronous, but also half-subharmonic (twice the forced-acceleration period) and non-periodic responses. Anyhow, the ﬂow was observed to tend to the quiescent state (diffusive thermal conditions) predicted by Gershuni’s approximation for θ = 0◦ as the frequency was increased; along these lines, Figure 8.12 shows that the threshold for the onset of convection increases dramatically as # → 103 (similar results were also obtained by Biringen and Danabasoglu, 1990, who recovered typical results of the thermovibrational theory for sufﬁciently high #). The periodicity is broadly summarized in Figure 8.12 as a function of both # and Raω . It can be seen that convective motion becomes more stable as # increases or as Raω decreases (regime ST) and that the motion is almost synchronous with the forced acceleration (regime SY) with the exception of three regimes: a half-subharmonic regime around # = 5 × 102 and Ra > 6 × 104 (regime SU) and two non-periodic regimes around # = 2 × 102 and Ra > 8 × 104 [regime NP(I)] and for # < 10 over a wide range of Ra [regime NP(II)]. The shaded area shows, in particular, where the ﬂuid is stationary everywhere over some intervals of time during each period (moreover, in the regime SY in this area a random alternation between the solutions with different symmetries often occurs). For rectangular enclosures, some interesting numerical analyses have also appeared in which vibrations (at ﬁnite frequencies) perpendicular to the applied temperature gradient (θ = 90◦ ) were considered. As an example for the square cavity, Thomson et al. (1995) and Kondos and Subramanian (1996) reported that also in this case there is a signiﬁcant qualitative difference between low and high frequencies (Thomson, in particular, considered the effect of both deterministic and stochastic acceleration modulations normal to the initial density gradient; in the latter case, the acceleration ﬁeld was modelled by narrowband noise deﬁned by a characteristic frequency, a correlation time and a given intensity).

Thermovibrational Convection

(a)

305

(b)

Figure 8.13 Critical vibrational Rayleigh number as a function of the nondimensional frequency # for the inﬁnite layer with walls at constant different temperatures and vibrations parallel to the plane of the layer, i.e. θ = 90◦ [vibrational control parameter deﬁned as Raω = bω2 βT Td 3 /να in (a) and Rav = (bωβT Td )2 /2να in (b), where T = γ d and d is the thickness of the layer]. After Gershuni et al. (1996); Reproduced with kind permission of Springer Science and Business Media

The idealized case of an inﬁnitely elongated rectangular cavity (inﬁnite layer with walls at constant different temperatures and vibrations parallel to the plane of the layer) was investigated by Gershuni et al. (1996). They carried out a linear stability analysis for the speciﬁc case Pr = 1 and a wide range of frequencies (Figure 8.13). In the limit # → 0, in particular, a basic solution in the form (plane-parallel ﬂow) Raω " 3 y # y − sin(#t) (8.53a) u= 6 4 T = −y (8.53b) was considered. It was found that for intermediate frequencies (Figure 8.13a), the basic oscillating parallel ﬂow with no mean convection can undergo shear instabilities of a hydrodynamic nature (see Section 1.5.4 for relevant theoretical background), whereas for sufﬁciently high # (Figure 8.13b), the source of instability is transferred again to the thermovibrational mechanism operating in the limit # → ∞ (as shown in Figure 8.13b, the curve gives for # ∼ = 104 a value very close to the critical vibrational Rayleigh number Ravcr = 2129 obtained in the framework of the averaged formulation and shown in Figure 8.5 for θ = 90◦ ). Despite this noteworthy progress, however, there are still many open questions. In particular, owing to computer performance limits there is still a signiﬁcant lack of numerical results concerning the system evolution at relatively low frequencies in ﬁnite-sized geometries (for both 2D and 3D conﬁgurations).

8.7.2 Reduced Equations and Related Ranges of Validity Among the possible improvements for ﬁlling the gaps mentioned above, one direction of research is the identiﬁcation of sets of reduced equations (and associated ranges of validity) for low and intermediate values of the frequency where the Gershuni (thermovibrational) theory is no longer applicable. Such an effort, in its broadest sense, may be of potentially great importance for the scientiﬁc community. Even though the governing equations for Newtonian ﬂuid dynamics, the unsteady Navier–Stokes equations, have been known for 150 years or more (see Chapter 1), the development of reduced

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Thermal Convection: Patterns, Evolution and Stability

forms of these equations is still an attractive area of research. This is particularly true for problems related to the effect of vibrations for which, in fact, current research is mainly focused on the construction and solution of the governing equations for different categories of problems and the study of various approximations to those equations. It is obvious that the motivation for further theoretical exploration along these lines lies in the fact that the aforementioned approximations can be used to obtain advantageous analytical and/or mixed numerical–analytical solutions of the problem. It is worth highlighting that, in general, the methods attempted by different investigators to yield such simpliﬁed solutions fall into two categories: (a) specifying series expansions, substituting them in the partial differential equations and writing a simpliﬁed algebraic set of equations for the coefﬁcients of the analytical expansions; and (b) ﬁnding some set of alternative equations by mathematical transformation and manipulation (a relevant example is Gershuni’s theory). For both cases, the probability of success is greatly improved by the simpliﬁcation (dropping some terms) of the initial set of nonlinear equations. Reduced sets of equations are provided here for Pr = 15 and two values of Rav (3.3 × 102 and 3.3 × 104 that correspond to the maximum values to be expected on the International Space Station). These different sets of equations, given by the omission of some terms, were originally obtained through an a posteriori order-of-magnitude analysis based on the direct numerical solution of the original thermal-convection equations Eqs (8.3), (8.7) and (1.61) by Savino and Lappa (2003a). It is worth noting that even though these authors considered a speciﬁc study case (a twodimensional cell, a single ﬂuid with high Prandtl numbers), their conclusions can apply to more general cases. The guiding principles at the basis of their approach (together with the related effective mathematical developments) are illustrated in detail in the following. Basically, such an approach foresees the nondimensional equations are rewritten in terms of the average values (V , T ) plus periodic oscillations (V , T ) : V = V + V , T = T¯ + T . They read as follows: ∇ ·V +∇ ·V = 0

(8.54)

∂V ∂V + + ∇p + ∇p + ∇ · V V + ∇ · V V + ∇ · V V + ∇ · V V ∂t ∂t = Pr(∇ 2 V + ∇ 2 V ) − #2 sin(#t)T nˆ − #2 T sin(#t)nˆ

(8.55)

∂T (8.56) + + ∇ · V T + ∇ · V T + ∇ · V T + ∇ · V T = ∇ 2T + ∇ 2T ∂t ∂t If the values of V (t) and T (t) are known at a given ﬁeld point, the average values for V and T can be calculated by ∂T

V =

# 2π

2π/ #

V dt, T = 0

# 2π

2π/ #

T dt

(8.57)

0

The time-dependent parts V and T at each instant of time can, therefore, be deﬁned as V = V −V, T = T −T

(8.58)

If steady conditions are reached for the averaged ﬁelds, that is, ∂V ∂T = 0 and =0 ∂t ∂t under these conditions, in terms of the following quantities: FV 1 =

(8.59)

∂V , FV 2 = ∇p, FV 3 = ∇p , FV 4 = ∇ · V V , FV 5 = ∇ · V V , FV 6 = ∇ · V V , ∂t

Thermovibrational Convection

307

FV 7 = ∇ · V V , FV 8 = − Pr ∇ 2 V , FV 9 = − Pr ∇ 2 V , FV 10 = #2 sin(#t)T n, ˆ FV 11 = #2 T sin(#t)nˆ

∂T FT 1 = , FT 2 = ∇ · V T , FT 3 = ∇ · V T , FT 4 = ∇ · V T , ∂t

FT 5 = ∇ · V T , FT 6 = −∇ 2 T , FT 7 = −∇ 2 T

(8.60)

(8.61)

Eqs (8.55) and (8.56) can be cast in compact form as 11 i=1 7

FV i = 0

(8.62)

FT i = 0

(8.63)

i=1

which represent the starting point for the application of the aforementioned a posteriori analysis of the order of magnitudes based on direct numerical solution. As a further step towards such an end, it should be pointed out that in the above equations, the generic term Fi can be seen as the sum of an average value F i (x, y) and a periodic oscillation Fi (x, y): # F i (x, y) = 2π

2π/ #

Fi (x, y) dt; Fi = Fi − F i

(8.64)

0

By deﬁnition Fi = 0, so that

and

FV i = 0 for i = 1, 3, 5, 6, 9, 10

(8.65)

FT i = 0 for i = 1, 3, 4, 7

(8.66)

FV i = 0 for i = 2, 4, 8

(8.67)

FT i

(8.68)

= 0 for i = 2, 6

Obviously, only the nonlinear periodic oscillating terms appearing in Eqs (8.62) and (8.63) that contain the product of oscillating quantities (FV 7 , FV 11 , FT 5 ) have non-zero values of the time average: FV 7 = F V 7 + FV 7 FV 11 = FT 5 =

F V 11 + FV 11 F T 5 + FT 5

(8.69a) (8.69b) (8.69c)

On averaging Eqs (8.62) and (8.63), one obtains 11 i=1 7

FV i = 0

(8.70)

FT i = 0

(8.71)

i=1

which taking into account Eqs (8.65) and (8.68) read ∇ ·V =0 (∇ p) + ∇ · [V V ] + ∇ · [V V ] = Pr(∇ 2 V ) − #2 T sin(#t)nˆ ∇ · [V T ] + ∇ · [V T ] = ∇ 2 T

(8.72) (8.73) (8.74)

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Thermal Convection: Patterns, Evolution and Stability

At this stage, simpliﬁed sets of the averaged equations Eqs (8.72)–(8.74) and also those of the equations in their original complete form, Eqs (8.54)–(8.56), can be obtained by dropping or retaining the various equation terms according to the parametric simulations of Savino and Lappa (2003a), which were performed in fairly wide ranges of the control parameters (10 ≤ # ≤ 105 and 3.3 ≤ Rav ≤ 3.3 × 104 ). Before doing so, however, some additional deﬁnitions are necessary. In fact, a procedure must be deﬁned for evaluating (starting from the numerical computations) the order of magnitude of each term of the equations at each ﬁeld point and then for properly averaging over the entire ﬂow ﬁeld. In practice, the order of magnitude of the terms with zero-average can be evaluated by computing the maximum (Fimax ) and the minimum (Fimin ) during the period 2π/ω:

(8.75) Fi (x, y) = Fi (x, y) max − Fi (x, y) min which represents the amplitude of the oscillatory part of the generic quantity Fi = Fi fi (t)/2 [where fi (t) is a periodic, zero time-average function, e.g. fi (t) = sin(ωt +φi )]. As a consequence, from a numerical point of view (N is the number of grid points, j refers to the generic mesh point), overall quantities can be deﬁned as (Fi )oν =

N 1 (Fi )j N

(8.76)

j =1

which leads, ﬁnally, to the possibility of evaluating the average and oscillatory terms as (F i )oν =

N 1 (F i )j N

(8.77)

N 1 (Fi )j N

(8.78)

j =1

(Fi )oν =

j =1

The major results of such analysis are shown in Figures 8.14–8.17. In particular, in these ﬁgures comparison of the order of magnitude of each term is provided through the additional quantities i and i deﬁned by the ratios between each term (i) and the leading term in the momentum (subscript V ) and energy (subscript T ) equations, as follows: V i =

(FV i )ov (FV 10 )ov

T i =

(FT i )ov (FT 4 )ov

V i =

(FV i )ov (FV 10 )ov

(8.79)

(FT i )ov (FT 4 )ov

(8.80)

T i =

2 is where (FV 10

)ov the driving term # T and (FT 4 )ov corresponds to the convective energy transport ∇ · V T . As illustrated in Figures 8.14 and 8.15, the overall relative importance of each term depends on # and to a lesser extent on Rav . For instance, at very high values of # only three terms in the momentum equation (V 1 , V 3 , V 10 ) and two terms in the energy equation (T 1 , T 2 ) prevail (regime 1; in such a case the equations at the basis of the thermovibrational theory are recovered, as will be shown later). Plots of the terms appearing in the averaged momentum and energy equations are shown in Figures 8.16 and 8.17. Remarkably, as a result of the comparison of the different terms, different regimes can be identiﬁed. Separation of the terms in those containing steady and oscillatory contributions shows that in the entire range of # and (or # and Rav ) the leading terms in the complete momentum equation is the driving action FV 10 = #2 sin(#t)(T )nˆ that is almost balanced by the

Thermovibrational Convection

309

(a)

(b)

Figure 8.14 Comparison between the order of magnitude of the different terms involved in the momentum equation for two values of the vibrational Rayleigh number: (a) Rav = 3.3 × 102 ; (b) Rav = 3.3 × 104 . After Savino and Lappa (2003a)

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Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 8.15 Comparison between the order of magnitude of the different terms involved in the energy equation for two values of the vibrational Rayleigh number: (a) Rav = 3.3 × 102 ; (b) Rav = 3.3 × 104 . After Savino and Lappa (2003a)

Thermovibrational Convection

311

(a)

(b)

Figure 8.16 Comparison between the order of magnitude of the different terms involved in the averaged momentum equation for two values of the vibrational Rayleigh number: (a) Rav = 3.3 × 102 ; b) Rav = 3.3 × 104 . After Savino and Lappa (2003a)

312

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 8.17 Comparison between the order of magnitude of the different terms involved in the averaged energy equation for two values of the vibrational Rayleigh number: (a) Rav = 3.3 × 102 ; b) Rav = 3.3 × 104 . After Savino and Lappa (2003a)

Thermovibrational Convection

313

Figure 8.18 Different regimes in the frequency-acceleration plane. The ﬁgure also shows lines with constant values of (dashed), # (dash-dotted) and Rav (solid) (M. Lappa)

pressure term FV 3 = ∇p . At relatively low frequency, the other term to be taken into account is FV 9 = − Pr ∇ 2 V , that at # > 104 can be neglected with respect to the term FV 1 = ∂V ∂t . In the

energy transport equation, the two leading terms are FT 1 = ∂T ∂t and FT 4 = ∇ · V T . Unless at 2 very low frequency (# < 10 ), in addition a third term must also be taken into account, FT 7 = −∇ 2 T . Therefore the full and averaged equations can be summarized as follows. First, the reduced sets of equations (in dimensional form) are shown for the case Rav = 105 /3 [regimes (a) in Figure 8.18]. Then the case Rav = 103 /3 is considered [regimes (b)]. T0 relates to reference conditions. For # > 104 (regime 1a): Complete equations: ∂V 1 + ∇p = bω2 βT T − T0 sin(ωt)nˆ ∂t ρ0

∂T + ∇ · V T = 0 ∂t

(8.81a) (8.81b)

Averaged equations:

1 ∇p + ∇ · V V = ν∇ 2 V + bω2 βT T sin(ωt)nˆ ρ0

∇ · V T = α∇ 2 T

(8.82a) (8.82b)

It is worth emphasizing how the simpliﬁed set of equations for this case corresponds to that used by Gershuni to develop the thermovibrational theory illustrated in Section 8.4. Equations (8.81) and (8.82), in fact, can be directly used for deriving Eq. (8.15). Using the Hodge theorem, the vector ﬁeld (T − T0 )nˆ can be decomposed into a solenoidal part w plus an irrotational part ∇η: T − T0 nˆ = w + ∇η (8.83) with ∇ · w = 0 and ∇ ∧ w = ∇T ∧ n. ˆ

314

Thermal Convection: Patterns, Evolution and Stability

According to Eq. (8.83), Eq. (8.81a) can be separated into two equations (for the potential and the solenoidal parts): ∂V = bω2 βT sin(ωt)w ∂t 1 ∇p = bω2 βT sin(ωt)∇η ρ0 Equation (8.84) can be easily integrated with respect to time, giving t V = βT w bω2 sin(ωt)dt = −βT wbω cos(ωt)

(8.84) (8.85)

(8.86)

0

Substituting Eq. (8.86) into Eq. (8.81b) and integrating with respect to time yields t T = βT w · ∇T bω cos(ωt)dt = βT w · ∇T b sin(ωt)

(8.87)

0

Taking into account Eqs (8.87) and Eq. (8.82a) and that −bω2 sin2 (ωt) = −bω2 cos2 (ωt)

(8.88)

Eq. (8.70) reduces to Eq. (8.15); this is an impressive example of the aforementioned possibility to use simpliﬁed equations (by dropping some terms) in order to introduce alternative sets of equations that can be solved with enhanced computational efﬁciency with respect to the original set of thermal-convection equations. For 103 < # < 104 (regime 2a): Complete equations: ∂V 1 (8.89a) + ∇p = bω2 βT T − T0 sin(ωt)nˆ + ν∇ 2 V ∂t ρ0

∂T + ∇ · V T + ∇ · V T = α∇ 2 T ∂t

(8.89b)

Averaged equations:

1 ∇p + ∇ · V V = ν∇ 2 V + bω2 βT T sin(ωt)nˆ ρ0

∇ · V T + ∇ · V T = α∇ 2 T For 102 < # < 103 (regime 3a): Complete equations: 1 ∇p = bω2 βT T − T0 sin(ωt)nˆ + bω2 βT T sin(ωt)nˆ + ν∇ 2 V ρ0

∂T + ∇ · V T + ∇ · V T = α∇ 2 T ∂t Averaged equations:

1 ∇p + ∇ · V V = ν∇ 2 V + bω2 βT T sin(ωt)nˆ ρ0

∇ · V T = α∇ 2 T

(8.90a) (8.90b)

(8.91a) (8.91b)

(8.92a) (8.92b)

Thermovibrational Convection

For 10 < # < 102 (regime 4a): Complete equations: 1 ∇p = bω2 βT (T − T0 ) sin(ωt)nˆ + bω2 βT T sin(ωt)nˆ + ν∇ 2 V ρ0

∇ · V T = α∇ 2 T

315

(8.93a) (8.93b)

Averaged equations: 1 ∇p = ν∇ 2 V + bω2 βT T sin(ωt)nˆ ρO

∇ · V T = α∇ 2 T

(8.94a) (8.94b)

Hereafter, the regimes corresponding to Rav = 10 /3 are denoted the (b) regimes. For # > 104 (regime 1b), the simpliﬁed set of equations corresponds to the Gershuni equations as for Rav = 105 /3. For 103 < # < 104 (regime 2b): Complete equations: 1 ∂V (8.95a) + ∇p = bω2 βT T − T0 sin(ωt)nˆ + ν∇ 2 V ∂t ρ0

∂T (8.95b) + ∇ · V T = α∇ 2 T ∂t Averaged equations:

1 ∇p + ∇ · V V = ν∇ 2 V + bω2 βT T sin(ωt)nˆ (8.96a) ρ0

(8.96b) ∇ · V T + ∇ · V T = α∇ 2 T 3

For 102 < # < 103 (regime 3b): Complete equations: 1 ∇p = bω2 βT T − T0 sin(ωt)nˆ + ν∇ 2 V ρ0

∂T + ∇ · V T + ∇ · V T = α∇ 2 T ∂t Averaged equations:

1 ∇p + ∇ · V V = ν∇ 2 V + bω2 βT T sin(ωt)nˆ ρ0

∇ · V T = α∇ 2 T For 10 < # < 102 (regime 4b): Complete equations: 1 ∇p = bω2 βT T − T0 sin(ωt)nˆ + ν∇ 2 V ρ0

∇ · V T = α∇ 2 T

(8.97a) (8.97b)

(8.98a) (8.98b)

(8.99a) (8.99b)

Averaged equations: 1 ∇p = ν∇ 2 V + bω2 βT T sin(ωt)nˆ ρ0

∇ · V T = α∇ 2 T

(8.100a) (8.100b)

9 Marangoni–B´enard Convection 9.1

Introduction

Studies on pattern formation driven by temperature gradients have so far been focused mainly on buoyancy-driven Rayleigh–B´enard (RB) convection for which a fairly comprehensive understanding has been reached in the last three or four decades. As illustrated in Chapters 4–7, these studies comprise the pattern formation in the inﬁnitely extended system (formulated in the Busse’s extensive theory) and the discovery of spiral-defect chaos (Chapter 4), the fascinating behaviour of systems transversely heated (Chapter 6), the combined effect of vertical and horizontal gradients (inclined layer convection, Chapter 7), and so on. As explained in Chapter 2, however, another signiﬁcant class of patterns and, in general, of convective phenomena emanates from another kind of driving force that arises as a consequence of the dependence of surface tension on temperature. The problem concerning thin ﬂuid layers with the upper free surface subjected to a perpendicular temperature gradient, in particular, falls into the category of phenomena generally known under the name Marangoni–B´enard (MB) convection. In this chapter, we will review the existing body of work on this subject from a historical standpoint, focusing on the most salient aspects that separate this kind of convection from all the thermogravitational variants considered in the earlier chapters. Let us start by considering the seminal experiments due to B´enard (1900, 1901), who observed suggestive and pleasing hexagonal patterns resembling the architecture created by bees for their honeycombs (Figure 9.1). These convective patterns were initially explained by Rayleigh (1916) in terms of buoyancy. Block (1956), however, gave a conclusive experimental demonstration of the role of surface tension in the formation of these planforms. In particular, he managed to observe cells at Ra < Racr when the thermogravitational mechanism is not active. On the theoretical side, treatments closely parallel to those developed for standard RB convection were used over the years to study both the linear and nonlinear properties of these ﬂows; both the absence of reﬂection symmetry with respect to the midplane of the layer and the location of the driving force at the free surface, however, were found to be sources of an increased complexity of MB convection with respect to RB convection. Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Thermal Convection: Patterns, Evolution and Stability

Figure 9.1 Sketch of classical Marangoni–B´enard convection with the honeycomb symmetry

Theoretical investigations of such a kind started with the landmark linear stability analysis of Pearson (1958) (which, among other things, explains why this convective motion is often referred to as Pearson’s instability). Resorting to the same mathematical framework already employed in earlier studies for the RB problem in the inﬁnite layer [the linear stability problem reduces to an eigenvalue problem for a system of ordinary differential equations in the vertical coordinate; see, e.g., Eqs (4.1)–(4.10)], Pearson neglected the presence of the gravitational term in the momentum equation and replaced the upper boundary conditions for velocity and temperature with those modelling a Marangoni stress and heat exchange with the ambient, respectively (Bi = 0 representing the limiting case of an adiabatic surface; see Figure 9.2). A combination of the two mechanisms was considered later in the linear stability analyses of Nield (1964) and Palmer and Berg (1971), who revealed that, in general, under normal gravity conditions where both driving forces act simultaneously, the two forces causing instability can reinforce one another. In particular, by taking into account both such forces, Nield (1964) provided important evidence of the fact that surface tension forces are the main drivers of Marangoni instability in layers with a depth of only a few millimetres. He illustrated, in fact, that the onset condition of motion by the mixed effect for a ﬂuid layer heated from below can be predicted according to the following correlation: Ra ∼ Ma (9.1) + =1 Macr Racr where Ra is the canonical Rayleigh number deﬁned as Ra = gβT T d 3 /να, Ma = σT T d/ µα, Macr is the critical Marangoni number for the onset of surface tension-driven convection (Pearson, 1958) and Racr is the critical Rayleigh number for the one induced by buoyancy (see Chapter 4). Remarkably, since Ma scales linearly with the height (d) of the layer, whereas Ra exhibits a cubic dependence on d, in general for depths of few millimetres and typical experiments, the

Marangoni–B´enard Convection

319

Figure 9.2 Marginal stability curves for pure Marangoni–B´enard convection (gravity absent) in inﬁnite layers (Ma deﬁned as σT T d/µα , where T is the temperature gradient perpendicular to the layer). After Pearson (1958); Reproduced by permission of Cambridge University Press

following conditions hold: Ra/Racr < 1 whereas Ma/Macr = O(1) or larger. Thereby, it becomes evident that this theory is in perfect agreement with experiments (for which in the case of shallow liquid layers investigators generally ascribe the onset of convection solely to exceeding Macr ) and the same can be assumed for the limiting case of zero gravity (zero-g, i.e. Ra = 0), for which the theory reduces exactly to the results of Pearson (1958). Other theoretical studies were based on the energy stability theory (see Section 1.6 for further details on this method). Davis (1969) proved that below Ma = 56.77 the system is unconditionally stable, while according to the linear stability analysis of Pearson (1958) it becomes unstable with respect to inﬁnitesimal perturbations with a wavenumber q = 1.99 above Ma = 79.607. Gaining information about the effective geometry of the convective cells when the disturbances saturate their amplitude required nonlinear analyses and application of the theory of bifurcations (Scanlon and Segel, 1967; Cloot and Lebon, 1984; Bragard and Lebon, 1993). The linear-theory problem for a static liquid layer (i.e. with initial thermal diffusive conditions) in fact, is degenerate (the planform of the instability cannot be determined as the form of the disturbances, i.e. hexagonal cells, is selected by nonlinear effects). It is worth remarking that in spite of all these efforts, the typical convective patterns in the B´enard experiment are still misinterpreted in some textbooks as a paradigm for a buoyant-convection structure. Morover, although the presence of convective cells induced by this type of ﬂow on the free surface of liquid layers can be detected by several possible experimental techniques (Figures 9.3), in comparison with RB convection, the experimental results in the ﬁeld of surface tension-driven convection are more sparse. This chapter may be regarded as a synthetic account of the state-of-the-art on this subject elaborated via a symbiotic analysis and comparison of the sparse theoretical, numerical and experimental data available in the literature. In particular, it will be shown how, although the physical

320

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

(c)

Figure 9.3 Photographs [plane (x, z)] of surface cells in silicone oils provided by means of different optical visualization methods: (a) shadograph image (courtesy of C. Piccolo); (b) Schlieren image (courtesy of C. Piccolo); (c) thermographic image (digitally enhanced) (M. Lappa)

understanding of surface tension-driven convection is far less advanced than that of buoyancydriven convection, the ﬁeld has reached a certain recognizable level of maturity.

9.2

High Prandtl Number Liquids: Patterns with Hexagons, Squares and Triangles

As anticipated in Section 9.1, for the transparent (high-Pr) ﬂuids traditionally employed in experiments, when the disturbances responsible for the onset of convection saturate their amplitude, the ensuing pattern is characterized by the well-known and suggestive presence of aesthetic hexagonal cells. As an example, Figure 9.3c shows a thermographic picture of the layer free surface; a sketch of the related ﬂow structure is given in Figure 9.1, and the computer simulations in Figure 9.4 allow direct comparison between velocity and temperature ﬁelds related to such a pattern. Although such a structure is a typical feature of MB convection, it should be remarked, however, that hexagonal cells are not an exclusive prerogative of this kind of ﬂow. They can be also observed in Rayleigh–B´enard convection, as widely discussed in Chapter 4 (the reader is referred, in particular, to Section 4.6). Anyhow, whereas for RB convection they are a special case emerging only when the vertical reﬂectional symmetry is violated (see Chapter 3) or occurring in particular ranges of the control

Marangoni–B´enard Convection

(a)

321

(b)

Figure 9.4 Hexagonal cells due to Marangoni–B´enard convection, observable on the free surface [plane (x, z)] of a layer of silicone oil of 1 cSt covered by an equal layer of air, having depth 1 mm and horizontal extension Lx = Lz = 2 cm (Ma = 200, Ma based on the depth Ly of the layer; zero-g conditions): (a) velocity ﬁeld; (b) temperature distribution (numerical simulations, M. Lappa)

parameters (Figures 4.20–4.22), in typical MB convection with transparent liquids they are the usual planform at the onset. These cells have proved to be fairly stable when the Marangoni number is further increased (Echebarria and P´erez-Garc´ıa, 2001). For moderate supercriticalities (Ma − Macr )/Macr ∼ = 1, the size of the hexagons decreases with Ma. At higher supercriticalities, such a decrease is reversed to a monotonic increase in the cell size (Cerisier et al ., 1987a). At a certain stage (Ma), however, a new secondary pattern emerges where the seemingly all-embracing hexagons are replaced by a ﬂow with another type of topology (Bestehorn, 1996). Such a new ﬂow is featured by square convective cells (Figure 9.5), which become the new persistent dominant feature of the ﬂow (see, e.g., Figure 9.6). As observed by Eckert et al . (1998), in particular, the transition from hexagons to squares is accompanied by an increase in the Nusselt number. From this observation, it follows that square cells are indeed a more efﬁcient mode of heat transfer than hexagonal cells (in approximately the same way as bimodal convection is more efﬁcient in transporting heat than two-dimensional rolls in RB systems, as explained in Section 4.2). Such square cells possess a wavenumber that is higher than that of hexagonal cells (see, e.g., Figure 9.7). Interestingly, the transition sequence arises via a local change in topology whereupon threefold vertices of the initial hexagonal cells become fourfold; an edge that separates two vertices shrinks to zero length and the two vertices coalesce to form the intersection of four edges. As the vertices coalesce, the angle between adjacent edges changes from 120◦ to 90◦ and the intersection of four edges becomes stable. Initially, this process forms pentagons (this means the transition from hexagons to squares is mediated by the occurrence of pentagonal cells). The threshold for this process depends on Pr and it is shifted towards higher Ma as Pr increases, as shown in Figure 9.8.

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(a)

(b)

Figure 9.5 Square cells due to Marangoni–B´enard convection, observable on the free surface [plane (x, z)] of a layer of silicone oil of 1 cSt covered by an equal layer of air, having depth 1 mm and horizontal extension Lx = Lz = 2 cm (Ma = 400, Ma based on the depth Ly of the layer; zero-g conditions): (a) velocity ﬁeld; (b) temperature distribution (numerical simulations, M. Lappa)

Figure 9.6 Fraction of cell class Pi as a function of (Ma − Macr )/Macr (Pi = Ni /N is deﬁned as the ratio between the number of cells Ni of a given planform and the number of complete cells N ; i = 4, 6 stand for square and hexagonal cells, respectively. After the experiments of Eckert et al. (1998), for Pr = 102 ; Reproduced by permission of Cambridge University Press

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Figure 9.7 Wavenumber as a function of (Ma − Macr )/Macr (Pr = 102 ). After Eckert et al. (1998); Reproduced by permission of Cambridge University Press

Figure 9.8 Condition for hexagon–square transition as a function of the Prandtl number. After Eckert et al. (1998); Reproduced by permission of Cambridge University Press

With a further increase in Ma, the ﬂow becomes time dependent (the interested reader will ﬁnd the state-of-the-art in the paper by Eckert et al ., 1998, and references therein). By virtue of such a behaviour (steady cells with square symmetry undergoing transition to a time-dependent regime), MB ﬂow relatively far from the onset again exhibits some interesting analogies with the case of thermogravitational convection.

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Experimentally, square patterns have been observed in RB with strongly dependent viscosity (Oliver, 1983; White, 1988) or at constant viscosity (Section 4.6) or in thermal convection with solidiﬁcation (Hadji et al ., 1990). None of these documented square patterns, however, evolve from a hexagonal planform as in the MB case, which can be regarded as a distinguishing mark of this type of ﬂow. Such square cells also emerge when the Prandtl number is reduced towards unit order. It is known, in fact, that at Pr = O(1) the primary mode of MB convection is given by square cells instead of hexagons, which coexist near the threshold of the primary instability with rolls. When Ma is increased, however, such structures (not shown) are taken over by patterns consisting of oscillating triangular cells as illustrated in Figures 9.9. Perhaps the most interesting property that the ﬂow displays under such conditions is that it looks similar to the spoke pattern convection discussed as a quaternary mode of RB convection in Section 4.5 for high-Pr ﬂuids. As is evident in Figure 9.9a, the lines bounding the (approximately) triangular cells organize themselves to form a complex network. Such lines, in particular, seem to originate from some special knots which behave as the centres of ‘ﬂower’ structures. The related overall pattern architecture is given by the coexistence of some (eight in Figure 9.9a) of these ﬂowers (partial overlap among them is possible). Radial spokes depart the aforementioned special knots, which exhibit a topological order p (p = number of departing spokes) larger than that of the others (for the case shown in Figure 9.9a, the topological order of such knots is on average about p = 8, in contradistinction to the values p = 3 and 4 for the patterns with hexagonal and square cells shown in Figures 9.4 and 9.5, respectively). Like spokes in the RB case, the pattern is time dependent, but it tends to retain a large-scale, nearly steady structure in which the location and spacing of the ﬂowers are surprisingly regular

(a)

(b)

Figure 9.9 Snapshot of oscillating triangular structures in Marangoni–B´enard convection, observable on the free surface [plane (x, z)] of a layer of KCl melt (Pr = 1), having depth 1 mm and horizontal extension Lx = Lz = 2 cm covered by a 1 mm layer of helium (Ma = 103 , Ma based on the depth Ly of the liquid layer; zero-g conditions): (a) velocity ﬁeld; (b) temperature distribution (numerical simulation, M. Lappa)

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(even if individual spoke structures vary considerably and the oscillations of the centres of ﬂowers have a random phase). The case with Pr < O(1) is even more complex. It is treated in the next section.

9.3

Liquid Metals: Inverted Hexagons and High-order Solutions

For Pr < O(1), due to lack of experimental results for liquid metals (most of the available experimental studies deal with transparent liquids), surface tension-driven MB convection in low-Pr ﬂuids has been studied especially by means of direct numerical simulation. Thess and Bestehorn (1995) predicted the occurrence (see also Parmentier et al ., 1996) of ‘inverted’ hexagons at the onset of convection in layers with a sufﬁciently small Prandtl number (Pr < 0.23). These hexagons were found to be cold (ﬂuid descends from the surface in the centre of the hexagon) and to have a distinctly skewed shape, in contrast to the case at high Pr (Figure 9.10). The subsequent possible transitions of this pattern when Ma is increased were studied later by Boeck and Thess (1999) for Pr ∼ = 0 (Pr = 0 and 0.005). They computed the ﬂow in a three-dimensional rectangular domain with periodic boundary conditions in both horizontal

Figure 9.10 Contour plot of surface temperature perturbation at the free surface [plane (x, z)] of a layer of liquid metal (Pr = 0, periodic domain 20 × 1 × 20, Ma = 80, Bi = 0 and Ra = 0; dashed lines correspond to negative values of the perturbation). This ﬁgure clearly shows the skewed shape of the hexagons. After Boeck and Thess (1999); Reproduced by permission of Cambridge University Press

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Figure 9.11 Emerging regimes of MB convection as a function of the Marangoni number for Pr ∼ = 0, Bi = 0 and Ra = 0

directions and a no-slip bottom wall using a method based on the poloidal–toroidal decomposition (see Thual, 1992, or Section 4.2.1 for further details) and a pseudospectral Fourier–Chebyshev discretization. According to these studies, as the Marangoni number is increased from the critical value for instability of the quiescent state to approximately twice this value, the initially stationary inverted hexagonal convection pattern quickly becomes time dependent and eventually reaches a state of spatiotemporal chaos. It has also been revealed that two-dimensional solutions can be observed in some intervals of Marangoni numbers. For the convenience of the reader, such important ﬁndings are described in detail in the following. After the threshold predicted by linear stability analysis (Macr = 79.607 for Bi = 0), inverted hexagons are stable up to Ma ∼ = 86. Beyond this value, ‘deformed’ hexagons appear. When the deformed hexagons cease to exist (Ma ∼ = 93), several possible solution branches exist beyond this point. Some of them correspond to stationary two-dimensional rolls either with the roll axes parallel to the short domain boundary or in two different oblique orientations. The other branches represent time-dependent solutions, namely three-dimensional travelling waves differing only in the direction in which they propagate. Such a sequence is summarized in Figure 9.11. Boeck and Thess (1999) also revealed that these three-dimensional travelling-wave solutions can become unstable with respect to purely two-dimensional travelling-wave solutions (namely rolls which travel sideways with axes again parallel to the short domain boundary).

9.4

Effects of Lateral Conﬁnement

In the light of all the results discussed in Sections 9.1–9.3, a fairly complete picture regarding MB pattern selection in inﬁnite layers can be sketched: There is a critical value Pr∗ ∼ = 0.23 where the ﬂow direction in the centre of the generic cell turns from downﬂow to upﬂow. Below Pr∗ , inverse hexagons are selected. At Pr ∼ = Pr∗ , rolls instead of hexagons are stable near the onset of convection. If Ma is increased, time-dependent states follow. For Pr Pr∗ , traditional hexagons are stable which are replaced by squares at larger Ma. Cerisier et al . (1987b) also showed that hexagons lose the competition with rolls at a certain distance from the instability threshold if the depth of the liquid layer is increased. When the ratio of the depth to the width of the container is relatively high, the morphology of the observed patterns and the cells’ symmetry, however, may exhibit a strong dependence on the shape of the lateral boundaries. Furthermore, the effects of buoyancy, which as illustrated in Section 9.1 can be neglected for shallow layers, tend to become important.

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Landmark terrestrial experiments on this type of convection under conditions for which the sidewall geometry has a signiﬁcant impact on the ﬂow structure were carried out by (to cite just a few) Koschmieder and Prahl (1990), Ondarcuhu et al . (1993a,b), Maza et al . (1996) and Ram´on et al . (1999). As an example, Koschmieder and Prahl (1990), using open cavities with circular and square sidewalls, found experimentally for transparent liquids [Pr > O(1)] that convective ﬂow exhibits the symmetry of the lateral walls with ﬂuid upwelling in the centre and ﬂowing down near the sides. In general, it is known that for circular boundaries, as the aspect ratio A = L/D (height/diameter) is increased, patterns with discrete rotational symmetries arise. For sufﬁciently small A, in particular, a distinct group of polygonal interior cells still form in the interior, which can exhibit clear hexagonal symmetry encircled by a distinct group of boundary cells at sidewalls. For larger A, sector-shaped cells are divided by downﬂow boundaries that extend radially from the centre of the container (as a consequence, the ﬂow must be characterized using two wavenumbers: m = azimuthal wavenumber and i = radial wavenumber); the number of cells (both i and m) decreases with increase in A. To study the ﬂow induced solely by surface tension effects and to discern the role potentially played by non-vanishing buoyancy effects, several theoretical analyses on the subject have also appeared. In the stability analysis of Rosenblat et al . (1982a), a circular geometry was considered for 1/6 ≤ A ≤ 1. The sidewalls were assumed to be adiabatic and impenetrable and for mathematical simplicity the liquid was allowed to slip on the sidewalls (the so-called ‘slippery’ sidewalls). A linear stability theory for heating from below was used to predict the critical Marangoni number as a function of cylinder dimensions, surface-cooling condition and Rayleigh number. Steady nonlinear convective states near Macr were also calculated using an asymptotic theory and the stability of these states was examined. Interestingly, it was found that the Prandtl number of the liquid inﬂuences the stability of axisymmetric states, distinguishing upﬂow at the centre from downﬂow. Near those aspect ratios corresponding to double eigenvalues (where two convective states of linear theory are equally likely), the nonlinear theory was found to predict sequences of transitions from one steady convective state to another as the Marangoni number was increased. These transitions were determined and discussed in detail. Time-periodic convection was also predicted in certain cases. Similar results were obtained for the case of rectangular containers with one horizontal dimension larger than either the other horizontal dimension or the depth (Rosenblat et al ., 1982b). Later, Dauby and Lebon (1996) and Dauby et al . (1997) reconsidered theoretically Marangoni– B´enard convection in rigid rectangular and circular containers, respectively, with the more realistic boundary conditions of no-slip lateral (adiabatic) walls. Square and rectangular containers were also theoretically examined by Dijkstra (1995a–c) and Bergeon et al . (1998, 2001). A common result is that the critical Marangoni numbers for ﬁnite boxes are always larger than the values corresponding to an inﬁnite domain, which means that rigid sidewalls are stabilizing (as expected). A common ﬁnding is also the stationary nature of the emerging convective structure. Fascinating non-steady behaviours, however, were sometimes reported in experiments. For instance, Ondarcuhu et al . (1993a) observed three kinds of oscillating patterns in a square container with an aspect ratio (width to height) of 4.6. Johnson and Narayanan (1996) observed, in a circular container with an aspect ratio (radius to height) of 2.5, a dynamic mode switching between two-cell patterns.

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9.4.1 Circular Containers Examples of linear stability results for circular geometry and adiabatic rigid sidewalls are plotted in Figure 9.12 for Bi = 0. This ﬁgure shows the curves corresponding to values of the azimuthal wavenumber m between 0 and 3 (Macr is the absolute minimum of these curves). It can be seen that the critical Marangoni number is a globally decreasing function of the radius to height ratio of the cylinder and tends (as expected) to the value predicted by Pearson in the limit as this ratio → ∞ (similar results were obtained by Dauby and Lebon, 1996, in the case of square and rectangular containers). The convective patterns appearing at threshold change according to the aspect ratio and, in particular, depend on the azimuthal wavenumber m giving rise to the absolute minimum of the different curves in Figure 9.12. In reality, as explained earlier, a precise classiﬁcation of these patterns requires also speciﬁcation of the aforementioned radial wavenumber i. Along these lines, it should be pointed out that each curve corresponding to a ﬁxed m in Figure 9.12 is made up of different regions in which the concavity is successively upwards and downwards (local minima and maxima appear). In practice, every region with upward concavity can be associated with a radial wavenumber i, which is obtained by counting the successive upward concavity regions from the left of the picture: on a curve with ﬁxed m, the ﬁrst upward concavity region has a radial wavenumber i = 0, the next corresponds to i = 1, and so on. In particular, for small radius/height ratios (i.e. large A), the critical mode is m = 1, i = 0 (1,0). A picture of the related convective structure is given in Figure 9.13a. An upﬂow is located in the right-hand half of the container and a symmetrical downﬂow in the left-hand part. For radius/height ratios around 2 (A = 0.25), the pattern is axisymmetric and consists of a circular roll centred in the middle of the box (Figure 9.13b). For radius/height close to 2.7 (A ∼ = 0.185), the critical mode is (1,1), represented in Figure 9.13c. The structure may be seen as made up by three transverse and somewhat deformed rolls.

Figure 9.12 Critical Marangoni number Macr as a function of the radius to height ratio for different values of the azimuthal wavenumber (open cavity with adiabatic no-slip side walls, Bi = 0, Ra = 0). Courtesy of G. Lebon

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(0,1)

(b)

(a) (2,1)

(1,1)

(c)

(d) (3,1)

(0,2)

(e)

(f)

(1,2)

(g)

Figure 9.13 Sketches of convective patterns in open cylindrical containers at the threshold for different aspect ratios. The vertical velocity w at mid-depth of the container is represented (solid line = positive w , dashed line = negative w )

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In a very small area around radius/height = 3.2 (A ∼ = 0.15), the critical pattern has the form given in Figure 9.13d; the marginally stable mode is (2,1). Near radius/height = 3.7 (A ∼ = 0.135), the critical mode is (0,2), which is also axisymmetric (Figure 9.13e) and consists of two concentric rolls. When the ratio is very close to 4 (A ∼ = 1/8), the mode (3,1), drawn in Figure 9.13f, is linearly unstable. The last mode, represented in Figure 9.13g, corresponds to a ratio = 4.3 (A ∼ = 0.11); the wavenumbers are in this case (m, i) = (1,2) and ﬁve deformed transverse rolls are displayed. The critical modes for larger values of the radius to height ratios are not discussed here because they can be easily derived when the wavenumbers are known. Moreover, as observed by Dauby et al . (1997) for large radius to height ratios, many modes are nearly critical at threshold and it is not always easy to determine which one would become unstable ﬁrst. The above results were obtained by Dauby et al . (1997) for Bi = 0 (and Ra = 0). In general, an increase in the Biot number makes the critical Marangoni number larger since perturbations may be dissipated in the upper gas layer. It is also worth noting that the critical Marangoni numbers for adiabatic sidewalls are always smaller than those for conducting sidewalls, because a temperature perturbation arriving at an insulating boundary is reﬂected towards the bulk of the ﬂuid while it is dissipated in the walls when these are conducting. For this reason, conducting sidewalls give rise to more stable systems. This argument is similar to that of Nield (1964), who interpreted the increase in Macr with Bi in inﬁnite layers. The patterns for conducting sidewalls will not be discussed in detail here since, for any value of the azimuthal wavenumber m, they are similar to those given in Figure 9.13. Note, however, that the succession of the critical modes may be different (for instance, in the conducting case, Dauby et al ., 1997, never observed modes m = 2 and 3 at the threshold). Most recently, the problem has been re-approached via experiments performed directly in the weightlessness of space. Schwabe (1999, 2006) considered silicone oils and aspect ratios A = 1/10, 1/8, 1/4, 1/3, 2/3, 1.3 and 1 (in particular, the experimental setup was realized in order to mimic the ‘slippery sidewalls’ originally considered by Rosenblat and co-workers). The onset of convection as a function of A was measured and compared with theory and ground-based experiments. Schwabe observed modes (m, i) with i = 1 due to the limited radial extent of the containers. Mode (0,1) was obtained for 1/2 ≤ A ≤ 1 (ﬂuid motion is toroidal: upﬂow at the centre of the container and downﬂow in the rim). For A = 1/4 and 1/8, he observed modes (1,1) and (2,1), respectively (Figure 9.14a and b) and for A = 1/10 mode (3,1) (Figure 9.14c) after relaxation of the pattern. It is worth noting that in such µg experiments, less convection cells (smaller mode numbers) were generally observed in the ﬁnal state in containers of comparable aspect ratios than by

(a)

(b)

(c)

Figure 9.14 Experimental patterns of MB convection in cylindrical containers ﬁlled with silicone oil under microgravity conditions (silicone oil 10 cSt; MAXUS 2 sounding rocket experiments) for three values of the aspect ratio: (a) A = 1/4; (b) A = 1/8; (c) A = 1/10. Courtesy of D. Schwabe

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Koschmieder and Prahl (1990) in their 1g experiments. This is consistent with other observations and theories that predict larger convection cells with increasing surface tension forces compared with buoyancy. For A = O(1) and µg, it is also worth considering the numerical simulations of Wagner et al . (1994), who investigated the case of water.

9.4.2 Rectangular Containers Regarding rectangular containers, let us start by observing that, in general, owing to the symmetry between the length and the width of the box, the solutions for such cases can be separated into four classes which are characterized by the parity of the unknown ﬁelds with respect to the coordinates x and z. The four classes are denoted EE, EO, OE and OO (the ﬁrst and second letter of each of these symbols refer to the parity of the x and z dependence of the temperature ﬁeld, respectively; this classiﬁcation is closely parallel to that already described in Section 4.9 for Rayleigh–B´enard convection). Figure 9.15 (Dauby and Lebon, 1996) is a three-dimensional picture of Macr determined via a linear stability analysis as a function of the aspect ratios Ax and Az (Ax,z = Lx,z /d, where d is the depth) for Bi = 0, Ra = 0 and adiabatic no-slip solid walls. It can be seen that the critical Marangoni number is a globally decreasing function of the lateral dimensions of the container. As in the circular case, as they grow, it tends to 79.607, which is the value found for the inﬁnite layer (Section 9.1). It is worth noting, however, that if only one of the two aspect ratios is allowed to become inﬁnite (the other one being ﬁxed), then Macr tends to a value that is higher than that for the inﬁnite layer (Dauby and Lebon, 1996). Notably, a similar trend was found by Davis (1967) for pure RB convection (Chapter 4). The convective structures corresponding to the thresholds shown in Figure 9.15 for the speciﬁc case of square containers (Ax = Az ) and sketched in Figure 9.16, can be described as follows. For very small cavities (Ax ≤ 2.2), one roll parallel to z and one roll parallel to x become simultaneously unstable at the threshold. For Ax = 2.5, a unique square cell appears (Figure 9.16a). For Ax = 4.4, there is still a square cell (Figure 9.16b), but its structure is somewhat different (a cell similar to that for Ax = 2.5 is visible in the middle but it appears rotated by 45◦ ). In practice, Figure 9.16a corresponds to the sum of two rolls perpendicular to x and two rolls perpendicular to z while Figure 9.16b may be regarded as the product of such rolls. In both cases, the rolls parallel to both sides combine, leading to a symmetric structure. For larger boxes (near Ax = 4.8), either three rolls perpendicular to x or three rolls perpendicular to z appear (Figure 9.16c). The pattern in the neighbourhood of Ax = 5.9 is given by Figure 9.16d and corresponds to an antisymmetric structure (which may be regarded as the product of three rolls parallel to x and three rolls parallel to z). In the vicinity of Ax = 6.2, the convective structure (Figure 9.16e) is featured by one square cell (similar to Figure 9.16a but with opposite sign) in the middle of the box, but a ring of sinking ﬂuid is also present along the sidewalls. For still larger values of the aspect ratio (Ax ∼ = 8), ﬁve rolls parallel to any side of the container occur (Figure 9.16f). The properties of the instability in these square boxes can be directly related to the symmetry of the convective patterns. For EE solutions (Figure 9.16a,b and e), the velocity ﬁeld is clearly invariant for a reﬂection about the medians of the box and also for a rotation of 90◦ around the centre of the container. Similarly, the OO solutions (Figure 9.16d) have the same symmetry. For the EO or OE classes, the structure is still invariant for reﬂections with respect to the medians, but the invariance is lost for a rotation of 90◦ . In fact, for square containers, the EO and OE modes are simultaneously unstable and the EO pattern can be obtained by simply rotating the OE pattern by 90◦ and vice versa.

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220 200 180 160 Macr 140 120 100

2

2 4

4 Ax

6

6 8

Az

8

Figure 9.15 Critical Marangoni number as a function of the geometric aspect ratios (rectangular open cavity with adiabatic no-slip sidewalls, Bi = 0 and Ra = 0). After Dauby and Lebon (1996); Reproduced by permission of Cambridge University Press

Following Dauby and Lebon (1994), a tentative physical interpretation of the geometric nature of these patterns may be based on the balance between two simple arguments. The ﬁrst one, introduced by Davis (1967), states that, due to dissipation at the no-slip walls, a structure with rolls parallel to the shorter side of the container dissipates less kinetic energy than a structure consisting of rolls parallel to the longer sides (as already mentioned in Chapter 4, this is the reason why patterns with rolls parallel to the shorter sides ate more likely to appear in experiments). The second argument is the fact that the system prefers a structure for which the rolls are not compressed too much or dilated but have a dimension approximately equal to the dimension they would have in an inﬁnite layer. As a concluding remark for this section, it is worth noting that the MB problem has also been investigated in spherical geometry (Wilson, 1994; Subramanian and Zebib, 2008). This subject complements from a theoretical point of view that of Rayleigh–B´enard convection in spherical shells treated in Section 4.13 and from a technological point of view it has application, for example, in the context of manufacturing small spherical shells by microencapsulation, used as laser targets in inertial conﬁnement fusion (McQuillan and Greenwood, 1999).

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(b)

(c)

(d)

(e)

(f)

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Figure 9.16 Sketches of convective patterns in open square containers at the threshold for different aspect ratios. The vertical velocity w at mid-depth of the container is represented (solid line = positive w , dashed line = negative w )

As an example, Wilson (1994) used a combination of analytical and numerical techniques to analyse the onset of steady Marangoni convection in a spherical shell of ﬂuid with an outer free surface surrounding a rigid sphere. It was found that if the free surface of the layer is non-deformable, then the layer is always stable when heated from the outside and is unstable when heated from the inside if the magnitude of the (positive) non-dimensional Marangoni number is sufﬁciently large.

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9.5

Temperature Gradient Inclination

The existing studies mentioned and discussed in the earlier sections were all concerned with the applied temperature gradient perpendicular to the interface. In general, however, it should be pointed out that in actual material processing applications and industrial processes the temperature gradient is often inclined to the surface. For this reason, some interesting analyses have also appeared in which thermocapillary convection induced by a temperature gradient inclined to the free surface was considered. From a theoretical point of view, this topic complements that of buoyancy convection in inclined systems treated in Chapter 7; moreover, the present section should be regarded as an effort to bridge the gap between the present chapter and situations in which the temperature gradient is primarily parallel to the free surface (Chapters 10 and 11). In such a context, the experimental work of Ueno et al . (2002) deserves special attention owing to the relatively wide range of experimental conditions examined and the interesting categorization of patterns observed. Silicone oils of 2, 5, 10 and 20 cSt (Pr = 27.9, 67.0, 111.9 and 206.8 at room temperature, respectively) were employed as the test ﬂuid. A thin liquid layer of the test ﬂuid was formed on a shallow cylindrical container 50 mm in diameter and up to 2 mm in depth, which gives an aspect ratio (depth/diameter) of A = O(10−2 ). A temperature gradient was applied to the liquid layer with an inclination to the free surface, that is, perpendicular and parallel temperature gradients were imposed on the free surface simultaneously. Accordingly, two types of nondimensional Marangoni numbers, the vertical and horizontal Marangoni numbers Mav and Mah , were introduced in order to characterize the induced ﬂuid motion: σT Tv d (9.2a) Mav = µα σT γ d 2 (9.2b) Mah = µα where Tv is the vertical component of the applied temperature gradient and γ is the rate of temperature increase along the horizontal direction. Five different ﬂow patterns were identiﬁed (Figure 9.17), depending on the relative magnitude of the perpendicular and parallel temperature gradients: (1) standard Marangoni–B´enard convection, (2) ﬂowing B´enard cells, (3) streak convection, (4) horizontal circulation and (5) thermal diffusive conditions (stagnation, i.e. no ﬂow), whose general properties are described in detail in the following and summarized in Figure 9.17. In such experiments with the vertical Marangoni effect prevailing over the horizontal one (Mav Mah ), the classical Marangoni–B´enard convection with honeycomb structures was found to emerge in the layer (for such a case the cellular motion is induced by ‘hot spots’ emerging on the free surface like those shown in Figure 9.4b; the resultant thermocapillary forces draw the ﬂuid from the hot spot to relatively colder regions). With increasing ratio Mah /Mav , a weak horizontal Marangoni effect was observed to drive the cells towards the cold end while still maintaining a recognizable underlying cellular pattern (see Figure 9.18a). Obviously, this can be explained by considering that the hot spots on the surface move towards colder region owing to the effects induced by an Mah = 0 (this phenomenon was referred to by Ueno et al ., 2002, as ﬂowing B´enard cells). With relatively increasing Mah , the underlying cellular pattern was found almost to disappear and ‘streak’ structures to emerge with the tracer particles accumulating in parallel lines along the horizontal temperature gradient with a constant interval λ (see Figure 9.18b). Careful observation revealed that such streaks are supported by pairs of counter-rotating streamwise vortices and that

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105

104

M-B conv. Flowing B. cell Streak conv. Horizontal circ. Stagnation

A

Mav 103

C

102

B D

101 10−1

100

101 Mah

102

103

Figure 9.17 Surface tension-driven convection in silicone oil thin liquid layers with temperature gradient inclined to the free surface: Conditions for the emergence of distinct ﬂow patterns as a function of Mav and Mah . Courtesy of I. Ueno

Figure 9.18 Examples of typical ﬂow patterns for surface tension-driven convection in silicone oil thin liquid layers with temperature gradient inclined to the free surface: (1) top view of the ﬂow ﬁeld; (2) enlarged view of the central part of the ﬂuid [dashed line region in (1)]; (3) sketch of ﬂow structure (experimental results). Courtesy of I. Ueno

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the resulting ﬂow is basically organized in a set of longitudinal rolls (rolls with axis parallel to the horizontal temperature gradient). When such a conﬁguration is attained, the ﬂow in the vicinity of the free surface directs towards the cold end wall, whereas the ﬂow returns in the bottom region of the layer along the streaks towards the hot end. With further increase in Mah , the convection was found to turn into a global return ﬂow in the container, i.e a horizontal circulation (see Figure 9.18c; the surface ﬂuid moves from the hotter to the colder end, and the subsurface ﬂuid from the colder to the hotter end). Neither cellular nor spiral motion was observed. On the basis of such observations, Ueno et al . (2002) partitioned the map of possible patterns into four main regions irrespective of the differences in liquid viscosity and liquid layer depth, with region A corresponding to the canonical Marangoni–B´enard convection, B to the streak convection, C to the horizontal circulation and D to stagnation (the ﬂow pattern of the ﬂowing B´enard cells appearing on the boundary between the regions A and B). In such a context, it should also be mentioned that, as illustrated theoretically by Nepomnyashchy et al . (2001), with further increase in the horizontal component of the temperature gradient (i.e. Mah ), a sixth possible pattern enters the dynamics as the global horizontal circulation mentioned earlier can undergo transition to the oblique hydrothermal waves originally predicted by Smith and Davis (1983) for the case of a ﬂuid layer subjected solely to a horizontal temperature gradient (i.e. Mah = 0 and Mav = 0). The peculiar properties of these waves will be discussed in Chapter 10, which is devoted entirely to surface tension-driven ﬂows induced by temperature gradients that are primarily parallel to the free surface. Notably, the streak convection (which, as mentioned earlier, exhibits longitudinal rolls parallel to the horizontal temperature gradient) could also be explained on the basis of some speciﬁc solutions originally predicted in the earlier analysis of Smith and Davis (in particular, the so-called ‘longitudinal stationary rolls’, which these authors obtained as a possible problem solution in conditions for which the basic horizontal ﬂow can be approximated as a linear velocity proﬁle). Anyhow, further details on the nature of these ﬂows can be found in the linear stability analysis of Nepomnyashchy et al . (2001), which was carried out for a water–air system (two layers superposed) of inﬁnite lateral extent subjected to the joint action of vertical and horizontal temperature gradients (see Figure 9.19). The main predictions of this linear stability analysis can be summarized as follows on the basis of the parameter b deﬁned as b=

Mav Mah

(9.3)

For relatively small values of this parameter (0 < b < bF ) and large value of Mah , the excitation of inclined hydrothermal waves is expected (line 3 in Figure 9.19). These waves move in the opposite direction to the ﬂow at interface. For relatively large values of b (b > bD ) and small values of Mah , the theory predicts the appearance of stationary (longitudinal) convective rolls due to the same mechanisms as the classical Pearson’s instability (lines 1 and 2 in Figure 9.19). This mode of convection is due to the same mechanism at the root of the hexagonal cells typical of standard Marangoni–B´enard convection, but it arises in the form of rolls due to interaction with the shear ﬂow induced by the horizontal temperature gradient. The axes of these rolls are ordered by the thermocapillary ﬂow along the direction of the horizontal temperature gradient. In this regard, the mechanism exhibits some notable qualitative analogies to the case of buoyancy convection in an inclined system (ILC) for which, as illustrated in Section 7.1, the patternless base state is characterized not only by a linear temperature gradient but also by a symmetry-breaking shear ﬂow, and the outcome is represented typically by steady longitudinal rolls (LRs) with their axes aligned with the shear ﬂow.

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Figure 9.19 Stability diagram of surface tension-driven ﬂow in a water–air system (Pr = 6.96) with inclined temperature gradient, expressed by the relation between the vertical Marangoni number Mav and the horizontal Marangoni number Mah . Line H–F marks the transition to oblique hydrothermal waves, line F–E that to transverse travelling rolls and lines 1 and 2 that to stationary longitudinal rolls (Mav = σT Tv d/µα, Mah = σT γ d 2 /µα , where Tv is the vertical component of the applied temperature gradient and γ is the rate of temperature increase along the horizontal direction). After Nepomnyashchy et al. (2001); Reproduced by permission of Cambridge University Press

In the present case of surface tension-driven phenomena, in practice, these theoretically-predicted longitudinal rolls would correspond to the spiral (streak) ﬂows observed by Ueno et al . (2002). According to the linear stability analysis of Nepomnyashchy et al . (2001), ﬁnally, for intermediate values of Mah , these rolls should become perpendicular to the imposed temperature gradient and drift with the thermocapillary ﬂow (unlike the hydrothermal waves, however, the drifting rolls should move in the same direction as the ﬂow at the interface). This additional regime, although predicted by linear stability analysis (Figure 9.19) and also reproduced in nonlinear numerical simulations, has not been observed yet in experiments. More recently, other studies have appeared in which the problem has been reconsidered for geometries with aspect ratio (height/diameter) A = O(1). These studies have been motivated expressly by potential applications in the ﬁeld of fusion welding technologies and for this reason were not limited to the case Pr > O(1). Most of them have resorted to models in which both the bottom and the (circular) lateral walls of the domain are assumed to be at a constant uniform temperature (the material melting point), while thermal input to the system is provided through a ﬁxed imposed axisymmetric heat ﬂux at the free surface (with a maximum Qmax at the centre and the minimum Qmin at the periphery), which obviously leads to temperature gradients acting along both directions parallel and perpendicular to the melt/gas interface. As an example, Schoisswohl and Kuhlmann (2007) investigated sufﬁcient conditions for an initial axisymmetric basic steady ﬂow (domain with A = 0.5) to become unstable to a nonaxisymmetric state and the possible underlying physical mechanisms driving the instability in a wide range of the Prandtl number (10−10 ≤ Pr ≤ 8; Figure 9.20).

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Figure 9.20 Neutral curves for surface tension-driven convection in a cylindrical domain with a ﬁxed axisymmetric heat ﬂux at the free surface (A = 0.5; lateral and bottom walls at constant temperature; Re deﬁned as Re = σT Qmax L/ρλν 2 , where L is the height of the cylinder, Qmax the heat ﬂux at the centre of the free surface and λ the thermal conductivity). Courtesy of H. Kuhlmann

Figure 9.21 Perturbation ﬂow (arrows, interpolated from the numerical data) and temperature ﬁeld (lines) on the free surface for A = 0.5 and Pr = 0.02 (m = 3 and Recr = 3.47 × 104 ; Re = Ma/Pr with Re deﬁned as in the caption of Figure 9.20; negative values are indicated by grey lines). Courtesy of H. Kuhlmann

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According to their ﬁndings, two different types of instability act according to the Prandtl number (which deserve some additional attention): a stationary instability for Pr ≤ 1 with a critical ﬂow three-dimensional and steady, and a time-dependent instability for Pr ≥ 2 leading to three-dimensional waves of the hydrothermal type like those originally predicted in the aforementioned theoretical study by Smith and Davis (1983). Concerning the related mechanisms, in particular, for low Prandtl numbers (e.g. liquid metals), the temperature ﬁeld tends to be nearly conducting and practically decouples from the stability problem (the basic temperature ﬁeld simply serves to drive a toroidal ring vortex, whereas the latter becomes unstable due to centrifugal effects arising in the region of curved streamlines originating from the cold corner). The critical mode creates surface temperature perturbations that result in Marangoni stresses counteracting the ﬂow that produces surface-temperature perturbations (Figure 9.21). In contrast, for Pr ≥ 2 (the instability is oscillatory), the perturbation ﬂow extracts thermal energy from the basic temperature ﬁeld leading to strong temperature extrema in the bulk. By conduction, the strong bulk extrema create weak temperature perturbations on the free surface that drive a thermocapillary perturbation ﬂow. In turn, the corresponding return ﬂow reinforces the bulk ﬂow which creates the internal extrema. As a concluding remark, it is worth mentioning that these instability mechanisms for low and high Prandtl numbers display notable similarity to those which are known to be operative in liquid bridges (see Chapter 10 for an exhaustive discussion).

10 Thermocapillary Convection The genesis of Marangoni convection [named after the Italian physicist Carlo Giuseppe Matteo Marangoni (Marangoni, 1871)], has already been discussed to a certain extent in Chapter 2 (the reader is referred, in particular, to Sections 2.2.3 and 2.3). As also highlighted in Chapter 3, this kind of convection (driven by differentially heated free surfaces) has very important effects during the growth of semiconductor crystals carried out by several techniques; for instance the Czochralski (see, e.g. Hurle and Cockayne, 1994; Nakamura et al., 1999; Azami et al., 2001a), the ﬂoating-zone and the open-boat (Hurle, 1994) methods. An undesired dopant distribution, in fact, has been reported in many circumstances. The dopant is often found concentrated in regular banded structures called striations, which are thought to be caused by oscillatory convection (see Section 3.1.2). For these reasons, much attention has been devoted to ﬂows in open rectangular cavities, liquid bridges and annular conﬁgurations, used, respectively, as models of the open-boat horizontal melt growth, of the ﬂoating-zone (FZ) method and of the Czochralski (CZ) technique. Apart from possible technological implications, it is clear that these phenomena have stimulated the interest of the scientiﬁc community also by virtue of the profound differences that they exhibit with respect to classical thermogravitational ﬂows induced by buoyancy forces. Indeed, such fundamental differences make Marangoni ﬂow driven by differentially heated free surfaces an important and independent branch of research in the realm of thermal convection. A distinction must be also invoked with respect to the somewhat related subject of Marangoni–B´enard (MB) convection (Chapter 9); even though the driving force is basically the same, the rotation of the direction of the applied temperature gradient by 90◦ induces signiﬁcant changes in the dynamics. As opposed to the MB problem, the quiescent state for these cases is not a solution; consequently, ﬂuid motion is present without the need to overcome a threshold value of any parameter. Remarkably, the secondary, tertiary and higher order modes of convection also depend signiﬁcantly on the situation considered. In typical terrestrial laboratory experiments, as explained in Section 2.3.2, this ﬂow can be emphasized with respect to thermogravitational convection by reducing the typical scale of the liquid volume. In fact, the ratio of Rayleigh number to Marangoni number (which measures the driving actions of the gravity and of the surface tension imbalance) increases quadratically with the linear dimension of the liquid zone [Eq. (2.26)], making thermal buoyancy convection less important for small liquid volumes. If this ratio is kept small by reducing the characteristic system length to few millimetres, the Marangoni effect can be effectively emphasized in comparison with the buoyancy effect. This approach provides a relevant alternative to the use of expensive Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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microgravity or other expensive means (e.g. virtual microgravity that can be obtained on Earth using intense gradients of magnetic ﬁelds, as will be explained in Chapter 13); from a historical point of view it has been largely used over the last three decades especially by scientists not having the opportunity to carry out research in true microgravity conditions, that is, on orbiting platforms. This chapter, however, expressly concentrates on pure Marangoni ﬂow (Ra = 0), that is, on true microgravity conditions. Some landmark experiments carried out in the weightlessness of space are reviewed. Given the limited number of such experiments, the discussion also proceeds with the fruitful support of theoretical arguments introduced by investigators over the years, available numerical results and critical comparison with other categories of phenomena (especially those treated in Chapter 6). On the one hand, such a philosophy, namely the explicit choice to omit here the discussion of ground-based experiments (microscale analyses and experiments with layers having a depth of few millimetres) and related theoretical studies, is justiﬁed by the subtle effect that gravity can exert in connection with Marangoni effects on the resulting system dynamics. This is specially true for high-order modes of convection and the related hierarchy of bifurcations, which make it worth considering these cases separately (we shall come back to them in Chapter 11). On the other hand, the critical, focused and ‘comparative’ study of pure Marangoni convection with respect to ﬂows of gravitational nature is in line with the general spirit of the present book, which is aimed from an ‘ideological’ synergetic point of view to making further progress in the understanding of pattern-forming systems of different nature (thermocapillary and thermogravitational ﬂows have so many differences that the study of the latter has often enriched our understanding of the former; experience has also shown that the recognition of the underlying dynamic concepts applicable to both is an excellent starting point for the study of either). Prior to expanding on such a theoretical framework for relatively complex geometries, the ﬁrst two sections of this chapter are devoted to a somewhat introductory historical perspective for the open-boat problem (HB), namely an upper surface tension-driven ﬂow, which implies Marangoni convection in a rectangular cavity with an insulated bottom wall and differentially heated sidewalls (see Sections 6.1 and 6.2 for the analogous problem of thermogravitational origin); then other classical models relating to techniques for the growth of crystal from the melts are considered, for example the annular conﬁguration (relevant to the CZ technique) and the liquid bridge (relevant to the FZ process).

10.1

Basic Features of Steady Marangoni Convection

Let us start the discussion by focusing on some basic features of this kind of ﬂow and accompanying important propaedeutical concepts. It is well known that for small temperature differences T , the ﬂow in shallow, elongated, open cavities is steady and simply unicellular (e.g. Yih, 1968). For two-dimensional models, it appears as a unique large roll , whose axis is perpendicular to the temperature gradient (e.g. Strani et al., 1983) and whose position changes according to the Prandtl number (at low Pr a recirculation roll develops near the cold wall, whereas at higher Pr the roll develops near the hot wall; see, e.g., Mercier and Normand, 2002, and references therein). It is also a well-established fact (Ben Hadid and Roux, 1990) that for low values of the control parameter, this ﬂuid conﬁguration reaches a plane-parallel ﬂow state in the central region (core ﬂow) of the cavity, with the exception of an upwind region (close to the hot sidewall) in which the ﬂow is accelerated and a downwind region (close to the cold sidewall) in which the ﬂow is decelerated (see Figure 10.1). Such core ﬂow admits an analytical expression (Section 2.4.2); in practice, it is an exact solution [Eqs (2.36)–(2.37)] of the thermal-convection equations generally referred to as the ‘return ﬂow’,

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Core flow (a)

Core flow Roll

Roll

(b)

(c)

Figure 10.1 Structure of pure Marangoni convection [plane (x, y)] in a two-dimensional open cavity with aspect ratio (ratio of the free surface extension to the liquid depth) A = 10 (Pr = 0.01; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions; free surface corresponding to the upper boundary) and three values of the Marangoni number: (a) Ma = 10; (b) Ma = 102 ; (c) Ma = 103 (Ma based on the extension of the free surface) (numerical simulation, M. Lappa)

derived ideally for a layer of inﬁnite extent with adiabatic or conducting horizontal boundaries and displaying a single component of velocity along the direction of the imposed temperature gradient. We shall come back to the properties of this solution later in this chapter as it has enjoyed widespread use in the literature as a paradigm model for establishing (in general) a theoretical foundation to the ﬁeld and (in particular) for explaining some of the typical manifestations of Marangoni ﬂows in practical situations. Like the Hadley ﬂow of gravitational nature, which was the main theme of Chapter 6, such a solution also allows us, in fact, to gain outstanding insights into the general stability behaviour of surface tension-driven ﬂows (along these lines, let us recall that the study of Marangoni ﬂow historically progressed by consideration of a study sequence within a hierarchy of increasingly complex models where each stage was built on the intuition developed by the precise analysis of simpler models). Like thermogravitational convection, as illustrated in Figure 10.1, in practical situations (i.e. for cavities of ﬁnite size) the region over which the ﬂow is parallel shrinks in length, making edge effects incrementally more important as the control parameter (Ma) is increased. In contradistinction to convection induced by buoyancy [for which a parallel ﬂow can be established even in the limit as the aspect ratio A = length/depth tends to zero (see Section 6.3), the reader is referred, in particular, to Eq. (2.61), also known as the ‘conduction-regime solution’], however, Marangoni ﬂow does not admit core parallel ﬂow in such a circumstance. Rather, for tall, open cavities (A < 1) a remarkable feature of this kind of convection is the emergence of several internal layers of vortices, with the strength of each layer decreasing approximately exponentially with the distance from the surface as A is decreased (see Figure 10.2). Anyhow, as for other types of convection (see, e.g., Section 6.4), one may expect the presence of sidewalls limiting the system also in the spanwise direction (i.e. along the third direction z) to induce a variety of three-dimensional effects. As a relevant example along these lines, Figure 10.3 shows the structure of Marangoni ﬂow in the classical three-dimensional cubical cavity (already considered in earlier chapters as a reference

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(a)

(b)

(c)

(d)

Figure 10.2 Structure of pure Marangoni convection [plane (x, y)] in two-dimensional open cavities (Pr = 0.01; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions; free surface corresponding to the upper boundary) with aspect ratio (length/depth) (a) A = 1; (b) A = 1/2; (c) A = 1/3 and (d) A = 1/4 (Ma = 10 based on the extension of the (1) (2) (3) = 9.3 × 10−2 , ψmax = 1.7 × 10−3 , ψmax = 3 × 10−6 , where the superscript refers to the free surface; ψmax considered vortex starting from the top of the container, that is, progressing from the free surface towards the interior) (numerical simulation, M. Lappa)

conﬁguration for gravitational convection with heating from below or from the side; see Sections 4.9 and 6.4, respectively). As pointed out by Babu and Korpela (1990), such a geometry may be speciﬁcally regarded as a paradigm case for the illustration of possible 3D boundary effects due to relatively close walls along all the spatial (x, y and z) directions (conversely, boundary effects induced by lateral constraints in the spanwise direction tend to be rather weak when Marangoni ﬂow is considered in shallow open cavities and pools; Lappa, 2005a). Figure 10.3, in particular, refers to the case with Pr = 0.01 and Ma = 100. It displays some very typical features of Marangoni convection in parallelepipedic geometries, hence it deserves some space for a relevant description. Obviously, vortex I represents the canonical unicellular convection discussed earlier. Vortices II and III (Figure 10.3a) are corner vortices, that arise owing to ﬂow separation close to solid (no-slip) boundaries (Moffatt, 1963). Vortex VI and its symmetrically located counterpart (Figure 10.3c) have a surface tension-driven origin. Since the velocity component along x (i.e. u) vanishes at the solid boundaries z = 0 and z = 1, the convective transport in the x direction at the free surface is reduced there with respect to the mid-section (z = 0.5). This results in larger temperature values at the mid-section at ﬁxed locations along x. Accordingly, Marangoni forces directed towards the sidewalls are induced in the spanwise direction, which, in turn, lead to the emergence of such vortices. This explanation holds, in general, for relatively small values of the Marangoni number [Ma ≤ O(102 )]. At large values of the Marangoni number [Ma = O(104 ), not shown], owing to the presence of thermal boundary layers (see Section 2.5 for general related concepts), compression of isotherms occurs close to the cold and hot walls. Such an effect is weakened in proximity of the z boundaries where, as mentioned earlier, no-slip conditions prevail; for this reason, the direction of the z

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Free surface z=0.5 Cold wall

Hot wall I

II III

(a)

Free surface x

z

x=0.5

Free surface

Cold wall

y=0.5 VI Hot wall

V

IV (b)

(c)

Figure 10.3 Differentially heated cubical cavity with adiabatic free surface (Pr = 0.01, Ma = 102 ): primary and secondary vortical structures shown in the three orthogonal coordinate planes. The roman numbers are referred to in the text (numerical simulation, M. Lappa)

component of the temperature gradient from the mid-section to the walls at z = 0 and z = 1 is reversed; accordingly, well-developed secondary vortices are still present, but their sense of rotation is reversed with respect to the case of smaller Ma. Finally, the other vortices IV and V (Figure 10.3b and 10.3c, respectively) in the vicinity of solid boundaries are Ekman vortices (see Sass et al., 1995), that is, (secondary) ﬂows speciﬁcally induced by ﬂuid recirculating in the primary main vortex I and related acceleration effects (Ekman, 1905).

10.2

Stationary Multicellular Flow and Hydrothermal Waves

In general, when the temperature gradient acting along the free surface reaches a critical value, various types of instabilities can be initiated in both two- and three-dimensional geometries. Along these lines, it is worth starting the discussion by mentioning the milestone linear stability analysis of Smith and Davis (1983), who considered a canonical ﬂuid layer with inﬁnite extent, exhibiting a free surface supporting Marangoni stresses, without gravity or heat exchange to the ambient (both top and bottom boundaries with adiabatic thermal conditions) and found two types of thermal instabilities in the case of an undeformable interface: stationary multicellular ﬂow (SMC) and oblique hydrothermal waves (HTW).

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The former takes the form of stationary longitudinal rolls (this steady multicellular state is made up of a series of steady corotating rolls with axis parallel to the applied temperature gradient) that become unstable in much the same way as the classical Marangoni layer heated from below and discussed in Chapter 9. The latter thermal instability takes the form of propagating waves.

10.2.1 Basic Velocity Proﬁles: The Linear and Return Flows The occurrence of SMC or HTW modes of convection depends essentially on the Prandtl number and on the basic ﬂow considered. Two parallel-ﬂow solutions were assumed, in fact, in the study by Smith and Davis (1983) as possible basic states reproducing the core region of the initial unicellular ﬂow, the ﬁrst referred to as ‘linear-ﬂow solution’ (Figure 10.4a) and the second corresponding to the aforementioned return-ﬂow solution shown in Figure 10.4b, whose general properties have been already illustrated in Section 2.4 (it is featured by a simple parabolic polynomial expression for the horizontal velocity proﬁle).

10.2.2 Linear Stability Analysis For application of the typical linear stability analysis protocols (see Section 1.5.3 for the necessary theoretical background and propaedeutical concepts), Smith and Davis (1983) assumed a basic state of the form   u0 (y) (10.1) V 0 = Ma  0  0 T0 = x + Maf (y)

(10.2)

(a)

(b)

Figure 10.4 Basic-state velocity proﬁles for an inﬁnite horizontal liquid layer bounded from below by a rigid wall: (a) linear ﬂow solution; (b) return ﬂow solution

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u0 (y)and f (y) being, as explained above, given polynomial expressions with constant coefﬁcients [see Section 2.4.2, Eqs (2.36) and (2.37a)]. Moreover, they considered normal modes of the form {δu, δv, δw, δp, δT } = {ud (y), vd (y), wd (y), pd (y), Td (y)} exp[i(qx x + qz z) + λt]

(10.3)

where qx and qz are the disturbance wavenumber in the x and z directions, respectively. In this form, the disturbance is a wave travelling in a direction qz (10.4) = tan−1 qx with respect to the positive x-axis and with an overall wavenumber ) q = qx2 + qz2

(10.5)

The resulting system of ordinary differential equations for the amplitudes ud , vd , wd , pd and Td is similar to Eqs (6.7)–(6.9) already derived in Chapter 6 for the Hadley ﬂow: Continuity equation: i(qx ud + qz wd ) + vd = 0

(10.6)

Pr(ud − q 2 ud ) − Ma(iqx u0 ud + u0 vd ) − iqx pd = λud Pr(vd − q 2 vd ) − Ma(iqx u0 vd ) − pd = λvd Pr(wd − q 2 wd ) − Ma(iqx u0 wd ) − iqz pd = λwd

(10.7a) (10.7b) (10.7c)

Momentum:

Energy: Td − q 2 Td − (iqx Ma u0 Td + ud + vd Maf ) = λTd

(10.8)

where the primes denote differentiation with respect to y. The related boundary conditions for velocity, however, are different with respect to those for the Hadley ﬂow due to the presence of surface stress; they read ud = 0 vd = 0 wd = 0

(10.9a) (10.9b) (10.9c)

ud = −Maiqx Td vd = 0 wd = −Maiqz Td

(10.10a) (10.10b) (10.10c)

for y = − 1/2 and

for y = 1/2, where Ma = σT γ d 2 /µα, with γ being the uniform rate of temperature increase along the x-axis and d the layer depth. These conditions must be supplemented with the thermal conditions, which for adiabatic horizontal boundaries (the case considered by Smith and Davis, 1983) simply read Td = 0 for y = ± 1/2. The results of the linear stability analysis (as they were obtained by these authors) are discussed in detail in the following. As usual, much space is devoted to explaining the underlying physical mechanisms, to providing critical and fruitful comparison with other types of phenomena and to placing the problem into a more general theoretical context. Prior to embarking on such a task, let us recall that some fundamental information and insights into the treatment of the stability properties of plane-parallel ﬂows have already been given in

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Section 1.5.4 for the limiting case Pr → 0, mentioning Squire’s theorem (about the two-dimensional nature of disturbances) and providing a general necessary condition for instability [Eq. (1.113)]. Remarkably, for the present case, given the absence of inﬂectional points in the basic solutions shown in Figures 10.4 (i.e. a point where ∂ 2 u/∂y 2 = 0), the instability problem for both the linear and return ﬂows in the limit as Pr → 0 (i.e. for Pr 1) does not admit solutions corresponding to two-dimensional waves, that is, no disturbances propagating along x are allowed.

10.2.2.1 The Linear Flow In agreement with the above statement, Smith and Davis (1983) found the preferred disturbance to be three-dimensional for Pr ∼ = 0. For the linear ﬂow, in fact, hydrothermal waves ‘oblique’, i.e. exhibiting an angle of propagation [see Eq. (10.4)] relative to the basic state (i.e. = 0◦ ) were shown to be the preferred mode of instability for Pr < 0.6, this mode being replaced by two-dimensional HTW ( ∼ = 0◦ ) for 0.6 < Pr < 1.6 and stationary longitudinal rolls ( = 90◦ ) for Pr > 1.6 (Figure 10.5). The properties of the hydrothermal waves will be discussed later (with regard, in particular, to the return-ﬂow solution for which they represent the preferred mode of convection over the whole range of Prandtl numbers 0 ≤ Pr ≤ ∞). Here attention is concentrated on the genesis of the longitudinal rolls that, as mentioned earlier, become unstable in much the same way as the classical Marangoni–B´enard layer. The basic-state temperature ﬁeld induced by the linear ﬂow contains, in fact, a ﬂow-induced vertical distribution corresponding to a layer heated from below; it is such a feature that can trigger a stationary convective instability, such as that described by Pearson (1958). In the classical case of a layer heated from below (Chapter 9), the mechanism for the instability is the following. If a local hot spot appears on the interface due to some random temperature perturbations, by continuity, a vertical velocity ﬁeld is established that convects hotter ﬂuid from

Figure 10.5 Critical Marangoni number as a function of the Prandtl number for the inﬁnite layer with linear ﬂow and adiabatic free surface (Marangoni number deﬁned as Ma = σT γ d 2 /µα , where γ is the uniform rate of temperature increase along the x -axis). After Smith and Davis (1983); Reproduced by permission of Cambridge University Press

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the interior of the layer to the hot spot on the interface. If the temperature gradient across the layer is sufﬁciently large, the transport of heat by convection will be sufﬁcient to balance or exceed the losses due to heat diffusion away from the interface. Hence the elevated temperature of the hot spot will be maintained or increased and the convection will continue. When Marangoni ﬂow induced by a free surface differentially heated is considered, however, the basic state will not be quiescent, but will display a shear ﬂow (a linear velocity proﬁle for the considered case); this means that the ﬂuid sucked towards the interface due to continuity will interact with such underlying shear ﬂow. In practice, in such a case the energy driving the instability comes from the vertical temperature gradient induced by a balance between horizontal convection and vertical conduction, as explained in detail below. Towards the aim of clarifying such a complex interplay, let us focus for simplicity on an initial disturbance having the shape of a hot ﬂuid line parallel to the x-axis (i.e. the direction along which the basic temperature gradient is applied) on the free surface of the layer. For the linear ﬂow, such a hot disturbance will produce a vertical velocity ﬁeld just as before, convecting warmer ﬂuid from the interior of the layer to the interface. As each particle of ﬂuid raises towards the hot line, it will move into a region where the basic-state velocity is higher than where it came from. In turn, this will produce a horizontal velocity perturbation under the line reducing the convective heating of the interface that occurs as a results of the horizontal basic-state temperature and velocity ﬁelds. If the temperature gradient in the x direction is large enough, the net convective heating of the interface by these (counteracting in terms of induced thermal effect) horizontal and vertical velocity perturbations will be sufﬁcient to balance or exceed the losses due to thermal diffusion away from the interface; hence the elevated temperature of the hot line will be maintained or increased and the perturbation convection will continue. These simple arguments provide the required explanation for the balance between the horizontal and vertical mechanisms, which, as mentioned above, is at the basis of the instability. Interestingly, among other things, they can be also used to justify the notable increase in the critical Marangoni number for decreasing Pr evident in Figure 10.5. Smith and Davis (1983), in fact, revealed that the magnitude of the horizontal velocity disturbances (induced, as illustrated earlier, by coupling between the vertical disturbance velocity and the horizontal basic-state velocity), which oppose the surface basic velocity convectively heating the initial hot spot, is proportional to Pr−1 ; this means that when Pr → ∞ it tends to zero, but for a ﬁnite relatively small Pr it acts to reduce the convective heating of the initial hot spot, hence, opposing the mechanisms that would maintain or amplify it; this provides a convincing explanation for the trend of Macr .

10.2.2.2 The Return Flow The return ﬂow in the layer does not exhibit stationary convective instabilities. Notably, its basic-state temperature ﬁeld contains, in fact, a ﬂow-induced vertical temperature distribution corresponding to a layer being cooled from below. If a temperature disturbance in the form of a hot line developed on the interface, the resulting vertical velocity ﬁeld would convect cooler ﬂuid from the interior of the layer to the surface; thereby, as illustrated by Smith and Davis (1983), the convection ﬁeld would tend to suppress the disturbance (this is the reason why unstable longitudinal rolls cannot form). For this case, the so-called hydrothermal waves always correspond to the preferred mode of instability. This oscillatory mechanism was revealed for the ﬁrst time by Smith and Davis (1983) (indeed, they were the ﬁrst to coin the related term hydrothermal waves). They found these waves to be oblique over the whole range of Prandtl number, that is, to have components along both the x- and z-axes, (in other words, to exhibit an angle of propagation relative to the basic state).

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Figure 10.6 Angle of propagation of the hydrothermal wave with respect to the direction of the imposed temperature gradient as a function of the Prandtl number for adiabatic free surface and adiabatic (solid line) or conducting (dashed line) bottom wall. After Priede and Gerbeth (1997a); Reproduced by permission of the American Institute of Physics

Such an angle of propagation [Eq. (10.4)] depends on Pr, as illustrated in Figure 10.6: it is nearly perpendicular to the basic state for low-Pr materials, that is, the disturbance propagates almost exactly in the spanwise direction z (the wave has a longitudinal wavefront in such a case and for this reason it is generally referred to as a longitudinal wave or disturbance), and nearly parallel to the surface ﬂow for high-Pr materials, that is, the disturbance propagates almost exactly in the upstream direction (the wavefront in such a case being transverse). A remarkable feature common to all cases is that, since the angle is always less than 90◦ , the disturbance always travels in a direction with a component in the direction opposite to that of the surface ﬂow . At Pr = ∞, in particular, the direction of propagation of the disturbances with respect to the imposed temperature gradient is only 7.90◦ , which means a two-dimensional wave (Figure 10.7c) could be regarded as a good approximation of the real phenomenon. The various possible situations are sketched in Figure 10.7 and the related neutral curves are 1 displayed in Figure 10.8. The critical Marangoni number Macr ∝ Pr /2 as Pr → 0, and when Pr → ∞, Macr approaches its largest possible value (∼400 based on the depth of the layer). It is also worth noting that Macr (Figure 10.8a) is always slightly larger than that for the linear ﬂow (Figure 10.5) and the critical wavenumber (not shown) is always slightly less for small values of Pr. Moreover, for large Pr, Macr is an order of magnitude greater than Macr for longitudinal rolls. Smith and Davis (1983) also elaborated important information about the related physical mechanisms; in particular, the waves were found to derive their energy from the imposed horizontal temperature gradient through horizontal convection when the Prandtl number of the liquid is small and from the vertical temperature gradient through vertical convection when it is large. Important extensions to the original study of Smith and Davis (1983) were provided by Priede and Gerbeth (1997a), who analysed the effect of various thermal boundary conditions on the linear stability of the return-ﬂow solution. They considered, in fact, the alternative cases in which the adiabatic bottom is replaced by a conducting boundary (which implies Td = 0 for y = 1/2 and Td = 0 for y = − 1/2) or the free surface is allowed to exchange heat with the external ambient

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(a)

(b)

(c) Figure 10.7 Sketch of travelling hydrothermal waves at the free surface: (a) disturbance propagating almost exactly in the spanwise direction [Pr < O (1)]; (b) disturbance propagating with two distinct components along x and z [Pr ∼ = O (1)]; (c) disturbance propagating nearly parallel to the basic state in the upstream direction [Pr > O (1)]

(Td = −BiTd for y = 1/2, where Bi is the Biot number) with the bottom wall retaining an adiabatic behaviour (Td = 0 for y = − 1/2). Interestingly, in addition to the usual linear stability technique, these authors devised an original approach to estimate the critical instability parameters based on an order of magnitude analysis, which allowed them to give useful insights into particular details of the instability mechanism as well as to yield simple analytic expressions (scalings) for the instability parameters.

