2. Experimental methods

Experiments were conducted on a flexible rectangular panel suspended in a recirculating water channel with flow speed $u$ . The chord $c$ , span $s$ , and thickness ${\it\delta}$ of the panel were 195 mm, 150 mm, and  0.11 mm, respectively. The panel had an elastic modulus $E=3.8$ GPa, resulting in a bending stiffness $b=6.9\times 10^{-5}$ $\text{N}~\text{m}^{2}$ , where $b\equiv Es{\it\delta}^{3}/(12(1-{\it\nu}^{2}))$ , and Poisson’s ratio ${\it\nu}$ was estimated to be 0.3. The elastic modulus was measured by tensile tests of the polyethylene from which the panel was made. A deflection test was also used to measure the bending stiffness of the panel, and the resulting estimate agreed with the tensile tests to within 10 %.

The panel was actuated at the leading edge by an aluminium spar (chord = 10 mm, thickness = 3 mm) undergoing heave and pitch oscillations. The oscillations were such that the lateral position $h$ and angle of attack ${\it\alpha}$ of the spar followed $h=a\;\text{sin}(2{\rm\pi}\,f\,t)$ and ${\it\alpha}={\it\alpha}_{0}\;\text{sin}(2{\rm\pi}\,f\,t-{\it\phi})$ , where $a$ is the heave amplitude, ${\it\alpha}_{0}$ is the maximum pitch angle, $f$ is the frequency of oscillation, $t$ is time, and ${\it\phi}$ is the phase lag between heave and pitch. The remainder of the panel deformed passively, finishing with a trailing-edge amplitude $a^{\prime }$ . The motions were produced by connecting the spar to a pitch motor (Pittman, GM8724S009) seated beneath a rack that heaved back and forth under the action of a second motor (MCG, IB23007) (see figure 1).

Figure 1.

Experimental arrangement. (a) Length scale definitions; (b) top view; (c) side view.

A six-axis force–torque sensor (ATI Inc., Nano-17 SI-50-0.5) was used to measure streamwise forces $\mathscr{F}_{x}$ , lateral forces $\mathscr{F}_{y}$ , and $z$ -torques $\mathscr{Q}_{z}$ . The true lateral ${\dot{h}}$ and angular $\dot{{\it\alpha}}$ velocities of the spar were recorded using two rotary encoders. Together, these force and velocity data give the power input to the fluid ${\wp}$ according to ${\wp}=-\mathscr{F}_{y}{\dot{h}}-\mathscr{Q}_{z}\dot{{\it\alpha}}$ . Time-averages were obtained by averaging over the largest number of oscillation cycles possible in a 10 s period of data acquisition. We will use ‘net thrust’ $\bar{{\it\tau}}$ and ‘net power’ $\bar{{\wp}}$ to refer to time-averaged streamwise force and power input to the fluid, respectively. Of particular importance to swimming performance is the propulsive efficiency ${\it\eta}\equiv \bar{{\it\tau}}u/\bar{{\wp}}$ . The average variance between trials for net thrust and power was $\pm 5$ mN and $\pm 1$ mW, resulting in $3\,\%$ uncertainty in the efficiency.

The chosen propulsive efficiency metric, often called the Froude efficiency, represents the fraction of energy transferred to the wake resulting in positive thrust. This metric is only defined when the thrust produced by the propulsor exceeds the drag incurred by the flow. Fully autonomous swimmers require that their net thrust be zero for steady swimming, and in this case the more appropriate metric is the cost of transport, $u/\bar{{\wp}}$ . Our treatment of efficiency is therefore best-suited for propulsors that operate with net thrust in order to drive drag-producing payloads. Given the $O(1)$ aspect ratio of our panel, our results may apply most directly to swimmers with broad flattened fins, such as rays and skates or batoid-inspired vehicles.

The oscillations of the leading-edge spar define four of the five input parameters to the system ( $f,{\it\phi},a$ , and ${\it\alpha}_{0}$ ), with the fifth input being the flow speed of the water channel ( $u$ ). The time to conduct full sweeps of the parameter space $t_{\mathit{exp}}$ depends exponentially on the number of input parameters $n$ , that is, $t_{\mathit{exp}}\propto \text{e}^{n}$ . Including multiple trials and resetting time, the proportionality constant for the current setup is approximately 1 h. This limitation precludes any full sweeps of the full five-dimensional (5D) parameter space. Instead, we will consider selected subspaces of the parameter space and employ gradient-based optimization to seek the global maxima of the full space.

