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PARTICLE COLLISION RATE AND SMALLSCALE STRUCTURE OF PARTICLE CONCENTRATION IN TURBULENT FLOWS by Kevin Cunkuo Hu A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 1998 ACKNOWLEDGMENTS I would like to take this opportunity to express my sincere thanks to my advisor, Professor Renwei Mei, for his patience, encouragement, and help. This work would be impossible without his guidance and insights. I appreciate my advisory committee members, Professor James F. Klausner, Professor Corin Segal, Professor Ulrich H. Kurzweg, and Professor Roger TranSonTay, for their suggestions and help. I also express gratitude to Professor Michael W. Reeks (in England), Professor Ibrahim K. Ebcioglu, Professor Otis Walton and Professor Nicolae D. Cristescu for their help. Thanks to Dr. Guobao Guo and my fellow graduate students Xueliang Zhang, Hong Shang, and others for their helpful discussions and friendship. I acknowledge the financial support of the Engineering Research Center (ERC) for Particle Science & Technology at the University of Florida, the National Science Foundation (EEC9402989), industrial partners of the ERC, and the ALCOA Foundation Award. I am deeply indebted to my wife, Rui, for her love, patience, and encouragement during my thesis work, which were more than I could ever have asked. TABLE OF CONTENTS Page ACKNOW LEDGM ENTS ........................................................................................ ii LIST OF TABLES...................................................................................................vi LIST OF FIGURES ................................................................................................ vii ABSTRACT............................................................................................................ xii CHAPTER 1 INTRODUCTION ........................................................................... 1 1.1 Smoluchowski's Prediction in Lam inar Shear Flow .......................................... 4 1.2 Various Theories for Collision Rate in Turbulence........................................... 5 1.3 Recent Development in Computer Simulation on Particle Collision Rate..... 7 1.4 Recent Development in Particle Concentration Nonuniformity ..................... 8 1.5 Scope of Present Study ........................................................................................ 11 CHAPTER 2 TURBULENCE AND PARTICLE COLLISION DETECTION ......................................................................................................... 13 2.1 Turbulence Representation ................................................................................. 13 2.1.1 Isotropic, Gaussian Turbulence........................................................................ 13 2.1.2 Rapidly Sheared Homogeneous Turbulence .................................................... 16 2.1.3 Isotropic Turbulence Generation in a Periodic Box......................................... 17 2.2 Numerical Integration of Particle Motion and Error Analysis ....................... 19 2.3 Particle Collision Treatm ent ............................................................................... 21 2.3.1 Collision Detection Scheme ............................................................................. 21 2.3.2 PostCollision Treatment.................................................................................. 24 2.4 Computer Simulation........................................................................................... 25 2.4.1 Time Step ......................................................................................................... 25 2.4.2 Turbulence Realization..................................................................................... 27 2.4.3 Validation of the Collision Detection Scheme ................................................. 27 iii CHAPTER 3 A NEW THEORETICAL FRAMEWORK FOR PREDICTING COLLISION RATE OF SMALL PARTICLES IN GENERAL TURBULENT F L O W S .............................................................................................. ............................... 33 3.1 Introduction........................................................................................................... 33 3.2 Analysis................................................................................................................. 37 3.3 Numerical Simulation.......................................................................................... 43 3.4 Results and Discussions ...................................................................................... 44 3.4.1 Comparison with Simulation Results in a Laminar Shear Flow ...................... 44 3.4.2 Collision Rate and Velocity in a Gaussian, Isotropic Turbulence.................... 47 3.4.3 Collision in a Rapidly Sheared Homogeneous Turbulence.............................. 52 3.5 Summary and Conclusions.................................................................................. 55 CHAPTER 4 EFFECT OF INERTIA ON THE PARTICLE COLLISION VELOCITY AND COLLISION ANGLE IN GAUSSIAN TURBULENCE .........59 4.1 Introduction .................................................................................................. 59 4.2 Saffnan & Turner's Analysis....................................................................... 62 4.3 Present Analysis ................................................................................................... 65 4.3.1 Asymptotic Analysis......................................................................................... 65 4.3.2 PDFBased Method..........................................................................................68 4.4 Results and Discussions................................................................................ 69 4.5 Summary and Conclusions.................................................................................. 76 CHAPTER 5 QUANTIFICATION OF PARTICLE CONCENTRATION NON UNIFORMITY IN TURBULENCE............................................................................. 77 5.1 Introduction .................................................................................................. 78 5.2 Quantification Method.................................................................................. 79 5.2.1 Inherent Noise of the Concentration Nonuniformity...................................... 79 5.2.2 Concentration Nonuniformity Parameter........................................................ 82 5.3 Results and Discussions................................................................................ 89 5.4 Summary and Conclusions.................................................................................. 96 CHAPTER 6 COLLISION RATE AT FINITE PARTICLE INERTIA AND SIZE IN GAUSSIAN TURBULENCE ........................................................................ 97 6.1 Introduction .................................................................................................. 98 6.2 Correction of Smoluchowski's Theory....................................................... 100 6.3 An Analysis on the Effect of Particle Inertia on the Collision Rate............ 106 6.3.1 Saffinan & Turner's Analysis ....................................................................... 106 6.3.2 Effect of Small Inertia on the Collision Kernel.............................................. 107 6.3.3 A Proposed Expression on Collision Kernel.................................................. 110 6.4 Comparison of the Analysis with Simulation Results.................................. 111 6.4.1 Effect of Particle Inertia................................................................................. 111 6.4.2 Effect of Particle Size..................................................................................... 113 6.4.3 Effect of Particle Mean Concentration........................................................... 114 6.4.4 Effect of Particle Gravitational Settling......................................................... 114 6.5 Collision Rate in Polydisperse Suspension..................................................... 115 6.5.1 Particle Size Spectrum.................................................................................... 122 6.5.2 Effect of Particle Inertia ................................................................................. 124 6.5.3 Effect of Particle Gravitational Settling ......................................................... 125 6.6 Collision Rate in Hard Sphere Suspension and Assessment of Collision M odels ................................................................................................................. 127 6.7 Summary and Conclusions................................................................................ 129 CHAPTER 7 SUMM ARY AND CONCLUSIONS .............................................. 137 REFERENCES...................................................................................................... 140 BIOGRAPHICAL SKETCH ................................................................................ 146 LIST OF TABLES Table 3.1 Comparison of the collision velocity among monosized particles between prediction and direct numerical simulation in lam inar shear flow................................................................................. 45 Table 3.2 Comparison of the collision velocity among monosized particles between prediction and direct numerical simulation in Gaussian, isotropic turbulence.............................................................. 50 LIST OF FIGURES Figure 2.1 The effect of time step on the cumulative average of the normalized collision kernel a', for inertialess particles in turbulence. D=0.05.............................. 26 Figure 2.2 Turbulence dissipation rate e against turbulence realization. (a) Variation of ensemble average. (b) Cumulative average .................................................................. 29 Figure 2.3 Turbulence small eddy shear rate (e / v) 1/2 against turbulence realization. (a) Variation of ensemble average. (b) Cumulative average......................................... 30 Figure 2.4 Inverse of total number of monodisperse particles against time in laminar shear flow F=1.0, D=0.02............................................ ............................................... 31 Figure 2.5 Multiple collisions of HARD SPHERE particles in laminar shear flow...... 32 Figure 3.1 Sketch of particle collision velocity. The separation distance is x2xI = Rij= (ri+rj) ................................................................................... 35 Figure 3.2 Variations of J(), (Q), 74() with i where fl= ), f2= G( ) and f3= ............................................................. ....... 36 Figure 3.3 Local coordinates for collision in laminar shear flow. 42 Figure 3.4 Comparison of the pdfs for the collision angles 9c between the theory and the simulation in laminar shear flow ...................................................... 47 Figure 3.5 Convergence of a*2 as Nk *oo ....................................................................... 50 Figure 3.6 Particle collision kernel a*, from direct numerical simulation in isotropic turbulence. (a) Variation of the ensemble average. (b) Cumulative time average....... 51 Figure 3.7 (a) Normalized particle collision kernel in a highly sheared homogeneous turbulence without contribution from the mean shear rate.............................................. 56 Figure 3.7 (b) Collision kernel in a rapidly sheared homogeneous turbulence. 6 denotes the dissipation rate of the isotropic state at t=0 ............................................................... 57 Figure 3.8 Collision velocity in a rapidly sheared homogeneous turbulence. so denotes the dissipation rate of the isotropic state at t=0 ............................................ 58 Figure 4.1 Sketch of particle collision scheme ............................................................... 63 Figure 4.2 Sketch of particle collision velocity and collision angles (012 & Oc ). The separation distance is x2 x1 = R = (r, + r2)..................................................... 63 Figure 4.3 Comparison for the increased collision velocity due to the particle inertia between numerical simulation and theoretical prediction, D/TI=0.957 .......................... 70 Figure 4.4 Effect of particle inertia on the probability density function of the particle collision velocity wr. (a) D/I=0.957. (b) D/rh=2.522.................................................. 71 Figure 4.5 Effect of particle inertia on the probability density function of the particle collision angle 012. (a) D/rl=0.957. (b) D/r9=2.522...................................................... 73 Figure 4.5 (c) Effect of particle inertia on the probability density function of the particle collision angle 0 D/ri=0.957 ...................................................................................... 74 Figure 4.6 Effect of particle inertia on the mean collision angle 012 in isotropic, G aussian turbulence ....................................................................................................... 74 Figure 4.7 Effect of particle inertia on the standard deviation of collision angle 012 in isotropic, Gaussian turbulence ....................................................................................... 75 Figure 4.8 Effects of particle inertia and size on the probability density function of the particle collision angle 012, pTk=1.023 ....................................................................... 75 Figure 5.1 The normalized variance of the concentration as function of cell size for inertialess particles in isotropic turbulence. PTtk =1.023 .............................................. 83 Figure 5.2 The expressions (5.17) & (5.18) of the concentration nonuniformity as function of cell size for inertialess particles in uniform shear flow. /F=oo ................. 84 Figure 5.3 The expressions (5.17) & (5.18) of the concentration nonuniformity as function of cell size for inertialess particles in isotropic turbulence. PT k = ............... 86 Figure 5.4 The concentration nonuniformity parameter AP as function of cell size in uniform shear flow. r=1.0, 03/F=l.0 ............................................................................ 87 Figure 5.5 The concentration nonuniformity as function of cell size for small inertia particles in isotropic turbulence ..................................................................................... 88 Figure 5.6 The concentration nonuniformity as function of cell size for various particle number densities at a given inertia in isotropic turbulence. p3T k =1.023........ 