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(a)

(b) Figure 10.8 Critical Marangoni number as a function of the Prandtl number for the inﬁnite layer with return ﬂow: (a) layer with both adiabatic boundaries; (b) layer with adiabatic free surface and conducting bottom wall (Marangoni number deﬁned as Ma = σT γ d 2 /µα , where γ is the uniform rate of temperature increase along the x -axis). After Priede and Gerbeth (1997c); Reproduced by permission of Cambridge University Press

The order of magnitude estimates, in particular, were used to solve the problem asymptotically for the case of small Prandtl numbers relevant to technological processes for the growth of semiconductor crystals from the melt. Several important instability characteristics were deduced directly from such order of magnitude considerations and the linearized equations Eqs (10.6)–(10.10) (in practice, from alternative forms

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of such equations obtained by eliminating the pressure). Anyhow, the order of magnitude analysis was speciﬁcally based on some fundamental principles, whose genesis, prior to expanding on the related results, must be (mandatorily) discussed in the following for the convenience of the reader. First, it was logically assumed that both the marginal Marangoni number and the corresponding frequency are deﬁned by a balance condition of viscous and thermocapillary stresses at the free surface, which has to be evaluated from the governing equations. Some additional considerations were elaborated for estimating relevant the length scales for characteristic velocity variations and for the evolution of the temperature disturbance. The length scale of variation of the characteristic velocity was assumed to depend on the frequency of the oscillatory disturbance. According to the their arguments, in particular, if the frequency is so low that the perturbation of shear stress can spread over the whole depth of the layer during the period of oscillation, the characteristic length scale has to be given simply by the thickness of the layer d. In the opposite case, when the oscillation period is considerably shorter than the characteristic viscous diffusion time over the depth of the layer (which is given by τν ∝ Pr−1 in dimensionless form assuming as reference time d 2 /α), the shear stress cannot penetrate throughout the layer during such a short time; hence the characteristic length scale has to be given by the thickness of oscillatory boundary layer (referred to here as a viscous skin layer) that is expected to form at the 1 free surface. This thickness was estimated as δ ∝ ω− /2 , where ω is the oscillation frequency. As anticipated, some ideas were also introduced for estimating a length scale for the evolution of the temperature disturbance. Since for both adiabatic walls there is no reference temperature ﬁxed at the boundaries, it was assumed that in such a case the disturbance wavelength rather than the depth of the layer represents a length scale relevant for the evolution of the temperature perturbation (note in this regard that although, in general, the temperature of long-wave perturbations relaxes within a characteristic thermal diffusion time τα ∝ 1 to its mean value over the depth of the layer, a further relaxation to the thermodynamic equilibrium is determined by the thermal diffusion over the wavelength that occurs within a characteristic time τα ∝ q −2 ). In contrast, for a perfectly conducting bottom having a ﬁxed temperature, the corresponding perturbation was assumed to relax to the equilibrium state already within the characteristic time τα independent of the wavelength. As expected, using these interesting concepts general scalings of the critical parameters were obtained for Pr < O(1). Notably, both such scalings and the asymptotic solutions obtained on their basis were veriﬁed with the results of the linear stability analysis. It was clearly shown that at low Prandtl numbers, the instability threshold is signiﬁcantly affected by the thermal boundary conditions considered (compare Figure 10.8a and b). The critical Marangoni number for conducting bottom wall is not only considerably higher than that in the adiabatic case, but it also increases (for the longitudinal mode) with a decrease in the Prandtl number. Interesting explanations for these differences were elaborated (in the framework of the aforementioned order of magnitude analysis and related concepts) as follows. For both boundaries adiabatic, the critical wavelength is much longer than the depth of the liquid layer. In this case, the minimum (critical value) of the marginal Reynolds number is attained when the thermal diffusion time over a wavelength becomes comparable to that of viscous diffusion over the depth of 1 the layer. This corresponds to the critical wavenumber scaling as qc ∝ Pr /2 . Since, furthermore, 1 1 Re ∝ q −1 , this leads to Re ∝ Pr− /2 and Ma ∝ Pr /2 , in good agreement with the ﬁndings of Smith and Davis (1983). In the presence of a nonadiabatic boundary, the most unstable mode occurs at the wavelength at which the effect of the heat diffusion across the wave becomes comparable to that over the depth of the layer. Thereby, the critical wavelength for the conducting bottom is comparable to the depth of the layer.

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Interestingly, Priede and Gerbeth (1997a) showed that in the latter case, unlike the conﬁguration with both boundaries adiabatic (for which, as highlighted by Smith and Davis, the mode with a longitudinal wavefront ( ∼ = 90◦ ) presents for low Prandtl numbers a fairly good approximation of the most dangerous oblique wave), the estimates obtained for the purely longitudinal disturbances are no longer applicable to the most unstable oblique waves. They ascribed this behaviour to the fact that in contrast to longitudinal disturbances, for oblique waves there is a characteristic length scale along the basic ﬂow. Therefore, if an oblique wave is advected by the basic ﬂow, then the oscillation frequency has to be determined by both the characteristic velocity of the advection and this length scale. As in the adiabatic case, with a decrease in Pr, the most dangerous oblique wave becomes nearly longitudinal. Nevertheless, the critical angle is such that the effect of advection renders scalings for both instability modes substantially different. This means that even though also for a conducting bottom and Pr 1, the direction of propagation of the disturbance is almost spanwise to the basic ﬂow (as in the case with adiabatic boundaries; see both solid and dashed lines in Figure 10.6 for Pr → 0); nevertheless, the approximation of purely longitudinal disturbances cannot be used for obtaining reliable estimates of the critical parameters. In such a context, it should be pointed out that, in general, estimates of the longitudinal mode remain valid for the most unstable oblique waves, provided that the boundaries are adiabatic. In this case, the asymptotic solution of Priede and Gerbeth (1997a) yields the wavevector of the most unstable wave directed at an angle = 77.8◦ with respect to the basic ﬂow (Figure 10.6). The 1 1 corresponding critical Marangoni number and wavenumber are Macr = 61.9Pr /2 and qc = 2.48Pr /2 , respectively. As explained earlier, however, under nonadiabatic boundary conditions the effect of advection of velocity disturbances by the basic ﬂow is so strong that it signiﬁcantly interferes with the mechanism of the longitudinal waves even at small deviations of the propagation direction from the spanwise direction (for the conducting bottom this effect becomes signiﬁcant at a propagation 1 angle scaling as − π/2 ∝ Pr /2 ). Additionally, this scaling determines also the direction of the most unstable oblique wave. For the case of the conducting bottom, the asymptotic solution shows that with a decrease in the Prandtl number the critical angle of propagation approaches that for the spanwise direction as 1 = 90◦ −82.3Pr /2 , whereas the critical wavenumber tends to a constant value qc = 1.61. The 1 corresponding critical Marangoni number is Macr = 229.5Pr /2 . For the other alternative case in which the bottom is adiabatic and the free surface is nonadiabatic, Priede and Gerbeth (1997a) found the critical wavenumber to depend on the Biot number as 1 qc ∝ Bi /2 , provided Pr < Bi < 1. It was shown that at small Prandtl numbers even a weak thermal coupling between the free surface and the ambient medium such that Bi ∼ = Pr can signiﬁcantly inﬂuence the instability threshold.

10.2.3 Weakly Nonlinear Analysis Additional interesting insights into the mechanisms at the basis of these phenomena were also provided by the weakly nonlinear analysis of Smith (1986), who, assuming again an adiabatic bottom wall and free surface, elucidated the process by which surface temperature disturbances can be ampliﬁed or damped according to the Prandtl number and their initial spatial ‘structure’. Some interesting information along these lines is given in the following (according to the same approach undertaken in other chapters of this book, in particular, ﬁrst the low-Pr case is considered, then the discussion is extended to ﬂuids with large Prandtl number).

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The assumed initial conditions correspond to the presence of a hot spot in the form of a line parallel to the x-axis (Figure 10.9a). As shown in Figure 10.9a, at the beginning such a spot induces an upward vertical ﬂow. Such an upﬂow can be regarded as the simple natural consequence of the combined effect of thermocapillarity (which brings interfacial ﬂuid from hot regions towards cold regions) and continuity. Since such an upﬂow, in turn, brings ﬂuid with a lower basic-state velocity towards the surface, it is responsible for the onset of an inertial force in the upstream direction (Figure 10.9b) that drives an upstream velocity perturbation. The upstream ﬂow induced by the aforementioned inertial force cools the initial hot spot because it brings cooler ﬂuid from downstream towards the hot spot (recall that the imposed surface temperature decreases downstream). Such a cooling reduces the temperature of the spot and, as a consequence, the intensity of the upﬂow. However, as long as there is an upﬂow, there is an upstream inertial force and so the upstream velocity perturbation increases. At a certain stage (see Figure 10.9c), the cooling exerted by this ﬂow suppresses the hot spot. When this occurs, the cooling upstream ﬂow (that is still present) causes the temperature of the spot to overshoot (i.e. the disturbance becomes a cold spot). This cold spot causes a surface ﬂow towards it as a result of thermocapillarity and a downﬂow beneath it due to continuity (Figure 10.9d). Such a downﬂow produces an inertial force in the downstream direction, which acts to mitigate the upstream ﬂow and, hence, reduce its cooling inﬂuence. As a consequence, the heating effect from the surrounding warm areas to the cold spot becomes dominant and the spot starts to increase in temperature. Just after the minimum temperature has been attained, the downstream inertial force reverses the upstream ﬂow. This new downstream ﬂow also heats the spot and so the net heating effect on the spot increases. When the temperature again becomes zero, the downﬂow is very small whereas the downstream ﬂow is large. Its heating effect causes the spot’s temperature to overshoot the zero point (it becomes a hot spot again and a new cycle can start). According to such a description, it is evident that the key to the mechanism is the inertially driven streamwise ﬂow. Inertial effects produce the proper phasing between this ﬂow and the vertical ﬂow so that, as outlined earlier (Section 10.2.2), it can extract energy from the horizontal basic-state temperature ﬁeld. Notably, in a different perspective, the stages of evolution illustrated in Figure 10.9 can also be used to explain why, as originally found by Smith and Davis (1983), the instability always involve two longitudinal hydrothermal waves that propagate in either normal direction to the basic ﬂow . Given the symmetry of the system with respect to an initial disturbance in the form of a line parallel to the x-axis, in fact (as shown in particular in Figure 10.9a and 10.9d), the original disturbance will generate disturbances with longitudinal wavefronts travelling spanwise along both senses (+z and −z). This feature is retained when larger values of the Prandtl number are considered [Pr ≥ O(1)]. As originally found by Smith and Davis (1983), in fact, for all values of Pr, the preferred mode of instability is always a couple of hydrothermal waves propagating obliquely at an angle ± with respect to the positive x-axis. In the case of large Pr, however, the ﬂow ﬁeld in the layer is dominated by viscous effects and the mechanisms responsible for the emergence of hydrothermal waves radically change. If a temperature perturbation in the form of a hot line parallel to the x-axis is considered, thermocapillarity and continuity again set up a cooling upﬂow directly beneath the spot. However, there will not be a streamwise ﬂow as in the small-Pr case because inertial effects are negligible. Hence there will be solely a cooling upﬂow (the interior of the layer being colder than the surface) that reduces the spot’s temperature to zero, thereby eliminating the initial disturbance. In practice, the disturbance has to be oriented along the z-axis to start the oscillatory behaviour.

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(a)

(b)

(c)

(d)

Figure 10.9 Schematic of the low-Pr mechanism for the spanwise travelling hydrothermal wave: (a) initial hot spot and upﬂow induced by thermocapillarity and continuity; (b) inertially induced cooling upstream ﬂow; (c) elimination of the hot spot and remaining cooling upstream ﬂow; (d) formation of the cold spot and accompanying downﬂow

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(a)

(b)

(c)

Figure 10.10 Schematic of the high-Pr mechanism for two-dimensional travelling hydrothermal wave

Clearly, also in such a case the temperature of the hot spot decreases as a result of the cooling upﬂow directly underneath. However, the downﬂow created by continuity in upstream and downstream positions with respect to the initial hot spots (see Figure 10.10a), moves surface ﬂuid with a relatively warmer basic-state temperature down to the interior, where the basic-state temperature is colder [this convective heating in the interior is the mechanism outlined earlier (in Section 10.2.2) by which energy can be transferred from the basic-state temperature ﬁeld to the disturbance]. However, hot spots do not form under the surface in both locations where disturbance downﬂow is present. The temperature perturbations induced by the vertical downﬂow, in fact, do interact with the basic-state velocity ﬁeld, which in the interior of the layer is directed from the cold side toward the hot side. For these reasons, as shown in Figure 10.10b, an interior hot spots forms solely to the right of the initial surface hot spot.

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Such an interior hot spot heats the interface of the layer upstream of the original surface spot by diffusion; hence, while the old hot spot decreases in temperature as a result of the cooling upﬂow, a new hot spot appears upstream of it as a result of diffusive transport of heat in the vertical direction. This new spot produces the same recirculating velocity ﬁeld as the old one; thereby, the process depicted above can start again with the initial location shifted slightly upstream with respect to the earlier one (resulting as time passes in the propagation of the initial hot spot in the upstream direction). It is evident that the key to this mechanism is the interior hot spot that drives the instability by conductively heating the interface (the energy for the spot is clearly extracted, as outlined earlier, from the vertical basic-state temperature ﬁeld by vertical convection). Transcending the speciﬁc details of cause-and-effect relationships driving the instability (differing in the more or less important role played by inertial effects and related orientation of the suitable disturbance in the form of hot line along x or z and ensuing direction of propagation of the waves according to the Prandtl number), from the foregoing arguments (and, in particular, Figures 10.9 and 10.10) its is clear that the presence of temperature disturbances on the free surface should be always regarded as an essential ingredient for instability incipience. The most interesting articulation of this observation is that the hydrothermal instability would be suppressed by a free surface behaving as a perfectly conducting boundary. In such a case, in fact, the thermal boundary condition for the disturbance linearized equations at y = 1/2 would read Td = 0 (which obviously excludes the possibility of having temperature disturbances at the free surface). Among other things, this explains why this type of boundary condition was not considered in the study by Priede and Gerbeth (1997a) and other investigations on the subject (see Chapter 11). Conversely, mechanisms such as those described by Smith (1986) can be retained if the free surface is allowed to exchange heat with the ambient as the related thermal boundary conditions at the free surface Td = −BiTd does not imply necessarily Td = 0 (even if, as illustrated by Priede and Gerbeth, 1997a, a strong inﬂuence of the heat exchange rate on the instability threshold and intrinsic mechanisms must be expected for such a case). As an example, Smith (1988) examined to a certain extent the inﬂuence of heat transfer at the free surface together with the Prandtl number in determining the scenario of superposition of the wave components along the positive and negative spanwise directions and related stability to amplitude perturbations (he studied the nonlinear behaviour of such a superposition to determine possible equilibrium waveforms for the instability when the critical point from the linear theory is slightly exceeded). At this stage, let us recall that the linear stability theory cannot provide any information about the effective amplitude of disturbances. As in the Marangoni–B´enard problem, the linear theory problem is degenerate (in that the planform of the instability cannot be determined; Scanlon and Segel, 1967; the form of the disturbance, e.g. the hexagonal cells, being dependent on nonlinear effects; see also the discussions in Section 9.1), in a similar way for the dynamic liquid layer considered here a degeneracy occurs for hydrothermal waves in that, as illustrated earlier, two sets of waves become unstable at the same time, one moving with a component in the positive z direction and the other in the negative z direction; hence the preferred form of the disturbance, i.e. an oblique right-moving wave, an oblique left-moving wave or some combination of the two, is generally selected by nonlinear interactions (which requires resorting to nonlinear studies). Studies along these same lines (but for both adiabatic boundaries) are due to Pavlovskii (1994). Secondary ﬂows arising after the onset of instability were determined from an analysis of the full nonlinear problem through expansion of the solution in a power series in terms of a supercritical state parameter in the vicinity of the bifurcation point. Three types of secondary ﬂows were investigated: plane two-dimensional waves propagating along the temperature gradient, plane

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waves travelling at a certain angle to the gradient and three-dimensional waves propagating along the gradient. Further analysis of the equilibrium states by Smith (1988) revealed that both states given by a single wave or by a combination of two linear waves can exhibit Eckhaus and Benjamin–Feir sideband instability and a corresponding phase instability (which means that they become modulated on long length and time-scales as the system develops; we shall come back to these aspects in Chapter 11). Other nonlinear studies (mostly based on the numerical solution of the Navier–Stokes and energy equations in geometries of ﬁnite extent) are discussed in Section 10.2.4.

10.2.4 Boundary Effects: 2D and 3D Numerical Studies One is often tempted to suppose that distant boundaries have a negligible effect on the process of pattern selection. Although this may be so for steady-state pattern-forming instabilities such as those treated in Chapters 4 and 9, for which the boundaries slightly shift the threshold for the primary instability and modify the pattern substantially only near the boundary where matching to the boundary conditions is effected, systems supporting propagating waves behave differently. In such systems, the wave is always in contact with the boundary and the boundaries can exert a profound inﬂuence. This is especially true when the waves have a preferred direction of propagation, such as those described in the preceding subsection. Along these lines, prior to embarking on the discussion of the dynamics for ﬁnite-sized geometries yielded via solution of the fully nonlinear time-dependent thermal-convection equations, it is convenient to recall here the main properties of the hydrothermal phenomena in a slightly different perspective, making them more suitable for a correct interpretation of typical results obtained by means of such an approach (Lappa, 2005b). Some general useful considerations (directly or indirectly gathered from theoretical studies for inﬁnite systems discussed in earlier subsections) can be drawn as follows. Since for low-Pr ﬂuids the hydrothermal wave predicted by Smith and Davis (1983), responsible for possible oscillatory ﬂow, has a signiﬁcant component perpendicular to the main ﬂow, it is expected to be suppressed by 2D models and/or computations; the same concept also applies to systems with a relatively small spanwise aspect ratio Az = Lz /d (all the possible disturbances with wavelength larger than the allowed extension along z, in fact, would be suppressed, leading naturally to a strong stabilization of the basic ﬂow). Among other things, as already outlined, the expected absence of such two-dimensional oscillatory solutions for Pr ∼ = 0 is also in line with the theoretical arguments elaborated in Section 1.5.4, given the absence of inﬂectional points in the basic velocity proﬁles [let us recall that, by contrast, such points are present for the analogous phenomena in which convection is induced by buoyancy forces (Chapter 6), which makes possible in those cases oscillatory two-dimensional instabilities of a hydrodynamic nature in the limit as Pr → 0, as illustrated in Figure 6.6]. To summarize, for small Prandtl numbers (Pr 1, semiconductor melts and liquid metals) increasing the Marangoni number sufﬁciently, oscillatory behaviours should occur only if the extension in the spanwise direction (i.e. along z) is sufﬁciently large, whereas in the opposite situation only patterns with stationary rolls should be the possible outcome. In contradistinction to low-Pr ﬂuids, for large Pr the existence of oscillatory Marangoni ﬂow should be allowed even if the ﬂow is constrained to be 2D or the system extension along z is very small since in this case the hydrothermal wave predicted by Smith and Davis is almost two-dimensional. These theoretical arguments have received validation to a considerable extent by several numerical and experimental studies that have appeared over the years for ﬁnite-sized open cavities and shallow liquid layers. For instance, the nearly two-dimensional nature of the hydrothermal wave for

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high Prandtl numbers is probably the reason why two-dimensional numerical computations of pure Marangoni convection in rectangular open cavities with various aspect ratios (streamwise aspect ratio Ax = Lx /d, generally simply referred to as A) have proved easily to capture oscillatory ﬂow (see, e.g., Peltier and Biringen, 1993; Xu and Zebib, 1998; Tang and Wu, 2005). In particular, Peltier and Biringen (1993) considered Pr = 6.78 within the range A ≤ 3.8 and Ma ≤ 105 . Xu and Zebib (1998) found two-dimensional supercritical oscillatory bifurcations over a wider range of Prandtl numbers (1 < Pr < 10) and aspect ratios (0 ≤ A ≤ 7). They also observed that the critical threshold tends to the corresponding value predicted by Smith and Davis (1983) for the hydrothermal wave case, in the limit as A → ∞. Some examples of such behaviours are shown in Figures 10.11 and 10.12–10.15 for Pr = 15 and A = 4 and 20, respectively. For both cases, a strong cell structure can be seen in the region close to the hot wall (such a more or less stationary roll existing near the right wall is basically maintained for relatively large supercriticalities by the strong temperature gradient established in the lateral boundary layer), while the travelling wave is manifested by the propagation of well-deﬁned convective structures from the cold side towards the hot side, that is, in the upstream direction (in agreement with the theoretical ﬁndings of Smith and Davis, 1983). In general, assuming a large (Az 1) or even inﬁnite aspect ratio (Az → ∞) in the spanwise direction, it is the streamwise aspect ratio A that determines the character of the ﬂow. As already

(a)

(b)

(c)

(d)

Figure 10.11 Oscillatory instability of Marangoni ﬂow [plane (x, y)] in a rectangular cavity (Pr = 15, A = 4, Ma = 2 × 105 , Ma based on the length; adiabatic free surface; cold side on the left, hot side on the right; the streamlines are shown in four snapshots evenly distributed within one oscillation period) (numerical simulation, M. Lappa)

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(a)

(b)

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(d)

(e)

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Figure 10.12 Oscillatory instability of Marangoni ﬂow in a liquid layer [Pr ∼ = 15, A = 20, Ma = 3 × 104 , Ma based on the depth; adiabatic free surface; cold side on the left, hot side on the right; the isolines of the stream function (ψmax = 43, ψ ∼ = 4) are shown in eight snapshots evenly distributed during one period of oscillation, plane (x, y )]. The location of the cells near the cold side at different time moments indicates the propagation of a wave to the right, that is, in the upstream direction; a single roll is steadily located near the hot side as a consequence of the strong temperature gradient established in the boundary layer adjacent to the right wall (for illustration purposes, in the ﬁgure the depth of the ﬂuid layer is two times its real dimension; the angular frequency of the wave is w ∼ = 46.2) (numerical simulations, M. Lappa)

illustrated in Section 10.1, for aspect ratios A of order unity, the basic ﬂow is dominated by a single vortex; if, on the other hand, the aspect ratio is very large (A → ∞), the outcome is a parallel shear ﬂow. As a natural consequence, the instabilities of these basic two-dimensional ﬂows on increasing the driving force tend to be very different in terms of threshold and related critical parameters (wavelength and wavenumber).

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(a)

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(h) 0

1

Figure 10.13 Hydrothermal travelling wave (temperature ﬁeld) for the same conditions as considered in Figure 10.12

As a relevant example of the control exerted by the geometry of the container on the instability, Xu and Zebib (1998) found (in agreement with the earlier study of Peltier and Biringen, 1993) that time-dependent motion is possible solely if the aspect ratio exceeds a critical value that increases with decreasing Prandtl number (A ∼ = 2.3 for Pr = 6.78, A ∼ = 2.6 for Pr = 4.4; see Figure 10.16); among other things, these studies also provided a theoretical justiﬁcation to earlier ﬁndings of other authors, who had examined the Prandtl number dependence and also structure and stability of the Marangoni ﬂow for A = 1 (square cavity) and found no evidence of unsteady behaviour for Ma up to O(105 ) (Carpenter and Homsy, 1990).

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(a)

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(h) Figure 10.14 Hydrothermal travelling wave (disturbance streamfunction) for the same conditions as considered in Figure 10.12 (ψmax = 14, ψ = 2.6)

Most interestingly, Xu and Zebib (1998) highlighted the possible existence of two stability limits in the parameter space (A, Ma), i.e. that the oscillatory two-dimensional ﬂow may come back to steady conditions upon a further increase in the characteristic number. It was revealed, in fact, that the unstable region with oscillatory Marangoni convection is bounded by two different critical Reynolds numbers (which means that as Re increases, the ﬂow ﬁrst changes from steady to oscillatory at Recr1 and then becomes stable again when Recr2 is reached; the dependence of these boundaries on the aspect ratio is displayed in Figure 10.16, where it is shown, in particular, that Recr2 increases monotonically with A whereas Recr1 does not).

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(a)

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(g)

(h)

Tdist:

−7.05Ε−02 −5.05Ε−02 −2.58Ε−02 −1.15Ε−03 2.35Ε−02 4.82Ε−02

Figure 10.15 Hydrothermal travelling wave (temperature disturbance) for the same conditions as considered in Figure 10.12

These authors also considered the three-dimensional case (Pr = 4.4 and 13.9), investigating what would happen if a Marangoni ﬂow in a two-dimensional cavity with a given streamwise aspect ratio was allowed to become three-dimensional. As expected, the dual effects of sidewalls located along z, which should have a damping effect, and the simultaneous possibility of a spanwise instability were found to be Pr dependent. For Pr = 4.4, in particular, they reported the coexistence of spanwise and streamwise ﬂuctuations to destabilize the ﬂow at Re lower than that required to trigger a pure two-dimensional streamwise instability at the same aspect ratio.

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Figure 10.16 Critical Reynolds number as a function of the aspect ratio for ﬂuids with Pr = 1, 4.4, 6.78 and 10 (Re based on the cavity depth; cavity with adiabatic horizontal boundaries). After Xu and Zebib (1998); Reproduced by permission of Cambridge University Press

Conversely, for Pr = 13.9, the sidewalls were shown always to exert a damping effect, whereas spanwise ﬂuctuations are much weaker than streamwise ﬂuctuations when Re is near its ﬁrst critical value. In practice, these ﬁndings provided some validation to the theoretical results obtained in the framework of the inﬁnite-layer approximation (see, in particular, Figure 10.6) for which at Pr = 4.4 the travelling waves should exhibit a signiﬁcant component along z (and, hence, should require a larger Re to emerge under the constraint of 2D computations), whereas for Pr = 13.9 they should propagate almost exactly in the plane of the basic ﬂow (hence limiting the inﬂuence of the longitudinal sidewalls to some weak mitigation of the ﬂow and small increase in the related threshold). For the opposite case of small Prandtl number, it is worth mentioning the two-dimensional numerical simulations of Ben Hadid and Roux (1990), who, in agreement with the predictions of Smith and Davis (1983), did not capture oscillatory instabilities of pure Marangoni ﬂow in ﬁnite two-dimensional long cavities and with small Pr. Similar results were reported by Chen and Hwu (1993) for Pr = 0.01. As an example, Figure 10.17 shows the changes in the pattern of Marangoni ﬂow for this speciﬁc value of the Prandtl number in a rectangular cavity (2D simulation) with A = 4 for increasing values of Ma. As expected, no hydrothermal waves arise with the effect induced by the increasing driving force being limited to the development of a stationary main recirculation roll near the cold wall and a secondary cell (with relatively small extension in the crosswise direction) embedded in the overall circulation. Let us recall at this stage that (as widely illustrated earlier) for low-Pr ﬂuids the hydrothermal waves tend to propagate almost exactly in the spanwise direction, that is, along z (see Figure 10.6). For further clarity, a prototype (3D) numerical example is shown in Figures 10.18 and 10.19 for a cavity having the same streamwise aspect ratio as considered in Figure 10.17, but a ﬁnite extension along the z-axis (the ﬁgures refer, in particular, to the case Ax = Az = 4, that is, a 4 × 1 × 4 cavity and Pr = 0.01). Such results can be regarded as a conﬁrmation of the expected intimate correspondence between the onset of oscillatory ﬂow and the allowance for disturbances

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(a)

(b)

(c)

Figure 10.17 Structure of pure Marangoni convection [plane (x, y)] for Pr = 0.01 in a two-dimensional open cavity with aspect ratio A = 4 (cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions) and three values of the Marangoni number: (a) Ma = 10; (b) Ma = 102 ; (c) Ma = 103 (Ma based on the length of the cavity) (numerical simulations, M. Lappa)

in the spanwise direction (of which the snapshots shown in such ﬁgures are just one effective realization). In particular, these ﬁgures also provide some lines of evidence supporting the widespread idea (we will come back to this concept later; it is one of the main subjects of Sections 10.3 and 10.4) that not too far from the onset of instability, the two counterpropagating waves predicted by the linear stability analysis of Smith and Davis (1983) combine, maintaining almost perfect symmetry in terms of relative strength (which, in other words, means superposition taking place with almost equal amplitudes). The disturbances shown in Figures 10.18 and 10.19, in fact, do not exhibit a preferred sense of propagation in the z direction. Rather, a ‘standing wave’ appears, that is, a spatiotemporal pattern retaining mirror symmetry with respect to the cavity midplane (z = Lz /2) and displaying pulsating cold and hot temperature spots at almost ﬁxed positions (i.e. disturbance nodes growing and shrinking alternately in time and moving back and forth in the streamwise direction, but not undergoing signiﬁcant spanwise displacement). Such a symmetry, however, tends to be broken as time passes (in particular, earlier for increasing values of the Marangoni number), leading to a well-deﬁned apparent single wave with disturbance nodes travelling in the positive or negative z direction, which should be regarded as a clear indication of the intrinsic transitory nature of the almost perfect initial balancing of the opposite wave components discussed earlier. The possible equilibrium waveforms for the instability in the nonlinear regime are not discussed further here (as mentioned above, we will come back to them and related intrinsic mechanisms later, in particular when discussing the typical properties of Marangoni ﬂows for other geometric conﬁgurations that attracted much attention over the years speciﬁcally for the study of these aspects; see, in particular, Sections 10.3 and 10.4.8–10.4.11).

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Figure 10.18 Temperature disturbances on the free surface [plane (x, z)] of a parallelepipedic cavity (Pr = 0.01, 4 × 1 × 4 cavity, Ma = 1.2 × 103 , Ma based on the streamwise length; geometry with adiabatic horizontal walls; the isolines are shown in four snapshots evenly distributed within one oscillation period). As a result given by the superposition of two waves travelling along z in opposite directions (with equal amplitude), a standing wave appears, that is, the ﬁeld exhibits cold and hot temperature nodes pulsating at given positions (cold and hot sides on the left and on the right of each frame, respectively; Min → −7 × 10−3 , Max → 7 × 10−3 , level = 10−3 ) (numerical simulation, M. Lappa)

The essentially spanwise nature of the disturbances in cavities with rectangular cross-section in the limit as Pr → 0 was also conﬁrmed in 2005 by Schimmel et al., who carried out a linear stability analysis (streamwise aspect ratio in the range 0.4 ≤ A ≤ 2.8) of the two-dimensional Marangoni ﬂow with respect to 3D perturbations assuming periodic boundary conditions along z in the zero Prandtl number limit (Schimmel et al., 2005). As a concluding remark for this section, let us observe that for Pr = 0.015 and 0.02, oscillations were found for mixed buoyant–Marangoni convection by means of two-dimensional numerical

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(a)

(b)

(c)

(d)

Figure 10.19 Temperature disturbance distribution (at x = 1, x = 2 and x = 3) for the same case as considered in Figure 10.18 The ﬁeld exhibits cold and hot temperature spots pulsating in cross-sections perpendicular to the basic ﬂow, while keeping mirror symmetry with respect to the (x, y) midplane (the isolines are shown in four snapshots evenly distributed within one oscillation period) (numerical simulation, M. Lappa)

computations by Ben Hadid and Roux (1989), Ohnishi et al. (1992) and Mundrane and Zebib (1994). Apparently, in contrast to all the considerations elaborated in this section, in practice these simulations highlighted that the presence of buoyancy convection can alter the mechanisms of Smith and Davis and that this can lead to the onset of two-dimensional oscillatory ﬂow (superimposed on the initial stationary rolls) also for low values of the Prandtl number (in practice, such a time-dependent ﬂow in 2D conﬁgurations and Pr 1 is due to the gravitational oscillatory hydrodynamic mode already discussed in Section 6.2.1 for pure buoyancy convection). Further information about mixed buoyancy–Marangoni ﬂows will be provided in Chapter 11, which is entirely devoted to such aspects.

10.3

Annular Conﬁgurations

Hydrothermal waves are not an exclusive prerogative of rectangular geometries with endwalls differentially heated to establish a unidirectional temperature gradient (mimicking more or less the idealized return-ﬂow model originally considered by Smith and Davis).

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Turning from rectangular cavities to rotationally symmetric annular gaps or cylindrical vessels (annular pools cooled or heated at the centre) does not change the nature of the primary bifurcation of Marangoni ﬂows. The azimuthal propagation of hydrothermal wave-like structures has been extensively reported for annular shallow layers subject to radial horizontal temperature gradients. Apart from the obvious general theoretical interest (of a fundamental nature), this geometry has attracted much attention speciﬁcally because it allows the elimination of the sidewalls in the spanwise direction in favour of periodic boundary conditions. Thereby, it (together with the liquid bridge, which is the main theme of Section 10.4) is more suitable for the investigation of the typical properties of hydrothermal waves and related supercritical waveforms (the advantage of this kind of geometry with respect to the rectangular one originates essentially from the avoidance of one pair of sidewalls, which, as already discussed in Section 10.2.4, can make the hydrothermal waves degenerate if the distance between them is smaller than the wavelength component in the spanwise direction). It is by virtue of such advantages that both the annular conﬁguration and the liquid bridge were used as fruitful alternatives to the less convenient rectangular layer in many of the space experiments expressly conceived for gaining insights into the typical secondary modes of Marangoni convection. As an example, experiments in microgravity with cylindrical containers ﬁlled with silicone oil (2 cSt) heated by a cylindrical cartridge heater placed at the symmetry axis (with the container sidewall maintained at a lower temperature) were carried out by Kamotani et al. (2000). Other investigators considered a different conﬁguration with the outer portion of the annulus heated with respect to the inner portion, introduced as an ‘approximate’ model of the Czochralski technique [see Section 3.1 for some speciﬁc information about this method for crystal growth from the melt; the annular geometry considered here (see Figure 2.4c) should be regarded as an approximate model of real systems for crystal growth as its inner rod reaches the bottom boundary, whereas it only ‘touches’ the melt surface in the real process]. In practice (see the linear stability analysis by Garnier and Normand, 2001), the problem is not invariant with respect to the exchange of the hot and cold sides. In particular, as demonstrated by the terrestrial experiments of Brunet et al. (2005), the conﬁguration with a basic ﬂow converging from periphery to centre is more unstable than that with the heating rod placed along the centreline of the container. For the latter model (heated from the outer wall), hydrothermal waves were clearly observed by Schwabe (2002a) in the case of a ﬂuid with Prandtl number Pr = 7 both under microgravity and under normal gravity (for high-Pr microgravity experiments, the reader may also consider Schwabe et al., 2003, and for related numerical simulations Sim et al., 2003; Li et al., 2003, 2004b; Shi and Imaishi, 2006, for Pr = 6.7). Although a larger amount of results is available for high-Pr liquids [experiments in space have been carried out mostly with transparent organic liquids, which are basically ﬂuids with Pr ≥ O(1); the interested reader is referred to Section 11.3 for a list of the physical properties that make the use of such substances particularly convenient], some relevant and important studies have also appeared for the low-Pr case. In such a context, there is no doubt that the numerical investigations (Pr = 0.01) by Imaishi and co-workers (e.g. Li et al., 2004a; Shi et al., 2009) deserve special attention owing to the number of novel results and related insights that they provided on the possible waveforms and related transitional stages of evolution. These authors were the ﬁrst to report clearly for a relatively shallow conﬁguration slightly above the critical condition, as shown in Figure 10.20a, travelling waves originating at a ﬁxed azimuthal position (ϕ = π in the ﬁgure) and travelling towards the opposite location (ϕ = 0) via two paths (one wave group propagating in the clockwise direction and the other in the counterclockwise direction), which might be regarded as an effective realization of the theoretical mechanisms described in Section 10.2.3 (on the basis of the weakly nonlinear analysis of Smith, 1986, according to which for a small-Pr ﬂuid an initial localized hot disturbance,

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(a)

(b)

(c)

Figure 10.20 Snapshots of surface temperature ﬂuctuation and spatiotemporal diagram of surface temperature along a circumference (r = 25 mm) of an annular layer for various depths and conditions (Pr = 0.01, a = 15 mm, b = 50 mm; adiabatic top and bottom boundaries): (a) d = 1.5 mm, Ma = 10.67; (b) d = 1.5 mm, Ma = 11.94; (c) d = 8 mm, Ma = 151.8 [Ma based on the reference length L = d 2 /(b − a)]. After Li et al. (2004a); Copyright Elsevier, 2004

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e.g. at ϕ = π , should lead to the unsteady propagation of induced disturbances in either normal direction to the basic ﬂow, as also sketched in Figure 10.9). It is known (N. Imaishi, Institute for Materials Chemistry and Engineering, Kyushu University, Fukuoka, Japan, personal communication, 2008; see also Shi et al., 2009) that increasing the distance from the onset this initial behaviour is usually taken over by a spoke pattern propagating circumferentially (with the HTW appearing on the free surface as many travelling curved arms; see, e.g. Figure 10.20b). Anyhow, for speciﬁc conditions (larger thickness of the low-Pr layer), these researchers also clearly observed as a transitory state (i.e. as a transitional regime preceding the ﬁnal rotating pattern) the emergence of standing waves (Figure 10.20c) with the instability appearing on the free surface as many broad and straight spots pulsating at ﬁxed azimuthal positions reminiscent of those already shown in this chapter for a parallelepipedic cavity (Figures 10.18, 10.19). As illustrated in Figure 10.20c, the peculiar properties of this state result in a spatiotemporal diagram of surface temperature along a given circumference (STD) looking like a checkerboard . Given the remarkable distinguishing features of this mode of convection with respect to the asymptotic one (with STDs simply displaying inclined lines of constant phase as shown in Figure 10.20b) in which disturbances clearly exhibit a preferred direction (the clockwise or anticlockwise azimuthal direction for the geometry with circumferential symmetry or the positive or negative spanwise direction for the shallow rectangular parallelepipedic cavity in the low-Pr case), Imaishi and co-workers devoted some special attention and effort to the description of this case and underlying mechanisms. A synthetic picture along these lines, including both the periodic steps of convection for a period of oscillation and the related cause-and-effect relationships, is shown in Figure 10.21. According to Figure 10.21 (showing computed snapshots of the temperature distribution on the free surface and related ﬂow structure on a circumference at r = 25 mm during one period τ of oscillation), this mode of convection should be regarded essentially as a set of radially extended (longitudinal) rolls periodically alternating their azimuthal sense of rotation. On the basis of the insights provided by Li et al. (2004a), in particular, the growth and decay behaviour of a sample pair of such oscillatory longitudinal rolls is shown schematically in Figure 10.21d (this sketch also provides interesting insights into the physics of the problem, as developed in the following). The assumed initial conditions at t = t0 correspond to the presence of a hot spot located in point A; moreover, it is assumed that TA > TB (as this theoretical condition ﬁts the effective computed velocity ﬁeld shown correspondingly in Figure 10.21c). At the beginning, in fact, such a hot spot induces surface ﬂuid motion towards both regions located on the right and on the left. Furthermore, as a simple natural consequence of the combined effect of thermocapillarity (which brings interfacial ﬂuid from hot regions towards cold regions) and continuity, an upward vertical ﬂow is established just beneath point A. Since such an upﬂow, in turn, brings ﬂuid with a lower temperature towards the surface (due to the effect of the basic return ﬂow by which a lower temperature is established near the bottom wall), it is responsible for a decrease in time of the temperature TA . When TA becomes lower than TB , the surface Marangoni effect will obviously work in the reversed direction. This may explain the emergence of the counterotating couple of vortices located near the free surface shown at t = t0 + τ /4. As time passes, such surface rolls will become stronger and expand along the depth, expelling the lower rolls. The new ﬂow generated by this process will cool the surface temperature at point B, leading ﬁnally to the emergence of surface rolls with a reversed sense of circulation. Notably, the repetition of these growth and decay processes may be regarded as the simple fundamental mechanism responsible for the steps of convection periodically displayed by the

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t0+t/4

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Figure 10.21 Evenly spaced snapshots of: (a) surface temperature ﬂuctuations; (b) circumferential view of temperature distribution; (c) ﬂow structure in the meridian plane for a couple of counter-rotating rolls; (d) sketch of the ﬂow structure shown in (c) (same conditions as in Figure 10.20c; snapshots correspond to four instances evenly distributed within one oscillation period τ ). After Li et al. (2004a); Copyright Elsevier, 2004

system in Figure 10.21a–c and, thereby, as the fundamental mechanism able to sustain a standing wave (at least for some time) in an annular conﬁguration. Additional insights into the dynamics by which standing or travelling waves can be established in geometries with rotational symmetry and about the interdependence between the wave components that create the recognizable identiﬁcation of such states will be provided in Section 10.4 for the speciﬁc case of liquid bridges and high-Pr liquids (Sections 10.4.8–10.4.11). Anyhow, before switching to the liquid-bridge problem, it should be pointed out that Imaishi and co-workers basically identiﬁed the same succession of stages of evolution observed for low-Pr ﬂuids (initial pattern with clearly distinguishable wave trains propagating in opposite azimuthal directions, possibly followed by the transitory appearance of the standing wave, ﬁnally replaced by a rotating

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0.01625

pattern as the equilibrium waveform) also for high-Pr ﬂuids (N. Imaishi, Institute for Materials Chemistry and Engineering, Kyushu University, Fukuoka, Japan, personal communication, 2008). Some examples of transitory patterns obtained by these investigators are shown in Figure 10.22 for different conditions (annular pools with a = 20 mm, b = 40 mm, Pr = 6.7 for different values of the depth in the range 1.5 ≤ d ≤ 17 mm and of the applied temperature gradient, see ﬁgure

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Figure 10.22 Snapshots of surface temperature ﬂuctuation and spatiotemporal diagram (STD) of surface temperature along a circumference (r = 25 mm) of an annular layer for various depths and conditions (Pr = 6.7, a = 20 mm, b = 40 mm; adiabatic conditions at the top and bottom): (a) d = 1.5 mm, T = 8 K, snapshot at t = 162 s, STD between t = 122 and 162 s; (b) d = 3.0 mm, T = 3 K, snapshot at t = 321 s, STD between t = 281 and 321 s; (c) d = 17 mm, T = 3 K, snapshot at t = 300 s, STD between t = 289 and 329 s [Ma based on the reference length L = d 2 /(b − a)]. Courtesy of N. Imaishi

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Figure 10.23 Typical stages of evolution of oscillatory Marangoni convection in an annular layer (Pr = 6.7, a = 20 mm, b = 40 mm, d = 1 mm; adiabatic conditions at the top and bottom). (a) Snapshots of surface temperature ﬂuctuations and temperature as a function of time at a given point (indicated by the arrow) for T = 6 K, Ma = 1 × 104 ; (b) snapshot of surface temperature ﬂuctuation and spatiotemporal diagram of surface temperature along a circumference at r = 25 mm for T = 12 K, Ma = 2 × 104 [Ma based on the reference length L = d 2 /(b − a)]. Courtesy of N. Imaishi

legend), and Figure 10.23a illustrates the typical temporal sequence of waveforms for a sample case (d = 1 mm). It is evident (Figure 10.23a) that at the beginning (t < 600 s) two waves with almost equal amplitude originate apparently from a speciﬁc source point and meet at the opposite point where they disappear (apparent sink); then, as time passes, the memory of the initial source point and opposite sink is lost and the waveform simply corresponds to an intricate pattern resulting from the interference of the two opposite wave trains having equal amplitude, which now extend to the overall cell; this leads to the recognizable emergence of a standing wave (700 < t < 800 s); ﬁnally, the symmetry in the amplitude of the counterpropagating waves tends to be broken (t = 900 s) with the system evolving towards a purely rotating pattern (t = 1000 s) displaying a single apparent travelling wave. Remarkably, such a similarity between the dynamics for low- and high-Pr liquids may lead to the conclusion that the scenario with initial pulsating patterns replaced as time passes by rotating ones is an intrinsic feature of Marangoni ﬂows (as also witnessed by, among other things, similar

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behaviours reported historically for liquid bridges; see, for instance, Figure 10.54; additional speciﬁc details will be provided in Section 10.4.10). Figure 10.23b is ﬁnally devoted to illustrating possible subsequent stages of evolution taken by the same conﬁguration considered as a representative example in Figure 10.23a when the Marangoni number is further increased. It basically shows the pattern emerging as a consequence of a new bifurcation to a secondary HTW with distinct wavenumber and independent frequency (with respect to the asymptotic HTW established through the primary bifurcation and shown in the last snapshot in Figure 10.23a); notably, such HTW coexists with the preceding equilibrium (rotating) waveform throughout the radial extension of the pool, but has a different travelling direction, which explains the peculiar aspect of the pattern. Inspection of this ﬁgure also reveals the presence of an outer rim in which the characteristic curved arms of the two coexisting HTWs exhibit signiﬁcant bending with respect to the general trend in the inner part. Such a behaviour could be explained by the interaction of the HTWs with the strong stationary roll located near the hot wall, whose genesis has been already discussed in Section 10.2.4 for rectangular cavities (this roll is maintained by the strong temperature gradient established in the vertical boundary layer for high-Pr liquids and fairly large Ma, which, among other things, may also justify why such an outer rim cannot be seen in Figure 10.23a and in the ﬁgures for low-Pr liquids).

10.4 The Liquid Bridge The remainder of this chapter is devoted to the popular liquid bridge, which has attracted great interest over the years as a paradigm for the study of the fundamental properties of Marangoni ﬂows, their stability and bifurcations. Due to intrinsic ﬂuid-dynamic complexity (up to the onset of full turbulence), it is worth remarking that, beyond possible application to the FZ technological process, many investigators have extensively used this model (see Section 2.3.3 for fundamental background information about it and Section 3.1.1 for potential applications) merely for testing their ability to predict the behaviour of nonlinear systems (leading to a rich variety of interesting results, which makes it worth considering the subject as a fully independent topic in the context of surface tension-driven ﬂows). The subject has always been widely open and has been approached from different directions and by distinct research groups with various backgrounds and perspectives. Most notably, this synergy has led over the years to the establishment of a common elegant theoretical framework that is currently more or less universally known as the liquid-bridge problem. Although the majority of results were obtained under terrestrial conditions, in the following (as mentioned at the beginning of this chapter) we will content ourselves with developing the subject solely for pure Marangoni ﬂow (zero-g conditions), and for a survey of ground-based studies (where the inﬂuence of gravity is more or less important depending on the size of the liquid volume) the reader is referred to Chapter 11 (see Section 11.8).

10.4.1 Historical Perspective As for annular conﬁgurations (Section 10.3), transparent organic liquids have enjoyed widespread use within the framework of these fundamental studies (typically, in fact, liquid semiconductors are opaque and at high temperatures, so that it is hard to visualize and to study experimentally Marangoni ﬂow and related transitions). Experiments performed with these ﬂuids in space showed that for sufﬁciently small values of the Marangoni number, the convection in liquid bridges is laminar, steady and axisymmetric, but

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when the Marangoni number exceeds a certain critical value (Macr ) depending on the Prandtl number of the liquid, on the geometry (height to diameter ratio AH and shape of the free surface) and on the thermal boundary conditions, the ﬂow undergoes a transition to a complex oscillatory three-dimensional pattern. The appearance of these instabilities in organic liquids was initially used by the investigators (in the 1980s) to provide a possible explanation for the undesired microscopic imperfections (see Section 3.1.2 for some background explanations) found in semiconductor crystals of technological interest obtained under microgravity conditions (see Eyer et al., 1984, 1985) and even under normal gravity conditions in situations where the Marangoni ﬂow is emphasized with respect to buoyancy forces (the experimental microscale technique described in Section 2.3.2). Additional research carried out in subsequent years revealed, however, that the choice of the proper model liquid is a very delicate aspect of the problem (Lappa, 2007d). As highlighted by many theoretical analyses (Neitzel et al., 1992; Kuhlmann and Rath, 1993a; Wanschura et al., 1995; Chen et al., 1997, 1999; Chen and Hu, 1998a), in fact, the instability of Marangoni ﬂows in liquid bridges is initiated through different mechanisms according to the liquid used. In particular, by means of an a posteriori energy analysis (which enables one to identify the physical processes that are most effective in transferring energy from the basic state to the disturbance) of the most dangerous three-dimensional modes predicted by the linear stability analysis, Kuhlmann et al. (1995) elucidated that the ﬁrst bifurcation of the ﬂow is due to an inertia instability of the axial shear layer below the free surface for small-Pr ﬂuids (semiconductor melts and liquid metals) and to the well-known hydrothermal wave for the large-Pr case (transparent organic liquids). In practice, the instability is hydrothermal in nature solely for Pr ≥ O(1) (for this case it is strictly related to the coupling and interplay between the velocity and temperature ﬁelds and the fundamental mechanisms are basically similar to those revealed in the landmark study by Smith and Davis, 1983, for the inﬁnite liquid layers and discussed in Section 10.2). For Pr < O(1) its origin becomes hydrodynamic, that is, it does not depend on the behaviour of the temperature ﬁeld, which in this case is limited to playing the role of driving force for the velocity ﬁeld. Notably, this instability persists in the limit Pr → 0 (when Pr goes to zero the critical Reynolds number remains ﬁnite and tends to a constant value, thereby indicating that thermal effects do not have a direct inﬂuence on the instability mechanism); it also persists for Pr small but =0 if the computations are repeated uncoupling the velocity and temperature ﬁelds (in this regard it is similar to the hydrodynamic mode characterizing the instability of the Hadley ﬂow for Pr 1 extensively described in Chapter 6; the reader is referred, in particular, to Sections 6.1.2 and 6.2.1). All such theoretical effort (mostly based on linear stability analyses) also revealed that, typically, for transparent organic liquids the transition to a 3D state comes with the emergence of oscillatory ﬂow (i.e. it is a Hopf bifurcation), whereas for semiconductor melts the instability breaks the spatial axisymmetry, but the ﬂow regime is still steady (which means that the bifurcation is stationary) prior to the onset of time dependence (Figure 10.24). Thereby, it can be concluded that despite the common aspects shared by liquid bridges and annular geometries (a differentially heated free surface extended circumferentially, the geometric rotational symmetry, an initial axisymmetric basic ﬂow with toroidal structure), the related dynamics in terms of emerging spatiotemporal patterns and nature of the most dangerous disturbance exhibit signiﬁcant differences [no stationary bifurcations occur, in fact, for the shallow annular model , discussed in Section 10.3, for which the primary instability is related to the onset of hydrothermal waves over the whole range of Prandtl numbers (0 ≤ Pr ≤ ∞) ]. Continuing with historical developments, it is worth mentioning that Camel et al. (1995) were the ﬁrst to demonstrate indirectly the existence of the (ﬁrst) non-oscillatory bifurcation mentioned above, examining experimentally the related macroscopic effect on the solidiﬁcation

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Figure 10.24 Critical Reynolds number (Re = σT T L/ρν 2 ) as a function of the Prandtl number for AH = 0.5 and S = 1. The ﬁgure clearly shows the existence of two branches with different properties. Data after Wanschura et al. (1995); Reproduced by permission of the American Institute of Physics

process of semiconductor crystals (this aspect had been ignored for a long time by other investigators mainly involved in the study of the oscillatory ﬂow and crystal defects on the microscopic scale length). Their experiments were carried out during the D-2 Mission (NASA Space Shuttle) and dopant concentration distributions inside the solidiﬁed specimen were measured by a posteriori spreading-resistance techniques. As no oscillations were detected in the melt during the ﬂight, on-the-ground analyses of the solidiﬁed samples did not show any striations; anyhow, undesired macroscopic radial segregation with a non-axisymmetric distribution was found and this effect was ascribed to the presence of ‘weak non-axisymmetric laminar convection’ in the liquid phase. Clear and direct experimental evidence of the existence of steady nonsymmetric temperature ﬁeld was provided later by JAXA for molten tin (Matsumoto et al., 2005; Kamotani et al., 2007) by means of direct temperature measurements along the circumferential extension of the free surface (see Section 10.4.2 for further elaboration of this aspect). The remainder of this chapter runs as follows. Sections 10.4.2–10.4.5 are focused on the ﬁrst (hydrodynamic) bifurcation for low-Pr liquids [Pr < O(1)], paying special attention to the inﬂuence of geometric parameters and the spatial structure of the related secondary ﬂows. Section 10.4.6 considers the subsequent bifurcation (providing also a synthetic and relevant classiﬁcation of the typical ensuing tertiary modes of convection). Sections 10.4.7–10.4.11 are concerned with the high-Pr case [Pr ≥ O(1)] for which disturbances of a hydrothermal nature are the typical outcome of the instability (as usual, emphasis is given to historical developments and the state-of-the-art; in particular, interesting information is elaborated and provided about the possible equilibrium waveforms produced in the nonlinear regime by the hydrothermal disturbances travelling circumferentially and about related physical mechanisms). Section 10.4.12 describes the special case in which the free surface of the high-Pr liquid bridge is not straight (and for which the dynamics seem to exhibit some interesting departures from the known dominant mechanism for the primary instability of Marangoni ﬂows in transparent organic liquids).

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Finally, Section 10.4.13 is devoted to the intermediate range of Prandtl numbers [joining the low and high limits, i.e. O(10−2 ) ≤ Pr ≤ O(1)], which displays an increased level of complexity due to a fascinating competition between hydrodynamic and hydrothermal modes.

10.4.2 Liquid Metals and Semiconductor Melts As outlined in the preceding section, for liquid metals the peculiar geometry of the liquid bridge changes the nature of the primary instability of Marangoni ﬂows with respect to other canonical geometries, making the ﬁrst bifurcation hydrodynamic and stationary (i.e. the supercritical state established when the disturbances saturate their amplitude is three-dimensional and steady). For such cases, in practice, temperature oscillations (known to be the source of undesired crystal striations typically reported in effective applications) appear only if the Marangoni number is further increased (this means that they correspond to a secondary bifurcation at Ma = Macr2 > Macr1 , where Macr1 is the threshold value for the stationary bifurcation; see, e.g. the numerical results of Rupp et al., 1989; Levenstam and Amberg, 1995; Imaishi et al., 1999, 2000, 2001; Yasuhiro et al., 2000; Leypoldt et al., 2000; Bazzi et al., 2001). Even though the ﬁrst numerical simulation of the stationary bifurcation dates back to 1989 (Rupp et al.), as anticipated in Section 10.4.1, a clear and convincing experimental proof has become available after about two decades (Matsumoto et al., 2005; see Figure 10.25). A reasonable justiﬁcation for such a delay should be ascribed to the considerable difﬁculties that arise when the detection of this critical point is attempted experimentally. One reason is that the critical temperature difference between the supporting disks (Tcr1 ) is relatively small (of the order of 1 K). The second reason is that since Macr1 = O(10), that is, the order of magnitude of the critical parameter is relatively small and at the same time Pr = O(10−2 ), heat transfer by diffusion tends to be particularly signiﬁcant at the time of the instability, thereby making temperature variations in the azimuthal direction very weak. JAXA researchers were the ﬁrst to succeed in detecting the transition directly in molten tin (Matsumoto et al., 2005) by using three thermocouples at different azimuthal positions. According to such landmark measurements shown in Figure 10.25b, at the beginning the three thermocouples placed on one side of the liquid bridge were seen to have nearly the same readings, but beyond a certain T to exhibit different but nearly steady outputs, which were regarded as a direct evidence of a transition from axisymmetric to non-axisymmetric stationary ﬂow. Similar results were also obtained by Matsumoto et al., 2006 (related numerical simulations are due to Li et al., 2005 and 2008). The next three subsections are devoted to an exhaustive theoretical treatment of the inﬂuence that different geometric parameters can have on the structure of such secondary 3D steady ﬂow, while the second bifurcation and related ensuing tertiary modes of convection are expressly examined in Section 10.4.6. Given the page limits, the companion problem in which the FZ process is modelled in the framework of the so-called ‘full zone’ (the ends of this geometric model are plane and isothermal as in the case of the classical liquid bridge, but the supporting disks are considered at the same temperature and the presence of a ring heater around the equatorial plane of the zone is simulated by imposing a speciﬁed heat-ﬂux distribution on the free surface with a maximum of the ﬂux in correspondence of the equatorial plane) is not treated in the present book. Typical ﬂow conﬁgurations obtained for small Marangoni numbers for such a ‘complete’ conﬁguration consist of two separate axisymmetric toroidal vortices in the lower and upper halves of the liquid zone with opposite senses of rotation. The dynamics are qualitatively similar to the liquid-bridge problem, although it should be pointed out that in many circumstances strong

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379

30

Critical Marangoni Number

25

20

15

10 Legend Numerical Results Experimental Results

5

0 0.30

0.40

0.50

0.60 0.70 0.80 An=Height/Diameter

0.90

1.00

1.10

(a) 0.12

Temperature [K]

0.10 0.08

TC1

0.06 TC2

0.04 0.02

TC3

0.00 −0.02 1460

1480

1500

1520 1540 Time [s]

1560

1580

1600

(b)

Figure 10.25 First critical Marangoni number (based on the axial extension of the liquid bridge) for Pr = 0.01 as a function of the aspect ratio (volume factor S = 1): (a) comparison between numerical results (Imaishi et al., 1999, 2000; Chen et al. 1999) and experimental results for molten tin (JAXA, data after Matsumoto et al., 2005); (b) typical thermocouple outputs near Macr1 (molten tin, AH = 0.9, Tcr = 1.7 K); Courtesy of Y. Kamotani and JAXA

interaction occurs between the toroidal convection rolls located in the upper and lower half, respectively (for relevant numerical studies the reader is referred, in particular, to Lappa, 2003a, 2004c, 2005d and to Chapter 4 in Lappa, 2004a).

10.4.3 The First Bifurcation: Structure of the Secondary 3D Steady Flow A theoretical framework for illustrating the typical properties of the secondary modes of Marangoni ﬂow in liquid bridges can be elaborated in its most fruitful form starting from the remark that in the axisymmetric state (prior to the transition) the ﬂuid-dynamic ﬁeld corresponds to a single axisymmetric toroidal vortex, while the primary bifurcation of the basic ﬂow breaks the spatial

380

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axial symmetry of the basic solution and replaces this solution of maximum symmetry with another steady ﬂow having a different pattern of symmetry. Investigators usually characterize this pattern by specifying the azimuthal wavenumber m (mode) of the instability. Following the general concepts provided in Chapter 1, in fact, after the bifurcation the ﬂow ﬁeld can be modelled as the superposition of a steady sinusoidal azimuthal disturbance (characterized by the appropriate wavenumber) to a reference axisymmetric state; this means that the generic ﬂow ﬁeld variable F (r, z,ϕ) can be expressed in a relatively simple way as F (r, z, ϕ) = F0 (r, z) + f (r, z)sin(mϕ + G)

(10.11)

where the subscript 0 refers to the basic (axisymmetric) ﬁeld, m is the aforementioned azimuthal wavenumber (from a physical point of view m represents the number of sinusoidal distortions in the azimuthal direction), f is the perturbation amplitude and G is a constant phase shift related to the azimuthal position with which the disturbances appear. Other important insights into the structure of the supercritical temperature and velocity ﬁelds can be provided as follows. For a given value of m (see Section 10.4.4 for possible factors that can determine the value of the wavenumber), multicellular structures are formed in the azimuthal direction; in particular, the generic section orthogonal to the z-axis (symmetry axis of the liquid bridge) will display 2m convective cells; at the same time, 2m temperature spots (m cold and m hot) are present on the lateral free surface (each surface spot having an angular extension of 360◦ /2m). The radial position of the nodal lines of the temperature perturbations in the cross-sections perpendicular to the liquid-zone axis corresponds to the position of the vortex centreline after the bifurcation. The position of the vortex core is deformed and, in particular, displaced sinusoidally along the perimeter of the toroidal convection roll. This is due to the occurrence of the aforementioned additional convective cells in the sections orthogonal to the bridge axis. The position of the vortex centreline also describes in space a sine curve having m maxima and m minima along z (Figure 10.26). Notably, for the speciﬁc case of low-Pr ﬂuids, the part of the toroidal vortex that is displaced downwards is also displaced away from the centre of the zone and the part that is displaced towards the centre is, at the same time, displaced upwards (in other words, the displacement up and down z

x

y

Figure 10.26 Sketch of toroidal vortex displaced sinusoidally along the circumferential perimeter of the liquid bridge in the supercritical state

Thermocapillary Convection

381

is coupled to displacement in and out, respectively). For this reason, each plane orthogonal to the axis of symmetry shows temperature nodes that correspond to the intersection of the distorted vortex centreline with the considered plane. Critical wavenumbers m = 1, 3, 5, 7, . . . belong to the class of asymmetric modes; m = 2, 4, 6, . . . are instead symmetric modes, such a terminology following from the simple observation that when the critical disturbance number (m) is odd, each meridian plane of the liquid bridge is characterized by two asymmetrical vortex cells (for m = 1, in particular, one of the two vortex cells in the section can prevail over the other and can be extended along the whole axial plane of the bridge; see, e.g., Figure 10.29); whereas for even critical wavenumbers, the ﬂow-ﬁeld structure is, on the whole, three-dimensional and depends on the azimuthal coordinate, but in each meridian plane the velocity and temperature display mirror symmetry with respect to the z-axis (see, e.g., Figure 10.31). Some relevant examples of these behaviours are illustrated in the next section, which has been expressly conceived (together with Section 10.4.5) to illustrate the intimate relationship between the azimuthal pattern and related controlling geometric parameters.

10.4.4 Effect of Geometric Parameters To understand truly the rich and wide-ranging convective phenomena displayed by liquid bridges and related technological processes, one must be well acquainted with the intricacies of multiple geometric parameters and related effects. Typically, the critical azimuthal wavenumber increases when the geometric aspect ratio of the bridge is decreased and, for a ﬁxed aspect ratio, it can be shifted to higher (lower) values by increasing (decreasing) the effective volume of liquid held between the supporting disks (measured by the nondimensional parameter S deﬁned in Section 2.3.3). For the speciﬁc case of straight cylindrical interface (S = 1), in particular, as shown by many authors, an interesting empirical correlation can be established between the geometric aspect ratio and the critical azimuthal wavenumber of the instability. Available numerical results on the subject are summarized in Figure 10.27. Rupp et al. (1989) found m = 2 for AH = 0.6 and for Pr ranging from 0.007 to 0.16. The linear stability analysis by Wanschura et al. (1995) gave the dependence of the most dangerous azimuthal wavenumber on the aspect ratio for Pr = 0.02 as follows: m = 1 for 1.5 > AH > 0.8, m = 2 for 0.75 > AH > 0.4, m = 3 for 0.35 > AH > 0.28 and m = 4 for 0.28 > AH > 0.25. By numerical solution of the nonlinear and time-dependent 3D thermal-convection equations, Levenstam and

Lappa et al. (2001)

4

3 4

2 3

2

1 2

2 4

0.0

0.1

0.2

3

0.3

1 Wanschura et al. (1995)

Rupp et al. (1989)

Levenstam and Abberg (1995)

2

0.4

0.5

0.6

1

0.7 AH

0.8

0.9

1.0

Chen, Lizee,Roux (1997)

1.1

1.2

1.3

1.4

1.5

Figure 10.27 Azimuthal wavenumber versus the aspect ratio AH of the liquid bridge (case S = 1 with straight surface; comparison among different results for liquid metals)

382

Thermal Convection: Patterns, Evolution and Stability

Amberg (1995) obtained m = 2 for AH = 0.5 and Pr = 0.01. Lappa and Savino (1999) reported m = 1 and 2 for AH = 1.0 and 0.6, respectively (Pr = 0.04). Imaishi et al. (2000) obtained m = 2 for 0.7 ≥ AH ≥ 0.5 and m = 1 for AH = 0.9 for Pr = 0.01. Lappa et al. (2001a) found m = 1 for AH ≥ 0.9, m = 2 for 0.85 ≥ AH ≥ 0.4, m = 3 for 0.35 ≥ AH ≥ 0.25 and m = 4 for AH = 0.2. All these efforts led to the conclusion that mAH ∼ =1

(10.12)

which represents the aforementioned correlation. In general, as explained in Section 2.3.3, cases with S = 1 are idealized situations. If the shape is not cylindrical (S = 1), the effect of the parameter S must also be taken into account (Lappa et al., 2001a; Shevtsova. 2005). With a synthesis of the effect of both AH and S on the instability threshold shown in Figure 10.28, some relevant examples along these lines are shown in Figures 10.29–10.34 for the canonical reference value of the Prandtl number (Pr = 0.01) already considered in many parts of this book. In particular, Figures 10.29, 10.31 and 10.33 refer to the case of straight conﬁgurations, whereas Figures 10.30, 10.32 and 10.34 illustrate the corresponding behaviour for the curved geometries (such an arrangement is adopted for the convenience of the reader and for emphasizing, by close comparison, the effect of the non-cylindrical volume on the 3D structure with respect to S = 1). Speciﬁc useful details deserving additional description (and also for clariﬁcation of the underlying mechanisms) are provided in the following. For AH = 1 and a cylindrical shape (S = 1) the wavenumber is m = 1. Two thermal spots are present on the liquid-bridge surface (Figure 10.29a). Moreover, in the generic cross-section orthogonal to the liquid bridge axis there are two azimuthal convective cells (see Figure 10.29d) and two thermal spots (Figure 10.29b). For S = 1.22 (fat liquid bridge) the wavenumber increases. Comparison of Figures 10.29 and 10.30 reveals, in fact, that in this case the convective cells and the temperature spots become four (10.30a,d). Moreover, comparing Figure 10.29e with Figure 10.30e and Figure 10.29f with Figure 10.30f, it is evident how in the ﬁrst case the ﬂow pattern in the generic meridian plane is asymmetric (due to an odd mode of convection), whereas in the latter it is symmetric (due to an even mode of convection). For AH = 0.75 and cylindrical shape the critical wavenumber is m = 2 (Figure 10.31). In this case, in contradistinction to AH = 1, the critical mode number does not change if the volume is

Figure 10.28 Critical threshold (Ma based on the axial extension) and wavenumber as a function of the aspect ratio and of the volume factor (Pr = 0.01). Courtesy of W.R. Hu

Thermocapillary Convection

383

T 1 6.12E−02 4.65E−02 3.18E−02 1.70E−02 2.29E−02 −1.24E−02 −2.72E−02 −3.76E−02 −4.14E−02

5 10 7 14 8

16

(a) 10 13 14

9

16

11

8

5

12 11 17

7

13 8 10

4

(b)

9

15 12 10 45 9 6 4 13

6

Level T 16 7.36E−02 15 6.57E−02 14 5.77E−02 13 4.97E−02 12 4.18E−02 11 3.38E−02 10 2.58E−02 9 1.79E−02 8 9.92E−02 7 1.95E−02 6 −6.01E−02 5 −1.40E−02 4 −2.19E−02 3 −2.99E−02 2 −3.79E−02 1 −4.19E−02

9 1211

10

Level V 18 1.59 17 1.42 16 1.25 15 1.08 14 0.90 13 0.73 12 0.56 11 0.39 10 0.21 9 0.04 8 −0.13 7 −0.30 6 −0.48 5 −0.65 4 −0.82 3 −0.99 2 −1.17 1 −1.34

7 14 15

8

8

6 5

11

13 109

9 13 11

Level T 18 1.86 17 1.64 16 1.42 15 1.20 14 0.99 13 0.77 12 0.55 11 0.33 10 0.12 9 −0.10 8 −0.32 7 −0.54 6 −0.76 5 −0.97 4 −1.19 3 −1.41 2 −1.63 1 −1.85

10 8

14 16 13 12 15 11 5

(c)

(d) 15

15

14 13 12

11 10

14 13 12 9

8

11 10

8

7

9

6 5

5

Level T 15 0.94 14 0.88 13 0.81 12 0.75 11 0.69 10 0.63 9 0.56 8 0.50 7 0.44 6 0.38 5 0.31 4 0.25 3 0.19 2 0.13 1 0.06

4

6

3

7

4 2

1

(e)

1

(f)

Figure 10.29 Structure of 3D Marangoni ﬂow with m = 1 (Pr = 0.01, AH = 1, S = 1, Ma = 35, Ma based on the axial extension of the liquid bridge) at the steady state: (a) temperature disturbances on the liquid-bridge surface; (b) temperature disturbances in the cross-section z = 0.5; (c) radial velocity component in the cross-section z = 0.5; (d) azimuthal velocity component in the cross-section z = 0.5; (e) velocity ﬁeld in a representative meridian plane; (f) temperature ﬁeld in the same meridian plane of frame (e) (numerical simulation, M. Lappa)

increased but it is shifted to a lower value (m = 1) if the volume is reduced (concave liquid bridge, S = 0.83). From inspection of Figures 10.31e and 10.32e, it is evident, in fact, how in the ﬁrst case the ﬂow pattern is symmetric whereas in the latter it is asymmetric. For AH = 0.35, the critical wavenumber becomes m = 3 for a cylindrical shape and it is shifted to m = 2 if S is reduced; consequently, in the generic cross-section orthogonal to the liquid-bridge

384

Thermal Convection: Patterns, Evolution and Stability

T 12

1.5E−02 1.1E−02 8.1E−03 4.8E−03 1.5E−03 −1.9E−03 −5.2E−03 −8.6E−03 −1.2E−02 −1.5E−02 −1.9E−02

10

14 13 7

6 10 11 1413 9

Level T 15 1.7E−02 14 1.4E−02 13 1.2E−02 12 9.2E−03 11 6.6E−03 10 4.1E−03 9 1.5E−03 8 −1.1E−03 7 −3.6E−03 6 −6.2E−03 5 −8.7E−03 4 −1.1E−02 3 −1.4E−02 2 −1.6E−02 1 −1.9E−02

34

(a)

(b)

6 9 10 2 3 7 12 6 4 5 5 13 8 79 5 4 12 4 2 15 6 11 10 7

9 8

11 14 8

9 7

6 5

1 3

Level V 15 0.40 14 0.35 13 0.29 12 0.23 11 0.17 10 0.11 9 0.05 8 −0.00 7 −0.06 6 −0.12 5 −0.18 4 −0.24 3 −0.30 2 −0.36 1 −0.41

6 3

8 109 6

3

4 11 5

17

7

9

13 12 11 8

12

10 6 7

24 9

(c)

(d) 15

14

13

12 9 7

8

11 10 8

6 5 4 2

3

(e)

Level VFI 13 0.32 12 0.27 11 0.21 10 0.16 9 0.11 8 0.05 7 −0.00 6 −0.05 5 −0.11 4 −0.16 3 −0.22 2 −0.27 1 −0.32

1

3

Level T 15 0.94 14 0.88 13 0.81 12 0.75 11 0.69 10 0.63 9 0.56 8 0.50 7 0.44 6 0.38 5 0.31 4 0.25 3 0.19 2 0.13 1 0.06

(f)

Figure 10.30 Structure of 3D Marangoni ﬂow with m = 2 (Pr = 0.01, AH = 1, S = 1.22, Ma = 30, Ma based on the axial extension of the liquid bridge) at the steady state: (a) temperature disturbances on the liquid-bridge surface; (b) temperature disturbances in the cross-section z = 0.5; (c) radial velocity component in the cross-section z = 0.5; (d) azimuthal velocity component in the cross-section z = 0.5; (e) velocity ﬁeld in a representative meridian plane; (f) temperature ﬁeld in the same meridian plane of frame (e) (numerical simulation, M. Lappa)

axis there are six azimuthal convective cells and six temperature spots and six thermal spots on the liquid-bridge surface for S = 1 (Figures 10.33), but for S = 0.915, the convective cells and the temperature spots are four and there are four thermal spots on the free surface (two hot and two cold, Figure 10.34). Moreover, comparison of Figures 10.33e and 10.34e leads to the conclusion that in the ﬁrst case the ﬂow pattern is asymmetric (owing to a critical wavenumber m = 3) whereas in the latter it is symmetric (due to a critical wavenumber m = 2).

Thermocapillary Convection

385

15 T

5

3.19E−02 2.80E−02 2.01E−02 1.23E−02 4.52E−03 3.30E−03 −1.11E−02 −1.89E−02 −2.67E−02 −3.45E−02 −4.24E−02

12 10 11 8

9

9

7

5

10

12 10

12

34

9

8 13 11

14 12

8 1 4 6

8

12 11 10 13 14

(b)

8

15 17

2 7 64

5

(a)

7 2 6 1113 14

Level T 15 3.1E−02 14 2.6E−02 13 2.1E−02 12 1.6E−02 11 1.1E−02 10 5.5E−03 9 4.2E−04 8 −4.7E−03 7 −9.8E−03 6 −1.5E−02 5 −2.0E−02 4 −2.5E−02 3 −3.0E−02 2 −3.5E−02 1 −4.0E−02

7

13

Level V 18 0.66 17 0.60 16 0.54 15 0.48 14 0.42 13 0.36 12 0.30 11 0.24 10 0.18 9 0.12 8 0.06 7 −0.00 6 −0.06 5 −0.12 4 −0.18 3 −0.24 2 −0.30 1 −0.36

10 8

15

Level VFI 20 0.73 19 0.65 18 0.57 17 0.50 16 0.42 15 0.35 14 0.27 13 0.19 12 0.12 11 0.04 10 −0.03 9 −0.11 8 −0.18 7 −0.26 6 −0.34 5 −0.41 4 −0.49 3 −0.56 2 −0.64 1 −0.72

18

11

3 10 2

12 10

2

13 15 16 10

7 11

9 12

8

(c)

42

(d) 15

14 13

12 11

11

10 9

8

5 4 2

(e)

9

8

6

7 3

7

2

1

3

Level T 15 0.94 14 0.88 13 0.81 12 0.75 11 0.69 10 0.63 9 0.56 8 0.50 7 0.44 6 0.38 5 0.31 4 0.25 3 0.19 2 0.13 1 0.06

(f)

Figure 10.31 Structure of 3D Marangoni ﬂow with m = 2 (Pr = 0.01, AH = 0.75, S = 1, Ma = 27, Ma based on the axial extension of the liquid bridge) at the steady state: (a) temperature disturbances on the liquid-bridge surface; (b) temperature disturbances in the cross-section z = 0.5; (c) distribution of radial velocity in the cross-section z = 0.5; (d) distribution of azimuthal velocity in the cross-section z = 0.5; (e) velocity ﬁeld in a representative meridian plane; (f) temperature ﬁeld in the same meridian plane of frame (e) (numerical simulation, M. Lappa)

A synthesis of the effect of both AH and S on the instability threshold is given in Figure 10.28 based on the studies of Hu and co-workers. For S = 1, it is well known that Macr increases if the aspect ratio is reduced (see, e.g., Imaishi et al., 1999, 2000, in the case of a cylindrical interface) and is characterized by a minimum value around S = 1 if the effect of the volume is taken into account (S = 1) (Chen and Hu, 1998b; Chen et al., 1999). The foregoing arguments supported by relevant numerical examples provide a single brief and focused survey on both the effects of the aspect ratio and of the volume factor under microgravity conditions. At this stage, it is also worth emphasizing that in the light shed on the problem by

386

Thermal Convection: Patterns, Evolution and Stability

T 2.19E−02 1.68E−02 1.18E−02 6.76E−03 1.71E−03 −3.33E−03 −8.37E−03 −1.34E−02 −1.85E−02 −2.35E−02

14 13

16

12

11 10 8

7 6 5 2

(a)

7 11 10

12

8 5

6

9

(b)

10

3 13 18 11

14 10

16

Level T 19 2.5E−02 18 2.2E−02 17 2.0E−02 16 1.7E−02 15 1.4E−02 14 1.2E−02 13 8.9E−02 12 6.2E−03 11 3.5E−03 10 8.4E−04 9 −1.9E−03 8 −4.6E−03 7 −7.3E−03 6 −1.0E−02 5 −1.3E−02 4 −1.5E−02 3 −1.8E−02 2 −2.1E−02 1 −2.3E−02

1918 17 15

13

Level V 19 0.91 18 0.80 17 0.69 16 0.58 15 0.47 14 0.37 13 0.26 12 0.19 11 0.15 10 0.09 9 0.04 8 0.00 7 −0.04 6 −0.07 5 −0.18 4 −0.29 3 −0.40 2 −0.51 1 −0.62

11 9

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(d) 15 14 13 12 10 9

8

7 6

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Level T 15 0.94 14 0.88 13 0.81 12 0.75 11 0.69 10 0.63 9 0.56 8 0.50 7 0.44 6 0.38 5 0.31 4 0.25 3 0.19 2 0.13 1 0.06

1

(e)

(f)

Figure 10.32 Structure of 3D Marangoni ﬂow with m = 1 (Pr = 0.01, AH = 0.75, S = 0.83, Ma = 33, Ma based on the axial extension of the liquid bridge) at the steady state: (a) temperature disturbances on the liquid-bridge surface; (b) temperature disturbances in the cross-section z = 0.5; (c) distribution of radial velocity in the cross-section z = 0.5; (d) distribution of azimuthal velocity in the cross-section z = 0.5; (e) velocity ﬁeld in a representative meridian plane; (f) temperature ﬁeld in the same meridian plane of frame (e) (numerical simulation, M. Lappa)

these studies, most of the scientiﬁc community is currently aware of the importance that the control of the boundary conditions and the geometry of the liquid zone may have on the threshold values and observed ﬂow patterns. Among other things, a direct numerical approach to the problem (through numerical solution of the unsteady nonlinear model equations) also provided some important conﬁrmation of the intrinsic nature of the instability predicted by earlier linear stability analyses. As an example, comparison of frames (b) and (d) in the series of Figures 10.29–10.34 proves that on the free surface the azimuthal

Thermocapillary Convection

2 5 8 6

Τ 8.88E−03 7.52E−03 6.15E−03 4.79E−03 3.43E–03 2.06E−03 6.99E−04 −6.65E−04 −2.03E−03 −3.39E−03 −4.75E−03 −6.12E−03 −7.48E−03 −8.85E−03 −1.02E−02 −1.16E−02 −1.29E−02 −1.43E−02

14 13 9 15 13 9

14 10 6 12 7 10 3 1 87

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Level V 15 0.86 14 0.76 13 0.66 12 0.56 11 0.46 10 0.36 9 0.26 8 0.16 7 0.06 6 −0.04 5 −0.14 4 −0.24 3 −0.34 2 −0.44 1 −0.54

9 7

Level T 15 1.0E−02 14 8.2E−03 13 6.4E−03 12 4.5E−03 11 2.7E−03 10 8.7E−04 9 −9.7E−04 8 −2.8E−03 7 −4.6E−03 6 −6.5E−03 5 −8.3E−03 4 −1.0E−02 3 −1.2E−02 2 −1.4E−02 1 −1.6E−02

(b)

9 13

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Level VFI 17 0.59 16 0.50 15 0.42 14 0.33 13 0.25 12 0.16 11 0.07 10 0.03 9 −0.01 8 −0.05 7 −0.10 6 −0.18 5 −0.27 4 −0.35 3 −0.44 2 −0.53 1 −0.61

(d)

(e) 15 13 11 9 7

7

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5 3 1

Level V 15 0.94 13 0.81 11 0.69 9 0.56 7 0.44 5 0.31 3 0.19 1 0.06

(f )

Figure 10.33 Structure of 3D Marangoni ﬂow with m = 3 (Pr = 0.01, AH = 0.35, S = 1, Ma = 45, Ma based on the axial extension of the liquid bridge) at the steady state: (a) temperature disturbances on the liquid-bridge surface; (b) temperature disturbances in the cross-section z = 0.5; (c) distribution of radial velocity in the cross-section z = 0.5; (d) distribution of azimuthal velocity in the cross-section z = 0.5 (e) velocity ﬁeld in a representative meridian plane; (f) temperature ﬁeld in the same meridian plane of frame (e) (numerical simulation, M. Lappa)

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Τ 6.4E−03 6.0E−03 5.4E−03 4.4E−03 3.3E−03 2.3E−03 1.2E−03 2.1E−04 −8.2E−04 −1.9E−03 −2.0E−03 −2.9E−03 −3.0E−03 −3.9E−03 −5.0E−03 −6.0E−03 −7.0E−03 −8.1E−03 −9.1E−03 −1.0E−02

5 4 7 10 9 15 14

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97

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Level V 13 0.54 12 0.44 11 0.33 10 0.23 9 0.13 8 0.02 7 −0.08 6 −0.19 5 −0.29 4 −0.40 3 −0.50 2 −0.61 1 −0.69

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Level T 16 1.2E−02 15 1.0E−02 14 8.6E−03 13 6.9E−03 12 5.3E−03 11 3.7E−03 10 2.0E−03 9 3.7E−04 8 −1.3E−03 7 −2.9E−03 6 −4.6E−03 5 −6.2E−02 4 −7.9E−02 3 −9.5E−02 2 −1.1E−02 1 −1.3Ε−02

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15

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(e) 15 13 11 9 7 5

5 3

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Level V 15 0.94 13 0.81 11 0.69 9 0.56 7 0.44 5 0.31 3 0.19 1 0.06

(f)

Figure 10.34 Structure of 3D Marangoni ﬂow with m = 2 (Pr = 0.01, AH = 0.35, S = 0.915, Ma = 40, Ma based on the axial extension of the liquid bridge) at the steady state: (a) temperature disturbances on the liquid-bridge surface; (b) temperature disturbances in the cross-section z = 0.5; (c) distribution of radial velocity in the cross-section z = 0.5; (d) distribution of azimuthal velocity in the cross-section z = 0.5; (e) velocity ﬁeld in a representative meridian plane; (f) temperature ﬁeld in the same meridian plane of frame (e) (numerical simulation, M. Lappa)

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ﬂow is directed from the cold spots towards the hot spots. This can be regarded as indirect evidence of the fact that the mechanisms of the instability are hydrodynamic. The azimuthal convective cells are not driven by the surface temperature gradients (on the contrary, in the hydrothermal case, the interplay between such a ﬂow and surface thermal spots in the azimuthal direction is of paramount importance; for a discussion on the role played by the surface-temperature disturbances in the case of high Prandtl number liquids, see Section 10.4.7 and subsequent sections).

10.4.5 A Generalized Theory for the Azimuthal Wavenumber In view of the propaedeutical ingredients provided in the preceding subsections and additional synergetic arguments (to be developed below) introduced by several investigators over the years, a general theory can be elaborated giving precise functional relationships between the azimuthal wavenumber of the ﬁrst bifurcation and a variety of controlling factors. Let us start this interesting discussion by observing that, in general, discrete wavenumbers of disturbances are selected from the full spectrum of disturbances because the convection roll is closed in a special zone geometry. Since (as explained earlier) the instability for liquid metals is hydrodynamic in nature, that is, it does not depend on the behaviour of the temperature ﬁeld (for this instability the temperature ﬁeld simply acts as a driving force for the velocity ﬁeld), the selection rule mentioned above is given simply by the constraint that the azimuthal wavelength must be an aliquot of the toroidal vortex core circumference (as stated by Chun and Wuest, 1979) and by the fact that the convection roll is axially limited (as stated by Xu and Davis, 1984). Concerning the latter aspect, it is also worth mentioning Wanschura et al. (1995), who found the feedback for the ampliﬁcation of the disturbance to be provided by the continuity equation in conjunction with the presence of the disks (in practice, these rigid boundaries are responsible for the return ﬂow during which axial momentum of the disturbance is transferred into radial momentum). If endwalls were not present, the possibility of a feedback from the axial to the radial velocity component would not be obvious; in fact, as an indirect conﬁrmation of this ﬁnding, Xu and Davis (1984) did not ﬁnd this type of instability in inﬁnitely long cylinders when Pr → 0 (i.e. this type of instability is suppressed when AH → ∞). On the basis of these guiding principles, one may be led to the conclusion that the critical wavenumber has to be directly related to the axial extension LV and to the diameter DV of the centreline of the convection roll. Additional arguments along these lines were provided by Preisser et al. (1983) through an experimental approach to the problem in the case of transparent liquids. They pointed out that for liquid bridges with AH = O(1), the radial penetration depth of the convection in the bulk of the liquid is approximately given by the extension along z of the toroidal convection roll (LV ); since the size of the azimuthal convective cells induced by the ﬂow instability in the cross-sections perpendicular to the liquid zone axis is approximately equal to the radial extension (LV ) of the toroidal vortex [in other words, the path that the disturbance takes around the free surface is comparable in length to the path of the disturbance in the (r, z)−plane], this means that the 2m convective cells cover a circumference 2mLV . Since the circumference of the toroidal vortex can be also calculated as π DV , these considerations can be ﬁnally used to introduce the following analytical dependence: π mAV ∼ (10.13) = 2 This equation provides a well-deﬁned relationship between the azimuthal wavenumber and the ‘aspect ratio’ of the toroidal vortex AV = LV /DV (hereafter simply referred to as the effective aspect ratio). Continuing with such theoretical developments, it is possible to link directly the behaviours described in the foregoing sections to the geometric aspect ratio in the case of a straight surface

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or to the effective aspect ratio for a non-cylindrical volume [among other things, such arguments also provide a solid basis for Eq. (10.12), which was introduced in Section 10.4.4 on an empirical basis without providing, however, an adequate theoretical justiﬁcation]. For liquid bridges with a straight surface (S = 1) and aspect ratio of O(1), since D ∝ DV and L ∝ LV , Eq. (10.13) reads mAH ∼ = constant. Since, as illustrated by Lappa et al. (2001a) for a convex or concave surface (S > 1 or S < 1, respectively), the diameter of the toroidal convection roll scales with the maximum diameter of the ﬂoating zone (and consequently with the volume factor S), the following relationship also holds: L AH = LV ∼ = L and DV ∝ D S → AV ∝ DS S

(10.14a)

and therefore Eq. (10.13) can be also written as mAH ∝ S

(10.14b)

The foregoing relationships explain why under microgravity conditions the azimuthal wavenumber of the half-zone m increases if the aspect ratio is reduced (and vice versa) and scales with the volume. Finally, it should be highlighted that the theory relying on Eq. (10.13) could also be used to provide a theoretical justiﬁcation for some effects induced by the dynamic change of the axial extension of the liquid zone that occurs during half-zone-based solidiﬁcation processes (the reader is referred to Lappa and Savino, 2002, for interesting and relevant results along these lines) or by the presence of boundary layers adjacent to the supporting disks in the high-Pr case (Section 10.4.7), i.e. it can also take into account dynamics related to variation of LV in addition to those originating from changes in DV .

10.4.6 The Second Bifurcation: Tertiary Modes of Convection As explained in Sections 10.4.1 and 10.4.2, the steady 3D ﬂows become unstable against time-dependent 3D disturbances and start oscillation solely if a secondary threshold (second critical Marangoni number Macr2 = PrRecr2 ) is exceeded. Notably, the route from the initial axisymmetric steady state to the oscillatory regime can be seen as a succession of two purely hydrodynamic instabilities (Levenstam and Amberg, 1995). Like the ﬁrst bifurcation, in fact, also the second one persists in the limit as Pr → 0, which can be regarded as a clear mark of its intrinsic nature. Along these lines, Figure 10.35 shows the stability map of Marangoni ﬂow in cylindrical liquid bridges (S = 1) for Pr = 0 obtained by Imaishi and co-workers via direct numerical simulation. These authors (Imaishi et al., 1999, 2000, 2001; Yasuhiro et al., 2000) also provided a synthetic and relevant classiﬁcation of the possible typical tertiary states of convection, which were categorized on the basis of the related spatiotemporal behaviour as follows: ‘m + 1 representing a time-dependent disturbance with m = 1 superimposed on an initial steady ﬂow having azimuthal wavenumber m (see Figure 10.36 and the description given in the caption); ‘mT ’ representing a torsional oscillation in the azimuthal direction emerging on an initial structure with wavenumber m (see Figure 10.37 and the description given in the caption); and ‘mR’ corresponding to a mere rotation in the azimuthal direction of an initial structure with wavenumber m. These states were determined in well-deﬁned ranges of the aspect ratio AH . For relatively large values of AH , an additional oscillatory state was also identiﬁed (see Figure 10.38 and the description given in the caption); it was referred to as ‘2T → 1T ’ owing to the presence of two characteristic frequencies: the shorter period corresponding to a torsional oscillation with m = 2 similar to that of Figure 10.37 and the longer period corresponding to an alternative, but incomplete, transition of the ﬂow mode between m = 2 and m = 1.

