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Multi-Scale Computational Fluid Dynamics with Interfaces

HIDE
 Title Page
 Dedication
 Acknowledgement
 Table of Contents
 List of Tables
 List of Figures
 Abstract
 Introduction to multi-scale computational...
 Continuum model: Navier-stokes...
 Error assessment of the lattice...
 Lbe method for immiscible two-phase...
 Summary and future work
 References
 Biographical sketch
 

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1 MULTI-SCALE COMPUTATIONAL FLUI D DYNAMICS WITH INTERFACES By JIANGHUI CHAO A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLOR IDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA 2006

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2 Copyright 2006 by Jianghui Chao

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3 To my parents, my wife and my son.

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4 ACKNOWLEDGMENTS I would like to express my sincere gratit ude to Drs. Wei Shyy and Renwei Mei for providing me the opportunity and flexibility to perform this work. I can not thank them enough for being so patient and understanding fo r years and pushing me to learn more. I would like to thank Drs. Siddharth Thakur a nd Alireza Haghighat for agreeing to serve in my thesis committee. Since the surroundings dictate the quality of life in several ways, I thank my research group members for helping with academic aspects while providing memorable company for the past five years. I thank my wife who has been the best frie nd for more than a decade and has been extremely patient and understanding during all thes e years, and my son who has been giving me lots of happiness. Finally I thank my parents w ho have been behind me every step of the way providing their unconditional support.

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5 TABLE OF CONTENTS page ACKNOWLEDGMENTS...............................................................................................................4 LIST OF TABLES................................................................................................................. ..........7 LIST OF FIGURES................................................................................................................ .........8 ABSTRACT....................................................................................................................... ............11 CHAPTER 1 INTRODUCTION TO MULTI-SCALE CO MPUTATIONAL DYNAMICS WITH INTERFACES..................................................................................................................... .....1 1.1 Motivation................................................................................................................. ..........1 1.2 Objectives................................................................................................................. ..........3 1.3 Structure of the Dissertation.............................................................................................. .4 2 CONTINUUM MODEL: NAVIER-STOKES EQUATIONS WITH MULTI-SCALES........5 2.1 Introduction to Computational Technique s for Solving Navier-Stokes Equations............5 2.2 Incompressible Viscous Flow Solvers................................................................................8 2.2.1 Introduction............................................................................................................. .8 2.2.2 Artificial Compressible Method (ACM)..................................................................9 2.2.3 Semi-Implicit Method for Pressure-Linked Equations (SIMPLE)...........................9 2.3 Computational Issues on Multi-scale Computations........................................................12 2.3.1 Stiffness................................................................................................................ ..12 2.3.2 Grid Requirements for Multi-scale Problems.........................................................14 2.3.3 Methods to Reduce Stiffness..................................................................................15 2.4 A Special Computation Example: Thermo-MEMS Computation Results.......................19 2.4.1 Introduction to Thermal Anemometry for Fluid Velocity and Skin Friction Measurement................................................................................................................19 2.4.2 Governing Equations..............................................................................................23 2.4.3 Numerical Schemes................................................................................................25 2.4.4 Computational Stiffness.........................................................................................25 2.4.5 Geometry and Grid Layout.....................................................................................25 2.4.6 Results and Discussion...........................................................................................30 2.4.7 Summary and Conclusion:.....................................................................................36 3 ERROR ASSESSMENT OF THE LA TTICE BOLTZMANN METHOD FOR VARIABLE VISCOSITY FLOWS........................................................................................46 3.1 Introduction to Lattice Boltzmann Method and Wall Boundary Condition.....................48 3.1.1 LBE BGK Method..................................................................................................48 3.1.2 Boundary Conditions..............................................................................................51

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6 3.2 Error Assessment of the LBE Method due to Variable Viscosity via a Fully Developed Channel Flow....................................................................................................55 3.2.1 Fully-developed Laminar Channel Flow with Variable Viscosity.........................55 3.2.2. The Lattice Boltzmann Equation Treatment.........................................................58 3.2.3 Assessment.............................................................................................................59 4 LBE METHOD FOR IMMISCIBLE TW O-PHASE FLOW COMPUTATION...................73 4.1 Overview of Immiscible Two-Phase Flow Computation.................................................73 4.2 Literature Review on LBE Method for Two-Phase Flow Computation...........................76 4.3 He et al.s Isothermal LBE Model for Two-Phase Flow..................................................80 4.3.1 The Boltzmann Equation for Non-Idea Fluids.......................................................80 4.3.2 Lattice Boltzmann Scheme for Multiphase Flow in the Near Incompressible Limit.......................................................................................................................... ...85 4.4 Code Validation: Raylei gh-Taylor Instability..................................................................86 4.4.1 Linear Analysis of Ra yleigh-Taylor Instability......................................................87 4.4.2 Rayleigh-Taylor Inst ability: LBE Results..............................................................88 4.4.2.1 A Single-Mode Growth Rate........................................................................89 4.4.2.2 Flow Field at Re=2048 and At=0.5..............................................................90 4.4.2.3 Flow Field at Re=256 and At=0.5................................................................91 4.5 Lee-Lins Implicit LBE Two-phase Mode l and He et al.s Model with Surface Tension........................................................................................................................ ........91 4.5.1 Lee-Lins Scheme for Large Density Ratio............................................................91 4.5.2 Diffusion of Lee & Lins Implicit LBE model.......................................................94 4.5.3 Modeling the Surface Tension................................................................................96 4.6.1 Surface Tension Calculation...................................................................................97 4.6.2 A Filter-based Technique for Surface Tension......................................................98 4.6.3 Volume Conservation and Mass Conservation....................................................100 4.7 Numerical Simulations...................................................................................................104 4.7.1 Stationary Bubble.................................................................................................104 4.7.2 Capillary Wave.....................................................................................................107 4.7.3 Rising Bubble.......................................................................................................108 4.8 Summary and Conclusion...............................................................................................110 5 SUMMARY AND FUTURE WORK..................................................................................127 5.1 Summary.................................................................................................................... .....127 5.2 Future Work................................................................................................................ ....128 LIST OF REFERENCES............................................................................................................. 129 BIOGRAPHICAL SKETCH.......................................................................................................138

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7 LIST OF TABLES Table page 4-1 Effect of grid resolution on computed pressure drop for Laplace number 100. The data were taken after 100 tim e steps and the density and viscosity ratios were set to 100 and 10 respectively...................................................................................................112 4-2 Effect of Laplace number on the ratio of numerical pressure drop to theoretical pressure drop: Bubble diameter is 40 lattices. The data were taken at nondimensional time = 100. Density ratio is 100 and dynamics viscosity ratio is 10...........112 4-3 Effect of density ratio on pressure drop: Bubble diameter is 40 lattice units. Viscosity ratio was set to 10 for Laplace number = 100 and the data were taken at nondimensional time = 100....................................................................................................112 4-4 Effect of viscosity rati o on pressure drop: Bubble di ameter is 40 lattice units. Laplace number = 100 and density ratio is 100. The data were taken at nondimensional time = 100....................................................................................................112

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8 LIST OF FIGURES Figure page 2-1 Geometry of channel flow with solid substrate.................................................................37 2-2 Reduced-domain geometry................................................................................................37 2-3 Relative reductions of residuals of singl e-phase and conjugate cases with specified sensor temperature............................................................................................................. 38 2-4 Relative reductions of resi duals of conjugate cases with specified sensor temperature solved by single-grid an d multi-grid solvers.....................................................................39 2-5 Shear stress comparisons for the coarse grid and refine grid computations......................40 2-6 Relative reductions of Pressure and te mperature residuals on the refined grids...............41 2-7 Temperature contours with 5 0 sGr ...............................................................................42 2-8 Temperature contours with 10 sGr ..................................................................................42 2-9 Temperature contours with 100 sGr ................................................................................42 2-10 Temperature distribution on a cross-section which origin ates from the middle point of the sensor to the top b oundary of the reduced-domain..................................................43 2-11 u velocity profiles of conjuga te cases on the cross-sections..............................................44 2-12 Wall shear stress distribution............................................................................................ .45 2-13 Shear stress variation.................................................................................................... .....45 3-1 Boundary nodes and their neighbor s using the square lattice............................................67 3-2 A boundary cell using the hexagonal (FHP) lattice (Noble et al. 1995)............................67 3-3 Absolute L2 norm errors of LBE with Nobles scheme....................................................67 3-4 A 2D 9-velocity lattice (D2Q9) model..............................................................................68 3-5 Absolute error of a fully-developed channel flow using Inamuro et. al.'s scheme............68 3-6 Two set of viscosity distri butions used in this study.........................................................68 3-7 The exact velocity profiles of the channel flows with different boundary layer thicknesses due to different viscosity distributions...........................................................69

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9 3-8 Square lattice di stribution in channel flow simulation......................................................69 3-9 Comparison of the LBE velocity profiles..........................................................................70 3-10 Dependence of the relative L2 -norm error on the lattice size h in the fully-developed channel flow with variable viscosity..................................................................................71 3-11 Comparison of L BEexactLBEEuu with FDexactFDEuu for H=200, and 0.0005,0.0102 .......................................................................................................72 4-1 The growth rate (measured in units of 1/3 2/ g) of a disturbance vs. its wave numbers k...................................................................................................................... ...113 4-2 The growth rate plot....................................................................................................... ..113 4-3 Evolution of the fluid interface from a single mode perturbation for At=0.5 and Re=2048........................................................................................................................ ...114 4-4 He et als results of the evolution of the fluid interface from a single mode perturbation................................................................................................................... ...115 4-5 Density profiles across the bubble and spike fronts at three different time steps............116 4-6 Evolution of the fluid interface from a single mode perturbation...................................117 4-7 Evolution of index function of a stat ionary droplet with zero velocity...........................117 4-8 Evolution of the capillary number with for the stationary bubb le simulation with density ratio 2 and dynamic viscosity ratio 2...................................................................118 4-9 Density profiles and pressure profiles at t=0 and t=2000 of the stationary bubble simulation..................................................................................................................... ....118 4-10 Surface tension profile for a stationa ry bubble computed from He-Chen-Zhang method......................................................................................................................... .....119 4-11 Surface tension calculated by Kims formulation............................................................119 4-12 Rising bubble with Eo=0.971, Mo=1.26e-3, density ratio=100, vi scosity ratio=10........119 4-13 The velocity profiles and stream line of the spherical bubble at t=20 in 4-12.................120 4-14 Time evolution of the volume (A) and velocity (B) of the rising bubble with volume correction in Figure 4-13B...............................................................................................120 4-15 The contours of space derivative of hydropr essure and density of a rising bubble with Eo=0.971, M=1.26e-3, Re=5.19, density ra tio=100, viscosity ratio=10.........................121

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10 4-16 Stationary bubble computation. (a) S purious currents of magnitude Ca=5.03 x 10-3, (b) comparison between computed a nd theoretical pressure jump..................................121 4-17 Density profiles of the stationary bub ble with diameter 40, density ratio 100, viscosity ratio 10, and La=100.........................................................................................122 4-18 Maximum spurious velocities for these two grid resolutions..........................................122 4-19 Initial interface profile for a capillary wave simulation...................................................123 4-20 Time evolution of the amplitude of a capillary wave with density ratio 100..................123 4-21 Shape diagram of Clift et al. (1978).................................................................................123 4-22 Computed bubble shapes. (A) Cylindrical, (B) Ellipsoidal, (C) Dimpled-ellipsoidal.....124 4-23 Time evolutions of rising bubbles (A ) Cylindrical, (B) Ellip soidal, (C) Dimpledellipsoidal.................................................................................................................... .....124 4-24 Time evolutions of bubble velocities...............................................................................124 4-25 Density profiles of rising bubbles (A) Cy lindricall, (B) Ellipsoidal, (C) Dimpledellipsoidal.................................................................................................................... .....125 4-26 Time evolutions of a rising bubbl es with Eo=97.1, M=0.971 and Re=31.2....................126

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11 Abstract of Dissertation Pres ented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy MULTI-SCALE COMPUTATIONAL FLUI D DYNAMICS WITH INTERFACES December 2006 Chair: Wei Shyy Cochair: Renwei Mei Major Department: Mechanical and Aerospace Engineering Fluid flow and heat transfer problems involvi ng interfaces typically contain property jumps and length and time scale disparities, resulting in computational stiffness, demanding resolution requirements due to nonlinearity and multiple physical mechanisms. In this dissertation, both continuum and kinetic approaches are investig ated to address the multi-scale thermo-fluid problems, especially those invo lving interfaces. A continuum m odel based on the Navier-Stokes fluid model and the Fourier law for heat transfer was employed to investigate the conjugate heat transfer problem. The problem is motivated by recent advancement in the micro-electromechanical systems (MEMS) devices for shear stress measurement. Due to the length scale disparity and large solid-fluid thermal conductiv ity ratio, a two-level co mputation is used to examine the relevant physical mechanisms and their influences on wall shear stress. The substantial variations in tran sport properties between the fl uid and solid phases and their interplay in regard to heat transfer and near-wa ll fluid flow structures are investigated. It is demonstrated that for the stateof-the-art sensor design, the buoya ncy effect can noticeably affect the accuracy of the shear stress measurement. A lattice Boltzmann equation (LBE) model de rived from the kinetic consideration is investigated to address (i) error behavior due to variable viscosit y, and (ii) the interfacial fluid dynamics of some two-phase flow problems. It is shown that the boundary treatment error does not have a significant interaction with the truncati on error associated with variable viscosity, and

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12 the LBE model closely matches the Navier-Stoke s model for fluid flows with large viscosity variation. Next, an improved interface lattice Boltz mann model is developed for two-phase interfacial fluid dynamics with large property vari ations. In order to suppress numerical instabilities associated with the presence of the interface, the following approaches were investigated: (i) a new surface tension formulation originated from the diffusion interface method was used to remove unphysical pressure wiggles across interfaces; (ii) a filter scheme was used in numerical gradient calculations to mainta in monotonic property vari ations; (iii) a volumecorrection procedure was devised. The performa nce of the improved LBE model was evaluated using the Rayleigh-Taylor instability proble m, stationary bubble under force equilibrium, capillary waves, and rising bubbles. The computati onal results demonstrated that this LBE twophase model is more robust than t hose reported in th e literature, capable of treating larger density ratio up to order O (102 ), while confining the interface thickness within 5-6 grids.

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1 CHAPTER 1 INTRODUCTION TO MULTI-SCALE CO MPUTATIONAL DYNAMICS WITH INTERFACES 1.1 Motivation Many fluid mechanics and heat transfer problems involve multiple scale phenomena, such as turbulence, multiphase flows, and conjugate heat transfer. The presence of the disparity in le ngth, time, and velocity scales is caused by the presence of different competing mechanisms, such as convection, diffusion, chemical reaction, body forces, and surface tension. Furthermore, these mechanisms are often coupled and nonlinear (Shyy et al. 1997b). Mathematically, issues related to multiscale problems can be illustrated by a system of ordinary differential equations whos e characteristic matrix has a large range of eigenvalues. These large range eigenvalues ph ysically represent different growth and decaying rates of different competing mechan isms. Numerically they make computations very sensitive to the stability of the numerical schemes (Lambert 1980; Shyy 1994; Lomax et al. 2001; Chapra and Canale 2002 ; Deuflhard and Bornemann 2002). If the linear algebraic equations discretized from the original stiff ODEs are written as (Miranker 1981) I hAxb (1.1) where I denotes a m-dimensional identity matrix, is a constant, h is integration step size, A is the matrix that has large range of eigen values, x is solution vector, and b is source term vectors that can include initial/boundary condi tions and other non-

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2 homogeneous sources, the condi tion number of matrix A is defined as (Chapra et al. 2002) 11 maxminCond AAA (1.2) where max and min are the smallest and la rgest eigen values of A respectively. From Eq. (1.2), it can be seen that the nume rical solution process can be ill-conditioned (Miranker 1981), which brings in computa tional difficulties in both stability and accuracy. These kinds of problems are thus called stiff problems. In order to understand the physical proce sses involving different scales, separate scaling procedures for each physical regime and the coupling need to be devised. However, different multi-scale problems invol ve different mechanisms and thus require different treatments. Generally, two classes of multi-scale problems exist. The first one is that the small scales are re stricted in distinct regions in space. Thus, the problems involved in this class require the matching and patching of the solutions in large and small scale domains. In the second class, the di sparate scales co-exit over the entire flow field and interact with each other. Resolving or modeling such disparate scales is the task of this class of multi-scale problems (Shyy et al. 1997a). At present, there are two major computat ional fluid dynamics solvers for modeling incompressible, viscous fluid flows. One is the well-established Na vier-Stokes equations solver. Another is the Lattice Boltzmann equa tion (LBE) solver (He et al. 1997; Chen et al. 1998; Succi 2001; Yu et al. 2003a). For mu lti-scale problems in fluid dynamics and heat transfer, if the continuity assumption is valid, all scales can be represented by Navier-Stokes equations. The issue in this ty pe of problem is how to resolve all the relevant scales based on Navier-Stokes equations.

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3 As an alternative computational fluid dyna mics solver, the LBE method offers a meso-scale framework for fluid dynamics com putations. It recovers the macroscopic fluid flow solution based on averaging of the particle distribution functions, obtained by solving the simplified form of the Boltzma nn equation. Despite some difficulties in compressible flows and heat transfer, the LB E method has been successfully used for isothermal complex flows. For interfaci al fluid dynamics, however, the numerical stability issues still exist from the large pr operty jumps and scale disparities in normal and tangential directions of interfaces. A study on how to stabilize the LBE two-phase computations for flows with large density ratio and prevent inte rface thickness from diffusion is still needed. In this dissertation, we will focus on th e interfacial problems solved by NavierStokes equations and lattice Boltzmann equati on since interfaces always result in multiscale processes (Shyy 1994; Shyy et al. 1997c) due to the sharp changes in the material properties across the interface. Because of the physical nature and numerical issues of multi-scale problems, they are very challenging and nontrivial. 1.2 Objectives One of the main objectives of the presen t research is to develop a two-level computational method based on the Navier-Stoke s equations solver to simulate a fluid flow and heat transfer problem involving a fluid-solid interface. Another main objective is to simulate a liquid-gas two-phase flow problem by using the lattice Boltzmann equation (LBE) method. Specifically, the objectives are listed below: To investigate the multi-scale issues involv ed in the fluid flow and heat transfer surrounding a MEMS-based thermal shear stress sensor; To develop an efficient, accurate numerical method to simulate the fluid flow and heat transfer surrounding the MEMS-based thermal shear stress sensor;

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4 To investigate the performa nce of the lattice Boltzmann equation method when it is applied to fluid flows with rapidly varyi ng viscosity over a thin boundary layer that is a typical multi-scale phenomenon; To develop a lattice Boltzmann method for im miscible liquid-gas tw o-phase flows with large density ratio. 1.3 Structure of the Dissertation The numerical techniques for solving th e Navier-Stokes equations are briefly presented in Chapter 2. The multi-scale and stiffness issues involved in the models represented by Navier-Stokes equations are pr esented. Finally, in Chapter 2, a two-level numerical technique is developed to simula te a multi-scale physical problem---the fluid flow and conjugate heat tran sfer surrounding a MEMS-based th ermal shear stress sensor. In Chapter 3 the method of Lattice Boltzmann equation (LBE) is briefly described, followed by the error assessment of the LBE method for variable viscosity. In Chapter 4, an improved interface latti ce Boltzmann model is developed for twophase interfacial fluid dynamics with la rge property variati ons. Some numerical approaches are investigated to suppress numer ical instability in the computations for large density ratio flows and to prev ent interface thickness from diffusion. This dissertation ends with Chapter 5, wh ich contains a summary and conclusion of the contributions of the present work, and the recommendations for future research efforts.

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5 CHAPTER 2 CONTINUUM MODEL: NAVIER-STOKES EQUATIONS WITH MULTI-SCALES 2.1 Introduction to Computational Techni ques for Solving Navier-Stokes Equations Before addressing the multi-scale flows desc ribed by the Navier-Stokes equations, it is necessary to introduce the differe nt numerical methods for fluid flow and heat transfer since different governing equation features requir e different numerical methods, and different numerical methods have different advantages on different flow and heat transfer problems. Generally, Navier-Stokes equations solvers can be classified into two groups: one is the pressure-based formulation, and another is the density-based formulation. The pressure-based formulation is often used for flows without disc ontinuities, such as inco mpressible flows without shock waves. In contrast, the density-based formulation is often used for flows with discontinuities, such as supersonic/hypers onic flows (Chung 2002). The algorithms for compressible and incompressible flows are differe nt, which can be understood from the role of pressure in compressible and incompre ssible flows (Moukalled et al. 2001). If fluid compressibility is zero (in other words, Mach number is zero), the thermodynamic pressure loses its sense for incompressible flows. Pressure gradient becomes a balance force to viscous, inertial and other body/surface forces. Ther efore, no state equati ons exist for pressure. In addition, because density does not change locally for incompressible flows, continuity equation can no longer be considered as the governing equation for density. Thus no explicit governing equation for pressure exists for inco mpressible flow. Also, for incompressible flow, the speed of pressure wave is infinity. If the pressure oscillation propaga ting at a finite speed could not be removed, then velocity obtained from the momentum equation could not satisfy continuity equation (Patankar 1980; Blosch et al. 1993; Shyy 1994). Therefore, for

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6 incompressible flow, mass conservation is satis fied through the driving action of pressure gradients on velocity. For compressible flow the continuity equation is never a function of ve locity alone but also includes the density variation. Fo r compressible flow with differe nt Mach number, the role of pressure is also different. In the hypersonic limit, velocity variation is relatively smaller than the velocity itself. Thus, pressure does not influence the mass c onservation very much through the velocity variation by the moment um conservation as compared to through the density variation by state equation. Therefore, pressure can be c onsidered to act on density through the state equation, and then the mass conservation is kept through the continuity equation. If Mach number is not as high as that of a hypers onic flow, the pressure has dual roles on mass conservation through the state equation and th e momentum equation. For subsonic flow, mass conservation is more readily satisfied by th e pressure acting on the velocity through the momentum equation than by the pressure act ing on the density through the state equation. However, for supersonic flow, mass conservation is more readily sa tisfied by the pressure acting on the density through the state equation than th e pressure acting on the velocity through the momentum equation (Moukalled et al. 2001). The different roles th at pressure plays in mass conservation can be thought of as the velocity scale disparity in the physical processes. Due to the different roles of pressure on th e mass conservation for incompressible flow and compressible flow, the pressure-based methods are originally designed for solving the incompressible flow and the density-based met hods are originally designed for solving the compressible flow. Also due to the dual roles of pr essure, pressure-based methods can be extended to solve a compressible flow (Shyy 1994) and the concepts of density-based NavierStokes solvers were borrowed to solve incompressibl e flows, such as the artificial compressible

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7 method (Chorin 1967) and preconditioning method (Turkel 1999) if the stiffness in the convection coefficient matrixes could be removed or reduced appropriatel y. If a single method is needed to simulate fluid flow at all speeds, the pressure-based method and preconditioning method are two popular frontiers. The artificial comp ressible method is seldom used for all speed flow solvers because of the stiff solution matrix which degrades convergence rate (Moukalled et al. 2001). Among the incompressible solvers, there is another popular kine tic method, called the lattice Boltzmann equation (LBE) method. In th is method, only one variable, the distribution function, is solved. Distribution f unction represents the mesoscopic information of the fluid flow. Navier-Stokes equations can be recovered from lattice Boltzmann equation on the condition of nearly incompressible assumption. Macroscopic variables can be obtained from the moment integrations of distribution function (Chen et al. 1998). The governing equation, namely the lattice Boltzmann equation, is linear (the non linear feature is hidden in the equilibrium distribution function) and whole co mputational process in one time step is localized. These give the LBE method an excellent capab ility to parallelize the computations. Also due to the simple boundary condition schemes that the LBE has, the LBE method is very easy to use for fluid flows with random complex geometri es (Chen et al. 1998; Succi 2001). In this dissertation, incompressible viscous flow solvers are used to examine the multiscale computational fluid dynamics problems with interfaces. Compressible Navier-Stokes solvers could be found in typica l computational fluid dynamics te xtbooks (Hirsch 1990; Fletcher 1991; Shyy 1994; Anderson 1995; Anderson et al 1997; Chung 2002; Ferziger et al. 2002).

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8 2.2 Incompressible Viscous Flow Solvers 2.2.1 Introduction Incompressible flow is a physical model of ge neral fluid flow on the condition of a very small Mach number. The time scales of the convec tion velocities are much larger than the time scale of sound speed as the Mach number becomes very small. If Mach number equals to zero, new issues are brought in for incompressible flow computations. As introduced in the previous section, the pressure waves have infinity propaga ting speed in incompressible flows, which leads more elliptic features to the gove rning equations, especially for the pressure fields (Kiris et al. 2003). Because of this characterist ic of the governing equations, the computational schemes have to be designed to couple the continuity and mo mentum equations through pressure in order to keep the pressure field from oscillating, by whic h the conservation of mass can be preserved as the sound speed becomes much higher than the convection velocity components. In dealing with incompressible flows describe d by the Navier-Stokes equations, two major approaches are generally used: the primitive variable methods and the vortex methods. The purpose of vortex methods is to remove the difficu lty of solving accurately the pressure field in incompressible flows. However, if some complex ities, such as chemical reaction, are needed to be added to the fluid flows, the primitive va riable methods can easily accommodate these requirements by adding additional equations to the stack (Blosch et al. 1993). The primitive variable approach includes th e artificial compressibility me thod (ACM) (Chorin 1967) and the pressure correction methods (PCM ). Pressure correction methods include the marker and cell (MAC) method, the semi-implicit method for pre ssure linked equations (SIMPLE) (Patankar 1980), and the pressure implicit with splitti ng of operators (PISO) (Issa 1985).

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9 2.2.2 Artificial Compressible Method (ACM) The incompressible Navier-Stokes system of eq uations are written in non-dimensionalized form as Continuity Equation: 0i iu x (2.1) Momentum Equation: 2 21 Re j iii j jiuuu p u txxx (2.2) In the ACM method, an artificial compressibility term is adde d into continuity equation as following 0i iu tx (2.3) where is the artificial density, which is equal to the product of artificial compressibility factor and pressure, 1 p (2.4) When steady state is reached, th e artificial density derivative with respect to time vanishes. How to choose the artificial factor is the key for ACM (Kwak et al. 1986). has to be maintained low enough (close to the convective ve locity) to avoid stiffn ess associated with a large range of eigen value magnitudes. But it has to be kept high enough su ch that the speed of sound can be as large as possible to achiev e the incompressible requirement (Chung 2002). 2.2.3 Semi-Implicit Method for Pressu re-Linked Equations (SIMPLE) The SIMPLE method was designed or iginally to solve the steady-state fluid flows. In this method, the velocity and pressure are solved se quentially in one iteration (which is called the

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10 outer iteration, the corre sponding inner iteration is the one fo r solving algebrai c equations). How to solve the pressure field is the key in this me thod. In one outer iteration, if pressure is known (which could be given as a guess for initial condition or obtained from previous iteration), velocity could be solved fr om the momentum equations. Th e velocity obtained from the momentum equation might not sati sfy continuum equation. This re quires correcting the pressure, and further correcting the velocity to make the continuum equation satisfied in this outer iteration. If we denote u, v, and p as the velocity and pressure which satisfy the momentum equations, the predictor-corrector procedure with successive pressure correction steps is given as p pp (2.5) where p is the actual pressure, and p is the pressure correction. The actual velocity components in two-dimensions are 'uuu (2.6) 'vvv (2.7) Since u, v, and p satisfy momentum equation, u, v and p also should satisfy the momentum equation, we have ''''eeenbnbnbPPEEeauuauubppppA (2.8) eenbnbPEeauaubppA (2.9) The subscriptions in these equations are same as those in Partankars book (Patankar 1980). Subtracting (2.9) from (2.8), we have '''' eenbnbPEeauauppA (2.10)

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11 From (2.10), it is clear that the velocity co rrections on any computational point have two components, one is induced from the difference of pressure correction on the neighboring points and the other comes from the velocity corrections of its neighboring points The former source of velocity correction, the difference of pressure co rrection on neighbor points, is the major factor. If the neighbor velocity correc tion impact is neglected (this is why this method is called semiimplicit method), the velocity correction equation (2. 10) could be written as ''' eePEeauppA (2.11) ''''' e ePEePE eA uppdpp a (2.12) Likewise, the velocity correction component in v direction is given as ''''' n ePNnPN nA vppdpp a (2.13) Substituting (2.12) and (2.13) into (2.6) and (2.7), and then substituting (2.6) and (2.7) into the discretized continuum equation, the pressure correction equation c ould be obtained as PPEEWWNNSSapapapapapb (2.14) EeeWwwNnnSssadyadyadxadx P EWNSaaaaa 0 PP wesnxy buuyvvx t Since the source term b represents the mass imbalance based on the velocity in the previous iteration, if its value is small enough, mass c onservation could be taken to be satisfied. Therefore, the value of b could be the criteri on for stopping iteration.

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12 Neglecting the velocity corre ction from neighboring points in the velocity correction equation does not influence the fi nal steady-state solution. In this sense, the iteration procedure for the pressure can be simplified such that it re quires only a few iterations at each time step. SIMPLE method has been modified to seve ral versions. People call them as SIMPLE family methods. These modified methods accelera te solving the pressure correction equation which is elliptic type equation. The detailed descriptions of these methods could be found in many CFD textbooks (Patankar 1980; Fletcher 1991; Shyy 1994; Ferziger et al. 2002). 2.3 Computational Issues on Multi-scale Computations 2.3.1 Stiffness Multi-scale computations in fluid dynamics a nd heat transfer means that widely different time or length scales phenomena need to be capt ured with expected accuracy. As described in Chapter 1, different scales ar ouse from a wide acting range of different forces in dynamics. These forces can interplay with each other such that the small scale pert urbations could amplify and propagate into large scale regions. Even th ough, for some cases, the small scale impacts can be restricted in sma ll local regions without amplification and propaga tion, the local phenomena restricted by the large scale dynamic mechanisms are sometimes interested, such as the heat and momentum transfer mechanisms surrounding a h eating element of hot-wire/film whose length scales are generally much smaller than the meas ured large scale fluid flows. These kinds of measurement devices are preferred because of thei r small sizes such that large scale fluid flows are disturbed as little as possible, and the measurement could be more accurate. Numerically, multi-scale problems are relative to stiff issue if the physical problems are modeled with sufficient accuracy by a coupled set of ODEs and these ODEs have unique and bounded solutions (Lambert 1980; Lomax et al. 2001; Deuflhard et al. 2002). Generally, the ODEs can be expressed as

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13 du Auft dt (2.15) where u is the dependent variable vector, t is the independent variable. If A is independent of t, Eq. (2.15) is linear, otherwise nonlinear. The second term on the right hand side is a forcing function which is determined by the inherent f eatures of dynamic system (Kinsler et al. 2000). The difference between the dynamic scales in physic al space is represented by the difference in the magnitude of the eigenvalues in the eigenspace of A. The forcing function can have its own scales. Usually its scale could be adequately re solved by the chosen computational step size. Thus, it is reasonable to assume that the forcin g function has no effect on the numerical stability of the homogeneous part in one computational step. However, fo r nonlinear problem, the matrix A is dependent on t. If the time scale of f t is needed to resolve, it has a limit on the discretized size of t which has an impact on the eigenvalues of A because the entries of A depend on the discretized size of t. This can be observed when simula ting complex flow phenomena such as turbulence, possibly in combina tion with chemical reactions. In these processes, additional source terms can be added into equations. The nature of these source terms results in an increasing stiffness of these pr oblems, which can reduce the conve rgence rate (St eelant et al. 1994). The major feature of stiffness is the interaction of computational accuracy and stability. Let us assume the independent variable t in Eq. (2.15) is time. The ODEs could be solved by using time-marching methods. In one time step the integration with respect to t is an approximation in eigenspace that is different for every eigenve ctor. Numerically, the eigenvectors corresponding to the small eigenvalue could be well resolved. In contrast, the eigen vectors corresponding to the large eigen values are no t (Shyy 1994; Lomax et al. 2001) The approximation to the eigenvectors associated with large eigenvalues re quires stable numerical schemes for the global

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14 computation procedures and the accuracy represente d by these eigen vectors has to be kept from ruining the complete solution. The wide range eigenvalues of multi-scal e problems bring string requirements on numerical schemes in terms of accuracy and stab ility. Accuracy and stability are always two contrary requirements for numerical schemes. As a result, the physical nature of multi-scale problems and the numerical nature of accur acy and stability make multi-scale problems nontrivial. Besides, grid requirement is anothe r factor making multi-scale problems harder to resolve. 2.3.2 Grid Requirements for Multi-scale Problems If the physical models accurately represent the physical processes, and the numerical schemes can be appropriately selected to cap ture the physics, the grid arrangement will determine the accuracy of solutio ns. Multi-scale phenomena in flui d flow and heat transfer are generally characterized by a coupl e of regions in which the spat ial gradients of the dependent variables are much higher than in other regions. Consequently, higher spatia l grid resolutions are required in those high gradient regions, and a lower grid resolution in others. This grid clustering can strongly affect the eigenvalues of the matrix A since A is calculated directly in terms of the grid sizes. This grid effect on the eigenvalues of the matrix A can be illustrated by the diffusion equation shown below (Lomax et al. 2001). The diffusion equation is 2 2uu tx (2.16) Using the three-point central di fferencing scheme to represent the second order derivative in space leads to the following ODE diffusion model 21,2,1 du B ubc dtx (2.17)

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15 with Dirichlet boundary conditions folded into the bc vector, where1,2,1Bis a banded matrix 21 121 1,2,1 121 12 B (2.18) The eigenvalues of the matrix B are 2 24 sin1,2,, 21mm mM xM (2.19) The stiffness ratio for this diffusion equation is 2 2 2 24 4 2 4Max r Minx x C x x (2.20) From this ratio, it is clearly seen that the sma ller the grid is, the stiffer the numerical equations would be (Atkinson 1989; Lomax et al. 2001). 2.3.3 Methods to Reduce Stiffness In order to achieve the expected accuracy for multi-scale problems, the truncation errors should be kept as small as possible. Reducing the grid size is a way to achieve a smaller truncation error if the adopted numerical schemes are consistent and stable. However, multi-scale problems often involve large gradients in spa ce or/and in time. For large time derivatives problems, one might suppose that the adaptive time step-size routines might offer a solution; that is, using small time steps during the rapid transients and large time steps otherwise. But, that is not the case in numerical practices. The reason is that the stability requ irement still necessitates very small steps over the enti re solution (Chapra et al. 2002) Therefore, A-stable methods

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16 (generally they are implicit methods) are preferred (Lambert 1980; Shyy 1994). For large variable gradients in space due to multi-length-scal es, the errors arising from insufficient grid resolution is a major c ontribution among the many types of er rors included in the overall solution (Celik et al. 2004). In such problems, grids should be clustered at locations where large gradients exist so that the truncation errors could be gene rally kept as small as possible and computational expenses could be reduced (Fletcher 1991; A nderson 1995; Ferziger et al. 2002). Besides clustering grids at the known loca tions where large gradients exist, adaptive meshing algorithms could be used to solve more complicated probl ems in which the locations of large gradients might not be known in advance (Bank 1990; Ande rson 1995; Park et al. 2004). However, the difficulty with adaptive meshing method lies in that finding the solution of the linear system, as the grid points, that are added for grid refineme nt are globally combined with the regular mesh (Park et al. 2004). The resulting linear system ma y also lose the banded structure or the positive definiteness of the uniform grids (Golub et al. 1996). If the physical geometries are very compli cated and the length scales have wide distributions in the computational domain, the inte nsity of clustering grid might be very severe. The aspect ratios of some grid cells might be ve ry large in some sub-re gions. Such computational stencils have poor grid quality si nce the order of truncation errors for such clustered grid can be lower because some terms in truncation errors c ould not be cancelled off due to non-uniform grid sizes. As a result, if the variati on of aspect ratio of computational stencils could not be kept as small as expected (generally 10% variation is accepted for practical computations suggested by Ferziger et al. (2002)), the trunc ation error could be larger than acceptable. Thus, for such multiscale problems with large space gradients, furt her improvement for reducing truncation error is necessary.

