Mach number doubt

Lattice Boltzmann Method LBM: Theory
#1

Hi everyone,

I have still a doubt, the Mach number in LB is defined as Ma=U_lb/cs, where U_lb is the characteristic velocity (LB units) for the system and cs is speed of sound, by definition cs=1/sqrt(3). As non-dimensional number, the Mach number can be also defined as Ma=U_p/u_sound for physical units §, where U_p[m/s] and u_sound is the speed of sound through the medium (air, water, etc). Once I have calculated both dx_P and dt_P, I can calculate the characteristic velocity U_lb. Now let us consider the simplest problem, a channel flow; I define the Re=U_P*h_P/vis_P. Perhaps I want to simulate water in a microchannel with Re=100, then I know vis_P, h_P and Re; from they I calculate the velocity U_P. If I want to have this velocity in terms of lattice units I use:

U_lb= U_P*(dt_P/dx_P)

Now the question, I want to calculate the Mach number, how?

Physical Units
1.- Ma=U_P/u_sound ------> where u_sound is the speed of sound in the medium (water), must it be 1484 m/s?

or

Lattice Units
2.- Ma=U_lb/cs -----> here cs is constant, or is it possible to modify cs to consider the water

(1 and 2 might be clearly equivalent, but they are not!!!)

what about the following
3.- since I have dx_P and dt_P, a characteristic velocity could be Uc_P=dx_p/dt_P, and then
Ma= Uc_P/u_sound

could somebody help me with this small problem?

thanks a lot

Anuhar

#2

Hello,

the Mach number is the velocity divided by the sound speed. The point is that there are different Mach numbers for the physical system and the numerical system. Why? We can do that because the Mach number is not important as long as it is small (<0.1 say). The Reynolds number on the other hand should be identical in both systems since it is much more important for the physics.
Whenever you read a LBM article and people tell you something about the Mach number, it is most probable that they talk about the numerical Mach number.
To say it differently: You have a numerical freedom in artificially increasing the Mach number for your simulations. There are also some discussions in the forum. Please have a look around.

Timm