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

Particles in Turbulent Flows

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

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

Particles in Turbulent Flows

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

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

V

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

XIII

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

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3 3.1 3.2 3.3 3.3.1 3.3.2 3.4

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

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

Contents

6.5 6.6

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

277

Subject Index

283

VII

IX

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

X

Preface

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

Preface

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

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

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XIII

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

XIV

Introduction

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

Introduction

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

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Introduction

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

Introduction

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

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

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

Introduction

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

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j1

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

1.1 Characteristics of Homogeneous Isotropic Turbulence

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

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

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2

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

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

Dt ¼ u0 T L ; 2

Ð1

ð1:2Þ

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

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

1.1 Characteristics of Homogeneous Isotropic Turbulence

C0 ¼

C01 Rel ; Rel þ C1

C 01 ¼ 7;

C 1 ¼ 32;

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

T L 2(Rel þ C1 ) ¼ : tk 151=2 C01

ð1:4Þ

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

tT ; TL

ð1:5Þ

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

ð1:6Þ

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

@ ¼ (2z2 1)1=2 :

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

a01 þ a01 Rel ; a02 þ Rel

a01 ¼ 11;

a02 ¼ 205;

a01 ¼ 7:

ð1:7Þ

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

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

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

Bij (0) ¼ u0 2 dij ;

ð1:8Þ

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

rirj ; r2

ð1:9Þ

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

1.1 Characteristics of Homogeneous Isotropic Turbulence

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

ð1:14Þ

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

ð1:15Þ

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

ð1:16Þ

0

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

ð1:17Þ

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

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

ð1:18Þ

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

er 2 ; 15n

Snn ¼

2er 2 : 15n

ð1:22Þ

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

1.1 Characteristics of Homogeneous Isotropic Turbulence

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

ð1:23Þ

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

g ij þ g ji 2

;

ð1:24Þ

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

wij ¼

g ij g ji 2

:

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

er 2 t exp ; 15n ts

SL nn ¼

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

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

eT Lr r 2 ; 15n

Drnn ¼

2eT Lr r 2 : 15n

ð1:27Þ

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

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

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

4 Snn ¼ C(er)2=3 ; 3

ð1:29Þ

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

A2 ¼ const:

ð1:30Þ

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

dSll : dr

ð1:32Þ

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

ð1:33Þ

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

ð1:34Þ

1.1 Characteristics of Homogeneous Isotropic Turbulence

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

ð1:35Þ

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

ð1:36Þ

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

ð1:37Þ

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

r qSll : 2 qr

ð1:39Þ

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

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

T Lr

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

TL ¼

TL : tk ð1:40Þ

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

tT T Lr : TL

ð1:41Þ

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

ð1:42Þ

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

ð1:43Þ

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

1.2 Motion of a Single Particle and Heat Exchange

1.2 Motion of a Single Particle and Heat Exchange

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

ð1:45Þ

dvp u(Rp ; t)vp ¼ þ F; dt tp

ð1:46Þ

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

tp0 ; j(Rep )

tp0 ¼

rp d2p 18rf n

;

ð1:47Þ

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

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

dqp q(Rp ; t)qp þ Q; ¼ dt tt

ð1:49Þ

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

C p rp d2p 6lNup

;

ð1:50Þ

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

where Pr is the Prandtl number of the ﬂuid.

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

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

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14

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

s ¼ Sþs ;

S ¼ Wt;

ðt

s ¼ vp0 (Rp (t1 ))dt1 : 0

ð1:51Þ

0

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

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

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

ð1:52Þ

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

ð1:53Þ

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

1 tp

ðt tð1 00

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

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

ð1:56Þ

ð1:57Þ

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

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

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

ð1:61Þ

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

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

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

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

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

1 ð

T Lp ¼

YLp (t) dt ¼ 0

1 0

my(t) t þ my(t) exp dt: 3T E TE

ð1:64Þ

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

f (StE ) ¼

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

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

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

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

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

ð1:66Þ

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

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

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20

t YLp (t) ¼ exp T Lp

ð1:67Þ

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

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

T lLp

1 ð

0

ð1:69Þ

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

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

= ¼ (1 þ g 2 )1=2 ; ¼

TE ; 1 þ mg

T lLp (StE ! 1) T nLp (StE ! 1) ¼

2 þ mg TE: 2(1 þ mg)2

ð1:70Þ

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

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

T lLp ¼

L ; W

T nLp ¼

L ; 2W

T lLp T nLp

¼ 2:

