Development of lattice boltzmann flux solvers and their applications

DEVELOPMENT OF LATTICE BOLTZMANN FLUX SOLVERS AND THEIR APPLICATIONS WANG YAN A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2014 DECLARATION I hereby declare that this thesis is my original work and it has been written by me in its entirety. I have duly acknowledged all the sources of information which have been used in the thesis. This thesis has also not been submitted for any degree in any university previously. Wang Yan 01 August 2014 i ACKNOWLEDGEMENTS First of all, I would like to express my deepest gratitude and heartfelt thanks to my supervisors, Professor Shu Chang and Dr. Teo Chiang Juay, for their foresight and sagacity in fluid mechanics and computational fluid dynamics, their invaluable and long-lasting guidance, great patience and endless support throughout my Ph. D study. Without them and their altruistic help, this dissertation could not have been finished. Secondly, I wish to express my great appreciation to the National University of Singapore for providing me the opportunity to complete this work. It provides various essential library resources, excellent study conditions and advanced computational facilities for me to the research work smoothly. I also wish to thank all the staff members in the fluid division for their kind help. My heartful appreciation will also go to all my friends, including Dr. Wu Jie, Dr Wang Junhong, Dr Shao Jiangyan, Dr. Ren Weiwei, Mr. Sun Yu, Dr. Wu Di, Dr. Zhang Xiaohu and many others, for their helpful instructions and discussions. Finally, I would like to express the deepest and heaviest love in the bottom of my heart to my family and my fiancee Liu Chenxi. Wang Yan ii TABLE OF CONTENTS DECLARATION i ACKNOWLEDGEMENTS ii TABLE OF CONTENTS iii SUMMARY ix LIST OF TABLES xi LIST OF FIGURES xiv NOMENCLATURE xxv Chapter Introduction 1.1 Background 1.2 Navier-Stokes solver 1.2.1 Vorticity-stream function approach 1.2.2 Artificial compressibility approach 1.2.3 Projection approach 1.2.4 Advantages and disadvantages of the N-S solver 1.3 Lattice Boltzmann equation solver 10 11 1.3.1 Origination and historical development of the LBE solver 12 1.3.2 Applications of the LBE solver 14 1.3.3 Advantages and disadvantages of the LBE solver 20 1.4 Motivations and objectives of the thesis 21 1.5 Organization of the thesis 24 iii Chapter Development of Lattice Boltzmann Flux Solver for Isothermal Incompressible Flows 29 2.1 Lattice Boltzmann method and Chapman-Enskog expansion analysis 30 2.1.1 Lattice Boltzmann method (LBM) 30 2.1.2 Chapman-Enskog expansion analysis 33 2.2 Lattice Boltzmann flux solver (LBFS) 36 2.2.1 Governing equations and finite volume discretization 36 2.2.2 Evaluation of f eq and f^ at cell interface by LBFS 38 2.2.3 Computational sequence 41 2.3 Numerical results and discussion 43 2.3.1 Decaying vortex flow 44 2.3.2 2D lid-driven flow in a square cavity 44 2.3.3 Viscous flow past a circular cylinder 48 2.3.4 Inviscid flow past a circular cylinder 51 2.3.5 3D lid-driven cavity flows 53 2.4 Conclusions 55 Chapter Development of Thermal Lattice Boltzmann Flux Solver for Simulation of Thermal Incompressible Flows 70 3.1 Simplified thermal lattice Boltzmann model 71 3.2 Thermal Lattice Boltzmann Flux Solver (TLBFS) 76 3.2.1 Governing equations and finite volume discretization 76 3.2.2 Evaluation of f eq and f^ at cell interface by LBFS 79 iv 3.2.3 Evaluation of h^ at cell interface 81 3.2.4 Computational sequence 84 3.3 Numerical results and discussion 85 3.3.1 2D natural convection in a square cavity 86 3.3.2 Natural convection in an 2D annulus 89 3.3.3 Mixed heat transfer from a heated circular cylinder 92 3.3.4 3D natural convection in a cubic cavity 95 3.4 Concluding remarks 97 Chapter Development of a Fractional Step-Lattice Boltzmann Flux Solver for Axisymmetric Flows 110 4.1 A fractional step-lattice Boltzmann flux solver 111 4.1.1 Governing equations and fractional-step discretization 111 4.1.2 Prediction of the intermediate flow field by TLBFS 116 4.1.3 Corrector step of the flow field 119 4.1.4 Computational sequence 120 4.2 Numerical examples of isothermal axisymmetric flows 122 4.2.1 Flow in a pipe 122 4.2.2 Taylor-Couette flow 125 4.2.3 Cylindrical cavity flow 126 4.3 Numerical examples for thermal axisymmetric flows 129 4.3.1 Natural convection in an annulus 129 4.3.2 Rayleigh-Benard convection in a vertical cylinder 130 v 4.3.3 Mixed convections in a tall vertical annulus 132 4.3.4 Wheeler’s benchmark problem 134 4.4 Concluding remarks 136 Chapter Multiphase Lattice Boltzmann Flux Solver for Incompressible Flows with Large Density Ratio 149 5.1 MLBFS for the flow field 150 5.1.1 Governing equations 150 5.1.2 Numerical discretization by the finite volume method 155 5.2 Cahn-Hilliard model for interface capturing 158 5.3 Computational sequence 160 5.4 Two-dimensional numerical examples 161 5.4.1 Immiscible two-phase