Murtazo Nazarov | Invariant domain preserving schemes

To make all the features of the solution visible we often plot Schlieren gray-scale diagram of the vertical momentum $\rho$ given by

\[ \sigma = \exp\left (-\beta \frac{|\nabla \rho|}{\max_{\Omega}(|\nabla \rho|)} \right), \]

where $\beta=10$.

Supersonic flow around circular cylinder

In this example we consider a supersonic flow past a circular cylinder of radius $0.25$ centered at the point $\boldsymbol{x} = (0.6, 1)$ in a two-dimensional wind tunnel of size $(x_1, x_2)\in [0, 4] \times [0, 2]$. We use the $\gamma$-law with $\gamma=\frac{7}{5}$. The initial data is $\rho=1.4$, $p=1$, $\boldsymbol v=(3,0)^\top$. The Mach 3 flow enters from the left boundary $\{x_1=0\}$, where Dirichlet boundary conditions are prescribed. The slip boundary condition is applied at the walls $\{x_2 = 0\}$ and $\{x_2 = 2\}$ and on the cylinder. Since the flow is supersonic, no boundary condition is applied at the outflow $\{x_1=4\}$.

Supersonic flow past circular cylinder. The mesh composed of $252820$ unstructured $\mathbb P1$ points and $\mbox{CFL}=0.3$ .

The flow enters the wind tunnel and hits the cylinder, then a strong bow shock develops at the front of the cylinder, while two attached oblique shocks develop from the back side of the cylinder. The flow separates at the points where the oblique shocks start. After separation the flow fluctuates and creates small scale vortices traveling downstream. The strong shocks travel towards the wall boundaries, reflect back in the tunnel, and pass through the small scale vortices.

Animation of the cylinder problem.