Many animals maneuver superbly underwater using their pectoral appendages. These animals range from sunfish, which have flexible, low aspect ratio fins,to penguins, which have relatively stiff, high aspect ratio wings. Biorobotics is a means of gaining insight into the mechanisms these animals use for maneuvering. In this study, experiments were carried out with models of abstracted penguin wings, and hydrodynamic characteristics – in particular, efficiency – were measured directly. A cross-flow vortex model of the unsteady force mechanism was developed that can compute instantaneous lift and drag forces accurately. This makes use of the steady characteristics of the fin and proposes that cross-flow drag vortices of bluff bodies in steady flow are analogous to dynamic stall vortices and that fin oscillation is a means for keeping the vortices attached to the fin. From what has been reported for sunfish with pectoral fins to our current measurements for single abstracted penguin wings, we infer that the maximum hydrodynamic efficiency has remained largely unchanged. A selection algorithm was used to rapidly find the fin oscillation parameters for optimum efficiency. Finally,we compared the measurements on the penguin-like relatively stiff fins and the reported flow visualization of flexible sunfish pectoral fins. The flexible pectoral fins of station-keeping sunfish exhibit a rich repertoire of capability such as the formation of dynamic stall vortices simultaneously on two leading edges during part of the cycle, changes in projected area in different planes, and the vectoring of jets. However, such fins may not be scalable to larger biorobotic vehicles and relatively stiff fins appear to be better suited instead, albeit with somewhat limited station-keeping ability.

The engineering implementation of biology-inspired fluid dynamics has received much attention since the work of Ellington and Dickinson showing how dynamic stall is used by flying animals to sustain themselves aloft in a low-density medium such as air (Ellington,1984a; Ellington,1984b; Ellington,1984c; Ellington,1984d; Ellington,1984e; Ellington,1984f; Dudley and Ellington,1990; Ellington,1991; Dickinson,1996; Dickinson et al.,1999; Ellington et al.,1996; Sane and Dickinson,2002; Birch et al.,2004; Triantafyllou et al.,2004; Bandyopadhyay,2004; Bandyopadhyay,2005). Similar investigations of the mechanisms of the appendages of swimming animals, however, are not as advanced(Lauder and Drucker, 2004),although progress is now being made(Lauder et al., 2007). The engineering significance of these largely model-based biorobotic'investigations is that it is indeed possible – after accounting for penalties – to sustain an entire platform based on the high-lift mechanism of dynamic stall, a phenomenon that has been known to engineers for a long time (Bandyopadhyay,2005; Ellington,1999). It is frequently cited that animals have wings and bodies that closely match NACA profiles, although the design and selection of the NACA profiles are based on steady-state flow. This led Bandyopadhyay, while at the Office of Naval Research, to commission a medical imaging survey of the cross-sections of the appendages of swimming animals. The close match with NACA profiles was found to be widely prevalent (for a collection of medical cross-sectional digital images of common dolphins, contact Prof. F. Fish,Westchester University, Weschester, PA, USA), but the role of the foil cross-section derived from studies on steady-flow section in the unsteady high-lift mechanism is unclear. The performance of swimming animals greatly depends on the hydrodynamic efficiencies of their active appendages, such as the pectoral fin. Estimates of performance based on hydrodynamic models tend to have large uncertainties and pertain to the animal as a whole, as opposed to their individual appendages (Fish,1993). The hydrodynamic efficiencies of pectoral appendages for cruising and maneuvering still remain unknown. The position and number of appendages and their morphological scaling depend on whether the animals are dexterous in cruising or in maneuvering(Bandyopadhyay et al., 1997). In this study, we explored how swimming efficiency could be rapidly optimized without any a priori knowledge of the hydrodynamic mechanisms. Finally, intrigued by the fact that large, open-water cruising animals (such as whales, dolphins and penguins) have fairly stiff wings, while smaller and more maneuverable and station-keeping animals (such as sunfish) have flexible pectoral fins, we examined the question: is dynamic stall universally present in both flexible and rigid high-lift appendages? What are the functional differences between relatively stiff and flexible fins?

The apparatus used to determine the hydrodynamic characteristics of rolling and pitching fins, which are similar to the pectoral fins and wings of swimming animals, is shown in Fig. 1. Three rigid fins with chord (c) of 10 cm in most of the span (s) and span/chord (s/c) ratios of 1, 2 and 3 were tested in a low-speed tow tank. The fin is imparted pitching and rolling motions using a set of two orthogonally placed motors. The pitch motor and fin assembly is attached to the roll motor shaft. The six forces and moments are read by the load cell that connects the roll motor to the tow carriage. Optical encoders on the motors give rolling and pitching positions and angular velocities, while torque sensors attached to each motor shaft used to measure the efficiency. The chord Reynolds numbers of the hovering and towing tests are in the 20 000 to 150 000 range, based on total speed Ut (the symbols are defined below, see List of symbols and abbreviations).