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

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

ð1:71Þ

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

TL

; (1 þ b2 g 2 )1=2

T nLp ¼

TL

; (1 þ 4b2 g 2 )1=2

b¼

T L u0 : L

ð1:73Þ

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

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

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

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

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

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

ð1:76Þ

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

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

f (St ) ¼

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

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

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

ð1:78Þ

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

ð1:79Þ

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

ð1:81Þ

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

T Ltp ¼

1 ð

YLtp (t) dt ¼ 0

0

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

ð1:82Þ

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

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

26

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

ð1:83Þ

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

T Et ; 1 þ mt (1 þ g 2 )1=2

T Ltp (StE ! 1) ¼

T Et : 1 þ mt g

ð1:85Þ

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

ð1:86Þ

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

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

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

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

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

ð1:88Þ

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

ð1:89Þ

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

1 tp

1 ð

t

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

ð1:90Þ

t

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

ð1:91Þ

where fu ¼

1 tp

1 ð

0

t YLp (t)exp I dt: tp

ð1:92Þ

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

ð1:93Þ

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28

In view of the relation 0

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

ð1:94Þ

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

at

t ¼ 0;

Bp ij ! 0

at t ! 1:

ð1:95Þ

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

Dp ij ¼

1 ð

Bp ij (t)dt ¼ 0

BLp ij (t)dt ¼ u0 T Lp ij : 2

ð1:96Þ

0

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

T Lp ij Dt : TL

ð1:97Þ

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

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

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

u0 2 2tp

1 ð

0

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

ð1:98Þ

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

ð1:99Þ

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

¼ d ij ;

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

¼

T Lp ij : tp

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

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

ð1:100Þ

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

ð1:101Þ

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

30

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

ð1:102Þ

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

ð1:103Þ

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

2

ð1:104Þ

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

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

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

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

Tp ¼

Yp (t)dt ¼ 0

T Lp : fu

ð1:106Þ

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

ð1:108Þ

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

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

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

32

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

ð1:109Þ

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

ð1:110Þ

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

1 ð

t

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

ð1:111Þ

t

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

ð1:112Þ

where ft ¼

1 tt

1 ð

0

t YLtp (t)exp dt: tt

ð1:113Þ

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

ð1:114Þ

In view of the fact that

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

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

ð1:115Þ

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

Bpt (t) ¼

hq0 i 2tt

0

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

ð1:116Þ

1.4 Velocity and Temperature Correlations for Particles in Stationary Isotropic Turbulence

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

2

ð1:117Þ

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

02

hq i

¼

v0 2 =u0 2

v0 2 =u0 2 ; þ ¡(1v0 2 =u0 2 )

¡¼

tt T Lp ; tp T Ltp

ð1:119Þ

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

hq0 i 2

1=2

hq0 i 2

tp v0 tt ¼ 1 1 0 : u

ð1:120Þ

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

j33

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

34

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

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

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

1.5 Particle Acceleration in Isotropic Turbulence

1.5 Particle Acceleration in Isotropic Turbulence

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

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

ð1:123Þ

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

2u0 2 dij a0 e3=2 d ij ¼ ; t2T n1=2

Ya (t) ¼

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

ð1:124Þ

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

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

ð1:125Þ

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

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

ð1:126Þ

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

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

Ypa (t) ¼

YLp (t)Yp (t) f u : ð1:127Þ 1f u

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

j35

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

36

ha0pi a0pj i ¼

ap0 e3=2 dij ; n1=2

ð1:128Þ

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

(1f u )Rel 151=2 St2

;

