# An implementation of smoothed particle hydrodynamic methods for fluids problems

## In this article, we present a numerical Smoothed Particle Hydrodynamic (SPH) method. In the SPH method for the Navier – Stokes equations the most widespread method to solve for pressure and mass conservation is the weakly compressible assumption (WCSPH). This article presents two important benchmark problems to validate the algorithm of SPH method. The two benchmark problems chosen are the Lid – driven cavity problem and Poiseuille flow problem at very low Reynolds numbers. The SPH results are also in good agreement with the analytical solution. | Vietnam Journal of Mechanics, VAST, Vol. 32, No. 1 (2010), pp. 37 – 46 AN IMPLEMENTATION OF SMOOTHED PARTICLE HYDRODYNAMIC METHODS FOR FLUIDS PROBLEMS Nguyen Hoai Son, Nguyen Duy Hung University of Technical Education Ho Chi Minh City Abstract. In this article, we present a numerical Smoothed Particle Hydrodynamic (SPH) method. In the SPH method for the Navier – Stokes equations the most widespread method to solve for pressure and mass conservation is the weakly compressible assumption (WCSPH). This article presents two important benchmark problems to validate the algorithm of SPH method. The two benchmark problems chosen are the Lid – driven cavity problem and Poiseuille flow problem at very low Reynolds numbers. The SPH results are also in good agreement with the analytical solution. 1. INTRODUCTION Smoothed particle hydrodynamic method (SPH) is a fully Lagrangian method, which does not require the use of any mesh. It was originally invented to simulate astrodynamics (Lucy 1977 , Gingold & Monaghan 1977 ). Since then the use of SPH has expanded in many areas of solid and fluid dynamics (involving large deformations, impacts, free-surface and multiphase flows). A major advantage of SPH over Eulerian methods is that the method does not need a grid to calculate spatial derivatives. Instead, they are found by summation of analytical differentiated interpolation formulae (Monaghan, 1992). The momentum and energy equations become sets of ordinary differential equations which are easy to understand in mechanical and thermodynamic terms. For example, the pressure gradient becomes a force between pairs of particles [4]. While Eulerian methods have difficulties to construct a mesh for the simulation domain when it has very complex interfaces, SPH is able to do it without any special front tracking treatment. Moreover, the convection term of Navier-Stokes equations can cause many problems in the Eulerian framework, which are only partially circumvented by introducing .

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