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

VI

Contents

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

XI

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

XVII

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Introduction

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.

XIX

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

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

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).


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:

j3


4

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.


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).

j5


6

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,


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.

j7


8

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


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:

j9


10

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

j11


12

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.


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

j13


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.


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

j15


16

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


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 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

j23


24

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


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

j25


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.


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


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

j29


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).


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.

j31


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


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


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


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

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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.


(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.

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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


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


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

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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


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

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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.


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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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


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

j49


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


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,

j51


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


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

j53


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.


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

j55


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


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.

j57


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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


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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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


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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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.


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


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


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
> :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


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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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


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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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


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).


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


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


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


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).


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


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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


86

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.


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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88

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 )


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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90

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.


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


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.


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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96

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


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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98

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


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


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.


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


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.


j103


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.


j105


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


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


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


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

j109


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


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

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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


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


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:


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


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


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


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¡


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


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


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


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


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


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


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


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


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


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


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


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


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


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).


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


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 macroscales 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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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


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.


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


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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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


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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150

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

j151


152

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 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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206

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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208

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.

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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);

j 6 Collision and Clustering of Bidispersed Particles in Homogeneous Turbulence

210

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.


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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212

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.


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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214

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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216

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.


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

j217


218

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


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


220

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

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