Unified semi-analytical wall boundary conditions applied to 2-D incompressible SPH

Abstract

This work aims at improving the 2-D incompressible SPH model (ISPH) by adapting it to the unified semi-analytical wall boundary conditions proposed by Ferrand et al. [10]. The ISPH algorithm considered is as proposed by Lind et al. [25], based on the projection method with a divergence-free velocity field and using a stabilising procedure based on particle shifting. However, we consider an extension of this model to Reynolds-Averaged Navier-Stokes equations based on the k − ǫ turbulent closure model, as done in [10]. The discrete SPH operators are modified by the new description of the wall boundary conditions. In particular, a boundary term appears in the Laplacian operator, which makes it possible to accurately impose a von Neumann pressure wall boundary condition that corresponds to impermeability. The shifting and free-surface detection algorithms have also been adapted to the new boundary conditions. Moreover, a new way to compute the wall renormalisation factor in the frame of the unified semi-analytical boundary conditions is proposed in order to decrease the computational time. We present several verifications to the present approach, including a lid-driven cavity, a water column collapsing on a wedge and a periodic schematic fish-pass. Our results are compared to Finite Volumes methods, using Volume of Fluids in the case of free-surface flows. We briefly investigate the convergence of the method and prove its ability to model complex free-surface and turbulent flows. The results are generally improved when compared to a weakly compressible SPH model with the same boundary conditions, especially * Corresponding author. tel: +33 (0)6 67 88 92 13Email addresses: agnes.leroy@edf.fr (A. Leroy), damien.violeau@edf.fr (D. Violeau),

Study on SPH Viscosity Term Formulations

“…The lid-driven flows have been widely used as a benchmark validation case for the different SPH viscosity force models. For example, Zheng et al [33] used an artificial viscosity, Leroy et al [39] used a turbulent dynamic viscosity, Aly et al [40] used a Morris formula, and Yildiz et al [41] derived a more general second-order derivative approximation based on the Taylor series expansion and the concept of second-and fourth-order isotropic tensors.…”

A review on approaches to solving Poisson’s equation in projection-based meshless methods for modelling strongly nonlinear water waves

“…Lo and Shao 2002;Gotoh and Sakai 2006) and unified semi-analytical wall boundary condition (e.g. Leroy et al 2014). To numerically implement the boundary condition on the free surface, the key issue is how to identify the particles on it.…”

“…The density may also be specified as the physical density (e.g. Asai et al 2012;Lind et al 2012;Leroy et al 2014) and thus should be the same at all particles, i.e. ρ i = ρ j .…”

Landslides and tsunamis predicted by incompressible smoothed particle hydrodynamics (SPH) with application to the 1958 Lituya Bay event and idealized experiment

“…Terms C μ , C ,1 , C ,2 , σ k and σ are constants specific to the k − model, taking the standard values presented in table 1. It is noted that equations (2.12) are solved explicitly with the right-hand side computed using values from time step n. More details on turbulence modelling in SPH can be found in Violeau [33,35], Leroy et al [37] and Ferrand et al [36].…”