I am running a simple LBM for decaying (unforced) isotropic homogeneous turbulent initial conditions, employing LBGK methodology. I have two questions/concerns.

In order to validate the model, I have energy spectrums generated by a spectral code for the same initial conditions and domain size, but I am unable to figure out how to calculate a reasonable physical time step in LBM. I need this in order to compare the energy spectrums at given times throughout the iterations seeing as there is no steady state solution.

My second question is about the time relaxation tau. I have been testing the code and found that to maintain a stable solution, tau must be greater than 1, but according to literature and every forum I have looked through, tau should be able to be decreased close to 0.5 and still have physical viscosities and stable solutions for some Re values. Could this be due to the nature of the turbulence I am simulating or is there some more fundamental LBM rule that I am missing.

I suggest that you search the forum for the term “unit conversion”; to summarize, you need to rescale the velocity in your initial condition. The maximum (or average) value of the velocity, as measured in lattice units, is proportional to the time step.

I don’t think that you are missing a fundamental LBM rule (you did remember that the velocity in lattice units must be substantially smaller than 1, didn’t you?). If your simulation is unstable in homogeneous isotropic turbulence (with periodic boundaries), then I’d say it’s likely that your Kolmogorov length is smaller than the size of a lattice cell, which means that the simulation is under-resolved and therefore invalid as a DNS. Once you’ve figured out the units in your simulation, you will probably understand what’s going on.

Is there any way to determine the maximum Reynold’s number for a given domain? In Succi’s book, he seems to mention Re = RMaN, but that is for the collisionality of the type of lattice boltzmann, correct? I was able to extract R = 6 from some convergent simulations, but I don’t know if that is the LGBK value and that gives the maximum Re value or if I am just crazy.

As I said, I think that the answer to your question comes from turbulence theory, and not from lattice Boltzmann theory. Here’s a paper which discusses this type of arguments:

If I remember right, if N is the number of lattice nodes in a given direction of the domain, the maximal Reynolds number is proportional to N^(4/3). Of course, this is only true if your simulation is turbulent and if you are doing a DNS (no turbulence model). And, additional arguments apply if you have walls, because the structures found in boundary layers tend to be even smaller than the structures in the bulk.