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

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

June 27 th 2007Dundee 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.

June 27 th 2007Dundee 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.

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

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

June 27 th 2007Dundee Smoothed Particle Hydrodynamics

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

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

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

June 27 th 2007Dundee Examples Some grow at infinity!!

June 27 th 2007Dundee 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.

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

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

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

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

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

June 27 th 2007Dundee Approximation in high dimension dCondition number d=100

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

June 27 th 2007Dundee 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.

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

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

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

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

June 27 th 2007Dundee 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.

June 27 th 2007Dundee d 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.

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

June 27 th 2007Dundee 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!!

June 27 th 2007Dundee 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.

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

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

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

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

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

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

June 27 th 2007Dundee 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

June 27 th 2007Dundee Decoupled viscosity from timestep Controlled viscosity

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

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

June 27 th 2007Dundee 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

June 27 th 2007Dundee Simulation looks good

June 27 th 2007Dundee Simulation of square cylinder

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

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

June 27 th 2007Dundee 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.

June 27 th 2007Dundee 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

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

June 27 th 2007Dundee Entropy Filtering Ehrenfest

June 27 th 2007Dundee 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.

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

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

June 27 th 2007Dundee Locating bifurcation point Re=7135

June 27 th 2007Dundee 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.