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Publication# Interaction fluide-structure par la méthode des éléments spectraux

Abstract

We elaborate in this thesis the numerical simulation of the fluid-structure interaction by the spectral element method. To this end we consider the Navier-Stokes equations for a viscous Newtonian incompressible fluid with an elastic solid the movement of which being described by the equations of the dynamics. The arbitrary Lagrangian Eulerian (ALE) formulation is introduced in the fluid governing equations to deal with the structure movement that is described in Lagrangian representation. The geometrical motion in each domain is built up by the mesh deformation. The space-time discretization of the full mathematical model rests upon the spectral element method. The solid is discretized in the space of polynomials of degree N, PN, while the fluid uses the PN - PN-2 approach. The efficiency of the ALE formulation is tested and validated through various applications. The full algorithm is based on a particular case of the staggered method. The simplified case of a solid immersed in a plane channel closes the thesis. It is then possible to draw the conclusions about the pros and cons of the proposed methodology.

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The research work reported in the present dissertation is aimed at the analysis of complex physical phenomena involving instabilities and nonlinearities occurring in fluids through state-of-the-art numerical modeling. Solutions of intricate fluid physics problems are devised in two particularly arduous situations: fluid domains with moving boundaries and the high-Reynolds-number regime dominated by nonlinear convective effects. Shear-driven flows of incompressible Newtonian fluids enclosed in cavities of varying geometries are thoroughly investigated in the two following frameworks: transition with a free surface and confined turbulence. The physical system we consider is made of an incompressible Newtonian fluid filling a bounded, or partially bounded cavity. A series of shear-driven flows are easily generated by setting in motion some part of the container boundary. These driven-cavity flows are not only technologically important, they are of great scientific interest because they display almost all physical fluid phenomena that can possibly occur in incompressible flows, and this in the simplest geometrical settings. Thus corner eddies, secondary flows, longitudinal vortices, complex three-dimensional patterns, chaotic particle motions, nonuniqueness, transition, and turbulence all occur naturally and can be studied in the same geometry. This facilitates the comparison of results from experiments, analysis, and computation over the whole range of Reynolds numbers. The flows under consideration are part of a larger class of confined flows driven by linear or angular momentum gradients. This dissertation reports a detailed study of a novel numerical method developed for the simulation of an unsteady free-surface flow in three-space-dimensions. This method relies on a moving-grid technique to solve the Navier-Stokes equations expressed in the arbitrary Lagrangian-Eulerian (ALE) kinematics and discretized by the spectral element method. A comprehensive analysis of the continuous and discretized formulations of the general problem in the ALE frame, with nonlinear, non-homogeneous and unsteady boundary conditions is presented. In this dissertation, we first consider in the internal turbulent flow of a fluid enclosed in a bounded cubical cavity driven by the constant translation of its lid. The solution of this flow relied on large-eddy simulations, which served to improve our physical understanding of this complex flow dynamics. Subsequently, a novel subgrid model based on approximate deconvolution methods coupled with a dynamic mixed scale model was devised. The large-eddy simulation of the lid-driven cubical cavity flow based on this novel subgrid model has shown improvements over traditional subgrid-viscosity type of models. Finally a new interpretation of approximate deconvolution models when used with implicit filtering as a way to approximate the projective grid filter was given. This led to the introduction of the grid filter models. Through the use of a newly-developed method of numerical simulation, in this dissertation we solve unsteady flows with a flat and moving free-surface in the transitional regime. These flows are the incompressible flow of a viscous fluid enclosed in a cylindrical container with an open top surface and driven by the steady rotation of the bottom wall. New flow states are investigated based on the fully three-dimensional solution of the Navier-Stokes equations for these free-surface cylindrical swirling flows, without resorting to any symmetry properties unlike all other results available in the literature. To our knowledge, this study delivers the most general available results for this free-surface problem due to its original mathematical treatment. This second part of the dissertation is a basic research task directed at increasing our understanding of the influence of the presence of a free surface on the intricate transitional flow dynamics of shear-driven flows.

Mathematical and numerical aspects of free surface flows are investigated. On one hand, the mathematical analysis of some free surface flows is considered. A model problem in one space dimension is first investigated. The Burgers equation with diffusion has to be solved on a space interval with one free extremity. This extremity is unknown and moves in time. An ordinary differential equation for the position of the free extremity of the interval is added in order to close the mathematical problem. Local existence in time and uniqueness results are proved for the problem with given domain, then for the free surface problem. A priori and a posteriori error estimates are obtained for the semi-discretization in space. The stability and the convergence of an Eulerian time splitting scheme are investigated. The same methodology is then used to study free surface flows in two space dimensions. The incompressible unsteady Navier-Stokes equations with Neumann boundary conditions on the whole boundary are considered. The whole boundary is assumed to be the free surface. An additional equation is used to describe the moving domain. Local existence in time and uniqueness results are obtained. On the other hand, a model for free surface flows in two and three space dimensions is investigated. The liquid is assumed to be surrounded by a compressible gas. The incompressible unsteady Navier-Stokes equations are assumed to hold in the liquid region. A volume-of-fluid method is used to describe the motion of the liquid domain. The velocity in the gas is disregarded and the pressure is computed by the ideal gas law in each gas bubble trapped by the liquid. A numbering algorithm is presented to recognize the bubbles of gas. Gas pressure is applied as a normal force on the liquid-gas interface. Surface tension effects are also taken into account for the simulation of bubbles or droplets flows. A method for the computation of the curvature is presented. Convergence and accuracy of the approximation of the curvature are discussed. A time splitting scheme is used to decouple the various physical phenomena. Numerical simulations are made in the frame of mould filling to show that the influence of gas on the free surface cannot be neglected. Curvature-driven flows are also considered.

Jean-Luc Desbiolles, Jacques Rappaz, Michel Rappaz

Micro-macrosegregation calculations have been performed for a rectangular cavity containing either a Pb-48 wt pct Sn alloy or a Sn-5 wt pct Pb alloy. The numerical results calculated with a finite volume method (FVM) and a finite element method (FEM) are compared with experimental results previously obtained by Hebditch and Hunt.([1]) The two methods are based on the same average conservation equations governing heat and mass transfer and the same assumptions: lever rule, equal and constant density of the solid and liquid phases (except in the buoyancy term), permeability of the mushy zone given by the Carman-Kozeny relation, and no transport of the solid phase. Although the same parameters are used in both calculations, small differences are observed as a result of the different formulations. In particular, the instabilities appearing in the mushy zone (channels) of the Sn-5 wt pct Pb alloy are more pronounced with the FVM formulation as compared with FEM, whereas the opposite trend is observed for the Pb-48 wt pct Sn alloy. Nevertheless, the final segregation maps at the end of solidification compare fairly well with the experimental findings.

1998