89 Figure 5.7 The concentration nonuniformity of sizeless particles as function of cell size for various inertia in isotropic turbulence ............................................................... 91 Figure 5.8 The concentration nonuniformity parameter AD of finitesize particles as function of cell size for various inertia in isotropic turbulence. (a) D/rn=0.957. (b) D /il=2.52................................................................................................................... 93 Figure 5.9 The maximum concentration nonuniformity parameter AP of finitesize particles as function of particle inertia in isotropic turbulence ...................................... 94 Figure 5.10 The concentration nonuniformity parameter AO as function of Ax/h for smallsize and zerosize particles in isotropic turbulence. P3 k =1.023 ......................... 94 Figure 5.11 The concentration nonuniformity of particles at a given inertia as function of Ax/D cell size for various sizes in isotropic turbulence. (a) Log scale in xaxis. (b) in norm al scale in xaxis ........................................................................................... 95 Figure 6.1 Collision kernel a ii against time in laminar shear flow. F=1.0, D=0.04. (a) Variation of ensemble average. (b) Cumulative average...................................... 105 Figure 6.2 The effect of particle inertia on the increased collision kernel Aa 1 for monodisperse particles for (a) D/rl=0.957. (b) D/rj=2.52 ........................................ 116 Figure 6.3 The effect of particle size on the collision kernel aot i for monodisperse particles at a very dilute system. Ou k =oo................................................................... 117 Figure 6.4 The effect of particle size on the collision kernel a 1 for monodisperse particles at 'not' very dilute system. p3 k = oo............................................................. 117 Figure 6.5 Variation of the collision kernel at 1 against particle number concentration for monodisperse particles. 13PTk = oo, D/TI=0.435...................................................... 118 Figure 6.6 The effect of gravitational settling on the collision kernel a*i for monodisperse particles. 3rk=1.023, D/rTI=2.52.......................................................... 118 Figure 6.7 The effect of gravitational settling for monodisperse particles. D/ri=2.52, PT k = 1.023. (a) on the collision velocity. (b) on the concentration nonuniformity... 119 Figure 6.8 Collision kernel a41 of polydisperse particles against time. Prtk = O, D/TI=0.957. (a) Variation of the collision kernel. (b) Cumulative average................. 120 Figure 6.9 Collision kernel cta of polydisperse particles with various inertia against turbulence realization. D/rp=0.957. (a) Variation of the collision kernel. (b) Cumulative average of the collision kernel............................................................ 121 Figure 6.10 Inverse of total number of particles against time for polydisperse particles. D/TI=0.957. (a) primary size (v0) particles. (b) secondary size (2 v0) particles......... 122 Figure 6.11 The evolution of particle size distribution at various instants for polydisperse particles. D/1=2.17, r k =1.023............................................................. 123 Figure 6.12 The effect of particle inertia on the increased collision kernel Aoal for polydisperse particles for (a) D/il=0.957. (b) D/I=2.52 ........................................... 126 Figure 6.13 The effect of gravitational settling on the collision kernel a*, for polydisperse particles. PTk=1.023, D/r1=2.52............................................................ 127 Figure 6.14 The effect of gravitational settling for polydisperse particles. D/rl=2.52, PT k=1.023. (a) on the collision velocity. (b) on the concentration nonuniformity.... 128 Figure 6.15 Collision kernel a 1 of hard sphere particles with various inertia against time in laminar shear flow. D=0.04. (a) Variation of the collision kernel. (b) Cum ulative average................................................................................................. 130 Figure 6.16 The effects of the collision schemes on the collision kernel a I1 against time in laminar shear flow. F=1.0, D=0.04. (a) Variation of the collision kernel. (b) Cumulative average of the collision kernel................................................................... 131 * Figure 6.17 The effects of the inertia on the collision kernel a11 against time for hard sphere particles in Gaussian turbulence. D/Ti=0.957. (a) Variation of the collision kernel. (b) Cumulative average of the collision kernel............................................... 132 Figure 6.18 The effects of the collision schemes on the collision kernel a* against time in Gaussian turbulence. D/Ti=2.174, Prk= 1.023. (a) Variation of the collision kernel. (b) Cumulative average of the collision kernel............................................................ 133 Figure 6.19 The effects of the collision schemes on the collision kernel a*,I of hard sphere particles in Gaussian turbulence. D/il=0.957, Pt k =0.292. (a) Variation of the collision kernel. (b) Cumulative average of the collision kernel................................ 134 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy PARTICLE COLLISION RATE AND SMALLSCALE STRUCTURE OF PARTICLE CONCENTRATION IN TURBULENT FLOWS by Kevin Cunkuo Hu May, 1998 Chairman: Renwei Mei Major Department: Aerospace Engineering, Mechanics and Engineering Science A study of particle collision and concentration nonuniformity in turbulent flows is carried out. A correction to the classical result of Smoluchowski for the collision rate of monodisperse particles in a laminar shear flow has been made. A theoretical framework has been developed to evaluate the collision rate and collision velocity of very small particles in general turbulent flows. The present approach differs significantly from the classical approach of Saffinan & Turner in that the ensemble average is taken after the collision rate for a given flow realization is calculated. This avoids the assumption of isotropy, as needed in the classical approach, and allows for the evaluation of the collision rate in general turbulent flows. Direct numerical simulations and analytical studies on particle collisions in an isotropic, Gaussian turbulence have been carried out. Several different collision models (or postcollision treatments) are implemented in the simulations and the difference among various collision models is examined. The effects of particle inertia, size, gravity, and meanconcentration on the collision rate, collision velocity, and collision angle are investigated. Asymptotic analyses have been developed to predict the effects of small inertia on the increases in collision rate and collision velocity. Asymptotic prediction for the increased collision velocity agrees well with the results of both numerical simulation and PDFbased method. The increased collision velocity is found to contain terms of both O((Ot k )1) and O( (Tk )[2 ), where P1 is the particle response time and tk is the kolmogorov time scale. The increases in particle size, gravitational acceleration, and meanconcentration in general reduce the normalized particle collision rate in isotropic turbulence for monosized particles as opposed to very small particles in the very dilute limit. A new method has been developed to quantify the concentration nonuniformity of particles in turbulent flows. The effects of particle size and inertia on the concentration nonuniformity are examined and elucidated. The concentration nonuniformity can be greatly overestimated if the effect of finite particle size is neglected. CHAPTER 1 INTRODUCTION Particles and turbulence are around us everywhere and greatly impact our daily life. Dusts, sand, rain drops, fine powders, glass beads, viruses and bacteria are all 'particles.' Most of the fluid flows occurring in nature such as air current and water flows in rivers and seas are turbulent flows. The particles in the atmosphere remain airborne for long periods, lowering visibility. The brown haze that is often seen over large cities in autumn, winter and spring is due mainly to the interactions between particles and the air. The improvement of particulatefluid systems requires a better understanding of the fundamental physics of the particleparticle and particlefluid interactions. The behavior of particulatefluid mixtures may vary widely depending upon the conditions under which the particlefluid system functions. Particleparticle interactions involve the collisions among particles which depend on interparticle forces, particle size distribution, and the concentration distribution. Particlefluid interactions include the hydrodynamic forces acting on particles by fluid and the feedback to fluid from particles, etc. Particle collision, coagulation, and aggregation are the examples of most important scenarios of those interactions that are determined by particle properties and flow structures. The fact that collisions do occur among the particles of size ranging from a few Angstroms to a few millimeters explains the vast amount of scientific and industrial applications of the collision process. For example, molecular collisions help explain the physical properties such as viscosity, diffusivity, and conductivity in gases and liquids. Numerous industrial applications can be cited: design of fine spray combustion nozzles, control of industrial emissions (Presser & Gupta 1993, Tu et al. 1996) and the understanding of the collision process among suspended particles in pneumatic transport systems or in gas cleaning chambers (Abrahamson 1975, William & Crane 1983), and in sewage disposal devices (Hunt 1982), where particle size distribution and concentration nonuniformity are important characteristics that depend on particle collision and coagulation. Most of previous studies on particulate removal have focused on the filters, with little attention to the particleparticle interactions and particlefluid interactions. Despite many years of research and development in particulate removal technologies, the collection of small particles with size less than 1 un is still a rather difficult task. It is extremely difficult to remove fine organic active particles (<0.03 mrn) from a waste stream because the particles are much smaller than the pores in existing filters. Typical collection efficiencies of the filtration devices decrease rapidly with the decrease of particle size in the range 0.230pm (Davies 1953). A good alternative is to aggregate fine particles which can be removed using conventional technologies. For the coagulation and aggregation to occur, particles have to collide with each other and adhere. It is very desirable that one can predict how the small particles are driven to collide with each other so they aggregate or grow in size and become large enough to be captured by the filters. Obviously, the efficiency of particle removal mainly depends on the control of particle collisions and concentration nonuniformity in filtration systems (Fathikalajahi et al. 1996). It is evident that particle collision is one of the most important interactions for understanding the fundamentals of particlefluid systems. Particle collision kernel and collision rate are critical quantities in the development of collision theories. The rate of flow induced particle collisions affects directly the particle size distribution in particle production. Examples of turbulence induced particle collision can be seen in the formation of rain drops in clouds, pulverized coal combustion, agglomeration of fine powders in gas flows, air filtration equipment, sewage disposal devices, fast fluidized beds, dust and spray burners, and so on. Particle collisions can take place through a variety of collision mechanisms. Particle collisions may be induced by centrifugal and gravitational forces, thermal forces in Brownian motion, and van der Waal forces. In an actual system, the scenario after every contact between particles may be different from one to another, depending on the underlying physics of the system. The collisions among raindrops enhance the growth of droplets in the air, since the contact between droplets may most likely result in the coalescence of two droplets. The growth of bubbles in a liquid flow is another example of particle coagulation upon collisions. The contacts between two solid particles may behave in various different ways. When two solid particles are brought together the colliding pair may stick or separate after collision upon contact. Many theoretical efforts were made to predict the collision kernel and rate. To validate a theoretical analysis, direct numerical simulations were carried out to obtain statistical average of the collision rate in a particle fluid system. Different interpretations of the immediate consequence after the contact between a pair of particles have resulted in different collision models or socalled "post collision treatment" schemes. To quantify the postcollision treatment, a statistical parameter called collision efficiency has been introduced. The collision efficiency is one if particles coagulate to form a larger one. Usually, it is less than one since some of the colliding particles may bounce back. Although many theoretical and experimental studies have been devoted to estimates of the collision efficiency for water droplets settling through still air (Klett & Davis 1973, Lin & Lee 1975, Beard & Ochs 1983) and droplets settling down through laminar shear flow (Jonas & Goldsmith 1972), even an approximate estimate of the collision efficiency in a complicated situation can be very difficult to obtain. At this stage, it is still not possible to obtain an accurate solution for the evaluation of collision efficiency in complex systems. To better focus on the issues of our interests, the collision rate, the collision efficiency is assumed to be either one or zero in this study. Although a unity collision efficiency will overestimate the size growth, however, theoretical and numerical exploration on the governing physics may still hold its significance for a better understanding of the interactions in particlefluid systems. Population balance equations are often used for predicting the evolution of particle size distribution in particle production systems or in studying particle coagulation. They are typically given in the form of dn1 d t anll1n1 a12nln2 a13nln3  dn2 dt 2 al jlnn a12nln2 CC22n2n2 232n3 . (1.1) dn3 dt = a12nln2 a13nln3 X23n2n3 c33n3n3  dn4 1 dt a13n1n3 + 2 22n2n2 "a14n1ln4 24n2n4  In the above, ni is the number concentration of the ith particle size group. For primary particles, i=l, with a volume v1, the particle in the ith group has a volume of iv1. The quantity aijninj, denoted as ncij in this thesis, ncij= aijninj (1.2) is called collision rate which represents the number of collisions among ith particle with jth particle per unit volume and per unit time. The coefficient aj is called the collision kernel or collision function, it depends on particle size, particle inertia and flow structure, and must be evaluated separately. The collision rate or collision kernel is a measure of the ability of particles to collide, coagulate, and aggregate. It is essential to determine the instantaneous particle size distribution and momentum transfer among particles in a particulate system. Over past 80 years, many efforts have been devoted to developing the particle collision theories for predicting the particle collision rate in fluid flows. A brief review is given below. 