Transient growth in the near wake region of the flow past a finite span wing

Abstract

We investigate optimal perturbation in the flow past a finite aspect ratio ($AR$) wing. The optimization is carried out in the regime where the fully developed flow is steady. Parametric study over time horizon ($T$), Reynolds number ($Re$), $AR$, angle of attack and geometry of the wing cross-section (flat plate and NACA0012 airfoil) shows that the general shape of linear optimal perturbation remains the same over the explored parameter space. Optimal perturbation is located near the surface of the wing in the form of chord-wise periodic structures whose strength decreases from the root towards the tip. Direct time integration of the disturbance equations, with and without nonlinear terms, is carried out with linear optimal perturbation as initial condition. In both cases, the optimal perturbation evolves as a downstream travelling wavepacket whose speed is nearly the same as that of the free stream. The energy of the wavepacket increases in the near wake region, and is found to remain nearly constant beyond the vortex roll-up distance in nonlinear simulations. The nonlinear wavepacket results in displacement of the tip vortex. In this situation, the motion of the tip vortex resembles that observed during vortex meandering/wandering in wind tunnel experiments. Results from computation carried out at higher $Re$ suggest that, even beyond the steady flow regime, a perturbation wavepacket originating near the wing might cause meandering of tip vortices.