# Multiple Relaxation-Time

Hi,

I have a basic question about multiple relaxation-time models.

As i understand it, relaxation time is related to viscosity, nu = 1/3(tau - 0.5). If I want to model a Newtonian fluid (constant viscosity), can I use a multiple relaxation-time model? From what I’ve found so far it seems to me the answer is yes, but I am looking for a little more understanding. Can anyone offer an explanation?

Thanks

Tim

Hi,

In multiple-relaxation time approaches, the collision matrix is diagonalized, or, equivalently, transformed into the space of the velocity moments of the particle populations. The eigenvalues are then interpreted as relaxation rates for the moments.

You can classify the velocity moments into three groups. The first group contains the conserved variables, density and velocity (and energy in thermal models). Their associated relaxation rates can be chosen arbitrarily, because the collision is independent of this choice (“density is the same in the pre-collision and in the post-collision state, no matter what relaxation time you take”). The second group is for the non-conserved, macroscopically relevant variables. For non-thermal flows, there is only the strain-rate tensor left in this group, with three independent variables in 2D, and six in 3D. Their relaxation rate is related to a transport phenomenon in the gas, which are observed macroscopically as fluxes. The relaxation parameters for the off-diagonal components of the strain-rate are related to the kinematic fluid viscosity through the formula you mention (all three of them must be equal in 3D), while the diagonal parts are connected to the bulk viscosity, and are relevant only if you consider compressibility effects (there is a lengthy discussion concerning the bulk viscosity in this forum).

The remaining moments are often called “ghost moments”, because they are not visible in the macroscopic interpretation of the lattice Boltzmann simulation. Conceptually speaking, the choice of the corresponding relaxation parameters is therefore irrelevant. In practice however, these relaxation parameters can affect numerical properties of the model, such as the stability. There exists a whole community of people who select these parameters carefully, for example through a linear stability analysis, in order to improve the numerical stability of their lattice Boltzmann simulations. I think that this is what is generally meant with the term “multiple-relaxation time models”.