Double Mach Reflection

References

"The Numerical Simulation of Two-Dimensional Fluid Flow with Strong Shocks", by Woodward, P.R., & Colella, P., JCP 54, 115 (1984)

Description

The problem is initialized by sending a shock wave diagonally into a reflecting wall. The system is equivalent to a shock that is moving horizontally and encounters a wedge that is inclined by some angle. In our simulation, the wedge is represented by a reflecting boundary starting some point along the x direction (see the description of the boundary conditions below). As the shock reflects off the lower wall, a jet of denser gas forms.

Set up

Domain 0.0 ≤ x ≤ 4.0, 0.0 ≤ y ≤ 1.0 Boundary conditions The inner x boundary is simply an "inflow" condition, in which the ghost zone fluid values are set by the initial conditions in the post-shock region. The outer x boundary is a simple outflow condition. The lower y boundary is constructed to mimic the wedge that the shock is being driven into. For x ≥ x0, this y boundary is a reflecting wall. For x < x0, the lower y ghost zone fluid values are set by the initial post-shock conditions. Here, we take x0 to be 1/6. The upper y boundary is constructed to follow the flow of the diagonal shock such that there is no interaction between the shock and this boundary. Given the initial condtions here, the intersection of the diagonal shock and the upper boundary at time t occurs at xs(t) = x0 + (1+20t)/31/2 (see "Fluid and Shock Properties" below). For x ≥ xs(t), the upper y ghost zone values are set by the initial pre-shock conditions. For x < xs(t), the ghost zone values are set by the initial post-shock conditions. Equation of state Adiabatic with γ = 1.4 Initial density The shock is initially set up to be inclined at an angle of 60º to the x-axis, and has a Mach number of 10. Therefore, the initial density is 8.0 for x < x0 + y(1/31/2) and is 1.4 for x ≥ x0 + y(1/31/2) Initial pressure P = 116.5 for x < x0 + y(1/31/2), P = 1.0 for x ≥ x0 + y(1/31/2) Initial velocity vx = 8.25 cos(30º) for x < x0 + y(1/31/2) vy = -8.25 sin(30º) for x < x0 + y(1/31/2) All velocities are zero for x ≥ x0 + y(1/31/2) MHD Components There is no MHD version of this test. Fluid and Shock Properties Given the initial conditions described above, the sound speed of the pre-shock gas is 1, and the shock speed is 10. Given the geometry, the intersection of this diagonal shock with the upper y boundary moves at a speed of 10/cos(30º). The movement of this intersection is the reason for the changing upper y boundary condition at the position xs(t) = x0 + (1+20t)/31/2 as described above. This shock is driven by a contact discontinuity that is initialized with velocity 8.25 pointing downward with respect to the horizontal by 30º.

Results

We ran the simulation on a grid of Nx = 1200 and Ny = 300. Plotted below are linear density maps. The top map occurs early in the simulation (t = 0.05). The density ranges from 1.4 to 21.3 in this image (higher values are redder). The lower map occurs at the final time of 0.25, with density ranging from 1.4 to 22.74. Click on the lower image for an animation of the density (size: 5.3 MB).

Here are linear color maps of the pressure, plotted at the same times as the corresponding density plots above. The top image has pressure ranging from 1 to 542.3, and the bottom image has pressure ranging from 1 to 557.8. The bottom image is an animation of the pressure over the whole simulation (size: 4.9 MB).

Finally, we plot density contours (30 levels) near the end of the simulation (t = 0.2) to compare with the plots of Woodward & Collela (1984). Note that we have only plotted up to x = 3 here for an easier comparison.

Summary The jet forms and grows as we expect. The contour plot of the density agrees well with the figures in Woodward & Collela (1984), though there are some differences. One major difference is the region of low density fluid that appears to be sort of "swept up" as the complex system moves to the right (see animation for example). When run at lower resolution, this feature goes away, and the corresponding density contour plot is closer to the plots in Woodward & Collela (1984).

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