An analytical-based method for studying the nonlinear evolution of localized vortices in planar homogenous shear flows

Recent experimental and numerical studies have shown that the interaction between a localized vortical disturbance and the shear of an external base flow can lead to the formation of counter-rotating vortex pairs and hairpin vortices that are frequently observed in wall bounded and free turbulent shear flows as well as in subcritical shear flows. In this paper an analytical-based solution method is developed. The method is capable of following (numerically) the evolution of finite-amplitude localized vortical disturbances embedded in shear flows. Due to their localization in space, the surrounding base flow is assumed to have homogeneous shear to leading order. The method can solve in a novel way the interaction between a general family of unbounded planar homogeneous shear flows and any localized disturbance. The solution is carried out using Lagrangian variables in Fourier space which is convenient and enables fast computations. The potential of the method is demonstrated by following the evolved structures of large amplitude disturbances in three canonical base flows, including simple shear, plane stagnation (extensional) and pure rotation flows, and a general case. The results obtained by the current method for plane stagnation and simple shear flows are compared with the published results. The proposed method could be extended to other flows (e.g. geophysical and rotating flows) and to include periodic disturbances as well.