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Modeling and numerical analysis of free surface flows

Abstract : In this thesis, we are interested by Finite Elements methods for the three-dimensional free surface Navier--Stokes equations under the ALE formulation. They enable to simulate geophysical flows. The initial and main goal is to analyse the existing limitations of these numerical methods and to provide perspectives of improvement, justified mathematically. This purpose helps us to present a review and improvement way for Telemac-3D, which is a hydrodynamics industrial software developed by the Laboratoire National d'Hydraulique et Environnement of EDF R&D. Therefore, we analyse precisely and we evaluate this algorithm, with respect to the recent scientific publications. This software solves the free surface Navier--Stokes equations with the decomposition of the pressure through a hydrostatic part and a dynamic part. A major limitation is that the velocity field of the fluid is not divergence-free. Furthermore, we highlight a time restriction on the time step. Moreover, alternative approaches are studied and compared. In particular, we focus on a numerical strategy which consists in advecting the free surface, in updating the domain and in solving the Navier--Stokes equations. Based on this strategy, we analyze a first order explicit scheme in time with a Finite Elements stabilization term. The numerical method allows to ensure important properties: the mass conservation of the water quantity and the weak free divergence condition. We demonstrate that this scheme is conditionally stable in time. Besides, we propose a new variational formulation allowing to obtain a semi-implicit scheme in time combined with the Finite Elements method, which is stable independently from the velocity of the mesh and without an exact free divergence velocity. Finally, in order to expand the hydrodynamic knowledges, some simplified models used in other software developed by EDF R&D are studied. In particular, we focus on the mild-slope equation solved in the software Artemis. It is an asymptotic model derived from the linear water wave equation. As a consequence, we study the hypothesis and the validity of the derivation. An approximate analytical solution is additionally derived for this purpose. Moreover, comparisons with other asymptotic models, such as the linear shallow water equation or the Helmholtz equation, are presented.
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Contributor : Pierrick Quemar <>
Submitted on : Tuesday, May 26, 2020 - 4:04:23 PM
Last modification on : Thursday, June 4, 2020 - 3:28:40 AM


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Pierrick Quemar. Modeling and numerical analysis of free surface flows. Numerical Analysis [math.NA]. Université Paris 13, 2019. English. ⟨tel-02626032⟩



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