PIV and water waves

COASTA ADVANCES \i ID OCEAN E PIV AND wflra wflucs John Grue Philip L.-F. Liu & Ceir K. Pcdersen VOLUME World Scient...

Author: John Grue | Philip L. F. Liu | Geir K. Pedersen | Philip L-F Liu

106 downloads 2403 Views 6MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!

Report copyright / DMCA form

DOWNLOAD PDF

COASTA

ADVANCES \i ID OCEAN E

PIV AND wflra wflucs John Grue Philip L.-F. Liu & Ceir K. Pcdersen

VOLUME

World Scientific

PIV O D WATER. WOJES

This page is intentionally left blank

ADVANCES IN COASTAL AND OCEAN E N G I N E E R I N G

PIV 01D WflTft WflUES VOLUME 9

€ditors

John Grue University of Oslo, Norway

Philip L.-F. Liu Cornell University, USfl

Geir K. Pedersen University of Oslo, Norway

\[p World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

PIV AND WATER WAVES Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-914-8 ISBN 981-238-949-0 (pbk)

Printed in Singapore.

Preface

Particle Image Velocimetry (PIV) denotes a group of laboratory techniques that extract velocity fields from particle displacements in subsequent digitized images of two-dimensional fluid sections. The velocities are found from cross-correlations or tracing of individual particles. The state of the art PIV employs digital camera technology, lighting by lasers and data processing on rapid computers. The advantage of PIV over competing techniques, as the Laser Doppler Velocimetry, is the simultaneous measurement of complete velocity fields that enables studies of complex flows, including vortical structures, flow separation, Reynolds stresses, turbulence characteristics and two-dimensional divergence at interfaces. Powerful experimental methods, like PIV, represent tools for advancement of theoretical computations, used for validation of models or as guidance in developing new theories where existing models either have shortcomings or do not exist. PIV and Water Waves grew out of a conference that was held on the subject in Peterhouse, Cambridge, UK, during 17-19 April 2002. The conference marked the end of a 4^ years long Strategic Program at the University of Oslo, entitled 'General analysis of Realistic Ocean Waves (GROW)'. In the project we developed a PIV system. Experiments on free surface and internal waves were performed and supported by theoretical and numerical modeling. With the wish to exchange results with other groups working with similar focus, we announced an open invitation to participate in this conference. As a result, more than twenty scientific papers on the subject were presented at the meeting. These papers are included as extended abstracts in part II, Chapters seven through nine, of this volume. Two additional papers that were presented at the meeting were later extended. They appear as Chapters three and five in part I of the book together with four invited review articles. The idea with this part is to review the status of the PIV technique and its various applications to free surface flows, interfaces, boundary layers, and internal waves. Chapter one gives

VI

Preface

an introduction to image based laboratory techniques with special focus on the use of image processing, digital data analysis and pattern matching in PIV. Various details of the method are discussed: resolution, displacement estimation, how to avoid peak-locking, true and false correlation peaks. The low-intensity mode of PIV, Particle Tracking Velocimetry, and how to extract higher order quantities from velocity fields, are described and illustrated by several examples. Chapter two describes a submersible PIV system for the measurement of turbulent dissipation, the Reynolds stresses and potential vortical structures in the bottom boundary layer of the coastal ocean. The system has the capacity to operate continuously for hours, and has been deployed at several sites in the coastal ocean. Wave induced, oscillatory boundary layer flow over a fixed ripple bed is discussed in Chapter three. PIV measurements of flow separation and vortex generation processes may serve as guidance in the choice of appropriate closure model of the turbulence equations, employed in simulations. Chapter four reviews PIV applied to flows in boundary layers and wakes of ships. Propulsion hydrodynamics, cavitation and free surface waves interacting with moving bodies are discussed. PIV measurements in the interface between water and air indicates regions of strong upwelling behind crests of microbreaking waves and regions of convergence in front of the crests (Chapter five). These processes are important for scalar exchange, e.g., green house gasses, between air and water. Field observations, laboratory measurements and theories are reviewed. Internal wave motion is the subject of Chapter six. Large amplitude solitary wave motion in experimental two-layer or stratified fluids compare excellent with fully nonlinear theory when the motion is smooth, but departs from theory when convective breaking or shear instability sets in. Solitary waves interacting with a submerged ridges or slopes may exhibit strong local deformation and cause strong currents at the sea floor. It is the hope that this volume will stimulate to enhanced interest in the PIV technique, and to further progress within the field of fluid mechanics, exploring the powerful combination between PIV and theory. We are indebted to the external referees and to Mrs. Dina Haraldsson for her willingness and proficient help in preparing parts of the text. The Strategic University Program was funded by the Research Council of Norway. The support is gratefully acknowledged. December 2003, The editors

Contents

Preface Part I

v Review Chapters

Chapter 1 Quantitative Imaging Techniques and Their Application to Wavy Flows J. Kristian Sveen and Edwin A. Cowen Chapter 2 PIV Measurements in the Bottom Boundary Layer of the Coastal Ocean W. A. M. Nimmo Smith, T. R. Osborn and J. Katz Chapter 3 Water Wave Induced Boundary Layer Flows Above a Ripple Bed Philip L.-F. Liu, Khaled A. Al-Banaa and Edwin A. Cowen

1

51

81

Chapter 4 Ship Velocity Fields Joe Longo, Lichuan Gui and Fred Stern

119

Chapter 5 The Air-Water Interface: Turbulence and Scalar Exchange S. Banerjee and S. Maclntyre

181

Chapter 6 Internal Wave Fields Analyzed by Imaging Velocimetry John Grue

239

Part II

Extended Abstracts

Chapter 7 Wave Breaking, Surface Motion, Surf Zone, Air-Sea Interaction and Wind Waves vii

279

viii

Contents

7.1. PIV Measurements of Accelerations in Water Waves A. Jensen, M. Huseby, D. Clamond, G. Pedersen and J. Grue

279

7.2. PIV Measurements of the Velocity Field in Steep Water Wave Events D. Clamond, J. Grue, M. Huseby and A. Jensen

282

7.3. LDA Laboratory Measurements in the Surf Zone for the RANS Depth-Averaged Shallow-Water Equations T. Feng and P. Stansby

285

7.4. Wave Acceleration Measurements Using PTV Technique Kuang-An Chang, Edwin A. Cowen and Philip L.-F. Liu 7.5. Two-Color LIF-PIV for Gas Exchange at Air-Water Interface Nobuhito Mori 7.6. Run-Up of Solitary Waves Geir Pedersen, Stefan Mayer and Atle Jensen 7.7. DPIV Measurements for Unsteady Deep-Water Wave Breaking on Following Currents Chin H. Wu, Aifeng Yao and Kuang-An Chang Chapter 8 PIV Methods, Boundary Layer Flows and Turbulence 8.1. A Digital High Speed Camera System Applied to PIV Measurements in Water Waves J. Kristian Sveen and Atle Jensen 8.2. Three-Dimensional Stereoscopic Particle Tracking Velocimetry for a Large Domain (3DSPTV-LD) and Its Application Yasunori Watanabe

288

291 294

297

301

301

304

8.3. Enhanced PIV Technique to Investigate Bottom Boundary Layer Dynamics Ahmed S. M. Ahmed and Shinji Sato

307

8.4. The Turbulent Boundary Layer Structure Beneath a Gravity Wave and Its Proper Decomposition Edwin A. Cowen and Stephen G. Monismith

311

Contents

8.5. Intermittency in Surf Zone Turbulence and Observations of Large-Scale Eddies Using PIV D. T. Cox and S. L. Anderson 8.6. Turbulence Distributions and Flow Structure in the Coastal Ocean from PIV Data W. A. M. Nimmo Smith, L. Luznik, W. Zhu, J. Katz and T. R. Osborn Chapter 9 Breakwaters and Internal Waves 9.1. Blocking Effects in Supercritical Flows Over Topography V. I. Bukreev, A. V. Gusev and V. Yu. Liapidevskii

ix

314

317

321 321

9.2. Study of the Flow Pattern in a Perforated Breakwater Using PIV Measurements Jean-Marc Rousset and Tom Bruce

324

9.3. Pseudo Image Velocimetry of a Regular Wave Flow Near a Submerged Breakwater Francisco Taveira-Pinto

327

9.4. Laboratory Modelling of the Motion of an Internal Solitary Wave Over a Ridge in a Stratified Fluid Y. Guo, P. A. Davies, J. Kristian Sveen, J. Grue and P. Dong

330

9.5. Measurements of Velocities During Run-Up of Long Internal Waves on a Gently Sloping Beach J. Kristian Sveen and John Grue

334

9.6. Comparison of Modelling of Strong Internal Wave Events with PIV Studies of the Phenomenon Deborah J. Wood, J. Kristian Sveen and John Grue

337

CHAPTER 1 QUANTITATIVE IMAGING TECHNIQUES A N D THEIR APPLICATION TO WAVY FLOWS

J. Kristian Sveen and Edwin A. Cowen Mechanics Division, Department of Mathematics, University of Oslo Box 1053 Blindern, N-0316 Oslo, Norway E-mail: [email protected] School of Civil and Environmental Engineering, Cornell University Ithaca, NY 14853-3501, USA E-mail: [email protected]

Quantitative imaging (QI) techniques are a general class of optically based laboratory measurement techniques used in the field of experimental fluid mechanics, which have seen rapid growth over the last two decades. They are particularly well suited for the study of wavy fluid flows which are characterized by unsteady free surfaces and internal motions. This paper presents an overview of QI techniques in general, with a particular focus on particle image velocimetry (PIV). We present QI methods in the context of the broader fields of pattern recognition and image processing techniques, which are currently used in a wide range of fields. In this review QI methods and their fundamentals are described in detail and recent developments, targeted at increasing accuracy and resolution, are described and put into perspective. More specifically we identify QI techniques as a digital data analysis (through software) set of issues built upon general principles of pattern matching. Throughout the paper we address the aspects that are particular to wavy flows although these issues can be argued to be important for any unsteady fluid flow of interest, e.g. turbulence. This review article thus serves as a general reference for the neophyte and experienced fluid mechanics experimentalist. 1. I n t r o d u c t i o n Perhaps the greatest challenge of making measurements in wavy free surface flows is measuring flow field characteristics near the dynamically moving free surface - i.e., between the trough and crest. Quantitative imaging (QI) techniques are a robust solution t o this problem, as demonstrated in Fig. 1, and hence QI techniques are becoming the methods of choice when 1

2

J. K. Sveen and E. A.

x (cm)

Cowen

x (cm)

Fig. 1. Incipient shoaling breaking wave. Left image with every 4 t h determined vector shown in each coordinate direction; right image is a magnified subregion of left image with every 2 n d determined vector in each coordinate direction.

attempting to interrogate laboratory wavy free surface flows. An important aspect to the emerging dominance of QI techniques is their ability to capture whole field properties - e.g., u(x,z) (the velocity field), u?y(x,z) (the vorticity field), u'w'(x,z) (the Reynolds stress field), and e(x,z) (the turbulent dissipation field). While many wavy free surface flows are periodic the reality is that due to reflections and to the difference between the phase velocity and the energy propagation velocity (group velocity) flows are often quasi-periodic at best. Researchers interested in spatial gradients have traditionally attempted to employ single-point measurement technologies at two or more spatial locations but on different experimental runs. Variability among experimental runs will lead to variability of induced rms velocities which is often on the order of the turbulence intensity itself making it extremely challenging to employ data from different runs for the determination of gradients. While QI techniques do not eliminate wave-to-wave variability they do capture a spatial field under the identical free surface conditions allowing accurate instantaneous gradients to be determined. Presently the most cited reason to work with single-point measurement technologies is a need for improved temporal resolution relative to QI techniques. However, over the last decade computers and image capture technologies have progressed sufficiently that QI techniques are now capable of reasonably high temporal as well as spatial resolutions, further

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

3

propelling the popularity of QI techniques. A challenge for these techniques is optical access, particularly for existing facilities that were not designed with PIV in mind. Recent advances with horoscopes and index-of-refraction material matching, however, demonstrate that optical access issues can usually be overcome with a bit of ingenuity. In order to visualize some of the methods presented in this paper, we have chosen to use the free, open-source PIV program MatPIV 1 and two example images taken from the paper by Jensen et al? This paper is organized as follows. Section 2 presents a brief introduction to some of the more commonly used QI techniques in the perspective of pattern matching. Section 3 is written to provide a basic overview of the fundamentals of Particle Image Velocimetry (PIV), while section 4 targets the more fundamental aspects of the technique. Section 5 gives a brief overview of Particle Tracking Velocimetry (PTV) and section 6 focuses on higher order measurements from velocity fields. Finally, Section 7 contains a short conclusion, and also provides an overview of a few other areas where the same principles of pattern matching are applied. 2. Quantitative Imaging Techniques QI techniques can be broken immediately into several fundamentally different types of techniques - flows seeded with discrete particles, flows seeded with continuous tracers (e.g., fluorescent dyes) and unseeded flows (to look at density differences). The former are generally employed for the determination of velocity while the second and third are generally used to determine a scalar field quantity (e.g., concentration or temperature). The literature is rife with various acronyms for these types of techniques and we will briefly introduce some of the more popular QI nomenclature here. 2.1. Particle

Based QI

Techniques

• Particle Streak Velocimetry (PSV) -A general class of techniques where the image exposure time is long relative to the time a particle occupies a point in space. The result is images of particle streaks. The length of the streak can be calculated to determine the velocity based on the known exposure time. • Laser Speckle Velocimetry (LSV) -A general class of techniques where the seeding density is sufficiently high that an image captures predominantly overlapping and interfering particle images, which can be thought of as an intensity texture or speckle field.

4

J. K. Sveen and E. A.

Cowen

Essentially no discrete particle images are seen. The velocity is extracted by correlating the speckle pattern in a small subregion with that in another subregion, either optically (Young's fringe analysis) or digitally (auto or cross correlation analysis). • Particle Image Velocimetry (PIV) -This term is sometimes taken to mean the entire broad class of discrete particle based techniques, however, its preferred definition is a general class of techniques where the seeding density is moderate such that the nearest neighbor distance of particle images is on the order of a few to perhaps ten times the particle diameter ensuring that all small subregions have several distinct discretely imaged particles within them and relatively few particle images overlap. The velocity field can be extracted in a number of ways, the most popular of which are digitally via auto or cross correlation analysis. • Particle Tracking Velocimetry (PTV) A general class of techniques where the seeding density is sufficiently low that an image captures predominantly non-overlapping or interfering particle images and the velocity can be extracted by tracking the motion of individual particles over known times.

Other common names for particle based techniques include: pulsed light velocimetry (PLV), particle image displacement velocimetry (PIDV), particle displacement velocimetry (PDV), digital particle image velocimetry (DPIV), digital particle tracking velocimetry (DPTV), correlation image velocimetry (CIV), spatial correlation velocimetry (SCV) and large-scale particle image velocimetry (LSPIV). We note that three dimensional QI techniques have seen rapid growth recently and are based on one of four fundamental approaches: holography 3 , stereoscopic imaging with multiple cameras 4 , depth-of-field with a single camera 5 , and scanning a light sheet through a volume 6 . We will restrict our review of QI techniques to two-dimensional implementations. Ron Adrian, in a survey through 19957, showed that the number of publications per year on particle based QI techniques grew exponentially. All indications are that the trend continues. The common principle to all particle based techniques is that the instantaneous fluid velocities can be measured by recording the position of images produced by small tracer particles, suspended in the fluid, at successive instants in time. The techniques listed above, as well as others, fall into two broad categories, each with different development paths: PIV and PTV.

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

5

LSV and PIV are different operating modes of the same technique. The velocity field is generally determined on an interrogation grid and each velocity vector is the average velocity over many tracers contained in a small volume of fluid. These techniques have their roots in solid mechanics. They were originally used to determine the in-plane displacement and strain of solids with diffusively scattering surfaces. PTV and PSV can also be thought of as different operating modes of the same technique. In contrast to PIV, velocity vectors are determined from the individual particle images or streaks produced by a single particle at random locations. These techniques have their roots in the field of flow visualization; particle streak photography and stroboscopic photography. Prandtl 8 was an early developer of particle tracking techniques, although not the first. For the remainder of this manuscript we will use PIV to indicate the class of particle based QI techniques where the velocity is extracted by looking at the movement of an ensemble of particle images. PTV will be used to indicate the class of particle based QI techniques where the velocity is extracted by looking at the movement of a single particle image.

2.2. Tracerless

QI

Techniques

This class of techniques relies on measuring either the concentration or displacement of a chemical tracer substance added to the flow. • Laser induced fluorescence (LIF) -A general class of techniques where a flow is seeded with a dilute fluorescent chemical tracer that will fluoresce proportionally with its local concentration. The local concentration field is often the objective of LIF measurements but careful choice of dye can allow the measurement of temperature, and pH as well as other scalar properties of the flow. • Scalar image velocimetry (SIV) -Really a specific analysis applied to LIF images where the gradient in intensity information recorded in images is treated like speckle in LSV and correlation based analysis is used to extract velocity field information. Another common name for chemical tracer based QI techniques is planar laser induced fluorescence (PLIF). As with the particle based techniques we note that three dimensional scalar QI techniques have seen growth recently and are generally based on scanning a laser light sheet through a volume 9 ' 10 .

6

2.3. Other QI

J. K. Sveen and E. A.

Cowen

Techniques

Other QI techniques do exist that do not rely on the use of passive tracers. In fact some of the oldest flow visualization 11 techniques do not, such as shadowgraphy, schlieren imaging and the Mach-Zendner Interferometer. Today many researchers utilize so-called Synthetic Schlieren or Quantitative Shadowgraph techniques using digital cameras and the very same principles of pattern matching as are used in for example Particle Image Velocimetry. We shall briefly return to this aspect in section 7.

2.4. QI Techniques — Image Processing Recognition by a Different Name

and

Pattern

Many newcomers to PIV in particular, and QI techniques in general, are often confused by the relatively large number of details that need to be addressed in order to apply these techniques to real fluid flow experiments. One of our primary goals in this review is to focus on the foundations upon which these techniques are built. PIV in particular has received a lot of attention within the last 10 to 15 years. As implied in section 2.1, authors often use different nomenclature for what are essentially identical approaches, perhaps with subtle implementation differences. We would like to stress that PIV relies on image processing and pattern recognition analysis and as such it should more properly be viewed as an interdisciplinary field between the experimental fluid mechanics research community and the image processing and pattern recognition (IP&PR) research communities. QI techniques are widely known and used by IP&PR researchers and in fact, experimental fluid mechanicians often re-invent analysis techniques as they ignore the previous efforts documented in the IP&PR literature. The basics for understanding PIV and pattern recognition may actually be found in most introductory books on image processing12. The details of applying QI techniques to fluid flows essentially consists of imaging a temporally and spatially varying pattern within the flow, generally by adding discrete tracer particles although we should not feel restricted to this particular case, illuminating the flow in a nearly two-dimensional slice as the majority of imaging devices capture two-dimensional information, and using IP&PR algorithms to extract the displacement in a known time of the imaged tracers. The entire QI measurement process can be divided into two fundamental components: (1) a hardware problem of experimental techniques, including illumination, seeding and image recording, and

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

7

(2) an analysis (software) problem of applying IP&PR techniques to extract displacement or other information of interest from the images. The latter point should be recognized as the fundamental application of QI methods, while the former is merely an experimental problem (which in many cases is the hardest part as the majority of QI experimentalists will agree the challenge is getting 'good' images - with 'good' images in hand it is simply a matter of finding the right IP&PR technique to extract the information one desires!). Hence we contend that QI analysis in general, and PIV as a particular example, is simply a sub-specialty within the broader fields of IP&PR. For our purposes QI/PIV are defined as extracting information of interest to fluid mechanicians by image processing and pattern recognition means. There are subtleties that we avoid by this generalization, but as a starting point this view is effective and allows us to break down a large problem into smaller, solvable pieces. We can view the more recent developments in QI and PIV analysis techniques as emerging to solve problems that arise in terms of accuracy and resolution in displacement estimates - to date the fundamental quantity that the vast majority of QI velocimetry techniques seek to extract from images. The reader should note that applications of pattern matching is currently an area of active research. For example, looking at the motion picture industry, the MPEG standards 13 for sound and video compression, storage and transfer are built on many of the same ideas we use in PIV. As an example we can consider the DVD standard (MPEG2), where local motion estimates are used in order to limit the amount of storage required. Instead of saving every single frame in a movie, only a few (typically every 8-16 frames) are stored as full frames and only the local motion is stored for the remaining images. There are many advances from this industry that have yet to be applied to PIV. For example, we should be able to perform motion estimation in the frequency domain within the PIV framework, much like is done in image processing14. In this way we should be able to avoid one fast Fourier transform (FFT), and hence save about one-third of the calculation time. In section 7 we present a brief overview of other research fields that use pattern matching approaches. 3. P I V — A General Overview Let us begin the description of the basics of PIV with a fluid flow that is seeded with particles that can be considered passive tracers, perhaps because they are very small or if they are not so small they are near neutrally

8

J. K. Sveen and E. A.

Cowen

buoyant with respect to the flowing fluid. A light source, generally a laser light source, is shaped with optical components into a light sheet to illuminate the particles. The light sheet is quasi-two-dimensional in the sense that it is thin in the direction orthogonal to the plane of motion that contains the two components of velocity we are interested in measuring while it is broad in the other two directions. An imaging device, in this case a digital camera, is equipped with appropriate optics (e.g., a lens) to collect images of the particles as they pass through the light sheet. In general, computer controlled timing signals are sent to the digital camera and laser light source (or the optical components that shape the light source into a light sheet) to synchronize the light source to the camera such that discrete images of particles (e.g., short time exposure images) are captured at desired times within each collected image. There are, of course, myriad ways in which particle images can be collected, which fall into two general categories: the single exposure of multiple images and the multi-exposure of single images. Again there are myriad ways that the mean displacement of particles in any sub-region (henceforth referred to as a subwindow) can be extracted from the images but the fundamental technique used in multi-exposed single images is autocorrelation analysis while the fundamental technique used in single-exposed multiple images is cross-correlation analysis. As cross-correlation analysis is more straightforward as well as more accurate, we will restrict our discussion of PIV basics to cross-correlation analysis of image pairs - that is the exposure of two sequential images, each individually, specifically for analysis by cross-correlation. Let's assume we have collected an image pair where the second image was captured a known time, At, after the collection time of the first image. The most straightforward approach to cross-correlation analysis is to define a square subwindow with side length N = 2" where n is an integer. N is typically taken as a power of 2 to take advantage of determining the crosscorrelation in the frequency domain via the fast Fourier transform (FFT). As will be described later N need not be restricted to these discrete values but as this restriction is frequently employed we will assume it for now. Let us assume a value of N = 32 and an image size of NR x NC pixels (an acronym for "picture element" that describes the smallest discrete unit of scattered light intensity measured by a digital camera, sometimes also referred to as a "pel"), where NR and NC are the number of rows and columns in the digital image, respectively. The simplest algorithm is to divide each image of the pair into non-overlapping N x N subwindows and to then perform the two-dimensional cross-correlation of each subwindow

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

9

pair in the image pair. The cross-correlation function is denned as N-1N-1

R S

(^ =^ E E

f

y ( M ' ) ^ ( < + *. 3+t)

(1)

where R is the cyclic cross-correlation between subwindows / , J in the first image of the image pair (F') and the second image of the image pair (F"), i,j is the pixel location within subwindow I, J, and s,t is the 2-D cyclic lag at which the cross-correlation is being computed. As indicated above R is often calculated in the spectral domain and hence equation (1) can be found as: R(S,t) = T~1 [^{F'^AiJ^^iF'/^i

+ s, j + t)}]

(2)

where T and T~x are the Fourier and inverse Fourier transform operators and the star denotes complex conjugate. This basic feature is known as the correlation theorem and can be found in most introductory level books on image processing 12 . For the purpose of visualization we consider the example images in Fig. 2a-b, which shows the image pair F' and F" along with the non-overlapping subwindow interrogation grid. Fig. 2c and 2d shows subwindow J = 10, J = 15 for each image in the pair along with panel e, which shows the resultant cyclic cross-correlation of this subwindow pair. The correlation plane contains a peak which has a maximum at (s, t) = (26,25). The displacement is measured from the center of the correlation plane to this peak. The integer displacement in our example is estimated to be dx = —6 and dy = —7. Our simple introduction here raises several issues. First we note that the maximum unambiguous displacement that can be resolved is N/2 pixels. If the displacement is larger than N/2 pixels (but less than TV pixels, so at least a few of the particles in the first image subwindow remain in the second image subwindow) the correlation peak will alias to the location — (N — £) where £ is the actual displacement. If the displacement is larger than N pixels then R represents the cross-correlation of two uncorrelated subwindows and the returned displacement estimate will be the result of a random noise peak (i.e., the lag where a maximum number of particles randomly align themselves between the two subwindows). Secondly, there is an implicit assumption that the particles are being translated without rotation or shear. If the particles undergo rotation and/or shear over the time At between the capture of the two images then we must be concerned about the effect of this rotation and/or shear on the existence of a usable correlation peak. Thirdly, if the flow is not two-dimensional where the outof-plane motion is identically zero then there is a finite non-zero probability

10

J. K. Sveen and E. A. Cowen

of a particle appearing in the first subwindow moving out of the light sheet (and hence the image) by the time the second image is captured. This outof-plane motion can clearly lead to a measurement bias if the out-of-plane motion is correlated to the velocity itself. And lastly, if N = 32 our maximum resolvable displacement is just 16 pixels. If we cannot extend this or resolve the displacements to sub-pixel accuracy (e.g., estimate the location of the correlation peak to fractional pixel values) then our maximum accuracy is greater than 3% and our typical accuracy is considerably higher than this. The above example also raises many other questions. With the above primer in PIV as our starting point we will now turn to the details of PIV. 4. P I V — The Fundamentals There are essentially three typical implementations of PIV: single exposure multiple image (cross-correlation based), double-exposure single image (auto-correlation based) and multi-exposure single image (auto-correlation based). There are several excellent fundamental references on QI techniques in general and PIV techniques in particular and the reader should explore these references for further details and perspectives on PIV 1 5 ' 1 6 ' 1 7 ' 1 8 , 1 9 . 4.1. Displacement

Estimates

— Correlation

Approaches

PIV techniques often rely on either auto or cross-correlation of subregions to extract the mean displacement of particles contained within the subregion. This methodology thus relies on estimating the auto or cross-covariance function between the subregions to extract the displacement estimate. For details on auto-correlation see Adrian 20 and Keane and Adrian 21 . The covariance can be determined in a number of ways but it is generally determined by either Fourier transforms 22 or the direct determination of the two-dimensional covariance function23 (also known as the correlation coefficient12). The expected value of the cross-covariance R(s, t) (see equation 1) is shown by Westerweel17 to be E{R(s,t)}=(l-^(l-lf)R(s,t)

(3)

and it is seen immediately to be biased (note the bias vanishes as N —> oo). The bias occurs because the shift over (s, t) results in only a part of F' correlating with F". Adrian 24 reported this as the result of in-plane loss-of-pairs. In the presence of strong gradients the interpolation region contains more particle pairs

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

11

b)

:

a; 3F«ttKtt#BKW»K«0##*&«VKKB8E8i8f««»»i#lffaft4 «£###»£

200 400 600

l£B8M»Kg8Btt«i(>*#t a«ssss*iK!»issa»B£sssiaK«i iBB8&tt8Sg8!«8!SS##8W*ft#««:8!£ Kg&BBBKM&WH*RM8Wai!

SSS«SffiKBiB!«lSSSSSSaf*«S(ftfSi KBBHBlBlBttSRMttWKSIIIKBi BRKfllBWMaKKiiltattBKMKI BKMttBBRMMHCaSBHSBI

800

BgKgttBB&£!8BttttB«!##i jasat -KBttgBBJgffBKttttttRaSii SSfSSi

1000 Hi

2()i

„ , , . . .._J«Httg88&»tttt$£&a*ttM«MK8stt&«ie«4ftt»K*«:l!iBE &»#g»aK«K££ggttttttB^8Hatt«#BHtttt&SS»K:iiagE#i«tt3£ te&S888«KStt«8im&ttHH#88»K8BKK88K&MBKI«!8:K8£ttttK f«Bg|«KWK8J«gKBK8KBBiSKS«SttgB«&»S:KKi«BBi8K8»&tt!g &M#H8»RHKKa«KK*a^aBMMg«£»Ma#*fe«tttt£##Hg&g; fwfigBttaMKJttKKBH* "— —-————

!KKB!BBBSSf*8f*SS*SSEWi lSfWW*»*«*cSSSBSK< isxsaHBasvKaMBiSiaisi

!HHK0MS8SH«»i8!S8SHIKWI lg:£MS3l»BaSKBS!g&M»#K&&*g :sasf*»BSSsspi«f!9i*saa»sfsiSiS9is :»«ssaiswsajiss as asssiasssssfsissass

200 400 600 80010001200

200 400 600 80010001300

d)

c)

e)

Fig. 2. Images with superimposed non-overlapping grid. Black cells marks the subwindows shown in c) and d). Correlation plane shown in e). Upper arrow shows the displacement. Lower arrows denote the displacements along each axis. Black vertical line marks the center of the plane.

with small displacements, therefore, this is an under bias. Prom the above equation it is apparent that the bias grows linearly with the shift size. Westerweel shows that two conditions arise from looking at the variance of the expected value of R. He finds that the noise in R due to random correlations

12

J. K. Sveen and E. A.

Cowen

is approximately a uniform random field and goes as 1/N2. The probability that the noise peak is greater than the correlation peak grows for increasing displacement (s,£). 4.1.1. Removing Effects of Correlation with the Mean Background As shown by Adrian 24 , the correlation between two images may be split into 3 contributions: (1) correlation of the mean background intensities, Rc(s,t), (2) correlation of the mean and the fluctuating intensities, Rf(s,t), (3) correlation of the fluctuating intensities, Rd(s^t).

and

In this way the correlation may be written as R = Rc + Rf + Rj. The latter part of the correlation plane will contain the displacement peak, and to avoid the other parts of the correlation, the mean is normally subtracted from each image prior to computing the correlation. The process is shown in Fig. 3 and also included in equation (4).

R

Rc A

Fig. 3.

Rf „

Rd

± <# •

Schematic figure showing the different contributions to the correlation plane.

4.1.2. Normalizing the Correlation Plane In many cases it may be favorable to be able to evaluate the degree of correlation between two subwindows. The maximal value in the correlation plane is a relatively important parameter to estimate since it is a direct

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

13

measure of the degree of match between the subwindow patterns. The ratio between the highest peak and the rest of the correlation plane will furthermore provide information of the quality of the match compared to the noise. In order to evaluate these quantities we need to normalize the correlation in equation (1). One function that calculates a normalized correlation is the full correlation coefficient or covariance function, which is defined as 12 p(

A -_

[S

'

l

£ < ^{F'^j) - F'][F"(i + s,j + t)- F"\ N2 £ . £,.([F'(i,j) ~ F'}2[F"(i + s,j + t)- F"]2y/2

'

[ }

where F' = mean(F'(i,j)) and F" = mean(F"(i + s,j + t)). We will return to this function later, but for now we note that the numerator easily can be calculated by the use of Fourier transforms. The denominator, however, is usually approximated by N2std(F'(i, j)) -std(F"(i, j)), where std() denotes standard deviation, which is defined as std(F') = E ( F ' - F ' ) 2 / ( N - l ) } 1 / 2 , and is calculated prior to R. In this way we can use the peak height in the correlation plane as a measure of how good the correlation was. A value close to 1 will indicate a very high degree of match between our windows, while a value of 0 will indicate the opposite. The underlying assumption for making this approximation is that our pattern is evenly distributed, meaning that the standard deviations do not change if we vary s and t during the calculation of R. An uneven distribution of the pattern may easily reduce the accuracy of our measurements, as we shall see later in the paper.

4.2. Minimizing

Covariance

Estimate

Induced

Bias

The source of the bias in estimating the displacement of an ensemble of particles can be minimized by insuring that the majority of particles captured in the first subregion are also found in the second subregion. There are two traditional approaches to minimizing this bias - dynamic subwindow location and adaptive subwindow sizes. The goal of dynamic subwindow location 22,25 is simply to insure that the mean displacement of particles contained within two subwindows is, to the nearest integer, zero pixels. Thus the second image subwindow (or both subwindows as will be discussed shortly) is dynamically located so that to the nearest integer the covariance estimate of the displacement is zero pixels. Cowen and Monismith 22 used this concept to remove spurious analysis regions. They argued that if a dynamic estimate of the integer pixel displacement does not converge to zero pixels in three iterations, the analysis of the subwindows is problematic and the subwindow pair is discarded.

14

J. K. Sveen and E. A.

Cowen

Fig. 4 (middle pair) shows the concept of this dynamic subwindow location. The interrogation region in the first frame is fixed in space, while in the second image it is shifted in integer steps found after an iterative process. Based on the information from two subwindows, the optimal estimate of displacement that can be obtained is second-order accurate. This is achieved by considering the location of the displacement estimate to be at the midpoint between the center of the first and second subwindows (this assumes the center of mass of the particles for each subwindow is at the center of the subwindow - a reasonable model but as subwindows get small there may be significant deviations from this assumption). Allowing the second subwindow to move dynamically with respect to the first subwindow results in a random perturbation to the originally intended interrogation grid due to the potentially random motion of the second subwindow. Wereley and Meinhart 26 eliminate this random perturbation by allowing both subwindows to move dynamically as shown in Fig. 4 (lower pair). This avoids the need to interpolate velocity vectors and ensures a truly second-order accurate spatial velocity field. The goal of an adaptive subwindow size 21 is to expand the second subwindow in area so as to ensure that all particles contained in the first subwindow will remain within the bounds of the second subwindow. If this is done in the absence of discrete subwindow offsets, the potential for the second subwindow to become large exists, and the signal to noise ratio of the image (ratio of the magnitude of the true correlation peak to the rms correlation peak) will decrease causing dropouts. An improved approach is likely to couple the two techniques, handling the mean displacement with a discrete offset and the effects of local turbulence induced shear by a slightly larger second image subwindow. An optimal approach involves the consideration of the image 1 subwindow as a system in the thermodynamic sense - a collection of mass of fixed identity. Tracking the system, and thus its boundaries, from its location in image 1 to its new location in image 2 results in no in-plane loss of particles but requires that the image 2 subwindow boundaries become deformed. Several researchers have developed such optimal methods, which have become known as particle image deformation, or PID, methods 27>28>29>30>31. piD methods are described in more detail in section 4.7.3.

4.3. Subwindow

Size

An important consideration is the size of the subwindow to be used for PIV analysis as this leads directly to issues of resolution. Prasad et al.32

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

15

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

260

280

300

320

340

Fig. 4. Schematic view of the three most basic dynamic subwindow location techniques. The upper pair shows fixed interrogation windows. The middle pair shows first order window shifting 2 2 , 2 5 and the lower pair shows second order accurate window shifting 26 .

investigated a range of subwindow sizes from N = 32 to N = 256 finding that a reduction of subwindow size from the traditional N — 256 to N = 128 resulted in no appreciable degradation of accuracy. Westerweel17 explored in detail the requisite subwindow size to obtain the desired information,

16

J. K. Sveen and E. A.

Cowen

namely the location of the correlation peak to sub-pixel accuracy. Westerweel17 sought the effective sampling rate needed to yield a good representation of an original QI image. A signal is band limited if it can be constructed from only signals with frequency components less than some maximum, i.e., its Fourier transform is non-zero over only a finite portion of the frequency domain. Therefore, F(F)(s,t)

= 0, for |*| > S, \t\ > T.

(5)

The bandwidth is defined as the maximum of S and T. A band limited continuous signal can be reconstructed exactly (given infinite samples) provided that the sampling rate is at least twice the bandwidth. This rate, known as the Nyquist rate, is max(2Sr, 2T). Goodman 33 has shown that the bandwidth of the image intensity for a thin spherical lens image system (aperture D, focal length / , coherent light with wave length A) is given by:

Xz0

\f(M + 1)

w

where ZQ = f(M + 1) is the image distance and M — ZQ/ZQ is the magnification (Zo is the object distance). Now, if an image is obtained with illumination wavelength A = 0.5 /xm, f/D = 8, and M = 1, then it follows that W = 125 m m - 1 and therefore that the Nyquist rate, 2W, is 250 m m - 1 , explaining the traditional subwindow size of 256 pixels and a subwindow area of 1mm2 early in the history of PIV. However, this is a significant hurdle for CCD based image acquisition. Most CCD's pixels are at least 10 /xm x 10 itm, therefore 2W = 100 m m - 1 is the best that can be done. However, the reality is that the goal of PIV interrogation is not to reconstruct the image exactly, but only to obtain the position of the displacement covariance peak. What is required is not the exact details of the particle shapes (edges) but just the details of their positions (lower wave number information). The covariance function has a spectral density function that is "nearly band limited" meaning its value vanishes for sufficiently large (s, t) but may not be exactly zero. Therefore F(s,t) ~ 0 for \s\ > S, \s\ > T. Parzen 34 showed that for a 1-D signal the bandwidth, Wp with a circularly symmetric spectral density function is defined as the width of a cylinder with the same volume, therefore Wp = (irs(0,0)y1/2-

(7)

It can be shown that if a square region is sampled instead of a circular region and the covariance of particle images with diameter d are of interest, the

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

17

covariance width, ah, is ah = ds/2/(2A4ir) which leads to Wp = l/(27rcr/,). Thus, given the same optical parameters as above and d = 20/xm (ah — 3.7 /xm), 2WP = 86 m - 1 , suggesting that 64 x 64 pixels over a bit less than 1 mm 2 is sufficient. Note that larger particle image diameters reduce the required sampling rate, so that 30/im images —>~ 32 x 32 pixels/mm 2 . Thus the dominant subwindow sizes in currently used PIV algorithms are N — 64 and N = 32. For more details on band limited signals see Westerweel17.

4.4. Sub-Pixel Displacement

Estimation

It can be shown that for any non-zero particle image diameter the width of the correlation peak will be greater than one pixel. By including values adjacent to the maximum in R, the center of the peak can be estimated to sub-pixel accuracy. We decompose the displacement (s,t) as s = sQ + es,

t = t0 + et

(8)

where So,io is the integer displacement and es,et is the fractional part. Therefore -0.5 < es < 0.5

and

- 0.5 < et < 0.5.

(9)

In the absence of an estimate for es and e* the error is ± A / 2 where A is one pixel. For N = 32 a displacement of eight pixels has an uncertainty of ±0.5/8 ~ 6%. If we digitize the same subwindow into 256 pixels (N = 256) the uncertainty is now ±0.5/64 ~ 1%. Therefore to work at small N, we need subpixel estimates of the center of the covariance peak. It can be shown that the covariance cov{e r , e*} = 0 (i.e., e^ei = 0), therefore, we can work with the 1-D problem without loss of generality. For narrow correlation peaks the covariance width is small enough that only the nearest pixels to the peak contain significant information. Since QI techniques generally satisfy the conditions for narrow covariance peaks, three-point sub-pixel estimators are generally sufficient. As an example we will consider the correlation peak shown in Fig. 2e. Fig. 5a shows a close up of this peak, while Fig. 5b shows the highest value, RQ, of the correlation along with the two nearest pixels in the vertical direction, i?_i and R+\. The three most commonly used estimators are Center-of-Mass, Parabolic fit and Gaussian fit.

J. K. Sveen and E. A.

Fig. 5.

Cowen

Close-up of the largest peak in the correlation plane in Fig. 2e.

Center-of-Ma88

(COM)

The sub-pixel estimation of the Center-of-Mass is calculated from .Y ._..

ii|-i "- R-i ,1Qx R i -I"- R-o "V it-fi This method ignores the need for a peak since R® can be less than i2__i or J?+x. Fig. 5 shows an example of a correlation peak and the positions of jfiL-i, R-\-i and RQ. Parabolic Another option is to assume that the peak has a certain shape, for example a parabola. We can then evaluate __ R-i -~ R+i , . €p 2(H__ 1 -2Ho + -R + i)" ' This formula actually requires a peak since |J?o| must be greater than \R i| and

|JR+I|.

Gaussian The most common assumption in PIV is to use a Gaussian peak-fit. This is mainly due to -the fact that spherical particles image as Airy functions and the central lobe of an Airy function is well approximated by a Gaussian curve. The log of a Gaussian curve is parabolic and therefore l n i i - i — lni? + i (12) €G 2(ln # _ i - 2 In R0 + In j?, +1 )'

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

19

Note, again that Ro must be a peak and R4 > 0 for i = —1,0,1. Clearly all three are a balance of R-\ and i?+i with a normalizing value in the denominator. Note that ec will always yield a result and ec has the most restrictions. Therefore we expect ec to be the most robust and ea the least. On the other hand, ec does not even acknowledge that we have a peak while ec acknowledges and uses shape information. Therefore we may expect ea to be the best performer and ec the worst. What is the expectation of these estimators? We begin with the bias by ignoring the fractional displacement for a moment. Therefore R-\ = R+i = ai-Ro where 0 < a\ < 1 and a\ is proportional to the width of the correlation peak. Westerweel17 shows ai

l + 2ai

m=

for ec

a\ for ep 4(1 - d )

AT-TOQ

(13)

1 for ec I 41n(l/ f l l ) Clearly the terms on the right hand side of the bracket are simply constants, but note that the first term is a function of the displacement mo (the integer pixel displacement). This is a negative bias, resulting from the probability of smaller velocities leading to higher pairing. If PIV is to operate in an unbiased manner this bias must be corrected. The correction is: R*{s,t) =

R(s,t) Fi(s,t)

(14)

where Fi

Hi N

1*1

(15)

N

We note that this could have been predicted looking at equation (3). We now consider the fractional part of the displacement. A particle will have a small imbalance in i?_i and -R+i; therefore we will model the particle image as: R.

,1+e

Ro = a\ and _R+i

,1-e

(16)

J. K. Sveen and E. A. Cowen

20

Note that this is not Gaussian. Westerweel17 shows that in this case sinh(elnai) e c

=

W~i

\ i i €-1'

(n>

cosh(emai) + | a a sinh(elnai)/2 piling m a i ^ / z , cosh(elnai) — a\~

«-5^-

(19)

The Gaussian sub-pixel fit is independent of a\, but COM and parabolic are functions of a±. If we look at the RMS error, f_1,2(error)2de, we find that the Gaussian performs the best under all particle diameters. We note that other peak fitting functions have been applied in the literature, for example various versions of spline-curve fits 23 . These, however, have the drawback that they are considerably more computationally intensive without significant gain in accuracy. As will be discussed in the next section, there are alternative methods to improve the sub-pixel fit. 4.5. Peak-Locking 17

and

Solutions

Westerweel has argued that the requisite bandwidth for a PIV image is constrained only by the need to locate the correlation peak. Hence, if this bandwidth criteria is met, we should be able to locate the correlation peak to subpixel accuracy in an unbiased manner. However, as demonstrated in section 4.4, we find in general a strong bias of subpixel fit estimators toward integer pixel locations. Looking at figure 6 we note that for actual subpixel displacements of—0.5 < e < 0 the bias error is positive - hence the determined displacement is biased towards a subpixel displacement of zero. We see the same effect for 0 < e < 0.5, namely the bias error is negative resulting again in a bias toward zero pixel displacements. This effect is often referred to as peak-locking as correlation peaks have a tendency to 'lock' onto integer-pixel displacements. The easiest way to identify the presence of peak-locking in a data set is to plot the histogram of the un-calibrated displacements (i.e., the displacements in pixels) as shown in figure 7. Depending on the quantity of interest peak-locking may not be a significant problem. For example, if one is interested in the mean velocity or even the variance, peak-locking contributes little to the error as long as the histogram spans at least two integer pixels of displacement. If the probability density function (PDF) of the underlying velocity field is of known shape then the histogram could be corrected by redistributing the displacements to have the assumed form of the PDF.

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

21

ccnter-of-itiosE parabolic fit Gaumian fit

I

i

i

i

i

L.

-^

Fig. 6. Sub-pixel bias. Comparison of different sub-pixel peak fits. Figure reprinted with permission from Westerweel 17 .

However, if the displacements range over less than two integer pixels the possibility of a strong bias in all calculated statistics exists. Further, if one is interested in derivative quantities (e.g., vorticity, acceleration) or other quantities involving the difference of two displacement calculations (e.g., turbulent structural functions) then errors induced by peak-locking effects may be considerable and must be monitored. There has been research on minimizing or even eliminating the effects of peak-locking. As there is no peak-locking effect if the subpixel displacement is truly zero the majority of efforts have focused on iterative solutions where the image subwindows are displaced dynamically in fractional steps (a process that is known as continuous window shifting as opposed to the more conventional discrete window shifting where subwindows are shifted an integer number of pixels). Continuous window shifting is generally achieved by interpolating the original image at subpixel locations. The idea is to dynamically shift the subwindow(s) until the subpixel displacement is identically zero. Gui and Wereley35 have found that continuous

22

J. K. Sveen and E. A. Cowen

2x10"

1.8 1.8 1.4 1.2

vector 1 count 0.8 0.6 0.4 0.2 0.

u (pixels/cm) Fig. 7.

Example of velocity histogram showing the peaklocking effect.

window shifting using simple bilinear interpolation and as few as four iterations provides a significant minimization of peak-locking effects. Fincham and Delerce29 employ a slightly different approach to continuous window shifting as they use a spline-thin-shell interpolator to analytically model the first image subwindow and then utilize a PIV result already calculated for the image pair to translate and deform this analytic function based on the velocity field estimate. This analytic function is then compared to the subwindow in the second image in a least-squares sense and iteratively adjusted to minimize the difference. They demonstrate that the final result reduces peak locking and gradient bias while enhancing the robustness of PIV in the presence of locally large shear. It must be pointed out that their approach is computational expensive without any obvious improvement relative to simple bilinear interpolation. As an alternative to interpolation in the spatial domain, Liao and Cowen36 have shown that interpolations for continuous window shifting can be performed more efficiently in the spectral domain taking advantage of the shift property of the Fourier transform. They demonstrate the elimination of peak locking effects at minimal computational cost, given the Nyquist particle image sampling criteria is met. We also note that bi-linear subpixel interpolation has been used earlier in other contexts of pattern matching which we will return to in section 7.2. Finally we note that the solutions to peak-locking problems are closely

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

23

connected to the evaluation methods in question, and readers may find several relevant publications cited in section 4.7.3. In one of the more recent works, Scarano and Riethmuller 30 give a general review and comparison of several iterative PIV methods. 4.6. Spurious

Vector

Detection

There are three fundamental approaches to test the validity of an individual vector: signal quality, spatial consistency (smoothness in space), and temporal consistency (smoothness in time). Signal quality based tests look at a specific property of the images, such as the magnitude of the correlation peak relative to the magnitude of the background noise in the correlation plane, to test the validity of a vector. Spatial consistency tests compare each vector with certain properties calculated from the local neighborhood of vectors (local filtering), or the entire vector field (global filtering). Temporal consistency tests are similar to spatial consistency tests with the exception that the vector whose quality is being tested is compared to either a statistic developed locally in time (e.g., acceleration) or globally in time (e.g., its distance from the mean relative to the standard deviation 22 ). Filtering using spatial statistics is the standard approach, however, and has the added advantage that it works on individual instantaneous vector fields. That said, the recent advances in camera speed have lead to temporally resolved data. As single-point measurement techniques can only rely on the temporal history of data to remove spurious points there is a well developed literature on the removal of spurious data points from temporal records (e.g., spike detection in acoustic Doppler velocimetry, which is essentially a local gradient (the acceleration) threshold in the time domain). The time domain is a relatively unexploited method for spurious vector detection in PIV which should be developed. The detection of spurious vectors is an essential part of any PIV measurement, and the topic is well covered in the literature 17 ' 37 ' 38 . Below is a review of the fundamental approaches currently in use. 4.6.1. Detectability It would be nice if we could simply look at a correlation-peak (PIV) and decide, based on some criteria, whether or not we think it is valid. Keane and Adrian 39 proposed the detectability, DQ, as such a measure. They define the detectability as the ratio of the peak value of the first correlation peak to the second and suggest that if DQ is greater than 1.2 — 1.5 that the vector is valid. The detectability criterion is often referred to as signal-to-

24

J. K. Sveen and E. A.

Cowen

noise ratio (snr) filtering. Westerweel37 demonstrates that any of the three methods based on the statistics of the measured vectors in a field perform better. These methods are known as global mean, local mean and local median filters.

4.6.2. The Global Mean Often during PIV measurements we observe that the data contains a few vectors that are very different from the whole ensemble of vectors. These vectors are typically the result of either i?_i or R+\ being close to zero, see equation (12). If Vij is the observed vector at i, j then the global mean v = jj J2r s Vr,sThe variance of this statistics will be a^ +cr^ where a% is the variance of the actual velocity field and a\ is the variance of the error. So, one possibility is to try to select the allowable variance of the signal and remove outliers (invalid vectors). However, of -C a% in practice so this is hard, hence, a user determined threshold is generally applied.

4.6.3. The Local Mean Let us now extend this idea by looking at individual vectors and comparing them with their closest neighbors. As opposed to looking at individual vectors compared to the global properties, this filter should be able to remove local "large" vector to vector variations. To do this we evaluate

NM

kJ^M

typically using a 3 x 3 region (NM = 8) around each vector, where M is the set of points in the neighborhood. The common approach is now to say that a vector is invalid if it is "very" different from the mean of its neighbors (for example twice as large). We immediately notice that in the cases where the neighborhood actually contains one or more outliers, our criterion for filtering is very difficult to determine, and actually spurious vectors are smoothed out with this method. It turns out we need to know a priori which vectors are bad to make reasonable decision criteria. If we can, perhaps using global mean, this performs fairly well.

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

25

4.6.4. The Local Median The obvious extension of the local filter is to use the local median value instead of the mean. The median value is the nth value in a 2n + 1 long sequence (data sorted in ascending order) and is well known to be robust to outliers for estimating the mean of data. It should be relatively obvious that spurious data normally will be sorted to either side in this sequence. The advantage is therefore that the median is much more robust to outliers and eliminates the need for a priori information on which vectors are bad. Westerweel37 finds that this performs the best. 4.6.5. Adaptive Gaussian Filtering (Temporal Filtering) Given the underlying stochastic nature of turbulence it is possible to exploit knowledge of the PDF of the velocity component statistics to remove spurious vectors. Cowen and Monismith 22 demonstrate such a technique for both PIV and PIV stray vector removal. The idea is to choose a robust estimator of the center of the distribution (accumulated in time as the temporal ensemble at a point), such as the median, and then iteratively test the data for lying within the assumed bounds of the PDF (in the case of Cowen and Monismith, a Gaussian) given the number of data points in the ensemble. The data that is retained is used to recalculate the center of the data, where the mean may be used after the first iteration in general, and the retained data is again tested for lying within the bounds of the assumed PDF. This processes is iterated until no data is removed and hence is an adaptive filter. 4.6.6. Filtering Example Fig. 8 shows a velocity field as measured by conventional FFT-based PIV and filtered using a global filter, a local median filter and a signal-to-noise ratio filter. The images in question are identical to the images shown in Fig. 2 and Fig. 11. Fig. 8a-c shows a part of the velocity field as measured from the example images. The three sub-panels depict the velocity field after first a snr-filter has been applied, followed subsequently by a global filter and a local filter. First the vector field has been filtered by a signal-to-noise ratio filter, discarding all vectors where the ratio of the tallest peak in the correlation plane to the second tallest is less than 1.3. Secondly the remaining vectors are filtered using a global filter with the criterion defined as Vij sg mean(v r>s ) ± k • std(iv iS ),

(20)

26

J. K. Sveen and E. A.

Cowen

where A; is a constant denned by the user (in the present example k = 3) and r = 1,...,R and s = 1 , . . . , S and (R, S) is the number of velocity vectors. Finally a local median filter is applied, and in this case the criterion used for validation is v(i,j) ^ median[i;(r, s)] ± k • std[v(r, s)],

(21)

where again A; is a user defined threshold (in the present example k = 3) and r — i — l,i,i+ 1 and s = j — l,j,j + 1 are the indexes of the 3 x 3 neighboring vectors.

4.6.7. Correlation Based Correction Another approach to vector validation was suggested by Hart 40 , and in fact his proposal is more a general processing technique than a filtering technique. The idea is to take a correlation plane and compare with one or more adjacent correlation planes. The comparison is performed via an element by element multiplication, and thus represents a zero-dimensional correlation (therefore the name "Correlation Based Correction" or CBC). The point here is that each correlation plane typically will consist of a peak that signifies the displacement plus several additional peaks which are due to noise. The noise-peaks are randomly distributed in the correlation plane, but the displacement peaks are not. Therefore the multiplication of two or more correlation planes cancels the noise peaks, while the displacement peak remains. The principle is shown in Fig. 9. In this way we are able to calculate the displacement of a vector centered between the adjacent regions used in the calculation. There are basically two ways to use CBC processing. The first is to use the signal to noise ratio in the combined correlation plane as an indicator, discarding vectors where this ratio is smaller than a certain threshold. The second method, suggested by Hart 40 , is to compare the peak in the combined correlation plane with the peaks in each of the correlation planes that produced it. If the peak in the combined plane exists as a peak in at least one of the other planes, it is assumed to represent the local displacement. Hart 40 concludes that the method improves sub-pixel accuracy and eliminates spurious vectors, reduces bias errors and improves vector yields. However, he provides no evidence of the reduction in bias errors with regard to peak locking and, in fact, tests performed in the preparation of this manuscript suggest that CBC processing may be sensitive to peak locking.

Quantitative

a)

Imaging Techniques and Their Application

to Wavy Flows

27

100 P5

_~

150 200

—^—^"V^-

s>£ .

:_;_:.

" z-~—Z-~.

_v:

300 350 400 450 500

550

600

650

700

750

800

850

900

950

1000

800

850

900

950

1000

800

850

900

950

1000

i b)

100 150 200 250

J 300 350

450 t 500

550

600

650

700

750

i

c) 1°° 150 200 250 3

300 350

500 500

550

600

650

700

750

i Fig. 8. Example of PIV velocity filtering, a) Velocity-field - black vectors removed by the snr-filter. Gray vectors are the remaining, b) shows the results after the global filter gray, valid, black, outliers, c) black vectors, outliers identified by the local median filter.

4.7. Alternative

Displacement

Estimation

Approaches

We have until now only discussed the "original" approach to PIV, namely by evaluation of the cross-correlation function via the use of FFTs. Further-

J. K. Sveen and E. A.

28

640

660

680

700

Cowen

640

660

680

700

Fig. 9. a) ana b), soiici square corresponds to panel c), dash-dot led s^uaAc U> panel d) and dashed square to panel e). Multiplication of correlation tables c), d) and e) produces the combined correlation plane f).

more we have seen that in order to normalize the correlation function, we must divide it by the covariance function. The latter is hard to implement using FFTs and therefore a first order approximation is applied by dividing the correlation plane with the standard deviations of the original images. in the following sections we will review a few different approaches that avoid the use of FFTs. Typically their downside is the additional computational time needed, but on the other hand often at improved accuracy. The cross-correlation function is used in many applications for pattern matching. By definition it is a measure of the "likeness" of two images (or subwindows). In the cases where there is a certain degree of match between the images, it will produce a plane which will contain a peak at the position of the highest match. In contrast we could have chosen to look for the position where the images are the least unlike. This can be done by looking at the minimum of the squared difference between the images. 4.7.1. MQD The use of the minimum squared distance (MQD) for PIV evaluations was first proposed by Gui and Merzkirch 41 and subsequently expanded in Gui

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

29

and Merzkirch 42 . The MQD method evaluates the function M-1N-1

where the location of the minimum value of R indicates the displacement between the windows. The method is reported to increase the accuracy compared to the cross-correlation function, and it is claimed that the reason for this is that "... the MQD method contains a term which accounts for a non-uniform distribution of the particle images and for a non-uniformly distributed illumination intensity...". If we turn to the basic mathematics of the method 43 , it should be fairly obvious that M-1N-1

E T,lF'(hJ)-F"(i + s, j + t)}2 = i=0 j = 0 M-liV-1

E

E [*"(«'J')' " *F'&3)F"{i + s,j + t) + F"(i + s,j+1)\

i=0

j=0

(22)

which in turn means that the MQD method equals the (zeroth order) autocorrelation of the first (sub-)image, plus the autocorrelation of the second (sub-)image (or a part of it, that is), minus two times the cross-correlation of image one and two. If the images now contain a nonuniform seeding or illumination, this will show up in both the auto-correlations and in the crosscorrelation, but will cancel each other out in the MQD function, roughly speaking. If, on the other hand, the images are uniformly seeded and the illumination is uniform, there is no need to evaluate the full MQD-function. The evaluation of the auto-correlation function can be understood as a measure of the non-uniformities within the overlapping regions. As an example we may consider each interrogation region as a wave-field, where the particles show up as small waves. A non-uniformity in the seeding density, or a background illumination gradient, will appear as a lower frequency wave in this field. Depending on the amplitude of this lower frequency, our PIV measurement may in some cases be overshadowed by it. We may argue that the comparison between MQD and cross-correlation is a little unfair since the MQD plane is not normalized. A proper normalization of the cross-correlation plane can only be achieved in the spatial domain, and this is the basis of the next evaluation method we shall look at.

30

J. K. Sveen and E. A.

Cowen

4.7.2. CIV The cross-correlation function, as defined in equation (1), may be used for pattern matching, but the reader may notice that the product will be sensitive to changes in the amplitude in the image matrices. If we double the values in one of the images, the correlation values will double. For this reason the function known as the correlation coefficient (also termed the covariance function earlier in this paper), defined as J?(.] { , )

1 S , S j [F'(i, 3) ~ F>) [F"(i + s,j+t)F"\ 2 2 N Ei Ej([F'(i,J) ~ F'] [F"(i + s,j + t)- F"P)i/2

^ '

12

is often used in pattern matching (with M = N). Here F" is the average value of image 2, which is evaluated only once, and F" is the mean value of image 1 which needs to be evaluated for every (s, t). As we already have seen, the denominator may be approximated by iV 2 std(F')std(F") in the cases where we have uniform illumination or seeding. If this is not the case, the standard deviation of the region in image 1, which overlaps image 2, will actually change as the values of (s, t) change. The method of direct calculation of correlations in PIV, has been suggested by a few authors, most notably Huang et al.27, but also Fincham and Spedding 23 . The latter authors suggest that there should be two main reasons for calculating the correlations in the spatial domain. Firstly they point out the argument about normalization that we have already presented. Secondly they claim that the use of FFTs place a limitation on the size of the interrogation windows. Traditionally interrogation windows needed to be of size 2™, but recent developments in FFTs have made this demand less stringent 44 . We note that the benefits of calculating the full correlation coefficient is well known within the image processing community 12 ' 43 and this fact can be found in many introductory level books on image processing. 4.7.3. PIP-Matching and PID The concept of Particle Image Pattern (PIP) was first introduced by Huang et al.27 and is basically a direct calculation of the full correlation coefficient, see equation (23). The authors vary the interrogation window sizes in order to maintain all particles in both frames. The works of Huang et al.27'28 also contain an approach to dealing with large gradients in flows, known as particle image deformation, or PID. Large gradients may lead to interrogation regions with a spatial variation in velocities, again leading to loss of correlation. To adjust for this they

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

31

distort the interrogation windows in an iterative manner where the rotation and shear is calculated from the velocity field and subsequently used to distort the images. These distorted windows are then used to calculate a new velocity, and the process is iterated upon until some predefined criteria is met. The window distortion is calculated from the shear components (du/dy, —dv/dx) and strain components (du/dx, dv/dy), neglecting the translation. A schematic drawing is shown in Fig. 10.

Fig. 10.

Schematic drawing of the interrogation window distortion in the PID method.

Scarano and Riethmuller 30 extended the PID method to also use interrogation window offsets, as well as distortion (Originally the PID method used larger interrogation regions in the second window to maintain a high correlation). Additionally they apply progressive grid refinements to maximize the spatial resolution, and in this respect they combine the results of Huang et al.27'28 with those of Westerweel et al.25 They report a decreased sensitivity with respect to peak-locking effects. Several authors 46 ' 47 ' 48 ' 49 have applied the PID concept or similar approaches and readers are advised to study these works individually. For the present review, the results by Lin and Perlin 47 are of particular interest since it deals with PIV measurements in water waves. More specifically they combine the PID methods with a particle tracking approach 50 in order to measure velocities in the boundary layer of gravity-capillary waves. 4.7.4. Other Methods for Displacement

Estimation

Some researchers have suggested the use of the phase correlation functions for pattern matching. This method is detailed in Kuglin and Hines 51 , and it is based on using the phase information in the images rather than the amplitude information. Finally we would like to mention that within the field of Optic (or Opti-

32

J. K. Sveen and E. A. Cowen

cal) Flow, another approach to motion estimation is often used. Under the assumption that the image intensity, E, is conserved and that the displacements between frames are small, one may solve Et(x,y,t)+v-VE(x,y,t) = 0,

(24)

where subscript denotes partial differentiation with respect to time, v is the velocity vector and V is the gradient operator. Interested readers may find a good starting point in the paper by Beghdadi et al.52 4.8. Comments Correlation

on the Normalization

of the

Cross

In the cases where the interrogation windows contain an unevenly distributed pattern or where we perhaps have non-uniformities in our illumination, the cross correlation may contain errors. In these cases the calculation of the full correlation coefficient may be more accurate. Since this is a time consuming task to calculate, an approximation is often used for the denominator. In PIV we normally use the standard deviations of the two sub-windows. For example in wavy flows, we often find a thick band of light at the fluid surface, where the light sheet leaves the fluid. For demonstrational purposes we will consider the work by Jensen et al.2, where errors in the PIV calculations are observed close to the free surface. Two images from those experiments are investigated (figure 4 b in that paper), and we focus our attention at this problematic region. Fig. l l a - b shows the original images taken with a time separation of At = 0.0012s. Fig. l l c - d shows two corresponding 64 x 64 pixel subregions, close to the free surface. The reader may notice that there is a gradient in the background illumination in these two sub-windows. Correlation via FFT produces the correlation plane shown in Fig. l i e and, although difficult to spot visually, the central peak is actually higher than the true displacement peak. The central peak corresponds to a correlation of the gradient in the background illumination within the sub-windows, and since this does not move between exposures, it shows up as a zero-displacement peak. If we, on the other hand, use direct calculation of the cross-correlation function, we get a correlation plane as shown in Fig. l i e . Here the true correlation peak is slightly taller than the false (zero-displacement) peak. We observe that the full correlation calculation is less sensitive to the background illumination gradient. This brief example has an interesting aspect if we compare with the results from the MQD algorithm. Fig. 12 shows the corresponding plane as calculated using the MQD method. Interestingly we can observe that due to the nature of

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

33

the algorithm, the false correlation peak has opposite sign of the true displacement peak and the true displacement may easily be identified as the minimum value (black color) in this figure. This indicates that the MQD method may be favorable in many cases where the image quality is not ideal. In other areas of pattern matching several approaches for calculating the normalized correlation plane are used. For example one method was developed for the motion picture Forest Gump43, which relies on using precomputed integrals of the image and image energy over the interrogation region.

5. P T V Particle Tracking Velocimetry is probably the oldest of the particle based QI techniques. In its simplest form it can be thought of as low particle image density PIV, where low particle image density means that the concentration of tracers is sufficiently low that the maximum displacement of a particle in some time, At, will always be less than the mean nearest neighbour spacing between particle images. In PTV's early incarnations particle images were manually digitized and tracked, but with the aid of computers the process is now automated. However, due to the difficulty in determining a particle's pair when the particle's mean spacing is on the order of, or less than the maximum expected displacement, the technique has remained, until the 1990's 22 ' 50 ' 53 ' 54 , a low seeding density technique. That decade saw multiple researchers working with a variety of computer algorithms for particle pairing - the process of identifying which particle image, in any particular image, resulted from the identical particle that produced a particle image in one or more other images (singly exposed multiple images) or a different particle image in the same image (multiply exposed single images). During the 1990's the restriction on displacements relative to the mean particle separation distance was relaxed. The specific seeding density limit depends on the physical parameters of the experiment and the procedure used for particle pairing, and ultimately Cowen and Monismith 22 demonstrated that the limit on seeding density was essentially set only by the requirement of having the majority of the particles produce non-overlapping images. While the low seeding density restriction on PTV was an early criticism of the technique, it was overcome. A second significant criticism of PTV is the resultant randomly distributed velocity vectors, which is still an important area of research. The traditional solution to dealing with randomly located PTV data is to interpolate it onto a regular grid. This interpola-

J. K. Sveen and E. A.

:u

400

800

Cowen

1280

True correlation peak False correlation peak Fig. 11. a)—b) original images, black squares indicate the position of subwindows shown in c)—d). e) FFT-based correlation plane, f) Direct calculation of cross correlation.

Quantitative

Imaging Techniques and Their Application

20

40

60

80

100

to Wavy Flows

35

120

Fig. 12. Figure showing result of MQD-interrogation of the subwindows in Fig. 11 c)-d). Note t h a t the displacement is found as the minimum value (black color) in this plane.

tion has an associated error, that while not a major problem relative to the mean velocity field, is problematic when considering the variance and the co-variance as well as the calculation of any velocity gradients. Several research efforts at determining optimal interpolation methods exist (e.g., Agui and Jiminez 57 , Spedding and Rignot 58 ) and Cowen and Monismith 22 demonstrate the success of a zeroth order interpolation scheme - simple data binning over small spatial regions. While the above criticism is generally restricted to PTV, the reality is that the majority of PIV techniques produce quasi-randomly distributed velocity vectors due to the perturbation to the position of the PIV interrogation grid as a result of using first-order displacement differences to estimate the velocity. As discussed, Wereley and Meinhart 26 have resolved this issue through the use of second-order accurate displacement differences that are centered on the interrogation grid node. It should be noted that in the limit of small subwindow size PIV converges to PTV which points to a remaining perturbation to the location of the second-order accurate displacement differences proposed by Wereley and Meinhart - the true location of the estimated average displacement between two subwindows must account for the location of particles in each subwindow (e.g., a particle weighting of the information in each subwindow) and as the subwindow gets small the probability that the center-of-mass of the particle location information is not at the center of the subwindow, as implicitly assumed in all subwindow techniques, increases.

36

J. K. Sveen and E. A.

Cowen

The fundamental PTV approach to the determination of the velocity is to measure the displacement of an individual particle, located at a random point in space and time, either in doubly-exposed single images or successive singly-exposed images. The velocity is calculated as: U

^'V' '

=

Ax(x,y,t) At

^ ^

Clearly in the limit, when At —> 0, we recover the fundamental definition of velocity. However, we must keep in mind that we are approximating the Eulerian velocity as the second-order difference (or in some cases even the first-order difference) of the Lagrangian track of a particle. In fact, it is important to remember that PTV essentially is a Lagrangian measurement technique. Dalziel 53 ' 54 developed an effective and relatively successful particle tracking algorithm, which was commercialized through the code Dig/mage and applied in many publications 59,60 ' 61 . In the Diglmage code, for example, the Lagrangian velocity is calculated by a least squares fit to more than two points in the particle path. By fitting a quadratic function we can obtain the Lagrangian acceleration directly, a concept also utilized by Chang et al.62 to obtain acceleration from a single-camera configuration.

5.1. Particle

Detection

Algorithms

All PTV based techniques employ some form of particle detection algorithm. This usually involves thresholding the image, leading to a binary image. The threshold may be set manually or determined automatically. In either case the ideal image will have a histogram that looks like the one shown in Fig. 13. If in fact two separate peaks exist (one for the background and one for the particles) it is a simple process to pick a threshold value between the two peaks. However, as discussed shortly, it is more likely that the histogram will decay monotonically (a negative exponential probability density function), making the threshold choice somewhat unclear. In the latter case a user defined threshold is often the simplest solution. Preconditioning of the images may, generally speaking, serve to enhance the particle images from the background. Such preprocessing may be accomplished by convolving the images with a top-hat function, which will serve to amplify the particle images. Subsequently thresholding is applied, followed by binarization. Once an image has been binarized, particles are generally defined in one of two ways, either as horizontally and vertically adjacent "hot" pixels or as horizontally, vertically and diagonally adjacent "hot" pixels.

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

37

m

su 60

count

40 30

20

to

0

••———

0

—

30

iMllM^^tetfM^tf^

™—

100

— —~—™

™»™™™»™™™™™™™,™™™.

150

200

250

Fixe] values, px Fig. 13.

Histogram of image intensities. Notice the small peak between 20 < px < 40.

Particle tracking may occur over multiple exposed or sequential singly pulsed images. Multiply exposed particle image based techniques can lead to higher accuracy since the probability of incorrectly drawing a vector decreases with each additional exposure. However, these techniques suffer from lower vector density as the probability of tracers remaining in an illuminated region is reduced the longer they are tracked. Multiply exposed image based techniques- can also be used to make higher order estimates of the velocity field (by retaining higher order terms in the Taylor Series expansion implied by equation 25) and the estimate of the acceleration field, which will be discussed further in section 6.4. 5.2. Locating

a Particle's

Center

Spherical particles tend to scatter light with an intensity pattern given by a rotated Airy wave. The central lobe of an Airy wave is well approximated as Gaussian. As discussed in section 4.4 as long as the particle image diameter is small ( 2 - 4 pixels) then a three-point Gaussian sub-pixel estimate of the particle's location is optimal. Unlike correlation peaks, however, we have no guarantee that the particle image diameter will be greater than one pixel. It is important in implementing the image acquisition and particle seeding that proper choices are made to insure that particle images will have diameters greater than one pixel and preferably less than four pixels 22 . Further, the same considerations about peak-locking as discussed in section 4.5 exist. In many cases, however, particle images may overlap (or nearly so) and in these cases the Gaussian fit may not work as well. In this case

J. K. Sveen and E. A.

38

Cowen

the particle image intensity centroid may be used as an estimate of the particle's position. 5.3. Particle

Pairing

After the individual particles are located at two different time steps, usually the next step will be to perform a pairing or matching of them. That is, we want to know which particle image in the second frame results from a particular particle that was imaged in the first frame. Hybrid PTV techniques have been developed that employ PIV based algorithms as predictors of the velocity field allowing particle pairing searches to be conducted in small search regions relative to the region to where a particle might have moved 22 ' 50 . This predictor approach overcomes the low seeding density limitation on PTV allowing particles to be tracked in flows where the mean spacing between particles is considerably less than the expected maximum displacements. Ultimately it must be remembered that these PIV predictor techniques suffer from the limitations of the PIV algorithms used to develop the predictive velocity field. The algorithm in Dig/mage 53 ' 54 is based on Operations Research. Particle pairing is accomplished via the use of a cost function. This cost function is dependent on individual particle factors, such as size, intensity, velocity history and shape. Particles are paired after a minimization process, where the cost of pairing particles in one frame with the particles in the next is minimized. The algorithm in Diglmage is similar to the so called transportation algorithm, although not quite the same since particles are allowed to leave and re-enter the light-sheet. There are many other algorithms that have been proposed for particle tracking that include fuzzy logic55 and simulated annealing 56 . An intriguing capability of several of these techniques is their ability to extract higherorder flow quantities, such as the local strain rate and rotation rate, directly from the analysis. 6. Higher Order Measurements from Velocity Fields Obtained by QI-Techniques In a great number of applications of Ql-techniques the velocity field information is only part of a complete description of the physical processes under study. For example many flows of interest are turbulent and derivative quantities of the velocity field may be of fundamental importance to describe the turbulence. In both turbulent and viscous flows the vorticity field is often an important descriptor of the flow physics. In stratified flows

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

39

it may be necessary to simultaneously measure the density field. In wavestructure interaction flows knowledge of the acceleration or pressure field may be of vital interest. In the present chapter we will turn our attention to the use of the velocity fields, obtained from Ql-imaging techniques, to determine higher order quantities. Such quantities may include shear, strain, vorticity, streamlines, acceleration, pressure and estimates of turbulence characteristics (e.g., the Reynolds stresses, turbulence dissipation, and spectra). 6.1. Vorticity,

Strain Rate and

Divergence

The calculation of vorticity, strain rate and divergence is often referred to as the calculation of differential quantities. These quantities are defined as dv

du

1 fdv

s

= A^

.„ . du\

+

-dy}

_ .

