Multiscale energy budget of inertially driven turbulence in normal and superfluid helium

In this paper we present a novel hydrodynamic experiment using liquid ⁴He. Lagrangian trajectories are obtained using two-dimensional particle tracking on hollow glass microspheres in a cryogenic liquid helium turbulent flow. The flow is forced inertially by a canonical oscillating grid, both below and above the superfluid transition. This allows for a direct comparison of the Lagrangian statistics in the normal (He I) and superfluid (He II) phases. The high temporal resolution allows us to resolve the velocity fluctuations at integral and inertial scales and, most importantly, assess the noise contribution. The careful analysis of velocity fluctuations, acceleration fluctuations, and pair dispersion, allows us to extract estimates of the energy-injection rate at large scale, the energy flux cascading through inertial scales, and the dissipation rate at small scale, and therefore build the energy budget in both the normal and superfluid phases. We find that, within experimental uncertainty, the statistical features of turbulence and the energy budget in superfluid helium is indistinguishable from those of normal helium, highlighting the importance of conducting experiments in both He I and He II to draw meaningful conclusions, because deviations from the theoretical predictions may arise from noise contributions or deviation from the homogeneous and isotropic approximations.