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

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