Introduction

Euler equations of inviscid flow in 2D displaying a time-independent rotating vortex whose centrifugal force is exactly balanced by a gradient of pressure.

Overly diffusive schemes, for instance due to too simple artificial viscosity terms, or schemes with low-order kernels, or generally that fail to conserve angular momentum, struggle to maintain the vortex’s profile and dissipate its angular momentum leading to its complete disappearance after only a few orbits.

Flow phenomena

Inviscid flow

Compressible and incompressible flow

Conservation of angular momentum

Preservation of symmetries – Pure rotation balanced by pressure gradient

Geometry

The test consists of a 1x1 domain with a rotating vortex placed at the center. The unit system is entirely arbitrary. For the fluid, an ideal gas equation of state with adiabatic index 5/3 is assumed but incompressible flow assumption can be adopted as well. The fluid has a constant density ⍴ = 1.

The angular velocity and pressure are a function of the radius r = (x²+y²)¹/².

The radial velocity is set to 0 everywhere. The pressure profile is given by:

As the system is set in equilibrium, this profile should be maintained for indefinite time periods.

Two simple extensions to this base setup can be made to test additional regimes:

- Add a constant pressure to the whole fluid. As the flow behavior only depends on the gradient of the pressure, the resulting vortex stability should be independent of this added pressure.

- Add a constant velocity in a random direction. This allows verification of the Lagrangian nature of the method.

Boundary conditions

Periodic boundary conditions.

Initial conditions

See the diagram in the Geometry section above.

Discretization

The initial setup consists of constant mass particles arranged either using a Cartesian grid or a glass-like configuration, i.e. a stable distribution of particles minimizing the SPH Lagrangian. The particles properties are obtained by applying the analytic profile described in the Geometry section.

The files attached give the initial conditions in ASCII format for 128² particles once using a Cartesian grid and using a glass configuration respectively.

For convenience, both the pressure and internal energy of the particles are provided.

Results specification

The setup is typically run until t = 1, close to one full rotation of the peak of the vortex profile.

The main result consists of the azimuthal velocity profile as a function of radius from the center of the vortex. Similarly, the pressure profile is measured as function of radius.

Both these quantities are then benchmarked against the analytic solution, available in the Geometry section. Since the setup is an equilibrium solution, the initial profile is time-independent.

Results format

The attached file contains the density, azimuthal velocity, radial velocity, internal energy, and pressure reported as a function of radius. This is a simple discretization in 200 bins of the setup described in the Geometry section provided here for plotting convenience.

Benchmark results

N/A.

Download

You can download the full test case below:

References

    Gresho & Chan, 1990, Int. J. Numer. Meth. Fluids, 11: 621-659. [Link]

    Liska & Wendroff, 2003, SIAM J. on Sci. Comp., 25, 3, 995-1017. [Link]