3.1 Propulsive performance

We performed computational experiments on five panels that differed in bending moduli and were actuated at a range of frequencies. In each of these simulations, the shape of the untethered panel and its swimming progression emerge from the full coupled system. For each of the five panels, figure 4(a) shows the normalized trailing edge amplitude achieved by the panel as a function of the effective flexibility as it is actuated at different frequencies. We see that peaks in trailing edge amplitude occur near certain effective flexibilities, and these basically coincide for each of the panels. Moreover, the trailing edge amplitude varied little for panels actuated with similar effective flexibilities even when their bending moduli differed. Following the classic definition (Den Hartog Reference Den Hartog1985) and more recently that by Quinn et al. (Reference Quinn, Lauder and Smits2014b ), we refer to the system as operating in resonance when the trailing edge amplitude reaches a local maximum at certain values of the effective flexibility $\unicode[STIX]{x1D6F1}$ (or, equivalently, the oscillation frequency $\unicode[STIX]{x1D719}$ ). Figure 4(a) shows that there are at least four such resonant peaks in the range of effective flexibilities that we studied.

The peaks in trailing edge amplitude correspond to peaks in average swimming speed. Figure 4(b) shows the average (dimensional) swimming speed $V_{avg}$ as a function of effective flexibility. Although the peaks in $V_{avg}$ and trailing edge amplitude occur at similar effective flexibilities, the resulting swimming speeds depend on the bending moduli of the panel. Stiffer panels had higher $V_{avg}$ compared to the more flexible panels heaved at similar effective flexibilities. Of course, different material panels must be actuated at different frequencies in order to achieve the same effective flexibility. Figure 4(c) shows the swimming speed divided by $c\unicode[STIX]{x1D719}$ , which corresponds to body lengths per heaving cycle ( $\bar{V}_{avg}$ ). We see that this non-dimensional velocity is better matched for panels actuated at similar effective flexibilities, but the most flexible panels still move more slowly.

In figure 5(a), we plot the average input power $\bar{P}_{avg}$ , and we note that local peaks of input power coincide with local peaks in velocity (figure 4 b) – the highest swimming speeds require the most power input. Figure 5(b) plots the swimming economy as a function of effective flexibility. While the peaks of average velocity and power occur at the same value of effective flexibility, the swimming economy, which is the ratio of these values, need not achieve a local maximum there. In fact, we find that the peak in economy occurs at slightly higher values of $\unicode[STIX]{x1D6F1}$ because the amplitude tapers off more slowly than power as $\unicode[STIX]{x1D6F1}$ increases. Similar shifts in efficiency peaks were noted in the model of Michelin & Llewellyn Smith (Reference Michelin and Llewellyn Smith2009). We also see that input power is higher for stiffer panels than more flexible panels near the same effective flexibility. This in turn leads stiffer panels to have peaks in swimming economy at lower effective flexibilities than more flexible panels and to have a lower swimming economy overall compared to more flexible panels, even at the same effective flexibility. Note that to achieve a fixed effective flexibility, a stiff panel must be heaved at a higher frequency than a more flexible panel. This would increase the power required to complete the heaving motion. However, stiffer panels have a higher inverse Strouhal number than more flexible ones figure 5(c). While each tailbeat results in greater forward motion when compared to the tailbeat of a more flexible panel, more energy is expended for the motion. We plot inverse Strouhal numbers rather than Strouhal numbers, because at very low swimming speeds, the Strouhal number approaches infinity. The stiffest panels actuated so that their effective flexibilities are near the first two resonant peaks reach Strouhal numbers in the range associated with peak propulsive efficiency, $0.2<St<0.4$ (Taylor et al. Reference Taylor, Nudds and Thomas2003).

3.2 Modal contributions

By choosing a cross-section down the middle of the panel and averaging over its thickness, we can represent its emergent, time-periodic oscillations by a one-dimensional curve. This curve can be decomposed into the eigenfunctions of the beam equation as discussed in § 2.3. For each simulation, the first five modal contributions of the heaving panel ( $\unicode[STIX]{x1D6E9}_{i}(t)$ , $i=1,2,3,4,5$ ) were calculated at every time step. Figure 6(a) shows the time-dependent modal contributions of a representative panel simulation ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.5~\text{s}^{-1}$ ) as it moves through eight heaving cycles. Each $\unicode[STIX]{x1D6E9}_{i}$ represents the weight of the eigenfunction, $\unicode[STIX]{x1D6F9}_{i}$ , with respect to the deflection of the panel. Following the initial heaving cycles, all of the panel’s modal contributions tend toward steady cycles. Although each of the modal contributions have the same frequency as the heaving frequency, they differ in both amplitude and phase.

