PRL-BathTub-Vortex-2003

PRL-BathTub-Vortex-2 - VOLUME 91 N UMBER 10 PHYSICA L R EVIEW LET T ERS week ending 5 SEPTEMBER 2003 A natomy of a Bathtub Vortex A Andersen,1,2 T

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Anatomy of a BathtubVortex A. Andersen, 1,2, * T. Bohr, 1 B. Stenum, 2 J. Juul Rasmussen, 2 and B. Lautrup 3 1 The Technical University of Denmark, Department of Physics, DK-2800 Kgs. Lyngby, Denmark 2 Risø National Laboratory, Optics and Fluid Dynamics Department, DK-4000 Roskilde, Denmark 3 The Niels Bohr Institute, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark (Received 11 March 2003; published 5 September 2003) We present experiments and theory for the ‘‘bathtub vortex,’’ which forms when a Fuid drains out of a rotating cylindrical container through a small drain hole. The fast down-Fow is found to be con±ned to a narrow and rapidly rotating ‘‘drainpipe’’ from the free surface down to the drain hole. Surrounding this drainpipe is a region with slow upward Fow generated by the Ekman layer at the bottom of the container. This Fow structure leads us to a theoretical model similar to one obtained earlier by Lundgren [J. ²luid Mech. 155 , 381 (1985)], but here including surface tension and Ekman upwelling, comparing favorably with our measurements. At the tip of the needlelike surface depression, we observe a bubble-forming instability at high rotation rates. DOI: 10.1103/PhysRevLett.91.104502 PACS numbers: 47.32.–y, 47.45.Gx The generation of strongly localized vorticity is a fas- cinating and complicating ingredient of a broad variety of Fuid Fows ranging from vortex shedding at solid surfaces (such as paddles, sand ripples, or insect wings) over Fows through turbines to large-scale tornadoes [1]. In general, these Fows are poorly understood, since the interplay between fast axial motion and intense, localized vorticity leads to dif±cult mathematical problems outside the com- fortable realm of classical subjects such as potential Fow or standard boundary layer theory. One of the most well-known examples of such Fows is the so-called ‘‘bathtub vortex,’’ which forms when water drains out of a container. The strong, localized deforma- tion of the free surface makes the vortex beautifully visible, and has made the bathtub vortex the prototype ‘‘vortex.’’ This popularity is in stark contrast to the attention which the phenomenon receives in the literature. The few classic papers about it either neglect the axial Fow [2] or consider the problem without a free surface [3]. Similarly, textbooks very seldom mention the bathtub vortex, and if they do [4,5] the Fow is modeled within potential theory (with the inclusion of an ad hoc viscous core [4]) without incorporating the axial motion. In the vortex core the axial velocity can be high, an essential ingredient of the strong ‘‘swirl’’ [6] which makes the Fow so fascinating. Our aim in this Letter is to provide basic understanding of the stationary bathtub vortex: the Fow structure, the shape of the free surface, and the interde- pendence of important characteristics such as the size of the central surface depression, the rate of the out-Fow, and the rotation rate in the vortex core.
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PRL-BathTub-Vortex-2 - VOLUME 91 N UMBER 10 PHYSICA L R EVIEW LET T ERS week ending 5 SEPTEMBER 2003 A natomy of a Bathtub Vortex A Andersen,1,2 T

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