Analysis and Numerical Simulation of Free Surface 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.