Figure 6.

(a) Plot of the modal contribution of the first five modes with respect to the heaving cycles of a panel ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.5~\text{s}^{-1}$ ). Starting from rest initially, the modal contributions of the panel deflection quickly follow a sinusoidal pattern, with each modal contribution varying in amplitude and peak point in the phase. (b) Plots of the modal contributions of the first (red) and second (magenta) mode for three different panels. (c) Plot of the maximum modal contribution of each mode with respect to the heaving panels’ effective flexibility. Higher mode contributions increase as effective flexibility increases.

The amplitude and phase of each mode depends on the effective flexibility. Figure 6(b) shows the shape of the panel when approximated only by the contribution of the first mode, $\unicode[STIX]{x1D6E9}_{1}$ (in red), over the phase $\unicode[STIX]{x1D709}\in [0,1)$ , for three panels of differing effective flexibilities ( $EI=1\times 10^{-6}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.125~\text{s}^{-1}$ ; $\unicode[STIX]{x1D6F1}=0.3827$ , $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.5~\text{s}^{-1}$ ; $\unicode[STIX]{x1D6F1}=4.8412$ and $EI=1\times 10^{-8}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.5~\text{s}^{-1}$ ; $\unicode[STIX]{x1D6F1}=45.9279$ ). For the two most flexible panels, $\unicode[STIX]{x1D6E9}_{1}$ has a similar amplitude and phase, while that for the third one is shifted and has a larger amplitude. Figure 6(b) also shows the contribution of the second mode to the panel’s shape, $\unicode[STIX]{x1D6E9}_{2}$ (in magenta). We see that the similarities between the more flexible panels are absent in the second mode, with significant differences in amplitude and phase in $\unicode[STIX]{x1D6E9}_{2}$ for all three panels.

As the effective flexibility increases, each of the higher-order modes emerge sequentially, and each new mode contributes to a resonant peak in amplitude and performance. Figure 6(c) shows the maximum contribution of each mode, $\max _{\unicode[STIX]{x1D709}}(\unicode[STIX]{x1D6E9}_{i}(\unicode[STIX]{x1D709}))$ , for all panels. At lower effective flexibilities, the modal contribution from the first mode dominates, indicating that the higher modes contribute little to the shape of the panel. As effective flexibility increases, the modal contributions of higher modes emerge sequentially. Their emergence is followed by a peak in the maximum modal contribution of that mode. The effective flexibilities at which a peak modal contribution occurs coincide with the effective flexibilities where peak trailing edge amplitudes and velocities occur (compare to figure 4).

3.3 Phase evolution

Not only are the amplitudes of the modal contributions similar when the effective flexibility is the same, but the evolution of the mode over the cycle is also similar. In figure 6(b), the maximum modal contributions of the first mode of two panels ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.5~\text{s}^{-1}$ and $EI=1\times 10^{-8}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.5~\text{s}^{-1}$ ) occurred at similar phase, while the maximum modal contributions for the third panel ( $EI=1\times 10^{-6}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.125~\text{s}^{-1}$ ) occurs at a different phase. To quantify the similarity or difference in the evolution of the mode shapes for different panels, we used (2.17) to compute a normalized correlation $Q^{pq}$ between panels $p$ and $q$ . This value is high when the phase evolution of the $i$ th mode is correlated in the phase between the two panels and has a large amplitude.

Figure 7.

$Q$ value from (2.17) associated with two panels, ordered by the two panels’ effective flexibility. Note that higher $Q$ values indicated that the modes of two panels are correlated and that the amplitude of modal contributions associated with both panels are substantial.

Figure 7 shows the value of $Q$ for all pairs of heaving panels arranged by effective flexibility. The high $Q$ values along the diagonal show that panels with good swimming performance and high amplitude modal contributions have a similar shape evolution over the phase as panels with similar effective flexibilities. These correspond to the peaks in trailing edge amplitude (figure 4 a) and swimming speed (figure 4 c). This suggests that panels with high $\bar{V}_{avg}$ and similar $\unicode[STIX]{x1D6F1}$ also have similar shape evolutions over the phase of the heaving cycle. The exact locations of the resonant peaks vary slightly as a function of the panel’s bending rigidity, accounting for relatively high $Q$ values off the diagonal at $\unicode[STIX]{x1D6F1}\approx 8$ .

Figure 8.

