Statistics of turbulence and intermittency enhancement in superfluid He4 counterflow

We report a detailed analysis of the energy spectra, second- and high-order structure functions of velocity differences in superfluid $⁴\mathrm{He}$ counterflow turbulence, measured in a wide range of temperatures and heat fluxes. We show that the one-dimensional energy spectrum ${E}_xz({k}_y)$ (averaged over the $xz$ plane, parallel to the channel wall), directly measured as a function of the wall-normal wave vector ${k}_y$, gives more detailed information on the energy distribution over scales than the corresponding second-order structure function ${S}₂({\ensuremath{\delta}}_y)$. In particular, we discover two intervals of ${k}_y$ with different apparent exponents: ${E}_xz({k}_y)\ensuremath{\propto}{k}_y^{\ensuremath{-}{m}_{\mathrm{C}}}$ for $k\ensuremath{\lesssim}{k}_{\ifmmode\times\else\texttimes\fi{}}$ and ${E}_xz({k}_y)\ensuremath{\propto}{k}_y^{\ensuremath{-}{m}_{\mathrm{F}}}$ for $k\ensuremath{\gtrsim}{k}_{\ifmmode\times\else\texttimes\fi{}}$. Here ${k}_{\ifmmode\times\else\texttimes\fi{}}$ denotes the wave number that separates scales with relatively strong (for $k\ensuremath{\lesssim}{k}_{\ifmmode\times\else\texttimes\fi{}}$) and relatively weak (for $k\ensuremath{\gtrsim}{k}_{\ifmmode\times\else\texttimes\fi{}}$) coupling between the normal-fluid and superfluid velocity components. We interpret these $k$ ranges as cascade-dominated and mutual-friction-dominated intervals, respectively. The general behavior of the experimental spectra ${E}_xz({k}_y)$ agrees well with the predicted spectra [L'vov and Pomyalov, Phys. Rev. B 97, 214513 (2018)]. Analysis of the $n\mathrm{th}$-order structure functions statistics shows that in the energy-containing interval, the statistics of counterflow turbulence is close to Gaussian, similar to the classical hydrodynamic turbulence. In the cascade- and mutual-friction-dominated intervals, we found some modest enhancement of intermittency with respect to its level in classical turbulence. However, at small scales, the intermittency becomes much stronger than in the classical turbulence.