Visualization of viscous and quantum flows of liquid ${}^4\mathrm{He}$ due to an oscillating cylinder of rectangular cross section

The motions of micrometer-sized solid deuterium particles in liquid ${}^4\mathrm{He}$, at temperatures between approximately 1.2 and 3 K, are visualized in the proximity of an oscillating cylinder of rectangular cross section (3 mm high and 10 mm wide). The cylinder is oscillating vertically, perpendicularly to its cross-section width, at frequencies between 0.05 and 1.25 Hz, and amplitudes of 5 and 10 mm, resulting in Reynolds numbers $Re$ up to ${10}^5$. The aim of the reported experiments is to investigate systematically the macroscopic vortical structures shed at the cylinder sharp edges, by tracking the deuterium particles. We find that large-scale, millimeter-sized vortices are generated in the surrounding fluid by the oscillating cylinder, both in viscous He I and superfluid He II. An estimate of the strength of the shed vortical structures reveals that, for $Re>{10}^4$, the corresponding magnitudes are approximately equal in He I and He II if, in He II, the kinematic viscosity is suitably defined. For $Re<{10}^4$, the strength of the large-scale vortices is smaller in He II than in He I. Although the outcome is partly affected by the larger scatter of the He I data and possibly also by the much larger heat conductivity of superfluid ${}^4\mathrm{He}$, we argue that the fundamental physical reason for observing this difference is that, at these Reynolds numbers, the experimentally probed length scales in He II are smaller than the average distance between quantized vortices—the quantum length scale of the flow. The result strongly suggests that, similarly to thermal counterflow, both viscous and quantum features can be observed in mechanically driven flows of He II, depending on the length scales at which the quantum flow is probed.

Figure 1

(Left) Picture of the 30 mm long plexiglass cylinder of rectangular cross section (3 mm high and 10 mm wide) and of the bottom part of its brass support. The cylinder section facing the camera and the bottom part of the support are painted in black to reduce possible laser-induced reflections. (Right) Picture of the step motor and of the crank mechanism, connected, on top of the cryostat, to the stainless steel shaft firmly linked to the top part of the cylinder support.

Figure 2

Particle trajectories, with at least five points, obtained within the ${15}^{\circ }$ phase interval centered at the lowest position of the cylinder oscillatory motion; frequency $f=0.5$ Hz and amplitude $a=5$ mm; 21.4 mm wide and 16.0 mm high field of view. (Left) He II, temperature $T=1.24$ K, Reynolds numbers $R{e}_n=11717$, Eq. (4), and $R{e}_{\kappa }=9428$, Eq. (5). (Right) He I, $T=2.18$ K and Reynolds number $Re=9092$, Eq. (3). The grey rectangles indicate the cylinder, at the lowest position of the cycle, while the white rectangles denote the corresponding mask used for data processing. See the text for further details on the three definitions of the Reynolds number.

Figure 3

Maps of the $\theta (\mathbf{r},\phi )$ parameter for (clockwise, from the top left panel) phase $\phi ={90}^{\circ }\pm 7.5^{\circ }$ (a), ${150}^{\circ }\pm 7.5^{\circ }$ (b), ${210}^{\circ }\pm 7.5^{\circ }$ (c), ${270}^{\circ }\pm 7.5^{\circ }$ (d), ${330}^{\circ }\pm 7.5^{\circ }$ (e), and ${30}^{\circ }\pm 7.5^{\circ }$ (f). Temperature $T=1.24$ K, frequency $f=0.5$ Hz, and amplitude $a=5$ mm. The grey rectangles indicate the cylinder positions, in the middle of the considered phase intervals, while the white rectangles denote the corresponding masks used for data processing. The green arrows between the panels show the direction of motion, e.g., in panel (a) the cylinder is at the lowest position of the cycle, see also the left panel of Fig. 2.

Figure 4

Maps of the $\theta (\mathbf{r},\phi )$ parameter, calculated at the lowest position of the oscillatory motion; amplitude $a=5$ mm. (a) Frequency $f=0.1$ Hz, temperature $T=1.26$ K, Reynolds numbers $R{e}_n=2443$ and $R{e}_{\kappa }=1886$; (b) $f=0.2$ Hz, $T=1.26$ K, $R{e}_n=4885$, and $R{e}_{\kappa }=3771$; (c) $f=0.5$ Hz and $T=1.24$ K, see also the left panel of Fig. 2; (d) $f=1$ Hz, $T=1.24,R{e}_n=23434$, and $R{e}_{\kappa }=18855$. Legend as in Fig. 3.

Figure 5

Maps of the $\theta (\mathbf{r},\phi )$ parameter, calculated at the lowest position of the oscillatory motion; frequency $f=0.5$ Hz. (a) He II, amplitude $a=5$ mm and temperature $T=1.24$ K, see also the left panel of Fig. 2; (b) He I, $a=5$ mm and $T=2.18$ K, see also the right panel of Fig. 2; (c) He II, $a=10$ mm, $T=1.40$ K, Reynolds numbers $R{e}_n=29973$ and $R{e}_{\kappa }=18855$; (d) He I, $a=10$ mm, $T=2.18$ K, and Reynolds number $Re=18184$. Legend as in Fig. 3.

Figure 6

Parameter $\langle {\theta }^2\rangle$ as a function of the Reynolds numbers $Re$ and $R{e}_n$. Red triangles: He I. Open black squares: He II. (Inset) Normalized standard deviation $\sigma$ of ${\theta }^2$ as a function of $Re$ and $R{e}_n$; symbols as in the main panel.

Figure 7

Parameter $\langle {\theta }^2\rangle$ as a function of the Reynolds numbers $Re$ and $R{e}_{\kappa }$. Red triangles: He I, as in Fig. 6. Open black squares: He II. (Inset) Normalized standard deviation $\sigma$ of ${\theta }^2$ as a function of $Re$ and $R{e}_{\kappa }$; symbols as in the main panel.

Figure 8

Parameter $\langle {\theta }^2\rangle$ as a function of the Reynolds numbers $Re$ and $R{e}_{\kappa }$. Red triangles: He I. Open black squares: He II. Each experimental condition is represented by at least 40 000 track points. At each Reynolds number the corresponding average of the obtained $\langle {\theta }^2\rangle$ values is plotted; the shown error bars denote the standard deviation of the $\langle {\theta }^2\rangle$ values obtained at that Reynolds number. (Inset) Normalized standard deviation $\sigma$ of ${\theta }^2$ as a function of $Re$ and $R{e}_{\kappa }$; symbols as in the main panel.

Figure 9

Ratio $R$ between the probed length scale $d$ and the flow scale $s$ as a function of the Reynolds numbers $Re$ and $R{e}_{\kappa }$. Red triangles: He I, with $s=\eta$, the Kolmogorov length scale, Eq. (11). Open black squares: He II, with $s=\ell$, the quantum length scale, Eq. (10). Open blue circles: He II, with $s=\ell$ calculated in the same fashion as $\eta$, i.e., $\nu$ is replaced by $\kappa /6$ in Eq. (11). Each experimental condition is represented by at least 40 000 track points. At each Reynolds number, the corresponding average of the obtained $R$ values is plotted; the shown error bars denote the standard deviation of the $R$ values obtained at that Reynolds number. See the text for further details on the used scales.