Statistical Microhydrodynamics

Emmanuil G. Sinaiski and Leonid I. Zaichik Statistical Microhydrodynamics Emmanuil G. Sinaiski and Leonid I. Zaichik...

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Emmanuil G. Sinaiski and Leonid I. Zaichik

Statistical Microhydrodynamics

Emmanuil G. Sinaiski and Leonid I. Zaichik Statistical Microhydrodynamics

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Statistical Microhydrodynamics

The Authors Prof. Dr. Emmanuil Sinaiski An der Kotsche 12 04207 Leipzig Germany Prof. Dr. Leonid I. Zaichik Nuclear Safety Institute Russian Academy of Science B. Tulskaya 52 115191 Moscow Russia

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 in the Internet at . # 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, India Printing betz-druck GmbH, Darmstadt Book Binding Litges & Dopf GmbH, Heppenheim

Printed in the Federal Republic of Germany Printed on acid-free paper ISBN:

978-3-527-40656-2

V

Contents Preface IX Nomenclature

XIII

Basic Concepts of the Probability Theory 1 Events, Set of Events, and Probability 1 Random Variables, Probability Distribution Function, Average Value, and Variance 4 1.3 Generalized Functions 5 1.4 Methods of Averaging 8 1.5 Characteristic Functions 12 1.6 Moments and Cumulants of Random Variables 14 1.7 Correlation Functions 16 1.8 Bernoulli, Poisson, and Gaussian Distributions 18 1.9 Stationary Random Functions, Homogeneous Random Fields 22 1.10 Isotropic Random Fields. Spectral Representation 25 1.11 Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation 28 1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations 31 1.12.1 Derivation of the Differential Chapman–Kolmogorov Equation 31 1.12.2 Discontinuous (‘‘Jump’’) Processes. The Kolmogorov–Feller Equation 35 1.12.3 Diffusion Processes. The Fokker–Planck Equation 35 1.12.4 Deterministic Processes. The Liouville Equation 40 1.13 Stochastic Differential Equations. The Langevin Equation 43 1.13.1 The Langevin Equation 43 1.13.2 The Diffusion Equation 44 1.13.2.1 The Diffusion Equation with Chemical Reactions Taken into Account 45 1.13.2.2 Brownian Motion of a Particle in a Hydrodynamic Medium 46 1.14 Variational (Functional) Derivatives 48 1.15 The Characteristic Functional 53 1 1.1 1.2

VI

Contents

2 2.1 2.2 2.3 2.3.1 2.3.2 2.4 2.5

Elements of Microhydrodynamics 59 Motion of an Isolated Particle in a Quiescent Fluid 61 Motion of an Isolated Particle in a Moving Fluid 70 Motion of Two Particles in a Fluid 78 Fluid is at Rest at the Infinity (v1 ¼ 0) 78 Fluid is Moving at the Infinity (v1 6¼ 0) 92 Multi-Particle Motion 95 Flow of a Fluid Through a Random Bed of Particles 98

3 3.1 3.1.1 3.1.2 3.1.3 3.2 3.3 3.4 3.5 3.6 3.7 3.8

3.11 3.11.1 3.11.2 3.12

Brownian Motion of Particles 109 Random Walk of an Isolated Particle 110 Isotropic Distribution 113 Gaussian Distribution 114 An Arbitrary Distribution tðrÞ in the Limiting Case N 1 115 Random Walk of an Ensemble of Particles 116 Brownian Motion of a Free Particle in a Quiescent Fluid 117 Brownian Motion of a Particle in an External Force Field 122 The Smoluchowski Equation 124 Brownian Motion of a Particle in a Moving Fluid 126 Brownian Diffusion with Hydrodynamic Interactions 130 Brownian Diffusion with Hydrodynamic Interactions and External Forces 136 High Peclet Numbers: Peij 1 139 Small Peclet Numbers, Peij 1 140 Particle Sedimentation in a Monodisperse Dilute Suspension 142 Particle Sedimentation in a Polydisperse Dilute Suspension, with Hydrodynamic and Molecular Interactions and Brownian Motion of Particles 151 Transport Coefficients in Disperse Media 157 Infinitely Dilute Suspension with Non-interacting Particles 161 The Influence of Particle Interactions on Transport Coefficients 164 Concentrated Disperse Media 167

4 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.10.1

Turbulent Flow of Fluids 183 General Information on Laminar and Turbulent Flows 183 The Momentum Equation for Viscous Incompressible Fluids 184 The Equations of Heat Inflow, Heat Conduction and Diffusion 187 The Conditions for the Beginning of Turbulence 189 Hydrodynamic Instability 190 The Reynolds Equations 192 The Equation of Turbulent Energy Balance 197 Isotropic Turbulence 202 The Local Structure of Fully Developed Turbulence 212 Turbulent Flow Models 223 Semi-empirical Theories of Turbulence 224

3.8.1 3.8.2 3.9 3.10

Contents

4.10.2 4.11 4.12

The Use of Transport Equations 230 Use of the Characteristic Functional in the Theory of Turbulence Intermittency in a Turbulent Flow 244

5 5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8 5.8.1 5.8.2 5.8.3 5.9

Particle Motion in a Turbulent Flow 251 The Eulerian and Lagrangian Approaches to the Description of Fluid Flow and Particle Motion 251 Lagrangian Statistical Characteristics of Turbulence 256 Turbulent Diffusion 267 A Semiempirical Model of Turbulent Diffusion 277 Models of Two-phase Disperse Turbulent Flows 282 Deposition of Particles from a Turbulent Flow 292 Interaction of Particles in a Turbulent Flow 304 Chemical Reactions in a Turbulent Flow 306 Concepts of Chemical Kinetics 309 Method of Moments 312 Approximations for Chemical Reaction Rates 319 The PDF Method 322

6 6.1 6.2 6.3 6.4 6.5 6.6 6.7

Coagulation and Breakup of Inertialess Particles in a Turbulent Flow 335 Kinetic Equations of Coagulation 335 Fundamental Features of the Coagulation of Particles 344 A Model of Turbulent Diffusion 349 Hydrodynamic, Molecular, and Electrostatic Forces 357 Conducting Particles in an Electric Field 365 Coagulation of Particles in a Turbulent Flow 370 Breakup of Particles 384

7 7.1 7.2 7.3 7.4

Motion and Collision of Inertial Particles in a Turbulent Flow 395 Motion of Particles without Mutual Collisions 396 Motion of Particles with Mutual Collisions 413 Frequency of Collisions of Particles 421 Preferential Concentration of Particles in Isotropic Turbulence 433

Author Index 455 Subject Index 459

234

VII

IX

Preface Practically all fluids in our everyday surroundings are to some degree disperse systems one of the following types: liquid-solid particles, liquid-gas particles, gasliquid particles, liquid-liquid particles. They are called accordingly suspensions, liquid-gas and gas-liquid mixtures, emulsions and consist of continuous or carrying phase, which could be liquid or gaseous, and dispersed phase, which contains solid particles of different size and form, bubbles and droplets. The latter will be henceforward called simply particles, adding when needed words solid, liquid or gaseous. Particle size changes in broad range from microscopic (submicron, micron) to macroscopic (millimeter, centimeter). Particles, which sizes do not exceed several microns, are called Brownian, since they are subjected to Brownian (thermal) motion. Small particles, which density slightly differs from that of surrounding medium, are called inertialess and passive when in addition their volume concentration is negligible small, so that they do not affect the motion of carrying flow. Increasing of particle size and (or) particle density relative the outer medium leads to rise of particle inertia, while buildup of volume concentration enhances particlefluid and particle-particle interactions, formation of particle aggregates and in the presence of external oriented force field production of oriented particle structures. The latter exerts influence on rheological properties and sometimes causes the nonNewtonian behavior of the medium. Particle shape may also be multiform. Sometimes it could be approached by form close to canonical: spherical, ellipsoidal with different ratio of semi axes (in limiting case oblate disk-shaped, prolate cigar-shaped) and cylindrical. Prime interest in engineering applications attracts determination of properties of dispersive media, separation of multiphase multicomponent mixtures subjected to different force fields (gravitational, centrifugal, electrical, magnetic) and motion of dispersive media in tubes, channels and through porous medium. To solve these problems it is necessary to know behavior of separate particle as well as ensemble of particles. One of the basic parameter affecting both properties and dynamic characteristics of medium is volume content (volume concentration) w of disperse phase. Disperse medium with w 1 are called dilute. For such a medium average spacing between particles is much more than the particle mean size and each particle in limiting case

X

Preface

of infinite diluted solution, that is at w ! 0, behaves as a single one, its motion is completely determined by external forces, including forces acting on particles from surrounding medium. To the latter belong regular (mean) viscous drag force and random force due to collisions of molecules of surrounding medium with the particle, which causes Brownian motion of the particle. Brownian motion is noticeable only for Brownian particles and does not substantially affect movement of larger particles. Random force arises besides by chaotic fluctuations of carrying fluid, when particle moves in turbulent flow. Enhancement of w reduces the mean distance between particles and requires to take into account interactions between particles. The motion of particle subject to interaction forces is called hindered motion. Among interaction forces are distinguished hydrodynamic, molecular and electrostatic forces. The first one is characterized by long-range interaction and depends on hydrodynamic parameters, geometrical properties of particles (size, shape, orientation in space) and mutual arrangement of particles in space (configuration). Hydrodynamic forces are most pronounced when the distance between particle surfaces (clearance) is equal or less than particle linear size. The molecular interparticle force (Van der Waals attractive force) manifests itself distinctly only when the clearance between approaching particles becomes much less than the particle size. This force keeps particles together and promotes coagulation of rigid or coalescence of liquid and gaseous particles. Electrostatic force is repulsion force due to thin charged double layers on particle surfaces. This force prevents particle collisions and stabilizes dispersive medium. The range of action of electrostatic force is small compared with hydrodynamic force, and so it is short-range as well as molecular force. When particle volume concentration exceeds 10% the average interparticle spacing is not great and combined action of all mentioned forces can lead to formation of ordered structures of particles causing anisotropy of transport coefficients and non-Newtonian properties of dispersive system. Great influence on particle behavior exert flow conditions of carrying phase. Particle in quiescent fluid settles under gravity with constant velocity called sedimentation velocity. At w ! 0 this velocity is determined by Stokes formula for rigid spherical particle and Hadamar-Rybczynski formula for liquid one. Increasing of w leads to noticeable influence of surrounding particles on sedimentation velocity. If besides the particle size is enough small, the Brownian motion also affects the sedimentation velocity. Inhomogeneity of velocity distribution in laminar flow can also influence particle motion. Since the particle size is small compared to characteristic linear scale of the flow region, the flow in the vicinity of the particle could be considered as sheared. Such a flow induces translation and rotation particle motions, which with regard to interaction forces brings to mutual approach and further collision and coagulation of particles. Particle motion in turbulent flow is more complicated problem, while particle random motion due to effects of chaotic turbulent fluctuations, which enhances collision frequencies, is superimposed on regular transport with mean velocity under the action of carrying medium and external forces. Hence the rate of particle coagulation increases and processes of heat and mass exchange become more intensive.

Preface

The size of particles and ratio of densities of particle and carrying medium determine the particle inertia. Dimensionless parameter that allows to distinguish inertial particle from inertialess one, is Stokes number St equal to the ratio of particle dynamic relaxation time to characteristic time of exposure to environmental factors on the particle. Inertialess particles (St 1, small particles which density differs slightly from that of carrying medium) are fully involved in the movement of the carrying flow and its motion is on the whole determined by the characteristics of continuous phase. Inertial particles (St 1, relative big particles which density considerably deviates from the density of carrying medium) are only partially involved in the motion of continuous phase. All this makes difficult to investigate dynamics of such particles, since it requires to take into account interparticle collisions (for high concentrated disperse medium this concerns inertialess particles too) and to recruit kinetic theory of gases. Presence of enormous amount of particles in a unit volume of disperse medium, action of random fluctuations of environmental factors and inverse influence of random motion of particles on the surrounding medium makes impossible the description of dispersive medium behavior through deterministic method. The most fruitful and productive method is statistical method. This method examines not the behavior of each particle but the behavior of particle ensemble by means of probability distribution function (PDF), which, is able to describe the change of particle ensemble configuration in space-time with regard for particle relative motion under action of different forces. Statistical characteristics of PDF permit to determine the macroscopic properties of dispersive medium. The content of the book stems from lecture courses given to students of Moscow State University of Oil and Gas. The aim of the book is to give foundation of statistical methods used in hydrodynamics of micro particles that is hydrodynamics of suspension, which contains suspended in fluid micro particles. The first two chapters provide an introduction to probability theory and microhydrodynamics. The theory of Brownian motion of micro particles taking into account particleparticle and particle-fluid interactions are described in chapter 3. The fourth chapter contains necessary information about turbulent flow and its statistical description. The motion of micro particles in turbulent flow forms the subject of chapter 5. Chapters 6 and 7 deals with interactions of inertialess and inertial particles. It should be made a remark about the title of the book. The notion of microhydrodynamics was first introduced by G.K. Batchelor (see Batchelor G.K. Developments in Microhydrodynamics/In theoretical and applied Mechanics. Ed. W.T. Koiter. - Amsterdam: North Holland, 1976. P. 33-55) and was defined as a part of hydrodynamics, which studies the motion of particles in fluid under low Reynolds numbers. In the book this notion is extended not only to small but also to finite Reynolds numbers, in order to cover micro particle motion in turbulent flow. It seems that this naturally reflects the increasing interest to the topic of the book. December 2007


XI

XIII

Nomenclature a a0 a1 ai aF ae A A A A A A,B, . . . A þ B!products A þ B!R, R þ B!P Au b bij bijk(r, t) bk1 k2 ...kN bLij(r, t) bLL(r, t) bLLL(r) bNN(r, t) buu(M1, M1) B kBk B B(0) B(M1, M2)

particle radius coefficient in the expression for correlation between acceleration fluctuations empirical constant semi-axes of an ellipsoid empirical constant in the e-F model constant in the k-e model set of events parameter in the equation of motion of a particle reaction constant drift vector (absolute) complement of a set A components (reactants) of a chemical reaction second order reaction two-step reaction particle acceleration particle radius mobility tensor third order structure function central moment Lagrangian two-point structure function longitudinal second-order structure function longitudinal third-order scalar function transverse second-order structure function two-point central moment of the second order; correlation function of fluctuations parameter in the equation of motion of a particle mobility matrix mobility tensor total energy of the field correlation tensor

XIV

Nomenclature

B(r) B1, B2, B3 Bab

correlation function of an isotropic random field empirical constants; dimensionless parameters correlation of concentration fluctuations at points Xa and Xb Bij(M1, M2) correlation function; two-point second-order moment Bijk(r, t) third-rank correlation tensor of isotropic turbulence Bk1 k2 ...kN moment of the order k1 þ k2þ . . . þkN BLa,b triple correlation of longitudinal velocity and concentration fluctuations at points Xa and Xb B0Li j ðtÞ Lagrangian single-point correlation function BLij(r, t) Lagrangian two-point correlation function BLL(r, t) longitudinal correlation function of isotropic turbulence transverse correlation function of isotropic turbulence BNN(r, t) BLL,L(r, t) scalar function of isotropic turbulence BLN,N(r, t) scalar function of isotropic turbulence BNN,L(r, t) scalar function of isotropic turbulence Buu. . .u(M1, M2, . . . MN) k-point moment Buv(M1, M2) mutual correlation function; two-point mixed moment c parameter of the distribution pðecr ; eÞ cp specific heat capacity at constant pressure specific heat capacity at constant volume cv C concentration of passive impurity C0 (X, t) concentration fluctuation h(C0 )2i intensity of concentration fluctuations C0 characteristic value of passive impurity concentration C1, C2 empirical constants; constants in the k-e model Ca0 concentration fluctuation at point Xa Ca resistance factor in the Archimedes force viscous resistance factor of a particle Cd Cd empirical constant CE ; CE0 empirical constants hCi0 DCi0 i moment that characterizes micromixing hðCi0 Þ2 i variance of concentration distribution of i-th component CL resistance factor in the lifting force arising due to the velocity shear Cm average mass concentration N binomial coefficient Cm Cp resistance coefficient of a particle Cw surface concentration Cz molar concentration dCz/dt specific rate of increase/decrease of matter concentration in a chemical reaction Ce concentration measurement error Cm parameter in the Reynolds stress equation Ca modified capillary number

Nomenclature

d d˜ d02 hdc2 i dcc(r) dij(t) ðrÞ

di j

dp D D D D Dbr Dbr D0br Drbr DF DE Dij Dij ð0Þ

Di j

Dri j DrLL Dm DrNN D pi j Dt ð0Þ Dt De DF DF Da DNS e

particle diameter ratio of particle diameter to pipe diameter dispersion of the average concentration distribution relative to the initial position of the source dispersion of the concentration distribution relative to the center of gravity structural concentration function of isotropic turbulence components of the fluid particle’s displacement dispersion tensor; components of the correlation tensor of displacement fluctuations components of the tensor of relative dispersion of two fluid particles particle diameter drag force on a particle diffusion coefficient dispersion tensor pipe diameter tensor of Brownian diffusion coefficient of Brownian diffusion coefficient of unhindered Brownian diffusion coefficient of rotational Brownian diffusion effective diffusion coefficient in equations for the e-F model effective diffusion coefficient in equations for the k-e model components of the dispersion tensor tensor of hindered relative (mutual) diffusion of two particles tensor of unhindered relative (mutual) diffusion of two particles tensor of relative diffusion of two fluid particles longitudinal component of the relative diffusion coefficient coefficient of molecular diffusion transverse component of the relative diffusion coefficient components of the tensor of turbulent diffusion of particles coefficient of turbulent diffusion coefficient of unhindered mutual turbulent diffusion function in the k-e model parameter in the three-parametric model of turbulence parameter in the three-parametric model of turbulence Damko¨hler number direct numerical simulation coefficient of restitution (COR)

XV

XVI

Nomenclature

e e (e1, e2, e3) ei ¼ u0k v0k =2 ek = ru kuk/2 ek ¼ u0k u0k =2 e0k ¼ r u0k u0k =2 e pk ¼ v0k v0k =2 e p ¼ v0k v0k =2 es ¼ rhui ihui i=2 E E E E(k, t) E(o) E1 EA E1 Ec(k) Ec(k, t) Ecr Eij Ei1j hEik(x, t)Ejn(x þ r, t)i E, F, . . . f f f f f (Xp(t), t) hfi f (V) h f (X, t)i h f (X, t)iT h f (X, t)iV h f (X, t)iTV f 1, f 2 _ _ f1 ; f2 fA fA fB feQ1

unit vector unit vector along the line of centers of two particles basis of a Cartesian coordinate system fluctuation energy of interfacial interaction kinetic energy density in the continuous phase intensity of turbulence; mean kinetic energy of fluctuational motion per unit mass in the continuous phase kinetic energy density of fluctuational motion turbulent energy of the disperse phase fluctuational energy of a particle kinetic energy density of averaged turbulent flow internal energy vector of electric field strength rate-of-strain tensor spectrum of average energy energy distribution over the frequency spectrum maximum principal value of rate-or-strain tensor activation energy rate-of-strain tensor at the infinity spectral density of concentration fluctuations intensity spectrum of concentration fluctuations critical value of external electric field strength rate-of-strain tensor components rate-of-strain tensor components at the infinity correlation function of rate-of-strain tensor components of the product of chemical reaction coefficient of particle’s involvement into fluctuational motion of the carrier flow (involvement coefficient) stochastic force external force density random force exerted by the flow on a fluid particle random Brownian force acting on a particle average (mean) value particle breakup frequency expectation value; ensemble average time average spatial average spacetime average functions in the k-e model dimensionless coefficients force acting on a particle due to the particle acceleration dimensionless force of molecular attraction between spherical particles Basset force resistance coefficient for particle’s translational motion

Nomenclature

fer fsr fsY h fAi0 pi h fBi0 pi fi fj fL h fLi0 pi fr

fu fm(Re) F F F F F(k) F[w] F(Xp(t), t) F2x F2z kF0 k FE FB Fe Fe

Ffl Fg Fm FAs

resistance coefficient for particle’s translational motion resistance coefficient for particle’s translational motion resistance coefficient for particle’s translational motion correlation between Archimedes force fluctuations and probability density function (PDF) of particles correlation between Basset force fluctuations and PDF of particles dimensionless coefficients in the expression for the force of electrical interaction between particles j-th component of the force acting on the unit surface area of a particle lifting force acting on a particle due to the velocity shear of the carrier flow correlation between lifting force fluctuations and PDF of particles coefficient of involvement of a pair of particles separated by the distance r into fluctuational motion of the continuous phase coefficient of involvement of a particle into fluctuational motion of the turbulent carrier flow parameter in the equation for Reynolds stresses in twoparametric turbulence models function in the e-F model systematic force particle interaction force generalized force vector spectrum of a homogeneous random field functional external force acting on a particle component of interaction force between two conducting charged particles perpendicular to their line of centers component of interaction force between two conducting charged particles along their line of centers generalized force matrix external force stochastic Brownian force external force component of the force exerted on a particle by the surrounding fluid; hydrodynamic force resulting from the particle’s proper motion fluctuating force gravitational force molecular interaction force force of molecular attraction between two spherical particles

XVII

XVIII

Nomenclature

Fab(k, t) Fcap Fg Fh Fh Fij(k) Fij,k(k) Fik(r, t) ðiÞ

Fk FLL(k, t) FMi FNN(k, t) F sR Fs Fih Fv g g gi(v/o) g1 gi1j jg jkj g jk T g jk gfl gu G G G G G(C, C0 , C00 )

S(v, o)

spectrum of concentration fluctuations surface tension force (capillary force) gravitational force force exerted on a particle by the surrounding fluid hydrodynamic resistance force acting on a particle spectral tensor spectrum of the third-rank correlation tensor of isotropic turbulence integral of two-point structure function of velocity fluctuations in a continuous medium, taken along the trajectory that describes the relative motion of a pair of particles force acting on i-th particle longitudinal spectrum of isotropic turbulence migration force caused by the interaction of particles with turbulent eddies of the carrier flow transverse spectrum of isotropic turbulence electrostatic repulsion force between two spherical particles Stokesian component of the force exerted on a particle by the carrier flow thermodynamic force viscous friction force acceleration of gravity coefficient of involvement of particles into fluctuational motion of the carrier flow single-modal density distribution of particle volume ratios in the course of particle breakup velocity gradient tensor at the infinity components of velocity gradient tensor at the infinity determinant of the matrix g jk matrix transposed matrix fluctuation of the rate of matter production or consumption in the course of a chemical reaction coefficient of involvement of a particle into fluctuational motion of the carrier turbulent flow dimensionless parameter Gibbs free energy spatial gradient of a physical parameter empirical function kernel of the micromixing operator in coalescencedispersion models; distribution density of intermediate concentrations that are formed during an elementary mixing event collision cross section of two particles of volumes V and o; normal cross-sectional area of the limiting flow tube

Nomenclature

hGi1 Gfl(X, t) Gik(r, t)

Gj Ga h h h h0 h0 hi hu hd H H(X) H(X) i (i1, i2, i3) i, j I[p] I I Ik IR j j(R1, R2) J0 J0(0) jfl J J J Jc

conditional average fluctuational term in the diffusion equation integral of the two-point structure function of velocity fluctuations in the continuous phase, taken along the trajectory of the relative motion of two particles parameter in the expression for coagulation kernel Galilei number coefficient of hydrodynamic resistance coefficient of hydrodynamic resistance for particles approaching each other along their line of centers distance between two planes minimum gap between surfaces of spherical particles hydrodynamic resistance coefficient for unhindered particle motion hydrodynamic resistance coefficient for i-th particle coefficient of particle’s involvement into fluctuational motion of a turbulent carrier flow asymptotic expression for the hydrodynamic resistance coefficient of a particle at small clearances between particles enthalpy Saffman function Heaviside function parameter of the gamma distribution basis of a local Cartesian coordinate system mutually orthogonal unit vectors in a plane perpendicular to the line of centers of two particles Boltzmann collision operator in the Enskog form unit tensor number of particle collisions in the absence of the force of electrostatic repulsion collisional term in the kinetic coagulation equation number of particle collisions, with electrostatic repulsion taken into consideration diffusion flux diffusion flux of particles of radius R2 toward a particle of radius R1 diffusion flux of unhindered particles diffusion flux per unit solid angle vector of diffusion flux fluctuation displacement vector dimensionless diffusion flux of particles of radius R2 toward a particle of radius R1 particle flux toward a test particle collision operator in the kinetic equation for PDF of velocity of a single particle

XIX

XX

Nomenclature

Jf Ji Ji Jij Jiaj Jik Jr Jw Jw k k k k k kB kc kf k kcab krab ktab kb kk km

K K K K(V,o)

K K c0 K r0 Kt K c 0 Kt K ti jk Kr

incident particle flux flux density of impurity along the Xi-axis moment of inertia of i-th particle term in the balance equation for turbulent stresses in the disperse phase, which are caused by particle collisions components of the Oseen tensor (stokeslet) for a-th particle i-th component of the diffusion flux of k-th component of the impurity particle flux reflected from the wall flux of matter toward the wall flux of particles depositing on the wall ratio of particle radii wave number permeability of a porous medium reaction rate constant unit vector along the line of centers of two colliding particles parameter in the equation of motion of a particle in a turbulent flow wave number of a scalar field coefficient of aerodynamic resistance the Boltzmann constant conjugate mobility tensor rotational mobility tensor translational mobility tensor wave number corresponding to the Batchelor scale wave number corresponding to the Kolmogorov scale wave number corresponding to the maximum size of inhomogeneity (the right end of the inertial-convective range, starting from which micromixing commences) empirical coefficient transport coefficient moment’s order kernel of the kinetic coagulation equation; collision frequency of two particles of volumes v and o in a unit disperse phase volume at unit particle concentrations; coagulation constant micromixing operator conjugate translational resistance tensor rotational resistance tensor translational resistance tensor conjugate rotational resistance tensor third-rank tensor of translational shear resistance components of translational shear resistance tensor third-rank tensor of rotational shear resistance

Nomenclature

kK0 k K t;r;c 0 Keff Kg Kirjk Kpr Ks Kt Ktg Kts l l l l1 li j lm lu l L

L L L L L Lc L ðiÞ

Lk Lki LL LN ðrÞ Lq ð j+jÞ Ls LES m m m m

generalized resistance matrix resistance matrices macroscopic (effective) transport coefficient collision kernel in the gravitational field components of third-rank rotational shear resistance tensor permeability matrix collision kernel in a shear flow collision kernel in a turbulent flow collision kernel in a turbulent flow with gravity taken into account collision kernel in a turbulent shear flow step length for a particle’s random walk Prandtl mixing length Prandtl–Nikuradze mixing length Taylor mixing length components of symmetric scale tensor mean free path of molecules coefficient of particle’s involvement into fluctuational motion of turbulent carrier flow unit vector characterizing particle’s orientation average distance between particles; characteristic linear size; spatial macroscale of turbulence; Eulerian spatial macroscale; integral spatial scale of turbulence linear operator torque linear macroscale of concentration field empirical function parameter in the equation of motion of a particle in a turbulent flow integral scale (macroscale) of concentration field hydrodynamic torque exerted on a particle by the surrounding fluid torque acting on i-th particle associated Laguerre polynomial longitudinal integral scale transverse integral scale Lagrange interpolation polynomial Stokesian component of torque exerted on a fixed particle by the surrounding fluid large eddy simulation mass empirical constant in the e-F model particle surface mobility parameter dimensionless parameter

XXI

XXII

Nomenclature

m _ m m0 m1 mi mi mk mu M Mi Ni n(X, a) n(v, X, t) n n n n n0 n0(v) ni N N N(t) N N hNi N Na hN c i ND

Nt

NW

Oh p p

virtual mass of a particle dimensionless moment of distribution zero order moment; number concentration of particles first order moment; volume concentration of particles mass of i-th particle mass of i-th component produced or consumed in the course of chemical reaction k-th order moment of the volume distribution of particles coefficient of particle’s involvement into fluctuational motion of turbulent carrier flow migration coefficient molecular mass of i-th component number of moles of i-th component distribution of particles over their radii distribution of particles over their volumes empirical constant in the e-F model total reaction order; kinetic order of reaction normal vector number concentration of particles concentration of particles far from the test particle initial distribution of particles over volumes order of i-th reaction spacetime point number of particles in a given volume number of particles in a unit volume (number concentration) number of steps (displacements) in particle’s random walk dimensionless parameter dissipation of concentration inhomogeneity scalar dissipation of concentration inhomogeneity number of particles of type a in a unit volume conditional average rate of scalar field dissipation at fixed concentration C ratio between the characteristic time of a process and the characteristic time of impurity transport via molecular diffusion ratio between the characteristic time of a process and the characteristic time of convective transport of impurity via turbulent motion ratio between the characteristic time of a process and the characteristic time of matter production or consumption in a chemical reaction Ohnesorge number pressure dimensionless parameter

Nomenclature

p p p0 p p(rjj1, r0, t, t0) p(u, C1, C2, . . ., CN; X, t) C1, C2, . . ., CN p(X) p(X1, X2, . . ., XN) pðX ; V jj1 ; j2 ; . . . ; jn ; t1 ; t2 ; . . . ; tm Þ pðX; Vjj1 ; j2 ; . . . ; jn ; t1 ; t2 ; . . . ; tm Þ pðX ; V ; ujj1 ; j2 ; . . . ; jn ; X 1; X 2; . . . ; X n; t1 ; t2 ; . . . ; tm Þ p(u, tju0, t0) pðV 1 ; V 2 ; . . . ; V n j j1 ; j2 ; . . . ; jn ; t1 ; t2 ; . . . ; tn Þ p(Yjt; j, t0) pðeÞ Dt _ Dt _ p e + e; e eÞ 2 2 _ pðe; eÞ ˙ pðecr ; eÞ hp0 u0 i p1 pc(u, C, X) peq pij(r) pp(X, v, t) P P = hpi P(AjB) PðA \ BÞ

unit vector dynamic probability density for particle ensemble perturbation (fluctuation) of pressure pressure averaged over the cross section PDF of distance between two particles; function of distance between neighbors joint single-point PDF of velocity and scalar quantities (concentrations of different passive impurities) at time t probability density function, PDF joint multi-particle PDF joint multi-dimensional (multi-particle) PDF of coordinates X and Lagrangian velocities V for a given set of time values t1, t2, . . ., tm and initial particle positions j1, j2, . . ., jm; the main Lagrangian statistical characteristic of turbulence joint PDF of Lagrangian (V) and Eulerian (u) quantities

conditional probability density joint PDF of Lagrangian velocities V(j1, t1), V(j2, t2), . . . V(jn, tn) of n fluid particles at different moments of time probability density distribution of particle displacements from the initial point j during the time t one-dimensional distribution of random specific energy dissipation joint distribution density of random values e and e_ at different moments of time joint distribution density of random values e and e_ at one and the same moment of time joint distribution density of ecr and e_ mixed correlation (correlator) of pressure and velocity fluctuations static pressure at the infinity joint PDF in a turbulent mixing layer equilibrium PDF pair PDF dynamic probability density in the phase space of particle coordinates and velocities probability velocity PDF of a system of particles conditional probability probability of a joint event

XXIII

XXIV

Nomenclature

P(r, w, t) P(v, o)

P(wr) P(YN) P0(v) Pr0 P2(v, v1) Pnm Pt Pe PeD PeT Pr Prp Prt q(r, t) q q q qe Q Q Qc Q aij;n r rw r (r, Y, z) (r, Y, F) r0 r0 (r0, Y0, F0) ri R R R R

PDF of a particle pair (probability of the relative velocity w separated by the distance r at the time t) probability of formation of a droplet with volume in the range (v, v þ dv) in the process of breakup of a droplet with volume in the range (o, o þ do) probability distribution of fluctuational component of the relative radial velocity probability distribution of a certain N-particle configuration Maxwellian velocity distribution associated vector of dynamic pressure in rotational motion PDF of velocities of two particles associated Legendre polynomial associated vector of dynamic pressure in translational motion Peclet number diffusional Peclet number thermal Peclet number Prandtl number Prandtl number of the disperse phase turbulent Prandtl number of the carrying phase Richardson function particle migration coefficient heat flux vector particle charge density of internal heat sources dynamic pressure directed flux of matter intensity of fluctuational energy dissipation via inelastic collisions of particles n-th order moment (multipole) of surface forces acting on the particle a distance between the centers of two particles pipe radius unit vector along the line of centers of two particles cylindrical coordinates spherical coordinates initial radius vector of the particle center radius vector of a point relative to the center O. initial coordinates of the particle center radius vector of the center of i-th particle gas constant resistance tensor tensor of generalized resistance due to the translational and rotational motion of the particle grand resistance matrix

Nomenclature

R R R R R/(R þ P) R12 Ra Rc Rij RFE RFU RFV RLE RLU RLV RSE RSU RSV Raij Rm Rp Re Recr Rep Ret Rel s s s s s hs 2i S S S S S(k) dS SA Sij Sij Saij

particle’s center of reaction particle radius pipe radius correlation coefficient of colliding particles selectivity parameter of a two-stage chemical reaction velocity correlation coefficient of two particles at the point of impact radius of a particle of type a coagulation radius components of the resistance tensor component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle component of the resistance tensor of a particle rotlet of particle a minimum radius of a breaking droplet coordinate vector of the particle’s center Reynolds number critical value of the Reynolds number Reynolds number of a moving particle Reynolds number corresponding to the spatial Taylor microscale local Reynolds number for fluctuations having scale l. dimensionless distance between particle centers minimum separation from a plane distance traveled by a particle in time t variable in the image domain of the Laplace transformation unit vector along the particle’s displacement vector r mean-square particle displacement surface entropy sedimentation coefficient symmetric part of the velocity gradient tensor element of sphere surface area jkj = k oriented surface element parameter of molecular interaction sedimentation coefficients intensity of force dipoles distributed over the particle surface surface force dipole (stresslet) of particle a

XXV

XXVI

Nomenclature

SBij SG ij SIi j Sk

Sk1 ;k2 ;...;kN SR SBT SET SH T Sc Scbr Sct St St0 t0 t12 t12 t21 tA tc tcom tD tE tew tew1 tsw ttE g

tE teff tik tL

contribution of Brownian diffusion to the sedimentation coefficient contribution of gravity to the sedimentation coefficient contribution of molecular interaction to the sedimentation coefficient specific source term of k-th component produced/ consumed in a chemical reaction in the mass conservation equation for passive impurity components cumulant (semi-invariant) electrostatic interaction parameter contribution of particles’ Brownian motion to the stress tensor contribution of particles’ motion driven by non-hydrodynamic forces to the stress tensor contribution of particles’ motion driven by interparticle hydrodynamic interactions to the stress tensor Schmidt number Schmidt number for Brownian diffusion turbulent Schmidt number Stokes number calculated with respect to the integral macroscale Stokes number calculated with respect to the Kolmogorov microscale initial moment of time time required for a perturbation to go through the wave number range (k1, k2) times of collisions of particle 1 with particles 2 times of collisions of particle 2 with particles 1 characteristic time of particle enlargement that takes into account molecular interactions between particles effective time between particle collisions characteristic time of change of system configuration characteristic time of a diffusion process characteristic enlargement time for conducting droplets in an electric field rotational resistance coefficient of a particle rotational resistance coefficient of a particle rotational resistance coefficient of a particle characteristic time of enlargement of conductive droplets in a turbulent flow in an external electric field characteristic time of enlargement of conductive droplets undergoing gravitational sedimentation in an electric field effective relaxation time of a particle components of the Maxwellian stress tensor characteristic non-stationarity time for an averaged flow

Nomenclature

tT tu tv tV tl T T T T0 T (E) Tfl T (L) ðLÞ T1

Ts Te

p

Tij TL0 ðLÞ

TL p ðtÞ ðLÞ

TN p ðtÞ ðLÞ

Tp

ðLÞ

Tr u u

u u(X, t) (u, v, w) u0 u0 u0 (X, t) u u0 u0

characteristic thermal relaxation time for a particle effective relaxation time that takes into account the effect of virtual mass characteristic time of dynamic relaxation for a particle (depends on Rep) characteristic time of viscous relaxation for a particle characteristic period of fluctuations having the scale l characteristic time scale stress tensor turbulent fluctuation period _ parameter of the distribution pðecr ; eÞ Eulerian time scale fluctuational component of the stress tensor Lagrangian correlation time; Lagrangian integral time scale of turbulence asymptotic expression for the Lagrangian integral scale of turbulence at high Re systematic component of the stress tensor component of hydrodynamic torque exerted on a particle by the surrounding fluid and caused by the particle’s proper motion contribution of particles suspended in the fluid to the stress tensor parameter in the parabolic exponential approximation of the autocorrelation function Lagrangian longitudinal (parallel to the average relative velocity of particles) integral time scale of fluid velocity fluctuations along the particle trajectory Lagrangian transverse (perpendicular to the average relative velocity of particles) integral time scale of fluid velocity fluctuations along the particle trajectory Lagrangian integral time scale two-point integral time scale velocity vector relative velocity of two particles approaching each other along their line of centers carrier flow velocity Eulerian velocity field at the point (X, t) velocity components velocity perturbation fluctuational component of the carrier flow velocity Eulerian field of velocity fluctuations mean flow rate velocity initial velocity characteristic turbulent velocity

XXVII

XXVIII

Nomenclature

u0 u0c(t) u02 ¼ hu0k u0k i=3 u u1 Dui(r, t) ui(S) uL um uN ul ul0 0 hu0i fBk i hu0i fLk0 i hu0i u0k i hu0i u0j i rhu0i u0j i hu0i q0 i U u (U, V, W) Ui U U ku0 k U ra U ta Um Uv v v v1 vp v p ; v p1 v0p ; v0p1 ˜v pi ðsÞ V V0 Vav

velocity of Stokesian flow around a particle effective path length of a particle in fluctuational motion turbulence intensity; intensity of velocity fluctuations of the continuous phase dynamic velocity velocity at the infinity difference of velocity fluctuations at two points Laplace transform image of carrying phase velocity projection of velocity onto r average molecular velocity projection of velocity onto a direction perpendicular to r rate of fluctuations of scale l rate of fluctuations on the Kolmogorov microscale l0 correlation between fluid velocity fluctuation and Basset force fluctuation correlation between fluid velocity fluctuation and lifting force fluctuation components of the correlation tensor of velocity fluctuations specific Reynolds stresses of the carrying phase Reynolds stresses in the carrying phase turbulent heat flux in the carrying phase velocity; averaged velocity; average velocity of turbulent flow generalized velocity vector, which includes both translational and rotational components of velocity velocity components degree of immiscibility of i-th reagent degree of immiscibility of the total reaction mean-flow-rate velocity matrix of generalized velocity angular velocity of particle a translational velocity of particle a average mass velocity sedimentation velocity of a particle of volume v velocity translational velocity of the particle center velocity at the infinity particle velocity velocities of two particles before the collision velocities of two particles after the collision Laplace transform image of particle’s velocity particle volume parameter of the volume distribution of particles average particle volume

Nomenclature

Vk Vm (vr , vy , vF) V V DV V V(j, t) V0 V 1, V 2 V r0 Vt p VA s VA Vij ð0Þ

Vij Vimj hViVji hVi0 V 0j i hVi0 q0j i Vp VRs w ¼ v p v p1 w w

wk0 wr wr0 hwr0 2i W W W jW0j Wg hWii

propagation speed of perturbations along the wave number axis minimum volume of breaking droplets components of the velocity vector in a spherical coordinate system spatial volume total interaction potential of two spherical particles volume element Velocity Lagrangian velocity particle velocity fluctuation potentials of charged particles rotational velocity tensor translational velocity tensor molecular interaction potential of two planes molecular interaction potential of two spherical particles relative velocity of two particles relative velocity of unhindered motion of two particles Lenard–Jones potential Lagrangian correlation function of components of the vector V turbulent stresses in the disperse phase turbulent heat flux in the disperse phase particle velocity electrostatic interaction potential of two particles relative velocity of colliding particles relative velocity of two particles term in the particle’s equation of motion that helps account for the hydrodynamic (collisionless) interactions of particles; represents a continuous random process random field of hydrodynamic interactions radial component of relative velocity of a particle pair fluctuational part of the radial component of relative velocity of a particle pair intensity of fluctuations of the radial relative velocity of a particle pair stability factor of a dispersed system collisional term in the Langevin equation of motion of a particle; discontinuous random process distribution of discontinuous jumps characteristic rate of chemical reaction difference of two particle’s sedimentation velocities average chemical reaction rate of i-th component

XXIX

XXX

Nomenclature

Wi Wr We x1 (X, Y, Z) X X(X1, X2, . . ., Xn) X(j, t) X 0a X sa ~ rab ; x~rab X ~ t ; x~t X ab ab X0(t) hXc2 i Xp(t) yþ ~ cab ; ~ycab Y ~ r ; ~yr Y ab ab ~ t ; ~yt Y ab ab Y(t) Y0 Z1, Z2, Z3 a a a aE a aj aL at ae b

source term in the diffusion equation, which results from a chemical reaction radial component of the average relative velocity of a particle pair Weber number dimensionless distance from the particle surface Cartesian coordinates generalized vector that includes spatial coordinates and orientation angles relative to a given coordinate system point in space random function – Eulerian coordinates of a fluid particle vector of spatial coordinates (position vector) of the center of particle a in a given coordinate system vector of orientation angles of particle a in a given coordinate system dimensionless resistance coefficients for rotational motion of particles about the line of centers dimensionless resistance coefficients for translational motion of particles along the line of centers coordinate of the center of mass variance of concentration distribution relative to a source particle coordinates at the moment t dimensionless distance to a wall dimensionless resistance coefficients for rotational motion of particles about the j-axis dimensionless resistance coefficients for rotational motion of particles about the i-axis dimensionless resistance coefficients for translational motion of particles perpendicular to the line of centers vector of particle’s displacement from the initial position during the time interval t fluctuation of particle displacement conservative variables; Schwab–Zeldovich variables empirical constant weight function _ parameter of the distribution pðecr ; eÞ constant in the three-parametric turbulence model empirical constant in the e-F model exponent in the coagulation kernel expressed as a power function of particle volumes constant dissipation rate of hydrodynamic field; inverse relaxation time constant in the three-parametric turbulence model constant in the k-e model correction factor

Nomenclature

b b b b1 b2 bE bj bL bt g g g g(t) g g˙ g˙ gf g0 gij(t) g˙ t gt gF G G(x) GF d d d d(X) d(X) dD D D(t) D1, D2 e e e e e e e e

dimensionless parameter empirical constant parameter defining the condition for heterogeneous reaction square of skewness excess (kurtosis) of the distribution constant in the three-parametric turbulence model exponent in the coagulation kernel expressed as a power function of particle volumes constant dissipation rate of a scalar field; inverse relaxation time constant in the three-parametric turbulence model empirical constant dimensionless parameter density ratio between a particle and the surrounding fluid Green’s function of the particle’s equation of motion function in the e-F model shear rate dimensionless hydrodynamic interaction parameter particle shape parameter constant in the three-parametric turbulence model components of velocity fluctuation gradient tensor average shear rate in small-scale fluctuations constant in the three-parametric turbulence model parameter in the three-parametric turbulence model Hamaker constant complete gamma function constant in the three-parametric turbulence model gap (clearance) between particle surfaces dimensionless parameter thickness of the viscous boundary layer delta function of a scalar quantity delta function of a vector quantity thickness of the diffusion boundary layer dimensionless gap (clearance) between particle surfaces effective diameter of a particle cloud dimensionless parameters dielectric permittivity relative fluctuation of distributed particle density rate of turbulent energy dissipation dissipation function small parameter permutation symbol parameter in the Lennard–Jones potential specific dissipation of turbulence energy

XXXI

XXXII

Nomenclature

e0 ecr ðVÞ heki eijk es z h q q0 hq0 u0 i q qp(t) y y(F) k k k k kp l l l0 lc lð0Þ c (lc)i lD lL lL lN lt L L L1 L1 Lij Lij L1 ij

specific dissipation rate of a scalar field critical value of specific dissipation of energy mean specific dissipation of energy of fluctuational motion components of the permutation (Levi–Civita) symbol specific dissipation of energy of averaged motion drift parameter characterizing the effect of intersection of particle trajectories porosity of a medium absolute temperature temperature perturbation (fluctuation) mixed correlation of temperature fluctuations and velocity fluctuations temperature of the carrying medium particle temperature orientation angle of the particle pair relative to the external electric field angle between the radius vector r0 of the particle center and a point on the limiting trajectory coefficient of heat conductivity empirical constant dimensionless parameter in the expression for the coefficient of mutual turbulent diffusion dimensionless parameter characterizing droplet stability in an external electric field coefficient of turbulent thermal conductivity dimensionless resistance coefficient scale of velocity fluctuations inner scale of turbulence; Kolmogorov spatial microscale differential scale of fluctuations inner scale of concentration; Batchelor microscale microscale of i-th component concentration Debye radius London wavelength longitudinal differential scale transverse differential scale Taylor spatial microscale vorticity tensor Loitsyansky invariant vorticity tensor at the infinity antisymmetric part of the undisturbed velocity gradient tensor components of the tensor of relative dispersion of particle cloud components of the vorticity tensor components of the vorticity tensor at the infinity

Nomenclature

hLik(x, t)Ljn(x þ r, t)i m m me meff mi mt n nA, . . ., nF nAA þ nBBþ. . .! nEE þ nFF. . . ne ni np nt j j(t) j jj Xi P P(X, t) r0 re ri rp s s s s2 sij YN S S S t t t t

correlation function of the vorticity tensor chemical potential ratio of viscosity coefficients of the internal and external fluids dynamic viscosity coefficient of the external fluid effective dynamic viscosity coefficient dynamic viscosity coefficient of the internal fluid turbulent dynamic viscosity number of steps in particle’s random walk stoichiometric coefficients kinetic scheme of chemical reaction kinematic viscosity coefficient of the external fluid kinematic viscosity coefficient of the internal fluid turbulent viscosity coefficient of the disperse phase turbulent viscosity coefficient of the continuous phase dimensionless gap (clearance) between particles degree of completeness of a chemical reaction ratio of particle diffusion fluxes with and without electric field Lagrangian coordinates; initial coordinates of a particle coefficient of virtual turbulent diffusion of a particle cloud in the Xi direction triadic (third-rank) tensor intermittency function density ratio of the carrying medium and the particle external fluid density internal fluid density particle density surface charge density parameter in the Lennard–Jones potential root-mean-square deviation distribution variance components of the viscous stress tensor element of the set of configurations of an N-particle ensemble specific surface area of the boundary between regions of different concentrations surface tension coefficient parameter characterizing the relative influence of gravity and turbulence on the collision kernel characteristic time of a process stress dimensionless time ratio of the particle radius to the double layer thickness

XXXIII

XXXIV

Nomenclature

tE t0 tc (tc)1 (tc)2 tD tf ti(ri) tij ð1Þ

ti j tm tt tt tw

t l0 tL u w w(r) w1 wm f f(s) f1, f2 fij kFk F(p)

F[(r, x)] Fp(r, t) x x x xt c c

characteristic time of decrease of the rate-of-strain tensor turbulent component of stress relaxation time of a concentration field; characteristic time of micromixing time of micromixing in the inertial-convective region time of micromixing in the viscous-convective region characteristic time of impurity transport via molecular diffusion friction force per unit surface area probability density distribution of i-th displacement components of the stress tensor components of the Reynolds stress tensor empirical constant characteristic time of turbulent convective impurity transport Taylor time microscale; characteristic time of hydrodynamic relaxation characteristic time of matter production/consumption in a chemical reaction characteristic period of fluctuations on scale l0; the Kolmogorov time microscale characteristic time of decrease of the vorticity tensor particle mobility volume concentration (content) of particles characteristic function; moment generating function empirical constant limiting volume concentration of closely packed particles force field/electric field potential function in the Laplace transform image of particle’s velocity particle surface potentials molecularinteractionpotentialbetweeni-thandj-thparticles shear resistance matrix correction factor introduced into the expression for the molecular attraction force in order to account for electromagnetic retardation characteristic functional probability density of particle’s displacement by the distance r during the time t reflection coefficient of a particle thermal diffusivity inverse Debye radius turbulent thermal diffusivity cumulant-generating function stream function

Nomenclature

c(t)

C(t) C(E)(r) C(E)(t) C(L)(t) Ccc ðEÞ CL ðr; tÞ

CL(t) CLp(t t1) ðLÞ

CL p ðtÞ

CLr(tjr) ðEÞ

CN ðr; tÞ ðLÞ

CN p ðtÞ

Cu(j) Cuu Cuv Cv CTv Cvt C kCk v v hvi v(X) v12 vl

function characterizing the particle’s effective free path resulting from its involvement into fluctuational motion of the carrier flow autocorrelation function Eulerian spatial autocorrelation function Eulerian time autocorrelation function Lagrangian autocorrelation function correlation coefficient of concentration fluctuations longitudinal (parallel to the average relative velocity vector) Eulerian spacetime autocorrelation function of fluid velocity fluctuations along the particle trajectory dimensionless autocorrelation function characterized by Lagrangian integral time scale two-time correlation function of carrier flow velocity fluctuations along the particle trajectory longitudinal (parallel to the average relative velocity vector) Lagrangian time autocorrelation function of fluid velocity fluctuations along the particle trajectory Lagrangian autocorrelation function characterizing the relative motion of two particles initially separated by the distance r = jrj transverse (perpendicular to the average relative velocity vector) Eulerian spacetime autocorrelation function of fluid velocity fluctuations along the particle trajectory transverse (perpendicular to the average relative velocity vector) Lagrangian time autocorrelation function of fluid velocity fluctuations along the particle trajectory autocorrelation function of fluid velocity fluctuations along the particle trajectory correlation coefficient correlation coefficient two-time autocorrelation function of fluid velocity fluctuations along the particle trajectory two-time autocorrelation function of fluid temperature fluctuations two-time autocorrelation function of fluid velocity fluctuations function appearing in the e-F and k-e models shear resistance matrix element of the event set vorticity vector average vorticity vector weight function frequency of collisions between particles of types 1 and 2 frequency of fluctuations on scale l

XXXV

XXXVI

Nomenclature

v0l V X X Vi X1 V1 i Vu

frequency; time period of fluid particle velocity recurrence set of all events angular velocity vorticity vector vorticity vector component vorticity vector at the infinity vorticity vector component at the infinity inertness parameter for a particle

Some mathematical notations [ union of sets \ intersection of sets 1 empty set 5 gradient operator r divergence operator 5 curl operator 5d gradient of the delta function D = jqXi/qjjj determinant of a Jacobian rr = rirj dyadic ab. . .c = aibj. . .ck polyadic product of tensor and vector Aa = Aijak Aa = Aijaj scalar product of tensor and vector A:B = AijBij double-dot scalar product of tensors 2 dX d X time derivatives ; X¨ ¼ 2 X˙ ¼ dt dt D @ @ substantial derivative ¼ +uk Dt @t @Xk dF½w functional derivative dwðX Þ dN F½rðX Þ N-th order functional derivative drðX1 ÞdrðX2 Þ . . . drðXN Þ R ðquÞ ¼ qðMÞuðMÞdM scalar product in the functional space d Dk ðMÞ ¼ dqk ðMÞ i Im erf(z)

functional derivative operator imaginary unit imaginary part probability integral

j1

1 Basic Concepts of the Probability Theory In order to formulate the theoretical concepts that will be crucial for the subsequent chapters, we must first mention some basic notions of the probability theory and their further ramifications. This chapter provides a brief outline. The interested reader will find a more detailed discussion of the relevant topics in textbooks and monographs on the probability theory and statistical physics (see the list of references at the end of the chapter).

1.1 Events, Set of Events, and Probability

Some of the typical situations considered in this chapter are as follows: a particle is in small volume (elementary volume) that includes a point X; N particles are in a small region of space; N1 particles of type 1 and N2 particles of type 2 are in a certain region. Speaking more generally, we shall be dealing with situations having to do with particle positions in space at different instants of time. Every such case can be thought of as a specific realization of some event. We shall define an event as an element of a certain space of events. In what follows, we will most often be using vector spaces defined by vectors X (X1, X2, . . ., Xn), where Xi are real numbers. The probability theory introduces the notion of a set of events. Then the condition that an event o belongs to a set of events A is written as o 2 A. Consider an event of finding a particle in some volume element DV centered at a point X. Let all such events form an ensemble of events denoted as A(DV, X). It turns out that it is possible to determine whether the particle is in the vicinity of X, but it makes no sense to determine whether the particle is exactly at the point X. Physically, the probability of finding a particle exactly at some given point is zero. As an idealization one can consider a probability distribution given by a delta function, in which case the probability can be a finite number, and it is associated with an infininesimal interval around this point as the interval tends to zero. Let o(X) denote the event that a particle is found in infinitesimal volume element centered at X. We can then ascribe a certain probability to the condition o(X) 2 A(DV, X). This is just the probability for the particle to be in the volume element DV centered at X.

2

j 1 Basic Concepts of the Probability Theory The probability P(A) of a set of events is defined as a function of A which satisfies the following probability axioms: 1. P(A) 0 for all A; 2. P(O) ¼ 1; 3. if {Ai} is a finite or countable sequence ofX non-overlapping sets, that is, (Ai \ Aj ¼ f), then Pð[ Ai Þ ¼ PðAi Þ. i

i

The two consequences of these axioms are: ¼ 1PðAÞ; 1. PðAÞ 2. P(f) ¼ 0. Here \ and [ denote, respectively, the intersection and union operations on sets; – the complement of a set A, that is set O is the set of all events; f – the empty set; A of all events not belonging to A. Coexisting with the notion of probability is the notion of frequency of an event. To understand the difference between probability and frequency of events, consider an event o selected at random from the full set O. The number of occurrences of o 2 A in N trials gives us the relative frequency of realization of the event o 2 A. When N is increased, the relative frequency goes to the limit P(A), which is defined as the probability of the event A. At N 1, it is safe to assume the relative frequency of an event to be equal to the probability of occurrence of this event presuming that relative frequency has been normalized. The axiom 3 is given for a finite or countable number of sets. But often it is necessary to deal with uncountable infinite number of sets. For instance, when studying the motion of particles under the action of external forces, one has to deal with sets of particle positions in spacetime. Let X is the position of a particle in space. The probability for the particle to be exactly at point X (it would then belong to a set consisting of only one element) is equal to zero, while the probability to find the particle in the vicinity of that point (that is, in the finite volume element DV centered at X) is nonzero. The region DV can be visualized as a union of an infinite number of one-element sets of the type X. A direct application of axiom 3 to this case would produce an uncertainty of the type 0 1. Therefore the axiom 3 is unsuitable for infinitive sequences of sets, and the probability for the event to belong to the set DV cannot be obtained as the sum of such probabilities for the sets X DV. Axiom 3 is applicable only to incompatible events, that is, mutually exclusive events that belong to non-overlapping sets. Consider now the case of intersecting sets and overlapping events, that is, events belonging to two or more sets at the same time. Such events are called joint. Consider two sets A and B, whose intersection A \ B is not empty. We say that o belongs to the intersection (o 2 A \ B) if o 2 A and o 2 B. Then the probability of the joint event o can be written as PðA \ BÞ ¼ Pðw 2 A and

w 2 BÞ:

ð1:1Þ

As examples of joint events, consider two situations, which will prove to be of interest further on:

1.1 Events, Set of Events, and Probability

1. At a given time, the volume element DV centered at the space point X contains N1 particles of type 1 (first event) and N2 particles of type 2 (second event). The probability of this happening is given by the joint probability of both events. 2. A volume element DV centered at a space point X contains N1 particles of type 1 and N2 particles of type 2 at the time t1 (first event) and n1 particles of type 1 and n2 particles type 2 at the time t2 (second event). The probability of the joint event is the joint probability of both events at times t1 and t2. Sometimes one is interested in the probability of an event given the occurrence of some other event. For example, we may want to know the probability of finding a particle in a volume element DV centered at the point X at the time t given that at the time t0 < t, it was located in a volume element DV0 centered at the point X0 6¼ X. Actually, we consider the set of all events C, where C denotes an event of finding the particle in the volume element DV at the time t. The particle could get into this element from any initial spatial position (with different probabilities), but we are interested only in some of those positions, that is, in a subset B of the set A. The probability of such an event is called a conditional probability. Conditional probability is defined as the probability of realization of an event o 2 A under the condition that o 2 B and is equal to PðAjBÞ ¼ PðA \ BÞ=PðBÞ:

ð1:2Þ

The theory of stochastic processes is based (to a considerable degree) on the notion of joint probability. In this context, let us mention an important property of the joint probability. Suppose the full set O is divided into non-overlapping subsets Bi, that is, Bi \ B j ¼ f

[ Bi ¼ W:

and

i

As far as UðA \ Bi Þ ¼ AIðU Bi Þ ¼ A \ W ¼ A i

i

and (see axiom 3) X PðA \ Bi Þ ¼ Pð [ ðA [ Bi ÞÞ ¼ PðAÞ i

i

i

we find from (1.2): X PðAjBi ÞPðBi Þ ¼ PðAÞ:

ð1:3Þ

i

Thinking of the subset Bi as a variable, one can see from the last relation that summation over all mutually exclusive possibilities (i.e. over all sets Bi) eliminates this variable from the outcome.

j3

4

j 1 Basic Concepts of the Probability Theory Yet another important notion is the notion of independent events. Two sets of events A and B are called independent if the probability for an event to belong to set A and the probability to belong to set B are not correlated. Then PðA \ BÞ ¼ PðAÞPðBÞ:

ð1:4Þ

1.2 Random Variables, Probability Distribution Function, Average Value, and Variance

The concept of a random variable is of primary importance in stochastic processes. A random variable F(X) is defined as a function of the element X of the space of probabilistic events X. An event is specified by X, so X now stands for the event previously denoted by o. The examples of random variables include position, momentum, and spatial orientation of a particle driven by random external forces (Brownian motion, motion in a turbulent flow). The introduction of a random variable notation simplifies operations with functions of random variables, calculations of random variable distributions, of averages and other statistical characteristics of distributions. Furthermore, the introduction of continuous random variables enables us to operate with stochastic differential equations and study the change of random variables in space and in time in the same way as we study deterministic systems by using differential equations. The frequency (or probability) of realization of a definite event is equal to some value between zero and one. If the events are mutually exclusive, the sum of probabilities must be equal to one. This means that one of the events will realize. Statistical mechanics is usually concerned with continuous random variables, that is, variables that can assume a continuous range of values. As far as the probability to get any given value from a continuum of possible values is zero, and the sum of all probabilities is one, it is necessary to look at the probability of realization of an event that is associated with an infinitesimal interval (set) of values rather than a single value. This probability is also an infinitely small quantity having the same order as the length of the interval (measure of event) and so is proportional to the measure of events, that is, to dX. Thus the probability that a random variable is contained in the interval (X, X + dX) can be represented as PðX 2 ðX ; X þ dX ÞÞ ¼ pðX ÞdX :

ð1:5Þ

The function p(X) is called the probability density function (PDF) or simply the probability density. The condition that the sum of probabilities for a continuous random variable is equal to one can be written in the integral form: ð pðX ÞdX ¼ 1;

ð1:6Þ

X

where X is the domain of the n-dimensional space in which X varies. The relation (1.6) can be interpreted as the normalization condition for the PDF.

1.3 Generalized Functions

The introduction of a PDF enables us to find statistical characteristics of the distribution of a random variable X. The most important of them is the average value (aka mean value, or expectation value) of a random variable or random function: ð h f i ¼ f ðX Þ pðXÞdX :

ð1:7Þ

X

If X is a vector in an n-dimensional coordinate space, then (1.6) and (1.7) can be written in the coordinate form: ð¥

ð¥ ð¥ ... ¥¥

f ðX1 ; X2 ; . . .; Xn Þ pðX1 ; X2 ; . . .; Xn ÞdX1 ; dX2 ; . . .; dXn ¼ 1: ¥

ð¥ ð¥ hf i ¼

ð¥ ...

¥¥

f ðX1 ; X2 ; . . .; Xn Þ pðX1 ; X2 ; . . .; Xn ÞdX1 ; dX2 ; . . .; dXn : ¥

Another statistical characteristic is the variance s2. For a one-dimensional space, the variance is defined as ð s2 ¼ ðf h f iÞ2 pðXÞdX:

ð1:8Þ

X

The square root s of the variance is called the standard deviation. Sometimes the PDF has the form of a function with a sharp peak at the point X ¼ X0. In limiting case it is infinite at X ¼ X0 and zero at X 6¼ X0. Such a case arises when we idealize a process. For example, we can choose to regard a mass that is continuously distributed in a small volume element centered at X0 as localized at one space point X0 (i.e. as ‘‘point mass’’). Than the density of the substance differs from zero only at this point and the integral (1.7) has the meaning of the total mass. A similar reasoning leads to the concept of a point force – the net force with which we replace a force that is continuously distributed over a small volume element. To ensure the existence of integrals of such functions, we have to extend the notion of a function, what is achieved by the introduction of generalized functions.


The simplest and most extensively used generalized function is Diracs delta function d(X X0), which can be defined as the limit of following sequence (sometimes referred to as ‘‘delta sequence’’):

j5

6

j 1 Basic Concepts of the Probability Theory dðX X 0 Þ ¼ lim

m!¥

m pffiffiffi p

n expðm2 ðX X 0 Þ2 Þ:

ð1:9Þ

Here n is the number of dimensions and, accordingly, X is an n-dimensional vector with components X1, X2, . . ., Xn. Eq. (1.9) can also be written for one-dimensional sequences of Xi Xi0 . Then the following identity will hold: dðX1 X10 ÞdðX2 X20 Þ. . .dðXn Xn0 Þ ¼ dðX X 0 Þ:

ð1:10Þ

The limit on the right-hand side of Eq. (1.9) is 0 at X 6¼ X0 and þ1 at X ¼ X0. Therefore Diracs delta function is not a function in the usual sense and should not be interpreted as giving the value of the dependent variable at each point. What is important, however, is that this function is still integrable, and behaves similarly to ordinary functions in its capacity as an integrand. In particular, the integral of the scalar product of Diracs delta function d(X X0) and an ordinary function j(X) equals ð ðdðX X 0 Þ; jðX ÞÞ ¼ dðX X 0 ÞjðX ÞdX ð n m pffiffiffi expðm2 ðX X 0 Þ2 ÞjðX ÞdX ¼ jðX 0 Þ m!¥ p

X

¼ lim

X

provided the domain contains the point X0. Thus, by its definition, the Delta function has two basic properties: dðX Þ ¼

0; 1;

for X ¼ = 0; for X ¼ 0;

ð1:11aÞ

ð dðX X 0 ÞjðX ÞdX ¼ jðX 0 Þ:

ð1:11bÞ

X

In the particular cases j(X) ¼ 1 and j(X) ¼ X one gets: ð dðX X 0 ÞdX ¼ 1:

ð1:12Þ

X

and ð dðX X 0 ÞX dX ¼ X 0 :

ð1:13Þ

X

Hence, according to Eq. (1.7), d(X X0) can be taken as a PDF such that the random variable X has the average value X0. For the one-dimensional case, the following equality can be written:


ðXX0 ÞdðX X0 Þ ¼ 0 or X dðXX0 Þ ¼ X0 dðX X0 Þ:

ð1:14Þ

Taking j(X) ¼ (X X0)2, we can write ð dðX X 0 ÞðX X 0 Þ2 dX ¼ ðX 0 X 0 Þ2 ¼ 0:

ð1:15Þ

X

The left-hand side of (1.23) coincides with the definition of the variance for the PDF d(X X0). Thus, its variance is zero, and the delta function describes the case when one knows for sure that hXi ¼ X0. The (one-dimensional) Cauchy sequence is not the only sequence converging to the delta function. For example, the sequence e

dðX X0 Þ ¼ lim

e ! 0 pðX X0 Þ2

ð1:16Þ

þ e2

can be used as an alternative representation of the delta function. The delta function is an infinitely differentiable function. Its derivative can be defined by differentiating the integral ð jðX Þ

d dðX X0 ÞdX dX

by parts and using the property (1.11): ð

ð d 0 ðX X0 ÞjðX ÞdX ¼ dðX X0 Þj0 ðX ÞdX ¼ j0 ðX0 Þ;

X

ð1:17Þ

X

from which there follows a useful symbolic equality d 0 ðXÞ ¼

dðX Þ ; X

ðX ¼ = 0Þ;

In the more general case, one can write d ðrÞ ðXÞ ¼ ð1Þr

dðX Þ ; Xr

ðX ¼ = 0;

r ¼ 0; 1; . . .Þ:

ð1:18Þ

If a function Y ¼ f(X) is single-valued, that is, if it can be solved with respect to X in a unique way, then X ¼ f1(Y) and

j7

8

j 1 Basic Concepts of the Probability Theory dðY f ðX ÞÞ ¼

dðX f 1 ðYÞÞ : jd f =dX j

ð1:19Þ

A similar relation takes place for a vector function Y ¼ f (X): dðY f ðX ÞÞ ¼

dðX f 1 ðY ÞÞ : D

ð1:190 Þ

where D is the determinant of the Jacobian ||@fi/@Xj||. The delta function can be connected with the unit step function (Heaviside function) defined as HðX Þ ¼

0; 1;

for X < 0; for X > 0

through the symbolic relation dðX Þ ¼

dH : dX

ð1:20Þ

If there is more then one independent variable, one has to use partial derivatives of the delta function. For example, if we take the delta function as a generalized vector function d(X X0), its gradient is defined as qdðX X 0 Þ rd ¼ ¼ qX

qd qd qd ; ; . . .; : qX1 qX2 qXn

ð1:21Þ

1.4 Methods of Averaging

When looking at the hydrodynamic characteristics of a turbulent flow or at the motion of particles under the action of random external forces, we notice one distinguishing feature shared by these two types of motion: the presence of random fluctuations. Because of fluctuations, the dependences of hydrodynamic field parameters on spacetime coordinates, and the configuration of particles in space at different moments look irregular and have a confusing pattern. If a process is repeated multiple times under the identical set of initial and boundary conditions, the observed values of field parameters and particle positions will be different. This necessitates the use of averaging methods in any study of random motions. Averaging allows us to make a transition from irregular characteristics to much more smooth and regular mean values. In practice the mean value is determined by averaging over the time interval,


h f ðX ; tÞiT ¼

1 T

T=2 ð

wðX ; tÞ f ðX ; t þ tÞdt;

ð1:22Þ

T =2

or by averaging over the considered spatial region, h f ðX ; tÞiV ¼

ð 1 wðx; tÞ f ðX þ x; tÞdx V

ð1:23Þ

V

or, most generally, by spacetime averaging, 1 h f ðX ; tÞiTV ¼ VT

ð ð T=2 wðx; tÞ f ðX þ x; t þ tÞdxdt;

ð1:24Þ

V T=2

where o(x, t) is the weight function. We can also introduce the autocorrelation function C(t), which is defined as follows: take a random function f(t) at one and the same point of space but at instances of time t and t + t, form the product, and find its average value over the time interval (0, T) for T ! 1: 1 T !¥ T

ðT

YðtÞ ¼ lim

f ðtÞ f ðt þ tÞdt:

ð1:25Þ

0

This function plays an important role in many applications. The three types of averaging mentioned above have one drawback, namely, they apply to only one instance of the process under consideration (turbulent velocity field, etc.). Another shortcoming is that one is faced with the problem of choosing the most convenient weight function. If the process is repeated multiple times under the same initial conditions, we are dealing with many instances of the same process. In this case one can talk about a statistical set of identical processes (flows, particle motions etc.) taking place under fixed initial and external conditions. Let one and the same experiment be replicated N times under the same conditions, yielding different values ui of one and the same parameter, for example, velocity u. By averaging the velocities ui observed in a discrete set of similar tests, we obtain the mean value huðX ; tÞi ¼

N 1X ui ; N i¼1

which is called the ensemble average. In many cases the ensemble average proves to be stable enough, in other words, the outcomes of a sufficiently large set of experiments show a very small variance.

j9

10

j 1 Basic Concepts of the Probability Theory Let a continuous random variable u (1 < u < 1) be characterized by the PDF p(u). If we are interested in the value of u at one and the same space point M, then p(u)du signifies the probability for u to be found in interval (u, u + du). Then the ensemble average of u is ð¥ huðX ; tÞi ¼

u pðuÞ du:

ð1:26Þ

¥

By analogy, the ensemble average of any function F is equal to ð¥ hF ðuðX ; tÞÞi ¼

FðuÞ pðuÞdu:

ð1:27Þ

¥

Now, let u be measured at different spacetime points M1 ¼ (X1, t1), M2 ¼ (X2, t2), . . . , MN ¼ (XN, tN). The resulting values of u are denoted by u1, u2, . . . , uN. We then introduce the N-dimensional PDF p(u1, u2, . . . , uN), where p(u1, u2, . . . , uN)du1du2 . . . duN means the probability of finding ui in the interval (ui, ui + dui) . We have used a common convention where the index stands for all N variations, that is ðui ; ui þ dui Þ ¼ ðu1 þ du1 ; . . . ; un þ dun Þ. The average of any function will be written as ð¥ ð¥ hF i ¼

ð¥ ...

¥¥

Fðu1 ; u2 ; . . . ; uN Þ pðu1 ; u2 ; . . . ; uN Þdu1 du2 . . . duN :

ð1:28Þ

¥

By introducing an N-dimensional vector u(u1, u2, . . . , uN), we can rewrite the relation (1.28) in a more compact form: ð¥ hF i ¼

FðuÞ pðuÞdu:

ð1:29Þ

¥

Multidimensional PDFs are especially important for studying the behavior of an N-particle system in a random field of external forces. If ui denotes the coordinate of the i-th particle, then the above-introduced PDF is called a multiparticle PDF. Oneparticle and two-particle PDFs are of particular interest in applications. Sometimes the two-particle PDF is also called ‘‘pair PDF’’ or ‘‘pair distribution’’. These PDFs can be derived from multidimensional PDFs by integrating them over all possible positions of the remaining particles. For instance, a single-particle PDF is obtained as ð¥ ð¥ pðX1 Þ ¼

ð¥ ...

¥8

pðX1 ; X2 ; . . .; XN ÞdX2 . . .dXN ¥

Such PDFs are also called marginal PDFs.

ð1:30Þ


If we consider spherical particles of different radii ai, than the PDF p(X1, . . ., XN, a1, . . . , aN) will be associated with the radius distribution in addition to the coordinate distribution, and p(X1, . . . , XN, a1, . . . , aN) dX1 . . . dXN da1 . . . daN will have the meaning of probability to find the N-particle system in the volume element (dX1 . . . dXN) with particle radii lying in the interval (a1 + da1), . . . , (aN + daN). The corresponding single-particle PDF is ð¥ ð¥ pðX 1 ; a1 Þ ¼

pðX1 ; X 2 ; . . .; X N ; a2 ; . . .; aN ÞdX 2 . . .dX N da2 . . .daN :

ð1:31Þ

¥ 0

If the radius a must be the same for all particles, then it is convenient to operate with particle distribution over the radius nðX ; aÞ ¼ N pðX ; aÞ;

ð1:32Þ

such that n(X, a)da is the probabilistic numerical concentration (aka number concentration) of particles with radius in the interval (a + da) in the volume element dX. The multidimensional PDF should satisfy the following properties: 1. Ðp(u) 0; 1 2. 1 pðuÞdu¼ 1; 3. pðu1 ; u2 ; . . . ; uN Þ ¼ pðui1 ; ui2 ; . . . ; uiN Þ, where the set {i1, i2, . . ., iN} is formed from the set {1, 2, . . ., N} by changing the order. 1 ð ð 1

4. pðu1 ; u2 ; . . . ; un Þ ¼

1 ð

... 11

pðu1 ; u2 ; . . . ; 1

un ; unþ1 ; . . . ; uN Þdunþ1 . . . duN for n < N; 5. For independent random variables u1 ; u2 ; . . . ; uN , there holds: pðu1 ; u2 ; . . .; uN Þ ¼ pðu1 Þ pðu2 Þ; . . .; pðuN Þ

ð1:33Þ

Property 3 is known as the symmetry property and property 4 – as the consistency property. It is now time to discuss the connection between different types of averaging. In practice, we use time or space averaging rather than ensemble averaging, because the latter requires a large number of experiments. In Statistical Mechanics, ensemble averaging, that is, averaging over the set of all possible states, is often replaced by time or space averaging, with the implicit assumption that by increasing the averaging interval we can always make the average values converge to the corresponding ensemble averages. This assumption is called the ergodic hypothesis, or, in those special cases when it can be rigorously proved, the ergodic theorem. When studying such problems as the flow of a disperse medium or the filtration of a fluid through a porous medium, one often uses the so-called Saffman step

j11

12

j 1 Basic Concepts of the Probability Theory function: HðX Þ ¼

0; 1;

for X for X

in rigid body; in fluid:

ð1:34Þ

This function depends on statistical parameters of the distribution of moving particles of the disperse phase or fixed particles of the porous medium. After averaging over the particle ensemble, we get hHðX Þi ¼ 1j;

ð1:35Þ

where j is volume concentration of particles. One can use the Saffman function to perform space averaging of hydrodynamic parameters. For example, the velocity of the fluid u will be averaged as ¼ hHui=hHi ¼ hui=ð1jÞ; u

ð1:36Þ

where u is the mean-flow-rate velocity through the microcapillaries of the porous medium. It should not be confused with the ensemble average hui, although for a highly permeable medium (j 1), the two velocities are equal: hui¼ u.

1.5 Characteristic Functions

Instead of using the PDF p(u1, u2, . . . , uN), it is often convenient to use its Fourier transform: ð¥ ð¥ jðr1 ; r2 ; . . .; rN Þ ¼

ð¥ ...

¥¥

¥

(

N X exp i rk uk

) pðu1 ; u2 ; . . .; uN Þdu1 du2 . . .duN

k1

or, in the vector form, ð¥ jðrÞ ¼

eiru pðuÞdu:

ð1:37Þ

¥

Here r is an N-dimensional vector with components (r1, r2, . . . , rN). The function j(r) is called the characteristic function or the moment-generating function. Because of Eq. (1.29), it can be represented as jðrÞ ¼ eiru :

ð1:38Þ

1.5 Characteristic Functions

If the characteristic function is known, then the PDF is obtained as the inverse Fourier transform: pðuÞ ¼

ð¥

1 ð2pÞN

eiru jðrÞdr:

ð1:39Þ

¥

So, the knowledge of the characteristic function is tantamount to the knowledge of the PDF. Hence the properties of the PDF are readily obtained from those of the characteristic function. The normalization condition for the PDF means that wð0Þ ¼ 1:

ð1:40Þ

For independent random variables we have, according to (1.33): jðrÞ ¼ jðr1 Þjðr2 Þ. . .jðrN Þ:

ð1:41Þ

The symmetry and consistency properties of characteristic function follow from properties 3 and 4 of the PDF: jðr1 ; r2 ; . . .; rN Þ ¼ jðri1 ; ri2 ; . . .; riN Þ;

ð1:42Þ

jðr1 ; r2 ; . . .; rn Þ ¼ jðr1 ; r2 ; . . .; rn ; 0; 0; . . .; 0Þ;

ð1:43Þ

where i1, i2, . . . , iN is any combination of non-repeating numbers 1, 2, . . . , N. In the last relation, n < N and the number of zeros is equal to N n. The property (1.57) allows us to obtain the characteristic function for a smaller number of dimensions (smaller number of particles) from the N-dimensional (N-particle) characteristic function, and then to get the corresponding marginal PDF by using the inverse Fourier transform (1.39). Therefore one can specify all PDFs, describing random variables at all possible points, through a single characteristic function known as the characteristic functional. In particular, for one-dimensional random function u(X) defined on a finite interval a X b, the characteristic functional is 8b 9+ 0;

ð1:63Þ

i¼1 j¼1

for all real Xi, non-negative integer n and any selection of points M1, M2, . . . , Mn. In particular, at n ¼ 2 the expression (1.63) becomes jBuu ðM1 ; M2 Þj jBuu ðM1 ; M1 Þj1=2 jBuu ðM2 ; M2 Þj1=2 :

ð1:64Þ

In addition to the above-mentioned two-point moments of one random function at different points, u(M1) and u(M2), one can consider two-point moments of different random functions, u(M1) and v(M2). A mixed two-point moment Buv (M1, M2) is called the mutual correlation function. Its properties are similar to those of ‘‘ordinary’’ moments. For example, the symmetry property still holds: Buv ðM1 ; M2 Þ ¼ Bvu ðM2 ; M1 Þ:

ð1:65Þ

j17

18

j 1 Basic Concepts of the Probability Theory Two-point moments of orders higher than two are referred to as correlation functions of higher order. By analogy, one can define two-point central moments of the second order: buu ¼ hðuðM1 ÞhuðM1 ÞiÞðuðM2 ÞhuðM2 ÞiÞi ¼ Buu ðM1 ; M2 ÞhuðM1 ÞihuðM2 Þi; buv ¼ hðuðM1 ÞhuðM1 ÞiÞðvðM2 ÞhvðM2 ÞiÞi ¼ Buv ðM1 ; M2 ÞhuðM1 ÞihvðM2 Þi

ð1:66Þ

ð1:67Þ

The variances of distributions of random variables u and v can be expressed through central moments: s2u ðMÞ ¼ buu ðM; MÞ;

s2v ðMÞ ¼ bvv ðM; MÞ:

ð1:68Þ

Two-point central moments of the second order relate the deviations of random functions from their mean values (i.e. fluctuations) at two different points. This is why they are also called correlation functions of fluctuations. Another important statistical parameter is the correlation coefficient, defined as Yuu ðMÞ ¼

buu ðM1 ; M2 Þ ; su ðM1 Þsu ðM2 Þ

Yuv ðMÞ ¼

buv ðM1 ; M2 Þ : su ðM1 Þsv ðM2 Þ

ð1:69Þ

As a consequence of the Schwartz inequality, these coefficients satisfy |cuu| 1 and |cuv| 1. If the correlation coefficient vanishes, the correlation between fluctuations at different spatial points is absent. An important property that follows from physical considerations is the damping of correlation between random variables at different spacetime points as we increase the distance between the points. As the distance goes to infinity, the correlation function will tend to zero. Of course, the ‘‘distance’’ that goes to infinity can be the geometrical distance, |X2 X1| ! 1 at t2 ¼ t1, the temporal distance (time interval), |t2 t1| ! 1 at X2 ¼ X1, or both.

1.8 Bernoulli, Poisson, and Gaussian Distributions

Let us consider the three distributions that are most frequently used in physical applications – the Bernoulli, Poisson, and Gaussian (normal) distributions. Bernoulli distribution To begin, let us formulate the random walk problem, which will be considered in more detail farther on, in the chapter dedicated to Brownian motion. A particle undergoes a sequence of random displacements along a straight line. Each displacement is a step of the same length 1, and each step can be directed


either forward or backward with the same probability of 0.5. The origin of the reference system is coincident with the initial position of the particle. Then the particles coordinate can assume only integer values . . . N, N þ 1, . . . , 0, 1, . . . , N 1, N . . .. The probability of finding the particle at a point m after N steps is given by the Bernoulli distribution, N Pðm; NÞ ¼ CðNþmÞ=2

N ¼ where CðNþmÞ=2

N 1 ; 2

N!

ð12ðNþmÞÞ!ð12ðNmÞÞ!

ð1:70Þ are binomial coefficients.

The mean and root-mean-square displacements of the particle are, respectively, pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi hm 2 i ¼ N :

hmi ¼ 0;

In the limiting case where N 1 and m N this reduces to an asymptotic formula Pðm; NÞ

2 pN

1=2

m2 exp : 2N

ð1:71Þ

Poisson distribution Let N particles be randomly distributed in a volume V. Then the probability of finding n particles in a volume element v, where v is a small part of V, is given by the Bernoulli distribution

PN ðnÞ ¼

v n

N! v Nn 1 : n!ðNnÞ! V V

ð1:72Þ

For a given N, V and v, the mean value of n equals hni ¼ N

v V

n:

In the limiting case where N ! 1 and V ! 1 but n remains finite, the distribution (1.72) tends asymptotically to the Poisson distribution PðnÞ ¼

nn en : n!

ð1:73Þ

If n is large and n is of the same order as n, the Poisson distribution is close to the distribution PðnÞ ¼

1 2pn

1=2

! ðnnÞ2 exp : 2n

ð1:74Þ

j19

20

j 1 Basic Concepts of the Probability Theory Gaussian (normal) distribution The distributions (1.88) and (1.92) are both special cases of the Gaussian (aka normal) distribution. In the general N-dimensional case this distribution has the following normalized form:

(

) N X N 1X pðu1 ; u2 ; . . .; uN Þ ¼ C exp g jk ðu j a j Þðuk ak Þ ; 2 j¼1 k¼1

ð1:75Þ

Here aj are real numbers; gjk are the elements of the positive definite matrix ||gjk||; C ¼ g1/2/(2p)N/2 is a constant that is given by the normalization condition for probability density (see Eq. (1.6); g ¼ |gjk| is the determinant of the matrix ||gjk||. The constants aj and gjk are related to the first and second moments of the distribution (1.75) (see Eq. (1.46) and Eq. (1.47): u j ¼ a j;

g jk b jk ¼ ðu j u j Þðuk huk iÞ ¼ g

ð1:76Þ

Here Gjk ¼ @g/@gjk is the algebraic complement of the element gjk in the determinant g. It means that the matrices ||gjk|| and ||bjk|| are mutually inverse. The Gaussian distribution can also represented in the matrix form: pðuÞ ¼

1 ð2pÞN=2 b1=2

1 ðuhuiÞT b1 ðuhuiÞ ; 2

ð1:77Þ

where b ¼ ||bjk|| and b ¼ |bjk|. The ordinary second-order moments Bjk can be expressed in terms of the normal distribution parameters according to Eq. (1.76): G jk þ a j ak : B jk ¼ u j uk ¼ G

ð1:78Þ

We see from Eq. (1.76) and Eq. (1.77) that the first two moments completely determine the PDF, and thereby the entire statistics of random variables, for a normal distribution. Hence the knowledge of mean values and correlation functions provides a complete statistical description of a random Gaussian field u(M) ¼ [u1(M), u2(M), . . . , uN(M)]. Central moments can be obtained from the property of normal distributions, which states that all central moments of an odd order are zero, whereas central moments of an even order are expressed through central moments of the second order: D E bk1 k2 ...kN ¼ ðu1 hu1 iÞk1 ðu2 hu2 iÞk2 . . .ðuN huN iÞkN X ¼ bi1 i2 bi3 i4 . . .bi2K1 bi2K ;

ð1:79Þ

where k1 þ k2þ . . . þkN ¼ 2K and subscript pairs are formed from numbers 1, 2, . . . 2K so that the first index is less than the second, for example,


b1111 ¼ hðu1 hu1 iÞðu2 hu2 iÞðu3 hu3 iÞðu4 hu4 iÞi ¼ b12 b34 þ b13 b24 þ b14 b23 : When studying random variables described by a normal distribution it is convenient to use characteristic functions because of their simple form (see Eq. (1.37)): ð¥ ð¥ jðr1 ; r2 ; . . .; rN Þ ¼

(

ð¥ ...

¥¥

(

exp i

N X

) rk uk

pðu1 ; u2 ; . . .; uN Þdu1 du2 . . .duN

k¼1

¥

) N X N 1X ¼ exp i ak rk b jk r j rk : 2 j¼1 k¼1 k¼1 N X

ð1:80Þ

Cumulants can be obtained from Eq. (1.53), using Eq. (1.80). For a Gaussian distribution, the cumulants of the first and second orders are respectively equal to aj and bjk whereas cumulants of higher orders are identically equal to zero. By using characteristic functions, one can prove that any linear combination of Gaussian random variables will also result in a Gaussian distribution. Gaussian distributions are of great importance in applications due to a number of reasons. First, the behavior of many random variables is well approximated by a Gaussian distribution. Secondly, according to the central limit theorem, a random variable that is a sum of a large number of independent components with arbitrary distributions (which is the most common situation in statistical mechanics) is Gaussian. Let us consider a one-dimensional Gaussian distribution 1 u2 pðuÞ ¼ pffiffiffiffiffiffi exp 2 ; ðs2 ¼ u2 Þ: ð1:81Þ 2s 2ps The characteristic function follows from the relations (1.37) and (1.52): 2 2 r s ; jðrÞ ¼ exp 2

r2 s2 2

ð1:82Þ

Sn > 2 ¼ 0:

ð1:83Þ

yðrÞ ¼

Then Eqs. (1.57)–(1.58) yield B1 ¼ S1 ¼ 0;

B2 ¼ S2 ¼ s2 ;

The recurrent relation (1.59) takes the form Bn ¼ ðn1Þs2 Bn2 ;

ð1:84Þ

from which there follows B2nþ1 ¼ 0;

B2n ¼ ð2n1Þ!!s2n :

ð1:85Þ

j21

22

j 1 Basic Concepts of the Probability Theory Consider yet another average value hXf (X)i, which is helpful in many applications. Here X is a Gaussian random variable given by Eq. (1.81) and f (X) is an arbitrary deterministic function. We make a further assumption that f (X) exp(X2/2s2) ! 0 for X ! 1 (i.e. the exponent dominates at large values of X). Then

1 hX f ðXÞi ¼ pffiffiffiffiffiffi 2ps

ð¥

X2 X f ðX Þexp 2 dX: 2s

¥

Integrating by parts, we arrive at the following expression:

s hX f ðX Þi ¼ pffiffiffiffiffiffi 2p

ð¥ ¥

d f ðX Þ X2 d f ðX Þ exp 2 dX ¼ s2 : 2s dX dX

ð1:86Þ

A similar expression can be obtained for a Gaussian random vector X ¼ (X1, X2, . . . , XN) with a multidimensional distribution given by Eq. 1.75:

h Xi f ðX Þi ¼ Bi j

d f ðX Þ ; dX

ð1:87Þ

where Bij ¼ hXiXji are components of the correlation matrix.

1.9 Stationary Random Functions, Homogeneous Random Fields

When discussing the problem of random variable averaging in Section 4, we mentioned the ergodic hypothesis, which states that as we increase the temporal or spatial averaging interval to infinity, the corresponding mean values tend to the ensemble average. For the ergodic hypothesis to be valid, some necessary conditions should be satisfied. We thus arrive to a special class of random fields u(X, t) satisfying the ergodicity conditions. These fields are frequently encountered in Statistical Mechanics, in particular, in problems that involve Brownian motion and turbulence. Consider first the time averaging of a function u(t), written for simplicity as a function of one variable because its dependence on space coordinates X is of no relevance to the problem. The time average will be denoted by huiT. Then, in accordance with Eq. (1.22), 1 huðtÞiT ¼ T

T=2 ð

uðt þ tÞdt: T=2

ð1:88Þ

1.9 Stationary Random Functions, Homogeneous Random Fields

According to the ergodic hypothesis, hu(t)iT should tend to the ensemble average hu(t)i at T ! 1. For this to happen, the following simple relation must take place: huðtÞiT ¼ U ¼ const:

ð1:89Þ

This condition can be derived by considering the difference between the average values of the random variable calculated at different moments, t and t1, where t1 > t:

huðtÞiT huðt1 ÞiT ¼

8 > T=2 ð 1
=

uðt þ tÞdt uðt1 þ tÞdt > T> ; : T=2 T=2 9 8 T=2þt > >T=2þt ð 1 ð 1 = 1< ¼ uðsÞds uðsÞds : > T> ; : T=2þt

ð1:90Þ

T=2þt

At T ! 1 the right-hand side of Eq. (1.90) goes to zero, thus giving rise to the condition (1.89). Similarly, by time-averaging the product u(t)u(t1) ¼ u(t)u(t + s), where s ¼ t1 t, and letting T go to 1, we conclude that the time average of the correlation function (Buu(t, t1))T can be equal to its ensemle average (Buu(t, t1)) ¼ hu(t)u(t1)i only if for any two instants of time t1 and t2, where t2 > t1, the following condition is satisfied: Buu ðt2 ; t1 Þ ¼ Buu ðt2 t1 Þ:

ð1:91Þ

For a moment of N-th order, this condition takes the form Buu...u ðt1 ; t2 ; . . .; tN Þ ¼ Buu ðt2 t1 ; . . .; tN t1 Þ:

ð1:92Þ

In order for the ensemble averages of random values u(t1), u(t2), . . . , u(tN) to be obtainable by time averaging, it is necessary to consider only those random functions u(t) for which the N-dimensional PDF (at any N and t1, t2, . . . , tN) will depend on N 1 parameters t2 t1, t3 t1, . . . , tN t1, rather than on N parameters t1, t2, . . . , tN. In other words, the PDF must satisfy the condition pi1 ;...;iN ðu1 ; u2 ; . . .; uN Þ ¼ pt2 t1 ;...;tN t1 ðu1 ; u2 ; . . .; uN Þ:

ð1:93Þ

It should be noted that the condition (1.93) leads to the conditions (1.98), (1.91) and (1.92), and if the random function is Gaussian, then from Eq. (1.89) and Eq. (1.91) one can derive the properties (1.92) and (1.93). The condition (1.93) describes a class of random functions whose PDF does not vary as we shift the time ti by any time interval. Such functions are called stationary random functions or stationary random processes. One example is a steady turbulent flow, whose average characteristics (velocity, pressure, temperature, etc.) do not change with time. Any hydrodynamic parameter u(u1(t), u2(t), . . . , uN(t)), for example, flow velocity

j23

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j 1 Basic Concepts of the Probability Theory at different space points, whose PDF for any set of ui1 ðt1 Þ; ui2 ðt2 Þ; . . . ; uiN ðtN Þ does not vary as we shift all the instants of time t1, t2, . . . , tN by one and the same value, represents a multidimensional stationary random process. Then all the mixed moments of functions u(t) will also depend only on the differences between the corresponding instants of time. For example, all mutual correlation functions Bjk (t1, t2) ¼ huj(t1)uk(t2)i depend only on the time difference t ¼ t2 t1. Consider now the space averaging of a random function u(X), where X(X1, X2, X3) is a space point. The space average (recall the definition (1.23)) is equal to 1 huðX ÞiABC ¼ ABC

A=2 ð B=2 ð C=2 ð

uðX1 þ x1 ; X2 þ x2 ; X3 þ x3 Þdx1 dx2 dx3 :

ð1:94Þ

A=2B=2C=2

By analogy with time averaging, we can find the conditions that must hold in order for hu(X)iABC to coinside with the ensemble average hu(X)i at A ! 1, B ! 1, C ! 1 (or when at least one of the intervals A,B,C goes to the limit). It is evident that the necessary conditions would be relations similar to (1.89), (1.91)–(1.93) with t replaced by X: huðX Þi ¼ U ¼ const;

ð1:95Þ

Buu ðX 1 ; X 2 Þ ¼ Buu ðX 2 X 1 Þ;

ð1:96Þ

pX1 ;X2 ;...;XN ðu1 ; u2 ; . . .; uN Þ ¼ pX2 X1 ;...;XN X1 ðu1 ; u2 ; . . .; uN Þ:

ð1:97Þ

where Buu (X1, X2) ¼ hu(X1)u(X2)i. A random field u(X) satisfying the conditions (1.95)–(1.97) is called a statistically homogeneous field. Thus, in order for the space averaging of a function of random variables to produce the same results as ensemble averaging, it is necessary for the field u(X) to be homogeneous. Parameters of a homogeneous turbulent flow (velocity, pressure, temperature, etc.), which do not depend on spatial coordinates, are good examples. It is clear that homogeneity of the flow cannot be realized in the entire flow region, because any flow is always restricted by boundaries, and the flow near the boundary is essentially inhomogeneous. In reality, the property of homogeneity can be realized only far enough from the boundary. It should be mentioned that generally speaking, the conditions of stationarity and homogeneity are not sufficient for the convergence of time and space averages to ensemble averages. The necessary and sufficient conditions are formulated by ergodic theorem. Namely, it is necessary and sufficient to ensure the fulfillment of the following condition for the correlation function of fluctuations buu(t): ðT 1 buu ðtÞdt ¼ 0: T !¥ T lim

0

ð1:98Þ

1.10 Isotropic Random Fields. Spectral Representation

The necessary averaging interval T can be estimated from the corresponding correlation time T1, which is given by

T1 ¼

ðT 1 buu ðtÞdt: buu ð0Þ

ð1:99Þ

0

For sufficiently large T, the following asymptotic formula for root-mean-square deviation of the time average from the ensemble average is valid: D E hui hui 2 2 T1 buu ð0Þ: T T

ð1:100Þ

Eq. (1.99) allows us to determine the minimum averaging time for a given deviation of huiT from hui. In the case of spatial averaging, a similar estimation for the rootmean-square deviation of huiV from hui gives D E hui hui 2 2 V1 buu ð0Þ: V V

ð1:101Þ

Here huiV is the spatial (volume) average, and V1 is the correlation volume equal to 1 V1 ¼ buu ð0Þ

ð¥ ð¥ ð¥ buu ðrÞdr1 dr2 dr3 :

ð1:102Þ

¥¥¥


A scalar random field u(X) is called isotropic when all finite-dimensional PDFs pX1 ;X1 ;...;XN ðu1 ; u2 ; . . . ; uN Þ corresponding to this field are invariant under rotations of points X1, X2, . . . , XN around the axis passing through the origin of the coordinate system and under mirror reflections of this set of points relative to planes passing through the origin. In applications, random fields that are both homogeneous and isotropic present the greatest interest. Henceforth these fields will be called simply isotropic. Thus the term ‘‘isotropic field’’ will imply a field whose PDF pX1 ;X2 ;...;XN ðu1 ; u2 ; . . . ; uN Þ is invariant under parallel translations, rotations, and specular reflections of the set of points X1, X2, . . . , XN. The homogeneity condition (1.95) for the field u(X) means that its average value hu(X)i should be constant. This constant is often made equal to zero by replacing the initial field u(X) with the field u0 (X) ¼ u(X) hu(X)i. The correlation function B(X, X 0 ) ¼ hu(X)u(X 0 )i of an isotropic field has the same values at any pair of points (X, X 0 ) and ðX1 ; X01 Þ that would coincide after a combination of parallel translation and rotation. If the distance between the points X and

j25

26

j 1 Basic Concepts of the Probability Theory X 0 is the same as the distance between X1 and X01 , then BðX; X0 Þ ¼ BðX1 ; X01 Þ. Hence the correlation function B(X, X 0 ) depends only on the distance r between the points X and X 0 ¼ X þ r. Here r ¼ |X 0 X| ¼ |r|, and correlation function can be written as huðX ÞuðX 0 Þi ¼ BðrÞ:

ð1:103Þ

Application of harmonic (Fourier) analysis to random processes and random fields, that is, expansion of random functions as Fourier series (for functions defined on a finite domain) or Fourier integrals (for functions defined on an infinite domain) has proved to be a very successful approach. For any stationary random functions or homogeneous random fields, which by their definition cannot decay on the infinity, it is possible to carry out Fourier expansion (another common term is ‘‘spectral representation’’ or ‘‘spectral expansion’’). It has a clear physical meaning: superposition of harmonic oscillations (for stationary random processes) or plane waves (for homogeneous random fields). The integral representation of the correlation function of a homogeneous random field is ð BðrÞ ¼ eikr FðkÞdk; ð1:104Þ ð 1 FðkÞ ¼ 3 eikr BðrÞdr; 8p

ð1:105Þ

where F(k) is called the spectrum of the homogeneous field, and k is the wave vector. For an isotropic field, the condition (1.103) holds, so the spectrum depends on k ¼ |k| rather than on k. If we represent x, y, z in terms of spherical coordinates as x ¼ r sin y cos F, y ¼ r sin y sin F, z ¼ r cos y, the relation (1.105) will take the following form: FðkÞ ¼

¼

ð ð¥ ðp ðp 1 1 ikr BðrÞdr ¼ eikr cos BðrÞr 2 sin ddFdr e 8p3 8p3 1 2p2

ð¥ ¥

0 p 0

sinðkrÞ BðrÞr 2 dr ¼ FðkÞ: kr

ð1:106Þ

Similarly, ð¥ BðrÞ ¼ 4p ¥

sinðkrÞ FðkÞk2 dk: kr

ð1:107Þ

Instead of looking at F(k), we can consider the following statistical characteristic: ð ð FðkÞdSðkÞ; ð1:108Þ EðkÞ ¼ jkj¼k

where S(k) is a surface element of the sphere |k| ¼ k.


Putting r ¼ 0 into Eq. (1.104) and recalling Eq. (1.103), we get D E ð¥ Bð0Þ ¼ ½uðX Þ 2 ¼ EðkÞdk:

ð1:109Þ

0

If u is the velocity (for instance, the velocity of a turbulent flow), then B(0) stands for the total energy of the field u(X). Therefore E(k)dk has the meaning of the energy of plane waves with wave numbers in the interval (k, k + dk). The derivations above can be generalized for the case of an isotropic multidimensional random field u(X) ¼ (u1(X), u2(X), . . . , uN(X)) characterized by the correlation matrix jjBi j jj ¼ ui ðXÞu j ðX þ rÞ

ð1:110Þ

The components of such a matrix are functions of r ¼ |r|. Hence the spectral representation of this field will be written as ð¥ sinðkrÞ Bi j ðrÞ ¼ 4p Fi j ðkÞk2 dk; kr

ð1:111Þ

0

Fi j ðkÞ ¼

ð¥ 1 sinðkrÞ Bi j ðrÞr 2 dr: 2p2 kr 0

The above-formulated definition of an isotropic random field is valid for scalar random functions, for example, pressure p(X), temperature W(X), one-dimensional velocity u(X) and so on. In the case of vector random fields such as three-dimensional velocity, as well as for the fields given by a set of vector and scalar hydrodynamic parameters (for example, a field of three-dimensional velocity, pressure, and temperature), isotropy is defined in the following way. A random vector field u(X) is called isotropic if the PDF of the components of the vector u(X) taken at an arbitrary set of points X1, X2, . . . , XN is invariant under parallel translations, rotations, and mirror reflections of this set of points accompanied by rotation or mirror reflection of the coordinate system. Using the theory of invariants of the rotation-reflection group, we may conclude from this definition that the correlation tensor Bij(r) should be a linear combination of the constant invariant tensor dij (‘‘Kroneckers delta function’’) and the tensor rirj. The coefficients in this linear combination will depend on the only invariant that can be built from components of the vector r, that is, on the length r ¼ |r|: Bi j ðrÞ ¼ A1 ðrÞri r j þ A2 ðrÞdi j :

ð1:112Þ

j27

28

j 1 Basic Concepts of the Probability Theory 1.11 Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation

The term ‘‘stochastic process’’ implies that the time evolution of a system is described probabilistically. This means that there is a certain time-dependent random variable. Examples of stochastic processes include Brownian motion of a particle driven by a random force and the motion of particles suspended in a turbulent flow. The random variable is the spatial position X of the particle at different instants of time. One can measure the values X1, X2, X3, . . . at the instants of time t1, t2, t3, . . . and assume that there should exist a joint PDF p(X1, t1; X2, t2; X3, t3; . . .) such that p(X1, t1; X2, t2; X3, t3; . . .)dX1, dX2, . . . would give the probability for the particle to be located in the interval (X1 þ dX1) at the instant of time t1, in the interval (X2 þ dX2) at the instant t2, and so on. When the particle moves under the action of a rapidly fluctuating random force (in the case of Brownian motion this random force is the sum of interaction forces between the particle and the molecules of the surrounding fluid, or, to use a more casual term, the sum of collision forces), it can change its direction millions times per second. In this context, when considering two successive particle positions Xi and Xiþ1 at the instants ti and tiþ1 such that the time increment Dti ¼ tiþ1 ti is much smaller than the characteristic time of the process but large as compared to the time between successive collisions of the particle with the surrounding molecules, it is natural to suggest a model where the particles position Xiþ1 at the instant tiþ1 is determined by its position Xi at the previous instant ti and does not depend on the earlier instants of time . . . ,ti2, ti1. In other words, in the proces of a chaotic small-scale random walk, the particle forgets its past very quickly. Such processes are known as Markovian processes. Hence a Markovian process is a stochastic process characterized by the independence of the future from the past, where the past is defined as the set of all events observed up to the present instant of time t. In other words, one has to deal with random functions whose variations are statistically independent from one another. A Markovian process can be described by using the concept of conditional probability. Let us consider an ordered sequence of times t1 t2 t3 . . . t1 t2 . . . , where t1, t2, . . . belong to the past and . . . t3, t2, t1 – to the future. Let Y1, Y2, . . . denote the values of a random variable at the past instants of time t1, t2, . . . and . . . X3, X2, X1 denote its values at the future instants of time . . . t3, t2, t1. The PDF of the events X1, X2, . . . under the condition that the events Y1, Y2, . . . have already occurred (the conditional PDF) is then written as pðX 1 ; t1 ; X 2 ; t2 ; X 3 ; t3 ; . . .jY1 ; t1 ; Y2 ; t2 ; . . .Þ: In accordance with the Markovian principle, we demand that the conditional probability must be completely determined by the state of the system at the most recent instant of time, that is, by the knowledge of the random variable at t1. Then the following equality must be valid:

1.11 Stochastic Processes. Markovian Processes. The Chapman–Kolmogorov Integral Equation

pðX 1 ; t1 ; X 2 ; t2 ; X 3 ; t3 ; . . .jY1 ; t1 ; Y2 ; t2 ; . . .Þ ¼ pðX 1 ; t1 ; X 2 ; t2 ; X 3 ; t3 ; . . .jY1 ; t1 Þ:

ð1:113Þ

This relation means that any conditional probability can be expressed through an ordinary conditional probability of the type p(X1, t1|Y1, t1). Indeed, from the definition (1.2) of conditional probability we have: pðX 1 ; t1 ; X 2 ; t2 jY1 ; t1 Þ ¼ pðX 1 ; t1 jX 2 ; t2 ; Y1 ; t1 Þ pðX 2 ; t2 jY1 ; t1 Þ: Applying the postulate (1.113) to the first factor on the right-hand side, we express the joint PDF through ordinary conditional PDFs: pðX 1 ; t1 ; X 2 ; t2 ; jY1 ; t1 Þ ¼ pðX 1 ; t1 jX 2 ; t2 Þ pðX 2 ; t2 jY1 ; t1 Þ:

ð1:114Þ

Continuing this procedure, we obtain for N successive events: pðX 1 ; t1 ; X 2 ; t2 ; . . .; X N ; tN Þ ¼ pðX 1 ; t1 jX 2 ; t2 Þ pðX 2 ; t2 jX 3 ; t3 Þ . . . pðX N1 ; tN1 jX N ; tN Þ:

ð1:115Þ

As one could expect, the Markovian principle results in the independence of conditional pairs of successive events. From the consistency property (see property 4 in Section 4) of the PDF for two successive events X2 and X1 and from the relation (1.2) for the conditional probability there follows: ð pðX 1 ; t1 Þ ¼

ð pðX 1 ; t1 ; X 2 ; t2 ÞdX 2 ¼

pðX 1 ; t1 jX 2 ; t2 ÞpðX2 ; t2 ÞdX 2 :

ð1:116Þ

A similar relation can be written for the conditional probability: ð pðX 1 ; t1 jX 3 ; t3 Þ ¼

pðX 1 ; t1 ; X 2 ; t2 jX 3 ; t3 ÞdX 2 ð

¼

pðX 1 ; t1 jX 2 ; t2 ; X 3 ; t3 Þ pðX 2 ; t2 jX 3 ; t3 ÞdX 2 :

As far as t1 t2 t3, the Markovian principle allows to drop the dependence on X3 in the first factor of the integrand: ð pðX 1 ; t1 jX 3 ; t3 Þ ¼

pðX 1 ; t1 jX 2 ; t2 Þ pðX 2 ; t2 jX 3 ; t3 ÞdX 2 :

ð1:117Þ

The integral equation (1.117) is called the Chapman–Kolmogorov equation. This equation forms the basis of the theory of stochastic processes.

j29

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j 1 Basic Concepts of the Probability Theory When considering a Markovian process, it is important to know whether the range of the random value is continuous or discrete and whether the trajectory X(t) is a continuous function of t. As an example, consider rarefied gas molecules characterized by the velocity V(t) and by the position X(t). In this example, the velocity range is obviously continuous, but the function V(t) can be discontinuious, which happens when the interactions between molecules are modeled by the elastic collisions of rigid spheres. However, even in such a model, the position of a gas molecule X(t) remains a continuous function. In reality, molecules do not interact as rigid spheres. There exists a molecular interaction between them that is characterized by some interaction potential (for example, the Lennard–Jones potential). If we account for this potential, we will find that the molecules trajectory deflects continuously in the process of collision. The characteristic time of molecular collisions is extremely short. It is much shorter then the time intervals that make up a Markovian chain. It can be said that the Markovian method circumnavigates the issue of continuity of a random variable by approximating the real process on a large-scale time grid. Hence, irrespective of how the collision process is modeled, on large-scale time grid, the collision will always be marked by a velocity jump. By the same token, the trajectories are not necessarily continuous on this time grid. Another example is a chemical reaction that involves production consumption of molecules of a certain substance. The characteristic time of a chemical reaction is also very short as a rule. Therefore the random value, for example, molecular concentration, changes discontinuously on the large-scale time grid during the reaction. In this context, the following continuity condition looks quite self-intuitive: if for any e > 0 uniformly in Z, t and Dt there holds 1 Dt ! 0 Dt lim

ð pðX ; t þ DtjZ; tÞdX ¼ 0;

ð1:118Þ

jX Zj > e

then the realization of X(t) is continuous function of t, with the probability 1. It means that the probability that the position X differs from Z by a finite amount at Dt ! 0 goes to zero faster than Dt. This is known as the Lindenberg continuity condition for a random function X(t). One can show that Einsteins solution of the Brownian motion problem, which is a Gaussian PDF written as ( ) 1 ðX ZÞ2 pðX ; t þ DtjZ; tÞ ¼ exp ; ð1:119Þ 4DDt ð4pDDtÞ1=2 satisfies the condition (1.118). On the other hand, the PDF pðX ; t þ DtjZ; tÞ ¼

Dt p½ðX ZÞ2 þ Dt2

;

ð1:120Þ

which describes a Cauchy process, does not satisfy this condition. Both distributions tend to the delta function d(X Z) at Dt ! 0 (see Eq. (1.9) and Eq. (1.16) and satisfy

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations

Eq. (1.117). So the Chapman–Kolmogorov equation allows for both continuous and discontinuous solutions (PDFs).

