# Compressible Navier-Stokes

The solver for compressible flows contains explicit time-stepping schemes for the Euler and Navier-Stokes equations. Supported options include classical Runge-Kutta schmes as well as a local time-stepping scheme introduced by Winters & Kopriva (2013). The convective fluxes are discretized using the HLLC flux, while the SIPG fluxes are used for the diffusive terms.

# Inviscid flow around a cylinder

We study the inviscid flow around a cylinder at Mach 0.2. Here, we use curved second order elements in conjunction with a second order polynomial approximation. Even on this coarse grid consisting of 32x32 cells, the pressure contours are almost perfectly symmetrical, which is characteristic for subsonic flows around smooth obstacles.

# Vortex shedding behind a cylinder

# Local time-stepping

The local time-stepping (LTS) scheme by Winters & Kopriva (2013) is based on a multistep Adams-Bashforth scheme. The grid is clustered into cells of similar size and each cluster is evolved with its maximum stable local time-step.

The scheme is verified with a time convergence study of an isentropic vortex. In the plot below, the time-step size Δt of the coarsest cluster is given for the LTS examples.

# Isentropic vortex in inviscid fluids with various equations of state

The isentropic vortex solution for the Euler equations can be visualized as follows:

Density contours and velocity vectors for an isentropic vortex in an ideal gas.

Typically, the isentropic vortex flow is considered for ideal gases. The solution can however be extended to more general equations of state like the covolume gas law

which imposes a maximum density

modeling the repellent intermolecular forces in highly compressed gases. The ideal gas law is recovered when the parameter b is set to zero. A particular solution for an isentropic vortex in such a covolume gas is given below.

The presented test case has also been simulated in BoSSS and the h-convergence study on the right shows that the expected rate of convergence could be obtained for a relatively stiff gas with a covolume of b = 0.1 which enforces a maximum non-dimensional density of ρ = 10.

**Müller, Björn** (2014):*Methods for higher order numerical simulations of complex inviscid fluids with immersed boundaries.*

Darmstadt, TU Darmstadt, [Online-Edition: http://tuprints.ulb.tu-darmstadt.de/3747],

[Dissertation]