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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.
Wind turbines play a crucial role in the revolution towards renewable resources. They need to be economically competitive to be sustainable. This still requires to lower the cost of energy (COE). To this end, nonlinear model predictive control (NMPC) is used within this paper. As known from literature, NMPC significantly improves the energy extracting performance as well as the mitigation of tower loads of wind turbines. As it is shown in this paper, the drivetrain fatigue strength drops disproportionally in parallel, which either increases the demands on the turbine design or decreases the lifetime of drivetrain components. Without additional boundary conditions to the underlying optimal control problems (OCPs), the application of energy-maximizing NMPC might so even raise the COE. Only penalizing axial torque oscillations by quadratic terms decreases energy-extracting performance below the level of classical wind turbine controllers. This makes more sophisticated conditions necessary. In this paper, the increase of the drivetrain damage by NMPC is analyzed, and appropriate boundary conditions are derived, to balance the two contradicting objectives of energy maximization and drivetrain load mitigation. An NMPC approach based on indirect methods is used, to obtain a solution of the OCPs very efficiently. It applies the Hamilton equations and Pontryagin's maximum principle. Its accuracy and efficiency to solve OCPs was presented over the last decades.