ð1:129Þ

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

2StL þ z2 ; 2StL þ 2St2L þ z2

ð1:130Þ

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

1 ; 1 þ StL

ð1:131Þ

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

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

1.5 Particle Acceleration in Isotropic Turbulence

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

j37

j39

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

j 2 Motion of Particles in Gradient Turbulent Flows

40

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

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

ð2:1Þ

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

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

we obtain

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

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

¼

ð2:5Þ

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

ðt vp (t1 )dt1 ; vp (t) ¼

1

1

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

ð2:6Þ

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

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

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

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

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42

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

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

ð2:9Þ

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

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

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

ð2:10Þ

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

ðt t1

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

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

ð2:11Þ

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

1 lij (x; t) ¼ tp

ðt

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

1

d jk tt1 exp tp tp

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

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

1 mij (x; t) ¼ tp

þ

ðt 1

ð2:12Þ

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

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

ð2:13Þ

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

t Dhu0i u0k i hu0i u0k i 2

Dt

YL kj (t);

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

ð2:14Þ

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

j43

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44

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

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

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

ð2:15Þ

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

hu0i u0k i

fu kj

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

mij ¼

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

hu0i u0k i

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

ð2:16Þ

ð2:17Þ

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

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

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

Nun ¼

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

ð2:18Þ

0

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

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

ð2:21Þ

then 1 F(s) ¼ (sI þ T1 Lp ) ;

(nþ1) ; Mun ¼ (I þ tp T1 Lp )

Nun ¼ (TLp =tp )nþ1 ;

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

ð2:22Þ

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

j45

j 2 Motion of Particles in Gradient Turbulent Flows

46

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

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

ð2:23Þ

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

2.2 Equations for Single-Point Moments of Particle Velocity

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

2.2 Equations for Single-Point Moments of Particle Velocity

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

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48

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

ð2:25Þ

where

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

Dp ij ¼ tp (hv0i v0j i þ mij )

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

þ Vk

qhv0i v0j i qx k

0 0 0

þ

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

2hv0i v0j i

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

ð2:27Þ

2.2 Equations for Single-Point Moments of Particle Velocity

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

hu0i u0k if kj þ hu0j u0k if ki 2

:

ð2:28Þ

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

ð2:29Þ

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

þ Vn

qhv0i v0j v0k i

0 0 Dp in qhvj vk i

tp

qx n

qx n þ

þhv0i v0j v0n i

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

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

ð2:30Þ

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

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50

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

q ln F : qx k

ð2:32Þ

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

ð2:33Þ

tp ! 0

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

cL ij ¼

ð1 0

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

ð2:34Þ

YL ij (t)tdt;

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

2.2 Equations for Single-Point Moments of Particle Velocity

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

ð2:38Þ

ð2:39Þ

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

a ¼ const:

ð2:40Þ

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

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52

hu0i u0j u0k i¼b

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

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

2.3 Algebraic Models of Turbulent Stresses

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

2.3 Algebraic Models of Turbulent Stresses

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

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

ð2:44Þ

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

R[P] ¼

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54

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

ð2:45Þ

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

ð2:46Þ

R Pð1Þ ¼ N P ð0Þ :

