Philippe Helluy

Numerical schemes for low Mach wave breaking

In this work, we describe a finite volume scheme for the computation of incompressible air–water flows. We use an artificial compressibility approach that permits us to use a completely explicit scheme. We describe successively the low Mach preconditioning of the scheme, the Riemann solver and then the non-conservative approach that is used to suppress velocity-pressure oscillations, the second order extensions and the parallel implementation. Then this is applied to the simulation of the breaking of a wave on a 15% slope.

Multi-Dimensional Two-Phase Flow Modeling Applied to Interior Ballistics

Complex phenomena occur in a combustion chamber during a ballistic cycle. From the ignition of the black powder in the primer to the exit of the projectile through the muzzle, two-phase gas-powder mix undertakes various transfers in different forms. A detailed comprehension of these effects is fundamental to predict the behavior of the whole system, considering performances and safety. Although the ignition of the powder bed is three-dimensional due to the primer’s geometry, simulations generally only deal with one- or two-dimensional problem. In this study, we propose a method to simulate the two-phase flows in 1, 2 or 3 dimensions with the same system of partial differential equations. A one-pressure, conditionally hyperbolic model [1] was used and solved by a nonconservative finite volume scheme associated to a fractional step method, where each step is hyperbolic. We extend our study to a two-pressure, unconditionally hyperbolic model [2] in which a relaxation technique was applied in order to recover the one-pressure model by using the granular stress. The second goal of this study is also to propose an improved ignition model of the powder grains, by taking into account simplified chemical kinetics for decomposition reactions in the two phases. Here we consider a 0th -order solid decomposition and an unimolecular, 2nd -order gas reaction. Validation of the algorithm on several test cases is presented.

A well-balanced approximate Riemann solver for variable cross-section compressible flows

We introduce in this paper a well-balanced approximate Riemann solver in order to compute Euler equations in one-dimensional variable cross section geometries. The scheme takes its roots on the early ideas of Greenberg and Leroux. In order to avoid expensive exact Riemann solvers through cell interfaces, the VFRoe-ncv approximate Godunov scheme is applied, while focusing on a specific change of variable. This choice preserves non-trivial steady states on any mesh size. Details on the scheme and its properties are exhibited, and numerical results are presented afterwards.

Pressure laws and fast Legendre transform

In this paper we investigate algorithms based on the Fast Legendre Transform (FLT) in order to compute tabulated Equation Of State (EOS) for fluids with phase transition. The equation of state of a binary mixture is given by an energy minimization principle. According to the miscible or immiscible nature of the mixture, the energy of the system is either a convex envelope or an inf-convolution of the energies of the two phases. Because these operations are closely linked to Legendre transform, it is possible to construct fast algorithms that compute efficiently these operations. In addition, it appears that the natural mathematical tool for studying mixture thermodynamics in the Legendre space is the max-plus algebra theory.

A well-balanced approximate Riemann solver for compressible flows in variable cross-section ducts

A well-balanced approximate Riemann solver is introduced in this paper in order to compute approximations of one-dimensional Euler equations in variable cross-section ducts. The interface Riemann solver is grounded on the VFRoe-ncv scheme, and it enforces the preservation of Riemann invariants of the steady wave. The main properties of the scheme are detailed. We provide numerical results to assess the validity of the scheme, even when the cross-section is discontinuous. A first series is devoted to analytical test cases, and the last results correspond to the simulation of a bubble collapse.

Asynchronous OpenCL/MPI numerical simulations of conservation laws

Hyperbolic conservation laws are important mathematical models for describing many phenomena in physics or engineering. The Finite Volume (FV) method and the Discontinuous Galerkin (DG) methods are two popular methods for solving conservation laws on computers. Those two methods are good candidates for parallel computing: • they require a large amount of uniform and simple computations, • they rely on explicit time-integration • they present regular and local data access pattern. In this paper, we present several FV and DG numerical sim- ulations that we have realized with the OpenCL and MPI paradigms. First, we compare two optimized implementations of the FV method on a regular grid: an OpenCL implementation and a more traditional OpenMP implementation. We compare the efficiency of the approach on several CPU and GPU architectures of different brands. Then we give a short presentation of the DG method. Finally, we present how we have implemented this DG method in the OpenCL/MPI framework in order to achieve high efficiency. The implementation relies on a splitting of the DG mesh into sub-domains and sub-zones. Different kernels are compiled according to the zones properties. In addition, we rely on the OpenCL asynchronous task graph in order to overlap OpenCL computations, memory transfers and MPI communications.