co-current flow in a 2D channel 161 5.4.2 Two-phase Taylor-Couette flows in two concentric cylinders 163 5.4.3 Droplet spreading on a flat plate 164 5.4.4 Rayleigh-Taylor instability 165 5.4.5 Droplet splashing on a thin film 167 5.5 Three-dimensional numerical examples 170 5.5.1 3D Laplace law 170 5.5.2 3D Droplet spreading on a flat plate with different wettability 171 5.5.3 Oscillating spherical droplet 172 5.5.3 Collision of binary droplets 173 5.6 Concluding remarks 175 vi Chapter Boundary Condition-enforced Immersed Boundary-Lattice Boltzmann Flux Solver and Its Applications for Moving Boundary Flows 6.1 Conventional immersed boundary method (IBM) 192 193 6.2 Boundary condition-enforced immersed boundary-lattice Boltzmann flux solver (IB-LBFS) 196 6.2.1 Governing equations and fractional-step discretization 197 6.2.2 LBFS for prediction of the flow field u* 198 6.2.3 Boundary condition-enforced IBM for velocity correction 200 6.2.4 Computational sequence and force calculations 202 6.3 Two-dimensional (2D) numerical examples 204 6.3.1 Flow past a stationary cylinder 204 6.3.2 Flow past a transverse oscillating cylinder 205 6.3.3 Flow past two counter-rotating cylinders 206 6.3.4 Sedimentations of one and two particles in a rectangular box 208 6.3.5 Vortex induced vibrations (VIV) of a circular cylinder 210 6.4 Three dimensional (3D) numerical examples 213 6.4.1 Flow past a stationary sphere 213 6.4.2 Flow past a torus 215 6.4.3 Flow past a transverse rotating sphere 218 6.4.4 Flow past a streamwise rotating sphere 219 6.5 Concluding remarks 221 Chapter Development of Arbitrary-Lagrangian-Eulerian-based IB-LBFS and vii Its Application for Freely Falling Flow Problems 7.1 ALE-based IB-LBFS 241 242 7.1.1 Governing equations 243 7.1.2 Prediction of the flow field u* by LBFS 244 6.2.2 Velocity correction by IBM 247 7.2 Rigid body dynamics 248 7.3 Computational sequence and numerical validation 250 7.4 Application to 2D freely falling plate 251 7.4.1 Fluttering mode at Re=1147 252 7.4.2 Tumbling at Re=737 and 837 254 7.5 Application to 3D freely falling disk 257 7.5. Motion of a falling disk with low aspect ratio 258 7.5.2 Motion of a falling disk with large aspect ratio 260 7.6 Concluding remarks 261 Chapter Conclusions and Recommendations 275 8.1 Conclusions 275 8.2 Recommendations 281 Reference 283 viii Summary Due to the complexity of fluid flows in different scales and regimes and the limited computational resources, developing simple, accurate and efficient numerical algorithms has been one of the primary and fundamental tasks of the Computational Fluid Dynamics (CFD) community. During the past several decades, the well-established and dominating approaches for simulating incompressible flows are the N-S solvers and the LBE solvers, which are respectively based on the macroscopic conservation laws and mescoscopic statistical physics theory. The roots in different theoretical foundations credit these two solvers unique and distinctive advantages as well as some intrinsic disadvantages. Up to date, many improved solvers have been proposed to eliminate their drawbacks. However, due to their independent developments within one theoretical framework, the improvements are constrained and the intrinsic drawbacks of the N-S solvers and the LBE solvers cannot be completely removed. One way to elaborate this constraint is to develop new numerical methods which start from the theoretical connections of these two solvers. This thesis is devoted to developing a series of unified solvers for incompressible flows in different regimes and also extending their applications for complex moving boundary and freely falling problems. Firstly, four consistent lattice Boltzmann flux solvers (LBFSs) have been proposed respectively for simulating isothermal, thermal, axisymmetric and multiphase flows. The LBFSs are finite volume schemes for direct updating the macroscopic flow ix Chen H, Chen S, Matthaeus WH. Recovery of the Navier-Stokes equations using a lattice-gas Boltzmann method. Physical Review A 1992; 45:R5339-R42. Chen D-J, Lin K-H, Lin C-A. Immersed Boundary Method Based Lattice Boltzmann Method to Simulate 2D and 3D Complex Geometry Flows. International Journal of Modern Physics C 2007; 18:585-94. Chen H, Teixeira C. H-theorem and origins of instability in thermal lattice Boltzmann models. Computer Physics Communications 2000; 129:21-31. Chen S, Doolen GD. Lattice Boltzmann method for fluid flows. Annual Review of Fluid Mechanics 1998; 30:329-64. Chen S, Tölke J, Geller S, Krafczyk M. Lattice Boltzmann model for incompressible axisymmetric flows. Physical Review E 2008; 78:046703. Chorin AJ. Numerical Solution of the Navier-Stokes Equation. Mathematics of Computations 1968; 22:745-762. Chorin AJ, On the Convergence of Discrete Approximations to the Navier-Stokes Equations. Mathematics of Computations 1969; 22:341-353. Chrust M, Bouchet G, Dušek J. Numerical