Particles in turbulent flows

Leonid Zaichik, Vladimir M. Alipchenkov, and Emmanuil G. Sinaiski Particles in Turbulent Flows Leonid Zaichik, Vladi...

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Leonid Zaichik, Vladimir M. Alipchenkov, and Emmanuil G. Sinaiski

Particles in Turbulent Flows

Leonid Zaichik, Vladimir M. Alipchenkov, and Emmanuil G. Sinaiski Particles in Turbulent Flows

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Leonid Zaichik, Vladimir M. Alipchenkov, and Emmanuil G. Sinaiski

Particles in Turbulent Flows

The Authors Prof. Dr. Leonid Zaichik Russian Academy of Sciences Nuclear Safety Institute Moscow, Russ. Federation Dr. Vladimir M. Alipchenkov Russian Academy of Sciences Institute for High Temperatures Moscow, Russ. Federation Prof. Dr. Emmanuil Sinaiski Leipzig, Germany

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate. Library of Congress Card No.: applied for British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek Die Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliograﬁe; detailed bibliographic data are available on the Internet at http://dnb.d-nb.de. # 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microﬁlm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not speciﬁcally marked as such, are not to be considered unprotected by law. Typesetting Thomson Digital, Noida, India Printing betz-druck GmbH, Darmstadt Binding Litges & Dopf GmbH, Heppenheim Printed in the Federal Republic of Germany Printed on acid-free paper ISBN: 978-3-527-40739-2

V

Contents Preface IX Introduction 1 1.1 1.2 1.3 1.4 1.5 2 2.1 2.2 2.3 2.3.1 2.3.2 2.4 2.5 2.6 2.6.1 2.6.2 2.6.3 2.7 2.8

XIII

Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence 1 Characteristics of Homogeneous Isotropic Turbulence 1 Motion of a Single Particle and Heat Exchange 11 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories 13 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence 27 Particle Acceleration in Isotropic Turbulence 35 Motion of Particles in Gradient Turbulent Flows 39 Kinetic Equation for the Single-Point PDF of Particle Velocity 40 Equations for Single-Point Moments of Particle Velocity 47 Algebraic Models of Turbulent Stresses 52 Solution of the Kinetic Equation by the Chapman–Enskog Method 53 Solution of the Equation for Turbulent Stresses by the Iteration Method 58 Boundary Conditions for the Equations of Motion of the Disperse Phase 62 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow 74 Motion of Particles in the Near-Wall Region 87 Near-Wall Region Including the Viscous Sublayer 87 The Equilibrium Logarithmic Layer 91 High-Inertia Particles 95 Motion of Particles in a Vertical Channel 96 Deposition of Particles in a Vertical Channel 107

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3 3.1 3.2 3.3 3.3.1 3.3.2 3.4

4 4.1 4.2 4.3 4.4 4.5 4.6 5 5.1 5.2 5.3 5.4 5.5 5.5.1 5.5.2 6 6.1 6.2 6.3 6.4

Heat Exchange of Particles in Gradient Turbulent Flows 115 The Kinetic Equation for the Joint PDF of Particle Velocity and Temperature 115 The Equations for Single-Point Moments of Particle Temperature 123 Algebraic Models of Turbulent Heat Fluxes 127 Solution of the Kinetic Equation by the Chapman–Enskog Method 127 Solving the Equation for Turbulent Heat Fluxes by the Iteration Method 130 Second Moments of Velocity and Temperature Fluctuations in a Homogeneous Shear Flow 132 Collisions of Particles in a Turbulent Flow 137 Collision Frequency of Monodispersed Particles in Isotropic Turbulence 138 Collision Frequency in the Case of Combined Action of Turbulence and the Average Velocity Gradient 149 Particle Collisions in an Anisotropic Turbulent Flow 151 Boundary Conditions for the Disperse Phase with the Consideration of Particle Collisions 159 The Effect of Particle Collisions on Turbulent Stresses in a Homogeneous Shear Flow 160 The Effect of Collisions on Particle Motion in a Vertical Channel 164 Relative Dispersion and Clustering of Monodispersed Particles in Homogeneous Turbulence 171 The Kinetic Equation for the Two-Point PDF of Relative Velocity of a Particle Pair 172 Equations for Two-Point Moments of Relative Velocity of a Particle Pair 177 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence 180 Inﬂuence of Clustering on Particle Collision Frequency 196 Relative Dispersion of Two Particles in Isotropic Turbulence 200 Dispersion of Inertialess Particles 202 Dispersion of Inertial Particles 205 Collision and Clustering of Bidispersed Particles in Homogeneous Turbulence 209 Collision Frequency of Bidispersed Particles in Isotropic Turbulence 209 Collision Frequency in the Case of Combined Action of Turbulence and Gravity 215 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow 217 Vertical Motion of a Bidispersed Particle Mixture 226

Contents

6.5 6.6

Equation for the Two-Particle PDF and its Moments 229 The Clustering Effect and its Inﬂuence on the Collision Frequency of Bidispersed Particles in Isotropic Turbulence 235 References 241 Notation Index 261 Author Index

277

Subject Index

283

VII

IX

Preface Two-phase dispersive ﬂows are found under many natural and technical conditions and practically all of them are always turbulent. Two-phase turbulent ﬂows currently represent one of the most developing divisions of mechanics and heat exchange. The aim of this book is the development and elaboration of continual statistic methods of modeling of hydrodynamics and mass exchange in two-phase turbulent ﬂows based on kinetic equations for probability density function (PDF) of velocity and temperature of dispersed phase particles. The main theoretical problems considered in the book consist of the investigation of particle interaction with turbulent eddies of carrier continuum (ﬂuid) and collisions of particles with each other in turbulent ﬂows. Particular emphasis has been placed on the accumulation (clustering) phenomenon of particles in the near-wall and homogeneous turbulence. A statistical approach based on distribution function in phase space is a very powerful tool for deriving theoretical models in different ﬁelds of physics whether it be molecular theory of gases and ﬂuids (Boltzmann equation and Bogolubov-BornGreen-Kirkwood chain of equations), motion of Bownian particles (Fokker-Planck equation), plasma physics (Vlasov kinetic equation), theory of coagulation (Smoluchowski-Müller equation) and so on. This method has also found applications in the theory of disperse turbulent ﬂows in the works of I.V. Derevich, V.I. Klyatskin, T. Elperin, M.W. Reeks, O. Simonin, K.E. Hyland, D.C. Swailes, J. Pozorski, J.-P. Minier, R.V.R. Pandya, F. Mashayek, L.X. Zhou and authors of this book. All these works are published in periodic journals. However in well-known monographs (e.g. Shraiber et al., 1987; Gorbis & Spokoyny, 1995; Crowe et al., 1998; Varaksin, 2003) devoted to two-phase turbulent ﬂows, the statistical method of modeling based on the probability distribution function practically was not embodied. The present book is the ﬁrst monograph dedicated to a consecutive presentation of statistical models of motion, mass-exchange and accumulation of particles in turbulent ﬂuid on the basis of PDF. The heart of the book is formed by results obtained by the authors. At the same time the most notable achievements of other scientists in the ﬁeld of statistical model construction of dispersed turbulent ﬂows are adequately covered.

X

Preface

In this book it is shown how different problems can be solved connected with motion, mass-exchange, dispersion and accumulation of inertial particles with the help of statistical models based on single-point and two-point PDF. One of the key problems is to develop a rational approach to model phenomenon of particle accumulation (clustering) in homogeneous and non-homogeneous ﬂows. As applications of submitted statistical models, the behavior of particles in isotropic homogeneous shear and near-wall turbulence is considered. It is reasonable that far from all questions connected with two-phase dispersed ﬂows are included here, because it is impossible in the framework of one book to embrace all aspects of such complex problems. In the introduction theoretical problems are considered, as well as chief lines of investigations and up-to-date methods of modeling two-phase dispersed turbulent ﬂows. Chapter 1 has an introductory character. It contains characteristics of isotropic turbulence and behavior of particles in isotropic turbulent ﬂuid. The central position is occupied by chapter 2, which describes methods to derive kinetic equation for single-point (single-particle) PDF of velocity distribution of particles in turbulent ﬂuid modeled by Gaussian random process. The kinetic equation obtained is used to construct continual transport (differential) and algebraic models able to calculate hydrodynamic characteristics (moments) of the dispersed phase. As examples of submitted models, the behavior of particles in homogeneous shear layer, near-wall turbulent ﬂow and ﬂow in vertical channel is considered. In chapter 3 mass-exchange between particles in gradient turbulent ﬂow is considered. Statistical methods of dispersed phase motion and mass-exchange developed in this chapter are based on kinetic for compatible PDF of velocity and temperature of particles. Continual transport and algebraic models to calculate mass-exchange in dispersed phase are presented. Methods of modeling collisions between particles in turbulent ﬂow are outlined in chapter 4, based on the assumption that compatible PDF of velocities of ﬂuid and particles is correlated Gaussian distribution. To account for anisotropy of particle velocity ﬂuctuations an extension of the Grad method known in kinetic theory of gases is proposed with the goal of accounting for the correlativity of motion of collided particles. Obtained statistical models permit analytical dependences for collision frequency of particles and collision terms in transport equations for the moments of the dispersed phase but do not account for the accumulation effect of particles. Chapter 5 is devoted to statistical descriptions of relative motion of two identical particles in homogeneous isotropic turbulence. The description of relative motion of particles requires invoking two-point statistical characteristics of the turbulence. The approach used here is founded on kinetic equation of PDF for relative velocity of two particles and develops the single–point (single-particle) statistical method outlined in chapters 2 and 3 on two-point (two-particle) one. The phenomenon of clustering in homogeneous turbulence is caused by a migration of particles due to the turbophoresis force in the space of relative motion of two particles. This force tends

Preface

to decrease the distance between particles and thus causes the particles to be attracted to each other due to their interaction with turbulent eddies of carried ﬂow. In chapter 6 the behavior of a bi-dispersed system consisting of particles of two sorts in turbulent ﬂow is considered. An analysis of bi-dispersed systems is of fundamental importance since it may be readily extended to the general case of poly-dispersed systems of particles. To describe characteristics of the dispersed phase, two approaches are invoked widening the statistical models presented in chapters 4 and 5 in the case of bi-dispersed particles. The book is intended for researchers specializing in the mathematical simulation of turbulent ﬂows, dynamics of multiphase media and mechanics of aerosols, as well as for graduate and postgraduate students. Moscow, Leipzig, 2008

L. I. Zaichik V. M. Alipchenkov and E. G. Sinaiski

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XIII

Introduction Theoretical and experimental research of two-phase dispersed turbulent ﬂows is the subject of books by Shraiber et al. (1990), Zhou (1993), Volkov et al. (1994), Gorbis and Spokoyny (1995), Crowe et al. (1998), Varaksin (2003) and reviews by Eaton and Fessler (1994), Elghobashi (1994), McLaughlin (1994), Crowe et al. (1996), Simonin (1996), Zaichik and Pershukov (1996), Loth (2000), Sommerfeld (2000), Mashayek and Pandya (2003). These publications cover a number of topics dealing with hydrodynamics and heat exchange in dispersed turbulent ﬂows. The purpose of the present book is to discuss the motion, heat exchange, dispersion and accumulation of small heavy particles in turbulent ﬂows without the consideration of the feedback action (the modulation effect) of particles on the turbulence. The latter clause means that turbulent characteristics can be taken as predeﬁned and unaffected by the presence of the disperse phase. The particles are assumed to be small (that is to say, their linear size is small as compared to the spatial microscale of turbulence), and sufﬁciently heavy (meaning that the density of particle material is much greater than that of the carrier medium). Motion of small heavy particles in turbulent ﬂows takes place in nature as well as in many technological processes. Examples of such processes include spreading of atmospheric aerosols, formation of raindrops, evolution of cumulus clouds, dynamics of sand storms, combustion of atomized solid and liquid fuel, separation of droplets and aerosols in cyclones, pneumatic transport of coal dust, and so on. Calculation of a two-phase ﬂows should involve the modeling of mass, momentum and heat transport of each phase as well as the interfacial interaction. The main challenges one faces when developing a theory of turbulent ﬂow of two-phase dispersed media stem from the necessity to take into consideration the turbulent operating conditions of the ﬂow and the interaction of particles between each other and with bounding surfaces. First, it should be pointed out that at the present moment, even the development of the theory of single-phase turbulent ﬂows is far from completion, even though the availability of numerous and rather effective models aiming to describe such ﬂows often makes it possible to carry out the required calculations without running into any serious difﬁculties. Although the ﬁrst work devoted to the theory of dispersed turbulent ﬂows was published a long time ago (Barenblatt, 1953), extensive development of this branch of mechanics has started in

XIV

Introduction

earnest only in the last 20 years. The main theoretical problems associated speciﬁcally with the modeling of two-phase dispersed turbulent ﬂows (as opposed to one-phase ﬂows) are due to the following physical phenomena: interaction of particles (drops, bubbles) with turbulent eddies of the continuous phase; mutual interaction of particles during particle collisions; time evolution of the size spectrum of particles as a result of combustion, phase transitions, coagulation and breakage; inﬂuence of turbulent ﬂuctuations on the rate of heterogeneous combustion and phase transitions; interaction of particles with the surface bounding the ﬂow and deposition of particles on this surface; feedback action of particles on the turbulence; dispersion and clustering (accumulation) of particles, and ﬂuctuations of particle concentration. As it we mentioned above, this book does not deal with the issue of the feedback action of particles on the turbulence, which is permissible under the assumption that the concentration of disperse phase in the ﬂow is small. We also disregard the effect of time evolution of the size spectrum of particles. The existing methods of modeling dispersed two-phase turbulent ﬂows can be divided into two groups. The ﬁrst group comprises methods that are based on the Lagrangian trajectory description of the disperse phase, in other words, on solving the Langevin equations for energy and momentum over the trajectories of individual particles, with the subsequent averaging of solutions thus derived over the ensemble of initial conditions. In the framework of this approach, the need to take into account the random character of particle motion that is due to the interaction of particles with turbulent eddies of the carrier ﬂow leads to a signiﬁcant increase of the amount of required calculations, since in order to obtain information that would be statistically accurate, one has to use a sufﬁciently representative ensemble of realizations. A deterministic Lagrangian description of disperse phases motion and heat exchange in a turbulent ﬂow based only on equations for the averaged quantities (i.e., with no consideration of particle interactions with random ﬁelds of velocity and temperature ﬂuctuations of the continuous phase) may be justiﬁed (with some reservations) only for very inertial particles whose relaxation time is much greater than the integral time scale of turbulence, so that they are only weakly involved in turbulent motion of the carrier ﬂow. As we decrease the size of particles, the representative number of realizations should increase because of the increasing contribution of particle interactions with eddies on smaller and smaller scale. The laboriousness of dynamic Lagrangian simulation is ampliﬁed even further when we deal with concentrated dispersed ﬂows and have to contend with the growing role of trajectory-crossing that results from particle collisions and with the variation of the number of particles as a consequence of generation and disappearance of particles in the process of coagulation, breakage, spontaneous nucleation, and so on. The Lagrangian trajectory approach gives us detailed information about the interaction of particles with turbulent eddies, with walls, and with each other, but it requires a considerable expenditure of time as we calculate the complex ﬂows taking place in either natural or industrial conditions. The other group comprises simulation methods based on the Eulerian continual description of both phases. This way of modeling two-phase turbulent ﬂows is known in the mechanics of interpenetrating heterogeneous media as the two-ﬂuid model.

Introduction

One essential advantage of the Eulerian continual approach as opposed to Lagrangian trajectory simulation is that we use the same type of balance equations for both phases and thus can apply a single solution algorithm to the whole system of equations. In addition, an attempt to describe the behavior of very small particles does not cause any major difﬁculties, because when the particle mass tends to zero, the problem under consideration reduces to the problem of turbulent diffusion of zero-inertia particles, that is, of a passive impurity. Moreover, in the framework of the continual approach, accounting for particle collisions and for the variation of the number of particles does not make calculations nearly as time-consuming and complicated as in the framework of Lagrangian simulation. Overall, the Lagrangian trajectory and Eulerian continual simulation methods complement each other. Each method has its advantages and disadvantages and consequently, its own ﬁeld of application. The Lagrangian method is applicable for sufﬁciently non-equilibrium ﬂows (high-inertia particles, rareﬁed dispersed media), while the Eulerian method is preferable for the ﬂows that are close to equilibrium (low-inertia particles, concentrated dispersed media). Since the disperse phase combines the properties of a continuous medium and the properties of discrete particles at the same time, the situation with these two approaches is somewhat similar to the well-known wave-particle duality in the micro-world. Pialat et al. (2005) have proposed a hybrid Lagrangian–Eulerian method combining the detailed description afforded by the Lagrangian approach with effectiveness of the Eulerian approach in describing the state of the disperse phase. The best way to obtain accurate and detailed information about the structure of turbulent two-phase ﬂow is to combine direct numerical simulation (DNS) of the disperse carrier medium with the Lagrangian stochastic approach. Direct numerical simulation can describe the entire spectrum of turbulent eddies, including smallscale eddies that are responsible for the dissipation of turbulent energy. But DNS takes quite a lot of time even when run on the fastest computers. This is why the applicability of the DNS method is usually limited to numerical experiments whose purpose is to validate and calibrate other, more efﬁcient methods of modeling turbulent ﬂows. In the so-called large-eddy simulation (LES) method, direct simulation is conﬁned to large eddies whose spatial scale exceeds the size of the computational grid, while the small-scale (under-grid) modes that lie beyond the resolution limit are described by semi-empirical relations. LES can only be used when simulating the behavior of particles whose dynamic response time is much greater than the time micro-scale of turbulence (Armenio et al., 1999; Boivin et al., 2000; Yamamoto et al., 2001; Kuerten and Vreman, 2005; Fede and Simonin, 2006). This restriction follows from the requirement that the contribution of under-grid ﬂuctuations (that is, of small-scale turbulence) to the disperse phase statistics must be negligible, and interaction of particles with large-scale energy-carrying turbulent eddies must play the primary role. Still, even the combination of LES for the continuum phase and the Lagrangian stochastic approach for the disperse phase can be too extensive for technical applications. Therefore some authors (Druzhinin and Elghobashi, 1998; Ferry and Balachandar, 2001, 2005; Pandya and Mashayek, 2002a; Rani and Balachandar, 2003; Kaufman et al., 2004; Moreau et al., 2005) have

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Introduction

developed other promising methods based on DNS and LES and using the two-ﬂuid formulation. One useful theoretical formalism, which for all intents and purposes forms the foundation of the two above-mentioned methods – direct two-ﬂuid and combined Lagrangian–Eulerian – of simulating particle-laden ﬂows, involves expansion of the particle velocity ﬁeld in the turbulent ﬂow into the correlated and quasiBrownian components (Simonin et al., 2002; Fevrier et al., 2005). The present book outlines several continual models of particle motion, heat transfer, dispersion and clustering in turbulent ﬂows, which are based on the probability density function (PDF) method. A statistical approach utilizing the kinetic equations for the PDF of velocity, temperature, and other characteristics of interest is the most direct way to develop a continual model, that is, to derive a set of hydrodynamic and heat- and mass-exchange equations for the disperse phase. Buyevich (1971a), who employed the Fokker–Planck equation to describe a pseudo-turbulent ﬂow of the disperse phase whose parameter ﬂuctuations are caused by random conﬁguration of particles, appears to be the ﬁrst author who used this line of attack. Introduction of the PDF presents an opportunity to obtain a statistical description of a particle ensemble (rather than dynamic description of individual particles) on the basis of equations of motion and heat transfer, which belong to the Langevin type of equations. However, statistical modeling that employs the PDF results in a partial loss of information concerning the individual features of particle behavior. Nevertheless, incomplete information about the dynamics of particle behavior is compensated by the additional information about the collective behavior of particles – statistical regularities of motion and heat transfer – that applies to the disperse phase as a whole. Application of the statistical method based on kinetic equations provides a uniform description of both interactions of particles with the turbulence and inter-particle interactions resulting from particle collisions. Interaction of particles with turbulent eddies is modeled by a second-order differential operator of the Fokker–Planck type, while inter-particle interactions due to collisions are described by an integral operator of the Boltzmann type. However, for most practical purposes, solving the kinetic equation is not only too arduous a task in terms of computational brainpower involved, but is also redundant because it is sufﬁcient to know the ﬁrst several moments of the PDF to derive the basic macroscopic properties of the ﬂow. In order to construct a closed set of moment equations, that is, to break the inﬁnite chain of equations for statistical moments, we employ different approaches, including the Chapman–Enskog and Grad methods, which lie in the basis of the derivation of all continual hydrodynamic models known today. When modeling particle motion in a rareﬁed dispersive medium, that is, when the volume fraction of the disperse phase is small, our attention should be focused on the interaction of particles with turbulent eddies of the carrier ﬂow, since the role of interparticle interactions is negligible. But the contribution of inter-particle interactions to momentum and energy transport in the disperse phase grows with volume fraction and size of particles. Chaotic motion of particles caused by their interactions has become known as pseudo-turbulence – a term that aims to distinguish this type of motion from the turbulent motion that results from particles participation in turbulent motion of the carrier ﬂow. The initiation of pseudo-turbulent ﬂuctuations

Introduction

may be caused by hydrodynamic interactions between particles, which are realized via the exchange of momentum and energy between particles and random velocity and pressure ﬁelds of the carrier ﬂow (Buyevich, 1972b; Koch, 1990), or by direct inter-particle interactions realized via collisions. With increase of particle fraction and size, momentum and energy exchange between particles via collisions assumes greater importance as compared to the role of hydrodynamic interactions. Thus, in a concentrated disperse medium, the dominant role in the formation of statistical properties belongs to inter-particle interactions, and hence the theoretical analysis of this problem is similar to the analysis of the kinetic problem in the molecular theory of rareﬁed gases (Chapman and Cowling, 1970; Lun et al., 1984; Jenkins and Richman, 1985; Ding and Gidaspow, 1990). The processes of particle–turbulence and particle–particle interactions may be considered as mutually independent only for high-inertia particles, whose dynamic response time is much greater than the characteristic time of their interaction with turbulent eddies. The relative motion of such particles is non-correlated and similar to the chaotic motion of molecules. In the case of low-inertia particles, it is necessary to take into account the interrelation of particle–turbulence and particle–particle interactions. As we pointed out above, the computational difﬁculties associated with Lagrangian trajectory simulation increase rapidly with volume fraction of the disperse phase. This fact, for the most part, is due to the necessity of simultaneous tracing of a large number of particles involved in the problem under consideration. One efﬁcient way to get around this difﬁculty, which was proposed by Oesterle and Petitjean (1991, 1993), Sommerfeld and Zivkovic (1992), and Sommerfeld (1999), is to replace the group of colliding particles by a model particle and introduce the probability density of collisions with ﬁctitious (virtual) particles. Another powerful technique is to simulate particle collisions with the Monte Carlo method (Tanaka et al., 1991; Fede et al., 2002; Moreau et al., 2004). However, in these two approaches, the number of trajectories that one needs in order to obtain a statistically reliable ensemble of realizations increases with the volume fraction of particles. Therefore the range of applicability of Eulerian continual simulation becomes wider as the fraction of the disperse phase increases and inter-particle collisions become more frequent. Formation of clusters, that is, of compact regions with signiﬁcantly higher concentration of the disperse phase surrounded by areas of low concentration, represents one of the most interesting and complex phenomena caused by the interaction of particles with turbulent eddies (Squires and Eaton, 1991c). We should distinguish between two types of ﬂows in which clusters may be formed: inhomogeneous and homogeneous turbulent ﬂows. The phenomenon of clustering (accumulation) of heavy particles in inhomogeneous turbulent ﬂows is explained by the effect of turbulent migration (turbophoresis) from the regions of high intensity of turbulent velocity ﬂuctuations to the regions of low turbulence (Caporaloni et al., 1975; Reeks, 1983). Clustering of inertial particles often takes place in homogeneous turbulence as well, where the gradients of velocity ﬂuctuations of the carrier ﬂow are zero and consequently, particle transport via turbophoresis does not take place in the conventional sense of the word. Fractal dimension of the resulting cluster structures may be less than the dimensionality of the physical space (Bec, 2003, 2005). In the

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majority of known theoretical models proposed for the calculation of particle collision, dispersion, sedimentation, and coagulation in turbulent ﬂows, particle clustering is not taken into account because these models are, as a rule, based on the assumption that particles are distributed in space uniformly and randomly, so that the effect of clustering can be ignored. However, despite the stochastic character of turbulence, the distribution of heavy inertial particles in turbulent ﬂows is not random, and the inertia of particles interacting with coherent vortex structures of the turbulent ﬂow may give rise to some signiﬁcant clustering. To illustrate the effect of clustering, Figure 1 shows the results of direct numerical simulation of instantaneous ﬁelds of particle distribution in homogeneous isotropic turbulence (Reade and Collins, 2000) for different values of Stokes number St, which characterizes particle inertia and is equal to the ratio between the response time of particles and the Kolmogorov time microscale. A local rise of concentration of heavy particles is observed in the regions of low vorticity due to the action of the centrifugal force and is caused primarily by the interaction of particles with small-scale vortex structures. Therefore the effect of clustering is most pronounced when particle response time coincides with the Kolmogorov time microscale of turbulence. The accumulation phenomenon and clustering of inertial particles in response to turbulent ﬂuctuations of concentration can take place in a number of physical processes – from combustion of solid and liquid fuels (Takeuchi and Douhara, 2005) to formation of planets from nebula (Cuzzi et al., 2001). Clustering plays a particularly important role in atmospheric processes at high Reynolds numbers. It

Figure 1 Effect of particle clustering in homogeneous isotropic turbulence (Reade and Collins, 2000).

Introduction

appears that only by taking the effect of clustering into proper consideration when calculating the rate of coagulation can we explain such phenomena as quick growth of droplets in rain-clouds (Pinsky and Khain, 1997; Falkovich et al., 2002) or the difference of actual radiation attenuation in a dusty medium from the one predicted by the Beer–Lambert exponential law (Shaw et al., 2002). One of the main objectives of the present book is to develop a rational approach that would help model the process of particle clustering in inhomogeneous and homogeneous turbulent ﬂows. In particular, as shown by Marchioli and Soldati (2002) and other authors, clustering (segregation) of inertial particles in near-wall ﬂows is caused by their interaction with coherent vortex structures. Hence it would be of interest to know whether the continual model is able to successfully reproduce and take into account the interaction of particles with coherent vortex structures. The main purpose of this book is to demonstrate the application of statistical models based on single-point and two-point PDF to solving physical problems that involve inertial particle motion. It should be pointed out that there exists a large number of phenomena such as, for example, particle clustering, increase of particle sedimentation rate in homogeneous turbulence, or inﬂuence of particles on turbulent energy dissipation, which in principle cannot be described on the basis of conventional single-point Eulerian models. It is evident that as particle inertia grows, this causes an increase of size of the spatial region in which a particle retains its memory of the past in the course of its interaction with turbulent ﬂuid. Therefore application of two-point models to describe inertial particle statistics becomes even more essential than in the theory of single-phase turbulence. A major criterion of validity of the models outlined in the book is their agreement with the known results of DNS and LES of the continuous phase combined with Lagrangian trajectory simulation of the disperse phase. This approach gives us a powerful tool for veriﬁcation of our models, since numerical experiments (in contrast to physical ones) make it possible to study the pure model of the phenomenon in question that is not distorted by extraneous factors. This book is intended for the readers familiar with the fundamentals of hydrodynamics and statistical physics. The book may be viewed as a natural sequel to Sinaiski and Zaichik (2007), which contains a description of statistical microhydrodynamics, that is, of statistical methods in the hydrodynamics of dispersed media consisting of small particles.

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j1

1 Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence The simplest and most extensively studied type of turbulence is statistically homogeneous isotropic turbulence of an incompressible ﬂuid. Consequently, any new models aiming to describe turbulent transfer of momentum or heat should be tested against the case of isotropic turbulence. It is evident that the notion of isotropic turbulence as applied to large-scale turbulence represents a mathematical idealization because such ﬂows are not occurring in nature or in technical devices. Nevertheless, it is well known that small-scale ﬁelds of velocity and temperature at large Reynolds numbers can be thought of as more or less homogeneous and isotropic. Following Monin and Yaglom (1975), we call them locally isotropic. Smallscale vortex structures are responsible for dissipation of turbulent energy and play a key role in the accumulation (clustering) of particles in a turbulent ﬂow. Study of isotropic turbulence is thus of fundamental importance for both single-phase and two-phase ﬂows.

1.1 Characteristics of Homogeneous Isotropic Turbulence

The present section lays out the key characteristics of homogeneous isotropic turbulence in an incompressible ﬂuid, which is a prerequisite for the subsequent discussion of statistical behavior of particles. In the course of this presentation, we shall be using both Lagrangian and Eulerian properties. Lagrangian correlations describe the connections between velocities and other characteristics of turbulence at various points along the mechanical trajectories of ﬂuid elements (ﬂuid particles). A Lagrangian single-particle (single-point) correlation moment of velocity ﬂuctuations in the ﬂuid is determined by BL ij (t) ¼ hu0i (x; t)u0j (R(tt); tt)jR(t) ¼ xi ¼ hu0i u0j iYL (t); hu0i u0j i ¼ u0 dij ; 2

ð1:1Þ where R is the position vector of a ﬂuid particle, YL(t) is the Lagrangian autocorrelation function, hu0i u0j i is the Reynolds stress tensor of the ﬂuid phase, and u0 2 hu0k u0k i=3 is the intensity of velocity ﬂuctuations in the ﬂuid. Here and

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2

afterwards, angle brackets will indicate the averaging over the ensemble of turbulent ﬁelds of velocity and temperature of the carrier ﬂuid. The turbulent diffusion tensor of ﬂuid particles is expressed through the Lagrangian correlation moment as ðt Dij (t) ¼ BL ij (x)dx: 0

At large values of time, the diffusion tensor obeys the asymptotic relation Dij ¼ Dt dij ;

Dt ¼ u0 T L ; 2

Ð1

ð1:2Þ

where Dt is turbulent diffusivity of an inertialess impurity and T L 0 YL (t)dt is the Lagrangian integral time scale. In scientiﬁc literature, the most commonly used approximation for the autocorrelation function is an exponential dependence of the form t YL (t) ¼ exp : ð1:3Þ TL Formula (1.3) is in good agreement with experimental data and with the DNS results at relatively high Reynolds numbers, except for the region of small values of t. The behavior of the autocorrelation function (1.3) in the vicinity of t ¼ 0 is incorrect, since Y0L (0) 6¼ 0. In isotropic turbulence, the Lagrangian integral scale TL may be expressed in terms of kinetic energy of turbulence k ¼ hu0n u0n i=2 and its dissipation rate e. This is accomplished by the relation TL ¼ 4k/3C0e, where C0 is Kolmogorovs constant which, generally speaking, depends on the Reynolds number Re but assumes a constant value C01 when Re is large. The DNS results shown on Figure 1.1 suggest that Kolmogorovs constant can be approximated as

Figure 1.1 Dependence of the Lagrangian integral scale on the Reynolds number: 1 – formula (1.4), 2 – Yeung and Pope (1989), 3 – Yeung (1997), 4 – Yeung (2001), 5 – Fevrier et al. (2001), 6 – Mazzitelli and Lohse (2004).

1.1 Characteristics of Homogeneous Isotropic Turbulence

C0 ¼

C01 Rel ; Rel þ C1

C 01 ¼ 7;

C 1 ¼ 32;

and the Lagrangian integral scale of turbulence divided by the Kolmogorov integral microscale of turbulence depends linearly on the Reynolds number: TL ¼

T L 2(Rel þ C1 ) ¼ : tk 151=2 C01

ð1:4Þ

Following Sawford (1991), we take the asymptotic value of Kolmogorovs constant at Rel ! 1 equal to 7. In the above-listed formulas, Rel (15u0 4/en)1/2 is the Reynolds number calculated for the Taylor microscale; tk (n/e)1/2 is the Kolvogorov time microscale; and n is the kinematic viscosity coefﬁcient of the ﬂuid. In order to describe YL for the entire range of t, including the vicinity of t ¼ 0, one can use the two-scale bi-exponential approximation (Sawford, 1991) 1 2t 2t (1þR)exp (1R)exp ; YL (t) ¼ 2R (1þR)T L (1R)T L R ¼ (12z2 )1=2 ; z ¼

tT ; TL

ð1:5Þ

where tT is the Taylor differential time scale 1=2 2 2Rel 1=2 ¼ tk : tT ¼ 00 YL (0) 151=2 a0

ð1:6Þ

Note that at 2z2 > 1, relation (1.5) may be represented as 1 t @t @t YL (t) ¼ exp 2 @cos 2 þ sin 2 ; @ z TL z TL z TL

@ ¼ (2z2 1)1=2 :

The quantity a0 in Eq. (1.6) represents the dimensionless amplitude of acceleration ﬂuctuations in isotropic turbulence via the relation hai aj i ¼ a0 e3=2 n1=2 d ij . According to the DNS data (Yeung and Pope, 1989; Vedula and Yeung, 1999; Gotoh and Fukayama, 2001) for the low and moderately high Reynolds numbers, to the experimental results of Voth et al. (2002) for the axial and transverse components of acceleration ﬂuctuations, and to the experimental studies at relatively high Reynolds numbers in the 900 to 2000 range (Voth et al., 1998), the dependence of a0 on Rel (see Figure 1.2) can be approximated by a0 ¼

a01 þ a01 Rel ; a02 þ Rel

a01 ¼ 11;

a02 ¼ 205;

a01 ¼ 7:

ð1:7Þ

Eulerian correlations express the connection between the parameters of a turbulent medium at ﬁxed spatial points. Thus, the Eulerian space-time correlation moment of ﬂuid velocity ﬂuctuations is deﬁned as follows: BE ij (r; t) ¼ hu0i (x; t)u0j (xr; tt)i:

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Figure 1.2 Dependence of a0 on the Reynolds number: 1 – formula (1.7); 2 – a0 ¼ 0.052 (Vedula and Yeung, 1999); 3 – a0 ¼ 7 (Voth et al., 1998); 4 – Yeung and Pope (1989); 5 – Vedula and Yeung (1999); 6 – Gotoh and Fukayama (2001); 7 – axial component (Voth et al., 2002); 8 – transverse component (Voth et al., 2002); 9 – Bec et al. (2006); 10 – Yeung et al. (2006).

Here and later, the Eulerian correlations are deﬁned in a coordinate system moving with the average velocity of the medium. The most common convention is to represent the second-order space-time correlation moment as a product of spatial and temporal correlations: BE ij (r; t) ¼ Bij (r)YE (t); Bij (r) ¼ hu0i (x; t)u0j (xr; t)i;

Bij (0) ¼ u0 2 dij ;

ð1:8Þ

where Bij(r) is the Eulerian two-point simultaneous correlation moment and YE(t) is the Eulerian single-point time autocorrelation function of velocity ﬂuctuations. In homogeneous, isotropic turbulence, any second-rank tensor can be represented as follows (Monin and Yaglom, 1975): Mij (r; t) ¼ M nn (r; t)dij þ [M ll (r; t)Mnn (r; t)]

rirj ; r2

ð1:9Þ

where Mll and Mnn are the longitudinal and transverse (with respect to the position vector r) components of the tensor, and r |r| is the distance between the two points. For an isotropic solenoidal ﬁeld such as the velocity ﬁeld u(x)of an incompressible ﬂuid, there exists the following relation between the longitudinal and transverse components of this tensor: r qMll (r; t) Mnn (r; t) ¼ Mll (r; t) þ : ð1:10Þ 2 qr According to Eq. (1.9) and Eq. (1.10), Bij(r) may be written as rirj r dF(r) 2 Bij (r) ¼ u0 G(r)dij þ [F(r)G(r)] 2 ; G(r) ¼ F(r) þ ; ð1:11Þ r 2 dr where F(r) and G(r) are the longitudinal and transverse Eulerian spatial correlation functions.

1.1 Characteristics of Homogeneous Isotropic Turbulence

Among the various common approximations of the Eulerian spatial and temporal correlation functions, the simplest ones are the exponential dependences r t F(r) ¼ exp ; YE (t) ¼ exp ; ð1:12Þ L TE characterized by integral spatial and temporal scales L and TE. The transverse spatial correlation that corresponds to Eq. (1.12) is r r exp : ð1:13Þ G(r) ¼ 1 2L L As it follows from Eq. (1.13), the transverse correlation becomes negative at large values of r. The existence of the negative tail is necessitated by the requirement that mass should be conserved in accordance with the continuity equation. To provide a description of turbulent ﬁelds that goes beyond the correlation moments, it is useful to introduce the so-called structure functions characterizing spatial and temporal increments of velocity or temperature. The Eulerian spatial structure function of the second order is deﬁned as Sij (r) ¼ hDu0i (r)Du0j (r)i ¼ h(u0i (x þ r; t)u0i (x; t))(u0j (x þ r; t)u0j (x; t))i ¼ 2(hu0i u0j iBij (r)):

ð1:14Þ

To describe the dispersion of a ﬂuid particle pair, we introduce the Lagrangian twoparticle (aka two-point) correlation moment and the Lagrangian structure function, BL ij (r; t) ¼ hu0i (R1 (t); t)u0j (R2 (tt); tt)i; R1 (t) ¼ x; R2 (t) ¼ x þ r; SL ij (r; t) ¼ h(u0i (R2 (t); t)u0i (R1 (t); t))(u0j (R2 (tt); tt)u0j (R1 (tt); tt))i ¼ 2(BL ij (t)BL ij (r; t)): The Lagrangian two-point correlation moment is associated with the Lagrangian single-point and Eulerian two-point correlation moments through the self-evident relations BL ij (0; t) ¼ BL ij (t); BL ij (r; 0) ¼ BL ij (r):

ð1:15Þ

The relative diffusion tensor of two ﬂuid particles may be represented as an integral of Lagrangian two-point correlations (Lundgren, 1981): ðt ðt Drij (r; t) ¼ 2 [BL ij (t1 )BL ij (r; t1 )] dt1 ¼ SL ij (r; t1 ) dt1 : 0

ð1:16Þ

0

A Lagrangian two-point correlation, in its turn, may be written as (Zaichik and Alipchenkov, 2003) BL ij (r; t) ¼ BL ij (t) þ [Bij (r)Bij (0)]YLr (tjr);

ð1:17Þ

where YLr (t|r) is the Lagrangian autocorrelation function characterizing the relative motion of two particles initially separated by the distance r. It is easy to see that once we require YLr (0) ¼ 1, expression (1.17) obeys Eq. (1.15).

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Approximation (1.17) makes it possible to represent the Lagrangian two-point structure function of velocity ﬂuctuations in the form of a product: SL ij (r; t) ¼ Sij (r)YLr (tjr);

ð1:18Þ

from which it follows that YLr(t|r) is a Lagrangian autocorrelation function of velocity ﬂuctuation increment of ﬂuid particles separated by a distance r. Substitution of Eq. (1.18) into Eq. (1.16) leads to the following formula for components of the relative diffusion tensor at large values of time: Drij (r) ¼ Sij (r)T Lr ; ð1:19Þ Ð1 where T Lr 0 YLr (t)dt is the two-point integral time scale characterizing the velocity ﬂuctuation increment of the two particles. By analogy with Eq. (1.3), the autocorrelation function of velocity ﬂuctuation increment can be approximated by an exponential dependence: t : ð1:20Þ YLr (tjr) ¼ exp T Lr Alternatively, following Eq. (1.5), we can approximate it by a two-scale bi-exponential dependence: 1 2t 2t YLr (t) ¼ (1 þRr )exp (1Rr )exp ; 2Rr (1 þRr )T L (1Rr )T L 1=2 tTr ; zr ¼ ; Rr ¼ 1;2z2r ð1:21Þ T Lr where tTr is the Taylor differential time scale of relative velocity of two ﬂuid particles. Let us now consider the behavior of the structure function Sij, the coefﬁcient of relative diffusion Drij , and the integral time scale of velocity ﬂuctuation increment TLr in the viscous, inertial, and external spatial intervals (in that order) within the framework of Kolmogorovs similarity hypothesis for small-scale turbulence (Monin and Yaglom, 1975). This hypothesis establishes universality of small-scale turbulence in the sense that the characteristics of turbulence in the viscous and inertial intervals at large Reynolds numbers are independent of the large-scale vortex structure. Such an assumption is permissible only if we disregard the intermittency of turbulence arising from the ﬂuctuations of the rate of turbulent energy dissipation (Monin and Yaglom, 1975; Kuznetsov and Sabelnikov, 1990; Pope, 2000). In the viscous interval (r h, where h (n3/e)1/4 is the Kolmogorov spatial microscale), the ﬁrst terms of Taylors expansion of the Eulerian longitudinal and transverse structure functions are equal to (Monin and Yaglom, 1975): Sll ¼

er 2 ; 15n

Snn ¼

2er 2 : 15n

ð1:22Þ

At small values of r, the difference of velocity ﬂuctuations at two points may be represented as a linear function of the vector connecting these points, namely,

1.1 Characteristics of Homogeneous Isotropic Turbulence

Du0i (r; t) ¼ u0i (x þ r; t)u0i (x; t) ¼ g ij (t)r j ;

ð1:23Þ

where g ij qu0i =qx j is the velocity ﬂuctuation gradient. In an isotropic linear ﬁeld, the correlation functions of the strain and rotation tensors have the form (Girimaji and Pope, 1990; Brunk et al., 1998) e 2 t hsik (x; t)sjn (x þ r; t)i¼ ; d ij dkn þ d in djk d ik djn exp 20n 3 ts sij ¼ hwik (x; t)wjn (x þ r; t)i ¼

g ij þ g ji 2

;

ð1:24Þ

e t (dij d kn d in djk )exp ; 12n tw

wij ¼

g ij g ji 2

:

As it follows from Eq. (1.24), the strain and rotation correlation functions decrease exponentially, their respective characteristic times ts and tw being proportional to the Kolmogorov microscale tk. Expressions for the Lagrangian two-point correlation functions can be derived from Eq. (1.23) and Eq. (1.24) provided that the distribution of the distance vector between the points ri and the distribution of the tensor of velocity ﬂuctuation gradients g ij are statistically independent: SL ll ¼

er 2 t exp ; 15n ts

SL nn ¼

er 2 1 t 1 t þ exp : ð1:25Þ exp 4n 5 ts 3 tw

By substituting Eq. (1.25) into Eq. (1.16) we obtain the longitudinal and transverse components of relative diffusion of two ﬂuid particles of the continuum (Brunk et al., 1997): ets r 2 e ts tw 2 r r ; Dnn ¼ þ r : ð1:26Þ Dll ¼ 15n 3 4n 5 At tw ¼ ts the relation (1.26) is consistent with the expression for the coefﬁcient of relative diffusion derived for the viscous interval by Lundgren (1981). On the other hand, from Eq. (1.19) and Eq. (1.22) there follows Drll ¼

eT Lr r 2 ; 15n

Drnn ¼

2eT Lr r 2 : 15n

ð1:27Þ

Comparing Eq. (1.26) and Eq. (1.27), we see that both expressions coincide at TLr ¼ tw ¼ ts . Consequently, in the viscous interval, the integral time scale of velocity ﬂuctuation increment TLr is equal to T Lr ¼ ts ¼ A1 tk

ð1:28Þ pﬃﬃﬃ Lundgren (1981) theoretically obtained the value 5 for the constant A1, which is in good agreement with the value 2.3 obtained by Girimaji and Pope (1990) by the DNS method.

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Consider now the behavior of characteristics of turbulence in the continuous phase in the inertial interval (h r L), where the effect of viscosity is negligible and the particulars of large-scale convection do not play any noticeable role. The well-known similarity hypothesis proposed by Kolmogorov leads to the following self-similar representation of second order structure functions: Sll ¼ C(er)2=3 ;

4 Snn ¼ C(er)2=3 ; 3

ð1:29Þ

where C 2.0 according to Monin and Yaglom (1975) and Sreenivasan (1995). It can be shown from similarity considerations that in the inertial interval, one can construct only one time scale of the order e1/3r 2/3, so the time scale TLr should be taken as T Lr ¼ A2 e1=3 r 2=3 ;

A2 ¼ const:

ð1:30Þ

In order to determine the constant A2, let us recall the relations for the third-order Eulerian structure function. In the case of isotropic turbulence, any tensor of the third rank may be presented in the following form (Monin and Yaglom, 1975): rirjrk rj ri rk d d d þ M (r; t) þ þ Mijk (r; t) ¼ Mlll (r; t)3Mlnn (r; t) : ij lnn jk ik r3 r r r For an isotropic solenoidal ﬁeld, the relation 1 qMlll (r; t) Mlnn (r; t) ¼ Mlll (r; t) þ r ; 6 qr holds, indicating that the third-rank tensor Mijk(r, t) is determined by just one longitudinal component Mlll(r, t). Third-order structure functions may be approximately expressed in terms of second-order structure functions via the relations (Zaichik and Alipchenkov, 2004, 2005) qSjk qSij T Lr qSik þ Sjn þ Skn Sin Sijk ¼ hDu0i Du0j Du0k i ¼ : ð1:31Þ 3 qr n qr n qr n Because of Eq. (1.31), the longitudinal third-order structure function of continuous phases velocity is equal to Slll ¼ T Lr Sll

dSll : dr

ð1:32Þ

In view of Eq. (1.29) and Eq. (1.30), it follows from Eq. (1.32) that 2 Slll ¼ A2 C 2 er: 3

ð1:33Þ

Then, taking into account the well-known Kolmogorovs relation for the inertial interval (Monin and Yaglom, 1975), we get 4 Slll ¼ er; 5

ð1:34Þ

1.1 Characteristics of Homogeneous Isotropic Turbulence

and comparing Eq. (1.33) with Eq. (1.34), ﬁnally obtain 2A2C2/3 ¼ 4/5, whence A2 ¼ 0.3 at C ¼ 2.0. At relatively large distances between the two points, ﬂuctuations of velocity at these points are statistically independent. Thus correlation functions vanish in the external interval (r > L), and structure functions become equal to Sll ¼ Snn ¼ 2u0 : 2

ð1:35Þ

Moreover, at large r the two-point time scale converts to the ordinary Lagrangian integral time scale T Lr ¼ T L ;

ð1:36Þ

and the tensor of relative (binary) diffusion is deﬁned by the expression Drij ¼ 2u0 T L dij ; 2

ð1:37Þ

In other words, it equals two times the turbulent diffusion tensor of individual ﬂuid particles (1.2). To provide a continuous description of the longitudinal structure function of velocity ﬂuctuations for the entire range of distances r between the two particles, we shall resort to the approximation proposed by Borgas and Yeung (2004), which combines relations (1.22), (1.29), and (1.35) for the viscous, inertial, and external spatial intervals: 4=3 1=6 r 153r 4 r 2 Sll ¼ 2u0 1exp ; r ¼ : 3 4 6 3=4 h 15 r þ (2Rel =C) (15C) ð1:38Þ The transverse structure function is expressed through the longitudinal one according to Eq. (1.10): Snn ¼ Sll þ

r qSll : 2 qr

ð1:39Þ

Figure 1.3 presents the longitudinal and transverse structure functions calculated by formulas (1.38) and (1.39) at C ¼ 2 and Rel ¼ 61. For comparison, we also provide structure functions obtained by Ten Cate et al. (2004) from the DNS with the help of the Lattice Boltzmann method. It is obvious that the longitudinal structure function tends monotonously to the limiting value of two times the intensity of velocity ﬂuctuations. On the other hand, the behavior of the transverse structure function, which tends to 2u0 2, is not monotonous. The maximum in the distribution of Snn manifests the presence of a negative loop in the distribution of the transverse spatial correlation function G(r). The integral time scale of velocity ﬂuctuation increment may be determined from the approximation similar to Eq. (1.38) that interpolates the relations (1.28), (1.30), and (1.36) for the corresponding characteristic intervals:

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Figure 1.3 Longitudinal (1, 2) and transverse (3, 4) Eulerian structure functions of the second order: 1 – Eq. (1.38); 3 – Eqs. (1.38)–(1.39); 2, 4 – Ten Cate et al. (2004).

T Lr

3=2 2=3 1=6 r 4 A2 r ¼ T L 1exp ; A1 r 4 þ ðTL =A2 )6

TL ¼

TL : tk ð1:40Þ

The Taylor time microscale of velocity ﬂuctuation increment tTr can be found from the assumption that the ratio between the macro- and the microscale is independent of the distance r, that is, tTr ¼

tT T Lr : TL

ð1:41Þ

A Lagrangian single-particle correlation moment of temperature ﬂuctuations in the continuous phase is deﬁned as BLt (t) ¼ hq0 (x; t)q0 (R(tt); tt)jR(t) ¼ xi ¼ hq0 iYLt (t); 2

ð1:42Þ

where YLt(t) is the Lagrangian autocorrelation function of temperature ﬂuctuations, and hq0 2i is the intensity of temperature ﬂuctuations in a turbulent ﬂuid. The Eulerian space-time correlation moment of temperature ﬂuctuations can be represented as the product of spatial and temporal correlations in the manner similar to Eq. (1.8), namely, BEt (r; t) ¼ hq0 (x; t)q0 (xr; tt)i ¼ hq0 iF t (r)YEt (t); 2

ð1:43Þ

where Ft(r) is the Eulerian two-point simultaneous correlation function of temperature ﬂuctuations and YEt(t) is the Eulerian single-point time autocorrelation function of temperature ﬂuctuations. By analogy with Eq. (1.12), the Eulerian spatial and time correlation functions are often approximated by exponential dependences: r t ; YEt (t) ¼ exp t ð1:44Þ F t (r) ¼ exp Lt TE with the appropriate integral spatial and temporal scales Lt and TEt.

1.2 Motion of a Single Particle and Heat Exchange

1.2 Motion of a Single Particle and Heat Exchange

The subject of the present book is the behavior of small solid spherical particles in a turbulent ﬂow. The density of particles is assumed to be much greater than that of the continuous carrier phase (ﬂuid), and the size of particles is assumed not to exceed the Kolmogorov spatial microscale. In this case equations of motion for the particles and for the carrier ﬂow can be written in the point-force approximation, with forces applied to the particles centers of mass (Boivin et al., 1998; Burton and Eaton, 2005). In addition, the behavior of particles in a turbulent medium under the postulated conditions is controlled primarily by the force of hydrodynamic resistance. As a rule, this force acts as the primary mechanism responsible for setting the particles in motion on the one hand, and for the opposite effect – decelerating or accelerating inﬂuence of the particles on the carrier ﬂuid ﬂow – on the other. Since the continuous phase density rf is much smaller than the density rp of particle matter, it is safe to disregard the forces arising from non-stationarity or inhomogeneity of the motion (the virtual mass effect), forces due to the acceleration or deceleration of the carrier ﬂow (the displaced mass effect), and the Basset force, which is due to the memory effect (Maxey and Riley, 1983; Michaelides, 2003). A careful study of the inﬂuence of these forces on the turbulent motion of particles for a wide range of density ratios, 2.65 rp/rf 2650 has been undertaken by Armenio and Fiorotto (2001), who used the DNS method. This study showed that in the entire considered range of rp/rf , the contribution of the virtual mass effect to the balance of forces acting on a particle is negligible, whereas the contributions of the ﬂow acceleration effect and the memory effect may be noticeable. Nevertheless, the inﬂuence of the two corresponding forces on the dispersion of particles in a turbulent ﬂow is still negligible compared to the resistance force, even at rp/rf ¼ O(1), and therefore need not be taken into account. The various effects resulting from the rotation of particles fall outside the scope of this book; bear in mind, however, that these effects can be essential for the motion of relatively large particles. Under these assumptions, the motion of a single heavy particle is described by the equation dRp ¼ vp ; dt

ð1:45Þ

dvp u(Rp ; t)vp ¼ þ F; dt tp

ð1:46Þ

where Rp and vp are the position and velocity of the particle and u(Rp, t) is the velocity of the ﬂuid at the point x ¼ Rp (t). The ﬁrst term on the right-hand side of Eq. (1.46) is the hydrodynamic resistance force the viscous ﬂuid exerts on the particle, and tp is the characteristic time of dynamic relaxation for the particle: tp ¼

tp0 ; j(Rep )

tp0 ¼

rp d2p 18rf n

;

ð1:47Þ

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where tp0 is the same relaxation time calculated using the Stokes approximation (that is, at Rep ! 0), dp is the particles diameter, and Rep dp |u vp|/n is the Reynolds number of the ﬂow that goes around the particle. The function j(Rep) in Eq. (1.47) describes the effect of the inertia force on hydrodynamic resistance of a spherical particle. One can ﬁnd in literature a lot of formulas approximating the standard resistance curve for a spherical particle. The most commonly used one is the Schiller–Neumann approximation (Clift et al., 1978) 1 þ 0:15Re0:687 at Rep 103 p j(Rep ) ¼ ð1:48Þ 0:11Rep =6 at Rep > 103 : If the size of the particle is much smaller than the spatial microscale of turbulence, the effect of velocity ﬂuctuations on the particles hydrodynamic resistance is nonexistent. Bagchi and Balachandar (2003) performed the DNS for a ﬂow that bypasses relatively large particles (whose diameter is 1.5–10 times greater than the Kolmogorov microscale) to see how turbulence affects particle resistance. It was shown that turbulent ﬂuctuations have but a small effect on the average resistance force described by the standard correlation (1.48). But when approximation (1.48) is applied to predict the instantaneous resistance force, its accuracy falls with increase of particle size. This result proves that when studying particle interaction with turbulent eddies of the continuous phase using the point-force approximation, it is necessary to satisfy the condition dp < h. Moreover, in the case of a ﬂow past a large particle at high Reynolds numbers, interaction of the wake formed behind the particle with turbulent eddies of the surrounding medium assumes increased importance (Wu and Faeth, 1994, 1995 Pan and Banerjee, 1997; Bagchi and Balachandar, 2004; Legendre et al., 2006). A thorough analysis of these effects is beyond the scope of present book. The second term on the right-hand side of Eq. (1.46) denotes a force of different physical nature – for example, gravity. In near-wall ﬂows, which are typically characterized by large gradients of all ﬂow parameters, one needs to be aware of the lift force (Saffman force) caused by the velocity shear (Saffman, 1965, 1968). But with increase of the density ratio rp/rf between the disperse and continuous phases, the role of the lifting force diminishes. In the case of a non-isothermal ﬂow, the motion of very small sub-micron particles near a heating or cooling surface may be strongly inﬂuenced by the thermophoretic force directed toward the cooler medium. Hence the symbol F encompasses not only the external mass force such as gravity, but also some other forces (Saffman force, thermophoretic force and so on). As opposed to the resistance force, the force F is considered deterministic because possible ﬂuctuations of parameters entering the expression for F are usually neglected, as the effect of these ﬂuctuations is, with rare exceptions, insufﬁcient to make an appreciable difference. When studying heat exchange, thermal inhomogeneity on the particle-size scale can be ignored in the majority of applied problems. Then, if we ignore heat exchange via radiation, the temperature change of a single particle is described by the equation

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

dqp q(Rp ; t)qp þ Q; ¼ dt tt

ð1:49Þ

where qp is particle temperature, q(Rp, t) – temperature of the ﬂuid at the point x ¼ Rp (t), and tt – the particles characteristic time of thermal relaxation. The ﬁrst term on the right determines the interfacial heat exchange that is taking place via conductive and convective mechanisms of heat transport in a ﬂuid. The quantity Q denotes the intensity of heat release inside the particle (e.g., as a result of combustion). The thermal relaxation time for the particle is found from the relation tt ¼

C p rp d2p 6lNup

;

ð1:50Þ

where Cp is heat capacity of the particles material and is l the coefﬁcient of heat conductivity of the ﬂuid. To calculate the Nusselt number Nup of the ﬂow past the particle, which enters Eq. (1.50), one can use a well-known relation (Ranz and Marshall, 1952) that is applicable for a wide range of Rep: 1=3 ; Nup ¼ 2 þ 0:6Re1=2 p Pr

where Pr is the Prandtl number of the ﬂuid.

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

The behavior of particles in a turbulent ﬂow is governed by their interactions with turbulent eddies of the continuous phase which these particles encounter on their way. Therefore any description of statistical characteristics of the disperse phase in a turbulent ﬂuid is critically dependent on the correlations of velocity and temperature of the ﬂuid along inertial particle trajectories. It is obvious that these correlations coincide with the corresponding Lagrangian correlations for ﬂuid particles in the limiting case of inertialess particles, that is, at tp ! 0. On the other hand, in the case of highly inertial particles that show weak response to turbulent ﬂuctuations of the continuous phase, correlations along particle trajectories should coincide with the corresponding Eulerian correlations in the ﬂuid, which express the statistical connection between ﬂuctuations of parameters at ﬁxed spatial points. Hence, in order to ﬁnd velocity and temperature correlations in the ﬂuid along inertial particle trajectories, it is necessary to know the relations between Lagrangian and Eulerian correlation moments in a turbulent ﬂow. This problem is closely linked to the problem of diffusion (dispersion) of a passive impurity (Lumley, 1962; Saffman, 1963; Kraichnan, 1964, 1970; Philip, 1967; Phythian, 1975; Lundgren and Pointin, 1976; Weinstock, 1976; Lundgren, 1981; Lee and Stone, 1983; Middleton, 1985; Squires and Eaton, 1991a; Kontomaris and Hanratty, 1993; Hesthaven et al., 1995; Stepanov, 1996). Derivation of theoretical relations between Lagrangian and Eulerian

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14

correlations is facilitated by Corrsins independence conjecture (Corrsin, 1959) – a hypothesis about independent averaging of random ﬁelds of particle displacements and Eulerian velocity ﬂuctuations. Error analysis and the domain of applicability of Corrsins independence conjecture is the subject of Weinstocks paper (1976). On the basis of this conjecture, Reeks (1977), Pismen and Nir (1978), and Nir and Pismen (1979) have established the correlations between ﬂuid velocity ﬂuctuations along particle trajectories and derived closed theoretical solutions for the problem of dispersion of heavy particles in a turbulent medium. The obtained solutions allow to describe the effect of diminishing correlativity of particle ﬂuctuations with increase of the average velocity slip (i.e., with increase of the drift velocity of particles relative to the ﬂuid) and the so-called crossing trajectory effect (Yudine, 1959; Csanady, 1963), and also to account properly for the inﬂuence of particle inertia on turbulent diffusion in the absence of average drift (the so-called inertia effect). Similar problems were later considered theoretically by Shraiber et al. (1990), Mei et al. (1991), Wang and Stock (1993), Mei and Adrian (1995), Stock (1996), Etasse et al. (1998), Pozorski and Minier (1998), Zaichik and Alipchenkov (1999), Derevich (2001), Gribova et al. (2003), Graham (2004). Numerical studies of particle dispersion in isotropic stationary and decaying turbulence by the DNS and LES methods were performed by Riley and Patterson (1974), Deutsch and Simonin (1991), Squires and Eaton (1991b), Yeh and Lei (1991), Elghobashi and Truesdell (1992), Mashayek et al. (1997), and experimental study of velocity correlations and turbulent diffusion of heavy particles is the subject of works by Snyder and Lumley (1971) and Wells and Stock (1983). The purpose of the present section is to work out a simple model that would result in suitable analytical expressions for velocity and temperature correlations in a turbulent ﬂuid along inertial particle trajectories. Lagrangian correlation moment of velocity ﬂuctuations of an element of the continuous phase (i.e., ﬂuid particle) calculated along an inertial particle trajectory has the following form (Reeks, 1977): BLp ij (t) ¼ u0 2 YLp ij (t) ¼ hu0i (x; t)u0j (Rp (tt); tt)jRp (t) ¼ xi Ð ¼ hu0i (x; t)u0j (xr; tt)d(rs(t))idr; 0

s ¼ Sþs ;

S ¼ Wt;

ðt

s ¼ vp0 (Rp (t1 ))dt1 : 0

ð1:51Þ

0

Here YLp ij(t) is the Lagrangian autocorrelation function of ﬂuid velocity along the particle trajectory and Rp is the position vector of a point on that trajectory. Displacement s of a particle relative to the moving ﬂuid is the sum of two components, which arise from two independent processes. The ﬁrst component is due to the drift, which is characterized by the drift velocity W (i.e. particles velocity relative to the average velocity of the surrounding medium), and the second (ﬂuctuational) component arises as a result of the particles involvement in turbulent motion. Note that, unlike the scalar Lagrangian function YL (t), the autocorrelation function of ﬂuid velocity ﬂuctuations YLp ij(t) is a tensor, because ﬂuid velocity ﬁeld along the particle trajectory is non-isotropic in view of the particles average drift relative to the ﬂuid.

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

In order to calculate the integral (1.51), we must employ Corrsins hypothesis about the possibility of independent statistical averaging of random ﬁelds of particle displacements and Eulerian velocity ﬂuctuations. In accordance with this hypothesis, we obtain the following: hu0i (x; t)u0j (xr; tt)d(rs(t))i ¼ hu0i (x; t)u0j (xr; tt)if(r; t); f(r; t) ¼ hd(rs(t))i;

ð1:52Þ

where f(r, t)is the probability density of particle displacement r at the moment t. The quantity f(r, t) usually has a Gaussian distribution in either the frequency space or the coordinate space, with the variance expressed through a Lagrangian autocorrelation function. Then Eq. (1.51) becomes non-linear and implicit because it contains YLp ij(t) on both the left-hand side and the right-hand side. Therefore its solution can be obtained only by numerical or iterative methods. In order to avoid the iteration procedure and still obtain a simple explicit expression for YLp ij(t) that could be easily used in further calculations, we shall take the probability density of particle displacement in the form of a d-function: f(r; t) ¼ d(rs(t)):

ð1:53Þ

Substituting Eq. (1.52) and Eq. (1.53) into Eq. (1.51) and taking into account the relations (1.8) and (1.11) for the Eulerian space-time correlation moment in isotropic turbulence, we arrive at the following expression for the Lagrangian autocorrelation function of ﬂuid velocity ﬂuctuations along the particle trajectory: hsi sj i YLp ij (t) ¼ G(s)d ij þ [F(s)G(s)] 2 YE (t); s ð1:54Þ hsi sj i ¼ W i W j t2 þ hs0i s0j i; s ¼ hsk sk i1=2 : The ﬂuctuational component of particle displacement is estimated by approximate integration of the equations of motion (1.45) and (1.46): s0 ¼

1 tp

ðt tð1 00

t1 t2 t u0 (Rp (t2 ))exp dt2 dt1 u00 t þ tp exp 1 : tp tp ð1:55Þ

Let us set the characteristic value of velocity ﬂuctuations in Eq. (1.55) equal to the root-mean-square value of ﬂuctuational velocity u0 : ju00 j ¼ u0 : Then relations (1.55) and (1.56) give us the following: u0 2 Y2 (t) t hs0i s0j i ¼ 1 : dij ; Y(t) ¼ t þ tp exp 3 tp

ð1:56Þ

ð1:57Þ

It is easily seen that approximation (1.54) together with Eq. (1.57) accounts for both the crossing trajectory effect, which is caused by the drift velocity W, and the inertia

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effect, which is characterized by the particle response time tp. Next, for the Eulerian temporal and spatial longitudinal correlation functions in Eq. (1.54) we shall take their exponential dependences (1.12). In doing so, we must keep in mind that these single-scale approximations, are, strictly speaking, justiﬁed only in the limit of high Reynolds numbers. But they can be adapted for use at ﬁnite Reynolds numbers by taking into account the dependence of the integral scales on Rel. Substitution of Eq. (1.12), Eq. (1.13), and Eq. (1.57) into Eq. (1.54) yields the following expressions for the autocorrelation tensor of ﬂuid velocity ﬂuctuations along the particle trajectory:

m 3g 2 t2 ei ej (3g 2 t2 þ 2y2 (t))dij YLp ij (t) ¼ dij þ 6(g 2 t2 þ y2 (t))1=2 T E ð1:58Þ t þ m(g 2 t2 þ y2 (t))1=2 exp ; TE where m TEu0 /L is the structure parameter of turbulence, g W/u0 – the drift parameter, W |W| – the absolute value of the drift velocity, ei Wi/W – components of the unit vector that points in the direction of the drift velocity. In the absence of drift (g ¼ 0), the tensor YLp ij(t) becomes isotropic, and YLp ij(t) YLp(t)dij, where in accordance with Eq. (1.58), my(t) t þ my(t) exp : ð1:59Þ YLp (t) ¼ 1 3T E TE In the limiting case of inertialess particles (tp ! 0), it follows from Eq. (1.59) that mt (1 þ m)t exp : ð1:60Þ YL (t) ¼ 1 3T E TE Formula (1.60) represents the Lagrangian autocorrelation function of ﬂuid particle velocity ﬂuctuations in terms of Eulerian variables, in other words, it is characterized by the Eulerian integral time scale TE and by the structure parameter m. From Eq. (1.60) there follows a simple relation between the Lagrangian and Eulerian temporal macroscales: TL 3 þ 2m ¼ ; T E 3(1 þ m)2

ð1:61Þ

which shows that the ratio of these scales, TL/TE, depends on structure parameter m, cannot exceed unity, tends to unity at m ! 0 and falls off with increase of m. Such behavior of TL/TE is in complete agreement with the results obtained by other authors, for example, Philip (1967), Lee and Stone (1983), Middleton (1985), Wang and Stock (1993), Stepanov (1996), Derevich (2001). Experimental data for isotropic grid turbulence give TLu0 /L 0.3 0.6 (Sato and Yamamoto, 1987). The DNS results predict TL/TE ¼ 0.72 0.06, m 1 (Yeung and Pope, 1989) and TL/TE 0.75, m 0.7 (Mazzitelli and Lohse, 2004) for stationary turbulence, and TL/TE 0.82 (Squires and Eaton, 1991a) for decaying turbulence. The method of kinematic simulation, which

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

Figure 1.4 Ratio of Lagrangian and Eulerian time macroscales vs. structure parameter of turbulence: 1 – (1.61); 2 – Philip (1967); 3 – Lee and Stone (1983); 4 – Stepanov (1996); 5 – Derevich (2001), 6 – Fung et al. (1992); 7 – Wang and Stock (1993); 8 – Oesterle (2004); 9 – Mazzitelli and Lohse (2004).

approximates non-stationary velocity ﬁeld of the continuum by a superposition of random Fourier modes, gives TL/TE ¼ 0.53 1.11, m ¼ 0.5 0.88 (Fung et al., 1992) and TL/TE 0.4, m 1.3 1.4 (Oesterle, 2004). Hence the scale ratio TL/TE in extensively studied isotropic ﬂows usually varies from 0.3 to 0.8 and the structure parameter is m 1. This is why Wang and Stock, 1993 adduce the results of calculations for m ¼ 1. For m ¼ 1, formula (1.61) gives TL/TE ¼ 0.417, which is slightly above the value TL/TE ¼ 0.356 given by Wang and Stock (1993). Figure 1.4 illustrates the correlation of Eq. (1.61) with theoretical dependences as well as the results of numerical calculations by other authors. It should be noted that the inequality TL
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Figure 1.5 Influence of the structure parameter of turbulence on the Lagrangian autocorrelation function: 1 – m ¼ 0 (1.3); 2 – Eq. (1.63) at m ¼ 1; 3 – m ! 1, i.e., Eq. (1.63).

at m ! 1, the maximum negative value of YL is equal only to 0.0061 at t/TL ¼ 6. Therefore the exponential dependence (1.3) is a good approximation for the Lagrangian autocorrelation function of velocity ﬂuctuations of ﬂuid (inertialess) particles at all possible values of structure parameter m. In view of expression (1.59) for the autocorrelation function, the integral scale of ﬂuid velocity ﬂuctuations along inertial particle trajectories in the absence of average slip (drift) of particles relative to the ﬂuid is given by 1 ð

1 ð

T Lp ¼

YLp (t) dt ¼ 0

1 0

my(t) t þ my(t) exp dt: 3T E TE

ð1:64Þ

The quantity TLp may be considered as the duration of particle interactions with energy-carrying turbulent eddies. It follows from Eq. (1.64) that TLp is determined by two parameters: the Stokes number StE tp/TE, which characterizes particle inertia with respect to the time macroscale of turbulence, and the structure parameter m. As one can see from Figure 1.6, the value of TLp increases monotonously from the Lagrangian to the Eulerian macroscale as the Stokes number StE grows from 0 to 1. It should be noted that the quantity u0 y(t) represents the effective free path of a particle undergoing ﬂuctuational motion. Decrease of the effective free path with increase of the Stokes number explains why the correlation between turbulent characteristics of ﬂuid particles moving along inertial particle trajectories changes from Lagrangian to Eulerian. The inﬂuence of the parameter m manifests itself through the ratio TL/TE. When the structure parameter is less than one (m 1), the integral (1.64) is approximated by the formula T Lp ¼ T L þ (T E T L ) f (StE );

f (StE ) ¼

StE 0:9mSt2E ; ð1:65Þ 1 þ StE (1 þ StE )2 (2 þ StE )

which asymptotically approaches the limiting relations at StE ! 0, StE ! 1 and m ! 0. It is easy to see that formula (1.65) gives a good approximation of Eq. (1.64)

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

Figure 1.6 Influence of the Stokes number on the duration of particle interaction with turbulent eddies at m ¼ 1: 1 – (1.64); 2 – (1.65); 3 – (1.66); 4, 5 – Oesterle (2004); 4 – Rel ¼ 38; 5 – Rel ¼ 112.

even at m ¼ 1. The dependence of TLp/TL on Stokes number StE, 0:644 TL ¼ 0:356; T Lp ¼ 1 TE; 0:4(1 þ 0:01StE ) TE (1 þ StE )

ð1:66Þ

which was obtained by Wang and Stock (1993) as an approximation of the general equations at m ¼ 1, is also shown on Figure 1.6. The ﬁgure suggests a qualitative agreement between the relations (1.64) and (1.66), even though the kinematic simulation data obtained by Oesterle (2004) is a better match for Eq. (1.66). The autocorrelation function (1.59) at m ¼ 1 and different values of StE is plotted against time on Figure 1.7a and 1.7b, time being divided by TE and TLp, respectively. Figure 1.7a demonstrates the growing correlativity of velocity of a ﬂuid particle moving along a solid inertial particle trajectory as its inertia increases. Figure 1.7b makes it clear that the dependence of variables yLp and t/TLp on Stokes number StE vanishes and the exponent

Figure 1.7 Autocorrelation function of fluid particles velocity along an inertial particle trajectory: 1 – StE ¼ 0; 2 – StE ¼ 1; 3 – StE ¼ 1 (1.67).

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20

t YLp (t) ¼ exp T Lp

ð1:67Þ

corresponding to the limiting case StE ! 1 is a good approximation for the dependence YLp(t/TLp) even at StE ¼ 1. In the presence of particle drift, the duration of particles interaction with energycarrying eddies is a tensor deﬁned as 1 ð T Lp ij ¼ YLp ij (t)dt: 0

In accordance with Eq. (1.58), the tensors of the autocorrelation function YLp ij and the duration of particle interaction TLp ij in isotropic turbulence may be written as 0 l 1 0 l 1 1 0 YLp 0 0 T Lp 0 ð B C B C YnLp 0 A; T Lp ij ¼ YLp ij (t) dt ¼ @ 0 T nLp 0 A; YLp ij (t) ¼ @ 0 n n 0 0 YLp 0 0 T Lp 0 ð1:68Þ where the top indexes l and n denote parallel (longitudinal) and transverse components of these tensors (here the terms parallel and transverse mean with respect to the vector of the average drift velocity of particles). Parallel and transverse components of the tensor TLp are written as my2 (t) t þ m(g 2 t2 þ y2 (t))1=2 ¼ 1 exp dt; TE 3(g 2 t2 þ y2 (t))1=2 T E 0 1 ð m(3g 2 t2 þ 2y2 (t)) t þ m(g 2 t2 þ y2 (t))1=2 T nLp ¼ 1 exp dt: TE 6(g 2 t2 þ y2 (t))1=2 T E

T lLp

1 ð

0

ð1:69Þ

As evidenced by Eq. (1.58) and Eq. (1.69), the reduction of velocity correlations in a turbulent ﬂuid resulting from the crossing trajectory effect is greater in the longitudinal than in the transverse direction. This fact was ﬁrst noticed by Csanady (1963) and was called the continuity effect, since it stems from the continuity equation (1.11) connecting the longitudinal and transverse spatial Eulerian correlation functions. In the limits of small and large values of the Stokes number StE Eq. (1.69) gives rise to the asymptotic relations T lLp (StE ! 0) ¼

3= þ m(2 þ 3g 2 ) 6= þ m(4 þ 3g 2 ) n TE; 2 T E ; T Lp (StE ! 0) ¼ 3=(1 þ m=) 6=(1 þ m=)2

= ¼ (1 þ g 2 )1=2 ; ¼

TE ; 1 þ mg

T lLp (StE ! 1) T nLp (StE ! 1) ¼

2 þ mg TE: 2(1 þ mg)2

ð1:70Þ

In another limiting case, when the crossing trajectory effect assumes a leading role (g ! 1), that is, when the drift velocity exceeds the intensity of turbulent ﬂuctuations by a sufﬁcientamount,Eq.(1.69)leadstothewell-knownrelations(Yudine,1959;Csanady,1963)

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

T lLp ¼

L ; W

T nLp ¼

L ; 2W

T lLp T nLp

¼ 2:

To make an approximate estimation of the duration of particles interaction with turbulent eddies while taking into account the crossing trajectory effect and the particle inertia effect, we can use interpolations that are based on Eq. (1.65) and Eq. (1.70):

l;n l;n l;n T l;n Lp ¼ T Lp (StE ! 0) þ T Lp (StE ! 1)T Lp (StE ! 0) f (StE ):

ð1:71Þ

Formula (1.71) approximates the integrals (1.69) in the parameter range m 1, 0 StE < 1, 0 g < 1 with the error of no more than 5%. In an effort to incorporate both effects – the inertia effect and the crossing trajectory effect – into their model, Wang and Stock (1993) employed the assumption that the parallel component of the autocorrelation function is constant over the ellipse t2 W 2 t2 ¼ const; 2 þ L2 T Lp where TLp denotes the duration of particle interaction with turbulence in the absence of drift (g ¼ 0). This a priori suggestion enables a smooth transition from a ﬂow with no drift to a ﬂow where drift is a major factor as g goes from 0 to 1, and allows to obtain the following simple formulas for the parallel and transverse components of the tensor TLp: 1=2 2 1 þ m2T g 2 mT g T Lp T Lp l n m; T Lp ; mT ¼ T Lp ¼ 1=2 ; T Lp ¼ 2 2 2 TE 2 1 þ mT g 1 þ mT g 2 ð1:72Þ where TLp is determined by Eq. (1.66). Using Csanadys (1963) relations for the diffusion coefﬁcient of particles in isotropic turbulence in the presence of gravity as their starting point, Deutsch and Simonin (1991) proposed the following simple formulas for the parallel and transverse components: T lLp ¼

TL

; (1 þ b2 g 2 )1=2

T nLp ¼

TL

; (1 þ 4b2 g 2 )1=2

b¼

T L u0 : L

ð1:73Þ

It is obvious that the dependences (1.73) include the crossing trajectory and continuity effects but ignore the inertia effect. By comparing experimental measurements (Wells and Stock, 1983) with their own numerical results, Deutsch and Simonin (1991) concluded that b2 ¼ 0.45. Figure 1.8 shows how the crossing trajectory effect inﬂuences the duration of particle interaction with turbulent eddies according to the relations (1.69) and (1.72) for m ¼ 1. For comparison, the same ﬁgure also presents the results of direct stochastic trajectory simulation of particle dynamics by Deutsch and Simonin, who employed the LES method to calculate the turbulent characteristics of the carrier ﬂow (Deutsch and Simonin, 1991). In this work, calculations were carried out for particles

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Figure 1.8 Influence of the average slip on the duration of particle interaction with turbulent eddies in parallel (a) and transverse (b) directions (with respect to the drift velocity): 1, 2 – (1.69); 3, 4 – (1.72); 1, 3 – StE ¼ 0.1; 2, 4 – StE ¼ 0.3; 5 – (1.73); 6–8 – Deutsch and Simonin (1991).

of diameters dp ¼ 45, 57, and 90 mm, the density ratio between the particle material and the carrier ﬂuid being equal to rp/rf ¼ 2000. As one can see from the graphs of l;n T Lp versus the drift parameter g shown on Figure 1.8, the dependences (1.69) and (1.72) are roughly similar, though not as close to each other as the corresponding dependences obtained by Deutsch and Simonin, whose results for relatively large values of g slightly favor the Eq. (1.69). It can also be seen that with increase of the average slip, the inﬂuence of particle inertia, that is, of the Stokes number, on TLp is l;n eliminated, and T Lp =T L is governed chieﬂy by g. Also shown on Figure 1.8 are the dependencies (1.73), which match the numerical data quite well except for the regions of small g – as one might have expected, because these dependences do not l;n take into account the inertia effect and the inﬂuence of StE on T Lp . l Figure 1.9a and 1.9b describe the autocorrelation functions YLp and YnLp calculated by the formula (1.58) at m ¼ StE ¼ 1 for different values of drift parameter g.

Figure 1.9 Autocorrelation functions at different drift parameter values: 1 – g ¼ 0; 2, 3 – YlLp ; 4, 5 – YnLp ; 2, 4 – g ¼ 1; 3, 5 – g ¼ 10; 6 – (1.74).

