Determination of the effective kinematic viscosity for the decay of quasiclassical turbulence in superfluid ${}^4\mathrm{He}$

The energy dissipation of quasiclassical homogeneous turbulence in superfluid ${}^4\mathrm{He}$ (He II) is controlled by an effective kinematic viscosity ${\nu }^{\prime }$, which relates the energy decay rate $dE/dt$ to the density of quantized vortex lines $L$ as $dE/dt=-{\nu }^{\prime }{\left(\kappa L\right)}^2$. The precise value of ${\nu }^{\prime }$ is of fundamental importance in developing our understanding of the dissipation mechanism in He II, and it is also needed in many high-Reynolds-number turbulence experiments and model testing that use He II as the working fluid. However, a reliable determination of ${\nu }^{\prime }$ requires the measurements of both $E\left(t\right)$ and $L\left(t\right)$, which was never achieved. Here we discuss our study of the quasiclassical turbulence that emerges in the decay of thermal counterflow in He II above 1 K. We were able to measure $E\left(t\right)$ by using a recently developed flow-visualization technique and $L\left(t\right)$ via second-sound attenuation. We report the ${\nu }^{\prime }$ values in a wide temperature range determined from a comparison of the time evolution of $E\left(t\right)$ and $L\left(t\right)$.

Figure 1

(a) Schematic diagram of experimental setup. (b) Typical images showing the deformation of the ${\mathrm{He}}_2^{*}$ molecular tracer lines in steady-state thermal counterflow. The white dashed lines indicate the initial locations of the tracer lines.

Figure 2

(a) Velocity probability density functions (PDFs) in decaying counterflow turbulence at 1.65 K with an initial heat flux of $426\mathrm{mW}/{\mathrm{cm}}^2$. The solid curves represent Gaussian fits to the data. (b) Streamwise velocity fluctuation $\mathrm{\Delta }{U}_z$ determined from the Gaussian fits of the velocity PDFs.

Figure 3

The calculated second-order transverse structure function at different decay times in decaying counterflow with an initial heat flux of $426\mathrm{mW}/{\mathrm{cm}}^2$.

Figure 4

The decay of the vortex line density $L\left(t\right)$ measured at different initial heat fluxes at 1.65 K.

Figure 5

The decay of the total turbulence energy density in decaying counterflow turbulence. The blue circles and black triangles are calculated based on the expression on the left-hand side of Eq. (3). The black solid curve and the red dashed curve represent the results calculated by using the integral on the right-hand side of Eq. (3). The best agreement of the calculated energy density at large decay times is achieved with ${\nu }^{\prime }/\kappa =0.46$.

Figure 6

Effective kinematic viscosity in units of $\kappa$. The blue triangles represent ${\nu }^{\prime }$ values calculated by using our new method via Eq. (3). The black squares are calculated by using our vortex density data via Eq. (2). The red solid circles represent ${\nu }^{\prime }$ values obtained by Stalp et al. in the towed-grid turbulence experiment [26] that are corrected by Chagovets et al. [8]. The black solid curve is the kinematic viscosity of He II calculated based on the normal-fluid viscosity alone [2].