The Kelvin–Helmholtz instability and smoothed particle hydrodynamics

Abstract

ABSTRACT We perform simulations of the Kelvin–Helmholtz instability using smoothed particle hydrodynamics (SPH). The instability is studied both in the linear and strongly non-linear regimes. The smooth, well-posed initial conditions of Lecoanet et al. (2016) are used, along with an explicit Navier–Stokes viscosity and thermal conductivity to enforce the evolution in the non-linear regime. We demonstrate convergence to the reference solution using SPH. The evolution of the vortex structures and the degree of mixing, as measured by a passive scalar ‘colour’ field, match the reference solution. Tests with an initial density contrast produce the correct qualitative behaviour. The $\mathcal {L}_2$ error of the SPH calculations decreases as the resolution is increased. The primary source of error is numerical dissipation arising from artificial viscosity, and tests with reduced artificial viscosity have reduced $\mathcal {L}_2$ error. A high-order smoothing kernel is needed in order to resolve the initial velocity amplitude of the seeded mode and inhibit excitation of spurious modes. We find that standard SPH with an artificial viscosity has no difficulty in correctly modelling the Kelvin–Helmholtz instability and yields convergent solutions.