1. 1 A different view of the Lattice Boltzmann method for simulating fluid flow Jeremy Levesley Robert Brownlee Alexander Gorban University of Leicester, UK Supported by EPSRC

2. June 27 th 2007Dundee 20072 Movie More Space Balls than Star Wars.

3. June 27 th 2007Dundee 20073 High Reynold’s Number Flow  Very low viscosity leads to complicated dynamics.  PDE model for such flows is the Navier-Stokes equation with small viscosity, the Euler equation for zero one.  Examples of shock tube, square cylinder, lid-driven cavity.

4. June 27 th 2007Dundee 20074 Talk  A new framework for looking at LBM. Not the Boltzmann equation.  It is close to the Navier-Stokes’ equations in some sense.  Stabilisation of method via targetted introduction of diffusion.  Filters and entropy limiters.

5. June 27 th 2007Dundee 20075 Fraud and hypocrite  Approximation theorist talking about “PDEs”  Numerical analysis conference – what is the order of convergence of your method?  Spot the hypocracy.

6. June 27 th 2007Dundee 20076 Little quiz. What is the condition number?

7. June 27 th 2007Dundee 20077 Smoothed Particle Hydrodynamics

8. June 27 th 2007Dundee 20078 The shock tube simulation Gas AGas B diaphragm Simulation of pressure with time.

9. June 27 th 2007Dundee 20079 Poorly modelled physics  Artificial viscosity  Slope limiters  Initial smoothing Continuum equations do not model the physics

10. June 27 th 2007Dundee 200710 Radial basis approximation univariate function A data set Y Approximate Low degree polynomial

11. June 27 th 2007Dundee 200711 Examples Some grow at infinity!!

12. June 27 th 2007Dundee 200712 Micchelli (1986, CA)  Interpolation problem is always solvable.  In all space dimensions  For any configuration of points (with some very mild restrictions).  A great challenge to find appropriate methods for solving real high dimensional problems.

13. June 27 th 2007Dundee 200713 Another representation Consider  x  =(  x+1  x  (x+   x)=|x|

14. June 27 th 2007Dundee 200714 More smoothness  Cubic B-splines from iterating twice. Shape to the data – partition of unity.

15. June 27 th 2007Dundee 200715 A Good Basis  Basis functions which match the shape of the data.  Discrete Laplacians formed using the data points.

16. June 27 th 2007Dundee 200716 Two or more dimensions (Beatson)  Generalised barycentric coordinates Sibson – Stone (boundary over distance) Mean value (Floater et. al.)

17. June 27 th 2007Dundee 200717 Simulation using B-splines Still need artificial viscosity!! Brownlee, Houston, Levesley, Rosswog, Proceedings of A4A5 (2005)

18. June 27 th 2007Dundee 200718 Approximation in high dimension dCondition number 15.1 8 21.4 6 43.5 4 81.3 4 164.4 3 323.4 3 642.8 3 d=100

19. June 27 th 2007Dundee 200719 Lattice Boltzmann Method  is the probability density function on phase space. This is a microscopic description.  Recover macroscopic variables via integration in phase space

20. June 27 th 2007Dundee 200720 Equilibrium distribution  Many different microscopic descriptions lead to the same macroscopic description.  For each macroscopic description M there is a distribution which maximises the entropy.  This is the quasi-equilibrium distribution.  Totality of these distributions is the quasi- equilibrium manifold.

21. June 27 th 2007Dundee 200721 Microscopic dynamics  Boltzmann equation, the collision operator Q conserves the macroscopic variables.

22. June 27 th 2007Dundee 200722 Popular choice of collision is Bhatnagar-Gross-Krook collision (BGK) (1954, PR) is a relaxation time and is viscosity parameter.

23. June 27 th 2007Dundee 200723 Lattice Boltzmann dynamics  Lattice Boltzmann method – break into a finite number of populations each moving with a fixed velocity.

24. June 27 th 2007Dundee 200724 Recover macroscopic variables  Sum rather than integrate  Operator form

25. June 27 th 2007Dundee 200725 Example – the shock tube  There are three velocities allowed  Excellent exposition on LBM by Karlin et al. (2006, CCP).  Three populations with two conservation laws to satisfy – density and momentum.  We can make trade between populations, conserving the macroscopic dynamics so as to control the introduction of diffusion.

26. June 27 th 2007Dundee 200726 2d Lattice in computational space – velocities allow us to move from one point in the lattice to a neighbouring one. Populations in phase space each moving in the direction of one of the arrows.

27. June 27 th 2007Dundee 200727 Numerical discretisation High Reynolds number has tending to 0.

28. June 27 th 2007Dundee 200728 Either  High viscosity and we can approximate the Boltzmann equation. Or  Low viscosity and we cannot let time step get less than  without incurring huge computational cost. Not approximating Boltzmann!!

