The interaction of a point vortex with a wall-bounded vortex layer

Abstract

The interaction of a point vortex with a layer of constant vorticity, bounded below by a wall and above by an irrotational flow, is investigated as a model of vortex–boundary layer interaction. This model calculates both the evolution of the interface which separates the vortex layer from the irrotational flow and the trajectory of the vortex. In order to determine the conditions which lead to sustained unsteady interaction, three cases are investigated where the mutual interaction between the vortex and interface is initially assumed to be weak. (i) When a weak point vortex lies outside the layer, the vortex moves with a horizontal speed that is small relative to the long-wave phase speed of interfacial waves. A uniformly valid solution is found for the interface evolution. This solution shows that for long times the interface and the vortex approach an equilibrium state. (ii) When a weak vortex lies inside the layer, the vortex is convected by the mean flow and moves with a horizontal speed which matches the phase speed of an interfacial wave. This results in a strong interaction between the vortex and the interfacial wave. On the interface, a monochromatic wavetrain forms upstream of the vortex and acts to attract or repel the point vortex. The displacement of the vortex due to the wavetrain results in the modulation of the amplitude and wavelength of the wavetrain. If the point vortex is attracted toward the interface the horizontal speed of the vortex slows and disturbances directly above the vortex focus and grow leading to the ejection of vorticity. (iii) When the point vortex lies close to the wall and it is sufficiently strong it propagates downstream with a large horizontal velocity. In this case, the amplitude of the interfacial disturbance is independent of the vortex strength. Again, the vortex and the interface approach an equilibrium state. The results of this paper indicate that when the horizontal speed of the vortex matches the phase speed of the interfacial disturbance, it is necessary to account for the vertical displacement of the vortex in order to predict the behaviour of vortex–boundary layer interactions.