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

Figure 1.9a shows the reduction of velocity correlativity of ﬂuid particles moving along inertial particle trajectories with increase of the drift parameter, which is just the physical manifestation of the crossing trajectory effect. One can easily notice the growing difference between the parallel and transverse components of correlation functions as the drift parameter gets larger. When the drift parameter and the Stokes number both tend to inﬁnity (g ! 1, StE ! 1), the autocorrelation functions tend to the asymptotic dependences Wt t Wt t ‘ n ; YLp (t) ¼ G ; ð1:74Þ YE YE YLp (t) ¼ F L TE L TE which express ﬂuid velocity correlations along an inertial particle trajectory as the product of Eulerian spatial and temporal velocity correlation functions of the ﬂuid. The negative loop in the transverse autocorrelation function becomes more visible with increase of the drift parameter. This fact, which is consistent with formula (1.74), has been discussed by Mei et al. (1991) and Squires and Eaton (1991b). Figure 1.9b illustrates the diminishing role of parameter g as well as the diminishing difference between YlLp and YnLp when the times entering these two parameters are scaled by their corresponding integral scales. Having said that, the distinction between the longitudinal and transverse components is still felt very clearly. Figure 1.9b also shows the exponential approximation of autocorrelations, YLp ij (t) ¼ exp tT 1 ð1:75Þ Lp ij : It is evident from Figure 1.9b that the dependence YlLp (t=T lLp )is accurately described by the exponential function (1.75). On the other hand, the deviation of YnLp (t=T nLp ) from the exponential function is more noticeable. By using stochastic simulation, Berlemont et al. (1995) found that the duration of interaction between the turbulence and the particles increases with frequency of interparticle collisions. At the ﬁrst glance this result seems counterintuitive because collisions should reduce the correlation between the motion of inertial particles and that of the ﬂuid and thereby shorten the interaction time. But in reality, the effect of increasing TLp with increase of collision frequency has the same explanation as the particle inertia effect and is related to the inequality TE > TL. To understand the ultimate effect of collisions, we should ﬁrst see how they change the effective mean free path u0 y(t) of a particle undergoing ﬂuctuational motion and then make the corresponding adjustments to Eq. (1.58). Suppose that collisions cause particles to lose all memory about their preceding involvement in ﬂuctuational motion of the carrier ﬂow and thereby – in the turbulent ﬂow. The end result is that whenever a collision occurs, the interaction of the particle with turbulent eddies starts again from the scratch. In this case it follows from the approximate solution of the equations of motion (1.45) and (1.46) that tc tntc y(t) ¼ t þ ntp exp 1 þ tp exp 1 ; tp tp ntc < t < (n þ 1)tc ; n ¼ 0; 1; 2; . . . ;

ð1:76Þ

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Figure 1.10 Effect of collisions on the duration of particles interactions with turbulent eddies (m ¼ 0.5): 1, 2, 3 – (1.64) with regard to (1.76); 1 – StE ¼ 0.25; 2 – StE ¼ 0.5; 3 – StE ¼ 1; 46 – Berlemont et al. (1995).

where tc is the time interval between collisions. Eq. (1.76) is saying that as the time interval between collisions becomes shorter (in other words, as the collision frequency increases), the effective mean free path of particles decreases as if we increased the particles inertia. The effect of collisions on the correlations of ﬂuid velocity ﬂuctuations along particle trajectories is determined by the dimensionless parameter StC tp =tc. This parameter can be considered as the Stokes number characterizing the inertia of particles in relation to the interval between collisions. When the average slip is absent (g ¼ 0), the autocorrelation function YLp(t) and the duration of particles interaction with the turbulence TLp coincide with the Lagrangian and Eulerian characteristics in the respective limiting cases StC ! 0 and StC ! 1. Figure 1.10 illustrates the inﬂuence of interparticle collisions on the duration of particles interactions with turbulent eddies in the absence of the average slip, when TLp is determined by expression (1.64) with proper account taken of Eq. (1.76). The same ﬁgure presents the results of direct stochastic simulation of particle motion in isotropic turbulent ﬁeld that were obtained by the LES method (Berlemont et al., 1995). The calculations were carried out for particles of diameter dp ¼ 656 mm and densities rp ¼ 50, 100, and 200 kg/m3 moving in air. As it follows from Figure 1.9, instead of being roughly consistent with the LES data, the dependence (1.64) together with Eq. (1.76) predicts stronger effect of collisions on TLp than suggested by the results of calculations made by Berlemont et al. (1995). It is also seen that when StC gets larger, the effect of inertia governed by parameter StE takes a smaller role, and the effect of collisions emerges as the predominant factor. For m 1 the integral (1.64) together with Eq. (1.76) can be approximated by an expression similar to Eq. (1.65): T Lp ¼ T L þ (T E T L ) f (St ); 1=5 St ¼ St5E þ 10 St5C ;

f (St ) ¼

St 0:9mSt2 ; 1 þ St (1 þ St )2 (2 þ St ) ð1:77Þ

1.3 Velocity and Temperature Correlations in a Fluid along the Inertial Particle Trajectories

where St denotes a parameter characterizing the effect of collisions on the duration of particle–turbulence interaction. The crossing trajectory effect inﬂuences TLp, which can be taken into account by replacing StE in Eq. (1.71) with the effective Stokes number St. The Lagrangian correlation moment of ﬂuid particles temperature ﬂuctuations along an inertial particle trajectory is represented in the form similar to Eq. (1.51): BLtp (t) ¼ hq0 iYLtp (t) ¼ hq0 (x; t)q0 (Rp (tt); tt)jRp (t) ¼ xi Ð ¼ hq0 (x; t)q0 (xr; tt)d(rsp (t))idr; 2

ð1:78Þ

where YLtp(t) is the Lagrangian autocorrelation function of ﬂuid temperature along the particle trajectory. Corrsins hypothesis about the possibility of independent statistical averaging of random ﬁelds of particle displacements and Euleruan characteristics of the continuum as applied to temperature ﬂuctuations yields hq0 (x; t)q0 (xr; tt)d(rs(t))i ¼ hq0 (x; t)q0 (xr; tt)if(r; t):

ð1:79Þ

Substituting Eq. (1.79) into Eq. (1.78) and employing the relations (1.43), (1.53), and (1.57), we arrive at the following expression for the Lagrangian autocorrelation function of ﬂuid temperature ﬂuctuations along the particle trajectory: qﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ ð1:80Þ YLtp (t) ¼ F t (s)YEt (t); s ¼ W 2 t2 þ u0 2 y2 (t): Exponential approximation of Eulerian spatial and temporal correlation functions of temperature (1.44) transforms Eq. (1.80) into t þ mt (g 2 t2 þ y2 (t))1=2 YLtp (t) ¼ exp ; T Et

ð1:81Þ

where mt u0 TEt/Lt is the temperature structure parameter of turbulence. The integral scale of ﬂuid temperature ﬂuctuations along the inertial particle trajectory is determined through the autocorrelation function (1.81) as 1 ð

T Ltp ¼

1 ð

YLtp (t) dt ¼ 0

0

t þ mt (g 2 t2 þ y2 (t))1=2 exp dt: T Et

ð1:82Þ

The autocorrelation function (1.81) and time microscale (1.82) take into account both the crossing trajectory effect and the particle inertia effect, which inﬂuence temperature ﬂuctuations of the continuum calculated along the particle trajectories. The inﬂuence of these two phenomena on velocity ﬂuctuations is governed by the parameters g and StE. Collisions between particles can be taken into account in Eq. (1.81) and Eq. (1.82) through the dependence (1.76), whereas the net result of interparticle collisions is characterized by Stokes parameter StC. In the limiting case of inertialess particles (StE ¼ StC ¼ g ¼ 0), there follows from Eq. (1.82) a relation

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j 1 Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence

26

between Lagrangian and Eulerian time macroscales of temperature ﬂuctuations in a turbulent ﬂuid: T Lt 1 ¼ : T Et 1 þ mt

ð1:83Þ

In accordance with Eq. (1.83) the ratio of scales TLt/TEt is governed only by the temperature structure parameter of turbulence mt u0 TEt/Lt similarly to the dependence (1.61) of the ratio of velocity ﬂuctuation scales TL/TE on m. A similar dependence of TLt/TEt on mt has been obtained by Derevich (2001) on the basis of Corrsins independence conjecture by using the spectral method. In the absence of drift, the integral (1.82) can be approximated on the interval mt 1 by the dependence similar to Eq. (1.65), StE 0:8mt St2E ; ð1:84Þ T Ltp ¼ T Lt þ (T Et T Lt ) 1 þ StE (1 þ StE )2 (2 þ StE ) the approximation being asymptotically exact at mt ! 0. In the limiting cases of very small and large values of the Stokes number StE, the following asymptotic relations result from Eq. (1.82): T Ltp (StE ! 0) ¼

T Et ; 1 þ mt (1 þ g 2 )1=2

T Ltp (StE ! 1) ¼

T Et : 1 þ mt g

ð1:85Þ

In the presence of drift and for the parameter range mt 1, 0 StE < 1, 0 g < 1, the integral (1.82) can be approximated by a dependence that is based on the relations (1.84) and (1.85), with the margin of error not exceeded 5% and the end result being similar to Eq. (1.71): T Ltp ¼ T Ltp (StE ! 0) þ [T Ltp (StE ! 1)T Ltp (StE ! 0)] StE 0:8mt St2E : 1 þ StE (1 þ StE )2 (2 þ StE )

ð1:86Þ

The effect of interparticle collisions on TLtp can be taken into account by replacing StE in Eq. (1.84) and Eq. (1.86) with the effective Stokes number St. Similarly to Eq. (1.67), the function YLtp of t/TLtp is rather accurately described by the exponential dependence t YLtp (t) ¼ exp ð1:87Þ T Ltp which is asymptotically exact in the limit St ! 1 and g ! 1. By making an appropriate preliminary choice of the autocorrelation functions, say, in the form given by the exponential functions (1.67), (1.75), and (1.87), we can reduce the problem of ﬁnding ﬂuid velocity and temperature correlations along inertial particle trajectories to the problem of ﬁnding the Lagrangian time scales TLp and TLtp characterizing the interaction of particles with turbulent eddies. It should be noted that the model proposed in the present section requires estimations for TLp and TLtp and has a semi-empirical character: not only does it hinge on the previously made assumptions, in particular, on Corrsins conjecture

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

employed in Eq. (1.52), Eq. (1.79) and especially on the suppositions we made when deriving Eq. (1.57) and Eq. (1.76), but it also critically depends on the structure parameters of turbulence m and mt that must be provided externally, that is, obtained from a physical experiment or from numerical simulation.

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

The present section considers the correlation moments for the velocity and temperature of particles in a stationary isotropic turbulent ﬁeld and presents the relations between the intensities of velocity and temperature ﬂuctuations in the disperse and continuous phases. First, let us deﬁne the mixed Lagrangian correlation moment of ﬂuid and particle velocity ﬂuctuations: Bfp ij (t) ¼ hu0i (x; t)v0j (Rp (tt); tt)jRp (t) ¼ xi;

ð1:88Þ

where Rp is the position vector of a point on the particles trajectory. From the particles equation of motion (1.46) there follows an equation for the mixed correlation moment (1.88): dBfp ij Bfp ij BLp ij ¼ ; dt tp tp

ð1:89Þ

where BLp ij (t) is the Lagrangian correlation moment of ﬂuid velocity ﬂuctuations along the particles trajectory (1.51). The solution of Eq. (1.89) obeying the condition Bfp ij ! 0 at t ! 1 is written in the matrix notation as Bfp (t) ¼

1 tp

1 ð

t

1 ð tx u0 2 tx BLp (x)exp I dx ¼ YLp (x)exp I dx; tp tp tp

ð1:90Þ

t

where I is the unit matrix. The relation (1.90) gives an expression for the mixed single-point moment of velocity ﬂuctuations of the continuous and disperse phases: hu0i v0j i ¼ Bfp ij (0) ¼ u0 f u ij ; 2

ð1:91Þ

where fu ¼

1 tp

1 ð

0

t YLp (t)exp I dt: tp

ð1:92Þ

The Lagrangian correlation moment of velocity ﬂuctuations for a particle along its trajectory looks as follows: Bp ij (t) ¼ hv0i (x; t)v0j (Rp (tt); tt)jRp (t) ¼ xi:

ð1:93Þ

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28

In view of the relation 0

0 d2 Bp ij dvi (t) dvj (tt) ¼ dt dt2 dt the particles equation of motion (1.46) leads to the following equation for the correlation moment of particle velocity ﬂuctuations: d2 Bp ij Bp ij BLp ij 2 ¼ 2 : dt2 tp tp

ð1:94Þ

The following boundary conditions apply for Eq. (1.94): dBp ij ¼0 dt

at

t ¼ 0;

Bp ij ! 0

at t ! 1:

ð1:95Þ

Integrating Eq. (1.94) and making use of the boundary conditions (1.95), we get an asymptotic expression for the tensor of turbulent diffusion of particles for large values of time: 1 ð

Dp ij ¼

1 ð

Bp ij (t)dt ¼ 0

BLp ij (t)dt ¼ u0 T Lp ij : 2

ð1:96Þ

0

In accordance with Eq. (1.96), the tensor of turbulent diffusion of inertial particles coincides with the corresponding quantity for ﬂuid particles moving along inertial particle trajectories. Comparing Eq. (1.2) and Eq. (1.96), we obtain Dp ij ¼

T Lp ij Dt : TL

ð1:97Þ

Thus the ratio of turbulent diffusion coefﬁcients for ﬂuid and solid (inertial) particles is equal to the ratio of integral scales of ﬂuctuation velocities of the continuous carrier medium calculated along the corresponding particle trajectories. If we ignore the distinction between Lagrangian integral scales of turbulence along the trajectories of ﬂuid and solid particles, in other words, if we assume TLp ij ¼ TLdij, then, as it follows from Eq. (1.97), the turbulent diffusion tensors for ﬂuid and solid particles will also coincide. This result was ﬁrst obtained by Chen (Hinze, 1975). From Eq. (1.97) it follows that in the absence of the average slip (drift) between the particles and the ﬂuid, when the duration of particle interaction with turbulent eddies TLp exceeds the Lagrangian macroscale TL, the coefﬁcient of turbulent diffusion turns out to be greater for solid particles than for ﬂuid particles. This phenomenon is called the inertia effect (Reeks, 1977; Pismen and Nir, 1978; Deutsch and Simonin, 1991; Squires and Eaton, 1991b; Elghobashi and Truesdell, 1992). With increase of the average slip, the duration of particle interaction with turbulent eddies decreases and thus the turbulent diffusion coefﬁcient decreases as well. This effect is referred to as the crossing trajectory effect (Yudine, 1959; Csanady, 1963). Besides, the time scale T lLp has turned out to be greater than T nLp , therefore the diffusion coefﬁcient of particles in the longitudinal direction (with respect to the drift velocity vector) Dlp

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

exceeds the diffusion coefﬁcient in the transverse directionDnp . This effect is called the continuity effect (Csanady, 1963). Taking Eq. (1.95) into account, we ﬁnd the solution of Eq. (1.94): Bp (t) ¼

u0 2 2tp

1 ð

0

jt þ xj jtxj I þ exp I YLp (x) dx: exp tp tp

ð1:98Þ

The expression (1.98) was ﬁrst obtained by Reeks (1977) by direct integration of equations of motion for the particles and invoking Corrsins hypothesis about independent averaging of particle displacement ﬁelds and ﬂuid velocity ﬂuctuations. In accordance with Eq. (1.98), a single-point second moment of particle ﬂuctuation velocities (turbulent stress tensor) is given by the expression similar to Eq. (1.91): hv0i v0j i ¼ Bp ij (0) ¼ u0 f u ij : 2

ð1:99Þ

The tensor fu ij in Eq. (1.91) and Eq. (1.99) characterizes the extent of particles involvement in ﬂuctuational motion of the turbulent carrier medium. Thus inertialess particles (tp ! 0) are completely involved in turbulent motion and their kinetic energy coincides with the turbulent energy of the carrier ﬂuid. If the particles are inertialess, then Eq. (1.92) gives us lim f tp ! 0 u ij

¼ d ij ;

since YLp ij (0) ¼ dij : The extent of particles involvement in ﬂuctuational motion decreases as their inertia gets higher, and thus kinetic energy of inertial particles in homogeneous isotropic stationary turbulence is always lower than turbulent energy of the ﬂuid. For highly inertial particles, if follows from Eq. (1.92) that lim f tp ! 1 u ij

¼

T Lp ij : tp

If the Lagrangian autocorrelation function of ﬂuid velocity along the particles path YLp ij(t)is described by the exponential approximation (1.75), the correlation moment of particle velocity ﬂuctuations (1.98) takes the form Bp (t) ¼

u0 2 t 1 1 (I þ tp T1 ) ) þ exp I exp(tT Lp Lp 2 tp t 1 þ (Itp T1 I ; exp(tT1 Lp ) Lp )exp tp

ð1:100Þ

and the tensor characterizing the extent of particles involvement in turbulent motion (1.92) becomes equal to 1 f u ¼ (I þ tp T1 Lp ) :

ð1:101Þ

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j 1 Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence

30

In the absence of the average slip of particles with respect to the turbulent ﬂuid, the Lagrangian correlation moments (1.88), (1.93) and the tensor of particles involvement in turbulent motion (1.92) turn out to be isotropic: Bfp ij (t) ¼ Bfp (t)d ij ; Bp ij (t) ¼ Bp (t)d ij ; f u ij ¼ f u dij ; 1 ð 1 t fu ¼ YLp (t)exp dt: tp tp 0

ð1:102Þ

It is evident that in this case the single-point moment of velocity ﬂuctuations of the continuous and disperse phases (1.91) and the single-point moment of velocity ﬂuctuations of particles (1.99) will be isotropic as well: hu0i v0j i ¼ hv0i v0j i ¼ v0 dij ; 2

ð1:103Þ

where v0 2 hv0n v0n i=3 is the intensity of particle velocity ﬂuctuations deﬁned as v 0 ¼ f u u0 : 2

2

ð1:104Þ

In the case of an exponential autocorrelation function the isotropic involvement coefﬁcient (1.102) simpliﬁes to tp 1 : ð1:105Þ fu ¼ 1þ T Lp The relation (1.104) for the intensity ratio of velocity ﬂuctuations of the disperse and continuous phases with the involvement factor given by (1.105) at TLp ¼ TL was ﬁrst obtained by Chen (Hinze, 1975). As evidenced by Figure 1.11, the dependence (1.104) with the involvement coefﬁcient given by (1.105) is in very good agreement with the results of numerical simulation by the LES method (Deutsch and Simonin, 1991).

Figure 1.11 Relation between the intensities of velocity fluctuations of the disperse and continuous phases: 1 – (1.104) and (1.105); 2 – Deutsch and Simonin (1991).

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

Deﬁne the Lagrangian autocorrelation function of particle velocities in the absence of the average drift as Yp(t) ¼ Bp(t)/Bp(0). Then, by virtue of Eq. (1.98) and Eq. (1.102) the Lagrangian integral scale for particles is 1 ð

Tp ¼

Yp (t)dt ¼ 0

T Lp : fu

ð1:106Þ

If we set the autocorrelation function of ﬂuid particles moving along the inertial particle trajectories equal to the exponential approximation of the autocorrelation function of inertial particle velocities, then Eq. (1.100) gives 1 t t Yp (t) ¼ exp þ exp 2 T Lp tp ð1:107Þ (T Lp þ tp ) t t exp exp ; þ 2(T Lp tp ) T Lp tp and the Lagrangian integral scale (1.106) becomes equal to T p ¼ T Lp þ tp :

ð1:108Þ

Figure 1.12 compares the formula (1.107) with the DNS results at Rel ¼ 53 (Simonin et al., 2002). The duration of particles interaction with turbulent eddies entering Eq. (1.107) was determined from Eq. (1.65). The DNS results suggest TL/TE ¼ 0.68, hence the theoretical curves corresponding to Eq. (1.107) correspond to the value of the structure parameter m ¼ 0.3, which due to Eq. (1.61) gives a value that is close to the above-mentioned ratio of the Lagrangian and Eulerian turbulence scales. The main conclusion following from Figure 1.12 is that the Lagrangian integral time scale of particle velocity ﬂuctuations grows with particle inertia, which is in good agreement with the formulas (1.106) and (1.108). For highly inertial particles, the Lagrangian macroscale Tp becomes equal to the relaxation time tp.

Figure 1.12 Autocorrelation function of particle velocity fluctuations: 1–5 – (1.107); 6–10 – Simonin et al. (2002); 1, 6 – StE ¼ 0.04; 2, 7 – StE ¼ 0.2; 3, 8 – StE ¼ 1.0; 4, 9 – StE ¼ 2.3; 5, 10 – StE ¼ 3.3.

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j 1 Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence

32

We now turn to the discussion of temperature ﬂuctuations. The Lagrangian mixed correlation moment of temperatures of the continuous and disperse phases along the particle trajectories is deﬁned as Bfpt (t) ¼ hq0 i (x; t)q0j (Rp (tt); tt)jRp (t) ¼ xi:

ð1:109Þ

The equation for Bfpt is derived directly from the heat exchange equation (1.49) for a single particle: dBfpt Bfpt BLtp ¼ ; dt tt tt

ð1:110Þ

where BLtp is the Lagrangian correlation moment of temperature ﬂuctuations of a ﬂuid particle moving along the inertial particle trajectory (see Eq. (1.77)). The solution of Eq. (1.110) has the form similar to Eq. (1.90): 1 Bfpt (t) ¼ tt

1 ð

t

ð 2 1 tx hq0 i tx BLtp (x)exp YLtp (x)exp dx ¼ dx: tt tt tt

ð1:111Þ

t

According to Eq. (1.111), the mixed single-point moment of temperature ﬂuctuations of the continuous and disperse phases is hq0 q0 i ¼ Bfpt (0) ¼ f t hq0 i; 2

ð1:112Þ

where ft ¼

1 tt

1 ð

0

t YLtp (t)exp dt: tt

ð1:113Þ

The Lagrangian correlation moment of temperature ﬂuctuations of a particle along its trajectory is equal to Bpt (t) ¼ hq0 (x; t)q0 (Rp (tt); tt)jRp (t) ¼ xi:

ð1:114Þ

In view of the fact that

d2 Bpt dq0 (t) dq0 (tt) ¼ 2 ; dt dt dt

Equation (1.49) yields the following equation for the correlation moment of particle temperature ﬂuctuations: d2 Bpt Bpt BLtp 2 ¼ 2 dt2 tt tt

ð1:115Þ

whose solution has the form similar to Eq. (1.98), ð 2 1

Bpt (t) ¼

hq0 i 2tt

0

jt þ xj jtxj exp þ exp YLtp (x)dx: tt tt

ð1:116Þ

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

Due to Eq. (1.116), second single-point moments of temperature ﬂuctuations of the disperse and continuous phases are connected by the relation hq0 i ¼ Bpt (0) ¼ f t hq0 i; 2

2

ð1:117Þ

where the coefﬁcient ft (see Eq. (1.113)) characterizes the susceptibility of particles to turbulent ﬂuctuations of the carrier ﬂuid temperature. If the Lagrangian autocorrelation function of ﬂuid temperature along the particle trajectory YLtp(t) is described by the exponential approximation (1.87), the correlation moment of the particles temperature ﬂuctuations (1.116) becomes 2 hq0 i tt 1 t t Bpt (t) ¼ exp þ exp 1þ T Ltp 2 T Ltp tt 1 tt t t exp exp ; þ 1 T Ltp T Ltp tt and the coefﬁcient of particles involvement in temperature ﬂuctuations of the continuum (1.113) reduces to 1 tt f t ¼ 1þ : ð1:118Þ T Ltp After some algebra, the formulas (1.104), (1.105), (1.117), and (1.118) give us the following relation between turbulent ﬂuctuations of temperature and velocities of the disperse and continuous phases: hq0 i 2

02

hq i

¼

v0 2 =u0 2

v0 2 =u0 2 ; þ ¡(1v0 2 =u0 2 )

¡¼

tt T Lp ; tp T Ltp

ð1:119Þ

where the parameter ¡ characterizes the relation between heat and the dynamic inertia of the particles. By performing a simple comparison of solutions of the equations for the velocity and particle temperature ﬂuctuations, Yarin and Hetsroni (1994) have obtained another relation between turbulent ﬂuctuations of temperature and velocities of the disperse and continuous phases:

hq0 i 2

1=2

hq0 i 2

tp v0 tt ¼ 1 1 0 : u

ð1:120Þ

Starting from Eq. (1.104) and Eq. (1.117) and employing step autocorrelation functions to determine the involvement coefﬁcients (1.102) and (1.113), Derevich (2001) also obtained a relation between turbulent ﬂuctuations of temperature and velocities of the disperse and continuous phases: 1 2 hq0 i v0 2 ¡ ð1:121Þ ¼ 1 1 0 2 ; 2 u hq0 i which is identical in form to the relation (1.120) but was derived more rigorously.

j33

j 1 Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence

34

Figure 1.13 Ratio of temperature fluctuation intensities of the disperse and continuous phases vs the ratio of velocity fluctuation intensities: 1 – ¡ ¼ 1; 2, 4, 6 – ¡ ¼ 0.5; 3, 5, 7 – ¡ ¼ 2; 2, 3 – (1.119); 4, 5 – (1.120); 6, 7 – (1.121).

Figure 1.13 presents the ratio of temperature ﬂuctuation intensities of the disperse and continuous phases as a function of the parameter ¡. It can be seen that the behavior of the dependences (1.119), (1.120), and (1.121) is qualitatively identical in the sense that all of them predict that hq0 2i/hq0 2i < v0 2/u0 2 at ¡ < 1 and that hq0 2i/ hq0 2i > v0 2/u0 2 at ¡ > 1. Figure 1.14 shows how the parameter ¡, which characterizes the ratio between heat inertia and dynamic inertia of the particle, inﬂuences the ratio of temperature ﬂuctuation intensities of the disperse and continuous phases when the ratio of velocity ﬂuctuation intensities remains ﬁxed (v0 2/u0 2 ¼ 0.71). It stands out that Eq. (1.119) is in very good agreement with the DNS results (Jaberi, 1998) whereas the dependence (1.120), when compared to the DNS data, predicts an excessively strong decrease of hq0 2i/hq0 2i with increase of ¡ – a fact that has been already noted by Jaberi (1998).

Figure 1.14 Ratio of temperature fluctuation intensities of the disperse and continuous phases vs the ratio between heat and the dynamic inertia of the particles: 1 – (1.119), 2 – (1.120), 3 – (1.121), 4 – Jaberi (1998).

1.5 Particle Acceleration in Isotropic Turbulence

1.5 Particle Acceleration in Isotropic Turbulence

In the present section we discuss the statistics of low-inertia particle acceleration under the assumption that the deviation of particle trajectories from those of ﬂuid (inertialess) particles can be neglected. Let us ﬁrst deﬁne the Lagrangian correlation moment of ﬂuid particle acceleration ﬂuctuations, 0

0 dui (x; t) duj (R(tt); tt) AL ij (t) ¼ ð1:122Þ jR(t) ¼ x ¼ ha0i a0j iYa (t); dt dt where ha0i a0j i is the variation of acceleration ﬂuctuations, and Ya(t) is the autocorrelation function of acceleration. As a consequence of the kinematic relation AL ij ¼ d2BL ij/dt2 and Eq. (1.1), the correlation of acceleration ﬂuctuations (1.122) in isotropic turbulence manifests itself as AL ij (t) ¼ Y00L (t)u0 dij : 2

ð1:123Þ

Taking into account the normalization condition Ya(0) ¼ 1, we derive from Eq. (1.123) the expressions for the variance and the autocorrelation function of acceleration ﬂuctuations: ha0i a0j i ¼

2u0 2 dij a0 e3=2 d ij ¼ ; t2T n1=2

Ya (t) ¼

Y00L (t) t2 Y00 (t) ¼ T L ; 00 YL (0) 2

ð1:124Þ

where tT is the Taylor differential time scale (1.6). The Lagrangian correlation moment of acceleration ﬂuctuations for inertial particles is deﬁned as Ap ij (t) ¼

0 dv0i (x; t) dvj (Rp (tt); tt) jRp (t) ¼ x ¼ ha0pi a0pj iYpa (t): dt dt

ð1:125Þ

We conclude from the self-evident relation Ap ij ¼ d2Bp ij/dt2 and from Eq. (1.47) that the correlation of the particles acceleration ﬂuctuations (1.125) is equal to Ap ij (t) ¼

BLp ij (t)Bp ij (t) [YLp (t)u0 2 Yp (t)v0 2 ]dij ¼ ¼ ha0pi a0pj iYpa (t): t2p t2p

ð1:126Þ

Expression (1.126) together with the condition Ypa(0) ¼ 1 gives us the variance of the ﬂuctuation as well as the autocorrelation function for the acceleration of inertial particles: ha0pi a0pj i ¼

(u0 2 v0 2 )dij (1f u )u0 2 dij ¼ ; t2p t2p

Ypa (t) ¼

YLp (t)Yp (t) f u : ð1:127Þ 1f u

By analogy with the ﬁrst relation (1.124), let us represent the ﬂuctuation variance of inertial particles acceleration as

j35

j 1 Motion of Particles and Heat Exchange in Homogeneous Isotropic Turbulence

36

ha0pi a0pj i ¼

ap0 e3=2 dij ; n1=2

ð1:128Þ

where ap0 is the dimensionless amplitude of acceleration ﬂuctuations. Comparing the ﬁrst relations in Eq. (1.127) and Eq. (1.128), we get ap0 ¼

(1f u )Rel 151=2 St2

;

ð1:129Þ

where St tp/tk is the Stokes number determined by the Kolmogorov time microscale. The coefﬁcient of particles involvement in turbulent motion entering Eq. (1.129) is deﬁned by Eq. (1.102). The autocorrelation function YLp(t) is taken to be equal to YL(t), which in its turn is given by the bi-exponential approximation (1.5). Accordingly, the involvement coefﬁcient takes the form fu ¼

2StL þ z2 ; 2StL þ 2St2L þ z2

ð1:130Þ

where StL tp/TL is the Stokes number determined by the Lagrangian time macroscale. As the Reynolds number gets larger, z ! 0 and the involvement coefﬁcient tends to fu ¼

1 ; 1 þ StL

ð1:131Þ

which corresponds to the exponential autocorrelation function (1.3). Substituting Eq. (1.130) into Eq. (1.129) and making use of Eq. (1.6), we get 1 151=2 a0 StðSt þ TL ) ap0 ¼ a0 1 þ : ð1:132Þ Rel Figure 1.15, which plots the dimensionless amplitude of particle acceleration against the Stokes number, compares the dependence predicted by formula (1.132)

Figure 1.15 Amplitude of particle acceleration fluctuations: 1, 2, 3 – (1.132); 4, 5, 6 – Bec et al. (2006); 1, 4 – Rel ¼ 65; 2, 5 – Rel ¼ 105; 3, 6 – Rel ¼ 185.

1.5 Particle Acceleration in Isotropic Turbulence

(with Eqs. (1.4) and (1.7)) taken into consideration) with the DNS results obtained by Bec et al. (2006). It is readily seen that Eq. (1.132) describes the inﬂuence of St as well as Rel with a sufﬁcient accuracy. It is obvious that since particle acceleration is governed primarily by small-scale turbulent structures, it makes sense to describe a particles inertia in terms of the relaxation time divided by the Kolmogorov microscale – in contrast to the effect of particle inertia on the intensity of velocity ﬂuctuations, which is better described in terms of the time macroscale of turbulence. As a consequence, we cannot avoid mentioning the fact at low values of St, insertion of the involvement coefﬁcient (1.131) into Eq. (1.129) rather than into Eq. (1.130) will produce results that are unacceptable even in the qualitative sense.

j37

j39

2 Motion of Particles in Gradient Turbulent Flows In the present chapter we take a look at particle motion in homogeneous and inhomogeneous gradient turbulent ﬂows. Concentration of the disperse phase is assumed to be small enough to neglect interparticle collisions and to ignore the feedback action of particles on the carrier continuum and the resultant modiﬁcation of turbulence. Therefore the average and ﬂuctuational characteristics of the turbulent carrier ﬂow are assumed to be known from the solution of the corresponding problem for a one-phase turbulent medium. Our goal is to develop a statistical method for modeling the motion of particles in a turbulent ﬂow that is based on the kinetic equation for the probability density function (PDF) of particle velocity distribution. This approach entails one major assumption, namely, that one can treat the velocity ﬁeld of the continuum as a Gaussian random process. In this context, we have to mention that a large body of experimental research and direct numerical calculations suggests that even isotropic turbulence is actually not Gaussian and the tail of the PDF distribution is noticeably different from the tail of a normal distribution (Monin and Yaglom, 1975; Kuznetsov and Sabelnikov, 1990; Pope, 2000). However, statistical arguments based on the central limit theorem warrant the conclusion that at high Reynolds numbers, the kernel of the PDF, where the energy of turbulence is predominantly concentrated, should be close to a normal distribution. In particular, the experimental data obtained by Tavoularis and Corrsin (1981) and Ferchichi and Tavoularis (2002) provides evidence that at least in a homogeneous shear ﬂow the kernel of the PDF is close to an elliptic Gaussian distribution. We are thus justiﬁed in employing functional calculus to solve the closure problem for the kinetic equation for the PDF of particle velocity in a turbulent ﬂow on the basis of the well-known Furutsu–Donsker–Novikov formula for the Gaussian random ﬁeld (Klyatskin, 1980, 2001; Frisch, 1995). We then use the kinetic equation obtained in this way to construct the continual transport (differential) and algebraic models that allow to calculate hydrodynamic characteristics (moments) of the disperse phase. As examples illustrating the application of these models, we consider the behavior of particles in three types of ﬂows: homogeneous shear ﬂow, ﬂow in the near-wall region, and ﬂow in a vertical channel.

j 2 Motion of Particles in Gradient Turbulent Flows

40

2.1 Kinetic Equation for the Single-Point PDF of Particle Velocity

The motion of a single particle in a turbulent medium is described by Eqs. (1.45) and (1.46). These are equations of the Langevin type in which the velocity u of the turbulent ﬂuid is regarded as a random process. In order to make a transition from a dynamic stochastic motion equation to a statistical description of particle velocity distribution, we have to introduce the dynamic probability density in the phase space of coordinates and velocities (x, v): p ¼ d(xRp (t))d(vvp (t));

ð2:1Þ

where d(x) is Diracs delta function. Differentiating Eq. (2.1) with respect to time and taking into account Eq. (1.45) and Eq. (1.46), we get the Liouville equation for the dynamic probability density of a single particle in the phase space uk vp k qp qp q þ þ F k p ¼ 0: þ vk ð2:2Þ tp qt qx k qvk Averaging Eq. (2.2) over the ensemble of random realizations of the ﬂuids turbulent velocity ﬁeld u, we get an equation for the statistical PDF of particle velocity distribution P ¼ hpi. A single-point, one-particle PDF P(x, v, t) is deﬁned as the probability density for a particle to have position x and velocity v at the instant of time t. Representing the actual velocity of the ﬂuid as a sum of the average and ﬂuctuational components u ¼ U + u0 and averaging Eq. (2.2) while keeping in mind that hvpi pi ¼ viP, we obtain qP qP q U k vk 1 qhu0k pi þ þ Fk P ¼ : ð2:3Þ þ vk tp qt qx k qvk tp qvk Terms on the left-hand side of Eq. (2.3) describe time evolution and convection in the phase space, whereas the right term characterizes the interaction of particles with turbulent eddies of the carrier ﬂow. To close the equation, it is necessary to determine the correlation between velocity ﬂuctuations of the continuous phase and the probability density of particle velocity hu0i pi. To this end, the velocity ﬁeld of the continuous phase should be taken as a Gaussian random process with given correlation moments. In this case, by using the Furutsu–Donsker–Novikov formula for a Gaussian random function (Klyatskin, 1980, 2001; Frisch, 1995), we get ðð dp(x; t) hu0i pi ¼ hu0i (x; t)u0k (x1 ; t1 )i ð2:4Þ dx1 dt1 ; duk (x1 ; t1 )dx1 dt1 In order to determine hu0i pi, we need to ﬁnd the functional derivative of p with respect to u. Making use of the self-evident equality q q f (xy) ¼ f (xy); qx qy

2.1 Kinetic Equation for the Single-Point PDF of Particle Velocity

we obtain

dp(x; t) duk (x1 ; t1 )dx1 dt1

dRp j (t) q p(x; t) qx j duk (x1 ; t1 )dx1 dt1 dvp j (t) q p(x; t) : qvj duk (x1 ; t1 )dx1 dt1

¼

ð2:5Þ

Consequently, it is necessary to determine the higher functional derivatives of particle coordinates and velocity with respect to velocity of the carrier ﬂuid. To this end, let us invoke the solutions of the equations of motion (1.45) and (1.46) written in the integral form, where the integrals are taken along the particle trajectory: ðt Rp (t) ¼

ðt vp (t1 )dt1 ; vp (t) ¼

1

1

tt1 u(Rp (t1 ); t1 ) exp þ F dt1 : tp tp

ð2:6Þ

Now, by applying the functional differentiation operator to Eq. (2.6), we get a system of integral equations with respect to the functional derivatives: dRp i (t) tt1 ¼ dij d(Rp (t1 )x1 ) 1exp H(tt1 ) tp duj (x1 ;t1 )dx1 dt1 ðt þ t1

dRp n (t2 ) tt2 qui (Rp (t2 );t2 ) dt2 ; 1exp tp qx n duj (x1 ;t1 )dx1 dt1 ð2:7Þ

d ij dvp i (t) tt1 ¼ d(Rp (t1 )x1 )exp H(tt1 ) duj (x1 ;t1 )dx1 dt1 tp tp ðt dRp n (t2 ) 1 tt2 qui (Rp (t2 );t2 ) þ exp dt2 ; qx n duj (x1 ;t1 )dx1 dt1 tp tp t1

ð2:8Þ where H(x) is the Heaviside function (H(x < 0) ¼ 0, H(x > 0) ¼ 1) and dij is the Kronecker symbol (d ij ¼ 1 at i ¼ j, d ij ¼ 0 at i 6¼ j). The Heaviside function enables the inception of ﬂuid action on the particle at the instant of time t ¼ t1. To solve the integral functional equation, we apply the iteration method (Zaichik, 1997, 1999). As the ﬁrst approximation for the solution of Eq. (2.7), we take the ﬁrst term on the right-hand side. We shall restrict ourselves to quasi-homogeneous ﬂows, where the average velocity gradients are either constant or nearly constant. Then, if we take only the ﬁrst two terms of the iteration expansion, the solution of Eq. (2.7) becomes

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42

dRp i (t) tt1 ¼ d(Rp (t1 )x1 )H(tt1 ) d ij 1exp duj (x1 ; t1 )dx1 dt1 tp þ

qui tt1 tt1 2tp þ (tt1 þ 2tp )exp : qx j tp

ð2:9Þ

A substitution of Eq. (2.9) into Eq. (2.8) gives d ij dvp i (t) tt1 ¼ d(Rp (t1 )x1 )H(tt1 ) exp tp tp duj (x1 ;t1 )dx1 dt1 þ

" qui tt1 tt1 qui qun tt1 3tp 1 1þ exp þ qx j tp tp qx n qx j

(tt1 )2 tt1 þ 3tp þ2(tt1 )þ exp : 2tp tp

ð2:10Þ

It is readily seen from Eq. (2.7) that the convergence rate of the iteration process is governed by the parameter DU e¼ Dx

ðt t1

tt2 DU 1 1exp dt2 T Lp ; 1Wp 1exp tp Dx Wp

tp Wp ¼ ; T Lp where DU/Dx is the characteristic the average velocity gradient of the carrier ﬂow, and TLp is the characteristic time of particle interactions with energy-carrying eddies. The parameter Wp describes the inertia of a particle in terms of the duration of its interaction with energy-containing eddies. The iteration process results in a rapid convergence when e is a small parameter. This condition is fulﬁlled for high-inertia particles, since e ! 0 at Wp ! 1. But for low-inertia particles (Wp ! 0) one runs into the same problem as in the theory of single-phase ﬂows since the parameter e ¼ TLDU/Dx is not small. Having said that, we shall still restrict ourselves to consider only the terms appearing in the expansions (2.9) and (2.10). Plugging Eqs. (2.9), (2.10) into Eqs. (2.4), (2.5) and averaging the obtained relations over the turbulent ﬂuctuation ensemble, we arrive at the following expression for the correlation between velocity ﬂuctuations of the continuous phase and the probability density of particle velocity: qP qP hu0i pi ¼ tp lij (x; t) þ mij (x; t) ; qvj qx j

ð2:11Þ

2.1 Kinetic Equation for the Single-Point PDF of Particle Velocity

1 lij (x; t) ¼ tp

ðt

hu0i (x; t)u0k (Rp (t1 ); t1 )i

1

d jk tt1 exp tp tp

" qU j qU j qU n tt1 tt1 tt1 3tp 1 1 þ exp þ þ qx k tp tp qx n qx k

(tt1 )2 tt1 þ 3tp þ 2(tt1 ) þ exp dt1 ; 2tp tp

1 mij (x; t) ¼ tp

þ

ðt 1

ð2:12Þ

tt1 hu0i (x; t)u0k (Rp (t1 ); t1 )i d jk 1exp tp

qU j tt1 tt1 2tp þ (tt1 þ 2tp )exp dt1 : qxk tp

ð2:13Þ

Here mij and lij denote the integrals containing the second correlation moment of continuous phase velocity ﬂuctuations along the particle trajectory. To calculate these integrals, one should determine the Lagrangian correlation moment of velocity ﬂuctuations of a ﬂuid particle moving along the inertial particle trajectory. But we ﬁrst determine the Lagrangian correlation moment of velocity ﬂuctuations for a ﬂuid particle moving along its own trajectory, BL ij (t) ¼ hu0i (x)u0j (R(tt); tt)jR(t) ¼ xi by using the relations BL ij (t) ¼

t Dhu0i u0k i hu0i u0k i 2

Dt

YL kj (t);

Dhu0i u0k i qhu0i u0k i qhu0i u0k i qhu0i u0k u0n i ¼ þ Un þ : Dt qt qx n qx n

ð2:14Þ

The tensor YL ij(t) in Eq. (2.14) denotes the Lagrangian autocorrelation function of continuous phases velocity ﬂuctuations. In contrast to relation (1.1) for the Lagrangian correlation moment of velocity ﬂuctuations of a continuum element, expression (2.14) contains an additional transport term Dhu0i u0k i=Dt. This term, which was ﬁrst introduced by Derevich (2000a), takes into account the transport of velocity ﬂuctuations of ﬂuid particles moving along their trajectories; this transport may occur via non-stationarity, convection, or diffusion. Thus approximation (2.14) implies the existence of two mechanisms of formation of the correlation moment BL ij(t) in non-stationary, non-homogeneous turbulence, in view of the fact that the

j43

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44

Lagrangian correlation moment: a) is determined locally through the Eulerian singlepoint second-order moment hu0i u0j i and b) takes into account the change of hu0i u0j i with time as well as the transport characterized by the average and ﬂuctuational velocities along a ﬂuid particle trajectory. It is clear that in stationary homogeneous isotropic turbulence, the second mechanism is absent. Inclusion of the transport term into Eq. (2.14) in some sense resembles Stratanovichs interpretation of stochastic integrals, according to which the integrand, in contrast to the Ito interpretation, is determined at the center and not at the boundary of the time interval (Gardiner, 1985). In the absence of the transport term relation (2.14) reduces to Popes expression for the Lagrangian correlation moments of ﬂuid particle velocity ﬂuctuations in anisotropic turbulence (Pope, 2002). By analogy with Eq. (2.14), the Lagrangian correlation moment of ﬂuid particle velocity ﬂuctuations along an inertial particle trajectory, BLp ij (t) ¼ hu0i (x)u0j (Rp (tt); tt)jRp (t) ¼ xi shows up as BLp ij (t) ¼

t Dp hu0i u0k i hu0i u0k i YLp kj (t); 2 Dt

Dp hu0i u0k i qhu0i u0k i qhu0i u0k i qhu0i u0k v0n i ¼ þ Vn þ ; Dt qt qx n qx n

ð2:15Þ

where YLp ij(t) denotes the autocorrelation function of continuous phases velocity ﬂuctuations determined along the particle trajectory. As we remarked in Section 1.3, it is only in the case of inertialess particles that the correlation moments of velocity ﬂuctuations in the carrier ﬂow calculated along particle trajectories coincide with the ordinary Lagrangian correlations determined along the trajectories of continuum elements (ﬂuid particles). Substituting Eq. (2.15) into Eqs. (2.12)–(2.13) and dropping third-order and higherorder terms when considering the average velocity gradient, we get after integration: lij ¼

hu0i u0k i

fu kj

qU j 1 Dp hu0i u0k i fu1 kj þ tp lu1 kn ; qx n 2 Dt

mij ¼

qU j qU n qU j þ lu kn þ tp mu kl tp qx n qx l qxn

hu0i u0k i

qU j tp Dp hu0i u0k i g u kj þ tp hu kn g u1 kj : qx n 2 Dt

ð2:16Þ

ð2:17Þ

Coefﬁcients fu ij, gu ij, lu ij, hu ij, mu ij, f1u ij, g1u ij, l1u ij, characterize the degree to which the particles are involved in turbulent motion of the continuum. In matrix notation, these coefﬁcients may be written as

2.1 Kinetic Equation for the Single-Point PDF of Particle Velocity

f u ¼ Mu0 ; gu ¼ Nu0 f u ; f u1 ¼ Mu1 ; lu ¼ gu f u1 ; hu ¼ Nu1 þ Mu1 2gu ; mu ¼ Nu1 þ 2Mu1 þ Mu2 3gu ; gu1 ¼ Nu1 f u1 ; f u2 ¼ 2Mu2 ; lu1 ¼ gu1 f u2 ; 1 ð 1 t (1)n dn F(s) n Mun ¼ nþ1 YLp (t)t exp I dt ¼ nþ1 ; n!tp n!tp tp dsn 0

Nun ¼

1 ð 1 (1)n dn F(s) YLp (t)tn dt ¼ nþ1 lim n ; nþ1 n!tp s!0 ds n!tp

ð2:18Þ

0

where F(s) denotes the Laplace transform of the autocorrelation function YLp(t), and s ¼ t1 p . The involvement coefﬁcient (2.18) satisﬁes the following limiting relations: f u ¼ f u1 ¼ I; f u2 ¼ 2I; gu ¼ lu ¼ Nu0 ; hu ¼ mu ¼ gu1 ¼ lu1 ¼ Nu1 at tp T1 L ¼ I; f u ¼ Nu0 ; gu ¼ f u1 ¼ Nu1 ; lu ¼ Nu2 ; gu1 ¼ f u2 ¼ 2Nu2 ; hu ¼ Nu3 ; lu1 ¼ 3Nu3 ; ð2:19Þ tp T1 L I:

mu ¼ Nu4 at ð2:20Þ Ð1 where TLp 0 YLp (t)dt is a matrix whose components have the meaning of microtimes of particle interaction with the turbulence. If the autocorrelation function is given as an exponential approximation, that is, YLp (t) ¼ exp(tT1 Lp );

ð2:21Þ

then 1 F(s) ¼ (sI þ T1 Lp ) ;

(nþ1) ; Mun ¼ (I þ tp T1 Lp )

Nun ¼ (TLp =tp )nþ1 ;

and the involvement coefﬁcients (2.18) take the form 1 1 1 2 1 f u ¼(Iþtp T1 Lp ) ; gu ¼(TLp =tp )(Iþtp TLp ) ; f u1 ¼(Iþtp TLp ) ; 2 2 1 2 lu ¼(TLp =tp )(Iþtp T1 Lp ) ; hu ¼(TLp =tp ) (Iþtp TLp ) ; 1 2 3 2 mu ¼(TLp =tp )2 (Iþtp T1 Lp ) ; gu1 ¼(TLp =tp ) (Iþtp TLp ) ; 1 1 3 3 2 f u2 ¼2(Iþtp T1 Lp ) ; lu1 ¼ (Iþ3tp TLp )(TLp =tp ) (Iþtp TLp ) ;

ð2:22Þ

where T1 Lp is the matrix inverse to TLp. It is evident that expression (2.22) obeys the asymptotic relations (2.19) and (2.20). If we neglect anisotropy of the duration of particle interactions with turbulent eddies, the tensors (2.18) and (2.22) become isotropic, in other words, their

j45

j 2 Motion of Particles in Gradient Turbulent Flows

46

components turn into scalars. Anisotropy of the involvement coefﬁcients (2.18) or (2.22) may be caused by two reasons: anisotropy of turbulence scales of the carrier continuum, and the crossing trajectory effect, which also takes place in isotropic turbulence. Anisotropy of turbulence scales is taken into consideration by Berlemont et al. (1990), Zhou and Leschziner (1996), Pascal and Oesterle (2000), Caballina et al. (2004), Carlier et al. (2004, 2005) within the framework of Lagrangian trajectory modeling, and also by Alipchenkov and Zaichik (2004), Zaichik, Oesterle and Alipchenkov (2004), who use the kinetic equations for the PDF as their starting point. A substitution of Eq. (2.11) into Eq. (2.3) leads to the following kinetic equation for the single-point PDF of particle velocity distribution in a turbulent shear ﬂow: qP qP q þ þ vi qt qx i qvi

U i vi q2 P q2 P þ F i P ¼ lij þ mij ; tp qvi qvj qx j qvi

ð2:23Þ

where mij and lij are determined by the relations (2.16) and (2.17). The terms on both sides of Eq. (2.23) describe, respectively, the convection and the diffusion of a singleparticle PDF in the phase space. Modeling turbulence by a Gaussian process, we can express the particles interaction with turbulent eddies in terms of a second order differential (diffusion) operator, which takes anisotropy of the time scale of turbulence into proper consideration through its involvement coefﬁcients. If we neglect anisotropy of turbulence scales and the contribution of transport terms containing Dp hu0i u0k i=Dt to Eqs. (2.16)–(2.17), then Eq. (2.23) reduces to the kinetic equation obtained by Zaichik (1997). In addition, when we neglect the terms containing average velocity gradients, there follows from Eq. (2.23) an equation for the PDF of particle velocity distribution in a homogeneous, shearless turbulent ﬂow (Derevich and Zaichik, 1988). It should be noted that some works, for instance, Kroshilin et al., 1985; Vatazhin and Klimenko, 1994), describe the interaction between particles and turbulent eddies of the continuum by the same diffusion operator in the velocity subspace that appears in the theory of Brownian motion, and the kinetic equation essentially coincides with the classic Fokker–Planck equation. However, the Fokker–Planck equation is suitable for describing a Markovian process that is dcorrelated in time, and consequently, it is suitable for modeling the motion of a particle driven by a random force of the so-called white noise type. Hence, a decision to use the Fokker–Planck equation is justiﬁed only for very inertial particles whose dynamic relaxation time by far exceeds the temporal microscale of turbulence. The kinetic equation (2.23) is obtained through a functional formalism based on the Furutsu–Donsker–Novikov formula for a Gaussian random ﬁeld. This method has been applied to this type of problems by Derevich and Zaichik (1988, 1990), Swailes and Darbyshire (1997), Hyland et al. (1999a), Zaichik (1999), Derevich (2000a), Pandya and Mashayek (2002b). The difference between these approaches is in the way of solving the system of integral equations for functional derivatives. In the present approach, the iteration method is employed to solve equations (2.7) and (2.8), which allows to get the kinetic equation in an explicit, closed form, whereas in the works of Swailes and Darbyshire (1997), Hyland et al. (1999a), Pandya and Mashayek (2002b), the closure problem for the kinetic equation reduces to that of

2.2 Equations for Single-Point Moments of Particle Velocity

solving an ordinary differential equation for the Green function. Both approaches become equivalent for times large enough compared to the integral scale of turbulence for quasi-homogeneous ﬂows. Several other methods of deriving kinetic equations for the PDF of particle velocity distribution have been described in literature. Thus, Reeks (1991) has obtained the kinetic equation using the principle of invariance with respect to random Galilean transformations (Kraichnan, 1965). In his follow-up work, Reeks (1992) derived a kinetic equation describing the motion of particles in non-homogeneous turbulence by summation of direct interactions using the Lagrangian method of renormalized perturbation theory (Lagrangian history direct interaction approximation – Kraichnan, 1965, 1977). This method was also used by Pandya and Mashayek (2003) to derive an equation for the joint PDF of velocity and temperature distribution for a particle. An alternative way to construct a closed kinetic equation was elaborated by Pozorski (1998) and Pozorski and Minier (1999), who carried out an expansion of the characteristic functional in a cumulant series (cumulant expansion – Van Kampen, 1992). When turbulence is modeled by Gaussian random ﬁelds, all three methods – functional formalism relying on the Furutsu–Donsker–Novikov formula, summation of direct interactions, and cumulant expansion – lead to the same kinetic equation for particles, the only difference being that the last two methods are not restricted to the case of a Gaussian distribution. In the works by Pozorski and Minier (1999), Peirano and Minier (2002), Minier and Peirano (2004), and Reeks (2005a), more general kinetic equations have been derived, in which the independent variables (i.e., phase space coordinates) include not only the ordinary coordinates and velocities of a particle but also the coordinates and velocities of continuum elements moving along ﬂuid particle trajectories as well as inertial particle trajectories. The motion of ﬂuid particles along their own trajectories is modeled by a linear stochastic equation, with the random function considered as Gaussian white noise. Such processes are known as Wiener random processes (Pope, 1994, 2000). The problem of modeling the turbulent characteristics of ﬂuid particles moving along inertial particle trajectories is more complicated, since one has to account for the effects of crossing trajectory, particle inertia, and continuity, which were discussed in Section 1.3. Simonin et al. (1993) incorporated these effects into the Langevin stochastic equation of motion in an effort to describe the transport of continuum elements along inertial particle trajectories. However, the problem of modeling the turbulent motion of the continuum is beyond the scope of the present book, and kinetic equations containing the description of ﬂuid particles will not be considered here.

2.2 Equations for Single-Point Moments of Particle Velocity

Direct solution of the kinetic equation presents a very difﬁcult problem owing to the high dimensionality of the phase space. Analytical and numerical solutions are obtainable in this way only for relatively simple ﬂows, which undoubtedly present a

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48

special interest in the context of analyzing particle behavior in some limiting cases, and also for the construction of closure relations and boundary conditions (Derevich and Zaichik, 1988; Derevich, 1991; Swailes and Reeks, 1994; Swailes et al., 1998; Devenish et al., 1999; Hyland et al., 1999b; Alipchenkov et al., 2001). However, it is more rational from the calculation standpoint to solve only the equations for the ﬁrst several moments appearing in the kinetic equations, even though it entails a loss of some statistical information about particle behavior. By integrating the kinetic equation (2.23) over the velocity subspace, we can obtain a system of continuous equations for the averaged single-point hydrodynamic characteristics of the disperse phase. It should be mentioned that determination of the averaged characteristics of the disperse phase by integrating the PDF over the velocity subspace is similar to the Favre averaging (which is well-known in the theory of one-phase compressible turbulent ﬂows), with density as the weighting function. The equation of mass conservation is qF qFV k ¼ 0; þ qx k qt ð ð 1 F ¼ Pdv; V i ¼ ð2:24Þ vi Pdv; F where F and Vi are, respectively, the average volume concentration and the disperse phases velocity. The balance-of-momentum equation has the form Dp ik q ln F qhv0 v0 i U i V i qV i qV i þ Vk ¼ i k þ þ Fi ; qx k qt qx k tp tp qx k

ð2:25Þ

where

ð 1 (vi V i )(vj V j )Pdv F are the turbulent stresses of the disperse phase caused by the involvement of particles in ﬂuctuational motion of the continuum. The last term in Eq. (2.25) describes turbulent diffusion of particles. The tensor of particles turbulent diffusion is deﬁned as hv0i v0j i ¼

Dp ij ¼ tp (hv0i v0j i þ mij )

qU j t2p Dp hu0i u0k i g u1 kj: ð2:26Þ ¼ tp hv0i v0j i þ hu0i u0k ig u kj þ t2p hu0i u0k ihu kn Dt qx n 2 The equation for the second moments of particle velocity ﬂuctuations (turbulent stresses of the disperse phase) is presented below: qhv0i v0j i qt

þ Vk

qhv0i v0j i qx k

0 0 0

þ

qV j 1 qFhvi vj vk i ¼ hv0i v0k i þ mik qx k qxk F

2hv0i v0j i

qV i hv0j v0k i þ mjk þ lij þ lji : qx k tp

ð2:27Þ

2.2 Equations for Single-Point Moments of Particle Velocity

Equation (2.27) describes convective and diffusive transport, generation of ﬂuctuations caused by the velocity gradient, generation of ﬂuctuations resulting from particles involvement in the irregular motion of the carrier ﬂow, and dissipation of turbulent stresses of the disperse phase due to the work done by the interfacial interaction force. For low-inertia particles, all differential terms that give rise to transport and generation of ﬂuctuations by the average velocity gradient can be omitted, and from Eq. (2.27) there follows a relation describing the local equilibrium between turbulent stresses in the disperse and continuous phases: hv0i v0j i ¼

hu0i u0k if kj þ hu0j u0k if ki 2

:

ð2:28Þ

In isotropic turbulence, expression (2.28) reduces to Eq. (1.99). Local equilibrium relations of the type (2.28) are often used in construction of simple continual models of particle motion and deposition in turbulent ﬂows (see, for example, Derevich and Zaichik, 1988; Johansen, 1991; Guha, 1997; Young and Leeming, 1997; Cerbelli et al., 2001; Slater et al., 2003; Zaichik et al., 2004). But models based on local equilibrium relations for turbulent stresses hold only for low-inertia particles and are not applicable to particles whose relaxation time exceeds the integral scale of turbulence of the continuum. To describe turbulent transport in the disperse phase consisting of inertial particles, one has to use non-local transport models. The system of equations following from Eq. (2.23) is similar to the well-known Friedmann–Keller chain of equations that we encounter in the theory of one-phase turbulent ﬂows (Monin and Yaglom, 1971). It is unclosed, since an equation for n-th moment contains (n + 1)-th moment. To get a closed system of equations, this chain should be broken by adding the closure relations. Thus, the equation for third-order moments of velocity ﬂuctuations may be closed with the help of the quasi-normal Millionshikov hypothesis, which postulates that fourth-order cumulants are equal to zero and represents fourth-order moments as sums of products of second-order moments: hv0i v0j v0k v0n i ¼ hv0i v0j ihv0k v0n i þ hv0i v0k ihv0j v0n i þ hv0i v0n ihv0j v0k i:

ð2:29Þ

In view of Eq. (2.29), we get from Eq. (2.23) a closed equation for third-order moments: qhv0i v0j v0k i qt þ

þ Vn

qhv0i v0j v0k i

0 0 Dp in qhvj vk i

tp

qx n

qx n þ

þhv0i v0j v0n i

qV j qV k qV i þhv0i v0k v0n i þhv0j v0k v0n i qx n qx n qx n

Dp jn qhv0i v0k i Dp kn qhv0i v0j i 3hv0i v0j v0k i þ þ ¼ 0: tp qx n tp qx n tp

ð2:30Þ

The system of equations (2.24)–(2.27)and (2.30) describes disperse phase motion in terms of third-order moments. To derive a system of equations for continual modeling of the disperse phase on the level of second-order moments, one has to get algebraic relations that would express third-order moments through second-order moments and their derivatives. Such relations could be obtained from Eq. (2.30) by

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50

ignoring the terms that describe time evolution, convection, and generation of thirdorder moments by the average velocity gradients: qhv0j v0k i qhv0i v0j i qhv0i v0k i 1 þ Dp jn þ Dp kn Dp in : ð2:31Þ hv0i v0j v0k i ¼ qxn qx n qx n 3 Formula (2.31) for third-order moments of particle velocity ﬂuctuations was obtained by Simonin (1991b) and Zaichik and Vinberg (1991) by simplifying the corresponding differential equations and independently by Swailes et al. (1998), who derived Eq. (2.31) by solving the kinetic equation by the Chapman–Enskog method. Wang et al. (1998) performed veriﬁcation of gradient relations of the kind (2.31) for the ﬂow in a planar vertical channel by comparing them with the results of LES calculations and ascertained that gradient relations predict roughly similar, though somewhat overrated, values for third-order moments. Equations (2.24)–(2.27) together with Eq. (2.31) allow to simulate the hydrodynamics of particles on the level of equations for second-order moments. Secondorder differential models were ﬁrst put to this purpose by Kondratev and Shor (1990), Derevich (1991), Simonin (1991a), Zaichik and Vinberg (1991). Consider the asymptotic behavior of this model in the limit of inertialess particles, that is, at tp ! 0. From the balance-of-momentum equation (2.25), it follows that the average velocity of the disperse phase consisting of inertialess particles is equal to V i ¼ U i Dik

q ln F : qx k

ð2:32Þ

Due to Eq. (2.32), the velocity of inertialess particles is a sum of convective and diffusional components, convective transport occuring with the average velocity of the carrier ﬂow. Recalling the ﬁrst relation (2.19), we see from Eq. (2.28) that turbulent stresses of inertialess particles coincide with Reynolds stresses of the continuum: lim hv0i v0j i ¼ hu0i u0j i

ð2:33Þ

tp ! 0

According to Eqs. (2.19), (2.26), (2.33), the components of the turbulent diffusion tensor for inertialess impurity (passive scalar) are as follows: Dij ¼ lim Dp ij ¼ hu0i u0k iT L kj þ hu0i u0k icL kn tp !0

cL ij ¼

ð1 0

qU j Dhu0i u0k i cL kj ; Dt qx n 2

ð2:34Þ

YL ij (t)tdt;

where, according to the deﬁnition (2.14), YL ij(t) is a Lagrangian autocorrelation function of continuous mediums velocity ﬂuctuations. The second term on the righthand side of Eq. (2.34), which was ﬁrst introduced by Riley and Corrsin (1974) for the case of homogeneous turbulence with a constant rate of shear, describes the direct effect of the average velocity gradient on the diffusion tensor of passive impurity. The third term accounts for the contribution of the transport effect arising as a result of non-stationarity as well as the convective and diffusion transport of

2.2 Equations for Single-Point Moments of Particle Velocity

turbulent ﬂuctuations. It follows from Eq. (2.34) that even if the autocorrelation function YL ij(t) is symmetric and consequently, TL ij and cL ij are also symmetric, the turbulent diffusion tensor of inertialess impurity in a shear ﬂow is non-symmetric (Riley and Corrsin, 1974). Ounis and Ahmadi (1991) were the ﬁrst to arrive to a similar conclusion in regard to the turbulent diffusion tensor of inertial particles in a ﬂow with constant shear. Substitution of Eq. (2.32) into Eq. (2.24) yields the diffusion equation for inertialess impurity: qF qFU k q qF ¼ þ Dik : ð2:35Þ qx k qt qx i qx k Equation (2.35) is the standard equation of passive scalar diffusion, in which the inﬂuence of the velocity gradient of the average motion is taken into account by the presence of the two last terms in Eq. (2.34). Hence the problem of turbulent diffusion at tp ! 0 involves a transition to the limiting case of passive impurity. In the inertialess limit, Eq. (2.31) together with Eq. (2.33) and Eq. (2.34) leads to the following expression for third-order moments of continuous mediums velocity ﬂuctuations: qhu0j u0k i qhu0i u0j i qhu0i u0k i 1 hu0i u0j u0k i ¼ lim hv0i v0j v0k i ¼ Din þ Djn þ Dkn : tp !0 qx n qx n qx n 3 ð2:36Þ Dropping the terms that are due to the average velocity gradient and to the inhomogeneity of turbulence, that is, the last two terms in Eq. (2.34), we reduce the expression (2.36) to the following relation (Cho et al., 2005): qhu0j u0k i qhu0i u0k i 1 þ hu0j u0l iT L ln hu0i u0j u0k i ¼ hu0i u0l iT L ln qx n qx n 3 qhu0i u0j i 0 0 þhuk ul iT L ln : ð2:37Þ qx n If we take the Lagrangian integral time scale tensor as isotropic, T L ij ¼ T L d ij ; then Eq. (2.37) reduces to qhu0j u0k i qhu0i u0j i qhu0i u0k i TL hu0i u0j u0k i ¼ þ hu0j u0n i þ hu0k u0n i hu0i u0n i : qx n 3 qx n qx n

ð2:38Þ

ð2:39Þ

At high Reynolds numbers, the time macroscale of turbulence is expressed through the turbulent energy and its rate of dissipation: k TL ¼ a ; e

a ¼ const:

ð2:40Þ

In view of Eq. (2.40), the relation (2.39) turns into the formula obtained by Hanjalic and Launder (1972) for third-order moments of velocity ﬂuctuations in single-phase turbulence,

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52

hu0i u0j u0k i¼b

qhu0j u0k i qhu0i u0j i qhu0 u0 i k a þhu0j u0n i i k þhu0k u0n i hu0i u0n i ; b¼ : qx n qx n qx n e 3 ð2:41Þ

Thus the second-order differential model for hydrodynamics of the disperse phase as given by Eqs. (2.24)–(2.27) and Eq. (2.31) enables a correct transition to the limiting case of inertialess particles and reproduces the well-established relations provided by the theory of one-phase turbulence. Therefore, in spite of the fact that the proposed model can be veriﬁed rigorously only in the limit of high-inertia particles, that is, at tp TLp when the velocity distribution in homogeneous turbulence is close to normal, it is safe to assume that the model is applicable in the entire range of particle inertia. By analogy with Eq. (2.31), third-order moments entering the transport term with Dp hu0i u0j i=Dt in Eq. (2.15), may be written as qhu0j u0k i qhu0i u0j i qhu0i u0k i 1 þ Dp jn þ Dp kn hu0i u0j v0k i ¼ : ð2:42Þ Dp in qx n qxn qx n 3 Second-order mixed correlation moments of velocity ﬂuctuations of the disperse and continuous phases may be determined directly from Eq. (2.11) by making use of Eq. (2.16) and Eq. (2.17): ð ð ð qV j 1 1 0 0 0 0 0 hui pivj dvV j hui pidv ¼ tp lij mik : hui (vj V j )pidv ¼ hui vj i ¼ qx k F F ð2:43Þ These correlation moments are needed primarily for the purpose of taking the feedback action of particles on the turbulence into proper consideration. As can be seen from Eq. (2.43), the tensor of mixed moments, in contrast to the tensors of turbulent stresses in the continuous and disperse phases hu0i u0j i and hv0i v0j i, is nonsymmetric. Simonin (1991a) and Simonin et al. (1993) have proposed to use the differential equations based on the Langevin equations of ﬂuid motion and inertial particle motion for modeling the mixed correlation moments hu0i v0j i and their convolution hu0k v0k i.

2.3 Algebraic Models of Turbulent Stresses

Computer simulation of complex three-dimensional single-phase ﬂows on the basis of a system of differential equations for all components of turbulent stresses is a very time-consuming operation even for fastest computers. This explains the popularity of algebraic models for the Reynolds stresses that use only the differential equations for turbulent energy; such models are commonly used in calculations of single-phase ﬂows. For all practical purposes, nonlinear explicit algebraic models offer the same accuracy as differential models for second and third moments of velocity and

2.3 Algebraic Models of Turbulent Stresses

temperature ﬂuctuations, which allows to run the calculations much faster while at the same time increasing stability of numerical schemes. Application of algebraic models toward the calculation of real-life two-phase or multiphase turbulent ﬂows is an even more urgent problem than the problem of calculating single-phase ﬂows. As a rule, the two-phase ﬂow is associated with a poly-disperse system consisting of particle of different sizes. The most general calculation method suitable for such a system is the fraction method, which aims to represent the whole particle system as consisting of separate mono-fractions, with subsequent modeling of mass, momentum, and heat transport in each fraction. It is obvious that any attempt to apply a second-order differential model to a system of particles divided into a large number of fractions would involve dramatically longer calculation times as compared to a onephase ﬂow. In this section we look at two different ways to construct algebraic models of turbulent stresses in the disperse phase. The two approaches are based, respectively, on solving the kinetic equation for the PDF by the Chapman–Enskog perturbation method and on solving the equation for turbulent stresses by the iteration method. 2.3.1 Solution of the Kinetic Equation by the Chapman–Enskog Method

The Chapman–Enskog expansion was devised as a way to solve the Boltzmann equation in the kinetic theory of gases (Chapman and Cowling, 1970). The zero-th term of the expansion, which corresponds to the equilibrium Maxwellian distribution of molecular velocities, leads to Eulers equations; the ﬁrst-order expansion gives birth to the Navier–Stokes equations; and the second-order expansion yields Barnetts hydrodynamic equations. Buyevich (1971b, 1972a) employed the Chapman–Enskog method to solve the Fokker–Planck equation describing the Brownian motion of particles as well as the motion of a pseudo-turbulent disperse medium whose parameter ﬂuctuations result from the random conﬁguration of particles. Later on, this method was invoked by Derevich and Yeroshenko (1990), Derevich (1991), Zaichik (1992a), Zaichik et al. (1997a), and Swailes et al. (1998) to solve the kinetic equations that model the dynamics of inertial particles in turbulent ﬂows. Let us apply the Chapman–Enskog perturbation method to the equation (2.23) presented in the operator form: R[P] ¼ N[P];

ð2:44Þ

where 1 2kp q2 P q(vi V i )P þ ; tp 3 qvi qvi qvi 2 2kp qP qP U i V i qP q P q2 P þ þ Fi þ dij lij mij : þ vi N[P] ¼ tp 3tp qt qx i qvi qvi qvj qx j qvi

R[P] ¼

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54

Here R[P] is an operator that realizes the Maxwellian distribution; the operator N[P] corresponds to the deviation of the actual particle velocity PDF from the equilibrium Maxwellian distribution; kp hv0k v0k i=2 is the turbulent kinetic energy of the disperse phase. We seek for the solution of Eq. (2.44) in the form of a series P(v) ¼ P ð0Þ (v) þ Pð1Þ (v) þ . . . ;

ð2:45Þ

where the functions P(0)(v) and P(1)(v) obey the equations R Pð0Þ ¼ 0;

ð2:46Þ

R Pð1Þ ¼ N P ð0Þ :

ð2:47Þ

Representation of the solution in the form (2.45) is valid only in the case when the ﬂow does not deviate much from its equilibrium state; the deviation is characterized by the parameter tp/Tu, where Tu is the characteristic time of change of the averaged parameters of the hydrodynamic ﬂow. This parameter is analogous to the Knudsen number in the kinetic theory of gases. The solution of Eq. (2.46) is a Maxwellian distribution 3v0k v0k 3 3=2 ð0Þ exp : ð2:48Þ P (v) ¼ F 4kp 4pkp Owing to hv0i v0j i0 ¼

ð 1 0 0 ð0Þ 2 v v P dv ¼ kp d ij ; F i j 3

ð2:49Þ

ð 1 0 0 0 ð0Þ v v v P dv ¼ 0: F i j k

ð2:50Þ

we can write hv0i v0j v0k i0 ¼

A substitution of Eq. (2.49) into Eq. (2.25) results in a closed equation of motion for the disperse phase that is similar to Eulers equation for a continuous medium. Let us now determine the right-hand side of Eq. (2.47) by using expression (2.48). The time derivatives qF/qt, qVi/qt, qkp/qt are determined, respectively, from Eqs. (2.24), (2.25), and (2.27), with Eqs. (2.49), (2.50) in mind. We obtain as the result: 3m V j;k 3lij v0 v0 1 3 v0i v0j k k dij V i;j þ ik N[Pð0Þ ] ¼ 3 2kp 2kp 2kp þv0i

0 0 3mij 3vk vk 5 dij þ kp;j Pð0Þ (v); 2kp 2kp

ð2:51Þ

where the subscript i denotes a spatial derivative with respect to coordinate xi.

2.3 Algebraic Models of Turbulent Stresses

Making use of Eq. (2.51), we arrive at the solution of Eq. (2.47): Pð1Þ (v) ¼

þ

3m V j;k 3lij tp 3 0 0 v0k v0k dij vi vj V i;j þ ik 2kp 2 3 2kp 2kp

3mij v0i 3v0k v0k 5 dij þ kp;j Pð0Þ (v): 3 2kp 2kp

ð2:52Þ

Then, in accordance with Eq. (2.48) and Eq. (2.52), we get ð 1 0 0 ð0Þ hv0i v0j i1 ¼ hv0i v0j i0 þ hv0i v0j ið1Þ ¼ vi vj P þ Pð1Þ dv F tp kp 2 2 ¼ kp dij V i;j þ V j;i V k;k d ij 3 3 3 tp tp 2 2 mik V j;k þ mjk V i;k mkl V k;l d ij þ lij þ lji lkk d ij ; 2 2 3 3 ð2:53Þ hv0i v0j v0k i1 ¼

ð 1 0 0 0 ð1Þ 2 vi vj vk P dv ¼ ðd ij Dp kn þ d ik Dp jn þ djk Dp in Þkp;n ; F 9 ð2:54Þ

Dp ij ¼ tp

2kp dij þ mij : 3

ð2:55Þ

Expression (2.53) suggests a linear dependence of turbulent stresses hv0i v0j i on the velocity gradient Vi,j. It should be noted that in addition to the terms containing the average velocity gradients Vi,j, which are routinely encountered in various turbulent viscosity models, the expression for hv0i v0j i also contains the terms lij that explicitly describe the interaction of particles with turbulent eddies of the carrier continuum. These terms are of special importance for low-inertia particles as they ensure a smooth transition to the limiting case of inertialess particles (2.33). Once Eq. (2.25) is closed by means of Eq. (2.53), it may be considered as the analogue of the Navier–Stokes equation for the continuum. Formula (2.54) for third-order moments of velocity ﬂuctuations would coincide with Eq. (2.31) if we plugged the isotropic representation (2.49) for turbulent stresses into Eq. (2.31). If, for the sake of simplicity, we neglect the terms containing mij and conﬁne oneselves to a single term in expression (2.16) for lij taken in the quasi-isotropic form lij ¼ f u hu0i u0k i=tp , then Eq. (2.53) reduces to tp kp 2 2 2k V i;j þ V j;i V k;k dij þ f u hu0i u0j i d ij : hv0i v0j i1 ¼ kp d ij 3 3 3 3

ð2:56Þ

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56

If we represent turbulent stresses of the continuum by the linear approximation 2 2 hu0i u0j i ¼ kdij nT U i;j þ U j;i U k;k d ij ; 3 3 and take note of the equality of the average velocities of the disperse and continuous phases (Vi Ui) in the inertialess particle limit, then expression (2.56) takes the form of a Boussinesq relation (Zaichik, 1992b): 2 2 ð2:57Þ hv0i v0j i1 ¼ kp d ij np V i; j þ V j;i V k;k d ij ; 3 3 np ¼ f u nT þ

tp kp : 3

ð2:58Þ

As evidenced by Eq. (2.58), the turbulent viscosity coefﬁcient of the disperse phase vp reduces to the turbulent viscosity coefﬁcient of the continuous phase (ﬂuid) vT in the inertialess limit, in other words, lim np ¼ nT :

ð2:59Þ

tp ! 0

In order to simplify Eq. (2.54), let us ignore the differential terms in the turbulent diffusion tensor of particles (2.55) and use an isotropic approximation for the turbulent stresses of both phases. In this approximation, the turbulent diffusion tensor of particles shows up as (Zaichik, 1992) 2 Dp ij ¼ Dp d ij ; Dp ¼ tp (kp þ g u k); 3

gu ¼

T Lp fu : tp

ð2:60Þ

In the inertialess approximation, the turbulent diffusion coefﬁcient for the particles coincides with the turbulent diffusion coefﬁcient for a passive scalar: 2 lim Dp ¼ DT ; DT ¼ kT L : 3

tp ! 0

ð2:61Þ

A substitution of Eq. (2.60) into Eq. (2.54) yields hv0i v0i v0k i1 ¼

2Dp ðdij dkn þ dik djn þ djk din Þkp;n : 9

ð2:62Þ

By solving the kinetic equation for high-inertia particles (tp/TLp > 1, where TLp is the characteristic time of particle interaction with energy-carrying turbulent eddies), Derevich and Yeroshenko (1990) have obtained expressions (2.57) and (2.62) for the second- and third-order moments of velocity ﬂuctuations with the following coefﬁcients of turbulent viscosity and diffusion: np ¼

tp kp ; 3

2 Dp ¼ tp kp : 3

ð2:63Þ

ð2:64Þ

2.3 Algebraic Models of Turbulent Stresses

It is evident that formulas (2.63) and (2.64) do not hold for inertialess particles, since instead of tending to Eq. (2.59) and Eq. (2.61), they predict zero values of viscosity and diffusion coefﬁcients in the inertialess limit. It should be noted that the range of applicability of the Boussinesq relation (2.57) with turbulent viscosity coefﬁcient given by Eq. (2.63) must be narrow enough, because it is limited on either side by the inequalities tp/Tu < 1 and tp/TLp > 1, quantities Tu and TLp having the same order. Owing to Eq. (2.27), the equation for the turbulent energy of the disperse phase has the form qkp qkp 2kp 1 qFhv0i v0i v0k i qV i þ ¼ ðhv0i v0k i þ mik Þ þ lkk : þ Vk qx k qt qx k 2F qx k tp

ð2:65Þ

In order to simplify Eq. (2.65), let us ignore the contribution of mij and conﬁne ourselves to just one term in the expression (2.16) for lij, using a quasi-isotropic representation lij ¼ f u hu0i u0k i=tp . If, in addition, we invoke the relations (2.57) and (2.62), then Eq. (2.65) will be rewritten as qkp qkp qkp 5 q 2 qV k qV k þ Vk ¼ kp þ np FDp qt qx k 9F qx k qx k qx k qx k 3 qV i qV k qV i 2ð fu kkp Þ þ np þ þ qx k qx i qx k tp

ð2:66Þ

The terms in Eqs. (2.65)–(2.66) have a clear physical meaning. They describe, respectively, the time rate of change, convection, turbulent diffusion, and generation (by the averaged motion) of disperse phases turbulent energy, as well as its generation and dissipation via interactions between particles and turbulent eddies. Relations (2.53), (2.56) and (2.57) together with the equation for the turbulent energy of particles (2.65) or (2.66) describe turbulent stresses of the disperse phase in the framework of a linear algebraic model. Linear models that include differential equations for kp have been proposed by Zhou and Huang (1988), Derevich and Yeroshenko (1990), Vinberg et al. (1992), Lain and Aliod (2000). The Chapman–Enskog perturbation method may be applied to ﬁnd the second approximation for the expansion (2.45) and construct on its basis a nonlinear algebraic model in the same way as it was done by Chen et al. (2004), who derived a quadratic model for the Reynolds stresses in single-phase turbulence by solving the simplest kinetic equation with a collisional relaxation term. However, such an approach becomes too cumbersome when applied to the kinetic equation (2.66). This is why it makes more sense to apply the iteration method to derive a nonlinear algebraic model.

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2.3.2 Solution of the Equation for Turbulent Stresses by the Iteration Method

The notion of an equilibrium state of the turbulent ﬂow is of fundamental importance to us when we construct algebraic models. Such a state is characterized by constant values of all components of the anisotropy tensor and other dimensionless correlation moments of higher orders. The possibility of realizing the equilibrium state has been demonstrated in a number of physical and numerical experiments involving two types of single-phase turbulent ﬂows: uniform ﬂows (e.g., a ﬂow with a constant shear rate) and ﬂows characterized by equal rates of generation and dissipation of turbulent energy (e.g., near-wall ﬂow with a logarithmic velocity proﬁle). To achieve an equilibrium state, it is necessary that the transport effects caused by nonstationary, convective, and diffusive transport of turbulent ﬂuctuations play an insigniﬁcant role. A two-phase ﬂow is said to be at equilibrium when convection and diffusion are unimportant and the anisotropy tensors of ﬂuid velocity ﬂuctuations bij, particle velocity ﬂuctuations bp ij, and correlations of velocity ﬂuctuations of the continuous and disperse phases bfp ij are unchanged in time. The anisotropy tensors are deﬁned as bij ¼

hu0i u0j i 2k

dij ; 3

bp ij ¼

hv0i v0j i 2kp

dij ; 3

bf p ij ¼

2hu0i v0j i kf p

dij ; 3

ð2:67Þ

where k hu0n u0n i=2 is the turbulent kinetic energy of the continuous phase, and kf p hu0n v0n i=2 is the kinetic energy of velocity correlations of the continuous and disperse phases. Following Rodi (1976), we can ﬁnd the turbulent stresses of the disperse phase in a state of equilibrium by using an approximation that expresses transport terms in the equation (2.27) for the second moments through transport terms in the equation for the turbulent kinetic energy of particles: Dp hv0i v0j i Dt

þ

1 Dp hu0i u0k i fu1 kj þ tp lu1 kn U j;n g u1 kn V j;n 2 Dt

0 0

1 Dp huj uk i fu1 ki þ tp lu1 kn U i;n g u1 kn V i;n þ 2 Dt hv0i v0j i Dp kp 1 Dp hu0k u0l i

; þ ¼ fu1 lk þ tp lu1 ln U k;n g u1 ln V k;n Dt kp 2 Dt

Dp hv0i v0j i Dt

¼

qhv0i v0j i qt þV k

þ Vk

qhv0i v0j i qx k

0 0 0

þ

1 qFhvi vj vk i ; qx k F

qkp 1 qFhv0i v0i v0k i þ : qx k qx k 2F

Dp kp qkp ¼ Dt qt ð2:68Þ

2.3 Algebraic Models of Turbulent Stresses

In view of Eq. (2.65) and Eq. (2.68), it follows from Eq. (2.27) that hv0i v0j i hu0 u0 iGlk 2kp hv0k v0l iV k;l þ k l ¼ hv0i v0k iV j;k hv0j v0k iV i;k tp kp þ

X ij þ X ji 2hv0i v0j i tp

;

X ij ¼ hu0i u0k i½ fu kj þ tp (lu kn þ tp mu kl U n;l )U j;n tp (g u kn þ tp hu kl U n;l )V j;n : ð2:69Þ Let us rewrite Eq. (2.69) as 2 1 2 hv0i v0j i ¼ kp d ij tp hv0i v0k iV j;k þ hv0j v0k iV i;k hv0k v0l iV k;l dij 3 2E 3 2 X ij þ X ji X kk dij ; 3 E¼

X kk tp hv0i v0k iV i;k : 2kp

ð2:70Þ

You can think of expression (2.70) as an algebraic model that enables us to ﬁnd turbulent stresses in the disperse phase; this model is an implicit one, since the unknowns hv0i v0j i appear on both sides of this equation. By its physical meaning, Eq. (2.70) is similar to the implicit model proposed by Rodi (1976) for turbulent ﬂows of a one-phase continuous medium. Models of this kind have been proposed earlier by Zaichik (1992b) and Lain and Kohnen (1999) for turbulent stresses in the disperse phase, and by Fevrier and Simonin (1998) for correlation moments of velocity ﬂuctuations of the continuous and disperse phases. Equation (2.70) is, strictly speaking, valid only for the equilibrium state, when all transport terms play an insigniﬁcant role. Nevertheless, the analogy with single-phase turbulence can sometimes be extended to the case two phases, and thus algebraic models ﬁnd some limited application in the theory of non-equilibrium turbulent two-phase ﬂows. The shortcoming of implicit algebraic models lies in the necessity to apply the matrix inversion operation in order to obtain the dependence of turbulent stresses on gradients of the average velocity. Consequently, the advantage afforded by the use of complete differential equations for turbulent stresses may be lost. From the computational standpoint, explicit algebraic models have a considerable advantage over implicit ones, because they relate turbulent stresses directly to the gradients of the average velocity. In order to solve implicit algebraic equations and obtain explicit algebraic models for single-phase ﬂows, we must invoke the theory of invariants and represent the turbulent stresses as expansions over an orthogonal tensor basis (Taulbee, 1992; Gatski and Speziale, 1993; Girimaji, 1996; Jongen and Gatski, 1998; Wallin and Johansson, 2000). But an attempt to carry out this representation procedure results in exceedingly cumbersome expressions for the turbulent stresses. Hence in our effort to derive explicit algebraic models that would be convenient in practical implementation, in the majority of cases we are

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60

constrained by the need to carry out the expansion over a truncated (as opposed to complete) basis. As a rule, we choose a trinomial basis; strictly speaking, the procedure is justiﬁed only for the two-dimensional case (Pope, 1975). A transition to two-dimensional ﬂows signiﬁcantly complicates the construction of explicit algebraic models due to the considerable enlargement of the tensor basis. For example, the number of basis functions necessary to represent turbulent stresses of the disperse phase in a two-dimensional averaged ﬂow with three-dimensional turbulence becomes equal to ﬁve (Mashayek and Taulbee, 2002). Turbulent stress models for the disperse phase derived by carrying out expansion over a small number of basis functions (in particular, over a trinomial basis) are too rough and result in signiﬁcant errors. Therefore, in order to solve Eq. (2.70) and obtain explicit algebraic models for turbulent stresses of the disperse phase, we are going to apply the iteration procedure (Zaichik and Alipchenkov, 2001b) instead of the expansion method. The solution of Eq. (2.70) can be represented as 2 1 2 hv0i v0j inþ1 ¼ kp d ij tp hv0i v0k in V j;k þ hv0j v0k in V i;k hv0k v0l in V k;l d ij 3 2En 3 2 X ij þ X ji X kk dij ; 3 En ¼

X kk tp hv0i v0k in V i;k ; 2kp

ð2:71Þ

where n is the number of the approximation. The condition for applicability of Eq. (2.71), that is, of the iteration procedure, is the requirement that tp(Vi, kVi, k)1/2 must be a small parameter. We shall take the isotropic approximation (2.49) as our zeroth approximation. Then a substitution of Eq. (2.49) into the right-hand side of Eq. (2.71) will yield the ﬁrst approximation for the turbulent stresses:

hv0i v0j ið1Þ

0 0 ð1Þ

hvi vj i 2 ; kp d ij þ E0 3 tp kp 2 1 2 ¼ V i; j þ V j;i V k;k d ij þ X ij þ X ji X kk d ij ; 3 3 2 3

hv0i v0j i1 ¼

E0 ¼

X kk tp V k;k : 2kp 3

ð2:72Þ

Expression (2.72) gives a linear dependence of turbulent stresses hv0i v0j i on the velocity gradient Vi,j. Here, similarly to Eq. (2.53), along with the ﬁrst term that is quite common for turbulent viscosity models, a second term is present in hv0i v0j ið1Þ that explicitly describes the interaction of particles with turbulent eddies of the carrier

2.3 Algebraic Models of Turbulent Stresses

continuum. This term plays an especially important role for low-inertia particles and ensures a smooth transition to the inertialess limit (2.33). A substitution of Eq. (2.72) into the right-hand side of Eq. (2.71) results in a nonlinear (quadratic) dependence of turbulent stresses of particles on the average velocity gradient: hv0i v0j ið1Þ hv0i v0j ið2Þ 2 hv0i v0j i2 ¼ kp dij þ þ ; E1 E0 E1 3 tp 2 0 0 ð1Þ 0 0 ð2Þ 0 0 ð1Þ 0 0 ð1Þ hvi vj i ¼ hvi vk i V j;k þ hvj vk i V i;k hvk vl i V k;l d ij ; 2 3 E1 ¼ E0

tp hv0i v0k ið1Þ V i;k : 2kp E0

ð2:73Þ

Thus relation (2.73) together with the equation for turbulent energy (2.65) represents a non-linear explicit algebraic model for turbulent stresses of the disperse phase. In addition to the terms that contain averaged velocity gradients of the disperse phase, this relation, as well as Eq. (2.53) and Eq. (2.72), contains other terms, which explicitly characterize the interaction of particles with the turbulent ﬂuid. It is evident that the accuracy of a non-linear model, though better than that of a linear model, gets lower as particle inertia increases because the contribution of transport effects grows with inertia. Smallness of the parameter tp/Tu can serve as the criterion of applicability of the algebraic model. With increase of tp/Tu, the accuracy of algebraic models falls in comparison with differential models. Thus the choice of a model for the description of turbulent stresses of the disperse phase should be the result of a compromise between accuracy and the required calculation time. Models that are based on the equations of turbulent energy balance, second moments and higher-order moments of velocity and temperature ﬂuctuations of the disperse phase are called non-local, since they take into account convective and diffusional mechanisms of spatial transport of ﬂuctuations of turbulent ﬂow characteristics. These models allow to calculate different types of two-phase ﬂows for a wide range of particle inertias. If we neglect convective and diffusional terms in equations for the second moments, these equations will lead to simple algebraic relations that directly express turbulent stresses in the disperse phase through characteristics of the continuous phase. Models based on algebraic relations that disregard transport mechanisms are called local models. Such models have found application in calculations of two-phase turbulent ﬂows (e.g. Elghobashi and Abou-Arab, 1983; Chen and Wood, 1986; Mostafa and Mongia, 1987; Shraiber et al., 1990; Derevich et al., 1989a; Rizk and Elgohbashi, 1989; Simonin and Viollet, 1990). But while these models have some practical value due to their relative simplicity, their range of applicability is conﬁned to the case of low-inertia particles. Local algebraic models are unable to predict a number of important physical effects that might be described by non-local differential models, in particular, the existence of regions in which turbulent energy of the disperse phase

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62

exceeds that of the continuous phase. It is obvious that the need to take into account mechanisms of spatial ﬂuctuation of ﬂow characteristics becomes more urgent as particle inertia grows, since particles have longer memory of their previous behavior.

