Correction for spiral channels helix angle

[dug20]

A correction needs to be made for the heat transfer rate into the channels for spiral channels. I have added this in the '0.2.1' branch, by using a 'corrected' coolant mass flow rate when calculating temperature rises. Instead of:

dQ = mdot cp dT

We have to do:

dQ = mdot cos(helix angle) cp dT

This has quite a significant impact on the temperature rise of the coolant. There has also been a corresponding correction made to the heat transfer from the fins, however this correction seems to have had a very small effect - it would be worth checking it in detail at a later date.

[dug20]

After further review this theory does not look correct, and it seems like the original way the code was formatted should be right (the total mass flow rate should be used for the temperature rise).

We are still attempting to validate the code, in particular there is a large discrepancy with data on the Vulcain nozzle extension (which uses spiralling channels - the light blue line is Bamboo):

But data from a NASA dump-cooled test case (also with spiralling channels) has the opposite issue:

[dug20]

Maybe there are compressible flow effects we're ignoring? Right now we don't distinguish between stagnation temperature and static temperature, but both these cases use gaseous hydrogen travelling at very high velocities (the NASA case seems to suggest the hydrogen hits 800 m/s with our current code, which is around M = 0.67 for hydrogen).

[dug20]

The issue was that the freestream gas temperature was used instead of the recovery temperature, with the convective heat transfer correlations.

The recovery temperature is defined as [1] [2]:

And for 'turbulent free boundary layers' (which are common in rocket engine exhausts), r = Pr^(1/3). I have assumed that Pr here is evaluated at the mean of the wall and freestream temperatures.

This is then the temperature used with the convective heat transfer equation, i.e. q = h(T_r - T_w). This has solved the discrepancy in the Vulcain nozzle extension data: