Development of an upwinding particle interaction kernel for simulating incompressible Navier‐Stokes equations

In this study, we are aimed to derive a kernel function, which accounts for the interaction among particles, within the framework of particle method. To get a computationally more accurate solution for the incompressible Navier-Stokes equations, determination of kernel function is a key to success in the developed interaction model. In the light of the underlying fact that the smoothed quantity for a scalar or a vector at a particle location is mathematically identical to its collocated value provided that the kernel function is chosen as the Dirac delta function, our guideline is to make the modified kernel function closer to the Dirac delta function as much as possible in flow conditions when diffusion dominates convection. As convection prevailingly dominates its diffusion counterpart, particle interaction at the upstream side should be favorably taken into account to avoid numerical oscillations resulting from the convective instability. The proposed particle interaction model featuring with the newly developed kernel function will be validated through several scalar transport and Navier-Stokes problems which have either analytical or benchmark solutions. The stability condition and the spatial accuracy order of the proposed particle interaction model will be also analyzed in details in this article for the sake of completeness. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 28: 1574–1597, 2012