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

@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?

Thanks in advance .

Hi dhouha,

Shi’s model is definitely the second of the two types I described: it uses a separate lattice grid to determine energy/thermal flows, while the first grid is used to solve for fluid flow only. The lattices can use the same number of lattice links, but they are effectively solved separately from each other, even if there is some form of coupling between them. (The first model - thermal lattices - uses extra links in a single lattice to give additional degrees of freedom to permit a definition of temperature based on extra moments of the distribution function.)

There is a distinction between the lattice speed of sound and the physical speed of sound for the second model. The speed of sound for D2Q9 lattices will always be 1/sqrt(3) in lattice-based units, but the physical speed of sound is the square root of the derivative of pressure with respect to density at constant entropy. The latter will depend on the equation of state for the fluid. By default, the equation of state for a lattice gas is p = rho * c_s^2 = 1/3 * rho (so the speed of sound would be c_s = 1/sqrt(3)) and this would still apply even if the thermal field were applied. However, if you apply a different equation of state to the lattice using some form of fluid interaction that also includes a temperature dependence, the physical equation of state (but not the lattice one) will alter according to the equation of state you apply.

Regards,

Michael