# Breakdown of Potential Flow to Turbulence Around a Sphere Oscillating in Superfluid \(^4\)He Above the Critical Velocity

## Abstract

The onset of turbulent flow around an oscillating sphere in superfluid \(^4\)He is known to occur at a critical velocity \(v_{\text {c}} \,\sim \sqrt{\kappa \,\omega }\) where \(\kappa \) is the circulation quantum and \(\omega \) is the oscillation frequency. But it is also well known that initially in a first up-sweep of the oscillation amplitude, \(v_{\text {c}}\) can be considerably exceeded before the transition occurs, thus leading to a strong hysteresis in the velocity sweeps. The velocity amplitude \(v_{\text {c}}^* > v_{\text {c}}\) where the transition finally occurs is related to the density \(L_0\) of the remanent vortices in the superfluid. Moreover, at temperatures below ca. 0.5 K and in a small interval of velocity amplitudes between \(v_{\text {c}}\) and a velocity that is about 2 % larger, the flow pattern is found to be unstable, switching intermittently between potential flow and turbulence. From time series recorded at constant temperature and driving force, the distribution of the excess velocities \(\Delta v = v_{\text {c}}^* - v_{\text {c}}\) is obtained and from that the failure rate. Below 0.1 K we also can determine the distribution of the lifetimes of the phases of potential flow. Finally, the frequency dependence of these results is discussed.

### Keywords

Quantum turbulence Oscillatory flow Intermittent switching Remanent vorticity## 1 Introduction

In a set of careful experiments with a vibrating wire, the Osaka group [3] showed in 2007 that no transition to turbulence occurred up to very large velocity amplitudes of ca. 1 m/s (corresponding to oscillation amplitudes of 100 \(\upmu \)m) if the superfluid was prepared in a state without any remanent vortices. Obviously, some initial vortices must exist in the fluid within the range of the oscillation amplitude for the vibrating object to produce turbulence.

## 2 Intermittent Switching Between Potential Flow and Turbulence

### 2.1 Temperatures Between 0.5 and 0.1 K

### 2.2 Temperatures Below 0.1 K

## 3 The Frequency Dependence of \(v_{\text {w}}\)

## 4 Discussion

Hysteresis and switching of the flow have been observed also with vibrating tuning forks and wires [8, 9, 10], and the significance of remanent vorticity for the critical velocity \(v_{\text {c}}^*\) at breakdown was demonstrated [3, 11]. In our work, we are relating \(v_{\text {c}}^*\) directly to the intervortex spacing of the remanent vortex density.

Discussing the distribution of the lifetimes of the potential flow, we note that because of the nonlinear dependence of \(\Delta v (t)\) in (3) the distribution is very complicated. However, the situation becomes much simpler at low temperatures, see Section 2.2, where \(\Delta v\) is proportional to the lifetime, see (4). The lifetimes are now also following a Rayleigh distribution, and the rms lifetime decreases with increasing driving force. This may be compared with the recent experimental result of the Lancaster group on intermittent switching of the flow around a vibrating tuning fork at mK temperatures [12]. These authors found that the average lifetimes of potential flow decrease toward larger flow velocities, in qualitative agreement with our rms value \(t_{\text {w}}\) (6).

In summary, our data can be analyzed even in more detail and without the assumption (8) when a theory of the transition to turbulence in oscillatory superflows will be available.

## Notes

### Acknowledgments

We are grateful to Jan Jäger and Hubert Kerscher for their co-operation. W.S. acknowledges discussions with Matti Krusius (Aalto University, Finland) and Shaun Fisher (Lancaster University, UK). R.H is supported by the Academy of Finland.

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