Coupled normal fluid and superfluid profiles of turbulent helium II in channels

We perform fully coupled two-dimensional numerical simulations of plane channel helium II counterflows with vortex-line density typical of experiments. The main features of our approach are the inclusion of the back reaction of the superfluid vortices on the normal fluid and the presence of solid boundaries. Despite the reduced dimensionality, our model is realistic enough to reproduce vortex density distributions across the channel recently calculated in three dimensions. We focus on the coarse-grained superfluid and normal fluid velocity profiles, recovering the normal fluid profile recently observed employing a technique based on laser-induced fluorescence of metastable helium molecules.

Figure 1

The structure of the fine (blue solid lines) and coarse grids (green solid lines) are illustrated on a particular portion of the computational domain, together with positive and negative vortices indicated with empty red and filled black circles, respectively.

Figure 2

Two-dimensional color plot of the longitudinal component of the mutual friction force ${\tilde{F}}^x/{\rho }_n$ [see Eq. (16)], interpolated on the fine grid on the same domain as Fig. 1. The axes of the plot are rescaled employing the scaling units defined in Sec. 3a.

Figure 3

Top: vortex distribution at $t=0$, red empty (black filled) circles indicate positive (negative) vortices; middle: coarse-grained profiles of positive vortex density ${\overline{n}}^{+}$ (solid red line), negative vortex density ${\overline{n}}^{-}$ (dashed black line), and total vortex density $\overline{n}$ (dot-dashed green line) at $t=0$. In the inset, the corresponding coarse-grained profile of the polarization magnitude $\overline{p}\left(y\right)$ is reported (solid magenta line); bottom: coarse-grained profiles of superfluid velocity ${\overline{u}}_s$ (solid red line), normal fluid velocity ${\overline{u}}_n$ (solid blue line), and counterflow velocity ${\overline{u}}_{ns}={\overline{u}}_n-{\overline{u}}_s$ (solid green line) at $t=0$.

Figure 4

Top: vortex distribution at $t=T_f$, red empty (black filled) circles indicate positive (negative) vortices; middle: coarse-grained profiles of positive vortex density ${\overline{n}}^{+}$ (solid red line), negative vortex density ${\overline{n}}^{-}$ (dashed black line), and total vortex density $\overline{n}$ (dot-dashed green line) at $t=T_f$. In the inset, the corresponding coarse-grained profile of the polarization magnitude $\overline{p}\left(y\right)$ is reported (solid magenta line); bottom: coarse-grained profiles of superfluid velocity ${\overline{u}}_s$ (solid red line), normal fluid velocity ${\overline{u}}_n$ (solid blue line), and counterflow velocity ${\overline{u}}_{ns}={\overline{u}}_n-{\overline{u}}_s$ (solid green line) at $t=T_f$. Red and blue dot-dashed lines indicate the initial laminar profiles of the superfluid and the normal fluid, respectively.

Figure 5

Top) coarse-grained profiles of positive vortex density ${\overline{n}}^{+}$ (solid red line), negative vortex density ${\overline{n}}^{-}$ (dashed black line), and total vortex density $\overline{n}$ (dot-dashed green line) at $t=6.8\times {10}^{-3}{T}_f$. In the inset, the corresponding coarse-grained profile of the polarization magnitude $\overline{p}\left(y\right)$ is reported (solid magenta line); bottom: coarse-grained profiles of superfluid velocity ${\overline{u}}_s$ (solid red line), normal fluid velocity ${\overline{u}}_n$ (solid blue line), and counterflow velocity ${\overline{u}}_{ns}={\overline{u}}_n-{\overline{u}}_s$ (solid green line) at $t=6.8\times {10}^{-3}{T}_f$. Red and blue dot-dashed lines indicate the initial laminar profiles of the superfluid and the normal fluid, respectively.

Figure 6

Coarse-grained profile of the longitudinal component of the mutual friction force ${\overline{F}}^x$ at different selected times: $t=0$ (dot-dashed green line); $t=6.8\times {10}^{-3}{T}_f$ (dashed green line); $t=T_f$ (solid green line).

Figure 7

Trajectories ${\mathbf{r}}_{+}\left(t\right)$ and ${\mathbf{r}}_{-}\left(t\right)$ of an antivortex pair, whose initial configuration is symmetric with respect to the midplane of the channel (in dashed blue line). The time interval between consecutive positions is constant, indicating an increasing streamwise velocity as the vortex points approach the walls. In an axisymmetric interpretation, the three-dimensional analog of this two-dimensional motion is a streamwise flow of an expanding vortex ring.

Figure 8

Distinct three-dimensional interpretations of vortex-points motion: (a) straight vortices model; (b) idealized axisymmetric vortex-ring interpretation described in Sec. 4a; (c) vortex-ring analog of the comprehensive two-dimensional motion.