Lattice Boltzmann Methods - Derivation of Navier-Stokes Equation From Discrete LBE

Derivation of Navier-Stokes Equation From Discrete LBE

Starting with the discrete lattice Boltzmann equation (also referred to as LBGK equation due to the collision operator used). We first do a order Taylor series expansion about the left side of the LBE. This is chosen over a simpler order Taylor expansion as the discrete LBE cannot be recovered. When doing the order Taylor series expansion, the zero derivative term and the first term on the right will cancel leaving only the first and second derivative terms of the Taylor expansion and the collision operator.

For simplicity, write as . The slightly simplified Taylor series expansion is then as follows where ":" is the colon product between dyads.

By expanding the particle distribution function into equilibrium and non-equilibrium components and using the Chapman-Enskog Expansion where is the Knudsen number, the Taylor expanded LBE can be decomposed into different magnitudes of order for the Knudsen number in order to obtain the proper continuum equations.

The equilibrium and non-equilibrium distributions satisfy the following relations to their macroscopic variables. These will be used later once the particle distributions are in the 'correct form' in order to scale from the particle to macroscopic level.

The Chapman-Enskog Expansion is then:

.

By substituting the expanded equilibrium and non-equilibrium into the Taylor expansion and separating into different orders of, the continuum equations are nearly derived.

For order, :

For order, :

Then, the second equation can be simplified with some algebra and the first equation into the following.

Applying the relations between the particle distribution functions and the macroscopic properties from above, the mass and momentum equations are achieved.

The momentum flux tensor, has the following form then.

Where is shorthand for the square of the sum of all the components of (i.e. ) and the equilibrium particle distribution with second order in order to be comparable to the Navier Stokes equation is:

.

The equilibirum distribution is only valid for small velocities or small Mach numbers. Inserting the equilibrium distribution back into the flux tensor leads to:

Finally, the Navier-Stokes equation is recovered under the assumption that density variation is small.

This derivation follows the work of Chen and Doolen.