Interpolated Pressure Laws in Two-Fluid Simulations and Hyperbolicity

We consider a two-fluid compressible flow. Each fluid obeys a stiffened gas pressure law. The continuous model is well defined without considering mixture regions. However, for numerical applications it is often necessary to consider artificial mixtures, because the two-fluid interface is diffused by the numerical scheme. We show that classic pressure law interpolations lead to a non-convex hyperbolicity domain and failure of well-known numerical schemes. We propose a physically relevant pressure law interpolation construction and show that it leads to a necessary modification of the pure phase pressure laws. We also propose a numerical scheme that permits to approximate the stiffened gas model without artificial mixture.

Highly gravity-driven flow of a NAPL in water-saturated porous media using the discontinuous Galerkin finite-element method with a generalised Godunov scheme

In this paper, we develop an implementation of gravity effects within a Global Pressure formulation in a numerical scheme based on the implicit pressure explicit saturation (IMPES) approach. We use the Discontinuous Galerkin Finite-Element Method (DGFEM) combined with a generalised Godunov scheme to model an immiscible two-phase flow with predominant gravity effects. The saturation profile of a displacing non-aqueous phase liquid (NAPL) in an initially water-saturated porous medium depends strongly on the ratio between the total specific discharge and the density difference between the NAPL and water. We discuss the solution of the nonlinear Buckley-Leverett equation for the general case in which the flux function is non-monotonic. Using a detailed functional analysis of the characteristics of the given hyperbolic equation, three limit cases are identified as significant for modelling the shock and rarefaction regions. The derived maximum (or entry) and front saturations of NAPL are functions of the viscosity ratio M and the gravity number G. We first test the developed numerical model in the case of a one-dimensional highly gravity-driven flow of NAPL within a homogeneous porous medium. The numerically calculated NAPL entry and front saturations of NAPL agree well with the theoretical values. Furthermore, the numerical diffusion of the shock front is lower than that of the calculated using a first-order Finite Volume method, which is generally used in reservoir engineering because of its robustness. Finally, we apply the developed DGFEM scheme to a 2D heterogeneous porous medium and analyse its capability of modelling the non-uniform saturation field using spatial moment analysis.

A Simple Model for Cavitation with Non-condensable Gases

In this paper, we propose a numerical method to model the dynamical behavior of a spherical bubble of vapor and air inside water. The air is assumed to be miscible with the vapor. Each phase is described by a stiffened gas law and the mixture pressure law is recovered by an entropy maximization process. We use an adaptive finite volume solver in order to solve the Euler equations. Numerical experiments are presented that validate our approach.

Conservative scheme for two-fluid compressible flows without pressure oscillations

Compressible two-fluid flows are difficult to numerically simulate. Indeed, classic conservative finite volume schemes do not preserve the velocity and pressure equilibrium at the two-fluid interface. This leads to oscillations, lack of precision and even, in some liquid-gas simulations, to the crash of the computation. Several cures have been proposed to obtain better schemes (see [1] and included references). The resulting schemes are generally not conservative. Based on ideas of [2], we propose a new Lagrange-Projection scheme. The projection step is based on a random sampling strategy at the interface. The scheme has the following properties: it preserves constant velocity and pressure at the two-fluid interface, it preserves a perfectly sharp interface and it is fully conservative (in a statistical sense). The scheme can be extended to higher space dimensions through Strang dimensional splitting. Finally, it is very simple to implement and thus well adapted to massively parallel GPU computations.The progress in the theory of hyperbolic conservation laws has always been and still is driven strongly by new fields of applications. The workshop addressed aspects of modelling, analysis and numerics for funda- mental problems at the interface between hyperbolic evolution and the emerg- ing mathematical theories of complex multiphasic materials. This includes problems in fluid and solid mechanics but also very recent applications in areas like swarm and traffic modelling.