simulation of the dynamics of freely falling discs. Physics of Fluids 2013; 25:044102. Daly BJ. Numerical Study of Two Fluid Rayleigh‐Taylor Instability. Physics of Fluids 1967; 10:297-307. De Vahl Davis G. Natural convection of air in a square cavity: A bench mark numerical solution. International Journal for Numerical Methods in Fluids 1983; 3:249-64. Dennis SCR, Chang GZ, Numerical Solutions for Steady Flow Past a Circular Cylinder at Reynolds Number up to 100, Journal of Fluid Mechanics 1970; 42: 471-489. Dennis SCR, Quartapelle L. Direct solution of the vorticity-stream function ordinary 285 differential equations by a Chebyshev approximation. Journal of Computational Physics 1983; 52:448-63. Ding H, Shu C, Yeo KS, Xu D. Simulation of incompressible viscous flows past a circular cylinder by hybrid FD scheme and meshless least square-based finite difference method. Computer Methods in Applied Mechanics and Engineering 2004; 193:727-44. Ding H, Shu C, Yeo KS, Xu D. Numerical computation of three-dimensional incompressible viscous flows in the primitive variable form by local multiquadric differential quadrature method. Computer Methods in Applied Mechanics and Engineering 2006; 195:516-33. Ding H, Spelt PDM, Shu C. Diffuse interface model for incompressible two-phase flows with large density ratios. Journal of Computational Physics 2007; 226:2078-95. Dubcova L, Solin P, Hansen G, Park H. Comparison of multimesh hp-FEM to interpolation and projection methods for spatial coupling of thermal and neutron diffusion calculations. Journal of Computational Physics 2011; 230:1182-97. Dutsch H, Durst F, Becker S, Lienhart H. Low-Reynolds-number flow around an oscillating circular cylinder at low Keulegan–Carpenter numbers. Journal of Fluid Mechanics 1998; 360:249-71. Ehrenstein U, Peyret R. A Chebyshev collocation method for the Navier–Stokes equations with application to double-diffusive convection. International Journal for Numerical Methods in Fluids 1989; 9:427-52. Feng Z-G, Michaelides EE. The immersed boundary-lattice Boltzmann method for solving fluid–particles interaction problems. Journal of Computational Physics 2004; 195:602-28. Feng Z-G, Michaelides EE. Proteus: a direct forcing method in the simulations of particulate flows. Journal of Computational Physics 2005; 202:20-51. Fernandes PC, Risso F, Ern P, Magnaudet J. Oscillatory motion and wake instability 286 of freely rising axisymmetric bodies. Journal of Fluid Mechanics 2007; 573:479-502. Fortes A, Joseph DD, Lundgren TS. Nonlinear mechanics of fluidization of beds of spherical particles, Journal of Fluid Mechanics 1987; 177: 467-483. Fujimura K, Hyun JM, Koyama HS. Time-Dependent Vortex Breakdown in a Cylinder With a Rotating Lid. Journal of Fluids Engineering 1997; 119:450-3. Fusegi T, Hyun JM, Kuwahara K, Farouk B. A numerical study of three-dimensional natural convection in a differentially heated cubical enclosure. International Journal of Heat and Mass Transfer 1991; 34:1543-57. Ghia U, Ghia KN, Shin CT. High-Re solutions for incompressible flow using the Navier-Stokes equations and a multigrid method. Journal of Computational Physics 1982; 48:387-411. Giacobello M, Ooi A, Balachandar S. Wake structure of a transversely rotating sphere at moderate Reynolds numbers. Journal of Fluid Mechanics 2009; 621:103-30. Gilmanov A, Sotiropoulos F, Balaras E. A general reconstruction algorithm for simulating flows with complex 3D immersed boundaries on Cartesian grids. Journal of Computational Physics 2003; 191:660-9. Glowinski R, Pironneau O. Finite element methods for Navier-Stokes equations, Annual Review of fluid mechanics 1992; 24:167-204. Guermond JL, Quartapelle L. A Projection FEM for Variable Density Incompressible Flows. Journal of Computational Physics 2000; 165:167-88. Guilmineau E, Queutey P. A Numerical Simulation of Vortex Shedding from an Oscillating Circular Cylinder. Journal of Fluids and Structures 2002; 16:773-94. Gunstensen AK, Rothman DH, Zaleski S, Zanetti G. Lattice Boltzmann model of immiscible fluids. Physical Review A 1991; 43:4320-7. Guo Z, Han H, Shi B, Zheng C. Theory of the lattice Boltzmann equation: Lattice Boltzmann model for axisymmetric flows. Physical Review E 2009; 79:046708. 