The largest subspace we considered spans the space of heave-only motions ( ${\it\alpha}_{0}=0$ ). Data were taken via grid search over 23 frequencies ranging from 0.6 to 2.8 Hz, 14 flow speeds ranging from 25 to $360~\text{mm}~\text{s}^{-1}$ , and four amplitudes ranging from 6 to 18 mm. This subspace was chosen to overlap with previous heave-only studies (Quinn et al. Reference Quinn, Lauder and Smits2014a ) and to demonstrate some general features of the efficiency manifold. The subspace covers Strouhal numbers and Reynolds numbers that are typical of fishes and bio-inspired propulsors with similar chord lengths to our panel (Lauder, Flammang & Alben Reference Lauder, Flammang and Alben2012; Shelton, Thornycroft & Lauder Reference Shelton, Thornycroft and Lauder2014). Two smaller subspaces were chosen to isolate the effects of pitch and phase offset. These subspaces correspond to conditions near the two optima discovered by the gradient-based optimization scheme. Here, the heave amplitude, frequency, and flow speed were fixed at their optimum values, while maximum pitch angle and phase offset were kept variable. Data were taken at 11 maximum pitch angles ranging from $0^{\circ }$ to $30^{\circ }$ and 16 phase offsets ranging from $0^{\circ }$ to $337.5^{\circ }$ . The upper bound for maximum pitch angle was a limitation based on load constraints of the force sensor.

Some optimization studies have used advanced learning algorithms (e.g. SNOPT by Gill, Murray & Saunders Reference Gill, Murray and Saunders2005 was used by Izraelevitz & Triantafyllou Reference Izraelevitz and Triantafyllou2014), but we chose an archetypal gradient-based algorithm in order to demonstrate the robustness of the technique. The algorithm is a form of iterated local search based on classical learning algorithms from Snyman (Reference Snyman2005). During each iteration, the inputs are shifted in the direction of the estimated gradient of the efficiency manifold, and the cycle is repeated until a minimum step size is reached. Further details of the method are given in appendix A.

At several sample conditions, PIV was conducted at the midspan. The laser sheet was generated by a continuous 10 W argon-ion laser (Coherent, Innova 70-C), and the particles were hollow silver-coated glass beads with an average diameter of $12~{\rm\mu}\text{m}$ . Images were taken at 250 Hz with $1024\times 1024$ resolution (Photron, FASTCAM 1024 PCI). The velocity field was calculated using Davis 8.1.3, the spatial cross-correlation algorithm developed by LaVision Inc. (Stanislas et al. Reference Stanislas, Okamoto, Kahler and Westerweel2005). Four passes with 50 % overlap were conducted on the data: two with $64\times 64$ pixel windows and two with $24\times 24$ . The resulting velocity field consisted of $86\times 86$ vectors. Phase-averaged velocity fields were calculated using the same number of cycles as the time-averaged force data. The PIV images were also used to calculate trailing-edge amplitude, which was taken to be the average difference between extrema of the trailing-edge position.

As a control case, a rigid NACA 0012 airfoil (67 mm chord) was run through the same testing procedure at sample conditions, chosen to overlap with those used for the flexible panel and with a previous NACA 0012 dataset taken in the same water channel (Lauder et al. Reference Lauder, Lim, Shelton, Witt, Anderson and Tangorra2011). For these cases, the heave amplitude, maximum pitch angle, and phase offset were fixed at 13 mm, $10^{\circ }$ , and $90^{\circ }$ , respectively. Data were taken at 11 frequencies ranging from 1.1 to 2.1 Hz and 11 flow speeds ranging from 30 to $180~\text{mm}~\text{s}^{-1}$ .

