Etude de methodes interatives pour la resolution des equations de Navier-Stokes

In the framework of developing software for the numerical simulation of turbulent flows in hydraulic turbines, this thesis presents an analysis of solution algorithms and strategies for Navier-Stokes equations with respect to iterative solvers. This study is required since the discretisation of Navier-Stokes equations in three-dimensional computational domains like hydraulic turbines components leads to a large system of equations. For these matrix systems, direct solvers cannot be used in combination with current computer resources since they require excessive amounts of memory. With iterative solvers, since the matrix only appears in matrix-vector products, relatively small amounts of memory are required for sparse matrices for which we only store non-zero terms. However, for iterative solvers applied to nonsymmetric matrices, convergence is not guaranteed.;High Reynolds flows are modeled by a Reynolds Averaged Navier-Stokes formulation coupled with a turbulence model. Matrix systems resulting from finite element discretisation using a Galerkin formulation are analysed in detail. This formulation leads to a system of equations, augmented by a set of linear equality constraints. Moreover, the resulting systems of equations are characterised by nonsymmetric matrices.;To solve linearly constrained problems, the mixed method, Uzawa's augmented Lagrangian method, the orthogonal projection method and SIMPLER-type algorithms are analysed. The behaviour of various iterative solvers is studied for typical Navier-Stokes problems. The analysis shows that the choice of the solution algorithm for Navier-Stokes equations has a major impact on the characteristics of matrix systems of equations. Hence Uzawa's augmented Lagrangian method, which is particularly suited to use of direct solvers, requires the solution of matrix systems with high condition number. The analysis also demonstrates that this formulation does not necessarily lead to accurate solutions with iterative solvers. It is demonstrated that the SIMPLER algorithm and its variants lead to the solution of two matrix systems; a symmetric positive definite matrix, and a nonsymmetric system, whose convergence properties have been improved by a pseudo-time term.;A segregated strategy is used to further decrease the need of memory storage for large scale systems. This strategy cannot be used efficiently with Uzawa's augmented Lagrangian and orthogonal projection method. However, it is particularly suited for SIMPLER method.;Following this work, recommendations are made about an analysis and an implementation of preconditioning techniques in order to produce more robust solution techniques based on iterative solvers. The analysis of SIMPLER algorithm and other schemes used with finite volume methods should be pursued to take advantage of the experience developed in this field. Semi-implicit projection methods used to compute transient solutions to Navier-Stokes equations should be also evaluated. Finally, least-squares variational formulations for Navier-Stokes need to be studied to evaluate their impact on the convergence behaviour of matrix solutions algorithms.