The roll motor is controlled to oscillate sinusoidally as follows:
$\ {\phi}(t)={\phi}_{0}\mathrm{sin}({\omega}t),$
(1)
where ϕ(t) is the instantaneous roll position, ϕ0is the roll amplitude, and ω is the flapping frequency in radians s–1. The pitch motor, which rides on the roll motor,oscillates as:
$\ {\theta}(t)={\theta}_{0}\mathrm{sin}({\omega}t+{\psi})+{\theta}_{\mathrm{Bias}},$
(2)
where θ(t) is the instantaneous pitch position,θ 0 is the pitch amplitude, Ψ is the phase angle between roll and pitch, and θBias is the pitch bias (which is 0 for zero-mean lift). Roll and pitch torque are measured from the output at each motor's gearbox. In conjunction with motor velocity data, these give the power applied by the motors to the fluid:
$\ P_{\mathrm{hydrodynamic}}(t)=({\tau}_{\mathrm{Roll}}{\dot{{\phi}}}+{\tau}_{\mathrm{Pitch}}{\dot{{\theta}}}),$
(3)
where τ is the instantaneous torque for the roll and pitch axes. This is a measurement of power independent of the specific actuators used in this experiment. To estimate inertial uncertainties, power time traces were measured in air. They were compared with the measurements made in water. The power trace in air leads that in water by 90°, because the resistance to motion in air is predominantly inertial, whereas in water it is predominantly hydrodynamic with velocity squared. The mean power in air calculated using torque and angular velocity is near zero, because power expended during acceleration is absorbed during deceleration. At the moment when the inertial effect in air is at maximum, it is about 15–25% of the hydrodynamic power in water for the cases shown in Fig. 8. At the moment of peak hydrodynamic power, the inertial component is zero, because the fin is at a point of peak velocity and zero acceleration.
Fig. 1.

Photograph showing a 30 cm-span fin attached to the roll/pitch mechanism.(Dimensions are in cm.)

Fig. 1.

Photograph showing a 30 cm-span fin attached to the roll/pitch mechanism.(Dimensions are in cm.)

Hydrodynamic efficiency ηhydrodynamic is defined as:
$\ {\eta}_{\mathrm{hydrodynamic}}=\frac{\overline{F}_{X}U}{\overline{P}_{\mathrm{hydrodynamic}}},$
(4)
where X is the cycle-averaged force in the forward direction, hydrodynamic is the cycle-averaged hydrodynamic power, and U is either the forward velocity of the tow carriage U or, if the carriage speed is zero, an estimate of the induced velocity through the swept area Uind. The induced velocity is defined as:
$\ U_{\mathrm{ind}}=\sqrt{\frac{\overline{F}_{X}}{2{\rho}A_{\mathrm{S}}}},$
(5)
where ρ is the fluid density and As is the area swept by the wing (Wakeling and Ellington,1997).

### Cross-flow vortex modeling of unsteady stall hydrodynamics

We propose that the dynamic stall vortices are analogous to cross-flow drag vortices of a fin placed normal to a steady uniform stream as a bluff body and as shown in Fig. 2. At shallower angles of attack, the vortices in Fig. 2 are simply the leading and trailing edge vortices resulting in the fin-normal component of force due to the cross-flow, of which lift and drag are resolved representations. If the angle of attack is constantly changing such that these drag vortices can be retained over the fin surface during the cycle, then stall can be prevented and lift enhanced at large angles of attack. Thus, unsteadiness does not do away with the basic steady lift and drag characteristics of the fin, but merely delays the occurrence of stall.

Fig. 2.

Schematic diagram of a fin positioned normal to uniform flow showing drag producing cross-flow bluff-body drag' vortices. These are formed as the angle of attack of the fin increases well above 0°. The 90° situation is shown. The rolling and pitching motion of the fin helps retain the vortices over the fin, thereby delaying stall and enhancing lift due to the low pressures in their cores.

Fig. 2.

Schematic diagram of a fin positioned normal to uniform flow showing drag producing cross-flow bluff-body drag' vortices. These are formed as the angle of attack of the fin increases well above 0°. The 90° situation is shown. The rolling and pitching motion of the fin helps retain the vortices over the fin, thereby delaying stall and enhancing lift due to the low pressures in their cores.