(27)

dw _ dv du dz dy dx' where u>, s and —dw/dz denote the (in-plane) vorticity, strain rate and two-dimensional divergence, respectively. The differentiation of the velocity is a straightforward calculation, using for example a second-order accurate central difference numerical scheme. However, QI velocity measurements contain errors which can lead to strong noise in the differential quantity. Furthermore the data is discrete, which also may introduce noise. In these cases many scientists apply higher order numerical schemes 63 , others apply smoothing 64 functions to deal with this noise, and Cowen and Monismith 22 argue for the careful choice of the differentiation length scale to maximize resolution and minimize noise amplification. Readers are advised to consult Raffel et al.lg, Nogueira et al.38 and Lourengo and Krothapalli 65 for further details. We also note that it is possible to measure differential properties directly from the images using image distortion techniques 66 ' 67 and other non-correlation based displacement extraction techniques 56 . 6.2. Streamlines

and Pressure

Estimation

Defining the streamfunction, ip, can be very useful in two-dimensional applications. The streamfunction is constant along a streamline, where the streamline is tangent to the velocity field at every point.

40

J. K. Sveen and E. A.

Cowen

Consider a two-dimensional incompressible flow, thus V • v = 0. The streamfunction can be obtained from dtp = u dy — v dx where u = dip/dy and v = —dtp/dx. This implies that the unknown scalar function tp at the location P is defined as

fP i>p = ipo + /

{udy — vdx),

(29)

Jo where tpo is the streamfunction at the origin O (typically set to zero 68 ). The path of the numerical integration can be freely chosen, but in order to minimize noise accumulation, the integrations are often performed on a staggered grid starting from the center of the velocity field and proceeding out towards the edges 68 . In order to estimate the pressure gradient we have to know the acceleration field (see section 6.4). If we make the assumptions that the flow is twodimensional, steady and incompressible, it is possible to estimate the pressure field from the 2D Navier-Stokes equations 68 or similar approaches 69 using the measured velocity field. 6.3.

Turbulence

As many fluid flows of interest are turbulent it is not surprising that QI techniques have a development history that reflects a desire to extract information on the turbulent statistics and turbulent structures of the flows. The application of QI techniques to turbulent flows presents unique challenges due to the range in space and time scales, which leads not only to a large requisit dynamic range in velocity to characterize the flow, but also the potential for strong strain rates, which historically are problematic for traditional PIV type techniques (e.g., 22 ). 6.3.1. Dissipation The determination of the dissipation of turbulence, e, is an important but challenging experimental goal. In theory it should be particularly straightforward using QI techniques based on its definition e = 2i/(sijSij), where Sij is the fluctuating strain rate tensor defined as

(30)

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

41

Hence, the in-plane fluctuating strain rate components can be directly calculated by second-order central differencing and the out-of-plane fluctuating strain rate components can be modeled by an understanding of the anisotropy ratio to the in-plane components 22 . However, presumably due to the noise level present in many velocity field calculations, relatively few researchers have reported dissipation results. Cowen and Monismith 22 report a dynamic algorithm that adjusts the length scale over which the velocity data is differentiated, arguing that the signal-to-noise ratio can be maximized by optimally choosing the length scale of the gradient calculation. Doron et al.74 discuss optimal assumptions about the anisotropy of the out-of-plane gradients impact on the total dissipation. Cowen et al.71 demonstrate several impacts of understating the dissipation structure in the swash zone. They show that the dissipation is in balance with the turbulent kinetic energy and that the turbulence is decaying as free turbulence during the swash uprush. They use the dissipation to estimate the friction velocity and the friction coefficient of the bed. The single point measurement fields with sufficient temporal resolution to capture temporal frequency based spectra (whose Ql-based calculation is described in the next section) have a well developed history of using the inertial subrange spectral energy measurements to estimate the turbulence dissipation. For temporally resolved QI measurements this is another approach to estimating the dissipation. For an example see Liao and Cowen75 and for a discussion of the errors with respect to direct measurements see Doron et al.74

6.3.2. Temporal and Spatial Spectra In unsteady flows in general and turbulent flows in particular, it is often of interest to look at flow statistics in the spectral domain. For single-point measurements this is only possible in the temporal frequency based sense. However, a significant advantage of QI techniques is the ability to determine instantaneous spatial information and hence spatial frequency based spectra. With typical QI hardware, say 30 frame per second cameras with 1024 x 1024 pixel imagers, we often find ourselves in the situation where the high spectral frequency content can only be determined in the spatial domain while the low spectral frequency content must be determined in the temporal domain. Typically a characteristic advective velocity is used to make the Galilean transformation from time to space and the temporal and spatial spectra are combined. In turbulent flows this transformation is known as Taylor's frozen turbulence hypothesis 70 and the characteristic

42

J. K. Sveen and E. A.

Cowen

velocity scale is simply the mean velocity. However, in turbulent flows one must be careful as Taylor's hypothesis requires (u' ) 1//2
(32)

where u is the instantaneous velocity vector at spatial location x and time t, u is the temporal mean velocity vector, u is the phase averaged velocity vector, and u' is the turbulent velocity perturbation vector. The turbulent perturbations are generally found by subtracting the temporal mean and phase averaged velocity vectors from the instantaneous velocity vector and hence all perturbations with respect to the temporal mean and phase averaged quantities are passed through to u'. From energy considerations gravity waves are known to propagate as groups and often contain reflection components. Hence at minimum one expects amplitude variations even in a monochromatic wave field. These amplitude variations, even if small, can lead to perturbations from the mean and phase averaged velocity fields that are on the order of the turbulent perturbations, yet are not the result of turbulence (and do not effect the flow in the same manner as turbulence, e.g., they do not enhance the mixing). To account for the amplitude and phase induced variability we can apply a wave-turbulence decomposition technique, originally proposed by Thais and Magnudet 72 for single-point temporal data, to spatially resolved velocity fields. A second-order Stokes wave can be fit to the velocity field in a least squares sense where the wave phase, wave length, and wave amplitude can all be determined by the least squares fit. The velocity field decomposition can be written u(x, t) = u(x) + u P ( x , t) + u f l (x, t) + u'(x, t),

(33)

where u p is the irrotational phased-dependent wave-induced velocity vector, UR is the rotational phase-dependent wave-induced velocity vector, and

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

43

u' is the turbulent velocity perturbation vector. Importantly, u p is used to separate the effects of amplitude and phase-induced variability in the waveinduced motions from the turbulence. An advantage of the decomposition is that it works independently of the choice of coordinate system. For example, a free-surface following coordinate system may be utilized to explore the free surface boundary layer structure (e.g., Cowen 73 ), or an Eulerian grid may be used, referenced to the bed, to look at the ensemble averaged phase dependent turbulence beneath the wave. 6.4.

Acceleration

Measurements of acceleration have not received the same attention in the literature as PIV itself. These measurements are important, for example, to determine pressure which is needed for calculating forces on structures that are subject to fluid forces. The acceleration of a fluid particle is defined as, du

a=

( du

+ U

du\

+

Du

W ( ^ ^J=DP

. „,

(34)

where D/Dt is the material derivative and consists of a temporally unsteady and an advective spatially unsteady term. The acceleration field arising from the temporal unsteadiness of the flow may be calculated from the difference between two consecutive velocity fields, as produced by conventional PIV, u2 - u i a = .

t2-h Here t\ and i 2 denote the time of measurement of the first and second velocity fields, respectively. However, to calculate the material derivative of the velocity, in addition to the temporal gradients, we also need to evaluate the spatial derivatives. In each calculation small differences in velocity, either in space or time, are normalized by small differences in either position or time, and may produce large errors. To enable proper control and accuracy of acceleration measurements typically between 2 and 4 consecutive images are required, depending on the choice of PIV-evaluation method (auto or cross-correlation, or a combination of the two). Furthermore we need to accurately control the timing of each exposure. Jensen et al? reported on acceleration measurements in periodic surface waves in a wave tank. They measured the temporal derivative, (u(x, £2) — u(x, £i))/(£ 2 — ^i) and used a stereo CCD-camera setup where each camera

44

J. K. Sveen and E. A.

Cowen

takes two single-exposed images. The exposure was controlled by the use of an Acousto-Optical Modulator. In this case careful camera alignment is crucial to avoid introduction of additional errors. Jensen et al.2 reported that their camera calibration typically introduced errors on the order of 1.9 — 2.6% compared to the displacement. Measurements were performed on accelerations due to water waves in a laboratory wave tank and the results were compared to the theoretical description of Stokes waves. In the best case they found a relative standard deviation of about 0.6% for the velocities and about 2% for the accelerations. Jensen et al.76 extended the acceleration measurements to also include the advective terms. They calculated the material derivative by tracing pseudo particles through a number of iterations. Their method thus uses an interpolated velocity in the second velocity field. They report the need for smoothing of the velocity fields to reduce the effects of spatial noise. Measurements were performed on runup of strongly non-linear waves on a steep beach. Additional results using this method are also published in Jensen et al.77 Jacobsen et al.78 reported on a 4-CCD-camera looking at the same field of view through the use of beamsplitters. The temporal component of the acceleration in standing waves and trains of irregular waves was measured, with errors being on the order of 20 — 100%. Other investigations 79 ' 80 use similar two CCD-camera setups, employing off-the-shelf PIV cameras that image approximately the same field of view. Dong et al.79 report a two camera system for measuring the temporal component of accelerations in a rotating flow using pairs of doubly exposed images. They report a relative uncertainty of less than 25%. A similar approach was used by Chang and Liu 81 . Christensen and Adrian 80 extended the acceleration calculation to also include the advective terms. They report that these terms are an order of magnitude smaller than the temporal part in their application, which is acceleration measurements in a turbulent channel flow. They calculate the convective derivative by shifting the evaluation region in the second velocity field by the horizontally averaged mean flow. We note that the work by Christensen and Adrian uses polarization of the laser illumination to avoid a multiple exposure on the second image at the first camera. Such a double exposure was a feature found in the system of Jensen et al.2 Chang et al.62 proposed an interesting 2D PTV based single-camera technique where they used one doubly exposed and one singly exposed image and PTV to calculate the substantial derivative in the Lagrangian reference frame.

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

45

La Porta et al.82 measured fluid particle accelerations in fully developed turbulence, using a 3D PTV method 83 with relatively advanced experimental techniques (recall from section 5 that PTV may give Lagrangian accelerations directly). 7. Conclusion In this work we have aimed at reviewing the foundations and fundamentals of Quantitative Imaging (QI) techniques in general with a particular focus on Particle Image Velocimetry (PIV) and specific details of its application to wavy flows. The QI techniques have been presented in the context of general pattern matching techniques with the aim of pointing out similarities and differences between the relatively large number of techniques available. With this in mind we conclude by tieing the specifics of QI techniques in fluid flows thus far presented to the more general field of pattern matching techniques. The thoughts below are only a minimal scratch at the surface of the variety of pattern matching techniques in use in quantitative research fields but we hope they give the reader a sense of the range of applications of these techniques. 7.1. Measurements

of

Deformation

The use of PIV techniques has received increasing attention within the field of solid mechanics where it is applied to measure deformation of various substances such as clay, soil or sand subject to pressure or forces 84 ' 85 . In these cases the addition of tracer particles is not as common as it is in fluid mechanics. Sand or soil, for example, will often provide a pattern suitable for matching, whereas a substance like clay can be (very gently) spray-painted on the clay surface to create a suitable pattern. 7.2. Measurements

of

Density

A light-ray that passes through a fluid may experience deflection due to changes in refractive index in the fluid. The method known as Schlieren11, is a well known quantitative method to determine gradients in a density field. By using a background pattern, any refractive index change will displace this pattern with a distance proportional to the density gradient. This is basically the idea behind the image processing versions of Schlieren, which are known as Synthetic Schlieren 86 ' 87 and Background Orientated Schlieren (BOS) 88,89 . The latter method is based on cross-correlation using FFT (i.e. equation 1) to locate the displacement, while the former method

46

J. K. Sveen and E. A.

Cowen

primarily calculates the full correlation coefficient (equation 23) but also the absolute difference, i.e. R(s,t) = \F'(i,j) — F"(i + s,j + t)\, and the minimum quadratic difference methods are used. In the sense that both methods rely on principles of pattern matching, we can say that they are identical. We should also emphasize that subpixel interpolation of the subwindows were implemented in these techniques. In fact, Dalziel et al.87 applied bilinear subpixel interpolations of their images, similar to that applied by Gui and Wereley35 in the context of PIV. 7.3. Measurements

of

Volume

In a vast number of industrial and scientific applications it is desirable to measure the volume of an object. As an example we can consider the inspection of manufactured parts where a lower limit tolerance exists with respect to accuracy. Manual inspection is in most cases impossible and the use of various 3D scanning systems are currently in use 90 . There are several, slightly different methods available, some of which are known as structured light or (computer) tomography. Applications include scanning of the human body 91 , automatic inspection of car parts 9 2 and fluid surface measurements 93 . The concept behind this class of methods is basically to illuminate a 3D object with a pattern. The pattern will thus be displaced compared to the scene without the object present. The displacement is directly proportional to the height of the object in the plane we are viewing. By rotating the object or using more than one camera, the 3-dimensional shape may be reconstructed. Acknowledgements J. Kristian Sveen gratefully acknowledges the financial support provided by the Research Council of Norway through the Strategic University Program "General Analysis of Realistic Ocean Waves" (GROW). Edwin A. Cowen gratefully acknowledges the financial support of the U.S. Office of Naval Research (grant No. N00014-98-1-0774, Dr. Keith Ward Program Officer) and the U.S. National Science Foundation (Grant No. CTS-0093794). Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author (s) and do not necessarily reflect the views of the Office of Naval Research or the National Science Foundation.

Quantitative Imaging Technigues and Their Application to Wavy Flows

47

References 1. J. K. Sveen, http://www.math.uio.no/~jks/matpiv/, (1998-2003). 2. A. Jensen, J. K. Sveen, J. Grue, J.-B. Richon and C. Gray, Exp. Fluids 30-5, 500 (2001). 3. H. Meng and F. Hussain, Fluid Dyn. Res. 8, 33 (1991). 4. A. K. Prasad and R. J. Adrian, Exp. Fluids 15, 49 (1993). 5. C. E. Willert and M. Gharib, Exp. Fluids 12, 353 (1992). 6. C. Briicker, Exp. Fluids 19, 255 (1995). 7. R. J. Adrian, TAM report, in cooperation with TSI, St. Paul, MN, (1996). 8. L. Prandtl and O. G. Tietjens, Applied Hydro and Aeromechanics, (Dover Publ. 1934). 9. G. W. Faris and R. L. Byer, Science 238, 1700 (1987). 10. P. J. W. Roberts, X. Tian and Y. Jung, Hydraulic Measurements and Experimental Methods 2002 - Proceedings of the Specialty Conference, EWRI and IAHR, July 30 - August 2, 2002, Estes Park, CO, (2002) 11. W. Merzkirch, Flow Visualization, (Academic Press, 1987). 12. R. C. Gonzales and R. E. Woods, Digital Image Processing (Addison Wesley 1992). 13. http://mpeg.telecomitalialab.com/ 14. H. Foroosh and S. Hoge, in Video registration, eds. M. Shah and R. Kumar, (Kluwer Academic, 2003). 15. L. Hesselink, Ann. Rev. Fluid Mech. 20, 421 (1988). 16. R. J. Adrian, Ann. Rev. Fluid Mech. 23, 261 (1991). 17. J. Westerweel, Ph.D. Dissertation, University Press, Delft, (1993). 18. J. Westerweel, Meas. Sci. Tech. 8, 1379 (1997). 19. M. Raffel, C. E. Willert and J. Kompenhans, Particle Image Velocimetry, (Springer, Berlin, 1998). 20. R. J. Adrian, Appl. Optics 25, 3855 (1986). 21. R. D. Keane and R. J. Adrian, Appl. Sci. Res. 6, 754 (1992). 22. E. A. Cowen and S. G. Monismith, Exp. Fluids 22, 199 (1997). 23. A. M. Fincham and G. R. Spedding, Exp. Fluids 23, 449 (1997). 24. R. J. Adrian, in Laser Anemometry in Fluid Mechanics - III, eds. R. J. Adrian etal, 115 (1988). 25. J. Westerweel, D. Dabiri and M. Gharib, Exp. Fluids 23, 20 (1997). 26. S. T. Wereley and C. D. Meinhart, Exp. Fluids 3 1 , 258 (2001). 27. H. Huang, H. Fiedler and J. Wang, Exp. Fluids 15, 168 (1993). 28. H. Huang, H. Fiedler and J. Wang, Exp. Fluids 15, 263 (1993). 29. A. Fincham and G. Delerce Exp. Fluids 29, 13 (2000). 30. F. Scarano and M. L. Riethmuller, Exp. Fluids 26, 513 (1999). 31. F. Scarano and M. L. Riethmuller, Exp. Fluids Suppl., S51, (2000). 32. A. K. Prasad, R. J. Adrian, C. C. Landreth and P. W. Offutt, Exp. Fluids 13, 105 (1992). 33. J. W. Goodman, Laser Speckle and Related Phenomena, ed. J. C. Dainty, (Berlin, Springer, 1984). 34. E. Parzen, Ann. Math. Statist. 28, 921 (1957). 35. L. Gui and S. T. Wereley, Exp. Fluids 32, 506 (2002).

48

J. K. Sveen and E. A. Cowen

36. Q. Liao and E. A. Cowen, Exp. Fluids submitted, (2004). 37. J. Westerweel Exp. Fluids 16, 236 (1994). 38. J. Nogueira, A. Lecuona and P. A. Rodriguez, Meas. Sci. Technol. 8, 1493 (1997). 39. R. D. Keane and R. J. Adrian, Meas. Sci. Technol. 1, 1202 (1990). 40. D. P. Hart, Exp. Fluids 29, 13 (2000). 41. L. Gui and W. Merzkirch, Exp. Fluids 21, 465 (1996). 42. L. Gui and W. Merzkirch, Exp. Fluids 28, 36 (2000). 43. J. P. Lewis, Proceedings of Vision Interface, 120 (1995). 44. M. Frigo and S. G. Johnson, I C A S S P conf. proc. vol. 3, 1381 (1998). 45. C. Willert and M. Gharib, Exp. Fluids 10, 181 (1991). 46. P. T. Tokumaru and P. E. Dimotakis, Exp. Fluids 19-1, 1 (1995). 47. H. J. Lin and M. Perlin, Exp. Fluids 25, 431 (1998). 48. J. Nogueira, A. Lecuona and P. A. Rodrguez, Meas. Sci. Technol. 12, 1911 (2001). 49. G. M. Quenot, J. Pakleza and T. A. Kowalewski, Exp. Fluids 25-3, 177 (1998). 50. R. D. Keane, R. J. Adrian and Y. Zhang, Meas. Sci. Technol. 6, 754 (1995). 51. C. Kuglin and D. Hines, Proc. 1975 Int'l Conf. on Cybernetics and Society, 163 (1975). 52. A. Beghdadi, M. Meshbah and J. Monteil, Image Vision Comp. 21, 383 (2003). 53. S. B. Dalziel, Appl. Scien. Res. 49, 217 (1992). 54. S. B. Dalziel, Dyn. Atmos. Oceans 20 127 (1993). 55. M. P. Wernet, Exp. Fluids 30, 434 (2001). 56. M Stellmacher and K Obermayer, Exp. Fluids 28, 506 (2000). 57. J. C. Agui and J. Jimenez, J. Fluid Mech. 185, 447 (1987). 58. G. R. Spedding and E. J. M. Rignot, Exp. Fluids 31, 417 (2001). 59. J. Grue, A. Jensen, P-O. Rusas and J. K. Sveen, J. Fluid Mech. 380, 257 (1999). 60. R. E. Hewitt, P. A. Davies, P. W. Duck and M. R. Foster, J. Fluid Mech. 389, 169 (1999). 61. V. Maderich, G. J. van Heijst and A. Brandt, J. Fluid Mech. 432, 285 (2001). 62. K.-A.Chang, E. A. Cowen and P. F.-L. Liu, 3rd International Symposium on Particle Image Velocimetry, Santa Barbara, USA, 16-18 Sept. (1999). 63. J. Grue, A. Jensen, P-O. Rusas and J.K. Sveen, J. Fluid Mech. 413, 181 (2000). 64. J. D. Luff, T. Drouillard, A. M. Rompage, M. A. Linne and J. R. Hertzberg, Exp. Fluids 26, 36 (1999). 65. L. Lourenco and A. Krothapalli, Exp. Fluids 18, 421 (1995). 66. X. Ruan, X. Song and F. Yamamoto, Exp. Fluids 30-6, 696 (2001). 67. S. Mayer, Exp. Fluids 33-3, 443 (2002). 68. K. Imaichi and K. Ohmi, J. Fluid. Mech. 129, 283 (1983). 69. R. Gurka, A. Liberzon, D. Hefetz, D. Rubinstein and U. Shavit, Proceedings to the 3rd International Workshop on Particle Image Velocimetry, Santa Barbara, California, September 16-18, (1999). 70. G. I. Taylor, Proc. Roy. Soc. Lond. (A) 164, 476 (1938).

Quantitative

Imaging Techniques and Their Application

to Wavy Flows

49

71. E. A. Cowen, I. M. Sou, P. L.-F. Liu and B. Raubenheimer, J. Eng. Mech. in press, (2003). 72. L. Thais and J. Magnaudet, J. Geophys. Res. 100-1, 741 (1995). 73. E. A. Cowen, Ph.D. Dissertation, Dept. Civil Engineering, Stanford Univ. (1996). 74. P. Doron and L. Bertuccioli and J. Katz and T. R. Osborn, J. Phys. Ocean. 3 1 , 2108 (2001). 75. Q. Liao and E. A. Cowen, Env. Fluid Mech. 2, 9 (2002). 76. A. Jensen, G. K. Pedersen and D. J. Wood, J. Fluid Mech. 486, 161 (2003). 77. A. Jensen, M. Huseby, D. Clamond, G. K. Pedersen and J. Grue, This volume, Chapter 7.4, (2004). 78. M. L. Jacobsen, T. P. Dewhirst and C. A. Created, Meas. Sci. Technol. 8, 1502 (1997). 79. P. Dong, T.-Y. Hsu, P. Atsavapranee and T. Wei, Exp. Fluids 30, 626 (2001). 80. K. T. Christensen, R. J. Adrian, Exp. Fluids 33, 759 (2002). 81. K.-A.Chang and P. F.-L. Liu, Phys. Fluids 10, 3390 (1998). 82. A. La Porta, G. A. Voth, A. M. Crawford, J. Alexander and E. Bodenschatz, Nature 409, 1017 (2001). 83. M. Virant and T. Dracos, Meas. Sci. Technol. 8, 1539 (1997). 84. D. J. White, W. A. Take and M. D. Bolton, Geotechnique 53-7, 619 (2003). 85. H. Wolf, D. Konig and T. Triantafyllidis, J. Struct. Geol. 25-8, 1229 (2003). 86. B. R. Sutherland, S. B. Dalziel, G. O. Hughes, and P. F. Linden, J. Fluid Mech. 390, 93 (1999). 87. S. B. Dalziel, G. O. Hughes and B. R. Sutherland, Exp. Fluids 28, 322 (2000). 88. G. E. A. Meier, Deutsche Patentanmeldung, DE 199 42 856 Al, (1999). 89. H. Richard and M Raffel, Meas. Sci. Technol. 12, 1576 (2001). 90. F. Arman and J. K. Aggarwal, ACM Comp. Sum. 25-1, 5 (1993). 91. H. Tapp and Z. Dunford, New Scientist 6, 14 (2003). 92. F. Couweleers, 0 . Skotheim, H. Schulerud and K. Kaspersen, Proc. of SPIE 5144, Optical Measurement Systems for Industrial Inspection III, 697 (2003). 93. T. Roesgen, A. Lang and M. Gharib, Exp. Fluids 25, 126 (1998).

This page is intentionally left blank

CHAPTER 2 PIV M E A S U R E M E N T S IN THE BOTTOM B O U N D A R Y LAYER OF THE COASTAL OCEAN

W. A. M. Nimmo Smith, T. R. Osborn and J. Katz The Johns Hopkins University 3400 N. Charles Street, Baltimore, MD 21218, USA E-mail: katz6jhu.edu

This chapter catalogs the development of a submersible PIV system at The Johns Hopkins University for the measurement of turbulence in the bottom boundary layer of the coastal ocean. Currently, this system acquires data simultaneously in two independently oriented sample areas, while operating continuously for several hours. A profiling platform enables sampling from the seabed, up to 10m above the bottom. This system has been deployed on several occasions at different locations on the east coast of the United States. Analysis methods for characterizing the turbulence and determining the Reynolds stress and dissipation rates are presented, along with some sample results. 1. I n t r o d u c t i o n Turbulence in t h e coastal ocean has a substantial impact on the distribution of sediment and biological particles, as well as on t h e dispersion of pollutants. T h e b o t t o m shear stress a n d t h e dissipation r a t e are key parameters in t h e m o m e n t u m balance, so successful modeling of transport, dispersion and biological processes in the coastal ocean requires detailed understanding of b o t h of these parameters. Several different methodologies have been previously adopted to t r y and quantify the b o t t o m shear stress and the dissipation rate. Indirect techniques rely either on assumed velocity profiles or turbulence characteristics. For example, assuming t h a t a constant stress layer exists, t h e friction velocity and b o t t o m roughness can be calculated using a least squares fit t o a measured velocity profile. Similarly, assuming t h a t there is a balance between t h e production and dissipation of turbulent kinetic energy, a n est i m a t e of the dissipation rate can be m a d e from a fit of t h e Kolmogorov - 5 / 3 spectral slope in the inertial range of t h e vertical velocity spectrum. 51

52

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

Various technologies have been used for measuring the turbulence in the coastal ocean. Doppler shift based methods include the AcousticDoppler Current Profiler, ADCP 1 ' 2 ' 3 ' 4 and the Acoustic Doppler Velocimeter, ADV 5 ' 6,7 . The latter has a higher spatial resolution and can measure all three velocity components, but only at a single point in the boundary layer. The BASS system (Benthic Acoustic Stress Sensor) is based on measuring the acoustic travel time 8 ' 9 ' 1 0 ' 1 1 ' 1 2 ' 1 3 . Electromagnetic velocity and vorticity sensors for measuring fine-scale fluctuations have been developed and deployed 14,15 as well as electromagnetic current meters 16 and Laser Doppler Velocimetry (LDV) 12 ' 17 ' 18 . Since these are all point measurement techniques, the turbulence statistics that they provide is adversely affected by contamination from surface waves. Although the waves typically have much larger spatial scales than the turbulence, the frequency of the orbital wave motion falls within the turbulent spectrum, since the wave speed is much higher than the typical flow. Thus, separating the wave-induced motion from turbulence in the signal of a point sensor is virtually impossible. Since the shear stress is a correlation between streamwise and vertical velocity components, the alignment of the instrument relative to the mean flow in an environment with waves is critical. When the instrument is misaligned, the stress is "contaminated" by the projection of the horizontal velocity component on the selected inclined coordinates. Consequently, Trowbridge 19 incorporates simultaneous measurements using two sensors spaced about 63cm apart and determines the stress from the covariance of the difference between the measured velocities. His underlying assumption is that the spacing is larger than the characteristic length scale of the turbulence but smaller than the length scale of the waves. In doing so he also overcomes uncertainties related to the alignment of the instrument, as long as the error in angle is small. His analysis clearly illustrates the advantage in performing simultaneous velocity measurements at multiple points, which makes Particle Image Velocimetry (PIV) a particularly attractive option for characterizing the flow and turbulence (including the Reynolds shear stresses) in the coastal ocean. Unlike all other techniques, PIV provides the instantaneous distribution of two velocity components over a sample area. Consequently, PIV enables measurements of turbulent spatial spectra without assumptions involving Taylor's Hypothesis. A time series of many repeated measurements (ensemble average) can be used for obtaining the turbulence statistics. However, like all other oceanic point measurement techniques in the coastal ocean, time series are adversely affected by unsteady flows associated with surface

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

53

waves. Methods to overcome this problem, based on the approach developed by Trowbridge 19 , will be addressed later in this paper. To this end, we have been developing, deploying and continuously improving a submersible PIV system for almost ten years. The design criteria and the various stages of the development system are detailed in the following section. Several successful deployments have been made at different locations and under various environmental conditions, as detailed in Section 3. Analysis techniques and some sample results, including mean profiles, spectra, dissipation rates and shear stress estimates are presented in Section 4.

2. Development of the Submersible P I V System The original submersible system, described by Bertuccioli et al.20 and Doron et al.21, was based on a single l K x l K , 15 frames/s, "crosscorrelation" digital camera (Kodak Megaplus XHF) and had a sample area of 20 x 20cm2. Schematics of the ship-board components and the submersible sampling platform are shown in Figs. 1 and 2 respectively. The camera was located in a submersible housing, and equipped with a remote focusing system. The images were streamed over 35m of cable to a Macron FT200 Board 272 MB RAM Control of Laser Data from Digital Camera Double Head Flashlamp Pumped Dye Laser:

Contrc I of Digital Camera 400um Core Diameter Optical Fiber - ^

^xn XI

Precision 3-Axis ^ Translation Stage

=5 :—•?

Fig. 1. Schematic of the original surface-mounted laser, control and image acquisition subsystems (from Ref 2 1 ) .

54

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

shipboard data acquisition system which had sufficient RAM to store 136 image pairs before requiring a break in recording to save the data to disk. Unlike typical laboratory applications that rely on artificial seeding with particles of known properties, natural seeding is sufficient for obtaining high quality PIV images in the ocean 20 . Visibility measurements performed during early deployments (measured with a SeaCat, 25cm path-length transmissometer) indicated that PIV data could be successfully acquired when the transmission was between 50 and 90%. A shipboard laser and optical fiber delivery system were chosen to illuminate the sample area for a number of reasons. Estimates of light scattering from particles, as well as laboratory experience, indicated that at least lOOmJ per exposure were necessary to sufficiently illuminate particles that may be smaller than 10/im in diameter, requiring the use of a highpower pulsed laser. One has two options to provide the required power, either to submerge a laser along with its power supply system, or leave the laser on-board ship and use optical fibers to transmit the light to the sample areas. We chose the second option, to keep the laser aboard ship, for reasons of safety and in order to minimize the size and required power (and associated cables) in the submerged system. Transmitting the required energy per pulse through a fiber without damaging it, while leaving some margin for variations in field test conditions, and keeping the fiber at a reasonable diameter (so that the beam can be refocused), eliminated NdYAG lasers as an option. As discussed elsewhere20, the short duration and high intensity of the Nd-YAG laser would have required large fibers and an unacceptable compromise in energy level. Consequently, we opted for a dual head, flash-lamp pumped dye laser (Cynosure Inc., Customized Model LFDL-3) that has a pulse-duration of 2/is, i.e. almost three orders of magnitude longer than the Nd-YAG, but still short enough for applications in water. Our system can generate 15pulse-pairs/s with essentially unlimited in-pair delay. Using Rhodamine 6G as dye, this laser can generate a maximum of 350mJ/pulse at 594nm. The two beams are focussed onto the same fiber using a double prism and a short focal length coupling lens (see Fig. 1). On exiting the fiber, the beams are re-collimated and then formed into a light sheet using optics contained within a small submerged probe 20 . The thickness of the light sheet varies between 2-4mm depending on application, wider than a typical laboratory system but still smaller than the interrogation window size. The housed camera and the light-sheet forming probe were mounted on a seabed platform comprising a turntable and scissor jack (see Fig. 2). The hydraulically operated scissor jack had a vertical travel of 1.5m, allowing the

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

55

sample area to be raised from close to the seabed into the water column. The turntable, driven by a submerged electric gear-motor, allowed the sample area to be aligned with the mean flow direction.

Sample Area: 20 cm x 20 cm

Vertical Travel: 150cm

Hydraulic Scissor Jack

Optical

Laser Probe

r

if -^-

\

^ i~i _ i Camera Control Jnit Dome Unit Dome

Light Sheet

t \

Kodak MegaPlusXHF:

IK x IK. Digital Camera K Max. 30 flames/sec.

Remote Control Focus Thermoelectric Cooling Biaxial Clinometer 4m 1.2 m 2m 700 kg

Turntable Motor ,

Power Supply Relays I/O Extension Iextension

TgTT^E

m

Overall dimensions: Length: Width: Height: Total Weight:

Fig. 2.

Timing pulley

M

Pressure Transducer: Depth & Wave Height Accuracy: ±6.8 mm Max. Depth: 58 m Digital Compass

Additional Instrumentation: Sea-Bird SEACAT CTD: Salinity, Temperature, Depth Dissolved oxygen, Light transmissivity

Schematic of t h e original submersible s a m p l i n g platform (from Ref 2 1 ) .

56

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

This system was deployed successfully in the New York Bight (see Section 3). However, analysis of the data collected there 21 highlighted the limitations in this system. As would be expected, it became evident that a sample area of 20 x 20cm was not sufficient to cover all the relevant turbulent length scales. Enlargement of the sample area without an increase in resolution was also unacceptable because that would mean an undesirable compromise at small scales (the spacing between vectors was already six times the Kolmogorov scale). Evaluation of the range of length-scales that should be covered indicated that we should extend the range of length scales to more than lm, preferably close to 1.5m. Consequently, we opted to develop a new system that utilizes two 2Kx2K cameras, each with a sample area of up to 50 x 50cm, operating simultaneously side by side. As described in Nimmo Smith et al.22, we had several objectives in enhancing the original PIV system: i) Continuous data acquisition while profiling the region of interest for at least an entire tidal cycle. At least 15mins of data at each elevation is a bare minimum in order to obtain converged estimates of mean flow and turbulence parameters, ii) The sample area must be large enough to cover the wavenumber range that would be "contaminated" by surface waves if one uses a time series of data and relies on Taylor's Hypothesis. Based on our previous results 21 , a wavenumber of 4.2rad/m, i.e. a wavelength of 1.5m, has been identified as a target scale. However, the enlarged sample area should not cause a reduced resolution in the high wavenumber range, iii) The system should enable us to record data in two different planes simultaneously in order to obtain statistics on all the three velocity components. When one of the light sheets is vertical and aligned with the flow direction, the second sheet can be horizontal. Such a capability would provide data on eight of the nine components of the velocity deformation tensor along the intersection line, which would greatly reduce the assumptions associated with estimates of dissipation. Stereo-PIV is also an option, which would provide three dimensional velocity distributions within the sample area, but not the cross-stream gradients, iv) Based on our past experience, a wide dynamic range for the cameras is an advantage due to variations in background lighting, water transmissivity, and particle characteristics. All of these features have been incorporated into the new PIV system. Schematic descriptions of the submersible system as it was deployed are presented in Fig. 3 and Fig. 4. The PIV system is the same in both figures, however Fig. 3 shows the old, limited range, sampling platform while Fig. 4 shows the new, extended range platform. Further details of the two platforms are discussed below. The PIV system enables us to simultaneously

PIV Measurements

in the Bottom Boundary

.Optical Fiber

SIDE

Compass

Layer of the Coastal Ocean

57

CTD Lifting Point

Laser Probe x 2

Turntable Motor Pressure Transducer

Light Sheet

Power Supplies

2 Digital Cameras: 4 Frames/s, 2K x 2K, Hardware Image Shift

PLAN

A

Camera FLOW

Direction Vane

Pressure Transducer

Power Supplies

Fig. 3. Schematic of the submersible components of the ocean PIV system mounted on the old, limited range platform (from Ref 2 2 ) . The platform is shown fully extended. The platform has a retracted height of about 1.75m and weighs about 700Kg.

operate two 2048 x 2048pixels2, 4frames/s, 12bit digital cameras (manufactured by SMD, now Dalsa) with hardware-based image shifters to overcome directional ambiguity. When these cameras record double-exposure images, the two overlapping frames are shifted on the CCD array relative to each

58

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

other by a prescribed number of pixels. The magnitude (in pixels) of the shift can be adjusted, using the computer controlling the camera, to be larger than all possible displacements in that direction. Consequently, all the particle traces appear to be moving in the same direction and the known fixed displacement is subtracted after processing the images. The CCD of each camera has 14x 14^m pixels, making it quite sensitive (equivalent to 800 ASA film). The increased resolution and sensitivity of these cameras, in comparison to the original camera, make it possible to increase the sample size up to 0.5x0.5m 2 without compromising the smallest scales (0.6-0.8cm). When these cameras were purchased and deployed, the 2Kx2K cross-correlation cameras (such as the Megaplus ES.4 available today) were not available. However, the dynamic range and sensitivity of the present image-shifted auto-correlation cameras are still considerable advantages in the ocean environment. A section of a sample image from the new PIV system captured at the LEO-15 site (see section 3) is shown in Fig. 5. The image has been enhanced 20 and the particle trace pairs are clearly visible, vertically offset from one-another by the hardware image-shifter. Each camera feeds data over 70m of cable to a high-speed disk array with a capacity of 240GB (six IDE disks, currently 40GB each, which can be replaced by larger disks) that can acquire data at a sustained maximum acquisition speed of 60MB/s (only 32MB/s are needed for the present system). When the disk arrays are full, the data is compressed without loss of resolution and then backed-up on another hard disk. Slower back-up tapes are also available. The storage capacity of this new system enables us to continuously acquire 16bit images at 0.5Hz for more than 15hrs (a maximum of 13hrs was actually recorded), i.e. we can cover an entire tidal cycle. Data can be collected at up to 4frames/s for proportionately shorter periods. As the system was originally designed, we opted to keep the laser on the ship and transmit the light to the bottom via optical fibers (see above). The primary motivations were: i) A desire to operate for many hours continuously, eliminating batteries as an option; ii) Reluctance to deploy large power supplies fed by high-energy lines due to logistics and safety concerns; and iii) Due to the image size, the distance between the camera and the light sheet and concerns about water transmission, about lOOmJ/pulse should be available for each laser sheet. As mentioned above, our laser can generate a maximum of 350mJ/pulse at 594nm, which is sufficient energy for pumping light through more than one fiber. Consequently, as Fig. 6 shows, we split the laser beams and focus them onto two 400//m diameter optical fibers that transmit the light to the submerged probes. The maximum output that we

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

co CD

•3 O

59

co

•a «J

bo a n

tf

CD no

d xi

•M

o

CM

•s

d 0

to CM

a

§

X»

43 i*

3 a CD

d XS CD

Tl

-M

~TlCD

CD

8 M (!) u m

a

O

CM „C}

o

in

CD

CU

o

JJ

0

XI

o

* M

2

bi)

cd

Ti CD

o

M4

^

J2 &o • CD °

fe H 8

• ^

g.3

. M

cS

B t< CD >^

X» u d a> CD

Xi

CD

o

S -M o a

.M

,;_» au Ca)O < a d o 0) a 43

CO

*d

>> H

sn a ft CD +3 CO

0

"S F) w

a)

o

fl

+3

P£3

03

new

60

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

Fig. 5. Sample enhanced PIV image from the new system (magnification is 1:1) Both exposures are recorded in the same frame, with the second exposure vertically offset from the first by, in this case, 17 pixels.

Fig. 6. Schematic of the shipboard laser system configured to provide two simultaneous light-sheets. A: 45° mirror, B: prismatic mirror, C: beam-splitter, D: coupling lens, E: fiber-holder and precision 3-axis translation stage, F: fiber to laser probe 1, G: fiber to laser probe 2.

can transmit without damaging the fibers (due to slight misalignments) is 120mJ/pulse. This setup enables us to orient the two cameras and laser sheets independently. They can be positioned to record images in the same plane, vertically or horizontally, near each other or apart, but can also be aligned in different planes. Fig. 4 shows the setup of cameras and sample areas used during most of the recent deployment tests. The submerged probes contain lenses for re-focusing and re-collimating the beam exiting from the fiber, and a cylindrical lens for expanding the

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

61

beam to a light sheet. The minimum thickness of the light sheet is limited by the diameter of the optical fiber, but the sheet also has to be thick enough to allow for out-of-plane displacement of the particles between the two exposures since the flow is three dimensional. Initially the thickness of the laser sheets varied between 3-4mm over the sample areas (see below). This width is much less than the vector spacing (here about 8mm), but still allows for variations in the mean flow direction of ±20 deg (with a typical particle displacement of 20 pixels). During the first experiments with the new PIV system, which were carried out using the old sampling platform (Fig. 3), the distance between the camera and the light sheet was 45cm. Since the field of view was 51X 51cm we had to use wide-angle lenses (Nikon ASF-28mm) and a circular dome as a front window to the waterproof camera housings. To calibrate the system and correct for the resulting image distortions we recorded images of a reference grid and created calibration maps that varied over the image and between cameras (the latter only slightly). In-house developed software 24,25 ' 26 with some modifications to the image enhancement procedures were used for computing the velocity. Constrained by the particle concentration, we used 64x64 pixel interrogation windows and 50% overlap between windows. Thus, each resulting vector map consists of a maximum of 63x63 velocity vectors with a vector spacing of 0.8cm. Increasing the resolution to 32x32 pixel windows, which would generate 127x127 velocity vectors/image (i.e. vector spacing of 0.4cm) would be possible for only some of the images depending on seeding, and may be performed only selectively for some of the data. After computing the velocity, the image shift was subtracted and the magnitude and location of each vector was corrected for image distortion. The resulting slightly irregular array was then interpolated to generate a regular, equally spaced array. The data quality varied depending on the particle concentration, size and spatial distribution of the particles, and the orientation of the laser sheet relative to the instantaneous flow. Typically we obtained 60-80% of the vectors that satisfied the accuracy criteria of the data analysis software 25 ' 20 (based on correlation magnitude, 0.2, and difference from neighboring vectors, 4pixels). Instantaneous distributions that contained less than 60% correct vectors were not used during subsequent analyses. Unfortunately, we found that the out-of-plane component of the velocity introduces significant bias into our measurements because of the combination of the large field of view, the small distance between the cameras and the light-sheets, and the thickness of the sheets (3-4mm). For example, let us consider a case where the out-of-plane motion is sufficient to displace

62

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

particles between exposures away from the camera by half the thickness of the light sheet (2mm - any greater displacement would typically result in loss of one of the particle-pairs). This motion results in an apparent inward, radial displacement of the particles on the image plane, which increases linearly in magnitude from zero at the center of the image up to a maximum of 4.5pixels at the edges (Fig. 7). This contamination is minimized for each vector map by first subtracting the map-mean velocity from each vector. Next, the radial component of velocity and the radial distance of each vector from the center of the map are calculated. From these, the mean radial gradient of the radial velocity is estimated using a least-squares method, setting the radial velocity to zero at the center of the map. This mean gradient is then used to construct a map of radial contamination which is subtracted from the original vector map. This procedure removes the instantaneous mean out-of-plane velocity from the measurements, but cannot account for any non-uniformity of the out-of-plane motion due to shear or turbulence.

Fig. 7. Schematic of the effect of the out-of-plane velocity component. Out-of-plane displacement of particles at A and B give apparent displacements on the image plane at A' and B' respectively.

Subsequently, the PIV system components were reconfigured to minimize the effects of the out-of-plane motion: the light sheets were made thinner (2.5mm), the separation between the cameras and the sample areas was increased to about lm and the sample areas were reduced in size

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

63

to about 35 x 35cm (see Fig. 4). Together, these alterations reduced the maximum apparent radial displacement at the edge of the images to about 0.5pixels (a reduction of 90%). To achieve the reduction in sample area together with the increased camera to light-sheet separation, we used longer focal length lenses (Nikon ASF-85mm) behind flat ports, which had the additional benefit of reducing the optical distortion within the images from about 10% (with the wide-angle lenses and dome ports) to less than 2%. After the re-configuration of the PIV system, the typical sample area coverage of vectors that satisfied the accuracy criteria of the data analysis software increased to 75-95%. Gaps within these distributions can now be more reliably filled by taking the average of surrounding good vectors. A sample instantaneous velocity distribution is shown in Fig. 8a. Here the instantaneous sample area mean horizontal and vertical velocities have been subtracted from each vector to reveal the turbulent fluctuations. The corresponding vorticity field, uy, calculated directly from the velocity field using center-differencing, is shown in Fig. 8b. In the laboratory, PIV measurements are usually carried out under controlled lighting conditions to improve the signal-to-noise ratio. Since it is impossible to control the background lighting in the natural environment, we experimented with using narrow-bandwidth interference filters to reduce its effects. However, we found that the interference filters introduced additional optical distortion and internal reflections within the camera housings, and so we finally opted simply to sample only during the hours of darkness. We have found that we can collect high quality PIV data from sunset to sunrise, particularly in deeper waters. During the first deployments of the new PIV system, the cameras and probes were mounted on the original sampling platform (Fig. 3), which allowed acquisition of data at various heights above the sea floor, up to a maximum elevation of 1.7m. The turntable was used to align the sample areas with the mean flow direction, monitored with a video camera directed towards an on-board vane that oriented itself to the flow. During the field tests we used a second submerged video camera to record images of the bottom topography. In the last two deployments, the original sampling platform was replaced with the new platform to allow measurements over an extended range, from the seabed up to 10m into the water column. The new platform (see Fig. 4) comprised a telescopic hydraulic cylinder mounted vertically on a weighted tripod base. The PIV system and auxiliary components were in turn attached to the top of the cylinder via rigid framework and a hydraulically operated turntable. Two hydraulic hoses linked the platform to a ship-board hydraulic power supply which used environmen-

64

W. A. M. Nimmo Smith,

(LUO) Z

T. R. Oshorn and J. Katz

• !

•0 '

.?]

; • . . ,

(i)

bfl 0

fl f,

u

:-•)

a>

•:i.

o r! CO

V " = T

i"

•

fi

•'.

m

0 +s T) CD r

••H

+3 CS

+3

=O : ' ft /: 8

-M>

siti

." :.:

1 00

bb

s .3

ovi

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

65

tally friendly, food-grade hydraulic oil. A remotely controlled manifold at the top of the cylinder was used to switch operation between the hydraulic turntable motor and the cylinder, reducing the required number of lines linking the platform to the surface. The submersible system also contains a Sea-Bird Electronics, SeaCat 19-03 CTD, optical transmission and dissolved oxygen content sensors, a ParoScientific Digiquartz, Model 6100A, precision pressure transducer (for measuring surface waves), an Applied Geomechanics, Model 900a biaxial clinometer, and a KVH C100 digital compass.

3. D e p l o y m e n t s The initial measurements using the original, single camera PIV system were performed in the New York Bight near the Mud Dump Site, 7 miles east of Sandy Hook, New Jersey, in June 199821. Data were collected at six different elevations ranging from 10cm to about 1.4m above the sea floor. Limited by storage capacity at that time, the data for each elevation consisted of 130s of image pairs recorded at 1Hz. The experiments using the two-camera PIV system took place during May, 2000 and September, 2001 at two sites on the east coast of the USA, one in the vicinity of the Longterm Ecosystem Observatory (LEO15), about 5km off the New Jersey coast, the other inside the wall of the harbor of refuge near the mouth of Delaware Bay 22 . The system was deployed from the RV Cape Henlopen (one of the UNOLS ships managed by the University of Delaware). As in the initial deployments, the ship was held in a fixed position using a three point mooring system. A ship-board ADCP provided us with data on the mean velocity distribution in the water column for the entire duration of the tests. We also acquired water column profiles of temperature and salinity and collected samples of particles at different elevations and from the seabed itself. During May, 2000 we recorded three data sets, two at the first site and one at the second. The exact position of the first site was 39°27'00"N and 74°14'15" W, i.e. about 0.5 nautical miles to the south-east of Node B (39°27'25"N, 74°14'45"W), and about 1.25 nautical miles to the south-east of Node A (39°27'41"N, 74°15'43"W) of LEO-15. The seabed consists of coarse sand ripples of approximately 50cm wavelength and 10cm height. The currents in the region are generally moderate, however the site is exposed to the oceanic swell. On the night of May 15-16 we recorded data at 0.5Hz for 9hrs continuously. On the night of May 16-17 we recorded data at 0.5Hz for 13hrs continuously. Thus, we obtained PIV images that will

66

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

enable us to study variations in flow structure and turbulence during an entire tidal cycle. During these tests the cameras were configured as illustrated in Fig. 3. Both were focused on the same vertical plane, had the same field of view (51cm), and their centers were located lm apart. The data was acquired at three different elevations, 9.5-60.5cm, 64-115cm and 118.5-169.5cm above the bottom. At each elevation we aligned the system to the mean flow direction (using the video images of the vane) and then acquired data, typically for thirty minutes, i.e. 2x900, 2Kx2K, 12bit PIV images. The elevation was then changed and the next image set was acquired. Since the flow direction changed with time and with elevation we had to re-align the platform for each data set. In a few cases the flow direction changed significantly during the run and we had to stop it early. Samples of the raw image data were examined at the end of each data series to check the data quality and if necessary the laser heads were re-aligned. On the night of May 19-20, 2000 we acquired a data set behind the wall of the harbor of refuge of Delaware Bay. Here, the flow is characterized by strong tidal currents (with surface velocities in excess of 1.5ms _1 ) and little wave motion. The seabed consists of fairly smooth sandy mud, with none of the sediment ripples observed at the LEO-15 site. This time the two sample planes were perpendicular to each other and the data was acquired at 3.3Hz. We recorded five data sets at different times and elevations, each consisting of 1000 image pairs. The latest version of the PIV system was deployed in May 2001 for testing, and in September 2001 for data collection. For the September 1-10 experiments, we again deployed the system near LEO-15, 5 nautical miles south-east of our previous sampling location at 39°23'37"N, 74°9'32"W, in 21m deep water. Here the water column was strongly stratified, with a sharp thermocline situated about 7m above the seabed. Data were collected for 20min periods at a time, sampling at either 2 or 3.3Hz. A total of 700GB of image data were collected at various elevations, up to 8m above the seabed and under differing conditions of mean flow and wave induced motion. Combined, we recorded over 1.2TB of images. 4. Analysis Techniques and Sample Results 4.1. Flow

Structure

Time series of the sample-area mean horizontal velocity ([/), obtained by averaging the velocity over the entire sample area of one camera, which are typical of the conditions encountered in Delaware Bay and near LEO-15 are shown in Fig. 9. At the Delaware Bay site (Fig. 9a), the mean flow is

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

67

strong, with only relatively weak orbital wave motion. In this 5min, 3Hz time series, the mean velocity is 35cms _ 1 and the orbital wave motion has a maximum amplitude of about 7cms _ 1 and a period of about 9s. Smaller amplitude (about 4cms _1 ) and longer period (about 150s) variations in the mean flow, or "beating", are also visible in the horizontal velocity record. Figures 9b and c show characteristic 20min time series, from the LEO-15 site. In Fig. 9b, the mean velocity is about 9cms _ 1 , with the orbital motion caused by surface waves, traveling in groups, superimposed. It is evident that the amplitude of the streamwise wave-induced flow, generated by the predominantly 10s period surface gravity waves, is comparable to the mean velocity. Thus, at the extreme, the flow direction is actually reversed. Fig. 9c shows a period with very little flow, consisting mostly of oscillatory wave motion.

300

1200

600

1200

Time (s) Fig. 9. Time series of the sample area mean horizontal velocity for each of the sampling periods, a Delaware Bay, Z=35cm, b LEO-15, Z=50.5cm, c LEO-15, Z=50.5cm.

The corresponding vertical distribution of mean (ensemble averaged over all measurements and over all the data at the same elevation) horizontal velocity for the period shown in Fig. 9a, and that of the adjacent 5min periods, is shown in Fig. 10. It can be seen that they have a nearly loga-

68

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

rithmic profile. We also observed a log layer in the New York Bight 27 under similar conditions of relatively high mean flow and low orbital wave motion. The vertical distributions of mean horizontal velocity measured at LEO-15, under conditions similar to the period shown in Fig. 9b, are presented in Fig. 11. Here, the vector maps forming each mean profile are conditionally sampled based on the phase of the wave-induced horizontal velocity. At no point is a logarithmic profile in evidence, consistent with the conclusions but not with the profile shape obtained in laboratory tests of an oscillatory boundary layer 28 . We do not have an explanation for this phenomenon. It may be related to the low turbulent Reynolds number of the LEO-15 data 23 , where the Taylor microscale Reynolds number is less than 100.

Fig. 10. Vertical distribution of mean horizontal velocity measured at the Delaware Bay site (from Ref 2 2 ) . The discontinuity in the profile is a result of performing measurements at different times and the "short" period of measurements at each of the three elevations.

To investigate the small-scale structure of the flow, Fig. 12 shows characteristic distributions of velocity and vorticity. During periods of no mean current and only oscillatory wave motion (Fig. 9c), the flow is very smooth, or "quiescent", with only very small turbulent fluctuations as shown in Fig. 12a. The quiescent periods also exist for some of the time when there is a moderate mean current (Fig. 9b). However, in between these quiescent periods, the character of the flow structure changes markedly. The more common pattern consists of powerful vortical structures with high core vorticity, of about 2cm in diameter, being advected through the sample area,

PIV Measurements

in the Bottom Boundary 1

20

r

U(z) 5/16/2000

•18 -16

•

....

'ID

E

Layer of the Coastal Ocean •

.

•

•

i

i

MaxU MinU Accel. Decel. j /

?f

j f

^o, -14 3

i

69

.-•/"

**^

-. >~

V'

/

i? •12

o

^ ^ > - ^

O

<0

. / /

,,,.-;**p-:> *

>^ ^ o

-10 .2

a E -8

^ • ^

.+

w £•.

.<-'•** *....<••'•• -

•

-

,

,

'

i

*/ *

• * - "

qj

5 -4

*•*

.'.*'-*

c -6

i

,^-

-^-

—

IV)

o

2

*

/

V*

' Height above bed, z c m ( 00) O^ IO - 4^- Dl- COO -M CT C O O O O O O O ^ o

Fig. 11. Conditionally sampled vertical distributions of mean horizontal velocity at different phases of the surface-wave induced motion (from Ref 2 2 ) . The data shown is from three consecutive 30min periods, one at each of the three elevations above the bed. LEO-15 data.

singly or in groups (see Fig. 12b), but with no obvious large-scale structure. These "transitional" regions can be short-lived or may persist for several seconds, before they either decay, or are carried from the sample areas. More intermittently, even larger vortical structures are carried through the sample areas (Figs. 12c and d). These events comprise of isolated regions of usually positive vorticity with diameters of up to 10cm. Again, these large vortices may appear singly, surrounded by smaller vortices, or in organized groups forming larger coherent structures. Strong shear layers exist between adjacent vortices (e.g. Fig. 12c). Groups of vortices, sometimes paired with vortices of opposite sign, may lie along regions of strong shear that exist at even bigger scales (e.g. Fig. 12d). When the mean current is very strong (Fig. 9a), the flow consists almost entirely of these large vortical structures. According to Adrian et al.2g, these structures are sections through hairpin vortices. We also find that these structures have a marked impact on the Reynolds stresses in the flow, and are strongly linked to the production of turbulent kinetic energy 23 . 4.2. Spatial

Spectra

Sample spatial spectra from the New York Bight deployment are presented in Fig. 13. The spectra are calculated using the procedures described in Doron et al.21. The horizontal spectra, E\i(ki) and E^{ki), are shown in

70

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

15

-10

-5

0 x(cm)

5

10

15

Fig. 12. Snapshots of the instantaneous flow structure, with velocity fluctuation vectors superimposed on the vorticity field (from Ref. ). a U=1.2 cm/s, W = 0 . 7 cm/s, b U=6.9 cm/s, W=-0.2 cm/s, c U=11.3 cm/s, W=-0.2 cm/s, d U=6.0 cm/s, W=0.6 cm/s.

the left-hand pane, while the vertical spectra, En(kz) and £'33(^3), are shown to the right. The error bars indicate the 95% confidence level range. The results are compared to the Nasmyth universal spectrum for isotropic turbulence 30 . The dissipation rates required as an input to the universal curves are chosen to match £11 (fci) at low wavenumbers. Several trends are clearly evident from these results. Firstly, in all cases, the data cover part of the inertial range of turbulence (slope ~ —5/3) but also extend well into

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

71

the dissipation range. Secondly, the turbulence is anisotropic over the entire range covered by these spectra. At low wavenumbers (k\ < 150rad s _ 1 ) E^siki), Eii(ks) and £33(^3) are clearly larger than the universal isotropic turbulence values. At high wavenumbers (fci > 150rad s _ 1 ), another feature is in evidence - a well defined "bump" or "hump" in the longitudinal spectra (.Eii(fci) and -£33(^3)). We attribute the presence of this "bump" to the "bottleneck effect"31 which is also observed experimentally in high Reynolds number boundary layer flow32. Qualitatively, this "bottleneck" is a result of the viscous suppression of high wavenumber (small) eddies, which makes the transfer of energy from larger scales less efficient, causing a "pile-up" of energy in eddies with scales at the transition between the inertial and viscous ranges. Spectral analysis of the more recent data (not shown) yields similar results 23 : the flow is always highly anisotropic and large "bumps" exist between the inertial and dissipation ranges. We find that the "bumps" are

Fig. 13. Measured true spatial energy spectra compared to the Nasmyth universal spectrum, n and 33 calculated using e = 8 X 1 0 ~ 6 m 2 s ~ 3 to match Bn(fci) in the inertial range (from Ref 2 1 ) . The experimental results are based on averages of three rows (or columns) from 130 instantaneous realizations.

72

W. A. M. Nimmo Smith,

T. R. Osborn and J. Katz

more pronounced in the streamwise spectra, and when the flow is weak and does not contain active eddy structures. The conclusion from the recent observations is that there is no clear relationship between the size of the "bump" and the component of the velocity field (i.e. the "bump" occurs in both the bottom-parallel and bottom-normal velocity components). Reasons for this phenomenon require further investigation. 4.3. Dissipation

Rate

Estimates

Doronei al.21 present several methods for estimating the dissipation rate from ocean PIV data. Three of these methods, which rely solely on the instantaneous spatial distribution of PIV data and do not resort to using Taylor's Hypothesis, are detailed below. 4.3.1. "Direct" estimate of the dissipation rate For a Newtonian fluid, the dissipation rate is defined as OXj

\OXj

OXi

where v is the kinematic viscosity and double indices denote summation over all three directions. Using our PIV data, we can measure five of the terms in this equation directly: du dx

2

/o.\2

/ Q \ 2

dw\'

/ c ...\ 2

(8w\

du

[dx]

dudw\ dz dx J

Using continuity we obtain 'dv\ ^dyj

( \

du dx

f^u\ \dxj

dw\ dz J

dudw\ dx dz J '

dw

Therefore, for isotropic turbulence, we can directly measure 8/15 of the total dissipation. The remaining terms are estimated assuming that all lateral fluctuations have similar average magnitudes: du\ dy)

/9u;\ \dy)

/dv\ \dxj

du\ dz J dudv dy dx

dw dv dy dz

(dv \dz

(dw \dx dudw dz dx

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

73

Introducing all these terms, we obtain a "direct" estimate of the dissipation rate, ejj: eD = 3;/

dx~)

+

+

\dz)

'dudw\ dz dx J

\!h)

+

\~dx~)

2 fdudw 3 \ dx dz

Here, the only assumption is that the cross-stream gradients have the same average magnitude as the measured in-plane cross-gradients. This assumption is supported by the similar magnitudes of the measured crossgradients, (du/dz)2 and (dw/dx)2. 4.3.2. Line fit in the inertial range Using an assumption of isotropic homogeneous turbulence, at least in the inertial range (which, as we have seen above is not accurate), the dissipation rate can be estimated from33

denoting this estimated dissipation as e^p. A least-square fit is used in the range that appears to have the correct slope. Since the spectra of the vertical velocity appear to have a more extended "inertial" range, we use 3/4E33(/ci) to estimate the dissipation rate. 4.3.3. Energy flux across the equilibrium range Large eddy simulation (LES) of turbulent flows is based on filtered NavierStokes equations 34 ' 35 : dui dt

_ dui J dxj

d dxj

%3

d2ui dxjdxj

where the tilde denotes a spatial filtering operation at a scale A, and the subgrid-scale stress tensor is defined as

If the scale of the filter corresponds to wavelengths within the inertial range, the condition of equilibrium suggests that the energy flux to smaller (subgrid) scales should provide an estimate of the dissipation rate 3 6 . Thus £SG =

—TijSij,

74

W. A. M. Nimmo Smith,

T. R. Oshorn and J. Katz

where Sij = l/2(dui/dxj + diij/dxi) is the filtered strain-rate tensor, can be used for estimating the dissipation rate. Since planar PIV provides only part of the tensor terms, we approximate the others, similar to the approach used in the estimation of e^. Inserting the known and approximated terms, TL2S12 — T23S23 = T13S1S T22S22 = -Z{T\ISH

+T33S33),

yields the subgrid-scale energy flux estimate of the dissipation rate: £SG

= --^{^TuSii

+ 3T33533 +

12TI3S,I3).

This estimate of the dissipation rate is calculated from the filtered strain rates, and the stresses are typically dominated by the largest subgrid scales. As a result, this approach is much less sensitive to high wave-number noise in the data. 4.3.4. Sample dissipation results Fig. 14 shows the vertical distribution of dissipation rate in the New York Bight calculated using the three different methods. It can be seen that, in general, there is good agreement between the estimation methods, and that the mean dissipation rate (approximately 6 x l 0 - 6 m 2 s - 3 ) is consistent with other oceanic dissipation observations 17 ' 37 . Some variations do exist however. Since the vector spacing in the PIV data is larger than the Kolmogorov scale, estimates based on velocity gradients (ec and CSG) are lower than the actual values. The values of CD remain most consistent between elevations, and probably provide the "best" estimate of the actual dissipation rate since this method relies on the least number of assumptions. There is more variability in the values of CSG since these require a much larger number of data points to converge36. There is also much variability in the values of €LF which is probably the result of the highly anisotropic nature of the flow and the "bottleneck effect". Dissipation estimates from the LEO-15 data show very different trends 23 . Here, under conditions of very weak mean flow which results in a Reynolds number based on the Taylor microscale of less than 100, esc can be more than an order of magnitude smaller than eo4.4. Estimating

the Reynolds

Stress

A method for calculating the Reynolds stresses which overcomes the problems introduced by unsteady wave motion, and takes advantage of the

PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

75

140

^

G

r> ^

120 100

^

'•""

<3(+S

SO

^ ^ . - - i-" -^

° S "S 60

-

' VI

D

•

•

•

^

. . 4 . . . ^ /

•

/

•

40

20 •

5

4

6

'"M

8

10

12

dissipation rate x 106 (rrfVs3)

Fig. 14. Estimates of dissipation rate at six measurement elevations, obtained using different estimation methods (adapted from Ref 2 1 ) . Note that variations with height represent both spatial and temporal variations due to the 15min time delay between successive measurement elevations.

2-D velocity distribution provided by PIV, is given by Nimmo Smithei al.22 and follows below. This method builds on procedures introduced by Trowbridge 19 . Briefly, the velocity is decomposed to Ui — Ui + Ui + u^, where Ui is the time average, Ui is wave induced motion and u^ is the turbulence contribution. The covariance of the difference in velocity between two points, also known as the second-order velocity structure function, Dij{rt,Xi), is Dij{ri, Xi) = cov [Am, AUJ] = [ui(xi + n) - m(xi)) [uj{xi + n) - Uj{xi where Aut = m(xi + n) -ufai)Assuming horizontal homogeneity, m(xi + ri)uj(xi + ri) = and Ui(xi)uj(xi + r») = Ui(xi + ri)uj(xi), then cov [Aui, AUJ] = 2ui(xi)uj(xi)

m(xi)uj(xi

- 2ui(xi + ri)v,j(xi).

Decomposing m, and assuming that there is no correlation between iii and 0 (discussion follows), one obtains cov [Aui,

AUJ]

=2

i>lllj

M

2 Uifa + Ti)uj{xi) + ufai

+ ri)u'Axi)

76

W. A. M. Nimmo Smith,

T. R. Oshorn and J. Katz

In flows with an integral scale, I, much smaller than the wavelength of surface waves, A, as long as r
^-s~

-0.1

-0.5

•0.4 « p

•008

X,

cov(Au,Av)(r)

i

Z = 50cm U = 34cm s" 1

•

Z = 50cm U = 34cm s" 1

* i

-J 0.02

-cov(Au,Aw)(r)

'°3 1 -CO

•o.osb

•

/

•

•0.1

/

r(cm

/

5

10

15

20

25

r(cm 30

35

40

45

y

5

10

15

20

25

30

35

40

45

Fig. 15. cov [Au, AV] (ri) (left) and —cov [Au, Aw] ( n ) (right) using data obtained using horizontally and vertically oriented light-sheets respectively, at the Delaware Bay site (from Ref 2 2 ) .

Using the PIV data and assuming horizontal homogeneity, it is possible to calculate cov [Au, Aw] ( n ) , where ri is the streamwise separation, using data obtained in a vertical plane, and cov [Aw, Av] ( n ) using the horizontal sheet data. Assuming horizontal homogeneity, one can average data obtained at different x (but the same z) as well as at different times. The distributions of cov [AM, AV] (ri) and cov [Au, Aw] (r\), using the Delaware Bay data, the only set recorded with perpendicular planes, are presented in Figs. 15a and b, respectively. Initially, the values of cov [Au, AM;] increase with r\, but then they start to asymptotically converge to constant values as ri becomes comparable to the distance from the bottom (^50cm). The condition of r
PIV Measurements

in the Bottom Boundary

Layer of the Coastal Ocean

77

there are differences in length scale between the vertical and cross-stream horizontal directions. A more detailed discussion of the application of these procedures to PIV data is currently in preparation 38 .

5. Conclusions A submersible PIV system has been developed to perform of turbulence measurements in the bottom boundary layer of the coastal ocean. The system has been deployed at several sites under contrasting environmental conditions of mean flow and oscillatory wave induced motion. Examination of individual velocity and vorticity distributions reveal that, under high and moderate current conditions, the flow is dominated by large, coherent vortical structures. These structures are persistent in time and are advected by the mean flow. At low mean flows and during substantial fractions of the moderate conditions, there is only low level weak background turbulence. Thus, wave-induced motion by itself produces very little turbulence in the boundary layer. Vertical distributions of the mean horizontal velocity show the presence of a logarithmic boundary layer only under conditions of strong mean flow and relatively weak wave motion. Several different analysis techniques are detailed for characterizing the turbulence. Spatial energy spectra reveal that the turbulence is highly anisotropic at all wavenumber scales. Also, a "bump" exists in the spectra between the inertial and dissipation ranges, most likely a signature of the "bottleneck effect". Three methods for estimating the dissipation rates are examined, including "direct" estimates that rely on all the available data (e£>), estimates based on curve fitting to the energy spectrum (€LF) in a range with —5/3 slope, and estimates based on the energy flux across the equilibrium range using filtered data (E^G)- In strong flows with weak oscillatory wave motion, the different methods provide consistent values for the dissipation rate. However, when the mean flow and turbulence level are weak, large differences appear between the different estimation methods 23 . Finally, using the covariance of the difference between velocity components, it is possible to estimate the Reynolds stress without being affected by unsteady phenomena related to surface waves.

Acknowledgments This project has been funded in part by NSF and in part by ONR. We are grateful to: Y. Ronzhes and S. King for their technical expertise in development of the equipment, the captain and crew of the RV Cape Henlopen,

78

W. A. M. Nimmo Smith, T. R. Osborn and J. Katz

D. R. Baldwin, L. Luznik, W . Zhu, A. Pricova and T. Kramer, for their invaluable assistance during deployments. References 1. A. Lohrmann, B. Hackett and L. P. Roed, J. Atmos. Oceanic Technol, 7, 19 (1990). 2. Y. Lu and R. G. Lueck, J. Atmos. Oceanic Technol, 16, 1568 (1999). 3. R. Lhermitte and U. Lemmin, J. Atmos. Oceanic Technol., 11, 1295 (1994). 4. H. van Haren, N. Oakey and C. Garrett, J. Marine Res., 52, 909 (1994). 5. N. C. Kraus and R. Lohrmann, A amd Cabrera, J. Hyd. Engng., 120, 406 (1994). 6. R. George, in Workshop on Microstructure Sensors in the Ocean, Timberline Lodge, Mt Hood Oregon, pp. 116-120 (1996). 7. G. Voulgaris and J. H. Trowbridge, J. Atmos. Oceanic Technol, 15, 272 (1998). 8. G. C. Johnson, T. B. Sanford and M. O. Baringer, J. Phys. Oceanogr., 24, 2084 (1994). 9. A. J. Williams, J. S. Tochko, R. L. Koegler, T. F. Gross, W. D. Grand and C. V. R. Dunn, J. Atmos. Oceanic Technol, 4, 312 (1987). 10. T. F. Gross, A. J. Williams and E. A. Terray, Continental Shelf Res., 14, 1239 (1994). 11. A. J. Williams, F. T. Thwairtes and A. T. Morrison, in Workshop on Microstructure Sensors in the Ocean, Timberline Lodge, Mt Hood, Oregon, pp. 38-43 (1996). 12. J. H. Trowbridge and Y. C. Agrawal, J. Geophys. Res., 100, 20729 (1995). 13. J. H. Trowbridge, A. J. Williams and W. J. Shaw, in Workshop on Microstructure Sensors in the Ocean, Timberline Lodge, Mt Hood, Oregon, pp. 143-150 (1996). 14. T. B. Sanford, J. A. Carlson, J. H. Dunlap, M. D. Prater and R. C. Lien, J. Atmos. Oceanic Technol, 16, 1647 (1999). 15. T. B. Sanford and R. Lien, J. Geophys. Res., 104, 1245 (1999). 16. D. P. Winkel, M. C. Gregg and T. B. Sanford, J. Atmos. Oceanic Technol, 13, 1046 (1996). 17. Y. C. Agrawal and D. G. Aubrey, J. Geophys. Res., 97, 20249 (1992). 18. Y. C. Agrawal, in Workshop on Microstructure Sensors in the Ocean, Timberline Lodge, Mt Hood Oregon, pp. 15-23 (1996). 19. J. H. Trowbridge, J. Atmos. Oceanic Technol, 15, 290 (1998). 20. L. Bertuccioli, G. I. Roth, J. Katz and T. R. Osborn, J. Atmos. Oceanic Technol, 16, 1635 (1999). 21. P. Doron, L. Bertuccioli, J. Katz and T. R. Osborn, J. Phys. Oceanogr., 3 1 , 2108 (2001). 22. W. A. M. Nimmo Smith, P. Atsavapranee, J. Katz and T. R. Osborn, Exp. Fluids, 33, 962 (2002). 23. W. A. M. Nimmo Smith, J. Katz and T. R. Osborn, J. Phys. Oceanogr. (submitted). 24. G. I. Roth and J. Katz, Meas. Sci. Technol, 12, 238 (2001).

PIV Measurements in the Bottom Boundary Layer of the Coastal Ocean

79

25. G. I. Roth, D. T. Mascenik and J. Katz, Physics of Fluids, 1 1 , 3512 (1999). 26. G. Sridhar and J. Katz, Physics of Fluids, 7, 389 (1995). 27. L. Bertuccioli, A study of the coastal ocean bottom boundary layer using a submersible PIV system, Ph.D. thesis, The Johns Hopkins University (1999). 28. B. L. Jensen, B. M. Sumer and J. Fredsoe, J. Fluid Mech., 206, 265 (1989). 29. R. J. Adrian, C. D. Meinhart and C. D. Tomkins, J. Fluid Mech., 422, 1 (2000). 30. N. S. Oakly, J. Phys. Oceanogr., 12, 256 (1982). 31. G. Falkovich, Phys. Fluids, 6, 1411 (1994). 32. S. G. Saddoughi and S. V. Veeravalli, J. Fluid Mech., 268, 333 (1994). 33. H. Tennekes and J. L. Lumley, A First Course in Turbulence (The MIT Press, 1972). 34. M. Lesieur and O. Metais, Annu. Rev. Fluid Mech., 28, 45 (1996). 35. R. S. Rogallo and P. Moin, Annu. Rev. Fluid Mech., 16, 99 (1984). 36. S. Liu, C. Meneveau and J. Katz, J. Fluid Mech., 275, 83 (1994). 37. R. K. Dewey and W. R. Crawford, J. Phys. Oceanogr., 18, 1167 (1988). 38. T. R. Osborn, W. A. M. Nimmo Smith, L. Luznik and J. Katz, J. Phys. Oceanogr. (in prep.).

