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

SUMMARY smoothed 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.

[1]  J. Monaghan.  Simulating Free Surface Flows with SPH , 1994 .

[2]  L. Libersky,et al.  Smoothed Particle Hydrodynamics: Some recent improvements and applications , 1996 .

[3]  L. Lucy.  A numerical approach to the testing of the fission hypothesis. , 1977 .

[4]  Benedict D. Rogers,et al.  Mathematical balancing of flux gradient and source terms prior to using Roe's approximate Riemann solver , 2003 .

[5]  Valerio Caleffi,et al.  High-order balanced CWENO scheme for movable bed shallow water equations , 2007 .

[6]  Javier Bonet,et al.  A corrected smooth particle hydrodynamics formulation of the shallow-water equations , 2005 .

[7]  Pilar García-Navarro,et al.  Flux difference splitting and the balancing of source terms and flux gradients , 2000 .

[8]  L. Hernquist,et al.  TREESPH: A Unification of SPH with the Hierarchical Tree Method , 1989 .

[9]  D. Causon,et al.  The surface gradient method for the treatment of source terms in the shallow-water equations , 2001 .

[10]  Petros Koumoutsakos,et al.  An immersed boundary method for smoothed particle hydrodynamics of self-propelled swimmers , 2008, J. Comput. Phys..

[11]  J. Bonet,et al.  Variational and momentum preservation aspects of Smooth Particle Hydrodynamic formulations , 1999 .

[12]  Roland W. Lewis,et al.  A variational formulation based contact algorithm for rigid boundaries in two-dimensional SPH applications , 2004 .

[13]  K. Steiner,et al.  A FINITE-VOLUME PARTICLE METHOD FOR COMPRESSIBLE FLOWS , 2000 .

[14]  J. Monaghan.  Smoothed particle hydrodynamics , 2005 .

[15]  J. Kuhnert,et al.  Finite Pointset Method Based on the Projection Method for Simulations of the Incompressible Navier-Stokes Equations , 2003 .

[16]  Rui Xu,et al.  Comparisons of weakly compressible and truly incompressible algorithms for the SPH mesh free particle method , 2008, J. Comput. Phys..

[17]  M. Vázquez-Cendón.  Improved Treatment of Source Terms in Upwind Schemes for the Shallow Water Equations in Channels with Irregular Geometry , 1999 .

[18]  J. Monaghan,et al.  Smoothed particle hydrodynamics: Theory and application to non-spherical stars , 1977 .

[19]  Timothy Nigel Phillips,et al.  A consistent reflected image particle approach to the treatment of boundary conditions in smoothed particle hydrodynamics , 2009 .

[20]  S. Shao,et al.  INCOMPRESSIBLE SPH METHOD FOR SIMULATING NEWTONIAN AND NON-NEWTONIAN FLOWS WITH A FREE SURFACE , 2003 .

[21]  Alfredo Bermúdez,et al.  Upwind methods for hyperbolic conservation laws with source terms , 1994 .

[22]  Azzeddine Soulaïmani,et al.  A stabilized SPH method for inviscid shallow water flows , 2005 .

[23]  Michael Dumbser,et al.  A new 3D parallel SPH scheme for free surface flows , 2009 .

[24]  Joe J. Monaghan,et al.  SPH particle boundary forces for arbitrary boundaries , 2009, Comput. Phys. Commun..

[25]  P. Cleary,et al.  Conduction Modelling Using Smoothed Particle Hydrodynamics , 1999 .

[26]  Qiuhua Liang,et al.  Adaptive quadtree simulation of shallow flows with wet-dry fronts over complex topography , 2009 .

[27]  F. Aureli,et al.  A weighted surface-depth gradient method for the numerical integration of the 2D shallow water equations with topography , 2008 .