Mass conservation in the LBM states that the sum over distribution-functions f[sub]i[/sub] equals the density. But what units do f[sub]i[/sub] have that make their sum equal the density, which has units of kg/m[sup]3[/sup]?
f is number distribution function which is correspoding to the number of particles which have the same velocity(between v and v+dv) and are in the same location(r and r+dr) at the same time. all particle have the same mass say it m. f set the mass of particles(mf gives kg/m^3), its momentum and etc. so in fact m is dropped from the both sides.
From basic kinetic theory, you can derive the units of f. Since the density (in kg/m[sup]3[/sup]) is found as the integral of f over all of velocity space (i.e. \rho = \int f du dv dw), you will see that f should have units of density divided by velocity cubed. Thus you get kg s[sup]3[/sup]/m[sup]6[/sup]