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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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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
0. What is a good numerical method ? • Performance • Cost (CPU) • Education (Human CPU)
Kinetic methods for fluid-dynamic equations Gas Kinetic Scheme Lattice Boltzmann Method Why Kinetic ? Boltzmann Eq. Euler, Navier-Stokes The path is INDIRECT !
Extraction of essence Subtraction rather than addition
Why Kinetic ? Case of compressible flows Kinetic gadget yields the flux for the Euler equations.
The discontinuities at cell-interfaces produce numerical dissipation, which suppresses spurious oscillations around shock waves……….. Shock Capturing ! Riemann Problem Riemann Problem Riemann Problem
Kinetic Flux Splitting Characteristics :
Prerequisite of Gas Kinetic Scheme is Taylor expansion !!!! Experiment using undergraduate students.
Gallery of Undergraduate Students ‘ Works Euler Navier-Stokes: Any asymptotic method is NOT employed. Blasius flow
Case of incompressible flows Gas kinetic Scheme Lax-Wendroff No kinetic ingredient !!!!
LBM -> Lattice Kinetic Scheme (LKS) No kinetic ingredient !! -> ACM LKS -> Lattice Scheme (LK)
Incompressible case: More difficult than compressible case !!!
LBM ! Poisson Free… • Poisson Free !!! • 2nd order accurate BB • Small time step • Parallel computation !!! Bouzidi, Firdaouss, Lallemand (2001) Ginzburg, d’Humières (2003)
LBM solves INSE via Artificial Compressibility Equations Prof. Asinari ’s morning lecture Chapman-Enskog expansion Hilbert expansion (diffusive scale) LBM
Artificial Compressibility Method (ACM) (Chorin,1967) (Témam, 1969) usually LBM
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)
Devil’s Project LBM-ACM ACM ACM LBM LBM Chapman-Enskog Expansion • Lattice Structure Finite Difference (Finite Volume) • Collocated Grid Kinetic
2. Numerical Computation of ACM Cartesian Grid D2Q9 Time Step Finite Difference Space : Central Difference Time : Semi-Implicit
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)
2D Generalized Taylor-Green Periodic Boundary
Comparison Uθ MRT ACM P MRT ACM
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
|div u|(Re=100) t=100 stream stream stream stream
|divu|(Re=100) t=100
* M. Schäfer, S. Turek, (1996) LBM: Mussa, Asinari, Luo, JCP 228 (2009)
Adaptive Mesh Refinement (Re=100) D: the diameter of the cylinder Poiseuille Flow D 0th-order extrapolation Simple Interpolation
Velocity u t=100
Velocity v t=100
Pressure t=100
2D Lid-driven Cavity Flow Top boundary
2D Lid-driven Cavity Flow CPU Time (129×129, 100000step) (intel Corei7, openMP) ×7.37 356 papers 4053 papers