Eulerian and Lagrangian second-order statistics of superfluid ${^4}$He grid turbulence

TL;DR

This study investigates the second-order statistics of superfluid helium turbulence using particle tracking velocimetry, revealing classical behavior at large scales and quantum effects at smaller scales, with implications for understanding quantum turbulence.

Contribution

It provides a detailed analysis of Eulerian and Lagrangian second-order statistics in superfluid helium turbulence, including the use of bridge relations to connect different statistical measures.

Findings

Eulerian energy spectra exhibit Kolmogorov-1941 scaling at large scales.

Lagrangian statistics show a transition from classical to quantum vortex-dominated behavior.

Reconstructed energy spectra from bridge relations differ from direct measurements in the quasi-classical range.

Abstract

We use particle tracking velocimetry to study Eulerian and Lagrangian second-order statistics of superfluid $^4$He grid turbulence. The Eulerian energy spectra at scales larger than the mean distance between quantum vortex lines behave classically with close to Kolmogorov-1941 scaling and are almost isotropic. The Lagrangian second-order structure functions and frequency power spectra, measured at scales comparable with the intervortex distance, demonstrate a sharp transition from nearly-classical behavior to a regime dominated by the motion of quantum vortex lines. Employing the homogeneity of the flow, we verify a set of relations that connect various second-order statistical objects that stress different aspects of turbulent behavior, allowing a multifaceted analysis. We use the two-way bridge relations between Eulerian energy spectra and second-order structure functions to reconstruct the energy spectrum from the known second-order velocity structure function and vice versa. The Lagrangian frequency spectrum reconstructed from the measured Eulerian spectrum using the Eulerian-Lagrangian bridge differs from the measured Lagrangian spectrum in the quasi-classical range which calls for further investigation.