Fix axisymmetric divergence for SST production

[bigfooted]

Proposed Changes

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the axisymmetric terms in the production term of the SST model are not correct and the current implementation is prone to error.

Related Work

Resolve any issues (bug fix or feature request), note any related PRs, or mention interactions with the work of others, if any.
#1783

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[pcarruscag]

Actually, can you elaborate on why the strain magnitude doesn't need adjustments?
Contrary to cartesian coordinates, a constant velocity in the radial direction causes some compression/expansion of the fluid and thus there should be a (v / y)^2 missing.
Disclaimer: I am not a maths person and my reference is for solid mechanics 😆

[bigfooted]

Actually, can you elaborate on why the strain magnitude doesn't need adjustments? Contrary to cartesian coordinates, a constant velocity in the radial direction causes some compression/expansion of the fluid and thus there should be a (v / y)^2 missing. Disclaimer: I am not a maths person and my reference is for solid mechanics 😆

Ah yes, I erroneously thought: 2D means only considering the r and z-directions and considered the 2D strain tensor S_ij. But you are right, the stress tensor component $S_{\theta \theta} = \frac{u_r}{r}$ contributes to the total strain magnitude.

[bigfooted]

OK, should this term be added to CFVMFlowSolverBase<V, R>::ComputeVorticityAndStrainMag? There is a call to GetStrainMag in SST and in SA, and in SetRoe_dissipation_NTS.

[pcarruscag]

Good point, I think we can include the axisymmetric term there. And there is something else... That strain magnitude is including the divergence and so it's closer to the norm of the stress tensor than to the norm of the stress tensor, and thus it is not consistent with the literature, again... 🤦

[bigfooted]

yes, I see it. Do you know why this is done? I think we should remove the Div term to make it compatible with literature. This will then probably change all turbulence regression tests.

[bigfooted]

The divergence term comes from the idea to write strain rate as [part without volume change] + [part with volume change]:
https://www.whoi.edu/cms/files/12.800_Chapter_3_%2706_25330.pdf (p 13)
Not sure what the advantage is of writing it like that in the code since the divergence terms should cancel again. I'm also not sure why we only add the divergence term when nDim==2. It looks wrong to me.

In any case,magnitude of strain rate is $S=\sqrt{2S_{ij}S_{ij}}$, which is, written out for 3D Cartesian coordinates,

$\frac{1}{2}S^2 = S^2_{11} + S^2_{22} + S^2_{33} + S^2_{12} + S^2_{21} + S^2_{13} + S^2_{31} + s^2_{23} + S^2_{32} $

It's symmetric, so:

$\frac{1}{2}S^2 = S^2_{11} + S^2_{22} + S^2_{33} + 2S^2_{12} + 2S^2_{13} + 2s^2_{23}$

and using $S_{ij}=\frac{1}{2}(\frac{\partial u_i}{\partial x_j} + \frac{\partial u_j}{\partial x_i})$, and $\frac{\partial u}{\partial x} = u_x$ etc. :

$\frac{1}{2}S^2 = u_x^2 + v_y^2 + w_z^2 + 2(\frac{1}{2}(u_y+v_x))^2 + 2(\frac{1}{2}(u_z+w_x))^2 + 2(\frac{1}{2}(v_z+w_y))^2$

$\frac{1}{2}S^2 = u_x^2 + v_y^2 + w_z^2 + \frac{1}{2}(u_y+v_x)^2 + \frac{1}{2}(u_z+w_x)^2 + \frac{1}{2}(v_z+w_y)^2$

For axisymmetry, all the terms with velocity component w and the gradients in z are zero, except that $w_z=\frac{v}{r}$

[pcarruscag]

Nice cleanup of the options 👍