Coflow turbulence of superfluid ${}^4\mathrm{He}$ in a square channel: Vortices trapped on a cylindrical attractor

We perform a numerical simulation of the dynamics of quantized vortices produced by coflow in a square channel using the vortex filament model. Unlike the situation in thermal counterflow, where the superfluid velocity ${\mathit{v}}_s$ and normal-fluid velocity ${\mathit{v}}_n$ flow in opposite directions, in coflow, ${\mathit{v}}_s$ and ${\mathit{v}}_n$ flow in the same direction. Quantum turbulence in thermal counterflow has been long studied theoretically and experimentally, and its various features have been revealed. In recent years, an experiment on quantum turbulence in coflow has been performed to observe different features of thermal counterflow. By supposing that ${\mathit{v}}_s$ is uniform and ${\mathit{v}}_n$ takes the Hagen-Poiseuille profile, which is different from the experiment where ${\mathit{v}}_n$ is thought to be turbulent, we calculate the coflow turbulence. Vortices preferentially accumulate on the surface of a cylinder for ${\mathit{v}}_s\simeq {\mathit{v}}_n$ by mutual friction; namely, the coflow turbulence has an attractor. How strongly the vortices are attracted depends on the temperature and velocity. The length of the vortices increases as the vortices protruding from the cylindrical attractor continue to wrap around it. As the vortices become dense on the attractor, they spread toward its interior by their repulsive interaction. Then, the superfluid velocity profile induced by the vortices gradually mimics the normal-fluid velocity profile. This is an indication of velocity matching, which is an important feature of coflow turbulence.

Figure 1

Time development of the VLD (a) and the anisotropic parameter (b) for $T=1.85$ K and ${\overline{\mathit{v}}}_n(={\overline{\mathit{v}}}_s)=1.2$ cm/s. Inset in (a) shows the time development of the VLD in the very early stage, 0 s $\le t\le 1.0$ s.

Figure 2

Snapshots of vortices at $t=20$ s in Fig. 1: (a) viewed along the flow direction and (b) viewed from the side of the flow direction.

Figure 3

Snapshots of the time development of the vortex tangle in the first stage viewed along the flow direction $(T=1.85$ K, ${\overline{\mathit{v}}}_n=1.2$ cm/s): (a) $t=0$ s, (b) $t=0.15$ s, (c) $t=0.25$ s, (d) $t=0.4$ s.

Figure 4

Region where mutual friction vanishes: the dark symbols show the region where ${\mathit{v}}_{s,a}={\mathit{v}}_n$, and the light symbols represent ${\mathit{v}}_s={\mathit{v}}_n$.

Figure 5

Typical snapshots of vortices in the diffusive state: (a) viewed along the flow direction and (b) viewed from the side of the flow direction.

Figure 6

(a) Typical development of $L$ with $T=1.55$ K for each state. (b) Phase diagram for the critical velocity.

Figure 7

Time development of $L_{\mathrm{in}}/L_{\mathrm{out}}$ for the parameters described in Fig. 6.

Figure 8

Spatial dependence of the superfluid velocity profile on $x=0$ mm at $t=40$ s with $T=1.95$ K and $\overline{\mathit{v}}=1.0$ cm/s.

Figure 9

Normal-fluid velocity profile and time development of superfluid velocity profile.