Hydrodynamic advantages of a low aspect-ratio flapping foil

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Abstract

A high aspect-ratio foil is known to be advantageous in terms of both thrust and efficiency in flapping propulsion. However, many species of fish have evolved a low aspect-ratio hydrofoil, which naturally leads one to search for its physical advantages in locomotion. Here we study the flow physics of a hydrofoil in angular reciprocating motion with negligible free-stream velocity to reveal the effects of an aspect ratio on hydrodynamic performance. By establishing a scaling law for the thrust of a foil of general shapes and corroborating it experimentally, we find that the thrust of an angularly reciprocating foil is maximized at a low aspect ratio of 0.7 while hydromechanical efficiency continuously increases with an aspect ratio. This result suggests that a low aspect-ratio foil can improve thrust produced by the foil when they start from rest, but at the expense of efficiency.

Introduction

Most of swimming and flying animals flap their fins or wings to propel through a surrounding fluid. The topics of flapping locomotion that have been addressed thus far include measuring the precise kinematics of animals Fish and Lauder (2006), Shelton et al. (2006), analyzing the flow structure and forces of a flapping foil von Ellenrieder et al. (2003), Dong et al. (2006), Buchholz and Smits (2008), Green and Smits (2008), Kim and Gharib (2013) and examining the role of an aspect ratio and flexibility of fish-like locomotion Dewey et al. (2013), Raspa et al. (2014), Feilch and Lauder (2015), Quinn et al. (2015), Yeh and Alexeev (2016). One of the important issues in the mechanics of flapping propulsion is the effect of an aspect ratio of the foil. Early studies have proposed that a high aspect-ratio foil is advantageous in terms of both thrust coefficient and hydromechanical efficiency Chopra (1974), Chopra and Kambe (1977), Cheng and Murillo (1984), Karpouzian et al. (1990), Dong et al. (2006), Buchholz and Smits (2008), Green and Smits (2008), Dewey et al. (2013).

However, many biological studies revealed clear distinction of the caudal fin aspect ratio between migratory and non-migratory fish Nursall (1958), Webb (1984), Domenici and Blake (1997), Flammang and Lauder (2009), Domenici and Kapoor (2010). The data on the aspect ratio are summarized in Fig. 1. The highly migratory fish, which are expected to give priority to efficiency improvement in cruising, have a high aspect-ratio caudal fin as shown in Fig. 1 a. On the other hand, non-migratory fish generally have a smaller aspect ratio than migratory fish. The comparison in normalized coordinates in Fig. 1 c,d clearly shows the stark difference of the caudal fin shape between migratory and non-migratory fish. We further compared the distribution of the caudal fin aspect ratio of highly migratory and non-migratory marine fish in Fig. 1 e (refer to Supplementary Material for the definition of an aspect ratio and the data of the aspect ratio of aquatic animals). In general, the former group has a caudal fin of a high aspect ratio, whereas the latter has a caudal fin of a much smaller aspect ratio.

The comparative studies on the fish caudal fins of diverse species found that a high aspect ratio is advantageous for efficiency improvement in cruising while a low aspect ratio is advantageous for thrust maximization in sprinting Nursall (1958), Webb (1984), Domenici and Blake (1997), Flammang and Lauder (2009), Domenici and Kapoor (2010), which is inconsistent with the conventional hydrodynamic argument that a high aspect ratio is advantageous in terms of both thrust and efficiency Chopra (1974), Chopra and Kambe (1977), Cheng and Murillo (1984), Karpouzian et al. (1990), Dong et al. (2006), Buchholz and Smits (2008), Green and Smits (2008), Dewey et al. (2013). Such controversy has not been treated or resolved by fluid-dynamic theory thus far. In this study, we investigate physical mechanisms underlying the advantages of a low aspect-ratio foil and resolve the inconsistent views on the role of an aspect ratio on thrust and efficiency.

In this study, we investigate the hydrodynamics of a flapping foil without steady forward motion as an elementary model of the situation where the caudal fin is used to propel from rest and especially focus on the effects of an aspect ratio on thrust and efficiency. Despite extensive fluid dynamic studies on flapping locomotion, most of them have considered heaving and/or pitching of a foil in steady forward motion or against incoming freestream. The hydrodynamics of a flapping foil without steady forward motion has drawn relatively less scientific interest. The vortex formation around a flapping foil that rotates from rest without forward motion has been investigated recently Ahlborn et al. (1997), Kim and Gharib (2011), DeVoria and Ringuette (2012). The theoretical model for the thrust of an angularly reciprocating rectangular plate has been suggested based on flow visualization (Lee et al., 2013). However, the effect of an aspect ratio on thrust and efficiency in such a motion has not been addressed by the previous studies.

Section snippets

Experimental apparatus

An angularly reciprocating foil as an elementary model of a flapping foil at the start of locomotion is shown in Fig. 2. The foil was immersed in a transparent water tank of 75, 55, and 33 cm in the x-, y- and z-directions, respectively. The foil is in a single degree-of-freedom sinusoidal oscillation about the z-axis, and the rotating axis is fixed in space. We employed rectangular, trapezoidal and cropped delta foils of various dimensions; the ranges of design dimensions are height h=[1.522.5]

Scaling law and added-mass theory

We first measured the thrust of a rectangular foil by varying an aspect ratio A over two orders of magnitude as shown in Fig. 3 a. The trend in thrust curves changes at the aspect ratio of about 0.7. Before the transition, the thrust rapidly increases with the aspect ratio. However, after the transition, the thrust decreases gradually with the aspect ratio. For a slender foil with an extremely low aspect ratio, the hydrodynamic force can be modeled as an added-mass force based on the potential

Discussion

From the experiment and modeling for thrust and efficiency, we are able to infer the following guidelines to determine the aspect ratio maximizing the propulsive performance at the start of locomotion. First, an aspect ratio below the slenderness limit (0.7) should be avoided because it is unfavorable in terms of both thrust and efficiency. Second, for the purpose of thrust maximization, the aspect ratio around the slenderness limit, i.e., 0.7, is preferred. On the other hand, the efficiency

Conclusion

In this study, by using scaling arguments for added-mass force and vortical impulse, we addressed a novel mechanism to explain the hydrodynamic advantage of a low aspect-ratio foil, which resolves the inconsistency of biological observations and conventional fluid dynamic viewpoints. The efficiency continuously improves as the aspect ratio increases, although there is a loss in thrust above the slenderness limit. On the other hand, for the purpose of thrust maximization, the aspect ratio around

Acknowledgments

This research was supported by National Research Foundation of Korea (Grant Nos. 2015R1C1A1A02037111, 2016901290 and 2016913167) and administered via SNU IAMD.

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