periodic boundary conditions and reflected waves

I am trying to validate LBM code by investigating the coexistence curve of P-R equation of state.
I am doing so by trying to repeat the validation (section 3.1) in the article of Shuai Gong and Ping Cheng, “A lattice Boltzmann method for simulation of liquid–vapor phase-change heat transfer” that was published at International Journal of Heat and Mass Transfer Volume 55, Issues 17–18, August 2012, Pages 4923–4927.

it is 2D rectangle lattice with 150 lattice units at each direction.
Relative density inside a circle of radius 10 at the center of the zone: 1.01 times the critical density.
Relative density outside the circle: 0.99 times the critical density.
Temperature is constant and is set to 0.9 times the critical temperature.
Boundary conditions are periodic at any direction.
The used eos is Peng–Robinson!
I used the same numerical scheme for the calculation of the acting force, with a vary similar definition of the potential ( p(r,T)- r*c^2/3) .

After a lot of cycles, a steady state is expected, with a drop of liquid at the center, and vapor all around.

My computation was exploded before reaching steady state: I think that something like sound waves were generated because of the difference of the densities. They reached the boundaries and reflected back, and so forth until the computation was unreasonable and unstable as well.
Since the explosion of the computation is caused because of the reflection of waves, and since it was not mentioned at all in the article, can somebody please tell me what might be the reason and solution for this, so that the computation will reach to steady state with stable drop in its center?

Thanks a lot


I can’t answer why these sound waves are generated, but if you want them to be absorbed quickly so that they don’t disturb your simulation too badly, you could try increasing the bulk viscosity of the fluid. This can be done using a MRT collision operator, or using the simpler method described in this article:


I will do!