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. 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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