Classical and Quantum-Mechanical Turbulence in He II Heat Flow

Abstract

The theory of the He II thermal counterflow process in wide ($d>\mathrm{}{10}^{-3\mathrm{}}$ cm) channels is investigated on the assumption that both the normal and superfluid components make a transition from a laminar to a turbulent type of flow. A critical heat current $W_0$ is identified with the superfluid transition. The superfluid turbulent state is taken to be essentially that described by Vinen in terms of quantized vortex line and has an associated mutual friction. A second critical heat current $W_c$ is identified with the normal-fluid transition. It is argued that this transition is essentially of a classical turbulent type, with the added condition that the critical value of the Reynolds number must depend on the extent of mutual-friction coupling. This interpretation is shown to be consistent with experimentally observed critical heat currents, as well as with critical-velocity effects found in other types of flow. The assumption of two critical heat currents defines three distinct flow regions. It is shown that these three regions are essentially the same as those found experimentally by Allen, Griffiths, and Osborne. On the basis of some simplifying assumptions regarding the normal-fluid turbulent state, the temperature and pressure gradients accompanying thermal counterflow are calculated. Comparison with experiment shows good qualitative and often quantitative agreement. It is also shown that the model developed can be successfully used to interpret experiments involving flows of a nonthermal counterflow type.