1.12 The Chapman–Kolmogorov, Chapman–Feller, Fokker–Planck, and Liouville Differential Equations 1.12.1 Derivation of the Differential Chapman–Kolmogorov Equation

When solving concrete problems, one uses the differential form of the Chapman– Kolmogorov equation, which can be derived from the integral equation (1.117) under some additional assumptions. An additional assumption of continuity of the random process leads us to the Fokker–Planck equation. But discontinuous processes can also take place, as was mentioned in the previous section. Thus the Chapman–Kolmogorov differential equation should be able to describe both continuous and discontinuous processes. To satisfy this general requirement, we shall demand realization of the following conditions: 1: lim

1

Dt ! 0 Dt

pðX ; t þ DtjZ; tÞ ¼ W ðX jZ; tÞ

ð1:121Þ

should take place in the region |X Z| e uniformly for all X, Z, t, and the limit should not depend on e; ð 1 ðXi Zi Þ pðX ; t þ DtjZ; tÞdX ¼ Ai ðZ; tÞ þ 0ðeÞ; ð1:122Þ 2: lim Dt ! 0 Dt jX Zj < e

1 Dt ! 0 Dt

ð ðXi Zi ÞðX j Z j Þ pðX ; t þ DtjZ; tÞdX

3: lim

jX Zj < e

¼ Di j ðZ; tÞ þ 0ðeÞ:

ð1:123Þ

The conditions 2 and 3 assume a uniform convergence with respect to Z, e, and t. Condition 1 is responsible for the continuity of the process. If W(X|Z, t) ¼ 0, the process can be described by continuous trajectories; otherwise the trajectories are discontinuous. To derive the differential equation, let us consider how the average value of some (twice differentiable) function f (X) varies with time. According to Eq. (1.7), the average is written as ð h f ðX Þi ¼

f ðX Þ pðX ; tjY ; t0 ÞdX :

j31

32

j 1 Basic Concepts of the Probability Theory Then qh f i ¼ qt

1 Dt ! 0 Dt lim

ð

f ðX Þ½ pðX ; t þ DtjY ; t0 Þ pðX ; tjY ; t0 Þ dX :

Let us now put X1 ¼ X, t1 ¼ t þ Dt, X2 ¼ Z, t2 ¼ t, X3 ¼ Y, t3 ¼ t0 into the Chapman– Kolmogorov integral equation (1.110). It means that in addition to the point Y at the time t0 and the point X at the time t þ Dt, we take yet another point Z (between these two) on the trajectory at the intermediate time t (t0 < t ðð "X X X 1 q2 f ðZÞ 1< q f ðZÞ ðXi Zi Þþ ðXi Zi ÞðX j Z j Þ ¼ lim Dt ! 0 Dt> qZi 2 qZi qZ j : j i i jX Z < e

pðX ; t þ DtjZ; tÞ pðZ; tjY; t0 ÞdX dZ ðð þ jX Zj2 RðX ; ZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ jX Zj < e

ðð

þ

f ðX Þ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj > e

ðð þ

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ

jX Zj < e

ð

9 > =

f ðZÞ pðZ; tjY ; t0 ÞdZ : > ; ð1:126Þ


Consider the terms in the left-hand side of Eq. (1.126) in the consecutive order. Since p(X, t þ Dt|Z, t) is the PDF, we note that ð pðX ; t þ DtjZ; tÞdX ¼ 1: With this in mind, and using the condition of uniform convergence that allows us to take the limit of the integrand, the last term in Eq. (1.126) can be rewritten as ð f ðZÞ pðZ; tjY ; t0 ÞdZ ð

0

¼

f ðZÞ pðZ; tjY ; t ÞdZ ðð

ð pðX ; t þ DtjZ; tÞdX

f ðZÞ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ:

¼

To transform the first term, it is necessary to exploit the conditions 2 and 3: 8 # > ðð "X X X 1 q2 f ðZÞ 1< q f ðZÞ ðXi Zi Þ þ ðXi Zi ÞðX j Z j Þ lim Dt ! 0 Dt> qZi 2 qZi qZ j : j i i jX Zj < e

pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZg # ð "X q f XX1 q2 f pðZ; tjY; t0 ÞdZ þ 0ðeÞ: Ai ðZ; tÞ þ Di j ðZ; tÞ ¼ qZi 2 qZi qZ j j i i The second term tends to zero because of the condition R(X, Z) ! 0 at e ! 0, since |X Z| ! 0. Carrying out integration in the last term over two subdomains, |X Z| e and |X Z| < e and taking into account Property 1, one can reduce the last three terms to 8 > ðð 1< lim f ðX Þ pðX ; t þ DtjZ; tÞ pðZ; tjY ; t0 ÞdX dZ Dt ! 0 Dt> : jX Zj > e

ðð þ


jX Zj < e

ðð


jX Zj > e

ðð


jX Zj < e

ðð ¼

9 > = > ;

½ f ðX ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ f ðZÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX dZ

jX Zj > e

j33

34

j 1 Basic Concepts of the Probability Theory Nothing will be changed if we swap the variables X and Z in the first term: ðð

f ðZÞ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX dZ:

jX Zj > e

Going to the limit e ! 0 in Eq. (1.126), one obtains the following relation: ð q f ðX Þ pðX ; tjY ; t0 ÞdX qt " # ð X q f XX1 q2 f þ pðZ; tjY ; t0 ÞdZ Ai ðZ; tÞ Di j ðZ; tÞ ¼ qZ qZ 2 qZ i i j j i ði ð þ f ðZÞdZ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX ; ð1:127Þ which is valid when the integral ð WðZjX ; tÞ pðX ; tjY ; tc ÞdX

ð1:128Þ

exists. The condition (1.121) determines the function W(Z|X, t) only if X 6¼ Z. But there are situations when X ¼ Z, for example, a Cauchy process given by Eq. (1.120). In this case the value of the integral (1.128) should be interpreted as the principal integral value. Because such singular cases are rare, we will not be using the principal integral value symbol in the discussion below. Integration by parts of the first term on the right-hand side of Eq. (1.127) yields ð

q pðZ; tjY ; t0 Þ dZ qt ( ð X q ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ dZ ¼ f ðZÞ qZi i

f ðZÞ

X X 1 q2 f ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ qZ 2 qZ i j j i ð þ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX þ . . . ;

þ

where the dots denote the surface integrals over the boundary enclosing the considered region. As far as the function f(Z) was chosen arbitrarily with the only requirement that it should be at least twice differentiable, we can impose an additional requirement that this function should vanish on the regions boundary. Then all surface integrals also vanish, and finally, we obtain the Chapman–Kolmogorov differential equation:


q pðZ; tjY ; t0 Þ qt X q ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ ¼ qZ i i X X 1 q2 f ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ qZ 2 qZ i j j i ð þ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX :

ð1:129Þ

þ

Consider now some particular cases of the Chapman–Kolmogorov equation. 1.12.2 Discontinuous (‘‘Jump’’) Processes. The Kolmogorov–Feller Equation

This equation follows from the Chapman–Kolmogorov equation at Ai ¼ Dij ¼ 0: ð qpðZ; tjY ; t0 Þ ¼ ½WðZjX ; tÞ pðX ; tjY ; t0 ÞWðX jZ; tÞ pðZ; tjY ; t0 Þ dX : qt ð1:130Þ If we take p(Z, t|Y, t0 ) ¼ d(Y Z) at t ¼ t0 as the initial condition, then for small values of Dt the solution will be approximately equal to ð pðZ; t þ DtjY; t0 Þ dðY ZÞ½1 WðX jZ; tÞDtdX þ WðZjX ; tÞDt:

It is implied by this solution that for any Dt, there is a finite probability ð 1 WðX jZ; tÞD td X

to find the particle at the initial position Y, and the distribution of particles leaving Y is given by the function W(Z|Y, t). Hence, the trajectory X(t) consists of linear segments X ¼ const alternating with jumps whose distribution is given by the function W(Z|Y, t). That is why the process has discontinuous character and the trajectories have discontinuities in a discrete set of points. 1.12.3 Diffusion Processes. The Fokker–Planck Equation

If the process is continuous, then W(Z|Y, t) ¼ 0, and the Chapman–Kolmogorov equation is reduced to the Fokker–Planck equation:

j35

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j 1 Basic Concepts of the Probability Theory X q qpðZ; tjY ; t0 Þ ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ ¼ qt qZ i i X X 1 q2 þ ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ 2 qZi qZ j j i

ð1:131Þ

Such a process is called the diffusion process. The vector A(Z, t) is called the drift vector. It is similar to the velocity vector in the convective term of the transport equation. The matrix D(Z, t) ¼ ||Dij(Z, t)|| is called the dispersion matrix. According to its definition (see Eq. (1.123)), it is non-negative, definite and symmetric. It can be shown that D is a tensor. It is known as the dispersion tensor. To understand the physical meaning of A and D, consider the initial phase of the process in the same manner as we just did for the Kolmogorov–Feller equation, with the same initial condition p(Z, t|Y, t0 ) ¼ d(Y Z) at t ¼ t0 . Assuming that during a small time Dt 1 the values of Aj and Dij. will not change much as compared to p, the equation transforms to X qpðZ; tjY ; t0 Þ q Ai ðZ; tÞ ½ pðZ; tjY ; t0 Þ ¼ qt qZ i i XX1 q2 þ ½ pðZ; tjY ; t0 Þ ; Di j ðZ; tÞ qZi qZ j 2 j i

ð1:132Þ

where tt0 ¼ Dt 1. On this small time interval, Ai ðY;tÞ and Di j ðY;tÞ are regarded as dependent on the initial position Y but independent of time t. The solution of Eq. (1.132) has the form pðZ; t þ DtjY ; tÞ ¼

1 ð2pÞN=2 jDðY ; tÞj1=2 Dt1=2

(

1 ½ZY AðY ; tÞDt T ½DðY ; tÞ 1 ½ZY AðY ; tÞDt exp 2 Dt

) ð1:133Þ

where D ¼ |D| is the determinant of the matrix D. Eq. (1.333) indicates that at the initial stage, the diffusion process is described by the Gaussian law (see Eq. 1.77) and that fluctuations with the correlation matrix D(Y, t)Dt are superimposed on the regular drift with the velocity A(Y, t). It means that at the initial stage, the systems trajectory can be represented as Zðt þ DtÞ ¼ Y ðtÞ þ AðY ðtÞ; tÞDt þ hðtÞðDtÞ1=2 ;

ð1:134Þ

where h(t) is a random vector with the mean value and the correlation matrix given by hðtÞhT ðtÞ ¼ DðY ; tÞ: ð1:135Þ hhi ¼ 0;


In a diffusion process, trajectories are continuous everywhere because Z(t þ Dt) ! Z(t) at Dt ! 0. They are also non-differentiable at any point because of the term proportional to (Dt)1/2. Since Z(t þ Dt) Y(t) ¼ DZ is a random increment of the particles position, we can divide both parts of Eq. (1.134) by Dt, obtaining DZ ¼ AðY ðtÞ; tÞ þ hðtÞðDtÞ1=2 : Dt

ð1:136Þ

Eq. (1.136) is a stochastic differential equation that has a fundamental role in describing the motion of particles driven by an external random force. In a three-dimensional case, the Gaussian distribution (1.133) can be written in a simpler form. Let us direct the Cartesian axes Z1, Z2, Z3, so that they would coincide with the principal directions of the dispersion tensor D. Let Dij be the principal values of the dispersion tensor matrix ||Dij||. In this coordinate system, the distribution (1.133) transforms into pðZ; t þ DtjY ; tÞ ¼

1 3=2

½D11 ðtÞD22 ðtÞD33 ðtÞ 1=2 ( ) ðZ1 Y1 Þ2 ðZ2 Y2 Þ2 ðZ3 Y3 Þ2 : exp 2D11 ðtÞ 2D22 ðtÞ 2D33 ðtÞ

ð2pÞ

ð1:137Þ

Let us now introduce the probability flux with components Ji ðZ; tÞ ¼ Ai ðZ; tÞ pðZ; t þ DtjY; t0 Þ

1X q ½Di j ðZ; tÞ pðZ; tjY ; t0 Þ : 2 j qZ j

Then the Fokker–Planck equation can be written in a compact, universally accepted form of a conservation equation: qpðZ; tjY ; t0 Þ X qJi ðZ; tÞ ¼ 0: þ qt qZi i

ð1:138Þ

Introduction of the probability flux allows us to formulate the boundary conditions for the Fokker–Planck equation. Consider the process in a region R bounded by the surface S. One can see the following possible types of boundary conditions. a) Absorbing boundary

It is assumed that as soon as a particle reaches the boundary, it vanishes, that is, it leaves the system, for example, adheres to the surface or reacts with the boundary surface. Hence the probability to find the particle at the boundary is equal to zero, and the boundary condition for the PDF is pðZ; tjY ; t0 Þ ¼ 0

at Z 2 S:

ð1:139Þ

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38

j 1 Basic Concepts of the Probability Theory b) Reflecting boundary

If the particle cannot leave the region R, then the probability flux in the n direction at the boundary surface should be equal to zero, that is, nJðZ; tÞ ¼ 0

at Z 2 S;

ð1:140Þ

where n is the normal to the boundary. c) Surface of discontinuity

Suppose that the coefficients Ai and Dij experience a jump at the surface S, but particles can cross the surface freely. Such a behavior is possible, when the surface is an interface between two media with different properties. At such a surface, the probabilities and the normal components of probability fluxes should be equal at both sides of the boundary surface: pðZ; tjY ; t0 ÞjSþ ¼ pðZ; tjY ; t0 ÞjS ;

nJðZ; tÞjSþ ¼ nJðZ; tÞjS :

ð1:141Þ

d) Conditions at infinity

If the process is considered in an infinite region, then, depending on the problem under consideration, one of the following two boundary conditions must be valid: pðZ; tÞ ! 0 or

1

and

q pðZ; tÞ !0 qZ

at jZj ! ¥:

ð1:142Þ

Of special interest is the one-dimensional case of the Fokker–Planck equation, in which the drift and dispersion coefficients are become scalar quantities A and D: q pðZ; tjY; t0 Þ q 1 q2 ½DðZ; tÞ pðZ; tjY; t0 Þ : ¼ ½AðZ; tÞ pðZ; tjY; t0 Þ þ qt qZ 2 qZ2 This equation can be rewritten as q pðZ; tjY; t0 Þ q 1 q 0 ¼ DðZ; tÞ pðZ; tjY; t Þ qt qZ 2 qZ q 1 qD AðZ; tÞ pðZ; tjY; t0 Þ : qZ 2 qZ

ð1:143Þ

Now it is possible to compare it with the molecular diffusion equation, which describes the change of concentration C of a substance in the solution due to the thermal motion of solvent molecules:


qC qðvCÞ q qC ¼ þ D ; qt qZ qZ qZ

ð1:144Þ

where v is velocity of the substance under the action of external force. A comparison of equations (1.143) and (1.144) shows that D/2 has the meaning of diffusion coefficient D and the drift A A¼vþ

1 qD 2 qZ

has the meaning of average velocity of particle displacement. The latter consists of two terms. The first term is the drift caused by external forces, and the second term is the drift caused by the inhomogeneneity of the medium. Another difference between equations (1.143) and (1.144) is the difference between the unknown variables: probability density p in (1.143) and concentration C in (1.144). But concentration can be obtained from the probability density by multiplying p by the number of particles N in a unit volume. Therefore if C/N is taken instead of C, then we can also take C/N instead of p under the condition that N ¼ const. A process described by the one-dimensional Fokker–Planck equation with A ¼ 0 and D ¼ 1, q pðZ; tjY; t0 Þ 1 q2 ½ pðZ; tjY; t0 Þ : ¼ qt 2 qZ2

ð1:145Þ

is called the Wiener process. Under the initial condition pðZ; tjY ; tÞ ¼ dðY ZÞ at

t ¼ t0

its solution is ) ðZYÞ2 pðZ; tjY; t Þ ¼ exp : 2ðtt0 Þ ð2pÞ1=2 0

1

(

ð1:146Þ

A multidimensional Wiener process is described by the multidimensional Fokker– Planck equation q pðZ; tjY ; t0 Þ 1 X q2 ½ pðZ; tjY ; t0 Þ ; ¼ qt 2 i qZi2 whose solution is

) ðZYÞ2 pðZ; tjY ; t Þ ¼ : exp 2ðtt0 Þ ð2pÞn=2 0

1

ð1:147Þ

(

ð1:148Þ

Sometimes the Wiener process is causally referred to as ‘‘Brownian motion’’.

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j 1 Basic Concepts of the Probability Theory 1.12.4 Deterministic Processes. The Liouville Equation

The Liouville equation follows from the Chapman–Kolmogorov equation at W(Z|Y, t)¼ 0 and Dij ¼ 0: X q q pðZ; tjY ; t0 Þ ½Ai ðZ; tÞ pðZ; tjY ; t0 Þ : ¼ qt qZ i i

ð1:149Þ

Eq. (1.149) describes deterministic motion, which is given by a single-valued function and is determined by the initial conditions. Indeed, consider a trajectory X(t) that is a solution of the characteristic equation dX ðtÞ ¼ AðX ðtÞ; tÞ dt with the initial condition X(Y, t0 ) ¼ Y. Here A is a vector with components Ai. Let us show that pðZ; tjY ; t0 Þ ¼ dðZX ðY ; tÞÞ is the solution of Eq. (1.149) with the initial condition pðZ; t0 jX ; t0 Þ ¼ ðZX Þ: Substituting it into Eq. (1.149), one obtains: qdðZX ðY ; tÞÞ qt XqdðZX ðY ; tÞÞ XqdðZX ðY ; tÞÞ dXi ðY ; tÞ Ai ðX ðY ; tÞ; tÞ ¼ ¼ qZi dt qZi i i X q X q ¼ fAi ðX ðY ; tÞ; tÞdðZX ðY ; tÞÞg ¼ fAi ðZ; tÞdðZX ðY ; tÞÞg: qZ qZi i i i The example above considers the motion of a single particle. In the case of many particles, the Liouville equation has a somewhat different form. Statistical Mechanics uses the Liouville equation to describe the motion of a particle ensemble as a set of mass points moving in accordance with Newtons second law, 2

d X X€ i ¼ 2 ¼ F i dt

or

dV i ¼ Fi; V_ i ¼ dt

dX i ¼ V i; X_ i ¼ dt

ð1:150Þ

where Xi, Vi, Fi are, respectively, the radius vector, the velocity, and the force exerted on a unit mass of i-th particle by other particles and the environment. Let X i ð0Þ ¼ X 0i ;

V i ð0Þ ¼ V 0i :

ð1:151Þ


be the initial positions and velocities of the points. Then in order to describe the time evolution of the state of an N-particle system, one has to integrate the system of equations (1.150) with the initial conditions (1.151). For N 1, this is practically impossible, so one has to use statistical methods. Statistical Mechanics usually studies the dynamics of a mass point system by introducing generalized coordinates, which are either the ordinary coordinates Xi and velocities Vi or the ordinary coordinates Xi and momenta miVi of particles. The state of an N-particle system is characterized by the joint probability density taken at one given instant of time, which determines the chance to find particle 1 in the generalized coordinate interval (X1 þ dX1, V1 þ dV1), AND to find particle 2 in the interval (X2 þ dX2, V2 þ dV2), . . . , and to find particle N in the interval (XN þ dXN, VN þ dVN). If particle trajectories are known, that is, the functions X i ¼ X i ðtÞ;

V i ¼ V i ðtÞ

are given, then the probability density is equal to zero if Xi 6¼ Xi(t) or Vi 6¼ Vi(t) for even one value of i. It means that probability reduces to certainty and the probability density is given by a product of delta functions: pðX ; V ; tÞ ¼

N Y dðX i X i ðtÞÞdðV i V i ðtÞÞ:

ð1:152Þ

i¼1

It is easy to show that this PDF satisfies the Liouville equation. Taking the derivative of the product of functions and using properties (1.10), (1.18), (1.21) of the delta function, we write: q pðX ; V ; tÞ qt 0

1

N B Y N C X B C dðX k X k ðtÞÞdðV k X_ k ðtÞÞCX_ j ¼ B @ A j¼1 k¼1 k„ j

0

1

N B Y N C X qdðX j X j ðtÞÞ B C dðV j X_ j ðtÞÞ B dðX k X k ðtÞÞdðV k X_ k ðtÞÞC @ A qX j j¼1 k¼1 k„ j

qdðV j X_ j ðtÞÞ : dðX j X j ðtÞÞX€ j qV j

j41

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j 1 Basic Concepts of the Probability Theory Now, using the relation (1.14) and the first equation (1.150), we get N X qdðX j X j ðtÞÞ q pðX ; V ; tÞ dðV k X_ k ðtÞÞ ¼ V j qX j qt j¼1

0

1

B Y C X _ B N C N _ k ðtÞÞC F j qdðV j X j ðtÞÞ dðX X ðtÞÞdðV X B k k k B C qV j @k ¼ 1 A j¼1 k„ j 0

1

B Y C B N C _ k ðtÞÞC: dðX dðX k X k ðtÞÞB X ðtÞÞdðV X k k k B C @k ¼ 1 A k„ j It is easy to verify that 0

1

N B Y C q pðX ; V ; tÞ qdðX j X j ðtÞÞ B C ¼ dðV k X_ k ðtÞÞ B dðX k X k ðtÞÞdðV k X_ k ðtÞÞC; @ A qX j qX j k¼1 k„ j

0

1

N B Y C q pðX ; V ; tÞ qdðV j X_ j ðtÞÞ B C ¼ dðX k X k ðtÞÞ B dðX k X k ðtÞÞdðV k X_ k ðtÞÞC @ A qV j qV j k¼1 k„ j

The Liouville equation finally reduces to N N X q pðX ; V ; tÞ q pðX ; V ; tÞ X q pðX ; V ; tÞ Fi : ¼ V i qt qX qV i i i¼1 i¼1

ð1:153Þ

If the generalized coordinates Zi are defined as coordinates Xi and momenta per unit particle mass Vi, then Eq. (1.153) will assume the following compact form: N X q pðZ; tÞ q pðZ; tÞ ; ¼ Ai qt qZi i¼1

where A is the generalized vector that includes Vi and Fi.

ð1:154Þ

1.13 Stochastic Differential Equations. The Langevin Equation


When a randomly and rapidly fluctuating function of time and (or) spatial coordinates appears in a differential equation, this equation is called stochastic. The presence of a random component in the equation means that the solution will also be a random function. One example of a stochastic differential equation is the Langevin equation describing random trajectories of a particle driven by a random force. Another example is the equation of diffusion, which takes into account chemical reactions that are responsible for fluctuations. Let us consider these equations in more detail. 1.13.1 The Langevin Equation

In the theory of Brownian motion one frequently encounters the following Langevin equation: dX ¼ aðX ; tÞ þ bðX ; tÞjðtÞ; dt

ð1:155Þ

where a(X, t) and b(X, t) are known functions and j(t) is a random fluctuating function. The problem of Brownian motion of a particle driven by a fluctuating external force (that is, the force resulting from collisions with molecules of the surrounding fluid) reduces to this equation. Every second, the particle experiences millions of collisions, each collision resulting in a microscopic motion in the direction of impact. Therefore the particles position at the time t is determined only by its position at previous instant of time and does not depend on the motion prehistory. The particles motion can be considered as a Markowian process that is taking place under the action of a random force j(t) with the average value of zero: hjðtÞi ¼ 0

ð1:156Þ

The force j(t) is a ‘‘pseudoforce’’ that models the real impact forces. There is no correlation between the forces at different instants of time, in other words, j(t) and j(t0 ) are mutually independent if t 6¼ t0 . Or, to use another term, they are deltacorrelated: hjðtÞjðt0 Þi ¼ jdðtt0 Þ:

ð1:157Þ

Equations (1.155) and (1.157) serve as idealizations of Brownian motion. The actual equation of motion of a particle is given by Newtons second law: m

d2 X dX ¼ h þFþf; 2 dt dt

ð1:158Þ

j43

44

j 1 Basic Concepts of the Probability Theory where hdX/dt is the drag force exerted on the particle by the surrounding fluid (see Example 4 for more details), F is a systematic force such as gravitational, centrifugal, or electrostatic force (in other words, an external force), and f is the stochastic force. The small size of the particle allows us to neglect its inertia in the first approximation, in other words, the left-hand side of Eq. (1.158) is set to zero. This immediately results in Eq. (1.155). The condition (1.157) is also an idealization, because at t ¼ t0 it gives an infinitely large dispersion, which is an obvious impossibility. The assumption (1.157) is similar to the idealization inherent in the concept of white noise in electrical engineering. The delta function representation of the correlation function is a natural idealization that helps us make the transition from a small time scale to a large scale typical for Brownian motion. Note, however, that differential equations that include perturbating terms given by random delta-correlated functions must be treated carefully, because the usual calculation rules will not always apply. 1.13.2 The Diffusion Equation

A deterministic description of processes in continuous media requires the use of conservation equations (conservation of mass, momentum, energy, and so on). But in order to get concrete solutions, the system of conservation equations must be complemented by constitutive relations or equations. As examples of such relations, we can mention Ficks law for diffusion (mass transport) processes, Fouriers law for heat transport processes, and the Navier–Stokes law describing the hydromechanics of viscous fluids. You may ask how these equations account for fluctuations of the relevant quantities. We shall answer this question by taking the diffusion equation as an example. According to Ficks law, the diffusive flux of the substance j(X, t) is proportional to the concentration gradient: jðX ; tÞ ¼ DrCðX ; tÞ:

ð1:159Þ

On the other hand, we have the equation of conservation of mass (the continuity equation): qC þ rjðX ; tÞ ¼ 0: qt

ð1:160Þ

Substitution of j(X, t) from (1.159) results in the standard diffusion equation: qC ¼ r½DrCðX ; tÞ : qt

ð1:161Þ

Fluctuations could be taken into account by adding a fluctuating term to the righthand side Eq. (1.159): jðX ; tÞ ¼ DrCðX ; tÞ þ j fl ðX ; tÞ:

ð1:162Þ


Just as we did for the fluctuating term in the Langevin equation, we assume that this term has the following statistical properties: D E D E fl fl j fl ðX ; tÞ ¼ 0; j i ðX ; tÞ jk ðX 0 ; t 0 Þ

¼ KðX ; tÞd jk dðX X 0 Þdðtt0 Þ:

ð1:163Þ

The second property says that different components of the fluctuating flux vector j fl taken at one and the same spatial point, as well as the values of one and the same component taken at different instants of time and/or at different points are assumed to be statistically independent. Or, to say it in fewer words, this property states that fluctuations have local behavior. Eqs. (1.160) and (1.162) give us a diffusion equation where the additional term is expressed as the divergence of a vector: qC ¼ r½DrCðX ; tÞ rj fl ðX ; tÞ qt

ð1:164Þ

It can be shown that this term possesses the following statistical properties: D

E rj fl ðX ; tÞ ¼ 0; D E rj fl ðX ; tÞr0 j fl ðX 0 ; t0 Þ ¼ rr0 ½K1 ðX ; tÞdðX X 0 Þ dðtt0 Þ:

ð1:165Þ

1.13.2.1 The Diffusion Equation with Chemical Reactions Taken into Account The deterministic diffusion equation with a source/sink term arising due to chemical reactions has the form

qC þ rjðX ; tÞ ¼ F½CðX ; tÞ ; qt

ð1:166Þ

where the rate of substance production/consumption in the course of the chemical reaction appears in the right-hand side. This term usually depends on the substance concentrationl; depending on the reaction kinetics, it could be a linear or a nonlinear function of concentration. Production/consumption of matter gives rise to fluctuations, which can be taken into consideration by adding a fluctuating term gfl(X, t) to the systematic term F[C(X, t)]. The new term must satisfy the following statistical conditions: fl g ðX ; tÞ ¼ 0; g fl ðX ; tÞg fl ðX 0 ; t0 Þ 0 ¼ K2 ðX ; tÞdðX X Þdðtt0 Þ:

ð1:167Þ

The second condition expresses locality (lack of correlation between fluctuations at different points) as well as the Markovian character of chemical reactions. Eqs. (1.162), (1.166), and (1.167) give us the stochastic diffusion equation that

j45

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j 1 Basic Concepts of the Probability Theory accounts for chemical reactions: qC þ rjðX ; tÞ ¼ F½CðX ; tÞ þ G fl ðX ; tÞ; qt

ð1:168Þ

where G fl ðX ; tÞ ¼ rj fl ðX ; tÞ þ g fl ðX ; tÞ: The fluctuating term has the following properties: fl fl G ðX ; tÞG fl ðX 0 ; t0 Þ G ðX ; tÞG fl ðX ; tÞ ¼ 0; 0 0 ¼ fK2 ðX X 0 ÞdðX X 0 Þ þ rr ½K1 ðX ; tÞdðX X Þ gdðtt0 Þ: Eqs. (1.164) and (1.168) are both Langevin equations. The main shortcoming of these equations is that the functions K1(X, t) and K2(X, t) are not known in advance. Information about these functions can be obtained by studying the process on the microscopic level, using the same approach that helped us derive the Chapman– Kolmogorov equation. This approach allows to interpret diffusion coefficients as components of a dispersion tensor or a correlation matrix. The difference between equations (1.168) and (1.164) is that F[C(X, t)] that appears in Eq. (1.168) is not a function but a functional. 1.13.2.2 Brownian Motion of a Particle in a Hydrodynamic Medium Slow motion of a particle in a fluctuating hydrodynamic medium (the medium is assumed to be at rest at the infinity) under the action of the average viscous force exerted by the surrounding fluid hF(t)i and the fluctuating force F f l(t) (a force induced by thermal hydrodynamic fluctuations in the fluid or by some other source of random forces) is described by the Langevin equation (a stochastic analogue of Newtons second law):

m

dU ¼ hFðtÞi þ F fl ; dt

U¼

dX : dt

ð1:169Þ

To find the statistical properties of the random force F f l, it is necessary to use the hydrodynamic equations describing the fields of velocity u and pressure p in the fluid. The particle motion occurs at low Reynolds numbers, so one can use Navier– Stokes equations in the inertialess approximation, (i.e., Stokes equations): ru ¼ 0;

rT ¼ 0;

ð1:170Þ

where T(r, t) ¼ T s þ T f l is the stress tensor in the fluid, which consists of a systematic component T s ¼ pI þ 2meE and a fluctuating component T f l; pI is the spherical tensor (the isotropic part of the stress tensor); 2meE is the deviator (the deviatoric part of the stress tensor); E ¼ 0.5(ru þ ruT) is the symmetric rate-of-strain tensor, me is the coefficient of dynamic viscosity of the carrier (external) fluid.


In a statistically homogeneous medium, the fluctuating stress tensor T f l is symfl metric and its trace is equal to zero, that is, T ii ¼ 0. The components Dof TEf l are fl usually assumed to have a Gaussian distribution with the average value T ik ¼ 0 and the correlation matrix D E 2 fl fl T ik ðr 1 ; t 1 ÞT lm ðr 2 ; t2 Þ ¼ 2kme ðd il d km þ dim dkl d ik dlm Þ 3 ð1:171Þ dðr 1 r 2 Þdðt1 t2 Þ: Equations (1.170) can then be rewritten in the coordinate form: qui ¼ 0; qXi

qTisj qX j

fl

¼

qTi j qX j

:

ð1:172Þ

Following the method of small perturbations, we shall be looking for the solutions in the form ~i ; ui ¼ hui i þ u

D E s Tisj ¼ Tisj þ T~ i j ;

ð1:173Þ

s where hui, hT si are the average values and u~ ,~ T are small fluctuating ‘‘additions’’ (perturbations). Substituting (1.173) into (1.172) and neglecting small terms of higher orders, we get the equations for the average values and for the fluctuating terms:

qhui i ¼ 0; qXi

D E q Tisj qX j

¼ 0;

q~ ui ¼ 0; qXi

~ qT ij s

qX j

qT~ i j

fl

¼

qX j

:

ð1:174Þ

Now, we must introduce two boundary conditions. First, the relative velocity at the ~i ¼ 0. interface between the fluid and the solid particle should be zero: huii ¼ Ui, u ~i ¼ 0. Secondly, the fluid must be at rest at the infinity: hui i ¼ u As a rule, the main purpose of hydrodynamic calculations is to find the force the surrounding fluid exerts on a moving particle. This force has systematic and fluctuating parts F and F f l, whose components are ðD E hFi i ¼ Tisj n j ds; s

ð

~ s n j ds; Fi ¼ T ij fl

ð1:175Þ

s

where ni are components of the outer normal vector and S is the surface of the particle. Solving the first two equations (1.174), we get the relation between the particles velocity and the drag force: hFi i ¼ Ri j U j ; where Rij are components of the resistance tensor R.

ð1:176Þ

j47

48

j 1 Basic Concepts of the Probability Theory ~ i , let us use In order to determine statistical characteristics of the random force F the relation that follows in a self-obvious way from Gausss theorem, equations (1.174), and the above-mentioned boundary conditions: ð V

D E1 ð D E q Tisj q s ~ s ~ @hui i A dV ¼ ~ ui ðhui iT i j ui Ti j ÞdV qX j qX j qX j 0

~ qT ij s

ð ¼

V

e s ~ ð h ui i T i j ui

s

ð ð D sE es qT ij e ~ s n j ds ¼ Ui F ~i; T i j Þn j ds ¼ hui i ds ¼ Ui T ij qX j s

s

where integration is carried out over the volume V occupied by the fluid. Using this relation together with the property (1.171), we finally get the two-time correlation between components of the fluctuating force:

~ j ðt0 Þ ¼ ~ i ðtÞF F

ð 1 2kTdðtt0 Þ hui i Ti j n j ds ¼ 2kTRi j dðtt0 Þ: Ui U j

ð1:177Þ

s

1.14 Variational (Functional) Derivatives

When considering random processes and random fields, one can see that the PDF depends on random variables, which, in turn, are functions of time and (or) spatial coordinates. Therefore, the variables in the PDF are random functions rather than random variables. It was shown in Sections 1.5 and 1.6 that knowing the characteristic function j and the characteristic functional, one can determine statistical parameters of random quantities such as moments and cumulants from the formulas (1.50) and (1.54). These formulas contain derivatives of j and lnj. When considering random processes and random fields, we treat these derivatives as derivatives with respect to functions and not as derivatives with respect to random variables. Hence, instead of ordinary differentiation we have to perform functional, or, to use another term, variational, differentiation. The corresponding derivatives are called functional (variational) derivatives. Let us recal the general definition of a functional. We say that a functional is given when there exists a rule that assigns a definite number to each function belonging to some set of functions. Some examples are given below: 1. - linear functional Xð2

F½jðX Þ ¼

aðX ÞjðX ÞdX ; X1

where a(X) is a given function and the limits X1 and X2 can be either finite or infinite;


2. - quadratic functional Xð2 Xð2

F½jðX Þ ¼

Bðt1 ; t2 Þjðt1 Þjðt2 Þdt1 dt2 ; X1 X1

where B(t1, t2) is a given function; 3. - function of functional F½jðXÞ ¼ f ðF½j Þ where f(X) is a given function and F[j(X)] is a functional. Consider the difference in values of one and the same functional taken for two functions j(t) and j(t) þ dj(t), where t lies in the interval t 2 (X 0.5DX; X þ 0.5DX). The difference (more strictly, the linear (with respect to dj(t)) part of that difference) is called variation of the functional: dF½j ¼ fF½j þ dj F½j g: The variational (functional) derivative is defined as dF½j dF½j : ¼ lim Ð DX ! 0 djðX Þ DX jðtÞdt

ð1:178Þ

The variational derivative of a functional F[j] is itself a functional of j(t) that also depends on the point X as a parameter. Thus the variational derivative itself has two different derivatives. One can differentiate it in the ordinary way with respect to the parameter X; or one can take the variational derivative with respect to j(t) at the ~ . The latter would be the second variational derivative of the initial point t ¼ X functional F[j]: d dF½j d 2 F½j ¼ : ~ Þ djðX Þ ~ ÞdjðXÞ djðX djðX

ð1:179Þ

The second variational derivative is also a functional of j(t), but, in contrast to the ~ . Variational first variational derivative, it now depends on two points: X and X derivatives of higher orders can be defined in a similar way. As examples, consider variational derivatives of the above-mentioned functionals. Xð2

dF ¼ F½j þ dj F½j ¼

X þ0:5DX ð

aðtÞdjðtÞdt ¼ X1

aðtÞdjðtÞdt: X 0:5DX

j49

50

j 1 Basic Concepts of the Probability Theory If the function a(t) is continuous on the interval DX, then, according to the mean value theorem, ð dF½j ¼ aðX 0 Þ djðtÞdt; DX 0

where X 2 (X 0.5DX; X þ 0.5DX). The definition (1.178) yields 0 1 ð ð dF½j djðtÞdtA ¼ aðXÞ: ¼ lim aðX 0 Þ@ djðtÞdt djðX Þ DX ! 0 DX

ð1:180Þ

DX

Applying the same approach to the quadratic functional, we get dF½j ¼ djðX Þ

Xð2

ðBðt; X Þ þ BðX ; tÞÞjðtÞdt;

ðX1 < t < X2 Þ:

ð1:181Þ

X1

Note that in many cases the function B(t, X) is a symmetric function of its arguments, that is, B(t, X) ¼ B(X, t). Finally, for a function of a functional, F½j þ dj ¼ f ðF½j þ dj Þ ¼ f ðF½j þ dFÞ qf ðF½j Þ qf ðF½j Þ ¼ f ðF½j Þ þ dF þ . . . ¼ F½j þ dF þ . . . qF qF and d qf ðF½j Þ dF½j f ðF½j Þ ¼ : djðX Þ qF djðX Þ

ð1:182Þ

Consider now some properties of variational derivatives. Let a functional be a product of two functionals: F[j] ¼ F1[j]F2[j]. The variation and variational derivative of this functional are: dF ¼ F½j þ dj F½j ¼ F1 ½j þ dj F2 ½j þ dj F1 ½j F2 ½j ¼ F1 ½j dF2 ½j þ F2 ½j dF1 ½j ; d dF2 ½j dF1 ½j ½F1 ½j F2 ½j ¼ F1 ½j þ F2 ½j : djðX Þ djðX Þ djðXÞ

ð1:183Þ

Of special interest is the functional of a Gaussian distribution, ( ) 1 ðtt0 Þ2 jðtÞdt: F½j ¼ pffiffiffiffiffiffi exp 2s2 2ps ð¥ 0

ð1:184Þ


Eq. (1.180) gives us its variatlional derivative: ( ) dF½j 1 ðtt0 Þ2 : ¼ pffiffiffiffiffiffi exp 2s2 djðtÞ 2ps

ð1:185Þ

Going to the limit s ! 0 in Eqs. (1.183), (1.184) and using the definition of the delta function Eq. (1.9) and its property (1.11), we obtain: lim F½j ¼ jðt0 Þ;

s!0

lim ¼

s!0

dF½j ¼ dðtt0 Þ: djðtÞ

Therefore, dj½t0 ¼ dðtt0 Þ: djðtÞ

ð1:186Þ

The relation (1.186) facilitates the derivation of formulas for some variational derivatives. As an example, let us derive the formula (1.181) for the quadratic functional: 8X X 9 ð2 ð2 = d < Bðt1 ; t2 Þjðt1 Þjðt2 Þdt1 dt2 ; djðX Þ: X1 X1

Xð2 Xð2

¼

Bðt1 ; t2 Þ

d ½jðt1 Þjðt2 Þ dt1 dt2 djðXÞ

X1 X1 Xð2 Xð2

¼

djðt1 Þ djðt2 Þ Bðt1 ; t2 Þ jðt2 Þ þ jðt1 Þ dt1 dt2 djðX Þ djðXÞ

X1 X1 Xð2 Xð2

¼

Bðt1 ; t2 Þ½dðt1 X Þjðt2 Þ þ jðt1 Þdðt2 X Þ dt1 dt2 X1 X1 Xð2

¼

½Bðt; XÞ þ BðX ; tÞ jðtÞdt;

ðX1 < t < X2 Þ:

X1

Another example is the variational derivative of the functional Xð2

F½jðtÞ ¼ X1

_ dt; L½ðt; jðtÞ; jðtÞ

_ jðtÞ ¼

djðtÞ : dt

j51

52

j 1 Basic Concepts of the Probability Theory We have: Xð2 qL qL d djðt0 Þ qL qL d þ dt ¼ þ dðtXÞdt qj qj_ dt djðX Þ qj qj_ dt X1 X1 d q q _ ; X 2 ðX1 ; X2 Þ: þ L½ðt; jðtÞ; jðtÞ ¼ dX qj_ qj

dF½j ¼ djðX Þ

Xð2

The functional F[j(t) þ Z(t)] can be expanded as a functional Taylor series in the vicinity of the point Z 0: ð dF½j hðX ÞdX F½jðtÞ þ hðtÞ ¼ F½jðtÞ þ djðXÞ ðð 1 dF½j hðX1 ÞhðX2 ÞdX1 dX2 þ . . .: þ 2! djðX1 ÞdjðX2 Þ

ð1:187Þ

Here and later, when the range of integration is not pointed out, it is assumed to be infinite. The Taylor series (1.187) could be written in the compact form ð F½jðtÞ þ hðtÞ ¼ exp dX hðX Þ

d F½jðtÞ djðX Þ

using the following operator notation: ð exp dX hðX Þ ðð

d djðX Þ

ð ¼ 1 þ dX hðX Þ

d djðX Þ

d2 þ. . . djðX1 ÞdjðX2 Þ ð 2 ð d 1 d ¼ 1 þ dX hðX Þ þ. . .: þ dX hðX Þ djðX Þ 2! djðX Þ 1 þ 2!

dX1 dX2 hðX1 ÞhðX2 Þ

Consider the transformation of variational derivatives as we change the functional variables. Let us replace the function j(t) by a new function c(t) given by the equality j(t) ¼ C[c(t); t], where c is the functional of a function c(t), which also depends on t. Then the functional F[j(t)] is a composite functional of c(t): F½jðtÞ ¼ F½Y½yðtÞ; t F1 ½yðtÞ : For such functional, there exists the following expression for the variational derivative: ð dF1 ½yðtÞ dF½jðtÞ dY½yðtÞ; X 0 ¼ dt0 : djðtÞ djðt0 Þ dyðtÞ

ð1:188Þ

1.15 The Characteristic Functional

A functional change of variables plays an important role in Fourier transforms: ð jðXÞ ¼ yðwÞeiwX dw ¼ j½yðwÞ; X : According to the formula (1.173), we have dj½yðwÞ; X 0 ¼ eiw X dyðw0 Þ and from (1.180), there follows ð dF1 ½j½yðwÞ; X dF½jðXÞ iw0 X 0 0 ¼ e dX : djðw0 Þ djðX 0 Þ

ð1:189Þ


It was shown in Section 1.5 that a random value u is completely determined by its characteristic function j(r) ¼ hexp(iru)i, which allows us to use the inverse Fourier transform to get the PDF, pðuÞ ¼

ð 1 eiru dr 2p

the moments, B n ¼ h un i ¼

1 d n jðrÞjr¼0 ; i dr

the cumulants, 1 d n ½lnjðrÞ r¼0 Sn ¼ i dr and other statistical parameters of the PDF. For a multidimensional random quantity u ¼ (u1, u2, . . . , uN), the complete description is contained in the characteristic function * jðrÞ ¼ jðr1 ; r2 ; . . .; rN Þ ¼ hexpðiruÞi ¼

expði

X

+ r k uk Þ :

ð1:190Þ

k

The corresponding joint PDF of the random values u1, u2, . . . , uN is the Fourier transform of the characteristic function j(r1, r2, . . . , rN):

j53

54

j 1 Basic Concepts of the Probability Theory ð

1

eirX jðrÞdr ð2pÞ ð X 1 expði rk Xk Þjðr1 ; r2 ; . . .; rN Þdr1 dr2 . . .drN ¼ N ð2pÞ k

pðX1 ; X2 ; . . .; XN Þ ¼

N

ð1:191Þ

¼ dðu1 X1 Þdðu2 X2 Þ. . .dðuN XN Þ: Consider now a random function u(X). For its complete statistical description, it is sufficient to know the characteristic functional

F½r ¼

ð exp i rðtÞuðtÞdt ;

ð1:192Þ

where the function r(t) is an arbitrary function replacing the set of numbers r1, r2, . . . , rN in (1.190). Given F[r], we can find statistical characteristics of the random function u(X), for example, the average value hu(X)i, N-point moments Buu. . .u ¼ hu(X1) . . . u(XN)i, etc. To find the variational derivative, let us use the results obtained in Section 1.14, keeping in mind that the averaging operator commutes with the operator d/dr(X): dF½r d ¼ exp drðX Þ djðX Þ

ð ð d i rðtÞuðtÞdt ¼ exp i rðtÞuðtÞdt djðXÞ ð

¼ i uðX Þexp i rðtÞuðtÞdt :

Similarly, we write

ð 1 d 1 d ... F ¼ uðX1 Þ. . .uðXn Þexp i rðtÞuðtÞdt : i drðX1 Þ i drðXn Þ

Setting r ¼ 0 in the last relation, we obtain: 1 dn ¼ huðX1 Þ. . . uðXn Þi ¼ Buu...u : F n i drðX1 Þ. . .drðXn Þ r¼0

ð1:193Þ

So, it is possible to find the multi-point moments for a given characteristic functional. Expanding the functional F[r] into a functional Taylor series according to Eq. (1.187) and taking into account Eq. (1.193), one can express the characteristic functional in terms of moments: F½r ¼

¥ nð X i

ð . . . Buu...u ðX1 ; X2 ; . . .; Xn Þ

n! rðX1 ÞrðX2 Þ. . .rðXn ÞdX1 dX2 . . .dXn : n¼0

ð1:194Þ


If the functional has the form F½r ¼ expðy½r Þ; then c is called the cumulant-generating function (see Section 1.6), and the cumulants themselves given by dn : S11...1 ðX1 ; X2 ; . . .; Xn Þ ¼ ðiÞ drðX1 Þ. . .drðXn Þ r¼0 k

ð1:195Þ

As an example, let us find the characteristic functional for a Gaussian random process u(X) with hu(X)i ¼ 0, under the additional assumption that the joint distribution for any two given values of X is also Gaussian. Let the random variable ð¥ hAi ¼

rðtÞuðtÞdt; ðrð¥Þ ¼ 0; ¥

have a Gaussian distribution ( ) 1 ðAhAiÞ2 pðAÞ ¼ pffiffiffiffiffiffi exp 2s2A 2psA with the parameters ð¥ A¼

rðtÞhuiðtÞdt ¼ 0; ¥

s2A ¼ A2 ðhAiÞ2 ¼

ð¥ ð¥ Bðt1 ; t2 Þrðt1 Þrðt2 Þdt1 dt2 ; ¥¥

where B(t1, t2) ¼ hu(t1)u(t2)i is a two-point correlation function. The characteristic functional of the random quantity A is 1 expðeiA Þ ¼ exp s2A : 2 Therefore the characteristic functional of a Gaussian random process is 8 9 < 1 ð¥ ð¥ = Bðt1 ; t2 Þrðt1 Þrðt2 Þdt1 dt2 : F½r ¼ exp : 2 ;

ð1:196Þ

¥¥

Characteristic functionals and variational derivatives are widely used in problems pertaining to particle motion under the action of random fluctuating forces, for

j55

56

j 1 Basic Concepts of the Probability Theory example, in the theories of Brownian motion, diffusion, turbulence, etc. Such problems are the subject of the statistical theory of dynamic systems with fluctuating parameters. These parameters (e.g., coordinates and velocities of particles suspended in a fluid), are described by ordinary or partial differential equations (stochastic equations). The main challenge is to obtain and then solve a closed system of equations. It turns out that many processes can be treated as Markowian processes. Also, the distribution of fluctuating parameters (random variables) has proved to be Gaussian. As an example, consider a simplified form of the Langevin equation (see Section 1.13) that describes the time rate of change of velocity V(t) of a particle driven by a fluctuating force x(t): dV ¼ hV þ xðtÞ; dt

X ð0Þ ¼ 0;

ð1:197Þ

where h is a constant and x(t) is a random function. For a given x(t), the solution of Eq. (1.197) has the form ði ð1:198Þ

VðtÞ ¼ xðtÞexpðhðttÞÞdt: 0

From Eq. (1.198), one can derive all statistical characteristics of the random process V(t). On the other hand, all statistical characteristics are contained in the characteristic function * jðrÞ ¼ ðexpðirVðtÞÞ ¼

8 t 9+ < ð = exp ir xðtÞexpðhðttÞÞdt : : ;

ð1:199Þ

0

Taking into account the relation (1.192), the last equation can be rewritten as jðrÞ ¼ FfrexpðhðttÞÞg:

ð1:200Þ

Finally, we should mention the Furutsu–Donsker–Novikov correlation formula for the product of a Gaussian random function X(t) and a functional R[X(t)] that may depend (either explicitly or implicitly) on X(t): ð¥ hX ðtÞR½X ðtÞ i ¼

dR½X ðtÞ dt1 : Bðt; t1 Þ dX ðt1 Þ

¥

This formula is the functional analogue of Eq. (1.86).

ð1:201Þ

References

References 1 Gardiner, C.W. (1985) Handbook of Stochastic Methods, 2nd ed., Springer– Verlag. 2 Klyatskin, V.I. (1975) Statistical Description of Dynamical Systems with Fluctuating Parameters, Nauka, Moscow, (in Russian). 3 Klyatskin, V.I. (1980) Stochastic Equations and Waves in Random Inhomogeneous Media, Nauka, Moscow, (in Russian). 4 Landau, L.D. and Lifshitz E.M. (1964) Statistical Physics, Nauka, Moscow, (in Russian). 5 Leontovitch, M.A. (1983) Introduction to Thermodynamics. Statistical Physics, Nauka, Moscow, (in Russian). 6 Monin, A.C. and Yaglom, A.M. (1971) Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 1, MIT Press, Cambridge, MA.

7 Monin, A.C. and Yaglom, A.M. (1975) Statistical Fluid Mechanics: Mechanics of Turbulence, Vol. 2, MIT Press, Cambridge, MA. 7 Feller, W. (1974) An Introduction to Probability Theory and Its Applications, Vol. I, II, Wiley. 8 Chandrasekhar, S. (1943) Stochastic Problems in Physics and Astronomy, Rev. Mod. Phys., 15, 1–89. 9 Cercignani, C. (1969) Mathematical Methods in Kinetic Theory, Macmillan. 10 Einstein, A. and Smoluchowski, M. (1936) Brownian Motion, ONTI, Moscow, (in Russian). 11 Pope, S.B. (2000) Turbulent Flows, Cambridge Univ. Press. 12 Saffman, P.G. (1971) On the Boundary Condition at the Surface of a Porous Medium, Studies in Appl. Math., 50 (2), 93–101.

j57

j59

2 Elements of Microhydrodynamics The branch of hydrodynamics studying the motion of microparticles in liquid at low Reynolds numbers is called microhydrodynamics. The present chapter is devoted to the fundamentals of low Reynolds number hydrodynamics which will be a prerequisite for the subsequent discussion. Low Reynolds number hydrodynamics is based on Stokes equations [1–5] r v ¼ 0;

r2 v ¼

1 r p: me

ð2:1Þ

The boundary conditions for these equations follow from the restrictions imposed on velocities at particle surfaces, at flow region boundaries (if any), and far away from the particles (‘‘at infinity ’’) if the fluid is unbounded. Exact solution of concrete boundary value problems presents considerable mathematical difficulties even for the single particle case. Still, there are some general properties common for this type of boundary value problems that prove to be very useful for derivation of solutions. For simplicity, we consider the behavior of particles in an unbounded fluid. The first property is called the reciprocity theorem. Let (v 0,T 0 ) and (v 00,T 00 ) be the velocity v and stress T fields corresponding to the flows of two fluids having different viscosities m0e and m 00e . Both flows are described by the same equations (2.1) and by the same boundary conditions. Then for any closed surface S bounding the fluid volume V (this surface could consists of several different surfaces, including surfaces of particles contained in the considered volume V ), there should hold ð ð m00e dS T 0 V 00 ¼ m0e dS T 00 V 0 : S

ð2:2Þ

S

Another important property, which is often used to determine the hydrodynamic force F and the torque L acting on a spherical particle that has translational velocity U and rotates with angular velocity V, is given by Faxen’s laws: F ¼ 6 pme aðv¥ j0 UÞ þ me pa3 ðrv¥ Þ0 ; 1 L0 ¼ 8pme a3 ðr v¥ Þ0 V : 2

ð2:3Þ

j 2 Elements of Microhydrodynamics

60

Here v¥ is the (translational) velocity of the liquid at the infinity. The subscript 0 indicates that the corresponding quantity is measured at the point occupied by the center of the sphere. When solving problems on slow motion of particles, it is important to estimate hydrodynamic field perturbations in the fluid that are induced by the moving particle. If the considered suspension is low-concentrated (dilute), then the effect of the particle on the fluid can be approximated by a point force applied at the particle center and equal to the drag force on the particle, but pointing in the opposite direction. The resultant problem is handled by writing Stokes equations and obtaining the fundamental solution. If we assume the volume concentration of particles j to be small, then the average distance between particles is large compared to their radius, and we need to consider only pair interactions between particles. Since the particles are far enough from each other, the hydrodynamic influence of particle B on particle A can be accounted for by replacing particle B with a point force FB applied at the particle center and equal to the drag force on particle B but having the opposite sign. The translational velocity of the fluid at the infinity is assumed to be constant: u¥ U0ex; we also assume that the line of centers of the particle pair is parallel to U0 and coincides with the X-axis. Then in order to find the drag force of particle A, it is necessary to solve the Stokes equations with the source term me Dur p ¼ FB dðX XB ÞdðYÞdðZÞex ;

r u ¼ 0

ð2:4Þ

and the boundary conditions u ¼ 0 on u ! U0

at

SA ; X ! ¥:

ð2:5Þ

The fundamental solution is ! FB 1 ðX XB ÞðX X B Þ ; u¼ ex þ 8pme jX X B j jX X B j3 p¼

! FB ðX XB Þ : 4p jX X B j3

ð2:6Þ

Owing to the linearity of the inhomogeneous boundary value problem, its solution can be obtained as a superposition of solutions of the corresponding homogeneous problems: Eqs. (2.4) at FB ¼ 0 with the boundary conditions (2.5); and Eqs. (2.4) with the boundary conditions (2.5) when U0 ¼ 0. The solutions of these particular problems are found as sums of spherical functions, while keeping in mind the axial symmetry of the problem. To determine the drag force acting on a particle of finite radius, one should average the velocity over the particle surface. The final expression for the force exerted on a single particle by the surrounding fluid will be given by the Stokes formula: FB ¼ 6pme bU0 :

ð2:7Þ

2.1 Motion of an Isolated Particle in a Quiescent Fluid

For a pair of particles, A and B, the resulting drag forces have the form 3 a3 a FB ; DA ¼ 6pme aU0 þ 2 3r 3 r

3 b3 b DB ¼ 6pme bU0 þ FA ; 2 3r 3 r

ð2:8Þ

where a, b are the particle radii and r is the distance between their centers. For identical particles (a ¼ b), DA ¼ DB ¼ 6pme u0 lða=rÞ;

ð2:9Þ

where l is a dimensionless resistance coefficient. Up to the order a/r, it is equal to l ¼ 1

3a : 2r

ð2:10Þ

With further applications in mind, we shall focus our attention on the motion of one particle, two particles, and, finally, many (i.e., more then two) particles in a quiescent fluid or in a shear flow of viscous incompressible fluid.


Consider a solid particle of an arbitrary shape moving with translational velocity U0 and rotating with angular velocity V. By its nature, such a motion is unstable. But if the Reynolds numbers derived from translational and rotational velocities U0 and W are small, that is, if Ret ¼ aU0re/me 1 and Rer ¼ a2Wre/me 1 where a is the characteristic linear size of the particle and re and me are the density and viscosity of the fluid, then it is legitimate to derive the velocity and pressure fields v and p induced in the fluid by the moving particle by using quasi-stationary Stokes equations r v ¼ 0;

r2 v ¼

1 r p: me

ð2:11Þ

Let 0 be a point attached to the particle, and let U0 be the instantaneous translational velocity of this point, and V – the instantaneous angular velocity of the particle. The particle is assumed to be solid, and thus the condition of zero velocity at the particle surface Sa yields v ¼ U0 þ V r0

at

Sa ;

where r0 is the radius vector measured with respect to 0.

ð2:12Þ

j61


62

The second boundary condition is that fluid should be at rest at the infinity: v!0

at r0 ! ¥:

ð2:13Þ

The solution of thusly formulated problem gives the hydrodynamic force F and the torque L acting on the particle [6]: ð

ð dS T;

F¼

L¼

Sa

r 0 ðdS TÞ:

ð2:14Þ

Sa

Here dS ¼ n dS is a particle surface element (directed towards the fluid), T is the stress tensor of an incompressible Newtonian fluid: T ¼ pI þ me ðrv þ rvT Þ;

ð2:15Þ

I is the unit tensor with components d ij, and rv is the velocity gradient tensor with components rivj. Due to the linearity of the boundary value problem defined by Eqs. (2.11)–(2.13), it is possible to represent the velocity, the pressure, and the stress tensor as sums of translational and rotational terms, ðvt0 ; pt0 ; T t0 Þ and ðvr0 ; pr0 ; T r0 Þ, each term describing its corresponding type of particle motion: v ¼ vt0 þ vr0 ;

p ¼ pt0 þ pr0 ;

T ¼ Tt þ Tr:

ð2:16Þ

As a result, the boundary value problem separates into the translational problem defined by r vt0 ¼ 0; vt0 ¼ U 0

r2 vt0 ¼

1 r pt0 ; me

Sa ;

vt0 ! 0

on

at

r0 ! ¥

ð2:17Þ

and the rotational problem defined by r vr0 ¼ 0; vr0 ¼ V r 0

r2 vr0 ¼ on

1 r pr0 ; me

Sa ;

vr0 ! 0

ð2:18Þ at

r0 ! ¥:

Using the formulas (2.14), we can obtained the hydrodynamic forces F t0 ; F r0 and torques Lt0 ; Lr0 induced by translational and rotational motions of the particle: ð ð t;r t;r F t;r ¼ dS T ; L ¼ r 0 ðdS T 0t;r Þ: ð2:19Þ 0 0 0 Sa

Sa

where T 0t;r are stress tensors corresponding to the translational and rotational particle motions: t;r t;r t;r T T t;r 0 ¼ p0 I þ ðrv0 þ ðrv0 Þ Þ:

ð2:20Þ


The resulting values of the hydrodynamic force F and torque L acting on the particle are F ¼ F t0 þ F r0 ;

L ¼ Lt0 þ Lr0 :

ð2:21Þ

Both translational and rotational fields, as well as forces and torques depend on the fluid’s viscosity me, on the length and orientation of vectors U0 and V relative to the particle’s coordinate system, and on the particle’s geometry. But such complex dependences are inconvenient to use. Let us try to separate these dependences. To this end, for each type of motion we shall introduce quantities that characterize only the geometrical properties of the particle, thus determining the geometrical dependence once and for all for any given class of geometrical shapes. For translational motion, there exists a velocity tensor V t (asymmetrical in the general case) and an associative dynamic pressure vector P t satisfying the relations vt0 ¼ V t U 0 ;

p0 ¼ me P t U 0

ð2:22Þ

and the equations r V t ¼ 0;

r2 V t rPt ¼ 0;

Vt ¼ I

Sa ;

on

Vt !0

at

r0 ! ¥;

ð2:23Þ

and independent of the fluid’s viscosity (at any point in the fluid), of the choice of reference frame (i.e., of the origin 0), and of the length and orientation of the vector U0. They depend only on the geometry of the particle’s surface and on the radius vector (which can be taken relative to any origin of coordinates attached to the particle). Consider a translational Stokesian flow that goes around a spherical solid particle of radius a when the fluid’s velocity at the infinity is U ¼ const. The stream function for this flow is expressed in spherical coordinates as 1 a 3 a y ¼ Ur 2 sin2 u 3 : 4 r r The velocity components and the dynamic pressure are given by

a2 a 1 qy 1 r ¼ U cosu 3 ; sinu qu 2 r r a

a a2 1 qy 1 v ¼ þ3 ; ¼ U sinu r sinu qr 4 r r

vr ¼

r2

3 cosu p p¥ ¼ ame U 2 ; 2 r where p¥ is the static pressure at the infinity.

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64

The velocity tensor and the pressure vector for such a flow are equal to Vr ¼

1 a3 3rr 3 a rr I ; þI 4r r 4 r r2

3 r Pr ¼ a 3 ; 2 r

ð2:24Þ

where r is measured from the center of the sphere. We see from Eq. (2.15) and Eq. (2.22) that the stress tensor can be written as T t ¼ me Pt U 0 ;

ð2:25Þ

where we have just introduced a tensor of the third rank (a triadic tensor) Pt ¼ I P t þ ðrV t þ T ðrV t ÞÞ:

T

The superscript T to the left of the tensor (it is known as a polyadic) means that ðabc. . .Þ ¼ bac. . .. Let us introduce the tensors K t and K c0 characterizing the given particle: ð Kt ¼

ð dS Pt ;

K c0 ¼

Sa

r 0 ðdS Pt Þ: Sa

Then expressons for the force and torque acting on the particle undergoing a translational motion can be written in a compact form: F 0 ¼ me K t U 0 ;

L ¼ me K c0 U 0 :

ð2:26Þ

The tensor K t is called the tensor of translational resistance. It depends only on the size and shape of the particle and does not depend on the particle’s velocity and orientation in space or on the properties of the fluid. The tensor of translational resistance has the dimensionality of length and characterizes resistance for the particle’s motion at low Reynolds numbers. Let us show that it is a symmetric tensor. Consider two motions of one and the same particle, in the same fluid, but with different velocities U 0 and U 00 . For these two motions we have the corresponding fields of velocities and stresses (v 0, T 0 ) and (v 00, T 00 ) in the surrounding fluid. As a consequence of the reciprocity theorem (2.2), we have for m0e ¼ m00e : ð S

ð dS T 0 v00 ¼ dS T 00 v0 :

ð2:27Þ

S

Suppose the surface of integration consists of a spherical surface S¥ of large radius (the whole sphere is still positioned inside the fluid) and the particle surface Sa which lies wholly inside the volume bounded by S¥. Then the integral (2.27) can


be written as ð

ð F dS ¼

ð F dS þ

S¥

S

F dS: Sa

2 Since dS r2, and for large r, the fluid velocity is vÐ r1 Ð and the stress is T r , 1 we have vTdS r for large r. Therefore at r ! ¥ , and Eq. (2.27) converts S Sa into

ð

dS T 0 v00 ¼

Sa

ð

dS T 00 v0 :

ð2:28Þ

Sa

On the particle surface S, the boundary conditions v0a ¼ U 0 and va00 ¼ U 00 must be satisfied. A substitution of these relations into Eq. (2.28) yields U 00

ð

dS T 0 ¼ U 0

Sa

ð

dS T 00 :

ð2:29Þ

Sa

Due to Eq. (2.14), we have ð

0

0

ð

dS T ¼ F ; Sa

dS T 00 ¼ F 00

Sa

and Eq. (2.29) reduces to U 00 F 0 ¼ U 0 F 00 :

ð2:30Þ

Let us now substitute (2.30) into (2.26). Then the forces acting on the particle in our two cases are equal to F 0 ¼ me K t U 0 ;

F 00 ¼ me K t U 00 :

ð2:31Þ

Note that one and the same tensor Kt enters the right-hand sides of both formulas, since it depends only on the size and shape of the particle. From Eq. (2.30) and Eq. (2.31) there follows U 00 K t U 0 ¼ U 0 K t U 00 : From the properties of scalar products of tensors and vectors we find that U 0 ðK t U 00 Þ ¼ U 0 ðU 00 ðK t ÞT Þ ¼ U 00 ðK t ÞT U 0

ð2:32Þ

j65


66

and, going back to Eq. 2.32, obtain the following: U 00 K t U 0 ¼ U 00 ðK t ÞT U 0 : Thus, K t ¼ ðK t ÞT :

ð2:33Þ

This equality means that the tensor K t is symmetric, that is, Kitj ¼ K tji . The tensor K c0 in the expression (2.26) for the torque L is called the conjugate tensor. Its dimensionality is the square of length, and it depends only on the particle’s geometry and on the position of the particle’s center 0. In general the tensor K c0 is neither symmetric nor antisymmetric. For rotational motion, we can also introduce the velocity tensor V r0 and the associative pressure vector P r0 satisfying the relations vr0 ¼ V r0 V;

pr0 ¼ me P0r V

ð2:34Þ

and the equations r V r0 ¼ 0; V r0 ¼ « r0

r2 V r0 rP r0 ¼ 0; on

Sa ;

V r0 ! 0 at

r0 ! ¥;

ð2:35Þ

where « is the so-called permutation symbol (Levi–Civita symbol) – a tensor with components

ei jk

8 1; > > < ¼ 1; > > : 0;

if

ði; j; kÞ are

cyclic;

if

ði; j; kÞ are

anticyclic;

otherwise:

These tensors do not depend on me and W at any point inside the fluid; they depend only on the choice of the origin 0 and on the surface geometry of the particle. If the origin 0 coincides with the geometrical center of the particle, then V r0 ¼

a3 r

« r;

Pr0 ¼ 0;

where r is measured relative to the particle’s center. The stress induced in the fluid by the particle’s rotational motion is determined by the tensor T 0r ¼ me P0r V; where we introduced a tensor of the third rank Pr0 ¼ I Pr0 þ ðrV r0 þ T ðrV r0 ÞÞ:


The force and torque acting on the particle during its rotational motion are ð ð F r0 ¼ dS T r0 ; Lr0 ¼ r 0 ðdS T r0 Þ: Sa

Sa

Now, if we define the tensors K c0 and K r0 that are invariable for the given particle by the relations

ð

K c0 ¼

ð dS Pr0 ;

Sa

K r0 ¼

r 0 ðdS Pr0 Þ; Sa

then expressions for the forces and torques experienced by the particle during its rotational motion can be written as

F ¼ me K c0 U 0 ;

L ¼ me K r0 V:

ð2:36Þ

The tensor K r0 is called the rotational resistance tensor and K c0 – its conjugate tensor. Their dimensionalities are, respectively, L3 and L2. The reciprocity theorem implies their symmetry, that is, K c0 ¼ ðK c0 ÞT . Thus it follows from Eqs. (2.21), (2.26), and (2.36) that an isolated particle undergoing translational and rotational motion of in a fluid that is quiescent at the infinity experiences the force and the torque that are given by F ¼ me ðK t U 0 þ ðK c0 ÞT VÞ;

L0 ¼ me ðK c0 U 0 þ K r0 VÞ:

ð2:37Þ

They depend on three independent resistance tensors K t, K r0 , and K c0 characterizing the particle’s inherent properties: surface shape and size. The tensors Kt and K r0 are symmetric, that is, K t ¼ ðK t ÞT and K r0 ¼ ðK r0 ÞT . If a is the characteristic linear size of the particle, then, introducing dimensionless resistance tensors by dividing each dimensional resistance tensor by its corresponding ak, we can rewrite the relations (2.37): ~ t U 0 þ aðK ~ c0 ÞT VÞ; F ¼ me aðK

~ c0 U 0 þ aK ~ r0 VÞ: L0 ¼ he a2 ðK

ð2:38Þ

~ r and K ~ c are called particle shape tensors be~ t; K The dimensionless tensors K 0 0 cause, in contrast to K t ; K r0 ; and K c0 , they do not depend on the particle’s size. The relations (2.37) imply two remarkable properties. The first one is that translational and rotational motions are related in the sense that rotational motion induces translational motion and vice versa. Secondly, the resistance matrices corresponding to tensors K c0 and ðK c0 ÞT , which are responsible for the connection between rotational and translational motions, consist of the same elements. The latter property results from the reciprocity theorem, which can be considered as a

j67


68

manifestation of the general mechanical reciprocity principle (another manifestation of this general principle is the well-known Onsager reciprocity principle). The relations (2.37) can be represented in a compact matrix form. Let us write the vectors of force, torque, translational and rotational velocities as 3 1 matrices, whose elements are the corresponding components in the chosen coordinate system with the vector basis e1, e2, e3: F1 jjFjj ¼ F 2 ; F3

L10 jjL 0 jj ¼ L20 ; L30

U10 jjU 0 jj ¼ U20 ; U30

W1 jjVjj ¼ W2 : W3

From these matrices, we can compose new matrices of generalized force and generalized velocity: kU 0 k kF k kF0 k ¼ : ; ku0 k ¼ kVk kL0 k In a similar manner, we can construct 3 3 resistance matrices t;r;c K 11 t;r;c t;r;c K ¼ K21 0 t;r;c K 31

t;r;c K12 t;r;c K22 t;r;c K32

t;r;c K13 t;r;c K23 K t;r;c 33

and the grand 6 6 resistance matrix T t kK kK c0 kK 0 k ¼ : K c0 K r0 With these notations, the relation (2.37) takes the form kF0 k ¼ me kK 0 k ku0 k:

ð2:39Þ

Since the rate of energy dissipation is ku0 kT kK 0 k ku0 k, the condition that it must be positive implies that the grand resistance matrix is positive definite. Resistance tensors K c0 and K r0 depend on the choice of origin 0. A transition to another coordinate system with the origin P transforms these tensors according to the rule K cp ¼ K c0 r 0 p K t ;

K rp ¼ K r0 r 0 p K t r 0 p þ K c0 r 0 p r 0 p ðK c ÞT ;

where r0p is the vector from 0 to P.


These formulas follow from the requirement that the force of hydrodynamic resistance does not depend on the choice of origin, while the torque and the translational velocity change as L p ¼ L0 r 0 p F;

U p ¼ U 0 þ V r0 p:

For particles whose shape possesses central symmetry, resistance tensors have the simplest form when the origin 0 lies on the particle’s symmetry axis. It is also possible to simplify the expression for the resistance tensor for a body of an arbitrary shape. It turns out that there exists a point called the center of reaction R, relative to which the conjugate resistance tensor is symmetric, that is, K cR ¼ ðK cR ÞT . If the tensor K c0 is known relative to some point 0 inside the particle, the position of the reaction center R can be determined from the relation r 0R ¼ ½ðI : K t ÞIK t 1 « : K c0 For some axisymmetric bodies (spheres, ellipsoids), the symmetric tensor is K cR ¼ 0. In this case the relations (2.37) have an especially simple form: F ¼ me K t U R ;

L0 ¼ me K rR V:

ð2:40Þ

It follows from the last relation that if the reaction center is taken as the origin, then translational and rotational motions are mutually independent. For spherical particles, the reaction center coincides with the center of the sphere, and resistance tensors are equal to K t ¼ 6pme aI;

K rR ¼ 8pme a3 I;

K cR ¼ 0:

For a particle with ellipsoidal surface, x12 x22 x32 þ þ ¼1 a21 a22 a23

ð2:41Þ

the reaction center coincides with the center of the ellipsoid, and in the coordinate system whose origin is placed at the point R and whose coordinate axes are parallel to the ellipsoid’s axes, the resistance tensors are K t ¼ 16p

K rR ¼

1 1 1 e e þ e e þ e e ; 1 1 2 2 3 3 c þ a21 a1 c þ a22 a2 c þ a23 a3

16p a22 þ a23 a23 þ a21 a21 þ a22 e e þ e e þ e e 1 1 2 2 3 3 ; 3 a22 a2 þ a23 a3 a22 a2 þ a21 a1 a21 a1 þ a22 a2

K cR ¼ 0;

ð2:42Þ (2.42)

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70

where ð¥ aj ¼ 0

ð¥ dX dX ; ð j ¼ 1; 2; 3Þ; c ¼ DðXÞ ða2j þ X ÞDðX Þ 0

DðX Þ ¼ ½ða21 þ X Þða22 þ X Þða23 þ X Þ 1=2 : If the force and torque acting on the particle are given, then Eq. (2.39) gives the particle’s velocity: 1 ku0 k ¼ m1 e kK 0 k kF0 k:

ð2:43Þ

The matrix kK 0 k1 ¼ kBk is called the mobility matrix and the corresponding tensor B – the mobility tensor.

2.2 Motion of an Isolated Particle in a Moving Fluid

Consider now the translational and rotational motion of a solid particle of an arbitrary shape in a viscous incompressible fluid whose velocity at the infinity is non-zero and equal to v¥. One encounters such problems when studying motion of fluids with suspended particles inside pipes. The velocity profile of the carrier fluid can be nonuniform over the pipe cross section; this is the case, for example, in Poiseulle and Couette flows. Since the particle size is small and the average distance between particles in low-concentrated mixtures is large as compared to the particle size, we assume a shear velocity profile of the fluid in the vicinity of the particle. Let us put the origin 0 of the coordinate system inside the particle. The translational velocity of the point 0 will be denoted by U0 and the rotational velocity of the particle about the axis passing through 0 will be denoted by W . The fields of velocity v and pressure p inside the fluid are described by Stokes equations (2.1): r v ¼ 0;

r2 v ¼

1 rp me

ð2:44Þ

with inhomogeneous boundary conditions v ¼ U0 þ V r0

on

Sa ;

v ! v¥

at

r0 ! ¥;

ð2:45Þ

where r0 is the radius vector of a point relative to the origin. The velocity field at the infinity, v¥(r), should also satisfy the appropriate hydrodynamic equations (for Poiseulle and Couette flows, these are Stokes equations


(2.44)). We assume that the unperturbed velocity profile at the infinity corresponds to shear flow. Let us introduce the tensor of velocity gradient at the infinity g¥ ¼ rv¥ with components gi¥j ¼ r j v¥i . It is evident that the symmetric part of this tensor E ¥ ¼ 0.5(g¥ þ (g¥)T) with components Ei¥j ¼ 0:5ðgi¥j þ g ¥ji Þ is the rate-of-strain tensor and the antisymmetric part L¥ ¼ 0.5(g¥ (g¥)T) with components L¥i j ¼ 0:5ðgi¥j g ¥ji Þ is the vorticity tensor. The latter is related to the vorticity vector V¥, which stands for the rotational velocity of the fluid in an unperturbed flow, by the expression V¥ ¼ 0.5e L, or, in the component form, V¥i ¼ 0:5ei jk Lk j . Then the unperturbed velocity, that is, velocity at the infinity, is written as v ¥ ¼ v 0 þ V¥ r 0 þ E ¥ r 0

ð2:46Þ

or, in the component form, v¥i ¼ v0i þ ei jk W¥i xk þ Ei¥j x j ;

ð2:47Þ

where v0 is the velocity of point 0 as if it were in the fluid, r0 is the radius vector (relative to point 0) of the point under consideration, and xj are the radius vector components. It should be noted that W¥ and E¥ are constants and do not depend on the choice of origin 0. To solve the boundary value problem defined by Eqs. (2.44), (2.45), one must exploit the linearity of the problem and seek the solution as a superposition of particular solutions ðv00 ; p00 Þ; ðv000 ; p000 Þ; and ðv000 ; p000 Þ; the first two of which are solutions of the boundary value problems r v00 ¼ 0; v00 ¼ E ¥ r 0

on

r2 v00 ¼ Sa ;

r v000 ¼ 0; v000 ¼ ðU 0 v¥ Þ þ ðVV¥ Þ r 0

1 r p00 ; me

v00 ! 0

r2 v000 ¼ on

at

r0 ! ¥;

1 r p000 ; me

Sa ;

v000 ! 0

at r0 ! ¥;

and the third one is equal to v000 ¼ v0 þ V¥ r 0 þ E ¥ r 0 ;

p000 ¼ 0:

in the entire region. It should be noted that the fields ðv00 ; p00 Þ and ðv000 ; p000 Þ depend on the choice of origin 0, while the field ðv000 ; p000 Þ and the net field v ¼ v00 þ v000 þ v000 ; p ¼ p00 þ p000 þ p000 do not depend on it. Having determined the velocity and pressure fields, one can find the stresses T 0, 00 T and T 000 . It is evident that T 000 ¼ 2me E ¥ in the whole region occupied by the fluid. It follows from Eq. (2.14) that the force and torque experienced by the particle can also

j71


72

be expressed as a sum of corresponding forces and torques: F ¼ F 00 þ F 000 þ F 000 ;

L0 ¼ L00 þ L000 þ L000 :

Their components are equal to F 00 ¼ me K t0 : E ¥ ;

L00 ¼ me K r0 : E ¥ ;

F 000 ¼ me ðK t ðU 0 v¥ Þ þ ðK c0 ÞT ðVV¥ ÞÞ; L000 ¼ me ðK c0 ðU 0 v¥ Þ þ K r0 ðVV¥ ÞÞ; F 000 ¼ 0;

L000 ¼ 0:

To summarize, the hydrodynamic force and torque acting on the particle are: F ¼ me ðK t ðU 0 v¥ Þ þ ðK c0 ÞT ðVV¥ Þ þ K t0 : E ¥ Þ; L0 ¼ me ðK c0 ðU 0 v¥ Þ þ K r0 ðVV¥ Þ þ K r0 : E ¥ Þ;

ð2:48Þ

A comparison with the relation (2.37) for the force and torque acting on a particle moving in a quiescent fluid shows that in a shear flow, additional terms appear in the relations for the corresponding force and torque. These terms are proportional to the rate-of-strain tensor of an undisturbed flow E¥, with third-rank tensors K t0 and K r0 being the proportionality factors. These tensors are called the tensors of translational and rotational shear resistance; they characterize the particle’s intrinsic properties. Currently, there exist exact solutions of the problem of motion of spherical and ellipsoidal particles in a shear flow [6]. Presented below are some results for the case when the origin 0 is made to coincide with the reaction center R. For an ellipsoidal particle (2.41), the tensors K tR and K rR are K tR ¼ 0; K rR

8p a23 a22 ¼ ðe1 e2 e3 þ e1 e3 e2 Þ 2 3 a2 a2 þ a23 a3 þ

a21 a23 a22 a21 ðe e e þ e e e Þ þ ðe e e þ e e e Þ : 2 3 1 2 1 3 3 1 2 3 2 1 a23 a3 þ a21 a1 a21 a1 þ a22 a2 ð2:49Þ

In the particular case of spherical particle, we have K tR ¼ K rR ¼ 0; and the other tensors are given in Section 2.1. Formulas (2.37) and (2.48) allow us to find the forces and torques acting on particles undergoing translational and rotational motion with given velocities. For a particle freely suspended in the fluid, the force and torque are equal to zero. From


these conditions, one can find the translational and rotational velocities of the particle: U 0 v¥ ¼ ½ðK t ðK c0 ÞT ðK r0 Þ1 K c0 1 ½K t0 ðK c0 ÞT ðK r0 Þ1 K r0 Þ : E ¥ ; VV¥ ¼ ½ðK t K c0 ðK t Þ1 ðK c0 ÞT Þ1 ðK r0 K c0 ðK t0 Þ1 K t0 Þ : E ¥ : The relation (2.48) can be written in a matrix form similar to Eq. (2.39). One complication is that K t0 and K r0 are third-rank tensors with elements K ti jk ; K ri jk and they cannot be represented in a matrix form as simply as other terms. Consider the components of vectors F 00 and L00 : F0i0 ¼ me Kitjk Ek¥j ;

L00i ¼ me Kirjk Ek¥j

ð2:50Þ

in more detail. For convenience’s sake, we have dropped the index 0 in the tensors K t0 and K r0 , and will follow this convention from now on. As far as (i, j, k) ¼ (1, 2, 3), each of the tensors Kt and Kr has 27 elements. Due to the symmetry of the rate-ofstrain tensor E¥, we have E ¥jk ¼ Ek¥j ; Kitjk ¼ Kikt j , and Kirjk ¼ Kikr j . Therefore each of the tensors Kt and Kr has only 18 independent components. To represent the last terms in Eq. (2.48) in the matrix form, let us proceed in the following way. First, we shall use the symmetry of tensors Kt and Kr with respect to the last two indices and the symmetry of the tensor E¥ to replace the pair of indices (jk) with one index l according to the rule below:

tensor indices matrix indices

( jk) (l)

11 1

22 2

33 3

23, 32 4

31, 13 5

12, 21 6

t t r r Furthermore, since K112 ¼ K121 ; K112 ¼ K121 ; and so on, let us denote 1 1 r r t t K112 ¼ K121 ¼ F16 ; K112 ¼ K121 ¼ t16 ; etc. 2 2 In the same way, we make a transition from the matrix E¥(Eij) to the matrix S. As a result, the tensor Kt with components Kitjk (enumerated in the table below)

t K111

i¼1 t t K112 K111 t K122

t K123 t K133

t K211

i¼2 t t K212 K213 t K222

t K223 t K233

turns into a matrix F with components Fij,

t K311

i¼3 t K312

t K313

t K322

t K323 t K333

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74

F11

1 F16 2

1 F15 2

F12

1 F14 2

F21

1 F26 2

1 F25 2

F22

1 F24 2

F13

F31

1 F36 2

1 F35 2

F32

1 F34 2

F23

F33

and the tensor E ¥ with components Eij, E11

E12

E13

E22

E23 E33

turns into a matrix S with components S1

S6

S5

S2

S4 S3

The transition from the tensor K r to a matrix t(tij) is handled in a similar manner. As a result, the relation (2.50) reduces to F 00i ¼ me Kitjk Ek¥j ¼ me Fil Sl ;

L00i ¼ he Kirjk Ek¥j ¼ me til Sl ;

ð2:51Þ

where i ¼ 1, 2, 3 and l ¼ 1, 2, . . ., 6. 0 Letusnow introduce a generalized force vector F0 written in the column form, so 0 0 0 T that F0 ¼ F 0 ; L0 ; a shear vector J, also written in the column form, so that || J||T ¼kS1, S2, S3, S4, S5, S6||; and a matrix of shear resistance ||Y||, F11 F21 F F31 kCk ¼ ¼ t t11 t21 t31

F12

F13

F14

F15

F22

F23

F24

F25

F32

F33

F34

F35

t12

t13

t14

t15

t22

t23

t24

t25

t32

t33

t34

t35

F16 F26 F36 : t16 t26 t36

The relation (2.51) may now be written in the matrix form: 0 F ¼ m Yk J k e 0

ð2:52Þ


Combining it with (2.39), we get kF0 k ¼ me ðkK 0 k ku0 k þ kCk k JkÞ:

ð2:53Þ

For a particle freely suspended in the fluid, we use (2.53) with F0 ¼ 0, obtaining both translational and rotational velocities of the particle: ku0 k ¼ kK0 k1 kCk k J k:

ð2:54Þ

Note that the matrices kK 0k and kFk depend only on the intrinsic properties of the particle, and for a given class of geometrical shapes, they can be determined once and for all. If the particle’s shape possesses some kind of symmetry, then the task of finding the elements of matrices K0 and F is considerably simplified. For example, consider an axisymmetric particle. Introduce a Cartesian coordinate system with the basis e1, e2, e3, where e3 is directed along the symmetry axis. For axisymmetric bodies, the tensors that were introduced in the previous discussion become K t ¼ aðIe3 e3 Þ þ be3 e3 ;

K c0 ¼ cð« e3 Þ;

K t0 ¼ f e3 e3 e3 þ gððIe3 Þ þ ðIe3 ÞT Þ;

K r0 ¼ dðIe3 e3 Þ þ ee3 e3 ;

K r0 ¼ hðð« e3 e3 Þ þ ð« e3 e3 ÞT Þ;

where the scalar factors a, b, . . . depend only on the geometrical properties of the particle. Using the definitions of polyadics abc ¼ aibjck and Aa ¼ Aijak, of the scalar product A a ¼ Aijaj and of the permuation symbol « ¼ e1 e2 e3 e1 e3 e2 þ e2 e3 e1 e2 e1 e3 þ e3 e1 e2 e3 e2 e1 we can rewrite these relations in the matrix form: a t K ¼ 0 0 d r K0 ¼ 0 0

0 a 0 0 d 0

0 0 ; b

0 c K0 ¼ c 0

0 0 ; e

0 c T ðK 0 Þ ¼ c 0

c 0 0

0 0 ; 0 c 0 0

0 0 : 0

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76

The grand resistance matrix ||K 0|| and the shear resistance matrix ||C|| are a 0 0 0 a 0 0 0 b kK 0 k ¼ d 0 0 0 d 0 0 0 e

0

c

c

0

0

0

0

c

c

0

0

0

0 0 0 ; 0 0 0

kqk kCk ¼ k0k

kCk ; k xk

where 0 kqk ¼ 0 0

0 0 0

0 0 ; kCk ¼ 2g 0 f þ 2g

2g

0

0 0

2h 0 0 ; kxk ¼ 0 0 0

0 2h 0

0 0 ; 0

and ||0|| is a 3 3 zero matrix. The inverse resistance matrix (mobility matrix) is m 0 0 1 kK 0 k ¼ 0 cmd1 0

0

0

0

cna1

m

0

cna1

0

0

b1

0

0

0

n

0

0

0

0

n

0

0

0

0

cmd

1

0 0 0 ; 0 0 1 e

where m¼

d ; adc 2

n¼

a : adc 2

Now, using Eq. (2.47), we can find the velocity of a freely suspended axisymmetric particle: kS2 0 U 0 v ¼ kS1 ; lS3 where k ¼

nS4 ¥ VV ¼ nS5 ; 0

2ðchgdÞ ; adc 2

l¼

f þ 2g ; b

ð2:55Þ

n¼

2ðgcahÞ : adc 2


It is readily seen that an axisymmetric particle freely suspended in a fluid rotates with the same angular velocity as the fluid in a shear flow at the infinity. Consider now an ellipsoidal particle with semi-axes a1, a2 and a3. Put the origin 0 into the center of the ellipsoid and direct the basis vectors {ei} along the axes of the ellipsoid. Using the corresponding tensors Kt, K rR ; K cR and the parameters a1, a2 and a3 (see Section 2.1), we get 0 0 0 16p 0 kCk ¼ 3 0 0

0

0

0

0

0

0

0

0

0

0

0

0

0

0

a23 a22 2 a3 a3 þ a22 a2

0

0

0

0

a21 a23 2 a1 a1 þ a23 a3

0

0

0

0

0 0 0 : 0 2 2 a2 a1 a22 a2 þ a21 a1 0

ð2:56Þ Similar relations for a spherical particle could be obtained from Eq. (2.42) and Eq. (2.56) by taking a1 ¼ a2 ¼ a3 ¼ a. For circular disk, we would have to take a1 ¼ a2, a3 ¼ 0. As an example, consider a circular disk freely suspended in a fluid. The origin of coordinates is placed into center of the disk, the basis i1, i2, i3 is fixed in space, and the unperturbed fluid velocity is given by 1 v¥ ¼ ˙gðx1 i1 þ x2 i2 2x3 i3 Þ: 2 For such flow, we have 1 v0 ¼ 0;

W¥ ¼ 0;

1 _ E ¥ ¼ gðI3i 3 i3 Þ: 2

By determining the matrices ||K 0||1 and ||F||, one can convince himself that U0 ¼ 0 (the translational velocity vanishes) and that the angular velocity is V ¼ S4 e1 S5 e2 : The basis vector e3 of the local coordinate system is normal to the plane of the disk. If q is the angle between the vector i3 and the plane of the disk, and j is the angle between the projection of i3 on that plane and one of the local axes, then 3 S4 ¼ E23 ¼ g_ sin q cos q sin j; 2

3 _ S5 ¼ E13 ¼ gsin q cos q cos j: 2

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78

2.3 Motion of Two Particles in a Fluid

Consider slow motion of two particles a and b in an unbounded viscouss incompressible fluid. The particles’ translational and rotational velocities are, respectively, Ua, Ub and Va, Vb. The Reynolds numbers of the particles are determined by their parameters and are assumed to be small, so the fluid flow is Stokesian and can be described by Stokes equations r v ¼ 0;

r2 v ¼

1 rp me

ð2:57Þ

with the boundary conditions v ¼ U a þ Va r a v!v

¥

at

on

Sa ;

v ¼ U b þ Vb r b

on

Sb ;

jr 0 j ! ¥:

ð2:58Þ

Here r0 is the radius vector of the considered point with respect to a local origin 0 that is fixed inside the fluid; ra and rb are the radius vectors of this point with respect to the centers 0a and 0b of particles a and b; v¥ ¼ v0 þ W¥ r0 þ E¥ r0 is the velocity of the fluid far from both the particles (at the infinity). Exact, approximate, and numerical solutions have been derived for slow translational and rotational motions of two spherical particles in a quiescent fluid and in a shear flow [7–31]. When one particle is much larger then the other, the motion of the smaller particle in close proximity to the larger one can be treated as motion near a plane. Particle motions in a quiescent fluid (v¥ ¼ 0) and in a moving flow (v¥ 6¼ 0) are considered separately. 2.3.1 Fluid is at Rest at the Infinity (v¥ ¼ 0)

The problem of obtaining the fields of velocity v and pressure p in the fluid that are induced by translational and rotational motions of two spherical particles of radii a and b reduces to the following boundary value problem: r v ¼ 0;

r2 v ¼

v ¼ U a þ Va r a

1 rp me on

Sa ;

v ¼ U b þ Vb r b

on Sb ;

at jr 0 j ! ¥: ð2:59Þ

Thanks to its linearity, it can be treated as a superposition of simpler problems. If the vectors of translational velocity are represented in terms of two components – along the line of centers and perpendicular to it, the problem is separated into three


particular problems: translational motion of particles with different velocities along their line of centers (axisymmetric problem); translational motion of particles with different velocities perpendicular to the line of centers (non-axisymmetric problem); and rotation of particles with different angular velocities. Each of these motions, in its turn, can be divided into two submotions where one of the particles is at rest. It should be noted that by setting the radius of one of the particles to infinity one obtains the solution for the problem of particle motion near a plane wall, which can be considered as an asymptotic solution of the problem of relative motion of two particles with substantially different sizes. If the particles are drops of fluid whose viscosity differs from that of the surrounding fluid, then particle rotation is a meaningless term, and we should assume Wa ¼ Wb ¼ 0. Boundary conditions on the particle surface differ from (2.59), because we must satisfy the condition of equality of velocities and stresses at the interface between the internal and external fluids. As opposed to the surface of a solid particle, drop surface deform under the action of nonuniform stresses, which become especially noticeable when the minimal clearance between drops becomes much smaller than the drop size. All of this greatly complicates the problem. The discussion that follows will apply to solid particles only. Exact and approximate solutions exist for the three above-mentioned problems, making it possible to determine the forces and torques acting on moving particles for two limiting cases: when the particles are relatively far from each other (the ‘‘far asymptotic region’’), and when the minimal clearance between particle surfaces is small compared to particle radii (the ‘‘near asymptotic region’’). An approximate solution suitable for all interparticle distances can be found by using the method of asymptotic expansions. To derive the exact solution, we introduce the stream function, which allows to make a transition from equations (2.57) to a single fourth order partial differential equation for the stream function. In a curvilinear bi-spherical coordinate system, the region between the two spheres transforms into a region between two parallel plates, and the new boundary value problem is solved by using the method of separation of variables. The variables are given by infinite series whose convergence rate decreases as the gap between particles gets smaller. This is why using an asymptotic solution is the preferred way of finding the hydrodynamic forces and torques acting on a sphere in the vicinity of another sphere (or a plane wall). In the case of more then two particles, as well as for two particles of non-spherical shape, it is impossible to find the appropriate coordinate system. This is why an exact solution exists only for a system of two spherical particles. The distinctive feature of the solution for two particles is the singularity of some elements of the resistance matrix (resistance factors) when the minimal interparticle distance tends to zero. This means that most of the influence particles exert on each other is confined to the near asymptotic region. This is why hydrodynamic forces and torques (together with molecular and electrostatic interparticle forces) play a key role in the phenomena of aggregation and coagulation of particles. When the particles are relatively far from each other, that is, when the interparticle distance is equal to several particle radii, elements of the resistance tensor are nearly

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equal to the corresponding values for an isolated particle. To derive hydrodynamic forces and torques for the case when the particles are relatively far from each other, we can use the regular method of successive approximations, which allows to solve the boundary value problem (2.59) to any approximation by considering at each step only the boundary conditions on one of the spheres. This method is called the reflection method. It was first applied by Smoluchowski to the problem of hydrodynamic interaction of n particles separated by large distances. Let us look at translational motion of n identical spherical particles with different velocities Uk(k ¼ a, b, c,. . .,n). Our goal is to find the forces Fk, and torques Lk acting on the particles and ensuring their translational motion with the given velocities Uk. The boundary value problem reduces to Eqs. (2.50) with the boundary conditions v ¼ Uk

on

Sk ;

ðk ¼ a; b; c; . . .; nÞ;

v!0

at r ! ¥:

Linearity of the problem allows us to seek the solution in the form of infinite sums: v¼

¥ X vðiÞ ; i¼1

p¼

¥ X

pðiÞ ;

i¼1

where each term (v(i), p(i)) obeys Eqs. (2.57) and the zero boundary conditions at the ðiÞ ðiÞ infinity and in its turn is represented as a finite sum of fields ðvk ; pk Þ; vðiÞ ¼

n X ðiÞ vk ; k¼b

pðiÞ ¼

n X

ðiÞ

pk ;

k¼b

also obeying Eqs. (2.57) and the zero boundary conditions at the infinity. Let now choose some particle, for example particle a, as the test particle and obtain the field (v(1), p(1)) that satisfies the boundary condition v(1) ¼ Ua on Sa. The reflection of this field from particle b is derived by using the boundary condition v(2) ¼ Ub v(1) on Sb. ð2Þ In general, the reflection of v(1) from the other n 1 particles is vk ¼ U k vð1Þ ; (1) where k ¼ b, c, . . ., n. Thus the total reflection of the velocity field v of particle Sa from the other n 1 particles is vð2Þ ¼

n X ð2Þ vk : k¼b

It is now possible to estimate (to the first approximation) the influence of (n 1) particles on the test particle via the velocity v(3) resulting from the reflection of the field v(2) from this particle. The field v(3) obeys the boundary condition v(3) ¼ v(2) on Sa. Therefore terms of the order a/l, where l is the distance between particle centers, appear in the expression for v(3). To summarize, the velocity in the vicinity of particle a is determined up to the terms of order a/l: v vð1Þ þ vð3Þ :


In the same way, one could find velocity to the first approximation for any other particle chosen as the test particle. The next approximation is derived by continuing the process of reflection, taking ð4Þ the field v(3) instead of v(1). Reflection vb from particle b, that is, from Sb, is equal to ð4Þ ð3Þ vb ¼ v , and so on. The net reflection is vð4Þ ¼

n X ð4Þ vk : k¼b

The second approximation of the velocity field v(5) in the neighborhood of particle a satisfies the boundary condition v(5) ¼ v(4) on Sa and ensures accuracy to the order of a2/l2. The next approximations give the velocity field as a power series of a/l. If the particles are far away from each other, one can get a good approximation by assuming the velocity field induced in the fluid by a particle to be the same as the velocity field induced by a point force and (or) by a point-pair force applied at the particle center. Then the resistance force exerted on the given particle by the reflected field can be obtained by considering this field as equivalent to a uniform velocity field with the same magnitude and direction as the velocity that would exist at the particle’s location if the particle itself were absent. The obtained approximations of velocity and pressure fields make it possible to find the force and torque on the particle from Eqs. (2.14) and (2.15): ð3Þ ð5Þ F a ¼ F ð1Þ a þ F a þ F a þ . . .;

ð3Þ ð5Þ La ¼ Lð1Þ a þ La þ La þ . . . :

In the case of two particles a and b separated by a large distance, forces and torques have the form of power series in l/a and l/b. Another limiting solution can be derived in the case when the gap between the surfaces is small as compared to the particles’ radii a and b. In this case there appears a small parameter equal to the ratio of the minimum clearance to the particle radius ( 1 and k ¼ b/a, b a), which allows to obtain a solution by employing the method of asymptotic expansions in a local cylindrical coordinate system (ar, q, az) with the origin at the intersection of the line of centers and the spherical surface Sb and the z-axis directed along the line of centers, while the r-axis lies in the plane tangential to Sb. In this coordinate system, the equations of spheres Sa and Sb become z ¼ 1 þ eð1r 2 Þ1=2 ;

z ¼ k1 þ k1 ð1k2 r 2 Þ1=2 :

If we now introduce new scaled coordinates Z ¼ r/e and R ¼ r/e1/2 instead of z and r, the equations of the spheres assume an asymptotic form: Z ¼ 1 þ 0:5R2 þ 0ðeÞ;

Z ¼ 0:5kR2 þ 0ðeÞ:

Stokes equations and boundary conditions at particle surfaces can be transformed in a similar way, after which we can extract the principal terms in asymptotic expansions of the velocity and pressure fields. The concrete form of these

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82

expansions will depend on the type of particle motion: translational motion parallel or perpendicular to the line of centers, or rotational motion around these axes. A distinctive feature of near asymptotic expansions is the singularity of almost all resistance coefficients in the grand resistance matrix, with the exception of coefficients corresponding to the rotation of the particles around their line of centers. Taking together the far asymptotic and near asymptotic solutions, we can use the method of matching of asymptotic expansions to get approximate solutions for all kinds of particle motions for the entire range of relative distances between the particles. In doing that, we make use of the fact that due to the linearity of Stokes equations, the general equation describing translational motion with velocities Ua and Ub and rotational motion with angular velocities Wa and Wb can be thought of as a superposition of simple motions in which one of the particles is assumed to be at rest. It was shown in Section 2.1 that forces Fa and torques La acting on the particles a ¼ 1, 2 (here and later, particles will be indicated by indices 1 and 2) are connected with their velocities through the relation t F1 K 11 F2 K t 21 ¼ L1 K c 11 L2 K c 21

t K12

c T ðK11 Þ

t K22

c T ðK21 Þ

c K12

r K11

c K22

r K21

c T ðK12 Þ c T ðK22 Þ r K12 r K21

U1 U2 : W1 W2

ð2:60Þ

The grand resistance matrix contains tensors K tab ; K cab and K rab , whose elements (where a, b ¼ 1, 2 are used to index the particles and i, j ¼ 1, 2 indicate the tensor elements) obey the conditions of symmetry, some of which are valid for particles of an arbitrary shape and some follow from the geometrical properties of a system of two spherical particles. The former type of symmetry conditions follows from the reciprocity theorem (see Section 2.1): ðabÞ Kij

ðabÞt

Ki j

ðabÞt

¼ K ji

;

ðabÞc T

ðKi j

ðbaÞc

Þ ¼ K ji

;

ðabÞr

Ki j

ðbaÞr

¼ K ji

:

ð2:61Þ

Let us prove the first condition. Consider two problems that involve translational motion of two spherical particles S1 and S2 in one and the same fluid. In the first problem particle S1 moves with the constant velocity U1 and the second particle is at rest: U2 ¼ 0. In the second problem, the opposite is true: the first particle is at rest, U1 ¼ 0, and the second one moves with the constant velocity U2. The velocity and stress fields in the two problems are designated as (v 0, T 0 ) and (v 00, T 00 ). From the reciprocity theorem (2.2), there follows that at m0e ¼ m00e ; ð S

ð ds v0 T 00 ¼ ds v00 T 0 :

ð2:62Þ

S

Form the surface S from the surface of a sphere S¥ with a large radius and the surfaces of two spheres S1 and S2 placed inside the sphere S¥. Since ds r2, while v 0 r1, v 00 r1, and T r2, we have for r ! ¥:


ð

ð ¼

S

ð þ

S¥

ð þ

S1

ð

S2

ð þ

S1

:

ð2:63Þ

S2

The following boundary conditions are valid at the surfaces S1 and S2: v0 ¼ U 1 ;

v00 ¼ 0

on

S1 ;

v0 ¼ 0;

v00 ¼ U 2

on

S2 ;

Therefore the relations (2.62) and (2.63) first reduce to ð ð U 1 ds T 00 ¼ U 2 ds T 0 S

S

and then, due to Eq. (2.14), to ð2Þ

ð1Þ

U 1 F1 ¼ U 2 F2 :

ð2:64Þ

ð2Þ

where F 1 is the resistance force acting on particle S1 and induced by the motion of ð1Þ particle S2, and F 2 – the resistance force acting on particle S2 and induced by the motion of particle S1. From Eq. (2.60), it follows that in the case of translational motion (W1 ¼ W2 ¼ 0) the forces acting on the particles S1 and S2 are ð2Þ

F 1 ¼ K t12 U 2 ;

ð1Þ

F 2 ¼ K t21 U 1 :

ð2:65Þ

A substitution of Eq. (2.65) into Eq. (2.64) yields U 1 K t12 U 2 ¼ U 2 K t21 U 1 :

ð2:66Þ

Using the properties of the scalar product of a tensor and a vector, one obtains: U 2 ðK t12 U 1 Þ ¼ U 2 ðU 1 ðK t21 ÞT Þ ¼ U 1 ðK t21 ÞT U 2 : and hence, U 1 K t12 U 2 ¼ U 1 ðK t21 ÞT U 2 and K t21 ¼ ðK t21 ÞT :

ð2:67Þ ðabÞt

This means that the tensor K t21 is symmetric, and its elements K i j conditions ðabÞt

Kij

ðbaÞt

¼ K ji :

The other relations in (2.61) can be proved in the same way.

obey the

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84

Geometry of the system of two spherical particles gives rise to a dependence of tensors Kab on the relative position of the particles, that is, on the vector r ¼ x2 x1 connecting the particle centers x1 and x2, and on the particle radii a1 and a2. It is obvious that physical essence of the problem does not change if we interchange the particle indices 1 and 2 and replace r with r. So, the conditions K ab ðr; a1 ; a2 Þ ¼ K ð3a;3bÞ ðr; a2 ; a1 Þ:

ð2:68Þ

must hold. Introduce a new coordinate system with the basis vectors e ¼ r/r (parallel to the line of centers) and i, j (perpendicular to each other and to the line of centers). Then the symmetry conditions (2.68) allow us to express components of the tensor Kab in terms of two scalar functions Xab and Yab: ðabÞt

t t ¼ Xab ei e j þ Yab ðdi j ei e j Þ;

ðabÞc

¼ ðK ji

ðabÞr

r r ¼ Xab ei e j þ Yab ðdi j ei e j Þ:

Kij Kij Kij

ðabÞc T

c Þ ¼ Yab ei jk ek ;

ð2:69Þ

where d ij is the Kronecker symbol and eijk – the permutation symbol. Let us replace dimensional tensors Kab with dimensionless ones: ~t ¼ K ab

K tab ; 3pðaa þ ab Þ

~c ¼ K ab

K cab 3pðaa þ ab Þ

2

;

~r ¼ K ab

K rab 3pðaa þ ab Þ3

;

Also, we introduce the dimensionless distance s between the particles and the ratio k of particle radii: s¼

2r ; a1 þ a2

k¼

a2 : a1

The symmetry conditions (2.61) and (2.68) will then rearrange to 1 ~ t ðs; kÞ ¼ X ~t ~ t ðs; kÞ ¼ X X ab ba ð3a;3bÞ ðs; k Þ; 1 ~ t ðs; kÞ ¼ Y ~t ~ t ðs; lÞ ¼ Y Y ab ba ð3a;3bÞ ðs; k Þ; 1 ~t ~ c ðs; kÞ ¼ Y Y ab ð3a;3bÞ ðs; k Þ; 1 ~ r ðs; kÞ ¼ X ~r ~ r ðs; kÞ ¼ X X ab ba ð3a;3bÞ ðs; k Þ; 1 ~ r ðs; kÞ ¼ Y ~r ~ r ðs; kÞ ¼ Y Y ab ba ð3a;3bÞ ðs; k Þ:

ð2:70Þ

The conditions (2.70) mean that the forces and torques acting on the particles are ~ cab and 3 functions for each of the specified by 16 scalar functions (4 functions for Y


~ tab ; Y ~ tab ; X ~ rab ; Y ~ rab ) that depend on s 2 [2,¥) and k 2 [0,1]. Each set of funcfollowing: X tions defined by superscripts t, c and r is responsible for a specific type of particle ~ tab are the resistance coefficients for translational motion of particles motion. Thus, X ~ t – resistance coefficients for translational motion along their line of centers; Y ab ~ c – resistance coefficients for rotational perpendicular to the line of centers; Y ab r ~ – for rotational motion around the line of centers motion around the axis j; X ab ~ r – for rotational motion around the axis i. (axis e); Y ab

Thus, scalar functions provide a suitable description of all kinds of relative motions of two solid spherical particles. Therefore, Eq. (2.60) allows to determine the forces and torques that act on particles moving in an arbitrary way as functions of translational and rotational velocities Ua and Wa, relative distance s between the particles, and the ratio k of particle radii. Both exact and asymptotic solutions of all the above-formulated problems about motion of two spherical particles in an unbounded fluid that is quiescent at the infinity are currently well-known. Of utmost interest are the approximate solutions obtained by splicing the far asymptotic solution and the near asymptotic solution because their relatively simple form comes in handy when we try to solve more complicated problems that involve, for example, the hydrodynamic behavior of a particle ensemble. Listed below are the expressions for dimensionless resistance functions that appear in Eq. (2.60), derived for various types of particle motions. Translational motion along the line of centers (axis e): Far asymptotic region (s ! ¥): ¥ X 1 1 ~ t11 ¼ X f2k ðkÞ ; 2k s2k ð1 þ kÞ k¼0 ~ t12 ¼ X

¥ 1 X 1 1 f2kþ1 ðkÞ ; ð1 þ kÞ k¼0 ð1 þ kÞ2kþ1 s2kþ1

where f0 ¼ 1; f1 ¼ 3k; f2 ¼ 9k; f3 ¼ 4k þ 27k2 4k3; f4 ¼ 24k þ 81k2 þ 36k3; f5 ¼ 72k2 þ 243k3 þ 72k4; f6 ¼ 16k þ 108k2 þ 281k3 þ 648k4 þ 144k5; f7 ¼ 288k2 þ 1620k3 þ 1515k4 þ 1629k5 þ 288k6; etc. Owing to the first condition (2.70), the following equality is valid: f2kþ1 ðkÞ ¼ k2kþ2 f2kþ1 ðk1 Þ: Near asymptotic region (s ! 2 or x ! 0): It is convenient to introduce the dimensionless distance (clearance) between the particles x¼

ra1 a2 ¼ s2: 0:5ða1 þ a2 Þ

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86

At x 1 and x k we have: ~ t11 ¼ g1 ðkÞ 1 þg2 ðkÞln 1 þ 0ð1Þ; X x x ~ t11 ¼ X

2 1 1 g1 ðkÞ þg2 ðkÞln þ 0ð1Þ; 1þk x x

where 2k2

g1 ðkÞ ¼

3

ð1 þ kÞ

;

g2 ðkÞ ¼

kð1 þ 7k þ k2 Þ 5ð1 þ kÞ3

:

Intermediate region (2 < s < ¥): ~ t11 ¼ X

g1 g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ ð14s2 Þ

þ f0 ðkÞg1 þ

¥ X m¼2 ðmevenÞ

1 2g2 4g3 fm ðkÞg1 þ m m m1 2m ð1 þ kÞm

m 2 ; s

1 2g1 sþ2 sþ2 t 2 ~ ð1 þ kÞX 12 ¼ g2 ln þ g3 ð14s Þ ln sð14s2 Þ 2 s2 s2

þ4

m ¥ X g3 1 2g2 4g3 2 þ þ f ðkÞg ; m 1 m m s m m m1 2 ð1 þ kÞ s m¼2 ðmevenÞ

where m1 ¼ 2dm2 þ ðm2Þð1dm2 Þ; g3 ¼

1 ð1 þ 18k29k2 þ 18k3 k4 Þð1 þ kÞ3 : 42

Translational motion perpendicular to the line of centers (axis i): Far asymptotic region (s ! ¥): ~t ¼ Y 11

¥ X k¼0

~t ¼ Y 12

f2k ðkÞ

1

1 2k s2k

ð1 þ kÞ

;

¥ 2 X 1 1 f2kþ1 ðkÞ ; ð1 þ kÞ k¼0 ð1 þ kÞ2kþ1 s2kþ1


where f0 ¼ 1;

3 f1 ¼ k; 2

f4 ¼ 6k þ

81 2 k þ 18k3 ; 16

f6 ¼ 4k þ 54k2 þ

f7 ¼ 144k2 þ

9 f2 ¼ k; 4 f5 ¼

f3 ¼ 2k þ

27 2 k þ 2k3 ; 8

63 2 243 3 63 4 k þ k þ k ; 2 32 2

1241 3 k þ 81k4 þ 72k5 ; 64

1053 3 19083 4 1053 5 k þ k þ k þ 144k6 ; 8 128 8

etc:

Near asymptotic region (x 1 and x k): ~ t ¼ g2 ðkÞ ln 1 þ 0ð1Þ; Y 11 x ~ t ¼ 2 g2 ðkÞ ln 1 þ 0ð1Þ; Y 12 1þk x where g2 ðkÞ ¼

4kð2 þ k þ 2k2 Þ 15ð1 þ kÞ3

:

Intermediate region (2 < s < ¥): ~ t11 ¼ g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ þ f0 ðkÞ Y

þ

¥ X

m¼2 ðmevenÞ

1 2g2 4g3 fm ðkÞ þ m m m1 2m ð1 þ kÞm

m 2 ; s

1 sþ2 sþ2 t 2 ~ ð1 þ kÞY 12 ¼ g2 ln þ g3 ð14s Þ ln 2 s2 s2 þ4

m ¥ g3 X 1 2g2 4g3 2 þ þ f ðkÞ ; m m s m¼2 2m ð1 þ kÞ m m m1 s ðmoddÞ

where g3 ðkÞ ¼

2 ð1645k þ 58k2 457k3 þ 16k4 Þð1 þ kÞ3 : 375

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88

Rotation around the axis perpendicular to the line of centers and to the direction of transverse translational motion (axis j): Far asymptotic region (s ! ¥): ~c ¼ Y 11

¥ X

f2kþ1 ðkÞ

k¼0

~c ¼ Y 12

1

¥ X

4 2

ð1 þ kÞ

1

ð1 þ kÞ2kþ1 s2kþ1

k¼0

f2k ðkÞ

1

; 1

ð1 þ kÞ2k s2k

;

where f0 ¼ f1 ¼ 0; f2 ¼ 6k; f3 ¼ 9k; f4 ¼

27 2 k ; 2

f5 ¼ 12k

81 2 k 36k3 ; 4

243 3 k 72k4 ; 8 8409 3 k 243k4 144k5 ; f7 ¼ 189k2 þ 16 f6 ¼ 108k2

etc:

Near asymptotic region (x 1 and x k): ~ c ¼ g2 ðkÞ ln 1 þ 0ð1Þ; Y 11 x 2 1 ~ c12 ¼ g ðkÞ ln Y þ 0ð1Þ; 2 x ð1 þ kÞ2 where g2 ðkÞ ¼

1 kð4 þ kÞ : 5 ð1 þ kÞ2

Intermediate region (2 < s < ¥): ~ c11 ¼ g2 ln s þ 2 þ g3 ð14s2 Þ ln s þ 2 þ 4 g3 Y s s2 s2 ¥ X 1 2g2 4g3 2 m þ þ ; m fm ðkÞ m m m m1 s 2 ð1 þ kÞ m¼2 ðmoddÞ

1 ~ c ¼ g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ ð1 þ kÞ2 Y 12 4 m ¥ X 1 2g2 4g3 2 þ f ðkÞg ; þ m 1 m m m1 s 2m ð1 þ kÞm m¼2 ðmevenÞ


where g2 ðkÞ ¼

1 ð3233k þ 83k2 þ 43k3 Þ : 250 ð1 þ kÞ2

Rotation around the line of centers (axis e): Far asymptotic region (s ! ¥): ~ r11 ¼ X

¥ X

f2k ðkÞ

k¼0

1 ð1 þ kÞ ¥ X

8

~ r12 ¼ X

1

ð1 þ kÞ3

2k s2k

;

f2kþ1 ðkÞ

k¼0

1

1


;

where f0 ¼ 1; f1 ¼ f2 ¼ 0; f3 ¼ 8k3; f 4 ¼ f5 ¼ 0; f6 ¼ 64k3; f7 ¼ 0; f 8 ¼ 768k5; f9 ¼ 512k6; etc. Near asymptotic region (x 1 and x k): k k2 1 z 3; x ln þ 0ðxÞ; 4ð1 þ kÞ 1þk x ð1 þ kÞ3 8k3 k 2k2 1 ¼ z 3; x ln þ þ 0ðxÞ; 4 6 1 þ k x ð1 þ kÞ ð1 þ kÞ k3

~r ¼ X 11 ~ r12 X

where zðz; aÞ ¼

¥ X ðk þ aÞz : k¼0

Intermediate region (2 < s < ¥): ~ r11 ¼ X

k2 k2 1 sþ2 lnð14s2 Þ þ ln þ1 2ð1 þ kÞ 1þks s2

( ¥ X þ k¼1

1 ð1 þ kÞ2k

f2k ðkÞ2

2kþ1

1 k2 kð2k1Þ 4ð1 þ kÞ

) 1 2k ; s

sþ2 8k2 ln lnð14s2 Þ þ 4 s2 ð1 þ kÞ ð1 þ kÞ4 ( ) ¥ X 8 1 1 k2 1 2kþ1 2kþ2 f ðkÞ2 : 2kþ1 kð2k þ 1Þ 1 þ k s ð1 þ kÞ3 k¼1 ð1 þ kÞ2kþ1

~ 12 ¼ þX r

4k2

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90

Rotation around the axis perpendicular to the line of centers and parallel to the direction of transverse translational motion (axis i): Far asymptotic region (s ! ¥): ~r ¼ Y 11

¥ X

f2k ðkÞ

k¼0

~r ¼ X 12

1

¥ X

8 3

ð1 þ kÞ

1

ð1 þ kÞ2k s2k f2kþ1 ðkÞ

k¼0

; 1

1


;

where f0 ¼ 1;

f1 ¼ f2 ¼ 0;

f6 ¼ 27k2 þ 256k3 ;

f3 ¼ 4k3;

f4 ¼ 12k; f5 ¼ 18k4 81 f7 ¼ 72k4 þ k5 þ 72k6 ; etc: 2

Near asymptotic region (x 1 and x k): ~ r11 ¼ g2 ðkÞ ln 1 þ 0ð1Þ; Y x

~ r12 ¼ g4 ðkÞ ln 1 þ 0ð1Þ; Y x

where g2 ðkÞ ¼

2k ; 5ð1 þ kÞ

g4 ðlÞ ¼

4k2 5ð1 þ kÞ4

:

Intermediate region (2 < s < ¥): ~ r11 ¼ g2 ln ð14s2 Þg3 ð14s2 Þ ln ð14s2 Þ þ f0 ðkÞ Y þ

¥ X

1 2g2 4g3 þ m fm ðkÞ m m m m1 2 ð1 þ kÞ

m¼2 ðmevenÞ

m 2 ; s

1 ~ r12 ¼ g4 ln s þ 2 þ g5 ð14s2 Þln s þ 2 ð1 þ kÞY 8 s2 s2 þ4

m ¥ g5 X 1 2g4 4g5 2 f ðkÞ ; þ þ m s m m m1 2m ð1 þ kÞm s m¼2 ðmoddÞ

where

g3 ðkÞ ¼

ð8 þ 6k þ 33k2 Þ ; 125ð1 þ kÞ

g5 ðkÞ ¼

4kð4324k þ 43k2 Þ 125ð1 þ kÞ4

:


When the forces and torques acting on the particles are known, one can find particle velocities via the mobility matrix: U 1 kt 11 U 2 kt 21 ¼ V1 kc 11 V2 kc 21

kt12

ðkc11 ÞT

kt22

ðkc21 ÞT

kc12

kr11

kc22

kr21

ðkc12 ÞT F1 c T F ðK22 Þ 2 : kr11 L1 kr21 L2

ð2:71Þ

The mobility matrix is the inverse resistance matrix, so, according of the reciprocity theorem, its elements ktab ; kcab ; and krab must obey conditions similar to Eq. (2.61), namely, ðabÞt

ðabÞt

¼ k ji ;

ki j

ðabÞc T

ðki j

ðbaÞc

Þ ¼ k ji

;

ðabÞr

ki j

ðbaÞr

¼ k ji

:

ð2:72Þ

They can also be represented in the form similar to (2.69): ðabÞt

t t ¼ xab ei e j þ yab ðd i j ei e j Þ;

ðabÞc

¼ ðk ji

ðabÞr

r r ¼ xab ei e j þ yab ðdi j ei e j Þ:

ki j ki j ki j

ðabÞc T

ð2:73Þ

c Þ ¼ yab ei jk ek ;

If we now introduce the dimensionless mobility tensors ~kt ¼ 3pðaa þ ab Þkt ; ab ab

~kc ¼ pðaa þ ab Þ2 kc ; ab ab

~kr ¼ pðaa þ ab Þ3 kr ab ab

and parameters s and k in the same way as before, the resulting 16 dimensionless scalar functions x~tab ; ~ytab ; ~ycab ; x~rab ;, and ~yrab depending on s 2 [2, ¥) and k 2 [0,1] will characterize all kinds of translational and rotational motions of two solid spherical particles in a fluid. Since the grand resistance and mobility matrices are mutually inverse, their components’ representations (2.69) and (2.73) in combination with the fact that axisymmetric translational motion (i.e., along the line of centers) is not mutually correlated with either non-axisymmetric motion (i.e., perpendicular to the line of centers) or axisymmetric rotation (i.e., around the line of centers), lead to the following dependence between the elements of both matrices:

t x11

2 xt 1 þ k 12

2 t t X11 x12 1þk ¼ 1 þ k 1 t t x22 X12 k 2

1 þ k t 1 X12 2 ; t kX22

j91


92

r x11

2 xr 1 þ k 12

2 r r x12 X11 1þk ¼ 1 r ð1 þ kÞ3 r x22 X12 k 8

t y12 2 t 1 þ k y12 3 c y 2 11 6 yc ð1 þ kÞ2 21

2 t y 1 þ k 12 1 t y k 22 6 ð1 þ kÞ2

6 ð1 þ kÞ2

6 ð1 þ kÞ2

c y 2 21 ð1 þ kÞ 3 c y 22 2 2k ¼ 6 r 3 y12 ð1 þ kÞ 3 r y22 3 4k 6

c y12

3 r y 4 11

c y12

3 c y 2k2 22

t Y12 2ð1 þ kÞY t 12 ¼ 3 c Y 2 11 ð1 þ kÞ2 c Y21 6

3 c y 2 11

1 ð1 þ kÞ3 r X12 8 ; r kX22

r y12

1þk t Y12 2

2 c Y 3 11

t kY22

ð1 þ kÞ3 c Y12 6

ð1 þ kÞ2 c Y12 6

4 r Y 3 11

2 2 c k Y22 3

ð1 þ kÞ3 r Y12 6

1 ð1 þ kÞ2 c Y21 6 2 2 c k Y22 3 : 3 ð1 þ kÞ r Y12 6 4 3 r k Y22 3

2.3.2 Fluid is Moving at the Infinity (v¥ 6¼ 0)

Let us now consider the motion of two spherical particles in an unbounded fluid whose velocity at the infinity is the one that is typical for a shear flow [32–38]: v ¥ ¼ v 0 þ V¥ r 0 þ E ¥ r 0 :

ð2:74Þ

The velocity and pressure fields are described by the boundary value problem (2.57), (2.58): r v ¼ 0;

r2 v ¼

1 r p; me

v ¼ U 1 þ V1 r 1 on S1 ; v ! v¥ at jr 0 j!¥: where v¥ is given by Eq. (2.74).

