Presentation on theme: "Artificial Compressibility Method and Lattice Boltzmann Method Similarities and Differences Taku Ohwada ( 大和田 拓） Department of Aeronautics & Astronautics,"— Presentation transcript:
Artificial Compressibility Method and Lattice Boltzmann Method Similarities and Differences Taku Ohwada ( 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 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
2 0. What is a good numerical method ? Performance Cost (CPU) Education (Human CPU)
Many paths to the summit
4 1. INTRODUCTION
5 Kinetic methods for fluid-dynamic equations Gas Kinetic Scheme Lattice Boltzmann Method Boltzmann Eq. Euler, Navier-Stokes The path is INDIRECT ! Why Kinetic ?
6 An indirect method is not always your best choice.
7 Extraction of essence Subtraction rather than addition
Case of compressible flows Why Kinetic ? 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
Kinetic Flux Splitting Characteristics :
11 Experiment using undergraduate students. Prerequisite of Gas Kinetic Scheme is Taylor expansion !!!!
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 !!!!
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)
40 M. Schäfer, S. Turek, (1996) The Flow Past a Cylinder D Non-Slip Boundary Poiseuille Flow 0th-order extrapolation Non-Slip Boundary 2.1D 2D
|div u| (Re=100) t=100 stream 41
|div u| (Re=100) 42 t=100
Re=100 (unsteady) 43
44 * M. Schäfer, S. Turek, (1996) LBM: Mussa, Asinari, Luo, JCP 228 (2009)
Adaptive Mesh Refinement 45 (Re=100) D Poiseuille Flow 0th-order extrapolation Simple Interpolation D: the diameter of the cylinder
Velocity u 46 t=100
Velocity v 47 t=100
Pressure 48 t=100
49 Top boundary 2D Lid-driven Cavity Flow
50 2D Lid-driven Cavity Flow CPU Time (129×129, step) (intel Corei7, openMP) ACMMRT sec sec × papers 4053 papers
51 3D Lid-driven Cavity Flow Lo D. C., Murugesan K., Young D. L., Int. J. Numer. Meth. Fluids (2005) Grid : 100³
52 Flow past a sphere Re=300
53 Comparisons of LBM and ACM Taylor-Green Vortex Flow Flow past a Cylinder Lid-driven Cavity Flow Circular Couette Flow ACM is capable of practical 3D simulation Performance of ACM in 3D problems Conclusion LBM – ACM is NOT decisively positive !!! Flow past a Cylinder Lid-driven Cavity Flow
57 3. Theory of ACM
Convergence of the solution of ACE to the solution of INSE in the limit of. Témam (1969) How fast ?
59 A mathematical analysis shows (Moise and Ziane, Journal of Dynamics and Differential Equations, Vol. 13, No. 2, 2001) Numerical observations show that the error is O(k) !!
60 Diffusive mode of ACE solution Assumption
INSE Oseen type The intrinsic error of ACM
62 Acoustic mode of ACE solution Characteristic time ~ Characteristic length / Speed of wave
Initial data for ACE = Initial data for INSE acts as an initial impact !!!
Richardson Extrapolation in the Mach Numbers 81 Diffusive mode Linear Combination
High order accurate ACM 82 Spatial discretization 4 th order accuracy Time-marching 2 nd order accuracy (Semi-implicit Runge-Kutta) (5point central finite difference)
4 th order accurate div u for compact stencil
2 nd order accuracy4 th order accuracy
85 3D Generalized Taylor-Green (3D-GTG)
86 Curved Solid Boundary LBM ACM: Macroscopic data Interpolation or extrapolation e.g. Interpolation Bounce-Back Bouzidi, Firdaouss, Lallemand (2001) Ginzburg, d’Humières (2003)
87 x N W S E P y B Solid Body Fluid W E P B Quadratic extrapolation x at B
88 x N W S E P y B Solid Body Fluid W E P B Quadratic extrapolation x at B at P
Why not ACM? 89 Since the ACM does not employ any kinetic theory gadget, it is much easier than the LBM. Up to now, any decisive inferiority of ACM to LBM has not been found. Conversely, superiority of ACM over LBM has been found in some fundamental test problems. Therefore, it is highly recommended to master ACM before learning LBM.
Richardson extrapolation = 累遍増約術 Its usefulness for practical computations can hardly be overestimated. Birkoff & Rota, Ordinary differential equations (1978). Lewis Fry Richardson, ``Approximate arithmetical solution by finite differences of physical problems including differential equations, with an application to the stresses in a masonry dam ‘’, Phil. Trans. Royal Soc. London, Series A 210: 307–357 (1910).
93 Finite Volume ACM
The initial data for must be compatible with INSE. The initial data for troublesome Excitation of acoustic mode