1.1 Smoluchowski's Prediction in Laminar Shear Flow A geometrical collision occurs when two particles reach a separation distance less than Rri+rj, in which ri and rj are the respective radii of two particle groups and R is the radius of the colliding surface. Theoretical analysis of geometrical collisions in a uniform shear flow was first carried out by Smoluchowski (1917). At a given instant the particle number concentrations per unit volume are n, and nj for each size group. Taking any particle of the ithgroup as a target particle, the volume flow rate of thejthgroup particles reaching the colliding surface of radius R of the given target particle is nj f ,, wrdS in which w is the relative velocity between ithgroup and jthgroup particles and Wr is the radial component of w. Thus the number of collisions occurring among ithgroup and jthgroup particles per unit time per unit volume is given by ncij = ninj L, owdS= ninj F(ri+rj)3 = a, nin (1.3) where F is the shear rate and aij= F(ri+rj)3 (1.4) is often called the collision kernel. For like particles, the number of collisions per unit time and per unit volume was often simply taken as 1 4 3 ncii= nini (2r,)3. (1.5) 2 3 The factor was introduced to avoid double counting of the collisions among the like 2 particles. When the above results are used in the population balance equations, for example, the rate of the "destruction" of ithgroup particles due to collisions is 2 n ci + 4 4 4 Y n = nini F(2ri)3 ninj r(ri+rj)3 since each collision among the like particles results in the loss of two particles of that size group. Equation (1.5) has been extensively used in the literature in calculating the particle collision rate and particle size evolution. In addition to Smoluchowski's work, Hocking & Jonas (1970) also calculated the collision rate for the sedimenting particles in a uniform shear flow using Stokesian theory. They studied the collision efficiency by which the particle coagulation was assumed if the surface distance between two neighboring particles was less than a prescribed fraction of particle diameter. 1.2 Various Theories for Collision Rate in Turbulence When particles collide in complex flows, such as turbulent flows, or when more than one collision mechanism are involved in the particulate system, it is quite difficult to derive an exact expression for collision rate. Numerous efforts have been made to estimate the collision rate by means of evaluating the collision velocity in turbulence during past decades. Camp and Stein (1943) employed the concept of mean velocity gradient related to the turbulent energy dissipation rate E and kinematics viscosity v and obtained the collision kernel for turbulence shearinduced collision rate based on Smoluchowski work (1917) in laminar shear flow. 4 R(__1/2 a = R' 2. (1.6) 3 v Saffman and Turner (1956) gave two predictions of particle collision kernel in an isotropic, Gaussian turbulence. In the first prediction, it was assumed that particles are inertialess so that they follow the fluid completely and particle size is small compared with the Kolmogorov length scale of the turbulence. The collision kernel was given ca = 1294 R3 (E)1/2. (1.7) IJ V Their second prediction was for the particles with small inertia and gravity. ai = 2(2n)12 R2 [(1 )2 t)2 ()2 PO Dt +(I )2i (T2)2g2 +R2 ]1/2. (1.8) 3 po 9 v However, collision kernel for like particles or the particles with same inertia is reduced to a, =1.67 R3 ( ) 1/2 (1.9) v Thus the effect of particle inertia and gravity disappeared in the monodisperse case, which is not physically sound. In addition, the result of their second analysis was inconsistent with the first one, as easily seen from (1.7) and (1.9). Abrahamson (1975) extended the concept of kinetic theory of gases for the collisions among large particles in a high Reynolds number turbulence. It was assumed that the velocities of colliding particles are uncorrelated, normally distributed in the absence of gravitational settling. The variance of particle velocity v was estimated by 2 "2 1+ 15 Tp 8/U2 where u2 is the meansquared fluctuating velocity of the fluid and Tp is the particle response time. The collision kernel, without gravitational settling, was given by 2 ( 2 > 2 > 1/2. aij = 5.0 R ( Balachandar (1988) performed an analysis on the collision kernel by assuming that the probability distribution function (PDF) of the velocity gradient in turbulence follows a lognormal PDF distribution. He found a correction to Saffminan & Turner's prediction for collision kernel in isotropic turbulence. Williams & Crane (1983) and Kruis & Kusters (1996) performed stochastic analysis to obtain the collision kernel with inclusion of the combined effects of local shear and particle inertia. Zhou et al. (1997) carried out a similar analysis and modified the Kruis & Kusters' result in turbulence. Those analyses on the collision kernels were obtained on the basis of twopoint velocity correlations without tracking the trajectories of individual particles in the turbulent flow (hereinafter referred as Eulerian method). However, we have found that the collection of the relative velocities (or collision velocities) upon collisions based on tracking the trajectories of individual particles in turbulent flow is a biased average in favor of those regions where the collision velocity is high. Hereinafter, this method based on particle tracking is referred as Lagrangian method. The detailed discussions on the difference between the results of Eulerian and Lagrangian methods on collision velocity will be presented in chapter IV. 1.3 Recent Development in Computer Simulation on Particle Collision Rate Recent developments of direct numerical simulations (DNS) in turbulence have greatly enhanced the understanding of the turbulenceinduced particle collisions. Balachandar (1988) also carried out DNS of particle collision in an isotropic turbulence. It was found that the probability distribution function of the velocity gradient follows a lognormal distribution instead of Gaussian distribution. The collision kernel was obtained in a frozen turbulence with a maximum of 500 particles in the flow field. Artificially large particles were used in order to increase the collision events within a reasonable computational time. It was concluded that the particle inertia does not affect the average collision kernel for large, monosize particles. Large discrepancy in the collision rate for monosize inertialess particles between his numerical results and Saffinan & Turner's prediction was reported. However it is not clear whether this discrepancy is due to the difference in the small scale turbulence structure or not. Chen et al. (1995) carried out DNS results for the aerosols in a turbulent channel flow. They mainly focused on the possible effects of collisions on the deposition rate of the droplets on the channel walls. They found that the particle concentration in the wall region can be significant while the bulk concentration is small. The collision kernel and rate were found to peak in the wall region, and particle depositions can be enhanced by the formation of larger particles. Wang et al. (1997) also carried out DNS to investigate the collision kernel in homogeneous turbulence. They allowed particles in the system to overlap in space which is unphysical. The effects of finite inertia on collision kernel were also reported in their DNS results (Zhou, Wexler & Wang 1997). Zhou et al. (1997) developed an analysis showing that the increases in particle collision velocity, the nonuniformity of particle concentration and the compressibility of particle velocity in the continuum sense are three possible mechanisms to affect the overall collision kernel for particles with finite inertia. However, they did not quantify these terms separately in their work. Thus how the collision velocity and the concentration nonuniformity play in enhancing the overall collision kernel remains unknown at this stage. 1.4 Recent Development in Particle Concentration Nonuniformity Recent studies have shown that the particle concentration in turbulence may be highly nonuniform. Heavy particles with finite inertia tend to accumulate in the regions of high strain rate and to move away from the vortex cores. The occurrence of particle accumulations is referred as concentration nonuniformity or preferential concentration. The coherent vortical structures are the mechanisms that cause the nonuniform distribution of particles. The coherent vortices and concentration nonuniformity have been observed by many numerical simulations and experimental visualizations. The numerical simulations for demonstration of the concentration nonuniformity were carried out in plane mixing layers (Crowe et al. 1988, Chien & Chung 1987, Wen et al. 1992, Tang et al. 1992, Wang 1992), in wake flows (Chien & Chung 1988, Tang et al. 1992), in jet flows (Chung & Trout 1988, Hansell 1992), in wallbounded flows (Kallio & Reeks 1989, McLaughlin 1989, Yonemura et al. 1993, Rouson & Eaton 1994, Tanaka et al. 1996), in homogeneous flows (Maxey & Corrsin 1986, Hunt et al. 1987, Fung & Perkins 1989, Squires & Eaton 1991, Wang & Maxey 1993, Fessler et al. 1993). The nonuniform distribution of particles has also been found by the experiments in plane mixing layers (Kobayashi et al. 1988, Kamalu et al. 1988, Lazaro & Lasheras 1992, Ishima et al. 1993), in wake flows (Tang et al. 1992, Yang et al. 1993, Bachalo 1993), in jet flows (Longmire & Eaton 1992), in wallbounded flows (Rashidi et al. 1990, Young & Hanratty 1991). Wang & Maxey (1993) have, through direct numerical simulations (DNS) in isotropic turbulence, shown that the preferential concentration achieves maximum at P3k =1 in which P3 is the reciprocal particle response time and Tk is the Kolmogorov time scale of the turbulence. One of the consequences of the preferential concentration is the enhancement of the settling velocity of heavy particles in the Stokes drag range (Maxey 1987, Mei 1994). One of the related industrial applications is that the particulate removal efficiency of the venturi type scrubber was affected by the nonuniform concentration distribution of droplets (Fathikalajahi et al. 1996). Particle trajectories, therefore individual particle positions, can be obtained accurately only in Lagrangian frame by solving the equations of particle motions. Particles have finite response times in general but they do not have an intrinsic diffusivity or viscosity associated with the motion of individual particles. Turbulence particle diffusivity is the consequence of ensemble averaging over many realizations and it is used for predicting the variation of average concentration field on a large scale. It cannot be used to describe the smallscale spatial variation of the particle concentration. From the computational point of view, a DNS using 1283 grid resolution for an isotropic turbulence becomes routine and DNS using 2563 or even 5123 have been carried out. For a reasonable contour representation of particle concentration, it is desirable to have at least, say, 10 particles in each cell so that the statistical noise does not become overwhelming. This requires 1.7x108 particles with a 2563 grid resolution. Due to the preferential concentration, there will be regions and cells that do not contain any particle or only one or two particles at a given instant. Thus, an accurate, threedimensional contour representation of instantaneous particle concentration with such high resolution will still be heavily contaminated by the inherent statistical noise. No such three dimensional concentration contour has been reported to date. Due to the computer power limitation, previous numerical simulations involved anywhere from 104 to 106 particles, which is far below the level of reaching a reasonable statistics. It has been difficult to quantify the extent/degree of particle concentration non uniformity induced by the smallscale turbulence structure. Squires & Eaton (1991) and Wang & Maxey (1993) showed in their DNS the similarity between the instantaneous particle concentration contours with the contours of the vorticity magnitude (or enstrophy) in the same plane at the same instant. To quantify the deviation of the particle concentration field from a statistically uniform distribution, Wang & Maxey (1993) used a statistical quantity (P(k) Pb(k))2 averaged over space, where P(k) is the probability of finding k particles in a given cell and Pb (k) follows the Bernoulli distribution for a random distribution of the particles. While this quantity shows that the preferential concentration maximizes near P3tk =1 (Wang & Maxey 1993), it varies with total number of particles in each of their simulations. The particleparticle interactions are neglected in their simulation and no systematic investigation on its dependence on the cell size was carried out. Hence, its usefulness is limited. Although numerous experimental studies have clearly shown the nonuniform particle distribution in turbulent flows, it is quite difficult to measure the 3D nonuniformity so as to isolate the effects of flow shear, particle inertia and gravity. Obviously, the local accumulation will affect the encounters among the particles and thus the overall collision rate is expected to be modified accordingly. However, the effect of the preferential concentration on the collision kernel has never been estimated or separated from other factors such as the increased particle collision velocity due to small particle inertia. 