Axisymmetric near-axis fixes using L’Hôpital-consistent formulations

[raghava-davuluri]

Proposed Changes

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• Improve axisymmetric source-term behavior near the symmetry axis by smoothly transitioning between the standard v/r formulation and a L’Hôpital-consistent ∂v/∂r formulation as r→0.
• Apply a L’Hôpital-consistent treatment for axisymmetric auxiliary gradient terms at the axis instead of zeroing them.

Related Work

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

• Addresses nonphysical behavior observed in axisymmetric stagnation-region solutions where the 1/r terms become singular near the axis. This PR was prepared for review/discussion with SU2 developers.
• The standard formulation is preserved away from the axis; the near-axis modification only affects the r→0 limit.

PR Checklist

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• I am submitting my contribution to the develop branch.
• My contribution generates no new compiler warnings (try with --warnlevel=3 when using meson).
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• I used the pre-commit hook to prevent dirty commits and used pre-commit run --all to format old commits.
• I have added a test case that demonstrates my contribution, if necessary.
• I have updated appropriate documentation (Tutorials, Docs Page, config_template.cpp), if necessary.

[bigfooted]

Hi @raghava-davuluri, thanks for this fix! Does this case need additional regression tests or is it covered by the existing regression tests for axisymmetric flow?
At the moment we have these testcases with axisymmetry, not too much so maybe add an additional one that you've worked on:

axisymmetric_rans/air_nozzle/air_nozzle_restart.cfg:16:AXISYMMETRIC= YES
flamelet/05_laminar_premixed_ch4_flame_cfd_axi/lam_prem_ch4_cfd_axi.cfg:29:AXISYMMETRIC= YES
nonequilibrium/visc_cylinder/cyl_ion_gy.cfg:14:AXISYMMETRIC= YES
nonequilibrium/visc_wedge/axi_visccone.cfg:18:AXISYMMETRIC= YES
py_wrapper/turbulent_premixed_psi/psi.cfg:67:AXISYMMETRIC= YES
vandv/species_transport/sandia_jet/validation.cfg:58:AXISYMMETRIC= YES

[Hanquist]

This is addressing #2438

It helps in most situations, but dips in axisymmetric heating cases can still be present for some meshes. So @raghava-davuluri is doing a pull request to help get more eyes on it.

[Hanquist]

@raghava-davuluri @bigfooted, reading up on this more, L'Hopital's rule may be more suitable for cell-centered and less so for vertex-centered codes. And that all the residuals should still be zero at the symmetry line. Here is an example of the axisymmetric sources along the surface. I would expect the energy and momentum-x to smoothly go to zero at theta=0 instead of a discontinuity