This page is intentionally left blank

CHAPTER 3 WATER WAVE INDUCED B O U N D A R Y LAYER FLOWS ABOVE A RIPPLE BED

Philip L.-F. Liu, Khaled A. Al-Banaa and Edwin A. Cowen School of Civil and Environmental Engineering, Cornell University Ithaca, NY 14853, U.S.A E-mail: [email protected]

Using Digital Particle Tracking Velocimetry (DPTV), we measured the oscillatory boundary-layer flow motions induced by surface gravity waves propagating over a fixed ripple bed. Spatial and temporal variations of the velocity field were recorded. Two sets of wave parameters, yielding two different Keulegan-Carpenter and Reynolds numbers, were used. The phase-dependant mean velocity and vorticity fields are presented in detail, illustrating the vortex generation and ejection processes. The bootstrap percentile technique is used to quantify the random error in the measurement. The wave-induced streaming profile is determined from the experimental data and its implications are discussed. The evolution of the Reynolds stress and turbulent kinetic energy are also presented. An estimate of the production and dissipation in the turbulent kinetic energy budget calculated within the turbulent boundary layer is illustrated.

1. I n t r o d u c t i o n Sand ripples are frequently found on the seafloor. T h e y affect the sediment t r a n s p o r t rate, cause energy dissipation and enhance mixing in the b o t t o m b o u n d a r y layer. T h e formation and stability of sand ripples are strongly influenced by boundary-layer flow motions, including t h e oscillatory and steady components. In recent years, many researchers have investigated analytically the relationships between b o u n d a r y layer flows and various ripple parameters, such as the ripple slope, e, Reynolds number, Re, and Keulegan-Carpenter number, a, (e.g., Longuet-Higgins 1 ; Vittori 2 ; H a r a and Mei 3 ; Blondeaux and Vittori 4 ; Vittori and Blondeaux 5 ). Although these studies have provided significant physical insights, they are limited t o two-dimensional laminar flows. To consider the effects of turbulence, 81

82

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

turbulence closure models are necessary, requiring numerical solution techniques (e.g., Sato 6 ; Tsujimoto 7 ). The task of developing an appropriate turbulence closure model for oscillatory boundary-layer flows is still challenging. To achieve this goal, accurate experimental data are essential. High quality experimental data are also needed to test assumptions employed in numerical models or analytical solutions. Most velocity measurements above ripple beds have been made with point measurement techniques, such as hot-film velocimetery (e.g., Sawamoto8) or laser-Doppler velocimetery (e.g., DuToit and Sleath 9 ; Sato 10 ; Ranasoma and Sleath 11 ; Horikawa and Ikeda 12 ; Hitching and Lewis 13 ). Because of the nature of point measurement techniques, the spatial resolution of reported data is generally low. Furthermore, each resolved point often requires a separate experiment be conducted leading to concerns about repeatability, which is a source of significant error. Consequently, these data can not provide accurate information on spatial gradient based quantities such as the vorticity field and components of the turbulent kinetic energy budget (TKEB). Recently, particle image velocimetry (PIV), a whole-field measurement technique, has been successfully employed in measuring the vorticity field in the vicinity of an object subject to either a steady current or an oscillatory flow (e.g., Lin and Rockwell 14 ' 15 ; Earnshaw and Greated 16 ). Chang and Liu 17 ' 18 have also reported PIV measurements of velocity and vorticity fields under breaking waves. The objective of this paper is to report the results of an experimental study of the flow field above a fixed ripple bed using digital particle tracking velocimetry (DPTV) (Cowen and Monismith 19 ). This technique uses cross-correlation based PIV as an estimate of the velocity field, allowing individual particles to be tracked accurately at traditional PIV seeding densities. The DPTV technique is able to obtain an order of magnitude more independent velocity vectors than traditional correlation based PIV. This allows us to investigate the generation and transport of vorticity and turbulence under various wave conditions. This chapter is structured as follows. The experimental set-up and procedures are described first in the next section, where the DPTV system, tank dimensions, and the wave parameters are presented. The errors associated with the measurements and the method used to quantify the random errors are then discussed. Experimental results, including the phaseaveraged velocity, vorticity, turbulence intensity, Reynolds stress and the turbulence production and dissipation fields are then presented and discussed.

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

83

2. Experimental Set-Up and Procedures The experiments were conducted in the wave tank in the DePrees Hydraulics Laboratory at Cornell University. The wave tank is 32 m long, 0.6 m wide and 0.9 m deep. It is constructed of plate glass sidewalls with a painted steel bottom. The middle test section has been fitted with a two-dimensional rippled bed made of painted plywood with the mean position between the ripple crest and trough aligned vertically with the steel bottom. The ripple crests are perpendicular to the centerline of the wave tank. The ripple bed consists of 60 ripples, each ripple is 1.2 cm in height and 3 cm in length. The removable section of the wave tank is 1.82 m long and 0.60 m wide. The configuration of the ripple at the measurement area is shown in Fig. 1. The wave tank is equipped with a piston-type wave generator, which is driven by a hydraulic servo-system and is capable of generating both regular and irregular waves. The downstream section of the tank was fitted with a 1:10 sloping beach and horsehair (a packing material) to reduce the reflected wave energy. The reflection coefficient was determined in terms of wave amplitude by using the method described in Hughes 20 and it was less than 8% in the experiments presented here. Fig. 2 shows a schematic sketch of the experimental set-up. In table 1, the wave parameters are listed and associated dimensionless parameters for this study are defined.

Table 1. Quantity,

Physical variables and dimensionless parameters.

Symbol (Unit)

Wave period, T (sec) Wave length, A (m) Water depth, h (m) Wave height, H (m) Horizontal particle displacement, A (m) Ripple wavelength, L (m) Ripple amplitude, n (m) Maximum velocity above the boundary layer, U0 (m/s) Boundary layer thickness, S = [TV/TT]1'2 (mm) Dimensionless Reynolds no., Re =

Case A

Case B

2.10 2.85 0.200 0.0244 0.030 0.030 0.006 0.090 0.82

2.19 3.12 0.220 0.0440 0.045 0.030 0.006 0.130 0.84

2700 0.20 0.010

5800 0.20 0.010

1.0

1.5

parameters [2-KA2/TV]

Ripple slope, £ = [rj/L] Wavelength ratio, /3 = [L/A] Keulegan-Carpenter no., a = [A/L]

84

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

Fig. 1. A typical image (1024 x 1024 pixels). The ripple shape is obtained from the images to sub-pixel accuracy.

The primary difference between the two experiments is the KeuleganCarpenter (K-C) number, a. In Case A the K-C number is unity and the maximum horizontal fluid parcel displacement outside of the boundary layer Is the same as the ripple wavelength. In Case B the K-C number Is Increased to 1.5 and therefore a fluid parcel travels 1.5 times the ripple wavelength over one half of the wave period. Moreover, the Reynolds number, which Is defined in table 1, In Case B is more than twice that of Case A. The boundary-layer flow above the ripple bed was measured using DPTV (Cowen and Monismith 19 ). Images were collected with a Silicon Mountain Designs (SMD) 1M60-10 digital CCD camera (1024 x 1024 pixel, 12-bit, 30 Hz). The camera was fitted with a 35mm extension tube and a Nlkkor 105mm f/2.8 lens. A typical Image (1024 x 1024 pixels) Is shown in Fig. 1. Also shown In the figure is the ripple shape, which is extracted to sub-pixel accuracy. The PTV illumination source was a Spectra-Physics PIV-200 twin-pulsed Nd:YAG system (140 mJ/pulse, 10 Hz on each laser head). A cylindrical lens (f = -50 mm) and a spherical lens (f — 1.0 m) were used to form a light sheet. This arrangement- yielded an approximately 1 mm thick light sheet centered on a ripple trough (between the 30th and the 31 s t ripple) and located 13 cm off the tank centerline. The camera Image

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

85

V Nd-YAQ Laser

Z

^^Mirrar

Wave Paddle

9*.1

CCD C a r n e r a _

F0V

1

t ^

^

Ul QB'O

1\ \

/1 -

Ripple bed ^ ^ M

*

^

r

E to d

X

Horsehair

\

4

-»« ^ < Fig. 2. Sketch of the experimental set up. The upper panel is the side-view and the lower panel is the top-view.

area was centered in this light sheet and the optical arrangement yielded a viewing area of 40.4 mm x 40.4 mm. The camera/laser/wave-maker system was controlled with National Instruments Labview. A single burst of 21 image pairs for Case A and 22 image pairs for Case B were collected at 10 Hz directly to host computer RAM. The time between illuminating the images in each pair (At) was 4 ms. This process was repeated 333 times. The DPTV algorithm tracked 7.7 million valid velocity vectors across all wave phases. The vectors at approximately 1027 active measurement locations were ensemble averaged at individual phases across the 333 waves into non-overlapping 1.26 mm x 0.98 mm bins. An average of 358 vectors were included in each bin. Near the boundaries the vector count in the bins drops, but remains greater than 200 vectors per bin in essentially all bins except those adjacent to the boundaries. The wave-maker was turned on for two hours before measurements were initiated. At which point the flow field was assumed to have reached a quasisteady state. The minimum spin-up time (two hours) was determined by monitoring the phase shift between the wave-maker driving signal and the

86

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

water surface elevation measured at the test location with a resistance type wave gage. After two hours the time history of the free surface elevation became periodic and the phase shift became constant. 2.1. Uncertainty

Analysis

The errors that contaminate experimental measurements can be categorized as bias or random errors. Bias errors are inherent in the system and are deterministic, such as errors associated with calibration. Bias errors do not contribute to the variance of the data but add a fixed mean to the data. In general, the DPTV results minimize bias errors as described in Cowen and Monismith 19 . Random errors have many sources such as thermal, electronic, light and CCD dark current noise, resulting in an overestimate of the variance of measured statistics. The random errors can be calculated from the measurements by using the bootstrap percentile technique as described in Efron and Tibshirani 21 . The Root-Sum-Square (RSS) technique is used to combine individual uncertainties to estimate the total error as described in Kline and McClintock 22 . In the bootstrap percentile technique, the confidence interval is estimated from the data set by resampling with replacement N times to calculate the probability distribution function for any statistic of interest. The 95% confidence interval is constructed from the sorted bootstrap generated statistics as the 2.5% lower bound and 97.5% upper bound statistic. It is suggested by Efron and Tibshirani 21 that N should be at least 1000 to have a good estimate of the confidence interval. The bootstrap technique is used to estimate the 95% confidence interval for the mean velocity, turbulent intensities and Reynolds stresses, which are presented in the results section. The uncertainty in the mean vorticity is determined from the uncertainties in the mean velocities using the RSS technique and a Taylor series expansion of the gradient. The uncertainty in the mean vorticity is a function of bin size after estimating the gradient by Taylor series expansion in the RSS technique. 3. Experimental Results and Discussions In this section, the experimental data for the phase-averaged velocity, vorticity, turbulence intensity, Reynolds stresses, and production-dissipation relationship will be presented. In Fig. 3 the time histories of the free surface displacements above the measurement area for both cases are shown. The circles and the associated letters represent the phases at which experimental data are analyzed and presented.

Water Wave Induced Boundary Layer Flows Above a Ripple Bed

w

87

o

-0.5

vji

o

Fig. 3. Time histories of free surface displacements at the measurement location for Case A and B, respectively. The "solid-circles" represent the wave phases where the data are analyzed and presented. The top panel is for Case A and the bottom panel is for Case B. £ is the free surface elevation normalized by the wave amplitude.

3.1. Phase-Averaged

Velocity

Since we assume that the flow field has reached a quasi-steady state, the velocity can be decomposed into its phase-averaged mean and turbulent

88

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

fluctuation. Thus, Ui=

(«<)+«<; i = l,2

(1)

where U\ = u and U2 = w denote the horizontal and vertical components of the velocity vector, respectively. The phase-averaged velocity is defined as i

N

(ui) = ^y£ui(t

+ jT),

(2)

j=i

in which T is the wave period and Ui(t + jT) is the i-component of the velocity obtained from j t h wave period of the measurements. In the present experiments the phase-averaged velocity was determined at 21 phases for CaseA (T = 2.10 s, sample rate = 10 Hz) and 22 phases for CaseB (T = 2.20 s, sample rate = 10 Hz). Each circle in Fig. 3 denotes a sampling point. In both cases 333 consecutive waves were sampled (i.e., N = 333 in equation 2). To plot the spatial variation of velocity vectors, the spatial domain is divided into non-overlapping bins of 1.26 mm x 0.98 mm. At a given phase all the velocity vectors inside a bin across 333 consecutive waves are averaged to give the phase-averaged velocity vector for the bin. We select the bin size such that at least one valid velocity vector occurs per bin, therefore 333 valid velocity vectors per bin are expected at each phase. However, near the solid boundary and along the edge of the field of view (FOV), data dropouts occur. Nevertheless, for most of the flow domain, there are more than 200 velocity vectors within each bin. An example of the DPTV determined valid number of velocity vectors in each bin across all wave phases for CaseA is shown in Fig. 4. A similar conclusion can be drawn for CaseB. For CaseA the spatial variations of the phase-averaged velocity field at different phases are shown in figures 5 - 10. In each figure the velocity vector field is plotted and profiles of the horizontal velocity component at four vertical cross sections are also shown. The 95% uncertainty intervals for the velocity data are displayed on the velocity profiles. Typical 95% uncertainty limits for the phase averaged velocity are (+0.095 cm/s, -0.092 cm/s) while the maximum 95% uncertainty limit is (+0.68 cm/s, -0.62 cm/s). At the phase ujt = 0.267T (Fig. 5), the flow field is accelerating as the wave crest is approaching the measurement area (see Fig. 3 for free surface locations). The velocity field above and away from the ripple bed appears to be uniform and unidirectional. However, close to the ripple bed the velocity profiles in the vicinity of the ripple crest are very different from those near the ripple

Water Wave Induced Boundary Layer Flows Above a Ripple Bed

4

^p?p5; '**'

-

•

:

: A

89

800

V-;||li|fi|^|^^^^^^^^^^^

3.5 " •••• ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ H

600

3

I

2.5 E'

&

2

N

fc^.

1.5

M \^.MlB •Hpr

^-

1

400

200

^ ^

0.5

n

• • = ^ = = r r r - f e r —

0

400

==3====^^™-^=

800

0

400

800

0

400

800

Average valid vectors per phase

Fig. 4. Average number of valid vectors per phase in each bin obtained from the D P T V technique for CaseA. The variations of the valid vectors at three vertical cross sections are also shown.

trough. Approaching the ripple crest from above (x = 0.6 cm in Fig. 5), the horizontal velocity increases first so as to satisfy continuity and then decreases and vanishes on the boundary. In the neighborhood of the ripple trough (x = 1.84 cm in Fig. 5), the velocity decreases monotonically. Consequently, fluid particles experience deceleration on the lee side (down slope) of the ripple crest and a reverse pressure gradient exists. These features suggest the possibility of flow separation, which indeed occurs and is shown in Fig. 6 (u)i = 0J5TT). It is interesting to point out that the flow separation appears before the wave.crest reaches the measurement area (see Fig. 3 for free surface locations), suggesting phase shifts between the velocity in the oscillatory boundary layer and the free surface elevation. Thus, the velocity reaches its maximum above the ripple bed before the wave crest arrives the measuring area and has a value greater than the maximum velocity at the outer edge of the boundary layer, i.e.,(w) maa ./t/o = 1.33. Because of the flow separation, a lee vortex is generated and is convected towards the ripple trough. As the wave crest passes the measurement area, the ambient velocity above the ripple bed decreases and the lee vortex continues its forward movement as shown in Fig. 7 (phase ut = 0.557r).

90

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

However, at wt = 0.83-K (Fig. 8), although the wave field is still moving forward (in the positive x-direction), the vortex has started to move backward. As the ambient velocity decelerates further, the vortex begins to rise (tut = 1.027T, Fig. 9). As the wave trough approaches the measurement area, the magnitude of the negative (backward) velocity near the ripple bed grows stronger, initiating the vortex ejection process. At wt = 1.217T flow separation has occurred on the weather side of the second ripple crest (Fig. 10). Thus, the processes of vortex generation, advection, and dissipation are repeated every one-half wave period. The ejected vortex is transported across the ripples during the next half period. This movement can be seen more clearly when the vorticity field is examined in section 3.2.

4 3.5 3 2.5

1

0.5 0 0.5 X = 0.60(cm)

1

1.5

2 X(cm)

X = 1.22(cm)

?3'

3

&2\

2

-20

2.5

3

X=1.84(cm)

3.5 X = 2.47(cm)

4I 3

J

0 20 - 2 0 0 20 - 2 0 0 20 - 2 0 0 20 (cm/s) (cm/s) (cm/s) (cm/s)

Fig. 5. The phase-averaged velocity field in CaseA at phase (a). Four vertical cross sections of the horizontal velocity component are also plotted at the same phase (see Fig. 3 for free surface location) with their respective 95% confidence intervals.

As the lee vortex moves from the ripple crest to the ripple trough, a thin viscous boundary layer develops between the vortex and the ripple bed surface. To illustrate this process, the vertical profile of the horizontal velocity at u)t = 0.557T (see Fig. 7) and x = 1.48 cm is replotted in Fig. 11. It is quite clear that the horizontal velocity approaches the ambient velocity at z> 4.0 cm, while the profile reflects the existence of the lee vortex.

Water Wave Induced Boundary

X = 0.60(cm) 4 _3 E

2.2

-20

Fig. 6.

t

0 (cm/s)

Layer Flows Above a Ripple Bed

X=1.22(cm)

X = 1.84(cm)

X = 2.47(cm)

4

4

4

3

3

3

2

2

2

1

1

0 20-20

^ 0 (cm/s)

n 20 -20

( 0 (cm/s)

91

1 n 20 -20

/ 0 20 (cm/s)

The phase-averaged velocity field in CaseA at phase (b). See caption for Fig. 5.

Underneath the vortex a very thin inner boundary layer, with the thickness of the order of 0.40 cm, is observed. This inner boundary layer has a characteristic length scale of order of {vT)1/2 (see table 1), which is equivalent to the Stoke's viscous layer thickness (Horikawa and Ikeda 11 ). Inside the inner boundary layer the direction of the horizontal velocity is the same as the free stream velocity. To ensure that this observation is not the result of the lack of resolution in data analysis, the averaging process has been repeated with smaller bins, 1.26 mm x 0.40 mm in size. The results with the refined bin size are very close to those obtained by the coarse bin size as shown in Fig. 11, confirming the boundary layer measurements.

3.1.1. Wave-Induced Steady Streaming Because of the nonlinearity in the wave field above the ripple bed, the period-averaged velocity does not vanish. The residual velocity, known as the steady streaming, is defined as:

(Ui)

1 ft+T (ui)dt;i -TJt '

=1,2

(3)

92

P. L.-F. Liu, K. A. Al-Banaa

 (cm/s)

Fig. 7.

 (cm/s)

and E. A.

 (cm/s)

Cowen

 (cm/s)

The phase-averaged velocity field in CaseA at phase (c). See caption for Fig. 5.

where T and (it*) are defined in equation 2. Thus, (ttj) = («») + (tij) and (ttj)' = 0. The magnitude and the direction of the steady streaming velocity have direct influence on near bed transport, such as sediment transport and the stability of the ripple bed. In Fig. 12, the spatial distribution of the steady streaming velocity vectors is presented. Two circulation cells with opposite direction appear above two adjacent ripple crests for CaseA. The circulation cells provide a mechanism for transporting sediment from the ripple trough to the crest and hence likely contribute to ripple growth. At the outer edge of the boundary layer, the vertical steady streaming vanishes as expected, while the horizontal component has a negative value; i.e., the steady streaming is in the direction opposite to that of wave propagation. Many researchers have also observed this negative steady streaming in laboratory wave facilities (e.g. Kemp and Simons 23 ; Nepf et al. 24 ), which is generally assumed to be the result of mass conservation in a closed wave tank. The circulation cells for the steady streaming are not symmetric with respect to the ripple trough. The asymmetry is partly the result of the negative wave-induced streaming just outside of the boundary layer. Consequently, the ripple-length-averaged steady streaming velocities (over one

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

93

X(cm) X = 0.60(cm)

4 _3 32

r

X = 1.84(cm) 4

3

3

2

2

1

1

0

c u > (cm/s)

Fig. 8.

X = 1.22(cm) 4

(

I

0

s u > (cm/s)

X = 2.47(cm) 4 3 2

}

1 n

 (cm/s)

< (cm/s)

The phase-aver aged velocity field in CaseA at phase (d). See caption for Fig. 5.

ripple wavelength, L), which can be defined as: Ui

--t

(ujdx,

1,2

(4)

do not vanish in the boundary layer. Fig. 13 plots the horizontal component of the ripple-length averaged streaming velocities above the ripple crest level for CaseA. In this case, the steady boundary layer has been established within the measurement area since the steady streaming velocity becomes essentially independent of the distance above the bed at « 20 5 from the ripple crest, where S is the laminar oscillatory boundary layer length scale as denned in Table 1. The velocity has been normalized by the maximum orbital velocity at the outer edge of the boundary layer, U0. Since the ripple-length-averaged steady streaming is positive in the vicinity of the ripple crest, the ripple bed would have been migrating in the direction of wave propagation, if the ripple bed were made of fine sediments. We also note that the steady streaming at the outer edge of the boundary layer, U/U0, is approximately -0.055 for CaseA and -0.080 for CaseB, which is in the same as order of magnitude of the wave slope (see table 1). As shown in Fig. 13, the steady streaming has a negative slope above the ripple bed, which has impacts on the asymmetric behavior in the phase-averaged vortex and hence on the vortex decay rate.

P. L.-F. Liu, K. A. Al-Banaa

94

x = o 60(cm)

X = 1.22(cm) 4

1

3 2

I

-20

Fig. 9.

0 (cm/s)

3 2

\ 0 (cm/s)

n 20 -20

I

I

0 (cm/s)

Cowen

X = 2 47(cm)

X = 1.84(cm) 4

1

1

n 20 -20

and E. A.

4 3 2 1 n 20 -20

( 0 20 
The phase-averaged velocity field in CaseA at phase (e). See caption for Fig. 5.

3.2. Phase-Averaged

Vorticity

One of the advantages of DPTV is the spatially resolved data, allowing the calculation of the velocity gradients at each bin. Consequently, the vorticity can be readily obtained as:

The central finite difference scheme is used to calculate the vorticity. It should be pointed out that because of the use of a central difference scheme and the relatively high data dropout rate near the solid boundary, the calculated vorticity in the bins immediately adjacent to the ripple bed should be interpreted with care. In figures 14 - 19 the evolution of the vorticity over one-half wave period for CaseA is presented. Negative and positive vorticity corresponds to the clockwise and counter-clockwise circulation, respectively. The time sequence for the figures is identical to that in figures 5 - 1 0 for the evolution of the velocity field. The uncertainty analysis determined the typical 95% confidence interval bound in the vorticity field is (+0.96 s _ 1 , -1.0 s - 1 ) , while the largest uncertainty limit in the vorticity field is (+8.05 s ~ \ -9.53 s" 1 ).

Water Wave Induced Boundary Layer Flows Above a Ripple Bed

95

X(cm)

J

1 (cm/s)

Fig. 10.

3 2 1

n

4

4

4

J s (cm/s)

x = 2.47(cm)

x = 1.84(cm)

X* = 1.22(cm)

X = 0.60(cm)

4 „3 E 3.2

3

3

2

2

1

n

1

N : u > (cm/s)

n

\ (cm/s)

The phase-averaged velocity field in CaseA at phase (f). See caption for Fig. 5.

As the wave crest approaches the measurement location, a vortex with strong negative vorticity is generated on the lee side of the ripple crest. A residual vortex with positive vorticity, which was generated and ejected from the previous wave period, appears on top of the newly generated vortex (Fig. 14). The pair of vortices is advected in the direction of wave propagation. The residual (counter-clockwise) vortex moves faster than the lee side (clockwise) vortex (Fig. 15). While the lee side vortex is initially confined to the trough region, the residual vortex travels a distance of two and half ripple wavelengths before its strength diminishes completely. As shown in Fig. 15, beneath the lee side negative vortex there is a thin layer of positive vorticity developing, which corresponds to the inner boundary layer illustrated in Fig. 6. As the ambient oscillatory flow starts to decelerate, the strength of the negative vorticity within the lee side vortex begins to decay, while the intensity of the positive vorticity in the inner boundary layer grows (Fig. 16 & 17). Once the ambient flow reverses its direction, both the positive and negative vortices are swept backwards (Fig. 18). Eventually, the positive vorticity within the inner boundary layer is separated from the ripple crest to form the weather side vortex for the adjacent ripple. In the

96

P. L.-F. Liu, K. A. Al-Banaa 4.5

I

O —

4

I

I

and E. A.

I

I

Cowen

\

i

F ine R e s o l u t i o n Coarse

Resolution

-

3.5 3 2.5 2

-

1 .5

-

Ripple Crest Level

1 0.5

__~r-fr—O

"

t^*^^

'

^ ^ ° .

^

CT

~ 4 6 (cm/s)

10

12

Fig. 11. The vertical profiles of the horizontal phase-averaged velocity at section x = 1.48 cm and the phase corresponding to Fig. 7. The "solid-line" represents the result using a coarse bin size for data analysis and the " circle" represents the result using a finer bin size.

mean time, the lee side vortex is ejected from the ripple crest and becomes the residual vortex for the adjacent ripple (Fig. 19). The process repeats itself during the second half of the wave period with opposite sense vortices. 3.3. Turbulence

Intensity

and Reynolds

Stress

A two-dimensional turbulent intensity, which is denned as (q')=((u'2)

+ (w'2))

1/2

(6)

with u' and w' being the velocity fluctuations in the x and z directions, respectively, and the Reynolds stress, —p(u'w'), can be calculated from the experimental measurements. The typical 95% uncertainty limit in the turbulent intensity is (+0.089 cm/s, -0.095 cm/s) and in case of the Reynolds stress the typical 95% uncertainty limit is (+0.017 cm2/s2, -0.015 cm2/s2). On the other hand, the largest 95% uncertainty limit is (+0.96 cm/s, -1.02

Water Wave Induced Boundary Layer Flows Above a Ripple Bed

97

2.5

u N

/ / s s ** * - » - — — " - v ** / / s s *• . -- V V 1 I / / s *• - «. V \ \

2

t t > \ 1 t t t t , / f r 1 S'SS ' ,

\ / \»- -~ \

2 (cm/s)

0.5

1

0.5

X(cm)

Fig. 12.

\\ \\

50

Wave induced steady-streaming for CaseA.

I

40

30

-

•

\^^^ 20

1

\

^~

—_

•

" \

Ripple crest level

\

10

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

0.0

Fig. 13. Vertical profile of the period averaged velocity over a ripple wavelength for CaseA. The x-axis is normalized by U0 and the z-axis by 5.

98

P. L.-F. Liu, K. A. Al-Banaa

1

).5 X = 0.60(cm) 4 _3 E 2-2 N 0 -100

[ <^ 0 0)(S _ 1 )

4

and E. A.

2 X(cm)

1.5 X = 1 22(cm)

2.5

X = 2.47(cm)

X = 1.84(cm) 4

3

3

3

2

2

2

n

100-100

0 0)(S" 1 )

n

100-100

{ 0 (0(S _ 1 )

3.5

3

4

( \

Cowen

n

100-100

< 0

100

0)(S _ 1 )

Fig. 14. Spatial variations of the phase-averaged vorticity field in CaseA at phase (a). The "solid-line" represents the positive vorticity and the "dotted-line" is for the negative vorticity. Four vertical cross sections of the vorticity field are plotted at the same phase (see Fig. 3 for free surface location) with their respective 95% confidence interval.

cm/s) in the turbulent intensity and is (+1.2 cm2/s2, -1.09 cm2/s2) in case of the Reynolds stress. The evolution of the turbulent intensity is presented in figures 20 - 25 for CaseA. The phase sequence for figures 20 - 25 is exactly the same as that for the evolution of the velocity field. The turbulent intensity is strongly correlated with vorticity in the boundary layer. When the lee-side vortex is being shed from the ripple crest, the production of turbulence can be observed in Fig. 20. The strength of the turbulence intensity grows as it moves downstream (Fig. 21). As the wave crest arrives at the measurement area, the turbulence spreads over the entire boundary layer between two adjacent ripple crests (Fig. 22). Some of the turbulence advects over the downstream ripple crest to the neighboring ripple as the wave is still in the forward motion mode (x-direction). When the wave starts to decelerate and then reverse its direction, dissipation dominates and most of turbulence decays (figures 23 & 24). In Fig. 25, the core of the turbulence intensity is moving back towards the upstream ripple and the mechanism is repeated for the next half-period.

Water Wave Induced Boundary Layer Flows Above a Ripple Bed

1

0.5 x = o.eo(cm)

4 E &2 1

J

x = 1.22(cm)

3

1

n -100

4

0

2 1

<

n 100-100

1

2 X(cm)

1.5

1 }0

2.5

4

3

3 2

t

1

1

_1

co(s~ )

X = 2.47(cm)

X = 1 84(cm)

n 100-100

0

0 100-100

_1

co(s )

3.5

3

4

2

99

i

0

100

o)(s_1)

oi(s )

Fig. 15. Spatial variations of the phase-averaged vorticity field in CaseA at phase (b). For rest of figure caption see Fig. 14.

Sleath 25 ' 26 and Jensen 27 have reported that for the mean velocity field the effects of a ripple bed can be considered as roughness elements with an equivalent roughness length, KS. They further demonstrated that the turbulence intensity is a function of Reynolds number and the bed roughness. If the turbulent boundary layer is fully developed, its dependency on the Reynolds number diminishes. Their findings are shown in Fig. 26 in terms of the maximum period-averaged turbulent intensity components, (U'2)1/2r

(«/,22^)

.„„ where {u'2)

1/2

M 1/2 ,

(u'2)1/2dt

= max

ft

= max

f l <»'2>

/2\l/2

(7)

dt

(8)

and the roughness parameter, A/ns. In the figure, the solid line represents the curve of the fully developed turbulent boundary layer. To include the present experimental results in the same figure, KS is estimated as twice the

100

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

4

-

3.5 3

•

-

2.5 2 1.5

*

> •_L"T

3=-

. • - : = .: = -: = : " ^ - ^ C ~ S ^ •' ^

1

_

s

^ j

y

\ \ ^ v - - - -y^ ^^^^x^^^

c

0.5

• - • • 1 5 ^ . * <S

#

#

1

D.5

~~

*

1.5

2 X(cm)

~~^ *

X=1.22(cm)

X = 0.60(cm)

2.5

X = 2.47(cm)

X = 1 84(cm)

4

4

4

4

3

3

3

3

2

2

2

1

1

F

i.2 N 1

n -100

i

n

0

100-100 1

o)(s" )

\

(1 100-100

0 w(s

_1

)

3.5

3

1

<

n 0

100-100 1

a>(s" )

\ 0

100

ci)(s" 1 )

Fig. 16. Spatial variations of the phase-averaged vorticity field in CaseA at phase (c). For rest of figure caption see Fig. 14.

ripple height (Sleath 25 ' 26 ). This is consistent with the KS values found by others in the literature, which are generally in the range of 2 to 4 sand grain diameters. Our results are plotted in Fig. 26. Even though the Reynolds numbers of the two cases measured in this study are much smaller than those presented by Sleath and Jensen, at least two orders of magnitude smaller, the present data fall on the same curve. Note that our results have extended the curve, which was presented originally by Jensen 27 , to smaller values of A/KS. The Reynolds stress, —p{u'w'), arises from the correlation between two components of turbulent fluctuations. In figures 27 - 32 the Reynolds stress field at several wave phases (same as those shown in Fig. 3) is shown for CaseA. Overall, the Reynolds stresses correlate strongly with the vortices. As the wave crest reaches the measurement area, the patterns of Reynolds stress distribution mimic those of the vorticity distribution (see Fig. 15 & 16 and Fig. 28 & 29). When the wave decelerates and reverses its direction, the Reynolds stress becomes relatively weak (Fig. 30 & 31), suggesting that turbulence dissipation is now the dominant process. Finally, as the wave trough approaches the measurement area (Fig. 32), two focused positive Reynolds stress regions reappear on top of the ripple crests, corresponding

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

1.5

2 X(cm)

X = 1,22(cm)

0 _1

o(s )

100-100

0

2.5 X = 1,84(cm)

100-100 -1

3

0

100-100 -1

101

3.5

X - 2.47(cm)

1 0

100

to(s~1)

Fig. 17. Spatial variations of the phase-averaged vorticity field in CaseA at phase (d). For rest of figure caption see Fig. 14.

to the vortices generated from the inner boundary layer (see Fig. 18). For a complete wave-cycle the reader may refer to Al-Banaa 28 .

3.4. Phase-Dependent Evolution of the Production Dissipation in the Turbulent Kinetic Energy

and Budget

The spatially resolved PTV data enables us to directly estimate each term in the 2-D turbulent kinetic energy budget TKEB equation, except the velocity-pressure fluctuation term. Here, the phase dependence of the production and dissipation terms are discussed above the ripple bed, in particular in the shear layer and in the region close to the ripple bed. The full TKEB equation can be found in Tennekes and Lumley 29 and Pope 30 and will not be presented here. The production, p, and dissipation, e, terms are defined, respectively, as P

(ulu'ASy

(9)

&1/\Sij Sij )

(10)

102

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

-

4 3.5

•

3

-

2.5

-

2

' —5

-5—

1.5

r^

1

e * 0.5

1

X = 0.60(cm) „ 3 E

o1— -100

0

100-100 1

o (s~ )

3

^CT Z-^ •* ~"—* "" * 1.5

X=1,22(cm)

i 0

_1

co (s )

2 X(cm)

2.5

X=1.84(cm)

L 0

Ms"1)

100-100

0

100

co(s"1)

Fig. 18. Spatial variations of the phase-averaged vorticity field in CaseA at phase (e) For rest of figure caption see Fig. 14.

where v is the kinematic viscosity, Sy and Sf, are the mean rate of strain and the fluctuation rate of strain tensors, respectively, and are defined as du d(Ui) , 0(u,->Y „ _ V ^ f i '^ lj dxt r ~ 22 \dxj + dxi ! ( dxj

(11)

A central finite difference scheme is used to evaluate the above equations. Equation 10 is multiplied by a constant; B = 1.4, which accounts for the missing out-of-plane terms and is based on a steady flat-plate boundary layer model as reported by Cowen31. The evolution of the in-plane components of p and e are presented for CaseA in figures 33 - 35. As the wave approaches the measurement area we can see that above the ripple crest p starts to gain strength due to flow separation and the value of e is roughly 25% of p at this phase. In the same figure, along the ripple trough, both p and e are nearly zero. We also note a significant level of negative p, which indicates the mean field is gaining energy at the expense of the turbulence, this is likely a result of the second-order steady streaming process. As phase progress and the wave crest approaches the measurement area, Fig. 33(b), p peaks above the ripple crest to reach its maximum value during the halfcycle. The e maintains its strength but spreads along the lee side of the ripple and has a value equal to roughly 12% of the p. In Fig. 34, as the

Water Wave Induced Boundary

-100

0 to(s_1)

100-100

0 io(s _1 )

Layer Flows Above a Ripple Bed

100-100

0 io(s"')

100-100

0 co(s"1)

103

100

Fig. 19. Spatial variations of the phase-averaged vorticity field in CaseA at phase (f). For rest of figure caption see Fig. 14.

wave decelerates e enhances along the ripple bed and the p spread across the ripple bed and behaves similar to Reynolds stress (Fig. 29). After separation, Fig. 34(d), The p and e are similar to that of a zero-pressure-gradient turbulent boundary layer where the p and e are of the same order of magnitude in both regions, close to ripple bed and in the high shear region, although the e peaks to its maximum close to ripple bed. As the flow starts to reverse its direction, Fig. 35, p diminishes in a region above the ripple trough, which is expected, since there is no mechanism to produce turbulence. At the same time, e is confined to a small region above the ripple bed and propagates upstream in the direction of the mean flow. In Fig. 35(e), the same mechanism repeats itself for the next half-cycle in the opposite direction. In figures 33 - 35 we note that there are two regions where the p and e play a central role in the TKEB equation. The first region is close to the ripple bed and the second is in the vicinity of the ripple crest where the separation enhances the p term in the TKEB equation. Two control boxes (CB) are used to calculate the terms in equation 9 and 10 by spatial averaging the values inside the CB at each phase. The location of the lower CB is 1.67 cm < x < 2.18 cm and 0.34 cm < z < 0.63 cm, whereas the upper CB location is 1.67 cm < x < 2.18 cm and 1.0

P. L.-F. Liu, K. A. Al-Banaa

104

and E. A. Cowen

X(cm) X=1.22(cm)

X=0.60(cm)

„3 E

1

4 3

\

2

f

1

(cm/s)

0

I \ / (cm/s)

X=2.47(cm)

X=1.84(cm) 4

4

3

3

2

2

1

1

n

/ (cm/s)

0

|

I

; (cm/s)

Fig. 20. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (a). Four vertical profiles of turbulence intensity are also plotted at the same phase (see Fig. 3 for free surface location) with their respective 95% confidence interval.

cm < z < 2.14 cm. The reason for selecting these locations is to illustrate the effects of the flow separation due to high shear on the estimation of p and e. In Fig. 36 the temporal evolution of p, e and the differences between them are plotted (see Fig. 3 for free surface profile) for the CB close to the bed. It is clear that the two mechanisms are balanced since the CB is close to the ripple bed and the viscous terms are important. Thus, the common assumption that the p and e are balanced is applicable (Le32 and Pope 30 ). We observe that both terms peak after the wave crest past the measurement area at ult = 0.837T. In the shear layer (the upper CB), Fig. 37, the p is relatively higher than the e. We also observe that the difference between p and e peaks at the same phase as that of p, i.e., ut = 0.557T. Le 32 suggested that this discrepancy could be due to the velocity-pressurefluctuation term in the TKEB equation. However, we observed that the advection by the mean velocity is also very important at the upper CB. In fact, we found that the advective term is twice the p at oot = 0.357T and 22% at ut = 0.557T. It is also interesting to point out that the p peak occurs at the same moment that the wave crest reaches the measurement area (see Fig. 3 for free surface location) and the flow starts to separate above the ripple crest (see Fig. 7). This separation corresponds to an increase in the

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

105

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s) (cm/s) (cm/s)

Fig. 21. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (b). The rest of figure caption remains the same as in Fig. 20.

0 10 -10 0 10 -10 0 10 -10 0 10 {cm/s) (cm/s) (cm/s) (cm/s)

Fig. 22. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (c). The rest of figure caption remains the same as in Fig. 20.

106

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s) (cm/s) (cm/s)

Fig. 23. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (d). The rest of figure caption remains the same as in Fig. 20.

0 10 -10 0 10 -10 0 10 -10 0 10 (cm/s) (cm/s) (cm/s) (cm/s)

Fig. 24. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (e). The rest of figure caption remains the same as in Fig. 20.

Water Wave Induced Boundary

(cm/s)

(cm/s)

Layer Flows Above a Ripple Bed

(cm/s)

107

(cm/s)

Fig. 25. The spatial variations of the phase-averaged turbulence intensity field in CaseA at phase (f). The rest of figure caption remains the same as in Fig. 20.

stresses, (u'^u1.•), in equation 9. Once the wave crest passes the measurement area, the p and e decrease rapidly. This mechanism repeats itself in the next half-cycle. 4. Concluding Remarks The DPTV technique was successfully employed to measure the mean and turbulent structure of water wave induced boundary layer flows above a ripple bed for two cases of differing Keulegan-Carpenter number. The phasedependant velocity, vorticity, turbulent intensity, Reynolds stress, turbulent production and dissipation fields were measured and analyzed. The technique allowed us to explore the evolution of these quantities in detail with high spatial and temporal resolution. The process of ripple-induced vortex generation, advection and ejection was also captured and discussed. As the wave crest approaches the measurement area, the magnitude of the positive horizontal velocity near the ripple bed grows stronger setting the stage for flow separation and vortex generation. At subsequent phases flow separation occurs and a new vortex is spawned. At separation the ratio of the phase averaged velocity to the free stream velocity is (u)/U0 « 1.33 in both cases. The ejected vortex advects across the ripple bed during the next half

108

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

o + 0

Sleath(1987,1988) I Jensen etat.(1988) Present results |

A

-

o\ 0

o o \* o

-

N^o

+

o

o o o

102

10'

103

104

A/K c

O + 0

Sleath(1987, 1988) Jensen etal.(1988) Present results

A/Kg

Fig. 26. The maximum mean turbulent velocity components as a function of bed normalized roughness, A/KS the upper panel is the streamwise component and the lower is the vertical component.

period where it reverses direction, travels back toward the crest where it was spawned, and rises until it decays completely. The second-order spa-

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

109

4 3.6 3 2.5 "E

1.5 1 0.5 0 0.5

1

X=0.60(cm)

1.5

2 X(cm)

X=1.22Jcm)

2.5

3

X=1.84(cm)

3.5 X=2.47(cm)

4I 3

-10

0

10 - 1 0

-(cm 2 /s 2 )

0

10 - 1 0

-(cm 2 /s 2 )

0

10 - 1 0

-(cm 2 /s 2 )

0

10

-(cm 2 /s 2 )

Fig. 27. The spatial variations of the phase-averaged Reynolds stress field in GaseA at phase (a). T h e "solid-line" represents the positive Reynolds stresses and the "dottedline" is for the negative Reynolds stresses. Four vertical cross sections of turbulence intensity are also plotted at the same phase (see Fig. 3 for free surface location) with their respective 95% confidence interval.

tially resolved Eulerian mean velocity (steady streaming) field is presented in Fig. 12. Two asymmetric circulation cells, with opposite sense, appear above adjacent ripple crests. The circulation cells provide a mechanism for sediment transport from the ripple trough to the crest and likely play an important role in ripple growth and stability. Our measurements indicate that the steady streaming is in the direction opposite to that of wave propagation outside of the oscillatory boundary layer. The DPTV technique provides two-dimensional spatially resolved data which allowed us to calculate the vorticity field at each bin. As the wave crest passes the measurement region, we recognize two sources of vorticity: bed generated vorticity, which diffuses vertically upwards into the flow, and the ripple crest induced flow separation generated vortices, which are advected out over the ripple trough before reversing direction and rising under their own self-induction. This latter vorticity differs in strength with half-period due to the wave-induced steady-streaming background mean velocity field. The negative vortex, which is generated under the wave crest, is stronger than the positive vortex, but decays faster than the positive

P. L.-F. Liu, K. A. Al-Banaa

110

and E. A.

Cowen

3 X=1.22(cm)

3.5 X=2.47(cm)

X=1.84(cm) 4I

_ 3 E •H.2

\ -10

10 -10

-(cm /s )

10 -10

-(cm /s )

10 -10

-(cm /s )

10

-{cm /s )

Fig. 28. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (b). The rest of figure caption is the same as figure 27.

(cm /s )

-(cm /s )

-(cm /s )

-(cm/s )

Fig. 29. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (c). The rest of figure caption is the same as figure 27.

Water Wave Induced Boundary

(cm /s )

Layer Flows Above a Ripple Bed

-(cm /s )

-(cm /s )

111

-(cm /s )

Fig. 30. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (d). The rest of figure caption is the same as figure 27.

4

•

-

3.5 3

•

2.5 2 1.5

•

:^-03

"5"

