# 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:

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]