A first remark, is that if you fix Re, then you will not be able to constrain freely Delta X, Ma, and tau. You will be able to fix two of them (in fact if you want to be able to have second order accuracy you have a relation between delta x and delta t, and therefore you have no freedom anymore) and the third will be a consequence of this choice.

I do not have any real idea on how to solve your question. But something that is sure is that it is completely problem dependent (as you said). When you do a LB simulation the computational cost (time of computation) T, for a given number of timesteps, N, on a grid with dimensions [0,nx-1] x [0,ny-1] x [0,nz-1]

T = alpha * N * nx * ny * nz = T0.

where alpha is the time needed to do a collide and stream step on a node. Now if you want to refine your grid by a factor 2, then in order to avoid compressibility errors (if you are in the incompressible regime) you have to refine the timestep by a factor 4. Therefore

T* = alpha * (4*N) * (2*nx) * (2*ny) * (2*nz) = 32*alpha*N*nx*ny*nz = 32*T.

This will result in a 4 times better accuracy result. (In 2D the factor will be 16.) The “law” will be more or less

T(T0,factor) = T0*factor^5 (if I haven’t done any mistake).

accuracy = accuracy0 / factor^2

Therefore for an initial setup of accuracy, accuracy0, and of total time simulation time T0,you can evaluate the potential gain for T=Tmax…

This is of course not a satisfactory answer, because you have to test a wide variety of “initial setups” to optimize your results (I guess that there are lots of books on this topics although I don’t know any)… But it may help you a bit.