Hi,
When I read the book and the papers about the LBM, all of them mentioned that c_s^2=1/3. I know it comes from the discretization of velocity space. But Is there any paper explicating the discretization details? Could you please give me some hints and some links of the references papers. Thank you. I am really appreciate for your help.
Glary
Hi,
I believe that the speed of sound comes from the conditions of lattice isotropy (section 2.1.1 in [1] and 4.3 in [2]). The only solution of these conditions for a D1Q3, D2Q9, D3Q15, or D3Q19 lattice is a set of lattice weights t_i and the speed of sound c_s^2 = 1/3. This means that this is the only possible value of the speed of sound which fulfils the isotropy conditions.
This speed of sound is then put into the equilibrium distribution equation (eq. 1.18 in [1] and section 4.4 in [2]), which I believe determines the actual speed of sound used in the LB model. I’m a little unclear on exactly why the equilibrium distribution determines the speed of sound, but I believe that it follows from the distribution’s derivation (which can be done from the Maxwell-Boltzmann velocity distribution).
I’d be really interested to know if my assumptions here are correct.
Now, if you actually measure c_s in the results of a lattice Boltzmann simulation, you get slightly different values. The theoretical value is 0.577, but I’ve measured values ranging from 0.576 to 0.607 [2]. While I didn’t know during the writing of my master’s thesis why this happened, I now believe that this is due to the dispersive effect of viscous media on monofrequency wave propagation (see for instancesection 8.2 in [3]).
Erlend M
[1] Jonas Lätt’s PhD thesis
[2] Erlend Magnus Viggen’s master’s thesis (that’s me)
[3] Kinsler, Frey, Coppens, and Sanders, “Fundamentals of Acoustics”, 4th ed
Dear Erlend,
This speed of sound is then put into the equilibrium distribution equation (eq. 1.18 in [1] and section 4.4 in [2]), which I believe determines the actual speed of sound used in the LB model. I’m a little unclear on exactly why the equilibrium distribution determines the speed of sound, but I believe that it follows from the distribution’s derivation (which can be done from the Maxwell-Boltzmann velocity distribution).
The speed of sound is determined by the Hermite quadratures. When you want to have the exact conservation laws, you want to have the same moments as for the continuous Maxwellian distribution function but only in the discrete space. That can be fullfilled for the Gaussian-Maxwellian types of functions, i.e. the integral of \int{exp(-u^2/2T) some coefficients here du}=\rho=\sum_i f_i. It is fullfilled only for certain velocities c_i in the discrete space. For the mentioned models, those velocities have the amplitude \sqrt{3} . During the renormalization procedure that exactly gives you the speed of sound.
Hopefully it’s clear,
Alex
Dear Glary,
there is a special article on the LBM derivation, where you’ll find exactly what you need: http://www.lbmethod.org/literature:he_97
Hope this helps,
Vasili