Propulsion efficiency of bodies appended with multiple flapping fins: When more is less

Abstract

Underwater animals propel themselves by flapping their pectoral and caudal fins in a narrow range of frequencies, given by Strouhal number St, to produce transitional vortex jets (St is generally expressed non-dimensionally as the product of flapping frequency and stroke (arc) length divided by forward speed). The organized nature of the selection of St and of the vortex jet is thought to maximize hydrodynamic efficiency, although the exact mechanism is not known. Our recent Stuart-Landau equation models, which have self-regulation properties, indicate that the fin and its jet vortices couple. Temporal maps of forces in single isolated fins show a bimodal behavior in certain ranges of the transitional Reynolds number; this behavior bears resemblance to neural bifurcation properties that owe their origin to the self-regulation mechanism. In view of our theoretical and biorobotic evidence of self-regulation in single flapping fins, we explore if this property is altered in a fin-appended body, the goal being to understand how the narrow selection of St, self-regulation, and maximization of hydrodynamic efficiency are related. Swimming vehicles of 1-m scale have been built where a rigid cylindrical body is appended with six flapping fins, three at each end. The fins are rigid, have a rounded leading edge and a laminar section (NACA 0012), and are hinged at one end. The planform is an abstracted version of the penguin wing; it has low aspect ratio and a chord Reynolds number that varies in the transitional range from 10 000 to 60 000. The fin geometry, Reynolds number range, and the nonflexible nature of the main body are in common with those in penguins, and the length and displacement volume are similar to those of sharks. The maximum hydrodynamic efficiency of the fin-appended body (0.40) is lower than that of the single fin (0.57), but is close to that of a fish using several fins. The propulsion density (kW/m3 of displacement volume) of the fin-appended cylinder is similar to that of a cruising shark. If we allow comparison of electrical versus thermal measurements, the total efficiency of the fin-appended body is similar to that of the damselfly and dragonfly, which are also based on vortex propulsion. The fin force fluctuations are modeled by a van der Pol oscillator. Measured phase maps of force fluctuation versus its time derivative correlate with the Strouhal numbers. Until stabilization, the maximum hydrodynamic efficiency of the fin-appended body increases with fin Reynolds number in a staircase pattern whose boundaries correlate with similar transitional sub-regimes in single fins, including the bimodal sub-regimes, thereby relating efficiency with the self-regulating jet vortex oscillators. At low Reynolds numbers, the peak of hydrodynamic efficiency remains flat over a wide range of St, becoming steeper at higher Reynolds numbers with the maximum occurring at lower values of St. The modeling shows that for self-regulation, future biorobotic design should focus on the reduction of structural damping and on a fin-body assembly that has reciprocal energetic interaction with the shed vortex.

Phase dynamics of freely swimming foils

“…However, the Strouhal number found in the present study is close in value to those reported as providing optimal propulsive efficiency in other work (Anderson et al 1998;Triantafyllou et al 2000;Read et al 2003;Eloy 2012). As pointed out by Bandyopadhyay & Leinhos (2013), limit cycle oscillators tend to self-regulate to a condition that minimises energy loss. An interesting avenue of future work would be to directly measure the propulsive efficiency of a freely swimming system and explore how it evolves in time as the system reaches steady-state cruising.…”

“…Likewise, at steady-state cruising, the phase spaces of the swimming data sets show a limit cycle structure (figures 10a,b), regardless of the initial position of the swimming system. Thus, the limit cycle structure appears to exist for each of the runs performed, suggesting that the coupling between the heaving motion of the foil and the force response of the fluid is a self-regulated process for each condition tested (Bandyopadhyay & Leinhos 2013). As shown in the following section ( § 3.2.2), the main difference between heaving foils operating near the F T = F D condition and those operating away from it is that the ratio of the instantaneous phase angles of the input heave position and the output fluidic force always approaches 1/2.…”

“…Recent work has suggested that free swimming may be a passive Bandyopadhyay & Leinhos 2013). A freely swimming system may passively tune (synchronise) itself, such that the oscillation frequency of the foil matches the resonant frequency of its wake, due solely to a combination of (i) the balance of energy input by the flapping foil and energy dissipated by viscous friction/vortex shedding and (ii) coupling between the forces applied to the fluid by the oscillating foil and the corresponding lift and drag forces exerted by the fluid on the foil.…”

Breakup and then makeup: a predictive model of how cilia self-regulate hardness for posture control

“…On the other hand, in inertia-dominated swimming animals, thrust is produced by reverse Karman vortex jets. Here, the hydrodynamic (thrust) efficiency is higher: a single penguin-like flapping fin has values of about 0.606, and a sunfish operating two pectoral flexible flapping fins has a value of 0.40 (due to Lauder & Madden—see Bandyopadhyay and Leinhos43); (this value has been reproduced in a robotic model whose propulsion power density matches that of a shark43). If improvement in efficiency was a motivation for the evolution of larger swimming animals, then the symmetry breaking44 of the drag vortex wake to a jet vortex wake was a significant milestone that accompanied the increase in the portfolio of motions.…”

“…The power consumed by this drag (output power) is , such that , where the above value assumes , , and C * = 24 (sphere). Taking our earlier estimate of cilium thrust efficiency of 0.125, and considering the paramecium's numerous appended cilia, we lower the thrust efficiency of the organism to, say, 0.08443. The power input to thrust from oxygen can be obtained from , with .…”