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

We study fluctuations of the local energy cascade rate ${\mathrm{\Phi }}_{\mathrm{\ell }}$ in turbulent flows at scales ($\mathrm{\ell }$) in the inertial range. According to the Kolmogorov refined similarity hypothesis (KRSH), relevant statistical properties of ${\mathrm{\Phi }}_{\mathrm{\ell }}$ should depend on ${\epsilon }_{\mathrm{\ell }}$, the viscous dissipation rate locally averaged over a sphere of size $\mathrm{\ell }$, rather than on the global average dissipation. However, the validity of KRSH applied to ${\mathrm{\Phi }}_{\mathrm{\ell }}$ has not yet been tested from data. Conditional averages such as $⟨{\mathrm{\Phi }}_{\mathrm{\ell }}\mathrm{\mid }{\epsilon }_{\mathrm{\ell }}⟩$ 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 (${\mathrm{\Phi }}_{\mathrm{\ell }}>0$) and inverse (${\mathrm{\Phi }}_{\mathrm{\ell }}<0$) 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.