pressure variations in incomprissible flow

Hi Dear friends;
I want to simulate an incompprissible flow in channel. we know that in incomprissible flow density is constant.if i want
to know pressure in channel in LBM we have P=(cs)**2*rho
so if rho be constatn(becouse of inncomprissibility) we have not any pressure
variention in channel?!!


the LBM is not an incompressible Navier-Stokes solver. It is only valid in the weakly compressible limit (low Mach number). Therefore small density variations are always present.
In my opinion, the pressure in an incompressible fluid is not the “thermodynamical sense”, since as you noticed p=c_s^2*rho and this would imply that pressure variations are forbidden. The pressure is just a “mathematical artefact” in order to be able to have divergence free velocities.


Thanks Malaspin for your quick answer, thus I can use LBM even for compressible flow.

Actually the standard LBM (D2Q9) is valid in the incompressible limit only. But the scheme is weakly compressible. If you want to do strongly compressible flows you have to use extended lattices. Such as those given by Shan et al. in [].