# Relaxation time tau: constant or dependent on mass density rho?

A question about whether relaxation time tau is constant or not.

In the standard LBM it holds that
tau = nu / c_s^2 + 1/2 (Eq 1)
where tau is the relaxation time, nu the kinematic dynamic viscosity, and c_s the speed of sound.
It also holds that
nu = mu / rho (Eq 2)
where mu is the regular (non-kinematic) dynamic viscosity and rho the macroscopic mass density.
Substituting Eq 2 in Eq 1 gives
tau = mu / rho / c_s^2 + 1/2 (Eq 3)
For an isothermal incompressible flow we assume the regular dynamic viscosity ‘mu’ is constant. We may not assume the kinematic dynamic viscosity ‘nu’ is constant. So, I would say the relaxation time ‘tau’ is not constant but depends on the locally computed mass density ‘rho’, just like Eq 3 makes clear. Still, most LBM papers state that the relaxation time ‘tau’ is constant. Can anybody shed light on this?

Dear Bart,

Even i am having the same doubts regarding the relaxation time.
would like some one experienced in non-isothermal lbm flow simulation to shed some light on the same

Regards

sandeep

Hi sandeep,

I found some confirmation for my thought that the relaxation time is not constant. The article ‘Lattice Boltzmann Simulations of Fluid Flows at Finite Knudsen Numbers’ of Calin Dan et al. (https://pure.tue.nl/ws/files/3722615/599608055576169.pdf) present a ‘relaxation time correction’ tau = 0.5 + (tau^ref - 0.5) * rho^ref / rho. Let me demonstrate this correction is effectively identical to the variable relaxation time I present in my first post.

We can write the dynamic viscosity as
mu = rho * c_s^2 (tau - 0.5)
and also as
mu = rho^ref * c_s^2 (tau^ref - 0.5)

So, we have
rho * c_s^2 (tau - 0.5) = rho^ref * c_s^2 (tau^ref - 0.5)

Some algebra:
rho * c_s^2 (tau - 0.5) = rho^ref * c_s^2 (tau^ref - 0.5)
rho * (tau - 0.5) = rho^ref * (tau^ref - 0.5)
(tau - 0.5) = rho^ref/rho * (tau^ref - 0.5)
tau = 0.5 + rho^ref/rho * (tau^ref - 0.5)
tau = 0.5 + rho^ref/rho * (tau^ref - 0.5)
which yields their ‘relaxation time correction’.

Although I have not read the article in detail, they conclude with:
“We introduced a correction of the relaxation time with
density tau(rho), which is critical to reproduce the physics
of fluid flows at finite Knudsen numbers.”

It would be nice to see some more confirmation, or a plausible argument why you should make the relaxation time constant. Be it from fellow forum users, or references to articles.