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17 One remedy is to use multi-level technique (Shyy et al. 1997a; Appukuttan et al. 2003b). This technique includes a global computation ma de for providing global information in the whole computational domain, which may not reso lve the small scale phenomena because of the low resolution in the vicinity of the small s cale process regions, and then a reduced domain computation, which includes the small scale proce ss regions, is carried ou t on a higher resolution gird with the boundary conditions extracted from the global comput ations. In this method, the length scale of the reduced domain is smaller than the global domain and larger than that of the small scale processes. The globa l computation can provide accura te boundary conditions for the reduced domain computation if the small scale pro cesses are mainly restricted to the interested regions, and their variations do not affect the fl ow and heat processes in the regions far away from them. In such cases, the reduced domain boundaries could be selected at such locations where no great impact comes from small scale regi ons. By this way, the length scale disparity could be reduced. Good grid qual ities for both global and reduced -region computations can be acquired. If small scale processes could be amplified and propagated over the global domain or if the physical geometries are very complex, a single gl obal grid of multi-level-grid technique may not be capable of accurately capturi ng the global phenomena. As a resu lt it can not provide accurate boundary conditions for the reduced-region co mputations. For such problems Domain Decomposition Methods (DDM) or multiblock techni ques can be used as alternative method to reduce the error due to grid re solutions (Shyy et al. 1997b). Unlike in multi-lev el method that has two separate computing processes, in multib lock methods, the computational domain is partitioned into several blocks according to differe nt physical scales in different blocks and the computational variables are solved simultaneous ly (without block loop (Thakur et al. 2002)) or

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18 iteratively (with block loop (Shyy et al. 1997b)). By reasonable domain partition, multiblock techniques can reduce the topolog ical complexity. Each block can be generated independently. Thus, grids can have expected hi gh resolutions in the required re gions. Also, grid lines do not need to be continuous across block interfaces. This makes the local grid refinement and adaptive redistribution easy to accommodate different physical length scales that exist in different regions. The poor quality of single gr ids on such complex problems is thus improved by block arrangement. Compared to the multi-grid-layout method, multiblock method can deal with more complicated problems, but, involves more work than multi-level method, in terms of specifying the block interface boundary conditions, data st ructure, interface region size effect on data exchange and convergence rate (Shyy 1994; Shyy et al. 1997b; Thakur et al. 2002). For very complex physical and geometrical problems, parallel computational techniques using on multiblock grids can improve convergence rate (Golub et al. 1993; Shyy et al. 1997b). Parallel computing techniques depend on both hard ware architecture (parallel architecture) and software (parallel algorithms). The above techniques are very useful fo r multi-scale problem computations. Block partition is determined in terms of different scale phenomenon regions, which is generally reflected by local dimensionless parameters in Navier-Stokes equations, such as Reynolds number. Grid characteristics (Grid st retching and clustering ) also yield to diffe rent physical scale distributions. In pressure-corre ction algorithms, the inner loop /iteration (solving algebraic equations) has a big impact on the outer loop /iteration that couples the unknown variables through conservation equations. The convergence ra te of the inner loop can influence the outer loop substantially as the number of grid poi nts increases (Shyy et al. 1997b). It is well-known that the eigenvalues of the algebraic equations of multi-scale problems might have a wide range

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19 of values (Lomax et al. 2001). Typical iterati on methods, such as point-Jacobi iteration on a single grid, do not have fast c onvergence rate for such equations. While, multigrid methods have the capability to acceler ate solving these kinds of equati ons (Shyy 1994; Shyy et al. 1997b; Tao 2001; Chung 2002). The large range of eigenvalues can be represented by the large range of frequency in terms of grid size s in Fouriers analysis. High fr equency error components damp off faster on fine grids. Multigrid methods resolv e different frequency error components on multilevel grids so that the low frequency error comp onents on fine grids become high frequency error components on coarse grids and they can be damped off faster on coarse grids. Because of this uniform (in terms of frequency) solution error treatment, multigrid methods are highly efficient to solve physical problems with disp arate multi-scales (Shyy et al. 1997b). 2.4 A Special Computation Example: Thermo-MEMS Computation Results In this section, a multi-scale problem with a sharp fluid-solid interface described by the Navier-Stokes is investigated. This problem is a conjugate heat transfer type resulting from a Micro-electromechanical Systems (MEMS)-based on thermal shear stress. In this problem, multi-scales come from (1) length scale disparity of characteristic length scale of a channel flow and the size of the MEMS-based th ermal shear stress sensor, and (2) large conductivity ratio of fluid to solid substrate on which the MEMS-based sensor is deposited. Before addressing the multi-scale issues in this simulation, an introduc tion about thermal anemometry and its issues are presented below. 2.4.1 Introduction to Thermal Anemometry for Fluid Velocity and Skin Friction Measurement Thermal anemometry has been widely used to measure the fluid velocity and the skin friction for decades (Winter 1977; Perry 1 982; Bruum 1995). All thermal sensors are temperature-resistive transducers that essentially measure heat-transfer rate. They are indirect

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20 sensors and thus require an empiri cal or theoretical corre lation, valid for very specific conditions, to relate the measured Joulean heating rate to the flow parameter of interest (i.e., velocity or wall shear stress). All indirect therma l shear-stress sensors possess se veral limitations when used for quantitative measurements. Specifically, they ar e limited by the difficulty in obtaining a unique calibration or relationship between heat transfer a nd flow parameters of in terest. In addition, the frequency-dependent conductive heat transfer into the support substrate (i.e., prongs for a hot wire and supporting substrate for a hot film) redu ces the sensitivity and complicates the dynamic response. Thermal sensors are also sensitive to the mean temper ature variations which can be difficult to correct. Finally, the thermal sensor is heated well above the ambient temperature during the operation, which can potentially influence th e near-wall flow structure and introduce measurement error. A numerical simulation can help interpret th e sensor output and co rrect the calibration formula. For example, Durst et al. (Durst et al. 2002) reported a numer ical study of conjugate heat transfer effect on the near-wall hot-wire co rrection. They found that the influence of the local length scale (or Reynolds number), Y+, on the heat transfer proc ess adjacent to hot-wire is very important, especially when the thermal conductivity of the wall is very small, such as that of glass or perspex. In the above, Y+ is defined as Y u Y (2.21) where u = w is the friction velocity, and Y is the wire-to-wall distance. Shi et al. (Shi et al. 2003) conducted two-dimensional numerical simulations of for ced convection from a microcylinder in a laminar cross-flow. Their numerical results showed that the heat diffusion from the wire is pretty large in the cas e of small wire-to-wall distance ( Y+<3). The reason is the

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21 modification of thermal boundary condition at the fluid-wall interf ace caused by property variations between phases. Thermal-based skin friction sensors, which rela te the convection from a thin heated film deposited on a substrate to the local wall shear stress, are limite d by the heat c onduction into the supporting substrate as well as the flow perturbations due to loca l fluid heating (Naughton et al. 2002; Sheplak et al. 2002; Appukuttan et al. 2003a ). Recently, researchers have developed silicon micromachined sensors because of the po ssibility of improved thermal isolation of the sensing element from the substrate by depositing the sensor on a thin membrane stretched over a vacuum cavity (Ho et al. 1998; Sheplak et al. 2002). While the vacuum cavity structure greatly improved the performance in terms of sensitivity with respect to conventional sensors, MEMSbased thermal sensors still suffer from the inherent limitations of localized flow heating and heat conduction into the supporting membrane (Naugh ton et al. 2002; Sheplak et al. 2002). The inherent small size of MEMS-based th ermal sensors invalidates the classical 1/3 power law of the hot-film theory, i.e., 3 / 1 wNu (Lin et al. 2000), which assumes that the thermal boundary layer resides entire ly within the linear region of the velocity profile and that the boundary layer approximation holds for the ener gy equation (Lin et al. 2000). Because of the small size of the MEMS-based sensors, diffusion can substantially affect the fluid flow and heat transfer in the vicinity of the MEMS-based thermal sensors (H o et al. 1998). Lin (Lin et al. 2000) thus included 2D heat conduction of fluid flow with unidirectiona l convection, re sulting in the following energy equation in the fluid phase: 2 2 2 2y T x T x T u (2.22)

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22 In addition, one-dimensional heat conduction in a solid membrane was employed to account for the conjugate heat transfer effect. Yoshino et al. (Yoshino et al. 2003) conducted a numerical analysis of frequency response of micro thermal flow sensor. In their simulations, no buoyancy force was taken into account and a lin ear fluid velocity was given. The heat conduction in the solid substrate was considered in their numerical simu lations. They found an optimum diaphragm size to frequency response for their thermal sensor, which is 200-300m long. Recently, mixed convection induced by ME MS-based thermal shear stress sensor was studied by Appukuttan et al (Appukuttan et al. 2003a). They disc ussed the flow and heat transfer surrounding a thermal shear stre ss sensor embedded on a wall of a channel. Buoyancy force effects induced by the thermal sensor on the shear stress were examined for different Grashof numbers. As they pointed out, buoyancy force has little impact on th e whole-domain flow structure in the channel, while its impact on shear stress measurement can be noticeably observed because the sensor performance depends on the near-wall velocity gradients. In the work of Appukuttan et al's, heat conduction in the solid substrate on which the thermal sensor is deposited was not considered. However, as Naught on and Sheplak stated in their review paper (Naughton et al. 2002), the heat co nduction in the substrate can noticeably influence the heat transfer process. In the above numerical studies on MEMS-based thermal sensors (Lin et al. 2000; Yoshino et al. 2003), velocity profiles were given. On ly energy equations were solved based on the assumed velocity profiles. Buoyancy force was not accounted for in the numerical simulations. Although one-dimensional (Lin et al. 2000) and twodimensional (Yoshino et al. 2003) conjugate heat transfer was considered in the energy equati ons, the impact of the solid heat transfer on the fluid velocity distribution thr ough the buoyancy forces was absent. In the hot-wire simulations of

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23 Durst et al (Durst et al. 2002) a nd Shi et al (Shi et al. 2003), th e temperatures of hot-wire heating elements were specified and conjugate heat tran sfer was involved, in order to identify the correction on the velocity profile. However it wa s found that the buoyancy force has very little effect on the velocity profile (D urst et al. 2002). Hence the buoyancy terms were neglected in the corresponding momentum equations. Sheplak et al (Sheplak et al. 2002) proposed th at sensor measurement is quite sensitive to the ambient temperature variation. In this disse rtation, we will investigate the conjugate heat transfer around the solid substrate and the su rrounding fluid. The effect of the buoyancy force which couples the momentum and energy equations is investigated because it plays an important role in establishing the velocity gradient at the fluid-solid surface. Two kinds of sensor boundaries are used, namely, specified temperatur e and specified heat flux. In both cases a single-phase (involving the gas phase only), and a conjugate heat transf er with coordinated thermal boundary conditions are cons idered to highlight the indivi dual heat transfer modes. The effect of heat transfer on th e MEMS-based wall shear stress sensor will be addressed. 2.4.2 Governing Equations In the previous work (Appukuttan et al. 2003a) 2-D steady state Navi er-Stokes equations with the Boussinesq approximation for buoyancy force were solved. In order to capture the buoyancy effect, velocities are scaled by th e buoyancy force term (Shyy 1994). The following dimensionless variables are employed to normalize the governing equations: TH g u u TH g v v T T T T Tsensor TH g P P H x x H y y (2.23) The non-dimensionalized Navier-Stokes equations s ubject to the above sca ling references are:

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24 *umomentum 2 2 2 21y u x u Gr x P y u v x u u (2.24) *vmomentum 2 2 2 21T y v x v Gr y P y v v x v u (2.25) Energy equation 2 2 2 2Pr 1 y T x T Gr y T v x T u (2.26) where Pr and 3 2 g TH Gr The last term in Eq. (2.25) represents the dimensionless buoyancy force. The current formulation corres ponds to a horizontal ch annel in which the gravity is perpendicular to the streamwise velocity. In the solid domain, heat conduction is the only transpor t phenomena. Without a heat source in the solid (the sensor is a heat source but treated via the bounda ry condition due to its very small thickness), the governing equation is, 02 2 2 2 y T x T ks (2.27) where *sk is the dimensionless solid thermal conductiv ity; here, in dimensionless form, it is the solid-fluid thermal conduc tivity ratio. It is noted that th e dimensionless fl uid viscosity and thermal conductivity are Gr1 and GrPr 1 respectively. For conveni ence, all asterisks are dropped hereafter unless specifically mentioned. In the present study, the Grashof numb er is defined based on the length (L) of the thermal sensor, even though the governing equations are normalized by the channel height (H). For the

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25 MEMS-based sensor cases, it is better to descri be the buoyancy effect by the sensor-length based Grashof number sGr which can be expressed as 3 H L Gr Grs (2.28) In the present study, Prandtl number is fixed as 0.71. 2.4.3 Numerical Schemes The well-established pressure-based approach wi th finite volume formulation is adopted to solve the governing equations. A second-order upwind scheme for the convective terms and a second-order central difference scheme for pre ssure and diffusion terms are used. Detailed information can be found in (S hyy 1994; Shyy et al. 1997b; Thakur et al. 2002). Code validation and numerical accuracy assessment were perf ormed by Appukuttan et al. (Appukuttan et al. 2003a). 2.4.4 Computational Stiffness Computational stiffness arises from the large difference in length and time scales, caused by variations in transport properties and sensor -to-channel sizes (Chapra et al. 2002). In this study, the dimensionless solid thermal conductivity is 1200, corre sponding approximately to the ratio between the MEMS-based thermal sensor materials and air. 2.4.5 Geometry and Grid Layout As shown in Figure 2-1, the 2-D channel is 25cm long and 1c m high. A sensor is located on the bottom wall of the channel, and is 20cm downstream from the channel inlet. The 2-D sensor is 200m in length, which is much smaller than the characteristic length scale of the channel flow. A solid substrate is below the bottom wall of the channel. The solid substrate and flow channel are symmetric with respect to the channel bottom wall.

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26 The grid layout is rela ted to the computational method whic h is used to overcome the scale disparity. If a single uniform grid is used to cap ture the entire physical fl ow and heat transfer process for the present problem, th e grid would be very fine in the vicinity of the sensor and coarse away from the sensor. This kind of grid can introduce a noticeable computational error if the grid aspect ratios in some parts of the co mputational domain are t oo large (Ferziger et al. 2002). In order to circumvent the length scale disp arity, two-level layout grids similar to those used by Appukuttan et al. (Appukuttan et al 2003a) are adopted. First, a whole-domain computation (Figure 2-1) is carried out, whic h generally does not provide the resolution necessary for the velocity field near the sensor Based on the whole-domain solution, a reduceddomain computation focusing on th e sensor region (Figure 2-2) with a much finer grid is performed. In the reduced-domain computation, the boundary conditions are obtained from the whole-domain computation by using a bilinear interpolation. Unless otherwise stated, the w hole-domain grid is meshed nonuniformly with finer grid near the vicinity of the sensor The fluid region has 342 grid points. The solid region has 342 grid points. Such arrangement is necessary because the temperature gradient in the fluid region is larger than th at in the solid region due to convective effects. Since the solid-fluid thermal conductivity ratio is large, this grid arrangement is also helpful for obtaining the numerical convergence. The relatio nship of convergence with grid size and thermal conductivity ratio was presented by Shyy and Burke (Shyy et al. 1994). As will be presented later, the grid refinement study supports the adequacy of the present grid system. The reduced-domain computation is employed on a uniform grid cons isting of 162 points which has much better resolution than the whole-domain gr id and is expected to capture the detailed heat and fluid flow

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27 structure surrounding the sensor. The issue of grid resolution will be discussed while presenting the results. The boundary conditions for the whole-domain co mputational domain (F igure 2-1) are as follows. (a) Boundary conditions for the momentum equations: Inlet: parabolic velocity profile is de termined by Reynolds number and Grashof number. (The average in let velocity is determined by Reynolds number, Reu H which is in the dimensional form. If this veloci ty is normalized by the velocity scale in Eq.(2.23), we have *Reavgu Gr This velocity value is used to specify the parabolic inlet velocity.) Channel walls: no-slip Outlet: velocity extrapolation (b) Boundary conditions for the energy equation: Inlet of channel: TT (c) Boundaries other than inle t and outlet in Figure 2-1: 02 2 n T (2.29) where n is the direction normal to the boundaries. This temp erature boundary condition is adopted from the energy equations because the te mperature and heat flux on these boundaries are unknown a priori. The boundary conditions of the reduced-dom ain (Figure 2-2) are as follows. (a) Inlet: velocity and temperature are interp olated from the solutions of the wholedomain computation by bilinear interpolation.

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28 (b) Outlet: velocity and temperature are ex trapolated from inner grid points. (c) Boundaries of the solid region (except fluid-solid interface): temperature is interpolated from the solution of the whole-domain computation by bilinear interpolation. At the solid-fluid interface (except the sensor surface on which the temperature or the heat flux is imposed), the interface conductivity ek (Patankar 1980) is empl oyed. This artificial conductivity arises from the non-homogeneity of th e materials on both sides of the interface. Let us consider two neighboring computational nodes ne xt to the interface, one in the fluid region represented by f and the other in the solid region represented by s. The temperatures on these two points are fT and s T, respectively. Since no computational node exists on the interface, the heat flux through the interface could be represent as fs e f sTT qk yy (2.30) wherefy is the normal distance from the node f to the solid-fluid interface, and s y is the normal distance from the point s to the solid-fluid interface. In Eq. (2.30), ek is the effective, interfacial conductivities If one-dimensional heat conducti on resistances between the nodes f and s are considered, the heat flux through th e interface can also be expressed as fs f s f sTT q y y kk (2.31) Combination of Eqs. (2.30)-(2.31) leads

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29 s s f f s f ek y k y y y k (2.32) The interfacial conductivity ek in Eq. (2.32) is the harmonic m ean of the fluid and solid thermal conductivities. Between the sensor and the solid substrate, adiabatic boundary condition is used to account for the vacuum cavity beneath the thermal se nsor. The thickness of the vacuum cavity is neglected because it is very sma ll (Sheplak et al. 2002). On the fluid side of the sensor surface, either one of the two thermal boundary conditions is adopted, namely, (a) specified sensor temperature, or (b) specified sensor heat flux. The specified sensor temperature condition is the same as the sensor temperature boundary conditi on of the single-phase co mputation previously reported in (Appukuttan et al. 2003a) The specified sensor heat fl uxes are calculated from the corresponding single phase cases with the same Gr ashof number. The values of Grashof numbers are selected based on the operati onal temperature range of the ME MS-based thermal shear stress sensor (Sheplak et al. 2002) used in the Interd isciplinary Microsystems Group in the University of Florida, which is about 20-400oC. For air, the Grashof numbers are in the range 0.05-0.8 based on the sensor length. Therefore, 0.08sGr and 0.5 are chosen. Numerically 10sGr and 100 are also used to examine larger Grashof numbers limit, which correspond small kinematic viscosity fluids. For the case with 0.08sGr only shear stress distri bution and variation are shown in the next section since th e flow structure variations in these cases are too small to be observed, while the shear stre ss variations are noticeable.

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30 2.4.6 Results and Discussion Computations are conducted for a horizontal channel flow in wh ich gravity is transverse to the streamwise direction. The Reynolds number is 500 with the length scale based on the channel height. Before discussing computational results, it is necessary to first discuss the numerical convergence because of the numerical stiffness en countered in the present investigation. The convergence might be problematic if iterative methods are used. To quantify the convergence behavior, residuals of the gove rning equations are used. Poor convergence is indicated by a lower rate of decrease of residuals. Furtherm ore, since small values of residuals do not necessarily represent small errors in the solutions for stiff problems, at least a three-order of magnitude drop in the residuals is ne eded for judging the convergence. Since the momentum and the energy equations are coupled by the buoyancy force, the stiffness in one equation also affects the accura cy and convergence of other equations. When the thermal conductivity ratio is fixed as a constant the stiffness arising from the large thermal conductivity ratio is simila r for all the cases with different Grashof numbers. This can be seen from the energy equations for the fluid and solid regions. Figure 2-3 shows the residua l reduction histories of u, v, pressure and temperature of the single-phase and conjugate cases using a logarithm scale with specified sensor temperature and5 0sGr. Both are solved using the algebraic multigrid technique, specifying the same number of inner loop iterations and relaxation factors. The re siduals of single-phase and conjugate cases are normalized based on the same normalization values. The four residual history plots show that the single-phase case co nverges faster than the corresponding conjugate

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31 case. These different convergence rates mainly re sult from the stiffness of thermal conductivity ratio of the conjugate case. To demonstrate the impact of the solution technique, Figure 2-4 shows u, v, pressure and temperature residual reduc tion history plots of a conjugate case with5 0sGr, using both single-grid and multigrid solvers. The multi-grid solver shows noticeabl y faster convergence than the single grid solver, indicating that the multigrid solver performs better for solving stiff problems than the single-grid solver. This phe nomenon is also observed in two-phase flow problems that generally have multi-scale issues and significant property jumps (Francois et al. 2004). In order to assess the resolution requirement for the current problem, grid refinement has been conducted for both the whole-domain com putation and the reduced-domain computation. For the whole-domain computation, the fluid region has 514 node s, compared to the baseline grid of 342 nodes. The solid region has 514 nodes, refined from the baseline grid of 342 nodes. For the reduced-domain computation, the refined grid is still uniform, which has 242 nodes in compar ison to the baseline grid of 162 nodes. The velocity and temperature fields computed on the coarse baseline grid are only marginally different from those on the refined grid. Figure 2-5 shows that the di fference between the shear stresses computed on these coarse and refined grids is small. As exp ected, the convergence rate s of the computation on the refined grid are noticeably slower than thos e on the coarse grid (see Figure 2-6), where the same multi-grid strategies have been adopted. Hen ce the baseline (coarse) grid is adopted in the rest of this study. Figure 2-7 Figure 2-9 show temperature contours for the single-phase and conjugate cases for different Grashof numbers. In these te mperature contour plots of the conjugate cases,

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32 similar to single-phase cases, the warmed-up re gions enhance with Grashof numbers, showing stronger buoyancy effect for higher Grashof number s. Comparing with si ngle phase temperature contours, near the sensor in the fluid region, temperatur e distributions of the conjugate cases are very similar to those of the single phase cases. Similarly distorted elliptical contours can be observed for both the single-phase and conjugate heat transfer cases with different sensor boundary conditions. This can be seen from the temperature contour line with the value 0.0625. In the outer fluid region, the single-phase temper ature contours have different patterns than the conjugate cases. The temperatur e contours of single phase cases originate from the sensor and expand asymptotically to the main stream, while the temperature contours of the conjugate cases in that region no longer c onnect with the sensor since the h eat conduction effect of the solid substrate raises the upstream a nd downstream fluid temperature, and thus, the effective heating length is enlarged due to this substrate heating. At the solid-fluid interface, temperature contour s have abrupt kinks, re flecting the effect of the large solid-fluid th ermal conductivity ratio. In the leading-edge regi ons of the conjugate cases, the temperature contours turn in different directions at the solid-fluid interface. Near the sensor, temperature contours from the solid region turn left into the fluid region. Beyond a small upstream distance from the sensor, the temperature contours turn right from the solid into the fluid region. In the vicinity of the sensor at the leading edge, h eat goes upstream in the fluid by diffusion near the sensor which heats up the solid s ubstrate. Since the heat diffusion effect in the fluid region at the lead ing-edge is restricted very close to the sensor by the fluid flow, the temperature gradient of the fluid in that region is larger than th at of the solid. Therefore, beyond a small upstream distance from the sensor, the temper ature of the solid is hi gher than that of the fluid at the solid-fluid interface. This leads to th e fact that the solid substrate heats up the fluid

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33 due to the heat conduction in th e solid region and the large solid -fluid thermal conductivity ratio. Then the heat transferred from the solid to the fluid is carried downstream by fluid convection. At the trailing edge, fluid temper ature is always higher than that of the solid since the heat diffusion in the fluid enforces heat convection. As a result, the temperature contours at the interface downstream of the sensor have similar shapes. In the solid region, heat flux is determin ed by solid thermal conductivity, solid-fluid interface thermal conductivity and c onvection effect of fluid at th e interface. The interface heat resistance results from both conduction and conv ection at the interface. The interface resistance of conduction can be indicated by the interface c onductivity (Eq.(2.32)). B ecause the solid-fluid thermal conductivity ratio is fixed, the inte rface thermal conductivity can be rewritten as s f s f s f f ek k y y y y k k (2.33) Therefore, the dimensionless interface condu ctivity varies with dimensionless fluid conductivity which is proportional to Gr1 In other words, the resistance from conduction increases with increasing Grashof number. The in terface heat resistance from the convection can also be indicated by Grashof number since the Reynolds number is fixed as 500 for the present study. The resistance from convection decreases w ith increasing Grashof number because larger Grashof numbers denote larger buoyancy forces wh ich enhance the convection effect. Thus, the two parts of the interface heat re sistances have different tendenci es of variation with increasing Grashof number. Dimensionless solid thermal conduc tivity also varies with dimensionless fluid conductivity in a similar way owing to fixed solid-fluid thermal c onductivity ratio. Hence, convective heat transfer enhancement in the fl uid region reduces the c onduction effect at the solid-fluid interface, and vice versa.

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34 Due to the interface resistances, the major part of the heat generated by the thermal sensor is carried away by convection. Th erefore, the wall has a function to isolate fluid and solid, which allows relatively small part of heat to transport into the solid substrate. This is the reason why, near the sensor, the temperatur e contours of conjugate cases do not vary so much from the single-phase cases. Comparing the conjugate cases with specified se nsor temperature, temperature contours of 5 0sGr case are very similar to those of 10 sGr. For the conjugate cases with specified sensor heat flux, at the same locati ons, temperatures of the case with 10sGr are lower than those of5 0sGr. While for the conjugate cases with100 sGr, temperatures in the solid are higher than the other cases. This might result from the different variation trends of conduction effect and convection effects at the in terface with regard to Grashof numbers. This opposite effect of Grashof number on the heat conduction and convection at the fluidsolid interface is shown more clearly in Figure 2-10. For the cases of the specified sensor temperature, Figure 2-10 shows the temperatur e distributions on a cro ss-section, which is perpendicular to the sensor surface and originates from the middle point of the sensor to the top boundary of the reduced-domain. Due to the oppos ite effect of Grashof number on the heat transfer on the fluid-solid interface, temper ature distributions of the cases with 5 0 sGrand 10sGrare very similar. For the case with 0.08sGr the temperature of the fluid far away from the sensor is the lowest of these four cases, which mainly results from the lower buoyancy force effect on heat convection in the fluid region. For the case with100sGr, temperature is higher in the fluid region far away from the sensor and smaller in the region closer to the sensor.

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35 It can be explained that, for the case with100 sGr, the convection effect is larger in the far field region and the diffusion effect is smaller in the region near the solid surface. Figure 2-11 presents the u-velocity profiles of conjugate cases with different sensor boundary conditions. Due to the wall isolation effect, the u-velocity profiles of conjugate cases do not have noticeable differences from those of the single-phase results (Appukuttan et al. 2003a). This is also because of th e isolation of the wall describe d previously. However, the wall effect on the shear stress can be clearly observed in Figure 2-12 and Figure 2-13. In Figure 2-13 shear stress variations are calcul ated based on the anal ytic shear stress of a developed channel flow without buoyancy force ac ting on the fluid. The an alytical shear stress is |wdu dy. If it is normalized by avgU H and avgUis the average velocity of the channel flow, the analytic shear stress becomes ** *(/) 6avg wduU dy (2.34) Comparing single-phase shear stresses (Appuk uttan et al. 2003a), for the small Grashof number cases, the shear stresses of conjugate cas es with the specified sensor heat flux have smaller variation than those of the specified se nsor temperature cases; for the larger Grashof numbers, their shear stress variations are high er than those of specified temperature cases. Therefore, different energy boundary conditions fo r the sensor lead to different shear stress distributions and variations for conjugate cases. This difference should be caused by the buoyancy force. Different sensor temperatur e conditions cause different temperature distributions surrounding the sensor, which genera te different buoyancy forces. Therefore, if a thermal sensor works in the consta nt temperature mode, its shear st ress behavior is different from those of thermal sensors working in constant current mode or constant voltage mode.

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36 2.4.7 Summary and Conclusion: The effects of conjugate heat transfer on shear stress variatio ns of a MEMS-based thermal shear stress sensor are investigated. In the very close vicinity of the sensor, temperature contours of conjugate cases with different sensor temperature conditions are similar to those of the singlephase cases, while the temperatur e contours in the region away fr om the sensor are different from the single-phase cases. For conjugate cases, the effective heating length is enlarged due to the substrate heating, which was emphasized in (Sheplak et al. 2002). Even though the velocity distributions of both single-phase and conjugate cases are very similar, shear stress distributions for the conjugate cases have observable devia tions from single-phase cases. Therefore, though buoyancy force does not change velocity profiles significantly, it noticeably influences velocity gradients near the thermal sensor, which introdu ces errors in the shear stress measurements.

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37 Figure 2-1. Geometry of cha nnel flow with solid substrate Figure 2-2. Reduced-domain geometry. 1, 2 a nd 3 represent the secti ons on which velocity profiles are plotted in Figure 2-11

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38 a ite r ationnu m b e r u r esidualno r m 0 1000 2000 3000 -5 -4 -3 -2 -1 0 1 2 single-phase conjugate iterationnumber v r esidualno r m 0 1000 2000 3000 -5 -4 -3 -2 -1 0 1 single-phase conjugate b c ite r ationnu m b e r p r essu r e r esidualno r m 0 1000 2000 3000 -8 -7 -6 -5 -4 -3 -2 -1 single-phase conjugate iterationnumber tempe r atu r e r esidualno r m 0 500 1000 1500 2000 2500 3000 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 single-phase conjugate d Figure 2-3. Relative reductions of residuals of single-phase and conjugate cases with specified sensor temperature, 0.5sGr. (a) u residual, (b) v residual, (c) pressure residual and (d) temperature residual

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39 a iterationnumber uresidualnorm 0 1000 2000 3000 4000 5000 6000 -5 -4 -3 -2 -1 0 1 2 singlegridsolver multigridsolver iterationnumber vresidualnorm 0 1000 2000 3000 4000 5000 6000 -5 -4 -3 -2 -1 0 1 singlegridsolver multigridsolver b c iterationnumber pressureresidualnorm 0 1000 2000 3000 4000 5000 6000 -7 -6 -5 -4 -3 -2 -1 0 singlegridsolver multigridsolver iterationnumber temperatureresidualnorm 0 1000 2000 3000 4000 5000 6000 -8 -7 -6 -5 -4 -3 -2 -1 0 singlegridsolver multigridsolver d Figure 2-4. Relative reductions of residuals of conjugate cases w ith specified sensor temperature solved by single-grid and multi-grid solvers, 0.5sGr (a) u residual, (b) v residual, (c) pressure residual and (d) temperature residual.

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40 x shearstress 0.1 0.2 0.3 5.9 5.95 6 6.05 6.1 6.15 6.2 6.25 6.3 coarsegrid finegrid U Y / (GrsH3/L3)/Re Figure 2-5. Shear stress comparisons for the coarse grid and refine grid computations (0.5sGrand Re500 )

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41 iterationnumber pressureresidualnorm 0 1000 2000 3000 4000 -9 -8 -7 -6 -5 -4 -3 -2 -1 coarsegrid finegrid iterationnumber temperatureresidualnorm 0 1000 2000 3000 4000 -7 -6 -5 -4 -3 -2 coarsegrid finegrid The whole-domain grid iterationnumber pressureresidualnorm 0 2000 4000 6000 -9 -8 -7 -6 -5 -4 -3 -2 -1 coarsegrid finegrid iterationnumber temperatureresidualnorm 0 1000 2000 3000 4000 5000 6000 7000 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 coarsegrid finegrid The reduced-domain grid Figure 2-6. Relative reductions of Pressure and temperature residuals on the refined grids (0.5sGr)

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42 0 0 0 0 5 0 0 0 8 8 0 0 6 2 5 0 1 2 5 0.375 0 7 5 x y 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25a 0 0 0 3 0. 004 0 0 0 5 0 0 0 6 0.0203 0 0 6 2 5 0.125 0.375 0 7 5 0.0103 0.0097 0 0 0 9 1 0.01 0 0 0 9 4 x y 0 0.1 0.2 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2b 0 0 0 6 5 0.0085 0.012 0.0174 0.0177 0 0 1 8 3 0 0 1 8 6 0.75 0 3 7 5 0.125 0 0 6 2 5 0.0203 0 0 1 8 x y 0 0.1 0.2 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2c Figure 2-7. Temperature contours with 5 0 sGr (a) Single-phase cases with specified sensor temperature; (b) Conjugate cases with speci fied sensor temperature; (c) Conjugate cases with specified sensor heat flux. 0.5 0 2 5 0 1 2 5 0 0 6 2 5 0 0 2 0 3 0 0 0 3 5 x y 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25a 0.003 0.004 0 0 0 5 0.006 0.008 0.0203 0 0 6 2 5 0.125 0 2 5 0.5 0 0 0 9 4 0 0 0 9 1 0.0103 0 0 1 0.0097 x y 0 0.1 0.2 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2b 0.0045 0 0 0 5 5 0.0065 0 0 0 8 5 0 0 1 0 3 0 0 2 0 3 0 0 6 2 5 0.125 0.25 0.5 0.0133 0.013 0.0127 0.0124 0.0136 x y 0 0.1 0.2 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2c Figure 2-8. Temperature contours with10 sGr (a) Single-phase cases with specified sensor temperature; (b) Conjugate cases with speci fied sensor temperature; (c) Conjugate cases with specified sensor heat flux. 0.625 0 1 8 7 5 0 1 2 5 0 0 6 2 5 0 0 0 2 5 3E-05 x y 0 0.1 0.2 0.3 0 0.05 0.1 0.15 0.2 0.25a 0 0 0 5 0 0 0 6 0 0 0 8 0 0 1 0 0 3 0 0 6 2 5 0.125 0.1875 0 6 2 5 0.0128 0.0125 0 0 1 3 1 0.0134 0 0 1 3 7 x y 0 0.1 0.2 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2b 0 0 0 8 0 0 1 0.0135 0 .0 3 0 0 6 2 5 0.125 0.1875 0.625 0 0 2 0 5 0.0202 0 0 1 9 9 0.0196 x y 0 0.1 0.2 0.3 -0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2c Figure 2-9. Temperature contours with100 sGr (a) Single-phase cases with specified sensor temperature; (b) Conjugate cases with speci fied sensor temperature; (c) Conjugate cases with specified sensor heat flux.

