A nonclassical law of the wall in superfluid helium-4

Abstract

The logarithmic law of the wall is one of the most robust scaling laws of classical turbulence, yet whether it survives in a quantum fluid such as superfluid $^4$He (He II) remains unknown. At a solid wall, the viscous normal-fluid and inviscid superfluid components of He II obey fundamentally different boundary conditions, yet their motions can be coupled through quantized vortices. How these competing effects organize the near-wall flow remains an open question in quantum-fluid hydrodynamics. Here we visualize the near-wall velocity field in mechanically driven He II pipe flows at Reynolds numbers up to $1.26\times10^6$ using molecular tagging velocimetry and independently determine the wall shear stress from pressure-drop measurements. We find that the normal-fluid velocity develops a robust logarithmic profile, showing that a law-of-the-wall description persists despite the two-fluid nature of He II. Strikingly, however, the resulting wall law is distinctly nonclassical: the effective Kármán constant is less than half its classical value and remains nearly unchanged across the temperature range studied. This invariance, despite substantial changes in the relative fractions of the two components, indicates strong dynamical coupling between them. These results not only reveal how a universal law of classical turbulence is transformed by quantum hydrodynamics, but also provide valuable guidance for the design and operation of cryogenic cooling systems involving He II transport.