Isocontour plots of the dimensionless $y$ -component of vorticity, $\bar{\unicode[STIX]{x1D714}}_{y}$ , at the start of the first through fourth heaving cycles (from left to right). The panel is of $\unicode[STIX]{x1D6F1}=19.3649$ ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=2.0~\text{s}^{-1}$ ) and $Re_{in}=500$ . Note that after the first heaving cycle, the deflection of the panel is nearly identical at the start of each heaving cycle, although the resulting vorticity field varies in time.

Figure 9.

Isocontour plots, cut out along the centre plane, of the (a) dimensionless $y$ -component of vorticity ( $\bar{\unicode[STIX]{x1D714}}_{y}$ ), (b) dimensionless $x$ -component of flow velocity ( $\bar{u}_{x}$ ) and (c) dimensionless pressure ( $\bar{p}$ ) during the start of the third heaving cycle of a panel with $\unicode[STIX]{x1D6F1}=6.0515$ ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ ).

3.4 Flow patterns

The fluid flow resulting from the panels’ heaving motion is examined via dimensionless vorticity, $\bar{\unicode[STIX]{x1D74E}}$ . Figure 8 shows an example of the development of the flow in the first four heaving cycles. Although the vortex structures present in the wake grow after each heaving cycle, the shape of the panel at the start of the cycle and the vortex structures in the immediate wake remain fairly constant following the initial heaving cycle.

Figure 10.

Isocontour plots of the dimensionless $y$ -component of vorticity, $\bar{\unicode[STIX]{x1D714}}_{y}$ , at the start of the third heaving cycle for three panels heaved at the same frequency, $\unicode[STIX]{x1D719}=1.5~\text{s}^{-1}$ , but with $EI$ equal to (a)  $1\times 10^{-6}~\text{N}~\text{m}^{2}$ ( $\unicode[STIX]{x1D6F1}=4.5928$ ), (b)  $1\times 10^{-7}~\text{N}~\text{m}^{2}$ ( $\unicode[STIX]{x1D6F1}=14.5237$ ), (c)  $1\times 10^{-8}~\text{N}~\text{m}^{2}$ ( $\unicode[STIX]{x1D6F1}=45.9279$ ).

Figure 11.

Isocontour plots of the dimensionless $y$ -component of vorticity, $\bar{\unicode[STIX]{x1D714}}_{y}$ , during the heaving cycle for three panels of similar effective flexibility, with $\unicode[STIX]{x1D6F1}$ equal to (a)  $5.7410$ ( $EI=1\times 10^{-6}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.625~\text{s}^{-1}$ , $Re_{in}=406.25$ ), (b)  $5.4127$ ( $EI=5\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.250~\text{s}^{-1}$ , $Re_{in}=312.5$ ), (c)  $6.0515$ ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ , $Re_{in}=156.25$ ). Snapshots are taken at $\bar{t}$ (from top to down) 3.0, 3.25, 3.50 and 3.75.

Figure 12.

Isocontour plots of the dimensionless $y$ -component of vorticity, $\bar{\unicode[STIX]{x1D714}}_{y}$ , at the start of the third heaving cycle for three panels that swim rapidly but have different effective flexibilities, $\unicode[STIX]{x1D6F1}$ equal to (a)  $0.5413$ ( $EI=5\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.125~\text{s}^{-1}$ , $Re_{in}=31.25$ ), (b)  $6.0515$ ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ , $Re_{in}=156.25$ ), (c)  $17.116$ ( $EI=5\times 10^{-8}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.250~\text{s}^{-1}$ , $Re_{in}=312.5$ ).

Figure 13.

Plots of the dimensionless $y$ -component of vorticity along the central plane at the end of the fourth heaving cycle for panels with $\unicode[STIX]{x1D6F1}$ equal to (a) 0.5143 ( $EI=5\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.125~\text{s}^{-1}$ ), (b) 6.0515 ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ ) and (c) 17.116 ( $EI=5\times 10^{-8}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.25~\text{s}^{-1}$ ). The three panels are representative of the panels with $\unicode[STIX]{x1D6F1}$ values associated with the first three resonant peaks. In (a) we note the presence of a 2S vortex shedding pattern, while in (b,c) the 2P vortex shedding pattern is present.

Figure 14.

Plot of the dimensionless vorticity magnitude, $|\bar{\unicode[STIX]{x1D74E}}|$ , near the point of peak panel swimming speed for a panel with $\unicode[STIX]{x1D6F1}=5.7410$ ( $EI=1\times 10^{-6}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.625~\text{s}^{-1}$ , $Re_{in}=312.5$ ). The tip vortices and trailing edge vortex are highlighted.

Figure 15.

