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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

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

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Many paths to the summit

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4 1. INTRODUCTION

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

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6 An indirect method is not always your best choice.

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7 Extraction of essence Subtraction rather than addition

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

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

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Kinetic Flux Splitting Characteristics :

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

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

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13 Pressure distribution along the wall Present

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

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15 Case of incompressible flows LBM

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

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

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

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

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

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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)

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

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

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26 Basic form of ACM

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4 th order accurate div u for compact stencil

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28 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)

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29 Periodic Boundary 2D Generalized Taylor-Green

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30 Time history of L1 error

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31 Convergence Rate (t=100)

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32 3D Generalized Taylor-Green (3D-GTG)

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33 R 5R Circular Couette Flow

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34 Error of Velocity U θ (ACM)

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35 Error of Velocity U θ (MRT)

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36 Error of Pressure P (ACM)

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37 Error of Pressure P (MRT)

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38 Comparison UθUθ P ACM MRT ACM MRT

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39 Convergence rate

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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

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|div u| (Re=100) t=100 stream 41

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|div u| (Re=100) 42 t=100

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Re=100 (unsteady) 43

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

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Adaptive Mesh Refinement 45 (Re=100) D Poiseuille Flow 0th-order extrapolation Simple Interpolation D: the diameter of the cylinder

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Velocity u 46 t=100

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Velocity v 47 t=100

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Pressure 48 t=100

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49 Top boundary 2D Lid-driven Cavity Flow

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50 2D Lid-driven Cavity Flow CPU Time (129×129, step) (intel Corei7, openMP) ACMMRT sec sec × papers 4053 papers

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51 3D Lid-driven Cavity Flow Lo D. C., Murugesan K., Young D. L., Int. J. Numer. Meth. Fluids (2005) Grid : 100³

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52 Flow past a sphere Re=300

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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

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54 Intermission

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55 Re=2000

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56 Re=500

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57 3. Theory of ACM

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Convergence of the solution of ACE to the solution of INSE in the limit of. Témam (1969) How fast ?

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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) !!

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60 Diffusive mode of ACE solution Assumption

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INSE Oseen type The intrinsic error of ACM

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62 Acoustic mode of ACE solution Characteristic time ~ Characteristic length / Speed of wave

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Initial data for ACE = Initial data for INSE acts as an initial impact !!!

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64 Magnitude of acoustic mode

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66 4. Improvement of ACM

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Suppression of acoustic mode 1. Bulk viscosity

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68 2. Dashpot Dashpot

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69 1. Bulk viscosity 2. Dashpot

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70 Time history of L1 error

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4.2 Suppression of Checkerboard Instability 71 Acoustic mode killer Checkerboard killer stability 1 3 2

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72 Collocated grid Staggered grid

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73 P-contour 2D-LID Re=5000

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74 P-contour *C.-H. Bruneau & M. Saad, Comput. & Fluids 35, (2006) ACM (s=0,2) (256×256) C.-H. Bruneau* (2048×2048)

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75 Stream line Pressure

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76 Stream line Pressure

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4.3 Enhancer of stability 77 Acoustic mode killer Checkerboard killer stability 1 3 2

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S=0S=2 78 Velocity modes Pressure mode Linear Stability Analysis The normalized wave number Checkerboard killer

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79 S=0S=2

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Stability in Lid-Driven 80 ACM (s=0) ACM (s=2) MRT (χ=0.0378) LBGK (χ=0.0002) 32×32 × ○ ×× 64×64 × ○ ×× 96×96 ○○○ × 128×128 ○○○ × 256×256 ○○○○ Grid

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Richardson Extrapolation in the Mach Numbers 81 Diffusive mode Linear Combination

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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)

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4 th order accurate div u for compact stencil

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2 nd order accuracy4 th order accuracy

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85 3D Generalized Taylor-Green (3D-GTG)

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86 Curved Solid Boundary LBM ACM: Macroscopic data Interpolation or extrapolation e.g. Interpolation Bounce-Back Bouzidi, Firdaouss, Lallemand (2001) Ginzburg, d’Humières (2003)

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87 x N W S E P y B Solid Body Fluid W E P B Quadratic extrapolation x at B

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88 x N W S E P y B Solid Body Fluid W E P B Quadratic extrapolation x at B at P

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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.

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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).

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91 谢谢您为您的关注

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有问题吗？

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93 Finite Volume ACM

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The initial data for must be compatible with INSE. The initial data for troublesome Excitation of acoustic mode

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