3. Scaling considerations

The five input parameters to the system are the flow speed $u$ , oscillation frequency $f$ , heave amplitude $a$ , maximum pitch angle ${\it\alpha}_{0}$ , and heave–pitch phase lag ${\it\phi}$ . In cases where PIV was conducted, the trailing-edge amplitude $a^{\prime }$ is also known. Additional constants are the density ${\it\rho}$ and viscosity ${\it\mu}$ of the water, and the chord $c$ , span $s$ , and bending stiffness $b$ of the propulsor. These 11 variables produce eight dimensionless parameters. One of them, the aspect ratio $AR\equiv s/c$ , is fixed for this experiment at 0.77. Another, the Reynolds number based on the chord, $Re_{c}\equiv {\it\rho}uc/{\it\mu}$ , ranges from 5000 to 70 000 in this study, and at these large Reynolds numbers pressure forces are assumed to dominate over viscous forces so that $Re_{c}$ will not be considered further. The dimensionless trailing-edge amplitude $a^{\prime }/a$ is an output of the system and will be used to verify the location of resonant frequencies.

The remaining five dimensionless parameters provide an alternative description of the five-dimensional input space over which we will optimize efficiency. First we have ${\it\alpha}_{0}$ and ${\it\phi}$ , which are already dimensionless. Next we have the amplitude–chord ratio $a^{\ast }\equiv a/c$ and the Strouhal number $St\equiv 2af/u$ ( $a$ is used as a proxy for the width of the wake since it is prescribed). The final parameter must incorporate the bending stiffness of the panel. By introducing an added-mass term into linear beam theory, it can be estimated that the first eigenfrequency of the fluid–panel system $\hat{f}_{1}$ is $(b/(\mathscr{M}{\it\rho}sc^{5}))^{1/2}$ , where $\mathscr{M}$ is a constant (see appendix B). We are thus motivated to introduce a second dimensionless frequency based on this time scale, the ‘flexural frequency’ $f^{\ast }\equiv f/\hat{f}_{1}$ , where $\mathscr{M}$ will be set to 1 for now (its actual value will be discussed later). Whereas $St$ compares the actuation frequency to time scales of the incoming flow, $f^{\ast }$ relates the actuation frequency to time scales of fluid–structure resonance.

Table 1 compares the range of dimensionless input parameters for several studies of propulsive flexible panels. Note that ${\it\phi}$ has been omitted because most studies examine heave or pitch only. The exceptions are Zhu (Reference Zhu2007), where ${\it\phi}$ was fixed at $90^{\circ }$ , and this study, where ${\it\phi}$ ranges from $0^{\circ }$ to $360^{\circ }$ . We will focus on low heave amplitudes and high flexibilities (high values of $f^{\ast }$ ) because this configuration can excite multiple resonant modes at modest frequencies ( $f\;(\text{Hz})=O(1)$ ).

Table 1.

Dimensionless parameters used in recent flexible panel studies. Type $\equiv$ N (numerical) or E (experimental); aspect ratio $AR\equiv s/c$ ; Reynolds number $Re_{c}\equiv {\it\rho}uc/{\rm\mu}$ ; amplitude–chord ratio $a^{\ast }\equiv a/c$ ; maximum pitch angle ${\it\alpha}_{0}$ ; Strouhal number $St\equiv 2a\,f/u$ ; flexural frequency $f^{\ast }\equiv f/\hat{f}_{1}$ .  $^{\text{a}}$ Designates $St$ values that used a direct measurement of $a^{\prime }$ in place of $a$ .

4. Propulsive efficiency results

Before introducing flexibility, we first consider the rigid airfoil control case. Scaling arguments based on added mass (e.g. Dewey et al. Reference Dewey, Boschitch, Moored, Stone and Smits2013 and Quinn et al. Reference Quinn, Moored, Dewey and Smits2014b ) suggest that for low-amplitude rigid airfoils swimming at high Reynolds numbers and moderate Strouhal numbers, the time-averaged net thrust $\bar{{\it\tau}}$ and power $\bar{{\wp}}$ scale according to

(4.1a,b ) $$\begin{eqnarray}\bar{{\it\tau}}\sim {\it\rho}sc\,f^{2}a^{2};\quad \bar{{\wp}}\sim {\it\rho}sc\,f^{2}a^{2}u.\end{eqnarray}$$