ð2:75Þ v ¼ U 2 þ V2 r 2

on

S2 ;

ð2:76Þ


The boundary value problem is handled in the same manner as in the case of single-particle motion (see Section 2.2). Think of the fluid velocity as a sum v ¼ v¥ þ v, where v¥, v satisfy Eqs. (2.75) and v also obeys the boundary conditions v ¼ ðU 1 v¥1 Þ þ ðV1 V¥ Þ r 1 E ¥ r1

v ¼

ðU 2 v¥2 Þ

v ! 0

at

¥

þ ðV2 V Þ r 2 E r 2 ¥

on

S1 ;

on S2 ;

jr 0 j ! ¥:

Here v¥1 and v¥2 are the unperturbed velocities at the particle surfaces. The fluid velocity disturbance is, in its turn, represented as a sum, v ¼ vt1 þ vr1 þ vs1 þ vt2 þ vr2 þ vs2 ; whose components with subscript 1 vanish on the sphere S1 and components with subscript 2 vanish on S2. The superscripts correspond to the categorization of the flows that has been done in Section 2, so t signifies translational, r – rotational,and s – shear flows and these flows obey the boundary conditions at particle surfaces Si (i ¼ 1, 2): vti ¼ U i v¥i ;

vri ¼ ðVi V¥ Þ ri ;

vsi ¼ E ¥ r i

on

Si :

Because of the linearity of the boundary value problem, the force and the torque acting on particles can be represented as sums Fi ¼

XX j

F kij þ K ti : E ¥ ;

Li ¼

XX j

k

Lkij þ K ri : E ¥ ;

ð2:77Þ

k

where i, j ¼ 1, 2; k ¼ t, r, and the summands are given by F ti j ¼ me ðK ti j ðU j v¥j ÞÞ;

F ri j ¼ me ððK cji ÞT ðV j V¥ ÞÞ;

Lti j ¼ me ðK ci j ðU j v¥j ÞÞ;

Lri j ¼ me ðK ri j ðV j V¥ ÞÞ:

For particles separated by large distances, the tensors K ti j ; K ri j ; K ci j and ðK cji ÞT are determined by the following relations:

K t11

9 a1 a2 1 þ 64 h2 ¼ 6pa1 0 0

0 1þ

9 a1 a2 64 h2 0

; 0 9 a1 a2 1þ 64 h2 0

j93


94

K r11

K t21

K c11

3 1 þ 3 a2 a1 64 h4 ¼ 8pa31 0 0 a 1 h 9 ¼ pa2 0 4 0

0 a1 h 0

0 4 9 a1 a2 ¼ p 3 1 16 h 0

0

1þ

3 a2 a31 64 h4 0

0 0 ; a1 h

K r21

0 0 0 0 ; 0 0

0 ; 0 1

1 3 3 1 a2 a1 ¼ p 3 0 2 h 0

K c21

0 1 0

0 0 ; 2

0 1 3 3 a1 a2 ¼ p 2 1 0 2 h 0 0

0 0 ; 0

where 2h is the distance between the particle centers and al, a2 are the radii of the particles. The other matrices are obtained from the ones above by permutation of indices 1 and 2 and (or) transposition of a matrix. Finally, one can obtain the grand resistance matrix t c T K i j ðK ji Þ kK k ¼ : r c K i j K i j

Let us introduce the shear resistance matrix, just like we did in Section 2.2: kCk ¼

kq1 k kq2 k k0k k0k

kC1 k kC2 k : kx1 k kx2 k

For the particles spaced far away from each other, the elements of this matrix are equal to

2.4 Multi-Particle Motion

0 0 0 0 kqi k ¼ 0 1 0 ; kCi k ¼ 2Gi 0 0 F0 0 i 2Hi 0 0 2Hi 0 ; kxi k ¼ 0 0 0 0

2Gi 1 0

0 0 ; 0 F i

where 15 a2 2 45 a1 3 3 a2 4 9 a1 5 þ þ þ ; 4 h 16 h 8 h 64 h 3 a2 4 9 a1 5 þ G1 ¼ pa1 a2 ; 16 h 128 h 5 2 a2 4 9 2 a1 5 þ H1 ¼ pa2 a a : 16 2 h 128 1 h F10 ¼ pa1 a2

The expressions for F20 and G2 are derived from F10 and Gi by interchanging the indices 1 and 2 and changing the sign, and the expression for H2 – by simply interchanging the indices 1 and 2. Introducing the generalized force vector F, the particle velocity u, and the shear rate J, as we did in Section 2.1, we can rewrite Eq. (2.77) in the matrix form: kFk ¼ me ðkK kkuk þ kCkkJ kÞ:

ð2:78Þ


Let us now generalize the problems examined in Sections 2.1–2.3, for the case of motion of more than two particles in an unbounded fluid [39]. Let there be n solid particles with arbitrary surface shapes and spaced far apart. We will be indicating i-th particle (i ¼ 1, 2, K, n) with the symbol Si, where Si will also imply the particle’s surface. The symbol 0i will stand for the center (local origin) of i-th particle; Ui will denote the translational velocity of the point 0i and Wi – the particle’s angular velocity. The velocity of the fluid at the infinity is v ¥ ¼ v 0 þ V¥ r 0 þ E ¥ r 0 ;

ð2:79Þ

where 0 is an origin of coordinates placed into an arbitrary point inside the fluid, r0 is the radius vector from the origin 0 to the given point, and v0 is the velocity of an undisturbed flow at the point 0.

j95


96

The velocity and pressure fields are represented as v ¼ v¥ þ

n X

vi ;

p¼

i¼1

n X

pi ;

ð2:80Þ

i¼1

where v¥, vi, pi obey Stokes equations and vi also satisfies the boundary conditions vi ¼ U i þ Vi r i v0 V¥ r 0 E ¥ r 0 vi ¼ 0

at

S j ð j „ iÞ;

vi ! 0

at

at Si ;

ð2:81Þ

jr i j ! ¥:

ð2:82Þ

Here ri is the radius vector connecting 0i with the given point. If v¥i is the value of the undisturbed velocity at the point 0i (i.e., the value the velocity would have if the particle were absent), then the condition (2.81) can be rewritten as vi ¼ ðU i v¥i Þ þ ðVi V¥ Þ r i E ¥ r i

at Si :

ð2:83Þ

In the next step we decompose (vi, pi) into summands corresponding to shear 000 ðv0i ; p0i Þ; translational, ðv00i ; p00i Þ and rotational ðv000 i ; pi Þ motions: vi ¼ v0i þ v00i þ v000 i ;

pi ¼ p0i þ p00 þ p000 i

where each component must obey Stokes equation and satisfy the boundary conditions vi0 ¼ E ¥ r i ;

v00i ¼ U i v¥i ;

¥ v000 i ¼ ðVi V Þ r i

on

Si :

ð2:84Þ

If the solutions of the corresponding boundary value problems are known, one can find the hydrodynamic force and the torque relative to the point 0 acting on i-th particle: ( ) n X ¥ ¥ t ¥ c T F i ¼ me ½K i j ðU i vi Þ þ ðK ji Þ ðVi V Þ þ Fi : E ; j¼1

( ) n X ¥ ¥ c ¥ r Li ¼ me ½K i j ðU i vi Þ þ K i j ðVi V Þ þ t i : E :

ð2:85Þ

j¼1

The tensors K ti j ; K ri j ; K ci j , and ðK cji ÞT are determined from the expression for the net stress at the particle surface Si induced by translational motion T 00i , rotational 0 motion T 000 i , and the shear flow T i:


ð

ð

K ti j ¼ P00j ds;

K ri j ¼ r i ðP000j dsÞ;

si

j¼1

K ci j ¼ r i ðP00j dsÞ

si

n ð X ds P0j ; Fi ¼

ð si

n ð X ti ¼ r i ðds P0j Þ: j¼1

si

si

Third-rank tensors P00j ; P000j (triadics) and the fourth-rank tensor (tetradic) P0j 000 0 0 characterize the properties of fields ðv00i ; p00i Þ; ðv000 i ; pi Þ, and ðvi ; pi Þ. The first two were determined in Section 2.1, and the third one is defined by the expression T 0i ¼ me P0j : E ¥ . The dyadics K ti j ; K ri j ; K ci j and the triadics Fi, ti characterize geometrical properties of the momentary configuration of a particle system: sizes, shapes, and positions of the centers 0i. To write the relations (2.85) in a matrix form, let us introduce 3n 1 matrices (column vectors) ||F||, ||L||, kU v¥k, and kW W¥k, and such that kF k ¼ kkF 1 kkF 2 k. . .kF 2 kkT ; which includes in consecutive order the parameters of n particles, and 3n 3n matrices ||Kt||, ||Kr||, ||Kc||: kK 11 k kK k ¼ . . . kK n1 k

kK 12 k . . . ... kK n2 k

kK 1n k . . . : kK nn k

t Each of the elementary matrices ||Kij|| is a 3 3 matrix, and the matrices K i j r and K i j are symmetric. The matrices ||Fi|| and ||ti|| are 3 6 matrices, from which we can form a 6n 6 combined matrix (see Section 2.2): kF1 k . . . kF k n : kCk ¼ kt1 k . . . ktn k The above-mentioned matrices allow us to construct the 6n 6n grand resistance matrix:

j97


98

t kK k kK k ¼ c kK k

c T ðK Þ ; r kK k

as well as generalized matrices of forces (6n 1), velocities (6n 1), and shear (6 1): kF k kFk ¼ ; kLk

kðUv¥ Þk kuk ¼ ; kðVV¥ Þk

S1 . k Jk ¼ .. : S6

Now the equations (2.85) can be written in a compact matrix form: kFk ¼ me ðkK k kuk þ kCk k J kÞ:

ð2:86Þ

If the particles are freely suspended in the fluid, then setting ||F|| ¼ ||0|| in Eq. (2.86), we obtain the velocities of particles: kuk ¼ kK k kCk kJ k:

ð2:87Þ

The grand resistance matrix ||K || is symmetric and non-singular, so there exists an inverse matrix and consequently, it is possible to obtain the velocities of suspended particles. The elements of matrices ||K || and ||C|| do not depend on dynamic parameters (velocities); they depend only on the geometric properties of the particles and on the configuration of a particle system. At the present time, they can be determined only by solving problems on hydrodynamic interaction of two spherical particles as a function of the relative distance between particles and the ratio of their radii. Some examples of pair interactions were given in Section 2.3.

2.5 Flow of a Fluid Through a Random Bed of Particles

Consider a slow Stokesian flow of viscous incompressible fluid through a layer of buoyant (cloud) or fixed in space (porous medium) rigid spherical particles with a given distribution of particles over radii [40–43]. Our goal is to determine the force of particle resistance and the effective viscosity of the disperse medium taking into account hydrodynamic interaction between particles, that is, the hindered motion of particles [44]. The velocity and pressure in the fluid are described by Stokes equations r u ¼ 0;

ð2:88Þ

r T ¼ me r2 urð prg rÞ ¼ 0;

ð2:89Þ


where T is the stress tensor with components ti j ¼ ð prg rÞd i j þ me

qui qu j þ qX j qXi

¼ 0:

ð2:90Þ

There are also the boundary conditions u ¼ us that must hold on particle surfaces; us is the particles’ velocity. The fluid’s velocity at the infinity is given. If the particles are at rest or their positions in space are fixed, then us ¼ 0. For buoyant particles, their velocity us is a sum of translational and rotational velocities and is found from the relations for the force and torque exerted on the particle by the surrounding fluid (see Section 2.1). Hydrodynamic interactions between particles can be taken into account by averaging over the particle ensemble. Following [42,43], we assume that u, p and r are defined in the whole region under consideration, including the particles. We further assume that that u is a continuous function, whereas p, r, and derivatives of u are discontinuous at the particle surface S. Introducing the Saffman function ( HðX Þ ¼

1;

if X is in fluid;

0;

if X is in particle:

ð2:91Þ

and averaging it over the particle ensemble, we get hHðXÞi ¼ 1j ¼ h;

ð2:92Þ

where j is the volume concentration (volume fraction) of particles and h is the porosity (volume fraction occupied by fluid) of the disperse medium. From now on, averaging over the particle ensemble will be designated by angle brackets, and the mean-flow-rate values of hydrodynamic parameters – by a horizontal bar on the top. The Saffman function gives the relation between these averages; in particular, the mean-flow-rate velocity and pressure in the fluid are equal to uðX Þ ¼

hHðX ÞuðX Þi huðX Þi ¼ ; h hHðX Þi

pðX Þ ¼

hHðX Þ pðX Þi h pðX Þi ¼ : h hHðX Þi

ð2:93Þ

Since velocity is a continuous function in the entire region, we can introduce the in addition to the ensemble average hui. For a average cross-sectional velocity u porous medium, this velocity is called the speed of filtration. For j 1 (a low hui. concentrated suspension or a highly permeable medium), u Before averaging Stokes equations, we must first average the stress tensor T. We have:

qui qu j þ : Hti j ¼ hHð prg rÞi > d i j þ me qX j qXi

ð2:94Þ

j99


100

Using the obvious equalities

H

qui qX j

¼

qui ð1HÞ qX j

q qH ; hHui i ui qX j qX j

q qH ¼ hð1HÞui i þ ui qX j qX j

ð2:95Þ

ð2:96Þ

and the property qH=qX j ¼ dðX X s Þ of the Saffman function (where Xs is a point on the particle’s surface), and adding the relations (2.95) and (2.96), we can write

H

qui qX j

qui q ui : þ ð1HÞ ¼ qX j qX j

Interchanging the indices,

q uj qu j qu j H ; þ ð1HÞ ¼ qXi qXi qXi and adding both relations, we obtain the following:

qui qu j qui qu j qhui i q u j H þ þ þ : þ ð1HÞ ¼ qXi qX j qXi qX j qXi qX j

ð2:97Þ

Inside rigid particles, the velocity remains constant, and in the fluid, H ¼ 1. Therefore the second term on the left-hand side vanishes and Eq. 2.97 becomes

qui qu j qhui i q u j H þ þ : ¼ qXi qX j qXi qX j

ð2:98Þ

At i ¼ j this equality reduces to hHr ui ¼ r hui, and we get the averaged continuity equation, r hui ¼ 0:

ð2:99Þ

Hence, Eq. (2.94) reduces to qhui i q u j Hti j ¼ ð1jÞðprg rÞdi j þ me þ : qXi qX j

ð2:100Þ


The ensemble-averaged momentum equation (2.89) has the form hHr T i ¼ 0:

ð2:101Þ

Using the well-known tensor equality r ðHTÞ ¼ Hr T þ T rH; we rewrite Eq. (2.101) as r hðHTÞihT rHi ¼ 0:

ð2:102Þ

Let us substitute Eq. (2.100) into Eq. (2.102) and then use the averaged continuity equation (2.99) to get me r2 huihT rHirð1jÞðprg rÞ ¼ 0:

ð2:103Þ

According to Eq. (2.91), the function H undergoes a jump at the particle surface, so rH is a delta function. In order to understand the meaning of the second term in Eq. (2.103), let us take an integral of this term over a finite volume V, transform it into a surface integral according to Gauss’s theorem, and use the property of the delta function. After some manipulations, we get

ð ð ð ð hT rHidr ¼ r hHT idr ¼ hHT i n dS ¼ T n dS ;

ð2:104Þ

where n is the unit normal vector. The quantity hHTi nds is equal to the force exerted by the fluid on a particle surface element, and Eq. (2.104) is the average force exerted on the fluid by the particles in the given volume. Hence, the second term in Eq. (2.103) has the meaning of volume force exerted on the fluid by the particles. Now, consider a disperse phase consisting of N identical spherical particles. The positions of their centers will be defined by radius vectors r1, r2, . . ., rN. Let pN (r1, r2, . . ., rN) be the N-particle PDF. If we assume the velocity u and function H to depend only on r1, r2, . . ., rN, then the ensemble average of a function G (r1, r2, . . ., rN), where the averaging is carried out over the ensemble of particle configurations, is given by the formula ð hGi ¼ Gðr 1 ; r 2 ; . . .; r N Þ pN ðr 1 ; r 2 ; . . .; r N Þdr 1 dr 2 . . .dr N :

ð2:105Þ

Of utmost interest are the one-particle PDF p1 (r1) and the two-particle PDF p2 (r1, r2). These are marginal PDFs that can be obtained from a given N-particle PDF pN (r1, r2, . . ., rN) using the formulas

j101


102

ð

p1 ðr 1 Þ ¼

pN ðr 1 ; r 2 ; . . .; r N Þdr 2 . . .dr N ;

ð2:106Þ

ð p2 ðr 1 ; r 2 Þ ¼

pN ðr 1 ; r 2 ; . . .; r N Þdr 3 . . .dr N :

ð2:107Þ

In the case of random independent single-particle distributions, the N-particle distribution is the product of single-particle distributions: pN ðr 1 ; r 2 ; . . .; r N Þ ¼ p1 ðr 1 Þ p1 ðr 2 Þ. . . p1 ðr N Þ: For identical particles, one can introduced the distribution of the number of particles, nðr 1 Þ ¼ N p1 ðrÞ;

ð2:108Þ

such that nd r is equal to the probable (i.e., ensemble-averaged) number of particles whose centers are located in the volume element d r. The Saffman function H for this case is N X Hðr; r 1 ; r 2 ; . . .; r N Þ ¼ 1 Hða rr j Þ;

ð2:109Þ

j¼1

where H (X) is the Heaviside function. Then ð hHi ¼ 1w ¼ H pN ðr 1 ; r 2 ; . . .; r N Þdr 1 dr 2 . . .dr N ð ¼ 1N Hðajrr 1 jÞ p1 ðr 1 Þdr 1 ð ¼ 1N

ð2:110Þ

ð p1 ðr 1 Þdr 1 ¼ 1

jrr 1 j < a

nðr 1 Þdr 1 ; jrr 1 j < a

Hence, the volume concentration of particles is ð nðr 1 Þdr 1 ;

j¼

ð2:111Þ

jrr 1 j < a

and for identical particles with a uniform distribution n ¼ const, j ¼ 4pa3 n=3:

ð2:112Þ

From the ensemble of probable particle configurations, we can choose a subensemble containing those configurations in which particle 1 occupies one and the same position in space. We can then introduce the PDF of all other particles. This


PDF is called the conditional PDF and is equal to pN ðr 1 ; r 2 ; . . .; r N Þ= p1 ðr 1 Þ:

ð2:113Þ

Averaging of functions that relies on the use of such conditional PDFs will be designated as ð pN ðr 1 ; r 2 ; . . .; r N Þ dr 2 . . .dr N : hGi1 ¼ Gðr 1 ; r 2 ; . . .; r N Þ p1 ðr 1 Þ

ð2:114Þ

The conditional two-particle PDF and the corresponding distribution of the number of particles of type 2 are, respectively, p2 (r1, r2)/p1 (r1) and n (r2) ¼ N p2 (r1, r2)/ p1 (r1). For a random and independent distribution of identical particles, we can take p2 (r1, r2) ¼ 0 at |r1 r2| < 2a; p2 (r1, r2) ¼ p1 (r1) p1 (r2) at |r1 r2| > 2a; and n (r2) ¼ 0 at |r1 r2| < 2a; n (r2) ¼ const at |r1 r2| > 2a. Such an approximation is possible for non-interacting particles only. The PDF for interacting particles is determined from Liouville or Fokker–Planck equations (for more details, see Sections 3.7–3.9). Going back to Eq. (2.103), let us look at the second term. We have ! N N X X rr j rH ¼ r 1 Hða rr j Þ ¼ rr j dð rr j aÞ j¼1

j¼1

and hT rHi ¼

N ð X j¼1

rr j dr 1 . . .dr N pN ðr 1 ; r 2 ; . . .; r N Þdð rr j aÞT rr j

ð ¼N

pN dðjrr 1 jaÞT

rr 1 dr 1 . . .dr N jrr 1 j

ð rr 1 dr 1 ¼ nðr 1 Þdðjrr 1 jaÞhT i1 jrr 1 j ð ¼ nðranÞhT i1 na2 dn: r1 ¼ran r fixed

ð2:115Þ The derivation relies on property 2 of the PDF (see Section 1.4) and also uses the fact that a unit normal vector to the particle surface is equal to n ¼ (r r1)/|r r1| and that dn is an element of a solid angle on a unit sphere. For n ¼ const, we have ð ð2:116Þ hT 1 rH i ¼ na2 hT i1 ndn: r1 ¼ran r fixed

j103


104

and now Eq. (2.103) takes the form ð me r2 huirð1jÞðprg rÞna2

hT i1 ndn ¼ 0:

ð2:117Þ

r 1 ¼ran r fixed

hT i1 is obtained from the relation (2.100), which, when reformulated for the case of conditional averaging, will be written as ! qhui i1 q u j 1 : ti j ¼ ð1jÞðp1 rg rÞd i j þ me þ qX j qXi

ð2:118Þ

When deriving the last relation, we remembered that j1 ¼ 0 at the surface of the sphere. The subscript 1 implies that we are considering the situation when particle 1 is fixed. The equation for hui1 is similar to (2.103): me r2 hui1 rð1j1 Þðp1 rg rÞhT rHi1 :

ð2:119Þ

The derivation of hT rHi1 is similar to the derivation of Eq. (2.115): ð hT rHi1 ¼

n1 ðranÞhT i1;2 na2 dn;

ð2:120Þ

r 2 ¼ran r 1 and r fixed

with the only difference that in the right-hand side, n is replaced by n1 and hT i1 is replaced by hT i1;2. Here hT i1;2 is the outcome of conditional averaging of stress with particles 1 and 2 fixed, and the dependence of hT rHi1;2 on hui1;2 has the same form as the dependence of hT rH i1 on hui1 . We can continue this process by fixing (in consecutive order) the positions of particles 1, 2, 3, etc. The result is an infinite system of equations, where each finite subsystem is unclosed. Considering the first approximation only, we shall assume ð na ¼ 2

hT i1 ndn ¼ FðhuiÞ

ð2:121Þ

r 1 ¼ran r fixed

to close the equations (2.99) and (2.117). The functional dependence (2.121) generalizes the assumption made in [40]. A medium which obeys the condition (2.121) is called the Brinkman medium. The form of the function FðhuiÞ is different for the cases when the particles are fixed in space (as in a porous medium) and when they are suspended in the fluid (as in a suspension). In the first case the function is assumed to have the form FðhuiÞ ¼ Ahui þ Br2 hui and in the second case – FðhuiÞ ¼ Br2 hui þ Cg.


Let us fix particle 1 and consider the fluid flowing around it. The presence of other particles is accounted for by introducing the effective fluid viscosity meff and the at the infinity. Both of these quantities have to be determined. upstream velocity u The velocity and pressure fields are described by the equations r hui1 ¼ 0; rð p1 rg rÞ þ meff r2 hui1 meff a2 hui1 ¼ 0;

ð2:122Þ

where meff a2 ¼ A/(1 j), meff ¼ (me B)/(1 j), and the parameter a has to be determined. These equations should be solved under the following boundary conditions: hui1 ¼ 0

at

r ¼ r 1 þ an;

hui1 ! u

at

r ! ¥;

ð2:123Þ

must obey the equations similar to Eqs. (2.122): The unperturbed velocity u ¼ 0; r u

meff a2 u ¼ 0; rp0 þ meff r2 u

ð2:124Þ

where p0 ¼ prg r. Solving equations (2.124) as discussed in [43], we get aa ¼

9j=4 þ 3ð8j3j2 Þ1=2 =4 13j=2

ð2:125Þ

and meff (j). The derivation of the latter dependence is lengthy and is not presented here. One can see from the second equation (2.122) that if the characteristic linear scale is negligibly small as compared L of the problem exceeds 1/a, then the term meff r2 u . The equation then reduces to the Darcy equation for a low-permeable to meff a2 u porous medium: rp

meff k

¼ 0; u

ð2:126Þ

where k ¼ 1/a2 is the permeability of the porous medium. In the second case, the problem reduces to the system of equations r hui ¼ 0; rð1jÞðprg rÞ þ me r2 huiFðhuiÞ ¼ 0

ð2:127Þ

r hui1 ¼ 0; r Þ þ me r2 hui1 Fðhui1 Þ ¼ 0 rð1jÞðp1 rg _

ð2:128Þ

j105


106

with the boundary conditions r V at hui1 ¼ V þ a _

r ¼ r1 þ a _ r ; hui1 ! hui at r ! ¥; ð2:129Þ

where r1 is the particle’s radius vector, and V and V are the translational and rotational velocities the test particle 1 freely suspended in the fluid. They are determined by the condition that the force and torque acting on the particle are both equal to zero: ð 2

a

4 r d_ r þ pa3 r p g ¼ 0; hT i1 _ 3

r¼r 1 þa _ r r 1 fixed

ð

_ rd_ r ¼ 0; r hT i1 _

r¼r 1 þa _ r r 1 fixed

ð2:130Þ The functional FðhuiÞ is given by FðhuiÞ ¼ Br2 hui þ Cg: The corresponding equations look simpler than equations (2.122) and (2.124). In particular, the second equation (2.122) assumes the form of a standard Stokes equation Cg r þ meff r2 hui1 ¼ 0; r p1 rg r þ 1j

ð2:131Þ

where meff ¼ (me B)/(1 j) is the effective viscosity. Skipping the solution of this equation, we give only the formula for the coefficient of effective viscosity: meff me

¼

1 : 15j=2

ð2:132Þ

At j 1, the equation (2.132) gives the well-known Einstein’s formula: meff me

5

1 þ j: 2

References 1 Lamb, H. (1945) Hydrodynamics, Dover, New York. 2 Levich, V.G. (1962) Physicochemical Hydrodynamics, Prentice– Hall,Englewood Cliffs, NJ.

3 Happel, J. and Brenner, H. (1983) Low Reynolds Number Hydrodynamics, Martinus Nijhof, The Hague. 4 Kim, S. and Karrila, S.J. (1991) Microhydrodynamics, Butterworth– Heinemann, Boston.

References

5 Probstein, R.F. (1995) Physicochemical Hydrodynamics, Wiley, New York. 6 Brenner, H. (1966) Hydrodynamic Resistance of Particles at small Reynolds Numbers. Adv. Chem. Eng., 6, 287–438. 7 Stimson, M. and Jefferey, G.B. (1926) The Motion of two Spheres in a viscous Fluid. Proc. Roy. Soc. A, 111, 110. 8 Brenner, H. (1961) The slow Motion of a Sphere through a viscous Fluid toward a plane Wall. Chem. Eng. Sci., 16, 242–251. 9 Goldman, A.J., Cox, R.G. and Brenner, H. (1966) The slow Motion of two identical arbitrary oriented Spheres through a viscous Fluid. Chem. Eng. Sci., 21, 1151–1170. 10 Goldman, A.J., Cox, R.G. and Brenner, H. (1967) Slow viscous Motion of a Sphere parallel to a plane Wall. Part I. Motion through a quiescent Fluid. Part. II. Couette Flow. Chem. Eng. Sci., 22, 637, 653. 11 O’Neil, M.E. and Stewartson, K. (1967) On the slow Motion of a Sphere parallel a nearby plane Wall. J. Fluid Mech., 27, 705–724. 12 Cooley, M.D. and O’Neil, M.E. (1968) On the slow Rotation of a Sphere about a Diameter parallel to a nearby plane Wall. J. Inst. Math. Applics., 4, 163–173. 13 O’Neil, M.E. (1969) On asymmetric slow viscous Flows caused by the Motion of two Spheres almost in Contact. Proc. Camb. Phil. Soc. A, 65, 543–556. 14 Cooley, M.D. and O’Neil, M.E. (1969) On the slow Motion generated in a viscous Fluid by the approach of a Sphere to a plane Wall or stationary Sphere. Mathematika, 16 (1), 37–49. 15 Cooley, M.D. and O’Neil, M.E. (1969) On the slow Motion of two Spheres in Contact along their Line of Centers through a viscous Fluid. Proc. Camb. Phil. Soc., 66, 407–415. 16 Davis, M.H. (1969) The slow Translation and Rotation of two

17

18

19

20

21

22

23

24

25

26

unequal Spheres in a viscous Fluid. Chem. Eng. Sci., 24, 1769–1776. O’Neil, M.E. (1970) Exact Solutions of the Equations of slow viscous Flow generated by the asymmetrical Motion of two equal Spheres. Appl. Sci. Res., 21, 452–466. O’Neil, M.E. and Majumdar, S.R. (1970) Asymptotic Flow viscous Motions caused by the Translation or Rotation of two Spheres. Part I. The Determination of Exact Solutions for any Values of the Ratio Radii and Separation Parameters. Part II. Asymptotic Forms of the Solutions when the Minimum Clearance between the Spheres approaches Zero. ZAMP, 21, 164, 180. Goren, S.L. (1970) The normal Force exerted by creeping Flow on a small Sphere touching a Plane. J. Fluid Mech., 41, 619–625. Goren, S.L. and O’Neil, M.E. (1971) On the hydrodynamic Resistance to a Particle of a dilute Suspension when in the Neighborhood of a large Obstacle. Chem. Eng. Sci., 26, 325–338. Wakiya, S. (1971) Slow Motion in Shear Flow of a Doublet of two Spheres in Contact. J. Phys. Soc. Japan, 31, 158– 1587. Wacholder, E. and Weihs, D. (1972) Slow Motion of a Fluid Sphere in the Vicinity of another Sphere or a Plane Wall. Chem. Eng. Sci., 27, 1817–1828. Haber, S., Hetsroni, G. and Solan, A. (1973) On the low Reynolds Number Motion of two Droplets. Int. J. Multiphase Flow, 1, 57–71. Zinchenko, A.Z. (1978) To Calculation of hydrodynamic Interactions of Droplets by Low Reynolds Number. Appl. Math. Mech., (5), 955–959. Zinchenko, A.Z. (1980) Slow asymmetric motion of two Droplets in viscous Medium. Appl. Math. Mech., (1), 49–59. Schmitz, R. and Felderhof, R. (1982) Mobility Matrix for two spherical Particles with hydrodynamic Interaction. Physica A, 116, 163–177.

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27 Jeffrey, D.J. and Onishi, Y. (1984) The Forces and Couples acting on two nearly touching Spheres in LowReynolds-Number Flow. ZAMP, 35, 634–641. 28 Jeffrey, D.J. and Onishi, Y. (1984) Calculation of the Resistance and Mobility Functions for two unequal Spheres in Low-Reynolds-Number Flow. J. Fluid Mech., 139, 261–290. 29 Kim, S. and Mifflin, R.T. (1985) The Resistance and Mobility Functions of two equal Spheres in Low Reynolds Number Flow. Phys. Fluids, 28, 2033– 2044. 30 Fuentes, Y.O., Kim, S. and Jeffrey, D.J. (1988) Mobility Functions for two unequal viscous Drops in Stokes Flow. Part 1. Axisymmetric Motions. Phys. Fluids, 31, 2445–2455. 31 Fuentes, Y.O., Kim, S. and Jeffrey, D.J. (1989) Mobility Functions for two unequal viscous Drops in Stokes Flow. Part 2. Nonaxisymmetric Motions. Phys. Fluids A, 1, 61–76. 32 O’Neil, M.E. (1968) A Sphere in Contact with a Plane Wall in a slow linear Shear Flow. Chem. Eng. Sci., 23, 1293–1298. 33 Lin, G.L., Lee, K.J. and Sather, N.F. (1970) Slow Motion of two Spheres in a Shear Field. J. Fluid Mech., 43, 35–47. 34 Nir, A. and Acrivos, A. (1973) On the creeping Motion of two arbitrary-sized touching Spheres in a linear Shear Field. J. Fluid Mech., 59, 209–223. 35 Hetsroni, G. and Haber, S. (1978) Low Reynolds Number Motion of two Drops

36

37

38

39

40

41

42

43

44

submerged in an unbounded arbitrary Velocity Field. Int. J. Multiphase Flow, 4, 1–17. Zinchenko, A.Z. (1983) Hydrodynamic Interactions of two identical Liquid Spheres in linear Flow Field. Appl. Math. Mech., (1), 56–63. Martinov, S.I. (1998) Hydrodynamic Interactions of Particles. Fluid Dyn., (2), 112–119, (in Russian). Martinov, S.I. (2000) Interactions of Particles in Flow with parabolic Velocity Profile. Fluid Dyn., (1), 84–91, (in Russian). Brenner, H. and O’Neil, M.E. (1972) On the Stokes Resistance of multiparticle Systems in a linear shear Field. Chem. Eng. Sci., 27, 1421–1439. Brinkman, H.C. (1947) A Calculation of the viscous Force exerted by a flowing Fluid on a dense Swarm of Particles. Appl. Sci. Res., A1, 27–34. Tamm, C.K.W. (1969) The Drag on a Cloud of spherical Particles in low Reynolds Number Flow. J. Fluid Mech., 38, 537–546. Saffman, P.G. (1971) On the Boundary Condition at the Surface of a Porous Medium. Studies in Appl. Math, 30 (2), 93–101. Lundgren, T.S. (1972) Slow Flow through stationary Random Beds and Suspensions of Spheres. J. Fluid Mech., 51, 273–299. Brenner, H. (1974) Rheology of dilute Suspension of axisymmetric Brownian Particles. Int. J. Multiphase Flow, 1, 195–341.

j109

3 Brownian Motion of Particles Chaotic motion of microparticles (whose size can range from 1 nm to 10 mm) suspended in a fluid was first discovered by Brown in 1827 and is now called Brownian motion. A systematic study of Brownian motion was initiated only in the early 20th century by Einstein, Smoluchowski, Perrin, Langevin, and Lorentz. Their works showed that Brownian motion is caused by very frequent collisions of particles with molecules of the surrounding fluid undergoing incessant random motion (thermal motion). The motion of molecules is so irregular that Brownian motion can be described only probabilistically, under the assumption of frequent and statistically independent impacts of molecules against a particle. More recently discovered phenomena similar to Brownian motion, for instance, fluctuations of current within conductors, are also caused by thermal motion of molecules and electrons. Brownian motion of a particle in a fluid is taking place under the action of two forces: a random force caused by collisions with molecules, and a systematic force caused by viscous friction. The statistical average of kinetic energy of thermal motion of a particle is equal to 3kW/2, where k is the Boltzmann constant and W – the absolute temperature. This follows from the law of uniform distribution of kinetic energy over degrees of freedom (equipartition theorem). Thus, when a particle is moving along a straight line, this energy equals kW/2. The first works on Brownian motion were devoted to the motion of an isolated particle. The results obtained in these works are applicable for suspensions with very low volume concentration of particles (infinitely dilute suspensions), when the average distance between particles is large compared to particle sizes, so that interparticle interactions (hydrodynamic interactions, molecular interactions, collisions) can be neglected. Later works on Brownian motion took into account pair interactions between particles, which made their results applicable for higher (but not too high!) volume concentrations j. The more recent works attempt to take into account multiparticle interactions as well. We begin our discussion of Brownian motion with the analysis of random motion of an individual particle in a quiescent fluid, neglecting external forces and

j 3 Brownian Motion of Particles

110

interactions with other particles and with the surrounding fluid. Such a motion is called random walk. Further on, we shall discuss particle motion under the action of systematic and random forces. Finally, we go to the case of particle motion affected by the flow of the surrounding fluid and by interparticle interactions. Works [1–12] will provide the reader with extra information about Brownian motion and the common mathematical techniques employed in research.

3.1 Random Walk of an Isolated Particle

The case of Brownian motion of an isolated particle boils down to the random walk problem, which can generally be formulated in the following way. A particle driven by some external influence undergoes successive displacements r1, r2, . . . , ri, . . . , each of which is independent from all previous ones (in terms of both absolute magnitude and direction). The probability for a displacement to lie in the interval (ri, ri þ dri) is specified by some probability density distribution ti(ri). (Possible types of distributions will be considered later on.) Our task is to find the probability that after N steps the particle will be located in the interval (R, R þ dR). Consider first the simplest variant of random walk – a series of steps of equal length along a straight line, where each step can be directed forward or backward with the same probability of 0.5. The step length is assumed to be 1, so after N steps the particle can wind up at any one of the points with coordinates N, N þ 1, . . ., 1, 0, 1, . . ., N 1, N. The probability P(m, N) that after N steps the particle will be found at the point m is given by the Bernoulli distribution Pðm; NÞ ¼

m CðmþNÞ=2

N 1 ; 2

ð3:1Þ

m where CðmþNÞ=2 ¼

N! is the binomial coefficient, and the numbers m ð12ðNþmÞÞ!ð12ðNmÞÞ! and N have the same parity. The average displacement and the root-mean-square deviation of the particle are, respectively,

hmi ¼ 0;

pffiffiffiffiffiffiffiffiffiffi pffiffiffiffi hm2 i ¼ N :

In the case when N 1 and m N, which is of primary interest to us, Eq. (3.1) gives the following asymptotic distribution: Pðm; NÞ ¼

2 pN

1=2

m2 : exp 2N

ð3:2Þ

Now, consider the same problem but with steps of length l. Let us introduce the particle displacement X ¼ ml and consider the intervals DX l along the


j111

straight line. Then the probability to find the particle in the coordinate interval (X, X þ DX) after N steps is DX 1 X2 exp DX: ¼ 2Nl2 2l ð2pNl2 Þ1=2

PðX ; NÞDX ¼ Pðm; NÞ

If the particle accomplishes n displacements in a unit time, the probability to find it in the interval (X, X þ DX) at the moment t is X2 DX : exp 4Dt 2ðpDtÞ1=2 1

PðX ; tÞDX ¼

ð3:3Þ

where D ¼ vl22. It was assumed up to this point that the straight line along which the particle is moving is unbounded. Consider now the case of a bounded straight line. Suppose a boundary is placed at the point (X1 ¼ m1l > 0) which either reflects or absorbs the particle. If the boundary is reflective, the probability for the particle to be at the point m < m1 after N steps is Pðm; N; m1 Þ ¼ Pðm; NÞ þ Pð2m1 m; NÞ: The second term on the right-hand side is the probability to find the particle at the mirror-reflected point 2m1 m after N steps; keep in mind that the probability P(m,n) is still determined by the formula (3.1). In the limiting case of N 1 and m N we arrive at the formula Pðm; NÞ ¼

2 pN

1=2

( )! m2 ð2m1 mÞ2 þ exp : exp 2N 2N

Introducing the same notation as the one implied by Eq. (3.3), we obtain the following: PðX ; t; X1 Þ ¼

1 2ðpDtÞ1=2

( )! X2 ð2X1 X Þ2 þ exp : exp 4Dt 4Dt

ð3:4Þ

Note that the distribution (3.4) obeys the well-known boundary condition ðqP=qXÞX ¼X1 that is responsible for the vanishing of the flux at the plane wall. For an absorbing boundary, the probability that after N steps the particle will be found at a point m < m1 is equal to Pðm; N; m1 Þ ¼ Pðm; NÞPð2m1 m; NÞ: The minus sign before the second term expresses the fact that the absorbing boundary prohibits the particle to get into a mirror-reflected point. The limiting case

j 3 Brownian Motion of Particles

112

N 1 and m N results in the formula PðX ; t; X1 Þ ¼

1 2ðpDtÞ1=2

( )! X2 ð2X1 X Þ2 exp : exp 4Dt 4Dt

ð3:5Þ

The distribution (3.5) satisfies the condition P(X1, t; X1) ¼ 0, which corresponds to a well-known boundary condition at the completely absorbing wall. The derived expressions make it possible to determine the probabilistic flux of particles deposited on the wall. In order to do so, it is necessary to find the probability that after N steps the particle will arrive at the point m1 and that while performing these N steps, it will never cross or touch the boundary, that is, the point m ¼ m1. This probability is A(m1, N) ¼ m1/NP(m1, N). In the limiting case of large N it is equal to X1 1 X2 DX ; ð3:6Þ exp AðX1 ; tÞ ¼ Nt ðpDtÞ1=2 4Dt where X1 ¼ m1l; N ¼ nt; D ¼ nl2/2; l is the step length; n is the number of displacements in a unit time. If the initial number of particles (t ¼ 0, X ¼ 0) is known, we can find the rate of deposition particles on the absorbing wall. At the moment t, the ratio of the number of deposited particles per unit time to the initial number of particles will be equal to qðX1 ; tÞ ¼

X1 1 X12 exp : t 2ðpDtÞ1=2 4Dt

ð3:7Þ

The distribution (3.7) obeys the condition qP ; qðX1 ; tÞ ¼ D qX X ¼X1

ð3:8Þ

where P is determined by the relation (3.5). If P stands for the concentration of a matter, then Eq. (3.8) can be interpreted as a probability flux or, from the physical viewpoint, as a diffusion flux of matter. Consider now the general random walk problem. The particle accomplishes N displacements, whose positions are determined by radius vectors r1, r2, . . . , rN, so P after N displacements the particle will be found at the point R ¼ N i¼1 r i . Each displacement is assigned its own probability density ti(ri). Thus ti(ri)dri is the probability for i-th displacement vector to belong to the range (ri, ri þ dri). In the coordinate form, this probability is written as ti ðr i Þdr i ¼ ti ðXi ; Yi ; Zi ÞdXi dYi dZi ;

ði ¼ 1; 2; . . .; NÞ:

The problem is to find the probability pN(R)dR that after N displacements the particle will be found in the interval (Ri, Ri þ dRi). The solution of this problem obtained by


j113

Markov’s method has the following form: 1 pN ðRÞ ¼ 3 8p AN ðrÞ ¼

N Y j¼1

Z1 AN ðrÞexpðirRÞdr; 1 Z1

ð3:9Þ

t j ðr j Þexpðirr j Þdr j ;

1

where r is an N-dimensional vector and AN(r) is the Fourier transform of the Q function Nj¼1 t j ðr j Þ. Hence, the solution of the general random walk problem depends on the form of probability density ti(ri) for each displacement. Let us take a look at some distributions commonly encountered in applications. 3.1.1 Isotropic Distribution

A distribution is isotropic if it does not depend on the direction of the displacement vector ri, and depends only on its length (or, which is the same, on the square of length jr 2i j ¼ r 2 if all displacements have the same length). Clearly, such a distribution possesses a spherical symmetry. That is why such distributions are sometimes called spherical distributions of displacement directions. Also, distributions of displacements must to be identical for all displacements, so that we can drop the index after t, writing t j ðr j Þ ¼ tðr 2 Þ:

ð3:10Þ

Then the second relation (3.9) yields 81 9N Recr it will become unstable, and small perturbations that always exist in a flow will eventually make it turbulent. So our task is to formulate the mathematical problem of stability of hydrodynamic equations (4.1) and (4.6) describing the laminar fluid flow in order to obtain the theoretical value of Recr. One valuable theoretical method of examining stability of the flow is the method of small perturbations, whose essence consists in the following. Let Ui(x, t) and

4.5 Hydrodynamic Instability

P(x, t) be particular solutions of the Navier–Stokes equations, and let u0i and p0 be small perturbations of these fields appearing in the flow at the initial moment, such that u0i Ui and p0 P. The resulting fields of velocity ui ¼ Ui þ u0i and pressure p ¼ P þ p0 also obey the equations (4.1) and (4.6). Consider a flow in the absence of external forces. Substituting ui ¼ Ui þ u0i and p ¼ P þ p0 into the equations and neglecting second-order terms, one gets the linear equations for perturbations u0 and p0i : qu0i ¼ 0; qXi

ð4:24Þ

qu0i qu0 qUi 1 q p0 þ Uk i þ u0k ¼ þ ne Du0i qt qXk qXk re qXi

ð4:25Þ

or, in the vector form, r u0 ¼ 0;

ð4:26Þ

qu0 1 þ ðUrÞu0 þ ðu0 rÞU ¼ r p þ ne Du0 : qt re

ð4:27Þ

The boundary condition on a rigid surface is u0 ¼ 0. Differentiating Eq. (4.25) with respect to xi, summarizing the result over i and using the continuity equation (4.24, we obtain Eq. (4.13). Therefore the general solution of equations (4.24) and (4.25) will be determined once we set the initial values of velocity perturbations u0i ðx; 0Þ. After solving Eqs. (4.24) and (4.25), we can find conditions under which perturbations will not decay in time. These are the hydrodynamic instability conditions, that is, the conditions for the transformation of the flow from laminar to turbulent. If the solutions u(x) and p(x), whose stability are investigated, are time-independent, the system of equations (4.26) and (4.27) allows a solution in the form u0 ðX ; tÞ ¼ eiwt fw ðX Þ;

p0 ðX ; tÞ ¼ eiwt gw ðX Þ:

ð4:28Þ

Here o is the complex frequency, fo and go – amplitudes that must be found by solving the eigenvalue problem for the system of linear partial differential equations. If the coefficients of this system do not depend on some spatial coordinate, the number of unknown variables can be reduced by assuming an exponential dependence of fo and go on this coordinate. Thus, when an unperturbed flow depends only on one coordinate X3, we can write fw ðX Þ ¼ eiðk1 X1 þk2 X2 Þ ~f ðX3 Þ;

gw ðX Þ ¼ eiðk1 X1 þk2 X2 Þ ~g ðX3 Þ:

The eigenvalue problem reduces to a system of ordinary differential equations, and we can find the eigenfrequencies o. In the flow region is finite, the eigenfrequencies

j191

j 4 Turbulent Flow of Fluids

192

form a discrete set. For the flow to be stable, it is necessary and sufficient that all eigenfrequencies should satisfy the condition Im(o) < 0. Since the Reynolds number enters the equations (4.26) and (4.27) written in the dimensionless form, the eigenfrequencies functionally depend on Re as parameter. Since at Re ! 0 (the rest state) the flow is stable, and at Re ! 1, it should be unstable, there always exists a Recr at which the stable flow changes to an unstable one. It means that as Re increases, imaginary parts of some eigenfrequencies should increase and become positive. Because different eigenvalues can change their sign at different values of Re, we should take the smallest of these critical Reynolds numbers as Recr. There are papers are devoted to the subject of to hydrodynamic stability that contain solutions of the stability problem for various flows: flow taking place between two rotating cylinders, convective flow in a fluid layer heated from below, plane-parallel flows, flow in pipes, boundary layer flows, and so on. Comparisons with the corresponding experimental data show that theoretical values of Recr do not always agree with experimental values. For example, the theory of stability of a plane Poiseuille flow gives a noticeably higher value of Recr than the one obtained in experiments on turbulent flows in a plane channel. A large discrepancy between theoretical and experimental values of Recr for this and other flows shows that the transition from a laminar flow to a turbulent one may not always be described by the linear perturbation theory. In the above-considered formulation of the problem, perturbations of hydrodynamic quantities were assumed to be small. But in the case of an unstable flow, initially small perturbations can become finite after a while, and then the small perturbation theory will no longer be applicable. Hence, the linear theory of small perturbations is capable of describing the initial stage of turbulence only, and cannot give a complete picture of the process. For finite perturbations, the stability problem reduces to a system of nonlinear equations. This is why the nonlinear theory of the beginning of turbulence is very complex and does not afford a complete solution of the problem as of today.

4.6 The Reynolds Equations

The primary characteristic feature of the turbulent flow is disordered (random) fluctuations of its hydrodynamic parameters. As a result, the dependence of these parameters on spatial coordinates at a given time, as well as the dependence on time at a given spatial point, is highly complex and difficult to handle. Besides, even if we reproduce one and the same flow under the same conditions, hydrodynamic parameters will still assume different values. In practice, one has no choice but to consider a set of similar flows, assuming that hydrodynamic parameters are random variables. This means that in a turbulent flow, an individual (deterministic) description of hydrodynamic fields of velocity, pressure, and so on, is practically impossible, and reliance on statistical methods becomes unavoidable. We can then define the


turbulent flow as the one for which there exists a statistical ensemble of similar flows characterized by known probability distributions, with continuous probability density functions (PDFs) for the hydrodynamic fields. In practice, it is quite unnecessary to know all the minutiae of hydrodynamic fields as we are primarily interested in their average characteristics. So we have a powerful reason to employ the averaging methods, which will allow us to operate with smooth and repetitive average values of the flow parameters. As we stated in Section 1.7, ensemble averaging can be replaced by averaging over the time or by spatial averaging thanks to the ergodic hypothesis. Further on, the validity of the ergodic hypothesis will be assumed by default. The simplest statistical characteristics of random hydrodynanic fields are their average values such as hui, hpi, and so on. We shall reserve the term ‘‘fluctuations’’ for the deviations of individual values from their averages, for example, u0 ¼ u hui, p0 ¼ p hpi, and so on. Then any hydrodynanic field can be expressed as a sum of the average value and the fluctuation: u ¼ hui þ u0 ;

p ¼ h p i þ p0 ;

etc:

ð4:29Þ

Average values behave in a rather smooth manner, while fluctuations are characterized by intense spatial and temporal ‘‘jumps’’. It is fluctuations that define turbulent inhomogeneities. Note that the scale and period of inhomogeneities can, generally speaking, be arbitrarily small. However, small-scale inhomogeneities must be accompanied by large velocity gradients, which requires high expenditures of energy to overcome friction forces that become quite considerable on such small scales. Thus the existence of microflows on very low scales is almost impossible. This is why turbulent motions should be characterized by minimum scales and minimum periods of inhomogeneities. For many turbulent flows inside pipes, the characteristic minimum scale of fluctuations ranges from 0.1 to 1 mm. On distances comparable to the minimum scale of fluctuations and on time intervals comparable to the minimum period of fluctuations, all hydrodynamic fields vary slowly and can be described by differentiable functions. Hence, a description of turbulent flows by means of differential equations is quite possible. But direct application of these equations proves to be very difficult and sometimes outright impossible, because hydrodynamic fields in a turbulent flow are non-stationary and depend on initial conditions, so even small perturbations will lead to unstable solutions. Therefore conventional hydrodynamic equations are all but useless for calculating individual hydrodynamic fields. But this does not mean that hydrodynamic equations cannot be used at all. Hydrodynamic equations have proved to be extremely useful for obtaining connections between statistical characteristics of turbulent hydrodynamic fields. The simplest of these connections were first established by Reynolds, who averaged the equations of motion of a viscous incompressible fluid. The resulting equations are known as the Reynolds equations. Before we proceed to average hydrodynamic equations, we must formulate the basic rules for the averaging of hydrodynamic fields; they were first established by Reynolds.

j193


194

h f þ g i ¼ h f i þ h g i;

ð4:30Þ

ha f i ¼ ah f i if a ¼ const;

ð4:31Þ

hai ¼ a if a ¼ const;

qf qf ¼ ; s coordinate or time; qs qs

ð4:32Þ ð4:33Þ

h h f ig i ¼ h f ih g i:

ð4:34Þ

Eqs. (4.30)–(4.34) are called the Reynolds rules. Taking in consecutive order g ¼ 1, g ¼ hui, and g ¼ u0 ¼ u hui, we derive additional rules from Eqs. (4.30)–(4.34): hh f ii ¼ h f i; h f 0 i ¼ h f h f ii ¼ 0; hh f iu0 i ¼ h f ihu0 i ¼ 0;

hh f ihuii ¼ h f ihui; ð4:35Þ

Let us proceed to average the Navier–Stokes equations (4.2) and (4.7), taking u ¼ hui þ u0 ;

p ¼ h p i þ p0 ;

ð4:36Þ

and using the continuity equation (4.2) to transform the second term on the lefthand side of Eq. (4.7) to uk

qui q ¼ ðui uk Þ: qXk qXk

Applying the Reynolds rules, we obtain: qhui i ¼ 0; qXi

ð4:37Þ

qhui i q 1 qh pi þ ne Dhui i þ h fi i: þ hui ihuk i þ u0i u0k ¼ qXk re qXi qt

ð4:38Þ

Eqs. (4.37)–(4.38) are called the Reynolds equations. The advantage of these equations as compared to Eq. (4.2) and Eq. (4.7) is that they operate with smoothly varying quantities. But at the same time they contain new unknown vari averaged ables u0i u0k characterizing fluctuational components of velocity; you should think of them as components of a second-rank correlation tensor (see Section 1.7). The appearance of new unknowns is the consequence of nonlinearity of the Navier– Stokes equations. The physical meaning of the terms with new unknowns becomes apparent if one carries them to the right-hand side of Eq. (4.38) and combines them with the


viscous term:

qhui i qhui i 1 qh pi q qhui i 0 0 ¼ þ ne ui uk þ h fi i: þ huk i re qXi qXk qt qXk qXk

ð4:39Þ

Comparing the resultant equation with Eq. (4.3), one concludes that the stress tensor is not represented by the viscous stress tensor sij ¼ 2meEij as in the laminar flow, but by the tensor D E ð1Þ ti j ¼ 2me Ei j re u0i u0j ¼ si j þ ti j : ð4:40Þ ð1Þ

It follows from Eq. (4.40) that turbulence gives rise to additional stresses ti j , which are induced by turbulent fluctuations. These additional stresses are called the Reynolds stresses. Turbulent stresses are examined using the same method that is commonly applied in hydromechanics. In particular, one can show that the quantiD E ties re u0i u0j stand for the normal components of turbulent stresses at i ¼ j and for the tangential components at i „ j. In this context, we should mention that the effect of turbulent mixing on the averaged flow is similar to the effect of viscosity, because turbulent fluctuations promote additional momentum transfer from one fluid volume to another in the same way as molecular viscosity forces promote the transport of momentum in the kinetic molecular theory. When considering the Reynolds stress tensor, we are primarily interested in tensor component describing the transfer of momentum transfer from the flow to the body placed into the flow, because momentum transfer characterizes the friction force (the drag force) acting on the body. Let us look at a simple case: a plane wall X3 ¼ 0 is placed into a turbulent flow, which is moving along the X1-axis and parallel to the wall. Then the friction force acting on a unit area of the wall is directed along the X1-axis and equal to qhu1 i qhu3 i þ : t0 ¼ ðhs13 ire u01 u03 ÞX3 ¼0 ; hs13 i ¼ re ne qX3 qX1 0 The zero relative velocity condition is satisfied at the wall surface, therefore ui , 0 ui , and their derivatives with respect to X1 are all equal to zero at X3 ¼ 0 and qhu1 i : ð4:41Þ t0 ¼ re ne qX3 X3 ¼0

Near the wall, the average flow velocity is directed parallel the wall, and the friction stress is equal to t ¼ re ne

qhu1 i re u01 u03 : qX3

ð4:42Þ

Suppose that qhu1 i re u01 u03 ¼ re nt ; qX3

ð4:43Þ

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The factor nt has the dimensionality of the kinematic viscosity coefficient (m2/s) and by analogy is called the coefficient of turbulent viscosity. In contrast to the ordinary (molecular) viscosity coefficient ne, the turbulent viscosity coefficient nt characterizes statistical properties of fluctuational motion, rather than physical properties of the fluid. In the general case nt does not remain constant but varies in space and in time. The turbulent viscosity coefficient is much larger than the molecular viscosity coefficient, since nt/ne Re 1. By virtue of Eq. (4.43), the friction stress near the wall can be written as t ¼ re ðne þ nt Þ

qhu1 i ; qX3

ð4:44Þ

while far away from the wall, we have nt ne and t tð1Þ ¼ re u01 u03 . Hence, the Reynolds equations (4.39) are essentially the equations of conservation of momentum for a turbulent flow, and the Reynolds stresses describe the turbulent transport of momentum. Similarly, we can derive the equations of conservation of other substances, such as heat and matter. Taking W ¼ hWi þ W0 in Eqs. (4.20)–(4.21) and averaging these equations, we get qhqi q þ hui ihqi þ u0i q0 ¼ cDhqi qt qXi or qhqi qhqi q qhqi 0 0 ¼ c ui q : þ h ui i qt qXi qXi qXi

ð4:45Þ

The last equation is written in the divergent form (4.39), and W can be either temperature or concentration of the passive impurity. In the former case w has the meaning of thermal diffusivity, and in the latter it is the diffusion coefficient. The equations of heat and mass transfer have the same structure as the Reynolds equation. In these two equations, there appears an additional flux caused by turbu 0 0 lent fluctuations: the heat flux c or the mass flux of passive impurity r q u p e i re q0 u0i . By analogy with Eq. (4.43), the additional heat flux can be written as qhqi ; c p r q0 u0i ¼ c p rct qXi

ð4:46Þ

where W is temperature and wt is turbulent thermal diffusivity. By the same token, we write for the mass transfer of passive impurity: qhqi ; r q0 u0i ¼ rDt qXi

ð4:47Þ

where W is the concentration of passive impurity and Dt – the turbulent diffusion coefficient.

4.7 The Equation of Turbulent Energy Balance

We have defined the coefficients of turbulent viscosity nt, thermal diffusivity wt, and diffusion Dt for the case of one-dimensional flow. For multidimensional flows, these coefficients will be tensor, rather than scalar, quantities. ð1Þ The appearance of additional terms containing Reynolds stresses ti j in the Reynolds equations means that the system of equations will no longer be a closed. In order to obtain a closed system of equations, one has to write additional equations ð1Þ that would describe ti j . A general method that aims to derive the necessary equations for the Reynolds stresses has been proposed by Keller and Friedman. However, in each of the newly derived equations, there still appear new unknowns, whose determination, in its turn, requires new equations. The resulting system of equations (the Keller–Friedman chain) becomes infinite because any finite subsystem turns out to be unclosed. ð1Þ Nevertheless, the equations for ti j still lead us to some important qualitative conclusions about the properties of turbulent flows.


Kolmogorov was the first to suggest using the D energy E balance equation in addition to the Reynolds equations. Since the quantity u0i u0j that we are trying to determine is a second order moment, let us employ the following general method that enables us to compile the equations for moments. Let u1, u2, . . ., uN be N hydrodynamic fields of the turbulent flow, and X1, X2, . . ., XN – N points in the volume filled by the fluid. The fields as well as the points might all be different, or some of them might be the same. Let us consider N-th order moment Bu1 u2 ...uN ðX 1 ; X 2 ; . . .; X N ; tÞ ¼ hu1 ðX 1 ; tÞu2 ðX 2 ; tÞ. . .uN ðX N ; tÞi: Differentiating this relation with respect to time and using the property (4.33), one obtains: qu1 ðX 1 ; tÞ u2 ðX 2 ; tÞ. . .uN ðX N ; tÞ qt

qu2 ðX 2 ; tÞ þ u1 ðX 1 ; tÞ . . .uN ðX N ; tÞ . . . qt

quN ðX N ; tÞ : ð4:48Þ þ u1 ðX 1 ; tÞu2 ðX 2 ; tÞ. . . qt

q Bu u ...u ðX 1 ; X 2 ; . . .; X N ; tÞ ¼ qt 1 2 N

Eliminating the derivatives qui(Xj, t)/qt on the right-hand side with the help of Eq. (4.9), we obtain a balance equation for the moment Bu1 u2 ...uN in the form of a combination of hydrodynamic fields and spatial coordinates. Let us first apply this method to the unaveraged quantities r uiuj. From the self-obvious

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equality qrui u j qu j qui ¼ rui þ ru j qt qt qt and from the momentum equation (4.9) that has been brought to the form qrui q þ ðrui uk þ pd ik sik Þ ¼ Fi ; qt qXk we have just used the continuity equation (4.2)), one obtains: qðrui u j Þ q þ ðrui u j uk þ ð pui d jk þ pu j dik Þðui s jk þ u j sik ÞÞ qt qXk qu j qui qu j qui ¼ ðrui F j þ ru j Fi Þ þ p þ þ s jk sik : qX j qXi qXk qXk

ð4:49Þ

If we introduce the density of kinetic energy ek ¼ rui ui =2; then at i ¼ j, the equation (4.49) turns into an equation for kinetic energy: qek q þ ðek uk þ puk ui ski Þ ¼ puk Xk re; qt qXk

ð4:50Þ

where

re ¼

me X qhul i qhum i 2 þ 2 l;m qXm qXl

is the dissipation of kinetic energy in a unit volume of the fluid in a unit time. It follows from this equation that the change of kinetic energy occurs due to the following factors: energy transfer via convective flux and via the work performed by pressure forces and molecular forces (second term on the left-hand side); work of the body force (first term on the right-hand side); viscous dissipation of energy, where the energy density e is given by Eq. (4.18). The averaged continuity equation (4.37) allows to bring the Reynolds equations to the form qrhui i q ðrhui uk i þ r u0i u0k þ h pid ik hd ik iÞ ¼ hFi i: þ qXk qt

ð4:51Þ


Application of this method to the moments rhuiuji yields D E qrhui i u j q þ ðrhui i u j huk i þ r u0i u0k u j þ r u0j u0k hui i qt qXk þ ðh pihui id jk þ h pi u j d ik Þðhui i d jk þ u j hd ik iÞÞ qhui i q u j ¼ ðrhui i F j þ r u j hFi iÞ þ h pi þ qXi qX j D E qhu i qhui i 0 0 q uj q uj i hdik i þ d jk þ r u0j u0k þ r ui u k : qXk qXk qXk qXk ð4:52Þ If we now introduce the density of kinetic energy of the averaged turbulent flow by the relation es ¼ rhui ihui i=2 Eq. (4.51) turns into an equation for the density of kinetic energy of the averaged turbulent flow: qes q þ ðes huk i þ r u0k u0i hui i þ h pihuk ihui ihski iÞ qt qXk qhui i ; ¼ rhuk ihFk ires þ r u0k u0i qXk

ð4:53Þ

where 1 qhul i ne X qhul i qhum i 2 ¼ þ es ¼ hslm i 2 l;m qXm r qXm qXl is the specific dissipation of energy of the averaged flow that occurs due to the viscous forces. The physical meaning of the terms in Eq. (4.53) is the same as the meaning of the terms in Eq. (4.50), with the exception of the term r u0k u0i hui i, which corresponds to the transport of energy via turbulent viscosity. The equation for components of the Reynolds stress tensor can be derived by subtracting Eq. (4.52) term by term from the averaged Eq. (4.49): D E D E qrhui i u j q þ ðr u0i u0j huk i þ r u0i u0k u0j þ ð p0 u0i d jk qt qXk D E D E D E D E þ p0 u0j d ik Þð u0i s0jk þ u0j s0ik ÞÞ ¼ r u0i F 0j D E qu0 qu0 qu0 qu0 j j i þ r u0j Fi0 þ p0 þ þ s0jk i s0ik qX j qXi qXk qXk D E q uj qhui i r u0i u0k þ r u0j u0k : qXk qXk

ð4:54Þ

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D E We see that in addition to the average velocity huii and Reynolds D stressesEr u0i u0j , Eq. (4.54) contains new unknowns: third order central moments r u0i u0k u0j ; second order moments D E D of fluctuations E of velocity and its spatial derivatives appearing in 0 u0i s0jk and s0jk ðqu =qX Þ i k ; and second order mutual moments of pressure and velocity fields p0 u0i and p0 ðqu0i =qX j Þ . The D latter moments can be Erepresented as two-point, third order moments of the type u0i ðX ; tÞu0j ðX 0 ; tÞu0k ðX 0 ; tÞ with the help of Eq. (4.14). From Eq. (4.54), one can obtain an equation for the average kinetic (turbulent) energy density of fluctuational motion: e0k ¼ r u0i u0i =2: Putting i ¼ j into Eq. (4.67), one gets: qe0k q 1 e0k huk i þ r u0i u0i u0k þ p0 u0k u0i s0ki þ qt qXk 2 0 0 qhui i ¼ r uk Fk rhek ir u0k u0i ; qXk

ð4:55Þ

where *

+ qu0l 1 ne X qu0m 2 0 qul þ ¼ slm hek i ¼ qXm 2 l;m qXm qXl r is the average specific energy of fluctuational motion under the action of viscous forces. The terms in Eq. (4.55) have the following physical meaning. The second term on the left-hand side expresses the change of turbulent energy flux density. The four summands in this term represent the contributions from energy transfer by the averaged flow, from turbulent viscosity, from pressure fluctuations, and from molecular viscosity, respectively. The term qhui i A ¼ r u0k u0i qXk

ð4:56Þ

describes energy exchange between the averaged and fluctuational motions. Thus Eq. (4.55) is the equation of turbulent energy balance. It follows from there that turbulent energy density at a given point inside the flow can change via the following mechanisms: transport of turbulent energy from other regions in the fluid; work performed by external force fluctuations; viscous dissipation of turbulent energy; and finally, transformations of energy of the averaged motion into turbulent energy and vice versa. One characteristic that we shall be using frequently in the subsequent discussion is the average kinetic energy of fluctuational motion per unit mass of the fluid,


ek ¼ e0k =r ¼ u0i u0i =2. The equation for ek can easily be obtained from Eq. (4.55): qhui i 1 0 0 Dek q 1 ¼ u0k u0i hek i þ u0i u0i u0k p uk Dt qXk 2 r qXk 0

0 q u q u i k þ þ ne u0i þ u0k Fk0 : qXk qXi

ð4:57Þ

This equation, in its turn, contains new unknown quantities u0i u0i u0k , p0 u0k , and heki. Therefore a system of equations containing the Reynolds equation (4.51) and either the equation for Reynolds stresses (4.54) or the equation for turbulent energy (4.57) will be unclosed. One can always try to construct new equations for the new unknowns, but the derived system will also be unclosed, because it will contain unknown moments of higher orders. Hence, construction of additional equations for higher-order moments gets us nowhere as we try to obtain a closed system of equations describing the turbulent flow. The Reynolds equations and the equation of turbulent energy balance only allow us to infer the existence of certain connections between different statistical characteristics of turbulence, but they cannot be solved. The only way out from this situation is to attempt to close the system of equations by making additional assumptions that are based on certain physical considerations and justified by their agreement with experimental data. In other words, we aim to specify the missing connections between statistical characteristics of turbulence irrespective of the available equations. The ‘‘easiest’’ way to close the system is to simply drop the higher-order moments. It turns out, however, that such a procedure attains its purpose only for relatively small Reynolds numbers that do not present any practical interest. We are interested specifically in the case of great Reynolds numbers, or, to use another term, in the case of fully developed turbulence. In a few cases, the form of additional connections between statistical characteristics can be guessed from dimensionality considerations; the expressions derived in this way are accurate up to a certain small number of empirical constants. But the dimensionality theory still stops halfway in solving the problem, because the resultant relations contain unknown functions and (or) constants, which then have to be determined experimentally. The total number of these functions and constants can be large, because different functions and constants are needed for different flows (flows in pipes, flows bypassing a solid body, flows in boundary layers, jet flows, etc.) Yet another closure method uses transport equations to find the characteristics of turbulence such as turbulent energy, turbulent viscosity, and the integral scale of turbulence. This method is the most popular as of today and is widely used in numerical calculations for different turbulent flows. It is quite natural that one would like to make his task simpler by finding the minimum required number of additional relations, functions, or constants that would applicable all at once to many different flows. Unfortunately, as of today, we are still lacking a universal theory that would describe all kinds of turbulent flows. Inevitably, all the existing turbulent flow models are valid for only one or several types of flows.

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Theories of turbulence that encompass relations found empirically or guessed from physical considerations and then proved by experiments (in addition to the available hydrodynamic equations) are referred to as semi-empirical theories. The existing models of turbulence will be examined in more detail in Section 4.10. But we must first discuss the internal structure of turbulence, and in particular, the underlying concept of isotropic turbulence.

4.8 Isotropic Turbulence

Turbulence is called homogeneous when all hydrodynamic fields are homogeneous random fields (see Section 1.9), and isotropic when all hydrodynamic fields are isotropic random fields (see Section 1.10). Isotropic turbulence is a mathematical idealization, which is suitable for approximate description of some special turbulent flows. In fact, turbulence can be isotropic only when the fluid occupies the entire space. Real flows always have boundaries, and this is where isotropy ends. The case of isotropic turbulence is the simplest one, yet it allows us to establish some distinguishing properties of turbulence. This explains why the concept of isotropic turbulence, which was first introduced by Taylor, has played such a crucial role in the development of the modern theory of statistical turbulence. Later on, Kolmogorov proposed a more general concept of locally isotropic turbulence, which embraced a greater variety of real flows and has since proved itself a powerful tool for the analysis of various turbulent flows. Consider isotropic turbulence in a viscous incompressible fluid in the absence of external forces. As evidenced by the discussion in Sections 4.5 and 4.7, we are primarily interested in the components of correlation tensors, which, according to the definition of a homogeneous, isotropic random vector field, depend on r ¼ X0 X and t, where X0 and X are two arbitrary points in space. For a homogeneous, isotropic random velocity field, the second order correlation tensor Bi j ðr; tÞ ¼ ui ðX ; tÞu j ðX þ r; tÞ is expressed in terms of two scalar functions, BLL(r, t) and BNN(r, t): Bi j ðr; tÞ ¼ ðBLL ðr; tÞBNN ðr; tÞÞ

ri r j þ BNN ðr; tÞdi j ; r2

ð4:58Þ

where BLL(r, t) ¼ huL(X, t)uL(X þ r, t)i and BNN(r, t) ¼ huN(X, t)uN(X þ r, t)i are the longitudinal and transverse correlation functions; uL and uN are projections of the velocity vector u onto the directions parallel and perpendicular to r; and r ¼ |r|. By the same token, the third order correlation tensor Bi j;k ðr; tÞ ¼ ui ðx; tÞu j ðx; tÞuk ðx þ r; tÞ


can be expressed in terms of three scalar functions, BLL,L(r, t), BLN,N(r, t), and BNN,L(r, t): ri r j rk r3

rj rk ri þ BNN;L ðr; tÞÞ d i j þ BLN;N ðr; tÞ d jk þ dik : r r r

Bi j;k ðr; tÞ ¼ðBLL;L ðr; tÞ2BLN;N ðr; tÞBNN;L ðr; tÞÞ

ð4:59Þ

The relations (4.58) and (4.59) can be considerably simplified if the velocity field is solenoidal or potential. The former case is true for an incompressible fluid (ru ¼ 0), and the second – for an ideal fluid (r · u ¼ 0). Since the fluid is assumed to be incompressible, we have a solenoidal velocity field. Then the continuity equation leads us to BNN ðr; tÞ ¼ BLL ðr; tÞ þ

r q ðBLL ðr; tÞÞ; 2 qr

ð4:60Þ

which is the so-called Karman equation. One can see that the second order correlation tensor of an isotropic solenoidal vector field can be expressed in terms of a single scalar function. The third order correlation tensor of an isotropic solenoidal vector field Bij,k(r, t) is also expressed in terms of a single scalar function, because 1 BNN;L ðr; tÞ ¼ BLL;L ðr; tÞ: 2 1 r q ðBLL;L ðr; tÞÞ: BLN;N ðr; tÞ ¼ BLL;L ðr; tÞ þ 2 4 qr

ð4:61Þ

We should mention yet another important property of isotropic turbulent vector fields. Any isotropic random vector field u(x) can be represented as a sum of two mutually uncorrelated fields, one of which is solenoidal and the other – potential. The corollary is that no scalar isotropic field can correlate with a solenoidal vector field. If we choose pressure to be our scalar field and velocity – our vector field, then this corollary reduces to the statement that BPi ðr; tÞ ¼ BPL ðr; tÞ

ri ¼ 0; r

ð4:62Þ

where BPi(r, t) ¼ hp(r, t)ui(r, t)i, BPL(r, t) ¼ hp(r, t)uL(r, t)i. It is now easy to derive the dynamic equation for the correlation tensor Bij(r, t). Let us apply the equations (4.9) (with f set to zero) to i-th component of velocity, ui, at the point X and to j-th component of velocity, u0j , at the point X þ r ¼ X0 . Multiply the first equation by u0j and the second – by ui, add both equations together and average the

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result. The outcome is D E D E D E E 0 D 1 0 q ui u0j q ui uk u0j q ui u0j u0k q p0 u0i A 1 @q pu j þ ¼ þ þ qX 0j r qt qXk qXk0 qXi E D E1 0 D q2 ui u0j q2 ui u0j A: þ þ ne @ qXk qXk qXk0 qXk0

ð4:63Þ

It follows from the homogeneity of turbulence that all two-point moments depend on r ¼ X0 X, therefore q/qXk and q=qXk0 are respectively equal to q/qrk and q/qrk. As a result, Eq. (4.63) reduces to qBi j ðr; tÞ q ¼ ðBik; j ðr; tÞBi; jk ðr; tÞÞ qt qrk q2 Bi j ðr; tÞ 1 qB p j ðr; tÞ qBi p ðr; tÞ : þ 2ne þ qr j r qri qrk qrk

ð4:65Þ

From the property of isotropy, one can deduce the relations (4.62), thus establishing that Bpj(r, t) ¼ Bip(r, t) ¼ 0, and the relations (4.60), (4.61), which mean that the tensors Bij(r, t), Bik,j(r, t), and Bi,jk(r, t) ¼ Bjk,i(r, t) can be expressed through scalar functions BLL(r, t) and BLL,L(r, t). After some algebra, we get the Karman–Howarth equation qBLL ðr; tÞ ¼ qt

q 4 þ qr r

qBLL ðr ; tÞ qBLL;L ðr; tÞ þ 2ne : qr

ð4:65Þ

Just like the Reynolds equations, Eq. (4.65) cannot be solved, because it contains two unknowns, BLL(r, t) and BLL,L(r, t). Consider some important corollaries that follow from the derived equations for correlation functions. These are equations for some functions of r and t, from which one can obtain certain numerical characteristics describing turbulence as a whole (in other words, these characteristics are independent of the distance r between the two points under consideration). To this end, it is sufficient to expand the functions that appear in these equations as a Taylor series over the powers of r and then equate the terms having the same power. We begin with the Karman–Howarth equation (4.65). The zeroth term of the expansion (i.e., r ¼ 0) gives us 2 dBLL ð0Þ q BLL : ð4:66Þ ¼ 10ne qr 2 r¼0 dt Since BLL(0) ¼ hu2i, Eq. (4.66) can be rewritten in the form d 3 2 15ne hu2 i ¼ 15ne u2 f 00 ð0Þ ¼ ; u 2 dt 2 lt

ð4:67Þ


where u is the velocity component along the X-axis, f(r) ¼ BLL(r)/BLL(0), and 2 lt ¼ 1= f 00 ð0Þ. Note that BLL(0) ¼ h(u(x))2i/3. Eq. (4.67) represents the balance of energy for isotropic turbulence. It describes the rate of decrease of the average kinetic energy of turbulence due to the action of viscous forces. The parameter lt has the dimensionality of a length and is called the Taylor microscale. It can be regarded as the smallest size of eddies, which are responsible for energy dissipation. As far as dhu2i/dt hu2i/tt, where tt is the characteristic time of hydrodynamic relaxation (the Taylor time microscale), Eq. 2 (4.67) gives tt lt =10ne . One can use the Karman equation to express the micro2 scale lt through the transverse correlation lt ¼ 2=g 00 ð0Þ, where g(r) ¼ BNN(r)/ BNN(0). The expressions for Taylor microscales can be brought to the form * + * + qu 2 qu 2 hu2 i h u2 i ¼ 2 ; ¼2 2 : ð4:68Þ qX qY lt lt The relations (4.68) make it possible to determine lt by finding the values of h(qu/ qX)2i and h(qu/qX)2i from the experiment. Obtaining lt and hu2i from independent measurements (they can be conducted, for example, behind the grate in a wind tunnel, at different distances from the grate), we can prove the relation (4.67) and establish the attenuation formula for hu2i. The averaged squares of all velocity components decrease with time in accordance with the ‘‘5/2 law’’, namely, 2 u ¼

C ðtt0 Þ5=2

;

ð4:69Þ 2

where t0 is some arbitrarily chosen initial moment (‘‘initial time reading’’), and lt increases linearly with time 2

lt ¼ 4ne ðtt0 Þ:

ð4:70Þ

Since BLL(r) is an even function of r, while BLL,L(r) is an odd function, both sides of Eq. (4.65) contain only even degrees of r. Equating the coefficients for r2 in the Taylor expansion, one gets the equation 1 d 00 7 000 7 B ð0Þ ¼ BLL;L ð0Þ þ ne BIV LL ð0Þ: 2 dt LL 6 3

ð4:71Þ

This equation is interpreted as the balance equation for a vortex, since the correlation tensor for a vortex is equal to Bwi wi ðrÞ ¼ DBii ðrÞ. If at r ! 1, the quantity BLL(r) goes to zero faster than r5, then from Eq. (4.65) there follows the relation ð¥ r 4 BLL ðrÞdr ¼ L ¼ const; 0

ð4:72Þ

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which has the form of a conservation law. The quantity L is called the Loitsyansky integral (or the Loitsyansky invariant). Similarly, by using Eq. (4.13) that expresses pressure in terms of velocity, we can study the statistical properties of a scalar hydrodynamic field – pressure, and by using the equations of heat conduction (4.20) and diffusion (4.21), we can study the statistical properties of scalar fields – temperature and concentration (see the end of this section). In addition to the above-mentioned Taylor microscale, the theory of turbulence introduces four other length scales: longitudinal and transverse differential scales BLL ð0Þ 1=2 l1 ¼ 00 ; 2BLL ð0Þ

BNN ð0Þ 1=2 l2 ¼ 00 2BNN ð0Þ

ð4:73Þ

and longitudinal and transverse integral scales ð¥ 1 L1 ¼ BLL ðrÞdr; BLL ð0Þ 0

ð¥ 1 L2 ¼ BNN ðrÞdr: BNN ð0Þ

ð4:74Þ

0

Comparing the relations (4.74) with the formulas (1.99) and (1,102), one can conclude that integral length scale has the meaning of characteristic correlation length, that is, the average distance that turbulent perturbations can travel. Since correlation between velocities at two different points decreases with increase of the distance between these points, the integral scale is equal by an order of magnitude to the maximum distance between these points at which the velocities still show a noticeable correlation. A further insight into isotropic turbulence can be gained by examining correlation functions in the wavenumber space. As was mentioned in Section 1.10, spectral representations of random functions have the meaning of superposition of harmonic oscillations for stationary random processes. For an isotropic turbulent field, the spectral representation looks especially simple. Representations for components of the correlation tensor Bij(r, t) and its spectral tensor Fij(k) are found from the definitions (1.104) and (1.105): ð¥ Bi j ðr; tÞ ¼ 4p

sinðkrÞ Fi j ðk; tÞk2 dk; kr

ð4:75Þ

0

ð¥ 1 sinðkrÞ Fi j ðk; tÞ ¼ 2 Bi j ðr; tÞr 2 dr; 2p kr

ð4:76Þ

0

The spectrum Fij(k) is symmetric and nonnegative, and its corresponding quadratic form is positive definite. The isotropy condition implies that Fij(k) can be


represented in the form (4.58), namely, Fi j ðk; tÞ ¼ ðFLL ðk; tÞFNN ðk; tÞÞ

ki k j þ FNN ðr; tÞd i j ; k2

ð4:77Þ

where FLL(k, t) and FNN(k, t) are the longitudinal and transverse spectra. The spectral representation of the average energy is ð¥ 1 1 2 u ðX ; tÞ ¼ Bii ð0; tÞ ¼ Eðk; tÞdk: 2 2

ð4:78Þ

0

In the isotropic case, the last relation transforms to Eðk; tÞ ¼ 4pk2

Fi j ðk; tÞ ¼ 2pk2 ðFLL ðk; tÞ þ 2FNN ðk; tÞÞ: 2

ð4:79Þ

The conditions of solenoidality and potentiality allow us to simplify the expressions for E(k, t) and Fij(k, t): Eðk; tÞ ¼

4pk2 FNN ðk; tÞ; for solenoidal field; 2pk2 FLL ðk; tÞ; for potential field;

8 ki k j EðkÞ > > < d ; ij 2 k2 Fi j ðk; tÞ ¼ 4pk > EðkÞki k j > : ; 2pk4

for solenoidal field;

ð4:80Þ

ð4:81Þ

for potential field;

For a solenoidal field, the longitudinal BLL and transverse BNN correlation functions are connected with E(k) through the relations ! ð¥ cos kr sin kr BLL ðrÞ ¼ 2 Eðk; tÞdk; þ ðkrÞ2 ðkrÞ2 0

! ð¥ sin kr cos kr sin kr Eðk; tÞdk; þ BNN ðrÞ ¼ 2 kr ðkrÞ2 ðkrÞ3

ð4:82Þ

0

whereas E(k) is expressed through BLL as

EðkÞ ¼

ð¥ 1 ðkr sin krk2 r 2 cos krÞBLL ðrÞdr: p 0

ð4:83Þ

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Similar relations exist for the spectrum of a third order correlation tensor of an isotropic field u(x): kj ki k j kk ki ; Fi j;k ðkÞ ¼ iFLN;N ðkÞ d jk þ d ik 2 3 k k k 1 FLN;N ðkÞ ¼ 2 8p

ð¥ 0

! 3cos kr 3sin kr BLL;L ðrÞr 2 dr: sin kr þ kr ðkrÞ2

The Karman–Howarth equation (4.65) has the following spectral representation: qFNN ðk; tÞ ¼ 2kFLN;N ðk; tÞ2nk2 FNN ðk; tÞ qt

ð4:84Þ

qEðk; tÞ ¼ 8pk3 FLN;N ðk; tÞ2nk2 Eðk; tÞ qt

ð4:85Þ

or

Equations (4.84) and (4.85) describe the time rate of change of the spectral distribution of isotropic turbulence energy. The second term on the right-hand side gives energy dissipation due to viscosity. The viscosity-related increase in the dissipation of kinetic energy of a perturbation with the wave number k is proportional to the intensity of this perturbation; 2nk2 is the proportionality coefficient. Hence, the energy of long-wavelength perturbations (small values of k) decreases under the action of viscosity at much slower rates than the energy of short-wavelength perturbations. The reason for this is that short-wavelength perturbations produce large velocity gradients, and the viscous friction force is proportional to the velocity gradient. The first term on the right-hand side of equations (4.84) and (4.85) describes the energy change of the spectral component of turbulence with the wavenumber k due to nonlinear inertial terms of hydrodynamic equations. This change leads to a redistribution of energy between spectral components without changing the total energy of turbulence. Hence, any change in the total energy of turbulence is caused exclusively by viscous forces, that is, ð¥ ð¥ q ui u j q ¼ Eðk; tÞdt ¼ 2ne k2 Eðk; tÞdt: qt 2 qt 0

ð4:86Þ

0

The first term on the right-hand side of Eq. (4.85) is negative at small values of k and positive at large values of k, therefore turbulent mixing leads to breakup of turbulent perturbations, that is, to energy transfer from large-scale to small-scale components, with energy being spent to overcome viscous friction. Hence viscosity becomes a major factor for small-scale components. This fact will be used in the next section as we examine the inner structure of developed turbulence. When looking at the inner structure of developed turbulence, we are not as much concerned with correlations between components of velocities at different points X þ r and X at a given moment of time (i.e., with components of the tensor Bij) as we


are with correlations between components of velocity differences Dru ¼ u(X þ r) u(r) at these points. For isotropic turbulence, the condition hDrui ¼ 0 must be valid (see Section 1.10). The corresponding symmetric tensor has components bi j ¼ ðui ðX þ rÞui ðrÞÞðu j ðX þ rÞu j ðrÞÞ ;

ð4:87Þ

known as the structure functions. For simplicity’s sake, we are considering stationary processes only, hence the omission of the time t in Eq. (4.87). Structure functions for an isotropic field can be written in the form similar to that of Eq. (4.58): bi j ðrÞ ¼ ðbLL ðrÞbNN ðrÞÞ

ri r j þ bNN ðrÞdi j ; r2

ð4:88Þ

where bLL(r, t) and bNN(r, t) are the longitudinal and transverse structure functions equal to D E bLL ðr; tÞ ¼ ðuL ðX þ rÞuL ðrÞÞ2 ;

D E bNN ðr; tÞ ¼ ðuN ðX þ rÞuN ðrÞÞ2 :

The longitudinal and transverse structure functions are connected with the corresponding correlation functions through the relations bLL ðrÞ ¼ 2ðBð0ÞBLL ðrÞÞ;

bNN ðrÞ ¼ 2ðBð0ÞBNN ðrÞÞ:

ð4:89Þ

Here B(0) ¼ BLL(0) ¼ BNN(0) ¼ hu2i/3. For a solenoidal field u(x) (incompressible fluid), the longitudinal and transverse structure functions bLL and bNN are mutually connected through an equation similar to the Karman equation (4.60): bNN ðr; tÞ ¼ bLL ðr; tÞ þ

r q ðbLL ðr; tÞÞ: 2 qr

ð4:90Þ

In addition to the two-point second order moments of velocity difference, bij, one can introduce two-point third order moments of velocity difference, bi jk ¼ ðui ðX þ rÞui ðrÞÞðu j ðX þ rÞu j ðrÞÞðuk ðX þ rÞuk ðrÞÞ ; which can be expressed through single scalar function bLLL(r) by virtue of the isotropy condition: bi jk ¼

1 qbLLL ðrÞ ri r j rk 1 qbLLl ðrÞ þ ðrÞ þ r bLLL ðrÞr b LLl r3 2 qr 6 qr

r rj rk i d jk þ dik þ di j : r r r

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Once again, the system of hydrodynamic equations for an isotropic turbulent flow turns out to be unclosed, which is evident, for example, from Eq. (4.90) that contains two unknown functions bNN(r, t) and bLL(r, t). We have to come up with additional hypotheses and relations in order to close this system. Our previous analysis for the isotropic vector field can be extended to the case of an isotropic scalar random field, for example, the field of passive impurity concentration. Let C(X, t) be the concentration of a substance in the fluid and C0 (X, t) – its fluctuation relative to the average value hCi. The fields of velocity (a vector field) and concentration (a scalar field) are assumed to be isotropic, so hui and hCi are constants. A theoretical examination of the turbulent scalar field can be performed in the same manner as for the vector field in the preceding discussion. Let Ca0 ¼ C0 ðX a ; tÞ and Cb0 ¼ C0 ðX b ; tÞ denote concentration fluctuations at the points Xa and Xb ¼ Xa þ r at one and the same instant of time. The correlation of these quantities is Bab ¼ Ca0 Cb0 and the corresponding correlation coefficient is 0 0 rcc ¼ Ca Cb =hðC0 Þ2 i. The quantity hðCa0 Þ2 i ¼ hðCb0 Þ2 i ¼ hðC0 Þ2 i is called the intensity of concentration fluctuations. Just as for the vector field (see Eqs. (4.73)–(4.74)), we can introduce two length scales for the scalar field: differential scale (microscale) lc and integral scale (macroscale) Lc: 2 q rcc ¼ ; 2 qr 2 r¼0 lc 2

ð¥ Lc ¼ rcc ðrÞdr:

ð4:91Þ

0

Having looked at correlations between the values of one and the same scalar quantity C, we may now ask about correlations between C and components of the velocity vector ui – either at one and the same point or at different points. It turns out that, due to the fact that no scalar isotropic field can correlate with a solenoidal vector field, these correlations are absent, that is, h(ui)aCai ¼ 0 and h(ui)aCbi ¼ 0. We can also introduce correlations of higher order, for example, third order correlations at two points h(ui)aCaCbi and h(ui)a(uj)bCbi. It is obvious that h(uiC2)ai ¼ 0 and ðu2i Þa Cb2 ¼ 0. The correlation function Bab(r, t) for an isotropic turbulent field satisfies a dynamic equation of the Karman–Howarth type: qBab q 2 qBab ¼2 : þ BLa;b þ Dm qt qr qr r

ð4:92Þ

where Dm is the coefficient of molecular diffusion; BLa,b ¼ h(uL)aCaCbi; (uL)a is the velocity component along the vector r connecting the points a and b. This equation is called the Corrsin equation. The Corrsin equation leads to the dynamic equation for the intensity of concentration fluctuations h(C0 )2i. One can derive it by going to the limit r ! 0 and expanding the functions entering Eq. (4.92) in a Taylor series similarly to the derivation


of Eq. (4.66). dhðC 0 2Þi Dm ¼ 12 2 hðC0 Þ2 i: dt lc

ð4:93Þ

One can see from Eq. (4.93) that the intensity of concentration fluctuations h(C0 )2i decreases with time, and furthermore, the characteristic time of this decrease is inversely proportional to the coefficient of molecular diffusion. So, in the final analysis, attenuation of intensity of concentration fluctuations is caused solely by the molecular diffusion, just as attenuation of turbulence is caused solely by the molecular viscosity. Introducing tc – the characteristic relaxation time (aka time of micromixing) of the scalar field – through the relation dh(C0 )2i/dt h(C0 )2i/tc, we 2 readily get tc lc =12Dm from Eq. (4.93). Scalar fields can be also represented in the spectral form. Let us introduce the spectral representation of correlations Bij according to the formulas (4.75) and (4.76), in which i and j should be replaced by a and b: ð¥ sin kr 2 ð4:94Þ Bab ðr; tÞ ¼ 4p k Fab ðk; tÞdk; kr 0

Fab ðk; tÞ ¼

ð¥ 1 sin kr 2 k Bab ðr; tÞdr: 2p2 kr

ð4:95Þ

0

At r ! 0, one gets the spectral representation of concentration fluctuation intensity h(C0 )2i: ð¥ D E 0 2 ðC Þ ¼ 4p k2 Fab ðk; tÞdk:

ð4:96Þ

0

By analogy with Eq. (4.79), we can introduce the function Ec(k, t) ¼ 4k2Fab(k, t). The relation (4.96) takes the form D E ð¥ ðC0 Þ2 ¼ Ec ðk; tÞdk

ð4:97Þ

0

and the relations (4.94)–(4.95) transform to

ð¥

sin kr Ec ðk; tÞdk; Bab ðr; tÞ ¼ kr 0

ð¥ 1 Ec ðk; tÞ ¼ kr sin kr Bab ðr; tÞdr: 2p 0

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The differential and integral length scales of the scalar field are expressed as 2 2

lc

¼

ð¥ 1 1 k2 Ec ðk; tÞdk; 3 hðC0 Þ2 i

Lc ¼

0

ð¥ p 1 Ec ðk; tÞ dk: 2 hðC0 Þ2 i k

ð4:98Þ

0

The spectral representation of the Corrsin equation gives rise to the following dynamic equation for the spectrum Ec(k, t): qEc ðk; tÞ ¼ Fc ðk; tÞ2Dm k2 Ec ðk; tÞ; qt

ð4:99Þ

where Fc(k, t) ¼ 8k2FLa,b(k, t); FLa,b(k, t) is the spectral representation of BLa,b in accordance with Eq. (4.94).

