Modelling Shear Flows with SPH and Grid Based Methods

TL;DR

This paper compares the evolution of Kelvin-Helmholtz instability in shear flows using SPH and grid-based methods, analyzing dissipation effects and validating numerical schemes against analytical predictions.

Contribution

It introduces a detailed analytical framework for KHI with viscosity and evaluates multiple numerical schemes against this model, highlighting their dissipation characteristics.

Findings

SPH with Balsara viscosity suppresses KHI at high density contrasts.

Alternative SPH with thermal conductivity matches analytical growth rates up to 10:1 contrast.

Grid codes FLASH and PLUTO show consistent non-viscous evolution aligning with theory.

Abstract

Given the importance of shear flows for astrophysical gas dynamics, we study the evolution of the Kelvin-Helmholtz instability (KHI) analytically and numerically. We derive the dispersion relation for the two-dimensional KHI including viscous dissipation. The resulting expression for the growth rate is then used to estimate the intrinsic viscosity of four numerical schemes depending on code-specific as well as on physical parameters. Our set of numerical schemes includes the Tree-SPH code VINE, an alternative SPH formulation developed by Price (2008), and the finite-volume grid codes FLASH and PLUTO. In the first part, we explicitly demonstrate the effect of dissipation-inhibiting mechanisms such as the Balsara viscosity on the evolution of the KHI. With VINE, increasing density contrasts lead to a continuously increasing suppression of the KHI (with complete suppression from a contrast of 6:1 or higher). The alternative SPH formulation including an artificial thermal conductivity reproduces the analytically expected growth rates up to a density contrast of 10:1. The second part addresses the shear flow evolution with FLASH and PLUTO. Both codes result in a consistent non-viscous evolution (in the equal as well as in the different density case) in agreement with the analytical prediction. The viscous evolution studied with FLASH shows minor deviations from the analytical prediction.