1.5 Scope of Present Study Particle collisions are extremely difficult to observe experimentally, even with today's sophisticated laser instrumentations. Direct numerical simulation, however, has its advantages of providing detailed information on particle trajectories, flow structures and is ideally suited for parametrical investigations of particle and fluid flow properties. Numerical implementation of particle interactions in a simulation, while posing a considerable numerical challenge, introduces no fundamental difficulties. In this work, extensive data obtained by numerical simulations on the collision velocity, collision angle, collision kernel and collision rate are vital to understanding of the fundamental physics in the particlefluid interactions. One of the major objectives of this thesis is to develop analytical and numerical methods for predicting the particle collision rate and concentration nonuniformity with respect to the effects of turbulent shear, particle inertia and gravity in particlefluid systems. As mentioned in 1.2, previous theories on the collision rate were restricted on assumed turbulent flow. In Chapter III, we have developed a new theoretical framework for prediction of the collision rate and collision velocity of small particles at zero inertia in general turbulent flows. The collision velocity is a critical variable related to the collision rate. However, the effects of particle inertia on the collision velocity and collision angle have never been predicted using Lagrangian tracking method. In Chapter IV, we carried out an asymptotic analysis to predict the effects of small particle inertia on the collision velocity and presented the effect of the inertia on collision angle in isotropic Gaussian turbulence. As pointed out in 1.4, no reliable method exists for quantifying the concentration nonuniformity and the effect of the preferential concentration on the collision rate has never been estimated. In Chapter V, a new quantification method has been developed to reliably quantify the concentration nonuniformity of particles in turbulent flows, and the effect of the preferential concentration on the collision rate is predicted in isotropic Gaussian turbulence. The first result of Chapter VI is our correction to the classical result of Smoluchowski for the collision rate of monodisperse particles in a laminar shear flow. The combined effects of both the increased collision velocity and increased concentration nonuniformity on the collision rate due to the inertia are then examined. The effects of particle size, gravity and meanconcentration on the collision rate are investigated through direct numerical simulations in Chapter VI. The effects of the collision models on the collision rate are also presented in Chapter VI. CHAPTER 2 TURBULENCE AND PARTICLE COLLISION DETECTION 2.1 Turbulence Representation 2.1.1 Isotropic. Gaussian Turbulence The following model for the energy spectrum developed in Mei & Adrian (1995) is assumed for turbulence, 3 2 k' 1 22 E(k) =2 uo Vk [l+(k/ko)217/6 exp(To k2) (2.1) where uo is the rootmeansquared turbulent velocity, k0 is a typical wave number and the dimensionless parameter ro oko is related to the turbulent Reynolds number Re,. Large Re. correspond to small rio and vice versa. The normalizing coefficient W in E(k) is determined from (2.1) by satisfying the total energy requirement, 0 4 k 2 (1k 2'd 7 /6 2 exp2(71 dk (2.2) 0 (1+k ) For small r10, E(k)~k4 when k = k/ko l and E(k)~k53, which is the scaling law in the inertial subrange, when 1 k /1"qo. For large 10o, the above energy spectrum recovers that of Kraichnan (1970). The relationship between the Eulerian integral length L, ko and ri (or Re.) is given in Mei & Adrian (1994). A turbulent eddy loses its identity as it is convected and dissipated; this behavior is described by the eddyself decay function D(x) which has an integral time scale TO. The Fourier transformation of D(T) gives the power spectrum D(o)). A composite form for the power spectrum D(o)) was constructed in Mei & Adrian (1995) as 2 exp(ri Io)2) () =A +(To(2.3) where TI, is intended for the viscous dissipation time scale if the turbulent Reynolds number is high, and T is close to To if ii I/To 1. The coefficient A is determined as T exp(rl /T2) (2.4) nt lerf(rTl/T) by satisfying J D(o) do) =1. The integral time scale To is related to T as To = 7tD(0) = oo 7lA . The relationship between L I and To is not known in general; it can be taken to have the form To = cE(Re)LI 1/UO, (2.5) where cE(Rex) was determined approximately in Mei & Adrian (1995) based on the experimental data of Sato & Yamamotto (1988) for the fluid dispersion and the value of high Reynolds number turbulent Prandtl number. Using random Fourier modes representation, an isotropic, Gaussian, pseudo turbulence N Ui (X, t) E bm) cos k(m) +(,(m) t u ( ,t = [bi (k x + .t ) m=! +cIm) sin(k(m) x + co( m t ) ] (2.6) is constructed to simulate the turbulent flow with a specified energy spectrum. In the above Nk is the number of the random Fourier modes in one fluid realization, k(m)and o)(m) are the wavenumber and frequency of the mth mode. The random coefficients b(m) and c(m) are chosen as follows, b(m) (m)(6 (m) k(m)/k (M) f(k, (nM) (2.7) i j qJ i j =b ((Si)ki W k ) fk^m ca (2.7) where b. follows a normal distribution with = 0 & = 8". (2.8) The factor (ijk(M) k(m) /k (M) in Eq (2.26) ensures the incompressibility Vu=0 for every Fourier mode. The scale factor f(k, o) depends on the energy spectrum and the probability density functions (pdf), pI(k) and p2(00), of k and co, f2(k, ) = E(k) D(o) (2.9) 2~,o = 5 (2.9) 47N2kpl(k)p2((O) Since E(k) decays as k5/3 in the inertia subrange, a Gaussian distribution for k would lead to a very slow statistical convergence for ui(x, t). To achieve convergence of the statistics involving the derivatives such as &', the following algebraic pdf axj p Ii(ki)= (1 + Iki)" fori=l, 2 & 3 (2.10Oa) 2 and P1(k) = P11(k1)P12(k2)Pi3(k3) (2.10Ob) is used to sample k, for ko= 1. The frequency om) is generated with the following pdf, 1 1 P2() = t +(/uko)2 (2.11) The scale factor is finally set to be E(k)D(co) f(k, (.) = E (2.12) 47rtNkk P l(kl)Pi2(k2)p13(k3)p2((O) It is noted that in the present studies on particle collisions, the collision rate is dictated by the spatial structure and the eddyself decay has no effect on the collision rate. In the direct numerical simulation of particle collision in isotropic turbulence, however, the eddy selfdecay can be easily incorporated. Hence, the dependence of b(m) and c(m) on (o is still included. 2.1.2 Rapidly Sheared Homogeneous Turbulence For an initially isotropic turbulence under rapid shear, F=a, the variation of b(m) with time was given in Townsend (1976) based on rapid distortion theory (RDT). Hunt & Carruthers (1990) gave an overview on the use of RDT for representing various types of homogeneous turbulence. Denoting the total shear S as S= Ft (2.13) the new wavenumber vector x(m) after the shearing is given by (M) = k(m) I I () = k(m) Sk(m) (2.14) x2 2 1 (M k(m) x k 3 3 where k(m) is the wavenumber vector of the mth mode before the mean shear is applied. The amplitude of each Fourier mode becomes b(f)(S) = b^m)(0) + alb(2)(0) b()(S) = a2 b(2m)(O) (2.15) b(m)(S) = b(3)(0) + a3b(m(0) where b(m)(0) is the corresponding amplitude of the mth mode before the rapid distortion. Dropping the superscript m for convenience, the coefficient ai for each mode m is given as Sk k22k2+Sklk2 al= 2 2 ak22 x k2k2 k2 k2Ski 2 23/2 [tanit ()tan] ( )] 2k23/ 2, 22 (k +k )3 k, l2i A; f 1 33k1k a2 = k2/x2 (2.16) Sklk3k22k2+Skjk2 a3 = k2 2 X2 1 ~3 k2k3 1 k2 k2Sk. ,2 ,.23/2 [tan ( ) tan( )] 2 2 32 2k22 where k= VkT+k2+k and x= 2 +X 2+ The coefficient c(m) is similarly obtained. It 1 2 3 2 3* is clear from the above that the amplitude of each mode is entirely determined by the total shear S=Ft. The dissipation rate s(S) can be evaluated as 1 N 2 e(S)/v= I < N {(Xm)2 3 [b()2 (S) +C(M)2(S)] } >" (2.17) m=l i=l Finally, the total velocity field is uj(x, t) = [ b(m)(S) cos((m) x + )(m) t) m=l + c( (S) sin((m) x +)(o Mt)] + Fx2 8ii. (2.18) Since the shear rate is high, the term o(m)t in the above acts merely as a random phase. 2.1.3 Isotropic Turbulence Generation in a Periodic Box For the direct numerical simulations of particle collisions in this work, the velocity field of an isotropic and Gaussian turbulence is generated by a random Fourier modes representation (2.6). To effectively conduct a computational study on particle collision, a region of finite size in which a large number of collisions occur should be used with periodic boundary conditions. To render the turbulence represented by random Fourier modes periodic in a box of volume R3, the random wave numbers, k( m), are rounded to the nearest even integers while all other quantities are held fixed so that the flow is periodic within the box of volume irt3. The periodic box of volume n3 yields virtually the same collision statistics when the volume is increased to (27t)3. Hence it3 is used throughout this study. The corresponding statistics (turbulence rmns velocity u0, Reynolds number Rex, Taylor micro length scale X, Kolmogorove length scale T, and Kolmogorov time scale Tk) are computed based on the modified wavenumbers. As will be demonstrated using the present analysis outlined in 2.1.1, the collision statistics, when normalized using R3( /v)1/2 for anI and <Wr>, the results are the same using the turbulence generated in an unbounded domain and in a periodic box. Unless specifically mentioned, most of the isotropic turbulence used in the numerical simulations throughout this study is within the periodic box of length nt and have the following characteristics: rootmeansquared turbulence velocity u0=l.O, integral length scale L, 11 =0.594, Taylor micro length scale X=0.277, Kolmogorov length scale r=O.023, Kolmogorov time scale Tk = (v / )I/2 =0.073, and Rex =40.1. The turbulence is imposed on a box it3 with periodic boundary conditions in three perpendicular directions. The periodic treatment of boundary conditions allows the simulation results by a finite volume to be extended to an infinite homogeneous system. Initially, a given number of monosize particles are randomly introduced in the flow field of volume it3. The periodic boundary conditions are also applied to the motions of the particles. The particles are advanced using ui(x, t) given by (2.6) with a time step At. A semiimplicit finite difference scheme is implemented to integrate the equation of particle motion and to detect collision. The scheme has an accuracy of second order. The detailed error analysis of the semiimplicit finite difference scheme will be given in next section. 2.2 Numerical Integration of Particle Motion and Error Analysis Major concerns in numerical simulation are the accuracy and robustness of the numerical scheme. Since the collision detection in the present simulation is based on the calculation of particle trajectories, the collision kernel/rate/velocity/angle are affected by the accuracy of the finite difference scheme. In this study, both tracking particles and detecting collision are carried out in a Lagrangian frame. A semiimplicit finite difference scheme is implemented to integrate the equation of particle motion and to detect collision. This scheme is unconditionally stable. With consistent treatment for both particle velocity and turbulence, The truncation error in the semiimplicit scheme is 0((At)2) and independent of 3. The accuracy analysis of the scheme is given as follows. The particle dynamic equation in the absence of (or weak) gravity is dv dt = P(u v). (2.19) The semiimplicit finite difference scheme for the particle dynamic equation (2.19) is Vn+i  n 1  n I (U.V)n+l +(UV)n] At 2 = p[(un+l + u(vn++v)]. (2.20) 2 Taylor series expansions for u"n+1 and vn+1 give ~~'=~ +AtdVn +1 2d2vn dv vn+ vn + dAt d + (At)2d + (At)3 3vn+ (2.21) dt 2 2 dt2 6 dt un+ =u AdU 1 Ad2u( 1 3d3un dt 2 dt62 d" Each side of equation (2.20) can be evaluated as dvn I d2vn 1 2d3Vn LHS= ++ (At) + (At + O(At)3 (2.23) dt 2 ( t dt2 6 dt3 I Atdun dvn 1 2 d~u dvn3 RHS=p(unvn)+PAt( dt )+IP(At)( d 2u 2 )+O(At)3. (2.24) 2 dt dt 4 dt dt Then equation (2.20) can be rewritten as Vn+l V 1 0 = n [(uv) n+' + (Uv) ] At 2 dv" 1 d2vn 2 d3vn +(At) +(At)+d dt 2 dt2 6 dt3 "  pi3(u v" I PAt( du dv" ) P(At)2 (d2un 2 dt dt 4 dt Thus, equation (2.20) is equivalent to dvn 1n d2vn dt 2) t dt2  (At)2 d3vn 6 dt3 I dun dvn I d2un d2vn + PAt( )+ (At)2( dt2) 2 dt dt 4 dt dt I dAt[2vn 2 dt2 dun dvnA dt dt 6 d3v 6At)2 dt3 1 2 d2un d2vn + I(At) (2 ) 4 dt2 dt2 Taking derivative with t repeatedly, we get d2vn dut dOt dt dvn dt 1 d3vn  At[ d n 2 dtT d2u d( d2vn)] dt2 A  (At)2 dt 6 dt4 + 4 dt3 d3vn dt3 d2un d2v) dt2 I d 4v At[  2 dt4 d3un d3 1 _dO _ __d _ 1 2d n + I p 2d4vn d4vn 6 dt 4 dt4 dt )t Repeatedly substituting above equations into the RHS of (2.26) results in dv n (A)2 d3vn 1(2 d3vn _+ (At)2_, + O((At)3) dt 6 d 4 dt 3 d2vT dt2 (2.25) (2.26) d 3vn dt3) dtO (2.27) (2.28) 1(At)2 d3vn (2.29) = M)2d+ O((At)3). (2.29) 12 dt3 It is noted that the truncation error in equation (2.29) is independent of P3. Thus motions of particles with very large P can be simulated without loss of accuracy. 2.3 Particle Collision Treatment 2.3.1 Collision Detection Scheme Collision detection between any pairing particles is given as follows (Chen et al. 1995). At nth time step to, these two particles are initially located at r1o =(x0, y1O, z10) and r20=(x20, Y20, z20) After a small time advancement At, at (n+J)th time step ti, new positions of these two particles are r11 =(x11, y11, z11) and r21 = (x21, Y21, z21 ). The particle position vectors at time t, which is between to and t I, can be expressed as r1(t) = {x1o0 + v1(tto), yio0 + Vy(t to), Zio0 + vz(t to)} (2.30) r2(t)= {X20 +v,2(tt0), Y20 +Vy2(tt0), z20 + Vz2(t t0)} (2.31) The distance between two particles, s(t), is s(t)= I r2 (t) ri (t). (2.32) These two particles will collide if the condition s(t) minimum relative distance occurring at time tm is determined by 9s, ast=t =0. (2.33) a~t ttm Combining (2.30)(2.33) gives D( tm =to  (2.34) D2 where DI = (xiO X2o)(Vxl Vx2) + (Yo y2o)(Vyi Vy2) +(ZIo Z2o)(vi Vz2) (2.35) and D2 =(Vxi v,2 )2 +(Vyi Vy2)2 + (vz v)2.) (2.36) Equation (2.33) does not guarantee that the moment, when the above mentioned minimum distance occurs, has to be within the time interval to and to + At. Three cases need to be considered for calculating the minimum distance within a time step. (i) if tm < to, then the minimum distance is at the initial time to. Smin = s(to) (2.37) i.e. Smin = {(X1io X20o)2 +(yio Y2o)2 +(Z10 Z20o)2}1/2. (2.38) (ii) if tm > to + At, the minimum distance must occur at the time to + At. Smin = s(to + At) (2.39) i.e. Smin = (s2 +Sy +s2)12 (2.40) where Sx= xl0 x20 +At(vxl Vx2) Sy = 0 Y20 +At(vyl Vy2) sz= zl0 z20 +At(vzl Vz2). (iii) if to < tm to + At, the minimum distance occurs during the time interval to and to + At. In this case, we have Smin= S(tm) (2.41) 2 2 2 .1/2 i.e. Smin = (sx + Sy + S )1/2 (2.42) where Smx = XlO X20 + (tm to)(Vxl Vx2) Smy = YI0 Y20 + (tm to)(Vyl Vy2) Smz ZIO Z20 + (tm to)(Vzl Vz2 ) . 