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Communication Dans Un Congrès Année : 2007

Orthogonal Helmholtz decomposition using divergence-free and curl-free wavelets. Application to the simulation of 2D turbulent flows.

Résumé

In many physical problems, like the study of incompressible fluids or in electromagnetism, the quantities have to fulfill a divergence-free condition. The Helmholtz decomposition consists in decomposing a vector field, into the (orthogonal) sum of its divergence-free component and its curl-free component.This decomposition is straightforward in the whole space thanks to the Leray projector which can be explicitly described in Fourier domain. The objective of the present talk is to propose an orthogonal Helmholtz decomposition in wavelet domain of any vector field. This decomposition will be based on divergence-free and curl-free wavelets originally designed by P.-G. Lemarié (1992) and K. Urban (2000). Since divergence-free and curl-free wavelets are biorthogonal bases, we will proposed an iterative algorithm to provide such decomposition, of which we will prove the convergence in arbitrary dimension, using Shannon wavelets. An important application of the Helmholtz decomposition is the Direct Numerical Simulation of turbulence. We will propose a new numerical scheme for solving the incompressible Navier-Stokes equations, based on divergence-free wavelets, and which only requires Fast Wavelet Transforms for the computation of the velocity and the pressure. Finally we show numerical results on the simulation of 2D homogeneous turbulent flows.
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Dates et versions

hal-00874985 , version 1 (20-10-2013)

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  • HAL Id : hal-00874985 , version 1

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Valérie Perrier. Orthogonal Helmholtz decomposition using divergence-free and curl-free wavelets. Application to the simulation of 2D turbulent flows.. Mathematical image processing meeting, Sep 2007, Marseille, France. ⟨hal-00874985⟩
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