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Figure 10.35 Stability map (Recr , AH ) of Marangoni ﬂow in cylindrical liquid bridges (S = 1) for Pr = 0 (Reynolds number based on the radius). The transition curve (Recr1 ) from stable axisymmetric ﬂow to stationary 3D ﬂow (solid line) is characterized by increasing values of the azimuthal wavenumber as the aspect ratio is reduced (m = 1, 2 and 3). The transition curve (Recr2 ) to oscillatory ﬂow (dashed line) is composed of three distinct branches categorized as follows: m + 1 (time-dependent disturbance with m = 1 superimposed on an initial steady ﬂow having azimutal wavenumber m), mT (torsional-oscillation in azimuthal direction emerging on an initial structure with wavenumber m), mR (structure with wavenumber m rotating azimuthally). The dashed-doted line denotes transition from one value of m to a different value and/or possible coexistence. Numerical simulations by Imaishi and co-workers; Courtesy of N. Imaishi

Interestingly, in 2007 Imaishi and co-workers gained additional insights into these behaviours (obtained via direct numerical integration of the unsteady thermal-convection equations) by means of a proper orthogonal decomposition (POD) analysis applied a posteriori to the numerical results (Li et al., 2007). Let us recall for the convenience of the reader that the POD analysis is a rigorous procedure for extracting a basis of characteristic modes from sampled time evolution signals. These modes are the eigenfunctions of an integral operator based on the spatial correlation function (in general, they form an orthogonal basis for the function space in which the process resides and represent this process in the most efﬁcient way; Li et al., 2007). Since the direct application of POD to a discretized three-dimensional ﬂow problem involves extremely heavy computational tasks (because the dimension of the spatial correlation matrix corresponds to the mesh number in the direct numerical simulation), in particular, these authors used a variant based on the so-called method of snapshots. This technique, which invokes the ergodic hypothesis, allowed them to reduce the computational task to a more tractable eigenproblem of a size equal to the snapshot number of the ﬂow ﬁeld obtained through direct numerical simulation (for an overview of POD and its applications, see Berkooz et al., 1993). Using this approach, Li et al. (2007) extracted from the velocity and temperature disturbances for each of the typical oscillatory states illustrated in Figures 10.36–10.38 the series of aforementioned characteristic modes (eigenfunctions); as a natural consequence, they also obtained spatial structures of such eigenfunctions, their oscillation frequencies, amplitudes and phase shifts. Most

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(a)

(b)

Figure 10.36 Tertiary mode of convection (corresponding to oscillatory ﬂow after the second bifurcation) for Pr = 0.01, AH = 0.5, S = 1, Ma = 90 (Ma based on the axial extension of the liquid bridge). The thermoﬂuid-dynamic ﬁeld (cross-section z = 0.5) is shown at ﬁve evenly distributed transitional stages of evolution for a 2 + 1 pulsating mode. From left to right: temperature ﬁeld (level = 2 × 10−2 ); distribution of azimuthal velocity (level = 0.25); projection of the velocity ﬁeld on the considered cross-section (the reference vector length corresponds to V = 5); velocity distributions on two orthogonal vertical planes (A and B) (the ﬂow ﬁeld in the generic cross-section moves from the left to the right and back again; the intersection of the toroidal vortex with a given plane ϕ = constant moves back and forth radially exhibiting a pendulum-like motion across the axis of the liquid bridge). Courtesy of N. Imaishi

interestingly, some features common to the different oscillatory tertiary states of Marangoni ﬂows in low-Pr liquid bridges were identiﬁed and, more speciﬁcally, it was found that the overall velocity ﬂuctuation energy can be captured by four (major) velocity eigenfunctions only (the interaction of other higher order modes only enriches the secondary spatiotemporal structures of the oscillatory disturbances); thereby, an intrinsic relative simplicity underlying the distinct oscillatory behaviours was revealed. Among other things, it was also conﬁrmed that the oscillatory disturbance, which is hydrodynamic in nature, originates primarily from the interior of the liquid bridge (in practice, from the shear layer created by the opposition of surface ﬂow and return ﬂow). It is worth highlighting at this stage that, compared with all of these interesting and relevant numerical studies, experiments on the ﬂow instability in liquid bridges of semiconductor materials are sparse. It is very difﬁcult, in fact, to conduct well-controlled experiments with low-Pr ﬂuids (mostly liquid metals) due to opacity, reactivity and high melting temperatures, especially in the microgravity environment. Hibiya and co-workers (Nakamura et al., 1995, 1998; Hibiya et al., 1998a,b; Hibiya and Nakamura, 1999) were the ﬁrst to investigate (during sounding rocket missions and other parabolic

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Figure 10.37 Tertiary mode of convection (corresponding to oscillatory ﬂow after the second bifurcation) for Pr = 0.01, AH = 0.7, S = 1, Ma = 91 (Ma based on the axial extension of the liquid bridge). The thermoﬂuid-dynamic ﬁeld (cross-section z = 0.7) is shown at four evenly distributed transitional stages of evolution for a 2T mode, that is, a torsional oscillation in the azimuthal direction with m = 2. From left to right: temperature ﬁeld (level = 2 × 10−2 ); distribution of axial velocity (level = 0.35); azimuthal velocity (level = 0.35); projection of the velocity ﬁeld on the considered cross-section (the torsional oscillation is featured by a back and forth motion of the pattern in the azimuthal direction). Courtesy of N. Imaishi

ﬂights) the structure of the supercritical ﬂow in liquid bridges of molten silicon. Most notably, they developed a novel technique based on X-ray radiography with zirconium-cored tracers to achieve direct visualization of the ﬂow ﬁeld in meridian planes. Temperature oscillation measurements revealed that the critical Marangoni number for transition from an oscillatory ﬂow with a single frequency to one with multiple frequencies is about 3 × 103 . Observation of the ﬂow’s structural instability also showed that the mode was either m = 1 or m = 2 depending on the aspect ratio of the liquid column (these experimental results gave some puzzling aspects and suggested that the oscillatory ﬂow in silicon melt becomes very complex at large temperature differences). It is also worth mentioning the subsequent experiments carried out by JAXA with molten tin (Matsumoto et al., 2005), which (as displayed in Figure 10.39) yielded fairly good agreement with earlier numerical results. For other similar experimental studies carried out in normal gravity conditions, see Chapter 11 (Section 11.2).

10.4.7 High Prandtl Number Liquids As anticipated in Section 10.4.1, the nature of the primary instability of Marangoni ﬂow in liquid bridges changes dramatically with respect to the low-Pr case when liquids with Pr ≥ O(1) (traditionally employed in experimental investigations on the subject due to transparency, low reactivity,

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Thermal Convection: Patterns, Evolution and Stability

Figure 10.38 Tertiary mode of convection (corresponding to oscillatory ﬂow after the second bifurcation) for Pr = 0.01, AH = 0.9, S = 1, Ma = 72 (Ma based on the axial extension of the liquid bridge). The thermoﬂuid-dynamic ﬁeld (cross-section z = 0.9) is shown at ﬁve evenly distributed transitional stages of evolution for a 2T → 1T mode, that is, a regime with two characteristic frequencies, the shorter period corresponding to a torsional oscillation with m = 2, the longer period representing an alternative, but incomplete, transition between m = 2 and m = 1. From left to right: temperature ﬁeld (level = 2 × 10−2 ); distribution of axial velocity (level = 0.36); azimuthal velocity (level = 0.18); projection of the velocity ﬁeld on the considered cross-section. Courtesy of N. Imaishi

etc.) are considered. The bifurcation, in fact, is no longer hydrodynamic and the related underlying mechanism is transferred to the well-known hydrothermal behaviour. Several authors investigated this category of phenomena in space using large transparent liquid bridges with a length of the order of several centimetres (Figure 10.40) (see, e.g., Schwabe et al., 1982; Schwabe and Scharmann, 1984; Monti, 1987; Monti et al., 1995, 1998b; Chun and Siekmann, 1995; Schwabe, 2002b, 2005). Stability analyses were also developed. In particular, Xu and Davis (1984) examined the idealized case with inﬁnite aspect ratio. They assumed for the basic ﬂow the class of similarity solutions valid in the core region away from the endwalls of the bridge introduced previously (Xu and Davis, 1983) and described in Section 2.4.5. It was shown that there is a critical value Pr∗ (∼ = 50) of the Prandtl number such that if Pr < Pr∗ , the mode m = 1 is preferred, whereas if Pr > Pr∗ , the mode m = 0 (two-dimensional) becomes the most dangerous disturbance (Figure 10.41). The effect of axial boundaries was introduced later for S = 1 (liquid bridges with straight surface) by Shen et al. (1990) and Neitzel et al. (1991) and used by Neitzel et al. (1992), Kuhlmann (1992), Kuhlmann and Rath (1993a,b) and Wanschura et al. (1995) to deﬁne, in the nondimensional parameter space, sufﬁcient conditions for stability and instability with respect to

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395

Figure 10.39 Critical Marangoni number (based on the axial extension) for the secondary bifurcation of Marangoni ﬂow in cylindrical liquid bridges (S = 1) as a function of the aspect ratio: comparison between experimental and numerical results [experimental results for molten tin (Pr ∼ = 0.01) obtained by JAXA, Matsumoto et al., 2005; numerical results (Pr = 0.01) by Imaishi and co-workers]. Courtesy of N. Imaishi

two- and three-dimensional disturbances (the studies by Shen et al., 1990, and Neitzel et al., 1991, were elaborated in the framework of an energy stability analysis whereas the others were performed through the standard linear stability analysis approach; the reader is referred to the theoretical background in Chapter 1 for fundamental differences between these two techniques in terms of guiding principles and applicability of results). In agreement with the experimental evidence, it was proven that for all cases three-dimensional disturbances (m = 0) are the most dangerous. There is no doubt that among the studies mentioned above, special attention should be devoted to the linear stability analyses of Kuhlmann and Rath (1993a) and Wanschura et al. (1995), where some revealing insights were gained into the vital mechanisms at the root of this kind of instability. The basic axisymmetric Marangoni ﬂow (for application of the typical protocols of linear stability analysis) was obtained by numerical solutions of the thermal-convection equations, together with the appropriate boundary and symmetry conditions, and the eigenvalue problem for the three-dimensional disturbances was solved over a range of Prandtl numbers and for aspect ratios close to unity. The results predicted the critical Marangoni numbers and the form of the most dangerous disturbances, characterized by the appropriate value of the critical wavenumber, in the neighbourhood of the neutral stability point (i.e. at the onset). Notably, these authors found the 3D supercritical state after the Hopf bifurcation threshold to be given by a superposition of two counterpropagating hydrothermal waves, similar to those already revealed in the milestone analysis of Smith and Davis (1983) for an inﬁnite liquid layer (see the discussions in Section 10.2 and various subsections, and also the interesting experimental study by Kawamura et al., 2007). Like the waves of Smith and Davis, which are ‘oblique’, that is, they exhibit an angle of propagation relative to the basic state (Section 10.2.2), in a similar way the waves in liquid bridges were found to be characterized by distinct axial and azimuthal components, that is, to be inclined

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Figure 10.40 Example of results of liquid-bridge experiments carried out during the Spacelab D-2 mission (the liquid motion in the meridian plane is visualized using tracers scattering the light generated by a laser diode forming a light sheet; the laser beam is oriented orthogonal to the main optical path of a CCD camera; the snapshots show Marangoni ﬂow at two different instants in the supercritical state). Courtesy of the Microgravity Advanced Research and Support Center

with respect to the geometric symmetry axis. Thereby, these studies provided a solid and convincing theoretical basis to the hydrothermal nature of the instability for the transparent liquids traditionally employed in experimental studies on the subject. Since all these studies considered liquid bridges of relatively limited axial extent, some years later Chernatinsky et al. (2002) focused (linear stability analysis) expressly on relatively long liquid columns (with aspect ratio near the Rayleigh limit, i.e. AH = 2.5) to investigate, in particular, the role played by the axial component of the hydrothermal waves on the emerging pattern in geometrically less restrictive systems than the usual liquid bridges with AH ∼ = 1 (the role of the azimuthal components will be discussed later in Section 10.4.8). This study, in particular, was

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Figure 10.41 Critical Marangoni number as a function of Pr for liquid bridges with inﬁnite axial extent (Marangoni number deﬁned as Ma = σT γ R 2 /µα , where γ is the constant rate of temperature increase along the bridge axis and R is its radius). After Xu and Davis (1984); Reproduced by permission of the American Institute of Physics

propaedeutical to the microgravity experiment by Schwabe (2005), who attempted to measure some typical features of the hydrothermal waves (such as the wave phase speed and the angle between the wavevector and the applied temperature gradient) under conditions for which the dynamics of such waves are thought not to be signiﬁcantly disturbed by edge effects (i.e. by relatively close disks and/or thick boundary layers adjacent to the walls). New, interesting results along the same lines were obtained by Xun et al. (2008), as discussed in detail in the following. They extended the original linear stability analysis for straight liquid bridges of Wanschura et al. (1995) (limited to Pr < 5) to the range of large Prandtl numbers (4 ≤ Pr ≤ 50) in which the boundary layers adjacent to the supporting disks are expected to become progressively thinner as Pr is increased (see Section 2.5 for relevant information, scaling laws and trends on such boundary layers; in general, they become thinner and the Reynolds number is increased for a ﬁxed Pr or as the Prandtl number is increased for a ﬁxed Re). For the same aspect ratio (AH = 0.5) as considered by Wanschura et al. (1995), they reported that for Pr ≥ 8 (which had been less studied in earlier works and where Recr was usually believed to decrease with increase in Pr), the critical Reynolds number Recr ﬁrst increases with increasing Prandtl number and then decreases, with a local maximum around Pr ∼ = 28 (Figure 10.42). Correspondingly, the critical azimuthal wavenumber m was found to change from m = 2 to 1 [with the critical Marangoni number Macr (Recr Pr) increasing approximately linearly with increasing Prandtl number for both Pr < 28 and Pr > 28 but with different slopes according to the value of m]. As expected, these interesting trends were explained on the basis of the role played in such processes by the aforementioned boundary layers adjacent to the supporting disks. In particular, from the computed surface temperature gradient in the considered range of values of the Prandtl number such boundary layers developed at both solid ends of liquid bridges were observed to have a twofold effect on the onset of instability and ensuing dynamics, which deserves some additional attention here (for the convenience of the reader, relevant synthetic information about these effects and dynamics is speciﬁcally elaborated in the following).

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Figure 10.42 Critical Reynolds number (based on the axial extension of the liquid bridge) as a function of the Prandtl number for three distinct values of the aspect ratio: AH = 0.3 (dotted line); AH = 0.5 (solid line); AH = 0.6 (dashed line). All the curves clearly show a decrease in the azimuthal wavenumber as Pr increases due to shrinkage of the wall boundary layers and ensuing increase of the effective aspect ratio. Courtesy of W.R. Hu

As already discussed, the boundary layers become thinner (with steeper temperature gradients inside) with increasing Prandtl number; correspondingly, the effective temperature gradient at the rest of the free surface decreases whereas the bulk region of the liquid bridge is uninﬂuenced by such edge effects increases, leading to expansion of the basic Marangoni cell in the axial direction. Taking into account that, as observed previously by Schwabe (2005), the most dangerous disturbances develop on the portion of the free surface out of the boundary layers and that, as illustrated in Section 10.4.5 (concepts elaborated therein also apply to the case of high-Pr ﬂuids), the azimuthal wavenumber is determined by the effective aspect ratio (based on the effective extension along z of the toroidal convection roll), this leads to the natural conclusion that it is the former effect (temperature gradient decrease on the portion of the free surface non affected by boundary layers) which is responsible for the increase in the critical Reynolds number, whereas the latter is responsible for the change in the azimuthal structure. Xun et al. (2008) also examined the inﬂuence exerted on such cause-and-effect relationships by AH . As shown in Figure 10.42, for AH = 0.6 the critical Reynolds number (based on the height of the liquid bridge) increases when 6 ≤ Pr ≤ 15; there is a local maximum around Pr ∼ = 15 and the critical azimuthal wavenumber changes from m = 2 to 1 with the increase in Pr around the same value of the Prandtl number (in particular, the peak of Recr in this region is sharper than that with AH = 0.5). This means that the aspect ratio affects the critical Reynolds number and the critical azimuthal wavenumber. As expected (see Section 10.4.5), the larger the aspect ratio, the smaller is m. Moreover, the region of increase of Recr and the location of the local maximum tend to be anticipated for larger aspect ratios (Xun et al., 2008, speculated that the early increase in Recr is sharper for larger aspect ratios, because of a quicker development of the boundary layers). The trend of the azimuthal wavenumber decreasing with increasing Pr was found to hold for all the aspect ratios investigated, thereby conﬁrming the signiﬁcant role played in the dynamics of liquid bridges with Pr ≥ O(1) by boundary layers developing close to the disks.

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10.4.8 Standing Waves and Travelling Waves Beyond effects induced by the presence of edges, related geometric parameters and Prandtl number, starting from the initial studies of Kuhlmann and Rath (1993a, b) many of the subsequent analyses also tried to shed some light on the mechanisms by which the azimuthal components of the counterpropagating hydrothermal waves determine the spatiotemporal features of the emerging pattern. Notably, on the basis of the properties of the waves provided by the linear stability analysis approach (the shape of the eigenfunction), two possible spatiotemporal modes of convection (waveforms) were predicted in principle: the standing wave (featured by disturbance nodes pulsating at ﬁxed azimuthal positions) and the travelling wave (with disturbances travelling circumferentially), also referred to as pulsating or rotating patterns, respectively. One of the most interesting outcomes of such theoretical efforts was the recognition (supported by precise mathematical arguments) of the possibility for the effective dynamics and related transitional stages occurring in real experiments to be largely determined by the relative amplitude with which the aforementioned counterpropagating waves interact (equal or different amplitudes). At this stage, the reader will remember that we have already introduced and developed to a certain extent such fascinating concepts in Sections 10.2.4 and 10.3; nevertheless, many of them were expressly conceived and further elaborated for the speciﬁc case of liquid bridges, which makes them worthy of additional detailed consideration in this part of the book (in particular, the reader may consider the discussion below, mostly developed on the basis of the arguments originally reported by Kuhlmann and Rath, 1993b). As we have mentioned before, the HTWs in liquid bridges have axial and azimuthal components, which means that the wavefront F of such waves is inclined with respect to the z-axis. In practice, following Kuhlmann and Rath (1993b), the surfaces of constant phase should be imagined as vertical planes that have been twisted around the vertical axis, the twist being given by a phase G(r, z), which leads to a general representation for each wave having an amplitude B(r, z) and angular frequency ω = 2πf of the type &

' F± = B(r, z) exp i ±mϕ − ωt + G(r, z)

(10.15a)

where the symbol ± indicates propagation in the clockwise or anticlockwise circumferential direction. By simple mathematical development, it is easy to verify that within such a theoretical framework a superposition of two counterpropagating waves with the same amplitude should result simply in

F = 2B(r, z) cos(mϕ) cos ωt − G(r, z)

(10.15b)

Since in this case the oscillatory term does not depend on ϕ, on the basis of Eq. (10.15b) this situation represents in principle a waveform characterized by maximum and minimum disturbances ﬁxed in space with the minimum being continually replaced by the maximum and vice versa {as soon as cos[ωt − G(r, z)] changes its sign}. Also, since these extrema in the disturbance distribution correspond to hotter and colder zones in the bridge, the three-dimensional temperature disturbance should simply consist of a number m of couples of spots (hot and cold) pulsating at the same azimuthal positions along the interface. In a similar way, mathematical development for the case in which the amplitude of the two hydrothermal waves is not the same [i.e. a superposition with an arbitrary amplitude ratio ε (0 < ε < 1)] gives F ∼ = B(r, z)a(mϕ) cos[b(mϕ, G) − ωt]

(10.16)

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where 1

a(mϕ) = [(1 + ε)2 cos2 (mϕ − ϕ0 ) + (1 − ε)2 sin2 (mϕ − ϕ0 )] 2 −1 1 − ε tan(mϕ − ϕ0 ) + G(r, z) b(mϕ, G) = tan 1+ε

(10.17a) (10.17b)

with the phase ϕ0 determining the location of the nodes in the case ε = 1 (Kuhlmann and Rath, 1993b). This means that for such a case the oscillatory term will depend on ϕ; accordingly, on the basis of Eq. (10.16), the minimum and maximum disturbances should travel in the azimuthal direction and the phase of the oscillations should depend continuously on ϕ. It is evident at this stage how such theoretical arguments can be used to deﬁne a simple but precise link among the properties of the waves, related amplitudes and the potential ensuing state (pulsating or rotating) established in the supercritical regime. Let us also recall that, among other things, these considerations led historically to the establishment of a precise terminology and a common elegant theoretical framework, which enjoyed widespread use over subsequent years (such terminology, in particular, was extended to other geometries traditionally used for the study of Marangoni ﬂows, e.g. the annular conﬁguration considered in Section 10.3, and, as highlighted in Section 4.10, even to other types of convection, as was used by Boronska and Tuckermann, 2006, for describing some peculiar states of Rayleigh–B´enard convection in cylindrical enclosures). Given the intrinsic limitations of the linear stability analysis [which, as discussed in Section 1.5.5, cannot provide quantitative data about the amplitude of the disturbances, i.e. about the nondimensional ratio ε appearing in Eqs (10.16)–(10.17)], determination through this approach of the effective spatiotemporal mode of convection emerging in liquid bridges at the onset of the instability and/or established eventually as a stable attractor (i.e. an equilibrium waveform when the disturbances saturate their amplitude) was not possible. The liquid-bridge problem was, therefore, reapproached over the years from new directions and perspectives by other research groups. As an example, there is a plethora of experimental results obtained in the framework of ground-based research (i.e. microscale experimentation; the reader is referred to Chapter 11 for a description of such results) that gave evidence for the existence of both aforementioned fundamental modes of convection (it was also shown that there exist, between the limiting cases, numerous intermediate situations). These states were also examined via direct numerical simulation (e.g. Lappa, 1995; Yasuhiro et al., 1997, 1999; Tang et al., 1997; Bazzi et al., 1999; Zeng et al. 1999a, 1999b, 2001b; Leypoldt et al., 2000; Shevtsova et al., 2001). Along these lines, and given the peculiar features of both waveforms with respect to equivalent ones for the annular geometry described in Section 10.3 (unlike annular slots for which the free surface is planar and the HTWs have a radial and an azimuthal component, for liquid bridges the free surface is cylindrical and one of the two wave components is axial , which makes the spatiotemporal aspect of these waves substantially different), the following pages are devoted to illustrating the typical properties of standing and travelling waves in liquid bridges as yielded by direct solution of the nonlinear thermal-convection equations. The ﬁgures shown hereafter, in particular, provide interesting examples of such behaviours for several values of the azimuthal wavenumber (1 ≤ m ≤ 4). The surface temperature spots and the consequently driven vortex cells in a cross-section z = constant (surface temperature spots are responsible for thermocapillary effects in the azimuthal direction, causing 2m counter-rotating vortex cells in the transversal section) are shown for the simplest case (m = 1) in Figures 10.43 and 10.48 (for the standing and travelling wave, respectively). Figures 10.45 and 10.50, respectively, illustrate the analogous stages of evolution for m = 2.

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Figure 10.43 Standing wave for m = 1 (Pr = 30, AH = 1.0, S = 1, Ma = 3 × 104 , Ma based on the axial extension of the liquid bridge; the thermoﬂuid-dynamic ﬁeld is shown at four evenly distributed instances within an oscillation period). From left to right: temperature disturbance on the liquid-bridge surface; projection of the velocity ﬁeld on the cross-section z = 0.5; sketch of surface-tension-driven convective cells (the dashed line denotes the plane of symmetry) (numerical simulation, M. Lappa)

10.4.8.1 The Standing Wave: a Pulsating Spatiotemporal Mode of Convection In the following, special attention is devoted, in particular, to m = 1; given the relative simplicity of the convective structure (a single couple of surface thermal spots and of azimuthal convective cells), this case, in fact, can be regarded as a good exemplar for illustrating some important basic features. As already explained, the temperature spots pulsate (Figure 10.43), that is, the cold spot grows in the axial direction during the shrinking of the hot spot and vice versa, but the azimuthal positions of these extrema do not change. Accordingly, the convective cells developed in the azimuthal direction change their sense of rotation periodically. Their intensity is not constant in time; if the considered convective cell is clockwise oriented during the ﬁrst half-period of oscillation, then during the second half it vanishes and ﬁnally reappears in the same azimuthal position anticlockwise oriented.

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(a) Apparent symmetric behaviour

(b)

Figure 10.44 Sketch of pulsating m = 1 mode: The distinguishing mark of this mode m = 1 is the mental divisibility of the ﬂow conﬁguration in two semicircular regions [such a divisibility can be sketched by introducing an ideal plane of symmetry (PS), separating the ﬂow conﬁguration into two sectors]

Most interestingly, the distinguishing mark of a pulsating mode m = 1 is the mental divisibility of the ﬂow conﬁguration in two semicircular regions (Frank and Schwabe, 1997); such a divisibility can be sketched (Figure 10.44) by introducing an ideal plane of symmetry AB (hereafter referred to as PS) that separates the ﬂow conﬁguration into two sectors. At one end of PS two opposite hydrothermal waves originate, and at the other end of PS they vanish. Assuming that the starting point of such counterpropagating waves is the point A (t = 0) on the left side, the two waves move towards the right side (point B) as t increases, one in a clockwise direction and the other anticlockwise. Both crests move with identical and constant velocity in opposing azimuthal directions. After a duration of one half of the essential oscillation frequency [t = 1/(2f )] both partial waves reunite at the opposite side of point A (point B). None of these waves carries on its motion; they reunite and die. At time t = 1/f a new peak of the hydrothermal waves is generated a point A and the same process is repeated. As a natural consequence of such a mechanism, the ﬂow exhibits mirror symmetry with respect to the geometric axis of the liquid bridge in any section perpendicular to PS, whereas the ﬂow will appear distorted in any section parallel to it (Figure 10.44). The computed temperature and velocity ﬁelds in a transversal cross-section for the standing-wave state and m = 2 (Pr = 30, AH = 0.4, Ma = 3.6 × 104 ) are illustrated in Figure 10.45, together with the pulsating temperature spots on the surface of the bridge. At this stage, it is also worth highlighting that the behaviour of the surface spots can be directly put in connection with the motion of the toroidal vortex centreline. As explained in Section 10.4.3 for the low-Pr case, in general, the position of the vortex centreline describes in space a sine curve having m maxima and m minima in the z direction. For high-Pr ﬂuids, during the ﬁrst half-period the maxima travel in z-direction towards the cold disk and the minima travel in the z direction towards the hot disk; in the second half the minima become maxima and vice versa. Since the displacement up and down of the toroidal convection roll is coupled to displacement out and in, respectively, with respect to the axis of symmetry of the liquid zone, this gives rise, as

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Figure 10.45 Standing wave for m = 2 (Pr = 30, AH = 0.4, S = 1, Ma = 3.6 × 104 , Ma based on the axial extension of the liquid bridge; the thermoﬂuid-dynamic ﬁeld is shown at four evenly distributed instances within an oscillation period). From left to right: temperature ﬁeld in the cross-section z = 0.75 (level 1 → −0.27, level 15 → 0.18, level = 0.03); projection of the velocity ﬁeld on the cross-section z = 0.5; temperature disturbances on the free surface (numerical simulation, M. Lappa)

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Figure 10.46 Standing wave for m = 3 (Pr = 30, AH = 0.25, S = 1, Ma = 3.8 × 104 , Ma based on the axial extension of the liquid bridge; the thermoﬂuid-dynamic ﬁeld is shown at four evenly distributed instances within an oscillation period): Radial velocity distribution (left column) in the cross-section z = 0.75 (level 1 → −17.2, level 15 → 154, level = 12.8) and temperature disturbances on the free surface (right column) (numerical simulation, M. Lappa)

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Figure 10.47 Standing wave for m = 4 (Pr = 30, AH = 0.2, S = 1, Ma = 3.9 × 104 , Ma based on the axial extension of the liquid bridge; the thermoﬂuid-dynamic ﬁeld is shown at four evenly distributed instances within an oscillation period): Radial velocity distribution (left column) in the cross-section z = 0.75 (level 1 → −7.3, level 15 → 115.2, level = 13.1) and temperature disturbances on the free surface (right column) (numerical simulation, M. Lappa)

406

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Figure 10.48 Travelling wave for m = 1 (Pr = 30, AH = 1.0, S = 1, Ma = 3 × 104 , Ma based on the axial extension of the liquid bridge; the thermoﬂuid-dynamic ﬁeld is shown at four evenly distributed instances within an oscillation period). From left to right: temperature disturbance on the liquid-bridge surface; projection of the velocity ﬁeld on the cross-section z = 0.5; sketch of surface-tension-driven convective cells (numerical simulation, M. Lappa)

an example, to the alternate expansion and contraction of the axes e1 and e2 of the elliptical inner region visible in the temperature ﬁeld in Figure 10.45 (ﬁrst column). The concept related to the mental divisibility of the ﬂow conﬁguration still holds for this case as it can be applied to the four sectors delimited by the aforementioned axes e1 and e2 . One complete period for the related time-dependent behaviour takes a time 1/f , where f is the characteristic frequency of the temperature oscillations measured at a ﬁxed point. The standing-wave modes of convection for AH = 0.25 (m = 3) and for AH = 0.2 (m = 4) are shown in Figures 10.46 and Figures 10.47, respectively. In this case, the stages of evolution are similar to those described for the case m = 2, the only difference being the number of spots and vortex cells in the azimuthal direction (notably, Figures 10.46 and 10.47 show a pulsation of inner

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triangular and quadrangular structures, respectively; the nodes of these polygons are replaced in time by the presence of sides and vice versa). This means, in general, that there are 2m pulsing extrema, engendering oscillation between two polygonal structures of opposite phases (m ﬁxed nodes identify this state as a standing wave with azimuthal wavelength 2π /m). At each instant, the ﬂow is invariant under rotation in ϕ by 2π /m. In addition, this ﬂow is also symmetric with respect to m different axes of reﬂection and, in general, one complete period for this time-dependent behaviour takes a time 1/f .

10.4.8.2 The Travelling Wave: A Rotating Spatiotemporal Mode of Convection For the travelling wave state, remarkable differences can be highlighted with respect to the prominent features of the standing wave. The waveform is not pulsating, but rotating. The surface temperature spots do not change their intensity and rotate around the perimeter of the liquid bridge. The vortex cells in the section orthogonal to the axis do not change their sense of rotation and their strength is constant in time (the cells never vanish). The time-dependent behaviour of the velocity ﬁeld is simply characterized by a full rotation of the entire ﬂow pattern in the azimuthal direction (Figures 10.48 and 10.50). For the paradigm case m = 1, in particular, as originally observed by Chun (1980a, b) resorting to a more spatial way of thinking, the spatiotemporal mode of convection could be imagined in 3D (see Figure 10.49) as determined by a steady circumferential rotation of the axis of the torous deformed due to the instability and inclined with respect to z (in the standing-wave state such an axis also undergoes continuous motion, but it is conﬁned to the symmetry plane AB; in other words, the inclined axis of the deformed Marangoni toroidal convection roll oscillates in this plane forward and backward like a pendulum). Inspection of Figure 10.50 related to the mode m = 2 reveals that the amplitude of the axes e1 and e2 is ﬁxed, whereas the inner elliptical region rotates with a centre of rotation corresponding to the axis of symmetry. In this case, one entire rotation needs a time 2/f . The rotating regimes for m > 2 (not shown) are merely characterized by a full rotation of the thermoﬂuid-dynamic ﬁeld as for m = 1 and 2 and the entire rotation of the conﬁguration needs a time m/f .

10.4.9 Symmetric and Asymmetric Oscillatory Modes of Convection At this stage, it should be mentioned that a detailed and exhaustive description of the thermoﬂuid-dynamic ﬁeld in the rotating regime also would require a proper categorization of the oscillatory state with respect to the behaviour in the generic meridian plane. In this plane, in fact, complex symmetric and asymmetric modes of convection appear according to the azimuthal wavenumber. Since these different oscillatory convective modes are caused by travelling of, respectively, symmetric or asymmetric structures of periodicity in the azimuthal direction, in practice it is possible to classify the aforementioned symmetric and asymmetric oscillatory modes simply according to the even and odd wavenumbers, respectively, like the case of low-Pr liquids (Section 10.4.3). If m is odd, the vortex in one half of the liquid-bridge section appears smaller than the opposite vortex. The time dependence is observed as a periodic interchange of the shape of the vortices in the left and right parts of the zone. After one half of the oscillation period, the small vortex and the large vortex change position (the branching streamline of the opposite vortices changes its inclination continuously during an oscillation period). This behaviour is illustrated in Figure 10.51 (m = 1), where the streamlines of the projected velocity vectors in the meridian plane ϕ = 0 are shown for eight subsequent moments in time during one cycle for the speciﬁc case AH = 1.0. The reader will clearly recognize that the characteristic sign of a mode m = 1 is a reciprocal variation

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Figure 10.49 Sketch of rotating m = 1 mode (steady rotation of the axis of the deformed torous inclined with respect to the z direction)

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Figure 10.50 Travelling wave for m = 2 (Pr = 30, AH = 0.4, S = 1, Ma = 3.6 × 104 , Ma based on the axial extension of the liquid bridge; the thermoﬂuid-dynamic ﬁeld is shown at four evenly distributed instances within an oscillation period). From left to right: temperature ﬁeld in the cross-section z = 0.75 (level 1 → −0.27, level 15 → 0.18, level = 0.03); projection of the velocity ﬁeld on the cross-section z = 0.5; temperature disturbances on the free surface (numerical simulation, M. Lappa)

of the diameters of the convection vortices on the left side and on the right side, that is, when the left vortex puffs up, the right roll contracts during a half-period and vice versa (see Monti et al., 2000a, and Savino et al., 2001b, for similar behaviours detected experimentally). For even critical wavenumbers, the time dependence is instead observed as a synchronous pulsation of two symmetrical vortices (see, e.g., Monti et al., 2000b).

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(a)

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Figure 10.51 Velocity ﬁeld (streamlines of the projected velocity vectors) in the meridian plane ϕ = 0 (AH = 1.0, S = 1, Pr = 30, Ma = 3 × 104 , Ma based on the axial extension of the liquid bridge; the ﬁeld is shown in eight snapshots evenly distributed within one oscillation period τ ) (numerical simulation, M. Lappa)

For the convenience of the reader, a survey of possible symmetric and asymmetric stages of evolution of the velocity ﬁeld in a generic meridian plane for several values of the aspect ratio is given in Figure 10.52 (for two moments in time t = 0 and t = τ /2, where τ is the oscillation period). For AH = 1.0 (m = 1) one of the two vortex cells in the section prevails over the other one and is extended along the whole axial plane of the bridge; the other cell contracts and is conﬁned in the hot

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(a)

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Figure 10.52 Streamlines of the projected velocity vectors in a generic meridian plane (Pr = 30, S = 1) for distinct values of the aspect ratio: (a) AH = 1.0, Ma = 3 × 104 ; (b) AH = 0.5, Ma = 3.5 × 104 ; (c) AH = 0.25, Ma = 3.8 × 104 ; (d) AH = 0.2, Ma = 3.9 × 104 (Ma based on the axial extension of the liquid bridge; the ﬁeld is shown at two evenly distributed instances within one oscillation period) (numerical simulation, M. Lappa)

corner, moreover, since in the boundary layer near the cold disk steep axial gradients of temperature are present, here a small Marangoni cell is still present driven by the surface-temperature distribution (see Figure 10.52a). For AH = 0.5 (m = 2) the velocity ﬁeld in the section of Figure 10.52b is symmetric and the convective cells travel axially up and down alternately. The situations for AH = 0.25 (m = 3) and AH = 0.2 (m = 4) are illustrated in Figure 10.52c and 10.52d, respectively. The liquid bridge in this case is very short and the two driving cells are conﬁned near the free surface, with other (weaker) counter-rotating vortex cells induced by continuity in the interior of the bridge (similar results were obtained using 2D computations by Rybicki and Floryan, 1987, who found, for short bridges, the emergence of several layers of vortices, with the strength of each layer decreasing approximately exponentially with the distance from the liquid/air interface). The overall conclusion from this discussion is that a characteristic feature of the asymmetric modes is a periodically ‘staggering ring’ of vortex core of the toroidal ﬂow pattern on a meridian section and also a periodically tilting branching streamline that divides the two vortices in the

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Figure 10.53 Sketch of the periodically tilting branching streamline characteristic of asymmetric modes in liquid bridges, leading to the recognizable identiﬁcation of two opposite nonsymmetric vortices in the generic meridian plane

generic meridian plane (see Figure 10.53), whereas for the symmetric modes two vortices on the meridian plane move up and down simultaneously and accordingly the tilting of the branching streamline disappears. It is also worth noting that for AH = 0.5 only one toroidal vortex exists (Figure 10.52b). When AH is reduced, however, as explained earlier, small additional vortices located around the axis of rotation and attached to the sidewalls emerge. These vortices provide some additional complexity to the dynamics. They are clearly visible in the bridges of aspect ratio AH = 0.25 and 0.2. For AH = 0.25 (Figure 10.52c), the critical wavenumber is odd (m = 3) and the additional counter-rotating vortex cells in the interior of the bridge are attached one to the hot disk and the other to the cold disk; for AH = 0.2 (Figure 10.52d), they are symmetric since the wavenumber is even (m = 4).

10.4.10 System Dynamic Evolution Lappa (1995) reported (for the speciﬁc case Pr = 30) that, even for slightly supercritical conditions, the standing wave and the travelling wave described in the preceding subsections correspond to two consecutive waveforms of the Marangoni ﬂow (in other words, ﬂow undergoing a ﬁrst transition from the axisymmetric, steady to a three-dimensional oscillatory state, characterized by the standing wave and, later, a second transition from the standing wave to the travelling wave), which means that these two states should be regarded as a secondary (transitional) and a tertiary (stable) mode of convection, respectively. Monti et al. (1996) examined the inﬂuence exerted on such dynamics by application of temperature ramps with different possible rates. Nonlinear studies based on numerical simulations also indicated that the time for the decay of one mode of oscillatory convection into the other depends on the aspect ratio of the liquid bridge as standing waves tend to be stabilized, that is, they become long-lasting, when (for a given diameter of the supporting disks and a given liquid) the aspect ratio is reduced (Lappa et al., 2001b). Motivated by these studies, some liquid-bridge experiments aimed at detecting such transition were speciﬁcally conceived under the acronym PULSAR (pulsating and rotating instabilities in oscillatory Marangoni ﬂows) and executed in microgravity (PULSAR I and II experiments) in the context of sounding rocket campaigns (MAXUS 3 in November 1998 and MAXUS 4 in April 2001, respectively).

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The major outcomes of these landmark experiments can be reported as follows. The aim of the PULSAR I experiment was the study of the oscillatory Marangoni convection in a cylindrical liquid bridge of silicone oil with kinematic viscosity of 5 cSt. The height of the bridge was equal to the disk diameter (20 mm) and the imposed temperature difference was set to 15 K during the ﬁrst 460 s and 20 K in the second part of the experiment, until the end of the microgravity period. The a posteriori analysis of the temperature proﬁles, measured by four thermocouples located at the same radial and axial coordinates but at different azimuthal coordinates (shifted at 90◦ ) and the surface temperature distribution, measured with an infrared thermocamera, showed that a pulsating and a mixed pulsating–rotating regime had been established during the experiment. The azimuthal wavenumber of the oscillatory regime, the oscillation period and the time for the onset of the oscillations were found to be in good agreement with the numerical predictions performed in the preﬂight analysis (Savino et al., 1999; Monti et al., 2000a). In the PULSAR II experiment, pulsating and rotating oscillatory regimes were clearly detected during the microgravity time at different temperature differences across the liquid bridge (T = 5, 10 and 20 K). At a certain stage (given the limited microgravity time available), T was increased from 10 to 20 K to accelerate the transition from the pulsating to the rotating regime. When the temperature difference was reduced again to 10 K, a rotating regime was ﬁnally established. For increasing values of the applied temperature difference, the following transitional stages of evolution were identiﬁed: T = 5 K → pulsating regime with fundamental frequency f = 3.3 × 10−2 Hz; T = 10 K → pulsating regime with fundamental frequency f = 4.4 × 10−2 Hz followed by a rotating regime with fundamental frequency f = 3.7 × 10−2 Hz; T = 20 K → rotating regime with fundamental frequency f = 4.3 × 10−2 Hz. The frequencies and the related wavenumbers in the different supercritical ﬂow regimes were studied through a Fourier analysis of the temperature proﬁles (Savino et al., 2001b). During the same period, the trend with an initial standing wave decaying into a travelling wave for increasing time and/or applied temperature difference was conﬁrmed numerically by other investigators (e.g. Zeng et al., 2001c, for Pr = 16 and Leypold et al., 2002, for 1.5 ≤ Pr ≤ 7; the latter authors expressly devised amplitude equations for quantitatively describing the instability of the standing-wave states in the weakly nonlinear regime). These efforts also led to the natural introduction of some simple criteria for discerning the state (pulsating or rotating), which enjoyed fairly widespread use in the literature in the framework of both experimental (e.g. the PULSAR experiments described above) and numerical studies and which deserve some discussion here also. In practice, for introducing the necessary rationale at the basis of such criteria, it is sufﬁcient to start from the observation that in the light of Eqs (10.15) and (10.16) a method for distinguishing standing from travelling waveforms could be based directly on the phase shift (hereafter denoted by φ) between probes properly located on the free surface of the liquid bridge (at the same axial position, but with a certain angular shift; Monti et al., 1997). If the oscillations are measured in points with the same axial and radial coordinates but at different azimuthal positions, for the standing wave, only two values of the azimuthal phase shift are allowed (φ = 0 or φ = π ); in contrast, for the travelling wave the possible values of the phases are not discrete. By monitoring this phase shift, therefore, it is possible to detect the regimes and in particular the transitions from one another. As an example, Figure 10.54 displays the signals provided by numerical thermocouples located on the free surface having the same axial coordinates (r = 1/2AH , z = 0.75), but different azimuthal positions (with a shift of 90◦ ) for a liquid bridge with Pr = 30, AH = 1 and Ma ∼ = 1.5Macr (for this case the azimuthal wavenumber is m = 1) undergoing the usual sequence of states: steady → standing wave → travelling wave.

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0.65 Pr=30, A=1, Ma=30 000 0.60

T1 T3

Temperature

0.55 0.50 0.45 0.40 0.35 0.30 0.18

Standing Wave Regime 0.19

0.20

Travelling Wave Regime

0.21 0.22 0.23 0.24 Non-dimensional time T2

T3

0.25

T2 T4 0.26

0.27

T1

T4

Figure 10.54 Signals provided by four numerical ‘probes’ (Pr = 30, AH = 1, S = 1, Ma = 3 × 104 , Ma based on the axial extension of the liquid bridge) (numerical simulation, M. Lappa)

During the early transient stage dominated by the pulsating behaviour, there is no phase shift between T1 and T4 and no phase shift between T2 and T3 , but T1 and T4 measure values with a phase shift of π with respect to T2 and T3 . As explained earlier, the temperature disturbance can be represented as spots on the liquid-bridge surface, pulsating while maintaining ﬁxed azimuthal positions. Since for m = 1 the azimuthal extension of each spot (360◦ /2m) is 180◦ and the thermocouples considered have an azimuthal shift of 90◦ , two thermocouples will be placed on a spot and the others on the second spot. This implies that φ = 0 (if the points are placed on the same spot) or φ = π (if the points belong to different spots). This simple argument explains why the signals in Figure 10.54 exhibit opposite phase initially. This ﬁgure also shows that the amplitudes of the temperature oscillations T1 , T2 , T3 and T4 are different, that is, the numerical probes with an angular shift of 90◦ do not measure the same maximum or the same minimum. This behaviour can also receive an interesting interpretation in terms of the relative position of surface temperature spots and thermocouples. In fact, since both the positions of the spots and of the probes are ﬁxed and the temperature distribution along each spot is not uniform (there is a peak in the ‘core’ of the spot; see Figure 10.43), each thermocouple will measure a maximum (minimum) value of the temperature depending on its local azimuthal position along the spot. When the pulsating mode of convection is taken over by the rotating mode, the amplitude of the temperature oscillations does not depend on the local azimuthal position of the thermocouples and it is the same for all the points having the same radial and axial positions (Figure 10.54). In this state, the surface spots are not ﬁxed, but rotate circumferentially, hence each of the four numerical thermocouples will measure the same maximum (minimum) temperature value when the hot (cold) spot passes on it. This is also the reason why in the rotating regime, the time–temperature proﬁles

Thermocapillary Convection 0.56

0.60

0.54

T3 versus T1

0.55

0.48

0.45 0.40

Azimuthally Standing Wave

0.46

0.35

0.44 0.44 0.46 0.48 0.50 0.52 0.54 0.56 T1

Azimuthally Travelling Wave 0.35 0.40 0.45 0.50 0.55 0.60 T1

0.60

Azimuthally Standing Wave

0.55

0.55

0.50 0.50

T2

T2

T3 versus T1

0.50

0.50

T3

T3

0.52

0.60

415

Azimuthally Travelling Wave

0.45 0.45 0.40 0.40

0.40 T2 versus T1 0.45

0.50 T1

0.55

0.60

T2 versus T1 0.35 0.35 0.40 0.45 0.50 T1

0.55

0.60

Figure 10.55 Oscillatory behaviours in the phase space (same case as considered in Figure 10.54) (numerical simulation, M. Lappa)

typically display a phase displacement depending continuously on the azimuthal coordinate (for the case chosen as an example, the critical wavenumber, in particular, is m = 1; the oscillations show a phase displacement of π /2 between two numerical thermocouples located at an angular distance of 90◦ and a phase displacement of π between two numerical thermocouples located at an angular distance of 180◦ ). Figure 10.55c and d are interesting as they indicate that the behaviours elucidated above in terms of phase shift and amplitude of the signals correspond to well-deﬁned attractors in the Ti –Tj plane. As outlined in Chapter 1, these plots are often used as idealized versions of the state of the considered system and its possible dynamic evolution. For instance, of particular interest is the interplay between T2 and T1 . During the ﬁrst regime, the system evolves along an ellipsoidal spiral (the initial point corresponds to the initial axisymmetric and non-oscillatory condition of the system), whereas in the subsequent rotating regime the attractor is given by a quasi-circle. This can be used as a distinguishing mark for detecting the oscillatory mode of convection. Between these two cases an intermediate (pulsorotating) regime occurs, with a phase shift increasing (decreasing) with time from zero (or π ) to π /2 (Figure 10.56). This case is interesting since it can be used to justify some unexpected behaviours reported in the literature (Velten et al., 1991). Using three thermocouples having the same axial and radial coordinates but different azimuthal positions, Velten et al. measured in many situations phase shifts changing continuously in time that, in the light of the above arguments, could be simply interpreted as a transition between pulsating and rotating modes of convection.

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0.60

0.55

T2

0.50

0.45

0.40

0.35 0.35

0.40

0.45

0.50

0.55

0.60

T1

Figure 10.56 Transition from the standing to the travelling-wave state in the phase space (Pr = 30, AH = 1, S = 1, Ma = 3 × 104 , Ma based on the axial extension of the liquid bridge; signals provided by two thermocouples with an angular shift of 90◦ ) (numerical simulation, M. Lappa)

Notably, the criterion (and related spatial interpretation in terms of temperature spots and signals) discussed in the foregoing for m = 1 focusing on T1 and T2 (at 90◦ ) can be generalized to the case m = 1 in a simple way by choosing thermocouples with an azimuthal shift of 90◦ /m. The study of such dynamic evolution and of the subsequent possible transitions to even more complex spatiotemporal behaviours for Ma Macr has attracted much attention over the years. As also mentioned at the beginning of Section 10.4, the liquid bridge with transparent liquids has often been used, in fact, as a mere pattern-forming dynamic system and as a paradigm for studying the progression from order to chaos (see, e.g., the space experiments of Chun and Siekmann, 1995, and the recent numerical investigations of Shevtsova et al., 2003b, for Ma up to ∼10 Macr ). In general, a precise hierarchy of bifurcations can be identiﬁed with a succession of bifurcations of increasing spatiotemporal complexity as Ma is increased. The new bifurcations generally correspond to the emergence of a new mode with its own wavenumber and independent frequency. As an example for Pr = 4 and AH = 0.5, Shevtsova et al. (2003b) found the basic two-dimensional steady ﬂow to become oscillatory with azimuthal wavenumber m = 2 as a result of the Hopf bifurcation at Ma = Macr with a second independent solution with wavenumber m = 3 appearing at Ma = Macr2 . Most interestingly, the two branches of travelling waves with m = 2 and 3 were observed to coexist for Ma > Macr2 . In that work, the onset of temporal nonperiodicity (chaos) was shown to be associated with the development of broadband noise in spectra (a set of incommensurate frequencies) preceded by a quasi-period state (two incommensurate frequencies). Also, they observed for some solutions intermittency phenomena with transition back to a periodic oscillatory behaviour at higher values of the Marangoni number. They concluded, however, that the Ruelle–Takens scenario prevails over others (see Section 1.8.1 for general background information about the routes that a system may take in transitioning to a chaotic state).

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Given the lack of space experiments, additional insights into these behaviours will be provided in Chapter 11 when discussing the outcomes of typical terrestrial investigations (microscale studies).

10.4.11 The Hydrothermal Mechanism in Liquid Bridges This section is devoted to a ‘reconstruction’ of the physical mechanism by which the aforementioned counteracting hydrothermal waves predicted by Smith and Davis (1983) and then by Kuhlmann and Rath (1993a) are initiated and lead to the subsequent stages of evolution with pulsating (and then rotating) temperature spots and azimuthal convective structures in liquid bridges. Before starting to deal with such a description, however, it is worth mentioning some arguments that have been made since 1980 by Chun and co-workers and which will prove very useful later. Chun and Wuest (1979) speculated that the growth or the damping of a small temperature disturbance must depend on the ratio of the rapidity of the heat transfers by conduction and by convection. For the extreme case of very high conductive heat transfer, a temperature disturbance should die diffusing very rapidly in all directions. In the other extreme case of very high convective heat transfer, the disturbances should move with the ﬂow, ‘frozen’ in the ﬂow motion. In practice, the ﬁrst case corresponds to low-Pr liquids for which the high thermal diffusivity tends to spread any possible temperature disturbance over a large extension in a very short time. Among other things, this may explain why the instability mechanism for Pr 1 is not based on the ampliﬁcation of temperature disturbances, but is essentially hydrodynamic in nature (see Section 10.4.1). The second case corresponds to Pr 1 for which the thermal diffusivity is relatively small compared with the kinematic viscosity. This is very interesting within the context of this section since it makes possible the introduction of a simple mechanism explaining the behaviours discussed in the preceding subsections. According to Chun (1980a), a temperature disturbance on the surface leads to a disturbance of the temperature gradient and thus to a disturbance of the surface tension gradient that induces a distortion of the velocity ﬁeld. The velocity distortion, in turn, is responsible for a distortion of the temperature distribution and this coupling mechanism between surface tension gradient and heat transfer may generate growth or damping of the initial temperature disturbance. Since, as explained earlier, the ampliﬁcation or damping of temperature disturbances on the free surface must depend on the relative importance between convection and diffusion (i.e. on the order of magnitude of the Marangoni number), the disturbances can grow and instability sets in if a critical temperature difference is exceeded (i.e. if the Marangoni number is large enough). The mechanism leading to the onset and ensuing development of hydrothermal waves illustrated in the text below is in line with this theory. Hereafter, in particular, following the same approach as undertaken in Section 10.2.3 for the inﬁnite layer and in Section 10.3 for the annular conﬁguration, the discussion is supported by some practical examples with hot or cold disturbances on the free surface of the liquid bridge. In such a context, let us start by observing that if a random hot disturbance is created on the free surface at a rather high position (not too distant from the hot disk), it will generate a surface azimuthal ﬂow resulting in two counterpropagating currents moving far away from the disturbance (Figure 10.57a). For continuity, a radial ﬂow will be generated and, owing to the presence of cold bulk ﬂuid, cold ﬂuid under the surface is carried at the interface, cooling the hot spot (Figure 10.57b). The phase difference between the temperature and the velocity disturbances will be responsible for further cooling of the free surface, resulting in a cold spot (Figure 10.57c). Such a mechanism is clearly visible if one compares frame by frame, for example, the snapshots in Figures 10.43. The hot disturbance causes a pair of counter-rotating vortex cells in the cross-section. Due to the phase difference between surface azimuthal- and radial-velocity

418

Thermal Convection: Patterns, Evolution and Stability Hot Spot

Hot Spot

Cold Spot Hot Spot

Hot Spot Cold liquid

(a)

(c)

(b) Cold Spot

Hot Spot

Hot Spot Cold liquid

(d)

(e)

Figure 10.57 Sketch illustrating the origin of counterpropagating hydrothermal waves on the liquidbridge free surface

components, when the hot spot disappears the radial ﬂow is different from zero and moves cold liquid to the surface; therefore, a cold spot is generated. If the initial disturbance starting the mechanism is not hot but cold, its main effect will involve a disturbance in the axial component of the surface velocity. The surface velocity directed from the hot disk to the cold one will increase locally (upstream with respect to the disturbance), leading to an overheating of the initial cold disturbances so that it is replaced by a hot spot (Zeng et al., 2004). At this stage, it is evident that, owing to coupling of these mechanisms, hot and cold disturbances can replace each other in a periodic fashion and lead to the emergence of a pulsating pattern. Superimposed on this is the fact that since for the case of high Prandtl number (as explained earlier), surface-temperature disturbances are somehow ‘frozen’; they can be convected on the free surface in the azimuthal direction without being destroyed by the effect of diffusion. Figure 10.57c shows, for instance, how, when cold ﬂuid is carried at the interface due to continuity effects, the hot disturbance, initially located in that position, in principle can be transported by Marangoni ﬂow in clockwise and counterclockwise directions towards new positions. In practice, by these effects, the pulsating mechanisms initiated by a random single disturbance at a certain azimuthal position can propagate along both the azimuthal directions on the surface (Figure 10.57d) until it affects the entire circumferential extension (Figure 10.57e). Among other things, such a discussion also illustrates how the counterpropagating azimuthal hydrothermal waves are initiated (precise mathematical arguments supporting the discussion above have been given in Section 10.4.8). At this stage, it also becomes evident that the emergence of the travelling wave simply corresponds to a subsequent azimuthal symmetry breaking in the mechanism described above (often the standing-wave state persists for such a long time that it might seem stable; however, a small

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reﬂection-symmetry breaking imperfection develops that eventually leads to the transition to the travelling wave). Available numerical results show, in fact, that during the standing-mode of convection, the whole body of the liquid bridge does not rotate (i.e. the azimuthally moving disturbance does not carry on its motion); standing waves do not have a detectable azimuthal mean ﬂow (i.e. the mean value of the oscillating azimuthal velocity is zero). In contrast, in the travelling wave state, a small azimuthal mean ﬂow is observed (see Figure 10.58, where at a certain distance from the onset, when the instability mechanism becomes dominated by a travelling wave, the velocity proﬁle loses its symmetry with respect to the zero value). Similar behaviours were reported experimentally by Schwabe et al. (1996), who under supercritical conditions found an additional clearly visible oscillation in the azimuthal direction. They observed the azimuthal oscillation of tracer particles captured in the vortex centre (a phenomenon that was referred to as ‘dynamic particle accumulation structure’ and is currently generally known under the acronym PAS) to be nonsymmetrical and characterized by a mean value of the azimuthal velocity (i.e. net ﬂow in the azimuthal direction). In particular, groups of tracer particles (trapped in the centre of the Marangoni vortex roll) were seen moving azimuthally back and forth asymmetrically, such that there was a net movement of the particles in the direction of propagation of the travelling wave. These initial ﬁndings (obtained using a vertical light sheet) were reﬁned 10 years later by Schwabe et al. (2006) and Tanaka et al. (2006), who provided a precise reconstruction of the path followed in space by particles through both top and side views. They observed the particles gathering along a closed spiral loop that winds itself around the toroidal vortex (observed from above, the spiral loop looks as if it is rotating azimuthally). The number of spirals was found to be equal to the azimuthal wavenumber of the travelling wave with each spiral consisting of one or two turns.

20.00

Non−dimensional azimuthal velocity

Standing wave regime Travelling wave regime

10.00

0.00 Mean value −10.00

−20.00 9.50

10.00

10.50 11.00 Non−dimensional time

11.50

12.00

Figure 10.58 Surface azimuthal velocity in a ﬁxed point as a function of time (Pr = 4, AH = 0.5, Ma = 4.8 × 103 , Ma based on the height of the liquid bridge). After Lappa et al. (2001b)

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It is also worth mentioning that some additional insights into this interesting phenomenon were provided a year later by Schwabe et al. (2007). They clearly proved that particles can accumulate in a dynamic string for certain aspect ratios of the liquid bridge and at, typically, two times the critical Marangoni number for the onset of time dependence (this was observed for particles with a density higher and lower than that of the ﬂuid and for the isodense case). In particular, PAS was observed to occur at a resonance between the azimuthally travelling wave and the ‘turnover time’ of the PAS-string in the thermocapillary vortex. Remarkably, another factor that can be used as a distinguishing feature between standing and travelling waves is the behaviour of the average Nusselt number on the cold and hot disks of the bridge. Available computations show that during the standing wave these average values are not constant in time but oscillate at a frequency that is double that of the temperature oscillations measured in a generic point. When the standing wave is replaced by the travelling wave, the average Nusselt number on both plates converges to a constant value that is smaller than the average Nusselt number related to the axisymmetric regime. This is shown in Figure 10.59. The Nusselt number oscillates with a frequency 2f during the pulsating regime but the amplitude of the oscillations decreases in time up to a constant value when the standing wave is taken over by the travelling wave (in practice, the decrease in the average Nusselt number can be explained on the basis of the amount of axial momentum that is converted in azimuthal momentum after the onset; the azimuthal ﬂuid motion, in fact, does not contribute to the axial heat transfer near the disks). As a concluding remark, let us observe that the oscillation frequency f = ω/2π does not depend signiﬁcantly on the waveform. Monti et al. (1995), in fact, found experimentally that, in general, the dimensionless frequency for liquid bridges with S ∼ = 1 and Pr > 1 (8 ≤ Pr ≤ 74 and aspect ratio 0.1 ≤ AH ≤ 1.6) can be expressed as f =

1 2 2 3 1 L2 α 1 Ma 3 L− 2 D − 2 = Ma 3 AH 2 2π α 2π

(10.18)

Figure 10.59 Average Nusselt number on the hot disk as a function of time (Pr = 4, AH = 0.5, Ma = 4.8 × 103 , Ma based on the axial extension of the liquid bridge). After Lappa et al. (2001b)

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10.4.12 Noncylindrical Liquid Bridges This section considers the case in which the free surface of the liquid bridge is not straight, that is, the liquid bridge is convex (fat) or concave (slender). Some theoretical studies have been reported in the literature, in fact, also addressing the role potentially played in the typical dynamics of high-Pr liquid bridges by non-unit values of the parameter S (for the case of low-Pr liquids, the reader is referred to Section 10.4.4). Most notably, such studies highlighted possible departure from the well-known hydrothermal mechanisms in some circumstances, which makes some discussion on this subject worthwhile. Chen and Hu (1998a) were the ﬁrst to carry out a linear stability analysis accounting for the effect of a non-cylindrical interface (S = 1) in zero-g conditions (considering three different values of the Prandtl number, Pr = 1, 10 and 50, a range of aspect ratios, 0.4 ≤ AH ≤ 1.4 for three values of S = 0.8, 1 and 1.2, and a wide range of nondimensional volumes 0.4 ≤ S ≤ 1.4 for the particular case AH = 0.6). They reported that for high-Pr liquids the critical values of the Marangoni number related to the classical hydrothermal instability are inﬂuenced signiﬁcantly by the liquid-bridge volume (e.g. for Pr = 10 a variation of 20% in the volume of the liquid bridge can lead to a variation of 100% in the critical Marangoni number; in particular, the ﬂow tends to be stabilized for 0.8 < S < 1, as also conﬁrmed by experiments, e.g. Hirata et al., 1997c). Later investigations on the subject for higher values of the Prandtl number (the linear stability analysis by Chen and Hu, 1999, for Pr = 100, S = 1.2 and AH = 0.6 and the numerical study by Tang et al., 2001, for a 10 cSt silicone oil), however, revealed some aspects overlooked in earlier investigations, namely the possible existence of a stationary bifurcation to 3D steady ﬂow preceding the well-known Hopf bifurcation of hydrothermal nature for some ranges of the nondimensional volume S [mostly for relatively fat (convex) liquid bridges]. Such a behaviour somehow resembling that occurring for low-Pr ﬂuids (Sections 10.4.2–10.4.6; see, e.g. Figure 10.35), with the existence of two consecutive transition at Macr1 and Macr2 leading the system from an initial steady and axisymmetric state to a ﬁnal 3D oscillatory mode, is clearly shown in Figure 10.60. For a ﬁxed aspect ratio, the liquid bridge exhibits different dynamics according to the volume of liquid held between the supporting disks. The ﬂow undergoes a transition directly to oscillatory ﬂow if the surface is concave and the volume small (i.e. Tcr1 = Tcr2 ), whereas in the case of larger volumes 3D ﬂow occurs prior to the transition to oscillatory ﬂow, that is, there is a ﬁrst instability with 3D non-oscillatory ﬂow and a second bifurcation to oscillatory convection if the Marangoni number is further increased (Tcr2 > Tcr1 ). The subject has been investigated especially by Hu and co-workers (10 cSt silicone oil). Interestingly, it has been shown that, in general, the nondimensional volume Sc at which there is the radical change in the system response depends on the aspect ratio considered (e.g. Sc ∼ = 0.9 for AH = 0.8 and Sc ∼ = 0.97 for AH = 0.5). Furthermore, for a ﬁxed volume (i.e. a ﬁxed value of S), there exists a critical value of the geometric aspect ratio Ac such that if the considered geometric aspect ratio AH is larger than the critical value Ac the stationary bifurcation is suppressed. For S = 1 (cylindrical liquid bridge) and a 10 cSt silicone oil (see Figure 10.61), such a value was found to be slightly larger than unity (Ac = 1.25), which may explain why the stationary bifurcation was not reported in microgravity experiments (many such experiments considered relatively large values of the aspect ratio which cannot be established under normal gravity conditions). Another reason may be related to the fact that none of the experiments performed in space was expressly conceived for detecting such a transition.

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The right part of Figure 10.61 is divided into three regions by two curves. The lower curve gives the ﬁrst critical temperature difference, while the upper curve gives the second critical temperature difference. As explained earlier, these curves converge towards a unique threshold for AH > 1.25 [(AH )−1 < 0.8 in the ﬁgure], which leads to the general interesting conclusion that there will exist a single-bifurcation transition for the onset of oscillations provided that both of the following conditions are true: (1) the liquid bridge volume ratio is smaller than the critical value Sc and (2) the aspect ratio is larger than the critical value Ac .

Figure 10.60 Stability map in the (T , S) plane (AH = 0.8, 10 cSt silicone oil). Transition from steady axisymmetric convection to 3D steady ﬂow (the dashed line represents the corresponding stationary bifurcation) and to 3D oscillatory ﬂow (solid line, Hopf bifurcation). Courtesy of W.R. Hu and Z.M. Tang

Figure 10.61 Stability map in the (T , AH ) plane (S = 1, 10 cSt silicone oil). Transition from steady axisymmetric convection to 3D steady ﬂow (dashed line, stationary bifurcation) and to 3D oscillatory ﬂow (solid line, Hopf bifurcation). Courtesy of W.R. Hu and Z.M. Tang

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Some experimental conﬁrmation of the existence of the stationary bifurcation has been obtained in the framework of ground-based studies (see Chapter 11). Nevertheless, the subject is still open and deserves additional investigation. A linear stability analysis for zero-g conditions was carried out recently by Chen and Liu (2008), who extended the earlier linear stability analysis of Chen and Hu (1998a), considering a larger value of the Prandtl number (Pr = 102 , 0.4 ≤ S ≤ 1.2 and AH = 0.75).

10.4.13 The Intermediate Range of Prandtl Numbers As widely illustrated in the foregoing text, the instability of Marangoni ﬂow displays different features depending on whether the Prandtl number is low [Pr < O(1)] or high [Pr ≥ O(1)]. As already explained, in the former case the instability is hydrodynamic in nature (the 3D ﬂow is initially steady and can become oscillatory as a consequence of a secondary instability), whereas the latter has a hydrothermal origin. It is well known that for both cases (in the present section the discussion is limited to the case of liquid bridges with a straight interface, i.e. S = 1), the critical Marangoni number for the ﬁrst bifurcation is an increasing function of the Prandtl number. From the analysis of Wanschura et al. (1995), the onset of stationary 3D ﬂow occurs at Re ∼ = 2 × 103 for aspect ratios close to AH = 0.5 and Pr < 0.03, which means that the order of magnitude of the critical Re for these low-Pr ﬂuids is nearly constant (accordingly, the Marangoni number Ma = RePr increases linearly with Pr). On the basis of the data summarized in the analysis of Kuhlmann (1995) for 5 < Pr < 100 and AH = 0.5, the critical Reynolds number for the onset of oscillatory ﬂow can be approximated in such a range by Re ∼ = 103 (although, for the sake of completeness, it should be mentioned that, according to a subsequent study by Xun et al., 2008, such a rule tends to overestimate the critical threshold Macr = Recr Pr for Pr > 50). Despite such fairly well-established knowledge and related general trends, the intermediate range of Prandtl numbers between approximately 0.07 and 0.8, which joins the low and high ranges, is complicated and has not been studied to the same extent. In particular, one striking feature is that the axisymmetric base state is much more stable in this intermediate range than at high or low Prandtl numbers. Within such a context, it is worth citing the analysis of Levenstam et al. (2000), who observed in this range (for the case AH = 0.5 and S = 1) the existence of four different oscillatory modes (see Figure 10.62) with different qualitative features (many of which were found to replace the stationary ﬁrst bifurcation that occurs for lower values of the Prandtl number). Speciﬁc important details deserving additional elaboration (and also clariﬁcation of the underlying physical mechanisms) are provided in the following. The range of Prandtl number between 0.07 and 0.8 (just below unity) is fairly complicated since there the low-Pr hydrodynamic instability competes with the high-Pr hydrothermal instability. Along these lines, for instance, Wanschura et al. (1995) found that the critical Reynolds number increases dramatically as the Prandtl number either increases from below towards the value 0.05 or decreases from above towards a value around 0.8 (they considered the ranges 0–0.05 and 0.5–4.8). Similarly, the critical Marangoni numbers computed by Rupp et al. (1989) showed two separate branches, one at low and one at high Prandtl numbers (in their plot of Marangoni number versus Pr, these two branches are separated by a discontinuity). It is known (Wanschura et al., 1995) that in the case of a pure hydrodynamic instability, the azimuthal ﬂow related to the 3D state is in the opposite direction to that which could be expected from thermocapillarity, that is, the azimuthal ﬂow is towards the hot spots on the free surface (see Section 10.4.4); hence thermocapillarity acts to suppress this ﬂow (i.e. it acts as a weak force counteracting the azimuthal ﬂow triggered by the hydrodynamic instability).

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(a)

(b)

Figure 10.62 Critical Reynolds number (a) and nondimensional frequency (b) as a function of the Prandtl number in the so-called intermediate range (AH = 0.5 and S = 1; Re based on the axial extension of the liquid bridge): S3D(2), stationary bifurcation with m = 2 for 0 < Pr < 0.057; OSA(3), oscillatory bifurcation with m = 3 for 0.057 ≤ Pr < 0.070; OSB(2), oscillatory bifurcation (standing wave) with m = 2 for 0.070 ≤ Pr < 0.183; OSC(3), oscillatory bifurcation (standing wave) with m = 3 for 0.183 ≤ Pr ≤ 0.840; OSD(2), oscillatory bifurcation (travelling wave) with m = 2 for Pr ≥ 0.840. After Levenstam et al. (2000); Reproduced by permission of the American Institute of Physics

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According to the study of Levenstam et al. (2000), for Pr < 0.02 this effect is too weak to suppress the instability. At Pr ∼ = 0.02, however, the critical Reynolds number starts to increase with increasing Pr; at these Prandtl numbers, the temperature disturbances on the free surface are large enough for the counteracting Marangoni force to inﬂuence the stability limit (the increase in critical Reynolds number is due to the opposing Marangoni force on the free surface). For Pr ≥ 0.0585, the axisymmetric state bifurcates directly to an oscillatory state. For 0.0585 ≤ Pr ≤ 0.0697, the instability is still of a hydrodynamic origin, but for 0.0697 ≤ Pr ≤ 0.183, the mechanism for the oscillatory instability is given by the interaction of the resisting Marangoni force on the free surface and the hydrodynamic instability responsible for the stationary bifurcation that occurs for Pr < 0.0585 (the hydrodynamic disturbances create a 3D state with hot and cold spots on the free surface; thermocapillary forces then become so strong as to restore the ﬂow ﬁeld to an axisymmetric state which then again becomes hydrodynamically unstable; as a consequence, hot and cold spots exchange places periodically). In the range 0.18 ≤ Pr < 0.84, the most critical branch is an oscillatory state that shares some features with the hydrothermal wave mechanism described in Section 10.4.11 (a standing wave-like regime appears). For Pr ≥ 0.84, the critical Reynolds number decreases rapidly with increasing Prandtl number (in this range the dominant instability mechanism is hydrothermal). These results are in fairly good agreement with those obtained by Chen et al. (1997), who found the ﬁrst bifurcation to be oscillatory for Pr ≥ 0.1 and a dramatic increase in the critical Marangoni number in the range 0.1 ≤ Pr < 1 with a peak at Pr = 0.7, and with those yielded by Gelfgat et al. (2005) for the full-zone model. As a concluding remark, it should be mentioned that despite these complex behaviours for the aforementioned intermediate range of Pr, many investigators have tried to introduce analytical relationships to relate the critical Marangoni number for the onset of the oscillatory instability (secondary in the case of Pr 1, primary in the case of Pr > 1) to the value of the Prandtl number. Rupp et al. (1989) for Pr > 1 obtained the following dependence of the critical Marangoni number on the Prandtl number (not fulﬁlled, however, for semiconductor melts): Macr = 2884 Pr0.638 for 1 ≤ Pr ≤ 49 and AH = 0.6, S = 1

(10.19)

Cr¨oll et al. (1998) found that the strong dependence of the critical Marangoni number Macr2 upon the Prandtl number can be expressed as Macr2 = 2.2 × 104 Pr1.32

(10.20)

Yang and Kou (2001) observed that a relationship able to ﬁt reasonably well the experimental data (Marangoni number for the onset of oscillatory ﬂow) of a number of materials (including Sn, Si, Bi, GaSb, GaAs, KCl, NaNO3 and C24 H50 ) is Macr = 2 × 103 Pr0.6 for 10−2 ≤ Pr ≤ 102 and AH = O(1), S = 1

(10.21)

11 Mixed Buoyancy–Marangoni Convection In general, numerous intermediate situations exist between the pure gravitational (buoyant) and Marangoni ﬂows. One of the most remarkable achievements in recent years was the discovery that these dynamics and the related transitional stages of evolution are largely determined by a sort of very complex dialectics between the characteristic modes of instabilities pertaining to the two limiting cases. Before expanding on such delicate (and in many cases counter-intuitive) interactions, let us start by recalling here some fundamental notions introduced in earlier chapters of this book. As widely illustrated in Chapters 4–7, buoyancy convection arises, in the presence of gravity, due to density gradients induced by temperature differences applied to the liquid; while Marangoni ﬂow (the reader is referred, in particular, to Chapter 10; as in that chapter, discussion here will be limited to systems in which the temperature gradient is imposed in a direction parallel to the interface) is the typical outcome of a free (liquid/gas or liquid/liquid) surface subjected to a temperature gradient. The latter is by deﬁnition a surface phenomenon and is generally conﬁned to regions near the free surface, whereas buoyancy drives ﬂuid motion basically involving the whole bulk of the considered ﬂuid. Marangoni ﬂow can be inhibited in the absence of free ﬂuid/ﬂuid surfaces and is a gravityindependent phenomenon (it does not disappear even if zero-g conditions are considered); in contrast, buoyancy convection can arise regardless of the presence of any free liquid surface (being induced by the presence of accelerations). Buoyancy forces are, in fact, volume driving actions [see the body force source term in Eq. (2.5)], whereas Marangoni stresses are surface driving actions [see Eq. (2.12)]. This chapter is entirely devoted to elucidating the possible interplay of such phenomena in situations where the two driving forces of different nature mentioned above are simultaneously responsible for the generation of ﬂuid motion (typically in terrestrial conditions). This subject (hybrid or mixed convection) is of particular importance as the identiﬁcation of the most dominant mechanism, and/or a proper understanding of the mutual interference of different mechanisms involved with a comparable intensity, may help researchers in elaborating rational guidelines relating to physical factors that can increase the probability of success in practical technological processes (as discussed in Chapter 3, in fact, in many terrestrial industrial applications ﬂuid motion is brought about by such coexisting mechanisms). Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Beyond practical applications, it is fairly clear that these problems have also attracted the interest of researchers and scientists as a consequence of the complexity of the possible stages of evolution, of the nonlinear behaviour and because they are a rich source of material propaedeutical to the development of new ideas concerning the way by which systems driven by more than a single force evolve. In normal gravity and in the presence of free surfaces, if the size of the system is not limited to a few millimetres, both Marangoni and buoyancy convection play an important role in determining the structure of the ﬂow and can interact in a very complex way even if the resulting ﬂow is in a steady state (Kirdyanshkin, 1984; Villers and Platten, 1987; Shyy and Chen, 1991). In the environment provided by microscale experimentation (if the liquid volume and/or the height of ﬂuid are small enough), Marangoni ﬂows, induced by the temperature dependence of the surface tension along the free surface of the liquid, tend to prevail over gravitational convection (the reader is referred, in particular, to Section 2.3.2, to the concept of microzone deﬁned therein and related theoretical bases). Anyhow, even under such conditions, the delicate interaction between surface tension-driven convection and residual gravitational effects may have a profound inﬂuence on the emerging patterns and especially on the ensuing hierarchy of bifurcations (this is the reason why these cases were not treated in Chapter 10). Along these lines, it is worthwhile to consider why, unlike pure buoyancy or Marangoni convection for which, due to the extensive body of theory and experiments described in the preceding chapters, a rigorous framework has been introduced (able to predict and elucidate a variety of experimental observations), a clear picture for the analogous problem of mixed ﬂows has not yet emerged. It is basically due to the increased number of parameters potentially affecting the system response (among them the possible static deformation of the free surface). In such a context, the principal objectives of this chapter are as follows: 1. The analysis of the succession of instabilities and the life of various structures in the course of evolution from an initial state affected by both gravitational and surface tension-driven phenomena. 2. The investigation of the mutual transformation of these structures as the control conditions are varied (three cases are possible in principle according to the nondimensional parameter W deﬁned in Chapter 2 [Eq. (2.26)]: (a) dominating Marangoni forces, (b) dominating buoyancy forces and (c) driving forces acting with a comparable intensity). 3. Understanding the cause-and-effect relationships at the root of the observed behaviours. Anyhow, advanced formal discussions are limited to a minimum. In line with the general spirit of this book, situations are selected that can be described with a relatively simple formalism (where additional layers of knowledge are particularly appropriate, reference is made to preceding chapters). Indeed, the chapter does try to provide a comprehensive review of the research literature available for each of the fundamental geometries deﬁned in Chapter 2. Each problem is used, in particular, as an opportunity for discussing fundamental issues that are shared among the two areas of thermogravitational and thermocapillary ﬂows and, therefore, can be said to unify the study of these subjects. For the emergence of such a uniﬁed picture (and also for breaking down the barriers between the two ﬁelds), it becomes necessary to introduce or adopt a new vocabulary (included in this new vocabulary are words that were not considered in one area while being extensively used in the other). Along these lines (and given the overall high degree of complexity of the subject), prior to entering the present chapter, the reader is strongly encouraged to undertake preliminary readings of Chapters 6 and 10 and become familiarized with related crucial concepts (symbolism, fundamental terminology, classiﬁcation of possible instabilities, related physical mechanisms, etc.). The order of presentation of the topics is a matter of taste. As outlined above, many of the sections are self-contained discussions of a particular system or question. The ﬁrst section,

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however, contains material that provides a foundation for the rest. Part of the role of this section is a review of concepts that will be used in later sections so that, despite the unavoidable reference to earlier chapters, the treatment is more self-contained (because of the interdisciplinary nature of the subject matter, the ﬁrst section is considered to have particular importance). Finally, let us note that the general approach is based on the evaluation of available results coming from ground-based research, scientiﬁc microgravity experiments (used for comparison) and especially numerical investigations. Indeed, such cross-comparisons are instrumental in discerning complex interwoven or overshadowed phenomena and unravelling the underlying delicate mechanisms.

11.1

The Canonical Problem: The Inﬁnite Horizontal Layer

The simplest model for this kind of ﬂows is the canonical layer of inﬁnite horizontal extent already examined as a paradigm reference case in Chapters 6 and 10 for pure buoyancy (the Hadley ﬂow) and Marangoni convection (the return-ﬂow solution), respectively. Following the same approach (linear stability analysis) employed in Sections 6.1 and 10.2, the related basic ﬂow is represented by 

 Rag1 (y) + Mag2 (y)  0 V0 =  0 T0 = x + Raf1 (y) + Maf2 (y)

(11.1) (11.2)

f1 , f2 , g1 and g2 being known polynomial expressions with constant coefﬁcients (see Section 2.4 for useful information about these analytical solutions of the thermal-convection equations and related properties), Ra = GrPr = gβT γ d 4 /να and Ma = RePr = σT γ d 2 /µα, respectively, where γ is the rate of uniform temperature increase along the x-axis. The resulting system of ordinary differential equations for the disturbance amplitudes ud , vd , wd , pd and Td [similar to Eqs (6.7)–(6.9) already derived in Chapter 6 and to Eqs (10.6)– (10.8) in Chapter 10] can be cast in compact form as follows: Continuity equation: i(qx ud + qz wd ) + vd = 0

(11.3)

Momentum: Pr(ud − q 2 ud ) − [iqx ud (Rag1 + Mag2 ) + vd (Rag1 + Mag2 )] − iqx pd = λud

(11.4a)

Pr(vd − q 2 vd ) − iqx vd (Rag1 + Mag2 ) − pd + PrRaTd = λvd

(11.4b)

Pr(wd − q 2 wd ) − iqx wd (Rag1 + Mag2 ) − iqz pd = λwd

(11.4c)

Energy: Td − q 2 Td − [iqx Td (Rag1 + Mag2 ) + ud + vd (Raf1 + Maf2 )] = λTd

(11.5)

where the primes denote differentiation with respect to y and q is the overall wavenumber q = ) qx2 + qz2 , with boundary conditions for velocity ud = 0 vd = 0 wd = 0

(11.6a) (11.6b) (11.6c)

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for y = − 1/2 (bottom wall), and ud = −Maiqx Td vd = 0 wd = −Maiqz Td

(11.7a) (11.7b) (11.7c)

for y = 1/2 (free surface). As usual, these conditions must be supplemented with the thermal conditions, which read Td = 0

(11.8a)

Td = 0

(11.8b)

Td = −BiTd

(11.8c)

for an adiabatic boundary,

for a conducting boundary and

for a boundary with a given heat exchange rate [measured by the relevant Biot number Bi deﬁned by Eq. (2.24) in Section 2.3]. Available results on these typical cases are discussed in detail in the following pages, giving emphasis, in particular, to the study of Gershuni et al. (1992) (already considered as a source of vital information in Section 6.1 for thermogravitational convection in the rigid–rigid case) and the analysis of Parmentier et al. (1993) [in practice, the former analysis concentrates on conducting boundary conditions and the latter addresses the conﬁguration with adiabatic horizontal surfaces; for the third case, deﬁned by Eq. (11.8c), the reader may consider the subsequent study by Mercier and Normand, 1996, for which additional details will be provided in Section 11.4]. Given the high sensitivity that hybrid convection has proven to exhibit to the type of thermal conditions deﬁned by the three variants of Eq. (11.8), to the Prandtl number and to the relative importance of buoyancy and surface tension-driven effects (measured by the aforementioned nondimensional parameter W = Ma/Ra), in the following the exposition is split into several subcases according to all the possible combinations of such factors (Pr, W , type of thermal conditions). Whenever possible, analogies and differences among these subcases are illustrated; also, as anticipated, considerable space is devoted to explaining the underlying physical mechanisms, to providing critical and fruitful comparisons (or links) with the parent phenomena (pure thermogravitational or Marangoni ﬂow) and to placing the problem into a more general theoretical framework. Some fundamental information about the properties of the plane-parallel ﬂows (which deﬁne a class of basic solutions for the present problem as the parameter W is varied in the range 0 ≤ W ≤ ∞) has already been provided in Sections 2.4.3 and 2.4.4. In particular, it has been illustrated therein that for 0 ≤ W ≤ 0.25 the velocity proﬁle has an inﬂection point whose position depends on W (as shown in Figure 11.1, it moves towards the upper free boundary when W increases from zero and it disappears when W > 0.25). Moreover, for the speciﬁc case of conducting boundaries (as in the rigid–rigid conﬁguration examined in Section 6.1.3), the temperature proﬁles display two zones of unstable stratiﬁcation near the boundaries and one zone of stable stratiﬁcation in the middle of the layer having location and thickness depending on W . In line with the above arguments, for conducting boundaries Gershuni et al. (1992) found the mechanisms leading to ﬂow instability for weak Marangoni ﬂow [W < O(10−1 )] to be essentially the same as in the pure buoyancy case with a free upper surface (oscillatory 2D hydrodynamic mode, OLR and Rayleigh mode). Given the sensitivity of the velocity and temperature proﬁles to W mentioned above, however, the values of the Prandtl number where the branches pertaining to these disturbance modes intersect

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Figure 11.1 Basic-state velocity and temperature proﬁle as a function of the nondimensional parameter W = Ma/Ra, for an inﬁnite horizontal liquid layer bounded from below by a rigid wall and from above by a ﬂat free surface (case with conducting boundaries)

(Pr∗ according to the symbolism used in Chapter 6) depend slightly on the speciﬁc value of W considered (e.g., for W = 0.025 as shown in Figure 11.2a, the oscillatory hydrodynamic mode is the dominant mechanism of instability for Pr < 0.045, being replaced by the 3D oscillatory longitudinal rolls for 0.045 ≤ Pr ≤ 0.85 and the Rayleigh mode for Pr > 0.85; the reader may compare these ranges with those provided in Section 6.1.2 for W = 0). For the companion conﬁguration with adiabatic boundaries (Eq. 11.8a), for which the Rayleigh mode is no longer allowed (see Section 6.1.3 for further explanations and details), interesting information was initially provided by Laure and Roux (1989). Notably, for such thermal boundary conditions and relatively small values of the Prandtl number (Pr = 0.02), they found surface Marangoni stresses to exert a profound stabilizing effect on the 2D hydrodynamic mode of gravitational nature already for W ≥ 0.02 (their analysis was focused on the range 0 ≤ W ≤ 0.04). Many other studies have appeared over the years specially concentrating on the stability properties of this mode (Pr 1) in open cavities (of ﬁnite lateral extent with ﬂat horizontal adiabatic boundaries) and attempting to discern the differences with respect to the analogous problem with upper solid wall (Section 6.2.1). As an example, Wang and Korpela (1989), Okada and Ozoe (1993) and Gelfgat et al. (1997) investigated 2D shallow cavities with aspect ratio (horizontal length to depth ratio) A = 4 of typical interest in the ﬁeld of crystal growth (the horizontal Bridgman technique; see Section 3.1.1); in such 2D numerical studies, the limit W → 0 (free surface with no surface tension-driven effects, i.e. stress-free conditions) was initially considered as a possible idealization of the practically important model which also includes the surface Marangoni stress. It was revealed that an upper stress-free surface (rigid–free case) in place of a solid wall (rigid–rigid case) leads to the suppression of the ‘multiplicity’ of solutions that is a typical feature of this mode of convection in closed cavity of ﬁnite lateral extent, as discussed in Chapter 6. Ben Hadid and Roux (1989, 1992) and Mundrane and Zebib (1994) elucidated (via 2D numerical simulation) the ability of the thermocapillary forces to inﬂuence these dynamics further; they basically conﬁrmed the stabilizing role played on the 2D hydrodynamic mode by Marangoni effects for Pr 1 in the range 0 < W ≤ 0.04 (in qualitative agreement with the earlier predictions yielded by Laure and Roux, 1989, in the framework of a linear stability analysis for the inﬁnite layer). This mode is completely suppressed when the strength of Marangoni convection is sufﬁciently increased in comparison to the thermogravitational counterpart [W ≥ O(10−1 )]. As an example, for Pr = 0.01 Gershuni et al. (1992) found Grcr → ∞ for W ∼ = 0.0675. In such a context, it should be also stressed that, as mentioned before, for W >0.25 the inﬂection point in the velocity proﬁle disappears (Figure 11.1), which means that two-dimensional disturbances can no longer be the most dangerous mode in the limit as Pr → 0 (see Section 1.5.4 for

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(a)

(b)

Figure 11.2 Minimal Grashof number (Grm = inf qx ,qz {Grcr }) as a function of the Prandtl number for various instability modes (plane-parallel ﬂow with conducting bottom wall and conducting free surface; Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the rate of uniform temperature increase along the x -axis; W = Ma/Ra = Gr/Re): (a) W = 0.025; (b) W = 0.25 (solid line) and W = 2.5 (dashed line). Courtesy of P. Laure

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additional theoretical background and relevant information about Squire’s theorem and related implications). The suppression of oscillatory convective states related to the 2D hydrodynamic mode was also reported for ﬁnite cavities by Mundrane and Zebib (1994): they showed by 2D numerical simulation for Pr = O(10−2 ) that when thermocapillarity dominates the convective pattern (in the limit W → ∞), the resulting ﬂow tends to be stable. The major outcomes of the study of Gershuni et al. (1992) for dominant Marangoni convection (for the speciﬁc case of conducting boundaries) are summarized in Figure 11.2b: in line with the considerations above, the most dangerous instability is either the 3D oscillating mode (OLR induced by waves of gravitational type) for Pr < 1 or the stationary longitudinal rolls of the Rayleigh mode for Pr > 1. Both modes appear earlier in terms of the Grashof number as W increases, which means that, as opposed to the 2D hydrodynamic mode, the thresholds for the onset of three-dimensional gravitational disturbances are lowered by the presence of Marangoni effects. Interestingly, even for W > 1 Gershuni et al. (1992) did not report waves with features of the hydrothermal type (such as those originally revealed in the landmark analysis of Smith and Davis, 1983). As extensively discussed in Chapter 10, the basic mechanism and feedback loops (see Figures 10.9 and 10.10) at the root of the HTW phenomena are intimately related to the coupling of surface stresses and surface temperature disturbances; as a natural consequence, they are suppressed if either W = 0 (no Marangoni effects) or the free surface is conducting [in such a case, in fact, according to Eq. (11.8b) no temperature disturbances are possible on the interface]. We shall return to these interesting and delicate aspects later (Section 11.4.1). By contrast, waves of such a kind were clearly identiﬁed in the subsequent linear stability analysis of Parmentier et al. (1993) for adiabatic horizontal boundaries and a wide range of variation of the Prandtl number, O(10−2 ) ≤ Pr ≤ O(10). Three kinds of behaviour, in particular, were observed in speciﬁc subranges of Pr [(a) 4 × 10−3 ≤ Pr ≤ 0.4, (b) 0.4 ≤ Pr ≤ 2.6 and (c) Pr > 2.6] on the basis of the trends displayed in the (Ra,Ma) and (Gr,Re) planes by the related neutral curves: the ﬁrst one was referred to as the ‘a-family’, the second as the ‘b-family’ and the third as the ‘c-family’ (see Figure 11.3; furthermore, some additional details on these modes of convection are provided in the following). For the curves relating to the a-family (Pr = 10−2 and 10−1 ), Macr (and Recr ) decreases monotonically with Racr (Grcr ); remarkably, this can be regarded as indirect evidence that both buoyancy and Marangoni effects are tightly coupled. For such branches, moreover, an increase in Pr raises the critical values Macr and Racr , whereas such an increase has the opposite effect on the critical Grcr and Recr thresholds, which decrease with Pr. It is also worth noting that the curves describing the a-family (Pr = 10−2 and 10−1 ) and the b-family (Pr = 1) intersect both the coordinate axes Racr = 0 and Macr = 0 (the differences between these two families is that the curves of the b-family pass through a maximum), whereas the c-family curves are characterized by unconditional stability in the case of pure buoyancy since they do not cut the Racr (Grcr ) axis. In particular, the branches for Pr = 3 and 7 admit two extrema with a weak dependence of Macr on Racr (Recr on Grcr ). In the last parts of these curves [Racr > 7 × 103 (Grcr > 103 ) for Pr = 7 and Racr > 5 × 104 (Grcr > 1.67 × 104 ) for Pr = 3], Macr (Recr ) increases monotonically with Racr (Grcr ) with a relatively weak dependence, indicating that buoyancy and thermocapillarity are loosely tied (it should be also pointed out that the curve Pr = 7 exhibits a bifurcation at Racr = 365, Grcr = 52). Perhaps the most interesting aspect of the study of Parmentier et al. (1993) is the related analysis of the angle [deﬁned by Eq. (10.4) and reported in Figure 11.4] with which the disturbances propagate relative to the x-axis, which allows interesting additional insights into the nature of the perturbations to be gained.

