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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),

In the SPH method for viscous fluids, the time step is subject to empirical stability criteria. We proceed to a stability analysis of the Weakly Compressible SPH equations using the von Neumann approach in arbitrary space dimension for unbounded flow. Considering the continuous SPH interpolant based on integrals, we obtain a theoretical stability criterion for the time step, depending on the kernel standard deviation, the speed of sound and the viscosity. The stability domain appears to be almost independent of the kernel choice for a given space discretization. Numerical tests show that the theory is very accurate, despite the approximations made. We then extend the theory in order to study the influence of the method used to compute the density, of the gradient and divergence SPH operators, of background pressure, of the model used for viscous forces and of a constant velocity gradient. The influence of time integration scheme is also studied, and proved to be prominent. All of the above theoretical developments give excellent agreement against numerical results.It is found that velocity gradients almost do not affect stability, provided some background pressure is used. Finally, the case of bounded flows is briefly addressed from numerical tests in three cases: a laminar Poiseuille flow in a pipe, a lid-driven cavity and the collapse of a water column on a wedge.

The numerical modelling of two-phase mixture flows with high density ratios (e.g. water/air) is challenging. Multiphase averaged models with volume fraction representation encompass a simple way of simulating such flows: mixture models with relative velocity between phases. Such approaches were implemented in SPH (Smoothed Particle Hydrodynamics) using a mass-weighted definition of the mixture velocity, but with limited validation. Instead, to handle high density ratios, a mixture model with a volumetric mixture velocity is developed in this work. To avoid conservation issues raised by the discretization of the relative material displacement contribution in the volume fraction equation, a formulation on phase volumes is derived following a finite volume reasoning. Conservativity, realizability, limit behaviour for single-phase flow are the leading principles of this derivation.Volume diffusion is added to prevent development of instabilities due to the colocated nature of SPH. This model is adapted to the semi-analytical SPH wall boundary conditions. Running on GPU, this approach is successfully applied to the separation of phases in a settling tank with low to high density ratios. An analytical solution on a two-phase mixture Poiseuille flow is also used to check the accuracy of the numerical implementation. Then, a Rayleigh-Taylor instability test case is performed to compare with multi-fluid SPH. Finally, a comparison with experimental and numerical data is made on a sand dumping case; this highlights some limits of this mixture model.

International audienceThis work aims at modelling buoyant, laminar or turbulent flows, using a 2D Incompressible Smoothed Particle Hydrodynamics (ISPH) model with accurate wall boundary conditions. The buoyancy effects are modelled through the Boussinesq approximation coupled to a heat equation, which makes it possible to apply an incompressible algorithm to compute the pressure field from a Poisson equation. Based on our previous work (Leroy et al., 2014), we extend the unified semi-analytical wall boundary conditions to the present model. The latter is also combined to a Reynolds-Averaged Navier-Stokes approach to treat turbulent flows. The k − turbulence model is used, where buoyancy is modelled through an additional term in the k − equations like in mesh-based methods. We propose a unified framework to prescribe isothermal (Dirichlet) or imposed heat flux (Neumann) wall boundary conditions in ISPH. To illustrate this, a theoretical case is presented (laminar heated Poiseuille flow), where excellent agreement with the theoretical solution is obtained. Several benchmark cases are then proposed: a lock-exchange flow, two laminar and one turbulent flow in differentially heated cavities, and finally a turbulent heated Poiseuille flow. Comparisons are provided with a Finite-Volume (FV) approach using an open-source industrial code

Due to the Lagrangian nature of SPH, treating inlet/outlet boundaries (that are intrinsically Eulerian) is a challenging issue. An extension to the Unified SemiAnalytical boundary conditions is presented to deal with unsteady open boundaries in confined and free-surface flows. The presented method uses Riemann invariants to calculate flow properties near the open boundaries, thus allowing the possibility to treat complex shapes. Furthermore, details are presented for a parallel implementation of this method, including particle creation and deletion, updating properties of vertices and segments, and additional constraints on the time step. Simple validation cases are then displayed to illustrate the performance of the proposed method as well as the ability to deal with complex problems such as generation of water waves and free outlets.

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