Forward and inverse energy cascade and fluctuation relation in fluid turbulence adhere to Kolmogorov's refined similarity hypothesis

TL;DR

This study confirms that local energy cascade fluctuations in turbulence follow Kolmogorov's refined similarity hypothesis, with data supporting the applicability of fluctuation relations from non-equilibrium thermodynamics to both forward and inverse cascades.

Contribution

The paper provides the first data-driven validation of KRSH for local energy cascade fluctuations, including both forward and inverse cascades, and demonstrates the relevance of fluctuation relations in turbulence.

Findings

Conditional averages of cascade rate match KRSH predictions.

Both forward and inverse cascade events follow KRSH.

Fluctuation relations consistent with non-equilibrium thermodynamics are observed.

Abstract

We study fluctuations of the local energy cascade rate $\Phi_\ell$ in turbulent flows at scales ($\ell$) in the inertial range. According to the Kolmogorov refined similarity hypothesis (KRSH), relevant statistical properties of $\Phi_\ell$ should depend on $\epsilon_\ell$, the viscous dissipation rate locally averaged over a sphere of size $\ell$, rather than on the global average dissipation. However, the validity of KRSH applied to $\Phi_\ell$ has not yet been tested from data. Conditional averages such as $\langle \Phi_\ell|\epsilon_{\ell}\rangle$ as well as of higher-order moments are measured from Direct Numerical Simulations data, and results clearly adhere to the predictions from KRSH. Remarkably, the same is true when considering forward ($\Phi_\ell>0$) and inverse ($\Phi_\ell<0$) cascade events separately. Measured ratios of forward and inverse cascade probability densities further show that a fluctuation relation adhering to the KRSH can be observed, raising the hope that important features of turbulence may be described using concepts from non-equilibrium thermodynamics.