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The onset of turbulent flow around an oscillating sphere in superfluid 4He is known to occur at a critical velocity where is the circulation quantum and is the oscillation frequency. But it is also well known that initially in a first up-sweep of the oscillation amplitude, can be considerably exceeded before the transition occurs, thus leading to a strong hysteresis in the velocity sweeps. The velocity amplitude where the transition finally occurs is related to the density of the remanent vortices in the superfluid. Moreover, at temperatures below ca. 0.5 K and in a small interval of velocity amplitudes between 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 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.
The flow of superfluid around a translationally oscillating sphere, levitating without mechanical support, can either be laminar or turbulent, depending on the velocity amplitude. Below a critical velocity that scales as and is temperature independent below 1 K, the flow is laminar (potential flow). Below 0.5 K, the linear drag force is caused by ballistic phonon scattering that vanishes as until background damping, measured in the empty cell, becomes dominant for K. Increasing the velocity amplitude above leads to a transition from potential flow to turbulence, where the large turbulent drag force varies as . In a small velocity interval % above , the flow is unstable below 0.5 K, switching intermittently between both patterns. From time series recorded at constant temperature and driving force, the lifetimes of both phases are analyzed statistically. We observe metastable states of potential flow which, after a mean lifetime of 25 min, ultimately break down due to vorticity created by natural background radioactivity. The lifetimes of the turbulent phases have an exponential distribution, and the mean increases exponentially with . We investigate the frequency at which the vortex rings are shed from the sphere. Our results are compared with recent data of other authors on vortex shedding by moving a laser beam through a Bose-Einstein condensate. Finally, we show that our observed transition to turbulence belongs to the class of "supertransient chaos" where lifetimes of the turbulent states increase faster than exponentially.
The flow of superfluid helium-4 at very low temperatures around an oscillating microsphere has been studied in detail. At small oscillation amplitudes the drag force is linear in velocity amplitude. Above a critical velocity amplitude vc a transition from potential flow to turbulent flow is signaled by a large and nonlinear drag force that scales as (v2 - v2 c ), where vc is found to be independent of temperature (below 1 K) and of driving force. Interestingly, the critical velocity is found to scale as vc ~ √κω, where κ = h/m ≈ 10−7 m2/s (h is Planck's constant and m is the mass of a helium-4 atom) is the circulation quantum of the superfluid vortices and ω/2π is the oscillation frequency ranging from about 120 Hz to 700 Hz. We also observe slightly above vc that the flow pattern is unstable and switches intermittently between potential flow and turbulence. From time series recorded at constant drive and temperature we have investigated the statistical properties of this switching phenomenon. In this talk the universal law vc ~ √κω will be discussed in detail. It can be derived theoretically in various ways: firstly, from a qualitative but very general argument based on the "superfluid Reynolds number" Rs = vl/κ where v is the flow velocity and l is a characteristic length scale, secondly, and in more detail, from Kopnin's equation of the time dependence of the vortex line density, and, finally, we have obtained this law rigorously by dynamical scaling of the equations of motion of vortex dynamics.