LBM Units

Lattice Boltzmann Method LBM: Theory
#1

HI,
I read this small article “Choice of units in lattice Boltzmann simulation” but stil I don’t get few things…

OK, I undersatnd that using Re number It’s a good idea for some test cases, but I need a little more universal way. In complicated systems it’s often dificult to set characteristic length and velocity. So my quastion is can I use such a way:

For simplicity let’s say that I have Lid driven cavity problem with additional gravity force:
So I have domian with


L{phys} = 0.1 [m]
N = 300 

dx = L/N = 3.333e-4

let’ assume that:


dt = 1e-3

then lattice constant c:


c = dx/dt
Cs = c/sqrt(3)

Viscosity:


vis{Phys} = 2e-6 [m^2/s]
vis{LBM} = vis{Phys} * (dt/dx^2)

Then parameters for simulation:


tau = (3 * vis{LBM})/c^2 + 0.5  =  0.97317 //relaxation time
ux{Phys} = 0.01
ux{LBM} = ux{Phys} * (dt/dx)     //top wall velocity for boundary conditions
gy{LBM} = -9.81 * (dt^2/dx)      //gravity force

I’m using this parameters {LBM} in the simulation. I’m computing equilibrium dystribution function with lattic constant c as falows:


c = dx/dt

Also discreet velocity set “e[sub]i[/sub]” is multiply by c

When I want to get a physical velocity in each cell then:


vel{Phys} = sqrt( (ux * (dx/dt))^2 + (uy * (dy/dt))^2 )
time{phys} = time{actual} + dt

And the problem…
Alwas when I’m changing time step dt (and everything else is the same) I get completly different results for given time{phys}.
Mayby I don’t get it… can I treat a lattice speed c as a scalling factor when dt<>dx ?

I’ll be greatfull for any help
Mike

#2

Hi,

When you try to verify your calculation. Did you compare the different results with the same physical time? Because you try to use different time step dt. So when compare the result, you need to calculate the physical time from the dimensionless time. I don’t know this is your problem or not. How difference of your result for the different dt right now?

Because you change the dt => you change the c => you change the Mach number

May be the difference is from the compression.

Lei.

#3

Hi.
I’m simulating thermal flow in a 2-d micro-couette using LBM. I have a serious problem with thermal unit conversion in LB.
I want to reproduce the results of the DSMC method in (XIAO-JUN GU et. al. “A high-order moment approach for capturing non-equilibrium phenomena in the transition regime”, J. Fluid Mech. (2009), vol. 636, pp. 177–216. c ,Cambridge University Press 2009, doi:10.1017/S002211200900768X) Part 6. Results and discussion, for thermal-micro-couette flow in the transition regime (namely, Knudsen number Kn>0.1). But I don’t know how to convert the values of the wall temperature (Tw) and the gas constant ® to lattice units.
as we know the sound speed in the LBM is Cs=C/SQRT(3) and C=SQRT(3RT). Then what should I do with the value of the R and T in the calculation of the C.
Please help me. I’m working on my thesis and I’m in a bad hurry.
I appreciate any help in advance.

#4

Hello,

The best way is to start with nondimensional numbers which govern problem physics and only after that match your LB parameters. What are the nondimensional numbers which describe the problem?

Cheers,
Alex