Theoretical Analysis of the No-Slip Boundary Condition Enforcement in SPH Methods

The aim of the present work is to provide an in-depth analysis of the most representative mirroring techniques used in SPH to enforce boundary conditions (BC) along solid profiles. We specifically refer to dunimy particles, ghost particles, and Takeda et al. [Prog. Theor. Phys. 92 (1994), 939] boundary integrais. The analysis has been carried out by studying the convergence of the first- and second-order differential operators as the smoothing length (that is, the characteristic length on which relies the SPH interpolation) decreases. These differential operators are of fundamental importance for the computation of the viscous drag and the viscous/diffusive terms in the momentum and energy equations. It has been proved that ciose to the boundaries some of the mirroring techniques leads to intrinsic inaccuracies in the convergence of the differential operators. A consistent formulation has been derived starting from Takeda et al. 1' boundary integrais (see the above reference). This original formulation allows implementing no-slip boundary conditions consistently in many practical applications as viscous flows and diffusion problems. Subject Index: 024 The Smoothed Particle Hydrodynamics scheme (hereinafter SPH) is a Lagrangian model based on a smoothing of the spatial differential operators of the fluid-dynamics equations and on their subsequent discretization through a finite number of fluid par­ ticles. The smoothing procedure (which is made at the continuum level) is performed by using a weight function (also called kernel function) with a compact support whose characteristic length is the smoothing length h. After the smoothed equations are discretized through fluid particles, the resolution of the discrete SPH scheme is a function of both the smoothing length and the mean particle distance dx. In this framework, the (continuous) equations of the fluid-dynamics should be recovered as both h and dx/h tend simultaneously to zero. 4) The SPH simulations in engineering involve usually solid boundary conditions (BC) for the velocity field and Dirichlet and Neumann type BC for other fields as, for instance, the temperature. In the SPH framework, these conditions are tackled in a number of ways: by using boundary forces-type models;5) by modifying the structure of the kernel in the neighborhood of the boundaries;6) by creating virtual