A turbulence model for Smoothed Particle Hydrodynamics

The aim of this paper is to describe a turbulence model for the particle method Smoothed Particle Hydrodynamics (SPH). The model makes few assumptions, conserves linear and angular momentum, satisfies a discrete version of Kelvin’s circulation theorem, and is computationally efficient. Furthermore, the results from the model are in good agreement with the experimental and computational results of Clercx and Heijst for two-dimensional turbulence inside a box with no-slip walls. The model is based on a Lagrangian similar to that used for the Lagrangian averaged Navier–Stokes (LANS) turbulence model, but with a different smoothed velocity. The smoothed velocity preserves the shape   of the spectrum of the unsmoothed velocity, but reduces the magnitude for short length scales by an amount which depends on a parameter ϵϵ. We call this the SPH-ϵϵ model. The effectiveness of the model is indicated by the fact that the second and fourth order velocity correlation functions calculated using the smoothed velocity and a coarse resolution, are in good agreement with a calculation using a resolution which is finer by a factor 2, and therefore requires 8 times as much work to integrate to the same time.