Steady interaction of a vortex street with a shear flow

Abstract

A one parameter family of explicit solutions of the Euler equations is presented comprising a steadily propagating point vortex street situated in a region of uniform vorticity below a periodically deformed vortex jump separating a region of irrotational flow from a uniform shear flow. Various features of the new solutions are described. The limiting solutions are such that the vortex jump develops a periodic sequence of cusps. The stability of the equilibria is investigated numerically using a cylindrical contour dynamics algorithm. The equilibria not too close to the limiting case are found to be structurally robust for a large range of parameter values.

‘H-states’: exact solutions for a rotating hollow vortex

“…They occur in various guises in fluid dynamics as surveyed by Crowdy (2005). We mention here that other classes of exact solutions – also viewable as quadrature domains – for steadily translating water waves with vorticity have been found by Crowdy & Nelson (2010).…”

A calculus for flows in periodic domains

“…It is straightforward to check that winding anti-clockwise around ζ = a ∞ yields an increase in f of P per loop. Type II mappings have previously been used to study the interaction of a vortex street with a shear flow in [32], and free surface Euler flows in [14]. In the simply connected case, [45] found an analogous form of the Schwarz-Christoffel formula to type II geometries, although the preimage domain in that case was the upper half-plane [ζ ] > 0.…”

“…a ∞ ± → a ∞ . In the non-periodic case, the flow is straining in both directions at the same angle so we write ± = and λ ± = λ and consider the sum of the two solutions in (32). Rescaling =˜ P 2 and P =P/(a + ∞ − a − ∞ ) and applying L'Hôpital's rule then yields…”

A calculus for flows in periodic domains

“…It is straightforward to check that winding clockwise around ζ = a ∞ yields an increase in f of P per loop. Type II mappings have previously been used to study the interaction of a vortex street with a shear flow in [30], and free surface Euler flows in [13].…”