4.9 The Local Structure of Fully Developed Turbulence

The concept of isotropic turbulence introduced in the previous section is a mathematical idealization that has little to do with real turbulent flows. Yet it would be a mistake to think that it has no practical importance. In the present section, we are going to introduce the concept of local isotropic turbulence, which makes it possible to examine the local structure of the turbulent flow with rather simple methods and, above all, has direct application to real turbulent flows at very high Reynolds numbers, that is, at Re Recr [12–15]. We reserve the term ‘‘developed turbulent flow’’ for this turbulent flow regime. It remains to note that it is precisely such flows that present the greatest practical interest in practical applications. A distinguishing feature of developed turbulence is the presence of fluctuational motions with various amplitudes that get superimposed on the averaged flow described by the velocity U. To describe turbulent fluctuations, one has to specify not only the absolute velocity values, but also the distances at which velocity can change noticeably. Such distances are called ‘‘motion scales’’ or ‘‘scales of eddies’’. The latter notion eludes precise definition, but for all practical purposes, it is acceptable to imagine a region of size l, within which the turbulent motion is localized. In the subsequent discussion, the term ‘‘motion scale’’ will be understood to refer to just such a region. The most rapid fluctuational motions have the largest motion scales. Their velocities are equal by the order of magnitude to the average flow velocity U and their motion scales – to the characteristic linear scale L of the flow. For example, if the fluid is flowing inside a pipe, then U is the average flow rate velocity and L is the diameter of the pipe. Such fluctuation are called ‘‘large-scale’’. There also exist small-scale fluctuations. Before we define the meaning of ‘‘small scale’’, let us remind you that in principle, the size of fluctuations can be as small as desired – up to the mean free path of a molecule. However, fluctuations that have a very small scale give rise to extremely large velocity gradients, which in their turn invoke


strong forces of viscous friction, causing a very rapid decay of such fluctuations. Hence, the size of fluctuations should be bounded from below by some scale l0 (we will talk about this motion scale later on). Fluctuations with scales l L are defined as small-scale fluctuations. Small-scale fluctuations of the size l l0 are accompanied by considerable energy dissipation, with subsequent conversion of energy into heat. Finally, there is an intermediate region with scales l0 l L. Hence, the entire spectrum of motion scales can be divided into three regions: the energy region l L, the inertia region l0 l L, and the viscous dissipation region l l0. To be sure, this classification is very inexact because it is impossible to establish sharply defined boundaries between these regions. It turns out that small-scale perturbations in a turbulent flow with a very high Reynolds number can be regarded as isotropic, and it is just this property of developed turbulence that we call local isotropy. This statement is based on the following qualitative model of developed turbulence. According to this scheme, developed turbulence consists of a set of disordered perturbations (eddies) that differ from each other by their scale l and velocity ul. As we gradually increase Re, the fluid flow accomplishes a transition from a laminar to a turbulent flow, and then, at a further increase of Re – to developed turbulence. Perturbations of different scales do not appear at the same moment. First, when Re becomes larger than Recr , large-scale fluctuations emerge. As Re keeps increasing, these fluctuations give birth to smaller-scale perturbations, transferring to them a part of their kinetic energy. Those perturbations, in their turn, give birth to even smaller perturbations, and so on. Eventually we get the entire spectrum of fluctuations, where each perturbation gets its kinetic energy from its larger-scale ‘‘parent’’. Perturbations can disintegrate because of their instability. Indeed, each perturbation (fluctuation) is characterized by its own Reynolds number Rel ¼ ull/ne. For the largest fluctuations, the Reynolds number is equal by the order of magnitude to the Reynolds number of the bulk flow, and since Re Recr, large fluctuations are unstable and disintegrate into smaller-scale fluctuations. The Reynolds number of these newly-generated fluctuations is still too large, so they too disintegrate into smaller fluctuations, and so on. The chain of ever-smaller fluctuations continues until the scale of resulting fluctuations approaches l0. This scale corresponds to the Reynolds number Rel0 1 and is called the inner (or Kolmogorov) scale of turbulence. Motions whose scale is l0 or less are hydrodynamically stable and do not disintegrate. For such fluctuations, viscous friction forces are essential. The energy of such fluctuations eventually dissipates into heat. Hence, instability of the averaged motion leads to a continuous flow of energy over the spectrum of fluctuations – from large-scale fluctuations to the fluctuations of minimum scale – with subsequent conversion into heat. In order to for developed turbulence to be sustainable, one has to continuously supply the averaged motion with energy from an external source. It is easy to see that the average specific dissipation energy e (average amount of energy per unit mass per unit time), is an important parameter characterizing the intensity of developed turbulence. The averaged fluid flow is generally inhomogeneous, anisotropic, and nonstationary. Because of the random character of energy transfer from large-scale

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motions to small-scale ones, the orientating influence of the averaged flow will have less and less effect on statistical characteristics of fluctuations as the scale decreases. It is therefore quite natural to assume that in the case of developed turbulence, all perturbations except the largest ones are isotropic. The change of the average flow velocity hui ¼ U with distance becomes noticeable only for distances of the order L. The distance has to be that large in order for inhomogeneity to affect the average flow velocity. Therefore inhomogeneity is only important for large-scale fluctuations, but does not affect small-scale fluctuations. Hence, the second assumption boils down to that of statistical homogeneity of small-scale fluctuations. As the fluctuation scale l decreases, so does its characteristic period tl ¼ l/ul. For small-scale fluctuations it becomes much shorter than the characteristic time tL ¼ L/U during which the averaged flow remains non-stationary. In other words, the change of the average velocity that is responsible for non-stationary character of the flow takes much longer than the change of statistical characteristics of small-scale fluctuations. Therefore small-scale perturbations can be regarded as stationary, or, more precisely, quasi-stationary. Recall that a quasi-stationary flow defined as a flow whose parameters do not explicitly depend on time, while the flow itself does change with time because of its dependence on the integral characteristics of the flow. Hence, the outlined mechanism of developed turbulence leads us to the logical assumption that the statistical regime of small-scale fluctuations (i.e., the ones with length scale l L and time scale tl TL) will be stationary, homogeneous, and isotropic over sufficiently small spacetime regions. This assumption forms the basis of the theory of local isotropic turbulence, which was first formulated by Kolmogorov. Though this assumption cannot be proved rigorously, many functional dependences that follow from the theory of local isotropic turbulence have been confirmed by numerous experiments. We shall now proceed to determine the general qualitative characteristics of developed turbulent flow, keeping in mind what we have just said about the pattern of developed turbulence and using some dimensionality consideration. We begin by considering large-scale fluctuations. In accordance with the preceding discussion, large-scale fluctuations are characterized by the following parameters: characteristic external integral length scale, equal by the order of magnitude to the characteristic length scale of the averaged flow L; characteristic velocity change DU of the most rapid fluctuations on the distance of equal to the scale of fluctuations l L (DU has the same order of magnitude as U); specific dissipation of energy e, equal to e 1 me X e ¼ ¼ re 2 re i; j

*

qui qu j þ qX j qXi

2 + ;

ð4:100Þ

and fluid density re. Since for large-scale fluctuations, the Reynolds number is large, Re 1, the coefficient of molecular viscosity me is not included in the list of characteristic parameters. Nevertheless, energy dissipation does take place, and by analogy with the formula (4.100), it should be characterized by the coefficient of turbulent viscosity mt. Since the expression inside the brackets in Eq. (4.100) has the same


order of magnitude as (DU)2/L2, we have e mt

ðDUÞ2 : L2

ð4:101Þ

On the other hand, in view of dimensionality considerations, the quantities e and e should be expressed through dimensional parameters L, DU, and r. Therefore, e

rðDUÞ3 ; L

e

ðDUÞ3 : L

ð4:102Þ

Thus Eqs. (4.101)–(4.102) give us the dynamic and kinematic turbulent viscosities: mt rDUL: nt ¼

mt ¼ DUL: r

ð4:103Þ ð4:104Þ

The ratio of molecular and turbulent viscosities is equal to ne ne 1 1: nt DUL Re

ð4:105Þ

So the coefficient of turbulent viscosity is much larger than the coefficient of molecular viscosity. The pressure change is approximately (i.e., by the order of magnitude) equal to D p rðDUÞ2 :

ð4:106Þ

Let us go on to small-scale fluctuations with the scale l L. We begin with the inertia region l0 l L where the motion can be considered as non-viscous. The velocity ul of fluctuations having the scale l does not depend on me or on the external parameters L and DU, because l L. Therefore ul can depend only on r, l, and e (or e). The only combination of these quantities that has the dimensionality of velocity is ðelÞ1=3 ¼ ðel=rÞ1=3 . Substituting e from Eq. (4.102) into this formula, we get 1=3 l : ð4:107Þ ul ðelÞ1=3 ¼ DU L We see from Eq. (4.107) that the change of fluctuation velocity on a small distance l is proportional to l1/3. This principle is known as the Kolmogorov–Obukhov law. It can be represented in a spectral form (‘‘spectral’’ here refers to the spatial spectrum) by assigning to each fluctuation its wave number k 1/l instead of l and the kinetic energy E(k)dk per unit mass contained in fluctuations with wave numbers in the interval (k, k þ dk). Since the dimensionality of E(k) is m3/s2, we should compose from the parameters e and k a combination that would have this dimensionality:

j215


216

EðkÞ e2=3 k5=3 :

ð4:108Þ

Integrating (4.10) over k from k to 1, one gets the total kinetic energy contained within fluctuations whose scale is l: ð¥ EðkÞdk

e2=3 ðelÞ2=3 u2l : k

k

Then u2l is equal by the order of magnitude to the total kinetic energy contained within such fluctuations. If we define the coefficient of turbulent viscosity as nt lul by analogy with the formula (4.104), the relation (4.107) will take the form

u 2 u3 e l nt l : ð4:109Þ l l Let us introduce the characteristic period of fluctuations, tl ¼ l/ul and determine the order of velocity change Dut at a given point in space during the time tl that is small compared to the characteristic external time tL L/U. The presence of the averaged flow leads to conclude that after the time tl any arbitrary point in space will be filled with the fluid that initially was separated from this point by the distance Utl. Therefore Dut can be derived from the formula (4.131), in which l should be replaced by Utl: Dut ðeUtl Þ1=3 :

ð4:110Þ

One should distinguish the quantity Dut from the change of velocity Du0t of a given volume element of the fluid (fluid particle) moving in space. Since the latter depends only on the parameters e and tl, we must compose a combination of this parameters that has the dimensionality of velocity dimension: Du0t ðetl Þ1=2 :

ð4:111Þ

It is readily seen that the change of velocity of a moving fluid particle is propor1=2 tional to Dut tl , while the change of velocity of the fluid at a given point in space 1=3 obeys another law: Dut tl , so when tl T, we have Du0t Dut . Now the formulas (4.107) and (4.110) can be represented as 1=3 ul l ; DU L

t 1=3 ut : l DU T

ð4:112Þ

The form of these relations shows that characteristics of small-scale fluctuations in different developed turbulent flows differ from each other only by their length and velocity (or length and time) scales. This statement forms the essence of the selfsimilarity property of the local isotropic turbulence. Let us determine the distance l0 at which viscous effects become significant. As we noted earlier, this distance corresponds to the local Reynolds number


Rel ¼ ull/ne 1. Substituting the relation (4.197), we write 4=3 4=3 DUl l Rel ¼ Re 1; L ne L1=3

ð4:113Þ

where Re is the Reynolds number of the average flow. This condition yields l0: 3 1=4 L n : l0 3=4 e e Re

ð4:114Þ

The characteristic velocity and characteristic time of such fluctuations are obtained from Eq. (4.107):

ul0

1=2 e DU ¼ l ; 0 ne Re1=4

tl0 ¼

1=2 l0 n : e ul 0

ð4:115Þ

The scales l0 and tl0 are respectively known as the Kolmogorov (or inner) spatial and temporal microscales. The values of l0 and ul0 decrease with increase of the Reynolds number of the average flow. At l l0, the motion of the fluid has viscous character. Turbulent fluctuations do not vanish suddenly; instead, they gradually decay subject to viscous forces. Since velocity changes rather smoothly in this region, it can be expanded in Taylor series over the powers of l. Let us keep only the first term of the series ul const l and determine the constant from the condition ul ul0 at l l0. Then ul

ul 0 DU l lRe1=2 : l0 L

ð4:116Þ

Scales of turbulent fluctuations are function as their spatial characteristics. In addition to those, we may consider the time characteristics of fluctuations, namely, the frequencies ol. The whole frequency spectrum can be divided into three intervals. The lower end of the spectrum, oL U/L, corresponds to the energy region; the upper end, wl0 U=l0 URe3=4 =L, corresponds to the dissipation region; and the intermediate interval wL wl wl0 – to the inertia region. The inequality ol oL means that the external (average) flow can be considered as stationary with respect to the local properties of small-scale fluctuations. Energy distribution over the frequency spectrum in the inertia region is derived from Eq. (4.108) by replacing k with ol/U: 5=3

EðwÞ ðUeÞ2=3 wl

:

ð4:117Þ

The frequency ol defines the repetition period of velocity at a fixed point in space. Together with ol, we can introduce another frequency w0l , which stands for the repetition period of velocity of a chosen fluid particle. The distribution of

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218

energy over the frequency spectrum for such particles does not depend on U but depends only on e and wl . We conclude from dimensionality considerations that Eðw0l Þ

e : w0l

ð4:118Þ

We now apply the results obtained for small-scale fluctuations to estimate the velocity of inertialess particles buoyant in the fluid. Turbulent mixing causes particles to gradually move away from each other. Consider two particles such that the initial interparticle distance does not exceed the size of fluctuations from the inertia region. We make this requirement because otherwise large fluctuations would just transport the two particles without changing the interparticle distance. Our assumption allows us to find the rate of change of interparticle distance d from the equation dd ul ðedÞ1=3 : dt

ð4:119Þ

By solving this equation at a given initial value of the interparticle distance d0, we find the time it takes for the two particles to move away from each other so that the gap between them reaches the value d1. In the limiting case d1 d0 this time is 2=3

t

d1 : e1=3

ð4:120Þ

Now consider the correlations of velocity differences of two neighboring particles at a fixed instant of time. These correlations were introduced as structure functions in Section 4.8. Even the formula (4.107) gives a qualitative correlation of velocities at two points separated by a distance l L, in other words, it provides a connection between velocity values at two neighboring points. Components of the correlation tensor bik(r, t) serve as quantitative characteristics of this correlation. In an isotropic vector field, these components depend on two scalar functions – longitudinal and transverse functions bNN(r, t) and bLL(r, t), where r ¼ |r2 r1| is the length of the radius vector between the points r1 and r2. In the case of local isotropic turbulence, we have l0 r L. The change of velocity at small distances is caused by small-scale fluctuations and is independent of the average flow. Therefore our analysis of correlation and structure functions can be simplified if we assume that isotropy and homogeneity take place not only at small scales, but at large scales as well. Then the average velocity can be taken to be zero (see Section 1.10), and we can take advantage of the relations between the functions bLL and bNN that have been established in Section 4.8. Because of Eq. (4.107), the difference of velocities over the distance r in the inertia region is proportional to r1/3, therefore bLL and bNN are proportional to r2/3. In other words, in any turbulent flow with a sufficiently high Reynolds number, the rootmean-square value of the difference of velocities at two points separated by the distance r (where r is neither too small nor too large) should be proportional to r2/3.


This law, which was first established by Kolmogorov, is one of the most important laws describing turbulent flows and is called ‘‘the law of two thirds’’. Switching to the spectral form, we can formulate a similar law for the energy spectrum: EðkÞ e2=3 k5=3 ;

ð4:121Þ

which is called ‘‘the law of five thirds’’. Let us now obtain the connection between bLL and bNN in the inertia region. First we transform Eq. (4.90) to the form bNN ¼

1 d 2 ðr bLL Þ: 2r dr

ð4:122Þ

Recalling that both bNN and bLL are proportional to r2/3, we can write bNN ¼ 4bLL =3;

ðl0 r < LÞ:

ð4:123Þ

In the dissipation region (l l0), the velocity difference at two neighboring points is proportional to r, as follows from Eq. (4.116). Then bLL and bNN are proportional to r2 and the formula (4.122) reduces to bNN ¼ 2bLL ;

ðr l0 Þ:

ð4:124Þ

The longitudinal and transverse functions bLL and bNN for small-scale fluctuations can be expressed in terms of specific dissipation of energy: bNN

2e 2 r ; 15ne

bLL

e 2 r : 15ne

ð4:125Þ

The above-considered case corresponds to the situation when the average fluid flow is absent, for example, the fluid has been subjected to intensive shaking and then left alone. Such motion decays with time, and small-scale fluctuations decay in accordance with the power law ul t5=4 :

ð4:126Þ

The adduced statistical characteristics have been examined using only considerations of similarity and dimensionality, which do not invoke any hydrodynamic equations. The main conclusion is that the statistical regime of small-scale components of turbulence at high Reynolds numbers is quite independent from the peculiarities of the macroscopic structure of the flow, which can affect only the value of e. Therefore dynamic equations for the characteristics of locally isotropic turbulence do not depend on the character of large-scale motions, and it is sufficient to consider the case of isotropic turbulence in unbounded space and find the connections between its local characteristics. The obtained characteristics will then be

j219


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the same for all turbulent flows sharing the same values of e and ne if Re 1. Hence, all the relations given above are universal for any locally isotropic turbulent flow. Note that in the case of local isotropy the system of dynamic equations is also unclosed, and we need additional hypothesis and relations to close such a system. Our analysis of the local structure of the velocity vector field can be repeated for a scalar field of passive impurity concentration C(X, t). Consider a developed turbulent flow of some fluid containing a passive impurity; the impurity does not influence the turbulent flow of the carrier fluid. An intense mixing of fluid volumes with different impurity concentrations occurs in a developed turbulent flow. Under the action of fluctuations with different scales, there occurs mixing of both small volumes (microvolumes) and relatively large volumes (macrovolumes or, to use a different term, moles). As was shown earlier, small-scale perturbations in a developed turbulent flow can be considered stationary and isotropic, that is, locally isotropic. It is natural to expect that perturbations of the field of concentrations in a small regions or space will also be stationary and isotropic, in other words, that the scalar field C(X, t) in such regions will be scalar isotropic. Gradual disintegration of fluctuations – starting from the largest-scale ones (whose size is equal by the order of magnitude to the characteristic linear size L of the flow region) and all the way down to the Kolmogorov microscale l0 – is the underlying process responsible for the formation of velocity fluctuation spectrum. Of all characteristics of large-scale motions, only specific energy dissipation e has an effect on small-scale motions. The same reasoning can be applied to the field of concentrations by replacing the Reynolds number Re ¼ DU L/ne with the diffusion Peclet number PeD ¼ LdU/Dm, assuming PeD 1 and, of course, keeping the condition Re 1. Here L is the characteristic linear scale of change of average concentration hC(X)i; dU is the change of average velocity on the distance L; Dm is the coefficient of molecular diffusion. If L > L, we should take DU as our dU. Because of their instability, large-scale fluctuations of concentration will rise to smaller and smaller fluctuations – all the way down to the minimum fluctuation, ð0Þ which has the inner concentration scale lc . Using the same reasoning as for the vector field, we arrive at the statement that in a spatial region of scale lc L, the field of concentrations will be locally isotropic so that the average concentration hCi can be considered constant. The degree of concentration inhomogeneity in these regions is given by the parameter characterizing the change of concentration fluctuations C0 ¼ C hCi. The meaning of this parameter is analogous to specific energy dissipation e, which is determined by the velocity gradient ru ¼ qui/qXj (see Eq. (4.100)) rather than velocity fluctuations. Therefore it is quite natural to assume that a quantity similar to (4.100), namely, D E hN i ¼ Dm ðrC0 Þ2 :

ð4:127Þ

will serve as a measure of concentration inhomogeneity. This parameter is called dissipation of concentration inhomogeneity. Since we have hCi ¼ const for small-scale (lc L) fluctuations of concentration, the formula


(4.128) can be rewritten as D E hN i ¼ Dm ðrCÞ2 :

ð4:128Þ

As far as there are two characteristic length scales L and L, let us introduce L0 ¼ min(L, L) and divide the entire spectrum of concentration fluctuations into two intervals: the interval of large-scale fluctuations with lc L0 and the interval of small-scale fluctuations with lc L0. For the first interval, the characteristic quantities are the length scale L0, the change of average velocity DL0 U, and the change of average concentration DL0 hCi. We can build a combination having the dimensionality of hNi from these parameters: hN i

DL0 UðDL0 hCiÞ : L0

ð4:129Þ

In practice, characteristic length scales L and L are equal by the order of magnitude, that is, L0 ¼ L L. Therefore we can introduce the coefficient of turbulent diffusion Dt by analogy with the formula (4.101): hN i ¼ D t

DhCi L

2 :

ð4:130Þ

Comparing the relations (4.129), (4.131) with (4.101), (4.104) and keeping in mind that L L, we get: Dt LDU nt :

ð4:131Þ

Thus the coefficients of turbulent diffusion and turbulent viscosity have the same order of magnitude. Now consider the small-scale interval lc L0. The velocity field of small-scale perturbations is characterized by two dimensional parameters: specific dissipation of energy e and kinematic viscosity coefficient ne. When examining the consideration field in this region, we should bring in two more parameters – the coefficient of molecular diffusion Dm, which enters the diffusion equation for the impurity, qC qC ¼ Dm DC þ uk qt qXk

ð4:132Þ

and the dissipation of concentration inhomogeneity hNi. The ratio between the convective term (second summand on the left-hand side) and the diffusional term (right-hand side) in Eq. (4.132) is equal by the order of magnitude to the diffusion Peclet number PeD ¼ DL0 L0 =Dm. Molecular diffusion plays a considerable role only for PeD < 1. For the averaged concentration, the Peclet number is usually PeD 1,

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so the effect of molecular diffusion is negligible for large-scale perturbations of concentration, and turbulent diffusion emerges as the main mechanism behind macroscopic mixing of regions with different impurity concentrations. Since PeD lc, a smaller scale of concentration fluctuations means a smaller PeD. When ð0Þ ð0Þ lc ¼ lc , the Peclet number is PeD ¼ 1, and when lc lc , the inequality PeD 1 is satisfied. Thus the interval of small-scale concentration fluctuations contains two ð0Þ subintervals: the convective interval lc lc < L, for which PeD 1, and the ð0Þ dissipation interval lc lc , for which Pe 1. In the convective interval, molecular diffusion does not play any noticeable role, therefore the parameter Dm does not figure among its governing parameters. The governing parameters in this interval are e, ne, and hNi. In the dissipative interval, where molecular diffusion plays a noticeable role, the governing parameters will include e; ne ; Dm , and hNi. Two combinations with the dimensionality of length can be built from these four parameters: the inner scale of turbulence (Kolmogorov scale) l0 ¼

3 1=4 ne L ¼ 3=4 e Re

and the inner scale of diffusion (Batchelor scale) ð0Þ lc

1=4 le D2m ¼ lb ¼ ¼ l0 Sc 1=2 ; e

ð4:133Þ

where Sc ¼ le/Dm is the Schmidt number. It should be noted that for fluids, Sc 10, whereas for infinitely dilute solutions, Sc 103. By its physical meaning, the convective interval should comprise those concentration scales for which molecular diffusion is negligibly small as compared to convection. But the presence of two quantities (ne and Dm) of the same dimensionality leads to the appearance of a new dimensionless number – the Schmidt number Sc – and to the dependence of the lower end of the interval upon Sc in accordance ð0Þ with Eq. (4.133). Therefore the condition l lc alone does not guarantee that the fluctuation belongs to the convective interval. To determine where the intervals where convection or molecular diffusion dominates, one should compare the transport coefficients ne and Dm. If they are of the same order of magnitude, then Sc 1 and the length scales l0 and lb are roughly the same. The cases of Sc > 1 and Sc < 1 require additional study. As long as Sc 1, we have lb l0, in other words, there exists within the viscous interval a visco-diffusional interval where molecular diffusion plays a significant role. In conclusion, we shall give an approximate expression for the structure function of the concentration field dcc(r) ¼ h[C(X þ r) C(X)]2i. For small-scale fluctuations of concentration, the dimensionality theory suggests the expression r ne dcc ðrÞ ¼ hN iðeÞ1=2 D1=2 F ; : ð4:134Þ m lb Dm

4.10 Turbulent Flow Models

In the dissipation region, at r lb, there exists the following representation: dcc ðrÞ

hN i 2 r : 3Dm

ð4:135Þ

In convective interval, at L r lb, the structure function is given by

dcc ðrÞ

hN i 2=3 r : 3ðeÞ1=3

ð4:136Þ

The formula (4.136) is called ‘‘the law of two thirds’’ for the concentration field. The spectral ‘‘law of five thirds’’ for a local isotropic concentration field has the form Ec ðkÞ

hN i : ðeÞ1=3 k5=3

ð4:137Þ


It was shown in Section 9.6 that the system of Reynolds equations describing a turbulent fluid flow is unclosed, because the number of unknowns is greater than the number of equations. Attempts to close the system by adding equations for higher-order moments were unsuccessful because those additional equations contain new moments of higher order. Therefore neither the Reynolds equations on their own, nor a system of Reynolds equations plus equations for higher-order moments (e.g., the energy equation discussed in Section 4.7), nor the simplified equations for isotropic (see Section 4.8) or locally isotropic turbulence (see Section 4.9) can be solved. All they can do for us is to establish certain connections between different statistical characteristics of turbulence. There exist several possible ways to close the system of Reynolds equations. The first way is to use experimental data to determine the functional connections between moments of some definite order and the lower-order moments. A second way is to deduce these connections from simple hypotheses that are well justified on physical grounds and are accurate up to some empirical constants. This method lies in the basis of all semi-empirical theories of turbulence. Finally, the third and currently the most widespread method is based on the use of transport equations for some characteristics of turbulence. It should be emphasized that there are no universal relations that would be applicable to all turbulent flows. Each of the existing approximations is suitable only for some type of flows, for example, flows in tubes, boundary layer flows, jet flows, flows past a body, and so forth. The present section offers a review of several models from which one can derive additional equations and thus close the system.

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4.10.1 Semi-empirical Theories of Turbulence

The failure of the Reynolds equations (4.37), (4.39) to form a closed system is explained by the presence of new unknowns – correlations of velocity fluctuations ð1Þ that appear in the Reynolds stresses ti j . The simplest way to close the Reynolds equations is to establish connections between the Reynolds stresses and the average hydrodynamic fields. Such methods are called local equilibrium algebraic methods and the corresponding relations are said to be of the gradient type. The methods based on approximating the Reynolds stresses with the help of parameters determined by the average velocity profile in the given cross section are well-developed and widely used for calculating different flows. The range of application of these methods is limited to the turbulent flows whose turbulence characteristics in a given cross section do not depend on their distributions in the preceding cross sections. The main difficulty is to find the range of applicability of these methods. Some of the resulting connections are adduced below. 1. The Boussinesq model: For simplicity’s sake, we consider a stationary fluid flow in a flat channel in the absence of external forces (fi ¼ 0). The average velocity has one component huxi ¼ U parallel to the channel wall and depending only on the transverse coordinate Z. Suppose that ux ¼ U þ u0 , uz ¼ w0, where u0 and w0 are fluctuations of the longitudinal and transverse velocity components. Then Eqs. (4.39) take the form qt qh pi ¼ ; qz qX

re

qhw 02 i q h pi ; ¼ qZ qZ

ð4:138Þ

where the stress t is equal to t ¼ re ne

dhui rhu0 w 0 i: dZ

ð4:139Þ

A new unknown function t0 ¼ rehu0 w0 i appears in the equations. Thus, in order to close the set of equations (4.138) and (4.139), it is sufficient to express t0 through U(Z). The Boussinesq hypothesis states that the following equality is valid: re hu0 w 0 i ¼ re nt

dU ; dZ

ð4:140Þ

where nt is a quantity with the dimensionality of viscosity; it is called the turbulent viscosity coefficient. Strictly speaking, the equality (4.140) does not constitute a closure relation, because in order to determine the new unknown nt, one needs to have experimental data or to formulate a supplementary hypotheses. The simplest way out is to take nt ¼ const. But then, in a notable analogy with Eq. (4.44), the introduction of nt will be tantamount to replacing the fluid viscosity ne with ne þ nt, and the Reynolds equations will be equivalent to equations for the laminar flow with a new viscosity


coefficient. In this case the obtained velocity profile will be a parabolic Poiseuille profile, while it is well known that such turbulent flows have a logarithmic, rather than parabolic, velocity profile. We conclude that nt cannot be a constant; instead, it should be a function of Z. Let us estimate the form of this function by using the dimensionality theory. Consider the flow near a flat wall. Let DU be the characteristic variation of velocity at the distance Z from the wall. Since no characteristic linear size has been assigned to our flow, we shall take Z as the characteristic linear size. The two governed parameters DU and Z can form the quantity nt ¼ DUZ

ð4:141Þ

that has the dimensionality of viscosity. From other hand, in the vicinity of the wall, we can assume DU ZdU/dZ. Then the friction force per unit area of the wall is 2 dU dU 2 dU t f ¼ mt ; ¼ re nt re Z dZ dZ dZ

ð4:142Þ

from which there follows 1=2 tf dU 1 : re dZ Z Since the value of tf at the wall has to be constant, we have U

1=2 tf lnZ: re

ð4:143Þ

Thus, a simple estimation shows that nt decays linearly as we get closer to the wall and that the longitudinal velocity profile has a logarithmic form. Actually, the structure of the flow in the vicinity of the wall is more complex. A detailed analysis of the flow structure, which takes into account the transformation of the turbulent boundary layer into a viscous boundary layer, shows that in fact, in the region adjacent to the wall, nt decays much faster (never slower than z3). In spite of the fact that the assumption nt ¼ const is inadmissible for turbulent flow inside pipes, there are some flows for which this simple model is acceptable, such as, for example, turbulent jet flows and turbulent flows in the open atmosphere. For such flows, nt should be considered as a parameter that varies for different flows and is determined from experiments. We should also mention that a similar model can be applied to the problems involving heat or passive impurity propagation in a turbulent flow once we introduce the coefficient of turbulent thermal diffusion wt and the diffusion coefficient Dt (see Eq. (4.46) and Eq. (4.47).

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2. The Prandtl model: The model proposed by Prandtl is based on the concept of mixing length. Prandtl attributes a physical meaning to the quantity L in Eq. (4.104), taking his hint from the analogy between the turbulent flow and the random molecular motion in the kinetic theory of gases. According to this theory, viscosity ne is defined by the same formula where DU is the average velocity and L is the mean free path of molecules. The velocity of molecular motion is certainly much greater then the average velocity of a turbulent fluid flow, whereas the mean free path of molecules is much shorter than the scale of fluctuations, so that the product of these quantities gives the difference between ne and nt that is in good agreement with the formula (4.105). Similarly to the exchange of momentum between molecules, in a turbulent momentum exchange, a finite fluid volume leaving the layer separated from the given layer by a certain distance conserves its average momentum until it reaches the given layer. There it mixes with the ambient fluid, transferring the entire momentum difference to this fluid. The average distance between the initial layer, from where the volume has started its journey, and the destination layer, where it mixes with the ambient fluid, is called the mixing length. That is why Prandtl’s theory is sometimes called the mixing length theory. As in the previous model, we shall consider a plane-parallel flow with the average velocity U along the X-axis. The Z-axis is perpendicular to the X-axis and is pointing in the upward direction. The adopted model says that the volumes coming from the lower layer Z l0 and from the upper layer Z þ l0 will reach the layer Z. If the mixing of the arriving volumes with the ambient fluid happens instantaneously, the volumes bring to the layer Z the same momentum which they held initially while inside the layers Z l0 and Z þ l0 . Such an exchange will lead to the emergence of fluctuations of the transverse velocity w0 which by their order of magnitude are equal to w 0 UðZ l0 ÞUðZÞ l0

dUðZÞ : dZ

ð4:144Þ

We may now determine the friction force per unit area exerted on the layer Z by the upper and lower layers. If we designate the momentum from the upper layer as positive and the one from the lower layer – as negative, then t f ¼ re hw 0 ðUðZ l0 ÞUðZÞÞi re hw 0 l0 i

dUðZÞ : dZ

Plugging in the expression (4.144) for w0 and designating l2 ¼ hl0 2i, we obtain

t f ¼ r e l2

dUðZÞ 2 : dZ

Since tf should be a positive quantity, the latter formula can be represented as


dUðZÞ dUðZÞ t f ¼ re l2 : dZ dZ

ð4:145Þ

Now, similarly to Eq. (4.42), if we take tt ¼ rnt dU/dZ and use the relation (4.145), we have dUðZÞ : nt ¼ l2 dZ

ð4:146Þ

For the problem of a flow in a plane channel near the wall, we can take l Z. Then, using the formulas (4.145) and (4.146), we obtain a logarithmic velocity profile near the wall but outside the viscous sublayer. However, in the region close to the symmetry axis of the channel, the obtained approach is unacceptable. In the latter case, it is better to take l ¼ const. In contrast to the Boussinesq model, the unknown parameter in the Prandtl model is the mixing length l, which depends on coordinates and must be obtained experimentally for any specific flow. Prandtl’s model, as well as Boussinesq’s one, is not applicable to all turbulent flows. 3. The Taylor model: Taylor has suggested his model, known as the theory of eddy transport, in an attempt to properly account for the influence of pressure fluctuations on fluid particles. The theory is similar to Prandtl’s in that it also uses the concept of mixing length, but unlike Prandtl, Taylor considers the mixing layer for the velocity vortex, and not for the momentum vortex. Consider a two-dimensional flow with the average velocity hui ¼ (U, W) in the (X, Z) plane. In this flow, the average vortex has only one component: qU qW wy ¼ O ¼ : qZ qX Let the average flow be parallel to the X-axis. Then hui ¼ (X, 0) and the turbulent component of stress t(1) ¼ rehu0 w0 i obeys the momentum equation qt q 0 0 qw0 qu0 þ w0 h u w i ¼ u0 ¼ qZ qZ qZ qZ

0 ð4:147Þ 0 qu qw 1 q ¼ w0 þ ðhu0 2ihw 0 2iÞ: qZ qZ 2 qX When deriving this equation, we used the continuity equation for velocity fluctuations, qu0 qw 0 þ ¼ 0: qZ qZ

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Let the flow be uniform along the X-axis. Then all derivatives with respect to X are equal to zero and the equation (4.147) transforms to D E qt ¼ re w 0 w0y : qZ

ð4:148Þ

We now introduce the mixing length l01 for the velocity vortex through the relation similar to Eq. (4.144): w0y ¼ l01

qO ; qZ

ð4:149Þ

where O ¼ hoyi ¼ dU/dZ. We also retain the expression (4.144) for transverse velocity fluctuations w0: w 0 ¼ l0

qU : qZ

ð4:150Þ

Then D E qU qO qt qU q2 U ; ¼ re w 0 w0y ¼ re l01 l0 ¼ re l21 qZ qZ qZ qZ qZ2

ð4:151Þ

where l1 ¼ ð l01 l0 Þ1=2 is the characteristic length that plays the same role in the Taylor model that l plays in the Prandtl model. Taking l1 ¼ const, we get from Eq. (4.151) 2 1 dU : t f ¼ re l21 2 dZ

ð4:152Þ

This p expression coincides with the formula (4.145) in the Prandtl theory if we take ffiffiffi l1 ¼ 2l. In all other respects, the Taylor theory is different from Prandtl theory, and makes different predictions. For example, the velocity profile for the channel flow predicted by the Taylor theory is in good agreement with experimental data all the way to the central axis of the channel, in a stark contrast with predictions of the Prandtl theory. As all other semi-empirical models, the Taylor model does not solve the closure problem entirely, because it reduces to a single empirical parameter – the vortex mixing length l1. The main shortcoming of the Taylor theory is it limited range of application – it is suitable only for two-dimensional problems. Finally, it is necessary to make the following note. The coefficient of turbulent viscosity nt in the Boussinesq model and the mixing lengths l and l1 in the Prandtl and Taylor models have been introduced purely formally (albeit with some supporting physical rationalizations) for plane-parallel fluid flows in an attempt to describe the simplest fluid flows – in pipes, channels, and boundary layers. We still


need to show now how these parameters can be introduced in the general case of an arbitrary spatial flow. Let us suppose that turbulence emerges as a result of transition of a part of the average flow energy into small scale perturbations. Then, according to the energy balance equation (4.55), the inequality A > 0 should hold, where A is a term in the energy equation (see Eq. (4.56) describing the exchange of energy between the averaged motion and the fluctuational motion. Indeed, the condition A > 0 means that the turbulent energy density ek at a given point increases at the expense of energy of the averaged flow. Then all statistical characteristics of turbulence, including the Reynolds stresses, should depend on the field of the average velocity. ð1Þ The Reynolds stresses ti j play the same role with respect to the averaged motion as viscous forces with respect to the laminar flow. Therefore, when deformation of fluid particles is not taken into account, the averaged flow is similar to the motion of a rigid body, and the Reynolds stresses are pointing along the normal to any surface element arbitrarily selected within the fluid. Then the tensor r u0i u0i is isotropic and can be represented as a spherical component of the rate-of-strain tensor (see Eq. (4.4)): D E r u0i u0j ¼ cdi j ;

2 1 c ¼ r u0k u0k ¼ rek : 3 3

ð4:153Þ

The turbulent energy rek is similar to –p in the incompressible liquid law (4.4). In the general D caseEthat takes into account the deformation of fluid particles, the stresses ti j ¼ r u0i u0j depend on the derivatives of the average velocity with respect ð1Þ to coordinates. Since the tensor ti j is symmetrical, it depends on the rate-of-strain tensor Eij (see Eq. (4.5)). In the case of small deformations, this dependence is linear and the proportionality coefficient has the meaning of turbulent viscosity coefficient, analogously to the Navier–Stokes law. Let us now dwell on the analogy with the kinetic theory of gases. This theory holds that the coefficient of molecular viscosity is equal to ne umlm, where um and lm are, respectively, the average velocity and the mean free path of molecules. Suppose that a similar relation is true for the turbulent motion, with the root-mean-square value of velocity fluctuation functioning as our um, and the integral scale of turbulence – as lm. In the Prandtl theory, this scale is the mixing length, which has the order of the integral scale of turbulence and, as we noted in Section 4.8, has the meaning of the average distance that turbulent fluctuations can travel. As we are concerned with spatial motions, the turbulence will be characterized by different scales assigned to different directions. The set of scales lij will then form a symmetric scale tensor. D Now, E using this tensor and taking advantage of the symmetry of the tensor r u0i u0j , we can assume D E 2 pffiffiffiffi ð1Þ ti j ¼ r u0i u0j ¼ rek d i j r ek ðlik Ek j þ l jk Eki Þ: 3

ð4:154Þ

This formula was first suggested by Monin. It can be thought of as a generalization of the Boussinesq and Prandtl models.

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230

Sometimes, in the first approximation, we can take li j ¼ ldi j : Then (4.154) takes the form pffiffiffiffi 2 ð1Þ ti j ¼ rek di j rl ek Ei j : 3 Defining the coefficient of turbulent viscosity as pffiffiffiffi n t ¼ l ek ; one gets 2 ð1Þ ti j ¼ rek di j 2rnt Ei j : 3

ð4:155Þ

4.10.2 The Use of Transport Equations

Following the papers [16–18], we make a brief review of several models of turbulence that are based on transport equations for various statistical characteristics of turbulent flows. In these models, the minimum possible number of parameters that can be used to describe turbulence is equal to three. The turbulent stress, the energy of turbulence, and a third parameter which, when combined with the energy of turbulence, would result in a quantity having the dimensionality of length, is the most common choice of parameters. These models have been tested for problems that involve flows in channels and boundary layers. The most influential publications that shaped the development of this models are [19–22]. Numerous models tailored for different types of turbulent flows have been proposed as of today. All of them can be classified as one-, two-, and three- parametric according to the number of transport equations employed. If a model contains less then three transport equations, it means that this model includes some algebraic relations between various characteristics of turbulence. An increase of the number of transport equations complicates the problem considerably, as we are faced with the necessity to measure the constants involved and determine the range of applicability of the model. The models listed below, as well as the corresponding equations, are written in the approximation of a stationary plane boundary layer for a homogeneous incompressible fluid (r ¼ const). The velocity u has two components, u1 ¼ u and u2 ¼ v along the X 4 Y coordinate axes. 1. One-parametric models: P 0 These models use one equation for the turbulence energy ek ¼ 0:5 ui 2 , for the ð1Þ Reynolds stress (shear stress) t0 ¼ t12 =r ¼ hu0 v0 i, or for the turbulent viscosity nt.


a. Kolmogorov [19] was the first to suggest to use the equation for the turbulence energy. In the stationary plane boundary layer approximation, this equation has the following form: Dek q qek qhui ¼ þ t0 DE e; Dt qY qY qY

ð4:156Þ

where D/Dt ¼ huiq/qX þ hviq/qY; e ¼ ne ðqu0i =qXk Þðqu0i =qXk Þ is the specific dissipation of energy; and DE is the effective diffusion coefficient. The respective terms on the right-hand side of Eq. (4.156) describe the processes of diffusion, production, and dissipation of energy. The diffusional term is written in the gradient form, with the effective diffusion coefficient DE. The parameters e and DE entering this equation were defined by Kolmogorov based on dimensionality considerations: e ¼ CE L1 e3=2 k ;

pffiffiffiffi DE ¼ CD L ek :

ð4:157Þ

Making a correction to account for the molecular viscosity, we rewrite Eq. (4.157) as 0 2 e ¼ CE L1 e3=2 k þ CE ne ek =L ;

pffiffiffiffi DE ¼ CD L ek þ ne :

ð4:158Þ

Here L is the integral scale of turbulence, and CE ; CE0 , and CD are empirical constants. A shortcoming of this model is the need to specify the integral scale of turbulence L, which depends on the flow pattern. b. Transport equation for the Reynolds stress t0 ¼ hu0 v0 i is derived in [23]. For a plane-parallel flow of an incompressible fluid in a boundary layer, it has the following form: pffiffiffiffiffiffi q Dt0 qhui t3=2 ¼ a1 t0 a1 tm a1 ðGt0 Þ: Dt L qY qY

ð4:159Þ

The presence of the empirical functions L and G together with the empirical constants a1 and tm is the main shortcoming of this model. It should be noted that, in contrast to the model a), where the transport equation (4.156) is parabolic because of the first term on the right-hand side, Eq. (4.159) is hyperbolic. c. Transport equation for turbulent viscosity nt was first proposed in [24] and later specified in more detail in [25]. For a plane boundary layer of incompressible fluid, it has the form qhui nt ðn þ bnt Þ Dnt q qnt g ¼ ant ; þ þ kn Þ ðn e t Dt qY qY s2 qY

ð4:160Þ

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where s is the minimum distance from the wall; a, b, g, and k are empirical constants. One-parametric models of turbulence use the transport equation to determine only one of the quantities characterizing the turbulent flow. As a rule, this quantity is either the energy of turbulence ek (see Eq. (4.156) or the turbulent viscosity nt (see Eq. (4.160). Somewhat less common is the one-parametric model that uses the transport equation (4.159) for the Reynolds stress. A serious disadvantage of one-parametric models is the necessity to specify the scale of turbulence L, which is not known beforehand and cannot be determined without additional hypotheses dependent on the type of the flow. For simple flows, the scale of turbulence can be determined through the governing parameters at a given point inside the flow. Thus, for a flow near a plane wall, L can be taken proportional to the distance between the given point and the wall, whereas for jet flows, L is usually taken proportional to the width of the jet. In complex turbulent flows, it is impossible to express L in terms of the governing parameters of the flow at a given point. Besides, in such flows the scale of turbulence, as well as other governing parameters, usually depends not only on their values at a given point but also on the entire prehistory of the flow (for example, on the conditions at the channel entrance plus the boundary conditions). d. Transport equation for the scale of turbulence L was proposed in [22], where it was used to calculate the shape of the turbulent boundary layer on a flat plate. The equation for F ¼ L2/2 has the form 2 2F qF qF q2 F nt qu 2 2F F F þ CF 1 2 j 2 eE ; þw ¼ ne 2 CL u ek qY ek qX qY qY s s qlnF qlnek ; ðne þ DE Þ w ¼ v þ 0:5ne qY qY ð4:161Þ 3=2

where ek is the energy of turbulence; s – distance from the plate; e ¼ jek =L, CL, CF, and j1 – empirical constants. 2. Two-parametric models: Turbulence models that use two transport equations to determine the characteristics of turbulence are called two-parametric models. The majority of such models involve a transport equations for the energy of turbulence ek anda transport equation for the specific energy dissipation e ¼ ne ðqu0i =qXk Þðqu0i =qXk Þ or for the function n F ¼ em k L . The first two-parametric model was proposed in [19]. This work considers transpffiffiffiffi port equations for ek and the combination ek =L. The Reynolds stress is determined by the relation pffiffiffiffi qhui t0 ¼ hu0 v0 i ¼ C ek L ; qY where C is constant.

ð4:162Þ


Paper [26] was the first to use a transport equation for e in a two-parametric model. As of today, the most popular two-parametric models are ones that describe turbulent flows by two transport equations for the functions ek and e. In many publications the energy of turbulence is denoted by k, hence the commonly used term ‘‘k e models’’. Consider two models: the e F model and the k e model. a. In addition to Eq. (4.156) for turbulent energy ek, the two-parametric e F n model contains the following equation for the function F ¼ em k L : pffiffiffiffi DF q qF ek qhui ¼ DF ðC ek L þ C1 ne Þ 2 þ gF þ Y; L Dt qY qY qY

ð4:163Þ

pffiffiffiffi 1=n ; aF and aF are empirical conwhere DF ¼ aF ek L þ aF n; L ¼ ðem k =FÞ 0 stants; g is a function of t , ek, and qhui/qY; Y is a function that depends on the sign of n: at n < 0, it is zero, while at n > 0, a special form of Y is required. The reason for such behavior of Y is that the first term on the right-hand side of Eq. (4.163) describes a diffusional process characterized by the diffusion coefficient DF. For positive DF, the maximum value of F must decrease with time, which happens only at n < 0. Therefore the case n > 0 requires the presence of such a function Y that DF would not change its sign. Accordingly, all e F models fall into two categories: those with n > 0 and pffiffiffiffi those with n < 0. The models with n > 0 use the function ekL or ek L for F. pffiffiffiffi The models with n < 0, on the other hand, use one of the functions ek =L, 3=2 er/L2, and ek =L. b. k e models use the transport equation for specific energy dissipation e (see Eq. (4.100). In the plane boundary layer approximation, this equation has the form e2 e qhui De q qe ¼ De þ Y: C1 f1 þ C2 f2 ek Dt qY qY ek qY

ð4:164Þ

pffiffiffiffi Here De ¼ ae ek L þ ae ne ; C1, C2, ae, and ae are constants; f1, f2, and Y are functions that depend on the governing parameters. Eq. (4.164) has some peculiarities, one of which has to do with the behavior of e near the wall. Two-parametric models rely on the following relation to determine the Reynolds stress: pffiffiffiffi qhui t0 ¼ hu0 v0 i ¼ Cm fm ðReÞ ek L : qY

ð4:165Þ

The scale of turbulence L that appears in this formula is defined differently in different models. The purpose of the function fm(Re) is to describe the effect of viscosity on t0 . This dependence is not universal but varies with the type of the flow.

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3. Three-parametric models: A distinguishing feature of three-parametric models that sets them apart from other models is that transport equations are written for all characteristics of turbulence that are employed by the model. Instead of introducing turbulent viscosity to find the Reynolds stress, these models rely on the corresponding transport equation whose structure is similar to that of the transport equation for the energy of turbulence. Models of this type are sometimes called Reynolds stress models. Three-parametric models include transport equations for the shear stress P 0 t0 ¼ hu0 v0 i, for the energy of turbulence ek ¼ 0:5 ui 2 , and for the parameter n F ¼ em k L . As far as equations for all three characteristics of turbulence have identical structure, they can be written in the general form DF F qhui q qhui ¼ DF 2 þ g F GF þ DF ; Dt L qY qY qY

ð4:166Þ

n where F successively assumes the values ek, t0 , and F ¼ em k L . The values of DF, g F, GF, and DF are different for each of these equations:

pffiffiffiffi DE ¼ aE ek L þ bE n; Gt ¼ ek ;

GE ¼ t0 ;

g E ¼ 1; t qhui g t ¼ m ng 0F sign ; Et qY

pffiffiffiffi Dt ¼ at ek L þ bt ne ;

where aE, bE, at, bt, gt, and g 0F are constants. A number of current publications present theoretical results for stationary turbulent flows obtained from three-parametric models. The experience with threeparametric models (as well as other, more sophisticated models) indicates that as we increase the number of differential equations (e.g., by using equations for thirdorder moments), we have to deal with ever-increasing number of empirical constants without any gain in accuracy or versatility of the models. Finally, it should be noted that the division of all existing models of turbulence into semi-empirical models and models that employ transport equations for the characteristics of turbulence is really a matter of convention because models of the second type also rely on experiments to garner the functions and constants involved and to determine the range of applicability. In this sense, all models are semiempirical.

4.11 Use of the Characteristic Functional in the Theory of Turbulence

If the flow is laminar, hydrodynamic equations uniquely determine the values of all hydrodynamic characteristics of the flow at any future instant of time, given the initial and boundary values of hydrodynamic fields. In the case of a turbulent flow, initial values of hydrodynamic fields also determine their future values in a unique


j235

way. But, in contrast to the laminar flow, these future values prove extremely sensitive to random uncontrollable perturbations of initial and boundary conditions. Besides, they have such complicated and tangled form that it is quite futile to attempt a rigorous derivation by solving the corresponding differential equations. Only probability distributions of hydrodynamic fields might be of any interest to us, but not the exact values of these fields. This is why hydrodynamic equations for turbulent flows are used to study the corresponding probability distributions and the statistical characteristics of turbulence that follow from these distributions. In the previous sections, we showed how to determine statistical characteristics of turbulence – average values and correlation functions of the velocity field – from hydrodynamic equations and algebraic relations. In this section, we are going to show how to get a complete statistical description of the velocity field from the probability distributions [7]. The fields of hydrodynamic parameters (velocity, pressure, temperature) in a turbulent flow are random fields, each field having its own probability density distribution (see Sections 1.2 and 1.7). For an incompressible fluid, the problem reduces to examining the velocity field. Consider the probability density distribution of velocity components uk(X, t), k ¼ 1, 2, 3, assuming that the fields uk(X, t) are random fields, that is, the values uk(X, t) ¼ uk(M) at any fixed point M ¼ (X, t) in spacetime are random quantities. Hence to each pair value (X, t) there should correspond some probability density distribution p(uk) dependent on the point M. Recall that the probability density distribution p(uk) is defined by the following equality: Pfuk < uk ðX ; tÞ < uk þ duv g ¼ pðuk Þduk :

ð4:167Þ

Such a distribution is called one-dimensional. Now, consider two spacetime points M(1) ¼ (X(1), t(1)), M(2) ¼ (X(2), t(2)) and the values of velocity uk at these points, ð1Þ ð2Þ uk ðM ð1Þ Þ ¼ uk , uk ðMð2Þ Þ ¼ uk . Then the two-dimensional probability density ð1Þ ð2Þ distribution pðuk ; uk Þ can be defined by a relation similar to Eq. (4.167): n o ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ P uk < uk ðM1 Þ < uk þ duk ; uk < uk ðM2 Þ < uk þ duk ð4:168Þ ð1Þ ð2Þ ð1Þ ð2Þ ¼ pðuk ; uk Þduk duk : Finally, for a system of N arbitrarily chosen spacetime points M(1) ¼ (X(1), t(1)), ð jÞ M ¼ (X(2), t(2)), . . ., M(N) ¼ (X(N), t(N)) and the corresponding values uk ðM ð jÞ Þ ¼ uk of the velocity field uk at these points, we can introduce the N-dimensional probað1Þ ð1Þ ðNÞ bility density distribution pðuk ; uk ; . . .; uk Þ according to the equality (2)

n ð1Þ ð1Þ ð1Þ ð2Þ ð2Þ ð2Þ P uk < uk ðM ð1Þ Þ < uk þ duk ; uk < uk ðMð2Þ Þ < uk þ duk ;. . .; o ðNÞ ðNÞ ðNÞ ð1Þ ð2Þ ðNÞ ð1Þ ð2Þ ðNÞ ¼ pðuk ; uk ; . . .; uk Þduk duk . . .uk : uk < uk ðM ðNÞ Þ < uk þ duk ð4:169Þ


236

This N-dimensional probability density distribution satisfies conditions 1–5 of Section 1.5. ð1Þ ð1Þ ðNÞ It was shown in Section 1.8 that instead of pðuk ; uk ; . . .; uk Þ, it is convenient to consider its N-dimensional Fourier transform ð1Þ ð2Þ ðNÞ jðrk ; rk ; . . .; rk Þ

ð¥ ð¥ ¼

ð¥ ...

(

N X ð jÞ ð jÞ exp i rk uk

)

j¼1 ¥ ð1Þ ð1Þ ðNÞ ð1Þ ð2Þ ðNÞ pðuk ; uk ; . . .; uk Þduk duk . . .uk :

¥¥

ð1Þ

ð2Þ

ðNÞ

ð1Þ

ð2Þ

ð4:170Þ

ðNÞ

Introducing vectors rk ¼ ðrk ; rk ; . . .; rk Þ, uk ¼ ðuk ; uk ; . . .; uk Þ and using the averaging rule (1.27), we can rewrite the relation (4.170) in a compact form: ð¥ expðirk uk Þ pðuk Þduk ¼ hexpðirk uk Þi:

jðrÞ ¼

ð4:171Þ

¥

The function j(r) is called the characteristic function. We outlined its properties in Section (1.5). The most important of them are jð0Þ ¼ 1;

ð4:172Þ

which follows from the normalization condition for probability density, and ð1Þ

ð2Þ

ðnÞ

ð1Þ

ð2Þ

ðnÞ

jðrk ; rk ; . . .; rk Þ ¼ jðrk ; rk ; . . .; rk ; 0; 0; . . .; 0Þ;

ð4:173Þ

where n < N and number of zeros equals N n. Taking the inverse Fourier transform

pðuÞ ¼

1 ð2pÞN

ð¥ expðirk uk Þjðrk Þdrk ;

ð4:174Þ

¥

of a given characteristic function, one can find the probability density distribution. The problem of finding the probability density distribution of velocity thus reduces to that of finding the characteristic function. In particular, using the property (4.173) and the relation (4.174), one can find the probability density distribution if the number of dimensions is smaller than N. The shortcoming of the characteristic function is that its range of application is limited a discrete system of points. For a continuous domain of points, it must be replaced with the characteristic functional r(X) (see Section 1.15). For a random function uk(X) of a single variable X (one-dimensional flow) defined on a finite interval (a, b), the characteristic functional is defined as (see Section 1.15)


9+ * 8 ða < = F½rk ðXÞ ¼ exp i rk ðX Þuk ðX Þdx : : ;

ð4:175Þ

b

If the characteristic functional F[rk(X)] is given, then, setting in Eq. (4.175) rk ðX Þ ¼

N X ð jÞ rk dðX X ð jÞ Þ j¼1

and recalling the properties of the delta function (see Section 1.3), we find that 9+ *8 ða < = ðkÞ ðkÞ F½rk ðXÞ ¼ exp i ri dðX X Þui ðX Þdx : ; * ¼ exp i

b N X

ð4:176Þ

+ ð jÞ ð jÞ

rk uk dX

ð1Þ

ð2Þ

ðNÞ

¼ jðrk ; rk ; . . .; rk Þ:

j¼1

So, the knowledge of the characteristic functional enables to find the multidimensional characteristic functional of a random function for any discrete system of points. Putting rk(X) ¼ 0 into Eq. (4.175) and using the property (4.172), one gets the following important property of the characteristic functional: F½rk ðXÞ rk ðX Þ 0 ¼ 1:

ð4:177Þ

Similarly, one can find the characteristic functional of a random function uk(X, t) that depends on coordinates X(X1, X2, X3) and time t: 8 ¥ ¥ ¥ 9+ < ð ð ð ð = exp i rk ðX ; tÞuk ðX ; tÞdX dt ; : ;

* F½rk ðX ; tÞ ¼

ð4:178Þ

¥¥¥

where dX ¼ dX1,dX2dX3. For several statistically correlated functions such as, for example, three components of velocity u(X, t) ¼ {u1(X, t), u2(X, t), u3(X, t)}, we can introduce the characteristic functional F½rðX ; tÞ ¼ F½r1 ðX ; tÞ; r2 ðX ; tÞ; r3 ðX ; tÞ 9+ * 8 ð ð¥ ð¥ ð¥ 3 < = X ¼ exp i rk ðX ; tÞuk ðX ; tÞdX dt : : ; k¼1 ¥¥¥

ð4:179Þ

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238

Let us now consider a system of points (X(1), t(1)), (X(2), t(2)), . . ., (X(N), t(N)) and plug into Eq. (4.179) the expression rk ðX ; tÞ ¼

N X

ð jÞ

rk dðX X ð jÞ Þdðttð jÞ Þ:

j¼1

The resultant values of the characteristic functional will coincide with the characteristic functions of probability density distributions for the values u(k) ¼ u(X(k), t(k)) of the velocity field u(X, t) at a finite set of spacetime points. One can see that all finite-dimensional probability density distributions of a field u(X, t) are uniquely determined by the values of the characteristic functional. A functional of the type (4.178) is called a spacetime characteristic functional. A less comprehensive but more simple statistical description of the random field is provided by the spatial characteristic functional 8 ¥ ¥ ¥ 9+ 3 < ð ð ðX = F½rðX Þ;t ¼ F½r1 ðX Þ;r2 ðX Þ;r3 ðX Þ ¼ exp i rk ðX Þuk ðX ÞdX : : ; k¼1 *

¥¥¥

ð4:180Þ This characteristic functional contains a complete statistical description of the velocity field u(X, t) at a fixed instant of time t but it cannot be used to calculate joint statistical characteristics of the velocity field at different instants of time. For simplicity’’s sake, we shall consider only spatial characteristic functionals. The knowledge of the characteristic functional enables us to find all statistical characteristics of the random field u(X, t). In order to determine them, we have to use variational (functional) derivatives (see Section 1.14). Recall that for a functional ðb ðb F½rðX Þ ¼

uðX1 ; X2 ÞrðX1 ÞrðX2 ÞdX1 dX2 ; aa

the first functional derivative is equal to (see Eq. 1.174) ðb dF½rðX Þ ¼ ðuðX2 ; X1 Þ þ uðX1 ; X2 ÞÞrðX2 ÞdX2 : drðX1 Þ a

It depends on the point X1, which plays the role of a parameter. The second functional derivative of this functional is d 2 F½rðX Þ ¼ uðX1 ; X2 Þ þ uðX2 ; X1 Þ: drðX1 ÞdrðX2 Þ


It depends on two points X1 and X2. For a functional of a more general form, we should define (if it exists) the N-th functional derivative d N F½rðX Þ ; drðX1 ÞdrðX2 Þ. . .drðXN Þ which depends on N points X1, X2, . . ., XN as on parameters. Let us apply the rule of functional differentiation to the characteristic functional (4.175). Since the operations of averaging and differentiation are interchangeable, we have 8b 9+ * Ce when the fluid is turbulent, that is, at t1 < t < t2. This temporal distribution will be repeated again after a while . Measurement data allow us to define the so-called intermittency function PðX ; tÞ ¼

0 1

at C < Ce ; at C > Ce :

ð4:201Þ

Mathematically, intermittency is defined as the average value of the intermittency function hP(X)i for stationary flows. It is equal to the fraction of time during which the flow is turbulent at the point under consideration. The intermittency function is important as the underlying concept behind the method of conditional sampling. The correlation of P(X, t) with some particular variable, for example, velocity ui(X, t), provides the average value of the velocity component ui during the time interval when only turbulent flow exists at the given point. Probability density distribution p(C, X) is characterized by a peak in the vicinity of C ¼ 0, by the effect of measurement error on the peak’s structure, and by the appearance of negative values as a result of these errors. For the purposes of theoretical analysis, this distribution is simulated by the delta function d(C) with the amplitude 1 hP(X)i. So, at any point where intermittency takes place, joint probability density distribution of velocity u and concentration C can be represented as a sum of two components, pðu; C; X Þ ¼ ð1hPðX ÞiÞ pc ðu; X Þ þ hPðX Þi pc ðu; C; X Þ:

ð4:202Þ

The first component corresponds to the external flow, and since the probability to observe the value C ¼ 1 of a random quantity (concentration of impurity) in a nonturbulent fluid is small, one can take pc(u, X) ¼ d(C). The second component corresponds to the mixing layer with the probability density distribution pc(u, C, X). Accordingly, we use the terms ‘‘non-turbulent’’ and ‘‘turbulent’’ for these two components. The expression (4.202) makes it possible to determine the correlation of flow parameters while taking intermittency into account. Thus, the correlation of velocity components is given by

ui u j ¼ ð1hPðX ÞiÞð ui u j Þ0 þ hPðX Þið ui u j Þ1 ;

ð4:203Þ

where (huiuji)0 and (huiuji)1 are the correlations in the external and internal flows.

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248

It must be emphasized that the formula (4.203) is valid in the limiting case of Re ¼ 1. But it is also used for approximate description of flows (with intermittency taken into account) at large but finite values of the Reynolds number. The problem of intermittency is discussed in more detailed in [10].

References 1 Prandtl, L. (1956) Fu¨hrer durch die Stro¨mungslehre, Braunschweig. 2 Batchelor, G.K. (1953) The Theory of Homogeneous Turbulence, Cambridge University Press, Cambridge. 3 Townsend, A.A. (1976) The Structure of Turbulent Shear Flow, Cambridge University Press,Cambridge. 4 Levich, V.G. (1962) Physicochemical Hydrodynamics, Prentice–Hall, Englewood Cliffs, N.J. 5 Abramovitch, G.N. (1960) The Theory of Turbulent Jets, Physmatgis, Moscow (in Russian). 6 Hinze, J.O. (1959) Turbulence, McGraw–Hill, New York. 7 Monin, A.S. and Yaglom, A.M. (1971) Statistical Fluid Mechanics: Mechanics of Turbulence, 1, MIT Press, Cambridge; MA. Monin, A.S. and Yaglom, A.M. (1975) Statistical Fluid Mechanics: Mechanics of Turbulence, 2, MIT Press, Cambridge, MA. 8 Loitsyansky, L.G. (1970) Mechanics of Fluid and Gas, Nauka, Moscow (in Russian). 9 Launder, B.E. and Spalding, D.B. (1972) Mathematical Models of Turbulence, Academic Press, London & New York. 10 Kuznetsov, Y.R. and Sabel’nikov, V.A. (1990) Turbulence and Combustion, Hemisphere, New York. 11 Landau, L.D. and Lifshitz, E.M. (1987) Fluid Mechanics, Pergamon Press, Oxford. 12 Kolmogorov, A.N. (1941) Local Structure of Turbulence in Incompressible Fluids at Very High

13

14

15

16

17

18

19

20

Reynolds Numbers. Dokl. Akad. Nauk SSSR, 30, (4), 299–303 (in Russian). Kolmogorov, A.N. (1941) Energy Dissipation in a Locally Isotropic Turbulence Dokl. Akad. Nauk USSR, 32 (1), 19–21 (in Russian). Kolmogorov, A.N. (1941) About Decay of Isotropic Turbulence in an Incompressible Viscous Fluid. Dokl. Akad. Nauk SSSR, 31 (6), 538–541 (in Russian). Taylor, G.I. (1935) Statistical Theory of Turbulence. I–IV, Proc. Roy. Soc. A, 151, (874), 421–478. Ginevski, A.S., Ioselevitch, V.A., Kolesnikov, A.V., Lapin, J.V., Pilipenko, V.N. and Sekundov, A.N. (1978) Methods of Calculation of Turbulent Boundary Layers Itogi Nauki i Techniki. VINITI. Mechanics of Fluid, and Gas Series. 11, 155–304 (in Russian). Lushik, V.G., Pavelev, A.A. and Jakubenko, A.E. (1988) Transport Equations for Turbulent Characteristics: Models and Results of Calculations Itogi Nauki i Techniki. VINITI. Mechanics of Fluid, and Gas Series, 22, 3–61 (in Russian). Lushik, V.G., Pavelev, A.A. and Jakubenko, A.E. (1994) Turbulent Flows. Models and Numerical Studies. Fluid Dyn. (4), 4–27 (in Russian). Kolmogorov, A.N. (1942) Turbulent Flow Equation for an Incompressible Fluid. Izvestia Acad. Sci. USS, Phys Series, 6, (1–2), 56–58 (in Russian). Rotta, J.C. (1951) Statistische Theorie nichthomogener Turbulenz, Z. Phys.,

References

B. 129, N. 5, S. 547–572, B. 131, N. 1, S. 51–77. 21 Glushko, G.S. (1965) Turbulent Boundary Layer on a Flat Plate in an Incompressible Fluid. Izvestia Acad. Nauk. SSSR, Fluid Dyn. Series, (4), 13–23 (in Russian). 22 Glushko, G.S. (1970) Differential Equation for the Scale of Turbulence and Calculation of the Turbulent Boundary Layer on a Flat Plate, Turbulent Flows: collected articles, pp. 37–44. Moscow, Nauka (in Russian). 23 Bradshaw, P., Ferris, D.H. and Atwell, N.P. (1967) Calculation of Boundary Layer Development using the Turbulent Energy Equation. J. Fluid Mech., 28 (3), 593–616.

24 Kovasznay, L.S.G. (1967) Structure of the Turbulent Boundary Layer. Phys. Fluids, 10 (9), 25–30. 25 Sekundov, A.N. (1971) Use of Differential Equations for Turbulent Viscosity in the Analysis of Plane Flows with no Self-Similarity. Izvestia Acad. Nauk SSSR, Fluid Dyn. Series, (5),114–127 (in Russian). 26 Davidov, B.I. (1961) About Statistical Dynamics of Incompressible Turbulent Fluids. Dokl. Akad. Nauk USSR, 136 (1), 47–50 (in Russian). 27 Hopf, E. (1952) Statistical Hydromechanics and functional Calculus. J. Rat. Mech. Anal., 1 (1), 87–123.

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5 Particle Motion in a Turbulent Flow 5.1 The Eulerian and Lagrangian Approaches to the Description of Fluid Flow and Particle Motion

In hydrodynamics, there are two approaches to the description of motion of a fluid and the particles suspended in the fluid – Eulerian and Lagrangian [1,2]. Eulerian Approach We are aiming to trace what happens at a given spatial point X at different instants of time t or what happens at different spatial points X at a given instant of time t. Coordinates of a point, X ¼ (X1, X2, X3), depend on the choice of coordinate system. We choose a fixed frame of reference (Xi), (i ¼ 1, 2, 3), for example, a coordinate system attached to a motionless boundary of the flow region. For each point X of the flow region V, we can see different points of the continuum passing through X. Motion of the continuum is said to be known when velocity, pressure, temperature, and other parameters are known functions of (Xi) and t. If t is fixed while (Xi) can vary, these functions give a spatial distribution of hydrodynamic parameters at this instant of time. If, on the contrary, (Xi) are fixed while t can change, these dependences give the time evolution of parameters at this spatial point. This formulation of the problem motion of a continuous medium is the essence of the Eulerian approach. Coordinates (Xi), time t, and hydrodynamic parameters given as functions of (Xi) and t are called Eulerian variables. The description of turbulence presented in Chapter 5 was based on the Eulerian approach. In this paradigm, the flow of an incompressible fluid is completely characterized by the velocity field u(X, t) that provides values of the velocity vector for all points X at different instances of time t. Hydrodynamic equations, in which pressure can be eliminated, provide the values of u(X, t) at any instance of time t when the initial velocity u(X, t0) ¼ u0(X) and the appropriate boundary conditions are specified. A serious disadvantage of the Eulerian approach is that it does not allow to trace the motion of small individual particles suspended in the fluid. Since spatial position of such particles varies in time as they move with the carrier fluid, their Eulerian coordinates (Xi) also vary in time.

j 5 Particle Motion in a Turbulent Flow

252

Lagrangian Approach The continuous medium is regarded as a continuous set of elementary volumes that are negligibly small compared to the total volume occupied by the fluid and at the same time large in comparison with the size of molecules. Smallness of elementary volumes makes it possible to consider them as material points (fluid particles) moving relative to a fixed coordinate system {Xi}. If the coordinates of a fluid particle (relative to a chosen coordinate system) change with time according to

Xi ¼ fi ðtÞ;

ði ¼ 1; 2; 3Þ;

ð5:1Þ

we say that the point is moving relative to a fixed coordinate system {Xi} and that Eq. (5.1) provides the law of its motion. However, it is insufficient to give the equation of motion relative to the coordinate system {Xi} for each fluid particle in the continuum in the form (5.1). One also has to individualize fluid particles that are identical from the geometrical viewpoint. The way to make the particles physically distinguishable is to notice that they have different spatial positions, that is, different coordinates, at some initial instant t0. Let the position of a point at the instant t0 be known and specified by the coordinates xj. Then the law of motion (5.1) will depend not only on t but also on xj: Xi ¼ fi ðt; x1 ; x2 ; x3 Þ;

ði ¼ 1; 2; 3Þ;

ð5:2Þ

If all xj are fixed and t varies, then (5.2) becomes the equation of motion for a given point of the continuum that had initial coordinates Xi ¼ xi. If, on the other hand, t is fixed and xj varies, then Eq. (5.2) gives the spatial distribution of various points of the continuum at a given instant of time t. Finally, when both xj and t vary, the relation (5.2) gives the law of motion for the whole volume of the continuum. Variables xj individualizing specific points of the continuum, grouped together with the time variable t, are called Lagrangian variables, and Eq. (5.2) is called the equation of motion of the continuum. If functions (5.2) and their partial derivatives are continuous and one-to-one functions at each instant of time, then D ¼ jqXi =qx j j „ 0 and Eq. (5.2) allows us to express xj through Xi: x j ¼ x j ðXi ; tÞ;

ð5:3Þ

functions xj(Xi, t) being continuous. Jakobian D has the meaning of the ratio of volumes of fluid elements as a result of transformation (5.2). For incompressible fluids, D ¼ 1. The set of values Xi specifies the spatial domain V occupied by the continuum at the moment t. Since xj 2 V0 are coordinates of some point M of the domain V0 at the initial time t ¼ t0, while Xi 2 V are coordinates of the same point, which now belongs

5.1 The Eulerian and Lagrangian Approaches to the Description of Fluid Flow and Particle Motion

to a domain V at the time t > t0, the relation (5.3) can be thought of as a one-to-one and continuous mapping of domains V and V0. It is a well-known topological property of this class of transformations that a volume is mapped onto a volume, a surface – onto a surface, a line – onto a line, and a closed line – onto a closed line. In addition to the fixed frame of reference {Xi}, one can introduce a moving frame {xi} attached to the fluid particle (an elementary volume around the point xi). In the mechanics of continuum, this frame is considered as a frame that is ‘‘frozen’’ into the elementary volume. As time goes on, it moves and continuously deforms together with the volume. Therefore Lagrangian coordinates can be regarded as an alternative set of coordinates for the same spatial points, and the coordinate system {xi} – as a moving, deformable, curvilinear coordinate system. The points that form an elementary volume of the continuum move relative to the fixed coordinate system {Xi}, but they are at rest relative to coordinate system {xi}. Use of coordinates xi and t as independent variables is the essence of the Lagrangian approach to the study of continuum motion. Comparing the two approaches, one comes to the following conclusion. Eulerian approach implies that one is interested in the change of parameters such as velocity, temperature, concentration, and so forth, at a given point visited by a continuous succession of different points of the medium. Lagrangian approach means that one is interested in the same parameters at a given individual point that is moving with the continuum. From the mathematical viewpoint, the two approaches differ in that in the former, the variables are the coordinates of spatial points Xi, while in the latter – the parameters xi (initial coordinates of fluid particles) individualizing different points of the continuum (plus the time variable t that is present in both approaches). Consider now how the parameters can be determined using either Eulerian or Lagrangian approach. Take a point of the continuum, whose radius vector relative to a motionless coordinate system at the time t is X. It depends on xi and t. Lagrangian approach requires the values of xi to be fixed, so the velocity of this point is equal to V ¼

qX ðj; tÞ qt

ð5:4Þ xj

or, in the component form, Vi ðx; tÞ ¼

qXi ðj j ; tÞ qt

; xj

where Vi(x, t) ¼ Vi(x1, x2, x3, t). The other parameters change with time in the same manner. The derivative (5.4) characterizes the change of parameters at a given point in the continuum; it is called substantial, material, or total derivative and is designated as D/Dt or d/dt. In the Eulerian approach, the velocity u and other parameters depend on X1, X2, X3 and t and are determined from hydrodynamic equations. In the Lagrangian approach these parameters can also be determined from hydrodynamic equations, if we write these equations in Lagrangian, rather than Eulerian, variables.

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254

But this transformation leads to cumbersome expressions, and solution of the equations thus obtained involves considerable mathematical difficulties. For this reason, one usually makes use of hydrodynamic equations in Eulerian variables. If the distribution of a parameter, for example, of temperature W, is given both as W ¼ W(x, t) (the Lagrangian approach) and W ¼ W(X, t) (the Eulerian approach), the substantial derivative follows from Eq. (5.2):

qJ DJ qJ qJ ¼ þVi : ¼ qt xi Dt qt Xi qXi

ð5:5Þ

Let us see how one can make a transition from Eulerian to Lagrangian variables. Let the velocity distribution ui ¼ ui(X, t) in Eulerian variables be known, for instance, from the solution of the Navier–Stokes equations. Then from Eq. (5.4) there follows a set of ordinary differential equations qXi ¼ ui ðX ; tÞ; qt

ði ¼ 1; 2; 3Þ

ð5:6Þ

and initial conditions X i ðt0 Þ ¼ xi :

ð5:7Þ

These equations describe trajectories Xi ¼ Xi(x1, x2, x3, t) of continuum points whose initial coordinates are equal to xi. Parameters xi individualize these points and can be regarded as Lagrangian variables. The relations Xi ¼ Xi(x1, x2, x3, t) allow a transition from Eulerian variables Xi to Lagrangian variables xi in all dependencies involving hydrodynamic parameters. Hereafter, Eulerian velocity and coordinates will be denoted through u(X, t) and X ¼ (X1, X2, X3), and Lagrangian velocity and coordinates – through V ¼ V(j, t) and j ¼ (j1, j2, j3). The relations (5.2) and (5.3) can then be represented in the form X ¼ X ðj; tÞ;

j ¼ jðX ; tÞ;

ð5:8Þ

with the initial condition j ¼ X ðj; t0 Þ:

ð5:9Þ

Eulerian u(X,t) and Lagrangian V(j, t) velocities are related to each other by Vðj; tÞ ¼ uðX ðj; tÞ; tÞ:

ð5:10Þ

So, the expression (5.4) can be interpreted as the relation between Eulerian and Lagrangian coordinates: qX ðj; tÞ ¼ uðX ðj; tÞ; tÞ: qt

ð5:11Þ

5.1 The Eulerian and Lagrangian Approaches to the Description of Fluid Flow and Particle Motion

Another way to establish connection between Eulerian and Lagrangian coordinates is based on writing the conservation condition for some continuum characteristic C, say, fluid particle mass. In the Lagrangian approach, it depends only on Lagrangian coordinates individualizing the point. Therefore C ¼ C(j). In Eulerian variables, on the other hand, its value would vary in time at a fixed spatial point. Using both ways of writing one and the same characteristic that is conserved during the motion of the continuum, we obtain YðjÞ ¼ yðX ðj; tÞ; tÞ:

ð5:12Þ

Eq. (5.12) provides an alternative relation between Eulerian and Lagrangian coordinates. The condition of conservation of C means that qYðjÞ DyðX ðj; tÞ; tÞ qy qy qy qðui yÞ ¼ ¼ 0: ¼ ¼ þ ui þ qt Dt qt qXi qt qXi In deriving this equation, we have used the relation (5.7) for the substantial derivative as well as the continuity equation @ui/@Xi ¼ 0. Hence the Eulerian field of any conserved characteristic of the continuum satisfies the following transport equation: qy qðui yÞ ¼ 0: þ qt qXi

ð5:13Þ

One example of conserved quantity is the unit mass concentrated initially at the point j0. Then C is given by C ¼ d(j j0), or, in Eulerian variables, Y ¼ yðX ; tÞ ¼ dðX X ðj0 ; tÞÞ:

ð5:14Þ

This function obeys Eq. (5.13) with the initial condition c(X,0) ¼ d(X j0) (see Section 1.12). When considering turbulent flow in the previous chapter, we assumed the Eulerian velocity field u(X, t) to be a random function at a fixed spatial point X. Then, owing to the relation (5.12), the Lagrangian velocity V(j, t) should be equal to the value of the random function u(X, t), but, in contrast to the Eulerian velocity field, the point X is a random point X(j, t) that corresponds to the random position of a continuum point at the time t, given the initial position X ¼ j of this point. It should be noted that all we said thus far refers not only to fluid particles of the carrier fluid, but also to particles of a foreign medium suspended in the fluid, provided that the size of particles and their volume concentration are sufficiently small so that their inertia and the hindered character of motion, that is, the effect of particle interactions, can be neglected, and that they move with the velocity of the carrier fluid.

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5.2 Lagrangian Statistical Characteristics of Turbulence

In a turbulent flow, both Eulerian and Lagrangian velocity fields are random fields. This means that coordinates of a fluid particle will also be random fields, since from Eq. (5.4) there follows ðt X ðj; tÞ ¼ j þ V ðj; tÞdt:

ð5:15Þ

t0

If we consider a finite set of n fluid particles of the continuum initially located at j1, j2, . . ., jn, then there should exist a multidimensional joint probability density distribution of coordinates X and velocities V of these particles at different instants of time t1, t2, . . ., tm. These distributions are functions of 3n þ m variables, which include the coordinates of Lagrangian variables j1, j2, . . ., jn and the times t1, t2, . . ., tm: pðX ; V jj1 ; j2 ; . . .; jn ; t1 ; t2 ; . . .; tm Þ:

ð5:16Þ

Multidimensional probability density distributions (5.16) are the basic Lagrangian statistical characteristics of turbulence. It should be noted that statistical characteristics of the type (5.16) depend only on Lagrangian variables. In addition to distributions (5.16), there can also exist probability density distributions for a set of quantities, some of which are Lagrangian and the others are Eulerian, for example, pðX ; V ; ujj1 ; j2 ; . . .; jn ; X 1 ; X 2 ; . . .; X n ; t1 ; t2 ; . . .; tm Þ: Some general relations exist between different Lagrangian statistical characteristics of turbulence. We shall list the most important of them. The first relation is a consequence of the transport equation (5.13). Since expression (5.14) is a solution of the transport equation, substituting it into Eq. (5.13) and noting that, according to Eq. (5.11), ui ðX ; tÞdðX X ðj; tÞÞ ¼ ui ðX ðj; tÞ; tÞdðX X ðj; tÞÞ ¼ Vi ðj; tÞdðX X ðj; tÞÞ; we obtain qdðX X ðj; tÞÞ qVi ðj; tÞdðX X ðj; tÞÞ ¼ 0: þ qt qXi

ð5:17Þ

Recalling the definition of averaging (see (1.26)) and in view of Eq. (5.16), we can write hdðX X ðj; tÞÞi ¼ pðX jj; tÞ; ð hVi ðj; tÞdðX X ðj; tÞÞi ¼ Vi ðj; tÞ pðX ; V jj; tÞdV :


The operation of averaging of Eq. (5.17) results in a statistical analog of the transport equation (5.13): ð q pðX jj; tÞ q þ Vi ðj; tÞ pðX ; V jj; tÞdV ¼ 0: qt qXi

ð5:18Þ

As was noted earlier, Lagrangian velocity V(j, t) is the value of the random function u(X, t) at a random point X(j, t). Introduce now the random function V(X1, j, t), which has the meaning of velocity of those fluid particles that were located at X ¼ j at the initial moment t ¼ t0 and were then found at the fixed point X1 ¼ X(j,t) at the later moment t > t0. Such a function corresponds to the probability density distribution p(V|X1, j, t) of the quantity V(j, t) under the condition X(j, t) ¼ X1. In the general case, the probability density of the conditional distribution for n fluid particles characterized by Lagrangian velocities V(j1, t1), V(j2,t2), . . ., V(jn,tn) at different instants of time can be written as pðV 1 ; V 2 ; . . .; V n jj1 ; j2 ; . . .; jn ; t1 ; t2 ; . . .; tn Þ ðð ð ¼ . . . pðV 1 ; V 2 ; . . .; V n jX 1 ; X 2 ; . . .; X n ; t1 ; t2 ; . . .; tn Þ pðX 1 ; X 2 ; . . .; X n jj1 ; j2 ; . . .; jn ; t1 ; t2 ; . . .; tn ÞdX 1 dX 2 . . .dX n :

ð5:19Þ

This formula follows from the theorem of total probability, which states that the probability for n fluid particles located initially at the points j1, j2, . . ., jn to have velocities V(j1, t1), V(j2, t2), . . ., V(jn, tn) at the instants of time t1, t2, . . ., tm is equal to the product of the conditional probability density of these velocities (under the condition that coordinates of these particles at the corresponding instants of time assume the values X(j1, t1) ¼ X1, X(j2, t2) ¼ X2, . . ., X(jn, tn) ¼ Xn) and the joint probability density distribution of Lagrangian random quantities X(j1, t1), X(j2, t2), . . ., X(jn, tn), integrated over all possible values of X1, X2, . . ., Xn. In the special case j1 ¼ j2 ¼ . . . ¼ jn ¼ j this formula contains probability the density distribution of coordinates and velocities of one and the same particle at different instants of time: pðV 1 ; V 2 ; . . .; V n jj; t1 ; t2 ; . . .; tn Þ ðð ð ¼ . . . pðV 1 ; V 2 ; . . .; V n jX 1 ; X 2 ; . . .; X n ; t1 ; t2 ; . . .; tn Þ pðX 1 ; X 2 ; . . .; X n jj; t1 ; t2 ; . . .; tn ÞdX 1 dX 2 . . .dX n :

ð5:20Þ

In the case of n ¼ 1 (one fluid particle), we have ð pðV jj; tÞ ¼

pðV jX ; tÞ pðX jj; tÞdX :

ð5:21Þ

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It follows from the last formula that the mean value of Lagrangian velocity hV(j, t)i may be represented as ð hV ðj; tÞi ¼ hV ðX ; j; tÞi pðX jj; tÞdX :

ð5:22Þ

After a certain amount of time, the fluid particle forgets the past. This time has the meaning of Lagrangian correlation time T (L), T ðLÞ ¼

ð¥ l0

ð dt YðLÞ ðtÞdt; hV 2 ðj; tÞihV 2 ðj; t0 Þi1=2 hVi ðj; tÞVi ðj; t0 Þi

ð5:23Þ

where C(L)(t) is the Lagrangian autocorrelation function. Hence, it should be anticipated that at the time t > T (L), the probability density distribution of Lagrangian velocity V(X, j, t) will, for all practical purposes, be independent of x, so at these values of t the random quantity V(X, j, t) can be taken equal to the Eulerian velocity u(X, t). Then the probability density distribution p(V|X, j, t) will be identically equal to the probability density distribution of V ¼ u(X,t), that is, of the Eulerian velocity field at a fixed spacetime point (X, t). The formula (5.22) will then take the form ð hV ðj; tÞi ¼ huðX ; tÞi pðX jj; tÞdX :

ð5:24Þ

In the case of n ¼ 2, the relation (5.20) at t ¼ t0 and t2 ¼ t gives the joint probability density distribution of the quantities V1 ¼ V(j,t0) and V2 ¼ V(j,t) at two successive instants of time: ð pðV 1 ; V 2 jj; t0 ; tÞ ¼

pðV 1 ; V 2 jX ; j; t0 ; tÞ pðX jj; tÞdX :

ð5:25Þ

It should be kept in mind that V1 ¼ V(j,t0) ¼ u(j,t0). From the formula (5.25) one can derive the expression for the Lagrangian correlation function hViVji – second-order moment for the components of the vector V: ð hVi ðj; t0 ÞV j ðj; tÞi ¼ hVi ðj; t0 ÞV j ðX ; j; tÞi pðX jj; tÞdX :

ð5:26Þ

In the more general case, the Lagrangian correlation function for the components of the vector V at different instants of time could be derived by using the formulas (5.20) and (5.25): ð hVi ðj; t1 ÞV j ðj; t2 Þi ¼ hVi ðX 1 ; j; t1 ÞV j ðX 1 ; X 2 ; j; t1 ; t2 Þi pðX 1 ; X 2 jj; t1 ; t2 ÞdX 1 dX 2 :

ð5:27Þ


where V(X1, X2, j, t1, t2) is the random Lagrangian velocity of a fluid particle initially located at the point X ¼ j and found at fixed points X1 and X2 at the instants of time t1 and t2. For sufficiently large values of t1, one can take V(X1, j, t1) ¼ u(X1, t1), whereas for t2 t1, it is safe to assume V(X1, X2, j, t1, t2) ¼ u(X2, t2). The motion of a continuum point (fluid particle) being initially at the space point j has been hitherto described by the vector X(j, t), which gives the random position of this particle at the moment t. This enabled us to derive those statistical characteristics of particle motion that are given by the relations (5.17)–(5.27). Consider now the other statistical characteristics of particle motion. A distinctive feature of the behavior of particles suspended in a turbulent flow is that the distance between particles changes in time. Section 4.9 has shown that two particles that were initially close together tend to separate as time goes on. The time it takes for the interparticle distance to reach a given value is estimated by Eq. (4.120). Therefore it makes sense to consider the displacement of particles from their initial position. First, let us consider the displacement of a single fluid particle. Instead of the vector X(j, t), we shall introduce the vector Y(t) of particle displacement from the initial position over the time interval t: l0ð þt

Y ðtÞ ¼ X ðj; t0 þ tÞx ¼

V ðj; tÞdt:

ð5:28Þ

l0

In order to determine statistical characteristics of the random displacement vector Y(t), it is necessary to specify the probability density distribution p(Y|t, j, t0). Let us start with the case when the time interval t is small compared to the Lagrangian correlation time T(L), that is, t T(L). In such a short time, the Lagrangian velocity will be practically unchanged, so Eq. (5.28) takes the form Y ðtÞ tV ðj; t0 Þ ¼ tuðj; t0 Þ:

ð5:29Þ

Consider now the probability density distribution p(u|j, t0) of the Eulerian velocity (u|j, t0) at a fixed point j at the instant of time t ¼ t0. Recalling that the vectors Y, u are three-dimensional and using the normalization condition for the probability density together with Eq. (5.29), we obtain pðY jt; j; t0 Þ ¼ t3 pðujj; t0 Þ ¼ t3 p

Y jj; t0 : t

ð5:30Þ

A large body of experiments shows that in a steady turbulent flow, the distribution p(u|j, t0) and thus the distribution p(Y|t, j, t0) is close to normal (Gaussian) at small values of t, that is, for t T (L), and at relatively large values of t, that is, for t T (L). The close resemblance of both particle displacement distributions to a normal one at t T (L) and t T (L) makes it reasonable to hypothesize that for the intermediate values of t, the particle displacement distribution will be normal as well. The assumption implies the absence of any radical rearrangement of the distribution

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(from the normal one at small t and again to a normal distribution at large t) in this intermediate region. The main advantage of this assumption is that it considerably simplifies the derivation of statistical characteristics of particle motion in a turbulent flow. Let us now consider the statistical characteristics of particle displacement. Because of the assumption we made, it will not be very different from Gaussian, so it is sufficient to determine only two first moments (see Section 1.8). In view of Eq. (5.29), the mean value of the vector Y(t) is l0ð þt

hY ðtÞi ¼

hV ðj; tÞidt:

ð5:31Þ

l0

For small values of t, we obtain from Eq. (5.31) hY ðtÞi Ut;

ð5:32Þ

where U is the average flow velocity. Let us introduce the fluctuations of displacement Y 0 and particle velocity V 0: Y 0 ðtÞ ¼ Y ðtÞhY ðtÞi;

V 0 ðj; tÞ ¼ V ðj; tÞhV ðj; tÞi:

Due to Eq. (5.31), 0

l0ð þt

hY ðtÞi ¼

V ðj; tÞhV 0 ðj; tÞidt:

ð5:33Þ

l0

The second central moments for components of the vector Y at one and the same instant of time t are components of the fluctuation correlation tensor, which is called the dispersion tensor of fluid particle displacements. Hitherto it has been denoted as bij (see Section 1.6). Since it is associated with the diffusion tensor in a turbulent flow, let us denote it further on as dij: di j ðtÞ ¼ hYi0 ðtÞY 0j ðtÞi ¼

l0ð þtl0ð þt

l0

hVi0 ðj; t1 ÞV 0j ðj; t2 Þidt1 dt2 :

ð5:34Þ

l0

The assumption that the displacement of a fluid particle is distributed in accordance with the Gaussian law means that first two moments characterize this distribution completely. At small values of t, we find from Eq. (5.34) while taking into account Eq. (5.29) ð0Þ

di j ðtÞ hu0i ðj; t0 Þu0j ðj; t0 Þit2 ¼ bi j t2 :

ð5:35Þ

5.2 Lagrangian Statistical Characteristics of Turbulence ð0Þ

Coefficients bi j ¼ hu0i ðj; t0 Þu0j ðj; t0 Þi are the correlation functions of fluctuations. In a steady turbulent flow, they depend only on and under the additional condition of homogeneity of Eulerian velocity they are constant. Consider the case of stationary homogeneous turbulence. Then all hydrodynamic fields are homogeneous random fields and stationary random functions. The mean velocity hui is constant in space and in time (see Section 1.9). In accordance with Eq. (5.10) and Eq. (5.32), we have hV ðj; tÞi ¼ huðX ðj; tÞÞi ¼ hui;

hY ðtÞi ¼ huit:

ð5:36Þ

The fluctuation velocity V 0 (X, t)of a fluid particle will possess the same statistical characteristics for all X, in other words, it will be a stationary random function. The following equation holds: ðLÞ

ðLÞ

hVi0 ðj; t1 ÞV 0j ðj; t2 Þi ¼ bi j ðt2 t1 Þ ¼ ðhu0i 2ihu0j 2iÞ1=2 Yi j ðt2 t1 Þ; ðLÞ

bi j

ðLÞ

where Yi j ¼

ðhu0i 2iÞðhu0j 2iÞ1=2

¼

hu0i u0j i ðhu0i 2iÞðhu0j 2iÞ1=2

ð5:37Þ

are correlation coefficients.

Here we have used the superscript symbol (L) to indicate that the corresponding function is a Lagrangian correlation function. Changing the variables in Eq. (5.37) according to s ¼ (t2 t1), t ¼ (t1 þ t2)/2 and substituting the result into Eq. (5.34), one gets s

ðt t0 þt ð 2 di j ðtÞ ¼

ðLÞ

ðLÞ

ðbi j ðsÞ þ b ji ðsÞÞdtds 0 t0 þ s 2 s

¼ ðhu0i 2ihu0j 2iÞ1=2

ðt t0 þt ð 2

ðLÞ

ð5:38Þ

ðLÞ

ðYi j ðsÞ þ Y ji ðsÞÞdtds 0 t0 þ s 2

¼

ðhu0i 2ihu0j 2iÞ1=2

ðt

ðLÞ

ðLÞ

ðtsÞ½Yi j ðsÞ þ Y ji ðsÞ ds: 0

The special case i ¼ j yields ðt ðt ðLÞ ðLÞ 0 dii ðtÞ ¼ 2 ðtsÞbii ðsÞds ¼ 2hui 2i ðtsÞYii ðsÞds: 0

ð5:39Þ

0 ðLÞ

ðLÞ

Lagrangian correlation coefficients Yii ðsÞ have the property that Yii ðsÞ ! 0 at s ! ¥, in agreement with our intuitive understanding that the correlation of LaðLÞ grangian velocities of a fluid particle at two different moments t2 t1 > > Ti should

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262

not be noticeable. With this in mind, we introduce the characteristic time ðLÞ Ti

ðt

ðLÞ

¼ Yii ðsÞds;

ð5:40Þ

¥ ðLÞ

which is simply the time that must pass before we get Yii ðsÞ 0. We can take as our Lagrangian correlation time T(L) either the maximum or the average time value from ðLÞ the set Ti . Note that this time is of the same order of magnitude as the Lagrangian correlation time introduced earlier by the formula (5.23). ðLÞ In many cases, especially for large Reynolds numbers, the function Yii can be approximated by an exponential dependence: ðLÞ

ðLÞ

Yii expðt=Ti Þ:

ð5:41Þ

Although this is not exactly the right function to describe the shape of the correlation curve, especially at t ! 0 and t ! 1, its use leads to satisfactory outcomes in many practical problems. ðLÞ For t > > Ti , the expression (5.39) can be replaced by the asymptotic relation dii ðtÞ ¼

2hu0i 2i

ð¥

ðLÞ

ðLÞ

ðtsÞYii ðsÞds ¼ 2hu0i 2iðTi tsi Þ;

ð5:42Þ

0

Ð ¥ ðLÞ where Si ¼ 0 sYii ðsÞds (assuming, of course, that this integral exists). ðLÞ Since t > > Ti , we have ðLÞ

dii ðtÞ 2hu0i 2iTi t:

ð5:43Þ

If i 6¼ j, we can still get a similar expression by introducing (instead of the characteristic time (5.40)) ð¥ ðLÞ ðLÞ ðLÞ Ti j ¼ ½Yi j ðsÞ þ Y ji ðsÞ ds:

ð5:44Þ

0

For large values of t, the dispersion of the fluid particle displacement equals ðLÞ

di j ðtÞ ðhu0i 2ihu0j 2iÞ1=2 Ti j t

ð5:45Þ

whereas for small values of t, in accordance with Eq. (5.35), it is equal to ð0Þ

di j ðtÞ ðhu0i 2ihu0j 2iÞ1=2 bi j t2 :

ð5:46Þ


For the intermediate values of t, the dependence dij(t) is more complex because it ðLÞ also depends on the form of Yi j . Hence, the dispersion of the fluid particle’s displacement in a turbulent flow is proportional to , given a sufficiently long time t T(L), whereas for short t T(L), the dispersion is proportional to t2. We must mention the analogy between the obtained result and the laws (3.30) and (3.32) that describe Brownian diffusion. As far as particle displacements are concerned, the difference between the Brownian and turbulent diffusion consists in the form of the proportionally coefficients, as well as in different characteristic times. The above-considered model of homogeneous, stationary turbulence is an idealization of real turbulence, therefore any application of the obtained results to spatial flows should be questioned. For example, the formula (5.42) can be used only if the homogeneity condition in the direction of the Xi-axis holds. One particular case when this condition is satisfied is that of a steady turbulent flow in a long pipe whose symmetry axis coincides with the Xi-axis of the coordinate system. Let Yi(t) denote the component of the fluid particle’s displacement vector Y along the Xi-axis. Then the corresponding Lagrangian velocity equals Vi ðj; t0 þ tÞ ¼

dYi ðtÞ : dt

ð5:47Þ

the mean-flow-rate velocity of the fluid in the direction of the Denote through U Xi-axis and assume that after a sufficiently long time , the initial position of the fluid particle will not have any noticeable effect on its motion and, in particular, on its Lagrangian velocity. Then Vi ðj; t0 þ tÞ U

ð5:48Þ

and one can see from Eq. (5.47) and Eq. (5.48) that at large values of t, the following equality for the average displacement of the fluid particle along the Xi-axis is valid: hVi ðtÞi Ut:

ð5:49Þ

The relation (5.48) yields the root-mean-square displacement of the fluid particle along the Xi-axis: ðLÞ

h½Yi ðtÞhYi ðtÞi 2 i 2hu0i 2iTi t: ðLÞ

The values p offfiffiffiffiffiffiffiffiffiffiffi hu0i 2i and Ti velocity u ¼ t0 =re, so that ðLÞ

hu0i 2iTi

ð5:50Þ

depend on the pipe radius R and on the dynamic

¼ cRu

where c is a universal constant, t0 – the frictional stress at the wall, re – the fluid density.

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Another example is the problem of particle motion in an unbounded turbulent shear flow with constant gradient of the average velocity ˙g, where the Eulerian field of velocity fluctuations u0 (X, t) is stationary and statistically homogeneous and the average velocity hu(X)i does not change in time and has a linear dependence on spatial coordinates. Let the average velocity be directed along the Xi-axis, so that ; hu1 i ¼ gX 3

hu2 i ¼ hu3 i ¼ 0:

Consider the motion of a fluid particle that was initially (t ¼ 0) located at the point x ¼ 0. At the moment t the particle is located at a point with coordinates Xi(t) and has Lagrangian velocity V(t). Then, in accordance with the formula (5.10), we have V ðtÞ ¼ hV ðtÞi þ V 0 ðtÞ ¼ huðX ðtÞÞi þ u0 ðX ðtÞ; tÞ or, in the component form, ðtÞ þ V 0 ðtÞ; V1 ðtÞ ¼ gX 3 1

V2 ðtÞ ¼ V20 ðtÞ;

V3 ðtÞ ¼ V30 ðtÞ:

Since V ¼ dX/dt, the last equations lead us to write ðt ðtÞ þ V 0 ðtÞ dt; X1 ðtÞ ¼ ½gX 3 1 0

ðt X2 ðtÞ ¼ V20 ðtÞdt;

ðt X3 ðtÞ ¼ V30 ðtÞdt:

0

0

Since u0 hX,ti ¼ 0, we have hV0 (t)i ¼ 0 and hX(t)i ¼ 0. The field u0 (X,t) is homogeneous and stationary, so V0 (X,t) is also a homogeneous and stationary function, and its correlation tensor has the form Vi0 ðt1 ÞV 0j ðt2 Þ ¼ bi j ðt1 t2 Þ: Let us now find the dispersion tensor dij(t) ¼ hXi(t)Xj(t)i. It is apparent that motion along the Xi-axis with the average velocity u(X) will not affect the particle’s displacement in the directions of X2- and X3-axes. Therefore the values d22(t), d33(t), and d23(t), which depend on Lagrangian velocities V2(t) and V3(t), have the form (5.38), just as in the case of u ¼ const. However, the components d11(t) and d13(t) will have a different form, because they depend on the form of the distribution hu(X)i, namely, d11 ðtÞ ¼

hX12 ðtÞi

ðt ðt h 2 hX ðt ÞX ðt Þi þ ghX ðt ÞV 0 ðt Þi ¼ ðtÞ 3 1 3 2 2 ˙ 3 1 00

i ðLÞ 0 þghX 3 ðt2 ÞV1 ðt1 Þi þ b11 ðt1 t2 Þ dt1 dt2 ; ðt ðt d13 ðtÞ ¼ hX1 ðtÞX3 ðtÞi ¼ 00

ghX3 ðt1 ÞV30 ðt2 Þi þ hV10 ðt1 ÞV30 ðt2 Þi dt1 dt2 :


Substituting the expressions for Xi(t) into these relations, using the formulas (5.38) for dii(t), we obtain after repeated integration by parts d11 ðtÞ ¼

ðt 2 ðt ðgÞ ðLÞ ðLÞ ð2t3 3t2 s þ s2 Þb33 ðsÞds þ g_ ðtsÞ2 b31 ðsÞds 2 0

0

ðt

ðt ðLÞ ðLÞ þg ðt2 s2 Þb13 ðsÞds þ 2 ðtsÞb11 ðsÞds; 0

0

ðt

ðt

0

0

h i ðtsÞbðLÞ ðsÞds þ ðtsÞ bðLÞ ðsÞ þ bðLÞ ðsÞ ds: d13 ðtÞ ¼ gt 33 13 31 We are primarily interested in the asymptotic expressions for components of the dispersion tensor at large values of time, t T(L), where, just as before, we take ðLÞ ðLÞ ðLÞ T ðLÞ ¼ maxðT1 ; T2 ; T3 Þ. As follows from Eq. (5.45), the components d22(), d33(), and d23() are proportional to , whereas d11() and d13() are given by 2 2 0 ðLÞ d11 ðtÞ ðgÞ hu3 2iT3 t3 ; 3

ðLÞ d13 ðtÞ g 2 hu03 2iT3 t2 :

ð5:51Þ

We conclude from these asymptotic expressions that the shear flow gives rise to anisotropy of particle displacement. The dispersion of particle displacement in the direction of the X1-axis grows with time much faster (t3) than in the transverse directions (t along the X3- and X1-axes), and the correlation of displacements along the X1- and X3-axes is different from zero. There exists for the limiting case of t T(L) the following asymptotic expression for the correlation coefficient: Y13 ¼

d13 ðtÞ d11 ðtÞd33 ðtÞ1=2

pffiffiffi 3 : 2

It should be noted that similar dependences exist for Brownian motion of particles as well (see Section 3.7). The above-described analysis of random motion of an elementary volume (fluid particle) can be applied to the study of behavior of a macrovolume (say, one mole) when the latter is visualized as a set of elementary volumes. Such a macrovolume gets deformed during its turbulent motion, because the distance between any two particles grows with time. Therefore, in order to find the extent of macrovolume deformation, one has to consider the relative motion of two fluid particles. If the two particles are initially spaced far apart, that is, the interparticle distance exceeds the integral scale of turbulence, the interaction between the particles can be neglected, and the statistical characteristics of each particle’s motion can be determined in the same manner as we did earlier for one fluid particle. If the initial distance between the fluid particles is small in comparison with the integral scale of turbulence, their motion will be mutually interrelated until they get farther apart.

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Therefore it should be anticipated that as time goes on, statistical characteristics of motion of two fluid particles that were initially close together will get closer to those of isolated particles. We see that the most interesting situation arises when the particles are initially close enough to each other, at the distance, say, of the order of the Kolmogorov scale of turbulence. The relative motion of particles under consideration occurs under the action of fluctuations, which have various scales. Smallness of particle sizes and smallness of the initial interparticle distance means that large-scale fluctuations will transport the fluid volume containing both particles as a whole without changing the interparticle distance. Small-scale fluctuations whose scale is of the order of the interparticle distance will change that distance. As the particles get farther apart with time, the larger-scale fluctuations will be able to affect the relative motion. It was shown in Section 4.9 that small-scale motion is isotropic. Therefore, at the initial stage the relative motion of two fluid particles takes place in an isotropic turbulent field, although the distance between the particles and the difference of their velocities are both functions of time. Consider two fluid particles a and b at the moment t, when they are located at their respective spatial points Xa and Xb. The radius vector connecting the two points is r ¼ Xa Xb. The relative motion of particles is characterized by the tensor of relative dispersion, whose components are ðrÞ

di j ¼ hri r j i ¼ hðXi X j Þb i þ hðXi X j Þa ihXbi Xa j ihXb j Xai i: Turbulence is assumed to be homogeneous, so hðXi X j Þb i ¼ hðXi X j Þa i ¼ hXi X j i and ðrÞ

di j ¼ 2hXi X j ihXbi Xa j ihXb j Xai i:

ð5:52Þ

Introduce a new coordinate system X ¼ X Xa, whose origin coincides with particle a. Then, in view of Eq. (5.28), we find that hXiXji ¼ hYiYji. At large values of t (t T(L)) the last two terms in Eq. (5.52) are small, and

1=2 ðrÞ ðLÞ Ti j t; di j 2hXi X j i ¼ 2hYi Y j i ¼ 2hYi0 Y 0j i ¼ 2di j ¼ hu0i 2ihu0j 2i

ð5:53Þ

where dij is the dispersion tensor given by the formula (5.45). If t T(L), the asymptotic relation (5.35) is valid: ðrÞ

ð0Þ

ð0Þ

ð0Þ

di j ¼ f2bi j bi j;ba þ bi j;ba gt2 ; ð0Þ

ð5:54Þ

where bi j;ab ¼ hu0i;b ðjðaÞ ; 0Þu0j;a ðjðbÞ ; 0Þi are components of the correlation tensor of fluctuations at the initial moment.

5.3 Turbulent Diffusion

Hence, the initial stage of relative motion of two fluid particles is characterized by ð0Þ the Eulerian correlation tensor of fluctuations bi j;ba that corresponds to the initial positions of these particles. For homogeneous isotropic turbulence, components of the tensor of relative dispersion can be determined from Eq. (4.88) and Eq. (4.90): ðrÞ

di j ¼ t2

qb ri r j LL r þ 2 1B d ½ ðrÞ þ ri r j ; i j LL r2 qr

ð5:55Þ

where bLL is the component of the tensor bij in the direction of r. A further simplification of Eq. (5.55) is possible if we make use of the relation (4.125), which is valid for locally isotropic developed turbulence:

ri r j 2 ðrÞ di j At2 4d i j 2 ðerÞ2=3 þ ri r j ; r 3 where A is a universal constant. Putting i ¼ j into the last formula, one obtains the root-mean-square displacement of one particle relative to the other at the initial stage of relative motion: ðrÞ ðdii r 2 Þ1=2

1=2 22

ðerÞ1=3 t: A 3

ð5:56Þ

A comparison of the relations (5.53) and (5.56) shows that the speed with which the particles recede from each other is different in the two cases. At the initial stage of the process, this speed is smaller than at t ! 1.


