Propulsive performance of biologically inspired flapping foils at high Reynolds numbers

The design of biologically inspired propulsion mechanisms for underwater vehicles continues to generate significant interest in the hydrodynamics of fish swimming. Fish and animals, such as aquatic penguins and turtles, are highly adapted to swimming in the ocean and serve as excellentmodels for developing novel propulsive devices for underwater vehicles. With the goal of designing compact, agile autonomous and unmanned underwater vehicles (AUVs and UUVs), engineers have turned to fish and their aquatic counterparts for inspiration (e.g. Bandyopadhyay,2005; Anderson and Chabra,2002; Triantafyllou et al.,2000; Anderson et al.,1998; Bandyopadhyay et al.,1997,). Biologists have studied the kinematics and morphology of swimming fish in great detail, revealing the superior agility of these creatures (e.g. Fish and Lauder,2006; Lauder,2000; Drucker and Lauder,1999; Videler,1993; Fish and Hui,1991). The undulatory nature of the fish body motion imparts flow control on the surrounding fluid generating a unique propulsive signature in the form of a reverse Kármán vortex street(Triantafyllou and Triantafyllou,1995). This simplified view of the wake structure does not account for three-dimensional effects that result from the geometry of the fish body and tail fin.

Flapping foils have been used as models for swimming fish fins to better understand fish swimming hydrodynamics. Heaving and pitching foils with moderate to high aspect ratios (Anderson,1996; Triantafyllou et al.,1993), and more three-dimensional motions of foils, modeling pectoral fins with lower aspect ratios, have also been investigated as alternative methods of propulsion for underwater vehicle technology(Bandyopadhyay, 2005; Lang et al., 2006; Kato, 2000). Extensive studies of the two-dimensional wake patterns behind flapping foils and live and robotic swimming fish have been performed both experimentally and numerically. Thorough reviews of this research on flapping foils can be found in the literature (Triantafyllou et al.,2004; Triantafyllou el al.,2005).

Typical wake patterns generated by the two-dimensional flapping foil motion consist of alternately rotating vortices organized to form a jet-like wake, or reverse Kármán street, which results in thrust generation over a range of Strouhal numbers and kinematic parameters. Strouhal number is defined as St=2h₀f/U, where 2h₀ is the peak-to-peak heave amplitude of the trailing edge, f is the flapping frequency (in Hz), and U is the incoming flow velocity relative to the foil. Optimal propulsive performance has been shown to be highly dependent on Strouhal number. Typically, flapping foils used for propulsion have high efficiencies in a range of Strouhal numbers 0.2–0.4 (Triantafyllou and Triantafyllou, 1995); higher thrust values can be achieved at higher values of Strouhal number but with less efficiency.

The unsteady flapping motion of a foil generates vorticity at the trailing edge and tips of the foil and also at the leading edge under certain conditions. These patterns vary with the amplitude and frequency of the motion as well as the shape of the kinematics employed (e.g. Hover et al., 2004; Anderson el al., 1998; Koochesfahani, 1989; Katz and Weihs, 1979). Experiments with heaving and pitching symmetric foils, with a maximum thickness that is 15% of the chord length (i.e. a NACA 0015 foil), at moderate Reynolds numbers [5000–12 000(Freymouth, 1990)] link the formation of a leading edge vortex, and its successive combination with the vorticity generated at the trailing edge, with high lift coefficients. Reynolds number (Re) is defined as the ratio of the inertial fluid forces to the viscous fluid forces: Re=(ρU²c^–1)/(μUc^–2)=ρUc/μ,where ρ is the density of the fluid, and μ the dynamic viscosity of the fluid. Re is often written in terms of kinematic viscosity:ν=μ/ρ; for water ν=10^–6 m²s^–1. This interaction is also important to achieve high efficiency for flapping foil swimming modes(Anderson et al., 1998).

