Thermally and mechanically driven quantum turbulence in helium II

In most experiments with superfluid helium, turbulence is generated thermally (by applying a heat flux, as in thermal counterflow) or mechanically (by stirring the liquid). By modeling the superfluid vortex lines as reconnecting space curves with fixed circulation, and the driving normal fluid as a uniform flow (for thermal counterflow) and a synthetic turbulent flow (for mechanically driven turbulence), we determine the difference between thermally and mechanically driven quantum turbulence. We find that in mechanically driven turbulence, the energy is concentrated at the large scales, the spectrum obeys Kolmogorov scaling, vortex lines have large curvature, and the presence of coherent vortex structures induces vortex reconnections at small angles. On the contrary, in thermally driven turbulence, the energy is concentrated at the mesoscales, the curvature is smaller, the vorticity field is featureless, and reconnections occur at larger angles. Our results suggest a method to experimentally detect the presence of superfluid vortex bundles.

Figure 1

Thermally induced turbulence. The evolution of the vortex line density $L$ (cm${}^{-2}$) vs time $t$ (s) at counterflow velocities (from top to bottom) $v_{ns}=1.25\mathrm{cm}/\mathrm{s}$ (black), $1.0\mathrm{cm}/\mathrm{s}$ (red), $0.75\mathrm{cm}/\mathrm{s}$ (blue), and $0.55\mathrm{cm}/\mathrm{s}$ (green).

Figure 2

Snapshots of vortex tangles ($y,z$ projections). Top: thermally driven by counterflow ($v_{ns}=0.75\mathrm{cm}/\mathrm{s}$, $L\approx 12000{\mathrm{cm}}^{-2}$); bottom: mechanically driven ($Re=208$, $L\approx 12000{\mathrm{cm}}^{-2}$).

Figure 3

Counterflow turbulence. Top: the perpendicular energy spectrum $E_{\perp }\left(k_{\perp }\right)$ (arbitrary units) vs wave number $k_{\perp }$ (${\mathrm{cm}}^{-1}$). Bottom: the parallel energy spectrum $E_{\parallel }\left(k_{\parallel }\right)$ (arbitrary units) vs wave number $k_{\parallel }$ (${\mathrm{cm}}^{-1}$). The vertical lines mark $k_{\ell }$ at increasing $v_{ns}$ from right to left.

Figure 4

Mechanically induced turbulence. Energy spectrum $E_k$ (arbitrary units) vs wave number $k$ (${\mathrm{cm}}^{-1}$) of vortex tangle driven by the synthetic turbulent flow of Eq. (3) with $M=188$ modes. The vertical dashed blue line marks $k_{\ell }$. The dashed red line shows the $k^{-5/3}$ Kolmogorov scaling. The effective Reynolds number of the normal fluid is $Re={(k_M/k_1)}^{4/3}=208$, defined by the condition that the dissipation time equals the eddy turnover time at $k_M$.

Figure 5

Probability density function (PDF) of the mean curvature per vortex loop $\overline{C}$. Solid black line: mechanically driven turbulence; dashed red line: thermally driven turbulence. Notice the larger curvatures present in thermally driven turbulence.

Figure 6

Smoothed vorticity $\omega$ sustained by a constant ${\mathbf{v}}_n^{\mathrm{ext}}$ (thermally driven turbulence).

Figure 7

Smoothed vorticity $\omega$ sustained by a turbulent ${\mathbf{v}}_n^{\mathrm{ext}}$ (mechanically driven turbulence). Notice the intense vortical regions compared to Fig. 6 which is plotted on the same scale.

Figure 8

Plot of ${\sigma }_{\text{DG}}/{\sigma }_r$ (ratio of Donnelly-Glaberson and vortex reconnection frequencies) vs wave amplitude $A$ (cm), for thermally (dashed, red line) and mechanically (solid line) driven turbulence; the (blue) dotted-dashed line represents ${\sigma }_{\text{DG}}={\sigma }_r$.

Figure 9

The probability density function (PDF) of the angle between reconnecting vortices $\theta$. Thermally driven turbulence: at $T=2.1$ K (dashed red line) and at $T=1.9$ K (solid red squares); mechanically driven turbulence: at $T=2.1$ K (solid black line) and at $T=1.9$ K (solid black line with solid black circles). Note that for thermally driven turbulence, the distribution peaks at large $\theta$, whereas for mechanically driven turbulence, it peaks at small $\theta$.

Figure 10

Probability density functions (PDF) of the fitting parameters $A$ (top) and $c$ (s${}^{-1}$) (bottom) of Eq. (7). Solid black line: mechanically driven turbulence; dashed red line: thermally driventurbulence.

Figure 11

Scatter plots of the fitting parameters $A$ (top) and $c$ (middle) of Eq. (7) vs the angle $\theta$ between the reconnecting vortices. The bottom figure shows the mean curvature $\overline{{\mathcal{C}}_r}$ of the reconnecting vortex segments vs the angle $\theta$. Solid black points: mechanically driven turbulence; open red circles: thermally driven turbulence.