Relating statistics to dynamics in axisymmetric homogeneous turbulence

The structure and the dynamics of homogeneous turbulence are modified by the presence of body forces such that the Coriolis or the buoyancy forces, which may render a wide range of turbulence scales anisotropic. The corresponding statistical characterization of such effects is done in physical space using structure functions, as well as in spectral space with spectra of two-point correlations, providing two complementary viewpoints. In this framework, second-order and third-order structure functions are put in parallel with spectra of two-point second- and third-order velocity correlation functions, using passage relations. Such relations apply in the isotropic case, or for isotropically averaged statistics, which, however, do not reflect the actual more complex structure of anisotropic turbulence submitted to rotation or stratification. This complexity is demonstrated in this paper by orientation-dependent energy and energy transfer spectra produced in both cases by means of a two-point statistical model for axisymmetric turbulence. We show that, to date, the anisotropic formalism used in the spectral transfer statistics is especially well-suited to analyze the refined dynamics of anisotropic homogeneous turbulence, and that it can help in the analysis of isotropically computed third-order structure function statistics often used to characterize anisotropic contexts.