The Navier-Stokes Equation

  Equation (3.26) is the continuity equation for a real fluid. Equation (3.27) is similar to the Navier-Stokes equation for a real fluid. We need to see if a fluid satisfying (3.27) satisfies the Navier-Stokes equation.

The equations for lattice gas hydrodynamics are only valid for , the speed of sound of the model, since we have assumed this in their derivation. In this regime we can follow the approach of Frisch et al. [8] and consider the density to have a constant value . The density is, however, allowed to fluctuate where it appears in the pressure term. This gives

 

and

 

for , and .

This equation differs from the standard Navier-Stokes equation in the following ways:

1. the nonlinear term is multiplied by a model and density-dependent function, ;
2. the viscosity is a function of density;
3. the pressure term has an extra term which is dependent on the density and the velocity.

It is possible to re-scale certain variables in equation (3.33) and to redefine the pressure in order to produce the Navier-Stokes equation. This can be done in two ways:

1. Following Frisch et al. [8] define

   

   

   and

   

2. Alternatively, following Wolfram [24], we can set

   

   and

   

In both cases equation (3.33) gives the Navier-Stokes equation in the rescaled variables

and

For both scalings we can define the Reynolds number for lattice gas simulations to be

 

where U and L are representative velocity and length scales of the flow and is the kinematic viscosity.