287 Guo Z, Zheng C, Shi B, Zhao TS. Thermal lattice Boltzmann equation for low Mach number flows: Decoupling model. Physical Review E 2007; 75:036704. Ha MY, Jung MJ. A numerical study on three-dimensional conjugate heat transfer of natural convection and conduction in a differentially heated cubic enclosure with a heat-generating cubic conducting body. International Journal of Heat and Mass Transfer 2000; 43:4229-48. Halliday I, Hammond LA, Care CM, Good K, Stevens A. Lattice Boltzmann equation hydrodynamics. Physical Review E 2001; 64:011208. Hardy J, Pomeau Y, de Pazzis O. Time Evolution of a Two-Dimensional Classical Lattice System. Physical Review Letters 1973; 31:276-9. Hatton AP, James DD, Swire HW. Combined forced and natural convection with low-speed air flow over horizontal cylinders. Journal of Fluid Mechanics 1970; 42:17-31. He X, Chen S, Doolen GD. A Novel Thermal Model for the Lattice Boltzmann Method in Incompressible Limit. Journal of Computational Physics 1998; 146:282-300. He X, Chen S, Zhang R. A Lattice Boltzmann Scheme for Incompressible Multiphase Flow and Its Application in Simulation of Rayleigh–Taylor Instability. Journal of Computational Physics 1999; 152:642-63. He X, Doolen G. Lattice Boltzmann Method on Curvilinear Coordinates System: Flow around a Circular Cylinder. Journal of Computational Physics 1997; 134:306-15. He X, Luo L-S, Dembo M. Some Progress in Lattice Boltzmann Method. Part I. Nonuniform Mesh Grids. Journal of Computational Physics 1996; 129:357-63. Higuera FJ, Jiménez J. Boltzmann Approach to Lattice Gas Simulations. EPL (Europhysics Letters) 1989; 9:663. Ho CJ, Tu FJ. An investigation of transient mixed convection heat transfer of cold 288 water in a tall vertical annulus with a heated rotating inner cylinder. International Journal of Heat and Mass Transfer 1993; 36:2847-59. Holdych DJ, Rovas D, Georgiadis JG, Buckius RO. An Improved Hydrodynamics Formulation for Multiphase Flow Lattice-Boltzmann Models. International Journal of Modern Physics C 1998; 09:1393-404. Huang H, Huang J-J, Lu X-Y, Sukop MC. On Simulations oF High-Density Ratio Flows Using Color-Gradient Multiphase Lattice Boltzmann Models. International Journal of Modern Physics C 2013; 24:1350021. Huang H, Lee TS, Shu C. Hybrid lattice Boltzmann finite-difference simulation of axisymmetric swirling and rotating flows. International Journal for Numerical Methods in Fluids 2007; 53:1707-26. Huang H, Lu X-y. Relative permeabilities and coupling effects in steady-state gas-liquid flow in porous media: A lattice Boltzmann study. Physics of Fluids 2009; 21:092104. Inamuro T, Ogata T, Tajima S, Konishi N. A lattice Boltzmann method for incompressible two-phase flows with large density differences. Journal of Computational Physics 2004; 198:628-44. Issa RI. Solution of the implicitly discretised fluid flow equations by operator-splitting. Journal of Computational Physics 1986; 62:40-65. Jeong J, Hussain F. On the identification of a vortex. Journal of Fluid Mechanics 1995; 285:69-94. Johnson TA, Patel VC. Flow past a sphere up to a Reynolds number of 300. Journal of Fluid Mechanics 1999; 378:19-70. Josserand C, Zaleski S. Droplet splashing on a thin liquid film. Physics of Fluids 2003; 15:1650-7. Karniadakis GE, Israeli M, Orszag SA. High-order splitting methods for the incompressible Navier-Stokes equations. Journal of Computational Physics 1991; 289 97:414-43. Kim D, Choi H. Laminar flow past a sphere rotating in the streamwise direction. Journal of Fluid Mechanics 2002; 461:365-86. Kim H-Y, Feng ZC, Chun J-H. Instability of a liquid jet emerging from a droplet upon collision with a solid surface. Physics of Fluids 2000; 12:531-41. Kim J, Kim D, Choi H. An Immersed-Boundary Finite-Volume Method for Simulations of Flow in Complex Geometries. Journal of Computational Physics 2001; 171:132-50. Kim J, Moin P. Application of a fractional-step method to incompressible Navier-Stokes equations. Journal of Computational Physics 1985; 59:308-23. Kim W-W, Menon S. An unsteady incompressible Navier–Stokes solver for large eddy simulation of turbulent flows. International Journal for Numerical Methods in Fluids 1999; 31:983-1017. Ku HC, Hirsh RS, Taylor TD. A pseudospectral method for solution of the three-dimensional incompressible Navier-Stokes equations. Journal of Computational Physics 1987; 70:439-62. Kuehn TH, Goldstein RJ. An experimental and theoretical study of natural convection in the annulus between horizontal concentric cylinders. Journal of Fluid Mechanics 1976; 74:695-719. Kumar R, Kalam MA. Laminar thermal convection between vertical coaxial isothermal cylinders. International Journal of Heat and Mass Transfer 1991; 34:513-24. Lee T, Lin C-L. A stable discretization of the lattice Boltzmann equation for simulation of incompressible two-phase flows at high density ratio. Journal of Computational Physics 2005; 206:16-47. Lee T, Liu L. Lattice Boltzmann simulations of micron-scale drop impact on dry surfaces. Journal of Computational Physics 2010; 229:8045-63. 