Robustness and accuracy of SPH formulations for viscous flow

Abstract

Numerous methods are available for the modelling of viscous stress terms in smoothed particle hydrodynamics (SPH). In this work, the existing methods are investigated systematically and evaluated for a range of Reynolds numbers using Poiseuille channel and lid-driven cavity test cases. The best results are obtained using two methods based on combinations of finite difference and SPH approximations, due to Morris et al. and Cleary. Gradients of high-valued functions are shown to be inaccurately estimated with standard SPH. A method that reduces the value of functions (in particular, pressure) before calculating the gradients reduces this inaccuracy and is shown to improve performance. A mode of instability in Poiseuille channel flows, also reported in other works, is examined and a qualitative explanation is proposed. The choice of boundary implementation is shown to have a significant effect on transient velocity profiles in start-up of the flow. The use of at least linear extrapolation for in-wall velocities is shown to be preferable to mirroring of velocities. Consistency corrections to the kernel are also found to result in significant accuracy and stability improvements with most methods, though not in all cases.