Energy cascade for cross-shear length scales in free-shear three-dimensional incompressible viscous flows

TL;DR

This paper rigorously analyzes the energy cascade process in free-shear three-dimensional viscous flows, establishing conditions under which energy transfer across scales occurs, consistent with classical turbulence theory.

Contribution

It provides the first rigorous proof of energy cascade phenomena in free-shear flows using statistical solutions and Fourier analysis, extending turbulence theory.

Findings

Proves energy cascade for wavenumbers above the viscous shear wavenumber.

Defines horizontal Taylor wavenumber for scale analysis.

Establishes conditions matching classical turbulence cascade scales.

Abstract

The phenomenon of energy cascade is addressed in the case of free-shear flows, modeled with the equations for incompressible Newtonian fluids with mixed periodic and free-slip boundary conditions driven by an imposed mean shear profile. The rigorous results are proved for ensemble averages with respect to stationary statistical solutions in the sense of Foias and Prodi. We obtain the energy-budget relations with an energy dissipation term, a shear-production term from the mean flow, and an energy flux term of the fluctuation field, based on a decomposition of the flow into high and low horizontal wavenumber components, corresponding to scales perpendicular to the mean shear gradient. We estimate the shear-production term exploiting the orthogonality of horizontal Fourier modes and look for an energy cascade of the fluctuation flux. For any given wavenumber, we define an associated horizontal Taylor wavenumber of the high-wavenumber band above the given wavenumber. We prove two energy cascade results, for the energy flux and for a restricted energy flux that discounts possible energy loss to flow singularities, both valid for wavenumbers whose associated horizontal Taylor wavenumber lies above the viscous shear wavenumber and whose low-wavenumber energy dissipation rate is negligible compared to the total energy dissipation rate. On heuristic grounds, these rigorous conditions correspond precisely to an energy cascade in the range from the classical Corrsin scale down to the Kolmogorov dissipation scale, as expected in the conventional theory of turbulent shear flows.