290 Lee TS, Huang H, Shu C. An Axisymmetric Incompressible Lattice Boltzmann Model for Pipe Flow. International Journal of Modern Physics C 2006; 17:645-61. Lemembre A, Petit JP. Laminar natural convection in a laterally heated and upper cooled vertical cylindrical enclosure. International Journal of Heat and Mass Transfer 1998; 41:2437-54. Li Q, He YL, Tang GH, Tao WQ. Lattice Boltzmann model for axisymmetric thermal flows. Physical Review E 2009; 80:037702. Li Q, He YL, Tang GH, Tao WQ. Improved axisymmetric lattice Boltzmann scheme. Physical Review E 2010; 81:056707. Liang SF, Vidal A, Acrivos A. Buoyancy-driven convection in cylindrical geometries. Journal of Fluid Mechanics 1969; 36:239-58. Lim CY, Shu C, Niu XD, Chew YT. Application of lattice Boltzmann method to simulate microchannel flows. Physics of Fluids 2002; 14:2299-308. Liu J-G, Liu J, Pego RL. Stable and accurate pressure approximation for unsteady incompressible viscous flow. Journal of Computational Physics 2010; 229:3428-53. Liu X-D, Osher S, Chan T. Weighted Essentially Non-oscillatory Schemes. Journal of Computational Physics 1994; 115:200-12. Liu Y, Numerical simulation of flows in Czochraski crystal growth and Taylor vortices. M. Eng. Thesis; National University of Singapore 1998. Malan AG, Lewis RW, Nithiarasu P. An improved unsteady, unstructured, artificial compressibility, finite volume scheme for viscous incompressible flows: Part I. Theory and implementation. International Journal for Numerical Methods in Engineering 2002; 54:695-714. Martin DF, Colella P, Graves D. A cell-centered adaptive projection method for the incompressible Navier–Stokes equations Computational Physics 2008; 227:1863-86. 291 in three dimensions. Journal of Marx YP. Time Integration Schemes for the Unsteady Incompressible Navier-Stokes Equations. Journal of Computational Physics 1994; 112:182-209. McNamara G, Alder B. Analysis of the lattice Boltzmann treatment of hydrodynamics. Physica A: Statistical Mechanics and its Applications 1993; 194:218-28. McNamara GR, Zanetti G. Use of the Boltzmann Equation to Simulate Lattice-Gas Automata. Physical Review Letters 1988; 61:2332-5. Miller CA, Scriven LE. The oscillations of a fluid droplet immersed in another fluid. Journal of Fluid Mechanics 1968; 32:417-35. Napolitano M, Pascazio G, Quartapelle L. A review of vorticity conditions in the numerical solution of the ζ–ψ equations. Computers & Fluids 1999; 28:139-85. Niazmand H, Renksizbulut M. Surface effects on transient three-dimensional flows around rotating spheres at moderate Reynolds numbers. Computers & Fluids 2003; 32:1405-33. Nieuwstadt F, Keller HB. Viscous flow past circular cylinders. Computers & Fluids 1973; 1:59-71. Niu XD, Chew YT, Shu C. Simulation of flows around an impulsively started circular cylinder by Taylor series expansion- and least squares-based lattice Boltzmann method. Journal of Computational Physics 2003; 188:176-93. Niu XD, Shu C, Chew YT, Peng Y. A momentum exchange-based immersed boundary-lattice Boltzmann method for simulating incompressible viscous flows. Physics Letters A 2006; 354:173-82. Nourgaliev RR, Dinh TN, Theofanous TG. A pseudocompressibility method for the numerical simulation of incompressible multifluid flows. International Journal of Multiphase Flow 2004; 30:901-37. Park J, Kwon K, Choi H. Numerical solutions of flow past a circular cylinder at Reynolds numbers up to 160. KSME International Journal 1998; 12:1200-5. 292 Patankar SV, Spalding DB. A calculation procedure for heat, mass and momentum transfer in three-dimensional parabolic flows. International Journal of Heat and Mass Transfer 1972; 15:1787-806. Patankar SV. Numerical Heat Transfer and Fluid Flow. Hemisphere, Washington D.C. and Mcgraw Hill, New York, 1980. Peng G, Xi H, Duncan C, Chou S-H. Lattice Boltzmann method on irregular meshes. Physical Review E 1998; 58:R4124-R7. Peng Y. Simulations of Incompressible Viscous Thermal Flows by Lattice Boltzmann Method, Ph. D. Thesis; National University of Singapore 2004. Peng Y, Shu C, Chew YT. Simplified thermal lattice Boltzmann model for incompressible thermal flows. Physical Review E 2003; 68:026701. Peng Y, Shu C, Chew YT. A 3D incompressible thermal lattice Boltzmann model and its application to simulate natural convection in a cubic cavity. Journal of Computational Physics 2004; 193:260-74. Peng Y, Shu C, Chew YT, Qiu J. Numerical investigation of flows in Czochralski crystal growth by an axisymmetric lattice Boltzmann method. Journal of Computational Physics 2003; 186:295-307. Peskin CS. Numerical analysis of blood flow in the heart. Journal of Computational Physics 1977; 25:220-52. Premnath KN, Abraham J. Simulations of binary drop collisions with a multiple-relaxation-time lattice-Boltzmann model. Physics of Fluids 2005; 17:122105. Qian YH, Humières DD, Lallemand P. Lattice BGK Models for Navier-Stokes Equation. EPL (Europhysics Letters) 1992; 17:479. Ramshaw JD, Mousseau VA. Accelerated artificial compressibility method for steady-state incompressible flow calculations. Computers & Fluids 1990; 18:361-7. 