The motion of a vortex near two circular cylinders

Abstract

The motion of a vortex near two circular cylinders of arbitrary radii-a problem of geophysical significance-is studied. The fluid motion is governed by the twodimensional Euler equations and the flow is irrotational exterior to the vortex. Two models are considered. First, the trajectories of a line vortex are obtained using conformal mapping techniques to construct the vortex Hamiltonian which respects the zero normal flow boundary condition on both cylinders. The vortex paths reveal a critical trajectory (i.e. separatrix) that divides trajectories into those that orbit both cylinders and those that orbit just one cylinder. Second, the motion of a patch of constant vorticity is computed using a combination of conformal mapping and the numerical method of contour surgery. Although the patch can deform, the results show that when the islands have comparable radii the patch remains remarkably coherent. Moreover, it is demonstrated that the trajectory of the centroid of the patch is well modelled by a line vortex. For the limiting case when one of the cylinders has infinite curvature (i.e. it becomes a straight line or wall) it is shown that the vortex patch, which propagates under the influence of its image in the wall, may undergo severe deformation as it collides with the smaller cylinder, with portions of the vortex passing around different sides of the cylinder.

Instabilities of the flow around a cylinder and emission of vortex dipoles

“…This calculation implies the use of the Green function G(x, y; x 0 , y 0 ), which can be interpreted as the stream function of a flow induced by a unit point vortex located at a point (x 0 , y 0 ) on the (x, y)-plane, and computation of contour integrals over the patch boundaries. The construction of the Green function in complex domains is often facilitated by the use of image vortices (Coppa, Peano & Peinetti 2002;Johnson & McDonald 2004;Elcrat, Fornberg & Miller 2005;Crowdy & Surana 2007). For a plane with a circular cutout of unit radius centred at the origin of the frame of reference, a Green function can be written as Makarov, L. Kamp and G. van Heijst (e.g.…”

Word representation of streamline topologies for structurally stable vortex flows in multiply connected domains

“…According to lemma 3.1, either B 0 or B 2 can follow the initial word O. Hence, the structurally stable streamline patterns in D ζ (2) are OB 0 and OB 2 , whose saddle connection diagrams are shown in figure 6a,c, respectively. Now, we consider the structurally stable streamline patterns with the 1-source-sink point constructed from the initial patterns I and II.…”

“…Hence, we have the maximal expression (3.2). (2). Owing to k = 1 for M = 1, we have three combinations of the indices (p, s 1 , t 1 ) = (1, 0, 0), (0, 1, 0) and (0, 0, 1), whose corresponding maximal II-words are IIA 0 , IIB 0 and IIB 2 , whose corresponding ss-saddle connection diagrams are shown in figure 10a-d, respectively.…”

Beach vortices near circular topography

“…The problem is tackled by finding the appropriate Green's function (Section III) from the conservation of vorticity with pressure and normal mass flux continuous across the topographic boundary. Unlike previous studies 7 in which the streamfunction was found by first ignoring the boundaries and then evaluating a complementary irrotational flow so the normal velocity of the combined flow is zero at the boundaries, a different method, avoiding these supplementary computations, is introduced. Section IV describes the form of the steadily propagating solutions and their robustness is tested using time-dependent contour dynamics.…”

“…5 Laboratory experiments 6 found that the cylinder diameter affects the vortex trajectories significantly. These observations motivated a study 7 of point vortex and vortex-patch motion around two impermeable circular cylinders where the point-vortex motion is shown to be governed by a Hamiltonian, derivable from complex variable techniques which extend directly 8 to point-vortex motion around multiple cylinders. Dipolar 9,10 and monopolar 11 surf-zone vortices (i.e., zero background rotation) have been studied analytically near a step of finite depth-ratio.…”