Characterizing vortex tangle properties in steady-state He II counterflow using particle tracking velocimetry

Historically, there has been little faith in particle tracking velocimetry (PTV) as a tool to make quantitative measurements of thermal counterflow in He II, since tracer particle motion is complicated by influences from the normal fluid, superfluid, and quantized vortex lines, or a combination thereof. Recently, we introduced a scheme for differentiating particles trapped on vortices (G1) from particles entrained by the normal fluid (G2). In this paper, we apply this scheme to demonstrate the utility of PTV for quantitative measurements of vortex dynamics in He II counterflow. We estimate $\ell$, the mean vortex line spacing, using G2 velocity data, and $c_2$, a parameter related to the mean curvature radius of vortices and energy dissipation in quantum turbulence, using G1 velocity data. We find that both estimations show good agreement with existing measurements that were obtained using traditional experimental methods. This is of particular consequence since these parameters likely vary in space, and PTV offers the advantage of spatial resolution. We also show a direct link between power-law tails in transverse particle velocity probability density functions (PDFs) and reconnection of vortex lines on which G1 particles are trapped.

Figure 1

Simple illustration of the experimental apparatus (not to scale).

Figure 2

Typical (a) streamwise velocity PDF $(T=1.85$ K, $q=38$ mW/${\mathrm{cm}}^2)$ and (b) transverse velocity PDFs $(T=2.00$ K, $q=113$ mW/${\mathrm{cm}}^2)$ obtained from PTV measurements of thermal counterflow at relatively low heat flux. The transverse velocity PDFs are normalized by standard deviation ${\sigma }_u$.

Figure 3

Prediction of mean vortex line spacing using G2 mean free path model and traditional second sound attenuation for (a) $T=1.70$ K, (b) $T=1.85$ K, and (c) $T=2.00$ K.

Figure 4

Linear fit to ${\sigma }_{G1}{ln}^{-1}\left(\ell /{\xi }_0\right)$ as a function of $v_{ns}-v_0$ at (a) $T=1.70$, (b) 1.85, and (c) 2.00 K. (d) Extracted values of $c_2$ as a function of temperature.

Figure 5

Selected G1 tracks that contribute to transverse PDF tails at (a) 1.70 K, (b) 1.85 K, and (c) 2.00 K. Blue circles indicate the beginning of each track and black arrows indicate the segment that contributes to the transverse velocity PDF tail. (d)–(f) Corresponding acceleration along the tracks. Dashed lines represent Eq. (3).