23 After Smin is evaluated within the time step, the examination of collision events is still needed because Smin can be either larger or smaller than R. Collision occurs if Smin < R (2.43) At the collision instant tc, s(t) = R (2.44) and s(tc)=I r2(t0) r(t.)1 (2.45) Hence the collision instant tc can be obtained solving (2.44) and (2.45) b ]b2 4(s R2)w2 tc =2to+ (2.46) 2w2 where b=2[(xio X20)(vx, vx2)+(YIO y20)(VyI Vy2) (ZIo z20)(vzl vz2) 2 22 SO {(x10x20) +(yIoy20)2 +(ZI0Z20)2}12 W = {(Vxi Vx2)2 + (Vy! Vy2)2 + (Vzl Vz2)2}/2 . With perfect sticking collision model, any collision can produce a new born particle (daughter particle). The daughter particle will inherit all possessions of parent particles, i.e., conservation of mass and momentum. Let the daughter particle be numbered 3, then the conservation law gives the size of daughter particle as d3 =(d + d32)/3 (2.47) where di (i=l 1, 2, 3) are particle radii. The velocity components of daughter particle are Vx3 d + 2d2 (2.48) d3 v = d3+ Vd3 Vy3 y 3y2 2 (2.49) d3 vzld3 + V z2d 3 vz3 1 2d (2.50) 3and the position components at the end of time step and the position components at the end of time step 3 3 X3 = {[xl0 + vxl(tc t0)]d1 + [x20 + Vx2(tc to)]d3} / d3 + v3(to t + At) (2.51) Y3 = {[Y0io +Vy(tc t0)]d + [Y20 + Vy2(tc to)]d} / d + Vy3(to tc + At) (2.52) Z3 = {[z 0 + Vzi(tc to)]d3 + [z20 + vz2(tc t0)]d} / d3 + vz3(to tc + At) (2.53) In general, the collision detection takes a lot of computer time since a large number of particles are considered. The primitive collision detection scheme searches the collision candidates by going through all particles N, so that the computing time is proportional to N2. The collision detection method in Chen et al. (1995) is an efficient scheme, and it is also implemented in this thesis. By dividing the computational domain into a number of small cells, the potential collision partners for a given particle in one cell are then searched within this cell and neighboring twentysix cells during one time step. This search only involves a small number of particles. The total computing time with this scheme can be reduced dramatically. Only binary collision is considered by assuming a negligible probability of multiple collisions in a small time step in a dilute condition. For any particle moving out of the computational box, it is reintroduced into the box by invoking periodicity. The collision rate or kernel can be determined, in principle, by counting the number of collisions per time step. Initially, a given number of particles are randomly distributed into the computational domain. The particles are evolved with turbulence for a period of time before collision detection turned on. The collision events are not counted until all particles with finite inertia have reached dynamical equilibrium in turbulence. 2.3.2 PostCollision Treatment The collision event is counted when two particles are brought into contact, as stated above. The collision among particles and its subsequent treatment are expected to disturb the original system and permanently impact the subconsequent collision evaluation. For the droplets in gas turbulence, the collision may, most likely, result in coalesce of two droplets and a larger droplet is then formed by the colliding droplets. Similar phenomena may be found in a system consisting of bubbles and fluids. The collision between two solid particles may behave in different ways. Two solid particles may stick together upon contact or the colliding solid particles will separate after collision. In summary, the issue dealing with immediate consequence of those contacts is referred as postcollision treatment. Several models with regard to postcollision treatment have been developed during past decades. In Chen et al. (1995) every collision results in the collision partners disappearing from their respective size groups to produce a larger particle. Mass and momentum conservation laws were applied for the birth of the new particle. This scheme is considered to be a physically realistic model assuming a unity collision efficiency. It is hereinafter referred as 'KEEP ALL' model. Balachandar (1988) used a different post collision scheme. Starting with monosize particles in system, after each collision, the resulting larger particle was discarded in the computation so that it does not contribute to future collisions. This postcollision treatment is hereinafter referred as the 'THROW AWAY' scheme. Sundaram & Collins (1997) employed a 'HARD SPHERE' collision model in which the particles were assumed to be rigid spheres and forced to bounce back after the contact. Momentum conservation is applied to the particles after collision. In contrast to the 'KEEP ALL' and 'THROW AWAY' schemes, the 'HARD SPHERE' model allows a particle to have multiple collisions with other particles in a system. Wang et al. (1997) tested three different schemes and presented one of their own. In their scheme, collision does not disturb the system. The colliding particles are allowed to overlap in space (hereinafter referred as 'OVERLAPPING' scheme), but no larger particles are formed after the collision. Two colliding particles can separate after their trajectories satisfy Ir2 (t) r (t)l>R. Thus, a particle can have multiple collisions with other particles. These four different postcollision treatments will, in principle, result in different collision rate/kernel. a(11 1.6 1..... Dt=0.005, simulation Dt=0.01, simulation 1.2 1.0. . .      0.8 0 .6  ' 0.0 0.5 1.0 1.5 2.0 2.5 3.0 T Figure 2.1 The effect of time step on the cumulative average of the normalized collision kernel ca I for inertialess particles in turbulence. Particle size D=0.05. 2.4 Computer Simulation 2.4.1 Time Step Three collision models: "THROW AWAY," "KEEP ALL" and "HARD SPHERE" are employed in this work. For the following simulations with '"THROW AWAY" and "KEEP ALL" collision models, the time step At=0.01O is used to advance the motions of particles. Figure 2.1 shows the effect of time step on the collision kernel a *, using "KEEP ALL" collision model. It can be seen that there is almost no difference between the results of At=0.01 and At=0.005. Similar results are obtained by '"THROW AWAY" scheme. The volume concentration of suspended particles is usually quite low in many natural systems such as spray burners and air scrubbers, which is within the interest of this work. In such dilute systems, the multiple collisions during a small time step may be negligible, thus binary collision mechanism is considered solely in this study. 2.4.2 Turbulence Realization The averages of statistical quantities are evaluated in two directions: average over all time steps and turbulence ensemble average over all turbulence realizations. The total time steps mainly depend on the particle response time and the number of collision pairs during whole time period. We start counting the time steps for collision detection after the motions of particles have been evolved for three times the particle response time. Ideally, to form the turbulence ensemble average, the numerical simulation should be repeated an infinite number of realizations with different initial particle distributions. An infinite number of turbulence realizations is impractical due to limited computer capacity and may be unnecessary from the viewpoint of the accuracy of numerical scheme. Figures 2.2 (a) & (b) show the turbulence dissipation rate against turbulence realizations. While the ensemble averaged turbulence dissipation rate varies with the realization by a large amplitude shown in figure 2.2 (a), the cumulative average of turbulence dissipation rate approaches a constant for the number of turbulence realizations greater than 30, as seen from figure 2.2 (b). Figures 2.3 (a) & (b) show the small eddy shear rate against turbulence realizations. It can also be seen that, although the ensemble averaged small eddy shear rate randomly varies with the turbulence realization in figure 2.3 (a), the cumulative average of small eddy shear rate is close to a constant by the 40th realization seen in figure 2.3 (b). Therefore, in this study forty turbulence realizations are used for turbulence ensemble average. 2.4.3 Validation of the Collision Detection Scheme Starting with monosize particles in system and using the "THROW AWAY" collision model for postcollision treatment. The evolution of the particle number n1 (t) follows the simple population balance equation: dn1/dt=a, In2 with n1(t = 0)=n0. (2.54) The solution is no / n1(t) = l +a n0t. (2.55) It should be mentioned that the collision rate of monosize particles (2.54) is only accurate for large number of particles in a system. For small number of particles, however, the collision rate (2.54) must be corrected. The detailed discussions on this issue will be given in 6.2. Since the statistics for the number of the remaining particles n1(t) is much more accurate, the average slope of the curve 1 / n1(t) gives the collision kernel, so it can be used to check the accuracy of the collision kernel obtained by simulation in laminar shear flow. To validate the collision detection, particle collision in a uniform shear flow of velocity gradient [= 1.0 is first considered. For large number of particles the theoretical prediction of the collision kernel aoij of Smoluchowski is accurate and is given by ac = 4 R ir / 3 For a small number of monosized particles the theoretical prediction of the collision rate of Smoluchowski has to be corrected, as mentioned above. Ten thousand particles of radius 0.075 are introduced into a box of volume 2x2x2. The time step is Dt=0.01 and a total of 800 time steps is used. The number of shear flow realizations is 80. Since no periodic condition may be imposed along vertical direction in uniform shear flow, periodic boundary conditions are employed in the streamwise and spanwise directions only. In such a case, a boundary correction of the collision rate has to be imposed (Wang et al. 1997). Another correction on the collision rate is also applied due to the inaccurate expression of the collision rate (6.2) Figure 2.4 shows the particle number evolution. The collision kernel a 11 can be obtained from the average slope of the curve. Very good agreement can be observed between simulation result and Smoluchowski's prediction. Hence, the collision detection scheme and the numerical implementation are validated. 9 8 (a) 7 c 6 o L 5 S S4 0 U C 1e 3 . ** p 2 1 *. *. 1 .* n , I . ** * * 0 5 10 15 20 25 30 35 40 Turbulence realizations C 2.0 (b) ' 1.6 i ** **.** .** . I 2 ** *12 U C S 0.8 I 0.4         0 5 10 15 20 25 30 35 40 Turbulence realizations Figure 2.2 Turbulence dissipation rate e against turbulence realization. (a) Variation of ensemble average. (b) Cumulative average. (S / v)1/2 25 (a) ** 20  *o 15 . .*** 4 * 10 * 5 0 I I "; ...; .. .. .. .......... ; .....I. .. . 0 5 10 15 20 25 30 35 40 Turbulence realizations (s / v)"2 14.2 * 14.0 13. (b) * 13.8 6 * 13.6 \ ^ 13.4 13.2 13.0 12.8 12.6  0 5 10 15 20 25 30 35 40 Turbulence realizations Figure 2.3 Turbulence small eddy shear rate (& / v) 1/2 against turbulence realization. (a) Variation of ensemble average. (b) Cumulative average. no / n 1.06 1.05 Smoluchow ski 1.04 ... Present Siulation 1.03 1.02 1.01 1.00 I I 0 1 2 3 4 5 6 7 8 T Figure 2.4 Inverse of total number of monodisperse particles against time in laminar shear flow. F=1.0, D=0.015. For "HARD SPHERE" particles, the time step for advancing the motions of particles is much less than 0.01. It is found that At=0.001 may generate almost same result as At=0.0005 for large inertia particles. For small inertia or zero inertia particles, the collision kernel strongly depends on time step due to multiple collision mechanism, which is shown in figures 2.5 for uniform shear flow. A pair of particles can have tens or hundreds of collisions depending on the size of time step before they completely separate. Similar multiple collisions have been found for particles in turbulence. Due to this technical difficulty, the "HARD SPHERE" collision model is limited to particles with large inertia. V2I V2 Figure 2.5 Multiple collisions of HARD SPHERE particles in laminar shear flow. CHAPTER 3 A NEW THEORETICAL FRAMEWORK FOR PREDICTING COLLISION RATE OF SMALL PARTICLES IN GENERAL TURBULENT FLOWS A new theoretical framework is developed to predict the collision rate and collision velocity of small, inertialess particles in general turbulent flows. The present approach evaluates the collision rate for small particles in a given instantaneous flow field based on the local eigenvalues of the rate of strain tensor. An ensemble average is applied to the instantaneous collision rate to obtain average collision rate. The collision kernels predicted by Smoluchowski (1917) for laminar shear flow and by Saffman & Turner (1956) for isotropic turbulence are recovered. The collision velocities presently predicted in both laminar shear flow and isotropic turbulence agree well with the results form direct numerical simulation for particle collision in both flows. The present theory for evaluating the collision rate and collision velocity is also applied to a rapidly sheared homogeneous turbulence. Using the mean turbulent shear rate (6/v)1/2 as the characteristic shear rate to normalize the collision rate, the effect of the turbulence structure on the collision rate and velocity can be reasonably described. The combined effects of the mean flow shear and the turbulence shear on the collision rate and collision velocity are elucidated. 