Consider a fin undergoing rolling and pitching motions in the Y–Z plane and moving forward in the direction X, as shown in Fig. 3. Forward velocity U and rolling velocity Uwing result in the total velocity Ut. The fin forces (lift Lfin and thrust Tfin) are transverse and in line with Ut, respectively, and N is normal to the fin surface and is the resultant of lift and thrust. Viscous drag is ignored. The angles are defined as follows: ϕ(t) is the roll angle, α(t) is the instantaneous angle of attack between the fin and Ut, and θ(t) is the pitch angle; t is time. For a steady fin at small angles of attack, lift changes much faster with angle of attack than drag does. Therefore, we assume that the coefficient of force normal to the fin surface is given by the same slope:
$\ m=\left(\frac{\mathrm{d}C_{N}}{\mathrm{d}{\alpha}}\right)\left|\right._{\mathrm{lim}{\alpha}{\rightarrow}0},$
(6)
as that of lift with an angle of attack near zero:
$\ C_{N,\mathrm{model}}[{\alpha}(t)]=m\mathrm{sin}[{\alpha}(t)].$
(7)
Resolving this orthogonally, lift and drag are given by:
$\ C_{L,\mathrm{model}}[{\alpha}(t)]=m\mathrm{sin}[{\alpha}(t)]\mathrm{cos}[{\alpha}(t)],$
(8)
$\ C_{D,\mathrm{model}}[{\alpha}(t)]=-m\mathrm{sin}^{2}[{\alpha}(t)].$
(9)
If the fin planform extends from radius ri to ro from the roll axis, the average radius of the fin swept area is defined as:
$\ R_{\mathrm{avg}}=\frac{\sqrt{r_{\mathrm{o}}^{2}+r_{\mathrm{i}}^{2}}}{2}.$
(10)
The wing velocity is given by:
$\ U_{\mathrm{wing}}(t)={\dot{{\phi}}}(t)R_{\mathrm{avg}}.$
(11)
The instantaneous angle of attack is given by:
$\ {\alpha}(t)=-\mathrm{atan}\left(\frac{U_{\mathrm{wing}}(t)}{U_{{\infty}}}\right)+{\theta}(t).$
(12)
The total speed is given by:
$\ U_{\mathrm{t}}(t)=\sqrt{U_{\mathrm{wing}}^{2}(t)+U_{{\infty}}^{2}}.$
(13)
Note that the angle of attack and the total velocity of the fin are continuously changing as given by its unsteady motion. The pre-stall steady fin characteristics can now be used to determine the unsteady lift and drag characteristics for any angle of attack. This modeling is shown below to hold for time signatures of forces. Because steady fin characteristics are used to model the unsteady behavior, this is also a quasi-steady model. However, we offer a physical basis of the inviscid character of drag and lift enhancement. This physical approach might provide a clue to the larger question as to how animals may have discovered that inviscid drag can be re-vectored to enhance lift and thrust.
Fig. 3.

Schematic diagram of the variables in a rolling and pitching fin. For definitions, see List of symbols and abbreviations.

Fig. 3.

Schematic diagram of the variables in a rolling and pitching fin. For definitions, see List of symbols and abbreviations.

### Method of optimizing fin efficiency

We implemented a downhill simplex method(Flannery et al., 1988), using frequency, roll amplitude, pitch amplitude, pitch bias, and phase between the roll and pitch sinusoids as the dimensions to be searched, and a user-chosen optimization parameter to minimize. This optimization parameter could be efficiency while hovering, defined from Wakeling and Ellington(Wakeling and Ellington, 1997)and subtracted from one:
$\ F_{\mathrm{opt}}=(1-{\eta})=\left(1-\sqrt{\frac{\overline{F}_{X}^{3}}{2{\rho}A_{\mathrm{swept}}P_{\mathrm{hydrodynamic}}^{2}}}\right).$
(14)
Alternatively, the optimization parameter could be a function of X, Y and power, such as:
$\ F_{\mathrm{opt}}=(\overline{F}_{X}-8)^{2}+(\overline{F}_{Y}-3)^{2}+kP^{2}_{\mathrm{hydrodynamic}},$
(15)
which would try to find a motion that would produce a mean X-force of 8 N and a mean Y-force of 3 N while minimizing the power spent. The weight k is added for priority.

When the search algorithm outputs a new set of motion parameters to try,the fin will change its motion to the new set and flap a user-defined number of times. The fin control program returns the mean forces, power and efficiency recorded during the routine, which are then sent back to the search algorithm, where they are translated into the optimization parameter of interest. A simulated annealing term was added to the method(Flannery et al., 1988), with a gradually reducing temperature', to avoid becoming stuck at a point where the repeatability level of the data resulted in an anomolous local minimum due to noise in data.

Fig. 4.

Comparison of the measurements of unsteady coefficients of lift (red) and drag (blue) with the cross-flow vortex model. The steady measurements are also included. Tow speed is 1.34 m s–1. Motion parameters areϕ 0=30°, θ0=15°, f=1.25 Hz, U=1.34 m s–1, span=20 cm,θ Bias=0°, St=0.26. St is Strouhal number defined as 2fϕ0Ravg/U,where f is the frequency of oscillation.

Fig. 4.

Comparison of the measurements of unsteady coefficients of lift (red) and drag (blue) with the cross-flow vortex model. The steady measurements are also included. Tow speed is 1.34 m s–1. Motion parameters areϕ 0=30°, θ0=15°, f=1.25 Hz, U=1.34 m s–1, span=20 cm,θ Bias=0°, St=0.26. St is Strouhal number defined as 2fϕ0Ravg/U,where f is the frequency of oscillation.

### Cross-flow vortex model

The cross-flow vortex model is compared with measurements in Figs 4 and 5 at a tow speed of 1.34 m s–1. The agreement is good. In Figs 6 and 7, a similar comparison is made but at a lower speed of 0.46 m s–1 and a higher pitch amplitude of 35° to retain a similar maximum angle of attack, but with other fin oscillation parameters remaining the same as in the case of the 1.34 m s–1 tow speed. At the lower speed, the model agrees with the time signatures of the force measurements as well. However, the measurements, and not the model, indicate a seeming hysteresis. This hysteresis is a direct measurement and is reminiscent of a Wagner effect as per which there is a time delay in the mechanism of force production in impulsively started lifting surfaces. Due to viscous effects, there is a delay in the development of the asymptotic value of the circulation around the lifting surface, that is, a delay in the establishment of the Kutta condition. The proximity of the starting vortex near the trailing edge in the early stages also affects this delay.