A closer look at performance of finite aspect ratio, three-dimensional flapping foils is warranted at higher Reynolds numbers, in order to design vehicles to swim with optimal kinematics and maximizing thrust while optimizing efficiency. In an effort to understand this trade-off, tests were performed using a three-dimensional, linearly tapered hydrofoil forced to move in roll and pitching motions, and the measured force and hydrodynamic efficiency data are presented, discussed and compared with data reported from previous tests (Flores, 2003; McLetchie, 2004; Polidoro, 2003; Read, 2000).

The three-dimensional (3D) flapping foil motion emulates that of an aquatic penguin wing or turtle fin. The 3D flapping foil actuator was constructed as a dual-canister design: two watertight canisters, one each for pitch and roll,house the motor and chain drive for each motion separately. The two canisters are coupled together and to the foil (see Fig. 1A). This apparatus is mounted in the Massachusetts Institute of Technology recirculating water tunnel, capable of flow speeds from 0.5 to 5.0 m s^–1 with turbulence levels under 3%. The tunnel's square test section is 0.5 m×0.5 m×1.25 m long. Constrained to move in roll ϕ(t)and pitch θ(t) only, a diagram of the motion coordinate system is given in Fig. 1B. This type of flapping apparatus presents underwater vehicle designers and engineers a simpler design problem, e.g. fewer sealing issues, than the heaving/pitching foil. The thesis by Lim (Lim,2005) presents further details of the housing design and operation.

The kinematic motion of the foil is measured through potentiometers mounted to each rotating shaft, which map the pitch and roll motion, and also through the encoders mounted on each motor. The encoder data are used for redundancy and kinematics validation. A torque sensor is coupled to the roll motor shaft to calculate the power input from the motor end during actuation; the power input can be used to determine the mechanical efficiency of the flapping foil. The forces and torques acting directly on the foil are measured with a six-axis, waterproof strain gauge sensor, which measures the three force components (F_x, F_y, F_z) and the three moment components(M_x, M_y, M_z). Hydrodynamic efficiency is calculated using the torque measurements from the six-axis sensor to calculate power input to the fluid by the foil, thus bypassing the mechanical losses in the drive mechanism. Cross-coupling between the six channels is eliminated using the factory supplied matrix, which has been validated extensively (Lim,2005).

The average thrust and lift forces are found by applying a force rotation matrix to the force traces and averaging the data over 10–15 cycles. Average force data presented in subsequent sections are given in terms of the non-dimensional force coefficient, based on the planform area of the foil,instead of the swept area as would be used for a propeller. The hydrodynamic efficiency is determined by comparing the power output, calculated from the product of average thrust and velocity, to the sum of the power input, applied to the pitch and roll axes.

The foil used in this experiment is fabricated with a NACA 0012 cross section. The NACA 0012 is a standard, symmetric foil profile with a maximum thickness that is 12% of the chord length(Abbott and von Doenhoff,1959). The test foil has a span of 24.6 cm from root to tip, and an average chord length, c̄, of 5.5 cm from leading edge to trailing edge. The foil has a linearly tapering trailing edge profile. Tests were performed over a range of kinematic parameters:heave/chord ratio, h_0.7/c={1.0, 1.5, 2.0};Strouhal number, St={0.2:0.1:0.6}; and maximum angle of attack,α _max={15:5:45}°. The experiments were conducted at Reynolds numbers 27 000 to 55 000. The lower speed was used to achieve higher Strouhal numbers. The speed was monitored at all times by a Laser Doppler Velocimetry system and maintained to within ±0.01 m s^–1.

The 3D flapping foil motion (roll/pitch) presents underwater vehicle designers with a more straightforward design problem compared to the heaving/pitching mechanism design and yields further insight into the performance of swimming animals such as aquatic penguins and turtles. To evaluate the performance of flapping foils sufficiently, accurate force and efficiency measurements are paramount. Force data are presented here for the 3D flapping foil mechanism for induced 2D heave-to-chord ratios(Eqn 4) h_0.7/c=1.0, 1.5 and 2.0, over a range ofα _max from 15° to 45°. Typical force traces obtained by the six-axis force sensor are presented in Fig. 3.

