Test Problems
We describe several test problems that users can test with HARD
Sod Shock Tube
The Sod shock tube is a standard test with a classical Riemann problem with the following initial parameters:
This leads to the development of a shock front, which propagates from high-density into low-density regions, and is followed by a contact discontinuity. A density rarefaction wave propagates into the high-density region.
Heating-Cooling
Temperature Induced Shock
Rayleigh-Taylor Instability
Rayleigh–Taylor Instability consists of a dense fluid over a lighter fluid in a gravitational field, and is perturbed to initiate the instability.
Gravitational acceleration acts in the x-direction:
g = g_xPressure at interface: \(p_0 = 2.5\)
Velocity perturbation (in the y-direction):
\[v(x, y) = v_0 \cdot \left( \frac{1 + \cos(4\pi x)}{2} \right) \left( \frac{1 + \cos(3\pi y)}{2} \right), \quad v_0 = 0.05\]Bottom fluid (light): - Density: \(\rho_L = 1.0\) - x-velocity: \(u_L = 0.0\)
Top fluid (heavy): - Density: \(\rho_H = 2.0\) - x-velocity: \(u_H = 0.0\)
The pressure is initialized to maintain hydrostatic equilibrium across the fluid interface. The pressure \(p(x)\) is given by:
At each point in the domain, the following quantities are initialized:
Density:
\[\begin{split}\rho(x) = \begin{cases} \rho_L, & x < 0.75 \\ \rho_H, & x \geq 0.75 \end{cases}\end{split}\]x-Momentum density:
\[(\rho u)(x) = \rho(x) \cdot u(x) = 0\]Gravitational source term (x-direction):
\[(\rho g)(x) = \rho(x) \cdot g\]Total energy density:
\[E(x) = \rho \cdot e(\rho, p) + \frac{1}{2} \rho \left( u^2 + v^2 \right)\]where \(e(\rho, p)\) is the specific internal energy computed from the equation of state.
Radiation energy density:
\[E_{\text{rad}} = 0\]
The setup supports 1D, 2D, and 3D simulations. A cosine-modulated perturbation in the y-velocity initiates the RTI at the fluid interface. The pressure profile ensures that the fluid is initially in hydrostatic equilibrium.