293 Reis T, Phillips TN. Modified lattice Boltzmann model for axisymmetric flows. Physical Review E 2007; 75:056703. Rogers SE, Kwak D. Upwind differencing scheme for the time-accurate incompressible Navier-Stokes equations. AIAA Journal 1990; 28:253-62. San O, Staples AE. A coarse-grid projection method for accelerating incompressible flow computations. Journal of Computational Physics 2013; 233:480-508. Shan X. Simulation of Rayleigh-Bénard convection using a lattice Boltzmann method. Physical Review E 1997; 55:2780-8. Shan X, Chen H. Lattice Boltzmann model for simulating flows with multiple phases and components. Physical Review E 1993; 47:1815-9. Shan X, Chen H. Simulation of nonideal gases and liquid-gas phase transitions by the lattice Boltzmann equation. Physical Review E 1994; 49:2941-8. Shapiro E, Drikakis D. Artificial compressibility, characteristics-based schemes for variable density, incompressible, multi-species flows. Part I. Derivation of different formulations and constant density limit. Journal of Computational Physics 2005; 210:584-607. Shapiro E, Drikakis D. Artificial compressibility, characteristics-based schemes for variable-density, incompressible, multispecies flows: Part II. Multigrid implementation and numerical tests. Journal of Computational Physics 2005; 210:608-31. Sharp DH. An overview of Rayleigh-Taylor instability. Physica D: Nonlinear Phenomena 1984; 12:3-18. Sheard GJ, Hourigan K, Thompson MC. Computations of the drag coefficients for low-Reynolds-number flow past rings. Journal of Fluid Mechanics 2005; 526:257-75. Sheard GJ, Thompson MC, Hourigan K. From spheres to circular cylinders: non-axisymmetric transitions in the flow past rings. Journal of Fluid Mechanics 2004; 506:45-78. 294 Shenoy AR, Kleinstreuer C. Influence of aspect ratio on the dynamics of a freely moving circular disk. Journal of Fluid Mechanics 2010; 653:463-87. Shi J-M, Gerlach D, Breuer M, Biswas G, Durst F. Heating effect on steady and unsteady horizontal laminar flow of air past a circular cylinder. Physics of Fluids 2004; 16:4331-45. Shu C. Application of differential quadrature method to simulate natural convection in a concentric annulus. International Journal for Numerical Methods in Fluids 1999; 30:977-93. Shu C, Chew YT, Liu Y. An efficient approach for numerical simulation of flows in Czochralski crystal growth. Journal of Crystal Growth 1997; 181:427-36. Shu C, Liu N, Chew YT. A novel immersed boundary velocity correction–lattice Boltzmann method and its application to simulate flow past a circular cylinder. Journal of Computational Physics 2007; 226:1607-22. Shu C, Niu XD, Chew YT. Taylor-series expansion and least-squares-based lattice Boltzmann method: Two-dimensional formulation and its applications. Physical Review E 2002; 65:036708. Shu C, Peng Y, Chew YT. Simulation of Natural Convection in A Square Cavity by Taylor Series Expansion- and Least Squares-based Lattice Boltzmann Method. International Journal of Modern Physics C 2002; 13:1399-414. Shu C, Richards BE. Application of generalized differential quadrature to solve two-dimensional incompressible Navier-Stokes equations. International Journal for Numerical Methods in Fluids 1992; 15:791-8. Shu C, Xue H. Comparison of two approaches for implementing stream function boundary conditions in DQ simulation of natural convection in a square cavity. International Journal of Heat and Fluid Flow 1998; 19:59-68. Shu C-W, Osher S. Efficient implementation of essentially non-oscillatory shock-capturing schemes. Journal of Computational Physics 1988; 77:439-71. 295 Shukla RK, Tatineni M, Zhong X. Very high-order compact finite difference schemes on non-uniform grids for incompressible Navier–Stokes equations. Journal of Computational Physics 2007; 224:1064-94. Singh SP, Mittal S. Vortex-induced oscillations at low Reynolds numbers: Hysteresis and vortex-shedding modes. Journal of Fluids and Structures 2005; 20:1085-104. Sotiropoulos F, Ventikos Y. Transition from bubble-type vortex breakdown to columnar vortex in a confined swirling flow. International Journal of Heat and Fluid Flow 1998; 19:446-58. Sousa E, Sobey IJ. Effect of boundary vorticity discretization on explicit stream-function vorticity calculations. International Journal for Numerical Methods in Fluids 2005; 49:371-93. Sui Y, Chew Y-T, Roy P, Low H-T. A hybrid immersed-boundary and multi-block lattice Boltzmann method for simulating fluid and moving-boundaries interactions. International Journal for Numerical Methods in Fluids 2007; 53:1727-54. Sussman M, Almgren AS, Bell JB, Colella P, Howell LH, Welcome ML. An Adaptive Level Set Approach for Incompressible Two-Phase Flows. Journal of Computational Physics 1999; 148:81-124. Suzuki K, Inamuro T. Effect of internal mass in the simulation of a moving body by the immersed boundary method. Computers & Fluids 2011; 49:173-87. Swift MR, Orlandini E, Osborn WR, Yeomans JM. Lattice Boltzmann simulations of liquid-gas and binary fluid systems. Physical Review E 1996; 54:5041-52. Tamamidis P, Zhang G, Assanis DN. Comparison of Pressure-Based and Artificial Compressibility Methods for Solving 3D Steady Incompressible Viscous Flows. Journal of Computational Physics 1996; 124:1-13. Tomboulides AG, Orszag SA, Karniadakis GE. Direct and Large Eddy Simulation of Axisymmetric Wakes. AIAA Paper 1993; 93-0546. Tryggvason