Steady Euler Solutions and the Effects of Dissipation
 

Hello all,

This being my first post, please don't rake me over the coals too much. I am an experienced structural dynamicist, but am relatively new to CFD. I have spent a lot of time the past few days reading posts on this forum and am already grateful for the help and advice that the community has offered. Any further help would be greatly appreciated.

I have am working on updating an existing coupled CFD-CSD (computational structural dynamic) code for the analysis of transonic flutter of airplane wings. The code uses the Euler equations discretized using finite elements over an unstructured tetrahedral mesh. Artificial dissipation is included via the method of Jameson (http://aero-comlab.stanford.edu/Pape...euler_1981.pdf).

Here is some quick background information on how this type of dissipation. There are two parameters that govern the dissipation: k2 and k4. k2 affects the dissipation in areas where pressure gradients are large (near shocks and stagnation points) while k4 affects areas where gradients are smaller. The terms are turned on and off using a pressure switch function. The ratio of k2/k4 defines a large pressure gradient.

Though the main focus of work is unsteady flows with a moving boundary, I am currently having a problem with the steady solution that is used to define the initial conditions to the unsteady problem. My solutions are not converging to steady state. By adjusting the dissipation parameters, I am able to produce solutions that fall into tow categories:

1. Solutions that begin to converge, but are eventually affected by reflections from the boundaries of the domain. This occurs when k4 is too small.
   
2. Undesirable oscillations on the wing surface near shock locations or stagnation points. This occurs when the ratio k2/k4 is too small.

The problem appears to be that as k4 is increased to damp reflections in the far field, the pressure switch magnitude is lowered, and k2 is essentially turned off in regions of the flow where it is needed. This results in oscillations at the leading edge stagnation point or on the wing upper surface where the shock is attached. Are there other pressure switch criteria that have been used in the past? I've considered changing the criteria, but I wanted to get the communities opinion first.

I am focusing on dissipation parameters, but if other areas of my analysis are in need of improvement/investigation, please don't be afraid to say so. Below are some plots, pictures, and videos that may help.

Mesh (attach wall and wing shown, cylindrical far field and opposite wall not shown):
http://s16.postimg.org/q651pr2wl/ONERA_mesh.png

Solutions are for ONERA M6 wing at Mach 0.84, Alpha=3.06. k2 dissipation parameter is 0.30.

Solution 1 - Small k4 (0.005) leads to boundary reflections
Convergence of aerodynamic parameters:
http://s8.postimg.org/nk74nqo85/Page..._25_k4_005.jpg
Convergence of residual:
http://s8.postimg.org/6iealn9d1/Page..._25_k4_005.jpg
Video:
http://www.youtube.com/watch?v=lHZX9AGpNdI

Solution 2 - Increased k4 (0.020) damps boundary reflections, but larger pressure switch value causes loss of dissipation at leading edge resulting in oscillations. Oscillations cannot be seen in video, but are present at a few locations on wing leading edge.
http://s29.postimg.org/ijoj6pvg7/Pag...time_plots.jpg

http://s29.postimg.org/6gj7j5kdz/Pag...time_plots.jpg

http://www.youtube.com/watch?v=sASSoF54z9Q

There can be many issues.
i) Using TVD schemes may help. As it uses the concept you are trying to use through pressure switch.
ii) Use unsteady equation with false transient concept to get steady state.
iii) Also, the boundary conditon has to be dealt in special manners, concerning hyperbolic system of equations. We also have non-reflecting type of boundary conditions.

You can go through CFD by Anderson book to understand these. And then search literature in these areas. There are many.