In Sections 5.1 and 5.2 we used the concept of a fluid particle – a volume that is small enough to be thought of as a material point moving together with the carrier fluid. One can use the Lagrangian approach to keep track of a given particle and to determine statistical Lagrangian characteristics of its random motion. In practice, in order for us to be able to observe the motion of fluid particles, these particles should somehow differ from the surrounding medium. However, the density of particles is usually the same or, at any rate, almost the same as that of the surrounding medium. Therefore a particle under consideration becomes distinguishable from the surrounding medium if it has a different color, chemical properties, or temperature. Such particles are known as impurities; sometimes they are also called ‘‘passive’’ because they do not exert any effect on the fluid motion. The motion of a passive particle has some important features. The first one is that its Lagrangian velocity at any instant of time coincides with Eulerian velocity of the surrounding fluid at the point where the particle is located at this very instant. The second feature is the so-called turbulent diffusion – a rapid spreading of the impurity

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injected into the flow at some spatial point. This spreading is caused mainly by the impurity transport via turbulent fluctuations, hence the origin of the term. Since molecules of the impurity differ from molecules of the surrounding medium, their spreading may be caused not only by turbulent diffusion, but also by molecular diffusion and Brownian motion. The characteristic time of the diffusion process can be estimated as tDL2/D, where D is the diffusion coefficient and L is the characteristic linear size of the region under consideration. Since the coefficient of turbulent diffusion Dt is much greater than the coefficient of molecular diffusion Dm (Dt/ Dm102 106), molecular diffusion in a region with length scale of the same order as the integral scale of turbulence requires much longer time than turbulent diffusion. Therefore when considering the process of impurity propagation in a macroscopic region, molecular diffusion can be neglected in comparison with turbulent diffusion. It is just this turbulent diffusion that is responsible for the rapid propagation of impurities in the atmosphere and in fluid streams. However, if one considers diffusion in a microscopic region whose size is of the same order as the microscale of turbulence, molecular diffusion can become noticeable. One example of such processes is the mixing of substances entering a chemical reaction, which will be discussed in Section 5.8. Impurity is usually injected into the flow as a liquid or gaseous admixture or as small solid particles. It can be thought of as a substance continuously distributed in space and characterized by a Eulerian concentration field C(X, t). For every separate instance of turbulent flow, the concentration field (in the absence of impurity sources) is described by the equation of convective diffusion qC qðui CÞ ¼ Dm DC; þ qt qXi

ð5:57Þ

where Dm is the coefficient of molecular diffusion, which is assumed to be constant, and ui are the velocity components. The assumption that the impurity is passive means that the velocity field u does not depend on C. The velocity u should then be known from the solution of the corresponding hydrodynamic problem, and one sees that Eq. (5.57) is linear. If any sources or sinks are present in the flow (this could happen, for example, due to homogeneous chemical reactions taking place in the given volume), the right-hand side of Eq. (5.57) will contain the corresponding term, which could be nonlinear with respect to C for certain kinds of reactions. In order to solve Eq. (5.57), one must provide the initial and boundary conditions, CðX ; t0 Þ ¼ C0 ðX Þ

ð5:58Þ

qC þ bC ¼ 0 qn

ð5:59Þ

and

Dm


The parameter b determines the conditions for the heterogeneous reaction at the surface S – the boundary surface of the flow region V. If b ¼ 1, then the surface is completely absorbing, and Eq. (5.59) reduces to the condition C ¼ 0. In the other limiting case, when b ¼ 0, the surface is completely impermeable for the impurity, and Dm@C/@n ¼ 0/. In the general case, b can depend on C. This dependence follows from the kinetics of the chemical reaction on the surface. Istantaneous sources of impurity at given spatial points are described by initial conditions that have the form of delta functions of time and spatial coordinates. Sources acting continuously are described by delta functions of spatial coordinates only. These delta functions appear in the boundary conditions if the sources are located at the boundary; otherwise, if the sources are located within the continuum volume, they appear as source–sink terms on the right-hand side of the diffusion equation (5.57). To summarize, we say that under the given homogeneous boundary conditions and in the absence of homogeneous chemical reactions, the concentration field of passive impurity C(X, t) is shaped exclusively by the turbulent transport, whose velocity is u(X, t), and by the molecular diffusion; the velocity field is determined independently (regardless of the concentration), by solving the corresponding hydrodynamic problem. The diffusion problem (5.57)–(5.59) thus reduces to a linear boundary value problem. When deriving the velocity field u(X, t) of the turbulent flow, one must supplement the hydrodynamic equations with the initial u(X, 0) ¼ u0(t) and boundary conditions. When studying a turbulent flow, the initial velocity field u0(x, t) is assumed to be random, in other words, there should exist a corresponding probability density distribution in the functional space of solenoidal vector fields. Then the passive impurity concentration C(X, t) that depends on the velocity distribution u(X, t) will also be a random quantity characterized by some probability density distribution. If the molecular diffusion is negligibly small compared to the turbulent one, Eq. (5.57) takes the form qC qðui CÞ ¼ 0: þ qt qXi

ð5:60Þ

Consider some general properties of this equation. First of all, it is linear, therefore, given the initial condition C(X, t) ¼ C0(X) and the linear boundary conditions (5.58), (5.59) with b ¼ const, its solution can be represented in the operator form as CðX ; tÞ ¼ L½u0 ðX ; tÞ; t C0 ðX Þ;

ð5:61Þ

where L is a linear operator that depends on the velocity u0(X, t), on the time t, and on the boundary conditions. Since u0(X, t) is a random function, L is also a random operator in the space of linear operators. Then, averaging Eq. (5.61) over all possible realizations of the initial vector field u0(X, t) assuming a fixed initial concentration distribution C0(X), one obtains hCðX ; tÞi ¼ hL½u0 ðX ; tÞ; t iC0 ðX Þ:

ð5:62Þ

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Since L is linear, the operator hLi is linear as well, and, given a fixed C0(X), the average concentration hC(X, t)i obeys some linear equation. Equations for correlation functions and higher-order moments of concentration C(X, t) can be derived from Eq. (5.57), but these equations will not be linear. In most cases, when considering turbulent diffusion of a non-reacting impurity, it is sufficient to look at equation (5.62) for the average concentration. If one decides to bring chemical reactions into the picture, it becomes necessary to look at the equations for moments as well. If an instantaneous impurity source of intensity Q was positioned at a point j at the initial moment, operator hLi can be found explicitly. Then C0 ðX ; tÞ ¼ QdðX jÞ: In the absence of molecular diffusion, the total mass of impurity concentrated initially within a fluid particle will remain inside that particle even as the particle undergoes random motion. After time t the particle will reach the point X ¼ X(x, t) where the impurity concentration is equal to CðX ; tÞ ¼ QdðX X ðj; tÞÞ:

ð5:63Þ

Let us bring into consideration the quantity p(X|j, t) – probability density distribution of finding a fluid particle at the point X at the instant of time t under the condition that the particle was initially located at the point j. Than, using the averaging rule, we obtain QhdðX X ðj; tÞÞi ¼ pðX jj; tÞ:

ð5:64Þ

Consider now an arbitrary initial concentration distribution C0(X). If we model the continuum as a set of fluid particles, each particle obeying condition (5.64), then, recalling that hLi is a linear operator, applying the superposition principle and replacing summation by integration, one gets ð hCðX ; tÞi ¼ hL½u0 ðX ; tÞ; t iC0 ðX Þ ¼

pðX jj; tÞC0 ðX ÞdX :

ð5:65Þ

This relation tells us that determination of the average concentration boils down to the problem of finding the probability density distribution p(X|j, t) for one fluid particle. It should be noted that, by virtue of Eq. (5.63) and Eq. (5.64), the distribution p(X|j, t) itself can be interpreted as a concentration field hC(X,t)i from an instantaneous source of unit intensity at the point X ¼ j. So, if the distribution p(X|j, t) is known, the formula (5.64) allows to determine the concentration field hC(X, t)i from different types of sources: an instantaneously acting source distributed in space; a continuously acting point source; or a continuously acting source distributed in space – in other words, from virtually any type of sources occurring in practice.


As we noted earlier in Section 5.2, in the case of stationary and homogeneous turbulence, the probability density distribution p(X|j, t) of fluid particle coordinates at any time t ¼ t ¼ t0 is close to a normal one. Let us direct the coordinate axes along the principle axes of the dispersion tensor dij (see Eq. (5.34)). Then pðX jj; t0 þ tÞ ¼

1 ð2pÞ3=2 ½d11 ðtÞd22 ðtÞd33 ðtÞ 1=2 ( ) ðX1 x1 Þ2 ðX2 x2 Þ2 ðX3 x3 Þ2 ; exp 2d11 ðtÞ 2d22 ðtÞ 2d33 ðtÞ

ð5:66Þ

where the values of dij(t) are given by the relations (5.39). Substituting Eq. (5.66) into Eq. (5.65), one obtains the average impurity concentration: ð¥ ð¥ ð¥ hCðX ; tÞi ¼

C0 ðX1 ; X2 ; X3 Þ ¥¥¥

(

) ðX1 x1 Þ2 ðX2 x2 Þ2 ðX3 x3 Þ2 exp dX1 dX2 dX3 : 2d11 ðtÞ 2d22 ðtÞ 2d33 ðtÞ

ð5:67Þ

To further simplify the expression (5.67), one can use asymptotic relations for principal values of the dispersion tensor for large values of time t ¼ (t t0) T(L); ðLÞ ðLÞ ðLÞ here T(L) is the maximum value of three Lagrangian time scales T1 , T2 , and T3 (see Eq. (5.40)), for which asymptotical expressions (5.43) are valid: ðLÞ

d11 ðtÞ ¼ 2hðu01 Þ2 it1 ;

ðLÞ


ðLÞ


where u01 ; u02 ; u03 are velocity fluctuation components. Let us see now how turbulent diffusion relates to molecular diffusion. At the beginning of this section, we mentioned that turbulent diffusion of an impurity is happening much faster than molecular diffusion. Does it imply that molecular diffusion can always be ignored? The special feature of turbulent diffusion that distinguishes it from molecular one is that the region initially occupied by the impurity can subsequently, in the course of its motion, be deformed in a most fanciful manner, but will still hold the total original volume of the impurity. Therefore, when the impurity is contained exclusively in the given fluid volume and is absent outside of the volume, the distribution of impurity concentration at a fixed point in space has the form of a step function: as long as pure fluid passes through the point, the concentration is equal to zero, but as soon as the observed fluid volume reaches this point, the concentration jumps to a constant value. In reality, of course, the distribution of concentration has no discontinuities. In a very short time after being injected into the fluid volume, the impurity diffuses into the surrounding fluid, which leads to an increase of the fluid volume, with

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smoothing down of concentration differences in the adjacent layers. This process is caused by molecular diffusion; the greater the coefficient of molecular diffusion, the faster it happens. Hence, when describing small-scale statistical structure of the field of concentrations, it is impermissible to neglect molecular diffusion: otherwise after a certain amount of time we would get an absolutely unrealistic picture of concentration distribution inside the flow. The role of molecular diffusion becomes evident if we consider the diffusion of a passive impurity from an instantaneous point source of unit intensity in the field of homogeneous turbulence with zero average velocity. Suppose the source is initially (t ¼ t0) placed the point X ¼ 0. The distribution of concentration C(X, t) is described by Eq. (5.57) with the initial condition C(X, t0) ¼ d(X). In a quiescent fluid with no turbulence, molecular diffusion causes the impurity to gradually spread out in a spherical cloud, where the concentration distribution exhibits spherical symmetry with the variance 2Dm(t ¼ t0). Let us examine the consequences of including turbulence into the picture. Consider the diffusion along the Xi-axis. The distinguishing characteristics of cloud spreading are as follows: – variance of the average concentration distribution hC(X, t)i relative to the initial position of the source, ð¥ d02 ðtÞ

¼

Xi2 hCðX ; tÞidX ;

ð5:68Þ

¥

– variance of the center of gravity of concentration distribution relative to the source,

hX02 ðtÞi ¼

*8 ð¥ < :

X i CðX ; tÞdX

¥

92 + = ;

;

ð5:69Þ

– variance of concentration distribution relative to the center of gravity, * ð¥ d02 ðtÞ

+ 2

¼

½Xi X0 ðtÞ CðX ; tÞdX :

ð5:70Þ

¥

To ensure conservation of the total mass of the impurity, we write the normalization condition, ð¥ CðX ; tÞdX ¼ 1; ¥


after which the equations above result in dc2 ðtÞ ¼ d02 ðtÞXc2 ðtÞ:

ð5:71Þ

In order to determine the parameters d02 ðtÞ, hXc2 ðtÞi and dc2 ðtÞ, one should switch from the fixed frame of reference X to a moving frame Y ¼ X X(0, t), whose origin at each instant of time coincides with the fluid particle’s position (X ¼ 0 being the initial position of the particle). Then, using the transformed diffusion equation (5.57) plus the relations (5.68)–(5.70), and expanding the functions d02 ðtÞ, hXc2 ðtÞi, and dc2 ðtÞ in Taylor series over the powers of t t0, one gets 2 dc2 ðtÞ ¼ 2Dm ðtt0 Þ þ hðrui Þ2 iðtt0 Þ3 . . .; 3

ð5:72Þ

1 d02 ðtÞ ¼ hXi2 ð0; tÞi þ 2Dm ðtt0 Þ hðrui Þ2 iðtt0 Þ3 . . .; 3

ð5:73Þ

hXc2 ðtÞi ¼ hXi2 ð0; tÞiDm hðrui Þ2 iðtt0 Þ3 . . .:

ð5:74Þ

From Eq. (5.72), it follows that at the initial stage of the process, when t t0 1, there should hold dc2 ðtÞ 2Dm ðtt0 Þ; that is, the impurity transfer relative to the center of mass is driven by molecular diffusion. As time goes on, the influence of the second term in Eq. (5.73) grows. This term characterizes turbulence through the average value of the square of the velocity gradient, h(!ui)2i. As a result, turbulence causes an acceleration of molecular diffusion, and thereby a faster spreading of the impurity. By and large, this effect is explained by significant deformation of the moving fluid volume in the turbulent flow. But in order to be noticeably deformed, the fluid volume should first undergo a significant expansion, which is possible only through the molecular diffusion mechanism. That is why molecular diffusion is vital at the initial stage, where it acts as a necessary prerequisite for the further spreading of the fluid volume. The formula (5.74) for Xc2 shows that the variance of the center of gravity is smaller than the variance of the fluid particle’s coordinate X12 ð0; tÞ. This time lag is due to the fact that, because of molecular diffusion, the impurity particle falls behind the fluid particle (on the average). As a result, molecular diffusion slows down the turbulent diffusion. The same conclusion can be made with respect to the formula (5.73) for d02. We conclude from the foregoing discussion that molecular diffusion exerts a major influence on the average concentration for a limited time, just after the beginning of the process. As time goes on, this influence becomes less and less important. So if we are interested in the spreading of the impurity in macroscopic

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volumes at high Reynolds numbers, it is safe to neglect molecular diffusion when determining the average concentration. Up to this point, we were concerned with turbulent diffusion of a single fluid particle. In practice, one is primarily interested in relative diffusion, defined as spreading of a macroscopic volume (say, one mole) of the impurity consisting of a large number of fluid particles. To solve such a problem, one must bring into consideration statistical characteristics, namely, multidimensional (multiparticle) probability density distributions of coordinates X and velocities V of n given particles at arbitrary instants of time t1, t2, . . ., tm under the condition that initial coordinates of particles are given: j1 ¼ X1(t01), j2 ¼ X2(t02), . . ., jn ¼ Xn(t0n), though the initial moments t01, . . ., t0n are not necessarily the same. Up till now, we have considered only probability density distributions of a single particle’s coordinates X (t). Consideration of multidimensional distributions is a very complicated problem. Let us therefore consider pair distributions only, that is, we shall be interested in the probability density distribution of coordinates X1, (t) and X2, (t) of two particles positioned different spatial points j1 and j2 at one and the same initial instant t0. Instead of the vectors X1 and X2, it is more convenient to use two other vectors: the displacement vector of the first particle Y(t) ¼ X1(t0 þ t)j1, and the vector r(t) ¼ X2(t0 þ t) X1(t0 þ t) connecting the particles at the moment t ¼ t0 þ t) and characterizing their mutual arrangement. The pair probability density distribution can then be written as p(Y, r|j1, r0, t, t0). This notation means that we are looking at the joint distribution of Y and r at the moment t after the beginning of the process, given that at the initial moment t0, the first particle was at j1 and the position of the second particle relative the first is described by the vector r. In the particular case of stationary, homogeneous turbulence, the probability density distribution does not depend on t0 and j1. Of primary interest is the distribution p(r|j1, r0, t, t0), equal to ð pðrjx1 ; r 0 ; t; t0 Þ ¼

pðY ; rjj1 ; r 0 ; t; t0 ÞdY :

ð5:75Þ

Following Richardson [3], who first introduced such a distribution, it is called the distance function between neighbors. We shall list the main properties of this distribution. 1. After a sufficiently long time t T(L), the two particles will get so far apart that their velocities V1(t0 þ t) ¼ u(X1(t0 þ t), t0 þ t) and V2(t0 þ t) ¼ u(X2(t0 þ t), t0 þ t) will become practically independent from each other. Then the distance function is derived from the distribution of displacements for a single particle: pðY ; rjj1 ; r 0 ; t; t0 Þ ¼ pðY jj1 ; t; t0 Þ pðY þ rr 0 jj1 þ r 0 ; t; t0 Þ;


where p(Y|j1, t, t0) is the displacement distribution for a single fluid particle, and Eq. (5.75) can be written as ð pðrjj1 ; r 0 ; t; t0 Þ ¼

pðY jj1 ; t; t0 Þ pðY þ rr 0 jj1 þ r 0 ; t; t0 ÞdY :

ð5:76Þ

If p(Y|j1, t, t0) satisfies the diffusion equation with constant diffusion coefficients Dij, then p(r|j1, r0, t, t0) obeys the same equation, except that the diffusion coefficients are now equal to 2Dij (see Eq. (5.95)). If we have r0 L at the initial moment, where L is the integral scale of turbulence, then the formulas (5.75) and (5.76) will be valid for any instant of time. Owing to the results obtained in Section 5.2, for large values of t we have the following asymptotic relations for the ðrÞ relative dispersion tensor di j ðtÞ and the mean square deviation 2 hr (t)i: ðrÞ

ðLÞ

di j ðtÞ 2hðu0i Þ2 iTi di j t;

hr 2 ðtÞi 2

X ðLÞ hðu0i Þ2 iTi t:

ð5:77Þ

i

2. If at the initial moment the particles were sufficiently close to one another, that is, |r0| L, then large-scale fluctuations with lL will only move the two particles as a whole, without changing their mutual arrangement. Therefore relative motion of particles is possible only under the action of small-scale fluctuations. To describe the behavior of the distance function for such a case, one should use the theory of local isotropic turbulence (see Section 4.9). Let us remind ourselves that when studying the local structure of developed turbulence, we classified small-scale motions with l L as belonging to one of the two intervals: inertial interval with l0 l L, where l0 is the inner (Kolmogorov) scale defined by the formula (4.114), and dissipative interval with l l0. Since the initial relative position of the two particles under consideration is characterized by the vector r0, the character of subsequent dependence of the vector r on time t is given by the relation between r0, l0, and L. As far as r(t) varies continuously, there always exists a time t1 such that the condition |r(t)| L is valid at t < t1. When t < t1, fluctuations with l L do not affect the mutual arrangement of particles, that is, the vector r. At t3 < t < t1, the interparticle distance is comparable with fluctuation scales in the inertial interval. In this time interval, it is possible to imagine the particles as moving in a straight line due to their inertia during the length of time t2. Finally, at t t3, the interparticle distance is comparable with the inner scale of turbulence.

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This division of time into separate intervals allows one to investigate the general properties of the distance function between neighbors p(r|x1, r0, t, t0) on the basis of the Kolmogorov hypotheses within the framework of the theory of local isotropic turbulence. Rather than focusing on minute details pertaining to the distance function, we refer the reader to [2]. We shall list here only two important formulas, 1 ðrÞ di j ¼ get3 d i j ; 3

hr 2 i ¼ g et3 ;

ð5:78Þ

which are valid in the interval t3 < t < t1, that is, for |r0|l0, t tl0 or for |r0| l0, t > t(r0), where g is a universal constant and e is the specific dissipation of energy. As an example, consider the relative diffusion accompanying the spreading of an impurity cloud consisting of a large number of particles. Let us select some particle from the cloud and call it a test particle. Then the distribution of concentration in the cloud relative to the test particle at the instant of time t ¼ t0 þ t coincides with the distribution of values r(t) for all possible pairs of particles that include the test particle. The spreading of a cloud of N particles is described by the Richardson function q(r, t) such that Nq(r, t)dr is equal to the number of particles for which the small volume surrounding the point Xi(t) þ r at the instant t0 þ t contains at least one cloud particle. This function obeys the normalization condition ð qðr; tÞdr ¼ 1

ð5:79Þ

and defines the relative number of pairs of cloud particles whose coordinates differ by r. The average value of the Richardson function q(r, t) is ð hqðr; tÞi ¼

pðrjr 0 ; tÞq0 ðr 0 Þdr 0 ;

where p(r|r0, t) is the probability density of the vector r(t) and q0(r) ¼ q(r, 0) is the initial value of q(r). For example, when the impurity initially fills a sphere of radius R, we have 4pR3q0(r) ¼ 1 3r/4R r3/16R3where r ¼ |r|. Let us introduce the tensor of relative dispersion of the cloud, having the components ð Li j ðtÞ ¼ hri r j i ¼ ri r j hqðr; tÞidr;

ð5:80Þ

which relates the quantities Lij(t) to components of the relative dispersion tensor ðrÞ di j ðtjr 0 jÞ ¼ hri r j i for a particle pair. The horizontal bar at the top denotes averaging over all particles pairs. Then the effective diameter of the cloud can be defined as DðtÞ ¼ ½Li j ðtÞ 1=2 :

ð5:81Þ

5.4 A Semiempirical Model of Turbulent Diffusion

The quantity X¼

1 dD2 ðtÞ 6 dt

ð5:82Þ

is called the virtual turbulent diffusion coefficient of the cloud, and Xi ¼

1 dLii ðtÞ 1 hdri2 ðtÞi ¼ 2 dt 2 dt

ð5:83Þ

is called the turbulent diffusion coefficient along the Xi-axis. If the initial distance between cloud particles satisfies the condition |r0| L, then for t tl0 and t3 < t < t1, where t1 and t3 are related to the initial cloud diameter D0, Eq. (5.78) gives us D2(t)t2, and the virtual turbulent diffusion coefficient of the cloud is equal to X ¼ aD4=3 :

ð5:84Þ

This law is called the law of four thirds or the Richardson law. Since this law is a consequence of general ideas about the structure of small-scale turbulence, we could also have derived it from dimensionality considerations in the same manner as we derived the relations that appear in Section 4.9. Indeed, the transport coefficient X in the inertial interval of fluctuations, that is, at l0 l L, should be defined in terms of dimensional quantities e and D. Since the dimensionality of X is m2/s, it is easy to obtain X ¼ ae1=3 D4=3 : Examination of relative motion of two fluid particles shows that the interparticle distance grows with time. So, turbulent motion causes the cloud (a mole, say) to change its shape in such a way that in the end, any small spherical volume stretches into a long thin band whose length, width, and thickness change exponentially with time, though with different values of coefficients in the exponent. Needless to say, this initial volume should be sufficiently small for all our original assumptions to be satisfied.


Consider the diffusion equation (5.57). If we neglect molecular diffusion, it takes the form

qC qhui Ci ¼ 0: þ qt qXi

ð5:85Þ

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The velocity u is equal to the sum of the average hui and fluctuational u0 velocities: u ¼ hui þ u0 . In a similar fashion, we shall represent concentration as a sum of its average value hCi and fluctuation C0 : C ¼ hCi þ C 0 :

ð5:86Þ

Averaging Eq. (5.85), we obtain qhCi qðhui ihCiÞ qðhu0i C0 iÞ ¼ : þ qt qXi qXi

ð5:87Þ

The expression Ji ¼ hu0i C0 i on the right can be interpreted as the flux density of the impurity in the direction of the Xi-axis. The structure of Eq. (5.87) is the same as that of Reynolds equations (see Section 4.6). It is unclosed, because it contains a new unknown. Therefore the closure problem (this time with respect to Eq. (5.87)) is just as imperative in the theory of turbulent diffusion as it is in the theory of turbulence. The simplest closure relation can be constructed by assuming that diffusion is isotropic and making use of the Taylor hypothesis [4] Ji ¼ hu0i C 0 i ¼ Dt

qhCi : qXi

ð5:88Þ

The coefficient Dt entering this equation is called the coefficient of turbulent diffusion. It should be noted that relation (5.88) is similar to the Boussinesq hypothesis (4.140). The Taylor hypothesis implies that impurity transport via random turbulent fluctuations is similar to the transport via molecular diffusion, since the impurity flux is proportional to the concentration gradient, just as in Fick’s law. The coefficient of turbulent diffusion Dt has nothing to do with the coefficient of molecular diffusion Dm. Indeed, Dt characterizes the impurity transport via chaotic turbulent motion, while Dm characterizes transport via chaotic molecular motion. The only common feature is the randomness of the relevant processes and the resulting need to employ statistical methods. Let us estimate Dt by the order of magnitude. As we noted in the previous section, turbulent diffusion is characterized by the scale of large-scale turbulence, namely, by a linear size L such as the pipe diameter; by the change of the average flow velocity DU; and by fluid density re. From these quantities one can construct a combination DUL, which has the dimensionality of the diffusion coefficient. Therefore, just as in the kinetic theory of gases, we can write Dt DUL:

ð5:89Þ


Note that Dt is equal by the order of magnitude to the coefficient of turbulent kinematic viscosity (see Eq. (4.104)). Therefore the coefficient of turbulent diffusion is akin to the coefficient of turbulent viscosity and has the same order of magnitude. Let us compare the coefficients of turbulent and molecular diffusion: Dt DUL Rene ¼ : Dm Dm Dm Since for fluids, ne107 m2/s and Dm109 m2/s, we have Dt/Dm102 Re 1. Such a large value of Dt ensures a rapid mixing of the fluid. So if a chemical reaction occurs at the surface, turbulent diffusion will quickly equalize concentration in the flow even at small distances from the wall. As a result, the concentration becomes practically constant in the whole region except for a thin layer adjacent to the wall. Near the wall, the characteristic linear scale is equal to the distance from the wall. If one coordinate axis, for example, Z, is directed perpendicularly to the wall, Z will be the characteristic linear scale in this region, and Dt Z2

qU : qZ

ð5:90Þ

So, as we approach the wall, the coefficient of turbulent diffusion decays as Z2. A more detailed analysis of the flow structure that takes into account the existence of a viscous sublayer shows that near the wall, Dt decays as Z4. The flux of matter toward the wall is by the order of magnitude equal to Jw Z 2

qU qhCi : qZ qZ

ð5:91Þ

It is well known that distribution of the average velocity near the wall obeys a logarithmic law (see Section 3.10). Then one can see from Eq. (5.91) that hCi is also distributed logarithmically. But in the immediate vicinity of the wall, the logarithmic law needs to be corrected in order to account for the viscous sublayer. In the anisotropic case, the diffusion coefficient is a second-rank tensor Dij. It is introduced through a relation that is similar to Eq. (5.88): Ji hu0i C 0 i ¼ Di j

qhCi : qX j

ð5:92Þ

As usual, summation over repeated indices is implied by default. When coefficients Dij depend on the coordinates Xi and time t, Eq. (5.87) takes the form qhCi qðhui ihCiÞ q qhCi ¼ Di j : þ qt qXi qXi qX j

ð5:93Þ

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Consider the case of stationary homogeneous turbulence. Then the tensor of fluctuation dispersions dij is defined by the expression (5.38). The average value of fluid particle’s displacement along the Xi-axis during the length of time t is hYi(t)i ¼ huiit (see Eq. (5.36)), where the constants huii are components of the average velocity. As was mentioned in Section 2.2, in the case under consideration the joint probability density distribution of particle displacements hYi(t)i for all t can be considered as a threedimensional Gaussian distribution (see Eq. (1.77)). Then the distribution p(X|j, t) of fluid particles initially located at X ¼ j over spatial coordinates X will also be Gaussian, with the mean value aj ¼ jj þ huji(t t0) and the matrix b ¼ jjdi j jj. The distribution p(X|, j t) obeys Eq. (5.94), where Dij should be expressed through components of the dispersion tensor in accordance with the relation

Di j ¼

ðt i 1 h ðLÞ 1d ðLÞ bi j ðtÞ þ b ji ðtÞ dt; di j ðtÞ ¼ 2 dt 2

ð5:94Þ

0

where t ¼ t t0. Since average concentration hC(X, t)i of the impurity obeys the same equation, the condition (5.94) means that fluid particle displacements in a stationary homogeneous turbulent field are distributed according to the normal (Gaussian) law. At t T(L), the asymptotic representation (5.45) is valid: ðLÞ

02 1=2 di j ðhu02 Ti j t i ihu j iÞ

so we can write the following approximate equality: 1 ðLÞ Di j ðtÞ ðhu02 ihu02j iÞ1=2 Ti j ; 2 i ð¥h i ðLÞ ðLÞ ðLÞ where Ti j ¼ Yi j ðtÞ þ Y ji ðtÞ dt.

ð5:95Þ

0

The discussion of Brownian motion in Chapter 5 was centered around the Fokker–Planck equation that describes the variation of probability density distribution p(X, t|j, t0), where p refers to the probability for the particle to arrive at the point X at the moment t given that it was at the point j at the moment t0. The derivation of the Fokker–Planck equation hinges on the assumption that the process is Markowian (see Section 1.11). Recall that our model of Brownian motion involved the division of particle trajectory into discrete intervals by the points X1, X2, . . . corresponding to the instants of time t1, t2, . . .. We required the time intervals t2 t1, . . . to be much longer than the time between successive collisions of the particle with molecules of the surrounding fluid, but at the same time much shorter than the characteristic time of the process under consideration. The probability density distribution p(X, t|j, t0) at two successive instants of time t0 and t is then described by the Fokker–Planck equation (1.124), whose solution is the Gaussian distribution (1.126).


In the turbulent flow, we have a similar situation. Consider the random motion of a fluid particle that is described using the Lagrangian approach. Let us identify the time between two successive collisions with Lagrangian correlation time T(L) and the characteristic time of Brownian motion – with the characteristic time of turbulence tt. We now divide the trajectory of the particle’s random motion in a turbulent flow into discrete intervals by spatial points X1, X2, . . . corresponding to the particle’s positions at the instants t1, t2, . . ., and assume that time intervals between successive particle positions are much longer than T(L) but much shorter than tt. Then, as was mentioned above, probability density distribution p(X, t|j, t0) obeys Eq. (5.93), whose solution (under the condition (5.95)) is a Gaussian distribution. We conclude that X(j, t) is a Markowian function and that p(X|j, t) satisfies the Fokker–Planck equation qp q q2 ðhVi ðX ; tÞi pÞ ¼ ðDi j ðX ; tÞ pÞ; þ qXi qX j qt qXi

ð5:96Þ

where hVii is the average velocity of a (fluid) impurity particle. This velocity is equal to the sum of the average flow velocity huii and the additional velocity @Dij/@Xj induced by the inhomogeneity of turbulent diffusion coefficients: hVi i ¼ hui i

qDi j : qX j

ð5:97Þ

To summarize, under the conditions we imposed above, the sequence of particle positions X(j, tk), as well as the process of turbulent diffusion at t T(L), wil be Markowian. Of course, representations (5.88) and (5.92) are but idealizations of a real process. Indeed, at t T(L) the dispersion is bijt2 (instead of being proportional to t), the condition (5.95) is no longer fulfilled, and the process is not Markowian. In this case it becomes impossible to use the semiempirical model, and in order to close the diffusion equation (5.87), one should employ other methods such as the method of moments. Hence, the semiempirical equation of turbulent diffusion (5.93) with effective turbulent diffusion coefficient (5.95) is useful when solving problems that involve diffusion of a passive impurity in a stationary homogeneous turbulent flow whose characteristic time exceeds the Lagrangian scale of turbulence (5.40). We should remark that in the problems that involve spreading of an impurity in the air, the Lagrangian time scale has the order of seconds. The turbulent diffusion equation (5.93) has helped to solve many practical problems on diffusion in homogeneous turbulent fields and in shear flow fields: impurities spreading away from sources; longitudinal spreading of impurities in pipes and channels; diffusion in free jets, in the atmosphere, and so on.

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5.5 Models of Two-phase Disperse Turbulent Flows

Two-phase disperse media are the media consisting of a continuous (carrier) phase (fluid, gas) and a disperse phase (solid particles of some mechanical impurity, droplets, bubbles) whose properties differ from those of the surrounding medium. Later on we shall call them simply ‘‘particles’’, providing a clarification as to their physical nature whenever necessary. An important parameter of such medium is the volume concentration of the disperse phase j, equal to the total volume of particles contained in a unit volume of the medium. A medium with j 1 is called dilute, or rarified. If, in addition, the particles are small and their properties are not much different from those of the continuous phase, the disperse phase has a weak influence on the hydrodynamic characteristics of the continuum, and the particles (as well the disperse phase as a whole) can be regarded as passive. To describe the behavior of a passive disperse phase, one can use the theory of turbulent diffusion, which essentially states that particle motion is defined by the turbulent motion of the continuous phase, whereas the inverse influence of the disperse phase on the continuous phase flow is absent. An increase of j (or, for small j, an increase of particle size) or a noticeable difference of particle properties (such as density or viscosity) from those of the continuous phase would make this reverse influence significant. In this case the problem becomes more complicated, as one is forced to take into account the motion and mutual interaction of both phases. Impurity particles are drawn into chaotic motion as a result of fluctuations of the viscous drag force, which depends on the particle’s velocity relative to the carrier phase. For Stokesian particles, this force is proportional to the difference of velocities V and u of the particle and the fluid. The difference V u by itself may be due to inertia and (or) to the action of an external force such as gravity. The influence of the external force is especially noticeable for large particles. In applications, we are mostly concerned with microparticles, that is, particles whose size does not exceed the microscale of turbulence. Relative motion of such particles obeys the Stokes resistance law, hence the term ‘‘Stokesian particles’’. Hydrodynamic aspects of the motion of such particles in a laminar flow have been discussed in Chapter 2. Another important parameter characterizing the particles’ inertia and affecting their motion is the viscous (dynamic) relaxation time. For spherical Stokesian particles, it is equal to tv ¼

d2 r p 18ne re

;

ð5:98Þ

where rp and re are the densities of the particle and the surrounding fluid, ne is the kinematic viscosity of the surrounding fluid, d is the diameter of the particle.


Dynamic relaxation time has the meaning of the time it takes a particle moving by intertia to be stopped by the viscous drag force. Therefore tv characterizes the influence of particle inertia on its motion relative to the fluid. For large particles and (or) for the relative motion of a particle under the action of large-scale fluctuations, dynamic relaxation time depends on the particle’s Reynolds number: tv ¼

4 d2 r p 1 : 3 ne re Re p C p

ð5:99Þ

where Rep is the Reynolds number determined by the particle’s parameters and Cp is the coefficient of resistance for the particle, equal to Cp ¼

24 ð1 þ 0:179Re0:5 p þ 0:013Re p Þ: Re p

In the absence of external forces, the relative motion of large particles is caused only by turbulent fluctuations of the carrier fluid, while the relative motion of small particles may also be caused by Brownian motion. If particle temperature differs from the temperature of the carrier phase, one can introduce the characteristic time of thermal relaxation tT, defined as the time it takes for the particle temperature to reach the temperature of the surrounding medium. Particle inertia is characterized by a dimensionless parameter called the Stokes number St, equal to the ratio between the particle’s dynamic relaxation time tv and the Lagrangian correlation time T(L): St ¼

tv : T ðLÞ

ð5:100Þ

Depending on how strongly the particles are involved into turbulent motion, they can be divided into two categories based on their inertia. The first category includes the particles with St > 1. These particles are called inertial. They exhibit only a weak response to small-scale fluctuations. Particles with St < 1 are called inertialess. Their motion is defined by the microstructure of turbulence. In the limit St ! 0 they become passive particles, whose motion is described by the turbulent diffusion model. At the present moment, the turbulent flow theory is far from complete, even for a single-phase medium, to say nothing of two-phase media. In Section 4.10, we discussed the existing turbulent flow models for single-phase media (fluids). Following the review [5], we now list some theoretical models of two-phase disperse turbulent flows. When simulating two-phase turbulent flows, we focus our attention on the following problems: interaction of particles with the turbulent flow of the continuous phase; the inverse effect – the action of the particles on the turbulent flow; interactions between particles; accounting for polydispersity (difference in particle sizes)

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and for the time evolution of the size spectrum of particles; the effect of turbulence on the rate of phase transitions, chemical reactions and combustion. Currently, there exist two simulation methods. The first one is based on the use of phenomenological or semiempirical description of the continuous phase, with introduction of algebraic and (or) differential equations as closure relations (see Section 4.10). The second method, known as the direct numerical simulation [32], relies on numerical solution of nonstationary equations of motion without the closure relations. This method requires the employment of supercomputers; it is still in the process of development and is far from perfection. Therefore we shall confine ourselves to the first method. Within this method, we can, in turn, identify two subcategories. The first one comprises techniques based on the combined Euler–Lagrange approach to the description of the medium’s motion. In this paradigm the momentum and energy equations for the continuous phase (for an isothermic flow of an incompressible fluid, it is possible to take the momentum equations only) are represented and solved in Eulerian variables, whereas equations for the disperse phase are written and solved in Lagrangian variables. This means that the motion of each particle is described by the Langevin stochastic equation in the relaxation approximation (see Section 1.13), dV p uðX p ðtÞ; tÞV p ðtÞ þ FðX p ðtÞ; tÞ þ f ðX p ðtÞ; tÞ; ¼ dt tv

V ¼

dX p dt

ð5:101Þ

whose right-hand side is a sum of the random viscous drag force, the external force, and the random Brownian force (in that order). The heat exchange equation for the particle in the relaxation approximation also has the form of the Langevin stochastic equation, dJ p JðX p ðtÞ; tÞJ p ðtÞ : ¼ dt tT

ð5:102Þ

where Wp(t) and W(Xp(t), t) are the temperatures of the particle and the carrier medium, tT is the characteristic time of thermal relaxation for the particle. The Langevin equations are integrated for different values of initial particle position Xp(0), velocity Vp(0), and temperature Wp(0), which corresponds to different particles. The obtained solutions are then averaged over the ensemble of initial data. This method is known as the method of stochastic simulation. A similar method for Brownian motion has been considered in Section 3.12. To have confidence in the obtained results, it is necessary to have a representative ensemble of the initial data, that is, the number of particles must be sufficiently large, though it comes at the cost of increased volume of computations. Besides, as we reduce the particle size, smaller-scale fluctuations also begin to interact with particles, which increases the number of possible variants of the initial data. Thus application of the method of stochastic simulation to discrete particles is sensible only for inertial particles satisfying the condition St > 1. It should be noted that in


the limiting case St 1 (highly inertial particles) it becomes possible to use the deterministic Lagrangian description based on equations for the average values only, disregarding interactions of particles with random fluctuations of the velocity and temperature fields. Another difficulty associated with stochastic simulation is that as we increase the volume concentration of particles j, the average distance between particles gets smaller, which means higher probability of particle collisions that may result in aggregation and/or breakup of particles. The difficulties associated with Lagrangian simulation become even more formidable when we take into account the possibility of nucleation of the disperse phase as a result of fluid boiling or condensation of supersaturated vapor. Another simulation method is based on the Eulerian continuous representation of momentum and energy equations for both the continuous and the disperse phase. Such models are called two-fluid models. The essence of this method is to model the disperse phase as yet another continuous phase with its own properties (density, viscosity, and so on) and phenomenological relation between the stress tensor with the rate-of-strain tensor. This brings us to the question of whether or not it is possible to describe the motion of a large group of particles by methods of continuum mechanics. To answer this question, one should turn to the so-called continuity hypothesis, which is commonly used in continuum mechanics, and states that the continuous approximation is applicable to scales that are sufficiently small compared to the characteristic scale of macroscopic flow parameter changes, yet large enough to be able to contain many particles. The appropriate scale can be estimated in the following way. Assuming statistically independent behavior of individual particles, the relative fluctuation of distributed density is of the order e N1/2, where N is the number of particles in the given volume. Consider a cubic volume with N spherical particles of diameter d lined up along the edge L. For a given volume concentration of particles j, we can make the approximate estimation L/d (2e2j)1/3. At j ¼ 103, e 102, and d 100 mm, this estimation yields L 1 cm. Thus, a disperse phase with such parameters can be modeled as continuous only on scales far larger than 1 cm. In the Eulerian approach, the main task is to determine the force and energy interactions between phases. The advantage of Eulerian two-fluid approach as compared to the Euler–Lagrange trajectory modeling is that equations of the same type (Reynolds equations plus the closure relations), as well as the same algorithm for solving the resulting system of equations, are used for both phases. Within the framework of the two-fluid approach, it is possible to go to the St ! 0 limit (the case of very small particles of passive impurity). Two approaches have emerged as the most popular methods of solving the problem of turbulent motion of a disperse medium. Namely, one can study turbulent motion based on semiempirical models of Prandtl mixing length, or one can use second-order one-point moments of velocity and temperature, by analogy with single-phase flows. In the framework of the first method, Prandtl mixing length models for a single-phase medium are generalized for the case of disperse medium.

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At of today, the second method, which involves equations for second moments of velocity and temperature fluctuations, is used more often. When describing motion and heat exchange of the disperse phase, it is necessary to determine turbulent stresses, as well as diffusion and heat fluxes resulting from the particle’s involvement into fluctuational motion of the carrier phase. Among the various methods of finding disperse phase characteristics, we should single out local equilibrium (aka algebraic) models. In one of these models, which relies on the locally homogeneous approximation, turbulent stresses in the disperse phase hVi0 V 0j i are connected with Reynolds stresses of the carrier phase hu0i u0 i through the relations hVi0 V 0j i

¼

ð¥

fv hu0i u0j i;

fv ¼

v ðtÞexp 0

t dt; tv

ð5:103Þ

where u0 and V0 are velocity fluctuations of the continuous and disperse phases, Cv is the two-time autocorrelation function (see Eq. (1.25)) of velocity fluctuations of the continuous phase along the particle trajectory. The relation between turbulent heat flux in the disperse phase hVi0 J0p i and turbulent heat flux in the continuous phase hu0i J0 i has a similar form: tv fvt þ tT fTv 0 0 hui J i; tv þ tT ð¥ t fvt ¼ YvT ðtÞexp dt; tv

hVi0 J0p i ¼

0

ð¥ t fTv ¼ YTv ðtÞexp dt; tT

ð5:104Þ

0

where J0p and q0 are temperature fluctuations in the disperse and continuous phases, tT is the heat relaxation time for a particle, and YvT and YTv are two-time autocorrelation functions of velocity and temperature fluctuations. The relations (5.103) and (5.104) will follow from differential equations for second moments of velocity and temperature fluctuations if we drop the terms in these equations that describe convective transport, diffusion, and production from the averaged motion; this is justified only for relatively small particles. Another way to determine turbulent characteristics of the disperse phase is to use expression of the gradient type according to the rules of the semiempirical theory. In this paradigm, turbulent stresses and heat flux of the disperse phase are represented in the form that is similar to the relations for a single-phase medium, namely, 2 qhVi i qhV j i 2 qhVk i hVi0 V 0j i ¼ ek p di j n p þ ; 3 qX j 3 qXk qXi hVi0 J0p i ¼

ð5:105Þ

0

n p qhJ p i ; Pr p qXi

ð5:106Þ


where ek p ¼ hVi0 Vi0 i=2 is the turbulent energy of the disperse phase, n p and c p are coefficients of turbulent viscosity and thermal diffusivity, respectively, and Pr p ¼ n p =c p is the Prandtl number of the disperse phase. Models of the gradient type are usually constructed on a purely phenomenological basis, therefore they contain additional empirical constants. Models of this type are valid at St < 1, that is, for particles with small inertia. In addition to local equilibrium algebraic models describing turbulent transfer of momentum and heat in the disperse phase, non-local differential models based on the energy balance equations and equations for the second moments of velocity and temperature are gaining broad recognition. Application of differential models (you can think of them as transport models by analogy with single-phase media, see Section 4.10) makes it possible to describe non-local effects of transport of velocity and temperature fluctuations by inertial particles, that is, turbulent transport of momentum and heat via convection and diffusion. These effects are especially important in the flow regions adjacent to the wall. We showed earlier in Section 4.11 that a complete statistical description of turbulence can be obtained from the probability density function (PDF) of particle positions and velocities. Knowledge of the PDF enables us to find the average values of the velocity and temperature fields as well as their correlations. So the main problem is to determine the PDF. As shown in the same Section 4.11, in the general case the PDF is described by the Hopf equation – equation in functional derivatives, whose solution still presents difficulties at the time of writing this book. In the theory of turbulence of disperse media, one comes up against the same difficulties. If the equation for the PDF is known, one can derive from it the systems of equations (moments equations) describing the motion and heat-and-mass-exchange of the disperse phase in context of the Eulerian approach in the same manner as one derives the Euler and Navier–Stokes equations from the Boltzmann equation in the kinetic theory of rarefied gases. As was shown earlier, in the Lagrangian approach particle motion is described by the Langevin equation, which enables us to keep track of each particle’s movements in a random force field. By examining the PDF we can obtain statistical description of the behavior of a particle ensemble, rather than dynamical description of individual particles. From the mathematical viewpoint, it means that the problem of integration of stochastic ordinary differential equations in a physical space is replaced by the problem of solving a deterministic partial differential equation in the phase space of coordinates, velocities, and temperature. Interaction of particles with turbulent fluctuations of the carrier medium is described by the same diffusion operator in the velocity space as the one we encountered when studying Brownian diffusion. For delta-correlated random fields (for a disperse phase, it means that the particles are inertial, that is, St 1), the equation for the PDF reduces to the Fokker–Planck equation. A more general form of the PDF equation is derived in [6–11]. Comparing the two approaches, we may conclude that some information about the behavior of individual particles is lost in the transition from the Lagrangian to the

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Eulerian approach, but, to compensate for that, we are gaining information about statistical regularities of particle motion. Besides (and one can argue that this is, in fact, the main advantage of the Eulerian approach), the equation for the PDF can be used to build a system of continual equations for the averaged values of hydrodynamic and heat-and-mass-transport characteristics of the disperse phase. The obtained system for the moments is similar to an infinite system of equations in the theory of single-phase turbulent flows (the Friedmann–Keller chain, see Sections 4.6 and 4.7). Any finite subsystem of these equations is unclosed because the equation for n-th moment already contains (n þ 1)-th moment. In order to close a finite subsystem, we must introduce closure relations similar to those introduced in Section 4.19 for single-phase media. In particular, we can use the equation for energy of the disperse phase ep; as for the turbulent stress and the heat flux, they can be obtained from the relations (5.105) and (5.106), in which we set n p ¼ Gv nt þ tv ek p =3;

Pr p ¼

ðtv þ tT Þð fv nt þ tv ek p =3Þ ; ðtv fvt þ tT fTv Þnt =Prt þ 2tv tT ek p =3

ð5:107Þ

where nt and Prt are, respectively, the turbulent viscosity coefficient and the turbulent Prandtl number for the carrier phase. We should mention one more method called the inertial diffusion model. It is used in hydrodynamical calculations of two-phase flows that contain a finely dispersed impurity consisting of particles whose inertia is small (St 1) but whose density is much higher than the density of the carrier phase. The essence of this method is to solve the diffusion equation for the impurity, taking into proper account some inertial transport mechanisms: turbulent migration of particles from regions of high turbulence to regions of low turbulence; the action of mass forces; and the deviation of particle trajectories from stream lines of the carrier medium due to their curvature and the nonstationary character of the flow. Fig. 5.1 shows how the range of applicability of the above-considered models of rarefied disperse media depends on the value of the Stokes number St. The analysis of these calculation models convinces us that for rarefied disperse media (j 1), it makes sense to use the Eulerian approach at St < 1, whereas the Lagrangian approach is more practical at St > 1. However, depending on the effectiveness and

Fig. 5.1 Regions of applicability of the models for rarefied disperse media.


complexity of a chosen model, its region of applicability could be widened or narrowed. The character of particle influence on the turbulent flow of the carrier phase is not definitely known. Depending on their inertia and concentration, the presence of particles could make the flow more laminar or more turbulent. The ‘‘feedback’’ effect of particles on turbulence is proportional to the mass concentration of particles g and becomes noticeable at g > 0.1. The mass concentration g is related to the volume concentration j as g ¼ jrp/re, where rp is the particle density and re is the density of the carrier medium. Therefore if condition rp/re 1, smallness of j does not guarantee that mass concentration g will be small. Therefore the often-used assumption that j 1 does not mean that the feedback effect of particles on the turbulence can be ignored. Relatively small particles with St 1 are not completely involved in fluctuational motion, and their velocity differs from that of fluctuations. As a result, this effect, called fluctuational phase slippage, produces additional energy dissipation and decreases the intensity of turbulent fluctuations. For very small particles with St 1, whose relaxation time is comparable to the inner (Kolmogorov) scale of turbulence, energy dissipation increases as a result of their interaction with highfrequency small-scale fluctuations of the carrier phase. As was shown in [12], this effect lowers the hydrodynamic resistance when a small admixture of impurity is injected into a turbulent flow. As the Stokes number St gets larger, additional dissipation caused by the fluctuational phase slippage attenuates and at St 1 becomes inessential. Some possible mechanisms of turbulence generation caused by the feedback influence of the disperse phase are: additional gradient production of turbulent energy by the averaged motion; formation of a nonstationary vortex wake behind large particles as they are bypassed by the flow; diffusional transport of particles as a result of non-uniform distribution of the disperse phase in space; generation of disturbances due to particle collisions. As we increase particle inertia, the laminarization effect of particles gets replaced by turbulization effect. To model a two-phase turbulent flow while taking into account the feedback effect of particles on the flow, we start from the equations for turbulent energy and its rate of dissipation, and bring in additional terms that describe the effect of the disperse phase; these terms have the form of integrals in the phase space of the PDF. The character of particle interactions with the surface depends on properties of the disperse phase (solid particles or droplets) and the surface (rigid wall or liquid film). Interactions between solid particles and the wall are described by heat and momentum recovery (or retention) coefficients and the rebound angle; these quantities characterize elasticity or inelasticity of the impact. They depend on velocity, angle of impact, roughness of the surface, and relative hardnesses of the touching surfaces. When droplets interact with a dry surface, we may observe droplet spreading or quasi-elastic droplet rebound, depending on the droplet velocity, angle of incidence, and wetting conditions. When a solid particle or a droplet collides with a liquid film, we may observe a splash accompanied by formation of secondary droplets. Besides,

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small droplets can get stripped away from the wave crest (dynamical ablation) due to the interfacial friction at the film surface. Deposition of particles on surfaces can be induced by different mechanisms: particle inertia; Brownian and turbulent diffusion; turbulent migration resulting from particle interactions with turbulent eddies of the carrier fluid; thermo- and electrophoresis; external mass forces; transverse force caused by the shear of the carrier fluid’s average velocity (the Saffman force); particle rotation (the Magnus force), and so forth. Besides, the presence of a liquid film on the wall might lead to evaporation or condensation at the film surface, initiating transverse flow of gas (the so-called Stefan flux), which, in its turn, will affect the deposition of particles on the wall. When considering particle interactions with the surface boundary of a two-phase flow, we formulate boundary conditions for the equations of motion and heat transfer for the disperse phase. As we use Eulerian continuous simulation to determine disperse phase’s parameters in the region adjacent to the wall, we lose information about the details of particles’ interaction with the surface because of summation of momenta, energies, and other parameters of incident and reflected particle fluxes. In the Eulerian paradigm, we determine the boundary conditions by finding the PDF of velocity and temperature near the wall from the corresponding kinetic equation (an analogy with the rarefied gas theory comes to mind). A more sophisticated treatment of this problem may be carried out on the basis of Lagrangian trajectory approach. The boundary conditions thus obtained describe interaction of particles with the surface in terms of the reflection coefficient (which is equal to the probability of rebound in a flow of particles colliding with the wall) and the velocity and temperature recovery coefficients. It follows from the boundary conditions that, due to the dynamic and thermal inertia, velocity and temperature ‘‘slipping’’ may take place at the wall surface (again, this phenomenon is akin to similar phenomena in the rarefied gas theory). When simulating particle motion in rarefied disperse gaseous media, that is, in media with low volume concentration of the disperse phase, the main attention is focused on particle interactions with turbulent eddies of the carrier phase, as the role of interparticle interactions is of little importance. As we increase concentration and particle size, interparticle interactions begin to make larger and larger contribution to the transport of momentum and energy in the disperse phase. Chaotic motion of particles caused by their interactions has come to be known as pseudoturbulent motion; the purpose of this term is to distinguish this motion from simple ‘‘turbulent motion’’ caused by the particles’ involvement into the turbulent flow of the carrier medium. Pseudoturbulent motion may arise from hydrodynamic interaction between particles (that is, through momentum and energy exchange with random velocity and pressure fields of the surrounding medium) as well as from direct interactions via collisions. Pseudoturbulent motion resulting from hydrodynamic (collisionless) interactions is anisotropic. Collisional interactions, on the other hand, result in an isotropic distribution similar to the Maxwellian distribution of velocity fluctuations. The role of momentum and energy exchange between particles via collisions becomes more important as we increase the size and concentration of particles. In concentrated


disperse media, collisions become a major factor defining statistical properties of the system. A theoretical investigation of this problem may be carried out in the same manner as in the kinetic theory of dense gases. The effective relaxation time teff describing particle interactions with the surrounding medium and with each other is found as te f

f

¼

tv tc ; tv þ tc

where tv is the dynamic relaxation time of particles and tc is the effective time between particle collisions, estimated as tc ¼

d p ð1j=j Þ1=3 1=2

ep j

;

where dp is the particle diameter, j* is the limiting volume concentration, and ep is the turbulent energy of the disperse phase. In a rarified disperse medium, tc tv and teff tv, while in a concentrated disperse medium, tc tv and teff tc. With the growth of volume concentration of particles, the Lagrangian approach runs into difficulties as one is forced to keep track of ever-larger number of particle trajectories, and the Eulerian approach becomes preferable. The boundary separating the regions of applicability of these two approaches can be estimated from the equality teff/T(L) 1. For teff/T(L) < 1, it is makes more sense to use the Eulerian approach, whereas for teff/T(L) > 1, the Lagrangian approach is preferable. Stokes number St (it tells us which of the two approaches is preferable) is shown qualitatively in Fig. 5.2 as a function of volume concentration of the disperse phase. Up till now we were dealing with monodisperse media, that is, media containing particles of the same size. The need to consider a polydisperse medium arises when we study combustion, phase transitions, breakup, coagulation, and other processes leading to variation of particle size spectrum. This spectrum is characterized by particle distribution over sizes n(v, t, X) such that n(v, t, X)dv is the number of particles whose volumes belong to the interval (v, v, dv) (here the size of a particle is

Fig. 5.2 Regions of applicability of the Eulerian approach and the Lagrangian approach.

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represented by its volume v). All currently existing methods fall into one of the two groups. Methods belonging to the first group aim to determine the distribution function of particles over sizes by calculating the time evolution of the entire system of particles. The second group consists of methods that involve partitioning of the whole particle spectrum into discrete fractions, with further consideration of the dynamics of each fraction. In the first group, we should single out the methods based on direct solution of the kinetic equations for the size distribution of particles and for the PDF moments. The applicability of such methods is limited to the case of small particles whose velocities and temperatures only slightly differ from those of the carrier medium. The methods from the second group are more universal and allow for a multi-speed and multi-temperature description of the disperse phase, in other words, they are capable of taking into account the differences in velocity and temperature between different fractions of the spectrum. Physical processes involving a change in the size spectrum of particles can be either continuous (combustion, condensation, evaporation) or discrete (breakup, coagulation). These processes are described, respectively, by differential and integral operators working on the mass coordinate in the phase space of the kinetic equation for the PDF. The birth of new particles in the flow volume may take place due to nucleation in the course of spontaneous homogeneous condensation of vapor in supercooled flows. A variation of the particle spectrum may also occur as a result of interaction of particles (or the flow itself) with boundary surfaces. The examples include deposition, breakup, and secondary ablation. These processes should be taken into account by writing the appropriate boundary conditions or by including source-type terms in the balance-of-mass equation. If we want to control the coagulation process in a turbulent flow, it is necessary to know beforehand the frequency of particle collisions that are caused by their chaotic motion. To calculate this frequency (i.e., the rate of coagulation), we can employ one of the two approaches. The first one focuses on finding the correlation coefficients of relative velocity and its derivatives for a pair of particles in a homogeneous turbulent flow by solving the equation of motion for the particles. The other approach is based on the diffusion model that is similar to the Smoluchowski model for Brownian diffusion. The most effective method, however, is to use the equation for the PDF of relative velocity of a pair of particles.

5.6 Deposition of Particles from a Turbulent Flow

When considering the problem of particle deposition onto a surface, we can model the disperse phase motion by using either Eulerian or Lagrangian approach. In the Eulerian continual approach, the form of equations depends on Stokes number St. In the most interesting case of St 1, the task of finding the concentration of particles boils down to solving some diffusion equation given by a diffusion or inertial diffusion model. The question about the appropriate boundary conditions at the surface remains open, because the Eulerian continual approach does not


provide information about interactions between the particles and the surface. So, in order to determine the boundary conditions, one has to find the PDF of velocity and temperature in the region adjacent to the wall, having solved the kinetic equation for the PDF. Lagrangian trajectory approach enables us to formulate the boundary conditions for the equations of motion and heat-and-mass-transfer equations written for the disperse phase. As an example, let us try to derive the kinetic equation for the PDF of particles in a turbulent flow, the diffusion equation and the boundary conditions that would allow us to handle the problem of particle deposition on the wall for the isothermal case. Consider the equation of motion for an individual solid spherical particle: dV p uðX p ðtÞ; tÞV p ðtÞ ¼ þ FðX p ðtÞ; tÞ þ f ðX p ðtÞ; tÞ; dt tv dX p ¼ V p; dt

tv ¼

d2p r p 18ne re

;

ð5:108Þ

where Xp(t) and Vp(t) are the coordinate and velocity of the particle at the instant t, u(Xt) – Eulerian velocity of the flow, dp – particle diameter, rp and re – densities of the particle and the carrier fluid. The first term on the right-hand side describes the force of viscous interaction between phases in Stoksian approximation. The second term is the external force, for example, gravity. The third term is the random force acting on a unit particle mass in the course of Brownian motion. This force must obey the condition (1.150), in other words, the random Brownian force f must be delta-correlated. The equation of motion (5.108) is written in the approximation of large difference between particle and fluid densities and smallness of the particle size so that additional forces acting on the particle could be ignored, specifically, the force caused by the pressure gradient, the force due to the virtual mass, and the Basset force, which arises when a nonstationary flow bypasses the particle. The Saffman and Magnus forces are also neglected. Additional information about forces acting on the particle can be found in [13–16]. Later on we shall need the solution of Eq. (5.108), which, when taken together with the initial conditions X(0) and V(0), can be represented in the integral form t V p ðtÞ ¼ V p ð0Þexp tv ðt uðX p ðt1 Þ; t1 Þ tt1 þFðX p ðt1 Þ; t1 Þ þ f ðX p ðt1 Þ; t1 Þ exp dt1 ; þ tv tv 0

ðt X p ðtÞ ¼ X p ð0Þ þ X p ðt1 Þdðt1 Þ: 0

ð5:109Þ

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Eq. (5.108) is the Langevin equation that depends on two random uncorrelated quantities: velocity u and force f. For simplicity’s sake, assume the mass concentration of particles to be small (g 1). Then the influence of particles on the characteristics of the carrier flow, as well as interparticle collisions can be neglected. Introduce the probability density distribution of particles p(X, V, t) over coordinatesX and velocities V in the (X, V) phase space (see Eq. (1.145)): pðX ; V ; tÞ ¼ hdðX X p ðtÞÞdðV V p ðtÞÞi:

ð5:110Þ

Here averaging is performed over all possible instances of the turbulent flow and the random force f. From Eq. (5.110), there follows hdðX X p ðtÞÞdðV V p ðtÞÞV p ðtÞi ¼ V p ðX ; V ; tÞ:

ð5:111Þ

Differentiating p(X, V, t) with respect to t and making use of Eq. (5.108), we write:

dX pk qp q dðX X p ðtÞÞdðV V p ðtÞÞ ¼ dt qt qXk

dV pk q dðX X p ðtÞÞdðV V p ðtÞÞ dt qVk

q ¼ dðX X p ðtÞÞdðV V p ðtÞÞV pk qXk

dV pk uk V pk q dðX X p ðtÞÞdðV V p ðtÞÞ þFk þ fk : dt tv qVk ð5:112Þ Velocity of the carrier phase is a sum of the average value Uk(X, t) and the fluctuational component u0k ðX ; tÞ: uk ðX ; tÞ ¼ Uk ðX ; tÞ þ u0k ðX ; tÞ;

ð5:113Þ

where, as usual, we take huk(X,t)i ¼ Uk(X,t) and hu0k ðX ; tÞi ¼ 0. Then from the relation (5.112) plus Eq. (5.111) and Eq. (5.113), one gets an equation for the probability density distribution (the Liouville equation): 0 puk qp qp q Uk V pk q þ þ Fk p þ þ h p fk i þ Vk tv tv qt qXk qVk qVk ¼ 0:

ð5:114Þ


Here h pu0k i ¼ hdðX X p ðtÞÞdðV V p ðtÞÞu0k i; h p fk i ¼ hdðX X p ðtÞÞdðV V p ðtÞÞ fk i: Eq. (5.114) is unclosed because it contains unknown correlations h pu0k i and hpfki. To derive these correlations, we assume the random fields u0k and fk to be Gaussian and use the relations (5.110) for p(X, V, t) and the Furutsu–Donsker–Novikov formula (see Eq. (1.201)):

ð dRðZðX ÞÞ hZðX ÞRðZÞi ¼ hZðX ÞZðX 1 Þi dX 1 ; dZðX 1 Þ

ð5:115Þ

where Z(X) is a random process in X-space, R(Z) – a functional dependent on the random process, and dR/dZ – the functional derivative. Let us substitute into Eq. (5.115) successively ZðX Þ ¼ u0j ðX ; tÞ, ZðX Þ ¼ f j0 ðX ; tÞ, and R(Z) ¼ p(X, V, t). Then h pu0i i ¼

ðð ðð

h p fi i ¼

d pðX ; V ; tÞ dX 1 dt1 ; du0k ðX 1 ; t1 Þ

ð5:116Þ

d pðX ; V ; tÞ dX 1 dt1 ; h fi ðX ; tÞ fk ðX 1 ; t1 Þi d fk ðX 1 ; t1 Þ

ð5:117Þ

hu0i ðX ; tÞu0k ðX 1 ; t1 Þi

where because of Eq. (5.109), the functional derivative in the integrand is equal to

dX p j ðtÞ d pðX ; V ; tÞ q dðX X ðtÞÞdðV V ðtÞÞ ¼ p p du0k ðX 1 ; t1 Þ du0k ðX 1 ; t1 Þ qX j

dV p j ðtÞ q : dðX X p ðtÞÞdðV V p ðtÞÞ 0 duk ðX 1 ; t1 Þ qV j Let us apply to Eq. (5.109) the operator of functional differentiation, using the equality dui(X, t)/duj(X1, t1) ¼ dijd(X X1), d(t t1) (see Eq. (1.186)); the condition of causality, which states that the solution of the stochastic equation (5.108) at the moment t is defined by the behavior of the random field f(t) on the time interval (0, t) and does not depend on the values of f(t) at t > t; and the absence of dependence of Vpi(0) and Xpi(0) on u0i . The result is dV pi ðtÞ 1 tt1 ; ¼ eb d i j dðX 1 X p ðt1 ÞÞ; b ¼ tv du0j ðX 1 ; t1 Þ tv ðt dX pi ðtÞ dV pi ðt2 Þ ¼ dt2 ¼ ð1eb Þdi j dðX 1 X p ðt1 ÞÞ: 0 du j ðX 1 ; t1 Þ du0j ðX 1 ; t1 Þ t1

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Now, putting the derived relations into Eq. (5.116), we get h pu0i i ¼ tv ghu0i u0k i

qp qp f hu0i u0k i ; qXk qVk

ð5:118Þ

where ghu0i u0k i

ð¥ 1 ¼ hu0i ðX ; tÞu0k ðX p ðt1 Þ; t1 Þið1eb Þdt1 ; tv 0

ð¥ 1 0 0 f hui uk i ¼ hu0i ðX ; tÞu0k ðX p ðt1 Þ; t1 Þieb dt1 ; tv

ð5:119Þ

0

hu0i u0j i are second single-point simultaneous moments of velocity fluctuation of the carrier flow, and g and f are the coefficients of the particles’ involvement into fluctuational motion of the carrier phase. Entrainment of particles by the turbulent flow can be characterized by the time T(L) (Lagrangian correlation time) of particle interaction with a macroscopic volume (a ‘‘turbulent mole’’). It is equal by the order of magnitude to the integral time scale of turbulence. Physically, T(L) stands for the length of time between the two given instants that is required for the single-point velocity correlation to disappear. In other words, the velocities measured at t1 and t do not correlate if (|t1 t| > T(L)). Therefore correlation functions at X ¼ Xp can be approximated by the step function hu0i ðX ; tÞu0j ðX p ðt1 Þ; t1 Þi

¼

8 < hu0i u0j i at jt1 tj T ðLÞ ; :0

at jt1 tj > T ðLÞ :

Substituting this relation into Eq. (5.119), one finds the particle involvement coefficients: T ðLÞ T ðLÞ T ðLÞ g¼ 1 þ exp ; f ¼ 1exp : ð5:120Þ tv tv tv Let us now turn to determination of hpfii. The random force f is a Brownian force, therefore its components are delta-correlated in time and, according to Eq. (3.50), obey the condition h fi ðX ; tÞ f j ðX p ðt1 Þ; t1 Þi ¼

1 0 D d i j dðtt1 Þ tv br

where D0br ¼ kJ=6pme a is the coefficient of unhindered Brownian diffusion, q – the absolute temperature of the carrier phase, k – the Boltzmann constant, a – the particle radius, me – the dynamical viscosity coefficient for the carrier phase.


From Eq. (5.117), there follows ðð h p fi i ¼

D0 d pðX ; V ; tÞ qp h fi ðX ; tÞ fk ðX 1 ; t1 Þi : dX 1 dt1 ¼ 2br d ik tv d fk ðX 1 ; t1 Þ qVk ð5:121Þ

Substituting the relations (5.118) and (5.121) into Eq. (5.114), we obtain a closed equation for probability density distribution of particles over coordinates and velocities in a turbulent flow: qp qp q Uk Vk þ þFk p þVk tv qt qXk qVk ¼

ghðu0i u0k Þi

D0 q2 p q2 p f q2 p þ hðu0i u0k Þi þ 2br : tv qVk qVk qXi qXk tv qVi qVk

ð5:122Þ

Plugging hu0i u0k i ¼ 0 into Eq. (5.122), we get the Fokker–Planck equation for Brownian particles in a laminar flow: D0 q2 p qp qp q Uk Vk þ þ Fk p ¼ 2br : þ Vk tv tv qVk qVk qt qXk qVk

ð5:123Þ

In the other limiting case – that of highly inertial particles with Stokes number St ¼ tv/T(L) 1 – one can neglect Brownian diffusion ðD0br ¼ 0Þ and take f 1/St 0 and g 1/(2St2) 0, thereby obtaining the Liouville equation for deterministic particle motion: qp qp q Uk Vk þ þ Fk p ¼ 0: þ Vk tv qt qXk qVk

ð5:124Þ

Eq. (5.122) is a multidimensional partial differential equation whose direct solution presents difficulties. Consider the widely used method of moments that yields equations for statistical characteristics of the disperse phase. Let us write disperse phase’s velocity as a sum of the average value hV i and the fluctuation V 0, V ¼ hV i þ V 0 and introduce the moments ð C¼

pðX ; V ; tÞdV ;

ð hVk i ¼ Vk pðX ; V ; tÞdV ;

hVi0 V 0j i ¼

ð 1 Vi0 V 0j pd V ; C

that have the meaning of concentration, average velocity of particles, and stress tensor components in the disperse phase, respectively. Integration of Eq. (5.122) over

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the entire velocity space V yields a balance-of-mass equation for the disperse phase: qC q ðChVk iÞ ¼ 0: þ qt qXk

ð5:125Þ

Multiplying (5.122) by Vk and integrating with respect to V, we then get an equation for the average velocity of the disperse phase: ~ ik hVi i qðlnCÞ qhVi0 Vk0 i Ui hVi i qhVi i qhVi i D ¼ þ þ Fi ; þ hVk i qXk qt qXk tv tv qXk ð5:126Þ ~ ik is the diffusion tensor equal to where D ~ ik ¼ tv hV 0 V 0 i þ ghV 0 V 0 i: D i k i k

ð5:127Þ

Terms on the right-hand side of Eq. (5.126) have the following meaning. The first term describes Brownian motion and generation of stresses in the disperse phase as a result of particles’ involvement into the turbulent motion of the carrier phase. The second term characterizes the action of the friction force exerted on the particles by the continuous phase. The third term stands for the external force, and the last term – for the thermodynamic diffusional force. Disregarding convective terms in Eq. (5.126), we get the following relation for the average velocity of the disperse phase: hVi i ¼ Ui þ tv Fi tv

qhVi0 Vk0 i ~ qðlnCÞ Dik : qXk qXk

ð5:128Þ

We get the equation for second moments hVi0 V 0j i by multiplying both sides of Eq. (5.122) by Vi0 V 0j and integrating it with respect to V. The result is qhVi0 V 0j i qt

þhVk i

qhVi0 V 0j i qXk

þðhVi0 Vk0 i þ ghu0i u0k i

qhVi i 1 q 2 þ ChVi0 V 0j Vk0 i ¼ qXk C qXk tv

qhV 0j i qXk

f hu0i u0j i þ

þðhVi0 Vk0 i þ ghu0i u0k iÞ Dbr di j hVi0 V 0j i : tv ð5:129Þ

The system of equations (5.125), (5.126), and (5.129) is unclosed because Eq. (5.129) contains a new unknown – the third moment hVi0 V 0j Vk0 i. The process of deriving higher-order moments can be continued, but new equations will also contain new unknowns. As a result, we get an infinite system of equations similar to the Friedman– Keller chain in the turbulence theory for a single-phase medium. Although the derived equations cannot be solved in the ordinary sense of the word, they still provide important information about statistical characteristics of the disperse phase.


In the special case when the random process is stationary and the turbulent flow is homogeneous, it follows from Eq. (5.129) that hVi0 V 0j i ¼ f hu0i u0j i þ

Dbr di j : tv

ð5:130Þ

We see that very small particles with St ¼ tv/T(L) ! 0 (and consequently, f ! 1 according to Eq. (5.120) are completely involved into the turbulent flow of the carrier phase, whereas relatively large particles with St ! 1 and f ! 0 do not get involved in fluctuational motion. After we substitute the relations (5.128) and (5.130) into the balance-of-mass equation (5.125), the latter turns into a diffusion equation: qC q q ~ ik qC þ C q ðqDik þ Dbr d ik Þ ; ððUk þ tv Fk ÞCÞ ¼ þ D qt qXk qXk qXi qXi ð5:131Þ ~ ik ¼ Dik þ Dbr dik , Dik is the coefficient of turbulent diffusion, and q ¼ Stf where D is the migration coefficient. Compare the diffusion equation (5.131) for particles with the turbulent diffusion equation (5.93) for a passive impurity. The first difference is the presence of an external force (gravity, for instance) that gives rise to an additional convective transport. Secondly, in addition to diffusional transport, there appears migrational transport (second term in parenthesis on the right-hand side) caused mainly by the inhomogeneity of the turbulent fluctuation field of the carrier flow. Migration coefficient q is proportional to Stokes number and increases as the particle size goes from the size of passive particles to the size of inertial ones. So, the deposition of small particles with tv/T(L) 1 is occurs mostly via turbulent and Brownian diffusion, while the deposition of relatively large particles with tv/T(L) 1 occurs (to a greater extent) via turbulent migration. To solve the diffusion equation (5.131), one has to formulate the boundary conditions at the surface that is partially or completely absorbing. To this end, let us examine the stationary solution of Eq. (5.122) in a thin kinetic layer near the wall. Suppose that in this layer, the only important terms are those associated with projections onto the normal direction to the wall (the Y-axis). Let the asymptotic relation (5.43), Dt ¼ T ðLÞ hðu0y Þ2 i, stand for the coefficient of turbulent diffusion. Then Eq. (5.122) reduces to ðqDt þ Dbr Þ q2 p qðVy pÞ þ tv qVy2 qVy qp qp q2 p ¼ tv Vy ð1qÞDt : þðUy þ tv Fy Þ qY qVy qYqVy

ð5:132Þ

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The solution will be obtained for the case of small deviations of p from the equilibrium value peq,

tv peq ¼ C 2pðqDt þ Dbr Þ

1=2 exp

tv Vy2 2ðqDt þ Dbr Þ

! ;

taking the right-hand side of this equation as the perturbing factor. Let us take p ¼ peq þ p0 , where p0 is a small perturbation of the distribution. Substituting this representation into Eq. (5.132) and ignoring small quantities of higher orders, one obtains an equation for p0 : ðqDt þ Dbr Þ q2 p0 qðVy p0 Þ þ qVy2 tv qVy ¼ tv Vy

q peq q peq q2 peq ð1qÞDt : þðUy þ tv Fy Þ qY qVy qYqVy

! 1=2 tv Vy2 tv Vy tv p¼C 1 exp 2pðqDt þ Dbr Þ 2ðqDt þ Dbr Þ qDt þ Dbr " # ðDt þ Dbr Þtv Vy2 dðlnCÞ 1 þ ð2qDt Dt þ Dbr Þ þ ðDt þ Dbr Þ 3ðqDt þ Dbr Þ dY 2 ) d lnðqDt þ Dbr ÞVy2 ðUy þ tv Fy Þ : dY

The distribution above enables us to find the incident Jf and reflected Jr particle fluxes: ð0 Jf ¼

Vy pdVy ¼ C

¥

þ

qDt þ Dbr 1=2 C ðUy þ tv Fy Þ 2ptv 2

ðDt þ Dbr Þ dC C dðqDt þ Dbr Þ þ ; 2 dY 2 dY

ð¥ qDt þ Dbr 1=2 C þðUy þ tu Fy Þ Jr ¼ Vy pdVy ¼ C 2ptu 2 0

ð5:133Þ

ð5:134Þ

ðDt þ Dbr Þ dC C dðqDt þ Dbr Þ : 2 dY 2 dY

The quantities on the right-hand side of relations (5.133) and (5.134) correspond to their values at the wall. A surface’s ability to reflect and absorb particles is characterized by the reflection factor w equal to the probability for the particle to get detached from


the wall, or by the absorption factor 1 w equal to the probability for the particle to adhere to the wall. The reflection coefficient is equal to the ratio between the reflected and incident particle fluxes, c¼

Jr : Jf

ð5:135Þ

Substitution of relations (5.133) and (5.134) for fluxes into Eq. (5.135) gives the following boundary condition connecting particle concentration at the wall Cw with the flux of particles deposited at the wall Jw ¼ Jf Jr: Cw ¼

1=2 ð1 þ cÞ ptv Jw ; ð1cÞ 2ðqDt þ Dbr Þ

ð5:136Þ

where Jw ¼ ðDt þ Dbr Þ

dC dðqDt þ Dbr Þ C CðUy þ tu Fy Þ: dY dY

It follows that for a completely reflecting surface, w ¼ 1 and Jw ¼ 0 whereas for a completely absorbing surface, w ¼ 0 and Cw 6¼ 0, contrary to a widespread opinion. It should be noted that when solving concrete problems, the boundary condition (5.136) is specified at some distance from the wall, outside the viscous sublayer, rather than at the wall itself. Sometimes this distance is taken to be equal to the particle radius. When considering the final stage of particle’s approach to the wall and the chance that it will be captured by the wall, it is necessary to take into account the interaction force between the particle and the surface at small values of clearance between the particle and the wall. Paper [17] performs numerical solution of the problem of plane-parallel turbulent flow of a two-phase disperse medium in the near-wall region and suggests an approximate expression for particle flux at the completely absorbing wall, which takes into account Brownian and turbulent diffusion, turbulent migration, convection, and the external force ð0:115=Scbr þ 2:5104 t2;5 þ Þu 3=4

Jw ¼ j

ð1 þ 103 t2;5 þ Þmax½0:61; minð1:320:27 lntþ ; 1Þ

! Uy tu Fy ;

where t+ ¼ tvu*2/ne; u* is the dynamic velocity; Scbr ¼ ne/Dbr is the Schmidt number for Brownian diffusion; Uy and Fy are the normal (i.e., perpendicular to the wall) components of velocity and external force-driven acceleration of the carrier phase (gas); j is the volume concentration of particles. The formula is valid at t+ < 100 for relatively small particles inside the logarithmic layer 30 < y+ < 100, where y+ ¼ yu*/ne. As an example, consider particle deposition from a stationary hydrodynamic developed turbulent flow in a plane-parallel or cylindrical channel. In the boundary

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layer approximation and in the absence of external forces, the diffusion equation (5.131) takes the form d a d dC d ðr Uy CÞ ¼ r a ðDt þ Dbr Þ þ C ðqDt þ Dbr Þ ; dX dr dr dr

ð5:137Þ

where X and r ¼ 1 Y are the longitudinal and transverse (again, with respect to the wall) coordinates; a ¼ 0 and 1 indicate plane-parallel and cylindrical channels, respectively. Integration of Eq. (5.137) over the channel cross section leads to the equation dðUm Cm Þ 2a ¼ Jw ; rw dX rðw dr Um ¼ 2a r a UX 1þa ; rw 0

rðw Cm ¼ 2a r a CUX 0

dr ; Um rw1þa

ð5:138Þ

where Um and Cm are the average mass velocity and particle concentration, rw is the radius of the channel. In the region of hydrodynamic stabilized flow one can take qðUX CÞ dðUm Cm Þ ¼ : dX dX Then in view of Eq. (5.138), Eq. (5.137) reduces to ðDt þ Dbr Þ

dC d r þ C ðqDt þ Dbr Þ ¼ Jw : dr dr rw

ð5:139Þ

Suppose the coefficient of turbulent diffusion is equal to the coefficient of turbulent viscosity of the carrier fluid Dt ¼ nt. The latter is taken in the form h

ky i nt k 1 þ ¼ yþ ð2y0 Þ þ ð1y0 Þ2 1exp 2 ; ne 3 A 2 which tends to the Reichardt formula far away from the wall and to the van Driest– Deissler formula near the wall. Here y0 ¼ Y/rw, y+ ¼ Yu*/ne, k ¼ 0.4 and A ¼ 26. When solving Eq. (5.139), the boundary condition (5.136) is taken at the distance from the wall equal to the particle radius a, and T ðLÞ ¼ 200ne =u2 is taken for the Lagrangian correlation time [18]. Look at Fig. 5.3a and fig. 5.3b to compare theoretical rates of deposition of suspended particles Vw ¼ Jw/Cw or J+ ¼ Vm/u* with experimental data. Curves 1 and 2 in Fig. 5.3a correspond to the experimental data [19] for the Reynolds numbers Re ¼ 2rwUm/ne ¼ 2.9105 and 5105 and curves 1–3 in Fig. 5.3b – to the experimental data [20] for the flow velocities Um ¼ 7.6; 17.6 and 26.6 m/s.


Fig. 5.3 Dependence of particle deposition rate on particle diameter.

The dependence of the rate of deposition on particle diameter has a minimum. The initial drop of the deposition rate is associated with the decrease of Brownian diffusion coefficient that accompanies the increase of the size of particles. In the region where particle size is smaller then 1 mm, Brownian diffusion becomes the predominant mechanism of particle deposition. Increasing the size of particles, we observe a significant increase of the velocity of turbulent migration due to the nonuniform distribution of turbulent fluctuation intensity over the channel cross section, which leads to further growth of the deposition rate. For particle sizes on the order of 100 mm, the dimensional deposition rate reaches its maximum value: J+ 0.2. The rate of deposition Vw increases with the flow velocity Um as a consequence of more intense particle deposition under the action of turbulent diffusion as well as turbulent migration. Fig. 5.4 demonstrates the dependence of the dimensional deposition rate Vw on the Reynolds number in a pipe of radius 2.5 mm for particles of diameters 0.01; 0.27; 2; and 8 mm (curves 1–5, respectively). The same figure also shows experimental

Fig. 5.4 Deposition velocity vs. Re.

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304

Fig. 5.5 Comparison with experiments.

points for particles of d ¼ 0.18 mm [21]. One can see that the rate of deposition of submicron particles practically does not depend on Re because the rate of deposition Vw of Brownian particles is proportional to the dynamic velocity u*. The correlation between numerical results and experimental data is shown in Fig. 5.5 as a dependence of J+ on the dimensional relaxation time of the particle tþ ¼ tþ u2 =ne . Curves 1, 2 correspond to the pipe radius rw ¼ 0.015 m and the flow velocities Um ¼ 10 and 30 m/s; curve 3 – to rw ¼ 0.3 m, Um ¼ 30 m/s; curve 4 – to rw ¼ 0.025 m, Um ¼ 30 m/s; the experimental data is taken from [16]. One can see that experimental results are in good agreement with theoretical ones only for the inertial particles (+ > 10), whereas for small (inertialess) particles, the particle size becomes essential. This is easy to understand once we realize that the intensity of particle deposition is determined by processes of turbulent nature, that is, turbulent diffusion and turbulent migration. For small (Brownian) particles with + < 1, the rate of deposition is determined not only by the intensity of turbulent transport, but also by Brownian diffusion, and thus essentially depends on the particle size.

5.7 Interaction of Particles in a Turbulent Flow

Particle interactions in a turbulent flow, as well as in a laminar flow, may be of two kinds: implicit interaction, that is, the influence exerted on the motion of the particle under consideration (test particle) by its neighbors through perturbation of hydrodynamic and concentration parameters, and explicit interactions, that is, direct particle collisions. The type of interaction depends on two parameters: volume concentration of suspended particles j and Stokes number St responsible for the inertia of particles. When we model particle motion in a rarefied disperse medium (j 1) containing particles with low inertia (St < 1), interparticle interactions are insignificant and

5.7 Interaction of Particles in a Turbulent Flow

result in a very small perturbation of hydrodynamic parameters of the carrier medium, which in its turn, exerts but a minuscule effect on the motion of particles. Moreover, the probability of particle collisions is low. We conclude that interparticle interactions can be neglected. In this statement of the problem, the main emphasis is on particle interactions with turbulent fluctuations of the carrier medium. When we increase volume concentration j and Stokes number St (i.e., particle size), both types of interparticle interactions begin to make a larger contribution to the transport of momentum and energy in the disperse phase. Chaotic motion of particles that arises from interparticle interactions is called pseudoturbulent and should be distinguished from particle motion arising from turbulent fluctuations of the carrier phase. The motion of particles in a turbulent flow of a concentrated disperse medium is affected by the medium itself and by interparticle interactions; the mechanism of these interactions can be collisionless hydrodynamic (implicit interaction) or collisional (explicit interaction). These mechanisms affect the motion of particles in different ways: particle motion caused by collisionless interactions is anisotropic, whereas collisional interactions result in isotropic motion whose distribution of velocity fluctuations in the disperse phase is close to Maxwellian. With an increase of size and concentration of particles, the role of momentum and energy exchange between particles is growing in importance as compared to that of hydrodynamic interactions. In concentrated disperse media, interparticle interactions assume a leading role in the formation of statistical properties. Theoretical investigation of this problem is performed in the same way as in the kinetic theory of dense gases: one writes the Boltzmann equation and solves it with the Enskog method. Various kinetic models whose aim is to calculate momentum transport in highly concentrated disperse media have been suggested in a number of works. Some of them that take into account particle interactions with turbulent fluctuations as well as with other particles. The motion of a heavy spherical particle in a turbulent flow is described by the Langevin equation (5.101), into which one should insert additional terms in order to account for both types of particle interactions: dV p uV p þ F þ w þ W; ¼ dt tv

dX p ¼ V p; dt

ð5:140Þ

where u and Vp are, respectively, the velocities of the carrier medium and the particle; Xp – particle coordinates; F – acceleration caused by the external force; tv – the particle’s dynamic relaxation time, which takes into account the inertia forces (they manifest themselves in deviations from the Stokes law) and the hindered character of particle motion, that is, dependence on j; w and W – terms that account for the hydrodynamic collisionless and collisional interactions between particles (the former term represents a continuous, and the latter – a discrete random process). In contrast to Eq. (5.101), Brownian motion is not taken into consideration in Eq. (5.140).

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Lagrangian approach can be used to model the behavior of the disperse phase while properly accounting for interparticle interactions. We start from Eq. (5.140) for a single particle and then proceed to the ensemble of a finite number of particles by varying the initial position X(0) and initial velocity V(0) of the particle. We can switch from Lagrangian trajectory description of particle motion (which is based on the Langevin equation (5.140)) to Eulerian description of the particle ensemble in the same manner as we did in Section 5.6. The probability density distribution of particles over coordinates and velocities p(X,V, t) ¼ hd(X Xp)d(V Vp)i will then be described by the equation similar to Eq. (5.114): [9] Uk V pk qp qp q þ þ Fk p þ Vk tv qt qXk qVk 1 q u0k p qhwk pi ¼ þ Ið pÞ; tv qVk qVk

ð5:141Þ

where Uk and u0k are the average and fluctuational components of velocity of the carrier phase, and I(p) is the Boltzmann collision operator written in the Enskog form. Suppose that the random fields of velocity fluctuations of the carrier phase u0 and of hydrodynamic interactions wk are Gaussian. Then, making the additional assumption that the field wk is delta-correlated and using the method of functional differentiation, we can obtain differential equations for the moments (average concentration and average velocity, second moments, and higher-order moments) of velocity fluctuations in the carrier phase. The interested reader will find more details and further verification of the model in [8,9,22].

5.8 Chemical Reactions in a Turbulent Flow

The problem of modeling the turbulent flow of a chemically reacting mixture can be divided into three parts [23–29]: – macromixing that results from turbulent diffusion on the distances on the order of the characteristic linear size of the flow region; – micromixing, or, to use another term, mixing to the molecular level, which results in the formation of ‘‘genuine’’ local concentrations and in the possibility of reactions between formely unmixed reagents; – calculation of average rates of chemical reactions with proper account taken of local fluctuations of reagent concentrations and temperature. Of greatest practical interest is the calculation of average rates of chemical reactions in a turbulent flow, because the composition of reaction products in chemical engineering devices (chemical reactors) corresponds to the value obtained by averaging over the volume in which the reaction is taking place.


Depending on the relation between the characteristic times of turbulent mixing and those of chemical reactions, reactions are classified as slow, fast, or very fast. If only a small fraction of reagents gets consumed in the reaction during the characteristic time of turbulent mixing (i.e., the time it takes for a uniform concentration to be established due to the action of turbulent mixing and molecular diffusion), we call such reaction slow. In other words, for slow reactions, the characteristic time of turbulent mixing is far shorter than the characteristic time of the reaction. This allows us to consider the two processes – turbulent mixing and chemical reaction – independently from one another. First, we solve the problem on turbulent mixing of the given substances; then, once the hydrodynamic and concentration fields have been obtained, we proceed to determine reaction rates. For fast reactions, the characteristic times of turbulent mixing are of the same order as the characteristic times of reactions. Then the local rate of chemical reaction strongly depends on temperature fluctuations and fluctuations of reagent concentration. Therefore the processes of turbulent mixing and chemical transformations should be considered jointly. The modeling of chemical reactions is complicated by the fact that turbulent mixing and chemical reactions affect each other. Therefore turbulent mixing occurs differently than it would in the absence of chemical reactions, because local concentration gradients depend on the peculiarities of chemical reactions. In the limiting case of very fast chemical reactions, the characteristic times of reactions are negligibly small compared to the characteristic time of turbulent mixing, and the reaction can be regarded as instantaneous. Then at each space point, we can expect to find one of the reagents involved in the chemical reaction, or the reaction products, because there is no chance to find reacting components simultaneously at the same point. In this case, the reaction zone changes into a surface of rather complicated topology. To illustrate the points made above, let us perform dimensional analysis of the convective diffusion equation (5.57), adding a source-type term describing the chemical reaction to the right-hand side: qCi þ urCi ¼ Dm DCi þ W i : qt

ð5:142Þ

The second term on the left and the two terms on the right correspond, respectively, to three different processes: convective transport via turbulent motion; transfer via molecular diffusion; production or consumption of i-th component in the course of chemical reaction. Each of them is described by its own characteristic time: tt ¼ L0 =u0 ;

tD ¼ L20 =Dm ;

tw ¼ C0 =jW0 j;

where L0, u0, C0, and |W0| are the characteristic values of the turbulent linear scale, velocity, passive impurity concentration, and reaction rate. Let t be the characteristic time of the process under consideration. If we replace t with dimensionless time t/t and divide all other dimensional quantities in Eq. (5.142) by their characteristic values, this will bring Eq. (5.142) to the dimensionless form. Conserving the old

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symbols, we can write qCi þ Nt u rCi ¼ ND DCi þ NW Wi : qt

ð5:143Þ

Here we have introduced dimensionless parameters Nt ¼ t/tt, ND ¼ t/tD, and NW ¼ t/tW . If any of the above-listed processes dominates, the time corresponding to this process should be taken as t. Let us consider these three cases successively. 1. Turbulent transport (macromixing) dominates. Then t ¼ tt and the dimensionless parameters in Eq. (5.143) are equal to Nt ¼ 1; ND ¼ tt =tD ¼ Dm =ðL0 u0 Þ ¼ Pe1 ; NW ¼ Da1 ¼ tt =tW ¼ jW0 jL0 =u0 C0 ; where Pe is the Peclet number and Da is the Damko?hler number. 2. Molecular diffusion (micromixing) dominates. Then t ¼ tD and the dimensionless parameters in Eq. (5.143) are equal to Nt ¼ tD =tt ¼ ðL0 u0 Þ=Dm ¼ Pe; ND ¼ 1; NW ¼ Da2 ¼ tD =tW ¼ Da1 Pe; 3. Chemical reaction dominates. Then t ¼ tW and the dimensionless parameters in Eq. (5.143) are equal to 1 Nt ¼ tW =tt ¼ Da1 1 ; ND ¼ tW =tt ¼ Da2 ; NW ¼ 1:

For slow chemical reactions, the conditions tW tD and tW tt should hold. Therefore in cases 1 and 2 we have NW 1 and the source term in Eq. (5.143) can be ignored. For very fast chemical reactions, the conditions tW tD and tW tt should be satisfied, and we also have Nt 1 and ND 1. Then the convective transport and molecular diffusion terms in Eq. (5.143) can be neglected. Since ND is a small parameter by high-order derivative, diffusion can be ignored everywhere except narrow regions adjacent to reaction surfaces. In case of fast chemical reactions we have tW tD, tW tt. Therefore no process is given a preference and all terms in Eq. (5.143) should be taken into account, which seriously complicates the solution of the problem. Mixing of components entering into a chemical reaction is a necessary prerequisite for this reaction. Insufficient mixing of components leads to concentration inhomogeneities (local concentration gradients) and thereby to inhomogeneous concentrations of reaction products. The main characteristic of the degree (level) of mixing of i-th reacting component is the variance of concentration distribution hðCi0 Þ2 i ¼ hðCi hCi iÞ2 i:


The degree of mixing depends on size of the volume over which the averaging is carried out. This size may vary from the characteristic macroscale of the volume filled with reacting mixture to the microscale that can be considered as a material point (or fluid particle) within the framework of the hydrodynamic approach. The quantity h(C0 )2i that depends on time and spatial coordinates will appear later on as a characteristic of micromixing. The principal methods for theoretical description of turbulent mixing accompanied by fast chemical reactions are the method of moments and the probabilistic method that employs the equation for probability density distribution (the PDFmethod). When using the method of moments, one usually considers only the equations for the first two moments of a scalar random quantity – concentration (and of temperature as well if the process is not isothermal). These two moments are hCii and h(Ci)2i. The corresponding equations are obtained by the averaging of the convective diffusion equation in the same manner as we did earlier for hydrodynamic equations. Just like in the turbulence problem, the equations thus obtained turn out to be unclosed. Therefore in order to close the system of equations for the moments, one needs additional relations, which, as a rule, have a semiempirical character. The attempt to account for chemical reactions compels us to insert into the diffusion equation an additional source-type term (reaction term), which, generally speaking, has a nonlinear dependence on the concentrations of reagents. Therefore in addition to the problem of correct representation of the terms describing turbulent transport and mixing, one encounters the closure problem for the reaction terms. The PDF method [30–32] is based on the equation for probability density distribution of fluid particles over coordinates, velocities, and concentrations. The knowledge of this distribution enables us to determine all statistical moments, that is, to solve the problem completely. Thus the PDF-method is more logical and informative in comparison with the method of moments. But solution of the problem is seriously complicated by multidimensionality of the distribution (it depends not only on time and spatial coordinates, but also on functions – velocities and concentrations). Before we turn our attention to the method of moments and the PDF method, let us dwell briefly on some basic concepts of chemical kinetics.

5.8.1 Concepts of Chemical Kinetics

The rate of chemical reaction is defined by the amount of substance z produced or consumed in a unit volume per unit time. In the general case, the volume of the system under consideration can vary in the course of the reaction. When the volume is constant, the reaction rate is given by the quantity dCz/dt, where Cz is the concentration of substance z measured in mol/m3. The plus sign corresponds to production and minus – to consumption of the matter. Therefore in chemical kinetics, the reaction rate is always positive. But when conservation equations are written in the general form (see Eq. (5.142)), the source term usually

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appears as W¼

dCz ; dt

ð5:144Þ

so that production or consumption of matter in the course of the chemical reaction is indicated by the sign of W. If a chemical reaction with n reagents A, B, . . ., E, F, . . . is taking place in a singlephase system, and A, B, . . . are the reacting components, while E, F, . . . are the products of reaction, the kinetic scheme of such a reaction is written as nA A þ nB B þ ! nE E þ nF F ;

ð5:145Þ

where nA, . . ., nF are the stoichiometric coefficients; they are negative for ‘‘reactants’’ (components that are being consumed) and positive for ‘‘products’’ of reaction (it is worth noting, however, that all coefficients must change sign if the reaction is viewed as going ‘‘from the right to the left’’). For example, the stoichiometric coefficients for the reaction 2NH3 !N2 þ 3H2 are nNH3 ¼ 2; nN2 ¼ 1; nH2 ¼ 3. The sum of stoichiometric coefficients on the left-hand side of Eq. (5.145) defines the so-called molecularity, which is defined as the number of particles involved in a single reaction step. It follows from the law of definite proportions that the increase (decrease) of mass mi of i-th component produced (consumed) in the reaction is proportional to its molecular weight Mi and stoichiometric coefficient ni for the given reaction. Therefore, denoting through Ni0 the number of moles of i-th component at the initial moment and through and Ni – at the current moment, and keeping in mind that mi ¼ NiMi, we get Ni Ni0 ¼ ni x;

ði ¼ A; B; ; E; F; Þ:

ð5:146Þ

where x(t) is the degree of reaction completion. At the initial moment, x ¼ 0. The value x ¼ 1 corresponds to the moment when nA, nB, . . . moles of components A, B, . . . are transformed into nE, nB, . . . moles of components E, F, . . . . If the system has made a transition from the state x ¼ 0 to the state x ¼ 1, we say that one reaction equivalent has occurred. From the equation of mass conservation there follows the stoichiometric equation X

ni Mi ¼ 0:

ð5:147Þ

i

Differentiating Eq. (5.146) with respect to t, we get dNA dNB dNF ¼ ¼ ¼ ¼ dx: nA nA nA

ð5:148Þ


The reaction rate is equal to W¼

dx ; dt

ð5:149Þ

and we see from Eq. (5.148) that this rate can be written as W¼

1 dNi ni dt

or, in terms of i-th component concentration Ci ¼ Ni/V, where V is the volume occupied by the reacting mixture, as W¼

dðCi VÞ : ni dt

If the reaction is not accompanied by a change of volume, the last relation gives us 1 dCA 1 dCB 1 dCE 1 dCF ¼ ¼ ¼ ¼ ¼ : nA dt nB dt nE dt nF dt

ð5:150Þ

The dependence of reaction rate on reagent concentration is called the kinetic equation of reaction. For one-stage reactions of type (5.145) taking place in a homogeneous system, this dependence is given by the law of mass action: dCA ¼ kðCA ÞnA ðCB ÞnB . . .: dt

ð5:151Þ

Constant k is known as reaction rate constant and has the meaning of specific reaction rate, that is, reaction rate for a unit concentration of reagents. Its dependence on temperature is given by the Arrhenius equation: EA ; k ¼ A exp RJ

ð5:152Þ

where EA is the activation energy, that is, the minimum energy a molecule must possess in order to enter into the reaction, A is a reaction constant, and R is the gas constant. X Exponent ni in the formula (5.151) is called the order of reaction, and n ¼ ni – the i total, or kinetic order of reaction. Kinetic order for single-stage reaction coincides with its molecularity. Among single-stage reactions, first- and second-order reactions are the most common. For multi-stage reactions, the order of reaction could be fractional. The simplest example of reaction requiring a prior mixing of components is a second order reaction of the form A þ B ! products:

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312

Among such reactions, one may single out (based on the reaction rate constant k) slow reactions with k < 103 1/mole s, fast reactions with 103 1/mole s < k < 106 1/ mole s, and very fast reactions with k > 106 1/mole s. It should be noted that dimensionality of the reaction rate constant is defined by the kinetic order of reaction in the formula (5.151). 5.8.2 Method of Moments

Let us consider the convective diffusion equation (5.57), adding a term of the form (5.144) to the right-hand side: qCi þ urCi ¼ Dm DCi þ Wi ; qt

ð5:153Þ

where Ci is i-th component’s concentration, u is the flow velocity, and Dm is a constant molecular diffusion coefficient of i-th component. Suppose the turbulent velocity field u is known from the solution of the corresponding hydrodynamic problem under the assumption that i-th component is a passive impurity. Since the velocity field is random, concentration Ci will also be a random variable. Let us write Ci as a sum of the average value and the fluctuation: Ci ¼ hCi i þ Ci0

ð5:154Þ

and perform the averaging of Eq. (5.153) in the conventional way (see Section 4.6). As a result, we get an equation for the first moment – the average concentration of ith component (see Eq. (4.45)): qhCi i þ huirhCi i ¼ Dm DhCi irhu0 Ci0 i þ hWi i; qt

ð5:155Þ

where hui and u0 are the average and fluctuational components of velocity. The average concentration hCii is of interest to us in the macromixing problem, when molecular diffusion can be ignored as compared to the turbulent diffusion (see Section 5.4). Equation (5.155) is not closed, since it contains the unknown function hu0 Ci0 i. To close the problem, we can use the semiempirical gradient hypothesis (5.88), according to which hu0 Ci0 i ¼ Dt rhCi i, where Dt is the coefficient of turbulent diffusion (see Section 5.4). Then Eq. (5.155) reduces to qhCi i þ huirhCi i ¼ rðDt DhCi iÞ þ hWi i: qt

ð5:156Þ

The problem of determination of hWii will be considered later. Let us now look at the equation for the second-order moment hðCi0 Þ2 i, which functions as the primary characteristic of micromixing. As was shown in Section 5.4,


when studying micromixing, it is essential to include molecular diffusion into the picture, along with small-scale turbulent fluctuations. The equation for hðCi0 Þ2 i is derived by successive multiplication of the equations (5.156) and (5.155) by 2Ci0, followed by term-by-term subtraction and subsequent averaging. The result is qhðCi0 Þ2 i þhuirhðCi0 Þ2 i ¼ rhu0 ðCi0 Þ2 i2hu0 Ci0 irhCi i qt

ð5:157Þ

þ2Dm hCi0 DCi0 i þ 2hCi Wi i: This equation, in its turn, is also unclosed because it contains unknown moments hu0 Ci0 i and hu0 ðCi0 Þ2 i associated with turbulent diffusion, the moment hCi0 DCi0 i characterizing micromixing, and the moments hWii and hCiWii containing reaction terms. For the moments hu0 Ci0 i and hu0 ðCi0 Þ2 i, one can use the semiempirical gradient hypothesis hu0 ðCi0 Þi ¼ Dt rhCi i;

hu0 ðCi0 Þ2 i ¼ D0t hðrCi0 Þ2 i:

ð5:158Þ

Here we have introduced an additional diffusion coefficient D0t , which, generally speaking, should differ from Dt, but is often taken equal to Dt. We should note that the second fluctuation moment hðCi0 Þ2 i is an important characteristic of mixing even in the absence of chemical reactions. As will be shown later, in a turbulent flow with chemical reactions taking place in the mixture, one can use the first and second moments to determine the average rates of chemical reactions hWii on the basis of the corresponding empirical hypotheses. Following Corssin, we represent the term that describes micromixing in the form 2Dm hCi0 DCi0 i ¼ 2Dm hðDCi0 Þ2 i:

ð5:159Þ

The derivation of this relation can be demonstrated for the case of a one-dimensional process, assuming that Ci0 depends on only one coordinate X. Since micromixing is considering in a microscopic region on small scale l, the random field Ci can be taken as homogeneous, that is, @hCii/@X ¼ 0. Then

Ci0

q2 Ci0 qX 2

ðl ðl 1 0 q2 Ci0 1 0 qCi0 l 1 0 qCi0 2 dX ¼ Ci C dX: ¼ Ci qX 2 l l qX 0 l i qX 0

Neglecting the first term, we obtain

Ci0

q2 Ci0 qX 2

* + qCi0 2 :

qX

0

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314

The expression (5.159) immediately follows from this relation. In Section 4.9, we introduced the parameter N ¼ Dm ðrC0 Þ2 (see Eq. (4.127)) called the dissipation of concentration inhomogeneities. One can see now that this parameter is the main characteristic of micromixing. Hence, the rate of micromixing is determined by local gradients of the random concentration field. Therefore in order to determine the rate of micromixing, we must know the correlation functions Bab ¼ hðrCi0 Þa ihðrCi0 Þb i relating the values of concentration fluctuations hCa0 i and hCb0 i at two neighboring points a and b with coordinates Xa and Xb ¼ Xa þ r (see Section 4.8). In the micromixing problem, the scalar field can be assumed locally isotropic. For such a field, the intensity of concentration fluctuations ðCi0 Þ2 obeys Eq. (4.93) that follows from the Corssin equation (4.92): dhðCi0 Þ2 i Dm ¼ 12 2 hðCi0 Þ2 i; dt ðlc Þi

ð5:160Þ

where Dm is the coefficient of molecular diffusion, (lc)i – microscale of scalar concentration field Ci (see Eq. (4.91)). Eq. (5.160) could in fact be considered as the definition of microscale – the characteristic distance on which the intensity of concentration fluctuation decays. Further on, we shall omit the index i. The microscale of a scalar field lc depends on the structure of random field inhomogeneities and for nonstationary process it, generally speaking, changes in the course of mixing. It follows from Eq. (5.160) that micromixing leads to dissipation of the scalar field, that is, to a decrease of h(C0 )2i. It is worth mentioning that this dissipation is caused solely by molecular diffusion. This can be illustrated by a simple example. Consider the Corssin equation (4.92) in the absence of molecular diffusion (Dm ¼ 0). In view of the gradient hypothesis (5.88) for the correlation BLa,b ¼ (uL)aCaCb, the Corssin equation reduces to qBab 1 q qBab ¼ 2 : r 2 Dt qt qr r qr

ð5:161Þ

Since hðC0 Þ2 i ¼ lim Bab , the change of hðC0 Þ2 i with time at Dm ¼ 0 is obtained from r !0 Eq. (5.161) by going to the limit in both sides of the equation. Suppose the mixture was completely segregated at the initial moment so that impurity concentration C assumes the value C0 in the region where the impurity is present and 0 in the region where it is absent. Since we are interested in the behavior of Bab at r ! 0, it makes sense to consider the points that lie in the vicinity of the interface. At small values of r, the quantity Bab can be represented to the first approximation as Bab ðr; 0Þ hðC0 Þ2 iSr;

ð5:162Þ

where S is the specific area of the initial interface between regions with different values of C. Transition to the limit r ! 0 takes us to the scale interval that belongs to


the dissipation region where, as we mentioned in Section 4.9, turbulent fluctuations are by their nature viscous. Thus the dependence of Dt on r is defined by Eq. (5.90), namely, Dt r2. It is easy to verify by substituting this dependence into Eq. (5.161) that for the initial distribution (5.162), the equality @Bab/@t ¼ 0 is valid, and hðC 0 Þ2 i remains constant under the condition that Dm ¼ 0. Another important characteristic of local structure of the scalar field is integral scale Lc (macroscale), equal by the order of magnitude to the average size of inhomogeneities and defined by the relation (see Eq. (4.91)) ð¥ Lc ¼ Ycc dr;

ð5:163Þ

0

where the integrand is the correlation coefficient Ycc ¼

hCa0 Cb0 i ðCi0 Þ2

:

In contrast to the microscale lc, the macroscale Lc varies even in the absence of molecular diffusion as the inhomogeneities get deformed by turbulent fluctuations. Thus the process of micromixing can be pictured as follows. The first stage is deformation of sufficiently large inhomogeneities by turbulent fluctuations. The integral scale of the scalar field L decreases, whereas h(C0 )2i remains practically unchanged because the specific surface area available for the transport of matter via molecular diffusion is small. With decrease of the integral scale, molecular diffusion begins to play a more important role. As a result, the quantity h(C0 )2i decreases, which indicates dissipation of the scalar field. Micromixing is takes place in small regions where turbulence is characterized by local isotropy. It was shown in Section 4.9 that, depending on the predominance of one or another physical mechanism, the spectrum of fluctuations can be divided into several intervals. The same approach is justified for the scalar field (see Section 4.9). Depending on the character of deformation of inhomogeneities by turbulent fluctuations and on the role of molecular diffusion in the mixing process, all scales lc of concentration inhomogeneities are divided into three subranges: inertial-convective with lc l0; viscous-convective with l0 lc lb, and viscous-diffusional with lb lc, where l0 ¼ ðn3e =eÞ1=4 is the inner (Kolmogorov) scale of turbulence, lb – the Batchelor scale (see Eq. (4.133)) lb ¼

nt D2m e

1=4 ;

ð5:164Þ

e – specific dissipation of turbulent energy; ne – kinematic viscosity coefficient of the carrier phase. The Batchelor scale is found from the condition that the characteristic time of diffusion is equal to that of velocity fluctuations, assuming that the equalization of concentration on this scale happens as a result of the molecular diffusion.