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43 T y 10-2 10-1 100 0 0.05 0.1 0.15 0.2 Gr_s=0.08 Gr_s=0.5 Gr_s=10 Gr_s=100 Figure 2-10. Temperature distri bution on a cross-section which or iginates from the middle point of the sensor to the top boundary of the reduced-domain (Specified sensor temperature cases)

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44 U y 0.25 0.5 0.75 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 Section1 Section2 Section3 (GrsH3/L3)/Re U y 0.25 0.5 0.75 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 Section1 Section2 Section3 (GrsH3/L3)/Re (a) Specified sensor temperature (b) Specified sensor heat flux 5 0sGr U y 0.25 0.5 0.75 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 Section1 Section2 Section3 (GrsH3/L3)/Re U y 0.25 0.5 0.75 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 Section1 Section2 Section3 (GrsH3/L3)/Re (a) Specified sensor temperature (b) Specified sensor heat flux 10sGr U y 0 0.25 0.5 0.75 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 Section1 Section2 Section3 (Gr s H3/L3)/Re U y 0 0.25 0.5 0.75 0.025 0.05 0.075 0.1 0.125 0.15 0.175 0.2 Section1 Section2 Section3 (GrsH3/L3)/Re (a) Specified sensor temperature (b) Specified sensor heat flux 100sGr Figure 2-11. u velocity profiles of conjugate cases on the cross-se ctions just before (section 1), after (section 2) and in the region of the sensor (section 3) at Re=500

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45 x Shearstress= 0 0.05 0.1 0.15 0.2 0.25 0.3 0 10 20 30 40 50 60 Gr_s=0.08 Gr_s=0.5 Gr_s=10 Gr_s=100 U/ Y (Gr_sH^3/L^3)/Re 0.05 0.1 0.15 0.2 5.9 6 6.1 6.2 6.3a x Shearstress= 0 0.05 0.1 0.15 0.2 0.25 0.3 0 20 40 60 Gr_s=0.08 Gr_s=0.5 Gr_s=10 Gr_s=100 U/ Y (GrsH3/L3)/Re 0.05 0.1 0.15 0.2 0.25 0.3 5.9 6 6.1 6.2 6.3b Figure 2-12. Wall shear stress di stribution: (a) Specified sensor temperature (b) Specified sensor heat flux GrashofNumber(Grs) shearstressvariation(%) 0 0.2 0.4 0.6 0 1 2 3 4 5 6 7 8 9a GrashofNumber(Grs) shearstressvaration(%) 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0 1 2 3 4 5 6 7 8 9b Figure 2-13. Shear stress variation: (a) Specified sensor temperature (b) Specified sensor heat flux

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46 CHAPTER 3 ERROR ASSESSMENT OF THE LATTICE BOLTZMANN METHOD FOR VARIABLE VISCOSITY FLOWS As an alternative CFD tool, lattice Boltzmann Equation (LBE) method has been developing for about two decades. From scale cla ssification, the LBE is a meso-scale method for fluid dynamics computations. Navi er-Stokes equations for compre ssible flows can be recovered from the LBE with the incompressible constraint. In other words, the LB E describes fluid flows with very low Mach number (Chen and Dool en, 1998; Succi, 2001; Yu et al. 2003a). Although there are some difficulti es for the LBE method in compressible flows and thermo-fluid flows, it has been successfully used for isothermal complex fluid flows, such as in terfacial dynamics (He et al. 1999), turbulent flows (Y u et al. 2005) and porous media fluid flows (Chen et al. 1991). Since large velocity gradients often accompa ny substantial viscosity variations due to either shear-thinning effects or turbulence models, errors associated with the variable viscosity model need to be addressed befo re exploring the accuracy for these problems. With variable viscosity such as for turbulent flows treated with the eddy viscosity models (Filippova and Hanel, 1998), or for fluids with flow-dependent properties, additional trun cation error appears in the macroscopic equations derived from the LBE (Hou et al. 1995). With the help of the truncation error, computationa l accuracy for those flows can be further understood. The truncation error behavior of the LBE with constant viscosity has been studied by Holdych et al. (Holdych et al. 2004). However, th e truncation error for variable viscosity problems has not been studied systematically in the literature. Like other computational models, the LBE method needs to addr ess the wall boundary conditions (Mei, et al., 1999). In particular, the no-slip wall boundary condition in the LBE method is based on the bounce-back concept (Chen and Doolen, 1998; Succi, 2001). Much work has been done to extend this simple scheme to arbitrary curvature boundaries with second order

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47 accuracy (Mei et al., 1999; Ladd, 1994; Yu et al., 2003; Bouzidi et al., 2001). These methods can handle geometrical complexities easily; however, they cannot exactly recover the no-slip boundary condition (He et al., 1997) at the mesoscopic level (Mei, et al., 1999). This inconsistency gives rise to the boundary condition error. Although the no-slip boundary condition of the LBE has been derived from hyd rodynamic conditions on walls by Noble et al. (1995) and Inamuro et al. (1995), these exac t no-slip boundary sche mes are incapable of handling geometrically complex boundaries because they require the computational nodes to coincide with the physical boundaries. The original bounce-back scheme and other im proved treatments have been successful in constant viscosity laminar flows with expect ed accuracy (Chen and Doolen, 1998; Yu et al., 2003; Mei et al., 1999; Mei et al ., 2000). However, for variable viscosity problems, the boundary condition error for no-slip walls may act on the truncation error. Thus, besides the truncation error behavior itself, whether the second-order accuracy of the bounce-back scheme can be maintained in the presence of the variab le viscosity is another open question. In this Chapter, these issues are investigat ed via a fully-developed laminar channel flow with a specified variable viscosity. With the help of the finite differenc e analysis, the truncation error of the LBE with variable viscosity is investigated. Two differ ent specified viscosity distributions, which lead to di fferent boundary layer characteristi cs, are employed to examine the errors associated with the variable viscosity, in the LBE model. The exact solution of this channel flow exists and can be used for error analysis. In the following, the LBE method and the bounda ry conditions will first be reviewed. We then present the variable viscos ity laminar channel flow equations along with the exact solution. Based upon the exact solution and the finite differe nce analysis, the error behaviors due to the

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48 variable viscosity and the boundary condition sche mes will be assessed. To separate the error associated with the variable viscosity from that with the boundary treatmen t, both Noble et al.s scheme (Noble et al. 1995), which is exact bu t restricted to straight boundaries, and bounceback-on-link (BBL) scheme, which is not exact but can handle irregular geometries, will be employed. 3.1 Introduction to Lattice Boltzma nn Method and Wall Boundary Condition 3.1.1 LBE BGK Method The Lattice Boltzmann equation with the singe r-relaxation-time (SRT) Bhatnagar-GrossKrook (BGK) model can be writ ten as (He and Luo, 1997a) eq1 ,,,,iiii f tttftftft xexxx (2.35) where f denotes ,, f txe, which is the distribution func tion in the direction of the th discrete velocity e, eq f is the corresponding equilibriu m distribution function in the discrete velocity space,/ t is the relaxation time, and ix represents computational nodes in physical space. The most popular lattice model for simulating two-dimensional flows is the nine-velocity square lattice model, which is often referred to as the 2-D 9-velocity (D2Q9) model (Qian et al., 1992). In this lattice model, the discrete velocities eare as follows: 00, cos1/4,sin1/4for=1,3,5,7 2cos1/4,sin1/4for=2,4,6,8c c e e e (2.36) where / cxt The corresponding equilibrium dist ribution functions are defined as 2 eq 222393 10,1,,8 22 fw ccc eu+euuu (2.37)

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49 where w is the weight coefficient given by 013572468411 9936 wwwwwwwww (2.38) The density and momentum fluxes can be obtained from the moments of th e distribution function as 8888 eqeq 0000 f fff uee (2.39) The kinematic viscosity associated with the D2Q9 lattice model can be expressed as 21 2sct (2.40) where s c is the speed of sound, which is equal to /3 c for D2Q9 lattice model. The corresponding equation of state is 2 s p c For a hexagonal latti ce model (Noble et al. 1995), th e discretized velocities are 00, cos12/6,sin12/6for=1,2,...,6c e e (2.41) The corresponding equilibrium di stribution functions are eq 00 2 2 eq 0 2422 for=1,2,...,6 22fd c DD d DD f bcbcbcb uu eueuuu (2.42) Where for the 2-D hexagonal lattice model, the dimension rank is 2D the number of lattice direction is 6b, the average rest particle density is 0/2d The kinematic viscosity for 2-D hexagonal lattice model is 221 8 x t (2.43)

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50 The LBE is usually solved in two steps: Collision step: ()1 ,,,,eq iiii f ttftftft xxxx (2.44) Streaming step: ,,ii f tttftt xex (2.45) where f represents the post-collision state. In order to get the Navier-S tokes equations, the Chapman-En skog expansion can be used with time and space being rescaled as 2 121 2 121,,, ,ttttxx tttxx (2.46) and the distribution function is expanded as (0)(1)2(2)3.ffffO (2.47) From Eq. (2.46) and (2.47), it is clear that the Champan-Enskog expansion is essentially a multiscale expansion (Frisch et al. 1987). In the incompressible flow limit, that is || 1u c, the NavierStokes equations can be recovered from the LBE (Eq. (2.35)) with the leading truncation error of 23, Oxu (Hou et al. 1995): 0,i iu x (2.48) 21ii ji jiuu p uu txx (2.49)

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51 3.1.2 Boundary Conditions In contrast to Navier-Stokes equations, the lattice Boltzmann equation has no physical wall boundary conditions for distribution function f The distribution functions are unknown and have to be constructed from inside information. Symmetric and periodic boundary conditions can be constructed without any ambiguities. Outlet boundary conditions are generally extrapolated from internal nodes (Yu et al. 2003a). Inlet boundary conditions can be extended directly from the boundary treatments for solid wall. Since a flow with a large Reyno lds number usually has boundary layer phenomenon near walls, which is a typical multi-scale pr ocess (Batchelor 1967) and sensitive to the wall boundary condition, the noslip wall boundary cond ition is thus the most important one among other boundary conditions in fluid dynamics. Historically, not only the models of the LBE evolved directly from its predecessor, the lattice-gas automata (LGA) (Frisc h et al. 1986), but also the wa ll boundary conditions, which is namely the bounce-back scheme (Lavallee et al 1991; Chen et al. 1998; Succi 2001). This boundary condition method has a very attractive feature: it is very easy to implement computationally, especially for complex geometri es. In comparison with the Navier-Stokes solvers, this amazing feature o ffers the LBE method a distinguished capability to handle flows with geometry complex (Chen et al. 1998; Mei et al. 1999; Mei et al. 2000). For the no-slip wall boundary condition, if computational boundary nodes are located on walls, the LBE method with the bounce-back sche me has first-order accuracy (Ziegler 1993). A slip wall velocity exits (He et al. 1997; He et al. 1997c) and increases with relaxation time (Noble et al. 1995). This bounce-back scheme with the computational nodes on walls can be referred as the bounce-back-on-node (BBN) scheme. While, if the boundary is shifted into the fluid by one half mesh unit, i.e. placing the wall between the computational nodes, second-order

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52 accuracy can be achieved by the bounce-back scheme (Ziegler 1993; Ladd 1994a; Ladd 1994b; Yu et al. 2003a). This bounce-back scheme w ith wall positions locating between computational nodes can be referred as the bounce-back-on-link (BBL) scheme (Mei et al. 1999). In the BBL scheme, after a particle distribution function f streams from a fluid node at f x to a boundary node at bx along the direction of e, the particle distribution function f reflects back to the node f x along the direction of ee, f f (2.50) which is illustrated in Figure 3-1. The BBL sc heme has second-order accuracy (Ziegler 1993; Ladd 1994a; Ladd 1994b), even through the slip wall velocity can not be exactly removed. The BBL scheme can be improved to arbitrar y curved wall boundaries with second-order accuracy, such as Filippova and Hnels scheme (F ilippova et al. 1998; Fillippova et al. 1998). Mei, Luo and Shyys scheme (designed as ML Ss scheme (Lai et al. 2001)) improved the numerical stability of Filippova an d Hnels scheme (Mei et al. 199 9). Bouzidi et al. (Bouzidi et al. 2001) presented a simpler bounce-back boundary condition for a wall located at an arbitrary position and no requirements for constructing fictit ious fluid points inside walls (Bouzidi et al. 2001; Yu et al. 2003a). Yu et al (Yu et al. 2003b) presented a unified boundary condition based on Bouzidi et al. and Mei et al .s works. Although the exact noslip velocity boundary condition could not be achieved by using the BBL and the improved BBLs, they are very attractive for flows with complex geometries (He et al. 1997c). Besides the bounce-back schemes, there are some other schemes for no-slip wall boundary condition, such as Noble et al.s hydrodynamic boundary condition scheme for hexagonal lattice (Noble et al. 1995), and Inamuros scheme fo r square lattice (Inamuro et al. 1995).

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53 Noble et. als scheme was derived for the he xagonal lattice from the macroscopic mass and momentum at a no-slip boundary without any assumptions. This scheme is illustrated in Figure 3-2. On the boundary nodes, the following expr essions of the unknown di stribution functions can be obtained by solving density and mass flux from the boundary nodes: 2301456 231456 2356222 2/3 fffffff ff u ffff ffvff (2.51) Through Eq.(2.51), the unknown distribution func tions on boundary nodes can be obtained from the distribution functions on the nodes of fluid side. This scheme can provide the exact no-slip wall boundary condition, which can be exhibited through simulations of a fully-developed channel flow with constant viscosity. The absolute2L-norm error of this simulation defined as 1/2 2 0 2H LBMexactuyuydy E H (2.52) is shown in Figure 3-3, which has the order of the round-off error. This demonstrates the exact no-slip boundary condition can be attained via Noble et al.s scheme. Inamuros scheme was developed for the 2-D s quare lattice model (D2Q 9). A lattice cell is shown in Figure 3-4. The definitions of the density and velocity on the no-slip wall are same as those on the internal nodes: 8888 eqeq 0000 ffff uee In Figure 3-4, the boundary node is on the wall. Thus, the distribution functions 015678,,,,, f fffffare known; while 234,, f ffare unknown and they need to be constructed from the known distribution functions. Because the slip wall velocity exists, Inamuro et. al. assume the unknown distribution functions 234,, f ff have similar forms as other known distribution

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54 functions except that there is a counter velocity wh ich is added to the slip wall velocity to force the wall velocity as zero. The unknown distribution2f,3f, and 4f are expressed as: 22 ''''2 2193 13 3622wwwwwwfuuvuuvuuv (2.53) 2 '2'2 3193 13 922wwwwfvvuuv (2.54) 22 ''''2 4193 13 3622wwwwwwfuuvuuvuuv (2.55) where wu ,wv are the slip wall velocity, 'u is the counter velocity, and is the density change due to the counter velocity. The velocity on the wall in the horizontal directionwu and vertical direction wv are calculated as 24861524111wx wwwufeffffffffA (2.56) 243687243111wy wwwvfefffffffffB (2.57) where 8615 A ffff and 678Bfff. In the above equations, the unknowns are 'u, andw With some algebraic manipulati ons, the unknowns can be obtained '1 63 13ww www wuA uuuv v (2.58) 26 133ww wwvB vv (2.59) 0156781 2 1w w f ffffff v (2.60)

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55 It can be seen that 'u and w are obtained from the known dist ribution functions and wall velocity. Substituting'u, and w back into(2.53), (2.54)and(2. 55), the unknown distribution functions on the wall can be obtained. Inamuro et. al.s scheme could enforce th e no-slip wall boundary condition. However, it could not remove all errors from the boundaries. This can be observed from simulations of a fully-developed channel flow with a constant viscosity. The absolute2L -norm error is shown in Figure 3-5. It is clearl y shown that the absolute2L -norm error is larger than the round-off error (15(10) O). The error in Inamuro et. al.s scheme might come from the counter velocity assumption which is the only hypothesis in th is scheme. Whether the unknown distribution functions really have the form (Eq. (2.53), (2.54) and (2.55)) or not is unknown. Through the comparison between Noble et al.s scheme and Inamuro et al.s scheme, it is clear that Noble et. al.s scheme can exactly realize no-slip boundary condition, while Inamuro et. al.s scheme does not. Noble et al.s sc heme is thus adopted in this chapter. 3.2 Error Assessment of the LBE Method due to Variable Viscosity vi a a Fully Developed Channel Flow 3.2.1 Fully-developed Laminar Channel Flow with Variable Viscosity In this section a 2D fully-developed steady lami nar channel flow with variable viscosity is adopted to assess the error behavior of the LBE with variable viscosity. The governing equation for this flow in a channel of height H is 0 /21 0and0,0total y yHdpddudu u dxdydydy (2.61) The viscosity total is modeled as

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56 0 2 2 3 23 3 0, / ,00.5, 1/ 1//totalst sty H (2.62) where 0 is a constant and the symmetry gives total for 0.51 The two dimensionless parameters and control the profile shapes for viscosity and velocity. Employing the boundary conditions and integrating Eq. (2.61) twice yield 3 2 32 0 20.5 for00.5 Hdp ud dx (2.63) Let 2/ and ) )( (2 2 3c b a It is easily seen that 32, aa ba and c a where the value of a can be obtained numerically in terms of and Integrating Eq. (2.63) results in the exact solution of this laminar channel flow, that is, 22 2 11 0 2221 ln/1lnln 22 122 tantan 2 444 B DAabcc Hdp u bb dx QbB cbcbcb (2.64) where D 2 2[(1/2)] 2 a A ac 2(12) (2) a B aac and () QAbBa In order to explore the trunc ation error due to variable viscosity under the condition of large velocity gradient over a short distance, the parameters and in Eq. (2.62) need to be chosen carefully so that boundary-layer like veloc ity profile can be obtained in Eq. (2.64). The first term in Eq. (2.64) in the curly bracket corr esponds to the parabolic ve locity profile with the constant viscosity 0 Other terms are due to vari able part of the viscosity s t For small values of and the fourth term 2lnln 2 B bcc does not change dramatically across the

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57 whole channel. The second term D is a negative linear term and the third term ln/1 Aa is positive across the channel, the sum of these two term s varies slowly in the near wall region. The last term in the near wall region varies rapidly because, for small and under further assumption of 3 2 (2.65) it is asymptotically equal to 111 2221221 tantantan 22 444 bb QbB cbcbcb (2.66) Thus, with small values of and the tan-1() term in Eq. (2.64) results in the boundary layer type of behavior. For comparison purpose, two sets of viscosity distributions, which satisfy Eq. (2.65) for small and small are used in this study: 0CaseI:,,0.004167,0.0005,0.0102 0CaseII:,,0.008333,0.002,0.0289 The corresponding viscosity profiles are shown in Figure 3-6. The resulting velocity distributions correspondi ng to these two viscosity distributi ons are shown in Figure 3-7. The boundary layer effect from the 1tan() terms in Eq. (2.64) or Eq. (2.66) for Case I is shown in the inset of Figure 3-7. Eq. (2.66) also gives a guideline for estimating the grid resolution required for resolving the boundary layers. Since2 / 1 ) ( tan / ) 1 ( tan1 1 it is seen that over a distance of the velocity reaches 50% of the maximum given by Eq. (2.66). Thus, for dimensionless grid size h =1/ H that is close to or larger than numerical solutions will not have sufficient resolution for this thin layer.

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58 3.2.2. The Lattice Boltzmann Equation Treatment In LBE method, the viscosity is as sociated with the relaxation time by Eq. (2.40). The variable viscosity in LBE can still be realized via a spatially varying relaxation time, and the total viscosity at a given lo cation can be expressed as 2(,)1 (,)forsquaregrid 6 2(,)1 (,)forhexagonalgrid 8total totalxy xy xy xy (2.67) Thus, the relaxation time can be represen ted by the local fluid viscosity as 6(,)1 ,forsquaregrid 2 8(,)1 ,forhexagonalgrid 2total totalxy xy xy xy (2.68) The computational setup for BBL scheme is s hown in Figure 3-8. The first computational node is half a lattice away from the channel wall. For Noble et al.s scheme the setup is the same except that the grid is hexagona l and the computational boundary nodes are on the channel walls. For comparison purpose, the computations on hexagonal grid and square grid should have the same grid resolution across the channel height. For the same grid resolution the channel height of the square grid setup is 2/3 times that of the hexagonal grid setup. In order to make absolute comparison among the velocity prof iles, the pressure gradients ar e accordingly adjusted in each computation so that 2dp H dx remains the same. Periodic boundary condition is used at the left and right side boundaries. The constant pressure gradient in Eq. (2.61) is treated as a body force, and is added to the distribution functions after th e collision step. The error analysis for the computations with both boundary conditions is ca rried out by examining the difference between the exact solution and the co mputational results at each y location.

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59 3.2.3 Assessment Since Case I exhibits thinner boundary layers an d sharper near-wall velo city gradients than Case II, we first discuss Ca se I. Figure 3-9 compares three velocity profiles for H=50 (h=1/H=0.02): a) exact solution; b) LBE soluti on using the standard BBL boundary condition and square lattice formulation; c) LBE soluti on using Nobel et als exact boundary condition and hexagonal lattice formulation. For the parameters considered, H=50 (h=0.02 >0102 0 ) does not resolve the thin boundary layer near the wall, as clearly show n in the inset of Figure 3-9. It is noted that due to the exactness of the boundary condition used, the hexagonal formulation gives only a slight overshoot in the velocity one full lattice away from the wall (at /~2 h ); however, it results in an overshoot for the re st of the lattices in the channe l. The LBE solution using square lattice formulation suffers from both the in accuracy of the bounce-back boundary condition and the insufficient near wall resolution and gives lo wer velocity throughout the entire channel. Both LBE velocity profiles have errors of similar magn itudes. This implies that when the velocity in the boundary layer is not sufficiently resolved, the exact boundary condition with the hexagonal lattice formulation would not offer any advant age compared to the approximate BBL condition using the square lattice formulation. For h ranging from 0.004 to 0.1 (H: 10 250 in lattice unit) the square lattice formulation with the BBL boundary scheme shows different over/underpredictions for the velocity pr ofiles from the hexagonal formulation comparing with Noble et al.s scheme. While the results are not shown here for brevity, it suffices to note that the magnitudes of the over-predictions in the hexagonal lattice case ar e comparable to those of the under-predictions in the square lat tice case. These velocity profiles indicate that the errors of the LBE solution using both boundary condition sche mes are comparable. Thus, the relative 2L norm error of the LBE solution, defined as

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60 1/2 2 0 2 1/2 2 0exactH LBMexact Huyuydy E uydy (2.69) is examined for both boundary condition schemes. The LBE computations with these two sets of viscosity distributions are carried out by using both Noble et als scheme and the BBL scheme for grid size h ranging from 0.004 to 0.1. Their relative 2L-norm errors in the velocity profiles with respect to grid resolution are shown in Figure 3-10a. As expected, the relative 2L-norm error curve of Case I shifts up with respect to that of Case II because Case II has a thicker boundary layer wh ich implies bette r computational resolution than in Case I for the same h (1/) H For sufficiently high grid resolution (h<0.01, points A-D shown in Figure 3-10a), both Noble et als scheme and the BBL scheme yield the asymptotic second order accuracy, which is consiste nt with the truncation error analysis for both square and hexagonal lattice schemes. For the present fully developed 2D channel flow with low speed, the velocity field satisfies 0 u and 3Ou, resulting in very small modeling error. Thus the errors in LBE computations are mainly from the truncation error (due to variable viscosity) and the boundary condition treatment. With sufficient grid resolution (points A-D in Figure 3-10a), the relative 2L norm errors of BBL scheme are about 15% larger than those of Noble et al.s scheme. Since Noble et al.s scheme does not contain bound ary condition-induced error, this 15% difference in error in the BBL scheme results from the boundary cond ition treatment. Thus comparing the results using Noble et al.s scheme and the BBL scheme, it can be inferred that in the presence of the strong velocity profile variation the truncation error contributes a significant part of the overall error.

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61 Because a substantial part of the overall error is from th e truncation error as opposed to the boundary condition error with highly variable viscos ity, the truncation error associated with the variable viscosity thus deserves close attention. However, the truncation error c ould have very complex form, even for the prob lems with constant viscosity (Holdych et al. 2004). It is recognized that there is a close relation between the LBE and the finite difference form of the momentum transport represented by the Navier-S tokes equation (He et al. 1997; He et al. 2002; Mei et al. 1998). For example, He et al.(1997) showed that the square lattice formulations for the particle distribution functions in a 2-D pressure driven channel flow with constant viscosity, after averaging, leads to a second orde r, central-difference formulation for the axial ve locity. For the present channel flow problem with the large vari ation in viscosity, such derivation could not be easily obtained. However, if the relative 2L-norm error of the finite difference method still behaves similarly to that of the LBE, it is reason able to expect that the truncation error behavior of the LBE is similar to that of the finite difference-based macroscopic model (in the present case, the Navier-Stokes equation). This hypothesis will be assessed first. In the Navier-Stokes model, the governing equa tion (Eq.(2.61)) is discretized by standard central difference scheme (with 1y lattice unit): 2 2 1/211/210()()()jjjjjjHdp uuuu dx (2.70) To compare the errors of the finite difference and LBE schemes, two finite difference solutions on different grid arrangements ar e obtained for the velocity prof ile. The first finite difference solution has the same grid arrangement as in the hexagonal lattice so that the first fluid node is one full mesh away from the wall. The second finite difference solution has the same grid arrangement as in the square latti ce with the BBL scheme so that the first fluid node locates half a mesh away from the wall. This second finite difference solution requir es an approximation for

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62 the velocity condition at the walls ; a linear extrapolation is used in conjunction with the no-slip condition at the wall. For completeness, a second or der extrapolation is also used to approximate the derivative at the wall in solving Eq. (2.61); the error is consistently larger than the linear extrapolation and hence the results are not presen ted. The viscosity dist ribution of Case I is chosen for the finite difference computations for Case I gives sharpe r near-wall velocity gradients than Case II. The results of the relative 2L-norm errors over a range of h=0.004 to 0.1 from these two finite difference solutions are s hown in Figure 3-10b. The results la beled as FD-H refer to the finite difference solution obtaine d on the hexagonal grid and the results labeled as FD-S refer to the finite difference solution obtained on the square grid. For small values of h (h<0.01), the O(h2) asymptotic behavior in error is clearly visible, which is similar to the LBE cases. This asymptotic error behavior is expected because the LBE scheme has second order accuracy and the boundary conditions are, depend ing on the specific scheme chos en, either exact or second order accurate, and the finite difference scheme with the central difference discretization gives global second order accuracy. As h increases close to or greater than 0.01, both the finite difference and the LBE errors increase faster than O(h2). Since the velocity profile given by the exact solution has a thin layer of thickness 0.0102 the error starts to increase more rapidly for h>0.01 when the resolution of the thin layer becomes inadequate. For 0.01
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63 Comparing the relative 2L-norm errors of the finite diffe rence solution with those of the LBE solution in Figure 3-10b, it is seen that the relative 2L-norm error of FD-H follows that of the LBE with Noble et al.s sc heme closely, and the relative 2L-norm error FD-S follows that of the LBE with the BBL scheme closely. This consistent behavior of the relative 2L-norm errors between both the LBE method and the finite diffe rence method suggests that one could obtain an insight on the truncation error of the LBE schemes by studying th e truncation error of its finite difference counterpart. For both LBE and finite difference schemes, th eir modified equations associated with the corresponding truncation errors can be represented as ()T.E. of the LBE methodLBELE (2.71) ()T.E. of the finite difference methodFDLE (2.72) where L is the differential operator totaldd dxdx L BEexactLBEEuu and F DexactFDEuu. Using Taylor series expansion the trun cation error in Eq. (2.72) is 23 432 2 432231111 T.E.H.O.T. 126824totaltotaltotal totalvvv uuuu v (2.73) Although the truncation error of the LBE is unknown, comp aring the solu tion errors, F DE and LB E E, on the left-hand-sides of Eq. (2.71) and (2 .72) can offer insight into the truncation errors on the right-hand-sides of these equation s while this indirect comparison avoids the tedious derivation of th e truncation error of the LBE. The value of F DE can be computed by subtracting FDu from the exact solution after the velocity FDu is solved from Eq. (2.70).

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64 To highlight the behavior of the leading te rm of the truncation error, the values of F DE are approximated by FD E obtained from solving Eq. (2.72) with small h. For LB E E it can only be determined directly by subtracting LB E u from the exact solution. Eq. (2.72) can be solved, with the boundary conditions 010FDFDEE by using the same central difference scheme given by Eq. (2. 70), and replacing the entire T.E. by the leading term of the T.E. provided that the resolution is sufficient and the high-order terms in the T.E. can be neglected. Caution must be exercised in inte rpreting the results from the numerical solution of Eq. (2.72). In this equation, even if the leadi ng term of the T.E. on the right hand side can be analytically evaluated, the variati on of the viscosity on th e left hand side is th e source of the T.E. in the first place. When solving for E the effect of the T.E. associated with the ODE (Eq. (2.72)) now is further compounded by the variation of the viscosity. The finite difference computation for Eq. (2.72) is carried out on a grid with the same grid arrangement and size as in the hexagonal lattice case so that the exact velocity boundary condition can be used. For Case I, th e thin wall layer has a length scale =0.0102 and can be adequately resolved using 200 / 1 / 1 H which is confirmed by the comparison between F DE and F DE in Figure 3-11. The curve representing F DE is almost on top of the curve representing F DE. Figure 3-11 also shows the variation of L BEE across the channel. Overall, F DE from Eq. (2.72) is smaller than L BEE across the whole channel which implies that these two methods indeed have quantitatively different truncation errors. Howe ver if we adjust the scale for F DE by a factor of 2, these two cu rves lie almost on top of each ot her, as shown in the inset of Figure 3-11. The matching of the error curves between two solutio ns demonstrate that the LBE truncation error behaves very similarly to that observed in the finite difference scheme in the

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65 presence of a strong vari ation in the viscosity. This is in ag reement with the ob servation on the relative 2L-norm error behavior shown in Figure 3-10b. Based on the foregoing discussions, it is seen that BBL scheme performs similarly as Nobels scheme in the presence of strong velocity profile variation. In vi ew of the simplicity and flexibility of the BBL scheme over the Noble et al.s scheme and comparable performance in accuracy, it is attractive to use the BBL scheme (or its extended version such as Bouzidi et al.(2001) and Yu et al. (2003a)) fo r handling problems of substantia l velocity profile variations caused either by complex geometry or variable viscosity. For complex 3D flows with curved boundaries, Navier-Stokes solvers currently have more flexibility on grid arrangement. For example, body-fitted coordinates and grid stretching can be more easily implemented to improve grid resolu tion near boundaries than in the LBE solvers. Although the recent developments in LBE met hod such as multi-block (Yu et al. 2003) and composite grid (Fillippova and Hanel 1998) techniques can allevi ate the difficulty in grid arrangement in LBE simulations to certain extent, further research efforts in LBE are needed. As the present study has indicated the boundary condition induced error may not be dominant, there is a great potential in improving the overall capability of the LB E solvers for complex flows with very strong velocity variation by focusing future research efforts on extending the grid flexibility of LBE schemes. 3.3 Summary and Conclusion In this chapter, the error behavior of the lattice Bo ltzmann equation (LBE) method for a flow with strong variation in vi scosity is studied. The variable viscosity in the LBE method is modeled through a variable relaxati on time. Solutions for a laminar channel flow w ith a specified variable viscosity are obtained using both the LBE method and the finite difference method to

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66 examine the truncation error behavior of th e LBE method for flows with strong varying viscosity. The effect of the boundary condition error of the bounce-back-on-link (BBL) scheme on the overall error is investigated via the compar ison of the error behavi or of the BBL scheme and that of Noble et al.s scheme. The results show that with rapid viscosity variation the boundary condition error of the BBL scheme does not induce noticeable, additional errors, and the overall error of such flows is dominated by the truncation erro r itself. The results also show that in the presence of strong variable viscosit y the truncation error beha vior of the LBE solution is consistent with that of the finite difference solution to Navier-Stokes solution.

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67 Figure 3-1. Boundary nodes and their neighbors using the square lattice Figure 3-2. A boundary cell using the he xagonal (FHP) lattice (Noble et al. 1995) H L 2 absolutee r r o r 20 40 60 80 100 10-22 10-21 10-20 10-19 10-18 10-17 10-16 10-15 10-14 Figure 3-3. Absolute L2 norm errors of LBE with Nobles scheme fo r fully-developed laminar channel flow with constant viscosity 2 3 4 1 3 f 2 f 4 f 1 f 5 f 6 f 5 6 Fluid ( f ) Boundary ( b ) Wall ( w ) Wall f x f x f x b x e e e e e e

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68 1 2 3 4 7 e2 e 3e1e4e5e6e7e8 5 6 8 Figure 3-4. A 2D 9-veloc ity lattice (D2Q9) model H L2absoluteerror 20 40 60 80 10 0 2E-11 3E-11 4E-11 5E-11 6E-11 Figure 3-5. Absolute error of a fully-developed channel flow using Inamuro et. al.'s scheme total/ 0 0 0.1 0.2 0.3 0.4 0. 5 -5 0 5 10 15 20 25 30 35 =0.0005, =0.0102(CaseI) =0.002, =0.0289(CaseII) Figure 3-6. Two set of viscosity distributions used in this study

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69 u 0 0.001 0.002 0.003 0.004 0.005 0.006 0.007 0 0.2 0.4 0.6 0.8 1CaseI CaseII u 0 0.002 0.004 -0.05 0 0.05 0.1 0.15 0.2 0.25CaseI CaseII tan-1terms inEq.(3.35)(CaseI) Figure 3-7. The exact velocity profiles of the channel flow s with different boundary layer thicknesses due to different viscosity distributions The parameters are: 0;;0.004167;0.0005;0.0102for Case I and 0;;0.008333;0.002;0.0289 for Case II Figure 3-8. Square la ttice distribution in ch annel flow simulation y x j=1 j=2 j=Ny-1 j=Ny 1 Inlet: Periodic boundary condition Outlet: Periodic boundary condition

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70 ve l oc i t y 0 0.1 0.2 0.3 0.4 0.5 0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006Exact u_Hex u_Square 0 0.02 0.04 0.06 0.08 0 0.0001 0.0002 0.0003 0.0004Exact u_Hex u_Square Figure 3-9. Comparison of the LBE velocity pr ofiles using square lattice (with bounce-back on the link boundary condition) and hexagonal la ttice (with Nobel et als exact boundary condition) with the exact so lution at H=50 lattice units. h RelativeL2no r me r r o r 0.02 0.04 0.06 0.08 0.1 0.12 10-5 10-4 10-3 10-2 10-1 100BBLscheme(CaseI) Nobleetal.'sscheme(CaseI) BBLscheme(CaseII) Nobleetal.'sscheme(CaseII) 1 2 2 1 A B C Da

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71 h RelativeL2normerror 0.02 0.04 0.06 0.08 0.1 0.12 10-5 10-4 10-3 10-2 10-1 100FD-S(CaseI) FD-H(CaseI) BBLscheme(CaseI) Nobleetal.'sscheme(CaseI) 1 2 2 1b Figure 3-10. Dependence of the relative L2 norm error on the lattice size h in the fullydeveloped channel flow with variable vi scosity. The viscosity parameters are: ,0.0005,0.0102for Case I, 0.002,0.0289for Case II. (a) The LBE with the boundary conditions of Noble et als sche me and BBL scheme for both Case I and Case II. (b) The finite difference and th e LBE with boundary conditions of Noble et als scheme and BBL scheme for Case I.

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72 error 0 0.1 0.2 0.3 0.4 0.5 0 2E-06 4E-06 6E-06 8E-06ELBEE* FDEFD ELBE E* FD 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0 2E-06 4E-06 6E-06 0 1E-06 2E-06 3E-06 4E-06 ELBEE* FD Figure 3-11. Comparison of L BEexactLBEEuu with FDexactFDEuu for H=200, and 0.0005,0.0102 F DE is the numerical approximation of F DE obtained from solving Eq. (2.72).

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73 CHAPTER 4 LBE METHOD FOR IMMISCIBLE TW O-PHASE FLOW COMPUTATION The characteristic feature of immiscible two/multi-phase flow s is the inter-phase boundary (interface) between different phases. The stru cture of any inter-phase boundaries is on the mesoscopic length scale level (Pis men 2001). The steep gradients of material properties in the normal direction of an interface result from the fact that the normal length scale approaches molecular level, while the length scale along th e interface is still on the macroscopic level (Rowlinson et al. 1982a; Carey 1992). As described in Chapter 1 these scale disparities make the computations more difficult. In this chapter, a LBE immiscible two-phase model is developed based on the framework of He et al.s mode l (1999). Several numerical techniques are implemented to the numerical stability so that tw o-phase flows with high liquid-gas density ratio can be simulated. This chapter begins with an overview of the immiscible two-phase flow computations and a literature review on the LB E two-phase models, followed by an introduction to He et al.s model (He et al. 1999). The Ra yleigh-Taylor instability is simulated and computational results are compared with analyt ical ones to validate the code. After this, the numerical issues in Lee-Lins implicit LBE model and He et al.s model are addressed. A filterbased LBE model along with a new surface force formulation and a volume-correction step is proposed. The stability and accuracy of this ne w model are tested by computing flows around a stationary bubbles, capillary wave s and rising bubbles. The results s how that the pr esent model is capable of handling the flows w ith large density ratio up to O(102). 4.1 Overview of Immiscible Two-Phase Flow Computation For macroscopic computations of immiscible two-phase flow the numerical methods can be classified into two categorie s. One is the interface-tracking me thod, in which interfaces have to be tracked numerically to separate materials of two different phases or properties (Shyy et al.

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74 1996). The other is the continuous interface method in which interfaces are not tracked explicitly (Anderson et al. 1998). Two approaches exist in terms of the grids used for computa tions. One is the moving grid (Lagrangian) approach. The other is the fixed gr id (Eulerian) approach (Shyy et al. 1996; Shyy 2004). These two approaches can also be combined to form a mixed approach that is called the Eulerian-Lagrangian approach. In the Lagrangian approach the interface is tracked explicitly by a body-fitted grid that deforms with the interface. No modeling is necessa ry to define the interf ace or its effect on the flow field. However, the body-fitted grid has to be regenerated as the interface deforms. This requires a set of equations that need to be solv ed for the grid regeneration at each time step. Moreover, Lagrangian approach is not easy to use when fluid flows involve complex topological changes of interfaces. The Eulerian approach is more robust in ge neral but it needs elaborate procedures to deduce the interface location based on the volum e fraction information. Typical Eulerian methods are the level-set (LS) method and the vol ume of fluid (VOF) met hod. Interface is often constructed after the field solution obtained. T hus the interface construc tion decouples from the field equation solver. Since the grid is fixed in the Eulerian approach, the grid generation required by the Lagrangian approach is obviated. Another advantage of the Eulerian approach over the Lagrangian approach is that the topo logical changes of interfaces are automatically taken into account in the Euleri an approach, which makes the Eulerian approach more suited for flows with complex interface deformations. Howe ver, the Eulerian approach suffers several disadvantages. The interface shape is smeared off. If high resolution information of interface is desired, an adaptive local grid refinement may be needed, which is another complicated process.