3

:—°- -\

i*

1 0.5

-oX

~~~

—

e

* 0.5

*

* '

»

X(cm) X=0.60(cm)

X=1.22(cm)

X=1.84(cm)

X=2.47(cm)

_3 E il2

10 -10 0 10 -10 0 10 -10 0 10 2 /e2\ _ ^ . i'...,^/om2/^2\ ..W~/~m2/*2\ . ~/,™2/.*2\ -(cmi /s ) -{cm/s ) -(cm /s ) -(cm /s )

Fig. 31. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (e). The rest of figure caption is the same as figure 27.

112

P. L.-F. Liu, K. A. Al-Banaa

and E. A. Cowen

4 3.5 3 2.5

f N

2

1.5 1 0.5 0 0.5

1

1.5

X=0.60(cm)

2 X(cm)

X=1.22(cm)

2.5

3

X=1.84(cm)

3.5 X=2.47(cm)

4 3

_ 3 E S.2

-10

0

10 - 1 0

-(cm 2 /s 2 )

0

10 -10

-(cm 2 /s z )

0

10 - 1 0

-(cm 2 /s 2 )

0

10

-(cm 2 /s 2 )

Fig. 32. The spatial variations of the phase-averaged Reynolds stress field in CaseA at phase (f). The rest of figure caption is the same as figure 27.

vortex. Interestingly we observe, as shown in Fig. 38, that the difference of the positive and negative vorticity peak's decay rate, as determined by a first order exponential curve fit, is twice the mean background vorticity induced at second-order, e.g., AC = 2^-, where AC is the difference in the determined positive and negative vortices decay rates. While we expected the background shear to play a role in setting the decay rate it appears that the effect of viscous and turbulent diffusion is adjusting itself to be an order one process. Following Sleath 25 ' 26 and Jensen 27 we determined that the two cases, A and B, are fully developed turbulent boundary layers. With these results we extended the curve presented by Sleath and Jensen to smaller values of the roughness parameter, A/KS. Also, we find that the maximum turbulent intensity is roughly 40% of the maximum phase average velocity, which is high compared to other flows. The Reynolds stress was found to peak as the wave crest approaches the measurement area as a result of the ripple crest-induced flow separation. As the flow decelerates, the Reynolds stress becomes relatively weak. The spatial distribution of the phase dependant p and e in the TKEB equation are estimated based on the DPTV results. We observed two funda-

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

3

1

f

2

T

•

p 2 n U" J

)

X(cm)

3

p 2 T, \X?

1

p 2 TI \T>

3

V [ J

\ p2llU"J

" 1 1 ^

•

2

3

«2>\U„

3

ja 2 TI U ^

X(cm)

113

*J2TlU"J

3

*>2nU;J

^ u l T

3

Fig. 33. The spatial variations of the p and e in CaseA at phases (a &; b). Four vertical cross sections of the p and e are also plotted at the same phase (see Fig. 3 for free surface location). The contours are normalized by U^/2TJ. The "solid-line" represents the positive contour and the "dash-dot-line" represents the negative contour. The left column is e and the right is p.

1

)

3

e 21! U" 3

X(cm)

l

2

3

3

e 2 TI U" 3

3

e 2 n IT3

e 2 n IT3

3

P __£«

a-

^.f®:Sp>-

1

'

-iSSS.

d

)

X(cm)

2

3

3

' E2TIU"J

)

3

e2TiU"J

3

*> 2 Ti U" d

2

X(cm)

3

I I '

p 2 r| LT3

P 2 T| U ^

3

| p 2 TI U" 3

Fig. 34. The spatial variations of the p and e in CaseA at phases (c & d). See caption for Fig. 33.

P. L.-F. Liu, K. A. Al-Banaa

114

and E. A.

Cowen

3

h

- ^

T40te

&

1 1 0

1

*

-0.0C-:

j£>

»

X(cm) 2

3 im—.

-0.1S

0

eZnU - 3

0.1S-0.15

0

e2nU-»

0.15-0.15

0

e2T]U^

0.15-0.15

0

'2<]U?

0.1

pinlT

P2t\U™

p2tlU: 3

»2nU^

Fig. 35. The spatial variations of the p and e in CaseA at phases (e & f). See caption for Fig. 33.

Fig. 36. The phase-dependent evolution of the p and e in the TKEB bed normalized by Ug/2r). "O", p; " 0 " , e; " + " , difference.

near the ripple

Water Wave Induced Boundary

Layer Flows Above a Ripple Bed

115

Fig. 37. The phase-dependent evolution of the production, p, and dissipation, e, in the TKEB away from the ripple bed normalized by U%/2r). "O", p; " 0 " , e; " + " , difference.

J

.

"\

>-\Y

•

T

V

' - 0+ \

^

V

\

> *\ \\T 1

x

T

v

\ \ o

ffl~

-

'V ^XK. /

•;

t.-*--*-.*-*X» 0.1

0.2

0.3

0.4

0.5 1/T

0.6

0.7

0.8

Fig. 38. The vortex strength in both Cases^l & B. "V" and "O" denote the experimental measurement of the vortex strength generated under wave crest and trough, respectively. The solid face symbols represent the measurements in CaseB. The "solidline" and "dash-line" represent the best-fit curve for the positive and negative vortex strength respectively. The " + " through the best-fit curves represent CaseB. The z-axis has been normalized by the maximum vorticity throughout the phases, uim.

116

P. L.-F. Liu, K. A. Al-Banaa

and E. A.

Cowen

mental regimes in the half-cycle characterized by different relative strengths of p a n d e. Consistent with other publications, our results indicate t h a t close t o the ripple trough, where the effect of viscosity is appreciable, e balances p in t h e TKEB equation during the half-cycle. Whereas in the shear layer above the ripple crest, p is an order of magnitude larger t h a n e and the e can be assumed negligible to leading order. It is this shear induced turbulence t h a t undergoes significant advection and on the life time-scale of t h e vortex the effect of e becomes significant. Therefore, qualitatively we could assume t h a t the difference between p , advection and e is the velocitypressure gradient t e r m in the TKEB equation.

Acknowledgments T h e authors wish t o acknowledge Kuwait Institute for Scientific Research (KISR) for the financial support of Khaled Al-Banaa, the National Science Foundation for the research grants (Grant No. CTS-8902407 and C T S 9302203) to Cornell University. We would also like to t h a n k Dr. Kuang-An Chang for his help in conducting t h e experiments.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19.

M. S. Longuet-Higgins, J. Fluid Mech., 107, 1 (1981). G. Vittori, J. Hydraulic Research, 27, 267 (1989). T. Hara, C. C. Mei, J. Fluid Mech., 211, 183 (1990). P. Blondeaux, G. Vittori, J. Fluid Mech., 226, 257 (1991). G. Vittori, P. Blondeaux, J. Fluid Mech., 314, 247 (1996). S. Sato,H. Uehara,A. Watanabe, Coastal Eng in Japan, 29, 65 (1986). G. Tsujimoto, N. Hayakawa, M. Ichiyama, Y. Fukushima, Y. Nakamura, Coastal Eng in Japan, 34, 177 (1991). M. Sawamoto, T. Yamashita, T. Kitamura, Coastal Eng., 1982, 282 (1982). C. G. DuToit, J. F. A. Sleath, J. Fluid Mech., 112, 71 (1981). S. Sato, K. Shimosako, A. Watanabe, Coastal Eng in Japan, 30, 89 (1987). K. I. Ranasoma, J. F. A. Sleath, Proc. 23rd Int. Conf. Coastal Eng., (ASCE 1992) p. 2383. K. Horikawa, S. Ikeda, Coastal Eng., 33, 661 (1990). E. Hitching, A. W. Lewis, J. Flow Measurement and Instrumentation, 10, 241 (1999). C. Lin, D. Rockwell, Phys. Fluids, 6, 2877 (1994). C. Lin, D. Rockwell, J. Fluid Mech., 302, 29 (1995). H. C. Earnshaw, C. C. Greated, Exp. Fluids, 25, 265 (1998). K.-A. Chang, P. L.-F. Liu, Phys. Fluids, 10, 327 (1998). K.-A. Chang, P. L.-F. Liu, Phys. Fluids, 11, 3390 (1999). E. A. Cowen, S. G. Monismith, Exp. Fluids, 22, 199 (1997).

Water Wave Induced Boundary Layer Flows Above a Ripple Bed

117

20. S. Hughes, in Physical Models and Laboratory Techniques in Coastal Engineering; Advance series on Ocean Engineering-volume 7, (World Scientific, Singapore, 1993), p. 568. 21. B. Efron, R. J. Tibshirani, in An Introduction to the Bootstrap , (Chapman & Hall, New York, 1993), p. 436. 22. S. J. Kline, F. A. McClintock, Mech. Eng., 3, Jan. (1953). 23. P. H. Kemp, R. R. Simons, J. Fluid Mech., 116, 227 (1982). 24. H. M. Nepf, E. A. Cowen, S. J. Kimmel, S. G. Monismith, J. Geophys. Res. C, 100, 16211 (1995). 25. J.F.A. Sleath, J. Fluid Mech., 182, 369 (1987). 26. J.F.A. Sleath, J. Waterw. Port. C, 114, 18 (1988). 27. B. L. Jensen, B. M. Sumer, J. Fredsoe, J. Fluid Mech., 206, 265 (1989). 28. K. A. Al-Banaa, in Experimental Study of the bottom Boundary Layer Induced by Water Waves Propagation Over a Rippled Bed, MS Thesis (Cornell University, 2001), p. 289. 29. H. Tennekes, J. Lumley, in A First Course in Turbulence, (MIT Press, Mass., 1974), p. 300. 30. S. Pope, in Turbulent Flows, (Cambridge University Press, New York , 2000), p. 771. 31. E. A. Cowen, I. M. Sou, P. L.-F. Liu, B. Raubenheimer, J. Eng. Mech., (In Press 2003) 32. H. Le, P. Moin, J. Kim, J. Fluid Mech., 330, 349 (1997). 33. J.F.A. Sleath, Sea Bed Mechanics, in Ref. 14 (Wiley 1984).

This page is intentionally left blank

CHAPTER 4 SHIP VELOCITY FIELDS Joe Longo", Lichuan Gui\ and Fred Stern" "IIHR-Hydroscience & Engineering 300 S. Riverside Drive The University of Iowa, Iowa City, IA, USA 52242 E-mail: [email protected] b

National Center for Physical Acoustics P.O. Box 1848, University, MS 38677

A review is made of application of PIV to ship velocity fields. An overview is given of PIV techniques, including measurement systems, tracer particles, laser, camera and recording medium, and evaluation methods along with recent advances. PIV data post processing and uncertainty assessment methods and procedures are described. Investigations from 1992-2002 are reviewed with application to boundary layer and wake without and with free surface, bow flow, wave hydrodynamics, propulsor hydrodynamics, appendages, and cavitation and bubbly flows in support of design through explication of physics, model development, and CFD validation. Discussions include YfW system; physics, geometry, and conditions; facilities; data acquisition, reduction, and locations; uncertainty assessment; and results. Overall conclusions are given as well as comments on future directions.

1. Introduction Knowledge of flow around ships is important for design, model development, and computational fluid dynamics (CFD) validation. Historically, five-hole pitot probes have been used for measuring time-mean velocity and pressure fields around ship hulls, e.g., Toda et al.1, Kim et al.2, and Longo and Stern3. Because of their intrusive nature, measurements in areas of high velocity gradient and turbulence contain biases not accounted for in usual calibration procedures (Gui et al.4). Additionally, multi-hole probes are unable to measure separated flows and Reynolds stresses. Transition to Laser-Doppler velocimetry (LDV) systems for measurement of ship velocity fields gained favor from reduction of probe-disturbance effects and expanded ability to measure separated flows and Reynolds stresses (Fry and Kim5; 119

120

J. Longo et al.

Hoekstra and Ligtelijn6; Chesnakas and Jessup7; Longo et al.8). Despite the success of multi-hole pitot and Laser-Doppler systems, they both require measurement of ship velocity fields with a time-consuming point-to-point technique that makes acquisition of large-scale coherent structures difficult. The inherent limitations of multi-hole pitot and LDV technology underscored the need for a new type of measurement system capable of rapid whole-field measurement of mean velocities and Reynolds stresses. The need was met with particle image velocimetry (PIV). In the past decade, researchers have made extensive use of PIV for measurement of ship-hydrodynamic flows. In fact, PIV is fast replacing techniques that rely on multi-hole pitot or LaserDoppler probes. PrV has gained widespread acceptance because it is minimally or non-intrusive and capable of measuring mean velocity and Reynolds stress fields and separated flows in a relatively large plane or volume. Current volume systems can obtain the full Reynolds-stress and strain-rate tensors. PIV temporal and spatial resolutions are improving with continuous advancements in digital camera framing rates and CCD resolution, respectively. The current fastest PIV systems are approaching data rates seen in LDV systems, and camera resolution improvements are enabling measurement of micro-scale flow structures so that new physics can be studied such as cavitation. Refinement of evaluation methods is ensuring that PIV measurement accuracy is at least as good as systems that rely on multi-hole pitot or Laser-Doppler probes. Recent development of three-dimensional (3D) planar or volume PrV systems opens up exciting new possibilities for model testing. A review is made of application of PIV for ship velocity fields. An overview is given of PIV techniques that have been developed for use in towing tank, rotating arm basin, water channel, water tunnel, and wind tunnel with specific application to ship hydrodynamics. PIV techniques include measurement systems, tracer particles, laser, camera and recording medium, and evaluation methods along with recent advances. Fundamental procedures for PIV steady- and unsteady-flow data post processing and uncertainty assessment (UA) methods and procedures are described. Many examples of PIV technique and its development are cited. Investigations from 1992-2002 are reviewed with application to boundary layer and wake without and with free surface, bow flow, wave hydrodynamics, propulsor hydrodynamics, appendages, and cavitation and bubbly flows in support of design through explication of physics, model development, and CFD validation. Discussions include PIV system; physics, geometry, and conditions; facilities; data acquisition, reduction, and locations; uncertainty assessment; and results. General PIV techniques and measurement procedures were reviewed by Raffel et al.9, Cenedese and Paglialunga10, Keane and Adrian11, Adrian12, Willert and Gharib13, Heckmann et al.14, Westerweel et al.15, and Grant16. PIV technology has been adapted for investigation of a variety of different flows. For example, micro

Ship Velocity Fields

121

PIV (MPIV) systems (Santiago et al.17; Wereley et al.18) were developed in recent years to measure fluid flow fields in sub-millimeter scale channels. In parallel with advances in desktop computing, PIV technology and capability has steadily evolved to facilitate measurements in three coordinates. For example, stereoscopic PIV (SPIV) systems apply two cameras to view particles in a light sheet at two different angles, so that the out-of-plane motion of particles can be determined to provide a two-dimensional (2D) distribution of all three velocity components. A recent review of SPIV systems was reported by Rrasad19. Holographic PIV (HPIV) systems on

o1

oo

(Thompson ; Meng and Hussain '; Zhang and Kate ) enable instantaneous measurements of a 3D distribution of all three velocity components. Defocusing digital particle image velocimetry (DDPIV) and multiple-sheet PIV technique (MSPIV) use a single camera to determine three velocity components of particles distributed in a volume. Technical details of DDPIV were reported by Willert and Gharib23. MSPIV .systems were suggested by Raff el et al.24 and used by many others. 2» Overview of PIV techniques An overview follows of PIV techniques used for ship hydrodynamic applications. The overview includes topics of PIV system setup, tracer particles, laser and cameras, and evaluation methods. 2.1* PIVsjstem setup In Fig. 1 a conventional 2D PIV system is illustrated. A 2D lightsheet is generated using a laser and a lens system to illuminate the measurement region. A camera views the measurement region and records images of tracer particles that follow the flow.

Single exposed recording

Double exposed recording

Fig. 1. Schematic illustration of a conventional 2D PIV system.

122

J. Longo et al.

The laser light sheet is usually required to be pulsed with a short duration and precisely controlled time interval. However, a continuous light sheet can also be used in conjunction with digital video or cinematic cameras with electric or mechanical shutters for exposure control. The displacements of particle images between two laser pulses are recorded in either a double-exposed PIV recording or a single-exposed PIV recording pair. In a double-exposed PIV recording the particle images of two different exposures may be overlapped, and the time sequence between exposures is unknown. Several image shifting techniques, e.g. translating camera and rotating mirror, can be used to avoid the unexpected particle image overlapping and to solve velocity ambiguity in double exposures. Nowadays, the double-exposed PIV recording technique is still used to measure high-speed flows where the single-exposed recording is impossible. In the case of steady flows, multiple exposures are used to increase the image number density in the PIV recording. Except for tracking individual particles in a time sequence, the multipleexposed PIV recordings are processed in the same way as the double-exposed PIV recordings. The displacements of particle images in the multiple/double-exposed recordings or single-exposed recording pairs are obtained in an image processing procedure that is usually referred to as an 'evaluation', and together with the known time interval between exposures and the camera magnification, determine the velocity distribution in the measurement region. Conventional 2D PIV systems with a range of configurations have been widely used in ship velocity field measurements in circulating water channels, cavitation tunnels, wind tunnels, rotating arm basins, and towing tanks. Conventional 2D PIV systems are capable of determining twodimensional distributions of the two velocity components in the light sheet plane. In order to determine the third, i.e. out-of-plane velocity component, the lightsheet and camera have to be reconfigured to an orthogonal plane. This is a cumbersome procedure for reconstructing the 3D flowfield with a 2D lightsheet, increases the workload by a factor of two, and produces redundant data as one component is measured twice. Another method is stereoscopic PIV. A SPIV system includes one more camera than a conventional 2D PIV system. The two cameras view particles in a light sheet at two different angles, so that the out-of-plane motion of the particles can be detected. Basically, there are two different configurations in SPIV systems, i.e., translation system and Scheimpfiug angular displacement system. In the translation system the lens axes of the two cameras are set parallel to each other and perpendicular to the laser light sheet as in the conventional 2D PIV system, but separated with a distance so that they view the same area at two different angles. Two 2D vector maps are obtained simultaneously with the two cameras that allow detection of the three velocity components in the measurement plane. In the Scheimpfiug angular displacement system the two camera axes are no longer parallel to each other but rotated such that the two axes intersect at the light sheet.

Ship Velocity Fields

123

In order that all the particles in the light sheet are in good focus in the image plane, the image plane should be also rotated such that the image plane, camera lens plane and the objective plane (i.e. the laser light sheet) intersect at a line. Since the imaging magnification in the Scheimpflug system is not uniform, calibrations are required in the image processing or data-post processing procedure. A recent review of SPIV systems was reported by Prasad19. Applications of SPIV systems in ship velocity fields were reported by Grant et al.25 and Calcagno et al.26. Another approach to realize 3D measurements with a single camera is the use of multi-layer laser sheets. The idea was suggested by Raffel et al. and applied by Kamoda et al.27 in ship velocity measurements. Instead of using a light sheet to illuminate particles, defocused PIV, micro PrV, and holographic PIV systems use volume illumination techniques. Willert and Gharib23 suggested a DDPIV system mat determines all three velocity components of particle movements with a single camera. The optical technique uses a special defocusing mask placed in front of the camera lens to obtain three images for each particle. This allows for accurate determination of the position of the particle in three dimensions along with its size estimate. Improvements on the original defocusing concept have recently produced a single lightweight camera pod containing three separate CCD cameras and lenses arranged in an equilateral triangle. This version is currently under development for future use in towed PIV experiments and full-scale trials. In MPIV systems, a microscope is installed between the measurement section and the camera so that particle images in microscale channels can be recorded. Since it is difficult to generate a light sheet in micro-scale channels, volume illuminations are usually adopted in MPIV systems. The focal plane of the microscope objective determines the measurement plane. Fluorescent particles and optical filters are used in MPIV systems to remove background noise. HPIV enables all three velocity components to be measured in a volume of flow. HPrV requires a precise illumination system and a complex, timeconsuming data processing procedure. HPYV details were reported by Thompson20, Meng and Hussain21, and Zhang and Katz22. No investigations were found that utilize MPrV and HPIV applications in ship velocity fields. 2.2. Tracer particles Tracer particles are an essential element of all PIV systems. They are chosen to accurately follow the flow and efficiently scatter laser light. Density and acceleration differences between the fluid and particles cause measurement biases. According to Stake's drag law, an additional velocity Ug is gravitationally induced for very small spherical particles as

124

J. Longo et al.

wherein g is the acceleration due to gravity, \i is the dynamic viscosity of the fluid, and dp is the diameter of the particle. pp and p are the density of particle and fluid, respectively. Equation (1) can be used to estimate the velocity lag (Us) of a particle in a continuously accelerating fluid flow as Us=Up-U=d2p¥j^fla,

(2)

wherein Up is the particle velocity, U is the fluid velocity, and "a" is the acceleration of the fluid flow. A rough estimation of the velocity bias with equations (1) and (2) is helpful for selection of particle material and size prior to experiments. When the particle size becomes too small, the effect of Brownian motion on the accuracy of PIV measurements should be taken into account (Santiago et al.17). For PIV experiments on ship velocity fields, silver-coated hollow glass spheres are widely used because they demonstrate very good light-reflectance for PIV image capture and adequate suspension capability. Fluorescent particles are also widely used which are excited in the presence of green laser light and emit light at a longer wavelength. A long-wave-pass optical filter can be used to reduce the light reflected from the background. Other particles, e.g., silicon carbide and Titanium dioxide, have also been used as tracers in ship flow PIV experiments. The choice of particles can be cost-dependent, especially for PIV in large facilities (Di Felice and Gregorio28). 2.3. Laser light sheet The dual-head Nd:YAG laser has become a standard light source for recent PIV systems. Its high pulse energy (e.g., up to 800 mJ/pulse; Judge et al.29,30), short pulse duration (as small as a few nanoseconds), and adjustable time interval between exposures makes it the ideal candidate for particle imaging. For many early studies, however, continuous lasers were used to generate pulsed illumination by chopping the light with rotating disks equipped with a single transparent section. Continuous lasers were also used as light sources in PIV systems with cinematic or video cameras where image exposures were controlled with mechanical or electric shutters. 2.4. Camera and recording medium Most early PIV systems applied photographic cameras and high-resolution films to capture and store PIV images. In ship flow field experiments, 35 mm still camera

Ship Velocity Fields

125

and Kodak TMAX 3200 film were widely used. PIV film recordings were either evaluated directly or after digitizing with scanning devices. Film-based PIV recordings have very high information density, e.g. Fu et al.31 digitized 35mm films to get digital recordings of 3072x2048 pixels. Multiple/double exposures of particle images were recorded in the film. Since the time between laser pulses, i.e. exposures, can be set to be very small, the system can be used to measure flow fields with very high speeds. However, since the time sequence of the exposures was not coded in the recording, there was an ambiguity to determine the velocity direction. For simple flows, previous knowledge of the flow was used to help determine velocity direction. In complex flow measurements, image-shifting techniques were adopted, which included the translation camera technique and the rotating mirror technique (Grant16). Color coding (Jaw and Kehr32), laser intensity difference coding (Grant et al.33), exposure duration coding (Shekarriz et al.34; Okuno et al.35) and numerical methods (Hong et al.36) were also used to solve die velocity ambiguity problem resulting from multiple/double exposures. In order to bypass complex and time-consuming procedures for processing film-based PIV recordings, video CCD cameras were adopted to replace photographic cameras (Yeung et al.37; Zhang et al.38; Siddiqui et al.39; Zhao et al.40). Since a large number of digital recordings can be captured with a video CCD camera and saved on tape for each test case, time-resolved investigations of unsteady flows and statistical analyses of turbulent flows become possible. However, video recordings were limited in digital resolutions (640x480 or 768x512 pixels) and time between frames (25 Hz or 30 Hz). The low frame rate limited the application of the video CCD-based PIV system to low-speed flow measurements. Another disadvantage of video recordings is that each frame contains an interlaced image of two half images separated with time intervals of 1/50 or 1/60 sec. Full-frame CCD cameras were used to overcome the interlace image problem. However, high digital resolution and framing rates could not be achieved simultaneously. Therefore, tradeoffs between camera resolution and framing rate have been necessary in recent investigations. High-resolution CCD cameras (e.g. 2048x2048 pixels by Gopalan et al.41) were used to capture multiple exposed PIV recordings so that the time between exposures were not limited by low framing rate, whereas low-resolution CCD cameras were used to capture single-exposed PIV recording pairs with a higher framing rate. When synchronizing the CCD camera with the double laser pulses of the dualheaded Nd:YAG laser, time interval between the two single-exposed PIV recordings in a pair can be controlled independent of the framing rate of the CCD camera. The lower limit of the time interval depends on the transfer time of the digital image from the CCD to the saving device. Frame transfer and interline transfer techniques reduce the image transfer time to several microseconds or even several hundredths

126

J. Longo et at

of nanoseconds, so that the single-exposed PIV recording pairs can be captured for high-speed flows. In ship flow field experiments, 1008x1018-pixel (Kodak MegaPlus ES 1.0) and 1280xl024-pixel CCD cameras have been frequently used to acquire single-exposed PIV recording pairs. 23. Recordingtypeand evaluation methods As discussed above, PIV recordings can be divided into two types according to the camera and recording medium, i.e., film or digital. According to the particle image number density PIV recordings were classified into three groups: laser speckle (LS) recording, low-image-density (LID) recording, and high-image-density (HID) recording. In LS recordings the individual particle images cannot be identified, e.g., Fig. 2a, whereas in LID recordings the individual particle images can be identified and tracked from frame to frame, e.g.. Fig. 2b. In HID recordings, e.g., Fig. 2c, the individual particle images can be identified but hardly tracked. Different processing methods [save been applied to different types of PIV recordings.

(a)

(b)

(c)

Fig. 2. Examples of LS (a), LID (b) and HID (c) PIV recordings. 2.5.L Young9s fringe amafysis method As discussed above, photographic cameras were used to capture multiple/doubleexposed recordings on photographic films. A traditional technique to evaluate photographic film recordings was Young's fringe analysis method. This method was initially adopted for LS and HID PIV recordings, m Fig. 3, a fine laser beam illuminates an interrogation spot on a film recording that includes a number of particle image pairs. The particle image pairs act as interfering point sources that form Young's fringes on a screen, e.g., a frosted glass. The distance between the fringes (SM) and the distance between the film and the screen (L) determine the magnitude of the mean displacement of particle images in the interrogation spot (SA). The direction of the particle image displacement was vertical to the direction of the fringes, i.e. f%rf»A=n/2. The computer-controlled analysis system included a

Ship Velocity Fields

127

Fig. 3. Young's fringe analysis system. 2-D automated traverse, a digital video camera, and a PC. It usually took several hours to complete evaluations of a PIV film recording. 2.5.2. Particle image tracking method Particle image tracking methods were used to determine displacements of individual particle images in LID PIV recordings of both photographic and digital form. PIV systems using particle image tracking techniques are also called particle-tracking velocimetry (PTV). Several different PTV systems and PTV algorithms have been reported. Examples of PTV in ship flow field measurements were reported by Shekarriz et al.34 and Lee et al.42. 2.5.3. Correlation-based interrogation algorithms The correlation-based interrogation algorithms have played a major role in evaluating digital PIV recordings since the first application of digital Prv. Correlation-based interrogation schemes and details of the respective procedures have been reviewed by Cenedese and Paglialunga10, Keane and Adrian11, Adrian12, Willert and Gharib13, Heckmann et al.14, and Westerweel et al.15. For a single exposed digital PIV recording pair, the correlation-based interrogation algorithm is applied to two small image fractions (samples) of the same size. Traditionally, the two evaluation samples are chosen, respectively, from the first and second images in the PIV recording pair at the same position in the recording coordinate. The evaluation function can simply be written as 0CWS(m,n)=

Ylgi(i,

j)- g2(i + m,j + n),

(3)

where gi(ij) and g2(i,j) are gray-value distributions of the two evaluation samples, which are restricted to a rectangular interrogation window of size of MxN pixels. Equation (3) is usually called the cross-correlation function. An example of the

128

J.LongoetaL

cross-correlation function is shown in Fig. 4(b). The mean particle image displacement vector in the interrogation window (Sx, Sy) is determined with the position of the maximal value of the cross-correlation function, i.e., s

=n'f

I

f°r ®crosVn'>n')

2

2

2

2\

For a multiple/double-exposed PIV recording, an auto-correlation function is defined as /

\

M

N

t

\

^aut0W,n)= £ lg{ij)'g{i

/

\

+ mj + n).

(5)

Wherein g(ij) is a jpray-value distribution of the AfxW-pixel evaluation sample. As shown in Fig. 4(a),- the auto-correlation function has a maximum at (m=0, n=0) and two symmetrically located secondary maxima at (&, Sy) and (-5*, -Sy), respectively.

Fig. 4. Example of auto-correlation (a) and cross-correlation (b) functions. The computations of the cross- and auto-correlation function are largely accelerated with the Fast Fourier Transformation (FFT) algorithm. In some publications the correlation function includes a denominator on the right-hand side of equations (3) and (4). If the denominator is constant, the particle image displacement determined by the position of the maximal cross-correlation function or the secondary maxima of the auto-conelation function is not changed. A variable denominator may produce different results of the particle image displacement, however, it has not been widely used because it cannot be directly accelerated with an FFT. When the computation of the correlation coefficient 0cms(mfn) is accelerated with FFT, gi(ij) and g^ij) are assumed as being distributed periodically in the i-j plane with periodicity M9 N. The assumption of periodicity enables a fast evaluation of the PIV recordings, and it makes the correlation-based interrogation algorithm insensitive to illumination differences between the two PIV recordings in. a pair (Gui and

129

Ship Velocity Fields

Merzkirch43). However, the use of FFT has disadvantages, such as large evaluation errors at large particle image displacements and large bias gradient around the zero displacement. These errors increase the measurement uncertainties of the mean velocities and Reynolds stresses, respectively, in turbulent flow measurements (Gui etal.44). The correlation function in equation (3) has also been used as a tracking criterion for determining the displacement of an image pattern that contains a group of particle images. The correlation-based tracking schemes were described and applied by Huang et al.45, Kemmerich and Rath46, Okamoto et al.47, and Fincham and Spedding48. When using the correlation-based tracking (CTR) algorithm, the gray value distribution g2(i,j) is defined in an area larger than the MxN-pixel interrogation window. Since the periodicity assumption is not necessary, a higher accuracy can be achieved for large particle image displacements, and the large bias error and bias gradient around the zero displacement can be avoided. To further increase the accuracy and reliability of the correlation-based tracking algorithm, the mean gray values are subtracted from the evaluation samples in the evaluation equation as 0ctr(m,n)= X TteAj)-gi\*\g2{i

+ m,j + n)-g2\.

(6)

i=lj=l

The computation of
130

J. Longo et al.

arbitrary sized interrogation window without padding, but no practical applications were reported in PIV. Arbitrary shaped interrogation windows can be achieved with some padding (Ronneberger et al.49) or digital mask techniques (Gui and Merzkirch52) to fit the flow boundary or accommodate the physics of the flow. 3.2. Adaptive discrete window shift (DWS) In order to avoid the large evaluation error of the correlation-based interrogation algorithm at large particle image displacement, a discrete window shift (DWS) is applied. The DWS is a multiple-pass evaluation procedure. It begins with an ordinary single-pass evaluation witfi the correlation algorithm. Then the second interrogation window, i.e., g2(i,j), is shifted by integer pixels in the flow direction based on the particle image displacements determined in the previous pass. The DWS usually includes several iterations, so that the magnitude of particle image displacement determined by the correlation function is within 0.5 pixel. The adaptive window shift method, as typically implemented, can be referred to as the forward difference interrogation (FDI), because the second interrogation window is shifted in the forward direction of the flow. Details of the discrete FDI technique are described by Keane and Adrian53, Willert54, Cowen and Monismith55, Westerweel et al.56, and Scarano and Riethmuller57. The adaptive window shift scheme also allows use of different size interrogation windows in different iteration cycles, i.e. large interrogation windows can be used in the early iteration cycles to obtain particle image displacements with high reliability, and then the size of the interrogation window is reduced along with the reduction of the particle image displacement to be determined, so that a high spatial resolution of the flow field can be achieved in the final iteration cycle. The multiple window size technique was described by Kemmerich and Rath46 and Hart58, and applied by Judge et al.29'30 in ship velocity field experiments. 3.3. Gaussian window mask In order to reduce the bias error and bias gradient of the correlation-based interrogation algorithm around the zero displacement, a Gaussian window mask is adopted by Gui et al.59. Using the Gaussian window mask in combination with DWS, a much higher accuracy was achieved for determining the Reynolds stresses in the wake flow of a surface-piercing flat plate in a towing tank experiment (Gui et al.44 3.4. Central difference interrogation (CDI) Although the FDI method leads to significant improvements in the evaluation quality of PIV recordings in many cases, there are still some potentially detrimental

Ship Velocity Fields

131

bias errors that cannot be avoided with FDI. For instance, the position deviation in flows with large velocity gradients and direction deviation in flows with curvature. A central difference interrogation (CDI) method was initially introduced by Wereley et al.60 and further developed and explored by Wereley and Meinharf61,62, to avoid the shortcomings of FDI and increase the accuracy of the PIV measurement. When using CDI, the first and the second interrogation windows are shifted backward and forward, respectively, each by half of the expected particle image displacement. As multi-pass adaptive spatial shifting techniques, both FDI and CDI require some iteration to achieve optimum results. When properly implemented, CDI does not require more computation time than FDI. 3.5. Continuous window shift (CWS) Recent studies (Sjodahl63; Sholl and Savas64; Scarano and Riethmuller65; Lecordier et al.66; Gui and Wereley67) indicated that the bias and random errors of the digital PIV recording evaluation using the correlation-based interrogation algorithms, i.e. auto- and cross-correlation, can be reduced further by continuously shifting the interrogation window according to previous evaluation results. Although the continuous window shift requires an interpolation of particle images that may add errors to the evaluation, the overall evaluation error shows significant reduction because the particle image displacement to be determined by the correlation function in the final iteration step is within 0.1 pixel. 3.6. Central difference image correction (CDIC) To account for the distortion of PIV image patterns in complex flow measurements, image correction techniques have been developed. The idea of image correction was first presented by Huang et al.68, and similar ideas were applied by Tokumaru and Dimotakis69, Fincham and Spedding48, Lin and Perlin70, Nogueira et al.71'72, and Scarano73. A central difference image correction (CDIC) method developed by Wereley and Gui74 combines ideas of central difference interrogation, adaptive continuous window shifting and image correction, and enables a most reliable and accurate evaluation of digital PIV recordings. The idea of CDIC is demonstrated in Fig. 5 by evaluating a synthetic PIV image pair near a strong vortex center with a 64x64-pixel interrogation window and a 50% overlap. The velocity distribution on the 3x3 interrogation grid (Fig. 5a) consists of simultaneous translation (Fig. 5b) and distortion (Fig. 5c). When using the traditional cross-correlation algorithm, the evaluation sample pair is obtained with an interrogation window centered at the evaluation point, e.g. Fig. 5d and 5e for the central point of the 3 x 3 grid. The traditional correlation method works well in cases of small particle image displacement and relatively simple flow. But in the

132

J. Longo et al

current case the image patterns in the sample pair do not match weE, so that the correlation function (Fig. 5f) does not show a dominant peak for the particle image displacement. Considering the translation movement of the particle images in the evaluation sample pair, the interrogation windows for the first and second recording can be shifted backwards and forwards, respectively, to realize a central difference interrogation, see Fig. 5g and 5h, so that the image patterns more closely match each other.

t—-t — f j 20 pixels

I

:

s r »»»^-—[> *

+ i

(a)

(b)

(d)

• * «

(e)

» (h)

(g)

\

/#

\

f

# " ^ # ## #^ % „« ^ l it **p ; (i)

^

€

• m

*ft». *

4

^

(k)

Fig. 5. Interrogation window shift and image pattern correction from Wereley and Gui74. However, the correlation function still does not present a very clear peak to reliably and accurately determine the particle image displacement (Fig. 5i). When considering the particle image distortion, i.e. pre-deforming the image patterns forward and backwards respectively for the first and second evaluation sample (Fig. 5j and 5k) based on the known displacements at the 9 grid points, a very good match of the image patterns is realized, so that a correlation function with a clear

133

Ship Velocity Fields

main peak is obtained (Fig. 5m). Although in this demonstration the window shifting and the image pattern correction are carried out in two steps, they can be realized in only one step in practical applications. 4. PIV data post processing and uncertainty assessment In most ship hydrodynamics PIV investigations, the post-processing of data obtained from PrV recording evaluations usually included validation of the displacement vectors, computation of instantaneous velocity vectors and other variables (i.e., vorticity), and statistical analysis of multiple samples of the velocity vectors for mean and turbulence variables. Some investigations of unsteady flows include descriptions of phase-averaging techniques for mean and turbulence variables. Detailed uncertainty assessment of PIV results with descriptions of specific UA methodology and tables of error sources has been outside of the scope of most studies, however, many investigators have reported bottom-line measurement uncertainties. Following is an overview of PIV post-processing techniques and an outline for application of the standards and guidelines of AIAA75 and ASME76 to PIV measurements. 4.1. Validation of the displacement vector maps The displacement vectors obtained from the PIV image evaluation procedure may include evaluation errors resulting from poor particle seeding, strong background noise, and improper choice of the evaluation parameters. Westerweel described a method that detected errors in the displacement vector at a given grid point by comparing its value to the mean of its neighboring vectors. Fujita and Kaizu78 applied a method to detect evaluation errors based on inviolable physics of fluid flow, i.e., continuity. A most simple and effective error detection method is the use of a median filter, however, there are many other filters such as range, average, and peak-height filters which can be used singularly or in groups for iterative filtering. When applying the median filter to a 2D data structure {Sx(i,j), Sy(i,j); i=\, 2,..., M; 7=1, 2,..., TV}, a median vector { S'x(i,j), S'y(i,j)} is determined in a (2r+l)x(2r+l) neighborhood such as S'x\i, j)=median\Sx(k,l), i -r
+ l, j — r
' . J Sy (i, j)= median\Sy (k,l),i-r
V + l, j - r
(7)

The validation criteria for vector {SJiJ), Sy(i,j)} is [Sx(i,j)-S'x(i,j)]2

+[sy(i,j)-S'y{i,j)]2

<e2 ,

(8)

134

J. Longo et al.

wherein e is the error tolerance. For consecutively recorded PIV recordings, a 3D median filter can be used to detect erroneous displacement vectors by considering the vector maps acquired before and after the current vector map, thus, 'time' is the third dimension in the 3D filter. The median vector of the 3D median vector is determined with equation (9) wherein '-' and '+' indicate that the vector maps are taken before and after the current vector map, respectively. Usually the 3x3 and 3x3x3 neighborhood, i.e. r=\, are chosen to determine the median vector of the 2D and 3D median filter, respectively. When a displacement vector, which is determined by the maximum of the cross-correlation function or the secondary maximum of the auto-correlation function, is detected as an erroneous vector, the position vectors at other maxima can be compared with the median vector using equation (8) to find a replacement. This procedure is called error correction. Error correction techniques can fail for some instances, such as when large blocks of grid points have null vectors resulting from insufficient seeding densities. In this case, the erroneous vector is simply removed without replacement.

median^'

(k,l), Sx (k,l),S^(k,l),i-r
+ l,j-r
+ r\ (9)

S'y(}j)= median^'

(k.l), Sy (k,l\Sy(k,l),i-r
+ l, j -r
j + r\

As an example, the 3D median filter was used to detect erroneous displacement vectors in the raw data of PIV measurements in the near wake of a surface-piercing flat plate. Details of the PIV system and test facilities were provided by Gui et al.44. In this test the carriage speed is 1.37 m/s, and 1022 pairs of single-exposed PIV recordings were acquired in several carriage runs. The most advanced evaluation technique, i.e. CDIC, was used to reduce the number of erroneous vectors, however, many could not be avoided in the high-turbulence region at the wake centerline, where the particle image pattern deformation is large. After the 3D median filter was applied, the percentage of the valid vectors in the total 1022 raw data samples were calculated and given in Fig. 6a. Near the trailing edge in the wake center, i.e. (x=0, y=0), nearly 20% of all samples at each grid point are detected as erroneous displacements and removed. 4.2. PIV data reduction After the error detection and correction procedure the valid particle image displacement vector S is used to further determine the velocity vector V. For the presentation, the subscript fc=l, 2, 3 indicates coordinates x, y, z, respectively, and

Ship Velocity Fields

It

lit «

M

Ml U

W

Valid vector percentage

*{ M

M

100.0

0-200

0*0

MOD

135 i ~n

B.TM

U/Ue

(b)

(a)

(c)

Fig. 6. Percentage of valid vectors (a) and mean velocity components (b), (c) in the wake near the trailing edge of a flat plate.

(a)

(c)

(b)

Fig. 7. Reynolds stress distributions in the wake near the trailing edge of a flat plate: (a) normal stress in x-direction, (b) normal stress in y-direction, (c) shear stress in the xy-plane. subscript i=\, 2 , 3,..., N represents the number of valid instantaneous measurement samples from a total number N. The data-reduction equation for deriving instantaneous velocity component Vki from instantaneous displacement component S^ is given as Vfcj = A• Sf,; with X-

-dbj

for k= 1,2,3; i = 1,2,---,N ,

(10)

Limg ' ^ where At is the time interval between exposures. The imaging magnification coefficient is usually determined by placing a ruler of length Lohj in the laser light sheet and reading the length in the digital image Limg. The mean velocity component Vk is determined by IN

IN

vk=-ivkii=x--zsk,i=x-sk, N i=i

N

i=J

(11)

136

J. Longo et at

wherein S* is the mean particle image displacement component. The mean square fluctuations of the instantaneous velocity components that usually quantify the normal Reynolds stresses of the turbulent flow are determined as ^

=^ f ( % - F , )

2

forfcl.2,3.

(12)

The shear stresses are obtained,with V™" = -L X (vMfi ~ Vm )• fyn4 - Vn ) for 111=1, 2, 3; ft=l, 2, 3; m*i.

(13)

In the flat plate example, the valid displacement components were used to determine the velocity components with equation (10), and mean velocity components (U, V) were computed with equation (11) and given in Fig. 6b and 6c, The Reynolds stresses are computed with equation (12) and (13) and shown in Fig. 7. Prior to the test program, several, measurements are taken with no model,..i.e., the uniform-flow condition, in order to establish the free-stream turbulence or noise level that is present and measurable with the towed PIV system. 4.3* Phase-averaged acquisition and analysis of PIV data Digital PIV systems enable phase-averaged measurements of. periodic flows. The most common applications are for propulsor flows (Cotroni et al.79; Di Felice et al.80; Judge et al.29'30; Calcagno et al.26; and Lee et al.42). Typically, shaft-mounted encoders supply trigger signals to the camera/laser synchronizer at pre-determined phase angles. An example of this type of experimental arrangement is shown in Fig. 8 for the investigation of a propeller wake using 3D stereo PIV in a cavitation tunnel

Fig. 8.

Schematic of experimental setup from Calcagno et al. .

Ship Velocity Fields

137

PIV recordings are acquired multiple times at pre-determined propeller phase angles. Statistical analysis of the propeller-induced flow is a matter of sorting the recordings on phase and computing mean and turbulence quantities grid-point by grid-point. For the case of Calcagno et al.26, 129 PIV recordings were taken at propeller phase angles between 0°-69° in 3° increments for measurement of the mean flowfield. Fig. 9 illustrates sample results of mean longitudinal velocity (U) at A/L=0.9997 and four phase angles. The detailed structures in U and the periodicity of the flow are evident. The authors were able to demonstrate measurement of coherent turbulence patterns with only 129 samples per phase angle.

Fig. 9. Longitudinal velocity U (m/s) at four phase angles (0°, 18°, 36°, 54°) for plane x/L=0.9997 (Calcagno et al.26). Other types of investigations require phase-averaged data acquisition and reduction. For instance, Longo et al.81 applied towed, digital PIV to study the forward-speed diffraction problem, i.e., measure the unsteady nominal wake of a restrained surface combatant advancing in regular headwaves. The data acquisition was phase-locked to wave measurements upstream of the model bow of a lowsteepness (Afc=0.025), long wavelength (/U4.572 m) incident wave as shown in Fig. 10. For this experimental setup, external triggering of the laser and camera was unnecessary because the incident wave height (and therefore, wave phase) was sampled simultaneously with each PIV recording. Phase angles were assigned to each PIV recording with equations (14) and (15) 7i,j=-ri,}+j2jc-^27t,

(14)

138

/. Longo et al. 1

(15)

fe ~2id

X

where YIJ is the lst»hamonic phase angle of the/-incident wave, D is the distance between the servo wave gage and middle of the PIV measurement area, 1 is the wavelength of the incident wave, tt is the time stamp of the ith PIV recording, Te is the encounter period, g is gravity, and Uc is the towing speed (see Fig. 10).

Servo- r Wavsmaker wave J —v probe I k..f. incident \ v

fUJ

-_;!"-#:**,.

*f s

......... I

PIV 2000 processor

Ur

V1

',

^m^^mr^m^

TV-

Measurement area--*

X= 4.572m, Ak = 0.025

(a)

(b)

Fig. 10. Experimental arrangement (a) and coordinate system (b) for phaseaveraged measurements of the unsteady nominal wake plane of a surface combatant in regular headwaves (Longo et al.81). 0.15

;

Sorted, unStsjsxJ; {+) 1 sirarajsIkdii', (-)1
w

-0.05

180

0.10

0.05

5?%-,

Pk .^1

-0.10

Y<°)

0.15 Istranga-fiSsrad '

— (-) 2nd range Stag'

3 0.00

-

fcz

0.10 hi

-0.15 ' 0

(c)

'--<*& 1

180

1

0.05 h 3 0.00 •0.05 -0.10 -0.15

(d)

0

180 YO

Fig. 11. Unsteady PIV data and data-reduction processes: (a) unsorted, unfiltered data; (b) phase-sorted, unfiltered data and lst-stage range-filter limits; (c) lst-stage range-filtered data and 2nd-stage range-filter limits; (d) range- and median-filtered data, least-squares average fit, and FS (Longo et al.81).

Ship Velocity Fields

139

Data was phase-sorted after ten carriage runs or 2000 PIV recordings (~5 recordings per 1° of phase angle) and analyzed on a grid-point by grid-point basis with FFT and Fourier series (FS) routines. Here, fewer PIV recordings are sampled per phase angle in comparison with Calcagno et al.26, but since the flow is periodic, enough recordings are acquired in the encounter period such that the harmonic content in the mean and turbulent flow are recoverable. Fig. 11 shows typical results of axial velocity ([/) through one encounter period for a random grid point at the nominal wake plane of model 5512. After range and median filtering the data, FS analysis provides the harmonic content (amplitudes and phases) in U. 4.4. PIV uncertainty assessment Assessment of quality of experimental fluid dynamics (EFD) results is imperative. Experimental uncertainty assessment has a long history dating to the early 1950's (Kline and McClintock82). Significant progress in the field has been accomplished in the previous two decades resulting in formalization (Coleman and Steele83) and then adoption of internationally recognized standards and guidelines (ASME76; AIAA75) with emphasis on simplification and practical application. Consistent and uniform use both in research and design at university, industry, and government laboratories continues to be a problem. As with fields of mechanical and aerospace engineering, ship hydrodynamics has also struggled to rectify this situation. Recent efforts include adoption by the 22 nd International Towing Tank Conference (ITTC) of the most recent standards and cooperative UA example for resistance test performed by most members of the Resistance Committee (ITTC84). Most studies herein (60%) do not report uncertainties while from the remaining 40%, only a few report uncertainties following established UA standards and guidelines. Explanation for the lack of emphasis on uncertainty assessment is difficult. In many cases, rigorous UA was substituted for order-of-magnitude analysis of the major PIV error sources. Nevertheless, measurement uncertainties are summarized chronologically and by flow group in Table 1 for studies reporting values. Table 1 includes entries for instantaneous velocity (0.6-10%), mean velocity (0.3-7.7%), turbulence (1.3-6.0%), and vorticity (3.9-27%). Most uncertainties are derived from estimates of the bias in the sub-pixel accuracy of the evaluation term (Ski) in equation (10). Clearly, there are other terms in equation (10) that contain sources of error, i.e., Lobj, Limg, and At. Additionally, spatial uncertainties in measurement positions, model geometry inaccuracies, and other factors (conceptual biases) can be significant but are not considered for most studies. Measurement variability (precision errors) observed through several measurement cycles for the same test conditions are another source of error contributing to total uncertainty, but again, are not considered for most studies. In general, the reported PIV values compare favorably with measurement uncertainties from other investigations

140

J. Longo et al.

employing multi-hole pitot (Toda et al.1; Kim et al.2; and Longo and Stern3) and Laser-Doppler probes (Fry and Kim5; Hoekstra and Ligtelijn6; Chesnakas and H

A

O

Q1 O f Q £

Jessup ; Longo et al. ). However, investigations of Gui et al. and Longo et al. ' which applied the latest measurement uncertainty standards and guidelines report generally higher mean and turbulence uncertainties than most studies in this review, no doubt due to a more comprehensive accounting of error sources. Detailed comparisons between PIV and five-hole pitot uncertainties for measurements in the nominal wake plane of a surface combatant (Gui et al.4) revealed lower uncertainties for the PIV technique by roughly 0.5-2.0%. For a few studies herein, the test design included rigorous UA of the PIV measurement results through application of the most recent UA standards and guidelines. To aid others in the application of the standards and guidelines, a general implementation of the 95% confidence, large-sample UA methodology recommended by the AIAA75 to PIV measurements is described in the following sections. The original work was presented by Gui et al.4 for steady mean and turbulent PIV measurements of the nominal wake of a surface combatant. Subsequently, Longo et al.81,85'86 extended the procedures for forced unsteady mean and turbulent PIV measurements of the nominal wake of a surface combatant. 4.4.1. Bias limits of mean velocity components According to data-reduction equations (10) and (11) and considering that the measured mean-velocity components are spatially dependent, Vk can be written as Vk = Vk [Lobj. Limg - M Sk, x, y, z).

(16)

The bias limit of Vk is determined with a root-sum-square (RSS) of the elementary bias limits, i.e. B

Vk=^9LobjliLobj+()LimgBLimg+9AtBAt+(>SkBSk+QxBx+QyBy+QzBz

'

(^)

where the sensitivity coefficients (6x) are defined as Bx = f p

X={Lobj,Limg,At,Sk,x,y,z).

(18)

For the first four variables in equation (17), 6X is evaluated analytically with equations (10) and (11). The remaining sensitivity coefficients for (x, y, z) are evaluated numerically after the final mean-flow results are computed. Bx (for X=L<^ Limg, At< $b x, y, z) are elementary bias limits.

Ship Velocity Fields

141

4.4.2. Bias limits of Reynolds stresses The bias limits of the normal and shear stresses are derived with statistical analyses. The measured instantaneous velocity Vki is the sum of the true value V°kh the bias error ft,, and the random error £fc„ i.e., n,-=V£+/5w+ew.

(19)

The measured and true mean velocity components are defined as IN

7 N

Vk=—ZVk;,VZ=—ZVh. (20) N i=i N i=1 If N is large enough, the sum of the random errors can be neglected, and the bias error of the mean velocity component equals the mean of the instantaneous bias error Pk=Vk-Vk°~-±-I.Bki.

(21)

N i=i

The mean bias error /3k can be shown to be a function of the true mean velocity component Vk. Therefore, a bias gradient can be defined as * * = % dVk°

(22)

The instantaneous bias error can approximately be determined by Pk4~Pk+*kkk°i-Vk)-

(23)

The measured and true normal stresses are defined as v*v* = - Z Wki ~ Vk J1 • v M = - I K i -Vk°J,

(24)

respectively, and are related by

^k=0+r k?W + ^ ^ h*M, -VkVjjlfl

•

(25)

The second term on the left side of equation (25) is a sum of the product of two independent random values, and can be neglected which produces vkvk=(l

+ TkYv°kv°k+e2k-

(26)

J. Longo et al.

142

Here, ek is the RMS value of the random error ekJ. The bias error for the normal stress can then be determined as •vkVk-vM

V*Vfc

VkVk +ej

=vkvk

^k^^k+^k

(l + r k f

(l + rkf

(27)

The bias error of the shear stress can also be derived in the same way as

P n

=

1-

v_._v__

1 (7+TJ-(;+T„)

km+*n)vmvn

—^

"

(7 + T m )-(7 + T„)"

(28)

Thus, the bias limits of the Reynolds stresses resulting from the PIV data reduction can be determined as

3

2Tkvkvk+ek

m

tt

Vf\ VmV "\

(29)

+ VV+M where Tk (k=l, 2, 3) is the limit of the bias gradient of the mean velocity component. According to the data reduction equations, x k can be determined with x

(30)

k ~~;~ +

The bias limit of A is determined by 'Lobj

X

V

L

obj

(B Limg

\2 (31)

At

-l"ng

The bias limit of the mean displacement depends on the evaluation methods and can be estimated with simulations (e.g., Gui et al.44). When the bias sources of the coordinate deviations are included, the bias limits of the Reynolds stresses are finally determined by

B2—=BL+lars,

J+(?;nBy J+(B™BZ

J,

(32)

where m=n-k for the normal stresses and rnfri for the shear stresses. The sensitivity coefficients for the coordinate deviation are ,mn _

^VmVn

dx

r,mn

®vmvn

amn

=

dvmvn

(33)

143

Ship Velocity Fields

for m=\, 2, 3 and n=\, 2, 3, and they are evaluated numerically after the final turbulence results have been computed. 4.4.3. Precision limits For a single test, the precision limit of a measured variable X is given by Px=Kax,

(34)

where K is the coverage factor and equals 2 for a 95% confidence level, ax is the standard deviation of the sample of N readings of the variable X and is defined as

ax=

\ihl\Xk-x^

(35)

X is the mean value and its precision limit is given by ^

=

^

,

(36)

4.4.4. Total uncertainty The total uncertainty (Ux) is defined as the RSS of the bias and precision limits, i.e.,

Ux=M+px

•

(37>

For PIV experiments 'X' in equations (34), (35), (36), and (37) can be replaced with mean velocity components and Reynolds stresses. 5. Application of PIV in ship velocity fields Ship hydrodynamics has many unique features in comparison to related applications due large Reynolds number (Re) ~ 109; small Mach number (incompressible flow); tanker, cargo/container, and combatant geometries; ballast, motions, maneuvering, restricted water, and ambient waves operating and environmental conditions; Froude number (Fr) and free-surface effects (waves, spray, breaking, near-surface turbulence, and boundary-layer and wake and vortex interactions); and propulsorbody interactions and cavitation. Detailed physics vary considerably depending on geometry and operating and environmental conditions. These features present grand challenges for theoretical, computational, and experimental methods for ship design. Theoretical methods are largely limited to simplified geometry and conditions and linear assumptions such that practical applications rely heavily on empirical data supported by ship model testing, which is subject to scale effects. CFD methods are

144

J. Longo et al.

fast developing and will ultimately provide simulation-based design; however, their development requires experimental data for model development and validation. Application of PIV to ship velocity fields is undertaken in support of design through explication of physics, model development, and CFD validation, as discussed in following sections, which for convenience of discussion are organized into topics of: boundary layer and wake without free surface; boundary layer and wake with free surface; bow flow; wave hydrodynamics; propulsor hydrodynamics; appendages; and cavitation and bubbly flows. Discussions include PIV system; physics; geometry and conditions; facilities; data acquisition, reduction, and locations; uncertainty assessment; and results. 5.1. Boundary layer and wake flows without free surface Shekarriz et al.34 apply double-exposed, film-based PIV with a 6-W argon laser and the particle image tracking method to study tip, wake, and junction vortex flow from a 2.72-m submarine model with a 23.2-cm chord length sail. The experiments are conducted in a 3(W)xl.5(D)x43(L)-m3 towing tank at the David Taylor Research Center. A schematic diagram (Fig. 12) of the experimental setup highlights the model, PIV system, and light-delivery method. Argon-ion Lasen

B

cM

Dia

^ b r a g m Spherical Lens Mirror

3—*HH£-^r--

Under Water Housing

Cylindrical Lens

Mirror

Fig. 12. Schematic of experimental setup from Shekarriz et al. The Reynolds number is /?e=9xl0 based on model length. Instantaneous vector maps in the crossplane are initiated as the carriage passes an electronic eye and recorded at x/c=0.44, 1.42, 2.41, 5.37, and 7.34 for the junction and wake vortices and x/c=2.%9, 3.89, and 6.89 for the tip vortex where x/c=0 is the trailing edge of the sail. The dataset includes 3-25 instantaneous vector fields for each crossplane from

145

Ship Velocity Fields

which vorticity and circulation are computed. Circulations are computed for each Yortex system with line integration of one or more vector fields. Fig. 13a shows a sample photograph of the tip vortex particle traces downstream of the sail and the relative circulation strengths of the three vortex systems. 1.00

0.10

0.01 0.001

-O— Tip vortex at 5 skta slip angie, (x/c = 2.89) -G - Junction vortex at 0° inddenca, (x/c = 2.41)' • A- - Wake vortex at 0° tnddsnee, (x/c = 3.40) Re = 7.2x10" ,

0.010

0.100

1.00

2

(Area)/(Thickness)

(a)

0>)

Fig. 13. Eaw image of vortex patterns at A/C=2.89 (a) and circulation values for three vortex systems (b) from Shekarriz et al.34. Analysis of results indicate that the junction vortex meanders in a direction parallel to the body, detaches with increasing x/c, and loses circulation strength with increasing x/c. The wake vortices are comparable in strength to the junction vortex. Circulation of the tip-vortex system is mostly unchanged over four chord lengths downstream of the sail with peak values nearly 12 times greater than for the junction vortex (Fig. 13b). The tip vortex circulation patterns follow the i"2 (solid-body rotation) and log r profiles for the inner and outer sections of the vortex core similarly as theoretical predictions by Hoffman and Joubert88. Shekarriz et al.87 use the same PIV measurement system and facility for study of near-field tip vortex flow from a 21.9-cm chord length sail attached to a 2.61-m submarine body. The experiments are conducted for a range of Re (3.6x10 » 7.2xl04, 2.2xl05) and angle of attack (5°, 10°) where the latter are achieved by yawing the model The dataset includes multiple instantaneous vector fields for a range of crossplanes from which circulation is computed. Results indicate substantial tip-vortex meandering in the spanwise (y) coordinate in comparison to the normal (z) coordinate. Overall circulation of the tip vortex remains constant through the range of x/c studied, i.e., 0<J/C<6.7. For a=5°, the gradient of the circulation increases rapidly with Re within the core of the tip vortex but much slower outside of the core (Fig. 14).

146

J. Longo et al. 0.10

- I

I

1

1

1

l__

1

1

1

1

1

1

0.06

l

-/s w 0.00 0.04

0.02

-

1

1

1

1

0.00

T

1

1

1

1

1

1

1

-

a = 5° -. x/c = 3.01 = __.- - Rec = 3.6x10 4 : — Rec = 7.2x10 ~= — Re c =2.2x10 5 ~-

0.08

1

1

1

1

1

0.02

1

1

1

1

1

1

1

1

I

1

0.04 A* 1

1

1

1

1

i

i

0.06

i

i

=

0.08

(a) 1

1

1

1

1

1

|

i— r •

a =10° x/c = 3.01 - - Rec = 3.6x10 4 -; - Re, = 7.2x10 4 5 Re, =2.2x10 0.06

0.08

(b) Fig. 14. Mean circulation curves at different Reynolds numbers and two angle of attack: (a) o=5°; (b) o=10° from Shekarriz et al.87. The same trends are not evident for a=10° which is probably due to transitional flow for the lower Re. The overall circulation from the average of multiple PIV recordings scales with Re and a. Measurements indicate expansion of the tip vortex due to adverse pressure gradients at the stern where contraction of the model occurs. Fu et al.31 utilize the same facility, measurement system, and PIV evaluation method as Shekarriz et al.34'87 for investigation of flow structures in the lee of an inclined 6:1 prolate spheroid as a function of boundary layer tripping (tripped versus untripped), model inclination (10°, 20°), and Re (0.42-2.1xl06). The dataset includes multiple instantaneous vector fields for several inclined crossplanes from which vorticity and circulation are computed. For both tripped and untripped cases, the lee-side flow is characterized as a region of open separation containing primary and secondary vortices that increase in size and strength with increasing axial location and incidence angle. Vorticity strength is not proportional to the incidence angle as is the case for most lifting surfaces. Fig. 15 is a qualitative description of the wake structure in the lee of the prolate spheroid inferred from PIV measurements and emphasizes vorticity sheets, counter-rotating secondary structures, and that a large portion of the vorticity is distributed outside of clearly defined vortices. Trippedflow data reveal a wider area of vorticity, increased number of secondary vortices, and higher fluctuation of circulation and forces acting on the body. Vorticity fields are computed from instantaneous vector maps and used to estimate lift and side forces and roll moment acting on the spheroid through application of the KuttaJoukowski theorem and integration of the crossplane vorticity fields circumferentially and radially. The estimated forces and moment from the PIV

Ship Velocity Fields

Vortex filament

Sting,.

x A'

Vorticity sheet \

JQ^J^^

^

Fig. 15. Qualitative description of the wake structure in the lee of a prolate spheroid inferred from PIV measurements from Fu et al.31. recordings show good agreement with directly measured values using force and moment block gages. Xu et al.89 also applies multiple-exposed film-based PIV and the autocorrelation evaluation method but with a 650 ml Nd:YAG laser for investigation of the flow around a submerged hydrofoil in a water channel. The focus of the study is measurement system development through an improved evaluation method for error reduction from particle image distortion. Fu et al.90 apply double-exposed digital PIV with a 120 ml Nd:YAG laser and the auto-correlation algorithm to study the flowfield of a 5.18-m submarine model no. 5484 (ONR Body-1) in a steady turn. The model has a NACA0014 sail and four identical NACA0018 stem appendages. The experiments are conducted in a 79.2-m diameter, 6.1-m deep rotating arm basin at the Naval Surface Warfare Center. Schematic diagrams (Fig. 16a3b) of the experimental setup highlight the rotating-arm facility, model, PIV system, and Eght-delivery method. i .iaser Sheet |

""pp Optics

Housings ONR Body-1

(a)

Laser "Sheet

(b)

Fig. 16. Schematics of experimental setup from Fu et al.90.

148

J. Longo et at

The model is towed at an angular velocity of r=0.163 rad/sec and a linear speed of 1.60 m/sec or Jte=8.3xl06 based on model length. The dataset includes 3-25 instantaneous vector fields for each crossplane from which vorticity and circulation are computed. Data acquisition of instantaneous crossplane velocities is initiated from a carriage-mounted photoeye and acquired at twenty axial stations along the body and through the near wake for fifty carriage runs. Fig. 17 illustrates crossplane vectors and contours of vorticity (looking aft) at four planes behind the sail

Y {IW|

Y Urn

Fig. 17. Velocity vectors and vorticity contours at planes 7, 8, 9, and 10 from Fu etaL90. The data captures features of a tip vortex detaching from the sail and moving laterally off of the centerplane toward the starboard -side of 'the model with increasing distance downstream. Positive (y=0.1 m, z=-0.24 m) and negative (y=0.33 m» z=~0.06 m) cells of vorticity near the body identify regions of flow separation commonly observed in bluff bodies in a crossflow. A low-vorticity region exists between the two cells. Force and moment measurements with block gages are used to aid explanation of the flow physics. Interaction of the sail trailing

Ship Velocity Fields

149

Yortex and the body produce negative vorticity flux which rotates the separation points clockwise toward the deck of the model while inducing a net upward force aft of the center of gravity and a downward pitching moment. Mechanisms leading to the observed forces and moments are typical of dominant cross-flow separation induced by a submarine in a turn. 5«2® B&mmdafy layer ami wakefl&w withfreesurface Grant et al.25,33 apply multiple-exposed, film-based PIV with a Nd:YAG laser to investigate the turbulent wake of a 1.2-m surface combatant with twin propellers. The tests are conducted in the Heriot-Watt University 25-m towing tank with a stationary PIV system for Fr=029 (ite=1.2xl06). PIV recordings are initiated from a wall-mounted ultrasonic transmitter/sensor pair and acquired in the vertical centerplane, one ship length aft of the stern and into the near wake. The dataset includes 35-50 instantaneous vector fields from which average vector fields are computed. The study focuses on development and the effectiveness of a novel concept for tagging PIV images with high- and low-intensity laser light to eliminate directional ambiguities in the flow during the evaluation phase of the data reduction. Pogozelski et al.91'92 apply multiple-exposed, film-based PIV with a 15-W copper vapor laser and the auto correlation evaluation method to investigate the instantaneous flow field of a large-draft, 0.914-m chord length, surface-piercing, wing-like body. The tests are done in the Naval Surface Warfare Center 3(W)xl.5(D)x43(L)-m3 towing .tank with a stationary PIV system at a. range of F^0.051-0.51 (Jfe=0.139~1.39xl06). A schematic diagram (Fig. 18) of the experimental setup highMghts the model, PIV system, light-delivery system, and measurement location.

Fig. 18. Schematic of experimental setup from Pogozelski et al."

150

J. Longo et ah

PIV recordings are acquired' at nearly two-dozen axial stations between the leading and trailing edges, of the model with emphasis on the bow- and shoulder-wave structures and boundary layer separation at Fr=0.255, a conditionforwhich there is no reverse flow within -the separated region. Instantaneous vector maps are used to compute vorticity fields. At the bow (J&C=0.22), the PIV data identifies considerable vorticity at the toe of the wave that may be associated with the formation of a 'necklace' vortex upstream of the leading -edge. Other instantaneous images along the backward face of the bow wave resolve the comer-flow details at the free surface-model juncture including significant negative vorticity in this region. Measurements in the shoulder-wave region reveal the unsteady nature of the flow and include multiple large-scale vorticity structures of opposite sign that entrain bubbles into the flow and trap them. The vorticity tends to be positive near the body and negative in the region of the shoulder-wave toe. Flow separation occurs near the free surface 70% from the leading edge. Sample PIV results at x/L=0.71 are shown in Fig. 19. 0.02

0.02

Uo=0.76 m/sec

z'/L

-0.18 L-^—i-^-J.

0.15 0.18 0.21 0.24 0.27 0.30 0.33 0.36 y'/L

(a)

-0.18 0.15 0.18 0.21 0.24 0.27 0.30 0.33 0.36 y'/L

(b)

Fig. 19. Velocity vectors (a) and vorticity contours (b) utx/L=0Jt from Pogozelski 91,92 etal. This station and other downstream stations contain several large-scale vortices with peak vorticity values that are 4-10 times greater than for the regions at the bow and shoulder wave. Considerable negative (shaded contours) and positive vorticity develops near the free surface and body, respectively. Combined quantitative data from the PIV measurements and video records of the flow are used to develop topographical sketches of the flow including demarcations for bow and shoulder waves, wave-breaking locations, separated-flow region, and topographies of node and saddle points within the separated-flow zone. The data is used to compare with numerical results of Zhang and Stem93. Comparisons show several similarities in the necklace vortex formation and wave profile near the trailing edge but the

Ship Velocity Fields

151

computations predict reverse flow for Fr<0.255 which is not observed in the experiments. Jaw and Kehr32 apply multiple-exposed, digital PIV with a 4-W argon laser to investigate the turbulent wake of a 0.55-m container model. The tests are performed in a 1.1-m circulating water tunnel with a stationary PIV system for Fr=0.26 (i?e=3.3xl05). Multiple instantaneous vector fields are acquired from the bow to the stern. The study focuses on development and the effectiveness of a new alternating color image anemometry (ACIA) technique for PIV image acquisition and evaluation. Zhang et al.38 apply multiple-exposed, digital PIV with a 5-W copper vapor laser and the auto correlation evaluation method to investigate the flow around a propeller-shaft bracket on a 4.2 m ship model with a 5-bladed propeller. The tests are conducted in the National Laboratory of Hydrodynamics 7.5(W)x4.5(D)xl50(L)-m3 towing tank with stationary and towed PIV systems. The study focuses on demonstrating the feasibility of applying towed or stationary PIV to ship hydrodynamics flows in a large facility. Di Felice and De Gregorio28 apply single-exposed, digital PIV with a 320 mJ Nd:YAG laser and the cross-correlation evaluation method to investigate the turbulent wake of a 5.41-m model ship with two 4-bladed 183-mm propellers. The tests were done in the 3.6(W)x2.3(D)xl0.0(L)-m3 INSEAN Circulating Water Channel with a stationary PIV system for Fr=0, 0.32 (Re=0, I.2xl07) and two propeller advance coefficients 7=0, 1.1. PIV recordings are synchronized with the propeller shaft rotation and acquired aft of the starboard propeller in the vertical, longitudinal plane of symmetry. Phase-averaged velocities and vorticity are computed from 30-35 image pairs at phase angles between 0°-90° in 10° increments. Instantaneous and phase-averaged results show the position and evolution of the tip vortex pattern. The tip vortices are subjected to strong diffusion effects and their trajectory moves upward as the slipstream contracts. Perturbation velocity and vorticity distributions indicate that the higher propeller loading generates larger tip vortex intensities. The detached flow and wake defect from the blunt end of the propeller hub and its interaction with the vorticity released from the blades is captured in the measurements. Gui et al.4 apply single-exposed, digital PIV with a 20 mJ NdrYAG laser to investigate the mean and turbulent nominal wake of a 3.048-m surface combatant (model 5512). The tests are performed in the IIHR 3.048(W)x3.048(D)xl00(L)-m3 towing tank with a worlds-first towed PIV measurement system at one Fr=0.28 (/?e=5.4xl06). A schematic diagram (Fig. 20) of the experimental setup highlights the model and trailer, PIV system, light-delivery system, and measurement location. The 3D nominal-wake flowfield (x/L=0.935) is constructed from measurements at several overlapping vertical and horizontal lightsheet positions. At each position,

J. Longo et al.

Fig. 20. Schematic diagram of experimental setup from Gui et al.4. 1200 instantaneous vector maps are acquired from which mean and turbulence statistics are computed as well as axial vorticity. Sample results are shown in Fig. 21 for three mean variables and five Reynolds stresses. Axial velocity contours show a low-velocity bulge (U ~ 65%UC ; Uc is the towing speed) near mid girth and relatively thin boundary layer near the center plane. Cross plane velocity vectors are generally upward and toward the centerplane with a region of weak out-board rotation in the region of the bulge and thin boundary layer. The normal-stress patterns correlate with the axial velocity but with largest values compressed closer to the hull. The turbulence is anisotropic since the axial stress is nearly twice the cross-plane stresses with largest values -Juu ~ 6%UC, yjvv ~ 4%UC, and V ww ~ 4%UC . The Reynolds shear stress wv is negative in regions of increasing dU/dy and positive in regions of decreasing 3t//3ywith largest values where gradient is largest (V«v = 3%UC). uw is similar but correlates with dU /dz with largest value V"w ~ 3%UC . The data is acquired with sufficient spatial density such that it is useful for revealing the underlying physics of the mean and turbulent nominal wake and as a test case for CFD validation. The same measurement system, facility, model, and test conditions (Fr=0.28; Re=5Axl06) are used in a subsequent study of the forward-speed diffraction problem, i.e., restrained model advancing in incident headwaves, through detailed measurement of the incident headwave flowfield (Longo et al.85'86) and then through measurement of the unsteady nominal wake plane (Longo et al.81). The incident headwave is a long wavelength (A=1.5L=4.572 m), low steepness (A&=0.025) regular wave and is chosen because this condition produces the most manageable linear response, especially, in the farfield region. Additionally, this condition enables comparison of EFD and CFD results since many free-surface solvers have difficulty with medium- and high-steepness waves that overturn and break. The phase-averaging techniques are described in section 4.3.

Ship Velocity Fields

-0.05

-0.04

-0.03

-0.02 -0.01

0.00

-0.05 -0.04 -0.03 -0.02 -0.0! y

y

(a) VW vectors and U contours

-0.05

-0.04

-0.03 -0.02 -0.01 y

153

0.00

(d) uu contours

0.00

-0.05 -0.04 -0.03 -0.02 -0.01 y

(b) uv contours

0.00

(e) w contours 0.00 -0.01

_,..., ,

.,.„,

T

.^_,

r-j-,

v0.0015 0.0014 0.0012 0.0010 0.0009 0.0008 0.0006 0.0004 0.0003 0.0001 0.0000

i^Sr^s

-0.02 N

-G.03 -0.04 -0.05

-0.04 -0.03 -0.02 -0.01 y

0.00

(c) uw contours

-0 05

-0.04 •0.03 -0.02

-0.01

0. 90

y

(f) ww contours

Fig. 21. Mean and turbulence data for the nomirial wake of model 5'512 from Gui etal.4. As per the steady case, the 3D unsteady nominal-wake flowfield (x/L=0.935) is constructed from measurements at several overlapping vertical and horizontal lightsheet positions. At each position, 2000 instantaneous vector maps are acquired from which Fourier series harmonic content for the mean and turbulence variables are computed. The UA methodology of section 4.4 was extended for unsteady PIV and applied to the unsteady nominal wake measurements. Sample results are shown in Fig. 22 for 1st- (UJf Vh Wj) and 2nd-harmonic amplitude (U2, F2s W2) of the mean velocities and reveal the periodic nature of the unsteady nominal wake.

J. Longo et ah

154

| I | I I j I i j

0.025 0.022 0.020 0.018 0.015 0.013 0.010 0.007 0.005 0.003 I 0,000

0.G7 0.06 0.05 0.04 0.03 0.02 0.01 0.00

(a) Uj contours

(d) I]2 contours i I | i I j i i | I I

st

(b) Vj contours

(e) V2 contours

(c) Wj contours

(f) W2 contours

0.025 0.022 0.020 0.018 0.015 0.013 0.010 0.007 0.005 0.003 0.000

nd

Fig. 22. 1 - and 2 -harmonic amplitude data for model 5512 in regular faeadwwes from Longo etal.81. Uj peaks at roughly 9% (of Uc) at the center of the low-momentum bulge. Vlf Wj are 3.5% near the free surface but small close to the centerplane. The 2nd-harmoiiic content is mostly insignificant except for I]2 at the center of the low-momentum bulge. The scope and detail of the tests make the results useful for detailed study of forced unsteady free surface flow physics and as a test case for CFD validation.

Ship Velocity Fields

155

53. Bowflow Dong et al.94,95 apply multiple-exposed, film-based PIV with, a 15-W copper vapor laser and the auto-correlation algorithm to investigate the structure of bow waves from a 3.05-m ship model (model no. 4817), The model has a fine bow and transom stem. The experiments were conducted in the David Taylor Research Center 3(W)xl.5(D)x43(L)-m3 towing tank with a 2D, stationary PIV system at two Fr=02$ (J?e=4.7xl06) and 0.45 (J?e=7.5xl06). A schematic diagram (Fig. 23) highlights the model, PIV system hardware, light-delivery system, and measurement locations.

(a)

(b)

Fig. 23. Model (a) and schematic of experimental setup (b) for Dong et al.94,95. Data is acquired at multiple axial stations including x^L=G.04, 0.1, 0.14, 0.196, 0.27 for FM).28 and xfcO.01, 0.08, 0.15, 0.24 for Fr=0.45. A carriage-mounted encoder was used to synchronize the laser pulses and camera firing with the location of the ship model. The dataset includes instantaneous vector fields for each crossplane from which vorticity is computed. The PIV results reveal the detailed structures within the bow wave at various axial and transverse locations for mild (Fr=0J2$) and steep (Fr=0.45) waves. Fig. 24 illustrates sample results for crossplane vectors and vorticity within the bow wave at Fi^0.28. For both wave cases, considerable negative vorticity is entrained into the flow at the toe of the bow wave. For the former, the vorticity originates at the free surface and for the latter, the boundary layer separates on the model and transports vorticity into the bow wave and close to the hull where it persists downstream of the bow wave. The momentum equation is used to derive the acceleration component tangent to the free surface of the bow wave. Through calculation of this variable, it is shown that peaks intikenear-surface vorticity occur in repons with high viscous flux of vorticity from the surface.

/. Longo et al.

156 &*.

*'„,,

lefts' -9

'""^ O

<§&Mjfr~

L •0.0 Mi

0.041

IJ .041

(b)

(a)

Fig. 24. Velocity (a) and vorticity (b) distributions at JK/L=0.10 and Fr=0.28 from Dongetal.94'95. H H "•'•_

;^t!;»r^^fe

0 015

o.ot§

ffl

dk

L

<XC8? 0.0&S O.G0S

IE

§.010

0.010

1 0 005

-

0 000

" I "': " ^ .1 . . "' '-„ "-. - ,- i "TX-""iI-:"::

**:; y~ .\ - , * - - „ - ~ 7 T i l -i.":::.:>:: ~ vr>.-\"*

0.005

•/ - v'v"-.!:-:""""'"-"* Dim

•QMS

-0.005

™JLa-ULu»i—l^-^JU^^^A^^.J^. G015 0 020 0.025 0 030 0.035 0.040

(a)

i • t . • , tJU. 0.015 0.020

_

0.025 0.039 J *

0.035

0.040

(b)

Fig. 25. Mean crossplane vectors (a) and w' /Ushi turbulence (b) distributions at x/L=QM90 and F^=0.30 from Roth et al96'97. Both'(a) and (b) also show the mean free surface shape measured from PIV images. Roth et al.96,97 apply single-exposed, digital PIV with a 27 mJ Nd:YAG laser and the cross-correlation algorithm to measure the flow structure and turbulence within a 7.01-m ship model (model no. 5422). The model has a bow sonar dome and transom stem. The experimental setup and procedures are very similar with those of Dong et al.94,95 in terms of facility,' particles, light delivery to the measurement region, and triggering of data acquisition. Differences include model, Fr, laser, and camera. The tests were done at one Fr=0.30 (/te=1.6xl07). Data is acquired at multiple axial stations between a location upstream of the bow stem and jt/L=0.1, however, the focus is on results at a single crossplane (x/L=O.069O). The dataset includes 92 instantaneous vector fields for each crossplane from which mean and turbulent statistics are computed. The PIV results reveal the detailed'structures of the bow wave at various axial and transverse locations for Fr=0.30 including free-

