Kelvin–Helmholtz instabilities in smoothed particle hydrodynamics

• Published in: Monthly notices of the Royal Astronomical Society Vol. 408; no. 1; pp. 71 - 86
• Main Authors: Valcke, S., De Rijcke, S., Rödiger, E., Dejonghe, H.
• Format: Journal Article
• Language: English
• Published: Oxford, UK Blackwell Publishing Ltd 11-10-2010; Wiley-Blackwell; Oxford University Press
• Subjects: Astronomy; Atoms & subatomic particles; Earth, ocean, space; Exact sciences and technology; Hydrocarbons; hydrodynamics; instabilities; methods: numerical; Stars & galaxies; Surface tension; methods: numerical; instabilities; hydrodynamics; Discontinuity; Mixing; Kelvin Helmholtz instability; Digital simulation; Smoothed particle hydrodynamics method; Energy density; Numerical method; Shock waves; Relaxation; Travelling waves; Initial conditions; Galaxy formation; Hydrodynamic model; Density gradient; Surface tension; Diffusion; Mach number
• ISSN: 0035-8711; 1365-2966
• DOI: 10.1111/j.1365-2966.2010.17127.x

In this paper we investigate whether smoothed particle hydrodynamics (SPH), equipped with artificial conductivity (AC), is able to capture the physics of density/energy discontinuities in the case of the so-called shearing layers test, a test for examining Kelvin–Helmholtz (KH) instabilities. We can trace back each failure of SPH to show KH rolls to two causes: (i) shock waves travelling in the simulation box and (ii) particle clumping, or more generally, particle noise. The probable cause of shock waves is the local mixing instability, previously identified in the literature. Particle noise on the other hand is a problem because it introduces a large error in the SPH momentum equation. The shocks are hard to avoid in SPH simulations with initial density gradients because the most straightforward way of removing them, i.e. relaxing the initial conditions, is not viable. Indeed, by the time sufficient relaxing has taken place the density and energy gradients have become prohibitively wide. The particle disorder introduced by the relaxation is also a problem. We show that setting up initial conditions with a suitably smoothed density gradient dramatically improves results: shock waves are reduced whilst retaining relatively sharp gradients and avoiding unnecessary particle disorder. Particle clumping is easy to overcome, the most straightforward method being the use of a suitable smoothing kernel with non-zero first central derivative. We present results to that effect using a new smoothing kernel: the linear quartic kernel. We also investigate the role of AC. Although AC is necessary in the simulations to avoid ‘oily’ features in the gas due to artificial surface tension, we fail to find any relation between using AC and the appearance of seeded KH rolls. Including AC is necessary for the long-term behaviour of the simulation (e.g. to get λ= 1/2, 1 KH rolls). In sensitive hydrodynamical simulations great care is however needed in selecting the AC signal velocity, with the default formulation leading to too much energy diffusion. We present new signal velocities that lead to less diffusion. The effects of the shock waves and of particle disorder become less important as the time-scale of the physical problem (for the shearing layers problem: lower density contrast and higher Mach numbers) decreases. At the resolution of current galaxy formation simulations mixing is probably not important. However, mixing could become crucial for next-generation simulations.