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.

Keywords:

• Divergence-free wavelets
• Curl-free wavelets
• Turbulent flows

Acknowledgments

The authors would like to thank G.-H. Cottet and G. Lapeyre for helpfully providing them with numerical turbulent flows for analyses. This work has been supported in part by the European Community's Improving Human Potential Programme under contract HPRN-CT-2002-00286, ‘Breaking complexity.’

Notes

† The Fourier transform of a functionf is defined by ξ) = ∫_−∞ ^+∞ f(x) e^ixξ α x