Hello All
In many papers in LBM, we see that gamma (ratio of specific heat) is defined by: gamma=(D+2)/D, and sometimes some authors directly use gamma=5/3 or 7/5 etc. and sometimes no need to use this.
Now, my question,
(i) when we use it and which value we should take for our parobelm.
(ii) if gamma=(D+2)/D, then whats the value of D? suppose, in d2q9 model, D=2 or 9??
Inform me Please. May it will help all new LBM researchers.
With all the best
Sincerely yours
MAT
Hi,
In many cases, I don’t think the precise value of gamma matters too much. In LBE you are usually simulating Navier-Stokes type flow,near incompressibility and vanishingly small mean free path. The D2Q9 LBE is also athermal. gamma is relevant when the mean-free-path and rarefaction are relevant. This relates to the Knudsen number, and then to tau in the LB model. That is, we can say (von-Karman’s relation)
Kn=sqrt(pi*gamma/2)*Ma/Re
where Kn is the Knudsen number, Ma the Mach number and Re the Reynolds number:
Ma=U/sqrt(gammaRT), Re=rhoUL/mu
and mu is the (dynamic) viscosity. The pressure is given by the equation of state P=rhoRT and we usually have RT=1/3 for the D2Q9 models. For an ideal, monotomic gas, gamma=5/3. Note that this is based on the mean free path being defined by lambda=musqrt(piRT/2)/P, because, Kn=lambda/L. Thus, gamma is effectively absorbed into tau.
The value gamma=7/5 is for diatomic molecules. The transport equation for these requires and energy equation as well as mass and momentum equations.
The ‘D’ in D2Q9 (D=2) refers to spatial dimensions of flow and the ‘Q’ (=9) is the dimensions of the discrete velocity space. However, I’m not sure if the formula you have for gamma is valid (the D is D2Q9 is for spatial dimensions in the equations of motion, whereas I think the D is the formula you have is related internal degrees of freedom of a real gas).