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Numerical Simulation of Non-Linear Waves with the Navier- Stokes Equations

Author(s): Jean-Michel Hervouet

Linked Author(s): Jean Michel Hervouet

Keywords: Navier-Stokes equations; Shallow water equations; Boussinesq equations; Finite elements; Nonlinear waves

Abstract: The hydroinformatic system Telemac, based on Finite Element techniques, addresses free surface and groundwater flows. The system includes the Saint-Venant or shallow water equations (Telemac-2D) and the Navier-Stokes equations in 3 dimensions with a free surface (Telemac3D). Recent advances, namely the adaptation to three dimensions of the pseudo wave equation already used for shallow water equations, have led to a robust algorithm for the treatment of the 3D Navier-Stokes equations with a free surface, which allows rapid flows, hydraulic jumps, wetting and drying (see Hervouet 2007). Though Telemac-2D also solves the Boussinesq equations, the question was then raised of using Telemac-3D in its non-hydrostatic option for simulating non-linear waves. Previous attempts (see Hervouet and Jankowski 2000) have shown that it could be advantageous as regards CPU time, provided that the number of planes on the vertical be limited in the 3D mesh. Other authors, see e. g. Li and Fleming 2001, have also published similar results with a finite difference model. However, some test cases with Telemac3D were not satisfactory, e. g. the case of a monochromatic non-linear wave over a bar (reference Garapon et al. 2002). It could be shown that the main reason was the fact that in the solution procedure, based on the fractional step method, the projection step that ensures a divergence-free velocity was done after the updating of the free surface. Knowledge of the dynamic pressure is indeed necessary when computing the new free surface, but to compute it, the final velocity is requested. The solution finally retained consists of estimating a first predictor value of the dynamic pressure. As the pseudo wave equation method consists of using the momentum equation to eliminate the velocity in the continuity equation, it requires an estimation of the final velocity. The idea is thus to use this estimated velocity to find an estimated dynamic pressure whose gradient will influence the new free surface. The final projection step is then only incremental. The tests show a dramatic improvement in the case of a monochromatic non-linear wave over a bar, but the accuracy is downgraded if too small a number of planes is chosen for building the mesh.


Year: 2007

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