I’m doing simulations of natural convection in a square cavity, using the model from . The velocity field and temperature field are solved using two independents BGK equations, coupled through a forcing term (boussinesq term).
My goal is to compare the results from my Lattice Boltzmann code to those of commercial CFD solvers (like Fluent for example).
To do these comparisons, a square cavity with a hot wall on one side and a cold wall on the other side is simulated for different Rayleigh number and Prandtl number.
So far, the simulations between the Lattice Boltzmann method and the others method don’t agree. But I think the only problem is that I’m not computing the Rayleigh number properly, as the results of my LBM simulation match those of Fluent but for different Rayleigh number.
Can someone help me in computing the correct Rayleigh number and the corresponding LB parameters?
This is how I compute the Rayleigh number at the moment:
The article  states that the kinematic viscosity nu is linked to the relaxation time tau through
nu = (tau-0.5)/3 * dx^2 / dt
and the thermal diffusivity is determined by
D = (tauT-0.5)/2 * dx^2 /dt
(where tauT is the relaxation time for the thermal lattice BGK)
The Rayleigh number Ra is defined by:
Ra = g*betta*deltaT*L^3 / (nu*D)
where g is the gravity force, betta the thermal expansion, deltaT the temperature difference between the hot and cold wall, and L the size of the cavity.
So in term of lattice units, if we suppose dx/dt = 1 and dx = 1/(Nx-1) (where Nx is the number of node along x axis) it should be:
Ra = g*betta*deltaT*(Nx-1)^2 * 24 / ( (2tau-1)*(2tauT-1))
and the Prandtl number Pr should be
Pr = nu/D = 2/3 * (2tau-1)/(2tauT-1)
What is wrong with this analysis, and how the Rayleigh number should be computed?
Note: I’ve already read the article “choice of units in lattice Boltzmann simulation” from Jonas latt, but it didn’t help.
 A coupled lattice BGK model for the Boussinesq equations. Guo et al.