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75 Thus, some physics may be lost if the complicated grid refinement is not available, such as the uniqueness of the shape interpre tation, and continuity and smoothness of the interface (Shyy 2004). The Eulerian-Lagrangian approach combines the advantages of both Lagrangian and Eulerian approaches. In this approach, the grid is fixed as the Eulerian approach. Grid refinement is not needed. The interface is explicitly tr acked by some marker points which are the Lagrangian components. This explicit interf ace-tracking can provide detailed interface information as the Lagrangian approach (Shyy et al. 1996). The Eulerian, Lagrangian and Eulerian-Lagra ngian approaches are numerical methods for two-phase flow computations with interfaces. Physically the interface can be modeled as a finite thickness line (2D) and surface (3D), or a zero thickness line (2D) and surface (3D). They are called the continuous interface method (CIM ) and the sharp interface method (SIM), respectively. At the macroscopic modeling level, the SIM is consistent with the concept of the continuum mechanics (Shyy 2004). Interface is tracked explicitly with some computational cells which are cut by the interface. Thus interface smear ing is not involved. In contrast to the SIM, the CIM does not track interfaces explicitly. Of these two methods, the SIM and CIM, if detailed interface information is desired, the SIM is a better choice since it can provide a second-order accuracy while the CIM can only provide a first-order accuracy in general. However, the CIM has some advantages which the SIM lacks. For in stance, the CIM can be easily adopted for threedimensional flows with topological changes of in terfaces; the CIM is part icularly useful for flows with phase changes; and the CIM is esp ecially appropriate for some problems that are currently tough for the SIM, such as contact line dynamics (Pis men 2001; Lee et al. 2005). More

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76 detailed information for moving interface comput ations based on the macroscopic computation point of view can be found else where (Shyy et al. 1996, Shyy 2004) The method of lattice Boltzmann equation is e ssentially a kinetic equation solver over mesoscopic length scale (Succi 2001). Physically the essential length scal e of this method is closer to microscopic level than macroscopic continuum-bas ed methods. Thus it is relatively easier for the LBE method to incorporate micr oscopic modeling for two-phase flows with interfaces. As a result, the LBE method has nume rous advantages that the molecular dynamics method has, which are especially useful for the simulations of pha se interfaces of non-ideal gases (Shan, et al, 1993; He, et al, 1999) or binary fluids and near wa ll treatment at micro-fluid level (Nie, et al, 2002). Phase segreg ation and interfacial dynamics can be achieved naturally by incorporating intermolecular inte ractions. Because the interface spreads over se veral lattices in the LBE method (He et al. 1999) most of the twophase LBE methods can be considered as CIM. As one of the continuous interface methods, th e LBE method has all the advantages of the CIM for two-phase flows. However, the nature of the modeling on the mesoscopic level may allow the LBE method to incorporate the inte rmolecular interaction more easily than the conventional CIM. Therefore, the LBE method ma y be a good alternative for two-phase flows simulations (Fang et al 2001; Lee et al. 2005). 4.2 Literature Review on LBE Method for Two-Phase Flow Computation The first lattice-based two-pha se flow model was the latti ce gas color particle model proposed by Rothman and Keller (Rothman et al. 1988). This model has not been widely used in practice since it suffers from the defects of th e lattice gas method, such as the computational noise. The LBE method overcomes the natural de fects of the lattice ga s method. Gunstensen et al. (Gunstensen et al. 1991; Gunstensen et al 1993) proposed the first LBE multiphase model developed from Rothman and Kellers lattice gas model. Although Gunstensen et als model

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77 possesses the essential f eatures of the LBE method, i.e., th e Galilean invarian ce and statistical noise free, it cannot simulate multiphase flows w ith different densities and viscosities. Grunau et al. (Grunau et al. 1993) improved Gunstensen et al.s model by using a single-time relaxation approximation and a proper equilibrium distribution function. This model can be used for flows with different densities and viscosities. Since these models were developed from Rothman and Kellers pioneering lattice gas color particle model, all require a re-color step to maintain the interfaces. Shan and Chen proposed a new two-phase LBE model hereinafter referred as (S-C model) in which the non-local microscopic particle inter action is incorporated (Shan et al. 1993; Hou et al. 1997). The molecular interactio n introduced into the ideal gas LBE model is represented by an interaction potential which models the multiphase separation and dynamics. Although the S-C model has performed much better for the multipha se flows than the R-K model (Hou et al. 1997), it possesses some anisotropy (He et al. 1998). The free-energy LBE model developed by Swift et al. (1995) uses an equilibrium distribution which ma kes the pressure tensor consistent with that of the free-energy function of non-uniform fluids. However th e Galilean invariance is not satisfied. Although the aforementioned models are based on different physical considerations, He et al. pointed out that they all originate from th e kinetic theory which can be represented by the continuous Boltzmann equation with certain approximations (He et al 1998). Subsequently He et al. proposed an improved LBE scheme using the single-time-relaxation approximation to simulate of multiphase flows in the incompressibl e limit (He et al. 1999a). In this new model, two distribution functions corresp onding to two evolution equations are employed. One acts as an index function to track interfaces implicitl y between different phases. The other is an

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78 evolution equation for pressu re distribution which satisf ies the mass and momentum conservation on the macroscopic level. Molecular interactions, such as the molecular exclusionvolume effect and the intermolecular attraction, are incorporated into this model. These interactions maintain the phase separation by th e mechanical instability on the supernodal curve of the phase diagram (He et al. 1999). Compared to the S-C model, this thermodynamics-based concept has much more flexibil ity for maintaining the phase separation. Furthermore, the numerical stability of He et al.s model is improved by reduci ng the numerical errors from O(1) to O(u) during the calculation of the molecular interactions. One common deficiency in all the aforementione d models is that they cannot be used for the multiphase flows with large density ratio (Lee et al. 2005). R-K model, S-C model, and Swift model can only be used for the two-phase flows w ith density ratio less than 2. He et al.s improved model brings the dens ity ratio to around 10. Chen et al. (1998) discretized the Boltzmann equation with a total variation dimini shing scheme. In Chen et al.s method, the second-order Runge-Kutta time-matching was us ed for the discretized Boltzmann equation (DBE), while the collision term and the intermolecular interaction term were treated explicitly as source terms. Although the use of TVD scheme im proved the stability of the LBE method for the large density ratio case, the accuracy is reduced in the presence of large gradients because of the additional numerical dissipati on of the TVD scheme. Inamuro et al. (2004) proposed another LBE model for the two-phase flows with large de nsity ratio. They also used two particle distribution function equations: one as an index function to indicate the phase separation, the other for the momentum evolution but without the pressure gradient. Velocity and pressure are coupled by an additional pressure Poisson equati on to satisfy the incompressibility requirement. The procedure is analogous to the fraction step method (Ferzige r 2003) which is used to solve

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79 the macroscopic Navier-Stokes equation. It is well known that it is time consuming to solve the Poisson equation (Shyy, 1997c). Although Inamuro et al.s were able to simulate multiphase flows with large density ratio us ing their scheme, such as risi ng bubbles in liquid, the natural simplicity of LBE method is lost due to the necessity of solving the pressure Po isson equation. Lee and Lin developed an implicit two-ph ase LBE model which can simulate large density/viscosity ratio problems without usi ng the iteration procedure (Lee and Lin 2005). Lee and Lins model is similar to He et al.s model but with a different pressure updating process. In He et al.s model, the pressure in calculating the intermolecular interaction term is updated from equation of state (EOS). Lee and Lin argued that this pressure updating process can lead to the spurious pressure fluctuations at the interfaces due to the EOS. Lee and Lin proposed a pressure evolution equation which can overcome this diffi culty and at the same time allow for large density difference across the phase interfaces wh en phase change occurs due to pressurization and depressurization (Lee et al. 2003). Using this model, Lee and Lin simulated 1-D advection equation with a source term, a stationary droplet, an oscillating droplet and a droplet splashing on a thin film at a density ra tio of 1000 with varying Reynolds numbers. However, very little information on the details of the flow field near the large gradient interf acial region was reported. In the following sections, He et al.s LBE mo del for two-phase flows will be reviewed. It will be applied to the Rayleigh-Taylor instabil ity problem for a code validation. The interface behavior of Lee and Lins implicit LBE is then studied. To maintain the interface thickness as sharp as the initial interface thic kness and improve numerical stabil ity, a filter-based two-phase LBE model is developed based on He et al.s model. Stationary liquid flow around a static bubble, the capillary wave caused by two superposed fluids with same viscosity, and flow

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80 induced by a rising deforming bubble ar e computed at de nsity ratio of O(102) to asses the efficacy of the proposed two-phase LBE model. 4.3 He et al.s Isothermal LBE Model for Two-Phase Flow 4.3.1 The Boltzmann Equation for Non-Idea Fluids The Boltzmann equation for non-ideal -fluid flows interm olecular interaction force is (He et al. 1998) eq eqfff ff tRT uFG (2.1) The single-relaxation-time BGK model is used in this equation. The derivation of Eq. (2.1) is based on the assumption that the distribution func tion gradient with respect to the particle velocity is equal to the equilibrium distribution function gradient with respect to the particle velocity (He et al. 1998). Compared to the standard LBE model for ideal gas flows, the last term on the RHS of Eq. (2.1) represents the e ffects of the intermolecular interaction F and gravity G. In Eq.(2.1), R is the gas constant and is the relaxation time. The Maxwellian distribution for an equilibrium state is 2 /2exp 2 2eq Df RT RT u (2.2) where D is the dimension of the space. The intermolecular attractive force originates from the van der Waals theory (Rowlinson et al. 1982b). This intermolecular force can be expressed as 22MFa F (2.3) The parameters a and result from the intermolecular pair-wise potential U 1 2rdaUrdr, 21 6rdrUrdr, (2.4)

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81 where d is the effective molecular diameter, dris the effective differential volume. Besides the intermolecular force, He et al. also included the exclusion-volume effect into the intermolecular interaction 22lnEVbRT F (2.5) which accounts for the extra volume effect of non-ideal fluids (Chapman et al. 1970). The parameter b is given by 32 3 d b m where m is the single molecular mass. The parameter in (2.5) is the increase in the collision probability due to the increasing fluid density. This parameter can be expressed as a polynomial in terms of the product of the density and the parameter b 235 10.28690.1103... 8 bbb (2.6) The overall intermolecular interaction is thus the sum of the intermolecular force and the exclusion-volume effect M FEV F FF (2.7) The intermolecular interac tion can be recast as s F F (2.8) where s F is associated with the surface tension, 2 sF. The potential term is associated with the pressure gradient as pRT (2.9) Thus the term depends on the equation of state (EOS) a nd it in turn determines the phase segregation. From the van der Waals theory, if th e temperature is lower than the critical point of a fluid, the phase segregation can be genera ted by the molecular attr action. The critical temperature is determined by the equation of st ate. For the van der Waals equation of state

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82 21 RT pa b (2.10) the critical temperature is 8 27cra T bR. If the hard sphere model is used for the repulsion interaction between molecules, the Carnahan-Sta rling equation of state is obtained (Carnahan and Starling 1969, 1972) 23 2 31 444 1 4 bbb pRTa b (2.11) and the critical temperature is 0.3773cra T bR where b is the co-volume of spheres. During the computation, the parameters a and b should be selected to sa tisfy the condition that the temperature is lower than the critical temperature. He et al. pointed out that in Eq. (2.8) is usually very large at the interfaces. As a result, the calculation of can involve to a substantial numerical error, which lends the numerical scheme unstable. In order to reduce this sensitivity, He et al. introduced another variable g to replace the di stribution function f, 0 gfRT (2.12) where 0 is the value of u evaluated at 0 u, and 2 /21 exp. 2 2DRT RT u u The derivative of g is then 0. D DggDf gRT DttDtDt (2.13) Since is a function of density, we can rewrite D Dt Dd Dtttdt +uuu (2.14)

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83 For incompressible flow ddd dtddt was then treated as zero in the wok of He et al (1999). Substituting Eqs. (2.1), (2.12) and (2.14) into (2.13) the evolution equation of g can be written as 0eq sggg g t uuFGu (2.15) The equilibrium dist ribution function of g is 0.eqgRTu (2.16) From g, the macroscopic variables pressure p, instead of the density and velocity u can be calculated pgd RTgd u (2.17) Note the function is calculated from the hydrodynamic pressure as hpRT (2.18) where the hydrodynamic pressure hp is obtained from Eq. (2.17), instead of equation of state (Eq. (2.11)) as was typically the case for single phase flow. The sensitivity of in Eq. (2.15) is alleviated by multiplying ( 0 u) which is proportional to the Mach number and is small in the incompressible limit. For multiphase flows, besides pressure and velocity which are given by (2.17), density is another important variable which describes the different phase di stributions. Density is only a single function of pressure for isothermal single-phase flows, but it cannot be uniquely determined by the pressure alone for isothermal mu ltiphase flows. Therefore, like other diffusive interface methods for multiphase incompressible fl ows, an index function evolution equation is needed for tracking different phases and then th e density or composition of fluids could be obtained from this index function. In the LBE models for two-phase flows, Eq. (2.1) can serve as

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84 the index function evolution equation. Since the gravity does not affect the mass conservation, He et al. dropped the gravity in the index function evolution equati on. It is given by, eq eqDffff ff DttRT u (2.19) It is noted that is the function of the index function instead of density Thus it is defined as tpRT (2.20) where the thermodynamic pressure tp is the thermodynamic pressure calculated from the equation of state Eq. (2.11), while the density is replaced by the index function 23 2 31 444 1 4tbbb pRTa b (2.21) To recapitulate, the evolution equations for f and g are eq eqDffff ff DttRT u (2.22) 0eq sggg g t uuFGu (2.23) where tpRT, and hpRT. The equilibrium distributions for f and g are eqfu (2.24) 0,eqgRTu (2.25) where 2 /21 exp. 2 2DRT RT u u The macroscopic variables can be obtained from the moments of f and g fd pgd RTgd u (2.26) Density and viscosity can be calculate d by the linear interpolation from

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85 ,l lhl hl (2.27) .l lhl hl (2.28) where l and h are the material densities of light and heavy fluids, respectively; l and h are the kinematical viscosities of light and heavy flui ds. The above formulation is equivalent to the following macroscopic equati ons (Zhang et al. 2000): pp t u (2.29) 1 0p RTt u (2.30) pG t u uuuu (2.31) 4.3.2 Lattice Boltzmann Scheme for Multiphase Flow in the Near Incompressible Limit The lattice Boltzmann equation can be obtai ned by discretizing th e Boltzmann equations described in the previous section. The discretization procedure is same as given in Chapter 3 for single phase flows. After discre tization in the phase space, the discretized Boltzmann equations are eq eqfff ff tRT eu e (2.32) 0eq sggg g t eeuuFGu (2.33) To improve the accuracy and maintain the exp licit computational scheme, He et al. further introduced two new variables, which are 2tff RT eu u (2.34) 1 0 2 s tgg euuFGu (2.35)

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86 The new evolution equations in term s of these two new variables are ,, ,,21 2tt eq tftft ftft RT xex xxeu u (2.36) ,, ,,21 0. 2tt eq s tgtgt gtgt xex xx euuFGu (2.37) where is still the non-dimensional relaxation time, /t The macroscopic variables can be calculated in terms of the ne w distribution functions as ,f (2.38) 1 2tpg u (2.39) 1 2 s tRTegRT uFG (2.40) In the D2Q9 model, the kinetic viscosity is inde pendent of surface force and is related to the nondimensional relaxation time as 0.5.tRT (2.41) Zhang et al. (2000), and McCracken et al. (2005) have used the fo llowing integral relationship to analytically rela te surface tension with the coefficient : 2kIakdz z (2.42) where z is a direction normal to a flat interface. 4.4 Code Validation: Rayleigh-Taylor Instability A two-dimensional D2Q9 LBE (Eq. (2.36)) code has been developed to solve Eqns (2.36)(2.41). To validate the code the Rayleigh-Taylor instability is simulated and the results are compared with analytical solutions. The Ra yleigh-Taylor instability problem is a very challenging computational task because it pos sesses great complexiti es including strong non-

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87 linearity associated with growth of the secondary Kelvin-Helm holtz instability and turbulent mixing at later stages. On the other hand, the Rayleigh-Taylor instability has been systematically studied in the past (Chandrasekhar 1981), and the re sults from previous studies can be used to validate the present LBE simulation. 4.4.1 Linear Analysis of Rayleigh-Taylor Instability The Rayleigh-Taylor (RT) instability occurs when a heavier fluid of density 2 superposes over a lighter fluid of density 1 and the lighter one accelerates the heavier one in the direction normal to the plane interface between these tw o fluids. The heavier fluid moves downward producing spikes and lighter flui d grows upward producing bubbles. The linear analysis is usually used to predic t the linear instability. In this approach, the perturbation analysis is used to linearize the non-linear convecti on terms. The resu lting pressure and density are functions of the vertical coordinate, say z, only (Chandrasekhar 1981). Chandrasekhar gave the relation between the grow th rate of the disturbance as a function of wave number: 2 211212 2 12 2 1122211212 3 2 112212 2At14 4 4 0, gkkT fqfqkkff g k fffqfqkff k ffqkqk (2.43) The Atwood number is defined as 21 12At= (2.44) where is growth rate, g is gravity, k iswave number, is viscosity, and 1 1 12f 2 2 12f 2 1 1qk 2 2 2qk

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88 4.4.2 Rayleigh-Taylor Instability: LBE Results The two-dimensional Rayleigh-Taylor instability with single-mode is simulated using the same physical parameters as in He et al.s work (He et al. 1999a). Alth ough He et al. presented the multi-mode Rayleigh-Taylor instability results the single-mode Raylei gh-Taylor instability is preferred to serve as the benchmark for the present numerical simulation. The computational domain is bounded by the top and bottom walls on which the no-slip boundary condition is applied. The periodic bounda ry condition is applied on the two lateral boundaries. Surface tension is neglected in this si mulation and the kinetic viscosities of the two fluids are assumed to be same. The function hpRT is calculated from the pressure which satisfies the Carnahan-Starling equation of st ate, i.e., Eq. (2.11). In the Carnahan-Starling equation, the parameter a is chosen to be 2bc, which induces sufficient molecular interaction to separate different phases. He et al. (1999), Zhang et al. (2000) and McCr acken et al. (2005) reported that 24 abc is sufficient to separate phases. With these values for a and b, the function I a in Eq. (2.42), that is 2 I adz z can be numerically determined for CanahanStarling fluid (Zhang et al. 2000) as 1.5 0.50.1518 13.385c caa Ia aa (2.45) where 3.53374ca is the critical value of Carnahan-Starling equation of state, below which a fluid cannot separate into differe nt phases. With these parameters, the range of the index function can be theoretically or numeri cally determined by the equation of state. For Canahan-Starling equation of state, Zhang et al (2000) obtained the range of th e index function as 0.02381-0.2508. In order to make meaningful comparisons two dimensionless numbers, Reynolds number and Atwood number, are used for this problem. Reynolds number is defined as

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89 Re UW (2.46) where the velocity scale is Wg and length scale is the domain width. The time scale is then WW T g Wg 4.4.2.1 A Single-Mode Growth Rate He et al. computed the singlemode growth rates of the Rayl eigh-Taylor instability with different Atwood numbers and wa ve numbers. Following their practice, we compare the LBE results with the linear analysis results described by Eq. (2.43). In the early stage, the magnit ude of the interfacial disturbance is much smaller than the wave length which is the domain width in our st udy. At this stage, th e perturbation of the interface exhibits an exponential growth 0 ththe (2.47) where h is the amplitude at time t, 0h is the initial amplitude, and is the growth rate of the interface perturbation. Fr om Eq. (2.43), we know the growth rate is a function of Atwood number and wave number 2 k W if the surface tension is neglected and the viscosities of the two fluids are same. For such cases, Eq. (2.43) reduces to 232 222 12121212 2 244 140. gbbb ffqbffqbffqbbff n nn (2.48) From this equation we can solve the growth rate measured in units of 1/3 2/ g in term of the wave number measured with 1/3 2/ g. The LBE simulations were car ried out over a range of the wave numbers using 64 x 128 grids for obtaining n. We choose three Atwood numbers: 0.2, 0.5 and 0.8. The theoretic and numerical results are shown in Figure 4-1, and they agree well with

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90 each other. This agreement indicates that He et al.s model could correctly capture the physics of the Rayleigh-Taylor instability a nd our code is thus validated. Figure 4-2 shows the same growth rate from He et al.s computation. Our results are slightly different from He et al.s results. The difference in the growth rate may result from the difference in the initial conditions used since He et al. did not provide th eir initial condition in their paper. 4.4.2.2 Flow Field at Re=2048 and At=0.5 The simulation was carried out on a 256 1024 grid. In order to ensure that the compressible effect can be neglected, the velocity magnitude must be small. For this case, He et al. chose the characteristic velocity U as 0.04 Wg Thus the desired acceleration can be obtained in the lattice units is 2 60.04 6.2510 g W (2.49) The Reynolds number is defined as ReUW (2.50) The viscosity could be obtained from the corresponding Reynolds number Re UW v In He et al.s study, the viscosities of both phases are take n as same. The time scale is chosen as /TWg. Figure 4-3 shows the evolution of the fluid interface from a 10% initial perturbation in which the time steps are counted in term of /TWg. The interface is plotted by 19 equally spaced density contours, which are same as in He et al.s work. Figure 4-5 shows the density variations in the vertical direction at two locations of the interface: one for the bubble front along the center line of the computational domain from the

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91 bottom boundary to the top boundary, and the other is for the spike front along the side line of the computational domain from the bottom bounda ry to the top boundary. From Figure 4-5, it seems that He et al.s model can maintain a sharper interface with just 3-4 grids, although a noticeable oscillation is present at the trailing edge of the interface for both the bubble and the spike. 4.4.2.3 Flow Field at Re=256 and At=0.5 He et al. also studied the effect of the vi scosity on the Rayleigh-Taylor instability by keeping all parameters same as those in Case 1, but reducing the Reynolds number down to 256. Figure 4-6 shows the interface evolu tion with time, which is also very similar to He et al.s results. As expected, as Reynolds number is re duced, the viscous effect between light and heavy fluid is enhanced compared to the high Re case. The side spikes rema in much smoother than those of the high Re case and no spiral vortices show up. 4.5 Lee-Lins Implicit LBE Two-ph ase Model and He et al.s Model with Surface Tension In the previous section, He et al.s model is introduced and the Rayleigh-Taylor instability is simulated as the code validat ion for this model. Although He et al.s model can capture the interface thickness as sharper as the initial one, this model cannot handle the two-phase flows with large density ratio. However large density ra tio is desired for practical purposes. Lee and Lin (2005) proposed an implicit LBE model for tw o-phase flows with large density ratio, which is similar to He et al.s model. In this s ection, Lee-Lins implicit LBE model will thus be introduced and reviewed first. Then the surface tension formulation in He et al.s model is assessed. 4.5.1 Lee-Lins Scheme for Large Density Ratio Large density ratio could cause substantial os cillations across inte rfaces (Appert et al. 1995; Qian et al. 2000). On the gas-liquid two-pha se interfaces, the molecular interactions are

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92 often modeled as the gradient of the non-ideal part of the thermodynamic pressure calculated from EOS and surface tension which is associated with the density gradients (Eq. (2.8). With very large density ratio, the accompanying pressure gradient could be very large at the interface. This can easily induce computational instabi lity as large pressure fluctuation can quickly propagate through the whole flow field. To s uppress the instability and extend the operating parameter range, Lee and Lin ( 2005) proposed an implicit two-pha se LBE model with a stable discretization formula for surfac e force calculation. They reported that their method can handle large density ratio as high as 1000:1. Their basic approach is very similar to that of He et al. The difference between these two models lies in that Lee-Lin used an implicit LBE scheme with some complicated formulations for the force calculation, which may induce s ubstantial amount of numerical diffusion. In Lee-Lins scheme, instead of computing the in dex function as in He et al.s model, they compute the density based on the corresponding distribution function f which is governed by 22 21 ,iiisij eq i iseuc ff eff txc u (2.51) The distribution function g is used for computing the pressure and velocity. The evolution for g is 2 2 21 0 ,iiis eq i i s iiikkjij seucp gg egg tx c eu c u u (2.52) where the notations are the same as t hose used in He et al.s model, and 4 2lh lh

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93 which is a cubic function of density, servi ng as the equation of state. The constant is associated with the surface tension intensity. Lee and Lin used an implicit scheme to discretize these two distribution functions, that is, pre-streaming collision-streaming-post-streaming collision process. The discretization can be described as follows: Pre-Streaming collision step: 2 2 , 2 ,1 ,,0 22 2iiis eq t s t iiikkjij s teuc t gtgtgg c eu t c x x xxxu u (2.53) 22 2 ,1 ,,. 22iiisik eq t s teuc t ftftff c x xxxu(2.54) Streaming step: ,, g tttgt xex (2.55) ,, ftttft xex (2.56) Post-streaming collision step: 2 2 2 ,1 ,, 21 2 0 212 2 212eq ttt iiis s ttt iiikkjij s tttgtttgtttgg euc t c eu t c xe xe xexexe u u(2.57) 22 2 ,1 ,, 21 2 212eq ttt iiisik s tttftttftttff euc t c xe xexexe u (2.58)

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94 The density, velocity, and hydrodynamic pressure are calculated afte r the streaming step as, f 2. 2s i ic t pgu x 2iikkjijt ug e(2.59) In the conventional CFD technique s, the implicit procedure is typically accomplished in an iterative manner. In Lee and Lins scheme, there is no such iteration. Their scheme appears to be more like a prediction-correction scheme. The discretization for the source term represents the discretization of the intermolecular force terms in the LBE. Lee and Lin used a s econd order central differe nce (2C), a second order upwind (2B) and their combinati on to discretize the source te rms. The second order central difference is 2, 2C x x etxet d et dx (2.60) and the second order upwind is 2243 2B x x etxetx d et dx (2.61) The combination of the 2C and 2B is 2222,if0,MBBC xxxxetdddd dxdxdxdx (2.62) 2222,if0.MCBC xxxxetdddd dxdxdxdx (2.63) 4.5.2 Diffusion of Lee & Lins Implicit LBE model Because of the coupling between the two dist ribution lattice Boltzma nn equations of the two-phase model and the highly non linear nature of the equations, it is difficult to directly quantify the amount of dissipation in Lee-Lins implicit LBE. It is noted that their first equation

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95 (2.51) provides the evolution for the index function. The surface tension and the exclusive volume force are calculated from the derivatives of this index functio n. The second equation (2.52) allows one to compute the velocity fiel d which is the only nece ssary input to equation (2.51) through the use of the equilibrium distribution functions eqf and u If the velocity is kept fixed, the kinematic equation, (2.51), for f is then decoupled from the dynamic equation, (2.52), for g. This allows for an easy and accurate assessment of the interfacial diffusion associated with the implicit LBE for the inde x function without involving the complicated procedures used for the inte rfacial force calculation. To illustrate the interfacial diffusion of the implicit LBE scheme, a stationary droplet is simulated as a test case. In this test, velocity is kept as zer o throughout the whol e computational period. The grid is 250250 lattices. A periodic boundary cond ition is used for all boundaries. The range of the index func tion is 0.02381 to 0.2508 for a = 4 in Eq. (2.11). Th e initial radius of the stationary droplet is 40 lattices. Both th e standard center difference and Lee-Lins hybrid scheme (Eq. (2.62)-(2.63)) are used in the imp licit scheme in this test. For comparison purpose, He et al.s explicit scheme is also implemented for this case. Figure 4-7 shows the index function evolutions by using both He et. als explicit scheme and Lee-Lins implicit scheme with fluid velocity in the entire flow field set to 0 for the entire duration of the computation. After 10000 time step s in lattice unit, the position of the droplet interface remains unchanged and little diffusion is obs erved when He et al.s explicit scheme is used. However, the interface smears off significan tly after only 1000 time steps in lattice unit when Lee-Lins implicit scheme is used. There is no significant differenc e by using the standard center difference and Lee-Lins h ybrid scheme for derivative cal culations. This test clearly demonstrates a large amount of num erical diffusion of the implicit scheme. Due to this attractive

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96 interface preserving feature, He et al.s model is adopted for high density ratio two-phase flow computation, although a significant improvement on the computational stability is required. 4.5.3 Modeling the Surface Tension In the previous section, the Rayleigh-Taylor instability was simulated by using He et al.s model for basic code validation. In those simulations, only the eff ect of gravity was considered, and the surface tension force is set to zero. Altho ugh He et al.s model can maintain the interface thickness within 3-4 lattices, it is not clear if such a feature is still maintained when the surface tension is included. Although Zha ng et al. (2000) tested He et al.s model with the surface tension, they did not show time evolution of interfaces for two-phase problems with surface tension. To test this model with the surface tension, a stationary bubble with a density ratio of 2 and a dynamic viscosity ratio of 2 is simulated. The computational domain consists of 256x256 lattices. The bubble diameter is 80 d lattices in length. The surface tension is 21.4610 The corresponding Laplace number defined as 1 2 1d La (2.64) is 23360, where 12 and 10.01 are density and dynamic visc osity for fluid outside the bubble, respectively; is the surface tension modeled using Eq. (2.8). The time scale is defined as chd t (2.65) Thus one dimensionless time step is equal to 55 lattice time steps. To monitor the magnitude of the spurious velocity, the capillary numb er is adopted here. It is defined as maxU Ca (2.66)

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97 where maxU is the maximum spurious velocity. Figur e 4-8 shows the capillary number evolution with respect to the dimensionless time t/tch. After t=2000, the capillary number does not change significantly, and its value is 49.8310 The maximum dimensionless spurious velocity (Ca) is 31.4410 at t=2000. Figure 4-9 shows the density pr ofiles and pressure profiles after 2000 dimensionless time steps. The density profiles indicate that the bubbl e interface remains as sharp as its initial one even after 55x2000=110,000 lattice time steps. Typical inte rface thickness is about 5-6 lattices wide. However the pressure prof ile at t=2000 is not monotonic. A big wiggle in the pressure profile across the bubble interface emerges that causes the pressure inside the bubble to deviate from the exact pressure value that is initially specified. 4.6 A Filter-based LBE Model with a New Surface Force Formulation and VolumeCorrection 4.6.1 Surface Tension Calculation In the previous section, a big wiggle in the pressure profile appears across the interface. For a stationary bubble, the pressure jump is bala nced by the surface tension in equilibrium. This wiggle in the pressure profile thus implies that there is also a sufficiently large wiggle in the surface tension profile across the in terface that created the large wiggle in the pressure field. To identify the wiggle in the surface tension profile, a stationary bubble is calculated here with a larger density ratio 100, a viscosity ratio 10, and its diameter 40. The surface tension profile is shown in Figure 4-10b along the vertical cen tral cross-section (Figure 4-10a). The surface tension variation shown in Fi gure 4-10 does have a big wiggle across the interface. This indicates that the surface tension fo rmulation in He et al. s model, from numerical point of view, cannot appropriate ly describe the real macrosc opic physics. Further computation

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98 and examination indicate that such large osci llation in the pressure and surface tension force often lead to computational instability at high density ratio. To eliminate this wiggle, a new surface tension formulation originated from the diffusion interface method (Kim, 2005) is employed in this work, sF (2.67) where is the coefficient calculated from Eq. (2.42) For the same case shown in Figure 4-10, the surface tension calculated using Kims formul ation is shown in Figure 4-11. No unphysical wiggles in the surface tension profile emerge. The use of Kims formulation for the surface tension can thus help to improve the computati onal stability by elimina ting the oscillation of pressure across the interface. 4.6.2 A Filter-based Technique for Surface Tension In order to ensure the monot onic variation of the index f unction across the interface and maintain the interface as shar p as its initial one, a filter based on the central di fference scheme is applied to He et al.s model. This filter idea is same as that used in solving the N-S equations (Shyy et al. 1992). If the index function has a lo cal extreme with respect to its neighbors, its value would increase (decrease) to the minimum (maximum) value of its neighbors. The filtering algorithm proceeds by first scanning the index function on a node and its neighbors. If it is an extreme with respect to its neighbors, a correction is added at this node, an d additional corrections are added to its neighbors to maintain index function conservation. Let (,) I J represents the index function on a tested node, (,)ij represents the index function on its eight neighbors, (,)minij represent the index function on a neighbor node with the minimum

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99 value, and (,)maxij represent the index function on a nei ghbor node with the maximum value. Then the filter technique can be described as if ((,) I J < all (,)ij on its neighbor points) distance=abs((,) I J (,)minij ) (,) I J = (,)minij all (,)(,)minijij on neighbor points = (,)ij distance/7.0 else if ((,) I J > all (,)ij on its neighbor points) distance=abs((,) I J (,)maxij ) (,) I J = (,)maxij all (,)(,)maxijij on neighbor points = (,)ij + distance/7.0 endif Even though there is no significant difference in overall computations in this dissertation by using these two filters, we use the conservative filter in our code to avoid unexpected nonconservative effect in index func tion for future applications. By adjusting the index functions on its neighbor points, this filter algorithm ensures the conservation of the index function. This filter can remove the local extremes and ensure monotonic variation of the index function. Since the filter is only implemented on the nearest and the next nearest neighbor points, the diffusion effect of the filter is restricted only to one latti ce. Comparing to Lee-Lins scheme, this filter is easier to implement and has much less computational overhead. Another nonconservative filter technique is also tested. The non-conservative filter process is: if ((,) I J < all (,)ij on its neighbor points)

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100 (,) I J = (,)minij else if ((,) I J > all (,)ij on its neighbor points) (,) I J = (,)maxij endif Even though there is no significan t difference in overall computa tions between using these two filters, the conservative filter is used to av oid unexpected non-conservative effect in index function for future applications. A central difference scheme is used for the di scretization of the deri vatives in the surface tension force, in which only the nearest neighb or points and next nearest neighbor points are involved: 1,1,1,11,11,11,1 ,1,11,11,11,11,1/4/3 /4/3ijijijijijij ijijijijijijx y (2.68) 4.6.3 Volume Conservation and Mass Conservation Besides the surface force calculation, another f actor which can lead to unphysical results is the volume/mass conservation in He et al.s model at high density ratio. In this two-distributionfunctions scheme, the density pr ofile is determined by the inde x function whose evolution is governed by Eq. (2.29). On the right hand side of Eq. (2.29), p is the thermodynamics pressure in term of the index function while p is the hydrodynamic pres sure in term of the density As the density ratio increases, the compre ssible effect across the interface may not be neglected, even though Eq. (2.29) is derived under the incompressible limit. This can result in a non-conservation in volume and ruin the computati onal accuracy when density ratio is not small.

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101 In order to ensure the volume conservation, another correction step for volume is required. Since the distribution functions have more freedoms (9 com ponents for D2Q9 model) than macroscopic variables, correcting the mesoscop ic distribution functions can be much more difficult than correcting the macroscopic volume. Thus the correction step is carried out on the macroscopic volume as in the level set method (Son, 2001) in the current study. In the current LBE model the volume-correct ion step (Son 2001) is adopted after the macroscopic variables are calculated 0VV (2.69) where V is the disperse phase vol ume before the correction, V0 is the initial volume of the disperse phase. The variable is an artificial time. Eq. (2.69) is computed till the steady state V=V0 is reached. Eq. (2.69) can be recast to another form as 0 u (2.70) where 0VV u. This is an advection equation. Thus some high-order resolution schemes can be used for this equation to avoid numerical oscillations across discon tinuities. In this study, a finite volume method is used to solve Eq. (2.70) with the second order essentially nonoscillatory (ENO) scheme for the advection term. The second order ENO scheme for the advection term discretization is illustrated at the east su rface of a computational cell: 12 12 12,if0 ,if0 minabs,PW EP ee PEEEE ee eeeu xxxx u xxxx xxx (2.71)

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102 where P is the lattice point and also the center of the finite volume; E is the east lattice point next to P; W is the west lattice point next to P. The time derivative is discreti zed using the first order Eulers scheme. To examine the volume conservation, a rising bubble is simulated with the dimensionless parameters given in the caption of Figure 4-12. Th e disperse phase is the gas phase inside the bubble. The dimensionless parameters for th is case are Eo=0.971, Mo=1.26e-3, density ratio=100, viscosity ratio=10. With these parame ters, the bubble shape should almost be a circle in a 2D simulation. However, without the volu me-correction, the bubble volume increases with time (Figure 4-12A). At later stages, the bubble shap e even changes from a circle to an ellipse due to larger rising velocity. The bubble thus cannot reach its steady state. When the volume correction is applied, the bubble volume remains constant and its shape remains almost a circle during the rising process, shown in Figure 4-12B. Although the bubble shapes look reasonable in Fi gure 4-12B, the streamline profiles inside the bubble (Figure 4-13A) show that there are some mass sources/sinks inside the bubble just next to the interface. The presence of the mass s ources/sinks might result from the compressible effect across interfaces. After the volume corr ection step, even though the index function governed by (2.29) is corrected to maintain the constant bubble volume, it just provides a density profile to the LBE for the force calculations, rather than the velocity correction. Thus the divergence free condition for the velocity may not be satisfied. To prevent the incorrect velocity profile from being develo ped, He et al.s model should be further modified. In He et al.s model, the inco mpressible assumption is used in Eq. (2.14), that is Dd Dtttdt +uuu

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103 in which 0 dddd dtddtdt u was assumed. Although 0 u is correct theoretically for individual phases, this condition is not sa tisfied during the computation across the interface. The effect of u which represents the compressibility therefore should remain in the LBE model. To include the effect of u in the present model, we need to simplify the expression hpRT given by Eq. (2.18)). The material derivative of has two components: one is the material derivative of the hydrodynamic pressure hdp dt and another is the product of () RT with the material derivative of density d dt In these material derivatives, the unsteady terms are generally much smaller than the convec tion terms in one lattice time step because, for macroscopic flows, one dimensionless time step is much larger than one lattice time step, and thus there is no significant cha nge contributed from the unsteady terms. Therefore the dominant terms in these two material derivatives are the convection ones. We then have hh hhdpp pp dtt d RTRTRTRT dtt uu uu (2.72) Comparison between th e convection term hp u and the convection term RT u can provide insight into which one is the dominant component in d dt Using a dimensional analysis, Lee and Lee found that the spatial de rivative of hydrodynamic pressure is much smaller than that of (RT ) (Lee and Lin 2005). Figure 4-15 shows the magnitude contours of RT and hp of a rising bubble with Eo=0.971, M=1.26e-3, Re=5.19, density ratio=100, and vi scosity ratio=10. It is observed that |RT| is about O(104) larger than | RT|. Thus we can safely neglect the hydrodynamic pressure term in d dt to obtain

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104 hdp ddd RTRTRTRTRT dtdtdtdttt uuu (2.73) Instead of assuming 0 d dtt u, we take 0 t u from the mass conservation law for compressible flow. Thus we have d RT dt u (2.74) The LBE for the distribution function g is finally written as 00eqggg guuRT t suFGu(2.75) With this modification, the mass sources/sinks in the two-phase model can be eliminated. For the same rising bubble case, Figure 413B shows the streamlines insi de and outside the bubble. The unphysical mass sources/sinks have been removed. With above improvements the volume of the rising bubble does not change with time (Figure 414A) and the velocity e volution indicates the bubble reaches its steady state at th e later stages (Figure 4-14B). 4.7 Numerical Simulations To systematically test the present LBE twophase model, computations for stationary bubbles, capillary waves and rising bubbles served as the test cases are conducted below. 4.7.1 Stationary Bubble (1) A Stationary Bubble with Diameter 40 Lattices The computational parameters for this stationary bubble are listed below: 2100h hd La 100h l 10h l The computation of this case was perfor med on a computational domain of 201x201 lattices. A circular bubble of diameter 40 is placed at the center of the domain with the periodic

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105 boundary condition for all boundaries. The dimensionl ess time is defined as in Eq. (2.65). For this case, one dimensionless time chd t is equal to 480 lattice time steps. The characteristic feature of a stationary bubbl e is its spurious velocity field which is created entirely due to the numer ical errors causing imbalance of interfacial stresses. Physically the velocity should be identically zero everyw here and the pressure drop across the interface balances the surface tension force di ctated by the Young-Laplace Equation: ,for 2D bubblesp R (2.76) For the continuous interface methods, because of numerical errors in surface tension, pressure and velocity calculations, the numeric al velocity is not zero (Torres and Brackbill, 2000). As one of the continuous interface met hod, the LBE method also has these numerical errors. The spurious velocity of this stationary bubble is shown in Figure 4-15A. In the pressure jump plot (Figure 4-15B), the si gnificant unphysical wiggle in He et al.s model is essentially removed by using the filter-b ased LBE two-phase model. The density profiles at dimensionless time t=0, and t=100 (48000 lattice time steps) are shown in Figure 4-17. The inte rface thickness is maintained within 5-6 grids throughout the entire computational period, and does not diffuse out with time. The computed density profiles are indeed monotonic, which is a significant improvement over He et al.s original model (see Figure 4-5). (2) Grid Resolution and Eff ect of Fluid Property Jump To compare the present numerical results with theoretical on es, a numerical pressure drop is adopted (Singh, 2006) 1111inoutNN numinoutij ij inout p pppp NN (2.77)

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106 where Nout is the number of lattice outside the interface ( 0.99 h) and Nin is the number of the other left lattices. Two stationary bubble cases with different grid resolutions inside the bubbles are computed. One bubble has 40 lattices in diameter and the other has 80 lattices. The evolutions of the maximum spurious velocity with time are shown in Figure 4-18. The maximum spurious velocities in both cases approach to constants, indicating the computational steady states are reached. For the case with lower grid resolution, the maximum spurious velocity has oscillations at early times; for the case with higher grid resolution, the maximum spurious velocity reaches the steady state much faster without any oscillations. In the computations using the Navier-Stoke s models, the capillary number is generally adopted to monitor if a steady st ate has been reached and to asse ss the overall accuracy of the solution. A large capillary number resulting fr om larger maximum spurious velocity suggests that the spurious velocity or computational e rror is larger. However it is not an appropriate parameter in the LBE computations. From the definitions of Laplace number and capillary number, we have max1h hCaUd La (2.78) in which Laplace number, density, and viscosity are kept same in these two computations. In the Navier-Stokes equation computations, the bubble diam eter is fixed when re solution is increased. The capillary number is thus determined onl y by the maximum spurious velocity whose magnitude will decrease as the grid is refine d. In the LBE computations, however, the bubble diameter increases as grid is refined. Thus th e capillary number is de termined not only by the maximum spurious velocity bu t also by the bubble diameter.

