Stabilizing a tensor-represented viscosity model for variationally consistent particle methods

Abstract

A tensor-represented viscosity model was developed for variationally consistent particle methods, which conserves linear and angular momentum and reduces instabilities related to particle distributions. The particle method adopted in this study can be interpreted as a Smoothed Particle Hydrodynamics (SPH) method except that it employs multiple kernels, including non-bell-shaped ones; therefore, it is termed the Multi-Kernel SPH (MK-SPH (MPH)) method. In this method, the kernels are chosen to avoid undesired particle agglomeration. In addition, two stabilization terms are proposed while maintaining variational consistency and momentum conservation. One is compensation viscosity, which reduces the oscillatory mode (e.g., zero-energy modes) with respect to the tensor-represented viscosity model. The other is regularization potential, which further suppresses particle agglomeration (e.g., tensile instability) even under negative pressure. Furthermore, a non-slip fixed particle boundary is proposed corresponding to the viscosity models. The present model was verified by calculating Kolmogorov flow, Taylor–Green flow, and lid-driven cavity flow, and its performance is demonstrated by calculating viscous rotating square patch, viscous square drop, and anisotropic compression. Specifically, the convergence with respect to particle size and the applicability of the two stabilization terms are investigated.