29. June 27 th 2007Dundee 200729 New idea  Simulate transport equation by Free flying for time t Equilibration  Macroscopic variables are transported by free flight.  Microscopic variable redistributed leaving macroscopic variables locally unchanged.  Smallness parameter is t.

30. June 27 th 2007Dundee 200730 “Nonlinearity is local, non-locality is linear” (Sauro Succi) Moreover, non-locality is linear, exact and explicit

31. June 27 th 2007Dundee 200731 Numerically  Numerical scheme is Free flightEquilibration

32. June 27 th 2007Dundee 200732 Stability problem is nontrivial: Entropic LBM does not solve it ELBM  LBGK  Shock tube 1D test {-c,0,c}.

33. June 27 th 2007Dundee 200733 Coarse-graining the Ehrenfests’ way  Formal kinetic equation  Microscopic dynamics

34. June 27 th 2007Dundee 200734 Macroscopic dynamics  Match the microscopic and macroscopic dynamics to order Euler Navier-Stokes

35. June 27 th 2007Dundee 200735 Summarise  Free fly for time and equilibrate populations f*  Integrate to recover macroscopic variables  Navier-Stokes’ equations to order with viscosity

36. June 27 th 2007Dundee 200736 Coupled steps – a scheme of LBM stabilization QE manifold Free flight steps  t Overrelaxation step Complete relaxation (Ehrenfests’ step) The mirror image f0f0 f1f1 f * -(f 1 -f * ) f * -(2β-1)(f 1 -f * ) f *f * f2f2

37. June 27 th 2007Dundee 200737 Decoupled viscosity from timestep Controlled viscosity

38. June 27 th 2007Dundee 200738 Shock tube 1D test {-1,0,1} LBGK  ELBM  magic steps

39. June 27 th 2007Dundee 200739 The Ehrenfests’ Step  Potential problem near shocks where we are too far from the quasi equilibrium. too big

40. June 27 th 2007Dundee 200740 Not enough artificial dissipation  LBGK and ELBM Step back from mirror. Not enough dissipation.  Ehrenfest Introduces dissipation in a very precise and targetted way. mirror dissipation

41. June 27 th 2007Dundee 200741 Simulation looks good

42. June 27 th 2007Dundee 200742 Simulation of square cylinder

43. June 27 th 2007Dundee 200743 Relationship for Strouhal number and Reynold’s number Okajima’s experiment (1982) …. LBM simulations, Ansumali et. al. (2004) Ehrenfest’s steps (2006)

44. June 27 th 2007Dundee 200744 Lid-driven cavity flow with ES (movie of vorticity) (k,δ)=(32,10 -3 )

45. June 27 th 2007Dundee 200745 Flux limiters  S.K. Godunov (1959) we should choose between spurious oscillation in high order non-monotone scheme and additional dissipation in first order scheme.  Flux limiter schemes are invented as the “formulas of compromise” to combine high resolution schemes in areas with smooth fields and first-order schemes in areas with sharp gradients.  The additional dissipation control is difficult.

46. June 27 th 2007Dundee 200746 Nonequilibrium entropy limiters for LBM  Entropy is a scalar quantity  Entropy trimming: we monitor local deviation of f from the correspondent equilibrium f*, and correct most nonequilibrium states (with highest ΔS(f)=S(f*)-S(f)); too big Ehrenfest

47. June 27 th 2007Dundee 200747 Positivity rule f *f * f f * +(2β-1)(f*-f) Positivity fixation Positivity domain

48. June 27 th 2007Dundee 200748 Entropy Filtering Ehrenfest

49. June 27 th 2007Dundee 200749 Median Filter  Choose a number of neighbouring points.  Arrange the non-equilibrium entropies in order of size.  Choose middle one.  Very robust and gentle in places where signal is smooth.  Preserves edges, but reduces oscillation.

50. June 27 th 2007Dundee 200750 Lid driven cavity  For Re < 7000 steady flow  For Re > 8500 periodic flow  Bifurcation point between Peng, Shiau, Hwang (2003) (100 by 100 grid)

51. June 27 th 2007Dundee 200751 Velocity at monitor point Reynolds’ number 7375

52. June 27 th 2007Dundee 200752 Locating bifurcation point Re=7135

53. June 27 th 2007Dundee 200753 Conclusions  Navier-Stokes’ equations arise naturally via free flight and equilibration in phase space.  The viscosity, both actual and artificial can be controlled precisely.  The appropriate notion of smallness is the free-flight time, which is a computational, not physical number.  Non-locality is exact and computable, non- linearity is local.  Reproduce statistics in some standard tests.  Flux limiting can be done via control of a scalar variable entropy.