Journal Article

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

Arakeri, Jaywant H., Shinde, Sachin Y.

2014

Overview

Thrust-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 (Marais et 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.

• Publisher: Cambridge University Press
• Subject: Applied sciences / Coherence / Computer science; control theory; systems / Control theory. Systems / Deflection / Exact sciences and technology / Flapping / Flaps / Flexibility / Fluid dynamics / Fluid flow / Fluid mechanics / Foils / Fundamental areas of phenomenology (including applications) / Jets / Physics / Robotics / Solid mechanics / Stiffness / Structural and continuum mechanics / Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...)