1 / 31

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

gilda
Download Presentation

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

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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

More Related