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(a)

(b)

Figure 11.3 Instability thresholds in the Marangoni–Rayleigh (a) and Reynolds–Grashof (b) planes at ﬁve different Prandtl numbers (inﬁnite layer with adiabatic horizontal boundaries; Marangoni and Rayleigh numbers deﬁned as Ma = RePr = σT γ d 2 /µα and Ra = GrPr = gβT γ d 4 /να , respectively, where γ is the rate of uniform temperature increase along the x -axis. The curve Pr = 7 is composed of two distinct solutions (solid line) and (dashed line), which intersect at Ra = 3.65 × 102 . After Parmentier et al. (1993); Copyright Elsevier, 1993

According to Figure 11.4b, in general, oblique waves are the most dangerous disturbance; there is, however, a trend towards waves with a longitudinal wavefront ( = 90◦ ) as Pr is decreased towards Pr = 0.01 for relatively small values of the Rayleigh number. Remarkably, this means that for low-Pr ﬂuids [Pr ≤ O(10−1 )] and adiabatic boundaries the 2D hydrodynamic mode of thermogravitational origin, which, as explained before, is strongly stabilized by the presence of Marangoni effects (Laure and Roux, 1989), is replaced by a mechanism with waves propagating in the spanwise direction. These waves reduce to the classical hydrothermal mechanism of Smith and Davis (1983) in the limit as Ra → 0 (as demonstrated by the good

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(a)

(b)

Figure 11.4 Critical wavenumber (a) and angle of propagation of the disturbances with respect to the x -axis (b) as a function of the Rayleigh and Prandtl numbers [as in Smith and Davis, 1983, the critical )

wavenumber is deﬁned as q = Copyright Elsevier, 1993

qx2 + qz2 and = tan−1 (qz /qx )]. Data after Parmentier et al. (1993);

agreement between the critical values for Pr = 0.01 and 0.1 on the Marangoni axis in Figure 11.3a and the corresponding critical values of Macr in Figure 10.8a) and to the OLR mode in the limit as Ma → 0 (proved by the excellent agreement of the critical values for Pr = 0.01 and 0.1 on the Grashof axis in Figure 11.3b with the corresponding critical values of Grcr in Figure 6.6). It is worth recalling that both limiting cases (OLR for Ma = 0 and HTW for Ra = 0) are featured by propagation along the z-axis of the disturbances (see Sections 6.1 and 10.2, respectively; in particular, as illustrated in Section 10.2.2, even if the hydrothermal wave of Smith and Davis is basically a phenomenon with oblique direction, for low-Pr ﬂuids, the disturbance propagates almost exactly in a direction perpendicular to the basic ﬂow, i.e. along z).

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Inspection of the curves for Pr = 0.01 and 0.1 displayed in Figure 11.4b also reveals that for both Ra = 0 and Ma = 0, the interacting disturbances of gravitational and thermocapillary origin maintain the approximately spanwise direction of propagation that they have in the respective ‘pure’ cases; furthermore, according to Figure 11.3, they reinforce each other, lowering the stability threshold with respect to the situation with pure buoyancy or Marangoni ﬂow [in Figure 11.3a, in fact, Racr (Ma) < Racr(0) and Macr (Ra) < Macr(0) , where Racr(0) and Macr(0) are the critical Rayleigh number and the critical Marangoni number in the pure cases]. When larger values of the Prandtl numbers are considered (Pr ≥ 1), however, the scenario exhibits a signiﬁcant departure from the behaviours reported earlier. Figure 11.4b illustrates that for Pr ≥ 1 and relatively small values of Ra (Ra < 103 ), the instability emerges as rolls propagating obliquely to the basic ﬂow, whereas for dominant buoyancy (larger Ra) the preferred mode of propagation is essentially along the direction of the imposed temperature gradient [in particular, purely transverse rolls ( = 0◦ ) appear above a critical value of the Rayleigh number that decreases when Pr is increased]. A separate discussion is required for Pr = 7, for which Parmentier et al. (1993) identiﬁed two distinct behaviours: for relatively large values of Ra (Ra > 3.65 × 102 , dashed branch in Figure 11.4b) they yielded a trend similar to that for Pr = 1 and 3, with a tendency of the pattern to evolve from an oblique spatiotemporal structure with > 90◦ towards = 0◦ for Ra = 4 × 103 (passing through the condition = 90◦ with longitudinal rolls when the Rayleigh number is almost exactly twice the corresponding value of the Marangoni number); for relatively small values of Ra (Ra < 3.65 × 102 , solid branch in Figure 11.4b), however, they recovered the classical oblique waves of Smith and Davis (1983) with ≤ 30◦ . In the light of all the foregoing arguments, some interesting concluding remarks can be made about the fundamental differences in terms of results between the two analyses by Gershuni et al. (1992) and Parmentier et al. (1993), differing in their choice of horizontal thermal boundary conditions. Following Gershuni et al. (1992) for the conducting conﬁguration, typical oscillatory perturbations with a thermogravitational genesis (2D hydrodynamic disturbances for W < O(10−1 ) or OLR for W ≥ O(10−1 ) and a three-dimensional (Rayleigh) mode in the form of stationary longitudinal rolls will be the preferred modes of instability for Pr 1 and Pr > 1, respectively, whereas for adiabatic boundaries, according to Parmentier et al. (1993), oscillatory (generally oblique) disturbances will be the typical outcome of the instability that reduces to the fundamental mechanisms of Smith and Davis (1983) in the limit as Ra → 0 for both low (Pr 1) and relatively high (Pr = 7) values of the Prandtl number. Additional useful information about the physical connections at the basis of these complex behaviours will be provided later. The related discussion requires, in fact, the introduction of additional notions and concepts and also the description of other illuminating results obtained over subsequent years by other investigators (see, in particular, Section 11.4.1).

11.2

Finite-sized Systems Filled with Liquid Metals

Following the same strategy as used in the preceding chapters, prior to expanding on the cases with Pr ≥ O(1) (Sections 11.3–11.8), the nature and structure of hybrid convection induced by the combined action of gravity and surface tension forces is discussed here using typical examples with relevance to the ﬁeld of crystal growth from the melt, that is, Pr < O(1). The present section is therefore devoted to providing a synthetic account of the most signiﬁcant results on the subject for the classical geometries (see Section 2.3.1) examined in Chapter 10 and (partially) in Chapter 6, namely the open cavity, the annular pool and the liquid bridge, generally assumed as models of the open-boat (horizontal Bridgman), the Czochralski (CZ) and ﬂoating zone

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techniques, respectively (see Section 3.1 for some background information about these well-known technological processes). The ﬁrst case has been discussed to a certain extent in Section 11.1, where a critical comparison was undertaken between ﬁndings obtained in the framework of linear stability analyses (for the idealized layer of inﬁnite horizontal extent) and available numerical (nonlinear) simulations for rectangular two-dimensional cavities. Some of these concepts are recalled here in a different perspective and such a theoretical melange (together with all the propaedeutical material provided in Chapters 6 and 10) is used to delineate a fairly exhaustive picture of the possible instability scenarios (which may help in discerning the effective physical factors that can induce oscillatory ﬂow in situations of practical interest). Such a fundamental knowledge is elaborated in the following, making a clear distinction between the idealized situation (often considered in numerical simulations) in which the ﬂow in the cavity is assumed to retain a 2D structure and the more realistic one in which such a constraint is removed. Let us start by observing that, in view of the arguments provided in Sections 6.2, 10.2.4 and 11.1, oscillatory regimes of convection eventually yielded via 2D numerical computations will be ascribed to the hydrodynamic mode of thermogravitational nature; in this case, in fact, the hydrothermal waves typical of Marangoni ﬂow have a signiﬁcant component perpendicular to the main ﬂow (for Pr 1 they have wavefront that is almost longitudinal, i.e. ∼ = 90◦ ) and are, therefore, suppressed by 2D simulations. In the more realistic situation in which the disturbances are allowed to propagate in the spanwise direction, the nature of the most dangerous oscillatory mode will exhibit high sensitivity to the speciﬁc conditions considered. For conducting horizontal boundaries, the basic ﬂow will be unstable to the gravitational 2D hydrodynamic mode or to OLR (whose general properties have been discussed in Section 6.1.2) according to whether W < O(10−1 ) or W ≥ O(10−1 ), whereas, when the conducting boundaries are replaced with adiabatic conditions, regardless of the order of magnitude of W , the source of instability is transferred to a synergetic superposition of the OLR mode with the classical hydrothermal waves of thermocapillary origin with both disturbances propagating primarily along the spanwise direction z . Anyhow, before switching to the other models with rotational symmetry mentioned earlier (annular slots and liquid bridges), it should be pointed out that even if the fairly exhaustive description developed above may provide an important theoretical basis for predicting, evaluating and placing in the proper context several effects, nevertheless, it does not account for the role potentially played by solid transverse walls (i.e. solid boundaries perpendicular to z). As an example, 3D computations by Bucchignani and Mansutti (2004) for a parallelepipedic 3D cavity (4 × 1 × 1) and typical liquid metals (Pr = 0.015) revealed that steady three-dimensional states can emerge for mixed buoyant–Marangoni convection (Ra = 1.5 × 102 , W ∼ = 1.3) prior to the onset of time-dependent ﬂow. The presence of such stationary three-dimensional effects in the simulations of these authors cannot be justiﬁed in the framework of the arguments illustrated earlier; other explanations must be invoked, which require resorting to other categories of phenomena. There exist, in particular, two fundamentally different possible explanations (which are worth at least mentioning). The steady 3D effects may be interpreted as a nonlinear effect coming from the interplay between some fundamental features of the Marangoni ﬂow induced by the presence of sidewalls limiting the system in the z direction (discussed in Section 10.1; the reader is referred, in particular, to Figure 10.3) and the possible (still poorly known) stationary bifurcation of buoyancy convection (described in Section 6.4; the reader is referred, in particular, to Figure 6.34). The latter tends to be the dominant cause responsible for the onset of 3D ﬂow for shallow open cavities and/or pools; for such a case, in fact, the mechanisms shown in Figure 10.3 are considerably weakened owing to the relatively large distance between the rigid walls along z (Lappa, 2005a).

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Steady 3D ﬂows eventually arising as ﬁnal states in 3D simulations, however, might also be ascribed to the stationary disturbances predicted for pure Marangoni ﬂow and cavities with rectangular transverse section by Schimmel et al. (2005). The problem is still open. Apart from the intrinsic complexity of the subject repeatedly mentioned in this chapter, most of these difﬁculties obviously stem from the lack of experimental results (due to the high opacity, reactivity and melting point of these materials), which prevent progress of knowledge driven by synergetic comparison of theoretical and experimental efforts. Notably, some outstanding experimental results are available for the laterally heated annular pool sketched in Figure 2.4c (given the relevance of this approximate model of the CZ technique, one of the most widespread processes for the production of semiconductor crystals and the ensuing attention that has been devoted to this topic especially by Japanese researchers). Resorting to this ‘simpliﬁed’ conﬁguration (the inner ﬁxed rod reaches the bottom boundary, whereas the growing crystal rotates and only ‘touches’ the melt surface in the real technological process), using a CCD camera Azami et al. (2001a) directly provided clear experimental evidence of wavy patterns on shallow silicon melts and found the number of related spokes to depend on the depth of the liquid, i.e. on W −1 . As elucidated in the preceding chapter of this book, for pure Marangoni ﬂow and annular slots with well-deﬁned and controlled thermal boundary conditions (bottom wall and free surface with adiabatic behaviour, lateral walls at ﬁxed temperatures; see Section 10.3), these phenomena tend to be a consequence of the hydrothermal waves originally predicted by Smith and Davis (1983). In the realm of Rayleigh–B´enard ﬂows (Chapter 4) and in the absence of surface tension-driven effects, similar patterns (featured by the macroscopic presence of spokes) are known to be a quaternary mode of convection for high-Pr ﬂuids (the reader is referred to Section 4.5 and to the studies of Busse and co-workers, who coined the term ‘spoke convection’). The latter case, however, has nothing to do with the conﬁgurations considered here where heating is not from below but from the side and Pr 1. For Pr ≥ O(1) (oxide melts) and real CZ systems, as also discussed in Section 3.1, wavy patterns are typically the manifestation of an instability driven by competition of buoyancy and baroclinic (induced by crystal rotation) effects. In practice, the emergence of similar waveforms in realistic CZ processing of semiconductor materials (with crystal rotation and strong heat exchange with the ambient at the free surface) is still a matter of investigation. For an interesting investigation of the subject in the speciﬁc case of low-Pr hybrid convection and simpliﬁed conﬁgurations such as that described earlier (with the outer portion of the annulus heated with respect to the inner cylindrical wall and adiabatic horizontal boundaries), it is worth considering the landmark studies of Li et al. (2004a, 2004c). According to their numerical results for a silicon melt (Pr = 0.01) and geometric parameters a = 15 mm, b = 50 mm, d = 8 mm (the same geometry as considered by Azami et al., 2001a), a small temperature difference in the radial direction generates steady axisymmetric thermocapillary–buoyancy ﬂow, whereas with a large temperature difference three-dimensional oscillatory convection characterized by a pulsating pattern travelling in the azimuthal direction emerges (such a hybrid pulsorotating space–time structure is shown in Figure 11.5). This pattern is different with respect to the case in which gravitational effects are absent (see Figure 10.20c for the same conﬁguration as in Figure 11.5 without buoyancy). Moreover, it is in good agreement with the related experimental ﬁndings of Azami et al. (2001a). There are many lines of evidence supporting the idea (anyhow, further theoretical studies on the subject would be required) that the origin of these travelling structures with temperature disturbances growing and decaying cyclically in time while moving in the azimuthal direction might be ascribed to the interplay between OLR gravitational modes and HTW phenomena; as explained earlier, such interplay is a typical feature of ﬂuids with Pr 1 when adiabatic conditions are

439

40s

Mixed Buoyancy–Marangoni Convection

2p −2.73K

2.73K (a)

1688.3Κ 1693.0Κ (b)

Figure 11.5 (a) Snapshot of surface temperature ﬂuctuation and (b) spatiotemporal diagram of surface temperature along a circumference (r = 20 mm) of an annular shallow layer [Pr = 0.01, a = 15 mm, b = 50 mm, d = 8 mm, Ma = 151.8; Ma based on the reference length L = d 2 /(b − a); adiabatic conditions at the top and bottom boundaries; 1g conditions]. After Li et al. (2004a); Copyright Elsevier, 2004

considered on the top and bottom boundaries (on the other hand this would also provide a reasonable justiﬁcation for the dependence of the number of spokes on W and the peculiar pulsorotating appearance of the pattern). An additional interesting study on the subject was reported by Shi et al. (2009), who for a similar conﬁguration (they considered a = 20 mm, b = 40 mm, 1 ≤ d ≤ 10 mm and Pr = 0.011) investigated both numerically and by linear stability analysis the primary ﬂow instability for microgravity and normal gravity conditions. They found that the buoyancy force decreased the critical azimuthal wavenumber. They also reported lower values of the critical Marangoni number in the presence of gravity, which, notably, may provide an additional line of evidence for the idea that instability is induced by the joint action of HTW and OLR mechanisms mentioned earlier (among other things, such a trend is also in qualitative agreement with the instability branch related to Pr = 0.01 in Figure 11.3a obtained by Parmentier et al., 1993, for an inﬁnite layer with adiabatic horizontal boundaries, which expressly refers to the case in which both such instability mechanisms are operative). For additional relevant and interesting numerical studies on Czochralski-like conﬁgurations also accounting for crystal rotation (baroclinic effects) and heat transfer at the melt/gas interface, the reader may consider the studies cited in Section 3.1 and references therein. The subject is not discussed further here due to page limits (the author apologizes to those whose work or publications are not described or cited). The remainder of this section is devoted entirely to the liquid bridge. Despite the similarities with the annular geometry (the circular symmetry, the axisymmetric toroidal structure of the basic ﬂow, etc.), as already explained in Section 10.4.1, the related dynamics (in terms of emerging spatiotemporal patterns and nature of the most dangerous disturbance) exhibit profound differences, which make it worth considering the liquid bridge problem for Pr < O(1) as a fully independent topic. In general, for pure Marangoni ﬂow (Ra = 0), the route from the initial axisymmetric steady state to the oscillatory regime is given by the succession of two purely hydrodynamic instabilities (see Section 10.4.6). This means that, unlike the classical hydrothermal mechanism, which gains energy from a feedback loop involving the thermal and velocity ﬁelds, the disturbances emerging in liquid bridges of liquid metals and semiconductor melts are basically due to an inertia instability of the axial shear layer below the free surface.

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Superimposed on this is the different relative orientation of gravity and free surface for the two geometric models (perpendicular for the annular conﬁguration, parallel for the liquid bridge), which make the static deformation of the interface induced by gravity (see Section 2.3.3) an additional signiﬁcant factor entering the system dynamics in the latter case. Unlike the case of high values of the Prandtl number, where a large amount of experimental results is available, only few results have appeared about oscillatory ﬂow in bridges of liquid metals (Tao and Kou, 1996; Han et al., 1996; Nakamura et al., 1999; Cheng and Kou, 2000; Takagi et al., 2001; Yang and Kou, 2001; Sumiji et al., 2001; Azami et al., 2001b; Hibiya et al., 2001, 2008; Okubo et al., 2005; Kamotani et al., 2007). The oscillations and related frequency in these conﬁgurations are usually detected by means of thermocouples inserted into the liquid since direct observation of the ﬂow is not possible. Special diagnostic techniques, however, have also been introduced to replace those (the usual light-cut technique-based visualization following tracers illuminated by a laser sheet in the generic meridian plane) traditionally employed for transparent liquids. Han et al. (1996), for instance, carried out experiments with mercury. They used a laser beam to detect the deformation and oscillation of the free surface; the reﬂected light from the curved liquid bridge surface was projected on a screen and the resultant enlarged interference image was monitored by a CCD camera (the interference pattern was found to be signiﬁcantly affected by the onset of oscillations). Similarly, the ﬂow transition was directly proved by surface-ﬂow visualization in the experiments of Takagi et al. (2001). Tin oxide particles, which were premixed with molten tin, were observed to repeat oscillatory movement along the circumferential direction with a ﬁxed interval after the onset of the oscillation. Analogous techniques were used by Azami et al. (2001b) and Okubo et al. (2005) for molten silicon. Most interestingly, Hibiya and co-workers elaborated a peculiar diagnostic technique based on direct visualization obtained via X-ray radiography with zirconium-core tracers (see, e.g., Nakamura et al., 1999; Hibiya et al., 2001). By combining this approach with temperature oscillation measurements and observation of surface oscillation through a spatial-phase measurement technique, they were able to determine the azimuthal ﬂow structure (i.e. the azimuthal wavenumber) in liquid bridges of molten silicon. In order to estimate gravity effects, however, both the case of a cylindrical shape (simpliﬁed model) and of a melt/air interface deformed by the effect of gravity (real conditions) must be considered from a theoretical point of view. In practice, the comparison among these situations is a theoretical craft that gives insight into the separate inﬂuences of buoyancy forces and of the gravity-induced departure of the free surface from the straight (cylindrical) conﬁguration. It is also worthwhile to stress how, within the context of these studies, the heating direction (bridge heated from above or from below) plays a critical role in the dynamics under investigation (Yasuhiro et al., 2001; Lappa et al., 2003a). As a ﬁrst step towards the understanding of these behaviours, Figure 11.6a (Lappa et al., 2003a) shows the values of the growth-rate constants (σ ) for the ﬁrst (stationary) ﬂow bifurcation in a liquid bridge with the same value of the Prandtl number considered earlier for the annular pool (Pr = 0.01), L = 1 cm, AH = 1, S = 1 [let us recall that AH is the geometric aspect ratio (height/diameter) and S is the volume factor deﬁned as the ratio of the effective volume of liquid held between the supporting disks to the volume of a straight cylinder with the same height and basis] as a function of the Marangoni number for different conditions [i.e. zero-g conditions (Ra = 0), normal gravity with heating from above (Ra < 0) and heating from below (Ra > 0)] under the assumption of a straight interface. The growth rates obtained for normal gravity and heating from above (Ra < 0) lie below the corresponding rates obtained in the case of microgravity (Ra = 0). Vice versa, the growth rates yielded for heating from below are above those corresponding to zero-g.

Mixed Buoyancy–Marangoni Convection

441

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0.80 s

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heating from below

1.00

heating from above

0.80 s

0.60 0.40 0.20 0.00 −0.20 −0.40 0

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20 30 40 Marangoni number (b)

Figure 11.6 Growth rates as a function of the Marangoni number and determination of the ﬁrst critical Marangoni number (silicon melt, Pr = 0.01, L = 1 cm, AH = 1, S = 1, 1g conditions): (a) cylindrical shape; (b) gravity-deformed shape (Ma based on the axial extension of the liquid bridge). Results after Lappa et al. (2003a)

Since all these cases share the same azimuthal structure (m = 1) and small differences in terms of Macr1 (Macr1 = 15.36 for heating from above and Macr1 = 14.92 for heating from below are slightly increased and decreased, respectively, with respect to microgravity conditions for which Macr1 = 15.24), a remarkable outcome of such simple observations is that, even if the height of the bridge of liquid metal is not limited to few millimetres, the ﬁrst instability of Marangoni ﬂow is affected weakly by the buoyancy forces.

442

Thermal Convection: Patterns, Evolution and Stability

Among other things, these trends are in agreement with the theoretical results of the linear stability analysis of Wanschura et al. (1995). They highlighted, in fact, that the efﬁciency of the process of ampliﬁcation of the azimuthal disturbances responsible for the aforementioned inertia instability is proportional to the radial gradient of the basic state axial velocity, i.e. the axial shear. Marangoni forces driving the basic ﬂow ﬁeld induce a strong axial shear ﬂow below the free melt/air interface. The higher the intensity of the axial share, the lower should be the critical Marangoni number. On the basis of this theory, the stabilization of the ﬂow ﬁeld in the case of heating from above simply follows from the decrease in the axial share intensity that in this case is reduced since buoyancy and Marangoni forces behave as counteracting effects. As a second step, in order to elucidate the overall effect of the gravity ﬁeld (i.e. buoyancy forces + surface deviation from the straight conﬁguration), the deformation of the shape must be taken into account. According to the results in Figure 11.6b, the growth rates obtained in the case of heating from above and a deformed shape lie below the corresponding rates for a straight shape, hence leading to stabilization of the ﬂow ﬁeld (Macr1 = 16.62). Despite the different value of the critical Marangoni number (Macr1 = 16.62 instead of 15.36), this trend is similar to that already discussed earlier. Nevertheless, as opposed to the case of a straight surface, if the surface deformation is not ignored, the ﬂow exhibits further stabilization for the heating from below condition. The corresponding values of the growth rate, in fact, are below those obtained in the opposite case (Macr1 = 18.78). Remarkably, this means that there is a signiﬁcant effect of the shape on the instability boundaries. If the liquid surface of the liquid bridge is allowed to deform, in fact, gravity always acts to stabilize the Marangoni ﬂow regardless of its direction (parallel or antiparallel to the z-axis). The combined effect of the shape and of the heating direction is also crucial in determining the azimuthal pattern. The structures of the ﬂow and temperature ﬁelds for the case under investigation (Pr = 0.01, AH = 1, S = 1), in fact, are characterized by m = 1 for heating from below and m = 2 if the direction of gravity is reversed. The different azimuthal structures of the thermoﬂuid-dynamic ﬁeld at the steady state, according to the gravity level and according to the heating condition (from above and from below), are shown in Figures 11.7 and 11.8. To summarize, the liquid bridge exhibits a different response according to the allowed bridge shape. If the shape is forced to be straight, the ﬂow ﬁeld is weakly stabilized in the case of heating from above and weakly destabilized if gravity is reversed. If the deformation is taken into account, gravity always exerts a stabilizing role regardless of its direction (parallel or antiparallel to the axis) and the three-dimensional ﬂow structure is different according to the heating condition (from above or from below). In the latter case, the critical Marangoni number is larger and the critical wavenumber is smaller compared with the opposite condition (the different structure of the ﬂow obviously being determined by the delicate interplay of shape deformation and heating direction in inducing instability of the shear layer below the free surface). Owing to this property, the stabilization of the ﬂow induced by the hydrostatic shape dramatically changes according to the substance under consideration (in fact, melts with different densities and surface tensions tend to be characterized by very different slopes of the amphora-like free surface). As an example, Figure 11.9 displays the free melt/air interface in the case of liquid gallium (Pr = 0.02) for the same conﬁguration considered earlier for a silicon melt (L = 1 cm, AH = 1.0, S = 1). The shape of the liquid bridge of gallium is more deformed than that of silicon because of the higher density of Ga and its lower surface tension (for Si and Ga the Bond number is 3.386 and 8.3344, respectively; see Section 2.3.2 for the related deﬁnition of this nondimensional parameter). Along these lines, Figures 11.10 and 11.11 show how the inﬂuence of the deformation of the interface on the basic ﬁeld becomes more signiﬁcant when gallium is considered.

Mixed Buoyancy–Marangoni Convection

15

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(b)

T. D. 0.030 0.015 0.000 −0.015 −0.030

(c)

(d)

Figure 11.7 Structure of 3D Marangoni ﬂow (steady state) with m = 2 (silicon melt, Pr = 0.01, L = 1 cm, AH = 1, S = 1, Ma = 40; Ma based on the axial extension of the liquid bridge; 1g , bridge heated from above): (a) velocity ﬁeld in the meridian plane ϕ = 3π/4; (b) azimuthal velocity in the cross-section z = 0.5; (c) 3D view of the isosurfaces of the temperature disturbance; (d) azimuthal velocity on the free surface. After Lappa et al. (2003a)

For heating from above, the most dangerous azimuthal wave number is m = 2. The value of the critical Marangoni number (Macr1 = 89.51) is signiﬁcantly increased with respect to the cases where the straight surface is assumed (m = 1 and Macr1 = 36.19). This trend is similar from a qualitative point of view to that already discussed for silicon melt. Anyhow, if the direction of gravity is reversed (heating from below), a completely different scenario arises: the value of the critical Marangoni number is increased up to Macr1 = 242.2; the bifurcation is no longer stationary and the instability threshold value is much higher than the second critical Marangoni number for a cylindrical liquid bridge of a Pr = 0.02 ﬂuid (Macr2 = 131.8 for g = 0). The ﬂow becomes unstable towards 3D disturbances directly through a Hopf bifurcation characterized by azimuthal wavenumber m = 1 and critical frequency f = 0.31 Hz (obtained by an extrapolation). Such a behaviour was reported for the ﬁrst time by Lappa et al. (2003a).

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Thermal Convection: Patterns, Evolution and Stability

13 11

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T. D. 0.030 0.015 0.000 −0.015 −0.030

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Figure 11.8 Structure of 3D Marangoni ﬂow (steady state) with m = 1 (silicon melt, Pr = 0.01, L = 1 cm, AH = 1, S = 1, Ma = 40; Ma based on the axial extension of the liquid bridge; 1g , bridge heated from below): (a) velocity ﬁeld in the meridian plane ϕ = 3π/4; (b) azimuthal velocity in the cross-section z = 0.5; (c) 3D view of the isosurfaces of the temperature disturbance; (d) azimuthal velocity on the free surface. After Lappa et al. (2003a)

Notably, the spatiotemporal stages of evolution are different with respect to typical tertiary modes of convection (emerging after the second ﬂow bifurcation) in liquid bridges of low-Pr ﬂuid that have been depicted in detail in Section 10.4.6. In those cases, oscillatory ﬂow is superimposed on a steady 3D ﬁeld established through the ﬁrst ﬂow transition. The case of gallium (Figure 11.12) represents a new type of transition directly from steady to oscillatory ﬂow without the intermediate state of steady, asymmetric ﬂow traditionally reported by many investigators for low-Pr materials (Levenstan and Amberg, 1995; Imaishi et al., 1999, 2000; Leypoldt et al., 2000). At an early stage, during a transient unsteady phase the spatiotemporal pattern is ‘rotating’. The temperature disturbances on the interface (surface temperature spots) rotate around the perimeter

Mixed Buoyancy–Marangoni Convection

445

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Figure 11.10 Basic state for a silicon liquid bridge (Pr = 0.01, AH = 1, S = 1, shape deformed by gravity): (a) surface-temperature distribution; (b) surface velocity proﬁle. After Lappa et al. (2003a)

of the liquid bridge (in Figure 11.13). The time dependence of the temperature ﬁeld is simply characterized by a full rotation of the entire ﬂow pattern in the azimuthal direction. After a transient phase, the initial rotating regime is taken over by a new spatiotemporal mode of azimuthal convection. During the rotating regime, the frequency is f = 0.449 Hz but after a transient time the frequency becomes f = 0.386 Hz. In this case, the pattern is ‘pulsating’. The pulsating temperature spots on the surface of the bridge are shown in Figure 11.14. These

446

Thermal Convection: Patterns, Evolution and Stability 5.00 Pr = 0.02 Ma = 40

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Figure 11.11 Basic state for a gallium liquid bridge (Pr = 0.02, AH = 1, S = 1, shape deformed by gravity): (a) surface-temperature distribution; (b) surface velocity proﬁle. After Lappa et al. (2003a)

Maximum azimuthal velocity

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Figure 11.12 Maximum surface azimuthal velocity as a function of time (gallium, Pr = 0.02, AH = 1.0, S = 1, shape of the liquid bridge shown in Figure 11.9, Ma = 3 × 102 ; Ma based on the height). After Lappa et al. (2003a)

temperature spots ‘pulsate’, that is, the cold spot grows in the axial direction during the shrinking of the hot spot and vice versa, but the azimuthal positions of these extrema do not change. This oscillatory ﬂow exhibits outstanding similarities with the secondary modes of Marangoni ﬂow of high Prandtl number liquids (standing waves and travelling waves; see Chapter 10 for further details) rather than with the tertiary modes of convection of low-Pr liquid bridges. Along these lines, it can be concluded that for ﬂuids with Pr = O(10−2 ) (at least for the circumstances considered here of typical technological interest for which the height of the liquid bridge is of the

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Figure 11.13 Secondary mode of convection corresponding to oscillatory ﬂow after the ﬁrst bifurcation for a liquid bridge of gallium (Pr = 0.02, AH = 1.0, L = 1 cm, S = 1, 1g with heating from below, Ma = 300; Ma based on the height). The snapshots of the thermoﬂuid-dynamic ﬁeld at four instances evenly distributed within one oscillation period show the transitional stages of evolution for an m = 1 rotating mode. From left to the right: temperature-disturbance isosurfaces; distribution of temperature disturbance (level 1 → −3.7 × 10−2 , level 20 → 5.2 × 10−2 , level = 4.7 × 10−3 ); azimuthal velocity in the cross-section z = 0.5 (level 1 → −2.59, level 13 → 3.65, level = 0.52). After Lappa et al. (2003a)

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Thermal Convection: Patterns, Evolution and Stability

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Figure 11.14 Secondary mode of convection corresponding to oscillatory ﬂow after the ﬁrst bifurcation for a liquid bridge of gallium (Pr = 0.02, AH = 1.0, L = 1 cm, S = 1, 1g with heating from below, Ma = 300; Ma based on the height). The snapshots of the thermoﬂuid-dynamic ﬁeld at four instances evenly distributed within one oscillation period show the transitional stages of evolution for an m = 1 pulsating mode. From left to the right: temperature-disturbance isosurfaces; distribution of temperature disturbance (level 1 → −6.5 × 10−2 , level 20 → 4.84 × 10−2 , level = 6 × 10−3 ); azimuthal velocity in the cross-section z = 0.5 (level 1 → −2.95, level 15 → 3.43, level = 0.46). After Lappa et al. (2003a)

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order of 1 cm), the static deformation of the liquid bridge induced by gravity may signiﬁcantly alter the instability threshold and even the type of bifurcation (stationary or oscillatory) in some speciﬁc cases (liquid heated from below). For similar results obtained by resorting to the model of ‘full zone’ (whose general features have been outlined in Section 10.4.2), the reader is referred to Lappa (2004b, 2004c), Chapter 4 in Lappa (2004a) and Savino and Lappa (2004). Beyond geometries with deformed interface, other numerical models have also appeared in the literature in which a more realistic reproduction of experimental boundary conditions (ramped temperature differences, i.e. time-dependent imposed temperatures on the supporting disks) was attempted (e.g., Yasuhiro et al., 2004a,b; Li et al., 2005, 2008a). The related details are not discussed here due to page limits.

11.3

Typical Terrestrial Laboratory Experiments with Transparent Liquids

Continuing with the description of the hybrid states of convection and related hierarchy of bifurcations for classical geometric models (set forth in Section 11.2 for low-Pr ﬂuids, i.e. liquid metals and semiconductor melts), hereafter attention is paid especially to liquids with Pr ≥ O(1), which are typically used in the framework of ground-based fundamental research. In such a context, it is worth noting that widespread use has been enjoyed by silicone oils and similar ﬂuids as they exhibit a number of characteristics that make their utilization particularly advantageous. Such properties can be summarized as follows. Silicone oils:

• • • • • • • • •

Are in a liquid state at ambient temperature. Exhibit a pronounced Marangoni effect. Are transparent to visible light, allowing direct observation of the dynamics. Are not transparent with respect to infrared diagnostic techniques (e.g. thermocameras with wavelengths 8–12 µm). thereby, such techniques can be employed to investigate the temperature distribution on the free liquid/gas interface. Are nonﬂammable. Are nontoxic. Are not reactive (they do not undergo chemical reactions when exposed to an external gas). Are available with a huge range of viscosities while retaining an almost constant thermal diffusivity, which allows a variety of possible values of the Prandtl number to be selected [Pr ranging from O(10) to O(104 ) and even larger values]. Can be used as relevant models of oxide melts (typically characterized by Prandtl numbers in the range 5 ≤ Pr ≤ 20) having very high melting temperatures (about 2000◦ C) that make their experimental study extremely difﬁcult.

The remainder of this chapter runs as follows. Section 11.4 is focused on the case in which the experimental setup is very close to the conditions considered theoretically in the landmark study by Smith and Davis (1983) for pure Marangoni ﬂows (i.e. shallow layers with rectangular section and similar conﬁgurations). Section 11.6 describes a special case with a vertical differentially heated cavity expressly conceived to investigate the possible interplay of gravitational and surface tension-driven effects in a variety of situations and combinations. Section 11.7 considers geometries with a rotational symmetry (radial temperature gradients) for which disturbances of a hydrothermal nature are expected to propagate circumferentially. Finally, Section 11.8 is devoted to the liquid-bridge problem in the context of microscale studies, to which so much attention has been devoted over the last 30 years.

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Thermal Convection: Patterns, Evolution and Stability

The Rectangular Liquid Layer

11.4.1 Waves and Multicellular Patterns Although a larger number of experimental results on this subject (liquid layers with rectangular section) is available for transparent liquids (in comparison with the companion problem treated in Section 11.2), since the problem of buoyant-thermocapillary convection in cavities is governed by a relatively large number of nondimensional parameters and there is consequently a large number of different types of ﬂow that can be found in this system, in general, these experiments have been limited to providing disjoint glimpses of a wide variety of qualitatively and quantitatively different behaviours in widely different parts of parameter space and a number of puzzling results. There have been many surprises with respect to theoretically predicted scenarios and not all experimental results have yet been satisfactorily explained. In general, however, three distinct, fairly well-deﬁned categories of spatiotemporal behaviour have been identiﬁed (as described in detail in the following for the convenience of the reader). Let us start the related discussion by noting that waves similar to those predicted by Smith and Davis (1983) were observed especially in experiments designed to mimic most closely the conditions of their linear stability analysis (some typical terrestrial results of this kind are shown in Figure 11.15).

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Figure 11.15 Travelling waves in a layer with horizontal extensions Lx = 3 cm, Lz = 25 cm and two values of the depth Ly = d (Pr = 10.3; on the left side, d < dc (d = 1.6 mm) and T = 5.6 K; on the right side, dc < d < dr (d = 3 mm) and T = 4.64 K): (a) shadowgraphic image [plane (x, z)] and (b) schematic drawing of the pattern in a horizontal plane; (c) spatiotemporal diagram. After Burguete et al. (2001); Reproduced by permission of the American Institute of Physics

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Perhaps the most interesting contribution in such a context was provided by Burguete et al. (2001), who were the ﬁrst to ﬁnd evidence for some peculiar features of these waves according to the considered layer thickness [let us recall here that for a given liquid, as the nondimensional parameter W deﬁned by Eq. (2.26) is related by a ﬁxed dependence to d, the liquid depth can be used as a relevant parameter in place of W for measuring the relative importance of gravitational and Marangoni effects; indeed, it has enjoyed widespread use in the literature for such a purpose]. For relatively large heights (d > dc ), the wave source was found to be a line and generally to evolve towards one end of the container leaving a single wave, whereas for smaller heights, the source was observed to look like a point and emit a circular wave (i.e. a wave with a varying propagation angle) turning into an almost planar wave at a certain distance from the source in both directions. In other words, depending on the ﬂuid depth d, two different types of sources of waves were clearly distinguished, as follows: 1. For relatively small values of d (d < dc ), a source looking like a point located on the cold wall and emitting a circular wave becoming almost planar far from the source as shown in Figure 11.15 (left side) due to the conﬁnement by the channel in the x direction. 2. For higher d (d > dc ), a source looking like a line almost parallel to the x-axis and emitting inclined planar waves (Figure 11.15, right side). We will return later to the properties of these waves, related departure from the classical theory for pure Marangoni ﬂow and associated possible explanations. Anyhow, it should be stressed that only a few of the experiments specially conceived for emphasizing Marangoni ﬂow in comparison with buoyancy convection provided direct evidence of waves. In many experiments, the basic ﬂow was observed to destabilize ﬁrst against a third type of pattern: a stationary multicellular instability with longitudinal rolls embedded in the main ﬂow all rotating in the same direction (generally indicated as SMC) before transitioning to an oscillatory regime. In practice, many investigators reported an initial transition from steady unicellular ﬂow to a steady multicellular ﬂow and then a subsequent (secondary) transition to a time-dependent state (we will expressly consider this second bifurcation and ensuing tertiary modes of convection in Section 11.4.2). The existence of these oscillatory patterns not corresponding to the propagation of waves was reported by Villers and Platten (1992), De Saedeleer et al. (1996) and many others for Pr > 1. Riley and Neitzel (1998) were the ﬁrst to provide experimental evidence of oblique waves in a rectangular pool Lx = 5.3 cm, Lz = 5.5 cm for silicone oil (Pr = 14) for sufﬁciently thin layers (d < 1.25 mm). Nevertheless, in line with the explanations above, for thicker layers they observed a steady multicellular ﬂow that becomes time dependent when the temperature gradient is increased. Most of the available results for transparent liquids (Villers and Platten, 1992, for Pr = 4.2; Ezersky et al., 1993a, for Pr = 60; Daviaud and Vince, 1993, for Pr = 10.3; Gillon and Homsy, 1996, for Pr = 9.5; De Saedeleer et al., 1996, for Pr = 15; Braunsfurth and Homsy, 1997, for Pr = 4.4; Garcimart`ın et al., 1997, for Pr = 10, 15 and 30; Pelacho and Burguete, 1999, and Pelacho et al., 2000, for Pr = 10.3; etc.) were summarized (together with the related experimental setups, i.e. length, width and height of the containers used) in the aforementioned analysis of Burguete et al. (2001), where it was highlighted how the emergence of a pattern with speciﬁc features depends essentially on geometric factors. In practice, according to the experiments, the two-dimensional basic ﬂow destabilizes into oblique travelling waves or longitudinal stationary rolls depending on the aforementioned height of the liquid and on the extension in the spanwise direction, the ﬁrst phenomenon being favoured for small thickness and large extension of the layer along z, and vice versa, the latter becoming dominant for thick layers and small extension along z.

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In such a context, it is also worth mentioning Pelacho et al. (2001), who performed a series of experiments expressly conceived to characterize the emergence of waves in rectangular containers whose dimensions could be continuously changed; in this way, they were able to investigate how boundaries affect the threshold for the instability, and also their consequences on other properties of the waves. An example of typical experimental behaviours is shown in Figure 11.16. Depending on the height of liquid, two regimes can be identiﬁed. The related Tcr depends on d (or on W −1 that is directly proportional to d 2 ). As explained earlier, when T > Tcr (d), for small d values, the system exhibits oblique travelling waves (waves similar to the type originally predicted by Smith and Davis, 1983), whereas for larger values of d, stationary longitudinal rolls are observed (in the latter case the basic ﬂow destabilizes in the form of a stationary pattern with wavevector perpendicular to the horizontal applied gradient, appearing at threshold only near the hot side and invading the rest of the cell when T is increased). As shown in the ﬁgure, the value dr which separates the waves from stationary rolls increases with Lz . At this stage, it is obvious that comparison of predictions of the theory of Smith and Davis (1983) with the above results would reveal a substantial disagreement. As mentioned before, according to available experiments, for sufﬁciently deep liquid layers, a stationary instability sets in before an oscillatory one, whereas the latter is predicted to be always the most unstable for pure Marangoni ﬂow. Superimposed on this is the fact that, although according to the theory for high-Pr liquids and zero-g conditions the oscillatory instability should be stabilized with increase of the depth of the layer (see, e.g., Figure 10.16), the experimentally found threshold of this instability in terrestrial conditions is signiﬁcantly higher than the predicted value (this disagreement sharply increasing with the depth of the layer), which means that the disagreement between experiments and existing theory for zero-g conditions is both qualitative and quantitative. There is no doubt that the existence of SMC states is not in line with the ﬁndings of Smith and Davis (1983) for pure Marangoni ﬂow. Moreover, such states were not mentioned in the theoretical analysis of Parmentier et al. (1993), where only oscillatory states were predicted as

Figure 11.16 Sketch of typical experimental stability diagram: temperature difference Tcr as a function of the height of liquid d (the transitional depth dr which separates HTW states from stationary longitudinal rolls increases with Lz , i.e. the system extension along the direction perpendicular to the imposed temperature gradient)

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unstable modes for coupled buoyancy and Marangoni ﬂows (both these linear stability analyses considered the limiting case of adiabatic horizontal boundaries). Some key information for the possible interpretation of the observed departure from the classical theories has been provided in Section 11.1. It has been illustrated therein that in the limiting case of conducting horizontal boundaries and for Pr > 1 the hydrothermal mechanism is essentially replaced by the 3D stationary Rayleigh mode as the most dangerous disturbance. Interestingly, an even more relevant and illuminating contribution along these lines was provided by Mercier and Normand (1996), who considered the more realistic situation in which the free surface is neither adiabatic nor conducting, but it is featured by a given heat exchange rate with the external environment (measured by the relevant nondimensional number Bi). By means of a linear stability analysis these authors found (Pr = 7 and inﬁnite layer bounded below by a plane whose temperature varies linearly along the direction of the thermal gradient) that for a given value of the Biot number at the free surface, the stationary longitudinal rolls are the preferred mode of instability provided that W does not exceed a certain value W0 that is an increasing function of the Biot number. Notably, in particular, two possible stationary modes (differing in the value of their respective horizontal wavenumbers) were identiﬁed, which require additional theoretical treatment. Related representative neutral stability curves are plotted in Figure 11.17 for two different values of W . The stability boundary Gr as a function of qz consists of two intersecting branches of a nearly parabolic shape, each of them having a local minimum at a given value of the wavenumber. For Bi = 10 and W = 0.03, the ﬁrst minimum Gr1 = 8.7 × 102 (that is also an absolute minimum) occurs on the left branch for qz1 = 7.0, while a secondary minimum Gr2 = 103 lies on the right branch for a higher wavenumber value qz2 = 11.5. For a higher value W = 0.1, the situation is just the opposite: the absolute minimum Gr2 = 5.4 × 102 occurs on the right branch for qz2 = 13.1, while a higher minimum Gr1 = 6.7 × 102 is located on the left branch at a lower value of the wavenumber qz1 = 6.7.

Figure 11.17 Stability curves of stationary modes for two values of the parameter W : W = 0.03 (solid line), W = 0.1 (dashed line) (inﬁnite layer with conducting bottom wall; Pr = 7, Bi = 10; Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 where γ is the rate of uniform temperature increase along the x -axis). After Mercier and Normand (1996); Reproduced by permission of the American Institute of Physics

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Those modes corresponding to a lower value of the wavenumber were found to be connected to the aforementioned Rayleigh modes previously identiﬁed by Gershuni et al. (1992) and involving the classical Rayleigh–B´enard instability mechanism. These Rayleigh modes can develop in regions where the basic temperature proﬁle is unstably stratiﬁed. As already discussed in Section 11.1, this condition arises in two sub-layers adjacent to the lower and upper boundaries. The heights of these sub-layers denoted hereafter hB and hT (for bottom and top) evolve with the parameters W and Bi (the most important evolution concerns the variation with the Biot number, since hB and hT shrink to zero when Bi = 0, thus preventing the possible onset of stationary longitudinal rolls for the case with adiabatic horizontal boundaries). The second type (second branch) of stationary modes was found to be characterized by a larger value of the critical wavenumber, indicating that the length scale of the instability is related to the height hT of the upper sub-layer. Mercier and Normand (1996) revealed that this region (bounded above by a free surface) can become unstable to either the above-mentioned Rayleigh–B´enard instability (as illustrated by Gershuni et al., 1992) or a Marangoni instability. This latter case requires additional clariﬁcation. As discussed in Chapter 10, Smith and Davis (1983) found hydrothermal waves to be the most critical disturbances when a return-ﬂow velocity proﬁle was assumed. They also predicted, however, that stationary longitudinal rolls can emerge as the preferred mode of instability for Pr > 1.6 when a linear base ﬂow is considered. Since in the upper region, the height of which is hT , the velocity proﬁle can be locally approximated by a linear ﬂow; according to Mercier and Normand (1996), this feature could trigger the onset of stationary longitudinal rolls. It is worth noting that such arguments also received some experimental conﬁrmation over subsequent years, for example by Ospennikov and Schwabe (2004), who revealed that the HTWs can be artiﬁcially suppressed (leading to stationary longitudinal rolls) in experimental conﬁgurations where the return ﬂow does not exist (interestingly, they suppressed the return ﬂow using channels and side channels with lower ﬂow resistance compared with that of the return ﬂow). To summarize, Mercier and Normand (1996) explained the SMC states according to the Rayleigh modes identiﬁed by Gershuni et al. (1992) in the upper and lower regions of potentially unstable thermal stratiﬁcation and/or the existence of a linear ﬂow-like solution in the upper sub-layer (potentially responsible for stationary thermocapillary disturbances as predicted by Smith and Davis, 1983, and later evidenced experimentally by Ospennikov and Schwabe, 2004). At this stage, is worth recalling that the analysis of Mercier and Normand (1996) was not limited to the stationary modes and related causes. As mentioned earlier, they also investigated oscillatory modes. Remarkably, for a ﬁxed value of the Biot number while varying the parameter W , they identiﬁed a value W0 above which the preferred modes of instability are oscillating at an angular frequency ω = 0. These unsteady modes were found to exist in the form of either waves with longitudinal front (qx = 0, qz = 0), oblique waves (qx = 0, qz = 0) or waves with transversal front (qx = 0, qz = 0). These oscillatory states were reminiscent of the hydrothermal waves identiﬁed by Smith and Davis (1983) in the case of a pure thermocapillary basic ﬂow and of the travelling rolls found by Parmentier et al. (1993) for a buoyant-thermocapillary basic ﬂow and a zero value of the Biot number. For Bi < 7, the curves for qc , the angle of propagation and the frequency as a function of W were found to be smooth and continuous. Nevertheless, for Bi = 10, the value WE = 0.17 was indicated as the characteristic point for an exchange of stability between two types of oscillatory modes (this exchange being accompanied by an abrupt change in the critical value of the characteristic parameters, as shown in Figures 11.18 for the wavenumber qc and the angle of propagation ).

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(a)

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Figure 11.18 Critical values of the characteristic parameters as a function of W (inﬁnite layer with conducting bottom wall, Pr = 7, Bi = 10): (a) critical wavenumber; (b) angle of disturbance propagation (the critical behaviour is represented by full lines). After Mercier and Normand (1996); Reproduced by permission of the American Institute of Physics

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Figure 11.19 States diagram showing the domain of existence of different critical modes in the (Bi, W) plane for Pr = 7 (inﬁnite layer with conducting bottom wall). After Mercier and Normand (1996); Reproduced by permission of the American Institute of Physics

As a consequence, for Bi > 7, a distinction was made between two types of oscillatory modes differing by the order of magnitude of their critical parameters qc , c and ωc and by the vertical extent on which the perturbed quantities take appreciable values (Figure 11.19). Mercier and Normand (1996) labelled modes of type I those modes characterized by small values of the wavenumber (3 < q < 5.5), low values of the frequency and a large range of variation of the angle of propagation (50◦ ≤ ≤ 120◦ , for Bi = 10) when W is varied. They labelled modes of type II those modes with higher values of the wavenumber (5.5 < q < 8), higher values of the frequency and a smaller range of variation of the angle of propagation (0◦ ≤ ≤ 20◦ , for Bi = 10) when W is varied. The pure transversal waves ( = 0◦ ) were included in modes of type II. The behaviour found for Bi = 10 was shown to remain qualitatively true for larger values of the Biot number, the main feature to be mentioned being that the domain of existence of oscillatory modes of type II increases with the value of Bi. As a concluding remark for this discussion, let us recall that (see Section 11.1) for dominant buoyancy (Ra > 4 × 103 ) Parmentier et al. (1993) found the preferred mode of oscillatory instability to be transverse rolls with c = 0◦ . Just above the threshold for oscillatory instability W > W0 Mercier and Normand (1996) reported that waves with transversal front are the most critical, later becoming oblique waves. A comparison for W ∼ = 0 between their results for Bi = 1 and the results of Parmentier et al. (1993) corresponding to Bi = 0 shows a qualitative agreement with modes of the b family.

11.4.2 Tertiary Modes of Convection: OMC and HTW with Spatiotemporal Dislocations When the applied temperature gradient is increased beyond the threshold of the primary instability, the system response and ensuing instabilities depend on the properties of the initial critical pattern. In this regard, and in view of the arguments developed in the preceding subsection, a coherent approach to the subject requires splitting the problem into three distinct subcases according to the depth of the layer (d < dc , dc ≤ d < dr or d > dr ).

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In the case d < dc , for which, as explained in Section 11.4.1, at the onset of hydrothermal disturbances the wave source is approximately in the middle of the channel, system evolution as the temperature difference is increased simply foresees the replacement of the initial critical state with a tertiary asymptotic homogeneous regime, where just a single wave is present (the related waveform is similar to that shown in Figure 11.15, right), which means the equilibrium supercritical waveform is a simple wave travelling at a ﬁxed angle with respect to the imposed temperature gradient. By contrast, for intermediate depth, dc ≤ d < dr , a well-deﬁned secondary instability occurs as a modulational instability of the critical single-wave pattern. As originally illustrated by Smith (1988), this instability is similar to the Eckhaus instability (see Section 4.2), but occurs with low (but ﬁnite) wavenumber. Spatially wavenumber modulations grow, which eventually lead to regular shedding of spatiotemporal dislocations in the ﬂow . These patterns are quasi-periodic in both frequency and space. In general, downstream of the dislocations, a lower wavenumber pattern is observed, which is either monoperiodic or chaotic according to d and the size of the lateral walls. Let us recall that the Eckhaus instability (see, e.g., Rashkeev et al., 2001) is one of the major secondary instabilities of nonlinear patterns (it has already been mentioned in Chapter 4 as one of the typical instabilities pertaining to the Busse balloon). Whereas the Eckhaus dynamics for steady patterns (e.g. Rayleigh–B´enard) are fast when dealing with local wavelength nucleation or annihilation, it presents a slow evolution of travelling modulated waves in the travelling wave pattern case. Some theoretical analyses have also appeared over the years to study these particular aspects. In such a context, it is worth mentioning that the concept of pattern ‘defects’, introduced in Chapter 4 (Section 4.3) with regard to the secondary and higher modes of Rayleigh–B´enard convection, is also generally applicable to the higher states of travelling waves for other types of convection (Coullet et al., 1989; Janiaud et al., 1992). As an example, Burguete et al. (1999) identiﬁed the possible defects of waves by means of topological arguments and studied them in the framework of Landau-type analysis. It was shown that they correspond to sinks, sources or dislocations of travelling waves and to dislocations of standing waves. It is also worth mentioning the theoretical study of Garnier and Chiffaudel (2002), who, assuming periodic boundary conditions, produced modulated wave patterns by increasing the control parameter or changing the discrete mean wavenumber of the waves. These patterns were observed to range from stable periodic phase solutions, due to the supercritical Eckhaus instability, to spatiotemporal defect chaos involving travelling holes and/or competition of counterpropagating waves, that is, travelling sources and sinks (further elaboration of this subject is beyond the scope of the present book). For relatively large depth d > dr , the stationary pattern (SMC) established through the primary instability has a secondary instability characterized by the onset of an oscillatory multicellular ﬂow (hereafter referred to as the OMC state). As shown in Figure 11.20 after the study by Burguete et al. (2001), in particular, for relatively shallow layers, such a state tends to displays a spatiotemporal diagram with oscillations of the rolls in phase opposition. Such a route with the succession of two instabilities, the ﬁrst in the form of a stationary bifurcation and the second as a Hopf bifurcation, was reported through numerical and experimental approaches by several investigators (e.g. Mundrane and Zebib, 1993, for Pr = 8.4; Gillon and Homsy, 1996, for Pr = 9.5; Braunsfurth and Homsy, 1997, for Pr = 4.4; Burguete et al., 2001, for Pr = 10.3). Apart from differences arising as a consequence of the speciﬁc setup (values of the streamwise and spanwise aspect ratios) considered in the experiments, it is important to shed some light on the

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Figure 11.20 Spatiotemporal diagram for unsteady evolution of longitudinal rolls (Pr = 10.3, Lx = 2 cm, d = 4 mm and T = 13 K). After Burguete et al. (2001); Reproduced by permission of the American Institute of Physics

fact that this oscillating multicellular ﬂow is also a time-dependent state, but is not a travelling wave one, since it is not characterized by a well-deﬁned wave speed [an important difference between the HTW and OMC states is that the former is preceded by a steady unicellular ﬂow that behaves as an underlying terrain for the propagation of the waves, whereas the latter is preceded by the steady multicellular ﬂow SMC (Bucchignani, 2004); let us also recall that similar behaviours have been observed in the literature for other convection experiments, e.g. Rayleigh–B´enard convection in layers (Dubois et al., 1989)].

11.5

Effects Originating from the Walls

In the preceding subsections, a fairly exhaustive theoretical framework has been developed containing all the necessary links between experimentally observed spatiotemporal patterns and possible explanations provided by linear stability analyses on the subject (resorting to the inﬁnite layer approximation). It should be pointed out, however, that some studies have also appeared in the literature where attention was expressly concentrated on the role potentially played in these dynamics by the presence of lateral limiting walls. Although the classical linear stability analyses for the inﬁnite layer have proved to provide several convincing insights into the fundamental mechanisms driving the phenomena under investigation, there are, however, some speciﬁc aspects that they could not explain, for example, some features of the SMC (and subsequent OMC) regimes and the intricate behaviour displayed by the hydrothermal waves close to the cold wall (Figure 11.15) according to the depth of liquid, which motivated further theoretical exploration of the subject.

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11.5.1 Lateral Boundaries as a Permanent Stationary Disturbance To obtain an improved ﬁt between experimental data and theory, some authors used the concepts of absolute instability, instead of simple convective instability. The principal idea at the basis of these theoretical studies (Priede and Gerbeth, 1997b; P´erez-Garc´ıa et al., 2004) is that distant lateral boundaries may be regarded as a permanent stationary disturbance of the homogeneous basic state assumed in conventional approaches and earlier studies for the inﬁnite layer. The fundamental problem at the basis of these studies was the determination of the degree of penetration of such perturbations into the homogeneous basic state. Priede and Gerbeth (1997b) were the ﬁrst to reveal that the lateral walls disturb the assumed uniformity of the basic ﬂow and to evaluate how far this perturbation can spread from a vertical boundary, showing that the effect due to the sidewalls can cause a stationary wave pattern extending through the whole layer when a zero-frequency mode becomes unstable. Unlike the stationary modes predicted for Pr > 1 in the earlier linear stability analyses of Gershuni et al. (1992) and Mercier and Normand (1996), which were found to be essentially longitudinal and to emerge solely for nonadiabatic free surface (conducting in the study of Gershuni et al., 1992; characterized by a given Bi in Mercier and Normand, 1996), these stationary patterns were predicted (Pr = 13.9) to be essentially transverse ( = 0◦ ) and to occur even in the limit Bi = 0 . Indeed, for the speciﬁc case of pure Marangoni ﬂow and adiabatic horizontal boundaries, Priede and Gerbeth (1997b) determined an effective possibility for stationary waves induced by the upstream (hot) endwall to be spatially ampliﬁed and spread over the whole layer regardless of its extent for Pr > 0.67 (for a sufﬁciently high value of the Marangoni number). In particular, for the speciﬁc case Pr = 13.9 and under the effect of buoyancy, the stationary perturbation due to the upstream endwall was found to become unstable and spread over the whole layer before the onset of oscillatory convection for W −1 > 0.2. Such results are summarized in Figure 11.21, showing that buoyancy effects may strongly stabilize the onset of hydrothermal waves and determine transition to stationary multicellular convection above a threshold value of W −1 .

Figure 11.21 Critical Marangoni numbers for the onset of both steady multicellular (SMC) and oscillatory convection depending on the nondimensional parameter W −1 = Ra/Ma for 1 cSt silicone oil (Pr = 13.9, top and bottom boundaries with adiabatic conditions, Marangoni and Rayleigh numbers deﬁned as Ma = σT γ d 2 /µα and Ra = gβT γ d 4 /να , respectively, where γ is the rate of constant temperature increase along the x -axis). Courtesy of J. Priede

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Figure 11.22 Critical Marangoni numbers for the onset of both steady multicellular (SMC) and oscillatory convection depending on the nondimensional parameter W −1 = Ra/Ma for 0.65 cSt silicone oil (Pr = 10, Bi = 1, Marangoni and Rayleigh numbers deﬁned as Ma = σT γ d 2 /µα and Ra = gβT γ d 4 /να , respectively, where γ is the rate of uniform temperature increase along the x -axis). Courtesy of G. Lebon

P´erez-Garc´ıa et al. (2004) obtained similar results for Pr = 10 for an inﬁnite layer with a bottom conducting wall and free surface with Bi = 1. In fairly good agreement with the earlier ﬁndings of Priede and Gerbeth (1997b), these authors reported stationary transverse corotating rolls to become the preferred mode of instability for W −1 ≥ 0.3 (Figure 11.22). Perhaps the most interesting new outcome of the latter study is that the curve for the oscillatory instability presents two branches (Figure 11.22) that could be related with the two distinct waveforms reported experimentally by Burguete et al. (2001) for rectangular layers (see Section 11.4.1) and by Garnier and Chiffaudel (2001) for annular conﬁgurations (as will be illustrated in Section 11.7), respectively.

11.5.2 Collision Phenomena of HTW and Wall-generated Steady Patterns In the analysis of Shevtsova et al. (2003a), the problem originally considered by Priede and Gerbeth (1997b) (Pr = 13.9 and adiabatic horizontal boundaries) was reinvestigated in the framework of direct numerical solution of the nonlinear time-dependent thermal-convection equations. Basically, this study proved that the nonlinear interplay between ﬁnite-amplitude boundary-induced steady patterns and hydrothermal waves is an essential ingredient in determining the system dynamics in practical situations and that nonlinear simulations of ﬂow regimes in a wide region of the values of W and Marangoni number are of crucial importance for understanding phenomena that cannot be predicted in the framework of linear theories. These authors carried out two-dimensional numerical simulations for a liquid layer with Pr ∼ = 14, aspect ratio A ∼ = 25 (to mimic the experiments of De Saedeleer et al., 1996) and both top (the free surface) and bottom walls thermally insulated, focusing on the different types of instabilities arising according to the variation of the two control parameters W and Ma. It was found that the problem related to the system evolution as the applied temperature difference T increases can be basically regarded as a fascinating competition between the aforementioned stationary wave spreading from the hot side towards the cold side responsible for

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multiroll steady convective structures SMC (such a wave is induced by the hot rigid lateral wall, as shown by Priede and Gerbeth, 1997b) and an HTW of the type predicted by Smith and Davis (1983) moving in the opposite direction (upstream). They identiﬁed, in particular, two characteristic values of the nondimensional parameter W , allowing the possible dynamics to be partitioned into three well-deﬁned subcases according to the two dominant instability mechanisms and related possible interaction. Such values (hereafter referred to as W1 and W2 ) were found to be W1 = 3.33 and W2 = 4, respectively (in fairly good agreement with the earlier theoretical predictions of Priede and Gerbeth, 1997b). For W > W2 the steady basic ﬂow becomes unstable to hydrothermal waves (HTWs) and for W < W1 the basic unicellular state is transformed into a steady multicellular state and then bifurcates to a time-dependent one (OMC) via a secondary instability. It is in the range W1 ≤ W ≤ W2 , however, that the investigation of Shevtsova et al. (2003a) provided the most interesting information, which deserve to be explicitly discussed and understood (as developed in the following). According to such simulations, with an increase in the temperature difference a few rolls (more than one) appear near the hot wall. The transition from one-roll to a multiroll ﬂow structure, as anticipated, is caused by the inﬂuence of the lateral wall that generates a wave, stationary in time, but spatially spreading towards the cold side with the increase in the applied temperature gradient. The subsequent transition corresponds to the birth of a hydrothermal wave at the cold wall. This hydrothermal wave, moving from the cold side, appears earlier than the rolls coming from the hot side occupying the whole cavity (let us recall that the hydrothermal wave itself looks as a succession of cells moving from the cold side towards the motionless rolls on the hot side, see Figure 11.24, as already discussed in Section 10.2.4 for pure Marangoni ﬂow). In this intermediate interval of W (W1 ≤ W ≤ W2 ), the parallel ﬂow is unstable with respect to the hydrothermal wave (HTW), but the multicellular periodic structure generated by the sidewall perturbation is fairly stable, so that the HTW decays in space when propagating on the background of the multicellular structure (in practice, the hydrothermal wave, moving from the cold side faces a strong ‘resistance’ of the steady rolls; the reader is referred, in particular, to the right side of Figure 11.23). On the one hand, under the impact of the powerful HTW, all the rolls that belong to the multicellular structure oscillate in time, but they resist (the amplitude of the temperature oscillations near the hot wall is many times smaller than that near the cold side). On the other hand, the HTW loses its power sharply due to the collision with the stationary structure. As a result a calm region (a practically vortex-free zone) is established in the central part of the cavity. Modulation of the hydrothermal wave basically arises due to its collision with the aforementioned powerful chain of vigorous rolls, which has its own characteristic frequency of oscillations. In practice, these arguments reveal some nontrivial features of the interaction between the instability-generated HTW and wall-generated steady patterns. An HTW typically decays in the region of steady pattern, however, it induces oscillations of the wall-generated system of vortices. Moreover, the HTW penetrating into the region occupied by the steady pattern keeps its frequency but changes its wavelength. Under the conditions of a coexistence of both types of motion, the HTW near the cold wall and oscillating multicellular structure near the hot wall, satellite frequencies of the fundamental oscillations appear, which lead to a more complicated, quasi-periodic, temporal behaviour of waves. For W > W2 , the HTW dominates. As soon as the applied temperature difference exceeds a certain value, the hydrothermal wave, generated in the cold part of the cavity, propagates from the left to the right (let us recall once again that the motion of rolls in the direction from the cold end to the hot end is a characteristic feature of the HTWs). For this set of parameters, the HTW collides with the single steady roll adjacent to the hot wall (as illustrated in Section 10.2.4, such more or less stationary rolls existing near the hot side are basically maintained by the strong temperature

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1.0 ∆ψmax = 3.34 t = τ/5

y 0.5 0.0 0.0 1.0

24.7 ∆ψmax = 3.28 t = 2τ/5

y 0.5 0.0 0.0 1.0

24.7 ∆ψmax = 2.76 t = 3τ/5

y 0.5 0.0 0.0 1.0

24.7 ∆ψmax = 2.23 t = 4τ/5

y 0.5 0.0 0.0 1.0

24.7 ∆ψmax = 3.43 t = 5τ/5

y 0.5 0.0 0

5

10

x

15

20

A

Figure 11.23 Oscillatory hybrid convection in a shallow rectangular cavity (Pr ∼ = 14, aspect ratio A∼ = 25, W = W1 = 3.3, Ma ∼ = 2.7 × 104 ; W and Ma based on the depth; adiabatic horizontal boundaries; cold side on the left, hot side on the right). The isolines of the stream-function of the disturbance are shown [plane (x, y )] in ﬁve snapshots evenly distributed within one period of oscillation. The location of the cells near the cold side at different time moments indicates the propagation of the wave to the right. The presence of some ‘ghosts’ in the right part reveals the oscillations of stationary rolls induced by the collision with the HTW. Courtesy of V. Shevtsova

gradient established in the boundary layer for relatively large supercriticalities). Anyhow, the hydrothermal wave still ‘feels’ the presence of the steady roll since it tends to decay in the region where it is located and/or to display a nonharmonic behaviour there. With decreasing W , the critical temperature difference Tcr , at which the HTW emerges on the cold wall, increases. Within such a context, it is worth noting that with an increase in T , the amount of steady rolls established near the hot side also increases. Moreover, the strength of each of them increases faster than that of the waves near the cold side. As a natural consequence, the region of the collision of the HTW and steady rolls tends to be slowly shifted away from the hot side towards the cold side. In the opposite case, when W becomes equal to W1 , there is the ﬁrst sign that the multirolls intend to control the heat and mass transfer in the cavity. For this combination of parameters, the strength of the second roll from the hot side is the same as the strength of the HTW. The HTW exists in a small region near the cold wall, followed by a region of decay. On the side of the multiroll structure there is also a small region of decay. In the central part, between the regions of the decay of both waves, there is a region of ‘shuttle’ motion. A roll, pushed by the HTW,

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moves to the right and, after collision with a stationary wave, goes back (see again the central part of Figure 11.23). The strength of both waves is similar, therefore, roll(s) moving with the fundamental frequency along the cavity look like the motion of a ball pushed from both sides by elastic walls. The HTW loses energy in favour of the stationary wave and all stationary rolls oscillate with time. For the ﬁnal case, in which W < W1 , the multiroll structure ﬁlls the cavity before the oscillatory instability sets in. The leading role is played by the ﬁrst roll near the hot side. Above the threshold of oscillatory instability, just this roll pushes the chain of vortices and the disturbances propagate from the hot to the cold side (Figure 11.24, intensive oscillations arise at the hot side and then they propagate to the other side decaying in space on the background of the system of vortices). As a concluding remark for this subsection, it should be noted that, beyond the results described in detail in the foregoing text and in Section 11.5.1, other analyses have also appeared where the possible interplay between the HTW and stationary rolls and/or other types of convection was

1.0

∆T = 12.7 ψmin = 9.46 t = τ/5

y 0.5 0.0 0.0 1.0

24.7 ψmin = 5.06 t = 2τ/5

y 0.5 0.0 0.0 1.0

24.7 ψmin = 4.41 t = 3τ/5

y 0.5 0.0 0.0 1.0

24.7 ψmin = 3.56 t = 4τ/5

y 0.5 0.0 0.0 1.0

24.7 ψmin = 3.56 t = 5τ/5

y 0.5 0.0 0

5

10

x

15

20

A

Figure 11.24 Oscillatory hybrid convection in a shallow rectangular cavity (Pr ∼ = 14, aspect ratio A ∼ = 25, W∼ = 1.5, Ma ∼ = 4.2 × 104 ; W and Ma based on the depth; adiabatic horizontal boundaries; cold side on the left, hot side on the right). The isolines of the stream-function of the disturbance are shown [plane (x, y )] in ﬁve snapshots evenly distributed within one period of oscillation. Comparison of this ﬁgure with Figure 11.23 clearly demonstrates that, as for high W the HTW spreads from the cold to the hot side, whereas for smaller W the oscillations propagate in the opposite direction, they are of a different nature (such a conclusion being also supported by the fact that the oscillation frequency of the multicellular ﬂow is many times higher than the frequency of the HTW). Courtesy of V. Shevtsova

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considered from slightly different perspectives. For instance, Nepomnyashchy et al. (2001) investigated the transition from the stationary Pearson (Marangoni–B´enard) instability mechanism (treated in Chapter 9) to oblique hydrothermal waves; they showed that the classical Marangoni–B´enard ﬂow structure disappears (it is replaced by the HTW) even for weak deviations of the mean temperature gradient from the vertical direction in thin liquid layers (for further important details, see Section 9.5).

11.5.3 Streaks Generated by a Lift-up Process and Instabilities of a Mechanical Nature Kuhlmann and Albensoeder (2008) provided a possible new explanation for the existence of stationary bifurcations reported in experimental results. Extending the earlier study of Xu and Zebib (1998) (limited to 2D disturbances and pure Marangoni ﬂow; see Section 10.2.4 and Figure 10.16) to the case of mixed convection and 3D disturbances, they performed a linear stability analysis of the buoyant–Marangoni ﬂow in open rectangular two-dimensional cavities with adiabatic horizontal walls and periodic boundary conditions in the spanwise direction; in particular, the following conditions were considered: Pr = 10, aspect ratio in the range 1.2 ≤ A ≤ 8 and a ﬁxed relationship A2 /W = 12.755 (such parameters corresponding to the material and geometry of the experiments of Daviaud and Vince, 1993). In agreement with the experimental results, they found the supercritical ﬂow to be steady and three-dimensional for A < 3 and oscillatory for larger aspect ratios (see Figure 11.25), identifying in the former case an instability mechanism of mainly a mechanical nature, never reported before in this kind of studies. To do justice to these interesting ﬁndings, the remainder of this section is devoted entirely to illustrating the major outcomes of this analysis (also highlighting, as usual, analogies and differences with respect to earlier results). Before becoming involved with such a description, however, let us start by recalling some fundamental properties of the initial basic 2D state, which these authors proved to play a vital role in the instability mechanisms (and which will be useful also for supporting the discussion to be developed in the present section). As illustrated in Chapter 10, the basic velocity ﬁeld for high-Pr ﬂuids and dominant Marangoni ﬂow is characterized by a single vortex that is strained elliptically. Moreover, as already discussed in Section 2.5, if the Marangoni number Ma = RePr is sufﬁciently high, the basic temperature ﬁeld exhibits a nearly isothermal region in the vortex core and thermal boundary layers near the cold and hot walls. Cold ﬂuid is transported from the cold wall along the insulating bottom of the cavity very close up to the hot corner, where it forms a cold ﬁnger. By a similar convective transport, a warm ﬁnger is created in front of the thermal boundary layer on the cold wall (see, e.g., Figure 2.17 for A = 2). For a value of the aspect ratio (A = 2.1) close to that considered for the example in Figure 2.17, Kuhlmann and Albensoeder (2008) found a stationary instability (among other things, it is also consistent with the previous nonlinear simulations of Mundrane and Zebib, 1993), which, as already outlined, can be ascribed to a process that is mainly mechanical . According to their analysis, this instability is characterized by ‘streaks’ in the shear layer of the free surface, which are triggered by a small vertical component of the perturbation velocity. The explanation of the related origin and ampliﬁcation mechanisms is not as straightforward as one would assume, which makes it opportune to open a detailed discussion on the subject. In practice, by horizontal advection of temperature from the aforementioned vertical hot and cold ﬁngers of the basic state, the streaks can produce free surface temperature extrema driving a spanwise Marangoni ﬂow supporting the streak formation. The thermal energy budget in such a

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(a)

(b)

Figure 11.25 Critical Reynolds number based on the cavity depth (a) and critical wavenumber qz (b) as a function of the aspect ratio A (Pr = 10 and A2 /W = 12.755; cavity with adiabatic horizontal boundaries and periodic boundary conditions in the spanwise direction). The dashed and the solid lines indicate the critical curves for the steady and the oscillatory instability, respectively; the dashed-dotted line is the neutral curve for purely two-dimensional perturbations with qz = 0. Courtesy of H. Kuhlmann

process is insigniﬁcant, because the thermal energy created by the convective action of the streaks on the cold-wall thermal boundary layer is essentially dissipated and the convection of thermal energy by the basic ﬂow merely leads to weak and small surface temperature perturbations near the hot wall. By contrast, the mechanical energy budget plays a crucial role. According to Kuhlmann and Albensoeder (2008), in fact, the major fraction of about 60% of the kinetic energy production is contributed by the term describing the ampliﬁcation of the horizontal perturbation ﬂow u by the

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vertical advection of horizontal momentum (u0 ) of the basic state. Since this term is peaked in the region of the maximum perturbation velocity, the perturbation ﬂow gains most of its energy from the strong basic shear ﬂow near the free surface, triggered by the vertical component of the perturbation ﬂow midway between the heated walls. This process leads to regions of very high absolute values of the horizontal (streamwise) perturbation velocity that represent the aforementioned ‘streaks’. While a streak (x component of V ) in the negative x direction is created by the downward y component of V , a streak in the positive x direction is created by the upward y component of V in an xy plane shifted by half a wavelength. The positive and negative streaks are connected by a turning ﬂow in the z direction. The reader may be interested in knowing that other authors found the same mechanism of ampliﬁcation of the perturbation energy to be operative in wall-bounded plane shear ﬂows (Hamilton et al. 1995; Waleffe, 1998). In such studies, the process of streak formation was referred to as ‘lift up’. It is a linear process that leads to an algebraic transient growth of the perturbation energy. While the linear perturbations ultimately decay in unbounded parallel shear ﬂows, they persist through an exponential instability in the laterally-bounded Marangoni-ﬂow system. This qualitative difference is due to the ﬁnite aspect ratio and aiding Marangoni forces which close the instability loop. Additional illuminating insights into such dynamics can be provided as follows. The streaks generated by the lift-up process act on the basic temperature ﬁeld and create relatively weak internal temperature perturbations. In the xy plane the perturbation ﬂow creates a weak cold perturbation-temperature spot in the interior by transporting cold ﬂuid from the cold ﬁnger to the centre of the cavity. This process also occurs near the free surface and gives rise to the large cold free-surface spot. An analogous mechanism creates large hot free-surface spots by convection from the hot ﬁnger. The Marangoni effect in the z direction due to the large but weak free-surface spots assists the mechanical instability by creating the required vertical perturbation ﬂow due to continuity in the region of strong shear. Kuhlmann and Albensoeder (2008) observed that the direct production rate of mechanical energy by the spanwise Marangoni effect contributes only roughly one-quarter to the total kinetic energy production. This is different from the hydrothermal wave mechanism in high-Pr liquids in which the Marangoni effect is responsible for almost the total mechanical energy production. They concluded that the stationary instability branch for A ≤ 3 and Pr = 10 shown in Figure 11.25 is due to a combined inertial and thermocapillary effect [the major cause (60% of the kinetic energy production) of the instability being provided by the lift-up process taking place in the shear ﬂow below the free surface and creating streaks in the ±x direction]. The basic temperature ﬁeld is mainly necessary to provide the driving of the basic shear ﬂow. The presence of the hot and cold ﬁngers in the basic temperature ﬁeld provides an additional mechanism (25% of the kinetic energy production) in support of the mechanical instability by enabling spanwise Marangoni forces resulting from a streamwise advection of the basic temperature due to the streaks. Let us recall at this stage that for inﬁnite layers, as discussed in Section 11.4.1, Mercier and Normand (1996) found that a sufﬁcient buoyancy level (W not exceeding a certain threshold value W0 depending on the Biot number) is required for the onset of stationary rolls. They speculated, moreover, that the thermal boundary conditions of a conducting bottom wall and a heat transfer at the free surface were decisive for a comparison with the experimental results of Daviaud and Vince (1993), even though it was not possible simultaneously to ﬁt the values of the Biot and W numbers to the experimental values. The work by Kuhlmann and Albensoeder (2008) proves, however, that experimental results can be well reproduced also if one assumes insulating top (zero Biot number) and bottom boundary conditions. Moreover, even if obviously the basic state depends to a certain extent on buoyancy

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(which means that buoyancy may be of importance indirectly as it inﬂuences the basic ﬂow and temperature ﬁelds), the instability mechanism that they ﬁnd under the condition W = A2 /12.755 is almost independent of gravity. Remarkably, this latter result is qualitatively different from the theoretical results discussed earlier (Section 11.4.1) in which stationary instabilities are usually associated with Pr > 1 and a vertical temperature proﬁle that is at least piecewise destabilizing in the sense of the Marangoni (Smith and Davis, 1983; Pearson, 1958) or the Rayleigh–B´enard instability (Gershuni et al., 1992), which means that the stationary 3D ﬂow identiﬁed by Kuhlmann and Albensoeder (2008) for A < 3 has nothing to do with the stationary longitudinal rolls of gravitational or thermocapillary nature repeatedly invoked in Section 11.4.1 as a possible reason for the departure of the experimental results from the typical HTW phenomena (see also Section 11.7.3 for some additional interesting discussions along these lines). For the speciﬁc conditions considered (A2 /W = constant, which means buoyancy force decreasing in comparison with the surface tension-driven force as A is increased) and relatively large aspect ratios (A > 3), Kuhlmann and Albensoeder (2008) recovered (with some interesting differences induced by the presence of lateral solid walls, which also deserve some discussion) the canonical phenomena related to the propagation of oblique waves of the hydrothermal type, as proven by the features shown in Figure 11.26. For A = 8, in particular, they found an angle with respect to the temperature gradient of ∼ = ◦ . This angle of wave propagation is smaller than the value ∼ 80◦ observed by Daviaud and 50 = Vince (1993). On the other hand, it is much larger than the critical angle (Pr = 10) = 21.4◦ for a hydrothermal wave at Ra = 0 in an inﬁnite thermocapillary layer (Smith and Davis, 1983; see Figure 10.6), which requires some theoretical justiﬁcation. In practice, for the slightly larger Prandtl number of Pr = 13.9 and for W −1 = 0.2, Priede and Gerbeth (1997) found an angle of 22◦ for plane layers also in the presence of gravity. By a three-dimensional numerical simulation in a large aspect ratio cavity for Pr = 13.9 and zero gravity, Xu and Zebib (1998) reported an angle of propagation of ∼ = 15◦ , which is even ◦ ∼ less than the value for plane thermocapillary layers = 17 (layer with Pr = 13.9).

Figure 11.26 Perturbation temperature in the plane y = 0.07 and isosurface of the total local thermal energy production. The background shows the basic streamlines and the foreground displays the perturbation velocity ﬁeld (Pr = 10, A = 8, W = 5). The wave propagates obliquely towards the hot wall and in the negative z direction. By comparing the intersection of the isosurfaces of the total local thermal energy production with the streamlines of the basic ﬂow (background of this ﬁgure), it can be seen that the disturbance thermal production attains a maximum in the regions of the hyperbolic stagnation points of the basic ﬂow where relatively large basic temperature variations exist; in these regions the perturbation ﬂow is relatively strong (ﬁgure foreground), which explains the structure of the thermal production; in particular, the local maxima of the production rate are well pronounced near the hot wall, whereas the production rate becomes smoother toward the cold wall and the related isosurfaces merge into a oblique varicose tube. Courtesy of H. Kuhlmann

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Anyhow, the result of Kuhlmann and Albensoeder (2008) for Pr = 10 is in good agreement with the ﬁnding = 46.6◦ of Mercier and Normand (1996) for inﬁnite thermocapillary-buoyant layers, with a slight heat loss characterized by a Biot number Bi = 1. Similarly, Burguete et al. (2001) for Pr = 10.3, A = 9.1 (d = 1.1 mm, Lx = 10 mm) and a similar value of W −1 = 0.15 obtained = 30◦ , which is also signiﬁcantly higher than for the pure Marangoni case. Along these lines, it is also worth mentioning that Pelacho et al. (2000) ascribed the rather different angle of propagation of ∼ = 80◦ found by Daviaud and Vince (1993) in comparison with their own experimental result to the three-dimensional geometry of the cavity. For a streamwise length Lx = 100 mm, a layer depth of 1.5 mm and Pr = 10, they found, in fact, that the angle of propagation of the wave is slightly larger than = 50◦ for a spanwise length Ly = 40 mm and that decays to about 40◦ on increasing Ly to 100 mm. In view of all these arguments, it may be concluded that, in addition to a possible dependence on the relative importance of buoyancy and Marangoni effects, the angle of propagation is very sensitive to the geometric constraints. Following Kuhlmann and Albensoeder (2008), in particular, the strong dependence on the streamwise aspect ratio A might be related to the fact that the instability is strongly inﬂuenced by the nonparallel structure of the basic ﬂow (let us recall that the basic convection does not correspond to a plane-parallel ﬂow as assumed in linear stability analysis for the inﬁnite layer, but rather it is roughly periodic in the x direction due to unavoidable embedded corotating vortices induced by the system ﬁnite size). As a concluding remark for this discussion, let us observe that, whereas for relatively shallow conﬁgurations boundary effects and gravity (the main themes of this section) can eventually lead to departure with respect to the idealized results obtained by Smith and Davis (1983) for pure Marangoni ﬂow, in geometries with A ∼ = 1 buoyancy-driven convection in the bulk can even ‘separate’ from the basic convection roll driven by thermocapillarity (see, e.g., the numerical simulations by Bueckle and Peri´c, 1992). These behaviours are treated in the next section, where some speciﬁc examples expressly conceived for such a purpose are considered.

11.6

The Open Vertical Cavity

Transcending speciﬁc aspects relating to ﬂow instabilities (which have been the main theme of preceding sections), the objective of this section is to provide the reader with an idea of how complex the interplay between gravitational and surface tension-driven effects may be if the characteristic system size is not limited to few millimetres and geometries with height comparable to the lateral extension are considered. In particular, the geometry of the problem used for the experiments considered in the present section as demonstrative exemplars (see Figure 11.27) is a parallelepipedic liquid volume (Lx = Lz = L = 5 cm, height Ly = 7.5 cm with a square cross-section 5 × 5 cm) ﬁlled by a 10 cSt silicone oil (Pr = 105) with a free ﬂat surface (after the experimental study by Savino et al., 2001a). As shown in Figure 11.27, each vertical wall displays three parts: a portion (height H = 5 cm), adjacent to the volume liquid, in fact, is thermally insulated from an upper smaller portion (height h = 5 mm) by a slab 2 cm high. Moreover, the temperatures Ti (i = 1, 2, 3, 4) of the four heaters shown in Figure 11.27 (T1 and T3 at the left side and the opposite T2 and T4 at the right side) can be controlled independently, while the bottom of the container is thermally insulated. Hereafter, the heaters at temperature T1 and T2 (whose thickness is 5 mm only to generate a temperature gradient parallel to the free surface solely in a thin layer of liquid) will be referred

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Figure 11.27 Paradigm model conﬁguration (sketch) expressly conceived for the investigation of possible terrestrial states of interacting buoyancy and Marangoni convection

to as ‘surface heaters’, while the term ‘volume heaters’ is adopted for the other two heaters at temperatures T3 and T4 . A similar conﬁguration was used in the landmark experiments of Schwabe and Metzger (1989) and Schwabe and D¨urr (1996). Notably, with this model conﬁguration, the liquid can be subjected simultaneously to temperature gradients parallel and perpendicular to the free surface. Obviously, gradients parallel to the free surface will induce a surface tension gradient, forcing the free surface to move, while gradients perpendicular to the free surface or generated in the horizontal direction at a certain depth (by the volume heaters) will be responsible essentially for buoyancy forces, leading to massive motion in the bulk. It is also obvious that the surface tension-driven forces on the free surface will have some inﬂuence on the convective structure in the bulk and, vice versa, buoyancy-driven circulation in the bulk (driven by horizontal and/or vertical temperature gradients) will affect the free-surface motion. This special conﬁguration has been expressly conceived to allow buoyant forces to be varied independently from Marangoni forces. In particular, in the following it will be shown how this artiﬁce can lead to a variety of possible variants of great interest, this being simply achieved by application of a temperature gradient along the free surface different from that in the bulk. Indeed, the gradients TL = T2 − T1 and TH = T4 − T3 can be parallel or antiparallel or one can be set equal to zero to investigate the separate effects of buoyancy and surface tension forces (e.g. TL = 0 and TH = 0 or vice versa). In this way, one may vary almost independently: the magnitude and direction of the buoyant-convection effects, measured by the appropriate Rayleigh number Ra = gβT TH H 3 /να, and the Marangoni convective effects, measured by the appropriate Marangoni number Ma = σT TL L/µα. It will be shown how, since the streamwise and spanwise aspect ratios (Lx /Ly and Lz /Ly ) are ∼ 1, the ﬂow tends to be steady and fairly stable (let us recall that high-Pr ﬂows in cavities are strongly stabilized by the presence of close sidewalls along both x and z; the reader is referred, in particular, to the last two paragraphs of Section 6.4 for the case of buoyant convection, to Section 10.2.4 for pure Marangoni ﬂow and to Section 11.5, for example Figure 11.25, for the hybrid case) and that, even if it is in a stationary state, the interplay between Marangoni and gravitational effects can be very complex.

