# The Kelvin-Helmholtz instability and smoothed particle hydrodynamics

## TL;DR

This paper demonstrates that smoothed particle hydrodynamics (SPH) can accurately simulate the Kelvin-Helmholtz instability, showing convergence and correct vortex evolution in both linear and non-linear regimes, with insights into error sources.

## Contribution

The study shows that standard SPH with artificial viscosity can correctly model the Kelvin-Helmholtz instability and achieve convergence, addressing previous challenges in simulating this phenomenon.

## Key findings

- SPH simulations match reference solutions in vortex evolution and mixing.
- L2 error decreases with increased resolution and reduced artificial viscosity.
- High-order kernels are essential for resolving initial conditions and preventing spurious modes.

## 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 L2 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 L2 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.