Simplest way to "upgrade" from Euler equations to Navier-Stokes equations in FV or FD framework

Question

I have quite a lot of experience solving unsteady Euler equations, including multi-component ones, with in house-coded finite-difference and finite-volume methods, including MacCormack and MUSCLE schemes and WENO flux reconstruction. Now I'm considering "upgrading" to laminar NS equations for compressible gases (i.e. without any turbulence model).

Is it generally considered a hard switch? What are the most complex parts to understand/implement regarding the difference between Euler and NS equations?

What would be the simplest way to incorporate viscosity in FV formulation? Same question with FD formulation.

What would be the simplest way to incorporate diffusion and heat conductivity?

In general, I'm looking for some good detailed literature/tutorials for NS equations similar to LeVeque's books (which are only for Euler equations, as far as I know).
EDIT: added info about compressible equations and comment on "laminar" term as  Spencer Bryngelson suggested in comment.

EMP · Answer

Use the alpha-damping formulation for the viscous fluxes, very simple to implement.

ConvexHull · Answer

Generally the step from compressible Euler equations to the Navier-Stokes equations is not that hard, at least the coding part.

If you want to implement it with an explicit scheme you have to consider the severe time step restriction of the parabolic contributions.
One tricky part, at least for a consistent FV implementation, is the calculation of the tangential gradients on the faces. These can't be calculated directly with the cell averages, especially on Cartesian meshes. Here you may have to use a Gauss-Green method.
Since you mentioned multi-component systems: Most CFD solvers only consider three parabolic contributions, the Fourier law, the Fick law and the Stokes law. However with multi-component systems other physical effects might become relevant. The effects are based on the Onsager reciprocal relations called, e.g. Dufour and Soret effects.

Regards

Simplest way to "upgrade" from Euler equations to Navier-Stokes equations in FV or FD framework

2 Answers

Add your own answers!

Ask a Question