Ship Velocity Fields

157

surface shape and the mean and turbulent flowfield. Fig. 25 illustrates sample results for mean velocity vectors and the vertical normal stress as well as the mean free surface shape at a single crossplane. As x/L increases, vector maps show a growing region of negative vorticity or shear layer originating at the toe of the wave and then penetrating into the wave before curling upward and remaining close to the forward face. Coincident with this region is concentrated distributions of normal and shear Reynolds stresses whose magnitudes are consistent with theoretical predictions. This experiment expands on the scope of Dong et al.94'95 through investigation of bow-wave turbulence, turbulence production, and Reynolds number effect. Significant efforts were also made to investigate convergence of mean and turbulence variables and effects of interrogation window size on turbulence structure and statistics. 5.4. Wave hydrodynamics Coakley and Duncan98 apply film-based PIV with a 200 mJ Nd:YAG laser and the auto correlation evaluation method to investigate the wave-breaking region induced by a towed, submerged hydrofoil in close proximity to the free surface. The tests are performed in a 1.2(W)xl(D)xl5(L)-m3 wave tank with a stationary PIV measurement system. Although a range of chord lengths and foil depths are tested, results are presented for one foil chord length (c=15 cm) and one foil depth (d/c=l.02). The foil towing speed for the 15-cm version corresponds to Fr=0.571. The dataset includes 10 instantaneous vector fields from which average values are computed. The results clearly reveal a thin shear layer above the underlying flow at the forward face of the breaking wave. The mean shear layer is thin at the toe of the wave and increases with thickness with increasing distance downstream. The authors have presented less than 10% of the data so the remainder may have enough information to derive turbulence statistics and increase understanding of the fundamental physics of breaking waves. Dabiri and Gharib" apply single-exposed, digital PIV with the cross correlation evaluation method to investigate flowfield structures in a 2D spilling breaking wave. The tests are conducted in the 0.152(W)x0.152(D)x0.61(L)-m3 Caltech water tunnel with a stationary PIV measurement system. A schematic diagram (Fig. 26) of the experimental setup highlights the water tunnel, PIV system hardware, light-delivery method, and measurement regions. Two cases are studied: (1) the first focuses on a breaking wave with small capillary waves upstream of the breaking region; and (2) the second focuses on a capillary-gravity wave with no breaking. The Froude and Reynolds numbers based on breaker and wave heights, respectively, are (Fr=2.04, 7370) and (Re=l.62, 1050). The PIV measurements are acquired in three and two overlapping sections for piecewise reconstruction of the area of interest for cases one and two, respectively. Measurements resolve the vector and vorticity fields near

158

J. Longo et al

the free surface for both cases and reveal that the dominant source of vorticity flux is from decelerating free-surface fluid in the breaking region. The capillary waves are not a source of significant vorticity flux. Sample vector and vorticity results from section 25 case 1 are shown in Fig. 27. The vectors identify a mixing layer accompanied by strong positive vorticity that transitions into a shear layer with increasing distance downstream.

Fig. 26. Schematic of experimental setup from Dabiri and Gharib . J 40 cm s*"1

I

x (cm)

'

|

'

x (cm)

(a) (b) Fig. 27. Vector and vorticity fields from section 2 and case 1 from Dabiri .and Gharib". Chang and Liu100'101 apply digital PIV with a 200 mJ Nd:YAG laser to investigate the mean and turbulent structures in a 2D spilling breaking wave. The tests are done in the Cornell University 0.6(W)x0.9(D)x30(L)»m3 wave tank with a stationary PIV measurement system. The breaking-wave height, wavelength, and phase velocity are 14.5 cm, 1.21 m, and 1.21 m/s, respectively. A schematic diagram (Fig. 28) of the experimental setup highlights the wave tank and wavemaker, PIV system hardware, and light-delivery method. Ensemble average velocities are computed from 20 multiple tests with a vertical lightsheet centered on the breaking region and a field-of-view that captures roughly one-third of a wavelength. Sample results are shown in Figs. 29 and 30 for mean velocity and turbulence intensity, respectively, at two discrete times during the wave-breaking event. The mean results show that the velocity beneath the aerated breaking region

159

Ship Velocity Fields IJaht sheet optics

To wavamak&r control psnel

—a^-A FOV

U

Ttmir^b«x_ Tolassrs

•^V^ecsH "H-' Mirror coniroiter

frame grabber

Fig. 28. Schematic of experimental setup from Chang and Liu100,101

80 85 90 95 100 105 110 115

80 85 90 95 100105 110 115

x (cm)

x(cm)

(a)

(b)

Fig. 29. Mean velocity of the third breaking wave at (a) 3.290 s and (b) 3.398 from Chang and Liu100'101.

(a) (b) Fig. 30. Turbulence intensity I (cm/s) of the third breaking wave at (a) 3.290 s and 100,101 (b) 3.398fromChang and Liu; is less than one-half of the phase velocity and matches closely with numerical results of Lin and Liu102. Turbulence intensity results indicate that in the trough region, the maximum ensemble-averaged value is roughly 10% of the phase velocity. Variation in the

160

J. Longo et al.

turbulence intensity is reasonably small for sample numbers greater than 16. The authors use the transport equation for turbulent kinetic energy to assess the transport processes under breaking waves. They find that under the trough level in a rectangular control volume, the diffusive transport term is an order-of-magnitude smaller than the advection and dissipation terms. They also find that the turbulence dissipation is greater than the turbulence production for all of the measurements. The authors publish a related wave-breaking study (Chang and Liu101) with the same facility and equipment to study bias errors on mean and turbulence measurements resulting from small ratio of particle image diameter to pixel size and correction procedures. Qiao and Duncan103 apply film-based PIV with a 200 mJ Nd:YAG laser and the auto correlation evaluation method to investigate the flowfield at the crests of gentle 2D spilling breakers. The tests are performed in a 1.22(W)xl(D)xl4.8(L)-m 3 wave tank with a stationary PIV measurement system. A schematic diagram (Fig. 31) of the experimental setup highlights the wave tank and wavemaker, PIV system hardware, and light-delivery method. Camera for surface profile pictures

Light sheet

Optics and the underwater mirror Wavemajcer Underwater mirror \ . Light sheet Mirror and cylindrical lerts Nd: YAG lasers

^

'

Cameras \ Two-dimensional wave structures

Plan View

Fig. 31. Schematic of experimental setup from Qiao and Duncan" The breaking-wave frequencies, wavelengths, and heights are (1.42, 1.26, 1.15 Hz), (77.4, 98.3,118.1 cm), and (3.83, 4.87, 5.87, 5.92 cm), respectively. In order to map the breaking wave structures, PIV recordings are acquired sequentially along the length of the wave tank by movement of the camera for successive waves. The dataset includes instantaneous vector fields from which vorticity is computed and maximum horizontal and wave-toe velocities are identified. PIV measurements track the evolution of the spilling breaker including initial formation of a bulge on the forward face of the wave. At this stage, differences in the wave-crest flowfield as a function of wave frequency are in the noise of the measurements. The next phase of the breaking process occurs when the leading edge of the bulge (toe) moves

161

Ship Velocity Fields

down the wave face. Fig. 32 shows sample results with identification of the toe region and a thin layer of vorticity distributed from the toe to the crest. Progressive extension of this region from the toe toward the crest suggests that vorticity enters the flow through tie free surface at a location near the toe. Although coarse PIV resolution precludes measurement of the thin shear layer on the forward face of the breaker, measurements near the free surface between the toe and wave crest are as high as 1.4 times the linear wave phase speed. JV

JU

s20 10 b

10

20

30

40"

X(mm)

(a)

"50

"0

10

20

30

40

50

X(mm)

(b)

Fig. 32. Instantaneous vectors (a) and vorticity contours (b) from Qiao and Duncan103. Sarpkaya and Merrill104 apply digital PIV to investigate the breakup of a bow sheet (i.e., liquid wall jet) into ligaments and droplets over smooth and sandroughened walls to gain insight into the underlying physics of bow-sheet breakup and spray from naval vessels. The tests are conducted in a 0.5(W)x0.5(D)x6(L)-m water tunnel with a stationary PIV measurement system at several Fr= 15-30, J?e=2.4xlO4-8.5xl04, and Weber number= 1500-7500. The emphasis of the study is flow visualization of free-surface structures (ligaments and drops) but PIV measurements of the underlying wall-bounded Equid jet provide evidence that the Reynolds stress normal to the wall increases with distance from the wall and gives rise to eruptions of slender ligaments through the free surface. Most ligaments lean backwards as they travel forward with the wall jet because the majority of ejections occur in the second quadrant, i.e., for (-K', + V'). The results are being used for numerical modeEng of spray sheets. §S« Propidsor hydrodynamics Jiang et al.105 apply multiple-exposed, film-based PIV with a 15-W copper vapor laser and the auto correlation evaluation method to investigate ring-vortex creation from propeller crashback with propeller no. 4381. The tests are done in the David Taylor Model Basin 24-inch water tunnel with a stationary PIV measurement system at two advance coefficients (J=-0.472, -0.732). The region of interest is confined to

162

J. Longo et at

a local area near the Made tip. The dataset includes instantaneous vector, fields at three time instances after crashback for both advance coefficients from which vorticity is computed. The data captures the detailed ring-vortex flow structures and their locations for propeller inflow, reverse flow, and during ring-vortex formation. The tip vortices are stronger for the case with higher propeller loading. Cotroni et al.79 apply digital PIV with a 200 mJ Nd:YAG laser and the cross correlation evaluation method to study propeller wake flow from a 0.274-rn diameter, four-bladed propeller with a uniform pitch (pitch/diameter=L49). The experiments are conducted in the 0.6(W)x0.6(D)x2.6(L)~m3 Italian Navy Cavitation Tunnel (CEQMM). A schematic diagram (Fig. 33) of the experimental setup highlights the cavitation tunnel test section, PIV system hardware, synchronizing hardware, light-delivery method, and propeller-wake geometry.

(a)

(b)

Fig. 33. Schematic diagram of experimental setup (a) and wake geometry and measurement positions (b) from Cotroni et al.79. The propeller inflow and revolution speed is 7.5 m/s and 25 rev/s, respectively, for an advance ratio of 1.1. The Reynolds number is 2.0xl0 6 based on propeller diameter. Instantaneous and phase-averaged (from 30 PIV image pairs in' 5° increments) data is acquired in a 15.2x15.2 cm2 window centered on the propeller shaft axis and positioned at the aft edge of the propeller blades. Sample instantaneous results are shown in Fig. 34. Contours of vorticity clearly ilustrate and locate the cross sections of the tip-vortex system captured in the field of view and the complex interaction between the vortex generated at the blade-hub juncture and the recirculation zone behind the blunt end of the propeller shaft. A secondary tip-vortex system is also measured but is an order-of-magnitode weaker than the primary tip-vortex system. The core of the primary tip-vortex system is mapped in xy-space with error bars, for the y position. The test procedures such as phaseaveraging, window offsetting (Westerweel56), and adaptive correlation (Hart58) are advanced and prove that PIV can be used to measure complex unsteady flows with improved accuracy..

163

Ship Velocity Fields

xm

xm

(a) (b) Fig. 34. Contours of instantaneous vorticity (a) and horizontal component of velocity (b) for a phase'angle of 10° (Cotroni et al.79). Di Felice et al.80 apply, digital PIV with the same facility and measurement system as Cotroni et al.79 (Fig. 33) to study the propeller wake flow of a smaller (D=0.227m; pitch/diameter=l.l) four-bladed marine propeler. The propeller inflow and revolution speed are 4.25 m/s and 25 rev/s, respectively, for an advance ratio of 0.748. The Eeynolds number is 1.12xl06 based on propeller chord length. Instantaneous and phase-averaged (from 65 PIV image pairs in 5° increments) data axe acquired in a 10x10 cm2 window centered on the propeller shaft axis and positioned at three overlapping positions aft of the propeller hub. The authors report uncertainties for the measurements (0.6%) away from the blade wake and tip vortex, two regions where strong centrifugal forces repel seed particles, create voids in the image recordings, and degrade the accuracy of the data. Phase-averaged PIV results for U and V are presented for six crossings of the tip vortex downstream of the propeller. Qualitatively, distributions of U and V are similar after rotation of the latter counterclockwise by 90°. The viscous blade wake is evident as a velocity defect mat is released from the blade and mostly dissipates within one propeller diameter downstream. Considerable deformation of the streamlines in the vicinities of the.tip and hub vortices is presented. The vorticity released from the Made trailing .edge weakens after one propeller diameter downstream but the tip vortex systems retains it strength through the measurement area. The measurements are made with sufficient spatial resolution to track the tip vortex system which highlights the slipstream contraction. Turbulence intensities are computed from 65 realizations at discrete phase angles. Results illustrate the tip-vortex system breakdown as a general diffusion of the turbulence patterns as they are converted downstream. Judge et al.29'30 apply digital PIV with an 800 ml Nd:YAG laser and the cross correlation evaluation method to investigate tip leakage vortices from a 0.8504»m diameter, three-bladed, ducted rotor (propeller no. 5206). The rotor has a blade

J. Longo et al

164

chord-length C=0.381 m across the span and a tip gap of 6.67 mm. The experiments are conducted in the Naval Surface Warfare Center 36-inch variable pressure water tunnel. A schematic diagram (Fig. 35) of the experimental setup highlights the water tunnel test section, ducted rotor, and light-delivery method. The propeller inflow and revolution speed are 6.89 m/s and 8.33 rev/s, respectively, for an advance ratio of 0.972. The Reynolds number is 7xl0 6 based on the propeller chord length. Instantaneous and phase-averaged (from 532 PIV image pairs) data are acquired in a 3x2.4 cm2 window centered on the tip leakage vortex. PIV and LDV data in the region of the tip vortex are in good agreement.

Fig. 35. Schematic diagram of experimental setup from Judge et al."1"''30. Instantaneous PIV recordings at the same Made positions show variability in location (2%-3% vortex core radii), strength (22% of average circulation), and vortex-core size (31% of average vortex core size). Fig. 36 presents tip-leakage vortex data for two blade phase angles separated by roughly 25° and reveals how the core structure wanders in space and reorganizes in time. As the tip-leakage vortex is convected downstream, its size and circulation strength increase smoothly and the average and maximum tangential velocities

(a)

(b)

Fig. 36. Velocity vectors and contours of vorticity for the tip-leakage vortex at two phase angles: S=6R/C=0.0296 (a) and S=0.5259 (b) from Judge et al.2930.

Ship Velocity Fields

165

decrease smoothly. The test section Met pressure is controlled to suppress cavitation but further tests at lower inlet pressure are examining the effect of the tipleakage vortex variability on cavitation inception. Calcagno et al.26 apply digital stereoscopic PIV with a 200 mJ Nd:YAG laser to investigate the propeller wake flow from a 6.096»m Series 60 Q=0.60 container/cargo ship model with a single 0.222-m, 5»bladed MAU. The experiments are conducted in the 3.6(W)x2.25(D)xl0(L)~m3 INSEAN Circulating Water channel. A schematic diagram (Fig. 8) highlights the water channel test section, propeller and hull geometries, asymmetric two-camera configuration for 3D stereoscopic PIV, and light-delivery method. The propeller inflow and revolution speed are 1.22 m/s (Fn=0.16) and 6.7 rev/s, respectively, for an advance ratio of 0.82. The Reynolds number is 2.7xl05 based on the propeller diameter. Instantaneous and phase-averaged (129 PIV image pairs from 0°-69° in increments of 3°) data are acquired in a 20x20 cm2 window at three separate measurement planes orthogonal to the propeller shaft and downstream of the propeller disk (Fig. 37;x£=0.9997,1.0,1.0187). Typical phase-averaged results (Fig. 38) reveal the absence of propeller wake symmetry inherent to isolated propeller studies. f f i l

26

Fig. 37. Ship model, propeller, and measurement locations from Calcagno et al.

''..."

1"

K i:

f

A

i

/44^"

^ l l § § a ' - "*••

1

" -**3r4ve

-&)

o

sa

i*»

ISC

(a) 0>) Fig. 38. Average velocity vectors and contours of vorticity at i=0° from Calcagno et al;26

166

J. Longo et al.

The largest longitudinal velocities are in the lower part of the propeller disk at radial positions of r/R=0.6-0.7. The tip vortex system is captured in the measurements at r/R=0.9 where strong velocity gradients create peaks in the vorticity contours. Turbulence statistics from 129 realizations of the flow are computed and indicate maximum values are produced at the tip and hub vortex cores. The turbulent wake of the hull emerges in the turbulence statistics and strong turbulence is created by the passage of the blades through the hull wake. The wake evolves longitudinally with a noticeable slipstream contraction between the first and second planes and strong diffusion and dissipation of the blade wakes. The hub and tip vortices are attenuated at the third measurement plane but still very apparent. The authors highlight the advantages of a 3D PIV system including time efficiency for data acquisition and capability of measuring three velocity components for highly threedimensional flows that have strong velocity gradients. This study is complementary to previous work by Toda et al. 106 where the velocity field was measured with a multi-hole probe. Lee et al.42 apply digital PIV with a 125 mJ Nd:YAG laser and the particle tracking evaluation metiiod to study propeller wake flow from a five-bladed propeller with a diameter of 54 mm. The experiments are conducted in a 0.3(W)x0.2(D)xl.2(L)-m 3 circulating water channel. The propeller inflow and advance ratios are 0.33 m/s and 7=0.59, 0.72, 0.88, respectively. The Reynolds number is 1.8xl05 based on propeller diameter. Instantaneous and phase-averaged (400 PIV image pairs from 0°-54° in increments of 18°) data are acquired in a 11.8x11.8 cm2 window that is centered on the propeller axis shaft. The field of view allows data acquisition two propeller diameters downstream of the blade trailing edges. The authors apply an adaptive hybrid 2-frame particle tracking velocimetry (PTV) evaluation method to improve spatial resolution and measurement accuracy while reducing computational time. The data is presented as a function of propeller loading. In general, as the propeller loading increases, phase-averaged axial velocity, vorticity, axial and vertical turbulence intensity, and Reynolds shear stress increase within the propeller slipstream. 5.6. Appendages Liu and Chang107 apply multiple-exposed, film-based PIV with a 15-W copper vapor laser and the auto-correlation evaluation method to investigate the structure of vortices from a towed submarine sailplane with a half-span of 70.4 cm, base chord length of 65.3 cm, tip chord length of 43.7 cm, and maximum thickness of 12.1 cm. The experiments are performed in the 6.4(W)x4.9(D)x514(L)-m3 David Taylor Model Basin high-speed basin with a 2D, stationary PIV system at one Fr=0.81 (Re=l.3xl06) based on the sailplane base chord length. Data is acquired at three angles of attack (5°, 10°, 15°) at ten discrete times corresponding to a range of 0.05-

Ship Velocity Fields

167

48 chord lengths downstream of the sailplane. The dataset includes instantaneous vector fields from which vorticity and circulation are presented. The PIV data captures the shear layer and tip vortex roll-up at the trailing -edge of the sailplane. Computed circulation strengths increase three-fold as the sailplane angle of attack is increased from 5° to 15°. Yeung et al.37 apply digital PIV with a 10-W argon laser and the cross correlation evaluation method to investigate vorticity fields from a 2.54-cm thick acrylic flat plate in forced roE motion, where the experimental setup approximates the flow generated by ship bilge keels. The experiments are conducted in the 2.44(W)xl.52(D)x61(L)-m3 University of CaMfomia, Berkeley towing tank with a 2D stationary PIV system. Draft, roll amplitude, and roll period are tested in the following ranges, (L=31.75 cm, 16.51 cm), (C4=5°-15°), and 0p=1.8 sec-4 sec), respectively, where the latter two are chosen to closely follow -typical roll resonances of ships. The field of view covers a local region at the bottom edge of the plate and the surrounding area where instantaneous vector fields are acquired at several instants of time. Vorticity contours are computed from the instantaneous vector fields. The Keulegan-Carpenter number {KC=2nmJJtp) is used to identify asymmetric and symmetric flow regimes. For KC>13, an asymmetric period steady state is reached in the flowfield with a vortex pair shed toward the side of initial plate swing. For KC<13, a vortex pair is generated with cells of opposite sign attached to either side of the bottom edge of the plate. The vorticity is not strong enough for the pair to detach and they are eventually absorbed by each other. The data is taken in support of CFD validation. Planquart and Riethmuller108 apply PIV to investigate the flow around the bulb appendage of a sailing yacht. The experiments are done in the 35 cm x 35 cm L2B wind tunnel at the von Karman Institute. Measurements are taken for three configurations of the bulb (Fig. 39a) in three separate planes as shown in Fig. 39b.

1IL ^s^^^^^^^ywP 8 ""^

^s^^^^^^^^pw"""""'

"^wnt^^^^^^^yww""™^""

(a) (b) Fig. 39. Bulb configurations and PIV measurement locations from Planquart and Reithmuller108.

168

J. Longo etal

The instantaneous Yelocity data is used to compute vortex sheddingfrequenciesfor the different bulb configurations by locating positive and negative cells of vorticity in the measurement planes. The model-scale data is then scaled to Ml scale. Kamoda et al.27 apply PIV to investigate the flow around an oscillating flat plate. The experiments are performed at the Osaka University water channel using an innovative multi-layer laser sheet with one camera to measure three components of velocity. A 1-mm thick laser sheet is traversed in 2-mm intervals. into the direction of the flow and images are recorded (Fig. 40a) with high-speed video at 500 frames/sec. Multi-layer leaser sheet

Observed direction

(a)

(b)

Fig. 40. Multi-layer laser sheet principle for 3D PIV (a) and oscillating-plate experiment (b) from Kamoda et al.27.

Li ;.! Wy: -10

80 70 f GO ?iS0 fr # 40 N

8

6 6 8 10

Wy-10 -H -6 4 8

:*o 20 1(5 ,it.x

-hi)

-25

0 X (mm)

:..$..1...*.&LU«I*&*J^

25

50

(a) (b) Fig. 41. Sample experimental results (a) of velocity vectors and vorticity and comparison with CFD (b) for an oscillating plate at t=0 sec. from Kamoda etal:27 Particle images blink on and off as they enter and leave the light sheet, respectively. The period of the image flashing determines the flow velocity in the direction of the light sheet traversing. The in-plane velocity components are determined with traditional 2D PIV evaluation methods. The light sheet is traversed fast enough to

Ship Velocity Fields

169

capture any regions of reverse flow. The authors applied the technique to the flow around an oscillating flat plate (Fig. 40b). S.ample results of vectors and vorticity contours are presented in Fig. 41 with comparisons to CFD. The authors report good agreement in the comparisons. Bourgoyne et al.109 apply digital PIV with an 800 ml Nd:YAG laser and the cross-correlation evaluation method to investigate the trailing-edge flow structures from the suction side of a large hydrofoil. The hydrofoil has a NACA0016 suction side, beveled trailing edge, chord length of 2.13 m, maximum thickness of 0.171 ms and fully spans the test section. A bolt-on trailing-edge insert is available to make the body more bluff and test for effects of edge geometry on vortex shedding patterns. The experiments are conducted in the U.S. Navy's 3.05(W)x3.05(D)xl3(L)-m3 William B. Morgan Large Circulating Channel (LCC) with a 2D, stationary PIV system for a range of I^c=0.25-61xl06 based on the foil chord length (C). A sectional drawing of the hydrofoil trailing edge (Fig. 42) highlights its overall size and shape and the details. y/C

/-[baseline]

[modified |

Fig. 42. Schematic of hydrofoil section and measurement locations from Bourgoyne etal.109. 320 PIV recordings are acquired with two digital cameras side-by-side at the trailing edge to cover a region where boundary-layer flow separation and vortex shedding are anticipated. The dataset includes instantaneous vector fields from which mean and perturbation velocity fields and vorticity are computed in order to analyze data trends. The data for the modified trailing edge displays the highest level of coherent vortex formation and shedding strength. An example of the instantaneous vorticity and velocityfluctuationsfor this geometry is provided in Fig. 43. Through analysis of the detailed measurements, the authors develop two hypotheses to explain the source of the vortex shedding strength: (1) direct correlation between vortex shedding strength and the magnitude of the suction side shear layer gradient; or (2) correlation between the vortex shedding strength and the symmetry of the pressureand suction-side shear layers with the highest symmetry corresponding to the greatest shedding strength. The test results are being used to validate simulations for design of new U.S. Navy propulsors.

170

J. Longo et al

Fig. 43. Mean streamwise contours (a) and instantaneous vorticity and velocity fluctuations (b) for the modified trailing edge from Bourgoyne et al.109. 5.7. Cavitation and bubbly flows Gharib-et al.110'111 and Pereira et al.112'113 apply digital PIV with Nd:YAG lasers to investigate the bubbly flow through a propeller. The experiments are done in a l~m3 water tank with a 3-CCD defocusing camera. The defocusing camera is a 3D imaging system for mapping two-phase bubbly flows but can also be used for volume PIV measurements using tracer particles. The defocusing method (Fig. 44) shifts two or more apertures away from the camera centerline to obtain multiple images of a scattering source. The image shift from off-axis apertures is used for the measurement of depth of location of the source points and the scattered light combined with the blurredness is

Fig. 44. Standard imaging system (a) and defocusing concept (b) from Pereira etal.113.

Ship Velocity Fields

171

used to recover the source size information. The effectiveness of the imaging system is tested for bubbly flow through a two-bladed propeller rotating at 12 rev/s. A dataset of 50 vector (volume) maps is acquired and averaged with outlier correction. Massless particles are numerically injected upstream of the propeller into the experimental mean dataset and their pathlines are shown in Fig. 45. The ratio of upstream and downstream bubble populations is 65%' which indicates coalescence as the bubbles are entrained through the propeller. The void fraction ratio of upstream and downstream bubble populations is close to 100% which demonstrates that the bubble-detection features of the imaging system are very promising. The authors conduct uncertainty assessment on the position and velocity detection errors of the camera but do not apply their UA methodology to measurements of bubbly flow through a propeller. The measurement uncertainty of the defocusing camera is on the same order as conventional 2D PIV cameras for the in-plane components. For Hie out-of-plane component (along the cameras longitudinal axis), errors in velocity are reportedly 15 times higher than for the inplane components. Recent improvements in the defocusing camera have helped achieve acceptable uncertainty levels for the out-of-plane velocity component.

Fig. 45. Pathlines of bubbles in the wake of a propeller from Pereira et al.113. Gopalan et al.41 apply multiple-exposed, digital PIV with a 30 mJ NdiYAG laser to investigate the flow structure and inception of tip leakage cavitation from the tip gap between a 50-mm chord length hydrofoil and the tunnel test section. The experiments are performed in the Johns Hopkins University 5.08(W)x6.35(D)x41(L) cm3 water tunnel with a 2D, stationary PIV system for one Jfec=2.5xl05 based on the foil chord length (Q. The tip gap clearance is adjusted to one of three values (0.6, 1.4, 2.6 mm) by increasing or decreasing the thickness of one test-section sidewall. Data acquisition occurs at a local area near one corner of

172

J. Longo et al.

the hydrofoils trailing edge with an inclined lightsheet. The dataset includes instantaneous vector fields from which circulation is computed with an area integral. High-resolution PIV data (180 (J,m vector grid spacing) is used to compute circulation strength and estimate the size and location of die tip-leakage vortex. Vortex core meandering is significant for all tip gap clearances. The tip-leakage vortex cores are 3-4 vector grid spacings. Increasing the gap size causes reduction in the tip-leakage vortex strength and increase in the vertical distance of the tip leakage vortex from the hydrofoil. 6. Conclusions and future directions Application of PIV to ship velocity fields mostly used stationary PIV measurement systems taking advantage of relative ease of system setup and lower cost and vibration, whereas fewer investigations used towed PIV measurement systems. In the past decade, several advancements in PIV hardware and software were realized including transition from continuous to pulsed lasers, film-based to digital cameras, low- to high-resolution digital cameras, and auto to cross correlation to adaptive correlation evaluation methods. New PIV techniques have emerged recently including methods for 3D measurements such as stereoscopic PIV, multiple-sheet PIV, and defocusing PIV which will enable more advanced model testing of increasingly complex flows. Many types of facilities have been used for ship hydrodynamics PIV studies including towing tank, rotating arm, water channel, water tunnel, and wind tunnel. Although some datasets were based on a limited number of instantaneous PIV recordings from which other variables such as vorticity and circulation were computed, most datasets conformed to an increasing trend of acquisition of instantaneous PrV recordings in sufficient quantity to enable statistical analysis of the data for mean and turbulence values and even phaseaveraged mean and turbulence variables for periodic flows. Only 40% of the investigations reported measurement uncertainties and just 18% from this population applied the latest standards and guidelines for uncertainty assessment methodology to the measurement results. Nevertheless, reported PIV uncertainties for ship hydrodynamics studies are comparable or better than those that have been reported for traditional measurement systems employing multi-hole pitot or Laser-Doppler probes. Detailed comparisons between PIV and five-hole pitot uncertainties for measurements in the nominal wake plane of a surface combatant (Gui et al.4) revealed lower uncertainties for the PrV technique by roughly 0.5-2.0%. Most investigations were undertaken for explication of flow physics (82%) and measurement system development (50%), whereas fewer but increasing number were done for support of model development or CFD validation (30%). For boundary layer and wake studies without free surface, the data showed the nature of flow around submarine bodies with and without midbody appendages for different

Ship Velocity Fields

173

angle of attack, Reynolds number, and angular velocity including tip, wake, and juncture vortices and 3D separation. Boundary layer and wake studies with free surface illustrated characteristics of flow around surface-piercing hydrofoils and ship models for different Froude number and steady and unsteady inflow conditions including necklace vortices, bow and shoulder waves, juncture flow, separated flow, and steady and unsteady mean and turbulent boundary layer. Bow-flow studies focused on the attached liquid sheet upstream of the point at which it detaches from a model for different Froude number and spatial locations near the bow and highlighted underlying physics of vorticity production, vorticity origin, shear layers, and juncture flow. For wave-hydrodynamics studies the data illustrated the nature of spilling breaking waves and spray sheets for different Froude and Weber numbers including mechanisms for mixing and shear layers, vorticity production, breaking wave structures, and breaking wave wakes. Propulsor-hydrodynamics investigations focused on the features of propeller wakes for steady and unsteady inflow and different Reynolds number and advance coefficients including ring and tip vortices, tip leakage, tip-vortex cavitation, tip-vortex tracking, secondary tip-vortex production, slipstream contraction, blade and hub vorticity interaction, and phaseaveraged mean and turbulent velocity fields. For appendage-flow studies the data illustrated the characteristics of flow around oscillating flat plates, bulb keel, and hydrofoil for different oscillation frequency, Reynolds number, and angle of attack including features of tip vortex, vortex production and vorticity fields, 3D separation, and turbulence. Finally, for cavitation and bubbly-flow studies, the data showed the qualities of two-phase flow through a propeller and flow around a hydrofoil including bubble distributions and trajectories, void fraction, cavitation inception, tip-leakage vortex cavitation, and tip-vortex cavitation. Significant progress in development and application of PIV technology to ship hydrodynamics research has occurred in the past decade as evidenced by the investigations herein reviewed. The increasing trend of data procurement for model development and CFD validation is consistent with the rapid development and widespread use of CFD technology over this same period. It is also consistent with change in testing at large-scale facilities from routine tests for global variables to detailed tests for local variables (ITTC84), as design methodology changes from model testing and theory to simulation based design. Detailed testing offers new opportunities for research institutes, as die amount and complexity of testing is increased. Development of simulation based design involves a new paradigm for hydrodynamics research in which CFD, EFD, and UA are conducted simultaneously using an integrated approach serving as an internal engine guaranteeing simulation fidelity (Stern et al.115). CFD is used to guide EFD, EFD is used for model development and validation, and lastly CFD is validated and fills in sparse EFD data for complete diagnostics and documentation of flow for specified benchmarks. This

J. Longo et al.

174

situation challenges both CFD and EFD as applications become increasingly complicated, including nonlinear conditions; ship motions, maneuvering, and capsizing; environmental conditions; and even full-scale testing. To keep pace, advancements in all areas of PIV technology and its application in ship hydrodynamics are expected, including smaller, faster, and higher resolution digital cameras; faster lasers; higher accuracy PIV software; more widespread use of towed systems; standard usage of 3D stereo or volume systems; PIV application to free surface measurements and correlation to subsurface measurements; and ever more detailed testing. Acknowledgement The authors' research was sponsored by the Office of Naval Research Grant N00014-01-1-0073, under the administration of Dr. Patrick Purtell. References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Y. Toda, F. Stern, and J. Longo, J. Ship Res. 36, 360 (1992). W.J. Kim, S.H. Van, and D.H. Kim, Exp. in Fluids 31, 567 (2001). J. Longo and F. Stern, Exp. in Fluids 32, 558 (2002). L. Gui, J. Longo, and F. Stern, Exp. in Fluids 31, 336 (2001). D.J. Fry and Y.H. Kim, 17'h Sym. Naval Hydro. (1988). M. Hoekstra and I.J.T. Ligtelijin, MARIN Rep. no. 410461-1-PV (1991). C. Chesnakas and S. Jessup, 22nd Sym. Naval Hydro. (1998). J. Longo, H.P. Huang, and F. Stern, Exp. in Fluids 25, 283 (1998). M. Raffel, C. Willert, and J. Kompenhans, Springer, (1998). A. Cenedese and A. Paglialunga, Exp. in Fluids 8, 273 (1990). R.D. Keane and R.J. Adrian, Meas. Sci. Tech. 1, 1202 (1990). R.J. Adrian, Ann. Rev. Fluid Mech. 23, 261 (1991). C.E. Willert and M. Gharib, Exp. in Fluids 10, 181 (1991). W. Heckmann, S. Hilgers, W. Merzkirch, and T. Schliiter, Opt. Meth. and Data Proc. In Heat and Fluid Flow (1994). J. Westerweel, A. Draad, J. Hoeven, and J. Oord, Exp. in Fluids 20, 165 (1996). I. Grant, Proc. Instn. Mech. Engr. 211, 55 (1997). J.G. Santiago, S.T. Wereley, C D . Meinhart, D.J. Beebe, and R.J. Adrian, Exp. in Fluids 25, 316(1998). S.T. Wereley, L. Gui, and C D . Meinhart, AIAA J. 40, 1047 (2002). A.K. Prasad, Exp. in Fluids 29, 103 (2000). B.J. Thompson, Proc. SPIE1136, 308 (1989). H. Meng and F. Hussain, Fluid Dyn. Res. 8, 33 (1991).

Ship Velocity Fields

175

22. J. Zhang and J. Katz, Proc. 8 Int. Sym. on Applications of Laser Techniques to Fluid Mech. (1996). 23. C.E. Willert and M. Gharib, Exp. in Fluids 12, 353 M (1992). 24. M. Raffel, M. Gharib, O. Ronneberger, and J. Kompenhans, Exp. in Fluids 19, 69 (1995). 25. I. Grant, E. Owens, J.N. Stewart, and A. Aroussi, Proc. 6Ih Int. Sym. on Applications of Laser Techniques to Fluid Mech. (1992). 26. G. Calcagno, F.D. Felice, M. Felli, and F. Pereira, 24th Sym. Naval Hydro. (2002). 27. H. Kamoda, T. Sasaki, T. Suzuki, and Y. Toda, Proc. Is' Asia Pacific Hydro. Sym. (2002). 28. F. Di Felice and F. De Gregorio, Proc. Int. Offshore and Polar Eng. Conf. 392 (2000) 29. C.Q. Judge, G.F. Oweis, S.L. Ceccio, S.D. Jessup, C.J. Chesnakas, and D.J. Fry, 4th Int. Sym. Cavitation (2001). 30. C.Q. Judge, G.F. Oweis, S.L. Ceccio, S.D. Jessup, C.J. Chesnakas, and D.J. Fry, 26th ATTC (2001). 31. T. Fu, A. Shekarriz, J. Katz, and T.T. Huang, /. Fluid Mech. 269, 79 (1994). 32. S.Y. Jaw and Y.Z. Kehr, 22nd Sym. Naval Hydro. (1998). 33. I. Grant, X. Wang, and T. Wagner, Proc. Int. Society for Opt. Eng. 2097, 214 (1994). 34. R. Shekarriz, T. Fu, J. Katz, H.L. Liu, and T.T. Huang, AIAA J. 30, 145 (1992). 35. T. Okuno, Y. Sugii, and S. Nishio, Meas. Sci. Tech. 11, 667 (2000). 36. F.W. Hong, F. Zhao, J. Zhang, and L.D. Zhou, J. Ship Mech. 6, 19 (2002). 37. R. Yeung, C. Cermelli, and S.W. Liao, 21s' Sym. Naval Hydro. (1996). 38. J. Zhang, J. Xu, F.W. Hong, and F. Zhao, J. Hydro. 4, 118 (1999). 39. K. Siddiqui, M. Loewen, C. Richardson, W. Asher, and A. Jessup, Phys. Fluids 13, 1891(2001). 40. F. Zhao, J. Zhang, J. Xu, and F.W. Hong, J. Ship Mech. 5, 6 (2001). 41. S. Gopalan, H.L. Liu, and J. Katz, 23rd Sym. Naval Hydro. (2000). 42. S. Lee, B. Paik, and C. Lee, 24th Sym. Naval Hydro. (2002). 43. L. Gui and W. Merzkirch, Exp. in Fluids 28, 36 (2000). 44. L. Gui, J. Longo, and F. Stern, Exp. in Fluids 30, 27 (2001). 45. H.T. Huang, H.E. Fiedler, and J.J. Wang, Exp. in Fluids 15, 168 (1993). 46. T.H. Kemmerich and H.J. Rath, Exp. in Fluids 17, 315 (1994). 47. K. Okamoto, Y.A. Hassan, and W.D. Schmidl, Exp. in Fluids 19, 342 (1995). 48. A.M. Fincham and G.R. Spedding, Exp. in Fluids 23, 449 (1997). 49. O. Ronneberger, M. Raffel, and J. Kompenhans, Proc. 9th Int. Sym. on Applications of Laser Techniques to Fluid Mech. (1998).

176

50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63. 64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82.

J. Longo et al.

L. Gui and W. Merzkirch, Exp. in Fluids 24, 66 (1998). R.C. Singleton, IEEE Tans AU (1996). L. Gui and W. Merzkirch, ERCOFTAC Bulletin 30, 45 (1996). R.D. Keane and R.J. Adrian, Laser Anem. Advances App., 5th Int. Conf. 2052, 477 (1993). C.E. Willert, Applied Sci. Res. 56, 79 (1996). E.A. Cowen and S.G. Monismith, Exp. in Fluids 22,199 (1997). J. Westerweel, D. Dabiri, and M. Gharib, Exp. in Fluids 23, 20 (1997). F. Scarano and M.L. Riethmuller, Exp. in Fluids 26, 513 (1999). D.P. Hart, Proc. 9* Int. Sym. on Applications of Laser Techniques to Fluid Mech. (1998). L. Gui, W. Merzkirch, and R. Fei, Exp. in Fluids 29, 30 (2000). S.T. Wereley, J.G. Santiago, C D . Meinhart, and R.J. Adrian, Proc. ASME/DSC (Micro-fluidics Sym.) 66 (1998). S.T. Wereley and C D . Meinhart, Proc. l(fh Int. Sym. on Applications of Laser Techniques to Fluid Mech. (2000). S.T. Wereley and C D . Meinhart, Exp. in Fluids 31, 258 (2001). M. Sjodahl, Applied Optics 33, 6667 (1994). M.J. Sholl and O Savas, 35th AIAA Aero. Sci. Meeting (1997). F. Scarano and M.L. Riethmuller, Exp. in Fluids 29, 51 (2000). B. Lecordier, D. Demare, L.M.J Vervisch, J. Reveillon, and M. Trinite, Meas. Sci. Tech. 12, 1382 (2001). L. Gui and S.T. Wereley, Exp. in Fluids 32, 506 (2002). H.T. Huang, H.E. Fiedler, and J.J. Wang, Exp. in Fluids 15, 263 (1993). P.T. Tokumaru and P.T. Dimotakis, Exp. in Fluids 19, 1 (1995). H.J. Lin and M. Perlin, Exp. in Fluids 25,431 (1998). J. Nogueira, A. Lecuona, and P.A. Rodriguez, Exp. in Fluids 27, 107 (1999). J. Nogueira, A. Lecuona, and P.A. Rodriguez, Meas. Sci. Tech. 12, 1911 (2001). F. Scarano, Meas. Sci. Tech. 13, Rl (2002). S.T. Wereley and L. Gui, Exp. in Fluids, 34 42 (2003). AIAA Standard, AIAA S-017A-1999, (1999). ASME, PTC 19.1, (1998). J. Westerweel, Exp. in Fluids 16,236 (1994). I. Fujita and T. Kaizu, J. Flow Vis. And Image Proc. 2, 173 (1995). A. Cotroni, D.F. Felice, G.P. Romano, and M. Elefante, Exp. in Fluids 29, S227 (2000). F. Di Felice, G. Romano, and M. Elefante, 23rd Sym. Naval Hydro. (2000). J. Longo, J. Shao, M. Irvine, and F. Stern, 24th Sym. Naval Hydro. (2002). S.J. Kline and F.A. McClintock, Mech. Engr. 75, 3 (1953).

Ship Velocity Fields

83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96. 97. 98. 99. 100. 101. 102. 103. 104. 105. 106. 107. 108. 109. 110. 111. 112. 113.

177

H. Coleman and G. Steele, Wiley 205 (1989). ITTC, 22nd International Towing Tank Conference (1999). J. Longo, J. Shao, M. Irvine, L. Gui, and F. Stern, 26th ATTC (2001). J. Longo, J. Shao, M. Irvine, L. Gui, and F. Stern, PTV and Modeling Water Wave Phenomena (2002). R. Shekarriz, T. Fu, J. Katz, and T.T. Huang, AIAA J. 31, 112 (1993). E.R. Hoffman and P.N. Joubert, 7. Fluid Mech. 16, 395 (1963). J. Xu, J. Zhang, B. Ni, M. He, and B. Xu, 7. Ship Mech. 2,1 (1998). T. Fu, P. Atsavapranee, and D. Hess, 24th Sym. Naval Hydro. (2002). E. Pogozelski, J. Katz, and T.T. Huang, 21s' Sym. Naval Hydro. (1996). E. Pogozelski, J. Katz, and T.T. Huang, Phys. Fluids 9, 1387 (1997). Z.J. Zhang and F. Stern, 7. Fluid Eng. 118, 546 (1996). R.R. Dong, J. Katz, and T.T. Huang, Laser Anemometry Conf. and Exhib. 229, 425(1995). R.R. Dong, J. Katz, and T.T. Huang, 7. Fluid Mech. 346, 77 (1997). G.I. Roth, D.T. Mascenik, and J. Katz, 25th ATTC (1998). G.I. Roth, D.T. Mascenik, and J. Katz, Phys. Fluids 11, 3512 (1999). D.B. Coakley and J.H. Duncan, 21s' Sym. Naval Hydro. (1996). D. Dabiri and M. Gharib, 7. Fluid Mech. 330, 113 (1997). K.A. Chang and P.L.-F. Liu, Phys. Fluids 11, 3390 (1999). K.A. Chang and P.L.-F. Liu, Exp. in Fluids 29, 331 (2000). P. Lin and P.L.-F. Liu, 7. Fluid Mech. 359, 239 (1998). H. Qiao and J.H. Duncan, 7. Fluid Mech. 439, 57 (2001). T. Sarpkaya and C.F. Merrill, AIAA 7. 39, 1217 (2001). C.W. Jiang, R.R. Dong, H.L. Liu, and M.S. Chang, 27 s ' Sym. Naval Hydro. (1996). Y. Toda, F. Stern, I. Tanaka, and V.C. Patel, 7. Ship Res. 34, 225 (1990). H.L. Liu and M.S. Chang, Laser Anemometry Sym.: Advances and Applications (1994). P. Planquart and M. Riefhmuller, 23rd Sym. Naval Hydro. (2000). D. Bourgoyne, J. Hamel, C. Judge, S. Ceccio, and D. Dowling, 24th Sym. Naval Hydro. (2002). M. Gharib, D. Modarress, D. Dabiri, F. Pereira, and F. Taugwalder, Proc. 9th Int. Sym. on Applications of Laser Techniques to Fluid Mech. (1998). M. Gharib, D. Modarress, F. Pereira, D. Dabiri, and F. Taugwalder, 25th ATTC (1998). F. Pereira, M. Gharib, D. Dabiri, and D. Modarress, Exp. in Fluids 29, S078 (2000). F. Pereira, M. Gharib, D. Modarress, and D. Dabiri, 9th Int. Sym. Flow Vis. (2000).

178

J. Longo et al.

114. D.P. Hart, Exp. in Fluids 29, 13 (2000). 115. F. Stern, R. Wilson, J. Longo, P. Carrica, T. Xing, Y. Tahara, C. Simonsen, J. Kim, J. Shao, M. Irvine, M. Kandysamy, S. Ghosh, and G. Weymouth, Proc. 8hlnt. Conf. Num. Ship Hydro. (2003).

Ship Velocity Fields

179

Table 1. Summary of uncertainties for PIV studies in ship hydrodynamics. Study

Instant.

Mean

Turb.

Vorticity

Boundary layer and wake without free surface Shekarriz et al.

34,87

Fu et al. 31

4.5-9.0%

NR

NR

NR

NR

NR

NR

10%

Boundary layer and wake with free surface 91 92

Pogozelski et al. '

1-3.5%

NR

Gui et al. 4 t

1.4-7.2%

2.4-7.7%

4.1-5.8%

NR

Longo et al. 8 1 , 8 5 ' 8 6 t

1.1-3.0%

1.6-3.8%

2.6-6.0%

NR

NR

NR

Bow flow 94 95

Dong et al. -

I

2%

NR

NR

10-20%

Rothetal. 96 ' 97

|

4.1%

NR

NR

26.6%

NR

NR

Wave hydrodynamics Coakley and Duncan

98

3%

NR

Dabiri and Gharib"

1.4%

NR

NR

3.9%

Chang and Liu 100 ' 101

1%

0.3%

1.3%

7.5%

1-3%

NR

NR

15%

3.5%

NR

NR

NR

Qiao and Duncan

103

Sarpkaya and Merrill104 t

Propulsor hydrodynamics Jiang et al.

105

Di Felice et al.

NR

NR

NR

0.6%

NR

NR

NR

NR

5%

NR

NR

2% 80

t

Calcagno et al. 26 t

Appendage flows Liu and Chang

107

2.5%

NR

NR

NR

Yeung et al. 37

NR

NR

NR

10-20%

Kamoda et al. 27

10%

NR

NR

NR

NR: not reported; t : UA follows established standards and guidelines; instantaneous velocity; Mean: mean velocity; Turb.: turbulent velocity

Instant.:

This page is intentionally left blank

CHAPTER 5 THE AIR-WATER INTERFACE: TURBULENCE SCALAR EXCHANGE

AND

S. Banerjee 1 and S. Maclntyre Department of Chemical Engineering, University of California, Santa Barbara E-mail: banerjee®engineering, ucsb. edu 2 Marine Science Institute, University of California, Santa Barbara E-mail [email protected]

Turbulence and scalar exchange at air-water interfaces have been investigated in the laboratory, in field studies, and with numerical simulations. Parameterizations and phenomena are reviewed, in particular surface divergence and renewal concepts for scalar exchange predictions. Microbreaking is seen to be important over a wide range of conditions but is not well understood. The effect of heat losses on near-interface transport processes also requires further study.

1. I n t r o d u c t i o n Transport processes at gas-liquid interfaces are of importance in a number of areas, such as for equipment like absorbers, condensers a n d boilers, a n d for environmental systems. For processes at the interface between the atmosphere and terrestrial water bodies, early interest related to t h e need t o understand gas transfer t o rivers and lakes t h a t had high biological oxygen demand due t o n a t u r a l causes or effluents, e.g. from paper mills. T h e desorption of dissolved substances, like P C B s , from inland and coastal water bodies, can also be an air quality concern and has been a subject of continuing interest. More recently, the transfer of greenhouse gases (e.g., CO2 and methane) and moisture, at the ocean surface have become important because of their impact on global warming. It is estimated t h a t approximately 30-40 percent of m a n - m a d e CO2 is taken u p by t h e oceans. These estimates are substantially affected by uncertainties in t h e prediction of gas transfer rates at the air-water interface. For example, oceanic CO2 uptake estimates vary by about a factor of three, depending on whether they are based on correlations by Liss and Merlivat 1 or Wanninkhof and McGillis 2 181

182

S. Banerjee and S.

Maclntyre

- the first yielding an uptake of 1.1 PgC/year, whereas the second gives about 3.3 PgC/year (Donelan and Wanninkhof 3 ). There is, therefore, considerable incentive to improve our state of knowledge regarding mass, heat and momentum transfer between the atmosphere and water bodies, and it is expected that the subject will continue to be an area of active research for the foreseeable future. The purpose of this review is to survey what is known about air-water transport processes and indicate areas where research would be beneficial. To set the stage for the discussion, we divide the subject into two broad categories. In the first, we will consider what is known about transport processes at an unsheared air-water interface, i.e., a situation in which the winds are light and the fluid motions and turbulence that occur near the air-water interface are generated elsewhere. For example, turbulence could be generated at the bottom boundary of a flowing stream or in the shear layer between subsurface currents flowing at different velocities, and these turbulence structures could then impinge on the air-water surface. An example is the "boils" that one sees at the surface of rivers. Turbulence can also be generated by heat losses that give rise to natural convective motions on the liquid side. In the second category, we consider situations with significant wind shear at the surface. In this case, the turbulence production occurs at the interface itself, rather than at some other, perhaps rather distant, location, giving rise to phenomena that are qualitatively different with regard to scalar exchange. Here again, there may be significant heat loss at the air-water interface. The turbulence production occurs at the interface, but since heat loss is often largest when latent heat fluxes (evaporation) rise due to high winds, the turbulence is not only due to shear, but also due to natural convection on the liquid side. In all cases, the interface is deformable and can form complex patterns in response to the fluid motions. This makes problems of measurement and simulation difficult, particularly in the near-interface region that controls transport processes. For this reason, progress in our understanding of these processes has been slow compared to our understanding of such problems at solid (non-deformable) boundaries. There have been several reviews of the subject, notably of laboratory studies and simulations by Banerjee4, of field studies and experimental methods by Maclntyre et al. 5 , by Jaehne and Haussecker6 and at a recent conference - see Donelan et al. 7 A further subdivision of transport processes at the air-water interface is possible, depending on whether the liquid side or the gas side controls the process. Typical concentration profiles for each case are shown in Fig. 1.

The Air- Water Interface:

Turbulence and Scalar

Exchange

183

For most sparingly soluble gases like CO2, He and CH4, it is liquid side motion that controls gas transfer. This is because the transfer velocity, as discussed in Sec. 2, depends on the molecular diffusivity raised to some exponent between 1/2 and 2/3, turbulent motions being damped at the interface. Molecular diffusivity of most gases is about two to three orders of magnitude less in water than in air, which explains why turbulence on the liquid side controls absorption or desorption. On the other hand, processes like evaporation draw species (in this case H2O) directly from the interface and therefore are controlled by air-side motions. A classification of the controlling side based on solubility and diffusivity is presented in Jaehne and Haussecker6. Suffice it to say here that the air-side behaves something like flow over a moving, solid (and wavy) boundary, whereas the motion on the liquid side is quite different. The exchange of sparingly soluble gases at the air-water interface will be the focus of this review, so it is the liquid side motions that will be considered in most detail here. To proceed, consider what happens as wind velocity is increased over a stretch of water. The wind imposes shear on the water surface and this, in turn, produces turbulence when the shear rate becomes high enough - which happens at relatively low wind velocities. Undulations start to form at the surface and grow into capillary and gravity waves as the wind velocity increases. When the wind velocity Uio (at 10 m above the surface) increases to about 3 to 5 m/s, these waves become quite steep and start to microbreak. By microbreaking we mean that plunging motions can be detected in the water below these waves, but there is no discernible air entrainment. These microbreaking waves have amplitudes ~ 0 ( 1 cm) and length ~ 0(10 cm). The character of the turbulence below the waves is different from that in lighter winds. Near-surface turbulence before commencement of microbreaking appears to be generated in much the same manner as near walls, i.e., due to shear in the near-interface region, and many of its characteristics are rather similar to wall turbulence. This means we see streamwise, streaky, high-speed/low-speed regions, as well as "bursts", which are intermittent (active) regions consisting of ejections of the fluid from the near-interface region into the bulk and sweeps in which bulk fluid is brought to the interface. These structures scale with the shear rate in a manner similar to what is observed in wall turbulence. However, on commencement of microbreaking there is a dramatic change in the near-interface turbulence structure. The surface velocity field indicates regions of strong upwelling behind the crests of microbreaking waves, with regions of convergence just in front of the crests, i.e. surface fluid does not appear to cross the microbreaking

184

S. Banerjee and S.

slightly

soluble gas

Maclntyre

soluble gas

Ca

°Ca jjr

Air Boundary

Ca

^

^

^

Water Boundary

Cw

Cw

Fig. 1. Schematic showing concentration profiles near the air-water interface. Here a is the Henry's Law constant that is used to calculate the equilibrium concentration of a sparingly soluble gas at the interface, when C 0 is the atmospheric concentration. Reproduced with permission from Fig. 3.1 on p. 53 in Maclntyre et al. 5

wave crests but to be entrained under them. Associated with this, there are intermittent bursts which plunge much deeper into the water and in this sense are "giant" bursts, compared to what one sees in the absence of microbreaking. As wind speed is further increased to about Uw ~ 1 2 - 1 3 m/s, breaking with air entrainment commences (wind-driven whitecapping) with water spray being entrained at the higher wind speeds. These phenomena also significantly affect gas transfer. However, while whitecapping and spray formation may be important in localized regions of high wind and near coastlines, the average wind speed over the ocean is ~ 7.5 m/s (Donelan and Wanninkhof 3 ), which suggests that microbreaking is widespread and of the most interest for scalar exchange between the water and air. A typical early correlation from Liss and Merlivat 1 for gas transfer rate as a function of wind velocity is shown in Fig. 2, together some data from Maclntyre et al. 5 The data are for inland water bodies, but the scatter shown is typical of data taken over oceans. Part of the scatter, of course, is due to difficulties in doing reproducible field experiments. There are a number of other factors that are important. The most important of these, as mentioned earlier, is that natural convection can arise due to the water surface being colder than the bulk liquid due to heat losses and evaporation

The Air- Water Interface:

Turbulence and Scalar

Exchange

185

from the water. These motions can interact with turbulence generated by shear and affect transport processes, as recently discussed by Eugster et al. 8 Thus, wind velocity is not the only variable that affects transfer rates, but factors like cloud cover, humidity and the diurnal cycle also play a role. Another factor contributing to the scatter arises from the experiments giving gas transfer velocities averaged over some period of time, and during these times the wind may vary, causing biases in the averaged measurements, as discussed later. Other factors include surface slicks, which can damp nearinterface turbulence, and the waves and wind being out of equilibrium, e.g., due to rising or falling wind conditions. In coastal regions and near shores, phenomena involving shoaling, wavebreaking and offshore flows of warm air over cold water complicate matters further.

15i

—

10'

o Mono lake A Crowley lake • Rockland lake 4.Lake302N O Pyramid lake — K=0.451M.6 — Use & Merllvat (1986)

5-

3

4

5

W(10) [ms-1]

Fig. 2. Data on gas transfer rates normalized to a Schmidt number of 600 against wind velocity at 10 m above the water surface. Liss and Merlivat's correlation is shown as the dotted line, and an empirical fit to the data k (cm/hr) 0.45 £/JQ 6 is shown as the solid line. Reproduced with permission from Fig. 3.3 on p. 58 in Maclntyre et al. s

In spite of these complications, and several other effects that add to the scatter, one can see from Fig. 2 that there appears to be a significant increase in the gas transfer rate in the range of wind velocities at which microbreaking commences. This is captured to some extent in the Liss and Merlivat 1 correlation by a change in its slope at a wind velocity of 3.6 m/s. There is a second change in slope at 13 m/s (not shown in the figure) when breaking of large waves with air entrainment may commence. In other correlations, such as that of Wanninkhof and McGillis2, these changes in

186

S. Banerjee and S.

Maclntyre

slope are captured by a cubic polynomial fit to gas transfer data in terms of the wind velocity. Because of the nonlinear nature of the gas transfer velocity vs. wind velocity curve, the averaging interval over which data is taken gives it a bias, since winds fluctuate. Thus a correlation based on averaging over a long period, e.g. a month, will be different from one in which averaging is done over a shorter period. This aspect must also be taken into account in interpreting data as different averaging periods for different data sets can lead to scatter. The rest of this review is organized as follows. We will first discuss modeling approaches in Sec. 2, followed, in Sec. 3, by consideration of situations in which the wind stress is light, so phenomena near the air-water interface are dominated by interactions with far-field turbulence, as well as by waves and natural convection. Laboratory studies and simulations will be considered first, followed by field experiments. In Sec. 4, we consider phenomena that arise when wind shear is significant, starting again with lab studies and simulations, followed by field studies. We end by summarizing the current understanding of processes at the air-water interface and the areas in which further work is desirable. 2. Scalar Exchange Models Before proceeding to discuss experimental results and simulations on scalar exchange and turbulence in the near-interface region, modeling approaches for such processes will be briefly reviewed. Early work by Lewis and Whitman 9 proposed that transport occurred across a film of laminar fluid, of thickness S, adjacent to the interface, with the bulk phase being well mixed and turbulent. This gave rise to mass or heat transfer coefficients, /3 (sometimes denoted by k in the literature, and also sometimes in this article) that were proportional to the diffusivity, D, i.e. (3 oc Z?2/3. However, it became clear from subsequent experiments that at fluid-solid boundaries, /3 oc Z3 2 / 3 , whereas at gas-liquid boundaries (3 oc D1'2, at least when Sc = vjD was high (where v is the kinematic viscosity). This led Higbie10 to speculate that turbulence brought fluid from the bulk to the interface, where unsteady absorption occurred into an essentially laminar fluid for some period, r, after which the surface element was replenished. Lewis and Whitman's theory gave (3 = D/6,

(1)

whereas Higbie's speculation gave (3=(D/(nT))1/2

(2)

The Air- Water Interface:

Turbulence and Scalar

Exchange

187

Subsequently, Danckwerts 11 allowed for a random distribution of surface ages for the renewed elements, which was more typical of what might be expected from a turbulent fluid, giving on the liquid side, 0 = (D/r)1/2

(3)

where f may be thought of as the mean time between surface renewals. Note that the Higbie-Danckwerts surface renewal model predicted the dependence of the liquid-side scalar transfer rate, /?, on D, as D1^2, which was also found in lab experiments. However, the quantity f, the time between renewals, remained unspecified. A number of researchers proposed various models for f in the 1960s, notably the "large-eddy" model of Fortescue and Pearson 12 and the "smalleddy model" of Banerjee et al. 13 The large-eddy model gave f ~ A/w,

(4)

leading to a mass transfer coefficient (5 Scl/2

= «Re t ~ 1 / 2

(4a)

where A (sometimes denoted as t) is the turbulence integral length scale and u is the integral velocity scale, and Re t = u\/v is the turbulent Reynolds numbers. On the other hand, the small-eddy (SE) model of Banerjee et al. 13 gave f ~ (v/e)1'2

(5)

where e was the turbulent energy dissipation rate close to the interface and v was the kinematic viscosity. Here e can be measured directly or calculated. For example, e ~ u3/A, so we have 0 So1'2 = uR%1/*

(5a)

There is some ambiguity in how to define the turbulent Reynolds numbers in (4a) and (5a). Strictly speaking, e is the dissipation rate near the interface, and it is better to use estimates of e directly. Be that as it may, both models agreed with limited sets of data, but gave very different results when it came to predictions of transfer rate under the same conditions. Theofanous et al. 14 resolved this discrepancy by showing that the large- and small-eddy models gave the asymptotic behavior of the transfer coefficient at small and large turbulent Reynolds numbers (Rej = uA/v), respectively. As Reynolds numbers in the field are variable, but often quite high, 101 to 105 for winds between 1 and 8 m s _ 1 in lakes, the small-eddy model

188

S. Banerjee and S.

Maclntyre

and the surface divergence model are appropriate, and many experiments are directed towards finding the near surface energy dissipation rate, e, as discussed later. It is also important to note that Banerjee et al.'s 13 SE model is quite general, and energy dissipation may arise from many factors, e.g. wind shear, natural convection, wave breaking, rain, etc. In fact, Banerjee et al. originally applied the model to estimating mass transfer in situations where vorticity generation by capillary waves was important. Banerjee4 derived a general form of the expression for the mass transfer coefficient for the case where there is no wind shear at the interface and the far-field turbulence is homogeneous and isotropic, based on the blocking theory of Hunt and Graham 15 . This theoretical result would appear to have a more restricted application than the SE model, but is worth considering. Using a result from McCready et al. 16 , Banerjee4 showed that for unsheared interfaces at which high Sc gas transfer occurs, £ = Sc- 1 /2 R e t -V2 [{du,/dx

+ d,//dy)*]

^

(6)

where the subscript int denotes the interface, and all quantities on the RHS of (6) have been nondimensionalized by u and A, the integral scales in the far field. The quantity in square brackets in (6) is the square of the surface divergence field due to the fluctuating motions. Here v! is the fluctuating velocity in the streamwise direction x, v' is the fluctuating velocity in the spanwise direction y - both directions being tangential to the interface. The expression arises in a straightforward way by noting that at a free surface the velocity fluctuations normal to the interface are given by w' ~ dw'/dz\zint

+ HOT

(7)

as a result of the boundary conditions (whereas at a solid surface w' ~ d2w'/d2z\bound(z2/2) + HOT). If (6) is written in dimensional terms P ~ [£7]1/2,

(6a) _1

where 7 is the dimensional surface divergence (s ) and D is the molecular diffusivity. The surface divergence cannot be predicted without a theory, and therefore the Hunt and Graham 15 blocking theory was used to relate it to the farfield turbulence characteristics when they are homogeneous and isotropic. Note, however, that Banerjee's result in (6) and (6a) is quite general and could be applied to situations where the surface divergence is governed

The Air-Water

Interface:

Turbulence and Scalar Exchange

189

by other phenomena, e.g. microbreaking waves which show regions of high surface divergence behind them. It could also be used for situations where surface divergence arose from natural convective motions in the liquid due to heat losses. In a sense it takes the place of the renewal parameter f in Danckwerts 11 , but is more easily measured - usually by scattering particles on the liquid surface and measuring their trajectories (see Kumar et al. 17 ). To proceed, for an unsheared interface with homogeneous isotropic farfield turbulence, the mass transfer coefficient was derived by Banerjee4 for high Sc as P/u ~ 5c- 1 / 2 Re t - 1 / 2 {0.3[2.83R4 / 4 - 2.14Re 2 7 3 ]} 1 / 4

(8)

with the proportionality constant ~ 0(1). This is sometimes called the surface divergence (SD) model. The quantity within the first set of parentheses is the square of the nondimensional surface divergence. The expression was asymptotic to Re t ' at small turbulent Reynolds numbers and to Re t at large turbulent Reynolds numbers, which was in line with Fortescue and Pearson's 12 models and Banerjee et al.'s 13 small-eddy model and validated Theofanous et al.'s hypothesis. Note also that the expression given above applies only to clean, unsheared interfaces with no effects due to surfactants or natural convection. If surfactants are present, the Schmidt number exponent may be somewhere between —1/2 and —2/3 (the value for a solid boundary), but the relationship between (3 and the surface divergence in Eq. (6a) is preserved, with a slightly different exponent. It should also be noted that (8) applies to a rigid slip surface because Hunt and Graham's theory can be applied to such a scenario. A free interface, however, is mobile and can deform in response to motions on the liquid side. The surface divergence with a deformable interface may be expected to be less, suggesting that the proportionality constant may be less than unity - in fact, 0.20 fits data quite well, when Re t is the turbulent Reynolds number in the bulk fluid. If wind shear is imposed, then, as mentioned earlier, the turbulence structure near the interface has characteristics somewhat similar to that of wall turbulence. The surface renewal (or surface divergence) models (3) and (6a) are still expected to apply for liquid-controlled transport processes, but the turbulence structure is now controlled by generation in the nearinterface region. The appropriate scaling variables are now related to the wind stress imposed on the water surface and the fluid kinematic viscosity (the so-called inner variables). A complication arises at a deformable interface in comparison to a rigid boundary in that part of the energy transferred from the wind to the water goes to generate waves, and part to generate

190

S. Banerjee and S.

Maclntyre

turbulence directly. Furthermore, the interfacial waves and turbulence can interact in complex ways. In any case, for situations in which waves break infrequently, the main near-interfacial turbulence structures are found to scale with the frictional drag (Df) at the interface (see De Angelis et al. 18 ). The appropriate velocity scale is then Ufrict ~ {Df/p)1/2, where p is the fluid density. Both the surface divergence, 7, and the time between renewals, f, are expected to scale with Mfrict and v (the kinematic viscosity). Indeed, both direct numerical simulations and experiments indicate that the turbulence events leading to surface divergence and renewal are "sweeps", i.e. events that bring bulk fluid to the interface. On the liquid side these are sometimes termed "upwellings" and on the gas side, "downdrafts" . If the time between sweeps is taken as the time between renewals, then experiments and direct numerical simulations indicate that this scales as Til*2

- ^ i

~ 100

(9)

leading to, with the subscript w denoting the liquid side (sometimes the subscript L is also used) /^Sc&Vufrict.w-O.l.

(10)

Banerjee4 showed that (10) agrees with the lab-scale data available at that time and De Angelis et al. 18 found agreement with wind-wave tank data taken subsequently. We will discuss these matters further when laboratory experiments are reviewed. For a solid (as is essentially the case for the gas-side) surface, the form of the parameterization changes as turbulence is more strongly damped with w' ~ z2 (where z is the distance from the surface). This leads to /?aSC2/3/
(11)

where the subscript a, or in some cases, G, denotes the gas side. Note that Mfrict is based on the frictional drag component, which is the total drag only if the surface is level. With interfacial roughness due to waves, the total or effective drag includes form drag. However, the expression in (10) requires estimation of the frictional drag component from which u| r i c t is calculated. An expression for u* like that in Charnock will overestimate Wfrict and give a value of f3 somewhat larger than found in experiments if (10) is used. This aspect will be discussed in more detail in Sec. 4. The various expressions for the transfer rate of sparingly soluble gases are summarized in Table 1, which also includes expressions from Csanady 19

The Air- Water Interface:

Turbulence and Scalar

Exchange

191

and Soloviev and Schluessel20. It should be mentioned that there have been several other forms of parameterization suggested, e.g. by Caussade et al. 21 and Coantic, 22 so Table 1 does not list all the parameterizations proposed. 3. Unsheared Air-Water Interface In this section we will consider turbulence phenomena and scalar exchange when wind shear at the interface is relatively small or nonexistent. In laboratory studies, this situation is reproduced in experiments using either a stirred tank or open-channel flow. Most stirred-vessel experiments are done by using an oscillating grid to generate a relatively homogeneous isotropic turbulence field in the bulk fluid. For open-channel flows, turbulence is generated by the shear at the bottom. In some situations, waves are superimposed on open-channel flows for investigations of wave-turbulence interactions. In most of these experiments, it is difficult to keep the liquid surface free of surfactants, and therefore the results should be treated with caution, unless precautions have been explicitly taken to keep the surface clean. This is especially true for stirred-vessel experiments, where the liquid surface is quite stagnant. Furthermore, most laboratory experiments have not investigated the effect of heat loss (or gain) by the liquid. As we will see later, natural convection associated with heat loss is an important factor in the field and clearly an area where further laboratory research is warranted.

3.1. Laboratory

Studies

and

Simulations

Stirred tank experiments Theoretical approaches to calculating turbulence structure near a wall moving at the mean fluid velocity were proposed by Hunt and Graham 15 . In their work, a thin viscosity-dominated layer was postulated to form near the wall. In the region outside this layer, but within one integral length scale from the wall, the turbulence structure was modified compared to the far field. In particular, the scales parallel to the surface showed a slight increase, whereas the normal component was suppressed. Theoretical prediction of the damping of the normal component is possible from the theory as a function of the wave number. Brumley and Jirka 23 conducted experiments near the free surface of grid-stirred tanks to check these results. They measured the turbulence energy spectrum as a function of distance from the surface for both the surface-normal and horizontal velocity fluctuations. They parameterized the fits using detailed calculations from the Hunt-Graham theory, and reference should be made to Brumley and Jirka's 23 paper or

192

S. Banerjee and S.

Maclntyre

Banerjee4 for the form of the energy spectra that fit the detailed calculations from the Hunt-Graham theory. Hunt-Graham theory appeared to provide a reasonable means of estimating the various parameters in the region of the surface and was used subsequently by Banerjee4 to obtain the form of the high Schmidt number mass transfer coefficient on the liquid side shown in Eq. (8) - based on surface divergence estimates. Turning now to gas transfer data, Chu and Jirka 24 measured the gas flux at the air-water interface of a grid-stirred tank. There is some concern that their data may have been affected by accumulation of surfactants at the water surface. This is discussed in more detail by McKenna et al. 25 , who also made simultaneous mass transfer and surface divergence measurements using DPIV. They found that the particles used for DPIV gave rise to surface-active effects unless they were thoroughly cleaned. In fact, the mass transfer rate data with particles that had not been washed fell below data without particles or with washed particles. The data with and without particles are compared with Chu and Jirka's data, and the predictions of the parameterization in Eq. (8) in Fig. 3. As evident from the figure, the particles reduce the gas transfer velocity, and Chu and Jirka' s data lies somewhere between the cases in McKenna et al., with and without particles. The Banerjee4 parameterization shown in Eq. (8) also gives a reasonable fit to the data, which is encouraging, since it was developed before the data were taken. The constant C ~ 0.20, which might be expected since Eq. (8) is based on a "rigid lid" approximation for the free surface, and in reality there will be some give which would reduce the surface divergence. Note also that the turbulent Reynolds number, Re t , in Eq (8) is half the turbulent Reynolds number, RenT, used by McKenna et al. Also, the point in McKenna's data shown in Fig. 3, where (3 (or k) ~ 11 cm/hr at RenT ~ 180, is out of line with all the other data and may be an outlier. Open-channel studies We turn now to considering turbulence generated by shear at the bottom of a channel flow. The structure of the overall fluid motions and their effect on phenomena near the free surface have been the subject of a number of studies, notably those of Komori et al. 26 , Rashidi and Banerjee 27 and Komori et al. 28 More recently, Kumar et al. 17 published the results of an extensive study of free-surface turbulence in channel flow using DPIV measurements of fluid velocity at the surface. They characterized the surface as consisting of spiral eddies, upwellings and downdrafts. They were able to show that the upwellings were caused by large active structures (bursts) that emanated

The Air- Water Interface:

Turbulence and Scalar

Exchange

193

from the bottom of the channel and impinged on the surface. Associated with these upwellings, and apparently formed by their action, were vortices (spiral eddies), typically attached to the free surface - something like whirlpools. These vortices persisted for long periods, as viscous dissipation was quite low in them and they were usually annihilated either by merging or by upwellings impinging on them. A typical merger of two vortices at the free surface is shown in Fig. 4, and annihilation by an upwelling is shown in Fig. 5, taken from Kumar et al. 17 11 • Particles present A Particles absent • Washed particles

10

9

* Chu&Jirka, 199?

8

*

; 7

,'

0 *

54 3 100

150

200 Re

250

300

350

400

HT

Fig. 3. Mass transfer coefficients at the unsheared surface of a grid-stirred vessel. Particles evidently give rise to surfactant-like effects. The data with the particles absent is considered the most reliable. The line is from Eq. 8 with C — 0.20 (also called C3 in Table 1).

They also measured the energy spectrum of the surface parallel velocity fluctuations and showed that the spectrum split into two ranges, viz. a —5/3 power range at low wave numbers and a —3 range at high wave numbers, as shown in Figs. 6 and 7 for both the streamwise and spanwise components. Also shown are some previous DNS results from Pan and Banerjee 30 that are in agreement with the experiments. Pan and Banerjee used a pseudospectral simulation technique for open-channel flows with a no-slip wall at the bottom and a slip surface at the top. They also turned off the shear (and hence turbulence generation) at the bottom wall for some of their studies to examine the dynamics of decaying free-surface turbulence. These results have been used to develop and test subgrid scale models for

194

S. Banerjee and S.

Maclntyre

Table 1. Various correlations for gas transfer velocity (/3 or k) for liquid-side at high Schmidt number. C i , C 2 , C3,d, C5 are constants. p\ is the fraction of the surface undergoing intense renewal. P2 is the probability distribution of renewal events, Rf is the flux Richardson number gHv/pCpuflict and RfCr is the critical value ~ 1.5x 1 0 - 4 . Ke is the Keulegan number v%rict(gv) and Kecr ~ 0.18. H is the surface heat flux obtained by summing latent, sensible and long wave radiation fluxes. (Modified from Maclntyre et al. 5 ). MODEL Large eddy Small eddy

kSl>* =

Eqn.no. 4

C*i«Re t " 1 / 2 C2uRe~1/4

5, 5a

References Fortescue and Pearson 1 2 Banerjee et al. 1 3 Lamont and Scott 2 9 Theofanous et al. 1 4

Two regime Surface divergence

Large eddy for Ret < 500. Small eddy applied for Ret > 500

Sc1'2^]1/2

6a

Banerjee 4

Surface divergence (no shear) Interfacial shear (liquid side) Interfacial shear (gas side) Eddy resolving analytical Surface processes

C 3 u[0.3(2.83Ref / 4 -2.14Re?/3)]i/4Re-1/2

8

Banerjee 4

10

Banerjee 4

11

Banerjee

a l

"Met,™

0-07 < i c t , G S c - V 3

Csanady 1 9

C

4Pluliict,w

C5P2Uiict!W(l x(l +

+

Rf/RfCr)V*

Soloviev and Schluessel 20

Ke/KeCr)-1/2

turbulence near free surfaces, but we will not consider these further here. To proceed, the —5/3 power range at the low wave numbers suggests quasi two-dimensionality of turbulence in the near-surface region. There is up cascading of energy from the wave number range at which the spectrum splits to the small wave numbers. Physically, this means that the smaller structures merge to form larger structures, which is what is seen in Fig. 4. These conclusions regarding quasi two dimensionality of the surface turbulence are supported by the direct numerical simulations of Pan and Banerjee. In fact, the two-dimensionality can be seen from the contours of the vortical structures hanging down below the free surface in Fig. 8. Several aspects need to be discussed related to these experiments. First, particles were scattered by Kumar et al. on the free surface in order to conduct the DPIV. As McKenna et al. have shown, this may give rise to some surfactant effects, even though such effects would be expected to be less apparent in open-channel flow than in grid-stirred tanks. Kumar et al.

The Air- Water Interface:

Turbulence and Scalar Exchange

195

were aware of this problem, however, and used rather low particle coverage of the free surface to minimize it. Details may be found in their paper. Nonetheless, these effects cannot be discounted. Second, the direct numerical simulations of Pan and Banerjee were in the "rigid lid55 approximation, i.e., the free surface was replaced by a slip surface. The interface had no give in the vertical direction in the simulations, and this might not be a good assumption. Third, while the comparison in the shapes of the energy spectra between the experiments and simulations is remarkable, nonetheless the coincidence in actual values is not significant, as the integrals are normalized to the same values over the same range of wave numbers, i.e. it is only the shapes that are similar. Finally, the experiments under consideration did not have significant interfacial waves, and the Reynolds numbers were quite low, though turbulent. The low Reynolds numbers for the flow could lead to an overestimate of the importance of surface renewal motions that originate at the bottom wall. At high Reynolds numbers, the channel flow might be quite homogeneous and isotropic near the free surface and the surface-region turbulence structure might be more like that for a grid-stirred system.

Fig. 4.

Merger of two vortices at the free surface of an open-channel flow. Top view.

For details of the DPIV technique used by Kumar et al. 17 , reference should be made to the original paper. As free surface turbulence involves motions with a wide dynamic range, some special image processing considerations are involved, as discussed briefly in what follows. The methodology - based on multi-grid image processing algorithms for rigid body motion analysis, estimates the displacement vectors at discrete particle locations. The essence of this technique is to estimate large-scale motions from image intensity patterns of low spatial frequencies and small-scale motions from

196

S. Banerjee and S.

Maelntyre

Fig. 5. Annihilation of a vortex at the free surface by an upwelling. The dark area in panel "d" is an upwelling where the visualizing particles have been swept outward.

intensity patterns of high spatial frequencies. Cross-correlation between a pair of time-separated particle images is implemented by the hierarchical computational scheme of Burt 31 . Each image is convolved with a series of band-pass filters and subsampled to obtain a set of images progressively decreasing in resolution and size. A coarse estimate of the displacement field obtained from pairs of lower resolution images is used to obtain more accurate estimates at the next (finer) level. Processing starts at the level of lowest resolution and stops at the highest resolution level, which contains the original image pair. Due to subsampling of low-resolution images, the match template size can be kept constant for all stages of computation, thus eliminating the dependence of the largest resolvable displacement on the size of match template. In the present work, the search area at each level is kept constant at 3 x 3 pixels and the match template size at 5 x 5 pixels for all levels of computation. The algorithm has been implemented using simple thresholding based on the confidence level of an estimated displacement vector, as suggested by Anandan 32 . However, the confidencelevel-based smoothing technique for rigid body motions (continuous velocity fields) could not be applied to displacement estimates obtained at discrete points, i.e., the particle locations. Instead, smoothing was performed over the area covered by each particle. The algorithm has been tested against

The Air- Water Interface:

Turbulence and Scalar Exchange

197

10° 10''

10*

10"

10

o """

1

(a

10

ioo

attik,

10°

•,

10-

ID"*

10

s

10

(b)

^\.«

„

0

\ \ ^ 1

• DNS HDPIV [Exp k" k"

10""

-

^^

-

1

10

100

2*hk,

Fig. 6. Energy spectra for the streamwise velocity component at the free surface (x streamwise direction, y — spanwise direction).