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

Determination of boundary conditions for equations of motion of the disperse phase is a central problem in the theory of two-phase ﬂows. Such boundary conditions should be obtained by considering particle interaction with the walls conﬁning the two-phase ﬂow. A detailed description of particle interaction with the wall can be achieved by using the method of Lagrangian continuous modeling. If we chose instead to use the method of Eulerian continuous modeling to determine disperse phases parameters in the near-wall region, this would cause some loss of information about the details of particle interaction with the surface as we sum up momentums, energies, and other parameters of the incident and reﬂected particle ﬂuxes. The overwhelming majority of existing models of solid particle collisions with a solid surface are based on Coulombs friction law (Matsumoto and Saito, 1970a, 1970b; Tsuji et al., 1985; Oesterle, 1989; Sommerfeld, 1990, 1992). This law allows to express components of particle velocity after the collision in terms of values of these components before the collision (see Figure 2.1). One can recognize two kinds of collisions: with and without particle slip relative to the point of contact with the wall. In both cases velocity components normal to the wall before and after the collision are connected through the relation v2y ¼ ey v1y ;

ð2:74Þ

where ey is the coefﬁcient of momentum restitution associated with the collision, y is the coordinate normal to the wall, and the indexes 1 and 2 refer to the parameters before and after the collision, respectively.

Figure 2.1 A collision of particle with the wall.

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

In the discussion below, we assume that the contribution of particle rotation to the components of translational velocity can be neglected. In such case the occurrence of a collision accompanied by slip is deﬁned by the condition 2 mt (1 þ ey )jv1y j < v1t ; 7

ð2:75Þ

where mt is the friction coefﬁcient and t denotes coordinates in the (x, z)-plane. The longitudinal velocity component after the collision, which is given by the condition (2.75), would be equal to v2t ¼ v1t mt (1 þ ey )jv1y j:

ð2:76Þ

It is seen from Eq. (2.75) that collisions accompanied by slip take place at small incident angles for the particles (tan at v1y/v1t ¼ 1). When the condition (2.75) is violated, a no-slip collision takes place, and the longitudinal velocity component after the collision is described by 5 v2t ¼ v1t : 7

ð2:77Þ

Relations (2.74), (2.76) and (2.77) between the velocities of incident and reﬂected particles are valid for a smooth wall. Models describing particle collisions with a rough wall have been proposed by Matsumoto and Saito (1970a, 1970b), Tsuji et al. (1985, 1987), Sommerfeld (1990, 1992), Sommerfeld and Zivkovic (1992), Schade and H€adrich (1998), Zhang and Zhou (2002, 2004). Sommerfelds model, in which a rough wall is modeled by a plane virtual surface whose slope is random and has a Gaussian distribution, appears to be the most promising. Further enhancements of this model have been performed in works by Derevich (1999), Khalij et al. (2004), Taniere et al. (2004), Khalij et al. (2005) and Konan et al. (2005) that demonstrate the possibility of using this model to construct boundary conditions for continuous equations of motion for the disperse phase. The most important result was obtained by Khalij et al. (2004) and Taniere et al. (2004), who derived the effective restitution and friction coefﬁcients as functions of the roughness parameter. Having introduced these effective coefﬁcients, we can then proceed to write the relation between incident and reﬂected particle velocities and the relation between the boundary conditions for disperse phases equations in the respective cases of smooth and rough wall (note the similar form of these two relations). One interesting peculiarity discussed in the works by Sommerfeld and Huber (1995, 1999), Derevich (1999), Khalij et al. (2004) and Taniere et al. (2004) is the possibility for the effective restitution coefﬁcient of the normal component of particle velocity to take values greater than unity (assuming that the wall is rough). Analogously to the derivation of boundary conditions in the kinetic theory of gases, derivation of boundary conditions for disperse phases equations of motion in the framework of continual modeling requires the knowledge of the PDF in the near-wall region. As of this writing, the typical way to address this problem is to employ one of

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the two approaches discussed below. The ﬁrst approach is based on solving the kinetic equation for the PDF by the iteration method (Derevich and Zaichik, 1988; Derevich, 1990, 2000; Zaichik, 1998; Alipchenkov et al., 2001). The second approach is based on the use of a PDF that is given a priori in the form of a binomial distribution (Zaichik et al., 1990; He and Simonin, 1993; Shraiber and Naumov, 1994). But before we dwell further on these approaches, let us present two simple relations between different correlation moments of particle velocity ﬂuctuations at the wall that are valid for any PDF (in the absence of particle deposition) for the case of collisions accompanied by slip. Let us deﬁne the average value of a quantity y as a weighted sum of its averaged values for the incident and reﬂected particles, with weights being equal to the corresponding concentrations: hyi ¼

F1 hyi1 þ F2 hyi2 : F

ð2:78Þ

From Eqs. (2.74), (2.76), (2.78) there follow the relations for the second and third moments of velocity ﬂuctuations at the impenetrable wall (Sakiz and Simonin, 1999): hv0t v0y i ¼ mt hv02 y i:

ð2:79Þ

2 0 hv0t v0y i ¼ 2mt hv0t v02 y imt hv y i: 2

3

ð2:80Þ

When deriving the boundary conditions for the disperse phase, we can restrict ourselves to the case of high-inertia particles whose relaxation time is much longer than the duration of their interaction with energy-carrying turbulent eddies (tp TLp). This simpliﬁcation is justiﬁed by small values of time scales of turbulence in the near-wall region. Moreover, the opposite limiting case – that of low-inertia particles (tp TLp) – is of little interest as far as the boundary conditions are concerned, because we know that the zero-slip boundary conditions must be realized in this case. Anisotropy of turbulence scales in the near-wall region, especially in the viscous sublayer, is strong enough, but, as we are going to show later, it does not have any signiﬁcant effect on the boundary conditions. Hence we shall conﬁne ourselves to the task of solving the kinetic equation for the PDF in the limit of high-inertia particles, assuming the time of particle–turbulence interaction to be isotropic so that Eq. (2.23) reduces to the Fokker–Planck equation qP qP q þ þ vk qt qx k qvk

T Lp U k vk q2 P þ F k P ¼ 2 hu0i u0j i : tp tp qvi qvk

ð2:81Þ

Suppose the averaged ﬂow in the near-wall region is stationary and planar, is directed along the longitudinal coordinate x, and all ﬂow characteristics can vary only in the y-direction, which is normal to the wall. The system of equations for the moments up to the second order follows from Eq. (2.81) and has the form

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

dFV y ¼ 0; dy 0 0

Vy

dV x U x V x 1 dFhvx vy i ¼ þ Fx ; dy tp F dy

Vy

02 dV y U y V y 1 dFhvy i ¼ þ Fy ; dy tp F dy

02 0 dhv0 2x i 1 dFhv x vy i 2 T Lp 0 2 0 0 dV x 02 þ hu x ihv x i ; þ ¼ 2hvx vy i Vy dy dy F tp tp dy

Vy

03 1 dFhv y i 2 T Lp 0 2 02 dV y 02 þ hu y ihvy i ; þ ¼ 2hvy i dy F dy tp tp dy

dhv02 y i

0 02 dhv0 2z i 1 dFhvy v z i 2 T Lp 0 2 02 hu z ihv z i ; þ Vy ¼ dy F tp tp dy

Vy

dhv0x v0y i dy

0 02

dV y 1 dFhvx vy i dV x hv0x v0y i ¼ hv02 y i dy dy F dy

þ

þ

2 T Lp 0 0 hux uy ihv0x v0y i : tp tp

ð2:82Þ

Since the average ﬂow is two-dimensional, it follows from the condition of symmetry with respect to the z-coordinate that hv0x v0z i ¼ hv0y v0z i ¼ 0:

ð2:83Þ

The solution of the kinetic equation (2.81) is sought in the form of an asymptotic expansion whose ﬁrst term is an equilibrium Gaussian distribution. With this in mind, let us present Eq. (2.81) in the operator form: R[P] ¼ N[P];

R[P] ¼ hv0 x i 2

0 2 2 q2 P qv0x P qvy P qv0z P 02 q P 02 q P þ hv i þ hv i þ þ þ ; y z qv2x qv2y qv2z qvx qvz qvy

ð2:84Þ

ð2:85Þ

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66

N[P] ¼

T Lp hv0 2x i tp

2 hu0 x i

T Lp 0 2 q2 P q2 P 02 þ hvy i hu y i tp qv2x qv2y

T Lp 0 2 q2 P T Lp 0 0 q2 P qP 2 þ hv0 z i hu z i 2 hu u i þ tp vy 2 tp tp x y qvx qvy qvz qy qP qP

þ U y V y þ tp F y : þ U x V x þ tp F x qvx qvy

ð2:86Þ

Operator R may be thought of as a basic operator and N – as the perturbation operator. We try to ﬁnd a solution of Eq. (2.84) in the form P ¼ Pð0Þ þ Pð1Þ þ . . . ;

ð2:87Þ

where P(0) and P(1) satisfy the equations R[Pð0Þ ] ¼ 0;

ð2:88Þ

R[Pð1Þ ] ¼ N[P ð0Þ ]:

ð2:89Þ

In view of Eq. (2.85), the solution of Eq. (2.88) has the form v02 F v0 2x v0 2z y : exp Pð0Þ (v) ¼ 0 2 1=2 2hv0 2x i 2hv02 2hv0 2z i (8p3 hv0 2x ihv02 y i y ihv z i)

ð2:90Þ

Notice that unlike the isotropic Maxwellian distribution (2.48), Gaussian distribution (2.90) takes into account the anisotropy of the diagonal components of particle velocity ﬂuctuations. This fact is of great importance because in the near-wall region, statistical characteristics of the disperse phase could be essentially anisotropic. From Eq. (2.90), it follows that 0 0 0 hv0x v0y i0 ¼ hv0 x v0y i0 ¼ hv0x v02 y i0 ¼ hv y i0 ¼ hvy v z i0 ¼ 0: 2

3

2

ð2:91Þ

Substituting Eq. (2.90) into Eq. (2.86), recalling Eq. (2.82) and taking into account the relations (2.91), we get 02 v0 2x dhv x i 1 N[P ] ¼ 0 2 02 sxy þ dy 2hv x ihvy i 2hv0 2x i hv0 2x i v0x v0y

ð0Þ

þ

v0y

02 02 02 dhvy i v0y vz dhv z i 1 3 ; þ 02 i 0 2 i hv0 2 i 2hv02 i hv dy dy 2hv y y z z v0y

v02 y

1 dV x T Lp 0 0 hu u i: þ sxy ¼ tp hv02 y i dy tp x y 2

ð2:92Þ

ð2:93Þ

For high-inertia particles in the near-wall region of the turbulent ﬂow, the second term on the right-hand side of Eq. (2.93) can be neglected compared to the ﬁrst term. Then in view of Eq. (2.83), the solution of Eq. (2.89) will be

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

0 0 02 02 tp vx vy dV x tp v0y vx dhv x i þ Pð1Þ (v) ¼ 1 dy 2hv0 2x i dy 6hv0 2x i hv0 2x i þ

tp v0y

v02 y

02 6hv02 y i hvy i

3

dhv02 y i dy

þ

02 v0 2z dhv z i ð0Þ P (v): 1 2 2 0 0 dy 6hv z i hv z i tp v0y

ð2:94Þ Insertion of Eq. (2.90) and Eq. (2.94) into Eq. (2.87) gives us an expansion that is correct to the ﬁrst two terms: 02 0 tp v0x v0y dV x tp v0y v02 dhvy i v0 2 dhv0 2x i tp vy y þ 0 2 1 0 x2 þ 02 3 02 P(v) ¼ 1 0 2 6hvy i hvy i dy dy 2hv x i dy 6hv x i hv x i v0 2z dhv0 2z i ð0Þ P (v): þ 0 2 1 0 2 6hv z i hv z i dy tp v0y

ð2:95Þ

Adopting the approximation (2.95), we obtain: hv0x v0y i ¼

tp hv02 y i dV x

hv0 x v0y i ¼ 2

dy

2

;

02 tp hv02 y i dhv x i

3

dy

ð2:96Þ

; hv0 y i ¼ tp hv02 y i 3

dhv02 y i dy

hv0x v02 y i ¼ 0:

; hv0y v0 z i ¼ 2

02 tp hv02 y i dhv z i

3

; dy ð2:97Þ

ð2:98Þ

Dynamic interaction of a particle with the wall during a collision may be described as follows: p1 ! 2 ¼ cd(v2x ex v1x )d(v2y þ ey v1y )d(v2z v1z );

ð2:99Þ

Coefﬁcient c in Eq. (2.99) characterizes the deposition of particles and is equal to the probability for the particle that has collided with the wall to recoil and get back into the ﬂow. If the particles are completely absorbed by the surface, then c ¼ 0; on the other hand, if there is no deposition, then c ¼ 1. The restitution coefﬁcient ex describes the loss of momentum in the direction of the average motion as the particle collides with the wall. Owing to Eqs. (2.75)–(2.77), this coefﬁcient is equal to

ex ¼

8 > > > 1xx <

at 0 xx <

5 > > > :7

at xx >

2 7

2 7

;

ð2:100Þ

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68

where xx mx(1 + ey)tan ax. Relation (2.100) shows that, in contrast to ey, the coefﬁcient of momentum restitution ex is not a physical parameter, as it depends on the angle of particle incidence at the wall ax. For the considered two-phase ﬂow, the coefﬁcient of momentum restitution in the transverse direction ez is equal to unity, since in order to obey the second relation (2.83), the friction coefﬁcient mz should be equal to zero at t ¼ z, as it follows from Eq. (2.79). The PDF (2.95) can be used to describe the velocity distribution of particles – for the entire ﬂow as well as for the particles incident upon the wall. We thus have P1 (vy < 0) ¼ P(vy ):

ð2:101Þ

The velocity distribution of reﬂected particles follows from Eq. (2.99): c vx vy ; ; vz : P2 (vx ; vy ; vz ) ¼ 2 P1 ex ey e x ey

ð2:102Þ

The normal component of particle velocity at the wall (deposition rate) is similar to Eq. (2.78) and is found by summation of corresponding velocities of incident and reﬂecting particles: 01 0 1 1 1 ð 1 ð ð 1 ð ð ð 1@ vy P1 dvx dvy dvz þ vy P2 dvx dvy dvzA: Vy ¼ F 1 1 1

1 0 1

Thus, in view of Eqs. (2.95), (2.101), and (2.102), the normal component is equal to Vy ¼

02 1=2 1c 2hvy i : 1þc p

ð2:103Þ

The boundary condition (2.103) was ﬁrst obtained by Razi Naqvi et al. (1982) for Brownian particles. Afterwards, by taking the assumption of a normal velocity distribution of particles as their starting point, Binder and Hanratty (1991) obtained the following expression for the deposition rate 02 1=2 2hvy i V y ¼ z ; p

ð2:104Þ

where z is the fraction of particles moving toward the wall (Binder and Hanratty took z ¼ 1/2). It is evident that the formulas (2.103) and (2.104) coincide at c ¼ (1 z)/ (1 + z) ¼ 1/3. 02 The boundary conditions for y ¼ Vx, hv0 2x i; hv02 y i; hv z i can be found by equating the ﬂux of one of these quantities in the near-wall region to the sum of the incident and reﬂected ﬂuxes: 1 ð

1 ð 1 ð ð 1

yvy Pdvx dvy dvz¼ 1 1 1

1 ð 1 ð 1 ð

ð0 1 ð

1 1 1

yvy P 1 dvx dvy dvzþ

yvy P 2 dvx dvy dvz :

1 0 1

ð2:105Þ

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

Setting in Eq. (2.105) y ¼ vx and taking into account Eq. (2.95) and Eq. (2.103), one obtains the boundary condition for the longitudinal velocity Vx: 1=2 dVx 1cex 1c 2 ¼2 Vx : ð2:106Þ tp dy 1 þ cex 1 þ c phv02 y i 02 Successive substitution of y ¼ vx0 2 ; v02 y ; vz into Eq. (2.105) gives the following boundary conditions for the diagonal components of disperse phases turbulent stresses: 1=2 dhvx0 2 i 1ce2x 1c 2 2 hvx0 i; ð2:107Þ ¼3 tp 1 þ ce2x 1 þ c phv02 i dy y

tp

tp

dhv02 y i dy

¼

2

1ce2y

1c 2 1 þ cey 1 þ c

02 1=2 2hvy i p

;

1=2 dhvz0 2 i 1ce2z 1c 2 2 hv0z i: ¼3 1 þ ce2z 1 þ c phv02 dy y i

ð2:108Þ

ð2:109Þ

The boundary conditions (2.103), (2.106)–(2.109) are true only for the near-equilibrium ﬂows, that is, when the PDF deviates a little from the Gaussian velocity distribution. 1=2 This requirement imposes a stringent constraint on the normal velocity Vy hv02 y i , according to which the average velocity normal to the surface should be much smaller than the ﬂuctuational component of this velocity. Hence in accordance with Eq. (2.103) the coefﬁcient of reﬂection c should not differ substantially from unity. In the case of a two-dimensional ﬂow, which can only be realized if ez ¼ 1, itfollows from Eq. (2.109) that dhvz0 2 i ¼ 0: dy

ð2:110Þ

The boundary conditions (2.103), (2.106)–(2.109) may also be obtained by means of Grads method that is used in the kinetic theory of gases and in the theory of disperse media for solving the Boltzmann equation (Grad, 1949; Jenkins and Richman, 1985). Grads expansion provides a solution of the kinetic equation that is equivalent to the ﬁrst approximation in the perturbation method (2.95). In the framework of this approach, the solution of Eq. (2.81) takes the form hvx0 2 v0y i q3 hvy0 3 i q3 hv0y vz0 2 i q3 q2 P(v) ¼ 1 þ hv0x v0y i qvx qvy 2 qv2x qvy 6 qv3y 2 qvy qv2z 02 0 hv0x v0y i v0y hvx vy i vx0 2 0 0 P (v) ¼ 1 þ vx vy 0 2 02 1 0 2 hvx ihvy i 2 hvx i hvx0 2 ihv02 y i ð0Þ

ð2:111Þ

03 0 02 v02 hvy i v0y hvy vz i vz0 2 y P ð0Þ (v): 3 0 2 1 2 2 0 02 02 6 2 hvx i hvy i hvz i hv02 y ihvz i

v0y

A substitution of Eq. (2.105) into Eq. (2.111) with the subsequent integration provides the boundary conditions for the quantities of interest. Obviously, the

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70

boundary condition for the velocity component normal to the surface is determined from relation (2.103). The boundary conditions for the longitudinal component of velocity and for the normal components of velocity ﬂuctuation intensities are as follows:

1cex 1c 1 þ cex 1 þ c

1ce2x 1c 1 þ ce2x 1 þ c

02 1=2 2hvy i p 02 1=2 2hvy i p

V x ¼ hv0x v0y i;

ð2:112Þ

hv0x i ¼ hv0x v0y i;

ð2:113Þ

2

2

02 3 1=2 1ce2y 2hvy i 1c 3 2 ¼ hv0y i; 2 1 þ cey 1 þ c p 02 1=2 2hvy i 1ce2z 1c 2 2 hv0 z i ¼ hv0y v0z i: 2 1 þ cez 1 þ c p

ð2:114Þ

ð2:115Þ

Taking hu0i u0j i ¼ 0 at the wall and recalling the relation (2.91) that corresponds to the equilibrium PDF, we ﬁnd from the last equation (2.82) that the tangential component of stresses is determined in the ﬁrst approximation by the expression (2.96). In a similar manner, we are able to show from Eq. (2.31) that in the ﬁrst approximation, third moments of particle velocity ﬂuctuations coincide with Eq. (2.97). Thus the boundary conditions (2.112)–(2.115) are the same as the boundary conditions (2.106)–(2.109). In the limiting case of inertialess particles (tp ! 0), Eqs. (2.106)–(2.109) lead to the following no-slip conditions: 0 V x ¼ hv0x i ¼ hv02 y i ¼ hv z i ¼ 0: 2

2

ð2:116Þ

In the case of absolutely elastic interaction of particles with the wall (ex ¼ ey ¼ ez ¼ 1) boundary conditions (2.106)–(2.109) transform into expressions similar to Eq. (2.110): 02

dVx dhv0x2 i dhvy i dhv0z2 i ¼ ¼ ¼ 0: ¼ dy dy dy dy

ð2:117Þ

It is evident that in the limit of high-inertia particles (tp ! 1), the boundary conditions tend to Eq. (2.117) as well. We shall take as our boundary condition for the tangential stress component the relation (2.79), namely, hv0x v0y i ¼ mx hv02 y i:

ð2:118Þ

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

In the inertialess limit (tp ! 0) or in the case of absolutely elastic collisions (mx ! 0), Eq. (2.118) gives rise to the sticking condition hv0x v0y i ¼ 0:

ð2:119Þ

Consider now the boundary conditions based on the bi-normal distribution of particle velocities in the near-wall region. The velocities of particles moving to or from the wall are described by the following distributions: v2y P1 (vy < 0) ¼ N 1 exp 2 ; 2hv1y i

v2y P2 (vy > 0) ¼ N 2 exp 2 : 2hv2y i

ð2:120Þ

It should be emphasized that the distributions (2.120) depend on the corresponding total components of normal velocity, in contrast to the expansion (2.95), which is a function of ﬂuctuational velocity components. Coefﬁcients N1 and N2 as well as the mean-square velocities of incident and reﬂected particles are determined from the relation (2.102) and the normalizing conditions ð0 F ¼ F1 þ F2 ; F1 ¼

1 ð

P1 dvy ; F2 ¼ 1

P2 dvy ; 0

which give Fey N1 ¼ c þ ey

sﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ 2 c ; N 2 ¼ 2 N 1 ; hv22y i ¼ e2y hv21y i: ey phv21y i

ð2:121Þ

Notice that in fact, it is necessary to get only the PDF of incident particles, since the PDF of reﬂected particles is coupled with the distribution of particles moving toward the wall through the relation (2.102). Equations (2.120)–(2.121) give rise to the following expression for the deposition rate: 2 1=2 1=2 2hvy i (1c)ey Vy ¼ ; ð2:122Þ p (1 þ cey )1=2 (ey þ c)1=2 where hv2y i is the total mean-square velocity of a particle at the wall: hv2y i ¼ V 2y þ hv02 y i¼

F1 hv21y i þ F2 hv22y i F

¼

ey (1 þ cey ) 2 hv1y i: ey þ c

Comparing the expressions (2.103) and (2.122) for the deposition rate, we can see the distinction between them from two different angles. First, as a consequence of Eq. (2.103), Vy is a function of c only, whereas Eq. (2.122) suggests that Vy depends on two parameters, c and ey, which is more justiﬁed from the physical standpoint. Secondly, formula (2.103) predicts the dependence of Vy on the ﬂuctuational

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72

component of energy hv02 y i, whereas Eq. (2.122) shows the dependence of Vy on the total energy hv2y i. In order to compare these formulas, we shall write Eq. (2.122) as Vy ¼

1=2 02 1=2 1=2 2hvy i 2(1c)2 ey (1c)ey : 1 1=2 1=2 p(1 þ cey )(ey þ c) p (1 þ cey ) (ey þ c) ð2:123Þ

It is easy to see that formulas (2.103) and (2.123) give close values for Vy at the values of parameters c and ey slightly different from unity. The difference between the deposition rates suggested by Eq. (2.103) and Eq. (2.123) increases as c and ey pﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ diminish, and their ratio reaches its maximum value p=(p2) 1:66 at c ¼ 0. In order to ﬁnd tangential stress at the wall, we adopt the assumption of statistical independence of the distributions over vx and vy in the incident and reﬂected fractions of particles. The tangential stress will then be equal to hvx vy i ¼

F1 V 1x V 1y þ F2 V 2x V 2y : F

ð2:124Þ

Taking the PDF in accordance with Eq. (2.120) and taking into account the relation V x ¼ (F1 V 1x þ F2 V 2x )=F; V 2x ¼ ex V 1x we obtain from Eq. (2.124): hvx vy i ¼

2 1=2 1cex 2ey (ey þ c)hvy i Vx: fy þ cex p(1 þ cey )

ð2:125Þ

Next, making use of hvx vy i ¼ V x V y þ hv0x v0y i and Eq. (2.122), we ﬁnd from Eq. (2.125) the ﬂuctuational component of tangential stress at the wall: 1=2 2ey (ey þ c)hv2y i 1cex 1c Vx: hv0x v0y i ¼ fy þ cex ey þ c p(1 þ cey )

ð2:126Þ

Hence, in view of Eq. (2.96), there follows from Eq. (2.126) a boundary condition for Vx: tp

1=2 1=2 1=2 2c(1ex )(1þey )ey 2(1c)2 ey dV x 2 Vx: ¼ 1 dy (ey þcex )(ey þc)1=2 (1þcey )1=2 p(1þcey )(ey þc) phv02 y i ð2:127Þ

The boundary conditions (2.106) and (2.127) coincide at c ¼ ey ¼ 1, but the difference between them gets larger as c and ey decrease. Furthermore, using the PDF (2.120), it is possible to express the triple correlation of the particles transverse velocity through the second moment:

hv3y i ¼ 1ce2y

8(ey þ c) pey (1 þ cey )3

1=2 hv2y i3=2 :

ð2:128Þ

2.4 Boundary Conditions for the Equations of Motion of the Disperse Phase

The fourth correlation moment of the transverse velocity component is expressed through second moments by Eq. (2.120) as

2 c 1e2y 4 ð2:129Þ hvy i ¼ 3 1 þ hv2 i2 : ey (1 þ cey )2 y It is readily seen that Eq. (2.129) turns into the well-known Millionshikov hypothesis connecting the fourth and the second moments only if the surface is completely absorbing (c ¼ 0) or if the collisions are absolutely elastic (ey ¼ 1). The boundary condition for hv02 y i stemming from the bi-normal distribution (2.120) could be presented in a particularly simple form in the absence of particle deposition at the wall (c ¼ 1), when the triple correlation of transverse velocity ﬂuctuations is deﬁned by Eq. (2.31) for high-inertia particles as hv0y i ¼ tp hv02 y i 3

dhv02 y i dy

:

ð2:130Þ

It should be noted that, as shown by Eq. (2.129), the relation (2.130) based on the Millionshikov quasi-normal hypothesis (2.29) should be considered only as an approximate one at c 6¼ 0 and ey 6¼ 1. A substitution of Eq. (2.130) into Eq. (2.128) yields the following boundary condition for hv02 y i: tp

1=2 2(1ey ) 2hv02 y i : ¼ 1=2 dy p ey

dhv02 y i

ð2:131Þ

Comparing Eq. (2.108) at c ¼ 1 and Eq. (2.131), we can convince ourselves that these expressions are coincident in the limit ey ! 1 but produce increasingly different results as ey gets smaller. When there is no deposition, the boundary condition for hv0x2 i may be obtained from Eq. (2.80) at t ¼ x: 2 0 hv0x v0y i ¼ 2mx hv0x v02 y imx hv y i: 2

3

ð2:132Þ

Triple correlations entering Eq. (2.132) contain ﬂuctuations of the longitudinal velocity component and can be obtained from Eq. (2.31) in the limit of high-inertia particles as follows: dhv0x v0y i tp dhv0 2x i 2 0 0 hv0x v0y i ¼ i v i hv02 þ 2hv ; ð2:133Þ y x y 3 dy dy hv0x v02 y i¼

dhv02 dhv0x v0y i tp y i i hv0x v0y i þ 2hv02 : y 3 dy dy

ð2:134Þ

If we set hv0x v0y i ¼ 0 in Eq. (2.133) and Eq. (2.134), these two expressions reduce to the relations given by Eq. (2.97) and Eq. (2.98). Substituting Eq. (2.130), Eq. (2.133), and Eq. (2.134) into Eq. (2.132) and recalling Eq. (2.118), we arrive at the following boundary condition for hv0x2 i:

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02

dhvy i dhv0x2 i ¼ m2x : dy dy

ð2:135Þ

In the limiting case of absolutely elastic collisions (mx ! 0, ex ¼ ey ! 1), the boundary conditions (2.127), (2.131), and (2.135) reduce to Eq. (2.117). Alipchenkov et al. (2001) have obtained a numerical solution of the one-dimensional kinetic equation taking into account the variation of the PDF only in the y-direction that is normal to the wall. A comparison of the numerical data with the results obtained on the basis of the two approaches described in the present section warrants the following conclusions: 1. The approach based on solving the PDF equation by the perturbation method is, strictly speaking, valid only for the values of parameters c, ex and ey large enough for the distribution of particle velocities to be close to the equilibrium Gaussian distribution. Nevertheless, this method gives boundary conditions that are qualitatively correct at all values of c, ex and ey . The accuracy of these boundary conditions decreases as the reﬂection and restitution coefﬁcients get smaller. 2. The approach involving the bi-nomial PDF remains valid in the entire range of parameter values c, ex and ey (i.e., from zero to unity), and its accuracy increases with particle inertia. While it leads to more cumbersome boundary conditions, overall, this approach is more accurate than a quasi-equilibrium one, since it is based on a more realistic distribution of particle velocities. Summarizing, we should note that the generalization to a three-dimensional average ﬂow is self-intuitive and is carried out by using the boundary conditions for the characteristics containing the velocity component vz, whose form is exactly the same as the form of the boundary conditions that involve vx.

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

Consider the behavior of disperse phases turbulent stresses hv0i v0j i and correlation functions of velocity ﬂuctuations of the continuous and disperse phases hu0i v0j i in a homogeneous turbulent ﬂow with constant shear rate in the absence of external forces. Owing to homogeneity of the velocity ﬁeld of the turbulent carrier ﬂow, triple moments of particle velocity ﬂuctuations vanish and consequently, one can obtain exact solutions of the equations for second moments. Therefore homogeneous ﬂows are of great importance in veriﬁcation of turbulent models of transport and dispersion of particles. An additional motivation to test the models by applying them to homogeneous ﬂows with a constant shear rate is the availability of numerical simulation results for Lagrangian characteristics of turbulence in a continuum (Sawford and Yeung, 2000, 2001) as well as for ﬂuctuational motion of particles (Yeh and Lei, 1991b; Simonin et al., 1995; Lavieville, 1997; Lavieville et al., 1997; Taulbee et al., 1999; Ahmed and Elghobashi, 2001; Pandya and Mashayek, 2003; Shotorban and Balachandar, 2006).

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

Due to inhomogeneity of the ﬂow, it follows from Eqs. (2.24), (2.25) that the spatial concentration of particles does not change, and the gradients in the continuous and disperse phases are equal. These gradients are given by the relations qU i qV i ¼ ¼ Sd i1 d j2 ; qx j qx j

ð2:136Þ

where S is the shear rate. In view of Eqs. (2.16), (2.17), and (2.136), we obtain from Eq. (2.27) the following system of equations for turbulent stresses of the disperse phase in a homogeneous shear ﬂow:

2 02 dhv02 0 0 1i hu1 ifu 11 ¼ 2S hv01 v02 i þ hu02 1 ifu1 12 þ hu1 u2 ifu1 22 þ dt tp 02

dhu1 i dhu01 u02 i þhu01 u02 ifu 21 hv02 i þ f f 1 u1 21 dt u1 11 dt þtp S

02 dhu1 i dhu01 u02 i fu2 12 þ fu2 22 ; dt dt

dhu01 u02 i dhv02 2 0 0 dhu02 02 2i 2i hu1 u2 ifu 12 þ hu02 if hv i þ ¼ f f ; 2 u 22 2 u1 12 dt tp dt dt u1 22

dhu02 dhv02 2 02 3i 3i hu3 ifu 33 hv02 ; ¼ f 3i dt tp dt u1 33

1 02 dhv01 v02 i 0 0 02 hu1 ifu 12 ¼ S hv02 2 i þ hu1 u2 ifu1 12 þ hu2 ifu1 22 þ dt tp

0 0 þhu01 u02 i fu 11 þ fu 22 þ hu02 2 ifu 21 2hv1 v2 i

1 dhu02 dhu01 u02 i dhu02 1i 2i fu1 12 þ ( fu1 11 þ fu1 22 ) þ fu2 21 2 dt dt dt tp S dhu01 u02 i dhu02 2i fu2 12 þ f : þ 2 dt dt u2 22

ð2:137Þ

Using Eq. (2.43) together with Eqs. (2.16)–(2.17), we ﬁnd that non-zero correlation moments of velocity ﬂuctuations of the continuous and disperse phases are determined by the relations

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02 0 0 0 0 hu01 v01 i¼hu02 1 if u 11 þhu1 u2 if u 21 tp S hu1 if u1 12 þhu1 u2 ifu1 22 tp dhu02 dhu01 u02 i 1i þ f f 2 dt u1 11 dt u1 21 t2p S dhu02 dhu01 u02 i 1i þ þ ; f f dt u2 12 dt u2 22 2 tp dhu01 u02 i dhu02 2i hu02 v02 i¼hu01 u02 ifu 12 þhu02 if þ f f ; 2 u 22 2 dt u1 12 dt u1 22 hu03 v03 i¼hu02 3 ifu 33

tp dhu02 3i ; f 2 dt u1 33

dhu02 dhu01 u02 i 1i þ ; f f 2 dt u1 12 dt u1 22

tp 0 0 hu01 v02 i¼hu02 1 ifu 12 þhu1 u2 if u 22

if tp S(hu0 u02 ifu1 12 þhu02 hu02 v01 i¼hu01 u02 ifu 11 þhu02 2 ifu1 22 ) 0 0 2 u 21 02 1 tp dhu1 u2 i dhu2 i þ f f 2 dt u1 11 dt u1 21 t2p S dhu01 u02 i dhu02 i þ fu2 12 þ 2 fu2 22 : dt dt 2

ð2:138Þ

At large values of time, an equilibrium state is reached, which is characterized by the independence of anisotropy tensors and other dimensionless ﬂow parameters on time. Therefore the equilibrium state may be interpreted as an asymptotic solution of the system of equations for large values of time, when the ﬂow becomes self-similar. This solution is independent from the initial conditions and requires the second moments of velocity ﬂuctuations and the dissipation rate of turbulence energy to grow exponentially with time (Speziale and Mac Giolla Mhuiris, 1989): hu0i u0j i hv0i v0j i hu0i v0j i e exp(ct); e P qU i c¼ ; 1 ; P ¼ hu0i u0j i qx j k e

ð2:139Þ

where P is the generation of turbulent energy. The involvement coefﬁcients in Eqs. (2.137), (2.138) are given by the relations (2.22). The tensor of Lagrangian time scales of turbulence, which is needed to calculate these coefﬁcients, is determined by the corresponding dependence (Pope, 2002) based on the DNS results (Sawford and Yeung, 2000, 2001) 0 1 0:44 0:06 0 k@ TL ¼ 0:11 0:22 0 A: e 0 0 0:24

ð2:140Þ

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

With the aim to analyze the effect of anisotropy on time scales on ﬂuctuational motion of particles, calculation results obtained by using the tensor (2.140) will be compared with those corresponding to the isotropic approximation of the Lagrangian scale of turbulence (2.38) and (2.40), namely, T L ij ¼ T L d ij ; T L ¼

T L kk ¼ 3

k a ; a ¼ 0:3: e

ð2:141Þ

In a homogeneous turbulent ﬂow with constant shear rate in the absence of bulk forces, the average velocities of both phases coincide. Therefore there are no trajectory intersections, and the only reason for the difference between velocity correlations in the continuum calculated along inertial and inertialess particle trajectories is the difference between the Lagrangian and Eulerian turbulence scales. Suppose that in the absence of the crossing trajectory effect, the inﬂuence of particle inertia on the interaction time is the same as for isotropic turbulence. Then the tensors of particle–turbulence interaction time and the Lagrangian macroscale should be related by the equation TLp ¼ F(StE )TL ;

ð2:142Þ

where the function F(StE) accounts for the effect of the Stokes number StE tp/TE that characterizes particle inertia. In accordance with Eq. (1.65) and Eq. (1.61), the dependence f (StE) at m 1 is written as T Lp TE StE 0:9mSt2E T E 3(1 þ m)2 ¼1þ 1 ¼ : ; F(StE ) ¼ 2 TL TL 1 þ StE (1 þ StE ) (2 þ StE ) TL 3 þ 2m ð2:143Þ The formula (1.66) obtained by Wang and Stock (1993) for m ¼ 1 gives F(StE ) ¼

T Lp T E (T E =T L 1) TE ¼ ¼ 2:81: ; TL T L (1 þ StE )0:4(1þ0:01StE ) T L

ð2:144Þ

The relaxation time for a particle is determined from Eq. (1.47), in which the Reynolds number associated with the ﬂow past the particle is 1=2 dp jUVj2 þ 2(k þ kp 2f u k) : ð2:145Þ Rep ¼ n Equation (2.145) describes with a sufﬁcient accuracy the contribution of the average and ﬂuctuational velocity slip of particles relative to the turbulent ﬂuid. The coefﬁcient fu here is determined by the isotropic time of particle–turbulence interaction corresponding to the Lagrangian isotropic scale of turbulence TL ¼TL kk/3. The discussion in the present section will be conﬁned to the ﬂows with zero average slip between the disperse and continuous phases (U ¼ V), so Rep will depend on ﬂuctuational slip only: 1=2 dp 2(k þ kp 2f u k) : Rep ¼ n

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To study the role of separate physical factors, we shall present the results obtained on the basis of several models – those including as well as those excluding such phenomena as anisotropy of the Lagrangian time scale, non-stationary turbulence in a continuous medium (in a homogeneous shear ﬂow, with convectional and diffusive transport absent and non-stationarity being the sole factor responsible for the transport), and the inﬂuence of particle inertia on the duration of interaction between turbulent eddies and the particles. Model 1 is the most comprehensive one as it includes all three factors: time scale anisotropy, non-stationarity, and inertia. Model 2 takes the Lagrangian scales (2.141) to be isotropic, but includes the non-stationarity of turbulence and the dependence of particle–turbulence interaction time on particle inertia. M@del 3 takes into account the anisotropy of turbulence scales in accordance with Eq. (2.140) and the transport occurring due to non-stationarity of the turbulent medium, but neglects the inﬂuence of particle inertia on the time of interaction with the turbulence ( f (StE) ¼ 1). Finally, Model 4 includes the anisotropy of scales and the dependence f (StE) but fails to account for the effect of non-stationarity, in other words, it neglects the transport term (2.15). First, consider two examples illustrating the results of direct trajectory modeling of disperse phase characteristics in homogeneous shear layers generated by the LES method (Simonin et al., 1995; Lavieville, 1997). The Reynolds stresses of the carrier continuum are taken from the calculations performed by Simonin et al. (1995) and Lavieville (1997). In both examples, the initial conditions corresponded to an isotropic state. The average velocity gradient was identical in both ﬂows and equal to 50 s1, but particle inertias were different. In Simonin et al. (1995), the density ratio between the disperse and continuous phases was rp/rf ¼ 2000, and particle diameter was dp ¼ 60 mm, whereas in Lavieville (1997), the density ratio was rp/rf ¼ 43 and particle diameter was dp ¼ 656 mm. Structure parameter of turbulence m was taken equal to unity in both examples. Figures 2.2 and 2.3 show the time evolution of normal and tangential components of turbulent stresses in the continuous and disperse phases. First of all, note that at m ¼ 1, the formulas (2.143) and (2.144) describing the effect of particle inertia on the duration of interaction with the turbulence give sufﬁciently close results. Therefore only the results corresponding to Eq. (2.144) are shown in the two ﬁgures. The reader can see that model 1, which takes into account all of the above-listed factors, provides the best agreement with the LES data. Inclusion of the anisotropy of turbulence time scales and the inertia effect brings about a larger anisotropy of velocity ﬂuctuations as the longitudinal component of ﬂuctuations increases. On the other hand, transport caused by the non-stationarity of turbulence results in a somewhat lower intensity of longitudinal ﬂuctuations of particle velocity and thereby smoothes the anisotropy of disperse phases velocity ﬂuctuations. Inclusion of time scale anisotropy and particle inertia in the numerical model enables us to reproduce an interesting phenomenon where the intensity of longitudinal ﬂuctuations of particle velocity hv02 1 i exceeds the intensity of longitudinal velocity ﬂuctuations of the carrier ﬂow hu02 1 i. As was noted by Liljegren (1993), Reeks (1993), Taulbee et al. (1999), Zaichik (1999), this phenomenon has to do with generation of ﬂuctuations of particles longitudinal velocity due to the shear of the average velocity, and is explained by the absence of small-scale dissipation

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

Figure 2.2 Time variation of longitudinal (a), normal (b), spanwise (c), and tangential (d) components of turbulent stresses of the disperse and continuous phases in a homogeneous shear layer: dp ¼ 60 mm; rp/r ¼ 2000; 1–5 – hv0i v0j i; 6 – hu0i u0j i; 1 – model 1; 2 – model 2; 3 – model 3; 4 – model 4; 5 and 6 – Simonin et al. (1995).

of turbulent energy in the disperse phase. Since the average ﬂow does not contribute to the creation of normal and spanwise velocity ﬂuctuations, these components of turbulent stresses for the particles are found to be much smaller than the longitudinal components; besides, they can not exceed the corresponding values of normal components of second-order moments of ﬂuid velocity ﬂuctuations. As a result of ﬂuctuation generation by the average ﬂow, tangential stresses in the disperse phase can exceed the corresponding quantities in the continuous phase by absolute value. Consider the time evolution of particles turbulent stresses and mixed correlation moments of velocity ﬂuctuations in the continuous and disperse phases in a homogeneous shear layer generated by the DNS method (Pandya and Mashayek, 2003) at tp0S ¼ 0.6. The time scale TLp is found from Eq. (2.142) complemented by Eqs. (2.140) and Eq. (2.143) at m ¼ 0.5. The relations (2.138) for hu0i v0j i, as opposed to the differential equations (2.137) for hv0i v0j i, are algebraic equations and therefore cannot satisfy the initial conditions corresponding to an isotropic state. Therefore

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Figure 2.3 Time variation of longitudinal (a), normal (b), transversal (c), and spanwise (d) components of turbulent stresses of the disperse and continuous phases in a homogeneous shear layer: dp ¼ 656 mm; rp/r ¼ 43; 1–5 – hv0i v0j i; 6 – hu0i u0j i, 1 – model 1; 2 – model 2; 3 – model 3; 4 – model 4; 5 and 6 – Lavieville (1997).

when Figure 2.5 shows the components hu0i v0j i as functions of time, the graph does not begin right away at t ¼ 0; instead, the ﬁrst point on the graph corresponds to some ﬁnite time t. Figures 2.4 and 2.5 show only the results suggested by models 1 and 2, since the effect of non-stationarity of continuous phases turbulence and of particle inertia on the duration of particle interactions with turbulent eddies described by models 3 and 4 is qualitatively the same as the one shown on Figure 2.2 and 2.3. One can see that Eqs. (2.137) and Eq. (2.138) reproduce the main features of the behavior of all components hv0i v0j i and hu0i v0j i with sufﬁcient accuracy. However, the relations (2.138) predict greater difference between tangential components hu01 v02 i and hu02 v01 i, that is, greater asymmetry of the tensor hu0i v0j i than the one we see from the DNS data. Similarly to Figure 2.2 and Figure 2.3, the inclusion of turbulence time scale anisotropy ensures better agreement with the DNS data for the longitudinal 0 0 components hv02 1 i and hu1 v1 i, but the inﬂuence of this factor on the other components 0 0 0 0 hvi vj i and hui vj i is insigniﬁcant. Shown in Figures 2.6–2.9 are the results of the asymptotic equilibrium solution (2.139) of equations (2.137) and (2.138) and the results of numerical calculations

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

Figure 2.4 Time variation of turbulent stresses of the disperse phase in a homogeneous shear layer: 1 – model 1; 2 – model 2; 3–6 – Pandya and Mashayek (2003).

obtained by the LES (Simonin et al., 1995; Lavieville, 1997) and DNS (Taulbee et al., 1999; Pandya and Mashayek, 2003) methods for the maximum values of time during which the ﬂow may be considered as being close to equilibrium. Turbulent characteristics of the continuum were taken from the experimental data for a homogeneous shear ﬂow (Tavoularis and Karnik, 1989): b11 ¼ 0.21, b22 ¼ 0.13, b33 ¼ 0.08, b12 ¼ 0.16, Sk/e ¼ 5, P/e ¼ 1.6. 02 As follows from Figure 2.6, the dependence of the ratio hv02 1 i=hu1 i on the product of relaxation time and shear intensity tpS is not monotonous and, as we already mentioned, the ratio of longitudinal components of velocity ﬂuctuations in the continuous and disperse phases may exceed unity. But at large values of tpS, this ratio 02 is less than unity and approaches zero as tpS increases. The ratio hv02 2 i=hu2 i decreases monotonously and also goes to zero with increase of tpS; this can be explained by decreasing involvement of particles in the irregular motion of the carrier continuum with increase of particle inertia, and by the lack of contribution from the average ﬂow.

Figure 2.5 Correlation moments of velocity fluctuations for the continuous and disperse phases in a homogeneous shear layer as functions of time: 1 – model 1; 2 – model 2; 3–7 – Pandya and Mashayek (2003).

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Figure 2.6 Ratios of turbulent stress components in the continuous and disperse phases in a homogeneous shear layer: 1 – model 1; 2 – model 2; 3 – Simonin et al. (1995); 4 – Lavieville (1997); 5 – Taulbee et al. (1999); 6 – Pandya and Mashayek (2003).

The behavior of the ratios of two other non-zero components of turbulent stresses 02 0 0 0 0 hv02 3 i=hu3 i and hv1 v2 i=hu1 u2 i is qualitatively similar to the behavior of the ratios 02 02 02 02 hv1 i=hu1 i and hv2 i=hu2 i. Figure 2.7 presents components of the anisotropy tensor of particle ﬂuctuation velocity as functions of parameter tpS. The values of bp ij at tpS ¼ 0 correspond to the values of components of the anisotropy tensor of velocity ﬂuctuations bij for the continuous medium. As it is seen from Figure 2.7a, the anisotropy of particle velocity ﬂuctuations increases with particle inertia, and ﬂuctuations of the longitudinal component of velocity of high-inertia particles considerably exceeds the corresponding quantities in the normal and spanwise directions. Increase of anisotropy is

Figure 2.7 Normal (a) and tangential (b) components of the anisotropy tensor of particle velocity fluctuations in a homogeneous shear layer: 1 – model 1; 2 – model 2; 3 – model 3; 4 – model 4; 5 – Simonin et al. (1995), 6 – Lavieville (1997); 7 – Taulbee et al. (1999); 8 – Pandya and Mashayek (2003).

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

Figure 2.8 Ratios of components of velocity fluctuation correlations of the continuous and disperse phases and components of the Reynolds stresses in the continuum: 1 – model 1; 2 – model 2; 3 – Simonin et al. (1995); 4 – Lavieville (1997); 5 – Taulbee et al. (1999); 6 – Pandya and Mashayek (2003).

caused by the contribution from the shear rate toward the generation of longitudinal velocity ﬂuctuations of the disperse phase, as well as by the lack of a mechanism that could level the difference between components of velocity ﬂuctuations (in contrast to the leveling that takes place in a turbulent ﬂuid due to pressure ﬂuctuations). Figure 2.7a also shows that anisotropy of time scales causes a small increase in anisotropy of velocity ﬂuctuations as compared to the values predicted by the quasiisotropic model 2, which neglects this effect. Figure 2.7b demonstrates a nonmonotonous variation of the tangential component of the anisotropy tensor with particle inertia: bp 12 initially decreases with increase of tpS, reaching its minimum value at tpS 1, and then increases, approaching its limiting value at tpS ! 1. Though the inﬂuence of time scale anisotropy on bp ij is insigniﬁcant, it is worthwhile to take this effect into account because it leads to a better agreement with the results of numerical simulation. It is also seen that the effect of particle inertia on the duration of particle interaction with turbulent eddies is almost completely eliminated. Figure 2.8 plots the ratios of components of velocity ﬂuctuation correlations of the continuous and disperse phases and the corresponding components of the Reynolds stresses in the continuum against the product of particle relaxation time and shear rate. Components of the anisotropy tensor of the continuous and disperse phases velocity ﬂuctuations are depicted in Figure 2.9. Comparison of Figure 2.6andFigure 2.8 showsa monotonous decrease of hu01 v01 i=hu02 1 i with increase of tpS, which is different from 0 0 02 the analogous dependence for hu02 v02 i=hu02 2 i. The behavior of hu2 v2 i=hu2 i and 0 0 0 0 02 02 0 0 0 0 hu2 v1 i=hu1 u2 i is similar to the behavior of hv2 i=hu2 i and hv1 v2 i=hu1 u2 i. Figure 2.9a demonstrates that while all the models that are qualitatively similar describe the difference between diagonal components bfp ij, the models that take into account the anisotropy of time scales predict greater anisotropy in correlations of velocity ﬂuctuations of the continuous and disperse phases, while at the same time showing better agreement with the numerical results. Figures 2.8b and 2.9b show that despite their

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Figure 2.9 Components of the anisotropy tensor of velocity fluctuations of the continuous and disperse phases in a homogeneous shear layer: 1 – model 1; 2 – model 2; 3 – model; 4 – model 4; 5 – Simonin et al. (1995); 6 – Lavieville (1997); 7 – Taulbee et al. (1999); 8 – Pandya and Mashayek (2003).

noticeable deviation from the results of numerical simulation, all models qualitatively reproduce the asymmetry of velocity ﬂuctuation correlation tensor of the continuous and disperse phases, which is manifest in the greater absolute value of the tangential component hu02 v01 i as compared to hu01 v02 i. Consider now the consequences of applying the linear and nonlinear algebraic models to the problem of calculating the turbulent stresses for particles in a homogeneous shear layer. The linear model (2.53) gives 2tp S 2 1 0 0 hv02 fu1 k2 hu0k u0n ifu nk 1 i1 ¼ kp þ hu1 uk i fu k1 3 3 3 0 0 tp dhu1 uk i 2tp S tp dhu0k u0n i fu2 k2 þ fu1 k1 fu1 nk ; 3 dt 2 dt 6 tp S 0 0 2 1 0 0 hv02 hu1 uk ifu1 k2 hu0k u0n ifu nk 2 i1 ¼ kp þ hu2 uk ifu k2 þ 3 3 3 0 0 0 0 tp dhu2 uk i tp S dhu1 uk i 1 dhu0k u0n i fu1 k2 þ fu2 k2 fu1 nk ; 3 dt dt 2 3 dt tp S 0 0 2 1 0 0 hv02 hu1 uk ifu1 k2 hu0k u0n ifu nk 3 i1 ¼ kp þ hu3 uk ifu k3 þ 3 3 3 tp dhu03 u0k i tp S dhu01 u0k i 1 dhu0k u0n i fu1 k3 þ fu2 k2 fu1 nk ; 3 dt dt 2 3 dt 2tp Skp 1 hu01 u0k ifu k2 þ hu02 u0k ifu k1 tp Shu02 u0k ifu1 k2 hv01 v02 i1 ¼ 3 2 0 0 0 0 0 0 tp dhu1 uk i dhu2 uk i dhu2 uk i fu1 k2 þ fu1 k1 tp S fu2 k2 : dt dt dt 4 ð2:146Þ

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

j85

The linear model (2.72) leads to ð1Þ 2 hv02 1i hv02 ; 1 i1 ¼ kp þ E0 3

ð1Þ 0 0 hv02 1 i ¼hu1 uk i

2tp S 1 fu k1 hu0k u0n ifunk ; f 3 3 u1k2

ð1Þ tp S 0 0 2 hv02 1 ð1Þ 0 0 2i hv02 ; hv02 hu0 u0 if ; hu u if 2 i ¼hu2 uk if u k2 þ 2 i1 ¼ kp þ E0 3 3 1 k u1k2 3 k n unk ð1Þ tp S 0 0 2 hv02 1 ð1Þ 0 0 3i hv02 ; hv02 hu0 u0 if ; hu u if 3 i ¼hu3 uk if u k3 þ 3 i1 ¼ kp þ E0 3 3 1 k u1k2 3 k n unk

hv01 v02i1 ¼ E0 ¼

tp Skp hv01 v02 i(1) 1 ; ; hv01 v02i(1)¼ (hu01 u0k ifu k2þhu02 u0kifu k1tp Shu02 u0kifu1k2 ) 3 2 E0

1 0 0 hu u if tp Shu01 u0k ifu1k2 : 2kp k n unk

ð2:147Þ

The quadratic model (2.73) for a homogeneous shear layer yields ð1Þ ð1Þ 2tp S 0 0 ð1Þ tp S 0 0 ð1Þ 2 hv02 2 hv02 1i 2i hv1 v2 i ; hv02 þ hv v i ; hv02 2 i2 ¼ kp þ 1 i2 ¼ kp þ 3E0 E1 3E0 E1 1 2 E1 E1 3 3 ð1Þ tp S 0 0 ð1Þ tp S 02 ð1Þ 2 hv02 hv01 v02 ið1Þ 0 0 3i i ¼ þ þ hv v i ; hv v i ¼ hv i ; k hv02 p 1 2 1 2 2 3 2 3E0 E1 2E0 E1 2 E1 E1 3

E1 ¼ E0

tp S 0 0 ð1Þ hv v i 2kp E0 1 2 ð2:148Þ

Equations for the turbulent energy of particles (2.65) in a homogeneous shear ﬂow are given below:

dkp 0 0 ¼ S hv01 v02 i þ hu02 1 ifu1 12 þ hu1 u2 ifu1 22 dt þ

þ

1 02 02 0 0 hu1 ifu 11 þ hu02 2 if u 22 þ hu3 if u 33 þ hu1 u2 i( fu 12 þ fu 21 )2kp tp 1 dhu02 dhu02 dhu02 1i 2i 3i fu1 11 þ fu1 22 þ f 2 dt dt dt u1 33

tp S dhu02 dhu01 u02 i dhu01 u02 i 1i fu1 12 þ fu1 21 þ fu2 12 þ fu2 22 : 2 dt dt dt ð2:149Þ

The turbulent energy equation (2.149) is solved together with the algebraic model for turbulent stresses. Figure 2.10 shows the time evolution of turbulent stress components in the disperse phase, which are calculated from the linear (2.147) and

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Figure 2.10 Comparison of algebraic models for turbulent stresses of the disperse phase with the DNS results: 1 – linear model (2.147); 2 – nonlinear model (2.148), 4–6 – Pandya and Mashayek (2003).

nonlinear (2.148) algebraic models. Calculations are carried out under the conditions that correspond to the DNS data (Pandya and Mashayek, 2003). We can see from Figure 2.10 that the nonlinear model reproduces the behavior of all stress components with rather high accuracy. The linear model overestimates the values of normal and spanwise components and underestimates that of the longitudinal component. In particular, the description of the tangential stress component by the linear model is grossly inadequate. Shown on Figure 2.11 are the components of the anisotropy tensor of particle velocity ﬂuctuations corresponding to the algebraic and differential models. One can see that the values bp ij predicted by the linear models (2.146) and (2.147) turn out to be very close to each other. At the same time linear models underestimate the anisotropy of normal components of turbulent stresses and do not reproduce the nonmonotonous dependence of tangential stress on parameter bp ij. The nonlinear model,

Figure 2.11 Components of the anisotropy tensor of particle velocity fluctuations in a homogeneous shear layer: 1 – differential model (2.137); 2 – linear model (2.146); 3 – linear model (2.147); 4 – nonlinear model (2.148).

2.6 Motion of Particles in the Near-Wall Region

on the other hand, predicts somewhat higher anisotropy of normal components of turbulent stresses as compared to the differential model. Overall, the quadratic model provides a description of all components of turbulent stresses that is qualitatively accurate, though its accuracy decreases noticeably with increase of parameter tpS. At tpS > 1 components of the anisotropy tensor obtained from Eq. (2.148) deviate noticeably from the corresponding values of bp ij given by the differential model (2.137). This deviation is due to the error we make in describing the contribution of transport terms to the balance of tangential stresses by approximation (2.67). Thus, as one would expect, algebraic models reduce the scope of calculations to a noticeable extent; the unavoidable side effect is some loss of accuracy when the particles possess high inertia.

2.6 Motion of Particles in the Near-Wall Region

Consider the dynamics of particles in the near-wall region of a stationary turbulent ﬂow. At high Reynolds numbers, we can distinguish in the near-wall region two zones with radically different characteristics: the viscous sublayer and the equilibrium logarithmic layer. In order to reveal the relevant phenomena, consider some model problems capable of illustrating the peculiarities of particle behavior in the viscous and logarithmic zones. 2.6.1 Near-Wall Region Including the Viscous Sublayer

In the viscous sublayer immediately adjacent to the wall, the role of viscous stresses turns out to be predominant as compared to the turbulent stresses. Accordingly, the kinematic viscosity coefﬁcient of the ﬂuid n and the dynamic velocity (friction velocity) u are the governing parameters in the viscous sublayer. In contrast, the contribution of viscous stresses to the total stresses in the continuum is insigniﬁcant outside the viscous sublayer. As the simplest approximation of ﬂuctuation structure of the carrier ﬂow in the near-wall region, the two-zone model has been suggested (Gusev and Zaichik, 1991); it considers a system consisting of a viscous sublayer with zero ﬂuctuation intensity and a turbulent zone with constant ﬂuctuation intensity hu0i u0j i ¼ Aij u2 H(yd);

ð2:150Þ

where y is the distance to the wall and Aij are constant coefﬁcients. The thickness of the viscous sublayer d is equal (by the order of magnitude) to d ¼ dþ

n ; u

d þ ¼ const:

ð2:151Þ

Suppose the Lagrangian time scale of turbulence near the wall is constant: TL ¼ Tþ

n ; u2

T þ ¼ const:

ð2:152Þ

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Suppose further that the average slip of particles relative to the carrier ﬂow is small so that the crossing trajectory effect does not inﬂuence the duration of particle interaction with turbulent eddies. Since particle collisions and the feedback action of particles on the carrier ﬂow are not taken into account, the equations in the system (2.24)–(2.27) are not coupled: concentration F and intensity of transverse ﬂuctuations hv02 y i could be found independently of the other hydrodynamic characteristics of the disperse phase. For a developed hydrodynamic ﬂow whose parameters vary only in the direction normal to the wall, Eqs. (2.24)–(2.27) with the consideration of Eq. (2.31) and Eq. (2.150) and in the absence of particle deposition give the following equations for F and hv02 y i: F

t2p

dhv02 y i dy

2 þ ðhv02 y i þ g u Ayy u H(yd)Þ

dF ¼ 0; dy

ð2:153Þ

dhv02 d y i 2 Fðhv02 þ 2Fð fu Ayy u2 H(yd)hv02 y i þ g u Ayy u H(yd)Þ y iÞ ¼ 0: dy dy ð2:154Þ

In a move consistent with the experimental data, we take Ayy ¼ 1 and introduce the dimensionless variables hv02 yþ i ¼

hv02 y i u2

; l¼

tp u tþ tp u2 y yþ yu ; t ¼ ¼ : ; yþ ¼ ; tþ ¼ ¼ n d dþ n d dþ

In terms of new variables, Eqs. (2.153)–(2.154) take the form F

t2

dh v02 yþ i dl

þ ðhv02 yþ i þ g u H(l1)Þ

dF ¼ 0; dl

ð2:155Þ

dhv02 d yþ i i þ g H(l1) Þ Fðhv02 þ 2Fð fu H(l1)hv02 yþ yþ iÞ ¼ 0: u dl dl

ð2:156Þ

The boundary conditions for Eqs. (2.155)–(2.156), with the consideration of Eq. (2.118) and in the absence of particle deposition on the wall (c ¼ 1), must be speciﬁed as t

dhv02 yþ i dl

¼2

1e2y 1 þ e2y

2hv02 yþ i

1=2

p

at l ¼ 0;

dhv02 yþ i dl

¼ 0; F ¼ 1 at l ¼ 1: ð2:157Þ

Ignoring the inﬂuence of particle inertia on the duration of particle interaction with turbulent eddies, let us set TLp equal to the Lagrangian scale (2.152), whereby T þ ¼ d þ. Then, according to Eq. (2.22), the involvement coefﬁcients are equal to fu ¼

1 ; 1 þ t

gu ¼

1 : t (1 þ t )

2.6 Motion of Particles in the Near-Wall Region

With Eq. (2.155) in mind, we rewrite Eq. (2.156) as t2 ðhv02 yþ i þ g u H(l1)Þ

d2 hv02 yþ i dl

2

þ 2ð fu H(l1)hv02 yþ iÞ ¼ 0;

ð2:158Þ

which allows us to obtain hv02 yþ i independently of F. Let us construct the solutions of Eq. (2.158) separately in the regions 0 < l < 1 and 1 < l < 1, and then match them at the boundary. In the viscous sublayer zone (0 < l < 1), Eq. (2.158) reduces to 2 02 d hvyþ i 2 i hv02 ¼ 0: ð2:159Þ yþ 2 t2 dl The solution of Eq. (2.159), in view of Eq. (2.157), is 02 hv02 yþ i ¼ 0 at 0 < l < l0 ; hvyþ i ¼

or 02 hv02 yþ i ¼ hvyþ (0)i þ

2(1e2y )

(ll0 )2 at l0 < l < 1 t2

02 1=2 2hvyþ (0)i

t (1 þ e2y )

p

ð2:160Þ

2

lþ

l at 0 < l < 1: t2

ð2:161Þ

The solution (2.160) is valid only at t < tcr, whereas Eq. (2.161) is true at t > tcr . The critical value tcr of the inertia parameter t represents the bifurcation point corresponding to the condition l0 ¼ 0. In the turbulent zone (1 < l < 1), Eq. (2.158) takes the form t2

d2 hv02 yþ i 2

dl

2 fu hv02 yþ i

þ 02 ¼ 0: hvyþ i þ g u

ð2:162Þ

In order to ﬁnd the analytical solution, Eq. (2.162) must be linearized by taking 02 hv02 yþ i ¼ hvyþ (1)i in the denominator of the second term. We obtain as a result the approximate solution 21=2 (l1) 02 i ¼ ð hv (1)if Þ exp þ fu at 1 < l < 1: ð2:163Þ hv02 yþ yþ u 1=2 t ðhv02 yþ (1)i þ g u Þ The matching conditions for solutions in the viscous and turbulent zones are as follows: 02 02 dhvyþ i dhvyþ i 02 02 02 (10)i¼hv (1þ0)i; hv (1)i ¼ ð hv (1)iþg Þ : hv02 yþ yþ yþ yþ u dl 10 dl 1þ0 ð2:164Þ Equations (2.160), (2.162), and (2.163) result in the following relations for hv02 yþ (1)i and l0 at t < tcr :

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1=2

02 3 1=2 1=2 fu hv02 ¼ ð2hv02 ; l0 ¼ 1t hv02 : yþ (1)i hvyþ (1)i þ g u yþ (1)i Þ yþ (1)i ð2:165Þ

Now, combining the equations (2.161), (2.163), and (2.164), we obtain the follow02 ing relations, from which we can ﬁnd hv02 yþ (1)i and hvyþ (0)i at t > tcr: 1=2 2(1e2y )

1=2

02 2 1=2 02 02 hv fu hv02 (1)i hv (1)i þ g ¼ hv (1)i þ (0)i ; yþ yþ yþ u t p1=2 (1 þ e2y ) yþ 1=2 1=2 1e2y 2(1e2y )2 2 1 1=2 02 hv02 (0)i ¼ þ þ hv (1)i : yþ yþ t (1 þ e2y ) p t2 p(1 þ e2y )2 t2 ð2:166Þ The critical inertia parameter is deﬁned from t2cr hv02 yþ (0)i ¼ 1. It is independent of momentum restitution coefﬁcient ey and is equal to 2.81. The distribution of particle concentration that obeys the condition F(1) ¼ 1 is found by integrating Eq. (2.155), which gives us (

02 1 02 at l < 1; hv02 yþ (1)i t hvyþ (1)i þ g u hvyþ i ð2:167Þ F¼

1 02 t hvyþ i þ g u at l > 1: Figure 2.12 shows the distributions of transverse velocity ﬂuctuations 1=2 (v0yþ ¼ hv02 ) and of particle concentration corresponding to Eqs. (2.160), yþ i (2.161), (2.163), (2.165), (2.166), and (2.167), assuming collisions with the wall to be elastic (ey ¼ 1). We can see that with increase of particle inertia, the intensity of particle velocity ﬂuctuations differs more and more from the intensity of ﬂuctuations in the continuum (2.150), and its distribution becomes homogeneous. Concentration of particles near the wall rises sharply, in other words, we observe accumulation of particles in the viscous sublayer zone.

Figure 2.12 Intensity distribution of transverse velocity fluctuation (a) and particle concentration (b) in the near-wall region including the viscous sublayer: 1–t ¼ 0.5; 2–1.5; 3–2.81; 4–5; 6–10.

2.6 Motion of Particles in the Near-Wall Region

Figure 2.13 Velocity fluctuation intensity (I) and particle concentration (II) at the wall vs. inertia parameter: 1 – ey = 0.5; 2 – 0.8; 3 – 1.0.

The phenomenon of particle accumulation that takes place in non-homogeneous turbulent ﬂows is explained by turbulent migration of particles (turbophoresis) from a region of highly intense turbulent velocity ﬂuctuations into a region of low turbulence, in particular, into the viscous sublayer adjacent to the surface of a body submerged into the ﬂow. A theoretical interpretation of this phenomenon has been proposed by Caporaloni et al. (1975) and Reeks (1983). Figure 2.13 shows the inﬂuence of particle inertia on the intensity of velocity ﬂuctuations and particle concentration at the wall. As one can easily see, ﬂuctuational energy of low-inertia particles (t < tcr) vanishes at the wall, unlike the ﬂuctuational intensity of inertial particles (t > tcr), which does not vanish. The phenomenon of non-zero velocity ﬂuctuations in a viscous sublayer and at the wall itself is caused by the transport of ﬂuctuations (via diffusion) from the turbulent region of the ﬂow by inertial particles. Notice the pronounced maximum on the graph of v0 yþ (0) versus t . The increase of v0 yþ (0) with t is explained by the increasing role of diffusive transport of ﬂuctuations from the turbulent region to the viscous sublayer zone. The reduction of v0 yþ (0) with further increase of t after reaching its maximum value is caused by decreased intensity of velocity ﬂuctuation in the disperse phase, because particles with higher inertia are more involved in the turbulent motion of the continuum. Particle concentration at the wall goes to inﬁnity at t < tcr and approaches unity at t ! 1. The intensity of ﬂuctuations falls off with decrease of momentum restitution coefﬁcient ey, while accumulation of particles in the viscous sublayer increases noticeably. 2.6.2 The Equilibrium Logarithmic Layer

At a certain distance from the wall, the equilibrium state of turbulent ﬂow characterized by the equal rates of production and dissipation of turbulent energy (P ¼ e) may be reached. In an equilibrium state, convective and diffusional mechanisms of ﬂuctuation transport become insufﬁcient, triple single-point correlations vanish, and the

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92

second moments of velocity correlations are constants. But on the whole, the region of turbulent ﬂow is not homogeneous, because spatial and temporal scales of turbulence depend linearly on the transverse coordinate y. Note that a sufﬁciently extent equilibrium near-wall region is observed only at high Reynolds numbers. For example, in a ﬂow along a tube (channel) of radius (half-width) R the logarithmic layer is located in a spatial region satisfying the condition v/u y R. The governing physical parameters in the near-wall equilibrium turbulent zone are u and y. Then, in accordance with the similarity theory, the gradient of the longitudinal velocity is equal to S¼

dU x ku ¼ ; dy y

ð2:168Þ

where k 0.4 is the Prandtl–Karman constant. Due to Eq. (2.168), the velocity distribution is described by the logarithmic dependence Uþ ¼

Ux ¼ k ln yþ þ B; u

B ¼ const;

ð2:169Þ

hence the word logarithmic in the name of this turbulent ﬂow region. Equation (2.27) leads to the following system of equations for turbulent stresses in the disperse phase of the equilibrium logarithmic layer: fu12 02 0 0 02 hv02 þf i¼hu if þhu u if t S hu i p 1 1 u11 1 2 u21 1 u112 2 f þf hu02 if þhu01 u02 i u11 u22 þfu122 þ 2 u21 2 2 þ

t2p S2 0 0 hu1 u2 i( fu12 þfu112 )þhu02 2 i( f u22 þfu122 ) ; 2

0 0 02 02 02 hv02 2 i¼hu1 u2 ifu12 þhu2 ifu22 ; hv3 i¼hu3 ifu33 ; 1 hv01 v02 i¼ ½hu02 if þhu01 u02 i( fu11 þfu22 )þhu02 2 if u21 2 1 u12

tp S 0 0 ½hu1 u2 i fu12 þfu112 þhu022 i fu22 þfu122 ; 2

ð2:170Þ

where the subscripts 1, 2, and 3 stand for the coordinates x, y, and z. The equations (2.170) look similar to the corresponding equations for a homogeneous shear ﬂow (2.137), once the terms containing time derivatives are eliminated from them. The equations for mixed moments of velocity ﬂuctuations for the continuous and disperse phases follow from Eq. (2.43): 0 0 02 0 0 hu01 v01 i ¼ hu02 1 ifu 11 þ hu1 u2 ifu 21 tp Sðhu1 if u1 12 þ hu1 u2 ifu1 22 Þ; 0 0 02 hu02 v02 i ¼ hu01 u02 ifu 12 þ hu02 2 ifu 22 ; hu3 v3 i ¼ hu3 ifu 33 ; 0 0 hu01 v02 i ¼ hu02 1 ifu 12 þ hu1 u2 ifu 22 ;

0 0 02 hu02 v01 i ¼ hu01 u02 ifu 11 þ hu02 2 ifu 21 tp S hu1 u2 if u1 12 þ hu2 if u1 22 :

ð2:171Þ

2.6 Motion of Particles in the Near-Wall Region

Figure 2.14 Ratio of turbulent stress components of the continuous and disperse phases in the equilibrium near-wall 02 02 02 02 02 region: 1 – hv02 1 i=hu1 i; 2 – hv2 i=hu2 i and hv3 i=hu3 i; 3 – hv01 v02 i=hu01 u02 i.

Equations (2.171) follow directly from equation (2.138) for velocity correlation ﬂuctuations of the continuous and disperse phases in a homogeneous shear ﬂow after we drop non-stationary terms in the latter equations. Figures 2.14 and 2.15 illustrate the behavior of all non-zero turbulent stress components of particles and correlation moments of velocity ﬂuctuations for the continuous and disperse phases divided by the corresponding turbulent stress components of the carrier ﬂow. Turbulent characteristics of the continuum are taken from the experimental data for the logarithmic layer of a turbulent ﬂow in a channel (Laufer, 1951): b11 ¼ 0.22, b22 ¼ 0.15, b33 ¼ 0.07, b12 ¼ 0.16, Sk/e ¼ 3.1, P/e ¼ 1. The results presented below have been obtained for the Lagrangian isotropic scale (2.141). It is evident from the comparison of Figure 2.6 and Figure 2.14 that the ratio 02 02 02 hv02 2 i=hu2 i, as well as the ratio hv3 i=hu3 i, shows nearly identical dependence on

Figure 2.15 Velocity fluctuation correlations for the continuous and disperse phases in the equilibrium near-wall region: 1 – 0 0 02 0 0 02 0 0 0 0 hu01 v01 i=hu02 1 i; 2 – hu2 v2 i=hu2 i, hu3 v3 i=hu3 i and hu1 v2 i=hu1 u2 i, 3 – hu02 v01 i=hu01 u02 i.

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94

parameter tpS in the homogeneous shear and logarithmic near-wall layers. On the 02 other hand, there is a noticeable difference in the behavior of hv02 1 i=hu1 i in these two 02 02 layers. For example, in the near-wall region, hv1 i=hu1 i increases unboundedly as tpS ! 1 instead of tending to zero, as it does in a homogeneous shear ﬂow. The ratio hv01 v02 i=hu01 u02 i approaches a ﬁnite limit as tpS ! 1 in the near-wall region, while tending to zero in the homogeneous shear layer. This difference in the behavior of longitudinal and tangential stresses is explained by the difference between stationary and non-stationary solutions for the two types of equilibrium ﬂows. Anisotropy of turbulent stresses grows with particle relaxation time tp and with velocity gradient S. The form of the dependence of anisotropy tensor components bp ij on parameter tpS in the logarithmic near-wall region and in the equilibrium state with a constant shear is qualitatively the same (Figure 2.7). As we see from the comparison of Figure 2.8 and Figure 2.15, the behavior of hu0i v0j i=hu0i u0j i in the homogeneous shear and logarithmic near-wall equilibrium layers is qualitatively similar. Asymmetry of the tensor hu0i v0j i is conﬁrmed by the results of experimental studies performed by Ferrand et al. (2003) in the region of maximum shear rate of an axisymmetric jet, where the ﬂow is close to the equilibrium state deﬁned by the condition P ¼ e. Let us compare the ratio between the longitudinal and transverse velocity ﬂuctuation intensities calculated on the basis of Eq. (2.170) to the experimental data obtained by Rogers and Eaton (1990). Experiments were carried out in the boundary layer developing in a rectangular channel; the disperse medium consisted of small glass balls of sizes 50 mm and 90 mm. In agreement with the experimental data, the following relations were speciﬁed between components of Reynolds stresses in the 02 1=2 carrier ﬂow: (hu02 ¼ 1:5 and hu01 u02 i=hu02 1 i=hu2 i) 2 i ¼ 1:0. The duration of particle interaction with turbulent eddies TLp was taken to be equal to the Lagrangian scale (2.141). In order to get an explicit dependence of TLp on the distance to the wall y, the following relations were invoked for the turbulent energy of the ﬂuid and its dissipation rate in the logarithmic layer: k¼

u2

; e¼ 1=2

Cm

u3 ; Cm 0:09: ky

ð2:172Þ

A substitution of Eq. (2.172) into Eq. (2.141) gives TL ¼

ky : u

ð2:173Þ

Figure 2.16 plots the ratio v01 =v02 calculated by Eq. (2.170) with the consideration of Eq. (2.168) and Eq. (2.173) against the distance from the wall y divided by the halfwidth of the channel R. For the economy of space, we have introduced the notations 1=2 0 1=2 0 1=2 0 1=2 u01 ¼ hu02 ; u2 ¼ hu02 ; v1 ¼ hv02 ; v2 ¼ hv02 . It is seen that in view of the 1i 2i 1i 2i experimental data, ﬂuctuational motion of particles is characterized by strong anisotropy. Furthermore, the ratio between the longitudinal and transverse ﬂuctuational velocities increases with particle diameter and also increases as the distance from the wall gets shorter.

2.6 Motion of Particles in the Near-Wall Region

Figure 2.16 Ratio of the longitudinal and transverse fluctuation intensities in the boundary layer: 1, 4 – u01 =u02 ; 2, 3, 5, 6 – v01 =v02 ; 2, 5 – dp ¼ 50 mm; 3, 6 – dp ¼ 90 mm; 4–6 – experiment (Rogers and Eaton, 1990).

2.6.3 High-Inertia Particles

Consider the behavior of high-inertia particles (t þ 1) in the near-wall region of the turbulent ﬂow. The behavior of very inertial particles in the near-wall region does not depend on the particulars of the ﬂow in the viscous sublayer and is essentially deﬁned by turbulent parameters of the ﬂow in the logarithmic layer. Hence in order to ﬁnd the distributions of concentration and ﬂuctuation intensity of the transverse velocity component for high-inertia particles, one should make a transition to self-similar variables in Eqs. (2.153)(2.154): y y h¼ ¼ þ tp u tþ and then obtain the asymptotic solution at t þ ! 1. Since the solution depends on the parameters of the logarithmic zone but not of the viscous sublayer, the Lagrangian time macroscale is described by formula (2.173) rather than by Eq. (2.152). The duration of particle interaction with turbulent eddies TLp is taken to be equal to the Lagrangian scale (2.173) and the coefﬁcient Ayy is taken to be equal to unity. In this formulation of the problem the presence of a viscous sublayer at the wall is taken into account only by the given boundary condition (2.108) at c ¼ 1. Thus determination of concentration distribution and ﬂuctuation intensity of transverse velocity of inertial particles in the near-wall region boils down to solving the problem of ﬁnding the self-similar relative relaxation time for a particle: dhv02

dF

dhv02

d yþ i 02 yþ i iþg ¼0; F hv02 þ hvyþ iþg u þ2F fu hv02 yþ yþ i ¼0; u dh dh dh dh 02 2 02 1=2 02 dhvyþ i 1ey 2hvyþ i dhvyþ i at h¼0; ¼2 ¼0; F¼1 h!1; 2 1þey dh p dh F

fu ¼

kh k 2 h2 : ; gu ¼ 1þkh 1þkh

ð2:174Þ

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Figure 2.17 Distributions of transverse velocity fluctuation intensity (") and concentration of inertial particles (b) in the nearwall region: 1 – ey ¼ 0.5; 2 – ey ¼ 0.8; 3 – ey ¼ 1.0; 4 – Eq. (2.192).

The obtained distributions of ﬂuctuation intensities of transverse velocity and particle concentration are shown on Figure 2.17. It is seen that at large values of h, when diffusive transport of velocity ﬂuctuations does not play a signiﬁcant role, intensities of velocity ﬂuctuations of the disperse and continuous phases are connected by a local homogeneous relation 0 hv02 yþ i ¼ f u hu yþ i: 2

ð2:175Þ

But this relation breaks down at small values of h owing to the governing role of the diffusional mechanism of ﬂuctuation transport, and hv02 yþ i tends to a limit as h ! 0. We can also see by looking at Figure 2.17b a monotonous increase of particle concentration with decrease of h. At h ¼ 0 for ey ¼ 1 we get hv02 yþ (0)i ¼ 0:076, F(0) ¼ 5:4. Similarly to the solution of the problem (2.155)–(2.157), reduction of ey at small h is accompanied by a decrease in intensity of velocity ﬂuctuations; on the other hand, particle concentration increases.

