Résumé : Ce travail de thèse est consacré à l'étude numérique de la forme instationnaire des équations de Navier--Stokes incompressibles en dimension trois d'espace. Ces équations sont étudiées dans leur formulation classique mais aussi dans leur formulation hydrostatique. Le schéma en temps est résolu à l'aide d'une méthode de projection, la méthode de projection incrémentale standard qui sert de référence à l'étude numérique des nouvelles méthodes proposées: la méthode de projection hydrostatique et la méthode de projection dynamique étendue. Ces méthodes sont combinées à la méthode de pénalisation de volume qui permet de tenir compte de la présence d'obstacles de forme quelconque dans le domaine de simulation en utilisant un maillage cartésien. Une nouvelle expression des équations de Navier--Stokes hydrostatiques issue de l'application de la méthode de pénalisation de volume est obtenue et étudiée. La discrétisation en espace s'effectue en utilisant la méthode des volumes finis sur une maillage décalé de type MAC. Un nouveau code de calcul baptisé VivAn'O a été conçu et validé durant ce travail. Il a permis l'étude numérique des différentes méthodes proposées et a également permis de réaliser des simulations dans la zone portuaire de Pointe-à-Pitre à partir de données bathymétriques réelles.
The present work is about the numerical study of the three dimensional form of the unstationnary Navier--Stokesequations. These equations are studied under their classical form but also under their hydrostatic one. Thetime--dependent problem is solved using a projection method called the standard incremental projection method which isused as the reference to the numerical study of the new projection methods we proposed : the hydrostatic projectionmethod and the extended dynamical projection method. These methods are combined with a volume penalization methodto take into account obstacles of any shape in the simulation domain using cartesian meshes. A new formulation of the hydrostatic Navier--Stokes equations is obtained from the use of the penalization method and studied. The space isdiscretized using the finite volume method on a staggered MAC mesh.A new computational fluid dynamic code nammed VivAn'O have been developped during this work. It had permitted thenumerical study of the proposed methods. It was also used to carry out simulations in the harbour area of Pointe--à--Pitre usine real bathvmetric data.

http://www.theses.fr/2013AGUY0631

Éditeur(s) : HAL CCSD Springer Verlag
Résumé : International audience
This article is intended as a preliminary report on the implementation of a finite volume multilevel scheme for the discretization of the incompressible Navier-Stokes equations. As is well known, the use of staggered grids (e.g., MAC grids [M. Perić, R. Kessler and G. Scheuerer, Comput. & Fluids 16 (1988), no. 4, 389-403; Zbl 0672.76018]) is a serious impediment for the implementation of multilevel schemes in the context of finite differences. This difficulty is circumvented here by the use of a colocated finite volume discretization [S. Faure, J Laminie and R. Temam, "Colocated finite volume schemes for fluid flows'', submitted; M. Perić, R. Kessler and G. Scheuerer, op. cit.], for which the algebra of multilevel methods is much simpler than in the context of MAC type finite differences. The general ideas and the numerical simulations are presented in this article in the simplified context of a two-dimensional Burgers equation; the two- and three-dimensional Navier-Stokes equations, which introduce new difficulties related to the incompressibility condition and the time discretization, will be considered elsewhere [see S. Faure, J Laminie and R. Temam, op. cit.; "Finite volume discretization and multilevel methods for the Navier-Stokes equations'', in preparation].
ISSN: 0885-7474

hal-00730193
https://hal.archives-ouvertes.fr/hal-00730193

Éditeur(s) : HAL CCSD Wiley
Résumé : International audience
In this paper, the generalized Stokes problem αu−νΔu+∇p=f, ∇u=0, is considered, where ν denotes the kinematic viscosity, α is a positive parameter proportional to the inverse of the time-step, and the source term f gathers the external forcing, the nonlinear term, and the explicit part of the linear term. The system is derived from a time discretization for the Navier-Stokes equation governing the motion of a two-dimensional incompressible fluid. A domain decomposition numerical method is developed for solving the system. The original domain is divided into several nonoverlapping subdomains, and in each subdomain the concerned equations are discretized by certain finite element methods. A new feature of the method is that the pressure p is required to be piecewise linear rather than piecewise constant, so that the nonphysical oscillations in the solution using piecewise linear pressure are eliminated. However, a difficulty in using piecewise linear pressure is that the Stokes problem in each subdomain involves Dirichlet boundary conditions on pressure at the interface, which is not a real Stokes problem. To avoid the difficulty, a modification is proposed which removes the continuity conditions on the pressure across the interface. In this way, the Stokes problem in each subdomain has only Dirichlet boundary conditions on the velocity. Formulations and discussions for both the methods are presented, which shows that the modified method is simpler than the original one. Numerical experiments are displayed to show the efficiency of the methods, especially that the modified method has the same accuracy as the original one.
ISSN: 0749-159X

hal-00730200
https://hal.archives-ouvertes.fr/hal-00730200

Éditeur(s) : HAL CCSD Global Science Press
Résumé : International audience
In this paper, the authors present two different time-discretization schemes combined with a finite-volume space discretization for the incompressible Navier-Stokes equations. They show that the collocated space discretization, usually used with a projection method, can also be extended to other time discretizations over previous projection methods, like the splitting methods. The main advantage of this splitting method is that the pressure approximation is truly of second order. In the first and second parts of this article, two different time-discretization schemes are considered with a collocated space discretization and it is explained how the unknown can be correctly coupled. Numerical simulations are presented in the last part. Tests of spatial and time accuracies are made. The numerical results when the Navier-Stokes problem is solved are shown, in the model problem of the driven cavity.
ISSN: 1815-2406

hal-00730191
https://hal.archives-ouvertes.fr/hal-00730191

Éditeur(s) : HAL CCSD Elsevier
Résumé : International audience
ISSN: 0377-0427

hal-00530820
https://hal.archives-ouvertes.fr/hal-00530820
DOI : 10.1016/j.cam.2008.08.014

Éditeur(s) : HAL CCSD EDP Sciences
Résumé : International audience
We show first results of 3D simulation of sea currents in a realistic context. We use the full Navier-Stokes equations for incompressible viscous fluid. The problem is solved using a second order incremental projection method associated with the finite volume of the staggered (MAC) scheme for the spatial discretization. After validation on classical cases, it is used in a numerical simulation of the Pointe à Pitre harbour area. The use of the fictious domain method permits us to take into account the complexity of bathymetric data and allows us to work with regular meshes and thus preserves the efficiency essential for a 3D code.
ISSN: 1270-900X
Guidel, France

hal-00728894
https://hal.archives-ouvertes.fr/hal-00728894
DOI : 10.1051/proc/201235025

Éditeur(s) : HAL CCSD
Résumé : International audience
We show first results of 3D simulation of sea currents in a realistic context. We use the full Navier-Stokes equations for incompressible viscous fluid. The problem is solved using a second order incremental projection method associated with the finite volume of the staggered (MAC) scheme for the spatial discretization. After validation on classical cases, it is used in a numerical simulation of the Pointe à Pitre harbour area. The use of the fictious domain method permits us to take into account the complexity of bathymetric data and allows us to work with regular meshes and thus preserves the efficiency essential for a 3D code.
ESAIM : Proceedings
Guidel, France

hal-00770270
https://hal.univ-antilles.fr/hal-00770270
DOI : 10.1051/proc/201235025