direct numerical simulations of turbulent flows when the flow field is known and particle images have been generated from these with the addition of noise. Both the accuracy of motion estimation and the computation time are seen to improve as compared to conventional PIV methods. Before moving on to scalar transfer results in open-channel flows, the effects of waves interacting with a turbulent stream will be considered. Early studies of mechanically-generated waves interacting with a turbulent current were those of van Hoften and Karaki 33 , Iwagaki and Asaro 34 , Kemp and Simons 35,36 , Simons et al. 37 , and more recently, Rashidi et al. 38 , Supharatid et al. 39 , Nan et al. 40 , and Nan 41 . The most extensive work is that of Nan, in which 3D laser Doppler anemometry (LDA) measurements were made as a function of flow depth for a wide range of wave-to-current conditions, albeit at relatively low depth-based Reynolds numbers - though still resulting in fully developed turbulence. Nan's data, over a range of

198

S. Banerjee and S.

Maclntyre

10°

10''

"s-

^ V . 10*

-

\

• DNS 1 HOPIV (Expl - - - k "4

--

10"

k

\ •

1

10 s 0

*

1

10

(C)

100

fcthk,

10°

10''

-

10 z

-

• '

r

^ < ! ! \

1

" ^ v

\ 10"4

" (d)

0

\

• DNS HDPIV (Exp k" — - k"'

1

10

100

2nhk,

Fig. 7. Energy spectra for the spanwise velocity component at the free surface (x streamwise direction, y — spanwise direction).

Reynolds numbers and wave strength parameters, indicates turbulence enhancement in the near-surface regions, the effect increasing with the wave strength parameter Ub/U which is the ratio of the wave-induced Stokes drift velocity to the mean velocity. This is shown for the streamwise, wall-normal and spanwise components in Figs. 9, 10, and 11. While there is some variation in the friction velocities measured at the bottom of the channel with and without waves, these changes are not large, so the figures do indicate the enhancement in turbulence intensities caused by the waves, even though the intensities, with and without waves, are nondimensionalized by the respective friction velocities. Note that the fluctuating velocities due to the wave motions themselves have been removed by ensemble averaging over many identical waves. Therefore, what is left are the turbulent fluctuations. Nan took data with essentially 2D waves, so that the spanwise fluctuations are not at all contaminated by the wave mo-

The Air- Water Interface:

Turbulence and Scalar Exchange

199

Fig. 8. Vortical structures hanging down from a free surface on which the instantaneous streamlines are shown (Pan and Banerjee 3 0 ). The surface is viewed from the side of a channel flow, looking up from below. The streamwise direction is 4wh and the spanwise direction is 2irh in length for a liquid layer thickness of 2/i in these calculations.

tion. (In. fact, the spanwise wave components of motion are negligible.) Nan attributed the increase in turbulence to- a wave-induced Reynolds stress, which was significant when waves interacted with the turbulent current. While this suggests that waves could have substantial effects on mass transfer, we are not aware of any experiments that have directly measured the effect of such mechanically generated waves on scalar exchange. However, there have been experiments on gas transfer in laboratory flumes and circular wind-wave facilities, as well as combinations of mechanically generated waves with wind to simulate such effects. These do indicate increases in gas transfer velocities when waves are present. We turn now to consideration of scalar transfer at the free surface of open-channel flow. A pioneering study in this regard was that of Komori et al. 28 They not only measured gas transfer rates (for C(>2)5 but also made simultaneous measurements of surface renewal events which emanated as "bursts" from the bottom of the channel. The data cover a range of water velocities of 5.9 to 23.5 cm/s and depth-based Re of about 2800 to 10,000. They found a remarkable correlation between the frequency of surface renewal events (f8) and the gas transfer velocity, ft. However, the measured

200

S. Banerjee and S.

$~* 3°

o (0

3

iyu

BJ+

I

0.33 0.30 0.25 0.29 0.42 0.57 0.54

0.24 0.27 0.37 0.35 0.35 0.36 0.44

-

§3 2 S

1

-

-

—

'

1

'

I

'

4

~ -

//

-

Run 1 3 5 6

8

.yy

i l l

*•

Maclntyre

i

3

T^r^-^

1

ill

E

3

n

0.0

i

i

0.2

0.4

,

t

06

,

i

0.8

1.0

y/h Fig. 9. Enhancement of streamwise turbulence intensity due to waves as a function of depth. The average position of the free surface is at y/h = 1.0. Here Uf,/U and Gi+ are parameters that are measures of wave strength and frequency. The mean depth of the flow is h, and y is measured from the bottom of the channel. The subscript w denotes channel flow with superimposed mechanically generated waves, and the subscript c denotes channel flow without waves, u* is the friction velocity of the channel flow. Depth based Reynolds number ~ 5000.

gas transfer velocity is about a third of what would be predicted by surface renewal theory, as pointed out by Banerjee4, i.e. (3 = 0.35 (Dfg)1/2. The discrepancy may have arisen because Komori et al. did not directly measure surface divergence, but rather viewed the structures that were ejected from the bottom of the channel, not all of which may have impinged on the free surface. Knowlton et al. 42 attempted to validate the surface divergence theory by calculating surface divergence directly from the velocity field measured by Kumar et al. 17 and then using it to solve the 3D concentration field equation from which they calculated the mass transfer coefficient. The only case for which Kumar et al.'s data coincided with one of Komori's cases was for depth-based Re = 2800. Knowlton et al. found remarkable agreement with Komori et al.'s data for this case. However, when Knowlton et al. calculated the gas transfer data based on a "rigid lid" direct numerical simulation, they found gas transfer velocities that were about 2-3 times higher. These results are also shown in Fig. 12. In Table 2, we show the predictions of Eq. (8) with C = 0.20 (which was the value for C that agreed with the stirred vessel gas transfer data of McKenna et al.) It is evident that

The Air- Water Interface:

1

t

-

XTs 3

E

—

Turbulence and Scalar

1

1

ub/U

TO+

Run

0.33 0.30 0.25 0.29 0.42 0.57 0.54

0.24 0.27 0.37 0.35 0.35 0.36 0.44

1 3 5 6 7

i

,

201

Exchange

.

I.

;

.

/

/ . / / / /

-

8 9

-

-

"

•

0.0

I

I

0.2

0.4

i

,

r

0.6

0.8

1.0

y/h Fig. 10. Enhancement of the vertical turbulence intensity due to waves as a function of depth. Same notation as in Fig. 9.

1

-

0.33 0.30 0.25 0.29 2 0.42 - — — 0.57 - — 0.54

0.24 0.27 0.37 0.35 0.35 0.36 0.44

1

'

1

/ / / /

/

.,

-

——

'—

;

i

Run 1 3 5 6 7 8 9

-

0.0

i

i

1

0.2

0.4

0.6

.

1

0.8

1.0

y/h Fig. 11. Enhancement of the turbulence intensity in the spanwise direction due to waves, as a function of depth. Same notation as in Fig. 9.

the agreement with these predictions is quite good. The velocity scale for Eq. (8) is taken to be the wall friction velocity, and the length scale was the depth. While these scales are reasonable, it is likely that the length scale is a weak function of the depth-based Reynolds number, i.e., (A/depth) varies

202

S. Banerjee and S.

Maclntyre

as Re~ ' , which is what would be expected in the core region of pipe flow. Making such an assumption would improve the agreement between Eq. (8) and the experimental data, which is rather over predicted at low Reynolds numbers and under predicted at high Reynolds numbers. Knowlton et al. concluded that a rigid lid approximation was poor for mass transfer calculations and that the "give" in the surface significantly reduced surface divergence at the air-water interface. This is also consistent with C ~ 0.20 in Eq. (8). These data have been taken in relatively shallow channels, and ejection events from the bottom boundary maintain their coherence up to the interface. For much higher Reynolds numbers and greater water depths, the bulk turbulence structure is expected to more closely approximate homogeneous, isotropic turbulence, where the surface divergence model given in Eq. (8) might again be expected to predict gas transfer velocity. In all this work the effect of heat loss from the liquid was not investigated. As demonstrated in the next section, such effects are important in the field - particularly at low wind velocities. We move on now to field experiments at relatively low wind velocities.

FULL DNS: 0.0750 Sc"° 5 DNS/DPIV (2h=1.5cm) EXPERIM: 0.0273 S c ~ 0 5

Schmidt Number

Fig. 12. Comparison of DNS with rigid slip surface as the interface in channel flow with experiment and calculations based on experimentally determined surface divergence field. Note that the rigid surface gives higher values for the gas transfer velocity.

The Air- Water Interface: Table 2. C = 0.2. Run No. I II III IV V VI VII VIII IX

Turbulence and Scalar Exchange

203

Comparison of Komori et al.'s 2 8 experimental data with Eq. (8), with u* cm/s

S (cm) 1.1 2.9 3.1 5.0 5.1 6.4 7.0 10.0 11.2

3.2. Field

(cm/s) 23.5 9.7 18.3 5.9 11.9 19.9 13.8 10.5 10.9

1.48 0.61 1.06 0.37 0.69 1.01 0.75 0.58 0.59

experimental P X 10 5 (m/s) 1.65 0.75 1.60 0.45 0.90 1.2 1.3 0.7 0.8

Eq. (8) prediction /3 x 10 5 (m/s) 1.97 0.81 1.22 0.49 0.78 1.01 0.8 0.56 0.55

Studies

Field studies using tracers to determine gas transfer coefficients have shown considerable scatter at low wind speeds (see summary in Maclntyre et al. 5 , Cole and Caraco 43 , Crucius and Wanninkhof 44 ). While shear will make a contribution to turbulence at low wind speeds, the contribution from heat loss and rain is significant (Shay and Gregg 45 , Anis et al. 46 , Maclntyre et al. 5 , Ho et al. 47 ). Eddy sizes and rates of dissipation of turbulent kinetic energy e in the upper water column including the air-water interface can be determined with temperature-gradient microstructure profiling. Such profilers ascend through the water column in free fall mode sampling temperature, temperature gradients, conductivity, fluorescence, and photosynthetically available radiation at 100 Hz. With an ascent rate of 0.1 m/s, mm scale resolution is obtained. While estimates of e have been obtained in numerous oceanographic investigations, the ship's wake confounds the values in the upper 10 m. Data obtained during deployments from small boats while the profiler is rising illustrate the turbulence at the air-water interface. Examples of such profiles in lakes include Maclntyre 48 ' 49 ' 50 , Robarts et al. 51 , Maclntyre et al. 52 , Maclntyre and Jellison 53 . Typically during low winds and when incoming radiation exceeds heat losses, the upper water column is stably stratified to the air-water interface or has small eddies (1 mm 20 cm scale) at the air-water interface. When heat loss exceeds incoming radiation, eddies can be much larger, typically meter scale, and, depending on depth of the water body, can extend to the bottom or at least through the weakly stratified upper mixed layer. In those cases, temperature profiles have a characteristic shape with low temperatures near the air-water interface, slightly higher temperatures just below and then a decrease of temperatures (Anis and Mourn 46 ). Rates of dissipation of turbulent kinetic

204

S. Banerjee and S.

Maclntyre

energy below the air-water interface during periods of heat loss depend on buoyancy flux (Shay and Gregg 45 , Anis and Mourn 46 ). Profiles illustrating dissipation rates at low speeds are included in (Maclntyre 48 ' 49 ' 50 ). During times when wind speeds were less than 3 m s _ 1 , values ranged from detection (10~ 10 m 2 s - 3 ) up to 10~ 7 m 2 s - 3 . At Mono Lake, CA, when winds were low but sometimes reached 4 m s _ 1 , dissipation rates occasionally exceeded 10~ 6 m 2 s - 3 (Maclntyre and Jellison 53 ). A recent study in the Soppensee, a small (25 ha) sheltered lake in Switzerland, employed thermistor chains, a microstructure profiler, measurements of surface meteorology, and eddy covariance techniques to determine key processes driving gas flux (Eugster et al. 8 ). Wind speeds were always less than 3 m s _ 1 . During times when heat inputs exceeded heat losses, the variance in temperature gradient in 15 minute bins, a proxy for turbulent mixing, at 4.4 m depth was negligible. In contrast, the variance was higher at times when heat losses exceeded heat gains. Similarly, bin averaged temperature gradients between 4.5 and 7 m depth increased at times of heat loss. At these times, gas flux was elevated 30% over stratified periods. Eddy covariance measurements at Toolik Lake, AK, showed gas fluxes were three times higher during periods of heat loss than during stratified periods (Maclntyre et al. 54 , Eugster et al. 8 ). Schladow et al. 55 describe the formation of thermal plumes and show the dependence of the gas transfer coefficient for oxygen on heat loss. These data clearly illustrate the importance of convective motions due to heat loss for gas exchange at low wind speeds and indicate that use of a surface renewal model to calculate gas transfer coefficients would more accurately represent the turbulence at the air-water interface than a model based on wind speed alone.

4. Sheared Air-Water Interface In Sec. 3, turbulence structure and scalar exchange were discussed for situations in which turbulence was generated by motions in the water column in the absence of significant wind shear. Field studies indicated the importance of heat loss from the water as a mechanism for enhancing turbulent mixing and hence gas transfer at the surface. The structures in the interfacial region therefore arose from interactions with far-field turbulence and thermal plumes generated due to cooling of the surface layer. Such mixed convective flows near the air-water interface have not, however, been studied in the laboratory or in simulations. In this section we turn to phenomena that occur when wind shear dominates in the generation of near-surface turbulence.

The Air-Water

4.1. Laboratory

Interface:

Studies

Turbulence and Scalar

and

205

Exchange

Simulations

A useful starting point is to consider a simple flow situation as shown in Fig. 13, which is a schematic of pressure-driven air and water streams between two horizontal plates. ^^r

Side view

yiiituruniiit

to pulsed current source

, , , , , , , , , , 1 1 1 } > J 1>I>!

1 >J 1 I i { I

. 25 Jim I t wire (vertical)

if

air interface

nuriamtal wire

. .

^j

W plauvkw

.

#

microtabWe tracer — lines

* • water

to pulsed current source

^ V S ' - V V U T O ; ivv'.i'H. v . v vvs. Vv s vv.v ^ v u v v u liorizoiilnl (spanwiK) Ptwirc

^

-microbubbte tracer lines

vertical wire _.^i

)A\\\\wl^i^^wiv^^v\.vvvv^^srgc7'

N* Fig. 13. Schematic of a co-current or countercurrent gas-liquid flow between two horizontal plates. This configuration defines a canonical experiment for investigations of turbulence and scalar exchange at gas-liquid interfaces. The wires are used in experiments to electrochemically generate microbubble lines that act as flow tracers - a device widely used to visualize turbulent flow structures.

Flat sheared interface First, let us consider the simplest situation where the air flow exerts shear on the liquid but not enough to form significant interfacial waves. Now consider what happens to the liquid near the bottom boundary, as this will serve as a reference. If the flow is visualized with lines of micro-bubble tracers which are generated by passing a pulsed current through a spanwise line, the streaky structures shown in Fig. 14 are seen to form close to the bottom boundary.

206

S. Banerjee and S.

Maclntyre

Fig. 14. Visualization of turbulence structure near the bottom boundary of turbulent channel flow. The flow tracers are microbubbles electrochemically generated by pulsing current through a spanwise (horizontal) wire lying a small distance above the channel floor. The flow structure consists of high-speed regions with alternating low-speed, "streaky", regions where the microbubbles accumulate. These regions form and reform and move around but their essential streaky character is always there.

These structures have been known since the pioneering experiments of Kline et al. 56 , who also observed that the low-speed regions would periodically be ejected in bursts, as shown in Fig. 15, which is a side view of the flow with the microbubble-generating wire placed vertically. It is known that these bursts or ejections, and the related sweeps that replace the ejected fluid, strongly influence heat and mass transfer at a wall. Bursts in boundary layers appear to be large-scale motions that contain within them several ejections away from the boundary with interspersed sweeps that bring fluid toward the boundary. A question that arises is whether such structures also occur near gas-liquid interfaces on the liquid side. They can be expected on the gas side, since the liquid surface, to a flowing gas, looks much like a wall. However, to the liquid, the interface is almost a slip surface, with a mean shear being impressed due to the gas flow. The boundary conditions are very different from those at solid walls, and it is not clear what is to be expected. To clarify this, Rashidi and Banerjee 57 carried out experiments in which they visualized the flow structure in a flowing stream of liquid, with shear imposed by the air flow. Fig. 16, upper two panels, shows structures close to the gas-liquid interface, on the liquid side, as the shear rate due to the gas motion is increased. One goes from a patchy structure at low shear to lowspeed, high-speed streaky structures that are qualitatively similar to those observed near a wall. Furthermore, if the flow is viewed from the side, one sees burst-like structures emanating from the gas-liquid interface, much like

The Air- Water Interface:

Turbulence and Scalar Exchange

207

Fi&. J.'i. Rule VH'-W of £». i-liannol flow, showing development of a burst which consbtf-; of an "active" period in which the low speed region shown in Fig. 14 appear to be lilted up in "ejections" followed by downdrafts of fluid called "sweeps" that bring high speed bulk fluid to the wall region. Flow is from right to left and the panels should be viewed starting on the right.

those seen in wall turbulence (see Fig. 16, lower panel). This led Rashidi and Banerjee 57 to conclude that shear rate was the primary determinant of structure in such situations, and the details of the boundary condition were secondary. There are, as discussed later, important quantitative differences in the interface low field on the liquid side from wall turbulence, though the qualitative features are similar. The criterion for transition from the patchy structures seen at low shear to the streaky structures was clarified through direct numerical simulations by Lam and Banerjee 58 . They showed that the criterion for formation of streaky regions near a boundary depends on a mean shear rate (non dimensionalized in such a. way that it is effectively a turbulence production to dissipation ratio)'being greater than one. Lombard! et al. 59 did a direct numerical simulation where the gas and liquid was coupled through continuity of velocity and stress boundary conditions at the interface. They artificially raised-the surface tension to maintain a fiat interface, as waves would introduce additional complexity. The interfacial plane itself shows regions of high shear stress and low shear stress, with low shear stress regions corresponding to the low-speed regions and the high shear stress to the high-speed regions. The low shear stress regions were streaky in nature with high shear stress islands. Lombard! et al. showed that the high shear stress regions occur below the sweeps on the gas side, i.e., the motions that bring high-speed fluid from the outer regions to the interfaces on the gas side. Conversely, ejections on the gas side, which take low-speed fluid away from the interface into the outer flow, strongly correlate with low shear stress regions. The liquid does not behave in this way and the ejections and sweeps do not correlate with shear stress on the interface. The difference between the gas and the liquid phases in the nearinterface region is further clarified by observing the velocity fluctuations

208

S. Banerjee and S.

Maclntyre

H$K;

Fig. 16, Upper panels: Structures at a gas-liquid interface with decreasing wind shear (a) to (d). Lower panel: Side view of a channel l o w with the l o o r at the bottom and the interface at the top of each panel. The top series of panels are for air flow countercurrent to the liquid. Qualitatively similar burst-like structures are seen at the interface, as well m at the channel l o o r . The bottom series of panels indicates bursts formed in cocurrent gas-liquid flow. Bursts are formed at the interface due to wind shear. Flow is from right to left, and time increases in the panels from right to left.

The Air- Water Interface:

Turbulence and Scalar

Exchange

209

on each side of the interface as shown in Fig. 17. The left panel of the figure is for the gas, whereas the right panel is for the liquid. Gas-side turbulence, as is evident, behaves much like flow over a solid wall, i.e., the fluctuations are almost identical to that at a solid boundary in all directions - streamwise, spanwise, and wall-normal. If the distance is measured from the wavy interface, then there is little effect of waves on intensity provided that everything is nondimensionalized with the friction velocity calculated by subtracting the form drag from the total drag. On the other hand, the liquid, as evident from the bottom figure, has the largest fluctuations in the streamwise and spanwise direction right at the interface itself. It sees the interface virtually as a free slip boundary, except for the mean shear. There is a somewhat greater effect of waves, but still rather small in the nondimensionalized form shown. These calculations are done with density for the gas and the liquid typical of air and water, i.e., ~ 1 : 1000. The cases with waves are at conditions where there is no microbreaking.

Fig. 17. Turbulence intensities on the gas side (left panel) and liquid side (right panel) as a function of nondimensional distance z+ from the interface, velocity U\-streamwise, U2spanwise, C/3-interface-normal. Case Do-flat interface; Di-wavy interface. All quantities nondimensionalized by the friction velocity, i.e., the shear velocity calculated on the basis of frictional drag(excluding form drag) at the interface.

Certain other aspects of turbulence on each side of a gas-liquid interface are worth considering. In the wall region, it is known that ejections and sweeps have a spacing in time that lies between a non-dimensional time of 30-100, with 100 being the upper limit. This last period corresponds to the time between assemblages of ejections and sweeps, i.e., bursts. This is shown in Fig. 18 for wall turbulence (i.e., at a solid boundary) as a function of the shear Reynolds number. Rashidi and Banerjee 57 found that on the liquid side of a sheared interface, this type of parameterization held. In Fig. 18

210

S. Banerjee and S.

Maclntyre

the time between interfacial bursts is also plotted as a function of the shear Reynolds number, and the behavior at a macroscopic level is very similar to that at a solid boundary. Thus, the qualitative behavior is similar, though within the bursts themselves there are substantial differences between the gas and liquid sides. I

' '•'

i " ""•

1

' • ' '"I

"

"I

100

•oo °

80 T

o

°

;

60

40 4n

"9

• O

intvriact tux K» wall bursts lit w i n I

2500

i* i

In

iini

5000 Re L

t

i i n* i ,». mmli

T500

10000

•

12500

Fig. 18. Scaling of interfacial and wall bursts. The time between bursts Tg is scaled with the local shear stress and kinematic viscosity.

Turbulence near nonbreaking wavy surfaces The previous discussion focused on turbulence phenomena for coupled gasliquid flows with fiat interfaces - a situation close to the experiments of Rashidi and Banerjee 57 . To understand the effect of waves on the gas side, flow over a surface with two dimensional waves (with spanwise crests and troughs) was investigated by direct numerical simulation (De Angelis et al. 60 ). The waves introduced a streamwise length scale, as indicated by the vortex structure shown in Fig. 19. However, when the phase average velocities are removed from the instantaneous field, then, typically, streaky regions of high speed-low-speed flow are seen, as indicated in Fig. 20. The streak spacing is ~ 100 in units nondimensionalized with the frictional drag (removing the form drag component from the total drag). This suggests that turbulence phenomena near the wavy surface may scale with frictional drag, rather than total drag. On examining the time between sweeps and bursts, a time non-dimensionalized

The Air- Water Interface:

Turbulence and Scalar

Exchange

211

a) Fig. 19. Quasi-streamwise vortical structures (shaded) over a wavy boundary with flow viewed from the side. The different shadings indicate different directions of rotation.

Fig. 20. Streaky structures over a solid wavy boundary with different wave lengths. The top left-hand and right-hand panels have the mean velocity removed and clearly show the streakiness of the high and low velocity regions very near the surface. The second row shows the velocities with the mean velocity included, and the third row shows the local shear stresses.

with the frictional drag, i.e. t+ — tDf/fi, is seen to be between 30 and 90, in agreement with what happens at a flat wall. Here Df is the frictional stress. These simulations are of interest for gas-side turbulence phenomena. However, liquid-side behavior may be expected to be substantially different, as discussed for the flat-interface case. De Angelis and Banerjee 61 have reported some details of DNS with a nonbreaking deformable interface between turbulent air and water streams. Further details are available in De Angelis62. The turbulence intensities and other qualitative features, e.g.

212

S. Banerjee and S.

Maclntyre

streak spacing and burst frequency, on both gas and liquid sides of the interface, were found to scale with upict and v, the kinematic viscosity. Fig. 21 shows an instantaneous snapshot of the interface configuration, together with the high-speed/low-speed streaky structures close to the interface on each side. It should be emphasized that these results only hold for situations where the interface does not break.

Fig. 21. Instantaneous configuration of the gas-liquid interface showing spanwise wave crests. The colors indicate velocity just above the interface (top panel). Red indicates high velocity, blue indicates low velocity, with yellow and green in between. The typical streaky structures seen in experiments are evident in these simulations as well. The bottom panel is just below the interface at the same instant and shows some differences in the local velocities.

Turbulence in microbreaking When L^io exceeds 3 to 5 m/s, relatively short waves steepen and start to microbreak. This is illustrated in Fig. 22, which shows a microbreaking wave viewed from the side. If microbreaking waves are viewed from the top, they are easily visualized by following the trajectories of particles scattered on the water surface. It is found that the liquid does not cross the crest of microbreaking waves, which consequently forms regions of convergence where particles gather. Therefore, an operational way of identifying microbreaking is to observe particles gathering at, and being convected with, the wave crests (see Fig. 23). Non-breaking waves do not show this behavior particles on the liquid surface pass through wave crests. Regions of high surface divergence form behind microbreaking waves, as illustrated in Fig. 23. This may be expected to significantly impact scalar exchange, as indicated in the "surface divergence" Eq. (6a). The subsurface motions associated with the convergence zones at the wave crests and the

The Air- Water Interface: Turbulence and Scalar Exchange

213

ffortzuntal Ptnftfcui (an) Fig. 22. A side Yiew of a microbreakiiig waYe showing typical dimensions.

divergence zones behind remain to be clarified. At this point, we have seen plunging motions associated with microbreaking that appear much larger than those associated with bursts that develop due to wind shear in the absence of microbreaking (Leifer et al. 63 ). This is illustrated in Fig. 24, which shows that the plunging motions associated with microbreaking are almost an order of magnitude larger than motions associated with the "usual" bursts. The impact of microbreaking structures on scalar exchange has yet to be studied. 4*2. Laboratory

Studies

and Simulations

— Scalar

Exchange

Simulation As simulations proved useful in clarifying aspects of turbulence structure near wavy (nonbreaking) air-water interfaces, they have been extended to studies of scalar exchange (De Angelis 62 ). The calculations for Pr or Sc ~ 0(1) are straightforward once the velocity field has been calculated. However, for higher Sc, the spacing of collocation points-normal to the interface must be rescaled, i.e., interface-normal mesh spacing must be reduced as a function of Sc to resolve concentration fluctuations (see De Angelis62 and De Angelis et al. 18 ). Contours of the instantaneous scalar fluxes at interfaces from such DNS are shown in Figs. 25 and 26, and compared with the shear stress at the interface for the gas and the liquid side, respectively. It is immediately

214

S. Banerjee and S.

0

10

20

Madntyre

30

40

Downwind Position (cm)

1 HI r 1

0

10

20

30

40

Downwind Position (cm) Fig. 23. Top view of a microbreaking wave, visualized by particles scattered in the water surface. The particles are swept along by the wave crests, which are clearly discernible in the top panel. The velocity vectors calculated for particle motion (by DP1V) are shown in the bottom panel. Note the convergence zone at the crest and the divergence zone behind.

evident that the gas-side luxes correlate well with the shear stress. This suggests that sweeps give rise to higher scalar exchange rates, as they also produce regions of high shear stress. On the other hand, the flux field on the liquid side shows a much finer structure (see Fig. 26) and no such correlation exists. It is of interest, therefore, to understand what processes control the liquid-side fluxes. To clarify what happens on the liquid side, recall the earlier discussion that liquid-side bursts (sweeps and ejections) do not affect interfacial shear stress, which is mainly determined by gas-phase motions.. Therefore, we must directly examine the relationship between liquid-side motions and

The Air- Water Interface:

Turbulence and Scalar Exchange

215

Fig. 24. Bursts formed by microbreaking shown in bottom 6 panels G-L, and the usual bursts in top 6 panels A-F. Scales are the same. Microbreaking leads to "giant" bursts. The vertical dimensions in the pictures are about 8 cm, and the liquid layer is 10 cm deep (the bottom is not shown).

liquid-side scalar transfer rates. The procedure is illustrated in Fig. 27. Here we plot the instantaneous interface-normal velocity fluctuations, together with the instantaneous Reynolds stress (t^t^), aDLd the instantaneous scalar flux (ui is the velocity fluctuation in the streamwise direction and tig is in the interface normal direction). It is clear from the figure that the scalar flux increases sharply when there is a high velocity (t4) towards the interface of

216

S. Banerjee and S.

Maclntyre

(a)

Fig. 25. (a): The mass l u x field on the gas side of the interface. Note close correlation of regions of high mass transfer with high shear stress. (b):The non-dimensional shear stress field on the liquid side of the interface. The time at which the field is shown is not exactly the same as in Fig. 25(a).

high velocity bulk fluid (high u^). This is a sweep. The figure illustrates that the sweep, which leads to a high surface divergence event, is strongly correlated with high liquid-side scalar flux, which decays like what might be expected of transient absorption into a batch of stagnant fluid between sweeps. This suggests a basis for the "surface renewal" model with sweeps providing the renewal events - as developed previously to parameterize the liquid-side scalar flux in Eq. (10). Scalar flux parameterization The previous discussion suggests a physical basis for the surface renewal theory used to parameterize scalar transfer rates on the liquid side. It is of interest to see how Eq. (10) compares with data and simulations. The

The Air- Water Interface;

Turbulence and Scalar Exchange

217

(a)

Fig. 26. (a): The shear stress field on the gas-side of the interface. The shear is nondimensionalized by pu*2f where u* is the average friction velocity. The streamwise dimension is 4wh and the spanwis© dimension is 2wh, where the gas-layer thickness is 2h. (b): The mass flux field on the liquid side of the interface. Notice the lack of correlation with the shear stress field. The mass flux field is much more fine grained. The time at which the field is shown is not exactly the same as in Fig. 25(b).

Schmidt number dependencies in Eqs. (10) and (11) are compared with, simulation results for two different Schmidt numbers in Fig. 28. It is clear that the Schmidt number dependence is correctly predicted at high Schmidt numbers, but there is some deviation at low Schmidt numbers. Also, the numerical value of the RHS of the equations is roughly correct. Eq. (10) is also compared with wind-wave tank data for SFg and OO2 transfer rates from Wanninkhof and Bliven64 and Ocampo-Torres et al. 65 in Figs. 29 and 30, respectively. The agreement is quite good, though the prediction always lies somewhat below the data. This could be due to a small effect related to turbulence from the channel bottom reaching the interface, which enhances the local turbulence. Turning now to the gas side,

S. Banerjee and S.

218

Maclntyre

ax

? D*"

-10.0 0,0

100.0

r

200.0

300.0

Fig. 27. Panels showing that a sweep, with high u i « 3 gives rise to a high mass transfer rate /3'L on the liquid side. Note t h a t the surface renewal model predicts the instantaneous and integrated value of 0'L quite well if the sweep is considered as a "renewing" event.

id-

ler 10' LIQUID SIDE Fig. 28. Gas transfer velocities nondimensionalized by the frictional velocities versus Schmidt number from direct numerical simulations with a flat interface. The solid lines are from Eqs. (11) and (10) for the gas and liquid sides, respectively. Left panel-gas side, right panel-liquid side.

the gas sees the liquid much like a solid surface, as discussed earlier. So, the surface renewal theory has to be modified somewhat for such applications, as in Eq. (11). This also leads to a different dependence on the Schmidt number. While this is encouraging, the issue of estimating the friction velocity

The Air- Water Interface:

Turbulence and Scalar

Exchange

219

60.0 •

4O.0

20.0 •

100.0 U., (ems') Fig. 29. Gas transfer velocity for SF6 desorption in the Delft wind-wave tank versus shear velocity (gas-side), from Wanninkhof and Bliven 6 4 . The open triangles may have been data taken in breaking conditions. The solid lines are from Eq. (10) and the same equation +50%.

10'

o

o •

o ° #~

o

M

10"

10"

10'

10'

10"

Re.

10'

10'

Fig. 30. Gas transfer velocity, non-dimensionalized by the shear velocity plotted against the shear Reynolds number (based on liquid depth and artificial shear velocity) from Ocampo-Torres et al. 6 5 The lines are Eq. (10) and the same equation +50%.

M

frict remains. This is a difficult estimate to make for field data. Form drag may be expected to increase with wave amplitude and steepness, therefore

220

S. Banerjee and S.

Maclntyre

estimating u* from total drag and using it in the proposed parameterization will significantly overestimate the scalar exchange rates. The effect of microbreaking has to be factored in as it is not clear whether these parameterizations will be applicable in such circumstances. Finally, as discussed in the next section, evaporation, and hence heat loss, from the water increases with wind speed, giving rise to natural convective effects that enhance windinduced turbulence. Such effects are important in field experiments, which are now discussed.

4.3. Field

Studies

In field settings, as wind speeds increase that increase shear, evaporation rates also increase, and at least two mechanisms for generating turbulence co-occur. Shear on the water side of the air-water interface is parameterized by the friction velocity u*w, where r = pwu2w = PO.CDU2 and pa and pw are density of air and water respectively, CD is the drag coefficient, and U is wind speed. A similar parameter exists for heat loss, the convective velocity scale w*, and is calculated from heat gains and losses into the actively mixing surface layer (Imberger 66 ). Only when heat gains exceed heat losses will the surface of the water body be influenced by shear alone. More often, heat loss and shear co-occur at the air-water interface of lakes, wetlands and oceans and contribute to turbulence production. Examples of profiles showing thermal structure and dissipation rates near the airwater interface for winds up to 12 m s _ 1 are provided in Maclntyre 48 ' 49 ' 50 , Robarts et al. 51 , Maclntyre et al. 52 , Maclntyre and Jellison 53 . A full surface energy budget was not completed for those studies, but dissipation rates in Mono Lake in 1995 reached 10~~5 m 2 s - 3 when winds were 12 m s"1 and significant white capping occurred. Surface meteorological data obtained from a meteorological station (http://ecosystems.mbl.edu/ARC/data_doc/lanwater/landwater default.htm) on Toolik Lake, Alaska, a 1.5 km 2 kettle lake in the foothills of the Brooks Range, illustrate the variability in surface forcing over aquatic ecosystems (Figs. 31-33). Data are from 12 and 14 July 2000 (day of year 194 to 197). Five minute averaged wind speeds were less than 4 m s _ 1 at night and increased in the afternoon to highs between 6 and 8 m s _ 1 , a range where microbreaking and shear are both important (Fig. 31 A). Solar insolation, which acts to stratify the lake, was barely detectable for a few hours at midnight and had daily maxima of 800 W m 2 (Fig. 31B). Intermittent cloudiness caused large fluctuations in irradiance. A surface energy budget was calculated as in Maclntyre et al. 67 Surface heat fluxes, which are

The Air- Water Interface:

Turbulence and Scalar

Exchange

221

the sum of heat fluxes from sensible, latent, and long wave radiation, were always negative, ranging from —100 to —400 W m - 2 (Fig. 31Ca). The effective heat flux (Fig. 31Cb) represents the actual heating of the upper surface layer. The surface layer is defined as the actively mixing layer near the airwater interface and operationally as having temperatures within 0.01 °C of those measured within the upper 5 cm of the lake. When the effective heat flux is greater than zero, the uppermost part of the water column will stratify; when it is less than zero, cooling occurs and turbulence can be induced not only by shear or microbreaking waves, but also by heat loss. Heat losses occur in the surface layer for nearly 12 hours of each 24-hour period and for an even longer period on cloudy days. Wind consistently induces some shear at the air-water interface; the shear velocity, u*, varied from 0.002 to 0.01 m s _ 1 (Fig. 32A). The convective velocity scale w„ is more intermittent due to solar heating. Values of w* are comparable to the friction velocity when winds are high and solar radiation low (e.g. day 194.8) and are zero when heat inputs exceed losses. At night, when wind speeds are low, tu* is often twice as high as it*, indicating heat losses will be the major source of turbulence at the air-water interface. Surface layer depths were calculated using data from Brancker TR1050 thermistors on a taut line mooring with loggers located at 0.05, 2.55, 3.3, 3.55, 3.8, 4.05, 4.3, 4.55, 4.8, 5.05, 5.3, 5.55, 5.8, 6.55, 7.55, and 8.55 m below the air-water interface (Fig. 32B). Due to insufficient thermistors in the upper water column, the surface layer was generally assumed to be 2 m deep during times of heating. The depth of the surface layer varied due to cooling and, during periods of high wind (e.g. Day 195.7), is confounded due to tilting of the thermocline. At these times, the surface layer deepens downwind; shallows upwind. The flux of energy into the mixed layer (Fig. 32C) was calculated following Imberger 66 as Fq — ql/2 — (wl + 1.333u»)/2, an approach that does not explicitly include the effects of microwave breaking, q represents the turbulent velocity scale due to both shear and heat loss. The rate of dissipation of turbulent kinetic energy (e) for the surface layer is estimated by dividing Fq by the depth of the actively mixing surface layer. Based on this approach, e rang ed from 1 0 - 1 0 to l O - 7 m s (Fig. 33A). Dissipation rates are underestimates during heating periods and when the thermocline is strongly tilted due to overestimates of the depth of the surface layer. Comparison of the flux of energy into the surface layer when winds dropped to 1 m s - 1 (u*w = 0.002 m s _ 1 ) but the lake was either cooling (Day 196.1, w* = 0.006 m s - 1 ) or gaining heat (Day 196.35, w* = 0) illustrates the contribution of heat loss to turbulence in the surface layer when shear is low. When cooling occurred (Day 196.35), Fq for the

222

S. Banerjee and S.

194

194.5

195

Maclntyre

195.5

196

196.5

197

196.5

197

196 5

197

1000

194.5

195.5

400

HeatFrluxes f

a-

200 0 -200 -400 194

194.5

195

195.5

196

Day of Year 2000 Fig. 31. (A): Five minute averaged wind speeds (m s _ 1 ) , B) solar insolation (W m - 2 ) and C) surface heat flux (a.) and effective heat flux (W m - 2 ) (b.) into Toolik Lake from July 12-15 (day of year 194 - 197) 2000 (unpublished data, Sally Maclntyre and George Kling).

The Air- Water Interface:

Turbulence and Scalar

Exchange

223

surface layer was 10~ 7 m 3 s - 3 and e was 10~ 8 m 2 s~ 3 , whereas when the surface layer was gaming heat (Day 196.1), Fq was 1 0 - 9 m 3 s - 3 . e was less than 10~ 9 m 2 s~ 3 . Generally, winds drop at night when cooling occurs, so the heat loss leads to continued production of turbulence and often to entrainment of deeper waters into the upper mixed layer. Temperature-gradient microstructure profiles illustrate the thermal structure, rates of dissipation of turbulent kinetic energy, and eddy sizes during periods with 7 m s _ 1 and considerable convective cooling (Fig. 34, 35) and during a period when winds were 4 m s " 1 but heating was occurring (Fig. 36). The temperature profile at 2014 h 12 July 2000 showed slightly cooler temperatures near the surface, a 0.02 °C increase in temperature by 4.5 m, and a gradual decrease in temperature to the base of the mixed layer at ca. 6.1 m (Fig. 34). Such a pattern is typical of conditions when convection due to heat loss occurs (Shay and Gregg 45 , Anis and Mourn 46 ). Temperature fluctuations of 0.01 °C and smaller were ubiquitous. Below the mixed layer, temperatures decreased rapidly. Eddy sizes are computed by calculating the displacement scale, the depth to which cooler water would descend to find water of similar density. In this case, the upper 50 cm may be considered an eddy, and dissipation rates are slightly higher within it (10~ 7 m 2 s~3) than in the water immediately below. A second eddy occurred between 0.5 and 3 m depth, and dissipation rates within it were ca. 3 x 10~ 8 m 2 s - 3 . A third eddy extended from 3 to 5 m and had similar dissipation rates. Despite this structure, the near uniformity in e and the overall pattern in temperature indicates this is a convectively mixed layer. A profile taken 15 minutes earlier had a similar temperature profile and average dissipation rate, but no accentuation of dissipation in near surface waters (not shown). While the profile at 2043 h (Fig. 35) was also obtained during a cooling period, the 6 m deep upper mixed layer does not have the classic structure of a convecting layer, but instead consists of 3 layers. In the uppermost one, temperature ranged from 11.97 to 12.01 °C and dissipation rates were 10~ 6 m 2 s - 3 at the surface to and decreased to 3 x 10~ 8 m 2 s - 3 at its base at 1.5 m. The higher values at the surface may have been due to the combination of shear, heat loss, and microbreaking. While the middle layer may have been an overturning eddy, the step changes associated with the top and bottom of the layers suggests lateral advection was occurring; the enhanced dissipation rates at the top and bottom of the lower layer support this interpretation. Dissipation rates estimated from the surface energy budget for the entire 6 m upper mixed layer were 10~ 7 m 2 s~ 3 (Fig. 33), which is

224

S. Banerjee and S.

Maclntyre

0.015 u. w»

A.

<0

E 0.01

f fe n 1 1 ll

ltf>$ y

M.

•a

« 0.005

v* • .. K , , UJJBL 194.5 194

' f*V-

II

•

•

•

•

1

195

1195.5 II

V : fe ? . , . , • , ,

196

:«A

, , ,

196.5

197

194

194.5

195

195.5

196

196.5

197

194

194.5

195

195.5

196

196.5

197

Day of Year 2000 Fig. 32. (A): Friction velocity in water u» and convective velocity wm, (B): depth of the surface layer, and (C): flux of energy into the surface layer for days of year 194-197, Toolik Lake, AK. (unpublished data, Sally Maclntyre and George Kling).

The Air- Water Interface:

'«

Turbulence and Scalar

Exchange

225

10

197

195

195.5 Day of Year 2000

196

196.5

197

Fig. 33. (A): Estimates of rates of dissipation of turbulent kinetic energy (e) based on surface energy budgets and depth of the surface layer derived from thermistor chains. Arrows indicate revised estimates of e when the depth of the surface layer was determined from high resolution temperature profiles in Figs. 35, 36(B).) Gas transfer coefficient calculated from surface divergence model (Eq. 8) ( • • • ) , and two empirical wind based models (Cole and Caraco 4 3 , gray, and Kling et al. 6 8 , black). Data for these models were derived from the meteorological data from Toolik Lake (unpublished data Sally Maclntyre and George Kling); calculations are described in the text. Gas transfer velocity is augmented over wind based models when heat losses are significant and reduced when surface waters gain heat (see Fig. 32A).

226

S. Banerjee and S.

Maclntyre

Temperature f C)

Fig. 34. Temperature profile and rates of dissipation of turbulent kinetic energy (histogram) at 2014 h on 12 July 2000, Toolik Lake, AK., show the structure within the 6.2 m deep upper mixed layer at times when the mixed layer is losing heat and winds reached 7 m s _ 1 . e is higher near the air-water interface and typically between 1 0 - 8 m 2 s - 3 and 1 0 - 7 m 2 s - 3 in the rest of the surface layer. Dissipation rates calculated as in Maclntyre et al. 5 2 .

in reasonable agreement with the measured values. However, recalculating e from the surface energy budget based on a surface layer depth of 0.80 m gives dissipation rates of 10~ 6 m 2 s - 3 which is representative of conditions at the surface (Fig. 33A). Four microstructure casts were obtained in the period from 1978 h to 2051 h 12 July, and the measured dissipation rates in surface waters ranged from 10~ 8 m 2 s - 3 to 10~ 6 m 2 s - 3 . This range reflects the dynamic nature of the processes at the air-water interface. The microstructure cast taken at 1618 h on 14 July was obtained while the 4.7 m upper mixed layer was gaining heat (Fig. 36). Surface temperatures had increased by a degree since noon; winds were 4 m s - 1 . While turbulence occurred throughout the 4.7 m upper mixed layer, dissipation rates were 10~ 6 m 2 s~ 3 in the uppermost 0.8 m. Eddies below it ranged in size from a few centimeters to 50 cm indicating the upper mixed layer did

The Air- Water Interface:

Turbulence and Scalar

227

Exchange

Temperature f C) 11.7 Qj

11.74 11.78 11.82 11.86 •

•

•

•

•

11.9 •

11.94 11.98 12.02 12.06 ^pqsss-| I

12.1

I

Fig. 35. Temperature profile and rates of dissipation of turbulent kinetic energy at 2043 h on 12 July 2000.

not mix as an entity. Dissipation rates were at least an order of magnitude lower than in the surface layer. Eddies embedded in stably stratified waters indicate shear induced turbulence. Estimates of dissipation from the surface energy budget (Fig. 33A) using a surface mixing layer of 0.25 m, based on the step change in temperature at that depth, were slightly less than the value of 1 0 - 6 m 2 s - 3 obtained while profiling. In summary, the distribution of turbulence in the upper mixed layer differs during times with and without cooling. In nearly all cases, turbulence is highest in a shallow layer near the surface. When heating occurs and winds are moderate, the turbulence is more accentuated in surface waters and decreases considerably by the base of the mixed layer. When cooling co-occurs with wind forcing, the turbulence in the upper mixed layer tends to be more homogeneous and eddies are larger. However, even that pattern is variable, possibly due to horizontal advection of different water masses.

228

S. Banerjee and S.

Maclntyre

Temperature f C) 11.14 11.28 11.42 11.56

11.7

11.84 11.98 12.12 12.26

io" £ (m 2 s 3 p '

12.4

10

Fig. 36. Temperature profile and rates of dissipation of turbulent kinetic energy at 1618 h on 14 July 2000 show the structure within the 4.7 m deep upper mixed layer, e is higher in the upper 0.8 m; shear induced eddies occur below.

Ho et al. 69 have calculated the flux of kinetic energy into the mixed layer due to rainfall ranging from 13.6 to 115.2 mm h~x. The fluxes ranged from 0.09 to 1.27 W m~ 2 . Assuming that the energy was initially introduced to a depth of 0.2 m, as supported by measurements of temperature change and SF6 distribution in experiments with rainfall of 68 and 76 mm ft.-1 (David Ho, personal communication), and converting to W k g - 1 (equivalent to m 2 s~3) by assuming a density of 1000 kg m - 3 , gives dissipation estimates on the order of 5 x 1 0 - 4 m 2 s - 3 . These values are an order of magnitude higher than has been observed with microstructure profiling at low to moderate winds and indicates that intense rainstorms could dominate the surface energy budget under those conditions. Dissipation rates under breaking waves in Lake Ontario ranged from 10" 5 to 10" 2 m 2 s" 3 (Agrawal et al. 70 , Terray et al. 71 ). That e did not follow law of the wall scaling in the near surface layer was attributed to the more intense turbulence during white capping and wave breaking.

The Air- Water Interface:

Turbulence and Scalar Exchange

229

5. Calculation of Gas Transfer Coefficients During Low and Moderate Wind Speeds Most attempts to relate gas transfer velocities to surface meteorology have used wind as the independent parameter (Liss and Merlivat 1 , Kling et al. 68 , Cole and Caraco 43 , Wanninkhof and McGillis2, Crusius and Wanninkhof 44 ). However, the surface renewal model takes into account the wide variety of processes occurring at the air-water interface (Crill et al. 72 , Soloviev and Schluessel20, Maclntyre et al. 5 , Eugster et al. 8 ). As we have seen above, dissipation rates in near-surface waters calculated from surface meteorological data and depth of the surface layer obtained from high resolution profiles are in good agreement, or are less than, those obtained from temperaturegradient microstructure profiles. Estimates might be improved were the contribution of microwave breaking to energy flux into the surface layer included. Gas transfer velocities (keoo) were calculated using the small eddy, SE, (Eq. 5, 5a this paper) the surface divergence, SD, (Eq. 8 this paper), and the Soloviev and Schluessel20, SS, versions of the surface renewal model (see Table 1) using five minute averaged meteorological data from Toolik Lake, AK. The 600 implies the values are normalized to CO2 at 20 °C. For the small eddy version of the surface renewal model, the gas transfer coefficient k = a\Dll2{e/v)xlA. D is molecular diffusivity of the gas. e is the rate of dissipation of turbulent kinetic energy and nu is kinematic viscosity. When written in terms of the Schmidt number and turbulent Reynolds number, Ret = ul/u, the expression becomes kSc1^2 = c\ uRe 4 . e is related to u and / through the expression e = u3/l (Taylor 73 ), e and I can be obtained from temperature-gradient microstructure profiles in which e is estimated from a least squares fit of the power spectral densities of the temperaturegradient signal to the Batchelor spectrum and I is the overturning scale. Here e is calculated from surface energy budgets (Maclntyre et al. 67 ) in which the energy flux into the surface layer Fq = (w3 + 1.33 3 u 3 )/2 and e = 0.82Fq/l and I is the depth of the surface layer (Imberger 66 , Maclntyre et al. 5 ). We let c\ = 0.56 as in Crill et al. 72 but recognize that this coefficient was obtained in laboratory experiments and has not been validated in the field. To correct for the somewhat lower estimates of e from surface energy budgets relative to microstructure profiles due to lack of sufficient selfcontained temperature loggers, e was increased five fold (e.g., Fig. 33A); mixed layer depths during low winds (< 3 m s _ 1 ) and heating (iv* < 0.003 m s _ 1 ) were assumed to be 0.3 m.

230

S. Banerjee and S.

Maclntyre

Following Soloviev and Schluessel20, k = 1 . 8 5 , 4 ^ * Sc~1/2(1 + Rf0/RfCI)lA(l

+

Ke/Kea)-1/2

where Ao, based on cool skin data, is 13.3, Rfo is a surface Richardson number, and RfCI is a critical Richardson number determined to be 1.5 x 10~ 4 . H is surface heat flux obtained from summing latent and sensible heat fluxes and long wave radiation flux (e.g. Fig. 31C, curve a.). Ke is the Keulegan number, v%/gv whose critical value is 0.18. a is the coefficient of thermal expansion, g is gravity, u is kinematic viscosity, and cp is specific heat. The spread in calculated gas transfer coefficients at each wind speed and for each method, as evidenced by the 9% confidence intervals (Fig. 37, Maclntyre, Shaw and Kling, to be submitted), is due in part to the enhancement of turbulence when heat loss is included and its decrease when stratification decreases heat loss (Fig. 32A). Curves for gas transfer velocities as a function of wind speed alone developed from field studies (Cole and Caraco (CC) 43 , Crusius and Wanninkhof (CW) 44 , Kling et al. (K) 68 ) are plotted for comparison. CC and SE give similar estimates at all wind speeds and are low in comparison to the other approaches at moderate winds. CW was forced to zero at low winds but is similar to SS and K at moderate winds. The data on which CC and CW are based only includes winds up to 5 and 6 m s _ 1 , respectively. Gas transfer coefficients obtained from experiments (Clark et al. 74 , Crusius and Wanninkhof44, Maclntyre et al. 5 ) are plotted with the five curves in Fig. 38. Considerable scatter is evident in the experimental data at both low and moderate wind speeds. At wind speeds below ca. 4 m s _ 1 , the majority of the estimates of k are less than 3 cm hr _ 1 . In a recent laboratory study of gas flux due to heat loss, gas transfer coefficients, normalized to CO2 at 20°C using a Schmidt number of 530 for oxygen, were 0.4 and 0.9 cm h~~l for outgoing heat fluxes of 400 and 570 W m~ 2 (Schladow et al. 55 ). These low values of k would support Crusius and Wanninkhof's 44 assumption of minimal flux at wind speeds less than 1 m s _ 1 . However, even at winds between 1 and 3 m s _ 1 , some of the data from Lake 302N and that from Sutherland Pond have values of k that exceed those predicted from CW and are within the range of estimates from SS. Field data of Cole and Caraco (not shown) indicate k ranges between 1 and 4 cm h r _ 1 at winds between 1 and 2 m s _ 1 . The low values in Schladow et al. 55 were obtained with an undisturbed surface; hence even the minor disturbances at low winds may contribute to vorticity that enhances gas flux (e.g. Maclntyre 75 ). At winds between 3 and 8 m s _ 1 , the gas transfer coefficients

The Air-Water

Interface:

Turbulence and Scalar Exchange

231

from Lake 302N, Sutherland Pond, Mono Lake, Pyramid Lake and Crowley Lake are in the range predicted from SS, SD, CW, and K. That the field data, which was not collected over time intervals defined to identify times of convection, microbreaking, or high winds, tends to follow these four models at moderate winds suggests these models better include the greater diversity of processes contributing to gas flux than do SE or CC. Clark et al.'s 74 results from Sutherland Pond further indicate that estimates of gas transfer coefficients based on the typical averaging schemes in tracer studies may be too low. During two of his sampling periods, Clark's estimates of k were obtained during steady winds, 3.2 and 3.4 m s _ 1 . These k values are better predicted by SS, SD, CW, and K than by CC or SE. Experimental results from a wide variety of lakes are well predicted by two approaches using surface renewal, SS and SD, and results with these models are similar in the mean to those obtained following Kling et al. 68 , who primarily used other data to develop their regression. To illustrate the importance of including a surface renewal approach for estimating gas flux, k is calculated over 3 diurnal cycles using the meteorological and time series temperature data from Toolik Lake (Fig. 33B). K and SD provide comparable estimates of k during heating periods, but K underestimates k when heat loss occurs. For this data set, CC underestimates k by ca. a factor of 2 under all conditions. These comparisons indicate the potential greater accuracy of models for predicting k that include the various mechanisms that induce turbulence at the air-water interface. Frequently, gas flux estimates from lakes used in regional carbon budgets are based on k values of 1 to 3 cm h r - 1 (Kling et al. 76 , Cole et al. 77 , Richey et al. 78 ). While many of the sites have low wind speeds, estimates are likely to be too low by a factor of at least 2. In tropical waters, k estimated by SE was two times higher than CC (Maclntyre et al. 54 ). If the trends observed here apply to the tropical data, gas flux estimates from those regions may be underestimated by a factor of four. Ho et al. 69 show gas transfer coefficients ranging from 16 to 70 cm h - 1 for rainfall ranging from 13 to 115 mm h _ 1 . Rainfall of such intensity is most likely in tropical environments and would only occur intermittently, but gas fluxes would be comparable to those anticipated when winds are high enough to induce considerable wave breaking. Clearly, consideration of the wide diversity of processes that lead to enhanced gas fluxes will have considerable impact on estimates of global carbon budgets. These approaches will also provide better estimates of k for biological studies estimating net community productivity (e.g. Cole et al. 79 ). These results strongly indicate that further work evaluating the surface renewal model in field situations is warranted, with special attention to heat loss, rain, and microbreaking.

232

S. Banerjee and S.

Maclntyre

Fig. 37. Estimates of gas transfer velocity obtained from 5 minute averaged meteorological data from Toolik Lake, Alaska, using three different surface renewal models: small eddy (SE, Eq. 5a this chapter calculated as in Crill 7 2 ), surface divergence (SD, Eq. 8 this chapter) and following Soloviev and Schluessel (SS, 1994 20 ). 95% confidence intervals are indicated. Gas transfer coefficients calculated from wind based expressions (Cole and Caraco 4 3 , CC, Crusius and Wanninkhof 44 , CW, Kling et al. 6 8 , K) are shown for comparison. Curves illustrate two to three fold variations in calculations of flux are possible dependent upon model selection. The range in values for each wind speed using surface renewal is due to diurnal heating and cooling of surface waters.

6. Conclusions Turbulence phenomena and scalar exchange at the air-water interface have been discussed in terms of the models used for prediction, numerical simulations, laboratory experiments and field observations. The focus of the discussion has centered on evasion of sparingly soluble gases from water bodies. These processes are controlled by liquid-side turbulence, and the main resistance to transfer lies in a very thin region at the interface. As it is difficult to make measurements and do computations of phenomena in the vicinity of a deforming, and perhaps breaking, interface, our understanding is still at a much earlier stage than for such processes at solid boundaries. Nonetheless, considerable progress has been made in understanding the mechanisms that control scalar exchange, when the interface remains continuous (does not break). In the absence of shear, and when the far-field turbulence approximates the homogeneous isotropic case, it has been found that the Hunt-Graham 15

The Air- Water Interface:

—

K CW

cc

. — 3D ,-.20

SE

Turbulence and Scalar

• * o « n A

Exchange

233

Mono Lake Crowley Lake Pyramid Lake ^ Rockkmd Late Sutherland Pond Laka302N

I | 10

i

Fig. 38. Data from tracer experiments (see text for description) with curves of Fig. 37 for comparison show that gas transfer coefficients at moderate wind speeds are well predicted by Soloviev and Schluessel 20 which explicitly includes waves. Similarities of mean values for each wind speed of SS and SD with K and CW and field data support these surface renewal models and the need for experimental work to define gas fluxes as a function of all processes inducing turbulence at the air-water interface.

blocking theory applies quite well, and predictions of the near-interface damping of the normal component of turbulence, and enhancement of the tangential components, are well predicted. Banerjee4 has applied the HuntGraham theory to calculate the surface divergence and developed the socalled surface divergence (SD) model, which predicts gas transfer across unsheared interfaces rather well. It has also been shown that this model predicts at low turbulent Reynolds number the same behavior as the Fortescue and Pearson 12 large-eddy (LE) model, and the Banerjee et al. 13 small-eddy (SE) model at high turbulent Reynolds numbers. In situations in which the shear rate imposed by the wind is high, turbulence is generated in the vicinity of the interface, much like near solid boundaries. For situations in which the air-water interface does not break, models for gas transfer based on scaling of active periods (such as sweeps and ejections) with interfacial frictional shear, have proved to be successful in predicting laboratory data. However, microbreaking (breaking of waves of amplitude ~ 1 cm. and length ~ 10 cm) starts to occur at L710 of about 3-5 meters/sec, and very little is currently understood about the effect of these phenomena on turbulence generation and scalar exchange. Labora-

234

S. Banerjee and S.

Maclntyre

tory experiments indicate strong regions of convergence in the surface fluid motion near wave crests and regions of divergence behind, suggesting that such waves would markedly increase the r.m.s. surface divergence and hence gas transfer. Clearly, considerable work still needs to be done to characterize such microbreakers, both in laboratory experiments and in the field, and to better understand their effects on turbulence and scalar exchange. Field observations also suggest that natural convective motions due to cooling of the interfacial layer play a significant role by affecting both turbulence and scalar exchange over a wide range of wind velocities. This suggests that parameterizations based on wind speed alone are inadequate for calculations of mass transfer. The field observations also suggest that surface divergence approaches are quite fruitful, as they can accommodate a variety of processes that may lead to turbulence generation. The SD model is found to predict field data rather well even at high wind speeds. This is surprising because the model was originally developed for a no-shear interface based on Hunt-Graham theory. On the other hand, the SE model significantly underpredicts gas transfer coefficients at moderate wind speeds - possibly because energy dissipation due to microbreaking was underestimated. These discrepancies indicate better estimates of energy dissipation and length scales of the turbulence from the near-surface regions are needed in the presence of microbreakers. These analyses illustrate the utility of extending the surface divergence/surface renewal approaches, which were derived on the basis of theory and laboratory experiments, to field settings where turbulence is induced by a variety of processes. Much work, however, remains to be done. In particular, the effect of natural convective motions observed in the field and microbreaking need investigation. To this end, more extensive and accurate field observations are of the highest priority, particularly in well-characterized conditions. Acknow

ledgement

We are grateful for continued support, extending from 1985, for SB's research program in this area by USDOE Basic Energy Sciences Contract #DE-FG03-85ER13314. References 1. P. S. Liss and L. Merlivat, in The Role of Air-Sea Exchange in Geochemical Cycling, Ed. P. Buat-Menard, 113-129, D. Reidel, Norwell, Mass. (1986). 2. R. Wanninkhof and W. R. McGillis, Geophys. Res. Letters 26, 1889 (1999). 3. M. Donelan and R. H. Wanninkhof, in Gas Transfer at Water Surfaces, Geophysical Monograph, 127, AGU (2002).

The Air- Water Interface: Turbulence and Scalar Exchange

235

4. S. Banerjee, Proc. Ninth Int. Heat Transfer Conf., Keynote Lectures, 1, 395 (1990). 5. S. Maclntyre, R. Wanninkhof and J. Chanton, in Biogenic Trace Gases: Measuring Emissions from Soil and Water, Eds P. Matson and Ft. Harriss, Blackwell, 52 (1995). 6. B. Jaehne and H. Haussecker, Ann. Rev. Fluid Mech. 30, 443 (1998). 7. M. A. Donelan, W. M. Drennan, E. S. Saltzman and R. Wanninkhof, Eds., Gas Transfer at Water Surfaces, (Geophysical Monograph 127, AGU, 2002) 8. W. G. Eugster, G. W. Kling, T. Jonas, J. P. McFadden, A. Wuest, S. Maclntyre and F. S. Chapin III, J. Geophys. Res. (Atmospheres). 108 (D12), 4362 (2003). 9. W. K. Lewis and W. G. Whitman, Ind. Eng. Chem. 16, 1215 (1924). 10. R. Higbie, Trans. Am. Inst. Chem. Eng. 31, 365 (1935). 11. P. V. Danckwerts, Ind. Eng. Chem. 43, 1460 (1951). 12. G. E. Fortescue and J. R. A. Pearson, Chem. Engr. Science 22, 1163 (1967). 13. S. Banerjee, E. Rhodes and D. S. Scott, Ind. Eng. ChE Fundamentals 7, 22 (1968). 14. T. J. Theofanous, R. N. Houze and L. K. Brumfield, Int. J. Heat Mass Transfer 6, 13 (1976). 15. J. C. R. Hunt and J. M. R. Graham, J. Fluid. Mech. 84, 209 (1978). 16. M. J. McCready, E. Vassilliadou and T. J. Hanratty, AIChE J. 32, 1108 (1986). 17. S. Kumar, R. Gupta and S. Banerjee, Phys. Fluids 10, 437 (1998). 18. V. De Angelis, P. Lombardi, P. Andreussi and S. Banerjee, "Micro-physics of scalar transfer at air-water interfaces", Proceedings of the IMA Conference: Wind-over-Wave Couplings, Perspectives and Prospects, Ed. S. G. Sajjadi, N. H. Thomas and J. C. R. Hunt, Oxford Press, 257 (1999). 19. G. T. Csanady, J. Geophys. Res. 95, 749 (1990). 20. A. V. Soloviev and P. Schluessel, J. Phys. Oceanogr. 24, 1319 (1994). 21. B. Caussade, J. George and L. Masbemat, AIChE Journal 36, 265 (1990). 22. M. Coantic, J. Geophys. Res. 9 1 , 3925 (1986). 23. B. H. Brumley and G. H. Jirka, J. Fluid Mech. 183, 235 (1987). 24. D. R. Chu and G. H. Jirka, Int. J. Heat Mass Transfer 35, 1957 (1992). 25. S. P. McKenna, W. R. McGillis and E. J. Bock, in TSFP-1, Eds S. Banerjee and J. K. Eaton, Begell House, NY, 455 (1999). 26. S. Komori, H. U-F. Ogino and T. Mizurhina, Int. J. Heat Mass Transfer 25, 513 (1982). 27. M. Rashidi and S. Banerjee, Phys. Fluids 31 (9), 2491 (1988). 28. S. Komori, Y. Murakami and H. Ueda, J. Fluid Mech. 203, 103 (1989). 29. J. C. Lamont and D. S. Scott, AIChE J. 15, 1215 (1970). 30. Y. Pan and S. Banerjee, Phys. Fluids 7, 1649 (1995). 31. P. J. Burt, Comput. Graphics Image Proc. 16, 20 (1981). 32. P. Anandan, Int. J. Comput, Vision 2, 283 (1989). 33. J. D. A. van Hoften and S. Karaki, 15th ICCE, Ch. 23, 404 (1976). 34. Y. Iwagaki and T. Asano, Coastal Eng. Jpn. 23, 1 (1980). 35. P. H. Kemp and R. R. Simons, J. Fluid Mech. 116, 227 (1982). 36. P. H. Kemp and R. R. Simons, J. Fluid Mech. 130, 73 (1983).

236

S. Banerjee and S. Maclntyre

37. R. R. Simons, A. J. Grass and A. Kyriacou, in Proc. of 21st ICCE, 363 (1988). 38. M. Rashidi, G. Hetsroni and S. Banerjee, Phys. Fluids 4 (12), 2727 (1992). 39. S. Supharatid, H. Tanaka and N. Shuto, Coastal Engineering in Japan 35 (1992). 40. X. Nan, D. Kaftori and S. Banerjee, in Proc. of FEDSM '98, ASME Fluids Engineering. Division Summer Meeting, Washington, D.C. (1998). 41. X. Nan, Ph. D. Thesis, Department of Mechanical Engineering, University of California, Santa Barbara (2003). 42. B. Knowlton, R. Gupta and S. Banerjee, TSFP-1, S. Banerjee, J. K. Eaton, eds., Begell House, NY, 461 (1999). 43. J. J. Cole and N. F. Caraco, Limnol. Oceanogr. 43, 647 (1998). 44. J. Crusius and R. Wanninkhof, Limnol. Oceanogr. 48, 1010 (2003). 45. T. J. Shay and M. C. Gregg, J. Phys. Oceanogr. 16, 1777 (1986). 46. A. Anis and J. N. Mourn, J. Phys. Oceanogr. 24, 2142 (1994). 47. D. T. Ho, R. Bliven, R. Wanninkhof and P. Schlosser, Tellus 49, 149 (1997). 48. S. Maclntyre, Limnol. Oceanogr. 38, 798 (1993). 49. S. Maclntyre, Jap. J. Limnol. 57, 395 (1996). 50. S. Maclntyre, in Physical Processes in Lakes and Oceans, Coastal and Estuarine Studies, AGU, Ed. J. Imberger, 539 (1998). 51. R. D. Robarts, M. Waiser, O. Hadas, T. Zohary and S. Maclntyre, Limnol. Oceanogr. 43, 1023 (1998). 52. S. Maclntyre, K. M. Flynn, R. Jellison and J. R. Romero, Limnol. Oceanogr. 44, 512 (1999). 53. S. Maclntyre and R. Jellison, Hydrobiologia 466, 13 (2001). 54. S. Maclntyre, W. Eugster and G. W. Kling, in Gas Transfer at Water Surfaces, Eds M. A. Donelan, W. M. Drennan, E. S. Saltzman, and R. Wanninkhof, AGU, (2001). 55. S. G. Schladow, M. Lee, B. E. Hurzeler and P. B. Kelly, Limnol. Oceanogr. 47, 1394 (2002). 56. S. J. Kline, W. C. Reynolds, F. A. Schraub and P. W. Runstadler, J. Fluid Mech. 70, 741 (1978). 57. M. Rashidi and S. Banerjee, Phys. Fluids 2, 1827 (1990). 58. K. Lam and S. Banerjee, Phys. Fluids 4 (2), 306 (1992). 59. P. Lombardi, V. De Angelis and S. Banerjee, Phys. Fluids 8 (6), 1643 (1996). 60. V. De Angelis, P. Lombardi and S. Banerjee, Phys. Fluids 9, 8 (1997). 61. V. De Angelis and S. Banerjee, "Heat and transfer mechanicsms at wavy gas-liquid interfaces", TSFP-1, S. Banerjee, J. K. Eaton, eds, Begell House, New York, 1249 (1999). 62. V. De Angelis, Ph. D. Dissertation, Department of Chemical Engineering, University of California, Santa Barbara, (1998). 63. I. Leifer, B. D. Piorek, W. C. Smith and S. Banerjee, in TSFP-3, Ed. N. Kasagi et al., Sendai, Japan, 2, 705 (2003). 64. R. H. Wanninkhof and L. F. Bliven, J. Geophys. Res. 96, 2785 (1991). 65. F. J. Ocampo-Torres, M. A. Donelan, N. Merzi and F. Fija, Tellus46B, 16 (1994). 66. J. Imberger, Limnol. Oceanogr. 30, 737 (1985).

The Air- Water Interface: Turbulence and Scalar Exchange

237

67. S. Maclntyre, J. R. Romero and G. W. Kling, Limnology and Oceanography 47, 656 (2002). 68. G. W. Kling, G. W. Kipphut and M. C. Miller, Hydrobiologia 240, 23 (1992). 69. D. T. Ho, W. E. Asher, L. F. Bliven, P. Schlosser and E. L. Gordan, J. Geophys. Res. 105, 24045 (2000). 70. Y. C. Agrawal, E. A. Terray, M. A. Donelan, P. A. Huang, A. J. Williams III, W. M. Drennan, K. K. Kahma and S. A. Kitaigorodskii, Nature 359, 219 (1992). 71. E. A. Terray, M. A. Donelan, Y. C. Agrawal, W. M. Drennan, K. K. Hahma, A. J. Williams III, P. A. Huang, and S. A. Kitaigorodskii, J. Phys. Oceanogr. 26, 792 (1996). 72. P. M. Crill, K. B. Bartlett, J. O. Wilson, D. I. Sebacher, R. C. Harriss, J. M. Melack, S. Maclntyre, L. Lesack and L. Smith-Morrill, J. Geophys. Res. 93, 1564 (1988). 73. G. I. Taylor, Proc. Roy. Soc. Lond. A151, 421 (1935) 74. J. F. Clark, P. Schlosser, R. Wanninkhof, J. H. Simpson, W. S. F. Schuster and D. T. Ho, Geophys. Res. Letters 22, 93 (1995). 75. S. Maclntyre, "Current fluctuations in the surface waters of small lakes", 125-131, W. H. Brutsaert and G. H. Jirka (eds.), Gas Transfer at Water Surfaces, D. Reidel Publishing Co., (1984). 76. G. W. Kling, G. W. Kipphut and M. C. Miller, Science 251, 298 (1991). 77. J. J. Cole, N. F. Caraco, G. W. Kling and T. K. Kratz, Science 265, 1568 (1994). 78. J. E. Richey, J. M. Melack, A. K. Aufdenkampe, V. M. Ballester and L. L. Hess, Nature 416, 617 (2002). 79. J. J. Cole, S. R. Carpenter, J. F. Kitchell and M. L. Pace, Limnol. Oceanogr. 47, 1664 (2002).

This page is intentionally left blank

CHAPTER 6 I N T E R N A L WAVE FIELDS ANALYZED BY IMAGING VELOCIMETRY

John Grue Mechanics Divsion, Department of Mathematics, University of Oslo P.O. Box 1053 Blindern, 0316 Oslo, Norway E-mail: [email protected]

We review recent laboratory measurents of internal waves using Particle Tracking Velocimetry and Particle Image Velocimetry. The methods enable accurate recordings of the wave induced velocity and vorticity fields. Global properties like wave length and wave speed are obtained from the measurements. Solitary waves and dispersive wave trains in twolayer fluids with an upper layer that is either homogeneous or linearly stratified, and a lower layer that is homogeneous, are investigated. The measurements provide references for mathematical models and are used to judge their applicability. Particular focus is paid to solitary waves that exhibit convective breaking, shear instability, how wave breaking changes the global wave properties, how solitary waves break at a submerged ridge or shelf, and how the wave breaking induces transport of suspended particles. Run-up of very long internal waves at a shelf-slope is found to introduce strong fluid velocities where the pycnocline meets the slope. The experiments are used to interpret observations in large scale. 1. I n t r o d u c t i o n A physical laboratory simulator represents a tool for investigating complex flow phenomenae where theoretical models have shortcomings or do not exist. L a b o r a t o r y experiments of internal waves (and other flows) m a y be used t o explain observations in the sea and may serve as guidance in developing theories of internal wave propagation where a breaking of the flow eventually is inherent. T h e experimental wave t a n k may be used t o study waves with very large amplitudes, waves t h a t propagate along thin or broad pycnoclines, conditions t h a t lead to breaking of t h e waves, properties of internal waves t h a t break and internal waves t h a t interact with a sub-sea ridge or shelf-slope. Laboratory experiments represent an important complement to 239

240

J. Grue

theoretical and numerical modelling of internal waves. Measurements may be used to confirm mathematical solutions and are useful to judge the range of validity and possible limitations of simulation models. Our main interest in studying internal waves was motivated by requests from industry, including the magnitude of the fluid velocities induced by internal waves, the vertical motion of pycnoclines in the deep ocean, and interpretation of sudden temperature or current spikes observed in the ocean close to the sea floor. Internal wave loading on pipelines, or on a submerged floating tunnel that was proposed as connection across a Norwegian fjord, represented other questions from industry. Marine biologists observed large vertical periodic motion of fish and navigation of fish along density contours in the ocean, and questioned if this could be explained by internal wave motion. Further, understanding and description of currents and waves in fjords, including the motion of interfaces in the sea, are important for a proper management of the ecosystem. The questions supported our general curiosity on the phenomenon of internal waves, the experimental measurements and the mathematical modelling of the waves. Descriptions of internal waves in large scale may be found in, e.g., Osborne, Burch and Scarlet 1 , Ostrovsky and Stepanyants 2 , Huthnance 3 . Both internal solitary waves of large amplitude (Pingree and Mardell 4 , Apel et al. 5 , Stanton and Ostrovsky 6 ) and dispersive wave trains (Gjevik and H0st 7 ) are observed. Mathematical description of the waves in terms of weakly nonlinear Korteweg-de Vries (KdV) and Benjamin-Ono equations may be employed for small and moderate wave amplitudes. Fully nonlinear theories are in general required for a proper modelling when the excursion of the isopycnals become comparable to their average depth (Holyer8, Meiron and Saffman9, Amick and Turner 10 , Turner and Vanden-Broeck 11 ' 12 , Evans and Ford 13 , Grue et al. 14 - 15 - 16 , Lamb 17 ). While several papers describe the mathematical modelling of the waves (solitary waves, periodic or unsteady wave trains), comparatively few experimental papers on the subject are published. The most important experimental investigations on nonlinear internal waves are summarized here. The amplitude-wave length relationship of interfacial solitary waves with small to moderate amplitude was experimentally determined by Koop and Butler 18 , Segur and Hammack 19 , Kao, Pan and Renouard 20 . Comparisons with weakly nonlinear theories were made. In the latter work the fluid velocity was measured using hot-film probes. Michallet and Barthelemy 21 measured profiles and speeds of solitary waves in a two-fluid system, with the amplitude ranging from small to relatively large values. Their attempts to investigate wave amplitudes close to the maximal one were prevented by

Internal

Wave Fields Analyzed by Imaging

Velocimetry

241

breaking during the wave generation process. Internal solitary waves of mode two propagating along the pycnocline were experimentally investigated by Davis and Acrivos 22 , Maxworthy 23 and Stamp and Jacka 24 . The small scale experiments in the latter work exhibited wave amplitudes that were up to five times the half-width of the pycnocline. Internal waves interacting with a shelf-slope show formation of turbulent boluses travelling up a slope, intense breaking where the pycnocline intersects the slope and partial reflection of the incoming waves (Cacchione and Southard 25 , Wallace and Wilkinson 26 , Kao et al. 20 , Helfrich27). Small particles may be transported along the pycnocline in the direction away from the slope as a consequence of the breaking (Michallet and Ivey 28 ). Internal waves interacting with a submerged ridge have been experimentally studied by Wessels and Hutter 29 , Vlasenko and Hutter 30 , Sveen et al. 31 , Guo et al. 32 . Up to now, measurements of internal waves were usually carried out using resistance wave gauges, ultrasonic probes or hot-film probes. We shall in this paper review recent measurements using the optical methods Particle Tracking Velocimetry (PTV) and Particle Image Velocimetry (PIV). Efficient processing of the relatively large data sets are enabled by powerful computers. It is indeed the computer efficiency and rapid storage techniques gained during the last decade that have made PTV and PIV tractable. Velocity profiles, local vorticity and global properties like propagation speed and wave length are derived from the velocimetry. Details of breaking processes are quantified by the methods. Comparisons with mathematical and numerical computations are performed whenever possible. The paper is organized as follows: The experimental wave tank and the PTV and PIV methods in use are summarized in section 2. An estimate of how close the particles follow the fluid flow is given. Wave propagation along a thin pycnocline is discussed in section 3. Measurements of solitary waves of depression with amplitudes ranging from a small value up to the maximal theoretical wave amplitude are discussed. Experiments with dispersive wave trains are included. Breaking and broadening of solitary waves propagating in a two-fluid system with a linear stratification and a homogeneous fluid in the upper and lower layer, respectively, are discussed in section 4. Features of waves breaking at a submerged ridge are reported in section 5. The motion of internal waves, the induced breaking and the runup at steep or mild-sloping beaches are discussed in section 6. Vertically induced velocities in the ocean identified by fish motion are described in section 7. Section 8 contains a description of the fully nonlinear mathematical/numerical reference models, while section 9 is a conclusion.

242

J. Grue

2. The experimental wave tank 2.1. The wave tank and

stratification

We shall mainly report experiments that were carried out in the Hydrodynamic Laboratory at the University of Oslo. Sections of a wave tank 0.5 m wide, 1 m deep and 6.2, 12.3 or 21.4 m long were used. Each experiment began by calibrating a two-fluid system where a lower layer of brine had density pi = 1.022...g/cm3 and depth hi. The top layer was either of fresh (homogeneous) water or had a linear stratification with density />2=0.999... g/cm 3 at the upper surface of the layer. The depth of the upper layer was /i2- The density profiles were recorded by a Yokogawa SC12 meter which determines the density from the local conductivity of the fluid. Reference measurements were taken at some selected locations using a Mettler-Toledo DA-300M density meter which determines the density to four decimal places. Some of the runs (described in sections 3.2, 6.2) were performed in a smaller wave tank with dimensions 0 . 3 m x 0 . 7 m x 7 m (width, depth, length). 2.2. Particle

tracking

and particle

image

velocimetry

For the benefit of the reader we briefly summarize the most important points of the PTV and PIV methods used in the experiments (Grue et al. 1 4 ' 1 5 , Sveen et al. 31 ). The PTV method traces individual particles in sequences of images. The method is ideal when the local fluid acceleration is small which is true for the slowly progressing internal waves in focus. Vertical sections of the wave tank were illuminated for subsequent recording of the wave motion. The light sheets were parallel to the side of the tank, 5 cm thick and had a distance of 10 cm from the glass wall. Crunched particles of pliolite VTAC with diameter in the range 0.8-1 mm were in the viewing sections seeded to the fluid. The density of the pliolite is approximately 1.0228 g/cm 3 . Particles were treated using a wetting agent to obtain an effectively neutral buoyancy in the entire range of the density profile. The viewing sections were typically 50 cm x 40 cm large. Monochrome COHU 4912 CCD cameras with a resolution 575x560 of pixels were used in the flow recordings. A particle at rest was normally covered by four pixels in the CCD-chip. The effective shutter speed (using a mechanical shutter) was usually 1/100 s. (In some initial experiments the shutter speed was 3/100 s.) The time between each frame is 1/25 s. The video recordings were digitized by a frame grabber card for subsequent analysis. In the experiments presented here we typically identified 800-4000 particles in each frame. Particles were traced during five frames us-

Internal

Wave Fields Analyzed by Imaging

Velocimetry

243

ing the Diglmage program developed and described by Dalziel 33 . A perspex plate with reference coordinates was submerged in the fluid and recorded after each set of runs. Reference points were mounted to the wave tank. The mapping coordinates between the recording section and the camera is linear. In some of the waves the acceleration was not always small. This is true when breaking of the flow was observed. We could not trace particles long enough for the PTV algoritm to be effective, for example. Data processing using PIV was then used to estimate the velocity field. For the PIV algorithm we implemented the method outlined by Willert and Gharib 34 using an interrogation window shifting procedure of Westerweel, Dabiri and Gharib 35 . The images were interrogated in three steps where the two first steps were used to estimate the window shift with integer accuracy. In the final step the displacement was estimated to sub-pixel accuracy using a three point Gaussian peak fit. Images were interrogated using windows of 32x32 pixels. In some experiments 64x64 pixels were used due to insufficient particle seeding. A signal to noise ratio filter was used to validate the final velocity vectors. A local median filter effectively removed vectors deviating significantly from their neighbours. The implementation is documented in full in Sveen36. 2.3. Relative

