Hydrodynamics Shock Tubes

These five shock tube tests are from Toro's book (Toro, 1999, p.225) and use the Euler Equations with dimensionless variables, and are listed in Table 2.1. They have varying levels of difficulty: test 1 is a modified version of Sod's test; test 2 is a strong double rarefaction; test 3 has a very strong shock; test 4 has three strong discontinuities; and test 5 is the same as test 3 except that the contact discontinuity is stationary. Results are shown in Fig. 2.1; the solid black line is the solution with 10,000 cells, and the lines with points are the solution obtained with 200 cells. All runs use a CFL number of $0.7$ and the artificial diffusion prescription of Falle (1998) which includes some shear and bulk viscosity to the momentum fluxes, using a viscosity parameter of $\eta _{\mathrm {v}}=0.05$. This only noticeably affects Test 2 by helping the calculation at $x=0.5$; without added viscosity there is a spike in the internal energy due to small errors in the (very small) density and pressure. It also fixes any entropy glitches in rarefactions. It is interesting to note that the shock and contact are much better resolved for Test 5 than for Test 3 because they are almost stationary with respect to the grid. This is a feature of shock-capturing schemes; adding extra diffusion would mitigate this.

Table 2.1: Shock Tube Test Problems. For these hydro tests, the three state variables are density, pressure, and normal velocity, $[\rho ,p_g,v_x]$, and all tests use an adiabatic equation of state with $\gamma =1.4$. Results are obtained using a second order accurate (in space and time) integration with a Courant (CFL) number of 0.7 and an artificial viscosity parameter $\eta _{\mathrm {v}}=0.05$.

Test	Left State	Right State	$t_f$	$x_0$
Toro 1	$[1,1,0.75]$	$[0.125,0.1,0.0]$	0.2	0.3
Toro 2	$[1,0.4,-2]$	$[1,0.4,2]$	0.14	0.5
Toro 3	$[1,1000,0]$	$[1,0.01,0]$	0.012	0.5
Toro 4	$[5.99924, 460.894, 19.5975]$	$[5.99242, 46.095, -6.19633]$	0.05	0.3
Toro 5	$[1,1000,-19.59745]$	$[1,0.01,-19.59745]$	0.012	0.8

Figure 2.1: 1D shock tube results for adiabatic hydrodynamics. The solid line shows the results using 10,000 grid cells, and the points show results using 200 cells. The figures show results for Toro's tests 1-5 from top to bottom (see Table 2.1).