Isocontour plots of the dimensionless $z$ -component of vorticity, $\bar{\unicode[STIX]{x1D714}}_{z}$ , near the point of peak panel swimming speed during the heaving cycle for three panels with similar effective flexibilities, $\unicode[STIX]{x1D6F1}$ equal to (a)  $5.7410$ ( $EI=1\times 10^{-6}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.625~\text{s}^{-1}$ , $Re_{in}=406.25$ ), (b)  $5.4127$ ( $EI=5\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=1.25~\text{s}^{-1}$ , $Re_{in}=312.5$ ), (c)  $6.0515$ ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ , $Re_{in}=156.25$ ).

Figure 16.

(a) Diagram of the planes from which the circulation of the trailing vortex (blue) and the tip vortices (red) were calculated. (b) Plot of the circulation, $\bar{\unicode[STIX]{x1D6E4}}$ , of the trailing edge vortices (blue) and the tip vortices (red) for the panels of figures 11 and 15 as a function of $Re$ .

In figure 9, we compare the isocontours along the centre plane for the spanwise ( $y$ ) vorticity ( $\bar{\unicode[STIX]{x1D714}}_{y}$ ), axial ( $x$ ) velocity ( $\bar{u}_{x}$ ) and pressure ( $\bar{p}$ ) at the start of the third heaving cycle for a panel with a substantial swimming speed ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ ). The trailing edge generates alternating vortex structures (figure 9 a). These vortices then generate a region of positive axial flow ( $\bar{u}_{x}$ ) near the trailing edge of the panel (grey region in figure 9 b). This zone of positive axial velocity corresponds to the presence of positive pressure near the trailing edge of the panel and negative pressure on the opposite side (figure 9 c), which combine to give the thrust force. We also find zones of positive and negative pressure on opposite sides of the panel due to the deflections of the panel.

Differences in the panel shape evolution yield differences in the resulting flow structures. Figure 10 compares the $\bar{\unicode[STIX]{x1D714}}_{y}$ isocontours at the start of the third heaving cycle for three panels, with differing passive material properties and identical heaving frequencies. The differences in the observed flow structures in turn account for differences in panel performance. Similar observations can be made when examining $\bar{\unicode[STIX]{x1D714}}_{y}$ isocontours of three panels with identical passive material properties but differing heaving frequencies. Movies of both comparisons can be found in the supplementary material available at https://doi.org/10.1017/jfm.2018.305.

Above we have shown that panels with similar effective flexibility $\unicode[STIX]{x1D6F1}$ have similar performance. This is because they produce similar flow patterns. Figure 11 shows phase-matched snapshots of the isocontours of $\bar{\unicode[STIX]{x1D714}}_{y}$ for three panels with similar effective flexibilities ( $\unicode[STIX]{x1D6F1}\approx 6$ ). Although of similar effective flexibility, the three panels all differ with respect to their heaving frequency and bending modulus. Because the deflections of the panels are similar throughout the phase of the heaving cycle, the dominant vortex structures found in the wake of each of the three panels are also similar. Minor variations in vortex structures of the wakes of the three panels correspond to differences in $Re_{in}$ , with the presence of more vortex structures in the higher $Re_{in}$ case (figure 11 a). Movies of these comparisons, as well as a comparison of other panels of similar effective flexibilities and high dimensionless swimming speeds, can be found in the supplementary material.

The structure of the wake changes for panels at values of $\unicode[STIX]{x1D6F1}$ that correspond to peaks in swimming speed. Figure 12 plots the $\bar{\unicode[STIX]{x1D714}}_{y}$ isocontours and figure 13 shows slices through the central plane, each for three swimmers with $\unicode[STIX]{x1D6F1}$ values associated with three different peaks in $\bar{V}_{avg}$ . The three panels each have high dimensionless forward swimming speeds but substantially differing deflections. The more flexible panels (figure 13 b,c) shed two pairs of vortices per cycle (a 2P wake structure; Williamson & Roshko Reference Williamson and Roshko1988), while the stiffer panel (figure 13 a) sheds two single vortices per cycle (a 2S wake).

To further explore how fluid effects allow for differences in swimming performance even when panels are of a similar $\unicode[STIX]{x1D6F1}$ value, we examined the differences in circulation for the panels of figure 11 ( $\unicode[STIX]{x1D6F1}\approx 6$ ). We found that the panels reached peak swimming speeds near phase $\unicode[STIX]{x1D709}=0.25$ during the heaving cycle. We also examined $|\bar{\unicode[STIX]{x1D74E}}|$ (figure 14) and found the presence of a vortex column at the trailing edge of the panel as well as tip vortices near the halfway point of the panels’ chordwise dimension.