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The relation between viscous and diffusional effects is characterized by the Schmidt number Sc ¼ ne/Dm. For liquids, the Schmidt number is on the order of 10–103, and thus there exists a sufficiently large interval of scales lc(l0 > lc > lb) where the flow can be regarded as viscous while the influence of diffusion is insignificant. In this region, convective transport of impurities happens mostly due to the deformation of fluid particles, and the quantity h(C0 )2i can be written as a sum of three components, hðC 0 Þ2 i ¼

3 X hðCi0 Þ2 i

ð5:165Þ

i¼1

corresponding to different scale subranges: inertial-convective, viscous-convective, and viscous-diffusional. Scalar field evolution is described by the Corrsin dynamic equation (4.92), which is unclosed because of the unknown third order moment BLa,b. One usually employs the spectral representation of Eq. (4.92) rather than Eq. (4.92) itself. This results in a transition from scales of concentration inhomogeneities lc to the corresponding 1 wave numbers kc lc . Then micromixing time gets reinterpreted as the time of motion of perturbations along the wave number axis. Spectral representation has proved to be more convenient for the purpose of formulating closure hypotheses for the dynamic equations. In the study of random hydrodynamic or scalar fields, there arises the problem of finding the spectral density of a random field given some assumptions about the term that is responsible for the transport of perturbations along the wave number axis. If turbulent mixing is accompanied by chemical reactions, we have to address the additional problem of estimating the mixing time to compare it with the characteristic time of the chemical reaction, so that we could judge whether the reaction is slow, fast, or very fast. While doing so, we should use the expected form of quasistationary spectral density of the scalar field for different wave number ranges. The relevant forms of the spectral density are as follows: – for the inertial-convective region with kc k0, Ec ðkc Þ ¼ A1ec ðeÞ1=3 kc5=3 ;

ð5:166Þ

– for the viscous-convective region with k0 < kc < kb, Ec ðkc Þ ¼ A2

1=2 ec ne k2 exp A2 c ; e kc kb

ð5:167Þ

– for the region with kc kb, Ec ðkc Þ A2

1=2 ec ne : e kc

ð5:168Þ


In the viscous-diffusional region (kc kb), molecular diffusion plays an important role, and the spectral function Ec(kc) decreases more and more rapidly as kc grows. In the relations above, A1 and A2 are constants approximately equal to 1, e is the specific dissipation of turbulent energy, ec – specific dissipation rate of the scalar field, kc kb and kb – wave numbers corresponding to the Kolmogorov and Batchelor scales. Consider the velocity of disturbance propagation vk along the wave number axis. Suppose it does not depend on the form of spectral distribution but instead is a function of a point on the wave number axis. This hypothesis is similar to the hypothesis about the spectral transport of turbulent energy [33]. Then the length of time t12 during which perturbations pass through the wave number range (k1, k2) equals kð2

t12 ¼ k1

dk : Vk

ð5:169Þ

For a quasi-stationary field, the velocity Vk can be expressed through the specific dissipation rate of the scalar field ec (it is connected with the dissipation of scalar and the introduced in Section 4.9 by the relation ec ¼ 2N) inhomogeneities N spectral distribution Ec(k): Vk ¼

ec ðkÞ ; Ec ðkÞ

ð5:170Þ

where ec ðkÞ and Ec(k) are related to one another by ð¥ ec ðkÞ ¼ 2Dm k2 Ec ðkÞdk

ð5:171Þ

k

and spectral distribution Ec(k) obeys the normalization condition ð¥

Ec ðkÞdk ¼ hðC0 Þ2 i:

ð5:172Þ

0

Substituting Eq. (5.170) into Eq. (5.169) and using the relations (5.166) and (5.167), we obtain expressions for the micromixing time tc for different wave number ranges. Thus, for the inertial-convective region we have 3 ðtc Þ1 ¼ A1ec ðeÞ1=3 k2=3 ; m 2

ð5:173Þ

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318

where km is the wave number corresponding to the maximum size of inhomogeneities (the right end of the inertial-convective region where micromixing begins). In the viscous-convective region, the micromixing time equals ðtc Þ2 ¼

A2 ne 1=2 ne ln : 2 e Dm

ð5:174Þ

The assumption that local velocity of perturbation propagation along the wave number axis must be independent from the general form of the spectrum is mostly applicable in the viscous flow region. In this region the equations of motion become linear, and interactions between fluctuations having different scales can be neglected. The above-mentioned relations make it possible, at least in principle, to estimate micromixing time, but in doing so we are facing the challenge of obtaining spectral distributions for concrete types of turbulent inhomogeneities. This prompts us to use various empirical equations for the dissipation rate of the scalar field ec ; it is related to the micromixing time c by ec ¼

hðC0 Þ2 i : tc

ð5:175Þ

As an example, we shall adduce two frequently-used relations, dec B1 B2 ec ; ¼ þ dt tt tc

ð5:176Þ

1=2 ec ee dec ec ; ¼ C1 þ C2 dt tc n

ð5:177Þ

where tt is the relaxation time of the hydrodynamic field, and B1, B2, C1, C2, are empirical constants. The form of equations depends on the procedure used to mix the reagents. For example, the mixing can take place in a turbulent flow behind a grid; this process is characterized by simultaneously dissipation of hydrodynamic and scalar fields. Another possible procedure is the mixing of two streams moving in a pipe, where the hydrodynamic field is assumed to be stationary, and the main focus is on the time evolution of the scalar field. Summarizing our efforts to describe turbulent transport and mixing on the basis of the method of moments, we can make the following conclusions. The equations (5.156) and (5.157) for first two moments of the scalar field are unclosed due to the presence of mixed moments in these equations. In most problems that involve turbulent transport and macromixing, one adopts the gradient hypotheses that is expressed by the relation (5.158). The disadvantages of this hypothesis are most evident at the initial stages of fast chemical reactions, when micromixing can


produce a noticeable change of mixture composition in the course of fluid particle’s motion. To describe micromixing, one should go beyond the scope of the gradient hypothesis, which necessitates the use of one additional equation for the mixed moments hu0 C0 i. One way to obtain such an equation is to multiply the convective diffusion equation for C0 by u0 and the equation of motion for u0 by C0 , add up the two equations and average the result. But the equation derived in this way turns out to be unclosed as well. The situation here is the same as for hydrodynamic equations. Therefore in order to close the obtained system of equations, one resorts to various semi-empirical approximations for the third-order mixed moments hu0i u0j C0 i and for the moment hC0 D p0 =re i that contains pressure. 5.8.3 Approximations for Chemical Reaction Rates

In the beginning of this section we said that inclusion of chemical reactions into the picture results in the appearance of an additional source-type term (reaction term) in the diffusion equation. This term defines the intensity of production or consumption of matter in course of the chemical reaction. In the general case, this term depends nonlinearly on reagent concentrations, and so in addition to the aboveconsidered problem of modeling the terms describing turbulent transport and mixing (in the two-moment approximation), we are now facing the closure problem for the reaction terms, in particular, for the moments hWii and hCiWii. In some cases it is convenient to switch from component concentrations to conserved scalar variables, so that our equations do not contain source-type terms that are due to the chemical reactions. Thus, for the reaction A þ nB ! ðn þ 1ÞP; where mass quantities (rather than molar quantities conventionally used in chemical kinetics) serve as concentrations (one should read this reaction as follows: 1 kg of matter A reacts with n kg of matter B, produce (n þ 1) kg of matter P), we have the following conserved variables: 1 Z1 ¼ CA CB ; n

Z1 ¼ CA þ

1 CP ; nþ1

Z3 ¼ CB þ

1 CP ; nþ1

ð5:178Þ

where CA, CB, and CP stand for the corresponding matter concentrations. According to the rule (5.145), the stoichiometric coefficients of this reaction are nA ¼ 1, nB ¼ n, nP ¼ n þ 1 and the equality (5.150) provides the following connection between the reaction rates of different components:

dCA 1 dCB 1 dCP ¼ ¼ : dt n dt n þ 1 dt

ð5:179Þ

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It is easy to convince ourselves in the validity of the relations dZi ¼ 0; dt

ði ¼ 1; 2; 3Þ:

ð5:180Þ

Thus, as we switch from the variables CA, CB, CP to the variables Z1, Z2, Z3, the source-type terms in the corresponding equations for Z1, Z2, Z3 will be absent in view of the expression (5.180). The variables (5.178) are referred to as Schwab– Zeldovich variables. They are widely used in the combustion theory. Suppose that diffusion coefficients of different components are identical. Then only one of these variables will be independent. If the reaction is also very fast, then substances A and B cannot be found simultaneously at one and the same point and their concentrations can be expressed through conserved variables. So, if Z1 is chosen as our independent variable, then 8 < CA at Z1 > 0; Z1 ¼ : 1 CB at Z1 < 0: n

ð5:181Þ

However, the use of conserved variables has one negative aspect – the difficulty of reverse transition from variables Zi to the concentrations. This difficulty arises because of the nonlinear or, to be more precise, piecewise linear, dependence (5.181). Therefore our knowledge of the moments hZii and hðZi0 Þ2 i gives no way to determine the corresponding moments of concentrations CA and CB. Further on we are going to show that in order to obtain the moments of concentrations, it is necessary to know the probability density distribution of the variables Zi. For a single-step reaction of the second order A þ B ! products the averaged values of reaction terms in the equations for the average concentrations hCAi and hCBi are equal to hWA i ¼ hWB i ¼ kðhCA ihCB i þ hCA0 CB0 iÞ;

ð5:182Þ

where k is the reaction rate constant. For isothermal reactions, the average reaction rate is calculated at constant temperature. If the reaction is accompanied by heat release, then local temperature is also a random scalar variable and the average reaction rate should be calculated with temperature fluctuations properly taken into account. One good example is combustion. Chemical reactions in this process are extremely exothermal, have very high activation energy and are therefore highly sensitive to local temperature changes [34]. If the reaction is taking place in a gas, it is necessary to take into account density fluctuations within the gas, which means we have to employ the equations of gas dynamics. Another useful procedure that helps account for density variations in a


turbulent gas flow is to average all parameters except pressure over the mass. Such averaging is called Favre averaging [35]. We see from Eq. (5.182) that if the kinetic order of reaction is higher than one, the average reaction rate cannot be expressed through the average concentrations. The second term inside the brackets on the right-hand side of Eq. (5.182) (this term characterizes the degree of local correlation between reagent distributions) is zero only in the case of a totally homogeneous mixture. Depending on the character of the correlation between concentrations of reagents A and B, the average reaction rate could be higher or lower than the reaction rate calculated on the basis of average concentration values. Additional equations for hCA0 CB0 i are therefore necessary. To this end, we construct semi-empirical relations that express this moment through the moments hCA0 i, hCB0 i, hðCA0 Þ2 i, and hðCB0 Þ2 i. But the equations for second moments hðCi0 Þ2 i will contain new unknown reaction terms, which in their turn will require new closure relations. The most common strategy is to calculate the second order fluctuation moments, taking hint from Toor’s hypothesis [25] that it might be safe to neglect the third-order moment containing the reaction term. To take into account the mutual correlation of reagents A and B, the average reaction rate in Eq. (5.182) is represented in the form WA ¼ WB ¼ kð1UÞhCA ihCB i;

ð5:183Þ

in agreement with the immiscibility model. Here U is the degree of immiscibility for the whole reaction, which is expressed through the degrees of immiscibility for each reagent and through the correlation between average concentration gradients of these reagents. The degree of immiscibility Ui of reagent i is defined as the fraction of time during which this reagent is absent at the given point: Ui ¼ 1

hðCi Þ2 i hðCi Þ2 i þ hðCi Þ2 i

:

ð5:184Þ

The sign of the moment hCA0 CB0 i depends on the direction of average concentration gradients of the reagents rhCAi and rhCBi. Suppose these gradients have opposite directions at the given point, as is the case, for instance, when we are mixing two previously segregated reagents A and B. Then we will still have partial segregation even as the reagents are being mixed together. Consequently, hCA0 CB0 i < 0. If, on the other hand, the previously mixed reagents A þ B are now being mixed with some inert solvent, then the two local gradients rhCAi and rhCBi are parallel and hCA0 CB0 i > 0. The first case is of greater interest to us, especially for fast reactions. As a rule, we consider this case when choosing the closure model. For example, according to Patterson’s model [27] for equimolar ratio of reagents A and B, the moment hCA0 CB0 i is represented as hC0 C0 i ¼

hðCA0 Þ2 ihðCB0 Þ2 i : hCA ihCB i

ð5:185Þ

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For more complicated two-step reactions of the form A þ B ! R;

R þ B!P

ð5:186Þ

we also have consider the selectivity of the reaction. Selectivity is defined as the parameter R/(R þ P) characterizing the relative product yield (note that selectivity depends on the intensity of mixing). The reaction terms for the two stages of reaction (5.186) are hW1 i ¼ k1 ðhCA ihCB i þ hCA0 CB0 iÞ;

hW2 i ¼ k2 ðhCR ihCB i þ hCR0 CB0 iÞ: ð5:187Þ

Note that one cannot use the same closure model for the moment hCR0 CB0 i as for the moment hCA0 CB0 i because local correlations between R and B differ from the corresponding correlations between A and B. Many possible closures have been suggested for the moment hCR0 CB0 i. One of them is hCR0 CB0 i ¼ KhCR CB i with the empirical coefficient K varying from –1 to 1. The eventual choice of the closure model for reaction terms can be made by running a comparison with experimental data or with results obtained from the PDF-method; remember that the PDF method makes it possible to calculate the rate of an arbitrary nonlinear reaction without any additional assumptions once the probability density distribution of concentrations is known.

5.9 The PDF Method

The PDF method is an abbreviation for the method that employs the equation for the probability density function (distribution) to describe the behavior of random hydrodynamic and scalar fields. The central idea is to represent the fluid as a large set of fluid particles, each particle characterized by a set of time-varying parameters: particle’s position in space, velocity, temperature, and composition (component concentrations). The currently existing methods of describing turbulence can be divided into two groups: – the method of moments, based on the use of transport equations and equations for the moments (which, in their turn, are obtained from transport equations) complemented by semi-empirical closure relations; – the PDF method, based on the equation for the probability density function (distribution) of random hydrodynamic and scalar fields.

5.9 The PDF Method

The main weakness of the first method is the need to use additional equations (closures)forhigher-ordermoments, sinceanysystem of equations forthefirstN moments turnsouttobeunclosedduetothepresenceofmomentsofevenhigherorder.Additional equations typically contain a large number of empirical constants and functions. To obtainreceivetheseconstantsandfunctions,onehastocarryoutamultipleexperiments, and even then the values obtained in this way are not universal, because they may be differentfordifferentflows.Theshortcomingsofsemiempiricalmodelsmanifestthemselves particularly strongly when one needs to describe turbulent flows of multi-phase, multi-component media accompanied by chemical reactions. The positive feature of semi-empirical models is that they reduce to a system of partial differential equations for velocities, concentrations, and temperature as functions of coordinates and time, whose solution for many types of flows such as flows inside channels, pipes, boundary layers, jet flows, flows bypassing a body, and so forth, can be obtained by well-developed numerical methods. The idea behind the PDF method is to introduce one single function – joint PDF p of several random functions (velocity vector u, concentrations Ci, enthalpy H or temperature W, etc.) and solve the Fokker–Planck equation for p. If the function p is known, we can find all statistical characteristics of the random functions (moments), which saves us the effort of constructing closure relations. Therefore the PDF method is more logical and informative then the method of moments. Its main disadvantage is the large number of dimensions in the problem (time t, coordinates X, functions u, C, W, . . .), which makes it difficult to obtain even a numerical solution, to say nothing of the analytical one. As of today, there are several different methods based on the models that use simplified equations for the PDF together with semiempirical models of turbulence such as the k e model. A comprehensive treatment of the PDF method with appropriate references can be found in [30–32]. The state of an incompressible fluid in an isothermal process is completely described by three components of the velocity vector u and by N scalar concentrations of passive impurities C1, C2, . . ., CN. Let us introduce a joint single-point PDF p(u, C1, C2, . . ., CN; X, t) at the point X at the instant of time t. By definition (see Section 1.2), the quantity p(u, C1, C2, . . ., CN; X, t) du dC1dC2,. . .,dCN is the probability that at the moment t, at the point X, we are going to find the velocity u in the interval (u, u þ du) and each of the concentration values Ci in its respective interval (Ci, Ci þ dCi). This function is non-negative and obeys the normalization condition ð pðu; C1 ; C2 ; . . .; CN ; X ; tÞ du dC1 dC2 . . .dCN ¼ 1; where the integral is taken over the entire phase space of velocities and concentrations. Having written the distribution p(u, C1, C2, . . ., CN; X, t), we can always derive distributions with fewer dimensions (marginal distributions) by integrating p over one variable or several variables,ð for example, pðC1 ; C2 ; . . .; CN ; X ; tÞ ¼

pðu; C1 ; C2 ; . . .; CN ; X ; tÞdu:

ð5:188Þ

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More importantly, we can find the average value of a random function F, ð hFi ¼ F pðu; C1 ; C2 ; . . .; CN ; X ; tÞdu dC1 dC2 . . .dCN ; and all moments of this function. Before we proceed to write the equation for p, let us write the system of conservation equations describing the isothermal flow of an incompressible multi-component fluid. It consists of the continuity equation qui ¼ 0; qXi

ð5:189Þ

the momentum equation r

Du j q p qti j þ þ r f j A j; ¼ Dt qX j qXi

ð5:190Þ

and the mass conservation equation for k-th substance r

Dmk qJ k ¼ i þ rSk Bk ; Dt qXi

ð5:191Þ

where D/Dt ¼ @/@t þ ui@/@Xi is the substantial derivative; fj – j-th component of the body force (for example, gravity); tij – components of the viscous stress tensor; Jik – i-th component of the diffusion flux of k-th substance; Sk – specific rate of production or consumption of k-th substance in the course of chemical reaction. One of the ways to derive the equation for p(u, C1, C2, . . ., CN; X, t) is to represent p as a product of delta functions (see Eq. (1.152)), N Y pðu; C1 ; C2 ; . . .; CN ; X ; tÞ ¼ dðuuðX ; tÞÞ dðCi Ci ðX ; tÞÞ: i¼1

Using the Eulerian approach, we write the transport equation for the PDFp in the form qp q q q ðu j pÞ ðhA j ðu; CÞi pÞ ¼ ðhBk ðu; CÞi pÞ; ¼ qt qX j qu j qCk

ð5:192Þ

where we have introduced the vector C ¼ (C1, C2, . . ., CN). We see from Eq. (5.192) that the variation of p is caused by convective transport in the physical space X with the velocity u and in the u C phase space with the velocities hAji along the uj-axis and hBki along the Ck-axis. An analytic solution of Eq. (5.192) can be obtained only for the simplest cases of no practical interest. Most situations that present any interest for applications allow for a numerical solution

5.9 The PDF Method

only. But the employment of numerical methods such as the finite difference method presents difficulties because of the excessively large number of dimensions in the problem. Even in the case of a single scalar quantity we have 8 dimensions (three velocity components, concentration, three coordinates, and time). To apply this numerical method, it is necessary to divide the spatial volume into N finite elements {Xk, uk, Ck} and to present p in the discrete form pN ¼

N 1X dðuuk ÞdðCC k ÞdðX X k Þ: N k¼1

The transition from the continuous to the discrete distribution is achieved with a statistical error proportional to N1/2, and the average value of any arbitrary quantity is calculated with the root-mean-square deviation N1/2 independently from the number of dimensions. So as N gets larger, the statistical error changes very slowly. For instance, to achieve a 10% error, one needs only 100 elements, but to decrease the error to 1%, one needs as much as 10000 elements. The discrete distribution for given values of t and N is defined by specifying N vectors and a set of 6N numbers in a 6-dimensional space. While in a 3-dimensional space one needs 1000 elements, in a 6-dimensional space 6000 elements are needed in order to achieve the same calculation accuracy. That is why the Monte Carlo method is the method of choice for a multidimensional case. As opposed to the Eulerian approach with its reliance upon Eq. (5.192), the Lagrangian approach uses the system of stochastic Langevin equations for fluid particles. As a rule, these are simply transport equations in the relaxation approximation (see Eq. (5.101)). In the case of a single scalar variable they have the form DX ðx; tÞ ¼ uðX ðx; tÞ; tÞ; Dt DuðX ðx; tÞ; tÞ ¼ aL ðuðX ðx; tÞ; tÞhuðX ðx; tÞ; tÞiÞ þ f ; Dt

ð5:193Þ

DCðX ðx; tÞ; tÞ ¼ bL ðCðX ðx; tÞÞhCðX ðx; tÞÞiÞ þ WðCÞ; Dt where it is assumed that initially X ¼ x, u ¼ v, C ¼ S. Here X(x, t) are the fluid particle’s coordinates at the moment t (x being the initial coordinate); u(X(x, t), t) is that particle’s Lagrangian velocity; aL and bL are dissipation rate constants (i.e., inverse relaxation times) of the hydrodynamic and scalar fields respectively; f is the random force exerted on the fluid particle by the flow. Using Eq. (5.193) and switching to Eulerian variables, we get the following equation for the PDF p(u, C; X, t): qp qp q q2 p q q þ ðaL ðui hui i pÞþh fi 2 i 2 þ ðbL ðChCi pÞ ðWðCÞ pÞ: ¼ ui qt qXi qui qC qui qC ð5:194Þ

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As you may guess from the formula (5.188), the equation for the PDF of concentration only, that is, for p(C; X, t) is derived by integrating both sides of Eq. (5.194) with respect to velocity from 1 to þ 1. Exploiting the condition @p/@ui ! 0 at |ui| ! 1 and introducing the coefficient of turbulent diffusion through the gradient hypothesis ð ðui hui iÞ pðu; C; X ; tÞdu ¼ Dt

qpðC; X ; tÞ ; qXi

ð5:195Þ

we reduce Eq. (5.194) to qp qp q qp þ Dt ¼ hui i qt qXi qXi qXi þ

q q ðb ðChCi pÞ ðWðCÞ pÞ: qC L qC

ð5:196Þ

The third term on the right-hand side of Eq. (5.196) bK ¼

q ðb ðChCi pÞ qC L

ð5:197Þ

describes the process of micromixing [24]. It turns out that the above-described Langevin model of micromixing does not always give satisfactory results. The need to introduce other models of micromixing, including nonlinear ones, is caused by the fact that, according to the Langevin model, all concentration values change simultaneously. Thus if the PDF p(C; X, t) has the form pðC; X ; tÞ ¼ gdðCC1 Þ þ ð1gÞdðCC2 Þ the Langevin model predicts that both peaks will eventually approach the average value hCi ¼ gC1 þ (1 g)C2, while in a real-life situation, partial mixing of two fluid elements (moles) having respective concentrations C1 and C2 will produce mixing layers containing all of the intermediate concentration values. The arbitrariness in the choice of phenomenological model for micromixing is explained by the fact that we are considering a one-point PDF, whereas micromixing is characterized by local gradients of the scalar field and thus requires the knowledge of spatial correlations that can be determined only from a two-point PDF. Another method is based on the idea of including local gradients into the PDF as independent variables. But such a procedure complicates the problem and, furthermore, it requires the corresponding new models that become necessary if we are to write an equation for the PDF that would properly account for the new variables. As the model of micromixing for the operator K, one typically chooses some model of the coalescence-dispersion type. Such models describe the process of micromixing as the outcome of random contact (coalescence) of two microvolumes

5.9 The PDF Method

with concentrations C1 and C2, hence the name used for these models. The probability of such interaction is taken to be proportional to the product p(C1; X, t) p(C2; X, t) in accordance with the so-called average field approximation. The contact and mixing of the microvolumes results in the formation of intermediate concentrations c (C1 < C < C2), whose distribution, in its turn, should be modeled beforehand. In a coalescence-dispersion model, the operator K reduces to the following expression [36]: ð1 ðc K ¼ 2 dC1 dC2 pðC1 ; X ; tÞ pðC2 ; X ; tÞGðC; C1 ; C2 Þ: 0

ð5:198Þ

c

The kernel of this equation G(C, C1, C2) is the distribution density of intermediate concentrations that are formed during the elementary mixing event. The choice of G is what distinguishes various coalescence-dispersion models from each other. For instance, Curl’s model proposed in [24] assumes that only one intermediate concentration C ¼ (C1 þ C2)/2 is formed during the contact of the microvolumes and that C1 þ C2 GðC; C1 ; C2 Þ ¼ d C : 2

ð5:199Þ

In the Nedorub model considered in [37], the intermediate concentration distribution is taken to be uniform and the kernel is expressed as GðC; C1 ; C2 Þ ¼ jC2 C1 j1 :

ð5:200Þ

We should note that the value of the constant b in Eq. (5.197) depends on the choice of the micromixing model and can be determined from the Corrsin equation (5.160) for the intensity of concentration fluctuations, dhðC0 Þ2 i ¼ ghðC0 Þ2 i dt

ð5:201Þ

with a given value of g. In the Langevin model, b ¼ g/2; in Curl’s model (5.199), b ¼ 2g; and in the Nedorub model (5.200) with a uniform distribution of intermediate concentrations, b ¼ 3g. So, models of the type (5.198) predict an exponential dependence of h(C0 )2i on time. In the general case, the dependence of contact frequency of fluid microvolumes with concentrations C1 and C2 on these concentrations is more complex. For example, in Frost’s model [29] the contact frequency is taken in the form bjC2 C1 j pðC1 ; X ; tÞ pðC2 ; X ; tÞ:

ð5:202Þ

For this model, the change of h(C0 )2i with time cannot be described by Eq. (5.201) with a constant value of g.

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In order to have criteria that a micromixing model should satisfy, we usually demand that the PDF of concentration at the early stage of mixing should be represented by a sum of two delta functions. We may also demand that for long mixing times the distribution should asymptotically approaches a Gaussian distribution. The proximity of an asymptotic distribution to a Gaussian one is usually estimated by the value of the excess b2, 0 4 ðC Þ b2 ¼ : ð ðC0 Þ2 Þ2

ð5:203Þ

For coalescence-dispersion models of the type (5.198), the excess is equal to b2 ¼ 1 þ 4ðexpðatÞ1Þ;

ð5:204Þ

where the parameter a is expressed through the moments of the distribution G(C, C1, C2). Consequently, for any coalescence-dispersion model, the excess tends to infinity when t ! 1, whereas for the Gaussian distribution, b2 ¼ 3. Notice that in the model (5.202), the asymptotic value of the excess is also close to 3, even though this is hard to explain from a physical point of view, because the contact frequency in this model increases with |C2 C1|. The form of the PDF at the initial stage of mixing can be estimated by considering the formation of a mixing layer caused by molecular diffusion between two fluid volumes with concentrations C1 and C2 as shown in Fig. 5.6. The diffusion equation gives a distribution with two maximum values that are close to C1 and C2 (Fig. 5.6 a). If we assume (as an approximation) that in a mixing layer of thickness d, concentration is distributed linearly, then the distribution of intermediate concentrations will also be linear, which corresponds to a uniform PDF (Fig. 5.6 b). Let us say a few words about the influence of chemical reactions on the process of mixing. The fact that such an influence exists becomes evident if we look at the

Fig. 5.6 Mixing layer between two fluid volumes.

5.9 The PDF Method

Fig. 5.7 Concentration profiles: (a) without chemical reaction; (b) with chemical reaction.

mutual diffusion of two substances A and B in a quiescent fluid. Concentration profiles for two limiting cases of mixing – without chemical reactions and with infinitely fast, irreversible chemical reactions of the type A þ B ! product – are shown schematically in Fig. 5.7. The effect of chemical reactions on the rate of micromixing is difficult to estimate, because turbulent fluctuations cause significant deformation of the surface where the mass transport is taking place. Of particular interest in this context is the structure of surfaces at which mass transport is observed in the course of turbulent mixing. In the case under consideration, these surfaces are isoconcentration surfaces. Theoretical studies suggest that isoconcentration surfaces formed in the course of mutual diffusion of substances A and B in a motionless fluid layer have a fractal structure, and that structure self-similarity exists in some scale interval. Fractal structure is characterized by the parameter d known as fractal dimensionality. Determination of d boils down to estimating the surface area by calculating number N(r) of cubes of linear size r that are needed to cover the whole surface. Then surface area is N(r)r2. As the cube size decreases, the area of surfaces one usually encounters in practical situations quickly reaches its limiting value no longer depends on r. For a fractal surface of dimensionality d, the number of cubes needed to cover the surface and the surface area increase as Nrd and Sr2d. For each system, there are constraints from above as well as from below, which must be satisfied by such power laws; note that the exponent may have different values in different intervals. Determination of surface structure and surface area is an important practical problem because the rate of mass exchange through any surface is given by the product of the molecular diffusion coefficient, the local concentration gradient, and the surface area (whose deformation due to the turbulent fluctuations should, of course, be taken into account). In particular, for scales that exceed the inner scale of turbulence l0 and thus belong to the inertial-convective region, fractal dimensionality of isoconcentration surfaces is equal to 7/3, which is confirmed by experiments that involve visualization of isoconcentration surfaces CA ¼ CB ¼ 0 during a very fast neutralization reaction between acid and alkali solutions. Estimates of fractal dimensionality of these surfaces in the viscous-convective scale interval give the value 3, in other words, such a surface would fill the entire volume under consideration. It is obvious that this scale subrange has to be bounded from below by the Batchelor

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scale lb, starting from which molecular diffusion leads to a completely uniform distribution of reagents. The above-considered models of micromixing have been developed for the PDF distributions of one variable – concentration. But there are many problems where several variables (velocity, concentration, temperature, etc.) are involved. Until recently, such problems were solved by considering Curl’s model [23] and using the Monte Carlo method to solve the corresponding stochastic equations. It should be noted that if we have than one variable, especially if these variables have different physical nature (e.g., concentration and velocity, concentration and temperature, and so on), we are facing two additional problems. Namely, we have to determine the regions where intermediate values of the variables arise during one elementary step of micromixing, and to find out how micromixing intensity for different variables depends on molecular transport coefficients, specifically, on diffusion, viscosity, and heat conductivity. When deriving Eq. (5.196) for the PDF, we started from the Langevin equations (5.193). Derivation of this equation directly from the convective diffusion equation has been attempted in [34] for the limiting case of strongly developed turbulence c i pÞ=qC2 was obtained for the term describing (Re ! 1). The expression – q2 ðhN c i is the conditionally averaged rate of dissipation of the scalar micromixing; here hN field for a given value of concentration C, so that the total dissipation rate ec is equal to ð c i pðC; X ; tÞ: ec ¼ dChN However, this results in an inverse parabolic equation for p(C; X, t), whose solution presents difficulties. When considering the Schwab–Zeldovich conserved variables Zi in Section 5.8, we mentioned that use of these variables makes it possible to exclude source-type reaction terms from the convective diffusion equations. Consequently, a transition to the Schwab–Zeldovich variables leads to the exclusion these terms from the equation for the PDF as well. We must know the PDF of conserved variables if we want to switch back to ordinary concentrations. Sometimes the form of conserved variable distributions is given a priori and one only needs to find the parameters of these distributions. Let us look at the examples of conserved variables for some reactions. For infinitely fast reactions of the type A þ B ! P the conserved variable is Z ¼ CA CB. Because reagents A and B cannot exist simultaneously at one and the same point, the reverse transition to concentrations CA and CB is accomplished by means of the relations Z ¼ CA for Z > 0 and Z ¼ CB for Z < 0. The model of micromixing (which is accompanied by an infinitely fast reaction) at the (CA, CB) plane is replaced by a one-dimensional model for the variable Z. For a two-step reaction of the type k

1 P1 ; A þ B !

k

2 A þ C ! P2 ;

References

in which the first step is considered as an infinitely fast reaction (k1 ! 1), Z ¼ CA CB and CC can be chosen as the independent variables. If both steps of the reaction are thought to be infinitely fast, that is, if k1 ! 1, k2 ! 1 and k1/ k2 ¼ const, then it is convenient to switch to new variables Z ¼ CB þ CC CA and CB and to describe the macrokinetics of this reaction with turbulent mixing models for these variables. The intermediate concentration boundary can be found from equations of chemical kinetics; one must keep in mind that each elementary step of micromixing is accompanied by an infinitely fast chemical reaction. For a two-step sequential-parallel reaction of the type A þ B ! R;

B þ R ! P;

when the first step is infinitely fast, the conserved variables are Z ¼ CA CB and CR. If both steps are infinitely fast, then the process is described with turbulent mixing models for the variables Z ¼ 2CA CB þ CR and CR. To conclude, the use of conserved variables allows to reduce the number of independent variables and thereby decrease the dimensionality of the problem, especially in cases when some reaction stages can be considered as infinitely fast. Finally, let us make a remark about the modeling of macromixing and turbulent transport in equations for the PDF. Turbulent transport in Eq. (5.196) is described by the second term on the right-hand side. This term is written in the gradient hypothesis approximation. When discussing the method of moments in Section 5.8, we already commented on the disadvantages of this approximation and on the need to go beyond the scope of the gradient hypothesis by using additional equations for the mixed moments, in particular, for huCi. Use of the joint PDF gives an exact expression for the moment huCi, namely, ð

ð

huCi ¼ du u dC C pðu; C; X ; tÞ:

ð5:205Þ

Transition to the joint PDF p(u, C, X, t) would certainly complicate the problem by increasing its dimensionality. For the sake of simplicity, one can use instead of p(u, C; X, t) the distribution p(C; X, t) for concentration only, which should be considered in combination with the equation for conditionally averaged velocity huci, ð huc i pðC; tÞ ¼ du u pðu; C; tÞ:

ð5:206Þ

References 1 Sedov, L.I. (1970) Mechanics of Continuum, Nauka, Moscow, 1, 2 (in Russian).

2 Monin, A.S. and Yaglom, A.M. (1971) Statistical Fluid Mechanics: Mechanics of Turbulence, 1, MIT Press, Cambridge,

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3

4

5

6

7

8

9

10

11

12

MA; Monin, A.S. and Yaglom, A.M. (1975) Statistical Fluid Mechanics: Mechanics of Turbulence, 2, MIT Press, Cambridge, MA. Richardson, L.E. (1926) Atmospheric Diffusion shown on a Distance– Neighbour Graph. Proc. Roy. Soc. A, 110 (756), 709–737. Taylor, G.I. (1921) Diffusion by Continuous Movements. Proc. London Math. Soc., 20 (2), 196–211. Zaichik, L.I. and Pershukov, V.A. (1996) Problems of Modeling Gas–Particle Turbulent Flows with Combustion and Phase Transitions. Review. Fluid Dynamics, 31 (5), 635–646. Derevich, I.V. and Zaichik, L.I. (1988) Particle Deposition from a Turbulent Flow. Fluid Dynamics, 23 (5), 722–729. Derevich, I.V. and Zaichik, L.I. (1990) An Equation for the Probability Density Velocity and Temperature of Particles in a Turbulent Flow Modeled by a Random Gaussian Field. J. Appl. Mathematics and Mechanics, 54 (5), 631–636. Zaichik, L.I. (1992) A Kinetic Model of Particle Transport in Turbulent Flows with Allowance for Collisions. Eng.-Phys. J., 63 (1), 44–50 (in Russian). Zaichik, L.I. and Pershukov, V.A. (1995) Modeling of Particle Motion in a Turbulent Flow with Allowance for Collisions. Fluid Dynamics, 30 (1), 49–63. Alipchenkoy, V.M. and Zaichik, L.I. (2000) Modeling of the Motion of Particles of Arbitrary Density in a Turbulent Flow on the Basis of a Kinetic Equation for the Probability Density Function. Fluid Dynamics, 35 (6), 883–900. Alipchenkov, V.M. and Zaichik, L.I. (2001) Particle Collision Rate in Turbulent Flow. Fluid Dynamics, 36 (4), 608–618. Buevich, Yu.A. (1970) Drag Reduction Model for Particle Injection into a Turbulent Viscous Fluid Stream. Fluid Dyn., 5 (2), 271–276.

13 Tchen, C.M. (1947) Mean Value and Correlation Problems Connected with the Motion of Small Particles Suspended in a Turbulent Fluid. Martinus Nijhoff, Haague. 14 Fuks, N.A. (1955) Mechanics of Aerosols, Acad. Nauk SSSR, Moscow (in Russian). 15 Hinze, J.O. (1975) Turbulence, 2nd ed.,McGraw–Hill, New York. 16 Mednikov, E.P. (1981) Turbulent Transport and Deposition of Aerosols, Nauka, Moscow (in Russian). 17 Gusev, I.N. and Zaichik, L.I. (1992) Numerical Modeling of Two-Phase Turbulent Flows in a Furnace. J. Appl. Math. Techn. Phys., 2, 116–122 (in Russian). 18 Kirillov, P.L. (1986) About the Effect of Thermophysical Properties of a Surface on Heat Transfer by Turbulent Motion. Eng.-Phys. J., 50 (3), 501–512 (in Russian). 19 Sehmel, G.A. (1973) Particle Eddy Diffusivities and Deposition Velocities for Isothermal Flow and Smooth Surfaces. J. Aerosol. Sci., 4, 125–138. 20 Friedlander, S.K. and Johnstone, H.F. (1957) Deposition of Suspended Particles from Turbulent Gas Stream. Ind. and Eng. Chem., 49 (7), 1151–1156. 21 Sehmel, G.A. (1970) Particle Deposition from Turbulent Air Flow. J. Geophys. Res., 75 (9), 1766–1781. 22 Alipchenkov, V.M. and Zaichik, L.I. (1998) Modeling the Dynamics of Colliding Particle in a Turbulent Shear Flow. Fluid Dynamics, 33 (4), 552–558. 23 Curl, R.L. (1963) Disperse Phase Mixing: I. Theory and Effects in Simple Reactors. AIChE J., 9 (2), 175. 24 Chung, P.M. (1969) A Simplified Statistical Model of Turbulent Chemical Reacting Shear Flows. AIAA J., 7, (1982). 25 Hill, J.C. (1976) Homogeneous Turbulent Mixing with Chemical Reaction. Ann. Rev. Fluid Mech., 8, 135.

References

26 Kompaneez, V.Z., Ovsyannikov, A.A. and and Polak, L.S. (1979) Chemical Reactions in Turbulent Flows of Gas and Plasma, Nauka, Moscow (in Russian). 27 Patterson, G.K. (1981) Application of Turbulence Fundamentals to Reactor Modelling and Scaleup. Chem. Engng. Commun., 8, 24. 28 Libby P.A. and Williams F.A. (eds.) (1980) Turbulent Reacting Flows, Springer, Berlin/New York. 29 Kaminski, V.A., Fedorov, A.J. and Frost, V.A. (1994) Calculation Methods for Turbulent Flows with Fast Chemical Reactions. Theor. Fund. Chem. Techn., 28 (6), 591–599 (in Russian). 30 Pope, S.B. (1985) PDF Methods for Turbulent Flows. Prog. Energy Combust. Sci., 11, 119–192. 31 Pope, S.B. (1994) Lagrangian PDF Methods for Turbulent Flows. Ann. Rev. Fluid Vtch., 26, 23–63.

32 Pope, S.B. (2000) Turbulent Flows, Cambridge Univ. Press, Cambridge. 33 Pao, Y-H. (1965) Structure of Turbulent Velocity and Scalar Fields at Large Wave Numbers. Phys. Fluids, 86, 1063–1075. 34 Kuznetsov, V.R. and Sabel’nikov, V.A. (1990) Turbulence and Combustion, Hemisphere, New York. 35 Favre, A. (1969) Statistical Equations of Turbulent Gases In Problems of Hydrodynamics and Continuum Mechanics, SIAM, Philadelphia p. 231. 36 Janicka, J. , Kolbe, W. and Kollman, W. (1979) Closure of the Transport Equation for the Probability Density Function of Turbulent Scalar Fields. J. Non-Equilib. Thermodyn., 4, 47–66. 37 Nedorub, S.A. (1979) Investigation of Models for Calculation of Probability Density Functions of Impurity Concentrations in Turbulent Flows, Dis. Thesis Cand. of Phys.-Math. Sci, MFTI, Dolgoprudni (in Russian).

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6 Coagulation and Breakup of Inertialess Particles in a Turbulent Flow 6.1 Kinetic Equations of Coagulation

When modeling particle motion in low-concentrated disperse media (j 1), we notice that the average distance between particles is large as compared to the particle size, and analysis of particle interactions can be restricted to pair interactions. If, in addition, the disperse phase consists of particles with low inertia (St < 1), the primary focus should be on particle interactions with turbulent fluctuations of the carrier phase. Examples of such a system include water–oil emulsions and gas–condensate mixtures, both of which are disperse media whose disperse phase consists of droplets sized from 0.1 mm to 100 mm. When particle coagulation is caused solely by pair collisions, and assuming a spatially homogeneous polydisperse system, the dynamics of the process is described by the following kinetic equation [1]: qnðV; tÞ ¼ Ik ; qt

Ik ¼

ð6:1Þ

ðV ðV 1 KðVw; wÞnðVw; tÞnðw; tÞdwnðV; wÞ KðV; wÞnðw; tÞdw; 2 0

0

where n (V, t) is the distribution of particles over volumes V at the instant of time t, and K (V, o) is the coagulation kernel having the meaning of collision frequency of particles of volumes V and o per unit volume of the disperse phase at unit concentration of these particles. The first term on the right-hand side of Eq. (6.1) corresponds to the rate of production of particles of volume V due to collisions between particles of volumes V o and o, and the second term – to the rate of decrease in the number of particles of volume V due to their coagulation with other particles.

j NaN Coagulation and Breakup of Inertialess Particles in a Turbulent Flow

336

By solving Eq. (6.1) for a given initial distribution n0 (V), we can observe how the volume distribution of particles changes with time and find the main parameters of this distribution: number concentration of particles, that is, the number of particles per unit volume of the medium 1 ð

NðtÞ ¼ nðv; tÞdv;

ð6:2Þ

0

volume concentration, that is, the volume of particles per unit volume of the medium 1 ð jðtÞ ¼ vnðv; tÞdv;

ð6:3Þ

0

and the average volume of particles Vav ðtÞ ¼ jðtÞ=NðtÞ:

ð6:4Þ

The kernel K(V, o) defines the mechanism of particle interaction. The reader should keep in mind that study of general properties of the kernel and determination its specific form for various processes should be treated as two separate problems. An important feature of the coagulation kernel is its symmetry with respect to the sizes of colliding particles, in other words, K (V, o) ¼ K (o, V). Multiplying both sides of Eq. (6.1) by V, integrating the result over V from 0 to 1 and taking into account the relation (6.3), we obtain dj 1 ¼ dt 2

1 ð ðV

VKðVw; wÞnðVw; tÞmðw; tÞdVdw 00

1 ð1 ð

VKðV; wÞnðV; tÞnðw; tÞdVdw:

ð6:5Þ

0 0

A change of variables, z ¼ V o, o ¼ o in the first integral (6.5) makes it possible to transform the region of integration 0 V < 1, 0 o < V into 0 z < 1, 0 o < 1. Then Eq. (6.5) takes the form dj 1 ¼ dt 2

1 ð1 ð

ðz þ wÞKðz; wÞnðz; tÞmðw; tÞdzdw 0 0

1 ð ð1

VKðV; wÞnðV; tÞnðw; tÞdVdw:

0 0

ð6:6Þ

6.1 Kinetic Equations of Coagulation

Let us change the variable of integration by replacing z with V. Then Eq. (6.6) will be rewritten as dj 1 ¼ dt 2

1 ð1 ð

ðwKðV; wÞVKðv; wÞÞnðV; tÞmðw; tÞdVdw:

ð6:7Þ

0 0

By interchanging the positions of V and o in the first term of the integrand in Eq. (6.7), which obviously does not affect the value of the integral, we get dj 1 ¼ dt 2

1 ð ð1

VðKðw; VÞKðV; wÞÞnðV; tÞmðw; tÞdVdw:

ð6:8Þ

0 0

Since we are considering only the coagulation of particles, the total volume of particles, that is, the volume concentration j, remains constant, and 1 ð ð1

VðKðw; VÞKðV; wÞÞnðV; tÞmðw; tÞdVdw ¼ 0:

ð6:9Þ

0 0

One can readily see from Eq. (6.9) that symmetry of the coagulation kernel K (V, o) ¼ K (o, V) is a sufficient condition for the constancy of the volume concentration of particles. Let us show that it is also a necessary condition. As a corollary of the kinetic equation (6.1), the relation (6.8) must be valid for any physically permissible particle distribution over volumes n (V, t), including bidisperse systems, that is, systems composed of particles of only two volumes V1 and V2. The particle distribution for such a system is nðV; tÞ ¼ n0 ðxdðVV1 Þ þ ð1xÞdðVV2 ÞÞ:

ð6:10Þ

where xn0 is the number of particles of volume V1, (1 x) n0 – the number of particles of volume V2 and d(x) – the delta function. Substituting Eq. (6.10) into Eq. (6.9) and recalling the property (1.11.b) of the delta function, we can write n20 ðV1 V2 Þxð1xÞðKðV2 ; V1 ÞKðV1 ; V2 ÞÞ ¼ 0:

ð6:11Þ

For a bi-disperse system, n0 6¼ 0, V1 6¼ V2, x 6¼ 0, and x 6¼ 1, and therefore KðV2 ; V1 Þ ¼ KðV1 ; v2 Þ:

ð6:12Þ

Since volumes V1 and V2 have been chosen arbitrarily, the equality (6.12) should be satisfied for all particle volumes present in the system.

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j 6 Coagulation and Breakup of Inertialess Particles in a Turbulent Flow

338

If the symmetry condition for the coagulation kernel is violated, it means that the volume concentration of particles does not remain constant, which is equivalent to saying that the system contains sources and sinks whose intensities depend on the degree and specific form of kernel asymmetry. Another consequence Eq. (6.12) is that when the sum of particle volumes is fixed, in other words, when V þ o ¼ z ¼ const, the function K (V, o) has a local extremum at V ¼ o. Coagulation kernel characterizes particle collision frequency and is usually determined by solving the particle interaction problem numerically and approximating the obtained numerical results. Being a necessary condition, the condition of symmetry thus imposes restrictions on the choice of approximating relations. An efficient method of achieving symmetrization of the kernel by using available numerical data is outlined in [2]. Kinetic Eq. (6.1) is nonlinear integro-differential equation, which solution presents greatly difficulties. Currently available exact solutions are based on use of operational method as applied to linear dependence of K (V, o) on particle volumes [3]. In order to solve Eq. (6.1) with kernels of more general form are used method of moments and numerical methods. The method of moments makes it possible to reduce Eq. (6.1) to a system of differential equations for the moments the distribution of particles over volumes. But the resulting system of equations is usually unclosed because, in addition to moments of integer orders, it contains fractional-order moments, which appear due to the power law dependence of the kernel K (V, o) on particle volumes. To close the system of equations, one needs additional relations or additional constraints on the form of the volume distribution. For example, the parametric method is based on the assumption that the volume distribution of particles belongs to some definite class of distributions (logarithmically normal distributions, gamma distributions, etc.) with time-varying parameters that need to be determined. Another method, known as the method of interpolation of fractional moments, is independent of the form of particle distribution but requires additional relations that express fractional moments in terms of integer ones. In the majority of practical cases one is interested not so much in the distribution n (V, t) as in its first several moments or their combinations, which have definite physical meaning and can be found experimentally with relative ease. Experimental data should tell us whether the chosen model of particle interaction is realistic, because the choice of particle coagulation mechanism is usually based on certain assumptions. Integer moments of particle distribution over volumes are defined by the following expressions: 1 ð

mk ¼ Vk nðV; tÞdV;

ðk ¼ 0; 1; 2; . . .Þ:

ð6:13Þ

0

The value k ¼ 0 corresponds to the zero-order moment m0 having the meaning of the number of particles per unit volume of the medium (i.e., number concentration).


The value k ¼ 1 corresponds to the first order moment m1 equal to the volume of particles per unit volume of the medium (i.e., volume concentration, aka volume content j). Several combinations of moments are worth noting: (3/4p)1/3(m1/3/m0) – mean particle radius; m1/m0 – mean particle volume; (36p)1/3m2/3 – total particle surface area per unit volume of the medium (i.e., specific area of the interface). Furthermore, information about the type of the given distribution n (V, t) can be garnered from the first five moments mi ði ¼ 0; 4Þ; by using a Pearson diagram [4] (see Fig. 6.1). The quantities b1 and b2 plotted along the coordinate axes of the diagram are respectively called the asymmetry square and the excess. These distribution parameters are expressed through the moments as follows:

b1 ¼

m3

!2

3=2 m2

; b2 ¼

m4 ; m22

m2 ¼

2 m2 m1 ; m0 m0

m3 ¼

3 m3 m2 m1 m1 3 þ 2 ; 2 m0 m0 m0

m4 ¼

4 m4 m3 m1 m2 m12 m1 4 þ 6 3 : 2 3 m0 m0 m0 m0

To make a transition from the kinetic equation (6.1) to moment equations, we multiply both parts by Vk and integrate the result with respect to V from 0 to 1:

Fig. 6.1 The Pearson diagram.

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340

01 1 1 ð ð ðV d@ k 1 V nðV; tÞdVA ¼ Vk KðVw; wÞnðVw; tÞdwdV dt 2 0

00 1 ð1 ð

Vk KðV; wÞnðV; tÞdwdV:

ð6:14Þ

0 0

Making a change of variables in Eq. (6.14) in the same way as in the derivation of Eq. (6.6), we rewrite this equation as dmk 1 ¼ dt 2

1 ð1 ð

1 ð ð1 k

Vk KðV; wÞnðV; tÞdwdV:

ðz þ wÞ Kðz; wÞnðz; tÞdwdz 0 0

0 0

Now, replace z in the first integral with V and rewrite this equation in the form that is regularly used in the method of moments: dmk ¼ dt

1 ð ð1

1 k k ðV þ wÞ V KðV; wÞnðV; tÞdwdV: 2

ð6:15Þ

0 0

The majority of known coagulation kernels have the form of power functions of particle volume. Since the kernel K (V, o) must be symmetrical with respect to V and o, it can be represented in the general case as KðV; wÞ ¼

l X

G j ðVa j wb j þ wa j Vb j Þ:

ð6:16Þ

j¼0

Let us substitute the relation (6.16) into Eq. (6.15) and expand the binomial in the integral on the right-hand side: l dmk X Gj ¼ dt j¼0

! k i 1X ðmiþa j mkiþb j þ miþb j mkiþa j Þ 2 i¼0 k !

ðmb j mkþb j þ ma j mkþa j Þ ;

ðk ¼ 0; 1; 2; . . .Þ:

ð6:17Þ

Expression (6.17) is an infinite system of ordinary differential equations for the moments mk. If aj and bj are nonzero, the number of equations in any finite subsystem is less than the number of unknowns, so all finite subsets of the system of equations are unclosed. Sometimes the exponent in the power function of volume appearing in the coagulation kernel will be fractional. Then fractional moments will appear in the


right-hand side of Eq. (6.17). In such a case regularization of the system (6.17) can be achieved by interpolating the fractional moments through the integer ones [2] or by using the parametric method. Let us consider both methods successively. Suppose we know the moments mk (0) at the initial instant t ¼ 0. Introduce _ dimensionless moments mk ðtÞ ¼ mk ðtÞ=mk ð0Þ, where k can be either integer or fractional. The left-hand side of Eq. (6.17) contains only integer moments, whereas the right-hand side may contain fractional moments as well. Our goal is to express fractional moments through integer ones. Consider a fractional moment of the order j þ , where j is the integral part and 0 < < 1, and integer moments ms, msþ1, . . ., msþr where s < j þ < s þ r. The logarithm of the fractional moment is sought in the form of a Lagrange interpolation polynomial _

lnðm jþx Þ ¼

sþr X _ LðrÞ q ð j þ xÞlnðmq ðtÞÞ;

ð j ¼ 1; 2; . . .Þ;

ð6:18Þ

q¼s ðrÞ

where Lq ð j þ xÞ are the coefficients of the interpolation polynomial when interpolating over r þ 1 nodes LðrÞ q ð j þ xÞ ¼

sþr Y j þ xr : q p p¼s

ð6:19Þ

p„q

With the help of Eq. (6.19), fractional moments can be represented as ðrÞ sþr Y m jþx ðtÞ mq ðtÞ Lq ð jþxÞ ¼ ; s j s þ r; 0 < x < 1 m jþx ð0Þ mq ð0Þ p¼s

ð6:20Þ

p„q

and thus the system (6.17) becomes regularized. Let us apply this method to a kinetic equation with the kernel KðV; wÞ ¼ GVa wa ;

0 a 1:

ð6:21Þ

We shall restrict ourselves to the first four moments in the system of equations (6.17); for the fractional moment entering the right-hand side, we shall make use of two-point interpolation, expressing the moments m j+a through the integer moments m j and m jþ1. This will give us a closed system of equations for the first four moments: dm0 1 m0 ðtÞ 22a m1 ðtÞ 2a 2 ; ¼ Gma ð0Þ dt 2 m0 ð0Þ m1 ð0Þ dm1 ¼ 0; dt

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342

dm2 m1 ðtÞ 22a m2 ðtÞ 2a 2 ¼ Gm1þa ð0Þ ; dt m1 ð0Þ m2 ð0Þ

ð6:22Þ

dm3 m1 ðtÞ 1a m2 ðtÞ 2a 2 2 ¼ 3Gm2þa ð0Þm1þa ð0Þ ; dt m1 ð0Þ m2 ð0Þ whose solution is as follows: – for a 6¼ 0.5: B0 t d m0 ðtÞ ¼ m0 ð0Þ 1 þ ; d

m1 ðtÞ ¼ m1 ð0Þ;

B0 t d m2 ðtÞ ¼ m2 ð0Þ 1 þ ; d

m3 ðtÞ ¼ m3 ð0Þ 1 þ

ð6:23Þ

1 !1a B3 B2 t 1þd 1þ ; 2B2 d

– for a ¼ 0.5: m0 ðtÞ ¼ m0 ð0ÞexpðB0 tÞ;

m1 ðtÞ ¼ m1 ð0Þ;

m2 ðtÞ ¼ m2 ð0ÞexpðB2 tÞ;

2 B3 m3 ðtÞ ¼ m3 ð0Þ expðB2 tÞ þ 1 ; B2

ð6:24Þ

where the following dimensionless parameters have been introduced:

t ¼ Gm12a t;

B2 ¼

B0 ¼

2 m1þa ð0Þ ; m2 ð0Þm12a

ma2 ð0Þ ; 2m0 ð0Þm12a

B3 ¼

d¼

1 ; 12a

2 m1þa ð0Þm2þa ð0Þ : m3 ð0Þm12a

Now, let us discuss the parametric method. It is based on the assumption that the sought-for distribution belongs to some definite class. The choice of this class is usually based on general physical reasonings about the possible shape of the distribution produced in a specific process. For example, when studying the developed turbulent flow of emulsion in a pipe, the distribution of droplets is assumed to be a logarithmically normal distribution or a gamma distribution. Consider these two distributions in sequence.


Let the particle distribution belong to the class of logarithmically normal distributions [1] N ln2 ðV=V0 Þ nðV; tÞ ¼ pffiffiffiffiffiffi exp ; 3 2psV 18 ln2 s

ð6:25Þ

where N is the number concentration of particles, s2 – the variance, and V0 – a parameter related to the average particle volume Vav through the expression vav ¼ v0 exp (1.5 ln2 s). The form of the distribution (6.25) does not change with time, although its parameters N(t), V0(t), and s(t) are time-dependent. We then substitute Eq. (6.25) into the moment equations (6.15) and take successively k ¼ 0, 1, 2. In the case of power dependence of K (V, o) on V and o the righthand side may contain other moments (fractional or integer) different from the first three. Then the obtained system of equations can be regularized by writing additional relations that follow from the property of the distribution (6.25) 9 2 2 mk ¼ m1 Vk1 exp 1Þln s : ðk 0 2

ð6:26Þ

As a result we obtain a closed system of equations for m1, V0, and s. In fact, there are only two equations, because during the process of coagulation, m1 ¼ const. Suppose now that the particle distribution belongs to the class of gamma distributions [1]

nðV; tÞ ¼

j ði þ 1Þ V i V exp ; V0 V0 V20 i!

ð6:27Þ

where j is the volume concentrations of particles, which remains constant during coagulation, and V0 and t are distribution parameters connected with p the average ffiffiffiffiffiffiffiffiffiffi volume and the variance through the relations V0 ¼ V=ði þ 1Þ; ¼ V= i þ 1. The form of the distribution does not change with time; only N and V0 change. The distribution (6.27) allows us to seek the solution of equations (6.15) as an expansion of n(V, t) in terms of associated Laguerre polynomials Lki [5]: nðV; tÞ ¼

V V0

i

1 V X V exp ak Lki : v0 k¼0 V0

The first two polynomials are L0i ¼ 1;

L1i ¼ i þ 1V=V0 ;

and other are defined by the recurrent relation ðn þ 1ÞLnþ1;i ðV=V0 2n þ i þ 1ÞLni þ ðn þ iÞLn1;i ¼ 0:

ð6:28Þ

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344

The orthogonally conditions for polynomials give us expansion coefficients: 1 ð k! V V Lki ak ¼ nðV; tÞd ; Gðk þ i þ 1Þ V0 V0

ð6:29Þ

0

where G(x) is the gamma function. In particular, the first two coefficients are equal to a0 ¼

1 ð k! V nðV; tÞd ; Gðk þ 1Þ V0

a1 ¼ 0:

0

Using the two-moment approximation, we can write nðV; tÞ ¼

i NðtÞ V V exp ; V0 ðtÞi! V0 V0

1 ð1 ð dðN=V0 Þ 1 N 2 wþV exp ¼ dt 2 V0 V0

ð6:30Þ

0 0

KðV; wÞ

V V0

i

w V0

i dwdV;

ði þ 1ÞNV0 ¼ j:

The system of equations (6.30) enables us to find the volume distribution of particles as a function of time, and to determine the parameters of this distribution. Practice has shown that the method of moments describes the kinetics of coagulation in a satisfactory manner only at the initial stage. When certain restrictions are imposed on the form of the coagulation kernel and on the initial distribution, there exists self-similar solution at t ! 1 [6]. In the general case, however, one has to rely on numerical methods to obtain the solution of the kinetic equation.

6.2 Fundamental Features of the Coagulation of Particles

We may distinguish two stages in the process of particle coagulation. During the first stage (the transport stage) particles approach each other and come into contact; during the second (kinetic stage) they coagulate if they are solid) or coalesce if they are liquid or gaseous. Experimental studies of droplet coalescence on a plane interface between two fluids [7] give grounds to suggest that the time of particle coalescence is much shorter than the time of their mutual approach. Thus it can be assumed that the duration of the coalescence process is limited by the transport stage. Then the

6.2 Fundamental Features of the Coagulation of Particles

coagulation kernel can be found from detailed analysis of the process of mutual approach of particles – all the way to their eventual collision, assuming that each collision of particles results in their coagulation or coalescence. The character of particles’ mutual approach (and thereby of interparticle collisions) depends on the hydrodynamic regime of the flow. In a laminar flow, or in the process of particle sedimentation in a quiescent fluid, collision frequency can be determined by studying trajectories of particles’ relative motion right up to the collision. In a turbulent flow, particle motion is caused by random turbulent fluctuations of the carrier fluid. The mutual approach and collision of particles can thus be considered as a random process as well. Let us examine both cases. Under the action of gravity, particles of different sizes settle with different velocities. As a result large particles overtake small ones, and collisions become possible. By studying particles’ trajectories relative to other particles, we can determine the collision frequency of particles. The volume concentration of particles is low, so the analysis can be restricted to the relative motion of two particles and to pair collisions. Examination of relative motion of particles is conducted in a spherical coordinate system (r, y, F) with the origin in center of the largest particle (Fig. 6.2). In this frame of reference the carrier fluid moves relative to the largest particle with a constant velocity U; on a large distance from the particle this velocity is equal to the

Fig. 6.2 Spherical coordinate system.

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Q1

sedimentation velocity for this particle. Another particle of smaller size moving together with the carrier flow goes around the large particle, taken as the frame of reference, either touching it or passing by. Due to the small size of particles, their motion should be regarded as inertialess. So when the small particle is sufficiently far from the large one, its trajectory coincides with the streamline of the carrier fluid. In the vicinity of the large particle, one can observe a deviation of the trajectory from the streamline due to hydrodynamic interactions with the carrier fluid and interactions between the particles. Interaction forces can be hydrodynamic, molecular, and electrostatic. Hydrodynamic forces are resistance forces acting on a particle that increase unboundedly as the gap between particle surfaces gets smaller. Molecular forces are attractive van der Waals forces acting only at relatively small distances between particles. Electrostatic forces are repulsive forces created by double electrical layers at particle surfaces and (or) interaction forces between electrically conductive particles (charged or uncharged) placed in an external electric field. To determine the relative trajectories of particles, it is necessary to know all of the above-mentioned forces; the evaluation of each force presents an independent hydrodynamical or physical problem. The whole set of trajectories of the smaller particle can be divided into two groups: trajectories that passing by the larger particle, and trajectories that lead to particle collision. These two trajectory groups are separated by the so-called limiting trajectories (Fig. 6.3). In the remote regions of the carrier flow (i.e., far away from the large particle) the limiting trajectories form a flow tube whose cross-sectional area G (V, o) is called the collision cross section of particles having volumes V and o (V > o). Then the collision frequency of particles with volumes V and o may be written as KðV; wÞ ¼ GðV; wÞjUV Uw j;

ð6:31Þ

where UV and Uo are velocities of unhindered particle sedimentation, that is, sedimentation velocities that would be observed in the absence of hydrodynamic interactions. These velocities can be assumed equal to the corresponding Stokesian velocities. Trajectories of a particle i are described by the equations of motion mi

dui X ðiÞ ¼ Fk ; dt k

Ji

dVi X ðiÞ ¼ Lk ; dt k

ui ¼

Fig. 6.3 Trajectories of particle 2 relative to particle 1.

dr i ; dt

ð6:32Þ

6.2 Fundamental Features of the Coagulation of Particles ðiÞ

ðiÞ

where mi is the particle mass; F k ; Lk are, respectively, forces and torques acting on the particle; Ji is the particle’s moment of inertia; and i ¼ 1, 2. If particle sizes are small enough and the density ratio of the particle and the external fluid is practically equal to unity, fluid flow can be taken as slow, that is, Stokesian, and particle motion – as inertialess. Then equations (6.32) reduce to the equations of inertialess motion in a quasistationary approximation, namely, X k

ðiÞ

F k ¼ 0;

X k

ðiÞ

Lk ¼ 0;

ui ¼

dr i : dt

ð6:33Þ

Assigning various initial positions to particle 2 and integrating equations (6.33), one obtains family of trajectories of particle 2 relative to particle 1. In a spherical coordinate system, the position of particle 2 is given by the initial coordinates of its center, r0, y0, F0. Consider a flow tube whose surface contains limiting trajectories, and let y* (F) be the angle between the vector r0 and the second vector that connects the particle center with a point on the surface of a fixed cross section of the flow tube (whose radius is therefore z ¼ r0 cosy*). At large distances from particle 1 (that is, at r0 ! 1 and y* ! 0), the flow tube’s cross-sectional area can be found from simple geometrical reasonings. Thus, in the general case when the cross section is not circular, its area equals 0 2p 1 ð 1 2 2 G ¼ lim @ r0 sin ð u ðFÞÞdFA: r0 ! 1 2 u !0

ð6:34Þ

0

If the fluid flow and the relative motion of particles are both axisymmetric, then y* does not depend on F, the cross section is circular, and G ¼ lim ðpr02 sin2 ð u ÞÞ: r0 ! 1 u ! 0

ð6:35Þ

Collision frequency is higher in a turbulent flow than in a laminar flow or during sedimentation in a quiescent fluid. Particles suspended in the fluid are carried along by turbulent fluctuations and move chaotically inside the flow region. Since their fluctuational motion is alike to Brownian motion, it can be characterized by the effective diffusion coefficient Dt and the problem of finding particles’ collision frequency can be reduced to a diffusion problem in the same manner as it was first done by Smoluchowski for Brownian motion [8]. Such an approach was attempted by Levich [9] in application to coagulation of hydrodynamically noninteracting particles in a turbulent flow. The decision to ignore hydrodynamic interactions ahs resulted in exaggerated values of collision frequency as compared to experimental data on turbulent motion of emulsions in pipes and agitators [10,11]. The influence of hydrodynamic interactions between particles on the value of particles’ interdiffusion coefficient has been investigated in [12–16]. It was shown

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that correct accounting for hydrodynamic interactions between particles assures good agreement with the experiment. When diffusion is the main mechanism responsible for the mutual approach of particles, the number of collisions between particles having radii R1 and R2 per unit time is equal to the flux of particles of radii R2 toward the particle of radius R1. If diffusion equilibrium is established much faster than concentration equilibrium, the problem reduces to that of solving the stationary diffusion equation in a force field [17,18] F r Dt rn n ¼ 0; ð6:36Þ h

Q2

where n is the number concentration of particles 2; F is the force of interaction between particles 1 and 2 (resulting from electrodynamic gravitational, molecular, or electrostatic fields); h is the coefficient of hydrodynamic resistance for particles approaching each other along their line of centers. Let u be the relative velocity of such particles and F – the force of hydrodynamic resistance experienced by particle 2. Then F h ¼ hu:

ð6:37Þ

Instead of the hydrodynamic resistance coefficient one sometimes works with mobility u ¼ h1, which helps express velocity in terms of force: u ¼ uF h :

ð6:38Þ

If every collision of fluid particles leads to their coalescence, the boundary conditions for Eq. (6.36) are n¼0

at r ¼ R1 þ R2 ;

n ¼ n0

at r ! 1:

ð6:39Þ

where n0 is the concentration of particles 2 far away from the test particle 1. By solving Eq. (6.36) under the conditions (6.39) we can find n and thereby the flux of particles 2 toward the test particle 1: ðð F J¼ Dt rn n n ds: ð6:40Þ h s

where n is the outward normal to the surface element ds. Hence in order to determine the collision frequency of particles, it is necessary to obtain the interaction forces acting between the particles and then to find particles’ trajectories or the diffusion flux of particles. In the latter case one needs to know the effective coefficient of turbulent diffusion. This procedure gives us the coagulation kernel. Sometimes the obtained kernel turns out to be asymmetric with respect to particles’ volumes, and then symmetrization of the kernel is in order. Only after that one can proceed to study the kinetics of coagulation, that is, to solve the kinetic equation of coagulation.

6.3 A Model of Turbulent Diffusion


Transport of small particles in a turbulent flow can be considered as diffusion and characterized by an effective diffusion coefficient. Before we proceed to determine this coefficient, it is worthwhile to recall the basic features of fluid particle motion in turbulent flows. For developed turbulence, these features have been studied extensively in [9,19–21]. Let us therefore restrict ourselves to those features that pertain to particle collisions. A turbulent flow means that a random fluctuational motion (characterized by a set of fluctuational velocities ul) is superimposed onto the average flow that has a definite speed U and a definite direction. Turbulent fluctuations are characterized not just by velocities, but also by the distances at which these velocities experience a noticeable change. These distances are known as fluctuation scales and denoted by l. The full set of l’s represents a spectrum of turbulent fluctuations, where l can vary from 0 to the maximum value, which is equal by the order of magnitude to the characteristic linear scale L of the flow region. Every fluctuational motion is characterized by its Reynolds number Rel ¼ lul/ve, where ve is the coefficient of kinematic viscosity of the carrier flow. Fluctuations with l L are called large-scale fluctuations. For such fluctuations, Rel 1, so the fluid flow induced by these fluctuations is inviscid. The Reynolds number decreases as the fluctuation scale gets smaller. At l ¼ l0, which is called the inner (or Kolmogorov) turbulence scale, the Reynolds number is approximately equal to Re ¼ Rel0 ¼ lul0 =ne 1:

ð6:41Þ

It means that fluctuations with l l0 are viscous, and fluctuational motion on such a scale is accompanied by energy dissipation. Fluctuations with l L are referred to as small-scale. They are induced by larger fluctuations. The energy is transmitted from large-scale fluctuations to small-scale ones and then dissipates into heat. Thus turbulent motion is accompanied by considerable dissipation of energy. Energy loss per unit mass per unit time e is called the specific dissipation of energy and is one of the most fundamental parameters of turbulence. Since energy is drawn from large-scale fluctuations, e depends on U and L. Thus e can be estimated from dimensionality considerations: e U 3 =L:

ð6:42Þ

The velocity of small-scale fluctuations with l0 l L can depend only on e and l. Therefore ul ðelÞ1=3 :

ð6:43Þ

Since at l l0, the condition Rel0 1 must hold, the inner scale of turbulence

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follows from Eq. (6.41) and Eq. (6.43): l0 ne =ul0 ðn3e =eÞ1=4 :

ð6:44Þ

According to the hypothesis put forward by Landau and Levich, viscous fluctuations with l < l0 attenuate rather gradually instead of disappearing abruptly. The motion of the fluid reduces to a set of mutually independent periodic motionsp whose ffiffiffiffiffiffiffiffiffi periods T are constant for all l < l0. The value of T can be estimated as T ne =e. The velocities of fluctuations with l < l0 are given by pffiffiffiffiffiffiffiffiffi ul l=T l e=ne :

ð6:45Þ

Combining the relations (6.43) and (6.45), one gets a general expression for the velocity of turbulent fluctuations: ( ul

ðelÞ1=3 lðe=ne Þ

1=2

at

l > l0 ;

at

l < l0

ð6:46Þ

Consider now the motion of small particles of radius R in a turbulent fluid flow. Volume concentration of particles is assumed to be small so that their influence on the carrier fluid can be ignored. Large-scale fluctuations with l R transport particles together with the adjacent fluid layers, whereas small-scale fluctuations with l R are not capable of drawing particles into their motion. In the latter case, particles behave as immovable bodies relative to the fluid. Fluctuations having intermediate scales can draw particles into their motion only to a moderate extent. Consider a situation that is typical for particles with low inertia, when densities of the particle ri and the surrounding fluid ri are roughly the same, and the particle radius is much smaller than the inner scale of turbulence, that is, R l0. For example, for water–in–oil emulsions, ri/re 1.1–1.5. Denote through u0 the fluid velocity at the point where the particle is located, and through u1 – the particle’s velocity relative to the fluid. If the particle were completely entrained by the surrounding fluid, the force acting on the particle would be the same as if the particle were composed of this fluid, that is, 43 pR3 re dudt0 . But because the entrainment is only partial, the particle experiences a resistance force F, which, given the conditions R l0 and u1 < ul0 ; is determined by the Stokes formula F ¼ 6prveRu1. The equation of motion for the particle takes the form

m

du1 4 3 du0 du1 4 du0 þ pR re þ ¼ pR3 re 6pri ne Ru1 ; dt dt dt dt 3 3

ð6:47Þ

where m ¼ 2pR3re/3 is the particle’s virtual mass. Let us estimate particle’s acceleration that appears in Eq. (6.47). The period T is constant for motions with scales l < l0, so


rffiffiffiffiffi e du1 u1 : u1 dt T ne

ð6:48Þ

To estimate the acceleration of the fluid at the point occupied by the particle, let us take the maximum acceleration of fluctuational motion in the interval of scales under consideration, that is, at l ¼ l0: 3 1=4 du0 ul0 l0 e e0 0 : dt T l ne

ð6:49Þ

Substitution of Eq. (6.48) into Eq. (6.47) and comparison with Eq. (6.49) shows that the vectors u1 and du0/dt are approximately collinear. Then, in view of Eq. (6.49), the approximate solution of Eq. (6.47) is given by u1

2R 2 jre ri jl0 e0 n1 1=2

2R2 ðri þ 0:5re Þe0 þ 9re ne

:

ð6:50Þ

The ratio u1 =ul0 tells us to which extent the particle is entrained by the fluctuation l. When u1 =ul0 1, the entrainment is complete, whereas at u1 =ul0 1, entrainment is absent. For fluctuations with scales l < l0, Eq. (6.46) and Eq. (6.50) give us 1=2 1=2

u1 2R2 jre ri je0 ne : ul R2 ðre þ 2ri Þe1=2 n1=2 þ 9re ne e 0

ð6:51Þ

For water-in-oil emulsions, where re 800 kg/m3, ri 1200 kg/m3, ne 105 m2/s, l0 104 m, e0 10 J/kg s, Eq. (6.51) gives u1 8R2 1: ul 28R2 þ 72103 So, water droplets of radius R < l < l0 are, for all practical purposes, fully entrained by fluctuations whose scale is l. Transport of particles of radii R l0 is defined by two parameters: velocity ul and fluctuation scale l. From these parameters, one can form a combination having the dimensionality of diffusion coefficient: ð0Þ

Dt lul :

ð6:52Þ

Substitution of fluctuational velocity (6.46) into Eq. (6.52) gives the diffusion coefficient for particles suspended in a turbulent flow: ( ð0Þ Dt

4 ðel Þ1=3

at

l > l0 ;

2 l ðe=ne Þ1=2

at

l < l0 :

ð6:53Þ

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Let us see how the obtained formulas relate to diffusional interaction of particles. Let r be the distance between particle centers. In the course of diffusion, resistance forces that obstruct the mutual approach of two particles are mostly confined to the region where the gap between the particles is very small: d < l0 (R1 þ R2). In this region, Eq. (6.53) gives the following diffusion coefficient: ð0Þ

Dt e1=2 l =n1=2 e nl =l0 : 2

2

2

ð6:54Þ

This still leaves open the question of the scale of fluctuations that might cause the particles to come closer together, because large fluctuations will transport the particle pair as a whole without changing the interparticle distance. It is evident that particles of the same size (R1 ¼ R2) can be brought closer together only by fluctuations with l r, and particles whose sizes are largely different (R1 R2) – only by fluctuations with l r R1, because the larger particle (when put next to the smaller one) can be considered as a plane wall, in whose vicinity small-scale fluctuations get attenuated. If the relative size of the two particles is in the intermediate zone, we can determine the relevant scale of fluctuations from the approximation that obeys both of the above-considered limiting cases together with the condition of symmetry of the diffusion coefficient with respect to particle radii, namely, ð0Þ ð0Þ Dt ðR1 ; R2 Þ ¼ Dt ðR2 ; R1 Þ. As a result we obtain rR1 R2 R1 R2 l ðR1 þ R2 Þ þ 2 : R1 þ R 2 R1 þ R22 R1 R2

ð6:55Þ

Going back to Eq. (6.54), we get the following expression for the interdiffusion coefficient for particles of radii R1 and R2: ð0Þ

Dt

¼b

ne

2 2 ðR 1 þ R 2 Þ

l0

2 rR 1 R 2 R1 R2 þ 2 ; R1 þ R2 R 1 þ R 22 R 1 R 2

ð6:56Þ

where we have introduced a correction factor b having the order of unity. This factor is required because the relations (6.54) and (6.55) are merely approximations accurate to the order of magnitude. The formula (6.56) was obtained under the assumption that particles are fully involved into relative motion by fluctuations having the scale l. So, this formula can be used only if the particles are spaced relatively far from each other. When they get closer together so that the gap d between particle surfaces is equal by the order of magnitude to the smaller particle’s radius, hydrodynamic resistance force begins to influence the particles’ relative velocity; the force goes to the infinity at d ! 0. To take this force into account, we employ the approach that is based on the Langevin equation [22,23]; this approach is also used in statistical physics when considering Brownian motion of a particle subject to an external random force. Driven by random turbulent fluctuations, the particle changes its direction many times during a short time interval. Thus it is next to impossible to trace its trajectory


visually. Besides, particle displacements cannot serve as useful characteristics of particle motion, because the average displacement during a finite time interval is equal to zero. Instead, the mean-square displacement emerges as the main characteristic of particle motion for finite time intervals. One should keep in mind that ‘‘mean’’ here implies ensemble averaging rather than time averaging. Consider two instants of time, t1 and t > t1. We make the assumption that during the relevant time intervals the particle will change its direction quite often, and, moreover, any two displacements taking place during non-overlapping time intervals are fully independent. For simplicity’s sake, we restrict ourselves to the case of one-dimensional motion. Let s denote the path traveled by the particle during the time interval (0, t). Similarly, the path s1 will correspond to the interval (0, t1) and the path s2 – to the interval (t1, t). From the condition of statistical independence of s1 and s2 and keeping in mind that positive and negative displacements should have equal probabilities, we derive the mean-square displacement: E 2 D s ¼ ðs1 þ s2 Þ2 ¼ s21 þ s22 :

ð6:57Þ

Denoting hs2 i ¼ yðtÞ; we rewrite the last equation as yðtÞ ¼ yðt1 Þ þ yðtt1 Þ:

ð6:58Þ

The solution of this equation has the form yðtÞ ¼ s2 ¼ 2Dt t;

ð6:59Þ

where Dt is the coefficient of turbulent diffusion that still needs to be determined. Consider now the equation of motion of a particle subject to a force exerted by the surrounding medium. This force consists of a systematical part – the friction force, and a random part – the force F whose average value is zero. The systematic force may also include the force of gravity; however, it is negligible for sufficiently small particles. Let us also ignore any other external forces. Then in the inertialess approximation the equation of motion of such a particle (i.e., the Langevin equation) along the X-axis has the form h

dx þ F ¼ 0; dt

ð6:60Þ

where h is the coefficient of hydrodynamic resistance. From Eq. (6.60), it follows that the particle’s displacement from its initial position X0 during the time t is equal to

X X0 ¼

ðt 1 FðtÞdt: h 0

ð6:61Þ

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The integral on the right-hand side of Eq. (6.61) stands for the impulse of the random force F during the time t. For any instant t1 in the range 0 < t1 < t, this integral can be written as ðt1

ðt

ðt

FðtÞdt ¼ FðtÞdt þ FðtÞdt: 0

ð6:62Þ

t1

0

Taking the square of both sides of Eq. (6.62), averaging the result and remembering that random impulses for any two non-overlapping time intervals should be independent, we obtain: 12 + *0t 12 + *0 t 12 + *0ðt ð1 ð @ FðtÞdtA ¼ @ FðtÞdtA þ @ FðtÞdtA : 0

0

ð6:63Þ

t1

This equation is similar to Eq. (6.57), so its solution is 12 + *0ðt @ FðtÞdtA ¼ 2Bt:

ð6:64Þ

0

The particle’s mean-square displacement is given by Eq. (6.61): 1 hðX X0 Þ2 i ¼ 2 h

12 + *0ðt @ FðtÞdtA :

ð6:65Þ

0

Substituting the relations (6.59) and (6.64) into this expression, we get B Dt ¼ 2 ; h

12 + *0ðt 1 @ B¼ FðtÞdtA : 2t

ð6:66Þ

0

In view of the fact that a similar approach is applied to Brownian diffusion, it is worthwhile to point out the main distinction of turbulent diffusion from Brownian one. In the course of Brownian diffusion, particles perform random thermal motion due to collisions with molecules of the surrounding medium. In [22], the corresponding force acting on the test particle is modeled as a quasi-elastic force F ¼ aX proportional to the displacement X. As a result the form of Eq. (6.60) changes: a term proportional to X appears in the equation, and the condition of thermodynamic equilibrium of the system leads us to B ¼ hkT;

Dbr ¼

kT : h

ð6:67Þ


Q3

Consequently, the coefficient of Brownian diffusion is inversely proportional to the first degree of the particle’s coefficient of hydrodynamic resistance h. In the case of turbulent diffusion, the situation is different. Particle motion driven by turbulent fluctuations is quite independent from any thermal fluctuations. Therefore B ¼ const and the coefficient of turbulent diffusion is inversely proportional to the second degree of the coefficient of hydrodynamic resistance. If the distance at which h can vary significantly is greater than the fluctuationdriven displacement of a particle, then h does not depend on displacement X. This is the case, for instance, when the particle is moving in a region adjacent to a wall or approaches another particle. Consider a pair of spherical particles with radii R1 and R2 (R1 R2) approaching each other along their line of centers. In the inertialess approximation, the equation of motion of one particle relative to the other (which is taken as motionless) is F 1r þ F r ¼ 0;

F 1r ¼ hr0 u0r ;

F r ¼ hr ur :

ð6:68Þ

where F1r is the force that would be exerted on the particle by the surrounding fluid if the particle were at rest; Fr is the drag force caused by the particle’s own motion; u0r is the fluid’s velocity at the point where the particle is located; ur is the particle’s velocity. When the interparticle distance is much larger than the size of each particle, both particles will be completely entrained by the fluid, so that hr0 ¼ hr ; u0r ¼ ur . A decrease of the interparticle distance leads to a change of the resistance coefficients hr0 and hr. While the first coefficient does not change much, the second one increases rapidly and becomes infinite when the particles touch. One can see from Eq. (6.68) that the motion of one particle in the vicinity of another is constrained and its velocity is equal to ur ¼

hr0 0 u : hr r

ð6:69Þ

Equation (6.68) for small displacements describes a motion that is similar to the motion of an unconstrained particle with the similarity coefficient hr0 =hr . This allows us to write the coefficient of turbulent diffusion (6.66) in a similar form, namely,

Dt ¼ ð0Þ

0 2 hr : hr

ð0Þ Dt ðrÞ

ð6:70Þ

where Dt ðrÞ is the coefficient of turbulent diffusion for particles performing unconstrained motion (see Eq. 6.56). The expression (6.70) does not take into account the influence of the second particle. To see how the influence of both particles will affect the velocity with which they approach each other, we shall proceed as follows. Let u be the velocity of one particle relative to the other, and u1 and u2 – particle velocities relative to a frame of

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356

reference whose origin is placed into a point on the line of centers of the two particles. Then u ¼ u1 u2. The forces Fi acting on the particles are equal in magnitude but opposite in sign. Then the mutual approach of these two particles will be characterized by the hydrodynamic resistance coefficient equal to hr ¼

F F h1 h2 ¼ ¼ ; u1 u2 F=h1 þ F=h2 h1 þ h2

ð6:71Þ

where hi is the hydrodynamic resistance coefficient of i-th particle (i ¼ 1,2). When the particles are still far from each other, their mutual influence is insignificant and hi ¼ 6prneRi. Then Eq. (6.71) yields hr ¼ hr0 ¼ 6prn

R1 R2 ; ðR1 þ R2 Þ

r R1 ; R2 :

ð6:72Þ

the resisAt relatively small distances d ¼ r R1 R2 between particlepsurfaces, ffiffiffi tance coefficient behaves as 1/d for solid particles [24] and as 1= d for droplets with mobile surfaces [25]. We shall restrict ourselves to the case of solid particles. Then we have for small clearances between the particles hr ¼ hr0

ðR1 þ R2 Þ : ðrR ðR1 þ R2 Þ 1 R2 Þ R1 R2

2

ð6:73Þ

Combining the far and near asymptotics (6.72) and (6.73), we get ! R1 R2 R1 R2 ðR1 þ R2 Þ hr ¼ 6prne 1þ : ðR1 þ R2 Þ ðR1 þ R2 Þ2 ðrR1 R2 Þ

ð6:74Þ

Let us now substitute the relations (6.56), (6.72), (6.74) into (6.70). As a result we obtain the coefficient of mutual turbulent diffusion for spherical particles that takes hydrodynamic interactions into account: Dt ¼ b

ne ðR1 þ R2 Þ2 ðs þ gÞ2 s2 2

l0

ðs þ kÞ2

:

We have introduced the following dimensionless parameters: D ¼ ðrR1 R2 Þ=ðR1 þ R2 Þ; k ¼ R1 R2 =ðR1 þ R2 Þ2 :

g ¼ R1 R2 =ðR21 þ R22 R1 R2 Þ;

ð6:75Þ

6.4 Hydrodynamic, Molecular, and Electrostatic Forces


Hydrodynamic forces acting on particles reveal themselves when the particles move relative to each other and to the surrounding fluid. Generally speaking, these forces can deform surfaces of particles (droplets, bubbles), especially when the gap between the particles becomes smaller than the particle size. Bur when the particles are sufficiently small and their surfaces are covered by adsorbed impurities which stabilize the surface, this deformation can be ignored. Henceforth, we shall assume the particles to be nondeformable and spherical. Among the factors influencing the value of the hydrodynamic force are the presence of neighboring particles and the velocity field of the carrier medium. For particles with radius up to 100 mm, it is safe to suppose that their motion is taking place at small Reynolds numbers. Thus, if both translational and angular velocities of a particle at each instant of time are given, the hydrodynamic force acting on the particle can be found from the Stokes equations. In this type of problems, one usually assumes that the disperse phase has a low volume concentration, so that it becomes permissible to consider pair interactions only. In the frame of reference placed at the center of the larger particle S1, the Stokes equations with the corresponding boundary conditions are ru ¼ 0;

me ru ¼ D p;

ð6:76Þ

ujSi ¼ vi þ Vi ðrr i Þ;

jrr i j ¼ Ri ;

u ! U 1 ¼ ðU 1 cos u;

U 1 sin uÞ at

ði ¼ 1; 2Þ;

ð6:77Þ

r ! 1;

where vi is translational velocity of the center of the particle Si, U1 – velocity of the carrier fluid far from the particle (further on, we shall simply take U ¼ const), Vi – angular (rotational) velocity of the particle Si, ri – radius vector of the particle center, and Ri – particle radius. The force and torque exerted on the particle by the surrounding fluid are, respectively [24], ðð F ih ¼

ðð dsð pI þ 2me EÞ;

si

Lih ¼

ðrri Þ ð pI þ 2me EÞds;

ð6:78Þ

si

where E is the rate-of-strain tensor, I is the unit tensor, and ds ¼ nds is the surface element pointing outward (i.e., toward the fluid). Hence the solution of the boundary problem (6.76)–(6.77) gives F 0ih and L0ih as functions of particle sizes and velocities, interparticle distance and viscosity me of the external fluid. If the particles are droplets with mobile surfaces, then in addition to the listed parameters there appears one extra parameter, namely, the internal fluid

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358

viscosity mi: F ih ¼ F i ðR1 ; R2 ; v1 ; v2 ; V1 ; V2 ; r 2 r 1 ; me ; mi Þ;

ð6:79Þ

Lih ¼ Li ðR1 ; R2 ; v1 ; v2 ; V1 ; V2 ; r 2 r 1 ; me ; mi Þ:

Linearity of the Stokes equations implies that the sought-for expressions for Fih and Lih can be found by superposition and approximation of particular solutions of two well-known problems: motion of particle S2 relative to particle S1 in a quiescent fluid; and bypassing of two immovable particles S2 and S1 spaced at a certain distance from each other by a flow whose velocity U at the infinity is given. We then conclude that the force Fh and torque Lh can be written as Fh ¼ Fs þ Fe;

Lh ¼ Ls þ L e :

ð6:80Þ

The terms Fs and Ls are Stokesian components which correspond to a fixed spherical particle S1 under the conditions v1 ¼ O1 ¼ 0 and U 6¼ 0. The terms Fe and Le are the respective contributions to the hydrodynamic force and torque from the motion of particle S2 (needless to say, in this case v2 6¼ 0 and O2 6¼ 0) in a fluid whose velocity at the infinity is zero: U ¼ 0. The terms Fs and Ls can be determined as follows. Denote through u0 the velocity of Stokesian flow that bypasses an isolated particle S1. Velocity components in the meridional section (F ¼ const) are [9] 3R1 R31 3R1 R31 u0r ¼ U 1 cos u 3 1 ; u0 u ¼ U 1 sin u 3 þ 1 : ð6:81Þ 2r 2r 4r 4r The characteristic distance at which the Stokesian velocity field (6.81) experiences a noticeable change is equal to R. Thus as long as the region under consideration is much smaller than R, the flow in the meridional section can be considered as an approximately quasi-planar flow resulting from the superposition of a uniform flow and a simple shear flow. The uniform flow induces the force Fs and the shear flow induces the torque Ls acting on the particle S2. On a sufficiently large distance from S2 they are equal to F s ¼ 6pme R2 u0 ;

Ls ¼ 4pme R32 r u0 :

ð6:82Þ

A decrease of the gap d between the particles S1 and S1 would cause a violation of the dependences (6.82). For narrow gaps between the particles, distortion of the velocity field can be taken into account by introducing the corresponding resistance coefficients into Eq. (6.82). While doing so, it is necessary to keep in mind that the form of resistance coefficients is different for the motion of the particle S2 along and perpendicular to the surface S1. Thus Fsr ¼ 6pme R2 fsr u0r ;

Fs u ¼ 6pme R2 fs u u0 u ;

Lsj ¼ 8pme R32 tsj G;

ð6:83Þ

where fsr and fsy are translational resistance coefficients that correspond to the motion perpendicular and parallel to the surface S1, tsj is a rotational resistance


coefficient, and G ¼ 0.5| · u0. Resistance coefficients in Eq. (6.83) depend on the relative gap between the particles, s ¼ (r R1 R2)/(R1 þ R2), and on the ratio of particle radii, k ¼ R2/R1. As evidenced by the solutions of the corresponding hydrodynamic problems [24– 36], the coefficients fsr, fsy, and tsj are not much different from unity when the distance from S1 is on the order of several radii of S2 or smaller, and remain finite as d ! 0. Since the analogous coefficients in the expressions for Fe and Le increase unboundedly as d ! 0, we can claim without sacrifice of precision that fsr, fsy, and tsj 1. Consider now the hydrodynamic force and torque resulting from particle S2’s own motion. This problem has been extensively studied, and the currently available solutions embrace practically all types of particle’s motion: along the line of centers; perpendicular to the line of centers in the meridional plane; rotation about an axis that is perpendicular to the meridional plane. In the case when one spherical particle moves in the vicinity of another particle or a plane, that is, when the gap is small compared to the smaller particle’s radius, the obtained numerical results are supplemented by near asymptotic and far asymptotic relations. The three types of particle motion we just mentioned give the following expressions for the components of force Fe in the meridional plane and for the torque Le acting on the particle: Fer ¼ 6pme R2 fer vr ;

Fe u ¼ 6pme R2 fe u v u þ 6pme R22 fe u1 W;

Lej ¼ 8pme R22 tej vj 8pme R32 tej1 W:

ð6:84Þ

Listed below are asymptotical expressions for the resistance coefficients of solid spherical particles written in dimensionless variables – the distance between the particle centers x1 ¼ (r R2)/R1 and the ratio of particle radii k ¼ R2/R1: – the near asymptotic 1 < x1 < 1.5 for k 1:

f e

2ð2 þ k þ 2k2 Þ 15ð1 þ kÞ3

f e1 f ej

15ð1 þ kÞ2 1 þ 4k

10ð1 þ kÞ2

f ej1 f er

2ð1 þ 4kÞ

2 5ð1 þ kÞ2

lnðx 1 1Þ þ 0:959;

lnðx 1 1Þ0:2526; lnðx 1 1Þ þ 0:29;

lnðx 1 1Þ þ 0:3817;

1 ð1 þ kÞ2 ðx1 1Þ

1 þ 7k þ k2 lnðx 1 1Þ þ 0:97: 5ð1 þ kÞ

ð6:85Þ

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360

– the far asymptotic x1 1/k for k 1:

fe u

9 1 1

1 þ ; 15x1 8x13

f ej1 1 þ

5 ; 16x 1

fe u1

f er 1 þ

1 ; 8x14

fej

1 ; 32x14

1:125 1:266 þ 2 15x 1 x1

ð6:86Þ

– finally, we have for k < 1: f er

1 12:25ðkx 1 þ 1Þ2

t ej

;

0:56k2 ðkx 1 þ 1Þ3 10:56kðkx 1 þ 1Þ2

f e

1 10:56kðkx 1 þ 1Þ2

; ð6:87Þ

:

In the intermediate region, the resistance coefficients are presented in the form of infinite series, and their numerical values can be found from tables. An analysis of relative motion of particles along their line of centers at low Reynolds numbers has been carried out in [35]. The approximate expression for the resistance coefficient of the fluid particle S2 is h ¼ 6pme R2 fer 6pme

R1 R2 R 1 þ R2

2

f ðmÞ ; d

ð6:88Þ

where d ¼ r R1 R2 is the gap between the particles and 1 þ 0:402m ; f ðmÞ ¼ 1 þ 1:711m þ 0:461m2

1 R 1 R 2 1=2 m¼ ; m R 1 þ R 2

¼ m

mi : me

The parameter m characterizes mobility of the particle surface. For m 1, the surface is, for all practical purposes, fully retarded, and the particle’s hydrodynamic behavior is that of a solid particle with resistance coefficient h 6pme

R1 R2 R1 þ R 2

2

1 : d

ð6:89Þ

This expression coincides with the first term of the asymptotic expansion (6.85) for fer. One can see from the form of the resistance coefficient that it has a nonintegrable singularity at d ! 0. Therefore particles with fully retarded surfaces cannot come into contact under the action of a finite force during any finite time interval.


The other limiting case, m 1, corresponds to a completely mobile surface. The particle behaves like a gas bubble, and its resistance coefficient is h 6pme

R1 R2 R1 þ R 2

2

R1 þ R2 1=2 0:872m : R1 R2 d1=2

ð6:90Þ

At d ! 0, the expression for h for a particle with a mobile surface has an integrable singularity. Therefore in a laminar flow (in particular, during gravitational sedimentation of particles), particles can come into contact even in the absence of the force of molecular attraction. In a turbulent flow, the coefficient of turbulent diffusion is Dt 1/h2, so in the absence of molecular attraction, any contact of particles with either immobile or fully mobile surfaces is impossible. Paper [25] derives an asymptotic expression for the resistance coefficient for the case of relative motion of two spherical particles (d ! 0) with different inner viscosð1Þ ð2Þ ities mi and mi suspended in a fluid whose viscosity is me: ð1Þ

h

ð2Þ

p2 k1=2 ðmi þ mi Þ ; 16 ð1 þ kÞ2 ðs2Þ1=2 ð1;2Þ

ð6:91Þ ð1;2Þ

i ¼ mi =me . where s ¼ 2r=ðR1 þ R2 Þ; m If one of the particles has a fully retarded surface, which corresponds to the case of relative motion of a droplet and a solid particle, the asymptotic expression for the particle’s resistance coefficient at d ! 0 is h

k 2ð1 þ kÞ3 ðs2Þ

þ

i 9p2 km 64ð1 þ kÞ2 ðs2Þ1=2

:

ð6:92Þ

Consider now the forces of molecular and electrostatic interactions between particles. In the absence of external forces (gravitational, centrifugal, electrical), uncharged particles and droplets dispersed in a quiescent fluid should be distributed homogeneously in space. But interaction always exists between particles, even in a quiescent fluid: molecular attraction (van der Waals–London forces) and electrostatic repulsion (for charged particles or for particles enveloped by electric double layers). Forces of electrostatic repulsion between particles with like charges help ensure a homogeneous distribution of particles. The ability of a system to maintain a homogeneous distribution of particles in the fluid for a long time characterizes its stability. In practice, in the majority of two-phase disperse systems, the number of particles decreases with time, while the particle size gets larger. Collision of two particles in an emulsion results in their coalescence with the formation of a single large particle. Colliding solid particles can form aggregates. The phenomena of coagulation, coalescence, and aggregation of particles occur due to the presence of van der Waals– London attractive forces. System in which aggregation, coagulation, or coalescence is taking place are called unstable. On the other hand, systems devoid of these processes are called stable.

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In the absence of external and hydrodynamic forces, stability of a disperse system depends on particle interactions that are due to surface forces: molecular attraction and electrostatic repulsion [37]. The potential of electrostatic interaction between two spheres of radii R1 and R2 (R1 > R2), whose centers are separated by the distance r is equal to [38] VRs

eR1 f21 R2 f2 1ech0 f22 2ch0 2 ln ¼ Þ ; þ 1 þ 2 lnð1e 4ðR1 þ R2 Þ f1 1 þ ekh0 f1 ð6:93Þ

where f1 and f2 are the potentials of particle surfaces, w is the inverse Debye radius, and h0 ¼ r R1 R2. The force of electrostatic repulsion between two particles is found from the following relation: FRs ¼

fR ¼

dVR eR1 f21 c ¼ fR ; dr 2

k ea e 2 22 a 2 1 þ e ; 1 k þ 1 1e2a 21

ð6:94Þ a ¼ 0:5R 1 cð1 þ kÞ

2r : R1 þ R2

This relation leads us to a frequently-used formula for the force of electrostatic repulsion between two identical particles of radius R with equal surface potentials f0: FRs ¼

eRf20 c : 2ð1 þ ea Þ

ð6:95Þ

The forces of molecular attraction between two parallel plates and between two identical spherical particles were derived in [39]. For parallel plates, the specific interaction potential is p

VA ¼

G ; 12ph2

ð6:96Þ

where h is the distance between the plates and G is the Hamaker constant equal to 1020–1010 J. For two identical spherical particles of radii R under the condition h0 R, where h0 is the minimum distance between their surfaces, this potential is equal to VAs ¼

RG : 12h0

ð6:97Þ


The formula (6.97) can be generalized for the case of an arbitrary distance r between the particle centers: V sA ¼

2 G 2R 2 2R 2 r 4R 2 þ þ ln r2 r2 6 r 2 4R 2

ð6:98Þ

and for the case of different radii R1 and R2: G V sA ¼

8k

ðs2 4Þð1þkÞ2

8k

!!

þ þln 6 ðs2 4Þð1þkÞ2 s2 ðkþ1Þ2 4ð1kÞ2 s2 ðk þ1Þ2 4ð1kÞ2

;

ð6:99Þ where s ¼ 2r/(R1 þ R2); k ¼ R2/R1. The force of molecular interaction between two arbitrary spherical particles is equal to FAs ¼

dVAs 2G fA ; ¼ dr 3R1

fA ¼

1 s ð8kð1 þ kÞ2 ðs2 4ÞÞ ð1 þ kÞ ðs2 4Þð1 þ kÞ2

þ

sð1 þ kÞ2 ðs2 ð1 þ kÞ2 4ð1kÞ2 Þ2

ð6:100Þ

! 2

ð8k þ s ð1 þ kÞ 4ð1kÞ 2

2

:

An approximate expression for the force of molecular interaction that takes electromagnetic retardation into account (leading to a decrease of the van der Waals force) is presented in [40] and has the form FAs ¼

2G fA Fð pÞ; 3R1

FðpÞ ¼

8 > > >
2:45 2:17 0:59 > > þ : 60p 120p2 420p3

ð6:101Þ

at

p 0:57;

at

p > 0:57;

where p ¼ 2p (r R1 R2)lL, and lL 103 A? is the London wavelength. The total potential energy of interaction between two spherical particles is equal to the sum of electrostatic and molecular potentials: V ¼ V A þ VR :

ð6:102Þ

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364

Q4

The energy of repulsion decreases exponentially with increase of h0 with the characteristic linear scale of energy variation lD. The energy of attraction falls off as 1/h0. Attraction therefore prevails on relatively small and relatively large distances between the particles, whereas repulsion dominates in the intermediate distance range (Fig. 6.4). Coagulation can be fast or slow. We speak of fast coagulation when only molecular interaction is considered. If, in addition, electrostatic repulsion is taken into account, we call such coagulation slow. It is a known fact that the electrolyte concentration C0 at which fast coagulation is initiated depends on the charge of counterions, that is, ions carrying the charge opposite in sign to the charge of particles. On the other hand, stability of the system is practically independent of the charge of ions or concentration of particles. This fact is in agreement with the Schulze–Hardy rule, according to which the valence of counterions is the main factor influencing the system’s stability. The values of molecular and electrostatic potentials of interparticle interaction for the boundary state of the system that separates stable states from unstable ones are determined from the following system of equations: V ¼ VA þ VR ¼ 0;

ð6:103Þ

dV dVA dVR þ ¼ 0: ¼ dr dr dr

ð6:104Þ

If the system is completely instable, in other words, if repulsion forces are not taken into account, each collision of two particles results in their coagulation. The presence of a stabilizer (electrolyte) in the system leads to the emergence of

Fig. 6.4 Potential energy of particle interaction.

6.5 Conducting Particles in an Electric Field

repulsion forces created by double electric layers on particle surfaces. Consequently, the rate of coagulation decreases, hence the name ‘‘slow coagulation’’. The rate of coagulation is characterized by the stability factor W equal to the ratio between the number of particle collisions in the presence and in the absence of electrostatic repulsion [41]: W¼

I : IR

ð6:105Þ

We then have IR ¼ I and W ¼ 1 for fast coagulation, and IR < I and W > 1 for slow coagulation.


Consider the behavior of conducting particles in an external electric field. When the particles are relatively far from each other, the influence of neighboring particles is small and our analysis can be restricted to a single particle in an unbounded fluid. Smaller interparticle distances would cause a distortion of the external electric field near the particle surface, which could noticeably affect the shape of a deformable particle (droplet). If the internal and external fluids are both ideal dielectrics and there are no free charges at the interface, or if the internal fluid is highly conductive and the external fluid is an insulator, the external electric field gives rise to a force distributed over the particle surface; this force emerges because the electric field is discontinuous at the interface [42]. This force is perpendicular to the interface and is directed away from the fluid with higher dielectric permittivity (conducting fluid) toward the fluid with lower permittivity (insulator). For the equilibrium shape of a motionless droplet in a quiescent fluid to be stable, it is necessary that the electrostatic surface force should be equal to the surface tension force. As a result, at static conditions, the shape of a droplet is that of a body of revolution – a prolate ellipsoid stretched along the direction of the external electric field. The theory of static equilibrium of droplets in an electric field (electrohydrostatics) is most thoroughly developed for ideal media (dielectrics and conductors) [43–49]. But the real fluids are media with finite conductivity and finite dielectric permittivity. The only exception is superconductivity, which takes place at very low temperatures for such fluids as, for example, liquid helium. When we take conductivity to be finite, the problem becomes very complicated both mathematically and physically, because the possible shapes of droplets are different from those of ideal conducting droplets. Thus a droplet may assume the shape of a prolate ellipsoid stretched in the direction parallel or perpendicular to the direction of the external electric field, or even be spherical [50]. A theoretical explanation of these phenomena is given in [51]. It is shown that a droplet with a finite conductivity accumulates electric charge in its surface layer, thereby giving rise to a non-uniform surface tangential electrical stress. This stress, in its turn, induces tangential hydrodynamic stresses in both the internal

j365


366

and external fluids, which affects droplet deformation. The magnitude of these stresses depends on the fluids’ properties and on the strength of the external electric field. Hence, depending on the relation between the electrical and hydrodynamic surface stresses, the droplet may take one of above-mentioned shapes. The solution of this problem carried out in [51] takes into account circulation of the internal fluid. In this work the flow is assumed to be slow and Stokesian, and deformations of the droplet surface are assumed to be small; with these assumptions, it becomes possible to get an approximate asymptotic solution. Let us restrict ourselves to simple estimates. Suppose an ideally conductive droplet is suspended in an ideal dielectric. The motion of internal as well as external fluid will be neglected. The external electric field E0 polarizes and deforms the droplet, which then takes the shape of an ellipsoid whose major axis is parallel to the field. Consider the behavior of an isolated conductive uncharged spherical droplet of radius R freely buoyant in a quiescent dielectric fluid with constant dielectric permittivity e in the presence of a uniform external electric field with constant strength E0. Since outside the sphere the electric charge is absent, electrostatic potential f obeys the Laplace equation: Df ¼ 0:

ð6:106Þ

The surface of the conductive droplet SR is equipotential, therefore f¼0

at

SR :

ð6:107Þ

On large distances from the droplet, the total electric field is equal to the external field: rf ! E 0

at

r ! 1:

ð6:108Þ

There is no electric field inside the droplet, and the electric charge induced by the external field is distributed over the surface so that the surface density s is equal to s¼

1 qf ; 2p qn

ð6:109Þ

where n is the outward normal to SR. The total charge of the droplet is zero, therefore ð

qf ds ¼ 0: qn

ð6:110Þ

SR

Suppose the droplet is slightly deformed. In the first approximation, its shape can be considered spherical. In spherical coordinates, the Laplace equation (6.106) for a spherically symmetrical problem is written as


sin u

q qf q qf r2 þ sin u ¼ 0: qr qr qu qu

ð6:111Þ

The boundary conditions are f¼0

at

r ¼ R;

qf ! E0 cos u qr

at

r ! 1:

ð6:112Þ

The solution of this boundary value problem is f ¼ E0

R3 r 2 cos u: r

ð6:113Þ

From Eq. (6.113) we obtain the strength of electric field at the droplet surface: qf ¼ 3E0 cos u: ER ¼ qr r¼R

ð6:114Þ

Let us now find the force acting on the conductive sphere. Momentum flux density in an electric field is defined by the Maxwell stress tensor t ik ¼

1 E2 d ik E i E k ; 4p 2

ð6:115Þ

and the force acting on an oriented surface element ds is tik dsk ¼ tik nk ds: where tiknk is the force acting on a unit surface area. At the surface of the droplet, the electrical stress is directed along the normal to the surface, so Ei Ek ¼ ER2 d ik . Consequently, the force acting from the inside on a unit surface area of the droplet is equivalent to the pressure Dp ¼

ER2 9E02 cos2 u: 8p 8p

ð6:116Þ

This pressure achieves its maximum value at y ¼ 0 and y ¼ p, that is, at the droplet poles lying along a straight line parallel to E0. As a result, the maximum deformation is observed in the vicinity of these points, and the droplet assumes the shape of a prolate ellipsoid stretched in the direction of the external electric field (Fig. 6.5). The equilibrium condition for the droplet boils down to the equality of two forces: the force that arises from electrical pressure (6.116) and the surface tension force. As long as electrical pressure dominates, the droplet keeps changing its shape until at

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368

Fig. 6.5 Deformation of a conductive droplet in an electric field.

some point the decrease of the principal curvature radii results in a stronger surface tension that can counterbalance the internal pressure. Note, however, that a considerable deformation of the droplet might cause a loss of stability and a breakup of the droplet. The critical strength Ecr of the external electric field can be estimated from the approximate equality 9Ecr2 2S ; 8p R

ð6:117Þ

from which it follows that droplet stability is characterized by the dimensionless parameter k2 ¼

electric force R ¼ E02 : force of surface tension S

ð6:118Þ

The simple estimation (6.117) gives kcr 1.77 for the critical value of the parameter k. A more accurate calculation that takes into account the deformation of the droplet [45,52] gives kcr ¼ 1.625. For example, for a water-in-oil emulsion, we have P ¼ 3102 N/m, and for a water droplet of 1 cm radius, the critical electric field strength is Ecr ¼ 2.67 kV/cm. Thus electric fields whose strength is E0 < 3 kV/cm will fail to produce any noticeable deformation of spatially separated droplets of radius R 1 cm. A different situation arises when droplets are located next to each other. Smaller distance between droplets leads to a considerable increase of the local electric field strength. It was shown in [53] that stability of a system of two conductive droplets is defined by the same parameter k, but in contrast with the case of an isolated droplet, it depends not only on E0, R, and S, but also on the relative clearance d/R between the particle surfaces. Table 6.1 lists the values of kcr for different values of d/R. From the adduced values of kcr, one can see that even very small droplets can lose their stability in relatively weak external electric fields as long as they are sufficiently close to each other. However, the relevant distances are so small that forces of

6.5 Conducting Particles in an Electric Field Table 6.1

d/R k cr

1 1.625

10 1.555

1 0.9889

0.1 0.0789

0.01 3.91 103

0.001 1.9 104

molecular interaction come into play, encouraging capture and coalescence of small droplets formed after breakup of the larger droplets. Consider now two motionless conducting spherical particles of radii R1 and R2 carrying the charges q1 and q2 in an external uniform electric field of strength E0 (Fig. 6.6) Denote through y the angle between the particles’ line of centers and the vector E0. The space between the particles is filled with a quiescent isotropic uniform dielectric medium having the permittivity e. Since there are no free charges outside the spheres, electrostatic potential f in the region outside the spheres obeys the Laplace equation Df ¼ 0:

ð6:119Þ

Far away from the particles, strength of the electric field tends to that of the external field: E ¼ rf ! E 0 ¼ rf0 :

ð6:120Þ

Since the frame of reference is chosen in such a way that the vector E0 lies in the XOZ-plane, f0 does not depend on y and is equal to f0 ¼ E0 ðZcos u þ X sin uÞ:

Fig. 6.6 Two conductive particles in an electric field.