These scalings demonstrate why ${\it\eta}$ ( $\equiv \bar{{\it\tau}}u/\bar{{\wp}}$ ) is $O(1)$ under typical swimming conditions, but to explain the variation in the efficiency ${\it\eta}$ , one must turn to potential flow models that include a wake. Such models reveal that efficiency tapers off with increasing $St$ and drops off sharply when $St\ll 1$ (Lighthill Reference Lighthill1975). In a real flow, viscous drag also causes ${\it\eta}$ to become negative below some threshold Strouhal number. In practice, these trends cause a local maximum in efficiency to occur at $St\approx 0.2{-}0.4$ for typical swimmers (Taylor, Nudds & Thomas Reference Taylor, Nudds and Thomas2003), although the range may be wider (Eloy Reference Eloy2012). These tendencies have been observed in a variety of experimental and numerical studies (see e.g. Triantafyllou, Triantafyllou & Yue (Reference Triantafyllou, Triantafyllou and Yue2000) for a review), and were reproduced as means of a control case for the present study (see figure 2).

Figure 2.

Propulsive efficiency as a function of Strouhal number for a pitching and heaving rigid NACA0012 airfoil. $a^{\ast }=0.19$ , ${\it\phi}=90^{\circ }$ .

4.1. Heave-only motions

We now compare the rigid case with the subspace of heave-only motions ( ${\it\alpha}_{0}=0$ ) for our flexible propulsor. With the additional time scale provided by flexibility, both $St$ and $f^{\ast }$ play a role in determining efficiency. This effect is illustrated in figure 3 by contour plots of efficiency over a variety of $St$ and $f^{\ast }$ values. Moving from low to high $St$ at a fixed value of $f^{\ast }$ shows the same sharp rise and slow taper observed in figure 2. Varying $f^{\ast }$ , however, reveals a multimodal response in efficiency as the fluid–panel system passes through resonant modes. The peaks are known to correspond to resonance because the same propulsor was previously shown to exhibit local maxima in trailing-edge amplitude at these particular frequencies (Quinn et al. Reference Quinn, Lauder and Smits2014a ). The global maxima in efficiency occur when the optimum Strouhal number aligns with fluid–structure resonance.

Figure 3.

Contour plots of propulsive efficiency for heave-only motions: (a) $a^{\ast }=0.03$ ; (b) $a^{\ast }=0.05$ ; (c) $a^{\ast }=0.07$ ; (d) $a^{\ast }=0.09$ . Labelled circles show conditions at which PIV was conducted (see § 5).

The efficiency also increases with amplitude, presumably because the relative contribution of viscous drag decreases. Alternatively, the same effect can be explained with actuator disc theory (Anderson et al. Reference Anderson, Streitlien, Barrett and Triantafyllou1998). High amplitudes must lead to a loss of efficiency, but no such effect was observed for the small amplitudes studied here. The global optimum over all heave-only conditions was found to be ${\it\eta}=0.23$ at $St=0.53$ , $f^{\ast }=57$ , and $a^{\ast }=0.09$ . However, within experimental error, this condition was equally efficient as the peak in the other mode where ${\it\eta}=0.21$ at $St=0.40$ , $f^{\ast }=27$ , and $a^{\ast }=0.09$ . We will refer to these two heave-only optima as $\mathscr{A}_{\mathit{3D}}$ and $\mathscr{B}_{\mathit{3D}}$ to signify that the optimization took place over three dimensions. The ratio between optimum $f^{\ast }$ values ( $=57/27=2.11$ ) is strikingly similar to the predictions of linear beam theory (see appendix B), strongly suggesting that for our propulsive system added mass plays the most important role in scaling the fluid–structure resonance.

The heave-only power data did not show the $u$ -dependence expected from (4.1), presumably because those predictions are based on prescribed kinematics, whereas our propulsor deforms passively. To try to resolve this issue, we replace $u$ with $c\,\hat{f}_{1}$ , a velocity based on the fundamental frequency. Guaranteeing the appropriateness of this particular replacement would require a separate study of panels with varying size and stiffness, but the replacement is chosen here because it uses the only remaining time scale in the problem. The modified scaling predictions can then be written in dimensionless form as

(4.2) $$\begin{eqnarray}\bar{{\it\tau}}^{\ast }\sim \bar{{\wp}}^{\ast }\sim (f^{\ast }a^{\ast })^{2},\end{eqnarray}$$

where thrust and power have been non-dimensionalized using the bending stiffness $b$ and the first-harmonic frequency $\hat{f}_{1}$ , that is,

