Spherical Blast Wave

References

Zachary, Malagoli, A., & Colella,P., SIAM J. Sci. Comp., 15, 263 (1994) Balsara, D., & Spicer, D., JCP 149, 270 (1999) Londrillo, P. & Del Zanna, L., ApJ 530, 508 (2000)

Description

This problem is initiated by an overpressured region in the center of the domain. The result is a strong outward moving spherical shock with rarified fluid inside the sphere. The blast wave should remain perfectly spherically symmetric at early times. Eventually, the periodic boundaries lead to interactions with neighboring blast waves, leading to complex shock-shock and shock-contact discontinuity interactions. These interactions should also create the Richtmyer-Meshkov instability in the very low-density region in the center of the grid (hydro case only). The "fingers" produced by this instability should be perfectly symmetric top-to-bottom and left-to-right.

Set up

Domain 2D: -0.5 ≤ x ≤ 0.5, -0.75 ≤ y ≤ 0.75. 3D: -0.5 ≤ x ≤ 0.5, -0.5 ≤ y ≤ 0.5., -0.5 ≤ z ≤ 0.5. Boundary conditions Periodic everywhere Equation of state Adiabatic with γ = 5/3 Initial density ρ = 1 everywhere Initial pressure P = 0.1 for r ≥ 0.1 and P = 10.0 for r < 0.1, where r = [x2 + y2]1/2. Initial velocity Zero everywhere MHD Components 2D: The MHD version of this test has Bx / (4π)1/2 = 1/(2)1/2 and By / (4π)1/2 = 1/(2)1/2. 3D: The MHD version of this test has Bx / (4π)1/2 = 1/(3)1/2, By / (4π)1/2 = 1/(3)1/2, and Bz / (4π)1/2 = 1/(3)1/2.

Results

2D Hydro We ran the 2D simulations on a grid of Nx = 512 and Ny = 768. Plotted below are linear density maps (higher values are redder). The left image is a frame of the hydro run at an early time of 0.2 with density ranging from 0.08 to 2.9. The right image is a frame from this simulation at time 3.75 with density ranging from 0.1 to 2.2. One can see the Richtmyer-Meshkov instability has produced a complex pattern of "fingers". Click on the right image to see an animation of the density over the whole simulation (size of animation: 18 MB).

2D MHD This simulation was run with the same parameters as the one described above, but with a magnetic field added in the x and y directions with Bx / (4π)1/2 = 1/(2)1/2 and By / (4π)1/2 = 1/(2)1/2. The images below are again linear density maps, shown at the same evolution times as for the hydro case. The left image has density ranging from 0.1 to 2.9, and the right image has density ranging from 0.1 to 2.3. The right image links to an animation of the density over the whole simulation (size: 23 MB)

3D simulation results coming soon.... Summary In the hydro case, the shock wave maintains spherical symmetry until it interacts with neighboring shocks from across the periodic boundaries. The only exception to this maintenance of symmetry are small grid-aligned features that appear early on in the evolution. These effects are due to the carbuncle instability and can be removed with a proper fix to the algorithm. Later in the evolution, the Richtmyer-Meshkov instability becomes apparent, and the border between the low- and high-density regions of the resulting "fingers" remains sharp. For the MHD simulation, the fluid is aligned with the magnetic field from early on in the evolution. Eventually, given the periodic boundary conditions, a sort of "wrapped" fluid structure appears. The magnetic field is evidently strong enough to maintain this diagonal structure for a long time. Furthermore, the Richtmyer-Meshkov instability is suppressed by this field, and no "fingers" form.

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