# Fluid Density

Dear all

How the density of the flow, (for example 1000 kg/m^3 for water) is implemented in the equations?

Kind Regards

I too have the same questions re LBM and further I am unsure as to how to set the density difference in MCMP type flows. Can anybody elucidate this point?

I suppose you have to scale density with a reference density, rho0, say. So your âLBâ density might be rhoLB=rho/rho0. If the fluid is incompressible, rhoLB will always be 1. If the density may fluctuate slightly then rhoLB will be âcloseâ to 1.

The LB density is related to the LB pressure by a (somewhat non-physical) ideal equation of state, pLB=âRTâ*rho, where pLB is the LB pressure (ie in lattice units - I say âsomewhat unphysicalâ because this standard definition depends on the scaling used). âRTâ is the gas constant in lattice units and usually taken to be 1/3.

The dimensional pressure, p, is related to the LB pressure by an appropriate numerical scaling like

pLB=p/(rho0*c^2)

where c=dx/dt is the grid, or particle, speed (dx=space-step, dt= time-step).

You can also relate the LB pressure to the dimensionless pressure P by, for example,

P=p/(rho0*U^2)

where U is the characteristic velocity of the flow.

You may want to make sure you have the correct value of a non-dimensioanl number, such as the Reynolds number. This would usually involve the kinematic (nu), or dynamic (mu) viscosity. So, depending on which of these you have/need, you may have to adjust the relaxation time tau accordingly (because nu=mu/rho, and thus Re=UL/nu=rhoU*L/mu, where nu~tau/3)

Good luck!

I think I understand how to scale from the âlattice unitsâ to physical quantities but Iâm still struggling to set up this MCMP as I wish. According to this paper we have

nu = (1/3)(tau-0.5) dx^2/dt

Now I can use this to set the kinematic viscosity of a fluid by setting tau (for a given lattice) or use it to calculate what physical time a single iteration represents. I wish to simulate an air/water system so after calculating dt based on nu for water it works out that for the air (kinematic viscosity =15 @ 300K) that tau is something ridiculous like 1.8E6. What can one do to simulate these mixtures?

Also can I just check, is the âomegaâ from the examples the same as âtauâ or is it something different?

Thanks

Perhaps itâs because youâve forgot the units and exponent of viscosity? At 300K, nu~15*10^-6 m/s^2 for air. So if you had a mesh with 101 points in every (unit) characteristic length so that dx=1/100 and wanted dt=dx^2 (diffusive scaling), ie c=10000, then tau~0.50005.This is close to the stability bound so you might want to increase the Ma number and add more points. For example, dx=1/1000, c=10. This would give tau~0.5045. If this were a single phase flow then the Re number would be large and an MRT scheme recommended to improve stability and efficiency.

Which omege are you referring to?

You are quite correct I missed the order of magnitude off (doh!). So I think the following is correct. For water phase we have nu = 1E-6 m^2/s, I have dx = 1E-4 m, and I choose tau=0.505 then using the formula above means each iteration is dt = 1.66âŚE-5 s of real time.

Now for the air phase I have nu, dx and dt specified so I use the same formula but this time to work out tau which gives tau = 0.575.

In the examples Iâve looked at there is a variable called omega which is passed into the ExternalMomentRegularizedBGKdynamics variable, am I right in thinking this is the relaxation time (aka tau)?

I donât know what âExternalMomentRegularizedBGKdynamics variableâ mean (if itâs something to do with palabos or pre-written codes I canât help as I donât use them); but people often use omega for either the relaxation time or the relaxation frequency (1/tau).

Yes itâs from the multicomponent2d example code from the palabos library. I think itâs just tau.

Is what I said regarding the tau for MCMP correct? Iâm finding that if they are too different the simulation becomes unstable, at the moment I am trying tau(water)=0.6 ==> tau(air)=2.0

If you have a question about palabos libraries then you could post it on one of the other forums on this webpage.

I donât know much about MCMP models but what you said seems correct: dx and dt are fixed, unless you have a non-uniform mesh, so you can easily find tau from nu. I imagine large density ratios are common problems for this model, especially if its based on Shan-Chen type LBEs. There is a viscosity jump, which has to be dealt with somehow. The model is also only first order accurate in time, so will probably require a lot of points and a small dt for large viscosity differences.

Hi
I did not get my answer!

In most references, the density is calculated as:

rho=summation of particle distribution function.

But the density of the desired flow (for example 1000 kg/m^3 for water) is not included.

Regards