# thermal relaxation time for high Pr

Hello All,

I’m trying to simulate the temperature field using the simplified thermal lattice Boltzmann model by Peng et al. (2003)
online: http://serve.me.nus.edu.sg/shuchang/Publications/Latest%20papers%20for%20web/Peng_Shu_Chew%20(PRE)%202003.pdf

and having problems with the relaxation time.
In the paper, the relaxation time is defined as: tau_v = 1/2 + 3 * viscosity,
and the thermal relaxation time as: tau_c = 3/2 * visc / Pr + 1/2

If I use ideal Gas with Re=10 and Pr = 0.7 everything works fine,
but when I use other Fluids with high Prandtl-Number, for example heating oil with Re = 10 and Pr = 12.4, the simulation becomes unstable, because tau_c comes to close to 0.5, or, if I change the viscosity so that tau_c isn’t near 0.5, tau_v becomes bigger than 10 and the simulation again unstable.

what I would like to ask, if someone had encountered the same problem and know the solution or if its possible to use this particular thermal model for Fluids with high Prandtl-Number.

Any help will be much appreciated. Thank you,

-Habib

Hi,

Have you tried using different boundary conditions? It is quite common with this type of problems that numerical instabilities are due to the choice of boundary condition, and not to the numerical model itself. Also, you may want to play around with the Matlab code for thermal fluids. It is based on the model by Guo and others which, it seems to me, is closely related to the model you are using.

Dear JLatt,

thank you for your reply, i haven’t tried different boundarys yet because I purposely want to apply the Neuman - Dirichlet boundary in the Simulation. If I may, i’d like to describe the problem a bit further.

Suppose I use an ideal Fluid with Re = 10 and Pr = 0.7 with speed u = 0.03 (Lattice units) and Length H = 150 (Grid size)

`````` visc = ( u * H) / Re = (0.03 * 150) / 10 = 0.45 (in lattice units)
``````

The relaxation times then becomes:

`````` tau_v = 1/2 + 3 * viscosity = 1/2 + 3 * 0.45 = 1.85
tau_c = 3/2 * visc / Pr + 1/2 = 3/2 * 0.45/ 0.7 + 1/2 = 1.465
``````

both of them are far from 0.5 so it’s ok and the simulation doesn’t become unstable.

Now if I use Heating Oil with Re = 10 and Pr = 12.4 with the same speed u and Length H the calculation becomes

`````` visc = ( u * H) / Re = (0.03 * 150) / 10 = 0.45 (stays the same)
tau_v = 1/2 + 3 * viscosity = 1/2 + 3 * 0.45 = 1.85 (stays the same)
tau_c = 3/2 * visc / Pr + 1/2 = 3/2 * 0.45/ 12.45 + 1/2 = 0.55 -------> Problem: to close to 0.5
``````

as indicated, tau_c becomes too close to 0.5 and the simulation becomes unstable.

What I would like to ask, if there is something false in the calculation, or if its possible to simulate the Heating Oil with this temperatur model.

I would be very grateful if you could help me out on this problem.