Divergence-free and curl-free wavelets in two dimensions and three dimensions: application to turbulent flows

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 for practically computing 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.