Forward and Inverse Energy Cascade in Fluid Turbulence Adhere to Kolmogorov’s Refined Similarity Hypothesis

We study fluctuations of the local energy cascade rate Φℓ in turbulent flows at scales (ℓ) in the inertial range. According to the Kolmogorov refined similarity hypothesis (KRSH), relevant statistical properties of Φℓ should depend on εℓ, the viscous dissipation rate locally averaged over a sphere of size ℓ, rather than on the global average dissipation. However, the validity of KRSH applied to Φℓ has not yet been tested from data. Conditional averages such as ⟨Φℓ|εℓ⟩ 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 (Φℓ>0) and inverse (Φℓ<0) cascade events separately. Measured ratios of forward and inverse cascade probability densities conditioned on εℓ also confirm the applicability of the KRSH to analysis of the fluctuation relation from nonequilibrium thermodynamics.