Three-dimensional transition after wake deflection behind a flapping foil

We report the inherently three-dimensional linear instabilities of a propulsive wake, produced by a flapping foil, mimicking the caudal fin of a fish or the wing of a flying animal. For the base flow, three sequential wake patterns appear as we increase the flapping amplitude: Bénard-von Kármán (BvK) vortex streets; reverse BvK vortex streets; and deflected wakes. Imposing a three-dimensional spanwise periodic perturbation, we find that the resulting Floquet multiplier |μ| indicates an unstable "short wavelength" mode at wave number β=30, or wavelength λ=0.21 (nondimensionalized by the chord length) at sufficiently high flow Reynolds number Re=Uc/ν≃600, where U is the upstream flow velocity, c is the chord length, and ν is the kinematic viscosity of the fluid. Another, "long wavelength" mode at β=6 (λ=1.05) becomes critical at somewhat higher Reynolds number, although we do not expect that this mode would be observed physically because its growth rate is always less than the short wavelength mode, at least for the parameters we have considered. The long wavelength mode has certain similarities with the so-called mode A in the drag wake of a fixed bluff body, while the short wavelength mode appears to have a period of the order of twice that of the base flow, in that its structure seems to repeat approximately only every second cycle of the base flow. Whether it is appropriate to classify this mode as a truly subharmonic mode or as a quasiperiodic mode is still an open question however, worthy of a detailed parametric study with various flapping amplitudes and frequencies.