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107 To assess the grid refinement effect on the co mputational accuracy, we can use the ratio of the numerical pressure drop to the theoretical pre ssure drop. Table 4-1 list s the relative error of the pressure drop, 1num exactp p for these two cases. The value for th e fine grid case is smaller than that for the coarse grid case, indicating that the computation carried out on the fine grid has better accuracy. The simulation for different Laplace numbers in the range of 100 to 1000 were found to have no appreciable effect on 1num exactp p as shown in Table 4-2. Dens ity and viscosity ratios (10 and 100 respectively) were also tested and foun d not to cause significant differences as shown Table 4-3 and Table 4-4, respectively. 4.7.2 Capillary Wave The second test case is a capillary wave, a small-amplitude motion of two superposed viscous fluids with same viscos ity (Prosperetti, 1981). In this test, gravity is not considered. Initially the interface between two stationary fluids is set up as a wave with a small-amplitude 0 H as shown in Figure 4-19. It wi ll damp off over time due to th e effects of viscous force and surface tension. The no-slip boundary condition is used on the top and bottom boundaries and the periodic boundary condition is used on the lateral boundaries. In this problem, the length scale is taken as 1k, in which k is the wave number defined as 2k NX and NX is the domain width. The time scale is taken as 1 0 in which 0 is the frequency defined as 3 0 12k Based on these time and length scales, th e dimensionless time and viscosity which characterize the wave motion are

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108 2 0 0,k tt (2.79) where is the kinematic viscosity of the fluid. With these dimens ionless time and viscosity, the theoretic solution for the dimensionless amplitude 0/aHH given by Prosperetti (1981) is 2' 2 2 4 0 '' 0 2 2 1 00 0414 exp 8141i i i i iizt z t aterfcterfcz Zz (2.80) where iz are the four complex roots of the following equation 3/22 4322 000004216413140 zzzz (2.81) and 4 1,iji jji Z zz. The parameter is given by 12 2 12 The test parameters taken here are -3 01.1310, 0.116 and 21/ is 100. The initial velocity is zero for the whol e domain and the distribution functions are assigned to the corresponding equilibrium values. The time evolutio n of the wave amplitude is shown in Figure 4-20. The time evolution of the dimensionless am plitude agrees well with that of the exact solution. The slight difference between them in the early stage may be caused by the numerical initial condition used in implem enting the present LBE model. 4.7.3 Rising Bubble A single bubble rising in an in itially quiescent flui d due to the buoyancy force is simulated and the results are presented in this section. In this problem the interaction of gravity and the surface tension determines the final steady bubbl e shape. Clift et al. (1978) gave a 3D bubble shape diagram (Figure 4-21) in terms of Eotvosnumber, Morton number and Reynolds number, defined as Raynolds number (Re)href hUd (2.82)

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109 4 3Morton number () h hg M (2.83) 2Eotvos number (Eo) = hgd (2.84) Although simulations are carried out for 2-D bubble in the present study, Clif t et al.s diagram is still used here to provide a rough, qualitative guidance. The computational domain is a rectangle (151 x 801). A circle bubble is initially placed at i=76, j=201 lattice with a radius R=20 lattices. The initial velocity is set to zero for the whole domain. The no-slip boundary condition is imposed on the top and bottom and the periodic boundary condition on the lateral boundaries. The de nsity ratio and viscosity ratio are 100 and 10 respectively here. Three groups of dimensionless parameters are chosen according to Clift et al.s diagram (1) Eo=0.971, M=1.26e-3, Re=5.19 (2) Eo=9.71, M=0.4, Re=6.92 (3) Eo=97.1, M=100, Re=9.78 The corresponding bubble shapes are cylindrical, el liptic, and dimpled-elliptic respectively. The computed bubble shapes are plotted in Figure 422, showing that the typical bubble shapes have been captured using the present 2D LBE two-ph ase model. Figure 4-23 shows the corresponding time evolutions of these bubbles. Due to larger shape deformation, the dimpled-ellipsoidal bubble rises faster. Figure 4-24 shows the velocity histories of thes e three bubbles. Because of the larger shape deformation comparing to cylindricall and el lipsoidal bubbles, the bubble velocity of the dimpled-ellipsoidal bubble exhibits larger oscillations at the early rising stages. At later rising

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110 stages, all three bubble velocities do not change significantly, indicating their steady states are reached. Figure 4-25 shows the density prof iles on the central cross sectio n of these three bubbles at T=0, 15 and 30 respectively. From these density pr ofiles, it can be observed that the interface does not diffuse out, and interface thickness is main tained with 5-6 grids. This relatively sharp interface is a desirable physical feature. Figure 4-26 shows the evolution of anothe r rising bubble during a period of in six dimensionless time with Eo=97.1, M=0.971, Re= 31.2, density ratio 100 and viscosity ratio 10. With these parameters, the bubble shape should be skirt-type in Clifts diagram and no bubble break occurs. However for this planar 2D si mulation, the bubble undergoes more complicated deformation process and breaks into two larger b ubbles with four satellite s at the dimensionless time t=6. Although this planar 2D bubble does not have the corresponding shape as the 3D bubble given same parameters, this simulation s hows the current code has the capability to automatically capture the complex dynamic process of bubble breaking-up. 4.8 Summary and Conclusion In this chapter, a new filter-based LBE method for immiscible two-phase flows is developed. First of all He et al.s model is introduced. The Rayleigh-Taylor instability is simulated for code validation by using this model. Lee-Lins implicit LBE two-phase is also studied in this chapte r. It was found that it possesses la rge computational diffusion across interfaces. To eliminate the unphy sical wiggles in pressure pr ofiles across inte rfaces, a filterbased LBE two-phase model is developed from He et al.s model along with a new surface tension formula, and a volume-correction step The present LBE model is tested by the computations of stationary bubbl es, a capillary wave and 2D ri sing bubbles. The computational

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111 results demonstrate that the present model can be us ed to simulate flows w ith large density ration up to O(102). The interface thickness is maintained within 5-6 grids.

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112 Table 4-1. Effect of grid resolution on co mputed pressure drop for Laplace number 100. The data were taken after 100 tim e steps and the density and viscosity ratios were set to 100 and 10 respectively. Grid resolution 1num exactp p Capillary number 1maxU Ca 40 lattice in bubble diameter 0.08 5.03-3 80 lattice in bubble diameter 0.04 6.31-3 Table 4-2. Effect of Laplace number on the ratio of numerical pressure drop to theoretical pressure drop: Bubble diameter is 40 lattic es. The data were taken at non-dimensional time = 100. Density ratio is 100 and dynamics viscosity ratio is 10. Laplace number 1num exactp p 100 0.08 1000 0.08 Table 4-3. Effect of density ra tio on pressure drop: Bubble diamet er is 40 lattice units. Viscosity ratio was set to 10 for Laplace number = 100 and the data were taken at nondimensional time = 100. Density ratio = 1 2 1num exactp p 1 0.06 10 0.07 100 0.08 Table 4-4. Effect of viscosity ratio on pressure dr op: Bubble diameter is 40 lattice units. Laplace number = 100 and density ratio is 100. The data were taken at nondimensional time = 100. Viscosity ratio = 1 2 1num exactp p 10 0.08 100 0.07

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113 k 0 0.5 1 1.5 0 0.1 0.2 0.3 0.4 0.5 theoreticalresults numericalsimulation At=0.8 At=0.5 At=0.2 Figure 4-1. The growth rate (measured in units of 1/3 2/g) of a disturbance vs. its wave numbers k (measured in units of 1/3 2/g). Figure 4-2. The growth rate pl ot of He et al .s computation

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114 t=0.0 t=1.0 t=1.5 t=2.0 t=2.5 t=3.0 t=3.5 t=4.0 t=4.5 t=5.0 Figure 4-3. Evolution of the fluid interface fr om a single mode perturbation for At=0.5 and Re=2048. The time is measured in units of Wg.

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115 Figure 4-4. He et als resu lts of the evolution of the fl uid interface from a single mode perturbation. The computational paramete rs are same as those in Figure 4-3.

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116 density 256 512 768 1024 0 1 2 3 4 density 256 512 768 1024 0 1 2 3 4 (a) t=1.0 density 256 512 768 1024 0 1 2 3 4 density 256 512 768 1024 0 1 2 3 4 (b) t=3.0 density 256 512 768 1024 0 1 2 3 4 density 256 512 768 1024 0 1 2 3 4 (c) t=5.0 Figure 4-5. Density profiles acro ss the bubble and spike fronts at th ree different time steps. The horizontal axis is the computational grid. Th e left panel shows the interface across the spike and the right panel shows the interface across the bubble. At=0.5 and Re=2048.

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117 t=0.0 t=1.0 t=2.0 t=3.0 t=4.0 Figure 4-6. Evolution of the fluid interface fr om a single mode perturbation. The At=0.5 and Re=256. The time is measured in units of Wg. X Y 50 100 150 200 250 50 100 150 200 250A X Y 50 100 150 200 250 50 100 150 200 250B Figure 4-7. Evolution of index function of a stat ionary droplet with zero velocity. (A) He et. al.s explicit scheme. (B) Lee and Lins implicit scheme. The red circles represent the initial index function contours, which indica te the droplet interface. The green circles represent the index function contours after 10000 tim e steps for (A) and 1000 time steps for (B) with the same contour values as red ones.

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118 t/tch Capillarynumber(Ca) 0 500 1000 1500 2000 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009 Figure 4-8. Evolution of the capillary number with for the stationary bubble simulation with density ratio 2 and dynamic viscosity rati o 2. The bubble diameter is 80 and Laplace number is 23360. Y Density 50 100 150 200 250 1 1.2 1.4 1.6 1.8 2densityprofilet=0 densityprofilet=2000 Y P r essu r e 50 100 150 200 250 -0.002 -0.0015 -0.001 -0.0005 0pressureprofilet=0 pressureprofilet=2000 Figure 4-9. Density profiles and pressure prof iles at t=0 and t=2000 of the stationary bubble simulation with density ratio 2 and dynamic viscosity ratio 2. The bubble diameter is 80. Laplace number is 23360.

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119 X Y 50 100 150 200 250 50 100 150 200 250a Y SurfaceTension 50 100 150 200 250 -0.01 0 0.01b Figure 4-10. Surface tension profile for a statio nary bubble computed from He et al.s method. Y S u r f ace T ens i on 50 100 150 200 250 -0.01 0 0.01 Figure 4-11. Surface tension calculated by Kims formulation. A B Figure 4-12. Rising bubble with Eo=0.971, Mo=1. 26e-3, density ratio=100, viscosity ratio=10. The time steps in this figure are t=0, 4, 8, 12, 16, and 20. Figure A shows the bubble shapes without correction on volume, figure B shows the bubble shapes with correction in volume.

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120 A B Figure 4-13. The velocity profiles and stream lin e of the spherical bubble at t=20 in Figure 4-12. T i m e B u b b l e V o l ume 0 5 10 15 20 0.8 1 1.2A Time BubbleVelocity 0 5 10 15 20 10-7 10-6 10-5 10-4 10-3B Figure 4-14. Time evolution of the volume (A) and velocity (B) of the rising bubble with volume correction in Figure 4-13B.

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121 6 0 0 E 0 3 5 0 0 E 0 3 2 5 0 E 0 3 1 0 0 E 0 3 1 0 0 E + 0 1 7 0 0 E + 0 0 1 0 0 E + 0 1 9 0 0 E + 0 0 RT hp Figure 4-15. The contours of space derivative of hydropressure and de nsity of a rising bubble with Eo=0.971, M=1.26e-3, Re=5.19, density ratio=100, viscosity ratio=10 Y Pressure 50 100 150 200 0 0.005 0.01 0.015 Exactpressureprofile Numericalpressureprofile (a) (b) Figure 4-16. Stationary bubble computation. (a) Spurious cu rrents of magnitude Ca=5.03 x 10-3, (b) comparison between computed and theoretical pressure jump.

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122 Y Density 50 100 150 200 0 20 40 60 80 100 densityprofileatt=0 densityprofileatt=100 Y Density 76 78 80 82 84 86 0 20 40 60 80 100 densityprofileatt=0 densityprofileatt=100 Figure 4-17. Density profiles of the stationary bubble with di ameter 40, density ratio 100, viscosity ratio 10, and La=100. t/tch M a ximumspu r iousvelocity(l a tticeunit) 0 20 40 60 80 100 0 0.0001 0.0002 0.0003 0.0004 Bubblediameter=40 Bubblediameter=80 Figure 4-18. Maximum spurious veloci ties for these two grid resolutions.

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123 Figure 4-19. Initial in terface profile for a capillary wave simulation. T i m e Amplitude 0 5 10 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 Simulation Theory Figure 4-20. Time evolution of the amplitude of a capillary wave w ith density ratio 100. wobbling spherical ellipsoid spherical-cap skirted Dimpled ellipsoidal-cap Eotvos numberReynolds number LOG M0(a) (b) (c) (d) Figure 4-21. Shape diagra m of Clift et al. (1978).

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124 (A) (B) (C) Figure 4-22. Computed bubble shapes. (A) Cylindr ical, (B) Ellipsoidal, (C ) Dimpled-ellipsoidal. Figure 4-23. Time evolutions of rising bubbles (A) Cylindrical, (B) El lipsoidal, (C) Dimpledellipsoidal. Time Bubblevelocity 0 5 10 15 20 25 30 0 0.0002 0.0004 0.0006 0.0008 dimpled-ellipsoildalbubble ellipsoildalbubble sphericalbubble Figure 4-24. Time evolu tions of bubble velocities.

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125 T= 0 Y Density 200 400 600 800 0 20 40 60 80 100 Y Density 200 400 600 800 0 20 40 60 80 100 Y Density 200 400 600 800 0 20 40 60 80 100 T=15 Y Density 200 400 600 800 0 20 40 60 80 100 Y Density 200 400 600 800 0 20 40 60 80 100 Y Density 200 400 600 800 0 20 40 60 80 100 T=30 Y Density 200 400 600 800 0 20 40 60 80 100 Y Density 200 400 600 800 0 20 40 60 80 100 Y Density 200 400 600 800 0 20 40 60 80 100 (A) (B) (C) Figure 4-25. Density profiles of rising bubbles (A ) Cylindricall, (B) Elli psoidal, (C) Dimpledellipsoidal.

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126 Figure 4-26. Time evolutions of a rising bubbles with Eo=97.1, M=0.971 and Re=31.2.

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127 CHAPTER 5 SUMMARY AND FUTURE WORK 5.1 Summary In chapter 2, the multi-scale computation ba sed on the Navier-Stokes solver has been illustrated by a conjugate heat transfer problem due to the fluid-solid interface in a MEMS-based thermal sensor. A two-level computational scheme was used to reduce the multi-scale effect due to the disparity in length scal es. Both grid refinement and th e convergence rate studies indicate that the multigrid solver has a good performance on this multi-scale problem. Through using the two-level scheme and the multigrid solver, the computational results show that in the very close vicinity of the sensor, temperatur e distributions of the conjugate cas es are similar to those of the corresponding single-phase cases, wh ile they are different in the region away from the sensor. For the conjugate cases, the effective heating lengt h is enlarged due to the substrate heating; shear stress distributions have obs ervable deviations from the si ngle-phase cases. This indicates that, though the buoyancy force does not change velocity profiles significantly, it noticeably influences velocity gradients near the thermal sensor, which introduces errors in the shear stress measurements. In chapter 3, the lattice Boltzmann equation me thod has been introduced as an alternative solver for the computational fluid dynamics. Its performance for variable viscosity has been tested through a lamina channel flow with a rapi d change of the variable viscosity near walls. Close analysis on the truncation error behavior due to the variable vi scosity and the boundary condition error shows that with rapid viscosity vari ations the boundary condition error of the BBL scheme does not induce noticeab le, additional errors, and the ove rall error of such flows is dominated by the truncation error itself, and the truncation error be havior of the LBE solution is consistent with that of the finite difference solution to the Navier-Stokes solution.

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128 In chapter 4, another multi-scale problem i nvolving interface, i.e. th e gas-liquid two-phase flow has been explored. A filter-based LBE two-pha se model is developed from He et al.s twophase model. In present model, a filter sche me along with a new surface force formulation originating from the diffusion-interface method is used to maintain the monotonic variation for index function. To preserve volume and mass c onservations, a volume-correction step and the numerical compressibility are adde d into present model. Stationa ry bubbles, a capillary wave and rising bubbles are tested by using present model, indicating that this filt er-based LBE model can capture interfaces just over 5-6 grids without unphysical wiggles; it extends the density ratio from O(10) up to order O(102) with conserved volume and mass. 5.2 Future Work For the multi-scale problems described by the Naiver-Stokes equations, if geometries are very complicated, structured grids may not ha ve good qualities due to large stretch of the computational cells. Adaptive structured grid or un structured grid may be used for these kinds of problems. For example, if the detailed geometry of the MEMS-based ther mal shear stress sensor needs to be considered, such as welding points at the end of the heatin g element, unstructured grid may be easier to use. The LBE model for two-phase fl ows is developed only for plan ar two dimensional flows in this dissertation. However, lots of two-phase flow problems are three di mensional in practice, such as rising bubbles, bubble mergence, droplet coll ision. Thus an obvious step of the future work is to extend the current 2D code to a 3D one. Although the volumecorrection can help to maintain the volume conservation, it is a macr oscopic computational step. The mechanism of non-conservation of volume in the mesoscopic level is not very clear. Th e assumption that the distribution function is equal to its equilibrium part in the inte rfacial force calculation should be examined since this assumption may be t oo rough when density ratio is not small.

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129 LIST OF REFERENCES Abe, T. 1997. Derivation of the lattice Boltz mann method by means of the discrete ordinate method for the Boltzmann equation. Journal of Computational Physics, 131: 241-246. Anderson, D. A., Tannehill J. C. and Pletcher R. H. 1997. Computational Fl uid Mechanics and Heat Transfer. McGraw-Hill, New York. Anderson, D. M., McFadden G. B. andWheeler A. A. 1998. Diffuse-interface methods in fluid mechanics. Annual Review of Fluid Mechanics 30: 139-165. Anderson, J. D., JR. 1995. Computational Fluid Dynamics: The Basics with Applications. McGraw-Hill, New York. Appert, C. and d'Humieres D. 1995. Density profiles in a diphasic lattice-gas model. Physical Review E 51: 4335-4345. Appukuttan, A., Shyy W., Sheplak M. 2003. Mi xed Convection Induced by MEMS-Based Thermal Shear Stress Sensors. Numerical Heat Transfer, Part A 42: 283-305. Aristov, V. V. 2001. Direct Methods for Solving the Boltzmann Equation and Study of Nonequilibrium Flows. Kluwer Academic Publishers, New York. Atkinson, K. E. 1989. An Introduction to Numerical Analysis. John Wiley & Sons Inc, New York. Bank, R. E. 1990. PLTMG: A Software Package for Solving Elliptic partial Differential Equations. SIAM Books, Philadelphia. Batchelor, G. K. 1967. An Introduction To Fluid Dynamics. Cambridge University Press, New York. Bhatnagar, P., Gross E. and Krook M. 1954. A mode l for collision processes in gases, I. small amplitude processes in charged and neutral one-component system. Physical Review 94: 511-525. Blosch, E., Shyy W. and Simth R. 1993. The Ro le of Mass Conservation in Pressure-Based Algorithms. Numerical Heat Transfer, Part B 24: 415-429. Bouzidi, M. 2001. Firdaouss M. and Lallemand P ., Momentum transfer of a Boltzmann-lattice fluid with boundaries. Physics of Fluids 13: 3452-3459. Bruum, H. H. 1995. Hot-Wire Anemometry, Prin ciples and Signal Analysis. Oxford University Press Inc., New York. Carey, V. P. 1992. Liquid-Vapor Phase-Change Phenomena: An Introduction to the Thermophysics of Vaporization and Condensation Processes in Heat Transfer Equipment. Taylor & Francis, New York.

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138 BIOGRAPHICAL SKETCH Jianghui Chao was born in 1972 in Daqing, Ch ina. He was received his Bachelor of Technology degree in Mechanical engineering from the Daqing Petroleum Institute, in 1994. Thereafter he joined Southeast University and gr aduated in 1997 with Mast er of Science degree in Thermal engineering. From 2001 he has been pursuing his Ph.D. degree in mechanical and aerospace engineering at the Univ ersity of Florida. His curren t research interests lie in computational modeling of two phase flows and heat transfer.


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Table of Contents
    Title Page
        Page i
        Page ii
    Dedication
        Page iii
    Acknowledgement
        Page iv
    Table of Contents
        Page v
        Page vi
    List of Tables
        Page vii
    List of Figures
        Page viii
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    Abstract
        Page xi
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    Introduction to multi-scale computational dynamics with interfaces
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    Continuum model: Navier-stokes equations with multi-scales
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    Error assessment of the lattice Boltzmann method for variable viscosity flows
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    Lbe method for immiscible two-phase flow computation
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    Summary and future work
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    References
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    Biographical sketch
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Full Text





MULTI-SCALE COMPUTATIONAL FLUID DYNAMICS WITH INTERFACES


By

JIANGHUI CHAO













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

2006

































Copyright 2006

by

Jianghui Chao

































To my parents, my wife and my son.









ACKNOWLEDGMENTS

I would like to express my sincere gratitude to Drs. Wei Shyy and Renwei Mei for

providing me the opportunity and flexibility to perform this work. I can not thank them enough

for being so patient and understanding for years and pushing me to learn more.

I would like to thank Drs. Siddharth Thakur and Alireza Haghighat for agreeing to serve in

my thesis committee. Since the surroundings dictate the quality of life in several ways, I thank

my research group members for helping with academic aspects while providing memorable

company for the past five years.

I thank my wife who has been the best friend for more than a decade and has been

extremely patient and understanding during all these years, and my son who has been giving me

lots of happiness. Finally I thank my parents who have been behind me every step of the way

providing their unconditional support.









TABLE OF CONTENTS



A C K N O W L E D G M E N T S ..............................................................................................................4

L IST O F T A B L E S ......................................................................................................... ........ .. 7

LIST OF FIGURES ............................................. ............ ...........................8

A B S T R A C T .......................................................................................................... ..................... 1 1

CHAPTER

1 INTRODUCTION TO MULTI-SCALE COMPUTATIONAL DYNAMICS WITH
IN T E R F A C E S .......................................................................................................................... 1

1.1 Motivation ............................................................................. 1
1 .2 O bje ctiv e s ................................................................................................ ..................... 3
1.3 Structure of the D issertation .................... ...............................................................4......

2 CONTINUUM MODEL: NAVIER-STOKES EQUATIONS WITH MULTI-SCALES ........ 5

2.1 Introduction to Computational Techniques for Solving Navier-Stokes Equations ............5
2.2 Incom pressible V iscous Flow Solvers........................................................... ...............8...
2 .2 .1 In tro d u ctio n ....................................................................................................... .. 8
2.2.2 A artificial Com pressible M ethod (A CM ) ........................................ ..................... 9
2.2.3 Semi-Implicit Method for Pressure-Linked Equations (SIMPLE) ...........................9
2.3 Computational Issues on Multi-scale Computations ............................. ..................... 12
2 .3 .1 Stiffn ess ........................................ ..... ............................................................ 12
2.3.2 Grid Requirements for Multi-scale Problems................................................... 14
2.3.3 M ethods to R educe Stiffness ................................................................ ............... 15
2.4 A Special Computation Example: Thermo-MEMS Computation Results....................19
2.4.1 Introduction to Thermal Anemometry for Fluid Velocity and Skin Friction
M e a su rem en t............................................................................................................... 19
2.4.2 G governing E quations ..................................................................... ................ 23
2 .4 .3 N um erical Schem es .. ...................................................................... ................ 25
2.4.4 C om putational Stiffness ......................................... ........................ ................ 25
2.4.5 G eom etry and G rid L ayout....................................... ...................... ................ 25
2 .4 .6 R results and D iscu ssion .......................................... ......................... ................ 30
2.4.7 Sum m ary and C conclusion: ......................................... ...................................... 36

3 ERROR ASSESSMENT OF THE LATTICE BOLTZMANN METHOD FOR
VARIABLE VISCO SITY FLOW S ............................................................. ................ 46

3.1 Introduction to Lattice Boltzmann Method and Wall Boundary Condition..................48
3.1.1 LBE BGK M ethod ... .. ................................. ......................................... 48
3.1.2 B oundary C conditions ...................................................................... ................ 5 1









3.2 Error Assessment of the LBE Method due to Variable Viscosity via a Fully
D developed C channel F low ............................................................................... .............. 55
3.2.1 Fully-developed Laminar Channel Flow with Variable Viscosity......................55
3.2.2. The Lattice Boltzmann Equation Treatment ....................................................58
3 .2 .3 A sse ssm en t ............................................................................................................. 5 9

4 LBE METHOD FOR IMMISCIBLE TWO-PHASE FLOW COMPUTATION ...................73

4.1 Overview of Immiscible Two-Phase Flow Computation ...................... ..................... 73
4.2 Literature Review on LBE Method for Two-Phase Flow Computation........................76
4.3 He et al.'s Isothermal LBE Model for Two-Phase Flow.............................................80
4.3.1 The Boltzmann Equation for Non-Idea Fluids ..................................................80
4.3.2 Lattice Boltzmann Scheme for Multiphase Flow in the Near Incompressible
L im it.................. ... .......................................................................... ......... 8 5
4.4 Code V alidation: Rayleigh-Taylor Instability ............................................. ................ 86
4.4.1 Linear Analysis of Rayleigh-Taylor Instability ................................................87
4.4.2 Rayleigh-Taylor Instability: LBE Results......................................... ................ 88
4.4.2.1 A Single-M ode G row th R ate................................................... ................ 89
4.4.2.2 Flow Field at R e=2048 and At=0.5......................................... ................ 90
4.4.2.3 Flow Field at R e=256 and A t=0.5........................................... ................ 91
4.5 Lee-Lin's Implicit LBE Two-phase Model and He et al.'s Model with Surface
T en sion ............... ....... .. .... ......... ..................................................................... . 9 1
4.5.1 Lee-Lin's Scheme for Large Density Ratio.......................................................91
4.5.2 Diffusion of Lee & Lin's Implicit LBE model..................................................94
4.5.3 M odeling the Surface Tension........................................................... ................ 96
4.6.1 Surface T ension C alculation .................. ............................................ ................ 97
4.6.2 A Filter-based Technique for Surface Tension .................................................98
4.6.3 Volume Conservation and Mass Conservation ....... ................. ...................100
4 .7 N um erical Sim ulations ................................................. ............................................ 104
4.7.1 Stationary B ubble .............................. ............................................ 104
4.7.2 Capillary Wave .................... .. ........... ............................... 107
4 .7.3 R ising B ubble .............. .. .................. .................. ................... ...... .. ............ 108
4.8 Sum m ary and C conclusion. ................................................................... ............... 110

5 SUMMARY AND FUTURE WORK .......................................................... 127

5 .1 Su m m ary ....................................................................................................... ....... .. 12 7
5 .2 F u tu re W o rk ................................................................................................................. ... 12 8

L IST O F R E F E R E N C E S ....................................................... ................................................ 129

B IO G R A PH IC A L SK E T C H .................................................... ............................................. 138









LIST OF TABLES


Table page

4-1 Effect of grid resolution on computed pressure drop for Laplace number 100. The
data were taken after 100 time steps and the density and viscosity ratios were set to
100 and 10 respectively .... .. .............................. ......... ........ ............... 112

4-2 Effect of Laplace number on the ratio of numerical pressure drop to theoretical
pressure drop: Bubble diameter is 40 lattices. The data were taken at non-
dimensional time = 100. Density ratio is 100 and dynamics viscosity ratio is 10.........112

4-3 Effect of density ratio on pressure drop: Bubble diameter is 40 lattice units. Viscosity
ratio was set to 10 for Laplace number = 100 and the data were taken at non-
dimensional time = 100 ..... ................ .......... .......................... ... 112

4-4 Effect of viscosity ratio on pressure drop: Bubble diameter is 40 lattice units.
Laplace number = 100 and density ratio is 100. The data were taken at non-
dim ensional tim e = 100 ............................................................. ...... .. ...... ..... 112









LIST OF FIGURES


Figure page

2-1 Geometry of channel flow with solid substrate ............... ....................................37

2-2 R educed-dom ain geom etry ...................................................................... ................ 37

2-3 Relative reductions of residuals of single-phase and conjugate cases with specified
sensor tem perature .............. .................................... ........ ....8.... .38

2-4 Relative reductions of residuals of conjugate cases with specified sensor temperature
solved by single-grid and m ulti-grid solvers ................................................ ................ 39

2-5 Shear stress comparisons for the coarse grid and refine grid computations...................40

2-6 Relative reductions of Pressure and temperature residuals on the refined grids ...............41

2-7 Tem perature contours w ith Gr = 0.5 ............................................................. ............... 42

2-8 Tem perature contours w ith Gr = 10 ............................................................. ................ 42

2-9 Tem perature contours w ith Gr = 100 ........................................................... ................ 42

2-10 Temperature distribution on a cross-section which originates from the middle point
of the sensor to the top boundary of the reduced-domain.............................................43

2-11 u velocity profiles of conjugate cases on the cross-sections.........................................44

2-12 W all shear stress distribution ........................................... .......................... ................ 45

2-13 Shear stress variation ........................................................................................... 45

3-1 Boundary nodes and their neighbors using the square lattice.......................................67

3-2 A boundary cell using the hexagonal (FHP) lattice (Noble et al. 1995).........................67

3-3 Absolute L2 norm errors of LBE with Noble's scheme ...............................................67

3-4 A 2D 9-velocity lattice (D 2Q 9) m odel ......................................................... ................ 68

3-5 Absolute error of a fully-developed channel flow using Inamuro et. al.'s scheme ............68

3-6 Two set of viscosity distributions used in this study .................................... ................ 68

3-7 The exact velocity profiles of the channel flows with different boundary layer
thicknesses due to different viscosity distributions. ..................................... ................ 69









3-8 Square lattice distribution in channel flow simulation .................................................69

3-9 Com prison of the LBE velocity profiles .................................................... ................ 70

3-10 Dependence of the relative L2 -norm error on the lattice size h in the fully-developed
channel flow w ith variable viscosity............................................................. ................ 71

3-11 Comparison of ELBE = -uLBE with E = u, -uFD for H=200, and
3 = 0 .000 5, 0 = 0 .0 102 ................................................. ............................................. 72

4-1 The growth rate a (measured in units of (g2 Iv)1/3) of a disturbance vs. its wave
num bers k ............................................................................. ....... ... ..................... 113

4-2 The grow th rate plot ............. .. .................. .................. ......................... .. ............ .. 113

4-3 Evolution of the fluid interface from a single mode perturbation for At=0.5 and
R e = 2 0 4 8 ................................................................................................ .................... 1 14

4-4 He et al's results of the evolution of the fluid interface from a single mode
perturb action .......................... ............................................... 115

4-5 Density profiles across the bubble and spike fronts at three different time steps..........116

4-6 Evolution of the fluid interface from a single mode perturbation. ..............................117

4-7 Evolution of index function of a stationary droplet with zero velocity .........................117

4-8 Evolution of the capillary number with for the stationary bubble simulation with
density ratio 2 and dynamic viscosity ratio 2............... .........................118

4-9 Density profiles and pressure profiles at t=0 and t=2000 of the stationary bubble
sim u nation ...................................................................................................... ....... .. 1 18

4-10 Surface tension profile for a stationary bubble computed from He-Chen-Zhang
m ethod .................................................................................................. .. 119

4-11 Surface tension calculated by Kim's formulation...... ........................................ 119

4-12 Rising bubble with Eo=0.971, Mo=1.26e-3, density ratio=100, viscosity ratio=10........119

4-13 The velocity profiles and stream line of the spherical bubble at t=20 in 4-12...............120

4-14 Time evolution of the volume (A) and velocity (B) of the rising bubble with volume
correction in Figure 4-13B .......................................................................... ............... 120

4-15 The contours of space derivative of hydropressure and density of a rising bubble with
Eo=0.971, M=1.26e-3, Re=5.19, density ratio=100, viscosity ratio=10 .......................121









4-16 Stationary bubble computation. (a) Spurious currents of magnitude Ca=5.03 x 10-3,
(b) comparison between computed and theoretical pressure jump............................... 121

4-17 Density profiles of the stationary bubble with diameter 40, density ratio 100,
viscosity ratio 10, and L a= 100 ..................................... ........................ ............... 122

4-18 Maximum spurious velocities for these two grid resolutions..................................122

4-19 Initial interface profile for a capillary wave simulation........................ ...................123

4-20 Time evolution of the amplitude of a capillary wave with density ratio 100 ................123

4-21 Shape diagram of Clift et al. (1978)...... ........... ........... ...................... 123

4-22 Computed bubble shapes. (A) Cylindrical, (B) Ellipsoidal, (C) Dimpled-ellipsoidal.....124

4-23 Time evolutions of rising bubbles (A) Cylindrical, (B) Ellipsoidal, (C) Dimpled-
ellip soid al...................................................................................................... ......... 12 4

4-24 Time evolutions of bubble velocities ....... ........ ........ ..................... 124

4-25 Density profiles of rising bubbles (A) Cylindricall, (B) Ellipsoidal, (C) Dimpled-
ellip soid al...................................................................................................... ....... .. 12 5

4-26 Time evolutions of a rising bubbles with Eo=97.1, M=0.971 and Re=31.2................. 126









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

MULTI-SCALE COMPUTATIONAL FLUID DYNAMICS WITH INTERFACES

December 2006

Chair: Wei Shyy
Cochair: Renwei Mei
Major Department: Mechanical and Aerospace Engineering

Fluid flow and heat transfer problems involving interfaces typically contain property jumps

and length and time scale disparities, resulting in computational stiffness, demanding resolution

requirements due to nonlinearity and multiple physical mechanisms. In this dissertation, both

continuum and kinetic approaches are investigated to address the multi-scale thermo-fluid

problems, especially those involving interfaces. A continuum model based on the Navier-Stokes

fluid model and the Fourier law for heat transfer was employed to investigate the conjugate heat

transfer problem. The problem is motivated by recent advancement in the micro-electro-

mechanical systems (MEMS) devices for shear stress measurement. Due to the length scale

disparity and large solid-fluid thermal conductivity ratio, a two-level computation is used to

examine the relevant physical mechanisms and their influences on wall shear stress. The

substantial variations in transport properties between the fluid and solid phases and their

interplay in regard to heat transfer and near-wall fluid flow structures are investigated. It is

demonstrated that for the state-of-the-art sensor design, the buoyancy effect can noticeably affect

the accuracy of the shear stress measurement.

A lattice Boltzmann equation (LBE) model derived from the kinetic consideration is

investigated to address (i) error behavior due to variable viscosity, and (ii) the interfacial fluid

dynamics of some two-phase flow problems. It is shown that the boundary treatment error does

not have a significant interaction with the truncation error associated with variable viscosity, and









the LBE model closely matches the Navier-Stokes model for fluid flows with large viscosity

variation.

Next, an improved interface lattice Boltzmann model is developed for two-phase

interfacial fluid dynamics with large property variations. In order to suppress numerical

instabilities associated with the presence of the interface, the following approaches were

investigated: (i) a new surface tension formulation originated from the diffusion interface method

was used to remove unphysical pressure wiggles across interfaces; (ii) a filter scheme was used

in numerical gradient calculations to maintain monotonic property variations; (iii) a volume-

correction procedure was devised. The performance of the improved LBE model was evaluated

using the Rayleigh-Taylor instability problem, stationary bubble under force equilibrium,

capillary waves, and rising bubbles. The computational results demonstrated that this LBE two-

phase model is more robust than those reported in the literature, capable of treating larger density

ratio up to order 0(102 ), while confining the interface thickness within 5-6 grids.














CHAPTER 1
INTRODUCTION TO MULTI-SCALE COMPUTATIONAL DYNAMICS WITH
INTERFACES

1.1 Motivation

Many fluid mechanics and heat transfer problems involve multiple scale

phenomena, such as turbulence, multiphase flows, and conjugate heat transfer. The

presence of the disparity in length, time, and velocity scales is caused by the presence of

different competing mechanisms, such as convection, diffusion, chemical reaction, body

forces, and surface tension. Furthermore, these mechanisms are often coupled and

nonlinear (Shyy et al. 1997b).

Mathematically, issues related to multi-scale problems can be illustrated by a

system of ordinary differential equations whose characteristic matrix has a large range of

eigenvalues. These large range eigenvalues physically represent different growth and

decaying rates of different competing mechanisms. Numerically they make computations

very sensitive to the stability of the numerical schemes (Lambert 1980; Shyy 1994;

Lomax et al. 2001; Chapra and Canale 2002; Deuflhard and Bornemann 2002). If the

linear algebraic equations discretized from the original stiff ODEs are written as

(Miranker 1981)

(I -hfA)= b (1.1)

where I denotes a m-dimensional identity matrix, /7 is a constant, h is integration step

size, A is the matrix that has large range of eigen values, x is solution vector, and b is

source term vectors that can include initial/boundary conditions and other non-









homogeneous sources, the condition number of matrix A is defined as (Chapra et al.

2002)

Cond(A)=AL A 1 A A axn (1.2)


where Amax and Amin are the smallest and largest eigen values of A, respectively. From

Eq. (1.2), it can be seen that the numerical solution process can be ill-conditioned

(Miranker 1981), which brings in computational difficulties in both stability and

accuracy. These kinds of problems are thus called stiff problems.

In order to understand the physical processes involving different scales, separate

scaling procedures for each physical regime and the coupling need to be devised.

However, different multi-scale problems involve different mechanisms and thus require

different treatments. Generally, two classes of multi-scale problems exist. The first one is

that the small scales are restricted in distinct regions in space. Thus, the problems

involved in this class require the matching and patching of the solutions in large and

small scale domains. In the second class, the disparate scales co-exit over the entire flow

field and interact with each other. Resolving or modeling such disparate scales is the task

of this class of multi-scale problems (Shyy et al. 1997a).

At present, there are two major computational fluid dynamics solvers for modeling

incompressible, viscous fluid flows. One is the well-established Navier-Stokes equations

solver. Another is the Lattice Boltzmann equation (LBE) solver (He et al. 1997; Chen et

al. 1998; Succi 2001; Yu et al. 2003a). For multi-scale problems in fluid dynamics and

heat transfer, if the continuity assumption is valid, all scales can be represented by

Navier-Stokes equations. The issue in this type of problem is how to resolve all the

relevant scales based on Navier-Stokes equations.









