Maximum Angle for Mach Stem Formation

After last week's conversation with Pat, I started testing his calculations for the maximum Mach stem formation angle. My recent paper on Mach stems verified the minimum angle. If we look at both equations graphically, we can define a region where Mach stem formation is possible:

Below is a table of the critical angles for a few cases of gamma.

gamma Min Angle Max Angle
5/3 37.38 41.81
1.4 41.65 58.05
1.2 47.67 69.73
1.01 68.73 85.92

For my paper, I had graphically determined the shape of the bow shock for a few cases of gamma (could not do this for gamma = 1.01). This allowed me to convert minimum angles into maximum separation distances.

I used Pat's formula to determine minimum separation distances. If the clumps are closer than this minimum distance, the bow shock becomes a smooth continuous shape with no Mach stem. Below is a table of critical separation distances in units of clump radii.

gamma Max Separation Min Separation
5/3 12.47 10.79
1.4 6.20 3.62
1.2 4.58 1.81

Note that the gamma = 1.2 case has a minimum separation distance < 2 which cannot be tested. A separation < 2 means that the clumps are overlapping.

So I plan on testing the gamma = 5/3 and 1.4 cases. I have a set of 8 runs queued up as follows:

run gamma separation Mach stem?
A 5/3 13 No, above maximum
B 5/3 12 Yes
C 5/3 11 Yes
D 5/3 10 No, below minimum
E 1.4 6.5 No, above maximum
F 1.4 5.5 Yes
G 1.4 4 Yes
H 1.4 3 No, below minimum

Below is what I have so far for run H. It seems consistent with what we were predicting, but we need the other runs to determine if Pat's formula is correct.

movie​

Pat also wanted to look at lateral Mach number plots. I generated a couple snapshots using two different scales:

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