SomeCommentsNeeded

Lattice Boltzmann Method LBM: Theory
#1

Hi Every one,

fi(x+ei,t+1)= fi(x,t)-[fi(x,t)- fieq(x,t)]/τ, (2)
fieq=ρwi(1+3(ei.u)+4.5(ei.u)2-1.5u2), (3)
ν=(2τ-1)/6. (4)

On above equations someone comments:
Dimension in Eq.(3) and Eq(4) are wrong. For example, in Eq(4), viscosity \nu should has identical dimension as \tau. On the other hand, from Eq.(2), it seems \tau is a non-dimensional parameter. Hence \nu seems a non-dimensional parameter. This is not correct.

What will be the possible answer of the above question.

Thanks

#2

Hi,

First of all, remember you are working with a numerical model. Your equations are written in so-called “lattice units”, where the grid spacing (dx) and the timestep (dt) are both unity (dx=dt=1).

More generally, t+1 is actually t+dt, x+ei is x+eidt, and ei is proportional to dx/dt. The kinematic viscosity becomes nu=(2tau-1)/6*dx^2/dt

#3

Beside the above, I suggest you to read this paper:
On pressure and corner boundary conditions with two lattice Boltzmann construction approaches

Mathematics and Computers in Simulation
Volume 84, October 2012, Pages 26–41

which also available
here

That paper explains those things, both for the standard LB model som for the Entropic model.