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Notably, in such a context four different fundamental situations can be considered: (a) TH > 0 and TL = 0 (dominant buoyancy ﬂow), (2) TH > 0 and TL > 0 (Marangoni and buoyant ﬂows concurrent), (3) TH > 0 and TL < 0 (Marangoni and buoyant ﬂows counteracting) and (4) TH = 0 and TL > 0 (no imposed bulk temperature gradient and only surface tension forces driving convection).

11.6.1 Volume Driving Actions and Rising Thermal Plumes For TL = 0 and TH > 0 [case (1) deﬁned earlier], the dominant feature of the ﬂow is the presence of a destabilizing vertical temperature gradient near the hot wall due to the different temperature of the heaters T4 (active) and T2 (at ambient temperature). Under such a condition, the hot ﬂuid heated by the element at temperature T4 rises (see Figure 11.28), giving rise to a strong convective cell in the upper right corner of the cavity; correspondingly, a hot region can be observed in the proximity of the heated wall on the free surface (Figure 11.29). Such a region is not homogeneous, but displays localized areas of higher temperature (hot spots), which provide indirect evidence that the rising ﬂuid adjacent to the right wall does not form a uniform sheet, but rather it breaks into distinct ﬁngers (or plumes) reaching the free surface. Remarkably, the number (m) of these ﬁngers changes on varying the magnitude of the bulk temperature gradient TH ; in fact, the free surface exhibits two spots for TH = 2.5 K, three spots for TH = 5 K, four spots for TH = 10 K, ﬁve spots for TH = 15 K and six spots for TH = 20 K (see Figure 11.29), which intrinsically means that the ﬁeld undergoes an increasing degree of complexity when the Rayleigh number is increased. It is worth noting that the role played in such dynamics by surface tension-driven effects can be assumed to be relatively weak, as witnessed by the relatively localized appearance of both the core of the convective cell in the upper right corner and corresponding surface temperature spots.

11.6.2 Aiding Marangoni and Buoyant Flows Figure 11.30 shows the streamlines and the velocity ﬁeld for TH = TL = 15 K [case (2)], that is, for concurrent surface and bulk temperature gradients. In such a situation, the pattern consists

Figure 11.28 Velocity ﬁeld (streamlines and velocity distribution, Vmax = 0.20 cm s−1 ) in the vertical midplane (z = 2.5 cm) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TH = 15 K and TL = 0 K: (a), (b) numerical simulation; (c) experimental result (M. Lappa)

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471

T ∆TH

0.785 0.729 0.673 0.617 0.561 0.505 0.449 0.392 0.336 0.280 0.224 0.168 0.112 0.056 0.000

(a) T ∆TH

0.801 0.744 0.687 0.630 0.572 0.515 0.458 0.401 0.343 0.286 0.229 0.172 0.114 0.057 0.000

(b) T ∆TH

0.821 0.762 0.703 0.645 0.586 0.528 0.469 0.410 0.352 0.293 0.234 0.176 0.117 0.059 0.000

(c) T ∆TH

0.840 0.780 0.720 0.660 0.600 0.540 0.480 0.420 0.360 0.300 0.240 0.180 0.120 0.060 0.000

(d) T ∆TH

0.854 0.793 0.732 0.671 0.610 0.549 0.488 0.427 0.366 0.305 0.244 0.183 0.122 0.061 0.000

(e)

Figure 11.29 Temperature distribution on the free surface (x, z) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TL = 0 and TH = (a) 2.5 K, (b) 5 K, (c) 10 K, (d) 15 K and (e) 20 K (numerical results on the left, corresponding experimental temperature ﬁelds provided by an Agema 900 infrared thermocamera on the right) (M. Lappa)

472

Thermal Convection: Patterns, Evolution and Stability

Figure 11.30 Velocity ﬁeld (streamlines and velocity distribution, Vmax = 0.24 cm s−1 ) in the vertical midplane (z = 2.5 cm) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TH = TL = 15 K: (a), (b) numerical simulation; (c) experimental result (M. Lappa)

essentially of a combined convection roll around the whole cavity driven by the joint action of surface tension and buoyancy forces. Such a roll is extended over the entire depth of the rectangular volume; anyhow, a roll mainly of a surface tension-driven nature is present with its own recirculation in the upper third of the liquid zone embedded into the larger buoyancy-driven circulation around the whole cavity. A small vortex located in the upper right corner is also present, whose shape (reminiscent of that observed in the absence on an imposed surface TL ) can be regarded as a clear signature of its gravitational origin. As in the case treated in Section 11.6.1, hot ﬂuid adjacent the heated wall at temperature T4 in the bulk (see Figure 11.27) is carried towards the free surface at a rather high position by buoyancy forces (due to the condition T2 < T4 ); after reaching the interface, however, it is accelerated by the Marangoni surface forces (induced by TL ) towards the cold wall at temperature T1 < T2 . When the ﬂuid reaches the surface heater at temperature T1 , since T1 < T3 , it is cooled and accelerated downwards again in the bulk by the buoyancy forces. These arguments provide a simple explanation for the convective structure visible in the midplane of the rectangular cavity. It is also evident that, owing to the surface forces that accelerate the ﬂuid from the side at temperature T2 to the side at temperature T1 , the surface temperature spots are stretched with respect to those shown in Figure 11.29 without an imposed temperature gradient, resulting in the presence of surface temperature ﬁngers protruding from the hot side toward the cold side (see Figure 11.31). Among other things, since for TH = 15 K the number of spots is m = 6 for TL = 15 K and m = 5 for TL = 0 K, such a comparison is also instrumental in understanding that the deﬁnition of any possible general laws of variation for m should take into account dependence on both TH and TL , i.e. m = f (TH , TL ).

11.6.3 Counteracting Driving Forces and Separation Phenomena When the surface temperature gradient is reversed [case (3), corresponding to TH > 0 and TL < 0], Marangoni and buoyant forces become opposing. As a natural consequence, a surface tension-driven roll appears in the upper part of the cavity, counter-rotating with respect to the volume buoyancy-driven roll.

Mixed Buoyancy–Marangoni Convection

473

T ∆TH 0.775 0.664 0.553 0.442 0.331 0.220 0.109 −0.001 −0.112 −0.223 −0.334 −0.445 −0.556 −0.667 (a)

(b)

Figure 11.31 Temperature distribution on the free surface (x, z) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TH = TL = 15 K: (a) numerical simulation; (b) experimental image (infrared thermocamera) (M. Lappa)

Figure 11.32 Velocity ﬁeld (streamlines and velocity distribution, Vmax = 0.17 cm s−1 ) in the vertical midplane (z = 2.5 cm) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TH = −TL = 15 K: (a) and (b) numerical simulation; (c) experimental result (M. Lappa)

Figure 11.32 shows, in particular, the velocity distribution and streamlines in the vertical midplane for TH = −TL = 15 K. Inspection of this ﬁgure reveals that there is a fully separated surface tension-driven roll in the upper third of the liquid zone rotating clockwise, whereas the buoyancy driven roll in the lower two-thirds of the ﬂuid volume rotates counterclockwise (in the contact region of the two rolls the ﬂow has the same direction for both rolls). Most interestingly, in this case no surface temperature spots are evident. To justify these behaviours, it is sufﬁcient to note that the hot ﬂuid adjacent to the bulk heater at temperature T4 cannot reach directly the upper free surface (where it could give rise to hot spots) owing to the existence of the surface counter-rotating convective cell. Indeed, the ﬂuid adjacent to the surface heater at temperature T2 < T4 is accelerated downwards by the buoyancy forces, hence preventing

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Thermal Convection: Patterns, Evolution and Stability

hot bulk ﬂuid (near the bulk heater at temperature T4 ) from reaching the surface directly. In practice, this hot ﬂuid tends to be entrained in the return ﬂow between the two vortex cells.

11.6.4 Surface Driving Actions and Vertical Temperature Gradients This last subsection is devoted to the conﬁguration in which only surface heaters are active (TH = 0, TL > 0). Apparently (relatively) simple from a conceptual point of view, this case is perhaps the best paradigm for illustrating the subtle effect played by buoyancy effects (in particular, in connection with localized vertical temperature gradients) in situations for which one would assume their inﬂuence to be negligible. Different situations may occur according to whether T1 = T3 and T2 > T4 or T1 < T3 and T2 = T4 . For T1 = T3 , T2 > T4 , a very thin Marangoni cell is conﬁned near the free surface (see Figure 11.33) and the ﬂuid in the bulk can be considered at rest. The ﬂow is apparently dominated by the Marangoni effect. The pattern is characterized by a full separation of the surface tension-driven roll from the bulk and beneath it only induced convection rolls are present (in practice, these weak vortex cells are secondary ﬂows induced by continuity in the interior of the cavity). The average temperature of the free surface [Tm = 1/2(T1 + T2 )] is signiﬁcantly above T3 = T4 . The mean temperature in the surface convection roll is higher than the mean temperature in the bulk ﬂuid. Remarkably, this means that although buoyancy is not actively propelling convection, it is the reason for the observed separation stabilizing hot liquid on top of colder (heavier) liquid (this separation should not occur in a microgravity experiment; there the pure surface tension-driven roll should penetrate fully into the liquid). For T1 < T3 , T2 = T4 , the ﬂow pattern is more complex (see Figure 11.34) and can be regarded as the outcome of a fully manifest combined effect of buoyancy and surface tension forces: the hot ﬂuid adjacent to the surface heater at temperature T2 is carried towards the side at temperature T1 due to the surface tension forces. When the ﬂuid reaches the cold side, it is cooled and accelerated downwards by the buoyancy effect. Due to the fact that the average temperature of the bulk ﬂuid is greater than T1 , the buoyancy forces give rise to a secondary inner vortex with the opposite sense

Figure 11.33 Velocity ﬁeld (streamlines and velocity distribution, Vmax = 0.12 cm s−1 ) in the vertical midplane (z = 2.5 cm) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TH = 0 and TL = 15 K (T1 = T3 and T2 > T4 ): (a), (b) numerical simulation; (c) experimental result (M. Lappa)

Mixed Buoyancy–Marangoni Convection

475

Figure 11.34 Velocity ﬁeld (streamlines and velocity distribution, Vmax = 0.21 cm s−1 ) in the vertical midplane (z = 2.5 cm) of the open vertical cavity shown in Figure 11.27 ﬁlled with 10 cSt silicone oil (Pr = 105) for TH = 0 and TL = 15 K (T1 < T3 and T2 = T4 ): (a), (b) numerical simulation; (c) experimental result (M. Lappa)

of rotation that counteracts the cold ﬂuid coming from the surface. The secondary buoyancy-driven roll occupies the lower third of the liquid zone and it is placed near the bulk heater at temperature T3 . In the contact region of the primary and the secondary roll, the two rolls meet to generate return ﬂow. As mentioned earlier, this last case is particularly signiﬁcant since it shows how on Earth buoyancy forces can inﬂuence Marangoni ﬂow in very unexpected ways.

11.7

The Annular Pool

After the cases of mixed buoyant–Marangoni convection in liquid rectangular layers, vertical deep cavities with imposed horizontal (and/or vertical) temperature gradients, this section is devoted to the thermoconvective instabilities appearing for Pr>1 in cylindrical annuli heated laterally (external boundary of the annulus) under conditions of normal gravity. Like the other geometric conﬁgurations, this system also has been the subject of intense investigation due to its aforementioned theoretical link with the CZ method applied to oxide melts (Hintz et al., 2001; Teitel et al., 2008). It is known that as soon as the temperature gradient reaches a critical value stationary or oscillatory bifurcations take place. Schwabe et al. (1992) were the ﬁrst to report the existence of HTWs on the ground in shallow annular liquid pools of ethanol (Pr = 17) with thickness ranging from 0.6 to 3.6 mm. They observed short-wavelength HTWs with curved arms in liquid pools with d < 1.4 mm and long-wavelength patterns in pools with d > 1.4 mm. For various conﬁgurations and a variety of experimental conditions, however, Benz and Schwabe (2001) observed the existence of three-dimensional stationary patterns consisting of pairs of counter-rotating longitudinal rolls (the liquids used were ethanol and silicone oil hexamethyldisiloxane). Interestingly, one implementation of the experiments was performed under microgravity (MAGIA experiment) and the other experiments under normal gravity. The experiment in microgravity indicated the absence of the three-dimensional stationary pattern. Additional insights into the potential effects of gravity on Marangoni ﬂow were provided a year later (silicone oil with Pr = 6.8) by Schwabe and Benz (2002), who found that (in contradistinction to the case of low-Pr ﬂuids discussed in Section 11.2, for which buoyancy has a destabilizing effect) gravity can signiﬁcantly stabilize the basic steady radial unicellular Marangoni ﬂow.

476

Thermal Convection: Patterns, Evolution and Stability

Hydrothermal waves in normal gravity conditions were also observed experimentally by Garnier and Chiffaudel (2001). These rotating waves were found to appear as spiralling patterns with many spiral branches (i.e. multi-armed spirals); however, they also identiﬁed a novel pulsating mechanism with a target-like wave pattern (almost coaxial circles travelling outward in the radial direction, i.e. an approximately radial wave) dominant only near the cold inner wall, which led them to separate the possible oscillatory ﬂows into two well-deﬁned classes. At relatively large ﬂuid depth, they observed planar waves in line with the convectional predictions of Smith and Davis (i.e. waves travelling with a clearly identiﬁable direction and angle with respect to the imposed temperature gradient), which were referred to as HW1 to distinguish them from the pulsating-like spatiotemporal behaviour (mentioned above) detected for smaller depths and referred to as HW2 . A clear distinction between these two time-dependent states was established in terms of speciﬁc considerations on the waveform, the ‘degrees of freedom’ displayed by the pattern and possible dynamic evolution of these features as a function of the applied radial temperature difference. Further important details about these aspects, which need to be explicitly discussed and understood, are given in the following two subsections.

11.7.1 Target-like Wave Patterns (HW2 ) Garnier and Chiffaudel (2001) considered an annular pool ﬁlled with a 0.65 cSt silicone oil (Pr = 10), having inner radius a = 4 mm, outer radius b = 67 mm and depth d = 1.2 mm. On the basis of their study, the main properties of the aforementioned waves of the HW2 type can be summarized as follows. These waves were found to propagate from the cold centre to the hot external boundary of the cell. In particular, they were observed to be emitted by a point source (the centre of the cell) with an almost radial wavevector at the onset (which makes them look like targets) with the pattern remaining localized around this source for sufﬁciently small T . For higher T , however, the pattern was observed to lose its azimuthal symmetry, as shown in the third and fourth frames of Figure 11.35. Even more interesting than such a patterning behaviour is the description of the properties of the related wavevector. Towards this end, let us recall that, for geometries with rotational symmetry, in general, such a vector is described by two cylindrical coordinates qr and qϕ = n/r, where r is the local radius and n is the number of wavelengths in the azimuthal direction. In practice, the most interesting property of the waves of the HW2 type is that there is no speciﬁc relation between qr and qϕ as they behave independently with T , which provides a justiﬁcation for making reference to these phenomena as a 2D hydrothermal instability (HW2 ) in contrast to the classical unidimensional hydrothermal wave (HW1 ) predicted by the linear stability theory of Smith and Davis (1983), discussed in Chapter 10. The classical waves of Smith and Davis (1983), in fact, are typically characterized by a direct proportionality between qr and qϕ that allows a relatively simple computation of the angle between the temperature gradient and the wavevector as = tan−1

qϕ qr

= tan−1

n rqr

(11.9)

For the HW1 this angle does not depend much on the applied T (it is a function of the ﬂuid only, i.e. Pr; the reader is also referred to the fundamental information in Section 10.2.2). For the HW2 , most interestingly, is not a ﬂuid-dependent-only quantity. Garnier and Chiffaudel (2001), in fact, observed the azimuthal wavenumber qϕ to exhibit a marked linear dependence on the difference T − Tcr (for the considered silicone oil with Pr = 10, the angle

Mixed Buoyancy–Marangoni Convection

477

(a)

(b)

(c)

(d) Figure 11.35 Shadowgraph images of an annular pool of high-Pr liquid (silicone oil, 0.65 cSt, Pr = 10, inner radius a = 4 mm, outer radius b = 67 mm, d = 1.2 mm) for increasing values of the applied radial temperature gradient: (a) T = 7.8 K (slightly above the onset purely radial time-dependent target pattern); (b) T = 8.5 K (azimuthal symmetry is slightly broken); (c) T = 9.5 K and (d) T = 14 K (the pattern looks like spirals, with left-turning and right-turning waves regions separated by sources and sinks). After Garnier and Chiffaudel (2001); Reproduced by permission of EDPS

at the given distance from the centre r = 11.6 mm was found to vary from 0◦ at onset to 45◦ at T = 20 K instead of being constant as in the case of HW1 . Remarkably, it is to this behaviour of qϕ that the qualitative changes of the pattern structure with increasing T (ﬁnally leading to the breaking of azimuthal symmetry shown in Figure 11.35c and 11.35d) should be ascribed: from a concentric (roll-like) pulsating pattern at the onset to a pattern with left-turning and right-turning waves regions separated by sources and sinks as the distance from the onset is increased (Figure 11.35). On increasing the temperature difference, the region occupied by the HW2 (initially localized near the centre) also increases (see Figure 11.35).

478

Thermal Convection: Patterns, Evolution and Stability

11.7.2 Waves with Spiral Pattern (HW1 ) When studying Marangoni instabilities in the same annular container but with a larger depth (d = 1.9 mm), Garnier and Chiffaudel (2001) showed that a different kind of hydrothermal wave appears, in good qualitative agreement with the theoretical features predicted by Smith and Davis (1983). In particular, following Garnier and Chiffaudel (2001), these waves may be characterized in a relatively simple way by observing that their equiphases almost ﬁt the law r ∝ ϕ, that is, Archimedean spirals (see Figure 11.36), although it should be pointed out (N. Imaishi, Institute for Materials Chemistry and Engineering, Kyushu University, Fukuoka, Japan, personal communication, 2008) that an even better ﬁt of their spiral arms might be achieved by using the so-called ‘logarithmic spiral’ for which the radius grows exponentially with the angle, that is, r ∝ exp(cϕ), c being a constant parameter. In practice, the logarithmic spiral can be distinguished from the Archimedean spiral by the fact that the distances between its turnings increase in geometric progression, whereas in an Archimedean spiral the separation between the windings is constant (because there is a simple linear relation between radius and the angle). It is such a logarithmic relation between radius and angle that leads to the known property of constant angle between the tangent and the radial line at any point (r, ϕ) on the spiral, which makes the logarithmic spirals particularly suitable to represent waves of the HW1 type. In fact, as also illustrated in Section 11.7.1, for such waves the angle between the temperature gradient and the wavevector is almost uniform in space (varying typically for the case considered by Garnier and Chiffaudel, 2001, from 60◦ to 40◦ along r) and independent of T . Hence n = rqϕ is proportional to qr and neither depends on the temperature gradient (the two components of the wavevector are proportional, their ratio is roughly constant anywhere in the domain and deﬁnes a constant angle between T and q).

Figure 11.36 Shadowgraph image of an annular pool of high-Pr liquid (silicone oil, 0.65 cSt, Pr = 10, inner radius a = 4 mm, outer radius b = 67 mm, d = 1.9 mm, T = 14.25 K): HW1 is observed as a multi-armed rotating spiral. After Garnier and Chiffaudel (2001); Reproduced by permission of EDPS

Mixed Buoyancy–Marangoni Convection

479

The related spiral pattern can then be described using only one wavenumber component qr or n, just as in the case of a unidimensional system. From a purely geometric point of view, this property can be expressed in differential terms as   1 −1  r  (11.10) = tan−1 = tan dr c dϕ where, by simple comparison of Eqs (11.9) and (11.10), it becomes evident that the constant c appearing in the equation of the logarithmic spiral should be regarded as the constant of proportionality between qr and qϕ . This justiﬁes making reference to these waves as ‘1D hydrothermal waves’ (HW1 ). In practice, for this mode of convection the constant angle between the imposed temperature gradient and the wavevector reduces the number of degrees of freedom for the description of the wavevector from the value 2 (two free components) to the value 1 (one free component). Another interesting way to explain the mechanism just stated is that HW2 are waves with wavenumber described by two independent spatial components (qr , qϕ ), whereas for HW1 one component is sufﬁcient (usually the component perpendicular to the temperature gradient, i.e. qϕ ). Another signiﬁcant difference between HW1 and HW2 lies in the fact that, as outlined earlier, the HW1 pattern is not localized near the cold endwall, but invades the whole cell as soon as the onset is crossed. The domains of existence of both of these patterns in the space of parameters are qualitatively shown in Figure 11.37, where it is evident that, as also explained in Section 11.7.1, waves of HW2 type are naturally taken over by waves with a spiralling pattern as the layer depth and/or the applied temperature gradient are increased. At this stage, the reader may be fascinated by realizing that the two possible phenomena (HW2 and HW1 ) exhibit a notable apparent similarity to the target and spiral patterns described in Sections 4.10 and 4.11 with regard to gravitational Rayleigh–B´enard convection. It is worth recalling, however, that, despite the fascinating commonalities in terms of patterning behaviour, the targets and spirals described in Chapter 4 have nothing to do with the present phenomena, for which patterns are not formed by self-organization of rolls but reﬂect dynamics related to the propagation of waves.

Figure 11.37 Sketch of phase diagram with regions of existence of HW1 and HW2 states

480

Thermal Convection: Patterns, Evolution and Stability

Perhaps a more relevant analogy could be established with respect to the experimental observations of Burguete et al. (2001) in rectangular layers. The ﬁnal stages of the HW2 mode (Figure 10.38c and 10.38d), in fact, look similar to the circular waves originating from a point-like source located on the cold wall observed by Burguete et al. (2001) for d < dc , while the HW1 mode could be successfully compared to inclined planar waves detected for dc < d < dr (see Section 11.4.1 and, in particular, Figure 11.15). Anyhow, it is opinion of the present author that experimentally reported waveforms for d < dr representing a deviation with respect to classical patterns with well-deﬁned inclined structures (inclined lines of constant phase) continuously travelling in a given direction might correspond in many circumstances to typical transitional stages of evolution of Marangoni ﬂow (similar to those already elucidated in Chapter 10) made somewhat long-lasting by gravity (the effect of which at this stage is not yet sufﬁciently clear; see, e.g., Section 11.7.6) or by the relatively small supercriticalities considered in the related experiments (the reader is also referred to the theoretical arguments elaborated for the parallelepipedic cavity in Section 10.2.4).

11.7.3 Stationary Radial Rolls Continuing with the fruitful analogy with the experiments of Burguete et al. (2001), it is also worth highlighting that as they observed the pattern to undergo transition to a multicellular state with steady rolls when the depth of the layer was further increased (d > dr ), in a similar way analogous behaviours have been reported for annular conﬁgurations. As anticipated at the beginning of Section 11.7, in fact, Benz and Schwabe (2001) for ethanol and silicone oil (hexamethyldisiloxane) reported experimentally on the existence of stationary longitudinal rolls, whose genesis and dynamics at this stage deserve some additional discussions and clariﬁcation. For relevant and illuminating numerical results, it is worth considering again the numerical studies of Imaishi and co-workers (already discussed in Section 10.3 for pure Marangoni ﬂows). Shi and Imaishi (2006), Li et al. (2006) and Peng et al. (2007), in fact, provided (numerical simulations) some possible explanation for the nature of such stationary radial rolls for Pr = 6.7. They conducted a series of unsteady three-dimensional numerical simulations of hybrid buoyancy–Marangoni ﬂow considering annular pools with different depths (1 ≤ d ≤ 11 mm) heated from the outer wall (radius b = 40 mm) and cooled at the inner cylinder (a = 20 mm) with an adiabatic solid bottom and adiabatic free surface. The simulation conditions were chosen to reproduce those in the experiments of Schwabe (2002a). The ensuing results with large Marangoni number predicted two types of three-dimensional ﬂow patterns: hydrothermal wave characterized by curved spokes in shallow thin pools (d < 5 mm); and a three-dimensional stationary ﬂow consisting of pairs of counter-rotating longitudinal rolls in deep pools (d ≥ 5 mm). In practice, these authors clearly proved that when the depth of the layer is increased (i.e. W = σT /ρgβT d 2 is decreased), transition occurs from HWT states (see, e.g., Figure 11.38a) to a state with steady radial rolls affecting the ﬁeld especially close to the external heated wall (see Figure 11.38b; similar structures were also observed by Benz and Schwabe, 2001). Interestingly, these numerical studies provided some evidence supporting the idea that (for the speciﬁc case of annular conﬁgurations with adiabatic horizontal boundaries), the onset of these rolls might be ascribed to the joint action of buoyancy and thermocapillary ﬂow; Li et al. (2006), in fact, illustrated that the resulting temperature ﬁeld tends to be featured by the presence of a zone close to the outer wall with a thermally unstable stratiﬁcation, which might be the source for convection of the Rayleigh–B´enard type there. A fairly exhaustive discussion of the possible convective states related to Rayleigh–B´enard convection in shallow cylindrical layers has been already given in Section 4.10; therefore, this subject is not discussed further in the present section.

481

10s

Mixed Buoyancy–Marangoni Convection

−0.01

0.01

2π

10s

(a)

2π (b)

Figure 11.38 Snapshot of surface temperature ﬂuctuation and spatiotemporal diagram of surface temperature along a ﬁxed circumference (r = 25 mm) of an annular layer of silicone oil (Pr = 6.7, a = 20 mm, b = 40 mm; adiabatic conditions at the top and bottom) for two different depths: (a) d = 1 mm, T = 8 K, W −1 = 0.125, many curved spoke patterns are shown, the ﬁelds correspond to the propagation along the counterclockwise azimuthal direction of an hydrothermal wave; (b) d = 6 mm, T = 10 K, W −1 = 4.5, straight spoke patterns are shown, the ﬁelds correspond to the presence of pairs of counter-rotating longitudinal steady rolls (of the Rayleigh–B´enard type) whose axes are parallel to the applied radial temperature gradient. Courtesy of N. Imaishi

Nevertheless, following the numerical studies by Imaishi and co-workers, some space is devoted here to elucidating the mechanism by which the aforementioned vertical (thermally unstable) temperature gradient is generated near the outer wall. In particular, the discussion progresses with the support of the arguments illustrated by Peng et al. (2007). Let us start by noting that the Marangoni effect generates an inward radial ﬂow (hereafter denoted the Ma-driven ﬂow) near the free surface. Accordingly, the temperature at the free surface is always higher than that at the bottom. Because the radial temperature drops are mainly concentrated in the vertical thermal boundary layers near the inner and outer walls, the ﬂow driven by the buoyancy force (the B-driven ﬂow) near the hotter wall carries low-temperature liquid to an area below the free surface, as shown in Figure 11.39a. At the same time, the return ﬂow carries high-temperature liquid from the free surface to the area below the B-driven ﬂow. This means the possible emergence of a region with an inverted temperature gradient (due to a turning back of the cold ﬁnger near the hot wall); its depth is denoted hi in Figure 11.39b.

482

Thermal Convection: Patterns, Evolution and Stability

Ma-driven flow B-driven flow 0.15

return flow

r = 37 mm

hi

0.10 dT

z 0.05 z 0.5

r

0.6

0.7

0.8

T (a)

(b)

cold layer hot layer z j (c) Figure 11.39 The mechanism leading to the emergence of stationary longitudinal rolls (Pr = 6.7, a = 20 mm, b = 40 mm, d = 6 mm, T = 10 K, adiabatic conditions at the top and bottom): (a) isotherms and pseudo-streamlines at ϕ = 0; (b) temperature as a function of z at ϕ = 0; (c) counter-rotating longitudinal rolls at r = 37 mm in the plane (ϕ , z). Courtesy of N. Imaishi

In such a layer with a reversed temperature gradient, Rayleigh–B´enard instability can be produced and extend to the free surface if the Rayleigh number exceeds its threshold value. For the conditions considered in the numerical simulations of Peng et al. (2007), the local Rayleigh number, deﬁned as Ra = gβT δT (hi )3 /να, was estimated to be about 728, which is larger than the critical value, Racr = 669 (see Section 4.1) predicted by the linear stability analysis for the occurrence of the Rayleigh–B´enard instability in an inﬁnitely extended ﬂuid layer with free upper adiabatic surface. Apart from these hints about the possible direct involvement of RB convection in the genesis of the stationary rolls, it is also worth mentioning that later ﬁndings (linear stability analysis, N. Imaishi, Institute for Materials Chemistry and Engineering, Kyushu University, Fukuoka, Japan, personal communication, 2009) for relatively deep conﬁgurations (a = 20 mm, b = 40 mm, 4 < d ≤ 20 mm), have also provided an alternative justiﬁcation for the emerging stationary pattern in terms of the instability of mechanical nature identiﬁed for rectangular cavities by Kuhlmann and Albensoeder (2008) for which (as discussed to a certain extent in Section 11.5.3) gravity has no direct role in the ampliﬁcation of disturbances (its effects being limited to the basic state). A separate discussion is required for intermediate values of the depth (2 ≤ d ≤ 4 [mm], for which Imaishi and co-workers observed a signiﬁcant bending of the HTW arms in proximity of the outer wall, as shown in Figure 11.40.

Mixed Buoyancy–Marangoni Convection

483

Figure 11.40 Snapshot of surface temperature ﬂuctuation for a layer of silicone oil with intermediate depth showing a travelling wave state (Pr = 6.7, a = 20 mm, b = 40 mm, d = 2 mm, T = 10 K, W −1 = 0.5; adiabatic conditions at the top and bottom surfaces). The outer rim displaying strong bending of the spiral arms indicates interaction of the HTW with the stationary toroidal roll located near the hot wall. Courtesy of N. Imaishi

For pure Marangoni ﬂow (see the ﬁnal part of Section 10.3), such a phenomenon is known to be the outcome of the interplay that occurs between the HTW and the steady roll maintained near the hot wall by the action of the strong (for sufﬁciently high values of the Marangoni number) temperature gradient established in the vertical boundary layer. Figure 11.40 proves that in the presence of gravity such a behaviour can become evident even for smaller supercriticalities in virtue of the increased strength of such roll driven by the joint action of surface tension-driven and buoyancy forces. Other interesting results on the subject were obtained by Wakitani (2007). An experimental study of thermocapillary convection of silicone oil (Pr = 18) was conducted in a shallow annular cavity with outer radius b = 55 mm and inner radius a = 15 or 27.5 mm for liquid heights d in the range 1 ≤ d ≤ 3 mm. The measurement of surface temperatures and observation of instability structures were made by using an IR thermography technique. As Ma exceeded a critical value, hydrothermal waves were observed for thin liquid layers, d ≤ 2.5 mm, of both the inner radii. For thicker layers, d > 2.5 mm, however, multicellular patterns were found to be dominant. At low Ma, the multicell patterns were steady, toroidal rolls. As Ma was increased, the rolls were observed to start rotating around the centre of the cavity. In the range 2.5 ≤ d ≤ 2.8 mm, hydrothermal waves and multicellular patterns were found to coexist.

11.7.4 Progression Towards Chaos and Fractal Behaviour The subsequent regimes (tertiary, quaternary, etc.) to which the system undergoes transition when the temperature difference is further increased were considered by several authors experimentally (e.g. Mukolobwiez et al., 1998), numerically (e.g. Shi and Imaishi, 2006) and theoretically (Garnier et al., 2003a, b). Shi and Imaishi (2006) found (by numerical simulation) that for rather shallow pools, a relatively small increase in T (i.e. increase in supercriticality) can stimulate additional wave trains with different wavenumbers and different travelling directions (with the ensuing appearance of interference between groups travelling in opposite directions, as shown in Figure 11.41). Among other things, comparison of such a ﬁgure with the equivalent one for pure Marangoni ﬂow (Figure 10.23b) also indicates that, in general, the role played by gravity in such dynamics should not be ignored (as witnessed by the different aspect of the pattern and related spatiotemporal diagram with respect to zero-g conditions).

484

Thermal Convection: Patterns, Evolution and Stability

t[s]

−2.1E-02 −7.0E-03 6.8E-03 2.1E-02 10

5

0

0

2π

Figure 11.41 Snapshot of surface temperature ﬂuctuation and spatiotemporal diagram of surface temperature along a circumference (r = 25 mm) of an annular shallow layer of silicone oil (Pr = 6.7, a = 20 mm, b = 40 mm, d = 1 mm, T = 12 K, W −1 = 0.125; adiabatic conditions at the top and bottom). Two groups of HTWs coexisting in the pool with different wavenumbers and different travelling directions are present; interferences between such trains occur throughout almost the entire volume of the pool [related values of the wavenumbers are m = 22 (counterclockwise direction) and m = 44 (clockwise direction)]. Courtesy of N. Imaishi

The possible coexistence of distinct groups of HTWs with different properties in normal gravity was also reported by Garnier and Chiffaudel (2001). For larger superciticalities (as also observed by Burguete et al., 2001, in the case of rectangular layers, Section 11.4.2), a travelling-wave pattern can undergo secondary bifurcations in the form of supercritical Eckhaus instabilities. In general, this can lead to a small-wavenumber phase-modulated nonlinear mode and show evidence of a nonlinearly saturated phase instability mode for travelling wave patterns. At higher forcing level, this secondary pattern is subject to a tertiary instability. This mode is an amplitude mode characterized by travelling hole patterns, that is, space–time defects that change the wavenumber (Figure 11.42). The progression towards fully developed chaos (fractal behaviour) has been studied, however, especially in systems where the initial pattern is relatively simple with respect to the complex multi-armed spiral conﬁgurations occurring in shallow circular pools (this allowed researchers to focus on the onset of time dependence and ensuing route to chaos while keeping a relatively simple basic spatial ﬂow structure). This has been obtained essentially using an open container with a depth comparable to the radial extension. In such circumstances, in fact, the waves travel mainly azimuthally, which makes the experimental investigation simpler (the radial component of the HTW, which makes it inclined with respect to the wall, is suppressed due to the geometric constraint; let us also recall that, in general, HTWs are as well degenerated if the gap between the hot and cold walls is smaller than their wavelength component in this direction). As an example, Brunet et al. (2005) considered an annulus with inner radius a = 0.63 mm, outer radius b = 3 mm, depth = 3 mm ﬁlled with silicone oil (Pr = 14), which may be expressly used as a paradigm for illustrating such behaviours. A summary of the most important ﬁndings of these researchers, in particular, can be provided as follows (Ma based on the outer radius). At Ma ≥ 1.15 Macr , the initial oscillatory ﬂow-regime (wavenumber m = 2) was found progressively to become purely rotating (travelling wave), still of mode 2 (Figure 11.43 shows the spatial structure of the dominant mode, and also successive snapshots of the rotating pattern obtained by image subtraction).

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Figure 11.42 Spatiotemporal diagram of local wavenumber showing a uniform travelling wave pattern densely invaded by travelling holes for T /Tcr ∼ = 4 (spatiotemporal defect chaos): These holes present a complex dynamics and interact together (they travel along the carrier wave in the same direction, but once they reach their minimal amplitude, very close to zero, they may reverse their direction of propagation). Courtesy of N. Garnier and A. Chiffaudel

Figure 11.43 Travelling wave (top left) with wavenumber m = 2 and three successive snapshots (time step = 1/5 s) of the related rotating ﬂow (Pr = 14, annular geometry with a = 0.63 mm, b = 3 mm, depth = 3 mm). The shadow circle represents light reﬂection on the non-perfect horizontal free-surface. After Brunet et al. (2005); Reproduced by permission of the American Institute of Physics

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At Ma ≥ 1.95 Macr , the periodic mode 2 state was observed to undergo a further bifurcation with the appearance of a subharmonic mode (this leading to a time-period doubling and constituting the ﬁrst step toward complexity). For Ma ≥ 2.28 Macr , the ﬂow ﬁnally entered a chaotic regime with the temperature ﬂuctuations showing suddenly very large bursts and also moments of lower amplitude. Along these lines, for additional clarity Figure 11.44 shows evolution of the ﬂow with increasing Ma, with plots of an extracted signal versus time, frequency power spectra and attractors built in a pseudo-phase space, in order to build a Poincar´e map. On the basis of this ﬁgure, the following speciﬁc considerations can be made:

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Figure 11.44 Extracts of time signals and reconstructed pseudo-phase space attractors (time step 1/20 s) for different increasing values of the Marangoni number (Pr = 14, same geometry as in Figure 11.42; Ma based on the outer radius): (a) oscillatory regime (Ma = 1.53 Macr ); (b) period doubling regime (Ma = 2Macr ); (c) further step to complexity (Ma = 2.10 Macr ) and (d) chaos (Ma = 2.8 Macr ). After Brunet et al. (2005); Reproduced by permission of the American Institute of Physics

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• At relatively small Ma (Figure 11.44a), the local temperature has a simple oscillating behaviour. The frequency power spectrum shows a dominant frequency around 1.2 Hz and its ﬁrst harmonics; the attractor is a simple loop corresponding to periodic oscillations. • At higher Ma (Figure 11.44b), a period-doubling bifurcation occurs as new peaks rise in the spectrum and the limit cycle is made of two loops. In the next set of plots (Figure 11.44c), this tendency is even clearer with the appearance of other peaks. The attractor becomes smoother and smoother, although keeping a pseudo-periodic structure. • At the highest Ma considered (Figure 11.44d), the ﬂow is chaotic: the power spectrum does not show any sharp peak, but a smooth bump around 1.3 Hz. At this stage, it is also worthwhile to mention that on comparison with typical experiments for the liquid-bridge problem, similar behaviours were reported: as in the observations by Brunet et al. (2005), the transition towards chaos in the classical liquid bridge heated from above seems to exhibit the generic scenario of successive period doubling. The reader is referred, in particular, to Section 11.8.3 for additional information about the typical routes to chaos of convection in liquid bridges under conditions of normal gravity (microscale experiments). It is also worth noting here that Ueno et al. (2003a) found a more complex set of successive regimes. In other liquid-bridge experiments (Frank and Schwabe, 1997), a transition by quasi-periodicity, with the appearance of incommensurable frequencies and the growth of a torus around the initial oscillatory attractor in the Poincar´e map, was observed. In view of these arguments, it is difﬁcult to conclude that, in general, a similar transition scenario is applicable to both annular and liquid-bridge geometries. Finally, it should be mentioned that some experimental studies dealing with the ‘reverse’ annular conﬁguration (with hot inner wall and cold outer boundary) have also appeared in the literature (e.g. Ezersky et al., 1993b; Kamotani et al., 1992, 1996, 1998; Shiomi et al., 2001). For such a case, the scenario and evolution towards chaos are also different (see Section 11.7.5).

11.7.5 The Reverse Annular Conﬁguration: Incoherent Spatial Dynamics For relatively shallow ‘reverse’ conﬁgurations, new unexpected patterns were reported by Garnier and co-workers (they considered experimentally the same geometric conﬁguration as already discussed in Sections 11.7.1 and 11.7.2, but reversing the direction of the applied radial temperature gradient, i.e. heating the centre with respect to the outside perimeter). They showed that, in general, when the direction of the radial gradient is changed, the dynamics are less coherent and more localized near the hot centre with respect to the case of centre cooled. For relatively small W (relatively large ﬂuid depth) and applied T , the basic ﬂow is stable. The ﬁrst instability is stationary and characterized by the emergence of corotative (circumferential) rolls. Such rolls appear on the hot side of the container, that is, around the inner cylinder for T > Tcr1 . As a consequence of a secondary instability (T > Tcr2 ), a time-oscillatory instability develops around the hot centre. In particular, at the onset, a rotating hexagon (Figure 11.45a) emerges. Most interestingly, when the temperature gradient is further increased, each corner of the hexagon moves away from the centre and the pattern takes the beautiful shape of a ﬂower (Figure 11.45b). On increasing T , an elongation of the petals occurs, then the emergence of additional petals (due to a modulational instability like the Eckhaus instability in the azimuthal direction). Furthermore, outside the ﬂower, a structure is present with the same azimuthal wavenumber evolving with T to form visible branches that rotate at the same angular frequency as the ﬂower. For T > Tcr3 , ﬁnally, waves of the HW1 type appear. Like the case discussed earlier for approximate models of the CZ technique (with heated outside perimeter), their radial propagation is from the cold side towards the hot side; accordingly, when the applied temperature gradient is

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Figure 11.45 Shadowgraph images of an annular pool of high-Pr liquid (silicone oil, 0.65 cSt, Pr = 10, inner radius a = 4 mm, outer radius b = 67 mm, d = 1.9 mm, hot centre) for increasing values of the applied radial temperature gradient: (a) T = 5.2 K, stationary rolls, with an additional hexagon turning around the centre; (b) T = 5.6 K, a six-petal ﬂower is turning right; (c) T = 7 K, an additional wavelength has appeared, and also branches in between the petals outside the ﬂower; (d) T = 20 K, hydrothermal waves of type 1 have appeared on top of the ﬂower pattern (because the radial component of the HW1 wavevector is pointing toward the centre of the cell, the spatial coherence of the resulting structure is small and the overall pattern is spatiotemporally chaotic). Courtesy of N. Garnier

reversed, the HW1 pattern lets energy ﬂow from the external perimeter to the centre. This situation is not comfortable and the structure exhibits a strong tendency to become incoherent, as explained in the next subsection.

11.7.6 Some Considerations About the Role of Curvature, Heating Direction and Gravity In general, the effect of system curvature (the circular geometry) on any generic disturbance is to constrain the wavevectors. Close to the centre, the azimuthal direction is not as extended as it is further from the centre. This implies that to keep a constant value of the wavenumber, the pattern

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has to increase the wavelength in the azimuthal direction and reduce the wavelength in the radial direction. When the centre is colder than the outside (CZ-like), the pattern propagates from the centre and, therefore, spreads in the azimuthal direction. This can be achieved while keeping spatial coherence. In the opposite case, when the outside perimeter is colder than the centre, the propagation is from the outside towards the inside and the information (or energy) of the structure has to converge from an extended region to a conﬁned region; in such a case, any inhomogeneity of the structure (wavenumber, frequency or amplitude) in the azimuthal direction will result in destruction of the coherence of the converging process (Figure 11.45d). By contrast, for rectangular layers, positive and negative temperature gradients are equivalent with respect to such geometric effects and one observes the same behaviours: HW2 for small depth and HW1 for larger depth, in agreement with the ﬁndings of Burguete et al. (2001) described in Section 11.4. From a theoretical point of view, the different ﬂuid-dynamic response of annular geometries with respect to the direction of the imposed T was expressly treated by Garnier and Normand (2001) via linear stability analysis. The horizontal temperature gradient was supposed to be applied as a temperature difference between the inner and the outer boundaries, considered as isothermal with temperatures Tinner and Touter , respectively; the bottom of the container was considered as perfectly conducting and a mixed thermal boundary condition such as that of Mercier and Normand (1996) was used for the top free surface to mimic effective conditions attained during the experiments. Obviously, both Marangoni and buoyancy driving forces were considered. As a result, the instability was predicted to appear primarily near the inner cylinder with the values of the corresponding wavenumbers being different according to the role (hot or cold) played by the inner cylinder (which is in agreement with both the experiments described in the preceding subsection with spiralling waves near the hot inner side and with those discussed in Section 11.7.1 where pulsating targets appear when the inner cylinder is the cold side). In particular, according to such linear stability results, for Touter > Tinner , the azimuthal wavenumber should vanish for a speciﬁc value of the curvature, which means that in cells with sufﬁcient curvature, hydrothermal waves should not only be localized near the centre, but should also propagate primarily in the radial direction at onset (in good qualitative agreement with the dynamics shown in Figure 11.35). For the reverse conﬁguration (Tinner > Touter ), the instability pattern should always display a non-vanishing azimuthal wavenumber, leading to a recognizable pattern localized near the centre of the cell (in good theoretical agreement with the experimental results in Figure 11.45). Apart from these interesting linear stability analyses providing a solid theoretical basis for many of the patterns and waveforms reported experimentally, it is also worth citing the recent theoretical investigations of Hoyas et al. (2002a, b, 2004, 2005). They considered a slightly different model in which the horizontal (radial) temperature gradient was imposed as a thermal boundary condition at the bottom wall, while the lateral circular walls and the free upper surface were assumed adiabatic and with a ﬁxed heat exchange (assigned Bi), respectively. In their studies (a = 1 cm, b = 3 cm, d = 1.7 mm), they explored the main mechanisms involved in the instabilities separately, by setting either the Marangoni or the Rayleigh number equal to zero, and their inﬂuence on the development of waves. Different kinds of spatially extended and localized structures were predicted (stationary or oscillatory) for Pr = 46. In particular for pure gravitational ﬂow (Ma = 0), stationary rolls were clearly identiﬁed close to the inner boundary (Figure 11.46), in qualitative agreement with the ﬂower structure reported by Garnier and co-workers (Figures 11.45). Also, competing solutions were found: stationary radial rolls with different wavenumbers, radial rolls with hydrothermal waves and hydrothermal waves with different wavenumbers.

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Figure 11.46 Growing perturbation at a stationary instability threshold (Pr = 46, Ma = 0, 1g conditions, Bi = 1.2 and T = 4 K, Tmax − Tamb = 16.22 K): (a) isotherms on the horizontal plane at z = 1; (b) same isotherms in the vertical plane; (c) corresponding velocity ﬁeld in the vertical plane. Courtesy of S. Hoyas

In particular, they discussed the role of horizontal gradients to determine the type of bifurcation both in experiments and in numerics approaching experimental conditions along with the role of vertical temperature gradients in comparison with previous theoretical work. Hoyas et al. (2004) revealed, in particular, that waves in driven ﬂows can be obtained for both thermocapillary (Ra = 0) and thermogravitational (Ma = 0) mechanisms, provided that heat-related parameters Bi and Tmax − Ta (Ta = ambient temperature) are conveniently tuned. Most notably, this work opened up a new perspective on the subject, showing that for high-Pr liquids, waves with features similar to those of the classical hydrothermal waves of pure Marangoni ﬂow can emerge in pure buoyancy ﬂow of the Hadley type (Figure 11.47). According to such an analysis, thermogravitational waves (Ma = 0) should appear for small Bi values (approximately Bi ≤ 0.3), whereas thermocapillary waves (Ra = 0) should appear for larger Bi values (approximately Bi ≤ 1.2). For the case Ma = 0, codimension-two stationary–oscillatory bifurcation points were obtained at smaller Bi values. Remarkably, the structure of the bifurcating solutions was found to be similar in both cases: for the speciﬁc boundary conditions considered by these authors, spiral structures were predicted to be possible both for only buoyant and capillary effects. Such structures differ, however, with respect to the region of the ﬂuid where they are localized (near the surface for the thermocapillary case and in the bulk for the thermogravitational mechanism). As a concluding remark, it is worth noting that the superposition of these thermogravitational waves with those of thermocapillary origin might be used as an additional argument for explaining the departure of the oscillatory dynamics observed in experiments from those expected theoretically for pure Marangoni ﬂow.

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Figure 11.47 Growing perturbation at an oscillatory instability threshold (Pr = 46, Ma = 0, 1g conditions, Bi = 0.2 and T = 4 K, Tmax − Tamb = 21.96 K): (a) isotherms on the horizontal plane at z = 1; (b) same isotherms in the vertical plane; (c) corresponding velocity ﬁeld in the vertical plane. Courtesy of S. Hoyas

11.8 The Liquid Bridge on the Ground A rich theoretical background for the salient aspects of pure Marangoni ﬂow in liquid bridges of high-Pr ﬂuids and the related dynamics (in terms of emerging spatiotemporal patterns and nature of the most dangerous disturbances) has been provided in Chapter 10 (Sections 10.4.7–10.4.12). The present section addresses the role played in such (already complex per se) a scenario by gravity (in terms of all the potential consequences that the presence of such a body force can have on the aforementioned dynamics). Towards this end, let us begin by observing that, unlike the CZ-like annular conﬁguration considered in the preceding section (Section 11.7), for which gravity is basically perpendicular to the imposed horizontal (radial) temperature gradient (which means thermogravitational convection primarily of the Hadley type), for the liquid-bridge problem these vectors are essentially parallel. This has remarkable consequences for the kind of ‘contamination’ exerted by the ensuing buoyancy forces on the base Marangoni ﬂow. For the liquid bridge, in fact, temperature gradients developed inside the liquid primarily along the vertical direction will lead to mechanisms which fall into the general theme of Rayleigh–B´enard convection. This consideration, in turn, leads to splitting of the subject into two subﬁelds, one in which ∇T is opposite to gravity and the other with g and ∇T concurrent. In the absence of Marangoni effects (the reader is referred to the abundant information provided in Chapter 4), in the ﬁrst case, gravity would have no effect (it does not induce convective ﬂow) for any value of T , whereas for the latter case ﬂuid motion arises only if the critical conditions for the onset of convection are exceeded.

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The latter case was investigated (linear stability analysis) by Wanschura et al. (1996) in the limiting case of a straight surface (undeformed liquid/gas interface and S = 1; see Section 2.3.3 for the deﬁnition of this nondimensional parameter), who showed the onset of pure buoyancy convection in such a geometric conﬁguration to be always three-dimensional (as shown in Figure 11.48, the neutral curves for m = 0 lie above those related to the most dangerous disturbances), with a complex dependence of Racr on the aspect ratio. In contrast, as extensively illustrated in Section 10.4, pure Marangoni ﬂow emerges without the need to exceed any threshold. It is initially axisymmetric and undergoes transition to an oscillatory three-dimensional state only if the Marangoni number is increased beyond a certain critical threshold. For Marangoni convection weakly affected by buoyancy forces, the behaviour is qualitatively similar. One has to keep in mind, however, that, due to the presence of gravity, the heating direction (from above or from below) and the possible static deformation of the free surface enter the dynamics as additional inﬂuencing factors (see Section 11.8.1; similar concepts also apply to the case with liquid metals – see Section 11.2).

11.8.1 Microscale Experiments Since the end of the 1970s (Chun and Wuest, 1979; Chun, 1980a,b), on the ground experiments with small-sized zones and high Prandtl number liquids (Pr 1) revealed that the Marangoni convection in liquid bridges heated from above or from below shows a direct transition from a steady axisymmetric toroidal ﬂow to a three-dimensional oscillatory ﬂow when a critical temperature difference between the liquid-bridge supports is exceeded. Schwabe and Scharmann (1983), Preisser et al. (1983), Kamotani et al. (1984), Velten et al. (1991), Schwabe et al. (1996), Frank and Schwabe (1998), Monti et al. (2000b), Ueno et al. (2003a,b, 2008) and Nishimura et al. (2005) systematically performed on-the-ground research over a wide range of experimental conditions. As in the case of pure zero-g conditions (space), they found the oscillatory Marangoni ﬂow to exhibit a different behaviour depending on the geometric aspect ratio of the zone, AH = height/diameter (Figures 11.49 and 11.50).

Figure 11.48 Curves of neutral stability for the onset of Rayleigh–B´enard convection in straight liquid bridges (Racr based on the cylinder height; 0.4 ≤ radius/height ≤ 3 corresponding to 0.16 ≤ AH ≤ 1.25; AH = height/diameter, liquid bridge with adiabatic free surface). Courtesy of H. Kuhlmann

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A number of diagnostic techniques were used for the experimental investigation, obtaining different glimpses of various aspects of the problem. Velten et al. (1991), measuring the temperature near the free surface by three thermocouples positioned at the same radial coordinate but at different azimuthal positions, detected two different non-axisymmetric spatial structures of Marangoni convection: (1) running waves with an azimuthal component (corresponding to the travelling wave regime already described in Section 10.4.8 for pure Marangoni ﬂow); and (2) axially running waves with a deformed wavefront (such a waveform corresponding, in practice, to the standing wave, as demonstrated by Kuhlmann and Rath, 1993b, in a subsequent analysis; see Section 10.4.8). Concerning the standing wave, most interestingly, Kamotani et al. (1984) used a piece of plastic to separate the liquid column into two semicircular regions and observed the oscillation to continue (in practice, they imprinted a pulsating standing mode); thereby, they demonstrated that this waveform does not involve transport of mass in the azimuthal direction. Schwabe et al. (1996) used a stereo microscope (rather than the usual light-cut technique-based visualization following tracers illuminated by a laser sheet in a generic meridian plane) and observed in the bulk ﬂuid dark inclined stripes moving in a horizontal direction from one side of the ﬂoating zone to the other. These stripes (moving horizontally) were interpreted as a direct manifestation of azimuthally travelling waves since they were generated by densely packed tracer particles bunched in a line wound around the toroidal convection roll moving azimuthally. Muehlner et al. (1997) investigated the time-dependent features of the instability using a thermographic system to visualize directly the temperature ﬁeld. They employed an infrared thermocamera with wavelengths centred at 4.61 µm. Since in the wavelength band centred at 4.61 µm the substance that they considered (tetradecamethylhexasiloxane with Pr = 35) is partially transparent,

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Figure 11.50 Examples of the modal structure (polygon-shaped stagnant region around the central axis of the liquid bridge) for various aspect ratios and S ∼ = 1. After Ueno et al. (2003); Reproduced by permission of the American Institute of Physics

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they observed an average temperature distribution over a depth of about 70% of the radius of the liquid bridge. Nevertheless, important insights into the structure of the hydrothermal wave were provided. A thermographic diagnostic system able to provide direct visualization of the surface temperature spots induced by the Marangoni ﬂow instability on the surface of the liquid bridge was used by Lappa et al. (2000); a thermocamera with wavelengths 8–12 µm, in fact, was employed that does not make the silicone oil transparent, thus providing an even clearer picture of the surface temperature ﬁeld. Frank and Schwabe (1998), by means of a light-cut technique, clearly observed standing and travelling waves. For this purpose, however, an unconventional setup with a transparent (quartz glass) upper disk heated by an electrical resistance was used for direct visualization of the azimuthal motion of tracers in cross-sections perpendicular to the axis of the bridge. Similar approaches were used by Kawamura and co-workers (see, e.g., the studies of Kawamura and Harada, 1998, Ueno et al., 2000, 2003a,b, 2008, and Nishimura et al., 2005, in which, among other things, a signiﬁcant effort was made to study the so-called PAS, i.e. the phenomena of dynamic particle accumulation by which tracer particles are captured in the vortex centre, as already discussed to a certain extent in Section 10.4.11 for the speciﬁc case of microgravity experiments) and by Hirata and co-workers (Arafune et al., 2000; Kinoshita et al., 2001). In Figure 11.50, the modal structure of the ﬂow is made evident by the polygon-shaped stagnant region (region free of tracers) established around the central axis of the liquid bridge (to be contrasted with the external shell rich in tracers; it is shown that tiny particles disperse in the region between the free surface and the outer boundary of the polygon and hardly break into the inside of the polygonal region once the modal structure is established). Examples of pulsating and travelling regimes are shown in Figures 11.51 and 11.52, respectively. All these experimental studies were concentrated primarily on liquid bridges with quasi-straight cylindrical surface (i.e. S ∼ = 1). The effect of volume (S = 1) was studied by several other groups. Terrestrial experiments (Hu et al., 1994, 1995; Hirata et al., 1997a–c) on transparent high-Prandtl liquids (various silicone oils) essentially revealed a non-monotonic behaviour of the stability limit versus the liquid-bridge relative volume; according to these studies, in fact, the stability diagram consists typically of two branches with an ‘overstability’ gap (i.e. very high values of the critical Marangoni number; see Figure 11.53) for intermediate values of the volume (S ∼ = 0.9), whereas the ﬂow is destabilized for large or small volumes (let us recall that, as discussed in Section 11.2, for Pr 1 the behaviour is just the opposite). In several experiments the overstability gap was found to correspond to 0.8 < S < 1.0. These trends were conﬁrmed by linear stability analyses (e.g. Ermakov and Ermakova, 2004, for a 5 cSt silicone oil, 0.5 ≤ S ≤ 1.4 and AH = 0.75) and numerical (solution of the nonlinear thermal-convection equations) studies (e.g. Tang and Hu, 1999, for a 10 cSt silicone oil, Pr = 105.6, 0.6 ≤ S ≤ 1.1 and AH = 0.8 and Sim and Zebib, 2002, for Pr = 27 and AH = 0.77). A study based on the energy theory was reported by Sumner et al. (2001). Energy theory was applied to the results provided by 2D simulations (5 cSt silicone oil, AH = 0.75, 0.8 ≤ S ≤ 1.4) to determine sufﬁcient conditions for stability (recall that, as illustrated in Chapter 1, the approach based on the linear stability analysis provides sufﬁcient conditions for instability). Abe et al. (2007) focused on the effects of the shape of the liquid bridge on the dynamics of particle accumulation (the aforementioned PAS phenomena). It is also worth mentioning that other authors (e.g. Hu and Tang, 2003, and Aa et al., 2005, by numerical and experimental investigation, respectively) found that the differences between slender (concave) and fat (convex) bridges might not be limited to the threshold for the onset of oscillatory ﬂow and the structures formed by accumulation of tracers. Their studies provided some evidence for the fact that whereas in the former case Marangoni ﬂow undergoes, as usual, transition from an initial axisymmetric state to an oscillatory state, in the latter such a transition may be preceded

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Figure 11.51 Evenly spaced snapshots of convection showing a standing wave state (2 cSt silicone oil, AH = 0.9, D = 4 mm, S ∼ = 1, T = 20 K): (a) meridian plane; (b) top view (liquid motion is visualized using tracers scattering the light generated by an He–Ne laser diode with a wavelength of 635 nm, forming a light sheet; the laser beam is oriented orthogonal to the main optical path of a CCD camera). After Monti et al. (2000b)

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Figure 11.52 Evenly spaced snapshots of convection showing a travelling wave state (2 cSt silicone oil, AH = 0.9, D = 4 mm, S ∼ = 1, T = 20 K): (a) meridian plane; (b) top view (liquid motion is visualized using tracers scattering the light generated by an He–Ne laser diode with a wavelength of 635 nm, forming a light sheet; the laser beam is oriented orthogonal to the main optical path of a CCD camera). After Monti et al. (2000b)

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Figure 11.53 Critical temperature difference for the onset of oscillatory ﬂow (a) and related frequency (b) as a function of the ratio between the minimum or maximum diameter of the liquid bridge Dm and the diameter of the supporting disks D (10 cSt silicone oil, AH = 1.0; normal gravity, liquid bridge heated from above). Courtesy of W.R. Hu

by the occurrence of a 3D asymmetric steady regime (as in the case of low-Pr melts discussed in Section 10.4; the reader is also referred to Section 10.4.12 for the case of pure Marangoni ﬂow). The aforementioned discontinuity gap disappears under microgravity conditions where the curve is a smooth, continuous function of S (as shown by the experiments of Hirata et al., 1997c, and the linear stability analysis of Chen and Hu, 1998a), which indirectly proves that gravity plays a crucial role in such dynamics. Some studies also focused on the sensitivity the stability threshold exhibits to the interaction between the liquid bridge lateral surface (generally considered adiabatic and not inﬂuenced by the external gas in the majority of existing studies) and the properties of the external environment (heat loss, air motion, etc.; see, e.g., Bennacer et al., 2002; Sim and Zebib, 2002; Kamotani et al., 2003,

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2007; Shevtsova et al., 2005, 2008; Selver, 2005; Irikura et al., 2005; Kousaka and Kawamura, 2006; Mialdun and Shevtsova, 2006; Melnikov and Shevtsova, 2006; Tiwari and Nishino, 2007). In the experimental analysis by Mialdun and Shevtsova (2006), the sensitivity of the ﬂow to the interfacial heat exchange was observed to be strongly dependent on the liquid-bridge volume. Slender liquid bridges (underﬁlled zone with respect to the straight cylinder) were found to be fairly stable to external disturbances. In contrast, fat liquid bridges are extremely sensitive to the thermal environment in the gas phase. Most of initial theoretical studies (linear stability analyses and numerical simulations; see, e.g., Sim and Zebib, 2002) for S = 1 predicted stabilization of the ﬂow (critical T changing by a factor of two or three by varying the air temperature relative to the cold wall temperature) for increasing values of the Biot number (increasing heat loss) in fairly good agreement with experimental investigations for moderate values of the Prandtl number (Pr < 10). Anyhow, for the speciﬁc case of a liquid bridge with Pr = 14 (AH = 0.9) and assuming the ambient air temperature to be equal to the cold rod, Melnikov and Shevtsova (2006) found numerically that the heat loss leads to destabilization of the ﬂow at small Biot numbers (Bi ≤ 2) and to stabilization at large Bi numbers (Bi ≥ 5). Similar results were obtained in the framework of a linear stability analysis by Li et al. (2008b) for 1 cSt silicone oil (Pr = 16 and AH = 0.9), where the change from destabilizing to stabilizing effect was explained by the competing inﬂuence of two effects, one inducing a larger temperature gradient on the middle part of the free surface (which tends to destabilize the thermocapillary ﬂow) and another weakening the temperature disturbances on the free surface. Notably, Wang et al. (2007) also considered (both experimentally and numerically) the opposite situation in which the free surface is heated by the external gas (heat gain). It was shown that the critical temperature difference undergoes a substantial decrease when the free surface heat transfer changes from loss to gain in the case of nearly straight liquid bridges, whereas it is not affected by the free surface heat transfer with concave liquid bridges. Most interestingly, the oscillatory ﬂow observed in such situations for conditions in which the ﬂow without surface heat exchange would exhibit no oscillatory instability was ascribed to the active role played in such dynamics by thermal buoyancy (enhanced by heat gain through the free surface). Although interesting, heat exchange through the liquid/gas interface of the liquid bridge is not discussed further here owing to page limits (the reader is referred to the studies cited above for further and detailed information).

11.8.2 Heating from Above or from Below Hereafter, for the sake of brevity and simplicity, the discussion is limited to some paradigm examples for S = 1 and an undeformed cylindrical interface (such approximations generally hold when the height of the liquid column is limited to a few millimetres and AH ≤ 0.4). Moreover, the free surface is assumed to be adiabatic. The case of counteracting or concurrent buoyancy and surface tension forces is elucidated by simply considering a bridge heated from above or from below, respectively. For the sake of clarity, ﬁrst, the case of stable and steady Marangoni ﬂow is examined, in order to elucidate the effect of gravity on the axisymmetric basic convection, then the transition to an oscillatory three-dimensional state is discussed. There are many numerical studies in the literature addressing the case of mixed convection in steady and axisymmetric conditions (e.g. Zeng et al., 2001a; Okano et al., 2001, Lappa et al., 2000). In the following, we refer to the numerical computations by Lappa et al. (2000). In the absence of thermal radial gradients and if thermocapillary forces were negligible, heating from above should yield a stable linear temperature distribution in the liquid bridge. It is obvious, however, that if thermocapillary effects were present in such circumstances, buoyancy effects

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could modify the distribution of the temperature and velocity inside the bridge with respect to pure Marangoni ﬂow. The numerical results for a typical silicone oil with Pr = 30 in Figure 11.54 show, in fact, that at the centre of the liquid bridge, the velocity in zero-g is larger than at 1g. The reason for this behaviour is that surface tension and buoyancy forces counteract in a liquid bridge heated from above. When gravity changes from zero-g to 1g, the centre of the vortex appears to move to the interface of the liquid bridge (the radial depth of the vortex cell is decreased; compare Figure 11.54a and 11.54c). The reason can be ascribed to the ﬂuid near the cool disk being slowed by the buoyancy forces (this effect reduces the radial extension of the convection roll). The inﬂuence of the buoyancy effect can also be seen in the temperature ﬁeld (Figures 11.55). In zero-g conditions a ‘cold ﬁnger’ is present near the hot disk, which is created by the return ﬂow that brings cool ﬂuid away from the cold wall along the symmetry axis. The cold liquid is carried toward the hot surface at a fairly high position. In normal gravity conditions, since the cold liquid, travelling in the return ﬂow, is slowed by the buoyancy forces, the temperature of this zone is weakened and consequently the radial temperature gradient near the hot disk is decreased. When the bridge is heated from below, Marangoni and buoyancy ﬂow act in the same direction along the liquid/air interface. The velocity on the free surface increases with respect to heating from above (see the streamfunction in Figure 11.54b) and for very short liquid bridges other counter-rotating vortex cells induced by buoyancy appear in the interior of the bridge. In practice, these additional vortex cells are Rayleigh–B´enard convection rolls. The critical Rayleigh number for the occurrence of these phenomena is about 1700, as pointed out in Chapter 4, and the Rayleigh numbers corresponding to the conditions considered in the present example are larger than this value. Due to the increase in the surface velocity (and of the return ﬂow) and owing to the presence of other buoyancy vortex cells in the inner part of the liquid bridge, the temperature ﬁeld is strongly deformed (see Figure 11.55b) with respect to the case of heating from above (this behaviour is particularly evident for AH ≤ 0.3). If the analysis for axisymmetric convection is extended to the case of the three-dimensional and oscillatory ﬂow that emerges when the Marangoni number is further increased, the inﬂuence of the combined effects of gravity and heating direction on the structure of the ﬂow becomes even more important. The surface temperature pattern (characterized by the existence of m cold spots and m hot spots on the free surface with oscillatory behaviour, as already illustrated for pure Marangoni ﬂow in Section 10.4.8) changes according to whether the bridge is heated from above or from below. This behaviour is evident by comparing the number of surface spots in Figure 11.56 with those in Figure 11.57, and those in Figure 11.58 with those in Figure 11.59. These intriguing phenomena correspond to the propagation of hydrothermal waves as already elucidated for the case of liquid layers and annular open cavities (Sections 11.4 and 11.7). Within the context of the present section, however, it is worth recalling that the main reason for the oscillatory instability for the liquid bridge of high-Pr ﬂuids is the existence of the aforementioned ‘cold ﬁnger’ near the hot disk, which is created by the return ﬂow that brings cool ﬂuid away from the cold wall along the symmetry axis. The cold liquid is carried toward the hot surface at a fairly high position. This cold zone (close to the free surface) plays a crucial role in the mechanism leading to the ampliﬁcation of surface-temperature disturbances and to the onset of surface hydrothermal waves. In previous work (Wanscura et al., 1995), it was pointed out that the instability of Marangoni ﬂow in high-Pr liquids is related to the convective radial heat transport coupled with the Marangoni effect. The possible ampliﬁcations of surface-temperature disturbances is based on the temporal interaction between the temperature distribution within the

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Figure 11.54 Velocity ﬁeld in liquid bridges of different aspect ratio for different environments (zero-g and 1g) and heating conditions (bridge heated from above or from below on the ground). Bridges with height L = 0.4 cm and L = 0.2 cm are shown on the left and right, respectively. After Lappa et al. (2000)

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Figure 11.55 Temperature ﬁeld in normal gravity conditions (bridge heated from above or from below). After Lappa et al. (2000)

ﬂow ﬁeld (in particular the cold ﬁnger established inside the liquid bridge near the hot disk) and the mechanism with which the surface generates ﬂow disturbances in response to temperature perturbations (this is why the surface temperature spots always appear near the hot disk as shown, for example, in Figures 11.56 and 11.57; therein the hot disk is located on the top of any frame). In line with these arguments, the different response that the system displays in the oscillatory state according to the gravity direction is a simple consequence of the fact that gravity alters the aforementioned mechanisms by modifying the radial temperature gradients inside the bridge and the magnitude of the surface axial velocity as shown in Figures 11.54 and 11.55 (Lappa et al., 2000). It is known that for moderate values of the Prandtl number, the ﬂow is generally stabilized by the effect of buoyancy forces with respect to zero-g conditions if the liquid is heated from above and further stabilization occurs if the imposed temperature gradient is reversed. The reader may also consider the parametric experimental studies reported by Velten et al. (1991) for additional details about the azimuthal structure of the ﬂow and the stability boundaries as a function of the heating direction and of the Prandtl number. They studied the inﬂuence of gravity on the transition point, frequency and spatial structure of the ﬂow ﬁeld, obtaining a large

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Figure 11.56 Evenly spaced snapshots within one oscillation period of temperature disturbance on the free surface (travelling wave with m = 2) of a liquid bridge with aspect ratio AH = 0.4 heated from above (Pr = 30, D = 4 mm, S = 1, Ma = 3.6 × 104 ; Ma based on the height): experimental thermographic visualization on the left, 3D numerical simulations on the right. After Lappa et al. (2000)

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Figure 11.57 Evenly spaced snapshots within one oscillation period of temperature disturbances on the free surface (travelling wave with m = 1) of a liquid bridge with aspect ratio AH = L/D = 0.4 heated from below (Pr = 30, D = 4 mm, S = 1, Ma = 4.3 × 104 ; Ma based on the height): experimental thermographic visualization on the left, 3D numerical simulations on the right. After Lappa et al. (2000)

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Figure 11.58 Two evenly spaced snapshots within one oscillation period of temperature disturbance on the free surface (standing wave with m = 3) of a liquid bridge with aspect ratio AH = L/D = 0.25 heated from above (Pr = 30, D = 4 mm, S = 1, Ma = 3.75 × 104 ; Ma based on the height): experimental thermographic visualization on the left, 3D numerical simulations on the right. After Lappa et al. (2000)

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Figure 11.59 Two evenly spaced snapshots within one oscillation period of temperature disturbances on the free surface (standing wave with m = 2) of a liquid bridge with aspect ratio AH = L/D = 0.25 heated from below (Pr = 30, D = 4 mm, S = 1, Ma = 4.2 × 104 ; Ma based on the height): experimental thermographic visualization on the left, 3D numerical simulations on the right. After Lappa et al. (2000)

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experimental database. The stabilizing effect of the heating from below condition was clearly observed over a large range of values of the Prandtl number (Pr = 1, 7, 49) and of the aspect ratio. The relatively weak stabilization of the bridge heated from above with respect to the situation of zero-g conditions may be explained by the weakened inﬂuence of the cold ﬁnger with respect to the case of zero-g conditions; in this case, as mentioned earlier, the temperature gradient along this cold, radially elongated zone is mitigated by the buoyancy effects, which slow the ﬂuid near the cool disk carried in the return ﬂow (it should be also pointed out that, as indicated by numerical simulations, e.g. Shevtsova et al., 2002, for Pr = 35, AH = 0.5 and W −1 = 1.227 and linear stability analysis, e.g., Li et al. 2008b, for Pr = 16, AH = 0.9 and W −1 ∼ = 3.2, the inﬂuence of buoyancy forces might be reversed in the case of heating from above, i.e. become destabilizing when relatively large values of the Prandtl number are considered). In the latter case (heating from below), strong stabilization of the ﬂow with respect to heating from above might be due to the fact that small surface temperature disturbances induced near the hot disk from inside by the cold, radially elongated zone have a weaker impact on the accelerations or deceleration of the main surface ﬂow (the sensitivity of the free surface is decreased by the strong surface velocities). Interestingly, Wanschura et al. (1997) explained these behaviours in terms of a delicate balance between some speciﬁc stabilizing and destabilizing buoyancy effects. They argued that since hydrothermal waves are characterized by strong axial vorticity whereas buoyant convection favours convection rolls with strong horizontal vorticity, therefore, both types of convection structures are incompatible in the sense that their respective transport mechanisms exclude each other, yielding a stabilization of the basic state with respect to pure Marangoni ﬂow. Precise mathematical arguments and computations in the framework of a linear stability analysis were provided to support such an explanation (which is worth detailed discussion in the text below). The limits of both large and small values of the parameter W were considered, showing that when W is high, the basic steady two-dimensional ﬂows and their instabilities are dominated by Marangoni ﬂow, whereas for small W , the stability behaviour is mainly due to buoyant convection, modiﬁed by weak surface tension-driven effects. The analysis was limited to a single aspect ratio (AH = 0.5, S = 1, straight cylindrical interface) and Pr = 4, but signiﬁcant insights were provided into the possible mechanisms occurring in ﬂuids with Pr > 1. The results for the case of high W , that is, dominating Marangoni ﬂow (the typical situation attained during microscale experimentation as discussed in the preceding pages) are shown in Figure 11.60. For Ra = 0, the critical mode is oscillatory with wavenumber m = 2 and develops continuously from a hydrothermal wave. Except for a small range of Rayleigh numbers (the minimum of the critical curve is located at Ra ∼ = 360), buoyancy forces act in a stabilizing manner. This means that critical Reynolds numbers obtained under weightlessness conditions (Ra = 0) should be smaller than those obtained in terrestrial experiments, apart from a small range of Rayleigh numbers (−700 ≤ Ra < 0 for Pr = 4) around the minimum of Recr (Ra), which is in agreement with results obtained during typical terrestrial experiments and with the physical considerations elaborated earlier. For buoyancy forces dominating the dynamics, the situation is completely different. With microscale experimentation, this situation occurs only in very special cases, namely for particular liquids characterized by very high values of the thermal expansion coefﬁcient and/or exhibiting a very weak dependence of the surface tension on temperature (e.g. non-pure water). As shown in Figure 11.61, the stability map becomes signiﬁcantly more complex and requires a detailed description; obviously, such a behaviour reﬂects an increased complexity of the underlying dynamics and related mechanisms.

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Figure 11.60 Stability map in the (Ra, Re) plane for dominating Marangoni ﬂow (Pr = 4, AH = 0.5, S = 1, straight cylindrical interface; characteristic numbers of driving forces based on the height of the liquid bridge, Ra < 0 → system heated from above, Ra < 0 → system heated from below): the linearly stable range is bounded from below by a stationary (m = 1) instability (solid line) and from above by an oscillatory instability (dashed line) with m = 2 or m = 1. Courtesy of H. Kuhlmann

Figure 11.61 Stability map in the (Ra, Re) plane for dominating buoyant ﬂow (Pr = 4, AH = 0.5, S = 1, straight cylindrical interface; liquid bridge heated from below; characteristic numbers of driving forces based on the height of the liquid bridge): Ra* as a function of Re (dotted line); linear stability boundaries of the strong state (solid line); linear stability boundaries of the weak state (dashed line). The capital letters denote A → Racr(m=2) = 1616, B → Racr(m=0) = 1825, C → Racr (m=2) = 3586, D → Racr (m=1) = 6557. Courtesy of H. Kuhlmann

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Figure 11.62 Nontrivial axisymmetric basic-ﬂow solutions related to hybrid convection in liquid bridges (for each sketch: axis of symmetry on the left, free surface on the right, heated and cooled disks on the bottom and on the top, respectively): (a) strong state; (b), (c) weak states

For the same liquid bridge (with aspect ratio AH = 0.5 and Pr = 4) already considered, for relatively small values of W up to three different two-dimensional basic ﬂow states enter the possible dynamics (Figures 11.62). In the light of this initial remark (there is a plethora of possible behaviours even if the investigation is limited to 2D ﬂow), following the same approach as Wanschura et al. (1997) it is convenient to discuss ﬁrst the stability behaviour considering only two-dimensional ﬂows (i.e. ignoring possible three-dimensional disturbances) and then examine the effective 3D behaviour. It is worth starting such a discussion by noting that for Re = 0 (pure RB convection) the initial diffusive state becomes linearly unstable with respect to a two-dimensional mode at Racr (m = 0) = 1825 via a perfect stationary bifurcation (this bifurcation is shown as point B in Figure 11.61). For Ra > Racr (m = 0) there exist two equivalent supercritical convective solutions representing ﬂow states with up- or downﬂow at the centre of the liquid bridge. If surface tension-driven effects are present (i.e. Re > 0), however, they alter the system symmetry (with respect to the boundary conditions) and, as a consequence, the bifurcation becomes imperfect. The related ﬂow ﬁeld consists of a single toroidal vortex with a sense of circulation (surface ﬂuid moving from the lower hot disk towards the upper cold one) supported by both buoyant and thermocapillary forces [hereafter referred to as strong state (a), see Figure 11.62a]. For Rayleigh numbers larger than Ra∗ (Re) > Racr (m = 0), which depends on Re [the Ra∗ (Re) curve is shown as a dotted line in Figure 11.61], there exist two more solutions, the weak state (b) and a state (c) (Figure 11.62), which are identical at Ra = Ra∗ . In both latter states, buoyant and Marangoni effects counteract, that is, they favour different directions of vortex motion. As a consequence, two additional small vortices appear in the hot and cold corners between the free surface and the supporting disks (where the buoyant convection is weak and surface tension forces are strong). The small corner vortices are surface tension driven in nature and the related ﬂow has an opposite sense of rotation with respect to the large vortex (of buoyant origin in the bulk). The weak state (b) is characterized by a strong internal toroidal vortex whose sense of rotation is determined by buoyancy forces. When the Rayleigh number is increased, both surface tension-driven corner vortices are suppressed and only very small corner cells remain. The larger of these corner vortices is always located downstream of the surface ﬂow due to the large buoyant vortex (Figure 11.62b). For very high Rayleigh numbers the corner vortices may eventually disappear.

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When Ra is increased beyond Ra∗ , the thermocapillary corner vortices for state (c) grow and ﬁnally merge to form a single Marangoni cell conﬁned to a layer below the free surface (Figure 11.62c). In the interior remains a weak counter-rotating buoyancy-driven vortex. The boundary Ra = Ra∗ (Re) of the parameter range for which all three nontrivial two-dimensional basic states (a), (b) and (c) exist is shown as a dotted line in Figure 11.61. The linear stability analysis of Wanschura et al. (1997) showed that the state (c) is always unstable, even with respect to two-dimensional perturbations. Moreover, all neutral modes of the basic states (a) and (b) are stationary along the neutral curves. When three-dimensional disturbances are considered, as also shown in Figure 11.48, the ﬁrst instability for Re = 0 is three-dimensional with m = 2 at Racr (m = 2) = 1616 (point A in Figure 11.61). Since Racr (m = 2) < Racr (m = 0) (points B and C in Figure 11.61), this instability corresponds to the instability of the strong state (a) [recall that the weak state can only exist for Ra > Racr (m = 0)]. When the Rayleigh number is increased while maintaining Re = 0, the linear growth rate of 3D inﬁnitesimal perturbations of the strong state becomes larger, reaches a maximum and vanishes again at Racr (m = 2) = 3586 > Racr (m = 0). The strong axisymmetric basic state becomes linearly stable immediately after Racr (m = 2). In Figure 11.61, both of the aforementioned bifurcation points at Re = 0 are connected in the (Ra,Re) plane by the critical curve for m = 2 of the strong state. Since the sign of the growth rate is preserved along both sides of the critical curve, the strong state is unstable with respect to m = 2 disturbances only inside the area delimited by this curve. Remarkably, the weak solution is always unstable in the range Ra∗ (Re = 0) < Ra < Racr (m = 2) and Re > 0. It becomes stable, however, in the area C–E–D. Also, the strong states are linearly stable in such an area, but they retain this property also outside it. Near Re = 0 and for Ra > Racr (m = 1), the roles of the strong and weak states are reversed again; therefore, the strong state becomes unstable to an m = 1 mode for Rayleigh numbers that are larger than the values indicated by the solid m = 1 curve in Figure 11.61. To summarize the major outcomes of the analysis of Wanschura et al. (1997) and related results plotted in Figure 11.61 there are two interesting general properties. The three-dimensional buoyant convection for Ra > Racr (m = 2) with basic mode m = 2 can be suppressed by relatively weak thermocapillary effects. Moreover, there is a range of Rayleigh numbers Ra > Racr (m = 2), for which the axisymmetric convection is linearly restabilized and where for Re = 0 two different axisymmetric states (a) and (b) exist. At this stage, it is worth stressing again that all the results discussed above in both limits of dominating (Figure 11.60) and weak (Figure 11.61) Marangoni ﬂow were obtained under the assumption of a straight cylindrical surface. For an extension of such results to the case in which the bridge is allowed to deform statically, it is worth citing the subsequent analysis by Nienh¨user and Kuhlmann (2002). For the same aspect ratio as examined by Wanschura et al. (1997), AH = 0.5, these authors considered two distinct values of the Prandtl number showing that the system response is markedly different according to whether Pr 1 or Pr > 1. According to such (linear stability) analysis for small Prandtl numbers (Pr = 0.02), the critical Reynolds number exhibits a smooth minimum near volume fractions which correspond approximately to the volume of a cylindrical bridge (S ∼ = 1). When the Prandtl number is large (Pr = 4), the intersection of two neutral curves results in a sharp peak of the critical Reynolds number similar to that shown in Figure 11.53. In practice, this means that since the instabilities for low and high Prandtl numbers are of a different nature (the reader is referred again to the discussion in Section 10.4), the inﬂuence of gravity leads to a distinctly different behaviour.

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While the hydrostatic shape of the bridge is the most important gravity effect on the critical point for low-Pr ﬂows (as already elucidated in Section 11.2), on the basis of the study of Nienh¨user and Kuhlmann (2002) buoyancy must be regarded as the dominating factor for the stability of the ﬂow in a gravity ﬁeld when the Prandtl number is high. For low Pr, the basic temperature ﬁeld practically decouples from the ﬂow. Buoyancy effects are small due to the high thermal conductivity compared with convective heat transport. The basic vortex, and hence its stability, are mainly determined by the hydrostatic interface shape, that is, by the hydrostatic pressure. On the other hand, for the same ratio of buoyant forces to hydrostatic forces (W = 3.7 in Nienh¨user and Kuhlmann, 2002), the hydrothermal wave instability mechanism for high Prandtl numbers is mainly affected by buoyancy rather than by the hydrostatic deformation of the free-surface shape: the basic temperature ﬁeld is crucial for the instability mechanism and it is susceptible to the buoyant convective transport. In other words, the instability of high-Pr ﬂows, caused by hydrothermal waves, is not altered qualitatively by the shape of the zone. As explained earlier, the prominent feature of these waves consists of pronounced internal temperature ﬂuctuations which arise in the region of large (radial) basic-state temperature gradients. In practice, the onset of oscillations depends on the interface shape (as shown, for instance, in Figure 11.53), because it affects the location and the magnitude of the energy-providing basic-state temperature gradients, mainly through the modiﬁed basic velocity ﬁelds and particularly through the return ﬂow in the bulk.

11.8.3 The Route to Aperiodicity For additional information on the behaviour of liquid bridges in normal gravity conditions (here the discussion is limited to microscale experimentation and liquid bridges heated from above) and the related route to aperiodicity, the reader may consider Chun (1984), Tang et al. (1995), Frank and Schwabe (1997), Ueno et al. (2003a) and Melnikov et al. (2004, 2005). In the remainder of this section, in particular, a number of outstanding results that are of particular interest or importance are mentioned. The list is not intended to be exhaustive, but rather to stimulate the interest of the reader in certain aspects, some longstanding, some new, where progress is needed. The transition of dominant Marangoni ﬂow from an initial steady state to chaos was carefully traced in experiments by Ueno et al. (2003a) for silicone oils of 1, 2 and 5 cSt (Pr = 16, 28.1 and 68.4, respectively). The ﬂow was visualized simultaneously by two video cameras and the time-dependent temperature was recorded by a thermocouple placed slightly inside the bridge at mid-height. They observed numerous bifurcations of the ﬂow on the way to chaos: 2D steady → standing wave → travelling wave → transition → new standing wave → new travelling wave → chaos → turbulence. Each of the regimes was identiﬁed through the observation of suspended particle motion, surface temperature variation, its Fourier spectrum and trajectories in a pseudo-phase space (PPS). Quasi-periodic structures (with two incommensurate frequencies) and also ﬂow regimes with a fractal structure were clearly detected (as discussed later). In particular, they focused on a ﬁxed aspect ratio (AH ∼ = 0.32, D = 5 mm, S ∼ = 1 corresponding to a value of W −1 based on the bridge height approximately in the range 0.36 ≤ W −1 ≤ 0.40 for the silicone oils considered) and classiﬁed the possible behaviour into eight distinct ﬂow regimes as shown in Figure 11.63 (see the caption). The PPS reconstruction was performed in the framework of a technique based on single time series of the fully developed surface temperature and the application of time-delayed coordinates [in Figure 11.63, T (t) is the surface temperature at time t and τ is a delay time]. The trajectories shown in this ﬁgure were reconstructed, in particular, with a constant delay time of τ = 0.1 s for Rg2–Rg6 and with τ = 0.06 s for Rg7 and Rg8).

