Compressible Navier-Stokes

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

by: Dr.-Ing. Björn Müller

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.

flow around cylinder 2nd order quadratic elements
Pressure distribution, pressure contours (black lines, top half) and mesh (black lines, bottom half) for the inviscid flow around a cylinder using second order polynomials as well as second order elements

Vortex shedding behind a cylinder

by: Stephan Krämer-Eis, M.Sc.

Vorticity or behind a cylinder at Re = 100 and Ma = 0.2 using 4th order DG
Vorticity or behind a cylinder at Re = 100 and Ma = 0.2 using 4th order DG

Local time-stepping

by: Stephan Krämer-Eis

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.

lts groups
Example of an automatic clustering: Cells of the same colour belong to one cluster

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.

Results of the time convergence study

Isentropic vortex in inviscid fluids with various equations of state

by: Dr.-Ing. Björn Müller

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

\[p = (\gamma – 1) \frac{\rho}{1 -b\rho} e\]

which imposes a maximum density

\[\rho_\text{max} = \frac{1}{b}\]

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.

vortex ideal vs covolume density
Density profile (ɣ=1.4, b=0.1)
vortex ideal vs covolume pressure
Pressure profile (ɣ=1.4, b=0.1)
ibm vortex convergence

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.

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