I run this code with the information above. The temperature and velocity profile obtained through this investigation corroborate the results obtained from classic CFD with some errors; however, the condition for stability does not satissfied with this collection:

?t???x?^2(not satisfied) because ?t=0.0086,?x=2.936101097573517e-04.

but I’ve run this this information:

lx=118,ly=118 (square enclosure),gr=0.000001.

with this selection, the condition for stability is satissfied :

?t???x?^2

however, the results are really bad.

I’d really appreciate that if you can tell me which gr is good for this problem.

actually I think that there is some confusion here. The condition delta t ~ delta x^2 is not a stability condition for the LBM (maybe for your alternative CFD solver…). For the LBM it is a condition to suppress the compressibility error terms.

Furthermore I am not sure to understand exactly your problem. In what sense are your results really bad?

The “rule” dx^2~dt is usefull only when you try to increase the lattice resolution. It only tells you that when you refine by a factor two dx you should refine by a factor four the time, in order to get rid of the compressibility errors.

Otherwise This rule has no real meaning. The two important conditions that must be satisfied are that the Mach number must be of the order 0.1 or smaller, and that the Knudsen number must also be small.