Theoretical analysis and numerical verification of the consistency of viscous smoothed-particle-hydrodynamics formulations in simulating free-surface flows.

The theoretical formulation of the smoothed particle hydrodynamics (SPH) method deserves great care because of some inconsistencies occurring when considering free-surface inviscid flows. Actually, in SPH formulations one usually assumes that (i) surface integral terms on the boundary of the interpolation kernel support are neglected, (ii) free-surface conditions are implicitly verified. These assumptions are studied in detail in the present work for free-surface Newtonian viscous flow. The consistency of classical viscous weakly compressible SPH formulations is investigated. In particular, the principle of virtual work is used to study the verification of the free-surface boundary conditions in a weak sense. The latter can be related to the global energy dissipation induced by the viscous term formulations and their consistency. Numerical verification of this theoretical analysis is provided on three free-surface test cases including a standing wave, with the three viscous term formulations investigated.

[1]  E. M. Lifshitz,et al.  Theory of Elasticity: Vol. 7 of Course of Theoretical Physics , 1960 .

[2]  M. Lighthill,et al.  Waves In Fluids , 2002 .

[3]  J. Monaghan,et al.  Shock simulation by the particle method SPH , 1983 .

[4]  P. Raviart,et al.  A particle method for first-order symmetric systems , 1987 .

[5]  J. Morris,et al.  Modeling Low Reynolds Number Incompressible Flows Using SPH , 1997 .

[6]  E. Grenier,et al.  The convergence of the SPH method , 1998 .

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

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

[9]  B. Ben Moussa,et al.  Convergence of SPH Method for Scalar Nonlinear Conservation Laws , 2000, SIAM J. Numer. Anal..

[10]  Andrew G. Glen,et al.  APPL , 2001 .

[11]  Pep Español,et al.  Smoothed dissipative particle dynamics. , 2003, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  Ericka Stricklin-Parker,et al.  Ann , 2005 .

[13]  Nikolaus A. Adams,et al.  A multi-phase SPH method for macroscopic and mesoscopic flows , 2006, J. Comput. Phys..

[14]  Maher Moakher,et al.  Fourth-order cartesian tensors: old and new facts, notions and applications , 2008 .

[15]  A. Colagrossi,et al.  Theoretical considerations on the free-surface role in the smoothed-particle-hydrodynamics model. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Salvatore Marrone,et al.  Fast free-surface detection and level-set function definition in SPH solvers , 2010, J. Comput. Phys..