The general trend of the thrust results(Fig. 4) compares well with those for the 2D case (Read,2000) and 3D case (McLetchie,2004). The thrust coefficients decrease with decreasing Strouhal numbers and α_max values, and increases with increasing values of both Strouhal number and α_max. At heave- to- chord ratio of 1.0 (h_0.7/c=1.0) a peak thrust coefficient of 1.6 is recorded at St=0.5 and α_max=30°. Extrapolating up to St=0.6, it appears that a higher peak thrust coefficient could be achieved. Such an increase is substantiated by McLetchie(McLetchie, 2004).

For the case of h_0.7/c=1.5, peak thrust coefficients were measured for St=0.6 and angle of attacks of 30°and 35°. At h_0.7/c=2.0, the maximum thrust coefficients now occur at the same Strouhal numbers (St=0.6) but have lower magnitudes at α_max=30° and 35°(Lim, 2005). Fig. 4 shows that for a given Strouhal number the thrust coefficient increases with the maximum angle of attack, but beyond a critical α_max value, the thrust coefficient is expected to decrease. This is similar to the data obtained for heaving/pitching foils at Re=750 and 1000 (Read, 2006). Increases in roll amplitude tend to have similar effects. The maximum thrust coefficient for these runs was recorded at h_0.7/c=1.5. At the higher roll amplitude, h_0.7/c=2.0, the peak thrust coefficient is lower, thus it is possible that an optimal roll amplitude can be found, between the heave-to-chord ratio of 1.5–2.0, for which the thrust coefficient can be maximized.

The left and bottom `borders' of the parametric space represent the boundaries for which the angle of attack profiles tend to corrupt, i.e. some regions of the foil would encounter negative angles of attack, resulting in drag instead of thrust production. Here, the results at St=0.2 show very low thrust production for all α_max values. This low thrust boundary represent the transition in the wake from drag- to thrust-producing vortices (Flores,2003). At high α_max, the transition to thrust does not occur until a Strouhal number of about 0.4. At experimental points for which low thrust values are measured (St⩽0.3;15°⩽α_max⩽25°), there are large errors associated with the results (>10% in certain cases). A more thorough analysis of the error from these tests is given in Lim(Lim, 2005). In addition, the thrust coefficients presented above were evaluated based on planform area. A more appropriate normalization, in keeping with that used in propeller performance analysis, might have been to use the projected swept area. This formulation would have produced numerically smaller values of C_T (Techet el al., 2005).

In a reverse fashion to the thrust coefficient results, efficiency peaks at the lower end of Strouhal and α_max values and then decreases with increasing St and α_max values(Fig. 5). This is as expected since, in low thrust regions, minimal energy is lost as a result of kinetic energy being imparted to the flow. The maximum efficiency recorded is about 0.7, centered at St=0.3 and α_max=20° for h_0.7/c=2.0. Its location corresponds with that reported by McLetchie (McLetchie,2004), although the magnitudes of efficiency differ. Read also shares the same peak location with measured efficiencies not exceeding 0.7,for 2D flapping foils (Read,2000).

Higher roll amplitudes result in greater energy expended in moving the foil through the large oscillations. It is only at higher frequencies where greater thrust is generated such that the efficiency appears to be improving. Figs 4 and 5 show that for lowα _max values, thrust generally increases with Strouhal number without significant penalty to efficiency. This is good news with respect to designing mechanisms for underwater vehicles, as an optimal point can be identified for relatively high thrust production without sacrificing efficiency – it is also good news for aquatic swimmers, as they can produce significant propulsive forces with minimal effort. For example, at h_0.7/c=1.5, we see relatively high efficiencies occurring at St=0.5, and α_max of 15° and 20°. This offers a good design point where efficiencies of more than 0.6 can be achieved with thrust coefficients ranging from 0.7 to 1.2.

The average range of thrust coefficient C_T is found to be in the order of ±0.026, or ±0.044 N in terms of absolute thrust; as such, the percentage error, which is taken as the fraction of standard deviation over the mean, is naturally higher at the lower thrust runs(Lim, 2005). In the regions where the thrust is low (St⩽0.3;15°⩽α_max⩽25°), the percentage errors associated with the measurements still remain significant and thus the efficiency measurements also require careful consideration and further validation. Likewise a larger percentage error for efficiency η also arises for these cases, since the error for η is a combination of errors from power input and thrust measurements.

