Investigation of a Vorticity-preserving Scheme for the Euler Equations

Abstract

We investigate the vorticity-preserving properties of the compressible, second-order residual-based scheme, "RBV2". The scheme has been extensively tested on hydrodynamical problems, and has been shown to exhibit remarkably accurate results on the propagation of inviscid compressible vortices, airfoil-vortex interactions on a curvilinear mesh, vortex mergers in an astrophysical accretion disk, and the establishment of a two-dimensional inverse cascade in high-resolution turbulent simulations. Here, we demonstrate that RBV2 sustains the analytic solution for a one-dimensional shear flow. We assess the fidelity by which the algorithm maintains a skewed shear flow, and present convergence tests to quantify the magnitude of the expected numerical dispersion. We propose an adjustment to the dissipation in the algorithm that retains the vorticity-preserving qualities, and accurately incorporates external body forces, and demonstrate that it indefinitely maintains a steady-state hydrostatic equilibrium between a generic acceleration and a density gradient. We present a novel numerical assessment of vorticity preservation for discrete-wavenumber vortical modes up to the Nyquist wavenumber. We apply this assessment to RBV2 in order to quantify the extent to which the scheme preserves vorticity for the full Euler equations. We find that RBV2 perfectly preserves vorticity for modes with symmetric wavenumbers, i.e., $k_x=k_y$, and that the error increases with asymmetry. We simulate the dynamical interaction of vortices in a protoplanetary disk to demonstrate the utility of the updated scheme for rendering astrophysical flows replete with vortices and turbulence. We conclude that RBV2 is a competitive treatment for evolving vorticity-dominated astrophysical flows with minimal dissipation.