Analytical and numerical study of the nonlinear interaction between a point vortex and a wall-bounded shear layer

Abstract

The unsteady interaction between a vortex and a wall-bounded vorticity layer is studied as a model for transport and mixing between rotational and irrotational flows. The problem is formulated in terms of contour integrals and a kinematic condition along the interface which demarcates the vortical and potential regions. Asymptotic solutions are derived for linear, weakly nonlinear and nonlinear long-wave approximations. The solutions show that the initial process of ejection of vorticity into the irrotational flow occurs at a stationary point along the interface. A nonlinear model is derived and shows that such a stationary point is more likely to exist when the circulation of the vortex is counter to the vorticity in the layer. A Lagrangian numerical method based on contour dynamics is then developed for the general nonlinear problem. Two sets of results are presented where for every initial height of the vortex its magnitude and sign are varied. In both sets, it is observed that when the magnitude of the vortex is held constant a much stronger interaction occurs when the sign of the vortex circulation is opposite to that of the vorticity in the layer. Moreover, when the horizontal velocity of the vortex is close to the velocity of the interfacial waves a strong nonlinear interaction between the vortex and the layer ensues and results in the ejection of thin filaments of vorticity into the irrotational flow. In order to study the dynamical consequences of strong unsteady interaction, the wall pressure distribution is computed. The results indicate that a significant rise in the magnitude of the wall pressure is associated with ejection of vorticity from the wall. The present analysis confirms that coherent vortical structures in the outer layer of a turbulent boundary layer can cause ejection of concentrated wall-layer vorticity and explains how and when this process occurs.