Dynamics of a 2D vortex doublet under external deformation

The influence of an external strain (or shear) field on the evolution of two identical vortices is investigated in a two-dimensional incompressible fluid. Using point vortex modeling, two regimes of the vortex doublet (co-rotation and irreversible separation) are determined; the critical intensity of the large scale flow separating these two regimes for a given initial separation of vortices, is calculated. Finite-area effects are then considered for the vortices. The steady states of piecewise constant vortices are computed algebraically and numerically; positive strain (or shear) favors vortex deformation. This deformation has a dominant elliptical component. An elliptical model of two vortices confirms the point vortex model results for centroid trajectories, and the steady state model results concerning the influence of positive strain on vortex deformation. It also provides an estimate of critical merger distance in the presence of large scale flow. Finally, the finite-time, nonlinear evolution of the vortex doublet is simulated with a numerical code of the 2D vorticity equation. The various regimes (stationarity, merger, co-rotation, ejection) are classified in the plane of initial vortex separation and of external deformation. These regimes are analyzed, and the critical merger distance is evaluated for negative and positive external strain; the results are in agreement with the elliptical model prediction. Merger efficiency, defined as the ratio of final to initial vortex circulation, is computed; for the same initial distance, it is smaller for negative strain. It also depends in a more complex way of the initial vortex distance.