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 0.79 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 2 TE max s f c 3 f /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.

Physics of unsteady thrust and flow generation by a flexible surface flapping in the absence of a free stream

“…Therefore, what appears as a vortex near the LE is a 'blob' of dye that is convected by the flow. This feature is clearly seen in the supplementary movie of [29]. (Online version in colour.…”

“…We sinusoidally oscillate the aerofoil in pitching at a fixed location and study the flow in a horizontal plane along the foil mid-span. Below we describe the experimental set-up and diagnostic techniques; however, as they are similar to those used in Shinde & Arakeri [29], we give only the essential details here. Additional information can be found in [38].…”

“…In the absence of free-stream flow, the jet generated by the flapping foil has the tendency to switch direction [20,22]. Shinde & Arakeri [29] have identified the two underlying mechanisms for suppressing the jet meandering and generating the coherent reverse Bénard-Kármán vortex street and the unidirectional thrust along the centreline: first, appropriate spatial and temporal shedding of the vortices from the flap tip, and, second, convection of the vortices away from the place of shedding. Further, they defined a non-dimensional parameter which includes the effects of fluid-structure interaction, namely 'effective stiffness' EI * = 8EI/(1/2 ρ V 2 TE max s f c 3 f ), which represents the inverse of the flap deflection due to the fluid-dynamic loading; in this expression, EI is the bending stiffness of the flap, ρ is the fluid density, V TE max is the maximum velocity of the rigid foil trailing edge, s f is the span and c f is the chord length of the flexible flap.…”

“…The undulating flow in the jet is due to the staggered array of vortices, originally created by the flap, which move under their own induced velocities. Additional details on vortex movement and spatial separation of vortices may be found in Shinde & Arakeri [29].…”

Biological Propulsion Systems for Ships and Underwater Vehicles

“…The rigid pitching foil produces a weak and meandering jet (and thrust), whereas the flexible foil produces a strong jet of the wake. Deformations of the flexible flap suppress meandering by increasing the initial gap between two successive vortices and by imposing a convective motion on the shed vortices even in the absence of free-stream velocity [4]. Four foils made of adhering layers of plastic sheets to produce discrete regions of high and low flexible stiffness of biologically-relevant magnitudes.…”