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Publication# Approximation numérique des écoulements turbulents dans des cuves d'électrolyse de l'aluminium

Abstract

Aluminium is a metal extracted from bauxite ore using electrolysis process in cells of big size. A huge electric current goes in the cell through an electrolytic bath and aluminium liquid. These currents generate strong magnetic forces that allow the bath and the aluminium to move. A good knowledge of these turbulent flows is very important to optimize the process. The purpose of this thesis is to study and simulate turbulent flows in the aluminium smelting process. These flows are solved numerically with a finite element method. In particular, the Navier-Stokes equations for bifluid flows with free moving interface are solved numerically. In the first part of this work, we develop some mixing-length models that take into account the effects of the wall. A theoretical mathematical study shows the validity of these models and we give some recommendation on the choice of the parameters of the computation. In the second part we study the resolution of the Navier-Stokes equations. The study focuses on algorithms that decouple the computation of the speed and pressure, commonly called projection method or Chorin-Temam algorithm. The final section provides answers on the relevance of wall modelling and projection methods in numerical simulation of turbulent flows in the aluminium smelting process. In particular, we obtain a numerical model that produces a realistic flow with a reasonable CPU time and we discuss the choice of certain parameters involved in the different models.

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The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems).

The Navier–Stokes equations (nævˈjeː_stəʊks ) are partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842-1850 (Stokes). The Navier–Stokes equations mathematically express momentum balance and conservation of mass for Newtonian fluids.

In fluid dynamics, turbulence or turbulent flow is fluid motion characterized by chaotic changes in pressure and flow velocity. It is in contrast to a laminar flow, which occurs when a fluid flows in parallel layers, with no disruption between those layers. Turbulence is commonly observed in everyday phenomena such as surf, fast flowing rivers, billowing storm clouds, or smoke from a chimney, and most fluid flows occurring in nature or created in engineering applications are turbulent.

Jacques Rappaz, Jonathan Rochat

Mixing-length models are often used by engineers in order to take into account turbulence phenomena in a flow. This kind of model is obtained by adding a turbulent viscosity to the laminar one in Navier-Stokes equations. When the flow is confined between two close walls, von Karman's model consists of adding a viscosity which depends on the rate of strain multiplied by the square of distance to the wall. In this short paper, we present a mathematical analysis of such modeling. In particular, we explain why von Karman's model is numerically ill-conditioned when using a finite element method with a small laminar viscosity. Details of analysis can be found in [1], [2].

2019