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

Abstract

AbstractThrust-generating flapping foils are known to produce jets inclined to the free stream at high Strouhal numbers$\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathit{St} = fA/U_{\infty }$, where$f$is the frequency and$A$is the amplitude of flapping and$U_{\infty }$is the free-stream velocity. Our experiments, in the limiting case of$\mathit{St} \rightarrow \infty $(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 (Maraiset al.,J. Fluid Mech., vol. 710, 2012, p. 659) have also shown that the flexibility increases the$\mathit{St}$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’,$EI^{*} = 8 \, EI/(\rho \, V_{{{TE_{{max}}}}}^2 \, s_{{{f}}} \, c_{{{f}}}^3/2)$, representing the inverse of the flap deflection due to the fluid-dynamic loading; here,$EI$is the bending stiffness of flap,$\rho $is fluid density,$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 \lesssim EI^{*} \lesssim 1)$generates a sustained, coherent, orderly jet. Relatively ‘stiff’ flaps ($EI^{*} \gtrsim 1$), including the extreme case of no flap, produce meandering jets, whereas highly ‘flexible’ flaps ($EI^{*} \lesssim 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.