Simulating Flows with SPH: Recent Developments and Applications

The chapter discusses recent theoretical developments and practical applications of the Smoothed Particle Hydrodynamics (SPH) method with specific concern to liquids. SPH is a meshless Lagrangian technique for the approximate integration of spatial derivatives, using particle interpolation over a compact support, without the usage of a structured grid. Its main related advantage is the capability of simulating the computational domain with large deformations and high discontinuities, bearing no numerical diffusion because advection terms are directly evaluated. SPH has recently become very popular for the simulation of fluid motion using computers, covering different fields, e.g. free surface flows, multiphase flows, turbulence modelling. In the following, recent theoretical achievements of SPH are firstly presented, concerning (1) numerical schemes for approximating governing equations, such as the Navier Stokes ones, most widely adopted in fluid dynamics, (2) smoothing or kernel function properties needed to perform the function approximation to the Nth order, (3) restoring consistency of kernel and particle approximation, yielding the SPH approximation accuracy. Secondly computation aspects related to the neighbourhood definition are discussed. Field variables, such as particle velocity or density, are evaluated by smoothing interpolation of the corresponding values over the nearest neighbour particles located inside a cut-off radius “rc”. The generation of a neighbour list at each time step takes a considerable portion of CPU time. Straightforward determination of which particles are inside the interaction range requires the computation of all pair-wise distances, a procedure whose computational time would be of the order O(N2), and therefore unpractical for large domains. Finally, some practical applications are presented, primariliy concerning free surface flows. The capability to easily handle large deformation is shown