ð2:47Þ

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

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

ð2:49Þ

ð 1 0 0 0 ð0Þ v v v P dv ¼ 0: F i j k

ð2:50Þ

we can write hv0i v0j v0k i0 ¼

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

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

ð2:51Þ

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

2.3 Algebraic Models of Turbulent Stresses

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

þ

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

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

ð2:52Þ

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

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

Dp ij ¼ tp

2kp dij þ mij : 3

ð2:55Þ

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

ð2:56Þ

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56

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

tp kp : 3

ð2:58Þ

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

ð2:59Þ

tp ! 0

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

gu ¼

T Lp fu : tp

ð2:60Þ

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

tp ! 0

ð2:61Þ

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

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

ð2:62Þ

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

tp kp ; 3

2 Dp ¼ tp kp : 3

ð2:63Þ

ð2:64Þ

2.3 Algebraic Models of Turbulent Stresses

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

ð2:65Þ

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

ð2:66Þ

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

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

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

hu0i u0j i 2k

dij ; 3

bp ij ¼

hv0i v0j i 2kp

dij ; 3

bf p ij ¼

2hu0i v0j i kf p

dij ; 3

ð2:67Þ

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

þ

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

0 0

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

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

Dp hv0i v0j i Dt

¼

qhv0i v0j i qt þV k

þ Vk

qhv0i v0j i qx k

0 0 0

þ

1 qFhvi vj vk i ; qx k F

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

Dp kp qkp ¼ Dt qt ð2:68Þ

2.3 Algebraic Models of Turbulent Stresses

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

X ij þ X ji 2hv0i v0j i tp

;

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

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

ð2:70Þ

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

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60

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

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

ð2:71Þ

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

hv0i v0j ið1Þ

0 0 ð1Þ

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

hv0i v0j i1 ¼

E0 ¼

X kk tp V k;k : 2kp 3

ð2:72Þ

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

2.3 Algebraic Models of Turbulent Stresses

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

tp hv0i v0k ið1Þ V i;k : 2kp E0

ð2:73Þ

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

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62

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

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

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

ð2:74Þ

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

Figure 2.1 A collision of particle with the wall.

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

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

ð2:75Þ

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

ð2:76Þ

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

ð2:77Þ

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

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

F1 hyi1 þ F2 hyi2 : F

ð2:78Þ

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

ð2:79Þ

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

3

ð2:80Þ

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

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

ð2:81Þ

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

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

dFV y ¼ 0; dy 0 0

Vy

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

Vy

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

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

Vy

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

dhv02 y i

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

Vy

dhv0x v0y i dy

0 02

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

þ

þ

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

ð2:82Þ

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

ð2:83Þ

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

R[P] ¼ hv0 x i 2

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

ð2:84Þ

ð2:85Þ

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66

N[P] ¼

T Lp hv0 2x i tp

2 hu0 x i

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

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

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

ð2:86Þ

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

ð2:87Þ

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

ð2:88Þ

R[Pð1Þ ] ¼ N[P ð0Þ ]:

ð2:89Þ

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

ð2:90Þ

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

3

2

ð2:91Þ

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

ð0Þ

þ

v0y

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

v02 y

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

ð2:92Þ

ð2:93Þ

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

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

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

tp v0y

v02 y

02 6hv02 y i hvy i

3

dhv02 y i dy

þ

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

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

ð2:95Þ

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

tp hv02 y i dV x

hv0 x v0y i ¼ 2

dy

2

;

02 tp hv02 y i dhv x i

3

dy

ð2:96Þ

; hv0 y i ¼ tp hv02 y i 3

dhv02 y i dy

hv0x v02 y i ¼ 0:

; hv0y v0 z i ¼ 2

02 tp hv02 y i dhv z i

3

; dy ð2:97Þ

ð2:98Þ

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

ð2:99Þ

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

ex ¼

8 > > > 1xx <

at 0 xx <

5 > > > :7

at xx >

2 7

2 7

;

ð2:100Þ

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68

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

ð2:101Þ

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

ð2:102Þ

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

1 0 1

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

02 1=2 1c 2hvy i : 1þc p

ð2:103Þ

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

ð2:104Þ

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

1 ð 1 ð ð 1

yvy Pdvx dvy dvz¼ 1 1 1

1 ð 1 ð 1 ð

ð0 1 ð

1 1 1

yvy P 1 dvx dvy dvzþ

yvy P 2 dvx dvy dvz :

1 0 1

ð2:105Þ

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

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

tp

tp

dhv02 y i dy

¼

2

1ce2y

1c 2 1 þ cey 1 þ c

02 1=2 2hvy i p

;

