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Artificial Compressibility Method and Lattice Boltzmann Method Similarities and Differences

知彼知己者 百戰不殆 If you know your enemies and know yourself, you can win a hundred battles without a single loss. Artificial Compressibility Method and Lattice Boltzmann Method Similarities and Differences. Taku Ohwada ( 大和田　拓 ） Department of Aeronautics & Astronautics, Kyoto University

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Artificial Compressibility Method and Lattice Boltzmann Method Similarities and Differences

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  1. 知彼知己者 百戰不殆 If you know your enemies and know yourself, you can win a hundred battles without a single loss Artificial Compressibility Method and Lattice Boltzmann MethodSimilarities and Differences TakuOhwada (大和田　拓） Department of Aeronautics & Astronautics, Kyoto University （京都大学大学院工学研究科航空宇宙工学専攻） Collaborators: Prof. Pietro Asinari, Mr. Daisuke Yabusaki May 4, 2011, Spring School on the lattice Boltzmann Method Beijing Computational Science Research Center May 2-6

  2. 0. What is a good numerical method ? • Performance • Cost (CPU) • Education (Human CPU)

  3. Many paths to the summit

  4. 1. INTRODUCTION

  5. Kinetic methods for fluid-dynamic equations Gas Kinetic Scheme Lattice Boltzmann Method Why Kinetic ? Boltzmann Eq.  Euler, Navier-Stokes The path is INDIRECT !

  6. An indirect method is not always your best choice.

  7. Extraction of essence Subtraction rather than addition

  8. Why Kinetic ? Case of compressible flows Kinetic gadget yields the flux for the Euler equations.

  9. The discontinuities at cell-interfaces produce numerical dissipation, which suppresses spurious oscillations around shock waves……….. Shock Capturing ! Riemann Problem Riemann Problem Riemann Problem

  10. Kinetic Flux Splitting Characteristics :

  11. Prerequisite of Gas Kinetic Scheme is Taylor expansion !!!! Experiment using undergraduate students.

  12. Gallery of Undergraduate Students ‘ Works Euler Navier-Stokes: Any asymptotic method is NOT employed. Blasius flow

  13. Pressure distribution along the wall Present

  14. Case of incompressible flows Gas kinetic Scheme Lax-Wendroff No kinetic ingredient !!!!

  15. Case of incompressible flows LBM

  16. LBM -> Lattice Kinetic Scheme (LKS) No kinetic ingredient !! -> ACM LKS -> Lattice Scheme (LK)

  17. Incompressible case: More difficult than compressible case !!!

  18. LBM ! Poisson Free… • Poisson Free !!! • 2nd order accurate BB • Small time step • Parallel computation !!! Bouzidi, Firdaouss, Lallemand (2001) Ginzburg, d’Humières (2003)

  19. LBM solves INSE via Artificial Compressibility Equations Prof. Asinari ’s morning lecture Chapman-Enskog expansion Hilbert expansion (diffusive scale) LBM

  20. Artificial Compressibility Method (ACM) (Chorin,1967) (Témam, 1969) usually LBM

  21. Considering the fact that the lattice Boltzmann method starts with the kinetic theory and has been derived to conserve high-order isotropy, the lattice Boltzmann method should be more accurate than the artificial compressibility method in capturing pressure waves. He, Doolen, Clark (JCP2002) ACM: Macroscopic (356 papers) LBM: Kinetic (4053 papers)

  22. Devil’s Project LBM-ACM ACM ACM LBM LBM Chapman-Enskog Expansion • Lattice Structure Finite Difference (Finite Volume) • Collocated Grid Kinetic

  23. 2. Numerical Computation of ACM Cartesian Grid D2Q9 Time Step Finite Difference Space : Central Difference Time : Semi-Implicit

  24. Basic form of ACM

  25. 4th order accurate div u for compact stencil

  26. Test Problems • Generalized Taylor-Green Vortex (2D, 3D) • Circular Couette Flow (2D) • Flow Past a Cylinder in a Channel (2D) • Lid-driven Cavity Flow (2D, 3D) • Flow Past a Sphere in Uniform Flow (3D)

  27. 2D Generalized Taylor-Green Periodic Boundary

  28. Time history of L1 error

  29. Convergence Rate (t=100)

  30. 3D Generalized Taylor-Green (3D-GTG)

  31. Circular Couette Flow 5R R

  32. Error of Velocity Uθ (ACM)

  33. Error of Velocity Uθ (MRT)

  34. Error of Pressure P (ACM)

  35. Error of Pressure P (MRT)

  36. Comparison Uθ MRT ACM P MRT ACM

  37. Convergence rate

  38. The Flow Past a Cylinder M. Schäfer, S. Turek, (1996) Non-Slip Boundary Poiseuille Flow 2.1D D 0th-order extrapolation 2D Non-Slip Boundary

  39. |div u|(Re=100) t=100 stream stream stream stream

  40. |divu|(Re=100) t=100

  41. Re=100(unsteady)

  42. * M. Schäfer, S. Turek, (1996) LBM: Mussa, Asinari, Luo, JCP 228 (2009)

  43. Adaptive Mesh Refinement (Re=100) D: the diameter of the cylinder Poiseuille Flow D 0th-order extrapolation Simple Interpolation

  44. Velocity u t=100

  45. Velocity v t=100

  46. Pressure t=100

  47. 2D Lid-driven Cavity Flow Top boundary

  48. 2D Lid-driven Cavity Flow CPU Time (129×129, 100000step) (intel Corei7, openMP) ×7.37 356 papers 4053 papers

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