Flexibility in flapping foil suppresses meandering of induced jet in absence of free stream

Abstract

Thrust-generating flapping foils are known to produce jets inclined to the free stream at high Strouhal numbers St=fA/U∞, where f is the frequency and A is the amplitude of flapping and U∞ is the free-stream velocity. Our experiments, in the limiting case of St→∞ (zero free-stream speed), show that a purely oscillatory pitching motion of a chordwise flexible foil produces a coherent jet composed of a reverse Bénard–Kármán vortex street along the centreline, albeit over a specific range of effective flap stiffnesses. We obtain flexibility by attaching a thin flap to the trailing edge of a rigid NACA0015 foil; length of flap is 0.79c where c is rigid foil chord length. It is the time-varying deflections of the flexible flap that suppress the meandering found in the jets produced by a pitching rigid foil for zero free-stream condition. Recent experiments (Marais et al., J. Fluid Mech., vol. 710, 2012, p. 659) have also shown that the flexibility increases the St at which non-deflected jets are obtained. Analysing the near-wake vortex dynamics from flow visualization and particle image velocimetry (PIV) measurements, we identify the mechanisms by which flexibility suppresses jet deflection and meandering. A convenient characterization of flap deformation, caused by fluid–flap interaction, is through a non-dimensional ‘effective stiffness’, EI∗=8EI/(ρV2TEmaxsfc3f/2), representing the inverse of the flap deflection due to the fluid-dynamic loading; here, EI is the bending stiffness of flap, ρ is fluid density, VTEmax is the maximum velocity of rigid foil trailing edge, sf is span and cf is chord length of the flexible flap. By varying the amplitude and frequency of pitching, we obtain a variation in EI∗ over nearly two orders of magnitude and show that only moderate EI∗ (0.1≲EI∗≲1) generates a sustained, coherent, orderly jet. Relatively ‘stiff’ flaps (EI∗≳1), including the extreme case of no flap, produce meandering jets, whereas highly ‘flexible’ flaps (EI∗≲0.1) produce spread-out jets. Obtained from the measured mean velocity fields, we present values of thrust coefficients for the cases for which orderly jets are observed.