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

ð2:108Þ

ð2:109Þ

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

ð2:110Þ

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

ð2:111Þ

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

v0y

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

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70

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

1cex 1c 1 þ cex 1 þ c

1ce2x 1c 1 þ ce2x 1 þ c

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

V x ¼ hv0x v0y i;

ð2:112Þ

hv0x i ¼ hv0x v0y i;

ð2:113Þ

2

2

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

ð2:114Þ

ð2:115Þ

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

2

ð2:116Þ

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

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

ð2:117Þ

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

ð2:118Þ

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

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

ð2:119Þ

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

v2y P2 (vy > 0) ¼ N 2 exp 2 : 2hv2y i

ð2:120Þ

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

1 ð

P1 dvy ; F2 ¼ 1

P2 dvy ; 0

which give Fey N1 ¼ c þ ey

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

ð2:121Þ

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

F1 hv21y i þ F2 hv22y i F

¼

ey (1 þ cey ) 2 hv1y i: ey þ c

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

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72

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

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

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

F1 V 1x V 1y þ F2 V 2x V 2y : F

ð2:124Þ

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

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

ð2:125Þ

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

ð2:126Þ

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

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

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

hv3y i ¼ 1ce2y

8(ey þ c) pey (1 þ cey )3

1=2 hv2y i3=2 :

ð2:128Þ

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

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

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

dhv02 y i dy

:

ð2:130Þ

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

1=2 2(1ey ) 2hv02 y i : ¼ 1=2 dy p ey

dhv02 y i

ð2:131Þ

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

3

ð2:132Þ

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

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

ð2:134Þ

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

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02

dhvy i dhv0x2 i ¼ m2x : dy dy

ð2:135Þ

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

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

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

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

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

ð2:136Þ

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

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

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

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

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

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

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

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

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

ð2:137Þ

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

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

tp dhu02 3i ; f 2 dt u1 33

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

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

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

ð2:138Þ

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

ð2:139Þ

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

ð2:140Þ

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

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

T L kk ¼ 3

k a ; a ¼ 0:3: e

ð2:141Þ

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

ð2:142Þ

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

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

ð2:144Þ

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

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

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

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

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

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

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

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

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

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

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

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

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

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

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

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

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

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

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

2.5 Second Moments of Velocity Fluctuations in a Homogeneous Shear Flow

j85

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

ð1Þ 0 0 hv02 1 i ¼hu1 uk i

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

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

hv01 v02i1 ¼ E0 ¼

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

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

ð2:147Þ

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

E1 ¼ E0

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

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

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

þ

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

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

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

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

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

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

2.6 Motion of Particles in the Near-Wall Region

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

2.6 Motion of Particles in the Near-Wall Region

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

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

ð2:150Þ

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

n ; u

d þ ¼ const:

ð2:151Þ

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

n ; u2

T þ ¼ const:

ð2:152Þ

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

t2p

dhv02 y i dy

2 þ ðhv02 y i þ g u Ayy u H(yd)Þ

dF ¼ 0; dy

ð2:153Þ

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

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

hv02 y i u2

; l¼

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

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

t2

dh v02 yþ i dl

þ ðhv02 yþ i þ g u H(l1)Þ

dF ¼ 0; dl

ð2:155Þ

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

ð2:156Þ

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

dhv02 yþ i dl

¼2

1e2y 1 þ e2y

2hv02 yþ i

1=2

p

at l ¼ 0;

dhv02 yþ i dl

¼ 0; F ¼ 1 at l ¼ 1: ð2:157Þ

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

1 ; 1 þ t

gu ¼

1 : t (1 þ t )

2.6 Motion of Particles in the Near-Wall Region

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

d2 hv02 yþ i dl

2

þ 2ð fu H(l1)hv02 yþ iÞ ¼ 0;

ð2:158Þ

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

or 02 hv02 yþ i ¼ hvyþ (0)i þ

2(1e2y )

