Dear Alex and Timm

Really nice discussion indeed.

I used the initialization procedure, previously posted, for two simple benchmarks: a Wormesly flow and a Taylor-Green vortex flow. In both cases, the initial velocity field was not only different from zero but also spatially non-uniform. Therefore, f_i^(1) had to consider the strain-rate information. Furthermore, since the momentum source term F_i is \epsilon^1 order it was also considered in the f_i^(1), more or less like you did in your post Alex.

I think the question here is: how the inclusion of F_i in f_i^(1) affects the proper initialization procedure?

To be honest I do not know a definite answer. If you choose to follow the Chapman-Enskog expressions I know for sure that you will be playing on safe grounds and that is generally the most sensible thing to do.

Yet, I did some tests without even initializing the f_i^(1) contribution at all, i.e. f_i^(neq) (x,t=0)=0, and only a few time steps after my LB velocity solution was reasonably matching the expected hydrodynamic solution.

Timm raised the question of oscillations in the LB solution introduced by a hydrodynamically inconsistent initialization. That makes total sense! However, I did not study that.

The inclusion of body forces in LB is in my opinion still a controversial issue because it affects several aspects of the LB scheme. It affects the bulk hydrodynamic solution, the boundary conditions construction and also introduces additional discretisation errors when the hydrodynamic body force is not exactly evaluated by the trapezoidal method, or any second-order quadrature scheme. What has the major impact over the final solution accuracy? I do not know. Yet, I insist, respecting Chapman-Enskog at least ensures that our model accuracy matches Navier-Stokes up to second order.

Finally, Timm commented about the physical significance of neglecting time in the initialization procedure. When initializing a mathematical problem, in case our initial solution is not known we have to solve for a time-independent problem to get it, right? Hence I think Alex’s reasoning is valid. Don’t you think so Timm?

Regards

Goncalo