The dimensionless dissipation rate and the Kolmogorov (1941) hypothesis of local stationarity in freely decaying isotropic turbulence

TL;DR

This paper derives an expression for the dimensionless dissipation rate in turbulence, examines the validity of the local stationarity hypothesis underlying Kolmogorov's 4/5 law, and discusses implications for turbulence theory and measurements.

Contribution

It provides a theoretical analysis of the local stationarity assumption in turbulence and proposes a method to quantify its error using the asymptotic dissipation rate.

Findings

Neglecting the time-derivative in the Kolmogorov equation cannot be justified solely by inertial range or high Reynolds numbers.

The hypothesis of local stationarity may lead to significant errors in certain conditions.

Scale invariance in wavenumber space holds at infinite Reynolds numbers, supporting the uncorrected -5/3 energy spectrum.

Abstract

An expression for the dimensionless dissipation rate was derived from the Karman-Howarth equation by asymptotic expansion of the second- and third- order structure functions in powers of the inverse Reynolds number. The implications of the time-derivative term for the assumption of local stationarity (or local equilibrium) which underpins the derivation of the Kolmogorov `4/5' law for the third-order structure function were studied. It was concluded that neglect of the time-derivative cannot be justified by reason of restriction to certain scales (the inertial range) nor to large Reynolds numbers. In principle, therefore, the hypothesis cannot be correct, although it may be a good approximation. It follows, at least in principle, that the quantitative aspects of the hypothesis of local stationarity could be tested by a comparison of the asymptotic dimensionless dissipation rate for free decay with that for the stationary case. But in practice this is complicated by the absence of an agreed evolution time for making the measurements during the decay. However, we can assess the quantitative error involved in using the hypothesis by comparing the exact asymptotic value of the dimensionless dissipation in free decay calculated on the assumption of local stationarity to the experimentally determined value (e.g. by means of direct numerical simulation), as this relationship holds for all measuring times. Should the assumption of local stationarity lead to significant error, then the `4/5' law needs to be corrected. Despite this, scale invariance in wavenumber space appears to hold in the formal limit of infinite Reynolds numbers, which implies that the `-5/3' energy spectrum does not require correction in this limit.