(ll0 )2 at l0 < l < 1 t2

02 1=2 2hvyþ (0)i

t (1 þ e2y )

p

ð2:160Þ

2

lþ

l at 0 < l < 1: t2

ð2:161Þ

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

d2 hv02 yþ i 2

dl

2 fu hv02 yþ i

þ 02 ¼ 0: hvyþ i þ g u

ð2:162Þ

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

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1=2

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

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

1=2

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

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

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

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

2.6 Motion of Particles in the Near-Wall Region

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

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

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

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92

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

dU x ku ¼ ; dy y

ð2:168Þ

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

Ux ¼ k ln yþ þ B; u

B ¼ const;

ð2:169Þ

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

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

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

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

ð2:170Þ

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

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

ð2:171Þ

2.6 Motion of Particles in the Near-Wall Region

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

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

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

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94

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

u2

; e¼ 1=2

Cm

u3 ; Cm 0:09: ky

ð2:172Þ

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

ky : u

ð2:173Þ

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

2.6 Motion of Particles in the Near-Wall Region

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

2.6.3 High-Inertia Particles

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

dF

dhv02

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

fu ¼

kh k 2 h2 : ; gu ¼ 1þkh 1þkh

ð2:174Þ

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

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

ð2:175Þ

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

2.7 Motion of Particles in a Vertical Channel

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

2.7 Motion of Particles in a Vertical Channel

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

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

ð2:176Þ

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

ð2:177Þ

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

dhv02

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

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

ð2:178Þ

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

ð2:179Þ

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

ð2:180Þ

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

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

ð2:181Þ

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

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

l12 ¼

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

l21 ¼

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

l22 ¼

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

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

m12 ¼ g nu hu01 u02 i

tp g nu1 Dp hu01 u02 i ; 2 Dt

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

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

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

tp g nu1 Dp hu02 2i : Dt 2

ð2:182Þ

2.7 Motion of Particles in a Vertical Channel

j99

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

dhu02

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

dhu01 u02 i 0 0

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

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

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

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

hv01 v02 i ¼ mx hv02 2 i at y ¼ 0

ð2:184Þ

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

ð2:185Þ

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

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

RV ¼ (12z2V )1=2 ; zV ¼

tT V ; V ¼ l; n: T Lp

ð2:186Þ

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

j 2 Motion of Particles in Gradient Turbulent Flows

100

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

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

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

3 2½ 2WV þ z2V 6W2V z4V 8W3V z2V ¼ ;

3 2WV þ 2W2V þ z2V

hVu ¼

24WV z2V 2W2V

V

V

þ 2f Vu þ fu1 ; g u1 ¼

g Vu ¼

2z2V 2W2V

1 V f V ; lV ¼ g Vu fu1 ; WV u u

V

V

V

V

fu1 ; lu1 ¼ g u1 fu2 ;

ð2:187Þ

V

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

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

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

T lLp ¼ T nLp

TE ¼

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

I ¼ (1 þ g 2 )1=2 ; f (StE ) ¼

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

ð2:188Þ

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

ð2:189Þ

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

2.7 Motion of Particles in a Vertical Channel

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

1=2

; fum ¼

ful þ 2fun : 3

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

j101

j 2 Motion of Particles in Gradient Turbulent Flows

102

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

ðR 1 2 T nLp hu0 i idy; tp R

i ¼ 2; 3:

ð2:190Þ

0

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

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

2.7 Motion of Particles in a Vertical Channel

j103

j 2 Motion of Particles in Gradient Turbulent Flows

104

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

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

2.7 Motion of Particles in a Vertical Channel

j105

j 2 Motion of Particles in Gradient Turbulent Flows

106

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

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

2.8 Deposition of Particles in a Vertical Channel

2.8 Deposition of Particles in a Vertical Channel

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

j107

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108

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

2.8 Deposition of Particles in a Vertical Channel

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

ð2:192Þ

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

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

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j 2 Motion of Particles in Gradient Turbulent Flows