A clear hysteretic unsteady behavior has not been directly evidenced in the literature on animal-inspired flying and swimming. In our measurements, such hysteresis is present in the cases of hovering and low speeds of tow, and this is eliminated as the speed is raised. The hysteresis increases at the extreme values of the angle of attack (Fig. 6), which coincide with the extreme roll positions of the fin. Our estimation of induced flow velocity – from an actuator disc method(Wakeling and Ellington, 1997)– is constant in time and space, which will lead to errors in the magnitude and direction of Ut. To resolve whether there truly is hysteresis in the low-speed fin behavior, the time signatures of fin torque were examined. This is because torque is an independent and direct diagnostic of the fin response to the roll and pitch oscillations where induced velocities do not have to be estimated. The torque versusroll velocity response is shown in Fig. 8 at tow speeds of 1.34 and 0.46 m s–1. The torque versus roll velocity is linear and elliptic', respectively,in Fig. 8 at the higher and the lower speeds. While there is no hysteresis at the higher speed, some is seen at the lower speed, but it is not as dramatic as the reduced data in Fig. 6 suggest. At 0.46 m s–1, for peak–peak roll torque of 8 N m, there is a maximum deviation from the mean value of about ±1 N m – a 12%effect. A more accurate method of estimating induced velocities during hover or low speeds of tow is needed.

Fig. 5.

Comparison of the measurements of unsteady time signatures of forces in the forward and transverse directions and of power with the cross-flow vortex model for the case shown in Fig. 4. Tow speed is 1.34 m s–1. The fin kinematics are shown at the top.

Fig. 5.

Comparison of the measurements of unsteady time signatures of forces in the forward and transverse directions and of power with the cross-flow vortex model for the case shown in Fig. 4. Tow speed is 1.34 m s–1. The fin kinematics are shown at the top.

Fig. 6.

Comparison of the measurements of unsteady coefficients of lift (red) and drag (blue) with the cross-flow vortex model. Tow speed is 0.46 m s–1. The steady measurements are also included. Motion parameters are ϕ0=30°, θ0=35°, f=1.25 Hz, U=0.46 m s–1,span=20 cm, θBias=0°, St=0.78.

Fig. 6.

Comparison of the measurements of unsteady coefficients of lift (red) and drag (blue) with the cross-flow vortex model. Tow speed is 0.46 m s–1. The steady measurements are also included. Motion parameters are ϕ0=30°, θ0=35°, f=1.25 Hz, U=0.46 m s–1,span=20 cm, θBias=0°, St=0.78.

The universal validity of the cross-flow vortex model for lift and drag over all oscillation parameters such as frequency, roll angle, pitch bias and pitch amplitude during hovering and towing, as well as for varying fin spans,is shown in Figs 9 and 10, respectively (roll 30°, 40°; pitch bias 0°; pitch 25°, 35°, 45°, 55°and 65°; roll–pitch phase –90°, 90°; frequency 0.75,1, 1.25 and 1.5 Hz; span 20, 30 cm; cruising speed –0.46, 0.46, 0.83 and 1.34 m s–1). The steady fin measurements are included for reference. The unsteady data indicate lift and drag forces averaged for bins at a given angle of attack for all cases. For low tow speeds, the averaged value of the hysteretic behavior is shown. The cross-flow vortex model describes the unsteady measurements well.

Fig. 7.

Comparison of the measurements of unsteady time signatures of forces in the forward and transverse directions and of power with the cross-flow vortex model for the case shown in Fig. 6. Tow speed is 0.46 m s–1. The fin kinematics are shown at the top.

Fig. 7.

Comparison of the measurements of unsteady time signatures of forces in the forward and transverse directions and of power with the cross-flow vortex model for the case shown in Fig. 6. Tow speed is 0.46 m s–1. The fin kinematics are shown at the top.

Fig. 8.

Measurements of the roll torque and roll velocity at a tow speed of 1.34 m s–1 for the case shown in Fig. 4 (blue) and a tow speed of 0.46 m s–1 for the case shown in Fig. 6 (green). Observe the presence of hysteresis in the latter in comparison with the former. The hysteresis is attributable to the slower development of the Kutta condition at lower speed.

Fig. 8.

Measurements of the roll torque and roll velocity at a tow speed of 1.34 m s–1 for the case shown in Fig. 4 (blue) and a tow speed of 0.46 m s–1 for the case shown in Fig. 6 (green). Observe the presence of hysteresis in the latter in comparison with the former. The hysteresis is attributable to the slower development of the Kutta condition at lower speed.

### Rapid optimization of hydrodynamic efficiency

The magnitude of the mean lift and thrust forces produced by the fin are functions of the flapping frequency, roll amplitude, pitch amplitude, phase difference between roll and pitch, and pitch bias, as well as the incoming flow speed and direction. To make the force production robust to disturbances in the incoming flow, such as that created by cross-flows and large-scale vortices, one would either have to study the fin in all sorts of flow fields and then measure those in practice, or enable the system to react to changing flows in such a way as to produce the desired forces nonetheless. As an initial step to a flow-adaptable fin, we demonstrated the fin searching through its parameter space – with no a priori knowledge of its hydrodynamic capabilities – in order to optimize between force production and power cost.

