Statistical properties of homogeneous and isotropic turbulence in He II measured via particle tracking velocimetry

Despite being a quantum two-fluid system, superfluid helium-4 (He II) is observed to behave similarly to classical fluids when a flow is generated by mechanical forcing. This similarity has brought up the feasibility of utilizing He II for high Reynolds number classical turbulence research, considering the small kinematic viscosity of He II. However, it has been suggested that the nonclassical dissipation mechanism in He II at small scales may alter its turbulent statistics and intermittency. In this work, we report our study of a nearly homogeneous and isotropic turbulence (HIT) generated by a towed grid in He II. We measure the velocity field using particle tracking velocimetry with solidified deuterium particles as the tracers. By correlating the velocities measured simultaneously on different particle trajectories or at different times along the same particle trajectory, we are able to conduct both Eulerian and Lagrangian flow analyses. Spatial velocity structure functions obtained through the Eulerian analysis show scaling behaviors in the inertial subrange similar to that for classical HIT but with enhanced intermittency. The Lagrangian analysis allows us to examine the flow statistics down to below the dissipation length scale. Interestingly, abnormal deviations from the classical scaling behaviors are observed in this regime. We discuss how these deviations may relate to the motion of quantized vortices in the superfluid component in He II.

Figure 1

Schematic diagram of the experimental apparatus.

Figure 2

(a) Ensemble-averaged vertical velocity $〈v_y〉$ profile following the passage of the grid. (b) The corresponding vertical velocity variance ${\sigma }_y$. The data were taken at 1.95 K with a grid velocity of 30 cm/s.

Figure 3

Horizontal (a) and vertical (b) particle velocity PDFs at different decay times as indicated. The data were taken at 1.95 K with a grid velocity of 30 cm/s.

Figure 4

(a) Time evolution of the turbulence kinetic energy $K\left(t\right)$ contribution from the horizontal and the vertical velocity components. (b) Decay of the quantized vortex-line density $L\left(t\right)$. The solid black curve represents the scaling $L\left(t\right)\propto {(t+t_0)}^{-3/2}$, where $t_0=0.27$ s is the virtual time origin [36]. The data were taken at 1.95 K.

Figure 5

(a1) and (a2) show the compensated third-order transverse and longitudinal structure functions. The dashed horizontal lines mark the constant value of 4/5. The shaded region indicates the inertial subrange. (b1) and (b2) show the second-order structure functions. The solid lines are power-law fits to the data in the shaded region.

Figure 6

Extended self-similarity plots of (a) transverse and (b) longitudinal velocity structure functions for $n=1\sim 5$ versus the third-order structure functions. The solid lines are power-law fits to the data in the inertial subrange.

Figure 7

Intermittency corrections to the scaling exponents of the transverse (a) and longitudinal (b) structure functions for He II grid turbulence. The corrections for classical fluids are also included for comparison [46].

Figure 8

Second-order (a) transverse and (b) longitudinal velocity structure functions obtained through both the Lagrangian and the Eulerian analyses of the particle trajectories for the data set obtained at 1.95 K and $t=4$ s.

Figure 9

Extended self-similarity plots of (a) transverse and (b) longitudinal structure functions based on both Lagrangian (empty symbols) and Eulerian (solid symbols) analyses of the particle trajectories for the data set obtained at 1.95 K and $t=4$ s. The solid lines represent the power-law fits to the Eulerian data shown in Fig 6.

Figure 10

Schematics showing the concept of the Eulerian and the Lagrangian velocity analyses.

Figure 11

Sample number as a function of $r$ for (a) Eulerian and (b) Lagrangian velocity statistical analyses of the particle trajectories for the data set obtained at 1.95 K and $t=4$ s. The analyses are conducted only in the shaded regions where the sample number is greater than ${10}^2$.