2.7 Motion of Particles in a Vertical Channel

As of today, there exists a large body of experimental and numerical research devoted to the subject of disperse turbulent ﬂows in vertical and horizontal channels and pipes. For the most part, this research is devoted to the feedback action of particles on characteristics of the turbulent carrier ﬂow, for example, Tsuji and Morikawa, 1982; Tsuji et al., 1984; Varaksin et al., 1998. Experimental data for turbulent structure of the disperse phase (intensity of ﬂuctuations and probability density of velocities, spatial distributions of particles and so on) in air and water ﬂows have been obtained by Lee and Durst (1982), Rogers and Eaton (1990), Young and Hanratty (1991), Sommerfeld (1992), Fessler et al. (1994), Kulick et al. (1994), Kaftori et al. (1995), Sato et al. (1995), Varaksin and Polyakov (2000), Khalitov and Longmire (2003), Righetti and Romano (2004), Hadinoto et al. (2005). Numerical studies of particle behavior in channels and

2.7 Motion of Particles in a Vertical Channel

pipes in the absence of particle deposition that employ the DNS and LES methods to model the continuous phases turbulent characteristics have been performed by Pedinotti et al. (1992), Rouson and Eaton (1994, 2001), Pan and Banerjee (1996, 1997), Wang and Squires (1996a), Simonin et al. (1997), Fukagata et al. (1998, 1999), Li et al. (1999), Wang et al. (1998), Li et al. (2001), Portela et al. (2002), Arcen et al. (2005), Picciotto et al. (2005), Kuerten (2006), Vance et al. (2006). A number of experimental and numerical works point to the possibility of formation of high particle concentration in the near-wall region of turbulent ﬂow, which was predicred theoretically by Caporaloni et al. (1975) and by Reeks (1983). Marchioli and Soldati (2002) have shown the governing role of coherent turbulent structures in particle accumulation and analyzed segregation mechanisms in the turbulent boundary layer. The question arises as to the possibility of adequately taking into account the contribution of these mechanisms and of particle interactions with coherent structures to the process of particle accumulation in the framework of the continuous statistical approach. Consider a hydrodynamic developed ﬂow in a planar vertical channel (Alipchenkov and Zaichik, 2006). All characteristics of the continuous and disperse phases are supposed to be self-similar with respect to the longitudinal coordinate x1 and dependent only on the coordinate x2 ¼ y normal to the wall. Channel walls are assumed to be impermeable, and particle deposition is assumed to be absent. Consequently, the normal components of average velocities of both phases are zero (U2 ¼ V2 ¼ 0), and, as follows from Eq. (2.25) and Eq. (2.26), the balance equations for the continuum written for the longitudinal and normal directions are

d ln F V 1 U 1 dhv01 v02 i 0 0 þgþ þ hv1 v2 i þ m12 ¼ 0; tp dy dy

ð2:176Þ

d ln F dhv02 2i þ hv02 ¼ 0; 2 i þ m22 dy dy

ð2:177Þ

where g is the acceleration of gravity (g > 0 for an upward ﬂow and g < 0 for a downward ﬂow). The last two terms on the left-hand side of Eq. (2.176) have the meaning of turbulent stress and contribution of diffusion to the balance of forces in the vertical direction. Eq. (2.177) expresses the balance of turbophoresis and diffusion-driving forces counterbalancing particle migration across the channel in the direction of turbulent ﬂuctuation velocity intensity decrease. We obtain from Eq. (2.27) and in view of Eq. (2.31) the following system of equations for non-zero components of disperse phases turbulent stresses:

dhv02

dhv01 v02 i 0 0 02 1 d 1i þ 2 hv1 v2 i þ m12 Ftp hv2 i þ m22 3F dy dy dy

dV 1 2hv02 i 2 hv01 v02 i þ m12 þ 2l11 1 ¼ 0; dy tp

ð2:178Þ

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1 d dhv02 2hv02 i 2i i þ m Þ Ftp ðhv02 þ 2l22 2 ¼ 0; 22 2 F dy dy tp

ð2:179Þ

1 d dhv02 2hv02 i 3i i þ m Þ Ftp ðhv02 þ 2l33 3 ¼ 0; 22 2 3F dy dy tp

ð2:180Þ

1 d dhv01 v02 i dhv02 0 0 2i i þ m Þ ð hv v i þ m Þ Ftp 2ðhv02 þ 22 12 2 1 2 3F dy dy dy ðhv02 2 i þ m22 Þ

dV 1 2hv0 v0 i þ l12 þ l21 1 2 ¼ 0: dy tp

ð2:181Þ

For the purpose of further simpliﬁcation, the Lagrangian time macroscale TL ij is assumed to be isotropic (2.38), despite the existence of theoretical and numerical studies pointing to a signiﬁcant anisotropy of TL ij in channels (Kontomaris et al., 1992; Bernard and Rovelstad, 1994; Wang et al., 1995; Mito and Hanratty, 2002; Rambaud et al., 2002; Iliopoulos et al., 2003; Ushijima et al., 2003; Arcen et al., 2004; Choi et al., 2004; Oesterle and Zaichik, 2004; Cho et al., 2005). And yet we shall still take into account the different durations of particle interaction with turbulent eddies in different directions – a phenomenon that arises as a consequence of the crossing trajectory effect. As a result, we shall still observe the distinction between longitudinal (in x1- direction) T lLp and transverse (in x2- and x3-directions) T nLp components of TL ij. In view of the adopted assumptions, the quantities lij and mij in Eq. (2.16) and Eq. (2.17) will be equal to l11

0 0 ful 02 1 l Dp hu02 n 0 0 dU 1 n Dp hu1 u2 i dU 1 1i ¼ hu1 i þ lu hu1 u2 i ; f þ tp lu1 tp dy dy 2 u1 Dt Dt

l12 ¼

fun 0 0 f n Dp hu01 u02 i hu1 u2 i u1 ; tp 2 Dt

l21 ¼

02 ful 0 0 dU 1 1 l Dp hu01 u02 i n Dp hu2 i dU 1 þ t ; hu1 u2 i þ lnu hu02 i l f p u1 2 Dt Dt tp dy dy 2 u1

l22 ¼

n n Dp hu02 Dp hu02 fun 02 fu1 fn fu1 2i 3i ; l33 ¼ u hu02 ; hu2 i 3 i tp 2 Dt tp 2 Dt

n 0 0 m11 ¼ g lu hu02 1 i þ tp h u hu1 u2 i

m12 ¼ g nu hu01 u02 i

tp g nu1 Dp hu01 u02 i ; 2 Dt

m21 ¼ g lu hu01 u02 i þ tp hnu hu02 2i m22 ¼ g nu hu02 2 i

dU 1 tp g lu1 Dp hu02 1i ; Dt dy 2

dU 1 tp g lu1 Dp hu01 u02 i ; Dt dy 2

tp g nu1 Dp hu02 2i : Dt 2

ð2:182Þ

2.7 Motion of Particles in a Vertical Channel

j99

In the case under consideration, the derivatives Dp hu0i u0j i=Dt entering the transport term of the Lagrangian correlation moment of ﬂuid particle velocity ﬂuctuations along inertial particle trajectories (2.15) are equal to 02 0

dhu02

dhu01 u02 i 0 0 Dp hu02 1d 1 i dhu1 v2 i 1i iþm v iþm ¼ tp hv02 þ2 hv ; ¼ 22 12 2 1 2 dy 3 dy dy dy Dt

dhu01 u02 i 0 0

dhu02 Dp hu01 u02 i dhu01 u02 v02 i 1d 2i ¼ iþm v iþm ¼ tp 2 hv02 þ hv ; 22 12 2 1 2 Dt dy 3 dy dy dy 02 0

dhu02 Dp hu02 d 2 i dhu2 v2 i 2i ¼ iþm ¼ tp hv02 ; 22 2 Dt dy dy dy 02 0 02

dhu02 Dp hu02 1d 3 i dhu3 v2 i 3i ¼ tp hv2 iþm22 : ð2:183Þ ¼ dy 3 dy dy Dt As our boundary conditions for turbulent stress components at the channel wall, we shall take the relations (2.107)–(2.109), (2.118) (while ignoring the deposition of particles (c ¼ 1) at ez ¼ 1), tp

1=2 1=2 1e2y 2hv02 dhv02 1e2x 2 dhv02 dhv02 02 1i 2i 2i 3i hv i; t ; ¼3 ¼ 2 ¼ 0; p 1 02 2 2 1 þ ex phv2 i 1 þ ey dy dy p dy

hv01 v02 i ¼ mx hv02 2 i at y ¼ 0

ð2:184Þ

and the symmetry condition on the channel axis, dF dhv02 i dhv02 i dhv02 i ¼ 1 ¼ 2 ¼ 3 ¼ hv01 v02 i ¼ 0 at y ¼ R; dy dy dy dy

ð2:185Þ

where R is the channel half-width. In view of the fact that even at large values of mass the average Reynolds number near the wall (in the viscous sublayer region) Rel is not large, we use for the coefﬁcient of particles involvement in the turbulent ﬂow of the carrier ﬂuid a twoscale bi-exponential autocorrelation function similar to Eq. (1.5): V YLp (t) ¼

1 2t 2t (1 þ RV )exp (1RV )exp ; V V 2RV (1 þ RV )T Lp (1RV )T Lp

RV ¼ (12z2V )1=2 ; zV ¼

tT V ; V ¼ l; n: T Lp

ð2:186Þ

Equation (2.186) takes into account the inﬂuence of particle inertia on the integral V time-scales T Lp only; the differential scale is taken to be equal to the Taylor time scale tT, which, in its turn, is assumed to be isotropic and is determined by the relations (1.6) and (1.7). The availability of isotropic relations to determine tT is supported by

j 2 Motion of Particles in Gradient Turbulent Flows

100

the DNS data for turbulent ﬂow in a channel (Choi et al., 2004), which is consistent with the fact that the amplitude a0 of ﬂuctuation accelerations is practically isotropic. Due to Eq. (2.18), the autocorrelation function (2.186) gives rise to the following involvement coefﬁcients in the expressions (2.182) and (2.183):

2 2WV þ z2V 2W2V z2V 2WV þ z2V V V ; fu1 ¼ fu ¼

2 ; 2WV þ 2W2V þ z2V 2WV þ 2W2V þ z2V V fu2

3 2½ 2WV þ z2V 6W2V z4V 8W3V z2V ¼ ;

3 2WV þ 2W2V þ z2V

hVu ¼

24WV z2V 2W2V

V

V

þ 2f Vu þ fu1 ; g u1 ¼

g Vu ¼

2z2V 2W2V

1 V f V ; lV ¼ g Vu fu1 ; WV u u

V

V

V

V

fu1 ; lu1 ¼ g u1 fu2 ;

ð2:187Þ

V

where WV tp =T Lp is the inertia parameter of the particle. At high Reynolds numbers (zV ! 0 at Rel (20 k2/3ev)1/2 ! 1), the involvement coefﬁcients (2.187) reduce to the relations (2.22) that are pertinent to the corresponding exponential autocorrelation function (2.21). Integral time scales of particle interaction with turbulent eddies are deﬁned by the model that is based on Eqs. (1.61), (1.65), (1.70), and (1.71):

3I þ m 2 þ 3g 2 3I þ m 2 þ 3g 2 1 ) TE; f (St

2 þ E 1 þ mg 3I(1 þ mI)2 3I 1 þ mI 6I þ m(4 þ 3g 2 ) 2 þ mg 6I þ m(4 þ 3g 2 ) ¼ þ ) TE; f (St E 6I(1 þ mI)2 2(1 þ mg)2 6(1 þ mI)2

T lLp ¼ T nLp

TE ¼

3(1 þ m)2 jV 1 U 1 j ; T L; g ¼ 3 þ 2m (2k=3)1=2

I ¼ (1 þ g 2 )1=2 ; f (StE ) ¼

StE 0:9 mSt2E : 1 þ StE (1 þ StE )2 (2 þ StE )

ð2:188Þ

The Lagrangian integral time scale is given by the approximation 2 1=2 10n 2 k TL ¼ þ a ; a ¼ C1=2 m ¼ 0:3: u2 e

ð2:189Þ

As a corollary of Eq. (2.189), which was also obtained independently by Kallio and Reeks (1989) and by some other authors by examining the characteristics of nearwall turbulence in the viscous sublayer, we get for the integral time scale: T Lþ T L u2 =n ¼ 10. As we get farther from the wall, the relation (2.189) turns into TL ¼ ak/e, which is consistent with Eq. (2.141). The relation between a and the Kolmogorov–Prandtl constant Cm follows from Eq. (2.172) and Eq. (2.173) for the logarithmic layer.

2.7 Motion of Particles in a Vertical Channel

The relaxation time for a particle is deﬁned by Eq. (1.47) where, due to Eq. (2.145), the Reynolds number of the ﬂow past the particle is estimated as dp ½(U 1 V 1 )2 þ 2(k þ kp 2fum k) Rep ¼ n

1=2

; fum ¼

ful þ 2fun : 3

It should be noted that if we neglect the crossing trajectory effect (i.e., neglect the dependence of time scales of particle interaction with turbulent eddies on drift parameter g), and the inﬂuence of particles Reynolds number Rep on their relaxation time (i.e., take tp ¼ tp0), than, provided that particles do not affect the characteristics of the carrier ﬂuid, the equations (2.176)–(2.185) can be decoupled so that the 02 02 0 0 variables hv02 2 i; hv3 i, and F can be determined independently of V 1 ; hv1 i, and hv1 v2 i. Shown below are the results of numerical solution of the system of equations (2.176)–(2.181), as based on the boundary conditions (2.184), (2.185) and on the relations (2.182), (2.183). Emphasis is placed on the effect of particle inertia and on the signiﬁcance of the coefﬁcients that appear in the boundary conditions. The structure parameter m was taken equal to 0.5 in all calculations. In order to compare the obtained results with the DNS and LES data, two sets of calculations were carried out – for low-inertia and high-inertia particles. The former were compared with the results obtained by Picciotto et al. (2005) and the latter – with those obtained by Rouson and Eaton (1994), Wang and Squires (1996) and Fukagata et al. (1998). In Picciotto et al. (2005), the gravity force was neglected and the Reynolds number Re þ Ru /v was taken equal to 150. Calculations in Rouson and Eaton (1994), Wang and Squires (1996) and Fukagata et al. (1998) were carried out with the force of gravity taken into account for the downward ﬂow at Re þ ¼ 180 but with no consideration of the feedback action of particles on the turbulence. In the cited works, the interaction of particles with the wall was assumed to be elastic and frictionless. Consequently, when running a comparison with the DNS and LES data, the coefﬁcients in the boundary conditions (2.184) were taken as follows: ex ¼ ey ¼ 1 and mx ¼ 0. When deﬁning the dimensional variables, kinematic viscosity coefﬁcient of the ﬂuid n and the dynamic velocity (friction rate) u were employed as follows: U1 þ ¼ U1/u , V1+ ¼ V1/u , u0 iþ ¼ hu0 2i i1=2 =u , v0 iþ ¼ hv0 2i i1=2 =u , y þ ¼ yu /n, tþ ¼ tp0 u2 =n, and the F=F(R). particle concentration was normalized by its value at the channel axis F Figure 2.18 demonstrates the distributions of average longitudinal velocity and all non-zero components of turbulent stresses and particle concentration over the channel cross-section at relatively small values of inertia parameter t þ . It is seen from Figure 2.18 that the average velocity of particles V1 slightly deviates from the velocity of the carrier medium U1 even at t þ ¼ 25. According to Picciotto et al. (2005), particle velocity slightly exceeds ﬂuid velocity in the region 1 < y þ < 10; the opposite is true for the region 10 < y þ < 60. The intensity of the longitudinal component of particle velocity ﬂuctuations v0 1þ in the near-wall region exceeds that of ﬂuid velocity ﬂuctuations u0 1þ and increases with t þ , though not as rapidly as suggested by the numerical simulation (see Figure 2.18b). It should be noted that, as follows from Eq. (2.178) and Eq. (2.181), there are two mechanisms of generation of longitudinal and tangential components of particle turbulent stresses: generation of ﬂuctuations due to the gradient of disperse phases average velocity, and generation resulting

j101

j 2 Motion of Particles in Gradient Turbulent Flows

102

from direct interaction between the particles and the turbulent eddies that is described by the terms lij. Thus growth of v01þ with t þ is explained by the gradient mechanism of generation of ﬂuctuations. Normal v02þ and spanwise v03þ components of ﬂuctuations decrease over the entire channel cross-section with increase of t þ (Figure 2.18c, d). One reason for this is the absence of the gradient mechanism of generation of normal and spanwise components of ﬂuctuations, as evidenced by Eq. (2.179) and Eq. (2.180); the other reason is the weaker involvement of particles in ﬂuctuational motion as particle inertia grows. As can be seen from Figure 2.18e, the absolute value of the tangential stress component ﬁrst rises with increase of t þ over the whole channel cross-section as a result of ﬂuctuation generation by the average velocity gradient and then falls off as a result of decreasing involvement of particles in ﬂuctuational motion. Figure 2.18f indicates increased concentration of inertial particles in the viscous sublayer adjacent to the wall, where the gradient of velocity ﬂuctuation intensity of the carrier ﬂow reaches its highest value. Notice the theoretical models prediction that F could have the singularity at y ! 0 for particles whose inertia is not very high. The decrease of F at y ! 0 for t þ = 1 and 5 that was obtained by Picciotto et al. (2005) is not physically understood; however, it might be connected with the lack of time needed for a stationary proﬁle of particle concentration to be established. Figure 2.19 shows distributions of disperse phases characteristics over the channel cross-section at large values of inertia parameter t þ . The key feature of the average velocity proﬁle is that it becomes more ﬂat as particle inertia increases (Figure 2.19"), which was pointed out in many experimental and numerical studies. From Figure 2.19b it follows that as t þ grows, the maximum of v0 1þ shifts toward the wall, and the intensity of the longitudinal velocity ﬂuctuation component at the wall increases. For high-inertia particles (t þ ¼ 810), we observe a monotonous increase of v0 1þ from the channel axis to the wall, where v0 1þ reaches its maximum value (even though the theoretical model (2.176)–(2.181) predicts smaller values of that maximum than suggested by the corresponding numerical models). With growth of t þ , the proﬁles of v0 2þ and v0 3þ are ﬂattened due to the intensive diffusive transport of velocity ﬂuctuations in the transverse direction and tend to homogeneous distributions. The asymptotic distributions of normal and spanwise components of velocity ﬂuctuations in the case of elastic interactions of particles with the wall follow from Eq. (2.179) and Eq. (2.180) at t þ ! 1: hv0 i i ¼ 2

ðR 1 2 T nLp hu0 i idy; tp R

i ¼ 2; 3:

ð2:190Þ

0

Tangential stress falls off with growth of t þ over the whole channel cross-section (Figure 2.19d), which is explained, ﬁrst, by weaker inﬂuence of the gradient ~

Figure 2.18 Distribution of low-inertia particle characteristics over the channel cross-section: 1 – fluid; 2–7 – particles; 2, 3, 4 – solution of equations (2.176)–(2.181); 5, 6, 7 – Picciotto et al. (2005); 2, 5 – t+ ¼ 1; 3, 6 – t+ ¼ 5; 4, 7 – t+ ¼ 25.

2.7 Motion of Particles in a Vertical Channel

j103

j 2 Motion of Particles in Gradient Turbulent Flows

104

mechanism of generation of ﬂuctuations due to the ﬂattening of the average velocity proﬁle, and secondly, by decreased involvement of particles in the turbulent motion. Figure 2.19e conﬁrms the decrease of the rate of particle accumulation near the wall with growth of t þ for high-inertia particles, which is due to the ﬂattening of the proﬁle of the transverse component of velocity ﬂuctuations and the resultant decrease of the turbophoretic driving force. As we see from Figures 2.18 and 2.19, the model (2.176)–(2.181) by and large reproduces all of the effects found in numerical experiments for both low-inertia and high-inertia particles. Figure 2.20 illustrates the action of the inertia parameter t þ on turbulent stresses and particle concentrations in the viscous sublayer at y þ = 1. We see that all stress components have a maximum. The rise of v0 1þ ; v0 2þ , and jhv01 v02 iþ j with t þ is explained by the greater role of diffusive transport of ﬂuctuations from the regions of high turbulence into the viscous layer zone. The decrease of v0 2þ with t þ that takes place after the maximum has been reached can be explained by the reduced intensity of disperse phases velocity ﬂuctuations, because the higher the inertia of particles, the less are they involved in turbulent motion of continuous phase. The decrease of v0 1þ and jhv01 v02 iþ j after they reach their corresponding maxima is also caused by the ﬂattening of the average velocity proﬁle, which reduces the role of the gradient mechanism of ﬂuctuation generation. Inertialess and very high-inertia particles are uniformly distributed in space, with concentra þ ) is observed at smaller values of t þ tion equal to unity. The maximum of F(t than the maxima of turbulent stresses. Figure 2.20 also shows the effect of the momentum restitution coefﬁcients ex, ey and the friction factor mx on velocity ﬂuctuations and particle concentration near the wall. First of all, we should note that for low-inertia particles (t þ 10), the inﬂuence of all coefﬁcients appearing in the boundary conditions is zero because the no-slip conditions are realized in such a case. As can be seen from Figure 2.20a, the value of v0 1þ for the collisions with no slip (ex ¼ 5/7) turn out to be smaller than for the collisions accompanied by slip with zero friction (ex ¼ 1) because of the momentum loss in the longitudinal direction. Decrease of momentum restitution factor in the transverse direction ey leads to a slight decrease of v0 1þ for moderately inertial particles and to increase of the same parameter for high-inertia particles, whereas the inﬂuence of the friction factor mx is insigniﬁcant. Since in the absence of particle collisions, the transverse component of velocity ﬂuctuations is rather weakly coupled with other variables, v0 2þ is highly dependent on ey but shows only a weak dependence on ex and mx (Figure 2.20b). Obviously, v0 2þ falls off as the loss of momentum by particles colliding with the wall increases, or, to use a different term, as the restitution factor ey gets smaller. The tangential stress shows a noticeable dependence on the ~

Figure 2.19 Distribution of characteristics of high-inertia particles over the channel cross-section: 1 – fluid; 2–9 – particles; 2, 3 – solutions of equations (2.176)–(2.181); 4, 5 – Wang and Squires (1996); 6, 7 – Rouson and Eaton (1994); 8, 9 – Fukagata et al. (1998); 2, 4, 6, 8 – t+ ¼ 117; 3, 5, 7, 9 – t+ ¼ 810.

2.7 Motion of Particles in a Vertical Channel

j105

j 2 Motion of Particles in Gradient Turbulent Flows

106

Figure 2.20 The influence of particle inertia on turbulent stresses and particle concentration at y+ ¼ 1: 1 – ex ¼ ey ¼ 1, mx ¼ 0; 2 – ex ¼ 1, ey ¼ 0.8, mx ¼ 0; 3 – ex ¼ 1, ey ¼ 0.8, mx ¼ 0.2; 4 – ex ¼ 5/7, ey ¼ 0.8, mx ¼ 0.2.

coefﬁcients ey and mx, decreasing as ey gets smaller and increasing with mx, whereas the dependence on ex is weak (Figure 2.20c). Concentration of particles near the wall is sensitive, ﬁrst of all, to the parameter ey that is responsible for particle interaction with the wall in the transverse direction. It follows from Figure 2.20d that if there is a momentum loss of particles colliding with the wall (ey < 1), the concentration of high-inertia particles near the wall is higher than in the case of elastic interaction with the wall. Increased accumulation of high-inertia particles in the case of inelastic collisions as compared to the case of elastic collisions has been observed by Fukagata et al. (1999) and explained in terms of the loss of particle momentum. To summarize, the outcome of the analysis performed thus far and a further comparison with direct numerical calculations warrant the conclusion that the outlined model is suitable for describing the statistics of the velocity ﬁeld and the phenomenon of particle accumulation in a vertical channel. In order to simplify the model to some extent, it is possible to neglect the contribution of the transport term in the approximation (2.15), or, in other words, to omit the derivatives Dp hu0i u0k i=Dt in the relations (2.182). Indeed, calculations prove that the role of these terms is extremely small and can be disregarded.

2.8 Deposition of Particles in a Vertical Channel

2.8 Deposition of Particles in a Vertical Channel

A large number of experimental and numerical studies are devoted to the deposition of aerosol particles and drops from a turbulent ﬂow on the bounding surfaces, which is only natural, considering the practical importance of this problem. Analysis and generalization of experimental studies of vertical and horizontal pipes and channels has been accomplished by McCoy and Hanratty (1977), Wood (1981), Papavergos and Hedley (1984). The ﬁrst theoretical models of particle deposition from turbulent ﬂow have been proposed by Friedlander and Johnstone (1957) and Davies (1966). Papavergos and Hedley (1984) have compiled a review of the semi-empirical models known at the time of publication that could be used to determine the rate of deposition. Cleaver and Yates (1975), Fichman et al. (1988), and Fan and Ahmadi (1993) have constructed Lagrangian models of particle deposition by considering the interaction of particles with two-dimensional eddies that served as models of organized (coherent) near-wall structures. Kallio and Reeks (1989) have tried to calculate the rate of deposition by using the stochastic Lagrangian approach based on the interaction of particles with random turbulent eddies having a Gaussian velocity distribution. Numerical study of particle deposition in a planar channel based on the method of trajectory modeling combined with the DNS and LES methods is the subject of works by McLaughlin (1989), Ounis et al. (1991, 1993), Brooke et al. (1992, 1994), Chen and McLaughlin (1995), Wang and Squires (1996b), and Wang et al. (1997), Zhang and Ahmadi (2000). Li and Ahmadi (1991, 1993) and Chen and Ahmadi (1997) have used the Gaussian random ﬁeld model (Kraichnan, 1970) to simulate generation of turbulent velocity ﬂuctuations of the continuous phase. Numerical simulation in a circular vertical pipe has been performed by Uijttewaal and Oliemans (1996) and Marchioli et al. (2003). In all of the cited works, the inﬂuence of gravity, direction (downward or upward) of the ﬂow, lifting force, and Brownian diffusion on particle deposition was taken into consideration. A numerical study of particle deposition by the DNS method has been performed by Zhang and Ahmadi (2000) for a horizontal channel and by van Haarlem et al. (1998) and Narayanan et al. (2003) for a planar channel with one open (free) wall. Lee et al. (1989), Binder and Hanratty (1991), and Mols and Oliemans have employed the diffusion model to determine the rate of deposition. Kroshilin et al. (1985) and Swailes and Reeks (1994) have analyzed particle deposition from a turbulent ﬂow by solving the kinetic equation for the velocity PDF. Derevich and Zaichik (1988), Johansen (1991), Zaichik et al. (1995), Guha (1997), Young and Leeming (1997), Slater et al. (2003) have constructed Eulerian models of turbulent deposition based on the local equilibrium relations between the intensities of normal components of velocity ﬂuctuations for the disperse and continuous phases of the type (2.28). However, the models based on local equilibrium relations for turbulent stresses are true only for low-inertia particles and unsuitable for particles whose relaxation time is comparable with the integral scale of turbulence for the continuous phase. Shin and Lee (2001), Shin et al. (2003) have suggested local equilibrium models that take the memory effect into account through the algebraic relations. Non-local transport models of turbulent deposition

j107

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108

based on differential equations for the second moments of particle velocity ﬂuctuations have been proposed by Zaichik et al. (1990), Derevich (1991), Gusev et al. (1992), Zaichik et al. (1995), Derevich (2000b). The present section attempts to analyze particle deposition in a vertical channel by employing the transport model based on the equations for the second moments that were described in Section 2.2. It is a common convention to describe the intensity of particle deposition from the turbulent ﬂow by the dependence of the deposition factor j þ Jw/Fmu on dimensional relaxation time tþ tp0 u2 =n, where Jw V2wFw is deposition ﬂux equal to the product of the normal component of velocity V2w and particle concentration at the wall Fw, where Fm is the average mass concentration of particles in the channel crosssection under consideration and u is the dynamic velocity. With the dependence of j þ on t þ acting as the primary mechanism describing the process of deposition, the entire range of particle inertia may be subdivided into three intervals: low-inertia, moderately inertial, and high-inertia particles. The deposition process for low-inertia particles (t þ < 1) is governed chieﬂy by Brownian and turbulent diffusion. In addition, some driving forces that cause transport of submicron particles (e.g., thermophoresis force in non-isothermic ﬂow) can play a signiﬁcant role. In a situation when the diffusional mechanism plays the leading role, the deposition factor j þ declines monotonously with increase of t þ as a result of the decrease of Brownian diffusion coefﬁcient with increase of particle size. The principal mechanism of deposition of moderately inertial particles (1 t þ 100) is turbulent migration (turbophoresis) of particles from the core of the ﬂow, characterized by high intensity of turbulent velocity ﬂuctuations, into the viscous sublayer adjacent to the wall. This particle inertia interval is characterized by strong dependence of j þ on t þ . McLaughlin (1989) and Kallio and Reeks (1989) were the ﬁrst to establish numerically the tendency of depositing particles to accumulate in the viscous sublayer under the action of turbophoresis. This result has been reproduced in numerous later works. In the given range of t þ , the lifting force arising due to the velocity shift can exert a noticeable inﬂuence on the deposition rate, provided the density ratio of the disperse and continuous phases rp/rf is not excessively high. Thanks to the lifting force, one can observe a difference between particle deposition rates in the downward and upward ﬂows, speciﬁcally, deposition rate in the downward ﬂow turns out to be greater than in the upward ﬂow (Zhang and Ahmadi, 2000). However, inclusion of the Saffman force in its classical form (Saffman, 1965, 1968), when applied to the conditions commonly realized in turbulent ﬂows, has proved to be not quite correct (Chen and McLaughlin, 1995; Wang et al., 1997). A more accurate treatment of the lifting force introduced by McLaughlin (1991, 1993) shows that its effect on particle deposition is not as signiﬁcant as the consequences of using the classical Saffman formula (Uijttewaal and Oliemans, 1996; Wang et al., 1997). Moreover, it should be kept in mind that due to the rapid increase of the deposition factor j+ with inertia parameter t þ in this region, the dependence j þ (t þ ) is highly sensible to aerosol polydispersity. As shown by Chen and McLaughlin (1995), even a small dispersion of particle sizes leads to a very noticeable increase of j þ (t þ ) averaged over the particle size spectrum. Therefore, one has to be careful

2.8 Deposition of Particles in a Vertical Channel

when working with the dependences j þ (t þ ) obtained by generalizing experimental data within the framework of semi-empirical models. High-inertia particles (t þ > 100) are weakly involved in turbulent motion of the carrier ﬂuid, which causes the deposition factor j þ in a vertical channel to decrease with growth of t þ . But the rate of deposition of high-inertia particles is deﬁned not only by the characteristics of near-wall turbulence, but also, to a great extent, by the external parameters of the ﬂow, in particular, the Reynolds number calculated for the given hydraulic diameter of the channel. In addition, the force of gravity, which manifests itself chieﬂy through the crossing trajectory effect, may exert a considerable inﬂuence on the deposition of high-inertia particles. Consider the deposition of particles in a vertical planar channel with the inertia parameter conﬁned to the range 1 t þ 105. Deposition of low-inertia particles is not considered because Brownian motion is out of the question. Besides, the lifting force is not taken into account. Consequently, the results presented in this section are true only for very large density ratios of the disperse and continuous phases rp/rf, when the role of the lifting force is negligible. Notice that within the framework of the conducted research, inclusion of Brownian motion as well as of the lifting force would not cause any additional difﬁculties. Exclusion of these factors is explained by the perceived advantage of keeping our focus exclusively on the inﬂuence exerted by particle interaction with turbulent eddies on the deposition rate. As in Section 2.7, the channel ﬂow is assumed to be hydrodynamically developed, so that all hydrodynamic parameters of the carrier ﬂuid are considered as being dependent only on the coordinate x2 ¼ y normal to the wall, and the normal velocity component of the ﬂuid U2 is zero. In this case there exists a self-similar solution for the disperse phase such that the average velocity components Vi, turbulent stresses hv0i v0j i, and third and higher moments of velocity ﬂuctuations are independent of the longitudinal coordinate x1 ¼ x. Consequently, all moments of particle velocity, starting from the ﬁrst, are assumed to depend only on y. In order for a self-similar solution to exist, it is necessary that the average concentration of the disperse phase has the following form: x F ¼ exp k j(y); ð2:191Þ R where k is a constant and j(y) is a function of y. The constant k is connected with the deposition factor j þ by the relation k ¼ j þ u /V1m, where V1m is the average mass velocity of particles in the longitudinal direction. Due to Eq. (2.191), it follows from the continuity equation that djV 2 kjV 1 ¼ 0: dy R

ð2:192Þ

Taking the longitudinal and normal projections of the balance-of-momentum equations (2.25), we arrive at

d ln j k hv02 dV 1 V 1 U 1 dhv01 v02 i 0 0 1 i þ m11 ¼ 0; V2 þgþ þ þ hv1 v2 i þ m12 R dy tp dy dy ð2:193Þ

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j 2 Motion of Particles in Gradient Turbulent Flows

110

V2

d ln j k hv01 v02 i þ m21 dV 2 V 2 dhv02 2i ¼ 0: þ i þ m þ þ hv02 22 2 R dy tp dy dy

ð2:194Þ

The ﬁrst terms in Eqs. (2.193) and (2.194) describe convective transport of momentum in the normal direction due to particle deposition. The last terms on the left-hand side of these equations are responsible for particle diffusion in the longitudinal direction. But the contribution of the latter is insigniﬁcant, since k is a small parameter. From Eq. (2.27) together with Eqs. (2.31) and (2.191) there follows a system of equations for the non-zero components of disperse phases turbulent stresses: V2

dhv02

dhv01 v02 i dhv02 1 d 1i 1i ¼ þ 2 hv01 v02 i þ m12 jtp hv02 2 i þ m22 dy 3j dy dy dy 0 0

dV 1 ktp hv1 v2 i þ m12 dhv02 2hv02 i 1i þ 2l11 1 ; 2 hv01 v02 i þ m12 R dy dy tp ð2:195Þ

V2

dhv02 dhv02 1 d 2i 2i i þ m ¼ jtp hv02 22 2 dy j dy dy

dhv01 v02 i 0 0

dhv02 ktp 02 2i 2 hv2 i þ m22 þ hv1 v2 i þ m12 3R dy dy

dV 2 2hv02 i 2 hv02 þ 2l22 2 ; 2 i þ m22 dy tp

ð2:196Þ

02

dhv02 ktp hv01 v02 iþm12 dhv02 dhv02 1 d 2hv02 i 3i 3i 3i ¼ jtp hv2 iþm22 þl33 3 ; V2 3R dy 3j dy dy dy tp ð2:197Þ

V2

dhv01 v02 i 0 0

dhv02 02 dhv01 v02 i 1 d 2i ¼ jtp 2 hv2 iþm22 þ hv1 v2 iþm12 dy 3j dy dy dy

dhv02 i

dhv01 v02 i ktp 02 hv2 iþm22 1 þ2 hv01 v02 iþm12 3R dy dy

dV 1 2hv0 v0 i þl12 þl21 1 2 : hv02 2 iþm22 dy tp

ð2:198Þ

A comparison of Eqs. (2.195)–(2.198) with Eqs. (2.178)–(2.181) allows us to conclude that deposition contributes to both convective and diffusive transport of second moments of particle velocity ﬂuctuations. But since k is a small parameter, we can ignore the contribution of diffusion. The quantities mij and lij in

2.8 Deposition of Particles in a Vertical Channel

Eqs. (2.193)–(2.198) are deﬁned, as previously, by Eq. (2.182). To the relations (2.183) for Dp hu0i u0j i=Dt we should add the convective terms, obtaining as a result

dhu01 u02 i 0 0 Dp hu02 dhu02 dhu02 1i 1i 1 d 1i i þ m ) v i þm tp (hv02 þ2 hv ; ¼ V2 22 12 2 1 2 dy 3 dy dy dy Dt

dhu01 u02 i Dp hu01 u02 i dhu01 u02 i 1 d ¼V2 tp 2 hv02 2 iþ m22 Dt dy 3 dy dy

dhu02 2i þ hv01 v02 i þ m12 dy

;

dhu02 Dp hu02 dhu02 d 2i 2i 2i i þm tp hv02 ; ¼ V2 22 2 dy dy dy Dt Dp hu02 dhu02 dhu02 3i 3i 1 d 3i tp (hv02 : ¼ V2 2 i þ m22 ) dy 3 dy dy Dt

ð2:199Þ

The rate of deposition of particles at the wall is found from the boundary condition (2.103): 1=2 1c 2hv02 2i : ð2:200Þ V 2w ¼ 1þc p y¼0 The boundary conditions for turbulent stress components at the wall are speciﬁed by Eqs. (2.107)–(2.109) and (2.118) for elastic interactions with no friction (ex ¼ ey ez ¼ 1, mx ¼ 0): 1=2 dhv02 dhv02 dhv02 1c 2hv02 1i 3i 2i 2i at y ¼ 0: ¼ ¼ hv01 v02 i ¼ 0; tp ¼ dy dy dy 1þc p ð2:201Þ At the channel axis, the symmetry conditions must be satisﬁed: dF dV 1 dhv02 dhv02 dhv02 1i 2i 3i ¼ V2 ¼ ¼ ¼ ¼ ¼ hv01 v02 i ¼ 0 at y ¼ R: dy dy dy dy dy

ð2:202Þ

All other relations deﬁning the response coefﬁcients, the times of particle interaction with turbulent eddies and so on are speciﬁed in the same manner as in the case when deposition is absent (Section 2.7). Deposition of particles is considered under the condition that they are completely adsorbed on the wall, which takes place at c ¼ 0, when the rate of deposition reaches its maximum value. The comparison of results without deposition (c ¼ 1) and with deposition (c ¼ 0) was performed at Re þ ¼ 180. The most conspicuous inﬂuence of deposition on the distribution of disperse phases characteristics over the channel cross-section is exhibited with respect to particle concentration. As is seen from Figure 2.21, deposition brings about a

j111

j 2 Motion of Particles in Gradient Turbulent Flows

112

Figure 2.21 Profiles of particle concentration in a channel: 1, 2, 3 – c ¼ 0; 4, 5, 6 – c ¼ 1; 1, 4 – t þ ¼ 1; 2, 5 – t þ ¼ 10; 3, 6 – t þ ¼ 100.

reduced relative concentration in the near-wall region, in other words, it weakens the accumulation effect near the wall. In the absence of deposition, a monotonous increase of concentration of moderately inertial particles is observed as we approach the wall, whereas in the case of non-zero deposition the maximum in the concentration proﬁle of these particles shifts to the viscous sublayer region. The most clear manifestation of the accumulation phenomenon occurs at t þ 10, which is in good agreement with the value reported by Chen and McLaughlin (1995) for the inertia parameter at which the maximum concentration is achieved. Continuous lines on Figure 2.22 show the dependence of the deposition factor on the inertia parameter obtained by solving the problem (2.191)–(2.202) – ﬁrst for the case of zero gravity (g ¼ 0) and then for a ﬁnite gravity force affecting the downward/ upward ﬂows. Also shown in the ﬁgure are the empirical correlations 8 4 2 > < 3:25 10 tþ at tþ < 22:9 at 22:9 < tþ< 14827; ð2:203Þ jþ ¼ 0:17 > : 1=2 20:7tþ at tþ> 14827 obtained by McCoy and Hanratty (1977), who carried out a generalization of just about the entire body of experimental data known by then. In addition, Figure 2.22 presents the experimental data by Liu and Agarwal (1974) for a downward ﬂow in a pipe. Also shown are the results of numerical simulation by McLaughlin (1989), Wang et al. (1997) for a planar channel and by Uijttewaal and Oliemans (1996) for a circular pipe; these simulations neglect gravitational and lifting forces. Calculations by McLaughlin (1989) and Wang et al. (1997) were performed, respectively, at Re þ ¼ 125 and 180, and those by Uijttewaal and Oliemans (1996) – at Re þ ¼ 180. We see two distinct regions on the graph of j þ (t þ ), corresponding to the increase and decrease of j þ with t þ ; the former is explained by the increased role of turbophoresis, and the latter– by the weaker involvement of particles in turbulent motion. At moderate values of t þ , the values of j þ obtained by numerical simulation are smaller than those obtained by experiment; the primary reason for this is the models failure to take into consideration the polydispersity of the particle system, which is accurately detected by the experiment. As we already mentioned, even small dispersion in particle sizes may

2.8 Deposition of Particles in a Vertical Channel

Figure 2.22 Dependence of the deposition factor on particle inertia: 1 – g = 0; 2 – downward flow; 3 – upward flow; 4 – asymptotic expression (2.208); 5 – empirical correlation (2.203); 6 – DNS (Uijttewaal and Oliemans, 1996); 7 – experiment. (Liu and Agarwal, 1974); 8 – DNS (McLaughlin, 1989); 9 – LES (Wang et al., 1997).

produce a noticeably higher value of j þ (t þ ) averaged over the size spectrum. Overall, the dependence j þ (t þ ) predicted by the model(2.191)–(2.202) is consistent with the experimental data by Liu and Agarwal (1974) and with direct calculations by McLaughlin (1989) and Wang et al. (1997). In addition, it is consistent with the direct calculations for a circular pipe by Uijttewaal and Oliemans (1996), which also predict a drop in j þ with increase of t þ for high-inertia particles. As we can see from Figure 2.22, in the absence of lift forces the inﬂuence of gravity and the distinction between downward and upward ﬂows becomes noticeable at t þ > 100. The force of gravity causes a reduction of j þ for very high-inertia particles due to the crossing trajectory effect, which is responsible for shorter duration of interaction of particles with turbulent eddies. It is worth noting that in a downward ﬂow there exists a region 102 t þ 103 in which the force of gravity enhances the deposition rate. At t þ ! 1, we can derive an asymptotic solution of the problem (2.191)–(2.202) that is similar to the solution obtained by Alipchenkov and Zaichik (1998a), Zaichik and Alipchenkov (2001a) for a circular pipe. As t þ grows, diffusive transport of velocity ﬂuctuations in the transverse direction is intensiﬁed, and the distribution proﬁles of concentrations, average axial velocity, and all components of particle velocity ﬂuctuations over the channel cross-section become more uniform. Hence in the limiting case of t þ ! 1, F ¼ Fw ¼ Fm ; V 1 ¼ V 1m :

ð2:204Þ

Due to Eq. (2.204), it follows from Eq. (2.192) that the average transverse velocity varies linearly over the channel cross-section: y V 2 ¼ 1 ð2:205Þ V 2w : R Integration of Eq. (2.196) with the consideration of the boundary conditions (2.200)–(2.202) and of the relations (2.204)–(2.205) yields an algebraic equation

j113

j 2 Motion of Particles in Gradient Turbulent Flows

114

for the intensity of the transverse component of velocity ﬂuctuations for high-inertia particles: hv02 2i

ðR 1c 2 1=2 3tp 02 3=2 1 hv i ¼ þ T nLp hu02 2 idy: 2R 2 1þc p tp R

ð2:206Þ

0

In the absence of deposition, that is, at c ¼ 1, Eq. (2.206) turns into Eq. (2.190). The 1=2 . Its solution can be expression (2.206) is a cubic equation with respect to hv02 2i obtained by the Cardan method. But in order to circumvent the difﬁculties that arise when we use this method, let us present the solution of Eq. (2.206) as an approximation that approaches Eq. (2.206) asymptotically at small and large values of inertia parameter t tpu /R: Ð1 2 hv0 2þ i

0

¼ t þ

T nLp hu0 22þ idy

3 1c 2=3 2 1=3 Ð1 2 1þc

p

0

4=3 1=3 t ; T nLp hu0 22þ idy

ð2:207Þ

where ¼ ¼ ¼ y=R; T nLp ¼ T nLp u =R. Deviation of Eq. (2.207) from the exact solution of Eq. (2.206) for t ranging from 0 to 1 does not exceed 3%. A substitution of Eq. (2.207) into Eq. (2.200) leads to the following asymptotic expression for the deposition factor for high-inertia particles: 1=2 1 1c 2 1=2 Ð n 02 T hu id y Lp 2þ 1þc p 0 jþ ¼ : ð2:208Þ 1=3 1=2 3 1c 2=3 2 1=3 Ð1 n 4=3 2 T Lp hu0 2þ idy t t þ 2 1þc p hv0 22þ i

2 02 hv02 2 i=u ; hu 2þ i

2 hu02 y 2 i=u ;

0

When calculating j þ in accordance with Eq. (2.208), we use the following asymptotic relations for the absolute value of the drift velocity W |U V|, drift parameter g, Reynolds number Rep, and time scale T nLp : W ¼ tp g; g ¼

tp g (2k=3)1=2

; Rep ¼

d p tp g TE ; T nLp ¼ ; n 2mg

which are true for very inertial particles in view of the action of the gravity force. As it is seen from Figure 2.22, the asymptotic dependence (2.208) is consistent with the solution of the problem (2.191)–(2.202) at large values of t þ . It is also evident that if we disregard the lifting force, the values of j þ for the downward and upward ﬂows will coincide as t þ gets large.

j115

3 Heat Exchange of Particles in Gradient Turbulent Flows The present chapter examines heat exchange of particles dispersed in a gradient turbulent ﬂow. For all the importance of mass exchange, in practice, the greatest theoretical difﬁculties in modeling turbulent ﬂows with heat exchange are associated with providing an adequate description of the ﬂow hydrodynamics, while heat transport models are taken by analogy with transport models for a continuous medium. In this chapter, we develop a statistical method to describe the ﬂow and heat exchange of the disperse phase. The method is based on the kinetic equation for the joint PDF of distributions of particle velocity and temperature in the ﬂuid. Velocity and temperature ﬁelds of the ﬂuid are modeled by Gaussian random processes. The kinetic equation is used to construct differential and algebraic models of heat exchange in the disperse phase.

3.1 The Kinetic Equation for the Joint PDF of Particle Velocity and Temperature

In order to derive a kinetic equation for the joint PDF of particle velocity and temperature, let us introduce a dynamic probability density in the phase space of coordinates, velocities, and temperature (x, v, q): p ¼ d(xRp (t))d(vvp (t)d(qqp (t)):

ð3:1Þ

The variation of particle temperature in a turbulent ﬁeld is described by Eq. (1.49) in which the temperature of the ﬂuid W, analogously to the velocity u, is considered as a random process. Differentiating Eq. (3.1) with respect to time and taking into account Eqs. (1.45), (1.46), and (1.49), we obtain the following equation for the dynamic probability density of a single particle: uk vp k Wqp qp qp q q þ þ Fk p þ þ Q p ¼ 0: ð3:2Þ þ vk tp tt qt qx k qvk qq Let us now average Eq. (3.2) over the ensemble of random realizations of velocity u and temperature W of a turbulent ﬂuid. As a result, we get an equation for the statistical PDF of velocity and temperature distributions for a particle, P ¼ hpi. The

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

116

single-point, single-particle PDF P(x, v, q, t) is deﬁned as the probability density for the particle to be at point x and to have velocity v and temperature q at the instant of time t. Let us represent the actual velocity and temperature of the ﬂuid in Eq. (3.2) as a sum of the average and ﬂuctuational components u ¼ U þ u0 and W ¼ T þ W0. Then in view of the obvious relations hvpipi ¼ viP and hqppi ¼ qP we get qP qP q U k vk q Tq þ þ Fk P þ þQ P þ vk tp qt qx k qvk qq tt ð3:3Þ 1 qhu0k pi 1 qhW0 pi ¼ : tp qvk tt qq To close the equation (3.3), one should ﬁnd the correlation between temperature ﬂuctuations of the continuous phase and the probability density hW0 pi in addition to the correlation hu0k pi. To obtain closed relations for these correlations, we shall model the velocity and temperature ﬁelds of the continuous phase by Gaussian random ﬁelds with known correlation moments. Involving the Furutsu–Novikov–Donsker formulas, (Klyatskin, 1980, 2001; Frisch, 1995), we obtain * + ðð dp(x; t) 0 0 0 hui pi ¼ hui (x; t)uk (x1 ; t1 )i dx1 dt1 duk (x1 ; t1 )dx1 dt1 ðð þ

0

hW pi ¼

ðð

* hu0i (x1 ; t1 )W0 (x; t)i ðð

þ where

*

+ dp(x; t) dx1 dt1 ; dW(x1 ; t1 )dx1 dt1

hu0i (x; t)W0 (x1 ; t1 )i

+ dp(x; t) dx1 dt1 duk (x1 ; t1 )dx1 dt1

+ dp(x; t) dx1 dt1 ; hW(x; t)W (x1 ; t1 )i dW(x1 ; t1 )dx1 dt1

*

*

0

dp(x; t) duk (x1 ; t1 )dx1 dt1

+

q ¼ qx j q qvj

*

dRp j (t) p(x; t) duk (x1 ; t1 )dx1 dt1

*

dvp j (t) p(x; t) duk (x1 ; t1 )dx1 dt1

dp(x; t) dW(x1 ; t1 )dx1 dt1

dqp (t) q p(x; t) ¼ : dW(x1 ; t1 )dx1 dt1 qq

ð3:5Þ

+

+

* + dqp (t) q ; p(x; t) duk (x1 ; t1 )dx1 dt1 qq

ð3:4Þ

ð3:6Þ ð3:7Þ

To determine the functional derivatives of particle temperature in Eq. (3.6) and Eq. (3.7), we shall present the energy equation (1.49) as an integral over the particle trajectory:

3.1 The Kinetic Equation for the Joint PDF of Particle Velocity and Temperature

ðt qp (t) ¼

tt1 W(Rp (t1 ); t1 ) exp þ Q dt1 : tt tt

ð3:8Þ

1

Then from Eq. (3.8) there follow the expressions for functional derivatives of particle temperature along the particle trajectory: ðt dqp (t) dRp n (t2 ) 1 tt2 qW(Rp (t2 ); t2 ) ¼ exp dt2 ; tt dui (x1 ; t1 )dx1 dt1 tt qx n dui (x1 ; t1 )dx1 dt1

ð3:9Þ

t1

dqp (t) 1 tt1 ¼ d(Rp (t1 )x1 )exp H(tt1 ): dW(x1 ; t1 )dx1 dt1 tt tt

ð3:10Þ

We use the iteration expansion (2.9) to ﬁnd the integral (3.9). As a result, the functional derivative of particle temperature with respect to ﬂuid velocity yields dqp (t) ¼ d(Rp (t1 )x1 )H(tt1 ) dui (x1 ;t1 )dx1 dt1 qW ¡ tt1 1 tt1 1þ exp exp tt tp qx i 1¡ 1¡ þ

qW qun ¡2 tt tt1 exp tt1 2tp tt þ tt qx n qx i (1¡)2

2tp 3tt 1 tt1 tt exp tt1 þ ; ¡¼ : þ 1¡ tp tp 1¡ ð3:11Þ We now substitute Eqs. (2.9), (2.10), (3.10), and (3.11) into Eqs. (3.4)–(3.7) and then average over the ensemble of realizations of turbulent ﬂuctuations with the consideration of quasi-uniformity of the proﬁles of average velocity and temperature of the continuous phase. As a result, we obtain, by analogy with Eq. (2.11), the following expressions for the correlation between ﬂuid velocity ﬂuctuations and the PDF of velocity and temperature of the particle, and for the correlation between ﬂuid temperature and the above-mentioned PDF: qP qP qP hu0i pi ¼ tp mij (x; t) þ lij (x; t) þ hi (x; t) ; qx j qvj qq ðt 1 hu0i (x;t)u0k (Rp (t1 );t1 )iH(tt1 ) hi (x;t) ¼ tp 1 qT ¡ tt1 1 tt1 exp exp 1þ tt tp qx k 1¡ 1¡

ð3:12Þ

j117

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

118

2tp 3tt qT qU n ¡2 tt tt1 1 þ exp tt1 þ tt1 2tp tt þ þ tt 1¡ qx n qx k 1¡ (1¡)2

ðt tt1 1 tt1 hu0i (x;t)W0 (Rp (t1 );t1 )iH(tt1 )exp exp dt1 þ dt1 ; tp tt tp tt 1

ð3:13Þ qP qP qP hW0 pi ¼ tt mi (x;t) þ li (x;t) þ h(x;t) ; qx i qvi qq

1 mi (x;t) ¼ tp tt þ

1 li (x;t) ¼ tt

ðt 1

tt1 hu0k (Rp (t1 );t1 )W0 (x;t)iH(tt1 ) d ik 1exp tp

qU i tt1 tt1 2tp þ(tt1 þ2tp )exp dt1 ; qx k tp

ðt 1

þ

1 tt

ð3:15Þ

dik tt1 hu0k (Rp (t1 );t1 )W0 (x;t)iH(tt1 ) exp tp tp

qU i tt1 tt1 qU i qU n 1 1þ exp þ tt1 3tp qxk tp tp qx n qx k

(tt1 )2 tt1 þ 3tp þ2(tt1 )þ exp dt1 ; 2tp tp

h(x;t) ¼

ð3:14Þ

ðt

hu0k (Rp (t1 );t1 )W0 (x;t)iH(tt1 )

1

qT ¡ tt1 1 tt1 1þ exp exp tt tp qxk 1¡ 1¡ þ

qT qU n ¡2 tt tt1 exp tt1 2tp tt þ tt qx n qx k (1¡)2

þ

2tp 3tt 1 tt1 exp dt1 tt1 þ 1¡ tp 1¡

ð3:16Þ

3.1 The Kinetic Equation for the Joint PDF of Particle Velocity and Temperature

1 þ 2 tt

ðt 1

tt1 hW0 (x;t)W0 (Rp (t1 );t1 )iH(tt1 )exp dt1 : tt

ð3:17Þ

The quantities mij and lij in Eq. (3.12) are calculated using Eq. (2.16) and Eq. (2.17), whereas in order to calculate hi, mi, li, and h using Eq. (3.13) and Eqs. (3.15)–(3.17), we should ﬁrst determine the Lagrangian correlation moments of joint ﬂuctuations of velocity and temperature of ﬂuid particles moving along inertial particle trajectories. But before to turn to this problem, let us determine the correlation moments along ﬂuid particle trajectories. By analogy with Eq. (2.14), let us assume BtL i (t) ¼ hu0i (R(tt); tt)W0 (x)jR(t) ¼ xi t Dhu0k W0 i YtL ki (t); ¼ hu0k W0 i 2 Dt Dhu0i W0 i qhu0i W0 i qhu0i W0 i qhu0i u0k W0 i þ ; ¼ þ Uk qx k Dt qt qx k

ð3:18Þ

where YtL ij (t) is the Lagrangian autocorrelation function of velocity and temperature ﬂuctuations of the continuum. Due to Eq. (3.18), the Lagrangian correlation moments of joint ﬂuctuations of ﬂuid particle velocity and temperature are represented as BtLp i (t) ¼ hu0i (Rp (tt); tt)W0 (x)jRp (t) ¼ xi t Dp hu0k W0 i YtLp ki (t); ¼ hu0k W0 i 2 Dt Dp hu0i W0 i qhu0i W0 i qhu0i W0 i qhu0i v0k W0 i ¼ þ ; þ Vk Dt qx k qt qx k

ð3:19Þ

where YtLp ij (t) is the autocorrelation function of velocity and temperature of the continuous phase deﬁned along the particle trajectory. Taking into account the transport term, we can write the Lagrangian correlation moment of temperature ﬂuctuations of the continuous phase (1.42) as BLt (t) ¼ hW0 (x)W0 (R(tt); tt)jR(t) ¼ xi ¼ DhW0 i qhW0 i qhW0 i qhu0k W0 i þ : ¼ þ Uk qx k Dt qt qx k 2

2

2

hW0 i 2

2 t DhW0 i YLt (t); 2 Dt

2

ð3:20Þ

By analogy with Eq. (3.20), the Lagrangian correlation moment of temperature ﬂuctuations of a ﬂuid particle along an inertial particle trajectory (1.77) is written as

j119

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

120

2 t Dp hW0 i 02 BLtp (t) ¼ hW (x)W (Rp (tt); tt)jRp (t) ¼ xi ¼ hW i YLtp (t); 2 Dt 0

0

Dp hW0 i qhW0 i qhW0 i qhv0k W0 i ¼ þ : þ Vk Dt qx k qt qx k 2

2

2

2

ð3:21Þ

Substituting Eqs. (2.15), (3.19), (3.21) into Eqs. (3.13), (3.15)–(3.17), taking into account the gradients of average velocity and temperature and retaining the terms up to the second order, we obtain after integration 0 0 tp Dp hui uj i qT qU n qT qT þ tp r u jk qu1 jk hi ¼ hu0i u0j i qu jk qx k qx n 2 Dt qx k qx k þ

hu0k W0 if tt ki ¡ Dp hu0k W0 i t f t1 ki ; Dt tp 2

ð3:22Þ

hu0k W0 i f tu ki qU i qU n qU i t t li ¼ þ lu kn þ tp mu kl ¡ tp qx n qx l qx n 1 Dp hu0k W0 i t qU i ; f u1 ki þ tp ltu1 kn qx n 2¡ Dt

ð3:23Þ

tp Dp hu0k W0 i t hu0k W0 i t qU i t g u1 ki ; g u ki þ tp hu kn mi ¼ ¡ qx n 2¡ Dt h¼

ð3:24Þ

tp Dp hu0i W0 i t hu0i W0 i t qT qU n qT qT þ tp r tu ik qu ik qu1 ik qx k qx n 2¡ Dt ¡ qx k qx k hW0 if t 1 Dp hW0 i f t1 : tt 2 Dt 2

2

þ

ð3:25Þ

The involvement coefﬁcients characterizing the response of particles to joint ﬂuctuations of velocity and temperature of the continuous phase can be written in the matrix form by analogy with Eq. (2.18): f tu ¼ Mtu0 ; gtu ¼ Ntu0 f tu ;

f tu1 ¼ Mtu1 ;

mtu ¼ Ntu1 þ 2Mtu1 þ Mtu2 3gtu ; f tu2 ¼ 2Mtu2 ;

ltu1 ¼ gtu1 f tu2 ;

ltu ¼ gtu f tu1 ;

htu ¼ Ntu1 þ Mtu1 2gtu ;

gtu1 ¼ Ntu1 f tu1 ; f tt ¼ Mtt0 ;

f tt1 ¼ Mtt1; ¡2 Mtt0 Mtu0 ; 1¡ 1¡

qu ¼ Nu0 þ

¡2 Mt0 Mu0 ; 1¡ 1¡

qtu ¼ Ntu0 þ

qu1 ¼ Nu1 þ

¡3 Mt1 Mu1 ; 1¡ 1¡

qtu1 ¼ Ntu1 þ

¡3 Mtt1 Mtu1 ; 1¡ 1¡

3.1 The Kinetic Equation for the Joint PDF of Particle Velocity and Temperature

ru ¼ Nu1 (2 þ ¡)Nu0 þ

¡4 Mt0 Mu1 (23¡)Mu0 þ þ ; (1¡)2 1¡ (1¡)2

rtu ¼ Ntu1 (2 þ ¡)Ntu0 þ

¡4 Mtt0 Mt (23¡)Mtu0 þ u1 þ ; 2 1¡ (1¡) (1¡)2

Mtun

j121

1 ð 1 It t n ¼ Y (t)t exp dt; Lp n!tpn þ 1 tp 0

1 It n Y (t)t exp dt; Lp tt n!ttn þ 1 1 ð

Mtn ¼

0

Mttn

1 It t n ¼ Y (t)t exp dt; Lp tt n!ttn þ 1 1 ð

Ntun

0

1 ð 1 ¼ YtLp (t)tn dt: n!tnp þ 1 0

ð3:26Þ The involvement coefﬁcients associated with the temperature response of a particle are given below: ft ¼

1 tt

1 ð

0

t YLtp (t)exp dt; tt

f t1 ¼

1 t2t

1 ð

0

t YLtp (t)t exp dt: tt

ð3:27Þ

If, by analogy with Eq. (2.21), the autocorrelation functions in Eq. (3.19) and Eq. (3.21) are given by the exponential approximations 1

; YtLp (t) ¼ exp t TtLp

YLtp (t) ¼ exp tT 1 Ltp ;

then 1 ðn þ 1Þ

ðn þ 1Þ ; Mtn ¼ I þ tt T1 ; Mtun ¼ I þ tp TtLp Lp

1 ðn þ 1Þ n þ 1 Mttn ¼ I þ tt TtLp ; Ntun ¼ TtLp =tp ; and the involvement coefﬁcients take the form 1 1

1 1 t 1 2 ; gtu ¼ TtLp =tp Iþ tp TtLp ; f u1 ¼ I þtp TtLp ; f tu ¼ Iþ tp TtLp

1 2 2 1 2 ltu ¼ TtLp =tp Iþ tp TtLp ; htu ¼ TtLp =tp Iþ tp TtLp ;

2 1 3 2 1 2 mtu ¼ TtLp =tp Iþ tp TtLp ; gtu1 ¼ TtLp =tp Iþ tp TtLp ;

1 3 t 1 2 1 3 f tu2 ¼ 2 I þtp TtLp TtLp =tp Iþ tp TtLp ;lu1 ¼ Iþ 3tp TtLp ; 1 1 t 1 2 f tt ¼ I þtt TtLp ;f t1 ¼ Iþ tt TtLp ;

1

1 qu ¼ TLp =tp I þtp T1 I þtt T1 ; Lp Lp

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

122

1 1 1 1 I þtt TtLp qtu ¼ TtLp =tp I þtp TtLp ;

2

2

2

2 qu1 ¼ I þ2 1 þ¡ tp T1 TLp =tp I þtp T1 I þtt T1 ; Lp þ 3tp tt TLp Lp Lp

1 2 qtu1 ¼ I þ2 1 þ¡ tp TtLp þ 3tp tt TtLp t

2 1 2 1 2 TLp =tp I þtp TtLp I þtt TtLp ;

2 1 2 I þtp T1 ¡4 I þtt T1 TLp TLp Lp Lp (2 þ¡) þ ru ¼ þ tp tp 1¡ (1¡)2 þ

1 (23¡) Iþ tp T1 Lp (1¡)2

; rtu ¼

1 1 ¡4 I þtt TtLp

TtLp

2

tp

1 2 I þtp TtLp

þ 1¡ (1¡)2 1 2 tt tt ; f t1 ¼ 1 þ : f t ¼ 1þ T Ltp T Ltp þ

t TLp (2þ ¡) tp þ

1 1 (23¡) I þtp TtLp (1¡)2

;

ð3:28Þ

At tt ¼ tp (¡ ¼ 1), the following relations take place: qu ¼ lu ; qu1 ¼ lu1 ; ru ¼ mu : Substituting Eq. (3.12) and Eq. (3.14) into Eq. (3.3), we arrive at the closed kinetic equation for the single-point PDF of particle velocity and temperature distributions in a turbulent ﬂow: qP qP q þ þ vk qt qx k qvk ¼ lij

U k vk q Tq þ Fk P þ þQ P tp qq tt

q2 P q2 P q2 P q2 P q2 P þ mij þ (hi þ li ) þ mi þh 2 : qvi qvj qxj qvi qvi qq qx i qq qq

ð3:29Þ

The left side of Eq. (3.29) describes time evolution and convective transport of the PDF in the phase space of coordinates, velocities, and temperature. Terms on the right-hand side of Eq. (3.29) determine the diffusive transport via dynamic and thermal interactions of particles with turbulent eddies. Integration of Eq. (3.29) over the temperature subspace leads to the kinetic equation for the PDF of particle velocity distribution (2.23). If we ignore the anisotropy of involvement coefﬁcients, the inﬂuence of transport terms in Eqs. (2.15), (3.19), (3.21), and the contribution of those terms that contain gradients of average velocities and temperature, then Eq. (3.29) reduces to the kinetic equation for the PDF of velocity and temperature of a particle in a homogeneous shearless turbulent ﬂow (Derevich and Zaichik, 1990). Equations for the joint PDF of particle velocity and temperature in a gradient turbulent ﬂow have been obtained by Zaichik and Alipchenkov (1998), Zaichik (1999), Pandya and Mashayek (2002b, 2003).

3.2 The Equations for Single-Point Moments of Particle Temperature

3.2 The Equations for Single-Point Moments of Particle Temperature

The task of solving the kinetic equation for the joint PDF of particle velocity and temperature presents an even greater challenge than solving the equation for the PDF of velocity distribution alone. As a consequence, models based on solving the system of equations for the ﬁrst several statistical moments of velocity and temperature of the disperse phase acquire even greater relevance. Integration of Eq. (3.29) over the velocity and temperature subspaces leads to a chain of equations for moments. Since equations for the moments of velocity have been introduced in Section 2.2, the present section will consider only equations for the moments of temperature of the disperse phase. The equation for the average temperature of the disperse phase has the form t

Dp k qlnF qhv0 q0 i TQ qQ qQ ¼ k þ þ Q ; þ Vk qx k tt qx k qt qx k tt where hv0i q0 i ¼

ð3:30Þ

ð 1 (vi V i )(qQ)Pdvdq F

is the turbulent heat ﬂux in the disperse phase resulting from the involvement of particles in ﬂuctuational motion of the continuum. The last term in Eq. (3.30) describes turbulent diffusive transport of heat. Components of the thermal diffusion vector are deﬁned by

Dtp i ¼ tt (hv0i q0 i þ mi ) ¼ tp ¡hv0i q0 i þ hu0k W0 ig tu ki þ t2p hu0k W0 ihtu kn

qU i tp Dp hu0k W0 i t g u1 ki : qx n 2 Dt 2

ð3:31Þ

The equation for second moments of joint velocity and temperature ﬂuctuations of a particle (turbulent heat ﬂux in the disperse phase) is written as qhv0i q0 i qhv0i q0 i 1 qFhv0i v0k q0 i þ þ Vk qx k qt qx k F qQ qV i 0 0 ¼ (hvi vk i þ mik ) (hv0k q0 i þ mk ) þ hi qx k qx k 1 1 þ li þ hv0i q0 i: tp tt

ð3:32Þ

The balance of intensities of temperature ﬂuctuations in the disperse phase is expressed by the equation qhq0 i qhq0 i 1 qFhv0k q0 i qQ 2hq0 i þ ¼ 2(hv0k q0 i þ mk ) þ 2h : þVk qt qx k F qx k qx k tt 2

2

2

2

ð3:33Þ

j123

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

124

Equations (3.32)–(3.33) describe joint ﬂuctuations of particle velocity and temperature, individual terms in these equations describing time variation, convection and turbulent diffusion of ﬂuctuations, generation of ﬂuctuations by the average motion due to the gradients of velocity and temperature, as well as generation and dissipation of ﬂuctuations due to dynamic and thermal interaction with the continuum. For lowinertial particles, differential terms describing transport of ﬂuctuations and generation of ﬂuctuations by the gradients become insigniﬁcant, and thus turbulent heat ﬂux and intensity of temperature ﬂuctuations of the disperse phase can be determined from the conditions of local equilibrium between generation and dissipation of ﬂuctuations via interfacial interactions: hv0i q0 i ¼

hu0k W0 i t tp f u ki þ tt f tt ki ; t p þ tt

ð3:34Þ

hq0 i ¼ f t hW0 i: 2

2

ð3:35Þ

The formula (3.35) coincides with the relation (1.117) between temperature ﬂuctuation intensities in the disperse and continuous phases for stationary homogeneous turbulence. Our next goal is to close the chain of equations stemming from Eq. (3.29) on the level of third-order moments. This closure is achieved with the help of the quasinormal hypothesis: hv0i v0j v0k q0 i ¼ hv0i v0j ihv0k q0 i þ hv0i v0k ihv0j q0 i þ hv0j v0k ihv0i q0 i; hv0i v0j q0 i ¼ hv0i v0j ihq0 i þ 2hv0i q0 ihv0j q0 i; 2

2

hv0i q0 i ¼ 3hv0i q0 ihq0 i: 3

2

ð3:36Þ

In view of Eq. (3.36), we obtain from Eq. (3.29) the following closed equations for third moments: qhv0i v0j q0 i qt þ

þ Vk

qhv0i v0j q0 i qx k

þ hv0i v0k q0 i

qV j qV i qQ þ hv0j v0k q0 i þ hv0i v0j v0k i qx k qx k qx k

0 t Dp ik qhv0j q i Dp jk qhv0i q0 i Dp k qhv0i v0j i þ þ þ tp qx k tp qx k tt qx k

2 1 þ hv0i v0j q0 i ¼ 0; tp tt ð3:37Þ

qhv0i q0 i qhv0i q0 i qQ 2 qV i þ hv0k q0 i þ 2hv0i v0k q0 i þ Vk qx k qt qxk qxk 2

2

t Dp ik qhq0 2 i 2Dp k qhv0i q0 i þ þ þ tp qx k tt qx k

1 2 2 þ hv0i q0 i ¼ 0; tp tt

3 3 3Dtp k qhq0 2 i 3hq0 3 i qhq0 i qhq0 i 2 qQ þ 3hv0 k q0 i þ þ ¼ 0: þ Vk tt qx k qt qxk qx k tt

ð3:38Þ

ð3:39Þ

3.2 The Equations for Single-Point Moments of Particle Temperature

If we neglect the terms describing time evolution, convection, and generation of third moments by the gradients of average velocity and temperature, then Eqs. (3.37)–(3.39) lead to the following algebraic relations expressing third moments through second moments and their derivatives: qhv0j q0 i qhv0i v0j i 1 qhv0i q0 i t ¼ þ ¡Dp jk þ Dp k ¡Dp ik ; qx k qx k 2¡ þ 1 qx k

hv0i v0j q0 i

hv0i q0 i ¼ 2

2 1 qhq0 i qhv0i q0 i þ 2Dtp k ¡Dp ik ; ¡þ2 qx k qx k

hq0 i ¼ Dtp k 3

qhq0 i : qx k

ð3:40Þ

ð3:41Þ

2

ð3:42Þ

The relations (3.40)–(3.42) for third moments have been obtained by Zaichik and Vinberg (1991). Equations (3.30)–(3.33) together with Eqs. (3.40)–(3.41) enable us to describe heat transport in the disperse phase within the framework of the differential model for second moments. Consider the asymptotic behavior of this continual model in the limit of inertialess particles (tp, tt ! 0). It follows from Eq. (3.30) that the average temperature of a disperse phase consisting of inertialess particles is equal to Q ¼ TDtk

qlnF : qx k

ð3:43Þ

Due to Eq. (3.43), the temperature of inertialess particles differs from the average temperature of the carrier ﬂuid because of the contribution of the diffusional component (a similar statement is true for the velocity of these particles, which is given by Eq. (2.32)). As we see from Eqs. (3.34)–(3.35), in the limit of inertialess particles, heat ﬂuxes and temperature ﬂuctuation intensities of the disperse and continuous phases coincide: lim hv0i q0 i ¼ hu0i W0 i;

ð3:44Þ

lim hq0 i ¼ hW0 i:

ð3:45Þ

tp ; tt ! 0

2

2

tp ; tt ! 0

We see by looking at Eq. (3.44) that the thermal diffusion vector of inertialess particles (passive scalar) (3.31) is equal to Dti ¼ lim Dtp i ¼ hu0k W0 iT tL ki þ hu0k W0 ictL kn tp ; tt ! 0

ctL ij ¼

ð1 0

YtL ij (t)t dt;

qU i Dp hu0k W0 i ctL ki ; Dt qx n 2 ð3:46Þ

j125

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

126

where YtL ij (t) is the Lagrangian autocorrelation function of joint ﬂuctuations of velocity and temperature of the continuum, which is characterized by the integral Ð1 time scale T tL ij 0 YtL ij (t)dt. Similarly to Eq. (2.38) and Eq. (2.40), we choose an isotropic representation for T tL ij: k ð3:47Þ T tL ij ¼ atu dij ; atu ¼ const: e If we drop the two last terms (terms that are due to the gradients of the average velocity and to the inhomogeneity of turbulence) in Eq. (2.34) and Eq. (3.46), then, going to the inertialess limit (tp, tt ! 0) and taking note of Eqs. (2.33), (3.44), we see from Eqs. (3.40)–(3.41) that qhu0j q0 i qhu0i q0 i k þ hu0j u0k i hu0i u0j W0 i ¼ Cu1 hu0i u0k i qx k qx k e 0 0 qhui uj i þ Ctu1 hu0k q0 i ; ð3:48Þ qxk 2 hu0i W0 i

02 0 0 k 0 0 qhq i t 0 0 qhui q i ¼ þ 2C u2 huk q i C u2 hui uk i ; e qx k qx k

Cu1 ¼

¡ 0 au ; 2¡0 þ 1

Ctu 2 ¼

atu ; ¡0 þ 2

Ctu1 ¼

atu ; 2¡0 þ 1

¡0 ¼ lim

tt

tp ; tt ! 0 t p

¼

C u2 ¼

¡0 au ; ¡0 þ 2

3Pr ; 2

ð3:49Þ

where Pr is the Prandtl number. The formulas (3.48) and (3.49) conform with the well-known gradient approximations for third moments of joint velocity and temperature ﬂuctuations of the continuum (Launder, 1976; Dekeyser and Launder, 1985). Thus the continual second-order differential model for heat transport in the disperse phase given by Eqs. (3.30)–(3.33) with the consideration of Eqs. (3.40)–(3.41) provides a correct transition to the limiting case of inertialess particles described by Eqs. (3.44)–(3.45) and reproduces the relations for third moments that we encounter in the one-phase theory. By analogy with Eqs. (3.40)–(3.41), the third moments of correlations that enter the 2 transport terms Dp hu0i W0 i=Dt and Dp hW0 i=Dt in Eq. (3.50) and Eq. (3.48) appear as hu0i v0j W0 i ¼

qhu0j W0 i qhu0i u0j i qhu0i W0 i 1 þ ¡Dp jk þ Dtp k ; ¡Dp ik qxk qx k qx k 2¡ þ 1

hv0i W0 i ¼

2 qhu0i W0 i 1 qhW0 i þ 2Dtp k : ¡Dp ik qx k ¡þ2 qx k

2

Mixed second-order moments of correlations of ﬂuctuations of velocity and temperature of the continuous and disperse phases are determined from Eq. (3.41) and Eq. (3.43):

3.3 Algebraic Models of Turbulent Heat Fluxes

ðð 1 qQ 0 0 0 0 hui piqdvdqQ hui pidvdq ¼ tp hi mik hui q i ¼ ; F qx k ðð ðð 1 qV i 0 0 0 0 hW pivi dvdqV i hW pidvdq ¼ tt li mk ; hvi W i ¼ qx k F ðð ðð 1 qQ 0 0 0 0 hW piqdvdqQ hW pidvdq ¼ tt hmk : hW q i ¼ F qx k ðð

ð3:50Þ ð3:51Þ ð3:52Þ

3.3 Algebraic Models of Turbulent Heat Fluxes

Analogously to the discussion of particle motion in Section 2.3, calculations of heat transport in the disperse phase may carried out on the basis of algebraic models for turbulent heat ﬂows (see, for example, Derevich et al., 1989b; Han et al., 1991; Vinberg et al., 1992; Zaichik et al., 1997b; Boulet et al., 1998; Derevich, 2002). As in the derivation of hydrodynamic models, we examine two algebraic models of heat transport. The ﬁrst approach is based on solving the kinetic equation by the Chapman–Enskog method, and the second – on solving the balance equation for turbulent heat ﬂuxes by the iteration method. 3.3.1 Solution of the Kinetic Equation by the Chapman–Enskog Method

The solution of Eq. (3.29) is presented in the form of an expansion P(v; q) ¼ Pð0Þ (v; q) þ Pð1Þ (v; q) þ . . . ;

ð3:53Þ

where the functions P(0)(v, q) and P(1)(v, q) obey the equations R Pð0Þ ¼ 0;

ð3:54Þ

R Pð1Þ ¼ N P ð0Þ ; 2 1 2kp q2 P q(vi V i )P 1 q(qQ)P 2 q P þ ; þ hq0 i 2 þ tp 3 qvi qvi qvi tt qq qq qP qP U i V i qP þ þ Fi N[P] ¼ þ vi tp qt qx i qvi R[P] ¼

2 2kp TQ qP q P q2 P þ þQ dij lij mij þ 3tp tt qq qvi qvj qx j qvi 2 02 2 q P hq i q P q2 P hi þ li h mi þ : 2 qvi qq tt qx i qq qq

ð3:55Þ

j127

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

128

It is evident that the expansion (3.53) may be valid only if the quantities tp/Tu and tt/Tt are small parameters. Here Tu and Tt are the characteristic times of variation of average hydrodynamic parameters and heat ﬂow parameters. The solution of Eq. (2.45) is an equilibrium distribution over velocities and temperatures: Pð0Þ (v; q) ¼ F

1=2

27

128p4 k3p hq0 i 2

2 3v0 v0 q0 : exp k k 2 4kp 2hq0 i

ð3:56Þ

Due to Eq. (3.56), hv0i q0 i0 ¼

ð 1 0 0 ð0Þ v q P dvdq ¼ 0; F i

hv0i v0j q0 i0 ¼

ð 1 0 0 0 ð0Þ v v q P dvdq ¼ 0; F i j

ð3:57Þ

hv0i q0 i0 ¼ 2

ð 1 0 0 2 ð0Þ v q P dvdq ¼ 0: F i ð3:58Þ

Hence the equilibrium PDF gives zero values for heat ﬂuxes (3.57) and for third moments of joint velocity and temperature ﬂuctuations (3.58), which corresponds to isotropic turbulence. Let us determine the right-hand side of Eq. (3.55) using Eq. (3.56). The time derivatives qF/qt, qVi/qt, qkp/qt, qQ/qt, qhq0 2i/qt are deﬁned respectively, by the equations (2.24), (2.25), (2.65), (3.30), (3.33) with the consideration of the relations (2.49)–(2.50) and (3.57)–(3.58). We obtain as a result 3m V j;k 3lij v0 v0 3 N Pð0Þ ¼ v0i v0j k k d ij V i; j þ ik 3 2kp 2kp 2kp þ

3mij v0i 3v0k v0k 3v0i q0 5 d ij þ kp; j þ 2 2kp 2kp 2kp 2kp hq0 i

2kp d ij þ mij Q; j þ mj V i; j (hi þ li ) 3 0 0 vk vk þ 1 kp;i 2 2kp hq0 i 2kp 3mi q0

þ

þ

3mij 2 1 d þ hq0 i; j ij 2 2 2kp 2hq0 i hq0 i v0i

mi q 0

q0

q0

2

2

2hq0 i hq0 i 2

2

2 3 hq0 i;i P ð0Þ (v; q):

ð3:59Þ

3.3 Algebraic Models of Turbulent Heat Fluxes

Making use of Eq. (3.59), we can solve Eq. (3.55): P ð1Þ (v; q) ¼

þ

þ

þ

3m V j;k 3lij 3tp 0 0 v0k v0k d ij vi vj V i; j þ ik 4kp 3 2kp 2kp

3mij tp v0i 3v0k v0k 5 dij þ kp; j 6kp 2kp 2kp

3tp tt v0i q0

2(tp þ tt )kp hq0 i 2

2kp d ij þ mij Q; j þ mj V i; j (hi þ li ) 3

v0k v0k 1 kp;i 2 2(tp þ 2tt )kp hq0 i 2kp 3tp tt mi q0

3mij 2 1 dij þ þ hq0 i; j 02 02 2k p 2(2tp þ tt )hq i hq i tp tt v0i

þ

tt mi q0

q0

2

6hq0 i hq0 i 2

2

q0

2

02 3 hq i;i Pð0Þ (v; q):

ð3:60Þ

According to Eq. (3.60), turbulent heat ﬂuxes are determined as ð 1 0 0 ð1Þ v q P dvdq F i 2kp d ij tp tt þ mij Q; j þ mj V i; j (hi þ li ) : ¼ tp þ tt 3

hv0i q0 i1 ¼ hv0i q0 ið1Þ ¼

ð3:61Þ

The expression (3.61) suggests a linear dependence of hv0i q0 i on the gradients of average velocity Vi, j and temperature Q,i of the disperse phase. Along with the gradient terms, there are also the terms hi and li, which directly describe particle interaction with turbulent eddies of the carrier continuum and play an especially important role for low-inertia particles, ensuring a smooth transition to the limiting case of inertialess particles (3.44). If we simplify the math by dropping the term with mij in Eq. (3.61) and the gradient terms in Eqs. (3.22)–(3.23) and assume the quasiisotropic representations hi ¼ f tt hu0i W0 i=tp , li ¼ f tu hu0i W0 i=tt , then the relation (3.61) will take the form hv0i q0 i1 ¼

2tp tt kp tp f tu þ tt f tt 0 0 hui W i: Q;i þ 3(tp þ tt ) tp þ tt

ð3:62Þ

If in addition we approximate turbulent heat ﬂuxes of the continuum by the expression hu0i W0 i ¼

nT T;i PrT

j129

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

130

and assume the equality of average temperatures of the disperse and continuous phases (Q T) in the limiting case of low-inertia particles, then the relation (3.62) with the consideration of Eq. (2.58) takes the form of Fouriers law (Zaichik, 1992b): hv0i q0 i1 ¼

np Q;i ; Prp

(tp þ tt )(3f u nT þ tp kp )PrT

; Prp ¼ t 3 tp f u þ tt f tt nT þ 2tp tt kp PrT

ð3:63Þ ð3:64Þ

where PrT and Prp are the turbulent Prandtl numbers of the continuous and disperse phases. In the inertialess limit, the turbulent Prandtl number of the disperse phase (3.64) approaches the corresponding value of the Prandtl number for the continuous phase: lim Prp ¼ PrT :

tp ; tt ! 0

Due to Eq. (3.60), the third moments containing temperature ﬂuctuations are given by ð 2tp tt 1 0 0 0 ð1Þ hv0i v0j q0 i1 ¼ dij mk kp;k ; ð3:65Þ v v q P dvdq ¼ 3(tp þ 2tt ) F i j hv0i q0 i1 ¼ 2

hq0 i1 ¼ 3

ð tp tt 2kp 1 0 0 2 ð1Þ 2 dik þ mik hq0 i;k ; vi q P dvdq ¼ (2tp þ tt ) 3 F

ð 1 3 2 q0 Pð1Þ dvdq ¼ tt mk hq0 i;k : F

ð3:66Þ

ð3:67Þ

The formulas (3.65), (3.66), and (3.67) coincide, respectively, with Eq. (3.40), Eq. (3.41), and Eq. (3.42) if we substitute the isotropic representations for turbulent stresses and heat ﬂuxes (2.49) and (3.57) into the latter three equations. 3.3.2 Solving the Equation for Turbulent Heat Fluxes by the Iteration Method

Similarly to Eq. (2.68), we express transport terms in the equation for turbulent heat ﬂuxes (3.32) through the transport terms in the equations for turbulent kinetic energy (2.65) and intensity of temperature ﬂuctuations (3.33):

Dp hv0i q0 i tp Dp hu0i u0k i þ qu1 kn T ;n g u1 kn Q;n 2 Dt Dt

1 Dp hu0k W0 i f tu1 ki tp t t t þ U i;n g u1 kn V i;n þ l ¡f t1 ki þ ¡ ¡ u1 kn 2 Dt ¼

hv0i q0 i Dp kp 1 Dp hu0i u0k i þ f u1 ki þ tp lu1 kn U i;n g u1 kn V i;n Dt 2kp 2 Dt

3.3 Algebraic Models of Turbulent Heat Fluxes

þ

2 2

tp Dp hu0k W0 i t hv0i q0 i Dp hq0 i Dp hW0 i t ; þ T g Q þ f q ;n ;n u1 kn t1 u1 kn 2 ¡ Dt Dt Dt 2hq0 i

Dp hv0i q0 i qhv0i q0 i qhv0i q0 i 1 qFhv0i v0k q0 i ¼ þ ; þ Vk Dt qx k qt qx k F Dp hq0 i qhq0 i qhq0 i 1 qFhv0k q0 i ¼ þ : þ Vk Dt qt qx k F qxk 2

2

2

2

ð3:68Þ

Substitution of Eq. (3.68) into Eq. (3.32) with the consideration of Eq. (2.65) and Eq. (3.33) gives hv0i q0 i ¼

tp tt 0 0 hv v iQ;k þ hv0k q0 iV i;k þ X i ; P(tp þ tt ) i k

X i ¼ hu0i u0j i g u jk þ tp hu jn U k;n Q;k qu jk þ tp r u jn U k;n T;k hu0j W0 i

þ

¡

f Et ¼

g tu jk þ tp htu jn U k;n V i;k ltu jk þ tp mtu jn U k;n U i;k

t t u ji ¡f t ji

; P¼

ˆtt hv0k q0 iQ;k 02

tt E þ tp Et ; tp þ tt

; ˆ ¼ hW0 if t 2

hq i

þ tp hu0i W0 i qtu ik þ tp r tu in U k;n T ;k g tu ik þ tp htu in U k;n Q;k :

ð3:69Þ

Equation (3.69) represents an implicit algebraic model for turbulent heat ﬂuxes because hv0i q0 i appears on both sides of the equation. We now solve this equation by applying the iteration procedure in the same manner as we did when solving Eq. (2.70): hv0i q0 in þ 1 ¼ Pn ¼

tp tt 0 0 hvi vk in Q;k þ hv0k q0 in V i;k þ X i ; P(tp þ tt )

tt En þ tp Etn ˆtt hv0k q0 in Q;k ; ; Etn ¼ 2 tp þ tt hq0 i

ð3:70Þ

where n is the order of the approximation. The isotropic approximation given by Eq. (2.49) and Eq. (3.57) is taken as the zeroth approximation. Then the ﬁrst approximation gives us a linear model for turbulent heat ﬂuxes: hv0i q0 i1 ¼ P0 ¼

hv0i q0 ið1Þ ; P0

hv0i q0 ið1Þ ¼

tt E0 þ tp Et0 ; tp þ tt

Et0 ¼

tp tt 2kp Qi þ X i ; tp þ tt 3

ˆ hq0 i 2

:

ð3:71Þ

j131

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

132

The region of applicability of the linear model (3.71) is deﬁned by the requirement that the parameters tp/Tu and tt/Tt must be small. Insertion of Eq. (2.72) and Eq. (3.71) into the right-hand side of Eq. (3.70) yields the second approximation (the non-linear model) for turbulent heat ﬂuxes: hv0i q0 i2 ¼

hv0i q0 ið1Þ hv0i q0 ið2Þ þ ; hv0i q0 ið2Þ P1 P0 P1

tp tt P0 0 0 ð1Þ 0 0 ð1Þ ¼ hv v i Q;k þ hvk q i V i;k ; tp þ tt E0 i k 0 0 ð1Þ tp tt hvi vk i V i;k hv0k q0 ið1Þ Q;k : P1 ¼ P0 þ 2 tp þ tt 2E0 kp P0 hq0 i

ð3:72Þ

The region of applicability of the non-linear model (3.72) is wider than that of the linear model (3.71), but its accuracy diminishes as the parameters tp/Tu and tt/Tt get larger. Notice that in addition to the algebraic relations for hv0i q0 i, the linear and non-linear models presented in this section also include differential equations for turbulent characteristics of the disperse phase. Thus, the linear models (3.61)–(3.64) include a differential equation for kp, whereas the linear model (3.71) and the non-linear 0 model (3.72) include equations for kp and hq 2i.