(4.3a,b ) $$\begin{eqnarray}\bar{{\it\tau}}^{\ast }\equiv \frac{\bar{{\it\tau}}}{b/c^{2}},\quad \bar{{\wp}}^{\ast }\equiv \frac{\bar{{\wp}}}{b\hat{f}_{1}/c}.\end{eqnarray}$$

For heave-only motions, figure 4 demonstrates that the proposed scaling gives the right trend for the power over a wide range of frequencies and amplitudes. The thrust, however, increases more slowly than predicted by the trend at low Strouhal numbers. At higher Strouhal numbers, added-mass forces dominate over viscous forces, and thrust approaches the slope predicted by the scaling. The bumps on the line to which thrust is converging are evidence of resonances, which lead to the local maxima in efficiency (figure 3). The power shows a better collapse than thrust because it scales with lateral forces at the leading edge, which are an order of magnitude higher than the viscous drag. Together, figures 3 and 4 demonstrate that efficiency is multimodal because of its sensitivity to slight changes in thrust at resonance, but that net thrust and power still scale predictably in the larger scope of the parameter space.

Figure 4.

Dimensionless performance coefficients for heave-only motions. Log–log scale. (a) Dimensionless net thrust ( $\bar{{\it\tau}}^{\ast }$ ); (b) dimensionless net power ( $\bar{{\wp}}^{\ast }$ ). Bold circles correspond to conditions where $St\geqslant 0.6$ .

4.2. Combined heave and pitch motions

We now consider leading-edge motions that include pitch, that is, where ${\it\alpha}_{0}\neq 0$ . To begin, the gradient-based optimization scheme was used to find the maximum efficiency conditions. Figure 5(a) shows the algorithm validated in two dimensions for the heave-only case. Note that $u/(c\,\hat{f}_{1})$ was used in place of $St$ for the optimization – this substitution was chosen to decrease the runtime of the algorithm by limiting changes in flow speed between steps. The substitution does not affect the dimensionality of the parameter space, and we will convert back to $St$ values in subsequent analyses. Figure 5(b) shows a 2D projection of the trajectories through the 5D state space as the algorithm converges on optimum conditions. The particular projection was chosen using multi-dimensional scaling (MDS). In this technique, the ${\it\Pi}_{1}{-}{\it\Pi}_{2}$ plane is chosen such that the L2 norm in the plane is most similar to the L2 norm in the 5D parameter space. In this way, distances in the ${\it\Pi}_{1}{-}{\it\Pi}_{2}$ plane are representative of distances in state space (see e.g. Borg & Groenen (Reference Borg and Groenen2005) for a full description of MDS).

The MDS projection illustrates that the optimization converges on two clusters in state space, and we will refer to the averaged conditions of these clusters as $\mathscr{A}_{\mathit{5D}}$ and $\mathscr{B}_{\mathit{5D}}$ . Table 2 shows the conditions at each optimum, as well as the conditions at the heave-only optima for comparison. The fact that the $f^{\ast }$ values are relatively unchanged is evidence that these newly discovered optima are the same resonant modes as those observed in the heave-only case. This observation is consistent with linear beam theory in that the boundary conditions for the perturbation component of the solution have not changed (one end clamped, one end free), and so the eigenfrequencies should be the same in both cases. The major changes between the two cases are: (i) the heave-only optima have higher $St$ values, meaning the heave-only propulsors need to heave faster to achieve the same zero net thrust flow speed; (ii) an equilibrium amplitude was reached in the trials with pitch ( $a^{\ast }=0.07$ ), suggesting that dynamic stall was a limiting factor in the 5D optimization; and (iii) introducing pitch nearly doubles the propulsive efficiencies. The errors reported are the standard deviations in the clusters of optimal design variables to which the procedure converged. These errors characterize the reproducibility of the procedure and the sensitivity to initial conditions.

Figure 5.

(a) Four gradient-based optimization routines demonstrated on a 2D subset of the parameter space. $a^{\ast }=0.05$ . Trajectories are superimposed on a contour plot of propulsive efficiency. Efficiency scaling as in figure 3. (b) A 2D projection of the 5D trajectories chosen by 15 gradient-based optimization routines; the ${\it\Pi}_{1}{-}{\it\Pi}_{2}$ plane was chosen via multi-dimensional scaling. Outlined circles denote the endpoints of the trajectories. The two double-outlined circles show projections of the heave-only optima ( $\mathscr{A}_{\mathit{3D}}$ and $\mathscr{B}_{\mathit{3D}}$ ) for reference.

