Lattice-Boltzmann method with Ehrenfests' steps (LBGK-ES)

A 1D MATLAB LBGK-ES code is available to download. Ehrenfests' steps belong to a class on nonequilibrium entropy limiters for lattice Boltzmann methods. The 1D code also implements other examples of limiters of nonequilibrium entropy as described by Brownlee et al in "Nonequilibrium entropy limiters in lattice Boltzmann methods" (to appear Physica A). A 2D FORTRAN LBGK limiter code is also available to download. The code solves the classical lid-driven cavity problem (see below) but can be easily adpated to solve other problems. Both codes are free software but are only made available under the proviso that no support will be be given. Freely distributable movies and figures produced using these codes are also available (see below).

1D isothermal shock-tube

The following applies to all of the 1d shock tube simulations below:

801 uniformly spaced lattice sites are used;
initial density discontinuity in ratio 2:1;
initial condition is (ρ_L,ρ_R,v_L,v_R,t_end)=(1.0,0.5,0.0,0.0,400);
initial condition is unsmoothed;
discrete relaxation frequency ω=2(1-10^-9);
bounce-back boundary condition are used.

Shock tube simulations:

LBGK: density movie [25s] / velocity movie [25s];
LBGK-ES (k,δ)=(∞,10^-3) [crosses indicate Ehrenfests' step]: density movie [25s] / velocity movie [25s];
LBGK-ES (k,δ)=(∞,10^-5) [crosses indicate Ehrenfests' step]: density movie [25s] / velocity movie [25s];
LBGK-ES (k,δ)=(8,10^-3) [crosses indicate Ehrenfests' step]: density movie [25s] / velocity movie [25s];

2D-flow around a square cylinder

Unless otherwise stated, the following applies to all of the 2d square cylinder simulations below:

characteristic length L=20 (side of cylinder);
601×501 uniformly spaced square lattice in used (301101 lattice sites in total);
channel blockage ratio of 4%;
free-stream velocity (u_∞,v_∞)=(0.05,0.0) (in lattice units);
constant inlet condition, zeroth-order extrapolation at outlet;
free-slip boundary condition on channel walls;
Maxwell diffusive boundary condition on cylinder;
simulation run for 125000 time steps;
Ehrenfests' tolerances (k,δ)=(10,10^-3);
only part of the computational domain is shown.

Square cylinder simulations:

Re=37: vorticity movie (close to the onset of vortex shedding) [1m18s];
Re=50: vorticity movie (vortex shedding has begun) [1m18s];
Re=110: vorticity movie (flow still laminar, one pronounced frequency) [1m18s];
Re=250: vorticity movie (transitional) [1m18s];
Re=500: vorticity movie [1m18s];
Re=1000: vorticity movie (onset of turbulence) (with Ehnrefests' steps histogram) [1m18s];
Re=3000: vorticity movie [1m18s] (with Ehnrefests' steps histogram) [1m18s];
Re=5000: vorticity movie [1m18s] (with Ehnrefests' steps histogram) [1m18s];
Re=10000: vorticity movie [1m18s] (with Ehnrefests' steps histogram) [1m18s];
Re=20000: vorticity movie [1m18s] (with Ehnrefests' steps histogram) [1m18s];
Strouhal–Reynolds relationship;
comparision of LBGK with LBGK-ES at Re=3000 for various values of the parameter pair (k,δ) [19s];
comparision of LBGK with LBGK-ES at Re=3000 for various values of the parameter pair (k,δ) showing position of Ehrenfests' steps [19s].

2D lid-driven cavity flow

The following applies to all of the lid-driven cavity simulations below:

characteristic length L=320 (length of lid);
321×321 uniformly spaced square lattice in used (103041 lattice sites in total);
fluid initially at rest;
Maxwell diffusive boundary condition on all sides of the cavity;
lid moving with constant velocity (u_∞,v_∞)=(0.075,0.0);
simulation run for 500000 time steps;
Ehrenfests' tolerances (k,δ)=(16,10^-3).

Lid-driven cavity simulations:

Re=1000: stream-function movie [13s];
Re=5000: stream-function movie [25s];
Re=15000: stream-function movie [25s];
Re=20000: vorticity movie [25s] / stream-function movie [25s].
Re=200000: vorticity movie [62s].