3.1 Introduction Smoluchowski (1917) considered collision rate among spherical particles in a laminar shear flow, (ux, uy, u.) = (Fy, 0, 0) with a constant gradient F. For a target particle of radius ri centered at an arbitrary position xi, any particle with a radius rj moving toward the target particle will cause a collision if Ixi xj < Rj = ri +rj (3.1) is satisfied. The collision rate can be evaluated as = ninj fwr where wr is the radial component of the relative fluid velocity between xi and xj, wr<0 indicates an impending collision, nln2wr is the particle flux moving inward to the target particle, and A is a spherical surface with radius Rjri +rj. Using standard spherical coordinates (R, 0, 4)) centered at particle i (see figure 3.1), wr on the spherical surface of R=Rij can be written as Wr = FycosO = Rij sinO cosO cos). With dA= Rj sinO dOdo, it is easily seen that 4li ~~~F R3 (3.3) 43 i ncij nnj 3 r Rij (3.3) or Fij RR. (3.4) 3 i Saffman & Turner (1956, hereinafter referred as ST) presented a classical theory for the collision rate of small droplets in Gaussian, isotropic turbulence. The collision rate among particle size group i and j is c= ninj fWr The ensemble average, denoted by < >, is necessary since turbulence is random. Because of the continuity of the fluid flow, the volume influx entering the spherical surface of radius Rj is equal to the efflux so that nc1= ninj < entire Iwrl dA>. (3.6) cij 2 n n entire sphere r ST further interchanged the integration with the ensemble average so that ncij can be evaluated in isotropic turbulence, ncij 2 ninj entire sphere Wr y 0 I, ~C u = Fy x / 4 , z Figure 3.1 Local coordinates for collision rate calculation in laminar shear flow. For isotropic turbulence, the above becomes ncij = ninj 27tRR where x is the local coordinate parallel to the line connecting the centers of the colliding particles as shown in figure 3.2. For particle size smaller than Kolmogorove length scale il, the relative velocity upon the collision was evaluated using the standard results of smallscale turbulence theory as where s is the dissipation rate of turbulence and v is fluid kinematic viscosity. This gives 3 8n s ]/2 3 6 1/2 ncij = ninjRij [ ]v =.2944nnR () (3.10) or aij=1.2944R (S 1/2 (3.11) or C,3 =1.2944 (3.12) oij (3!) * where aii is normalized collision kernel. V2 x VIX V 2x = Wx Figure 3.2 Sketch of particle collision velocity. The separation distance is x2xI = Rij= (ri+rj). It needs pointed out that from equation (3.6) to equation (3.7), the interchange between the integration with the ensemble average is only an approximation even in a Gaussian, isotropic turbulence. Furthermore, the interchange between these two operations severely limits the extension of ST's result and approach to other turbulent flows such as the near wall region where the mean flow gradient contributes to course, in the theory of Smoluchowski for laminar shear flow, this issue does not arise since the ensemble average is not needed. In a typical industrial facility where particle collision leads to the desirable growth of particles, the inhomogeneity of the turbulent flow structure is common. Due to the effects such as turbopheresis, particles tend to be driven toward wall regions to cause a further inhomogeneity in the particle concentration distribution. It was not clear how one can predict the collision rate of small particles in the presence of both strong mean flow shear and high turbulence intensity (Kontamaris, 1995). In this chapter, a new theoretical framework is presented for the collision rate of inertialess particles, whose sizes are smaller than the Kolmogorove length scale, in general turbulent flows. The present analysis starts from equation (3.5). Using the leading order term, (xixj)Vu, in the Taylor series expansion to approximate the relative fluid velocity, w, at the collision instant, the collision rate for an arbitrarily given Vu is evaluated first. The ensemble average is then carried out in a Gaussian, isotropic turbulence to obtain the collision rate and collision velocity. Direct numerical simulations are also performed to obtain the collision rate and collision velocity in a Gaussian, isotropic turbulence. Good agreement is obtained between the present * prediction and the simulation for the collision kernel cai Excellent agreement is obtained for the average collision velocity between the prediction and the simulation. While the present result for the collision rate agrees with the prediction of ST, the present theory predicts an average collision velocity <Wr> that is 1.58 times the value given by ST. The difference is caused by the bias of the collision toward higher values of collision velocity which was not considered in previous studies on flow induced particle collisions. Using the rapid distortion theory (RDT) for a rapidly sheared homogeneous turbulence and the random Fourier modes representation, the dependence of particle collision kernel aij on the mean flow shear rate and the turbulence structure in homogeneous shear flow is studied under the present theoretical framework. It is found that the collision kernel, after normalized by particle volume and the turbulence mean shear rate characterized by the dissipation rate, mainly depends on the ratio of the mean flow shear rate to the turbulence mean shear rate. 3.2 Analysis The relative fluid velocity, w = u1 u2, between target particle centered at x, and the colliding particle centered at x2 in space with Ix, x21 = Irl2 = R less than the Kolmogorove length scale q can be expressed using Taylor series expansion as w = U2U1 ; r12.Vu +... (3.13) r12 = X2 Xl (3.14) to the leading order. The collision rate, ncl for a given Vu at a given instant is thus nc12 =nn2 Wr< wndA (3.15) where n is the outward normal of the surface given by Ix2 Xi = Rj = R and w= wn. For a spherical surface, n= r12/R so that J 1 wndA = f wJrl2 dA wr; SR Ow, 12 r1 r2 2dA. (3.16) Representing Vu as the sum of a symmetrical part (rateofstrain tensor) and an anti symmetrical part (vorticity tensor), it can be easily seen that the antisymmetrical part of Vu does not contribute to wr*. Physically, it is simply because a rigid body rotation does 1 not contribute to normal velocity. For a symmetric tensor E = I (Vu +(Vu)T) or a symmetric matrix Eij, a linear transformation, which is a rotation of the coordinate system from (x, y, z) to (x, z ), can be easily found to reduce Eij to a diagonal form as =[eR 0 0  E = 0 ey 0 (3.17) 0 0 e2_ where (ek ey eZ ) are the eigenvalues of E and are the principal values of the rateof strain tensor. Due to incompressibility, ez + ey + ez = 0 so that there are only two independent parameters (say ez and ey ) in E in order to evaluate the integral in (3.16). Arranging the eigenvalues in the order of ez =(ex + ey) _ _y ewith 0.5<4<1 (3.19) the integrand in (3.16) can be expressed as r12 Vurl2 = (ez R 2 + ey y 2 + e j 2) = eR [X 2 + Cy 2 (1+g j 2] = ex R2[cos2O+ sin20cos2( l+)sin20sin2o] (3.20) where a local spherical coordinate system (x =RcosO, Y =Rsin0cos+, z =Rsin0sino) has been introduced. Hence 2itr J Wr0wndA= R3ez ff <0[cos20+ 4sin20cos20(1+)sin2Osin2O] sinO dO do W <0 JjWj.<0 fr W 0 0 =R3e e (9). (3.21) Again, Wr<0 in the integration limit implies that only the nonpositive values inside the square bracket are evaluated in the integration. While an analytical expression for J9() is difficult to obtain due to the dependence of the integration limits on arbitrary 4, accurate numerical integration for Y(Q) can be easily carried out (using Simpson's rule). The following interpolation is then constructed for (4) based on numerical integration results, 8 8( [0.90725 + 0.2875(0.5+4)2 + 0.333(0.5+4)4] <0 8 ,j [I + 0.554 + 0.7240712 0.9687443 + 0.739244 0.230355]. >0 (3.22) F 8 Consider the laminar shear flow for example. In this case, 4=0, eR = 2, H(0) = 3, so that 1 33 nc2 = n1n2 FR = nln2 FR3 which is the same as given by Smoluchowski (1917). Equation (3.22) is valid for 1_<40.5 as well since it is based on (3.21). The restriction for 4>0.5 results only from equation (3.18) when the eigenvalues are arranged in such a manner. For 1<_< 0, JY() is symmetrical with respect to C =0.5. For example, 4=0 corresponds to a 2D stagnation flow in (Rx z ) plane and C=1 corresponds to a 2D stagnation flow in (R Y ) plane so that the collision rates are completely equivalent. It is also interesting to note that c=0.5 is equivalent to 4=1 since both correspond to an axisymmetric stagnation flow with 4=0.5 for an axisymmetric contraction and C=l for an axisymmetric expansion; the only difference is the scale ek With these examples in mind, a simple analysis for the flow field quickly shows that M1(_<0) can be obtained from 1;2f0) as 1y )= I(1 1)1+y :() for 4>0. (3.23) Finally, the collision rate for small inertialess particles in a turbulent flow is simply nC12 = nn2R3 < ej 9(4) > (3.24) where the ensemble average is taken over all possible values of the rateofstrain tensor. It is emphasized that this result is applicable to general turbulent flow. Needless to say, the isotropy assumption is not needed in the present framework. When the detailed knowledge of Vu is available, such as from DNS or random Fourier modes representation (Kraichnan, 1970) of turbulence, the ensemble average in (3.24) can be carried out. The foregoing analysis can also be extended to evaluate the average of the normal (or radial) component of the collision velocity <Wr>. For a given position with arbitrary eR and 4 at any given instant, the instantaneous average of the normal collision velocity can be calculated by integrating over the spherical surface of radius R as 0wr In the above, the superscript 4. on the left hand side denotes the areaaverage, (Wr)dA in the numerator can be interpreted as proportional to the probability of a particle striking the target surface at a position (0, i) over an area dA and wr in the numerator is the relative radial velocity of the collision at (0, 0) on the target surface. The denominator is the normalizing factor for the probability. The product, wr(wr), clearly indicates that Suppose that NR realizations of turbulence be taken in a turbulent flow, the total number of collisions will be Nc = NRnIn2< .r< (wr)dA> per unit time using equation (3.5). The chance of a particle striking the target surface over an area dA at (0, 4) relative to the axis of the principal direction R of the instantaneous rateofstrain tensor in a given realization is then nin2(wr)dAw<0/Nc. Integrating over the target surface and summing over NR realizations, the ensemble average, Snin2(Wr) Wr W> =R < Wr Nc d> = The above can be simplified as <2 9(e ) > r where 2n ( = f f [cos20+ 4sin20cos20(l+)sin2Osin2w]2r<0 sinO dO do. (3.28) 0 0 The integration can again be evaluated numerically and represented in the following piecewise interpolations "G =0.4+1.0777(0.5+4)2 0.64767(0.5+4)4 for 4<0 =0.62831 + 0.68495 0.0910832 +0.09030143 0.0346584 for C>O. (3.29) 2 Similarly, the mean square value of the radial component velocity < wr > can be evaluated as < e3 Wf) > with =0.17114+l1.2643(0.5+)20.48123(0.5+;)4 for 4<0 =0.45714 + 0.976554 + 0.396062 + 0.072430(3 0.0291C4 for _>0. (3.31) The functions 4), q(4)/I(), and 7{)/4( ) are shown in figure 3.3. Take a laminar shear flow for example: ez =172 and 4=0 so that G/H= 0.62832 (=d5) and G/H=16/35. It is easily seen that F2 n8 18 7t 2 2F3 168 18 4 , The results of these two moments of wr can be used to validate the present theoretical formulation by comparing with direct numerical simulation for particle collisions. 0.04 0.5 0.0 0.5 Figure 3.3 Variations of i(, G(), K(Q. ) with where fl38 ](),f2= ) andf3= . To apply the above results to turbulent flows, we consider the following two cases: i) Gaussian isotropic turbulence; and ii) rapidly sheared homogeneous turbulence under a mean flow gradient F. In both cases, the turbulence can be represented using random Fourier modes and ensemble averages can be obtained without resorting to the direct numerical solutions to the Navier Stokes equations for the fluid flows. Pertinent comparison can be made with ST's prediction in the first case. Insights on the effect of mean flow shear can be gained by examining the collision rate in the rapidly sheared turbulence. 3.3 Numerical Simulation The detailed representation of the isotropic turbulence and rapidly sheared homogeneous turbulence are given in Chapter 2. In short, the isotropic turbulence is represented in the form of N Ui(X, t)= Y[b m) cos(k(m) .x +tom) ( t ) m=! +clm) sin(k(m). x + o(m) .t ) ] (2.6) where k(m)and ((m) are the wavenumber and frequency of the mth mode. The random coefficients b1m) and cm) are their corresponding amplitudes. For sheared homogeneous turbulence, the flow field is given by N Ui(X, t)= E[bim)(S)cos( (m) x+to)(m)t ) m=! + cm)(S)sin(X(m) x +o(m) t) + Fx2 8i (2.18) where X(m) is the new wavenumber after the distortion by the mean flow shear, S = Ft is the total shear by the mean flow, and b7m)(S) is the amplitude of the mth random mode after the rapid shearing. The scheme developed in Chen et al. (1995) is employed here to search for particle collisions. Detailed description on the collision algorithm is given in Chapter 2. Periodic boundary conditions are employed for both turbulent flow and particle motion. A number of particles (Np) are randomly introduced in the flow field of volume n3. The average collision velocity is obtained based on all the collision events for whole time period and all turbulence realizations. As discussed in Chapter 2, four different postcollision treatments will give different collision rate and it is not clear which treatment is the best. In the study throughout this chapter, the 'THROW AWAY' scheme is implemented. However, an extremely low particle volume concentration is used. The initial particle volume concentration is chosen carefully so that the average number of collisions in each turbulence realization is less than 1 The postcollision treatment thus has little practical effect on the collision statistics. A large number of turbulence realizations are used to obtain sufficient ensemble average. A subsequent cumulative time average smoothes out statistical noise in the collision rate. 3.4 Results and Discussions 3.4.1 Comparison with Simulation Results in a Laminar Shear Flow In an earlier paper dealing with particle collision a laminar shear flow (Hu & Mei, 4 . 