accuracy

of the optical

method

The difference AV between the fluid velocity and the velocity of a seeded particle may be analysed using the equation of momentum for the particle, assuming that the Stokes drag is much less than the time derivative of the particle momentum, i.e. \mV\ > > |67rp^i?AV|. Here a dot denotes time derivative, v the kinematic viscosity and R, V, m = (4ir/3)pR3 the radius, velocity and mass of the particle (assuming neutral boyuancy), respectively, p denotes fluid density. This gives |AV| < < a|V| where a — m/Girpi'R = 2R2/9u. The acceleration is estimated by |V| ~ | V m | / T where | V m | denotes a typical maximal velocity and T a typical period of the flow. Characteristic values of | V m | and T in the present experiments are 10 cm/s and 5-10 s, respectively. With R = 0.5 mm and v = 10~ 6 m 2 /s we find IAVI L | «

2.R2 _ -0.005-0.01.

(1)

The corresponding particle Reynolds number Re = 2R\AV\/i> is less than unity. The seeded particles thus quite closely follow the fluid motion, with an accuracy indicated by (1).

244

J. Grue

The accuracy of the digital tracing of a particle may be estimated as follows: We assume that the position and displacement of each particle is determined with an accuracy better than the length of a pixel which corresponds to about 1 mm in physical space in our case. (Dalziel33 found in his experiments that that the position of a particle was determined with an accuracy of about 0.2 times the pixel length.) A particle with a typical velocity of 10 cm/s, traced during a sequence of five frames, i.e. a time interval of 0.2 s, moves a distance of 2 cm. This means that the relative error in the particle displacement and the velocity is at most lmm/2cm=5%. This is a much larger figure than the error in (1). We have found that a relative error of the measured velocities is approximately in agreement with the error analysis just described. We find, for example, that the measured fluid velocity induced by a solitary wave, relative to the linear long wave speed Co of the two-layer fluid, has a deviation of about 7-8% at most, from a line fitted to all experiments that were made, see Fig. 2c. Our results include analysis using both PTV and PIV, preferring the method giving the best accuracy for the actual run. We note that in all practical applications it is possible to obtain the same accuracy using PTV and PIV. 3. Motion along a thin pycnocline 3.1. Solitary

waves

Investigations of large amplitude solitary waves propagating in a two-layer fluid represented a main focus of the laboratory studies that were undertaken. The waves were measured in the wave tank (dimensions are previously given) using the procedure described above: A two-layer fluid with a lower layer of brine and an upper layer of homogeneous fresh water was calibrated. In most of the runs the lower layer depth was hi = 62 cm and the upper layer depth /12 = 15 cm which means a depth ratio of /11//12 = 4.13. The thickness of the pycnocline was 1-2 cm. Vertical sections of the wave tank were illuminated for subsequent recording of the wave motion. Solitary waves were produced by trapping a volume of light fluid behind a gate that was lowered in one end of the wave tank. Upon removal of the gate the initial depression rather quickly was transformed into a solitary wave propagating along the tank. A single solitary wave was in each experiment produced by a combined adjustment of the position of the gate and the added volume (see Fig. la). The wave amplitude ranged from a rather small value to about the maximal possible one, as indicated by the measurements presented in Fig. 2c. The video cameras placed along the tank, at positions 4.5 m and 10.4 m from the end

Internal

Wave Fields Analyzed by Imaging Velocimetry

245

of the tank where the waves were generated, recorded the wave induced motion of the particles that was seeded into the fluid. From the recordings we determined the induced velocity profiles, speed and shape of the waves. The velocity measurements of the individual runs were scaled by the linear long wave speed of the two-fluid system given theoretically by 2 _ ghih2{p\ - P2) 0 P2/11 + Pih2

(2)

where g denotes the acceleration of gravity. The effect on the wave and fluid velocities due to small variations of the relative density jump in the different runs was then ruled out. The wave speed c was estimated by the elapsed time of the wave propagation between the two viewing sections (at 4.5 and 10.4 m, including accurate reference points) divided by distance. The confined pycnocline is very visible due to a higher concentration of particles there than elsewhere. The pycnocline motion was determined by visual inspection of video images. The wave speed scaled by the theoretical linear wave speed determined from eq. (2) is visualized in Fig. 2a. The wave speed is seen to grow at a rate that is decreasing with amplitude. At maximal amplitude the growth is zero. The experimental wave amplitudes are found to undergo a small reduction during the propagation due to the effect of viscosity. The decrease of the amplitude between the recording stations is indicated by the horizontal bars in the figure. (The largest amplitude refers to the first recording station.) Wave profiles were obtained from the video images by first tracing the vertical motion of the middle position of the pycnocline, as function of time. The recorded motion was then plotted versus time multiplied by the computed (and measured) wave speed c. Supplementary estimates of the pycnocline motion were obtained by tracing the jump in the velocity profile (across the pycnocline). The two methods give the same result, see Fig. lb. The main output from the velocimetry is the velocity field induced by the wave motion. The velocities in any position of the wave were determined. An example of the velocity profile at the wave crest is visualized in Fig. 2b. Results from a total of 10 different runs with slightly different pycnocline thicknesses are included. The consistent results illustrate the robustness and accuracy of the experiments. We were also interested in measuring the induced fluid velocities in the layers as function of wave amplitude. Results for the horizontal fluid velocity in the upper layer are obtained for the whole range of possible wave amplitudes (Fig. 2c). The maximal theoretical

246

J. Grue

h2

""-

-^^

I

!

i _=pLJu——

-0.25 i r]_ -0.75 h2 -1.25h -1.75. -25

D

an

\

c) i

-15

-5

i

15

25

-ct/h2 Fig. 1. Solitary waves propagating along a thin pycnocline. Sketch of experiment (upper). Profiles r)(ct) of waves with amplitude a/hi = 1.23 (mid) and a/h.2 = 1.51 (lower). Squares: measured pycnocline. Triangles: measured jump in velocity. Solid line: fully nonlinear theory. Dashed line: KdV theory. h\/h,2 = 4.13. From Grue et al. 1 5 . Reproduced with permission by J. Fluid Mech.

amplitude in these experiments is a/h2 = 1.55 (accounting accurately for the slightly small difference between the densities p\ and p2). The corresponding theoretical maximal fluid velocity is U/CQ = 0.77. In the figure the function {u/CQ)/{a/h2) is plotted. For very small a/h2 the function is close

Internal

Wave Fields Analyzed by Imaging

Velocimetry

247

to unity. At maximal amplitude the value of (u/co)/(a//i2) is about 0.5 for the actual depth ratio between the layers. The experimental solitary waves may be computed employing a theoretical two-layer model where in the latter the thin pycnocline is replaced by an interface separating the motion in the upper and lower layer. An interesting aspect of the campaign is that the fully nonlinear model may be used to check the PTV method. The high precision computations provide a theoretical reference when the experimental conditions fit with the assumptions of the theory. Vice-versa, a fit between theory and experiment documents the applicability of the theoretical interface model. Fully nonlinear computations of solitary waves were performed using the theoretical-numerical two-layer interface model developed by Grue et a j 14,15 rp n e r e s m t s exhibit a surprisingly good agreement between measurements and theory for the whole range of wave amplitudes. The agreement is good even for a pycnocline which is not very sharp. The range up to the highest waves is covered. We were particularly interested in investigating experimental solitary waves with amplitudes as close as possible to theoretical maximum, which in the Boussinesq limit (Ap/p < < 1) means that the pycnocline moves to a position that is mid-way in the fluid. (This implies the simple formula amaxjhi = \{h\/h2 — 1) for the maximal wave amplitude, with numerical value 1.56 when h\jhi = 4.13.) We observe a good correspondence between theory and experiment for the profile of the maximal wave, in the leading part of the wave (Fig. lc). In this experiment the minimal value of the Richardson number, estimated from velocimetry, was 0.07. Local breaking of the flow due to a shear instability was then expected. Indeed, breaking is observed in the tail of the wave. The shear instability is very visible. Instability and subsequent breaking of the flow cannot be predicted by the theoretical two-layer model. The results in Fig. 2c indicate a relative accuracy of the measurements of about 7-8%. This represents the maximal deviation of U/CQ from the theoretical line and from a fitted line of the recordings. We have here studied the motion of solitary depression waves. The polarity of the solitary waves is determined according to the sign of the coefficient h\pi — h\p2- The waves are depression or elevation waves when this factor is negative or positive, respectively. Korteweg-de Vries theory provides a supplementary reference when the wave amplitude is small. The assumption of a weak nonlinearity is violated for moderate and large wave amplitudes, however. The amplitude range where KdV theory is valid is found to be rather small (a/h2 less than about 0.4).

248

J. Grue

a)

0.8

1.2

a/h2

b)

c)

Fig. 2. Solitary waves, a) Excess speed vs. wave amplitude, b) velocity profile above and below crest (a//i2 = 0.78), c) nondimensional velocity in upper layer (at level y = — /12/2) vs. nondimensional amplitude a/h?. /11//12 = 4.13. From Grue et al. 1 5 . Reproduced with permission by J. Fluid Mech.

Internal

Wave Fields Analyzed by Imaging

Velocimetry

249

b)

x/h2 Fig. 3. Fully nonlinear wave profile (square), KdV (dots) and Intermediate-long-wave equations (solid line), a) 0//12 = 0.15, b) a//i2 = 0.7. /11//12 = 4, P2/P1 = 0.9. From Grue et al. 1 4 . Reproduced with permission by J. Fluid Mech.

It may be questioned if the Intermediate-long-wave-equations 19 may be more relevant than KdV theory in the present examples, where the wave length L//12 ~ 10—20 is not much longer than the depth ratio /11//12 = 4.13. The Intermediate-long-wave-equations have very limited range of validity and represent a poorer approximation than KdV theory, however (Fig. 3). This is true even if the wave length is comparable to the total depth of the fluid. Extensions based on the KdV and the BO equations, valid for solitary waves with large amplitude, are discussed by Ostrovsky and Grue 37 .

3.2. Dispersive

wave

trains

Experiments were also performed with a portion of brine trapped by the gate. This corresponds to an initial condition where a part of the interface has an elevation. The resulting wave system is a dispersive wave train propagating along the tank with the longer waves travelling faster than the shorter ones. The mechanical energy per wave length is relatively much smaller than the wave energy transported by a single solitary wave. The induced fluid velocities are correspondingly smaller (Fig. 4). The experimental waves are adequately modelled by the (linear) Cauchy-Poisson problem modified to interfacial flows (Carlin 38 , Hald 39 ). The experiments and theory

J. Grue

250

document wave systems that may be generated upon release of a volume of heavy water into a two-layer system. Such wave trains are observed in large scale in Skagerak off the southern coast of Norway. During summer a thermocline is developed in the sea with an average depth of 15 m, a relative jump in density of 3 . 7 x l 0 - 3 , with corresponding linear long wave speed Co — 70 cm/s. Wave groups show leading waves that typically are 1.5 km long, see Gjevik and H0st 7 . The wavenumber at any position of the wave train is implicitly determined by: d/dk[(x/t)k — uj(k)] = 0, where LO2 = g'k/(cothkhi + cothfc/12), k the wavenumber, x horizontal position, t time, u! wave frequency, g' = gAp/p, g acceleration of gravity, Ap/p relative jump in density. The laboratory results and the mathematical model predictions give an indication of the magnitude of the fluid velocities that are induced by the waves in large scale. (Nondimensional velocity in the upper layer, U/CQ ~ 0.1 corresponds to nondimensional interfacial vertical excursion, a/h2 ~ 0.15.)

4. Solitary waves that break Heating from sun may result in a linear stratification of the upper part of the ocean. Such a stratification is season-dependent and is observed at many locations. The thickness of the linearly stratified layer typically ranges from some tens of meters to a few hundreds. Examples are represented by the Sulu Sea (Fig. 5) and the Knight Inlet in British Columbia (Farmer and Smith 40 ). Another example is off the northern coast of Norway, see section 7. Internal wave propagation along such a stratification may be studied in the laboratory wave tank by preparing a fluid where an upper part of the undisturbed water column has a constant Brunt-Vaisala frequency and a lower part is homogeneous. The properties of internal solitary waves propagating along such a stratification were investigated experimentally and theoretically by Grue et al. 16 . The waves were generated using the same procedure as for the two-layer fluid with constant density in the layers (sections 2, 3). In all experiments reported here the thickness h\ of the lower homogeneous fluid is 4.13 times the thickness h2 of the upper fluid, where the latter has a linear stratification (and the density is continuous in the transition between the layers). The recorded velocities in the experiments were scaled by the linear long wave speed of the two-fluid system determined by 16 N0h2 Co

cot

N0h2 Co

h2 1- 7— = 0, hi

(3)

Internal

Wave Fields Analyzed by Imaging

Velocimetry

251

191199 251199--A&B 291199 -A&B, 031299-A&B 071299-A&B 0.15

-

0.1

/.'' I

0.05

•''i'.l * i]

<

o •« 3

0

•!•

-0.05 /•I

I I\

f\

i*''\

/•SI'*

I

\'

J "*" 1

r

.1

/ •!• *

/*** 1

rr A:* A.

1. .y /.*

T V*

t J*

\ : /?'

'l-

-*J -0.1

60

0.06

1

I

70

80

90

100

110 120 ^ 2

130

140

150

160

170

191199, 251199-A&B, 291199-A&B, 031299-A&B, 071299-A&B

b)

-0.06 60

70

80

90

100

110 120 ^ 2

130

140

150

160

170

Fig. 4. a) Horizontal velocity in upper layer and b) below pycnocline. Dispersive waves. P T V measurements (dots), linear theory - solution of the Cauchy-Poisson problem for two-layer interfacial flows (solid line). Recording position 6.95 m from end of wave tank. Elevation of pycnocline behind gate: 5 cm. hi = 10 cm, /12 = 60 cm. Position of gate at 0.42 m. Adapted from Hald 3 9 .

252

J. Grue

where N§ = (Apg)/(p0h2) denotes the Brunt-Vaisala frequency at rest and Co is obtained for N0h2/c0 in the interval (TT/2,TT). The linear long wave speed becomes c0 = N0h2/1.711.. in the experiments discussed, where the ratio h\jh2 equals 4.13.

32

33

•

•

S

0

10

Salinity (opt) 34 35 1 —

»

36 .

Temperature { C) 15 20 2S

i

30

250

500

f a

750 Mean SS3

1000

1250

1500 21

22

23

24 25 SIGMA-T

26

27

Fig. 5. Linear stratification in the Sulu Sea (Apel et al. 5 , Fig. 3), reprinted from J. Phys. Oceanogr., 15.

First we study the solitary wave speed c. This is taken as the (horizontal) speed of a vertical line through the approximate centre of the wave, characterized by a vanishing vertical fluid velocity, and is estimated by the median of the vectors that are available in the velocity maps. The measurements of c exhibit an almost linear increase with wave amplitude. (For a/h2 « 1 the fully nonlinear, fully dispersive computations give c/co — 1 ~ 0.584 a/h2, while KdV theory gives c/c0 - 1 ~ 0.605 a/h2.) The correspondence between experiment and theory is good (Fig. 6). The wave speed plot does

Internal

Wave Fields Analyzed by Imaging

253

Velocimetry

1.9 1.8

• • ' '

-

1.7 1.6 C_

• On

1.5

CO

•

A^

1.4 1.3 1.2 1.1

-

^

D Camera 1-2 <J Camera 2-3 Theory ^ Inverted model, Camera 2-3

> 0.5

; 1.5

a//i2 Fig. 6. Propagation speed c vs. wave amplitude. Symbols: measurements. Solid line: theory. From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

not indicate that a saturation of the wave, in the form of a maximal value of the amplitude, exists. This is unlike what was observed for a two-layer fluid with constant density in each layer. Another important difference from the previous two-layer fluid is that an abrupt velocity transition across the thin pycnocline is replaced by a smooth velocity field. The maximal fluid velocity is now occurring at the top of the water column and is significantly larger than with the other two-fluid system, for otherwise same wave amplitude. The variation of the horizontal velocity profile during the passage of a solitary wave is visualized in Fig. 7. The induced fluid velocity in the upper part of the upper layer is rather pronounced despite the relatively modest nondimensional wave amplitude of a/h2 = 0.4. There is again a good correspondence between experiment and theory. The wave motion introduces a nonzero vorticity field in the stratified fluid. Vorticity profiles are obtained from the particle imaging measurements using a finite difference operator, i.e., df/dxi = (2/j+2 + fi+i —

254

J. Grue

o

o

o

o

Fig. 7. Time series of passage of wave. Horizontal velocity U/CQ as function vertical coordinate. Symbols: measurements. Solid line: theory. From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

Internal

Wave Fields Analyzed by Imaging

255

Velocimetry

/i_i — 2/i_2)/(10AX). The experimental vorticity estimates are supported by the fully nonlinear computations for nonbreaking waves (Fig. 8a). Breaking of the waves is manifested by pronounced spikes in the vorticity field. A pronounced negative vorticity is observed at a level of J///12 = —0.3 in Fig. 8b, for example. The spike corresponds to a vortex seen in the velocity field and is caused by convective breaking. The maximal value of the vorticity in the series of experiments reported here is |w|/(co//i2) — 3, a result that is true for the largest waves. With NQH^/CO = 1.711.., the maximal value of the vorticity relative to the Brunt-Vaisala frequency becomes, about, OJ/NQ ~ 1.7.

%*-

b )

M.

c0/h2

c0/h2

Fig. 8. Vorticity (UJ) at crest of wave vs. vertical coordinate, a) Without breaking, b) with breaking. Solid line (theory, a//i2 = 0.65), other lines (experiments). From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

For increasingly larger solitary waves, both the wave speed c and the induced fluid velocity relative to c grow until a saturation is reached. The limiting mechanism of the waves is found to be convecting breaking taking place when the induced fluid velocity attempts to exceed the wave speed. The convective breaking is observed in a region in the centre of the wave in the upper fluid. In the breaking region the fluid velocity has the form v = ci + v ' , where ci denotes the wave velocity and v' a velocity field where |v'| < < c. (The waves transport a body of mass.) The vector i denotes the horizontal unity vector along the direction of the wave propagation. The horizontal velocity profile above wave crest is characterized by u/c being

256

J. Grue

unity in a vertical range corresponding to almost the thickness of the upper layer (Fig. 9a-b). The experimental waves exhibit broadening when breaking occurs (Fig. 9c). The nondimensional wave amplitude then exceeds about 0.80.9. (The theory predicts u/c = 1 for a//i2 = 0.855.) The broadening effect found here is entirely different from the mechanism taking place in a two-fluid system with constant density in the layers. Then the limiting amplitude and limiting wave speed, and thereby the broadening of the waves, are determined by a finite total depth of the fluid (Amick and Turner 10 , Turner and Vanden-Broeck 12 ), a result which is experimentally confirmed by the results in section 3. We note that in the present two-layer fluid a solitary wave (of mode one) always exhibits an excursion out of the layer with linear stratification. Observations in the wave tank, and computations, confirm this result (results not shown). The polarity of the solitary waves may be determined analytically from the sign of the coefficient of the quadratic nonlinear term in weakly nonlinear KdV- or Intermediate-long-wave theory. This shows that the polarity is always the same for a two-layer fluid where one layer is linearly stratified and the other layer is homogeneous (and the stratification is continuous), see also Grue et al. 16 eq. (8.1). The results exhibit good correspondence between experiment and theory up to breaking. The fully nonlinear inviscid theory does neither explain the breaking nor the broadening of the waves observed in the experiments, however.

5. Wave-breaking at a submerged ridge A solitary wave that interacts with a submerged ridge undergoes a local distortion that causes a scattering of the wave. Mechanical energy may be lost if breaking occurs. The case of a solitary wave incident normal to a long-crested ridge was experimentally investigated using PTV and PIV by Sveen et al. 31 . A stably stratified two-fluid system had homogeneous fluids in each layer. Conditions leading to breaking at the ridge and transmission of the solitary waves represented the main objectives. We summarize the features that is characteristic for strong interaction between wave and ridge. The incoming internal solitary waves break during the deformation if the amplitude relative to the local depth at the top of the ridge exceeds a certain level. The breaking process is illustrated in Fig. 10. The incoming wave is in this case interacting with a relatively wide ridge. More specifically, the parameters in the experiment were: lower layer

Internal

Wave Fields Analyzed by Imaging

257

Velocimetry

te//l2

b)

o

12.5

o

-

-

10

c)

o X

o-g.

0

o 7.5

O

~^*~-^*_ =^t^^

"

X

Fig. 9. a) Wave induced horizontal velocity at the free surface as funtion of time (initial volume 100 dm 3 , measurements at several positions along the wave tank), b) horizontal velocity profile at crest in breaking waves, c) wave length A//12 = ( l / " m o i ) f^° u dx //12 vs. Umax/c obtained in experiments (symbols) and in theory (solid and dashed lines). From Grue et al. 1 6 . Reproduced with permission by J. Fluid Mech.

258

J. Grue

depth outside ridge h\ = 45 cm, upper layer depth /12 = 10 cm, incoming (nondimensional) wave amplitude a//i2 = 0.59, width of ridge 6.2 m, height of ridge above tank floor 30.5 cm, constant ridge slope ±0.1 and top of ridge positioned 7.83 m from the endwall of the wave tank where the wave was generated. (There is a smooth transition at the top of the ridge between the two slopes.) The length of wave tank was 21.5 m. The top of the ridge was in all experiments below the pycnocline at rest, and the upper layer was always thinner than the lower layer. The incoming waves were solitary waves of depression. During a strong interaction between wave and ridge the excursion of the pycnocline typically became lower than the mid depth of the water column. A significant distortion of the wave was observed as it encountered the turning point, i.e. the horizontal position (on the slope) where the pycnocline is half-way between the bottom and top of the water column. The incident depression wave was then transformed into a shorter leading depression and a subsequent dispersive wave train, where the first of the waves typically has a relatively high elevation. This process is the same as the one that occurs to a solitary wave of depression travelling up a sloping beach, beyond the turning point, described theoretically in Djordjevic and Redekopp 43 , Knickerbocker and Newell44, Malomed and Shrira 45 and experimentally by Helfrich, Melville and Miles46. The process on the ridge develops a breaking of the wave in the form of mixing between salt and fresh water at the pycnocline, in the rear of the leading depression, and in the leading part of the subsequent wave of elevation. The mixing is visible as a fog in the upper picture in Fig. 10. A second breaking event occurs when the tail of the wave collapses due to the local strong steepness there, see the lower picture in Fig. 10. Estimates indicate a magnitude of the accelerations of about 0.55' during the collapse. This means that a Rayleigh-Taylor instability is not the cause of the breaking. Dispersion has a weak or strong effect on the incoming wave, depending on the slope of the ridge. If the slope is relatively small, dispersion has a stronger effect than indicated in Fig. 10, on the incoming depression solitary wave, as it arrives at the turning point. The wave is in the small slope case transformed into a relatively short leading depression that is decreasing in time, and a subsequent dispersive wave train, with a number of waves that gains in time. The dispersive wave train represents the origin of boluses or vortex blobs - travelling up the slope, see the forthcoming section 6.2. If the ridge slope is relatively steep, and the waves have sufficiently large amplitude, wave breaking in the form of overturning is observed at the obstacle.

Internal

t

Wave Fields Analyzed by Imaging

Velocimetry

259

-0.6

X

h2

b)

£ ^0.5

X

h2 Fig. 10. Waves breaking at a ridge. Time difference 2.04 seconds. From Sveen et a!.3 Reproduced with permission by J. Fluid Mech.

260

J. Grue

For a closer examination of the local process at the top of the ridge, prior to wave breaking, vertically averaged horizontal velocities in the upper and lower layer, u2 and u\, respectively, are measured. Local thicknesses of the layers at the top of the ridge are defined by h\^, where the sum h\ + h,2 equals the total depth of the fluid at the top of the ridge, see Fig. 11a. The values of hi and h2 are obtained using u\h\ 4- u2h2 = 0, given h\ + h2. The local thicknesses of the layers define the local nonlinear shallow-water speed at the top of the ridge, serving as a reference speed, i.e. Co = [g'hih2/(hi + h.2)]1/2, where g' = gAp/p, and the maximal value of h,2 and the minimal value of hi are used. The measured M2 above the ridge show good correspondence with the fluid velocity in the upper layer due to the undisturbed incoming wave (results not shown). On the other hand, due the thin local depth of the lower layer below the wave crest, at the top of the ridge, the magnitude of the velocity ui in the lower layer may become large. This is readily seen from the equation of continuity, i.e., U\hi+U2h2 = 0, which means that if ^i//i2 becomes small, |ui/«2| becomes correspondingly large. This is characteristic for waves with large negative excursions. An example is displayed in Fig. l i b . Measurements of ui^/co show that breaking of incoming solitary waves of depression always occur when |ui|/co exceeds the value of 0.7, i.e. somewhat less than critical flows (Fig. l i e ) . This is a result from a total of 56 runs where the parameters in the experiments were: wave amplitude in the range 0.2-1.9 times the depth of the upper layer, ratio between the lower and upper layer depth in the range 3-8.5, and ridge slope in the range 0.10.33. Incipient wave breaking happens either in the form of local spilling or violent overturning of the waves. Shear instability, which plays no role in the initial part of the wave breaking process, is observed during the later stages in some of the breaking events. In all cases of breaking, vortices are shed on the up-slope side of the ridge. The (almost critical) lower layer flow at the slope is attached close to the top of the ridge, but separates at some point, as indicated in the photos in Fig. 10. The initial separation point is on the up-slope side. As a result of the separation, tracer particles are distributed vertically across the lower layer. The transmitted waves exhibit a leading solitary wave with a subsequent dispersive wave train. The amplitude of the transmitted solitary wave corresponds in general to that of the incoming wave if breaking does not occur at the ridge. In the case of wave breaking the solitary wave undergoes a significant reduction in amplitude, however. The breaking process leads locally to a mixing between fresh and salt water at the ridge. Portions of mixed fluid that locally widens the pycnocline may subsequently generate

Internal

Wave Fields Analyzed by Imaging

Velocimetry

261

solitary waves of mode two propagating along the pycnocline, see Fig. 12. The induced velocity within the pycnocline, due to the mode two wave, is as large as 30 % of the horizontal fluid velocity due to the leading solitary wave of mode one, with nondimensional amplitude a/h,2 = 0.35 in the actual example. Generation of waves of mode two at a ridge has also been studied by Vlasenko and Hutter 30 . A corresponding interaction between an incoming solitary wave and a ridge is true also if the undisturbed upper layer has a linear stratification and the lower layer is homogeneous. A strong interaction between wave and ridge exhibits a pronounced increase of the lower layer velocity at the top of the ridge, a separation of the flow on the upwave side of the ridge and subsequent breaking of the wave. The reduction of the solitary wave amplitude due to the wave-ridge encounter is somewhat less pronounced if the upper layer is linearly stratified as compared to when the fluid is homogeneous (Guo et al. 32 ). 6. Wave breaking and run-up at a shelf-slope 6.1. Steep slope. Strong local breaking and

mixing

Internal solitary waves that interact with a relatively steep shelf-slope (continuing through the entire water column) represented one of the focuses in the work by Kao et al. 20 . They ascribed the breaking of the waves at the shelf, observed in their experiments, to shear instability. Michallet and Ivey 28 studied solitary waves of large amplitude that broke as they encountered the shelf-slope. In the latter work PIV was used to quantify the velocity fields. The mixing efficiency at the shelf and the magnitude of a reflected solitary wave travelling away from the beach were measured for wave slopes in the range 0.07-0.2. Michallet and Ivey suggested that the mechanism leading to breaking of the waves at the shelf was due to gravitational instability (this is not supported by our measurements), following the separation of the flow in the lower layer. The mixing efficiency was estimated from the difference between the initial density profiles and the density profiles at the shelf-slope after breaking. The main part of the mixing was found to occur in the lower layer. Their and our results for solitary depression waves interacting with a submerged ridge, presented in the previous section, are conform on this point. The reflected to incident wave energy ratio (ER/EO) as function of the wave length to slope length ratio (Lw/Ls) was quantified. The results showed that ER/EO — Lw/Ls for Lw/Ls < 0.6 and that a maximum value of ER/EQ ~ 0.65 was reached for larger ratios of Lw/Ls- A strong

J. Grue

a)

b)

10

15

20

tc0/h2

1.25 •

T~

0.75 •

0.3

0.4

0.5

0.6

0.7

0.8

(a + h2)/(h1 + h2 - hr) Fig. 11. Waves breaking at a ridge, a) Sketch, b) horizontal velocities at top of the ridge, c) in plot: breaking (filled squares), strong local deformation (open circles), weak interaction (dots). From Sveen et al. 3 1 . Reproduced with permission by J. Fluid Mech.

Internal

Wave Fields Analyzed by Imaging

263

Velocimetry

1

0.5

h2 -0.5

-1

-0.05

0

0.05

0.1

U/CQ

Fig. 12. Velocity profile in wave of mode two propagating along the pycnocline. From Sveen et al. 3 1 . Reproduced with permission by J. Fluid Mech.

flow up the slope in the form of a bolus could emerge from the interaction between the incident wave and the slope. After breaking, an off shelf flow was observed in the lower layer. Suspended particles were transported along the pycnocline. Particles that initially were located in the breaking region (at the slope) were spread far offshore.

6.2. Run-up

at a

shelf-slope

Relatively short incoming periodic waves or solitary waves of depression interacting with a weakly sloping beach show a transformation into boluses propagating up-slope (Cacchione and Southard 25 , Wallace and Wilkinson 26 , Helfrich27). A solitary wave of depression may split into a shorter leading depression with a subsequent dispersive wave system, each individual wave developing into a bolus. This results in a sequence of boluses propagating up the slope (Fig. 15). We shall here extend previous work, reporting experiments where the initial elevation or depression of the interface is very long. Our particular focus is strong run-up and strong currents that are generated where the pycnocline intersects the slope. The experiments are motivated by strong current events that have been observed at the sea floor at the offshore gas

264

J. Grue

field Ormen Lange. This is located in water 850 m deep on the slope of the Norwegian shelf. A thermocline at an average depth of about 500 m represents a characteristic feature of the ocean at the location. The thermocline separates the warm northward Norwegian Atlantic Current with temperature about 8 C° from cold water masses in the deep ocean with temperature -0.5 C°. Outside the shelf-slope, the lower cold layer has a thickness that is 1-2 times the depth of the upper warm water. (At the location of Ormen Lange the upper layer is thicker than the lower.) The relative density jump across the thermocline is 0.5 x 1 0 - 3 . Fig. 13 shows a recording of a strong landward current event at the sea floor at the actual location. The recorded current speed has a maximum of 0.5 m/s (Fig. 13a). The flow direction is up the shelf-slope. There is only a very small velocity component along the shelf-slope. Prior to the current event the temperature at the sea floor exhibited a slow build-up phase reaching a maximum of 3 C° (Fig. 13c). This indicates that the thermocline had moved downward the slope prior to the event, beyond the measurement position. The sudden return of the temperature coincides with the rapid increase of a current at the sea floor. The duration of the event is about 16-24 hours. It is convenient to put the observed current on nondimensional form. As reference speed we use the linear long wave speed of the two-fluid system determined by CQ = [g'hifi2/(hi + /12)]1/2 ~ 1-3 m/s where g' = gAp/p and we have put h2 = 500 m for the upper layer depth and the lower layer depth is determined by h\/li2 = 2. The maximum recorded speed of 0.5 m/s corresponds to about 40 % of the linear long wave speed. Further, the duration of the event corresponds to a nondimensional time ico//i2 in the range 150-224. Observations of the thermocline motion in the ocean indicate that a strong current event may occur due to very long wave motion along the thermocline. Typical periods may be in the range 1-2 days. A corresponding estimated wave length is 100-200 km which means 200-400 times the upper layer thickness. A set of model experiments in scale 1 : 5000 were undertaken in a small wave tank of length 7 m to shed light on the wave events observed at Ormen Lange (Sveen and Grue 47 ). The upper layer depth in the wave tank was h2 = 10 cm, the lower layer depth in the deep water part h\ = 20 cm and the slope factor was 0.3 m / 7 m corresponding to 2.5 degrees. The lower layer of brine and upper layer of fresh water was separated by the pycnocline being about 1 cm thick. With a relative jump in density of 0.02 the linear long wave speed in the laboratory was 13 cm/s.

Internal

Wave Fields Analyzed by Imaging

Velocimetry

265

Currents and temperature at position II 5m asb

20

21

22

23

24

25

26

27

28

29

1

1

1

b)

40 Component normal to max velocity 20

0

1

0 -_

20

1

"\IM

l>' '

^

20 20

ufll [•Hs

kj,: TVU JWT—-"^ ^

I

I

I

23

24

25

: . . . ~sr^.:

23

1

1 n>JlAAywS

p^"

1

nAJ^wft,

^ : W

^J"

1

I

29

:\

24 25 26 Dates November 1999

jT.... V J U A < X W - :

28

-

29

30

_

30

Fig. 13. Strong current event. Measurements 5 m above sea floor. Reproduced with permission by Norsk Hydro ASA.

Waves were generated by trapping a volume either of brine or fresh water behind the gate, elevating or depressing, respectively, the pycnocline relative to the level elsewhere in the tank (Fig. 14a). In the interest of investigating the response due to very long initial disturbances, the gate was placed 2 m from the end of the wave-generating part, corresponding to 20 times the upper layer thickness. Both an initial elevation and depression leads to the formation of a long bore at the point where the pycnocline intersects the shelf-slope. The bore propagates up the slope in the form of

266

J. Grue

g a i e linear stratification

a)

•>r"Mk3Slli

b)

c)

Fig. 14. Internal run-up on slope, a) Sketch of experiment, b) velocity field during internal wave run-up, c) induced horizontal velocities in lower (u%) and upper (U2) layer.

Internal

Wave Fields Analyzed by Imaging

Velocimetry

267

an internal run-up, with a leading part of the bore resembling the head of a gravity current (Simpson and Britter 41 , Simpson 42 ), see Fig. 14b. The induced fluid velocity in the head of the lower layer flow shows a maximal value of about 40 % of the linear long wave speed (U/CQ ~ 0.4), i.e. the same nondimensional speed as observed in the field (Fig. 14c). In the figure is plotted vertically averaged horizontal velocities in the layers. The duration of the wave event in laboratory of tco/h2 = 40 nondimensional periods is about 3 times shorter than in large scale. This difference is probably due to the relative difference of the input wave length in laboratory and large scale. The most important point with the two-dimensional laboratory investigation was to estimate the magnitude of the fluid velocities that may be introduced during run-up of very long internal waves. The laboratory measurements may be used, e.g., as reference for simplified ocean model simulations in two dimensions with the rotation (of the earth) terms switched off. Three-dimensional and rotation effects should be included in full simulations of internal run-up.

7. Fish as markers of internal waves In-situ regsistrations of internal wave motion were made by the Norwegian Institute of Marine Research in a small campaign off the Norwegian coast during August 1998. While the main objective was to register fish schools by means of echo-sounders, the data logs showed more than just the appearance of fish. A pronounced oscillatory motion of the schools became evident and were observed from both ship and buoy. The observations were made in the Norwegian Sea north of S0r0ya in Finnmark at position 71°N 22°10'E. A part of the record from the buoy is shown in Fig. 16 (God0 48 ). It was evident that the fish was eating plankton that was drifting with the flow in the ocean. The motion of the individual fishes were traced by the data processing routines and are visible in the plot by small streaks. It can be concluded that the fishes serve as markers of an oscillatory vertical fluid motion in the sea, with an average level at 100 m depth, amplitude of 20 m at a depth of the sea of 146 m. Hydrographical data from the area show that a roughly linear stratification in the upper part of the sea during late summer and autumn is developed annually. The thermocline is 200 m thick and has a relative density jump of 1 0 - 3 . The deep part of the sea has a constant density. Syntetic Aperture Radar (SAR) images from satellite were obtained during autumn 2000 and revealed occurrence of internal wave motion in the actual area in

268

J. Grue

the form of dispersive wave trains propagating from shallow to deep water. Wave lengths (from several images) were in the range 300-500 m. We interpret the set of observations on basis of the experiments and theories of waves moving along the linearly stratified upper layer and the homogeneous lower fluid. Relevant for the actual observation we assume that the homogeneous lower layer is relatively thin compared to the thickness of the upper layer. The linear long wave speed is estimated using eq. (20) (see section 8.2 below) with h2 » hi, giving c0 ~ N0h2/Tr ~ 30 cm/s. The Brunt-Vaisala frequency is determined by N$ = gAp/(ph2) with h2 = 146 m. The equations (18)-(20) show that a wave length of A = 300 m corresponds to a wave speed of 30 cm/s. The wave induced oscillatory vertical velocity at level 100 m becomes 12 cm/s at maximum, i.e. 40 % of the wave speed. The linear formulae predicts a horizontal surface velocity that somewhat exceeds the wave speed. The estimated wave period becomes 16 minutes (frequency 0.39 m i n - 1 ) which is significantly longer than indicated by the raw data from the echo-sounder (period 5 min, frequency 1.26 m i n - 1 ) . (The buoy was drifting with the unmonitored surface current.) Recent measurements 49 in the area (70° 27,6'N, 17° 13,9'E) reveal a maximal current speed of 65 cm/s, contributing to the frequency of encounter by an amount Uk = 0.82 m i n - 1 . This may explain the difference between the observed and estimated wave periods. The wave trains are most truly tidally driven. The Atlas of the tides of the shelves of the Norwegian and Barents Seas (Gjevik, N0st and Straume 50 ) exhibts dominating M2 and S2 modes with maximal currents at the location of about 25 cm/s and 10 cm/s, respectively. At spring tide the tidal current has a maximal speed of 35 cm/s. This corresponds to the magnitude of the linear long wave speed of the stratified fluid at the location. It is widely documented that a pronounced internal wave making may take place when the flow is close to critical, i.e. the bulk flow velocity is close to the linear long wave speed of the stratified fluid, see e.g., Grue at al. 14 . 8. Fully nonlinear reference models For completeness were include a brief documentation of the fully nonlinear solitary wave models used in the reference computations. 8.1. Interface

model

Several works on modelling fully nonlinear internal solitary waves are published (e.g. Amick and Turner 10 , Turner and Vanden-Broeck11, Evans and

Internal

Wave Fields Analyzed by Imaging

Velocimetry

269

Ford 13 ). Our method is based on the fully nonlinear transient interfacial model Grue et al. 14 , solving the Laplace equation in the layers, integrating the prognostic equations resulting from the kinematic and dynamic boundary conditions at the (moving) interface. The model accounts for full nonlinearity, full dispersion, any density or depth ratio, and puts no assumption on the wave length. The model was developed with the primary goal to compute transient waves. A version of the method to obtain waves of permanent form is derived (Grue et al. 15 , section 3.1). The coded method is fully documented and may be downloaded, see Rusas 51 . The main steps of the method which is used in the present computations are outlined. Solitary waves of permanent form are modelled in the frame of reference moving with the wave speed c. In this frame of reference the interface is "frozen". There is a horizontal current in the far field with velocity —c. A coordinate system O — xy is introduced with the x-axis at the level of the interface in the far field and the y-axis pointing upwards. We apply complex analysis and introduce the complex variable z = x+iy and complex velocity 52(2) = u2(x,y) — c — iv2(x,y) in the upper layer and qi(z) = u\{x,y) — c — ivi(x,y) in the lower. The components Uk — c and Vk (k = 1,2) determine the horizontal and vertical components of the velocity field, respectively. At the top and bottom of the fluid layer the vertical velocity is zero, qi + c and q2 + c, being analytic functions, are found using Cauchy's integral theorem, giving (the fluid in each layer is assumed incompressible, homogeneous and inviscid, and the motion irrotational)

Js

z'-z

Jsz*

+ 2ih2-z>

\

27ri(g2 + c), z' e f i 2 ( 4 ) -7ri(g2 + c ) , z' € S

in the upper layer, where q'2 = q2(z'), and PV f

qi z)

^

+ C

dz 4- f

Q1{Z)

+ C

dz* - f

27d(
'

Z

'

e

fil

(5)

in the lower, where q[ = qi(z'). Here an asterix denotes complex conjungate, S the interface and fife the domain of fluid number k. The interface S is parameterized by z(£). (Scaled) tangential velocities are defined by T2 = q2dz/d£ on the upside of the interface and Ti = qidz/d£ below. From (4) an integral equation to determine F2 is derived:

-„« + v4) . pvfte (£BL) « -j* ( ; ^ f g ^ ) « +

jMvh

+

s+2L-J){u+CXi)di

<e)

270

J. Grue

where z% = dz/d£, x$ = dx/d£, etc. A similar equation is derived from (5) for determination of Ti. The dynamic boundary condition at the interface gives:

^~{p2T\-piTl)-^{p1-p2)c2

+ {Pl-p2)gy

=0

at

S

(7)

where g denotes the acceleration of gravity. In the computations an equidistant distribution in x is applied. /
8.2. Fully nonlinear internal waves propagating linear upper stratification

along a

Weakly or fully nonlinear models assuming a continuous and smooth density profile are developed in several works (Long 52 , Yih 53 , Tung et al. 54 ; Turkington et al. 55 , Brown and Christie 56 ). Here, the fully nonlinear integral equation method used in the computations included in section 4, is outlined. Waves of permanent form are modelled in the frame of reference moving with the waves (with a current with speed — c at infinity). Assuming incompressible, inviscid fluid, and conservation of mass, gives, V • v = 0, v- Vp = 0 where v = (u — c, v) denotes the fluid velocity. If a pseudo-velocity v' is introduced by v' = ( / o/p 0 ) 1 ^ 2 v, where po is a reference density, and a pseudo-stream function ty' satisfies v' = Vvl/' x k where k = Va; x Vy, it follows that p = p(V). The equation of motion gives „

dH

d¥

dp

2 T /

/dp =

dH(W) c2 dp

\ dy

{dy

+P9

)d¥

+

Jd¥

dp + 9y

d¥-

(9)

The vertical component of the equation of motion becomes in the far-field Py+P9 = 0) which means that the first term on the right of (9) is zero. The pseudo-stream function is then decomposed by \&' = ^ ^ + ip', where \I/^0 satisfies dV^/dy = - c ( p / p 0 ) 1 / 2 , giving V 2 * ^ = (c2/2p0)(dp/d^'). Since dp/dW is constant along each streamline, (8) becomes P o V y + <7(tf-|feo)j£=0,

(10)

Internal

Wave Fields Analyzed by Imaging

271

Velocimetry

where y and y^ are vertical coordinates on the same streamline, with y ^ in the far-field. Applying the Boussinesq approximation (Ap/p « 1) we find \I>^ = -cy[l + 0(Ap/p)}, giving V-V00=il>'/c.

(11)

Furthermore we have g dp

g dp dy

N2

n(KnlM

70w = y0dyd¥ " v [ 1 + 0(Ap//5)L

/10x

(12)

where N2 = — (g/po)(dp/dy) determines the Brunt-Vaisala frequency. Within the Boussinesq approximation we may replace the pseudo-stream functions (\P', tp') by (ty, ip). The motion is thus governed by the Helmholtz equation, i.e. V 2 ^ + ^-ip = 0. A two-layer model where the undisturbed fluid has an upper layer with linear density profile and a lower layer with constant density is employed: ,» p{y)

=

(p0~Apy/h2, \po,

for for

0 < y < h2, -fc1
,

.

(13)

The line y = 0 separates the two layers. At rest, the upper layer extends in the interval 0 < y < h2 and the lower layer in the interval —h\ < y < 0. The Brunt-Vaisala frequency becomes constant in the upper layer, and is there determined by NQ — (Apg)/'{poh2)• In the lower layer the BruntVaisala, frequency becomes zero. Let tp = tp2 in the upper layer and tp = ip\ in the lower. tp2 satisfies the Helmholtz equation in the upper layer, i.e. V 2 ^2 + -$~il>2 — 0. The stream function in the lower layer satisfies the Laplace equation, i.e. V 2 ^ i = 0. The upper boundary of the upper layer and the lower boundary of the lower layer are approximated by rigid lids. The two layers are separated by the streamline with vertical coordinate 77 where 77 —> 0 for x —> ±00. The kinematic boundary condition requires that the fluid velocity is continuous at the boundary between the layers, i.e. that V(V>i - cy) - V(V>2 - cy) at y = rj. The stream functions are determined by

v 1,2

J

<TI,2(*')GI,2(X,

y, x'(s'), y'(s'))ds',

(14)

where (T\(s) and a2(s) are distributions, / the contour y = 77 and s arclength along the streamline y = rj. The Green function G\ is a pole at (x, y) = (x',y') and satisfies the Laplace equation, i.e. Gi(x,y,x',y') — ln(r/ri). The other Green function is a pole at (x,y) = (x',y') and satisfies the

272

J. Grue

Helmholtz equation, i.e. G2(x, y, x',y') = (7r/2)[Y0(Kr) - Y0(Kr2)}, where Yo denotes the Bessel function of second kind of order zero and K = NQ/C. Furthermore, r = [(x-x')2 + (y-y')2}1/2, n = [(x-x')2 + (y+y'+2h1)2}1/2, 2 2 1 2 ri = [(x — x') + (y + y' — 2h2) ] / . The kinematic boundary condition at y ~ r\ gives

™/*«%«-%-o.

(15)

Pvf^^-^0,

(16)

-TT[(7I( S ) + 7ra2(s)] + J ( W ) ^

1

" V2(s')^)ds'

= 0,

(17)

where s denotes the arc length along the streamline y = r\ and n the normal pointing out of the lower fluid layer. The linear long wave solution of the mathematical problem is given by V2 = ac0

sm(NQ(y-h2)/co) . sm(N0h2/c0)

cos vx ,

tpi = — aco(l + y/hi)cosvx,

, . (18) (19)

where v denotes the wavenumber and CQ the linear long wave speed. The latter is obtained by putting dipi/dy = dtp2/dy at y = 0, giving N0h2

N0h2 cot

Co

C0

^h2 h — = 0,

(20)

All

where CQ is obtained for N0h2/co in the interval (7r/2,7r). 9. Conclusion Particle Tracking Velocimetry and Particle Image Velocimetry facilitate efficient and detailed recordings of velocity fields induced by internal wave motion. Global properties like wave speed and wave length are derived from the velocimetry. The laboratory measurements serve as references for fully nonlinear theoretical studies and confirm the applicability of the theories for a wide range of parameters. This is true for wave properties in a two-fluid system with constant density in the layers, for example. The experimental results, put on nondimensional form, may be used to predict internal wave

Internal

Wave Fields Analyzed by Imaging

Velocimetry

273

propagation and internal wave induced velocity fields in large scale. The wave tank experiments are used to explain observations of internal wave motion in the sea (Skagerak and the Norwegian Sea), indicating that the induced velocities may be rather strong. The laboratory experiments we refer to here were run with Reynolds number in the range 0.5xl0 4 -3xl0 4 , where the Reynolds number is estimated from a typical velocity of the flow times a characterisitic length scale, taken to be the thickness of the upper layer, divided by the kinematic viscosity of the water. The laboratory simulations of internal wave motion are particularly useful beyond the range of validity of theoretical models. This is true for breaking flows induced by internal waves. Convective breaking of solitary waves propagating along a linear stratification in the upper part of the water column introduces a broadening of the waves, for example. This type of broadening differs from the one that takes place in a two-fluid system with constant density in the layers, where the waves are conjugate flow limited. The latter type of broadening is true for all depth ratios when the solitary waves are depression waves. Such a broadening is also true for solitary waves of elevation in the Boussinesq limit. Solitary waves of depression may interact with a ridge that is located in the bottom of a two-fluid system with constant density in the layers. In such cases the PIV measurements show that breaking occurs when the induced fluid velocity in the lower layer, at the top of the ridge, exceeds 0.7 times the local (nonlinear) long wave speed. This conclusion is based on totally 56 experiments with wave amplitude in the range 0.2-1.9 times the upper layer depth, the ratio between the lower and upper layer depth in the range 3-8.5, and ridge slope in the range 0.1-0.33. An important subject is interaction of thermoclines with a shelf-slope. Run-up of very long internal waves induces strong currents where the pycnocline intersects the slope. It is found that the maximal speed may be up to 40 % of the linear long wave speed of the two-fluid system. Such strong currents is a concern for industies operating in deep water. Breaking of the flow at the sloping bottom results in significant local mixing. It is observed that suspended fluid particles may be transported along the pycnocline, far off the shelf. Three-dimensional analysis of the motion of a stratified fluid at the shelf-slope represents a future challenge. An investigation combining precise laboratory experiments and theoretical ocean modelling may prove to be efficient. The effect of the earth rotation should be included in realistic predictions of internal run-up. Breaking wave effects and the effect of shear instability represent important subjects where the use of PTV and PIV can still enhance our

274

J. Grue

knowledge. Experiments can be used to quantify the value of the Richardson number that corresponds to the onset of shear instability. For such experiments measurements of the density profile in the pycnocline (using e.g. Head probes or Schlieren technique) are required in addition to the velocimetry. Theories can be used to investigate the limit where u/c exceeds unity, indicating onset of convective breaking, and to compute the conjugate flow limit, indicating saturation of the wave amplitude and wave speed. Vertical transport processes in the ocean represents another subject where more investigations are required. Supply of nutrient water masses from deep ocean to depleted water in shallow regions are important for the production taking place there, for example. We have in the experimental studies referred here, estimated the vertical transport of particles that may occur during breaking of the waves or in post breaking flows. We have found that the vertical transport is always very small, however. It may be true that vertical transport phenomenae occurring in the ocean cannot be properly modelled in the laboratory. One reason may be that the Reynolds number in the laboratory is relatively small. A more important reason may be that the time scale of the vertical processes taking part in the ocean is much longer than the time scale of the processes that can properly be modelled in laboratory scale. Acknowledgments This work was conducted under the Strategic University Programme 'General Analysis of Realistic Ocean Waves' funded by the Research Council of Norway. The data of the oscillatory fish motion provided by Dr. O. R. God0, the assistance with the figures by J. K. Sveen and the technical assistance in the laboratory by A. Kvalheim and S. Vesterby are gratefully acknowledged. References 1. 2. 3. 4. 5.

A. R. Osborne, T. L. Burch and R. I. Scarlet, J. Pet. Tech. 30, 1497 (1978). L. A. Ostrovsky and Yu. A. Stepanyants, Rev. Geophys. 27, 293 (1989). J. M. Huthnance, Geophys. Astrophys. Fluid Dyn. 48, 81 (1989). R. D. Pingree and G. T. Mardell, Prog. Oceanogr. 14, 431 (1985). J. R. Apel, J. R. Holbrook, A. K. Liu and J. Tsai, J. Phys. Oceanogr. 15, 1625 (1985). 6. T. P. Stanton and L. A. Ostrovsky, Geophys. Rev. Lett, 25, 2695 (1998). 7. B. Gjevik and S. E. H0st, Naturen 6, 209 (1984). 8. J. Y. Holyer, J. Fluid Mech. 93, 433 (1979).

Internal

9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39. 40. 41. 42. 43. 44. 45. 46. 47. 48. 49. 50. 51.

Wave Fields Analyzed by Imaging

Velocimetry

275

D. I. Meiron and P. G. Saffman, P. G. J. Fluid Mech. 120, 213 (1983). C. J. Amick and R. E. L. Turner, Trans. Am. Math. Soc. 298, 431 (1986). R. E. L. Turner and J.-M. Vanden-Broeck, Phys. Fluids 29, 372 (1986). R. E. L. Turner and J.-M. Vanden-Broeck, Phys. Fluids 31, 2486 (1988). W. A. B. Evans and M. J. Ford, Phys. Fluids 8, 2032 (1996). J. Grue, A. Friis, E. Palm and P.-O. Rusas, J. Fluid Mech. 351, 223 (1997). J. Grue, A. Jensen, P.-O. Rusas and J. K. Sveen, J. Fluid Mech. 380, 257 (1999). J. Grue, A. Jensen, P.-O. Rusas and J. K. Sveen, J. Fluid Mech. 413, 181 (2000). G. K. Lamb, J. Fluid Mech. 451, 109 (2002). C. G. Koop and G. Butler, J. Fluid Mech. 112, 225 (1981). H. Segur and J. L. Hammack, J. Fluid Mech. 118, 285 (1982). T. W. Kao, F.-S. Pan and D. Renouard, J. Fluid Mech. 159, 19 (1985). H. Michallet and Barthelemy, J. Fluid Mech. 366, 159 (1998). R. E. Davis and A. Acrivos, J. Fluid Mech. 29, 593 (1967). T. Maxworthy, J. Fluid Mech. 96, 47 (1980). A. P. Stamp and M. Jacka, J. Fluid Mech. 305, 347 (1995). D. A. Cacchione and J. B. Southard, J. Geophys. Res. 79, 2237 (1974). B. C. Wallace and D. L. Wilkinson, D. L. J. Fluid Mech. 191, 419 (1988). K. R. Helfrich, J. Fluid Mech. 243, 133 (1992). H. Michallet and G. N. Ivey, J. Geophys. Res. 104, 13467 (1999). F. Wessels and K. Hutter, J. Phys. Oceanogr. 26, 5 (1996). V. I. Vlasenko and K. Hutter, Nonl. Proc. Geophys. 8, 223 (2001). J. K. Sveen, Y. Guo, P. A. Davies and J. Grue, J. Fluid Mech. 469, 161 (2002). Y. Guo, J. K. Sveen, P. A. Davies, J. Grue and P. Dong, J. Environm. Fl. Mech. (2004, to appear). S. B. Dalziel, Applied Scientific Research 49, 217 (1992). C. E. Willert and M. Gharib, Exp. in Fluids 10, 181 (1991). J. Westerweel, D. Dabiri and M. Gharib, Exp. in Fluids 23, 20 (1997). J. K. Sveen, (1998). L. A. Ostrovsky and J. Grue, Phys. Fluids, 15, No. 10, 2934 (2003). A. Carlin, Master thesis in Mechanics, Dept. of Math., Univ. of Oslo (2000). K. Hald, Master thesis in Mechanics, Dept. of Math., Univ. of Oslo (2000). D. M. Farmer and J. D. Smith, Deep-Sea Res. 27A, 239 (1980). J. E. Simpson and R. E. Britter, J. Fluid Mech. 94, 477 (1979). J. E. Simpson, Ann. Rev. Fluid Mech. 14, 213 (1982). V. D. Djordjevic and L. G. Redekopp, J. Phys. Oceanogr. 8, 1016 (1978). C. J. Knickerbocker, A. C. Newell, Phys. Lett. A 75, 326 (1980). D. Malomed and V. Shrira, Physica D 53, 1 (1991). K. R. Helfrich, W. K. Melville and J. Miles, J. Fluid Mech. 149, 305 (1984). J. K. Sveen and J. Grue, This volume, Chapter 9.5, (2004). O. R. God0, Personal communication, (2000). V. Bakhanov, Personal communication, (2003). B. Gjevik, E. N0st and T. Straume, F&U report 90012, Statoil, Stavanger, Norway 74 pp. (1990). P.-O. Rusas, http://bukharin.hiof.no/~por/internalwaves/ (2000).

276

J. Grue

52. 53. 54. 55. 56.

R. R. Long, Q. J. R. Met. Soc. 84, 159 (1958). C.-S. Yih, J. Fluid Mech. 9, 161 (1960). K. K. Tung, T. F. Chan and T. Kubota, Stud. Appl. Math. 16, 1 (1982). B. Turkington, A. Eydeland and S. Wang, Stud. Appl. Math. 85, 93 (1991). D. J. Brown and D. R. Christie, Phys. Fluids 10, 2569 (1998).

Internal

Fig. 15.

Wave Fields Analyzed by Imaging

Velocimetry

Boluses running up slope. Free surface is at top of the figures.

277

278

J, Grue

im/m

1

IKW

P*

<^>

Fig, 16.. Motion of fish school. Registrations by echo-sounders. Time variable along horizontal axis. Depth in meters are indicated. Total water depth at location: 146 m.

CHAPTER 7 WAVE BREAKING, SURFACE MOTION, SURF ZONE, AIR-SEA INTERACTION A N D W I N D WAVES

7 . 1 . P I V m e a s u r e m e n t s o f a c c e l e r a t i o n s in w a t e r w a v e s A. Jensen , M. Huseby, D. Clamond, G. Pedersen and J. Grue Mechanics Division, Department of Mathematics, Box 1053 Blindern N-0316 Oslo University of Oslo E-mail: [email protected]

Introduction T h e P I V technique has proven t o b e efficient for velocity measurements for a wide spectrum of applications. Apart from cases involving turbulence, aeration etc., velocities are often reported to be measured within 1-2% error, say. Accelerations, on the other hand, are much more difficult to obtain from P I V d a t a and are reported less frequently in the literature i- 2 - 3 - 4 ' 5 ' 6 . T h e accelerations can be extracted from two, or more, velocity fields at slightly different times by computing temporal as well as spatial gradients. Clearly, such procedures will yield results t h a t are very sensitive to the noise t h a t is inherent in t h e velocity measurements. We have investigated a series of cases where acceleration measurements have been a t t e m p t e d , with a variable degree of success; periodic waves of small amplitude 4 , solitary waves, leading p a r t of wave trains 7 , focusing waves, run-up on beaches 8 and extreme events in irregular seas. Herein, we will focus on cases involving nearly periodic waves. Theory Irregular wave trains are investigated and generated from a J O N S W A P spectrum with significant wave height Hs = 0.065 m, mean wave period 279

280

A. Jensen, M. Huseby, D. Clamond,

G. Pedersen and J. Grue

T\ = 0.78 s and a peak enhancement factor 7 = 3.3. As a analytical reference we have compared the experiments with thirdorder Stokes theory; + 0(ak)4,

(j)= — exp(ky)sin(kx-ujt) w2=gk(l

+ a2k2).

a=^+g(ak)2e2kyj

+ 0(ak)\

(I) (2) (3)

where y is vertically upwards. The accelerations are made dimensionless by the acceleration of gravity (g). The Stokes theory gives a convective term for the vertical component, 3. All the horizontal components are zero under the crest of the wave. The local acceleration field dv/dt may be estimated from two velocity fields by evaluating v(x,f2)-v(x,*i) ,4t2 — h and the particle acceleration field Dv/dt is estimated using an implicit expression r2 = f i + that is solved for f2 particles.

8

Example:

wave

irregular

2

1

(vi(fi) + v2(f2))

(5)

. This corresponds to a simple tracing of pseudo

train

The wave train experiment is repeated three times. An image showing the surface and measured velocities at t = 283,4 s are shown in the left panel of Fig. 1, while accelerations are shown the right panel. At this instant the local wave steepness is fc£max = 0.31, where Cmax is the maximum surface elevation. Computed 9 local (dv/dt) and particle (Dv/dt) accelerations are displayed with solid and dashed lines, respectively, while measured accelerations are shown with circles (local) and squares (particle). Even though there is scatter, significant acceleration pattern do emerge from the figure. At the workshop, corresponding results were presented also for the other applications mentioned in the introduction. It should be noted that we got good overall results when we had high repeatability in our measurements. The time variables in the image acquisition must be carefully tuned to gain high accuracy in the acceleration measurements.

PIV Measurements

of Accelerations

in Water Waves

281

Fig. 1. Measured velocities/accelerations in a irregular wave train, feCniax = 0.31. Left panel: Velocity vector plot on a PIV-image at t = 283,4s. Right panel: Acceleration profiles as explained in the text. The vertical component is recognized by its negative value, whereas the horizontal component is centered around zero. The dotted line is third order Stokes theory.

References 1. K.-A. Chang and P. L.-F. Liu, Physics of fluids 10, 327 (1998) 2. K.-A. Chang, E. A. Cowen and P. L.-F. Liu, 3th International Symposium on Particle Image Velocimetry, Santa Barbara, USA, 16-18 Sept. (1999). 3. M. L. Jacobsen, T. P. Dewhirst and T. P. Greated, Meas. Set. TechnoL 8, 1502'(1997). 4. A. Jensen, J. K. Sveen, J. Grue, J.-B. Richon and C. Gray, Experiments in Fluids 30, 500 (2001). 5. K. T. Christensen and R. J. Adrian, 4th International Symposium on Particle Image Velocimetry; Gottingen, Germany, 17-19 Sept (2001). 6. P. Dong, T.-Y. Hsu, P. Atsavapranee and T. Wei, Experiments in Fluids 30, 626 (2001). 7. M. Huseby, A. Jensen and J. Grue, Proceedings 15th International Workshop on Water Waves and Floating Bodies fed. T. Miloh & G. Zilman), 75 (2000). 8. A. Jensen, G. K. Pedersen and D. J. Wood, J. Fluid Mech. 486, 161 (2003) 9. D. Clamond and J. Grue, J. Fluid Mech. 447, 337 (2001).

282

D. Clamond, J. Grue, M. Huseby and A.

Jensen

7 . 2 . P I V m e a s u r e m e n t s of t h e v e l o c i t y field i n s t e e p w a t e r wave events D. Clamond, J. Grue , M. Huseby and A. Jensen Mechanics Divsion, Department of Mathematics, University of Oslo P.O. Box 1053 Blindern, 0316 Oslo, Norway E-mail: johngmath.uio.no1

T h e velocity field induced by steep waves is the focus of the present study. We compare the velocity fields due t o sets of irregular wave trains produced from wave spectra, large wave events due t o focusing waves, the unsteady leading p a r t of a periodic wave train, and the velocity in a large wave event of a wave group. T h e study is motivated by needs within the offshore and engineering industry. T h e kinematics induced by storm waves are requested. T h e velocities and accelerations are required for subsequent analysis of the wave induced loads on e.g. ships, offshore platforms, tension legs and risers. Previous studies are devoted to large scale observations of storm waves (Buckley and Stavovy 1 , Forristall 2 and laboratory experiments using Laser Doppler Anemometry/Velocimetry ( L D A / V ) (Skjelbreia et al. 3 , Kim et al. 4 , Longridge, Randall and Zhang 5 , Baldock, Swan and Taylor 6 ) . Standard analysis techniques include second order theories with extensions. Shortcomings include, among others, a lack of resolution in (experimental) wave fields above mean water level, particularly in t h e top and near the tip of the wave, uncertainties of t h e velocities in steep waves. W i t h this motivation we study the velocity profiles under crest of a total of 62 different steep wave events, in deep water, measured in laboratory using Particle Image Velocimetry (PIV) (Jensen et al. 7 ) . T h e waves take place in the leading unsteady part of a wave train, focusing wave fields and r a n d o m wave series. Complementary fully nonlinear theoretical/numerical wave computations are performed using the the simulation program of Clamond and G r u e 8 , 9 . T h e velocities have been p u t on a nondimensional form in the following way: from the wave record (at a fixed point) the (local) trough-to-trough period, TTT, and t h e maximal elevation, r]m, of an individual large wave event are identified. T h e local wave slope, e, and the wavenumber, k, are estimated from u2/(gk) = 1 + e 2 , kr/m = e+ | e 2 + | e 3 , where to = 2TT/TTT and g denotes the acceleration of gravity. A reference fluid velocity, e^/g/k, is then defined. Waves with a fluid velocity u p t o 75 % of the estimated wave speed are measured. A corresponding wave slope is 0.46. A strong collapse of the nondimensional experimental velocity profiles

PIV Measurements

of the Velocity Field in Steep Water Wave Events

283

is found. This is also true with the fully nonlinear computations of transient waves. There is excellent agreement between t h e present measurements and previously published Laser Doppler Anemometry (LDA) d a t a . A surprising result, obtained by comparison, is t h a t t h e nondimensional experimental velocities fit with the exponential profile, i.e. eky, y t h e vertical coordinate, with y = 0 in the mean water level. Results from one of the r a n d o m wave series are given in Fig. f. T h e experiments were performed in a wave t a n k 24.6 m long, 0.5 m wide, 0.72 m deep. Waves were produced using hydraulic piston wave-maker with movements controlled by a computer. For data-acqusition we use the cards ATM I 0 1 6 E - 1 and AT-MIO-64E-3 from National instruments. In the example, t h e J O N S W A P spectrum was used, with significant wave height, Hs = 6.55 m and t h e peak period Tp = 0.939 s. T h e corresponding values of the wave slope is UpHs/(2g) = 0.15, where up = 2ir/Tp. Recordings of the wave field were made for times 124.45, 133.24, 165.27, 214.01, 283.43 s. T h e series had three repetitions, all with the same triggering of the P I V system. T h e experiments were very repeatable. T h e velocity d a t a from the repeated runs exhibit a vanishingly small spread. T h e results show an all over collapse of the velocity profiles when they are put on nondimensional form. Moreover, t h e measurements become very close t o t h e theoretical curves. T h e correspondence is particularly good in the top of the wave. T h e good correspondence is t r u e b o t h for wave events t h a t are early and late in t h e series. T h e events in series 1 are relatively steep, and the water is deep (0.19 < ak < 0.34, 3.9 < kh < 4.6). For more examples, see Grue et al. 1 0 .

Acknowledgments This work was conducted under the Strategic University P r o g r a m m e 'General Analysis of Realistic Ocean Waves' funded by the Research Council of Norway. T h e technical assistance in t h e Hydrodynamics laboratory by Mr. Arve Kvalheim and Mr. Svein Vesterby are gratefully acknowledged.

References 1. W. H. Buckley and A. B. Stavovy, Proc. Extreme Loads Response Symp. The Ship Structure Committee and the Soc. of Nav. Arch, and Mar. Engng. (SNAME), USA (1981). 2. G. Z. Forristall, Proc. 20th Coastal Eng. Conf. Taipei, Taiwan, 208 (1986). 3. J. E. Skjelbreia, O. T. Gudmestad, R. D. Ohmart G. Berek, Z. K. Bolen, J. C. Heideman, N. Spidsoe and A. T0rum, OMAEI-A, Offshore technology (1991).

D. Clamond, J. Grue, M. Huseby and A. Jensen

284

ak= 0.24966, f=1.2976, a= 3.9144

0

ak= 0.238, f=1.2271, a= 4.1503

0.5 1 1.5 ak= 0.33922, f=1.2722, a= 5.807

2

0.5 1 1.5 ak= 0.31046, f=1.2182, a= 5.6993

2

0

0.5 1 1.5 ak= 0.18515, f=1.2397, a= 3.0964

2

Fig. 1. Horizontal velocity profiles below wave crest. 5 wave events (o). Solid line: Velocity profile in 'freak' wave computation with ak = 0.29. Dotted line: eky.

4. C. H. Kim, E. Randall, S. Y. Boo and M. J. Krafft, J. Waterway, Port, Coastal and Ocean Engng. 118(2), 147 (1992). 5. J. K. Longridge, R. E. Randall and J. Zhang, Ocean Engng. 23, 277 (1996). 6. T. E. Baldock, C. Swan, C. and P. H. Taylor, Phil. Trans. R. Soc. Lond. A 354, 649 (1996). 7. A. Jensen, J. K. Sveen, J. Grue, J., J.-B. Richon and C. Gray, Exp. in Fluids 30, 500 (2001). 8. D. Clamond and J. Grue, J. Fluid Mech. 447, 337 (2001). 9. D. Clamond and J. Grue, C. R. Acad. Sci. Paris. 330, 575 (2002). 10. J. Grue, D. Clamond, M. Huseby and A. Jensen, Appl. Ocean Res. 26 (2004).

LDA Laboratory Measurements

in the Surf Zone

285

7.3. LDA laboratory measurements in the surf zone for the R A N S depth-averaged shallow-water equations T. Feng and P. K. Stansby Manchester Centre of Civil and Construction Engineering, UMIST, Manchester M60 1QD, UK Flow in the surf zone is a major area of the hydrodynamics research of concern to coastal engineers. The flow has complicated features due to the wave breaking as well as other periodic motions. In the present study a three-component, fibre-optic LDV has been used to measure the velocity field in the surf zone. The measurements covered a length along the beach covering plunging to well-developed spilling breakers. The data are used for the evaluation of each term appearing in RANS depth-averaged shallowwater equations. Apparatus

and Experimental

Results

An inclined beach of slope 1:20 has been built in an existing wave flume (11 meters long, 0.3 meters wide and 0.45m deep). The side-walls of the flume and the bottom plate of the beach were made with Perspex to allow the laser beam or laser sheet to shine through for LDV or PIV measurement. Regular waves with period, T = 2.4s, were generated by a piston wave paddle. The still water depth was set up to 0.342m. A wave height of 0.12m was obtained before the beach. In these conditions, the waves break at 5.1m from the wave paddle. The breaking locations are only approximately repetitive with a variation of ±0.02m found in the data analysis. The fibre-optic, three-component LDV system has a 4 Watt Argon-Ion laser source (a TSI product, from ESPRC equipment loan pool). Surface elevation was measured with wave probes made from stainless steel wires. By running the FIND programme (a TSI product) on a PC, the simultaneous measurements of three components of velocity and surface elevation can be made in a multi-channel manner (TSI DataLink). LDV measurements were conducted at 12 horizontal locations. The measurements were along vertical lines throught the depth, at 15-60 vertical locations depending on the depth variation. The green line of laser (514nm wavelength) was used to measure horizontal velocity, u, and the blue line (488nm) vertical velocity, w. A sampling time of 15 minutes was used and with the wave period, T = 2.4 seconds, 375 wave cyles were measured in a coincident data rate up to 400 Hz. Hence an average size of data set of 6.0 millions (4 channels) was obtained for

286

T. Feng and P. K. Stansby

evaluating the turbulent velocity at each point. The phase average for N samples within a period is defined, for example, t+T

as u = ^ J udt' where r = T/N (N = 128 here). Assuming the transform, t

Ax = —dAt, is valid with celerity c locally uniform, velocity vector maps (u — c, w) have been constructed with surface elevation, r/, to demonstrate the complete process from plunging to spilling breaker. At location (Lx = 5540mm, distance from the paddle) the wave is steepening. At location (Lx — 57377mm) plunging has occurred and the roller is growing. At the next two locations (Lx = 5887 and 5962mm) the breaker becomes spilling and is bore-like at location (Lx = 6412mm). At location (Lx = 7200mm) the strength of the bore has clearly decreased. Since wave motion after breaking is rather complicated, the boundaries of this classification are not too distinct. RANS

Depth-Averaged

Shallow-Water

Equations

Consider 2-D vertical plane flow at a point with periodic and turbulent components of horizontal velocity u, vertical velocity w and pressure p: u = u + u', w = w + w', p — p + p'. The phase-averaged, continuity equation is given by | ^ + ^ = 0 and the phase-averaged, horizontal force/momentum equation by du/dt + d(u2)/dx

+ d(uw)/dz

= -(l/p)dp/dx

- d(u'2 )/dx - d(u'w'

)/dz

with the overbar indicating phase averaging of turbulent quantities. The phase-averaged kinematic free-surface boundary condition is given by ws — dfjdt + usdfj/dx + u'dr)'/dx and the last term is usually ignored. After integrating over depth, from bed b to surface s, where for example _

s

uh = fudz,

with several applications of the Leibnitz theorem, we obtain

b

for horizontal force/momentum , d(u2h) dx

d(uh) dt

(I)

__ _ah^R " dx

(II)

"

J

dx

(IV)

a z

(III) u

w

s +UW

(V)

b

(VI)

d(f(u-u)2dz)

d(fZ^dz)

dx

dx

(VII)

(VIII)

"I""

sdx

(IX)

LDA Laboratory Measurements

in the Surf Zone

287

where pr> is the dynamic pressure (total-hydrostatic) and h is depth. The dynamic pressure term (IV), the surface and bed shear stress terms (V, VI) require modelling; this also applies to the 'dispersion' term (VII). The normal turbulent stress terms, (VIII), (IX), are not usually included but will be evaluated. Note terms in d/dx may be transformed to —\§i in periodic conditions where the wave speed c is assumed to be locally uniform. The non-hydrostatic pressure may be determined from the RANS vertial force/momentum equation. After some manipulation and putting p£>s = 0 we obtain for term (IV) v v 1 f dpD , 1 9 / . , PDb dzb p J ox peat J p ox b

b

More complete derivations will be given elsewhere. Matlab programmes were written for the evaluation of the terms I to IX for each horizontal location. The turbulence terms, IV, V, VIII, IX were found to be small compared to the other terms. The profile of each term is quite complex although the overall balance is achieved to a close approximation. A 128 point spline interpolation was used. Conclusions Turbulent velocity fields in the surf zone have been measured with laser Doppler velocimetry. The measurement area covers the region from plunging to spilling breaking. Using the LDV data, evaluation of the terms in the RANS, depth-averaged, shallow-water equations was undertaken. It is found that the turbulence stress terms are negligible but that non-hydrostatic (dynamic) pressure and dispersion terms are significant. Numerical and physical modelling work is ongoing and more detailed results will follow. Funding from the UK EPSRC grant GR/N21741 is gratefully acknowledged by the authors.

288

K.-A.

Chang, E. A. Cowen and P. L.-F.

Liu

7.4. W a v e a c c e l e r a t i o n m e a s u r e m e n t u s i n g P T V t e c h n i q u e Kuang-An Chang 1 , Edwin A. Cowen 2 and Philip L.-F. Liu 3 Department of Civil Engineering Texas A&M University, College Station, TX 77843, U.S.A.1 School of Civil and Environmental Engineering Cornell University, Ithaca, NY 14853, U.S.A.2'3

A particle tracking velocimetry ( P T V ) based measurement technique is developed for determination of wave acceleration. Based on the illumination of seeding particles three times over two frames and continuously tracking an individual particle over the three distinct illuminations, two velocity fields and a Lagrangian acceleration are obtained. T h e technique not only provides a method for acceleration measurement b u t also reduces t h e constraint on the time separation between adjacent illumination pulses. T h e restriction imposed by the camera framing rate in a single pulsed tripleframe technique can be significantly reduced to a much shorter camera specific frame transfer time. Monte Carlo simulations are used t o test the technique for a deep water wave. T h e most simple and direct method for determining acceleration is to analyzing three successive images. This gives two velocity fields so the acceleration can be calculated. T h e time separation, At, is basically one half (or greater) of the frame time. However, most cameras with reasonable high resolution (e.g., I K x I K pixels) have a framing rate of about 30 or less. For example, a typical 30 frame per second (fps) camera gives At > 17 ms. This renders At being too large t o determine acceleration of interest in water wave studies. In addition t o the time separation issue, traditional P I V gives Eulerian acceleration if two velocity fields are used. Finite differences are needed t o compute the convective acceleration, u • V u . Errors will inevitably be magnified, especially when t h e spatial grid increments, Aa; and Ay, between velocity vectors are large. On the other hand, P T V tracks an individual particle so Lagrangian acceleration is obtained - no need t o calculate t h e convection t e r m in acceleration determination. Therefore, P T V is used in this study. To overcome the time separation problem mentioned above, a two-frame double-pulsed/single-pulsed technique is introduced for acceleration measurement. T h e hybrid technique is based on three illumination pulses in two successive frames. Unlike the three-frame single-pulsed method (trigger signal is shown in Fig. 1(a)), t h e first frame is double-pulsed a n d second frame is single-pulsed, as shown in Fig. 1(b). T h e double-pulsed first frame

Wave Acceleration

Using PTV

Technique

289

(a) Camera Laser

Measurement

I _

^

L

J

(b) Camera Laser

Fig. 1. Trigger timing diagrams, (a) the single-pulsed image-triple trigger signal, (b) the double-pulsed/single-pulsed trigger signal; and (c) a typical cross-correlation result of (b).

is correlated with the single-pulsed second frame resulting in two correlation peaks. The two peaks are of the same order of magnitude, which allows their identification above the background noise level set by random particle image correlations as seen in Fig. 1(c). Assuming At is small and the particle displacement is unidirectional (a reasonable assumption given the physics and the enhanced statistical probability of randomly getting unidirectional displacement), the individual particles can be tracked based on the displacements estimated from the two correlation peaks. The definition of "unidirection" is that the distance between the second and third particle images, |d23 [, is shorter than that between the first and third particle images, |di 3 |, i.e., |d 2 3 | < |di 3 |

(1)

This means that the correlation peak closer to the origin is from the correlation of the second and third particle images, while the other correlation peak is from the correlation of the first and third particle images. This allows the flow direction to be identified unambiguously from the correlation such as in Fig. 1(c). The acceleration measurement technique is tested by using the simulation under linear water waves. Linear wave theory is used to calculate the particle displacement in synthetic images. A deep water sinusoidal wave is tested. The wave has wave parameters as follows: water depth h — 20 cm, wave period T = 0.5 s, wavelength A = 39 cm, and wave height H = 4 cm. The field of view (FOV), with a 1024 x 1024 pixel resolution, is 6x6 cm 2 , from trough level at y = — 2 cm downward to y = — 8 cm, and At is 5 ms.

290

K.-A.

Chang, E. A. Cowen and P. L.-F.

Liu

Fig. 2. Vector fields of the deep water wave, (a) velocity from the 2nd and 3rd pulses, (b) velocity from the 1st and 3rd pulses, (c) acceleration.

The simulated velocity fields are shown in Fig. 2(a,b) with the resulting acceleration shown in Fig. 2(c). The r.m.s. error in acceleration is about 0.15 pixel in each direction (0.0018 g in physical unit) . Note that the error has strong dependence on velocity gradient. The wave parameters were chosen to represent a case that has relatively large velocity gradients. If a shallow water wave is used, the error is typically smaller and about 0.1 pixel due to smaller velocity gradients.

Two-Color LIF-PIVfor Gas Exchange at Air-Water Interface

291

7.5. Two-color LIF-PIV for gas exchange at air-water interface Nobuhito Mori Department of Fluid Mechanics, Central Research Institute of Electric Power Industry (CRIEPI), 1646 Abiko, Abiko, Chiba 270-1194, JAPAN Email: [email protected]

Introduction An accurate estimation of gas transfer velocity at the air-sea interface is very important to understand environmental mechanisms of the earth. A wind dependent gas transfer model by Liss and Merlivat1 has widely used the last decade. The gas transfer velocity of the Liss and Merlivat model increases rapidly when the wind speed exceeds 13m/s. However, the quantitative role and the detail mechanisms of the rapid gas transfer enhancement are not well known due to the lack of detailed measurements. Particle image velocimetry (PIV) and laser-induced fluorescence (LIF) are powerful tools in the study of turbulent flows. Several kinds of PIV and LIF coupling techniques have been demonstrated to measure velocity and scalar fields2'3. The LIF method can measure pH through change of fluorescence intensity using pH dependent fluorescent dye 4 ' 5 . In the study, two-color LIF and PIV technique are employed to visualize the gas transfer at air-water interface in the wind-wave fields. Method of

Analysis

The LIF method is able to measure pH through the fluorescence intensity change using pH dependent fluorescent dye. The fluorescence intensity of fluoresceine sodium dye depends on the HCO^ ion between pH = 4-8. On the other hand, pH independent fluorescent dye is used to measure the local laser intensity to eliminate attenuation and vignetting effects. To combine the two methods, the concentration of CO2 enables measuring the ratio of the local fluorescence intensity of the pH dependent dye to the pH independent one. In this study, Fluoresceine sodium: FNa and Rhodamine WT: RWT are used as pH dependent and independent fluorescent dyes, respectively.

292

N. Mori

Using FNa and RWT, the local fluorescence intensity ratio RFR can be obtained, _ FFNa(x,y,t,pH)

FRwr{x,y,t)

_

aFNa$FNaCFNa(x,y,t,pH)

aRWT$RWTCRwT(x,y,t)

where F is the local fluorescence intensity, a is fluorescent efficiency,
Setup

