Alexandre Caboussat

Numerical Simulation of Two-Phase Free Surface Flows

SummaryFree surface flows are of most interest in many engineering or mathematical problems and many methods have been developed for their numerical resolution in various fields of the physics or the engineering. In this work, the volume-of-fluid method is used for the numerical simulation of two-phase free surface flows involving an incompressible liquid and a compressible gas and taking into account the surface tension effects. The incompressible Navier-Stokes equations are assumed to hold in the liquid domain, while the dynamical effects in the ideal gas are disregarded. A time splitting scheme is used together with a two-grids method for the space discretization. An original algorithm is introduced to track the bubbles of gas trapped in the liquid. Numerical results are presented in the frame of mold filling and bubbles and droplets flows. Some theoretical results concerning free boundary problems are also summarized.

An augmented Lagrangian approach to the numerical solution of a non-smooth eigenvalue problem

Abstract In this article, we address the numerical solution of a non-smooth eigenvalue problem, which has implications in plasticity theory and image processing. The smallest eigenvalue of the non-smooth operator under consideration is shown to be the same for all bounded, sufficiently smooth, domains in two space dimensions. Piecewise linear finite elements are used for the discretization of eigenfunctions and eigenvalues. An augmented Lagrangian method is proposed for the computation of the minima of the associated non-convex optimization problem. The convergence of finite element approximations of generalized eigenpairs is investigated. Numerical solutions are presented for the first eigenvalue and eigenfunction. For non-simply connected domains, the augmented Lagrangian method also captures larger eigenvalues as local minima. Bifurcation between the first and second eigenvalues is investigated numerically.

Numerical simulation of two-phase flow with interface tracking by adaptive Eulerian grid subdivision

A numerical model for the simulation of three-dimensional liquid-gas flow with free surfaces is presented. The incompressible Navier-Stokes equations are assumed to hold in the liquid domain, while the surrounding gas is assumed to be compressible, with constant pressure in each bubble of gas. An implicit time splitting scheme couples a method of characteristics for the solution of advection problems, the continuum surface force model for the computation of surface tension effects, and an implicit scheme for the solution of a time dependent Stokes problem. A two-grid method that couples a structured grid of small cells and a finite element mesh of tetrahedrons is used. A novel interface tracking technique involving local adaptive mesh refinement around the interface is detailed to obtain a more accurate approximation of the free surfaces and the surface forces. Numerical experiments, including sloshing and oscillations problems, illustrate the accuracy improvement when using the adaptive Eulerian grid subdivision

A numerical method for interface reconstruction of triple points within a volume tracking algorithm

A numerical method for the reconstruction of interfaces in finite volume schemes for multiphase flows is presented. The computation of the triple point at the intersection of three materials in two dimensions of space is addressed. The determination of the normal vectors between pairs of materials is obtained with a finite element approximation. A numerical method for the localization of a triple point is described as the minimum of a constrained minimization problem inside an interfacial cell of the discretization. For given volume fractions of materials in the cell, an interior-point/Newton method is used for the reconstruction of the local geometry and the localization of the triple point. Initialization of the Newton method is performed with a derivative-free algorithm. Numerical results are presented for static and pure advection cases to illustrate the efficiency and robustness of the algorithm.

A phase equilibrium model for atmospheric aerosols containing inorganic electrolytes and organic compounds (UHAERO), with application to dicarboxylic acids

Computation of phase and chemical equilibria of water-organic-inorganic mixtures is of significant interest in atmospheric aerosol modeling. A new version of the phase partitioning model, named UHAERO, is presented here, which allows one to compute the phase behavior for atmospheric aerosols containing inorganic electrolytes and organic compounds. The computational implementation of the model is based on standard minimization of the Gibbs free energy using a primal-dual method, coupled to a Newton iteration. Water uptake and deliquescence properties of mixtures of aqueous solutions of salts and dicarboxylic acids, including oxalic, malonic, succinic, glutaric, maleic, malic, or methyl succinic acids, are based on a hybrid thermodynamic approach for the modeling of activity coefficients (Clegg and Seinfeld, 2006a, 2006b). UHAERO currently considers ammonium salts and the neutralization of dicarboxylic acids and sulfuric acid. Phase diagrams for sulfate/ammonium/water/dicarboxylic acid systems are presented as a function of relative humidity at 298.15 K over the complete space of compositions.

Solving Optimization-Constrained Differential Equations with Discontinuity Points, with Application to Atmospheric Chemistry

Ordinary differential equations are coupled with mixed constrained optimization problems when modeling the thermodynamic equilibrium of a system evolving with time. A particular application arises in the modeling of atmospheric particles. Discontinuity points are created by the activation/deactivation of inequality constraints. A numerical method for the solution of optimization-constrained differential equations is proposed by coupling an implicit Runge-Kutta method (RADAU5), with numerical techniques for the detection of the events (activation and deactivation of constraints). The computation of the events is based on dense output formulas, continuation techniques, and geometric arguments. Numerical results are presented for the simulation of the time-dependent equilibrium of organic atmospheric aerosol particles, and show the efficiency and accuracy of the approach.

On the modeling and simulation of non-hydrostatic dam break flows

The numerical simulation of three-dimensional dam break flows is discussed. A non-hydrostatic numerical model for free-surface flows is considered, which is based on the incompressible Navier–Stokes equations coupled with a volume-of-fluid approach. The numerical results obtained for a variety of benchmark problems show the validity of the numerical approach, in comparison with other numerical models, and allow to investigate numerically the non-hydrostatic three-dimensional effects, in particular for the usual test cases where hydrostatic approximations are known analytically. The numerical experiments on actual topographies, in particular the Malpasset dam break and the (hypothetical) break of the Grande-Dixence dam in Switzerland, also illustrate the capabilities of the method for large-scale simulations and real-life visualization.

Numerical Methods for the Vector-Valued Solutions of Non-smooth Eigenvalue Problems

In this article, we address the numerical solution of non-smooth eigenvalue problems coming from continuum mechanics. These problems have applications in plasticity theory, since the smallest eigenvalue of the non-smooth operators under consideration appears in the estimation of the cut-off time of some Bingham flows. Three vector-valued eigenvalue problems are investigated. The case of divergence free functions is included. Piecewise linear finite elements are used for the discretization of the eigenfunctions. An augmented Lagrangian method is proposed for the solution of the associated non-convex optimization problem. Numerical solutions are presented for the first eigenpair of these problems and convergence orders are discussed.

Numerical approximation of electromagnetic signals arising in the evaluation of geological formations

The integration of complicated oscillatory functions arises in computational electromagnetics when evaluating signals produced by the propagation of electromagnetic radiation through the anisotropic layers of a geological formation. The computation of exact integrals involves the evaluation of Sommerfeld integrals. The matrix-pencil method is used for the numerical approximation of such signals. Numerical results show accuracy and robustness of the method for the approximation of these signals, and efficiency in their numerical integration. Sampling frequency is discussed and numerical efficiency is improved by down-sampling.

A Numerical Method for Fluid Flows with Complex Free Surfaces

A numerical method for the simulation of fluid flows with complex free surfaces is presented. The liquid is assumed to be a Newtonian or a viscoelastic fluid. The compressible effects of the surrounding gas are taken into account, as well as surface tension forces. An Eulerian approach based on the volume-of-fluid formulation is chosen. A time splitting algorithm, together with a two-grids method, allows the various physical phenomena to be decoupled. A chronological approach is adopted to highlight the successive improvements of the model and the wide range of applications. Numerical results show the potentialities of the method.