Numerical Study on Entrance Length in Thermal Counterflow of Superfluid $^4$He

TL;DR

This study uses 3D numerical simulations to analyze how mutual friction influences entrance lengths of normal and superfluid components in thermal counterflow of superfluid helium-4, revealing that mutual friction shortens entrance lengths.

Contribution

It introduces a coupled HVBK model with a parameterized mutual friction coefficient to investigate entrance lengths and flow conditions in superfluid helium-4.

Findings

Mutual friction shortens entrance lengths of both fluids.

Normal fluid entrance length is affected by superfluid flow near hot end.

Superfluid entrance length depends on mutual friction strength.

Abstract

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.