Relationship between scale and structure in turbulence analyzed using three-dimensional wavelet transforms

Both the physical space and Fourier space descriptions of homogeneous turbulence are valuable for different purposes and from each of these two viewpoints insight into the nonlinear dynamic development of turbulence can be attained. As a class of Fourier spectral space filters with simultaneous physical space interpretations, the wavelet transform is useful for relating these two views. We study the relationship between structure in physical space and scale in Fourier space by applying three-dimensional wavelet transforms to homogeneous turbulence data generated by direct numerical simulations.;Three types of turbulence are analyzed: isotropic turbulence, shear-dominated anisotropic turbulence and a coherently forced nonisotropic turbulence. In isotropic turbulence, vortical tube-like structures exist at the small scales with all directions represented equally, corresponding in Fourier space to uniform distributions of vorticity, energy and phase on spheres centered at the origin. Mean shear modifies this structure such that the vortex tubes become aligned at an angle below 45;We also find that localized grouping of Fourier modes in Fourier space and phase relationships among the modes may be directly correlated with subsets of aligned vortex tubes in physical space. The aligned vortex tubes correspond to 3D toroidal regions in Fourier space with the axis of the torus aligned with the average direction of the vortex tube axes. The Fourier modes within the toroidal region which contribute to the higher enstrophy vortex tubes are strongly phase correlated. In the transition to shear-dominated turbulence, the "optimal" toroidal region becomes elliptical with further subgrouping of Fourier modes describing the anisotropic vortex structures.