1998), comparison for the collision kernel, 1 FR3, based on Smoluchowski's prediction '3 and the direct numerical simulation was presented; excellent agreement was obtained. To confirm the present analyses in the laminar shear flow, we compare the collision velocity given by equations (3.323.33) with that based on the direct numerical simulations. Zero inertial particles of radius 0.009, 0.0125 and 0.02, respectively, are introduced into a cubic box with length Lx=2, Ly=2.0, and Lx=2.0, with the restriction 0.1 uniform shear flow with a shear rate r=I.0 is imposed and the flow velocity field is given by v=(u, v, w) = (Fy, 0, 0). Periodic boundary conditions are employed in the streamwise and spanwise directions. Particles near y=0.1 and y=1.9 have less collisions because there are no particles in regions of y<0.1 and y>l .9 in the simulation. A correction due to this boundary effect (Wang et al. 1997) was needed for the collision rate (Hu & Mei, 1998). However, this boundary correction has little effect on the collision velocity since the decrease in the collision kernel affects both the numerator and denominator in equation (3.25). Three hundred time steps with a step size At=0.01 are used to advance particles and a total of 40 realizations were employed to further increase the statistical accuracy. Ten thousand particles are used for r=0.009, 0.0125 while four thousand particles are used for larger size with r=0.02. Table 3.1 shows the comparisons for both <Wr> and for these two moments indicates that the basis of the present formulation is sound. It is worth noting that a naive, Eulerianbased estimation of the average collision velocity 4 based on the incoming particle flux, i FR3ni, and the available collision area, 2nR , would lead an erroneous value of FR instead of n7 FR for <wr>. Table 3.1 Comparison of the collision velocity among monosized particles between prediction and direct numerical simulation in laminar shear flow. Particle <Wr> <Wr> r r radius simulation prediction simulation prediction 0.009 5.6421x103 5.6549x103 3.7018xl05 3.7029x105 0.0125 7.8638x103 7.8540x103 7.1529x105 7.1429x105 0.02 1.2425x102 1.2566x103 1.8012x104 1.8286x104 In a laminar shear flow, Wr=UxCOS0 and the collision angle Oc between the collision velocity w and the axis connecting the center of colliding particle pair, n/2_ easily evaluated so that the pdf for 0c is readily obtained. Since (Wr)wr< sin0dOdo is proportional to the probability of the particle striking the target surface at (0, )) with a solid angle do)=dOd), it is readily seen that (wr)dA = FR3sin2OcosO cos4) d0d4) for (0, 4)E Q, = 0 elsewhere (3.35) where Q is the region defined by (2 <0_ Wr<0. Interpreting sin20cos0sino as proportional to the probability density function of incoming particles striking the target particle near (0, )), further integrating over 0 from 0 to 27t, treating 0 p(Oc) = B sin2 ccos0c for in<_O =0 for 0<09 where B is a normalizing factor which is 3 for 0 in radian and t/60 for 0 in degree. The average angle is 7t2 simulation gives prediction and the direct numerical simulation using ten thousand particles of radius 0.009 over 40 realizations of initial particle positions. Excellent agreement is observed for the whole region it/2<0c< The present theoretical framework is thus validated in the laminar shear flow. 0.025 ...., 0.020 0.015 0.010 Analytical prediction 0.005 Simulation 0.000 # , 90 100 110 120 130 140 150 160 170 180 Collision angle 0 c Figure 3.4 Comparison of the pdfs for the collision angles 0c between the theory and the simulation in a laminar shear flow. 3.4.2 Collision Rate and Velocity in a Gaussian, Isotropic Turbulence For particle collision in isotropic turbulence which is represented using equation (2.6), the dimensionless collision kernel aIl S R3(e/v)12 (s/v)1/2 (3.37) is evaluated using NR=500 realizations. The dissipation rate E is based on the ensemble average of the same NR realizations. In each realization, ez W/(4) and E/v are computed in the flow field with 654 statistically independent data. It is found that a1I varies slightly with the number of the Fourier modes, Nk, used in the turbulence representation. Furthermore, it weakly depends on the power g in equation (2.10) with which the random * wavenumber is chosen. Figure 3.5 shows the variation of an as Nk increases from 100 to 4096 for t=1.2 and 1.4 for ri 0= 0.05 (Rex=53.8). The result seems to converge after * Nk=2048; a, 1(P.=1.2) converges to 1.2935 while a, 1(t=1.4) converges to 1.2945 with a difference less than 0.1%. They are both very close to 1.2944, the value predicted by ST. It is worth commenting that due to the rapid decrease of the energy spectrum E(k) in the high k range, not all wavenumbers that are randomly generated need to be used in equation (2.10). For those high wavenumber modes that correspond to very small amplitudes, they can be simply neglected without compromising the accuracy. Thus the effective number of the random modes is smaller than the specified Nk. From the simulations, the effective number of the modes for g=1.4 is roughly three times that for g=1.2 for the same Nk. Hence the fact that it takes larger values of Nk for p.=1.2 than for ji=1.4 to get a converged result is not surprising. It is also interesting to note that, using Nk=1024 and g=1.2, a higher value of Rex (=124.4 using fi 0=0.01) gives a,1 =1.2899 as * opposed to al =1.2912 at Rex=53.8. This clearly shows that shape of the energy * spectrum E(k) in isotropic turbulence has little effect on a, With Nk=200 and .=1.2, * a1 =1.2756 which is only 1.45% less than the converged result. Thus in the following direct numerical simulation for particle collision, Nk=200 and p=1.2 will be used. To validate the present analyses, direct numerical simulations of particle collision in isotropic turbulence are performed. The turbulence in the periodic box of length n has the following characteristics: rootmeansquared turbulence velocity uo=l1.0, integral length scale L1 =0.594, Taylor micro length scale X=0.277, Kolmogorov length scale 71=0.023, Kolmogorov time scale Tk=(v/)l/2=0.073, and Re,=40.1. With the modified Fourier modes in the periodic box, the dimensionless collision kernel based on the * prediction is ac =1.2711 as opposed to a11 =1.2756 without the modification of the wavenumbers. Initially, five hundred particles of radius r=0.005 (r/ril=0.217) are randomly distributed in the box of volume n3 with a particle volume concentration of 8.44xl0O6. These inertialess particles are advanced using local fluid velocity. A total time period of T=6 (i.e. T/Tk=82.2) is simulated with time step At=0.01. The average number of collisions within the entire period of the simulation 0 To obtain a reliable statistics, a total of 10,000 realizations is used to generate an * estimated collision pairs Nc=4271. Figure 3.6 (a) shows the simulation result for ctaI in 0 numerical simulation data due to the small number of collisions within each time step despite the fact that 10,000 realizations have been used. A cumulative average is applied to reduce the noise. Figure 3.6 (b) shows the cumulative time average of cc1 from the simulation. The direct numerical simulation is less than the predicted value (1.2711) by about 2.5%. This difference may be caused by a small effect of particle size, since the theoretical predictions are expected to be accurate only in the small size limit, i.e. r/r< size on the collision rate will be explored in Chapter 6. The agreement for o 1 between the simulation and the prediction is quite good. Table 3.2 shows the results of the prediction and the numerical simulation for the two moments of particle collision velocity, <Wr> and predicted values are based on =1.2 and Nk=200 in the periodic box with NR=400 realizations. The particle sizes are r/rI = 0.2175 and 0.4875, respectively. Satisfactory agreements are obtained. On the other hand, ST (1956) gave an expression, equation (3.9), for the Eulerianbased average particle radial collision velocity. Using that <Wr> expression, one would obtain j expression, one would obtain /v)2 0.206. The present prediction for the periodic domain with NR=400, after ensemble average, gives a value of 0.328 for this dimensionless radial collision velocity which agrees rather well with the direct numerical simulation show in Table 2. Since the direct numerical simulation is based on the tracking of particle trajectories, the probabilitybased prediction for <w,> is consistent with the Lagrangian approach. The Eulerianbased approach for <Wr> does not take into account the bias of the collision probability toward the high values ofwr; hence it gives a lower value for <Wr>. cci 1.30 1.29 1.28 1.27 1.26 1.25  1.OE+2 1.0E+3 1.OE+4 Figure 3.5 Convergence ofacq as Nk +8. Table 3.2 Comparison of the collision velocity among monosized particles between prediction and direct numerical simulation in Gaussian, isotropic turbulence. Particle <Wr> <Wr> r r radius (r/rhi) simulation prediction simulation prediction 0.217 4.5430x102 4.5611x102 2.7356x103 2.7296x103 0.4875 1.0093x 10 1 1.0040x101 1.3405x102 1.3224x102 oa11 2.5 (a)  siulation ....... Prediction 2.0 1.5 1.0 0.5 0 10 20 30 40 50 60 70 80 90 t( / v)1/2 * a11 1.8 1.7 (b)  Prediction 1.6 Cumulative average of simtlation result 1.5 1.4 1.3 1.2 0 10 20 30 40 50 60 70 80 90 t(e / v)/2 Figure 3.6 Particle collision kernel ax from direct numerical simulation in Isotropic turbulence. (a) Variation of the ensemble average. (b) Cumulative time average. 3.4.3 Collision in a Rapidly Sheared Homogeneous Turbulence The statistics of the turbulence, such as four nonzero components of the Reynolds stress tensor, generated using RDT and the random Fourier modes was first checked against the analytical values at short times (Ftnl) given by RDT (Townsend, 1976, p.84) to ensure the correct implementations. It was found through simulation that the development of the sheared turbulence with time does not depend on the Reynolds number, Rex, of the initial state or the energy spectrum at t=0. The turbulence part of the flow in the RDT is determined by the total shear S=Ft. The complete flow field, sheared turbulence plus the mean shear flow, thus depends on two parameters: F and S=Ft. Since the collision rate depends on the spatial structure of the turbulence while the turbulence structure becomes increasingly anisotropic as the total shear S increases, finding an appropriate representation of the collision rate in the homogeneous turbulent shear flow is not trivial. To develop a better understanding on the dependence of the collision rate on the turbulent flow, it is instructive to examine the collision rate due to the sheared turbulence but without the contribution from the mean shear F; this part of the collision kernel will be denoted as ai1. Another word, a jj is evaluated using the flow field given by equation (2.18) but without the last term [x25il which is deterministic. Following the * definition for all in the isotropic turbulence case, we use the dissipation rate (s/v)1/2, which increases with S, as the turbulence characteristic shear rate to normalize t I I, * an! c" = R3(/V) 1/2 (without contribution from mean shear F). (3.38) * It is easily seen that a,1 depends on the total shear S=Ft only. Figure 3.7 (a) shows the * variation of &1 on S. In the high S limit, the simulation results suggest that a11 approach a constant of 1.1676 in the form of a1 1.1676+0.659/S forS>10. (3.39) In this rage of 0 turbulence intensity becomes dominated by the longitudinal component in the x1 direction. Based on RDT prediction which neglects the nonlinear interaction among different scales, the longitudinal component of the turbulence makes up 92.7% of the total kinetic energy at S=Ft=20. The anisotropy invariant IlIb reaches a value of 0.25 at S=20. At such large values of total shear, S, the RDT results may become a little inaccurate in comparison with DNS result. However, for the present work, since the RDTbased turbulence is strongly anisotropic, it serves our purpose to evaluate the effect of such anisotropy in the turbulence on particle collision rate. When the collision kernel is normalized using (s/v)"2, the maximum difference in an is only about 15% at the * extreme limit of S+co. If the isotropic result is used, &a, ~1.2944, only a maximum error of 10% will result from the limiting value of aE at very large total shear. This * encouraging result will be used to develop an approximation for a1 in the presence of strong mean shear. Now consideration is given to the particle collision kernel with contribution from both the turbulence part and the mean flow shear part as the flow field is given by the * entire righthandside of equation (2.18). Figure 3.7 (b) shows aI as a function of the ratio of the mean shear rate to the turbulence shear rate, F7/( )/2, for a range of shear V rates. For a given F, (s/v)1/2 increases as t or S increases so that the abscissa mainly reflect the effect of t or S. To see the effect of mean flow shear rate, F is made dimensionless by the turbulence mean shear rate at t=0 in the isotropic state (eo/v)1/2. Four values of F/(eo/v)1/2 are used: 0.87, 2.18,4.36, and 8.72; they correspond to the solid symbols in figure 3.7 (b). The open symbols specifically correspond to the cases when S=0 (or t=0) so that the turbulence is still isotropic. The dimensionless values of aoI seem to collapse reasonably well. At large values of r/FI( )1/2, the curve becomes linear V as the collision rate is increasingly dominated by the mean shear so that aI scales linearly with F as Smoluchowski's theory predicts. A small degree of scattering exists for F/( )/2<2. This is caused by the 15% maximum variation of a11I over the whole range of total shear S, as already demonstrated in figure 3.7 (a). There are two possibilities for F/(& )1/2 1: i) both F and S=Ft are very small even comparing with (So/v)1/2 of the initial isotropic turbulence; ii) F is finite but S=Ft is large so that (E/v)1/2 become large comparing with (so/v)/2. In the first case, the turbulence is closer to the isotropic state _* * so that a11 1.2944. In the second case, the turbulence is highly anisotropic and a&, + 1.1676 for very large values of S. No attempt is made to correlate d& with the total shear S since S is not relevant in practical applications such as in turbulent pipe flow. _* Using ,11 =1.2944 in the isotropic state to represent the turbulence part of the collision kernel, a simple interpolation is proposed to represent the effect of mean shear and turbulence on the collision rate, 2.2 22 1.3333 22 1/2.2 ax 1 R3 (/v)1/2 1 {1.2944 + [ (/v)1/2 I 2 (3.40) This interpolation is shown by the solid line in figure 3.7 (b). A satisfactory agreement can be observed. It is also worth commenting that when F is normalized using the eddy 2 turn over time, 3u0 /&o, of the initial isotropic turbulence, the shearrate parameter F* F3u/o0, (3.41) corresponds to F*=36.98 and 92.45 for F/(Eo/v)1/2 =0.87 and 2.18. In a low Reynolds number turbulent channel flow, the maximum value of F* is around 35 at a location of 10 wall units away from the wall (Kim et al. 1987). Hence, the parameters used in figure 3.7 (b) are relevant to practical situations. Although a 10% error exists for a11 given by (3.40), the advantage of this simple approximation is that the total shear S=Ft is absent and it depends only on the spatial structure of the turbulence characteristics by the scalar quantity (e/v)"12. Figure 3.8 shows the radial collision velocity, <Wr>, as a function of F/( )1/2, for (V different values of F/(' )1/2. Similarly, the data seem to collapse approximately into one V curve in the range studied. The following interpolation fits the radial collision velocity well, <Wr> 1.7 1 r F 17 1/17 (.1 Reiv)/ {0.32451 + 1 ./v)12 } 41) (eiv'/ 6v~l2](. For the rootmean squared radial collision velocity, r that of <Wr>. While the data is not shown here, the following interpolation, which fits the data quite well, is provided, r____ 1 7 0.3381 F 17 1/1.7 3 R(e/v)"2 {0.3681 + (e/v) } (3.42) The application of the above expressions to turbulent channel flow, which is different from the homogeneous, rapidly sheared turbulence, is yet to be tested. However, since the rapidly sheared turbulence at large total shear S considered in this work is already drastically different from the isotropic turbulence, there are reasons to expect a similar dependence ofoa 1, <Wr>, and which is of significant industrial importance. 