Modélisation numérique des écoulements à surface libre avec bancs couvrants-découvrants par les volumes finis et la décomposition orthogonale aux valeurs propres

Abstract

Flooding events caused by dam breaks, long rains and tsunamis have reached such violence that more critical preventive measures should be considered. Preventive measures against floods generally consist on flow managing and on facilities such as protective dykes, flood managing dams or pipes of evacuation. The implementation of such measures of protection requires the intervention of the hydraulic engineer in modeling and predicting the flow dynamics that characterize the floods. The main objective of this thesis is to propose a numerical model for the simulation of free surface flows in general and floods specifically and to build a reduced-order model (ROM) that will allow significant accelerations in the calculations. This is achieved through the discretization of the shallow water equations based on the finite volume technique and the reduction of those equations by the Galerkin projection onto a subspace spanned by some bases through the proper orthogonal decomposition (POD) technique. The proposed finite volume numerical model is well-adapted to the simulation of wetting and drying processes which commonly characterize flooding event. Dealing with wetting and drying processes on highly irregular real bathymetries has long been a challenge. Indeed, at a wet and dry interface, the water depth tends to zero and forms a discontinuity zone that may be internal (appearance of islets) or external. The treatment of this discontinuity problem is very often a source of numerical instabilities that can lead to the generation of negative water depths or some unrealistic velocities, especially for variable bathymetry. In this study, the Saint-Venant equations are considered in their basic form in order to avoid the integration of the source term of geometry. This term is obtained currently by splitting the term that contains the forces of gravity (which does not respect the strict divergent form) in a term of pressure and a source term of the variation of the geometry. A local approximation of the source term of gravity is applied so as to obtain a divergent form to be included in the interface flux. A Lax-Friedrichs scheme with an artificial dissipation term is used to calculate the flux at the interfaces. The calculation of the flux is adapted to the wet or dry state of the calculated cell. The local free surface correction technique is used to circumvent the generation of a water level gradient in the vicinity of dry areas which is a phenomenon responsible for non-physical fluxes and numerical instabilities. This technique preserves the condition of the fluid at rest or the C-property. In addition, the symmetrical calculations at the interfaces ensure the global mass conservation overall the computational domain. The construction of a reduced order model for the simulation of free surface flows answers the necessity of accelerating the calculations while handling the uncertainties involved in the physical parameters. Indeed, parameters such as the bathymetry, the inflow flux, the friction coefficient, that governs real flows cannot be defined precisely because of their natural variability. In effective flood risks managements, repetitive calculations have to take into account the uncertainties in these physical parameters and this, through a probabilistic analysis. This can be very time consuming when a high fidelity model with thousands of degrees of freedom is used. The reduced order model (ROM) proposed is based essentially on the reduction of the finite volume scheme through the Galerkin projection of the discretized equations. The equations are projected onto a subspace spanned by bases obtained from the proper orthogonal decomposition of the snapshots matrices of the variables of interest. The snapshots matrices are obtained by recording a number of numerical solutions of the considered problem during the simulation time. Approximations of the nonlinear terms related to the convection flux and the wave velocity are applied to derive an effective reduced order model. The CPU time with the ROM depends mainly on the size of the POD reduced bases. The finite volume model proved accurate and robust through validation tests and mainly in the simulation of wetting and drying transitions on a real complex bathymetry. On the other hand, the results obtained from the ROM are quite close to those of the finite volumes model for the reproduction phase. During the operation phase, the sensitivity analysis showed that for reasonable perturbations on the initial conditions and the physical parameters (less than 50% for the initial free surface level), the ROM simulates with a quite good accuracy each new scenario. The speedups are very significant with respect to the finite volume scheme. It appears that the reduced order model proposed here can be of great help for the engineer in the simulation of hypothetical flows and possibly for the definition of flood maps.

Date	20 Dec 2011
Original language	French
Awarding Institution	École de technologie supérieure
Supervisor	Azzeddine Soulaïmani (Supervisor)