511

Amplitude

Mixed Buoyancy–Marangoni Convection

(a) Frequency [Hz] T(t + 2τ)

Amplitude

0.4

0 0.4

−0.4 −0.4

(b)

T(t)

0.4 −0.4

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0 T(t + τ)

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Frequency [Hz] T(t + 2τ)

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0.4

0 0.4

−0.4 −0.4

(c)

0

Frequency [Hz]

Amplitude

T(t + 2τ) 1 0 −1

(d)

−1

Frequency [Hz]

0

−1

T(t) 1

1 0 T(t + τ)

Amplitude

T(t + 2τ)

(e)

2 0

Frequency [Hz]

−2 −2

2 0

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−2

0 T(t + τ)

T(t)

3

−3

0 T(t + τ)

T(t)

4

−4

0 T(t + τ)

T(t)

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−5

0 T(t + τ)

T(t)

Amplitude

T(t + 2τ)

(f)

3 0

Frequency [Hz]

−3 −3

3 0

Amplitude

T(t + 2τ) 4 0

Frequency [Hz]

(g)

−4 −4

4 0

Amplitude

T(t + 2τ)

(h)

5 0

Frequency [Hz]

(1) Top view

(2) Surface temperature

(3) Fourier spectrum

−5 −5

5 0

(4) Reconstructed PPS

Figure 11.63 Typical ﬂow regimes in liquid bridges of silicone oil (AH ∼ = 0.32, D = 5 mm, S ∼ = 1) for increasing values of the Marangoni number in terrestrial conditions (0.36 ≤ W −1 ≤ 0.40, Ma and W based on the height): (a) Rg1 steady ﬂow (Ma = 2.6 × 104 ); (b) Rg2 pulsating ﬂow I (Ma = 3.2 × 104 ); (c) Rg3 rotating ﬂow I (Ma = 3.4 × 104 ); (d) Rg4 transition (Ma = 6.4 × 104 ); (e) Rg5 pulsating ﬂow II (Ma = 6.2 × 104 ); (f) Rg6 rotating ﬂow II (Ma = 7.7 × 104 ); (g) Rg7 chaotic ﬂow I (Ma = 8.4 × 104 ); and (h) Rg8 chaotic ﬂow II (turbulence, Ma = 9.6 × 104 ). Column (1) top view of the ﬂow ﬁeld, (2) time series of surface temperature variation, (3) its Fourier spectrum and (4) reconstructed pseudophase space (PPS). After Ueno et al. (2003); Reproduced by permission of the American Institute of Physics

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According to such results (Ueno et al., 2003a), a standing-wave-type pulsating ﬂow emerges in Rg2 (Figure 11.63b) and a travelling wave-type rotating ﬂow in Rg3 (Figure 11.63c). With further increase in Ma from Rg3, the ﬂow becomes spatially disordered (Rg4; see Figure 11.63d); however, it cannot be regarded as a chaotic regime as evidenced by the regular time-dependent behaviour of the corresponding surface temperature. An ordered structure reappears in Rg5 (Figure 11.63e) and Rg6 (Figure 11.63f), which leads to the conclusion that the regime Rg4 is a transitional state (mixed mode) between Rg3 (rotating mode) and Rg5 (pulsating mode). Notably, the regimes of Rg5 and Rg6 display different kinds of pulsating and rotating ﬂows with respect to Rg2 and Rg3, respectively. In Figure 11.63e, the particles are trapped within the 1/6 azimuthally divided cells and seldom move to the neighbouring one, but the pattern observed through the top rod is different from that in the ﬁrst pulsating ﬂow Rg2. The ﬂow, in fact, never exhibits any polygonal structure; rather, it forms a particle-free region with 6 (= 2m) branches (two branches lie in every other cell and oscillate within it). Also, the new rotating regime Rg6 displays remarkable differences compared with Rg3. In this new regime, in fact, the particles tend to accumulate along a single closed path (PAS phenomenon) as shown in Figure 11.63f (let us recall that, as illustrated in Section 10.4.11, if looked at in a snapshot, the string formed by such a phenomenon tends to be wound m times around the thermocapillary vortex as a deformed spiral; if one looked at the full dynamics, it would be seen that the spiral string is rotating around its ring-shaped axis). Up to the regime of Rg6, the surface temperature varies with a fundamental frequency and its higher harmonics. In the regime of Rg7, however, the ﬂow becomes apparently chaotic (Figure 11.63g, the fundamental frequency is buried in the broadband noise). Interestingly, the visualized ﬂow in Rg8 shows no distinct difference with respect to Rg7. Nevertheless, applying the PPS reconstruction from the surface-temperature ﬂuctuation and the estimation of the correlation dimension and maximum Lyapunov exponent (see Chapter 1 for the related deﬁnition) from the reconstructed time-delayed coordinate, an evident difference can be established between these two regimes, as will be illustrated in detail later (Figure 11.67). For examination of spatiotemporal behaviour of the liquids from the laminar ﬂow state up to the onset of chaotic motion, Frank and Schwabe (1997) used another approach. In addition to the optical observations (views from above and from the front), up to 15 thermocouples were placed around half of the free surface without touching it. This allowed them to recognize different spatial reasons for quasi-periodic and period-doubling temporal behaviour and to identify various spatiotemporal chaotic structures. Most notably, all the results developed from the observation of temporal ﬂow structures were interpreted in a more spatial way of thinking, i.e. ideas of the spatial ﬂow conﬁgurations were formed on the basis of the temporal observations. They focused, in particular, on possible reasons for the existence of a second incommensurate frequency in the Fourier spectrum and for such a purpose considered four possible cases categorized according to the spatiotemporal behaviour related to the ﬁrst frequency (pulsating or rotating) and the value of the azimuthal wavenumber (m = 1 or m = 2) as summarized in the following: 1. Pulsating mode with m = 1. In this case, the appearance of a second frequency f2 in the spectrum is induced by the onset of an overall azimuthal rotation of the initial ﬂow structure pulsating at a frequency f1 (a 2π rotation of the f1 structure requires a period of τ = 2/f2 ). 2. Pulsating mode with m = 2. Here the second essential frequency visible in the frequency spectrum (f2 ) is again called forth by an azimuthal rotation of the ﬂow pattern pulsating with the frequency f1 (one complete rotation of the ﬂow conﬁguration in such a case, however, needs a time 4/f2 ).

Mixed Buoyancy–Marangoni Convection

513

3. Rotating mode with m = 1. This regime is featured in the case of a simply periodic behaviour by rotation around the geometric axis of an eccentric geometric area of lower temperature (such a rotation originates the essential frequency f1 ). In the experiments of Frank and Schwabe (1997), this area was found to correspond approximately to a region without tracer particles. The onset of a second incommensurate frequency was ascribed to a slow periodic variation of the distance between the centre of the ‘empty’ region and the geometric axis of symmetry (see Figure 11.64; such a periodic distance variation produces a modulation of the amplitude of f1 and, therefore, the appearance of f2 in the frequency spectrum). 4. Rotating mode with m = 2. For an initial rotating mode with m = 2, a similar description can be used; a modulation of the oscillation amplitude in such a case results from a variation of the axes e1 and e2 , that is, a slowly ‘pulsating’ elliptical region inside the ﬂuid (made evident by the absence of tracers in the experiments or by a lower temperature in numerical simulations) is responsible for the appearance of f2 by modulating the oscillation amplitude of f1 (see Figure 11.65a). Interestingly, Frank and Schwabe (1997) also elaborated an interesting geometric interpretation for the ﬁrst period doubled state arising from a pure rotating regime with m = 2. They observed, in fact, an eccentric position of the inner elliptical region with respect to the geometric axis of symmetry (Figure 11.65b). For the liquids with Pr = 7, 49 and 65, these authors reported a variety of phenomena as splitting of subharmonics in the Fourier spectrum, locking of quasi-periodic modes, the presence of only odd harmonics and frequency skips (see Section 1.8.1 for additional details about the mechanisms

Figure 11.64 Sketch of mode m = 1 rotating and undergoing at the same time periodic expansion and contraction of the distance between the centre of the inner circular region at a lower temperature and the geometric axis of symmetry

e1

(a)

e1

f1

e2

e2

(b)

Figure 11.65 Sketch of rotating mode m = 2 (a) with superimposed periodic expansion and contraction of the inner elliptical region axes; (b) with eccentric inner elliptical region

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Thermal Convection: Patterns, Evolution and Stability

leading to chaos and the various theories proposed over the years to explain the possible transitional scenarios). Period doubling bifurcations and the Feigenbaum universal law were also observed in the earlier experiments of Tang et al. (1995). It is also worth discussing similar behaviours obtained via numerical solution of the nonlinear thermal-convection equations. As an example Melnikov et al. (2004) reported for Pr = 18.8 and AH = 0.5 (W −1 = 0.9) the following scenario (in qualitative agreement with the experiments of Ueno et al., 2003a): 2D steady → standing wave (m = 1) → travelling wave (m = 1) → standing wave (m = 1 + 2) → travelling wave (m = 1 + 2) → chaos (dominant m = 2). Even more interesting are the numerical simulations carried out 5 years later by Shevtsova et al. (2009) for a larger value of W −1 (a liquid bridge, Pr = 14.3, AH = 0.9, W −1 = 3.11) for which an exotic behaviour was observed in certain ranges of the Marangoni number never detected before for pure Marangoni ﬂows (which provides indirect conﬁrmation of the potential role played in such dynamics by the presence of perturbing gravitational mechanisms). These authors reported novel oscillatory ﬂow (quasi-periodic) states created by the coexistence and interaction of two modes, one propagating vertically from the cold towards the hot side (m = 0) and the other travelling in the azimuthal direction (m = 1). The wave with m = 0 was found to appear at Ma = 1.416 × 104 and to persist up to 1.3 Macr . Far above the ﬁrst bifurcation, at Ma > 1.76 × 104 , a ﬂow state in the form of an azimuthal travelling wave with the mode m = 1 was also observed. A novel type of ﬂow organization was found in the intermediate range, at 1.652 × 104 < Ma < 1.76 × 104 . As shown in Figure 11.66, this type of oscillation does not correspond to either m = 0 or travelling/standing wave. Starting from a surface (disturbance) temperature distribution with almost four symmetrical temperature spots (Figure 11.66a), such spots transform to a pattern typical of azimuthal TW (Figure 11.66b). Further (Figure 11.66c), each spot splits into three disturbance nodes and the wave in the upper part becomes inclined whereas the spots on the cold side are almost straight. The most remarkable situations are shown in Figure 11.66d and 11.66e, when cells

1.8

0.0 0.0

0.5

ϕ/2π

1.0 0.0

ϕ/2π

0.5

(a)

1.0 0.0

0.5

(b)

ϕ/2π

1.0

(c)

1.8

0.0 0.0

0.5 (d)

ϕ/2π

1.0

0.0

0.5 (e)

ϕ/2π

1.0

Figure 11.66 Snapshots of temperature disturbance ﬁeld on the unrolled interface of a liquid bridge displaying an exotic quasi-periodic behaviour created by interaction of two modes one propagating vertically from cold towards hot side (m = 0) and another travelling in the azimuthal direction (m = 1) (Pr = 14.3, AH = 0.9, Ma = 1.72 × 104 , W −1 = 3.11; Ma and W based on the height). Courtesy of V. Shevtsova

Mixed Buoyancy–Marangoni Convection

515

Figure 11.67 Correlation dimension D 2 as a function of the Marangoni number (silicone oils, AH = 0.32). Data after Ueno et al. (2003); Reproduced by permission of the American Institute of Physics

near cold and hot walls have a different inclination angle and tend to expand essentially along the axial direction. It is worth concluding this discussion by noting that in the study of Ueno et al. (2003a) mentioned earlier, the problem of transition to chaos was even approached in terms of the maximum Lyapunov exponent λ and the fractal correlation dimension (see Sections 1.8.4 and 1.8.5 for a proper deﬁnition of these quantities). The correlation dimension for Ma = 8.4 × 104 (regime Rg7 in Figure 11.63) was found to have a fractional value of 2.2 (that indicates that the ﬂow ﬁeld for such a regime has a fractal structure). The regime with Ma = 9.6 × 104 (Rg8), on the other hand, has an even larger value of the dimension (in the range 2.1 ≤ D2 ≤ 2.6). Figure 11.67 clearly proves that both of these two regimes contain fractal structures in their dynamics. The maximum Lyapunov exponent up to Ma = 8 × 104 was found to have values very close to zero, which corresponds to a simple periodic behaviour of the attractor; in the last two regimes, however, this exponent was observed deﬁnitely to become positive (which provides clear evidence for the fact that these regimes possess sensitivity to initial conditions, i.e. SIC). Given the current level of knowledge (mostly coming from limited experimental results), it is not possible to assess the role played in all such complex dynamics by gravity. A proper investigation of this area will require new experimental parametric investigations focusing on the route to chaos in liquid bridges for several values of the parameter W (i.e. for several sizes), and also comparison of numerical results mimicking effective experimental conditions with equivalent ones obtained artiﬁcially setting Ra = 0.

12 Hybrid Regimes with Vibrations This chapter is focused entirely on the hybrid regimes of convection in which one of the driving forces involved is represented by mechanical vibrations (equivalent to a periodic time-varying acceleration; following the same concepts as deﬁned in Chapter 8, hereafter the terms gravity modulation, periodic acceleration, container vibration, periodic forcing, shaking and g-jitter will be used as synonyms). In addition to the possible technological applications, the interaction of thermovibrational convection with standard thermogravitational convection or with thermal Marangoni ﬂow (or both) complements, from a theoretical point of view, the similar topic of mixed buoyancy–Marangoni convection extensively treated in the preceding chapter. As already suggested at the beginning of Chapter 11, owing to the high degree of complexity of the topic, prior to entering the present discussions, the reader is strongly encouraged to undertake preliminary readings of many earlier parts of this book (especially Chapters 4, 6, 9 and 10) to become familiarized with important propaedeutical concepts related to classical buoyancy and surface tension-driven ﬂows (terminology, classiﬁcation of possible instabilities and physical mechanisms, etc.). For the pure thermovibrational convection (ﬂuid motion induced by vibrations or zero mean value periodic accelerations and the related nondimensional parameters and characteristic numbers), in particular, the reader is referred to Chapter 8. Some fundamental key aspects of these types of ‘pure’ ﬂows can be summarized as follows. When steady gravity is opposite to ∇T , it has no effect (does not induce convective ﬂows) for any value of its magnitude; when g is concurrent with ∇T , then TFD distortions arise only if the critical conditions for the onset of convection are exceeded; if these vectors are perpendicular, ﬂuid motion arises without the need to overcome a threshold value of any parameter (similar considerations apply to surface tension-driven ﬂows by replacing the direction of gravity with the direction perpendicular to the free surface). The effect of high-frequency vibrations, such as that of a steady acceleration, strongly depends on the shaking direction relative to the temperature gradient. In particular, vibrations parallel to the temperature gradient tend to maintain initial thermal diffusive conditions, whereas perpendicular vibrations lead to TFD distortions in ﬁnite-sized geometries for any value of T . In such a context, it is also worth mentioning that, in general (see Birikh et al., 1993), the mean vibration force is a bulk driving action induced by temperature gradients normal to the vibration axis. A local feature of this force is that, if temperature distortions, with respect to the purely diffusive case, are induced by another type of convection, average vibrational ﬂows arise in such a

Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

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Thermal Convection: Patterns, Evolution and Stability

way as to permit the isotherms to turn and again become perpendicular to the vibration direction, which provides an interesting concept for elaborating a possible ﬂow control strategy. These problems have been the focus of signiﬁcant attention over recent decades. The response of hydrodynamic systems to periodic modulation of the driving force has been of considerable interest since the seminal experiments of Donnelly et al. (1962) and the theoretical work by Simonenko and Zen’kovskaja (1966). The importance of these studies was emphasized later by many authors (see, e.g., Davis, 1976, and Ostrach, 1982). Over subsequent years, the case of combined buoyancy (induced by a static acceleration) and thermovibrational convection has continued to attract interest, in particular in the context of studies devoted to carefully planning, executing and analysing space experiments; it corresponds, in fact, to the real situation occurring on the ISS and other orbiting platforms, where, due to unavoidable disturbances, in many circumstances both steady and oscillatory levels of residual acceleration are present (see Section 2.2.4 for the relevant theoretical background). As will be shown later in this chapter, it is possible to take advantage of the interplay between these two types of convection in space to induce a mitigation of the resulting convective disturbances. Along these lines, time-periodic forcing has also been proposed as a possible means of dynamic control of ﬂow instabilities in terrestrial experiments (or processes of practical interest). As anticipated, the character of natural buoyant convection in nonuniformly heated, rigidly contained ﬂuids, in fact, can be drastically altered by vibration of the container; thereby, vibrational induced ﬂow can potentially be used to inﬂuence and even control transport in some crystal situations (Davis, 1976) and/or to affect the morphological stability of the crystal/melt solidiﬁcation interface (which is a very important factor in these processes; for further details, the reader is referred to, e.g., the theoretical studies of Schulze and Davis, 1994, 1995, Murray et al., 2000, Volfson and Vinals, 2001, and Lyubimova et al., 2007). In general, the study of these aspects is based on the introduction of a general gravitational forcing g, ˜ which may have arbitrary spatial and temporal dependence. Moreover, a speciﬁc nondimensional parameter characterizing these situations is usually deﬁned as G=

Raω bω2 = g Ra

(12.1)

where b and ω are the amplitude and angular frequency, respectively, of the vibrations, g is the considered level of static acceleration (g = g0 = 9.81 m s−2 in terrestrial conditions) and Ra and Raω are the usual characteristic numbers for thermogravitational and thermovibrational convection deﬁned by Eqs (2.7) and (8.4), respectively. This leads the aforementioned generalized gravity to be expressed simply as ˆ g˜ = g0 (i g + Gn)

(12.2)

where nˆ and i g are the unit vectors along the directions of shaking (vibrations) and static gravity, respectively (it is obvious that if G 1 and nˆ = i g , then the periodic forcing may be simply regarded as a perturbation of the mean gravity). Similar concepts apply to the possible mitigation and stabilization of ﬂows induced by surface tension gradients for which, by analogy with Eq. (2.26), a speciﬁc parameter measuring the relative importance of vibrational and Marangoni effects can be deﬁned as bω2 ρβT 2 Raω L = (12.3) (Wω )−1 = σT Ma where Ma is the usual Marangoni number deﬁned by Eq. (2.17). Remarkably, like magnetic ﬁelds (Chapter 13), vibrations allow contactless control of the ﬂow. It is a fairly new and a so far less well investigated technique that can be used more universally because it is not restricted to electrically conductive melts as is the case for magnetic ﬁelds (Lappa, 2005e).

Hybrid Regimes with Vibrations

519

These factors have combined to make the possible technological application of vibrations a subject of increasing interest, especially in recent years. Accordingly, a number of studies are currently available for both low- and high-frequency cases (see Chapter 8 for fundamental information about the differences between these two conditions) employing both linear and nonlinear theories to investigate the effect of modulation on convection, using numerical or a combination of numerical and asymptotic solution techniques. Superimposed on this is the undeniable appeal that this topic has exerted on researchers and scientists as a natural outcome of the richness of patterning behaviours that can be obtained (when the speciﬁc parameters of vibration, direction, frequency and amplitude, are varied), even in the case of ﬂows retaining a simple two-dimensional structure. In practice, the subject becomes even more interesting when three-dimensional conﬁgurations are considered. One of the most exciting achievements made in recent years, in fact, is the discovery of new exotic 3D patterns never reported before for conditions in which the driving forces (static gravity or surface tension-driven forces on the one hand and mechanical vibrations on the other) act separately. These exotic states are composed of two distinct spatial scales, each displaying a different temporal dependence; most notably, they pertain to a rare kind of complexity, known as complex order, which has been used by some investigators to shed some new light on the possible dialectics (that can be established in the general context of nonlinear phenomena) between order and disorder, self-organization and chaos (the reader is also referred to the discussion at the beginning of Chapter 1). Many of these analyses, in particular, investigated the possible existence of ‘resonances’, i.e. of special convective (resonant) states in which the frequency of the mechanical vibrations is almost equal to the ‘natural’ frequency related to an oscillatory instability of the ‘base’ ﬂow that occurs in the unmodulated case, namely the ﬂow induced by steady accelerations (it is worth noting that, of course, there is no possibility of resonance in systems subjected to periodic acceleration with zero mean value, i.e. in the absence of a steady acceleration). Special types of spatiotemporal resonances arise in the case in which the system preserves the original isotropy of the ‘base’ ﬂow (the most relevant example along these lines is represented by vertically oscillated RB convection, which will be treated in Section 12.2). This condition permits easy selection of different interacting spatial length scales from a range of wavenumbers. Accordingly, a wide ﬂexibility in possible spectral resonances (which can form) is allowed leading to the possible onset of states that display dynamics pertaining to the aforementioned ‘complex order’.

12.1

RB Convection with Vertical Shaking

Starting from the landmark study of Simonenko and Zen’kovskaja (1966), many investigations have focused on Rayleigh–B´enard (RB) convection subjected to gravity modulation. Gresho and Sani (1970) were the ﬁrst to consider the stability of a horizontal layer of ﬂuid heated from above or below with a time-dependent buoyancy force generated by shaking vertically the ﬂuid layer (causing a sinusoidal modulation of the gravitational ﬁeld). A linearized stability analysis was performed to prove that gravity modulation can signiﬁcantly affect the stability limits of the system. Also, they introduced an interesting analogy to the case of a simple pendulum governed by the Mathieu equation (reducing the number of parameters affecting the system behaviour from six (ﬂuid transport) parameters to three (Mathieu) parameters). It was proved, in particular, that vibrations can destabilize a stably stratiﬁed ﬂuid (uniformly heated from above) and stabilize an unstably stratiﬁed ﬂuid (in practice, in analogy with an inverted pendulum stabilized by oscillating

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Thermal Convection: Patterns, Evolution and Stability

its pivot, unstable/stable density stratiﬁcations can be stabilized/destabilized by vibrating the layer; for similar considerations, see also Gershuni and Zhukhovitskii, 1963, and the study by Ahlers et al., 1985a). Later, Meyer et al. (1988) found by experimental investigation that new types of patterns (among them hexagonal structures) can arise in temporally modulated RB convection. Two years later, Biringen and Peltier (1990) considered ﬂuid motion between horizontal parallel walls with different constant temperatures and with vibrations parallel to the vertical steady Earth’s gravity. Unlike earlier theoretical work, however, they expressly took into account the effects of three-dimensionality in addition to full nonlinearity. In this numerical study, in particular, both randomly and sinusoidally modulated gravitational ﬁelds imposed on RB convection were investigated in an effort to understand the effects of random vibrations; the time-dependent Navier–Stokes equations and the energy equation with Boussinesq approximations were solved by a semi-implicit, pseudo-spectral procedure. Random modulations were found to be less stabilizing than sinusoidal modulations and were shown to impose local three-dimensionality on the ﬂow for some parameter ranges under both terrestrial and zero gravity conditions. For the case of sinusoidally modulated forcing, they recovered the results of Gresho and Sani (1970). Figure 12.1a, taken from their work, shows the stability curve for a ﬂuid with Pr = 7 and for G = bω2 /g0 = 19.6. The critical Rayleigh number and Racr and wavenumber qc have been plotted as a function of the modulation frequency # [deﬁned according to Eq. (8.5) assuming L = d = thickness of the layer]. The solid line represents neutral stability; above the curve the ﬂuid moves and below it, it is quiescent. It is shown that the critical Rayleigh number increases with increasing # for # less than a certain value (#∗ ), while it decreases with increasing the modulation frequency for # > #∗ . The most interesting feature displayed by the neutral curves, perhaps, is that the emerging convective regime can be subdivided into two spatiotemporal categories according to the sign of # − #∗ : in fact, for 0 < # < #∗ (#∗ = 1250 for Pr = 7 and G = 19.6) the system response to the imposed vibration is synchronous (or harmonic) (temperature and velocity ﬁelds oscillating at the same frequency of the imposed sinusoidal forcing), whereas for # > #∗ such a response is subharmonic (see, e.g., Figure 12.2). The synchronous response regime was found to be characterized by simple, repeated orbits in the phase space and by the existence of a viscous boundary layer near the solid surfaces, whereas the regime with subharmonic response was characterized by simple, repeated orbits in phase space like the synchronous one, but with a greater number of direction reversals. The viscous boundary layer (clearly evidenced for the synchronous regime) was not observed, however, by Biringen and Peltier (1990) for the subharmonic response. These authors also revealed that for # < #∗ with increase in the parameter G = bω2 /g0 (i.e. with increasing Raω for a ﬁxed Ra), the evolution of the ﬂuid system is from synchronous response to subharmonic response and ﬁnally to chaos. Continuing with the review of the existing literature on the subject, it is also worth mentioning the numerical studies of Clever and co-workers. Clever et al. (1993a), in particular, extended the stability analysis of Biringen and Peltier (1990) to a much larger region of parameter space. Introducing the Froude number as Fr = b

G α2 = 2 g0 d 4 #

(12.4)

where α is the thermal diffusivity. They observed that the crossover frequency #∗ decreases with increase in Fr (see Figure 12.1b, which was obtained by these authors for the same value of the Prandtl number as considered by Biringen and Peltier, 1990). They also highlighted many other effects related to Fr, which can be summarized as follows. The value of Racr at #∗ also decreases with increasing Fr. Synchronous

Hybrid Regimes with Vibrations

521

(a)

(b)

Figure 12.1 Critical Rayleigh number Ra (based on the thickness of the layer) for the onset of convection in a modulated ﬂuid layer (heated from below and cooled from above subjected to vertical vibrations) as a function of # for the speciﬁc case Pr = 7 and the following conditions: (a) ﬁxed value of the parameter G = bω2 /g0 (G = 19.6) (after Biringen and Peltier, 1990; Reproduced by permission of the American Institute of Physics); (b) different values of the parameter Fr (Courtesy of F.H. Busse)

convection extends to larger # for smaller values of Fr, whereas subharmonic convection (an example of its spatiotemporal appearance is shown in Figure 12.2) occurs at smaller # for larger values of Fr. Moreover, for sufﬁciently small #, Racr for synchronous convection becomes independent of Fr (the increase in Racr with # for synchronous convection being limited to a narrow range of # near #∗ ). In general, synchronous convection occurs at smaller values of qc than subharmonic convection for any value of Fr. Moreover, for synchronous convection the asymptotic value for # → 0, qc =

522

Thermal Convection: Patterns, Evolution and Stability

Figure 12.2 Subharmonic convection [plane (x, y)] (Pr = 7, # = 316.2, Ra = 4 × 104 , G = 3.162, q = qc = 2.9; the snapshots of streamlines and isotherms cover a full modulation period; the left and right boundaries of each frame correspond to periodic boundary conditions). Courtesy of F.H. Busse

3.117, is independent of Fr; qc decreases with increase in # in a narrow range of # near #∗ ; there is a discontinuous change in qc at #∗ ; then the critical wavenumber for subharmonic convection generally increases with increasing # (although for Pr = 7 it has a local minimum at Fr = 10−3 ). For sufﬁciently large #, qc for subharmonic mode tends to become independent of Fr as in the opposite case of synchronous convection and # → 0. These behaviours were also found to depend on the Prandtl number (Figure 12.3), as discussed in detail in the following.

Hybrid Regimes with Vibrations

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Figure 12.3 Critical Rayleigh number for the onset of convection in a modulated ﬂuid layer heated from below and cooled from above as a function of # for Fr = 10−4 and different values of the Prandtl number (synchronous branches on the left and branches with subharmonic response on the right). Courtesy of F.H. Busse

At sufﬁciently large #, in particular, Racr was found to obey the asymptotic equation of Gresho and Sani (1970): 0.4739 −1 as # → ∞ (12.5) Fr Pr As an example, Figure 12.1b shows that for Pr = 7, the asymptotic regime in which Eq. (12.5) is valid lies at values # > 6 × 103 . Figure 12.3 illustrates the Pr dependence of Racr versus ﬁnite # for Fr = 10−4 . It is clear that increasingly large values of # are required to reach the asymptotic state given by Eq. (12.5) as Pr increases. Concerning the wavenumber of the subharmonic mode, as already shown in Figure 12.1, it increases with # at large #; the large-frequency limit of qc , however, depends on Pr. In particular, in the limit as # → ∞, qc decreases with increase in Pr, as does Racr (Figure 12.3). At lower frequencies, the dependence of the subharmonic qc and Racr on Pr are non-monotonic (not shown). A similar behaviour occurs for the synchronous mode. In agreement with earlier studies, Clever et al. (1993a) also reported that for sufﬁciently large #, convective instability is possible even for heating from above, as can be seen in Figure 12.4, which shows Racr as a function of # for different values of Fr for the same case as considered before (Pr = 7). On the basis of this ﬁgure, some interesting general conclusions can be drawn. At a given value of Fr, convection with heating from above cannot occur until # exceeds a value #L for which Racr →

524

Thermal Convection: Patterns, Evolution and Stability

Figure 12.4 Critical Rayleigh number for the onset of convection in a modulated ﬂuid layer heated from above and cooled from below as a function of # for different values of the parameter Fr (Pr = 7; SY and SU denote synchronous and subharmonic response, respectively). Courtesy of F.H. Busse

convection emerges as a synchronous mode. The related Racr decreases with increasing #. As for heating from below, there exists, however, a #∗ such that for # > #∗ a subharmonic mode of convection replaces the synchronous mode as the preferred mode of convection at the onset. The smaller the value of Fr, the larger # must be for convection to emerge in either the synchronous mode or the subharmonic mode. As # → ∞, Racr tends to a constant value, which exhibits an inverse dependence on Fr, similar to Eq. (12.5). In a subsequent study (Clever et al., 1993b), nonlinear 3D solutions were obtained for the speciﬁc Prandtl number of air (0.71) and for two Rayleigh numbers above the value for onset of oscillatory convection of the RB type (see Section 4.2 for fundamental information about the secondary oscillatory instabilities of pure RB convection). Multiples of the fundamental frequency of oscillatory convection were chosen in order to study the effects of possible resonances of the frequency of gravitational modulation. They found that modulation causes a transition from travelling wave convection, which persists in the unmodulated case (travelling wave convection for Pr ∼ = 1 and pure RB convection has been shown in Figure 4.10), to standing wave convection. Along these same lines, Volmar and M¨uller (1997) reported (numerical simulation) unstable modes in harmonic or subharmonic temporal resonance with the mechanical drive depending upon the modulation frequency and amplitude. Similar results were obtained by Rogers et al. (2000a,b, 2003, 2005). They, in particular, investigated the possible emergence of new and unexpected ﬂow patterns driven by resonance mechanisms between the synchronous and subharmonic modes in speciﬁc ranges of G and Ra. These analyses focused, in fact, on situations where modes in harmonic and subharmonic temporal resonance with the mechanical drive compete. Rogers et al. (2000a), in particular, were the ﬁrst to ﬁnd for such a condition and in the case of compressed CO2 a new star-like structure located on a long-length scale square lattice [referred to as square superlattice (SQS)]. This pattern was found to bifurcate supercritically from conduction at the codimension-two (bicritical) point where linear modes of two disparate wavenumbers simultaneously become marginally stable.

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Given the astonishing variety of patterns observed in this type of study, it is worth opening a brief but detailed discussion on this subject. For a deeper understanding of the studies by Rogers and co-workers mentioned above, the reader is referred, in particular, to the following section.

12.2

Complex Order, Quasi-periodic Crystals and Superlattices

The majority of pattern formation studies in modulated RB convection mentioned in the preceding section were focused on the case when a single wavenumber q is accessible at onset. By allowing for multiple and distinct accessible wavenumbers near the onset, some investigations have extended this focus and, in the process, found a number of exotic patterns including quasi-periodic crystals, superlattices and domain coexistence. Many of these patterns display complex spatial structure, often on distinct length scales, described by relatively few spectral modes. Due to such characteristics, as anticipated at the beginning of this chapter, they are generally referred to as ‘complex order’. Prior to expanding on this subject and related results provided over the years by the aforementioned investigators, some propaedeutical concepts must be provided to help the reader in the understanding of the descriptions and elaborations given later. Among other things, these deﬁnitions complement and enrich from a theoretical point of view the general ones already provided at the beginning of this book in Section 1.1. For such purposes, it is worth making reference to the theoretical work of Pismen and Rubinstein (1999), where the concept of complex order was treated in an exhaustive and precise way, as brieﬂy discussed in the following. Spontaneous symmetry breaking and pattern formation are universal phenomena observed in a wide variety of non-equilibrium systems. As widely illustrated in the preceding chapters of this book, most commonly, patterns have a simple basic structure, usually stripes or hexagons, which is distorted on a long scale, so that the pattern has disordered local orientations and contains various defects such as domain walls, dislocations or disclinations (complexity of patterns is usually understood in the sense of disorder, which may be regarded as a fascinating but uncontrollable kind of complexity). A rarer kind of complexity, however, is complex order: a well-controlled structure ordered in a nontrivial way. Quasi-crystalline patterns belong to this variety: they have a complicated spatial structure that never repeats itself, but are well ordered in the Fourier space. In principle, understanding such patterns is easy: in two dimensions they can be formed just by superposing four or more non-collinear modes. Near a symmetry-breaking transition from a homogeneous to a patterned or crystalline state, these modes appear formally as degenerate, neutrally stable eigenmodes of linearized macroscopic equations and may admit, in various contexts, different physical interpretations. Since a symmetry-breaking transition in an isotropic system implies a preferred wavelength but no preferred direction, an indeﬁnite number of modes may be excited, with the wavevectors q j having the same absolute value but arbitrarily directed. The emerging pattern or crystalline structure is selected by nonlinear interactions. In complex order, these interactions are resonant, that is, phase dependent. The phases of the interacting modes always become locked in such a way that the interaction is destabilizing. Further reﬁnement of these concepts is possible according to the criteria originally given by Lifshitz (1997): these states, in fact, can be further distinguished in quasi-periodic crystals and superlattices.

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Thermal Convection: Patterns, Evolution and Stability

As also illustrated by Rogers et al. (2003), the essential difference between these two sub-categories is the number of basic (indexing) vectors required to map out the dominant spectral modes. If the number of indexing vectors is larger than the spatial dimension of the pattern, then the complex state is a quasi-periodic crystal. A superlattice exists when the number of indexing vectors equals the pattern’s spatial dimension. Coming back to the original problem of interest here, namely vertically oscillated RB convection, it is worth pointing out that such a system preserves the original isotropy of RB systems; furthermore, it permits easy selection of different interacting length scales from a range of wavenumbers, and thereby wide ﬂexibility in possible spectral resonances (which can form) is allowed. In particular, as explained in Section 12.1, a system consisting of a ﬂuid layer experiencing forcing from both a vertical temperature difference and vertical time-periodic oscillations, depending on the parameters, can produce ﬂuid motion with either a harmonic (H) or a subharmonic (S) temporal response. Along these lines, Rogers and co-workers reported and analysed complex patterns observed in a combination of standard pattern-forming experiments related to these two behaviours. These exotic states were found to be composed of two distinct spatial scales, each displaying a different temporal dependence. Over a parameter range where the mechanisms (H and S) have comparable inﬂuence, the spatial scales associated with both responses were found to coexist, resulting (as outlined at the end of Section 12.1) in complex, highly ordered patterns. Unlike the earlier studies by Gresho and Sani (1970) and Biringen and Peltier (1990) in which the focus was on the system behaviour as a function of # for ﬁxed values of the nondimensional parameter G = bω2 /g0 , Rogers and co-workers, in particular, focused on the case of ﬁxed Pr (= 0.93) and # (= 98) by varying the remaining control parameters Ra and G = bω2 /g0 = Fr#2 (as in these studies # was assumed to be ﬁxed, hereafter arguments and descriptions will be provided directly in terms of Fr, until the end of this section). The problem was approached via both a linear stability analysis for the idealized case of an inﬁnite layer and experimental investigation of shallow conﬁgurations (to mimic the theoretical conditions of a system with inﬁnite extent) with circular or square sidewalls. They also performed numerical simulation (solution of the nonlinear Navier–Stokes equations). The most interesting outcomes are discussed in the following. According to the results of the linear stability analysis shown in Figure 12.5, modulated ﬂows with a long-length scale and harmonic (H) temporal response (ω) arise for small Fr. Short-length scale ﬂows displaying subharmonic (S) temporal response (ω/2) occur when Fr is sufﬁciently high. At a bicritical point (Frcr2 , Racr2 ), H and S modes emerge simultaneously with distinct S H critical wavenumbers q2c and q2c (the linear stability curves for harmonic and subharmonic onsets bisect at a critical point where both q H and q S are accessible; for the conditions considered (Pr = 0.93 and # = 98) the bicritical point occurs at Fr2c = 3.768 × 10−4 , Racr2 = 4.554 × 103 S H with wavenumbers q2c = 1.742 and q2c = 5.173). In general, harmonic ﬂows are more stable than unmodulated convection, that is, the critical Rayleigh number for harmonic convection RaH cr (solid line in Figure 12.5) is expected to be larger than Racr in the absence of modulation (Racr = 1707). In contrast, subharmonic ﬂows may be either more stable or less stable. The critical wavenumber of harmonic patterns qcH is signiﬁcantly less than the critical wavenumber of subharmonic patterns qcS and also less than the unmodulated value (3.117). A third region of possible behaviours originates from the bicritical point; as a consequence, above the onset of convection three distinct regions can be identiﬁed: purely harmonic, purely subharmonic and the central region in which coexistence of the two types of system response leads to complex order and related patterns.

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Figure 12.5 Phase diagram for harmonic (H), subharmonic (S) and coexistence states: comparison of linear stability predictions (marginal stability curves) with experiments and numerical simulations for Pr = 0.93 and # = 98 (Rayleigh number based on the thickness of the layer). The diagram contains the regions of diffusion (quiescent state), convection with H(ω) and S(ω/2) modulations as well as coexisting H–S patterns (the arrow indicates the codimension-two point at the intersection of the H and S linear stability curves represented by the solid and dashed lines, respectively). The dashed circle indicates the presence of experimental patterns displaying mixed harmonic cellular symmetries (square, rhombic and hexagonal symmetry). After Rogers et al. (2003); Reproduced by permission of IOP Publishing Limited

12.2.1 Purely Harmonic Patterns As already outlined before, purely harmonic patterns occur when the driving (Fr) is relatively small. They are qualitatively similar to modes of convection usually observed in canonical RB convection. In proximity of the primary onset, the observed patterns depend on both the validity of the Boussinesq symmetry (see Chapter 4 for the related deﬁnition) and the intensity of sidewall forcing (see Section 4.10.2, in particular, for the meaning of this concept). When the imposed temperature gradient is relatively small, the Boussinesq symmetry is valid and striped states are found near the onset. In particular, if the sidewalls result in relatively weak forcing, parallel rolls are the striped state (Figure 12.6a). If the sidewall forcing is increased, then the outcomes are targets (Figure 12.6b; the reader is referred, in particular, to Section 4.10 for the deﬁnition of targets and other patterns related to RB convection). If the Boussinesq symmetry is lost, hexagons form at the onset (Figure 12.6c); typically they form in the vicinity of the bicritical point (Figure 12.5) where T is several times larger than the onset value required in the absence of vibrations. Each type of harmonic onset planform (parallel rolls, targets and hexagons) tends to display a different evolution scenario as the experiment moves away from the onset and into the harmonic parameter region. Although onset patterns are relatively regular, various kinds of irregularities or defects arise as the system moves away from the onset (some general theoretical background on this type of behaviour has been provided in Section 4.3). Parallel strips undergo dislocations and curvature about two sidewall foci (Figure 12.6a). Increasing the distance from the onset conditions, the number of dislocations and stripe curvature also

528

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

(c)

(d)

(e)

(f)

(g)

(h)

(i)

(j)

(k)

(l)

Figure 12.6 Representative purely harmonic [(a)–(f)] and purely subharmonic [(g)–(l)] patterns Near harmonic observed in experiments for Pr = 0.93 and # ∼ = 98, unless noted otherwise. onset: (a) parallel strips (Ra = 4529, Fr = 3.49 × 10−4 ); (b) targets (Ra = 4050, Fr = 3.59 × 10−4 ); (c) hexagons (Ra = 4858, Fr = 3.62 × 10−4 ); away from onset parallel strips display (d) defects (Ra = 3926, Fr = 3.36 × 10−4 ); and sufﬁciently into the harmonic convective region (e) spiral defect chaos (Ra = 4384, Fr = 2.59 × 10−4 ); similar states (f) are observed with square sidewalls Near subharmonic primary onset: (g) parallel strips (Pr = 0.893, Ra = 5025, Fr = 3.17 × 10−4 ). (Ra = 4857, Fr = 3.74 × 10−4 ); (h) giant spirals (Ra = 4108, Fr = 4.12 × 10−4 ); and (i) giant convex dislications (Ra = 4385, Fr = 4.31 × 10−4 ); away from the onset: (j) transverse modulated stripes (Ra = 6128, Fr = 4.33 × 10−4 ); and (k) radial stripes (Ra = 6639, Fr = 4.31 × 10−4 ); similar results are observed with (l) square sidewalls (Pr = 0.894, Ra = 5037, Fr = 4.40 × 10−4 , # = 99.8). After Rogers et al. (2003); Reproduced by permission of IOP Publishing Limited

increase, resulting in grain boundaries (Figure 12.6d) and additional sidewall foci. Sufﬁciently far from the onset, spiral defect chaos (SDC) (Figure 12.6e) occurs (see Sections 4.11 and 4.12 for relevant information about this special state and the related dynamics). In contrast, a target will begin this transition as a defect moves to the centre of the target, thereby creating a one-arm spiral. Increasing the distance form the onset, additional defects enter the spiral, leading to the formation of a multi-arm spiral (up to six arms have been observed in

Hybrid Regimes with Vibrations

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the experiments). Then, the spiral core moves off-centre and the number of defects increases. Eventually, the spiral core will annihilate with the sidewalls and again SDC will occur. Lastly, regular hexagons break down to bistable domains of stripes and hexagons relatively close to the onset. The stripes will then ﬁll the pattern and the transition to SDC proceeds as it does for the case of parallel stripes at the onset. Each of these onset planforms and transitions to SDC are qualitatively similar to those reported for standard RB convection for Pr ∼ = 1 as illustrated in Sections 4.11 and 4.12. Harmonic patterns qualitatively similar to those described above also occur in experiments with square lateral walls (Figure 12.6f).

12.2.2 Purely Subharmonic Patterns At the subharmonic onset (dashed line in Figure 12.5), striped patterns are typically observed. In the majority of cases, parallel stripes occur (Figure 12.6g), although signiﬁcant sidewall forcing can lead to spirals (Figure 12.6h). In proximity of the onset, dislocations and giant convex disclinations (Figure 12.6i) are commonly found. When the distance from the onset is increased, however, more point defects (dislocations and disclinations) enter the subharmonic pattern until the abrupt occurrence of transverse modulations (Figure 12.6j) that propagate from the sidewalls down the length of the rolls. Two and three sidewall foci are common in transverse modulated stripes. Continuing through the region of transverse modulations (Figure 12.5), the base striped pattern is gradually replaced by radial stripes (Figure 12.6k). Eventually, the radial stripes undergo instability and plumes form throughout the experiment. With square sidewalls, parallel rolls (Figure 12.6l) are typically observed both near to and far from the primary onset.

12.2.3 Coexistence and Complex Order As repeatedly explained earlier, coexisting harmonic and subharmonic modes can interact and lead to several novel patterns in the coexistence parameter range. In the experiments by Rogers and co-workers, coexisting patterns were found over a wide parameter range bounded by the harmonic and subharmonic linear stability curves (Figure 12.5). These patterns can be split into three broad categories: harmonic dominated, subharmonic dominated and those with relatively equal power spectrum contributions from both temporal responses (Figure 12.7). Harmonic dominated coexistence can be found close to the harmonic parameter region: to the left of the bicritical point in Figure 12.5. Interestingly, according to this ﬁgure, when the system moves from the region of harmonic patterns into coexistence, subharmonics do not appear as soon as the subharmonic stability curve (dashed line) is crossed. Initial subharmonic formations are localized as rolls pinned perpendicular to the sidewalls or forming as patches about defects in the harmonic pattern. Moving further into the coexistence region, subharmonic stripes begin to form perpendicular to the base harmonic stripes (Figure 12.7a). The harmonic component in these patterns contributes more than 60% of the pattern’s total spectral power at an average q H approximately the same as that of nearby purely harmonic states. The subharmonic contribution to the patterns grows slowly over this parameter region and is at an average wavenumber q S that is higher than that of purely subharmonic patterns (Rogers et al., 2003, 2005). Subharmonic dominated coexistence occurs near the subharmonic parameter region (Figure 12.5). Unlike the harmonic–coexistence boundary discussed earlier, harmonics appear in

530

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

(c)

(d)

(e)

(f)

Figure 12.7 Comparison between representative coexisting patterns at Pr = 0.93 and # ∼ = 98 observed in experiments [(a)–(c)] and simulations [(d)–(f)]. Harmonic dominated coexistence (a) at Ra = 6679, Fr = 3.50 × 10−4 ; subharmonic dominated coexistence (b) at Ra = 6660, Fr = 3.76 × 10−4 ; square quasi-periodic crystal (c) displaying domains with different orientations at Ra = 6560, Fr = 3.55 × 10−4 ; harmonic dominated coexistence (d) at Ra = 6600, Fr = 3.54 × 10−4 ; subharmonic dominated coexistence (e) at Ra = 6800, Fr = 4.061 × 10−4 ; square quasi-periodic crystal displaying two orientation domains (f) at Ra = 6600, Fr = 3.748 × 10−4 . After Rogers et al. (2003); Reproduced by permission of IOP Publishing Limited

the immediate vicinity of the harmonic marginal stability curve (solid line in Figure 12.5) and are found to be present throughout the subharmonic base state (Figure 12.7b). The emerging harmonic component typically is responsible for the formation of parallel stripes containing several domains of different orientations. Moreover, paralleling the emergence of a harmonic component is the gradual increase in q S from its value to the transition to coexistence. At this subharmonic–coexistence boundary q H is again similar to the values displayed by purely harmonic and coexistence patterns near the harmonic–coexistence boundary. Moving further into the coexistence region, q H steadily decreases. Sufﬁciently far into the coexistence region, spectral power contributions from harmonic and subharmonic patterns abruptly become relatively comparable and complex order patterns arise. Correspondingly, q H will reach its minimum whereas q S attains its maximum as well-deﬁned modes will start to dominate the power spectrum. At the beginning, such patterns exhibit domains of ordered structures with various orientations (Figure 12.7c). Near the transition to complex order (Figure 12.5), such domains continually change as defects emerge and annihilate, altering the domain orientations.

Hybrid Regimes with Vibrations

531

All of the experimental patterns described in these transitions have been reproduced in the numerical simulations performed by these authors: examples at similar parameter values to the experimental patterns are shown in Figure 12.7d, e and f. Moving further into the region of complex order (Figure 12.5) and away from the transition boundary, the ordered domains coalesce into a single domain featured by a precise mode locking of the wavenumbers (q S /q H ∼ = 3). By virtue of the isotropy of the considered system, the single domain does not exhibit a preferred orientation. An example of such a square superlattice (SQS) found in simulations with periodic boundary conditions is shown in Figure 12.8a. This pattern can be regarded as a superlattice since the harmonic and subharmonic spectral modes require solely two indexing vectors to map out the dominant modes [the pattern, in fact, is periodic with respect to translations by only two lattice sites in the square (harmonic) sublattice]. These SQS states have been typically observed in numerical simulations for a range of Fr from Ra 1.38 Racr2 up to Ra = 2 Racr2 (Figure 12.5). Over the SQS region q H remains relatively H H to 0.94 q2c whereas q S decreases monotonically with increasing constant with values from 0.91 q2c S S ∼ at Ra ∼ Ra from 0.92 q2c at Ra = 1.38 Racr2 to 0.78 q2c = 1.96 Racr2 . For 1.2 Racr2 Ra 1.38 Racr2 and over a narrow range of Fr, the uniform square H sublattice found in SQS, however, is replaced by an experimental disordered sublattice containing domains of locally square, rhombic and hexagonal symmetries (dashed circle in Figure 12.5) (Rogers et al., 2000b). Near the bicritical point, simulations (Rogers et al., 2000a) also revealed the existence of another supercritical lattice state for Fr Fr2c . In this case, the initial diffusive state becomes unstable to pure H rolls at the onset. With increasing Ra, a secondary instability leads to a state that Rogers et al. (2000a) named rolls superlattice (Figure 12.8b). It is (at least linearly) stable over a narrow parameter range before the transition to SQS at larger Ra. For instance, at Fr = 3.732 × 10−4 roll superlattices have been observed for 1.033 Racr2 Ra 1.079 Racr2 . There exists partial bistability with SQS; for example, SQS can arise for Ra 1.063 Racr2 at Fr = 3.732 × 10−4 . Patterns qualitatively similar to SQSs obtained via numerical simulation, which require more than two spectral indexing vectors, were observed in experiments (Figure 12.8c). In view of the arguments provided earlier, they have to be referred to as square quasi-periodic crystals.

(a)

(b)

(c)

Figure 12.8 Complex ordered patterns observed for Pr = 0.930 and # ∼ = 98: square superlattice (simulation) (a) at Ra = 4750, Fr = 3.75 × 10−4 ; roll superlattice (simulation) (b) at Ra = 4795, Fr = 3.732 × 10−4 ; square quasi-periodic crystal (experiment) (c) at Ra = 7030, Fr = 3.88 × 10−4 . After Rogers et al. (2003); Reproduced by permission of IOP Publishing Limited

532

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 12.9 Complex ordered patterns observed for Pr = 0.930. Rhombic quasi-periodic crystal (experiment) (a) observed at Ra = 5180, Fr = 8.92 × 10−4 and # = 50.4; qualitatively similar pattern found in simulations (b) at Ra = 3800, Fr = 16.74 × 10−4 and # = 33 with a characteristic spectral structure composed of four harmonic and four subharmonic modes. After Rogers et al. (2003); Reproduced by permission of IOP Publishing Limited

It should be pointed out that all of the observed quasi-periodic crystals and superlattices in this system rely on qualitatively different four-wave resonances (resonant tetrads) to form. These resonant tetrads always involve two harmonic and two subharmonic modes. Finally, it is worth highlighting that a variety of other complex-ordered patterns exist at # values away from # ∼ = 102 in part because q S is strongly dependent on #. Some beautiful examples are shown in Figures 12.9 and 12.10.

Figure 12.10 Another form of the square quasi-periodic crystal (simulation) found for Pr = 0.93, Ra = 7800, Fr = 1.167 × 10−4 and # = 300, in this case displaying four harmonic spectral modes and 12 subharmonic modes. After Rogers et al. (2003); Reproduced by permission of IOP Publishing Limited

Hybrid Regimes with Vibrations

12.3

533

RB Convection with Horizontal or Oblique Shaking

The studies discussed in the preceding section were all related to the case of gravity and vibrations acting along the same direction (it is worth recalling that the case of an extended horizontal layer in the presence of a periodic vertical acceleration has been of special interest since it preserves the property of horizontal isotropy). Some theoretical analyses, however, have also appeared in which a horizontal ﬂuid layer periodically accelerated in its plane has been considered. Jacqmin (1990) studied the stability of a stratiﬁed ﬂuid in a periodic gravitational ﬁeld, either parallel or perpendicular to the density gradient; he addressed the linear stability of a base state in which the velocity ﬁeld oscillated in time with the same frequency as the imposed gravitational ﬁeld. With the Boussinesq approximation and for the case of an unbounded ﬂuid, transformation to Lagrangian coordinates was used to reduce the problem to an ordinary differential equation for each three-dimensional wavenumber. The problem was studied as a function of three parameters: the nondimensional amplitude of the base-state oscillation, the nondimensional level of background steady acceleration and the Prandtl number Pr. Thevenard and Ben Hadid (1991), assuming low Grashof numbers and low Prandtl numbers, compared a linearized analysis with fully nonlinear simulations for two-dimensional convection between vertical adiabatic walls with horizontal vibrations perpendicular both to the walls and to the imposed temperature gradient. Interesting experiments along the same lines were carried out by Ishikawa and Kamei (1993). They also simulated their experiments using a model similar to that originally introduced by Lorenz (see Section 1.8.3). As the frequency was increased at ﬁxed modulation amplitude at values of Ra of about Racr , they found transitions from two-periodic to quasi-periodic and chaotic ﬂows. Some studies also appeared treating these problems in the limit of Gershuni’s formulation (vibrations with high frequency and small displacement; for the necessary theoretical background, see Sections 8.2–8.4). As an example, stability borders in terms of Ra and Rav for different inclination angles of the vibration axis were provided by Cisse et al. (2004). In particular, they investigated the possible existence (in the presence of the oscillating force) of quasi-equilibrium solutions, that is, states in which the mean velocity is zero, but the pulsational component is not in general (states in which the time-averaged body force is compensated by the pressure gradient; see also related concepts given in Chapter 8). For the inﬁnite layer, they recovered the results of Gershuni and co-workers (see Figure 12.11, θ being, as usual, the angle between the imposed vertical temperature gradient and the vibration direction). Obviously, the critical point on the Ra = PrGr axis of this ﬁgure is the solution of the classic RB problem (as illustrated in Chapter 4, this critical Rayleigh number Racr = 1707 is connected with the most dangerous perturbation of wavenumber qc = 3.117). The points on the Rav axis correspond to the case of pure weightlessness. In the range of inclination angles, the stability curves intersect the Rav axis and go to negative Ra (heating from above). This means that (as expected) a speciﬁc vibrational mechanism of instability initiation also operates during heating from above. According to this ﬁgure, mechanical equilibrium (a state in which the time-averaged velocity is zero, i.e. time-averaged thermal diffusive conditions are maintained) is always possible for arbitrary inclinations: for each inclination, in fact, a well-deﬁned threshold (for the onset of time-averaged convection) can be identiﬁed. Interestingly, the critical Rayleigh number Ra behaves as an increasing function of Rav for θ < θc (π/6 < θc < π/4, θc = 23π/100), which means that under such conditions high-frequency vibrations tend to exert a stabilizing effect on the classical RB convection, whereas for θ > θc

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Thermal Convection: Patterns, Evolution and Stability

Figure 12.11 Stability borders in the plane (Ra, Rav ) for different inclination angles of the vibration axis [inﬁnite layer; θ is the angle between the imposed vertical temperature gradient and the vibration direction; gravitational and vibrational Rayleigh numbers deﬁned as Ra = gβT T d 3 /να and Rav = (bωβT T d)2 /2να , respectively]. After Cisse et al. (2004); Copyright Elsevier, 2004

Ra decreases as Rav is increased, which means that vibrations have a destabilizing effect (in particular, the destabilizing effect becomes stronger as the inclination of vibrations is increased towards θ = π/2, i.e. when vibrations tend to become perfectly parallel to the walls of the layer). Obviously, for the case in which there is no steady gravity (Ra = 0), one recovers the results for pure thermovibrational ﬂow, that is, for θ = π/2, time-averaged convection emerges for Rav = 2129, which is the minimal value possible; this threshold increases as θ is decreased and tends ideally to inﬁnite as θ → 0 (in agreement with the ﬁndings of Gershuni and Zhukhovitskii, 1981; see Figure 8.5 in Section 8.5.3). At this stage, it is worth noting that very recently, Pesch et al. (2008) presented a new theoretical study based on a multi-technique approach to the problem. Starting from the interesting remark that for such a case (shaking in a ﬁxed horizontal direction) the original isotropy of the layer is broken as in the case of inclined layer convection treated in Chapter 7, they performed a linear stability analysis, a weakly nonlinear analysis and direct numerical solution of the thermal-convection equations written in the framework of the Boussinesq approximation, for which detailed consideration (given in the text following) is really worthwhile. They considered horizontal vibrations (θ = π/2) and relatively small values of # for which the averaged formulation is no longer applicable. The velocity ﬁeld V was decomposed as follows: V = (RaGx Vx + Ux )i x + (RaGz Vz + Uz )i x + v

(12.6)

with v = V − V , where, as usual, the overbar indicates the horizontal average, and Gx =

bx ω 2 g0

(12.7a)

Gz =

bz ω 2 g0

(12.7b)

Hybrid Regimes with Vibrations

535

bx and bz being the displacement components along the x and z directions, respectively (gravity parallel to the negative y direction). Assuming an inﬁnitely extended layer, they used the analytical expressions for Vx and Vz , previously determined for ﬁnite frequencies by Gershuni et al. (1996): Pr [y cos(#t) + Y1 cos(#t) + Y2 sin(#t)] # Pr Vz (y, t) = [−y sin(#t) − Y1 sin(#t) + Y2 cos(#t)] #

Vx (y, t) =

(12.8a) (12.8b)

where Y1 =

cosh(l) cos(m) − cosh(m) cos(l) 2[cos(n) − cosh(n)]

(12.9a)

Y2 =

sinh(l) sin(m) − sinh(m) sin(l) 2[cos(n) − cosh(n)]

(12.9b)

and

l=n m=n . n=

1 +y 2 1 −y 2

(12.10a)

# 2 Pr

(12.10b)

(12.10c)

It is worth noting that in the limit # → 0, such a solution for the basic ﬂow corresponds to a time-periodic cubic shear proﬁle: 1 " 3 y# y − sin(#t) (12.11a) Vx (y, t) = 6 4 1 " 3 y# y − cos(#t) (12.11b) Vz (y, t) = 6 4 that exhibits a clear similarity to the time-independent solution for inclined layer convection [Eq. (2.60); see Section 2.4.6]. In this case, in fact, a base ﬂow proﬁle that is cubic in y also exists. The equations for the components Ux and Uz accounting for non-zero time-averaged contributions to the velocity ﬁeld were obtained by horizontal averaging of the momentum equation. The general toroidal–poloidal decomposition was used (see Section 4.2.1 for further details about the derivation of the related governing partial differential equations), leading to [∇ 2 2 ϕ + Gx sin(#t)∂z − Gz cos(#t)∂x ] Pr = ξ · [(ηφ + ξ ϕ) · ∇(ηφ + ξ ϕ)] +[(RaGx Vx + Ux )∂x + (RaGz Vz + Uz )∂z + ∂t ]2 ϕ −{[∂y (RaGx Vx + Ux )]∂z − [∂y (RaGz Vz + Uz )]∂x }2 φ

(12.12a)

2 2 [∇ 4 2 φ + Gx sin(#t)∂xy + Gz cos(#t)∂yz − 2 ] Pr = η · [(ηφ + ξ ϕ) · ∇(ηφ + ξ ϕ)]

+[(RaGx Vx + Ux )∂x + (RaGz Vz + Uz ) ∂z + ∂t ]∇ 2 2 φ 2 2 (RaGx Vx + Ux )]∂x + [∂yy (RaGz Vz + Uz )]∂z }2 φ −{[∂yy

(12.12b)

with no-slip conditions at y = − 1/2 and 1/2 and periodic boundary conditions at the lateral boundaries.

536

Thermal Convection: Patterns, Evolution and Stability

Linear stability results were obtained in the speciﬁc case of vibrations acting along the x-axis (Gz = 0) with regard to the onset of both longitudinal and transverse rolls (rolls aligned parallel to the direction of vibration and with their axis perpendicular to this direction, respectively). Longitudinal roll solutions with wavevector q = (qx = 0, qz = p) were found to be time independent. In practice, this solution is identical with the two-dimensional RB solution in the absence of shaking except for an additional oscillating component ux (y, z, t) of u arising from the toroidal potential (its sinusoidal dependence on time reﬂects the buoyancy of the horizontal acceleration and the advection of the vertical velocity component of the longitudinal rolls by the oscillatory base ﬂow). Moreover, the onset of these rolls does not depend on the parameters G and # and simply reﬂects the aforementioned critical values for canonical unmodulated RB convection Racr = 1707 (qc = 3.117) indicated by the thin short-dashed line in Figure 12.12. Unlike longitudinal rolls, transverse roll solutions with q = (qx , 0) were found to display a neutral and a critical wavenumber qx , which become complicated functions of the parameters Gx , Pr and #. Figure 12.12 shows the critical Rayleigh numbers Ra for the onset of such transverse rolls plotted as functions of Gx and # for Pr = 1. The curves for Ra always start at Gx = 0 with the RB value of Ra = 1707 and increase with increasing Gx as long as # is sufﬁciently small (this is due to a stabilizing inﬂuence exerted by the shear related to the basic symmetry-breaking ﬂow; at the same time, the wavenumber q decreases in order to diminish the effect of the shear on the convection rolls; Pesch et al., 2008). Nevertheless, for sufﬁciently high values of Gx , the time-dependent shear becomes destabilizing at Prandtl numbers of the order of unity or less and leads to decreasing values of the critical Ra with increasing Gx . As shown in Figure 12.12, the transition from the stabilizing to the destabilizing role of the shear (point corresponding to the maximum of the curves) is shifted to increasing values of Gx with increasing #.

Figure 12.12 Critical Rayleigh numbers Ra for the onset of transverse roll convection at Pr = 1 as functions of Gx for different values of # as indicated; also indicated by the thin short-dashed line is the value Racr = 1707 for the onset of longitudinal rolls. After Pesch et al. (2008); Reproduced by permission of Cambridge University Press

Hybrid Regimes with Vibrations

537

For each #, moreover, there exists a value G∗ (corresponding to the intersection of the considered curve with the thin short-dashed line) such that if G > G∗ → Racr < 1707 (whereas for G < G∗ the effect of vibrations is stabilizing). For values of # in excess of 50, however, this scenario is modiﬁed in that Racr < 1707 already at small values of Gx , as can be seen for # = 53. Notably, similar results were obtained in the earlier linear stability analysis carried out for ﬁnite frequencies by Gershuni et al. (1996), where it was clearly highlighted that in contrast to the case # → ∞ in which the vibrational mechanism with vibrations perpendicular to gravity always exerts a destabilizing inﬂuence on RB convection (see the branch for θ = π/2 in Figure 12.11), at ﬁnite vibration frequencies destabilization and stabilization of the equilibrium state due to parametric generation are both possible, the choice between the two outcomes depending on amplitude and frequency. As already outlined, Pesch et al. (2008) went beyond such earlier study by also considering a weakly nonlinear analysis. With this approach, they examined the stability of primary (longitudinal or transverse) patterns established at the onset. The stability of longitudinal rolls, in particular, was found to be dominated near onset by a wavy instability. According to such results, longitudinal rolls at moderate values of Pr become unstable almost at onset, that is, at very small values of Ra/Racr − 1. Such a wavy instability of longitudinal rolls is induced by the steady component of shear ﬂow. It is worth recalling along these lines that similar results were found for inclined layer convection (Clever and Busse, 1977), as already illustrated in Section 7.1. Concerning transverse rolls, it is obvious that they are unstable against the longitudinal rolls in the case of unidirectional shaking for not too large Gx since their critical Rayleigh number is larger than 1707. Conversely, for larger Gx , transverse rolls tend to be stable near the onset. Pesch et al. (2008) also investigated these behaviours by means of direct numerical simulation. Since the direct simulations consume large amounts of computer resources, in particular, they focused on a speciﬁc parameter combination (Pr = 1, Gx = 3.6 and # = 50) for which competition between longitudinal and transverse modes should be expected. According to Figure 12.12, for such a case the longitudinal rolls bifurcate at Ra = 1707 with q c = (0, pc ) and pc = 3.117 whereas the onset of transverse rolls is at Ra = 1773 for q c = (2.27, 0). First, numerical simulations were carried out for Ra ∼ = 1800 where, according to the preceding weakly nonlinear analysis, longitudinal rolls should be unstable against the wavy instability. Indeed, starting the simulations from random initial conditions they did not ﬁnd longitudinal rolls. Instead, the system was found to lock into an oblique roll pattern with wavevector q with |q| = 2.6 which includes an angle χ = π/4 with the x-axis (see Figure 12.13a).

(a)

(b)

(c)

(d)

Figure 12.13 Representative snapshots of the temperature modulation in the (x, z) plane showing the route towards a chaotic dynamic state in which patches with wavevectors oriented mirror symmetrically with respect to the z-axis coexist (Pr = 1, # = 50, Gx = 3.6, Gy = 0, Ra = 1.05 × Racr = 1862, qc = 2.32. After Pesch et al. (2008); Reproduced by permission of Cambridge University Press

538

Thermal Convection: Patterns, Evolution and Stability

They conﬁrmed the stability of this pattern for Ra 1800 by the Galerkin analysis. At Ra = 1810, however, the pattern was observed to become unstable against long-wavelength modulations with a wavevector S which includes with q an angle ψ slightly less than π /4. To study the impact of this instability, they performed a numerical simulation at Ra = 1862; it was found that when the simulation is started with the oblique roll pattern obtained at Ra = 1800 and superimposed noise, at the beginning the modulational instability becomes visible (see Figure 12.13b); continuing the simulation, however, a steady dynamic state is established which is characterized by the coexistence of patches with q vectors that are oriented mirror symmetrically with respect to the z direction (as demonstrated in Figure 12.13c and 12.13d, the patches change their shape and location chaotically). To recapitulate, according to a critical comparison of ﬁndings obtained in the framework of several possible approaches to the problem, the following scenario can be deﬁned (at least for the speciﬁc case Pr = 1 considered by these authors): At onset of convection and at small acceleration, longitudinal rolls are the expected pattern (where the roll axis aligns parallel to the acceleration direction); with increasing acceleration amplitude, a shear instability takes over and transverse rolls with the axis perpendicular to the shaking direction nucleate at onset. In the nonlinear regime, the longitudinal rolls tend to become unstable against transverse modulations very close to the onset, which leads to a kind of domain chaos between patches of symmetry-degenerated oblique rolls.

12.4

Laterally Heated Systems and Parametric Resonances

12.4.1 The Inﬁnite Horizontal Layer There have also been studies on the effect of time-modulated gravity (or vibrations) on systems with basic buoyant convection induced by vertical static gravity and horizontal temperature gradients. The simplest model for this kind of ﬂow is the canonical layer of inﬁnite horizontal extent already examined as a paradigm reference case in Chapter 6 for pure buoyancy. By analogy with the analogous problem treated in Section 12.3 for mixed RB–vibrational convection, let us start the discussion by observing that under the effect of high-frequency vibrations (in the framework of the Gershuni’s formulation) this system admits solutions corresponding to quasi-equilibrium, that is, states in which the mean velocity is zero. As already illustrated in Section 8.5 for pure thermovibrational ﬂow, these states are made possible by a perfect balance of the static component of the body force and pressure gradient established in the liquid. The related mathematical (necessary) conditions of existence can be obtained setting V = 0 in the averaged momentum equation, assuming that the equilibrium ﬁelds T 0 , w0 and p0 do not depend on time and taking the curl of the resulting equation. This leads to ˆ ∧ ∇T 0 + Ra∇T 0 ∧ i g = 0 Rav ∇(w0 · n)

(12.13a)

∇2T 0 = 0

(12.13b)

where, as usual, nˆ = cos(θ )i x + sin(θ )i y is the unit vector along the direction of vibrations [here we assume both the temperature gradient and the vibrational axis to lie in the xy plane, θ being the angle between them, Ra = gβT γ d 4 /να and Rav = (bωβT γ d 2 )2 /2να, respectively, with γ being the uniform rate of temperature increase along the x-axis].

Hybrid Regimes with Vibrations

539

Such equations, supplemented with those for the auxiliary function w [Eqs (8.16)], are generally known as vibrational hydrostatic conditions. For the speciﬁc case considered in this section, that is, vertical steady gravity and horizontal temperature gradients ∇T 0 = i x , they reduce to Rav cos(θ ) sin(θ ) − Ra = 0

(12.14)

When requirements for mechanical quasi-equilibrium are not satisﬁed, time-average ﬂuid motion arises. Such convective states are two-dimensional and admit analytical solution in the form of plane-parallel ﬂow. Following the general concepts introduced in Section 2.4, these solutions can be represented as   u0 (y) V0 =  0  (12.15) 0 T o = x + f (y)

(12.16)

p0 = p0 (x, y)   w0 (y) w0 =  0  0

(12.17) (12.18)

with u0 (y)andf (y) known polynomial expressions (Birikh, 1990): $ R1 cosh(2ξy) sin(2ξy) sinh(ξ ) cos(ξ ) − sinh(2ξy) cos(2ξy) cosh(ξ ) sin(ξ ) u(y) = − 16 ξ 3 [sinh(ξ ) cosh(ξ ) + sin(ξ ) cos(ξ )] (12.19) $ 1 R1 sinh(2ξy) cos(2ξy) sinh(ξ ) cos(ξ ) + cosh(2ξy) sin(2ξy) cosh(ξ ) sin(ξ ) f (y) = 2y − 2 R2 ξ [sinh(ξ ) cosh(ξ ) + sin(ξ ) cos(ξ )] (12.20) for the case with adiabatic walls and $ R1 cosh(2ξy) sin(2ξy) sinh(ξ ) cos(ξ ) − sinh(2ξy) cos(2ξy) cosh(ξ ) sin(ξ ) u(y) = − 16 ξ 2 [sinh2 (ξ ) cos2 (ξ ) + cosh2 (ξ ) sin2 (ξ )] (12.21) 1 R1 sinh(2ξy) cos(2ξy) sinh(ξ ) cos(ξ ) + cosh(2ξy) sin(2ξy) cosh(ξ ) sin(ξ ) 2y − f (y) = 2 R2 sinh2 (ξ ) cos2 (ξ ) + cosh2 (ξ ) sin2 (ξ ) (12.22) for the case with conducting walls (see Figure 12.14), where 1 1 ξ = √ (R2 ) 4 2 2

(12.23)

R1 = Rav cos(θ ) sin(θ ) − Ra

(12.24)

R2 = Rav cos (θ )

(12.25)

2

Like other plane-parallel ﬂow solutions (see, for instance, Section 6.1 for similar concepts elaborated for the case of pure buoyancy ﬂow, i.e. the canonical Hadley ﬂow), this type of convection also can undergo instabilities of two- or three-dimensional nature.

540

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 12.14 Exact solution for the case of mixed thermogravitational–thermovibrational (average) convection in inﬁnite layer with top and bottom solid conducting walls (θ = 0◦ , Ra = 1): (a) velocity proﬁle u(y); (b) temperature proﬁle f (y)

Following the ideas illustrated in Chapter 1 about the linear stability theories and related tools of analysis, generic 3D disturbances potentially arising on such a basic state can be represented as 

 ud (y) δV =  vd (y)  eλt ei(qx x+qz z) wd (y)

(12.26)

δT = Td (y)eλt ei(qx x+qz z)

(12.27)

δp = pd (y)eλt ei(qx x+qz z)   $1d (y) δw = $2d (y) eλt ei(qx x+qz z) $3d (y)

(12.28) (12.29)

where qx and qz are the wavenumbers for the x and z directions, respectively, and the quantities with the subscript d are the disturbance amplitudes (see Section 1.5.3). Substituting these disturbances into Eqs (1.59)–(1.61) [supplemented with Eqs (8.16) for the auxiliary potential function introduced by the Gershuni’s formulation] and linearizing with respect to the perturbation quantities (i.e., neglecting all products and powers of the increments higher than the ﬁrst while retaining only terms that are linear in them), the linear-stability problem reduces to ∇ · (δV ) = 0 ∂ ∂t δV

(12.30)

+ V 0 · ∇(δV ) + δV · ∇(V 0 ) + ∇(δp) = Pr ∇ 2 (δV ) + δF b

(12.31)

+ V 0 · ∇(δT ) + δV · ∇(T 0 ) = ∇ 2 (δT )

(12.32)

∂ ∂t δT

∇ 2 (δw) = −∇ ∧ [∇(δT ) ∧ n] ˆ

(12.33)

Hybrid Regimes with Vibrations

541

where δF b (the change induced in the body forces acting on the system by the disturbance) can be determined taking into account that for the present case the body force is given by the sum of the contributions related to static gravity − Pr RaT i g and vibrations Pr Rav [(w · ∇T )nˆ − w · ∇w]; hence, following the general concepts explained in Section 1.5.3, one obtains δF b = − Pr RaδT i g + Pr Rav {[w0 · ∇(δT ) + δw · ∇T 0 ]nˆ − [w0 · ∇(δw) + δw · ∇w 0 ]} (12.34) Then, the eigenvalue problem can be formulated as a system of ordinary differential equations for the amplitudes ud , vd , wd , pd , Td and $1d , $2d and $3d as follows: Continuity equation: i(qx ud + qz wd ) + vd = 0

(12.35)

Pr(ud − q 2 ud ) − (iqx u0 ud + u0 vd ) − iqx pd + Pr Rav {[cos(θ ) − iqx w0 ]$1d ' +[cos(θ )f − w0 ]$2d + iqx cos(θ )w0 Td = λud

(12.36a)

Momentum:

Pr(vd

− q vd ) − 2

(iqx u0 vd ) − pd

+ Pr RaTd

+ Pr Rav {sin(θ )$1d + [sin(θ )f − iqx w0 ]$2d + iqx sin(θ )w0 Td } = λvd Pr(wd − q 2 wd ) − (iqx u0 wd ) − iqz pd − Pr Rav (iqx w0 $3d ) = λwd

(12.36b) (12.36c)

Energy: Td − q 2 Td − (iqx u0 Td + ud + vd f ) = λTd

(12.37)

Auxiliary potential function:

where

q2

=

qx2

+

− q 2 $1d = cos(θ )Td − iqx sin(θ )Td − qz2 cos(θ )Td $1d

(12.38a)

− q 2 $2d = −iqx cos(θ )Td − q 2 sin(θ )Td $2d

(12.38b)

$3d

(12.38c)

qz2 ,

− q $3d = 2

−iqz sin(θ )Td

+ qx qz cos(θ )Td

with boundary conditions at y = ± 1/2: ud = 0 vd = 0 wd = 0

(12.39a) (12.39b) (12.39c)

$1d = 0 $2d = 0 $3d = 0

(12.40a) (12.40b) (12.40c)

for velocity, and

for the auxiliary potential function. As usual, these conditions must be supplemented with the thermal ones; such conditions read Td = 0

(12.41a)

Td = 0

(12.41b)

for adiabatic boundary and

for conducting boundary.

542

Thermal Convection: Patterns, Evolution and Stability

Figure 12.15 Minimal Grashof number (Grm = inf qx ,qz {Grcr }) as a function of the Prandtl number for the Hadley ﬂow with top and bottom conducting walls subjected to high-frequency vibrations parallel to the walls: Grv = 1.6 × 104 (solid lines) and no vibrations, i.e. Grv = 0 (dashed line) [gravitational and vibrational Grashof numbers deﬁned as Gr = gβT γ d 4 /ν 2 and Grv = Rav /Pr = (bωβT γ d 2 )2 /2ν 2 , respectively, where γ is the uniform rate of temperature increase along the x axis]. After Birikh and Katanova (1998); Reproduced with kind permission of Springer Science and Business Media

The speciﬁc case in which vibrations are parallel to the temperature gradient (θ = 0◦ ) was investigated by Birikh and Katanova (1998) for conducting boundaries (Eq. 12.41b). Related ﬁndings are summarized in Figure 12.15. Such results show that, in the presence of high-frequency vibrations parallel to the temperature gradient (at least for the case shown in Figure 12.15 with Grv = Rav /Pr = 1.6 × 104 ), the modes of instability affecting the ﬂow are basically the same as already discussed for pure buoyancy ﬂow in Section 6.1: the hydrodynamic two-dimensional disturbances and the OLR mode for Pr < 1 and the Rayleigh modes for Pr O(1). High-frequency vibrations, however, increase signiﬁcantly the stability threshold for all Prandtl numbers (compare the solid and dashed lines) and affect the domain of existence of the different modes (moreover, for different instability modes the stabilizing effect is different). It is worth starting the related description observing that for both pure Hadley ﬂow (Grv = 0) and hybrid ﬂow, results provided by linear stability analysis are in agreement with Squire’s theorem (Section 1.5.4) as in the limit as Pr → 0 the most dangerous disturbances are always planar (both most dangerous dashed and solid branches in proximity of the Grashof axis are related to the two-dimensional hydrodynamic disturbance). Such a behaviour persists for Pr = 0 until the hydrodynamic disturbances are replaced by OLR modes (as the most dangerous ones) when Pr exceeds a value Pr∗ (the value of the Prandtl number at which curves related to hydrodynamic and OLR disturbances intersect). This value is affected by the presence of vibrations. As illustrated in Section 6.1.2 (in the absence of vibrations) Pr∗ = 0.114 for the conducting case; however, it shifts to Pr∗ = 0.07 for

Hybrid Regimes with Vibrations

543

Grv = 1.6 × 104 (which means the region where the hydrodynamic disturbance is the most dangerous mode tends to be reduced when the vibrational Grashof number increases). The same concept applies to the overall region of existence of this mode. As explained in Section 6.1, it is known that the trend of the hydrodynamic instability branch becomes fairly steep when the Prandtl number approaches a certain value (PrL ) representing a limiting value above which hydrodynamic disturbances are suppressed. This value, found to be PrL = 0.2 for the conducting case in the absence of vibrations, becomes 0.1 for Grv = 1.6 × 104 . A similar behaviour holds for the region of existence of the OLR mode. This mode disappears at a value of Pr (PrL2 ) at which the related branch seems to admit vertical asymptotes (which means again it is strongly stabilized or suppressed when such a value is exceeded). The value PrL2 = 0.45 in the absence of vibrations is reduced to PrL2 = 0.34 for Grv = 1.6 × 104 , which clearly indicates that as the vibrational parameter is increased, the extent of the existence domain of this three-dimensional oscillatory disturbance in the direction of high Prandtl number also decreases. At approximately the same Prandtl number, the oscillatory 2D Rayleigh mode becomes the most dangerous. Let us recall that in the absence of vibrations for the range of Prandtl numbers Pr O(1) the Hadley ﬂow with conducting boundaries may undergo an instability of RB type (see Section 6.1.3) that emerge in two relatively thin, unstably stratiﬁed ﬂuid layers in the neighbourhood of the rigid boundaries (the related critical perturbations have a small wavelength and, being localized in the neighbourhood of the edge of the layer in zones with nonzero average velocity, are entrained by the main stream). In the absence of vibrations, these disturbance are known to be of stationary type and lead typically to the emergence of longitudinal rolls (with the axis parallel to the temperature gradient), whereas disturbances of the same type leading to oscillatory transverse (with the axis perpendicular to the imposed temperature gradient) two-dimensional rolls are less dangerous. Vibrations have a double effect on this type of instability. As shown in Figure 12.15, they make the 2D oscillatory Rayleigh mode more dangerous than the 3D stationary mode and extend the existence domain of this oscillatory Rayleigh mode towards lower Prandtl numbers. The 3D stationary Rayleigh mode and the 2D oscillatory mode intersect at Pr = 0.26 and for Pr > 0.26 the latter becomes the most critical. In the remainder of this subsection, for the convenience of the reader, a theoretical framework for explaining the mechanisms underlying the observed stabilization of convection over the whole range of Prandtl numbers is elaborated on the basis of relatively simple physical arguments and guiding principles already developed in earlier chapters (in particular, Chapters 6 and 8). Starting with the remark that in the absence of the static component of the acceleration ﬁeld (i.e. for Gr = 0) the present conﬁguration (θ = 0◦ ) would maintain pure (time-averaged) thermal diffusive conditions and no mean convection (i.e. it would correspond to a stable state of mechanical equilibrium; see Section 8.5), it becomes obvious that, when such horizontal vibrations act in combination with vertical static gravity, they are basically responsible for a mitigation of the basic ﬂow [Eqs (12.21)–(12.25)] with respect to the one that would be induced by steady gravity alone [Eqs (2.34)]. Accordingly, the increase in the threshold for 2D hydrodynamic disturbances can be ascribed to the weakening of the basic ﬂow and, as a logical consequence, of the induced shear, which (Section 6.1.2) is at the root of this type of instability. Similarly, an equivalent explanation for the increase in the threshold for the OLR and Rayleigh modes can be obtained taking into account the thermal effect that vibrations parallel to the imposed temperature gradient exert on the zones of unstable stratiﬁcation produced by the basic plane-parallel ﬂow.

544

Thermal Convection: Patterns, Evolution and Stability

As mentioned at the beginning of this chapter, if temperature distortions are induced by convection, average vibrational ﬂows arise in such a way as to permit the isotherms to turn and again become perpendicular to the vibration direction. This means that the considered vibrations essentially tend to suppress the zones of unstable stratiﬁcation; this leads naturally to a signiﬁcant increase in the critical Grashof number for all the disturbance modes that have a signiﬁcant thermal component.

12.4.2 Domains with Vertical Walls Important studies have also been performed for the case in which the system is limited along the horizontal direction by vertical walls (Forbes et al., 1970; Fu and Shieh, 1992; Farooq and Homsy, 1994, 1996; Liz´ee and Alexander, 1997; Chen and Chen, 1999; Ishida and Kimoto, 2000; Kim et al., 2001, 2002; Yan et al., 2005); these cases are usually referred to as ‘lateral heating’, as illustrated in Chapter 2.2, and generally involve enclosures with vertical walls at ﬁxed temperatures and horizontal walls with adiabatic conditions. Forbes et al. (1970) were the ﬁrst to present an experimental study of the inﬂuence of vibrations on heat transfer and ﬂuid dynamic behaviour of a differentially heated vertical slot. The direction of vibration was perpendicular to the temperature gradient. They found that there is a possibility of resonance, which leads to heat transfer augmentation. Similar results were obtained by Fu and Shieh (1992), who identiﬁed the existence of several possible regimes of heat transfer depending on the vibration frequency and relative importance of vibrational and gravitational ﬂows. According to their results, the possible regimes of modulated thermal convection for lateral heating can be divided into ﬁve regions: (i) quasi-static convection, (ii) vibration convection, (iii) resonant vibration convection, (iv) intermediate convection and (v) high-frequency vibrational convection. They pointed out that the strength of the resonant interactions is dependent on the modulation amplitude. The speciﬁc case of resonant convection and related intrinsic mechanisms was investigated in detail 2 years later by Farooq and Homsy (1994), who considered over a wide range of the Prandtl number a square cavity where a lateral temperature gradient interacts with a constant-gravity ﬁeld modulated by small harmonic oscillations. The Boussinesq equations were expanded by regular perturbation in powers of the parameter G = bω2 /g0 (assumed to be less than 1). The resulting hierarchy of equations was solved by ﬁnite differences to investigate the O(G) and O(G2 ) ﬁelds and their parametric dependence on the Rayleigh number Ra, Prandtl number Pr and forcing frequency #. The problem was examined in the limit as # → 0 and for ﬁnite (relatively low) #. In practice, introducing the stream function ψ, the full time-dependent thermal-convection equations were cast in compact form as ∂T ∂ 2 ∇ ψ + (ψ, ∇ 2 ψ) = Pr ∇ 4 ψ − Pr Ra [1 + G sin(#t)] ∂t ∂x ∂T + (ψ, T ) = ∇ 2 T ∂t

(12.42) (12.43)

∂ ∂m ∂ ∂m with the operator deﬁned as ( , m) = ∂y ∂x − ∂x ∂y and, as mentioned above, in the G 1 limit the unknowns appearing in these equations were expanded in powers of G:

ψ = ψ (0) + Gψ (1) + G2 ψ (2) + · · · T =T

(0)

+ GT

(1)

+G T 2

(2)

+ ···

(12.44) (12.45)

Hybrid Regimes with Vibrations

545

Substitution of these expansions into Eqs (12.42) and (12.43) generated the aforementioned hierarchy of equations to be solved at each order of G [obviously, at O(G0 ) one recovers the steady Boussinesq equations for the stationary gravitational ﬁeld]. It was demonstrated that, under certain parametric conditions of ﬁnite frequency and moderate Pr, resonance (the strength of which increases as Ra increases) is caused by the interaction of convective motion of vibrational origin with the fundamental instabilities of the base ﬂow related to the constant component of the acceleration ﬁeld. According to such a study, in particular, the mechanism for parametric resonance should be regarded basically as a resonance between the forced oscillation of the basic ﬂow and the free oscillations of stable perturbations of the time-averaged ﬂow. In such a context, it should be pointed out that from a theoretical point of view there should exist two potential modes of resonance according to whether resonance occurs with respect to the instability of the lateral boundary layers or with respect to the propagation of internal waves (see Section 6.2.2 for additional details on these mechanisms). Farooq and Homsy (1994) found, however, that the resonant frequency over the ranges 1 Pr 1 102 and 104 Ra 107 scales as (RaPr) /2 like the Brunt–V¨ais¨al¨a frequency [the maximum possible frequency that can be supported by a stable oscillating stratiﬁed ﬂuid; see, in particular, Eq. (6.27)], which provided indirect evidence that resonance occurs with the internal waves. No evidence of any interaction with the boundary-layer instability was found. Moreover, no resonance was found for Pr 1 because there is no stratiﬁcation to begin with and, hence, no internal waves can be formed. In a later study, Farooq and Homsy (1996) conﬁrmed that parametric resonance in a gravity-modulated differentially heated slot occurs in association with excitation of internal waves and leads to instability in the ﬂow. The calculated stability boundary was found to depend on the frequency and magnitude of the modulation amplitude. The minimum value of the modulation amplitude at which instability was found, in a remarkable analogy to the stability boundaries of the Mathieu equation, was shown to correspond approximately to forcing frequencies of 2ωR , ωR , 2ωR /5, and so on, where ωR corresponds to the fundamental resonant frequency of the system. The possible existence of resonant states has been discussed in several other papers, for example, by Chen and Chen (1999) and Kim et al. (2001, 2002), in which the geometry was a rectangular cavity (vibrations perpendicular to the temperature gradient). Kim et al. (2002) found experimentally that the resonant frequency increases with Ra and it is little affected by an increase in the forcing amplitude. Also, in agreement with the earlier ﬁndings by Farooq and Homsy (1996), they showed that the resonant frequency might be well predicted by the Brunt–V¨ais¨al¨a frequency. Other authors (e.g. Liz´ee and Alexander, 1997, and Ishida and Kimoto, 2000, for a square cavity and Pr = 0.71) focused on the system evolution up to the onset of chaos. Liz´ee and Alexander (1997), in particular, considered gravity perpendicular to both the applied temperature gradient and to g-jitter and observed period-doubling transitions, periodic windows, strange attractors and intermittencies as a function of the Rayleigh number with the evolution occurring through a sequence of bifurcations of the Feigenbaum type (see Section 1.8 for further relevant information about this kind of route to chaos). They pointed out that in the absence of vibration, such behaviour would only be expected at values of the Rayleigh number several orders of magnitude higher than those considered in their work (let us recall that in the case of air, i.e. Pr = 0.71, transition to chaotic ﬂow in the differentially heated cavity with horizontal adiabatic boundaries is expected to take place at Rayleigh numbers in excess of 108 , as discussed in Section 6.2.2), thus proving that in the speciﬁc range of nondimensional frequencies considered (10 # 103 ) horizontal vibration of the cavity can lead to an early transition to chaos (notably, in the limit of the Gershuni’s

546

Thermal Convection: Patterns, Evolution and Stability

Figure 12.16 Possible ﬂow regimes in a square cavity as a function of # and the parameter F = #2 = PrRaω (Pr = 0.71 and Ra = 104 ; gravity perpendicular to both the applied temperature gradient and to g -jitter; cavity with adiabatic horizontal walls): I, quasisteady regime; II, oscillatory regime; III, asymptotic regime. Courtesy of J. Iwan D. Alexander

formulation, i.e. large values of #, the effect is expected to be the opposite given the stabilizing action exerted by vibrations parallel to the temperature gradient, as also discussed in Section 12.4.1 for the inﬁnite layer). Their results for Ra = 104 and Pr = 0.71 are summarized in Figure 12.16 as a function of # and the parameter F = #2 = PrRaω . Observed ﬂow regimes included a quasi-steady regime (I), where inertial effects are negligible, an oscillatory regime (II) where the velocity is out of phase with the driving force and an asymptotic regime (III) where inertial effects dominate and the ﬂow oscillates with small amplitude about the mean ﬂow. A fourth regime, which ﬁrst appears at F F∗ = 4.36 × 104 for 106 < # < 112, was found to be characterized by subharmonic cascades and chaotic behaviour. The edge of a second region of instability was also detected at larger F values for # = 87. These interesting behaviours deserve some additional discussion as developed in the following. For F < F∗ , the response in regimes I–III is characterized by single-frequency oscillations. For F > F∗ , there is a band of frequencies for which marked transitions in the ﬂow are observed. This band increases in width as Rav (and thus F) increases. For a given frequency in this band (regime IV in Figure 12.16), the region of instability is contained between two values of F. As the region is entered from below, there is a sequence of subharmonic bifurcations leading to chaos, which exhibit properties characteristic of a Feigenbaum-type scenario. At the upper limit of the region, the behaviour is typically characterized by intermittencies (these ﬂows show characteristics corresponding to a Pomeau–Manneville type I intermittency). Upon exiting the unstable region, the ﬂow is again monoperiodic and has a frequency equal to the forcing frequency. Liz´ee and Alexander (1997) also considered evaluation of the possible ‘self-similar structure’ related to such bifurcations (i.e. the Lyapunov exponents and the Hausdorff dimension deﬁned in Chapter 1). For a speciﬁc value of # (# = 152) in region IV, the authors detected the possible existence of a strange attractor in the range 5 × 104 Rav 5.8 × 104 . For Rav = 5.8 × 104 , a positive Lyapunov exponent was determined; the existence of a negative Lyapunov exponent was

Hybrid Regimes with Vibrations

547

Figure 12.17 Snapshots of temperature harmonic mode [plane (x, y)] at #t = 0, π/2, 3π/4, π , 3π/2 and 7π/4 (Pr = 0.71, A = 1, # = 152 and Rav = 4.2 × 104 ; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions). A travelling wave moves from the corners toward the centre. Courtesy of J. Iwan D. Alexander

also detected (let us recall that, as explained in Section 1.8.4, a positive Lyapunov exponent is the signature of a chaotic state whereas a negative Lyapunov exponent causes a contraction of the attractor in phase space; their simultaneous existence is characteristic of a strange attractor). An estimate of the attractor’s fractal dimension, the Hausdorff dimension D0 , was computed at two different locations. Liz´ee and Alexander (1997) found D0 = 2.11 and 2.2, which was regarded as a clear signature of fractal behaviour. In the interval 5.9 × 104 < Rav < 6 × 104 , moreover, they observed a period-3 window. Each of the three points within the window was found to undergo a sequence of period-doubling bifurcations and a reverse cascade (this mimics the behaviour of the bifurcation diagram on a larger scale and reﬂects its self-similar nature). As Rav was further increased there were more periodic windows and chaotic regimes. Finally, at Rav = 105 , the aforementioned type I intermittencies were detected. For the speciﬁc value of # = 152 mentioned above, these authors also isolated the ﬂow and temperature modes (using Fourier time-series analysis at each spatial location) and studied in detail the thermal harmonic (#) and subharmonic (#/2) modes. For Rav = 4.2 × 104 , the # mode was found to consist of two waves that travel towards the cavity centre from the upper cold and lower hot corners (see Figure 12.17; the cavity centre is a node and the diagonal connecting the upper hot corner with the lower cold corner is roughly a nodal line). The thermal #/2 mode, in contrast, was found to rotate around the cavity in an anticlockwise sense (Figure 12.18). As Rav is increased above 3.94 × 104 , perturbations to the thermal # mode lead to the transfer of hot and cold packets across the #-mode nodal line. These packets rotate around the cavity, requiring two periods of the #-mode oscillation to complete their cycle. However, when the cavity was driven at # = 76, they observed that the thermal #-mode appears as a standing wave with an approximately vertical nodal line that cuts the cavity in half. They conjectured that, at the ﬁrst period-doubling transition, this lower frequency mode is parametrically excited and the resulting nonlinear interaction with the # mode leads to a disturbance that travels around the cavity.

