Mach Stem Runs

Following work done by Kris, (see ​Mach Stems), I redid some of my old calculations.

The formulas Kris and I used in our calculations were for bow shocks from jets, but they should be applicable to bow shocks formed by a wind and clump interaction.

The theory gives a critical angle based entirely on . Mach stems should form at angles equal to and greater than the critical angle. This does not depend on how the bow shock is formed…

critical angle (deg)
5/3 37.4
1.4 41.7
1.2 47.7
1.1 53.5

The set up I am doing right now is for two stationary clumps. Their symmetric bow shocks will interact exactly in between the clumps and either form or not form a mach stem. The angle formed between the bow shocks decreases with increasing separation distance. So if the critical angle is a minimum for mach stem formation, then critical separation distance is a maximum.

Based on the formula for the shape of a bow shock, you can get a critical separation distance for each critical angle. Note that the separation distance is measured from clump center to clump center. These are given in units of clump radii…

critical angle (deg) critical separation (rclump)
5/3 37.4 2.76
1.4 41.7 2.64
1.2 47.7 2.52
1.1 53.5 2.45

Since I am simulating one of the bow shocks with a reflecting wall, my domain has to be half the size of this critical separation. I think it will suffice to do two runs for each gamma: one above critical and one below. Here are my proposed runs:

Run critical separation (rclump) simulation separation (rclump) Mach Stem?
A 5/3 2.76 3.036 no
B 5/3 2.76 2.484 yes
C 1.4 2.64 2.904 no
D 1.4 2.64 2.376 yes
E 1.2 2.52 2.772 no
F 1.2 2.52 2.268 yes
G 1.1 2.45 2.695 no
H 1.1 2.45 2.205 yes

If I use 4 levels of AMR and aim for at least 64 cells per clump radii, I can use these resolutions:

Run Mach Stem? Domain Size (x, y, z) Base Grid (mx, my, mz)
A no 1.518, 3.036, 0.759 8, 16, 4
B yes 1.242, 2.484, 0.621 6, 12, 3
C no 1.452, 2.904, 0.726 6, 12, 3 ←— lowest physical resolution at 66 cells/rclump
D yes 1.188, 2.376, 0.594 6, 12, 3
E no 1.386, 2.772, 0.693 6, 12, 3
F yes 1.134, 2.268, 0.567 6, 12, 3
G no 1.3475, 2.695, 0.67375 6, 12, 3
H yes 1.1025, 2.205, 0.55125 6, 12, 3 ←— highest physical resolution at 87 cells/rclump

Comments

No comments.