G. Numerical simulations of the Rayleigh-Taylor instability. Journal of 296 Computational Physics 1988; 75:253-82. Turkel E. Preconditioned methods for solving the incompressible and low speed compressible equations. Journal of Computational Physics 1987; 72:277-98. Van Doormaal JP, Raithby GD. Enhancements of the Simple Method for Predicting Incompressible Fluid Flows. Numerical Heat Transfer 1984; 7:147-63. Van Kan J. A Second-Order Accurate Pressure-Correction Scheme for Viscous Incompressible Flow. SIAM Journal on Scientific and Statistical Computing 1986; 7:870-91. Venkatachalappa M, Sankar M, Natarajan AA. Natural convection in an annulus between two rotating vertical cylinders. Acta Mechanica 2001; 147:173-96. Wan D, Turek S. Direct numerical simulation of particulate flow via multigrid FEM techniques and the fictitious boundary method. International Journal for Numerical Methods in Fluids 2006; 51:531-66. Wang J, Wang M, Li Z. Lattice Poisson–Boltzmann simulations of electro-osmotic flows in microchannels. Journal of Colloid and Interface Science 2006; 296:729-36. Wang XY, Yeo KS, Chew CS, Khoo BC. A SVD-GFD scheme for computing 3D incompressible viscous fluid flows. Computers & Fluids 2008; 37:733-46. Wang Y, Yu B, Cao Z, Zou W, Yu G. A comparative study of POD interpolation and POD projection methods for fast and accurate prediction of heat transfer problems. International Journal of Heat and Mass Transfer 2012; 55:4827-36. Weinan E, Liu J-G. Vorticity Boundary Condition and Related Issues for Finite Difference Schemes. Journal of Computational Physics 1996; 124:368-82. Wheeler AA. Test Problems for the Numerical-Simulation of Flow in Czochralsko Crystal-Growth. Journal of Crystal Growth 1990; 102:691-5. Williamson CHK. Oblique and parallel modes of vortex shedding in the wake of a circular cylinder at low Reynolds numbers. Journal of Fluid Mechanics 1989; 297 206:579-627. Wu J, Shu C. Implicit velocity correction-based immersed boundary-lattice Boltzmann method and its applications. Journal of Computational Physics 2009; 228:1963-79. Wu J, Shu C. Particulate flow simulation via a boundary condition-enforced immersed boundary-lattice Boltzmann scheme. Communications in Computational Physics 2010; 7: 793-812. Wu J, Shu C. An improved immersed boundary-lattice Boltzmann method for simulating three-dimensional incompressible flows. Journal of Computational Physics 2010; 229:5022-42. Wu J, Shu C. Simulation of three-dimensional flows over moving objects by an improved immersed boundary–lattice Boltzmann method. International Journal for Numerical Methods in Fluids 2012; 68:977-1004. Xu D, Shu C, Khoo BC. Numerical simulation of flows in Czochralski crystal growth by second-order upwind QUICK scheme. Journal of Crystal Growth 1997; 173:123-31. Xu K. A Gas-Kinetic BGK Scheme for the Navier–Stokes Equations and Its Connection with Artificial Dissipation and Godunov Method. Journal of Computational Physics 2001; 171:289-335. Yan YY, Zu YQ. A lattice Boltzmann method for incompressible two-phase flows on partial wetting surface with large density ratio. Journal of Computational Physics 2007; 227:763-75. Yu H, Girimaji SS, Luo L-S. DNS and LES of decaying isotropic turbulence with and without frame rotation using lattice Boltzmann method. Journal of Computational Physics 2005; 209:599-616. Zheng HW, Shu C, Chew YT. A lattice Boltzmann model for multiphase flows with large density ratio. Journal of Computational Physics 2006; 218:353-71. 298 Zheng L, Shi B, Guo Z, Zheng C. Lattice Boltzmann equation for axisymmetric thermal flows. Computers & Fluids 2010; 39:945-52. Zhong H, Lee C, Su Z, Chen S, Zhou M, Wu J. Experimental investigation of freely falling thin disks. Part 1. The flow structures and Reynolds number effects on the zigzag motion. Journal of Fluid Mechanics 2013; 716:228-50. Zhou JG. Axisymmetric lattice Boltzmann method. Physical Review E 2008; 78:036701. 299 VITA NAME: WANG YAN DATA OF BIRTH: 25, MARCH, 1987 I was born in Jiangsu province of China in 1987. I obtained my B. Eng. Degree in 2009 from Nanjing University of Aeronautics and Astronautics. I studied as a Ph. D. student from 2010 to 2014 in the department of Mechanical Engineering in the National University of Singapore. The following publications are related to my Ph. D. work: 1. C. Shu, Y. Wang, C.J. Teo and J. Wu, Development of lattice Boltzmann flux solver for simulation of incompressible flows, Adv. Appl. Math. Mech., 6: 436-460 (2014). 2. Y. Wang, C. Shu, C.J. Teo, Development of LBGK and incompressible LBGK-based lattice Boltzmann flux solvers for simulation of incompressible flows, Int. J. Numer. Meth. Fluids, 75: 344-364 (2014). 3. Y. Wang, C. Shu and C.J. Teo, Thermal lattice Boltzmann flux solver and its applications for simulation of incompressible thermal flows, Comput. Fluids, 94, 98-111 (2014). 