Fig. 9.

Comparison of the averaged measurements of instantaneous lift forces with the cross-flow vortex model. The steady fin measurements are also included. Blue and green data indicate tests with a 20 or 30 cm span, respectively. The inset expands the data up to an angle of attack of 20° to clarify the validity of the model up to angles where stall occurs in the steady case.

Fig. 9.

Comparison of the averaged measurements of instantaneous lift forces with the cross-flow vortex model. The steady fin measurements are also included. Blue and green data indicate tests with a 20 or 30 cm span, respectively. The inset expands the data up to an angle of attack of 20° to clarify the validity of the model up to angles where stall occurs in the steady case.

Fig. 10.

Comparison of the averaged measurements of instantaneous drag forces with the cross-flow vortex model. The steady fin measurements are also included. Blue and green data indicate tests with a 20 or 30 cm span, respectively. The inset expands the data up to an angle of attack of 20° where stall occurs in the steady case (Fig. 9) to clarify that the cross-flow vortex model does not account for viscous drag.

Fig. 10.

Comparison of the averaged measurements of instantaneous drag forces with the cross-flow vortex model. The steady fin measurements are also included. Blue and green data indicate tests with a 20 or 30 cm span, respectively. The inset expands the data up to an angle of attack of 20° where stall occurs in the steady case (Fig. 9) to clarify that the cross-flow vortex model does not account for viscous drag.

The optimization method was successful in converging to optimized points within 40–50 cycles, in approximately 4 min, depending on the initial random set of motion parameters. This is shown in Fig. 11. The optimized motion parameters were within the range seen in a previous matrix study of the fin performance versus input parameters. This is shown in Fig. 12.

Fig. 11.

Time sequence trace of the search for the highest efficiency during hovering. The green circles track the highest level of efficiency reached as yet during the scheme. Two flapping cycles were tested for each oscillation parameter set. Total search time is ∼4 min.

Fig. 11.

Time sequence trace of the search for the highest efficiency during hovering. The green circles track the highest level of efficiency reached as yet during the scheme. Two flapping cycles were tested for each oscillation parameter set. Total search time is ∼4 min.

Fig. 12.

Measurements of efficiency versus coefficient of X-force. The blue data were first collected in a systematic matrix study over a predefined range of oscillation parameters. This data gathering was spread over about 1.5 years, which is common in conventional experimental procedures where the hydrodynamic characteristics and the models of control laws are determined before a vehicle design is carried out. The symbols denote the carriage speed, where × is U=0, o is 0.46, +is 0.83, and * is 1.34 m s–1. The numbered dots denote trial numbers from the random search algorithm; green dots denote trials with a pitch bias greater than 5°, and red dots denote runs after which the bias had converged to less than 5°. Observe that from an arbitrary starting point, the algorithm rapidly reaches the point of highest efficiency for hovering as denoted by the × symbols. The search for maximum efficiency converges with any initial random selection of oscillation parameters. The rapid search method also works well when the fin is towed.

Fig. 12.

Measurements of efficiency versus coefficient of X-force. The blue data were first collected in a systematic matrix study over a predefined range of oscillation parameters. This data gathering was spread over about 1.5 years, which is common in conventional experimental procedures where the hydrodynamic characteristics and the models of control laws are determined before a vehicle design is carried out. The symbols denote the carriage speed, where × is U=0, o is 0.46, +is 0.83, and * is 1.34 m s–1. The numbered dots denote trial numbers from the random search algorithm; green dots denote trials with a pitch bias greater than 5°, and red dots denote runs after which the bias had converged to less than 5°. Observe that from an arbitrary starting point, the algorithm rapidly reaches the point of highest efficiency for hovering as denoted by the × symbols. The search for maximum efficiency converges with any initial random selection of oscillation parameters. The rapid search method also works well when the fin is towed.

The measurements of efficiency in Fig. 12 contain two sets of data – one for hovering and one for cruising. The thrust was normalized as CX,wing=X/(1/2ρU2wingAplanform),and the efficiency was defined asη hydrodynamic=XU/hydrodynamicfor U>0 andη hydrodynamic=XUinduced/hydrodynamicfor U=0, where the induced velocity was estimated using the disk method of Wakeling and Ellington(Wakeling and Ellington,1997), meaning that the ηhydrodynamic values during hovering have higher uncertainties than those during cruising. All data with CX,wing>1.5 are for hovering, and most of the data for CX,wing<1.5 are for cruising. Using the best means available for comparison, the best efficiency for hovering is nearly half that during cruising. This suggests that propulsive efficiency when maneuvering is less efficient than when cruising. Note that the highest efficiency of about 0.6 is similar to the 0.5–0.6 measured for two-dimensional fins (Read et al.,2003). Thus, increasing the aspect ratio of fins beyond the maximum of 3 in the present work does not offer an increase in efficiency. A private communication with G. V. Lauder and P. G. A. Madden in 2006 indicated that the highest efficiency of a sunfish with a pair of flexible pectoral fins during station keeping in a stream is 0.42. If we allow that the efficiency of a fish is lower than that of its appendages, then the sunfish pectoral fin efficiency is probably higher than 0.42 and may be closer to 0.6. This suggests that the efficiency of pectoral appendages from penguins to sunfish is similar.