3.4 Second Moments of Velocity and Temperature Fluctuations in a Homogeneous Shear Flow

Consider the behavior of a turbulent heat ﬂux, the intensity of temperature ﬂuctuations of the disperse phase, and mixed correlation moments of velocity and temperature ﬂuctuations of the continuous and disperse phases in a homogeneous turbulent ﬂow with constant gradients of average velocities and temperature. Due to the homogeneity of the ﬂow, all triple single-point correlations of velocities and temperature vanish, and thus the chain of equations for the moments is terminated on the level of second-order moments. In this section, the solutions of the equations for second moments presented in Sections 3.2 and 3.3 will be compared with the DNS data obtained by Pandya and Mashayek (2003). This section will also examine the ﬂow in a layer that is thermally and hydrodynamically homogeneous, assuming equal average velocities of the continuous and disperse phases. As in Section 2.5, gradients of average velocities are given by the relations (2.136). Time rate of change of turbulent stresses for the particles and time rate of change of mixed moments of velocity ﬂuctuations of the continuous and disperse phases are shown, respectively, in Figure 2.4 and Figure 2.5. Pandya and Mashayek (2003) ran their simulation for three types of temperature gradients superimposed on the ﬂow ﬁeld (2.136). Temperature gradients of the ﬁrst

3.4 Second Moments of Velocity and Temperature Fluctuations in a Homogeneous Shear Flow

type correspond to velocity gradients and are applied in the x2-direction, which is transverse to the main ﬂow: qT qQ ¼ ¼ S; qx 2 qx 2

ð3:73Þ

where S is the temperature gradient. Temperature gradients of the second type are applied in the transverse direction x3: qT qQ ¼ ¼ S: qx 3 qx 3

ð3:74Þ

Finally, temperature gradients of the third type are applied in the longitudinal direction x1: qT qQ ¼ ¼ S: qx 1 qx 1

ð3:75Þ

In their calculations, Pandya and Mashayek (2003) used the same value for the temperature gradient S for all three cases. Since the cases (3.73) and (3.75) are symmetrical in all essential aspects, the analysis is conﬁned to the ﬁrst two cases. The behavior of the turbulent heat ﬂux hv0i q0 i, intensity of temperature ﬂuctuations hq0 2i, and the mixed correlation moments hu0i q0 i, hv0i W0 i, hW0 q0 i should be described, respectively, by the equations (3.32), (3.33), and (3.50)–(3.52) simpliﬁed for the case of homogeneous ﬂows under consideration. The involvement coefﬁcients in these equations are deﬁned by Eq. (3.28). The correlation tensor for the durations of velocity and temperature ﬂuctuations TtLp is taken equal to TLp and is calculated using Eqs. (2.140), (2.142), and (2.143). The integral time scale of ﬂuid temperature ﬂuctuations along inertial particle trajectories TLtp is calculated using the dependences (1.82) and (1.83) for isotropic turbulence, in which the Lagrangian scale of temperature ﬂuctuations TLt is taken equal to the average diagonal scale of velocity ﬂuctuations TL ¼TLkk/3. The temperature structure parameter of turbulence mt as well as the hydrodynamic parameter m is taken equal to 0.5. Time rate of change of the turbulent heat ﬂux, of the intensity of temperature ﬂuctuations, and of mixed correlation moments of velocity and temperature ﬂuctuations is shown on Figures 3.1–3.4 for the cases (3.73) and (3.74). The isotropic state characterized by zero values of all components of turbulent heat ﬂuxes hu0i W0 i, hv0i q0 i, hu0i q0 i, and hv0i W0 i is taken as the initial state. As time goes on, the anisotropy of the turbulent heat ﬂux increases. Only two components of the heat ﬂux are nonzero in the case (3.73), as evidenced by Figures 3.1 and 3.2, and only one component is nonzero in the case (3.74), as evidenced by Figure 3.3 and Figure 3.4. Since the relations (3.50)–(3.52) for hu0i q0 i, hv0i W0 i, and hW0 q0 i, as opposed to the differential equations (3.32) and (3.33) for hv0i q0 i and hq0 2i, are algebraic, they cannot satisfy the initial conditions corresponding to the isotropic state. Therefore the time dependences hu0i q0 i, hv0i W0 i and hW0 q0 i shown on Figure 3.2 and Figure 3.4 start at some distance from t ¼ 0 similarly to the time dependence for mixed correlation moments of velocity ﬂuctuations of the continuous and disperse phases hu0i v0j i shown

j133

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

134

Figure 3.1 Time rate of change of the turbulent heat flux and of the intensity of temperature fluctuations of the disperse phase in a homogeneous shear layer for the temperature gradient (3.73): 1 – model I; 2 – model II; 3 – linear model; 4 – non-linear model; 5–7 – Pandya and Mashayek (2003).

on Figure 2.5. Model I stands for the differential model for turbulent stresses and heat ﬂuxes that takes into consideration the anisotropy of Lagrangian time scales of turbulent ﬂuid velocity ﬂuctuations (2.140) and (2.141). Model II stands for the differential model for turbulent stresses and heat ﬂuxes that does not factor in the anisotropy of Lagrangian time scales. These models are similar to the corresponding hydrodynamic models presented in Section 2.5. We see that in all cases, the effect of time scale anisotropy on the turbulent heat ﬂux and especially on the intensity of temperature ﬂuctuations is not signiﬁcant. Overall, the models presented here are in a good agreement with the DNS data. Also shown on Figures 3.1 and 3.2 are the turbulent heat ﬂux and the intensity of temperature ﬂuctuations obtained on the basis of the algebraic models (3.71) and (3.72) with the consideration of anisotropy of Lagrangian time scales. The

Figure 3.2 Time rate of change of mixed correlation moments of velocity and temperature in a homogeneous shear layer for the temperature gradient (3.73): 1 – model I; 2 – model II; 3–5 – Pandya and Mashayek (2003).

3.4 Second Moments of Velocity and Temperature Fluctuations in a Homogeneous Shear Flow

Figure 3.3 Time rate of change of the turbulent heat flux and of the intensity of temperature fluctuations of the disperse phase in a homogeneous shear layer for the temperature gradient (3.74): 1 – model I; 2 – model II; 3 – linear model; 4 – non-linear model; 5, 6 – Pandya and Mashayek (2003).

Figure 3.4 Time rate of change of mixed correlation moments of velocity and temperature in a homogeneous shear layer for the temperature gradient (3.74): 1 – model I; 2 – model II; 3, 4 – Pandya and Mashayek (2003).

calculation results given by the linear algebraic models (3.61) and (3.71) are nearly identical, therefore the distributions corresponding to Eq. (3.61) are not presented here. One can see that after a small initial time lapse, the distributions of heat ﬂuxes and temperature ﬂuctuations based on the differential and non-linear algebraic models become sufﬁciently close to one another, but may deviate noticeably from the corresponding distributions obtained by using the linear model.

j135

j137

4 Collisions of Particles in a Turbulent Flow Turbulence is one of the principal mechanisms that cause particles to collide with each other. In their turn, collisions of particles can affect their interaction with turbulent eddies of the continuum. In order to calculate the collision frequency of particles and derive the collisional terms entering the transport equations for macroscopic characteristics of the disperse phase, one usually assumes that the main contribution to the collision terms comes from pair collisions and that collisions are Marcovian random processes, that it to say, they are independent of the preceding collisions. In view of these assumptions for the collision frequency, it is necessary to determine the PDF of particle-pair velocity distributions. For this purpose, by analogy with the molecular chaos hypothesis that is employed in the kinetic theory of gases and lies in the basis of the Boltzmann equation, one can use the concept of noncorrelativity (statistical independence) of motion of the colliding particles. In accordance with this assumption, the two-particle PDF is written as a product of singleparticle PDFs, and the process of particle collision is in fact described in the same manner as molecular collisions within the framework of the solid sphere model in the kinetic theory of gases (Chapman and Cowling, 1970; Lun et al., 1984; Jenkins and Richman, 1985; Ding and Gidaspow, 1990). However, such an approach (the noncorrelative model) may work only for collisions of sufﬁciently inertial particles whose dynamic relaxation time tp by far exceeds the characteristic time of interaction with turbulent eddies TLp (tp/TLp 1) so that their relative motion is non-correlative and similar to the chaotic motion of molecules. When modeling the collision process between moderately inertial particles (tp/TLp 1) one should take note of the correlativity of their ﬂuctuational motion caused by their interaction with turbulent eddies of the carrier ﬂow. Hence the two processes – particle–turbulence interactions and inter-particle collisions – may be considered independently from one another only for tp/TLp 1, whereas for tp/TLp 1 it is necessary to take their interplay into consideration. The method for modeling particle collisions in turbulent ﬂows that will be outlined in the present chapter is based on the assumption that the joint PDF of ﬂuid and particle velocities is correlated by a Gaussian distribution (Lavieville et al., 1995; Lavieville, 1997). To take into account anisotropy of particle velocity ﬂuctuations, a generalization of the procedure known in the kinetic theory of gases as Grads

j 4 Collisions of Particles in a Turbulent Flow

138

method is proposed, with the aim to extend the procedure to allow for the correlativity of motion of colliding particles. Statistical models presented in this chapter enable us to obtain analytical dependences for the collision frequencies of particles and for the collisional terms in the transport equations for moments of the disperse phase but without regard for the effect of particle accumulation (clustering). Particles are assumed to be identical (monodisperse), and the disperse phase is assumed to be low-concentrated (F 0.01 where F is volume concentration) so that it is sufﬁcient to consider pair collisions only while neglecting the direct contribution of inter-particle collisions to the stresses and ﬂuctuation energy ﬂux in the disperse phase. In addition, when deriving collision frequency and collisional terms, the process is treated as Markovian, that is to say, the prehistory of the process can be disregarded.

4.1 Collision Frequency of Monodispersed Particles in Isotropic Turbulence

The problem of determining collision frequency of solid particles or liquid drops in turbulent ﬂows is of interest in the analysis of many industrial and meteorological processes. This acute practical interest has stimulated an extensive body of theoretical and experimental research whose subject is the rate of particle collisions due to turbulence. Relatively simple solutions may be obtained for this problem only within the framework of the isotropic stationary turbulence approximation. When the number of particles in the system under consideration is large, the frequency of collisions of a test particle with other particles wc is proportional to the number concentration of particles (average number of particles in a unit volume) N: wc ¼ bN; where the quantity b is called the collision kernel due to the fact that it appears as the integrand (kernel) in the kinetic equation that describes the evolution of the size distribution of particles as it changes as a result of particle coagulation. In the present Section we consider collisions that are due to the interaction of particles with turbulent eddies of the continuum, without regard for gravitational sedimentation, Brownian motion, and the action of hydrodynamic and molecular forces. The reader will ﬁnd detailed information about these effects in the books by Sinaiski and Lapiga (2006); Sinaiski and Zaichik (2007). Interaction of inertial particles with turbulent eddies involves two statistical phenomena that contribute to the collision kernel, namely, relative motion of neighboring particles that is characterized by their relative velocity (the so-called turbulent transport effect), and inhomogeneous distribution of particles in space (the particle accumulation effect, also known as the clustering effect). The clustering effect manifests itself in the tendency of particles to congregate in the regions of low vorticity due to the action of the centrifugal force (Squires and Eaton, 1991c). This local rise of concentration is caused by deviation of particle trajectories from stream lines of the carrier ﬂuid and can lead to a noticeable increase of the collision rate and to further coagulation in

4.1 Collision Frequency of Monodispersed Particles in Isotropic Turbulence

homogeneous turbulence (Reade and Collins, 2000; Wang et al., 2000; Zaichik et al., 2003; Chun and Koch, 2005). The best-known analytical solutions for the problem of particle collisions in a turbulent ﬂow have been obtained for the cases of small inertialess particles and relatively large high-inertia particles (Saffman and Turner, 1956; Abrahamson, 1975). The theory proposed by Saffman and Turner (1956) is true for particles whose relaxation time is less than the Kolmogorov time microscale. Such particles obediently follow every turbulent velocity ﬂuctuation of the carrier continuum and become involved in the motion of the smallest eddies responsible for the dissipation of turbulent energy. Therefore the collision rate of inertialess particles is deﬁned by their interaction with small-scale turbulent eddies. The solution obtained by Abrahamson (1975) corresponds to the other limiting case when particle relaxation time is much greater than the time macroscale of turbulence. In such a case, the particles become statistically independent, that is, their motion is fully noncorrelative and similar to the motion of molecules in the kinetic theory of rareﬁed gases, for which the molecular chaos hypothesis is valid. For such particles, it is sufﬁcient to take into account their interactions with energy-carrying eddies only, while ignoring the contribution of interactions with small-scale turbulence to the collision kernel. It should be noted that both inertialess particles and high-inertia particles are distributed in space independently and randomly, which results in the absence of clustering. The greatest difﬁculties arise when we try to ﬁnd the collision kernel for the case of ﬁnite ratios of particle relaxation time to the micro- and macro- scales of turbulence, that is, when tk tp TL, where tk is the Kolmogorov time microscale, tp – particle relaxation time, TL – the Lagrangian integral scale. In this case it is necessary to take into account the interaction of particles with the entire spectrum of turbulent eddies as well as the correlativity of motion of neighboring particles and the effect of clustering. In an effort to determine the relative velocity of particles, Saffman and Turner (1956), Williams and Crane (1983), and Kruis and Kusters (1996) have considered two mechanisms of particle collisions. The ﬁrst mechanism is the result of velocity shear of the carrier ﬂow; the second is caused by ﬂuctuations of relative velocity between the particles and the ﬂuid due to the acceleration of particles. The velocity shear mechanism is thus associated with collisions resulting from the motion of particles with the ﬂuid, and the acceleration mechanism – with collision resulting from the motion of particles relative to the ﬂuid. It is evident the breaking down of a single process of interaction between the particles and the turbulent eddies into two separate mechanisms that is entailed by this approach is largely a matter of convention. An expression for the collisional term due to acceleration that also factors in the correlativity of particle velocities was ﬁrst obtained by Williams and Crane (1983) by using the spectral method. Because analytical solutions existed only for very small and very large particles, while the goal was to write an expression for the collision kernel that would be applicable for the whole range of particle inertias, the authors obtained their expression for the kernel by interpolating the limiting solutions. However, Williams and Crane ignored the interaction of particles with small-scale turbulence. As a consequence,

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j 4 Collisions of Particles in a Turbulent Flow

140

the velocity shear mechanism was excluded from their analysis and their expression did not reduce to the Saffman–Turner solution in the inertialess particle limit. Kruis and Kusters (1996) have used the Williams–Crane approximation to combine the solutions for velocity shear and acceleration. Yuu (1984) has also obtained a solution combining the two collision mechanisms – particle shear and acceleration, but his model is valid for small particles only, since the correlation coefﬁcient of velocities of the two particles was derived incorrectly, yielding the wrong value (equal to 1) for identical particles. In view of this, particle interactions with energy-contained eddies do not make any contribution to the collision kernel, so the model cannot be used for large particles. By solving the diffusion equation that stems from the kinetic equation for the probability density of velocities of two particles, Derevich (1996) has found the collision kernel, taking into account the contributions from particle interactions with both energy-carrying eddies and small-scale turbulent eddies. But his model suggests that the correlation coefﬁcient should approach unity (instead of zero) with increase of particle inertia, and thus it fails to reduce to Abrahamsons dependence in the limiting case. Note that the analytical models by Williams and Crane (1983), Yuu (1984), Kruis and Kusters (1996), and Derevich (1996) are based on the cylindrical formulation of the problem – a practice that is common in Statistical Mechanics. In the framework of the cylindrical formulation, the collision kernel is related to the full relative velocity of the two particles h|w|i and the collision radius s (which for identical particles is equal to their diameter dp), by the formula b ¼ ps2 hjwji: As shown by Wang et al. (2000), spherical formulation is more appropriate for this type of problems, in particular, for the problem of ﬁnding the collision rate of low-inertia particles in a turbulent ﬂow. In this formulation the collision kernel b is expressed through the average radial component of relative velocity h|wr|i. The difference between the two formulations for small particles is due to the difference of longitudinal and transverse structure functions at small distances between the two points (see Eq. (1.22)), or, to use another term, with the difference in the intensity of ﬂuctuations of relative velocity of colliding particles in different directions. For high-inertia particles, both formulations lead to identical results. Wang et al. (2005) have improved the cylindrical formulation of the collision problem by explicitly including the dependence of the probability distribution of relative velocity w between the two touching particles on the orientation vector r connecting the particle centers. For the collisions of a test particle with other particles, a spherical formulation of the problem that does not consider the effect of clustering gives the following deﬁnition for the collision kernel: b ¼ 2ps2 hjwr (s)ji;

ð4:1Þ

where h|wr(s)|i is the average radial component of relative velocity of two particles at contact. The relation (4.1) is true under the assumption that the relative velocity w is

4.1 Collision Frequency of Monodispersed Particles in Isotropic Turbulence

incompressible in the sense that the inward and outward ﬂuxes through the surface of the collision sphere are equal. Based on a simple model of particle interactions with eddies whose lifetime is equal to a constant value, Zhou et al. (1998) have obtained an analytical expression for the kinetic energy of particles and the correlation coefﬁcient of velocities of two particles, which is needed in order to determine the collision kernel. This model ignores particle interaction with small-scale eddies and models large eddies as a Monte-Carlo process with a triangular autocorrelation function. In an effort to include the contribution of velocity shear mechanism to the relative motion of two particles, Wang et al. (2000) have modiﬁed the model proposed by Kruis and Kusters (1996) by switching from a cylindrical formulation to a spherical one. The same authors proposed a semi-empirical model of collisions, based on approximate dependences between the DNS results for radial velocity and the radial distribution function of a particle pair. Let us consider the statistical model of turbulent collisions (Lavieville et al., 1995; Lavieville, 1997, 2003; Zaichik et al., 2003) that takes into account particle interaction with the entire spectrum of turbulent eddies and is valid in the entire range of particle inertia. This model is based on the assumption that a single-particle PDF of ﬂuid and particle velocities in isotropic turbulence is represented as the normal distribution 3=2 0 0 F 1x2uv uk uk v0k v0k xuv u0k v0k 1 P (u(x); v(x)) ¼ exp þ 02 0 0 ; uv (2pu0 v0 )3 2v (1x2 ) 2u0 2 ð0Þ

u0 ¼ 2

hu0k u0k i 2k ¼ ; 3 3

v0 ¼ 2

hv0k v0k i 2kp ¼ ; 3 3

x¼

hu0k v0k i huk uk i1=2 hvk vk i1=2

:

ð4:2Þ

Here u0 2 and v0 2 are the intensities of ﬂuid and particle velocity ﬂuctuations, x – the correlation coefﬁcient of ﬂuid and particle velocities. Volume concentration of particles appears in Eq. (4.2) as a result of normalization of the PDF of particle velocity. In view of Eq. (4.2), the separate velocity PDFs for the ﬂuid and the particle obey the Maxwellian distribution 0 0 0 0 uk uk v vk 1 1 ð0Þ Pð0Þ (u) ¼ exp (v) ¼ exp k 02 : ; P 2u0 2 2v (2pu0 2 )3=2 (2pv0 2 )3=2 Note that the joint PDF of ﬂuid and particle velocities (4.2) is the simplest function satisfying the Maxwellian distributions for ﬂuid and particle velocities and giving the covariance of their velocities hu0k v0 k i. The two-point PDF of ﬂuid velocity is also assumed to be Gaussian: Pð0Þ (u1 (x); u2 (x þ r)) ¼

1 1G(r)2 (2pu0 2 )3 aij u01i u01j aij u02i u02j aik Bkj u01i u02j exp þ ; 2u0 2 2u0 2 u0 4 1F(r)2

1=2

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142

0

1F(r)2 B aij ¼ @ 0 0

0 1G(r)2 0

11 0 C 0 A ; 2 1G(r)

ð4:3Þ

where Bij(r) is the Eulerian two-point correlation moment and the functions F(r) and G(r) denote the longitudinal and transverse components of the Eulerian two-point correlation function of velocity ﬂuctuations (1.11). To determine the correlation between velocities of two neighboring particles, we now introduce the joint two-point PDF of ﬂuid and particle velocities (Lavieville et al.,1995) P ð0Þ (u1 (x); v1 (x); u2 (x þ r); v2 (x þ r)) ¼ Pð0Þ (v1 (x)ju1 (x))Pð0Þ (v2 (x þ r)ju2 (x þ r))P ð0Þ (u1 (x); u2 (x þ r));

ð4:4Þ

where P(0)(v|u) is the probability density of particle velocity, provided that ﬂuid velocity is u. The conditional probability density is equal to P ð0Þ (vju) ¼ Pð0Þ (u; v)=Pð0Þ (u);

ð4:5Þ

where P (u) is the Maxwellian distribution of ﬂuid velocity. Then, making use of Eqs. (4.2)–(4.5), we are able to determine the PDF of a particle pair by integrating Eq. (4.4) over the subspace of ﬂuid velocities: (0)

Ð

Pð0Þ (v1 (x); v2 (x þ r)) ¼ P ð0Þ (u1 ; v1 ; u2 ; v2 )du1 du2 1=2 1 1x4 G(r)2 F2 1x4 F(r)2 ¼ (2pv0 2 )3 0 exp@ 0 B bij ¼ @

bij v01i v01j 2v0 2

bij v02i v02j 2v0 2

þ

x2 bik Bkj v01i v02j u0 2 v0 2

1x F(r)2

4

0

0

0

1x4 G(r)2

0

0

0

1 A;

11 C A :

ð4:6Þ

1x4 G(r)2

The distribution (4.6) factors in the correlation of velocities of two neighboring particles. The phenomenon of velocity correlation plays a signiﬁcant role for particles whose relaxation time is less than the integral scale of turbulence, because it is in this case that the velocities of approaching particles become correlated as they interact with the same eddies. On the other hand, when particle relaxation time is much larger than the turbulent macroscale, the particles approach each other with randomly distributed and independent velocities. As follows from Eq. (4.6), in the absence of particle interaction with the ﬂuid (x ¼ 0), the two-particle PDF becomes equal to the product of two single-particle distributions P(0)(v1,v2) ¼ P(0)(v1)P(0)(v2), which corresponds to the molecular chaos hypothesis in the kinetic theory of gases.

4.1 Collision Frequency of Monodispersed Particles in Isotropic Turbulence

Let us now switch to new velocities characterizing, respectively, the motion of the particle pair as a whole and the relative motion of the two particles: q¼

v1 þ v2 ; 2

w ¼ v2 v1 :

Integration of the two-particle PDF (4.6) over q results in the following distribution of relative velocity of the particle pair w: c ij w 0i w0j 1=2 1 2 1x2 G(r) exp Pð0Þ (w) ¼ (2pv0 )3=2 1x2 F(r) ; 2v0 2 0 B c ij ¼ @

1x2 F(r)

0

0

0

1x G(r)

2

0

0

11 C A :

ð4:7Þ

2

1x G(r)

0

Further integration of Eq. (4.7) over the relative velocity component wn normal to the line of centers leads to the following PDF for the distribution of the radial component of relative velocity wr : 1 w 02 r Pð0Þ (w r ) ¼ ð4:8Þ 1=2 exp 2hw02 i ; r 2phw 02 i r

where the intensity of ﬂuctuations of radial velocity of the two particles as they come into contact is equal to 0 hw02 r i ¼ 2v (1z12 ); 2

2

z12 ¼ x F(s);

ð4:9Þ

and z12 is the correlation coefﬁcient of two particles colliding due to their interaction with the turbulence. In Eq. (4.9), the distance r between the two particles is taken equal to the collision radius s. Distribution (4.8) allows to express the average relative velocity of particles through the intensity of radial velocity ﬂuctuations: 2 02 1=2 hjwr ji ¼ : ð4:10Þ hw r i p As it was shown in Section 1.4, in homogeneous isotropic stationary turbulence with no average velocity slip of particles, the variance of particle velocity ﬂuctuations and the covariance of ﬂuid and particle velocity ﬂuctuations are related to the variance of ﬂuid velocity by the following expression: hv0k v0k i ¼ hu0k v0k i ¼ fu hu0k u0k i;

ð4:11Þ

where fu is the coefﬁcient (1.102) characterizing the involvement of particles in turbulent motion. In view of Eq. (4.11), the correlation coefﬁcient of ﬂuid and particle velocities x takes the form x ¼ fu1=2 :

ð4:12Þ

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From Eq. (4.9) and Eq. (4.12) there follows a relation for the correlation coefﬁcient of velocities of two particles at the moment when they come into contact: z12 ¼ fu F(s):

ð4:13Þ

Due to Eq. (4.13), the correlation coefﬁcient of velocities of two particles at contact is equal to the product of the coefﬁcient of particles involvement in turbulent motion fu and the correlation function F(s) that takes into account spatial correlativity of ﬂuid velocities over the distance of one collision radius s. If particle diameter is less than the Kolmogorov spatial microscale h and consequently s belongs to the viscous interval, the correlation function, in accordance with Eqs. (1.11), (1.14), (1.22), is equal to es2 s 2 FðsÞ ¼ 1 ; 2 ¼ 1 1=2 0 30u n 60 Rel

s s ¼ ; h

Rel ¼

15u0 4 en

!1=2 :

ð4:14Þ

In view of Eqs. (4.9), (4.10), (4.13), and (4.14), the average relative radial velocity of the two particles is represented as 1=2 1=2 2v0 2u0 s 2 hjwr ji ¼ 1=2 1fu F(s) ¼ 1=2 fu 1fu 1 1=2 : p p 60 Rel

ð4:15Þ

Plugging Eq. (4.15) into Eq. (4.1), we arrive at the following expression for the collision kernel: 1=2 2 1=2 s ¼ 4p1=2 s2 v0 1fu F(s) b ¼ 8phw02 r i ( " !#)1=2 2 s ¼ 4p1=2 s2 u0 fu 1fu 1 1=2 : 60 Rel

ð4:16Þ

Consider several limiting cases following from Eq. (4.16). For inertialess particles (tp ¼ 0, fu ¼ 1), the formula (4.16) reduces to the kernel obtained by Saffman and Turner (1956): bST ¼

8pe 1=2 3 s : 15n

ð4:17Þ

Now, let us examine the relation (4.16) when the relaxation time belongs to the inertial interval (tk tp TL) in the limit of high Reynolds numbers (Rel ! 1). In accordance with Kolmogorovs local similarity theory (Monin and Yaglom, 1975), the Lagrangian structure function in the inertial interval is represented as D E u0i (R; t)u0i (R(tt); tt) u0j (R; t)u0j (R(tt); tt) ¼ 2 hu0i u0j iBL ij (t) ¼ C01 etd ij at tk t T L ;

SL ij (t) ¼

ð4:18Þ

4.1 Collision Frequency of Monodispersed Particles in Isotropic Turbulence

where C01 is the value of Kolmogorovs constant at Rel ! 1. The formula (4.18) leads to the following relations for the Lagrangian autocorrelation function and involvement coefﬁcients in the inertial interval: YL (t) ¼ 1

C01 et 2u0 2

at tk t T L ;

fu ¼ 1

C01 etp 2u0 2

at tk tp T L : ð4:19Þ

Substituting Eq. (4.19) into Eq. (4.16), we get the collision kernel b ¼ b1 (etp )1=2 s2 ;

b1 ¼ (8pC 01 )1=2

ð4:20Þ

corresponding to Kolmogorovs local similarity theory as applied to particle relaxation times lying within the inertial interval. In the limiting case of high-inertia particles (tp ! 1, fu ! 0), the formula (4.16) provides a transition to the kinetic collision kernel obtained by Abrahamson (1975), bA ¼ 4p1=2 v0 s2 ;

ð4:21Þ

which is similar to the collision kernel of molecules in the kinetic theory of gases. In order to determine the involvement coefﬁcient fu in Eq. (4.16), it is necessary to give the Lagrangian autocorrelation function of ﬂuid velocity along the particle trajectory YLp(t). To this end, we shall use a two-scale bi-exponential approximation similar to Eq. (1.5) and Eq. (2.186): 1 2t 2t )exp YLp (t) ¼ (1 þ Rp )exp (1R ; p V V 2Rp (1 þ Rp )T Lp (1Rp )T Lp 1=2 Rp ¼ 12z2p ;

zp ¼

tTp ; T Lp

ð4:22Þ

where tTp and TLp are the differential and integral time scales, respectively. Approximation (4.22) leads to the following dependence for the involvement coefﬁcient: fu ¼

2Wp þ z2p 2Wp þ 2W2p

þ z2p

;

Wp ¼

tp : T Lp

ð4:23Þ

At small and large values of relaxation time tp, it follows from Eq. (4.23) that lim fu ¼ 1

tp ! 0

lim f tp ! 1 u

¼

2t2p t2Tp

;

T Lp : tp

ð4:24Þ

ð4:25Þ

At high Reynolds numbers (zp ! 0 at Rel ! 1), the involvement coefﬁcient (4.23) reduces to the expression (1.105), fu ¼

1 ; 1 þ Wp

corresponding to the exponential autocorrelation function.

ð4:26Þ

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146

It is obvious that when one uses the involvement coefﬁcient (4.26) instead of (4.23), the error will be maximal for low-inertia particles whose interaction with the turbulence is characterized by the differential (rather than integral) scale of turbulence according to Eq. (4.24). For low-inertia particles, the collision kernel (4.16) with the consideration of Eq. (4.24) becomes 1=2 tp 8p e 1=2 3 30a0 St2 b¼ s 1þ ; St ¼ ð4:27Þ tk 15 n s 2 if we neglect the inﬂuence of particle inertia on the Taylor scale of turbulence (1.6). Here particle inertia is characterized by the Stokes number St equal to the ratio of particle relaxation time to the Kolmogorov time microscale. For high-inertia particles, it follows from Eq. (4.16) with the consideration of Eq. (4.25) that 1=2 T Lp b ¼ 4p1=2 s2 u0 : ð4:28Þ tp In the limit of high Reynolds numbers, the collision kernel is described by the asymptotic dependence b ¼ 4p1=2 R2 u0

W1=2 p 1 þ Wp

;

which we obtain by plugging the relation (4.26) into Eq. (4.16). On Figure 4.1, the formula (4.27) with the consideration of Eq. (1.7) for a0 is compared with the DNS results obtained by Zhou et al. (1998) for low-inertia particles. It is readily seen that Eq. (4.27) describes the DNS data with sufﬁcient accuracy, predicting an increase of the normalized collision kernel with increase of the Reynolds number and with decrease of particle diameter to the ratio of spatial microscales.

Figure 4.1 Collision kernel of low-inertia particles: 1–6 – formula (4.27); 7–12 – Zhou et al. (1998) [16]; 1, 4, 7, 10 – Rel ¼ 45; 2, 5, 8, 11 – Rel ¼ 59; 3, 6, 9, 12 – Rel ¼ 75; ¼ 1; 4–6, 10–12 – s ¼ 0:5; 13 – formula (4.17). 1–3, 7–9 – s

4.1 Collision Frequency of Monodispersed Particles in Isotropic Turbulence

Figure 4.2 The influence of inertia on the correlation coefficient of ¼ 1: 1–3 – formula (4.13); 4–6 – Wang et al. particle velocities at s (2000); 1, 4 – Rel ¼ 24; 2, 5 – Rel ¼ 45; 3, 6 – Rel ¼ 58.

Figure 4.2 plots particle velocity correlation coefﬁcient (4.13) calculated with the consideration of Eq. (4.14) and Eq. (4.23) against the ratio of particle relaxation time tp to the time macroscale of turbulence T e ¼ u0 2 =e. Integral and differential time scales are determined from Eqs. (1.4), (1.6), (1.7). The inﬂuence of particle inertia on these time scales is not taken into account. It is easy to see that the dependence of the correlation coefﬁcient on particle inertia predicted by Eq. (4.13) and the DNS results obtained by Wang et al. (2000) are in qualitative agreement: both methods state that z12 diminishes with increase of tp. Also, even though the dependence (4.13) gives smaller values for the correlation coefﬁcient than the results by Wang et al., we see that both methods concur in their predictions about the inﬂuence of the Reynolds number on the character of the dependence z12(tp/Te). Figure 4.3 compares the radial relative velocity calculated by Eq. (4.15) with the DNS results obtained by Wang et al. (2000). The dependence of hj w r ji on tp is characterized by the existence of a maximum. The initial growth of hj w r ji is due to the decrease of the correlation coefﬁcient z12 with increase of particle inertia (that is, with increase of tp). After reaching it maximum value, hj w r ji diminishes because the involvement coefﬁcient fu gets smaller as particles become less mobile and less responsive to the turbulent motion of the carrier ﬂuid. For low-inertia particles (tp/Te 1), the formula (4.15) gives exaggerated values of relative radial velocity as compared to the DNS results. Having said that, we must mention that Wang et al. (2000) have performed their calculations of particle motion for the case of frozen turbulence, when particles are injected into the ﬂow once the ﬂow has reached a statistically stationary state so that its characteristics do not change with time, as though they were frozen. However, when a particle pair is moving in forced turbulence, which means that a random force was employed to support a statistically homogeneous state, the ﬂuctuation intensity of relative motion of the two particles appears to be higher than in frozen turbulence (Fede and Simonin, 2005) and thus better conforms to the formula (4.15). Figure 4.4 shows how particle inertia affects the ratio of the kinetic energy of particles kp ¼ 3v0 2 =2 to the turbulent energy of the carrier ﬂow k ¼ 3u0 2 =2 (this ratio

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Figure 4.3 The effect of inertia on the relative velocity of the ¼ 1: 1–3 – formula (4.15); 4–6 – Wang et al. (2000); particles at s 1, 4 – Rel ¼ 24; 2, 5 – Rel ¼ 45; 3, 6 – Rel ¼ 58.

is found from Eq. (4.11) with the consideration of Eq. (4.23)) and the ratio of the collision kernel (4.16) to the kinetic collision kernel (4.21). We see a good agreement between the analytical dependences (4.11) and (4.16) and the DNS results obtained by Sundaram and Collins (1997). The ratio b/bA approaches unity at large St, illustrating the diminishing role of correlativity of particle motion as particle inertia increases. Figure 4.5 compares the ratio of the collision kernel b to the collision kernel for inertialess particles bST given by Eq. (4.17) with the DNS data by Wang et al. (2000). In spite of the fact that the theoretical model predicts exaggerated values of relative radial velocity as compared to the DNS results, by ignoring the effect of particle clustering, we get smaller maximum values for the collision kernel at St 1 than those suggested by the DNS data. Finally, note that a decision to use the single-scale involvement coefﬁcient (4.26) that follows from the exponential autocorrelation function (1.67) instead of the

Figure 4.4 The influence of particle inertia on the kinetic energy of ¼ 0:36: particles and the collision kernel at Rel ¼ 54.2 and s 1, 3 – kp/k; 2, 4 – b/bA; 1 and 2 – formulas (4.11) and (4.16); 3, 4 – Sundaram and Collins (1997).

4.2 Collision Frequency in the Case of Combined Action of Turbulence and the Average Velocity Gradient

Figure 4.5 The influence of particle inertia on the collision kernel ¼ 1: 1–3 – formula (4.16); 4–6 – Wang et al. (2000); at s 1, 4 – Rel ¼ 45; 2, 5 – Rel ¼ 58; 3, 6 – Rel ¼ 75.

two-scaled involvement coefﬁcient (4.23) that corresponds to the bi-exponential autocorrelation function (4.22) would result in a signiﬁcant error when calculating the relative velocity as well as the collision kernel of low-inertia particles.

4.2 Collision Frequency in the Case of Combined Action of Turbulence and the Average Velocity Gradient

Let us try to determine the collision frequency and collision kernel resulting from the combined effects of turbulence and the average velocity gradient of the ﬂow, in other words, we are going to consider both the ﬂuctuational and the average components of relative velocity between the particles (Alipchenkov and Zaichik, 2001). In this case, due to the presence of the ﬂuctuational and average components of relative velocity between the particles, it is necessary to obtain h|wr|i in Eq. (4.1). To this end, we should average the relative velocity over the random distributions of wr and the spatial angle characterizing orientation of the relative velocity vector w with respect to the vector r connecting the centers of colliding particles: 1 hjwr ji ¼ 4p

2ðp ð p 1 ð

jwr jP(w r )sinfdydfdw r ;

ð4:29Þ

0 0 1

where f is the polar angle between the vector r and the vertical z-axis; y is the azimuthal angle orthogonal to z in the (x, y)-plane. Let us integrate Eq. (4.29) over wr , taking note of the fact that the ﬂuctuational component of relative radial velocity is described by the Gaussian distribution (4.8): 2p # 1=2 ð ðp " 1 2hw 02 W 2r Wr r i exp þ W r erf hjw r ji ¼ 1=2 2hw 02 4p p (2hw 02 r i r i) 0 0

sinf dy df:

ð4:30Þ

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In the absence of the average velocity (Wr ¼ 0) the expression (4.30) reduces to Eq. (4.10). We can assume a shear proﬁle of average ﬂuid velocity in the vicinity of colliding particles: U ¼ (Sz,0,0). Let us also assume that the particles are completely involved in the average motion of the carrier ﬂow, in other words, V ¼ U. Then the radial component of the average relative velocity of two particles at contact will be equal to W r ¼ Ss cosy sin f cos f:

ð4:31Þ

Making use of Eq. (4.31), integrating Eq. (4.30) and substituting the result into Eq. (4.1), we obtain the collision kernel for the case of combined action of turbulence and the average velocity shear: ( " 2 2 n 1 X G(2n þ 1)G(n þ 1=2) S s n 02 1=2 2 s (1) b ¼ 8phw r i 3n þ 1 2 02 i hw 2 G (n þ 1)G(2n þ 3=2) r n¼0 G(2n þ 3)G(n þ 3=2) þ 3n þ 3 (2n þ 1)G(n þ 1)G(n þ 2)G(2n þ 7=2) 2

S2 s2 hw 02 r i

n þ 1 #) ;

ð4:32Þ

where G(x) is the gamma function. At (Ss)2 =hw 0 2r i ! 0, Eq. (4.32) gives rise to the following turbulent collision kernel: 1=2 bt ¼ 8phw 02 : r i

ð4:33Þ

At (Ss)2 =hw 0 2r i ! 1, the series (4.32) converges to the well-known Smoluchowski solution (Smoluchowski, 1917): bs ¼

4Ss3 : 3

ð4:34Þ

Figure 4.6 compares the dependence (4.32) with the results of direct numerical calculations performed by Mei and Hu (1999) for inertialess particles when the turbulent component of the collision kernel is deﬁned by the formula (4.17). As we see, Eq. (4.32) is in good agreement with the numerical results. The expression (4.32) may be approximated by the simple formula b ¼ (b2t þ b2s )1=2 ;

ð4:35Þ

where bt is the component of the collision kernel (4.33) that is due to turbulence, whereas the component bs that is due to the velocity shear is deﬁned by Eq. (4.34). It is seen from Figure 4.6 that the dependences (4.32) and (4.35) are, for all practical purposes, indistinguishable from each other. The dependences (4.32), (4.34), and (4.35) are true not only for inertialess particles, when the collision kernel is deﬁned by the Saffman–Turner formula (Saffman and Turner, 1956) where S stands for the average velocity gradient of the carrier ﬂow, but also for inertial particles, when the turbulent kernel is described by the dependence (4.16) and S is interpreted as the average velocity gradient of the disperse phase.

4.3 Particle Collisions in an Anisotropic Turbulent Flow

Figure 4.6 Collision kernel in a turbulent flow with a uniform velocity shear: 1 – Eq. (4.32), 2 – Eq. (4.35), 3 – Mei and Hu (1999).

4.3 Particle Collisions in an Anisotropic Turbulent Flow

Within the framework of the solid sphere model, the velocities of two particles after b the collision vpb, vp1 are related to the velocities of the same particles before the collision vp, vp1 by the expression b

vp ¼ vp þ

1 (1 þ e)(wp l)l; 2

1 b vp1 ¼ vp1 (1 þ e)(wp l)l; 2

ð4:36Þ

where e is the coefﬁcient of momentum restitution associated with this collision, wp vp1 vp is the relative velocity of the colliding particles, and l is the unit vector (measured at the moment of collision) directed along the line of centers and pointing toward the second particle. With collisions taken into account, the kinetic equation for the single-point (singleparticle) PDF P(x, v, t) is written as qP qP q U k vk 1 qhu0k pi qP þ þ Fk P ¼ þ : ð4:37Þ þ vk tp qt qx k qvk tp qvk qt coll The ﬁrst term on the right-hand side of Eq. (4.37) describes the interaction of particles with turbulent eddies, whereas the second term is associated with the contribution of particle collisions. If the duration of particle interaction with energycarrying turbulent eddies TLp is much less than the characteristic time lapse between two successive collisions tc, the effect of collisions on the particle–turbulence interaction may be neglected. In this case, when modeling the velocity ﬁeld of the continuum by a Gaussian process, the correlation hu0 i pi between ﬂuid velocity ﬂuctuations and the probability density of particle velocity is deﬁned by Eq. (2.11) with the consideration of Eqs. (2.16)–(2.17). When the condition TLp tc is violated, the correlation hu0i pi will be, as previously, deﬁned by the expressions (2.11), (2.16), and (2.17), and the effect of collisions, which becomes noticeable at these values of

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TLp, will be accounted for by using the isotropic homogeneous turbulence approximation as we did in Section 1.3. When the particles are considered as solid spheres – the assumption that is baked into Eq. (4.36), the collision operator can be represented as the Boltzmann integral (Chapman and Cowling, 1970): ðð qP b b P(x; v ; x þ sl; v1 ; t)P(x; v; x þ sl; v1 ; t) (w l)dl dv1 ; ¼ s2 qt coll wl<0

ð4:38Þ where the inequality wl < 0 is telling us that integration is performed over such l and v1 for which a collision of particles can take place. The relation (4.38) contains a two-particle PDF that is deﬁned as follows: P(x; v; x1 ; v1 ; t) ¼ hd(xRp (t))d(vvp (t))d(x1 Rp1 (t))d(v1 vp1 (t))i: From Eq. (4.37) complemented by Eq. (2.11) and Eq. (4.38) there follows a chain of equations for single-point statistical moments of the single-particle PDF. Breaking this chain on the level of third moments and exploiting the quasi-normal hypothesis (2.29), we obtain the following system of equations: qF qFV k ¼ 0; þ qx k qt

ð4:39Þ

Dp ik q ln F qhv0 v0 i U i V i qV i qV i þ Vk ¼ i k þ þ Fi ; qx k qt qx k tp tp qx k

ð4:40Þ

qhv0i v0j i qt

þ Vk

qhv0i v0j i qx k

þ

0 0 0 qV j 0 0 qV i 1 qFhvi vj vk i ¼ hv0i v0k i þ mik hvj vk i þ mjk qx k qx k qx k F

þ lij þ lji

qhv0i v0j v0k i qt

þ Vn

qhv0i v0j v0k i qx n

¼ hv0i v0j v0n i

tp

þ C ij :

ð4:41Þ

qV j qV k qV i hv0i v0k v0n i hv0j v0k v0n i qxn qx n qx n 0 0

2hv0i v0j i

0 0

Dp in qhvj vk i Dp jn qhv0i v0k i Dp kn qhvi vj i tp qx n tp qx n tp qx n 3hv0i v0j v0k i tp

þ C ijk :

ð4:42Þ

One can see that collisions do not make any contribution to the equations (4.39) and (4.40) because the mass and the momentum of a particle system are conserved during a collision. The quantities C ij and C ijk denote the contribution of collisions to

4.3 Particle Collisions in an Anisotropic Turbulent Flow

the balance equations for the second and third moments of velocity ﬂuctuations of the disperse phase. As previously, the turbulent diffusion tensor Dp ij entering Eq. (4.40) and Eq. (4.42) is deﬁned by the expression (2.26). At small volume concentrations of particles (F 0.01), it is safe to ignore the direct contribution of inter-particle collisions to the stress and the ﬂux of ﬂuctuational energy of the disperse phase. It means that collisional terms enter the system of equations for the moments only as sources and never as ﬂuxes. Hence the collisional terms in the equations (4.41) and (4.42) are represented as (Jenkins and Richman, 1985) ð ððð 1 0 0 qP 6 C ij ¼ dv ¼ (w l)P(x; v; x þ sl; v1 ; t)½½v0i v0j dl dv dv1 ; vi vj F qt coll psF wl<0

ð4:43Þ ð 1 0 0 0 qP vi vj vk F qt coll

C ijk ¼ dv ¼

6 psF

ððð

(w l)P(x; v; x þ sl; v1 ; t)½½v0i v0j v0k dl dv dv1 ;

ð4:44Þ

wl<0

where ½½y ¼

y b þ y1byy1 : 2

In accordance with the particle collision model (3.53), we have ½½V i ¼ 0;

ð4:45Þ

1 ½½v0i v0j ¼ (1 þ e)(w0 l) (1 þ e)(w0 l)li lj (w 0i lj þ w 0i lj ) ; 4

ð4:46Þ

1 ½½v0i v0j v0k ¼ (1 þ e)(w0 l) li (v0j v0k v01j v01k ) þ lj (v0i v0k v01i v01k ) 4 (1 þ e)2 þ lk (v0i v0j v01i v01j ) þ 8 (w0 l)2 li lj (v0k þ v01k ) þ li lk (v0j þ v01j ) þ lj lk (v0i þ v01i ) :

ð4:47Þ

The equality (4.45) expresses the conservation of momentum during particle collisions. As follows from Eqs. (4.43) to (4.44), in order to determine the collisional terms, it is necessary to obtain the PDF of velocities of the two particles at the moment of collision. To this end, we shall assume non-correlativity (statistical independence) of particle motions by analogy with the molecular chaos hypothesis in the kinetic theory of gases. As a result of this assumption the two-particle PDF is represented as a product of single-particle PDFs. The resulting expressions that describe particle collisions in a turbulent ﬂow turn out to be similar to the corresponding relations

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154

encountered in the kinetic theory of gases (Abrahamson, 1975; Louge et al., 1991; Simonin, 1991a; Sommerfeld and Zivkovic, 1992; Zaichik and Pershukov, 1995). However, such an approach is valid only for high-inertia particles whose relaxation time by far exceeds the integral time scale of turbulence and whose relative motion is non-correlated and similar to the chaotic motion of molecules. Therefore in order to describe the collisions of particles having arbitrary inertia, we proceed in the same manner as in Section 4.1, more speciﬁcally, we base our derivation of collision frequency in isotropic turbulence upon the assumption of correlated Gaussian distribution of particle velocities (Lavieville et al., 1995; Lavieville, 1997). This approach, coupled with Grads expansion, enables us to obtain explicit expressions for the collisional terms while taking into account the correlativity of particle motion. The correlated Gaussian PDF of two-particle velocities in isotropic turbulence is described by the distribution (4.6). If particle size does not exceed the Kolmogorov spatial microscale, the longitudinal and transverse components of the Eulerian twopoint correlation function of ﬂuid velocity ﬂuctuations over the distance equal to the collision sphere radius s are deﬁned kas FðsÞ ¼ 1

s 2 ; 60 Rel 1=2

GðsÞ1

s 2 1=2

15

Rel

:

ð4:48Þ

The discussion below will focus on the case of small-particle collisions at high Reynolds numbers. Then the correlation functions (4.48) can be taken equal to unity, as a consequence of which the distribution (4.6) is simpliﬁed and with the consideration of Eq. (4.12) takes the form ! v0k v0k þ v01k v01k 2fu v0k v01k F2 P (v; v1 ) ¼ ; exp (2pv0 2 )3 (1fu2 )3=2 2v0 2 (1fu2 ) ð0Þ

ð4:49Þ

where the superscript (0) shows that the Gaussian PDF is the zeroth term of the expansion describing the distribution of particle velocities in isotropic turbulence. In order to take into account the anisotropy of particle velocities, we employ Grads method to represent the two-point PDF as an expansion over Hermitian polynomials (Zaichik and Alipchenkov, 1997; Alipchenkov and Zaichik, 1998b). As compared to other basis functions, Hermitian polynomials are unique in that the resulting expansion coefﬁcients can be expressed through PDF moments whose orders do not exceed the order of the expansion term under consideration. The two-particle PDF is thus represented as 2 Rij q q q q2 P(v; v1 ) ¼ 1 þ Ri þ þ þ 2! qvi qvj qv1i qv1j qvi qv1i 2 2 Rijk Qij q q q3 q3 þ þ þ þ 2! qvi qv1j qv1i qvj 3! qvi qvj qvk qv1i qv1j qv1k

4.3 Particle Collisions in an Anisotropic Turbulent Flow

þ

Qijk 3!

þ

q3 q3 q3 þ þ qv1i qv1j qvk qvi qv1j qv1k qv1i qvj qv1k

q3 q3 q3 þ þ qvi qvj qv1k qvi qv1j qvk qv1i qvj qvk Pð0Þ (v; v1 );

ð4:50Þ

where P(0)(v, v1) is deﬁned by Eq. (4.49). It is worth noting that Lavieville (1997) and Lavieville et al. (1997) have tried an approach that is somewhat more general than the expansion (4.50) in their effort to determine P(v, v1) in anisotropic turbulence. Their approach is based on constructing Grads expansion for the joint PDF of ﬂuid and particle velocities P(u, v) with subsequent determination of P(v, v1) through an integration procedure analogous to the one in Eq. (4.6). In this way they directly took into account the contribution of velocity correlations of both phases to P(v, v1) in addition to the contribution of anisotropy of particle velocities. However, P(v, v1) obtained by this means includes only quadratic terms of the expansion over particle velocity, which makes it possible to obtain C ij . The inclusion of cubic terms that are needed to determine C ijk is a very cumbersome procedure. Therefore in order to obtain reasonably simple analytical expressions for C ij as well as for C ijk we shall use the expansion (4.50) as our starting point. The coefﬁcients Ri , Rij , Rijk in Eq. (4.50) are found from the conditions 1 F2 1 F2

ðð ðð

v0i P(v; v1 )dvdv1

¼

hv0i i

1 F2

¼ 0;

ðð

v0i v0j P(v; v1 )dvdv1 ¼ hv0i v0j i;

v0i v0j v0k P(v; v1 )dvdv1 ¼ hv0i v0j v0k i:

These equations together with Eq. (4.50) give us Ri ¼ 0;

Rij ¼ hv0i v0j iv0 dij ; 2

Rijk ¼ hv0i v0j v0k i:

ð4:51Þ

Since spatial correlation functions are taken to be equal to unity (F(s) ¼ G(s) ¼ 1), the correlation coefﬁcient of velocities of two particles (4.13) takes the form z12 ¼ fu :

ð4:52Þ

Due to Eq. (4.52) we should take Qij ¼ fu Rij ;

Qijk ¼ fu Rijk ;

which yields hv0i v01j i ¼ fu hv0i v0j i;

hv0i v0j v01k i ¼ hv0i v01j v0k i ¼ hv01i v0j v0k i ¼ fu hv0i v0j v0k i:

ð4:53Þ

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In view of Eqs. (4.49), (4.51), and (4.53), the two-particle PDF (4.50) is equal to P(v; v1 ) ¼

0 0 vk vk þ v01k v01k 2fu v0k v01k F2 exp (2pv0 2 )3 (1fu2 )3=2 2v0 2 1fu2 hv0i v0j iv0 2 dij 0 0 1þ vi vj þ v01i v01j fu (v0i v01j þ v01i v0j ) 2v0 4 (1fu2 ) þ

hv0i v0j v0k i

6v0 6 (1fu2 )3

(1fu )2 (1 þ 2fu )(v0i v0j v0k þ v01i v01j v01k )

fu2 (1fu )2 (v0i v0j v01k þ v0i v01j v0k þ v01i v0j v0k þ v01i v01j v0k

(1 þ 2fu )hv0i v0k v0k i 0 0 þ v0i v01j v01k þ v01i v0j v01k ) (v þ v ) : i 1i 2v0 4 (1 þ fu )2

ð4:54Þ

To make the calculations below less cumbersome, let us introduce the velocities q¼

v1 þ v ; 2

w ¼ v1 v;

ð4:55Þ

which characterize, respectively, the combined motion of the particle pair as a whole, and the relative motion of the two particles. Then, in these new variables, the PDF (4.54) will be written as P(q; w) ¼

q0k q0k w 0k w 0k F2 exp v0 2 (1 þ fu ) 4v0 2 (1fu ) (2pv0 2 )3 (1fu2 )3=2 hv0i v0j iv0 2 dij 4(1fu )q0i q0j þ (1 þ fu )w 0i w 0j 1þ 2 4 0 4v (1fu ) þ

hv0i v0j v0k i 12v0 6 (1fu2 )3

4(1fu )3 (1 þ 3fu )q0i q0j q0k þ (1fu2 )2

(1 þ 2fu )hv0i v0k v0k i 0 (q0i w0j w 0k þ q0j w0i w 0k þ q0k w 0i w 0j ) q i : v0 4 (1 þ fu )2

ð4:56Þ

Making a transition to the variables (4.55) in the equations (4.43) and (4.44) and integrating them while taking into account Eqs. (4.46), (4.47), (4.56), we arrive at the following expression for the collisional terms: 2(1e2 )(1fu )kp 2(1fu ) 2 0 0 C ij ¼ d ij hvi vj i kp dij ; 9tc tc1 3 C ijk ¼

3(1fu ) 0 0 0 hvi vj vk iE hv0i v0n v0n id jk þ hv0j v0n v0n idik þ hv0k v0n v0n idij ; tc1

ð4:57Þ

ð4:58Þ

4.3 Particle Collisions in an Anisotropic Turbulent Flow

1=2 1=2 s 2p 5s 2p ; t ¼ c1 1=2 3k 1=2 3k 16F(1fu ) 8(1 þ e)(3e)F(1fu ) p p 10tc ; ¼ (1 þ e)(3e) 1 þ 3e E¼ ; ð4:59Þ 18(3e) where tc, tc1 are the characteristic times between particle collisions, and kp is the turbulent kinetic energy of particles. Notice that the formula for tc coincides with the 2 =Rel 1 because the time between expression that follows from Eq. (4.15) at s collisions tc is inversely proportional to the collision frequency wc: tc ¼

1 tc ¼ w1 c ¼ (bN) :

As we can see from Eqs. (4.41), (4.42) and Eqs. (4.57), (4.58), the contribution of terms that describe the interaction of particles with turbulent eddies and the interparticle interaction via collisions is characterized by the ratio of the two times tc/tp. At tc/tp 1 the role of collisions is negligible, whereas at tc/tp 1 collisions play a decisive role in the development of the statistics of particle velocity ﬁeld. According to Eq. (4.57), the effect of collisions on turbulent stresses of the disperse phase manifests itself in two phenomena: ﬁrst, in dissipation of particle velocity ﬂuctuations via inelastic collisions and second, in redistribution of ﬂuctuational energy between different components. The latter effect is proportional to the degree of anisotropy of particle velocity ﬂuctuations and is responsible for the tendency of hv0i v0j i to approach an isotropic state. This effect is similar to the effect of pressure ﬂuctuations on turbulent stresses in the ﬂuid (Rotta,1951). Lavieville (1997) and Lavieville et al. (1997) have obtained an expression for the collisional term C ij for the case of elastic collisions (e ¼ 1), taking into account not only the anisotropy of particle velocities but also the anisotropy of ﬂuid velocities and the correlations of velocities of both phases: 2 2 2 hv0i v0j i kp dij fu hu0i v0j i þ hu0j v0i i hu0k v0k id ij C ij ¼ tc1 3 3 2 þ fu2 hu0i u0j i kd ij : ð4:60Þ 3 Note that if we assume local equilibrium relations between second moments of the disperse and continuous phases, hv0i v0j i ¼ hu0i v0j i ¼ fu hu0i u0j i; then the relation (4.60) will reduce to the second term in Eq. (4.57). In the framework of Grads 13-moment approximation, we have the following relation between triple correlations and their convolutions (Jenkins and Richman, 1985): 1 hv0i v0j v0k i ¼ hv0i v0n v0n id jk þ hv0j v0n v0n id ik þ hv0k v0n v0n idij : ð4:61Þ 5

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In view of Eq. (4.61), the expression (4.58) takes the form C ijk ¼ tc2 ¼

3(1fu )(15E) 0 0 0 3(1fu ) 0 0 0 hvi vj vk i ¼ hvi vj vk i; tc1 tc2

180tc : (1 þ e)(4933e)

ð4:62Þ

Equations (4.58) and (4.62) show that collisions cause a decrease in triple correlations of velocity ﬂuctuations in the disperse phase. At fu ! 0, the formulas (4.57), (4.58), and (4.62) reduce to the relations for the collisional terms obtained by Jenkins and Richman (1985) for non-correlated chaotic motion of particles. As the involvement coefﬁcient fu gets larger (i.e., as particle inertia gets smaller), correlativity of particle motion grows, which reduces the inﬂuence of collisions on the disperse phases turbulence. Sakiz and Simonin (2001) have obtained an analytical expression for the collisional term C ij for a model of collision that is more general than the one associated with Eq. (4.36). This model is true for collisions in the absence of slip. In addition to the normal restitution coefﬁcient of momentum e, it includes the tangential restitution coefﬁcient, while also taking into account the rotation of colliding particles. If we neglect the terms in Eq. (4.42) that describe time evolution, convection, and generation of third moments by the gradients of the average velocity, we will get the following algebraic gradient relations for the third moments of velocity ﬂuctuations: 1 qhv0j v0k i 1 þ (1fu )tp =tc2 qhv0i v0k i 0 0 0 hvi vj vk i ¼ þ Dp jn Dp in qx n qx n 3 0 0 qhvi vj i : ð4:63Þ þ Dp kn qx n When collisions play a minor role, that is, when particle relaxation time is much shorter than the time interval between collisions (tp/tc2 1), the relation (4.63) reduces to Eq. (2.31). The formula (4.63) for the triple moments of velocity ﬂuctuations in the absence of correlativity of particle motions (fu ¼ 0) was ﬁrst obtained by Simonin (1991a, 1991b) for elastic collisions (e ¼ 1) and later generalized for the case of inelastic collisions by Simonin (1996). According to Eq. (4.63), as particle inertia gets smaller and correspondingly, the correlativity of particle motion increases, the effect of collisions on the diffusive transport of velocity ﬂuctuations decreases. It follows from Eq. (4.63) that if we dont take into account the variation of second moments hv0i v0j i, collisions should lead to a decrease of triple correlations and consequently, to a reduced role of the diffusive mechanism in the transport of particle velocity ﬂuctuations. However, as we take into account the inﬂuence of collisions on hv0i v0j i, the above-mentioned phenomenon turns to its opposite: as shown in the experiment carried out by Caraman et al. (2003), collisions bring about an increase of the triple moments and thereby enhance turbulent transport of velocity ﬂuctuations in the disperse phase. In a highly concentrated disperse medium

4.4 Boundary Conditions for the Disperse Phase with the Consideration of Particle Collisions

(F 0.1), particle collisions also enhance turbulent diffusive transport by making a direct contribution to the ﬂuxes of velocity ﬂuctuations (Zaichik, 1992a; Zaichik and Pershukov, 1995).

4.4 Boundary Conditions for the Disperse Phase with the Consideration of Particle Collisions

As was noted in Section 2.4, when deriving boundary conditions for the disperse phase we may limit our attention to sufﬁciently inertial particles whose relaxation time by far exceeds the duration of particle interaction with energy-carrying eddies, since the opposite limit – that of low-inertia particles – is of no interest because sticking conditions apply in this case. At tp TLp, it is possible to ignore the correlativity of particle motions when considering the effect of inter-particle collisions on the boundary conditions for the disperse phase. However, since inclusion of this phenomenon does not really complicate the analysis, the inﬂuence of correlativity of particle motions on the collisional terms will be taken into account. The kinetic equation for high-inertia particles (2.81) in the presence of collisions is written as qP qP q þ þ vk qt qx k qvk

T Lp U k vk q2 P qP þ F k P ¼ 2 hu0i u0j i þ : tp tp qvi qvk qt coll ð4:64Þ

The solution of Eq. (4.64) can be presented in the form of Grads expansion (2.111), where P(0)(v) is the equilibrium Gaussian distribution (2.90). The boundary conditions are derived in the same way as in Section 2.4 – by integrating Eq. (2.105) with the consideration of Eq. (2.111). The normal component of velocity at the wall determined in this way has the form (2.103), just as before. In other words, collisions do not exert any inﬂuence on the boundary condition for Vy . The expressions (2.112)–(2.115) for the longitudinal velocity and for the normal components of velocity ﬂuctuation intensities at the wall also will not change. With the consideration of the collisional term (4.57) instead of Eq. (2.96), the relations for the tangential stress will be as follows: hv0x v0y i ¼ tp1 ¼

tp1 hv02 y i dV x 2

dy

tp tc1 : tp (1fu ) þ tc1

;

ð4:65Þ

ð4:66Þ

Insertion of Eq. (4.65) into Eq. (2.112) results in the boundary condition for the longitudinal velocity Vx:

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160

tp1

!1=2 dV x 1cex 1c 2 Vx ¼2 dy 1 þ cex 1 þ c phv02 y i

ð4:67Þ

It follows from Eq. (4.63) that in the current approximation, the third moments of particle velocity ﬂuctuations with the consideration of collisions are determined by the expressions 0 hv02 x vy i ¼

02 tp2 hv02 y i dhvx i

hv0y v02 zi¼ tp2 ¼

dy

3

02 tp2 hv02 y i dhvz i

3

dy

02 hv03 y i ¼ tp2 hvy i

;

dhv02 y i dy

;

tp tc2 tp (1fu ) þ tc2

; ð4:68Þ ð4:69Þ

rather than by Eq. (2.97). Substitution of Eq. (4.68) into Eqs. (2.113)–(2.115) results in the following boundary condition for the diagonal components of turbulent stresses in the disperse phase: tp2

tp2

tp2

!1=2 dhv02 1ce2x 1c 2 xi hv02 ¼3 x i; 1 þ ce2x 1 þ c dy phv02 y i dhv02 y i dy

¼

! !1=2 2hv02 1c y i ; 2 1 þ ce2y 1 þ c p 1ce2y

!1=2 dhv02 1ce2z 1c 2 zi hv02 ¼3 z i: 1 þ ce2z 1 þ c dy phv02 y i

ð4:70Þ

ð4:71Þ

ð4:72Þ

As we see from Eq. (4.67) and Eqs. (4.70)–(4.72), the role of collisions boils down to the replacement of tp in the boundary conditions by the effective relaxation times (4.66) and (4.69) whose values get smaller as collision frequency increases. Particle collisions have no effect on the boundary condition for the tangential stress component, and therefore the condition (2.118) still holds.

4.5 The Effect of Particle Collisions on Turbulent Stresses in a Homogeneous Shear Flow

Consider the effect of particle collisions on turbulent stresses of the disperse phase in a homogeneous shear ﬂow. The behavior of turbulent stresses in the absence of collisions was discussed in Section 2.5 and described by Eq. (2.137). With the consideration of the collisional terms (4.57), the system of equations for non-zero

4.5 The Effect of Particle Collisions on Turbulent Stresses in a Homogeneous Shear Flow

components hv0i v0j i in a homogeneous layer with the velocity gradients given by Eq. (2.136) is represented as dhv02 0 0 1i ¼ 2S hv01 v02 i þ hu02 1 if u1 12 þ hu1 u2 if u1 22 dt 2 02 þ hu1 ifu 11 þ hu01 u02 ifu 21 hv02 1i ; tp 02 02 dhu1 i dhu01 u02 i dhu1 i dhu01 u02 i fu1 11 þ fu1 21 þ tp S fu2 12 þ fu2 22 dt dt dt dt 2 02 2(1e )(1fu )kp 2(1fu ) 2 dhv2 i 2 0 0 hu1 u2 ifu12 ¼ hv02 1 i kp ; 9tc tc1 3 dt tp dhu01 u02 i dhu02 2i 02 þ hu02 if hv i þ f f 2 u 22 2 u1 12 dt dt u1 22 2(1e2 )(1fu )kp 2(1fu ) 2 i k hv02 p ; 2 9tc tc1 3 dhu02 i 2(1e2 )(1fu )kp dhv02 2 02 3i 3 hu3 ifu 33 hv02 ¼ fu1 33 3i 9tc dt tp dt 2(1fu ) 2 hv02 3 i kp ; tc1 3 dhv01 v02 i 0 0 02 ¼ S hv02 2 i þ hu1 u2 if u1 12 þ hu2 ifu1 22 dt 1 02 0 0 hu1 ifu 12 þ hu01 u02 i fu 11 þ fu 22 þ hu02 2 if u 21 2hv1 v2 i tp dhu02 1 dhu02 dhu01 u02 i 1i 2i þ f fu1 11 þ fu1 22 þ f 2 dt u1 12 dt dt u2 21 tp S dhu01 u02 i 2(1fu ) 0 0 dhu02 2i þ hv1 v2 i: fu2 12 þ fu2 22 tc1 dt dt 2 þ

ð4:73Þ

The discussion below will focus on an example of a ﬂow in which the contribution of the average velocity gradient to the collision kernel bs is negligible compared to the turbulent collision kernel bt. In this case characteristic times between collisions in Eq. (4.73) are deﬁned in accordance with Eq. (4.59) as tc ¼

dp 16F(1fu )1=2

2p 3kp

1=2 ; tc1 ¼

10tc : (1 þ e)(3e)

By analogy with Eq. (2.142), the tensor of particle–turbulence interaction time is determined by the expression

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TLp ¼ F(St )TL ;

ð4:74Þ

where TL is given by Eq. (2.140) and the function F(St) takes into account the contribution of collisions and in accordance with Eq. (1.77) is equal to TE St 0:9mSt2 T E 3(1 þ m)2 F(St ) ¼ 1 þ ; 1 ¼ ; 2 TL 1 þ St (1 þ St ) (2 þ St ) TL 3 þ 2m St ¼ (St5E þ 10 St5C )1=5 :

ð4:75Þ

The involvement coefﬁcient fu entering the collisional terms is taken in the isotropic approximation (1.105), where TLp ¼ TLp kk/3 is the average diagonal time scale. Particle collisions do not have any direct effect on the correlation moments of velocity ﬂuctuations in either the continuous or the disperse phase. Therefore the components hu0i v0j i are, as before, described by the system of equations (2.138). However, collisions can inﬂuence hu0i v0j i indirectly by affecting the duration of particle interaction with turbulent eddies. This mechanism, whereby particle collisions inﬂuence the quantity hu0i v0j i, is presumably described by the relations (4.74) and (4.75). As an example, consider the time evolution of a uniform shear ﬂow whose turbulent characteristics have been obtained by Lagrangian simulation of separate particle motions using the LES method (Moreau et al., 2004). The initial conditions correspond to isotropic turbulence. The average velocity gradient is 50 s1, particle diameter is dp ¼ 656 mm, the density ratio of the disperse and continuous phases is rp/rf ¼ 85.5, the volume concentration of particles is F ¼ 0.0125. The force of gravity is absent, and particle collisions are assumed to be elastic (e ¼ 1). The structure parameter of turbulence m is taken equal to unity. Figure 4.7 shows the time evolution of turbulent stresses in the disperse phase, and Figure 4.8 the time evolution of correlation moments of velocity ﬂuctuations in

Figure 4.7 The effect of collisions on turbulent stresses of the disperse phase in a homogeneous shear layer: 1 and 2 – solutions of Eq. (4.73); symbols – Moreau et al. (2004).

4.5 The Effect of Particle Collisions on Turbulent Stresses in a Homogeneous Shear Flow

Figure 4.8 The effect of collisions on the correlation moments of velocity fluctuations of the continuous and disperse phases in a homogeneous shear layer: 1 and 2 – solution of Eq. (2.138); symbols – Moreau et al. (2004).

the continuous and disperse phases under the action of the rate of shear. Both ﬁgures trace how the system evolves from the isotropic to the equilibrium state. The results obtained by solving the equations (4.73) and (2.138) in the presence and in the absence of particle collisions are depicted by curves 1 and 2. The data obtained by direct numerical simulation in the presence and in the absence of particle collisions are depicted by solid and open symbols, respectively. Turbulent stresses of the disperse phase hv0i v0j i on Figure 4.7 are normalized by the initial kinetic energy of particles kp(0), and correlations of velocity of the continuous and disperse phases hu0i v0j i on Figure 4.8 – by the initial kinetic energy of these correlations kfp(0). Figure 4.7 reveals strong anisotropy of particle velocity ﬂuctuations in the absence of collisions. This anisotropy manifests itself in signiﬁcantly higher intensity of the longitudinal component as compared to the transverse components. There are in this case two mechanisms responsible for the generation of the longitudinal component of velocity of the disperse phase: via the average velocity gradient and via the interaction of particles with the turbulent eddies of the continuum. On the other hand, ﬂuctuations in the transverse directions are generated only via the interaction of particles with the turbulent ﬂuid. Collisions tend to equalize the intensities of particle ﬂuctuations in different directions, decreasing the intensity of the longitudinal component of velocity ﬂuctuations and increasing the intensities of the transverse components; in other words, the net effect of collisions is the isotropization of turbulent characteristics of the disperse phase. Notice the opposite effect of 0 0 collisions on the longitudinal components hv02 1 i and hu1 v1 i. As one can see from Figure 4.8, the theoretical model based on the equations (2.138) with the consideration of the relations (4.74) and (4.75) concurs with the numerical simulation data, predicting an increase of hu01 v01 i due to collisions (although this increase is less pronounced the one obtained by Moreau et al., 2004). The effect of collisions on hu0i v0j i manifests itself through the dependence of the duration of particle interactions with the turbulence on the time interval between particle collisions tc. A comparison

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with the data obtained by Moreau et al. (2004) shows that the above-described model reproduces all the essential features of the dependence of the character of disperse phases velocity ﬂuctuations as well as correlations of velocity between the two phases on particle collisions.

4.6 The Effect of Collisions on Particle Motion in a Vertical Channel

There is a large number of papers devoted to theoretical and numerical research of the inﬂuence of particle collisions on statistical characteristics of the disperse phase in turbulent ﬂows in vertical and horizontal channels and pipes (see, for example, Louge et al., 1991; Tanaka and Tsuji, 1991; Oesterle and Petitjean, 1991, 1993; He and Simonin, 1993; Sommerfeld, 1995, 2003; Zaichik and Pershukov, 1995; Lun and Liu, 1997; Chen et al., 1998; Fohanno and Oesterle, 2000; Lun, 2000; Li et al., 2001; Sakiz and Simonin, 2001; Wassen and Frank, 2001; Yamamoto et al., 2001; Sommerfeld and Kussin, 2003; Kartushinskii et al., 2004; Rani et al., 2004; Vance et al., 2006). Particle collisions can be taken into account by running a direct simulation of collisions or they can be incorporated into a statistical model. As conﬁrmed by the results of numerical and experimental studies (Simonin et al., 1997; Fukagata et al., 2001; Caraman et al., 2003), the inﬂuence of collisions becomes signiﬁcant even for very small volume concentrations of particles (F < 104). The present section examines the effect of particle collisions on the characteristics of particle motion in a channel based of the statistical model discussed in Section 4.3. Consider the motion of particles in a vertical planar channel at a sufﬁcient distance from the inlet. The mathematical statement of the problem will be the same as in Section 2.7. The system of equations for the ﬁrst and second moments of particle velocity that also factors in the collisional terms (4.57) and the relations for third moments (4.63) is given below: d ln F V 1 U 1 dhv01 v02 i 0 0 þgþ þ hv1 v2 i þ g nu hu01 u02 i ¼ 0; tp dy dy

ð4:76Þ

d ln F dhv02 n 02 2i hv02 ¼ 0; 2 i þ g u hu2 i dy dy

ð4:77Þ

dhv02 dhv01 v02 i 0 0 1 d n 02 n 0 0 1i i þ g hu i v i þ g hu u i Ftp2 hv02 þ 2 hv 2 1 2 u 2 u 1 2 3F dy dy dy dV 1 dU 1 2 l 02 þ 2lnu hu01 u02 i þ 2 hv01 v02 i þ g nu hu01 u02 i f hu ihv02 1i dy dy tp u 1 2(1e2 )(1fu )kp 2(1fu ) 2 02 hv2 i kp ¼ 0; ð4:78Þ 9tc tc1 3

4.6 The Effect of Collisions on Particle Motion in a Vertical Channel

dhv02 1 d 2 n 02 n 02 2i fu hu2 ihv02 i þ g hu i Ftp2 hv02 þ 2 2i u 2 F dy dy tp 2(1e2 )(1fu )kp 2(1fu ) 2 i hv02 ¼ 0; k p 2 9tc tc1 3

j165

ð4:79Þ

02 dhv02 1 d 2 n 02 n 02 3i f hu ihv02 Ftp2 hv2 i þ g u hu2 i þ 3i 3F dy dy tp u 3

2(1e2 )(1fu )kp 2(1fu ) 2 i hv02 ¼ 0; k p 3 9tc tc1 3

ð4:80Þ

dhv01 v02 i 0 0 dhv02 02 1 d n 02 n 0 0 2i Ftp2 2 hv2 i þ g u hu2 i þ hv1 v2 i þ g u hu1 u2 i 3F dy dy dy l dV 1 f þ f n hu01 u02 i2hv01 v02 i dU 1 n 02 hv02 þ lnu hu02 þ u u 2i 2 i þ g u hu2 i dy dy tp

2(1fu ) 0 0 hv1 v2 i ¼ 0: tc1

ð4:81Þ

Remember that collisions do not make any contribution to the momentum equations (4.76) and (4.77), and are present only in the equations (4.78)–(4.81) expressing the balance of stresses. Therefore, as a result of collisions, all components of turbulent stresses become interconnected. As we mentioned at the end of Section 2.7, it is quite safe to neglect the contribution of the transport term to the diffusive transport of turbulent ﬂuctuations in the channel. Therefore for the sake of simplicity the terms containing Dp hu0i u0k i=Dt are not included in the equations (4.78)–(4.81). The boundary conditions for stresses at the channel wall are given in accordance with Eqs. (4.70)–(4.72) with the consideration of c ¼ 1 and ez ¼ 1 as well as Eq. (2.118): tp2

dhv02 1e2x 1i ¼3 1 þ e2x dy

tp2

1e2y dhv02 2i ¼2 1 þ e2y dy

hv01 v02 i ¼ mx hv02 2i

2 phv02 2i 2hv02 2i p

1=2

hv02 1 i;

1=2 ;

dhv02 3i ¼ 0; dy

at y ¼ 0:

ð4:82Þ

Just as before, we require that symmetry conditions should hold at the channel axis: dF dhv02 dhv02 dhv02 1i 2i 3i ¼ ¼ ¼ ¼ hv01 v02 i ¼ 0 dy dy dy dy

at y ¼ R:

j 4 Collisions of Particles in a Turbulent Flow

166

Figure 4.9 Distribution of the average axial velocity of particles over the channel cross-section in the absence of collisions (a) and with collisions (b): 1–3 – solutions of Eqs. (4.76)–(4.81); 4–6 – Vance et al. (2006); 1, 4 – t þ ¼ 29; 2, 5 – t þ ¼ 117; 3, 6 – t þ ¼ 468.

The involvement coefﬁcients that appear in Eqs. (4.76)–(4.81) are deﬁned by the relations (2.187). The integral time scales of particle interaction with turbulent eddies are determined by the expressions (2.188), where in order to take into account the effect of collisions, the function f(StE) should be replaced with f (St) in accordance with Eq. (1.77). The Lagrangian integral time scale is described by the approximation (2.189). The involvement coefﬁcient fu in the collisional terms is taken in the isotropic approximation given by Eq. (4.23) with the Lagrangian integral time scale given by the arithmetic average T Lp ¼ (T lLp þ 2T nLp )=3. The effective relaxation time tp2 is given by the relation (4.69). Figures 4.9–4.11 compare the solution of equations (4.76)–(4.81) with the data obtained by Lagrangian trajectory modeling of particle dynamics in a turbulent ﬂow using the LES method (Vance et al., 2006). The calculations in this paper were

Figure 4.10 Distribution of intensity of transverse velocity fluctuations over the channel cross-section in the absence of collisions (a) and with collisions (b): 1–3 – solutions of Eqs. (4.76)–(4.81); 4–6 – Vance et al. (2006); 1, 4 – t þ ¼ 29; 2, 5 – t þ ¼ 117; 3, 6 – t þ ¼ 468.

