Thrust generation from pitching foils with flexible trailing edge flaps

Abstract

In the present experimental study, we investigate thrust production from a pitching flexible foil in a uniform flow. The flexible foils studied comprise a rigid foil in the front (chord length $c_{R}$) that is pitched sinusoidally at a frequency $f$, with a flexible flap of length $c_{F}$ and flexural rigidity $EI$ attached to its trailing edge. We investigate thrust generation for a range of flexural rigidities ($EI$) and flap length to total chord ratio ($c_{F}/c$), with the mean thrust ($\overline{C_{T}}$) and the efficiency of thrust generation ($\unicode[STIX]{x1D702}$) being directly measured in each case. The thrust in the rigid foil cases, as expected, is found to be primarily due to the normal force on the rigid foil ($\overline{C_{TN}}$) with the chordwise or axial thrust contribution ($\overline{C_{TA}}$) being small and negative. In contrast, in the flexible foil cases, the axial contribution to thrust becomes important. We find that using a non-dimensional flexural rigidity parameter ($R^{\ast }$) defined as $R^{\ast }=EI/(0.5\unicode[STIX]{x1D70C}U^{2}c_{F}^{3})$ appears to combine the independent effects of variations in $EI$ and $c_{F}/c$ at a given value of the reduced frequency ($k=\unicode[STIX]{x03C0}fc/U$) for the range of $c_{F}/c$ values studied here ($U$ is free-stream velocity; $\unicode[STIX]{x1D70C}$ is fluid density). At $k\approx 6$, the peak mean thrust coefficient is found to be about 100 % higher than the rigid foil thrust, and occurs at $R^{\ast }$ value of approximately 8, while the peak efficiency is found to be approximately 300 % higher than the rigid foil efficiency and occurs at a distinctly different $R^{\ast }$ value of close to 0.01. Corresponding to these two optimal flexural rigidity parameter values, we find two distinct flap deflection shapes; the peak thrust corresponding to a mode 1 type simple bending of the flap with no inflection points, while the peak efficiency corresponds to a distinctly different deflection profile having an inflection point along the flap. The peak thrust condition is found to be close to the ‘resonance’ condition for the first mode natural frequency of the flexible flap in still water. In both these optimal cases, we find that it is the axial contribution to thrust that dominates ($\overline{C_{TA}}\gg \overline{C_{TN}}$), in contrast to the rigid foil case. Particle image velocimetry (PIV) measurements for the flexible cases show significant differences in the strength and arrangement of the wake vortices in these two cases.

How shape and flapping rate affect the distribution of fluid forces on flexible hydrofoils

“…Aside from their pertinence in the fish world, the experimental parameters (AR, Re and St) are similar to those selected in previous research about fluid-structure interactions of flapping foils (Lauder et al 2005;Godoy-Diana et al 2008;Dai et al 2012;Marais et al 2012;Dewey et al 2013;Quinn, Lauder & Smits 2014;Shinde & Arakeri 2014;Quinn, Lauder & Smits 2015;David et al 2017;Floryan et al 2017;Liu et al 2017;Rosic et al 2017;Zhu et al 2017). We explore a specific portion of the flow parameter space to understand how propulsive efficiency and surface distributions of hydrodynamic forces depend on shape and frequency at relatively low Reynolds numbers, a topic of high interest in the field of fin propulsion and for the practical design of underwater vehicles relying on bio-inspired undulating membranes.…”

“…As a measure of propulsive energy, we use η, the ratio between the energy employed to produce thrust and the sum of that energy with the energy employed to produce lateral forces (2.8). Figure 8 allows us to compare the efficiency thus defined for all foils with their respective flapping rates, as a function of St r and St. David et al (2017) have raised the question of whether or not the Strouhal number calculated from the tip excursion amplitude is a good metric for the width of a vortex wake. They showed that for highly flexible foils, the width of the wake is overestimated by the large tip excursion, and that a rigid projection of the pitching peduncle approximates better the vortices spacing.…”

“…Many authors have calculated the denominator of (2.8) for pitching fins (input power) as the period-averaged product of torque and angular velocity at the base (Dewey et al 2013;Lucas et al 2015Lucas et al , 2017Quinn et al 2015;Egan, Brownell & Murray 2016;David et al 2017;Floryan et al 2017;Rosic et al 2017;Zhu et al 2017). This efficiency metric is appropriate for hydrodynamic experiments where forces and torque sensors are placed at the attachment rod of the flapping propulsor.…”

“…Methods to quantify the fluid velocity field are typically based on imaging tracer particles seeded into the flow corresponding to particle imaging velocimetry (PIV) or particle tracking velocimetry (PTV) (Maas, Gruen & Papantoniou 1993; Dracos 1996; Raffel et al 1998; Pereira et al 2006). The vortex wakes and thrust production of flapping foils with various geometries and flexibilities have been extensively characterized in the literature (Triantafyllou, Techet & Hover 2004; Godoy-Diana, Aider & Wesfreid 2008; Bohl & Koochesfahani 2009; Kim & Gharib 2010; Green, Rowley & Smits 2011; David et al 2012; Marais et al 2012; Shinde & Arakeri 2014; David, Govardhan & Arakeri 2017; Lucas, Dabiri & Lauder 2017; Muir, Arredondo-Galeana & Viola 2017). Furthermore, many researchers have resorted to particle velocimetry experiments in order to quantify the flow field of aquatic animal appendages and bioinspired synthetic fins (Blickhan et al 1992; Stamhuis & Videler 1995; Müller et al 1997; Drucker & Lauder 1999; Lauder 2000; Müller, Stamhuis & Videler 2000; Drucker & Lauder 2001; Müller et al 2001; Nauen & Lauder 2002 a , b ; Drucker & Lauder 2005; Müller & Van Leeuwen 2006; Tytell 2006; Lauder & Madden 2007; Tangorra et al 2007; Müller, van den Boogaart & van Leeuwen 2008; Tytell, Standen & Lauder 2008; Tangorra et al 2010; Flammang et al 2011 a , b ; Dewey, Carriou & Smits 2012; Esposito et al 2012; Ren et al 2016 a , b ; Mwaffo et al 2017).…”

Hydrodynamic stress maps on the surface of a flexible fin-like foil

“…The desire to reproduce the elegant complexity of nature has inspired the fabrication of elaborate fins models [19][20][21][22][23][24]. Countless studies have resorted to particle imaging or particle tracking velocimetry (PIV and PTV) to investigate the flow topology and hydrodynamic forces of bio-inspired or authentic fish fins [25][26][27][28][29][30][31][32][33][34][35][36][37]. Euler-Bernoulli beam theory has also been employed to describe the fluid-structure interactions of fins and hydrofoils, addressing concepts such as foil compliance, damping effects, resonance frequency and efficiency optimization [38][39][40][41][42][43].…”