# Choice of discretization to ensure stability

Hello,

I am performing 3d simulations in a pipe with a contraction which causes a nine-fold increase in velocity. The Reynolds number is 500 based on the velocity and diameter inside the contraction. Due to the shape of the domain the diameter of the contraction is about 1 hundredth of the length of the domain.

My problem is that the lattice viscosity (Nu_LB) is determined uniquely by the diameter (d_LB) and mean velocity (u_LB) inside the contraction by

Nu_LB = u_LB*d_LB/Re.

As u_LB can’t be above 0.1 and even on a very large grid d_LB is only 20 lattice units this leads to very low values of Nu and therefore a relaxation parameter very close to 0.5 (max 0.512). This was OK using a D3Q19 lattice and I got some reasonable results but the simulations fairly quickly become unstable on a D3Q15 lattice and almost immediately on a D3Q27 lattice. I’d be really grateful if anyone has any ideas about how I might be get a stable simulation without using an unreasonably large grid.

Thanks,
Alex White

p.s. I want to use the D3Q27 and D3Q15 lattices to see if they will correct some problems which I saw in the results from the D3Q19 lattice.

Dear Alex,

You certainly need to use MRT model or TRT with Lambda=1/4 to ensure stability for this small velocity. It of course depends on what problem you want to simulate, on boundary conditions, etc. Please take a look at the paper by I.Ginzburg in Journal of Statistical Physics for TRT explanation and stability limits.

My suggestion is to play a bit with TRT (Lambda=1/4) to see whether it will be better for your simulations.

Hopefully it will help,
Alex

Thanks Alex, The MRT model has improved stability greatly, even on small grids.