I’m new to the topic LBM. I’ve already successfully implemented a LBM method here in our group, and it’s working quite well. Presently we’ve coupled it with a finite difference thermal model which already exists. Since there is the possibility to use a thermal LBM also, I was asking myself, what will be the difference. Ok, the methods have a completely different point of view, but in the end, what will be the advantages of one or the other? Please, could someone with some more experience give me some hints on this topic?
I think that is very good question. I don’t know much about thermal LBE but I think there are arguably two approaches, but perhaps only one of them is practical. (As always I may well be wrong so I hope others can correct me.)
One method is to use an MRT-type scheme with an extended velocity set to get the correct energy equation (D2Q9 is not sufficient). I think this becomes quite difficult, if at all possible, and it is unclear if there are any benefits.
The other way is to couple an LBE for the fluid flow with another LBE for the advection/diffususin of the temperature field. This is certainly a lot easier than the above as LBEs for passive scalars are nice and simple.
One thing to be aware of is if you use an LBE for the temperature field, a full Chapman-Enskog expansion will reveal error terms which I think are order Ma^2, but I’m not sure (It may be that they are Ma^3 when the velocity is constant and Ma^2 otherwise, I can’t remember). This can be implemented with a 4-velocity square lattice (which is cheaper), but I think the D2Q9 will have a positive effect on the error terms.
Is it better to use LBE or FD here? This is the interesting bit and I’m not sure of the answer! If it were me, I’d use an LBE for both equations because I like to evaluate things on the same stencil, if possible, because we can subject them to the same analysis, and because we can implement them neatly and retain the LBE advantages. However, it may be the case that the FD scheme is more accurate, because of the sources of error mentioned above (some of the multiphase people use a FD scheme for the phase field for similar reasons).
Perhaps someone else with experience in this field can shed some light?
One other quick point. In my opinion, LBE and FD are not as different as some of the literature suggests.
thanks for your answer! In the meantime I’ve read some stuff, and I found at least one item (Book of A.A.Mohammad) where it was pointed out, that Thermal LBM would need less timesteps compared with FD… ? Well, I’m not quite sure if this statement holds!?
Please, everyone is invited to send me his experience/feelings/… about this issue
Francois, both your implementation and the 2nd method mentioned by pleb01 fall under the category where an isothermal LBM scheme for fluid flow is coupled to the transport of temperature treated as a passive scalar. While this works for many flow situations, it seems to me inherently inconsistent at a deeper level to couple an “isothermal” LBM such as D2Q9 to temperature dynamics. This inconsistency will probably show up in highly compressible flows or large temperature variations, I am not sure. I’ll point out one specific issue: the speed of sound remains independent of temperature in such a coupled scheme. For example, if using D2Q9 for the fluid, cs^2 = c^2/3, no matter what the temperature is (note that c is the lattice speed which isn’t determined by the simulated tempr). In other words, the eqn of state is p=\rho c^2/ 3 and not p= \rho R T