Mode selection in trailing vortices: harmonic response of the non-parallel Batchelor vortex
In the present study, the response of model trailing vortices subjected to a harmonic forcing is studied. To this purpose, a globally stable non-parallel Batchelor vortex is considered as the baseflow. Direct numerical simulations (DNS) show that a large variety of helical responses can be excited and amplified through the domain when a harmonic inlet forcing is imposed. The spatial shape of the responses strongly depends on the forcing frequency, with the appearance of modes with progressively higher azimuthal wavenumber m as the frequency increases. The mode-selection mechanism is shown to be directly connected to the local stability properties of the flow, and is simultaneously investigated by a WKB (Wentzel, Kramers, Brillouin) approximation in the framework of weakly non-parallel flows and by the global resolvent approach. In addition to the excellent agreement between the two (local and global) approaches for the computation of the linear response to harmonic forcing at the inlet, the usual WKB analysis is extended to a suitably chosen type of harmonic body forcing, showing also good agreement with the corresponding global results. As expected, the gain of the nonlinear response is significantly lower than that of the linear response, but the mode selection observed in the DNS as a function of the forcing frequency can be predicted fairly accurately by the linear analysis. Finally, by comparing the linear and nonlinear results in terms of energy content for different m, we suggest that the origin of the meandering observed in trailing-vortex experiments could be due to a nonlinear excitation stemming consistently at m = 1 from the competition between the leading linear modes.