ð6:121Þ

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370

The condition of equipotentiality of particle surfaces gives us fs1 ¼ V1 ;

fs2 ¼ V2 ;

ð6:122Þ

where V1 and V2 are constant surface potentials. In order to determine them, we must use the condition that the charge of a conducting particle is distributed over the particle surface:

ð e qf ds ¼ qi 4p qn

ði ¼ 1; 2Þ:

ð6:123Þ

si

where n is the outward normal to the particle surface. The projection of the force acting on particle 2 onto the unit vector p is equal to Fp ¼

e 8p

ð

qf qn

2 npds:

ð6:124Þ

s2

Without going into details (see [1] for a comprehensive treatment of the problem), we adduce the expression for the electric field strength near the point A in the interparticle region (Fig. 6.6): EA ¼

1 ðE1 q1 þ E2 q2 Þ þ E3 E0 cos u; eR22

ð6:125Þ

and the longitudinal and transverse (with respect to the line of centers) components of the force acting on particle 2: E2z ¼ eR22 E02 ð f1 cos2 u þ f2 cos2 uÞ þ E0 cos uð f3 q1 þ f4 q2 Þ þ

1 ð f5 q21 þ f6 q1 q2 þ f7 q22 Þ þ E0 q2 cos u; eR22

ð6:126Þ

E2x ¼ eR22 E02 f8 sin2 u þ E0 sin uð f9 q1 þ f10 q2 Þ þ E0 q2 sin u: The coefficients Ei and fi are dimensionless quantities that depend on the relative clearance between the particle surfaces D ¼ (r R1 R2)/R1 and on the ratio of particle radii k ¼ R2/R1. The dependences of E3 and f1 on D and k are shown in Fig. 6.7 and Fig. 6.8.

6.6 Coagulation of Particles in a Turbulent Flow

Q5

Consider the coagulation of particles in a developed turbulent flow. It is assumed that particles are spherical, non-deformable, and inertialess, and that their size is


Fig. 6.7 E3 vs. D and k: 1 – k ¼ 1; 2 – k ¼ 0.01.

much smaller than the inner scale of turbulence, that is, R l0. Under these conditions the particles’ interdiffusion coefficient (with hydrodynamic interactions taken into account) is given by the expression (6.70). To determine the frequency of collisions between particles of radii R1 and R2 (R1 < R2), one has to solve the diffusion equation (6.36) with the boundary conditions (6.39). Let us place the origin of a spherical coordinate system (r, y, F) at the center of the larger particle R1. If the forces of particle interaction possess spherical symmetry, Eq. (6.36) with the conditions (6.39) takes the form 1 d dn2 F 2 n2 r Dt ¼ 0; dr h r 2 dr n2 ¼ 0 at

r ¼ R 1 þ R2 ;

n2 ¼ n20

ð6:127Þ at

r ! 1;

ð6:128Þ

where n2 is the number concentration of particles of radius R2, Dt is the interdiffusion coefficient, F is the force of interaction (molecular, electrostatic, hydrodynamic) between the particles, and h is the coefficient of hydrodynamic resistance to particle

Fig. 6.8 f1 vs. D and k: 1–7 – k ¼ 1; 0.5; 0.2; 0.1; 0.05; 0.02; 0.01.

j371


372

motion. Eq. (6.127) has a general solution 0 1 1 ð egðrÞ gðrÞ @ C1 C2 drA; n2 ðrÞ ¼ e rD2t ðrÞ

ð6:129Þ

r

1 ð

gðrÞ ¼ C2 r

FðrÞ dr: hðrÞDt ðrÞ

The boundary conditions (6.28) enable us to find the constants C1 and C2: 0 C1 ¼ n20 ;

1 ð

B C2 ¼ n20 @

R1 þR2

11 gðzÞ

e C dzA ; z2 Dt ðzÞ

resulting in the solution 0

1 ð

B n2 ðrÞ ¼ n20 egðrÞ @1

r

gðzÞ

e dz z2 Dt ðzÞ

1

1 ð

gðzÞ

R1 þR2

e C dzA: z2 Dt ðzÞ

ð6:130Þ

The diffusion flux of particles 2 toward the test particle 1 is determined from the expression (6.40), which in the spherically symmetric case reduces to dn2 F j ¼ 4pðR1 þ R2 Þ2 Dt n2 : dr h

ð6:131Þ

Substitution of Eq. (6.130) into Eq. (6.131) gives 0 B jðR1 ; R2 Þ ¼ 4pn20 @

1 ð

R1 þR2

dr r 2 Dt ðrÞ

01 ð exp@ r

FðzÞ hðzÞDt ðzÞ

111 AC A :

ð6:132Þ

Let us now consider the interaction of particles, consecutively introducing hydrodynamic, molecular, electrostatic, and electric forces. We start with the case when particle surface is fully retarded, in other words, particles can be thought of as rigid and non-deformable; coagulation occurs the under the joint action of turbulent fluctuations and forces of molecular attraction. Molecular attraction between two spherical particles is described by Eq. (6.100), which says that this force is defined by the distance between particle surfaces and does not depend on the orientation of the particle pair, that is to say, the force possesses spherical symmetry relative to the center of particle R1. Since the molecular force manifests itself only when the gap D between the particles is small, it is


possible to take its asymptotic expression at D ! 0: FA

GR1 R2

1 ; D 6ðR1 þ R2 Þ 2

ð6:133Þ

3

where D ¼ (r R1 R2)/(R1 þ R2) is the dimensionless gap between the particle surfaces and G is the Hamaker constant. For the coefficient of hydrodynamic resistance we take the approximation ! R1 R2 R1 R2 ð0Þ ; hð0Þ ¼ 6pne re h¼h 1þ ; ð6:134Þ R1 þ R 2 ðR1 þ R2 Þ2 D

Q6

which is obtained by combining the far asymptotic and near asymptotic expressions for the force of hydrodynamic interaction, when the particles approach each other along the line of centers. The coefficient of particle interdiffusion is ð0Þ

Dt ¼ Dt

hð0Þ h

2 ;

ð6:135Þ

where ð0Þ

Dt

¼

ne

ðR1 þ R2 Þ2 2

l0

rR1 R2 R1 R2 þ 2 R1 þ R 2 R1 þ R22 R1 R2

2 :

Eqs. (6.132)–(6.135) yield the dimensionless flux: 2

J¼

j1 1

l0 jðR1 ; R2 Þ ¼ ð1 þ kÞ3 j1 ; 4pne R31 n20 1 ð

¼ 0

ðD þ D1 Þ2 D2 ð1 þ DÞ2 ðD þ D2 Þ2

ð6:136Þ 0 exp@SA

1 ð

A

1 ðD1 þ zÞdz A dD; z3 ðz þ D2 Þ3 2

where D1 ¼ R1 R2 =ðR1 þ R2 Þ2 ; D2 ¼ R1 R2 =R21 þ R22 R1 R2 ; and SA ¼ Gl0 =36pre n2e ðR1 þ R2 Þ3 is the parameter of molecular interaction. For water–in–oil emulsions, where G 1020 J, l0 103 m, ne 5 105 m2/s, re 103 kg/m3, this parameter equals SA 41023 (R1 þ R2)3. For particles with R1, R2 10 mm, we get SA 108. Smallness of SA makes it possible to find an asymptotic expression for j1. For particles of comparable sizes, the quantity D (1, D1, D2) gives the major contribution to the integral in the expression for j1 1 . Expanding the integrand as a power series of D and keeping the dominant terms, we get 0 1 1 1 ð 2 ð D1 D dz 1 1 exp@SA 3 3 AdD: ð6:137Þ j1 ¼ D2 D22 z D2 0

0

j373


374

We can take this integral by using the method of quadratures: j1 1 ¼

3=2

D1 D2

p 2SA

1=2 :

ð6:138Þ

Substitution of Eq. (6.138) into Eq. (6.136) gives an asymptotic expression for the diffusion flux at SA 1: jðR1 ; R2 Þ ¼

pffiffiffi 2 2G1=2 ðR1 þ R2 Þ9=2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 n20 : l0 re R1 R2 ðR1 þ R22 R1 R2 Þ

ð6:139Þ

The obtained expression for the diffusion flux corresponds to pair interaction of particles freely buoyant in the fluid. It is interesting to compare this flux with the flux caused by pair interactions when one of the particles is fixed: j1 ðR1 ; R2 Þ ¼

pffiffiffi 2 2G1=2 ðR1 þ R2 Þ11=2 n20 : pffiffiffiffiffi 1=2 1=2 l0 re R31 R1 R2

ð6:140Þ

The ratio between the fluxes (6.140) and (6.139) is estimated as j1 = j ðR31 þ R32 ÞR3 1 for R1 R2, with j1/j 1 for R1 R2 and j1/j 2 for R1 R2. 2. So the fixed particle approximation is pretty accurate: the resulting flux and consequently the resulting collision frequency is exaggerated (as compared to the case of free particles) by the factor of 2 at most. The main flaw of the Levich model of turbulent coagulation is that it overstates the particle collision frequency by 1–2 orders of magnitude. We can demonstrate that accounting for the interaction forces in a proper manner would eliminate this disadvantage. Let j0 be the diffusion flux in the case of unconstrained particle motion (i.e., in the absence of interactions). Let us take h ¼ h0 ¼ 6preneR, FA ¼ 0, and SA ¼ 0. Then Eq. (6.136) yields j0 ¼

4pne ðR1 ; R2 Þ3 2

l0

1 2 2 ln D2 þ D2 1 ðD2 1Þ2 ðD2 1Þ3

!1 n20 :

ð6:141Þ

At R1 ¼ R2 the flux (6.141) coincides with the flux obtained by Levich. To estimate the effect of hydrodynamic forces on the frequency of collisions, we compose the ratio j0/j, where j0 obeys Eq. (6.141) and j obeys Eq. (6.139) for identical particles R1 ¼ R2 ¼ R. Then pffiffiffi j0 3 2pne 3=2 ¼ 1=2 R : j G l0

ð6:142Þ

Taking the parameter values that are typical for water–in–oil emulsions, l0 103 104 m, R 105 m, ne 105 m2/s, re 103 kg/m3, G 1020 J, we get


j0/j 10 102. It means that the hydrodynamic resistance of particles reduces the collision frequency by 1–2 orders of magnitude, which is consistent with experimental data; thus the main disadvantage of the Levich model is eliminated. The effect of hydrodynamic interaction on the diffusion flux decreases as the particle size gets smaller. For R 108 m, we have j0/j 1, and for R 108 m, hydrodynamic interaction becomes insignificant. For particles that small, the range of action of molecular forces exceeds the particle radius. It should be noted that in the case of Brownian coagulation, hydrodynamic interactions also reduce the collision frequency, though but as strongly (the attenuation factor is 1.5–2 for Brownian coagulation vs. 10102 for turbulent coagulation). There are two factors that explain why the effect of hydrodynamic interactions on the collision frequency is so different in these two cases: the difference in characteristic particle sizes (particles involved in Brownian motion are much smaller than those involved in turbulent motion) and different character of dependence of diffusion coefficients on the hydrodynamic resistance coefficient (Dbr h1 for Brownian motion and Dt h2 for turbulent motion). The asymptotic expression (6.139) for diffusion flux is suitable for particles of roughly the same size. For particles whose sizes differ by a lot (k ! 0), we have the following expression:

j1 1 ¼

1 ð

0

0 1 1 ð 2 1=3 1 dz @SA AdD ¼ 3 Gð4=3Þl0 ; exp 1=3 z4 D2 SA ne ðR1 þ R2 Þ3

ð6:143Þ

0

where G(x) is the gamma function. The dependence (6.136) of j1 on k ¼ R2/R1 and SA is shown in Fig. 6.9. The dotted line illustrates the corresponding asymptotic dependence (6.143).

Fig. 6.9 Dependence j1 on k and SA.

j375


376

Going back to Eq. (6.136), we get the following asymptotic expression for the diffusion flux of small particles toward a larger particle (R2 R1): 24=3 p Gne jðR1 ; R2 Þ ¼ ðR1 þ R2 Þ2 4 3Gð4=3Þ pre l0

!1=3 n20 :

ð6:144Þ

Hence, for particles of different sizes the diffusion flux turns out to be proportional to the surface of the larger particle, not to the third power of radius as is the case for particles of comparable sizes. Let us now find the coagulation kernel K(V, o). When particles coagulate via diffusion, the coagulation kernel is equal to the flux of particles of radius R2 (at a unit concentration) toward the particle of radius R1. Making use of equations (6.139) and (6.144) for the fluxes of particles having the same size and different sizes, respectively, we get K10 ðV; wÞ ¼

K20 ðV; wÞ

pffiffiffi 1=2 1=3 2G ðV þ w1=3 Þ9=2 ðVwÞ1=6 pffiffiffiffiffiffiffiffiffiffi 2=3 l0 3re pðV þ w2=3 V1=3 w1=3 Þ

1 Gne ¼ 1=3 ðV1=3 þ w1=3 Þ2 4 3 Gð4=3Þ pre l0

at

V w;

ð6:145Þ

!1=3 at

V w:

ð6:146Þ

We can match the asymptotic expressions (6.145) and (6.146) by introducing a weight function a ¼ 4Vo/(V þ o): K 0 ðV; wÞ ¼ aK10 ðV; wÞ þ ð1aÞK20 ðV; wÞ:

ð6:147Þ

Because the coagulation kernel (6.147) is inconvenient to use in calculations, we replace it with the approximate expression " K ðV; wÞ ¼ 16 0

#1=2

G 4

3pre l0

ðV1=3 w1=6 þ V1=6 w1=3 Þ:

ð6:148Þ

that has the form (6.16) and easily renders itself to the method of interpolation of fractional moments, which is the preferred way to solve the kinetic equation of coagulation [1]. Determination of collision frequency for particles with mobile surfaces (droplets, bubbles) having viscosity mi that is different from the viscosity of the carrier medium me is carried out in a similar manner [1]. One major difference from the case considered above is the ability of the particle (droplet) surface to deform, which is especially noticeable when particles come close together (to the distances that are small compared to their sizes). The other difference is in the form of the resistance coefficient. If the particles are located far from each other, the coefficient of


hydrodynamic resistance h(0) that pertains to the particles’ relative motion is given by Eq. (6.71) in which the two coefficients h1 and h2 are determined from the Hadamar– Rybczynski formula hð0Þ ¼

ð0Þ ð0Þ

h1 h2

ð0Þ h1

þ

ð0Þ h2

¼ 6pme

R1 R2 2 þ 3m : R1 þ R 2 3 þ 3 m

ð6:149Þ

When the gap d between the droplets is small, one can use the following asymptotic expressions (see Eq. (6.188)): hd ¼ 6pme

m¼

R1 R2 R1 þ R2

2

1 R1 R2 1=2 ; R1 þ R 2 m

f ðmÞ ; d

f ðmÞ ¼

1 þ 0:402m ; 1 þ 1:711m þ 0:461m2

ð6:150Þ

d ¼ rR1 R2 ;

!1 ¼ mi =me. The case m where we just introduced a dimensionless parameter m corresponds to droplets with fully retarded surfaces (rigid particles) and the case ¼ 0 – to gas bubbles. m The resistance coefficient can be approximated by the expression f ðmÞ R1 R2 3 þ 3m ; h ¼ hð0Þ 1 þ d R1 þ R2 2 þ 3m

ð6:151Þ

which correctly describes the behavior of h for large and small gaps between the particles. We can introduce a modified capillary number Ca ¼

me uR1 R2 ; SðR1 þ R2 Þd

ð6:152Þ

which tells us how strongly the particles (droplets) are deformed. Here S is the surface tension coefficient of a droplet, and d – the clearance between droplet surfaces; droplets approach each other with the speed u. If Ca 1, deformation of the droplet surface is negligible. But at d ! 0 this condition may no longer be true. However, at such narrow gaps between droplet surfaces, the force of molecular attraction becomes important, which helps to promote coalescence of droplets at the final stage. Therefore surface deformation of interacting droplets can be noticeable only at the final stage of their approach and while it can reduce the rate of coalescence, it does not essentially affect the collision frequency of particles. In the case under consideration, with molecular and electrostatic interaction forces taken into account, the dimensionless diffusion flux of particles 2 toward

j377


378

particle 1 is given by 2

J¼

j1 1

2l0 j ¼ ð1 þ kÞ3 j1 ; pne R31 n20 1 ð

¼ 2

ð6:153Þ

2

3 1 ð 1 þ m hðxÞ exp4 ðSA Fð pÞ fA ðxÞ þ SR t fR ðxÞÞdx 5ds: Þ ðxbÞ2 kð2 þ 3m s2 ðsbÞ2 h2 ðsÞ

s

The following dimensionless parameters have been introduced: SA ¼ 2 2 2Gl0 =3pR31 n2e re – the parameter of molecular interaction; SR ¼ ef21 l0 =2pR31 n2e re – the parameter of electrostatic interaction; t ¼ R1 c ¼ R1 =lD – the ratio of the droplet ¼ mi =me – radius to the thickness of the double layer on the droplet surface ðt 1Þ; m the ratio of viscosity coefficients of the internal end external fluids; p ¼ g(1 þ k)(s 2); g ¼ pR1/lL; b ¼ 2(1 k)2/(1 þ k2 k). A detailed analysis of the numerical solution and the appropriate results are presented in [1]. Let us now determine the collision frequency of uncharged conductive spherical particles in a turbulent flow of dielectric fluid in the presence of a uniform external electric field. It is assumed that the turbulent flow is developed and that particle sizes are much smaller than the inner scale of turbulence. The particles are supposed to be non-deformable, which is possible when external electric field strength E0 does not exceed the critical value Ecr and the particle size is sufficiently small. Under these conditions, the coefficient of mutual diffusion of particles of two types 1 and 2 in the presence of hydrodynamic interactions can be taken in the form (6.135), where the drag coefficients h and h(0) are approximated by the expressions (6.134) that correspond to particles with fully retarded surfaces. We shall also take into account molecular and electric interactions of particles. Consider the relative motion of two particles of radii R1 and R2 subject to hydrodynamic, electric, and molecular forces. Introduce a spherical frame of reference (r, y, F) connected to the center of the larger particle 1, angle y being measured from the direction of the electric field strength vector E0. The expression (6.125) corresponds to the radial electric force acting on particle 2. If both particles are uncharged, then q1 ¼ q2 ¼ 0 and _

_

F2r ¼ eE02 R1 R2 ð f 1 cos2 u þ f 2 sin2 uÞ:

ð6:154Þ

The force F2r is_represented in a form that is symmetric with respect to particle radii. Coefficients f i are expressed through the coefficients fi introduced in Eq. (6.125) as follows: _

f i ðk; DÞ ¼ k fi ðk; Dð1 þ k1 ÞÞ;

where k ¼ R2 =R1 ;

ði ¼ 1; 2Þ;

D ¼ ðrR1 R2 Þ=ðR1 þ R2 Þ:

ð6:155Þ


The total force acting on the particle is equal to the sum of molecular and electric forces: F2 ðR1 ; R2 ; r; uÞ ¼ F2A ðR1 ; R2 ; rÞ þ F2r ðE0 ; R1 ; R2 ; r; uÞ:

ð6:156Þ

The force of molecular interaction is taken in the approximate form (6.133). In contrast to the above-considered case of particle coagulation in the absence of electric fields, when the force of particle interaction possessed spherical symmetry, in the present case the electric field causes the force to be dependent on the orientation of the particle pair relative to the external electric field vector E0 and thus possesses axial, rather than spherical, symmetry. As a result, a second-order partial derivative with respect to y appears in the diffusion equation (6.36), which not only complicates the solution but also causes rotation of the particle pair relative to the direction of the electric field. Let us therefore try to estimate the diffusion flux using the following procedure. We first determine the diffusion flux J0(0) per unit solid angle, assuming the interaction force to be purely radial. Then we integrate the resulting expression over the surface of a sphere of radius R ¼ R1 þ R2 (the coagulation radius); while doing so, we must keep in mind the dependence of the interaction force on y – the orientation angle of the particle pair relative to the direction of electric field strength. The flux determined in this way can be regarded as the first approximation for the total diffusion flux j of particles of radius R2 toward the particle of radius R1. Carrying out all these steps, we obtain j0 ð0Þ ¼

ne 2 l0

_

ðR1 þ R2 Þj 1 ð uÞn0 ; p=2 ð

j ¼ 4pðR1 þ R2 Þ

2

j0 ð0Þsin u d u;

ð6:157Þ

0 _

where n0 is the number concentration of particles 2 and j 1 ð uÞ is determined by analogy with Eq. (6.136). Making use of Eq. (6.134) for h, Eq. (6.135) for Dt, and Eq. (6.156) for the interaction force between particles, and introducing the same dimensionless variables as earlier, we get

j ¼ 4pn0

ðR1 þ R2 Þ3 ne 2

l0

p=2 ð

j1 ;

j1 ¼

_

j 1 ð uÞ sin u d u;

0

2

1 ð 4SA ðD1 þ zÞdz exp z3 ðz þ D2 Þ3 D2 ð1 þ DÞ2 ðD þ D2 Þ2 0 D 3 _ _ 1 ð ðD1 þ zÞð f 1 cos2 u þ f 2 sin2 uÞ 5 dz ; SE zðz þ D2 Þ2

_

1

ðj 1 ð uÞÞ

D

1 ð

¼

ðD þ D1 Þ2

ð6:158Þ

j379


380

D1 ¼

R1 R2 ðR1 þ R2 Þ2

; D2 ¼

R1 R2 ; R21 þ R22 R1 R2

2

SA ¼

2

Gl0 36pre n2e ðR1 þ R2 Þ

; 3

SE ¼

eE02 l0 : 6pre n2e

Another method of estimating the diffusion flux j is based on the natural assumption that in the course of mutual approach of the two particles, the orientation of the pair relative to electric field will change a number of times. This will result in the blurring of the concentration profile n2, which can be determined by averaging over the angle y. Accordingly, we average the relations (6.156) and (6.154) over the spherical surface, getting the average force acting on particle 2: ðp hF2 ðR1; R2 ; DÞi ¼ F2 ðR1; R2 ; D; uÞsin ud u 0

¼ F2A ðR1; R2 ; rÞ þ hF2r ðE0; R1; R2 ; r; uÞi;

ð6:159Þ

_ _ 1 hF2r ðE0; R1; R2 ; r; uÞi ¼ eE02 R1 R2 ð f 1 þ 2 f 2 Þ: 3

and substitute these relations into Eq. (6.132). Then the flux j has the form (6.158), but the expression for j1 changes to

j1 1

1 ð

¼ 0

2

1 ð ðD1 þ zÞdz 4 exp S A 2 2 2 z3 ðz þ D2 Þ3 D ð1 þ DÞ ðD þ D2 Þ

ðD þ D1 Þ2

D

1 ð

SE

_

_

ðD1 þ zÞð f 1 þ 2 f 2 Þ

D

3zðz þ D2 Þ2

3

ð6:160Þ

dz5:

Numerical calculations have shown that while the two methods give different expressions for j1, the resulting diffusion fluxes are roughly the same: the maximum difference between the two fluxes is about 5% for a wide range of parameters SA and SE. The influence of the external electric field on the collision frequency decreases with growth of SA and reduction of SE. Since SE does not depend on particle size and SA increases as particles get smaller, we conclude that the smaller the particle size, the less influence electric field has on the collision frequency. The effect of the electric field on the collision frequency is characterized by the ratio between diffusion fluxes with and without the electric field: x¼

jðSA ; SE Þ : jðSA ; 0Þ

ð6:161Þ


Taking the parameter values typical for for water–in–oil emulsions in a turbulent flow, e 2e0, E0 0.9 kV/cm, ne 105 m2/s, l0 103 m, r ¼ 900 kg/m3, G 1020 J, R1 þ R2 105 m ¼ 10 mm we get SE 1, SA 107, and x 30. Hence, the collision frequency of conductive water droplets in dielectric oil in the presence of an electric field having the strength 0.9 kV/cm is 30 times higher than the same frequency in the absence of the electric field. Since SA R3, an increased radius of a droplet results in a higher collision frequency (see Fig. 6.10). That being said, even sufficiently strong electric fields are not able to fully cancel out the effect of hydrodynamic interactions on collision frequency of particles in a turbulent flow. To estimate this effect, consider the ratio of the flux in the absence of hydrodynamic interactions to the flux when such interactions are present: j0 3 ¼ : j j1

ð6:162Þ

At SE 5, this ratio is j0/j 10, and at SE 1 we already have j0/j 50. Thus even in strong electric fields where SE is large, hydrodynamic interactions accompanying the mutual approach of particles with fully retarded surfaces decrease the collision frequency by one order of magnitude as compared to the case of noninteracting particles. It should be noted that for particles with mobile surfaces (droplets), the role of hydrodynamic interactions is not as strong, because at small clearances between the particles, hydrodynamic resistance increases slower: (1/D1/2) for droplets as opposed to (1/D) for rigid particles. Numerical treatment of the problem leads to the following relation that approximates Eq. (6.158): 2

1=3

2=3

1=3

2=3

j 0:24pSE ne l0 ðR1 R 2 þ R2 R 1 Þn20 :

ð6:163Þ

The diffusion flux of particles of radius R2 (at number concentration n20 ¼ 1) toward the particle of radius R1 has the meaning of kernel K (V, o) of the kinetic equation (6.1) (coagulation kernel). Replacing radii in Eq. (6.163) with volumes, we get 1 1=9 2=9 Kðv; wÞ ¼ 0:01eE 20 n1 w þ w1=9 V 2=9 Þ3 : e re ðV

ð6:164Þ

It is now possible to consider the dynamics of the process of aggregation of particles with fully retarded surfaces. If the electric field is absent and the particles are uncharged, the coagulation kernel is given by Eq. (6.148). We employ the method of moments to solve the kinetic equation (6.1). Eq. (6.15) gives us the following equation for the zero-order moment (i.e., for the number concentration of particles): dm0 1 ¼ G½m1=3 ðtÞm1=6 ðtÞ þ m1=6 ðtÞm1=3 ðtÞ ; dt 2 2

where G ¼ 16ðG=ð3pre l0 ÞÞ1=2 :

ð6:165Þ

j381


382

Fig. 6.10 Dependences of j1 and x on k, SE and SA (a) SA ¼ 108; (b) SA ¼ 106; (c) SA ¼ 104; (d) SE ¼ 1.

The right-hand side or this equation contains fractional moments. They can be determined by the method of interpolation of fractional moments (see Section 6.1). Adopting the two-point interpolation scheme, we express them through the integer moments m0 and m1 according to Eq. (6.20). The result is

6.6 Coagulation of Particles in a Turbulent Flow _

m1=3 ð0Þm1=6 ð0Þ _ 3=2 _ 1=2 dm0 ¼G m0 ðtÞm1 ðtÞ; m0 ð0Þ dt

ð6:166Þ

where we have introduced the dimensionless moments _

m0 ðtÞ ¼

m0 ðtÞ ; m0 ð0Þ

_

m1 ðtÞ ¼

m1 ðtÞ m1 ð0Þ

and mk(0) is the initial value of k-th moment. In the course of coagulation, the volume concentration does not vary with time _ _ and m1 ðtÞ ¼ 1. Then the solution of Eq. (6.166) with the initial condition m0 ðtÞ ¼ 1 is _

m0 ðtÞ ¼

m1=3 ð0Þm1=6 ð0Þ 2 t 1 þ 8G : m0 ð0Þ

ð6:167Þ

Estimating the initial values of fractional moments in the same manner as above, we obtain: 1=2

m1=3 ð0Þm1=6 ð0Þ ¼ ðm0 ð0ÞÞ3=2 m1

ð6:168Þ

Then Eq. (6.167) and Eq. (6.168) give us the number concentration the and average particle volume as functions of time: _

m0 ðtÞ ¼ ð1 þ 8Gðm0 ð0ÞÞ1=2 m1 tÞ2

ð6:169Þ

Vav ðtÞ 1 ¼_ ¼ ð1 þ 8GðVav ð0ÞÞ1=2 m1 tÞ2 ; Vav ð0Þ m ðtÞ 0

ð6:170Þ

1=2

and

where Vav ð0Þ ¼ m1 =m0 ð0Þ: Defining the characteristic time of coagulation tA as the time it takes for the average particle (droplet) volume to increase by a factor of e, we write ! pffiffi 2 1=2 ð e1ÞV1=2 av ð0Þ 3pre l0 tA : G 128m1

ð6:171Þ

Let us take m1 ¼ 0.01; l0 ¼ 5 104 m; G ¼ 1020 J; re ¼ 850 kg/m3; VaV ¼ 4 10 m3 (the average particle radius is 10 mm). Then the characteristic time of particle enlargement is tA 100 s. The decision to take hydrodynamic interactions into account decreases the diffusion flux by two orders of magnitude as compared to the case of turbulent coagulation in the absence of hydrodynamic interactions, so the characteristic time of coagulation increases by the same order of magnitude. 15

j383


384

Consider now the dynamics of enlargement of conductive particles in an external electric field. The field strength E0 is supposed to be less than the critical value at which the breakup of approaching particles (droplets) becomes possible. In this case the coagulation kernel has the form (6.164) and the equation for the zero-order moment is written as _

m1=3 ð0Þm2=3 ð0Þ þ 3m4=9 ð0Þm5=9 ð0Þ _ _ dm0 ¼ G1 m 0 m1 : m0 ð0Þ dt

ð6:172Þ

1 where G1 ¼ 0:01eE 20 n1 e re . The solution of this equation is

m0 ðtÞ ¼ m0 ð0ÞeBt ;

where

t ¼ G1 m1 t

ð6:173Þ

and

B¼

m1=3 ð0Þm2=3 ð0Þ þ 3m4=9 ð0Þm5=9 ð0Þ : m0 ð0Þ

The characteristic time of coagulation follows from Eq. (6.173): tE

1 25me : ¼ G1 m1 B eE02 m1

ð6:174Þ

Let us take m1 ¼ 0.01; me ¼ 5 103 kg/m s; E0 ¼ 1 kV/cm; e 2e0. Then tE 6 s. If we increase electric field strength to 2 kV/cm with all other conditions being the same, the coagulation time decreases to 1.5 s. A comparison of the characteristic times of particle enlargement in a turbulent flow ttE with the same characteristic time for gravitational sedimentation in an g g electric field tE shows that tE =ttE 2 3

6.7 Breakup of Particles

In Section 6.5 we broached the subject of deformation of conductive droplets and the possibility of their breakup. The present section will discuss breakup of nonconducting droplets. Stokesian motion of small droplets in the carrier flow does not lead to any noticeable deformation of the droplets. Droplet breakup is always preceded by a considerable deformation of its surface, which is possible when in the fluid layers adjacent to both sides of the droplet there exist large gradients of velocity and pressure that can overcome surface tension of the interface. In order to describe droplet deformation, it is necessary to take into account the joint action of inertial and viscous effects and surface tension forces. Data on particle deformation and breakup in viscous fluids at Re 1 are presented in [54], while [55] deals with the case Re 1.


Breakup of small droplets in a turbulent flow of emulsion should be considered as a random process characterized by the following parameters: breakup frequency f (V) of droplets whose volume lies in the interval (V, V þ dV); probability P (V, o) that a particle whose volume lies in the interval (V, V þ dV) will be formed after breakup or a larger droplet whose volume lies in the interval (o, o þ do); minimum droplet radius Rm for which breakup becomes possible. The kinetic equation describing the dynamics of particle distribution over volumes n(V,o) during the process of particle coagulation/breakup is given below: 1 ð ðV qnðV; tÞ 1 ¼ KðVw; wÞnðVw; tÞnðw; tÞdwnðV; wÞ KðV; wÞnðw; tÞdw qt 2 0

0

1 ð

ð6:175Þ

f ðwÞPðV; wÞnðw; tÞdw f ðVÞnðV; tÞ:

þ V

Let us start with the estimation of the minimum particle radius. To this end, we must estimate the forces acting on a particle in a turbulent flow that are capable of deforming the particle. A droplet suspended in a homogeneous, isotropic turbulence field is subject to the following forces exerted by the carrier fluid: dynamic pressure Q kfreu2/2, where kf 0.5 is the resistance coefficient; re and u – the density and velocity of the surrounding fluid relative to the droplet; Fv me g_ t – the force of viscous friction, where me is the viscosity coefficient of the carrier fluid and g_ t ¼ ð4e=15pne Þ1=2 is the average shear rate of small-scale fluctuations; e – specific dissipation of energy; ne ¼ me/re – the coefficient of kinematic viscosity. In addition to the forces mentioned above, the droplet surface is subject to the force of P surface tension Fcap ¼ 2 /R, where S is the surface tension coefficient and R – the particle radius. There are two possible mechanisms of droplet breakup; which mechanism will actually be employed in the breakup process depends on the dominant force. Suppose dynamic pressure is the major force [9]. Then particle deformation is caused by the difference between dynamic pressures applied to the opposite sides of the droplet: Q

k f re 2 2 ðu1 u2 Þ; 2

ð6:176Þ

where u1 and u2 are the velocities at the opposite poles of the droplet separated by the distance 2R. First, consider a droplet of size R > l0, where l0 is the inner scale of turbulence. Then large-scale fluctuations (l0 l L), which vary slightly over distances of the order of the droplet size, fail to affect a noticeable impact on the droplet. It means that droplet deformation and breakup can be caused only by small-scale fluctuations.

j385


386

For such fluctuations, Eq. (6.43) gives the change of fluctuational velocity ul over a distance of the order of 2R: ul ðelÞ1=3 ðe0 2RÞ1=3 :

ð6:177Þ

Then it follows from Eq. (6.176) and Eq. (6.176) that Q

k f re 2=3 e ð2RÞ2=3 : 2

ð6:178Þ

If the difference of dynamic pressures exceeds the surface tension force, a noticeable droplet deformation takes place and the droplet can be broken. So the breakup condition for the droplet may be written as k f re 2=3 2S e ð2RÞ2=3 : 2 R

ð6:179Þ

Substituting the expression (6.42) for e, we obtain a relation for the minimum radius: pffiffiffi Rm 2

L2=5 S3=5 3=5 3=5

k f re U 6=5

;

ð6:180Þ

where L is the characteristic linear scale of the flow region and U is the mean flow velocity. It should be noted that the formula (6.180) was obtained under the assumption that the inner and outer fluids have roughly the same density, as is the case for water–in–oil emulsions. Otherwise it would be necessary to take into account the dynamic pressure from the inner fluid. Let us introduce the Weber number We ¼

dynamic thrust 2Rre U 2 : ¼ S force of surface tension

ð6:181Þ

Then Eq. (6.180) can be rewritten as Rm 211=4 S3=5 We3=2 ¼ CWe3=2 : 3=2 L kf

ð6:182Þ

Now consider droplets whose sizes is smaller than the inner scale of turbulence R l0. Clearly, breakup of such droplets may cause fluctuations with scales l < l0, that is, fluctuations characterized by considerable viscous friction forces. Therefore the force of viscous friction at the droplet surface emerges as the principal mechanism of droplet deformation, and equality between the forces of viscous friction and


surface tension becomes the criterion of strong deformation of the droplet: me

4e 1=2 2S ; 15pn R

ð6:183Þ

which gives the following minimum droplet radius [56]: pffiffiffiffiffiffiffiffi Rm 15p

S re ðneeÞ1=2

C

S re ðneeÞ1=2

:

ð6:184Þ

Introducing another dimensionless parameter – the Ohnesorge number Oh ¼

me ð2Rere SÞ1=2

;

ð6:185Þ

we rewrite the relation (6.184) as Rm ¼ L

15p 1=4 3=4 1 Re Oh ; 4

ð6:186Þ

where Re ¼ UL/ne is the Reynolds number. We now proceed to determine the breakup frequency of droplets f (V), adopting the model of droplet breakup in a locally isotropic developed turbulent flow suggested in [57]. The model is based on the assumption that the possibility of breakup of an isolated droplet is completely predetermined by fluctuations of energy dissipation in the vicinity of the droplet. Droplet breakup takes place when the value of energy dissipation averaged over the fluid volume that is equal by the order of magnitude to the droplet volume V exceeds some critical value ecr ðVÞ. For a given droplet size, the critical value of energy dissipation should be equal to the value of e in the formula for the minimum radius of breaking droplets. When viscous friction is the dominant force that is responsible for the deformation and breakup of droplets, one can use Eq. (6.136) to derive !2 1=3 4p S 1 ecr ðVÞ C : 3 r ne V2=3

ð6:187Þ

Suppose the distribution of energy dissipation in the vicinity of the droplet is uniform and has the average value eðtÞ. We can then interpret breakup frequency (i.e., the number of breakup events per unit time) as the relative probability of attaining a certain (constant) level eðtÞ in a random process eðtÞ. Consider two instants of time t and t þ Dt. Suppose that at the moment t the dissipation energy is eðtÞ < ecr ðVÞ, whereas at the moment t þ Dt it equals eðt þ DtÞ > ecr ðVÞ. Then the

j387


388

breakup frequency may be represented as f ðVÞ ¼ lim

Dt ! 0

1 PðeðtÞ < ecr ðVÞ; eðt þ DtÞ < ecr ðVÞÞ : Dt PðeðtÞ < ecr ðVÞÞ

ð6:188Þ

The right-hand side gives the probability for the average dissipation energy to be less than the critical value at the time t while exceeding the critical value at the time t þ Dt. Expanding eðt þ DtÞ in series over the powers of Dt and taking random process as stationary, we can transform the relation (6.188) as 1 eðcr ð pede; f ðVÞ ¼ e_ pðecr ; e_ Þde_ 0

ð6:189Þ

0

where pðe; e_ Þ is the joint distribution density of the random variables eðtÞ and e_ ðtÞ at one and the same instant of time t, and pðeÞ is a one-dimensional distribution of the random variable eðtÞ. According to [58], the joint distribution density pðecr ; e_ Þ is expressed through the joint distribution density of one and the same random variable e taken at different instants of time: Dt Dt pðecr ; e_ Þ ¼ lim Dtp e þ e_ ; e e_ : Dt ! 0 2 2

ð6:190Þ

To determine the one-dimensional distribution pðeÞ, we shall draw on the result obtained in [59], which states that the distribution is well approximated by the logarithmic normal distribution 1 1 pðeÞ ¼ pffiffiffiffiffiffi exp 2 ðlnðkeÞÞ2 ; 2a 2pae

ð6:191Þ

where a2 ¼ lnðd2 =hei2 þ1Þ; k ¼ expða2 =2Þ=hei2 ; and hei and d 2 are the mean value and the variance of the distribution of specific energy dissipation. It has been shown in [60] that the assumption that fourth moments of velocity gradients are connected with second moments in the same fashion as in the case of the normal distribution (the Millionschikov hypothesis) leads us to d 2 ¼ 0:4hei2

ð6:192Þ

and the joint distribution density is equal to ! _ 2 2 T T 1 e 0 2 0 exp 2 2 ðlnðkeÞÞ ; pðecr ; e_ Þ ¼ 2c e 2a 2pace2

ð6:193Þ


pffiffiffiffiffiffiffiffiffiffiffiffiffi where T0 ¼ ne =hei and c ¼ 1expða2 Þ: Substitution of Eq. (6.191) and Eq. (6.193) into Eq. (6.189) gives us the breakup frequency: c d ðln FðyÞÞ; f ðVÞ ¼ pffiffiffiffiffiffi 2pT0 a dy

ð6:194Þ

where 1 FðyÞ ¼ pffiffiffiffiffiffi 2p

ðy

ey dy; 2

y¼

1

1 ln ðkecr ðVÞÞ: a

Putting ecr taken from Eq. (6.187) into this formula and making use of Eq. (6.192), we finally get f ðvÞ ¼

1 d pffiffiffiffiffiffi ðln FðxÞÞ; 2:03 2pT 0 dx

ð6:195Þ

where v ; x ¼ 1:1ln 1;3 vm

vm ¼

4p 2 R : 3 m

It remains for us to find the probability of droplet formation P(V, o). Experiments have been set up to study how the process of breakup of isolated droplets depends on hydrodynamic conditions. Various types of droplet breakup were observed. Most often it resulted in two almost identical droplets (daughter droplets) and several droplets of smaller sizes (satellites). None of the publications has mentioned any correlation between the type of droplet breakup and the size of the splitting droplets. This gives grounds to believe that at fixed parameters of the internal and external fluids, the probability of breakup depends only on the ratio between the volume o of the splitting droplet and the volume V of the daughter droplets and has the form Pðv; wÞ ¼ k

1 v g : w w

ð6:196Þ

Since the total volume of the droplets must be conserved, we may write ðw VPðV; wÞdV ¼ w 0

ð6:197Þ

j389


390

which together with Eq. (6.196) leads us to conclude that the mean value of the distribution density g(y) equals ð1

1 hyi ¼ ygðyÞdy ¼ : k 0

Based on the this brief discussion of droplet disintegration, we can assume that g(y) is a bimodal function with two well-defined maxima, one maximum in the region of daughter droplet sizes and the other – in the region of satellite droplet sizes. Then breakup probability can be written as a sum of two weighted singlemodal distribution densities g1(y) and g2(y) defined in the domain (0, 1) and having the average values hy1 i ¼ V1 =w and hy2 i ¼ V2 =w: PðV; wÞ ¼ k1

1 V 1 V g1 þ k2 g2 : w w w w

ð6:198Þ

The condition of conservation of total volume (6.197) requires that k1 y1 þ k2 y2 ¼ 1:

ð6:199Þ

In the limiting case when the daughter droplets are identical and all satellites have the same size, the variances of the distributions g1(y) and g2(y) are zero and Eq. (6.150) takes a simple form PðV; wÞ ¼ k1 dðVy1 wÞ þ k2 dðVy1 wÞ;

ð6:200Þ

where d(x) is the delta function. In the case of a multimodal breakup probability, the breakup probability has the form PðV; wÞ ¼

n X

ki dðVyi wÞ;

i¼1

n X ki yi ¼ 1:

ð6:201Þ

i¼1

In the simplest case when breakup leads to the formation of two identical droplets, the breakup probability is Pðv; wÞ ¼ 2dðv0:5wÞ:

ð6:202Þ

Substituting the breakup probability (6.202) into the kinetic equation (6.175) and neglecting the first two integrals (i.e., droplet breakup is the only process taken into account), we get the following equation: qnðV; tÞ ¼ 2 f ð2VÞnð2V; tÞ f ðVÞnðV; tÞ: qt

ð6:203Þ

References

j391

A solution of this equation is given in [57] as a sum of independent particular solutions with discrete spectra. This paper also discusses two special cases of initial distributions: a monodisperse distribution, and a distribution that is uniform in a certain interval. The obtained solutions make it possible to determine the first four moments of the distribution, and one can show from the Pearson diagram that as time goes on, the solution tends to a logarithmic normal distribution. This conclusion is consistent with the result obtained in [59] for a constant breakup frequency f (V) that does not depend on the size of particles.

References 1 Sinaiski, E.G. and and Lapiga, E.J. (2007) Separation of Multiphase, Multicomponent Systems, Wiley-VCH, Weinheim. 2 Loginov, V.I. (1979) Dewatering and Desalting of Oils, Chemistry, Moscow (in Russian). 3 Voloshuk, V.M. (1984) Kinetic Theory of Coagulation, Hydrometeoizdat, Moscow (in Russian). 4 Hahn, G.J. and and Shapiro, S.S. (1967) Statistical Models in Engineering, Wiley & Sons, New York. 5 Gradstein, J.S. and Ryszik, I.M. (1971) Tables of Integrals, Sums, Series, and Products, Nauka, Moscow. 6 Ruckenstein, E. (1975) Condition for the Size Distribution of Aerosols to be Self-Preserving, J. Colloid Interface Sci., 50 (3), 508–518. 7 Jeffreys, G.V. and Davies, G.A. (1971) Coalescence of Droplets and Dispersions in Recent Advances in Liquid–Liquid Extraction (ed. K. Hanson), Pergamon Press, Oxford. 8 Einstein, A. and Smoluchowski, M. (1936) Brownian Motion, ONTI, Moscow (in Russian). 9 Levich, V.G. (1962) Physicochemical Hydrodynamics. Prentice–Hall, Englewood Cliffs, N.J. 10 Delichatsips, M.A. and Probstein, R.P. (1975) Coagulation in Turbulent Plow: Theory and Experiment, J. Colloid Interface Sci., 51, 394–405.

11 De Boer, G.B.J., Hoedamakers, G.F.M. and Thones, D. (1989) Coagulation in Turbulent Flow, Part I, Chem. Eng. Res. and Des., 67 (3), 301–307. 12 Entov, V.M., Kaminski, V.A. and Lapiga, E.J. (1976) Calculation of Emulsion Coalescence Rate in a Turbulent Flow. Izvestia Acad. Sci. USSR, Fluid Dyn., 3, 47–55 (in Russian). 13 Sinaiski, E.G. and Rudkevich, A.M. (1981) Coagulation of Bubbles in a Turbulent Flow of Viscous Fluid. Colloid J., 43 (2), 369–371 (in Russian). 14 Sinaiski, E.G. (1993) Coagulation of Droplets in a Turbulent Flow of Viscous Fluid. Colloid J., 55 (4), 91–103 (in Russian). 15 Sinaiski, E.G. (1994) Brownian and Turbulent Coagulation of Droplets in a Viscous Fluid and Stability of Emulsions. Colloid J., (1), 105–112 (in Russian). 16 Sinaiski, E.G. (1994) Coagulation of Droplets with Different Viscosities in a Turbulent Flow of Viscous Fluid. Colloid J., (2),222–225 (in Russian). 17 Tunizki, A.N. , Kaminski, V.A. and Timashov, S.F. (1972) Methods of Physicochemical Kinetics, Chemistry, Moscow (in Russian). 18 Kaminski, V.A. (1976) Kinetics of Turbulent Coagulation. Colloid J., 5, 907–912 (in Russian).

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19 Hinze, J.O. (1959) Turbulence, McGraw–Hill,New York. 20 Monin, A.S. and Yaglom, A.M. (1971) Statistical Fluid Mechanics: Mechanics of Turbulence, Vol 1, MIT Press, Cambridge, MA Monin, A.S. and Yaglom, A.M. (1975) Statistical Fluid Mechanics: Mechanics of Turbulence, Vol 2, MIT Press, Cambridge, MA. 21 Landau, L.D. and Lifshitz, E.M. (1987) Fluid Mechanics, 2nd ed., Pergamon Press, Oxford. 22 Leontovich, M.L. (1983) Introduction to Thermodynamics. Statistical Physics, Nauka, Moscow (in Russian). 23 Russel, W.B. (1981) Brownian Motion of Small Particles Suspended in Liquids. Ann. Rev. Fluid Mech., 13, 425–455. 24 Happel, J. and and Brenner, H. (1965) Low Reynolds Number Hydrodynamics, Prentice–Hall, Englewood Cliffs. 25 Zinchenko, A.Z. (1978) Calculation of Hydrodynamic Interactions of a Droplet at Low Reynolds Numbers. Appl. Math. Mech., (5),955–959 (in Russian). 26 Davis, M.H. (1969) The Slow Translation and Rotation of Two Unequal Spheres in a Viscous Fluid. Chem. Eng. Scl., 24, 1769– 1776. 27 O’Neil, M.E. and Majumdar, S.R. (1970) Asymmetrical Slow Viscous Fluid Motion Caused by the Translation or Rotation of Two Spheres, Part I, II. ZAMP, 21, 164. 28 Wacholder, E. and Weihs, D. (1972) Slow Motion of a Fluid Sphere in the Vicinity of Another Sphere or a Plane Boundary. Chem. Eng. Scl., 27, 1187– 1827. 29 Haber, S., Hetsroni, G. and Solan, A. (1973) On the Low Reynolds Number Motion of Two Droplets. J. Multiphase Flow, 1, 57–71. 30 Rushton, E. and Davies, G.A. (1973) The Slow Unsteady Settling of Two

31

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35

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37

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39

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Fluid Spheres Along Their Line of Centers. Appl. Sci. Res., 28, 37–61. Reed, L.D. and Morrison, F.A. (1974) The Slow Motion of Two Touching Fluid Spheres Along Their Line of Centers, J. Multiphase Flow, 1, 573– 583. Hetsroni, G. and Haber, S. (1978) Low Reynolds Number Motion of Two Drops Submerged in an Unbounded Arbitrary Velocity Field. J. Multiphase Flow, (4), 1–17. Zinchenko, A.Z. (1980) Slow Asymmetric Motion of Two Drops in a Viscous Medium. Appl. Math. Mech., 1, 30–37 (in Russian). Fuentes, Y.O., Kim, S. and Jeffrey, D.J. (1988) Mobility Functions for Two Unequal Viscous Drops in Stokes Flow. Part I. Axisymmertric Motions. Phys. Fluids. A., 31, 2445–2455. Fuentes, Y.O., Kim, S. and Jeffrey, D.J. (1989) Part II. Nonaxisymmertric Motions. Phys. Fluids. A., 1, 61–76. Davis, R.H., Schonberg, J.A. and Rallison, J.M. (1989) The Lubrication Force Between Two Viscous Drops. Phys. Fluids. A., 1, 77–81. Zinchenko, A.Z. (1981) Calculation of Short-Range Interaction of Droplets with Regard to Internal Circulation and Slippage. Appl. Math. Mech., 45, 759–763 (in Russian). Kruyt, H.R. (ed.) (1952) Colloid Science. Irreversible Systems, Elsevier, Amsterdam, Vol 1. Hogg, R., Healy, T.W. and Fuerstenau, D.W. (1966) Mutual Coagulation of Colloidal Dispersions. Trans. Faraday Soc., 62, 1638–1651. Hamaker, H.C. (1937) The London–van der Waals Attraction Between Spherical Particles. Physica., 4, 1058–1078. Shenkel, J.N. and Kitchener, J.A. (1960) A Test of the Derjaguin– Verwey–Overbeek Theory with a

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Colloidal Suspension. Trans. Faraday Soc., 56, 161–173. Fuks, N.A. (1955) Mechanics of Aerosols, Acad. Sci,USSR, Moscow, (in Russian). Melcher, J.R. and Taylor, G.I. (1969) Electrohydrodynamics: a Review of the Role of Interfacial Shear Stress. Ann. Rev. Fluid. Mech., 1, 111–146. O’Konski, C.T. and Thacher, H.C. (1953) The Distortion of Aerosol Droplets by an Electric Field. J. Phys. Chem., 57, 955–958. Garton, C.G. and Krasucki, Z. (1964) Bubbles in Insulating Liquids: Stability in an Electric Field. Proc. Roy. Soc. Lond. A., 280, 211–226. Taylor, G.I. (1964) Disintegration of Water Drops in an Electric Field. Proc. Roy. Soc. Lond. A., 280, 383–397. Rosenklide, C.E. (1969) A Dielectric Fluid Drop in an Electric Field. Proc. Roy. Soc. Lond. A., 312, 473–494. Miskis, M.J. (1981) Shape of a Drop in an Electric Field. Phys. Fluids, 24, 1967–1972. Adornato, P.M. and Brown, R.A. (1983) Shape and Stability of Electrostatically Levitated Drops. Proc. Roy. Soc. Lond. A., 389, 101–117. Basaran, O.A. and Scrlven, L.E. (1989) Axisymmetric Shape and Stability of Charged Drops in an Electric Field. Phys. Fluids A., 1, 799–809. Allan, R.S. and Mason, S.G. (1962) Particle Behaviour in Shear and Electric Fields. I. Deformation and Burst of Fluid Drops. Proc. Roy. Soc. Lond. A., 267, 45–61. Taylor, G.I. (1966) Studies in Electrohydrodynamics. I. The

52

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Circulation Produced in a Drop by an Electric Field. Proc. Roy. Soc. Lond. A., 291, 159–166. Ausman, E.L. and Brook, M. (1967) Distortion and Disintegration of Water Drops in Strong Electric Fields. J. Geophys. Res., 72, 6131–6141. Brazier-Smith, P.R. (1971) Stability and Shape of Isolated Pair of Water Drops in an Electric Field. Phys. Fluids, 14, 1–6. Stone, H.A. (1994) Dynamics of Drop Deformation and Breakup in Viscous Fluids. Ann. Rev. Fluid Mech., 26, 65–102. Nigmatullin, R.I. (1987) Dynamics of Multiphase Media, Nauka, Moscow, Vol 1 (in Russian). Sherman, P. (ed.) (1968) Emulsion Science, Academic Press,London–New York. Loginov, V.I. (1985) Dynamics of Dropping Liquid in a Turbulent Flow. Appl. Mech. Techn. Phys., (4), 66–73, (in Russian). Tichonov, V.I. (1970) Overshoots of Random Processes, Nauka, Moscow (in Russian). Kolmogorov, A.N. (1941) On Logarithmic Normal Low of Particle Distribution During Breakup. Dokladi Acad. Nauk SSSR, 31 (2), 99–101 (in Russian). Golizin, G.S. (1962) Fluctuations of Energy Dissipation in a Locally Isotropic Turbulent Flow. Dokladi Acad. Nauk SSSR, 144 (3), 520–523 (in Russian).

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7 Motion and Collision of Inertial Particles in a Turbulent Flow Inertness of a particle is characterized by the Stokes number St ¼ tv/T(L), where tv is the characteristic relaxation time for the particle, and T(L) is the Lagrangian time scale (integral scale of turbulence). Particles for which St 1 are called inertial. This class of particles includes sufficiently large particles whose density by far exceeds the density of the carrier phase, for instance, solid particles in a gas. Another important parameter is the volume concentration of particles j. The region j 1 corresponds to a rarefied disperse medium, in which the average distance between the particles is much greater than their average size. Under these conditions, collisions between particles play a minor role, and our attention can be restricted to interactions of particles with turbulent fluctuations (eddies) of the carrier flow. In contrast, when we are interested in the behavior of particles in sufficiently dense disperse media, particle collisions play the key role. Our goal in both cases is to describe the dynamics of the disperse phase and to determine the collision frequency. The method most suitable for this task is the PDF (Probability Density Function) method, which is based on the use of the kinetic equation for the PDF of particle velocity; this equation takes into account particle interactions with each other as well as with turbulent fluctuations of the carrier phase (gas). An equation for the PDF of particle velocity and temperature has been derived in [1,2] (this equation neglects particle collisions). For relatively large particles in an isotropic turbulent flow this equation reduces to the Fokker–Planck equation for Brownian motion. The effect of particle collisions on the turbulent transport of disperse phase momentum at small values of j was taken into account in [3.4]. The latter publication deals with the limiting case of inertial particles, when their interactions with turbulent fluctuations can be neglected. In highly concentrated disperse media, particle collisions play the leading role, and the equation for the PDF reduces to the Boltzmann equation [5]. Two kinetic models of particle transport have been proposed in [6–8]: one based on the solution of the PDF equation by the method of perturbations, and another one representing a modification of the Enskog method of solving the Boltzmann equation for dense gases, specially tailored for the case of colliding particles.

j 7 Motion and Collision of Inertial Particles in a Turbulent Flow

396

Simulation of motion of particles with an arbitrary density based on the kinetic equation for the PDF in inhomogeneous turbulent flows is presented in [8–12]; these papers also attempt to derive the collision frequency and the rate of coagulation.

7.1 Motion of Particles without Mutual Collisions

When particles are moving in sufficiently rarefied, disperse media (j 1), the role of particle collisions is insignificant and the main consideration should be given to particle interactions with turbulent fluctuations of the carrier flow. If the characteristic time of dynamic relaxation tv for the particles is much shorter than the Lagrangian integral time scale of turbulence T(L) that defines the time of decay of fluctuations carrying large energies, the particles are well involved into the fluctuational motion of the carrier flow. When the density of the carrier phase is much lower than that of particles, one can neglect the forces arising due to the pressure of a pressure gradient in the fluid, the virtual mass forces, and the Basset forces associated with the particles motion relative to the carrier fluid and arising due to the instability of the flow that bypasses the particle. If, in addition, the particles have a low volume concentration j, one can also neglect interparticle interactions resulting from the infrequent collisions, and the feedback influence of the particles on the parameters of the carrier flow. Then the equation of motion of an isolated solid spherical particle has the form of the Langevin equation (5.101), which depends on two random uncorrelated fields describing the velocity of the turbulent carrier flow u and the Brownian force f. In general, the motion of an isolated, sufficiently large (i.e., non-Brownian) spherical particle of arbitrary density in a sufficiently rarefied turbulent liquid or gaseous medium is described (neglecting interparticle interactions) by the following equation: dv p 3re r Du r Du dv p ¼ CD juv p jðuv p Þ þ g þ e g þ CA e dt 4r p d p r p Dt r p Dt dt þ

9re r pd p

rffiffiffiffiffiðt ne dðuv p Þ dt1 r pffiffiffiffiffiffiffiffiffi þCL e ðuv p Þ ðr uÞ; p rp dt1 tt1

ð7:1Þ

0

dR p D q q ¼ v p ; ¼ þ uk ; dt Dt qt qXk where t is time; vp and Rp are the velocity components and coordinates of a particle; u is the velocity of the carrier flow; re and rp are the densities of the carrier phase and the particle; dp is the particles diameter; ne is the coefficient of kinematic viscosity.


It should be noted that, unlike Eq. (5.101), Eq. (7.1) is not limited to low Reynolds numbers Rep of a particle but is valid for larger values of Rep as well, owing to the dependence of resistance coefficients on the Reynolds number. The terms on the right-hand side of Eq. (7.1) describe the corresponding specific forces of viscous resistance, gravity, and buoyancy; the effect of the virtual mass (in the form that is commonly used in modeling particle motion in a non-viscous fluid); the Basset force (written in the low Reynolds numbers approximation); and an additional lifting force due to velocity shear in the carrier flow. The resistance coefficient CD and the coefficient CL in the lifting force depend on the Reynolds number associated with the particles motion relative to the carrier fluid, on the velocity shear in the carrier flow, and on other parameters. It is convenient to represent Eq. (7.1) in the relaxation form dv p uv p ¼ þ f A þ f B þ f L þ Fg ; dt tu

ð7:2Þ

where

fA ¼A

du ; dt

ðt f B ¼ kB 0

1r0 Fg ¼ g; 1 þ C A r0 A¼

ð1 þ CA r0 Þr0 ; 1 þ CA r0

dðuv p Þ dt1 pffiffiffiffiffiffiffiffiffi; dt1 tt1

tu ¼ tv ð1 þ CA r0 Þ; kB ¼

9r0 d p ð1 þ CA r0 Þ

f L ¼ Lðuv p Þ ðr uÞ; tv ¼

4d p ; 3r0 CD juv p j

rffiffiffiffiffi ne ; p

L¼

CL r0 ; 1 þ CA r0

r0 ¼

re : rp

Here tv is the particles characteristic time of dynamic relaxation, which is a function of the Reynolds number Rep = |u vp|dp/ne, and tu is the characteristic relaxation time that takes into account the effect of the virtual mass. A decision to take into account all of the forces entering Eq. (7.2) would unduly complicate the problem. However, in many practical applications some of the forces can be neglected. Thus, for rp re, the forces associated with fluid acceleration fA, memory fB, and velocity shear fL are of no importance and can be excluded from consideration. In order to make a transition from dynamic stochastic description of discrete particles (Eq. (7.2)) to simulation of statistical behavior of a particle ensemble (i.e., of the disperse phase), we introduce the dynamic probability density in the phase space of particle coordinates X and velocities v p p ðX ; v; tÞ ¼ dðX R p ðtÞÞdðvv p ðtÞÞ;

ð7:3Þ

where averaging is performed over all possible instances of the turbulent flow and the random force field f.

j397


398

The procedure used to derive the kinetic equation has been described in Section 5.6. Differentiating Eq. (7.3) with respect to time and taking into account Eq. (7.2), we get the Liouville equation for dynamic probability density in the phase space: q pp q p p uk v pk þ fAi þ fBi þ fLi þ Fgi p p ¼ 0: þ v pk ð7:4Þ qt qXk tu Low volume concentration of particles makes it possible to consider them as independent. Then the dynamic probability density of a particle ensemble can be introduced as p¼

vX p p; V p

where v is the particles volume and V is the volume of the spatial region under consideration. Let us now introduce the PDF of velocities for a system of particles P ¼ hpi and represent the velocity of the carrier medium as a sum of the average and fluctuational components: u ¼ U þ u0 . Averaging Eq. (7.4) over the ensemble of random realizations of the turbulent velocity field and using the fact that hvpipi ¼ viP, we obtain: qP qP q ui vi þ þ FAi þ FBi þ FLi þ Fgi P þ vi tu qt qXi qvi 0 q hui pi 0 0 0 ¼ þ h fAi pi þ h fBi pi þ h fLi pi ; qvi tu

ð7:5Þ

where FAi ¼ A

D0 Ui ; Dt ðt

FBi ¼ kB 0

D0 q q ¼ þUi ; Dt qt qXk

dðUi Vi Þ dt1 pffiffiffiffiffiffiffiffiffi; dt1 tt1

qU j qUi : FLi ¼ LðUi Vi Þ qXi qX j

The terms on the right-hand side of Eq. (7.5) stand for the interactions of particles with turbulent eddies, which are described by the interfacial forces appearing on the right-hand side of Eq. (7.2). These terms need to be determined. We start with establishing the correlation between velocity fluctuations in the carrier flow and the particles PDF hu0i pi. We shall model the velocity field of the carrier flow by a Gaussian process with a given autocorrelation function. Then, using the Furutsu– Donsker–Novikov formula (1.201) for Gaussian random processes, we obtain hu0i pi ¼

ðð

d pðX ; tÞ dX 1 dt1 ; u0i ðX ; tÞu0k ðX 1 ; t1 Þ duk ðX 1 ; t1 Þ

ð7:6Þ


where d pðX ; tÞ q ¼ duk ðX1 ; t1 Þ qX j

dR p j ðtÞ dv p j ðtÞ q : pðX ; tÞ pðX ; tÞ qv j duk ðX1 ; t1 Þ duk ðX1 ; t1 Þ

To find the functional derivatives here, one needs to solve the equation of motion (7.2) for the particle. The main difficulty is that the expression for fA contains Du/Dt – a substantial derivative of fluid velocity along the trajectory of the carrier flow. We take it to be equal to the derivative along the particles trajectory du/dt. We also assume that the averaged velocity slip at the interface is zero and the transverse force fA is negligibly small. The first assumption is based on the hypothesis that by knowing the duration of the particles interaction with turbulent eddies, we can take into account the effect of the averaged slip at the interface on the fluctuational motion of the disperse phase (the socalled effect of trajectory intersection [13]) in the proper manner. As for the second assumption, insignificance of the liftingforce caused by the velocity shear can be proved rigorously only for the case of low Reynolds numbers. Contribution of this effect to the correlation hu0i pi will still be insignificant at finite, but moderate, values of Rep. Applying the Laplace transform to Eq. (7.2), we obtain for the particles velocity image: pffiffiffiffiffi 1 þ Atu s þ kB tu ps pffiffiffiffiffi u ~i ðsÞ ¼ ðsÞ~ ~v pi ðsÞ ¼ ui ðsÞ: ð7:7Þ 1 þ tu s þ kB tu ps The transition to the original is carried out by using the convolution formula ðt ðt v pi ðtÞ ¼ gðtt1 Þui ðR p ðt1 Þ; t1 Þdt1 ; R pi ðtÞ ¼ v pi ðR p ðtÞdt; 0

ð7:8Þ

0

where g(t) is the Green function for Eq. (7.2). Applying functional differentiation to the second equation (7.8), one obtains a system of integral equations in functional derivatives: dv pi ðtÞ ¼ d i j dðX 1 R p ðt1 ÞÞgðtt1 ÞHðtt1 Þ du j ðX 1 ; t1 Þ ðt þ gðtt2 Þ t1

qui ðR p ðt2 Þ; t2 Þ dR pn ðt2 Þ dt2 ; qXn du j ðX1 ; t1 Þ

ð7:9Þ

ðt dR pi ðtÞ ¼ d i j dðX 1 R p ðt1 ÞÞ gðtt2 Þdt2 du j ðX 1 ; t1 Þ t1

ðt ðt þ

gðtt3 Þdt3 t1 t2

qui ðR p ðt2 Þ; t2 Þ dR pn ðt2 Þ dt2 ; qXn du j ðX1 ; t1 Þ

where H(x) is the Heaviside function.

ð7:10Þ

j399


400

Integral equations (7.9) and (7.10) are solved by the iterative method. As the first approximation, we take the first term on the right-hand side of the equation. This results in a solution that corresponds to a homogeneous, shearless flow. The second approximation takes into account the inhomogeneity of the flow up to the first order spatial derivatives. This yields dR pi ðtÞ ¼ du j ðX 1 ; t1 Þ

dR pi ðtÞ dR pi ðtÞ þ ; du j ðX 1 ; t1 Þ 1 du j ðX 1 ; t1 Þ 2

ðt dR pi ðtÞ ¼ di j dðX 1 R p ðt1 ÞÞ gðtt2 Þdt2 du j ðX 1 ; t1 Þ 1 t1

ðt ðt ðt2 dR pi ðtÞ qui ðR p ðt2 Þ; t2 Þ ¼ dðX 1 R p ðt1 ÞÞ gðtt3 Þdt3 gðt2 t4 Þdt4 dt2 : du j ðX 1 ; t1 Þ 2 qX j t1 t2

t1

ð7:11Þ Then Eq. (7.9) takes the form dv pi ðtÞ ¼ d i j dðX 1 R p ðt1 ÞÞgðtt1 ÞHðtt1 Þ du j ðX 1 ; t1 Þ qui ðR p ðt2 Þ; t2 Þ dR pn ðt2 Þ þ gðtt2 Þ dt2 qXn du j ðX1 ; t1 Þ 1 ðt

ð7:12Þ

t1

ðt þ gðtt2 Þ t1

qui ðR p ðt2 Þ; t2 Þ dR pn ðt2 Þ dt2 : qXn du j ðX1 ; t1 Þ 2

Approximating the last term in Eq. (7.12) with the accuracy up to the first order spatial derivatives and making use of the second equation (7.8), we obtain ðt gðtt2 Þ t1

qui ðR p ðt2 Þ; t2 Þ dR pn ðt2 Þ qv pi ðR p ðt2 Þ; t2 Þ dR pn ðtÞ dt2 ¼ : qXn du j ðX1 ; t1 Þ 2 qXn du j ðX1 ; t1 Þ 2 ð7:13Þ

We now introduce a two-time autocorrelation function of velocity fluctuations in the carrier flow along the particle trajectory, YL p ðtt1 Þ ¼

hu0i ðX ; tÞu0j ðR p ðt1 Þ; t1 Þi hu0i ðX ; tÞu0j ðX ; tÞi

:

In view of rapid decrease of the function cLp(x) with increase of x, we assume that the major contribution to the integrals comes from the region in the vicinity of x ¼ 0.


Then it follows from Eq. (7.6), Eqs. (7.11)–(7.13)(see [12]) that qP qP qUn qP þ tu gu þ tu lu hu0i pi ¼ hu0i u0k i fu qXk qvn qvk qXk qUn qP qUn qVj qP þt2u hu þ t2u hu : qXk qXn qXk qXk qvj

ð7:14Þ

The first two terms on the right-hand side describe the interactions of particles with turbulent eddies in a homogeneous shearless carrier flow, and the last three terms describe the influence of the velocity gradient. The coefficients in Eq. (7.14) are called the involvement factors. They are equal to ð¥ fu ¼ yL p ðxÞgðxÞdx; 0

lu ¼

ðx ð¥ 1 gu ¼ y ðxÞ gðx1 Þdx1 dx; tu L p 0

0

xð1 ðx ð¥ 1 yL p ðxÞ gðxx1 Þ gðx1 x2 Þdx2 dx1 dx; tu 0

0

ð¥

ðx ðx

0

0 x1

1 hu ¼ 2 yL p ðxÞ tu

ð7:15Þ

0

xð1

gðxx3 Þdx3 gðx1 x2 Þd2 dx1 dx: 0

The relations (7.14) and (7.15) are valid for times t much greater than the LaðLÞ grangian integral time scale T p of fluctuations of the carrier flow velocity along the particle trajectory. This time can be considered as the characteristic time of particle interaction with highly energetic turbulent eddies of the carrier flow. The coefficients fu, gu, lu, hu tell us how strongly the particle is involved into fluctuational motion of the turbulent carrier flow. In order to obtain these coefficients, one needs to know the autocorrelation function cLp(x). It is often approximated by the expression yL p ðxÞ ¼ expðx=T ðLÞ p Þ: Then, in view of Eq. (7.7), the involvement factors (7.15) are 1 fu ¼ fðs ¼ ðT ðLÞ p Þ Þ¼

1 þ AWu þ BWu ; 1 þ Wu þ BWu

ðLÞ

lu ¼

f2 ðs ¼ ðT p Þ1 Þ ; Wu

ðLÞ

gu ¼

fðs ¼ ðT p Þ1 Þ ; Wu

ðLÞ

hu ¼

f2 ðs ¼ T p Þ ; W2u

Wu ¼

tu

; ðLÞ

Tp

B ¼ kB

ð7:16Þ qffiffiffiffiffiffiffiffiffiffiffi ðLÞ pT p :

Eq. (7.14) gives us the mixed correlation moments between velocity fluctuations of the continuous and disperse phases:

j401


402

ð ð 1 hu0i piv j dvV j hu0i pidv j qU j qV j 0 0 0 0 ¼ fu hui u j i þ tu hui uk i lu gu ; qXk qXk ð ð 1 j ¼ pdv; vi ¼ vi Pdv; j

hu0i v0j i ¼

ð7:17Þ

where j and vi are, respectively, the averaged volume concentration and the averaged velocity of the disperse phase. The expression (7.16) connects the mixed correlation moments between velocity fluctuations of the continuous and disperse phases and the Reynolds stresses. In the case of a homogeneous turbulent flow, the expression (7.17) simplifies and, in view of Eq. (7.16), takes the form hu0i v j i ¼ fu hu0i u0j i ¼

1 þ AWu þ BWu 0 0 hu u i: 1 þ Wu þ BWu i j

ð7:18Þ

With the Basset force neglected, B = 0, it follows from Eq. (7.18) that due to the action of non-stationary forces caused by the pressure gradient in the fluid (the buoyant force) and by the effect of the virtual mass, the involvement factor fu for light particles (at A > 1) could be greater than 1 and, consequently, the mixed correlation moments could exceed the Reynolds stresses in the fluid. The Basset force increases the instantaneous resistance to the flow and thus exerts a smoothing effect, and in doing so it increases fu for heavy particles and decreases it for light ones. Having determined the correlation hu0i pi associated with the resistance force, we now need to find the correlation moments of the particle PDF with the other forces acting on particles. We shall neglect the third- and higher-order derivatives of the PDF. Making use of the continuity equation for incompressible fluid quk/qXk ¼ 0, we can write the correlation moment due to the fluctuations of the buoyancy force and the virtual mass force:

D0 hu0i pi Dp qUi qhu0i u0k pi h fAi0 pi ¼ A þ u0i þ hu0i pi : qXk qXk Dt Dt

u0i

ð7:19Þ

uk v pk Dp 0 q þ fAk þ fBk þ fLk þ fgk p ¼ ui tu Dt qvk

qhu0i pi qp Uk Vk 0 þFAk þ FBk þ FLk þ Fgk þ ui ðuk v pk Þ ¼ tu qXk qXk 0 0 0 0 q hui uk pihui vk pi 0 0 þhu0i fAk pi þ hu0i fBk pi þ hu0i fLk0 pi tu qvk

qp qp qp þðUk Vk Þ u0i þ u0i u0k u0i v0k : qXk qXk qXk ð7:20Þ


Dropping the terms in Eq. (7.20) that contain second- and higher-order derivatives of p and neglecting contributions from the correlation moments of fluid velocity 0 fluctuations with fluctuations of the Basset force hu0i fBk i and from the buoyant force 0 0 hui fLk i, we get

u0i

0 0 hui uk ihu0i v0k i Dp qP 0 0 þ hu0i fAk i þ hu0i fBk i þ hu0i fLk0 i ¼ tu Dt qvk þ

0 hu0i fAk i¼

ðhu0i v0k ihu0i v0k iÞ

ð7:21Þ

qP : qXk

qu0 qu0 qu0 u0 qUk A u0i k þ Un u0i k þ hu0i u0n i þ u0i k n : qt qXn qXn qXn ð7:22Þ

hu0i u0k pi ¼ hu0i u0k iP:

ð7:23Þ

When determining hu0 pi in the first and the third term in Eq. (7.19), we shall confine ourselves to the first term in Eq. (7.14). Then, in view of Eqs. (7.21)–(7.23), the expression (7.19) takes the form hu0 u0 ihu0i v0k i qP D0 qP h fi 0 pi ¼ A fu hu0i u0k i þ i k tu Dt qvk qvk

0 0 0 qP 0 quk 0 quk 0 0 qUk 0 quk þ A ui þ ui þ Un u i þ hui un i qt qXn qXn qXn qvk qP qUi qP qhu0i v0k i þ hu0i v0k i fu hu0k v0n i þ P : ð7:24Þ qXk qXk qvn qXk To simplify our analysis of the correlation, we can drop the term h fi 0 pi in Eq. (7.5), because the Basset force is, as a rule, insignificant and, as opposed to non-stationary forces caused by the pressure gradient in the carrier flow and by the virtual mass acceleration, does not cause any qualitative changes but only slightly attenuates nonstationary effects. Then the correlation between fluctuations of the buoyant force and the particle PDF is

0 qu j qu0i qU j qUi h fi 0 pi ¼ L ðU j V j Þ p þ hu0j pi qXi qX j qXi qX j

0 qu j qu0i þ ðu0j v0j Þ p : qXi qX j

ð7:25Þ

At large Reynolds number of the turbulent carrier flow, one can neglect any correlations between fluid vorticity fluctuations and velocities of the continuous

j403


404

and disperse phases. Then, retaining only the first term in the relation (7.14) for hu0i pi, we can represent Eq. (7.25) as h fLi0 pi ¼ L fu hu0j u0k i

qU j qUi qP : qXi qX j qvk

ð7:26Þ

Using the above assumptions and the relations (7.14), (7.24), (7.26), we obtain from Eq. (7.5) a closed kinetic equation for the PDF of particle velocity in a turbulent flow: qhu0i u0k i qP qP q Ui vi þ þA þFAi þ FBi þ FLi þ Fgi P þvi qXk tu q qXi qvi q qP qP qUn qP 2 2 qUn qP hu0i u0k i fu þtu gu þtu lu þt h ¼ qXk qvn u u qXk qXn qvi qvk qXk hu0 v0 ihu0i u0k i qP D0 2 qUn qV j qP 0 0 qP þtu hu fu hui uk i þA þ i k tu qXk qXk qv j Dt qvk qvk

0 0 0 0 qu qu qu u qUk qP þ u0i k n A u0i k þ Un u0i k þ hu0i u0n i qt qt qXn qXn qvk qU j qUi qP qP qUi qP hu0i u0k i þ fu hu0k u0n i þL fu hu0j u0k i : qXk qvn qXi qX j qvk qXk ð7:27Þ Terms on the left-hand side describe the convection of the PDF in the phase space of velocities and coordinates, while terms on the right-hand side characterize the diffusional transport due to particle interactions with turbulent eddies of the carrier flow in the same phase space. The equation (7.27) enables us to obtain a system of continuum equations for the averaged characteristics (moments) of the disperse phase. Integrating over the whole volume of the velocity space, we get the mass conservation equation qj qjVk ¼ 0: þ qXk qt

ð7:28Þ

Multiplication of both sides of Eq. (7.27) by vi and a further integration over v yields a balance equation for the momentum: qVi qVi q ¼ ðhv0 v0 iAhu0i u0k iÞ þ Vk qt qXk qXk i k D pik qlnj Ui Vi þFAi þ FBi þ FLi þ Fgi ; tu tu qXk ð 1 hv0i v0j i ¼ ðvi Vi Þðv j V j ÞPdv; j þ

ð7:29Þ


where hv0i v0j i are turbulent stresses in the disperse phase caused by the participation of particles in the fluctuational motion of the carrier flow. The first term on the righthand side of Eq. (7.29) describes turbulent migration of particles caused by the emergence of turbulent stresses in the disperse phase and by the action of turbulent stresses in the carrier flow. The relation between these two terms defines the direction of turbulent migration. The last term describes turbulent diffusion of particles, which is characterized by the diffusion coefficient 0 0 0 0 0 0 qU j 0 0 D pi j ¼ tu hvi v j i þ gu hui u j i þ tu hu hui uk i Ahui u j i : ð7:30Þ qXk It should be noted that the particles turbulent diffusion tensor, as well as the mixed correlation moment of velocity fluctuations in the continuous and disperse phases (7.17), is non-symmetric in shear flows. The equation for the second moments of velocity fluctuations is obtained from Eq. (7.27) by multiplying both sides of this equation by vivj and integrating over v: qhv0i v0j i

0 0 0 qV j 1 qjhvi v j vk i qVi ¼ hv0i v0k i hv0j v0k i qXk qXk qt qXk j qXk qU j qVi qUi qVi þð1 þ AÞhu0i u0k i lu gu gu þ ð1 þ AÞhu0j u0k i lu qXk qXk qXk qXk 2 2ð fu 1Þ 0 0 hui u j i þ fu þ ð fu hu0i u0j ihv0i v0j iÞ þ A tu tu qU j qV j qUi qVi 2 D0 0 0 0 0 ð fu jhu0i u0j iÞ þ hui uk i þ hu j uk i þ qXk qXk qXk qXk j Dt 0 0 0 D0 hu0i u0j i qU j qUi qhui u j uk i þhu0i u0k i þhu0j u0k i þ A Dt qXk qXk qXk qU qUk j þL ð fu hu0i u0k ihv0i v0k iÞ qX j qXk qUk qUi þð fu hu0j u0k ihv0j v0k iÞ : qXi qXk

þVk

qhv0i v0j i

þ

ð7:31Þ Eq. (7.31) describes convective and diffusional transport, emergence of fluctuations within the averaged shear flow, generation of fluctuations due to the involvement of particles into fluctuational motion of the carrier flow, and dissipation of turbulent energy of the disperse phase due to the work of interfacial interaction forces. In a uniform shearless turbulent flow, or for small particles in the framework of the locally homogeneous approximation, there follows from Eq. (7.31) a simple algebraic expression for the stress tensor in the disperse phase: hv0i v0j i ¼ ð fu ð1 þ AÞAÞhu0i u0j i ¼

1 þ A2 Wu 0 0 hui u j i: 1 þ Wu

ð7:32Þ

j405


406

The relation (7.32) is identical to a known formula for the fluctuational energy of a particle in a homogeneous, isotropic turbulent flow [14]. Under these conditions, the coefficient of particles turbulent diffusion (7.30) takes the form 0 0 D pi j ¼ tu ðhv0i v0j i þ gu hu0i u0j i þ A fu hu0i u0j iÞ ¼ T ðLÞ p hui u j i;

ð7:33Þ

which corresponds to the solution obtained in [14]. In view of Eq. (7.32), we can describe acceleration resulting from the migration force that is caused by particle interactions with turbulent eddies of the carrier flow by the following expression: FMi ¼

q q ðhv0 v0 iAhu0i u0k iÞ ¼ Mhu0i u0k i; qXk i k qXk

M¼

ð1AÞð1A2 Wu Þ : 1 þ Wu

The migration coefficient M is defined by the particle inertia parameters Ou and A, which, in their turn, depend on the density ratio r0 and on the virtual mass factor CA commonly taken to be equal to 0.5. For heavy particles (A ! 0), we have M > 0 and consequently, these particles move under the action of turbulent migration from highly turbulent regions into regions of low turbulence. But at A ¼ 1 and A ¼ W1 u the sign of M and thereby the direction of the migration force changes. Thus, at A < 1, migration of large (inertial) particles and at A > 1 – migration of small (inertialess) particles (bubbles) is directed from the regions of low turbulence toward the regions of high fluctuation level. It is interesting to note that turbulent migration will displace large bubbles into low-turbulence regions. At A ¼ 1 (r0 ¼ 1), both the factor M and the migration force vanish. One of the most important characteristics of particle behavior in a turbulent flow is the duration of particle interaction with high-energy eddies. For very small inðLÞ ertialess particles, the interaction time T p coincides with Lagrangian scale of turbulence T(L) measured along the trajectories of liquid particles. For large particles, ðLÞ ðLÞ the time T p may significantly differ from T(L), and the ratio T p =T ðLÞ may be greater or smaller than 1 depending on the values of parameters responsible for the particles inertia and for the averaged interfacial velocity slip. To get an explicit connection between Lagrangian and Eulerian characteristics of turbulence, one should use the Corrsin hypothesis [15] about the possibility of independent representation (and consequently, of averaging) of random velocity fluctuations and fluid particle displacement fields. In accordance with this hypothesis, the Lagrangian time autocorrelation functions of fluid velocity fluctuations calculated along the particle trajectories are connected with Eulerian spacetime autocorrelation functions in a stationary homogeneous isotropic turbulent field by the following relations: ð ð ðLÞ ðEÞ ðLÞ ðEÞ YL p ðtÞ ¼ YL ðr; tÞF p ðr; tÞdr; YN p ðtÞ ¼ YN ðr; tÞF p ðr; tÞdr:

ð7:34Þ

Here L and N indicate the directions parallel and transverse to the particles relative velocity vector W = V U, and Fp(r, t) is the probability density of particle


displacement over the distance r during the time t that is given by the delta function u0 yðtÞ F p ðr; tÞ ¼ d rW t pffiffiffi s ; 3

ð7:35Þ

where u0c(t) is the effective free path of a particle in its random motion, s is a unit vector in the direction r, u20 ¼ hu0k u0k i=3 is the intensity of turbulence. Eulerian spacetime correlation function is represented as ðEÞ

ðEÞ

Yi j ðr; tÞ ¼ Yi j ðrÞYðEÞ ðtÞ;

ð7:36Þ

and if the turbulence is isotropic, then ðEÞ

Yi j ðrÞ ¼ ½ f ðrÞgðrÞ gðrÞ ¼ f ðrÞ þ

ri r j þgðrÞdi j ; r2

r 0 f ðrÞ; r ¼ jrj: 2

ð7:37Þ

Combining Eq. (7.34) with Eqs. (7.35)– (7.37), we see that " ðLÞ YL p ðtÞ

ðLÞ

¼

# f ðrÞgðrÞ u0 yðtÞ 2 Wt þ pffiffiffi þgðrÞ YðEÞ ðtÞ; r2 3

f ðrÞgðrÞ u20 y2 ðtÞ p ffiffi ffi þgðrÞ YðEÞ ðtÞ; r2 3 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u0 yðtÞ 2 2 2 2 þ u0 y ðtÞ; W ¼ jW j: r¼ Wt þ pffiffiffi 3 3

YL p ðtÞ ¼

ð7:38Þ

The function c(t) characterizes the particles effective free path associated with its involvement into fluctuational motion of the carrier flow; it can be determined by solving the particles equation of motion (7.2). Retaining only the two most essential terms on the right-hand side of this equation, one obtains the approximation yðrÞ ¼ t þ ð1AÞtu ½expðt=tu Þ :

ð7:39Þ

For Eulerian correlation functions, one commonly uses the following exponential dependences: f ðrÞ ¼ expðr=LÞ; YðEÞ ðtÞ ¼ expðt=T ðEÞ Þ;

ð7:40Þ

where L and T(E) are the Eulerian spatial integral and time integral scales. The formulas (7.40) give a good approximation of correlation functions at large Reynolds numbers, even though they are not correct when r and t tend to zero.

j407


408

Eqs. (7.38)–(7.40) can be used to determine Lagrangian time integral scales of fluid velocity fluctuations along the particle trajectories: ðLÞ TL p

ð¥ ¼

ðLÞ YL p ðtÞdt;

ðLÞ TN p

ð¥

ðLÞ

¼ YN p ðtÞdt:

0

ð7:41Þ

0

Formulas (7.41), together with (7.38) and (7.39), tell us how the duration of particle interaction with turbulent eddies is influenced by the particles inertia, by the density ratio of the carrier and disperse phases, and by the effect of trajectory intersection due to the averaged interfacial velocity slip. As a consequence, there follows the relation between the Lagrangian and Eulerian integral time scales of turbulence: T ðLÞ 3 þ 2m ¼ ; T ðEÞ 3ð1 þ mÞ

m¼

u0 T ðEÞ : L

ð7:42Þ

It follows from Eq. (7.42) that the Eulerian time macroscale defined in a reference frame that is moving with the average velocity of the carrier flow is greater than the corresponding Lagrangian scale. Numerical solution allows to trace the influence of the Stokes number St ¼ tu/T(E) and the parameter A on the duration of particle ðLÞ interaction with turbulent eddies (Fig. 7.1). For heavy particles (A < 1), T p inðLÞ ðLÞ creases monotonously with St; at A ¼ 1, we have T p ¼ T ; and for light particles ðLÞ (bubbles) (A > 1), the value of T p decreases as the Stokes number St gets larger. The effect of trajectory intersection is characterized by the parameter z ¼ W/u0. For all values of A, an increase of the average interfacial slip (that is, an increase of z), ðLÞ causes T p to fall (see Fig. 7.2). A decrease of the density ratio of the carrier and disperse phases (i.e., an increase of A) would be qualitatively similar to increased averaged slip.

Fig. 7.1 Effect of St on the duration of particles interactions with turbulent eddies: 1–5 A ¼ 0.05, 1, 2, 3; 6 – [16].


Fig. 7.2 Effect of the averaged slip on the duration of particles interaction with turbulent eddies: 1–5 A¼0.05, 1, 2, 3; I, II – directions parallel and transverse to the relative velocity vector.

In the limit of extremely inertial particles, that is, at St ! 1, we find from Eqs. (7.37)–(7.42): ðLÞ

TL p

T ðEÞ

¼

ðLÞ

TN p T ðEÞ

¼

pffiffiffi 1 þ 0:5 mN þ mðz þ A= 3Þ2 ð2NÞ1 ð1 þ mNÞ2 1 þ 0:5mN þ mA2 ð6NÞ1

ð1 þ mNÞ2 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A2 A 2 N¼ þ z þ pffiffiffi : 3 3

;

;

ð7:43Þ

In the other limiting case, specifically, that of strong influence of the effect of trajectory intersection (in other words, for large values of the drift parameter z), we have for all values of A ðLÞ

TL p ¼

L ; W

ðLÞ

TN p ¼

L : 2W

ð7:44Þ

Based on the obtained equation of motion for the disperse phase, we can now examine the distribution of bubbles over the cross section of a long vertical tube of diameter D. The flow is assumed to be stationary and hydrodynamically developed. The disperse phase is assumed to have a low volume concentration so that we can neglect the feedback influence of bubbles on turbulent characteristics of the carrier flow as well as the effect of particle collisions. In the present case we can take r0 ! 1. Parameters of a hydrodynamically developed flow vary only with the radial coordinate r and are independent of the longitudinal coordinate X, while the averaged radial components of velocity of the continuous and disperse phases are equal to zero, Ur ¼ Vr ¼ 0. Then the distribution of bubbles over the tubes cross section can

j409


410

be found by solving Eq. (7.29) projected onto the r-direction. We represent fluctuational energy and the coefficient of turbulent diffusion in Eq. (7.29) in the framework of the locally homogeneous approximation (7.32) and (7.33). The intensities of turbulent fluctuations of the radial and azimuthal velocity components of the disperse phase are assumed to be equal. Then it follows from Eq. (7.29) that ðLÞ

Tp dlnj dMhu0 2i CL dUx hu0r 2i ðUx Vx Þ : ¼ þ tu CA dr dr dr

ð7:45Þ

It is readily seen that the concentration profile of bubbles in the cross section of the tube is shaped by turbulent migration and by the buoyant force caused by velocity shear. To simplify the analysis, we assume that the average phase slip (drift velocity) in Eq. (7.45) is defined by the buoyant force only, while the effect of turbulent transport can be neglected in the first approximation. Then Ux Vx ¼ 7 gtv ;

ð7:46Þ

where the 7 signs relate to the upward and downward flows respectively. In the bubble size range for which the resistance factor CD is approximately constant, we have instead of Eq. (7.46) sffiffiffiffiffiffiffiffiffiffi 4d p g Ux Vx ¼ 7 : 3CD

ð7:47Þ ðLÞ

According to Eq. (7.44), the duration T p of bubbles interaction with turbulent eddies under the condition |Ux Vx| u* (where u* is the dynamic velocity) can be written as T ðLÞ p ¼

L ; 2j Ux Vx j

L ¼ 0:1 D ;

ð7:48Þ

where D is the diameter of the tube. The parameter Wu ¼

tu ðLÞ

Tp

¼

80CA ~ d; 3CD

~d ¼ d p D

ð7:49Þ

is responsible for the bubbles inertia and depends only on the ratio of diameters of the bubble and the tube. The gradient of the averaged fluid velocity in a hydrodynamically developed flow is given by Ux u2* r ¼ ; dr Rðne þ nt Þ

ð7:50Þ

where nt is the coefficient of turbulent viscosity and R = D/2 is the radius of the tube.


In view of Eq. (7.47) and Eq. (7.50), we can write Eq. (7.45) in the dimensionless form as ~ C dMh~ u0 2 i ¼ Wu 7 d~r d~r

2 d lnj h~ u0 r i

qffiffiffiffiffiffiffiffiffiffiffiffi d0p Ga Wu~r ~t 1þn

;

jð0Þ ¼ j0

ð7:51Þ

pffiffiffiffiffiffiffiffiffi ~ ¼ CL =ðCA 3CD Þ, and Ga = gD3/n2 is ~t ¼ nt =n, C where ~r ¼ r=R, h~ u0 2r i ¼ hu0r 2i=u2* , n the Galileo number. The solution of Eq. (7.51) is ~¼ j

j ~M j ~L; ¼j j0 "

ðMh~ u0 2r iÞr¼0 ~M ¼ j Mh~ u0 2r i

#Wu M

2 ~ ~ L ¼ exp4 7 CGa j

1=2

2 ~r 3 ð dW M u 2 0 exp4 lnMh~ u ri d~r 5; d~r 0

ð7:52Þ

3 ð~r ~ 1=2 ðd p Þ Wu~r d~r 5: ~t Þ h~ u0 2r ið1 þ n 0

The expression for bubble concentration is represented as a product of two factors, one of which describes turbulent migration, while the other is responsible for the lifting effect (buoyancy) due to the velocity shear. The extent to which the concentration profile is affected by turbulent migration depends strongly on the bubble size, which determines the sign of the migration factor M and thereby the direction of the migration force; on the other hand, the direction of bubbles motion (upward or downward) is inessential. If the size of bubbles does not vary over the ~ M in Eq. (7.52) simplifies to tube cross section, the expression for j "

ðMh~ u0 2r iÞr¼0 ~M ¼ j Mh~ u0 2r i

#Wu M ð7:53Þ

In contrast, the extent to which the concentration profile is affected by the lifting force is qualitatively independent from the bubble size and is primarily determined by the direction of motion. In calculations, one can use the approximate formula for turbulent viscosity (the van Driest–Reichard formula) 1 ~t ¼ n 6

(rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ) h

y i2 þ 2 1 ð1 þ ~ k 2 yþ 1 þ 4 1exp r 2 Þ; A

where yþ ¼ yu* =ne ¼ ð1~r ÞRþ , Rþ ¼ Ru* =ne , k ¼ 0:4, A ¼ 26.

ð7:54Þ

j411


412

The intensity of turbulent fluctuations of the radial component of fluid velocity and the Lagrangian scale of turbulence are determined by the relations Dt nt n 2 h~ u0 r i ¼ ¼ ; TL ¼ 2 TL Sct TL u* l ¼ Rð0:140:08~r 2 0:06~r 4 Þ ;

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 l u* 100 þ ; ne

ð7:55Þ

where l is the Prandtl–Nikuradze mixing length, and Sct is the turbulent Schmidt number for the diffusion of an inertless (passive) impurity in a fluid; the latter is taken to be equal to 0.9. Shown in Figures 7.3 and 7.4 are the distributions of bubbles over the tube cross section calculated on the basis of Eq. 7.52 while taking into account Eq. (7.54) and Eq. (7.55), and the experimental data for the two bubble flow regimes (downward and upward) taken, respectively, from [17] and [18] for different reduced velocities jf and jg of the fluid and the gas. The lifting force factor CL depends on Rep as well as on the parameter of velocity shear. However, as distinct from the case of low Reynolds numbers, in a flow bypassing a particle at moderate and high Reynolds numbers, these dependencies become sufficiently weak [19]. With increase of Rep, the factor CL approaches 0.5 for a non-viscous fluid. Thus, for the case under consideration, if the flow bypassing a bubble has a relatively high Reynolds number of Rep 100, then CL ¼ const will be an acceptable approximation. In calculations, the experimental value CL ¼ 0.05 [17] has been used. Hence, the distribution of particles over the tube cross section is non-uniform, with bubble concentration reaching its maximum on the tubes symmetry axis in a downward flow, whereas in an upward flow, the peak concentration is observed near the walls of the tube.

Fig. 7.3 Calculated (1) and measured (2) in [17] bubble distribution in a downward flow (jf ¼ 0.71 m/s, jg ¼ 0.10 m/s).

7.2 Motion of Particles with Mutual Collisions

Fig. 7.4 Calculated (1) and measured (2) in [18] bubble distribution in an upward flow (jf ¼ 1.391 m/s, jg ¼ 0.180 m/s).


In concentrated disperse media, particle collisions often play a considerable and sometimes a crucial role. The influence of particle collisions on turbulent transport of momentum of the disperse phase at low volume concentrations has been taken into account in [3,4,10], to which we should add that [4] is devoted to the case of inertial particles whose interaction with turbulent fluctuations can be ignored. The best way to describe disperse phase dynamics while taking into account particle collisions and particle interactions with turbulent fluctuations of the carrier flow is to use the kinetic equation for the PDF of particle velocity. This kinetic equation is derived in the same way as in the previous section. To simplify the analysis, we shall limit ourselves to consideration of particle collisions. The motion of a heavy particle subject to an external force and to collisions with the other particles in a turbulent flow is described by the Langevin equation dv p uv p ¼ þ F þ W; dt tv

ð7:56Þ

dR p ¼ v p; dt where vp is particles velocity, u is the velocity of the carrier fluid, and F is the external force. The first term on the right-hand side of Eq. (7.56) defines the force of interfacial hydrodynamic resistance. The dynamic relaxation time tv for a particle depends on the Reynolds number of the flow bypassing this particle, and thereby takes into account the influence of both the averaged and the fluctuational component of interfacial slip on the flow regime. The last term describes particle interactions caused by random collisions.

j413


414

We shall consider only pair interactions, which is consistent with the case of low volume concentration j of particles. The velocities of two particles after the collision, v0p , v0p1 , are related to the same velocities before the collision, vp, vp1 by 1 v0p ¼ v p ð1 þ eÞðwkÞk; 2

1 v0p1 ¼ v p1 þ ð1 þ eÞðwkÞk; 2

ð7:57Þ

where e is the restitution coefficient of the relative velocity (momentum) in the collision of two particles; dependence of the material of the colliding particles is baked into this coefficient (e ¼ 1 for elastic impact, e ¼ 0 for perfectly inelastic impact, and 0 < e < 1 otherwise). Also present in this equation are the relative velocity of the colliding particles w ¼ vp vp1 and the unit vector k directed from the center of the first particle to the center of the second one. In order to affect a transition from dynamic trajectory modeling of the individual particles on the basis of stochastic equations (7.56) and (7.57) to continual statistical modeling of the disperse phase as a whole, we introduce the PDF P(X, v, t) in the phase space of particle coordinates and velocities. An equation for a single-particle PDF is derived from Eq. (7.56) in the same manner as in Section 7.1: qP qP q þ þ vk qt qXk qvk

Uk vk 1 qhu0k pi þ Fk P ¼ þ Jc ; tv tv qvk

ð7:58Þ

where Uk and u0k are the averaged and fluctuational components of the carrier phase velocity. The terms on the right-hand side of Eq. (7.58) describe, respectively, particle interactions with turbulent eddies of the carrier flow and particle interactions due to collisions. Assuming the random velocity field of the continuous phase to be Gaussian, one obtains an explicit expression for the correlator hu0k pi in an inhomogeneous turbulent flow, accurate to the first-order spatial derivatives: hu0i pi ¼ hu0i u0k i þt2v hu

fu

qP qP qUn qP þ tv gu þ tv lu qXk qvn qvk qXk

qUn qP qUn qV j qP þ t2v mu ; qXk qXn qXk qXk qv j

ð7:59Þ

where fu ¼ fu0 ; gu ¼ Yu0 fu ; lu ¼ gu fu1 ; hu ¼ fu1 þ Yu1 2gu ; mu ¼ fu2 þ Yu1 þ 2 fu1 3gu ; fun ¼

ð¥ ð¥ 1 x 1 n Y ðxÞx exp ¼ Yu ðxÞxn dx: dx; Y u un n!tvnþ1 tv n!tnþ1 v 0

0


The coefficients fu, gu, lu, hu, mu characterize the response of particles to turbulent velocity fluctuations of the carrier flow and are defined by the autocorrelation function of velocity fluctuations of the continuous phase along the particle trajectory ðLÞ cu(x). A common convention is to take Yu ðxÞ ¼ expðx=T p Þ. Then the involvement coefficients become fu ¼ hu ¼

1 ; 1 þ Wu

gu ¼

1 W2u ð1

2

þ Wu Þ

;

1 ; Wu ð1 þ Wu Þ mu ¼

lu ¼

1 W2u ð1

þ Wu Þ

3

;

1 Wu ð1 þ Wu Þ2 fu1 ¼

; 1

ð1 þ Wu Þ

2

;

Wu ¼

tv ðLÞ

Tp

:

ð7:60Þ ðLÞ

The Lagrangian time scale T p for an inertialess impurity coincides with the Lagrangian integral time scale of turbulence T(L), whereas for inertial particles, ðLÞ especially in the presence of the averaged velocity slip, T p may differ substantially (L) from T . The last three terms on the right-hand side of Eq. (7.59) play an important role in shear flows because they are closely related to derivatives of the averaged velocity. In the absence of these terms the expression for hu0 pi reduces to a corresponding relation obtained in [1,2] for an unbounded homogeneous flow. The condition j 1 makes it possible to neglect the direct contribution of particle collisions to the stresses and the fluctuational energy flux of the disperse phase. Also, the particles are assumed to be sufficiently small to neglect any change of the averaged characteristics of the flow over distance of the order of one particle size. _ 1=2 The is the case when d p ge p < < 1, where dp is the particle diameter, g_ – the characteristic shear rate, e p ¼ hv0k v0k i=2 – the particles fluctuational energy density. Under these conditions, the collision operator in Eq. (7.58) has the form (see [6,20]) Jc ¼

d2p 4

ðð P2 ðv; v1 ÞðwkÞdk dv1 ;

ð7:61Þ

where P2(v, v1) is the two-particle PDF of velocity. Integration of the kinetic equation (7.58) over the velocity subspace of the phase space gives us a set of continual equations for moments of the PDF: the continuity equation, the equation of conservation of momentum, and the balance equations for turbulent stresses in the disperse phase: qj qjVk ¼ 0: þ qXk qt

ð7:62Þ

D pik qlnj qhv0 v0 i Ui vi qVi qVi ¼ i k þ þ Fi ; þ Vk qXk qt qXk tv tv qXk

ð7:63Þ

0 0 0 0 0 0 qU j ¼ tv hvi v j i þ gu hui u j i þ tv hu hui uk i : qXk

ð7:64Þ

D pi j

j415


416

qhv0i v0j i

0 0 0 qV j 1 qjhvi v j vk i qVi ¼ hv0i v0k i hv0j v0k i qXk qXk qt qXk j qXk qV j qU j qV j qU j qUn 0 0 hui uk i gu lu þtv hu mu qXk qXk qXk qXn qXn qVi qUi qUn qVi qUi 0 0 lu þtv hu mu hv j vk i gu qXk qXk qXk qXn qXn

þVk

qhv0i v0j i

þ

ð7:65Þ

2 þ ð fu hu0i u0j iÞvhv0i v0j iÞ þ Ji j ; tv ð ð ð 1 1 j ¼ Pdv; Vi ¼ vi Pdv; hv0i v0j i ¼ ðvi Vi Þðv j V j ÞPdv: j j Here Dpij is the tensor of turbulent diffusion of particles. Eq. (7.65) includes terms that describe the time evolution of the system: convection, diffusion, generation of fluctuations by the averaged shear flow, generation of fluctuations due to the involvement of particles into fluctuational motion of the carrier phase, and dissipation of turbulent energy of the disperse phase that goes into work of the hydrodynamic resistance force. The last term Jij describes particle collisions. From the expression (7.59) for the correlator hu0 v0 i one can determine the correlation moment of velocity fluctuations of the continuous and disperse phases, which is necessary in order to calculate the feedback action of particles on the turbulent parameters of the carrier flow. Thus, it follows from Eq. (7.59) that hu0i v0j i

ð ð ð 1 1 0 0 0 ¼ hui iðv pi Vi ÞPdv ¼ hui piv p j dvV j hui pidv j j qU j qV j qU j qV j qUn ¼ fu hu0i u0j i þ tv hu0i u0k i lu gu þtv mu hu : qXk qXk qXk qXn qXn ð7:66Þ

It remains for us to find the collisional term Jij. To this end, we need to determine the two-particle PDF of velocity at the instant when the collision takes place. By analogy with the molecular chaos hypothesis in the kinetic theory of gases that leads us to the Boltzmann equation, we adopt the assumption that all particle motions are statistically independent. According to this assumption, the two-particle PDF is represented as a product of two single-particle PDFs, and the resulting expressions describing particle collisions in a turbulent flow turn out to be similar to the corresponding relations in the kinetic theory of gases [3,10,21]. But this approach works only for relatively big particles whose dynamic relaxation time is much longer than the integral scale of turbulence, meaning that their relative motion is similar to the chaotic motion of molecules and is uncorrelated. So we resort to a different approach first suggested in [22] that yields a simple explicit expression for the collisional term when used in combination with the Grad expansion [5].


Let us represent the two-particle PDF as a generalized Grad expansion: P2 ðv; v1 Þ ¼ P20 ðv; v1 Þ þ P21 ðv; v1 Þ;

ð7:67Þ

where the first term is the correlated normal distribution [22.23] P20 ðv; v1 Þ ¼

N2

3 4pe p

3

ð1R4 Þ3=2 3 0 0 2 0 0 0 0 ex p ðv v 2R v v þ v v Þ ; k k k 1k 1k 1k 4ð1R4 Þpe p

R ¼ ei =ðe p ek Þ1=2 ;

ek ¼ hu0k u0k i=2;

ei ¼ hu0k v0k i=2;

ð7:68Þ

ð7:69Þ

N is the number of particles per unit volume (number concentration), R is the correlation coefficient of colliding particles, ek is the fluctuational energy density of the continuous phase, and ei is the fluctuational energy density of interfacial interactions. It should be noted that the expression (7.69) for the correlation coefficient of colliding particles is a local parameter, because it takes into account only that correlation of colliding particles motions that is caused by particle interactions with the velocity field of the carrier flow, and fails to account for the spatial correlation of velocities of particles moving toward the point of collision along different trajectories. Nevertheless, the expression for R correctly describes the behavior of the correlation coefficient in the limiting cases of small and large particles, that is, at Ou ! 0 and at Ou ! 1, and is consistent with the results obtained from direct numerical calculations [22]. It follows from Eq. (7.68) that ð ðð 1 1 2 P20 ðv; v1 Þdv1 ¼ P0 ðvÞ; v0i v0j P20 ðv; v1 Þdvdv1 ¼ e p di j ; N N2 3 ðð 1 2 v0i v01 j P20 ðv; v1 Þdvdv1 ¼ R2 e p d i j ; N2 3 where P0(v) is the Maxwellian velocity distribution for a single particle. The second term in Eq. (7.67) represents an expansion over Hermit–Sonin polynomials (which are widely used in the kinetic theory of gases), with only one expansion term being retained [24]: P21 ðv; v1 Þ ¼ P20 ðv; v1 Þ

v0k v0k 9 2 0 0 0 0 hv d v i d v e v p ij ij i j i j 3 8e2p ð1R2 Þ 3

0 0 vi v1 j þ v0j v01i v0k v01k v0 v0 di j : þ v01i v01 j 1k 1k d i j 2R2 3 3 3 ð7:70Þ

j417


418

The distribution P21(v, v1) satisfies the conditions 1 N2 1 N2

ðð ðð

2 v0i v0j P21 ðv; v1 Þdvdv1 ¼ hv0i v0j i R2 e p d i j ; 3 2 v0i v01 j P21 ðv; v1 Þdvdv1 ¼ R2 hv0i v0j i R2 e p d i j : 3

It then follows from Eqs. (7.67)–(7.70) that 16ð1e2 Þjk p 2e p 1=2 ð1R2 Þ3=2 di j Ji j ¼ 3d p 3p 16ð1 þ eÞð3eÞje p 3e p 1=2 2 ¼ ð1R2 Þ3=2 hv0i v0j i e p di j : 5d p 3p 3

ð7:71Þ

The first term in Eq. (7.71) describes dissipation of turbulent fluctuations of the disperse phase via inelastic collisions, and the second term represents the redistribution of turbulent energy between different components via collisions. This redistribution reflects the tendency of the system to reach an isotropic state. At R ¼ 0 the expression (7.71) reduces to a relation for uncorrelated chaotic motion of particles [20]. It is readily seen from Eq. (7.71) that the presence of correlation between particle motions results in an increase of fluctuational energy dissipation and thereby of the rate at which the system is approaching its isotropic state. Thus the role of particle collisions diminishes as their motions become more correlated. Let us now apply the obtained equations to the problem of time evolution of a homogeneous shear layer. Consider a flow along the X-axis with a constant velocity gradient along the Y-axis (dUx/dY ¼const). Then it follows from Eq. (7.62) and Eq. (7.63) that the volume concentration of particles in a homogeneous shear layer in the absence of external forces should be constant (j ¼ const), and the velocity of particles should coincide with that of the carrier flow (Vx ¼ Ux). It is known [25] that stationary solutions of equations for the Reynolds stresses in the continuous phase cannot be obtained for a homogeneous shear layer. Therefore the analysis of turbulent stresses in the disperse phase should be carried out on the basis of nonstationary equations for the second moments. Then from Eq. (7.65) combined with Eq. (7.71) there follows a system of equations for stress tensor components: dhv0x 2i dUx ¼ 2ðhv0x v0y i þ fu1 hu0x u0y iÞ dY dt

2 2 2 2 hv0x 2i e p ; þ ð fu hu0x 2ihv0x 2iÞ Qc tv 3 tc 3 dhv0y 2i 2 2 2 2 ¼ ð fu hu0y 2ihv0y 2iÞ Qc hv0y 2i e p ; dt tv 3 tc 3

ð7:72Þ


dhv0x v0y i dhv0z 2i 2 2 2 2 hv0z 2i e p ; ¼ ð fu hu0z 2ihv0z 2iÞ Qc dt tv 3 tc 3 dt dUx 0 0 ¼ ðhvy 2i þ fu1 huy 2iÞ dY 2 2 þ ð fu hu0x u0y ihv0x v0y iÞ ðhv0x v0y i; tu tc Qc ¼ tc ¼

8ð1e2 Þjk p 2e p 1=2 ð1R2 Þ3=2 ; dp 3p 1=2 5d p 2p ð1R2 Þ3=2 ; 8ð1 þ eÞð3eÞje p 2e p

where Qc is the intensity of fluctuational energy dissipation due to particle collisions, and tc is the effective time between particle collisions. The expressions for mixed correlation moments of velocity fluctuations of the continuous and disperse phases (7.66) take the form hu0x v0x iÞ ¼ fu hu0x 2itv fu1 hu0x u0y i

dUx 0 0 ; huy vy iÞ ¼ fu hu0y 2i; dY

hu0z v0x iÞ ¼ fu hu0z 2i; hu0x v0y iÞ ¼ fu hu0x u0y i; hu0y v0x iÞ ¼ fu hu0x u0y itv fu1 hu0y 2i

ð7:73Þ

dUx : dY

The results of numerical solution of these equations are presented in [24]. The Reynolds stresses in the disperse phase are determined by the method of large eddy simulation (LES) [26–28]. The initial conditions correspond to an isotropic state. The average velocity shear is 50 s1, the particle diameter is dp ¼ 60 mm, the density ratio between particles and the carrier flow (gas) is rp/re ¼ 2000. Thanks to the absence of averaged velocity slip, the duration of eddy–particle interactions (particle interactions with eddies with a high energy content) is taken to be equal to the integral ðLÞ Lagrangian scale T p ¼ ek =ð2:075eÞ, where ek is the turbulent energy of the continuous phase and e is the rate of turbulent energy dissipation. The dynamic relaxation time and the Reynolds number are taken in the form tv ¼

r p d2p 18me

ð1 þ 0:15Re0:687 Þ; p

Re p ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðek þ e p 2ei Þd p =ve :

ð7:74Þ

The numerical results are depicted in Fig. 7.5–Fig. 7.8. Fig. 7.5–Fig. 7.7 show the solutions of Eq. (7.72) and Eq. (7.73) (curves 4, 5) juxtaposed to the normal and tangential components calculated in [26]. Fig. 7.8 demonstrates the effect of particle collisions on turbulent stresses in the disperse phase. The stresses are derived from Eq. (7.72) for the case of elastic particles (e ¼ 1) with dp ¼ 656 mm, rp/re ¼ 85, and the average velocity shear of 50 s1. The solution of Eq. (7.72) is illustrated by the curves

j419


420

Fig. 7.5 Variation of longitudinal velocity fluctuations (m2/s2) with time (s) 1 – hu0x 2i; 2,4 – hv0x 2i; 3,5 – hu0x v0x i:

Fig. 7.6 Variation of transverse velocity fluctuations (m2/s2) with time (s) 1 – hu0y 2i; 2,4 – hv0y 2i; 3,5 – hu0y v0y i:

Fig. 7.7 Variation of tangential velocity fluctuations (m2/s2) with time (s) 1 – hu0x u0y i; 2,4 – hv0x v0y i; 3,5 – hu0x v0y i:

7.3 Frequency of Collisions of Particles

Fig. 7.8 The effect of collisions on velocity fluctuations (m2/s2) 1 – hv0x 2i; 2,4 – hv0y 2i; 3,5 – hv0x v0y i:

1 (no collisions), 2 and 3 (collisions are taking place, j ¼ 0.0125). The direct stochastic simulation [27] is illustrated by the curves 4 (no collisions) and 5 (collisions are taking place, j ¼ 0.0125).


