Small-scale universality of particle dynamics in quantum turbulence

We show that, in turbulent two-fluid flows of superfluid ${}^4\mathrm{He}$, the statistically described dynamics of small particles suspended in the liquid does not depend on the type of imposed flow, at scales smaller than the average distance between quantized vortices, the quantum length scale of the flow. More precisely, regardless of the mechanically or thermally driven nature of the large-scale flow, the tails of the observed particle velocity statistical distributions, indicating the occurrence of large-magnitude events in the proximity of quantized vortices, display the same power-law shape, characteristic of the quantum description of superfluid ${}^4\mathrm{He}$. This property of quantum turbulence can be linked to the small-scale universality observed in classical turbulent flows of viscous fluids.

Figure 1

Sketches of the three different flow generators (dimensions in mm; see the main text for further details). Only the bottom part of our experimental volume (the innermost part of the cryostat optical tail) is shown, for the sake of clarity (the cyan rectangles represent three of the five tail optical ports). The laser sheet, perpendicular to the camera, is depicted in green, while the camera field of view is indicated by the black dashed line. The red line represents the flat heater, placed on the bottom of the counterflow glass channel (displayed in black). Left: bulk counterflow. Middle: counterflow in the proximity of a wall (dark blue line). Right: the oscillating cylinder (shown in gray) is attached to a metallic shaft (black); a perpendicular view of the cylinder is displayed at the bottom of the sketch.

Figure 2

Probability density function (PDF) of $(u-u_m)/u_{sd}$. Trajectories with at least five particle positions; total number of points: see the inset of Fig. 4; the area below the data curves is normalized to unity. Circles and triangles denote counterflow data obtained in the proximity of a wall and in the bulk, respectively, with $t_1=0.025$ s (filled symbols), $t_1=0.25$ s (open black and red circles, open green triangles), and $t_1=0.05$ s (open blue triangles). Black circles: $T=1.95$ K, $q=293\mathrm{W}/{\mathrm{m}}^2,v_{ns}=2.4$ mm/s, $V_{abs}=1.5$ mm/s, ${\ell }_{\mathrm{q}}=145\mu \mathrm{m}$; red circles: $T=1.95$ K, $q=587\mathrm{W}/{\mathrm{m}}^2,v_{ns}=4.9$ mm/s, $V_{abs}=1.7$ mm/s, ${\ell }_{\mathrm{q}}=73\mu \mathrm{m}$; green triangles: $T=1.77$ K, $q=612\mathrm{W}/{\mathrm{m}}^2,v_{ns}=6.8$ mm/s, $V_{abs}=2.4$ mm/s, ${\ell }_{\mathrm{q}}=70\mu \mathrm{m}$; blue triangles: $T=1.77$ K, $q=608\mathrm{W}/{\mathrm{m}}^2,v_{ns}=6.8$ mm/s, $V_{abs}=3.9$ mm/s, ${\ell }_{\mathrm{q}}=70\mu \mathrm{m}$, deuterium particles (the other data are obtained with hydrogen particles). Squares indicate oscillating cylinder data, with $t_1=0.01$ s and $a=5$ mm: see the legend and Table 1. Orange lines: power-law distributions, $C/|(u-u_m)/u_{sd}{|}^3$, to guide the eye, with $C$ equal to 0.055 (top panel), 0.15 (bottom panel), and 0.03 (insets); magenta lines: Gaussian distributions. Insets: Log-log plots of the PDF of $|(u-u_m)/u_{sd}|$ for $R<1$; symbols as in the main panels.

Figure 3

PDF of $(u-u_m)/u_{sd}$ for $R<1$ (core region); symbols as in the main panels of Fig. 2.

Figure 4

Flatness of the $(u-u_m)/u_{sd}$ distribution as a function of $R$. Circles and triangles denote counterflow data obtained in the proximity of a wall and in the bulk, respectively, as in Fig. 2 (note, however, that the same symbols in Fig. 2 represent solely the data at the smallest $R$). Squares indicate oscillating cylinder data obtained in He II: see Table 1. Diamonds denote oscillating cylinder data obtained in He I: see Table 3. Magenta line: flatness of the Gaussian distribution. Inset: total number of particle positions as a function of $R$; symbols as in the main panel.

Figure 5

PDF of $(v-v_m)/v_{sd}$. Circles and triangles denote counterflow data obtained in the proximity of a wall and in the bulk, respectively, as in Fig. 2. Orange line: power-law distribution, $0.03/|(v-v_m)/v_{sd}{|}^3$, to guide the eye. Magenta line: Gaussian distribution.