Table 2.

Input parameters and efficiencies at optimum conditions for heave-only motions ( $\mathscr{A}_{\mathit{3D}}$ and $\mathscr{B}_{\mathit{3D}}$ ) and heave/pitch motions ( $\mathscr{A}_{\mathit{5D}}$ and $\mathscr{B}_{\mathit{5D}}$ ).

To explore the effects of phase offset, we consider two projections onto the ${\it\phi}{-}{\it\alpha}_{0}$ plane in the vicinity of $\mathscr{A}_{\mathit{5D}}$ and $\mathscr{B}_{\mathit{5D}}$ . Figure 6 shows the projections for net thrust and efficiency, and it demonstrates that introducing pitch is not always advantageous. For some values of ${\it\phi}$ , introducing pitch can decrease thrust and efficiency, and can even push the propulsor into a drag-producing regime.

An important parameter when interpreting these plots is the effective angle of attack of the leading edge,

(4.4) $$\begin{eqnarray}{\it\alpha}_{\mathit{eff}}={\it\alpha}+\text{arctan}\frac{{\dot{h}}}{u},\end{eqnarray}$$

which includes the geometric angle of attack and the relative angle of attack caused by motion perpendicular to the incoming flow. The effective angle of attack is minimized when ${\it\phi}=90^{\circ }$ and maximized when ${\it\phi}=270^{\circ }$ , and in general lower values have been associated with higher efficiencies (Kang et al. Reference Kang, Aono, Baik, Bernal and Shyy2013). The trailing-edge amplitude, however, is maximized near ${\it\phi}=0^{\circ }$ and minimized near ${\it\phi}=180^{\circ }$ , and the higher trailing-edge amplitudes at low values of ${\it\phi}$ can increase thrust and therefore shift the phase for optimum efficiency to values less than $90^{\circ }$ . This effect has been observed for rigid airfoils (Anderson et al. Reference Anderson, Streitlien, Barrett and Triantafyllou1998) and is consistent with the thrusts and efficiencies found near $\mathscr{A}_{\mathit{5D}}$ (figure 6 a,b). The phase at $\mathscr{B}_{\mathit{5D}}$ , however, is indistinguishable from $90^{\circ }$ within the experimental error. Conditions near $\mathscr{B}_{\mathit{5D}}$ have higher ${\it\alpha}_{\mathit{eff}}$ values than those near $\mathscr{A}_{\mathit{5D}}$ , and it appears that this feature makes minimizing ${\it\alpha}_{\mathit{eff}}$ more important than maximizing trailing-edge amplitude near $\mathscr{B}_{\mathit{5D}}$ .

Figure 6.

Contour plots of dimensionless net thrust and efficiency in the ${\it\alpha}_{0}{-}{\it\phi}$ plane: (a) net thrust near $\mathscr{A}_{\mathit{5D}}$ ; (b) efficiency near $\mathscr{A}_{\mathit{5D}}$ ; (c) net thrust near $\mathscr{B}_{\mathit{5D}}$ ; (d) efficiency near $\mathscr{B}_{\mathit{5D}}$ . Labelled circles show conditions at which PIV was conducted (see § 5).

5. Flow field results

To investigate the origins of the efficiency maxima, we turn to PIV images of the flow surrounding the panel. All of the images shown are phase-averaged snapshots of vorticity at the moment when the leading edge passes the $y$ -axis headed toward negative $y$ -values since this point best distinguishes the various cases. Figure 7 shows the flow field for optimum conditions $\mathscr{A}_{\mathit{3D}}$ and $\mathscr{A}_{\mathit{5D}}$ . Both flows are characterized by a thin layer of vorticity along the panel and a 2S reverse von Kármán street in the wake. The efficiency measurements, however, showed that $\mathscr{A}_{\mathit{5D}}$ demonstrates nearly double the efficiency of $\mathscr{A}_{\mathit{3D}}$ . A primary difference between the two cases is the presence of a leading-edge vortex in $\mathscr{A}_{\mathit{3D}}$ (figure 7 a). It appears that by decreasing the effective angle of attack, the non-zero pitch angle reduces leading-edge separation. Introducing pitch also leads to a lower Strouhal number at optimum conditions ( $St=0.26$ compared to 0.40), which is known to be in the range of Strouhal numbers that optimize wake stability (Triantafyllou, Triantafyllou & Grosenbaugh Reference Triantafyllou, Triantafyllou and Grosenbaugh1993). This feature is reflected by the greater coherence of the vortex street for $\mathscr{A}_{\mathit{5D}}$ (figure 7 b).

