Heat flux in axisymmetric simulations

[Hanquist]

When running axisymmetric simulations, we get poor heat flux predictions (and trends) near the leading edge:

Zoomed in:

The symmetric (analogous to an infinite cylinder) shows the correct trend where the surface heating is highest at the stagnation point (x = 0). The axisymmetric case (analogous to a hemisphere) shows a significant decrease in heating at this stagnation region.

If you look at the flow field below where y>0 is the symmetric case (infinite cylinder) and y<0 is the axisymmetric case, you will oberve that there is too much thermal diffusion near the symmetry line. Note that these are two different simulations and I mirrored the axisymmetric case so we can see them on the same figure. This thicker "thermal boundary layer" is not physical and creates the smaller heat flux value and poor trend seen above:

Discussing with @bigfooted and @pcarruscag, we think it may have to do with the axisymmetric source terms. Interestingly enough we do not see this artifact when running NEMO simulations so it is handled better there. @WallyMaier @jtneedels @CatarinaGarbacz do you recall doing anything special in the NEMO implementation of axisymmetric source terms?

[bigfooted]

were there experiments or simulations from other codes for this or a similar testcase?

[pcarruscag]

@Hanquist please upload any SU2 files that can be used to replicate the results.

[Hanquist]

Hi @bigfooted, these run conditions are arbitrary (Mach ~6 at 25 km altitude). We have gotten good agreement with experimental data for the Standard SU2 solver for full 3D shapes (see Figs. 18 and 19 here: https://arc.aiaa.org/doi/epdf/10.2514/6.2025-2769) and good agreement with experiments/other codes for axisymmetric NEMO (see Figs. 9 and 10 here: https://arc.aiaa.org/doi/epdf/10.2514/6.2022-1636). After we get this sorted out, we can add a tutorial/testcase for aerodynamic heating.

Hi @pcarruscag, we are attaching the mesh and cfg files. This is a coarse mesh but I would still not expect to see the trends shown above. The only difference in the cases above is:

AXISYMMETRIC= YES/NO

AeroHeatingAxiBug.zip

[CatarinaGarbacz]

@Hanquist I'm not aware of the details for axisymmetric source terms, it would probably be worth looking into it if NEMO looks ok

[bigfooted]

From the book of Hoffmann and Chiang, vol II:

[Hanquist]

The only difference I notice between how NEMO handles the axisymmetric formulation and the standard solver is how they handle the singularity at the symmetry line:

NEMO (nemo_sources.cpp):

 if (Coord_i[1]!= 0.0) yinv = 1.0/Coord_i[1];
 else yinv = 0.0;

then the jacobians is a multiple of yinv. For example:

jacobian[iVar][jVar] *= yinv*Volume;

Standard solver (flow_sources.cpp):

 if (Coord_i[1] > EPS)
...
else {

    for (iVar=0; iVar < nVar; iVar++)
      residual[iVar] = 0.0;

    if (implicit) {
      for (iVar=0; iVar < nVar; iVar++) {
        for (jVar=0; jVar < nVar; jVar++)
          jacobian[iVar][jVar] = 0.0;
      }
    }

Since EPS is pretty small, I would expect these to be consistent with each other, unless more than the symmetry line is being captured by this. I don't think this is the reason for us because we are still using a course mesh. But I could see both of these formulations being a challenge if for some crazy reason someone did an axisymmetric flow not in the tradiational horizontal plane.

As I write this, I also see a potential difference in thermal conductivity:

NEMO:

const su2double heat_capacity_cp_i = V_i[RHOCVTR_INDEX]/rho + Ru/Mass;

Standard:

Thermal_Conductivity_i = V_i[nDim+6];

[bigfooted]

There is even still an issue in the regular compressible solver.
I noticed that discrepancies on the symmetry axis are probably due to the function
CSourceAxisymmetric_Flow::ComputeResidual
You can test this with a simple case, like the Testcases/navierstokes/cylinder testcase.
With AXISYMMETRIC=YES, it leads to an accumulation of energy on the symmetry axis. You can also switch to the EULER solver, the problem is still there. If you put the source terms in CSourceAxisymmetric to zero, there are no discrepancies at the symmetry axis.

[bigfooted]

I am still confused about the axisymmetry problem. Here's a zoom of the leading edge of a NAVIER_STOKES flow around a sphere (2d axisymmetric). Density and energy show the biggests issues. incompressible flow looks fine for a similar setup. Density is clearly not behaving properly on the symmetry line. No improvement with the new Jacobian fix.

[Hanquist]

I have had a chance to look at this a little closer. We have been running a lot of 3D heat transfer cases as well, and have been getting an unrealistically high heat flux at the stagnation point:

The 3D in this image was ran with SU2 8.0. When we run it with SU2 8.1, this high jump goes away. We are running more cases on this and will start a new thread on that to show more details.

But this gave us the idea to look at different versions of SU2 and how they handle the axisymmetric heating:

Versions 7.0 and 7.5 have the same prediction. There is a small dip but I believe this magnitude of dip is acceptable and could be improved with grid quality, etc. The 8.1 dip is massive though. So something happened between 7.5 and 8.0/8.1..

Here is the temperature contour near this point where the top is SU2 version 7.5 and bottom is 8.1: