How to choose the relaxation time in the Lattice Boltzmann Method?

We know that the relaxation time is very important in LBM. I have searched lost of papers, but can't find some systematic introductions about the choice of relaxation time in SRT LBM. Could you give some ways or papers to get a suitable relaxation time?

The dimensionless relaxation time $\bar{\tau}$ should be chosen as to obtain the correct fluid viscosity. That is : $$\bar{\tau} = \frac{\mu}{\rho c^2 \Delta t} + \frac{1}{2}$$

where $c$ is the lattice celerity, $\Delta t$ the time step and $\rho$ the density. Generally $\rho$ is chosen as unity in LBM Therefore, generally, you know the physical property you want to simulate ($\mu$ and $\rho$) and you use a value of $\tau$ that you choose. Then, with this choice of value of $\tau$ (which is generally 1 or lower), you obtain the value of the time step $\Delta t$.

You can find some explanation on this issue in this paper : http://www.sciencedirect.com/science/article/pii/S0021999115001916

However, many books also treat the subject, like this one : http://www.worldscientific.com/worldscibooks/10.1142/8806

• In fact, I don't know the physical property. Because I simulated the case from paper, there is always no physcial property in the paper. So I have doubt about the relaxation time. Dec 15 '16 at 13:11
• Well what is the Reynolds number? If you know which Reynolds number you want to obtain, since $Re=\frac{\rho u L}{\mu}$, if you know the length of the geometry, the approximative speed, then you can fix $\rho$ and then choose $\mu$ to obtain the right Reynolds number. You must at least have some information about the flow (such as a dimensionless number), otherwise your reference paper does not make sense.
– BlaB
Dec 15 '16 at 13:13
• Yes, I agree with you. Another question, if the paper was computated by the FVM, but I use the LBM. So how can I find the relaxation time? Dec 15 '16 at 13:21
• You can either chose the time step, this will decide the relaxation time, or you can chose the relaxation time (usually ]0.5, 1]), then this will impose the time step. You just need to chose the combination of all parameters in order to reproduce the correct Reynolds number of your simulation. I believe you should take some time to review the bases of LBM, since this is really fundamental to the method.
– BlaB
Dec 15 '16 at 13:33
• Thank you. I don't understand why the time step effects. Because in the code, the time step is only related to the relaxation. Such as, in my code, dt=dx=1,and I iterate 200,000 times. So where is time steps? Dec 15 '16 at 13:38