4. Y. Wang, C. Shu and C.J. Teo, A fractional step axisymmetric lattice Boltzmann flux solver for incompressible swirling and rotating flows, Comput. Fluids, 96, 204-214 (2014). 5. C. Shu, Y. Wang, L.M. Yang and J. Wu, Lattice Boltzmann Flux Solver, an efficient approach for numerical simulation of fluid flows, Transactions of Nanjing University of Aeronautics and Astronautics, 31, 1-15 (2014). [...]... proposed solvers have been validated by simulating a variety of 2D and 3D flows Numerical simulations have verified that the LBFSs not only successfully eliminate the drawbacks of LBE solvers, such as mesh uniformity, tie-up between time step and mesh spacing, limited to viscous flows and complicated implementation of boundary conditions, but also combine the advantages of the N-S solvers and LBE solvers. .. both of the N-S solver and the mesoscopic LBE solver have their unique advantages as well as disadvantages Many improved solvers have been proposed in each individual group which eliminate the drawbacks of each solver However, due to their independent developments within one theoretical framework, the improvements seem to be constrained and the intrinsic drawbacks of the N-S solvers and the LBE solvers. .. circular cylinder 65 Fig 2.16 u and v velocity profiles along the vertical centerline of cubic 66 cavity for 3D lid-driven cavity flow at Reynolds numbers of 100, 400 and 1000 Fig 2.17 Streamlines and pressure contours on the mid-plane of x=0.5 67 for 3D lid-driven cavity flow at Reynolds numbers of 100, 400 and 1000 Fig 2.18 Streamlines and pressure contours on the mid-plane of 68 y=0.5 for 3D lid-driven... devoted to simulating and analyzing physical behaviors and mechanics of fluid flows On the one hand, with the continuous emergence of more and more powerful yet inexpensive computers, CFD is now able to simulate more sophisticated flows in academic research and industrial applications, ranging from microfluidics in micro-electromechanical systems (MEMS), aerodynamics in aviation and automobile industry... Finite element FVM Finite volume method IBM Immersed boundary method IB-LBM Immersed boundary- lattice Boltzmann method LBE Lattice Boltzmann equation LBFS Lattice Boltzmann flux solver LBM Lattice Boltzmann Method MEMS Micro-electromechanical systems N-S Navier-Stokes TLLBM Taylor-series expansion based- and least square based LBM PA Projection method PDEs Partial differential equations VIV Vortex-Induced... completely removed In view of this, it is natural to ask whether we can develop a solver to combine their advantages, and in the meantime, to remove their drawbacks This motivates the present work The primary purpose of this thesis is to develop a series of new solvers for isothermal, thermal, axisymmetric and multiphase flows and more complex flows with moving boundaries and freely falling objects... Reynolds numbers of 100, 400 and 1000 Fig 2.19 Streamlines and pressure contours on the mid-plane 69 of z=0.5 for 3D lid-driven cavity flow at Reynolds numbers of 100, 400 and 1000 Fig 3.1 Local construction of 2D LBM solution at an interface 102 between two control Fig 3.2 Local construction of 3D LBM solution at an interface 102 between two control cells Fig 3.3 The computational domain and corresponding... meteorology On the other hand, due to the complexity of fluid flows in different scales and regimes and the limited computational resources, new challenges in accuracy and efficiency for the available numerical methods are also continuously imposed In this regard, developing simpler, more accurate and efficient numerical approaches has been one of the primary and fundamental tasks of the CFD community In... cells 178 Fig 5.2 3D Flux evaluation at an interface between two control cells 178 Fig 5.3 Configuration and computational grid of the two-phase 178 co-current flows Fig 5.4 Velocity profiles of the two-phase co-current flows with 179  H /  L  10, 20, 100 and 1000: forces on Fluid 1 Fig 5.5 Velocity profiles of the two-phase co-current flows with 179  H /  L  10, 20, 100 and 1000: forces on Fluid... equations recovered by the LBE models The fluxes of the LBFSs are modeled at each interface by local reconstruction of the standard LBE solutions, where the theoretical connections between the macroscopic fluxes and the microscopic density and/ or internal energy distribution functions are utilized Additional source terms, including external forces and those of axisymmetric effects, are conveniently . DEVELOPMENT OF LATTICE BOLTZMANN FLUX SOLVERS AND THEIR APPLICATIONS WANG YAN A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF MECHANICAL. 3 Development of Thermal Lattice Boltzmann Flux Solver for Simulation of Thermal Incompressible Flows 70 3.1 Simplified thermal lattice Boltzmann model 71 3.2 Thermal Lattice Boltzmann Flux. Advantages and disadvantages of the LBE solver 20 1.4 Motivations and objectives of the thesis 21 1.5 Organization of the thesis 24 iv Chapter 2 Development of Lattice Boltzmann Flux Solver