Mach Stems

2D Stationary Clumps

This problem set up has bow shocks from two identical, stationary clumps. I test different gammas and different separation distances. To reduce computation, I am only using one clump with a reflecting wall on the lower r boundary to simulate the second clump's bow shock. I also only simulate ¼ of the clump due to symmetry. In the context of this problem, critical separation distance is equivalent to 2 times the critical radius.

Thinking in terms of critical radius versus critical angle can get confusing. Traditionally, we think of mach stem formation in terms of critical angle. Anything above this angle means a mach stem will form. The critical angle corresponds to a critical radius, but now we must remember that anything below this radius means a mach stem will form.

Below is a table of the resolutions used for all of the runs. These were determined by keeping the proportions of the domain constant and shooting for a minimum of 32 cells per clump radius. As a result of doing it this way, some runs will have higher resolution, but it basically stays between 32 and 38 cells per clump radius. For the 'lower r boundary' column, crit = critical, acrit = above critical, bcrit = below critical. Mach stems should form at crit and bcrit, but not at acrit.

gamma	lower r boundary	Base Grid (r,z) [cells]	Domain Size (r,z) [cu]	Cells / Rclump
1.6667	crit	12, 18	1.46, 2.19	32.88
1.6667	acrit	14, 21	1.66, 2.49	33.73
1.6667	bcrit	12, 18	1.26, 1.89	38.10
1.4	crit	12, 18	1.39, 2.085	34.53
1.4	acrit	14, 21	1.59, 2.385	35.22
1.4	bcrit	10, 15	1.19, 1.785	33.61
1.2	crit	12, 18	1.32, 1.98	36.36
1.2	acrit	14, 21	1.52, 2.28	36.84
1.2	bcrit	10, 15	1.12, 1.68	35.71
1.1	crit	12, 18	1.27, 1.905	37.80
1.1	acrit	12, 18	1.47, 2.205	32.65
1.1	bcrit	10, 15	1.07, 1.605	37.38

gamma = 1.6667

left boundary	Image	Movie
bcrit		movie​
crit		movie​
acrit		movie​

gamma = 1.4

movie​

3D Runs

Simulation Setups

All simulations are done in a reference frame moving at M = 7. Simulation times are shown in computation units, but the total physical time is approximately 75 years.

2-clump setup (runs A - F): Clumps separated by a distance d measured in clump radii. Clump velocities given in terms of their Mach numbers.

3-clump setup (runs G - L): See diagrams below. The clumps are labeled by their Mach numbers. The clump separation d (measured in clump radii) and angle a are the defining parameters that change for each run. The point labeled O marks the location of the observer from which a line of sight is drawn to generate the emission maps. The position of the observer is determined by two angles: the rotation angle phi is denoted in the top view diagram, and the inclination angle i is denoted in the side view diagram.

multi-clump setup (run M): In x and z, the clumps are arranged in a 3-4-3 lattice-like structure. Each clump is given a random position in y and an appropriate velocity based on that y-position so that the clump does not run off the grid during the simulation.

Run	# Clumps	Defining parameters	Machine	Frames (out of 50)	Emission Angles (out of 9)
A	2	d = 2.5, M = 7, 10	Gordon	50	9
B	2	d = 4.5, M = 7, 10	Gordon	50	1
C	2	d = 6.5, M = 7, 10	Gordon	50	0
D	2	d = 2.5, M = 7, 15	Stampede	50	6
E	2	d = 4.5, M = 7, 15	Stampede/BlueStreak	50	4
F	2	d = 6.5, M = 7, 15	Stampede/Comet	50	8
G	3	d = 2.5, a = 30	BlueHive	50	9
H	3	d = 2.5, a = 60	BlueHive	50	9
I	3	d = 2.5, a = 90	BlueHive	50	9
J	3	d = 4.5, a = 30	BlueStreak	50	9
K	3	d = 4.5, a = 60	BlueStreak	50	9
L	3	d = 4.5, a = 90	BlueStreak	50	9
M	10		BlueStreak	50	9

Individual Runs

A

B

C

D

E

F

G

H

I

J

K

L

M

Separation Comparisons

A/B/C

D/E/F

G/J

H/K

I/L

Orientation Comparisons

G/H/I

J/K/L

Velocity Comparisons

A/D

B/E

C/F