Damien Violeau

Application of weakly compressible and truly incompressible SPH to 3-D water collapse in waterworks

Two algorithms of the SPH Lagrangian numerical method, the first weakly compressible, the second truly incompressible, are presented and applied to two free-surface three-dimensional flows. The first (schematic) case consists of a water column collapsing in a rectangular tank with a central rectangular obstacle, and allows the comparison and validation of both algorithms. It appears that the incompressible method is superior to predict the total strength experienced by the obstacle, while the weakly compressible method shows weaknesses under this criterion. The second application case, very close to an industrial study, represents a “ski-jump” spillway connecting the reservoir of a river dam to a valley with complex bottom shape. The global flow pattern is compared to laboratory observations from a physical model, leading to satisfactory conclusions which prove SPH has the potential to be a promising method for the design of complex waterworks.

Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future

ABSTRACT This paper assesses some recent trends in the novel numerical meshless method smoothed particle hydrodynamics, with particular focus on its potential use in modelling free-surface flows. Due to its Lagrangian nature, smoothed particle hydrodynamics (SPH) appears to be effective in solving diverse fluid-dynamic problems with highly nonlinear deformation such as wave breaking and impact, multi-phase mixing processes, jet impact, sloshing, flooding and tsunami inundation, and fluid–structure interactions. The paper considers the key areas of rapid progress and development, including the numerical formulations, SPH operators, remedies to problems within the classical formulations, novel methodologies to improve the stability and robustness of the method, boundary conditions, multi-fluid approaches, particle adaptivity, and hardware acceleration. The key ongoing challenges in SPH that must be addressed by academic research and industrial users are identified and discussed. Finally, a roadmap is proposed for the future developments.

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

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-@e 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 in terms of pressure prediction.

Investigation of wall bounded flows using SPH and the unified semi-analytical wall boundary conditions

Abstract Semi-analytical wall boundary conditions present a mathematically rigorous framework to prescribe the influence of solid walls in smoothed particle hydrodynamics (SPH) for fluid flows. In this paper they are investigated with respect to the skew-adjoint property which implies exact energy conservation. It will be shown that this property holds only in the limit of the continuous SPH approximation, whereas in the discrete SPH formulation it is only approximately true, leading to numerical noise. This noise, interpreted as a form of “turbulence”, is treated using an additional volume diffusion term in the continuity equation which we show is equivalent to an approximate Riemann solver. Subsequently two extensions to the boundary conditions are presented. The first dealing with a variable driving force when imposing a volume flux in a periodic flow and the second showing a generalization of the wall boundary condition to Robin type and arbitrary-order interpolation. Two modifications for free-surface flows are presented for the volume diffusion term as well as the wall boundary condition. In order to validate the theoretical constructs numerical experiments are performed showing that the present volume flux term yields results with an error 5 orders of magnitude smaller then previous methods while the Robin boundary conditions are imposed correctly with an error depending on the order of the approximation. Furthermore, the proposed modifications for free-surface flows improve the behavior at the intersection of free surface and wall as well as prevent free-surface detachment when using the volume diffusion term. Finally, this paper is concluded by a simulation of a dam break over a wedge demonstrating the improvements proposed in this paper.

Unified semi-analytical wall boundary conditions in SPH: analytical extension to 3-D

Solid wall boundary conditions have been an area of active research within the context of Smoothed Particle Hydrodynamics (SPH) for quite a while. Ferrand et al. (Int. J. Numer. Methods Fluids 71(4), 446–472, 2012) presented a novel approach using a renormalization factor in the SPH approximation. The computation of this factor depends on an integral along the boundary of the domain and in their original paper Ferrand et al. gave an analytical formulation for the 2-D case using the Wendland kernel. In this paper the formulation will be extended to 3-D, again providing analytical formulae. Due to the boundary being two dimensional a domain decomposition algorithm needs to be employed in order to obtain special integration domains. For these the analytical formulae will be presented when using the Wendland kernel. The algorithm presented within this paper is applied to several academic test-cases for which either analytical results or simulations with other methods are available. It will be shown that the present formulation produces accurate results and provides a significant improvement compared to approximative methods.

On the maximum time step in weakly compressible SPH

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 discretisation. 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.

Explicit algebraic Reynolds stresses and scalar fluxes for density-stratified shear flows

An explicit algebraic model for Reynolds stresses and active scalar turbulent fluxes is proposed for simply stratified, stable shear flows such as atmospheric boundary layers of estuaries, including buoyant effects due to density variations (temperature, salinity). Under equilibrium assumptions, these models lead to explicit functions of the gradient Richardson number Ri that model the damping of the eddy viscosity and eddy diffusivity coefficients. The present model is compared to other existing theoretical approaches, as well as experimental observations, with satisfactory agreement. The evolution of the turbulent Prandtl number as a function of Ri is correctly predicted and its neutral value is consistent with existing values in the literature. The behavior of the model predicts that internal waves are responsible for momentum diffusion at large Ri, consistently with recent publications. The properties of turbulence anisotropy are briefly investigated.

Optimal time step for incompressible SPH

A classical incompressible algorithm for Smoothed Particle Hydrodynamics (ISPH) is analyzed in terms of critical time step for numerical stability. For this purpose, a theoretical linear stability analysis is conducted for unbounded homogeneous flows, leading to an analytical formula for the maximum CFL (Courant-Friedrichs-Lewy) number as a function of the Fourier number. This gives the maximum time step as a function of the fluid viscosity, the flow velocity scale and the SPH discretization size (kernel standard deviation). Importantly, the maximum CFL number at large Reynolds number appears twice smaller than with the traditional Weakly Compressible (WCSPH) approach. As a consequence, the optimal time step for ISPH is only five times larger than with WCSPH. The theory agrees very well with numerical data for two usual kernels in a 2-D periodic flow. On the other hand, numerical experiments in a plane Poiseuille flow show that the theory overestimates the maximum allowed time step for small Reynolds numbers.

Foreword: SPH for free-surface flows

(2010). Foreword: SPH for free-surface flows. Journal of Hydraulic Research: Vol. 48, No. sup1, pp. 3-5.

A new open boundary formulation for incompressible SPH

Abstract In this work a new formulation for inflow/outflow boundary conditions in an incompressible Smoothed Particles Hydrodynamics (ISPH) model is proposed. It relies on the technique of unified semi-analytical boundary conditions that was first proposed for wall boundary conditions in 2013, then extended to open boundaries in the framework of weakly-compressible SPH (WCSPH). An ISPH model relying on that formulation for solid boundaries was then proposed, which is the one considered here. It includes a buoyancy model for temperature effects and a k − ϵ turbulence closure. There are two main requirements for the imposition of open boundaries in ISPH: an algorithm to let particles enter and leave the domain, and the correct imposition of open boundary conditions on the fields. Regarding the algorithm for particles creation/destruction, it relies on the variation of mass of the particles located at the open boundaries. When the mass of such a particle reaches a threshold, a new particle is released. On the other hand, the imposition of open boundary conditions on the fields is done by prescribing the value of the boundary terms appearing in the semi-analytical formulation. The formulation was first validated in 2-D on a cut dam-break, a case of propagation of a solitary wave and a Creager weir. It was then extended to 3-D and tested on a 3-D circular pipe. A preliminary application case consisting of two connected pipes at different temperatures was then simulated. The results are promising since in all cases the fluid enters and leaves the domain as prescribed and generating none or very few reflected waves.