4.6 The Effect of Collisions on Particle Motion in a Vertical Channel

Figure 4.11 Distribution of particle concentration over the channel cross-section in the absence of collisions (a) and with collisions (b): 1–3 – solutions of Eqs. (4.76)–(4.81); 4–6 – Vance et al. (2006); 1, 4 – t þ ¼ 29; 2, 5 – t þ ¼ 117; 3, 6 – t þ ¼ 468.

carried out for a ﬂow in a planner channel at the Reynolds number Re þ ¼ 180, in the absence of gravity and assuming no feedback action of the disperse phase on the carrier continuum. Collisions of particles with channel walls and with each other were assumed to be elastic. Accordingly, the values ex ¼ ey ¼ 1, mx ¼ 0 were plugged into the boundary conditions (4.82); also, the restitution coefﬁcient e associated with collisions was taken equal to 1. The calculations were preformed for a ﬁxed volume concentration of particles in the channel Fm ¼ 2.3104. The dimensionless size of particles d þ dpu/n was taken equal to 1 and the relaxation time tp was allowed to vary as the density ratio between the disperse and continuous phases rp/rf changed. As follows from Eq. (4.76), collisions strongly inﬂuence the proﬁle of the average longitudinal velocity, making the proﬁle more ﬂat and causing a considerable increase of V1 in the near-wall region. This role of collisions is explained by the strengthening of transport mechanisms, which causes an increase of tangential turbulent stresses in the disperse phase. We must mention the essential discrepancy between the theoretical results and the data V1 obtained by Vance et al. (2006), especially for the case of no collisions (Figure 4.9a). Thus, according to Vance et al., the axial velocity near the wall (y þ 1) in the absence of collisions is all but equal to the ﬂuid velocity even for highinertia particles (t þ ¼ 468), in a stark contradiction with numerical simulation data by Rouson and Eaton (1994), Wang and Squires (1996) and Fukagata et al. (1998), which is evident from Figure 2.19a. However, the inﬂuence of collisions on the axial velocity proﬁle that is predicted by the model (4.76)–(4.81) is in satisfactory agreement with the results of numerical simulation obtained by Vance et al. (2004), although the theoretical model predicts a somewhat smaller effect of collisions as compared to the direct calculations. The two transverse components of particle velocity ﬂuctuations are especially strongly inﬂuenced by collisions. In the absence of collisions, the only mechanism responsible for generation of these components of ﬂuctuations is direct interaction with the corresponding components of the continuous phase, which means that v02 and v03 get smaller as t þ increases. In the presence of collisions, the generation of v02 and v03 mostly occurs via redistribution of the longitudinal component of ﬂuctuations. This

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mechanism explains why, in contrast to the signiﬁcant decrease of intensity of transverse velocity ﬂuctuations of non-colliding high-inertia particles (curve 3 and symbols 6 in Figure 4.10a), transverse ﬂuctuations of colliding high-inertia particles assume rather large values over the entire cross-section of the channel (curve 3 and symbols 6 in Figure 4.10b). The decisive role of collisions in generation of transverse (radial) velocity ﬂuctuations of the disperse phase is conﬁrmed by the results of experimental studies for a circular pipe (Caraman et al., 2003). A comparison of Figure 4.11a and b shows the role of collisions in the distribution F=F(R) over the channel crosssection. The main of relative concentration F conclusion is that collisions result in reduced clustering in the near-wall region. Indeed, the model (4.76)–(4.81) concurs with the data obtained by Vance et al. (2006) in predicting that collisions may cause the concentration of particles at the wall to be even smaller than the average concentration at the channel axis. The ﬂattening of the particle distribution over the channel cross-section is explained by intensiﬁcation of turbulent transport of velocity ﬂuctuations of the disperse phase due to collisions. Figure 4.12 shows how the inertia parameter t þ and collisions affect turbulent stresses and particle concentration in the viscous sublayer at y þ ¼ 1. It is seen

Figure 4.12 The influence of inertia and collisions on the turbulent stresses and concentration of particles at y þ ¼ 1: 1 – in the absence of collisions, 2 and 3 – in the presence of collisions for e ¼ 1 and e ¼ 0.5.

4.6 The Effect of Collisions on Particle Motion in a Vertical Channel

from Figure 4.12a that collisions result in a shift of the position of the maximum of v0 1 þ toward the region of lower particle inertias. As a result, the intensity of longitudinal component of velocity ﬂuctuations in the near-wall region increases low-inertia particles and decreases for high-inertia particles. The increase of v0 1 þ for low-inertia particles is explained by the transport of ﬂuctuations from the ﬂow core to the wall due to collisions, whereas the decrease of v0 1 þ for high-inertia particles is explained by the redistribution of ﬂuctuation intensity from the longitudinal component to the transverse components. Figure 4.12b shows an increase of intensity of transverse velocity ﬂuctuations due to collisions for all values of the inertia parameter. As we see from Figure 4.12c, tangential turbulent stresses of the disperse phase also increase at all values of t þ in yet another manifestation of the intensiﬁcation of transport of velocity ﬂuctuations of the disperse phase due to collisions. As evidenced by Figure 4.12, particle concentration is especially strongly inﬂuenced by collisions, which results in a rapid decrease of clustering in the near-wall region. As we already mentioned, collisions may result in a lower concentration of high-inertia particles near the wall than the average concentration in the ﬂow. A comparison of the results obtained for elastic (e ¼ 1) and inelastic (e ¼ 0.5) collisions shows that the dissipation of turbulent energy of particles due to inelastic collisions manifests itself in a decreased intensity of the transverse component of velocity ﬂuctuations. The inelasticity of inter-particle collisions may also bring about an increase of concentration in the near-wall region – a phenomenon that reminds us of the similar inﬂuence of inelasticity on the concentration of particles in the case of particle collisions with the wall (Figure 2.20d).

j169

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5 Relative Dispersion and Clustering of Monodispersed Particles in Homogeneous Turbulence The present chapter is devoted to statistical description of relative motion of two identical particles in homogeneous isotropic turbulence. Description of relative motion of two particles (particle pair) necessitates the use of two-point statistical characteristics of turbulence. The approach presented below is based on the kinetic equation for the PDF of relative velocity of a particle pair and generalizes the singlepoint (single-particle) statistical method outlined in Chapters 2 and 3 for the twopoint case. One can observe in homogeneous turbulence (as well as in inhomogeneous turbulent ﬂow) the phenomenon of accumulation (clustering) of inertial particles, that is, formation of compact regions with above-average concentration of the disperse phase (preferential concentration) surrounded by regions of low concentration. Clustering of particles taking place in the inhomogeneous turbulent ﬂow is explained by particle migration (turbophoresis) from the regions of high turbulence energy into the regions of low turbulence energy such as the viscous sublayer near the channel wall. In homogeneous turbulence, the gradient of turbulent energy is zero, and thus the mechanism of particle transport due to turbophoresis in its conventional formulation does not develop. Clustering of particles in homogeneous turbulence represents one of the most interesting phenomena associated with the interaction between particles and turbulent eddies and can lead to an increase of the sedimentation rate (Wang and Maxey, 1993; Yang and Lee, 1998; Aliseda et al., 2002; Yang and Shy, 2003, 2005) as well as to higher collision rate, and faster formation of the coagulation kernel (Reade and Collins, 2000; Wang et al., 2000; Zaichik et al., 2003; Bec et al., 2005; Chun and Koch, 2005). The phenomenon of particle clustering in homogeneous turbulence attracts considerable interest and is the subject of numerous publications (e.g. Squires and Eaton, 1991c; Wang and Maxey, 1993; Sundaram and Collins, 1997; Reade and Collins, 2000; Wang et al., 2000; Balkovsky et al. 2001; Fevrier et al., 2001; Hogan and Cuzzi, 2001; Kostinski and Shaw, 2001; Elperin et al., 2002; Sigurgeirsson and Stuart, 2002; Bec 2003; Klyatskin and Elperin, 2002; Klyatskin, 2003; Boffetta et al., 2004; Collins and Keswani, 2004; Falkovich and Pumir, 2004; Oesterle, 2004; Chun et al., 2005; Wood et al., 2005; Chen et al., 2006). The phenomena of concentration ﬂuctuations and spatial segregation of inertial particles are closely related to the compressibility of the disperse phases

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velocity ﬁeld even in an incompressible carrier medium, in other words, there exists a strong correlation between the velocity divergence ﬁeld and the regions of increased particle concentration (Maxey, 1987; Elperin et al., 1996; Picciotto, Reeks et al., 2005; Reeks, 2005b). In Zaichik and Alipchenkov (2003, 2005) and Chen et al. (2006), the phenomenon of clustering is interpreted as the result of particle migration under the action of the turbophoretic driving force in the region of the particle pairs relative motion. This force tends to decrease the distance between the two particles, that is, it generates mutual attraction of particles due to their interaction with turbulent eddies. Thus, statistical characteristics of inertial particle distributions in turbulent ﬂows are not random, despite the stochastic nature of turbulence. It should be noted that at Reynolds numbers not exceeding some critical number, one can observe the formation of clusters whose fractal dimension is smaller than the dimensionality of the physical space (Bec, 2003, 2005); if we picture trajectories of the attractor evolving in the phase space, clustering of particles can be interpreted as the convergence of these trajectories at one point (Bec et al., 2005). For quantitative characterization of particle accumulation, different criteria can be used. Since heavy particles are concentrated in the regions of low vorticity (with great extension), the second invariant of the continuum velocity gradient tensor is suited for this purpose (Squires and Eaton, 1991c). Another method that is widely used (e.g. Squires and Eaton, 1991c; Fessler et al., 1994; Hogan and Cuzzi, 2001; Sigurgeirsson and Stuart, 2002; Larsen et al., 2003; Fede et al., 2004) is the method of accumulation – a description based on the deviation of the probability density of the number of particles to be found in ﬁxed space cells from the Poisson distribution corresponding to a statisticaly independent random distribution of particles. But the one characteristic of the clustering phenomenon that is the most convenient to incorporate into the computational models of various processes (collision and coagulation of particles, radiation scattering on aerosols, and so on) is the radial distribution function, equal to the ratio of the probability density of particle pair formation to the same quantity in a homogeneous suspension. The two-point statistical approach presented in this chapter is precisely the right approach for determining the radial distribution function.

5.1 The Kinetic Equation for the Two-Point PDF of Relative Velocity of a Particle Pair

Consider the motion of two identical particles in isotropic turbulent ﬁeld in the absence of gravity under the condition that the size of the particles does not exceed the Kolmogorov spatial macro-scale. Equations describing the motion of each particle in accordance with Eq. (1.45), Eq. (1.46) are given below: dRpa ¼ vp ; dt

dvpa u(Rpa ; t)vp ; ¼ dt tp

ð5:1Þ

where Rpa and vpa are the coordinates and velocities of the particles, u(Rpa,t) is the velocity of the continuum at point x ¼ Rpa(t), and a is the particles index (a ¼ 1, 2).

5.1 The Kinetic Equation for the Two-Point PDF of Relative Velocity of a Particle Pair

From Eq. (5.1) there follow equations for the relative motion of the particle pair: drp dwp Du(rp ; t)wp : ¼ wp ; ¼ dt dt tp

ð5:2Þ

Here rp Rp2 Rp1 is the distance between the two particles, wp vp2 vp1 is the relative velocity. It is evident that Eq. (5.2) belong to the Langevin type of equations, since the turbulent velocity ﬁeld of the carrier continuum is treated as a random process. In order to make a transition from the stochastic equations (5.2) to a statistical description of the relative velocity distribution, we need to introduce a twopoint PDF of the particle pair: P ¼ hpi ¼ hd(rrp (t))d(wwp (t))i:

ð5:3Þ

The operation of averaging in Eq. (5.3) is performed over the ensemble of random instances of the continuum velocity ﬁeld. The function P(r, w, t) is deﬁned as the probability to observe the relative velocity w at the time t for a pair of particles separated by the displacement r. Representing actual velocity increments of ﬂuid and inertial particles as sums of the average and ﬂuctuational components, Du ¼ U þ Du0 and w ¼ W þ w0, differentiating Eq. (5.3) with respect to time and taking into account both Eq. (5.2) and the relation hwpipi ¼ wiP, we obtain the transport equation for the PDF of relative velocity of the particle pair: qP qP 1 q(DU k wk )P 1 qhDu0k pi þ ¼ : þ wk qt qr k tp qw k tp qw k

ð5:4Þ

The left-hand side of Eq. (5.4) includes the terms describing the evolution in time and convection in the phase space (r, w), whereas the right-hand side characterizes the interaction between particles and turbulent eddies of the continuum. To determine the correlations hDu0k pi describing the particle–turbulence interaction, the relative velocity ﬁeld of the continuum is modeled by a Gaussian random process with given correlation moments. Then, using the Furutsu–Donsker–Novikov formula for Gaussian random functions (Klyatskin, 1980, 2001; Frisch, 1995), we obtain hDu0i pi ¼

ðð

hDu0i (r; t)Du0k (r1 ; t1 )i

dp(r; t) dDuk (r1 ; t1 )dr1 dt1

dp(r; t) dr1 dt1 ; dDuk (r1 ; t1 )dr1 dt1

dr p j (t) q p(r; t) qr j dDuk (r1 ; t1 )dr1 dt1 dwp j (t) q p(r; t) : qw j dDuk (r1 ; t1 )dr1 dt1

ð5:5Þ

¼

ð5:6Þ

To ﬁnd the functional derivatives in Eq. (5.6), we should invoke the solutions of Eqs. (5.2) describing the relative motion of the particle pair: ðt ðt 1 tt1 wp (t1 )dt1 ; wp (t) ¼ Du(rp (t1 ); t1 )exp ð5:7Þ dt1 : rp (t) ¼ tp tp 1

1

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The use of functional differentiation as applied to Eq. (5.7) results in the following expressions for the functional derivatives: dr p i (t) tt1 ¼ d ij d(rp (t1 )r1 ) 1exp H(tt1 ) tp dDuj (r1 ; t1 )dr1 dt1 ðt tt2 qDui (rp (t2 ); t2 ) þ 1exp tp qr n t1

dr p n (t2 ) dt2 ; ð5:8Þ dDuj (r1 ; t1 )dr1 dt1 dij dw p i (t) tt1 ¼ d(rp (t1 )r1 )exp H(tt1 ) dDuj (r1 ; t1 )dr1 dt1 tp tp ðt dr p n (t2 ) 1 tt2 qDui (rp (t2 ); t2 ) exp dt2 : þ qr n dDuj (r1 ; t1 )dr1 dt1 tp tp

t1

ð5:9Þ To solve the system of Eqs. (5.8) and (5.9), apply the iteration procedure similar to the one used for the system of Eqs. (2.7) and (2.8). Since the contribution of the gradient of the average ﬂuid particle velocity increment qDUi/qrj is small relative to the contribution of the average velocity gradient qUi/qxj (in the majority of cases of interest we have DU ¼ 0), we retain only one term in Eq. (5.8) and two terms in the iteration expansion (5.9). Then the functional derivatives are reduced to dr p i (t) tt1 ¼ dij d(rp (t1 )r1 ) 1exp ð5:10Þ H(tt1 ); tp dDuj (r1 ; t1 )dr1 dt1 dij dw p i (t) tt1 ! ! ¼ d(r (t )r )H(tt ) exp 1 1 p 1 dDuj (r!1 ; t1 )dr!1 dt1 tp tp þ

qDui tt1 tt1 1 1 þ exp : qrj tp tp

ð5:11Þ

Substituting Eq. (5.10) and Eq. (5.11) into Eq. (5.5) and Eq. (5.6) and averaging over all instances of turbulent ﬂuctuations, we arrive at the following expression for the correlation between ﬂuctuations of ﬂuid velocity increments at the two points and the probability density of relative velocity of the two particles located at these points: qP qP þ Gik (r; t) ; ð5:12Þ hDu0i pi ¼ tp F ik (r; t) qw k qr k d jk tt1 exp tp tp 1 qDU j tt1 tt1 1 1 þ exp dt1 ; þ qx k tp tp

1 F ij (r; t) ¼ tp

ðt

hDu0i (r; t)Du0k (rp (t1 ); t1 )iH(tt1 )

ð5:13Þ

5.1 The Kinetic Equation for the Two-Point PDF of Relative Velocity of a Particle Pair

Gij (r; t) ¼

1 tp

ðt 1

tt1 hDu0i (r; t)Du0j (rp (t1 ); t1 )iH(tt1 ) 1exp dt1 : tp

ð5:14Þ Quantities Gij and Fij denote the integrals containing second correlation moments of ﬂuid velocity ﬂuctuation increments determined along the trajectories of particle pair motion. To calculate these integrals, it is necessary to determine the Lagrangian two-point structure function of velocity ﬂuctuations of ﬂuid particles moving along inertial particle trajectories, and the Lagrangian two-point structure function of velocity ﬂuctuations of ﬂuid particles moving along their own trajectories deﬁned by Eq. (1.18). In order to account for the transport effect, the Lagrange two-point structure function of velocity ﬂuctuations of ﬂuid particles moving along inertial particle trajectories is deﬁned as SLp ij (r; t) ¼ h(u0i (Rp2 (t); t)u0i (Rp1 (t); t))(u0j (Rp2 (tt); tt)u0j (Rp1 (tt); tt))i t Dp Sij ¼ Sij ð5:15Þ YLrp (tjr); 2 Dt fp

r ¼ Rp2 (t)Rp1 (t);

qSijk Dp Sij qSij qSij ¼ þ Wk þ ; Dt qt qr k qr k

fp

Sijk (r; t) ¼ hDu0i Du0j w 0k i ¼ h(u0i (x þ r; t)u0i (x; t))(u0j (x þ r; t) u0j (x; t))(v0k (x þ r; t)v0k (x; t))i; where YLrp(t|r) is the Lagrangian autocorrelation function of velocity ﬂuctuation of a pair of ﬂuid particles moving along inertial particle trajectories and separated by the distance r ¼ |r|. Similarly to Eq. (2.15), the relation (5.15) contains a transport term that takes into account time evolution, convective and diffusive transport of velocity ﬂuctuations in the continuum along the trajectories of relative motion of the two particles. In view of Eq. (5.15), the expressions (5.13) and (5.14) in the given approximation take the following form: F ij ¼

qDU j f r1 Dp Sij fr ; Gij ¼ g r Sij ; Sij þ lr Sik tp qr k 2 Dt

ð5:16Þ

1 ð T Lrp 1 t fr ¼ YLrp (t) exp f r ; dt; g r ¼ tp tp tp 0

f r1

1 ð 1 t ¼ 2 YLrp (t)t exp dt; lr ¼ g r f r1 ; tp tp 0

ð5:17Þ

Ð1 where T Lrp 0 YLrp (t)dt is the integral time scale of velocity ﬂuctuation increments of ﬂuid particles determined along the relative motion trajectories of inertial particles.

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The factors fr , gr , fr1, lr characterize the involvement of a particle pair separated by a distance r in turbulent motion of the continuum. With increase of the distance between particles, one can use the asymptotic relations YLrp (tjr) ¼ YLp (t); lr ¼ lu

T Lrp ¼ T Lp ;

f r ¼ f u;

gr ¼ gu;

f r1 ¼ f u1 ;

at r ! 1;

expressing the asymptotic convergence between two-point autocorrelation functions, integral time scale and involvement coefﬁcients and the corresponding single-point quantities. At high Reynolds numbers, the autocorrelation function can be given by an exponential dependence YLrp ¼ exp(t=T Lrp ); and accordingly the involvement factors (5.17) become fr ¼

T Lrp ; tp þ T Lrp

gr ¼

T 2Lrp tp (tp þ T Lrp )

;

f r1 ¼

T 2Lrp (tp þ T Lrp )

2

;

lr ¼

T 3Lrp tp (tp þ T Lrp )2 ð5:18Þ

Substitution of Eq. (5.12) into Eq. (5.4) brings us to a closed kinetic equation for the two-point PDF of relative velocity of the particle pair in a turbulent medium: qP qP 1 q(DU i w i )P q2 P q2 P þ ¼ F ij þ Gij : þ wi qt qr i tp qw i qw i qw j qr j qw i

ð5:19Þ

Terms on the left-hand side of equation (5.19) describe the diffusive transport in the phase space (r, w) caused by the interaction of particles with turbulent eddies of the continuum. The modeling of turbulent velocity ﬂuctuations in the continuum by a Gaussian random process makes it possible to express the particle–turbulence interaction in the kinetic equation in the form of a second-order Fokker–Planck operator. It should be noted that in many studies, for example, in Monin and Yaglom, 1975; Kuznetsov and Sabelnikov, 1990; Praskovsky and Oncley, 1994; Wang et al., 2000; Ten Cate et al., 2004), we ﬁnd that the PDF of relative velocity of ﬂuid or low-inertia particles is not a normal distribution. It is well known that the deviation of the PDF of two-point velocity difference from the Gaussian distribution is the result of intermittency phenomena. But for the most part, the deviation from the normal distribution becomes really noticeable only at high absolute values of relative velocities, that is, on the tails of the PDF distribution, whereas at small values of relative velocities the normal distribution provides an accurate approximation for the PDF (Van de Water and Herweijer, 1999; Verzicco and Camussi, 2002). Since the contribution of the tails of the PDF to the average characteristics is generally insigniﬁcant, the Gaussian random process predicts the two-point moments of relative velocities with a sufﬁcient accuracy (Wang, L.-P. et al., 1998).

:

5.2 Equations for Two-Point Moments of Relative Velocity of a Particle Pair

It is obvious that Eq. (5.19) resembles in its appearance a kinetic equation for a single-point PDF (2.23). However, one-point and two-point kinetic equations have only a superﬁcial resemblance, because the single-point kinetic model describes a single-particle PDF in the phase space and thus cannot take into account the spatial correlativity of particle pair motion. The two-point approach, on the other hand, allows us to consider the correlative motion of particles due to their interaction with turbulent eddies and thereby is capable of describing clustering phenomena. Ignoring the average velocity increment of ﬂuid particles and the transport term, that is, setting DU ¼ DpSij/Dt ¼ 0 in Eq. (5.16), we reduce Eq. (5.19) to the kinetic equation obtained by Zaichik and Alipchenkov (2003).

5.2 Equations for Two-Point Moments of Relative Velocity of a Particle Pair

As a result of integrating the kinetic Eq. (5.19) over the velocity subspace, a system of equations can be obtained for two-point statistical moments of the PDF of relative velocity of a particle pair. Thus, the equation for the number concentration of particle pairs N12 is qN 12 qN 12 W k ¼ 0; ð5:20Þ þ qt qr k ð ð 1 N 12 ¼ Pdw; W i ¼ w i Pdw: N 12 The balance-of-momentum equation for the relative motion of particles has the form r

qSp ik DU i W i Dp ik qlnN 12 qW i qW i þ Wk ¼ þ ; qt qr k qr k tp tp qx k

ð5:21Þ

where

ð 1 (w i W i )(w j W j )Pdw N 12 is the Eulerian structure function of the second order characterizing the intensity of ﬂuctuations of the velocity difference between two particles located at different points. The last term in Eq. (5.21) describes relative turbulent diffusion of a particle pair. The tensor of relative turbulent diffusion of particles is deﬁned as Sp ij ¼ hw 0i w 0j i ¼

Drp ij ¼ tp (Sp ij þ g r Sij ):

ð5:22Þ

The equation for second moments of ﬂuctuation increment of two particles (structure functions of the second order) has the following form: qSp ij qSp ij qW j 1 qNSp ijk qW i þ Wk þ ¼ (Sp ik þ g r Sik ) (Sp jk þ g r Sjk ) qt qr k qr k qr k N qr k qDU j Dp Sij qDU i 2 ; þ lr Sik þ Sjk þ (f r Sij Sp ij )f r1 qr k qr k Dt tp

ð5:23Þ

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Sp ijk ¼ hw 0i w0j w 0k i ¼

ð 1 (w i W i )(w j W j )(w k W k )Pdw: N 12

Derivation of the equation for the PDF in phase space does not solve the closure problem completely, since it gives rise to an inﬁnite chain of equations for the moments, the equation for n-th moment containing (n þ 1)-th moment. In order to get a closed system of equations for moments whose order does not exceed three, we must invoke the quasi-normal hypothesis about the vanishing of fourth-order cumulants. Similar to Eq. (2.29), this hypothesis allows us to express structure functions of the fourth order as sums of products of second-order structure functions. As a result, there follows an equation for the third order structure function: qSp ijk qSp ijk qW j qW k qW i þ Sp ijn þ Sp ijkn þ þ Sp jkn þ Wn qt qr n qr n qr n qr n þ

Drp in qSp jk tp

qr n

þ

Drp jn qSp ik tp

qr n

þ

Drp kn qSp ij tp

qr n

þ

3 Sp ijk ¼ 0: tp

ð5:24Þ

Equations (5.20)–(5.24), describe two-point statistics of a particle pairs relative velocity in terms of third moments. In order to represent the process in terms of second moments in Eq. (5.24), we shall neglect the terms characterizing time evolution, convection, and generation of ﬂuctuations due to the average velocity gradient. This procedure results in the following algebraic relation for the third-order structure function: qSp jk qSp ik qSp ij 1 Sp ijk ¼ þ Drp jn þ Drp kn Drp in : ð5:25Þ qr n qr n qr n 3 Triple correlations describe the diffusional mechanism of velocity ﬂuctuation transport. Expression (5.25) is similar by its appearance to the relation (2.31) for single-point third moments. The system (5.20)–(5.23), which also factors in the relation (5.25), allows to model the two-point statistics of relative velocity of a particle pair on the level of second moments. For the third-order structure function of ﬂuid particle pair velocities, from Eq. (5.25) there follows relation (1.31): 1 r qSjk r qSik r qSij Sijk ¼ lim Sp ijk ¼ þ Djn þ Dkn Din tp ! 0 qr n qr n qr n 3 ¼

qSjk qSij T Lr qSik þ Sjn þ Skn Sin ; 3 qr n qr n qr n

ð5:26Þ

where Drij denotes the tensor of relative turbulent diffusion of ﬂuid particles (1.19). Similarly to Eq. (5.25), triple correlations appearing in the transport term with DpSij/Dt in Eq. (5.15) are represented as qSjk qSij 1 qSik fp Sijk ¼ þ Drp jn þ Drp kn Drp in : ð5:27Þ qr n qr n qr n 3

5.2 Equations for Two-Point Moments of Relative Velocity of a Particle Pair

The mixed second-order structure function of ﬂuid and particle velocities can be determined directly from Eq. (5.12) that also factors in Eq. (5.16): fp

Sik ¼

ð ð ð 1 1 hDu0i piw k dwV k hDu0i pidw hDu0i (w k W k )pidw ¼ N 12 N 12

¼ f r Sik þ tp Sin

tp f Dp Sik qDU k qW k g r : lr r1 qr n qr n 2 Dt

The isotropic turbulence is characterized by spherical symmetry – dependence of relative velocities and particle pair probability density only on the modulus r of the position-vector r, not on its orientation. The average relative velocity vector in isotropic turbulence may be expressed through its radial component as follows: Wi ¼

ri Wr r

and thereby the system of equations (5.1)–(5.4) with the consideration of Eqs. (5.25) and (5.27) reduces to qN 12 1 q þ 2 (r 2 N 12 W r ) ¼ 0; qt r qr

ð5:28Þ

qW r qW r 2(Sp nn Sp ll ) qSp ll DU r W r qlnN 12 þ Wr ¼ þ ; (Sp ll þ g r Sll ) qt qr qr tp qr r ð5:29Þ

qSp ll qSp ll qSp ll tp q 2 þ Wr ¼ 2 r N 12 (Sp ll þ g r Sll ) qt qr r N 12 qr qr 4tp qSp nn 2 (Sp ll þ g r Sll ) þ (Sp nn þ g r Snn )(Sp ll Sp nn ) 3r qr r qSll qSll tp q 2 qSll f r1 þ Wr 2 r (Sp ll þ g r Sll ) qt qr r qr qr 4tp qSnn 2 þ þ (Sp nn þ g r Snn )(Sll Snn ) (Sp ll þ g r Sll ) 3r qr r 2(Sp ll þ g r Sll )

qSp nn qSp nn tp þ Wr ¼ 4 qt qr 3r N 12 þ2

qW r qDU r 2 þ 2lr Sll þ ( fr Sll Sp ll ); qr qr tp

qSp nn q 4 r N 12 (Sp ll þ g r Sll ) qr qr

q 3 [r N 12 (Sp nn þ g r Snn )(Sp ll Sp nn )] qr

ð5:30Þ

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tp q 4 qSnn qSnn qSnn þ Wr 4 r (Sp ll þ g r Sll ) qt qr 3r qr qr

2tp q 3 Wr 4 r (Sp nn þ g r Snn )(Sll Snn ) 2(Sp nn þ g r Snn ) 3r qr r

f r1

þ 2lr Snn

DU r 2 þ ( fr Snn Sp nn ); r tp

ð5:31Þ

where Sp ll and Sp nn are the longitudinal and transverse components of structure functions of particle velocity ﬂuctuations. Thus in isotropic turbulence, simulation of two-point statistics and dispersion of particles on the level of second moments boils down to solving the system of equations for the number concentration of particle pairs N12, the radial component of relative velocity Wr , and the structure functions Sp ll and Sp nn.

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Consider a stationary suspension in which the total number of particles does not vary with time and the average relative convective transport with the velocity of the continuum is absent (DU ¼ 0). The condition of stationarity suggests a balance of particle ﬂuxes toward and away from the origin of coordinates and consequently, the zero average relative radial velocity Wr . Hence the equation for particle pair concentration (5.28) is identically satisﬁed and the system of equations (5.29)–(5.31) reduces to the following system written in dimensionless variables: 2ðSp ll Sp nn Þ dSp ll d ln G þ ðSp ll þ g r Sll Þ þ ¼ 0; d r r d r

ð5:32Þ

dSp ll dSp nn 4 1 d 2 r GðS p ll þ g r S ll Þ ðS p ll þ g r Sll Þ St 2 d r d r 3 r r G dr 2 1 d 2 dSll þ ðSp nn þ g r Snn ÞðSp ll Sp nn Þ þ St2 f r1 2 r ðS p ll þ g r Sll Þ d r r r d r nn 2 4 dS þ ðSp nn þ g r Snn ÞðSll Snn ) ðS p ll þ g r Sll Þ dr 3 r r

2

þ 2ð fr Sll Sp ll Þ ¼ 0; St2 3r4 G

ð5:33Þ

dSp nn d 4 d 3 þ2 r GðSp ll þ g r Sll Þ r GðSp nn þ g r Snn ÞðSp ll Sp nn Þ d r d r d r

St2 f r1 d 4 dSnn d 3 þ þ2 r ðSp ll þ g r Sll Þ r ðSp nn þ g r Snn ÞðSll Snn ) 3r 4 d r d r d r þ 2ð fr Snn Sp nn ) ¼ 0:

ð5:34Þ

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Here the transition to dimensionless variables is accomplished through the Kolmogorov length scale h (n3/e)1/4, time scale tk (n/e)1/2, and velocity scale uk (ne)1/4; St tp/tk is the Stokes number based on the Kolmogorov time microscale, and G N12/N2 is the radial distribution function describing the effect of particle clustering. G is equal to the ratio of the probability of ﬁnding a given number of particles in an inﬁnitely thin shell of radius r centered at the test particle to the same probability in a homogeneous suspension. By analogy with Eq. (2.177), the equation (5.32) can be said to express the balance between the turbophoretic driving force pushing the particle toward the origin of coordinates, (i.e., in the direction of structure function decrease), and the diffusion driving force. The ﬁrst force is caused by the interaction of particles with turbulent eddies and tends to reduce the distance between two particles; one can say that it provokes mutual attraction between the particles. The boundary conditions for Eqs. (5.32)–(5.34) are speciﬁed as follows: r ¼ 0; r! 1;

dSp ll dSp nn ¼ ¼ 0: d r d r 2f Re Sp ll ¼ Sp nn ¼ u 1=2l ; 15

ð5:35Þ G ¼ 1:

ð5:36Þ

Conditions (5.35) express the balance between particle ﬂuxes directed toward and away from the origin of coordinates, and they hold in the case when the particle size is much smaller than the Kolmogorov microscale h. Conditions (5.36) reﬂect the fact that at large distances r between the particles their motion becomes non-correlated, that is, relative velocities of the particles no longer depend on r, so the spatial distribution of particles becomes random, or, to use another term, they are distributed uniformly. Assuming that the relative velocity of a particle pair is distributed normally, the relation between the average modulus of relative velocity and the longitudinal structure function takes the form 1=2 2 0 2 1=2 2 hjwr ji ¼ ¼ : ð5:37Þ hw i Sp ll p r p Wang et al. (2000) have shown that the PDF of relative velocity depends on particle inertia and becomes Gaussian only at large values of the Stokes number. Nevertheless, according to the direct numerical calculations (Wang, L.-P. et al., 1998), even for 0 2 1=2 inertialess particles pﬃﬃﬃﬃﬃﬃﬃﬃ the quantity hjw r ji=hw r i is equal to 0.77, which is rather close to the value 2=p ¼ 0:798 predicted by the normal distribution. Henceforth we shall be using formula (5.36) to express the quantity h|wr|i. The longitudinal and transverse structure functions of turbulent ﬂuid velocity ﬂuctuations are deﬁned by the relations (1.38) and (1.39). The integral time scale TLrp of continuum velocity ﬂuctuation increment along the trajectories of inertial particles relative motion is taken equal to the corresponding scale TLr along ﬂuid particle trajectories. The latter is assumed to be described by approximation (1.40), which is strictly valid only in the limit of inertialess particles and stops working for high-inertia

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particles. But one can hope that in the absence of the crossing trajectory effect which is caused by the average relative velocity between the particles and the ﬂuid, this shortcoming of Eq. (1.40) will not be of great importance. Consider now some asymptotic solutions of the problem (5.32)–(5.35). At St ¼ 0, inertialess particles are fully involved in ﬂuctuational motion of the continuum and their distribution in space is homogeneous: Sp ll ¼ Sll ;

Sp nn ¼ Snn ;

G ¼ 1:

ð5:38Þ

In the case of low-inertia particles (St 1), the solution at high Reynolds numbers should be in the vicinity of the point r ¼ 0. The solution of equations (5.32)–(5.34) obeying only the boundary conditions (5.35) is sought in the form Sp nn ¼ br2 ;

Sp ll ¼ ar2 ;

G ¼ grc :

ð5:39Þ

The longitudinal and transverse structure functions of ﬂuid velocity ﬂuctuations and the time scale of velocity increment in the vicinity of r ¼ 0, are represented in accordance with Eq. (1.22) and Eq. (1.28) as r2 Sll ¼ ; 15

2r Snn ¼ ; 15 2

Lr ¼ A1 : T

ð5:40Þ

Substitution of Eq. (5.39) and Eq. (5.40) into Eqs. (5.32)–(5.34) yields a system of algebraic equations for the coefﬁcients a, b and c: g 4a2b þ c a þ r ¼ 0; 15 g 4 g St2 a(5 þ c) a þ r (2ab) b þ r 15 15 3 St2 f r1 f (7a þ 4b þ g r ) þ r a ¼ 0; 45 15 St2 (2ab) g 7f 2f (7 þ c) b þ r þ r1 þ r b ¼ 0; 15 15 15 3 þ

ð5:41Þ

where we have in view of Eq. (5.18) fr ¼

A1 ; A1 þ St

gr ¼

A21 ; St(A1 þ St)

f r1 ¼

A21 : (A1 þ St)2

Due to linearity of the equations (5.32)–(5.34) with respect to G, the coefﬁcient g in Eq. (5.38) cannot be found from the local solution in the vicinity of the point r ¼ 0, because in order to do so, one needs the boundary condition (5.36). The correct way to ﬁnd this coefﬁcient is by matching with the solution of the boundary-value problem (5.32)–(5.36). In the case of St 1 the solution of Eq. (5.40) may be found as a Taylor series expansion over a small parameter St:

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

a¼

1 2A1 1 1 14A21 14 þ þ St2 þ O(St3 ); St þ 45 15A1 2025 225 15 15A21

b¼

2 2 2 28A21 St þ þ St2 þ O(St3 ); 2025 15 15A1 15A21

8 280 3 c ¼ St2 þ St þ O(St4 ): 3 75A1

ð5:42Þ

It follows from Eq. (5.39) and Eq. (5.42) that the radial distribution function of lowinertia particles has a singularity at the zero point, in other words, G diverges at r ! 0. The singularity and the power law for G was ﬁrst established by Reade and Collins (2000) by the DNS method. Quadratic dependence of the exponent c on Stokes number St (c / St2) at St 1 has been derived analytically by Balkovsky et al. (2001), Zaichik and Alipchenkov (2003), Chun et al. (2005). The unbounded increase of the radial distribution function when the distance between the particles tends to zero can be interpreted as the clustering phenomenon and be taken as the criterion of clustering. Figure 5.1 compares numerical solution results (Eq. (5.41)) with the asymptotic solution (Eq. (5.42)) at A1 ¼ 51/2. It is readily seen that there is an ambiguity present in the solution of Eq. (5.41). The second solution corresponds to greater absolute values of c and thereby to greater singularity of the radial distribution function, but the possibility of its physical realization requires additional study. It is important to notice that a physically meaningful solution of Eq. (5.41) exists only within the Stokes number range St < Stcr , where Stcr ^ 0.95. The existence of the critical Stokes number in clustering phenomena was ﬁrst established by Bec (2003, 2005), who arrived at his result by considering Lyapunov exponents for a stochastic dynamic system of equations describing particle motion in a random Gaussian

Figure 5.1 Dependence of the coefficients in Eq. (5.39) on the Stokes number: 1, 4 – a; 2, 5 – b; 3, 6 – c; 1, 2, 3 – solution of Eq. (5.41); 4, 5, 6 – (5.42).

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ﬁeld. Bec (2003, 2005) also showed that at values of St not exceeding the critical value Stcr , the formation of clusters with fractal dimension less than the dimensionality of the physical space can take place. Consider now the solution of the problem (5.32)–(5.34) with the boundary conditions (5.35) in the limiting case Rel ! 1 when the inertial space interval becomes unbounded. Discovery of particle clustering regularities at high Reynolds numbers is of great scientiﬁc and practical importance. The reason is that the Reynolds numbers achievable at present in the DNS method and in laboratory experiments are smaller than the typical values encountered in atmospheric ﬂows by at least one order of magnitude. Besides, sizes and relaxation times that are typical for raindrops and atmospheric aerosols are so small in comparison with the corresponding turbulent microscales that their realization presents severe difﬁculties both in DNS simulation and in experiments. The feasibility of obtaining a universal asymptotic solution in problems involving relative dispersion and clustering of two particles (or a group of particles) in a turbulent ﬁeld at high Reynolds numbers is closely connected with the intermittency phenomenon caused by ﬂuctuations of turbulent energy dissipation rate (Monin and Yaglom, 1975; Kuznetsov and Sabelnikov, 1990; Pope, 2000) and with the violation of self-similarity of the two-point PDF of ﬂuid velocity ﬁeld (Tatarskii, 2005). If intermittency can be neglected, Kolmogorovs similarity hypotheses for small-scale turbulence (Lundgren, 2003) may hold. These hypotheses establish versatility of small-scale turbulence, in other words, they claim that turbulence characteristics in the viscous and inertial intervals at high Reynolds numbers do not depend on large-scale eddy structures. On the other hand, the intermittency phenomenon leads to a dependence of small-scale turbulence characteristics on the Reynolds number. Therefore derivation of versatile (i.e., self-similar over the whole range of Reynolds numbers) solutions is possible only for relatively low-inertial particles whose relaxation time falls within the viscous interval or within the inertial interval, with intermittency playing a minor role. The DNS results (Collins and Keswani, 2004) conﬁrm the presence of such versatile dependence for the radial distribution function at least for those particles whose relaxation time lies in the viscous interval. Presented solutions at high Reynolds numbers are obtained within the framework of the classical hypotheses of local similarity and the effect of intermittency is disregarded. Lundgren (2003) has established the validity of the classical Kolmogorov theory of local similarity for the inertial interval at Rel ! 1, but he found that the intermittency factor characterizing anomalous scaling tends to zero very slowly (by a logarithmic law). Because sensitivity of the procedure to intermittency increases with the order of the statistical moment of characteristic turbulent ﬁeld ﬂuctuations, the results obtained without consideration of the intermittency phenomenon can be true only for sufﬁciently low-order moments of the PDF. The longitudinal structure function of continuum velocity ﬂuctuations and the time scale of velocity increment are given by the approximations following from Eqs. (1.38), (1.40) and combined with the relations (1.22), (1.29) and (1.28), (1.30) for the viscous and inertial spatial intervals:

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Figure 5.2 Influence of the Stokes number on the values of the structure functions and the radial distribution function at r ¼ 0 p (0); 2, 4 – G(0); 1, 2 – solution for high Reynolds numbers: 1, 3 – S of Eqs. (5.32)–(5.35); 3, 4 – Eq. (5.46).

Sll ¼ Cr 2=3 1exp

r (15C)3=4

4=3 ;

C ¼ 2;

3=2 2=3 Lr ¼A2 r 2=3 1exp A2 T r ; A1

A1 ¼ 51=2 ;

A2 ¼ 0:3:

ð5:43Þ

The solution of the problem (5.32)–(5.35) that also factors in the relations (5.43) is valid for particles whose relaxation time lies within the viscous interval or the L ). The involvement coefﬁcients are deﬁned by the inertial interval (St T relations (5.18). Figure 5.2 shows the values of the structure functions and the radial distribution function at r ¼ 0. The inference about the existence of a critical Stokes number is conﬁrmed. At St < Stcr the solution in the vicinity of r ¼ 0 has the form (5.39), while Sp ll(0) ¼ Sp nn(0) ¼ 0 and G(0) ¼ 1, which indicates that particle clustering is taking place. At St > Stcr the longitudinal and transverse structure functions have identical ﬁnite values at the zero point: Sp ll(0) ¼ Sp nn(0) ¼ Sp(0). Since G(0) also assumes a ﬁnite value in accordance with Eq. (5.32), no clustering (in the above-mentioned sense) is taking place at this point. The critical Stokes number obtained by solving the problem (5.32)–(5.35) with the consideration of Eq. (4.47) is found to be somewhat greater than the value obtained from Eqs. (5.41) and is close to 1.8. Thus the obtained results justify the conclusion that the critical Stokes number is the bifurcation point of the system (5.32)–(5.34), because at St < Stcr the solution in the vicinity of r ¼ 0 has the form (5.38), whereas at St > Stcr it takes the form Sp ll ðrÞ ¼ Sp nn ðrÞ ¼ Sp ð0Þ;

GðrÞ ¼ Gð0Þ;

p (0) and G(0) depend on St. where the values of S Let us ﬁnd the asymptotic solution of the problem (5.32)–(5.35) at high Reynolds where L ¼ L=h) for particles that numbers in the inertial spatial interval (1 r L, L ). In this statement of the problem, belong to the inertial time interval (1 St T

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taking into account Eq. (1.29), Eq. (1.30), and Eq. (5.18), we can rearrange Eqs. (5.32)–(5.35) as follows: 2(sp ll sp nn ) dsp ll d ln G þ (sp ll þg r sll ) þ ¼ 0; dr r dr dsp ll dsp nn 1 d 2 4 G(s þg s ) r (sp ll þ g r sll ) p ll ll r dr dr r2 G dr 3r 2 1 d 2 dsll þ (sp nn þ g r snn )(sp ll sp nn ) þf r1 2 r (sp ll þg r sll ) dr r r dr 4 dsnn 2 þ (sp nn þ g r snn )(sll snn ) (sp ll þg r sll ) dr 3r r þ 2( fr sll sp ll ) ¼ 0;

dsp nn 1 d 4 d 3 þ 2 G(s þ g s ) G(s þ g s )(s s ) r r p nn p nn p ll p ll r ll r nn dr 3r4 G dr dr f þ r14 3r

d 4 dsnn d 3 þ2 r (sp ll þ g r sll ) r (sp nn þ g r snn )(sll snn ) dr dr dr

þ 2( fr snn sp nn ) ¼ 0; r¼0

ð5:44Þ

dsp ll dsp nn ¼ ¼ 0; r! 1 sp ll ¼ f r sll ; sp nn ¼ f r snn ; G ¼ 1; dr dr

4 A2 r2=3 A22 r4=3 sll ¼ Cr2=3 ; snn ¼ Cr2=3 ; f r ¼ ; gr ¼ ; 2=3 3 1 þA2 r 1 þA2 r2=3 f r1 ¼ sp nn ¼

Sp ll A22 r4=3 r ; ; r ¼ 3=2 ; sp ll ¼ 2 2=3 St (1þ A2 r ) St Sp nn Sll Snn ; sll ¼ ; snn ¼ : St St St

The problem (5.44) is self-similar with respect to the Stokes number. There is a certain analogy with the problem of particle motion in the near-wall logarithmic layer (2.174). Figure 5.3 shows the structure functions of the continuum and the particles and the radial distribution function obtained by solving the problem (5.44); Figure 3.3 shows also the same situation except that the transport term in the approximation (5.15) has been neglected, that is, the value fr1 ¼ 0 has been plugged into Eq. (5.44). At large distances (r 1), when the diffusional mechanism of velocity ﬂuctuation transport does not play any signiﬁcant role, the longitudinal and transverse structure functions of the particles are smaller than the corresponding structure functions of the continuum and are approximately deﬁned by the local homogeneous relations sp ll ¼ frsll, sp nn ¼ frsnn. However, at small distances

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Figure 5.3 The structure functions (a) and the radial distribution function (b) in the inertial interval at Rel ! 1: 1 – problem (5.44); 2 – problem (5.44) with fr1 ¼ 0.

between the particles, their structure functions exceed those of the ﬂuid. This effect is caused by the diffusional mechanism of velocity ﬂuctuation transport and takes place only for sufﬁciently inertial particles. It also follows from Figure 5.3a that with decrease of r the values of sp ll and sp nn get closer to each other and approach a certain limit. Figure 5.3b shows a monotonous increase of radial distribution function with decrease of r, which is qualitatively similar to the behavior of particle concentration in the near-wall logarithmic layer (Figure 2.17b). Comparing the solution of Eq. (5.44) with the solution of the same problem at fr1 ¼ 0, we see that the transport term in approximation (5.15) does not exert any signiﬁcant inﬂuence on the proﬁles of particles structure functions; on the other hand, it does have a noticeable effect on the radial distribution function, and this effect becomes more signiﬁcant as r gets smaller. At the origin of coordinates, we have the following values: sp ll (0) ¼ sp nn (0) ¼ 0:69; G(0) ¼ 4:0 and sp ll (0) ¼ sp nn (0) ¼ 0:57; G(0) ¼ 3:4 at f r1 ¼ 0:

ð5:45Þ

The asymptotic dependences at high St are depicted in Figure 5.2 by the dotted lines, and in accordance with Eq. (5.45) we have Sp (0) ¼ 0:69 St; G(0) ¼ 4:0:

ð5:46Þ

In the case of high-inertia particles whose dynamic relaxation time by far exceeds L ), the problem (5.32)–(5.36) the Lagrangian integral scale of turbulence (St T with the consideration of Eqs. (1.35)–(1.36) has a simple asymptotic solution. Due to the intensive diffusive transport of velocity ﬂuctuations, proﬁles of the longitudinal and transverse structure functions of particles coincide and become uniform, and the radial distribution function is equal to unity:

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L Re 2T Sp ll ¼ Sp nn ¼ 1=2 l ; 15 St

G ¼ 1:

ð5:47Þ

We now proceed to solve the problem (5.32)–(5.36) for ﬁnite Reynolds numbers Rel. To this end, the longitudinal structure function of velocity ﬂuctuations in the continuum and the time scale of velocity increment are approximated by the expressions (1.38) and (1.40) combining the corresponding dependences for the viscous, inertial, and external spatial intervals: 4=3 1=6 2Rel r 153r 4 ¼ ; C ¼ 2; 1exp Sll 153r 4 þ (2Rel =C)6 (15C)3=4 151=2 3=2 2=3 1=6 A2 r 4 r ; T Lr ¼ T L 1exp L =A2 Þ6 A1 r 4 þ ðT A1 ¼ 51=2 ; A2 ¼ 0:3:

ð5:48Þ

To calculate the coefﬁcient of particle involvement in turbulent motion of the ﬂuid, we use the two-scale bi-exponential autocorrelation function (1.21), according to which we can take 2StLr þz2r 1 ; gr ¼ f ; StLr r 2StLr þ 2St2Lr þz2r

2 2StLr þ z2r 2St2Lr z2r f r1 ¼

2 ; lr ¼ g r f r1 ; 2StLr þ2St2Lr þz2r fr¼

ð5:49Þ

(notice the similarity with Eq. (2.187)),where StLr ¼ tp/TLr. The Lagrangian integral L is given by the dependence (1.4) and the Taylor time time scale of turbulence T microscale of velocity ﬂuctuation increments tTr is deﬁned by Eq. (1.41). At Rel ! 1, Eq. (5.48) and Eq. (5.49) reduce to Eq. (5.43) and Eq. (5.18) respectively, under the condition that TLrp ¼ TLr. We solve the problem (5.32)–(5.36) with the consideration of Eq. (5.48) and Eq. (5.49) and compare our results with numerical calculation data obtained by various authors. Figure 5.4 shows the structure functions and the radial distribution function of particles obtained by solving the problem (5.32)–(5.36) at Rel ¼ 75. The comparison of Fig. 2.18c, d, e and Fig. 2.19c, d, e suggests a qualitative analogy between the behavior of transverse velocity ﬂuctuation intensity and particle concentration in the near-wall turbulence on the one hand, and the behavior of structure functions and the radial distribution function in isotropic turbulence on the other hand. With increase of the Stokes number St , structure functions of particles deviate more and more from structure functions of the continuum (curves 1) approaching the asymptotic uniform distributions for high-inertia particles (5.47). As opposed to structure functions of the continuum, which are equal to zero at r ¼ 0, structure functions of sufﬁciently inertial particles are non-zero at the origin of coordinates due

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Figure 5.4 The longitudinal (a) and transverse (b) structure functions and the radial distribution function (c) at Rel ¼ 75: 1 – St ¼ 0; 2 – 1; 3 – 3; 4 – 10; 5 – 100; 6 – 1000.

to the diffusive transport. In accordance with Eq. (5.39), the radial distribution function of low-inertia particles for c < 0 has a singularity at r ¼ 0. With increase of particle inertia, this singularity at the origin disappears and G tends to 1 because of the decrease of structure function gradients and consequently of the turbophoretic driving force. Figure 5.5 demonstrates the effect or particle inertia on relative radial velocity p ll using Eq. (5.37) for a wide Stokes number range and shows a determined from S comparison with the DNS results (Wang et al., 2000) at r ¼ 1. It is seen that the dependence of hj w r ji on St has a maximum. The initial growth of hj w r ji is caused by declining motion correlativity of the two particles with increase of their inertia. The subsequent decrease of hj wr ji with increase of St is explained by decreased intensity of particle velocity ﬂuctuations as particles become more inertial and thereby less capable of getting involved in turbulent motion of the continuum. Note that in the case of low-inertia particles the inﬂuence of the Reynolds number on hj wr ji can be ignored and the calculated results for relative velocities presented on Figure 5.5 may well be generalized by using the Kolmogorov microscales. On the other hand, relative velocities of high-inertia particles are governed by the macroscales of turbulence.

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Figure 5.5 Influence of the Stokes number on the relative radial velocity of particles at r ¼ 1: 1–3 – model (5.32)–(5.36); 4–6 – model (5.32)–(5.36) with fr1 ¼ 0; 7 – Eq. (5.50); 8–10 – Eq. (5.51); 11–13 – (Wang et al., 2000); 1, 4, 8, 11 – Rel ¼ 45; 2, 5, 9, 12 – Rel ¼ 58; 3, 6, 10, 13 – Rel ¼ 75.

In the limit of inertialess particles (St ! 0), Eq. (5.37) and Eq. (5.40) give us 2 1=2 r: ð5:50Þ hj wr ji ¼ 15p For high-inertia particles, as St increases, hj w r ji asymptotically approaches the dependence 1=2 TL Rel 2 hj wr ji ¼ 1=4 ; ð5:51Þ p St 15 which follows from Eq. (5.47). As indicated by Figure 5.5, the values of hj w r ji predicted by the model (5.32)–(5.36) within the range 0.1 < St < 10 noticeably exceed those obtained by numerical integration. This difference is explained ﬁrst of all by strong sensitivity of the statistics of particles relative velocity ﬂuctuations to the type of turbulence. As noted in Section 4.1, the values of hj w r ji obtained in the approximation of «frozen turbulence» (Wang et al., 2000) have proved to be noticeably smaller than the DNS results for «forced turbulence» (Fede and Simonin, 2005). The results obtained by approximating the Lagrangian two-point structure function by Eq. (5.15) with no consideration of transport phenomena (i.e., by solving the problem (5.32)–(5.36) with fr1 ¼ 0) are more consistent with the DNS data for «frozen turbulence». Figure 5.6 demonstrates the inﬂuence of the Stokes number on the ratio of the longitudinal and transverse structure functions for different Reynolds numbers at r ¼ 1. The results obtained from the model (5.32)–(5.36) together with the DNS data indicate only a marginal effect of Rel on Sp nn(1)/Sp ll(1). It is seen that the ratio Sp nn(1)/Sp ll(1), being equal to two for the continuum in accordance with Eq. (1.22), falls and tends to unity with increase of St. It should also be noted that the model taking into account the transport effect when approximating the Lagrangian two-point structure function with Eq. (5.15) predicts a sharper drop of Sp nn(1)/Sp ll(1)with increase of St and is in better agreement with the DNS data.

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Figure 5.6 The ratio of the transverse and longitudinal structure functions: 1–4 – model (5.32)–(5.36); 5–8 – model (5.32)–(5.36) with fr1 ¼ 0; 9–12 – Wang et al. (2000); 1, 5, 9 – Rel ¼ 24; 2, 6, 10 – Rel ¼ 45; 3, 7, 11 – Rel ¼ 58; 4, 8, 12 – Rel ¼ 75.

The values of the radial distribution function calculated for different Stokes numbers, inter-particle distances, and Reynolds numbers are presented on Figures 5.7–5.9 together with numerical simulation data obtained by Reade and Collins (2000), Wang et al. (2000), and Février et al. (2001). In the limiting cases of lowand high-inertia particles, the respective equations (5.38) and (5.47) suggest that the concentration ﬁeld is statistically homogeneous, and consequently, G ¼ 1. According to the results of numerical simulation, as the particle relaxation time increases, the radial distribution function goes through its maximum. As evidenced by Figure 5.7, the position of the maximum at small inter-particle distances is well scaled by Kolmogorov microscales, which conﬁrms that small-scale turbulent structures play a crucial role in the formation of particle clusters. It is readily seen from Figures 5.7–5.9 that as we increase the inter-particle distance, the peak value of Ggets smaller, and the

Figure 5.7 Effect of the Stokes number on the radial distribution function at small inter-particle distances: 1–4 – model (5.32)–(5.36); 5, 6 – model (5.32)–(5.36) with fr1 ¼ 0; 7, 8 – Reade and Collins (2000); 9, 10 – Wang et al. (2000); 1, 5, 7 – r ¼ 0:025, Rel ¼ 37; 2, 8 – r ¼ 0:025, Rel ¼ 82; 3, 6, 9 – r ¼ 1, Rel ¼ 24; 4, 10 – r ¼ 1, Rel ¼ 75.

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Figure 5.8 Effect of particle inertia and inter-particle distance on the radial distribution function, written in a dimensionless form using the Kolmogorov microscales: 1–3 – model (5.32)–(5.36); 4–6 – Fevrier et al. (2001); a – r ¼ 6; b – r ¼ 12; c – r ¼ 18; d – r ¼ 24; 1, 4 – Rel ¼ 53; 2, 5 – Rel ¼ 69; 3, 6 – Rel ¼ 134.

position of the maximum is shifted toward the larger values of particle relaxation time. As shown by Février et al. (2001), the particles whose relaxation time is much greater than the Kolmogorov time microscale (St 1), also tend to cluster in space. However, since the motion of high-inertia particles is governed by large-scale turbulent structures, this clustering effect is better described by using integral scales of turbulence rather than Kolmogorov microscales. A comparison of Figrue 5.8 to Figure 5.9 makes this conclusion self-apparent. So the effect of clustering is especially pronounced for the particles whose relaxation time is close to the Kolmogorov time scale (i.e., at Stokes numbers St of the order of unity). At the same time, clustering also takes place for high-inertia particles at large inter-particle distances, though in a rather weaker form. It is also seen from Figure 5.7 that a decision to neglect the additional diffusive transport due to the transport term in Eq. (5.15) brings about a smoother, more homogeneous distribution of particles, reducing clustering and shifting the maximum of the radial distribution G to the right (i.e., toward larger values of St).

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Figure 5.9 Effect of particle inertia and inter-particle distance on the radial distribution function written in a dimensionless form using the integral macroscales: 1–3 – model (5.32)–(5.36); 4–6 – Fevrier et al. (2001); a – r/L ¼ 0.05; b – r/L ¼ 0.1; c – r/L ¼ 0.2; d – r/L ¼ 0.3; 4 – Rel ¼ 53; 2, 5 – Rel ¼ 69; 3, 6 – Rel ¼ 134.

It is interesting to note that, as one might expect from the results of numerical simulation, the model (5.32)–(5.36) predicts strengthening of the clustering effect with increase of the Reynolds number when dimensionless parameters are introduced using Kolmogorov microscales (Figures 5.7 and 5.8). On the other hand, Figure 5.9 shows weakening of the clustering effect with increase of the Reynolds number when dimensionless parameters are introduced via macroscales. However, the dependence of the radial distribution function on the Reynolds number turns out to be less pronounced than the one suggested by calculations performed by Reade and Collins (2000), Wang et al. (2000) and Fevrier et al. (2001). While the classical similarity theory requires that the Reynolds number should have negligible inﬂuence on the characteristics of small-scale turbulence in the limit of Rel ! 1, the rather strong inﬂuence of Rel on G revealed by the DNS data appears to be explainable by the intermittency phenomenon. But the effect of intermittency on particle clustering is beyond the scope of the present book and will not be discussed here.

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Figure 5.10 Longitudinal two-point correlation function normalized by the total kinetic energy of particles (a) and its correlated component (b) at Rel ¼ 134: 15 – solution of the problem (5.32)–(5.36); 6–10 – Fevrier et al. (2005); 1, 6 – StL ¼ 0.05; 2, 7 – StL ¼ 0.3; 3, 8 – StL ¼ 1.47; 4, 9 – StL ¼ 3.4; 5, 10 – StL ¼ 4.83.

Consider how the behavior of two-point correlations of particle velocity ﬂuctuations is affected by particle inertia. The two-point Eulerian correlation moment is related to the structure function by the equation Sp ij (r) 2 Bp ij (r) ¼ hv0i (x; t)v0j (x þ r; t)i ¼ kp d ij ; 3 2

kp ¼

hv0k v0k i Sp kk (1) ¼ : 2 4

Figure 5.10a plots the longitudinal correlation function 3Bpll/2kp against the interparticle distance normalized by the spatial macroscale. The theoretical data points were obtained from the statistical model (5.32)–(5.36), whereas the results of numerical simulation obtained by the DNS and LES methods were taken from Fevrier et al. (2005). We immediately notice that, in a contradiction with a well-known result for ﬂuid particles, the correlation function for inertial particles normalized by the total kinetic energy does not approach unity at r ! 0. This peculiar feature points to the fact that due to inertia, the velocities of the two particles are not completely correlated even at r ¼ 0. Therefore, in contrast to the turbulence in a continuum, where the halfconvolution of the spatial correlation moment Bkk/2 approaches the turbulent kinetic energy k at r ! 0, the function Bp kk/2 does not approach the kinetic energy of particles kp even as the distance between them gets smaller. The phenomenon of discontinuity of the spatial correlation moment at the zero point has been conﬁrmed experimentally by Khalitov and Longmire (2003) for the turbulent ﬂow in a channel. The two-point statistical model presented in this chapter is closely related to the theoretical formalism developed by Simonin et al. (2002) and Fevrier et al. (2005), which is based on the decomposition of particle velocity ﬁeld in a turbulent ﬂow into two components. Since the disperse phase simultaneously possesses the properties of a continuum on the one hand, and a collection of discrete particles on the other, these two components are interpreted in the context of the Eulerian and Lagrangian simulation methods. The ﬁrst (correlated) component takes into account spatial correlativity of particle motion, whereas the second component is associated with

5.3 Statistical Properties of Stationary Suspension of Particles in Isotropic Turbulence

Figure 5.11 Influence of particle inertia on the correlated and quasi-Brownian components of kinetic energy of the particles: 1–6 – model (5.32)–(5.36); 1–3 – ~kp =kp ; 4–6 – dkp/kp; 7–9 – Fevrier et al. (2005); 1, 4, 7 – ReL ¼ 110; 2, 5, 8 – ReL ¼ 140; 3, 6, 9 – ReL ¼ 700.

non-correlated (quasi-Brownian) velocity that is caused by statistically independent motion of particles. In accordance with this approach, the disperse phases kinetic energy is represented as the sum kp ¼ ~kp þ dkp ;

ð5:52Þ

where ~kp and dkp are, respectively, the correlated and quasi-Brownian components. These parts of kinetic energy of the particles are expressed in terms of twopoint structure functions as follows: ~kp ¼ Sp kk (1)Sp kk (0) ; 4

dkp ¼

Sp kk (0) : 4

ð5:53Þ

Figures 5.11 and 5.12 compare numerical simulation results based on the decomposition (5.50) to the theoretical model (5.32)–(5.36) complemented by Eq. (5.53). The

Figure 5.12 Influence of particle inertia on the quasi-Brownian kinetic energy: 1–3 – model (5.32)–(5.36); 4–6 – Fevrier et al. (2005); 1, 4 – ReL ¼ 110; 2, 5 – ReL ¼ 140; 3, 6 – ReL ¼ 700.

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inﬂuence of particle inertia on the correlated and quasi-Brownian components (with both components normalized by the total kinetic energy of the disperse phase) is shown in Figure 5.11. One can see that the main contribution to the kinetic energy of low-inertia particles comes from the correlated component, whereas the role of the quasi-Brownian component increases with particle inertia. Hence, we already mentioned, the motion of high-inertia particles in a turbulent ﬂow is statistically independent and similar to the behavior of Brownian particles. Figure 5.12 presents the quasi-Brownian kinetic energy of the particles normalized by the turbulent energy of the ﬂuid. It is readily seen that the statistical model (5.32)–(5.36), which predicts the existence of a maximum of the dependence of quasi-Brownian kinetic energy on particle inertia, is consistent with the results obtained by Fevrier et al. (2005). A rather interesting physical effect is worth mentioning: when St ! 1, ~kp decreases faster than dkp. It follows from Figure 5.10b that the spatial correlation function normalized by the correlated component of kinetic energy of the particles behaves in the same manner as in the continuum case and approaches unity as the distance between the two points gets smaller. However, particle velocity ﬁeld is correlated at large distances between the particles as compared to the velocity ﬁeld of a turbulent ﬂuid, and the spatial scale of velocity correlations increases with particle inertia. Thus the size of the region in which the particle retains the memory about its interaction with the turbulent ﬂuid gets expanded as particle inertia increases.

5.4 Influence of Clustering on Particle Collision Frequency

Interaction of inertial particles with turbulent eddies gives rise to two physical effects that determine collision frequency: relative motion of neighboring particles (turbulent transport effect), and inhomogeneous distribution of particles (clustering effect). In Section 4.1, we presented a statistical model of turbulent collisions that takes into account the relative motion of neighboring particles but neglects particle clustering. In the present section, we consider a collision model that takes into account both of these two effects and is based on the results of the previous section. In the framework of the spherical formulation, collision kernel is deﬁned by the following formula (Reade and Collins, 2000; Wang et al., 2000): b ¼ 2ps2 hjwr (s)jiG(s);

ð5:54Þ

where the radial distribution function G(s) at the particles point of contact describes the phenomenon of clustering. In view of Eq. (5.37), the kernel (5.54) is equal to b ¼ (8pSp ll (s))1=2 s2 G(s):

ð5:55Þ

Thus due to Eq. (5.55) the turbulent collision kernel is determined by the values of the longitudinal structure function and the radial distribution function at the distance equal to the collision sphere radius s. Let us ﬁrst list some simple analytical dependencies for b that follow as the limiting cases of Eq. (5.55).

5.4 Influence of Clustering on Particle Collision Frequency

According to Eq. (5.38), inertialess particles are fully involved in ﬂuctuational motion of the carrier medium and consequently, they are uniformly distributed in the surrounding space. Substitution of Eq. (5.39) into Eq. (5.55) leads to the familiar Saffman–Turner formula (4.17) for the collision kernel of inertialess particles whose diameter does not exceed the Kolmogorov microscale. With particle relaxation time varying in the inertial interval (tk tp TL), which is the case at high Reynolds numbers, we conclude (in view of the solution of the problem (5.44) that the collision kernel (5.55) reduces to b ¼ b2 (etp )1=2 s2 ;

b2 ¼ (8psp ll (0))1=2 G(0);

which is consistent with Eq. (4.19). Taking Eq. (5.47) into consideration, we deﬁne the collision kernel for high-inertia particles (tp TL) as follows: pT L 1=2 0 2 us ; b¼4 tp which is in good agreement with the kinetic collision kernel obtained by Abrahamson (4.21). Though in the limiting cases of inertialess particles and high-inertia particle, collision rate is governed exclusively by the turbulent transport mechanism, the inﬂuence of clustering becomes signiﬁcant when particle relaxation time falls within the inertial interval. The inﬂuence of clustering on the collision rate and thereby on the rate of particle coagulation is the most pronounced when particle relaxation time is close to the Kolmogorov temporal microscale of turbulence, that is, when the radial distribution function reaches its maximum. Shown on Figure 5.13 is the collision kernel determined by solving the problem (5.32)–(5.36) with the consideration of the relations (5.48) and (5.49). The model employs the assumption that the structure function Sp ll and the radial distribution

Figure 5.13 Influence of the Stokes number on particle collision kernel in isotropic turbulence: 1–3 and 4–6 – model (5.32)–(5.36) with and without (G ¼ 1) the consideration of the effect of clustering; 7–9 – DNS (Wang et al., 2000); 1, 4, 7 – Rel ¼ 45; 2, 5, 8 – Rel ¼ 58; 3, 6, 9 – Rel ¼ 75.

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function G determined for inﬁnitely small particles are relevant for the estimation of the collision kernel for particles of ﬁnite size s. The collision kernel is related to the corresponding value for inertialess particles (4.16) and is in agreement with the DNS ¼ 1. As we see from the ﬁgure, the model results obtained by Wang et al. (2000) at s shows a satisfactory correlation with the DNS results, though it bears mentioning that the maxima of b predicted by the model exceed the values obtained by the DNS method. This discrepancy is caused by the shift of the G maximum relative to the DNS results (Figure 5.7). Figure 5.13 shows that the inﬂuence of the Reynolds number disappears for low-inertia particles, and according to the DNS results, the collision kernel is well generalized by using the Kolmogorov microscales. It is also seen, on the other hand, that the collision rate is governed by the macroscales of turbulence. In order to visualize the contribution of clustering to the collision kernel, we also depict on Figure 5.13 the ratio b/bST calculated for the case when Sp ll is determined by solving the problem (5.32)–(5.36) and G in Eq. (5.55) is taken equal to unity. Figure 5.14 plots the ratio of the collision kernel to the corresponding quantity for inertialess particles (4.17) against the Stokes number for Reynolds numbers ranging from 50 to 5000. One can see that the dependence predicted by the model (5.32)–(5.36) ¼ 1 has only one maximum, which corresponds to the maximum of the relative at s ¼1 radial velocity h|wr|i (Figure 5.5). The existence of only one maximum b/bST at s for high Reynolds numbers contradicts the empirical model by Wang et al. (2000), which predicts two peaks corresponding to the maxima of G and h|wr|i, as can be seen from the curves corresponding to high Rel on Figure 5.14. However, the model by Wang et al. (2000) is based on the DNS data for relatively small Reynolds numbers (Rel < 75), the data for high Reynolds numbers being obtained by extrapolation. As evidenced by Figure 5.14, the statistical model based on Eqs. (5.32)–(5.36) does predict the existence of two maxima that depend on St in a manner consistent with the . Besides, the maximum empirical model at high Rel, but only for small values of s corresponding to the peak of G is not as sharp as in the empirical model. So, the behavior of the collision kernel at high Reynolds numbers requires further study.

Figure 5.14 Collision kernel of particles at high Reynolds numbers: 1–6 – model (5.32)–(5.36); 7–9 – model by Wang et al. ¼ 0:1; 4–9 – s ¼ 1; 1, 4, 7 – Rel ¼ 50; 2, 5, (2000); 1–3 – s 8 – Rel ¼ 500; 3, 6, 9 – Rel ¼ 5000.

5.4 Influence of Clustering on Particle Collision Frequency

Figure 5.15 Influence of the density ratio between the particle and the fluid on the collision kernel in the near-axis zone of the channel (dp ¼ 9.96 mm): 1 – model (5.32)–(5.36); 2 – Chen et al. (1998).

The statistical model considered above can be applied to determine the collision kernel of particles in the near-axis zone of the channel where the characteristics of turbulence are close to isotropic. Figures 5.15 and 5.16 compare numerical results for the turbulent collision kernel given by the model (5.32)–(5.36) with the DNS data obtained by Chen et al. (1998) for a planar channel at Rel ¼ 29.7. It is seen that the ratio of particle and ﬂuid densities rp/rf, as well as particle diameter dp, effectively determine the collision kernel, since due to relation (1.47), particle relaxation time tp increases with rp/rf and dp. When the density ratio drops, the relaxation time decreases, and in view of the Saffman–Turner formula (4.17) b no longer depends on rp/rf, which is consistent with Figure 5.15. For comparison, Figure 5.16 also plots b against dp as predicted by the Saffman–Turner and Abrahamson theories, respectively. It is evident that the Saffman–Turner formula, which does not take into account

Figure 5.16 Dependence of the collision kernel in the near-axis zone of the channel on particle diameter (rp/rf ¼ 713): 1 – model (5.32)–(5.36), 2 – Saffman and Turner (1956), 3 – Abrahamson (1975), 4 – Chen et al. (1998).

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particle inertia, understates the collision kernel provides good agreement with the data only for very small particles. The Abrahamson theory, which does not factor in the velocity correlativity of colliding particles, overstates the collision kernel. The statistical model based on the solution of the problem (5.32)–(5.36) provides a sufﬁciently good agreement with the DNS data.

5.5 Relative Dispersion of Two Particles in Isotropic Turbulence

Relative dispersion (diffusion) of an inertialess particle pair (passive impurity) in a turbulent ﬂow has long drawn attention of researchers beginning from the classical work of Richardson (1926). The statistics of relative motion of a particle pair is of interest in many physical problems including biophysics of microorganisms (Mann et al., 2002). A solution to the problem of relative dispersion would provide an answer to the question of how the distance between two particles that were initially close to each other varies with time. Richardson (1926) has established the third-power law for the time dependence of the mean square distance between the particles (which serves as the measure of dispersion). Richardsons law is closely related to the well-known Kolmogorov hypotheses of local similarity for the inertial interval of turbulence at high Reynolds numbers (Monin and Yaglom, 1975). In contrast to a single dispersion governed primarily by large-scale (energy-carrying) eddies, relative diffusion of a particle pair depends on the velocity difference (increment) on distances equal to the distance between these particles. Therefore relative dispersion of a particle pair at the initial stage (for time intervals shorter than the integral scale of turbulence) reﬂects the universal character of small-scale turbulence, that is, its weak dependence on largescale ﬂow structure. Numerous publications, for example, Obukhov (1941), Batchelor (1952a), Kraichnan (1966), Durbin (1980), Lundgren (1981), Borgas and Sawford (1994), Thomson (1996), Borgas and Yeung (2004) have been devoted to theoretical research of turbulent relative dispersion of passive impurity. Numerical solutions of the problem of relative diffusion of two inertialess (ﬂuid) particles in isotropic turbulence through the DNS and kinematic simulation methods have been obtained, respectively, by Yeung (1994, 1997), Yeung and Borgas (2004), Biferale et al. (2005) and Fung et al. (1992), Nicolleau and Vassilicos (2003), Nicolleau and Yu (2004), Thomson and Devenish (2005), and fundamental experimental studies have been carried out by Ott and Mann (2000). One of the most important outcomes of these studies was veriﬁcation of Richardsons third power law for particle dispersion within the inertial interval. The reader will ﬁnd a review of the obtained information about relative turbulent diffusion of passive impurity in Sawford (2001). There appears to be just one publication (by Chronopoulos and Hardalupas, 2004) devoted to the subject of relative dispersion of inertial particles and presenting the results of experimental studies as well as numerical solutions. The experiments have shown a much slower dependence of particle dispersion on time (hr 2i t0.56) than in Richardsons law (hr2i t3). The deviation from Richardsons law can be explained by the incomplete involvement of inertial particles in ﬂuctuational motion of the turbulent

5.5 Relative Dispersion of Two Particles in Isotropic Turbulence

carrier ﬂuid or, in other words, by the slippage of particles through turbulent structures. Calculations were performed on the basis of the Lagrangian trajectory method for freely buoyant particles in a turbulent ﬂuid described by the standard k e model of turbulence. It was shown that Richardsons law is true only for very low-inertia particles and as particle inertia increases, their relative dispersion becomes more and more ﬂattened with time. Thus the question of validity of Richardsons law and its range of applicability for inertial particles is still of prime interest. Consider the motion of particles in a homogeneous, isotropic, stationary and incompressible turbulent ﬂow having the zero average velocity in the absence of gravity. As our starting point for the analysis of relative dispersion, we shall use the kinetic equation for the PDF of a particle pair (5.19), in which we are going to drop the last transport term containing the derivative DpSij/Dt that is of no importance for the subsequent analysis. Integration of the kinetic equation (5.19) over the velocity subspace results in a system of equations for the spatial distribution of particle pair density Pr(r,t|r0,0), deﬁned as the probability density for two particles to be separated by a distance r at a time t if they were separated by a distance r0 at t ¼ 0. The density distribution for the particle pair satisﬁes the initial condition Pr(r,0) ¼ d(r r0). Using spherical symmetry of the problem and the relation (5.25) for triple correlations, and assuming the condition DU ¼ 0, we can write the system of equations in terms of second moments as follows: qP r 1 q þ 2 (r 2 Pr W r ) ¼ 0; qt r qr

ð5:56Þ

qW r qW r 2(Sp nn Sp ll ) qSp ll W r q ln P r þ Wr ¼ ; (Sp ll þ g r Sll ) qt qr qr tp qr r

ð5:57Þ

qSp ll qSp ll qSp ll tp q 2 þ Wr ¼ 2 r P r (Sp ll þ g r Sll ) qt qr r Pr qr qr

4tp qSp nn 2 þ (Sp nn þ g r Snn )(Sp ll Sp nn ) (Sp ll þ g r Sll ) 3r qr r

2(Sp ll þ g r Sll ) qSp nn qSp nn tp þ Wr ¼ 4 qt qr 3r Pr

qW r 2 þ ( fr Sll Sp ll ); qr tp

ð5:58Þ

qSp nn q 4 r Pr (Sp ll þ g r Sll ) qr qr

q 3 þ2 r N 12 (Sp nn þ g r Snn )(Sp ll Sp nn ) qr 2(Sp nn þ g r Snn )

Wr 2 þ ( f r Snn Sp nn ); r tp

ð5:59Þ

Integration of Eqs. (5.56)–(5.59) with respect to r leads to a chain of ordinary differential equations for the moments of probability density Pr(r, t), where probability density is expressed as a function of inter-particle distance. It is assumed that Pr

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approaches zero at r ! 1 faster than any power function, namely, lim r n Pr ¼ 0 at

n ¼ 0; 1; 2; . . .

r !1

Integration of Eqs. (5.56)–(5.59) with the normalizing condition yields 1 ð dhr 2 i ¼ 2 r 3 P r W r dr; dt

Ð1 0

r 2 Pr dr ¼ 1

ð5:60Þ

0

d dt

1 ð

1 ð

r P r W r dr ¼ 0

1 ð

r

3

2

P r W 2r dr

þ

0

0 1 ð

Pr

þ

0 1 ð

1 r Pr (Sp ll þ 2Sp nn )dr tp

1 ð

r 3 Pr W r dr

2

0

q(r 3 g r Sll ) dr; qr

ð5:61Þ

1 ð qSp ll d 2 qW r r P r Sp ll dr ¼ 2 rP r (Sp ll þ g r Sll ) tp þr dr qr qr dt 0

0

2 þ tp d dt

1 ð

r P r f r Sll dr r 2 Pr Sp ll dr ; 1 ð

2

0

ð5:62Þ

0

1 ð

1 ð

r P r Sp nn dr ¼ 2

rP r (Sp nn þ g r Snn )W r dr

2

0

0

2 þ tp

1 ð

1 ð

r Pr f r Snn dr 2

0

r Pr Sp nn dr ; 2

ð5:63Þ

0

where the dispersion hr2i is equal to 1 ð

hr i ¼

r 4 Pr dr:

2

ð5:64Þ

0

Before analyzing inertial particle dispersion, we look at the behavior of equations (5.56)–(5.64) for the case of inertialess particles. 5.5.1 Dispersion of Inertialess Particles

In the limit of inertialless (ﬂuid) particles (tp ! 0), Eqs. (5.57)–(5.59) yield the relations W r ¼ T Lr Sll

q ln Pr ; qr

Sp ll ¼ Sll T Lr Sll

qW r ; qr

ð5:65Þ Sp nn ¼ Snn T Lr Snn

Wr : r

ð5:66Þ

5.5 Relative Dispersion of Two Particles in Isotropic Turbulence

According to Eq. (5.65), the average non-zero relative velocity between inertialess particles appears due to the diffusive transport mechanism. In view of Eq. (5.65), it follows from Eq. (5.66) that diffusive transport also contributes to the structure functions of ﬂuid particles (ﬂuctuation rate of particle pairs relative velocity). A substitution of Eq. (5.65) into Eq. (5.56) leads to the diffusion equation qP r 1 q 2 r qPr ¼ 2 : r Dll qt qr r qr

ð5:67Þ

where Drll Sll T Lr is the longitudinal component of the relative turbulent diffusion tensor of a passive impurity (1.19). Thus it follows from Eqs. (5.56) to (5.59) that the probability density of an inertialess particle pair obeys the equation of relative (binary) diffusion (Richardson, 1926; Sawford, 2001). From Eq. (5.61), it follows that at gr ! TLr/tp with tp ! 0 we have 1 ð

1 ð

r P r W r dr ¼

Pr

3

0

0

q(r 3 Drll ) dr: qr

ð5:68Þ

Substitution of Eq. (5.68) into Eq. (5.60) gives the following equation for the dispersion: dhr 2 i ¼2 dt

1 ð

Pr 0

q(r 3 T Lr Sll ) dr: qr

ð5:69Þ

Hence Eq. (5.69) shows that the time variation of dispersion, that is, increase of the distance between ﬂuid particles, depends on the behavior of the longitudinal component of the relative turbulent diffusion tensor. Consider now the dispersion of inertialess particles successively in the viscous, inertial, and external spatial regions. In the viscous interval (r h), the coefﬁcient of relative diffusion is deﬁned by Eq. (1.27) and Eq. (1.28), which lead to a closed equation for the dispersion: dhr 2 i 2A1 2 hr i: ¼ 3tk dt

ð5:70Þ

The solution of Eq. (5.70) is hr i ¼ 2

r 20 exp

2A1 t : 3tk

ð5:71Þ

The exponential increase of dispersion in the viscous interval (5.71) was ﬁrst discovered by Batchelor (1952b) and later – by Lundgren (1981). Within the inertial interval (h r L), the coefﬁcient of relative diffusion, according to Eqs. (1.19), (1.29), and (1.30), is equal to Drll ¼ CA2 e1=3 r 4=3 :

ð5:72Þ

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Expression (5.72) represents the well-known Richardson four-thirds law for the coefﬁcient of relative diffusion of a passive impurity in the inertial interval. The selfsimilar solution of Eq. (5.67) with the consideration of Eq. (5.72) is Pr ¼

2187 9r 2=3 exp : 4CA2 e1=3 t 560p1=2 e3=2 (CA2 t)9=2

ð5:73Þ

Substituting the probability density distribution of inter-particle distance (5.73) into Eq. (5.64), we arrive at the cubic power law for the dispersion: hr 2 i ¼ CR et3 ;

CR ¼

1144(CA2 )3 ; 81

ð5:74Þ

where CR is the Richardson–Obukhov constant. It should be noted that the calculated values of this constant given in literature vary from 0.1 up to 4. The difﬁculty of experimental determination of CR lies in the necessity of simultaneous measurement of both the relative dispersion of particles and the rate of turbulent energy dissipation. According to Eq. (5.74), at C ¼ 2 and A2 ¼ 0.3 we have CR ¼ 3.05, which is very close to the value CR ¼ 3.06 obtained by Lundgren (1981). For Richardsons law to be realized, several conditions must be fulﬁlled simultaneously, among which are the absence of the viscosity effect (high Reynolds numbers), insufﬁcient inﬂuence of large-scale convection, and loss of memory about the initial distance between the particles. As was established in the experiments by Bourgoin et al. (2006), Richardsons law is valid at t (r 20 =e)1=3 , whereas at smaller time values, the dispersion of ﬂuid particles depends on the initial distance r0 between the particles. In what follows, the initial distance r0 between the particles is supposed to be equal to the spatial microscale h by the order of magnitude. When the distance between the two points is large compared to the spatial macroscale (r > L), ﬂuctuations at these points are statistically independent and the coefﬁcient of relative diffusion is deﬁned by relation (1.37). The self-similar solution of Eq. (5.67), which also factors in the Eq. (1.37), is as follows: 1 r2 Pr ¼ 5=2 : ð5:75Þ exp 8T L u0 2 t 2 p1=2 (T L u0 2 t)3=2 Then in view of Eq. (5.75), the dispersion law takes the form hr 2 i ¼ 12u02 T L t:

ð5:76Þ

It is seen from Eq. (5.76) that for times large compared to the temporal macroscale (t > TL), the relative (binary) dispersion increases linearly with time and consequently the distance between the two particles changes according to the parabolic law. For comparison, consider the dispersion of a single ﬂuid particle in isotropic turbulence at t > TL, when the relation (1.2) for the coefﬁcient of turbulent diffusion is valid, that is, when the Taylor diffusion is taking place. Because of spherical symmetry of the system, we can write the following corollary of the diffusion equation (2.35) for the homogeneous isotropic turbulence with the zero average velocity:

5.5 Relative Dispersion of Two Particles in Isotropic Turbulence

qP x 1 q qPx ¼ 2 ; x 2 Dt qt qx x qx

ð5:77Þ

where Px(x, t) is the probability density for the particle to be at the point x at the moment of time t provided that it was located at the point x ¼ 0 at t ¼ 0. The probability density satisﬁes the initial condition Px(x, 0) ¼ d(x), and the solution of Eq. with the consideration of Eq. (1.2) and of the normalizing condition Ð 1 (5.77) 2 x P dx ¼ 1 is x 0 1 x2 exp Px ¼ : ð5:78Þ 4T L u0 2 t 2p1=2 (T L u0 2 t)3=2 From Eq. (5.78), we obtain the spatial dispersion of a passive impurity from a point source: 1 ð 2 ð5:79Þ hx i ¼ x 4 Px dx ¼ 6u0 2 T L t: 0

Expression (5.79) shows that in isotropic turbulence at large values of time, the time dependence of dispersion obeys a linear law (Taylor, 1921; Batchelor, 1949) similar to the corresponding law for Brownian diffusion. Comparison of Eq. (5.76) with Eq. (5.79) shows that because of non-correlativity of ﬂuid particle pair motion under the conditions r > L and t > TL, the relative dispersion of two particles is twice as large as the dispersion of a single particle, in an analogy with the relation between the coefﬁcients of binary and single diffusion. Thus the model based on the system of equations (5.56)–(5.59) correctly reproduces all of the major results known in the theory of turbulent relative dispersion of a passive impurity. 5.5.2 Dispersion of Inertial Particles

Here we shall ﬁrst examine particle dispersion within the inertial interval, where the relations (1.29) and (1.30) are valid. To this end, we introduce the dimensional variables t¼

t r ; r¼ ; 3=2 tp e1=2 tp

sll ¼

Sll Snn ; snn ¼ : etp etp

w¼

Wr ; (etp )1=2

sp ll ¼

Sp ll ; etp

sp nn ¼

Sp nn ; etp

Switching to new variables and making use of Eqs. (1.29), (1.30), and (5.18), we rewrite the system of equations (5.56)–(5.59) as follows: qP r 1 q þ 2 (r2 Pr w) ¼ 0; qt r qr

ð5:80Þ

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qPr sp ll qP r w 1 qr2 Pr w2 2P r (sp ll sp nn ) qP r þ Pr wg r sll ; ¼ þ 2 qr qr qr r qt r ð5:81Þ

qP r sp ll qsp ll 1 qr2 Pr wsp ll 1 q þ 2 ¼ 2 r2 P r (sp ll þ g r sll ) qt qr qr r r qr qsp nn 4P r (sp ll þ g r sll ) 3r qr þ

2 (sp nn þ g r snn )(sp ll sp nn ) r

2P r (sp ll þ g r sll ) qP r sp nn 1 qr2 P r wsp nn 1 þ 2 ¼ 4 qt qr r 3r

qw þ 2P r ( fr sll sp ll ); qr

ð5:82Þ

qsp nn q r4 P r (sp ll þ g r sll ) qr qr

þ2

q 3 r Pr (sp nn þ g r snn )(sp ll sp nn ) qr

2Pr (sp nn þ g r snn )

Wr þ 2P r ( f r snn sp nn ); r ð5:83Þ

sll ¼ Cr2=3 ;

4 snn ¼ Cr2=3 ; 3

fr ¼

A2 r2=3 ; 1 þ A2 r2=3

gr ¼

A22 r4=3 : 1 þ A2 r2=3 ð5:84Þ

One can see from Eqs. (5.80)–(5.84) that introduction of dimensional variables makes it possible to eliminate particle relaxation time from the list of relevant parameters. Next, we ﬁnd the asymptotic solution of the problem (5.80)–(5.84) for the values r 1;

t 1;

ð5:85Þ

which, when plugged into Eq. (5.84), give us the following values for the involvement coefﬁcients: fr ¼ 1;

g r ¼ A2 r2=3 :

ð5:86Þ

Then, in view of Eq. (5.84) and Eq. (5.86), it follows from Eqs. (5.82) and (5.83) that sp ll ¼ O(r2=3 );

sp nn ¼ O(r2=3 ):

ð5:87Þ

Taking into account Eqs. (5.86) and (5.87), we obtain from Eq. (5.81) w ¼ CA2 r4=3

q ln Pr : qr

ð5:88Þ

5.5 Relative Dispersion of Two Particles in Isotropic Turbulence

self-similar solution (5.80), in view of Eq. (5.88) and the normalizing condition Ð 1The 2 r P r dr ¼ 1, takes the form similar to Eq. (5.73), namely, 0 2187 9r2=3 exp : Pr ¼ 4CA2 t 560p1=2 (CA2 t)9=2

ð5:89Þ

From Eq. (5.89) there follows the expression for the dispersion hr2 i ¼ CR t3 ; which reduces to Richardsons law (5.74) when we switch back to the coordinates r and t. So, far from being limited to ﬂuid particles only, Richardson relative dispersion law (5.74) can also describe a system of inertial particles. However, both its spatial and temporal applicability ranges become narrower with increase of particle inertia. Thus, in the case of inertialess particles Richardsons law holds in the range h r L;

tk t T L ;

however, if the particles are inertial, we must require that, apart from Eq. (5.85), the following conditions should be met: St3=2 h r L;

Sttk t T L :

ð5:90Þ

Conditions (5.90) clearly show the narrowing of the range of applicability of the dispersion law (5.74) with increase of the Stokes number. For high-inertia particles, that is, at large values of the Stokes number St, Richardsons law holds only in the limit of very high Reynolds numbers, when the spatial and temporal inertial intervals are sufﬁciently extended. We now turn to the case of relative dispersion of inertial particles for r > L. Since the motion of particles in this case is non-correlated, the Lagrangian integral time scale of two particles velocity increment TLrp becomes equal to the Lagrangian integral time scale for a ﬂuid particle calculated along the inertial particle trajectory (the so-called time of particle interaction with turbulent eddies) TLrp ¼ TLp. Then in view of Eq. (1.35), from the equations (5.57) and (5.58) there follows an asymptotic expression for t ! 1, Sp ll ¼ Sp nn ¼ 2f u u0 ¼ 2

2u0 2 T Lp ; tp þ T Lp

ð5:91Þ

and Eq. (5.57) complemented by Eq. (5.91) yields W r ¼ Drp

qlnP r ; qr

Drp ¼ 2Dp ¼ 2T Lp u0 ; 2

ð5:92Þ

where Drp and Dp are, respectively, the coefﬁcients of binary and single diffusion of inertial particles in isotropic turbulence for large values of time.