As an alternative computational fluid dynamics solver, the LBE method offers a

meso-scale framework for fluid dynamics computations. It recovers the macroscopic

fluid flow solution based on averaging of the particle distribution functions, obtained by

solving the simplified form of the Boltzmann equation. Despite some difficulties in

compressible flows and heat transfer, the LBE method has been successfully used for

isothermal complex flows. For interfacial fluid dynamics, however, the numerical

stability issues still exist from the large property jumps and scale disparities in normal

and tangential directions of interfaces. A study on how to stabilize the LBE two-phase

computations for flows with large density ratio and prevent interface thickness from

diffusion is still needed.

In this dissertation, we will focus on the interfacial problems solved by Navier-

Stokes equations and lattice Boltzmann equation since interfaces always result in multi-

scale processes (Shyy 1994; Shyy et al. 1997c) due to the sharp changes in the material

properties across the interface. Because of the physical nature and numerical issues of

multi-scale problems, they are very challenging and nontrivial.

1.2 Objectives

One of the main objectives of the present research is to develop a two-level

computational method based on the Navier-Stokes equations solver to simulate a fluid

flow and heat transfer problem involving a fluid-solid interface. Another main objective

is to simulate a liquid-gas two-phase flow problem by using the lattice Boltzmann

equation (LBE) method. Specifically, the objectives are listed below:

To investigate the multi-scale issues involved in the fluid flow and heat transfer
surrounding a MEMS-based thermal shear stress sensor;

To develop an efficient, accurate numerical method to simulate the fluid flow and heat
transfer surrounding the MEMS-based thermal shear stress sensor;









To investigate the performance of the lattice Boltzmann equation method when it is
applied to fluid flows with rapidly varying viscosity over a thin boundary layer that
is a typical multi-scale phenomenon;

To develop a lattice Boltzmann method for immiscible liquid-gas two-phase flows with
large density ratio.

1.3 Structure of the Dissertation

The numerical techniques for solving the Navier-Stokes equations are briefly

presented in Chapter 2. The multi-scale and stiffness issues involved in the models

represented by Navier-Stokes equations are presented. Finally, in Chapter 2, a two-level

numerical technique is developed to simulate a multi-scale physical problem---the fluid

flow and conjugate heat transfer surrounding a MEMS-based thermal shear stress sensor.

In Chapter 3 the method of Lattice Boltzmann equation (LBE) is briefly described,

followed by the error assessment of the LBE method for variable viscosity.

In Chapter 4, an improved interface lattice Boltzmann model is developed for two-

phase interfacial fluid dynamics with large property variations. Some numerical

approaches are investigated to suppress numerical instability in the computations for

large density ratio flows and to prevent interface thickness from diffusion.

This dissertation ends with Chapter 5, which contains a summary and conclusion of

the contributions of the present work, and the recommendations for future research

efforts.









CHAPTER 2
CONTINUUM MODEL: NAVIER-STOKES EQUATIONS WITH MULTI-SCALES

2.1 Introduction to Computational Techniques for Solving Navier-Stokes Equations

Before addressing the multi-scale flows described by the Navier-Stokes equations, it is

necessary to introduce the different numerical methods for fluid flow and heat transfer since

different governing equation features require different numerical methods, and different

numerical methods have different advantages on different flow and heat transfer problems.

Generally, Navier-Stokes equations solvers can be classified into two groups: one is the

pressure-based formulation, and another is the density-based formulation. The pressure-based

formulation is often used for flows without discontinuities, such as incompressible flows without

shock waves. In contrast, the density-based formulation is often used for flows with

discontinuities, such as supersonic/hypersonic flows (Chung 2002). The algorithms for

compressible and incompressible flows are different, which can be understood from the role of

pressure in compressible and incompressible flows (Moukalled et al. 2001).

If fluid compressibility is zero (in other words, Mach number is zero), the thermodynamic

pressure loses its sense for incompressible flows. Pressure gradient becomes a balance force to

viscous, inertial and other body/surface forces. Therefore, no state equations exist for pressure.

In addition, because density does not change locally for incompressible flows, continuity

equation can no longer be considered as the governing equation for density. Thus no explicit

governing equation for pressure exists for incompressible flow. Also, for incompressible flow,

the speed of pressure wave is infinity. If the pressure oscillation propagating at a finite speed

could not be removed, then velocity obtained from the momentum equation could not satisfy

continuity equation (Patankar 1980; Blosch et al. 1993; Shyy 1994). Therefore, for









incompressible flow, mass conservation is satisfied through the driving action of pressure

gradients on velocity.

For compressible flow the continuity equation is never a function of velocity alone but also

includes the density variation. For compressible flow with different Mach number, the role of

pressure is also different. In the hypersonic limit, velocity variation is relatively smaller than the

velocity itself. Thus, pressure does not influence the mass conservation very much through the

velocity variation by the momentum conservation as compared to through the density variation

by state equation. Therefore, pressure can be considered to act on density through the state

equation, and then the mass conservation is kept through the continuity equation. If Mach

number is not as high as that of a hypersonic flow, the pressure has dual roles on mass

conservation through the state equation and the momentum equation. For subsonic flow, mass

conservation is more readily satisfied by the pressure acting on the velocity through the

momentum equation than by the pressure acting on the density through the state equation.

However, for supersonic flow, mass conservation is more readily satisfied by the pressure acting

on the density through the state equation than the pressure acting on the velocity through the

momentum equation (Moukalled et al. 2001). The different roles that pressure plays in mass

conservation can be thought of as the velocity scale disparity in the physical processes.

Due to the different roles of pressure on the mass conservation for incompressible flow and

compressible flow, the pressure-based methods are originally designed for solving the

incompressible flow and the density-based methods are originally designed for solving the

compressible flow. Also due to the dual roles of pressure, pressure-based methods can be

extended to solve a compressible flow (Shyy 1994) and the concepts of density-based Navier-

Stokes solvers were borrowed to solve incompressible flows, such as the artificial compressible









method (Chorin 1967) and preconditioning method (Turkel 1999) if the stiffness in the

convection coefficient matrixes could be removed or reduced appropriately. If a single method is

needed to simulate fluid flow at all speeds, the pressure-based method and preconditioning

method are two popular frontiers. The artificial compressible method is seldom used for all speed

flow solvers because of the stiff solution matrix which degrades convergence rate (Moukalled et

al. 2001).

Among the incompressible solvers, there is another popular kinetic method, called the

lattice Boltzmann equation (LBE) method. In this method, only one variable, the distribution

function, is solved. Distribution function represents the mesoscopic information of the fluid flow.

Navier-Stokes equations can be recovered from lattice Boltzmann equation on the condition of

nearly incompressible assumption. Macroscopic variables can be obtained from the moment

integration of distribution function (Chen et al. 1998). The governing equation, namely the

lattice Boltzmann equation, is linear (the nonlinear feature is hidden in the equilibrium

distribution function) and whole computational process in one time step is localized. These give

the LBE method an excellent capability to parallelize the computations. Also due to the simple

boundary condition schemes that the LBE has, the LBE method is very easy to use for fluid

flows with random complex geometries (Chen et al. 1998; Succi 2001).

In this dissertation, incompressible viscous flow solvers are used to examine the multi-

scale computational fluid dynamics problems with interfaces. Compressible Navier-Stokes

solvers could be found in typical computational fluid dynamics textbooks (Hirsch 1990; Fletcher

1991; Shyy 1994; Anderson 1995; Anderson et al. 1997; Chung 2002; Ferziger et al. 2002).









2.2 Incompressible Viscous Flow Solvers


2.2.1 Introduction

Incompressible flow is a physical model of general fluid flow on the condition of a very

small Mach number. The time scales of the convection velocities are much larger than the time

scale of sound speed as the Mach number becomes very small. If Mach number equals to zero,

new issues are brought in for incompressible flow computations. As introduced in the previous

section, the pressure waves have infinity propagating speed in incompressible flows, which leads

more elliptic features to the governing equations, especially for the pressure fields (Kiris et al.

2003). Because of this characteristic of the governing equations, the computational schemes have

to be designed to couple the continuity and momentum equations through pressure in order to

keep the pressure field from oscillating, by which the conservation of mass can be preserved as

the sound speed becomes much higher than the convection velocity components.

In dealing with incompressible flows described by the Navier-Stokes equations, two major

approaches are generally used: the primitive variable methods and the vortex methods. The

purpose of vortex methods is to remove the difficulty of solving accurately the pressure field in

incompressible flows. However, if some complexities, such as chemical reaction, are needed to

be added to the fluid flows, the primitive variable methods can easily accommodate these

requirements by adding additional equations to the stack (Blosch et al. 1993). The primitive

variable approach includes the artificial compressibility method (ACM) (Chorin 1967) and the

pressure correction methods (PCM). Pressure correction methods include the marker and cell

(MAC) method, the semi-implicit method for pressure linked equations (SIMPLE) (Patankar

1980), and the pressure implicit with splitting of operators (PISO) (Issa 1985).









2.2.2 Artificial Compressible Method (ACM)

The incompressible Navier-Stokes system of equations are written in non-dimensionalized

form as

Continuity Equation:

= 0 (2.1)


Momentum Equation:

au, au, 8p 1 2U,
-+u +--2 (2.2)
at 9x x 9x, Re ax2

In the ACM method, an artificial compressibility term is added into continuity equation as

following

S+ L = 0 (2.3)


where p is the artificial density, which is equal to the product of artificial compressibility

factor / and pressure,

~P = p8-l (2.4)

When steady state is reached, the artificial density derivative with respect to time vanishes.

How to choose the artificial factor/ is the key for ACM (Kwak et al. 1986). ,/ has to be

maintained low enough (close to the convective velocity) to avoid stiffness associated with a

large range of eigen value magnitudes. But it has to be kept high enough such that the speed of

sound can be as large as possible to achieve the incompressible requirement (Chung 2002).

2.2.3 Semi-Implicit Method for Pressure-Linked Equations (SIMPLE)

The SIMPLE method was designed originally to solve the steady-state fluid flows. In this

method, the velocity and pressure are solved sequentially in one iteration (which is called the









outer iteration, the corresponding inner iteration is the one for solving algebraic equations). How

to solve the pressure field is the key in this method. In one outer iteration, if pressure is known

(which could be given as a guess for initial condition or obtained from previous iteration),

velocity could be solved from the momentum equations. The velocity obtained from the

momentum equation might not satisfy continuum equation. This requires correcting the pressure,

and further correcting the velocity to make the continuum equation satisfied in this outer

iteration.

If we denote u v and ) as the velocity and pressure which satisfy the momentum

equations, the predictor-corrector procedure with successive pressure correction steps is given as

p =P+ (2.5)

where p is the actual pressure, and p is the pressure correction. The actual velocity components

in two-dimensions are

u =+u (2.6)

v = +v (2.7)

Since u, v, and p satisfy momentum equation, u, v and p also should satisfy the momentum

equation, we have

ae(e +u)= ab(unb u)+b+[(P,+p,)(PE +EA, (2.8)

ael = Zanbnb + b +[p -E ]Ae (2.9)

The subscriptions in these equations are same as those in Partankar's book (Patankar 1980).

Subtracting (2.9) from (2.8), we have

aeue = anbu +(p PE )A (2.10)









From (2.10), it is clear that the velocity corrections on any computational point have two

components, one is induced from the difference of pressure correction on the neighboring points

and the other comes from the velocity corrections of its neighboring points. The former source of

velocity correction, the difference of pressure correction on neighbor points, is the major factor.

If the neighbor velocity correction impact is neglected (this is why this method is called semi-

implicit method), the velocity correction equation (2.10) could be written as

aeue = ( -p)p eA (2.11)


u = A pp p = d( pp p) (2.12)
ae

Likewise, the velocity correction component in v direction is given as


v A (pP -, p = d (pP N p1) (2.13)

Substituting (2.12) and (2.13) into (2.6) and (2.7), and then substituting (2.6) and (2.7) into the

discretized continuum equation, the pressure correction equation could be obtained as

app' aPEp +awp' +aNpN + asp' + b (2.14)

aE PedAy aW = p dAy aN =pdnAx as = p~dAx

ap =aE +aW +aN +aS


b (p= pp ) AxAy (pv) A
At w e A +

Since the source term b represents the mass imbalance based on the velocity in the previous

iteration, if its value is small enough, mass conservation could be taken to be satisfied.

Therefore, the value of b could be the criterion for stopping iteration.









Neglecting the velocity correction from neighboring points in the velocity correction

equation does not influence the final steady-state solution. In this sense, the iteration procedure

for the pressure can be simplified such that it requires only a few iterations at each time step.

SIMPLE method has been modified to several versions. People call them as SIMPLE

family methods. These modified methods accelerate solving the pressure correction equation

which is elliptic type equation. The detailed descriptions of these methods could be found in

many CFD textbooks (Patankar 1980; Fletcher 1991; Shyy 1994; Ferziger et al. 2002).

2.3 Computational Issues on Multi-scale Computations

2.3.1 Stiffness

Multi-scale computations in fluid dynamics and heat transfer means that widely different

time or length scales phenomena need to be captured with expected accuracy. As described in

Chapter 1, different scales arouse from a wide acting range of different forces in dynamics.

These forces can interplay with each other such that the small scale perturbations could amplify

and propagate into large scale regions. Even though, for some cases, the small scale impacts can

be restricted in small local regions without amplification and propagation, the local phenomena

restricted by the large scale dynamic mechanisms are sometimes interested, such as the heat and

momentum transfer mechanisms surrounding a heating element of hot-wire/film whose length

scales are generally much smaller than the measured large scale fluid flows. These kinds of

measurement devices are preferred because of their small sizes such that large scale fluid flows

are disturbed as little as possible, and the measurement could be more accurate.

Numerically, multi-scale problems are relative to stiff issue if the physical problems are

modeled with sufficient accuracy by a coupled set of ODEs and these ODEs have unique and

bounded solutions (Lambert 1980; Lomax et al. 2001; Deuflhard et al. 2002). Generally, the

ODEs can be expressed as









di= Aii- j(t) (2.15)
dt

where ii is the dependent variable vector, t is the independent variable. If A is independent of t,

Eq. (2.15) is linear, otherwise nonlinear. The second term on the right hand side is a forcing

function which is determined by the inherent features of dynamic system (Kinsler et al. 2000).

The difference between the dynamic scales in physical space is represented by the difference in

the magnitude of the eigenvalues in the eigenspace of A. The forcing function can have its own

scales. Usually its scale could be adequately resolved by the chosen computational step size.

Thus, it is reasonable to assume that the forcing function has no effect on the numerical stability

of the homogeneous part in one computational step. However, for nonlinear problem, the matrix

A is dependent on t. If the time scale of (t) is needed to resolve, it has a limit on the discretized

size of t which has an impact on the eigenvalues of A because the entries of A depend on the

discretized size of t. This can be observed when simulating complex flow phenomena such as

turbulence, possibly in combination with chemical reactions. In these processes, additional

source terms can be added into equations. The nature of these source terms results in an

increasing stiffness of these problems, which can reduce the convergence rate (Steelant et al.

1994).

The major feature of stiffness is the interaction of computational accuracy and stability. Let

us assume the independent variable t in Eq. (2.15) is time. The ODE's could be solved by using

time-marching methods. In one time step, the integration with respect to t is an approximation in

eigenspace that is different for every eigenvector. Numerically, the eigenvectors corresponding

to the small eigenvalue could be well resolved. In contrast, the eigen vectors corresponding to

the large eigen values are not (Shyy 1994; Lomax et al. 2001). The approximation to the

eigenvectors associated with large eigenvalues requires stable numerical schemes for the global









computation procedures and the accuracy represented by these eigen vectors has to be kept from

ruining the complete solution.

The wide range eigenvalues of multi-scale problems bring string requirements on

numerical schemes in terms of accuracy and stability. Accuracy and stability are always two

contrary requirements for numerical schemes. As a result, the physical nature of multi-scale

problems and the numerical nature of accuracy and stability make multi-scale problems

nontrivial. Besides, grid requirement is another factor making multi-scale problems harder to

resolve.

2.3.2 Grid Requirements for Multi-scale Problems

If the physical models accurately represent the physical processes, and the numerical

schemes can be appropriately selected to capture the physics, the grid arrangement will

determine the accuracy of solutions. Multi-scale phenomena in fluid flow and heat transfer are

generally characterized by a couple of regions in which the spatial gradients of the dependent

variables are much higher than in other regions. Consequently, higher spatial grid resolutions are

required in those high gradient regions, and a lower grid resolution in others. This grid clustering

can strongly affect the eigenvalues of the matrix A since A is calculated directly in terms of the

grid sizes. This grid effect on the eigenvalues of the matrix A can be illustrated by the diffusion

equation shown below (Lomax et al. 2001). The diffusion equation is

au 2u
-v (2.16)
at ax2

Using the three-point central differencing scheme to represent the second order derivative in

space leads to the following ODE diffusion model


S B(1, -2,1)i+bc) (2.17)
dt AX2









with Dirichlet boundary conditions folded into the (bc) vector, where B(1,-2,1)is a banded

matrix

-2 1
1 -2 1
B(1,-2,1)= '. (2.18)
1 -2 1
1 -2

The eigenvalues of the matrix B are

-4v mF 1
A = sin m =1,2,...,M (2.19)

The stiffness ratio for this diffusion equation is

( 4v "(AX'D 2
Cr 2 4 (2.20)
C, M, n I -4v AX2
Ax2

From this ratio, it is clearly seen that the smaller the grid is, the stiffer the numerical equations

would be (Atkinson 1989; Lomax et al. 2001).

2.3.3 Methods to Reduce Stiffness

In order to achieve the expected accuracy for multi-scale problems, the truncation errors

should be kept as small as possible. Reducing the grid size is a way to achieve a smaller

truncation error if the adopted numerical schemes are consistent and stable. However, multi-scale

problems often involve large gradients in space or/and in time. For large time derivatives

problems, one might suppose that the adaptive time step-size routines might offer a solution; that

is, using small time steps during the rapid transients and large time steps otherwise. But, that is

not the case in numerical practices. The reason is that the stability requirement still necessitates

very small steps over the entire solution (Chapra et al. 2002). Therefore, A-stable methods









(generally they are implicit methods) are preferred (Lambert 1980; Shyy 1994). For large

variable gradients in space due to multi-length-scales, the errors arising from insufficient grid

resolution is a major contribution among the many types of errors included in the overall solution

(Celik et al. 2004). In such problems, grids should be clustered at locations where large gradients

exist so that the truncation errors could be generally kept as small as possible and computational

expenses could be reduced (Fletcher 1991; Anderson 1995; Ferziger et al. 2002). Besides

clustering grids at the known locations where large gradients exist, adaptive meshing algorithms

could be used to solve more complicated problems in which the locations of large gradients

might not be known in advance (Bank 1990; Anderson 1995; Park et al. 2004). However, the

difficulty with adaptive meshing method lies in that finding the solution of the linear system, as

the grid points, that are added for grid refinement are globally combined with the regular mesh

(Park et al. 2004). The resulting linear system may also lose the banded structure or the positive

definiteness of the uniform grids (Golub et al. 1996).

If the physical geometries are very complicated and the length scales have wide

distributions in the computational domain, the intensity of clustering grid might be very severe.

The aspect ratios of some grid cells might be very large in some sub-regions. Such computational

stencils have poor grid quality since the order of truncation errors for such clustered grid can be

lower because some terms in truncation errors could not be cancelled off due to non-uniform grid

sizes. As a result, if the variation of aspect ratio of computational stencils could not be kept as

small as expected (generally 10% variation is accepted for practical computations suggested by

Ferziger et al. (2002)), the truncation error could be larger than acceptable. Thus, for such multi-

scale problems with large space gradients, further improvement for reducing truncation error is

necessary.









One remedy is to use multi-level technique (Shyy et al. 1997a; Appukuttan et al. 2003b).

This technique includes a global computation made for providing global information in the

whole computational domain, which may not resolve the small scale phenomena because of the

low resolution in the vicinity of the small scale process regions, and then a reduced domain

computation, which includes the small scale process regions, is carried out on a higher resolution

gird with the boundary conditions extracted from the global computations. In this method, the

length scale of the reduced domain is smaller than the global domain and larger than that of the

small scale processes. The global computation can provide accurate boundary conditions for the

reduced domain computation if the small scale processes are mainly restricted to the interested

regions, and their variations do not affect the flow and heat processes in the regions far away

from them. In such cases, the reduced domain boundaries could be selected at such locations

where no great impact comes from small scale regions. By this way, the length scale disparity

could be reduced. Good grid qualities for both global and reduced-region computations can be

acquired.

If small scale processes could be amplified and propagated over the global domain or if the

physical geometries are very complex, a single global grid of multi-level-grid technique may not

be capable of accurately capturing the global phenomena. As a result it can not provide accurate

boundary conditions for the reduced-region computations. For such problems Domain

Decomposition Methods (DDM) or multiblock techniques can be used as alternative method to

reduce the error due to grid resolutions (Shyy et al. 1997b). Unlike in multi-level method that has

two separate computing processes, in multiblock methods, the computational domain is

partitioned into several blocks according to different physical scales in different blocks and the

computational variables are solved simultaneously (without block loop (Thakur et al. 2002)) or









iteratively (with block loop (Shyy et al. 1997b)). By reasonable domain partition, multiblock

techniques can reduce the topological complexity. Each block can be generated independently.

Thus, grids can have expected high resolutions in the required regions. Also, grid lines do not

need to be continuous across block interfaces. This makes the local grid refinement and adaptive

redistribution easy to accommodate different physical length scales that exist in different regions.

The poor quality of single grids on such complex problems is thus improved by block

arrangement. Compared to the multi-grid-layout method, multiblock method can deal with more

complicated problems, but, involves more work than multi-level method, in terms of specifying

the block interface boundary conditions, data structure, interface region size effect on data

exchange and convergence rate (Shyy 1994; Shyy et al. 1997b; Thakur et al. 2002).

For very complex physical and geometrical problems, parallel computational techniques

using on multiblock grids can improve convergence rate (Golub et al. 1993; Shyy et al. 1997b).

Parallel computing techniques depend on both hardware architecture (parallel architecture) and

software (parallel algorithms).

The above techniques are very useful for multi-scale problem computations. Block

partition is determined in terms of different scale phenomenon regions, which is generally

reflected by local dimensionless parameters in Navier-Stokes equations, such as Reynolds

number. Grid characteristics (Grid stretching and clustering) also yield to different physical scale

distributions. In pressure-correction algorithms, the inner loop/iteration (solving algebraic

equations) has a big impact on the outer loop/iteration that couples the unknown variables

through conservation equations. The convergence rate of the inner loop can influence the outer

loop substantially as the number of grid points increases (Shyy et al. 1997b). It is well-known

that the eigenvalues of the algebraic equations of multi-scale problems might have a wide range









of values (Lomax et al. 2001). Typical iteration methods, such as point-Jacobi iteration on a

single grid, do not have fast convergence rate for such equations. While, multigrid methods have

the capability to accelerate solving these kinds of equations (Shyy 1994; Shyy et al. 1997b; Tao

2001; Chung 2002). The large range of eigenvalues can be represented by the large range of

frequency in terms of grid sizes in Fourier's analysis. High frequency error components damp off

faster on fine grids. Multigrid methods resolve different frequency error components on multi-

level grids so that the low frequency error components on fine grids become high frequency error

components on coarse grids and they can be damped off faster on coarse grids. Because of this

uniform (in terms of frequency) solution error treatment, multigrid methods are highly efficient

to solve physical problems with disparate multi-scales (Shyy et al. 1997b).

2.4 A Special Computation Example: Thermo-MEMS Computation Results

In this section, a multi-scale problem with a sharp fluid-solid interface described by the

Navier-Stokes is investigated. This problem is a conjugate heat transfer type resulting from a

Micro-electromechanical Systems (MEMS)-based on thermal shear stress. In this problem,

multi-scales come from (1) length scale disparity of characteristic length scale of a channel flow

and the size of the MEMS-based thermal shear stress sensor, and (2) large conductivity ratio of

fluid to solid substrate on which the MEMS-based sensor is deposited. Before addressing the

multi-scale issues in this simulation, an introduction about thermal anemometry and its issues are

presented below.

2.4.1 Introduction to Thermal Anemometry for Fluid Velocity and Skin Friction
Measurement

Thermal anemometry has been widely used to measure the fluid velocity and the skin

friction for decades (Winter 1977; Perry 1982; Bruum 1995). All thermal sensors are

temperature-resistive transducers that essentially measure heat-transfer rate. They are indirect









sensors and thus require an empirical or theoretical correlation, valid for very specific conditions,

to relate the measured Joulean heating rate to the flow parameter of interest (i.e., velocity or wall

shear stress). All indirect thermal shear-stress sensors possess several limitations when used for

quantitative measurements. Specifically, they are limited by the difficulty in obtaining a unique

calibration or relationship between heat transfer and flow parameters of interest. In addition, the

frequency-dependent conductive heat transfer into the support substrate (i.e., prongs for a hot

wire and supporting substrate for a hot film) reduces the sensitivity and complicates the dynamic

response. Thermal sensors are also sensitive to the mean temperature variations which can be

difficult to correct. Finally, the thermal sensor is heated well above the ambient temperature

during the operation, which can potentially influence the near-wall flow structure and introduce

measurement error.

A numerical simulation can help interpret the sensor output and correct the calibration

formula. For example, Durst et al. (Durst et al. 2002) reported a numerical study of conjugate

heat transfer effect on the near-wall hot-wire correction. They found that the influence of the

local length scale (or Reynolds number), Y+, on the heat transfer process adjacent to hot-wire is

very important, especially when the thermal conductivity of the wall is very small, such as that of

glass or perspex. In the above, Y+ is defined as


Y+ = uY (2.21)
V

where u,= is the friction velocity, and Y is the wire-to-wall distance. Shi et al. (Shi et al.


2003) conducted two-dimensional numerical simulations of forced convection from a micro-

cylinder in a laminar cross-flow. Their numerical results showed that the heat diffusion from the

wire is pretty large in the case of small wire-to-wall distance (Y<3). The reason is the









modification of thermal boundary condition at the fluid-wall interface caused by property

variations between phases.

Thermal-based skin friction sensors, which relate the convection from a thin heated film

deposited on a substrate to the local wall shear stress, are limited by the heat conduction into the

supporting substrate as well as the flow perturbations due to local fluid heating (Naughton et al.

2002; Sheplak et al. 2002; Appukuttan et al. 2003a). Recently, researchers have developed

silicon micromachined sensors because of the possibility of improved thermal isolation of the

sensing element from the substrate by depositing the sensor on a thin membrane stretched over a

vacuum cavity (Ho et al. 1998; Sheplak et al. 2002). While the vacuum cavity structure greatly

improved the performance in terms of sensitivity with respect to conventional sensors, MEMS-

based thermal sensors still suffer from the inherent limitations of localized flow heating and heat

conduction into the supporting membrane (Naughton et al. 2002; Sheplak et al. 2002).

The inherent small size of MEMS-based thermal sensors invalidates the classical 1/3

power law of the hot-film theory, i.e., Nu oc rz3 (Lin et al. 2000), which assumes that the

thermal boundary layer resides entirely within the linear region of the velocity profile and that

the boundary layer approximation holds for the energy equation (Lin et al. 2000). Because of the

small size of the MEMS-based sensors, diffusion can substantially affect the fluid flow and heat

transfer in the vicinity of the MEMS-based thermal sensors (Ho et al. 1998). Lin (Lin et al. 2000)

thus included 2D heat conduction of fluid flow with unidirectional convection, resulting in the

following energy equation in the fluid phase:

OT (02 T 02 T
u- = a + (2.22)
Ox 2 2y )









In addition, one-dimensional heat conduction in a solid membrane was employed to

account for the conjugate heat transfer effect. Yoshino et al. (Yoshino et al. 2003) conducted a

numerical analysis of frequency response of micro thermal flow sensor. In their simulations, no

buoyancy force was taken into account and a linear fluid velocity was given. The heat

conduction in the solid substrate was considered in their numerical simulations. They found an

optimum diaphragm size to frequency response for their thermal sensor, which is 200-300m

long. Recently, mixed convection induced by MEMS-based thermal shear stress sensor was

studied by Appukuttan et al. (Appukuttan et al. 2003a). They discussed the flow and heat transfer

surrounding a thermal shear stress sensor embedded on a wall of a channel. Buoyancy force

effects induced by the thermal sensor on the shear stress were examined for different Grashof

numbers. As they pointed out, buoyancy force has little impact on the whole-domain flow

structure in the channel, while its impact on shear stress measurement can be noticeably

observed because the sensor performance depends on the near-wall velocity gradients. In the

work of Appukuttan et al's, heat conduction in the solid substrate on which the thermal sensor is

deposited was not considered. However, as Naughton and Sheplak stated in their review paper

(Naughton et al. 2002), the heat conduction in the substrate can noticeably influence the heat

transfer process.

In the above numerical studies on MEMS-based thermal sensors (Lin et al. 2000; Yoshino

et al. 2003), velocity profiles were given. Only energy equations were solved based on the

assumed velocity profiles. Buoyancy force was not accounted for in the numerical simulations.

Although one-dimensional (Lin et al. 2000) and two-dimensional (Yoshino et al. 2003) conjugate

heat transfer was considered in the energy equations, the impact of the solid heat transfer on the

fluid velocity distribution through the buoyancy forces was absent. In the hot-wire simulations of









Durst et al (Durst et al. 2002) and Shi et al (Shi et al. 2003), the temperatures of hot-wire heating

elements were specified and conjugate heat transfer was involved, in order to identify the

correction on the velocity profile. However it was found that the buoyancy force has very little

effect on the velocity profile (Durst et al. 2002). Hence the buoyancy terms were neglected in the

corresponding momentum equations.

Sheplak et al (Sheplak et al. 2002) proposed that sensor measurement is quite sensitive to

the ambient temperature variation. In this dissertation, we will investigate the conjugate heat

transfer around the solid substrate and the surrounding fluid. The effect of the buoyancy force

which couples the momentum and energy equations is investigated because it plays an important

role in establishing the velocity gradient at the fluid-solid surface. Two kinds of sensor

boundaries are used, namely, specified temperature and specified heat flux. In both cases a

single-phase (involving the gas phase only), and a conjugate heat transfer with coordinated

thermal boundary conditions are considered to highlight the individual heat transfer modes. The

effect of heat transfer on the IVIMEMS-based wall shear stress sensor will be addressed.

2.4.2 Governing Equations

In the previous work (Appukuttan et al. 2003a), 2-D steady state Navier-Stokes equations

with the Boussinesq approximation for buoyancy force were solved. In order to capture the

buoyancy effect, velocities are scaled by the buoyancy force term (Shyy 1994). The following

dimensionless variables are employed to normalize the governing equations:

__ u v T* T-To
p u U v T T T
p gf.ATH gfATH Tv ... To

P x (2.23)
P x =_ y* =_ (2.23)
p .gfATH H H

The non-dimensionalized Navier-Stokes equations subject to the above scaling references are:









u momentum


u -- +v --= -- + +p* I+2U* (2.24)
S* y* Ox* 7 ax*2 y *2

v* momentum

S* v v *P 1 2 2 *
u* +v* -= -+ + +T (2.25)
&* y* y* .7r\Ox 2 y* )

Energy equation

._ _T .* OT 1 2 T* 2 T*
+ -+v _+ (2.26)
Ox" -y" Pr xGr Ox*2 ay*2)


where Pr and r gfA TH3 The last term in Eq. (2.25) represents the dimensionless
a V

buoyancy force. The current formulation corresponds to a horizontal channel in which the

gravity is perpendicular to the streamwise velocity.

In the solid domain, heat conduction is the only transport phenomena. Without a heat

source in the solid (the sensor is a heat source but treated via the boundary condition due to its

very small thickness), the governing equation is,


k a2+ = 0 (2.27)


where ks is the dimensionless solid thermal conductivity; here, in dimensionless form, it is the

solid-fluid thermal conductivity ratio. It is noted that the dimensionless fluid viscosity and

1 1
thermal conductivity are and --- respectively. For convenience, all asterisks are
Gr PrVGr

dropped hereafter unless specifically mentioned.

In the present study, the Grashof number is defined based on the length (L) of the thermal

sensor, even though the governing equations are normalized by the channel height (H). For the









MEMS-based sensor cases, it is better to describe the buoyancy effect by the sensor-length based

Grashof number Gr, which can be expressed as


Gri = Gr -j (2.28)


In the present study, Prandtl number is fixed as 0.71.

2.4.3 Numerical Schemes

The well-established pressure-based approach with finite volume formulation is adopted to

solve the governing equations. A second-order upwind scheme for the convective terms and a

second-order central difference scheme for pressure and diffusion terms are used. Detailed

information can be found in (Shyy 1994; Shyy et al. 1997b; Thakur et al. 2002). Code validation

and numerical accuracy assessment were performed by Appukuttan et al. (Appukuttan et al.

2003a).

2.4.4 Computational Stiffness

Computational stiffness arises from the large difference in length and time scales, caused

by variations in transport properties and sensor-to-channel sizes (Chapra et al. 2002). In this

study, the dimensionless solid thermal conductivity is 1200, corresponding approximately to the

ratio between the MEMS-based thermal sensor materials and air.

2.4.5 Geometry and Grid Layout

As shown in Figure 2-1, the 2-D channel is 25cm long and 1cm high. A sensor is located

on the bottom wall of the channel, and is 20cm downstream from the channel inlet. The 2-D

sensor is 200[pm in length, which is much smaller than the characteristic length scale of the

channel flow. A solid substrate is below the bottom wall of the channel. The solid substrate and

flow channel are symmetric with respect to the channel bottom wall.









The grid layout is related to the computational method which is used to overcome the scale

disparity. If a single uniform grid is used to capture the entire physical flow and heat transfer

process for the present problem, the grid would be very fine in the vicinity of the sensor and

coarse away from the sensor. This kind of grid can introduce a noticeable computational error if

the grid aspect ratios in some parts of the computational domain are too large (Ferziger et al.

2002). In order to circumvent the length scale disparity, two-level layout grids similar to those

used by Appukuttan et al. (Appukuttan et al. 2003a) are adopted. First, a whole-domain

computation (Figure 2-1) is carried out, which generally does not provide the resolution

necessary for the velocity field near the sensor. Based on the whole-domain solution, a reduced-

domain computation focusing on the sensor region (Figure 2-2) with a much finer grid is

performed. In the reduced-domain computation, the boundary conditions are obtained from the

whole-domain computation by using a bilinear interpolation.

Unless otherwise stated, the whole-domain grid is meshed nonuniformly with finer grid

near the vicinity of the sensor. The fluid region has 342x36 grid points. The solid region has

342x 12 grid points. Such arrangement is necessary because the temperature gradient in the fluid

region is larger than that in the solid region due to convective effects. Since the solid-fluid

thermal conductivity ratio is large, this grid arrangement is also helpful for obtaining the

numerical convergence. The relationship of convergence with grid size and thermal conductivity

ratio was presented by Shyy and Burke (Shyy et al. 1994). As will be presented later, the grid

refinement study supports the adequacy of the present grid system. The reduced-domain

computation is employed on a uniform grid consisting of 162x202 points which has much better

resolution than the whole-domain grid and is expected to capture the detailed heat and fluid flow









structure surrounding the sensor. The issue of grid resolution will be discussed while presenting

the results.

The boundary conditions for the whole-domain computational domain (Figure 2-1) are as

follows.

(a) Boundary conditions for the momentum equations:

Inlet: parabolic velocity profile is determined by Reynolds number and Grashof

number. (The average inlet velocity is determined by Reynolds number, u = Rev which
H

is in the dimensional form. If this velocity is normalized by the velocity scale in

Eq.(2.23), we have =-Re. This velocity value is used to specify the parabolic inlet
aveu 7Gr7


velocity.)

Channel walls: no-slip

Outlet: velocity extrapolation

(b) Boundary conditions for the energy equation:

Inlet of channel: T =

(c) Boundaries other than inlet and outlet in Figure 2-1:

2T
S= 0 (2.29)
an2

where n is the direction normal to the boundaries. This temperature boundary condition is

adopted from the energy equations because the temperature and heat flux on these boundaries are

unknown a priori.

The boundary conditions of the reduced-domain (Figure 2-2) are as follows.

(a) Inlet: velocity and temperature are interpolated from the solutions of the whole-

domain computation by bilinear interpolation.









(b) Outlet: velocity and temperature are extrapolated from inner grid points.

(c) Boundaries of the solid region (except fluid-solid interface): temperature is

interpolated from the solution of the whole-domain computation by bilinear

interpolation.

At the solid-fluid interface (except the sensor surface on which the temperature or the heat

flux is imposed), the interface conductivity k, (Patankar 1980) is employed. This artificial

conductivity arises from the non-homogeneity of the materials on both sides of the interface. Let

us consider two neighboring computational nodes next to the interface, one in the fluid region

represented byf and the other in the solid region represented by s. The temperatures on these two

points are T7 and T,, respectively. Since no computational node exists on the interface, the heat

flux through the interface could be represent as

Tf-T
q=k 'f
q- S (2.30)

whereCo y is the normal distance from the node f to the solid-fluid interface, and y, is the

normal distance from the point s to the solid-fluid interface. In Eq. (2.30), k, is the effective,

interfacial conductivities. If one-dimensional heat conduction resistances between the nodes f

and s are considered, the heat flux through the interface can also be expressed as

T Ts
i y (3 (2.31)
kf ky


Combination of Eqs. (2.30)-(2.3 1) leads










ke = y + y (2.32)
3Yf 9y
+ +
kf k

The interfacial conductivity ke in Eq. (2.32) is the harmonic mean of the fluid and solid thermal

conductivities.