548

Thermal Convection: Patterns, Evolution and Stability

Figure 12.18 Snapshots of temperature subharmonic mode [plane (x, y)] at #t = 0, π/2, π , 2π , 5π/2 and 3π (Pr = 0.71, A = 1, # = 152 and Rav = 4.2 × 104 ; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions). The disturbance rotates around the cavity in an anticlockwise sense. Courtesy of J. Iwan D. Alexander

12.4.3 The Inﬁnite Vertical Layer This subsection is focused on the speciﬁc case in which the cavity aspect ratio tends to zero (i.e. the differentially heated inﬁnite vertical layer discussed in Section 6.3 for which a basic state with a linear temperature proﬁle known as conduction regime exists; see Figure 6.24). As illustrated in Section 6.3, in the absence of vibrations (only static gravity), for ﬂuids with Pr < 12.45, the primary instability is shear dominated (hydrodynamic) and onsets in a steady convection mode; for ﬂuids with Pr > 12.45, the primary instability is buoyancy dominated and emerges in an oscillatory mode (temperature waves). The same problem with the addition of vertical vibrations was approached in the framework of the averaged formulation by Gershuni and co-workers (Gershuni and Zhukhovitskii, 1988). The stability curves in Figure 12.19 taken from their work describe the interaction of the different instability mechanisms, that is, the hydrodynamic and wave mechanisms on the one hand and the vibrational convective mechanism on the other (see Chapter 8, Section 8.5.3, Figure 8.5 for the case of pure thermovibrational ﬂow). Figure 12.19 clearly shows, in particular, that for the case of vibrations parallel to gravity at all Prandtl numbers their presence leads to destabilization of the ﬂow of gravitational origin (Gr is a decreasing function of Rav over the whole range of Rav and Pr). At the same time, the gravitational effect exerts a destabilizing action of the thermovibrational one for Pr < 0.27, whereas for Pr > 0.27 the effect is stabilizing for relatively low values of Gr (Gr < Gr∗ , with Gr∗ depending on Pr) and then destabilizing for Gr > Gr∗ . For Rav = 0 or Gr = 0, the stability limits reduce to the respective values for the pure cases shown in Figures 6.29 and 8.5, respectively. It should also be pointed out that disturbances for the mixed vibrational–gravitational state are always planar (two-dimensional) as in the respective pure cases.

Hybrid Regimes with Vibrations

549

Figure 12.19 Stability map in the Gr–(Rav )1/2 plane for the inﬁnite vertical layer with walls at different constant temperatures (vibrations parallel to gravity, i.e. θ = 90◦ , where θ is the angle between the temperature gradient and the direction of vibrations; gravitational and vibrational Rayleigh numbers deﬁned as Ra = gβT T d 3 /να and Rav = (bωβT T d)2 /2να , respectively): the solid lines denote steady convection (hydrodynamic) modes, the dashed lines oscillatory modes (temperature waves). After Gershuni and Zhukhovitskii (1988); Reproduced with kind permission of Springer Science and Business Media

Subsequently, Chen and Chen (1999) reconsidered such a problem for ﬁnite frequency. They examined the effect of gravity modulation on the stability characteristics of basic convection for both fundamental categories of ﬂuids with Pr < 12.45 and Pr > 12.45 (in particular, Pr = 1 and Pr = 25, respectively). For Pr = 1, at a relatively small nondimensional oscillation frequency, the critical state was observed to alternate between synchronous and subharmonic modes. At higher frequencies, all critical states were found to occur in the synchronous mode. For Pr = 25, with a modulation amplitude ratio of 0.5, resonant interaction was reported in the neighbourhood of # = 2#c , where #c is the oscillation frequency of the instability at the critical state under steady gravity (this resonant interaction being destabilizing, with the critical Grashof number being reduced by approximately 20% from that at steady gravity). Their analysis of the rate of change of the perturbation kinetic energy revealed that, for Pr = 1, the instability is shear dominated regardless of the mode of oscillation, synchronous or subharmonic. Similarly, for Pr = 25, the instability is buoyancy dominated whether it is in the quasi-periodic or subharmonic mode. Other authors (e.g. Farooq and Homshy, 1996; Christov and Homsy, 2001) examined the convective ﬂow in differentially heated vertical layers with a vertical temperature gradient introduced to mimic the boundary-layer regime. Let us recall that as already illustrated in Chapter 6, when a boundary-layer regime is attained the basic state is no longer represented by the conduction regime cited earlier; rather, the steady base natural convection is featured by a stably stratiﬁed ﬂuid between vertical surfaces maintained at different uniform temperatures. Thin boundary layers exist along the lateral walls, whereas in the

550

Thermal Convection: Patterns, Evolution and Stability

central core the temperature is uniform in horizontal planes and increases in the vertical direction (this means that the buoyancy-driven boundary-layer ﬂow between inﬁnite vertical planes generally involves the presence of both a constant vertical temperature gradient and a constant horizontal temperature difference). Christov and Homsy (2001) took gravity to consist of both a mean and a harmonic modulation at a given frequency and amplitude. The ﬂow was investigated for a range of Prandtl numbers from Pr = 1000, when ﬂuid inertia is insigniﬁcant and only thermal inertia plays a role, to Pr = 0.73, when both are signiﬁcant and of the same order. The presence of g-jitter was found to render the ﬂow susceptible to new modes of parametric instability (not discussed here due to page limits).

12.4.4 Inclined Systems Interesting studies have also appeared where the effect of vibrations was considered for the ‘inclined’ systems of inﬁnite extent considered in Chapter 7 (where gravity may have components in both directions parallel and perpendicular to the applied temperature gradient). In the limiting case of high-frequency vibrations (averaged formulation), the related mathematical conditions for the existence of mechanical quasi-equilibrium read Ra[cos(φT ) sin(φg ) + sin(φT ) cos(φg )] +Rav [sin(φT ) cos(φv ) − cos(φT ) sin(φv )] cos(φT ) cos(φv ) = 0

(12.46)

where φg , φT and φv are the angles between the direction of the solid walls (x-axis) and, respectively, gravity, imposed temperature gradient and vibrations (all assumed to lie in the same plane). The linear stability of these states was investigated theoretically by Birikh (1990), Demin et al. (1996) and Gershuni and Demin (1998). The layer was oriented in an arbitrary direction with respect to the vertical. Each of the two relevant vectors (the temperature gradient and the axis of vibration) was assumed to have one of the four orientations: vertical (i.e. parallel to steady gravity), horizontal (i.e. perpendicular to steady gravity), longitudinal (i.e. parallel to the walls) and transversal (i.e. perpendicular to the walls). Thus a total of 16 situations were studied. The stability of the plane-parallel ﬂows (which are established when quasi-equilibrium is not possible) with respect to inﬁnitesimal disturbances was considered by Shklyaev (2001), Smorodin (2003) and Demin (2005). In particular, Smorodin (2003) examined the stability problem with respect to two-dimensional (qz = 0, i.e. transverse rolls) and spiral (qx = 0, i.e. longitudinal rolls) perturbations. It was shown that, at ﬁnite frequencies, there are parametric instability regions induced by 2D perturbations. Depending on their amplitude and frequency, vibrations were found either to stabilize the unstable ground state or to destabilize the liquid ﬂow, while the stability boundary for spiral perturbations is independent of vibration amplitude and frequency. In the case of high-frequency vibrations, Demin (2005) found that with heating from below the perturbations leading to longitudinal rolls are always the most dangerous for vibrations perpendicular to the layer. For vertical vibration, the stability limit determined by three-dimensional perturbations was observed to depend in a complicated way on the angle of inclination of the layer and the vibrational Rayleigh number. For interesting experiments, the reader may consider Gabdrakhmanov and Kozlov (2002). The above topics are not discussed further here due to page limits.

12.5

Control of Thermogravitational Convection

As anticipated at the beginning of this chapter, the peculiar properties of the thermovibrational effect can be used to elaborate possible strategies for the control of classical thermogravitational

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551

convection (induced by steady acceleration). The next two subsections provide some interesting examples along these lines, the ﬁrst, in particular, with regard to some typical space experiments with transparent (high-Pr) liquids and the latter for some situations of practical interest in the ﬁeld of crystal growth from the melt.

12.5.1 Cell Orientation as a Means to Mitigate Convective Disturbances on Orbiting Platforms The considerations provided in the preceding sections suggest that undesired convective disturbances on orbiting platforms (see Section 2.2.4 for relevant information about the typical acceleration disturbances occurring on the International Space Station) can be weakened by appropriate orientation of the experiment container with the density gradient along the steady residual g or along the high-frequency vibration according to the prevailing effect. The inﬂuence of the facility orientation is discussed here for the same study case considered in the earlier chapters, consisting of a two-dimensional square test cell bounded by rigid walls, completely ﬁlled with liquid, in the presence of a prescribed temperature difference. Although the quantitative results are restricted to this particular study case, conclusions could be extended to different classes of microgravity and also terrestrial experiments (e.g. Bridgman crystal growth, directional solidiﬁcation; see Bouhallab et al., 1996, and Fedoseyev and Alexander, 2000) in which the temperature gradient direction is well deﬁned and for which the facility orientation can be properly changed to reduce undesirable convection disturbances. Figures 12.20–12.23 summarize the numerical results (Gershuni’s formulation) for a reference study case with Pr = 15. Figure 12.20 compares the computed streamlines, vector plots and temperature distributions for the US-Lab and the ESA-COF environments, when only g-jitter is present with direction perpendicular to the temperature gradient (i.e. under the assumption that the steady residual-g is zero and the cell is subjected solely to the vibrations typical of the US-Lab or of the ESA-COF). Since the predicted vibrations are larger for the US-Lab module (shown in Figure 2.2), the convective disturbances and the temperature distortions are larger for the US-Lab. Accordingly, for this case the ﬂow exhibits the secondary pattern of symmetry with a single dominant cell already discussed in Section 8.6 (in contrast, for the ESA-COF module the ﬂow is characterized by the primary structure with four vortex cells). Figure 12.21 shows the numerical results obtained in the other extreme situation that only static residual-g is present (0.5 µg for the US-Lab and 1.6 µg for the ESA-COF) without any g-jitter. The situation is reversed, that is, the convective disturbances are large in the ESA-COF, due to the larger value of the residual gravity. The results obtained assuming that both residual-g and g-jitter are present with direction perpendicular to the temperature gradient are illustrated in Figure 12.22. In this case, the overall convective disturbances are comparable in value, but in the US-Lab the major cause of disturbances is the g-jitter, whereas in the ESA-COF it is the residual-g. If residual-g and g-jitter are not parallel, orienting the experimental cell with the temperature gradient along the residual-g or along the g-jitter can help in mitigating the convective disturbances. Figures 12.23 displays the different thermoﬂuid-dynamic ﬁelds obtained by changing the orientation of the residual-g and g-jitter vectors. Notably, the TFD distortions in the US-Lab are minimized when the cell is oriented with the temperature gradient parallel to the g-jitter direction (main source of disturbances); in this case, the stabilizing effect of the g-jitter can also be beneﬁcial in minimizing the destabilizing effect of a residual-g parallel to the temperature gradient (i.e. like a cell heated from below on Earth).

552

Thermal Convection: Patterns, Evolution and Stability

US LAB

1.00 0.95 0.89 0.84 0.79 0.74 0.68 0.63 0.58 0.53 0.47 0.42 0.37 0.32 0.26 0.21 0.16 0.11 0.05 0.00

g-jitter

Thot

Tcold

Vmax/Vα= 7.15

ESA-COF

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

g-jitter

Thot

Tcold

Vmax/Vα= 0.0021

Figure 12.20 Thermoﬂuid-dynamic ﬁeld [time-averaged ﬁelds, steady state, plane (x, y)] in a square cavity ﬁlled with a high-Pr ﬂuid subjected to typical ISS acceleration disturbances (silicone oil, Pr = 15, A = 1, T = 50 K; US-Lab and ESA-COF disturbances, g -jitter only). After Monti et al. (2001)

In the European COF, the convective disturbances are minimized only when the residual-g vector is antiparallel to the temperature gradient (i.e. stabilizing), even in the presence of an orthogonal g-jitter. These simple demonstrative simulations illustrate how combined steady residual-g and oscillatory g-jitter may be beneﬁcial to approaching pure diffusion conditions on orbiting platforms. These principles can be applied also to the case in which the ﬂuid container has a cylindrical shape (e.g. Monti et al., 1998a). The use of vibrations artiﬁcially imposed on the considered system has been also proposed by many authors as a means to ‘control’ the ﬂow intensity in normal gravity conditions. This subject is treated in the following subsection.

Hybrid Regimes with Vibrations

553

US-LAB (g = 0.5 mg)

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

Residual-g

Tcold

Thot

Vmax / Va = 5.42 ESA-COF (g = 1.6 mg)

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

Residual-g

Tcold

Thot

Vmax / Va = 12.31

Figure 12.21 Thermoﬂuid-dynamic ﬁeld [time-averaged ﬁelds, steady state, plane (x, y)] in a square cavity ﬁlled with a high-Pr ﬂuid subjected to typical ISS acceleration disturbances (silicone oil, Pr = 15, A = 1, T = 50 K; US-Lab and ESA-COF disturbances, static residual-g only). After Monti et al. (2001)

12.5.2 Control of Convection Patterning and Intensity in Shallow Enclosures In this section, some two-dimensional numerical results concerning the possible spatial structure of the ‘mixed’ states of buoyant (static g) and vibrational convection in shallow enclosures are discussed for various possible relative orientations of the imposed T and vibrations (gravity is assumed to be perpendicular to the longest side of the box).

554

Thermal Convection: Patterns, Evolution and Stability

US LAB 1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

g-jitter Residual-g

Tcold

Thot

Vmax / Va = 11.3 ESA- LAB

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

g-jitter Residual-g

Tcold

Thot

Vmax / Va = 12.82

Figure 12.22 Thermoﬂuid-dynamic ﬁeld [time-averaged ﬁelds, steady state, plane (x, y)] in a square cavity ﬁlled with a high-Pr ﬂuid subjected to typical ISS acceleration disturbances (silicone oil, Pr = 15, A = 1, T = 50 K; US-Lab and ESA-COF disturbances, residual-g and g -jitter acting along the same direction). After Monti et al. (2001)

Figures 12.24–12.27, in particular, show the interplay between vibrational (in the limit of Gershuni’s formulation, that is, high frequency and small amplitude of the vibration) and gravitational convection induced by steady gravity for different cases (system heated from the side or from below) in a canonical case in which the presence of lateral walls is expected to play a crucial role in determining the structure of convection [a rectangular cavity with aspect ratio A = 4 ﬁlled with silicon melt (Pr = 0.01), already considered as a paradigm model in earlier chapters]. The related discussion illustrates that the interaction between these two types of convection, apart from its effect on the magnitude of the resulting ﬂow (enhanced or damped according to the relative direction of the vibrations and of the imposed temperature gradient) can also be used as a means to control ‘convection patterning’, that is, the spatial structure of the convection ﬁeld (accordingly, values of the Rayleigh and vibrational Rayleigh number for which the ‘pure’ states exhibit a comparable ﬂow strength are selected for the examples shown in the ﬁgures since it is expected that in

Hybrid Regimes with Vibrations US LAB

ESA-COF

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

Tcold

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

Stabilizing residual-g Orthogonal g-jitter

Thot

Tcold

Vmax / Va = 2.99

Vmax /Va = 0.00175

US LAB

ESA-COF

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

Tcold

555

1.0000 0.9474 0.8947 0.8421 0.7895 0.7368 0.6842 0.6316 0.5789 0.5263 0.4737 0.4211 0.3684 0.3158 0.2632 0.2105 0.1579 0.1053 0.0526 0.0000

Orthogonal residual-g

Tcold

Thot

Thot

Thot

Stabilizing g-jitter

Vmax / Va = 1.67

Vmax /Va = 12.8

Figure 12.23 Thermoﬂuid-dynamic ﬁeld [time-averaged ﬁelds, steady state, plane (x, y)] in a square cavity ﬁlled with a high-Pr ﬂuid subjected to typical ISS acceleration disturbances (silicone oil, Pr = 15, A = 1, T = 50 K; US-Lab and ESA-COF disturbances, residual-g and g -jitter having different orientations). After Monti et al. (2001)

these cases, signiﬁcant ‘competition’ in determining the ﬁnal convection pattern can occur between the different convective modes). In the case of heating from the side and a vertical steady acceleration component, when high-frequency vibration is applied in the vertical direction, a main diagonal cell with two secondary vortices located in the upper left and in the lower right corners, respectively, appears for the considered parameters (Ra = 1 × 103 and Rav = 5 × 104 ; Figure 12.24b). This structure is similar to that obtained in the case of pure thermal gravitational convection (Figure 12.24a), the difference being given by the oblique diagonal orientation of the main roll and by the absence of minor corotating rolls embedded in the overall recirculation pattern (in Figure 12.24b, in practice, two minor cells still exist but they are of a vibrational nature, counter-rotate with respect to the main one and, in addition, are conﬁned to the corners). In Figure 12.24c (vibrations parallel to the temperature gradient), as expected (let us recall that, as discussed in Sections 8.5 and 8.6, in the absence of steady gravity, this situation would

556

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

(c) Figure 12.24 Convection [streamlines, plane (x, y)] induced by coupled steady (vertical) and high-frequency oscillatory (vertical or horizontal) accelerations in a laterally heated enclosure (Pr = 0.01, A = 4, Ra = 1 × 103 and Rav = 5 × 104 ; Ra and Rav based on the depth of the cavity; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions): (a) steady acceleration only (max = 0.678); (b) steady and oscillatory accelerations both acting in the vertical direction (max = 0.62); (c) steady vertical acceleration and oscillatory acceleration acting along the horizontal direction (stabilizing effect, max = 0.134) (numerical simulations, M. Lappa)

correspond to a state of mechanical equilibrium), strong ﬂow mitigation occurs (see the maximum value of the stream function given in the caption). Moreover, the twisted cell with three embedded minor rolls that characterizes the pure thermal gravitational ﬂow in Figure 12.24a is replaced by an elongated cell with straight streamlines almost parallel to the horizontal walls and only two corotating embedded anticlockwise rolls near the left and right edges. The mitigation and the opposite effect are clearly reﬂected in the TFD distortion distribution plotted in Figure 12.25 (εT = 3.55 × 10−2 , 3.6 × 10−2 and 7.3 × 10−3 in Figure 12.25a, b and c, respectively). In the case of RB convection with superimposed high-frequency vibrations (Ra = 2 × 103 and Rav = 5 × 104 ; see Figure 12.26), in an unexpected way, the application of acceleration in the horizontal direction (i.e. in direction perpendicular to the imposed temperature gradient) does not

Hybrid Regimes with Vibrations

557

10 9

14 12

10

5 1

6

5

3

(a) 12

9 14 12

10

5

5 1

7

2

7 (b)

15 12

5 5

15

13

12 6

12

12

9 5

1

4 1

4

(c) Figure 12.25 TFD distortions δT = (T − Tdiff )/T [εT = max(δT )] for the same conditions as in Figure 12.24: (a) level 1 → T = −3.15 × 10−2 , level 15 → T = 3.15 × 10−2 , level = 0.45 × 10−2 , εT = 3.55 × 10−2 ; (b) level 1 → T = −3.15 × 10−2 , level 15 → T = 3.15 × 10−2 , level = 0.45 × 10−2 , εT = 3.6 × 10−2 ; (c) level 1 → T = −6.4 × 10−3 , level 15 → T = 6.4 × 10−3 , level = 0.92 × 10−3 , εT = 7.3 × 10−3

affect the qualitative structure of the velocity ﬁeld (compare Figure 12.26a and b), i.e. the pattern. Thermovibrational ﬂow simply strengthens the intensity of the basic RB mode with four rolls. This is reﬂected in both the stream function distribution and in the temperature ﬁeld where the distortion with respect to the diffusive ﬁeld becomes more signiﬁcant (see Figure 12.27; εT = 4.3 × 10−2 and 4.9 × 10−1 in Figure 12.27a and b, respectively). When the modulation is considered in the same direction of steady gravity (Figure 12.26c), it leads to complete suppression of convection and consequently purely thermal diffusive (average) conditions are established (εT = 0). The effect of vertical vibrations on the threshold of RB convection as a function of the cavity aspect ratio was studied in detail by Cisse et al. (2004). The most interesting outcome of such an analysis was that the inﬁnite layer cannot perfectly represent the physical phenomena observed in long rectangular cavities. According to their study, in fact, when lateral walls (adiabatic in their study) are considered, their contribution is not negligible; the presence of sidewalls, in particular, implies that mechanical equilibrium (discussed in Section 12.3) becomes impossible when the axis

558

Thermal Convection: Patterns, Evolution and Stability

2

2

3

3

12

14

14

13

(a) 5

5 3

12

9 5

8

14

9

(b) 1

1

2

3

5

6

7

8 10

14

12

13

15 (c)

Figure 12.26 Convection and related temperature ﬁeld [plane (x, y)] induced by coupled steady and high-frequency oscillatory accelerations in an enclosure heated from below and cooled from above (Rayleigh–B´enard conﬁguration; Pr = 0.01, A = 4, Ra = 2 × 103 and Rav = 5 × 104 ; Ra and Rav based on the depth of the cavity; vertical walls with adiabatic conditions; level 1 → T = 0.0625, level 15 → T = 0.9375, level = 0.0625): (a) absence of oscillatory acceleration (max = 0.337); (b) oscillatory accelerations acting along the horizontal direction (max = 11.3); (c) oscillatory acceleration acting along the vertical direction (stabilizing effect with suppression of convection) (numerical simulations, M. Lappa)

of vibrations is not along the temperature gradient (i.e. when θ = 0◦ , the convective ﬂow sets in at inﬁnitely small values of temperature difference). When the axis of vibration is perfectly vertical, however, vibrations can still preserve the thermal diffusive regime above the classical threshold of RB convection as in the case of an unbounded layer and clearly proved by Figure 12.26c. In such a case, the mechanical equilibrium is linearly stable up to a critical value of the unique stability parameter, which depends on the vibrational ﬁeld [which means, in agreement with earlier studies, that Cisse et al. (2004) clearly conﬁrmed that high-frequency vertical oscillations with θ = 0◦ can delay convective instabilities and, in this way, reduce the convective ﬂow]. Along these lines, Figure 12.28 clearly proves that the neutral curve for Grcr as a function of the aspect ratio is translated up and dilated horizontally when Grv is increased. For similar studies considering RB convection in vertical cylindrical containers of inﬁnite length with applied temperature gradient and direction of g-jitter both along the axis, the reader may consider Wadih and Roux (1988). Zharinov et al. (1990) and Wheeler et al. (1991) focused on directional solidiﬁcation problems and Murray et al. (1992, 1993) included thermosolutal effects (see also Alexander et al., 1991, 1997).

Hybrid Regimes with Vibrations

6

10

12

4

8

6

11

559

10

1

12 10 3

9

10

5

9 (a) 9

10 14

15 8

7 9

1

14

6

9

9

7 4

1

9

(b)

Figure 12.27 TFD distortions δT = (T − Tdiff )/T [εT = max(δT )] for the same conditions as in Figure 12.26: (a) level 1 → T = −4 × 10−2 , level 15 → T = 4 × 10−2 , level = 0.55 × 10−2 , εT = 4.3 × 10−2 ; (b) level 1 → T = −4.3 × 10−1 , level 15 → T = 4.3 × 10−1 , level = 0.6 × 10−1 , εT = 4.9 × 10−1

Figure 12.28 Critical Grashof number Grcr as a function of the aspect ratio A for several values of Grv (Pr = 1; Grv = Rav /Pr). After Cisse et al. (2004); Copyright Elsevier, 2004

12.5.3 Modulation of Thermal Boundary Conditions In the preceding sections, modulation of gravitational convection has been presented as a situation in which the driving force varies periodically in time. A number of studies, however, have also appeared where ‘modulation’ was introduced in terms of boundary thermal effects. These studies are brieﬂy reviewed in the present section.

560

Thermal Convection: Patterns, Evolution and Stability

Basically, these studies considered a temperature gradient that has both a steady and a time-periodic component (i.e. in addition to a steady temperature difference between the walls, a time-dependent sinusoidal perturbation is applied to the wall temperatures). In general, the consequences of such a modulation are different in comparison to those related to physical vibrations of the overall system, for several reasons: ﬁrst, the acceleration affects the momentum balance equations, whereas the temperature enters into the energy equation; second, gravity modulation (a body force) does not change the (linear) symmetry of the diffusive state, whereas temperature modulation induces a nonlinear diffusive proﬁle that may alter the condition for the onset of convection (in the RB case); third, unlike body forces that affect the overall system, thermal variations on a boundary take some time to propagate through the system and for the high-Pr case they may be ‘ﬁltered’ by the thermal boundary layers. For the speciﬁc case of RB convection, in particular, a variety of effects have been observed, including shifts in the threshold, subharmonic bifurcations over certain ranges of the temperature modulation amplitude and frequency, changes in the nonlinear properties such as the Nusselt number and the patterns above the onset (Venezian, 1969; Rosenblat and Herbert, 1970; Rosenblat and Tanaka, 1970; Yih and Li, 1972; Finucane and Kelly, 1976; Roppo et al., 1984; Ahlers et al., 1985a,b; Niemela and Donnelly, 1987; Swift and Hohenberg, 1987; Or, 2001; Soong et al., 2001; Bhadauria, 2002, 2003; Bhadauria and Bhatia, 2002; Sarris et al., 2002; Semma et al., 2005, 2006). In this context, it is worth mentioning, in particular, the work of Semma et al. (2005), who considered two fundamental cases for the initial state of the melt ﬂow, steady or time dependent, and can therefore be regarded as a good exemplar for this kind of study. They reported the existence of a characteristic modulation frequency allowing the reduction of the average intensity of an initial steady ﬂow. A critical modulation frequency allowing the control of the amplitude of an oscillating ﬂow was also identiﬁed. For the case of heating from the side (systems of the Hadley type), the reader may consider the interesting investigations of Li (1994), Dabiri and Gharib (1996), Antohe and Lage (1996), Kwak et al. (1998), Chung et al. (2001), Shu et al. (2005) and Kalabin et al. (2005a,b). Li (1994) studied experimentally natural convection in vertical slots with cold-wall temperature oscillation. Interestingly, he found that at sufﬁciently large Rayleigh numbers a travelling-wave instability occurs, but only in the region close to the cold wall (when the hot wall is maintained at the initial temperature) and appears in both cold and hot regions when both hot and cold wall temperatures are changed simultaneously and symmetrically. These instabilities were ascribed to the leading-edge effect induced by the wall temperature oscillation and selectively ampliﬁed by the natural convection ﬂow. Dabiri and Gharib (1996) studied three different possible cases for a cubical cavity (Pr = 12.4) with sinusoidally forced boundary conditions at the Brunt–V¨ais¨al¨a frequency: the ﬁrst with heating between the two walls in phase, the second with heating between the two walls 180◦ out of phase and the third with heating between the two walls 90◦ out of phase. They illustrated that different oscillatory convective structures (with one cell, two cells or a combination of both structures) emerge depending on the heating phase shift between the two opposite walls. Thermal boundary layers were clearly observed together with many thermal ‘islands’ or pockets of ﬂuid where the temperature is different with respect to its surroundings. Some numerical studies have also appeared specially focusing on the possible onset of resonance phenomena and ensuing increase of heat transfer through the system (Antohe and Lage, 1996, Kwak et al., 1998, Chung et al. 2001). Most interestingly, Shu et al. (2005) considered the role played in these mechanisms by the Prandtl number. They revealed that for a ﬂuid with a small Prandtl number typical of molten metals and semiconductor melts, modulated gravity and thermal gradients produce almost the same ﬂow ﬁeld both in structure and in magnitude.

Hybrid Regimes with Vibrations

12.6

561

Mixed Marangoni–Thermovibrational Convection

Like buoyancy ﬂow, the thermal Marangoni ﬂow can also be strongly affected by imposed vibrations. Such a case is considered in this section (surface tension-induced ﬂow interacting with vibrational effects in the absence of a static acceleration component, i.e. Ma = 0, Raω or Rav = 0 and Ra = 0). Apart from the possible technological application, the interaction of thermal Marangoni ﬂow and thermovibrational convection complements, from a theoretical point of view, the similar topics of mixed buoyancy–Marangoni convection (treated in Chapter 11) and mixed buoyancy–thermovibrational ﬂow (examined in the earlier sections of this chapter). In this case, new mechanisms of ﬂow generation arise and by varying the vibrational parameters (i.e. amplitude or vibration frequency or simply the vibrational Rayleigh number in the context of the Gershuni’s model), it is possible to change the ﬂow characteristics (not only to damp the melt ﬂow but also to generate a speciﬁc one) over a wide range.

12.6.1 Basic Solutions As usual (see, e.g., Section 12.4), a theoretical framework for explaining these behaviours is elaborated here in its simplest form by resorting to the model of a layer of inﬁnite extent for which relatively simple analytical solutions can be introduced. Some interesting results along these lines are due to Grassia and Homsy (1998a,b), who considered inﬁnite parallel Marangoni ﬂow subjected to gravitational modulation at low frequencies (where Gershuni’s model is not applicable) in various directions (Figure 12.29) (they assumed as base unmodulated Marangoni ﬂow the popular ‘return ﬂow solution’ introduced in Section 2.4.2 and employed a quasi-steady approach, in the limit of very low forcing frequency). Notably, for gravity modulations in the plane of the basic ﬂow, analytical solutions for velocity and temperature proﬁles were obtained for vibrations of arbitrary amplitude. As explained in Chapter 8 (Section 8.3), generally, under such conditions the time-averaged departure from the basic solution induced by the vibrational effect is very small with respect to the oscillatory component of such a departure (let us recall that, in general, the TFD distortions induced by a periodic zero-mean value acceleration are made up of an average contribution plus a time-periodic component; with

Figure 12.29 Sketch of ﬂuid layer of inﬁnite extent with upper free surface subjected to vibrations s(t) = b sin(ωt)% n with arbitrary direction

562

Thermal Convection: Patterns, Evolution and Stability

the former prevailing for high frequency and small displacement amplitude and the latter becoming dominant in the opposite situation; see, in particular, Section 8.2). Focusing expressly on the oscillatory part, Grassia and Homsy (1998a) highlighted that when low-frequency vibrations are applied in the plane of the basic 2D ﬂow, the ensuing distortions involve signiﬁcant temporal modiﬁcations to the earlier plane-parallel ﬂow solutions, which depend basically on the intensity of the temperature gradient (externally imposed and/or determined by the action of the basic ﬂow) established in a direction perpendicular to the considered vibrations. They found, in fact, vibrations in the vertical direction (θ = 90◦ ) to generate time-dependent vorticity due to coupling with the applied horizontal temperature gradient (this alternately cooperating or competing with the steady basic ﬂow over a cycle of the modulation), while vibrations applied along the layer (θ = 0◦ ) to produce vorticity only when coupled to vertical convected temperature gradients (in practice, only when the basic ﬂow is sufﬁciently strong). It was reported, in particular, that when a strong periodic acceleration is considered, the ﬂow proﬁle alternates during the modulation cycle between boundary-layer structures and vertically stacked cells (the type of structure selected depending, as mentioned above, on the sense of the horizontal thermal stratiﬁcation with respect to the shaking direction). They also showed that in that part of the cycle where this stratiﬁcation is unstable with respect to the onset of RB modes (heating from below), there are particular amplitudes of the acceleration that can give strong cellular motions or runaways (these runaways representing a resonant interaction with stationary RB cells). In a companion study (1998b), they also considered the basic state perturbed by vibrations imposed in the spanwise direction (normal to the plane of the basic ﬂow, i.e. along z). The resulting ﬂuid motion was found to be three-dimensional with ﬂow and temperature ﬁelds displaying a simple functional dependence on streamwise and spanwise coordinates, but retaining a complicated dependence on the vertical coordinate (they obtained polynomial solutions in vertical coordinate for the various ﬁelds). Some interesting insights into these complex behaviours were provided in terms of possible cause-and-effect relationships. Following these arguments, at ﬁrst order for the spanwise vibration, there is a time-periodic spanwise–streamwise circulation around the slot. As this circulation also advects heat, it produces spanwise temperature gradients, enabling thermocapillarity to generate subsidiary spanwise ﬂows. At next order, parallel streamwise ﬂows emerge along with streamwise and vertical temperature gradients (in most parameter regimes these secondary effects were found to be opposed to the ﬂow and temperature ﬁelds in the basic state). A year later, Suresh et al. (1999) re-examined the condition with vibrations parallel to the temperature gradient (θ = 0◦ ), considering a small but ﬁnite frequency of the vibrations in order to moderate some singularities affecting the earlier quasi-steady model of Grassia and Homsy (1998a). The problem was reapproached in terms of the RB modes that are excited when the system is temporarily subjected to an acceleration acting along the positive x direction (i.e. in the part of the cycle when the applied stratiﬁcation is unstable with respect to the acceleration); it was highlighted that in this conﬁguration the body force, which acts to displace the hot, light ﬂuid by the cold, heavy ﬂuid, can enhance the basic Marangoni return ﬂow; in turn, the usual symmetry of the low-frequency vibrational convection tends to be broken by the Marangoni effects. Various possible situations in terms of system response were reported (together with the related functional dependences) according to the considered frequency and Prandtl number. For the case in which ﬁnite-frequency vibrations are perpendicular to the layer (θ = 90◦ ), it is also worth considering the subsequent study by Suresh and Homsy (2001), who provided a nice exact solution to the problem, for which it is worth providing all the necessary details (remarkably, such a solution has attracted some interest in the literature as a basic state for application of linear stability analysis, as will be illustrated in Section 12.6.3).

Hybrid Regimes with Vibrations

563

Following the general concepts introduced in Section 2.4, such a solution can be expressed as the superposition of two components, the steady Marangoni component proportional to Ma (σT γ d 2 /µα) and a periodic vibrational component proportional to Raω (bω2 βT γ d 4 /να = #2 /Pr, where γ is the rate of uniform temperature increase along the x-axis): u = MagM (y) + Raω gB (y) exp(i#t)

(12.47)

T = x + MafM (y) + Raω fB (y) exp(i#t)

(12.48)

where, as already reported in Section 2.4.2, for adiabatic horizontal walls 1 1 2 3y + y − gM (y) = − 4 4 fM (y) = −

1 3 3 5 3y 4 + 2y 3 − y 2 − y − 48 2 2 16

(12.49)

(12.50)

and, as analytically determined by Suresh and Homsy (2001): gB (y) = mc1 exp(mη) − mc2 exp(−mη) + c3 −

η m2

(12.51)

and fB (y) = d1 exp(nη) + d2 exp(−nη) + d3 exp(mη) + d4 exp(−mη) + d5 − η/m2 n2

(12.52a)

for Pr = 1 and fB (y) = (d1 + d3 η) exp(mη) + (d2 + d4 η) exp(−mη) + d5 − η/m4 for Pr = 1, with

1 η= y+ 2 m=

i# Pr

(12.52b)

(12.53a)

1 2

(12.53b)

1

n = (i#) 2

If Pr = 1: d1 =

1 2n sinh(n)

(12.53c)

c1 =

exp(−m)(m2 /2 − 1) − m + 1 2m4 [−m cosh(m) + sinh(m)]

(12.54a)

c2 =

exp(m)(−m2 /2 + 1) − m − 1 2m4 [−m cosh(m) + sinh(m)]

(12.54b)

c3 = m(c2 − c1 )

(12.54c)

c4 = −(c1 + c2 )

(12.54d)

$ m2 1 − exp(−n) [(c + c ) exp(−n) − c exp(m)−c exp(−m)] + 1 2 1 2 n2 − m2 m2 n2 (12.55a)

564

Thermal Convection: Patterns, Evolution and Stability

1 d2 = 2n sinh(n)

1 − exp(n) m2 [(c1 + c2 ) exp(n) − c1 exp(m)−c2 exp(−m)] + n2 − m2 m2 n2

$ (12.55b)

mc1 d3 = 2 n − m2 −mc2 d4 = 2 n − m2 c3 d5 = 2 n If Pr = 1: d1 =

d2 =

d3 = d4 = d5 =

(12.55c) (12.55d) (12.55e)

$ 1 2[1 − exp(−m)] 2 sinh(m)c1 + m[c1 exp(m) − c2 exp(−m)] + 4m sinh(m) m4 (12.56a) $ 1 2[1 − exp(m)] −2 sinh(m)c2 + m[c1 exp(m) − c2 exp(−m)] + 4m sinh(m) m4 (12.56b) c1 − (12.56c) 2 c2 − (12.56d) 2 c3 (12.56e) m2

0.50

0.50

0.40

0.40

0.30

0.30

0.20

0.20

0.10

0.10

0.00

y

y

with the horizontal boundaries located at y = ± 1/2. Such a solution (it is plotted in Figure 12.30) reduces, obviously, to the quasi-static analytical model elaborated by Grassia and Homsy (1998a) in the limit as m → 0 and n → 0 (i.e. # → 0). The response of the inﬁnite parallel Marangoni ﬂow to the application of external vibrations in the opposite situation in which the frequency is high and the displacement amplitude is small (the

−0.10

−0.20

time

−0.30 −0.50 −0.40

0.00 −0.10

−0.20 −0.40

time

−0.30 gw −0.20 0.00 Velocity (a)

gw

−0.40

0.20

−0.50 −0.01

gw

0.00

gw

0.01 0.02 Temperature (b)

0.03

Figure 12.30 Exact solution for the case of mixed Marangoni–thermovibrational convection in an inﬁnite layer with adiabatic boundaries (Pr = 1, Ma = 1, θ = 90◦ , Raω = 5, # = 1)

Hybrid Regimes with Vibrations

565

framework for the application of the Gershuni’s formulation; see Section 8.4) was considered for the ﬁrst time by Birikh et al. (1992, 1994). As explained in Sections 8.2 and 8.3, for such a case the situation is reversed with respect to the condition in which # is small. The most signiﬁcant departure from the basic solution, in fact, is not due to oscillatory zero mean value components of temperature and velocity ﬁelds, but must be ascribed to the time-averaged vibrational effect. The problem for the determination of the basic ﬂow is formally equivalent to that developed in Section 8.5.3 for pure thermovibrational ﬂow. Birikh and co-workers, in fact, reduced the original Gershuni’s partial differential equations for momentum, energy and the auxiliary potential function [Eqs (8.15), (1.61) and (8.16), respectively] to a system of ordinary differential equations assuming a generic plane-parallel ﬂow solution in the form   u0 (y) (12.57) V = 0  0 T = x + f (y) (12.58)   w0 (y) (12.59) w= 0  0 where w is the auxiliary potential function. With such assumptions, the governing equations, Eqs (8.15), (1.61) and (8.16b), can be rewritten as u 0 + Rav w0 cos(θ ) = 0 f + u0 = 0 w0 = f cos(θ ) − sin(θ )

(12.60) (12.61) (12.62)

(where θ is, as usual, the angle between the direction of vibrations and the imposed temperature gradient and the primes denote differentiation with respect to y), which on combining Eq. (12.60) with Eq. (12.62) reduce to the condensed form: u 0 + R2 f = R1 f + u0 = 0

(12.63) (12.64)

( with Rav = (bωβT γ d 2 )2 2να, where γ is the rate of uniform temperature increase along the x-axis, d the layer depth, R1 = Rav cos(θ )sin(θ ), R2 = Rav cos2 (θ ) and the following boundary conditions hold (adiabatic boundaries): u = − Ma and f = 0 for y =

with Ma = σT γ d

( 2

1 2

1 u = 0 and f = 0 for y = − 2

(12.65a) (12.65b)

µα and the additional constraint

1 2 1 −2

u(y)dy = 0 and

1 2 1 −2

w(y)dy = 0

(12.66)

Numerical solution of the above system of equations (unfortunately it does not admit exact solutions like those illustrated in Section 12.4.1 for the Hadley ﬂow) demonstrated the possibilities for providing effective control of basic ﬂow in an inﬁnite ﬂuid layer with a constant temperature gradient along its surface. Vibrations or arbitrary orientation θ were considered and it was found that vibrations with θ ∼ = 0◦ can deform signiﬁcantly the velocity proﬁles of the basic parallel ﬂow (some examples are shown in Figure 12.31).

566

Thermal Convection: Patterns, Evolution and Stability

0.50

0.50 0.40 (2)

0.40

0.20

0.20

0.10

0.10

0.00 −0.10 −0.20

−0.20

−0.30

−0.30

−0.40

−0.40 −0.40

−0.20

0.00

0.20

(3)

0.00 −0.10

−0.50 −0.60

(4)

(1)

0.30

(1)

y

y

0.30

(3)

(4)

−0.50 −0.30

(2)

−0.20

−0.10

Velocity

Velocity

(a)

(b)

0.00

0.10

Figure 12.31 Velocity proﬁle u(y) for mixed Marangoni-thermovibrational (average) convection in an inﬁnite layer with adiabatic free surface and adiabatic bottom wall: (a) R2 = 1; (1) R1 = 0, Ma = 1, (2) R1 = 1, Ma = 1, (3) R1 = 1, Ma = 1/2, (4) R1 = 1, Ma = 0; (b) R1 = 1, Ma = 1; (1) R2 = 1, (2) R2 = 10, (3) R2 = 102 , (4) R2 = 103

High-frequency vibrations were found to affect the Marangoni ﬂow seriously also in the numerical study of Jue and Ramaswamy (1992), who focused on the case of rectangular cavities of ﬁnite extent and low Prandtl number ﬂuids. Some interesting examples along these lines are illustrated in the next subsection. Following the same approach as in Section 12.5.2, attention is paid to some cases of practical interest in the ﬁeld of semiconductor crystal growth (Pr = 0.01, A = 4).

12.6.2 Control of Convection Patterning and Intensity in Shallow Enclosures Similarly to Figures 12.24–12.27 in Section 12.5.2, as relevant examples of possible ﬂow patterns Figures 12.32–12.35 show mixed two-dimensional Marangoni–thermovibrational and mixed buoyant–Marangoni–thermovibrational convective states for high-frequency imposed vibrations (Gershuni’s approximation), heating from the side and various conditions. Figure 12.32a shows the pure Marangoni ﬂow that occurs for Ma = 1 × 103 with a strong anticlockwise vortex near the cold wall (as discussed in Section 10.2.4, when the Marangoni number is increased above 1 × 103 , the cell, which for lower values is extended throughout the horizontal length of the cavity, tends to be conﬁned near the cold side). When perpendicular vibrations are applied to the cavity, the strength of the main Marangoni is slightly reduced (due to the nonlinear interplay between the two types of convection) and a clockwise roll of a thermovibrational origin appears on the other side of the cavity (i.e. near the hot wall; see Figure 12.32b). As a consequence of the thermovibrational effect, however, the TFD disturbances increase (see the value of εT in Figure 12.33). With (mitigating) vibrations parallel to the interface, the Marangoni ﬂow is considerably weakened by the counteracting action of thermovibrational ﬂow and the TFD disturbance is reduced to about half of its value in the absence of forcing. The case with three distinct driving forces (buoyancy due to steady g, vibrations and Marangoni effect) acting all together is illustrated in Figures 12.34. Without vibrations, a Marangoni cell is located near the cold wall and a corotating anticlockwise circulation cell induced by buoyancy

Hybrid Regimes with Vibrations

567

(a)

(b)

(c)

Figure 12.32 Mixed steady Marangoni–thermovibrational convection [streamlines, plane (x, y)] in a laterally heated open cavity (Pr = 0.01, A = 4, Ma = 1 × 103 and Rav = 5 × 104 ; zero-g conditions; Ma based on the length and Rav based on the depth of the cavity; cold and hot sides on the left and on the right of each frame, respectively; upper and lower boundaries with adiabatic conditions): (a) absence of oscillatory acceleration (max = 1.03); (b) oscillatory acceleration acting in the vertical direction (max = 1.01); (c) oscillatory acceleration acting along the horizontal direction (stabilizing effect, max = 0.45) (numerical simulations, M. Lappa)

forces arises in the middle of the cavity. Both cells are anticlockwise oriented and display a similar strength for the considered values of the Marangoni and Rayleigh numbers. Interestingly, in Figure 12.34b (with gravity modulation in the vertical direction), three rolls of different nature are simultaneously present in the cavity: the aforementioned fairly strong Marangoni roll clustered near the cold wall, a central vortex due to classical thermal buoyancy and a lateral small cell of vibrational origin conﬁned in the corner between the bottom adiabatic wall and the right hot side. When forcing is considered in the horizontal direction, in a unexpected way, for the considered parameters (see the ﬁgure caption), the action of vibrations is not limited to a simple damping of the intensity of the Marangoni ﬂow; total suppression of the central cell of gravitational (steady) origin, in fact, occurs.

12.6.3 Control of Hydrothermal Waves Some theoretical analyses have also appeared in the literature expressly focusing on the effect of vibrations (in both limits of low and high frequency) on the typical primary instabilities of Marangoni ﬂow discussed in Chapter 10.

568

Thermal Convection: Patterns, Evolution and Stability

10

14

6

6 4

9

8

3

11

6 (a)

14

9

3

8

5 3

8

3

1

12

(b) 4

14

8

4

1

11

7 9

2

2

6 10

(c) Figure 12.33 TFD distortions δT = (T − Tdiff )/T [εT = max(δT )] for the same conditions as in Figure 12.32: (a) level 1 → T = −7.18 × 10−2 , level 15 → T = 1.97 × 10−2 , level = 0.65 × 10−2 , εT = 7.8 × 10−2 ; (b) level 1 → T = −8.2 × 10−2 , level 15 → T = 1.73 × 10−2 , level = 0.7 × 10−2 , εT = 8.9 × 10−2 ; (c) level 1 → T = −3.66 × 10−2 , level 15 → T = 7.0 × 10−3 , level = 3.25 × 10−3 , εT = 4.1 × 10−2

As a relevant example, using the analytical solution with Eqs (12.47)–(12.56) as the basic state, Suresh and Homsy (2001) investigated (by linear stability analysis) the combined vibrational–thermocapillary instability in a ﬂuid layer of inﬁnite extent (with constant temperature gradient applied along its length and adiabatic boundaries) for a zero mean value harmonic acceleration perpendicular to the layer and for various Prandtl numbers. Interestingly, they found that either pure thermocapillary (of the hydrothermal type; see Section 10.2 for some necessary propaedeutical background) or vibrational instabilities are possible and found regions in the parameter space where the two instability mechanisms either reinforce or oppose each other (the latter case refers to regions of stability in the (Raω , Ma)-plane where ﬂows are stable due to the combined effect of vibrations and thermocapillarity, whereas a purely vibrational or a purely thermocapillary ﬂow at the same conditions would be unstable). Shear-driven inﬂectional instabilities (of the hydrodynamic type) were shown not to be important (see Section 1.5.4 for fundamental information on this different kind of instability). Their study, however, was limited to two-dimensional situations and the combined mechanisms of instability were examined at a single frequency.

Hybrid Regimes with Vibrations

569

(a)

(b)

(c) Figure 12.34 Structure of steady mixed buoyant–Marangoni–vibrational convection [streamlines, plane (x, y)] in a laterally heated open cavity (Pr = 0.01, A = 4; cold and hot sides on the left and on the right of the ﬁgure, respectively; upper and lower boundaries with adiabatic conditions) for Ra = 1 × 103 and Rav = 5 × 104 (Ra and Rav based on the depth of the cavity) and Ma = 1 × 103 (Ma based on the horizontal extension of the cavity): (a) absence of oscillatory acceleration (max = 1.64); (b) oscillatory acceleration acting in the vertical direction (max = 1.54); (c) oscillatory acceleration acting along the horizontal direction (stabilizing effect, max = 0.53) (numerical simulations, M. Lappa)

Because the hydrothermal waves are known to be three-dimensional (see again the discussions in Section 10.2), Zebib (2005) re-examined the problem allowing for three-dimensional disturbances. It was found that the vibrational instabilities are also three-dimensional and that in general three-dimensional modes are preferred everywhere in the (Raω , Ma)-plane. In particular, it was shown that with Pr 1, modulation stabilizes the three-dimensional thermocapillary branch for all the acceleration amplitudes and frequencies, whereas at larger Pr and at low frequencies, there are regions of alternating stability and instability as the acceleration amplitude (gω = bω2 ) increases from zero. Some examples of the Zebib’s results are shown in Figures 12.36–12.38 for the speciﬁc case # = 10. Towards the end to bridge the gap between the present complex hybrid scenario and those holding for the respective pure cases, the related description (developed in the following) progresses, giving some emphasis to both limits of large and small values of the control parameters, that is, to the two limiting situations in which the instability is dominated by Marangoni ﬂow

570

Thermal Convection: Patterns, Evolution and Stability

14

8

10

8 4 1

6

1

2

4 9

(a)

8

10

14

1 11

9

5 2

6 1

6

7

11

(b) 15

6 9

9

2 3

1

10 3

6

11

(c) Figure 12.35 TFD distortions δT = (T − Tdiff )/T [εT = max(δT )] for the same conditions as in Figure 12.34: (a) level 1 → T = −8.64 × 10−2 , level 15 → T = 5.06 × 10−2 , level = 1 × 10−2 , εT = 9.6 × 10−2 ; (b) level 1 → T = −1.14 × 10−1 , level 15 → T = 2.75 × 10−2 , level = 1 × 10−2 , εT = 1.24 × 10−1 ; (c) level 1 → T = −5.1 × 10−2 , level 15 → T = 9.85 × 10−3 , level = 4.36 × 10−3 , εT = 5.5 × 10−2

weakly affected by vibrational convection or vice versa it is mainly related to vibrational ﬂow slightly modiﬁed by thermocapillarity. Figure 12.36 shows, in particular, the two- and three-dimensional critical curves for Pr = 1. It is evident that for Raω = 0, the results of Zebib (2005) are the same as those of Smith and Davis (1983) (with the oblique three-dimensional waves preferred and good agreement in the values of the critical Marangoni number; compare, e.g., the value given by the intersection of the solid line and the Ma axis with the corresponding value for Pr = 1 in Figure 10.8a). According to this ﬁgure, as anticipated, 3D instabilities (solid line) are more critical everywhere in the (Ma, Raω ) -plane. Interestingly, while a small amplitude modulation destabilizes the two-dimensional hydrothermal waves, it stabilizes the preferred three-dimensional modes (the related branches in Figure 12.36 are decreasing and increasing functions of Raω , respectively). Concerning the reverse effect, small thermocapillarity (Ma 10 and Ma 35) seems to stabilize both the purely vibrational two- and three-dimensional modes at the considered small value of #. Other interesting results obtained by this author can be summarized as follows. Three-dimensional modes continue to be preferred with increasing #. For # = 100 (not shown here) the effect of vibrations on the two-dimensional hydrothermal waves, however, is

Hybrid Regimes with Vibrations

571

Figure 12.36 Stability boundaries in the (Raω , Ma) plane for Pr = 1 and # = 10 (inﬁnite layer with adiabatic boundaries; zero-g conditions; Marangoni and Rayleigh numbers deﬁned as Ma = RePr = σT γ d 2 /µα and Raω = bω2 βT γ d 4 /να = #2 /Pr, respectively, where γ is the rate of uniform temperature increase along the x -axis). After Zebib (2005); Reproduced by permission of Cambridge University Press

Figure 12.37 Stability boundaries in the (Raω , Ma) plane for Pr = 0.01 and # = 10 (inﬁnite layer with adiabatic boundaries; zero-g conditions; Marangoni and Rayleigh numbers deﬁned as Ma = RePr = σT γ d 2 /µα and Raω = bω2 βT γ d 4 /να = #2 /Pr, respectively, where γ is the rate of uniform temperature increase along the x -axis). After Zebib (2005); Reproduced by permission of Cambridge University Press

572

Thermal Convection: Patterns, Evolution and Stability

Figure 12.38 Stability boundaries in the (Raω , Ma) plane for Pr = 10 and # = 10 (inﬁnite layer with adiabatic boundaries; zero-g conditions; Marangoni and Rayleigh numbers deﬁned as Ma = RePr = σT γ d 2 /µα and Raω = bω2 βT γ d 4 /να = #2 /Pr, respectively, where γ is the rate of uniform temperature increase along the x -axis). After Zebib (2005); Reproduced by permission of Cambridge University Press

no longer destabilizing. Moreover, while thermocapillarity continues to stabilize the purely 3D vibrational instability up to # = 200, above such a threshold the trend is reversed. Figure 12.37 gives the stability boundaries for Pr = 0.01. In agreement with Smith and Davis (1983), for pure Marangoni ﬂow (Raω = 0) there are no two-dimensional hydrothermal waves (let us recall that in the light of Squire’s theorem stated in Section 1.5.4, given the absence of inﬂectional points in the basic exact solution for pure Marangoni ﬂow, the instability problem cannot admit disturbances corresponding to two-dimensional waves in the limit as Pr → 0; the reader is also referred to the explanations provided in Section 10.2.2). Such waves, however, become possible for pure vibrational ﬂow (the dashed line intersects the Raω axis at Raω ∼ = 5.7×103 ), which indirectly indicates that shear-driven inﬂectional instabilities can occur in this case (anyhow, as shown in the ﬁgure, such a phenomenon is sensitive to the presence of Marangoni effects, as indicated by the increase in the related threshold with Ma). Concerning 3D modes, in this case the preferred region of three-dimensional stability is very small with modulation stabilizing the hydrothermal waves and thermocapillarity stabilizing the purely vibrational modes. Figure 12.38 with Pr = 10 displays other possibilities. Weak thermocapillarity is shown to stabilize and destabilize the two- and three-dimensional vibrational modes, respectively. With regard to the reverse inﬂuence, modulation initially stabilizes both two- and three-dimensional thermocapillary waves with the effect of alternating with increasing Raω . Notably, Zebib (2005) also considered the limit of high frequencies (i.e. Gershuni’s approximation; see Section 8.4), for which assuming a basic ﬂow as in Eqs (12.15)–(12.18) and generic 3D disturbances in the form given by Eqs (12.26)–(12.29), the eigenvalue problem reduces to the

Hybrid Regimes with Vibrations

573

following system of ordinary differential equations (where the primes denote differentiation with respect to y) for the amplitudes ud , vd , wd , pd , Td and $1d , $2d and $3d : Continuity equation: i(qx ud + qz wd ) + vd = 0

(12.67)

Momentum: Pr(ud − q 2 ud ) − (iqx u0 ud + u0 vd ) − iqx pd − Pr Rav [(iqx wo )$1d + (w0 )$2d ] = λud (12.68a) Pr(vd − q 2 vd ) − (iqx u0 vd ) − pd + Pr Rav [$1d + (f − iqx w0 )$2d + iqx w0 Td ] = λvd Pr(wd − q 2 wd ) − (iqx u0 wd ) − iqz pd − Pr Rav (iqx w0 $3d ) = λwd

(12.68b) (12.68c)

Energy: Td − q 2 Td − (iqx u0 Td + ud + vd f ) = λTd

(12.69)

Auxiliary potential function: − q 2 $1d = −iqx Td $1d

(12.70a)

$2d

2

− q $2d = −q Td

(12.70b)

$3d

− q $3d =

−iqz Td

(12.70c)

2 2

where q 2 = qx2 + qz2 , with boundary conditions: Velocity:

for y =

− 1/2

ud = 0

(12.71a)

vd = 0

(12.71b)

wd = 0

(12.71c)

ud = −Maiqx Td

(12.72a)

vd = 0

(12.72b)

wd

= −Maiqz Td

(12.72c)

$1d = 0

(12.73a)

$2d = 0

(12.73b)

$3d = 0

(12.73c)

Td = 0

(12.74)

and

for y = 1/2. Auxiliary potential function:

for y =

± 1/2

and for adiabatic boundaries

574

Thermal Convection: Patterns, Evolution and Stability

Figure 12.39 Critical Wv∗−1 for the reversal of the direction of propagation of waves as a function of the Prandtl number. Courtesy of A.M.G. Zebib

Within the framework of such a formulation, Zebib (2005) found modulation to stabilize the hydrothermal branch for any value of Pr (let us recall that, as discussed in Section 8.5, in the absence of a Marangoni effect, the situation considered here, with a temperature gradient applied along the inﬁnite layer and vibrations perpendicular to the boundaries, would correspond to a state of mechanical equilibrium). The inﬂuence of vibrations on the hydrothermal waves was reported as a function of a dimensionless parameter Wv−1 that is proportional to Rav /Ma: Wv−1 =

Ra2ω Pr 2 #2 Rav = = 2 2Ma# 2Ma Pr Ma

(12.75)

The hydrothermal waves were observed to be stabilized with increasing Wv−1 . In particular, the relationship between Ma and Wv−1 was found to be almost linear in speciﬁc ranges of Wv−1 : Macr = 6.3 + 4.22 × 10−2 Wv−1 for Pr = 0.01 and 0 < Wv−1 <10 Macr = 150 + 8.3Wv−1 for Pr = 1 and 4 < Wv−1 <10 Macr = 100 + 375Wv−1 for Pr = 10 and 2 < Wv−1 <10

(12.76a) (12.76b) (12.76c)

Most interestingly, the waves were observed to reverse their direction of propagation, that is, to travel in the same direction as the free surface ﬂow (recall pure hydrothermal waves propagate in the upstream direction as extensively discussed in Chapter 10) at particular values of Wv∗−1 that decrease with increasing Pr (Figure 12.39). As in Section 12.5.2, the present section also is concluded with a short survey of the existing studies on the cylindrical geometry (the liquid bridge). The stability of the parallel axisymmetric Marangoni ﬂow in a cylindrical geometry with free lateral surface was initially investigated by Birikh et al. (1992), who considered the case of the Marangoni effect in an inﬁnite liquid bridge with an axial temperature gradient and axially imposed high-frequency vibrations. Ramachandran and Winter (1992) and Birikh et al. (1993) carried out some numerical simulations for the ﬁnite-sized system. The presence of axial boundaries (a liquid column supported by two end rods with one rod vibrating) was also addressed later by Lee et al. (1996) and Lee (1998). These analyses were based on a one-dimensional nonlinear model of a viscous liquid.

Hybrid Regimes with Vibrations

575

Nicol`as et al. (1997) proved theoretically that mechanical vibrations can be used to annihilate almost completely Marangoni ﬂows in low-Pr ﬂuids. Lappa (2005e) investigated numerically the stabilizing action exerted by high-frequency vibrations on the ﬁrst (stationary) bifurcation of Marangoni ﬂow for low-Pr ﬂuids (Pr = 0.01). The effect on the hydrothermal instability in liquid bridges with high Pr was examined by Tang and Hu (1995) and Tang et al. (1996) in the case of silicone oils. The solidiﬁcation process and the resulting microstructure during ﬂoat zone processing of eutectic alloys [NaNO3 –Ba(NO3 )2 with and without vibrations] were considered by Anilkumar et al. (1993), Grugel et al. (1994) and Shen et al. (1996). These authors veriﬁed by temperature measurements in the zone under normal and vibrational growth conditions that vibration essentially eliminates radial temperature gradients ahead of the growth interface (thereby eliminating microstructural segregation).

12.7 Modulation of Marangoni–B´enard Convection For the ﬁnal case of interacting Marangoni–B´enard convection and vibrations (this case complements from a theoretical point of view that of modulated RB convection treated in Section 12.1), interesting studies are due to Skarda (2001) and Or and Kelly (2002). Skarda (2001), in particular, examined the stability of the canonical horizontal layer of inﬁnite extent heated from below with adiabatic surface (Bi = 0) subjected to vertical vibrations (whose general stability properties for the unmodulated (pure) case have been discussed in Chapter 9). The analysis was limited to a ﬁxed value of the acceleration amplitude (Raω = 5 × 103 ); however, a variety of interesting results and insights into the subject were reported, for which it is worth opening a rich and exhaustive discussion. As a ﬁrst important result, Skarda (2001) observed the system response at the onset of convection to be synchronous with the forcing (harmonic response) for most of the conditions considered [O(10−2 ) Pr O(102 )]. Moreover, for all Pr values investigated (0.01, 7.1, 10 and 100), the effect of vibrations was found to be initially destabilizing (for relatively small values of #), then stabilizing (with Macr reaching a maximum at some ﬁnite value of # depending on Pr) and ﬁnally becoming negligible with Macr approaching the unmodulated critical value of Ma = 79.607 in the limit as # → ∞ (Figure 12.40). In addition to these general trends, some unexpected behaviours were detected. At certain values of Pr and #, multiple minima were observed, making the neutral stability curves highly distorted. As for the case of modulated RB convection treated in Section 12.1, closed regions of subharmonic instability were also determined (i.e. alternating regions in the space of parameters of synchronous and subharmonic instability separated by thin stable regions). Such results, which were obtained in the speciﬁc case Pr = 1, deserve some additional discussion (they are shown separately in Figure 12.41). This level of complexity, in fact, was not observed in the neutral stability plots at the other Pr values investigated (Skarda ascribed the absence of subharmonic behaviour for Pr values away from 1 to larger damping with respect to Pr = 1). In Figure 12.41, the unmodulated neutral stability curve (already reported in Chapter 9 as Figure 9.2) is added to each frame as a reference point to compare and contrast directly the effect of modulation on the neutral stability boundaries. Figure 12.41a reveals the existence of two small local minima (hereafter referred to as humps) near the bottom of the neutral stability for # = 7. The critical Marangoni number Macr is associated with hump 3 in Figure 12.41a and shifts to hump 2 for # = 10 indicated in Figure 12.41b. This suggests that a double minima exists along the synchronous curve for 7 < # < 10. In Figure 12.41c where # = 14.5, hump 2 forms a narrow ﬁnger that extends below the unmodulated Macr of 80, thus having a destabilizing effect on the unmodulated neutral stability curve. Another local minimum, hump 1, is located near q = 1 in Figure 12.41c.

576

Thermal Convection: Patterns, Evolution and Stability

Pr = 7.1 10 100

120

80 120

Macr 40 Macr

80 40

0

0 0.1 −40

20

40

60

80

100

Ω 5

200

400

600

800

1000

Ω

Figure 12.40 Critical Marangoni number as a function of the nondimensional frequency # for the synchronous branch at different values of Pr (Bi = 0, zero-g conditions, Raω = 5 × 103 ; Marangoni and Rayleigh numbers deﬁned as Ma = σT T d/µα and Raω = bω2 βT T d 3 /να , respectively; the inset shows critical curve for Pr = 0.01). After Skarda (2001); Reproduced by permission of Cambridge University Press

300 200 Ma 100 hump 2

0

hump 3

hump 3

hump 2

(a)

hump 2

hump 2

Ω = 10

Ω=7

−100

hump 1

hump 1 Ω = 14.5

(b)

Ω = 17

(c)

(d)

300 200 Ma 100

hump 1

hump 2

0 −100

hump 2

hump 1

hump 1

Ω = 18

0

2

4

Ω = 26

6 0

2

4

Ω = 40

6 0

2

4

Ω = 100

6 0

2

4

q

q

q

q

(e)

(f)

(g)

(h)

6

Figure 12.41 Neutral stability curves for Pr = 1, Bi = 0, zero-g conditions, Raω = 5 × 103 : modulated neutral stability (solid line), unmodulated neutral stability (dashed line). After Skarda (2001); Reproduced by permission of Cambridge University Press

Hybrid Regimes with Vibrations

577

Raw × 10−3

Examination of Figure 12.41d–f indicates that hump 1 continues to grow as hump 2 recedes with increasing #, suggesting the presence of another double minima. Hump 2 eventually disappears with increasing modulation frequency. For 17 < # < 18, a subharmonic closed neutral stability branch forms. This subharmonic region of instability grows with increasing modulation frequency until reaching an # of approximately 35. The subharmonic branch then begins to shrink and shifts to higher wavenumbers and lower Marangoni numbers. The subharmonic branch continues to shrink and eventually disappears as # increases to a value of 100, where only the synchronous branch is evident. For # 100, the subharmonic loop shifts out of the range of Ma values explored by Skarda, while modulation is observed to have a stabilizing effect on the synchronous branch. The stabilization effect, however, diminishes with further increase in the modulation frequency. The shapes of the neutral stability curves also change with modulation. To gain additional insights into these intriguing behaviours, Skarda (2001) examined the system response in the plane (#−1 , Raω ) for increasing values of the Marangoni number. The related interesting ﬁndings are discussed in detail in the following. A sequence of stability boundaries in (#−1 , Raω ) space is shown in Figure 12.42a–l. The value of Ma increases for each snapshot (graph) in the ﬁgure set, proceeding from (a) to (l)

8 4

Raw × 10−3

0

8

Ma = 0

0.1 (a)

4

0.2 0

Ma = 1.01Ma0

0.1 (b)

4

0.2 0

8

Ma = 1.1Ma0

0.1 (c)

4

0.2 0

8

8

8

8

4

4

4

4

Ma = 1.3Ma0 0

Raw × 10−3

8

0.1 (e)

Ma = 1.4Ma0 0.2 0

0.1 (f)

Ma = 1.45Ma0 0.2 0

0.1 (g)

0.2 0

8

8

8

4

4

4

4

0

0.1 1/Ω (i)

Ma = 1.5Ma0 0.2 0

0.1 1/Ω (j)

Ma = 1.6Ma0 0.2 0

0.1 1/Ω (k)

0.1 (d)

0.2

Ma = 1.47Ma0

8

Ma = 1.48Ma0

Ma = 1.2Ma0

0.1 (h)

0.2

Ma = 2.0Ma0 0.2 0

0.1 1/Ω (l)

0.2

Figure 12.42 Sequence of stability boundaries in (#−1 , Raω ) space for varying Ma (Pr = 1, Bi = 0, q = 2, zero-g conditions, Ma0 = 79.607; hatched, subharmonic response; cross-hatched, synchronous response). After Skarda (2001); Reproduced by permission of Cambridge University Press

578

Thermal Convection: Patterns, Evolution and Stability

(the wavenumber q is assumed to be equal to 2, approximately the critical wavenumber for the unmodulated Marangoni–B´enard problem). In Figure 12.42a, the steady acceleration level is g0 = 0, corresponding to a zero gravity condition and Ma = 0. Therefore, both steady driving forces potentially driving the onset of convection, namely static buoyancy and surface tension variation, are zero (this means that the related ﬂow is pure vibrational convection). Alternating tongues of subharmonic and synchronous instability can be observed in this ﬁgure for sufﬁciently large modulation amplitudes (Raω ). As the modulation amplitude increases and the modulation frequency decreases, the stable regions separating the subharmonic and synchronous instability tongues become fairly thin. The minimum amplitude required for the instability is Raω = 4.426 × 103 and occurs for a modulation frequency of 24.5. Above this minimum amplitude, a subharmonic instability develops. It is interesting at this stage to stress the qualitative analogy that could be established between this ﬁgure and the results presented in Chapter 8 in Figure 8.12 obtained by Hirata et al. (2001) (for the case of a square cell) with regions of alternating synchronous and nonsynchronous response in the plane (#, Raω ). When Ma exceeds Ma0 (where Ma0 = 79.607 is the threshold for the unmodulated case), however, a region of instability develops for small modulation amplitudes. For example, when Ma = 1.01 Ma0 , a very thin region of instability is observed at the bottom of Figure 12.42b. Obviously, this region corresponds to the emergence of the fundamental Marangoni–B´enard mode of convection. On increasing the modulation amplitude, Raω suppresses the fundamental instability (white regions in the ﬁgures). As the modulation amplitude is further increased, regions of alternating subharmonic and synchronous instability are eventually reached. The modulation amplitude necessary to suppress the fundamental instability increases for larger values of Ma, as observed in Figure 12.42b–g. Comparison of the snapshots in Figure 12.42a–h also reveals that the synchronous ﬁngers grow with increasing Ma. For a sufﬁciently large Ma value, the synchronous tongues extend to lower Raω than the subharmonic tongues. The lowest modulation amplitude to incite instability occurs in the ﬁrst subharmonic ﬁnger when Ma 1.2 Ma0 , while the second tongue (synchronous) holds the minimum amplitude threshold for Ma values in the interval 1.3 Ma0 Ma 1.47 Ma0 . Upon further increasing Ma, the region of fundamental instability and the synchronous instability tongues merge and surround the subharmonic tongues. The value of Ma where the synchronous tongues and fundamental instability region initially meet occurs for Ma in the range 1.47 Ma0 Ma 1.48 Ma0 . Beyond 1.48 Ma0 , the fundamental instability region and the synchronous tongues are indistinguishable and the stable regions surrounding the subharmonic instability regions continue to shrink. Careful inspection of Figure 12.42a–l also reveals that the subharmonic tongues recede with increasing Ma although this occurs much more slowly than the growth of the synchronous tongues. All these interesting ﬁndings led Skarda to the conclusion that as is typical of the modulated RB problem (treated in Section 12.1), both synchronous and subharmonic regions of instability can be observed in modulated Marangoni–B´enard convection for certain values of the modulation frequency. As illustrated above, in particular, for some speciﬁc values of Pr (Pr = 1), the synchronous branch can become distorted with local minima and narrow tongues forming at certain values of #. Sometimes both local minima can be above the unmodulated Macr , thus resulting in a stabilizing effect. At other times one local minimum can extend below the unmodulated Macr (maximum stabilization occurring at some ﬁnite value of modulation frequency).

Hybrid Regimes with Vibrations

579

Moreover, on the basis of a comparative analysis of the results of Skarda with those reported by Gershuni and co-workers (Gershuni and Zhukhovitskii, 1963), the following conclusions can be also drawn. In the (#−1 , Raω ) space, the modulated Marangoni–B´enard problem exhibits a fundamental instability when Ma exceeds its corresponding unmodulated neutral stability value. This is consistent with the RB instability behaviour. At larger Raω , resonant instability regions or tongues are attained. A fundamental difference between the stability boundaries, however, occurs when comparing in detail the modulated Marangoni–B´enard and RB problems. Well-deﬁned alternating regions of harmonic and subharmonic instability are observed in the (#−1 , Raω ) space for the modulated RB studies (Gershuni and Zhukhovitskii, 1963). However, in the case of the Marangoni–B´enard problem, the fundamental instability boundary and the synchronous tongues tend to merge for sufﬁciently large Ma values. For additional results on modulated Marangoni–B´enard convection the interested reader may consider the theoretical studies of Zen’kovskaya and Shleikel’(2002) and Zen’kovskaya et al. (2007).

13 Flow Control by Magnetic Fields In the realm of thermal convection and many related technological domains, ﬂow control is typically regarded as a synonym for the ensemble of possible strategies that can be employed (in general) to alter a natural ﬂow state or evolutionary path (i.e. the sequence of transitional stages of evolution between states) and (in particular) to damp the intensity of ﬂuid motion (i.e. to induce a reduction of the rate of convective transport), to somewhat delay (or suppress completely) undesired instabilities and/or to obtain a ﬂow with a well-deﬁned structure. As illustrated in the last two chapters, the interplay between some speciﬁc effects of a different origin, naturally present in the considered environment (i.e. buoyancy, surface tension gradients) or induced artiﬁcially (e.g. mechanical vibrations), can lead in some speciﬁc circumstances to the stabilization of certain types of ﬂow and even to establish patterns with desired features (see, in particular, Chapter 12); there is no doubt, however, that among the technological strategies elaborated over the years towards these ends, the cases in which such control is attained via the use of magnetic ﬁelds deserve special attention. From a historical standpoint, this approach was originally conceived within the framework of studies aimed at improving the quality of crystals grown from the melt. Solidiﬁcation from the melt is the most commonly used means of manufacturing to near net shape for most materials, especially metals, binary alloys and semiconductors (see Section 3.1.1). The quality of the ﬁnal product implicitly depends on the process of solidiﬁcation. In general, convective transport in the melt may lead to small-scale spatial oscillations of the crystal’s dopant composition, which are called striations or microsegregation (as already outlined in Chapter 3, during crystal growth, heat ﬂux ﬂuctuations from the melt to the crystal produce a cycle of crystallization and remelting at the interface; this cycle produces dislocations and other microscopic defects in the crystalline structure of the material; see Section 3.1.2), and/or large-scale variations of the crystal’s dopant composition, which are called macrosegregation. Since magnetic ﬁelds can lead to a mitigation of the ﬂow or to the control of possible oscillatory convective instabilities (the ﬁrst effect being important with respect to the macroscopic homogeneity and the latter with respect to the formation of microstructures), stabilization techniques based on their application have enjoyed widespread use. Indeed, this strategy of ﬂow control has been a prospective ﬁeld, since it offers to the engineer and scientist numerous technical potentialities. This is the reason why, during all these years, this ﬁeld has gained a well-established theoretical and technical framework and remains full of possible technological variants and promising applications. Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

582

Thermal Convection: Patterns, Evolution and Stability

Since all metals and most semiconductor melts have a relatively high electrical conductivity and are subject to electromagnetic forces when moving in a static and uniform (hereafter referred to as constant ) magnetic ﬁeld, many researchers initially used this simple concept to damp convection (Series and Hurle, 1991). It was clearly proved that the application of constant magnetic ﬁelds can stabilize both ﬂow and temperature oscillations in the melt and, thereby, represents an effective opportunity to obtain an improved crystal quality. However, owing to some drawbacks (a constant magnetic ﬁeld suppresses thermosolutal ﬂow, but macrosegregation patterns are not signiﬁcantly affected; any signiﬁcant convection damping requires prohibitively large magnetic ﬁelds; the magnetic ﬁeld must be oriented in a speciﬁc direction relative to the bulk ﬂow to take effect), to circumvent these problems different types of magnetic ﬁelds were also investigated. As an example, rotating magnetic ﬁelds (RMFs) were found to be an effective tool to organize well-deﬁned ﬂow structures in liquid metals (but to have little effect on the suppression of the ﬂuctuations in temperature and concentration leading to striation patterns in the material). Other authors considered the possibility of using gradients of magnetic ﬁelds [the theoretical foundation to this approach being provided by the last term in the second member of Eq. (1.185) in Section 1.9]. Most interestingly, they demonstrated experimentally that a magnetic gradient could be used effectively to mitigate convection in any material, conducting or non-conducting, thereby spreading the use of this technique from its traditional heartlands (semiconductor crystal growth and the processing of binary metal alloys) into new areas where the considered ﬂuids are not necessarily low-Pr [Pr < O(1)] liquids. The following sections are devoted to illustrating the fundamental principles at the root of these strategies. Emphasis is given to the effect of magnetic ﬁelds on the typical geometric models, related convective phenomena and hierarchy of bifurcations treated in the earlier chapters; in some cases, however (given the page limits), discussion is limited to just providing fundamental concepts and notions (the abundance of publications on the subject forces one to select the material carefully, presenting only the results that clarify some salient aspects); since any complication in the statement of the problem makes the set of possible regimes and structures much wider, the description of some details (in particular those arising from the application of rotating magnetic ﬁelds and magnetic gradients) lies beyond the scope of this short chapter or is touched on more or less cursorily. For the sake of brevity, it is also assumed that readers of this chapter are familiar with the classical Maxwell equations and related concepts (see Section 1.9).

13.1

Static and Uniform Magnetic Fields

13.1.1 Physical Principles and Governing Equations The motion of an electrically conducting melt under a magnetic ﬁeld induces electric currents. Lorentz forces, resulting from the interaction between the electric currents and the magnetic ﬁeld, affect the ﬂow. The complete theoretical description is rather complex; it is comprehensively treated in the monograph of Chandrasekhar (1961) on magnetohydrodynamics (for different possible degrees of approximation for the related magnetohydrodynamic equations, the reader may also consider Baumgartl and Muller, 1992, and Dulikravich and Lynn, 1997). In the following, as stated in the title of the present section, the ‘physics’ of the problem and the introduction of the corresponding model equations are considered for the simple and representative case of constant (static and uniform) magnetic ﬁelds. A uniform magnetic ﬁeld with magnetic ﬂux density (also referred to as magnetic induction) B0 generates a damping Lorentz force through the aforementioned electric currents induced by the motion across the magnetic ﬁeld that in dimensional form can be expressed as F L = J f ∧ B,

Flow Control by Magnetic Fields

583

where J f is the electrical current density. Like other body forces (e.g. the buoyancy forces), an account for this force can be obtained simply by adding a relevant term to the momentum equation. Following this approach, scaling the magnetic ﬂux density with B0 and the electric current density J f with σe Vref B0 , the momentum equation [Eq. (1.60)], including the Lorentz force, can be written in nondimensional form (assuming as usual Vref = α/L) and in the absence of phase transitions as ∂V (13.1) = −∇p − ∇ · [V V ] + Pr∇ 2 V − PrRaT i g + PrHa2 (J f ∧ i B0 ) ∂t where Ha is the Hartmann number: 1 σe 2 (13.2) H a = B0 L ρν σe is the electrical conductivity and i B0 the unit vector in the direction of B 0 (see Chapter 6 for the other quantities and parameters). It should be pointed out that, in the derivation of Eq. (13.1), the magnetic ﬂux density vector B, which can be generally expressed as B = B 0 + b (where B 0 = µp H , H being the imposed magnetic ﬁeld strength, µp the magnetic permeability and b the melt’s magnetic response), has been set equal to B 0 (the applied ﬁeld). This assumption holds in many circumstances of practical interest since, as will be shown later, the induced ﬁeld b is small. Towards this end, let us start by observing that in general, the relationship between the magnitudes of B and B 0 can be written as |B| = |B 0 | + O(Rem )

(13.3)

where Rem is the magnetic Reynolds number: Rem = µp σe Vref L

(13.4)

This parameter represents the characteristic ratio of the induced to imposed ﬁelds and according to some authors (see, e.g., Baumgartl et al ., 1990), in practice, may be regarded as a measure of the ‘bending’ of ﬁeld lines of the original applied undisturbed magnetic ﬁeld B 0 by the ﬂuid ﬂow with velocity V . In many situations of technological relevance (semiconductor melts and metal alloys), the deviation of the disturbed ﬁeld B from the original applied ﬁeld B 0 , as mentioned above, can be estimated to be very small since the order of magnitude of the magnetic Reynolds number is O(10−3 ). Additional insights into this approximation can be provided by the following arguments: the magnetic diffusion coefﬁcient (µp σe )−1 is about four orders of magnitude greater than the thermal diffusivity α, which in turn is 102 times higher than the kinematic viscosity ν (values given for liquid silicon); hence on a time-scale deﬁned by α or ν which gives a natural time-scale for transport processes in the ﬂuid (if the ﬂow velocity is not too high), the magnetic ﬁeld B always retains its steady-state value. Accordingly, the nondimensional electric current density J f can be expressed via Ohm’s law for a moving ﬂuid as J f = E + V ∧ i B0

(13.5)

where E is the electric ﬁeld normalized by Vref B0 . Since, as mentioned earlier, the unsteady induced ﬁeld b is negligible, in particular, the electric ﬁeld can be written as the gradient of an electric potential: E = −∇e

(13.6)

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Thermal Convection: Patterns, Evolution and Stability

The conservation of the electric current density gives ∇ · Jf = 0

(13.7)

which combined with Eq. (13.6) leads to a Poisson equation for the electric potential: ∇ 2 e = ∇ · (V ∧ i B0 ) = i B0 · ∇ ∧ V

(13.8)

Finally, the momentum equation can be rewritten as ∂V = −∇p − ∇ · [V V ] + Pr∇ 2 V − PrRaT i g + PrHa2 (−∇e + V ∧ i B0 ) ∧ i B0 ∂t

(13.9)

13.1.2 Hartmann Boundary Layers For the sake of completeness (the reader is also referred to the general theoretical arguments provided in Section 2.5), it should be pointed out that in many circumstances of practical interest, regardless of the type and nature of basic ﬂow considered (induced by gravity, surface tension gradients or both), the electromagnetic force changes the ﬂow pattern so that the convective circulation and the largest values of the temperature gradient are typically located in thin Hartmann layers, which are boundary layers adjacent to the walls normal to the magnetic ﬁeld. The nondimensional thickness of the Hartmann boundary layers decreases proportionally to Ha−1 as Ha increases: (normal)

δHa

∝ Ha−1

(13.10)

It is also known that due to continuity effects, boundary layers are also formed near the walls parallel to the magnetic ﬁeld (see, e.g., Ben Hadid et al ., 1997a). The (nondimensional) depth of 1 such layers behaves as Ha− /2 : (parallel)

δHa

1

∝ Ha− 2

(13.11)

To summarize, the electromagnetic damping leads to the development of Hartmann boundary layers, in addition to boundary layers adjoining the walls parallel to the magnetic ﬁeld. These layers, whose 1 depth decreases proportionally to Ha−1 and Ha− /2 , respectively, participate actively in determining the overall structure of the thermoﬂuid-dynamic ﬁeld and the related stability properties.

13.2

Historical Developments and Current Status

A number of theoretical studies have appeared in recent years focusing on the effect of static uniform magnetic ﬁelds on the ‘basic’ ﬂow motion in several geometric models of widespread semiconductor growth techniques (the term ‘basic’ is used here to indicate the various types of steady or unsteady two- or three-dimensional buoyant, Marangoni or ‘mixed’ convection without the presence of magnetic ﬁelds that have been discussed in detail for various geometric conﬁgurations and heating conditions in the earlier chapters).

13.2.1 Stabilization of Thermogravitational Flows The effect of a constant magnetic ﬁeld on the electrically conducting liquid metal ﬂows of gravitational origin in heated cavities has been mainly studied for the lateral heating condition corresponding to horizontal Bridgman crystal growth conﬁgurations. Obviously, the simplest model in such a context is the ﬂow in an inﬁnite planar liquid metal layer conﬁned between two horizontal solid walls driven by a horizontal temperature gradient (the canonical inﬁnite Hadley ﬂow whose general properties and stability have been extensively treated in Chapter 6).

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585

y

Lx x Lz ∆T

Vertical B Ly = d Spanwise B

Streamwise

z Figure 13.1 Sketch of liquid layer under the effect of a horizontal temperature gradient and a magnetic ﬁeld with vertical, streamwise or spanwise direction

Notably, for electrically insulating horizontal walls, such a ﬂow admits analytical expression even in the cases in which it is subjected to a magnetic ﬁeld with direction vertical or parallel to the applied temperature gradient or even directed spanwise to the basic ﬂow (see Figure 13.1). In the case of a vertical ﬁeld, the analytical solution (see, e.g., Kaddeche et al ., 2003) reads sinh(Hay) Ra − y (13.12) u= Ha2 2 sinh(Ha/2) y3 Ra 1 sinh(Hay) − + ηy (13.13) T =x+ 6 Ha2 2Ha2 sinh(Ha/2) with the horizontal boundaries located at y = ± 1/2 and Ra = gβT γ d 4 /να (γ being the rate of 1 uniform temperature increase along the x-axis and d the depth of the layer), Ha = B0 d(σe /ρν) /2 and the parameter η is given by η=

1 cosh (Ha/2) − 8 2Ha sinh (Ha/2)

(13.14a)

1 1 − 24 Ha2

(13.14b)

for adiabatic boundaries and η=

for conducting boundaries. If a horizontal magnetic ﬁeld parallel to the applied temperature gradient (magnetic ﬁeld applied along the x direction) is considered, there is no direct effect of the ﬁeld on the parallel ﬂow in the layer as the velocity is parallel to the ﬁeld direction (V ∧ i B0 = 0), which [taking into account Eqs (13.5)–(13.8)] leads the production term in Eq. (13.9) to vanish. The basic ﬂow is, hence, the ﬂow without magnetic ﬁeld described in Section 2.4.1 [Eqs (2.34) and (2.35)]. For a magnetic ﬁeld still horizontal but directed spanwise to the basic ﬂow (i.e. a magnetic ﬁeld applied along the z direction), the basic parallel ﬂow in the layer is still the ﬂow without magnetic ﬁeld given by Eqs (2.34) and (2.35).

