Prove the refined similarity hypothesis in turbulence

Prove the refined similarity hypothesis of Kolmogorov and Oboukhov for hydrodynamic turbulence by establishing that the small-scale statistics of velocity increments are governed by the locally averaged energy dissipation rate rather than a globally averaged one, thereby providing a rigorous foundation for the hypothesis that currently remains conjectural.

Background

Kolmogorov’s 1941 theory (K41) assumes universality of small-scale turbulence statistics based on viscosity and a globally averaged dissipation rate. Experimental evidence of intermittent, spatially inhomogeneous dissipation led to the refined similarity hypothesis (K62/Oboukhov–Kolmogorov), which replaces the global average with a locally averaged dissipation rate to account for intermittency.

While refined similarity has been extensively used and is consistent with many observations, its status remains conjectural without a rigorous proof. Establishing its validity would provide a fundamental theoretical underpinning for a wide range of intermittency models and analyses in fluid turbulence and inform analogous developments in magnetohydrodynamic turbulence.

References

It should be emphasized that these descriptions of intermittency are physically plausible and consistent with observations, but the refined similarity hypothesis itself remains a useful conjecture \citep{wang1996examination,chen1997refined} though still unproven to our knowledge.

— Evaluation of scale-dependent kurtosis with HelioSwarm (2407.06679 - Pecora et al., 9 Jul 2024) in Section 1: Introduction