Inertial Range Flow And Turbulent Cascades Classic Reprint PDF Download

Turbulent flows are chaotic and highly fluctuating, with a large number of interacting scales of motion, which makes them difficult to predict. One approach (known as Large Eddy Simulation) is to resolve all scales of motion down to a cutoff scale of interest and to capture the effect of all omitted smaller scales by modeling the unclosed unresolved turbulent deviatoric stress. This stress appears in the energy equation of the resolved scales of motion in the scale-to-scale energy flux term. The energy flux term is a dot product of two symmetric tensors: the unresolved turbulent stress tensor and the resolved strain rate tensor. Thus, we may interpret it as mechanical work done by the large resolved scales of motion on the smaller unresolved ones. This work is not only a function of the magnitudes of these two tensors but also of the relative alignment of their eigenframes. For no work can be done when the direction of force (stress) is orthogonal to the displacement (strain). A new quantity that characterizes this alignment is the cascade efficiency. Although one of the defining characteristics of the turbulent cascade is the scale-to-scale energy transfer, this transfer is found to be on average inefficient in the inertial range (around 25\%) due to a poor alignment between the stress and strain rate tensors. Another finding is that the embedding dimension of the flow plays a strong role in setting the direction of the scale-to-scale energy flux, for when geometric alignments between the stress and strain rate tensors encountered in three-dimensional turbulence are projected onto a two-dimensional configuration, the direct energy cascade towards smaller scales of motion reverses to a two-dimensional inverse energy cascade towards larger scales. The efficiency proves to be an effective metric in revealing the broken time-reversal symmetry of the turbulent cascade, for it is seen that the geometry of the turbulent stress lags that of the strain rate in three-dimensional turbulence, which is the opposite of what is observed in two dimensions. However, the time lags required for peak alignment do not properly scale with filter size when considering Eulerian quantities lagged in-place or along particle trajectories. It is only when expressing the scale-to-scale energy flux in a fully Lagrangian manner based on the right Cauchy-Green strain rate tensor and the second Piola--Kirchhoff stress tensor, that the times required for peak efficiency along fluid eddies follow the expected 2/3 Kolmogorov scaling.