Divergence-free and curl-free wavelets in 2D and 3D, application to turbulent flows

Abstract. We investigate the use of compactly supported divergence-free wavelets for the representation of solutions of the Navier-Stokes equations. After reviewing the theoretical construction of divergence-free wavelet vectors, we present in detail the bases and corresponding fast algorithms for two and three-dimensional incompressible flows. We also propose a new method to practically compute the wavelet-Helmholtz decomposition of any (even compressible) flow: this decomposition, which allows the incompressible part of the flow to be separated from its orthogonal complement (the gradient component of the flow) is the key-point for developing divergence-free wavelet schemes for Navier-Stokes equations. Finally, numerical tests validating our approach are presented.