Implement fast recycling

[mikekryjak]

Turns out fast recycling has a big impact. This PR does a simple implementation of this:

$$\Gamma_{E_{n}} = R \times (R_{f} \alpha_{E} \Gamma_{E_{i}}^{sheath} + (1 - R_{f}) T_{R} \Gamma_{N_{i}})) $$

Where $R$ is the recycle multiplier, $R_{f}$ is the fast reflection fraction, $\alpha_{E}$ is the energy reflection factor,
$\Gamma_{E_{i}}^{sheath}$ is the incident heat flux from the sheath boundary condition, $T_{R}$ is the recycle energy and $\Gamma_{N_{i}}$ is the incident ion particle flux.

This is implemented for the SOL and the PFR and is fully compatible with the neutral pump.

In addition, the example cases haven't been corrected to reflect the new recycling flags - this is now done in this PR for all examples.

Tasks:

• Implement for ylow and yup targets
• Test
• Run case for complex test
• Make compatible with neutral pump
• Add documentation

[mikekryjak]

There is an issue here. The formula for the energy of recycled neutrals is using a weighting of temperature, not heat flux. Yes, the returning thermal neutrals are all 3eV and their heat flux calculation is therefore very straightforward. However, the ions are a drifting maxwellian going through sheath_boundary.cxx. I have to use the ion heat flux as calculated there. I think the easiest way would be to store the sheath heat flux somehow and pick it up in fast recycling.

Probably the easiest way would be to add it to EnergyFlow_d+_ylow which already exists and is the total species heat flux by definition.

[mikekryjak]

Corrected the formulation to use the actual ion energy flows leaving the model, extended to SOL/PFR, fixed some bugs and improved code clarity. Documentation next

[mikekryjak]

Commit fe6161c causes a segmentation fault in the 1D-recycling integrated test.

[mikekryjak]

@bendudson ready for review. Unfortunately it seems to arbitrarily fail the documentation build test again. I have no idea why since all I've changed is some text... it would be good to finally get this merged and reduce the amount of live PRs. Do you think the issue is with the test or with the documentation itself?

This is the error:

Configuration error:
There is a programmable error in your configuration file:

Traceback (most recent call last):
File "/home/docs/checkouts/readthedocs.org/user_builds/hermes3/envs/176/lib/python3.11/site-packages/sphinx/config.py", line 358, in eval_config_file
exec(code, namespace) # NoQA: S102
^^^^^^^^^^^^^^^^^^^^^
File "/home/docs/checkouts/readthedocs.org/user_builds/hermes3/checkouts/176/docs/sphinx/conf.py", line 152, in
import sphinx_rtd_theme
ModuleNotFoundError: No module named 'sphinx_rtd_theme'

[bendudson]

Thanks @mikekryjak ! Readthedocs is acting up at the moment, will sort out at some point.