Improved SPH methods for simulating free surface flows of viscous fluids

In this paper we present two enhanced variants of the smoothed particle hydrodynamics (SPH) method for the numerical simulation of free surface flows of viscous fluids. Improvements are achieved by deriving a new set of general discrete SPH-like equations under an energy-based framework and applying a corrected (high-order) or coupled particle approximation scheme for function derivatives. By doing so, we ensure that the enhanced variants retain the conservative nature of SPH which is important for the stability of long-term simulations. Among various corrected approximations, we here implement the one obtained by the so-called finite particle method (FPM) within the framework to produce a higher-order SPH method which conserves both linear and angular momentums. In order to improve the efficiency of the higher-order variant, a coupled approach with the idea of using the SPH approximation for the interior particles and the FPM approximation for the exterior particles is also proposed and tested in this paper. Three prototype tests concerning free deformation of a viscous fluid patch with free surface are presented with comparisons between different methods to demonstrate the performance of the two proposed methods. Numerical results show that both the higher-order version using FPM and the coupled version using FPM/SPH outperform the original version of SPH in respect of accuracy and stability.