1D HD Test: Sod’s Shock Tube

This is the 1D shock tube problem [Sod, 1978]. Because the initial condition has a discontinuity, a shock wave, a contact discontinuity, and a rarefaction wave are generated. This problem is often used to verify numerical schemes that solve the fluid equations.

The test problems provided in MISO run the 1D problem in each of the \(x\), \(y\), and \(z\) directions, but this page describes only the problem in the \(x\) direction.

Location

demo/hd1d_shock_tube/

Geometry

\(0 \leq x \leq 1\).

Initial Conditions

The initial condition is described by the left and right states separated at \(x=0.5\). The ratio of specific heats is \(\gamma = 1.4\).

\[\begin{split} \begin{align*} \begin{pmatrix} \rho_\mathrm{L} \\ p_\mathrm{L} \\ v_\mathrm{L} \end{pmatrix} &= \begin{pmatrix} 1.0 \\ 1.0 \\ 0.0 \end{pmatrix} \\ \begin{pmatrix} \rho_\mathrm{R} \\ p_\mathrm{R} \\ v_\mathrm{R} \end{pmatrix} &= \begin{pmatrix} 0.125 \\ 0.1 \\ 0.0 \end{pmatrix} \end{align*} \end{split}\]

Boundary Conditions

A symmetric boundary condition is imposed on all variables.

Results

You can visualize the results with Python and compare the results in the \(x\), \(y\), and \(z\) directions. The image files are saved in demo/hd1d_shock_tube/figs.

cd demo/hd1d_shock_tube
python plot_data.py

This Python program also computes the analytical solution and compares it with the numerical solution.