# Please help : If the speed of sound cs=RT in LBM then why it is computed cs=RT=1/3 as constant in a thermal model?

Hi everyone,
I’m struggling with thermal LBM. I’ve found that the speed of sound cs= RT=1/3 in LBM. My question is :
-Why it is computed as 1/3 in a thermal LBM ,while it is “thermal” so logically it depends on T value ?
help !

Hi there,

The value of the speed of sound (cs) is actually 1/sqrt(3) for most lattices used in LBM, but RT = cs^2 = 1/3. The standard forms of lattice assume the fluid is isothermal (or, more accurately, athermal) and thus the speed of sound is a constant. The equation of state would also happen to be p = cs^2 * rho, which is essentially the same as an ideal gas.

There are two forms of thermal LBM I am aware of: one based on changing the lattice scheme itself, and the other that makes use of an additional lattice to solve the conductive thermal equation. If it is the former (thermal lattices), the speed of sound should indeed be a direct function of temperature, which is one of the properties that comes out of summed moments of distribution functions for those lattices.

If it is the latter (it uses an additional lattice for heat transfer), the fluid is normally still technically isothermal. However, if a multiple phase model with a dependence on temperature is also applied (e.g. Shan/Chen pseudopotentials or a free-energy approach), the equation of state will change and because the speed of sound is equal to the square root of the derivative of pressure w.r.t. density, this property for the fluid will change. However, the sonic speed of the lattice itself will not change from cs = 1/sqrt(3).

I hope this helps clear things up.

Regards,

Michael

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@Mikeoas: Thanks for the reply ! I’m still a little bit confused. I’m using the model of Shi and al here “http://journals.aps.org/pre/abstract/10.1103/PhysRevE.70.066310” which i don’t know according to your description is the first or the second model. What do you mean by “it uses an additional lattice for heat transfer”? Do you mean that it uses for example D2Q4 model for temperature and D2Q9 model for momentum (two different types) ? . In Shi’s article, the lattice is the same but there is an additional distribution function “g” for temperature.

My question is : According to you, in Shi’s model, the speed of sound must be function of T ? Can you please give me any reference for further readings?