Fig. 13.

PIV and laser cross-sectional end view of sunfish pectoral fin station keeping in a stream of speed 8.5 cm s–1 (from G. V. Lauder and P. G. A. Madden, personal communication, 2005); total fish length is 17 cm. The fin is in abduction phase. Observe the formation of contrarotating vortices at the fin tips. We hypothesize that they are cross-sections of two different stall vortices as shown in Fig. 14.

Fig. 13.

PIV and laser cross-sectional end view of sunfish pectoral fin station keeping in a stream of speed 8.5 cm s–1 (from G. V. Lauder and P. G. A. Madden, personal communication, 2005); total fish length is 17 cm. The fin is in abduction phase. Observe the formation of contrarotating vortices at the fin tips. We hypothesize that they are cross-sections of two different stall vortices as shown in Fig. 14.

### Biorobotic rigid fins and flexible pectoral fins

A convergence between the above results of the rigid biorobotic fin and those of the sunfish flexible pectoral fin was sought to explore the force production mechanism of sunfish pectoral fins. An explanation of how sunfish maintain station within close tolerance has been lacking. Lauder et al.(Lauder et al., 2007) have argued that the sunfish pectoral fin has two leading edges because, due to the cupping of the fin during outstroke, both the leading and trailing edges become positioned at the same axial station during station keeping in a laboratory flow tunnel. If so, then, where are the trailing edges? A still frame showing the formation of two clear vortices at the fin edges is given in Fig. 13 and a synthesis is shown in Fig. 14. If both edges are leading, then the contrarotating vortices in Fig. 13 must be from the leading edges of two symmetric fins. We then attach importance to the fact that similar contrarotating vortex pairs away from the leading edge vortices are not produced, as would be expected had there been two trailing edges. This implies that there are no trailing edges at all, but only two leading edges. Accordingly, we synthesize these arguments to arrive at the hypothesis that the sunfish pectoral fin acts as a conjoined symmetric biplane during station keeping in a uniform stream during the outstroke half of the fin beat. The flexibility is a means of making two fins out of one in order to balance unsteady forces in the vertical direction. The formation of the two attached edge vortices shows that the dynamic stall mechanism is present in sunfish pectoral fins although the projected fin area is much reduced from the maximum possible.

Fig. 14.

Proposed dynamic stall vortex pairs shown in color in the sunfish pectoral fin formed during outstroke when the fin is undergoing cupping' motion. The fin picture is from Lauder et al. (Lauder et al., 2007). The total fish length is 17 cm and the stream speed is 8.5 cm s–1. The fish is maintaining its station in auniform' stream, shown by the vertical arrow, while the fin is turning upstream and the spanwise edges are curling inward. The stall vortices locally augment the pressure difference across the two leading edges and could help cancel perturbations in the vertical directions. The co-flowing jets on both sides of the fish could be vectored appropriately to hold station laterally and provide some thrust. The vertical arrow shows the stream direction.

Fig. 14.

Proposed dynamic stall vortex pairs shown in color in the sunfish pectoral fin formed during outstroke when the fin is undergoing cupping' motion. The fin picture is from Lauder et al. (Lauder et al., 2007). The total fish length is 17 cm and the stream speed is 8.5 cm s–1. The fish is maintaining its station in auniform' stream, shown by the vertical arrow, while the fin is turning upstream and the spanwise edges are curling inward. The stall vortices locally augment the pressure difference across the two leading edges and could help cancel perturbations in the vertical directions. The co-flowing jets on both sides of the fish could be vectored appropriately to hold station laterally and provide some thrust. The vertical arrow shows the stream direction.

To achieve station keeping, it is necessary to be able to promptly produce forces and moments along stream and also normal and cross-stream in amplitudes that are just enough to cancel the perturbations. The flexible fin has the ability to control the projected surface area in all planes as indeed we see in the movie frames in Lauder et al.(Lauder et al., 2007). The fin folding also changes the local angles of attack. The twist of the fin can also be controlled at the root, thereby controlling the vector of the co-flowing jet (Fig. 14). These three traits, with the aid of sensitive lateral line sensors and an instantaneously deployable controller, could dynamically cancel force and moment perturbations in all directions and axes. The rigid fin measurements of force vectors at each instant were examined in three-dimensional fields. Rigid fins produce large transverse periodic forces, which may be undesirable when holding position (Beal and Bandyopadhyay,2007). To achieve station keeping in the vertical plane, two symmetric fins are needed to balance the instantaneous vertical (Y-)forces. Such data show quantitatively that station keeping can be achieved at all times by symmetric biplanes, conjoined or not. The symmetric fins would not have to be mirror images; their angles of attack to the flow and flapping speed could be different as long as they cancel undesired unsteady forces. A sample of the sunfish pitch angle time history from Lauder et al.(Lauder et al., 2007) is compared with a similar sinusoidal history of the rigid fin in Fig. 15, where a qualitative similarity is seen to exist. The sum of the pitch angles between the dorsal and ventral rays is within 0° to 20° during abduction, while it is closer to zero during the adduction phase. The pitch angles in the biorobotic rigid fins, of course, sum exactly to zero at all times. In the absence of detailed kinematics of the sunfish pectoral fin, this zero sum is taken as an indication of real-time station keeping.

