On energy cascades in non-homogeneous 3D Navier-Stokes equations

TL;DR

This paper demonstrates that Kolmogorov's dissipation law, combined with a small Taylor length scale, ensures energy cascades in forced Navier-Stokes equations, and establishes scaling laws relating energy metrics to the Grashof number.

Contribution

It introduces new sufficient conditions for energy cascades in non-homogeneous 3D Navier-Stokes equations using physical scales and averages, improving previous criteria.

Findings

Energy cascades are guaranteed under Kolmogorov's law and small Taylor scale.

Scaling laws for energy metrics are derived in terms of Grashof number.

Conditions for forced cascades are refined using physical scale analysis.

Abstract

We show - in the framework of physical scales and $(K_1,K_2)$-averages - that Kolmogorov's dissipation law combined with the smallness condition on a Taylor length scale are sufficient to guarantee energy cascades in the forced Navier-Stokes equations. Moreover, in the periodic case we establish restrictive scaling laws - in terms of Grashof number - for kinetic energy, energy flux, and energy dissipation rate. These are used to improve our sufficient condition for forced cascades in physical scales.