Figure 7.

Phase-averaged density plots of vorticity: (a) first optimum condition in the heave-only subspace ( $\mathscr{A}_{\mathit{3D}}$ ),  $St=0.40$ , $f^{\ast }=27.0$ , $a^{\ast }=0.09$ ; (b) optimum condition in the full parameter space ( $\mathscr{A}_{\mathit{5D}}$ ),  $St=0.26$ , $f^{\ast }=24.8$ , $a^{\ast }=0.07$ , ${\it\alpha}=30^{\circ }$ , ${\it\phi}=76^{\circ }$ . Labels correspond to labelled circles in figures 3 and 6.

To examine the effect of Strouhal number more closely, we consider perturbations from an optimum condition, this time from $\mathscr{B}_{\mathit{3D}}$ . Figure 8 shows the flow field at $\mathscr{B}_{\mathit{3D}}$ as well as at non-optimal values of $St$ and $f^{\ast }$ . Strouhal numbers less than and greater than optimal will be referred to as $St^{-}$ and $St^{+}$ , respectively, and other parameters will follow the same convention. When $St$ is lower than optimal ( $St^{-}$ , figure 8 b), the wave along the body is too slow to keep pace with the incoming flow, and a large vortex detaches from the body just downstream of the first peak in lateral position. At $St^{+}$ , no separation occurs along the body, but the vortices in the wake break down farther upstream. The thrust and power data confirm that these conditions are associated with lower efficiencies.

Figure 8.

Phase-averaged density plots of vorticity: (a) second optimum condition in the heave-only subspace ( $\mathscr{B}_{\mathit{3D}}$ ),  $St=0.53$ , $f^{\ast }=56.8$ , $a^{\ast }=0.09$ ; (b) non-optimum $St$ conditions near $\mathscr{B}_{\mathit{3D}}$ , $St=0.39$ for $St^{-}$ and $St=0.86$ for $St^{+}$ ; (c) non-optimum $f^{\ast }$ conditions near $\mathscr{B}_{\mathit{3D}}$ , $f^{\ast }=44.6$ for ${f^{\ast }}^{-}$ and $f^{\ast }=61.9$ for ${f^{\ast }}^{+}$ . All other variables were held at their optimum values. Labels correspond to labelled circles in figure 3. Vorticity scaling as in figure 7.

Variations in $f^{\ast }$ lead to less significant differences in the flow field, as seen in figure 8(c). The ${f^{\ast }}^{-}$ case shows the beginnings of a transition to a 2P wake, though this may be due more to the lower flow speed required to keep $St$ constant when decreasing $f^{\ast }$ . The more significant difference is believed to be the lower trailing-edge amplitudes observed at non-optimal $f^{\ast }$ values. Compared with $a^{\prime }/a=1.68$ at $\mathscr{B}_{\mathit{3D}}$ , the low and high $f^{\ast }$ conditions give $a^{\prime }/a=1.66$ and 1.52, respectively. At the same $f^{\ast }$ values but with the flow speed fixed at its respective optimal value, the decreases were more pronounced, with $a^{\prime }/a=1.63$ and 1.51. An error analysis of the edge tracking technique suggests that $\pm 0.02$ is a proper confidence interval for $a^{\prime }/a$ , so these decreases are significant. The zero in $\partial a^{\prime }/\partial f^{\ast }$ at optimal conditions confirms that $f^{\ast }$ is associated with resonance, and that the efficiency is locally maximized at resonant frequencies of the fluid–structure system.