Fig.·15.

Comparison of measurements of the variation in fin pitch angle with time during one cycle (red and blue diamonds) in bluegill sunfish pectoral fins(Lauder et al., 2007) with our rigid fin data. Sunfish pectoral fin: roll amplitude, 40.8°; frequency,1.0·Hz; pitch amplitude, 44.8°.

Fig.·15.

Comparison of measurements of the variation in fin pitch angle with time during one cycle (red and blue diamonds) in bluegill sunfish pectoral fins(Lauder et al., 2007) with our rigid fin data. Sunfish pectoral fin: roll amplitude, 40.8°; frequency,1.0·Hz; pitch amplitude, 44.8°.

What can we learn by comparing the force production in sunfish due to their flexible pectoral fins and the present relatively stiff penguin wing-like biorobotic fins? Lauder et al. (Lauder et al., 2007) have given the time sequences of the flexible fin contortions and the distributions of local velocity variations about the freestream velocity for a sunfish holding station in a stream. We propose the following mechanism to be in play in the flexible fin. During the outstroke,while the spanwise edges of the fin are folding inward, due to separation and the ensuing pressure difference, two symmetric leading-edge vortices are formed (Fig. 14). The projected area of the fin is much reduced from its maximum value during the cupping process, which suggests that its thrust production role is smaller. The two leading-edge vortices coalesce to form a downstream-pointing jet. The spanwise bone structure and the chordwise fin corrugations would assist this spanwise jet flow. The jet is inclined rearward to the body and is a source of thrust. The outstroke takes the fin to a position just short of being normal to the body. The fin subsequently expands, which causes the twin leading-edge vortice jet to expand into a diffuser, allowing pressure recovery, at the end of the outstroke. During the return stroke, the fin concaves with the fin tip facing upstream. Due to stiffness, the spanwise edges of the fin this time do not cup inward and no significant leading-edge vortex is probably produced,although a stall vortex could still form at the tip. Conceivably, this tip vortex inclined cross-stream and parallel to the body could interact with the caudal fin downstream, providing a means for precision streamwise control of the fish body to station itself in the face of perturbations, because it is sinusoidal and small in amplitude compared with net thrust(Bandyopadhyay et al., 1998). During the sweeping motion of the return stroke, the fin acts like a row. The sweep motion is analogous to rolling in rigid fins and the cupping of the fin changes pitch – the angle of attack constantly changes during both motions. In Lauder and colleague's (Lauder et al., 2007) experiment, pitch bias may be zero because the fish is not steering (Fig. 15). Because rowing is inefficient, and the fin edges would break at higher thrust levels, flexible fins are not seen in larger animals. The flexible pectoral fins are probably more suitable in low speed and smaller swimming animals. They would be difficult to scale up for biorobotic application. Because flying insects need to produce steady lift as well as thrust – unlike fish,which are nearly neutrally buoyant – they do not use similar flexible fins, which are optimal for station keeping. Although the cupping motion during the outstroke produces a lower thrust peak than the return stroke(Lauder et al., 2007), it can nevertheless cancel vertical perturbations to allow exquisite station keeping. Therefore, the outstroke cupping motion may assist in station keeping with regard to vertical perturbations while the return stroke sweeping motion is important for thrust production because the projected fin area approaches its maximum value. What is fascinating about the flexible fin is that it can conceivably cancel force and moment perturbations in all directions and axes and produce thrust. Clearly, there is a need to investigate how in a sunfish the controller uses lateral lines to actuate this unsteady hydrodynamics of the flexible fin. At the very least, the control and sensing of the flexible fin hydrodynamics are shaping up to be far richer than those in rigid fins.

This work identifies the common underlying principles in steady and unsteady fin dynamics and in rigid and flexible fins. The effectiveness of the cross-flow vortex model helps to place the role of unsteadiness in the context of classical steady-foil theories, which have a firm theoretical foundation– unsteadiness, while being novel, is inclusive rather than an entirely exclusive phenomenon, from an engineering view point. On the other hand, in nature, unsteadiness is not novel and steadiness is rare. The validity of the model points to intriguing notions; for example, would it be possible to infer the habitats of ancient species from the fossils of their pectoral fins?

Swimming and flying animals and their biorobotic renditions are reported to have efficiencies that vary widely. Direct measurements have tended to be on the low side. On the one hand, Fish (Fish,1993) has modeled bottlenose dolphin efficiency to be 81%. On the other, Wakeling and Ellington (Wakeling and Ellington, 1997) have reported the measurements of dragonfly mechanical efficiencies to be 9–13% based on heat production after flight. Anderson et al. (Anderson et al.,1998) have reported the measurements of peak hydrodynamic efficiencies of 87% for two-dimensional heaving and pitching foils. However,later measurements from the same laboratory have reported an efficiency plateau of 50–60% (Read et al.,2003). The present measurements are in this range, as are the estimates for sunfish by G. V. Lauder and P. G. A. Madden (personal communication, 2006).

