Bow Shock

References

We did not use any specific reference to develop this test problem. This particular problem was developed for Athena3D by Jake Simon and David Nidever (University of Virginia).

Description

This problem is initiated by a low density gas surrounding a spherical region of very high density. The low density gas is moving to the right at a supersonic velocity (except for behind the high density sphere). When this high velocity gas hits the stationary sphere, a bow shock forms. This problem tests the algorithm's ability to handle a stationary shock that is not aligned with the grid in all directions. The shock should also be symmetric about the x axis, which makes this a good symmetry test. Finally, this problem can test the robustness of the hydrodynamics solver used because of the factor of 10,000 density jump at the interface between the low- and high-density gas.

Set up

Domain -1.0 ≤ x ≤ 1.0, -1.5 ≤ y ≤ 1.5 Boundary conditions The outer x boundary and both y boundaries have outflow conditions. The inner x boundary has the ghost zone fluid values set by the innermost physical zone. Since the initial condition in the innermost physical zone is that of a supersonic flow to the right, the inner x boundary condition is inflow. Equation of state Adiabatic with γ = 5/3 Initial density ρ = 100.0 for r ≤ 0.0625 and ρ = 0.01 for r > 0.0625 where r = [(x+0.75)2 + y2]1/2. Initial pressure P = 0.0015 everywhere Initial velocity There is a uniform supersonic x velocity in all regions except for within the overdense region and to the right of this overdense region. Specifically, vx = 0 for the region defined by r ≤ 0.0625 and also defined by x > -0.75, |y| ≤ 0.0625. Everywhere else, vx = 1.0. MHD Components There is no MHD version of this test.

Results

We ran the simulation on a grid of Nx = 800 and Ny = 1200. Plotted below are logarithmic (base 10) density maps (the logarithm of the density is needed, given the huge range in density values). Higher values are redder. The left image occurs early on in the evolution (time 1.0) when the bow shock is still forming. The right image is the final frame (time 10.0). In both images, the logarithm of density ranges from -2.4 to 2.4. Click on the right image to see an animation of the density logarithm over the whole simulation (size of animation: 13 MB).

Summary The evolution shows that the shock forms quickly around the small high density region and then remains relatively stationary for the duration of the simulation. It also maintains good symmetry about the x axis. Note that the low-density gas does push some of the high-density gas off of the sphere, creating tails. This process also serves to flatten the sphere. The tails are not exactly symmetric about the x axis even though the bow shock and flattened sphere maintain good symmetry. The reason for this behavior is currently unknown. The algorithm evolves the system without crashing, showing that it can handle the strong density contrasts and supersonic velocities present.

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