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arXiv:1211.5869 · Procedia IUTAM

Modulation of a Quantized Vortex Street with a Vibrating Obstacle

Hiroki Saito, Kenta Tazaki, Tomohiko Aioi

Published Jan 1, 2013• 7 citations
arXivHTMLPDF

Abstract

Dynamics of a superfluid flow past an obstacle are investigated by solving the Gross–Pitaevskii equation numerically. For appropriate velocity and size of the obstacle, quantized vortices are periodically generated in the wake, which form a Bénard–von Kármán vortex street. It is found that vibration of an obstacle modulates the vortex street, breaking a symmetry. The parameter dependence of the modulation dynamics of vortices is investigated.

DOI
10.1016/j.piutam.2013.09.011

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Dynamics of a superfluid flow past an obstacle are investigated by solving the Gross–Pitaevskii equation numerically. For appropriate velocity and size of the obstacle, quantized vortices are periodically generated in the wake, which form a Bénard–von Kármán vortex street. It is found that vibration of an obstacle modulates the vortex street, breaking a symmetry. The parameter dependence of the modulation dynamics of vortices is investigated.
We show that an elliptical obstacle moving through a Bose–Einstein condensate generates wakes of quantum vortices which resemble those of classical viscous flow past a cylinder or sphere. The role of ellipticity is to facilitate the interaction of the vortices nucleated by the obstacle. Initial steady symmetric wakes lose their symmetry and form clusters of like-signed vortices, in analogy to the classical Bénard–von Kármán vortex street. Our findings, demonstrated numerically in both two and three dimensions, confirm the intuition that a sufficiently large number of quanta of circulation reproduce classical physics.
We show that an elliptical obstacle moving through a Bose–Einstein condensate generates wakes of quantum vortices which resemble those of classical viscous flow past a cylinder or sphere. The role of ellipticity is to facilitate the interaction of the vortices nucleated by the obstacle. Initial steady symmetric wakes lose their symmetry and form clusters of like-signed vortices, in analogy to the classical Bénard–von Kármán vortex street. Our findings, demonstrated numerically in both two and three dimensions, confirm the intuition that a sufficiently large number of quanta of circulation reproduce classical physics.
Bénard-von Kármán vortex street in dipolar Bose-Einstein Condensate (BEC) trapped by a square-like potential is investigated numerically. In the frame of mean-field theory, the nonlinear dynamic of the dipolar BEC can be described by the so-called two-dimensional Gross-Pitaevskii (GP) equation with long-range interaction. In this paper, we only consider the case that all the dipoles are polarized along the z-axis, which is perpendicular to the plane of disc-shaped BEC. Firstly, the stationary state of the BEC is obtained by the imaginary-time propagation approach. Secondly, the nonlinear dynamic of the BEC, when a moving Gaussian potential exists in such a system, is numerically investigated by the time-splitting Fourier spectral method, in which the stationary state obtained before is set to be the initial state. The results show that when the velocity of the cylindrical obstacle potential reaches a critical value, which depends on interaction strength and the shape of the potential, the vortex-antivortex pairs will be generated alternately in the super-flow behind the obstacle potential. However, in general, such a vortex-antivortex pair structure is dynamically unstable. When the velocity of the obstacle potential increases to a certain value and for a suitable potential width, a stable vortex structure called Bénard-von Kármán vortex street will be formed. While this phenomenon emerges, the vortices in pairs created by the obstacle potential have the same circulation. The pairs with opposite circulations are alternately released from the moving obstacle potential. For larger potential width and velocity, the shedding pattern becomes irregular. We also numerically investigate the effects of the dipole interaction strength, the width and the velocity of the obstacle potential on the vortex structures arising in the wake flow. As a result, the phase graph is presented by lots of numerical calculations for a group of given physical parameters. Thirdly, the drag force on the obstacle potential is also calculated and the mechanical mechanism of vortex pair is analyzed. Finally, we discuss how to find the phenomenon of Bénard-von Kármán vortex street in dipolar BEC experimentally.
The Gross-Pitaevskii equation is widely used for vortex dynamics, but finite domains with hard walls or confining potentials distort bulk behavior through vortex-image effects or induced flows. Periodic boundaries reduce wall artifacts yet cannot realize finite net vorticity because of topological obstruction, so bulk simulations with nonzero circulation are typically unavailable. Hence, we impose quasi-periodic boundary conditions that keep the superfluid's density periodic while enforcing phase windings consistent with a net prescribed total vorticity. This setting conserves the net number of vortices and enables long-time tracking of vortex trajectories in settings that finite containers cannot capture. This allows us to study vortex depinning and nucleation leading to the creation of Kármán vortex streets and perfectly periodic vortex arrays. The framework also provides a toy model for studying vortex dynamics in the bulk of neutron stars, free of possible limitations induced by confining potentials.
The dynamics of pseudo-spin-1/2 Bose-Einstein condensates with weak spin-orbit coupling through a moving obstacle potential are studied numerically. Four types of wakes are observed and the phase diagrams are determined for different spin-orbit coupling strengths. The conditions to form Bénard-von Kármán vortex street are rather rigorous, and we investigate in detail the dynamical characteristics of the vortex streets. The two point vortices in a pair rotate around their center, and the angular velocity and their distance oscillate periodically. The oscillation intensifies with increasing spin-orbit coupling strengths, and it makes part of the vortex pairs dissociate into separate vortices or combine into single ones and destroys the vortex street in the end. The width b of the street and the distance l between two consecutive vortex pairs of the same circulation are determined by the potential radius and its moving velocity, respectively. The b/l ratios are independent of the spin-orbit coupling strength and fall in the range 0.19-0.27, which is a little smaller than the stability criterion 0.28 for classical fluids. Proper b/l ratios are necessary to form Bénard-von Kármán vortex street, but the spin-orbit coupling strength affects the stability of the street patterns. Finally, we propose a protocol to experimentally realize the vortex street in ^{87}Rb spin-orbit-coupling Bose-Einstein condensates.