In many technological and meteorological processes, it is essential that we should know the rate of particle coagulation. When we try to determine this rate, the main challenge is to find collision frequency (or the average time between particle collisions) in a turbulent flow. There exists a large body of theoretical works devoted to particle collisions and coagulation in turbulent flows. Relatively simple (and best known) solutions of this problem have been obtained in the approximation of homogeneous, isotropic turbulence for the limiting cases of very small (inertialess) [29] and very large (inertial) particles [30]. The solution [29] is valid for particles whose dynamic relaxation time tv is less than the Kolmogorov microscale of turbulence ðtv < tl0 Þ; such particles are fully involved into the turbulent motion of the carrier flow. In this case the effort to determine collision frequency o12 can be restricted to particle interactions with small-scale turbulent fluctuations that are responsible for turbulent energy dissipation. The solution [30] holds for the opposite limiting case and refers to particles whose dynamic relaxation time is much greater than Lagrangian time macroscale of turbulence, that is, tv T(L). Motions of such particles are statistically independent. It means that their relative motion is uncorrelated and is similar to chaotic motion of molecules in the kinetic theory of rarefied gases. In this case it is sufficient to consider particle interactions with turbulent eddies with high energy content (large-scale fluctuations), ignoring the contribution of particle interactions with small-scale turbulence to the collision frequency.

j421


422

The difficulties arise when we try to determine the collision frequency for the case of finite ratios of particle relaxation time to the microscale and macroscale of turbulence (in other words, at tl0 < tv < T ðLÞ ), when one needs to take into account particle interactions with both small-scale and with large-scale fluctuations (eddies) while being aware of the correlativity of particle motions. A solution that takes these factors into account has been obtained in [31], but its description of particle interactions with small-scale eddies is not sufficiently consistent and therefore does not provide a smooth transition to the solution obtained in [29] for inertialess particles. Simultaneous interaction of particles with small-scale and large-scale (i.e., energycarrying) eddies has been considered in [32]. However, the correlation coefficient of velocities of two particles was not determined correctly, and its suggested value goes to 1 when the particles are identical. As a consequence, particle interactions with energy-carrying eddies do not contribute to the collision frequency of identical particles, and a transition to the limit of very large particles (solution [30]) cannot be made. Correct expressions for the correlation coefficient and the time interval between particle collisions have been obtained in [22] for the case of identical inertial particles (tv T (L)) under the assumption that the carrier flow velocity and the disperse phase velocity are both normally distributed. Determination of collision frequency with due consideration of particle interactions with both large-scale and small-scale eddies is carried out in [23] by solving the diffusion equation that follows from the kinetic equation for the two-particle PDF of velocity. However, the solution thus obtained predicts a correlation coefficient that approaches 1 (rather than 0) as the particle inertia gets larger, and thus does not allow for a continuous transition to the solution [30]. Interactions of particles with both large-scale and as small-scale eddies has been considered in [33] to obtain the time interval between collisions. This paper assumes normal distributions of velocities and their derivatives. The solution obtained in [33], as well as the one obtained in [32], is based on the so-called cylindrical formulation, which is commonly used in the kinetic theory of gases to calculate the collision frequency of molecules o12 and relate it to the average relative velocity of a particle pair. As shown in [34], a more suitable way to find the collision frequency of lowinertia particles that results from their involvement into small-scale turbulent motion of the carrier flow is the so-called spherical formulation of the problem, which expresses o12 in terms of the average radial component of the particles relative velocity. It should be noted that both formulations lead to identical results for those characteristics of inertial particle collisions that are caused by the involvement of particles into large-scale turbulent motion. If the particles are small, the outcomes of the two formulations are no longer identical. This is explained by the fact that the longitudinal and transverse correlation functions as well as spatial scales for the two-point velocity field are going to be different to these two formulations. This difference manifests itself as we try to determine the longitudinal and transverse components of the turbulent energy dissipation tensor [14]. The respective


expressions for the collision frequency of low-inertia particles and highly inertial particles have been derived in [35] based on separate treatment of particle interactions with small-scale eddies and energy-carrying eddies; while being qualitatively correct, these expressions are not very accurate. As of today, we do not have any single analytic expression for the collision frequency that would be valid in the whole spectrum of values of particle inertia, and that would provide a correct continuous transition to the limiting case solutions [29] and [30]. Numerical solutions of the problem under consideration for relatively small particles with finite (nonzero) relaxation time based on the DNS method [28] are presented in [35,36] (for homogeneous isotropic turbulence) and in [37] (for a flat channel flow). The collision velocity of small particles in a turbulent flow with a constant transverse velocity gradient is calculated in [38]. If there is a velocity shear, the flow cannot be called isotropic. However, because of the local isotropy of its small-scale structure, it presents some interest for theoretical studies of the combined effect of turbulence and velocity shear on the collision frequency of inertialess particles. To simulate collisions of large particles in a turbulent flow tv T (L); interactions with small-scale fluctuations do not play any considerable role), the method of large eddy simulation (LES) may be employed as being less cumbersome and costly as compared to the DNS method. This method was used in [22] and [39]; the latter publication considered a binary mixture, where relative drift of particles of different densities in the gravitational field plays an important role. In this section, we present an analytic model that can be used to determine the collision frequency of particles in the entire range of particle inertias. It is assumed that particle density is much higher than the density of the carrier phase (e.g., gas). In addition, the model is generalized for the case when the contribution of the averaged velocity component caused by the relative motion of particles having different inertias (e.g., in the gravitational field or in the velocity field of a shear flow) becomes of special importance. We shall consider only those collisions that are induced by turbulent velocity fluctuations or by the averaged relative motion of particles. Brownian motion and hydrodynamic, molecular, and electrostatic interactions between the particles will not be taken into account. Let us look at the collisions of spherical particles of radii R1 and R2 caused by turbulent fluctuations of the carrier flow velocity. In the framework of the spherical formulation, the collision frequency of particles of types 1 and 2 is defined by the following relation [33]: v12 ¼ 2pR2c hjwr ðRc ÞjiN2 ¼ KN2 ;

ð7:75Þ

where Rc ¼ R1 + R2 is the so-called coagulation radius; Ra is the radius of the aparticle; Na is the number of a -particles per unit volume; wr ¼ wr is the radial component of relative velocity w ¼ v1 v2 of two particles of types 1 and 2; va is the velocity of an a -particle; r is a unit vector pointing from the center of the first particle to the center of the second one; K is the probability of collision of two particles (collision kernel); a ¼ 1, 2.

j423


424

By convention, probability density of the fluctuational component of radial relative velocity is defined by the Gaussian distribution 1 w0 2 ; Pðwr Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp r 0 2hwr 2i 2pðwr0 2Þ

ð7:76Þ

where wr ¼ Wr þ wr0 , and Wr, wr0 are the averaged and fluctuational components of radial velocity. According to Eq. (7.76), in the absence of relative drift of particles (Wr = 0), the average absolute value of the radial relative velocity is related to its root-mean-square by the equation 1=2 ð¥ 2 hjwr ji ¼ jwr jPðwr Þdwr ¼ hwr0 2i : p

ð7:77Þ

0

It follows from here that the turbulent component of the collision kernel in Eq. (7.75) is equal to Kt ¼ ð8phwr0 2iÞ1=2 R2c :

ð7:78Þ

If particle diameters are smaller than the inner scale of turbulence, (the Kolmogorow spatial microscale), the mean square of fluctuations of radial relative velocity is represented as [32.33] hwr0 2i ¼ hv01r i þ hv02r i2hv01r v02r i

0

0

0 qv1r qv2r qv1r qv02r þ R22 þ 2R1 R1 ; þ R21 qr qr qr qr

ð7:79Þ

where r is the distance between the particles centers. In a homogeneous, isotropic turbulence the quantity hv01r i is directly related to the turbulent energy of the carrier flow ek ¼ hu0k u0k i=2 (see [40]): hv01r i

ð¥ 2 2 1 x dx; ¼ e p ¼ f u ek ; f u ¼ Yu ðxÞexp 3 3 tv tv

ð7:80Þ

0

where e p ¼ hv0k v0k i=2 is the kinetic fluctuational energy of a particle, fu is the particles involvement factor, and c(x) is the Lagrangian autocorrelation function of velocity fluctuations in the carrier flow along the particle trajectory. The expression for the involvement factor fu in Eq. (7.80) is valid for heavy particles whose density is much higher than the density of the carrier medium (gas), so the only essential force among the interfacial forces is that of aerodynamic resistance. In order to calculate the correlation moment of two particles


velocities hv01r v02r i, the joint PDF of the particle velocity and the gas velocity is taken in the form of a Gaussian distribution [22]. In accordance with this assumption, the joint PDF at the point of collision is described by the correlated normal distribution Pðv1 ; v2 Þ ¼

27N1 N2 64p3 ð1 fu1 f2 Þ1=2 " !# 1=2 1=2 3 v1k v1k fu1 fu2 v1k v2k : 1=2 1=2 4ð1 fu1 f2 Þ k p1 e e p1

ð7:81Þ

p2

Then the correlation moment of two particles velocities is 2 1=2 1=2 hv01r v02r i ¼ R12 e p1 e p2 ; 3

1=2

1=2

R12 ¼ fu1 fu2 :

ð7:82Þ

The correlativity of particle velocities at the point of collision through their interaction with the velocity field of gas at this point is taken into account by the coefficient R12. The contribution of terms in Eq. (7.79) that contain velocity derivatives is important only for very small particles whose relaxation time tv is of the same order as the microscale tl0 . In the interests of simplicity, we determine these terms under the assumption that the particles are fully involved into small-scale motion of the carrier flow. Then *

qv0r qr

2 +

¼

qv1r qv2r qr qr

* + qu0r 2 : ¼ qr

For isotropic turbulence, we have [41] *

qu0r qr

2 + ¼

e ; 15ne

ð7:83Þ

where e is the specific dissipation of turbulent energy. In view of Eq. (7.79), Eq. (7.80), and Eqs. (7.82)–(7.84,) there follows from Eq. (7.78) an expression for the turbulent component of the collision kernel: Kt ¼

1=2 1=2 e 8p 1=2 1=2 R2c 2ðe p1 þ e p2 2R12 e p1 e p2 Þ þ R2c : 3 15ne

ð7:84Þ

The first two terms in square brackets account for the contribution from the effect of particle involvement into large-scale motion, whereas the third and fourth terms describe the contribution from small-scale motion. As far as stochastic motion of large particles with tv T(L) is uncorrelated (R12 ¼ 0) and their involvement into small-scale turbulent motion does not produce a noticeable contribution to the

j425


426

kernel, Eq. (7.84) leads us to the following expression for the turbulent collision kernel that is applicable to large inertial particles [30]:

p1=2 Kt0 ¼ 4 R2c ðe p1 þ e p2 Þ1=2 : 3

ð7:85Þ

For very small inertialess particles, which can be completely involved into fluctuational motion of the carrier flow, we have fu ¼ R12 ¼ 0, and the contribution of the first two terms is small. In this case the collision kernel takes the form [29] Kt ¼

8pe 15ne

1=2 R3c :

ð7:86Þ

To determine the involvement coefficient fu, one has to know the autocorrelation function cu(x). It is usually represented by a single-scale exponential function x Yu ðxÞ ¼ exp tv

ð7:87Þ

which, generally speaking, could be also used at large Reynolds numbers for sufficiently inertial particles with tp > tt, where tt is the Taylor time microscale. To determine fu in the region of small values of tp/tt at moderate Reynolds numbers, we can use the two-scale parabolic exponential function [42] 8 x2 > > x > x0 ; > < 1 t2 ; t Yu ðxÞ ¼ > > 2x T 2 xx0 > : 02 L0 exp ; x > x0 ; TL0 tt which at x0 ¼

ð7:88Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ t2 T TL0 L0 satisfies the conditions t

Yu ðx0 0Þ ¼ Yu ðx0 þ 0Þ;

Y0u ðx0 0Þ ¼ Y0u ðx0 þ 0Þ:

In view of Eq. (7.88), the Lagrangian integral scale is

T

ðLÞ

ð¥ ¼ Yu ðxÞdx ¼ 0

2 t30 2t0 TL0 þ : 2 2 3tt 3tt

ð7:89Þ

The Taylor time microscale is equal to tt ¼

2Ret ne 1=2 pffiffiffiffiffi ; 15a0e

ð7:90Þ


where Ret ¼ ð20e2k =3ene Þ1=2 is the Reynolds number determined by the Taylor spatial microscale, and a0 is connected with the amplitude of acceleration fluctuations in 3=2 isotropic turbulence trough the relation hai a j i ¼ a0 ðeÞ1=2 ne di j . In the range of values 20 < Ret < 100, one can use the formula a0 ¼ 0:13 Re0:64 [43], which approxt imates the results of DNS [44]. The parameter TL0 is taken as TL0

ek 17 ¼ 0:19 1þ ; e Ret

ð7:91Þ

so that at moderate Reynolds numbers Ret, the DNS results presented in [44] would be consistent with the formula (7.89) for the Lagrangian integral scale, and at Ret ! 1 – with the asymptotic expression T¥ðLÞ ¼ 4ek =3C0e (where C0 ¼ 7 in accordance with [43]). In view of Eq. (7.88), the involvement coefficient in Eq. (7.80) is equal to " ! ! # pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 þ t2 T 2 þ t2 T TL0 TL0 2t2v L0 L0 t t fu ¼ 1 þ 2 ¼ exp 1þ 1 : tv tv þ TL0 tt ð7:92Þ The asymptotic relations for small and large values of tv follow directly from Eq. (7.192): 2t2 lim fu ¼ 1 2v ; tv ! 0 tt

2T 3 lim fu ¼ 2L0 tv ! ¥ 3tt tv

" 3=2 # t2t 1þ 2 1 : TL0

ð7:93Þ

At high Reynolds numbers Ret ! 1 and tt/TL0 ! 0, Eq. (7.92) provides an expression for the involvement coefficient fu associated with the autocorrelation function (7.87): fu ¼ 1 þ

tv : T ðLÞ

ð7:94Þ

For identical particles, the collision kernel (7.84), in view of Eq. (7.80) and Eq. (7.82), takes the form 1=2 e 2 1=2 8p 2 Rc 4 fu ð1 fu Þek þ R : Kt ¼ 3 5ne c

ð7:95Þ

For low-inertia particles at tv tt, the expression (7.84) reduces to " 2 2 #1=2 8p e 1=2 3 tv l0 Rc 1 þ 30a0 ; Kt ¼ tk Rc 15 ne

ð7:96Þ

j427


428

Fig. 7.9 Collision kernel of low-inertia particles: 1–6 – formula (7.117); 7–12 – results of [35]; 1,7 – Ret ¼ 45, l0/Rc ¼ 1; 2,8 – Ret ¼ 59, l0/Rc ¼ 1; 3,9 – Ret ¼ 75, l0/Rc ¼ 1; 4,10 – Ret ¼ 45, l0/ Rc ¼ 2; 5,11 – Ret ¼ 59, l0/Rc ¼ 2; 6,12 – Ret ¼ 75, l0/Rc ¼ 2; 13 – formula (7.86)

where tk ¼ ðne =eÞ1=2 is the Kolmogorov time microscale and l0 ¼ ðn3e =e1=4 – the Kolmogorov spatial microscale. Fig. 7.9–Fig. 7.12 compare the expression (7.95) with the DNS results for an isotropic turbulent field [35,36]. It should be noted that among the outcomes of comparison of Eq. (7.95) with numerical data from [35] there are variants that fall outside of the models applicability range Rc/l0 < 1. Therefore a comparison with variants corresponding to Rc/l0 > 1 is not quite correct. Fig. 7.9 presents the above comparison for the case of low-inertia particles tv tt, when the collision kernel has the form (7.96). It can be seen that the dependence shown in Fig. 7.9 gives a qualitatively accurate description of the DNS results [35], showing an increase of Kt with tv that is close to linear at 30a0 ðtv =tk Þ2 ðl0 =Rc Þ2 1. According to the data [35], the dependence (7.96) predicts growth of the collision kernel with the decrease of the ratio between the particle diameter and the spatial microscale of turbulence (i.e., of Rc/l0). Fig. 7.10 shows the dependence of Kt on the ratio of the particles relaxation time tv to the temporal Kolmogorov scale tk, whereas Fig. 7.11 shows the dependence of Kt on the ratio between tv and the Eulerian integral scale T ðEÞ ¼ 2ek =3e. Fig. 7.12 demonstrates the influence of particle inertia on the kinetic energy ep related to the energy of turbulence ek, and on the collision kernel Kt. The latter is normalized by the collision kernel Kt0 obtained from Eq. (7.85) by neglecting motion correlativity, that is, by using the kinetic theory approach. One can see that analytical relations (7.95) and (7.80) are in good agreement with numerical results [36]. The fact that Kt =Kt0 tends to unity reflects a decrease of particle motion correlativity with the growth of particle inertia.


Fig. 7.10 Effect of particle inertia on the collision kernel at Ret ¼ 24 and l0/Rc ¼ 1.78: 1 – formula (7.96); 2,3,4 – results of [35]; 5 – formula (7.96)

Fig. 7.13 compares the dependence (1.95) with the results of DNS [37] in the nearaxis zone of a flat channel, where the flow characteristics are close to those observed in isotropic turbulence. The relaxation time and the particle diameter are made dimensionless by introducing tþ ¼ tv u2* =ne and R+ ¼ Rcu*/ve. Let us now determine the collision frequency and collision kernel of particles due to the combined effect of turbulence and of the averaged component of particles relative velocity induced by the velocity shear of the carrier flow or by the force of gravity. To calculate the average radial component of relative velocity h|wr|i in Eq. (7.75), it is necessary to average this equation over the random distribution |wr| and over the solid angle that characterizes spatial orientation of

Fig. 7.11 Effect of particle inertia on the collision kernel at Ret ¼ 45 and l0/Rc ¼ 1: 1 – formula (7.95); 2,3,4 – results of [35]; 5 – formula (7.86); 6 – analytical dependence [35].

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430

Fig. 7.12 Effect of particle inertia on the kinetic energy of particles and on the collision kernel at Ret ¼ 54.2 and l0/Rc ¼ 0.36: 1,3 – ep/ ek; 2,4 – Kt =K 0t ; 1,2 – formulas (7.80) and (7.95); 3,4 –

results of [36].

the velocity vector w relative to the vector r connecting the centers of colliding particles:

hjwr ji ¼

1 4p

2p ð ðp ð¥

jwr jPðwr ÞsinFdydFdwr ;

ð7:97Þ

0 0 ¥

where j is the polar angle between the vector r and the Z-axis pointing upward, and F is the azimuthal angle in the (X, Y)-plane.

Fig. 7.13 Effect of particle inertia on the collision kernel in the near-axis zone of the channel: 1,2 – formula (7.95); 3,4 – results of [37]; 1,3 – R+ ¼ 0.498; 2,4 – R+ ¼ 0.840; 5 – formula (7.86).


Taking wr to be normally distributed (see Eq. 7.76), we obtain 1 hjwr ji ¼ 4p

2p ð ðp "

2hwr0 2i p

0 0

þWr erf

Wr2 2hwr0 2i

1=2

Wr2 exp 2hwr0 2i

sinFdydF:

ð7:98Þ

In the absence of the averaged relative velocity (wr = 0), Eq. (7.98) reduces to Eq. (7.77). Suppose the velocity field of the carrier flow is a uniform shear field ˙ 0; 0Þ and the particles are fully involved into the averaged motion of the U ¼ ðgZ; carrier flow, in other words, V ¼ U. Then the averaged radial component wr resulting from the velocity shear and/or gravity will be ˙ c cosysinFcosF þ Wg cosF; Wr ¼ gR

ð7:99Þ

where Wg = |tv1 tv2|g is the difference between the sedimentaion velocities of the two particles. In the absence of gravity (Wg ¼ 0), the equations (7.75) and (7.98), together with Eq. (7.99), lead to the following expression for the collision kernel in a shear flow (shear coagulation kernel): (

"

2 2 n Gð2n þ 1ÞGðn þ 1=2Þ g˙ Rc Kts ¼ ð8phw 2iÞ 3nþ1 G2 ðn þ 1ÞGð2n þ 3=2Þ hw 0 2i 2 r n¼0 2 2 nþ1 #) Gð2n þ 3ÞGðn þ 3=2Þ g˙ Rc þ 3nþ3 : ð2n þ 1ÞGðn þ 1ÞGðn þ 2ÞGð2n þ 7=2Þ hwr0 2i 2 0

1=2

R2c

¥ X ð1Þn

ð7:100Þ ˙ c Þ2 =hw 0r 2i ! ¥, the series (7.100) converges to the Smoluchowski soluWhen ðgR tion [45] 4 3 ˙ : Ks ¼ gR 3 c

ð7:101Þ

The expression (7.100) is cumbersome and inconvenient to use. It can be approximated by the simple formula Kts ¼ ðKt2 þ Ks2 Þ1=2 ;

ð7:102Þ

where Kt is the component of the collision kernel defined by Eq. (7.85), Eq. (7.86), and Eq. (7.95) or Eq. (7.96).

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432

Fig. 7.14 Collision kernel in a turbulent flow with uniform shear: 1, 2 – formulas (7.100) and (7.102); 3 – numerical calculations [38].

Fig. 7.14 shows the dependence (7.100) next to the results of numerical calculation for low-inertia particles when the turbulent component of the collision kernel is defined by the formula (7.86). One can see that Eq. (7.100) is in good agreement with numerical results. In the absence of shear ðg_ ¼ 0Þ, we get the following relation for the collision kernel that describes the combined effect of turbulence and gravity: Ktg ¼ ð8phw

0

2iÞ1=2 R2c

1 1 pffiffiffi 1 2 expðS Þ þ erf ðSÞ ; p Sþ 2 2 2S

ð7:103Þ

where S ¼ Wr =ð2hwr0 2iÞ1=2 is a parameter that characterizes the relative importance of gravity and turbulence in terms of their effect on the collision kernel. When the influence of gravity is weak, in other words, at small values of S, the expression (7.103) transforms into [29] Ktg ¼ Kt ð1 þ S2 =3Þ:

ð7:104Þ

At S ! 1 the relation (7.103) reduces to the collision kernel for the particles subject to gravity only: Kg ¼ pR2c Wg

ð7:105Þ

The obtained formulas for the collision kernel allow us to estimate the time interval between collisions of a particle of type 1 with particles of type 2, and the time interval between collisions of a particle of type 2 with particles of type 1: 1 t12 ¼ w1 12 ¼ ðKN2 Þ ;

1 t21 ¼ w1 21 ¼ ðKN1 Þ :

ð7:106Þ

7.4 Preferential Concentration of Particles in Isotropic Turbulence

Fig. 7.15 Time between particle collisions in a binary mixture: 1,2,3,7 – t12; 4,5,6,8 – t21; 1,4 – Eq. (7.106), Eq. (7.103); 2,5 – Eq. (7.106), Eq. (7.85); 3,6 – Eq. (7.106), Eq. (7.105); 7,8 – results of (39).

In Fig. 7.15, the time interval between collisions of particles of different types calculated on the basis of Eq. (7.103) is compared with the results of direct numerical integration of stochastic equations of motion of particles in a turbulent field using the LES method [39]. The considered mixture contains two types of particles of equal radii ra ¼ 0.325 mm but having different densities rp1 ¼ 117.5 kg/m3 and rp1 ¼ 235 kg/m3. Volume concentration ja ¼ 4pra3 Na =3 is fixed at j1 ¼ 1.3102 for particles of type 1 and is variable for particles of type 2. Since the considered particles are sufficiently large, the correlativity of their motion and their interaction with small-scale turbulence do not play any noticeable role, and therefore we can set in Eq. (7.103)hwr0 2i ¼ 2ðe p1 þ e p2 Þ=3. Also shown in Fig. 7.15 is the time interval between collisions found from the expressions (7.85) by neglecting the relative drift, and from Eq. (7.105) by taking into account the gravity force while ignoring the contribution from turbulence. It is readily seen that each one of these two dependencies overestimates t12 and t21 as compared to the values calculated in [39]. In Eq. (7.85), this overestimation occurs at low volume concentrations, and in Eq. (7.105) – at relatively large values, say, j1 ¼ 1.3102, as shown in Fig. 7.15. It should be noted that as we increase j2, the collision frequency o12 also increases, and the difference between the average velocities of particles of types 1 and 2 (that is, the relative drift velocity Wg) gets smaller as a result. Thus Eq. (7.105) is more suitable for the estimation of t12 and t21 at small j2, whereas Eq. (7.85) works better at great values of j2. The equation (7.103), which takes into account the effects of turbulence and gravity, is well consistent with the results of [39].


In many experimental and theoretical studies the phenomenon of increased particle concentration in certain regions of gradient turbulent flows (e.g., in the near-wall

j433


434

region of a channel), has been observed. The phenomenon of non-uniform particle distribution in inhomogeneous turbulent flows is explained by the turbulent migration (turbophoresis) of particles from the regions of highly intense turbulent velocity fluctuations to the regions of low turbulence [46]. DNS results [34,36,47–50] show that the tendency of particles to prefer certain regions, which leads to their accumulation and further clustering and coagulation, is also discernible in homogeneous turbulence, where there the carrier flow has zero gradient of velocity fluctuations, and particle transport via turbophoresis is not possible. Calculations show that particles are accumulated in the regions of high vorticity because of the action of centrifugal forces. Increased concentration of particles can lead to a noticeable increase of their settling velocity [48] as well as their coagulation frequency [51] in a homogeneous turbulence field. In view of this phenomenon, there arises a pressing need to find the appropriate methods for modeling of binary dispersion medium and for modeling the process of accumulation of inertial particles in an isotropic turbulent field with given statistical parameters. Our attention will be limited to small particles whose sizes are smaller than the Kolmogorov fluctuation scale. For such particles, small-scale turbulence, which can be considered as local, isotropic, and homogeneous, exerts a dominating influence. Then the turbulent field can also be assumed isotropic, homogeneous, stationary, incompressible, and having the zero average velocity. Thus the chosen model can be used in calculations of binary dispersion or coagulation of particles whose relaxation time is of the same order as the microscale of turbulence in real turbulent flows. In order to describe particle interactions with turbulence, we need to determine second-order single-point and two-point correlation moments of velocity fluctuations in the carrier flow. Lagrangian single-point correlation function is defined as B0Li j ðtÞ ¼ hui ðx; tÞu j ðRðt þ t; t þ tÞÞi ¼ u20 YL ðtÞd i j ;

RðtÞ ¼ x;

ð7:107Þ

where R is a vector describing the trajectory of a volume element of the continuous medium, u20 is the intensity of velocity fluctuations of the continuous phase, cL(t) is the dimensionless autocorrelation function characterized by the Lagrangian integral time scale T

ðLÞ

ð¥ ¼ YL ðtÞdt:

ð7:108Þ

0

Euleruian single-time, two-point correlation and structure functions are introduced through the expressions Bi j ðrÞ ¼ hui ðx; tÞu j ðx þ r; tÞi;

Bi j ð0Þ ¼ u20 d i j ;

bi j ðrÞ ¼ hui ðx þ r; tÞui ðx; tÞihu j ðx þ r; tÞu j ðx; tÞi ¼ 2½Bi j ð0ÞBi j ðrÞ : ð7:109Þ


To describe the behavior of a particle pair, it is necessary to introduce Lagrangian two-point correlation and structure functions, BLi j ðr; tÞ ¼ hui ðR 1 ðtÞu j ðR 2 ðt þ t; t þ tÞi;

R 1 ðtÞ ¼ x;

R 2 ðtÞ ¼ x þ r;

bLi j ðr; tÞ ¼ hðui ðR 2 ðtÞ; tÞu j ðR 1 ðtÞ; tÞÞðu j ðR 2 ðt þ t; t þ tÞÞ u j ðR 1 ðt þ t; t þ tÞÞÞi ¼ 2½Bi j ð0ÞBi j ðrÞ : ð7:110Þ Lagrangian two-point correlation function is connected with Lagrangian singlepoint correlation function and Eulerian two-point correlation function through the relations BLi j ð0; tÞ ¼ B0Li j ðtÞ;

BLi j ðr; 0Þ ¼ Bi j ðrÞ:

ð7:111Þ

Therefore it can be approximated by BLi j ðr; tÞ B0Li j ðtÞ þ ½BLi j ðrÞBLi j ð0Þ YLr ðtjrÞ;

ð7:112Þ

where cLr(t|r) is the Lagrangian autocorrelation function characterizing relative motion of two particles initially separated by the distance r ¼ |r|. Together with the condition cLr(0) ¼ 1, the approximation (7.112) satisfies the relations (7.111). For simplicitys sake, this autocorrelation function is given by the expression ( YLr ðtjrÞ ¼ exp

t ðLÞ

Tr

) ;

ð¥ TrðLÞ ¼ YLr ðtÞdt;

ð7:113Þ

0 ðLÞ

frequently used in the theory of turbulence. Here Tr is the two-point time integral scale. Because the approximation (7.113) is not correct at r ! 0, it should be considered only in the region of relatively large values of t. The approximation (7.113) makes it possible to represent the Lagrangian twopoint structure function of velocity fluctuations as the product bLi j ðr; tÞ ¼ bLi j ðrÞYLr ðtjrÞ:

ð7:114Þ

Since the coefficient of relative diffusion of two particles has the form of an integral of Lagrangian two-point correlations [52],

Dri j

ðt ðt 0 ¼ 2 ½BLi j ðt1 ÞBi j ðr; t1 Þ dt1 ¼ bLi j ðr; t1 Þdt1 ; 0

0

ð7:115Þ

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436

substitution of Eq. (7.114) into Eq. (7.115) leads to an expression for the coefficient of relative diffusion of two particles at large values of t: Di j ðr; t ! ¥Þ ¼ bi j TrðLÞ :

ð7:116Þ

ðLÞ

To determine Tr , consider the behavior of structure functions and of the coefficient of relative diffusion in three spatial intervals: viscous, inertial, and external. It should be noted that in the field of isotropic homogeneous turbulence, any secondrank tensor quantity can be represented as M i j ðr; tÞ ¼ M NN ðr; tÞd i j þ ½M LL ðr; tÞM NN ðr; tÞ

rir j r2

ð7:117Þ

(recall Eq. 7). Here MLL and MNN are the longitudinal and transverse (with respect to the vector r) components of tensor M. In the viscous interval (r < l0), the first terms of the Taylor series expansion of Eulerian structure functions are as follows [55]: bLL ¼

er 2 ; 15ne

bNN ¼

2er 2 ; 15ne

ð7:118Þ

where e is the specific energy dissipation and ne is the coefficient of kinematic viscosity. At small values of r the difference between velocity fluctuations at two points can be represented as a linear function of the vector connecting these points, namely, Dui ðr; tÞ ¼ ui ðx þ r; tÞui ðx; tÞ ¼ g i j ðtÞr j ;

ð7:119Þ

where gij ¼ riuj = qui/qxj is the gradient of velocity fluctuations. In a linear isotropic field, the correlation functions of strain and rotation tensors E and L are [53,54] e 2 t hEik ðx; tÞE jn ðx þ r; tÞi ¼ d i j d kn þ d in d jk dik d jn exp ; 20ne 3 tE e t ½d i j d kn din d jk exp ; hik ðx; tÞL jn ðx þ r; tÞi ¼ 12ne t Ei j ¼

g i j þ g ji 2

;

Li j ¼

g i j g ji 2

: ð7:120Þ

The two correlation functions decrease exponentially with their respective characteristic times tE and tL, which are proportional to the inner temporal microscale tl0 ¼ ðne =eÞ1=2 . An expression for the Lagrangian two-point correlation function follows from Eq. (7.119) and Eq. (7.120) under the assumption that the distribution


of the separation vector between the considered points ri is statistically independent from the distribution of the velocity fluctuation gradient tensor gij: er 2 t exp ; 15ne tE 2er 2 t 1 t ¼ exp þ exp : 15ne tE 3 tL

bLLL ¼ bLNN

ð7:121Þ

Substituting Eq. (7.121) into Eq. (7.115), we obtain the expressions for the longitudinal and transverse components of the relative diffusion coefficient: DrLL ¼

etE r 2 ; 15ne

DrNN ¼

e ntE tL o 2 þ r : 3 4ne 5

ð7:122Þ

At tE ¼ tL, the relation (7.122) is in agreement with the expression for the relative diffusion coefficient obtained in [52] for the viscous interval. On the other hand, from Eq. (7.116) and Eq. (7.118) there follows DrLL ¼

eTrðLÞ r 2 ; 15ne

DrNN ¼

ðLÞ 2eTr r 2 : 15ne

ð7:123Þ

A comparison of Eq. (7.122) with Eq. (7.123) shows that both expressions coincide ðLÞ ðLÞ at Tr ¼ tL ¼ tE . So, in the viscous interval, the two-point time scale Tr is defined as TrðLÞ ¼ tE ¼ A1 tl0 :

ð7:124Þ

pffiffiffi The constant A1 ¼ 5 is obtained theoretically in [52] and is close to the value 2.3 found in [53,54] by the DNS method. Consider now the behavior of turbulence characteristics of the carrier flow in the inertial interval (l0 < r < L), where L is the spatial macroscale. The Kolmogorov similarity hypothesis [55] leads to the following self-similar representation for the second-order structure function: bLL ¼ CðerÞ2=3 ;

4 bNN ¼ CðerÞ2=3 ; 3

ð7:125Þ

where C = 2 according to [55,56]. As evident from dimensionality considerations applied to the inertial interval, the only temporal scale that can be constructed from the available physical quantities is ðLÞ ðeÞ1=3 r 2=3 . Therefore the temporal scale Tr should be equal to TrðLÞ ¼ A2 ðeÞ1=3 r 2=3 ;

A2 ¼ const:

ð7:126Þ

j437


438

On the other hand, the two-point temporal scale may be taken in the form 1=2

TrðLÞ ¼ lbLL ;

ð7:127Þ

where l is some length scale analogous to the mixing length for near-wall turbulence. In the viscous and inertial intervals, this scale will be taken proportional to the distance between the two points l ¼ ar;

a ¼ const;

ð7:128Þ

which, again, is justified by dimensionality considerations. Then, comparing Eq. (7.124) with Eq. (7.127) in the viscous interval and making use of Eq. (7.118), Eq. (7.125), Eq. (7.126), and Eq. (7.128), we write a¼

A1 ; 151=2

A2 ¼

A1 ð15CÞ1=2

:

ð7:129Þ

A substitution of Eq. (7.125) and Eq. (7.126) into Eq. (7.116) leads to the following expressions for the coefficient of relative diffusion in the inertial interval: DrLL ¼ CA2 ðeÞ1=3 r 4=3 ¼ 0:816ðeÞ1=3 r 4=3 ; pffiffiffi C ¼ 2; A2 ¼ 1= 6;

4 DrNN ¼ DrLL ¼ 1:09ðeÞ1=3 r 4=3 ; 3

which are in good agreement with theoretical dependences obtained in [52]: DrLL ¼ 2C 1=2 f0 ðeÞ1=3 r 4=3 ¼ 0:854ðeÞ1=3 r 4=3 ; C ¼ 1:77;

f0 ¼ 0:321:

5 DrNN ¼ DrLL ¼ 1:42ðeÞ1=3 r 4=3 ; 3

In the external interval (r > L), the distance between the two test points where fluctuations are observed is sufficiently large to consider the fluctuations as independent. Hence the correlation functions vanish in the external interval, and the structure functions are equal to bLL ¼ bNN ¼ 2u20 :

ð7:130Þ

In addition, at large values of r the two-point temporal scale converts into an ordinary Lagrangian temporal scale, TrðLÞ ¼ T ðLÞ ;

ð7:131Þ

and the coefficient of relative diffusion transforms into Dri j ¼ 2u20 T ðLÞ d i j :

ð7:132Þ


According to DNS data [44], Lagrangian integral time scale in the Reynolds number range of Ret ¼ 38 93 is approximated by the expression ðLÞ T0

T ðLÞ ¼ ¼ 0:06Rel þ 3; tl0

15u20 Rel ¼ ene

1=2 :

ð7:133Þ

Let us now turn to the derivation of the kinetic equation for the PDF of relative velocity of a particle pair. Consider the motion of two identical heavy particles in an isotropic turbulent field in the absence of gravity. Equations describing the motion of each particle are dR pa ¼ v pa ; dt

dv pa uðR pa ; tÞv pa ; ¼ dt tv

ð7:134Þ

where Rpa and vpa are the position and velocity of the particle, u(Rpa, t) is the velocity of the continuous phase at the point x ¼ Rpa (t), tv is the dynamic relaxation time for the particle, and a denotes the particles index (a ¼ 1, 2). Equations (7.134) hold for particles whose density is way above the density of the continuous phase while their size is smaller than the Kolmogorov microscale. In this case, the only substantial interfacial force is that of viscous resistance. The equations for the particle pair follow from Eq. (7.134): dr p ¼ w p; dt

dw p Duðr p ; tÞw p ¼ ; dt tv

ð7:135Þ

where rp ¼ Rp2 Rp1, wp ¼ vp2 vp1. Since the turbulent velocity field is considered as a random process, Eq. (7.135) is an equation of the Langevin type. In order to make a transition from the stochastic equation (7.135) to a statistical description of relative velocity distribution, let us introduce the PDF of a particle pair, Pðr; w; tÞ ¼ h pi ¼ hdðrr p ðtÞÞdðww p ðtÞÞi:

ð7:136Þ

The operation of averaging of Eq. (7.136) is carried out over the ensemble of random realizations of the continuous phases velocity field. The function P(r, w, t) is defined as the probability of finding the relative velocity vector equal to w for two particles separated by the distance r at the moment t. Differentiating both parts of Eq. (7.136) with respect to t, one obtains the transport equation for the PDF of the particle pair: dP qP 1 qwk P 1 qhDuk Pi ¼ : ¼ wk dt qrk tv qwk tv qwk

ð7:137Þ

The left-hand side of Eq. (7.137) describes the time evolution of the distribution and the convection in the (r, w) phase space, whereas the right-hand side describes

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440

interaction of particles with turbulent eddies of the continuous phase. To determine the correlation hDu pi describing the interaction of particles with the turbulence, we model the field of relative velocities of the carrier flow by a Gaussian random process with known correlation moments. Then, in view of the Furutzu–Donsker–Novikov formula [57], we obtain:

d pðr; tÞ drdr1 ; dDuk ðr 1 ; t1 Þ

dr p j ðtÞ dw p j ðtÞ d pðr; tÞ q q pðr; tÞ pðr; tÞ ¼ : dDuk ðr 1 ; t1 Þ qr j qw j dDuk ðr 1 ; t1 Þ dDuk ðr 1 ; t1 Þ ðð

hDui pi ¼

hDui ðr; tÞDuk ðr 1 ; t1 Þi

ð7:138Þ In order to obtain the functional derivatives entering Eq. (7.138), one should use the solution of Eq. (7.135), ðt r pi ðtÞ ¼ r pi ð0Þ þ w pi ðt1 Þdt1 ; 0

ðt t 1 tt1 w pi ðtÞ ¼ w pi ð0Þexp Dui ðr p ðt1 Þ; t1 Þexp þ dt1 : tv tv tv

ð7:139Þ

0

Applying the operator of functional differentiation to Eq. (7.139) and remembering that the initial conditions rpi(0) and wpi(0) are independent of Dui, we write dr pi ðtÞ tt1 ¼ d i j 1exp dðr 1 r p ðt1 ÞÞHðtt1 Þ; tv dDu j ðr 1 ; t1 Þ di j dw pi ðtÞ tt1 ¼ exp dðr 1 r p ðt1 ÞÞHðtt1 Þ; dDu j ðr 1 ; t1 Þ tv tv

ð7:140Þ

where H(x) is the Heaviside function. In view of Eq. (7.140), the expression (7.138) transforms to ðt tt1 qP dt1 hDui pi ¼ hDui ðr; tÞDuk ðr p ðt1 Þ; t1 Þi 1exp tv qrk 0

1 tt1 qP hDui ðr; tÞDuk ðr p ðt1 Þ; t1 Þiexp dt1 tv tv qwk ðt

0

¼ tv Gik ðr; tÞ

qP qP Fik ðr; tÞ ; qrk qwk

ð7:41Þ


where Gik ðr; tÞ and Fik ðr; tÞ stand for the integrals of the two-point structure function of continuous phases velocity fluctuations; integration is carried out along the trajectory that describes the relative motion of the two particles. Using the approximation (7.114) for bLij(r, t), we can write these functions in the explicit form: ðt bik ðrÞ t Gik ðr; tÞ ¼ YLr ðtjrÞ 1exp dt; tv tv 0

ð7:142Þ

bik ðrÞ t Fik ðr; tÞ ¼ YLr ðtjrÞexp dt: tv tv ðt 0

After the substitution of Eq. (7.142) into Eq. (7.141), the correlator hDuipi takes the form qP qP hDui pi ¼ bik tv gr þ fr ; qrk qwk

ð7:143Þ

ð¥ 1 t YLr ðtjrÞ 1exp dt; gr ¼ tv tv 0

ð7:144Þ

1 t fr ¼ YLr ðtjrÞexp dt: tv tv ð¥ 0

The expression (7.143) is valid for times much greater than the Lagrangian integral time macroscale T(L). The coefficients gr and fr characterize the involvement of a particle pair separated by the distance r into fluctuational motion of the carrier flow. If the autocorrelation function is given by the exponential dependence (7.113), these coefficients take the form ðLÞ

gr ¼

ðLÞ

Tr ðTr Þ2 ; fr ¼ ðLÞ tv tv ðtv þ Tr Þ

ðLÞ

fr ¼

Tr

ðLÞ

tv þ Tr

:

ð7:145Þ

Substitution of Eq. (7.143) into Eq. (7.137) leads to a closed kinetic equation for the PDF of relative velocity of the two particles in an isotropic homogeneous turbulence dP qP 1 qwk P q2 P fr q2 P ¼ bik gr þ : þ wk dt qrk tv qwk qri qwk tv qwi qwk

ð7:146Þ

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442

Terms on the right-hand side of Eq. (7.146) describe the diffusional transport in the (r, w) phase space that is caused by the interaction of particles with turbulent eddies of the carrier flow. Modeling turbulent velocity fluctuations by a Gaussian random process, we can describe this interaction by a second-order operator of the Fokker–Planck type. It is true that, according to the DNS results [51,58], the PDF of relative velocity v for a particle pair differs substantially from the normal distribution, especially at high values of v. However, the tail of a distribution does not significantly affect its average characteristics. Therefore a Gaussian random process predicts the two-point moments of relative velocities with a satisfactory accuracy [34]. It should be remarked that, on the face of it, Eq. (7.146) resembles the kinetic equation for a single-point PDF of velocity in a homogeneous turbulent flow [1,2]. But this resemblance between the single-point and two-point kinetic equations is only apparent, because a single-point kinetic model operates with a single-point PDF in the phase space and thus is unable to take into account spatial correlativity of motions of the two particles. In contrast, the two-point approach allows to examine the correlated motion of particles resulting from their interaction with the turbulent eddies and is thus capable of describing the phenomenon of coagulation (clustering). Integration of the kinetic equation (7.146) over the velocity space yields a system of equations for two-point moments of the particle pair velocity PDF. The equations for the density of particle pairs, for the average relative density, and for the second-order structure function of particle velocity fluctuations are given below: dN qNWk ¼ 0; þ qrk dt

ð7:147Þ k

qb pik Wi D pik qlnN dWi qNWi þ Wk ¼ ; dt qrk qrk tv tv qrk qb pi j qb pi j 1 qNhwi0 w 0j wk0 i qWi þWk þ ¼ ðb pik þ gr bik Þ qt qrk qrk qrk N qWi 2 ðb p jk þ gr b jk Þ þ ð fr bi j b pi j Þ; qrk tv ð N ¼ Pdw;

wi ¼

ð 1 wi Pdw; N

ð7:148Þ

ð7:149Þ

ð b pi j ¼ hwi0 w0j i ¼ ðwi wi Þðw j w j ÞPdw;

Drpi j ¼ tv ðb pi j þ gr bi j Þ: Equations (7.147)–(7.149) do not form a closed system, because the third equation contains a third-order moment. An attempt to close the system by adding equations for higher-order moments would be pointless, because each new equation would introduce moments of still higher orders. The result would be an infinite chain of equations. To close any finite subsystem of moment equations, one needs closure


relations. For example, a system of equations for third-order moments can be closed by using a well-known quasi-normal hypothesis stating that the fourth-order cumulants are equal to zero, which allows to represent the fourth-order moment as a sum of products of second-order correlations. In doing so, we get the following equation for the third-order structure function: qhwi0 w 0j wk0 i qt þ

þWn

qhwi0 w 0j wk0 i qrn

Drpin qhw 0j wk0 i tv

qrn

þ

þhwi0 w 0j wn0 i

Drp jn qhwi0 wk0 i tv

qrn

þ

qW j qWk qWi þhwi0 w 0j wn0 i þhwi0 w 0j wn0 i qrn qrn qrn Drpkn qhwi0 w 0j i tv

qrn

3 þ hwi0 w 0j wk0 i ¼ 0: tv ð7:150Þ

Equations (7.147)–(7.150) form a closed system and describe the two-point statistics of relative velocity of a particle pairs in terms of the third moments. In order to limit ourselves to a second-order approximation, we shall ignore the terms in Eq. (7.150) that are responsible for time evolution, convection, generation due to the gradients of averaged velocity. As a result, we obtain an algebraic equation for the third-order structure function: hwi0 w 0j wk0 i ¼

qhw0j wk0 i qhwi0 w 0j i qhwi0 wk0 i 1 þ Drp jn þ Drpkn Drpin : qrn qrn qrn 3

ð7:151Þ

Triple correlations describe the diffusional transport of velocity fluctuations. The form of Eq. (7.151) is consistent with the relations obtained in [59] for triple correlations in a single-phase turbulent flow and in [60,61] for the single-point third-order moments of particle velocity fluctuations in a two-phase turbulent flow. The system of equations (7.147)–(7.149) and (7.151) allows us to model the two-point statistics of the relative velocity of a particle pair by equations for second-order moments. The averaged relative velocity vector Wi in an isotropic turbulence can be written in terms of its radial component Wr: W i ¼ r i W r =r; and thanks to the spherical symmetry, which implies that the relative velocities as well as the particle pair PDF may depend only on the absolute value of the vector r, rather than on its spatial orientation, the system of equations (7.147)–(7.149) and (7.151) reduces to qN 1 q 2 þ ðr NWr Þ ¼ 0; qt r 2 qr qb pLL Wr qWr qWr 2ðb pLL b pNN Þ qlnN ¼ þ Wr þ ðb pLL þ gr bLL Þ ; r qt qr qr tv qr

j443


444

qb pLL qb pLL qb pLL tv q 2 þWr ¼ 2 r Nðb pLL þ gr bLL Þ qt qr r N qr qr qb pNN 2 4tv ðb pLL þ gr bLL Þ þ ðb pNN þ gr bNN Þðb pLL bNN Þ 3r qr r 2ðb pLL þ gr bLL Þ

qWr 2 þ ð fr bLL b pLL Þ: qr tv

ð7:152Þ

qb pNN qb pNN qb pNN tv q 4 þWr ¼ 4 r Nðb pLL þ gr bLL Þ qt qr 3r N qr qr q Wr þ2 ðr 3 Nðb pNN þ gr bNN Þðb pLL b pNN ÞÞ 2ðb pNN þ gr bNN Þ r qr 2 þ ð fr bNN b pNN Þ: tv As an application of the above-proposed model and its ramifications, consider a stationary disperse medium with a fixed total number of particles. The stationarity requirement implies a balance of particle fluxes toward and away from the origin, in other words, the averaged radial relative velocity Wr must vanish. Introduce the dimensionless variables: the ratios r=l0 ; v=ul0 ; NðrÞ=Nð¥Þ; and so on. Here l0 is the Kolmogorov microscale and N(r) is the particles radial distribution function. Dimensionless variables will be denoted by a tilde placed over the corresponding symbol. As a result of this transition to dimensionless variables, there appears a dimensionless parameter St0 ¼ tv =tl0 called the Stokes number and characterizing particles inertia. The problem reduces to the following system of ordinary nonlinear differential equations for the radial distribution function of particle pairs N and the longitudinal bpLL and transverse bpNN structure functions of particle velocity fluctuations: ~ 2ð~b pLL ~b pNN Þ d~b pLL qlnN þ ð~b pLL gr ~bLL Þ ¼ 0; þ ~r d~r q~r ( ! d~b pLL q~b pNN 1 d 2 ~ 4 ~ ~r Nðb pLL þ gr ~bLL Þ St20 2 ðb pLL þ gr ~bLL Þ ~ d~r d~r q~r 3~r ~r N 2 þ 2ð fr ~bLL ~b pLL Þ ¼ 0; þ ð~b pNN þ gr ~bNN Þð~b pLL ~b pNN Þ ~r ( ! ~b pNN d St20 d 4 ~ ~ ~r Nðb pLL þ gr ~bLL Þ d~r 3~r 4 N d~r þ2

d 3~ ~ ð~r Nðb pNN þ gr ~bNN Þð~b pLL ~b pNN ÞÞ þ 2ð fr ~bNN ~b pNN Þ ¼ 0: d~r ð7:153Þ


The boundary conditions for the equations (7.153) are as follows: d~b pLL d~b pNN ¼ ¼0 d~r d~r ~b pLL ¼ fr ~bLL ;

at ~r ¼ 0;

~b pNN ¼ fr ~bNN

ð7:154Þ ;

~ ¼1 N

at ~r ! ¥:

ð7:155Þ

The conditions (7.154) describe the balance of particle fluxes directed toward and away from the origin of the coordinate system and are valid for particle sizes much smaller than l0. The conditions (7.155) reflect the lack of correlation between particle motions when the particles are spaced far apart, that is to say, when the particles are randomly distributed in space and their relative velocities no longer depend on r. Structure functions of velocity fluctuations in the continuous phase ~bLL , ~bNN in Eq. (7.153) are given by the approximations that combine the relations (7.18), (7.125), and (7.130): 1 ¼ ~bk LL

1 ~bk NN

pffiffiffiffiffi !k k 15 1 15 þ þ ; 2Rel ~r 2 ðC~r 2=3 Þk

pffiffiffiffiffi !k k k 15 3 15 ¼ þ þ : 2Rel ~r 2 4C~r 2=3

ð7:156Þ

The two-point time scale is approximated in a similar way by the relation that combines the expressions (7.27) and (7.31):

1 ~ ðLÞ Þk ðT r

¼

~b1=2 LL a~r

!k þ

1 ~ ðLÞ k

ðT

Þ

:

ð7:157Þ

pffiffiffi The constants in Eq. (7.156) and Eq. (7.157) are equal to C ¼ 2 and a ¼ 1= 3. In order for the results to be independent of k, the latter should be much greater than 1. Numerical estimations show that the results cease to depend on k at k > 10. When doing the calculations, the value k ¼ 20 was used. Before we discuss the numerical results, consider some asymptotic solutions. In the case of inertialess particles (St0 ¼ 0), it follows from Eq. (7.153) that ~b pLL ¼ ~bLL ;

~b pNN ¼ ~bNN ;

~ ¼ 1: N

ð7:158Þ

According to (7.158), inertialess particles are fully involved into fluctuational motion of the carrier flow, which is why the particle distribution in space is uniform. A solution of the system (7.153)–(7.157) can be obtained for low-inertia particles (0 < St0< 1) by doing a series expansion over the small parameter St0. The first terms

j445


446

of this expansion are 2 ~b pLL ¼ 1 þ St0 ~r 2 ; 15 75 2 ~ ð~r Þ4St0 =5 : N

~b pNN ¼

2 2St0 28St20 2 ~r ; p ffiffi ffi þ 675 15 15 5

ð7:159Þ

~ where L ~ ¼ L=l0 ). Suppose furConsider the inertial space interval (1 ~r L, thermore that the particle relaxation time belongs to the inertial interval ~ ðLÞ Þ, and take the limiting case Rel ! ¥, meaning that both T ~ ðLÞ and ð1 St0 T ~ go to infinity. Then the problem (7.153)–(7.155) will reduce to L ~ 2ðs pLL s pNN Þ ds pLL qlnN þ þðs pLL gr sLL Þ ¼ 0; r dr qr ds pLL qs pNN 1 d 4 2 ðs pLL þ gr sLL Þ Nðs þ g s Þ r pLL r LL 2 ~ dr qr dr 3r r N 2 þ ðs pNN þ gr sNN Þðs pLL s pNN Þ þ 2ð fr sLL s pLL Þ ¼ 0; r

ð7:160Þ

ds pNN 1 d 4~ þ g s Þ r Nðs pLL r LL ~ dr dr 3r4 N

d 3~ þ2 ðr Nðs pNN þ gr sNN Þðs pLL s pNN ÞÞ þ 2ð fr sNN s pNN Þ ¼ 0: dr

ds pLL ds pNN ¼ ¼ 0 at dr dr

r ¼ 0;

ð7:161Þ

s pLL ¼ fr sLL ;

s pNN ¼ fr sNN ;

~ ¼1 N

sLL ¼ Cr2=3 ;

4 sNN ¼ Cr2=3 ; 3

fr ¼

r¼

~r 3=2 St0

;

s pLL ¼

~b pLL ; St0

s pNN ¼

at

r ! ¥:

A2 r2=3 ; 1 þ A2 r2=3

~b pNN ; St0

sLL ¼

ð7:162Þ

gr ¼ ~bLL ; St0

A22 r4=3 ; 1 þ A2 r2=3

sNN ¼

~bNN : St0

In the case of highly inertial particles, when their dynamic relaxation time is much ðLÞ greater than the Lagrangian integral scale of turbulence ðSt0 > T~ Þ, the particles longitudinal and transverse structure functions become equal and uniform as a result of intensive diffusional transport, whereas the radial distribution function is almost identically equal to unity: ~ ðLÞ Rel T ~b pLL ¼ ~b pNN ¼ 2p ffiffiffiffiffi ; 15St0

~ ¼ 1: N

ð7:163Þ

The particles collision frequency depends on h|wr|i. As shown in [51], the form of the PDF of relative velocity depends on particle inertia, and it is only at large values of


the Stokes number that the PDF becomes Gaussian. But even for inertialess particles the quantity hjwr ji=hjwr0 2ji1=2 is equal to 0.77 (according to the results of DNS [27]), pffiffiffiffiffiffiffi ffi which is sufficiently close to the value 2=p ¼ 0:798 predicted by the normal distribution. Therefore we can state that the relative velocity of a particle pair is distributed according to the normal law. Then hjwr ji ¼

1=2 1=2 2 2 ¼ : hjwr0 2ji b pLL p p

ð7:164Þ

In general, equations (7.153) with the boundary conditions (7.154) and 7.155() are solved numerically and the results are compared with those obtained in [49–51] by the DNS method. In Fig. 7.16, the particles structure functions, structure functions of the continuous phase, and the radial distribution function are plotted against the parameter r obtained by solving Eq. (7.160) with the boundary conditions (7.161) and (7.162). One can see that at r1, when diffusional transport of velocity fluctuations does not play any significant role, the longitudinal and transverse structure functions of the particles are smaller than the respective structure functions of the continuous phase, and appoximately are valid relations spLL ¼ fr sLL , spNN ¼ fr sNN . At small distances the relations structure functions of particles exceed structure functions of the continuous phase. This effect is caused by the diffusional mechanism of velocity fluctuation transport, and it takes place only for sufficiently inertial particles. The

Fig. 7.16 Structure functions (a) and radial distribution function (b) of inertial particles in the inertial spatial interval at Rel ! 1: 1 – sLL; 2 – sNN; 3 – spLL; 4 – spNN.

j447


448

values of spLL and spNN approach each other as r gets smaller, and the radial distribution function approaches some limit. At the origin of coordinates, we have s pLL ð0Þ ¼ s pNN ð0Þ ¼ 0:37;

~ Nð0Þ ¼ 3:13:

ð7:165Þ

Structure functions and the radial distribution function of particles obtained by solving Eqs. (7.153)–(7.155) at a fixed Reynolds number and for different values of the Stokes number are shown in Fig. 7.17.

Fig. 7.17 Longitudinal (a) and transverse (b) structure functions and radial distribution function (c) at Rel ¼ 75: 1 – St0 ¼ 0; 2 – St0 ¼ 1; 3 – St0 ¼ 2; 4 – St0 ¼ 4; 5 – St0 ¼ 10; 6,9 – St0 ¼ 100; 7,10 – St0 ¼ 1000; 8,11 – St0 ¼ 10000.


It is easy to see that as St0 increases, structure functions of the particles deviate more and more from the structure functions of the continuous phase (curve 1) and approach the asymptotic uniform distributions (7.163) for highly inertial particles (curves 9–11). Although bLL ¼ bNN ¼ 0 at ~r ¼ 0, the diffusional transport causes structure functions of sufficiently inertial particles at the origin of coordinates to differ from zero. Fig. 7.17, c shows that the radial distribution function of low-inertia particles becomes singular at ~r ¼ 0. With increase of particle inertia, the singularity ~ ! 1. at the origin disappears, and N ~ on particle inertia, that is, on the Stokes The dependences of hj~ w r ji and N number St0 (Figs. 7.18–7.21) present the greatest interest. At small values of St0, the function hj~ w r ji increases with particle inertia, which corresponds to the solution (7.159). Then it reaches its maximum value at a certain St0 ¼ St00 . This maximum reflects the growth of hj~ wr ji with relaxation time, since the motions ~ r ji begins to of particles become less correlated. As St0 grows further, St0 > St00 , hjw decrease, which is explained by the lower intensity of particle velocity fluctuations (particles become more inertial and less capable of participating in turbulent motion of the carrier flow). With growth of Rel the results get closer to the asymptotic relation that follows from Eqs. (7.160)–(7.162) when the relaxation time belongs to the inertial interval, hj~ w r ji ¼

2 s pLL ð0Þ p

1=2

1=2

St0 :

~ show that in the limiting cases of Calculations of the radial distribution function N low-inertia particles and highly inertial particles the concentration field is statistically ~ ¼ 1. However, according to the DNS data [49–51], the radial homogeneous and N distribution function peaks sharply at St0 / 1. As shown in Fig. 7.19, at small

˜ r ji: 1–3 – calculations; Fig. 7.18 Effect of Rel on hjw 9–11 – [51]; 1,6 – Rel ¼ 45; 2,7 – Rel ¼ 58; 3,8 – Rel = 75.

j449


450

˜ : 1–4 – calculations; 5,6 – [49]; 7,8 Fig. 7.19 Effect of St0 on N

– [51]; 1,5 – ~r ¼ 0:025, Rel = 37; 2,6 – ~r ¼ 0:025, Rel = 82; 3,7 – ~r ¼ 1, Rel = 24; 4,8 – ~r ¼ 1, Rel = 75.

˜ 1–3 – calculations; 4–6 – Fig. 7.20 Effect of St0 and ~r on N: [50]; (a) ~r ¼6; (b) ~r ¼–24; 1,4 – Rel ¼ 53; 2,5 – Rel ¼ 69; 3,6 – Rel ¼ 134.


˜ 1–3 – calculations; 4–6 – Fig. 7.21 Effect of St0 and ~r on N: [50]; (a) r/L¼0.05; (b) r/L¼–0.3; 1,4 – Rel ¼ 53; 2,5 – Rel ¼ 69; 3,6 – Rel ¼ 134.

distances between the particles, the position of the maximum matches neatly with the Kolmogorov microscale, which indicates the crucial role of small-scale turbulent structures in the formation of particle clusters. With increase of interparticle dis~ falls in intensity and shifts to the larger values of particle tances, the peak of N relaxation time. As shown in [7], the particles can also form spatial clusters at St0 1. But since the motion of highly inertial particles is controlled by large-scale turbulent structures, it is better to use integral scales of turbulence rather then Kolmogorovs microscales. This conclusion is readily supported by the comparison of Fig. 7.20 to Fig. 7.21. The obtained results, as well as the DNS data, prove that the phenomenon of particle clustering manifests itself most strongly for the particles whose relaxation time is close to the Kolmogorov temporal microscale, but is also noticeable, albeit to lesser degree, for highly inertial particles separated by large distances [62,63].

j451


452

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32 Yuu, S. (1984) Collision Rate of Small Particles in a Homogeneous and Isotropic Turbulence. AIChE J., 30 (5), 802–807. 33 Zaichik, L.I. (1998) Estimation of Time Between Particle Collisions in Turbulent Flow. High Temperature, 36 (3), 433–437. 34 Wang, L.-P., Wexler, A.S. and Zhou, Y. (1998) Statistical Mechanical Description of Turbulent Coagulation. Phys. Fluids, 10 (10), 2647–2651. 35 Zhou, Y., Wexler, A.S. and Wang, L.-P. (1998) On the Collision Rate of Small Particles in Isotropic Turbulence. II. Finite Inertia Case. Phys. Fluids, 10 (5), 1206–1216. 36 Sundaram, S. and Collins, L.R. (1997) Collision Statistics in Isotropic Particle-Laden Turbulent Suspension. Part 1. Direct Numerical Simulation. J. Fluid Mech., 335, 75–109. 37 Chen, M. and Kontomaris, K. (1998) Direct Numerical Simulation of Droplet Collision in a Turbulent Channel Flow. Part II: Collision Rates. Int. J. Multiphase Flow, 24 (7), 1105–1138. 38 Mei, R. and Hu, K.C. (1999) On the Collision Rate of Small Particles in Turbulent Flows. J. Fluid Mech., 391, 67–89. 39 Gourdel,C., Simonin,O.and Brunier, E. (1998) Modelling Simulation of GasSolid Turbulent Flows with a Binary Mixture of Particles. Proc. 3-rd Int. Conf. on Multiphase Flow 1–8, Lyon. 40 Zaichik, L.I. and Pershukov, V.A. (1996) Problems of Modelling Gas– Particle Turbulent Flows with Combustion and Phase Transitions Review. Fluid Dynamics, 31 (5), 635–646 (Ref. 5 in Chapter 5). 41 Landau, L.D. and Lifshitz, E.M. (1987) Fluid Mechanics, Pergamon Press, Oxford. 42 Derevich, I.V., Yeroshenko, V.M. and Zaichik, L.I. (1995) The Influence of Particles on Turbulent Flow in

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53 Girimaji, S.S. and Pope, S.B. (1990) A Diffusion Model for Velocity Gradients in Turbulence. Phys. Fluids A, 2 (2), 242–256. 54 Drunk, B.K., Koch, D.L. and Lion, L.W. (1998) Turbulent Coagulation of Colloidal Particles. J. Fluid Mech., 364, 81–113. 55 Monin, A.S. and Yaglom, A.M. (1975) Statistical Fluid Mechanics: Mechanics of Turbulence, MIT Press,Cambridge, MA. 56 Sreenivasan, K.R. (1995) On the Universality of the Kolmogorov Constant. Phys. Fluids, 7 (11), 2778– 2784. 57 Klyatskin, V.I. (1980) Stochastic Equations and Waves in Random Inhomogeneous Media, Nauka, Moscow (in Russian) (Ref. 3 in Chapter 1). 58 Kuznetsov, V.R. and Sabel’nikov, V.A. (1990) Turbulence and Combustion, Hemisphere, New York. 59 Hanjalic, K. and Launder, B.E. (1972) A Reynolds Stress Model of Turbulence and its Application to Thin Shear Flows. J. Fluid Mech., 52, 609–638. 60 Zaichik, L.I. (1999) A Statistical Model of Particle Transport and Heat Transfer in Turbulent Shear Flows. Phys. Fluids, 11 (6), 1521–1534. 61 Wang, Q., Squires, K.D. and Simonin, O. (1998) Large Eddy Simulation of Turbulent Gas–Solid Flows in a Vertical Channel and Evolution of Second-Order Models. Intn. J. Heat Fluid Flow, 19 (5), 505–511. 62 Alipchenkov, V.M. and Zaichik, L.I. (2003) Particle Clustering in Isotropic Turbulent Flow. Fluid Dynamics, 38 (3), 417–432 (Ref. 12 in Chapter 7). 63 Zaichik, L.I., Simonin, O. and Alipchenkov, V.M. (2003) Two Statistical Models for Predicting Collision Rates of Inertial Particles in Homogeneous Isotropic Turbulence. Phys. Fluids, 15 (10), 2995–3005.

j455

Author Index a Abrahamson, J. 421, 422, 423, 426 Abramovitch, G. N. 184 Acrivos, A. 92 Adornato, P. M. 434 Alipchenkov, V. M. 287, 396, 401, 417, 419, 451 Allan, R. S. 365 Atwell, N. P. 231 Ausman, E. L. 368 Auton, T. N. 412

b Bankoff, S. G. 412 Baranov, V. E. 168 Basaran, O. A. 365 Batchelor, G. K. 130, 136, 142, 151, 157, 184 Bird, R. B. 395, 416 Boivin, M. 419 Bossis, G. 168, 170 Bradshaw, P. 231 Brady, J. F. 168, 170 Brazier-Smith, P. R. 368 Brenner, H. 59, 62, 72, 78, 95, 98, 110, 356, 357, 359 Brinkman, H. C. 98, 104 Brook, M. 368 Brown, R. A. 365 Brunier, E. 423, 433 Buevich, J. A. 289

c Cercignani, K. 58 Chandrasekhar, S. 110 Chen, M. 423, 429, 430 Chepurnity, N. 395, 415, 455 Chung, P. M. 326, 327 Collins, L. R. 428, 430, 447, 449, 450 Cooley, M. D. 78

Corrsin, S. 406 Cox, R. G. 78 Crane, R. J. 422 Curl, R. L. 306, 330 Curtiss, C. P. 395

d Davidov, B. I. 233 Davies, G. A. 344, 351, 421, 423, 426 Davis, M. H. 78, 82, 95 Davis, R. H. 360 De Boer, G. B. J. 347 Delichatsips, M. A. 347 Derevich, I. V. 278, 344, 395, 415, 422, 426, 442 Deutsch, E. 416, 417, 419, 425 Ding, J. 451 Drunk, B. K. 395, 451, 455 Durbin, P. A. 418 Durlofsky, L. 170

e Eaton, J. K. 434 Einstein, A. 59 Entov, V. M. 347 Eroshenko, V. M. 426

f Favre, A. 321 Fedorov, A. J. 327 Felderhof, R. 78 Feller, W. 451 Ferris, D. H. 231 Fevrier, P. 450, 451 Friedlander, S. K. 302 Frost, V. A. 327 Fuentes, Y. O. 78, 359 Fuerstenau, D. W. 362 Fuks, N. A. 293, 425

j Index

456

g Gardiner, C. W. 59 Garton, C. G. 365 Gidaspow, D. 451 Ginevski, A. S. 230 Girimaji, S. S. 436, 437 Glushko, G. S. 230, 232 Goldman, A. J. 78 Golizin, G. S. 338 Goren, S. L. 78 Gourdei, C. 423, 433 Gradstein, J. S. 283 Gusev, I. N. 301

h Haber, S. 48, 92, 359 Hahn, G. J. 339 Hamaker, H. C. 362 Hanjalic, K. 443 Happel, J. 356, 357, 359 Healy, T. W. 362, Hetsroni, G. 78, 92, 359 Hill, J. C. 321, 306 Hinze, J. O. 184, 293, 349, 406, 422 Hirschfelder, J.O. 395, 416 Hoedamakers, G. F. M. 347 Hogg, R. 362 Hopf, E. 243 Hu, K. C. 432, 423

i Ioselevich, V. A. 230

j Jakubenko, A. E. 230 Janicka, J. 327 Jeffrey, D. J. 157, 359, 395, 415 Jefferey, G. B. 78 Jeffereys, G. V. 344 Jenkins, J. T. 413, 418 Jones, O. C. 412 Johnstone, H. F. 302

k Kaminski, V. A. 327, 347, 348 Karrila, S. J. 59 Katayama, Y. 126 Kim, S. 59, 78 Kirillov, P. L. 302 Kitchener, J. A. 363 Klyatskin, V. I. 440 Koch, D. L 436, 437, 442 Kolbe, W. 327 Kolesnikov, A. V. 230 Kollman, W. 230

Kolmogorov, A. N. 212, 230, 231, 232, 388, 391 Kompaneez, V. Z. 306 Kontomaris, K. 429, 423 Koplik, J. 168 Kovasznay, L. S. G. 231 Krasucki, Z. 365 Kruyt, H. R. 362 Kuznetsov, V. R. 320, 330, 442 Kynch 168

l Lahey, R. T. Jr. 412 Lamb, H. 59, 85, 91 Landau, L. D. 110, 184, 349, 395, 425 Lapiga, E. J. 335, 343, 347, 370, 376, 378 Lapin, J. V. 230 Launder, B. E. 184, 443 Lee, K. J. 92 Lee, S. J. 412 Legendre, D. 449, 447 Leontovich, M. A. 110 Levich, V. G. 59, 184, 347, 349, 358, 385 Libby, P. A. 306 Lifshitz, E. M. 110, 184, 349, 425 Lin, G. L. 170 Lion, L. W. 436, 437 Liu, T. J. 412, 413 Loginov, V. I. 338, 341, 387, 391 Loitsyansky, L. G. 184 Louge, M. Y. 395, 413 Lun, C. K. K. 395, 415 Lundgren, T. S. 98, 99, 105, 435, 437, 438 Lushik, V. G. 230

m Majumdar, S. R. 78, 359 Martinov, S. I. 92, 168 Mason, S. G. 365 Mastarakos, E. 395, 413 Maxey, M. R. 399 Mazur, P. 168 Mednikov, E. P. 293 Mei, R. 423 Melcher, J. R. 365 Mifflin, R. T. 78 Miskis, M. J. 365 Monin, A. S. 184, 235, 251, 276, 349, 437 Morrison, F. A. 359

n Nedorub, S. A. 327 Nigmatullin, R. I. 384 Nir, A. 92

Index

o O’Brien, R. W. 157 O’Konski, C. T. 365 O’Neil, M. E. 78, 92, 95, 359 Onishi, Y. 78 Ovsyannikov, A. A. 306

p Pao, Yin-Ho 31 7 Patterson, G. K. 306, 321 Pavel’ev, A. A. 230 Pershukov, V. A. 283, 287, 306, 424 Phung, T. H. 170 Pilipenko, V. N. 230 Polak, L. S. 306 Pope, S. B. 284, 309, 323, 427, 436, 437, 439 Prandtl, L. 184 Probstein, R. P. 347

r Rallison, J. M. 360 Read, W. C. 447, 449, 450 Reed, L. D. 359 Reeks, M. W. 395, 415, 434, 442 Richardson, L. E. 274 Richman, M. W. 415, 418 Riley, J. J. 399 Rosenklide, C. E. 365 Rotta, J. C. 230 Ruckenstein, E. 344 Rudkevich, A. M. 347 Rushton, E. 359 Russel, W. B. 352 Ryszik, I. M. 343

s Sabel’nikov, V. A. 184, 248, 320, 330, 442 Saffman, P. G. 98, 99, 421, 422, 423, 426, 432 Sather, N. F. 92 Savage, S. B. 395, 415 Sawford, B. L. 427 Schmitz, R. 78 Schonberg, J. A. 360 Scriven, L. E. 365 Sedov, L. I. 251 Sehmel, G. A. 302, 304 Sekundov, A. N. 230, 231 Shapiro, S. 344 Shenkel, J. N. 363 Sherman, P. 387 Simonin, O. 395, 413, 416, 417, 419, 421, 422, 423, 425, 443, 447 Sinaiski, E. G. 335, 343, 370, 376, 378

Smoluchowski, M. V. 110, 347, 431 Solan, A. 359 Sommerfeld, M. 416 Spalding, D. B. 184 Squires, K. D. 443 Sreenivasan, K. R. 437 Stewartson, K. 78 Stimson, M. 78 Stock, D. E. 408 Stone, H. A. 384 Sundaram, S. 423, 428, 430, 434

t Tamm, C. K. W. 98 Taylor, G. I. 212, 278, 365, 366, 368 Tchen, C. M. 293 Terauti, R. 126 Thacher, H. C. 365 Thones, D. 347 Tichonov, V. I. 388 Timashov, S. F. 110, 348 Tomsend, A. A. 184 Tunizki, N. N. 110 Turner, J. S. 421, 422, 423, 426, 432

v van Kampen, N. G. 110 van Saarloos, W. 168 Voloshuk, V. M. 338

w Wacholder, E. 359 Wakiya, S. 78 Wang, L.-P. 408, 412, 422, 423, 428, 429, 434, 442, 446, 449 Wang, Q. 443 Weihs, D. 359 Wen, C.-S. 151 Wexler, A. S. 422, 423, 428, 429, 434, 442, 446 Williams, F. A. 306 Williams, J. J. 422

y Yaglom, A. M. 235, 276, 436, 437 Yeung, P. K. 427, 439 Yuu, S. 422, 424

z Zaichik, L. I. 283, 287, 301, 306, 395, 396, 401, 415, 417, 419, 422, 423, 424, 426, 442, 443, 451 Zhou, Y. 422, 423, 428, 429, 434, 442, 446 Zinchenko, A. Z. 356, 361 Zivkovic, G. 416

j457

j459

Subject Index a Ablation of particles from a surface 292 Acceleration of a particle 118 Accumulation 434 Aggregation 285, 361 Approach – Eulerian 251 – Lagrangian 251 – Lagrangian trajectory 290 Approximation, two-moment 344 Asymmetry 15, 338 Averaging 8 – ensemble 9, 11 – Favre 321 – of fields 8 – of hydrodynamic parameters of the disperse phase 12 – of Navier–Stokes equations 185 – over time interval 8, 25 – probabilistic theoretical, see ensemble averaging – rules 236, 270 – spacetime 9 – spatial 9

b Balance equations: – for energy 287 – for mass 292 – for turbulent stresses 415 Boltzmann constant 109, 296 Boltzmann operator 306 Boundary – absorbing 37, 111 – conditions: – at the discontinuity surface 38 – at the infinity 38, 47 – at the interface 47 – at the rigid boundary 186 – at the wall 301

– for zero relative motion at the interface 47 – reflecting 38 Breakup of – bubbles 376, 377 – droplets – as random process 385 – at large Reynolds numbers 408 – at small Reynolds numbers 357 – bimodal 390 – conductive 384 – fluctuations 351 – frequency 376 – in electric field 365 – in turbulent flow 246 – minimum radius 387 – multimodal 390 – probability 385 – single-modal, sum of 390 – turbulent perturbations 208

c Charge 346, 367 Chemical reactions: – degree 310 – degree of completeness 310 – equivalent 310 – exothermal 320 – fast 307 – heterogeneous 269 – homogeneous 268 – in turbulent flow 349 – isothermal 320 – kinetic equation 311 – kinetics 309 – kinetic scheme 310 – molecularity 310 – multi-stage 311 – order 311 – kinetic 311

j Subject Index

460

– total 296 – rate – approximations 319 – constant 320 – local 307 – mean 321 – specific 311 – reaction term 309, 319 – closure problem 309 – selectivity 322 – single-stage 311 – slow 307 – source term 309 – two-stage 322 – very fast 307 Cloud – effective diameter 276 – relative dispersion 266, 267 – tensor of 275 – spreading 272 Coagulation 335 – Brownian 280 – fast 364 – features 344 – gravity 431 – identical particles 374 – inertialess particles 395 – inertial particles 407 – kernel 376, 381, 384 – approximation 400 – asymptotic 79 – in electric field 365 – in turbulent flow 335, 347, 370 – power function 340 – Smoluchowski 431 – symmetrization 348 – symmetry 336 – kinetics 344, 348 – kinetic stage 344 – low-inertia particles 433 – rate 365, 396 – slow 364, 365 – transport stage 344 – velocity shear 410 Coalescence 345, 361 Coefficient – absorption 301 – binomial 19 – correlation 18 – diffusion 46 – Brownian 121, 129 – in a shear flow 281 – in a quiescent fluid 117, 129 – relative 172

– hindered 130 – two particles 135, 265, 415 – unhindered 135, 296 – molecular 188 – turbulent 188, 189, 196, 229, 263, 268, 326, 349, 378 – damping in the near-wall region 301 – hindered 135 – mutual 345 – symmetry 352 – unhindered 346 – external interval 439 – inertial interval 277, 439 – viscous interval 437, 438 – virtual 277 – heat conductivity 159, 162 – effective 162 – turbulent 183 – involvement 286, 296, 402 – for a particle pair 406 – lifting force 396 – migration 299, 407 – mobility 151 – reflection 290 – resistance 335, 336, 356, 358 – rotational 85 – single particle 60, 274, 365 – translational 85, 358 – two particles 81, 95, 131, 133 – restitution, of velocity 415 – sedimentation 151, 152, 154, 156, 160 – stoichiometric 310, 319 – thermal diffusivity 188 – in disperse phase 287 – turbulent 196 – transport, effective 183 – in disperse medium 98 – in turbulent flow 183 – virtual mass 398 – viscosity: – carrying medium 46 – disperse medium 98 – turbulent 196, 197, 199, 200, 201, 215, 216, 221, 279 Collisions 290 – of particles 361, 364 – conductive, in electric field 365 – cross section 346 – frequency 346 – in cylindrical coordinates 465 – in laminar flow 292, 327, 335, 338, 345 – in turbulent flow 345 – inertial 396 – inertialess 416, 424

Subject Index – in spherical coordinates 366 – with mobile surface 381 – with retarded surface 378, 381 – probability, see Coagulation, kernel – pair 415 Concentration – electrolyte 364 – impurity 188 – intermediate 327 – limiting, of particles 329 – mass 294 – numerical (aka number) 11 – preferential 434 – volume 12, 60 Configuration of a particle system 97, 98 Convergence to the mean 24, 31 Coordinates – Eulerian 251 – generalized 42 – Lagrangian 252 Correlation 17 – gradients 305 – mutual 321 – of velocity fluctuations: – in disperse phase 305 – in continuous phase 286 – reagent concentrations 314 – spacetime 18 – spatial 18 – time 9 – velocity difference 209 Cumulant 14–16, 21 Curl 185

d Deposition in gravitational field, see sedimentation of isolated particles 170 Derivative – functional 48 – change of variables 53 – of Gaussian distribution 50 – of linear functional 48 – of quadratic functional 51 – properties 50 – substantial 123, 254 – variational, see functional Deviation, root-mean-square 25, 110 Diagram, Pearson 339 Dielectric 365 Diffusion, see also Coefficient, diffusion – Brownian 129 – anisotropy 130 – hindered 130

– in a force field 348 – rotational 164 – from point source 270 – molecular 165 – turbulent 271 – characteristic scale 285 – semi-empirical model 234 Dipole 174 – surface force 177, 362 Dispersion – distribution 12 – of concentration 297 – relative to center of gravity 272 – relative to the source 272 – of particle displacement 259 Displacement – dispersion tensor 260 – fluctuation 260 – of a particle 283 Dissipation – of concentration field – rate 307 – relation to the time of micromixing 302 – specific 276 – of concentration inhomogeneities 245 – of energy – kinetic 187 – of fluctuational motion, see also Energy, fluctuational – empirical equations 318 – mean specific 199 – rate 285 – scale 193 – specific 199 – of averaged turbulent motion 199 – viscous 188 – specific 213 Distribution – Bernoulli 18, 110 – Boltzmann 141 – discontinuities 35 – Gaussian 37, 47, 114, 280 – multidimensional 22 – one-dimensional 21 – marginal 323 – Maxwell 119, 290 – normal, see Gaussian – correlated 418 – of bubbles 410 – of particles 167 – orientation 163 – radius 11 – volume 338

j461

j Subject Index

462

– evolution 440 – gamma 338 – logarithmic normal 388, 391 – mean 339 – of probability density 114, see also PDF – multidimensional 11 – properties of 11 – multiparticle 143 – of displacements 113, 274 – isotropic 113 – pair, see two-particle – single-particle 102 – two-particle 103 – particle configuration probability 138 – Poisson 19 – random quantity 15 – statistical characteristics 15 Droplets – conductive 336 – daughter 398, 390 – deformation 368 – finite conductivity 365 – in electric field 366, 367 – satellites 398 – surface charge 381

e Energy – activation 311, 320 – fluctuational 407 – Gibbs free 131 – internal 131, 187 – kinetic 109, 184, 187 – of averaged turbulent motion 199 – of fluctuational motion 200 – of plane waves 27 – of thermal motion 109 – of dissipation, specific 187 – critical value 387 – distribution density 390 – in averaged flow 229 – potential 363, 364 – redistribution in collisions 419 – turbulent 183 – for the disperse phase 284, 410 Enlargement of particles 384 Ensemble 99, 184 Enskog form, see Boltzmann operator Entropy 131 Equations – Boltzmann 287, 396 – Chapman–Kolmogorov 28, 29 – conservation 37

– of mass 44 – of momentum 196, 416 – continuity 44, 100, 101, 116, 184, 191, 194, 198, 415 – continuum 405 – convective heat conduction 188 – Corrsin 210, 212, 327 – diffusion 44, 45, 299, 309, 319, 328 – convective 268, 307 – in force field 348 – molecular 38, 188, 210, 211 – with chemical reactions 45, 313 – Euler 245 – Fokker–Planck 31, 116 – see also Boundary, conditions – for fluid particle trajectories 288, 291 – for moments 197 – heat inflow 187 – Hopf 243, 244 – Karman 203, 205, 209 – Karman–Howarth 204, 208 – Keller–Friedman chain 197, 241 – kinetic – of coagulation 335 – for PDF 287, 395 – of particle velocity 396, 405, 414 – of relative velocity of two particles 441 – Kolmogorov–Feller 35, 36 – Langevin 43 – of heat exchange 284 – for particle pair 435 – for single particle 306 – Laplace 366, 369 – Liouville 40, 397 – Navier–Stokes 185, 186 – Newton 169 – of Balance, see Balance equations – of motion 168, 189, 193, 346, 355 – Reynolds 192, 193 – Smoluchowski 124, 125 – stochastic – differential 4, 43 – of diffusion, with chemical reactions 45 – stoichiometric 310 – Stokes 46, 56, 70, 134, 194 – transport – for PDF of particle pair 439, 442 – for Reynolds stresses 201 – for specific energy dissipation 233 – for turbulent energy 201, 289 – for turbulent viscosity 231 – relaxation approximation 325 – scale of turbulence 232 Events 1

Subject Index – continuum of 4 – incompatible 2 – independent 3 – possible 1 – realization 2, 3, 4 – relative frequency 2 – set 1 Evolution of particle system 284, 292 Excess 328 Expansion, multipole 177

– scale 193 – small-scale 213 – stability 190 Fluid – incompressible 61 – Newtonian 62 – non-turbulent 246 – potential 246 – turbulent 246 – viscous and incompressible 184 – vorticity 403 Flux of – charge 157 f – heat 187, 196, 286 Field – matter 112 – electric 346, 365 – particles – Gaussian 36 – diffusion 324, 348 – homogeneous 24, 36 – asymptotic 359 – isotropic 26, 27, 202, 206, 208 – hindered 305 – potential 126, 207 – unhindered 346 – random 193, 202, 238, 239, 287, 295, 306 – from a wall 279 – solenoidal 203, 207 – to a wall 279 – spectral density 316 – probability 38, 111, 112 – spectral expansion 26 Force – spectrum 26 – Archimedes 142 – structure, local 214 – Basset 293, 396 – vector 202 – Brownian 169 Filtration 11, 99 – central 137 Flow – diffusion-driving 132 – Couette 70, 126 – dynamic pressure 385 – granular 170 – external 2, 43 – inhomogeneous 396 – fluctuating 46, 48 – isothermal 184 – generalized 68 – laminar 183, 190 – gravity 136, 433 – non-isothermal 187 – hydrodynamic 59, 62 – near wall 433 – interfacial 398 – Poiseuille 126, 189 – lifting 397 – rate, mean 99 – Magnus 290 – shearless homogeneous 400 – of particle interaction 371 – stationary 186 – attractive, see molecular – through particle layer 98 – electrostatic 44, 346, 357 – turbulent 183 – hydrodynamic 346 Fluctuations 14 – molecular 141, 346 – frequency 217 – repulsive, see electrostatic – hydrodynamic field 192 – van der Waals 130, 137, 346, 361 – intermediate 213 – point 5, 60 – large-scale 212 – perturbations in fluid 177 – of velocity 14, 200 – quasi-elastic 354 – period 193, 214, 216 – random 46 – scalar concentration field 314 – resistance 81 – correlation 211 – interfacial 398 – dynamic equation 212 – particle 60 – intensity 210, 211 – viscous 46, 189, 208 – length scale 210, 212, 214 – Saffman 290 – spectral representation 211

j463

j Subject Index

464

– stochastic 44, 170 – surface tension 365 – systematic 44 – thermodynamic 131 – virtual mass 396 – viscous friction 109, 147, 386 Formula – Einstein 106, 129 – Furutsu–Donsker–Novikov 56, 295 – Gauss–Ostrogradski 55 – Hadamar–Rybczynski 377 – Maxwell 162 – Stokes 60, 142 – Taylor 162 – van Driest–Reichard 411 Functional 48 – characteristic 53, 54, 234, 236 – dynamic equation 242 – expression through moments 52 – Gaussian random process 55 – spacetime 238 – spatial 238 – function of 50 – linear 48 – quadratic 49, 50 – Taylor series 52 – variation 48, 49 Function – autocorrelation 9 – Lagrangian, for relative motion of two particles 435 – spacetime, Eulerian 406 – two-time – temperature fluctuations 286 – velocity fluctuations 286, 400 – characteristic 12, 13 – correlation 23, 24 – Eulerian 435 – fluctuation 24 – higher order 21 – isotropic field 25 – Lagrangian 435 – mutual 24 – properties of 17 – strain tensor 62, 64 – time 23 – two-point 55 – vorticity tensor 71 – delta 5–7 – dissipation 187 – field – Eulerian, in interval – external 438 – inertial 437

– viscous 437 – single-time two-point 434 – isotropic 203 – scalar concentration 314 – second order 202 – third order 208 – vector solenoidal 210 – velocity 210 – locally isotropic 220 – Lagrangian two-point 435 – Saffman 11, 12, 99 – stream 63 – weight 9 – gamma 344 – generalized 5 – Green 399 – Heaviside 8 – mobility scalar 179 – of distance between neighbors 274, 276 – of functional 49 – of particle orientation 161 – probability density, see PDF – PDF 4 – conditional 28 – dynamic 397 – joint 323, 331 – of orientation and configuration 161 – multidimensional 10, 11 – multiparticle 10 – of coordinates and velocities 414 – and concentrations 309 – pair 10, 133 – particle displacements 116, 128 – two-particle 10, 101, 415 – particle pair 442 – random 5 – average value 5 – delta-correlated 44 – multidimensional 14 – one-dimensional 16 – stationary 22, 23 – Richardson 276 – structure 196 – behavior in spatial interval: – external 436, 438 – inertial 436 – longitudinal 436 – transverse 436 – viscous 436

g Gas – constant 311 – rarefied 30

Subject Index Gradient of – chemical potential 132 – particle concentration 131 – physical parameter 157 – pressure 198 – temperature 157 Gravity 136

h Hamaker constant 362, 373 Heat capacity 187 Heat conduction 188 Hydrodynamics 192 Hypothesis – Boussinesq 224 – continuity 285 – Corrsin 407 – ergodic 11 – Millionschikov 388 – molecular chaos 417 – semi-empirical gradient 312 – Taylor 278 – Toor 321

– multiparticle 168 – of particles – in a turbulent flow 304 – with a surface 289, 290 – with turbulent flow 283, 395 Intermittency 244 Interpolation, two-point 341 Invariant, Loitsyansky 206 Isotropy, local 213

j Jacobian 8

k Kinematics, chemical 119, 183 Kronecker symbol 84, 127

l

Law – Arrhenius 311 – Darcy 105 – Faxen 59 – Fick 44 – Fourier 44, 187 i – motion 252 Immiscibility 321 – Navier–Stokes 44, 147, 229 Impurity – of energy conservation 187 – concentration 188, 210 – of mass action 311 – passive 188, 210 – of 2/3, 219, 223 – source 269 – of 4/3, 277 – instantaneous 270 – of 5/2, 219, 223 – intensity 270 – of 5/3, 219, 223 – point 270, 272 – Richardson 277 – spreading 276, 281 Layer Inequality, Schwartz 18 – boundary Inertia of a particle 304 – diffusion 140, 189 Instability, hydrodynamic 190 – thermal 189 Intensity of – viscous 186 – concentration fluctuations 210, 211, 314, – double electrical 346 327 – mixing 328 – production (consumption) of matter 45 – shear, evolution of 418 – turbulence 407 Lindenberg conditions 30 Interaction Line of centers 60 – explicit 304, 305 m – implicit 304, 305 Macromixing 306, 308 – interfacial 405 Macromolecules, see Molecules, compound – interparticle 304, 305 Macroscale 210 – collisional 290, 305 Matrix – collisionless 290, 305 – correlation 22, 27 – electrostatic 130 – dispersion 36 – hydrodynamic 99, 130 – generalized, of forces 68 – molecular 136, 137, 151 – generalized, of velocities 68 – pair 60, 131, 335 – mobility 70 – with surface 289

j465

j Subject Index

466

– permeability 173 – resistance 67, 68, 76, 97, 174 Mechanics, statistical 4, 11 Medium – Brinkman 104 – carrying, see continuous – continuous 251, 252 – disperse, two-phase 282 – concentrated 167, 291, 305 – coarsely dispersed 170 – dense 395 – infinitely dilute 117, 222 – low-concentrated 60 – macroscopic parameters 167 – microstructure 168 – monodisperse 142 – non-interacting particles 161 – polydisperse 151 – rarefied 288, see also low-concentrated – rheological properties 173 – porous 12, 98, 105 Method – closure 201 – Enskog 305 – Markov 113 – Monte–Carlo 325 – of asymptotic expansions 79 – of boundary integral equations 168, 176 – of Brownian dynamics 170 – of conditional sampling 246, 247 – of deterministic Lagrangian description 285 – of direct numerical simulation (DNS), 284 – of interpolation of fractional moments 338 – of large eddy simulation (LES), 419, 423 – of molecular dynamics 167 – of moments 176, 281, 297, 312, 340 – of perturbations 141 – of small perturbations 141, 190 – of stochastic simulation 284 – of Stokesian dynamics 167 – parametric 338, 342 – PDF 309, 322, 395 – reflection 80 Microhydrodynamics 59 Micromixing 306, 308, 315 Microparticles 59, 282 Microscale 205 Migration of particles 288, 405 Mixing of substances 268 Mixture, segregated 314 Model – droplet breakup 387 – immiscibility 321

– micromixing 327, 328 – coalescence–dispersion 327, 328 – Curl 327 – Frost 327 – Langevin 326, 327 – Nedorub 327 – Patterson 321 – turbulent coagulation 374 – turbulent diffusion 267, 277, 349 – turbulent flow 223 – two-phase turbulent flow 289 – algebraic 286, 287 – continual equations 288 – differential 287 – gradient 287 – inertial diffusion 288 – kinetic 305 – semi empirical 228, 234 – two-fluid 285 Modeling – of chemical reaction 307 – of macromixing 331 – of turbulent flow: – disperse phase 285 – monodisperse 291 – polydisperse 291 – trajectory 285 – two-fluid 285 Mole 220 Molecules, compound 169 Moment – central 14, 16, 18, 20 – correlation 401–403, 416, 425 – fractional 338, 341 – generating function 12 – hydrodynamic 69 – integer 338 – mixed 16, 17, see also Moment, correlation – integer 338 – of distribution 328, 338 – order 14 – single-point 16 – spacetime 17 – surface force density 177 – two-point 16–18 – type 16 Monopole 177 Motion – Brownian 18, 109, 117 – constrained, see Motion, hindered – hindered 98 – of free particle in a fluid 117

Subject Index – of isolated particle 109 – in a hydrodynamic medium 46 – moving 70, 126 – quiescent 61 – in a shear flow 78, 127 – rotational 62, 66 – translational 63 – of many particles 40 – of particles in turbulent flow 349 – of two particles in a fluid 78 – moving 78, 79 – quiescent 61 – relative 133, 265–267, 277, 283 – pseudo turbulent 290 – quasi-stationary 214 – relative, of two fluid particles 265–267 Multipole 177

n Noise, random Gaussian white 127 Normal, outer to a surface 47 Nucleation 285, 292 Number – capillary, modified 377 – Ohnesorge 387 – Peclet 138–140, 165, 172, 188, 221 – diffusion 189, 221 – heat 189 – Prandtl 189, 287, 288 – Reynolds 61 – Reynolds, critical 190, 192 – Reynolds, local 216 – Schmidt 189, 222, 301, 316, 412 – Stokes 283, 444 – wave 27, 206, 208, 316 – Batchelor scale 222, 315 – Kolmogorov scale 213 – inertial-convective region 316, 317 – viscous-convective region 316, 318 – viscous-diffusional region 317 – Weber 386

o Operator – collision 306, 415 – micromixing 326

p Parameter – drift 409 – inertia 406 – interaction – electrostatic 361 – hydrodynamic 172

– molecular 363 – macroscopic 158, 161, 167 – particle shape 172 – stability of droplet 368 Particles – acceleration of 118 – axisymmetric 75, 76 – Brownian 109, 122, 126 – center of reaction 69 – concentration – mass 289, 294 – volume 12, 285, 291, 301 – conductive 346 – distribution over radius 11 – drift, relative 424 – effect on the flow: – laminarization 289 – momentum 4 – orientation 4 – turbulization 289 – fluid 267 – free path, effective 407 – inertial 415, 421, 449 – inertialess 283, 335, 346 – passive 283 – position in space 1, 2 – random walk 110 – size spectrum 292 – Stokesian 282 – test 139 PDF – consistency 13 – symmetry 13 Permeability 105 Permittivity, dielectric 365, 366 Perturbation, of hydrodynamic – parameters 60, 192 Point – spacetime 10, 18 – spatial 14 Polyadic 64 Polynomial – Hermite–Sonin 417 – Lagrange, interpolation 341 – Laguerre, associated 343 Porosity 99 Potential – chemical 131–133 – electrostatic 364, 369 – external force 132 – interaction 130 – Lenard–Jones 168 – molecular – between parallel plates 362

j467

j Subject Index

468

– between particles 138, 361 Principle, Markovian 28 Probability – axioms 1 – conditional 3 – density 4 – joint 3 – multidimensional 10, 14 – multiparticle 10, 14 – of event 1 – density function, see PDF – of a set of events 1 – of particle position 2 – theory 1 Process – Cauchy 30, 34 – continuous 292 – combustion 291, 292, 320 – condensation 291, 292 – evaporation 292 – deterministic 40 – diffusion 35 – discontinuous 35 – Markovian 28 – random 24 – stochastic 3, 28, 29 – Wiener 39 Product – dyadic 153 – scalar (double-dot), of tensors 171, 187

q Quadrupole 177

r Radius – effect on coagulation 423 – Debye inverse 362 Radius vector 61, 63 Random walk of – particle ensemble 99 – single particle, along a straight line 110 – bounded 111 – flux – asymptotic 374 – diffusion 112 – distribution 111 – probability 112 – unbounded 111 – with absorbing boundary 111 – with reflecting boundary 111 Reactions, see Chemical reactions Reactor, chemical 306 Relation, constitutive 44, 183

Retardation, electromagnetic 363 Reynolds conditions 196 Reynolds stress, see Stress Rotlet 177 Rule, Schulze–Hardy 364

s Scale – concentration inhomogeneity 245 – energy region 213 – inertial region 275 – Kolmogorov 222 – motion 212 – scalar field – Batchelor 222, 315 – differential 206 – Eulerian 428 – inner 222 – integral 201, 395 – Kolmogorov 213 – of concentration 315 – inner, see Batchelor – integral 315 – macroscale 210, 309, 315, 408 – spatial 217, 309 – viscous dissipation region 213 – Taylor 205, 427 – time, two-point, in viscous interval 437 – turbulence 206 – integral – Eulerian 408 – in external interval 439 – in inertial interval 437 – Lagrangian 396, 401, 415 – two-point 438 Sedimentation 136, 137 – of particles 142, 151, 361 – velocity 137, 142 Semi-invariant, see Cumulant Set 25 – complementary 2 – countable 2 – empty 2 – finite 2 – infinite 2 – intersection 2 – non-overlapping 2, 3 – of events 1 – of spatial arrangements 3 – overlapping 2 – union 2 Shear rate 95, 385, 415 Similarity criterion 186 Size, characteristic linear 220

Subject Index Slip, interfacial 399, 406, 407, 408, 413 Solution – Blasius 189, 245 – self-similar 344 Space – n-dimensional 4 – phase 125 – probability 3, 4 – vector 1 Spatial – displacement 125 – elementary volume 5. 11 – event 1 Stability – criterion 190 – factor 364, 365 – hydrodynamic 191, 192 – of conductive droplets 368 – of disperse system 362 Statistical characteristics, Lagrangian 256 Stokeslet 176 Stress 195 Stresslet 177 Surface – area, of interface 339 – density of charge 366 – discontinuity at 38 – element 62 – fully absorbing 301 – fully reflecting 301 – isoconcentration 329 – mass transfer at 329 – of a particle – equipotential 370 – free 374 – fully mobile 361 – fully retarded 360 – mobility 360 Suspension, see Medium, disperse Symbol, alternating 95 System – bidisperse 337 – colloid 169 – deterministic 4 – granular 170 – of coordinates 251

t Temperature, absolute 109 Tensor – correlation 27 – deviatoric 46 – diffusion 116, 260, 405 – dispersion 36, 37, 46, 260

– fluctuation correlation 240, 260 – mobility 91, 133 – Oseen 176 – particle shape 68 – rate-of-strain 46, 71 – resistance 47 – conjugate 69 – of axisymmetric particle 77 – rotational 67 – shear 72, 74, 76 – translational 64, 72 – scale 229 – second rank 17 – spherical 46 – stress 46 – fluctuating component 46 – Maxwellian 305 – of disperse phase 170, 395 – Reynolds 195 – rotational 66 – systematic component 46 – translational 64 – viscous 147 – third rank 64, 66, 97 – trace 47 – turbulent energy dissipation 423 – unit 62 – velocity 63, 66 – velocity gradient 62 – vorticity 71 Tetradic 97 Theorem – central limit 21 – ergodic 11 – reciprocity 64 Theory – DLVO 169 – kinetic of gas 278, 385, 390 – kinetic molecular 183 – of locally isotropic turbulence 202 – semi-empirical 202 Time, characteristic – between particle collisions 291, 410, 419 – Lagrangian correlation 258, 283 – of Brownian diffusion 172 – of change of configuration 172 – of chemical process 307, 316 – of interaction with turbulent eddies 399, 408 – of micromixing 211, 309, 314 – of non-stationarity of averaged flow 214 – of relaxation – dynamic 396, 397

j469

j Subject Index

470

– for chemical reaction 319 – for scalar field 211 – thermal 283 – viscous 119, 282 – of turbulent mixing 307 Trajectory – in continuous process 30, 32 – in discontinuous process 31 – intersection of 399, 408, 409 – limiting 346 Transform – Fourier 12, 13 – inverse 13 – isotropic turbulent field in spectral – of average energy 207 – of correlation tensor 202 – of Karman–Howarth equation 204, 208 – of PDF 13 – of potential field 207 – of solenoidal field 207 – representation 206 – spectrum 206 – Laplace 399 Transport, turbulent: – convective 286 – diffusional 289, 299 – momentum 195, 287 – of charge 157 – of heat 158, 196, 286 – of matter 183, 196 Transpose 17 Triadic 64, 97 Turbophoresis 434 Turbulence – attenuation 211 – beginning 189 – criterion 186, 190 – developed 201 – full stastistical description 287 – local structure 212 – scheme, qualitative 213 – homogeneous 202 – isotropic 202 – local isotropic 121 – models 223 – algebraic 224 – e-F, 233 – k-e, 233 – one-parametric 230 – with transport equations for – Reynolds stresses 286 – turbulence scale 349 – turbulent energy 287 – turbulent viscosity 287, 288

– semi-empirical 224 – Boussinesq 224 – mixing length 226, 228 – Monin 229 – Prandtl 226 – Taylor 227 – three-parametric 234 – two-parametric 232, 233 – semi-empirical theory 224

u Uncertainty

2

v Variables – conserved 319 – see also Schwab–Zeldovich – continuous 35 – Eulerian 251 – Lagrangian 25, 252 – random 10 – Schwab–Zeldovich 320 Variance, see dispersion Vector – associative, of pressure 63, 66 – n-dimensional 10, 12 – drift 36, 38, 39 – fluctuating 45 – generalized 42 – heat flux 187 – of particle displacement 259 – of particle orientation 161 – shear 74 – vorticity 71, 185 – wave 26 Velocity – angular 59 – drift 410 – Eulerian 254, 293 – generalized 68 – Lagrangian 254, 255 – of particles 98 – axisymmetric 75 – colliding 395 – ellipsoidal 77 – free 72, 75, 77 – spherical 77 – profile, see Shear rate – relative 406, 415 Viscosity 105 – dynamic 117 – effective 98 – kinematic 119 Volume, elementary 1

Statistical Microhydrodynamics

Statistical Explanation and Statistical Relevance

Statistical Physics

Statistical Physics

Statistical Methods

Statistical distributions

Statistical Models

Statistical Mechanics

Statistical Mechanics

Statistical Distributions

Statistical Mechanics

Statistical mechanics

Statistical Design

Statistical mechanics

Statistical Inference

Statistical Mechanics

Statistical Inference

Statistical mechanics

Statistical Mechanics

Statistical distributions

Statistical Optics

Statistical Inference

Statistical bioinformatics

Statistical mechanics

Statistical physics

Statistical Mechanics

Statistical methods

Statistical Mechanics

Statistical Optics

Statistical Mechanics

Statistical Mechanics

Statistical Microhydrodynamics

Statistical Explanation and Statistical Relevance

Statistical Physics

Statistical Physics

Statistical Methods

Statistical distributions

Statistical Models

Statistical Mechanics

Statistical Mechanics

Statistical Distributions

Statistical Mechanics

Statistical mechanics

Statistical Design

Statistical mechanics

Statistical Inference

Statistical Mechanics

Statistical Inference

Statistical mechanics

Statistical Mechanics

Statistical distributions

Statistical Optics

Statistical Inference

Statistical bioinformatics

Statistical mechanics

Statistical physics

Statistical Mechanics

Statistical methods

Statistical Mechanics

Statistical Optics

Statistical Mechanics

Statistical Mechanics

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