I’m rather confused about the use of rho and rhoBar in the AdvectionDiffusionDynamics code. In the advection-diffusion lattices, rhoBar is the sum of the populations and rho=rhoBar+1. From the literature, then, it seems like the BGK equilibrium term should be
f[sub]i[/sub][sup]eq[/sup] = t[sub]i[/sub] * rhoBar * (1 + ([b]u[/b]*[b]v[sub]i[/sub][/b])/c[sub]s[/sub][sup]2[/sup]).
But actually, in the code, it’s given as
f[sub]i[/sub][sup]eq[/sup] = t[sub]i[/sub] * (rhoBar + (rho * [b]u[/b]*[b]v[sub]i[/sub][/b])/c[sub]s[/sub][sup]2[/sup]).
So it seems that rhoBar distributed to one term, and rho the other. Or am I mistaken about something here? Is there any meaning in the distinction between rho and rhoBar other than its stated (in the user’s guide) use for increased numerical accuracy?
This distinction is a problem for me because it seems that, even when I initialise part of the system at rho=1, rhoBar=0 (so all populations are 0), over time, the advecting velocity seems to “create rho” in places where there should be no populations (nothing has diffused there). And that (at least to me) seems due to the fact that the full rho = 1 is used as a coefficient in the equilibrium term. It doesn’t make sense to me that the equilibrium distribution of a node initialised at all 0 populations should be at all non-zero, is that an incorrect intuition on my part? I would greatly appreciate an explanation, thank you.