An Explosion Problem inside a Tube

The following example is a two-dimensional version of an example in three dimensions suggested by Langseth and LeVeque.

The setup is as follows (see the figure). A circle with radius 0.2 is centered at (0.0,-0.1) in a tube with solid walls at y=+/-0.5. The gas is initially at rest and has unit density everywhere. Inside the circle the pressure is 5.0 and outside it is 1.0. Due to symmetry, we use computational domain [0.0,2.0]x[-0.5,0.5.0] with a reflective boundary at x=0.

The solutions are computed on a 400x200 grid. To reach final computational time 1.0 we have used 400 equally spaced time steps. We present a Schlieren movie of the density, including the symmetric part. (The images have been reduced by a factor 2 in each direction). A full resolution movie is also available, which takes longer to down-load and runs slower on the computer.

The circular high pressure region results in a strong outward moving shock wave and contact discontinuity and an inward moving rarefaction wave. The outer shock reflects at the lower boundary. Before that the solution is cylindrically symmetric and afterwards we have symmetry around the plane x=0. The rarefaction wave implodes on the origin and produces a second outward moving shock wave. This shock interacts with the reflected shock, and so on. As a special feature of the solution, note the low density region at the origin. (It can be seen by the increase in wave speed as a shock moves through it).