Speed of sound

what expression connects speed of sound in LBM c[sub]s[/sub] = 1/sqrt (3)=0.577:

  1. with real speed in air c[sub]ph[/sub]=333 m/s?
  2. with formal speed of sound in the Navier-Stokes model of incompressible fluid where c[sub]inc[/sub] = infinity?

Hi Kaff,

Your question is related to the one posted by Spartan, and I will therefore answer both at the same time. To summarize, Spartan was wondering if one needs to compute the density rho for a BGK collision step, even in case of an uncompressible fluid in which rho is supposedly constant.

Lattice Boltzmann can be used to solve the incompressible Navier-Stokes equations as well as some versions of the compressible Navier-Stokes equations. The former case is much more frequent though, and the latter (compressible) case requires careful considerations and often the use of a more complicated collision term than BGK.

As far as solutions to the incompressible Navier-Stokes equations are concerned, common LB models like BGK are part of the family of quasi-compressible fluid solvers. This means that although they aim at solving incompressible fluid flow, they enter a compressible regime (but stay always close to the limit of incompressibility). This allows them to solve the pressure equation “on the go” by solution of the continuity equation. Unlike other solvers, they don’t need to solve a non-local Poisson equation for the pressure, which can provide a big computational advantage. In practice, this means that the density changes in space and time, because it reflects pressure variations (in BGK, the relation between pressure and density comes from the ideal gas equation of state p=c_s^2 rho = 1/3 rho). So much for Spartan’s question: yes, rho needs to be computed anyway, and included into the collision step. However, in order to recover an asymptotically incompressible fluid flow, you must make sure that the Mach number (the fluid velocity divided by the speed of sound) is sufficiently small. Fortunately, the error due to fluid compressibility effects is known to decrease as the square of the Mach number, which allows you to control this source of error without needing to use too small velocities (you can find a more detailed discussion of this issue in my thesis[/url]). In practice, people often use a velocity around u=0.02, as measured in lattice units, for systems of size around 200x200 or so. To be sure your Mach number is small enough, simply decrease the velocity a bit, and verify that you still obtain the same result, within the accuracy you wish to obtain. When doing this, be sure to perform the correct unit conversions so that you model the same physics in both simulations (a discussion of unit conversions is found in [url=http://www.lbmethod.org/howtos:main]the LBMethod.org document on unit conversions).

The BGK model can also be used to simulate compressible fluid flows, but only at small Mach number. It cannot represent velocities close to or superior to the sound speed. If this is what you want, you need more advanced models like the one described in Shan e.a., 2006[/url], with an extended neighborhood. If low Mach number is fine with you, be aware of other limitations such as the impossibility to adjust the bulk viscosity (which may be overcome for example with P. Dellar’s model). If this is still OK, go ahead and use BGK. To convert the speed of sound to physical units, simply restore its units in the same way you would restore the units of any other quantity, knowing that it has the dimensions of a velocity (dx/dt). See again the paper on unit conversions mentioned above. You don’t really have the option of adjusting the speed of sound to its physically adequate value, though. Indeed, for setting up a simulation, you only have two degrees of freedom, which are the space step dx, determined by the choice of the grid resolution, and the time step dt, which is determined by the choice of the Mach number. For some purposes, it may be irrelevant if the speed of sound is right or not, and you may find it sufficient to have a simulation which gets the Mach number right. If on the other hand you insist on having an additional degree of freedom on the speed of sound, BGK is insufficient for you. There exist several LB models with adjustable speed of sound, such as the one presented in the book by Chopard and Droz and summarized in the [url=http://www.lbmethod.org/openlb/techreports.html]OpenLB technical report TR2.