Journal of Fluid Mechanics


Polygon formation and surface flow on a rotating fluid surface

R. BERGMANNa1 c1, L. TOPHØJa1, T. A. M. HOMANa1a2, P. HERSENa3, A. ANDERSENa1 and T. BOHRa1

a1 Department of Physics and Center for Fluid Dynamics, Technical University of Denmark, DK-2800 Kgs Lyngby, Denmark

a2 Physics of Fluids Group and J. M. Burgers Centre for Fluid Dynamics, University of Twente, 7500 AE Enschede, The Netherlands

a3 Laboratoire Matière et Systèmes Complexes, CNRS and Université Paris Diderot, Paris 75013, France


We present a study of polygons forming on the free surface of a water flow confined to a stationary cylinder and driven by a rotating bottom plate as described by Jansson et al. (Phys. Rev. Lett., vol. 96, 2006, 174502). In particular, we study the case of a triangular structure, either completely ‘wet’ or with a ‘dry’ centre. For the dry structures, we present measurements of the surface shapes and the process of formation. We show experimental evidence that the formation can take place as a two-stage process: first the system approaches an almost stable rotationally symmetric state and from there the symmetry breaking proceeds like a low-dimensional linear instability. We show that the circular state and the unstable manifold connecting it with the polygon solution are universal in the sense that very different initial conditions lead to the same circular state and unstable manifold. For a wet triangle, we measure the surface flows by particle image velocimetry (PIV) and show that there are three vortices present, but that the strength of these vortices is far too weak to account for the rotation velocity of the polygon. We show that partial blocking of the surface flow destroys the polygons and re-establishes the rotational symmetry. For the rotationally symmetric state our theoretical analysis of the surface flow shows that it consists of two distinct regions: an inner, rigidly rotating centre and an outer annulus, where the surface flow is that of a point vortex with a weak secondary flow. This prediction is consistent with the experimentally determined surface flow.

(Received June 10 2010)

(Revised December 17 2010)

(Accepted March 24 2011)

(Online publication May 24 2011)

Key words:

  • rotating flows;
  • vortex flows;
  • waves/free-surface flows


c1 Present address: Instrumentation and Controls Department, German–Dutch Wind Tunnels, Emmeloord, The Netherlands. Email address for correspondence: