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Analysis of turbulence in the orthonormal wavelet representation

A decomposition of turbulent velocity fields into modes that exhibit both localization in wavenumber and physical space is performed. We review some basic properties of such a decomposition, the wavelet transform. The wavelet-transformed Navier—Stokes equations are derived, and we define new quantities such as e(r, x), t(r, x) and π(r, x) which are the kinetic energy, the transfer of kinetic energy and the flux of kinetic energy through scale r at position x. The discrete version of e(r, x) is computed from laboratory one-dimensional velocity signals in a boundary layer and in a turbulent wake behind a circular cylinder. We also compute e(r, x), t(r, x) and π(r, x) from three-dimensional velocity fields obtained from direct numerical simulations. Our findings are that the localized kinetic energies become very intermittent in x at small scales and exhibit multifractal scaling. The transfer and flux of kinetic energy are found to fluctuate greatly in physical space for scales between the energy containing scale and the dissipative scale. These fluctuations have mean values that agree with their traditional counterparts in Fourier space, but have standard deviations that are much larger than their mean values. In space (at each scale r), we find exponential tails for the probability density functions of these quantities. We then study the nonlinear advection terms in more detail and define the transfer T(r|r′, x) between scale r and all scales smaller than r′, at location x. Then we define πsg(r, x), the flux of energy caused only by the scales smaller than r, at x, and find negative values for πsg(r, x), at almost 50% of the physical space at every scale (backscatter). We propose the inclusion of local backscatter in the phenomenological cascade models of intermittency, by allowing some energy to flow from small to large scales in the context of a multiplicative process in the inertial range.