Dependence of the asymptotic energy dissipation on third-order velocity scaling

TL;DR

This paper investigates how the third-order velocity increment scaling influences the energy dissipation in turbulence, establishing conditions under which dissipation remains finite or vanishes in the inviscid limit, thus addressing the zeroth law of turbulence.

Contribution

It demonstrates the critical role of third-order absolute velocity increment scaling in determining whether turbulence exhibits anomalous dissipation or not.

Findings

Third-order velocity increment scaling must not exceed one for anomalous dissipation.

If the scaling exceeds one, dissipation vanishes asymptotically.

The velocity field becomes symmetric when dissipation vanishes.

Abstract

The asymptotic energy dissipation is connected to the third-order scaling of the longitudinal velocity increment magnitude in three-dimensional turbulence via the Kolmogorov $4/5$ law. It is shown that the third-order longitudinal absolute velocity increment scaling should not exceed unity for anomalous dissipation to occur, that is for non-vanishing average dissipation in the inviscid limit -- also known as the ``zeroth law" of turbulence. Conversely, if the third-order longitudinal absolute velocity increment scaling exceeds unity then the average dissipation must asymptotically vanish and the velocity increment field will becomes symmetric at least at the level of its skewness. This work highlights the importance of the third-order absolute velocity increment scaling in assessing the status of the ``zeroth-law" of turbulence.