In summary, the 3D flapping foil, similar to a pectoral fin, holds the most promise for implementation in an underwater vehicle due to the reduced complexity over the heaving/pitching mechanism. To further extend our understanding of flapping foil performance at higher Reynolds numbers the 3D rolling/pitching foil apparatus was studied in a water tunnel at Reynolds numbers on the order of Re∼10⁴. These tests, using force sensors to measure thrust and efficiency, with the 3D flapping foils are highly valuable from a design point of view.

Clearly indicated in the data presented here is that for a fixed Strouhal number, there is a critical maximum angle of attack value beyond which the thrust coefficient will start to decrease and performance is reduced. A peak planform area thrust coefficient of 2.09 was measured at h_0.7/c=1.5, St=0.6 andα _max=30°. Increasing the heave-to-chord ratio from 1.5 to 2.0 does not appear to improve C_T values, but further iterations are still warranted. Combined with data from McLetchie(McLetchie, 2004), the data presented herein for the 3D foil suggests a useful performance trend where,for low α_max, high thrust and high efficiency can be gained at sufficiently high Strouhal numbers (St=0.6 and possibly higher). In particular, for higher roll amplitudes, these large oscillations produce high thrust with relatively little power loss. Higher thrust and efficiencies can be achieved through pre-shaping algorithms applied to the heaving/pitching foil motions for 2D flapping (Hover et al., 2004). Further investigation is warranted to determine whether pre-shaping would improve the performance of the 3D foil, but data suggests that this is a reasonable assumption.

Further comprehensive and systematic investigations of foil geometry on flapping foil performance are warranted; for example, it was suggested that an optimal aspect ratio is nearer to 4.0 for flapping wings(Polidoro, 2003). The effect of chordwise or spanwise flexibility on the performance of flapping foils is also still not well understood. Live fish employ active flexure of their fins,especially their pectoral fins, suggesting that they can optimize their fin shape for maximized performance (Fish and Lauder, 2006). Further experiments in tandem with numerical simulations are necessary to further augment our understanding of flapping foil performance as seen in nature.

List of symbols and abbreviations

 
• c

  foil chord

 
• c̄

  mean chord

 
• C_T

  thrust coefficient

 
• f

  flapping frequency (Hz)

 
• F_x, F_y,F_z

  force components

 
• F̄ _x0

  x-force component translated into reference frame of test section

 
• ḣ

  heave velocity(ḣ=dh/dt)

 
• h₀

  heave amplitude

 
• h_0.7

  heave amplitude at 0.7 span location

 
• h(t)

  heave position

 
• M_x, M_y,M_z

  moment components

 
• r₀

  radius from base of foil

 
• r_0.7

  radius from base of foil shaft to 0.7 span

 
• P_in

  power input

 
• P_out

  power output

 
• Re

  Reynolds number

 
• S

  foil span

 
• St

  Strouhal number

 
• t

  time

 
• T

  thrust

 
• U

  free stream velocity

 
• α

  angle of attack

 
• αmax

  maximum angle of attack

 
• Ψ

  phase angle between pitch and roll

 
• ϕ(t)

  roll angle

 
• ϕ0

  roll amplitude

 
• η

  efficiency

 
• μ

  dynamic viscosity of the fluid (kg m^–1s^–1)

 
• ν

  kinematic viscosity of the fluid [ν=μ/ρ (kg m^–1 s^–1)]

 
• ρ

  fluid density (kg m^–3)

 
• θ(t)

  pitch angle

 
• θ0

  pitch amplitude

 
• θbias

  pitch bias angle

 
• ω

  circular frequency (rad s^–1)

The author would like to thank K. Lim for his assistance with experiments and Dr Franz Hover for his insightful discussions related to data processing and angle of attack effects. The development of the flapping foil mechanism was made possible through the support of Admiral Paul Sullivan, NAVSEA, and the MIT Sea Grant program (Grant NA16RG2255).