Change dynamic viscosity in FD hybrid central / upwind blending to kinematic viscosity

[clarkpede]

Proposed Changes

In both the original paper on DDES and the paper by @EduardoMolina on hybrid RANS/LES in SU2, the calculation for r_d uses kinematic viscosity. In SU2, the dynamic viscosity is currently used. This leads to poor behavior if density is not close to 1. This PR exchanges the dynamic viscosity for the kinematic viscosity.

Related Work

This is part of the changes originally proposed in PR #552

PR Checklist

• I am submitting my contribution to the develop branch.
• My contribution generates no new compiler warnings (try with the '-Wall -Wextra -Wno-unused-parameter -Wno-empty-body' compiler flags).
• My contribution is commented and consistent with SU2 style.
• I have added a test case that demonstrates my contribution, if necessary.

[clarkpede]

The behavior can be seen in the backward facing step case, as shown in a snapshot below. As currently set up, Roe_Dissipation is equal to f_d from the original DDES paper. So Roe_Dissipation should:

• Go to 0 in attached boundary layers
• Go to 1 in regions with resolved turbulent fluctuations

Here's a comparison of the Roe_Dissipation before and after this PR:

Here's the same case, but zoomed in at the step:

With this PR, the model does indeed go to 0 near the walls and go to 1 in the highly unsteady ("LES") region after the step.

[vdweide]

I don't know too much about this topic, but the first thing that comes to my mind when I read this is dimensional consistency. So did somebody do a proper dimensional analysis on the implementation?

Edwin

[clarkpede]

@vdweide It's easy to do, so I'll show the dimensional analysis here.

I'll use L for length, T for time, and M for mass.

The original paper defined r_d, which is supposed to be a nondimensional parameter. (r_d gets raised to the 3rd power and put into a hyperbolic tangent, so it would be messy if it were dimensional).

In the code you'll see:

 r_d = (nu + nu_t)/(uijuij*k2*pow(val_wall_dist, 2.0));

where nu is the kinematic viscosity, nu_t is the kinematic eddy viscosity, uijuij is the magnitude of the velocity gradient tensor, k2 is the square of the von-Karman constant, and val_wall_dist is the wall distance.

• nu has units of L^2/T
• nu_t has units of L^2/T
• uijuij has units of 1/T
• val_wall_dist has units of L

So if you do the math, r_d has units of L^/T in the numerator and L^2/T in the denominator. r_d is nondimensional.

If you switch nu and nu_t from kinematic viscosities to dynamic viscosities, you get that r_d has units matching density, or M/L^3. That's obviously not correct.

[EduardoMolina]

LGTM and ready to merge.

[vdweide]

Merging things in. I kept the branch, but delete it if you don't need it anymore.