A stabilising diffusion-based shifting algorithm for incompressible smoothed particle hydrodynamics

The incompressible smoothed particle hydrodynamics (ISPH) method with projection-based pressure correction has been shown to be highly accurate and stable for internal flows and, importantly for many problems, the pressure field is virtually noise-free in contrast to the weakly compressible SPH approach (Xu et al. (2009)). However for almost inviscid fluids instabilities at the free surface occur due to errors associated with the truncated kernels. A new algorithm is presented which remedies this issue, giving stable and accurate solutions to both internal and free-surface flows. Generalising the particle shifting approach of Xu et al. (2009), the algorithm is based upon Fick’s law of diffusion and shifts particles in a manner that prevents highly anisotropic distributions and the onset of numerical instability. The algorithm is validated against analytical solutions for an internal flow for Reynolds numbers as large as 106, the flow due to an impulsively started plate, and highly accurate solutions for wet bed dam break problems at small times. The method is then validated for progressive regular waves with paddle motion defined by linear theory. The accurate predictions demonstrate the effectiveness of the algorithm in stabilising solutions and minimising the surface instabilities generated by the inevitable errors associated with truncated kernels. The test cases are thought to provide a more thorough quantitative validation than previously undertaken.