Between the sensor and the solid substrate, adiabatic boundary condition is used to account

for the vacuum cavity beneath the thermal sensor. The thickness of the vacuum cavity is

neglected because it is very small (Sheplak et al. 2002). On the fluid side of the sensor surface,

either one of the two thermal boundary conditions is adopted, namely, (a) specified sensor

temperature, or (b) specified sensor heat flux. The specified sensor temperature condition is the

same as the sensor temperature boundary condition of the single-phase computation previously

reported in (Appukuttan et al. 2003a). The specified sensor heat fluxes are calculated from the

corresponding single phase cases with the same Grashof number. The values of Grashof numbers

are selected based on the operational temperature range of the MEMS-based thermal shear stress

sensor (Sheplak et al. 2002) used in the Interdisciplinary Microsystems Group in the University

of Florida, which is about 20-400C. For air, the Grashof numbers are in the range 0.05-0.8

based on the sensor length. Therefore, Gi, =0.08 and 0.5 are chosen. Numerically G, =10 and

100 are also used to examine larger Grashof numbers limit, which correspond small kinematic

viscosity fluids. For the case with Gi, =0.08, only shear stress distribution and variation are

shown in the next section since the flow structure variations in these cases are too small to be

observed, while the shear stress variations are noticeable.









2.4.6 Results and Discussion

Computations are conducted for a horizontal channel flow in which gravity is transverse to

the streamwise direction. The Reynolds number is 500 with the length scale based on the channel

height.

Before discussing computational results, it is necessary to first discuss the numerical

convergence because of the numerical stiffness encountered in the present investigation. The

convergence might be problematic if iterative methods are used. To quantify the convergence

behavior, residuals of the governing equations are used. Poor convergence is indicated by a

lower rate of decrease of residuals. Furthermore, since small values of residuals do not

necessarily represent small errors in the solutions for stiff problems, at least a three-order of

magnitude drop in the residuals is needed for judging the convergence.

Since the momentum and the energy equations are coupled by the buoyancy force, the

stiffness in one equation also affects the accuracy and convergence of other equations. When the

thermal conductivity ratio is fixed as a constant, the stiffness arising from the large thermal

conductivity ratio is similar for all the cases with different Grashof numbers. This can be seen

from the energy equations for the fluid and solid regions.

Figure 2-3 shows the residual reduction histories of u, v, pressure and temperature of the

single-phase and conjugate cases using a logarithm scale with specified sensor temperature

andGr, =0.5. Both are solved using the algebraic multigrid technique, specifying the same

number of inner loop iterations and relaxation factors. The residuals of single-phase and

conjugate cases are normalized based on the same normalization values. The four residual

history plots show that the single-phase case converges faster than the corresponding conjugate









case. These different convergence rates mainly result from the stiffness of thermal conductivity

ratio of the conjugate case.

To demonstrate the impact of the solution technique, Figure 2-4 shows u, v, pressure and

temperature residual reduction history plots of a conjugate case with Gr = 0.5, using both

single-grid and multigrid solvers. The multi-grid solver shows noticeably faster convergence

than the single grid solver, indicating that the multigrid solver performs better for solving stiff

problems than the single-grid solver. This phenomenon is also observed in two-phase flow

problems that generally have multi-scale issues and significant property jumps (Francois et al.

2004).

In order to assess the resolution requirement for the current problem, grid refinement has

been conducted for both the whole-domain computation and the reduced-domain computation.

For the whole-domain computation, the fluid region has 514x53 nodes, compared to the baseline

grid of 342x36 nodes. The solid region has 514x18 nodes, refined from the baseline grid of

342x 12 nodes. For the reduced-domain computation, the refined grid is still uniform, which has

242x302 nodes in comparison to the baseline grid of 162x202 nodes. The velocity and

temperature fields computed on the coarse baseline grid are only marginally different from those

on the refined grid. Figure 2-5 shows that the difference between the shear stresses computed on

these coarse and refined grids is small. As expected, the convergence rates of the computation on

the refined grid are noticeably slower than those on the coarse grid (see Figure 2-6), where the

same multi-grid strategies have been adopted. Hence the baseline (coarse) grid is adopted in the

rest of this study.

Figure 2-7 Figure 2-9 show temperature contours for the single-phase and conjugate

cases for different Grashof numbers. In these temperature contour plots of the conjugate cases,









similar to single-phase cases, the warmed-up regions enhance with Grashof numbers, showing

stronger buoyancy effect for higher Grashof numbers. Comparing with single phase temperature

contours, near the sensor in the fluid region, temperature distributions of the conjugate cases are

very similar to those of the single phase cases. Similarly distorted elliptical contours can be

observed for both the single-phase and conjugate heat transfer cases with different sensor

boundary conditions. This can be seen from the temperature contour line with the value 0.0625.

In the outer fluid region, the single-phase temperature contours have different patterns than the

conjugate cases. The temperature contours of single phase cases originate from the sensor and

expand asymptotically to the main stream, while the temperature contours of the conjugate cases

in that region no longer connect with the sensor since the heat conduction effect of the solid

substrate raises the upstream and downstream fluid temperature, and thus, the effective heating

length is enlarged due to this substrate heating.

At the solid-fluid interface, temperature contours have abrupt kinks, reflecting the effect of

the large solid-fluid thermal conductivity ratio. In the leading-edge regions of the conjugate

cases, the temperature contours turn in different directions at the solid-fluid interface. Near the

sensor, temperature contours from the solid region turn left into the fluid region. Beyond a small

upstream distance from the sensor, the temperature contours turn right from the solid into the

fluid region. In the vicinity of the sensor at the leading edge, heat goes upstream in the fluid by

diffusion near the sensor which heats up the solid substrate. Since the heat diffusion effect in the

fluid region at the leading-edge is restricted very close to the sensor by the fluid flow, the

temperature gradient of the fluid in that region is larger than that of the solid. Therefore, beyond

a small upstream distance from the sensor, the temperature of the solid is higher than that of the

fluid at the solid-fluid interface. This leads to the fact that the solid substrate heats up the fluid









due to the heat conduction in the solid region and the large solid-fluid thermal conductivity ratio.

Then the heat transferred from the solid to the fluid is carried downstream by fluid convection.

At the trailing edge, fluid temperature is always higher than that of the solid since the heat

diffusion in the fluid enforces heat convection. As a result, the temperature contours at the

interface downstream of the sensor have similar shapes.

In the solid region, heat flux is determined by solid thermal conductivity, solid-fluid

interface thermal conductivity and convection effect of fluid at the interface. The interface heat

resistance results from both conduction and convection at the interface. The interface resistance

of conduction can be indicated by the interface conductivity (Eq.(2.32)). Because the solid-fluid

thermal conductivity ratio is fixed, the interface thermal conductivity can be rewritten as



3)Af + g2 kf
k,= k s + k (2.33)


Therefore, the dimensionless interface conductivity varies with dimensionless fluid

1
conductivity which is proportional to In other words, the resistance from conduction


increases with increasing Grashof number. The interface heat resistance from the convection can

also be indicated by Grashof number since the Reynolds number is fixed as 500 for the present

study. The resistance from convection decreases with increasing Grashof number because larger

Grashof numbers denote larger buoyancy forces which enhance the convection effect. Thus, the

two parts of the interface heat resistances have different tendencies of variation with increasing

Grashof number. Dimensionless solid thermal conductivity also varies with dimensionless fluid

conductivity in a similar way owing to fixed solid-fluid thermal conductivity ratio. Hence,

convective heat transfer enhancement in the fluid region reduces the conduction effect at the

solid-fluid interface, and vice versa.









Due to the interface resistances, the major part of the heat generated by the thermal sensor

is carried away by convection. Therefore, the wall has a function to isolate fluid and solid, which

allows relatively small part of heat to transport into the solid substrate. This is the reason why,

near the sensor, the temperature contours of conjugate cases do not vary so much from the

single-phase cases.

Comparing the conjugate cases with specified sensor temperature, temperature contours of

G, = 0.5 case are very similar to those of G, = 10. For the conjugate cases with specified

sensor heat flux, at the same locations, temperatures of the case with Gi = 10 are lower than

those ofGr = 0.5. While for the conjugate cases withGr = 100, temperatures in the solid are

higher than the other cases. This might result from the different variation trends of conduction

effect and convection effects at the interface with regard to Grashof numbers.

This opposite effect of Grashof number on the heat conduction and convection at the fluid-

solid interface is shown more clearly in Figure 2-10. For the cases of the specified sensor

temperature, Figure 2-10 shows the temperature distributions on a cross-section, which is

perpendicular to the sensor surface and originates from the middle point of the sensor to the top

boundary of the reduced-domain. Due to the opposite effect of Grashof number on the heat

transfer on the fluid-solid interface, temperature distributions of the cases with G, = 0.5 and

Gi = 10 are very similar. For the case with Gr = 0.08, the temperature of the fluid far away

from the sensor is the lowest of these four cases, which mainly results from the lower buoyancy

force effect on heat convection in the fluid region. For the case withGr = 100, temperature is

higher in the fluid region far away from the sensor and smaller in the region closer to the sensor.









It can be explained that, for the case withGr, = 100, the convection effect is larger in the far field

region and the diffusion effect is smaller in the region near the solid surface.

Figure 2-11 presents the u-velocity profiles of conjugate cases with different sensor

boundary conditions. Due to the wall isolation effect, the u-velocity profiles of conjugate cases

do not have noticeable differences from those of the single-phase results (Appukuttan et al.

2003a). This is also because of the isolation of the wall described previously. However, the wall

effect on the shear stress can be clearly observed in Figure 2-12 and Figure 2-13.

In Figure 2-13 shear stress variations are calculated based on the analytic shear stress of a

developed channel flow without buoyancy force acting on the fluid. The analytical shear stress

du U
is-p A l. If it is normalized by p---, and U, is the average velocity of the channel flow, the
dy H

analytic shear stress becomes

d(u* /IU )
= dy =6 (2.34)

Comparing single-phase shear stresses (Appukuttan et al. 2003a), for the small Grashof

number cases, the shear stresses of conjugate cases with the specified sensor heat flux have

smaller variation than those of the specified sensor temperature cases; for the larger Grashof

numbers, their shear stress variations are higher than those of specified temperature cases.

Therefore, different energy boundary conditions for the sensor lead to different shear stress

distributions and variations for conjugate cases. This difference should be caused by the

buoyancy force. Different sensor temperature conditions cause different temperature

distributions surrounding the sensor, which generate different buoyancy forces. Therefore, if a

thermal sensor works in the constant temperature mode, its shear stress behavior is different from

those of thermal sensors working in constant current mode or constant voltage mode.









2.4.7 Summary and Conclusion:

The effects of conjugate heat transfer on shear stress variations of a MEMS-based thermal

shear stress sensor are investigated. In the very close vicinity of the sensor, temperature contours

of conjugate cases with different sensor temperature conditions are similar to those of the single-

phase cases, while the temperature contours in the region away from the sensor are different

from the single-phase cases. For conjugate cases, the effective heating length is enlarged due to

the substrate heating, which was emphasized in (Sheplak et al. 2002). Even though the velocity

distributions of both single-phase and conjugate cases are very similar, shear stress distributions

for the conjugate cases have observable deviations from single-phase cases. Therefore, though

buoyancy force does not change velocity profiles significantly, it noticeably influences velocity

gradients near the thermal sensor, which introduces errors in the shear stress measurements.













Inlet 1 Fluid Outlet


1 Substrate /
Focused region boundaries


-20
Figure 2-1. Geometry of channel flow with solid substrate

0.32

inlet

inle( 13 2 0.2 Fluid outlet


/ 0.2 Substrate
sensOr



-- 0.1-
Figure 2-2. Reduced-domain geometry. 1, 2 and 3 represent the sections on which velocity
profiles are plotted in Figure 2-11


















-- single-phase
- conjugate


-- single-phase
- conjugate


U -'

2 2 I "-







-3-
-- -- s e N




-33
-3 N- -3


-4 -4 -


-5 -5 .
0 1000 2000 3000 0 1000 2000 3000
a iteration number iteration number b


0

-1 -1
s-1 inl-1 epsingle-phase
single-phase - conjugate
---2 conjugate -2 -
-2

-3-






7 -7 5


-7 N
-6-
-8-
-7-
S -9

-8 -10

0 1000 2000 3000 0 500 1000 1500 2000 2500 3000
C iteration number iteration number d

Figure 2-3. Relative reductions of residuals of single-phase and conjugate cases with specified

sensor temperature, GCr = 0.5. (a) u residual, (b) v residual, (c) pressure residual and

(d) temperature residual
















0
0


-- smgle grid solver
- multi grid solver


25 0 1'II 201 3000 4 5000 6000 00 200 00 4000 00 100
0


















-1 smgle grid solver smgle grid solver
2 2 -
32 -3 -

-3 4 -












-5 5
-3 -4 -
-4 \ -5


0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
a iteration number iteration number b

0 0

-1 solved bysingle grid solver solvers,1 Gr = 0.5. (a) u residual, (b)d solver
(c) pressure residual angrid solved) temperature multi grid solver
2

-3
-3
~-4


1 -5\
\ ~-6\




-7 \-8

0 1000 2000 3000 4000 5000 6000 0 1000 2000 3000 4000 5000 6000
C iteration number iteration number d
Figure 2-4. Relative reductions of residuals of conjugate cases with specified sensor temperature

solved by single-grid and multi-grid solvers, Gr, = 0.5. (a) u residual, (b) v residual,

(c) pressure residual and (d) temperature residual.














6.31


- coarse grid


a- 6.25 --- fine grid

6.2

6.15

6.1

6.05

6

5.95

5.9
0.1 0.2 0.3
x

Figure 2-5. Shear stress comparisons for the coarse grid and refine grid computations

(Gr, =0.5 and Re= 500)
















--- coarse grid
fine grid


-- coarse gnd
fine grid


-3 -3 N

-4 -

1 -5 \ I "4 \


-6 -- -
\ 2 -5- \

o


N
-9 -N

-7-7
0 1000 2000 3000 4000 0 1000 2000 3000 4000
iteration number iteration number

The whole-domain grid


0
-1
coarse grid -1 -- coarse grid
-2 - fine grid - fine grid
-2
-3
0 \ -3 \
-4 -
-4
-5
o/ = -5
-6 -
S-6
S-1-


-7
9 -


0 2000 4000 6000 0 1000 2000 3000 4000 5000 6000 7000
iteration number iteration number


The reduced-domain grid

Figure 2-6. Relative reductions of Pressure and temperature residuals on the refined grids

( Gr = 0.5 )















025

02 -

0 15


-01 9 / 01 00177
0 0 5 0 0 62 5 I Z "
S01 02 03 0 1 02 03 b 0 01 02 03
0 375


a 0 b 1 c

Figure 2-7. Temperature contours with Gr, = 0.5 (a) Single-phase cases with specified sensor

temperature; (b) Conjugate cases with specified sensor temperature; (c) Conjugate

cases with specified sensor heat flux.


-01 1 127 /
0 [5 00625
S125 015 015 0012

0 0 -020 01 02 03 -02
1 02 03 a 0 b 0 l 02 03

Figure 2-8. Temperature contours withGr5 =10 (a) Single-phase cases with specified sensor

temperature; (b) Conjugate cases with specified sensor temperature; (c) Conjugate

cases with specified sensor heat flux.


'5





3E


)5 00625

o-iS^,'26


02 a


00625
S 0125

000205
00202


00196

01 02 03
x C


Figure 2-9. Temperature contours withG, = 100 (a) Single-phase cases with specified sensor

temperature; (b) Conjugate cases with specified sensor temperature; (c) Conjugate

cases with specified sensor heat flux.


S0 1















Gr s=0 08
0.15 Gr s=0 5
Gr s=10
----- Gr s=100


a 0.1




0.05




10 10 10
T

Figure 2-10. Temperature distribution on a cross-section which originates from the middle point
of the sensor to the top boundary of the reduced-domain (Specified sensor
temperature cases)














02 02

0175 0 S- Un1 0175 S1:

015 Secton3 015 on

0125 0- 125

A 01 01

0 075 0- 075

005 005

0025 0025

025 05 075 025 05 075
U4(GH L)/Re U (Grf/L)/R0


(a) Specified sensor temperature (b) Specified sensor heat flux


Gr, =0.5

02 02 ]

0175 s 0175 S

015 015

0125 0125-

A 01 01

0075 0075 *

005 005

0025 0 025

025 75 025 05 075
UL (G^H/I/)R
(a) Specified sensor temperature (b) Specified sensor heat flux


Gr, = 10


02 02 -

0175 0 175 -

015 015

0 125 0- 125 *

A 01 A 01

0 075 0075 -

005 005 *

0025 0025 -

0 025 075 0 025 075

(a) Specified sensor temperature (b) Specified sensor heat flux

Gr, = 100

Figure 2-11. u velocity profiles of conjugate cases on the cross-sections just before (section 1),

after (section 2) and in the region of the sensor (section 3) at Re=500




















631-

6 2

61-



59
0 05 0 1


<-I


n n n;


Figure 2-12. Wall shear stress distribution: (a) Specified sensor temperature (b) Specified sensor

heat flux


, I II I I I I I
0 02 04
Grashof Number (Gr )


0 01 02 03 04 05 06
Grashof Number (Gro)


Figure 2-13. Shear stress variation: (a) Specified sensor temperature (b) Specified sensor heat

flux


' '


01 015 02 025


40









CHAPTER 3
ERROR ASSESSMENT OF THE LATTICE BOLTZMANN METHOD FOR VARIABLE
VISCOSITY FLOWS

As an alternative CFD tool, lattice Boltzmann Equation (LBE) method has been

developing for about two decades. From scale classification, the LBE is a meso-scale method for

fluid dynamics computations. Navier-Stokes equations for compressible flows can be recovered

from the LBE with the incompressible constraint. In other words, the LBE describes fluid flows

with very low Mach number (Chen and Doolen, 1998; Succi, 2001; Yu et al. 2003a). Although

there are some difficulties for the LBE method in compressible flows and thermo-fluid flows, it

has been successfully used for isothermal complex fluid flows, such as interfacial dynamics (He

et al. 1999), turbulent flows (Yu et al. 2005) and porous media fluid flows (Chen et al. 1991).

Since large velocity gradients often accompany substantial viscosity variations due to

either shear-thinning effects or turbulence models, errors associated with the variable viscosity

model need to be addressed before exploring the accuracy for these problems. With variable

viscosity such as for turbulent flows treated with the eddy viscosity models (Filippova and

Hanel, 1998), or for fluids with flow-dependent properties, additional truncation error appears in

the macroscopic equations derived from the LBE (Hou et al. 1995). With the help of the

truncation error, computational accuracy for those flows can be further understood. The

truncation error behavior of the LBE with constant viscosity has been studied by Holdych et al.

(Holdych et al. 2004). However, the truncation error for variable viscosity problems has not been

studied systematically in the literature.

Like other computational models, the LBE method needs to address the wall boundary

conditions (Mei, et al., 1999). In particular, the no-slip wall boundary condition in the LBE

method is based on the bounce-back concept (Chen and Doolen, 1998; Succi, 2001). Much work

has been done to extend this simple scheme to arbitrary curvature boundaries with second order









accuracy (Mei et al., 1999; Ladd, 1994; Yu et al., 2003; Bouzidi et al., 2001). These methods can

handle geometrical complexities easily; however, they cannot exactly recover the no-slip

boundary condition (He et al., 1997) at the mesoscopic level (Mei, et al., 1999). This

inconsistency gives rise to the boundary condition error. Although the no-slip boundary

condition of the LBE has been derived from hydrodynamic conditions on walls by Noble et al.

(1995) and Inamuro et al. (1995), these exact no-slip boundary schemes are incapable of

handling geometrically complex boundaries because they require the computational nodes to

coincide with the physical boundaries.

The original bounce-back scheme and other improved treatments have been successful in

constant viscosity laminar flows with expected accuracy (Chen and Doolen, 1998; Yu et al.,

2003; Mei et al., 1999; Mei et al., 2000). However, for variable viscosity problems, the boundary

condition error for no-slip walls may act on the truncation error. Thus, besides the truncation

error behavior itself, whether the second-order accuracy of the bounce-back scheme can be

maintained in the presence of the variable viscosity is another open question.

In this Chapter, these issues are investigated via a fully-developed laminar channel flow

with a specified variable viscosity. With the help of the finite difference analysis, the truncation

error of the LBE with variable viscosity is investigated. Two different specified viscosity

distributions, which lead to different boundary layer characteristics, are employed to examine the

errors associated with the variable viscosity, in the LBE model. The exact solution of this

channel flow exists and can be used for error analysis.

In the following, the LBE method and the boundary conditions will first be reviewed. We

then present the variable viscosity laminar channel flow equations along with the exact solution.

Based upon the exact solution and the finite difference analysis, the error behaviors due to the









variable viscosity and the boundary condition schemes will be assessed. To separate the error

associated with the variable viscosity from that with the boundary treatment, both Noble et al.'s

scheme (Noble et al. 1995), which is exact but restricted to straight boundaries, and bounce-

back-on-link (BBL) scheme, which is not exact but can handle irregular geometries, will be

employed.

3.1 Introduction to Lattice Boltzmann Method and Wall Boundary Condition

3.1.1 LBE BGK Method

The Lattice Boltzmann equation with the singer-relaxation-time (SRT) Bhatnagar-Gross-

Krook (BGK) model can be written as (He and Luo, 1997a)


fa (x, +e.9t, t +9t) f (x,, t) =- x (x,,t) f eq (x,,t)] (2.35)

where f, denotes f(x, e, t), which is the distribution function in the direction of the

ath discrete velocity ea, f(eq) is the corresponding equilibrium distribution function in the

discrete velocity space, r =A/ t, Ais the relaxation time, and x) represents computational

nodes in physical space.

The most popular lattice model for simulating two-dimensional flows is the nine-velocity

square lattice model, which is often referred to as the 2-D 9-velocity (D2Q9) model (Qian et al.,

1992). In this lattice model, the discrete velocities (ea) are as follows:

eo = 0,
e =c(cos((a-1);r/4),sin((a-1);r/4)) for a=1,3,5,7 (2.36)

e = ,c (cos((a-1);r/4),sin((a-1);r/4)) for a=2,4,6,8

where c = 8x /St. The corresponding equilibrium distribution functions are defined as

3f(eq) W 3- u+9 U)2 3 U
f[Ieq) 2 er 2+ (e 2C.21 u I a= 0,1,...,8 (2.37)
c/ 2-c- 2-c










where w, is the weight coefficient given by


1
W1 = W3 = 5 = W7 =- W2 = 4
9


1
W6 8 =W
36


The density and momentum fluxes can be obtained from the moments of the distribution function

as


8

a=0


8

a=0


8O
f(eq)
a-=0


-f (eq)
a=0


(2.39)


The kinematic viscosity associated with the D2Q9 lattice model can be expressed as


V= )T c23t
I 2 S


(2.40)


where c is the speed of sound, which is equal to c/l3 for D2Q9 lattice model. The


corresponding equation of state is p = pc2.

For a hexagonal lattice model (Noble et al. 1995), the discretized velocities are


e = 0,

e= c(cos((a-1)2;r/6),sin((a-1)2;r/6)) for a= 1,2,...,6


(2.41)


The corresponding equilibrium distribution functions are


f (eq)- do _- (u.u)
c (2.42)

f (eq) =p d pD +pD (D +2) 2 pD
S + 2(e .- u) + (e u) (u -. u) for a= 1,2,...,6
b c2b 2c4b 2c2b

Where for the 2-D hexagonal lattice model, the dimension rank is D = 2, the number of lattice

direction is b = 6, the average rest particle density is do = p/2. The kinematic viscosity for 2-D

hexagonal lattice model is


=2r -1 3x2
8 8t


(2.43)


4
w =-
9


(2.38)








The LBE is usually solved in two steps:

Collision step:

(xt+3t)= f(x t)- [f(x,t)- f (xt)] (2.44)

Streaming step:

f. (x + e. t, + 8t)= (xx,, + 9t) (2.45)

where f, represents the post-collision state.

In order to get the Navier-Stokes equations, the Chapman-Enskog expansion can be used

with time and space being rescaled as

t = t, t2 = '2t, X, = EX,
a a 2 a a a (2.46)
-t 8t8 8t2
at at, at2 a 1 &:

and the distribution function is expanded as

f = fl0 f(1) 2 (2) O( 3). (2.47)

From Eq. (2.46) and (2.47), it is clear that the Champan-Enskog expansion is essentially a multi-

scale expansion (Frisch et al. 1987). In the incompressible flow limit, that is -<<1, the Navier-
C

Stokes equations can be recovered from the LBE (Eq. (2.35)) with the leading truncation error of

O (3x)2 (u)3 I(Hou et al. 1995):

=0,
(2.48)

chu chu 1 p
a- +u 1 =-- +V2u (2.49)
at 8x, p 8x









3.1.2 Boundary Conditions

In contrast to Navier-Stokes equations, the lattice Boltzmann equation has no physical wall

boundary conditions for distribution function f,. The distribution functions are unknown and

have to be constructed from inside information. Symmetric and periodic boundary conditions can

be constructed without any ambiguities. Outlet boundary conditions are generally extrapolated

from internal nodes (Yu et al. 2003a). Inlet boundary conditions can be extended directly from

the boundary treatments for solid wall. Since a flow with a large Reynolds number usually has

boundary layer phenomenon near walls, which is a typical multi-scale process (Batchelor 1967)

and sensitive to the wall boundary condition, the no-slip wall boundary condition is thus the most

important one among other boundary conditions in fluid dynamics.

Historically, not only the models of the LBE evolved directly from its predecessor, the

lattice-gas automata (LGA) (Frisch et al. 1986), but also the wall boundary conditions, which is

namely the bounce-back scheme (Lavallee et al. 1991; Chen et al. 1998; Succi 2001). This

boundary condition method has a very attractive feature: it is very easy to implement

computationally, especially for complex geometries. In comparison with the Navier-Stokes

solvers, this amazing feature offers the LBE method a distinguished capability to handle flows

with geometry complex (Chen et al. 1998; Mei et al. 1999; Mei et al. 2000).

For the no-slip wall boundary condition, if computational boundary nodes are located on

walls, the LBE method with the bounce-back scheme has first-order accuracy (Ziegler 1993). A

slip wall velocity exits (He et al. 1997; He et al. 1997c) and increases with relaxation time

(Noble et al. 1995). This bounce-back scheme with the computational nodes on walls can be

referred as the bounce-back-on-node (BBN) scheme. While, if the boundary is shifted into the

fluid by one half mesh unit, i.e. placing the wall between the computational nodes, second-order









accuracy can be achieved by the bounce-back scheme (Ziegler 1993; Ladd 1994a; Ladd 1994b;

Yu et al. 2003a). This bounce-back scheme with wall positions locating between computational

nodes can be referred as the bounce-back-on-link (BBL) scheme (Mei et al. 1999). In the BBL

scheme, after a particle distribution function f streams from a fluid node at xf to a boundary

node at xb along the direction of e, the particle distribution function f, reflects back to the node

xf along the direction ofe, (= -e),

f = fa, (2.50)

which is illustrated in Figure 3-1. The BBL scheme has second-order accuracy (Ziegler 1993;

Ladd 1994a; Ladd 1994b), even through the slip wall velocity can not be exactly removed.

The BBL scheme can be improved to arbitrary curved wall boundaries with second-order

accuracy, such as Filippova and Hanel's scheme (Filippova et al. 1998; Fillippova et al. 1998).

Mei, Luo and Shyy's scheme (designed as MLS's scheme (Lai et al. 2001)) improved the

numerical stability of Filippova and Hanel's scheme (Mei et al. 1999). Bouzidi et al. (Bouzidi et

al. 2001) presented a simpler bounce-back boundary condition for a wall located at an arbitrary

position and no requirements for constructing fictitious fluid points inside walls (Bouzidi et al.

2001; Yu et al. 2003a). Yu et al (Yu et al. 2003b) presented a unified boundary condition based

on Bouzidi et al. and Mei et al.'s works. Although the exact no-slip velocity boundary condition

could not be achieved by using the BBL and the improved BBLs, they are very attractive for

flows with complex geometries (He et al. 1997c).

Besides the bounce-back schemes, there are some other schemes for no-slip wall boundary

condition, such as Noble et al.'s hydrodynamic boundary condition scheme for hexagonal lattice

(Noble et al. 1995), and Inamuro's scheme for square lattice (Inamuro et al. 1995).









Noble et. al's scheme was derived for the hexagonal lattice from the macroscopic mass and

momentum at a no-slip boundary without any assumptions. This scheme is illustrated in Figure

3-2. On the boundary nodes, the following expressions of the unknown distribution functions can

be obtained by solving density and mass flux from the boundary nodes:

f2 + A3 = P-(fO + fl + f4 + f + f6 )
f2 f3 = 2pu -(2f,- 2f4 -f5+ f6) (2.51)
f2 + f3 =(2/ )pv+(f5 +f6)

Through Eq.(2.51), the unknown distribution functions on boundary nodes can be obtained from

the distribution functions on the nodes of fluid side. This scheme can provide the exact no-slip

wall boundary condition, which can be exhibited through simulations of a fully-developed

channel flow with constant viscosity. The absolute L2 -norm error of this simulation defined as


H [ULB (y) u.,t (y)] dy\
E2 = [IILBM H f- (2.52)

is shown in Figure 3-3, which has the order of the round-off error. This demonstrates the exact

no-slip boundary condition can be attained via Noble et al.'s scheme.

Inamuro's scheme was developed for the 2-D square lattice model (D2Q9). A lattice cell is

shown in Figure 3-4. The definitions of the density and velocity on the no-slip wall are same as

those on the internal nodes:
8 8 8 8
p = i/f = e.e pi = 4cf = fq
a=0 a=0 a=0 a=0
In Figure 3-4, the boundary node is on the wall. Thus, the distribution functions

f f, f, fs, f6, f8 are known; while f2, f, f4 are unknown and they need to be constructed from

the known distribution functions. Because the slip wall velocity exists, Inamuro et. al. assume the

unknown distribution functions f2, f3, f4 have similar forms as other known distribution








functions except that there is a counter velocity which is added to the slip wall velocity to force

the wall velocity as zero. The unknown distribution /2, f, and /4 are expressed as:


f2 = P 1+3(UW+u +vW)+ (u,+u +v) -_- [(u +u) +v2, (2.53)


f3= p{1+3vw + 3[(u+ +)2v+ (2.54)


f4 P 1+3 uw-u, +vw )+ UW -u +vW u +L +V)2 (2.55)

where uw, vw are the slip wall velocity, u is the counter velocity, and p is the density change due

to the counter velocity. The velocity on the wall in the horizontal directionuw and vertical

direction v, are calculated as

u =--fe2 (A 4 +f8 6 +f 5f) =-(f2 4 +A) (256)
Pwa Pw Pw
vw= Z f/e,= (=f2+f4+f3 -6 -8 -7)= (f2+f4+f+ B) (2.57)
pw a Pw Pw

where A = f +f / and B =-1f-f- f1 In the above equations, the unknowns are u', p

and pw. With some algebraic manipulations, the unknowns can be obtained


u = [6 u 3uwvw (2.58)
1+ 3v, p

p'6 p v B
1+3v, +3v2
v1 [( (2.59)


S -vW (2.60)









It can be seen that u', p and p, are obtained from the known distribution functions and wall

velocity. Substitutingu', p and p back into(2.53), (2.54)and(2.55), the unknown distribution

functions on the wall can be obtained.

Inamuro et. al.'s scheme could enforce the no-slip wall boundary condition. However, it

could not remove all errors from the boundaries. This can be observed from simulations of a

fully-developed channel flow with a constant viscosity. The absolute L2-norm error is shown in

Figure 3-5. It is clearly shown that the absoluteL2 -norm error is larger than the round-off error

(O(10-15)).

The error in Inamuro et. al.'s scheme might come from the counter velocity assumption

which is the only hypothesis in this scheme. Whether the unknown distribution functions really

have the form (Eq. (2.53), (2.54) and (2.55)) or not is unknown.

Through the comparison between Noble et al.'s scheme and Inamuro et al.'s scheme, it is

clear that Noble et. al.'s scheme can exactly realize no-slip boundary condition, while Inamuro

et. al.'s scheme does not. Noble et al.'s scheme is thus adopted in this chapter.

3.2 Error Assessment of the LBE Method due to Variable Viscosity via a Fully Developed
Channel Flow

3.2.1 Fully-developed Laminar Channel Flow with Variable Viscosity

In this section a 2D fully-developed steady laminar channel flow with variable viscosity is

adopted to assess the error behavior of the LBE with variable viscosity. The governing equation

for this flow in a channel of height H is

+- I d o 0 and u- = 0, du =0 (2.61)
p dx dyv dy) o dy yH2

The viscosity vo,,, is modeled as









Total V0 VSt,
S2 ( )2 0 < = < 0.5, (2.62)
vo 02(1+13/3) [1+(1/70)031/3' H

where vo is a constant and the symmetry gives vo,, for 0.5
parameters a and 0 control the profile shapes for viscosity and velocity. Employing the

boundary conditions and integrating Eq. (2.61) twice yield

u H2dp j 0.5)( 4 ,J) for 0<4<0.5 (2.63)
pvdxo + +,5


Let a = /02 and r73 +ar/2 + = (r7+a)(1r2 +br +c). It is easily seen that a3 -aa2 =6, b =a-a

and c =-, where the value of a can be obtained numerically in terms of a and a Integrating Eq.
a

(2.63) results in the exact solution of this laminar channel flow, that is,


(q2 -q)-Dq+Aln(ql/a+1)+B -[In(q2 +bq+c)-ln(c)]
H2 dp 2 2 .6
uM(*) = (2.64)



a[(1/2+a)a2 +3] B a(1+2a-a)
where D= -a, A= a[( -,)a2 +] B = 3(+2-a) 8 and Q= -(Ab+Ba).
a2 + 2c a(a2 + 2c)

In order to explore the truncation error due to variable viscosity under the condition of

large velocity gradient over a short distance, the parameters 3 and 0 in Eq. (2.62) need to be

chosen carefully so that boundary-layer like velocity profile can be obtained in Eq. (2.64). The

first term in Eq. (2.64) in the curly bracket corresponds to the parabolic velocity profile with the

constant viscosity v0. Other terms are due to variable part of the viscosity vt For small values

of a and 0, the fourth term -[in(r72 +b7+c)-ln(c)] does not change dramatically across the









whole channel. The second term -Dr is a negative linear term and the third term Aln(Q7/a+1) is

positive across the channel, the sum of these two terms varies slowly in the near wall region. The

last term in the near wall region varies rapidly because, for small 8 and 0, under further

assumption of

0 >> 52 (2.65)

it is asymptotically equal to

I Bl b2 [tan b _tan1 b 1-l0 tan '- (2.66)
I 2 2 W {2 0

Thus, with small values of 8 and 0, the tan- (...) term in Eq. (2.64) results in the boundary layer

type of behavior.

For comparison purpose, two sets of viscosity distributions, which satisfy Eq. (2.65) for

small 3 and small 0, are used in this study:

Casel: (o,3,0)= (0.004167,0.0005,0.0102)

CaseHl: (vo, ,0) =(0.008333,0.002,0.0289)

The corresponding viscosity profiles are shown in Figure 3-6. The resulting velocity

distributions corresponding to these two viscosity distributions are shown in Figure 3-7. The

boundary layer effect from the tan-' (...) terms in Eq. (2.64) or Eq. (2.66) for Case I is shown in

the inset of Figure 3-7. Eq. (2.66) also gives a guideline for estimating the grid resolution

required for resolving the boundary layers. Since tan-1 (1)/tan -1(oc) = 1/2, it is seen that over a

distance of 7 = 0, the velocity reaches 50% of the maximum given by Eq. (2.66). Thus, for

dimensionless grid size h=1/H that is close to or larger than 0, numerical solutions will not have

sufficient resolution for this thin layer.









3.2.2. The Lattice Boltzmann Equation Treatment

In LBE method, the viscosity is associated with the relaxation time r by Eq. (2.40). The

variable viscosity in LBE can still be realized via a spatially varying relaxation time, and the

total viscosity at a given location can be expressed as

2r(x, y)-1
total (x, y) = (xy)- for square grid
6 (2.67)

total (x, y) = for hexagonal grid
8

Thus, the relaxation time can be represented by the local fluid viscosity as


r (x, 6v (x, y)l for square grid
2 (2.68)

(x, y) = 8vtrtl (x, y) +1 for hexagonal grid
2

The computational setup for BBL scheme is shown in Figure 3-8. The first computational

node is half a lattice away from the channel wall. For Noble et al.'s scheme, the setup is the same

except that the grid is hexagonal and the computational boundary nodes are on the channel walls.

For comparison purpose, the computations on hexagonal grid and square grid should have the

same grid resolution across the channel height. For the same grid resolution the channel height of

the square grid setup is 2//3 times that of the hexagonal grid setup. In order to make absolute

comparison among the velocity profiles, the pressure gradients are accordingly adjusted in each


computation so that H2 dp remains the same. Periodic boundary condition is used at the left and
dx

right side boundaries. The constant pressure gradient in Eq. (2.61) is treated as a body force, and

is added to the distribution functions after the collision step. The error analysis for the

computations with both boundary conditions is carried out by examining the difference between

the exact solution and the computational results at each y location.









3.2.3 Assessment

Since Case I exhibits thinner boundary layers and sharper near-wall velocity gradients than

Case II, we first discuss Case I. Figure 3-9 compares three velocity profiles for H=50

(h=1/H=0.02): a) exact solution; b) LBE solution using the standard BBL boundary condition

and square lattice formulation; c) LBE solution using Nobel et al's exact boundary condition and

hexagonal lattice formulation. For the parameters considered, H=50 (h=0.02 >0 = 0.0102) does

not resolve the thin boundary layer near the wall, as clearly shown in the inset of Figure 3-9. It is

noted that due to the exactness of the boundary condition used, the hexagonal formulation gives

only a slight overshoot in the velocity one full lattice away from the wall (at h/O- 2 ); however,

it results in an overshoot for the rest of the lattices in the channel. The LBE solution using square

lattice formulation suffers from both the inaccuracy of the bounce-back boundary condition and

the insufficient near wall resolution and gives lower velocity throughout the entire channel. Both

LBE velocity profiles have errors of similar magnitudes. This implies that when the velocity in

the boundary layer is not sufficiently resolved, the exact boundary condition with the hexagonal

lattice formulation would not offer any advantage compared to the approximate BBL condition

using the square lattice formulation. For h ranging from 0.004 to 0.1 (H: 10 250 in lattice unit)

the square lattice formulation with the BBL boundary scheme shows different over/under-

predictions for the velocity profiles from the hexagonal formulation comparing with Noble et

al.'s scheme. While the results are not shown here for brevity, it suffices to note that the

magnitudes of the over-predictions in the hexagonal lattice case are comparable to those of the

under-predictions in the square lattice case. These velocity profiles indicate that the errors of the

LBE solution using both boundary condition schemes are comparable. Thus, the relative L2 -

norm error of the LBE solution, defined as










{ [I0 ,LM (y)-u_,, (y)]2 dY
E U 1y 2 (2.69)
ru2o (Y)dy
_0

is examined for both boundary condition schemes.