586

Thermal Convection: Patterns, Evolution and Stability

A justiﬁcation for such behaviour is not as straightforward as one might assume and deserves some speciﬁc additional insights. First, it has to be noted that as long as we consider the basic ﬂow as being horizontally homogeneous, the electric ﬁeld induced by the liquid motion is uniform and directed perpendicular to the plane of the layer. Zero circulation of this ﬁeld (related to the fundamental irrotational nature of electric ﬁelds when steady magnetic ﬁelds are considered) implies that no electric currents closing within the liquid layer can be induced, but the electric impermeability of the horizontal boundaries also precludes the possibility of any electric current passing normally through the layer; thereby, insulating horizontal walls lead to a separation of electric charges over the depth of the layer, giving rise to an electrostatic ﬁeld which cancels that induced by the liquid motion. As a result, there is no electric current induced by a horizontally uniform basic ﬂow in a coplanar magnetic ﬁeld and, therefore, there is no direct inﬂuence of the magnetic ﬁeld on the basic ﬂow. Stability analysis results pertaining to any of these three fundamental situations are discussed in the following with some emphasis, as usual in this book, on historical developments and groundbreaking work that have appeared in the literature. Let us start by considering the stabilizing effect exerted by a constant vertical magnetic ﬁeld on the typical instabilities that affect the Hadley ﬂow for Pr < O(1) [i.e. the stationary 2D hydrodynamic instabilities with transverse rolls and the oscillatory 3D instabilities with longitudinal rolls (OLR)] as reported by Kaddeche et al . (2002a). Interestingly, these authors proved the vertical magnetic ﬁeld to have a great stabilizing effect on both types of instability with variations of the thresholds (critical Grashof numbers) as Grcr − Grcr(Ha=0) ∝ Ha2 for the OLR and as Grcr ∝ exp(Ha2 /21.6) for the transverse instabilities. In a companion paper (Kaddeche et al ., 2002b), they considered the stabilizing effects exerted by a constant horizontal (coplanar) magnetic ﬁeld with its direction being either parallel or perpendicular with respect to the imposed temperature gradient. Even if, as explained earlier, when the magnetic ﬁeld is coplanar to the liquid layer (as deﬁned by the relation B · i y = 0) it has no inﬂuence on the basic ﬂow; however, it does affect the disturbances, and, consequently, the stability of the ﬂow. In particular, it was shown that the spanwise ﬁeld only affects the 3D modes and the streamwise ﬁeld the 2D modes. A year later, Kaddeche et al . (2003) re-examined the problem by means of both a linear stability analysis and energy considerations, conﬁrming that the vertical magnetic ﬁeld stabilizes the ﬂow most efﬁciently and could be effectively used to suppress both transverse (2D) modes and longitudinal (3D) disturbances. A synthesis of such studies (vertical magnetic ﬁeld) is shown in Figure 13.2 through neutral stability curves. Such curves give the evolution of the critical Grashof number, Grcr , as a function of Pr for different Hartmann numbers with thermal boundary conditions along the walls either adiabatic (Figure 13.2a) or conducting (Figure 13.2b). Both ﬁgures give qualitatively similar results: an increase in Ha shifts the onset of the instabilities to higher Grashof numbers. Moreover, with regard to the dependence on the Prandtl number (see Figure 6.5 for the case without a magnetic ﬁeld), for the two-dimensional hydrodynamic modes on increasing Pr the critical Grashof number ﬁrst remains almost constant (at least up to Pr ∼ = 0.01 for Ha ≤ 5) and then starts to increase; whereas for the three-dimensional oscillatory modes, Grcr undergoes a rapid decrease as Pr is increased (Pr−1 dependence), then reaches a minimum before beginning to increase (for speciﬁc insights into these trends, the reader is referred to the general discussions in Section 6.1). The main difference that appears when Ha is increased is that for both disturbance modes the observed increase occurs for smaller values of Pr. The system response to the application of the magnetic ﬁeld, however, is not exactly the same for both types of disturbances. From Figure 13.2, it can be also seen, in fact, that for any Ha, the curves for the two- and three-dimensional instabilities intersect at a certain value of the Prandtl

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587

Ha Ha

Ha

Ha

(a)

Ha Ha

Ha Ha

(b)

Figure 13.2 Variation of the thresholds Grcr as a function of Pr and different values of Ha (vertical magnetic ﬁeld) for typical two- and three-dimensional instabilities of the Hadley ﬂow: (a) thermally adiabatic boundaries; (b) thermally conducting boundaries (Grashof number deﬁned as Gr = gβT γ d 4 /ν 2 , where γ is the rate of uniform temperature increase along the x -axis; Hartmann number deﬁned as 1 Ha = B0 d(σe /ρν) /2 ). After Kaddeche et al. (2003); Reproduced by permission of Cambridge University Press

number (referred to as Pr∗ in Section 6.1.2), which decreases quickly with Ha. This means that two-dimensional modes are better stabilized than the three-dimensional modes (the domain where two-dimensional modes are critical shrinks when the magnetic ﬁeld is applied, corresponding, for example, for the adiabatic case to Pr ≤ 0.001 at Ha = 10 compared with Pr ≤ 0.034 at Ha = 0).

588

Thermal Convection: Patterns, Evolution and Stability

Since as Ha increases the range of Prandtl numbers over which the three-dimensional oscillatory instabilities are the preferred modes extends progressively towards lower values of the Prandtl number, the most interesting outcome of such a discussion, perhaps, is that in practical situations the 3D instabilities, less stabilized, should prevail. In this regard, it should be noted that the effect exerted by the vertical magnetic ﬁeld is qualitatively similar to that induced by high-frequency vibrations with direction parallel to the temperature gradient (which has been elucidated in Section 12.4.1). As for high-frequency vibrations, the strong stabilization of the 2D modes by a vertical magnetic ﬁeld is directly connected to a strong reduction in the shear energy (let us recall that, as explained in Section 6.1.2, the 2D mode has a hydrodynamic nature, i.e. is driven by the mean shear stress; this is the reason why it is often referred to as ‘shear instability’ and the related disturbances as hydrodynamic ones). For the three-dimensional (OLR) instabilities, which are of thermal origin (see Section 6.1.2), the effect of shear is less important, which explains why they are less stabilized. As an additional distinguishing mark, for the OLR modes the wavenumber q increases with Ha (which means that the typical size of the rolls shrinks), the opposite of the case of two-dimensional instabilities (in practice, such an increase in the wavenumber with the intensity of the applied magnetic ﬁeld should be regarded, in general, as an intrinsic feature of all disturbance modes that have a signiﬁcant thermal component, as also occurs in Rayleigh–B´enard convection; see, e.g., Zierep, 2000). Concerning the effect of the thermal boundary conditions on these dynamics, the related inﬂuence may be summarized as follows: (Grcr )conducting < (Grcr )adiabatic for the two-dimensional instabilities and (Grcr )conducting >(Grcr )adiabatic for the three-dimensional instabilities. A consequence is that the value of Pr∗ is smaller in the adiabatic case than in the conducting case for any value of Ha. All the above considerations apply to the case of a vertical ﬁeld. Horizontal magnetic ﬁelds, in fact, have far less impact (as mentioned earlier, the spanwise ﬁeld only affects 3D modes, whereas the streamwise ﬁeld only acts on 2D modes). For these cases, the stabilization is very weak at small Ha, but then reaches an asymptotic behaviour corresponding to Grcr ∝ Ha, this asymptotic stabilization again being related to a decrease in the destabilizing shear energy term induced by the expansion of the cell length in the horizontal direction (the related wavenumbers decrease as Ha−1 ; Kaddeche et al ., 2003). For numerical studies focused on geometries of ﬁnite extent, the reader is referred to Ozoe and Okada (1989), Ben Hadid and Henry (1996, 1997b) and Ben Hadid et al . (1997a), who investigated numerically the steady states for cubical, cylindrical and parallelepipedic cavities, respectively. Different magnetic ﬁeld orientations were considered. Moreover, different situations were speciﬁcally examined in the parallelepipedic case, namely buoyancy-driven convection in a closed cavity or in an open cavity with a stress-free surface at the top boundary and Marangoni convection in cavities with the upper boundary subjected to surface tension variation. In the case of relatively long cavities (e.g. A = horizontal length/vertical height = 4), the magnetic damping was found to be more effective with a vertical ﬁeld (which is in qualitative agreement with the theoretical studies and analytical solutions for the system of inﬁnite extent). In general, good agreement was found for relatively elongated conﬁgurations with the analytical estimates obtained by Garandet et al . (1992) under the assumption of parallel core ﬂow. Subsequent studies along the same lines are due to Aleksandrova and Molokov (2004) and Kenjereˇs and Hanjali´c (2004), who shed some additional light on the structure and development of Hartmann boundary layers and jets according to the different orientation, strength and penetration depth of the magnetic ﬁeld. In particular, analysis of the convective heat transfer for low values of the characteristic number showed that either the vertical or the longitudinal ﬁeld is the most efﬁcient in damping of the convective heat transfer, depending on the dimensions of the cavity.

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Other interesting numerical results were reported by Gelfgat and Bar-Yoseph (2001); they investigated the inﬂuence of a constant magnetic ﬁeld belonging to the same plane of the basic velocity ﬁeld (i.e. no spanwise component) with different magnitudes and orientations on the stability of the two distinct hydrodynamic (shear) 2D branches (with a single-cell or a two-cell pattern), which, as illustrated in Section 6.2.1, occur for A = 4 (Figure 6.12). In qualitative agreement with the ﬁndings for the hydrodynamic mode obtained in the framework of the inﬁnite-layer approximation by Kaddeche and co-workers, it was proved that a vertical magnetic ﬁeld provides the strongest stabilization. Most interestingly, however, it was reported that the electromagnetic effect can suppress the multiplicity of steady states (so that at a certain ﬁeld level only the single-cell ﬂow remains stable) and that hysteresis loops can exist in certain ranges of the Hartmann number. The dependence of the critical Grashof number on the Hartmann number and on the orientation of the ﬁeld as determined by Gelfgat and Bar-Yoseph (2001) for Pr = 0.015 is summarized in Figure 13.4 (several ﬁxed orientations were considered, namely δ = 0, 30, 45, 60 and 90◦ , with δ = 180◦ − θ , θ being the angle shown in Figure 13.3). It is evident that the marginal stability curves (Grcr as a function of Ha) are non-monotonic and consist of several continuous branches corresponding to different types (stationary or oscillatory) of perturbations. Figure 13.4a represents the results for the single-cell solution. All marginal stability curves in this ﬁgure corresponding to δ ≥ 30◦ show a rapid increase in the critical Grashof number with rather deep hysteresis loops in intervals of the Hartmann number located according to the angle of the magnetic ﬁeld. Interestingly, inside these loops the critical Grashof number is almost halved with an increase of the Hartmann number, which means that, in some circumstances, moderate ﬁeld magnitudes, i.e. at Ha ≤ 10, could yield a better stabilization of the 2D hydrodynamic mode than the higher ones (larger Ha). According to Figure 13.4a, in particular, when the ﬁeld orientation changes from vertical (δ = 90◦ ) to horizontal (δ = 0◦ ), the hysteretic behaviour of the critical Grashof number is postponed to higher Ha values. Changes in the critical parameters are relatively small (i.e. the stability curves are relatively close to one another) at values of δ close to 90◦ and become larger below 30◦ . The unexpected destabilization of the single-cell ﬂow in certain ranges of (relatively low) values of Ha was ascribed by Gelfgat and Bar-Yoseph (2001) to an undesired effect due to the presence of Hartmann boundary layers (i.e. to the onset of an instability in the boundary layers near the vertical walls).

Figure 13.3 Sketch showing possible relative orientations of temperature gradient, gravity and imposed magnetic ﬁeld

590

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 13.4 Stability diagrams in the (Gr, Ha) plane for buoyancy convection in a rectangular cavity (Pr = 0.015, A = 4; horizontal walls with adiabatic conditions): (a) steady single-cell ﬂows; (b) steady two-cell ﬂows [Gr and Ha based on the cavity height; δ = 0◦ (∇), δ = 30◦ (), δ = 45◦ (), δ = 60◦ (), δ = 90◦ (O)]. After Gelfgat and Bar-Yoseph (2001); Reproduced by permission of the American Institute of Physics

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In the two-cell ﬂow case, the maximum magnitudes of the horizontal and vertical velocities do not differ as much as in the case of single-cell ﬂows. Accordingly, as is evident in Figure 13.4b, the effects of different orientations of the magnetic ﬁeld are similar. For each orientation, the two-cell ﬂows are stable inside the corresponding marginal stability curves, the lower part of these curves representing transition from the basic unicellular (Hadley) ﬂow to steady bicellular convection, whereas the upper continuous part corresponds to a transition from steady to oscillatory two-cell ﬂow due to a supercritical Hopf bifurcation. At small values of the Hartmann number, the stabilization effect is not very strong. However, Gelfgat and Bar-Yoseph (2001) found that for sufﬁciently high values of the ﬁeld magnitude these ﬂows were destabilized and transformed into single-cell ﬂows, subject to their own stability properties, thereby clearly indicating the possibility of using the magnetic ﬁeld for control not only of the stability (threshold) but also of the ensuing ﬂow structure. Perhaps the most signiﬁcant outcome of this study is that, in general, the role potentially played in these dynamics by the lateral walls, in terms of both the emerging pattern and stabilizing effect exerted by the magnetic ﬁeld, cannot be ignored. Some results along these lines of research are also due to Hof et al . (2005). Limiting the investigation to the onset of oscillatory ﬂow (given the obvious difﬁculties in detecting stationary disturbances in ﬂuids that are opaque to visible light), they considered experimentally a rectangular container ﬁlled with liquid gallium subjected to a horizontal temperature gradient and a uniform magnetic ﬁeld applied separately in three directions. The natural logarithm of the rescaled critical Grashof number for the onset of oscillatory ﬂow, Gr/Grcr(Ha=0) , was found to scale in proportion to exp(Ha3 ) and exp(Ha2 ) for the vertical and the horizontal (streamwise or spanwise) orientations of the magnetic ﬁeld, respectively (see Figure 13.5). Moreover, the vertical and spanwise ﬁeld were observed to damp oscillations more efﬁciently than the horizontal magnetic ﬁeld parallel to the applied temperature gradient.

Figure 13.5 Critical threshold for oscillatory ﬂow determined experimentally as a function of the Hartmann number (rectangular container 5 × 1 × 1.3 ﬁlled with liquid gallium; Gr = gβT γ d 4 /ν 2 , where γ = T /L, T and L being the imposed temperature gradient along x and the related system extension, respectively; Ha based on the cavity depth d). After Hof. et al. (2005); Reproduced by permission of Cambridge University Press

592

Thermal Convection: Patterns, Evolution and Stability

Although in qualitative agreement with earlier studies, these experiments revealed an unexpected strong stabilization related to the spanwise ﬁeld, which may be explained by a preference of the system for OLR modes (which, as discussed in Section 6.4, tend to be the most dangerous perturbation in ﬁnite-sized parallelepipedic geometries for values of the Prandtl number close to that of gallium and, as mentioned earlier, are sensitive to the application of magnetic ﬁelds with spanwise direction) or by the peculiar geometry considered (with comparable ﬁnite extension along the y- and z-axes). In this regard, the problem was re-examined theoretically by Henry et al . (2008a). They reported different laws of variation (as a function of the intensity of the magnetic ﬁeld) for the thresholds, from exponential increases to clear decreases (with hysteresis phenomena similar to those detected by Gelfgat and Bar-Yoseph, 2001, for the 2D cavity), depending on the effective geometry of the conﬁguration (extended layer, transversally conﬁned layer, 3D cavity), the type of instability and the orientation of the magnetic ﬁeld. Energy budgets at the threshold revealed that, in most cases, the variations observed should be ascribed to changes in the spatial distribution of the dominant destabilizing shear term. Again through budgets of ﬂuctuating kinetic energy associated with the time-periodic disturbances at threshold , Henry et al . (2008b) gained additional insights into the problem, considering a box ﬁlled with gallium with relative dimensions 4 × 1 × 2. According to such analysis, at the threshold of the instability the oscillatory transition is dominated by the kinetic energy produced by the vertical shear of the basic ﬂow, and when a magnetic ﬁeld is applied, such kinetic energy must increase in order to balance the stabilizing magnetic energy. In other words, this means that as the destabilizing effect of the velocity ﬂuctuations strongly decreases when Ha is increased (due to the decay of the velocity ﬂuctuations in the bulk accompanied by the appearance of steep gradients localized in the Hartmann layers), an increase in the shear of the basic ﬂow at Grcr is required in order to sustain the instability, which leads to an increase in Grcr depending, in turn, on the way in which the considered ﬁeld (vertical or horizontal) affects the shear energy term at threshold conditions. Along these lines, it is also worth mentioning Rachdia et al . (2007), who further examined the problem by considering an external magnetic ﬁeld subjected to a slight polar deviation with respect to the initially selected direction. They investigated the effects of the weak deviation on the stability characteristics of the two-dimensional stationary and three-dimensional oscillatory instabilities. The ﬂow was found to exhibit some interesting and unexpected stability characteristics, the effect of the deviation being either stabilizing or destabilizing depending essentially on the initially selected magnetic ﬁeld orientation. New features and behaviours were found for some orientations of the magnetic ﬁeld. As usual, these studies were extended over the years also to other geometries, not necessarily with rectangular or parallelepipedic shape. As an example, for the problem of horizontal differentially-heated cylinders with a vertical magnetic ﬁeld the reader may consider Davoust et al . (1999). Similar studies concerning the effect of static and uniform magnetic ﬁelds in laterally heated vertical cylindrical vessels (introduced as models of the vertical Bridgman technique) were carried out by Kim et al . (1988), Baumgartl et al . (1990), Uspenskii and Favier (1994), Ben Hadid et al . (1997c) and Gelfgat et al . (2001) for the speciﬁc case of an axial direction of the magnetic ﬁeld. Baumgartl et al . (1990) proposed the use of magnetic ﬁelds as a means to suppress undesired thermogravitational convection as a possible alternative to the expensive microgravity and a detailed critical comparison of the disadvantages and advantages provided by these two different approaches was presented. Towards the same end, Uspenskii and Favier (1994) compared the efﬁciency of ﬂuid motion damping under magnetic ﬁelds and under high-frequency vibrations (see Chapter 12 for relevant

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background on the latter strategy of ﬂow control). For the horizontal Bridgman geometry in conditions of terrestrial gravity, they observed the damping to be more efﬁcient with the magnetic ﬁeld for semiconductor melts with relatively high electroconductivity; for the vertical Bridgman geometry with horizontal vibration, interestingly, a signiﬁcant braking of ﬂuid movement near the bottom of the crucible could be achieved (particularly in the region near the solid/liquid interface where conditions for purely diffusive mass transfer growth can be met). Owing to the experimental evidence that neither magnetic ﬁelds nor microgravity alone can produce perfect ideal crystals, Ma and Walker (1996, 1997) also addressed the use of magnetic ﬁelds as a possible means to damp the typical convective disturbances of the microgravity environment (g-jitters) in laterally heated cylinders (thereby opening up a new, interesting subﬁeld with respect to the topics covered in the present book, i.e. mixed vibrational–magnetic convection that, however, is not discussed further here due to page limits; the reader is also referred to the interesting work of Okano et al ., 2006). For the usual conditions of terrestrial gravity and convection damping via axial magnetic ﬁelds, perhaps the most interesting results were provided by Gelfgat et al . (2001), who showed that the electromagnetic stabilization tends to be stronger in lower cylinders and also illustrated that after the magnetic ﬁeld exceeds a certain value, the type of primary instability in these systems (see Section 6.5 for the parent problem without magnetic ﬁelds) changes such that the transition to oscillatory ﬂow (Hopf bifurcation) is replaced by a stationary bifurcation from the initial steady axisymmetric ﬂow to a steady asymmetric convective state. These authors performed calculations for a ﬁxed value of the Prandtl number, Pr = 0.015, and three ﬁxed values of the aspect ratio (height/diameter), A = 0.5, 1 and 1.5. For each value of the aspect ratio, a family of curves Grcr (Ha, m) was calculated for the azimuthal wavenumber m varying from 0 to 20. The ﬁnal marginal stability curves obtained as the lower envelope of all curves Grcr (Ha, m) are shown in Figure 13.6 as a dependence of Grcr on Ha (the critical azimuthal wavenumbers m are indicated on each continuous branch of the curves).

Figure 13.6 Critical Grashof number as a function of Ha for Pr = 0.015 and three values of the aspect ratio (Gr and Ha based on the cylinder height). After Gelfgat et al. (2001); Copyright Elsevier, 2001

594

Thermal Convection: Patterns, Evolution and Stability

On the basis of Figure 13.6, some general considerations can be made. The stabilization of convective ﬂow by the external magnetic ﬁeld, that is, the growth of Grcr with increase in Ha is monotonic (except for a short interval 20.5 < Ha < 22 at A = 1.5, where a tiny decrease in the Grcr number is connected with the switch of the dominant perturbation mode). At the same time, as mentioned earlier, in lower cavities (A = 0.5) the stabilization is considerably stronger than in taller cavities (A = 1.5). These trends were explained by the authors by a delicate interaction of the magnetic ﬁeld and the distinct components of velocity along different directions, as follows. Since the electromagnetic force in Eq. (13.9) is generated by the interaction of the axial magnetic ﬁeld with the radial and azimuthal components of velocity (the latter appears only as a part of three-dimensional perturbation), and not with the axial velocity (the axial velocity is affected only via continuity), hence in taller cylinders, where the axial velocity of the convective roll is larger than the radial and the azimuthal velocities, the electromagnetic suppression of the base ﬂow must be weaker (which leads to the natural conclusion that a stronger magnetic ﬁeld is required to reach a certain stabilization). For all values of the aspect ratio considered, Gelfgat et al . (2001) found the instability of ﬂuid ﬂow to appear as a Hopf bifurcation (oscillatory instability) at low values of Ha, whereas at larger Ha as a transition from steady axisymmetric to steady nonaxisymmetric ﬂow. The switch from Hopf to steady bifurcation was observed to take place at Ha ∼ = 10 for A = 0.5 (see, e.g., Figure 13.7, showing the steady temperature perturbation ﬁeld for Gr = 3.24 × 106 and Ha = 20), Ha ∼ = 31 for A = 1 and Ha ∼ = 20.5 for A = 1.5. Another observed effect of the increasing magnetic ﬁeld, which is also worth mentioning, concerns the growth of the critical azimuthal wavenumber mcr , as evident in Figure 13.6 for A = 0.5 starting from Ha = 12.5 and A = 1 starting from Ha ∼ = 32 (for A = 1.5 the growth of mcr takes place for values of Ha larger than 60).

Figure 13.7 Isolines of temperature perturbation at the cross-section z = 0.95 for the axisymmetrybreaking steady bifurcation (Pr = 0.015, A = 0.5, Ha = 20, mcr = 10, Grcr = 3.24 × 106 ). After Gelfgat et al. (2001); Copyright Elsevier, 2001

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Similar studies have been also developed over the years for the Czochralski (CZ) conﬁguration and related models (annular geometry; see, e.g., Khine and Walker, 1995, and Kakimoto and Liu, 2006). In addition to the studies for a laterally heated cylindrical container ﬁlled with liquid metal (which, as mentioned earlier, has extensive background applications in the ﬁeld of crystal growth), some work also concerned the classical conﬁguration in which the cylinder is heated from below and cooled from above (Rayleigh–B´enard convection; see Section 4.10.1 for the case without magnetic ﬁelds). Relevant and interesting numerical studies are due to Touihri et al . (1999b) for constant axial and horizontal magnetic ﬁelds and to Grants and Gerbeth (2004) in the case of both static and rotating ﬁelds (see Section 13.3 for further information on the latter case). Touihri et al . (1999b), in particular, illustrated that a vertical magnetic ﬁeld does not change the symmetries that occur in the pure thermal case, whereas a horizontal magnetic ﬁeld reduces the number of symmetries (suppressing, e.g., the axisymmetric mode m = 0). Stabilization was found to be much stronger with a vertical magnetic ﬁeld. Further details on such interesting dynamics are developed below. Since the ﬂow with a vertical magnetic ﬁeld has the same symmetries as that without a magnetic ﬁeld, similar convective modes (m = 0, 1 and 2) occur. However, they are not equally stabilized (Figure 13.8). For A = 1, convection sets in with azimuthal wavenumber m = 1 followed by the modes m = 2 and 0. All the primary thresholds increase with the Hartmann number, conﬁrming the phenomenon of stabilization of the convection by application of the magnetic ﬁeld. The case with horizontal magnetic ﬁeld deserves even more attention. As expected, the use of a horizontal magnetic ﬁeld can change the symmetries of the physical situation. In fact, only three reﬂections are kept: a reﬂection with respect to the horizontal midplane and two reﬂections with respect to vertical midplanes, the plane parallel to the direction of the magnetic ﬁeld and the plane orthogonal to this direction. This has two important consequences. First, the axisymmetric mode

Figure 13.8 Effect of a vertical magnetic ﬁeld on the onset of RB convection in a cylindrical enclosure. Evolution of the primary thresholds Racr , corresponding to the modes m = 0, 1 and 2, as a function of the Hartmann number Ha (A = 1; Ra and Ha based on the cylinder height). After Touihri et al. (1999b); Reproduced by permission of the American Institute of Physics

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Thermal Convection: Patterns, Evolution and Stability

disappears because the azimuthal invariance is no longer valid [this mode is transformed into a new oriented mode consisting of two counter-rotating rolls and referred to by Touihri et al . (1999b) as m = 02, i.e. a combination of the m = 0 and m = 2 modes]. Second, each of the azimuthally asymmetric modes (m = 1 and m = 2), which in the absence of a magnetic ﬁeld were invariant by azimuthal rotation, can display two possible orientations under the effect of the horizontal ﬁeld (either parallel or perpendicular to B). Thus, a total of ﬁve distinct modes are possible (m = 02 and m = 1 or m = 2 with orientation parallel or perpendicular to the imposed ﬁeld). These modes are differently stabilized, weakly if the axis of the rolls is parallel to B and strongly if the axis is perpendicular, as shown in Figure 13.9. It is also worth mentioning the subsequent study by Houchens et al . (2002), who presented two linear stability analyses, the ﬁrst based on a hybrid approach combining an analytical solution for the Hartmann layers adjacent to the top and bottom walls with a numerical solution for the rest of the liquid domain, and the second involving an asymptotic solution for large values of the Hartmann number. By such an approach, they were able to obtain numerically accurate predictions of the critical Rayleigh number in the speciﬁc case of a vertical magnetic ﬁeld for Hartmann numbers ranging from zero to inﬁnity. For parallelepipedic geometries and rectangular layers, there is an even larger amount of publications and interesting results about the symmetries broken or retained according to the direction of the ﬁeld (let us recall that, as illustrated in Section 4.9, a variety of possible symmetries of the emerging pattern is possible for RB convection in parallelepipedic or cubical enclosures). Here we will content ourselves with giving only some ﬂavour of the subject by discussing some recent examples.

Figure 13.9 Effect of a horizontal magnetic ﬁeld on the onset of RB convection. Evolution of the primary thresholds Racr , corresponding to the most important modes, as a function of the Hartmann number Ha (A = 1; Ra and Ha based on the cylinder height). The axisymmetric mode disappears giving an asymmetric mode m = 02, i.e. a combination of the m = 0 and m = 2 modes, whereas the asymmetric modes (m = 1 and m = 2), which were invariant by azimuthal rotation without a magnetic ﬁeld, now have two possible orientations, either parallel (solid lines) or perpendicular (dashed lines) to the applied magnetic ﬁeld B . After Touihri et al. (1999b); Reproduced by permission of the American Institute of Physics

Flow Control by Magnetic Fields

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Solutions of the small perturbation equations for a horizontal ﬂuid layer of inﬁnite extent heated from below with an applied magnetic ﬁeld either in vertical or in horizontal direction were presented by Zierep (2000). It was conﬁrmed that the critical Rayleigh number increases with increasing Hartmann number while the corresponding wavelength decreases. The general inﬂuence of a horizontal magnetic ﬁeld was found to be larger than that of a vertical ﬁeld. An interesting numerical study of three-dimensional natural convection in parallelepipedic enclosures (of ﬁnite horizontal extent) driven by vertical temperature gradients subjected to stationary magnetic ﬁelds of arbitrary direction is due to M¨ossner and Muller (1999). More recently, Burr and Muller (2001, 2002) considered both theoretically and experimentally Rayleigh–B´enard convection in liquid metal layers under the inﬂuence of vertical and horizontal magnetic ﬁelds. Their investigation was based on experiments with eutectic sodium–potassium Na22 K78 (Pr ∼ = 0.02) in a container 20 × 1 × 10 and O(103 ) ≤ Ra ≤ O(105 ) and a linear stability analysis resorting to two-dimensional model equations. The linear stability analysis showed that for the case of horizontal magnetic ﬁelds, the critical Rayleigh number for the onset of convection in the form of quasi-two-dimensional ﬂow is shifted to signiﬁcantly higher values due to Hartmann braking at walls perpendicular to the magnetic ﬁeld (this ﬁnding was conﬁrmed experimentally by measured Nusselt numbers). Furthermore, by experimental investigation, it was found that adjacent to the onset of convection a signiﬁcant region of stationary convection exists with signiﬁcant convective heat transfer prior to the onset of time-dependent convection. Estimates of the local isotropy properties of the ﬂow by a four-element temperature probe demonstrated that further increase in convective heat transport is accompanied by the formation of strong non-isotropic time-dependent ﬂow in the form of large-scale convective rolls aligned with the magnetic ﬁeld displaying a simpler temporal structure compared with ﬂow obtained under the same conditions but without a magnetic ﬁeld and being responsible for effective heat transport. Beyond speciﬁc details, these authors basically provided evidence that when a horizontal magnetic ﬁeld is imposed on a heated liquid metal layer, the electromagnetic forces can be used to drive a transition of a pre-existing three-dimensional convective roll pattern into a quasi-two-dimensional ﬂow pattern in such a way that convective rolls become more and more aligned with the magnetic ﬁeld (this study was extended to the case of a vertical slot by Burr et al ., 2003).

13.2.2 Stabilization of Surface Tension-driven Flows The use of constant magnetic ﬁelds as a means to damp ﬂuid motion and exert some control on the related hierarchy of bifurcations has attracted considerable interest also in the context of ﬂows induced by variations of the surface tension force. An important example is the book of Chandrasekhar (1961) in which much space was devoted to evaluating the inﬂuence of vertical magnetic ﬁelds on convection arising in a ﬂuid layer heated from below with a free upper surface (the canonical Marangoni–B`enard problem treated in the absence of magnetic ﬁelds in Chapter 9).Via a linear analysis approach, it was basically demonstrated that the primary thresholds for the onset of convection from initial thermal diffusive conditions vary as Ha2 . Other interesting studies are due to Nield (1966), Sarma (1979), Wilson (1993) and Maekawa and Tanasawa (1989). A study based on a weakly nonlinear approach was carried out, in particular, by Miladinova and Slavtchev (2001). Concerning Marangoni ﬂow driven by an imposed temperature gradient parallel to the free surface, some linear stability analyses and theoretical studies have appeared considering the model of an inﬁnite planar liquid metal layer with a free surface supporting Marangoni stresses in the absence of steady gravity (in practice, the canonical inﬁnite ‘return ﬂow’ solution whose general stability properties without magnetic ﬁelds have been extensively treated in Chapter 10).

598

Thermal Convection: Patterns, Evolution and Stability

Notably, like the Hadley ﬂow discussed earlier, also such surface tension-driven ﬂow admits analytical expression under the effect of a magnetic ﬁeld. In particular, if the magnetic ﬁeld is parallel to the free surface, that is, satisﬁes B·i y = 0 (hereafter referred to as coplanar ﬁeld), then, following the same arguments as provided in Section 13.2.1 for inﬁnite Hadley ﬂow, it can be concluded that the ﬁeld has no inﬂuence on the basic ﬂow and consequently the parallel ﬂow remains the same as without it [this is exactly the return ﬂow given by Eqs (2.36) and (2.37) in Section 2.4.2]. By contrast, if the magnetic ﬁeld is perpendicular to the free surface, the analytical solution (neglecting exponentially small terms for strong magnetic ﬁeld Ha 1) becomes: $ 1 1 1 1 1 exp − exp − − y Ha + + y Ha − Ha 2 2 Ha2 Ha2 1 y y +1 1 + T = x − Ma − 2 exp − − y Ha 2 2 Ha Ha3 $ 1 y 1 exp − + y Ha + 3 + 2 Ha Ha4

u = −Ma

(13.15)

(13.16)

with the horizontal boundaries located at y = ± 1/2, Ma = σT γ d 2 /µα (γ being the rate of uniform 1 temperature increase along the x axis and d the depth of the layer) and Ha = B0 d(σe /ρν) /2 . Assuming this solution as the basic state, Priede et al . (1995) were the ﬁrst to perform a linear stability analysis to assess the inﬂuence of a magnetic ﬁeld perpendicular to the free surface on the typical hydrothermal instability of Marangoni ﬂow (see Section 10.2 for fundamental information about this type of instability). Limiting their investigation to waves with a longitudinal wavefront (i.e. disturbance propagating spanwise to the basic ﬂow, along the z direction in Figure 13.1), they found that the critical Reynolds number increases with Ha2 , whereas the wavelength of the most unstable mode is inversely proportional to the ﬁeld strength. The companion case with a coplanar magnetic ﬁeld directed spanwise to the basic parallel ﬂow was treated by this group in a separate analysis (Priede and Gerbeth, 1995). Under the same constraint as already invoked for the vertical ﬁeld, that is, waves with a longitudinal wavefront, they found the critical Marangoni number and the wavelength of the most unstable mode to increase directly with the strength of the applied magnetic ﬁeld. Although longitudinal disturbances, that is, waves propagating exactly spanwise to the basic ﬂow (qx = 0, qz = 0), are not the most unstable ones, as explained in Section 10.2 for low Prandtl numbers they are rather close to the most dangerous waves (Figure 10.6). For this reason, some space is devoted in the following to illustrate the salient outcomes of these studies (analysis of waves with a longitudinal front can be useful for understanding the more complicated instability mechanism of oblique waves); in particular, ﬁrst attention is paid to the case of a coplanar ﬁeld directed spanwise to the basic ﬂow, then the ﬁeld perpendicular to the free surface is considered. Notably, for these two limiting cases, even more interesting information was provided by the same group in two subsequent studies (Priede and Gerbeth, 1997c; Priede et al ., 2000), where in addition to the aspects already treated in the previous two analyses, oblique disturbances were considered and interesting theoretical arguments based on an order of magnitude analysis (and ensuing analytical relationships among the parameters of interest) were elaborated to support the discussion. Furthermore, two distinct situations were investigated in terms of thermal boundary conditions: adiabatic free surface and bottom wall with adiabatic or conducting behaviour (let us recall that, as elucidated in Section 10.2.3, the hydrothermal mechanism is suppressed if the free surface is perfectly conducting, which explains why the case with a conducting free surface was not considered as a possible variant).

Flow Control by Magnetic Fields

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Ha

(a) Ha

(b)

Figure 13.10 Critical Marangoni number (a) and wavenumber (b) of longitudinal waves (qx = 0) as a function of the Prandtl number at various Hartmann numbers (magnetic ﬁeld imposed spanwise to the basic ﬂow) for Bi = 0 and both insulating or conducting bottom (c denotes conducting, i insulating; zero-g conditions; Marangoni number deﬁned as Ma = σT γ d 2 /µα , where γ is the uniform rate of temperature 1 increase along the x -axis; Hartmann number deﬁned as Ha = B0 d(σe /ρν) /2 ). After Priede and Gerbeth (1997c); Reproduced by permission of Cambridge University Press

As mentioned above, in the following we begin to discuss such results for the conﬁguration with coplanar spanwise ﬁeld. The critical Marangoni number and the wavenumber for the case of perfectly longitudinal disturbances are plotted versus Pr and several Ha in Figures 13.10a and b, respectively. A synthesis of the behaviours displayed in these ﬁgures can be provided as follows. At different Hartmann numbers and adiabatic boundary conditions and for long-wave disturbances (q 1), the marginal Marangoni number decreases as q −1 with increasing wavenumber (not shown) and attains its minimal (critical) value (Figure 13.10a) at a critical wavenumber qc 1 which scales for low Prandtl numbers as qc ∝ Pr /2 (see Figure 13.10b).

600

Thermal Convection: Patterns, Evolution and Stability

When the magnetic ﬁeld is sufﬁciently strong, it causes a shift of the most unstable perturbation to lower wavenumbers, scaling for Ha 1 as q ∝ Ha−1 . This relation means a stretching of the critical wavelength proportionally to the strength of the magnetic ﬁeld. Similarly, the corresponding critical Marangoni number is increased directly with the strength of the magnetic ﬁeld (Figure 13.10a). It is also worth noting that according to these results, within the range of Prandtl number inﬂuenced by the magnetic ﬁeld, the critical Reynolds number (Ma/Pr) for adiabatic boundaries is almost independent of the Prandtl number. Also, for sufﬁciently small Prandtl numbers, both the 1 critical wavenumber and corresponding Marangoni number, vary as Pr /2 and are weakly affected by the magnetic ﬁeld. Interestingly, such asymptotic dependences evident from the numerical results were also deduced by Priede and Gerbeth (1997c) directly from the order of magnitude estimates of the linearized disturbance equations. Such estimates allowed many more characteristics of the instability to be revealed than those obvious from the computed data. Numerical results also proved that there is a signiﬁcant difference concerning the critical wavelength between the cases of a conducting and an adiabatic bottom. It is evident, in fact, in Figure 13.10b that for a conducting bottom the critical wavelength is comparable to the depth of the layer, whereas in the adiabatic case it is substantially longer. Furthermore, the critical Marangoni number becomes considerably higher than that for an adiabatic bottom and does not change much with increasing Prandtl number, in contrast to the marked increasing characteristic of the previous case (Figure 13.10a). Thereby, it was concluded that for longitudinal waves the magnetic ﬁeld necessary to obtain a signiﬁcant inﬂuence on the instability depends strongly on the thermal boundary conditions. In particular, the instability is more sensitive to the magnetic ﬁeld for a non-adiabatic bottom. For the more general case in which the disturbance is allowed to be oblique (qx = 0, qz = 0), scalings of the longitudinal disturbances were shown also to remain valid for the most unstable oblique waves when the boundaries are adiabatic. However, this was not the case for non-insulating boundary conditions, where the advection of perturbations by the basic 2D ﬂow was found to interfere signiﬁcantly in the mechanism of the longitudinal waves at very small deviations of the propagation direction from the spanwise direction (the reader is also referred to the propaedeutical arguments provided in Section 10.2.2). Perhaps the most interesting result that these authors obtained for oblique waves is that, in general, a coplanar magnetic ﬁeld stabilizes all disturbances, except those with wavevectors perpendicular to the ﬁeld (i.e. wavefronts aligned with the ﬁeld), this effect resulting in the alignment of the wavefront of the most dangerous disturbance along the magnetic ﬁeld with increasing strength. As an interesting articulation of this ﬁnding, the critical Marangoni number was found to increase with the magnetic ﬁeld only until the wavefront of the most dangerous disturbance aligned along the ﬁeld. These intriguing behaviours are not as straightforward as one would assume and deserve some additional discussion, given in the following. Priede and Gerbeth (1997c) observed the most effective stabilization for oblique waves still to be provided by the ﬁeld spanwise to the basic ﬂow. The curve for the minimal Marangoni number for different strengths of magnetic ﬁeld spanwise to the basic ﬂow, however, was found to consist of two intersecting branches, where one branch develops from a transverse mode (2D hydrothermal wave) whereas the other evolves from the longitudinal mode. According to their analysis, for a sufﬁciently weak magnetic ﬁeld, the most unstable mode resides on the longitudinal branch (as already mentioned, when no magnetic ﬁeld is considered the most dangerous disturbances are almost longitudinal). An increase in the magnetic ﬁeld, however, causes the critical Marangoni number to rise, while the minimal Marangoni number of the purely transverse wave mode remains unaffected by the magnetic ﬁeld

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of the given orientation. At a certain strength of the magnetic ﬁeld, the local minimum of the longitudinal wave branch rises above the critical value for the transverse waves. At this point, the global minimum jumps to the purely transverse mode. Since the latter has a wavefront aligned with the magnetic ﬁeld, in the light of the arguments provided above nothing can change with a further increase in the ﬁeld strength. This leads to the natural conclusion that, remarkably, exactly the transverse mode yields the maximum possible critical Marangoni number over all directions of propagation. The numerical results of Priede and Gerbeth (1997c) showed, in particular, that the alignment for a conducting bottom takes place at lower Hartmann numbers than that for an adiabatic bottom. However, the maximum attainable critical Marangoni number for the conducting case was observed to be practically indistinguishable from that of the adiabatic case (this was ascribed to an intrinsic property of the transverse mode that, in contrast to the longitudinal mode, is almost insensitive to the thermal properties of the bottom). The later analysis by Priede and Gerbeth (2000) was concentrated on the case of a magnetic ﬁeld perpendicular to the free surface (previously considered by Priede et al ., 1995). It was reported that more effective stabilization is provided by a magnetic ﬁeld perpendicular to the liquid layer that, in contrast to the coplanar one, effectively retards the basic ﬂow [as shown by Eqs (13.15) and (13.16)]. They calculated numerically thresholds of convective instability for both longitudinal and oblique disturbances and also asymptotically by considering the Hartmann and Prandtl numbers as large and small parameters, respectively. It was elucidated that the magnetic ﬁeld has a stabilizing effect on the ﬂow with the critical temperature gradient for the transition from steady to oscillatory convection increasing as the square of the ﬁeld strength (as also does the critical frequency), whereas the critical wavelength reduces inversely with ﬁeld strength (Figure 13.11). Interestingly, for longitudinal waves such asymptotic dependences were straightforwardly explained according to the effect of the magnetic ﬁeld on the basic ﬂow, which in a strong magnetic ﬁeld (see also Section 13.1.2) forms a jet at the free surface having the thickness of the Hartmann layer Ha−1 (in fact, if this is the effective length scale in the strong magnetic ﬁeld, then the wavenumber, being proportional to the characteristic length, will scale as Ha, whereas both the Marangoni number and the frequency, being proportional to the square of the characteristic length, will scale as Ha2 ). The instability of this jet was studied separately and the related stability properties were found to be in good agreement with those of the whole layer (see the solid line in Figure 13.11). The ﬁeld strength at which these asymptotics become effective, that is, at which the behaviour of the Hartmann layer can be regarded as a good approximation of the whole layer, in particular, was found to depend on the Prandtl number: the smaller the Prandtl number, the higher is the ﬁeld strength needed (this was ascribed to the ﬁnite depth of the layer that, in addition to the thickness of the Hartmann layer, becomes a signiﬁcant length scale at sufﬁciently small Prandtl numbers). On the basis of the arguments elaborated by these authors, additional necessary insights into these behaviours can be provided as follows. For the thickness of the Hartmann layer to be the only relevant length scale, disturbances should decay over a distance shorter than the depth of 1 the layer, which is the case provided by Ha Pr− /2 . Obviously, in this case the instability is insensitive not only to the depth of the layer but also to the actual thermal boundary conditions at 1 the bottom (this changes at Ha ∼ = Pr− /2 where disturbances begin to extend over the whole depth). In contrast to the base ﬂow, the critical disturbances, having a long wavelength at small Prandtl numbers, extend from the free surface into the bulk of the liquid layer over a distance exceeding 1 1 the thickness of the Hartmann layer by a factor O(Pr− /2 ); therefore, for Ha ≤ Pr− /2 the instability tends to be inﬂuenced by the actual depth of the layer.

Thermal Convection: Patterns, Evolution and Stability

Rescaled Marangoni number, Macr/Ha2

602

101

Conducting 100 Ha = 10 30 100 Hartmann-layer mode

Insulating 10–1 10–5

10–4

10–3

10–2

10–1

100

Rescaled Wavenumber, qc/Ha

Pr (a)

10–1 Conducting Ha = 10 30 100 Hartmann-layer mode

10–2

Insulating 10–3 10–5

10–4

10–3

10–2

10–1

100

Pr (b)

Figure 13.11 Rescaled critical Marangoni number (a) and wavenumber (b) of longitudinal waves (qx = 0) as a function of the Prandtl number at various Hartmann numbers (vertical magnetic ﬁeld) for adiabatic or conducting bottom (zero-g conditions; Marangoni number deﬁned as Ma = σT γ d 2 /µα , where γ is the 1 uniform rate of temperature increase along the x -axis; Hartmann number deﬁned as Ha = B0 d(σe /ρν) /2 ). After Priede and Gerbeth (2000); Reproduced by permission of Cambridge University Press

For such moderate magnetic ﬁelds the instability threshold is sensitive to the thermal properties of the bottom of the layer and the dependences of the critical parameters on the ﬁeld strength are more complicated. Various instability modes become possible depending on the thermal boundary conditions and the relative magnitudes of Prandtl and Hartmann numbers. Although interesting, these aspects are not discussed further here due to page limits. A subsequent linear stability analysis of the subject is due to Morthland and Walker (1999). Henry et al . (2005) further extended these studies by taking into account the presence of gravitational convection (mixed buoyancy–Marangoni ﬂows with magnetic ﬁelds). Like the case of thermogravitational convection discussed in Section 13.2.1, other studies have also appeared expressly focused on ﬁnite-sized geometries relevant to techniques of crystal growth from the melt. As an example, interesting numerical results for Marangoni ﬂow in Czochralski

Flow Control by Magnetic Fields

603

conﬁgurations/models (annular geometries) are due to Khine and Walker (1994); later, Jing et al . (2000) analysed the combined effect of buoyancy forces, Marangoni forces and static magnetic ﬁelds. For the liquid bridge, it is worth mentioning the linear stability analysis by Prange et al . (1999). Before embarking on the related discussion, let us recall that, in general, for axisymmetric geometries axial magnetic ﬁelds are known to cause a concentration of the Marangoni ﬂow near the cylindrical free surface (see, e.g., Lappa, 2005e, and references therein). Since Lorentz forces are purely radial, in fact, the ﬂow tends to avoid radial momentum. As a result, the radial extent of the Marangoni vortex decreases with increasing Hartmann number (among other things, this also leads to the formation of weak secondary vortices inside the bulk; for instance, Prange et al ., 1999 observed that these secondary vortices are at least two orders of magnitude weaker than the primary vortex). It is also known that the width of the primary Marangoni vortex is approximately proportional 1 to Ha− /2 for sufﬁciently large Hartmann numbers (this behaviour is in agreement with the general considerations in Section 13.1.2). Interestingly, Prange et al . (1999) reported that for liquid bridges the surface velocity also 1 scales with Ha− /2 if the Hartmann number exceeds a given threshold.

Figure 13.12 Marginal Reynolds number and wavenumber for the ﬁrst instability of Marangoni ﬂow in liquid bridges as a function of the aspect ratio and Ha (Pr = 0.02, Gr = 0; adiabatic free surface; Re and Ha based on the axial extension). After Prange et al. (1999); Reproduced by permission of Cambridge University Press

According to their analysis, for the low-Prandtl-number ﬂuids considered, the most dangerous disturbance is a nonaxisymmetric steady mode of hydrodynamic nature as in the case of pure Marangoni ﬂow (let us recall that unlike the shallow layers examined earlier, the primary instability of low-Pr Marangoni ﬂow in liquid bridges is not of a hydrothermal nature; see Section 10.4.1). These authors found the axial magnetic ﬁelds to stabilize the basic state, the stabilizing effect increasing with the Prandtl number and decreasing with the zone height, the heat transfer rate at the free surface and buoyancy when the heating is from below. The magnetic ﬁeld was also observed to inﬂuence the azimuthal symmetry of the most unstable mode.

604

Thermal Convection: Patterns, Evolution and Stability

Some of the results obtained by these authors are shown in Figure 13.12. As already mentioned, the stabilizing inﬂuence of Lorentz forces strengthens with increasing the radius to height ratio. Concerning the azimuthal structure of the steady supercritical ﬂow, magnetic ﬁelds actually favour modes with larger wavenumbers if radius/height > 1, as can be seen by the shift of the intersection of the critical curves for different wavenumbers in Figure 13.12. Nevertheless, for radius/height < 1 the effect is the opposite, i.e. axial magnetic ﬁelds favour m = 1 over m = 2. Prange et al . (1999) ascribed this trend to a stabilizing effect induced by azimuthal thermocapillary forces. For experimental investigations on the subject, the reader may consider, for example, Kimura et al . (1983) and Dold et al . (1998). It is also worth citing Lan and Yeh (2005), who carried out numerical simulations for both axial and transverse (perpendicular to the zone axis) ﬁelds, focusing on the secondary (Hopf) ﬂow bifurcation. They found that a transversal direction of the ﬁeld is more efﬁcient (oscillatory ﬂow can be suppressed at lower values of Ha), but asymmetric zone shapes are induced (the melt ﬂow on the meridian plane perpendicular to the magnetic ﬁeld is not suppressed like that on the parallel meridian plane; this leads to a longer zone length there and can also inﬂuence signiﬁcantly the resulting azimuthal wavenumber; the reader is also referred to the earlier study by Robertson and O’Connor, 1986).

13.3

Rotating Magnetic Fields

Transverse rotating magnetic ﬁelds (RMFs) follow a different approach with respect to the static uniform ﬁelds treated in the preceding section: remarkably, the guiding principle is not to damp the ﬂuid motion, but to dominate the irregular ﬂow structure in axisymmetric geometries (i.e. cylinders or annular conﬁgurations) by overlaying it with fast, azimuthal, axisymmetric ﬂows. For the typical rotation frequencies and ﬁeld strengths used in crystal-growth experiments, an RMF produces, in fact, a steady, axisymmetric, azimuthal Lorentz force on the melt. This body force drives an azimuthal velocity in the melt in the direction that the RMF rotates. Mathematical models and numerical procedures to study RMFs and their interaction with liquid metals have been improved over the years. For a detailed review of earlier studies on MHD ﬂows driven by the RMF the reader may consider Gelfgat and Priede (1995). In practice, the interaction of an RMF with a conducting liquid depends on several parameters, including the material properties of the liquid (kinematic viscosity ν, density ρ, electrical conductivity σe , magnetic permeability µp ), the characteristics of the RMF (the magnetic induction B0 , the angular frequency of the magnetic ﬁeld ω = 2πf , where f is the frequency of the alternating current used to generate it, the ratio of the number of poles to the number of phases in the current source p) and the geometry of the system (height L and radius R = D/2). These parameters can be grouped into the following dimensionless quantities: Ha = B0 R

σe 2ρν

1 2

(13.17)

that is, the Hartmann number based on the radius R of the considered geometry; the rotation Reynolds number deﬁned as Reω =

ωR 2 pν

(13.18)

and the relative frequency of the magnetic ﬁeld (also referred to as the shielding parameter): ω¯ = µp σe ωR 2

(13.19)

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605

Ha and Reω can be uniﬁed leading to the so-called magnetic Taylor number: Ta = Ha2 Reω =

σe ωB02 R 4 2ρν 2 p

(13.20)

As explained earlier, the RMF induces a Lorentz force, F L , driving the liquid in rotation. In general, the induced F L is composed of two parts: a mean axisymmetric contribution, which is time independent, and a time-dependent contribution, oscillating with a frequency doubled with respect to the angular frequency of the RMF. The latter component has a minor effect on the ﬂuid ﬂow and is, therefore, neglected in crystal growth traditional applications (Witkowski et al ., 1999). In general, in fact, inertia limits the response to the periodic body force with frequency 2ω, so that the ﬂow driven by the periodic, nonaxisymmetric part of the electromagnetic body force is negligible compared with the ﬂow driven by the steady, axisymmetric part. The discussion below is focused on the canonical model used in the literature to deal with RMFs in typical cases of technological interest, that is, RMFs with low induction and low frequency. The low-induction condition means that the current induced by the ﬂuid ﬂow is negligible in comparison with the current generated by the RMF, which is satisﬁed for ω #c

(13.21)

where #c is the ‘core rotation rate’ (Moffatt, 1965): #c =

2 2 1 3 ν Ta 3 2 c R

(13.22)

where c is a constant (c = 0.675). Equation (13.21) can be rewritten as (Priede and Gelfgat, 1996) Ha4 c2 Reω

(13.23)

The low-frequency condition reads (Priede and Gelfgat, 1996): ω¯ = µp σe ωR 2 < 1

(13.24)

which implies that the secondary magnetic ﬁeld can be neglected. The nature and origin of such secondary ﬁeld are not obvious; some additional explanations are provided in the following. In addition to the primary periodic magnetic ﬁeld produced by the inductors at equally spaced azimuthal positions around the cylindrical liquid zone, there is a secondary magnetic ﬁeld that is produced by the periodic electric currents in the liquid and which partially cancels the primary magnetic ﬁeld in the interior of the liquid. The intensity of this secondary magnetic ﬁeld depends on the value of the aforementioned shielding parameter, µp σe ωR 2 . When it is <1 the secondary magnetic ﬁeld is negligible, that is, the low-frequency condition implies that the so-called ‘skin’ depth of the RMF is larger than the radius of the considered cylindrical geometry. From a practical point of view, as anticipated at the beginning of this section, the effect of the RMF on a conductive ﬂuid in the absence of other types of convection (i.e. pure magnetic ﬂow) is characterized by the generation of a primary azimuthal axisymmetric (often referred to as ‘swirling’) ﬂow (Moffatt, 1965; Davidson and Hunt, 1987). Several authors, for example Priede and Gelfgat (1997), Kaiser and Benz (1998) and M¨ossner and Gerbeth (1999), carried out simulations to study in detail the structure of such ﬂows. It was understood that they can also undergo transition to higher modes of convection when the characteristic number is sufﬁciently increased. These higher modes of convection are, in general, axisymmetric like the primary one. However, Grants and Gerbeth (2002) found that the ﬁrst linearly unstable mode can be three-dimensional (see Figure 13.13) over a fairly wide range of aspect ratios; in fact, this ﬂow instability was observed

606

Thermal Convection: Patterns, Evolution and Stability

(a)

(b)

Figure 13.13 Taylor–Gortler-type ¨ vortices for two values of the Taylor number: (a) 4.5 × 104 and (b) 3 × 105 . Courtesy of J. Stiller

to set in prior to its axisymmetric counterpart with relatively low frequency at diameter-to-height ratios between 0.5 and 2. These axisymmetric and three-dimensional instabilities were found to have similar characteristic features (they originate in the cross-section of the horizontal and vertical rotating boundary layers and excite inertial waves in the inviscid core). RMFs have been applied over the years to a variety of materials science problems. Many of the existing studies in the ﬁeld of crystal growth from the melt considered vertical Bridgman-like conﬁgurations as in the case of static uniform ﬁelds. For the FZ case, relevant information is available in the analyses of Fischer et al . (1999), Dold et al . (2001) and Walker et al . (2003), who approached the problem in terms of time integration of the full time-dependent three-dimensional governing equations, experimental investigation and linear stability analysis, respectively. They illustrated that the axial variation of the centrifugal force due to the azimuthal velocity drives steady, axisymmetric, circulations with radially inward ﬂows near the planar liquid/solid interfaces and with radially outward ﬂow near the plane midway between these interfaces. For the FZ process, these circulations due to the RMF essentially reinforce the base Marangoni circulations. Walker et al . (2003), in particular, reported that, as the strength of the RMF is increased from zero, its effects are initially destabilizing, that is, the critical value of the Reynolds number for the instability of Marangoni ﬂow, Recr , decreases; however, as the RMF strength is further increased, Recr reaches a minimum and then increases to values which are considerably higher than the Recr without an RMF. In practice, this stabilization emerges once the azimuthal velocity produced by the RMF becomes large enough that the coupling of the radial convection of angular momentum and the centrifugal force due to the azimuthal velocity can signiﬁcantly alter the basic Marangoni convection. For application of RMFs to the travelling solvent method, the reader may consider Jaber and Saghir (2006) and references therein. Some studies have also appeared addressing the effect of RMFs on fundamental Rayleigh– B´enard convection in cylinders heated from below, cooled from above, with adiabatic sidewall (Friedrich et al ., 1999; Walker et al ., 2004).

Flow Control by Magnetic Fields

13.4

607

Gradients of Magnetic Fields and Virtual Microgravity

As illustrated in the earlier sections of this chapter, magnetic ﬁelds are widely applied to control the ﬂow of electrically conducting melts during solidiﬁcation processing, such as bulk single-crystal growth in the semiconductor industry. In such cases, the main reason for the use of an external magnetic ﬁeld is attributed to the Lorentz force induced by moving electrically conducting ﬂuids. Owing to the low electrical conductivity of aqueous solutions, typically four and ﬁve orders of magnitude smaller than that of the semiconductor melt [for example, the electrical conductivity of 25% by weight salt (NaCl) solution at 25 ◦ C is about 21#−1 m−1 , while it is 1.29 × 106 #−1 m−1 for silicon melt], the application of magnetic ﬁelds in solution crystal growth has not attracted similar attention. With the recent development of high magnetic ﬁeld technology, however, the effects of high magnetic ﬁelds on solution crystal growth have begun to attract interest. As an example, the possible utilization of such a strategy for the protein crystallization problem was suggested in the pioneering work of Wakayama et al . (1997) and Lin et al . (2000). Their studies proved experimentally that in an inhomogeneous (not uniform) magnetic ﬁeld, the magnetic body force, not related to the Lorentz effect [the reader is referred to the general theoretical background provided in Section 1.9 and, in particular, to Eq. (1.185)], can be exerted on both diamagnetic and paramagnetic materials to obtain effective ﬂow control. Along these lines, Evans et al . (2000) showed that a magnetic gradient can be used to damp convection in any material, conducting or non-conducting. Unlike the Lorentz force, in fact, it does not require the material to be electrically conductive and possibly applies to electrically nonconducting ﬂuids, such as melts of inorganic oxides and organic solvents and aqueous solutions with lower conductivity. When this force [which is intimately related to the density of magnetization of the medium as expressed by Eq. (1.193), and, hence, is a magnetization force] acts on either an aqueous solution of a paramagnetic salt or a protein solution, which is diamagnetic, convection can be suppressed in the solution. The present section provides some relevant theoretical information in such a context. The discussion is carried out within the framework of the brilliant ideas and concepts pointed out by Qi and Wakayama (2004). Generally, the magnetic forces acting on a ﬂuid element can be derived from the Helmholtz free energy, which is stored in the ﬂow medium by a magnetic ﬁeld. According to Landau and Lifshitz (1960), the total magnetic force F in dimensional form can be expressed as ∂µp 1 H2 F = ∇ H 2ρ (13.25) − ∇µp + µp [J f ∧ H ] 2 ∂ρ 2 where the ﬁrst term on the right-hand side represents magnetostriction forces caused by magnetic ﬁeld gradients, the second term represents the dependence of the magnetic force on the gradients of magnetic permeability µp and the last term accounts for the well-known Lorentz force F L in electrically conducting ﬂuids. Moreover, H is the magnitude of the magnetic ﬁeld intensity vector H (also known as the magnetic ﬁeld strength to distinguish it from the magnetic ﬂux density B that is given by µp H ), ρ is the ﬂuid density and J f is the electrical current density vector. Obviously, for electrically nonconducting ﬂuids, J f = 0 and the last term vanishes. The magnetic permeability µp can be expressed as follows: µp = µ0 [1 + ρχg (T )]

(13.26)

where χg (T ) ∝ T −1 for paramagnetic materials χg (T ) = constant for diamagnetic materials

(13.27a) (13.27b)

608

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where µ0 is the permeability of vacuum, T is the temperature and χg is the mass magnetic susceptibility. If χg > 0, the substance is paramagnetic (e.g. oxygen gas and paramagnetic salts). Its magnetic susceptibility follows Curie’s law, χg = c/T , where c is a constant, characteristic of the substance. Most substances are diamagnetic, χg < 0 (e.g., water and protein crystals). For a diamagnetic substance, χg is a negative constant and independent of temperature. Substituting Eq. (13.26) into Eq. (13.25) (Qi and Wakayama, 2004), the ﬁrst term on the right-hand side of Eq. (13.25) becomes ∂µp 1 1 1 1 = ∇(µ0 ρχg H 2 ) = µ0 ρχg ∇(H 2 ) + µ0 H 2 ∇(ρχg ) (13.28) ∇ H 2ρ 2 ∂ρ 2 2 2 and since the second term on the right-hand side of Eq. (13.25) reads 1 H2 ∇µp = − µ0 H 2 ∇(ρχg ) 2 2 then Eq. (13.25) can be rewritten as −

F = Fm + FL

(13.29)

(13.30)

where 1 (13.31) µ0 ρχg ∇(H 2 ) 2 is the aforementioned magnetization force not related to the Lorentz effect. For most cases except anisotropic susceptibility existing in some materials, F m depends only on the gradient of magnetic ﬁeld magnitude, ∇(H 2 ), but ﬁeld direction does not play any effect . As explained earlier, in nonconducting ﬂuids this force can be used as a means to control convection instead of the Lorentz force. Along these lines, in particular, it is worth highlighting that as the magnetization force F m expressed by Eq. (13.31) is a body force formally analogous to the gravitational force F g = ρg (with the volumetric magnetic susceptibility χ = ρχg and ∇(H 2 ) playing the same role of the density and gravity acceleration, respectively), this leads to the effective possibility of using this force to establish a virtual environment of partially reduced gravity (Wakayama et al ., 2005). For instance, when an upward magnetization force F m acts in the vertical direction (z) homogeneously within the solution, the total force F under the one-dimensional magnetic ﬁeld gradient can be written as dH (13.32) i g = ξρgi g F = F g + F m = ρ g 0 + µ 0 χg H dz Fm =

which means that the ﬂuid in such conditions will behave as if it were experiencing a gravity acceleration reduced by a factor µ0 χg dH (13.33) H ξ =1+ g0 dz The foregoing discussion supports by simple theoretical arguments the idea that it is possible to establish a condition with g g0 (ξ 1) on Earth by simply tuning the intensity of the magnetic ﬁeld and its gradient along the vertical direction. It is a rather new and a still not well known technique, made possible by the recent advances in the manufacturing processes of magnets, that deserves further investigation.

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Index

actinoform clouds 112 advection force 134 aesthenosphere 104–105 amplitude equations 26 angular frequency 32 annular conﬁguration 85, 368–375 annular liquid pool 475–490 annular liquid pool, 2D hydrothermal wave (HW2) 476–477 annular liquid pool, chaos and fractal behaviour 483–487 annular liquid pool, reverse annular conﬁguration 487–488 annular liquid pool, role of curvature, heating direction and gravity 488–490 annular liquid pool, spiral pattern (HW1) 478–480 annular liquid pool, stationary radial rolls 480–483 anticentrosymmetric mode (ACS) 241, 242 antisymmetric–antisymmetric mode 151 antisymmetric–symmetric mode 151 aperiodic ﬂows 25 arbitrary amplitude, disturbances of 36 Archimedean spiral 175 Arnoldi’s method 240 aspect ratio 77 aspect ratio, moderate 165–170 aspect ratio, mall 170–173 asymmetric modes 381 asymmetric oscillatory modes of convection 407–412

asymmetric squares 145, 146, 147 asymptotic core-ﬂow solution 85 atmospheric convection 108–117 attractors 3, 47–48 augmentation 544 averaged formulation 293 azimuthal wavenumber 31, 389–390 balance equations, incompressible formulation of 18–19 balance equations, microscopic derivation 6–7 baroclinic instabilities 101 baroclinic waves 108 basic state 28 bifurcation 3, 25–26, 188, 319 bimodal convection 128, 129, 130, 131, 137 Biot number 74, 83 bistability problem 142 blobs, thermal 131 bond number 75, 77, 78 bound charge 61 boundary condition (BC) 4, 44 boundary layer exit jets 90 boundary-layer regimes 89–94 boundary-layer structures 562 boundary, system 3 Boussineq equations 544 Boussinesq approximation 290 Boussinesq model 64–66 Boussinesq symmetry 527 Bridgman crystal growth 551 Bridgman method 96–97

Thermal Convection: Patterns, Evolution and Stability Marcello Lappa  2010 John Wiley & Sons, Ltd

660

Index

Brunt-V¨ais¨al¨a (BV) frequency 123, 124, 238, 240, 545 budget of internal energy 13 Buisse balloon 127, 128, 130, 134, 135 buoyancy convection 64–66 buoyancy velocity 65 Busse balloon 124–132, 179, 457 Busse oscillations 276 butterﬂy effect 48–51 Callisto (Jupiter satellite) 107–108 capacity dimension 55 capillary number 72 cats-eye patterns and temperature waves 247–253 centrosymmetric mode (CS) 241, 242, 244 Chandrasekhar functions 124 chaos theory 46 chaos, transition to 25–26 chaotic length scale 182 chaotic states, universal properties 46–58 characteristic number 25 charge density 59 Chebyshev polynomial expansions 191 Childress theory 205–208 circular containers 328–331 circular cylindrical problem 165–173 circular cylindrical problem, azimuthal structure 165–170 circular cylindrical problem, effect of lateral boundary conditions 165–170 circular cylindrical problem, targets and PanAm textures 170–173 cloud patterns 110–112 cloud streets 113 codimension-two point 273–275 cold ﬁnger 464, 500 combination resonance 226 compatibility relations 44 competition of disturbances of convection 225–227 complex order 519, 525–532 conditional stability 27–28 conductive regimes 88, 89–94, 343

consistent pressure Poisson equation (CPPE) 44 container vibration 289, 517 containerless processing 95 continental drift 105 continuation method 241 continuity equation 9–10 convection 5 convection patterning 554 convective dynamo 188 convective ﬂuxes 7 core 79 Coriolis effect 192 Coriolis force 193 corner vortices 344 correlation dimension 57, 180 Cowling’s theorem 189 crawling rolls 276 critical condition 32 cross-roll instability 125, 127, 128, 129, 130 crystal growth from the melt 96–102 cubic symmetry 183, 184, 186 cubical box 160–161 curl operator 16, 17, 41, 42 current density J f 59 cylindrical geometries, heating and 262–269 Czochralski (CZ) method 99–101, 341 damped driven harmonic oscillator 123 deep convection 110 density of entrophy 16 density of magnetization of the medium 61 density of polarization of the medium 61 density of viscous heating 18 deterministic systems 5 detrimental effects induced by convective phenomena 101–102 diamagnetic magnetization 61 diamagnetic materials 607 diffusive–viscous regime 197–198 dimension density 182 directional solidiﬁcation 96–97 dislocation dynamics 132, 133–135

Index

dissipative parallel shear ﬂow 35 dissipative part of the stress tensor 14 dissipative structures 3, 21–25, 119 disturbance growth rate 32 dodecadhedron, symmetry of 183, 184 domain coexistence 525 doped semiconductors 96 dynamic particle accumulation structure (PAS) 419–420 dynamic viscosity 14 Earth, core convection 106–107 Earth, core 103 Earth, crust 103 Earth, layered’ structure 103 Earth, mantle 103 Earth, mantle convection 104 Eckhaus instability 125, 127, 128, 134, 226, 457 EEA 67 Eigenvalue problem 28–30 Ekman vortices 345 electric dipole moments 60 electric displacement ﬁeld 62 energy 9 energy dissipation rate 23 energy equality 21–25 energy stability theory 36–39 energy theory 36 equation of evolution 28 ESA MSDB 67 ESA-COF, 551, 552, 553, 554, 555 Europa (Jupiter satellite) 107–108 evolution 2 excitation source 237 exponential matrix 28–30 extensive chaos problem 179–182 extremum problem 39 feedback loop, nonlinearity as 4 Feigenbaum bifurcations 545 Feigenbaum scenario 46–47, 54 ferromagnetism 62 ﬁeld of pattern formation 2 ﬁnite element method 158

661

ﬁxed-point attractors 142 ﬂoating-zone (FZ) Method 95, 97–99, 341 Floquet (disturbance) wavenumbers 125, 137 ﬂow stability 25–26 ﬂower pattern 324–325, 487 ﬂux density 8, 10 ﬂywheel mechanism 202–208 Fourier law 14 Fourier spectrum 47 Fourier-Chebyhev discretization 326 fractals 5, 54 fractional-step method 42 free-fall 67, 69 frequency locking 47 frequency skip 47 Froude number 520–522, 524 fundamental problem (FP) 23, 24 g-jitter 25, 69, 70, 71, 289, 292, 293, 294, 517, 551–553, 554 Galerkin expansion 124 Galerkin spectral method 163, 538 Ganymede (Jupiter satellite) 107–108 general balance equation 8 generalized dimension 56 generalized disturbance energy, global budget 36–38 geodynamo problem 188–193 Gershuni’s formulation 293, 303, 304, 305, 533, 538, 540, 545–546, 572 Glatzmaier-Roberts dynamo 192 global thermodynamic equilibrium (GTE) 9 gradient Richardson number 114 gradient-freeze technique 95 Grashof number 66, 200, 218, 220–225, 228, 229, 230, 231, 249, 250–252, 259, 267, 268 gravity 517 gravity modulation 289 Hadley ﬂow 96–97, 100, 108–110, 80, 215–269, 279 harmonic patterns, purely 526–529

662

Index

Hartmann boundary layers 584 Hartmann number 583 Hausdorff dimension (Hausdorff – Besicovitch dimension) 55–56, 57, 180 heat transfer (internal energy) 9 heteroclinic cycles 183–185 hexagonal patterns 142–149, 320–325 hexagonal patterns, inverted 325–326 hexagonal patterns, rotating 487 high-order solutions 325–326 high Prandtl number liquids 320–325 Hodge theorem 313 Hopf bifurcation 30, 46, 457 horizontal Bridgman (HB) technique (open boat) 96, 97 horizontal shear, sinuous instabilities created by 200–201 hybrid pulsorotating pattern 438, 439 hydrodynamic modes 219–223 hydrothermal waves 345–368, 395–396 hydrothermal waves, control of 567–575 hydrothermal waves, oblique 345, 349–351 hydrothermal waves, with spiral pattern (HW1) 478–480 hydrothermal waves, 2D 476–477 hysteresis loops 589 Icy Galilean Satellites 107–108 IDEA 67 ideal straight rolls (ISRs) 119–123, 142, 143 inclined layer convection 272–279 inclined side-heated slots 279–281 inclined systems 86–89, 550 induced shear 543 industrial setups, small-scale 95–103 inﬁnite horizontal layer 215–227, 429–436, 538–544 inﬁnite layer, solutions for 297–299 inﬁnite vertical layer 247–253, 548–550 inﬁnitesimal disturbances 27–28 information dimension 57 instability 2 intermediate velocity 45

internal energy 12–13, 15 internal gravity waves 122–124 internal waves 115, 123, 235 International Space Station requirements curve 70, 71 interphase layer 70 interrelation 2–4 intrusion layers 90 inverse cascade 46 inverse theorem of calculus 43, 60 inversion symmetry 111 Inviscid–Diffusive Regime (IVD) 198–200, 208 Inviscid–Nondiffusive Regime (IVND) 200, 201, 208 Io (Jupiter moon) 107–108 irregular 1 ISRDB 67, 68 Jupiter satellites 107–108 Kaplan – Yorke equation 53, 57 Kelvin–Helmholtz waves 114, 201 kinematic dynamo problem 189 kinetic energy 15 kinetic energy, degradation 18 knot convection 136 knot instability 125, 128, 130 knots 130 Kolmogorov entropy 52, 53 laboratory setups, small-scale 95–103 Ladyzhenskaya decomposition theorem 45 lateral conﬁnement, effects of 327–333 lateral heating 544 lateral walls, convection with 149–150 laterally heated systems and parametric resonances 538–550 law of conservation of charge 59 l’Hospital’s rule 57 linear stability analysis 39, 40, 295–297, 318, 319, 326, 346–354, 492, 542, 597 linear stability analysis, principles and methods 27–36

Index

linear stability analysis, weaknesses and limits of 35–36 linear-ﬂow solution 346 liquid bridge 75–78, 98–99, 339, 375–425, 491–515 liquid bridge, aperiodicity 510–515 liquid bridge, azimuthal wavenumber 389–390 liquid bridge, geometric parameters 381–389 liquid bridge, heating from above or below 499–510 liquid bridge, high Prandtl number liquids 393–398 liquid bridge, historical perspective 375–378 liquid bridge, hydrothermal mechanism 417–420 liquid bridge, inﬁnitely long 85–86 liquid bridge, intermediate range of Prandtl numbers 423–425 liquid bridge, liquid metals and semiconductor melts 378–379 liquid bridge, microscale experiments 492–499 liquid bridge, noncylindrical 421–423 liquid bridge, secondary three-dimensional steady ﬂow 379–381 liquid bridge, standing waves and travelling waves 399–407 liquid bridge, symmetric and asymmetric oscillatory modes of convection 407–412 liquid bridge, system dynamic evolution 412–417 liquid bridge, tertiary modes of convection 390–393 liquid metals 325–326, 378–379, 436–449 lithosphere 105 local Richardson number 201 local thermodynamic equilibrium (LTE) 9 logarithmic spiral 478 logistic map 54

663

longitudinal rolls 216, 217, 218–219 Lorentz effect 607 Lorentz force 193 Lorenz attractor 54, 57 Lorenz model 48–51, 180, 181 Lyapunov dimension 53, 57 Lyapunov exponent 52, 57 Lyapunov spectrum 51–53, 180 Lyapunov stability 27 macrosegregation 101–102 MAGIA experiment 475 magma plumes 105 magnetic diffusivity 190 magnetic dipole moments 60 magnetic ﬁeld intensity vector 62 magnetic ﬁelds, ﬂow control by 581–608 magnetic ﬁelds, gradients of 607–608 magnetic ﬁelds, rotating 604–606 magnetic ﬁelds, stabilization of thermogravitational ﬂows 584–597 magnetic ﬁelds, static and uniform 582–584 magnetic ﬁelds, terrestrial, secular variations 192 magnetic gradient 607 magnetic induction equation 189 magnetic Prandtl number 190 magnetic Reynolds number 583 magnetic Taylor number 605 magnetization 60, 188 magnetization force 607 magnetohydrodynamic dynamo problem 189 magnetohydrodynamics 183, 582 magnetostriction 607 Mandelbrot set 53–58 Manneville–Pomeau scenario 46–47 mantle convection, second mode 106 Marangoni–B´enard convection 74, 148, 317–339 Marangoni–B´enard convection, modulation of 575–579 Marangoni convection 68, 70, 342–345 Marangoni effect 75 Marangoni ﬂow 70–78, 80–82

664

Index

Marangoni number 72, 74, 85, 439, 441 Marangoni–thermovibrational convection, mixed 561–575 Marangoni–thermovibrational convection, mixed, basic solutions 561–566 Marangoni–thermovibrational convection, mixed, control of convection patterning and intensity in shallow enclosures 566–567 Marangoni–thermovibrational convection, mixed, control of hydrothermal waves 567–575 Marangoni velocity 74 mass density 9 mass magnetic susceptibility 62 mass velocity 9–10 mass 9 MAXUS 412 Maxwell equations 58–62, 189 mean ﬂow 172 mean vibration force 517 memory loss 52 mental divisibility of the ﬂow conﬁguration in two semicircular regions 402 mesoscale cellular convection (MCC) 110 mesoscale shallow cellular convection 110–112 MICREX 67–68 microgravity 68–69, 429, 439–441, 474, 475 microgravity, deﬁnition 66–67 microgravity, virtual 607–608 microgravity containerless processing 98 microscale experimentation 75 microsegregation 102 microzone apparatus 75 microzone facilities 75 mixed buoyancy–Marangoni convection 83, 427–515 mixed buoyancy–Marangoni convection, annular pool 475–490

mixed buoyancy–Marangoni convection, annular pool, 2D hydrothermal wave (HW2) 476–477 mixed buoyancy–Marangoni convection, annular pool, chaos and fractal behaviour 483–487 mixed buoyancy–Marangoni convection, annular pool, reverse annular conﬁguration 487–488 mixed buoyancy–Marangoni convection, annular pool, role of curvature, heating direction and gravity 488–490 mixed buoyancy–Marangoni convection, annular pool, spiral pattern (HW1) 478–480 mixed buoyancy–Marangoni convection, annular pool, stationary radial rolls 480–483 mixed buoyancy–Marangoni convection, ﬁnite-sized systems ﬁlled with liquid metals 436–449 mixed buoyancy–Marangoni convection, inﬁnite horizontal layer 429–436 mixed buoyancy–Marangoni convection, liquid bridge 491–515 mixed buoyancy–Marangoni convection, liquid bridge, aperiodicity 510–515 mixed buoyancy–Marangoni convection, liquid bridge, heating from above or below 499–510 mixed buoyancy–Marangoni convection, liquid bridge, microscale experiments 492–499 mixed buoyancy–Marangoni convection, open vertical cavity 468–475 mixed buoyancy–Marangoni convection, open vertical cavity, aiding Marangoni and buoyant ﬂows 470–472 mixed buoyancy–Marangoni convection, open vertical cavity, counteracting driving forces and separation phenomena 472–474

Index

mixed buoyancy–Marangoni convection, open vertical cavity, surface driving actions and vertical temperature gradients 474–475 mixed buoyancy–Marangoni convection, open vertical cavity, volume driving actions and rising thermal plumes 470 mixed buoyancy–Marangoni convection, rectangular liquid layer 450–458 mixed buoyancy–Marangoni convection, rectangular liquid layer, OMC and HTW with spatiotemporal dislocations 456–458 mixed buoyancy–Marangoni convection, rectangular liquid layer, waves and multicellular patterns 450–456 mixed buoyancy–Marangoni convection, terrestrial laboratory experiments with transparent liquids 449 mixed buoyancy–Marangoni convection, walls, effects originating from 458–468 mixed buoyancy–Marangoni convection, walls, effects originating from, HTW collisions and wall-generated steady patterns 460–464 mixed buoyancy–Marangoni convection, walls, effects originating from, lateral boundaries as permanent stationary disturbance 459–460 mixed buoyancy–Marangoni convection, walls, effects originating from, streaks generated by lift-up process and mechanical instabilities 464–468 mixing zone 204 modulation 517 momentum 9, 15 momentum equation 10–11 monochromatic disturbance 289 mountain waves 115 multi-armed spirals 176–179 multiplicity of solutions 3

665

multiplume conﬁgurations, discrete sources of buoyancy and 208–213 natural patterns 4 natural measure 56 natural swaying motion 202 Navier-Stokes equations 6–21, 189, 217, 305 Navier-Stokes equations, linearization 30–32 Navier-Stokes equations, numerical integration 40–45 Navier-Stokes equations for thermal problems 78–89 Navier-Stokes equations, weak formulation 23 neutral stability 32 Newhouse – Ruelle – Takens theorem 47 Newtonian ﬂuids 13–15 Newton’s iteration 240 no-slip isothermal horizontal boundaries 120 non analytical dependence 53 nonconﬁned ﬂuid layers 119–123 nonconservative form 19 nondimensional form of thermal equations 19–21 nondimensional numbers 71–74 nonlinear dynamics 1–6 nonlinearities 4 nonperiodic response 303–305 number density 58 numerical boundary condition (NBC) 44 numerical experiments 4 Nusselt number 271, 321, 420 objectives of research 2 oblique hydrothermal waves 336 oceanic phenomena 108–117 Ohnesorge number 72 open-boat horizontal melt growth 341 open vertical cavity 468–475 optimal average wavenumber 134

666

Index

orbiting platforms, acceleration disturbances on 68–70 order of magnitude analysis (OMA) 89–90, 92, 94 ordered undulations (OU) 277, 278 ordinary differential equations, linearized stability 30 Orr-Sommerfeld equation 200 orthogonality condition 29 oscillatory bimodal 195 oscillatory blob mode 125, 128 oscillatory instability 125, 128, 133 oscillatory knot 141, 195 oscillatory longitudinal instability 284–286 oscillatory longitudinal rolls (OLR) 219–223 oscillatory skewed-varicose instability 125, 128, 132 oscillatory transversal long-wavelength instability 280, 284 oscillon 135, 177 overstability gap 495 PanAm structures 170–173 parallel ﬂow with an inﬂectional velocity proﬁle, stability of 32–35 paramagnetic materials 607 paramagnetism 62 parametric resonance 538–550 parcel theory for the atmosphere 122 parcels of ﬂuid 7 partial differential equations (PDEs) 4–6 pattern as prototype 3 pattern as repetition 3 pattern formation 1–6 pattern per se 2 pattern, deﬁnition 2–4 Peach–Koehler force 134 perfect 1 periodic acceleration 289, 517 periodic forcing 517 phase locking 47 phase trajectories 47–8 physical boundary condition (PBC) 43–44

plane disturbances 218 plane-parallel convective ﬂow 32, 298, 302 planetary boundary layer 112 planetary structure and dynamics 103–108 plate tectonics theory 104–106 plume regimes, free 196–202 plume regimes, free, geometric constraints 201–202 plume regimes, free, geometric effects 204–205 plume regimes, free, sinuous instabilities created by horizontal shear 200–201 plume regimes, free, upwelling and downward jets and alternating eruption of 203–204 polarization 60 poloidal–toroidal decomposition 125–127 Pomeau-Manneville type I intermittency 546 postulated Maxwell equations 58, 59 power input due to the volume force/ﬂow interaction 23 power-down method 95 Prandtl number 20–21, 74, 95, 103, 127, 522–524 Prandtl number, intermediate range 423–425 Prandtl – Rayleigh space 92 pressure 9 pressure-correction method 42 primary convective modes 119–122 primitive-variables methods 42–45 production density 8 projection method 42 propagating waves 346 proper orthogonal decomposition (POD) 391 provisional velocity 45 pseudo-phase space (PPS) 510 PULSAR 412–413 pulsating pattern 399, 445

Index

pulsating spatiotemporal mode of convection 401–407 quadrupolar ﬁeld 300 quasi-equilibrium states 294–299 quasi-periodic crystals 525–532 quasi-periodic ﬂows 25 quasi-steady residual 69 quaternary modes of convection 135–138 radial spokes 100 random behaviour 5 Rayleigh mode 223–225 Rayleigh necessary condition for instability 35 Rayleigh number 65, 66, 75 Rayleigh-B´enard convection 66, 92, 97, 100, 103, 119–193 Rayleigh-B´enard convection, cloud patterns and 110, 113, 114 Rayleigh-B´enard convection, dislocation dynamics 133–135 Rayleigh-B´enard convection, distinct and possible symmetries 151–155 Rayleigh-B´enard convection, higher and oscillatory regimes 155–157 Rayleigh-B´enard convection, nonconﬁned ﬂuid layers and ideal straight rolls 119–123 Rayleigh-B´enard convection, tertiary and quaternary modes 135–138 Rayleigh-B´enard convection, thermal diffusion in 208 Rayleigh-B´enard convection, thermal plumes and 202 Rayleigh-B´enard convection with horizontal or oblique shaking 533–538 Rayleigh-B´enard convection with vertical shaking 519–525 Rayleigh’s equation 34 reboost 69 recirculating pocket 237 rectangular containers 331–333 rectangular liquid layer 450–458 reﬂectional symmetry 147

667

Renyi dimension 56–57 Renyi exponent 56 representative elementary volume (REV) 9 residual-g 551–553, 554 resonance 544 return ﬂow 346, 349–354 reversals 192 Reynolds number 72, 74 Reynolds–Orr equation 37 roll relaxation 155 roll superlattice 531 rotating hexagon 487 rotating patterns 399 rotating spatiotemporal mode of convection 407 route to chaos 26 Ruelle–Takens scenario 46–47, 415 Saturn, thermal convection around 116–117 scale 2–4 scaling 3 secondary modes 125–133 self-organization 1 self-similarity 54 semiconductor melts 378–379 sensitivity to initial conditions (SIC) 47–48, 51, 52 shallow convection 110 shape factor 77 shape of system boundary 4 shear instability 219, 282 shielding parameter 604 sidewall forcing 172 signature of ﬂuid 19 single plumes 196 sixth problem of the millennium 23 slippery sidewalls 327, 330 Smale’s problems 49 snapshots, method 391 solitons 135 soxi-cell solutions 183 space, convection in 66–70 spatially localized structures 135 spatiotemporal chaos 4

668

Index

spectral decomposition 29 spherical shell, three-dimensional convection 182–193 spiral core instability 176–179 spiral defect chaos (SDC) 142–149, 171, 179, 528–529 spiral patterns 170, 171, 478–480 spiral wavenumber 175–176 spirals 135, 173–179 spoke pattern convection 100, 112, 138–141, 195, 438 square patterns 142–149, 320–325 square superlattice (SQS) 524, 531 Squire’s theorem 33, 115, 216, 542 stability 2, 3 standard bifurcation theory 190–191 standing longitudinal waves 256 standing waves 169, 399–407, 524 state-space form 27 state variables 27 stationary bifurcation 30 stationary disturbance 134 stationary longitudinal long-wavelength instability (SLL) 280, 281–282 stationary longitudinal rolls (SLR) 219, 346, 453 stationary longitudinal short-wavelength instability 284 stationary multicellular ﬂow (SMC) 345–368 stationary multicellular instability 451 stationary pattern 121 stationary radial rolls 480–483 stationary transversal instability 282–284 statistical mechanical theory of transport processes 7–9 steady residual acceleration 69 stochastic behaviour 5 Stokes theorem 16 strange attractors 5, 47–48, 54 streak convection 334–335, 336 streamfunction – vorticity methods 41 stress 9 stress-free isothermal horizontal boundaries 120

stress-free no-slip isothermal lower boundary 120 stress-free thermally insulated upper boundary 120 stress tensor 11 striations 102, 341 strong roll curvature gradients 134 subduction 106 subharmonic cascade 46 subharmonic patterns, purely 529 subharmonic resonance 226 subharmonic response 303–305 sun, thermal convection around 117 superlattices 525–532 superplumes 201 surface tension-driven ﬂows 68, 597–604 symmetric modes 381 symmetric oscillary instability 137–138 symmetric oscillatory modes of convection 407–412 symmetric-antisymmetric mode 151 symmetric-symmetric mode 151 symmetry breaking 5 symmetry-breaking shear ﬂow 272 synchronous response 303–305 symmetry in pattern formation 5 T-shaped rolls 158 temperature 9 temperature gradient inclination 334–339 temperature waves, cats-eye patterns and 247–253 tensor, internal energy 9 tertiary modes of convection 135–138, 225–227, 275–279 tetrahedron, symmetry of 183, 184 thermal convection equations 25, 189 thermal effect 543 thermocapillary convection 341–425 thermocapillary convection, boundary effects 359–368 thermocapillary convection, linear ﬂow 346, 348–349 thermodynamic ﬁeld (TFD) distortion 49, 66, 74, 79

Index

thermogravitational convection 80, 119–193 thermogravitational convection, in inclined systems 271–288 thermogravitational convection, control of 550–560 thermogravitational convection, control of cell orientation, orbiting platforms and 551–552 thermogravitational convection, control of shallow enclosures 553–559 thermogravitational convection, control of thermal boundary conditions 560 thermogravitational convection, magnetic ﬁelds and 584–597 thermovibrational convection 70, 289–314 thermovibrational convection, distortions 291–292 thermovibrational convection, equations and relevant parameters 289–290 thermovibrational convection, ﬁelds decomposition 290–291 thermovibrational convection, high frequencies 293 thermovibrational convection, medium and low frequencies 303–315 thermovibrational convection, primary and secondary patterns of symmetry 299–302 thermovibrational convection, quasi-equilibrium states and related stability 294–299 thermovibrational convection, reduced equations and related ranges of validity 305–315 three-dimensional asymmetric steady regime 498 three-dimensional Parallelepipedic Enclosures 157–165, 253–262 three-dimensional streamfunction 42 three-dimensional vorticity/vector–potential formulation 42

669

time dependence, onset of 161–165 time periodic ﬂows 25 Tollmien-Schlichting theory 200, 240 total (internal and kinetic) energy 15 total energy equation 11–13 total kinetic energy 23 transition regimes 89–94 transition to chaos 25–26 transverse rolls 216, 217 travelling waves 169, 399–407 triangles 320–325 truncation error 41 turbulence 202–208 turbulent Rayleigh–B´enard Convection , thermal diffusion in 208 two-dimensional hydrothermal wave 476–477 two-dimensional crystal lattice 130 two-dimensional horizontal enclosures 228–247 two-dimensional horizontal enclosures, boundary layers and patterns with internal waves 235–247 two-dimensional horizontal enclosures, geometric constraints 228–235 two-dimensional models 151–157 undamped harmonic oscillator equation 123 undulation chaos 276, 277, 278 undulations patterns 275 uniqueness 3 US LAB 551, 552, 553, 554, 555 velocity–vorticity formulations 42 vertical Bridgman (VB) technique 96, 97 vertical motion 110 vertical walls, domains with 544–547 vertically stacked cells 562 vibrational ﬂows, disturbances on 68–70 vibrational hydrostatic conditions 294–295, 539 Villermaux theory 205–208 virtual microgravity 607–608 viscous stress tensor 14

670

Index

viscous–nondiffusive regime (VND) 198 volume factor 77 vortex-pairing instability 276 vorticity 9 vorticity, dynamics of 15–18 vorticity balance equation 87 vorticity methods 41–42

wavy oscillatory instability 138 weak formulation 22 weakly nonlinear analysis 354–359 weightlessness 67 work (ﬂux of kinetic energy) 9 Young-Laplace equation 77

warm ﬁnger 464 wavenumbers 31 wavy instability 537

Zeldovich’s theorem 189 zig-zig instability 127, 128, 130, 134