Journal of Low Temperature Physics

, Volume 196, Issue 1–2, pp 35–41 | Cite as

Numerical Study on Entrance Length in Thermal Counterflow of Superfluid \(^4\)He

  • Hiromichi KobayashiEmail author
  • Satoshi Yui
  • Makoto Tsubota


Three-dimensional numerical simulations in a square duct were conducted to investigate entrance lengths of normal fluid and superfluid flows in a thermal counterflow of superfluid \(^4\)He. The two fluids were coarse-grained by using the Hall–Vinen–Bekharevich–Khalatnikov (HVBK) model and were coupled through mutual friction. We solved the HVBK equations by parameterizing the coefficient of the mutual friction to consider the vortex line density. A uniform mutual friction parameter was assumed in the streamwise direction. Our simulation showed that the entrance length of the normal fluid from a hot end becomes shorter than that of a single normal fluid due to the mutual friction with the parabolically developed superfluid flow near the hot end. As the mutual friction increases, the entrance length decreases. Same as that, the entrance length of the superfluid from a cold end is affected by the strength of the mutual friction due to the parabolically developed normal fluid flow near the cold end. Aside from the entrance effect, the realized condition of a tail-flattened flow is discussed by parameterizing the superfluid turbulent eddy viscosity and the mutual friction.


Entrance length Superfluid Thermal counterflow HVBK model Two-fluid model Mutual friction 



This work was supported by JSPS KAKENHI Grant Number JP18K03935.


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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of PhysicsKeio University, Hiyoshi CampusYokohamaJapan
  2. 2.Department of PhysicsOsaka City UniversityOsakaJapan

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