The LBE computations with these two sets of viscosity distributions are carried out by

using both Noble et al's scheme and the BBL scheme for grid size h ranging from 0.004 to 0.1.

Their relative L2 -norm errors in the velocity profiles with respect to grid resolution are shown in

Figure 3-10a. As expected, the relative L2 -norm error curve of Case I shifts up with respect to

that of Case II because Case II has a thicker boundary layer which implies better computational

resolution than in Case I for the same h (= 1/H). For sufficiently high grid resolution (h<0.01,

points A-D shown in Figure 3-10a), both Noble et al's scheme and the BBL scheme yield the

asymptotic second order accuracy, which is consistent with the truncation error analysis for both

square and hexagonal lattice schemes.

For the present fully developed 2D channel flow with low speed, the velocity field satisfies

Vu = 0 and O(u3), resulting in very small modeling error. Thus the errors in LBE computations

are mainly from the truncation error (due to variable viscosity) and the boundary condition

treatment. With sufficient grid resolution (points A-D in Figure 3-10a), the relative L, norm

errors of BBL scheme are about 15% larger than those of Noble et al.'s scheme. Since Noble et

al.'s scheme does not contain boundary condition-induced error, this 15% difference in error in

the BBL scheme results from the boundary condition treatment. Thus comparing the results

using Noble et al.'s scheme and the BBL scheme, it can be inferred that in the presence of the

strong velocity profile variation the truncation error contributes a significant part of the overall

error.









Because a substantial part of the overall error is from the truncation error as opposed to the

boundary condition error with highly variable viscosity, the truncation error associated with the

variable viscosity thus deserves close attention. However, the truncation error could have very

complex form, even for the problems with constant viscosity (Holdych et al. 2004). It is

recognized that there is a close relation between the LBE and the finite difference form of the

momentum transport represented by the Navier-Stokes equation (He et al. 1997; He et al. 2002;

Mei et al. 1998). For example, He et al.(1997) showed that the square lattice formulations for the

particle distribution functions in a 2-D pressure driven channel flow with constant viscosity, after

averaging, leads to a second order, central-difference formulation for the axial velocity. For the

present channel flow problem with the large variation in viscosity, such derivation could not be

easily obtained. However, if the relative L2-norm error of the finite difference method still

behaves similarly to that of the LBE, it is reasonable to expect that the truncation error behavior

of the LBE is similar to that of the finite difference-based macroscopic model (in the present

case, the Navier-Stokes equation). This hypothesis will be assessed first.

In the Navier-Stokes model, the governing equation (Eq.(2.61)) is discretized by standard

central difference scheme (with Ay = 1 lattice unit):

0 = d 1(A)2 }+1/2 (+1 +--) v-1/2 (u -1). (2.70)


To compare the errors of the finite difference and LBE schemes, two finite difference solutions

on different grid arrangements are obtained for the velocity profile. The first finite difference

solution has the same grid arrangement as in the hexagonal lattice so that the first fluid node is

one full mesh away from the wall. The second finite difference solution has the same grid

arrangement as in the square lattice with the BBL scheme so that the first fluid node locates half

a mesh away from the wall. This second finite difference solution requires an approximation for









the velocity condition at the walls; a linear extrapolation is used in conjunction with the no-slip

condition at the wall. For completeness, a second order extrapolation is also used to approximate

the derivative at the wall in solving Eq. (2.61); the error is consistently larger than the linear

extrapolation and hence the results are not presented. The viscosity distribution of Case I is

chosen for the finite difference computations for Case I gives sharper near-wall velocity

gradients than Case II.

The results of the relative L2 -norm errors over a range of h=0.004 to 0.1 from these two

finite difference solutions are shown in Figure 3-10b. The results labeled as "FD-H" refer to the

finite difference solution obtained on the hexagonal grid and the results labeled as "FD-S" refer

to the finite difference solution obtained on the square grid. For small values of h (h<0.01), the

O(h2) asymptotic behavior in error is clearly visible, which is similar to the LBE cases. This

asymptotic error behavior is expected because the LBE scheme has second order accuracy and

the boundary conditions are, depending on the specific scheme chosen, either exact or second

order accurate, and the finite difference scheme with the central difference discretization gives

global second order accuracy. As h increases close to or greater than 0.01, both the finite

difference and the LBE errors increase faster than O(h2). Since the velocity profile given by the

exact solution has a thin layer of thickness 0= 0.0102, the error starts to increase more rapidly

for h>0.01 when the resolution of the thin layer becomes inadequate. For 0.01
the numerical resolution is insufficient, all the error curves are quite close, indicating that the

need for a better resolution of the boundary layer exceeds the need for a better boundary

condition. One also notes that when h exceeds the boundary layer thickness, truncation errors

associated with high order velocity gradients may not be small.









Comparing the relative L2 -norm errors of the finite difference solution with those of the

LBE solution in Figure 3-1Ob, it is seen that the relative L2 -norm error of"FD-H" follows that of

the LBE with Noble et al.'s scheme closely, and the relative L2 -norm error "FD-S" follows that

of the LBE with the BBL scheme closely. This consistent behavior of the relative L2 -norm errors

between both the LBE method and the finite difference method suggests that one could obtain an

insight on the truncation error of the LBE schemes by studying the truncation error of its finite

difference counterpart.

For both LBE and finite difference schemes, their modified equations associated with the

corresponding truncation errors can be represented as

L(ELBE) = T.E. of the LBE method (2.71)

L(EFD) = T.E. of the finite difference method (2.72)


where L is the differential operator d vota ELBE = t -uLBE and ED = u,t u,. Using
dx dx

Taylor series expansion the truncation error in Eq. (2.72) is

1 04U _I VtotI_ 2 tota 2 I V total C2
T.E.= -v- +1 a -_+ a (A)2 +H.O.T.(2.73)
12 o 4 6 ao] o]3 8 a]2 Or2 24 r73 ar]

Although the truncation error of the LBE is unknown, comparing the solution errors, EFD

and ELBE, on the left-hand-sides of Eq. (2.71) and (2.72) can offer insight into the truncation

errors on the right-hand-sides of these equations while this indirect comparison avoids the

tedious derivation of the truncation error of the LBE. The value of EFD can be computed by

subtracting uFD from the exact solution after the velocity uFD is solved from Eq. (2.70).









To highlight the behavior of the leading term of the truncation error, the values of EFD are

approximated by ED obtained from solving Eq. (2.72) with small h. For ELBE it can only be

determined directly by subtracting ULBE from the exact solution.

Eq. (2.72) can be solved, with the boundary conditions ED (0) = ED (1) = 0, by using the

same central difference scheme given by Eq. (2.70), and replacing the entire T.E. by the leading

term of the T.E. provided that the resolution is sufficient and the high-order terms in the T.E. can

be neglected. Caution must be exercised in interpreting the results from the numerical solution of

Eq. (2.72). In this equation, even if the leading term of the T.E. on the right hand side can be

analytically evaluated, the variation of the viscosity on the left hand side is the source of the T.E.

in the first place. When solving for E(77) the effect of the T.E. associated with the ODE (Eq.

(2.72)) now is further compounded by the variation of the viscosity.

The finite difference computation for Eq. (2.72) is carried out on a grid with the same grid

arrangement and size as in the hexagonal lattice case so that the exact velocity boundary

condition can be used. For Case I, the thin wall layer has a length scale 0=0.0102 and can be

adequately resolved using Aq = 1/H = 1/200, which is confirmed by the comparison between

ED and E in Figure 3-11. The curve representing ED is almost on top of the curve

representing E Figure 3-11 also shows the variation of LBE. across the channel. Overall, E

from Eq. (2.72) is smaller than LBEL across the whole channel which implies that these two

methods indeed have quantitatively different truncation errors. However if we adjust the scale for

E* by a factor of 2, these two curves lie almost on top of each other, as shown in the inset of

Figure 3-11. The matching of the error curves between two solutions demonstrate that the LBE

truncation error behaves very similarly to that observed in the finite difference scheme in the









presence of a strong variation in the viscosity. This is in agreement with the observation on the

relative L2 -norm error behavior shown in Figure 3-10b.

Based on the foregoing discussions, it is seen that BBL scheme performs similarly as

Nobel's scheme in the presence of strong velocity profile variation. In view of the simplicity and

flexibility of the BBL scheme over the Noble et al.'s scheme and comparable performance in

accuracy, it is attractive to use the BBL scheme (or its extended version such as Bouzidi et

al.(2001) and Yu et al. (2003a)) for handling problems of substantial velocity profile variations

caused either by complex geometry or variable viscosity.

For complex 3D flows with curved boundaries, Navier-Stokes solvers currently have more

flexibility on grid arrangement. For example, body-fitted coordinates and grid stretching can be

more easily implemented to improve grid resolution near boundaries than in the LBE solvers.

Although the recent developments in LBE method such as multi-block (Yu et al. 2003) and

composite grid (Fillippova and Hanel 1998) techniques can alleviate the difficulty in grid

arrangement in LBE simulations to certain extent, further research efforts in LBE are needed. As

the present study has indicated the boundary condition induced error may not be dominant, there

is a great potential in improving the overall capability of the LBE solvers for complex flows with

very strong velocity variation by focusing future research efforts on extending the grid flexibility

of LBE schemes.

3.3 Summary and Conclusion

In this chapter, the error behavior of the lattice Boltzmann equation (LBE) method for a

flow with strong variation in viscosity is studied. The variable viscosity in the LBE method is

modeled through a variable relaxation time. Solutions for a laminar channel flow with a specified

variable viscosity are obtained using both the LBE method and the finite difference method to









examine the truncation error behavior of the LBE method for flows with strong varying

viscosity. The effect of the boundary condition error of the bounce-back-on-link (BBL) scheme

on the overall error is investigated via the comparison of the error behavior of the BBL scheme

and that of Noble et al.'s scheme. The results show that with rapid viscosity variation the

boundary condition error of the BBL scheme does not induce noticeable, additional errors, and

the overall error of such flows is dominated by the truncation error itself. The results also show

that in the presence of strong variable viscosity the truncation error behavior of the LBE solution

is consistent with that of the finite difference solution to Navier-Stokes solution.












Xf Xf


Wall


Figure 3-1. Boundary nodes and their neighbors using the square lattice


3 \ / 2 2


Fluid
(f)


1 Boundary





\Wall
5 6 (w)


Figure 3-2. A boundary cell using the hexagonal (FHP) lattice (Noble et al. 1995)


20 40 60 80 100


. Absolute L2 norm errors of LBE wi
channel flow with constant viscosity


th Noble's scheme for fully-developed laminar


|eh
e_ ee e.


e,+
Rx a x


10

10

10

10

10

10

10

10

10


Figure 3-3



















































40 60 80 100


H
Absolute error of a fully-developed channel flow using Inamuro et. al.'s scheme


4 3


7 8

A 2D 9-velocity lattice (D2Q9) model


6
Figure 3-4.


6E-11 -

5E-11 -


4E-11 -



.2 3E-11 -
0


2j
2E- 11


Figure 3-5.


20


I I I I I


Figure 3-6. Two set of viscosity distributions used in this study















0.8 se- ii.


025 -

0.6 0- 2 Case I
- Case II
015 tan terms
S in mEq (3 35) (Case )


0. ,
0.4 -
00 5 -







0
0 0002 0004 / .




0 0.001 0.002 0.003 0.004 0.005 0.006 0.007
u

Figure 3-7. The exact velocity profiles of the channel flows with different boundary layer
thicknesses due to different viscosity distributions. The parameters are:
(v;8; O) = (0.004167; 0.0005; 0.0102) for Case I and

(vo0;;0) =(0.008333;0.002;0.0289) for Case II


1


Outlet: Periodic
boundary condition


Figure 3-8. Square lattice distribution in channel flow simulation


j=Ny

j=Ny-1


Inlet: Periodic
boundary
condition




j=2

j=i












0 0006 Exact
Exact
u Hex
00005 u_Square


0 0004 -



o 00003 -



0 0002000








0 01 02 03 04 05
11

Figure 3-9. Comparison of the LBE velocity profiles using square lattice (with bounce-back on
the link boundary condition) and hexagonal lattice (with Nobel et al's exact boundary
condition) with the exact solution at H=50 lattice units.


p
-0--

C


BBL scheme (Case I)
Noble et al.' s scheme (Case I)
BBL scheme (Case II)
Noble et al.' s scheme (Case II)


I I I I I I I I I I I I II li li


0.02
h


0.04 0.06 0.08 0.1 0.12

a


















10' **

21O-

10-2


10 -
2 ----- FD-S (Case I)
1 FD-H (Case I)
----- BBL scheme (Case I)
10 -- Noble et al.' s scheme (Case I)



10 I I i I I I I
0.02 0.04 0.06 0.08 0.1 0.12
h b

Figure 3-10. Dependence of the relative L2 -norm error on the lattice size h in the fully-
developed channel flow with variable viscosity. The viscosity parameters are:
(3,0) = (0.0005,0.0102) for Case I, (0.002,0.0289) for Case II. (a) The LBE with the
boundary conditions of Noble et al's scheme and BBL scheme for both Case I and
Case II. (b) The finite difference and the LBE with boundary conditions of Noble et
al's scheme and BBL scheme for Case I.












8E-06 E- E

G- E



6E-06 4E-06


6E-06 --- E_ 3E-06



4E-06 :4E-06 2E-06


2E-06 E-06


2E-06 0 -0

0 002 004 006 008 01 012 014
'1






0 0.1 0.2 0.3 0.4 0.5

T]

Figure 3-11. Comparison of ELBE = c, -uLBE with E = ue,, -uF for H=200, and

8 = 0.0005,0 = 0.0102, E*, is the numerical approximation of ED obtained from

solving Eq. (2.72).









CHAPTER 4
LBE METHOD FOR IMMISCIBLE TWO-PHASE FLOW COMPUTATION

The characteristic feature of immiscible two/multi-phase flows is the inter-phase boundary

(interface) between different phases. The structure of any inter-phase boundaries is on the

mesoscopic length scale level (Pismen 2001). The steep gradients of material properties in the

normal direction of an interface result from the fact that the normal length scale approaches

molecular level, while the length scale along the interface is still on the macroscopic level

(Rowlinson et al. 1982a; Carey 1992). As described in Chapter 1 these scale disparities make the

computations more difficult. In this chapter, a LBE immiscible two-phase model is developed

based on the framework of He et al.'s model (1999). Several numerical techniques are

implemented to the numerical stability so that two-phase flows with high liquid-gas density ratio

can be simulated. This chapter begins with an overview of the immiscible two-phase flow

computations and a literature review on the LBE two-phase models, followed by an introduction

to He et al.'s model (He et al. 1999). The Rayleigh-Taylor instability is simulated and

computational results are compared with analytical ones to validate the code. After this, the

numerical issues in Lee-Lin's implicit LBE model and He et al.'s model are addressed. A filter-

based LBE model along with a new surface force formulation and a volume-correction step is

proposed. The stability and accuracy of this new model are tested by computing flows around a

stationary bubbles, capillary waves and rising bubbles. The results show that the present model is

capable of handling the flows with large density ratio up to 0(102).

4.1 Overview of Immiscible Two-Phase Flow Computation

For macroscopic computations of immiscible two-phase flow the numerical methods can

be classified into two categories. One is the interface-tracking method, in which interfaces have

to be tracked numerically to separate materials of two different phases or properties (Shyy et al.









1996). The other is the continuous interface method in which interfaces are not tracked explicitly

(Anderson et al. 1998).

Two approaches exist in terms of the grids used for computations. One is the moving grid

(Lagrangian) approach. The other is the fixed grid (Eulerian) approach (Shyy et al. 1996; Shyy

2004). These two approaches can also be combined to form a mixed approach that is called the

Eulerian-Lagrangian approach.

In the Lagrangian approach the interface is tracked explicitly by a body-fitted grid that

deforms with the interface. No modeling is necessary to define the interface or its effect on the

flow field. However, the body-fitted grid has to be regenerated as the interface deforms. This

requires a set of equations that need to be solved for the grid regeneration at each time step.

Moreover, Lagrangian approach is not easy to use when fluid flows involve complex topological

changes of interfaces.

The Eulerian approach is more robust in general but it needs elaborate procedures to

deduce the interface location based on the volume fraction information. Typical Eulerian

methods are the level-set (LS) method and the volume of fluid (VOF) method. Interface is often

constructed after the field solution obtained. Thus the interface construction decouples from the

field equation solver. Since the grid is fixed in the Eulerian approach, the grid generation

required by the Lagrangian approach is obviated. Another advantage of the Eulerian approach

over the Lagrangian approach is that the topological changes of interfaces are automatically

taken into account in the Eulerian approach, which makes the Eulerian approach more suited for

flows with complex interface deformations. However, the Eulerian approach suffers several

disadvantages. The interface shape is smeared off. If high resolution information of interface is

desired, an adaptive local grid refinement may be needed, which is another complicated process.









Thus, some physics may be lost if the complicated grid refinement is not available, such as the

uniqueness of the shape interpretation, and continuity and smoothness of the interface (Shyy

2004).

The Eulerian-Lagrangian approach combines the advantages of both Lagrangian and

Eulerian approaches. In this approach, the grid is fixed as the Eulerian approach. Grid refinement

is not needed. The interface is explicitly tracked by some marker points which are the

Lagrangian components. This explicit interface-tracking can provide detailed interface

information as the Lagrangian approach (Shyy et al. 1996).

The Eulerian, Lagrangian and Eulerian-Lagrangian approaches are numerical methods for

two-phase flow computations with interfaces. Physically, the interface can be modeled as a finite

thickness line (2D) and surface (3D), or a zero thickness line (2D) and surface (3D). They are

called the continuous interface method (CIM) and the sharp interface method (SIM),

respectively. At the macroscopic modeling level, the SIM is consistent with the concept of the

continuum mechanics (Shyy 2004). Interface is tracked explicitly with some computational cells

which are cut by the interface. Thus interface smearing is not involved. In contrast to the SIM,

the CIM does not track interfaces explicitly. Of these two methods, the SIM and CIM, if detailed

interface information is desired, the SIM is a better choice since it can provide a second-order

accuracy while the CIM can only provide a first-order accuracy in general. However, the CIM

has some advantages which the SIM lacks. For instance, the CIM can be easily adopted for three-

dimensional flows with topological changes of interfaces; the CIM is particularly useful for

flows with phase changes; and the CIM is especially appropriate for some problems that are

currently tough for the SIM, such as contact line dynamics (Pismen 2001; Lee et al. 2005). More









detailed information for moving interface computations based on the macroscopic computation

point of view can be found elsewhere (Shyy et al. 1996, Shyy 2004)

The method of lattice Boltzmann equation is essentially a kinetic equation solver over

mesoscopic length scale (Succi 2001). Physically the essential length scale of this method is

closer to microscopic level than macroscopic continuum-based methods. Thus it is relatively

easier for the LBE method to incorporate microscopic modeling for two-phase flows with

interfaces. As a result, the LBE method has numerous advantages that the molecular dynamics

method has, which are especially useful for the simulations of phase interfaces of non-ideal gases

(Shan, et al, 1993; He, et al, 1999) or binary fluids and near wall treatment at micro-fluid level

(Nie, et al, 2002). Phase segregation and interfacial dynamics can be achieved naturally by

incorporating intermolecular interactions. Because the interface spreads over several lattices in

the LBE method (He et al. 1999) most of the two-phase LBE methods can be considered as CIM.

As one of the continuous interface methods, the LBE method has all the advantages of the

CIM for two-phase flows. However, the nature of the modeling on the mesoscopic level may

allow the LBE method to incorporate the intermolecular interaction more easily than the

conventional CIM. Therefore, the LBE method may be a good alternative for two-phase flows

simulations (Fang et al. 2001; Lee et al. 2005).

4.2 Literature Review on LBE Method for Two-Phase Flow Computation

The first lattice-based two-phase flow model was the lattice gas color particle model

proposed by Rothman and Keller (Rothman et al. 1988). This model has not been widely used in

practice since it suffers from the defects of the lattice gas method, such as the computational

noise. The LBE method overcomes the natural defects of the lattice gas method. Gunstensen et

al. (Gunstensen et al. 1991; Gunstensen et al. 1993) proposed the first LBE multiphase model

developed from Rothman and Keller's lattice gas model. Although Gunstensen et al's model









possesses the essential features of the LBE method, i.e., the Galilean invariance and statistical

noise free, it cannot simulate multiphase flows with different densities and viscosities. Grunau et

al. (Grunau et al. 1993) improved Gunstensen et al.'s model by using a single-time relaxation

approximation and a proper equilibrium distribution function. This model can be used for flows

with different densities and viscosities. Since these models were developed from Rothman and

Keller's pioneering lattice gas color particle model, all require a "re-color" step to maintain the

interfaces.

Shan and Chen proposed a new two-phase LBE model hereinafter referred as (S-C model)

in which the non-local microscopic particle interaction is incorporated (Shan et al. 1993; Hou et

al. 1997). The molecular interaction introduced into the ideal gas LBE model is represented by

an interaction potential which models the multiphase separation and dynamics. Although the S-C

model has performed much better for the multiphase flows than the R-K model (Hou et al. 1997),

it possesses some anisotropy (He et al. 1998). The free-energy LBE model developed by Swift et

al. (1995) uses an equilibrium distribution which makes the pressure tensor consistent with that

of the free-energy function of non-uniform fluids. However the Galilean invariance is not

satisfied.

Although the aforementioned models are based on different physical considerations, He et

al. pointed out that they all originate from the kinetic theory which can be represented by the

continuous Boltzmann equation with certain approximations (He et al. 1998). Subsequently He et

al. proposed an improved LBE scheme using the single-time-relaxation approximation to

simulate of multiphase flows in the incompressible limit (He et al. 1999a). In this new model,

two distribution functions corresponding to two evolution equations are employed. One acts as

an index function to track interfaces implicitly between different phases. The other is an









evolution equation for pressure distribution which satisfies the mass and momentum

conservation on the macroscopic level. Molecular interactions, such as the molecular exclusion-

volume effect and the intermolecular attraction, are incorporated into this model. These

interactions maintain the phase separation by the mechanical instability on the supernodal curve

of the phase diagram (He et al. 1999). Compared to the S-C model, this thermodynamics-based

concept has much more flexibility for maintaining the phase separation. Furthermore, the

numerical stability of He et al.'s model is improved by reducing the numerical errors from 0(1)

to 0(u) during the calculation of the molecular interactions.

One common deficiency in all the aforementioned models is that they cannot be used for

the multiphase flows with large density ratio (Lee et al. 2005). R-K model, S-C model, and Swift

model can only be used for the two-phase flows with density ratio less than 2. He et al.'s

improved model brings the density ratio to around 10. Chen et al. (1998) discretized the

Boltzmann equation with a total variation diminishing scheme. In Chen et al.'s method, the

second-order Runge-Kutta time-matching was used for the discretized Boltzmann equation

(DBE), while the collision term and the intermolecular interaction term were treated explicitly as

source terms. Although the use of TVD scheme improved the stability of the LBE method for the

large density ratio case, the accuracy is reduced in the presence of large gradients because of the

additional numerical dissipation of the TVD scheme. Inamuro et al. (2004) proposed another

LBE model for the two-phase flows with large density ratio. They also used two particle

distribution function equations: one as an index function to indicate the phase separation, the

other for the momentum evolution but without the pressure gradient. Velocity and pressure are

coupled by an additional pressure Poisson equation to satisfy the incompressibility requirement.

The procedure is analogous to the fraction step method (Ferziger 2003) which is used to solve









the macroscopic Navier-Stokes equation. It is well known that it is time consuming to solve the

Poisson equation (Shyy, 1997c). Although Inamuro et al.'s were able to simulate multiphase

flows with large density ratio using their scheme, such as rising bubbles in liquid, the natural

simplicity of LBE method is lost due to the necessity of solving the pressure Poisson equation.

Lee and Lin developed an implicit two-phase LBE model which can simulate large

density/viscosity ratio problems without using the iteration procedure (Lee and Lin 2005). Lee

and Lin's model is similar to He et al.'s model but with a different pressure updating process. In

He et al.'s model, the pressure in calculating the intermolecular interaction term is updated from

equation of state (EOS). Lee and Lin argued that this pressure updating process can lead to the

spurious pressure fluctuations at the interfaces due to the EOS. Lee and Lin proposed a pressure

evolution equation which can overcome this difficulty and at the same time allow for large

density difference across the phase interfaces when phase change occurs due to pressurization

and depressurization (Lee et al. 2003). Using this model, Lee and Lin simulated 1-D advection

equation with a source term, a stationary droplet, an oscillating droplet and a droplet splashing

on a thin film at a density ratio of 1000 with varying Reynolds numbers. However, very little

information on the details of the flow field near the large gradient interfacial region was reported.

In the following sections, He et al.'s LBE model for two-phase flows will be reviewed. It

will be applied to the Rayleigh-Taylor instability problem for a code validation. The interface

behavior of Lee and Lin's implicit LBE is then studied. To maintain the interface thickness as

sharp as the initial interface thickness and improve numerical stability, a filter-based two-phase

LBE model is developed based on He et al.'s model. Stationary liquid flow around a static

bubble, the capillary wave caused by two superposed fluids with same viscosity, and flow









induced by a rising deforming bubble are computed at density ratio of 0(102) to asses the

efficacy of the proposed two-phase LBE model.

4.3 He et al.'s Isothermal LBE Model for Two-Phase Flow

4.3.1 The Boltzmann Equation for Non-Idea Fluids

The Boltzmann equation for non-ideal-fluid flows intermolecular interaction force is (He et

al. 1998)

f Vf f feq (+ -u) (F +G) fq (2.1)
+(- Vf_= + Ieq (2.1)
8t A pRT

The single-relaxation-time BGK model is used in this equation. The derivation of Eq. (2.1) is

based on the assumption that the distribution function gradient with respect to the particle

velocity is equal to the equilibrium distribution function gradient with respect to the particle

velocity (He et al. 1998). Compared to the standard LBE model for ideal gas flows, the last term

on the RHS of Eq. (2.1) represents the effects of the intermolecular interaction F and gravity G.

In Eq.(2.1), R is the gas constant and 1 is the relaxation time. The Maxwellian distribution for an

equilibrium state is


feq = ) ex (e R u)2 (2.2)
(27RT)D/2 2RT

where D is the dimension of the space.

The intermolecular attractive force originates from the van der Waals theory (Rowlinson et

al. 1982b). This intermolecular force can be expressed as

F,, = pV 2ap+ AV2p) (2.3)

The parameters a and K result from the intermolecular pair-wise potential U

1I
a =- U (r)dr, K =- r2U (r)dr, (2.4)
2 r>d 6 r>d









where d is the effective molecular diameter, dr is the effective differential volume. Besides the

intermolecular force, He et al. also included the exclusion-volume effect into the intermolecular

interaction

FE, = -bp2RTzVln(p2z) (2.5)

which accounts for the extra volume effect of non-ideal fluids (Chapman et al. 1970). The

2 d3
parameter b is given by b = 3 where m is the single molecular mass. The parameter, in (2.5)
3m

is the increase in the collision probability due to the increasing fluid density. This parameter can

be expressed as a polynomial in terms of the product of the density p and the parameter b


(p)= l+bp +0.2869(bp)2 +0.1103(bp)3 +... (2.6)


The overall intermolecular interaction is thus the sum of the intermolecular force and the

exclusion-volume effect

F = FM, + FV (2.7)

The intermolecular interaction can be recast as

F = -V V + F (2.8)

where F, is associated with the surface tension, F = KpVV2p The potential term VVi is associated

with the pressure gradient as


= p pRT (2.9)

Thus the term VVi depends on the equation of state (EOS) and it in turn determines the phase

segregation. From the van der Waals theory, if the temperature is lower than the critical point of

a fluid, the phase segregation can be generated by the molecular attraction. The critical

temperature is determined by the equation of state. For the van der Waals equation of state










pRT ap2 (2.10)
1-bp

the critical temperature is T =a If the hard sphere model is used for the repulsion
27bR

interaction between molecules, the Carnahan-Starling equation of state is obtained (Carnahan

and Starling 1969, 1972)


1+bp +(bp 2bp3
p = pRT 4ap2 (2.11)
(1-^


and the critical temperature is Tr = 0.3773-a, where b is the co-volume of spheres. During the
bR

computation, the parameters a and b should be selected to satisfy the condition that the

temperature is lower than the critical temperature.

He et al. pointed out that vIy(p) in Eq. (2.8) is usually very large at the interfaces. As a

result, the calculation of VI,(p) can involve to a substantial numerical error, which lends the

numerical scheme unstable. In order to reduce this sensitivity, He et al. introduced another

variable g to replace the distribution function

g= fRT + (p)F (0) (2.12)


where F(0)is the value of F(u) evaluated at u=0, and F(u)- (exp (- ) The
(2T/1/)' 2RT

derivative of g is then

g g =Vg =RT +F(O)) (2.13)
Dt 9t Dt Dt

Since y(p) is a function of density, we can rewrite DV
Dt

DyI V/ + = +u.V+V. -u.V = d +(y-u).Vi (2.14)
Dt at at dt









For incompressible flow dy =y -p was then treated as zero in the wok of He et al (1999).
dt dp dt

Substituting Eqs. (2.1), (2.12) and (2.14) into (2.13) the evolution equation of g can be written as

g+ +.-Vg- gg +( -u){[F(u)(F1 +G)-(F(u)-F(O))VV(p)] (2.15)


The equilibrium distribution function of g is

geq = pRTF(u)+V(p)F(0). (2.16)

From g, the macroscopic variables pressure p, instead of the density p, and velocity u can be

calculated

p= fgdf, pRTu = fgd. (2.17)

Note the function V/ is calculated from the hydrodynamic pressure as

(p)= p, pRT (2.18)

where the hydrodynamic pressure p, is obtained from Eq. (2.17), instead of equation of state

(Eq. (2.11)) as was typically the case for single phase flow. The sensitivity of V/(p) in Eq.

(2.15) is alleviated by multiplying (F(u)-F(0)) which is proportional to the Mach number and is

small in the incompressible limit.

For multiphase flows, besides pressure and velocity which are given by (2.17), density is

another important variable which describes the different phase distributions. Density is only a

single function of pressure for isothermal single-phase flows, but it cannot be uniquely

determined by the pressure alone for isothermal multiphase flows. Therefore, like other diffusive

interface methods for multiphase incompressible flows, an index function evolution equation is

needed for tracking different phases and then the density or composition of fluids could be

obtained from this index function. In the LBE models for two-phase flows, Eq. (2.1) can serve as









the index function evolution equation. Since the gravity does not affect the mass conservation,

He et al. dropped the gravity in the index function evolution equation. It is given by,

-= +Vf=-- +( )-V )feq (2.19)
Dt 8t 2 pRT

It is noted that / is the function of the index function 0, instead of density p Thus it is defined

as

(0) = p, ORT (2.20)

where the thermodynamic pressure p, is the thermodynamic pressure calculated from the

equation of state Eq. (2.11), while the density is replaced by the index function 0


1+ +
p, = ORT 3 a02 (2.21)

4

To recapitulate, the evolution equations forf and g are

D.f -Vf f + (-U) VI(0) feq (2.22)
Dt 8t 2 pRT

g+ Vg = -- +( -u).[F(u)(F1 +G)-(F(u)-F(O))VV((p)] (2.23)


where V/ () = p, ORT, and /(p) = p, pRT. The equilibrium distributions forfand g are
f q = F (u) (2.24)

geq = pRTF(u)+V(p)F(O), (2.25)


where F(u)= 1 exp L -u )2. The macroscopic variables can be obtained from the
(2RT)D 2 2RT

moments off and g

S= ffde, p = gde, pRTu = f gde. (2.26)

Density and viscosity can be calculated by the linear interpolation from 0









P () = P, + -P,), (2.27)




where p, and p, are the material densities of light and heavy fluids, respectively; v, and V, are

the kinematical viscosities of light and heavy fluids. The above formulation is equivalent to the

following macroscopic equations (Zhang et al. 2000):

+ .(u)= -V p(p)- Vp()V (2.29)

1 p +V.u=0 (2.30)
pRT Ot

S +]t(V) =-Vp+ V[pv(Vu+uV)]+V2p+G (2.31)

4.3.2 Lattice Boltzmann Scheme for Multiphase Flow in the Near Incompressible Limit

The lattice Boltzmann equation can be obtained by discretizing the Boltzmann equations

described in the previous section. The discretization procedure is same as given in Chapter 3 for

single phase flows. After discretization in the phase space, the discretized Boltzmann equations

are

a a+e .V (e-u) ) (2.32)
Qt A pRT

-+e Vg a- q +(e -u).[F (u)(F +G)-(Fa (u)-F (0))VV(p)] (2.33)


To improve the accuracy and maintain the explicit computational scheme, He et al. further

introduced two new variables, which are


7 = -(e u)V (0) (u),, (2.34)
2RT

a ga -(e. -u).[F (u)(F] +G)-(F,(u)-Fa(O))Vw(p)]a1 (2.35)
2g 7c









The new evolution equations in terms of these two new variables are

(x+e?, ,t + ,)- (x,t)


( (x,t) {(xxt)(2r_ 1) 1){( u).y() F (2.36)
T 2z RT

-X +(x +e6,,t + 8 )--' (x,t)
= t) -_'q(xt) +(2z-- ---(e, u). [F, (u)(F + G) -(F (u) -F (0))VV((p)] (2.37)

where is still the non-dimensional relaxation time, r=/,. The macroscopic variables can be

calculated in terms of the new distribution functions as

S=Z (2.38)

p = u-VV/(p),, (2.39)


pRTu= e, + RT(F +G)6,. (2.40)

In the D2Q9 model, the kinetic viscosity is independent of surface force and is related to the non-

dimensional relaxation time as

v = (- 0.5)RT8,. (2.41)

Zhang et al. (2000), and McCracken et al. (2005) have used the following integral relationship to

analytically relate surface tension a with the coefficient ic :

cr=kl(a) = k dz (2.42)

where z is a direction normal to a flat interface.

4.4 Code Validation: Rayleigh-Taylor Instability

A two-dimensional D2Q9 LBE (Eq. (2.36)) code has been developed to solve Eqns (2.36)-

(2.41). To validate the code the Rayleigh-Taylor instability is simulated and the results are

compared with analytical solutions. The Rayleigh-Taylor instability problem is a very

challenging computational task because it possesses great complexities including strong non-









linearity associated with growth of the secondary Kelvin-Helmholtz instability and turbulent

mixing at later stages. On the other hand, the Rayleigh-Taylor instability has been systematically

studied in the past (Chandrasekhar 1981), and the results from previous studies can be used to

validate the present LBE simulation.

4.4.1 Linear Analysis of Rayleigh-Taylor Instability

The Rayleigh-Taylor (RT) instability occurs when a heavier fluid of density p2 superposes

over a lighter fluid of density p and the lighter one accelerates the heavier one in the direction

normal to the plane interface between these two fluids. The heavier fluid moves downward

producing spikes and lighter fluid grows upward producing bubbles.

The linear analysis is usually used to predict the linear instability. In this approach, the

perturbation analysis is used to linearize the non-linear convection terms. The resulting pressure

and density are functions of the vertical coordinate, say z, only (Chandrasekhar 1981).

Chandrasekhar gave the relation between the growth rate of the disturbance as a function of

wave number:


2iAt+ k2 +1 }(f +fq,-k)-4kf


+4k(fl v2)[(f2q -flq2)+k(f, f)] (2.43)
4k3
+ (fv f)2) (q-k)(q -k)=0,

The Atwood number is defined as

At= 2 p (2.44)
P1 + p2

where a is growth rate, g is gravity, k iswave number, v is viscosity, and

P 2 1 Aq2 2 +a
f= A, f2k2+-, q2 = k2 .
PA 2 Al + A V, V 2









4.4.2 Rayleigh-Taylor Instability: LBE Results

The two-dimensional Rayleigh-Taylor instability with single-mode is simulated using the

same physical parameters as in He et al.'s work (He et al. 1999a). Although He et al. presented

the multi-mode Rayleigh-Taylor instability results, the single-mode Rayleigh-Taylor instability

is preferred to serve as the benchmark for the present numerical simulation.

The computational domain is bounded by the top and bottom walls on which the no-slip

boundary condition is applied. The periodic boundary condition is applied on the two lateral

boundaries. Surface tension is neglected in this simulation and the kinetic viscosities of the two

fluids are assumed to be same. The function V/(0)= p,,- RT is calculated from the pressure

which satisfies the Carnahan-Starling equation of state, i.e., Eq. (2.11). In the Carnahan-Starling

equation, the parameter a is chosen to be bc2, which induces sufficient molecular interaction to

separate different phases. He et al. (1999), Zhang et al. (2000) and McCracken et al. (2005)

reported that a =bc2= 4 is sufficient to separate phases. With these values for a and b, the


function I(a) in Eq. (2.42), that is I(a)= f (c \dz, can be numerically determined for Canahan-


Starling fluid (Zhang et al. 2000) as

0."1518(a-a5 (2.45)
I(a)= 05 (2.45)
l+3.385(a-a )5

where a, = 3.53374 is the critical value of Carnahan-Starling equation of state, below which a

fluid cannot separate into different phases. With these parameters, the range of the index function

can be theoretically or numerically determined by the equation of state. For Canahan-Starling

equation of state, Zhang et al. (2000) obtained the range of the index function as 0.02381-0.2508.

In order to make meaningful comparisons two dimensionless numbers, Reynolds number

and Atwood number, are used for this problem. Reynolds number is defined as