# Incompressible or isothermal

When using a D2Q9 LBGK with the local Maxwellian distribution function expanded up to second order, am I correct to say that it in principle, it simulates an isothermal flow and that it only approximates incompressible flow due to the small Mach number expansion as for Mach < 0.3 the density is nearly constant?

I ask this, because I would like to simulate flow over an airfoil and calculate lift and drag from it, so I need the pressure distribution. If density is indeed varying, then could I calculate the pressure distribution from the isothermal relation
p = cs[sup]2[/sup] rho
where cs[sup]2[/sup] = RT is the isothermal speed of sound and the density is known from the simulation?
Actually what happened to gamma, the ratio of specific heats that also appears in the speed of sound in ‘classical’ aerodynamics?

Elrohir

Dear Elrohir,

You are right about restoration but not right about connection with physics. The speed of sound 1/3 is obtained through the discretization of the velocity space and is not related to the real speed of sound. It happened that this coefficient goes with gradient of density and we can think in terms of macroscopic equations as pressure = c_s^2 \rho. That means you need to be really careful when you connect your physical reality with LBM one. Usually it’s done through non-dimensional parameters. Those are that make sense. Some parameters in LBM are just coming from discretization but are not connected with physics.

Cheers,
Alex

Hello Alex,

What exactly do you mean by ‘restoration’?

About the lattice speed of sound, I understand that it is obtained in the first place through the lattice velocities that are used in the LBM. But I read in Succi, ‘The Lattice Boltzmann Equation for Fluid Dynamics and Beyond’ that it actually does relate to the physical sound speed via Cs(physical) = cs(lattice)* delta x / delta t
Just to organise my thoughts, does it follow from cs[sup]2[/sup] = RT that p = rho RT = rho cs[sup]2[/sup] or rather the other way around?

Hi Elohir,

I have a few thoughts on your original post. The LBM may be used to simulate both incompressible and (weakly) compressible flow.

As you are working on an incompressible problem, the speed of sound is not relevant and may be seen as an artefact in the method. Since each node is linked only to its neighbours when streaming particles, it necessarily takes some time for information to propagate from one point in your simulation to another. (For a perfectly incompressible “real” fluid, the speed of sound would be infinite and this information propagation would occur instantaneously.)

For incompressible flows, the lattice speed of sound may thus be seen as the speed of information propagation in the model instead of a physically relevant speed of sound. If your simulation reaches a steady state, the information does not propagate around any more, and the “speed of sound” is no longer relevant at all.

On the other hand, the full capacity of the LBM is as a solver for compressible flow (I can give references if you like), although with error terms which grow with Ma[sup]3[/sup]. These error terms mean that you’re restricted to small velocities, or weak compressibility. Compressible effects at low fluid velocities include acoustics, which works reasonably well in the LBM. If you go this path, the speed of sound is physically relevant and becomes an additional restriction when choosing your units.

Anyway, the density will be varying along your airfoil as you suggest. I don’t think you need to know the physical pressure along the surface, only that it is proportional to the density. So, if you know the density in the far field from the airfoil in the simulation, you may take that as your atmospheric pressure, and find the pressure along the airfoil from the ratio between the density there and the density far away. Mathematically, p / p_0 = rho / rho_0, where each side of the equation may be in lattice or physical units.

I hope that helps,
Erlend M

Thanks Erlend, that confirms what I understood about incompressible flow simulation from what I’ve read in literature. Indeed, as you suggest, by using p / p[sub]0[/sub] = rho / rho[sub]0[/sub] does not require the speed of sound anymore.

However, for my thesis report, I would like to give a clear description of the parameters that normally appear in the equations and relate them to the more familiar thermodynamic variables, as LBM is still quite new to my department.

I’m very interested in references on compressible flow, so if you could direct me to some of those I’d gladly read them. By the way, you mention an error that grows with Ma[sup]3[/sup]. Does this have to do with the small velocity expansion used in e.g. the equilibrium distribution of the LBGK or are there other sources that contribute to this as well?

Sure, I can give you some quick references. The compressible Navier-Stokes equation is derived from the LB dynamics in Jonas Lätt’s PhD thesis [1], and the same derivation is done slightly more explicitly in my own Master’s thesis [2]. Paul Dellar wrote a good article [3], which derives the LBM and its correspondence with the compressible Navier-Stokes equation from kinetic theory.

The last article states that the O(Ma[sup]3[/sup]) terms may be eliminated by expanding the equilibrium distribution and using an extended lattice. You might want to look at the articles it refers to for that, I haven’t had the time to do so myself.

There are more ways of removing these terms. For instance, Prasianakis & Karlin wrote an article [4] where they derive an extended LBM which conserves momentum and energy without these error terms. Their method does not need to use a more complicated lattice. In a later article [5], they show that this method gives good results for a number of compressible problems also.

Hope that helps!
Erlend M

Thanks Erlend for these references. I will definitely have a look at them.

Elrohir