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Solving Eq. (5.56) and taking Eq. (5.92) into consideration, we obtain the distribution 1 r2 ; exp Pr ¼ 5=2 8T Lp u0 2 t 2 p1=2 (T Lp u0 2 t)3=2

ð5:93Þ

which, in turn, generates the following expression for the relative dispersion of particles: hr 2 i ¼ 12T Lp u0 t: 2

ð5:94Þ

The expressions (5.93) and (5.94) will coincide with Eq. (5.75) and Eq. (5.76) if in the latter pair of equations the Lagrangian scale of turbulence TL is replaced with the time of particle interaction with turbulent eddies TLp. Since TLp can exceed TL, the relative dispersion of inertial particles may exceed the relative diffusion of passive impurity, as is the case for the coefﬁcient of turbulent diffusion. Substitution of Eq. (5.93) into Eq. (5.92) results in the expression for the average relative velocity of the particles motion: Wr ¼

r : 2t

ð5:95Þ

In order to determine the region of validity of the distribution (5.93) (and thus of the corresponding dispersion (5.94) and average relative velocity (5.95)), let us estimate the contribution of the unrecorded terms in Eqs. (5.57)–(5.59) that we ignored when deriving the relations (5.91) and (5.92). The terms containing Wr in Eqs. (5.57)–(5.59) will be small as compared to the leading terms (5.91) if fu

T Lp : t

ð5:96Þ

The contribution of the unrecorded terms in Eq. (5.57) will be small as compared to Eq. (5.92) if t tp :

ð5:97Þ

Finally, combining the equations (5.96) and (5.98), we obtain the validity condition for the solutions (5.93) and (5.94): t T Lp þ tp :

ð5:98Þ

Hence, according to Eq. (5.98), the lower limit of the time interval t TLp þ tp for which the linear dependence (5.94) is valid increases with particle inertia.

j209

6 Collision and Clustering of Bidispersed Particles in Homogeneous Turbulence The behavior of a bidispersed system containing two types of particles that is placed in a homogeneous turbulent ﬂow is the subject of this chapter. The theory of the bidispersed system has special importance because it can easily be extended to cover the general case of a polydispersed particle mixture. Two approaches can be employed to describe the characteristics of the disperse phase. The ﬁrst approach is similar to the approach for monodispersed particles presented in Chapter 4. It is based on the joint PDF of ﬂuid and particle velocities, which is given a priori in the form of a correlated Gaussian distribution, and utilizes the Grad method to take into account the anisotropy of particle velocity ﬂuctuations. This approach allows to determine the collision frequency and the collisional terms in the transport equations for hydrodynamic (macroscopic) characteristics of the disperse phase; the effect of particle clustering is neglected. The second approach, which is discussed in last two sections of this chapter, is based on the kinetic equations for the two-particle PDF and thus generalizes the kinetic model presented in Chapter 5 for the case of bidispersed particles.

6.1 Collision Frequency of Bidispersed Particles in Isotropic Turbulence

The number of collisions between particles of different types per unit volume per unit time calculated without consideration of the clustering effect is deﬁned by the relations S12 ¼ wc12 N 1 ¼ wc21 N 2 ¼ bN 1 N 2 ;

b ¼ 2ps2 hjw r (s)ji;

ð6:1Þ

where wc12, wc21 are the collision frequencies of particles of type 1 with particles of type 2 and of particles of type 2 with particles of type 1, respectively; s ¼ r1 þ r2 is the radius of the collision sphere equal to the sum of radiuses of the two particles; ra is the radius of particles of type a; Na is the average number concentration of particles of type a; a ¼ 1, 2; wr is the radial component of relative velocity of two particles of different types. The expression for the collision kernel in Eq. (6.1) is assumed to be identical to the collision kernel for monodispersed particles (4.1);

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this assumption holds if the relative motion of the particle pair possesses spherical symmetry, which is the case for the turbulence that is both homogeneous and isotropic. Simple analytical solutions for the problems of collisions of bidispersed (as well as monodispersed) particles in turbulent ﬂows have been obtained for the limiting cases of small (inertialess) and large (inertial) particles (Saffman and Turner, 1956; Abrahamson, 1975). Zhou et al. (2001) have carried out an extensive study of bidispersed particle collisions in isotropic turbulence by the DNS method and compared the obtained results with predictions of the models proposed by Abrahamson (1975), Williams and Crane (1983), Kruis and Kusters (1996), and Zhou et al. (1998). All of these models determine only the relative velocity of two particles, which is responsible for turbulent transport regardless of the clustering effect. This comparison showed that Abrahamsons (1975) model results is a signiﬁcant overstating of the collision kernel for low-inertia particles, since it fails to take into account the correlation between velocities of colliding particles. The models by Williams and Crane (1983) and Kruis and Kusters (1996), on the other hand, predict understated values for the collision kernel as compared to the DNS data. The model by Zhou et al. (1998) generalized for the case of collisions of bidispersed particles by Zhou et al. (2001) is in good agreement with the numerical data except when the relaxation times tp1 and tp2 for particles of types 1 and 2 are close to the Kolmogorov time microscale tk, in which case the effect of clustering becomes signiﬁcant. An empirical model for the collision kernel obtained by approximating the dependencies suggested by the DNS results for the relative velocity and radial distribution function of colliding particles is also presented by Zhou et al. (2001). The present section generalizes the analytical model for the turbulent collision kernel of monodispersed particles considered in Section 4.1 for the case of a bidispersed system (see Fede and Simonin, 2003; Zaichik et al., 2006). The distinguishing feature of this model is a uniform treatment of particle interactions with all turbulent eddies over the whole range of particle inertia. The model is based on the assumption that the one-point PDF of ﬂuid and particle velocities is represented by the normal distribution (4.2) Pð0Þ (ua (x);va (x)) ¼

v0 a ¼ 2

hv0ak v0ak i ; 3

2 3=2 Fa 1xa (2pu0 v0a )3 0 0 uak uak v0ak v0ak xa u0ak v0ak 1 exp þ ; u0 v0a 2u0 2 2v0 2a 1x2a

xa ¼

hu0ak v0ak i hu0k u0k i1=2 hv0ak v0ak i1=2

ð6:2Þ

;

where ua is the velocity of the ﬂuid at the point coinciding with the center of particle a; va – the velocity of particle a; Fa and v0 2a – respectively, the volume concentration and intensity of velocity ﬂuctuations of particle a; xa – the correlation coefﬁcient for the velocities of particle a and the ﬂuid.

6.1 Collision Frequency of Bidispersed Particles in Isotropic Turbulence

The two-point PDF of ﬂuid velocity is assumed to be Gaussian and deﬁned by Eq. (4.3). The velocity PDF of a particle pair is deﬁned by analogy with Eq. (4.6), making use of relations (4.4) and (4.5). Integration over the subspace of ﬂuid velocities with the consideration of Eq. (4.3) and Eq. (6.2) yields ð Pð0Þ (v1 (x); v2 (x þ r)) ¼ Pð0Þ ðu1 ; v1 ; u2 ; v2 )du1 du2 1=2 1 1x412 Gðr)2 F1 F2 1x412 Fðr)2 ¼ (2pv01 v02 )3 bij v01i v01j bij v02i v02j x212 bik Bkj v01i v02j exp þ ; ð6:3Þ 2v0 21 2v0 22 u0 2 v01 v02 0 B bij ¼ @

1x412 F(r)2

0

0

0

1x412 G(r)2

0

0

0

11 C A ;

x12 ¼ (x1 x2 )1=2 :

1x412 G(r)2

Further integration of the PDF (6.3) with respect to the velocity components v1n and v2n normal to the line of centers of the two particles gives us the PDF of particles radial velocity distribution: ðð Pð0Þ (v1r (x); v2r (x þ r)) ¼ Pð0Þ (v1 (x); v2 (x þ r))dv1n dv2n ¼

(1z212 )1=2 2pv01 v02 02 1 v 1r v0 22r 2z12 v01r v02r exp þ ; v01 v02 v0 22 2(1z212 ) v0 21

z12 ¼ x1 x2 F(r);

ð6:4Þ

where z12 is the correlation coefﬁcient of the radial velocity components of two particles arising due to their interaction with the turbulence. The two-particle PDF (6.4) leads to the distribution (4.8) for the radial component of particle pairs velocity, with ﬂuctuation intensity at the point of contact equal to hw0 r i ¼ v0 1 þ v0 2 2z12 v01 v02 : 2

2

2

ð6:5Þ

Due to the relations (4.11) for the variance of particle velocity ﬂuctuations and the covariance of ﬂuid and particle velocity ﬂuctuations in homogeneous isotropic turbulence, the correlation coefﬁcient of radial velocity components of colliding particles takes the following form: z12 ¼ ( fu1 fu2 )1=2 F(s):

ð6:6Þ

Hence the correlation coefﬁcient of particle velocities at the point of contact is the product of the geometric mean of the coefﬁcients fua that characterize particles involvement in turbulent motion, and the correlation function F(s), which helps us take into account spatial correlativity of ﬂuid velocity over the distance equal to the collision radius s.

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In view of Eqs. (4.10), (6.5), and (6.6), the radial relative velocity of colliding particles is equal to 1=2 0 2 hjwr ji ¼ u: ð6:7Þ fu1 þ fu2 2fu1 fu2 F(s) p Substitution of Eq. (4.14) and Eq. (6.7) into Eq. (6.1) gives us the following formula for the collision kernel of particles whose diameters are smaller than the Kolmogorov spatial microscale: 1=2 1=2 b ¼ 8phw0 2r i ¼ (8p)1=2 s2 v0 21 þ v0 22 2z12 v01 v02 1=2 2 s 1=2 2 0 ¼ (8p) s u fu1 þ fu2 2fu1 fu2 1 1=2 : 60 Rel ð6:8Þ For monodispersed particles, formula (6.8) reduces to Eq. (4.15). Consider some limiting cases that follow from Eq. (6.8). For inertialess particles of type 2 (tp2 ¼ 0, fu2 ¼ 1), Eq. (6.8) reduces to 2 1=2 fu1 s : ð6:9Þ b ¼ (8p)1=2 s2 u0 1fu1 þ 1=2 15 Rel If particles of type 1 are inertialess too, then expression (6.9) reduces to the Saffman–Turner formula (4.17). Suppose now that particle relaxation time lies within the inertial interval (tk tpa TL) and consider the case of high Reynolds numbers (Rel ! 1). According to Eqs. (4.18)–(4.19), the collision kernel (6.8) equals b ¼ 2s2 [pC01 e(tp1 þ tp2 )]1=2 : Finally, in the limiting case of high-inertia particles (tpa ! 1) when the motion of particles is non-correlated (z12 ! 0), formula (6.8) turns into the Abrahamson collision kernel (Abrahamson, 1975) 2 2 1=2 bA ¼ (8p)1=2 s2 v0 1 þ v0 2 : ð6:10Þ If only the particles of type 2 possess high inertia (tp2 ! 1, fu2 ! 0), then Eq. (6.8) leads to the expression b ¼ (8pfu1 )1=2 s2 u0 :

ð6:11Þ

The involvement coefﬁcients fua entering Eq. (6.8) are deﬁned by the relations (4.23) ensuing from the bi-exponential autocorrelation function (4.22). Figures 6.1–6.3 compare the above-described analytical model with the numerical simulation data obtained by Zhou et al. (2001). The results are presented as dependences involving the ratio of the relaxation time of particles 2 to the time macroscale of turbulence Te ¼ u0 2/e, relaxation time of particles 1 (i.e., parameter tp1/Te) being ﬁxed. In order for the analytical results to be germane to the data presented by Zhou et al. (2001), we run the comparison at r ¼ s ¼ 1 and Rel ¼ 45.

6.1 Collision Frequency of Bidispersed Particles in Isotropic Turbulence

Figure 6.1 The correlation coefficient of velocities of a particle pair at s ¼ 1 and Rel ¼ 45: 16 – formula (6.6); 58 – Zhou et al. (2001); 1, 5 – tp1/Te ¼ 0.1; 2, 6 – tp1/Te ¼ 0.4; 3, 7 – tp1/Te ¼ 1; 4, 8 – tp1/Te ¼ 2.

Figure 6.2 Relative radial relative velocity of colliding particles at r ¼ 1 and Rel ¼ 45: 1, 2 – bidispersed system; 3 – monodispersed system; 1 – formula (6.7); 2, 3 – model (6.49)–(6.52); 4 – Zhou et al. (2001); a – tp1/Te ¼ 0.1; b – tp1/Te ¼ 0.2; c – tp1/Te ¼ 1; d – tp1/Te ¼ 2.

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Figure 6.3 Collision kernel at s ¼ 1 and Rel ¼ 45: 1, 2 – bidispersed system; 3 – monodispersed system; 1 – formula (6.8); 2, 3 – model (6.49)–(6.52); 4 – Zhou et al. (2001); a – tp1/Te ¼ 0.1; b – tp1/Te ¼ 0.2; c – tp1/Te ¼ 1; d – tp1/Te ¼ 2.

Figure 6.1 presents the correlation coefﬁcient of radial components of velocity of colliding particles. We see that everywhere except for a narrow range of values tp2/Te, Eq. (6.6) describes the variation of z12 with acceptable accuracy, predicting a monotonous decrease of the correlation coefﬁcient between particle velocities with increase of both tp1 and tp2. Such behavior of z12 is explained by the decreased involvement of heavier particles in ﬂuctuational motion of the carrier medium, which means greater independence of particles from the turbulence. Comparison with the DNS results shows that Eq. (6.6) fails to reproduce the small spike in the correlation coefﬁcient at small values of tp2/Te and exaggerates the actual drop of the correlation coefﬁcient at large values of tp2/Te. Figure 6.2 demonstrates the inﬂuence of the inertia parameter on the average relative radial velocity of colliding particles divided by the ﬂuid velocity ﬂuctuation. Due to Eq. (6.7), the quantity h|wr|i/u0 grows monotonously with tp2/Te at small values of parameter tp1/Te and decreases with tp2/Te at large values of this parameter. The increase of h|wr|i/u0 is caused by the decreased correlation between particle velocities as particle inertia grows; the drop of h|wr|i/u0 is associated with the decreased

6.2 Collision Frequency in the Case of Combined Action of Turbulence and Gravity

involvement of heavier particles in turbulent motion of the surrounding ﬂuid. It is evident from Figure 6.2 that formula (6.7) fails to explain the trough on the graph of h|wr|i/u0 versus tp2/Te that was discovered by Zhou et al. (2001), which becomes more pronounced as the parameter tp1/Te gets smaller. Figure 6.3 compares the ratio of the collision kernel to the corresponding value for inertialess particles (4.17) with the results of numerical simulation. Similarly to Eq. (6.7) for h|wr|i, formula (6.8) predicts a monotonous variation of b with tp2/Te at ﬁxed values of tp1/Te. We see that Eq. (6.8) replicates the DNS results with sufﬁcient accuracy except for the region around tp1/Te ¼ 0.2 characterized by the presence of a maximum. Overall, the considered analytical model shows much better agreement with the DNS results for the collision kernel than with the DNS results for the relative radial velocity. It appears that this accuracy improvement can be attributed to the fact that the inaccuracy in the description of the effect of turbulent transport and the error we made when neglecting the effect of clustering cancel each other.

6.2 Collision Frequency in the Case of Combined Action of Turbulence and Gravity

Let us try to determine the frequency of collisions resulting from the combined effects of turbulence and the average relative velocity (drift) between particles of different types that is produced by the action of some external force, for example, gravity. Similarly to the case of combined action of turbulence and the average velocity gradient of the ﬂow (see Section 4.2), in order to calculate h|wr|i in Eq. (6.1) we must perform averaging over the random distributions of wr and the spatial angle characterizing orientation of the relative velocity vector w about the vector r connecting the centers of colliding particles (see Eq. (4.29)). By integrating the Gaussian distribution for the ﬂuctuational component of radial relative velocity, we obtain expression (4.30). The radial component of the average relative velocity of two particles arising due to the force of gravity is equal to W r ¼ W g cosj;

ð6:12Þ

where Wg ¼ |tp2 tp1|g is the difference of sedimentation rates of the two particles, and g is the acceleration of gravity. Integration of Eq. (4.30) with the consideration of Eq. (6.12) and the subsequent insertion of the result into Eq. (6.1) gives the collision kernel describing the combined action of turbulence and gravity: 2 p1=2 1 0 2 1=2 2 exp(S ) b ¼ 8phw r i s Sþ þ erf S ; ð6:13Þ 2 2 2S where S ¼ W g =(2hw 0 2r i)1=2 is a parameter characterizing the interrelation between the effects of gravity and turbulence on the collision kernel. At small values of S there follows from Eq. (6.13) an expression for b that coincides with the solution obtained by Saffman and Turner (1956) when the effect of gravity is weak:

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S2 b ¼ bt 1 þ ; 3 where bt is the turbulent component of the collision kernel (6.8). At S ! 1, Eq. (6.13) reduces to the well-known expression for the collision kernel of particles under the action of the gravitational force: bg ¼ ps2 W g :

ð6:14Þ

Dependence (6.13) offers an accurate description of the effect of average relative velocity on the collision kernel in isotropic turbulence not only in the presence of gravity force but also in the presence of any other body force that causes relative motion of particles. Formula (6.13) was ﬁrst obtained by Abrahamson (1975) and later reproduced by Gourdel et al. (1999), Alipchenkov and Zaichik (2001), and Dodin and Elperin (2002). Figure 6.4 compares time intervals between collisions of particles of different types found from Eq. (6.13) with the results of Lagrangian trajectory simulation of particle motion in a turbulent ﬁeld employing the LES method (Gourdel et al., 1999) for a mixture of particles of two types having the same radius ra ¼ 325 mm but different densities (rp1 ¼ 117.5, rp2 ¼ 235 kg/m3). Volume concentration of particles 1 was set equal to F1 ¼ 1.3102, while the volume concentration of particles 2 varied (volume and number concentration of particles are connected by the relation Fa ¼ 4pr 3a N a =3). The time intervals between collisions of particles 1 with particles 2 and of particles 2 with particles 1 are respectively equal to 1 t12 ¼ w1 12 ¼ (bN 2 ) ;

1 t21 ¼ w1 21 ¼ (bN 1 ) :

Since the particles under consideration are highly inertial, correlativity of their motion and their interactions with small-scale turbulence do not play a signiﬁcant role. Therefore in order to determine bt, one can use Abrahamsons formula (6.10).

Figure 6.4 Time intervals between particle collisions in a binary mixture: 1, 2, 3, 7 – t12; 4, 5, 6, 8 – t21; 1, 4 – Eq. (6.13) with the consideration of Eq. (6.10); 2, 5 – Eq. (6.10); 3, 6 – Eq. (6.14); 7, 8 – Gourdel et al. (1999).

6.3 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow

Also shown on Figure 6.4 are the time intervals between collisions calculated by the formula (6.10) that describes the effect of turbulence while neglecting the relative drift and by the formula (6.14) that considers the force of gravity while neglecting the contribution of turbulence. One can see that both formulas overstate t12 and t21 as compared to the values calculated by Gourdel et al. (1999); this is especially true for Eq. (6.10) at small concentrations and for Eq. (6.14) at large values of F2. We must also mention that as F2 grows, the increased collision frequency w12 results in a smaller difference between the average velocities of particles 1 and 2, that is, in a smaller relative drift velocity Wg. Therefore t12 and t21 are better described by Eq. (6.14) at small F2 and by Eq. (6.10) at large F2. Formula (6.13), which includes both of these two effects – turbulence and gravity – ensures reasonably good agreement with the data obtained by Gourdel et al. (1999), especially if we consider the fact that due to the presence of a vertical component of the average velocity, the intensity of particle velocity ﬂuctuations is not isotropic.

6.3 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow

Dynamic interaction of particles of different types will be deﬁned by a relation that generalizes Eq. (4.36) for the case of different masses of colliding particles m1 and m2: b

vp1 ¼ vp1 þ

m2 m1 b (1þe)(wp l)l; vp2 ¼ vp2 (1þe)(wp l)l; m1 þm2 m1 þm2

ð6:15Þ

Statistical description of motion of each type of particles is based on the kinetic equation for the single-point (single-particle) PDF that is similar to Eq. (4.37): qP a qP a q þ þ vk qt qx k qvk ¼

U k vk þ F ak Pa tpa

1 qhu0k pia qPa a qPa b þ þ ; qt qt tpa qvk coll

ð6:16Þ b 6¼ a:

coll

The last two terms on the right-hand side of Eq. (6.16) take into account, respectively, the collisions of particles of the type under consideration and the collisions of particles of the other type. Integration of the kinetic equation over the velocity subspace yields a chain of equations for single-point moments. The discussion below will be restricted to homogeneous ﬂows, so the third moments vanish and the chain of equations is broken on the level of second moments. Besides, insofar as the volume concentration of particles of both types is supposed to be small, direct contribution of inter-particle collisions to the stress and the ﬂux of ﬂuctuational energy of the disperse phase can be neglected, and consequently, collisional terms appear in the system of equations for the moments only as sources and never

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appear as ﬂuxes. Hence the system of conservation equations for particles a has the following form: qFa qFa V ak þ ¼ 0; qt qx k

ð6:17Þ

Dp ik q lnFa qhv0 v0 i U i V ai qV ai qV ai ¼ ai ak þ þF ai þC ai ; þ V ak qt qx k tpa tpa qx k qx k a

qhv0ai v0aj i qt

þ V ak

qhv0ai v0aj i qx k

¼ (hv0ai v0ak i þ maik ) a

a

þ lij þ lji

ð6:18Þ

qV aj qV ai (hv0aj v0ak i þ majk ) qx k qx k

2hv0ai v0aj i tpa

ab þ C aa ij þ C ij ;

b 6¼ a; ð6:19Þ

where Fa, Vai, hv0ai v0aj i denote the volume concentration, average velocity, and a turbulent stresses of particles a, respectively. The quantities lij , maij and the a coefﬁcients of turbulent diffusion Dp ik for particles a are deﬁned, as before, by the expressions (2.16), (2.17), and (2.26). It is evident that, as follows from Eq. (6.17), collisions do not change the volume concentration of particles. But collisions with particles of the other type contribute to the equation of momentum conservation through the term C ai in Eq. (6.18). Obviously, collisions between particles of one and the same type do not inﬂuence the average velocity of these particles. The terms ab C aa ij and C ij in Eq. (6.19) characterize the contribution of particles of one and the same type and of particles of different types to the balance of turbulent stresses. To close the system (6.17)–(6.19), we only need to determine C ai and C ab ij , since the quantity C aa was already determined in Section 4.3 by Eq. (4.57). In order to ﬁnd ij the collisional terms, it is necessary to know the PDF of velocities of a particle pair. As in Section 4.3, we use Grads method to represent the two-point PDF as an expansion: P(v1 ; v2 ) ¼ Pð0Þ (v1 ; v2 ) þ Pð1Þ (v1 ; v2 ):

ð6:20Þ

The zeroth term of the expansion (6.20) represents the normal equilibrium distribution, which may take place in isotropic turbulence. If we restrict ourselves to the collisions of relatively small particles (s h) at high Reynolds numbers, when, according to Eq. (4.48), spatial correlation functions can be taken equal to unity, then Eq. (6.3) for bidispersed particles leads us to F1 F2 3 3=2 2pv01 v02 1z212 0 0 v1k v1k v02k v02k 2z12 v01k v02k 1 exp þ 02 ; v01 v02 v0 21 v1 2 1z212

P ð0Þ (v1 ; v2 ) ¼

ð6:21Þ

6.3 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow

j219

where the correlation coefﬁcient of the two particles is z12 ¼ ( fu1 fu2 )1=2 :

ð6:22Þ

The ﬁrst term of the expansion (6.20) takes into account the anisotropy of particle velocities. Following Grads procedure, we represent it as R1ij q2 R2ij q2 q q ð1Þ P (v1 ; v2 ) ¼ R1i þ R2i þ þ 2 qv1i qv1j 2 qv2i qv2j qv1i qv2i 2 2 Qij q q þ þ P ð0Þ (v1 ; v2 ): 2 qv1i qv2j qv2i qv1j ð6:23Þ Coefﬁcients Rai and Raij in Eq. (6.23) are found from the conditions ðð ðð 1 1 0 0 vai P(v1 ;v2 )dv1 dv2 ¼ hvai i ¼ 0; v0ai v0aj P(v1 ; v2 )dv1 dv2 ¼ hv0ai v0aj i; F1 F2 F1 F2 ðð 1 v0ai v0bj P(v1 ;v2 )dv1 dv2 ¼ hv0ai v0bj i; b 6¼ a; F1 F2 from which, in accordance with Eqs. (6.20), (6.21), and (6.23), it follows that Rai ¼ 0; Raij ¼ hv0ai v0aj iv0 a d ij ; 2

Qij ¼

hv01i v02j i þ hv01j v02i i 2

z12 v01 v02 dij :

ð6:24Þ ð6:25Þ

Assume a linear relationship between the correlation moments of velocity ﬂuctuations of different particles and the corresponding moments for identical particles: hv01i v02j i þ hv01j v02i i 2

¼ C1 hv01i v01j i þ C2 hv02j v02i i:

ð6:26Þ

Then we have due to the expression (6.21) for the PDF hv01k v02k i ¼ 3z12 v01 v02 ; and the trace of (6.26) is written as C1 v0 1 þ C2 v0 2 ¼ z12 v01 v02 : 2

2

ð6:27Þ

Since C1 and C2 are connected by a single relation (6.27), we still have one degree of freedom when choosing these coefﬁcients. We thus impose an additional constraint by demanding that the form of the ﬁrst term of the expansion P(1)(v1,v2) should be as simple as possible. With this in mind, we require that the terms containing velocities of particles of one type should be independent of the characteristics of particles of the other type. This condition is satisﬁed if we take C1 ¼

z12 v02 ; 2v01

C2 ¼

z12 v01 : 2v02

ð6:28Þ

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In view of Eqs. (6.24)–(6.26) and Eq. (6.28), the anisotropic component of the PDF (6.20) takes the form R1ij v1i0 v1j0 R2ij v2i0 v2j0 R1ij R2ij z12 (v1i0 v2j0 þ v1j0 v2i0 ) Pð1Þ (v1 ; v2 ) ¼ þ þ 0 2v10 v21 v 0 41 v 0 42 v 0 21 v 0 22 Pð0Þ (v1 ; v2 ) : 2 1z212

ð6:29Þ

Let us now switch to new variables characterizing the motion of a bidispersed system and the relative motion of the two types of particles: q ¼ k 1 v2 þ k2 v1 ; w ¼ v2 v1 ; v 0 2 z v 0 v 0 v 0 2 z v 0 v 0 k 1 ¼ 0 2 1 0 2 12 1 2 0 0 ; k 2 ¼ 0 2 2 0 2 12 1 2 0 0 : v 1 þ v 2 2z12 v1 v2 v 1 þ v 2 2z12 v1 v2 Using Eq. (6.21) and Eq. (6.29), we can represent the PDF (6.20) in these new variables: P(w; q) ¼ P ð0Þ (w; q) þ P ð1Þ (w; q); w 0 2 qk0 qk0 w k0 wk0 F1 F2 ð0Þ P (w; q) ¼ ; 3=2 exp 2 1z212 v 0 21 v 0 22 2w 0 2 (2pv10 v20 )3 1z212

ð6:30Þ

P ð1Þ (w; q) ¼ Aij qi0 qj0 þ Bij w i0 w j0 þ Cij (qi0 w j0 þ qj0 w i0 ) Pð0Þ (w; q); where

R1ij R2ij 1 v01 v02 Aij ¼ 1z þ 1z ; 12 0 12 0 v2 v1 v0 41 v0 42 2 1z212 2 k 1 R1ij k 22 R2ij 1 k 2 v01 k1 v02 1 þ z þ 1 þ z ; Bij ¼ 12 12 k 1 v02 k2 v01 v0 41 v0 42 2 1z212

R2ij R1ij 1 z12 (k 2 k1 )v02 z12 (k 1 k 2 )v01 þ þ k k ; Cij ¼ 2 1 2v01 2v02 v0 42 v0 41 2 1z212 w 0 2 ¼ v0 21 þ v0 22 2z12 v01 v02 :

First of all, let us employ Eq. (6.30) to derive the average relative radial velocity between particles of different types that we need to know in order to determine the collision kernel: hjwr ji ¼ hjw r jið0Þ þ hjw r jið1Þ ; hjwr ji

ð0Þ

1 ¼ 2pF1 F2

ððð wl<0

(w l)Pð0Þ (w; q)dldwdq ¼

ð6:31Þ W F 0 (z); 2

ð6:32Þ

6.3 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow

hjwr jið1Þ ¼

1 2pF1 F2

ððð

(w l)Pð1Þ (w; q)dldwdq ¼

wl<0

Bij W i W j W Y0 (z); 2

ð6:33Þ

pﬃﬃﬃ exp(z) 1 F 0 (z) ¼ pﬃﬃﬃﬃﬃ þ erf z 1 þ ; 2z pz pﬃﬃﬃ 3 3exp(z) 1 : Y0 (z) ¼ 1=2 5=2 þ erf z 8z3 4z2 4p z Here z denotes the parameter of the average relative drift of particles of different types, which is deﬁned as the ratio of the average and ﬂuctuational components of relative velocity: z¼

W2 ; W i ¼ V 2i V 1i ; W ¼ (W k W k )1=2 : 2w0 2

Due to Eq. (6.1) and Eq. (6.31), we arrive at the following expression for the collision kernel that also factors in the relative drift between the particles: b ¼ 2ps2 hjwr ji ¼ 2ps2 hjw r jið0Þ þ hjw r jið1Þ ¼ bð0Þ þ bð1Þ :

ð6:34Þ

The collision kernel (6.34) is a sum of two terms, the ﬁrst of which is responsible for isotropic ﬂuctuations of particle velocities and the second – for anisotropy of ﬂuctuations. In view of the equalities Wg ¼ W, hw 0 2r i ¼ w 0 2 , and S2 ¼ z, b(0) coincides with Eq. (6.13). Since BijWiWj ¼ O(z), it is easy to get lim

bð1Þ

z!0 b

¼ O(z); lim

ð0Þ

bð1Þ

z ! 1 bð0Þ

¼ O(z1 ):

Hence in the limiting cases of small and large values of the average drift the contribution of particle velocity ﬂuctuations to the collision kernel will be insigniﬁcant. It appears that anisotropy can play a noticeable role only at z ¼ O(1). Making use of the collision equation (6.15) and the PDF (6.30), we express the collisional term in the momentum conservation equation for particles a as follows: að0Þ

C ai ¼ C i að0Þ

Ci

að1Þ

þ Ci

;

ð6:35Þ ððð

¼

s2 N b mb (1 þ e) Fa Fb (ma þ mb )

¼

ps N b mb (1 þ e)W W i F 1 (z); 2(ma þ mb ) 2

(w l)2 li P ð0Þ (w; q)dldwdq

wl<0

ð6:36Þ

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að1Þ

Ci

ððð

¼

s2 N b mb (1 þ e) Fa Fb (ma þ mb )

¼

Bjk W i W j W k ps N b mb (1 þ e)w0 4 W Y (z) (z) ; Y WB 2 ik k 1 2W (ma þ mb )W 2

(w l)2 li P ð1Þ (w; q)dldwdq

wl<0

2

pﬃﬃﬃ exp(z) 1 1 1 1þ þ erf z 1 þ 2 ; F 1 (z) ¼ pﬃﬃﬃﬃﬃ 2z z 4z pz Y1 (z) ¼ 4F 0 (z)3F 1 (z);

Y2 (z) ¼ Y1 (z)

2Y0 (z) : z

ð6:37Þ

When the correlation coefﬁcient of particle velocities vanishes, the relation (6.36) reduces to the expression obtained by Gourdel et al. (1999). In the limiting cases of small and large values of the drift parameter, it follows from Eqs. (6.36)–(6.37) that að1Þ

lim

Ci

z ! 0 C að0Þ i

að1Þ

¼ O(1); lim

Ci

z ! 1 C að0Þ i

¼ O(z1 ):

So the contribution of anisotropy of particle velocity ﬂuctuations to C ai is more important in the case of small values of the drift parameter. Since the quantity C aa ij in the balance equation for turbulent stresses (6.19) is deﬁned by the relation (4.57), let us now proceed to derive the collisional term C ab ij . Calculating collision integrals appearing in the collision law (6.15) and in the expression for the PDF (6.30), we obtain abð0Þ

C ab ij ¼ C ij abð0Þ ¼ C ij

abð1Þ

þ C ij

;

s2 N b mb (1þe) Fa Fb (ma þmb )

ð6:38Þ ðð ð wl<0

mb (1þe) 3 2 (wl) li lj k a (wl) (w i lj þw j li ) (ma þmb )

að0Þ að0Þ Pð0Þ (w;q)dldwdqþk a C i W j þC j W i ¼ps2 N b W

mb (1þe) mb (1þe) W2 W iW j þ dij F 2 (z) 3 ma þmb 4(ma þmb )

W2 3F 1 (z) 2F 0 (z) W2 3F 1 (z) d ij dij k a W i W j W iW j ; W iW j 3 3 2z z 2z F 2 (z)¼F 1 (z)þ

2F 0 (z) z

ð6:39Þ

6.3 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow ab(1)

C ij

¼

s2 N b mb (1þe) Fa Fb (ma þmb )

ððð wl<0

mb (1þe) (wl)3 li lj ka (wl)2 (w i lj þw j li ) (ma þmb )

að1Þ að1Þ P ð1Þ (w;q)dldwdqþka C i W j þC j W i

¼

ps2 N b W(1þe)2 m2b 4(ma þmb )2

4 2 2w 0 Bij Y3 (z)þw 0 [dij W k W n Bkn

þ2(W i Bjk þW j Bik )W k ] ps2 N b W

Y4 (z) Y5 (z) W i W j W k W n Bkn 2z 4z2

(1þe)mb Y2 (z) 4 2 ka 2w 0 Bij Y3 (z)w0 dij W k W n Bkn ma þmb 4z2

þw0 (W i Bjk þW j Bik )W k 2

v0 2a v0 2b 1z212

Y3 (z)¼ F 1 (z)þ

w0 4

Y6 (z) Y7 (z) W i W j W k W n Bkn 2z 2z2

Y1 (z) 2w Cij F 1 (z)þw (W i Cjk þW j Cik )W k 2z 04

02

;

2Y1 (z) Y2 (z) 7Y2 (z) ; Y4 (z)¼Y1 (z) ; Y5 (z)¼Y1 (z) ; z 2z 2z

Y6 (z)¼ Y1 (z)

Y2 (z) Y2 (z)þY5 (z) ; Y7 (z)¼ : z 2

ð6:40Þ

The convolution (6.39) gives the collisional term in the balance equation for abð0Þ the ﬂuctuational energy of particles C kk =2 that coincides with the expression obtained by Gourdel et al. (1999) if the correlation of particle velocities is neglected (z12 ¼ 0). If there is no average drift (z ¼ 0), we obtain from Eqs. (6.39)–(6.40) abð0Þ

¼

abð1Þ

¼

C ij

C ij

8(2p)1=2 s2 N b mb (1 þ e)w 0 3 mb (1 þ e) ka dij ; 3(ma þ mb ) 2(ma þ mb ) 16(2p)1=2 s2 N b mb (1 þ e)w0 5 m a þ mb v0 2a v0 2b 1z212 mb (1 þ e) 2ka Bij þ C ij : 5 10(ma þ mb ) 3w 0 4

ð6:41Þ

ð6:42Þ

The convolution (6.41) produces the same collisional term in the balance equation for the ﬂuctuational energy of particles that was obtained by Fede and Simonin (2003). The physical meaning of Eqs. (6.41)–(6.42) becomes especially clear when we neglect particle velocity correlations, expressing dissipative processes and exchange processes in an explicit form (Reade and Collins, 1998; Fede and Simonin, 2005).

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Equations (6.41)–(6.42) acquire a particularly simple form when we neglect the correlation of particle velocities (Fede and Simonin, 2005): 1=2 8(2p)1=2 s2 N b mb v0 2a þ v0 2b abð0Þ C ij ¼ 3(ma þ mb )2 mb (1e2 ) v0 2a þ v0 2b 2 2 þ (1 þ e)(mb v0 b ma v0 a ) d ij ; 2 1=2 8(2p)1=2 s2 N b mb (1 þ e) v0 2a þ v0 2b (3e)mb abð1Þ C ij (Raij þ Rbij ) ¼ ma þ mb 10(ma þ mb ) þ

mb v0 2b ma v0 2a mb Rbij ma Raij (Raij þ Rbij ) þ 2 2 1=2 3(ma þ mb ) 0 0 (ma þ mb ) v a þ v b

If the particles are of the same type, the relations (6.38)–(6.40) transform to abð0Þ

¼

4p1=2 s2 N b (1e2 )v0 3 (1z12 )3=2 dij ; 3

abð1Þ

¼

4p1=2 s2 N b (1 þ e)(3e)v0 (1z12 )3=2 Rij ; 5

C ij C ij

which is identical to Eq. (4.57). When the drift parameter goes to inﬁnity, a comparison of Eq. (6.39) to Eq. (6.40) yields að1Þ

lim

C ij

z ! 1 C að0Þ ij

¼ O(z1 ):

Thus contribution of particle velocity anisotropy to the collisional term in the balance equation for turbulent stresses becomes essential only when the relative drift is small. Consider the contribution of collisional terms to the balance equations for the characteristics of the entire bidispersed system that consists of particles of both types. It follows from Eqs. (6.36)–(6.37) that collisions do not exert any inﬂuence on the momentum V i (rp1 F1 V 1i þ rp2 F2 V 2i )=(rp1 F1 þ rp2 F2 ) of a bidispersed system. Collisional terms in the equation for the total turbulent energy kp (rp1 F1 kp1 þ rp2 F2 kp2 )=(rp1 F1 þ rp2 F2 ) of the bidispersed system are represented as I C kp ¼ C Ikp þ C II kp ; C kp ¼

22 N 1 m1 C 11 kk þ N 2 m2 C kk ; 2(N 1 m1 þ N 2 m2 )

C II kp ¼

21 N 1 m1 C 12 kk þ N 2 m 2 C kk : 2(N 1 m1 þ N 2 m2 )

Quantities C Ikp and C II kp characterize the respective contributions of collisions between particles of one and the same type and of different types to the average turbulent energy of a mass-dispersed system. Due to Eq. (4.57), C Ikp vanishes for elastic collisions and is negative for inelastic collisions, in other words, collisions of identical particles can lead only to dissipation of ﬂuctuational kinetic energy of the

6.3 Collisions of Bidispersed Particles in a Homogeneous Anisotropic Turbulent Flow

j225

disperse phase. In the case of collisions of different particles the situation is not so unambiguous. Due to Eqs. (6.38)–(6.40), the collisional term C II kp is equal to IIð0Þ

C II kp ¼ C kp IIð0Þ

¼

C kp

IIð1Þ

C kp

IIð1Þ

þ C kp ;

ð6:43Þ

21=2 ps2 N 1 N 2 m1 m2 (1þe)w 0 3 (1þe)z3=2 F 2 (z) P(z); P(z) ¼ 2z1=2 F 0 (z); (m1 þm2 )(N 1 m1 þN 2 m2 ) 2 ð6:44Þ

¼

21=2 ps2 N 1 N 2 m1 m2 (1 þ e)w 0 3 W k W n Bkn 8(m1 þ m2 )(N 1 m1 þ N 2 m2 ) (1 þ e) 7Y4 (z) þ Y5 (z) Y7 (z) Y6 (z) 3Y2 (z) þ þ : 2z1=2 z1=2 4z3=2

ð6:45Þ

Expressions for the two limiting cases readily follow from Eqs. (6.44)–(6.45): IIð1Þ

lim

C kp

z!0 C IIð0Þ kp

IIð1Þ

¼ O(z);

lim

C kp

z!1 C IIð0Þ kp

¼ O(z1 ):

Therefore for the purpose of qualitative examination of the dependence of C II kp on the IIð1Þ drift parameter z, we can think of the contribution of C kp to the sum (6.43) as IIð0Þ negligible compared to C kp . Thus the effect of collisions between particles of different types on the kinetic energy of a bidispersed system is deﬁned by the dependence P(z) entering Eq. (6.44). This dependence is shown on Figure 6.5 for different values of the restitution coefﬁcient e. One can see that the combination of elastic collisions and noticeable relative drift results in the generation of ﬂuctuational energy, that is to say, it causes turbulization of the bidispersed particle ﬂow. The kinetic energy of ﬂuctuations comes from the average motion due to the difference of average velocities of particles of different types. The outcome of the combination of slow drift and inelastic collisions is kinetic energy dissipation; as we increase the drift parameter, the laminarization effect of collisions changes to that of turbulization. Hence collisions can lead either to the

Figure 6.5 The effect of the drift parameter on the behavior of the II (0) collisional term C kp : 1 – e ¼ 0; 2 – e ¼ 0.5; 3 – e ¼ 1.

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226

dissipation (when the drift is slow) or to the generation (when the drift is signiﬁcant) of ﬂuctuational kinetic energy of a bidispersed system of particles.

6.4 Vertical Motion of a Bidispersed Particle Mixture

Consider the settling of a bidispersed system of particles in a turbulent ﬂuid under the action of the gravitational force. Such a ﬂow represents a model of a pseudoboiling bed. Turbulent ﬁeld of the ﬂuid is assumed to be homogeneous and isotropic, with the zero average velocity. But due to gravity, ﬂuctuations of particle velocities are not isotropic, because they are different in the vertical and horizontal directions. Under those conditions, equations (6.18) express the conservation of momentum of particles of each type in the vertical direction x: V ax U x þ g ¼ C ax ; tpa

a ¼ 1; 2;

ð6:46Þ

where C ax is deﬁned by Eqs. (6.35)–(6.37). Since the ﬂow is homogeneous and its characteristics in the horizontal plane are isotropic, the system of equations (6.19) reduces to the balance of second moments of particle velocity ﬂuctuations in the vertical (x) and horizontal (y) directions: n 02 l 02 2 fua u hv0 2y ia 2 fua u hv0 2x ia aa ab ab þC xx þC xx ¼ 0; þC aa yy þC yy ¼ 0; b 6¼ a; tpa tpa ð6:47Þ where the collisional terms are given by the relations (6.38)–(6.40). In what follows, we shall consider high-inertia particles, whose involvement coefﬁcients are deﬁned by Eq. (2.22) as V

V fua ¼

T Lpa V

tpa þ T Lpa

;

V ¼ l; n:

ð6:48Þ

The crossing trajectory effect is the main factor responsible for the difference between integral time scales of interactions of large heavy particle with turbulent eddies and the Lagrangian turbulence scale in the gravitational ﬁeld, therefore T lLp and T nLp will be described by the simple relations (1.73) at b2 ¼ 0.45. The response (involvement) coefﬁcients in the collisional terms are taken as the arithmetic averages of the three perpendicular components: m fua ¼

l n fua þ 2fua : 3

Equations (6.46) and (6.47) should be solved under the conditions that correspond to the outcome of direct Lagrangian trajectory simulation in a turbulent ﬁeld carried out by Gourdel et al. (1999) by using the LES method. Gourdel et al. (1999) considered the motion of bidispersed particles having the same size (ra ¼ 325 mm) but different densities (rp1 ¼ 117.5, rp2 ¼ 235 kg/m3) in the air ﬂow whose density was

6.4 Vertical Motion of a Bidispersed Particle Mixture

Figure 6.6 Average velocities of particles: 1 – V 1 1x ; 2, 6 – V1x; 3 – V1 2x ; 4, 7 – V2x; 5, 8 – Vx; 2, 4, 5 – model (6.46)–(6.47); 6, 7, 8 – Gourdel et al. (1998).

rf ¼ 1.17 kg/m3. Volume concentration of particles 1 was ﬁxed (F1 ¼ 1.3102), while volume concentration of particles 2 varied. Collisions were assumed to be elastic (e ¼ 1). Figure 6.6 shows the average velocities of particles of both types Vax and the average mass velocity of a bidispersed system Vx (rp1 F1 V 1x þ rp2 F2 V 2x )=(rp1 F1 þ rp2 F2 ) depending on the concentration of heavy particles F2. Also shown on Figure 6.6 are sedimentation rates of particles V 1 ax corresponding to their free fall in the absence of collisions. The velocities shown on Figure 6.6 satisfy the inequalities 1 V1 1x < V1x < Vx < V2x < V 2x . As a result of collisions there occurs a transport of momentum from the heavy particles having the greater sedimentation rate to the lighter particles, resulting in the reduction of the difference between the velocities V2x and V1x with increase of F2. The average mass velocity of the bidispersed system 1 as a whole Vx is close to V 1 1x at small values of F2 and tends to V 2x at large F2. There is a good agreement between the results obtained on the basis of the model (6.46)–(6.47) and the outcome of numerical simulation by Gourdel et al. (1999). Figure 6.7 demonstrates the effect of F2 on the kinetic energy of particles of both types. Notice the sharp maximum in the dependence of kinetic energy of light

Figure 6.7 Kinetic energy of particles: 1, 3 – kp1; 2, 4 – kp2; 1, 2 – model (6.46)–(6.47); 3, 4 – Gourdel et al. (1998).

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228

particles kp1 on the concentration F2 of heavy ones. The presence of the maximum is explained by the fact that at small concentrations F2 the kinetic energy of light particles is not affected by collisions and depends solely on the interaction with the turbulence. As F2 increases, generation of kp1 intensiﬁes due to the collisions with heavy particles. At relatively large values of F2 the average velocity slip between light and heavy particles resulting from the collisions becomes insigniﬁcant and consequently, the contribution of collisions to the kinetic energy of particles becomes smaller. It is worth noting that at small values of F2, kp2 exceeds kp1, whereas at relatively large values of F2 the opposite is true, that is, kp2 is smaller than kp1. The inequality kp2 > kp1 that holds at small F2 is explained by the fact that the concentration F1 is left unchanged and therefore the kinetic energy of heavy particles kp2 is mostly determined by their collisions with light particles. As F2 grows, the role of energy generation due to collisions diminishes, and therefore kp2 becomes smaller than kp1 because the heavy particles are less involved in turbulent motion than the light ones. Velocity ﬂuctuation intensities for particles of both types are shown on Figure 6.8. Figure 6.8a presents velocity ﬂuctuation intensities obtained by taking into account abð0Þ only the collisional term C ij (6.39) corresponding to the isotropic Gaussian PDF (6.20), while Figure 6.8b takes into account the general collisional term C ab ij (6.38) that also factors in the anisotropic component of the PDF (6.29). It is important to remember that generation of particle velocity ﬂuctuation is strongly anisotropic. Indeed, the direction of the average drift velocity between particles of different types, that is, in the vertical direction, is singled out because it is in this direction that the ﬁrst collisional mechanism acquires a major role as we see from Eqs. (6.38)–(6.40). The second collisional mechanism is caused by the continuity effect (see Section 1.3), according to which the duration of particle–turbulence interaction and thereby the degree of particle involvement in ﬂuctuational motion of the turbulent ﬂuid is greater in the vertical than in the horizontal direction. The two generation mechanisms add up, enhancing the anisotropy of particle velocity

Figure 6.8 Velocity fluctuation intensities: 1, 5 – hv0 21x i; 2, 6 – hv0 21y i; 3, 7 – hv0 22x i; 4, 8 – hv0 22y i; 14 – model (6.46)–(6.47); 58 – Gourdel et al. (1998).

6.5 Equation for the Two-Particle PDF and its Moments

ﬂuctuations in one and the same (vertical) direction, which leads to a much greater intensity of particle velocity ﬂuctuations in the vertical direction than in the horizontal direction for all values of F2. One can see by comparing Figure 6.8a; abð1Þ with Figure 6.8b that the decision to take into account the collisional term C ij given by Eq. (6.40) leads to a considerable decrease of velocity ﬂuctuation anisotropy for particles of both types.

6.5 Equation for the Two-Particle PDF and its Moments

We now want to provide a statistical description of the motion of a particle pair in a homogeneous shearless turbulent ﬁeld in the absence of gravity while neglecting the contribution of collisions. We shall treat the problem based on the kinetic equation for the two-point, two-velocity PDF. (Alipchenkov and Zaichik, 2005; Zaichik et al., 2006.) P(x1 ; v1 ; x2 ; v2 ; t) ¼ hpi ¼ hd(x1 Rp1 (t))d(v1 vp1 (t))d(x2 Rp2 (t))d(v2 vp2 (t))i: ð6:49Þ The transport equation for the two-particle PDF with the consideration of the equation of motion (5.1) for individual particles has the form 2 2 X qP X qP 1 q(U k vak )P 1 qhu0k (xa ; t)pi þ : þ vak ¼ qvak qt qx ak tpa qvak t a¼1 a¼1 pa

ð6:50Þ

Modeling the turbulent ﬂuid velocity ﬁeld by a Gaussian random process with given correlation moments as outlined in Sections 2.1 and 5.1, we can express the correlation between velocity ﬂuctuations in the continuous medium and probability densities of particle velocities as follows: hu0i (xa ; t)pi ¼

2 1X 2 b¼1

ðð

dp(x1 ; v1 ; x2 ; v2 ; t) hu0i (xa ; t)u0k (xb0 ; t0 )i dxb0 dt0 ; ð6:51Þ duk (xb0 ; t0 )dxb0 dt0

dp(x1 ; v1 ; x2 ; v2 ; t) duk (xb0 ; t0 )dxb0 dt0

2 X dRpa j (t) dRpa j (t) q q ¼ p þ p : qx aj qvaj duk (xb0 ; t0 )dxb0 dt0 duk (xb0 ; t0 )dxb0 dt0 a¼1 ð6:52Þ

The factor ½ in Eq. (6.51) was introduced in order to avoid double-counting when summing up the contributions from all particle pairs. To ﬁnd the functional derivatives entering Eq. (6.52) in the same manner as it was done in Section 2.1, we apply the

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230

operation of functional differentiation to the solutions of equations (5.1). We then obtain for the shearless turbulent ﬁeld dRpa j (t) tt0 ¼ djk d(Rpa (t0 )xb0 ) 1exp H(tt0 ); tpa duk (xb0 ; t0 )dxb0 dt0 d jk dvpa j (t) tt0 ¼ d(Rpa (t0 )xb0 )exp H(tt0 ): ð6:53Þ tpa duk (xb0 ; t0 )dxb0 dt0 tpa In view of Eqs. (6.52)–(6.53), the correlations (6.51) between ﬂuid velocity ﬂuctuations and probability densities of particle velocities can be written as hu0i (xa ; t)pi ¼

t 2 ð X

tt0 qP dt0 tpb qvbk

hu0i (xa ; t)u0k (Rpb (t0 ); t0 )iexp

b¼1 1 ðt

þ 1

tt0 qP hu0i (xa ; t)u0k (Rpb (t0 ); t0 )i 1 exp dt0 : tpb qx bk ð6:54Þ

In order to calculate the integrals in Eq. (6.54), we should ﬁrst determine the Lagrangian two-particle (two-point) correlation moment of ﬂuid velocities BL ij (r; t) ¼ hu0i (Ra (t); t)u0j Rb (t þ t); t þ t i; Ra (t) ¼ x; Rb (t) ¼ x þ r; where Ra and Rb the position vectors describing trajectories of continuum elements. We can then invoke approximation (1.17), representing the correlation moment as BL ij (r; t) ¼ hu0i u0j iYL (t)

Sij (r) YLr (tjr); 2

ð6:55Þ

where hu0i u0j i are single-point moments of ﬂuid velocity ﬂuctuations, Sij(r) – Eulerian two-point structure function of the second order, YL(t) – Lagrangian autocorrelation function of ﬂuid velocity ﬂuctuations, YLr(t|r) – Lagrangian autocorrelation function of ﬂuid velocity ﬂuctuation increment. The two relations below connect the moments and the autocorrelation functions of ﬂuctuations and increments: lim Sij (r) ¼ 2hu0i u0j i;

ð6:56Þ

lim YLr (tjr) ¼ YL (t):

ð6:57Þ

r!1 r!1

We now assume that velocity ﬂuctuation correlations calculated along ﬂuid trajectories will be identical to those calculated along inertial particle trajectories, in other words, we are going to ignore the effect of inertia on velocity correlations in the continuum. In addition, in contrast to the previously made approximations (2.14), (2.15), and (5.15), we now try to avoid the cumbersome algebra by neglecting the contribution of the transport term to the approximation for the two-particle correlation moment BL ij (r; t). Note that the assumptions just made do not necessarily act as

6.5 Equation for the Two-Particle PDF and its Moments

constrains on the two-particle kinetic model and the excluded effects may in fact be taken into account. The approximation (6.55) allows to present the correlation (6.54) in the form qP qP Sik hu0i (xa ; t)pi ¼ hu0i u0k i fua þ tpa g ua hu0i u0k ifub frb 2 qvak qx ak qP Sik qP tpb hu0i u0k ig ub g rb ; b 6¼ a; ð6:58Þ 2 qvbk qx bk fra ¼

1 tpa

1 ð

0 1 ð

fua ¼ t1pa 0

t YLr (tjr) exp dt; tpa

t YL (t) exp dt; tpa

g ua ¼ g ra ¼

TL f ; tpa ua

T Lr f : tpa ra

ð6:59Þ

The factors fua and gua deﬁne the degree of participation of particles a in ﬂuctuational motion, whereas the factors fra and gra characterize the involvement of a pair of particles of type a separated by distance r in the ﬂuids turbulent motion. Due to Eq. (6.57), the following relation holds for these factors: lim f r!1 ra

¼ fua ;

lim g r!1 ra

¼ g ua :

ð6:60Þ

Plugging Eq. (6.58) into Eq. (6.50), we get a closed kinetic equation for the twopoint, two-velocity PDF of a particle pair in homogeneous shearless turbulence: 2 qP X qP 1 q(U k vak )P þ þ vak qt a¼1 qx ak tpa qvak 2 X fua q2 P q2 P 1 Sik 0 0 0 0 ¼ þ g ua hui uk i þ hui uk ifub f rb tpa qvai vak 2 qx ai vak tpa a¼1

tpb q2 P Sik q2 P þ g rb hu0i u0k ig ub ; 2 qvai qvbk tpa qxbi qvak

b 6¼ a:

ð6:61Þ

The respective terms on the left-hand side of Eq. (6.61) describe convective and diffusive transport in the twelve-dimensional phase space (x1, v1, x2, v2). If we model turbulent velocity ﬂuctuations in the continuum by a Gaussian random process, it is possible to express particle–turbulence interaction in the kinetic equation as the Fokker–Planck diffusion operator of the second order. It should be noted that the kinetic equation for the two-particle PDF in a turbulent ﬂow was ﬁrst obtained by Derevich (1996). The main distinction of the current approach from the one by Derevich is that in order to derive Eq. (6.61), we made use of the Lagrangian twopoint autocorrelation function of velocity ﬂuctuation increments YLr(t|r) instead of using only the Lagrangian single-point autocorrelation function of ﬂuid velocity ﬂuctuations YL(t). It follows from Eq. (6.61) with the consideration of Eq. (6.56) and Eq. (6.60) that as the distance between the particles goes to inﬁnity (r ! 1), the

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two-particle PDF becomes equal to the product of two one-particle distributions P(x1, v1, x2, v2, t) ¼ P1(x1, v1, t)P2(x2, v2, t) obeying the kinetic equation (Derevich and Zaichik, 1988, 1990; Reeks, 1991): f q2 P a qP a qPa 1 q(U k vk )P a q2 Pa þ vk þ ¼ hu0i u0k i ua þ g ua : qt qx k qvk tpa qvi vk qx i vk tpa If we bring into play the bi-exponential autocorrelation functions (1.5) and (1.21) in order to get the involvement coefﬁcients (6.59), these coefﬁcients will take the form fua ¼

2Wa þ z2 ; 2Wa þ 2W2a þ z2

f ra ¼

2Wra þ z2r ; 2Wra þ 2W2ra þ z2r

g ua ¼

2Wa þ z2 z2 Wa ; Wa (2Wa þ 2W2a þ z2 )

g ra ¼

Wa ¼

2Wra þ z2r z2 Wra ; Wra (2Wra þ 2W2ra þ z2r )

tpa ; TL

Wra ¼

tpa : T Lr ð6:62Þ

If the characteristics of turbulence hu0i u0j i, Sij(r), TL, TLr , tT, tTr are known, Eq. (6.61) complemented by Eq. (6.62) will completely determine on the kinetic level both the single-point and two-point statistics of a pair of particles in a homogeneous shearless turbulent ﬂow. By integrating Eq. (6.61) over the velocity subspace (v1, v2) one can derive a chain of equations for the two-point PDF moments of a particle pair. Thus the number concentration of particles 2 in a bidispersed suspension N12 is given by the following equation: qN 12 qN 12 V 1k qN 12 V 2k þ ¼ 0; þ qt qx 1k qx 2k ðð ðð 1 N 12 ¼ Pdv1 dv2 ; V ai ¼ vai Pdv1 dv2 : N 12

ð6:63Þ

The equation for average velocities of particles Vai is given below: 0

0

qV ai qV ai qV ai U i V ai qhv0ai v0ak i qhvai vbk i þ V bk ¼ þ V ak qx bk qt qx ak qx bk tpa qx ak

Da ik q lnN 12 Dab ik q lnN 12 ; b 6¼ a; tpa qx ak tpa qx bk

ð6:64Þ

Da ij ¼ tpa hv0ai v0aj i þ g ua hu0i u0j i ; Dab ij ¼ tpa hv0ai v0bj i þ tpb g ub hu0i u0j ig rb Sij =2 : The equation for second moments of velocity ﬂuctuations of particles 2 is written as qhv0ai v0aj i qt

þ V ak

qhv0ai v0aj i

qx ak qN 12 hv0ai v0aj v0ak i

qhv0ai v0aj i

1 þ qx bk N 12 qN 12 hv0ai v0aj v0bk i

þ V bk

þ qx bk qx ak Da ik qV aj Da jk qV ai 2 0 0 0 0 f hu u ihvai vaj i ¼ tpa qx ak tpa qx ak tpa ua i j Dab ik qV aj Dab jk qV ai ; b 6¼ a: tpa qx bk tpa qx bk

ð6:65Þ

6.5 Equation for the Two-Particle PDF and its Moments

The equation for second moments of velocity ﬂuctuations of particles of different types is written as qhv0ai v0bj i

qhv0ai v0bj i

qhv0ai v0bj i

1 þ qt qx ak qxbk N 12 qN 12 hv0ai v0bj v0ak i qN 12 hv0ai v0bj v0bk i þ qx ak qx bk ¼

þ V ak

fub tpa

þ

þ V bk

f rb Sij fua f 1 1 þ ra þ hu0i u0j i hv0ai v0bj i tpb tpa tpb 2 tpa tpb

Da ik qV bj Db jk qV ai Dab ik qV bj Dba jk qV ai ; tpa qx ak tpb qx bk tpa qx bk tpb qx ak

b 6¼ a:

ð6:66Þ

In the discussion below we are going to need equations describing the relative motion of a pair of particles, therefore let us introduce the variables (r, w) characterizing the relative motion of particles of two types: r ¼ x2 x1 ;

w ¼ v2 v1 :

Switching to new variables in Eqs. (6.63)–(6.66) and taking into account the homogeneity of turbulence, we obtain the following system of equations: qN 12 qN 12 W k þ ¼ 0; qt qr k

ð6:67Þ

0 0 0 0 qhvai vak i qhvai vbk i qV ai qV ai U i V ai þ Wk ¼ (1)a qr k qt qr k tpa qr k (1)a

Da ik Dab ik q lnN 12 ; tpa qr k

ð6:68Þ

0 0 0 0 (1)a qN 12 hvai vaj (vbk vak )i qr k N 12 qt qr k Da jk Dab jk qV ai D D 2 a ik ab ik qV aj f hu0 u0 ihv0ai v0aj i (1)a ¼ (1)a ; tpa qr k tpa qr k tpa ua i j

qhv0ai v0aj i

þ Wk

qhv0ai v0aj i

ð6:69Þ qhv0ai v0bj i qt ¼

0 0 0 0 (1)a qN 12 hvai vbj (vbk vak )i N 12 qr k qr k f Sij f f 1 1 rb þ ua hu0i u0j i þ ra þ hv0ai v0bj i tpb tpa tpb 2 tpa tpb

þ Wk

fub

tpa

(1)a

qhv0ai v0bj i

Db jk Dba jk qV ai Da ik Dab ik qV bj þ (1)a ; tpa qr k tpb qr k

b 6¼ a; W i ¼ V 2i V 1i :

ð6:70Þ

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234

Consider a stationary quiescent suspension in the ﬁeld of homogeneous stationary turbulence with the zero average velocity. Then the equalities U i ¼ V 1i ¼ V 2i ¼ W i ¼ 0

ð6:71Þ

should hold and the balance equation for the numerical density of particles (6.67) can be excluded from consideration, since it is satisﬁed automatically. From Eq. (6.68) with the consideration of Eq. (6.71) there follow the equations r qSp ik Dp ik q ln G þ ¼ 0; qr k tp qr k

ð6:72Þ

Sp ij ¼ hw 0i w0j i ¼ hv01i v01j i þ hv02i v02j ihv01i v02j ihv02i v01j i; D1 ij D12 ij D2 ij D21 ij þ Drp ij ¼ tp tp1 tp2 ¼ tp Sp ij þ tp ¼

2 tp1 g r1 þ t2p2 g r2 2(tp2 tp1 )(tp1 g u1 tp2 g u2 ) hui uj i þ Sij ; tp1 þ tp2 tp1 þ tp2

2tp1 tp2 ; tp1 þ tp2

G¼

N 12 ; N1N2

where Sp ij is the second-order structure function for particle velocity ﬂuctuations, Drp ij – the coefﬁcient of relative diffusion of two particles, tp – the effective relaxation time of two particles, G – the radial distribution function, Na – the average number concentration of particles a. Equation (6.72) can be interpreted as expressing the balance of the turbophoretic driving force caused by the gradient of intensity of relative velocity ﬂuctuations and the diffusional driving force in the phase space of relative motion of the two particles. The equation for the structure function Sp ij is obtained from Eqs. (6.69)–(6.70) with the consideration of Eq. (6.71): qGSp ijk 2G (tp2 tp1 )( fu1 fu2 ) 0 0 (tp1 fr1 þ tp2 fr2 ) þ hui uj i Sij ¼ 0; Sp ij qr k tp tp1 þ tp2 tp1 þ tp2 ð6:73Þ Sp ijk ¼

tp1 þ tp2 0 0 0 tp1 þ tp2 0 0 0 hv1i v1j (v2k v01k )i þ hv2i v2j (v2k v01k )i 2tp2 2tp1 h(v01i v02j þ v02i v01j )(v02k v01k )i:

In order to close the system of equations (6.72) and (6.73) one needs to determine the third-order correlation Sp ijk. For monodispersed particles, this quantity is just a third-order structure function Sp ijk hw 0i w 0j w 0k i. In Section 5.2, Sp ijk was derived from the corresponding differential equation while neglecting the terms characterizing time evolution, convection and generation due to the average velocity gradient. The result was an algebraic equation for the third-order structure function of velocity ﬂuctuations for monodispersed particles (Eq. (5.25)). Suppose the relation (5.25) is

6.6 The Clustering Effect and its Influence on the Collision Frequency

also true for bidispersed particles. Then Eqs. (6.72)–(6.73) with the consideration of Eq. (5.25) allow us to model the two-particle statistics of relative velocity of a particle pair in a bidispersed suspension on the level of second moments.

6.6 The Clustering Effect and its Influence on the Collision Frequency of Bidispersed Particles in Isotropic Turbulence

In isotropic turbulence, spherical symmetry implies independence of relative velocities and particle distribution density on spatial orientation of the vector r, so the task of solving equations (6.72) and (6.73) with the consideration of Eq. (5.25) reduces to the problem r 2ðSp ll Sp nn Þ dSp ll Dp ll d ln G þ þ ¼ 0; r d r St d r

ð6:74Þ

4G r dSp nn 2 r 1 d r dSp ll 2 St 2 þ Dp nn ðSp ll Sp nn ) r G Dp ll Dp ll r r d r d r d r 3 r (St2 St1 )( fu1 fu2 )Rel St1 f r1 þ St2 f r2 þ 2G S S þ ll p ll ¼ 0; St1 þ St2 ð6:75Þ 151=2 (St1 þ St2 )

d d 3 r r dSp nn 4 r G Dp ll þ 2 ½r G Dp nn ðSp ll Sp nn Þ d r d r d r (St2 St1 )(fu1 fu2 )Rel St1 f r1 þ St2 f r2 þ 2G Snn Sp nn ¼ 0; þ St1 þ St2 151=2 (St1 þ St2 )

St 3r 4

ð6:76Þ

2(St2 St1 )(St1 g u1 St2 g u2 )Rel (St21 g r1 þ St22 g r2 ) r Sll ; Dp ll ¼ St Sp ll þ þ St1 þ St2 151=2 (St1 þ St2 ) 2(St2 St1 )(St1 g u1 St2 g u2 )Rel (St21 g r1 þ St22 g r2 ) r Snn ; Dp nn ¼ St Sp nn þ þ St1 þ St2 151=2 (St1 þ St2 ) St ¼

2St1 St2 : St1 þ St2

The bar over the variables in Eqs. (6.74)–(6.76) denotes dimensionless variables determined using the Kolmogorov length and velocity scales; Sta tpa/tk is the Stokes number characterizing particle inertia; Sp ll and Sp nn are the longitudinal and transverse components of structure functions of particle velocities Sp ij. Hence, just as in the case of a monodispersed particle system, calculation of statistical characteristics of bidispersed particles boils down to solving a non-linear system of three ordinary differential equations with respect to the radial distribution function and the longitudinal and transverse structure functions of particle velocity ﬂuctuations. For

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236

monodispersed particles, the system of equations (6.74)–(6.76) will transform into Eqs. (5.32)–(5.34) if we remove the contribution of the transport effect from the latter system of equations, which is achieved by setting fr1 ¼ 0. The boundary conditions for Eqs. (6.74)–(6.76) are speciﬁed as follows: r ¼ 0

dSp ll dSp nn ¼ ¼ 0; d r d r

r ! 1 Sp ll ¼ Sp nn ¼

( fu1 þ fu2 )Rel 151=2

;

G ¼ 1: ð6:77Þ

The longitudinal structure function of ﬂuid velocity ﬂuctuations and the time scale of velocity ﬂuctuations are given by the approximations (5.47) with the consideration of Eqs. (1.4), (1.6), (1.7), (1.41), whereas the transverse structure function is deﬁned by Eq. (1.39). The coefﬁcients describing the involvement of particles in the turbulent ﬂow of the ﬂuid are given by the relations (6.62). The collision kernel is equal to ð6:78Þ

b ¼ (8p Sp ll (s))1=2 s2 G(s):

This formula is valid provided that the radial relative velocity is normally distributed; accordingly, we have h|wr|i ¼ (2Sp ll/p)1/2. As opposed to Eq. (6.1), the expression (6.78) takes into account the contribution of clustering to the collision frequency. Equations (6.74)–(6.76) with boundary conditions (6.77) have been solved numerically and the obtained results compared with the DNS data (Zhou et al., 2001). But let us ﬁrst consider the asymptotic solutions of this problem for the cases of low-inertia and high-inertia particles. For inertialess particles of type 2 (St2 ¼ 0, fu2 ¼ fr2 ¼ 1), Eqs. (6.74)–(6.76) give us (1fu1 )Rel (1fu1 )Rel Sp ll ¼ þ f r1 Sll ; Sp nn ¼ þ f r1 Snn ; 151=2 151=2

G ¼ 1:

ð6:79Þ

Substitution of Eq. (6.79) into Eq. (6.78) leads to the following dependence for the collision kernel for particles whose size does not exceed the Kolmogorov spatial microscale: b ¼ (8p)1=2 R2 u0 1fu1 þ

2 1=2 f r1 R

151=2 Rel

:

ð6:80Þ

Comparing Eq. (6.9) with Eq. (6.80), we see that at sufﬁciently high Reynolds numbers, these formulas practically coincide. If particles 1 are also inertialess, the formula (6.80) reduces to the Saffman–Turner formula (4.17). In the limiting case of high-inertia particles (tpa ! 1), we get the Abrahamson collision kernel (6.10). If only particles 2 are highly inertial (St2 ! 1, fu2 ¼ fr2 ! 0) then Eqs. (6.74)–(6.76) give us f Re Sp ll ¼ Sp nn ¼ u1 1=2l ; 15

G ¼ 1;

ð6:81Þ

and the substitution of Eq. (6.80) into Eq. (6.78) yields the collision kernel (6.11).

6.6 The Clustering Effect and its Influence on the Collision Frequency

Figures 6.2, 6.3, 6.9, 6.10 compare the predictions of the above-considered model with numerical simulation data obtained by Zhou et al. (2001). The ratio of relaxation time of particles 2 and time macroscale of turbulence Te ¼ u0 2/e is plotted as a function of time, with relaxation time of particles 1 ﬁxed. Figure 6.2 demonstrates the inﬂuence of particle inertia on the average radial velocity divided by the ﬂuid velocity ﬂuctuation. We see that according to Eq. (6.79), at tp2 ¼ 0 the quantity h|wr|i/u0 grows monotonously with tp1/Te because the correlation between particle velocities decreases with increase of particle inertia. On the other hand, according to Eq. (6.81), at large values of tp2/Te the quantity h|wr|i/u0 decreases monotonously with increase of tp1/Te as a result of diminishing involvement of heavier particles in turbulent motion of the surrounding continuum. It is also seen from Figure 6.2 that according to the DNS data, as tp2/Te gets larger, the relative radial velocity decreases at ﬁrst, then, upon reaching its minimum, begins to increase, and ﬁnally approaches a constant value predicted by Eq. (6.81). The minimum on the graph of h|wr|i/u0 versus tp2/Te becomes especially sharp at small values of parameter tp1/Te. As observed by Zhou et al. (2001), it is evident from the comparison with the results for a monodispersed system that the difference of particle inertias brings about an increase of the relative velocity between the particles. Therefore the quantity h|wr|i/u0 is always greater in a bidispersed system than in a monodispersed one, in other words, it is bounded from below by the corresponding value for a monodispersed system (curve 3). Thus the minima in Figure 6.2 correspond to tp2 ¼ tp1. For all practical purposes, the obtained values of h|wr|i/u0 coincide with the results by Zhou et al. (2001) for low-inertia particles (St1, St2 1); the agreement is somewhat worse, however, for high-inertia particles (St1, St2 1). This mismatch is probably due to the fact that we have assumed the identity of velocity ﬂuctuation correlations calculated along continuum element trajectories and along inertial particle trajectories. Overall, the model presented in this section, as opposed to the model described in Section 6.1, accurately predicts the main features of behavior of relative radial velocity for the entire range of relaxation times for both types of particles.

Figure 6.9 The effect of the Reynolds number on the radial relative velocity of particles: 1, 2 – solution of the problem (6.74)–(6.77); 3, 4 – Zhou et al. (2001); 1, 3 – Rel ¼ 45; 2, 4 – Rel ¼ 58.

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Figure 6.9 illustrates the effect of the Reynolds number on the radial relative velocity of particles at r ¼ 1 and St1 ¼ 1, and compares this dependence with the results of Zhou et al. (2001) for two different values of Rel. As one can see from the ﬁgure, there is a good agreement between the theory and the DNS results, especially for low-inertia particles. It should be noted that the model accurately predicts the decrease and increase of h|wr|i with Rel at small and relatively large values of St2, respectively. The theoretical results and the DNS data are presented on Figure 6.10, which plots the radial distribution function against St2 at ﬁxed values St1. As follows from Eq. (6.79) and Eq. (6.81), in the limiting cases when particles of one type possess very low or very high inertia, the relative distribution of particles of the other type is spatially homogeneous, and consequently, G ¼ 1. According to the DNS data, as St2 grows, the radial distribution function reaches its maximum at some point, though the maxima obtained from the model (6.74)–(6.77) are somewhat shifted toward the higher values of St2 as compared to the DNS results. The maxima coincide when both types of particles have the same relaxation times, that is, at St1 ¼ St2. Also shown on Figure 6.10 are the radial distribution functions for a monodispersed system of particles corresponding to the maxima of the dependence G(St2) for a bidispersed suspension and bounding them from above. So the clustering effect is most pronounced in a monodispersed suspension and becomes less noticeable with increase of the difference between particle inertias. It follows from Eq. (6.78) that, just like in a monodispersed system, interaction of inertial particles with turbulent eddies gives rise to two physical phenomena that determine the frequency of particle collisions, namely, the non-zero relative velocity between neighboring particles (turbulent transport effect) and the inhomogeneous distribution of particles in space (clustering effect). The ratio of the collision kernel obtained by solving the problem (6.74)–(6.77) with the consideration of Eq. (6.78) to the collision kernel for the case of inertialess particles (4.16) is compared with the results of numerical simulation on Figure 6.3. We see that the dependence of b/bST

Figure 6.10 Radial distribution function at r ¼ 1 and Rel ¼ 45: 1–6 – solution of the problem (6.74)–(6.77); 7–12 – Zhou et al. (2001); 1, 7 – St1 ¼ 0.5; 2, 8 – 1.0; 3, 9 – 1.17; 4, 10 – 2.34; 5, 11 – 11.7; 6, 12 – monodispersed particles.

6.6 The Clustering Effect and its Influence on the Collision Frequency

on tp2/Te that is predicted by the model presented in the current section can be nonmonotonous, which is consistent with the presence of a minimum and a maximum in the corresponding dependences of h|wr|i and G. According to the DNS results by Zhou et al. (2001), in the case when the clustering effect is negligibly small (St1, St2 1 or St1, St2 1), the collision kernel of a bidispersed suspension is greater than the collision kernel of a monodispersed system due to the increase of the average relative velocity between particles of different types. On the contrary, when the clustering effect turns out to be signiﬁcant (St1, St1 1), the collision kernel of a bidispersed suspension becomes smaller than that of a monodispersed system. So as a result of particle clustering, the collision kernel of monodispersed particles can exceed the collision kernel of bidispersed particles.

