Kolmogorov scaling bridges linear hydrodynamic stability and turbulence

TL;DR

This paper reveals that the -5/3 Kolmogorov energy spectrum, traditionally associated with turbulence, also appears in the linear dynamics of stable perturbation waves in shear flows, suggesting a fundamental link between linear and nonlinear turbulence features.

Contribution

The study demonstrates that the -5/3 energy spectrum emerges in linear wave dynamics, bridging the understanding between linear stability analysis and turbulence energy distribution.

Findings

Residual energy spectrum follows -5/3 scaling after transient phase.

Scaling depends only on wavenumber, not on flow inclination or Reynolds number.

Linear dynamics exhibit turbulence-like energy distribution patterns.

Abstract

The way in which kinetic energy is distributed over the multiplicity of inertial (intermediate) scales is a fundamental feature of turbulence. According to Kolmogorov's 1941 theory, on the basis of a dimensional analysis, the form of the energy spectrum function in this range is the -5/3 spectrum. Experimental evidence has accumulated to support this law. Until now, this law has been considered a distinctive part of the nonlinear interaction specific to the turbulence dynamics. We show here that this picture is also present in the linear dynamics of three-dimensional stable perturbation waves in the intermediate wavenumber range. Through extensive computation of the transient life of these waves, in typical shear flows, we can observe that the residual energy they have when they leave the transient phase and enter into the final exponential decay shows a spectrum that is very close to the -5/3 spectrum. The observation times also show a similar scaling. The scaling depends on the wavenumber only, i.e. it is not sensitive to the inclination of the waves to the basic flow, the shape-symmetry of the initial condition and the Reynolds number.