Smoothed Particle Hydrodynamics: Approximate zero‐consistent 2‐D boundary conditions and still shallow‐water tests

Abstract

SUMMARYsmoothed particle hydrodynamics; boundary conditions; shallow water equations; source terms; virtual boundary particles; still waterIn this paper, an approximate modified virtual boundary particle method (MVBP) for solid boundary conditions in a two‐dimensional (2‐D) smoothed particle hydrodynamics (SPH) model is presented; this is a development of the original VBP method recently proposed by Ferrari et al. (Comput. Fluids 2009; 38(6): 1203–1217). The aim is to maintain the zeroth moment of the kernel function as closely as possible to unity, a property referred to as zero‐consistency, for particles close to solid boundaries. The performance of the new method in approximating zero‐consistency in the presence of complicated boundaries is demonstrated where we show that the MVBP method improves the accuracy of the zeroth moment by almost an order of magnitude. Shallow‐water flows are an important two‐dimensional (2‐D) application and provide the simple test case of still water. The shallow‐water equations (SWEs) are thus considered in SPH form and the zero‐consistency approximation is tested for still water in domains with different boundaries: a circle and two squares, one with an additional internal angle of 300∘ and one with four internal angles of 345∘. We demonstrate that for an internal angle of 300∘, the MVBP method demonstrates numerical convergence to still‐water conditions whereas both mirror particles and the VBP method cannot. The method is also demonstrated for the dynamic case of a circular dam break interacting with an outer circular wall where conventional mirror particles fail to prevent particles passing through the solid wall. The SPH SWEs are further generalized through a new method for discretizing the bed source term allowing arbitrarily complicated bathymetries. The resulting formulation is tested by considering many different bed shapes in still water: submerged and surface‐piercing humps, a submerged step, a submerged and surface‐piercing parabolic bed. Copyright © 2011 John Wiley & Sons, Ltd.