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Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

Taylor Series Expansion- and Least Square- Based Lattice Boltzmann Method C. Shu Department of Mechanical Engineering Faculty of Engineering National University of Singapore. Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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Standard Lattice Boltzmann Method (LBM) Current LBM Methods for Complex Problems

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  1. Taylor Series Expansion- and Least Square- Based Lattice Boltzmann MethodC. ShuDepartment of Mechanical EngineeringFaculty of EngineeringNational University of Singapore

  2. Standard Lattice Boltzmann Method (LBM) • Current LBM Methods for Complex Problems • Taylor Series Expansion- and Least Square-Based LBM (TLLBM) • Some Numerical Examples • Conclusions

  3. 1. Standard Lattice Boltzmann Method (LBM) Streaming process Collision process • Particle-based Method (streaming & collision) D2Q9 Model

  4. Features of Standard LBM • Particle-based method • Only one dependent variable Density distribution function f(x,y,t) • Explicit updating; Algebraic operation; Easy implementation No solution of differential equations and resultant algebraic equations is involved • Natural for parallel computing

  5. Limitation---- Difficult for complex geometry and non-uniform mesh

  6. 2. Current LBM Methods for Complex Problems • Interpolation-Supplemented LBM (ISLBM) He et al. (1996), JCP Features of ISLBM • Large computational effort • May not satisfy conservation Laws at mesh points • Upwind interpolation is needed for stability

  7. Differential LBM Taylor series expansion to 1st order derivatives Features: • Wave-like equation • Solved by FD, FE and FV methods • Artificial viscosity is too large at high Re • Lose primary advantage of standard LBM (solve PDE and resultant algebraic equations)

  8. 3. Development of TLLBM • Taylorseries expansion P-----Green (objective point) Drawback: Evaluation A----Red(neighboring point) of Derivatives

  9. Idea of Runge-Kutta Method (RKM) Taylor series method: Runge-Kutta method: n+1 n Need to evaluate high order derivatives n n+1 Apply Taylor series expansion at Points to form an equation system

  10. Taylor series expansion is applied at 6 neighbouring points to form an algebraic equation system A matrix formulation obtained: [S] is a 6x6 matrix and only depends on the geometric coordinates (calculated in advance in programming) (*)

  11. Least Square Optimization Equation system (*) may be ill-conditioned or singular (e.g. Points coincide) Square sum of errors M is the number of neighbouring points used Minimize error:

  12. Least Square Method (continue) The final matrix form: [A] is a 6(M+1) matrix The final explicit algebraic form: are the elements of the first row of the matrix [A] (pre-computed in program)

  13. Features of TLLBM • Keep all advantages of standard LBM • Mesh-free • Applicable to any complex geometry • Easy application to different lattice models

  14. Flow Chart of Computation Input Calculating Geometric Parameter and physical parameters ( N=0 ) N=N+1 No Convergence ? Calculating YES OUTPUT

  15. Boundary Treatment Non-slip condition is exactly satisfied

  16. 4. Some Numerical Examples Square Driven Cavity (Re=10,000, Non- uniform mesh 145x145) Fig.1 velocity profiles along vertical and horizontal central lines

  17. Square Driven Cavity (Re=10,000, Non-uniform mesh 145x145) Fig.2 Streamlines (right) and Vorticity contour (left)

  18. Lid-Driven Polar Cavity Flow Fig. 3 Sketch of polar cavity and mesh

  19. Lid-Driven Polar Cavity Flow 1 — Present 4949 – – – Present 6565 — – — Present 8181 ur uθ 0.75 uθ 0.5 ■ Num. (Fuchs ▲ exp. Tillmark) 0.25 0 ur -0.25 -0.5 r-r0 0.2 0.4 0.6 0.8 1 0 Fig.4 Radial and azimuthal velocity profile along the line of q=00 with Re=350

  20. Lid-Driven Polar Cavity Flow Fig. 5 Streamlines in Polar Cavity for Re=350

  21. Flow around A Circular Cylinder Fig.6 mesh distribution

  22. 5 L 4 Re=40 3 2 Re=20 1 0 0 4 8 t 12 16 20 24 Flow around A Circular Cylinder Fig.7 Flow at Re = 20(Time evolution of the wake length ) Symbols-Experimental (Coutanceau et al. 1982) Lines-Present

  23. Flow around A Circular Cylinder Fig.8 Flow at Re = 20 (streamlines)

  24. T=5 Re=3000 Flow around A Circular Cylinder Fig. 9 Flow at Early stage at Re = 3000(streamline)

  25. T=5 Re=3000 Flow around A Circular Cylinder Fig.10 Flow at Early stage at Re = 3000(Vorticity)

  26. Flow around A Circular Cylinder Fig.11 Flow at Early stage at Re = 3000 (Radial Velocity Distribution along Cut Line) Symbols-Experimental (Bouard &Coutanceau) Lines-Present T=1 T=2 T=3 Re=3000

  27. Flow around A Circular Cylinder Fig. 12 Vortex Shedding (Re=100) t = 3T/8 t = 7T/8

  28. Natural Convection in An Annulus Fig. 13 Mseh in Annulus

  29. Natural Convection in An Annulus Fig. 14 Temperature Pattern

  30. 5. Conclusions • Features of TLLBM • Explicit form • Mesh free • Second Order of accuracy • Removal of the limitation of the standard LBM • Great potential in practical application • Require large memory for 3D problem Parallel computation

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