Energy cascade and the four-fifths law in superfluid turbulence

TL;DR

This study confirms the 4/5-law of turbulence in superfluid helium through experiments and simulations, revealing an effective viscosity matching the normal fluid's viscosity despite superfluid conditions.

Contribution

It demonstrates the validity of the 4/5-law in superfluid turbulence and introduces an empirical approach to estimate effective viscosity in superfluids.

Findings

The 4/5-law holds in superfluid turbulence at high Reynolds numbers.

An effective viscosity in superfluid turbulence matches the normal fluid's viscosity.

The Kármán-Howarth equation describes deviations from the 4/5-law at small scales.

Abstract

The 4/5-law of turbulence, which characterizes the energy cascade from large to small-sized eddies at high Reynolds numbers in classical fluids, is verified experimentally in a superfluid 4He wind tunnel, operated down to 1.56 K and up to R_lambda ~ 1640. The result is corroborated by high-resolution simulations of Landau-Tisza's two-fluid model down to 1.15 K, corresponding to a residual normal fluid concentration below 3 % but with a lower Reynolds number of order R_lambda ~ 100. Although the K\'arm\'an-Howarth equation (including a viscous term) is not valid \emph{a priori} in a superfluid, it is found that it provides an empirical description of the deviation from the ideal 4/5-law at small scales and allows us to identify an effective viscosity for the superfluid, whose value matches the kinematic viscosity of the normal fluid regardless of its concentration.