Overview

In the previous chapters, the essential formulations and the smoothing functions for the SPH method were provided. It is seen that the SPH kernel and particle approximations can be used for discretization of partial differential equations (PDEs).

This chapter describes the detailed formulation of the SPH method and some interesting applications to general fluid dynamic problems. The SPH formulation is derived by discretizing the Navier-Stokers equations spatially, leading to a set of ordinary differential equations (ODEs) with respect to time. This set of ODEs can then be solved via time integration. The SPH formulation can be employed to simulate general dynamic fluid flows. The following numerical aspects in implementing the SPH formulations will be discussed.

• artificial viscosity

• artificial heat

• physical viscosity

• variable smoothing length

• symmetrization of particle interaction

• zero-energy mode problem

• artificial compressibility

• solid boundary treatment

• choice of time step

Some modifications for these numerical procedures are made to the conventional SPH method to better suit the needs of dynamic fluid flow simulations. Nearest neighboring particle searching algorithms and the pair interaction technique for particle interactions are described. A three-dimensional SPH code is developed and applied to simulate different interesting fluid flow problems, which include incompressible flows with solid boundaries, free surface flows, and compressible flows. The SPH results are validated by comparisons with those obtained using other methods.