Three-Dimensional Coupled Dynamics of the Two-Fluid Model in Superfluid ${}^4\mathrm{He}$: Deformed Velocity Profile of Normal Fluid in Thermal Counterflow

The coupled dynamics of the two-fluid model of superfluid ${}^4\mathrm{He}$ is numerically studied for quantum turbulence of the thermal counterflow in a square channel. We combine the vortex filament model of the superfluid and the Navier-Stokes equations of normal fluid. Simulations of the coupled dynamics show that the velocity profile of the normal fluid is deformed significantly by superfluid turbulence as the vortices become dense. This result is consistent with recently performed visualization experiments. We introduce a dimensionless parameter that characterizes the deformation of the velocity profile.

Figure 1

The quantized vortices and normal fluid velocity profile in thermal counterflow are obtained by numerical simulation of the coupled dynamics at ${\overline{v}}_n=1.0\mathrm{mm}/\mathrm{s}$ and $T=2.0\mathrm{K}$ ($\mathrm{Re}=300$). (Top) The black curves show vortex filaments that represent quantized vortices. (Bottom) The color map shows the $x$ component of the normal fluid velocity. Here, the mean velocity ${\overline{v}}_n$ of the normal fluid is constant.

Figure 2

(a) The central velocity $v_{n,c}$ of a normal fluid as a function of time at different mean velocity ${\overline{v}}_n$ of the normal fluid. The values fluctuate around the constant values in the early stage, and decrease after the onset time shown by the arrows. (b) Mutual friction force $\varphi$ of Eq. (3) as a function of time. The values significantly increase after the deformation begins, which is marked by the green circles.

Figure 3

Simultaneous dynamics of the two fluids: profiles at $z=0.5\mathrm{cm}$ through the center of the channel cross section at $t=20,40\mathrm{s}$ for ${\overline{v}}_n=1.0\mathrm{mm}/\mathrm{s}$. (a) The normal fluid velocity $v_n$ and the superfluid velocity $v_s$. (b) The local vortex line density $L^{\prime }$ scaled by $L$.

Figure 4

Results for ${\overline{v}}_n=0.7\mathrm{mm}/\mathrm{s}$ with a different scale factor $c$ of the mutual friction force. (a) The central velocity $v_{n,c}$ of a normal fluid as a function of time. (b) Amplified mutual friction force $c\varphi$ as a function of time. (c) Snapshots of velocity profiles of normal fluid in the deformation state at $z=0.5\mathrm{cm}$ through the center of the cross section.