110

V2

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

ð2:194Þ

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

dhv02

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

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

V2

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

dhv01 v02 i 0 0

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

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

ð2:196Þ

02

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

V2

dhv01 v02 i 0 0

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

dhv02 i

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

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

ð2:198Þ

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

2.8 Deposition of Particles in a Vertical Channel

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

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

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

dhu02 2i þ hv01 v02 i þ m12 dy

;

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

ð2:199Þ

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

ð2:202Þ

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

j111

j 2 Motion of Particles in Gradient Turbulent Flows

112

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

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

2.8 Deposition of Particles in a Vertical Channel

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

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

ð2:204Þ

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

j113

j 2 Motion of Particles in Gradient Turbulent Flows

114

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

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

ð2:206Þ

0

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

0

¼ t þ

T nLp hu0 22þ idy

3 1c 2=3 2 1=3 Ð1 2 1þc

p

0

4=3 1=3 t ; T nLp hu0 22þ idy

ð2:207Þ

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

2 02 hv02 2 i=u ; hu 2þ i

2 hu02 y 2 i=u ;

0

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

tp g (2k=3)1=2

; Rep ¼

d p tp g TE ; T nLp ¼ ; n 2mg

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

j115

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

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

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

ð3:1Þ

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

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

116

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

0

hW pi ¼

ðð

* hu0i (x1 ; t1 )W0 (x; t)i ðð

þ where

*

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

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

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

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

*

*

0

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

+

q ¼ qx j q qvj

*

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

*

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

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

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

ð3:5Þ

+

+

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

ð3:4Þ

ð3:6Þ ð3:7Þ

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

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

ðt qp (t) ¼

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

ð3:8Þ

1

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

ð3:9Þ

t1

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

ð3:10Þ

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

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

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

ð3:12Þ

j117

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

118

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

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

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

1 mi (x;t) ¼ tp tt þ

1 li (x;t) ¼ tt

ðt 1

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

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

ðt 1

þ

1 tt

ð3:15Þ

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

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

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

h(x;t) ¼

ð3:14Þ

ðt

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

1

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

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

þ

2tp 3tt 1 tt1 exp dt1 tt1 þ 1¡ tp 1¡

ð3:16Þ

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

1 þ 2 tt

ðt 1

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

ð3:17Þ

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

ð3:18Þ

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

ð3:19Þ

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

2

2

hW0 i 2

2 t DhW0 i YLt (t); 2 Dt

2

ð3:20Þ

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

j119

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

120

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

0

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

2

2

2

ð3:21Þ

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

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

ð3:22Þ

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

ð3:23Þ

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

ð3:24Þ

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

2

þ

ð3:25Þ

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

f tu1 ¼ Mtu1 ;

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

ltu1 ¼ gtu1 f tu2 ;

ltu ¼ gtu f tu1 ;

htu ¼ Ntu1 þ Mtu1 2gtu ;

gtu1 ¼ Ntu1 f tu1 ; f tt ¼ Mtt0 ;

f tt1 ¼ Mtt1; ¡2 Mtt0 Mtu0 ; 1¡ 1¡

qu ¼ Nu0 þ

¡2 Mt0 Mu0 ; 1¡ 1¡

qtu ¼ Ntu0 þ

qu1 ¼ Nu1 þ

¡3 Mt1 Mu1 ; 1¡ 1¡

qtu1 ¼ Ntu1 þ

¡3 Mtt1 Mtu1 ; 1¡ 1¡

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

ru ¼ Nu1 (2 þ ¡)Nu0 þ

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

rtu ¼ Ntu1 (2 þ ¡)Ntu0 þ

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

Mtun

j121

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

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

Mtn ¼

0

Mttn

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

Ntun

0

1 ð 1 ¼ YtLp (t)tn dt: n!tnp þ 1 0

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

1 tt

1 ð

0

t YLtp (t)exp dt; tt

f t1 ¼

1 t2t

1 ð

0

t YLtp (t)t exp dt: tt

ð3:27Þ

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

; YtLp (t) ¼ exp t TtLp

YLtp (t) ¼ exp tT 1 Ltp ;