The experiments were conducted in a wind-wave channel that is 8.0 m long, 0.25 m wide, and 0.55 m high. Double pulsed Nd:YaG laser with a wavelength of 532nm was used as the light source. Three 10-bit CCD cameras with a resolution of l K x l K pixels were used to capture the images. Two CCD cameras were used for LIF and one CCD camera was used for PIV. One CCD camera for LIF was installed with a high cut filter (shorter than 520nm), another CCD camera for LIF was installed with a red filter, and the rest of the CCD camera for PIV was installed with a low cut filter (higher than 470nm). Using the filters, fluorescence of FNa and RWT and the Mie scattering of particle images could be captured individually. The wind flume experiments were conducted under the wind condition of 1.22m/s. To increase CO2 gas concentration in the air, three small outlet tubes were installed lm upstream of the measurement point. CO2 gas was injected lOmin before the experiments to stabilize the gas concentration in the wind flume with very low wind velocity. The concentration of CO2 gas was checked by CO2 meter and the experiments were started when the concentration of CO2 gas was regarded as constant in time. Fig. 1 shows the results of the wind flume experiments. The size of measurement area was 9.0cmx 13.0cm. The contour in the figure indicates pH (white color:

Two-Color LIF-PIV for Gas Exchange at Air-Water Interface

6 S .v:i.\iMwin|

Fig. 1.

10

293

12

Instantaneous velocity field superimposed over pH concentration.

p H = 9 , black color: pH=6). T h e horizontal white b a n d in t h e figure is t h e location of free surface. Below t h e white b a n d is t h e water area and t h e above t h e gas phase. T h e local downward velocity currents were measured every sequence with radical p H variation.

References 1. P. Liss and L. Merlivat, The role of air-sea exchange in geochiiical cycling, pp. 113-129, D. Reidel, Norwell, Mass. (1986). 2. K. Hishida and J. Sakakibara, Proc. of the third Int Workshop on PIV'99 1, 21 (1999). 3. E. A. Cowen, K.-A. Chang and Q. Liao, Experiments in Fluids 3 1 , 63 (2001). 4. S. Someya, B. Chen, H. Akiyama, M. Nishio, K. Okamoto, J. Flow Visualization and Image Processing 6, 234 (1999). 5. N. Mori and M. Imamura, The 10th Inter. Sympo. on Flow Visualization, Kyoto, Japan, F0241 (2002). 6. L. Gui, W. Merzkirch and R. Fei, Experiments in Fluids 29, 30 (2000).

294

G. Pedersen, S. Mayer and A.

Jensen

7.6. R u n - u p of s o l i t a r y w a v e s Geir Pedersen , Stefan Mayer and Atle Jensen Mechanics Division, Department of Mathematics, University of Oslo ' Department of Mechanical Engineering, Section of Maritime Engineering, Technical University of Denmark? E-mail: [email protected] , [email protected], [email protected]

An extended P I V system is employed t o measure the velocity and acceleration distributions for waves running u p a beach. T h e experiments were performed in a wave t a n k at the Hydrodynamic Laboratory, Department of Mathematics, University of Oslo. T h e wave t a n k is 1 m high, 0.5 m wide and a beach, a n d an equilibrium depth of 0.2 m was employed for the experiments reported herein. At one end of t h e tank, waves were generated by a hydraulic piston attached t o a vertical paddle. This was separated by a constant d e p t h region of length 3.75 m from a beach of inclination 10.54°. We employ a P I V system 1 with two high-sensitive C C D camera with a resolution of 1280 x 1024pixels and 12-bit digital o u t p u t s and a 10W continuous wave (CW) laser and a scanning b e a m system. T h e experiments have been compared to several theoretical models. Herein, we will briefly discuss the performance of a Navier-Stokes model, with t h e V O F technique for t h e free surface 2 ' 3 . T h e simulations were performed in a numerical wave t a n k t h a t was an exact match of the one in the laboratory, including a vertical paddle behaving according t o the measured paddle motions of t h e experiments. T h e computations were performed on irregular grids ranging from 256 x 32 t o 1536 x 128 cells. Following, in p a r t , Grilli 4 we may identify an intermediate regime between breaking and non-breaking waves on a beach, where t h e wave front becomes vertical, or nearly vertical, when the wave reaches t h e equilibrium shoreline. T h e steep front may propagate some distance up-beach, but will never form a plunger and will be transformed into a smooth, thin wedge of fluid t h a t climbs the beach. T h e steepest wave in our run-up experiments 5 , t h a t is close to a soliton of amplitude A/h — 0.66, belongs t o this intermediate regime. Measurements and numerical results for this case are shown in Fig. 1 for t = 3.05s. At this stage, the wave has just started t o climb the beach and we can see an almost vertical front and a nearly uniform velocity behind it. T h e agreement between model and experiment is excellent for the velocities, while the computed surface is less advanced and slightly less steep t h a n t h e the experimental one. T h e difference in t h e position corresponds t o a 0.007 s time delay, t h a t is of the same order as the experimental

Run- Up of Solitary

295

14

14

VOF 12

Waves

PIV>

VOF

PIV >

12

10

10

8

8

6

6

4

4

2

2

0

0

/'

. VOF --2

l

-6

-A

i

— t

PIV —

0 2 x(cm)

Fig. 1. Comparison PIV and N S / V O P simulation at t = 3.05 s. Left panel: velocities, reference arrow 140 cm/s. Right panel: residual velocities (no = —125 cm/s), reference arrow 21 cm/s. The origin is at the equilibrium shoreline.

errors due to inaccuracy in equilibrium depth etc. To reveal the spatial velocity distribution, that determines the evolution of the shape of the run-up front we subtract a uniform "translation" velocity according to Vd = V — UQ(I — tanflj). The choice of UQ is somewhat arbitrary. In the left panel of Fig. 1 it is close to a mean value for the measurements. We observe some differences in va between the experiments and the numerical results. The deviations may partly be explained by experimental scatter. It is noteworthy that the agreement for the velocity pattern, i?d, is substantially reduced if the model solution is advanced by the 0.00? s that yields the best match for the surface position. The theoretical and experimental residual fields display the similar qualitative pattern, pointing to further steepening. This imply that the experimental wave front, at least, will be overturning shortly. However, the acceleration pattern in Fig. 2 opposes the development of breaking and indicates a tendency towards the formation of a fluid tongue at the toe of the wave front. This behavior is in many respects analogous to initial stages of the dam-break problem and is characteristic for the run-up of nearly breaking waves. The overall agreement between the Navier Stokes solution and the experiments is very good, but further work is needed to investigate the reasons for the less steep run-up front obtained in the simulations.

296

G. Pedersen, S. Mayer and A. Jensen

9 "" /

- .>

/

!

/ /

£ o V

6h

>s 4.

* / / i I

S S J / / J i „/ S ,/
\

iMM\

[_

]-"-i-:-r'^:-i'"r--'l.'--V*."--'*' '-" *

L..„..„i„™„I JUZ^*-J_____J______X__--J_J

-4

~-2

0 2 x(crn)

0,25 OLtS OttJ

w-

am \ am

o x(cm)

Fig. 2. Accelerations at t = 3.05s. Left panel: Particle accelerations. Reference arrow corresponds to the acceleration of gravity (g). Right panel: Standard deviation of the x component relative to g.

References 1. A. Jensen, J. K. Sveen, J. Grae, J. B. Richon and C. Gray, Experiments in fluids 30, 500 (2001). 2. S. Mayer, A. Garapon and L. S. S0rensen, Int. J. Num. Meih. Fluids 28, 293 (1998). 3. S. Mayer and P. A. Madsen, In ICCE 96} 25}th Coastal Engineering Conference, Sydney, Australia (2000) 4. S. Grilli, I. Svendsen, and R. Subramanya 5 J. Wtrwy., Port, Coast, and Oc. Bngrg. 123 (3), 102 (1997). 5. A. Jensen, G. K. Pedersen and D. J. Wood, J. Fluid Meek 486, 161 (2003)

DPIV Measurements

for Deep-Water

Wave Breaking on Following Currents

297

7.7. D P I V measurements for unsteady deep-water wave breaking on following currents Chin H. Wu1, Aifeng Yao2 and Kuang-An Chang3 ' Dept. of Civil and Environmental Engr., University of Wisconsin, Madison Department of Civil Engineering, Texas A&M University E-mail: [email protected] , [email protected] The co-existence of wave breakings on currents is very common in marine environments. It is well known that wave breaking provides an important conduit for the exchange of gas, water vapor, energy, and momentum between air-water interface. The existence of currents may further enhance these exchanges. Extensive reviews of breaker type classification, the mechanisms of breaking, and the evolution of wave geometry are given by Peregrine 13 for shallow-water waves and by Melville2 for deep-water waves. For non-breaking surface waves with co-existent currents, the wave-current interaction (Peregrine 1 , Jonsson 3 , and Thomas and Klopman 4 ) is usually considered as an extension of the fundamental wave theories. A handful of detailed experiments (Thomas 5 ' 6 , Lee 7 , Skyner and Easson 8 , and Swan et al. 9 ) have been conducted to show the significant effects of current profiles and directions on steady or slowly modulated surface waves. However, it is unclear that to what extent currents can affect the stongly nonlinear waves, and even unsteady breaking waves. Therefore, details of kinematic flow measurements is essential to provide insight on the effects of the following currents on wave breaking processes. A Digital Particle Image Velocimetry (DPIV) system with a temporal resolution of a frequency of 15 Hz, and a spatial resolution of 1024 x 1024 pixels per second were used to measure velocity field of unsteady deep-water breaking waves on different strengths of following currents. In addition, surface wave displacements were also measured by an array of 6 wave gages. The experiments were conducted in a newly renovated two-dimensional wave-current flume in the Environmental Fluid Mechanics Laboratory at the University of Wisconsin-Madison. The flume has a dimension of 46 m long, 0.91 m wide and 1.22 m deep, and was filled with 0.6 m water depth. The flume was leveled and a smooth bed condition was maintained. The waves were generated by two servo-controlled flat, bottom-hinged paddles located at the upstream of the flume. At the downstream end of the flume, a beach with a slope of 1:10 was constructed using high absorption horsehair material. Less than 5% reflection coefficient for the beach was achieved.

298

C. H. Wu, A. Yao and K-A.

Chang

Three steady current speeds (5 cm/s, 9 cm/s and 15 cm/s) were generated by a controllable speed pump and a two-pipe system. The water was pumped from at the end of the flume then was directed to an upstream settling well sitting behind the wavemaker. The following current was supplied by a bottom opening in front of the wavemaker . The current profiles of the three stations (x = 6 m, 10 m, and 14 m, where x is defined at the neutral position of the wavemaker) were measured to be nearly uniform with less than 3% variance using a Sontek 16Mhz Micro-Acoustic Doppler Velocimeter. A single, unsteady and deep-water wave breakings on the following currents was generated by a wave focusing method (Rapp and Melville10) with further consideration of Doppler shift relation in the presence of currents. The modified wave focusing method used a specific range of wave components and predetermined phases from the Doppler shift dispersion relationship, a — u> + kllc (where a is an apparent wave frequency relative to a stationary observer, k is the wavenumber, and u> is the intrinsic wave frequency observed in a moving frame at velocity Uc), to allow constructive interference so that all wave components traveling on the currents were able to focus at one spatial location. To examine the effect of currents on wave breaking, the same apparent frequency band with 32 frequency components evenly spaced between 0.686 Hz and 1.274 Hz was used for our experiments. Using the modified method, a very repeatable single wave breaker on the different strength of the following currents was able to be generated at the same location. To reveal the onset of wave breaking, both geometric criteria (based on the geometry of wave shape, Kjeldsen and Myrhaug 11 ) and kinematic criteria (based on the characteristics of wave motions, particle velocity and wave phase speed) were used. It was observed that the geometric criteria of wave breaking on different strength of following currents could vary a wide range of values, suggesting that the existence of following currents in the field can change surface steepness at breaking. These effects may explain a wide range of local wave shape criteria reported from field observations (Kjeldsen and Myrhaug 11 ) and illustrate why geometric wave shape parameters have not proved to be stable limits for ocean wave breaking. To examine the kinematic criteria, the particle velocity at the crest of wave breaking on the following currents was measured using the DPIV techniques. The images were analyzed by an interrogated 64 x 64 pixels window

DPIV Measurements

for Deep-Water

Wave Breaking on Following Currents

299

with 50% overlapping using a hybrid digital particle tracking velocimetry technique (Cowen and Monismith 1 2 ). T h e results show t h a t the combined velocity field for spiller breakers with a 5 c m / s and 10 c m / s following currents reached 1.83 m / s and 2.24 m / s , respectively. Wave phase speeds (1.81 m / s for a 5 c m / s current and 2.14 m / s for a 10 c m / s current) were obtained by using a pair of the consecutive images near the onset of wave breaking. Similar to wave breaking without currents, the ratio of the particle velocity to t h e wave phase speed for waves on following currents exceed unity at t h e onset of wave breaking. T h e results indicate t h a t currents can affect the geometric steepness b u t not the kinematic criteria of unsteady wave breaking. Future study for the effects of opposing currents on the unsteady wave breaking is under way.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

D. H. Peregrine, Adv. Appl. Mech. 16, 9 (1976). W. K. Melville, Annu. Rev. Fluid Mech. 28, 279 (1996). I. G. Johsson, In the Sea, J. Wiley and Sons, 65 (1990). G. P. Thomas and G. Klopman, Computational Mechanics Publication, 215 (1997). G. P. Thomas, J. Fluid Mech. 110, 457 (1981). G. P. Thomas, J. Fluid Mech. 216, 505 (1990). B. W. Lee, MS thesis, University of Liverpool, Liverpool, (1990). D. J. Skyner and W. J. Easson, J. of Waterway, Port, Coast, and, Ocean Eng., ASCE 124, 1 (1998). C. Swan, I. P. Cummins, and R. L. James, J. Fluid Mech. 428, 273 (2001). R. J. Rapp and W. K. Melville, Phil. Trans. R. Soc. London A331, 735 (1990). S. P. Kjeldsen, and D. Myrhaug, Proc. 11 Offshore Tech. Con}., Houston, Texas, 3646 (1979). E. A. Cowen and S. G. Monismith, Experiments in Fluids 22, 199 (1997). D. H. Peregrine, Nonlinear Topics in Ocean Physics, 449 (1991).

This page is intentionally left blank

CHAPTER 8 P I V M E T H O D S , B O U N D A R Y LAYER FLOWS A N D TURBULENCE

8.1. A digital high speed c a m e r a s y s t e m applied t o P I V m e a s u r e m e n t s in w a t e r waves J. Kristian Sveen and Atle Jensen Department of Mathematics, Mechanics Division, University of Oslo, PB 1053, Blindern, 0316 Oslo, Norway

High speed camera

system

This abstract is concerned with the application of an affordable, high speed, digital camera system for PIV. The system is capable of capturing and saving 262 images per second with storage on a RAID-0 hard-drive. The system comprises a PC with 64 bit PCI-bus, a 64 bit frame grabber card and a Dalsa CA-D6-0512 videocamera. The system is applied to the measurement of velocities during run-up of internal waves on a beach. Particle Image Velocimetry (PIV) has over the last two decades developed into one of the most important measurement methods available to fluid dynamicists. The rapid developement of PIV is much owed to the simultaneous developement of hardware. The last 20 years have seen a change from analoge camera technology to digital cameras and storage. With the introduction of digital images came the software for digital processing of PIV (in many cases referred to as DPIV), where pioneering work was performed a little more than a decade ago 1 . A more comprehensive review of PIV may be found in previous works 2 ' 3 ' 4 .

301

302

Hard- and

J. K. Sveen and A.

Jensen

software

One of the really challenging tasks today is to provide consistently high speed capture of images and storing them. Many high speed cameras today use onboard memory to store images, which ultimately implies a severe limitation on the recording length. One true challenge is to acquire a sustained high temporal frame rate, with high resolution and with storage directly on the hard-drive of a computer. One such system has recently been setup in the Hydrodynamics Laboratory at the University of Oslo. The system comprises a Dalsa CA-D6-0512 digital camera of 542 x 530 pixels resolution and 8-bit color-depth. The camera is capable of capturing 262 images per second (aka frames per second, hereafter fps) and these images are transferred to a computer and stored on a harddrive. The image transfer is done via a Coreco Imaging PC-DIG-L framegrabber in a Dual Pentium III computer with a 64 bit motherboard (SuperMicro PIIIDR3, Dual, U160). This motherboard has an onboard Ultral60 SCSI controller, which is needed to cope with the large amount of data which the camera transfers through the fiber. The images are saved on a software controlled RAID-0 hard-drive system consisting of 4 fast SCSI-disks. Tests were also performed using a hardware RAID controller, but this did not provide us with the required writing speed. Repeated tests using the onboard SCSI controller have shown the system capable of throughputting between 90 and 130Mbyte/s to the hard-drive, depending on which disk-sector that is being written to. This implies that the system is well capable of storing the continuous data-flow from the camera operating at full speed (~ 75Mbyte/s). Further tests using the video camera at both 262fps, lOOfps and 50fps has shown the system to be very stable and reliable. Over the recent few years a software toolbox for PIV has been developed at the department of Mathematics, University of Oslo. The toolbox, named MatPIV 5 , is made to run in MATLAB a . It is mainly written following the references1'6'3. The toolbox is distributed as "freeware" under the GNU General Public License, and it is supplied with open source code. It is therefore easy to make modifications to the code or write additions. Furthermore the code is well suited for use in batch-processing of huge amounts of data. Application Experiments are performed on internal waves in a laboratory wave tank of dimensions 7.3 x 0.5 x 0.25m (length, heigth and width respectively). a

b y T h e M a t h w o r k s Inc.

Digital High Speed Camera System for PIV Measurements

in Water Waves

303

In one end of the tank a beach is installed. Measurements are in this case performed using 50 fps recording rate. Waves are generated on the interface between salt (lower layer) and fresh water (upper layer) with densities pi and p2 and thicknesses hi and h2 respectively. The measurements are performed as the internal waves shoal and run-up on the beach. The internal wave in this case has in initial length of about 22 times the upper layer thickness, and the runup produces a gravity current. An experiment typically produced more than 2400 velocity fields. As an example we show (Fig. 1) horizontal velocities at the location given by the intersection of the pycnocline and bottom in the equilibrium state. To demonstrate the overall flow-pattern we display the averaged quantities close to the surface and bottom, respectively. In spite of a very high turbulence level, the mean flow is still dominated by a strong onslope flow in the lower layer, and a counterbalancing flow in the upper layer.

-0.5 l

-

-

0

-

-*•

10

'

20

'--

30

l

40

~

—

J

50

L

-

60

tco/h<2 Fig. 1. (Mack)

Time trace of the velocities close to the free surface (gray) and the bottom

Conclusion An affordable, digital, high speed, camera system was designed, to record and store 262 fps to a hard-drive in real-time. The system was thoroughly tested and found to deliever the required performance. The system is highly flexible and will allow the users to give a more elaborate temporal description of a flow field. References 1. C. E. Willert and M. Gharib. Exp. Fluids 10, 181 (1991). 2. R. J. Adrian, Ann. Rev. Fluid Mech. 23, 261 (1991). 3. M. Raffel, C. E. Willert, and J. Kompenhans, Springer Verlag, flrst edition, (1998). 4. J. Westerweel, PhD thesis, Delft University of Technology, The Netherlands, (1993). 5. J. K. Sveen, http://www.math.uio.no/~jks/matpiv. (2003). 6. J. Westerweel, D. Dabiri, and M. Gharib, Exp. Fluids 23, 20 (1997).

304

Y. Watanabe

8.2. Three-dimensional stereoscopic particle tracking velocimetry for a large domain (3DSPTV-LD) and its application Yasunori Watanabe Hydraulic Engineering Laboratory, Hokkaido University N13 W8, Sapporo, Japan E-mail: yasunori©eng. hokudai. ac.jp

Introduction In the surf zone and vicinity of a coastal structure, three-dimensional largescale vortex structures are formed that promote suspension and transport of sediments. The imaging techniques, such as 3DSPIV, make it possible to know the instantaneous 3D velocity field. However, only a portion of the large-scale 3D motion can be captured in this PIV system because of the limited flow plain illuminated by a Laser sheet, and then most volumetric velocity distribution in the instantaneous vortex structure is lost. In this study, a new 3D PTV algorithm is developed to acquire Lagrangian fluid motion involved in the large-scale vortex structure on less costs, based on the stereoscopic method for three cameras. The accuracy of this system is examined via theoretical and experimental comparisons. 3D Lagrangian fluid motion under breaking waves and particle motion of settling solid particles are measured by the present system. The PTV

Algorithm

and System

Set- Up

The particle position on the real coordinate system is determined by three sets of the three image coordinates, based on the stereoscopic method. The third order polynomials, rather than the conventional perspective projection, are employed to approximate the real position from the image coordinates to avoid errors occurring by a gap of the reflective index at a glass wall and an aberration. The application of the third-order polynomials significantly improves accuracy over 20%, compared with the conventional method. The coefficients in the polynomials are determined by the method of least squares during a calibration to relate the real and image positions. Applying the polynomials, the image positions of all particles over all frames for three cameras are converted into the real positions, and then all the particles in real space are linked over consecutive frames on the basis of the minimum acceleration principle along the particle trajec-

3D SPTV-LD

and Its

305

Application

tories. The experiments are conducted in a laboratory wave flume that is 24m long, lm height and 0.6m wide. Fluorescent neutrally buoyant particles were put into the recording domain (L: 60cm, H: 45cm, W: 50cm) as tracking makers. Three digital video cameras (640*480 pixels) in the flume were mounted onto stages, recording particle motion through a glass wall of the flume. Black lights were used as a source of illumination to get high reflection from the fluorescent particles. The estimated particle trajectories under progressive waves have high correlation over 95% with the small amplitude wave theory, which demonstrates the present PTV system has good accuracy to estimate the largescale particle trajectory.

Settling

Velocity of Solid

Particles

Fig. l(left) presents Lagrangian velocity vectors of settling particles (number of particles; 6, specific gravity; 2.6). The particles exhibit linear settling trajectories with almost uniform velocity because of relatively small number of particles released, ^f the number of particles released increases (see Fig. 1 middle), the settling particles tend to have zigzag motion with much more diffusion than in Fig. l(left). The frequency range of the fluctuation of the trajectories is much lower than the frequencies of a particle oscillation caused by separation. The grouping effect occurring due to particle interactions can be found to induce this low-frequency fluctuation of the trajectories because it depends on number density of particles. The drag coefficient {Co) calculated for the solid particles during the present experiment (representative velocity is ensemble-averaged over all particles) can be identically plotted on a well-known relationship of the CD to the particle Reynolds number (See Fig. 1 right), which suggests that the present system correctly estimates particle motion. Three-Dimensional

Fluid Velocity under Breaking

Waves

The turbulent flow field after wave breaking contains a large-scale vortex structure, which consists of numbers of large-scale vortices such as horizontal roller and obliquely descending eddies. Fig. 2 shows the typical 3D vortical motion in the vortex structure. The particles descend with vortical motion, rotating around a vertical-oblique axis, when the first wave passes over the particles. The vortical trajectories spirally advecting downward can be found to be originated from the large-scale turbulence in the obliquely descending eddy. These results demonstrate that 3DSPTV, as proposed in

306

Y.

Watanabe

Fig. 1. Lagrangian velocity vectors of the settling particles: num. of particles: 6 (left), num. of particles: 16 (middle), and the drag coefficient for spherical particles vs Reynolds number (right).

Fig. 2. Lagrangian velocity vectors under breaking waves (breaking wave height; 15cm, breaking water depth; 40cm, wave period; 1.2sec, recording duration; 3sec).

this paper, can visualize the vortex structure and estimate quantitatively 3D fluid velocity in large-scale vortices in the surf zone. Conclusions The low-cost 3DSPTV system for measuring large-scale fluid motion in a large domain has been developed. The 3DSPTV make it possible to estimate quantitative instantaneous velocity in large-scale vortices as well as solid particle behaviour in two-phase turbulent flows, ft has been confirmed that this system is accurate enough to apply to various 3d flows in coastal engineering.

Enhanced PIV Technique to Investigate Bottom Boundary Layer Dynamics 8.3. Enhanced P I V technique to investigate bottom boundary layer dynamics Ahmed S. M. Ahmed1 and Shinji Sato2 Coastal Engineering Lab, Department of Civil Engineering The University of Tokyo, 7-3-1 Hongo, Bunkyo-Ku Tokyo 113-8656, Japan ahmed© coastal, t. u-tokyo. ac.jp sato ©coastal, t. u-tokyo. ac.jp

Introduction Many researchers measured the flow field in the bottom boundary layer utilizing Laser-Doppler Velocimeter (LDV). Owing to the difficulty of measuring the flow field in the presence of the sand that blocks the laser beams, the measurements were conducted over an artificial rippled bed, over a rough fixed plane bed, or over a natural plane bed at which no influences from the sand concentration. On the other hand, since LDV is a technique of measuring the velocity components at one point, the description of the detailed flow field require many points, even up to 200 points Sato 1 or 316 points Ikeda et al?. This means much efforts and time consumption. Utilizing the advanced technology, the efforts and time consumption are decreased without losing the accuracy. In the present work, owing to the lack of information of particle velocity at high-concentration in which LDV cannot be used, the oscillatory boundary layer flow field over a natural rippled bed and a natural plane bed consisting of fine sand was measured by means of an enhanced approach of PIV technique. Comparison of the PIV technique results with other measurements verified the enhanced technique. Once the validity of the technique is established, the net volume flux of the sand transported in the sheetflow under asymmetric oscillation was estimated and compared with measurements.

Experimental

Conditions

Cases conducted in this work are given in Table 1. T is the oscillation period; umax stands for the maximum onshore velocity and (umax/u) is the asymmetric index. Case 1 is conducted to verify the flow field over rippled bed. It should be mentioned that to conduct the present rippled bed experiment, case 1, the water excursion was identical to Sato's case so that

307

308

A. S. M. Ahmed and S. Sato

Table 1. Experimental conditions Case

Type of bedforms

(umJu)

T(s)

1

Natural rippled bed

0.60

3.00

2

Sheetflow

Sinusoidal

3 4

Sheetflow Sheetflow

0.60 0.60

3.60 3.00 3.00

Wmax

(m/s) 0.48 1.27 1.20 1.40

the present maximum velocity is higher than that utilized by Sato 1 . Case 2 is conducted to examine a correlation relationship between brightness and sand concentration. The fine sand has 0.20 mm median diameter. Cases 3 and 4 are conducted to measure net volume flux. Cases 1, 3, 4 and additional 9 experiments were conducted under asymmetric oscillation. The nine experiments were conducted to verify the net volume flux measured by PIV technique. Results

and

Discussion

It is prescribed in the calculation process that the pixels located below the instantaneous unmoved bed are excluded, while pixels just above the unmoved bed that encompass the moving pattern, are included. Fig. 1 shows that the velocity vectors highlighted by circles were improved after enhancement. Fig. 2 reveals the comparison between the measured velocity by PIV and the measured velocity by Sato 1 for rippled bed. The measured velocity by PIV technique showed good agreement with the velocity measurements. To estimate the net volume flux of sheetflow under

-

1

0

1

2

3

4

5

6

7

x(cm) Fig. 1.

Velocity field before enhancement (left) and after enhancement (right).

Enhanced PIV Technique to Investigate

Bottom Boundary

Layer Dynamics

309

Fig. 2. Comparison between the flow field measured by PIV (right) and the flow field (left) measured by Sato 1 at phase 1.9 7r.

I

000 -

~"~*X»

S -0.40 u -0.80

^ —•— Measured after 30 cycles •

-i on 110

V

120

^^

Measured by enhanced PIV technique 130

140

150

160

u ^ (cm/s)

170

180

190

Fig. 3. Comparison between net volume flux measured by the PIV technique and measured by mass difference.

asymmetric oscillation, a correlation relationship between brightness values, b, of an image and sand concentration, c, measured by Horikawa et al. 3 which can be established as Eq. 1. c(g/cm3) = 0.0003 e,0.16446

(1)

Both velocity and concentration of sand were measured and the net volume flux was calculated. Fig. 3 displays excellent agreement between the net volume flux measured by the PIV technique and the one measured by the difference in the mass of the transported sand. Conclusions An enhanced approach of the PIV is developed based on a modified Minimum Quadratic Difference approach. Information of pixels below the instantaneous unmoved level is expelled from the calculation process so that

310

A. S. M. Ahmed and S. Sato

t h e velocity close t o the bed is measured accurately. It is verified t h a t t h e technique is capable of measuring t h e flow field over a n a t u r a l rippled bed. T h e comparison between the measured volume flux by P I V technique and t h e measured volume flux shows t h a t the technique could predict the net onshore volume flux accurately.

References 1. S. Sato, Doctor Thesis, University of Tokyo, Japan, 135 (1987) 2. S. Ikeda, K. Horikawa, H. Nakamura and K. Noguchi, Coastal Engineering in Japan, 34 (1), 15 (1991) 3. K. Horikawa, A. Watanabe and S. Katori, Proc. ltfh ICCE, ASCE, 1335 (1982)

Turbulent Boundary Layer Beneath a Gravity Wave and Its Decomposition

311

8.4. The turbulent boundary layer structure beneath a gravity wave and its proper decomposition Edwin A. Cowen and Stephen G. Monismith School of Civil and Environmental Engineering, Cornell University Ithaca, NY 14853-3501, USA; E-mail: [email protected] Department of Civil and Environmental Engineering, Stanford University Stanford, CA 94305-4020, USA; E-mail: [email protected] A major question in wave-turbulence research is how the turbulent velocity field at a moving air-water interface should be separated from the non-steady but essentially irrotational wave-induced motions. Cowen and Monismith 1 developed a hybrid digital particle tracking velocimetry (DPTV) technique, based on a dynamic subwindow first-order accurate (in space) PIV technique. The PIV algorithm described in Cowen and Monismith 1 has been extended to second-order accurate (in space) following the work of Meinhart and Wereley2. This modified algorithm, which allows both image subwindows to move dynamically about the desired velocity interrogation point, was applied to an image set, described in detail in Cowen3, of a single phase of a gravity wave propagating on a turbulent current (wave number, k, 4.16 rad/m; wave amplitude, a, 15.8 mm; water depth, h, 0.354 m; mean current U0, 63.6 mm/s; bulk fluid column streamwise turbulence intensity level, u'0, 3.5 mm/s ). The imaged area is approximately 25.8 x 31.0 mm and the phase is approximately at the crest (defined as kx — at = 0). The PIV analysis, finishing with 16x16 pixel subwindows overlapped at 50%, yielded up to 5165 vectors per image pair (115 points normal to the free surface) at a maximum resolution of 244 [im. A traditional phase-averaged turbulence analysis of the 2-D velocity fields from the PIV analysis can be defined as: u(x,t) = u(x) + u ( x , i ) + u / ( x , f )

(1)

where u is the instantaneous velocity vector at spatial location x and time t, u is the temporal mean velocity vector, u is the phase averaged velocity vector, and u' is the turbulent velocity vector. The turbulent perturbations are found by subtracting the temporal mean and phase averaged velocity vectors from the instantaneous velocity vector and hence all perturbations with respect to the temporal mean and phase averaged quantities are passed through to u'. From energy considerations real waves are known to propagate as groups and often contain reflection components. Hence at minimum one expects amplitude variations even in a monochromatic wave field. These amplitude variations lead to perturbations from the phase averaged veloc-

312

E. A. Cowen and S. G.

Monismith

ity fields that are on the order of the turbulent perturbations yet are not the result of turbulence (and do not effect the flow in the same manner as turbulence, e.g., they do not enhance the mixing). We have extended the wave decomposition technique of Thais and Magnaudet 4 , developed originally for single-point time history data, to spatially resolved data fields. This extended velocity field decomposition can be written as follows: u(x, t) = u(x) + u P ( x , t) + u f l (x, t) + u'(x, t)

(2)

where u p is the irrotational phased-dependent velocity vector and u # is the rotational phase-dependent velocity vector. Importantly, u p is used to separate the effects of amplitude and phase-induced variability in the wave-induced motions from the turbulence. A second-order accurate PIV analysis was carried out on 1500 image pairs of the above described image set and both velocity decomposition algorithms applied to the results. The u p field was determined through a least-squares fit to a second-order Stokes wave with the free parameters limited to a and t0, the time of the phase being measured. Fig. 1 shows two typical examples of the turbulent perturbation vector fields produced using the newly developed decomposition (Eq. 2). Fig. 2 shows the vertical profile of the streamwise and vertical turbulence intensity field under each decomposition algorithm compared with the quiescent free surface case. Removing the wave-induced potential flow reveals canonical turbulent velocity structures. Additionally, as shown in Fig. 2, the proper decomposition of the velocity field reveals statistics that are in better agreement with the free-surface boundary conditions, namely the vertical component of the turbulence should decay toward the free surface. The horizontal component of turbulence also decays suggesting that the enhanced straining by the wave field has increased the rate at which the background turbulent field, created by grids at the inlet of the channel, decays. E. Cowen gratefully acknowledges the financial support of the National Science Foundation (Grant Nos. OCE-0095834 & CTS-0000675) References 1. E. A. Cowen and S. G. Monismith, Exp. Fluids 22, 199 (1997). 2. C. D. Meinhart and S. T. Wereley, Exp. Fluids 31, 258 (2001). 3. E. A. Cowen, An Experimental Investigation of the Near-Surface Effects of Waves Traveling on a Turbulent Current, (Ph.D. Dissertation, Dept. Civil Engineering, Stanford Univ., 1996). 4. L. Thais and J. Magnaudet, J. Geophys. Res. 100(C1), 741 (1995).

Turbulent Boundary

Layer Beneath a Gravity Wave and Its Decomposition

313

0.08 , free surface 0.06 0.04 0.02 H

0 -0.02

0.02

-0.06 -0.04 -0.02 Ax-or (radians)

flkl,'",ii»hS 'II ,! ffV

MfflBffi*

-0.04

-0.06 -0.04 -0.02 0 kx-Gt (radians)

0.02

Fig. 1. Perturbation velocity field where the vectors are the u', w' field resulting from the decomposition as defined by Eq. 2. The first field shows an eddy interacting with the free surface while the second field shows a separatrix.

0 Si

\A \

-0.02

I -0.04

*®~>

\r /O

\

/

\ /

X

f,

-0.1

A

-0.02

•

-0.04

^ \

t

0.5

4

/

i

/

*t

«'/«'„

i o

U> ^4 u

-0.08 -0.1

* 4

h

-0.06

1.5

<' *

4

/

J?

if

*

0

u 0

/

0

r

/

F

k 'r

f

?\

?

9?

V

-0.08

M'

/

A

N -0.06

"~?

1A

0

l

0.5

i ? ? 1.5

«'/«',

Fig. 2. Profiles of the vertical and horizontal components of the rms turbulence intensity field. In each plot dashed line is quiescent free surface case where o is the horizontal component and A is the vertical component. Left plot solid line is turbulence as resolved by Eq. 1; right plot solid line is turbulence as resolved by Eq. 2.

314

D. T. Cox and S. L. Anderson

8.5. Intermittency in surf zone turbulence and observations of large-scale eddies using P I V D. T. Cox1 and S. L. Anderson2 Department of Civil Engineering, Texas A&M University College Station, Texas, USA 77843 E-mail: dtctamu.edu TSI, Incorporated Fluid Mechanics Instrument Division P.O. Box 64204, St. Paul, MN 55164 USA

Introduction Turbulence produced by wave breaking in depth-limited conditions is one of the most important nearshore processes. Our understanding of wave breaking turbulence is rudimentary, however, despite its fundamental importance to a number of other nearshore processes, including wave height variation, currents, mixing and dispersion, and sediment suspension. Nadaoka, Nino and Koyano 1 presented pioneering work on the structure of the turbulent flow field under breaking waves in the surf zone in which they identified large "obliquely descending eddies" (ODE). These eddies occur below trough level, have vertical axes inclined in the cross-shore direction, are intermittent in both space and time, and are thought to be a mechanism by which the intense turbulence due to wave breaking is brought to the bottom to suspend sediments. Experimental

Setup and

Procedure

The experiment was conducted at Texas A&M University in a 35 m long by 0.90 m wide by 1.2 m high glass-walled wave flume with a smooth, impermeable, 1:35 plane slope positioned with a still water depth of d = 0.498 m in the flat section of the flume (Cox and Anderson 2 ). The still water depth at this location was d — 0.132 m. A mirror angled at 45 degrees was positioned below the slope, and the PIV camera was normal to the glass sidewall of the flume. The laser sheet of the PIV system was in the horizontal plane. This combination of camera and laser sheet provided a "fixed sand grain's view" of the wave passing overhead. Intermittency

and Large

Eddies

Using PIV, Fig. 1 (left) shows the instantaneous flow in the horizontal plane at a depth of z = —0.097 m (0.035 m from the bottom) for three

Intermittency

in Surf Zone Turbulence and Large-Scale Eddies with PIV

315

successive frames with 1/15 s between each frame. The large scale vortex is clearly visible in these figures and has a nominal diameter of approximately 0.05 m. Fig. 1 (right) shows the vorticity estimated using a finite difference scheme without smoothing. The maximum vortex strength is about 30 1/s in the first two frames and is about 50 1/s in the third frame. This later value is estimated from only a few velocity vectors. Similar vortices were observed when the light sheet was positioned at z = —0.067 m. There was no preferred direction of rotation, and it was not possible to identify vortex pairs, possibly because the field of view was limited to 0.10 m by 0.10 m. It is still an open question of whether these structures exist in pairs, and the complex mechanisms of their formation are still not well understood. Conclusions

and

Outlook

This work shows the importance of intermittency in surf zone turbulence through the use of a point measurement technique and quantifies quantifies coherent eddies using PIV. Further work is necessary to elucidate the mechanisms by which these large vortices are generated and the possible mechanisms, which may be responsible for the intermittent behavior of surf zone turbulence. This work was supported by NSF-Career Award CTS-9734109. References 1. K. Nadaoka, M. Hino, M. and Y. Koyano, J. Fluid Mech., 204, 359 (1989). 2. D.T. Cox and S.L. Anderson, Coastal Engng. Journal, 43 (2), 121 (2001).

316

D. T. Cox and S. L. Anderson

x [mm]

Fig. 1. Left: Instantaneous velocity in the horizontal plane after the passage of a breaking wave. The plane is oriented at z = —0.097 m in a water depth of d = 0.132 m, and wave propagation is from left to right. The first frame occurs approximately i.2 s after the passage of the wave crest, and the time between frames is 1/15 s. Right: Instantaneous vorticity. Contour lines are in 10 1/s increments. Minimum vorticity is —^30 1/s in top and middle panel and —50 1/s in bottom panel. The maximum vorticity is 20 1/s in top panel and 20 1/s in middle and bottom panel. Contours are not smoothed.

PIV Data of Turbulence Distributions and Flow Structure in Coastal Ocean

317

8.6. T u r b u l e n c e d i s t r i b u t i o n s a n d flow s t r u c t u r e i n t h e coastal o c e a n from P I V data W. A. M. Nimmo Smith 1 , L. Luznik, W. Zhu, J. Katz and T. R. Osborn Department of Mechanical Engineering, The Johns Hopkins University, 3400 N. Charles Street, Baltimore, Maryland USA E-mail: [email protected]@

A submersible P I V system has been developed to measure flow structure and turbulence distributions in the coastal ocean. T h e present version of the submersible system comprises two 2Kx2K pixels, 12bits/pixel digital cameras operating simultaneously, each with a sample area of u p to 0.5x0.5m. We record two exposures within each frame of the digital cameras. A hardware based 'image shifter' creates a known fixed offset between exposures on the C C D array to remove directional ambiguity. T h e cameras can capture u p to 4 frames/s, requiring a total image acquisition rate of 6 4 M b / s . T h e d a t a is stored using ship-board h a r d disk arrays. W h e n the two sample areas are aligned horizontally in the same plane, and spaced l m apart, they enable us to resolve turbulent scales ranging from about 8 m m (the vector spacing) t o 1.5m. T h e light source of t h e P I V system is a pair of flashlamp-pumped dye lasers located at the surface, whose beams are transferred t o each of the sample areas using two independent optical fibres. Submerged probes are used for expanding t h e beams into light sheets. Naturally occurring particles are used as tracers.

^ P

( Camera (x2)

/

Light Sheet (x2)

weighted Tripod Base (2m Dia.)

D a t a analysis consists of enhancing each image usFig. 1. The new ing a modified histogram equalization method followed 10m sampling platby auto-correlation analysis. T h e calculated velocity dis- form. tributions are then corrected for optical distortions in the original images. Errors associated with the out-ofplane component of the velocity are minimised by limiting the thickness of the light sheets (to 2.5mm), restricting t h e sample areas (to a b o u t 35cm square), setting a minimum of the camera t o light sheet separation of about l m , and by subtracting the resulting contamination of t h e radial velocity component.

318

W. A. M. Nimmo Smith et al.

The components of the PIV system are mounted on a rigid sea bed platform, which enable us to align the sample areas with the direction of the mean current and to profile them vertically from very close to the bottom up to 10m above the bed (see Fig. 1). The submersible PIV system was deployed at two loU=10.7cms~ ,W=0.0cms -*1cms"' cations close to the Longterm Ecosystem Observatory (LEO15) site off the coast of New Jersey in regions with depths of 12 and 20m. Data were collected at different elevations and under different mean flow and wave conditions for periods in excess of 20min each, and at rates of up to 3.3Hz. Fig. 2 shows a sample distribution of the instantaneous fluctuation velocity (calculated by subtracting the instantaneous -15 -10 -5 0 5 10 15 Streamwise distance (cm) mean velocity from each vector). The flow contains vortical Fig. 2. Sample distribution of the instantaneous structures which are advected velocity fluctuations. The values of the mean horizontal and vertical velocities subtracted to give by the mean flow and coherent the fluctuation velocities are shown at the top. for periods in excess of 5 seconds. 1

1

Fig. 3 shows sample time series of the mean streamwise velocity (of each vector map) and vorticity intensity for two 20min periods under conditions of moderate and low overall mean current, plotted in black and grey respectively. In both series the amplitude of the wave induced motion is about 8cms _ 1 with a period of about 10s. Wave groups are also in evidence. The time series of vorticity intensity for the moderate flow case shows that the flow is characterised by extended periods of relatively quiescent flow, interspersed by short intense burst, or 'gusts', of high vorticity (sample 'gusts' are marked as 'X'). The vorticity intensity for the low overall mean current case is, on average, about 70% of the moderate current case, with only a few distinct 'gust' structures. The obvious exception is structure 'Y', which is most probably a fish wake. Vertical distributions of mean streamwise velocity in the coastal boundary layer are shown in Fig. 4. It can be seen that the vertical profile from

PIV Data of Turbulence Distributions

i X

1:1

and Flow Structure in Coastal Ocean

i Y

i

X

X

J

hr\k\wiwjik 1 jjy*|Llyj^jiim% kA^.kt jkiMAm

m * W mmWWW

1

Run 57 Run 66

319

_

HWM* *MW

Time (s)

Fig. 3. Sample time series of instantaneous mean streamwise velocity of each vector map (top) and normalised RMS vorticity (bottom) during 20min periods of moderate mean flow (black, 7.7cms - 1 ) and low mean flow (grey, 0 . 9 c m s - 1 ) .

the site with high mean flow and low wave motion has a logarithmic form. In contrast, the profile from the site with low mean flow and high wave motion does not exhibit a logarithmic form, regardless of the phase of the wave induced motion. This observation is consistent with laboratory experiments. A chapter containing further information, results and interpretation can be found in the main section of this volume. Funded in part by ONR and in part by NSF.

40

"36

.

; /

_

•

.

.

U(z) 5/20/2000

36 - 3

/

a'8 34

S

^

streamwi

CD

'

y

c

-30

S

Height above bed, z (cm) 28

8

S

S

8 8 S* i l l

Height above bed, z (cm) £ o> oo o o o

_,_,_._._, g M S qj oo o S o o o o S

Fig. 4. Vertical distributions of longterm mean streamwise velocity from (a) a site with a longterm mean current of 3 5 c m s _ 1 , and wave-motion amplitude of 5 c m s _ 1 and (b) a site with a longterm mean current of 1 2 c m s - 1 , and wave-motion amplitude of l O c m s - 1 . The distributions in (b) have been conditionally sampled according to the phase of the wave induced motion.

This page is intentionally left blank

CHAPTER 9 B R E A K W A T E R S A N D I N T E R N A L WAVES

9.1. Blocking effects in supercritical flows over topography V. I. Bukreev, A. V. Gusev, V. Yu. Liapidevskii Lavrentyev Institute of Hydrodynamics, Novosibirsk, RUSSIA E-mail: [email protected]

Introduction Dynamics of large amplitude surface and internal waves is closely related with wave breaking and following turbulence development at the fronts of nonlinear perturbations. Recent experimental studies of this process including the Particle Image Velocimetry (PIV) flow mapping technique reveal a complicated flow patterns in breakers and turbulent patches at wave fronts 1,2,3,4 -T^g mathematical models of spilling breakers and turbulent bores are developed in Longuet-Higgins 5 and Svendsen and Madsen 6 . In the present paper the structure of a stationary blocked zone upstream an obstacle in one- and two-layer supercritical flows is considered. This type of flow is interesting from many points of view. The class of flows is wide enough because hydraulic jumps and bores can be realized as stationary for suitable downstream control. The steady blocked zone upstream the obstacle is governed by mixing and dispersion effects at the wave front. It is very sensitive to changes in upstream flow parameters, therefore it may be considered as a good test for mathematical models of stationary and nonstationary motions. In the paper the mathematical model developed in Liapidevskii7 is adapted to the problem on stationary blocked zone in oneand two-layer flow. The model takes into account the dispersion and mixing effects on nonlinear wave structure. 321

322

V. I. Bukreev, A. V. Gusev and V. Yu.

Mathematical

Liapidevskii

model

Our attention will be focused on two limiting cases of flow. The first leads to the classic problem on the hydraulic jump structure in open channel flows of homogeneous fluid. The second is the stratified flow of miscible fluid in Boussinesq approximation. Shallow water approximation will be used for both cases. The main problem under consideration is to describe the mixing and nonhydrostatic effects on the stationary bore, generated and controlled by the obstacle. Two-layer shallow water equations with mixing and nonhydrostatic effects may be written in the form7: ht + (hu)x = -aq ut + uux + b(h + z)x + br]x + p~ = 0,

^

Vt + (vv)x = kaq vt + vvx +b(h + r) + z)x = —(u — kv), V _ qt + vqx = — (w2 - 2uv + kv2 - (k + 6)q2 - 2{k - 1)677) 2r) P

=

^

Jt = 8t+Ud-X

p-=ab(h-C)

(2)

(3)

(4)

Here h, r) are the depths of the lower layer and the turbulent mixing layer, u, v are the corresponding mean horizontal velocities, q is the velocity of large eddies and b — b/k is the buoyancy in the turbulent layer, z = z{x) is the bottom level, C, is the instant depth of the lower layer, which is the internal flow characteristic and is necessary for nonhydrostatic flow description. The coefficients a, 0, a, /3 are constants. Equations (l)-(3) describe the nonhydrostatic and mixing effects for two types of flow: A. k = 1. Surface turbulent layer dynamics in open channel flows. B. k = 2. Bottom layer dynamics. The main subsystems of (l)-(3) are: I. Hydrostatic model (l)-(2) (a = 0, p~ = 0). II. Dispersive model (1), (3) (no mixing, a = 0, 77 = 0). The experimental and calculated free surfaces (the model IA) are shown in Fig. 1 for downstream controlled flow in homogeneous fluid. The main conclusion here is that the model IA can be used for a repetitive sequence of long breaking waves.

Blocking Effects in Supercritical Flows Over Topography

100

323

X , CM

Fig. 1. The stationary bore of maximal amplitude in homogeneous fluid (ho — 2.9 cm, uo = 150 cm/sec). Free surface positions are obtained by an author's experiment (bold points) and by the model IA (Eqs. (l)-(2), a = 0.15, 9 = 2). The boundary of the surface turbulent layer calculated by the model IA is shown as the broken lines.

Conclusions T h e hierarchy of the shallow water models describing the dispersion and mixing for nonlinear wave propagation in homogeneous and stratified fluids is considered. T h e models are applied t o the problem on blocking effects in supercritical flow over topography. T h e stationary partially blocked zone u p s t r e a m the obstacle is investigated. It is shown t h a t the flow features for a stationary bore in homogeneous and stratified fluid are different. T h e hyperbolic shallow water equations with mixing and dispersion effects give satisfactory description of this difference. T h e work was supported in p a r t by Russian Foundation for Basic Research under grants No. 01-01-00846 and No. 00-01-00850.

References 1. 2. 3. 4. 5. 6. 7.

J. Grue, A. Jensen, P. O. Rusas, J. K. Sveen, J.Fluid Mech. 413, 181 (2000). J. A. Battjes and T. Sakai, J.Fluid Mech. 380, 257 (1999). J. W. Hoyt and R. H. J. Sellin, J. Hydraul. Eng. 115(12), 1607 (1989). K. Nadaoka, M. Hino, and J. Koyano, J.Fluid Mech. 204, 359 (1989). M. S. Longuet-Higgins and J. S. Turner, J. Fluid Mech. 63, 1 (1974). I. A. Svendsen and P. A. Madsen, J.Fluid Mech. 148, 73 (1984). V. Yu. Liapidevskii, J.Appl. Mech. Tech. Phys. 40, 238 (1999).

324

J.-M. Rousset and T. Bruce

9 . 2 . S t u d y o f t h e flow p a t t e r n i n a p e r f o r a t e d b r e a k w a t e r using P I V measurements Jean-Marc Rousset

and Tom Bruce

Division of Engineering, The University of Edinburgh Sanderson Building, UK-EH9 3JL Edinburgh, Scotland E-mail: [email protected]

Introduction Despite their increased complexity a n d cost of construction as compared to plain caissons, perforated caisson breakwaters are becoming more and more popular as anti-reflective quaywalls or external dykes. Most are composed of a vertical perforated (or slotted) screen in front of a vertical plain wall, forming a chamber of typically about 10 m width. Perforated vertical breakwaters are intended to absorb p a r t of the wave energy t h r o u g h various mechanisms, such as turbulence, viscous friction and wave resonance. T h u s their hydraulic performance is a complex function of wave and geometric parameters. However t h e processes dissipating wave energy are still poorly described, mainly due to t h e lack of experimental d a t a . T h e present paper presents a preliminary experimental study of the flow p a t t e r n s in the chamber of a slotted caisson.

Description

of the

experiments

T h e experiments were carried out in a 22 m long wave flume with absorbing paddle. A slotted caisson model was placed on a horizontal Perspex plate at the end of the flume. Fig. 1 shows the model and main dimensions. Four 0.025 m

H S1 S2

tt

S3

• Free surface 0.30 m

0.024 m 1 0.036 m *

Slope (1.15)

Model basement (Perspex) 0.40 m Flume bottom (glass)

Fig. 1. Main dimensions of the experiment and cross-section of the model caisson.

Study of the Flow Pattern in a Perforated

Breakwater

325

regular wave periods and four wave heights were generated (T=l.3-1.9 s, H=0.045-0.100 m) with 0.30 m water depth at the caisson toe. Three wave gauges were also deployed close the vertical walls to record the free surface evolution. PIV was employed to obtain 2D velocity fields in the caisson chamber and close to the perforated screen, the vertical laser sheet illuminating a section of the model through the bottom of the wave flume. The image acquisition was performed by a Kodak MegaPlus ESI camera (1008x1016 pixels) connected to Dantec Dynamics PIV processor. The image analysis was performed by proprietary software and by the modified MatPiv toolbox. 1 Both analyses employed cross-correlation and advanced interrogation techniques (multi-grid interrogation process, subpixel accuracy method, . . . ) . 2 Two illumination systems were used successively: a scanning beam system with Argon-Ion laser 3 to get the whole chamber velocity fields (delay between single frames Ai=40 ms) and Nd:Yag laser to study the jets generated by the perforated screen (pulse delay=300-900 fis and repetition rate=12.5-15 Hz). The PIV system was triggered by monitoring wave gauge Sl(Fig. 1).

Preliminary

results

Fig. 2-left shows the flow pattern as an incoming crest reaches the slotted screen (T=1.3 s and H=0.05 m). The 30x20 vector map is superimposed onto the original image. The caisson chamber is filled by the flow through the slotted screen with maximum horizontal velocity of U=0.25 m/s±6 mm/s. It leads also to the elevation of the free surface close to the plain back wall with a maximum ascending velocity of 0.20 m/s. During the opposite phase (trough at slotted screen), the previous wave is reflected by the back wall and the free surface level decreases. It generates a flushing effect which occurs with lower maximum speeds (U>-0.15 m/s and W>-0.12 m/s, W vertical velocity). Fig. 2-right shows an area close to the inner side of the slotted screen and centered on a jet. This figure emphasises the horizontal velocities and the vorticity, with distances normalised with the screen thickness (0.036 m). The maximum velocities reach 0.80 m/s in the jet while the velocities are null or sligthly negative in the shadow of the bars. The jet velocity decreases with the distance from the slots and, at a dimensionless distance ~2 screen thicknesses, the velocity is almost the same than deeper in the chamber. As vorticity measurement shows, pairs of conter-rotating vortices appear also behind the bars, their size varying with the wave phases. Their greatest diameter is of the horizontal bar thickness.

326

J.-M. Rousset and T. Bruce

'Ajg.r

s

0

05 Ufm/s]

1

Vorticity

m. ^

Fig. 2. Flow pattern as a crest crosses through the slotted sreen (T=1.3 s, H=0.05 m) -leftflowin the chamber, -right horizontal jet in the chamber.

Conclusions The main conclusion of this preliminary work is that vortices are produced and disappear only in the vicinity of the slotted screen, and thus that the wave energy dissipation occurs mainly at the perforated wall. The next point of our study will be its experimental assessment and the implementation of these dissipation terms in an existing numerical code. Our experimental study may be considered as a step forward to a better understanding of the complex flows generated by coastal structures. This research was supported through a EC Individual Marie Curie Fellowship (contract HPMF-CT-2000-01010). The EC is not responsible for any view or results expressed. References

1. J. K. Sveen, http://www.math.uio.no/~jks/matpiv. (2003). 2. M. Raffel, C. Willert and J. Kompenhans, Particle Image Velocimeiry: a practical guide (3rd edition), Springer, Berlin (2001). 3. C. Gray, C. A. Greated, D. R. McCluskey and W. J. Easson, Mem. Sci TechnoL 25 717 (1991).

Pseudo Image Velocimetry of Regular Wave Flow Near Submerged Breakwater

327

9.3. Pseudo image velocimetry of a regular wave flow near a submerged breakwater Francisco Taveira-Pinto Faculty of Engineering of the University of Porto, Rua do Dr. Roberto Frias, 4200-465 Porto, Portugal E-mail: [email protected] Several types of costal structures are now employed for coastal protection, against wave action, such as detached breakwaters, groins, longitudinal block mounds and submerged breakwaters. Submerged breakwaters are one of the possible solutions that can be used for coastal protection against wave destruction. As experience shows, they afford greater protection of the coastline and also decrease the risk of erosion and help that sand retention could occur. It has also been shown that they permit coastal protection solutions, but it is not entirely understood how they work and how they interact with the oncoming waves. It is shown in this investigation that they work by modifying the wave flow characteristics in a fairly efficient way. This was the aim of this work as well as to document the submerged breakwater interaction with the wave flow. As the data show, the interaction is complex but very efficient and it could explain the high efficiency of submerged breakwaters in coastline protection. The measurements were carried out in the old unidirectional wave tank of the Hydraulics Laboratory of the Faculty of Engineering of the University of Porto. The wave tank is 4.8 m wide and 24.5 m long with a maximum water depth of 0.60 and 0.40 m near the wave generator and at the test section, respectively. A window in the lateral wall of the tank allows visual access to the test section. A piston-type wave generator installed on the tank allows the wave generation of the used regular waves. A wave-absorbing beach built with sand and gravel and placed at the opposite end to the wave generator was used to reduce reflected waves to a minimum. A thin dividing wall placed on the test section was used in order to avoid three-dimensional effects on the wave flow. Wave probe modules allow water elevation measurements at different points of the test section. One of these probes was always located at the section where velocity measurements were being carried out. For velocity measurements, the used light source was a Spectra-Physics Stabilite 2017S argon-ion laser operating in single mode with a power of 2 W. The optical system consisted of a 55X modular LDA optics based on a Dantec one-component fibre-optic system, with a 60 mm probe, working in

328

Francisco

Taveira-Pinto

the backscatter configuration. A three-dimensional traversing table located in front of the tank window supported the fibre-optic probe and allowed the control volume to be positioned at each required point on the test section. To improve the alignment of the optics and to reduce the size of the control volume, a 55 x 12 beam expander (Dantec) was placed before the 600 mm front lens. The scattered light was collected by a photomultiplier (PM). A burst spectrum analyser (BSA) processed the signal from the PM. An AT-MIO-10 card, interfaced the BSA and the wave probe module located at the measuring section. In order to obtain A/D data coincident with LDA samples, the analogue data acquisition was performed at a sampling frequency approximately 2-3 times the mean data rate of LDA samples. Coincidence filtering was used to match both AD and LDA data. Measurements of horizontal and vertical velocity components, at different locations around the structure, were defined, allowing phase-dependent flow patterns to be defined. Phase averaged r.m.s. values of velocity fluctuations for both the horizontal and vertical velocity components were also evaluated, allowing a detailed picture to be obtained of the wave interaction with submerged breakwaters. Measurements were carried out for a water depth of 0.22 m and for regular waves, with height and period of 3.5 cm and 1.25 s, respectively, which when duly scaled correspond to common situations of the sea state on the Portuguese coast. The celerity C of the wave, defined as the ratio between the wavelength and the wave period, was equal to 1.3 m/s. An impermeable breakwater model with a smooth surface and trapezoidal section was used and a crest width of 0.10 m. For each of the measured 33 profiles and for each water level value (z), 50 mean velocity values corresponding to 50 intervals of the wave period were evaluated. For each profile the phase-dependent flow field was defined. Water elevation and velocity values were normalized by the still water level (d) and by the wave celerity (C), respectively. The data obtained allowed the definition of the velocity field around the submerged breakwater in 50 different time-steps, corresponding to 50 different wave phase values. In Figure 1 one of these situations is presented. Velocity fluctuations were also evaluated for both horizontal and vertical velocity components. The calculated r.m.s. values (u' and v') of velocities at each profile and each level were normalized by the mean local corresponding velocity value (u and v). As for the mean velocity values, 50 different situations corresponding to 50 different wave phase values were analysed. Flow motions near the submerged breakwater can be observed during the wave phase, allowing the definition of recirculation regions in front of and behind the structure. Velocity fluctuation flow fields show that the

Pseudo Image Velocimetry of Regular Wave Flow Near Submerged Breakwater —- Velocity/Celerity = 1

Wave Phase = (10/50)*360

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Profile Coordinate - X (m) —' Velocity/Celerity = 1

0

2.2 2.4 2.6

Wave Phase = (34/50)*360

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.1 Profile Coordinate - X (m) Fig. 1.

329

2.2 2.4 2.6

Examples of velocity flow fields.

presence of the breakwater leads to an increase in turbulence, the location of the higher turbulence intensity region being dependent on the wave phase value. The author wishes to thank to FCT (Foundation for Science and Technology), sponsor of the Project PRAXIS/C/ECM/11303/1998.

330

Y. Guo et al.

9.4. Laboratory modelling of the motion of an internal solitary wave over a ridge in a stratified fluid Y. Guo1, P. A. Davies2, J. Kristian Sveen3, J. Grue4 and P. Dong5 ' ' Department of Civil Engineering, University of Dundee, Dundee DD1 4HN, UK 34 ' Department of Mathematics, University of Oslo, Oslo N0316, Norway Email: [email protected] ; [email protected]; ; [email protected] ; [email protected] ; [email protected] Internal solitary waves (ISWs), observed at many oceanic locations, can be generated when tidal currents encounter bottom topography in stratified oceans. Such waves are known to produce significant hydrodynamic loading on offshore structures in deep water oceanic conditions and their properties in a two-homogeneous-layer system of constant water depth have been extensively investigated in the past decades. The shoaling processes of ISWs over a sloping bottom have been examined by several investigators, though comparatively few investigations have been undertaken on the propagation of ISWs of depression over a bottom ridge. The present study considers the motion of ISWs over a bottom ridge in a stratified (upper layer) fluid by means of laboratory experiments that were carried out in a channel with dimensions 6.4 m long x 0.4 m wide x 0.6 m deep. The channel was filled with salt water of density pi to a depth h2. Then an upper layer of linearly-stratified saline fluid was filled to a depth h\ with buoyancy frequency NQ being NQ = (g/po)(Ap)/h\, where g is the gravitational acceleration, po and Ap are respectively the density at mid-depth and the density difference between the top and bottom of the upper layer fluid. A watertight movable gate was installed at one end of the channel to allow internal solitary waves with a prescribed amplitude a (realized by adjusting the location of the gate and the volume of lighter salt water behind the gate) to be generated by using the step-like pool techniques 1 . Density and velocity data were obtained by means of microconductivity probe arrays and the Digimage particle tracking technique at three camera locations, namely 3.2 m (CI), 4.05 m (C2) and 4.9 m (C3) downstream of the gate. Reference runs without the presence of the ridge were performed for comparison purposes. Three different ridges (placed at C2 with height hr and slope angle a) were used in the experiments. Parameter values investigated in the present study are: 0.65 < N0 < 1.41 s _ 1 ; 3.1 < h2/hi < 4.1; 30 < H (total water depth) < 39 cm; 11.3 < a < 18.40; 10 < hr < 15 cm; 0.1 < a/h2 < 0.27.

The Motion of an ISW Over a Ridge in a Stratified

Fluid

331

Observations show that the wave patterns characterizing the resulting encounter with the ridge depend primarily upon (i) the amplitude of the wave, (ii) the height of the ridge and (iii) the buoyancy frequency of the upper layer. For sufficiently small wave amplitude, the ridge has little influence on the propagation of wave. No mixing occurs in the upper layer as the wave propagates over the ridge though the vertical density structures measured through the wave trough are stretched as a result of the passage of the wave, thereby weakening the upper layer stratification. For such weak encounters, the measured velocity profiles for experiments with and without the ridge for identical initial input conditions agree well with the nonlinear theory 2 . With moderate and large wave amplitude, vortices are initially formed at the back and middle part of the wave and roll and move forward, causing mixing and overturning. This also happens for the cases in which no ridge is present provided that the wave amplitude is sufficiently large, though the vertical extent of mixing and overturning is smaller than those with the presence of ridge. For such strong encounters, the velocity data show that the velocity values measured at C3 in both the upper and lower layers for the ridge case are appreciably smaller than those without the ridge for the identical incoming wave upstream, indicating that the ridge applies a significant blocking effect on the propagation of the wave, resulting in a local reduction of the values of the velocity. The values of the velocity in the region close to the free surface are almost uniform since mixing and overturning occurring there make the local density structure more or less homogeneous. This is also confirmed by the density measurement (see Fig 1) taken at C2 for cases with and without the ridge and nominally identical input conditions in which the solid line represents the density profile taken before experiment. It is seen that the partially mixed region extends further into the lower layer when the ridge is present. In order (i) to quantify the vigor of the overturning events described above and (ii) to determine their vertical distribution, it is convenient to compute the buoyancy anomaly profiles gl(yi) from the measured density profiles p0(j/») as gl(yi) = g{p(yi) - PT{yi)}/pm, where pT{Vi) is the corresponding Thorpe-ordered density profile and pm the mean density. Fig 2 shows the buoyancy anomaly profile deduced from the density profiles of Fig 1, in which the buoyancy anomaly with no ridge case is offset by 0.4 cm/s 2 . The non-zero value of the buoyancy anomaly at a particular vertical position indicates that overturning occurs there. The greater the magnitude of the buoyancy anomaly, the stronger the overturning processes.

332

Y. Guo et al.

1046

1048 density (kg/mA3) Fig. 1.

1050

Density profiles for strong encounter

0

0.4

0.8

g'(cm/sA2) Fig. 2.

Buoyancy anomaly deduced from Fig 1

Thus, the data shown in Fig 2 demonstrate that the degree and excursion of the mixing are enhanced by the presence of the ridge. As seen, the attenuation of the wave amplitude due to the combined effects of viscous damping and the local loss due to the bottom ridge blockage can be conveniently evaluated from the measured density profiles. Summary results are plotted in Fig 3 in which the dimensionless integrated wave amplitude attenuation is calculated as (ai — a-s)/ai with 03 being the wave

The Motion of an ISW Over a Ridge in a Stratified Fluid

333

o

2 -

^

<§> ° v v

• •

0

Z^

o

_ •

ridge I

ridge DI

•

•ff-B 0.1

0.2

0.3

0.4

a1/h2

Fig. 3.

Wave amplitude attenuation (ai — a 3 ) / a i versus ai//i2

amplitude measured at C3. It is seen that there is an increase in the wave amplitude damping for high wave amplitude where the mixing occurs and such damping can be significantly enhanced by the presence of the bottom ridge. However, for low wave amplitude, the ridge has little influence on the wave amplitude attenuation. The authors are grateful for the support of this work by the UK Engineering and Physical Science Research Council (EPSRC). J. Grue et al., J. Fluid Mech. 413, 181 (2000). J. K. S. Sveen, Y. Guo, P. A. Davies and J. Grue, J. Fluid Mech. (469), 161 (2002).

J. K. Sveen and J. Grue

334

9.5. Measurements of velocities during run-up of long internal waves on a gently sloping beach J. Kristian Sveen and John Grue Dept. of Mathematics, University of Oslo, PB 1053, Blindern, 0316 Oslo Norway

Introduction Internal waves are a well known feature in fluids where two, or more, layers of different densities are found. Recent work 1 ' 2 ' 3 , has revealed several of the properties of propagation of internal waves and their subsequent runup on a slope. However, none of these works have been able to explain a recent finding at the offshore platform Ormen Lange, where a relatively large velocity was measured close to the bottom of the ocean. The measurement is shown in Fig. 1 and the max velocity can be found between the 26 and 27 of November to be around 0.5m/s. This velocity can be made non-dimensional with respect to the linear long wave speed defined by Co = (g'hih,2/hi + /12)1/2, where g' = Ap/p and hi and /12 denote the thickness of the lower and upper layers respectively. At the position of the Ormen Lange platform, there is an upper layer of thickness /12 = 500m and a lower layer of thickness hi ~ 1 — 2/12 and a relative density difference of about Ap/p ~ 0.5 • 10~ 3 resulting in Co = 1.1 — 1.3m/s. This means that the maximum velocity is measured to be Umax/co ~ 0.39 — 0.45. Experiments To investigate this phenomenon a laboratory experiment was set up. The physical setup followed that of an earlier investigation 1 and the layer ratios Currents and tempaiBtur* at peMen II am Mb

23

Fig. 1.

24 25 28 MMNovtflMrlflM

27

28

29

Velocity at position 5m above the sea floor.

30

Velocities in Rmn- Up of Long Internal

•4,43

—4.27

Fig. 2.

-2.24

Waves on a Gently Sloping Beach

335

©

Schematic drawing of experimental setup.

that of the Ormen Lange conditions. In one end of the wave-tank a beach with slope 0,3m/6.24m (height/length) was installed, and upon run-up of the internal wave, velocities was measured in the fluids using Particle Image Veloeimetry (PIV). Experiments were performed for a few different lengths of the initial volume and for waves of both elevation and depression. A schematic drawing of the experimental setup is shown in Fig. 2. Images were recorded using a Dalsa DA-512 video-camera, recording 50 frames per second. Velocity fields were then calculated from these recordings using PIV.. Finally, velocities were extracted from two small regions, close to the bottom and close to the free surface. The two regions were then monitored during wave propagation. The variation of the velocity in the upper and lower layer with time, in a selected experiment, is depicted in Fig. 3ci. The incident wave had in this case a length of ~ 2.24m and an elevation of «o/^2 = 0.5, with hi = 10cm. It is observed that the maximum velocity close to the beach is H2/C0 ~ 0.4. Repeated experiments (not shown here) confirm that the maximum velocity is about H2/C0 ^ 0 . 4 — 0.5. It is observed that the duration of the gravity current is about 40 non dimensional periods in the laboratory. In the in-situ measurements this period is about 160. A possible explanation for this is that in the real-scale phenomenon the incident wave may be longer than in the laboratory. At the Ormen Lange location the waves may be generated by two high and low pressure regions in the nordic sea implying that the incident wave may be as long as 100 — 200/&2- I n the laboratory the longest waves are ~ 20/i2* A similar experiment was also performed where the length of the initial volume was reduced to 5^2- In this case the result from the run-up on the beach was notably different than the previous case. A series of so-called boluses was observed to run up the beach slope. A total of 6 boluses can' be seen from the velocity trace depicted in Fig. 36. It is noted that the maximum velocity in the lower layer in this case is about U2/C0 ~ 0.05.

336

J. K. Sveen and J. Grue

Conclusion T h e findings of the initial experiments indicate t h a t run-up of relatively short internal waves produce boluses as shown in earlier works 2 . These boluses induce velocities on the order of M2/C0 ~ 0.05. Run-up of relatively long internal waves may produce a turbulent gravity current with m a x i m u m velocity close to the b o t t o m of about M2/C0 ~ 0.5. T h e same extreme nondimensional speed was observed in the laboratory as in t h e field observation. Also the durations of the extreme events are similar in the laboratory and in large scale. This suggests t h a t pronounced gravity currents may cause the extreme currents observed at the actual location of the Ormen Lange.

'

0.5 "1,2 CO

!

!

!

!

30

40

50

iW" : - l i .-... ° .. - -TM4

-0.5 0.2 0.1 "1,2 CO ° -0.1

-0.2

0

10

20

!

'

!

•

60

7

!

1

-i

A, „J4 Ului wwm

Mm

^Bl^WBW

;

,

i

1

!

;

!

0

10

20

30

40

50

60

70

tc0/h2 Fig. 3. Time trace of the mean velocity in the upper(+ and solid line) and lower layer (• and solid line). Dashed lines indicate the standard deviation. Upper panel: initial wave of elevation. Lower panel: initial wave of depression.

This work was conducted under the Strategic University Programme "General Analysis of Realistic Ocean Waves", funded by the Research Council of Norway. The authors are grateful for the additional funding provided by Norsk Hydro. References 1. J. Grue, A. Jensen, P.-O. Rusas, and J. K. Sveen, J. Fluid Mech. 380, 257 (1999). 2. K. R. Helfrich, J. Fluid Mech. 243, 133 (1992). 3. H. Michallet and G. N. Ivey, J. Geophys. Res. 104, 13467 (1999).

Strong Internal

Wave Events with PIV Studies of the Phenomenon

337

9.6. Comparison of modelling of strong internal wave events with P I V studies of the phenomenon Deborah J. Wood, J. Kristian Sveen, and John Grue Mechanics division, Department of Mathematics University of Oslo, P.O. Box 1053, Oslo, 0316, Norway E-mail: [email protected]

High amplitude

internal

waves

A region of fluid in an ocean or fjord may consist of layers of different density fluid caused by changes in salinity or temperature. At these regions in the ocean an internal wave may be generated and travel along the interface between different density fluids. We simulate sets of experiments which were carried out at the University of Oslo where high amplitude internal waves were generated and the velocities were measured using PIV, see Grue et al.1 In the wave tank the fluid has a shallow upper layer with either constant density or linear stratification, and a deep lower layer with constant density, relevant to conditions in the ocean at several locations. A volume of light water is trapped behind a gate in one end of the wave tank and when the gate is removed a mode-one solitary wave is generated after a while. This wave propagates along the tank and it is this propagation together with the actual wave generation which we simulate using the NS solver (FLUENT). The thermocline in the experiments is simulated in the NS solver by linking density to temperature, hence an initial thermocline of any form can be created by imposing suitable temperature conditions. Initially the solver was tested on small amplitude waves2 which reproduced the experimental results very well. We also considered a large amplitude wave where strong breaking was observed in experiment but for which there is no existing theory. The NS solver agreed well with experiment in the following conclusions (linear stratification case): for the larger waves the NS solver predicts the upper bound on fluid velocity of c + 5c where c is wave speed and 5c « c (Fig. 1); as the volumes of less dense fluid increase behind the gate the NS solver predicts a broadening of the wave form; the curved feature at the top of the velocity plot (Fig. 1) corresponds to an area of breaking; no re-circulation zone was seen. In addition it was also found that: general stratifications could be modelled; a slightly less volume of lighter fluid was needed in the NS solver than in experiment thought due to the NS solver not modelling mixing; NS solver does not fully reproduce overturning motion; no problems due to false boundary layers as seen previously when using FLUENT for surface waves3.

338

Internal

D. J. Wood, J. K. Sveen and J. Grue

waves travelling

over a beach

In nature it is common that an internal wave may be travelling over a sloping bottom. When the internal wave reaches a sloping bottom it may develop into a gravity current. These gravity currents can cause high velocities close to the sea-bed and hence damage any risers or pipe-lines which may be present there. The experiments run at University of Oslo use similar thermoclines to those in field measurements by Norsk Hydro ASA except scaled down to laboratory scale with the top layer being 0.09m (corresponding to 500m in the ocean) and the stratification of thickness 0.02m. We simulate the experiments using the NS solver in a similar manner to the high amplitude waves. The details are given in Wood 4 . In particular we reference all velocities to that of the linear wave speed, CQ given by: (?0 = 9 1 /^l^f , where hi and pi are the height and density of the bottom layer respectively, hi and P2 are the height and density of the top layer respectively, and g is gravitational acceleration. In general the form of a gravity current is that of a 'head' of dense fluid followed by a thinner tail of denser fluid. This was simulated by the NS solver as shown in Fig. 2. The form of the wave is similar to that found experimentally with high density gradients at the front and top of the wave with a 'core' of small density changes at the centre of the head. Other results found were as follows: the head of the density current is formed by intermediate density fluid; as the gravity current forms the thermocline becomes part of the front part of the wave; as the wave develops the thermocline is at its thickest at the middle of the head of the gravity current; the maximum horizontal velocity is 0.5co which is a similar figure to that found experimentally. This work was conducted under the Strategic University Programme 'General Analysis of Realistic Ocean Waves' funded by the Research Council of Norway, with additional financial support from Norsk Hydro. The discussions with Dr. Didier Clamond, Dr. Atle Jensen and Prof. Geir Pedersen are gratefully acknowledged. References 1. J. Grue, A. Jensen, P.-O. Rusas and J. K. Sveen, J. Fluid Mech. 413, 181 (2000). 2. D. J. Wood and J. Grue, Appl. Ocean Res. 24, 331 (2002). 3. D. J. Wood, G. Pedersen, A. Jensen, Ocean Engng. 30, 625 (2003). 4. D. J. Wood, 21st Conf. on Offshore Mech. and Arctic Engng., Oslo (2002).

Strong Internal

Wave Events with PIV Studies of the Phenomenon

1' -4h

I

339

1

u/c Fig. 1. Horizontal velocity (n), normalized by wave velocity (c) profile of a high amplitude internal wave at a vertical cross-section at the centre of the wave (h is depth of top layer). Dots are experimental data and the solid line is NS solver data.

Fig. 2. Contours of density (kg/m 3 ) for internal wave travelling over a beach at time i = 36s. Solid white line shows the beach.

ADVANCES IN C O A S T A L AND OCEAN E N G I N E E R I N G

-

PIV flHD WATER. WflUCS This volume introduces particle image velocimetry (PIV), a technique for water wave measurement in the laboratory and in the open ocean. It discusses the turbulent dissipation, Reynolds stresses and vortical structures in boundary layers of the sea bed, as well as ships, ship wakes, propulsion hydrodynamics, cavitation and free surface waves. Upwelling behind crests of micro-breaking ocean surface waves (important for the exchange of greenhouse gases between air and water) and large amplitude internal solitons in the ocean are measured. The book includes velocities and accelerations in breaking ocean waves, run-up, interaction between strong waves and breakwaters, as well as a concise description of the state-of-the-art PIV technique.

This book has its origins in a meeting on PIV and water waves which was held in Cambridge in 2002. The main body of the book consists of six overview ox in-depth articles by invited authors w h o are specialists in their respective fields, as well as practitioners of PIV. A complete set of abstracts from the meeting is enclosed. The book is well suited for scientists who want to acquaint themselves with current experimental hydrodynamics, as well as for researchers and graduate students who are already working in the field or plan to do so.

Key Features • State-of-the-art overview of PIV applied to water waves Invited articles, in carefully chosen fields, providing profound insight into the role of PIV in important applications « Updated overviews of the PIV method that will benefil newcomers to the field

ISBN 981 238-914 8

www.worldscientific.com 5581 he

9 "789812"389145"