3.5 Summary and Conclusions A theoretical framework has been developed to evaluate the collision rate and collision velocity of small particles in general turbulent flows. The classical results for the collision kernels by Smoluchowski (1917) for laminar shear flow and by Saffman & Turner (1956) for isotropic turbulence are recovered. The present approach differs significantly from that of Saffmnan & Turner in that the ensemble average is taken after the collision rate for a given flow realization is calculated. This avoids the assumption of isotropy, as needed in Saffmnan & Turner, and allows for the evaluation of the collision rate in general turbulent flows. The collision velocity predicted in both laminar shear flow and isotropic turbulence agrees well with the direct numerical simulation for particle collisions. The present theory is used to evaluate the collision rate and velocity of small particles in the rapidly sheared homogeneous turbulence. It is found that the effect of the turbulence structure on the collision rate and velocity can be mainly captured by using the mean turbulent shear rate (F/v)112. The combined effects of the mean shear and the sheared turbulence on the collisions are elucidated. Simple interpolations are developed to evaluate the collision rate and velocity for arbitrary mean flow shear rate and turbulence mean shear rate. 1.4 1.3  1.2  I ........ ........ I. 10 1 100 102 S= Ft Figure 3.7 (a) Normalized particle collision kernel in a highly sheared homogeneous turbulence without contribution from the mean shear rate. * 0 * 0* 0* * * 1.1676 6 5 4 4 x ("" /v) 1/2 (S ) 3 0 = 0.87 elOx 2.18 2 "X x 4.36 8.72 1 0 S=O =0or t=0 Interpolation 0 1 2 3 4 (d/v) 1/2 Figure 7 (b) Collision kernel in a rapidly sheared homogeneous turbulence. cO denotes the dissipation rate of the isotropic state at t=0. 1.5 *** S1.0 ~F ^ >^ r 71/2 S 0C /V~ 0.5 0.87 2.18 X^5 x 4.36 8.72 0 S=0 or t=0  Interpolatior 0.0 ,,, i 0 1 2 3 4 F (E/V) 1/2 Figure 3.8 Collision velocity in a rapidly sheared homogeneous turbulence. c0 denotes the dissipation rate of the isotropic state at t=0. CHAPTER 4 EFFECT OF INERTIA ON THE PARTICLE COLLISION VELOCITY AND COLLISION ANGLE IN GAUSSIAN TURBULENCE Numerical simulation of particle collisions in isotropic, Gaussian turbulence is carried out to study the effects of particle inertia on the turbulence induced collision velocity and collision angle. An asymptotic analysis is developed to predict the effect of the small particle inertia on the collision velocity. A PDFbased method is also implemented to predict the effect of the small particle inertia on the collision velocity. The analysis predicts that the increase in the collision velocity is proportional to the terms (P3Tk)I and (0Ck )2, in which p1 is the particle response time and Tk is the Komogorov time scale. Good agreement is obtained among three different methods: the numerical simulation, asymptotic prediction, and the PDFbased method, for the collision velocity of particles with small inertia. The effects of both particle inertia and size on the collision angle have also been investigated. It is found that collision angle can be significantly affected by both inertia and size. Average collision angle and the variation of collision angle increase as the inertia increases, while the decrease in the size will also reduce both the average collision angle and the variation of collision angle. 4.1 Introduction Particle collisions can take place through a variety of collision mechanisms. Particles may be brought together by inertial centrifugal and gravitational forces. The velocity gradient is responsible for collisions among particles in laminar shear flow and is a dominant factor for particle collisions in turbulence. Brownian motion and Van der Waal forces can also affect the collisions. One of the common characteristics of these collision mechanisms is the ability to generate a negative relative velocity between two particles along the line connected with two particle centers. Of significance are the statistics of the relative velocity of two particles at contact (hereinafter referred as collision velocity) and the relative angle between the velocities of two particles at contact (hereinafter referred as collision angle). For solid particles, collision causes momentum exchange among particles and may influence the local and global particulate flow conditions. The suspended solid particles are produced during the manufacturing process of materials, the combustion of heavy fuel and pulverized coal. The collisions among suspended particles will greatly affect the properties of particulate flow and thus influence the performance of the systems. Adhesion may occur when two colliding solid particles have small collision velocity, while large collision velocity will produce large restitutive rebounce velocities for the colliding particles and may overcome the bonding force determined by cohesive surface energy. The collision also affects the angular momentum of particles. Tangential velocity components will be imposed to the colliding particles if the collision angle 012 is not equal to zero. The particle rotation will be either enhanced or reduced depending on the collision angle 012, which determines the orientation of the particle tangential velocities upon collision. Gunaraj et al. (1997) performed experiments measuring the collision angle 012 between heavy particles (silica sand particles with 10001500 gm in diameter) in turbulent channel flow. The collision velocity is also a critical quantity in determining the collision kernel and collision rate. In uniform shear flow, Smolochowski (1917) integrated the volume flow rate using collision velocity along the spherical surface of collision radius R (= ri + rj, where ri and rj are the radii of collision pair respectively) and obtained collision kernel for uniform shear flow. In turbulent flow, particle collisions are affected by local turbulence fluctuations. In the absence of any other force creating relative motion between particles, these local fluctuations entirely control relative velocity distribution which determines the collision kernel. In most of the previous theories, the collision kernel in a particleladen system is linearly related to the collision velocity, the first moment of the relative velocity distribution of colliding particles. Thus the scale and form of these particle relative velocity distributions directly determine the collision rate. The fact that the collision rate and collision kernel in previous studies are all proportional to the collision velocity explains the critical importance of the prediction of collision velocity (Saffinan & Turner 1956, Williams & Crane 1983, Kruis & Kusters 1996, et al). The particle collision has been studied in two different ways. One is based on Eulerian method, which is favorable to theoretical prediction and was followed by previous studies on the theoretical evaluation of both collision velocity and collision kernel. The other is to use Lagrangian tracking in direct numerical simulation of particle collisions. For inertialess particles in both uniform laminar shear flow and isotropic turbulence, the collection of collision velocities using Lagrangian method shows a biased average in favor of those regions where wr is high, as shown clearly in Chapter 3. Saffman & Turner (1956) obtained the collision velocity using average integration on the spherical collision surface by Eulerian method without considering the influence of the biased average. Although their formulation correctly predicts the velocity correlation between two points for turbulent flow, it has been incorrectly interpreted as the collision velocity upon collisions. The effect of particle inertia on the collision velocity was recently studied also by Sundaram and Collins (1997). They used two statistical variables: g(R), the particle radial distribution function at contact, and P(w/R), the conditional relative velocity probability density function at contact, where w is the relative velocity of two particles at contact. It is noted that in Sundaram and Collins' work, in additional to the assumption of the exponential form for P(w/R), the g(R) was evaluated based on their numerical extrapolation form for distances smaller than the first numerically evaluated point. However, the usefullness of g(R) and P(w/R) is rather questionable. In this work, we study the effect of particle inertia on increasing particle collision velocity and collision angle by carrying out numerical simulation of particle collisions in an isotropic, Gaussian turbulence, and developing an asymptotic analysis to predict the effect of small particle inertia on the collision velocity based on Lagrangian description. An integration method is also implemented to predict the effect of the small particle inertia on the collision velocity. The increase in the particle collision velocity due to inertia predicted by both the asymptotic analysis and integration method is in agreement with the numerical simulation results. The analysis predicts that the increase in the collision velocity contains terms that are proportional to (PT k) and (pxrk )2. Good agreement is obtained by the numerical simulation, asymptotic prediction and the integration method for the collision velocity of particles with small inertia. The effects of particle inertia and size on the collision angle have also been investigated. It is found that collision angle are significantly affected by both inertia and size. Average collision angle and the variation of collision angle increase as the inertia increases, while the decease in the size will reduce both the average collision angle and the variation of collision angle. 4.2 Saffman & Turner's Analysis In Saffman & Turner's (1956) prediction of the collision rate among inertialess particles, the volume flow rate of particles towards a target particle was evaluated by an integration, The integration was carried out over the entire spherical surface of radius R, as shown in figure 4.1. Because of the continuity of fluid flow, = Entire sphere <wrI> dS. (4.1) For isotropic turbulence, which is true at small scales for anisotropic turbulence, the above becomes where x is the local coordinate parallel to the line connecting the centers of the colliding particles as shown in figure 4.2 and < > denotes ensemble averaging. 63 A y SCollision area Sfor the target F particle x Figure 4.1 Sketch of particle collision scheme W = V V2 wx = VIx V2x Figure 4.2 Sketch of particle collision velocity and collision angles (012 & 0 J). The separation distance is x2 xi = R = (r, + r2). In the limit of zero particle inertia, particle velocity v follows the fluid velocity u very well so that the radial component of relative velocity w = vi v2 upon collision is Iwxllulx U2xlRl = R( 2 )/2 (4.3) Ox 157t v and Sax 15 v where wx is the collision velocity. For particles with inertia, the volume flow rate of particles to a target particle was expressed as Jffentire sphere Wl P(w)dw, where P(w) is the probability density function (PDF) of w and contains the effects of the spatial variations of velocity in the fluid, the turbulent accelerations due to the relative motion with the turbulence, and gravity. The volume flow rate can be integrated assuming P(w) to be a normal distribution function. For monodisperse particles, JJentire sphere wl P(w)dw = 2 R3 2 12 (4.5) 3 v Compared with the equation (4.2), the equivalent collision velocity based on equation (4.5) can be evaluated iWxl1 2R 3 ( 2rt)12/(2R2) = 2R(6_)l12 (4.6) 3 v 3 27tv and 9 v It is noted that equation (4.7) is independent of the inertia parameter 13. For the small inertia limit 13+oo, comparing equations (4.6) & (4.7) with equations (4.3) & (4.4), an inconsistency that (4.6) & (4.7) can not recover (4.3) & (4.4) occurs, which is physically incorrect. 4.3 Present Analysis 4.3.1 Asymptotic Analysis To understand and predict the effect of inertia on the collision velocity wx, it is assumed that the particle size is much less than the Taylor microlength scale, and the gravitational force is negligible. For simplicity it is assumed that turbulence is isotropic and Gaussian. The particle dynamic equation in the absence of(or weak ) gravity is dv dt =3(uv). (4.8) The above can be written as I dv vau (4.9) 13 dt For large values of P, repeatedly substituting the left hand side into the right hand side gives I du I d2V I Du I( ) Vu+ I d2v vu + u+(uv)UVu+ P3dt D2 dt2 P3Dt 13 1P2 dt2 I Du I  +()(4.10) f3Dt p3 The variance of w = vi V2x in which particle 1 and particle 2 are separated by a distance of I x2 x, I= R is 1 Duix 1 Du2x var(wx) var(Ulx U2x Du ) Pi Dt P2 Dt =<(Ulx U2x)2>1x2x=R +<(I Dux 1 Du2x )2> x=R P13 Dt P32 Dt x2xI=R 1 Dux 1 Du2x 2 <(Ulx U2x)( Dt ^ Dt)>lx2xl=R 01 Dt f02 Dt = term I + term II + term III. (4.11) The first term can be expanded as term I=<(ul U2x)2 >lx2x=R = =2 where u2 is the mean square intensity of turbulence and f(r) is the standard longitudinal velocity correlation coefficient. If the particle size is much less than the Taylor micro length scale X, the above can be simplified to term I= <(ul U)2 >x2x=R 2 (2 = IXR2v U~,1~1,R 2U "2X 15 v for R< I u1D,~ Di _!I(Dux)2>~(X term II <( L 2 )2 =R 2 <(DUI)> + 1 <(Du2t> D 'P1 Dt P2 Dt I x 1R p Dt / 2 vDt 2 1 13P2 Dt Dt x2x=+R44) where DIwhere < D U2X >1 X _X =R is the spatial correlation of ux in the xdirection with a Dt Dt x2 x iDt separation distance of R. Noting that var( Du) = var( DU2 ) and expanding the Dt Dt correlation for small R, we obtain DUIDU 2xI=R <(x )2>[l+R2 a2fD(o) + ] (4.15) Dt Dt 2 ax2 "'" where fD is the correlation coefficient of Dux. Defining the micro length scale of the Dt fluid acceleration Du" as Dt I I a2fD() (4.16) ?! 2 x2( term II can be expressed as term 1<(DUx)2>( l + 1 2 <(DUx)2>(1R2 Dt t F2 0P12 Dt 7 ) _<(DU)2>( I_ 1)2+ 2<(DU)2>R for R< (4.17) Dt 1 0i ) + 0 Dt)T f D Finally, equations (4.13) & (4.17) lead to var(wx) IlR2 +<(Dux)2>( 1 1 2 <(DU)2 R2 15 v Dt 01 02 P1P2 Dt TD 2<( u, )( DUX )>. (4.18) 1 Dt P2 Dt For the like particles, 31 = 02 = 1, I 2c 2 /Dux Du2 2 (Dux 2 R2 var(w.) 15R21  <(UxU2x)( )> <(__ > DDt Dt (i) (ii) (iii) (4.19) where term (i) is the meansquared collision velocity for inertialess particles, terms (ii) & (iii) are contributed by the inertia. The net increase of the meansquared collision velocity due to the inertia can be expressed as 2 AWD U lx Du2x 2 Dux 2 ( 2 SDt Dt
(ii) (iii) 