then 1 ðn þ 1Þ

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

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

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

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

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

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

1

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

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

122

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

2

2

2

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

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

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

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

1 (23¡) Iþ tp T1 Lp (1¡)2

; rtu ¼

1 1 ¡4 I þtt TtLp

TtLp

2

tp

1 2 I þtp TtLp

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

t TLp (2þ ¡) tp þ

1 1 (23¡) I þtp TtLp (1¡)2

;

ð3:28Þ

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

U k vk q Tq þ Fk P þ þQ P tp qq tt

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

ð3:29Þ

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

3.2 The Equations for Single-Point Moments of Particle Temperature

3.2 The Equations for Single-Point Moments of Particle Temperature

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

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

ð3:30Þ

ð 1 (vi V i )(qQ)Pdvdq F

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

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

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

ð3:31Þ

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

ð3:32Þ

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

2

2

2

ð3:33Þ

j123

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

124

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

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

ð3:34Þ

hq0 i ¼ f t hW0 i: 2

2

ð3:35Þ

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

2

hv0i q0 i ¼ 3hv0i q0 ihq0 i: 3

2

ð3:36Þ

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

þ Vk

qhv0i v0j q0 i qx k

þ hv0i v0k q0 i

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

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

2 1 þ hv0i v0j q0 i ¼ 0; tp tt ð3:37Þ

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

2

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

1 2 2 þ hv0i q0 i ¼ 0; tp tt

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

ð3:38Þ

ð3:39Þ

3.2 The Equations for Single-Point Moments of Particle Temperature

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

hv0i v0j q0 i

hv0i q0 i ¼ 2

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

hq0 i ¼ Dtp k 3

qhq0 i : qx k

ð3:40Þ

ð3:41Þ

2

ð3:42Þ

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

qlnF : qx k

ð3:43Þ

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

ð3:44Þ

lim hq0 i ¼ hW0 i:

ð3:45Þ

tp ; tt ! 0

2

2

tp ; tt ! 0

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

ctL ij ¼

ð1 0

YtL ij (t)t dt;

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

j125

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

126

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

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

Cu1 ¼

¡ 0 au ; 2¡0 þ 1

Ctu 2 ¼

atu ; ¡0 þ 2

Ctu1 ¼

atu ; 2¡0 þ 1

¡0 ¼ lim

tt

tp ; tt ! 0 t p

¼

C u2 ¼

¡0 au ; ¡0 þ 2

3Pr ; 2

ð3:49Þ

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

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

hv0i W0 i ¼

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

2

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

3.3 Algebraic Models of Turbulent Heat Fluxes

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

ð3:50Þ ð3:51Þ ð3:52Þ

3.3 Algebraic Models of Turbulent Heat Fluxes

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

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

ð3:53Þ

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

ð3:54Þ

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

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

ð3:55Þ

j127

j 3 Heat Exchange of Particles in Gradient Turbulent Flows

128

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

1=2

27

128p4 k3p hq0 i 2

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

ð3:56Þ

Due to Eq. (3.56), hv0i q0 i0 ¼

ð 1 0 0 ð0Þ v q P dvdq ¼ 0; F i

hv0i v0j q0 i0 ¼

ð 1 0 0 0 ð0Þ v v q P dvdq ¼ 0; F i j

ð3:57Þ

hv0i q0 i0 ¼ 2

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

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

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

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

þ

þ

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

mi q 0

q0

q0

2

2

2hq0 i hq0 i 2

2

2 3 hq0 i;i P ð0Þ (v; q):

ð3:59Þ

3.3 Algebraic Models of Turbulent Heat Fluxes

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

þ

þ

þ

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

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

3tp tt v0i q0

2(tp þ tt )kp hq0 i 2

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

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

3mij 2 1 dij þ