Hi everyone, as Andrew mentioned I am the other collaborator on this project. I wanted to note some analysis that I did on a pattern we saw in the paint after the flight. There was a short boat tail on the aft end that was painted green, and the paint was applied without primer two days before the launch, so it did not have the full adhesive strength as the white paint applied to the rest of the rocket.
After the flight, we noticed that there were some interesting triangular patterns on the boat tail behind the fins where some of the paint was removed:
Those marks are probably either regions of high turbulence in the wake of the airfoil (vorticity/recirculation=paint removal) or the trailing edge shock wave. I would guess the latter because the wake doesn’t fan out at an angle like that.
Examining a diamond airfoil, we have the following scenario. I pulled this from someone’s CFD results of a foil with half angle 10 deg at mach 3:
https://iopscience.iop.org/article/10.1088/1742-6596/2272/1/012003/pdf
From this photo, I measure the included trailing edge shock angle to be approximately 27 degrees:
The angle of the first oblique shock can be calculated analytically assuming a weak shock and a theta of 10 degrees (in the above case) from a freestream mach number of 3. From here the shock angle beta and post shock mach number (conditions in the red zone) are computed from the oblique shock relations.
In this zone, the local mach number is lower than that of the freestream. As the flow crosses the mid-chord of the foil, it undergoes a prandtl-meyer expansion through an angle equal to twice the foil half angle. The mach number downstream of the expansion (blue zone) is computed using the prandtl meyer function. In this region, the local mach number is higher than that of the freestream.
Finally, due to the symmetry of the foil and assuming no angle of attack, the flow undergoes another compression at the trailing edge (with the creation of another oblique shock) at a half angle equal to the foil half angle. The same oblique shock relations are used as before to find the shock angle, but with two modifications. First, the “upstream” pre-shock mach number is not equal to the freestream mach, it is the higher value (blue zone) calculated from the PM function. Secondly, the shock angle beta is not the same as the geometric half angle measured from the photo because the shock angle is relative to the top surface of the foil, not the horizontal. So, the foil half angle must be subtracted from beta to get the shock half angle. When I do this for the CFD case above (mach 3, half angle 10 degrees), I get an included angle of 28.4 deg which is very close to the crude measurement of the CFD result.
Running the calculation for a 5 degree foil half angle at mach 3 gives an included angle of 32.8 deg, which compares to the 26.5 deg that I measured from the rocket. Lots of sources of error including angle of attack deviation, changes in the shock angle with mach number, 3D effects of finite span etc, so it seems to be in reasonable agreement given those factors.
