Oscillating pendulum decay by emission of vortex rings

Abstract

We have studied oscillation of a pendulum in water using spherical bobs. By measuring the loss in potential energy, we estimate the drag coefficient on the sphere and compare to data from liquid-helium experiments. The drag coefficients compare very favorably illustrating the true scaling behavior of this phenomenon. We also studied the decay of amplitude of the pendulum over time. As observed previously, at small amplitudes, the drag on the bob is given by the linear Stokes drag and the decay is exponential. For larger amplitudes, the pendulum bob sheds vortex rings as it reverses direction. The momentum imparted to these vortex rings results in an additional discrete drag on the bob. We present experiments and a theoretical estimate of this vortex-ring-induced drag. We analytically derive an estimate for a critical amplitude beyond which vortex ring shedding will occur as well as an estimate of the radius of the ring as a function of amplitude.

Experimental test of unsteady Stokes’ drag force on a sphere

“…Table 1 summarizes the measurements of the drag force on spheres. The results of Jäger et al [18], reanalyzed by Bolster et al [11], have not been included because some information is missing: following Bolster et al, the Reynolds number is larger than 20 (respectively 40) when the helium temperature is 2.1 K (respectively 2.2 K) in these experiments. As far as we know, no experiment with oscillating spheres has explored Reynolds numbers smaller than about 20.…”

“…These experiments use micron-sized probe particles and very sensitive position-measurement techniques. Some experiments are active, with the application of forces or torques on the probe particle, while other experiments Table 1 The information collected in this table is taken from the papers of Baily [8], Gupta et al [10], Bolster et al [11]. From the published data, whenever it was necessary, we have calculated the values of the angular frequency ω and of the viscous penetration depth δ, the Stokes are passive with the probe particle motion due to random thermal fluctuations i.e.…”

“…Table 2 For each sphere, we give the rounded value in millimeters of its radius a, the measurements in seconds of the damping time constant measured under atmospheric pressure, τ exp1 in the preliminary set-up and τ exp2 in the set-up described above, their mean τexp, τ nh measured under vacuum and τ h calculated thanks to eq. (11) .…”

Slowing down of vortex rings in Bose-Einstein condensates

“…Vortex ring dynamics is still a very active area of research [3][4][5][6]. Unfortunately, real fluids are viscous, and vortex rings with a thick core are unstable [7]: vorticity diffuses and the core becomes distorted.…”

The motion of a 2D pendulum in a channel subjected to an incoming flow

“…Recently some attention has been devoted to cases where the pendulum interacts with a surrounding fluid. For instance, Bolster, Hershberger & Donnelly (2010) have studied the large-amplitude oscillation of a spherical pendulum immersed in a fluid at rest, quantifying the additional dissipative mechanisms due to vortex ring shedding.…”