Although I have read very good explanations about unit conversion from Physical system to LB system, specially the paper by Jonas Latt posted in this forum, I have still some doubts about converting the body force Fx.
I just want to summarize the information from the simulation I am trying to carry out:
Channel flow with some cavities driven by a constant body force included in the equilibrium.
The physical values (subscript p):
Pr= 0.72 (air at 0°C)
nu_p= 1.3324 x10-5 [m^2/s]
h_p= 0.005 [m] ------> height of the cavity
Dh_p= 6*h_p [m] ------> hydraulic diameter
l_p= 0.22 [m] ------> length of the channel
From the Reynolds number, which is approximated as Re= (u_p * Dh_p)/nu_p , one gets the velocity u_p= 0.0444 [m/s].
With these parameters I can already do the conversion as follows:
I want to have a resolution of -----> h_p (0.005 m) / # of cells (20)= 0.00025 [m], which is actually dx= 0.00025 [m].
Now I calculate the LB viscosity (nu_LB) from the following equation, imposing a fixed relaxation time 1/tau= 1.98:
nu_LB= (tau - 0.5)/3 = 0.001683
with dx, nu_LB and nu_p, it is possible to calculate the time (dt) using the equation:
dt= (nu_LB/nu_p)dx^2= (0.001683/1.3324 x10-5)(0.00025)^2 ----> dt= 7.89694x10-6 [s]. This means that I require around Niter= 1/dt= 126631 iterations for simulating 1 s. (I hope this last is correct).
Now one can calculate the lattice velocity u_LB using:
u_LB= u_p * (dt/dx)= 0.0444 * (7.89694x10-6 / 0.00025) ------> u_LB= 0.0014
Now the question is how to know the body force required in this problem in order to get a Re=100 under the conditions explained above.
Does anybody know how to relate the body force or pressure drop with the Reynolds number? It think if this possibility exists, then I can know both the physical and LB body forces by means of dx, dt, nu_LB, etc.
I followed a paper (Guo et al. 2002, Physical Review E 65 046308) where the pressure gradient was expressed as
dp/dx= -rho*G_p = Fx_p [kg/m^2 s^2],
where rho is density and G_p is the acceleration due to the body force density Fx_p. The equation for the maximum velocity in a Poiseuille flow is given as
uo= (G_pL_p^2) / 2nu_p —> with L_p being half width of the channel
If we consider u_p as uo we could relate the Re to G_p as follows:
Re= (uoDh_p)/nu_p= (G_p Dh_p^3)/(2*nu_p^2) or in terms of the body force:
Re= (Fx_pDh_p^3)/(2rho*nu_p^2) !!!
Is this last possible?, I am a bit skeptical
I will be very very grateful for your help.