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

International audienceThis 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- 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

Efficient asymptotic preserving schemes for BGK and ES-BGK models on cartesian grids

This work is devoted to the study of complex flows where hydrodynamic and rarefied regimes coexist. This kind of flows are found in vacuum pumps or hypersonic re-entries of space vehicles where the distance between gas molecules is so large that their microscopic behaviour differ from the average behaviour of the flow and has be taken into account. We then consider two models of the Boltzmann equation viable for such flows: the BGK model dans the ES-BGK model. We first devise a new wall boundary condition ensuring a smooth transition of the solution from the rarefied regime to the hydrodynamic regime. We then describe how this boundary condition (and boundary conditions in general) can be enforced with second order accuracy on an immersed body on Cartesian grids preserving the asymptotic limit towards compressible Euler equations. We exploit the ability of Cartesian grids to massive parallel computations (HPC) to drastically reduce the computational time which is an issue for kinetic models. A new approach considering local velocity grids is then presented showing important gain on the computational time (up to 80$\%$). 3D simulations are also presented showing the efficiency of the methods. Finally, solid particle transport in a rarefied flow is studied. The kinetic model is coupled with a Vlasov-type equation modeling the passive particle transport solved with a method based on remeshing processes. As application, we investigate the realistic test case of the pollution of optical devices carried by satellites due to incompletely burned particles coming from the altitude control thrusters

Multiscale modeling of rapid granular flow with a hybrid discrete-continuum method

Both discrete and continuum models have been widely used to study rapid granular flow, discrete model is accurate but computationally expensive, whereas continuum model is computationally efficient but its accuracy is doubtful in many situations. Here we propose a hybrid discrete-continuum method to profit from the merits but discard the drawbacks of both discrete and continuum models. Continuum model is used in the regions where it is valid and discrete model is used in the regions where continuum description fails, they are coupled via dynamical exchange of parameters in the overlap regions. Simulation of granular channel flow demonstrates that the proposed hybrid discrete-continuum method is nearly as accurate as discrete model, with much less computational cost

A non-hybrid method for the PDF equations of turbulent flows on unstructured grids

In probability density function (PDF) methods of turbulent flows, the joint PDF of several flow variables is computed by numerically integrating a system of stochastic differential equations for Lagrangian particles. A set of parallel algorithms is proposed to provide an efficient solution of the PDF transport equation, modeling the joint PDF of turbulent velocity, frequency and concentration of a passive scalar in geometrically complex configurations. An unstructured Eulerian grid is employed to extract Eulerian statistics, to solve for quantities represented at fixed locations of the domain (e.g. the mean pressure) and to track particles. All three aspects regarding the grid make use of the finite element method (FEM) employing the simplest linear FEM shape functions. To model the small-scale mixing of the transported scalar, the interaction by exchange with the conditional mean model is adopted. An adaptive algorithm that computes the velocity-conditioned scalar mean is proposed that homogenizes the statistical error over the sample space with no assumption on the shape of the underlying velocity PDF. Compared to other hybrid particle-in-cell approaches for the PDF equations, the current methodology is consistent without the need for consistency conditions. The algorithm is tested by computing the dispersion of passive scalars released from concentrated sources in two different turbulent flows: the fully developed turbulent channel flow and a street canyon (or cavity) flow. Algorithmic details on estimating conditional and unconditional statistics, particle tracking and particle-number control are presented in detail. Relevant aspects of performance and parallelism on cache-based shared memory machines are discussed.Comment: Accepted in Journal of Computational Physics, Feb. 20, 200

A numerical investigation into the correction algorithms for SPH method in modeling violent free surface flows

A quantitative comparison of the usual and recent numerical treatments which are applied to the Smoothed Particle Hydrodynamics (SPH) method are presented together with a new free-surface treatment. A series of numerical treatments are studied to refine the numerical procedures of the SPH method particularly for violent flows with a free surface. Two dimensional dam-break and sway-sloshing problems in a tank are modeled by solving Euler's equation of motion utilizing weakly compressible SPH method (WCSPH). Initially, the dam-break benchmark problem is studied by adopting only conventional basic equations of SPH without any numerical remedy and then by considering numerical treatments of interest one after another. In the WCSPH method, the precise calculation of the densities of the particles is vital for the solution, accordingly a density correction algorithm is presented as a basic numerical treatment. Subsequently, Monaghan's (1994) [1] XSPH velocity variant algorithm, artificial particle displacement (APD) algorithm (Shaldoo et al., 2011) [2], and a hybrid combination of velocity updated XSPH (VXSPH) and APD algorithms are implemented separately, but all with the density correction algorithm as a default treatment. The effects of each of these treatments on the pressure and on the free surface profiles are analyzed by comparing our numerical findings with experimental and numerical results in the literature. After the detailed scrutiny on the dam-break problem, sway-sloshing problem is handled with the VXSPH+APD algorithm which has been noted to provide the most reliable and accurate results in the dam-break problem. For the sway-sloshing problem, the time histories of free surface elevations on the left side wall of the rectangular tank are compared with experimental and numerical results available in the literature. It was shown that the VXSPH+APD treatment significantly improves the accuracy of the numerical simulations for violent flows with a free surface and lead to the results which are in very good agreement with experimental and numerical findings of literature in terms of both the kinematic and the dynamic point of view

Analysis of the incompressibility constraint in the Smoothed Particle Hydrodynamics method

Smoothed particle hydrodynamics is a particle-based, fully Lagrangian, method for fluid-flow simulations. In this work, fundamental concepts of the method are first briefly recalled. Then, we present a thorough comparison of three different incompressibility treatments in SPH: the weakly compressible approach, where a suitably-chosen equation of state is used; and two truly incompressible methods, where the velocity field projection onto a divergence-free space is performed. A noteworthy aspect of the study is that, in each incompressibility treatment, the same boundary conditions are used (and further developed) which allows a direct comparison to be made. Problems associated with implementation are also discussed and an optimal choice of the computational parameters has been proposed and verified. Numerical results show that the present state-of-the-art truly incompressible method (based on a velocity correction) suffer from density accumulation errors. To address this issue, an algorithm, based on a correction for both particle velocities and positions, is presented. The usefulness of this density correction is examined and demonstrated in the last part of the paper