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Disproportionate entrance length in superfluid flows and the puzzle of counterflow instabilities

J. Bertolaccini

E. Lévêque

P.-E. Roche

  • Université Lyon, ENS de Lyon, CNRS, Laboratoire de Physique, F-69342 Lyon, France

  • Université Lyon, Ecole Centrale de Lyon, CNRS, Laboratoire de Mécanique des Fluides et d'Acoustique, F-69134 Ecully cedex, France

  • Institut NEEL, CNRS, and Université Grenoble Alpes, F-38042 Grenoble, France

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Phys. Rev. Fluids 2, 123902 – Published 26 December, 2017

DOI: https://doi.org/10.1103/PhysRevFluids.2.123902

Abstract

Systematic simulations of the two-fluid model of superfluid helium (He-II) encompassing the Hall-Vinen-Bekharevich-Khalatnikov (HVBK) mutual coupling have been performed in two-dimensional pipe counterflows between 1.3 and 1.96 K. The numerical scheme relies on the lattice Boltzmann method. A Boussinesq-like hypothesis is introduced to omit temperature variations along the pipe. In return, the thermomechanical forcings of the normal and superfuid components are fueled by a pressure term related to their mass-density variations under an approximation of weak compressibility. This modeling framework reproduces the essential features of a thermally driven counterflow. A generalized definition of the entrance length is introduced to suitably compare entry effects (of different nature) at opposite ends of the pipe. This definition is related to the excess of pressure loss with respect to the developed Poiseuille-flow solution. At the heated end of the pipe, it is found that the entrance length for the normal fluid follows a classical law and increases linearly with the Reynolds number. At the cooled end, the entrance length for the superfluid is enhanced as compared to the normal fluid by up to one order of magnitude. At this end, the normal fluid flows into the cooling bath of He-II and produces large-scale superfluid vortical motions in the bath that partly re-enter the pipe along its sidewalls before being damped by mutual friction. In the superfluid entry region, the resulting frictional coupling in the superfluid boundary layer distorts the velocity profiles toward tail flattening for the normal fluid and tail raising for the superfluid. Eventually, a simple analytical model of entry effects allows us to re-examine the long-debated thresholds of T1 and T2 instabilities in superfluid counterflows. Inconsistencies in the T1 thresholds reported since the 1960s disappear if an aspect-ratio criterion based on our modeling is used to discard data sets with the strongest entry effects. Furthermore, it is observed that entry effects can spuriously reproduce the signature of a T2 transition with a normal flow remaining laminar.

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Phys. Rev. Fluids 2, 123902– Published 26 December, 2017
  • Received 25 July 2017

©2017 American Physical Society

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