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

Thrust-generating flapping foils are known to produce jets inclined to the free stream at high Strouhal numbers $ #1{} #1{ {#1}} ≤ = ≤ = ≥ = ≥ = {Pr} {Fr} {Re}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 $0.79\, c$ 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* = 8 \, EI/(ρ \, V_TE_max² \, s_f \, c_f³/2) , 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, V_TE_max is the maximum velocity of rigid foil trailing edge, s_f is span and c_f 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.