Disentangling temperature and Reynolds number effects in quantum turbulence

Abstract: The interplay between viscous and frictional dissipation is key to understanding quantum turbulence dynamics in superfluid $^4$He. Based on a coarse-grained two-fluid description, an original scale-by-scale energy budget that identifies each scale's contribution to energy dissipation is derived. Using the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) model to further characterize mutual friction, direct numerical simulations at temperatures $1.44 \, \mathrm{K} \lesssim T \lesssim 2.16 \, \mathrm{K}$ indicate that mutual friction promotes intense momentum exchanges between the two fluids to maintain a joint energy cascade despite their viscosity mismatch. However, the resulting overall frictional dissipation remains small (compared to the viscous dissipation) and confined to far-dissipative scales. This remarkable feature allows us to define an effective Reynolds number for the turbulence intensity in a two-fluid system, helping to disentangle the effects of Reynolds number and temperature in quantum turbulence. Thereby, simple physical arguments predict that the distance $\ell$ between quantized vortices (normalized by the turbulence integral scale $L_0$) should behave as $\ell/L_0 \approx 0.5 \, \mathrm{Re}_\kappa^{-3/4}$ with the Reynolds number based on the quantum of circulation $\kappa$. This law is well supported by a large set of experimental and numerical data within the temperature range of the HVBK model. Finally, this new approach offers the possibility of revisiting the ongoing controversy on intermittency in quantum turbulence. It is shown that observed changes in intermittency arise from Reynolds number effects rather than from temperature variations, as proposed in recent studies.