Multiple critical velocities in oscillatory flow of superfluid ${}^4\mathrm{He}$ due to quartz tuning forks

We report recent investigations into the transition to turbulence in superfluid ${}^4\mathrm{He}$, realized experimentally by measuring the drag forces acting on two custom-made quartz tuning forks with fundamental resonances at 6.5 kHz and 55.5 kHz, in the temperature range 10 mK to 2.17 K. In pure superfluid in the zero temperature limit, three distinct critical velocities were observed with both tuning forks. We discuss the significance of all critical velocities and associate the third critical velocity reported here with the development of large vortical structures in the flow, which thus starts to mimic turbulence in classical fluids. The interpretation of our results is directly linked to previous experimental work with oscillators such as tuning forks, grids, and vibrating wires, focusing on the behavior of purely superfluid ${}^4\mathrm{He}$ at very low temperatures.

Figure 1

Diagram of the principal measurement scheme used in Prague. To achieve the full range of velocities, the applied voltage was either (a) directly fed to the tuning fork, (b) attenuated by one or more inline attenuators, or (c) amplified by a transformer with a voltage ratio of 7.31 at the 6.5 kHz fork's fundamental resonance and 2.00 at the overtone. The transformer's output was constantly monitored by a Keithley Model 2000 digital multimeter. An I/V converter [43] with a conversion ratio of 1000 V/A was used to convert the current signal into voltage prior to detection with the SR-830 lock-in.

Figure 2

Peak velocity of the 6.5 kHz tuning fork oscillating at its fundamental frequency (bottom) and at the first overtone (top) as a function of the applied force in helium at various temperatures.

Figure 3

Analysis of linear drag forces acting on the 6.5 kHz tuning fork with changing temperature. The $\lambda$ coefficients for the fundamental mode and overtone (at the specified frequencies) are plotted vs temperature in the top panel. The low-temperature part of each dependence fits well to a sum of ballistic phonon drag ($\propto T^4$) and intrinsic TF damping (constant), while at higher temperatures, deviations occur due to viscous damping from the normal component gradually taking over the part of the ballistic phonon drag. In the lower panel, the ratio ${\lambda }_{\mathrm{o}}/{\lambda }_{\mathrm{f}}$ is shown, highlighting the frequency dependencies of the linear damping in different regimes. The frequency-independent ballistic limit, as well as the viscous limit with square root frequency dependence [39], is clearly indicated.

Figure 4

Selected frequency sweeps of the 6.5 kHz tuning fork at the indicated driving voltages at 20 mK for the fundamental mode (bottom) and overtone (top). The vertical solid blue lines mark the position of the resonance peak at low drives. The red dotted lines estimate the velocity at which the frequency softening sets in for the fundamental mode and the frequency shift becomes apparent for the overtone; the black dashed lines mark estimated velocities at which nonlinear damping becomes apparent. Note that the behavior of the overtone at low drives has almost systematic variations of the resonant frequency. This is not fully understood at this point and is likely related to very weak coupling to resonances of the fork's mechanical support or to the acoustic resonances of the capillary enclosing the fork. See also Fig. 5 for comparison of the resonance frequencies and Fig. 6 in the next section for the corresponding drag coefficients.

Figure 5

Resonant frequency shift plotted vs tuning fork peak velocity for the 6.5 kHz tuning fork (outliers removed). The red dotted lines mark the onset of the shifts in frequency and coincide with those in Fig. 4; the black dashed lines from Fig. 4 are included for comparison. The horizontal solid blue lines mark zero frequency shift with respect to the resonance at low drives. The onset of the frequency shifts seems to be independent of temperature for either mode in the investigated range. Examples of individual frequency sweeps taken at 20 mK are shown for comparison in Fig. 4, and corresponding drag coefficients can later be found in Fig. 6. The red dotted lines mark the estimated first deviation of the resonance frequency.

Figure 6

Dimensionless drag coefficient as a function of peak velocity for the 6.5 kHz tuning fork at various temperatures—fundamental mode (bottom) and overtone (top). Each point is obtained from a full frequency sweep at constant drive. The green dash-dotted horizontal line highlights the value $C_{\mathrm{D}}=1$, while the red dotted and black dashed vertical lines mark the first two critical velocities, respectively, and correspond to the same lines in Figs. 4 and 5. The blue solid line in the overtone drag coefficient (top) shows the value of a third hydrodynamical critical velocity $v_{\mathrm{co}3}=1.5$ m ${\mathrm{s}}^{-1}$. The existence of such a third critical velocity for the fundamental mode (bottom) remains unclear, as higher velocities cannot be attained with this tuning fork using the fundamental resonance—the displacement becomes comparable to the prong spacing and at higher drive the two prongs start hitting each other at $\approx 1.8$ m ${\mathrm{s}}^{-1}$.

Figure 7

Peak nonlinear drag force (see text) as a function of velocity for the 6.5 kHz tuning fork at the temperatures indicated—fundamental mode (bottom) and overtone (top), with outliers removed. The meaning of lines is the same as in Fig. 6. Additionally, a slanted straight line is used as a guide for the eye; its slope corresponds to a power law with an exponent of approximately 2.3.

Figure 8

Top: drag coefficient as a function of velocity measured by amplitude sweeps with resonance tracking in Lancaster and full frequency sweeps in Prague for the 6.5 kHz tuning fork. While the frequency sweeps are unable to measure the hysteresis at the onset of nonlinear dissipation and the amplitude sweeps detect it clearly, the overall reproducibility of the data is surprisingly good, including the values of the second critical velocity. Bottom: drag coefficient as a function of velocity measured by amplitude sweeps with resonance tracking in Lancaster using a 55.5 kHz custom-made tuning fork (dimensions $T$, $W$, and $D$ the same as the 6.5 kHz fork, $L=1.2\mathrm{mm}$) displaying the third critical velocity in its fundamental mode, $v_{\mathrm{cf}3}=1.9\mathrm{m}{\mathrm{s}}^{-1}$, which again appears almost independent of temperature below 1 K. For this tuning fork, the first two critical velocities were determined as $v_{\mathrm{cf}1}=0.021\mathrm{m}{\mathrm{s}}^{-1}$ and $v_{\mathrm{cf}2}=0.16\mathrm{m}{\mathrm{s}}^{-1}$. In both panels, the green horizontal dash-dotted line marks the value of unity expected for classical flows. Note that at higher temperatures close to the lambda point, the drag coefficient approaches this value at comparatively lower velocities.

Figure 9

Comparison of the low temperature drag coefficient velocity dependence of the fundamental mode of the 6.5 kHz tuning fork measured by amplitude sweeps and a series of frequency sweeps. The second critical velocity is indicated by the vertical dashed line. Note that the data measured using the amplitude sweeps show hysteresis.