Since the efficiency increases with $a^{\ast }$ through the full space of $a^{\ast }$ values ( $0.03{-}0.09$ ), it may be more revealing to consider the effects of varying $a^{\ast }$ around $\mathscr{B}_{\mathit{5D}}$ , where the optimum $a^{\ast }$ value was 0.07. Figure 9 shows the flow field at non-optimal values of $a^{\ast }$ , as well as at non-optimal values of ${\it\phi}$ and ${\it\alpha}$ . For the $a^{\ast }$ perturbations, flow speed was kept constant instead of $St$ , because higher flow speeds reduce the amplitude along the body and interfere with conclusions about amplitude effects. In the case of ${a^{\ast }}^{+}$ , amplitudes higher than optimal lead to detached flow along the body. Presumably the lateral pressure gradients become sufficiently adverse for the boundary layer to separate from the panel. The ${a^{\ast }}^{-}$ case shows no such separation, while the optimal case shows the beginnings of separation. The efficiency data indicated that in this low-amplitude regime, higher amplitudes are associated with higher efficiencies. It appears, therefore, that the efficiency is optimized when the amplitude is as high as possible but low enough for the flow to stay attached along the body.

Figure 9.

Phase-averaged density plots of vorticity: (a) second optimum in the full parameter space ( $\mathscr{B}_{\mathit{5D}}$ ),  $St=0.33$ , $f^{\ast }=50.8$ , $a^{\ast }=0.07$ , ${\it\alpha}=30^{\circ }$ , ${\it\phi}=96^{\circ }$ ; (b) non-optimum $a^{\ast }$ conditions near $\mathscr{B}_{\mathit{5D}}$ , $a^{\ast }=0.05$ for ${a^{\ast }}^{-}$ and $a^{\ast }=0.09$ for ${a^{\ast }}^{+}$ ; (c) non-optimum ${\it\phi}$ conditions near $\mathscr{B}_{\mathit{5D}}$ , ${\it\phi}=41^{\circ }$ for ${\it\phi}^{-}$ and ${\it\phi}=141^{\circ }$ for ${\it\phi}^{+}$ ; (d) non-optimum ${\it\alpha}$ conditions near $\mathscr{B}_{\mathit{5D}}$ , ${\it\alpha}=15^{\circ }$ for ${\it\alpha}^{-}$ and ${\it\alpha}=0^{\circ }$ for ${\it\alpha}^{--}$ . Labels correspond to labelled circles in figure 6. Vorticity scaling as in figure 7.

Changing ${\it\phi}$ has severe consequences for the surrounding flow field (figure 9 c). In the ${\it\phi}^{-}$ condition, the effective angle of attack is approximately twice what it is in the optimum condition, and the resulting periodic separation is evident to the side of the panel. In addition, heave and pitch are working against each other such that $a^{\prime }/a$ drops to 0.99 from 1.55 at $\mathscr{B}_{\mathit{5D}}$ . The result is a severe drop-off in efficiency going counterclockwise from $90^{\circ }$ in figure 6(d). In the ${\it\phi}^{+}$ condition, heave and pitch are working constructively, and $a^{\prime }/a$ increases to 2.01. This high amplitude helps to explain the large thrust seen at low values of ${\it\phi}$ in figure 6(c). As figure 6(d) shows, however, the higher effective angles of attack still cause lower efficiencies as ${\it\phi}$ increases beyond $90^{\circ }$ , albeit at a slower rate than when ${\it\phi}$ is decreased. The result is that the optimum ${\it\phi}$ remains at $90^{\circ }$ .

Finally, we consider variations in maximum pitch angle ${\it\alpha}_{0}$ (figure 9 d). Since the pitch angle converged to the boundary of the range considered ( $30^{\circ }$ ), we use $15^{\circ }$ ( ${\it\alpha}^{-}$ ) and $0^{\circ }$ ( ${\it\alpha}^{--}$ ) as perturbed conditions. Both cases lead to increased separation along the panel, which is consistent with the fact that the effective angle of attack is increasing. Because ${\it\alpha}^{--}$ is a heave-only case, it can also be referenced on the contour plots in figure 3. From this figure, we understand that ${\it\alpha}^{--}$ corresponds to a heave-only motion with a suboptimal Strouhal number (note the similarity with the $St^{-}$ case in figure 8 b). With this in mind, it would be misleading to say lower values of ${\it\alpha}_{0}$ necessarily lead to separated flow along the body. It is more precise to say that under these conditions, maintaining attached flow while decreasing ${\it\alpha}_{0}$ requires a higher Strouhal number, and this higher Strouhal number leads to efficiencies that are globally non-optimal.