The space and scale dependencies of the self-similar structure of turbulence

Abstract Two new techniques for analysing the space and scale dependencies of the self-similar structure of turbulence are introduced. Both methods are based on the wavelet transform, but use different wavelets. First, the concept of a ‘turbulence eddy’ is defined in terms of the Mexican hat wavelet transform of the velocity signal in such a way that the ‘eddy’ has a size and a location in space. A new scaling exponent, the eddy capacity DE (0 ≼ DE ≼ 1), is defined in terms of the zero crossings of this wavelet transform. DE is a measure of the space-fillingness of eddies and of their scaling in real space whereas the Hausdorff dimension D'H and the Kolmogorov capacity D'K are scaling exponents that are sensitive to the scaling and space-fillingness in wavenumber space. DE differs from both D'K and D'H in that it is particularly sensitive to phase correlations. For any random phase signal, DE = 1, whereas D'H and D'K depend on the energy spectrum. For well defined spiral signals, and for signals with a fractal intermittency in space, DE = D'K. Secondly, we propose a practical test for discriminating between H-fractals and K-fractals (i. e. between a dense and a sparse distribution of singularities). This method is based on successive averages of wavelet transforms, and the Morlet wavelet is used. The technique is shown to be robust with respect to large amounts of phase noise. Both methods are applied to a one-dimensional (1D) turbulence velocity signal of very high Reynolds number Reλ = 2720. It is found that DE = 1 for the inertial scales larger than the Taylor microscale λ, thus indicating some degree of phase scrambling of the 1D turbulence velocity signal at the larger inertial scales. There is some indication of a spiral-type structure in this signal at scales below λ. Finally, we measure the spatial fluctuation of wavelet energy, comparing an experimental turbulence signal with random phase and spiral-like signals. The magnitude of spatial fluctuations is set by the scrambled phase part of the turbulence, but the increase of energy fluctuation at small scales may be due to locally self-similar small-scale structures.