Table of Contents

*Main author: Erlend Magnus Viggen*

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In Jonas Lätt's PhD dissertation, he derives that the basic lattice Boltzmann method's dynamics give a behaviour consistent with his formulation of the compressible Navier-Stokes equations with one constant set to 0. But this formulation is different than the usual formulation of compressible Navier-Stokes. We show here that the two formulations are equivalent.

Due to limitations of this wiki's math formatting systems, some quantities have to be written differently. The operator and are vectors, while is a matrix.

The compressible Navier-Stokes equation with no external force (Wikipedia et al):

The compressible Navier-Stokes equation with no external force (Lätt, page 7):

We make no assumptions yet about what the viscosities and represent. After all, they are merely constants in one equation, which we are trying to show that is equivalent with another equation with other constants.

We rewrite the viscosities in Lätt's formulation from kinematic viscosities () to dynamic viscosities (). The dynamic viscosities are assumed a constant material property.

We rewrite the terms in Lätt's formulation of the equation. This is done by using index notation, but we will only write the results here.

which, using the continuity equation, becomes

We insert the rewritten terms into Lätt's formulation to get:

This looks pretty good already. The three first terms are identical to the ones in the standard formulation. The fourth term has a different name for the constant. The fifth term is dissimilar in the parenthesis, but let's take a look at our and viscosities.

For the fourth term to match up, we see that . Therefore we see that must be the dynamic shear viscosity and the kinematic shear viscosity .

Now we take a look at the fifth term to determine what is. For the two equations to match up, we demand that . This means that , giving us

which is the formulation that we have been trying to reach. Success!

To sum up, the two formulations of the compressible Navier-Stokes equation can be shown to be equivalent when and and all these are assumed constant.

This conclusion might be considered puzzling, as Lätt says in his thesis that is to be considered the bulk viscosity. But this is not wrong, it is just one of the many definitions of bulk viscosity. It could be useful to take a look at a paper written by Paul Dellar on bulk viscosities in lattice Boltzmann, where he talks a little about the differences in bulk viscosity terminology. At least the introductory chapter is highly recommended to read.

This paper also uses a formulation of Navier-Stokes similar to Lätt's, but Dellar initially defines the bulk viscosity in a way that is consistent with the definition in Wikipedia et al.'s formulation.