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j261

Notation Index Upper-Case Roman

A1 A2 Aij B BE ij(r, t) BEt(r, t) Bfp ij(t) Bfpt(t) Bij Bij(r) BLij(t) BL ij ðr; tÞ BLp ij(t) BLt(t) BLtp(t) BtL i ðtÞ BtLp i ðtÞ Bp(t) Bp ij(t) Bpt(t)

constant constant constant coefﬁcient in Eq. (2.150) constant Eulerian space-time correlation moment of ﬂuid velocity ﬂuctuations Eulerian space-time correlation moment of temperature ﬂuctuations Lagrangian mixed correlation moment of ﬂuid and particle velocity ﬂuctuations Lagrangian mixed correlation moment of temperatures of the continuous and disperse phases along particle trajectories parameter in Eq. (6.30) Eulerian two-point simultaneous correlation moment Lagrangian single-particle (single-point) correlation moment of ﬂuid velocity ﬂuctuations Lagrangian two-particle (two-point) correlation moment Lagrangian correlation moment of velocity ﬂuctuations of a ﬂuid particle along an inertial particle trajectory Lagrangian single-particle correlation moment of temperature ﬂuctuations of the continuous phase Lagrangian correlation moment of temperature ﬂuctuations of a ﬂuid particle along an inertial particle trajectory correlation moment along a ﬂuid particle trajectory Lagrangian correlation moment of joint ﬂuctuations of velocity and temperature for a ﬂuid particle correlation moment tensor of particle velocity ﬂuctuations Lagrangian correlation moment of particle velocity ﬂuctuations along the particle trajectory correlation moment of particle temperature ﬂuctuations

j Notation Index

262

C C0 C0¥ C1 C2 Cij C ai að0Þ

Ci

að1Þ

Ci

C ij

C aa ij C ab ij abð0Þ

C ij

abð1Þ

C ij C ijk

C kp C Ikp C II kp IIð0Þ

C kp

IIð1Þ

C kp Cp CR Cu1 Ctu1 Cu2 Ctu2 Cm Cm Dti Dij

constant Kolmogorov constant at ﬁnite Reynolds numbers universal Kolmogorov constant corresponding to high Reynolds numbers constant constant parameter in Eq. (6.30) collisional term in the equation of conservation of momentum for particles a component of the collisional term in the equation of conservation of momentum for particles a component of the collisional term in the equation of conservation of momentum for particles a collisional term denoting the contribution of collisions to the balance equation for second moments of velocity ﬂuctuations of the disperse phase contribution of collisions between particles of the same type to the balance equation for turbulent stresses contribution of collisions between particles of different types to the balance equation for turbulent stresses component of the collisional term C ab ij component of the collisional term C ab ij collisional term denoting the contribution of collisions to the balance equations for the moments of velocity ﬂuctuations of the disperse phase collisional term in the equation for the turbulent energy of a bidispersed system component of the collisional term in the equation for the turbulent energy of a bidispersed system component of the collisional term in the equation for the turbulent energy of a bidispersed system component of the collisional term C II kp component of the collisional term C II kp heat capacity of particle material Richardson–Obukhov constant constant in Eq. (3.49) constant in Eq. (3.49) constant in Eq. (3.49) constant in Eq. (3.49) Kolmogorov–Prandtl constant constant in Eq. (2.172) components of the vector of thermal diffusion for inertialess particles (passive scalar) tensor of turbulent diffusion of ﬂuid particles

Upper-Case Roman

Drij ðr; tÞ Drll Drnn Dp Dlp Dnp Dp ij Drij Dap ik Drp ij Dtp i Dt DT E En Et E t0 F(r) F F(s) F(s) F(s)

F0(z) F1(z) Fij

Ft(r) G(r) G(s)

Gij

H(x) I Jw = L L

tensor of relative diffusion of two ﬂuid particles longitudinal component of relative diffusion of two ﬂuid particles transverse components of relative diffusion of two ﬂuid particles coefﬁcient of particles turbulent diffusion coefﬁcient of particle diffusion along the vector of drift velocity coefﬁcient of particle diffusion over a plane perpendicular to the vector of drift velocity components of the tensor of turbulent diffusion of particles tensor of relative turbulent diffusion of ﬂuid particles coefﬁcients of turbulent diffusion for particles a tensor of relative turbulent diffusion of a particle pair components of the vector of thermal diffusion of the disperse phase turbulent diffusivity of an inertialess impurity coefﬁcient of turbulent diffusion for a passive scalar parameter in Eq. (2.70) parameter in Eq. (2.71) parameter of Eq. (3.69) parameter in Eq. (3.71) longitudinal Eulerian spatial correlation function deterministic external force acting on the particle Laplace transformation of the autocorrelation function YLp(t) function in Eq. (4.9) longitudinal component of the Eulerian two-point correlation function of ﬂuid velocity ﬂuctuations over a distance equal to the collision sphere radius s function in Eq. (6.32) function in Eq. (6.36) integral in Eq. (5.12) containing second correlation moments of ﬂuid velocity ﬂuctuation increments along the trajectories of relative motion of a particle pair Eulerian two-point simultaneous correlation function of temperature ﬂuctuations transverse Eulerian spatial correlation function transverse component of the Eulerian two-point correlation function of ﬂuid velocity ﬂuctuations over a distance equal to the collision sphere radius s integral in Eq. (5.12) containing second correlation moments of ﬂuid velocity ﬂuctuation increments along the trajectories of relative motion of a particle pair Heaviside function unit matrix deposition ﬂux at the wall parameter in the formula (1.70) spatial integral scale dimensionless spatial integral scale

j263

j Notation Index

264

Lt Mij(r,t) Mll Mnn Mijk(r,t) Mlll(r,t) N½P

spatial macroscale of temperature ﬂuctuations second-rank tensor second-rank tensor component parallel to the position vector r second-rank tensor component perpendicular to the position vector r components of a third-rank tensor longitudinal component of a third-rank tensor operator that corresponds to the deviation of particle velocity PDF from the equilibrium Maxwellian distribution N number concentration of particles N1 coefﬁcient in the relation (2.120) N2 coefﬁcient in the relation (2.120) N12 number concentration of particle pairs Na average number concentration of particles a @ parameter in the two-scale bi-exponential approximation of the Lagrangian autocorrelation function Nup Nusselt number of the ﬂow past a particle P PDF P(x, v, x1, v1, t) two-particle PDF P generation of turbulent energy P(0)(v) term in the Chapman–Enskog expansion (2.45) P(1)(v) term in the Chapman–Enskog expansion (2.45) qPa a collision operator for particles a qt coll qP collision operator in the kinetic equation for a single-point (singleqt coll particle) PDF Pr(r, t|r0, 0) probability density distribution for a particle pair in space Pr(r, t) probability density distribution of inter-particle distance Pr Prandtl number Prp turbulent Prandtl number of the disperse phase PrT turbulent Prandtl number of the continuous phase Q intensity of heat release inside a particle coefﬁcient in Eq. (4.50) for a two-particle PDF Qij Qijk coefﬁcient in Eq. (4.50) for a two-particle PDF R channel half-width R position (radius) vector of a ﬂuid particle Rp particle coordinate, position vector of a particle R½P operator that realizes a Maxwellian distribution R parameter in the two-scale bi-exponential approximation of the Lagrangian autocorrelation function Ri coefﬁcient in Eq. (4.50) for a two-particle PDF Rij coefﬁcient in Eq. (4.50) for a two-particle PDF Rijk coefﬁcient in Eq. (4.50) for a two-particle PDF Rp parameter in the Lagrangian autocorrelation function of ﬂuid velocity along the particle trajectory

Upper-Case Roman

RV Re Re þ Rep Rel S S Sij(r) Sijk(r, t) fp

Sik

Slll(r, t) SL ij(t) SL ij ðr; tÞ Sp ij Sp ll Sp nn p ll S p nn S Sp ijk SLp ij ðr; tÞ StC St StE Tþ TE TEt TL L T TL

TL TL þ TL ij T tL ij

dimensionless parameter in Eq. (2.186) Reynolds number Reynolds number Re þ in a planar channel calculated for the velocity u Reynolds number of the ﬂow past a particle. Reynolds number calculated from the Taylor microscale shear rate average particle displacement relative to the moving ﬂuid Eulerian spatial structure function of the second order third-order structure functions of velocity of the continuous phase mixed second-order structure function of ﬂuid and particle velocities longitudinal third-order structure function of velocity of the continuous phase Lagrangian structure function Lagrangian two-particle (two- point) structure function Eulerian structure function of the second order longitudinal component of the structure function of particle velocity ﬂuctuations transverse component of the structure function of particle velocity ﬂuctuations dimensionless longitudinal component of the structure function of particle velocity ﬂuctuations dimensionless transverse component of the structure function of particle velocity ﬂuctuations Eulerian structure function of the third order Lagrangian two-point structure function of velocity ﬂuctuations of ﬂuid particles moving along inertial particle trajectories Stokes number that characterizes particle inertia as related to the time interval between collisions effective Stokes number characterizing the inﬂuence of collisions on the duration of particle–turbulence interaction Stokes number constant in Eq. (2.152) integral time scale Eulerian temporal macroscale of temperature ﬂuctuations Lagrangian integral time scale dimensionless integral time scale of velocity ﬂuctuation increment dimensionless Lagrangian integral scale of turbulence divided by the Kolmogorov integral microscale of turbulence tensor of Lagrangian macroscales dimensionless parameter Lagrangian scale of turbulence integral time scale of joint ﬂuctuations of velocity and temperature in a continuum

j265

j Notation Index

266

TLp TLp TtLp 1 TLp

T lLp

T nLp V

T Lp TLp ij TLr TLrp TLt TLtp Tp Tu U DU Uþ V Vai V1m Vi Vi,j Vy W W Wg Wi Xi Xij

Lagrangian integral time scale of particle interactions with energycarrying turbulent eddies tensor (matrix) of particle–turbulence interaction times correlation tensor for the durations of velocity and temperature ﬂuctuations inverse of the matrix TLp longitudinal (with respect to the drift velocity) component of the tensor of particle interaction times transverse (with respect to drift velocity) components of the tensor of particle interaction times integral time scale components of the tensor of particle interaction times integral time scale of velocity ﬂuctuation increment integral time scale of velocity ﬂuctuation increments of ﬂuid particles along the trajectories of relative motion of inertial particles Lagrangian time macroscale of temperature ﬂuctuations integral scale of ﬂuid temperature ﬂuctuations along an inertial particle trajectory integral scale for particles characteristic time of change of the average parameters of the hydrodynamic ﬂow average velocity increment of the average ﬂuid velocity dimensionless velocity particle velocity, velocity of the disperse phase average velocity of particles a average mass velocity of particles in the longitudinal direction. components of average velocity of the disperse phase subscript j denotes spatial derivative with respect to the coordinate xj (average velocity gradient) particle deposition rate drift velocity particle drift velocity relative to the average velocity of the surrounding ﬂuid difference between sedimentation rates of two particles component of the average relative velocity vector parameter in Eq. (3.69) parameter in Eq. (2.69)

Lower-Case Roman

a0 a01, a02, a0¥ ai

amplitude of acceleration ﬂuctuations constants appearing in the dependence of a0 on Rel components of acceleration ﬂuctuations

Lower-Case Roman

aij b1 b2 bfp

ij

bij bij bp ij cij dþ dp ei ex ey ez f (StE) fr fr1 ft ft f tt f tu ft1 f tt1 f tu1 fu fu fu ij fu1 f1u ij

elements of the matrix appearing in the Gaussian distribution (4.3) constant in Eq. (4.20) constant components of the anisotropy tensor of velocity ﬂuctuation correlations in the continuous and disperse phases components of the anisotropy tensor of ﬂuid velocity ﬂuctuations elements of the matrix appearing in the Maxwellian distribution (4.6) components of the anisotropy tensor of particle velocity ﬂuctuations components of the matrix appearing in the distribution (4.7) dimensionless size of particles particle diameter component of the unit vector parallel to the drift velocity coefﬁcient of momentum restitution in the x-direction coefﬁcient of momentum restitution describing a particles collision with the wall coefﬁcient of momentum restitution in a transverse direction function in Eq. (2.142) involvement coefﬁcient of a particle pair involvement coefﬁcient of a particle pair coefﬁcient characterizing the responsiveness of particles to turbulent ﬂuctuations of temperature of the carrier ﬂuid involvement coefﬁcients describing the temperature response of particles matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase involvement coefﬁcients describing the temperature response of particles matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase isotropic involvement coefﬁcient matrix of involvement coefﬁcients coefﬁcient of particles involvement in turbulent motion of the continuum matrix of involvement coefﬁcients coefﬁcient of particles involvement in turbulent motion of the continuum

j267

j Notation Index

268

fu2 f tu2 g gr gu gtu gu ij gu1 gtu1 g1u ij hu htu hu

ij

jþ ka kp ~kp dkp kfp k ¼ hu0 n u0 n i=2 l lr lu ltu lu

ij

lu1 ltu1

matrix of involvement coefﬁcients matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase acceleration of gravity involvement coefﬁcient of a particle pair matrix of involvement coefﬁcients matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase coefﬁcient of particles involvement in turbulent motion of the continuum matrix of involvement coefﬁcients matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase coefﬁcient of particles involvement in turbulent motion of the continuum matrix of involvement coefﬁcients matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase coefﬁcient of particles involvement in turbulent motion of the continuum deposition factor parameter of a bidispersed particle system turbulent kinetic energy of the disperse phase correlated component of particles kinetic energy quasi-Brownian component of particles kinetic energy kinetic energy of velocity correlations between the continuous phase and the disperse phase kinetic energy of turbulence unit vector along the line of centers of colliding particles involvement coefﬁcient of a particle pair matrix of involvement coefﬁcients matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase coefﬁcient of particles involvement in turbulent motion of the continuum matrix of involvement coefﬁcients matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase

Lower-Case Roman

l1u

ij

m mu mu ij mtu m mt mT n p q qu qtu qu1 qtu1 r |r| r r r hr 2 i ra rp ru rtu s ¼ tp 1 s s0 t

coefﬁcient of particles involvement in turbulent motion of the continuum particle mass matrix of involvement coefﬁcients coefﬁcient of particles involvement in turbulent motion of the continuum matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase structure parameter of turbulence temperature structure parameter of turbulence parameter in the formula (1.72) order of approximation dynamic probability density velocity characterizing the motion of a particle pair matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase distance between two points dimensionless distance between points (particles) dimensionless radius vector connecting the centers of colliding particles dispersion radius of particle a vector connecting the centers of two particles matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase matrix of involvement coefﬁcients describing particles response to joint ﬂuctuations of velocity and temperature of the continuous phase reciprocal time of particle relaxation displacement of a particle relative to the moving ﬂuid ﬂuctuation of particles displacement relative to the moving ﬂuid time

j269

j Notation Index

270

u(x) u(Rp, t) Du ua u0 Du0 u0 u0 y(t)

actual velocity of the ﬂuid velocity of the ﬂuid at the point x ¼ Rp (t) increment of the actual ﬂuid velocity ﬂuid velocity at the point coinciding with the center of particle a ﬂuctuation of ﬂuid velocity ﬂuctuational component of ﬂuid velocity increment ﬂuctuational velocity of the continuous phase effective mean free path of a particle undergoing ﬂuctuational motion u0 i components of velocity ﬂuctuations of the ﬂuid phase hu0 i u0 j i Reynolds stress tensor of the ﬂuid phase u0 2 hu0 k u0 k i=3 intensity of ﬂuid velocity ﬂuctuations hu0 i v0 j i second-order mixed correlation moments of velocity ﬂuctuations of the disperse and continuous phases space increment of velocity Du0 i ðrÞ u dynamic velocity (friction rate) hv0 i v0 j i turbulent stresses of the disperse phase hv0 i v0 j ia turbulent stresses of particles a v0 2 intensity of particle velocity ﬂuctuations hv0 i v0 j v0 k i third-order moment hv0 i v0 j v0 k v0 n i fourth-order moment (cumulant) v0 y þ dimensionless variable turbulent heat ﬂux in the disperse phase hv0 i q0 i va velocity of particle a vp particle velocity, velocity of a particle before the collision b vp velocity of a particle after the collision v0 p particle velocity ﬂuctuation v0 t velocity ﬂuctuation in the direction parallel to the wall w relative velocity of two particles wp relative velocity of colliding particles radial component of relative velocity wr wr(s) radial component of relative velocity of two particles at contact w0 r ﬂuctuation of the radial component of the relative velocity vector hw0 2r i intensity of ﬂuctuations of radial velocity of two particles at contact x point in space x longitudinal coordinate y a coordinate normal to the wall yþ dimensionless transverse coordinate in a planar channel z dimensionless time z average relative drift parameter for particles of different types z vertical axis pointing upward zr dimensional parameter in Eq. (5.49) zz dimensionless parameter in Eq. (2.186)

Upper-Case Greek

Upper-Case Greek

G G(x) ¡0 Q P P(z) P0 P1 S S S12 F F Fa Fm Fw Y0(z) Y1(z) Y2(z) Y3(z) Y4(z) Y5(z) Y6(z) Y7(z) YE(t) YEt(t) YL(t) YtL ij ðtÞ YtLp ij ðtÞ YlLp YnLp V YLp ðtÞ YLp ij(t) YLr(t|r)

radial distribution function of particles gamma function constant in Eq. (3.49) average temperature of the disperse phase parameter in Eq. (3.69) function in Eq. (6.44) parameter in Eq. (3.71) parameter in Eq. (3.72) temperature gradient parameter characterizing the interrelation between the actions of gravity and turbulence on the collision kernel number of collisions between particles of different types per unit volume per unit time average volume concentration dimensionless particle concentration volume concentration of particles a average mass concentration of particles in a channel cross-section particle concentration at the wall function in Eq. (6.33) function in Eq. (6.37) function in Eq. (6.37) function in Eq. (6.40) function in Eq. (6.40) function in Eq. (6.40) function in Eq. (6.40) function in Eq. (6.40) Eulerian single-point time autocorrelation function of velocity ﬂuctuations Eulerian single-point time autocorrelation function of temperature ﬂuctuations Lagrangian autocorrelation function Lagrangian autocorrelation function of joint ﬂuctuations of velocity and temperature of the continuum Lagrangian autocorrelation function of velocity and temperature of the continuous phase along a particle trajectory longitudinal autocorrelation function transverse autocorrelation function two-scale bi-exponential autocorrelation function Lagrangian autocorrelation function of ﬂuid velocity along a particle trajectory Lagrangian autocorrelation function of two particles initially separated by the distance

j271

j Notation Index

272

r YLrp(t|r)

YLt(t) YLtp(t) Yp(t) Wp Wr WV

(Lagrangian autocorrelation function of velocity ﬂuctuation increment of ﬂuid particles separated by a distance r) Lagrangian autocorrelation function of velocity ﬂuctuations of a pair of ﬂuid particles moving along inertial particle trajectories and separated by a distance r ¼ |r| Lagrangian autocorrelation function of temperature ﬂuctuations of the continuous phase Lagrangian autocorrelation function of ﬂuid temperature along the particle trajectory Lagrangian autocorrelation function of particle velocities when the average velocity is zero parameter characterizing particle inertia in terms of the duration of particle interaction with energy-carrying eddies dimensionless parameter parameter of particle inertia

Lower-Case Greek

a a a a a au atu b b b b bA bg bs bST bt g g ij d(x) d dþ d ij e e z

factor in Eq. (5.39) parameter in the relation (2.141) constant in the formula (2.40) constant in Eq. (2.189) particles number (a ¼ 1, 2) constant in Eq. (3.49) constant in Eq. (3.47) factor in Eq. (5.39) constant in the formula (2.41) dimensionless parameter in the formula (1.73) collision kernel Abrahamson collision kernel collision kernel of particles under the action of the gravity force Smoluchowski collision kernel Saffman–Turner collision kernel, collision kernel of inertialess particles turbulent collision kernel drift parameter components of the tensor of velocity ﬂuctuation gradient Diracs delta function thickness of the viscous sublayer constant in Eq. (2.151) Kronecker symbol rate of kinetic energy dissipation convergence rate parameter fraction of the number of particles moving toward the wall

Lower-Case Greek

z12 h h h hi q q0 (x,t) qp 2 hq0 i 02 hq i k k c c c cL ij ctL ij l l li lij a

lij

mi mij maij

mt n np nT x z12 xa xx r rf rp }

correlation coefﬁcient of two particles colliding owing to their interaction with the turbulence Kolmogorov spatial microscale self-similar variable function in Eq. (3.17) function in Eq. (3.13) actual temperature of the ﬂuid ﬂuid particle temperature ﬂuctuation particle temperature intensity of temperature ﬂuctuations of the disperse phase intensity of temperature ﬂuctuations of the continuous phase Prandtl–Karman constant constant in Eq. (2.191) coefﬁcient of particle deposition parameter in the relation (2.139) factor in Eq. (5.39) parameter in the formula (2.34) parameter in Eq. (3.16) coefﬁcient of thermal conductivity of the ﬂuid dimensionless variable function in Eq. (3.16) integral containing the second correlation moment of velocity ﬂuctuations of the continuous phase along the particle trajectory integral containing the second correlation moment of velocity ﬂuctuations of the continuous phase along the trajectory of particle a function in Eq. (3.15) integral containing the second correlation moment of velocity ﬂuctuations of the continuous phase along the particle trajectory integral containing the second correlation moment of velocity ﬂuctuations of the continuous phase along the trajectory of particle a the coefﬁcient of friction kinematic viscosity coefﬁcient of the ﬂuid turbulent viscosity coefﬁcient of the disperse phase turbulent viscosity coefﬁcient of the continuous phase (ﬂuid) correlation coefﬁcient of ﬂuid and particle velocities coefﬁcient of correlation of radial velocity components of two particles due to their interaction with the turbulence correlation coefﬁcient of velocities of the ﬂuid and the particle a parameter in Eq. (2.100) dimensional variable density of the continuous phase density of particle material parameter in Eq. (3.69)

j273

j Notation Index

274

s s sij(x, t) sll snn sp ll sp nn sxy t t t tþ t12 tc tc1 tc2 tcr tk tp tp/Tu tp0 tp1 tp2 tt tT tTp tTr tTr ts tw j j(Rep) j(y) f(r, t) y y(t) syt w wc w12

collision radius dimensionless collision radius components of the strain tensor dimensional variable dimensional variable dimensional variable dimensional variable shear stress time dimensionless variable dimensionless variable dimensionless variable; dimensional relaxation time time intervals between collisions of particles 1 with particles 2 characteristic time interval between particle collisions characteristic time interval between particle collisions characteristic time interval between particle collisions critical value of inertia parameter t corresponding to the bifurcation point Kolmogorov time microscale characteristic dynamic relaxation time of a particle particle inertia parameter characteristic dynamic relaxation time of a particle in the Stokes approximation effective relaxation time of the disperse phase effective relaxation time of the disperse phase characteristic thermal relaxation time of a particle Taylor differential time scale differential time scale Taylor differential time scale of relative velocity of two ﬂuid particles Taylor time microscale of velocity ﬂuctuation increment characteristic time of decrease of strain correlation function characteristic time of decrease of rotation correlation function polar angle Schiller–Neumann approximation for the resistance factor of a spherical particle function in Eq. (2.191) probability density of particle displacement r at the moment t azimuthal angle in the (x, y) plane orthogonal to the z-axis function in the formula (1.57) operator in Eqs. (4.43)–(4.44) dimensionless variable frequency of collisions of a particle with the other particles frequency of collisions of particles 1 with particles 2

Abbreviations

wc12 wik(x, t)

frequency of collisions of particles 1 with particles 2 components of the rotation tensor

Abbreviations

PDF DNS LES

probability density function direct numerical simulation large-eddy simulation

j275

j277

Author Index a Abou-Arab, T. W. 61 Abrahamson, J. 139, 145, 154, 199, 210, 212, 216 Adou, J. 63 Adrian, R. J. 14, 23, 107 Agarwal, J. K. 112, 113 Ahluwalia, A. 171, 183 Ahmadi, G. 51, 107, 108 Ahmed, A. M. 74 Alexander, J. 3, 4 Aliod, R. 57 Alipchenkov, V. M. 14, 41, 46, 49, 50, 53, 59, 64, 107, 108, 127, 139, 141, 171, 210, 229 Aliseda, A. 171 Arcen, B. 63, 97, 98 Armenio, M. XV, 11 Ayala, O. 198

b Bagchi, P. 12 Balachandar, S. XV, 12, 74 Balkovsky, E. 171, 183 Banerjee, S. 12, 96, 97 Barenblatt, G. I. XIII Bark, F. H. 97, 101, 104, 106, 164, 167 Barré, C. 74, 78, 81, 82, 83, 84 Batchelor, G. K. 200, 203, 205 Bazile, R. 94 Bec, J. XVII, 171, 172, 183, 184 Bedat, B. XV Berg, J. 204 Berlemont, A. 23, 24, 46, 74, 78, 81 Bernard, P. S. 98 Biferale, L. 4, 36, 200 Binder, J. L. 68, 107 Bodenschatz, E. 4, 204 Boersma, B. J. 107

Boffetta, G. 4, 36, 183, 200 Boivin, M. XV, 11, 47, 52, 91, 78, 81, 82, 84 Borée, J. 94, 154, 164, 168 Borgas, M. S. 9, 200 Botto, L. 127 Boulet, P. 127 Bourgoin, M. 204 Brooke, J. W. 107 Brunier, E. 216, 217, 222, 223, 226, 227 Brunk, B. K. 7 Burton, T. M. 11 Buyevich, Y. A. XVI, 53

c Caballina, O. 46 Camussi, R. 176 Cantrell, W. 172 Caporaloni, M. XVII, 91, 97 Caraman, N. 158, 164, 168 Carlier, J.-Ph. 46 Cartellier, A. 171 Celani, A. 4, 36, 171, 200 Cencini, M. 4, 36, 172 Cerbelli, S. 49 Chapman, S. XVII Chang, Z. 74, 141, 142 Charnay, G. 94 Chen, C. P. 61 Chen, H. 57 Chen, L. 164 Chen, M. 57, 164, 171, 172, 199 Chen, Q. 108 Chepurniy, N. XVII, 137 Cho, W. C. 51, 98 Choi, J. 98, 100 Chronopoulos, E. 200 Chun, J. 139, 171, 183 Chung, J. N. XIII

j Author Index

278

Chung, M. K. 51, 98, 100 Cleaver, J. W. 107 Clift, R. 12 Collins, L. R. 139, 148, 171, 183, 184, 191, 193, 196, 223 Corrsin, S. 14, 71, 72 Cota, P. 97 Cowling, T. G. XVII, 137, 152 Crane, R. T. 139, 140, 210 Crawford, A. M. 4 Crowe, C. T. IX, XIII Csanady, G. T. 14, 20, 21, 28 Curtis, J. S. 96 Cuzzi, J. N. 172

d Darbyshire, K. F. F. 46 Davies, C. N. 107 De Lillo, F. 171 Dekeyser, I. 126 Derevich, I. V. 14, 16, 17, 26, 33, 43, 46, 48, 49, 50, 53, 56, 57, 61, 63, 64, 107, 108, 122, 127 Derksen, J. J. 9, 176 Desjonqueres, P. 46 Deutsch, E. 21, 41, 74, 78, 80, 81, 82, 84, 137, 154, 155, 157 Devenish, B. J. 48, 200 Ding, J. XVII, 137 Dobrovolskis, A. XVIII Dodin, Z. 216 Donzis, D. A. 4 Douhara, N. XVIII Druzhinin, O. A. XV Durbin, P. A. 200 Durst, F. 96

Février, P. XVI, 22, 191, 192, 196, 193, 194, 218, 219, 220 Fichman, M. 107 Fiorotto, V. XV, 11 Fohanno, S. 164 Fouxon, A. XIX, 196, 208 Frank, Th. 164 Friedlander, S. K. 107 Frisch, U. 39, 40, 116, 173 Fukagata, K. 97, 101, 104, 106, 164, 167 Fukayama, D. 4 Fung, J. C. H. 17, 200

g Gamba, A. 171 Gardiner, C. W. 44 Gatski, T. B. 59 Gavin, L. B. XIII, 14, 64 Gidaspow, D. XIII, 137 Girimaji, S. S. 7, 80 Givi, P. 14 Gorbis, Z. R. IX, XIII Goto, S. 164 Gotoh, T. 4 Gourdel, C. 216, 222, 223, 226, 227, 228 Gouesbet, G. 24 Grace, J. R. 12 Grad, H. 69 Graham, D. I. 14 Gribova, E. Z. 14 Guha, A. 49, 107 Giusti, A. 49, 107 Gusev, I. N. 46, 49, 53 Guseva, E. I. 87, 108 Gutﬁnger, C. 107

h e Eaton, J. K. XIII, XVII, 11, 14, 23, 94, 95, 97, 101, 138, 167, 171, 172 Elghobashi, S. E. XIII, XIV, 14, 28, 61, 74 Elperin, T. 171, 172 Etasse, E. 14

f Faeth, G. M. 12 Falkovich, G. XIX, 171, 183 Fan, F. 107 Fede, P. XV, XVII, 147, 162, 163, 164, 172, 190, 210, 223, 224 Ferchichi, M. 39 Ferrand, V. 94 Ferry, J. 15 Fessler, J. R. XIII, 96, 172

Hadinoto, K. 96 Han, K. S. 127 Hädrié, Th. 63 Häinaux, F. 171 Hanjalic, K. 51 Hanratty, T. J. 13, 14, 23, 68, 96, 98, 107, 112 Hanzawa, A. 96 Hardalupas, Y. 200 He, J. 64, 164 Hedley, A. B. 107 Herweijer, J. A. 176 Hesthaven, J. S. 13 Hetsroni, G. 33, 96, 97 Hinze, J. O. 28, 30 Hishida, K. 96 Hogan, R. C. XVIII, 171, 172 Hu, K. C. 150, 151

Author Index Hu, Y. 98 Huang, X. 57 Huber, N. 63 Hunt, J. C. R. 17, 200 Hwang, W. 171 Hyland, R. E. 46, 48

Kulick, J. D. 96, 172 Kurdyumov, V. N. 48 Kussin, J. 164 Kusters, K. A. 139, 140, 210 Kuznetsov, V. R. 6, 39, 176, 184

l i Iliopoulos, I. 98

j Jaberi, F. A. 14, 34 Jeffrey, D. J. XVII, 137 Jenkins, J. T. XVII, 69, 137, 153, 154, 157, 158, 164 Johansen, S. T. 49, 107 Johansson, A. V. 59 Johnstone, H. F. 107 Jones, E. W. 96 Jongen, T. 59

k Kaftori, D. 96 Kajishima, T. XV, 164 Kallio, G. A. 100, 107, 108 Kannosto, J. 172 Karnik, U. 81 Kartushinskii, A. I. 164 Kaufman, A. XV Keswani, A. 171, 184 Khain, A. P. XIX Khalij, M. 46, 63 Khalitov, D. A. 96, 194 Kheiri, A. 46 Kim, D. S. 107 Kim, K. 51, 98 Kiribayashi, K. XVII Kitoh, O. 98 Kleeorin, N. 171, 172 Klimenko, A. Ju. 46 Klyatskin, V. I. 39, 40, 116, 171, 173 Konan, A. 63 Koch, D. L. XVII, 139, 171, 183 Kohnen, G. 59 Kondo, S. 97, 106, 164 Kondratev, L. V. 50 Kontomaris, K. 13, 97, 98, 107, 164 Kostinski, A. B. XIX, 171, 172 Kozelev, M. V. 53, 127 Kraichnan, R. H. 13, 47, 107, 200 Kroshilin, A. E. 46, 107 Kruis, F. E. 139, 140, 141, 210 Kukharenko, V. N. 46, 107 Kuerten, J. G. M. XV, 97

La Porta, A. 3 Lain, S. 57, 59 Lakehal, D. 107 Lamorgese, A. G. 4 Lanotte, A. 4, 36, 37, 200 Lanterman, D. D. XIX Lapiga, E. J. 138 Lapinova, S. A. 14 Larsen, M. L. 172 Lasheras, J. C. 171 Laufer, J. 93 Launder, B. E. 51, 126 Laviéville, J. XV, 31, 78, 80, 97, 142, 154, 164, 172, 194 Lee, C. 98, 100 Lee, J. T. 13, 16, 17 Lee, J. W. 107 Lee, M. M. 107 Lee, S. L. 96 Lee, U. 171 Leeming, A. D. 49, 107 Legendre, D. 12, 171, 191, 192, 193 Lei, U. 14, 76 Leschziner, M. A. 46 Li, A. 113 Li, C. 97 Li, Y. 97, 164 Liljegren, L. M. 78 Lion, L. W. 7 Liu, B. Y. H. 112, 113 Liu, H. S. 164 Lohse, D. 2, 16, 17 Longmire, E. K. 94, 196 Loth, E. XIII Louge, M. Y. 154, 164 Lumley, J. L. 13, 14 Lun, C. K. K. XVII, 137, 164 Lundgren, T. S. 5, 7, 13, 184, 200, 203, 204 Lvov, V. S. 171

m Mac Giolla Mhuiris, M. 76 Maeda, M. 96 Magnaudet, J. 12 Malik, N. A. 17, 200 Mann, J. 200

j279

j Author Index

280

Marchioli, C. XIX, 97, 101, 102, 107, 172 Mariotti, G. 97 Marshall, W. R. 13 Mashayek, F. XIII, XV, 14, 46, 47, 74, 78, 81, 82, 83, 84, 134, 135 Mastorakos, E. 154, 164 Matsumoto, S. 62, 63 Maxey, M. R. 11, 171, 172 Mazzitelli, I. M. 2, 16 McCoy, D. D. 107, 112 McKee, S. 46, 48 McLaughlin, J. B. XIII, 98, 107, 108, 112, 113, 164, 199 Mei, R. 14, 23, 150, 151 Meneveau, C. 14 Merle, A. 12 Michaelides, E. F. 11, 164 Middleton, J. F. 13, 16 Miller, R. S. 14 Minier, J.-P. 14, 47, 52 Mito, Y. 98 Mols, B. 107 Mongia, H. C. 61 Monin, A. S. 1, 4, 6, 8, 39, 49, 176, 184, 200 Moreau, M. XV, XVII, 162, 163, 164 Morikawa, Y. 62, 63, 96 Mork, K. J. 68 Mostafa, A. A. 61 Mosyak, A. 97 Musacchio, S. 4, 36, 171, 172

n Nakatani, M. 63 Nakatsukasa, N. 63 Narayanan, C. 107 Naumov, V. A. XIII, 14, 61, 64 Nicolleau, F. 200 Nielsen, A. H. 13 Nieuwstadt, T. M. 107 Nigmatulin, B. I. 46, 107, 108 Nir, A. 14, 28

o Obukhov, A. M. 200 Oesterlé, B. XVII. 17, 19, 46, 62, 63, 97, 98, 127, 164, 171 Oliemans, B. V. A. 97, 107, 108, 112, 113 Oncley, S. 176 Orszag, S. A. 57 Oshima, T. 62, 63 Ott, S. 200 Ouelette, N. T. 204 Ounis, H. 51, 107

p Pan, Y. 12, 97 Pandya, R. V. R. XIII, XV, 46, 47, 74, 79, 81, 82, 83, 84, 86, 122, 132, 133, 134, 135 Papavergos, P. G. 107 Paque, J. XVIII Parker, A. 48, 50, 53 Pascal, P. 46 Patterson, G. S. 14 Pécseli, A. L. 13, 200 Pedinotti, S. 97 Peirano, E. 47 Perkins, R. J. 17, 200 Pershukov, V. A. XIII, 53, 57, 107, 108, 127, 154, 159, 164 Petitjean, A. XVII, 164 Philip, J. R. 13, 16, 17 Phythian, R. 13 Pialat, X. XV Picciotto, M. 97, 101, 102, 172 Pinsky, M. B. XIX Piomelli, U. XV Pismen, L. M. 14, 28 Pnueli, D. 107 Poinsot, T. XV, 14 Pointin, Y. B. 13 Polezhaev, Yu. V. 96 Polyakov, A. F. 96 Pope, S. B. 2, 3, 4, 6, 7, 16, 39, 44, 47, 60, 76, 184 Portela, L. M. 9, 10, 97, 164, 176 Potthoff, M. XV, 164 Pozorski, J. 14, 47 Praskovsky, A. 176 Pumir, A. 171

r Rambaud, P. 98 Rani, S. L. XV, 164, 171, 183 Ranz, W. E. 13 Rasmussen, J. J. 13 Razi Naqvi, K. 68 Reade, W. C. XVIII, 139, 171, 183, 191, 193, 196, 223 Reeks, M. W. XVII, 14, 28, 29, 46, 47, 48, 78, 91, 97, 100, 101, 102, 107, 108, 172, 232 Richardson, L. F. 200, 203 Richman, M. W. XVII, 69, 137, 153, 157, 158 Righetti, M. 96 Riley, J. J. 11, 14, 50, 51 Rizk, M. A. 61 Rogachevskii, I. 171, 172 Rodi, W. 58, 59 Rogers, C. B. 94, 95, 96

Author Index Romano, G. P. 96 Rotta, J. C. 157 Rouson, D. W. I. 97, 101, 104, 167 Rovelstad, A. L. 98 Rudi, Yu. A. 164

s Sabelnikov, V. A. 6, 39, 176, 184 Saffman, P. G. 12, 13, 108, 139, 140, 144, 150, 199, 200, 215 Saichev, A. I. 14 Saito, S. 62, 63 Sakiz, M. 64, 158, 164 Sato, Y. 16, 96 Salvetti, M. V. 107 Satyanarayan, K. 3, 4 Savage, S. V. XVII, 137 Sawford, B. L. 3, 74, 76, 200, 203 Schade, K.-P. 63 Sergeev, Y. 48, 50, 53 Shaw, R. A. XIX, 171 Shin, M. 107 Shiomi, H. 96 Shor, V. V. 50 Shotorban, B. 74 Shraiber, A. A. XIII, 14, 61, 64 Shy, S. S. 171 Sigurgeirsson, H. 171, 172 Simonin, O. XIII, XV, XVI, XVII, 2, 11, 16, 21, 22, 23, 24, 28, 30, 31, 47, 50, 52, 59, 61, 63, 64, 74, 78, 79, 81, 82, 83, 84, 97, 101, 102, 137, 139, 141, 142, 147, 154, 155, 157, 158, 162, 163, 164, 166, 167, 168, 171, 172, 190, 191, 192, 193, 194, 195, 196, 210, 216, 217, 222, 223, 224, 226, 227, 228, 229 Sinaiski, E. G. XIX, 138 Skibin, A. P. 49 Slater, S. A. 49, 107 Smoluchowski, M. V. 150 Sokoloff, D. 171 Soldati, A. XIX, 49, 97, 101, 102, 107, 172 Snyder, W. H. 14 Soloviev, S. L. 49 Sommerfeld, M. IX, XIII, XVII, 23, 24, 62, 63, 96, 154, 164 Speziale, C. G. 59, 76 Spokoyny, E. E. IX, XIII Sreenivasan, K. R. 8 Squires, K. D. XV, XVII, XIX, 11, 13, 14, 16, 23, 28, 50, 97, 98, 107, 108, 112, 113, 138, 164, 166, 167, 171, 172, 194, 195, 196 Staroselsky, I. 57 Stepanov, A. S. 13, 16, 17 Stepanov, M. G. XIX

Stock, D. E. 14, 16, 17,19, 21, 77 Stone, G. L. 13,16, 17 Stuart, A. M. 171, 172 Succi, S. 57 Sundaram, S. 148, 171 Sung, H. J. 51, 98, 127 Swailes, D. C. 46, 48, 50, 53, 107

t Takeuchi, S. XVII Tampieri, F. XVII, 91, 97 Tanaka, T. XV, XVII, 63, 164 Taniere, A. 97, 98, 127 Tatarskii, V. I. 184 Taulbee, D. B. 59, 60, 74, 78, 81, 82, 83, 84 Tavoularis, S. 39, 81 Taylor, G. I. 205 Ten Cate, A. 9, 10, 176 Thomson, D. J. 200 Toschi, F. 4, 36, 37, 200 Trombetti, F. XVII, 91, 97 Truesdell, G. S. 14, 28 Troutt, T. R. XIII Trulsen, J. 200 Tsuji, Y. XIII, XVII, XVII, 62, 63, 96, 164 Turner, J. S. 139, 144, 150, 199, 210, 215

u Uijttewaal, W. C. J. 107, 108, 112,113 Ushijima, T. 98

v Van de Water, W. 176 Van den Akker, H. E. A. 9, 10, 176 Van Haarlem, B. 107 Van Kampen, N. G. 47 Vance, M. W. 97, 164, 166, 167, 168 Vanka, S. P. 164 Varaksin, A. Yu. XIII, 96 Vassilicos, J. C. 171, 172, 200 Vatazhin, A. B. 46 Vedula, P. 3, 4 Verzicco, R. 176 Villedieu, P. XV, XVII, 162, 163, 164 Vinberg, A. A. 50, 53, 57, 108, 127 Viollet, P. L. 61 Vittori, O. XVII, 91, 97 Volkov, E. P. XIII Voth, G. A. 3, 4 Vreman, A. W. XV

w Waldenstrøm, S. 68 Wallin, S. 59

j281

j Author Index

282

Wang, L.-P. 14, 16, 17, 21, 50, 77, 97, 98, 101, 104, 107, 108, 112, 113, 139, 140, 141, 147, 148, 149, 167, 171, 176, 181, 189, 190, 191, 193, 196, 197, 198, 212, 213, 214, 215, 236, 237 Wang, Q. 50, 97, 98, 104, 107, 108, 112, 113 Wassen, E. 164 Weber, M. E. 12 Weinstock, J. 13, 14 Wells, M. R. 14, 21 Wexler, A. S. 50, 97, 139, 140, 141, 146, 148, 149, 171, 176, 181, 189, 190, 191, 193, 196, 197, 198, 210, 212, 213, 214, 215, 236, 237, 238, 239 Williams, I. J. E. 139, 140, 210 Winkler, C. M. 164 Wood, A. M. 171 Wood, N. B. 171 Wood, P. E. 61 Wu, J.-S. 12 Wu, X. 98

x Xu, H. 204 Xue, Y. 140

y Yaglom, A. M. 1, 4, 6, 8, 39, 49, 144, 176, 184, 200 Yamamoto, K. 16 Yamamoto, Y. XV. 164 Yang, C. Y. 171

Yang, T. S. 171 Yarin, L. P. 33 Yates, B. 107 Yatsenko, V. P. XIII, 14, 61 Yeh, F. 14, 74 Yeo, K. 98, 100 Yeroshenko, V. M. 53, 56, 57, 61, 127 Young, J. B. 49, 96, 107 Yeung, P. K. 2, 3, 4, 16, 74, 76, 200, 203 Yu, G. 200 Yudine, M. I. 14, 20, 28 Yurteri, C. 96 Yuu, S. 140

z Zahrai, S. 97, 101, 104, 106, 164, 167 Zaichik, L. I. XIII, XIX, 5, 8, 14, 17, 19, 41, 46, 48, 50, 53, 56, 57, 59, 60, 64, 74, 78, 87, 97, 98, 107, 108, 113, 122, 125, 127, 130, 139, 141, 149, 154, 159, 164, 171, 177, 183, 210, 216, 229 Zhang, H. 63, 107, 108 Zhou, L. XIII Zhou, L. X. 57, 63, 107, 108, 141, 210 Zhou, Q. 46 Zhou, Y. 50, 139, 140, 141, 146, 147, 148, 149, 171, 176, 181, 189, 190, 191, 193, 196, 197, 198, 210, 212, 213, 214, 215, 236, 237, 238, 239 Zhukova, I. S. 14 Zivkovic, G. XVII, 63, 154

j283

Subject Index a Acceleration – autocorrelation function of 35 – of low-inertia particles 35 – mechanism of particle collisions 139 Accumulation, see Clustering Accuracy of – boundary conditions 74 – non-linear model 61 Actual velocity of the ﬂuid 40 Adsorption at the wall 111 Aerosols – deposition 107 – polydispersity 108 – separation XIII – spreading XIII Algebraic model – for hydrodynamic characteristics of the disperse phase 39 – of heat transport, two approaches 127 – implicit, for turbulent heat ﬂuxes 131 – implicit, for turbulent stresses of the disperse phase 59 – for Reynolds stresses 52 – for turbulent heat ﬂuxes 127 – for turbulent stresses of the disperse phase 59 Algebraic relation 49 – for third-order structure function 178 Amplitude of acceleration ﬂuctuations in isotropic turbulence 3 Analytical solutions of particle collision problem 138 Anisotropy – of action of time scales on ﬂuctuational motion of particles 77 – of duration of particle interaction with turbulent eddies 46

– of involvement coefﬁcients 46 – of particle velocity ﬂuctuations 137, 209 – tensor 58 – of turbulence scales 46 – of turbulence scales in the near-wall region 64 Approximate integration 15 Approximating – the axial and transverse components of acceleration ﬂuctuation 3 – integral time scale of velocity ﬂuctuation increment over the entire distance range 9 – the Kolmogorov constant 2 – the longitudinal structure function of velocity ﬂuctuations over the entire range of inteparticle distances 9 Asymptotic – behavior of the continual model in the limit of inertialess particles 50, 125 – expression – for deposition factor of high-inertia particles 114 – for diffusion tensor at large values of time 2 – for the Kolmogorov constant at high Reynolds numbers 3 – for turbulent diffusion tensor of particles at large values of time 28 – solution for high-inertia particles in the near-wall region 95 Atmospheric processes XVIII Attractor 172 Autocorrelation function – of acceleration 35 – of continuous phases velocity and temperature along the particle trajectory 119

j Subject Index

284

– of ﬂuid velocity ﬂuctuations along a particle trajectory 14, 16 – as a tensor 16, 20, 133 – triangular 141 Average – characteristics of gradient turbulent ﬂow 39 – radial component of relative velocity 140 – relative radial velocity of two touching particles 140 – slip 21 – temperature of a disperse phase consisting of inertialess particles 125 – velocity gradient 42 – velocity of a disperse phase consisting of inertialess particles 50 – velocity slip relative to the ﬂuid 14 – volume concentration 48 Averaging over the ensemble of – initial conditions XIV – random realizations 40 – velocity and temperature ﬁelds of the carrier ﬂuid 2 Azimuthal angle 149

b Balance equations 48, 97, 123, 177 Barnett equations 53 Basic operator 66 Beer–Lambert exponential law XIX Bidispersed system of particles 209 Bifurcation point 89 Binomial distribution 64 Boltzmann equation 53, 137 Boltzmann integral 152 Boltzmann integral operator XVI Boundary conditions for – absolutely elastic particle interactions with the wall 70 – average longitudinal velocity of particles 70 – Brownian particles 68 – diagonal components of turbulent stresses in the disperse phase 69, 160 – disperse phase with the consideration of particle collisions 159 – equations of motion of the disperse phase 62 – ﬂuctuation intensities 73 – high-inertia particles 70 – inertialess particles 70 – longitudinal velocity 68 – longitudinal velocity with the consideration of particle collisions 159

– normal components of velocity ﬂuctuation intensities 70 – particles with bi-normal velocity distribution in the near-wall region 71 – sticking 70 – tangential stresses 159 Boundary layer – turbulent 96 Boussinesq relation 56 Breakage of particles XIV Brownian diffusion – coefﬁcient 108 – effect on particle deposition 107 Brownian motion 53, 138

c Central limit theorem 39 Chain of equations for moments 123 Chaotic motion of molecules 8 Chapman–Enskog – expansion 53 – perturbation method XVI, 53, 127 Characteristic gradient of average velocity of the carrier ﬂow 42 Characteristic time of – dynamic relaxation of a particle 11 – in the Stokesean approximation 12 – particle interactions – with energy-carrying eddies 42 – with turbulence XVI – with turbulent eddies of the carrier ﬂow 137 – thermal relaxation of a particle 13 – variation of average hydrodynamic ﬂow parameters 54 – variation of heat ﬂow parameters 127 Characteristics of – gradient turbulent ﬂow 39 – turbulence along ﬂuid element trajectories 52 – turbulence of the continuous phase in the inertial interval 8 Closed kinetic equation for – single-point PDF of particle velocity and temperature 122 – two-point PDF of relative velocity of a particle pair 176 Closed set of moment equations XVI Closure problem for the kinetic equation for particle velocity PDF 39 Closure relations 49 Clustering XVII – asymptotics for particle pairs in the inertial spatial interval 185

Subject Index – criteria 172 – decrease in the near-wall region due to collisions 168 – of inertial particles XVII, 171 – in the near-wall ﬂow XIX – inﬂuence on the collision frequency 196, 235 – in the near-wall region 91 – in non-homogeneous turbulent ﬂow 91 – of small heavy particles XIII Coagulation – in homogeneous turbulence 139 – kinetic equation 138 – of particles 138 – rate XIX Coefﬁcient of – heat conductivity of the ﬂuid 13 – momentum restitution 62 – particle involvement in temperature ﬂuctuations of the continuum 33 – particle responsiveness to temperature ﬂuctuations of the carrier ﬂuid 32 – turbulent diffusion of particles in the inertialess approximation 56 Coherent structures 107 Collision kernel 138 – for bidispersed particles 212 – for high-inertia particles 145 – in the inertial interval 212 – for inertialess particles, see Saffman–Turner kernel – for low-inertia particles 146 – for particles at high Reynolds numbers 146 – in the presence of average velocity gradient and turbulence 149 – in the presence of gravity 215 – in the presence of gravity and turbulence 215 – spherical formulation 140 – turbulent component of 150 Collision problem 140 Collisional term – due to acceleration 140 – in the transport equation 137 Collisions, see Particle collisions Combined Lagrangian–Eulerian simulation method XVI Combining solutions for velocity shear and acceleration 140 Combustion 13 – of atomized solids and liquid fuels XVIII – effect of turbulent ﬂuctuations on the rate of XIV – heterogeneous XIV

Concentration – of disperse phase 39 – ﬂuctuations of XVII, 171 – of inertial particles in the near-wall region 95 – of particles–inﬂuence of inertia 168 – preferential 171 Conservation equations 218 Continual model – differential of the second order 126 – of dispersion XVI – of heat transport XVI – hydrodynamic XVII – of particle clustering XVI – of particle motion 7 – of transport 39 Continual model 50, 125 Continuity effect 20 Continuity equation 5 Continuous phase XIX, 58 – velocity correlations 58 Convection of one-particle PDF in the phase space 46 Convective transport 49 – of momentum in the transverse direction due to particle deposition 109 Correlations – of acceleration ﬂuctuations in isotropic turbulence 35 – of ﬂuid velocity and temperature along inertial particle trajectories 13 – between probability density of velocity and particle temperature 117 – between temperature ﬂuctuations of the continuous phase and probability density 116 – between velocity and turbulent diffusion of heavy particles 14 – between velocity ﬂuctuations and ﬂuid temperature 117 – between velocity ﬂuctuations of the continuous phase and PDF of particle velocities 42 Correlation coefﬁcient of – ﬂuid and particle velocities 141 – radial velocity components of two particles 211 – two particles 143 Correlation function – Lagrangian 1 – of strain tensor 7 – of rotation tensor 7 – of velocity ﬂuctuations of the continuous and disperse phases 74

j285

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286

Correlation moment – of transverse component of velocity 73 – of velocity ﬂuctuations of the continuous and disperse phases 75 Corrsins hypothesis 14 Coulomb friction law 62 Covariance of ﬂuid and particle velocities 141 Crossing trajectory effect 14, 28 Cumulant 49 Cylindrical formulation of the collision problem 140

d Density of – ﬂuid 11 – particle material 11 Deposition XIV, 49 – of aerosol particles and drops on bounding surfaces 107 – coefﬁcient 67 – dependence on gravity, lifting force, and ﬂow direction 107 – factor 108 – rate 68 – in a vertical channel 107 Deterministic Lagrangian description of – heat exchange XIV – motion of the disperse phase XVI Differential operator of the Fokker-Planck type XVI Diffusion – Brownian 107, 108 – coefﬁcient 28, 108 – equation for an inertialess impurity 50 – for one-particle PDF in the phase space 46 – of passive impurity 50 – turbulent – of particles 49 – tensor 1, 28, 56 – of zero-inertia particles XV Diffusive transport 49, 123 Diracs delta function 15, 40 Direct – interactions – resulting in collisions XVII – summation using the Lagrangian method 47 – numerical simulation (DNS) XV – stochastic trajectory simulation of particle dynamics 21 – trajectory simulation of disperse phases characteristics in a homogeneous shear layer 78 Disperse phase

– fraction of XVII – hydrodynamic, heat exchange, and mass exchange equations for XVI – kinetic energy 54, 194 – low-concentrated 138 – turbulent stresses in 49 – velocity 48 – velocity correlations 58 Dispersion of – ﬂuid particle pair 5 – heavy particles 14 – inertial particles 205 – inertialess particles 202 – passive impurity 13 – small heavy particles XIII Displaced mass 11 Dissipation – rate in isotropic turbulence 2 – of turbulent energy XV, 139 – inﬂuence of particles XIX – of turbulent stresses 49 Distance between two particles 143 Distribution – binomial 64 – bi-normal 71 – of the distance vector between two points 7 – elliptic 39 – Gaussian 15, 39, 137, 141, 149, 209 – Maxwellian 53, 141 – of ﬂuid velocity 142 – of particle size 138 – Poisson 172 – of relative concentration 168 – of temperature 47 – of velocity 39, 46, 68, 141 Drift – parameter 16 – velocity 13 Droplets XIII, XIX Dynamic – Lagrangian simulation XIV – probability density in the phase space of coordinates, velocities, and temperature 115 – relaxation time 137 – response time XVII – stochastic equation of motion 40 – velocity 87 Dynamics of sand storms XIII

e Effect of diminishing correlativity of particle ﬂuctuations 13 Effective

Subject Index – free path of a particle 18 – friction coefﬁcient 63 – restitution coefﬁcient 63 – Stokes number 25 Energy – equation 116 – kinetic, see Kinetic energy – Langevin equation for XIV – turbulent dissipation 139 Equation for – average temperature of the disperse phase 123 – balance of momentum for the relative motion of particles 177 – correlation moment of particle velocity ﬂuctuations 28 – dynamic probability density for a single particle 115 – joint PDF of velocity and temperature distribution for a particle 47 – Lagrangian mixed correlation moment of temperatures of the continuous and disperse phase along particle trajectories 32 – mixed correlation moment 27 – mixed moment of velocity ﬂuctuations of the continuous and disperse phase 92 – number concentration of particle pairs 177 – PDF of particle velocity distribution in a homogeneous non-shear turbulent ﬂow 46 – relative motion of a particle pair 173 – second moments of ﬂuctuation increment of a particle pair 177 – second moments of particle velocity ﬂuctuations 48 – exact solution 74 – single-point moment of particle temperature 123 – single-point single-particle PDF of velocity temperature 116 statistical PDF of particle velocity distribution 40 – temperature variation 12 – third-order moment of velocity ﬂuctuations 49 – turbulent heat ﬂux 130 – turbulent stresses of the disperse phase in the equilibrium state 58 – two-particle PDF and its moments 229 – two-point statistical moments of relative velocity PDF of a particle pair 177 – velocity moments 123 Equations of motion for

– particle 173 – approximate integration of 15 – particle and the carrier ﬂow 11 – single heavy particle 11 Equilibrium – logarithmic layer 87 – state of turbulent ﬂow 91 – two-phase ﬂow 91 Euler equations 52 Eulerian – continual – approach XV – description XIV – simulation XV – integral time scale 16 – longitudinal structure function 6 – second order space-time correlation moment of ﬂuid velocity ﬂuctuations 3 – single-point time autocorrelation function of temperature ﬂuctuations 10 – single-point time autocorrelation function of velocity ﬂuctuations 4 – space-time correlation moment – in isotropic turbulence 15 – of temperature ﬂuctuations 10 – spatial structure function of the second order 5 – structure function – second-order 8 – third-order 8 – transverse structure function 6 – two-point correlation moment 5, 142, 230 – two-point simultaneous correlation function of temperature ﬂuctuations 10 – two-point simulatenous correlation moment 4 Evolution of cumulus clouds XIII Expansion over orthogonal tensor basis 59 Exponential dependence – approximation of the autocorrelation function 31 – of autocorrelation function of velocity ﬂuctuation increment 6

f Favre averaging 48 Feedback action of particles on the turbulence 52 Flow – concentrated XIV – core 169 – of disperse phase XIV – dispersed XIII, XIV – downward, in a vertical channel 97

j287

j Subject Index

288

– homogeneous, see Homogeneous ﬂow – hydrodynamically developed in vertical planar channel 109 – macroscopic properties XVI – in the near-wall region 39 – non-equilibrium XV – non-isothermal 12 – pseudo-turbulent XVI – shear 39, 46, 51, 74, 78 – turbulent XIV–XVI – two-phase 1 – uniform 58 – upward, in a vertical channel 97 – in vertical channels 39, 97 Fluctuational characteristics of gradient turbulent ﬂow 39 Fluctuations – generation – from averaged motion due to the velocity gradient 49 – via dissipation of turbulent stresses of the disperse pahse 49 – due to the involvement of particles in the motion of the carrier ﬂow 49 – of particle concentration 90 – of pressure 167 – of relative velocity between particles and ﬂuid due to particle acceleration 149 – of velocity, redistribution between components 157, 167 Fluid – element 1 – incompressible 1 – particle 1 – temperature 14 – velocity 14 Fokker-Planck equation 46, 53 Force – balance 11 – Basset 11 – centrifugal 138 – deterministic 12 – diffusion driving 97, 181 – gravitational 12, 107, 215 – of hydrodynamic resistance 11 – hydrodynamic 138 – interfacial 12 – lifting 12, 107 – mass 12 – molecular 138 – point 11 – random 46 – Saffman 12, 108 – thermophoretic 12

– turbophoretic driving 97, 172 Forced turbulence 148 Formation of – clusters XVIII – compact regions with enhanced concentration 172 – planets from nebula XVIII – raindrops XIII Furutsu-Donsker-Novikov formula for a Gaussian random ﬁeld 39, 116 Fourier law 130 Fourier modes superposition 18 Fractal dimension of a cluster structure XVIII, 184 Fraction method 53 Friction coefﬁcient 61 Friction velocity 87 Friedmann–Keller chain of equations in the theory of one-phase turbulent ﬂows 49 Functional – calculus technique 39 – derivative of particle temperature 40 – along a particle trajectory 118 – with respect to ﬂuid velocity 118

g Gamma function 150 Gaussian – distribution 14, 150 – correlated 154, 210 – elliptic 39 – of relative radial component of velocity ﬂuctuation 149 – random ﬁeld model 107 – random process 39 Grads method XVI Grads 13-moment approximation 158 Gradient approximation for the third moment of joint ﬂuctuations of velocity and temperature of the continuum 126 Gradient relations 51 Gravitational sedimentation 139 Greens function 47

h Heat – capacity of particle material 13 – diffusion, see Thermal diffusion – exchange – equation for a single particle 33 – interfacial 14 – of particles in gradient turbulent ﬂow 173 – of small heavy particles XIII – ﬂux 123, 130

Subject Index – transport XIII, 13, 14, 116, 123, 125 Heaviside function 40 Hermitian polynomials 154 High-inertia particles XV, XVII – in near-wall region 95 Homogeneous ﬂow – anisotropic 218 – gradient 39 – non-shear 47 – shear 39, 75, 132, 137 – shearless 123 – turbulent XIV, 39, 123, 216 – with constant shear rate 75 Horizontal pipes and channels 107, 164 Hybrid Lagrangian-Eulerian method XV Hydrodynamic resistance 11 – inertia dependence 12 – of a spherical particle 12 Hydrodynamics of dispersed turbulent ﬂow XIII–XIV

i Impenetrable wall 63 Incident particles 63 Increment 6 Inertia effect 14 Inertia parameter 208 Inertial – particle dynamics in turbulent ﬂows 52 – particles XIV–XVI – spatial interval 6, 204 Inertialess impurity 2, 51 Inertiality 14 Inﬁnite chain of equations for statistical moments XVI, 176 Inhomogeneous – gradient turbulent ﬂow 40 – spatial distribution of particles 139 – turbulent ﬂow XIV Integral scale – of ﬂuid temperature ﬂuctuations along an inertial particle trajectory 25 – of ﬂuid velocity ﬂuctuations along inertial particle trajectories 171 – spatial 5 – temporal 5 – of ﬂuid temperature ﬂuctuations along an inertial particle trajectory 132 – of velocity ﬂuctuation increment 6 – of ﬂuid particles along trajectories of relative motion of inertial particles 176 – in the viscous interval 7 Intensity of – ﬂuid velocity ﬂuctuations 8

– heat release inside a particle 14 – radial velocity ﬂuctuations 138 – of two particles at contact 138 – temperature ﬂuctuations 11 Interaction of particles, see Particle interactions Interfacial interaction XIV Intermittency phenomenon 177, 184 Invariance principle 46 Involvement coefﬁcient 31, 144, 45 – associated with particles response to joint ﬂuctuations of velocity and temperature 45 – associated with temperature response of particles 45 – in the inertial interval 145 Involvement factor, see Involvement coefﬁcient Involvement of – particle in ﬂuctuational motion of the carrier ﬂow 24 – particle pair in turbulent motion of the continuum 176 Isotropic – approximations for turbulent stresses 56 – involvement coefﬁcient 31 – Lagrangian integral time scale tensor 52 – Lagrangian scale of turbulence 77 – particle–turbulence interaction time 77 – stationary turbulence approximation 137 – turbulence 8 Iteration procedure 14 Ito interpretation of stochastic integral 39

k

ke model of turbulence 201 Kinematic simulation method 16 Kinematic viscosity coefﬁcient of ﬂuid 3 Kinetic energy – of the continuous phase 58 – of the disperse phase 54, 195 – of inertial particles in homogeneous isotropic stationary turbulence 29 – of isotropic turbulence 2 – turbulent 54, 58 – of velocity correlations 58 Kinetic equation – for high-inertia particles 56 – in the presence of collisions 167 – for joint PDF of particle velocity and temperature 115 – for PDF of particle velocity and temperature in a homogeneous shearless turbulent ﬂow 122 – for PDF of particle velocity distribution 39

j289

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290

– for PDF of relative velocity of a particle pair 171 – for PDF of velocity, temperature, and other characteristics XVI – for single-point PDF of particle velocity distribution in a turbulent shear ﬂow 46 – for single-point (single-particle) PDF in the presence of collisions 151 – solving with the Chapman-Enskog method 127 – for two-particle PDF for bidispersed particles 209 – for two-point PDF of relative velocity of a particle pair 172 Knudsen number 54 Kolmogorov–Prandtl constant 100 Kolmogorov – constant 2, 145 – at high Reynolds numbers 2 – similarity hypotheses 6 – spatial microscale 6, 181 – theory of local similarity 144 – time microscale XVII, 139, 181 – velocity scale 181 Kronecker symbol 41

l Lagrangian – continual simulation method 62 – correlation function 1 – correlation moment of – ﬂuid particle acceleration ﬂuctuations 35 – ﬂuid particle temperature ﬂuctuations along an inertial particle trajectory 25, 119 – ﬂuid particle velocity ﬂuctuations calculated along an inertial particle trajectory 14 – ﬂuid velocity ﬂuctuations 1 – joint ﬂuctuations of velocity and temperature of a ﬂuid particle 119 – particle acceleration ﬂuctuations 35 – temperature ﬂuctuations of the continuous phase 119 – temperature ﬂuctuations of a particle along its own trajectory 32 – history direct-interaction approximation 47 – integral – scale for particles 31 – time scale 2 – method in the renormalized perturbation theory 47 – mixed correlation moment of – ﬂuid and particle velocity ﬂuctuations 27

– temperature of the continuous and disperse phase along a particle trajectory 29 – model of particle deposition 107 – simulation XIV – single-particle correlation moment 1 – of temperature ﬂuctuations of the continuous phase 10 – stochastic approach XV – structure function 5 – in the inertial interval 144 – time scale anisotropy 78 – trajectory – approach XV – description of the disperse phase XIII – simulation XV – two-particle correlation moment 5 – two-point structure function of velocity ﬂuctuations 56 – of ﬂuid particles moving along inertial particle trajectories 175 – of ﬂuid particles moving along their own trajectories 175 Lagrangian autocorrelation function 1 – of ﬂuid temperature along a particle trajectory 25 – of ﬂuid velocity along a particle trajectory 15, 145 – in the inertial interval 145 – of joint ﬂuctuations of velocity and temperature of the continuum based on the integral time scale 126 – of relative motion of two particles 6 – of temperature ﬂuctuations 10 – of velocity ﬂuctuations of the continuous phase 43 – of velocity ﬂuctuations of a ﬂuid particle pair along an inertial particle trajectory 52 – of velocity ﬂuctuation increment of ﬂuid particles 5 – of velocity and temperature ﬂuctuations in the continuum 119 Langevin stochastic equation of motion 47 Laplace transform of the autocorrelation function 21 Large-eddy simulation (LES) XV Lattice-Boltzmann method 9 Lifting force 107 Limiting relations for involvement coefﬁcients 45 Linear – algebraic model for turbulent stresses of the disperse phase 58 – approximation for turbulent stresses of the continuum 56

Subject Index – dependence of – Lagrangian integral scale on the Reynolds number 2 – turbulent heat ﬂux on average velocity gradients and temperature of the disperse phase 134 – model for turbulent heat ﬂuxes 131 Liouville equation for dynamic probability density of a single particle in the phase space 34 Local – disperse phase model 62 – equilibrium relation between turbulent stresses of the disperse and continuous phase 49 Locally isotropic turbulence 1 Logarithmic velocity proﬁle 58 Longitudinal – component of – Eulerian two-point correlation function of velocity ﬂuctuations 142 – relative diffusion of two ﬂuid particles 7 – second-rank tensor 5 – third-order structure function of velocity of the continuous phase 8 – third-rank tensor in isotropic turbulence 8 – turbulent stress 79 – Eulerian spatial correlation function 10 – structure function at small distances between two points 140 Low-turbulence zone XVII Lyapunov exponents 183

m Markovian process 137 – d-correlated in time 46 Mass conservation equation 48 Matching of solutions in the viscous and turbulent zones 89 Matrix of – involvement coefﬁcients 45 – particle interaction micro-times 45 Mechanics of interpenetrating heterogeneous media XIV Memory effect 107 Modeling of – heat transport XIII, 115 – mass transport XIII – momentum transport XIII – particle collision with a rough wall 63 – rigid particle collisions with a rigid surface 62

– rough wall as a plane virtual surface with random slopes having a Gaussian distribution 66 – turbulent stresses for the disperse phase 59 – two-phase, dispersed turbulent ﬂows XIII Modulation effect, see Feedback action of particles on the turbulence Molecular – chaos hypothesis in the kinetic theory of gases 115 – theory of rareﬁed gas XVI Moments – correlation, see Correlation moments – fourth-order 52 – correlation, of transverse component of velocity 73 – mixed – of temperature ﬂuctuations of the continuous and disperse phase 32 – of velocity ﬂuctuations of the continuous and disperse phase 27 – of velocity and temperature ﬂuctuations of the continuous and disperse phase 126 – second-order 126 – third-order – expressed through second-order moments and their derivatives 50 – incorporating temperature ﬂuctuations 130 – relation for 125 – of velocity ﬂuctuations in a single-phase turbulence 51 Momentum – balance equation 48 – Langevin equation XIV – restitution coefﬁcient 151 Monte Carlo method XVII Motion of – particle pair as a whole 143 – particles in a vertical channel 40 – single particle in turbulent media 39 – small heavy particles in a turbulent ﬂow XIII Mutual interaction of particles resulting from their collisions XIV

n Navier-Stokes equations 53 Near-wall ﬂow 58 Near-wall region 62 Non-correlative model 137 Nonlinear – algebraic model for turbulent stresses of the disperse phase 57

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292

– explicit algebraic model 52 – model for turbulent heat ﬂux 131 Non-local – differential model XIV – disperse phase model 61 – transport models 49 Non-stationarity 78 Normal distribution 39 Normalization condition 35 Number concentration of particles 137 Numerical solution of one-dimensional kinetic equation 74 Nusselt number of the ﬂow past a particle 13

o Operator – basic 66 – of deviation of particle velocity PDF from the equilibrium Maxwellian distribution 54 – differential XVI, 47 – of functional differentiation 40 – integral XVI – of perturbation 69 – realizing a Maxwellian distribution 54 – second-order 46 Orientation vector 140

p Pair collisions 137 Parameter – characterizing the relation between heat and dynamic inertia of particles 33 – of convergence rate of the iteration process 41 – of interrelation between the effects of gravity and turbulence on the collision kernel 215 – of particle inertia 42 Particle collisions XV, 137 – acceleration mechanism of 139 – of bidispersed particles in homogeneous anisotropic turbulent ﬂows 217 – contribution to energy ﬂux of the disperse phase 138 – contribution to stress 138 – elastic 157 – frequency 23, 138, 209 – inﬂuence of particle clustering 197 – inelastic 158 – of inertialess particles 139 – inﬂuence on statistical characteristics of the disperse phase in turbulent ﬂows 164

– inﬂuence on turbulent stresses and particle concentration 169 – intensity 138 – as Marcovian random processes 138 – of particles in anisotropic turbulent ﬂows 150 – probability density XVII – radius 140 – rate 197 – role of 24 – in the distribution of relative concentration along the channel axis 168 – solid sphere model analogy 138 – time interval between 24, 151 – velocity before and after 151 Particle interactions – with bounding surface XIII – with coherent vortex structures XIX – due to collisions XVI – with each other XIII, XVI – hydrodynamic XVI – with turbulent eddies XIII, XVI, 137, 138 – duration of 19, 23 – role of inertia 77 Particle motion – non-correlated XVII Particle pair – concentration 177 – diffusion 177 – dispersion 5 – line of centers 151 – motion 109, 143 – PDF 141 – relative velocity 138, 142, 172, 177, 212 – in a turbulent continuous medium 177 Particles – clustering 138 – in homogeneous and inhomogeneous turbulent ﬂows XVII – in a vertical channel 107 – collisions of, see Particle collisions – concentration distribution 90 – coordinates 11 – diameter 12 – diffusion XV, 48 – dispersion in isotropic stationary turbulence 13 – displacement 5 – dynamic description based on the Langevin equations for motion and heat transfer XVI – effective mean free path 23 – ﬂuid 1 – generation and disappearance XIV

Subject Index – heavy XII, XVII, 11, 14 – high-inertia XV, 29, 139 – boundary conditions 69 – collision kernel 145 – deposition factor 114 – kinetic equation for 56 – in the near-wall region 95 – velocity ﬂuctuations 73 – viscosity 56 – incidence angle 63 – incident 68 – inertialess XV, 139 – boundary conditions for 69 – collision kernel 144 – collisions 139 – diffusion 56, 125, 200 – of the disperse phase 50, 125 – limiting case of 126 – interaction time tensor 20 – interaction with walls 62 – low-inertia XV, 146, 183 – migration across the channel 97 – moderately inertial 108 – monodispersed 138 – reﬂectred 63, 68 – size 11, 138 – size spectrum XIII, 108, 139 – small 139 – small solid spherical in turbulent ﬂow 11 – spherical 12 – temperature 13 – thermal relaxation time 13 – transport via turbophoresis XVII – turbulent diffusion tensor 49 – velocity 12 Passive impurity XV, 50 Passive scalar 50, 51 – standard diffusion equation 51 PDF – of incident particles 71 – of interparticle distances 183 – one-particle 46 – of a particle pair 142, 172, 200 – of ﬂuid and particle velocities 137 – of ﬂuid and particle velocities, twopoint 142 – of radial velocity distribution 211 – of reﬂected particles 71 – of relative radial velocity distribution 137 – of relative velocity 171 – of velocity distributions for a particle pair 137, 210 Phase space 40, 46, 115

Phase transitions XIV Physical space 15 Pneumatic transport of coal dust XIII Point-force approximation 11 Polar angle 149 Polydisperse mixture of particles 209 Position vector 4 – of a ﬂuid particle along its trajectory 1 – of a point on the particle trajectory 14 Prandtl–Karman constant 92 Prandtl number 126 – of a ﬂuid 13 Pressure ﬂuctuations 81 Probability density – conditional 142 – dynamic, in the phase space of coordinates and velocities 40 – function (PDF) 40 – of particle displacement 14 – of particle velocity 142 Pseudo-turbulence XVI

q Quadratic model for Reynolds stresses in single-phase turbulence 57 Quasi-Brownian component of disperse phases kinetic energy 194 Quasi-Brownian components of particle velocity ﬁeld XVI Quasi-homogeneous ﬂow 41 Quasi-isotropic form 55, 129 Quasi-isotropic model 81 Quasi-normal Millionshikov hypothesis 49

r Radial distribution function – with the consideration of particle clustering 196 – of low-inertia particles 183 – of particle concentration 180 Radius vector, see Position vector Random – conﬁguration of particles 53 – ﬁeld 14 – ﬁeld of velocity and temperature ﬂuctuations of the continuous phase XIV – Galilean transformation 47 Rareﬁed – disperse media XV – gas 139 Reﬂection coefﬁcient 69 Region of high intensity of turbulent velocity ﬂuctuations XVII Relation between

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– Lagrangian and Eulerian correlation moments 14 – Lagrangian and Eulerian time macroscales 16 – Lagrangian and Eulerian time macroscales of temperature ﬂuctuations 25 – longitudinal components of a second-rank tensor in a homogeneous isotropic random ﬁeld 14 – second single-point moments of temperature ﬂuctuations of the disperse and continuous phase 32 – temperature and velocity ﬂuctuations of the disperse and continuous phase 33 – transverse components of a second-rank tensor in a homogeneous isotropic random ﬁeld 14 – velocities of incident and reﬂected particles in the case of a smooth wall 63 Relative (binary) diffusion – coefﬁcient 6 – in the viscous, inertial, and external intervals 203 – tensor – components for large values of time 6 – in the external interval 6 – of two ﬂuid particles 5 Relative dispersion (diffusion) – asymptotics of two particles in the inertial spatial interval 185 – of inertial particles 200 – of inertialess particles 201 Relative motion of – neighboring particles 138 – two particles in homogeneous isotropic turbulence 171 Relative turbulent diffusion of particle pairs 177 Relative velocity ﬂuctuation intensities of colliding particles 140 Relative velocity of – colliding particles 151 – particle pair 142 – particles 139 – two particles at contact 140 Relaxation time XIV, 108 Response coefﬁcient, see Involvement coefﬁcient Reynolds number – calculated using the Taylor microscale 3 – of the ﬂow past a particle 12 – high 29 – moderate 29 Reynolds stress tensor of the ﬂuid phase 1

Reynolds stresses, see Turbulent, stresses Richardsons law 200 Richardson-Obukhov constant 204 Root-mean-square value 15 Roughness parameter 63

s Saffman–Turner kernel 144 Schiller–Neumann approximation 12 Second invariant of continuum velocity gradient tensor 171 Second moment of velocity and temperature ﬂuctuations in a homogeneous shear ﬂow 132 Second-order – differential (diffusion) operator 46 – differential model 50 – for hydrodynamics of the disperse phase 52 – mixed correlation moments of velocity ﬂuctuations of the disperse and continuous phase 53 Second-rank tensor in homogeneous isotropic turbulence 4 Sedimentation rate 215 – in homogeneous turbulence XVII, 171 Segregation of inertial particles 171 Self-similar – ﬂow 76 – representation of the second order structure function 8 – variable 95 Semi-empirical model 107 – of collisions 140 Semi-empirical relations XV Separation of droplets and aerosol in cyclones XIII Shear – component of the collision kernel 150 – ﬂow 39, 46, 74, 132, 160 – mechanism of particle collisions 139 – rate 74 Single-phase turbulent ﬂow XIII Single-point single-particle PDF 39 Slippage of particles through turbulent structures 200 Smoluchowski kernel 150 Solenoidal isotropic ﬁeld 4 Spectral space 15 Spectrum of turbulent eddies XV Spherical formulation of the collision problem 140 Spontaneous nucleation XIV Stationary suspension 180

Subject Index Statistical – behavior of particles 1 – description of a particle ensemble XVI – method based on the kinetic equations XVI – method for modeling particle motion in a turbulent ﬂow 39 – microhydrodynamics XIX – model of turbulent collisions 140 – models based on – single-point PDF XIX – two-point PDF XIX – properties XVI Stokes number XVII, 18 – calculated using the Kolmogorov time microscale 181 – characterizing particle inertia in relation to the interval between collisions 24 – critical value for clustering 183 – determined using the Lagrangian time macroscale 36 Stratanovich interpretation of the stochastic integral 44 Structure function 5 – in the external interval 8 – mixed second-order, of ﬂuid and particle velocities 179 – third-order 8 – of ﬂuid particle pair velocities 178 – transverse, expressed through the longitudinal 9 Structure parameter of turbulence 16 System of equations – continual, for average single-point hydrodynamic characteristics of the disperse phase 48 – continual, for the disperse phase on the level of second-order moments 49 – describing motion of the disperse phase on the level of third-order moments 49 – integral 40 – for moments 64 – for PDF of a particle pair in space 200 – for single-point statistical moments with the consideration of particle collisions 152 – stochastic dynamic 183 – for turbulent stresses of the disperse phase – in the equilibrium logarithmic layer 91 – in a homogeneous shear ﬂow 74

t Tangential stress 72 – with the consideration of the collisional term 159 – ﬂuctuational component 72

Taylor differential time scale of relative velocity of two ﬂuid particles 6 Taylor time microscale of velocity ﬂuctuation increment 9 Temperature – ﬂuctuations 123 – gradient 132 – as a random process 115 – response 120 – structure parameter 28 – time increment of 5 Tensor – correlation, for the durations of velocity and temperature ﬂuctuations 132 – describing particle involvement in turbulent motion 30 – of Lagrangian macroscales 76 – of mixed moments 52 – of particle–turbulence interaction times 76 – of particle–turbulence interaction times with the consideration of collisions 162 – of relative turbulent diffusion of ﬂuid particles 178 – of relative turbulent diffusion of a particle pair 177 – third-rank – in isortopic turbulence 8 Test particle 138 Theory of – Brownian motion 46 – invariants 59 Thermal diffusion – vector components 123 – vector, of inertialess particles 125 Thermal inhomogeneity 12 Time evolution – in a homogeneous shear layer 79 – of mixed correlation moments of velocity ﬂuctuations of the continuous and disperse phases 79 – of particle turbulent stresses in a homogeneous shear layer 79 Time scale – anisotropy 78 – Taylor differential 3 Transition from Lagrangian to Eulerian correlations 18 Transport – diffusive 123 – of heat, see Heat, transport – resulting from the non-stationarity of turbulence 78 – turbulent 123 Transport equation 137, 173

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Transverse component – of Eulerian two-point correlation function of velocity ﬂuctuations 141 – of relative diffusion of two ﬂuid particles 7 – of second-rank tensor 5 – of turbulent stress 79 Transverse – Eulerian spatial correlation functions 5 – structure function at small distances between two points 140 – velocity ﬂuctuation intensity of inertial particles in the near-wall region 95 Triple correlation of – transverse velocity 72 – transverse velocity ﬂuctuations of high-inertia particles 73 Turbophoresis XVII, 171 Turbulence – anisotropic 44 – component of collision kernel 151 – decaying 16 – frozen 147 – integral timescale XIV – intermittency 6 – isotropic XVIII, 1 – large-scale 1 – modeling by a Gaussian process 46 – modiﬁcation of 39 – nonstationarity of continuum 77 – small-scale XV – spatial microscale XIII – time microscale XIV Turbulent – diffusion tensor 1, 28, 56 – diffusivity of an inertialess impurity 2 – eddies XIV, 45, 78, 137 – energy-carrying XV – large-scale XVI – energy 53, 57 – balance equation 61 – dissipation XV, XIX, 139 – equation for the disperse phase 56 – of a ﬂuid 28 – generation of 76 – heat ﬂux 123, 129 – migration effect, see Turbophoresis – momentum transfer 1 – Prandtl number of – continuous phase 130 – disperse phase 130 – stress tensor 29 – stresses 50 – in the disperse phase 48 – inﬂuence of inertia 168

– inﬂuence of particle collisions 160 – tangential component 79 – transport 138 – diffusive 123 – in the disperse phase 49 – of heat 1, 123 – viscosity coefﬁcient – for high-inertia particles 56 – of the continuous phase XIV – of the disperse phase 56 – zone of constant ﬂuctuation intensity 87 Two-ﬂuid model XIV Two-particle PDF 137, 152 Two-phase dispersed turbulent ﬂow XIII Two-point – integral time scale of velocity ﬂuctuation increment of two particles 6 – PDF of a particle pair 172 – statistical model 194 – time scale in the external interval 9 Two-scale – bi-exponential approximation 3 – bi-exponential dependence of the autocorrelation function of velocity ﬂuctuations 3 Two-zone model 87

u Unit matrix 32 Unit vector along the line of centers of a particle pair 151 Unit vector parallel to the drift velocity 16

v Variance 15 Variation of acceleration ﬂuctuations 35 Variation of particle temperature in a turbulent ﬁeld 115 Velocity – correlated component XVI – distribution of particles moving toward the wall and away from the wall 71 – ﬁeld of a continuum 39 – ﬁeld of particles XVI – ﬂuctuation gradient 7 – ﬂuctuation gradient tensor 7 – gradient 42 – mean-square, of a particle at the wall 71 – of shear 139 – of slip 14 – spatial increment 5 – of turbulent ﬂuid, as a random process 40 Vertical pipes and channels 107, 164 Virtual mass 11

Subject Index Viscous – interval 144, 203 – spatial interval 6 – sublayer 64 – sublayer with zero ﬂuctuation intensity 87

w Wave-particle duality in the microworld XV Weighted sum 64 White noise 46 Wiener random process 47

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