The efficiency optimization work shows that the process is remarkably rapid and seemingly effortless. This approach makes a cumulative accounting of past experience in the sense of learning' and is akin to the spirit of evolution. This suggests that once the species discovered the high-lift mechanism, the optimization may have evolved quickly. As a consequence, it may be that rolling and pitching appendages of all species are of generally high efficiency.

Neuroscience posits that mobility demands richness in neuronal abilities,and the higher density of neurons is a precursor of an ability to perform complex motions. If we look at specialized maneuvering as more complicated than cruising, then was the evolution of species endowed with such richness the result of a step increase in neuron density? Our experience indicates that the roll and pitch motions of the fin are useful for both cruising and maneuvering, whereas pitch bias is useful only for maneuvering. These classes of behaviors might give us a framework for understanding the relationship between hydrodynamics and control in different species of swimming animals because they are proving crucial in the development of biorobotic underwater vehicles (Menozzi et al., 2007).

The following conclusions can be drawn from our controlled hydrodynamic experiments in a laboratory with biorobotic abstracted penguin pectoral wings.

1. A cross-flow vortex model of unsteady lift and drag forces has been proposed that is reasonably accurate, instant to instant. It follows that the rolling and pitching motions of unsteady pectoral appendages may be viewed as a means of trapping' what are essentially inviscid drag vortices manifested at the fin leading and trailing edges over the fin surface for the augmentation of lift. From an engineering viewpoint, unsteady hydrodynamics is seen as a manifestation of the steady-state characteristics in the regime studied and not as a totally new phenomenon. However, unsteadiness in nature is old and steadiness is rare.

2. Measurements of hydrodynamic efficiency have been given for hovering and for forward motion. The peak efficiency for cruising is similar to that for two-dimensional foils and in the neighborhood of that for the pectoral fins of sunfish. Maneuvering is less efficient, hydrodynamically speaking, compared with cruising, and in the best cases it is twice as costly.

3. An optimization method has been shown to rapidly determine the fin oscillation parameters required for highest efficiency at a given forward speed, or during hovering for a given coefficient of force. The method does not require any prior knowledge of any steady or unsteady characteristics of the fin.

4. The biorobotic measurements of the rigid fin were used to explore the station-keeping mechanism of sunfish pectoral fins in a stream. It is proposed that a dynamic stall mechanism is present in the flexible fins of sunfish pectoral fins and that such fins fold to act essentially as pairs of symmetric biplanes that are conjoined at their trailing edges in order to balance unsteady force byproducts. The flexible fin appears to have a rich portfolio of control schemes for station keeping in streams of low speeds. Scaling up of rigid fins appears to be more feasible than scaling up of flexible fins.

List of symbols and abbreviations

• As

fin swept area

•
• Aplanform

fin planform area

•
• c

fin chord

•
• CD,model

coefficient of drag force; CD,model=D/(1/2ρUt2Aplanform)

•
• CL,model

coefficient of lift force; CL,model=L/(1/2ρUt2Aplanform)

•
• CN,model

coefficient of normal force; CN,model=N/(1/2ρUt2Aplanform)

•
• CX,wing

coefficient of cycle-averaged force in X-direction

•
• f

frequency of flapping (Hz)

•
• Fopt

optimization function

•
• F̄X

cycle-averaged force in direction of U

•
• F̄Y

cycle-averaged force in horizontal direction transverse to U

•
• Lfin

fin lift relative to instantaneous inflow angle

•
• m

slope of N with α near α=0

•
• N

normal force on fin

•
• electric

cycle-averaged electrical power

•
• Phydrodynamic

power put into the fluid by the fin

•
• hydrodynamic

cycle-averaged hydrodynamic power

•
• Ravg

average radius of the fin swept area

•
• ri

•
• ro

•
• s

fin span

•
• St

Strouhal number

•
• t

time

•
• Tfin

fin thrust relative to instantaneous inflow angle

•
• U

flow velocity, defined as U or Uind

•
• Uind

induced velocity through As

•
• U

velocity of tow carriage

•
• Ut

absolute wing inflow velocity at Ravg

•
• Uwing

flapping-induced fin velocity at Ravg

•
• X, Y, Z

coordinate system attached to tow carriage

•
• X0, Y0, Z0

coordinate system attached to Ravg

•
• α

instantaneous angle of attack of Ut on fin

•
• ηhydrodynamic

efficiency defined using hydrodynamic

•
• ηelectric

efficiency defined using electric

•
• θ

pitch position of fin

•
• $${\dot{{\theta}}}$$

pitch velocity

•
• θ0

pitch amplitude of the fin

•
• θBias

fixed pitch amplitude of the fin

•
• ρ

fluid density

•
• τRoll

roll torque measured at the motor output shaft

•
• τPitch

pitch torque measured at the motor output shaft

•
• ϕ

roll position of fin

•
• $${\dot{{\phi}}}$$

roll velocity

•
• ϕ0

roll amplitude of fin

•
• ψ

phase between roll and pitch sinusoids, defined as positive for pitch leading roll

•
• ω

The authors acknowledge the sponsorship of the Cognitive and Neurosciences Program of the Office of Naval Research, Program Officer Dr Thomas McKenna and NUWC ILIR Program, Program Officer Mr Richard Philips. Discussions with Professor George Lauder are appreciated.

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