Structural evolution of intermittency and anisotropy at different scales analyzed using three-dimensional wavelet transforms

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Abstract
The three‐dimensional Mexican hat wavelet transform is used as a Fourier‐spectral space filter to study (1) the process by which the nonlinearities in the equations of motion create intermittent regions of concentrated vorticity and (2) the process by which isotropic turbulence becomes anisotropic by mean shear. In the first study, the three‐dimensional wavelet transform is applied to direct numerical simulations of the transition from Gaussian initial conditions to isotropic turbulence. During this transition, the initially noncoherent vorticity field is transformed by the Navier–Stokes nonlinearities into coherent regions of concentrated vorticity. Analysis of the wavelet‐transformed enstrophy field suggests (a) that small scales are always more intermittent than large scales, (b) that the level of intermittency at the small scales grows more rapidly than at the large scales, and most importantly (c) that the structural development of turbulence at different scales is correlated both with the evolution of global statistical measures that dominate at those scales (energy and dissipation rate) and with the evolution of the energy and dissipation‐rate (or enstrophy) spectra. In the second study, the three‐dimensional wavelet transform is applied to simulations of the transition from isotropic to homogeneous but anisotropic shear‐dominated turbulence. It is found that the effect of mean shear on large scales is very different from the effect of shear on small scales. Enstrophy structures rotate continuously to the mean flow direction with time at all scales, while the structures are elongated by mean strain rate dominantly in the principal strain‐rate direction. Small scales elongate more rapidly than large scales. However, scales both smaller and larger than the enstrophy peak had lower inclination angles relative to the mean flow direction. Furthermore, the anisotropic structure of the enstrophy field at scales smaller than the peak in the enstrophy spectrum is dominantly ‘‘rodlike,’’ whereas large‐scale structure is more ‘‘disklike.’’

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