The presence and sign of $\bar{\unicode[STIX]{x1D714}}_{z}$ in the tip vortices (figure 15) indicate a source of additional drag on the panel as the flow generated by $\bar{\unicode[STIX]{x1D714}}_{z}$ is against the swimming direction of the panel. We computed the circulation (figure 16 a) around a contour enclosing the trailing edge vortex and lying on the centre plane $y=0$ as well as the circulation of the tip vortices around a contour lying on the $xy$ -plane enclosing the tip vortices. Plotting $\bar{\unicode[STIX]{x1D6E4}}$ of the trailing edge vortex as a function of $Re$ at $\unicode[STIX]{x1D709}=0.25$ (figure 16 b), we found that this circulation was approximately equal for the three panels. However, the circulation of the tip vortices was larger for panels with larger $Re$ (figure 16 b). This suggests that the tip vortices play a role in the observed differences in the swimming economy, where stiffer panels that move faster (higher $Re$ ) are less efficient than slower, more flexible panels of a similar $\unicode[STIX]{x1D6F1}$ .

3.5 Spanwise variation

In order to gain insight into the effects of tip vortices on swimming performance, we chose a reference panel and varied its span length. From (2.9) and (2.14), we note that $\unicode[STIX]{x1D6F1}$ is not dependent on the span of the panel,

(3.1) $$\begin{eqnarray}\unicode[STIX]{x1D6F1}=\sqrt{\frac{12\unicode[STIX]{x1D70C}c^{4}\unicode[STIX]{x1D719}^{2}(1-\unicode[STIX]{x1D708}^{2})}{Ew^{3}}}.\end{eqnarray}$$

A high performing panel with a $\unicode[STIX]{x1D6F1}$ value associated with the second mode resonance was chosen as the reference case ( $EI=1\times 10^{-7}~\text{N}~\text{m}^{2}$ , $\unicode[STIX]{x1D719}=0.625~\text{s}^{-1}$ ). We compare the performance of the reference panel with aspect ratio $s/c=0.6$ , and two others of aspect ratios 0.2 and 1.0, each with the same effective flexibility. Additionally, a simulation of a heaving filament in a two-dimensional fluid domain was performed, corresponding to the case free of tip vortices and infinite aspect ratio.

Figure 17.

Plots of (a) the dimensionless swimming speed ( $\bar{V}_{avg}$ ) and (b) the dimensionless circulation of the tip vortices ( $\bar{\unicode[STIX]{x1D6E4}}$ ) for panels of differing aspect ratio. Here an aspect ratio of $\infty$ represents a two-dimensional panel. As the panel aspect ratio increases, the swimming speed increases although the strength of the tip vortices remains relatively constant.

As the span of the panel increases, the swimming performance increases. In figure 17(a) the swimming speed was found be highest for the two-dimensional panel, with swimming speed for the three-dimensional cases approaching this level as the span length increased. Measuring the circulation of the tips for the three-dimensional cases (figure 17 b), we found the circulation measured increases slightly as the span decreases, where the panel with an aspect ratio of 0.2 has an 8 % increase circulation relative to the reference panel. This suggests that the additional drag generated from the tip vortices plays a relatively larger role as the span decreases.

3.6 Comparison between tethered and free-swimming panels

Do the peaks in the swimming speeds of an actuated, untethered panel correspond to the peaks in the thrust generated by the same actuated panel that is tethered and not free to swim? To examine this, we performed a set of simulations of tethered panels at the same heaving frequencies and five bending moduli as in untethered simulations. In this set of simulations, a tethered body force was applied to the leading edge and the panel was heaved until its motion reached a steady state. Figure 18(a) shows the maximum modal contributions of the first five modes computed from the emergent geometries of the tethered panels. We find that the peaks in these modal contributions correspond to the peaks observed for the freely swimming panels shown in figure 6(c). In addition, for these tethered panels the resulting forces ${\mathcal{F}}$ from the heaving motion were recorded and used to compute the resulting non-dimensional forward thrust $\bar{T}$ . This thrust was then averaged over a heaving cycle ( $\bar{T}_{avg}$ ). Figure 18(b) shows the thrust developed by each of the five tethered panels as a function of the effective flexibility. We find that peaks in $\bar{T}_{avg}$ occur near the same values of effective flexibility that displayed peaks in forward swimming speed for the corresponding untethered simulations (see figure 4 c). These peaks in turn correspond to local peaks in the modal contributions.

Figure 18.

(a) Plot of the maximum modal contribution of each mode as a function of the tethered, heaving panels’ effective flexibility. Higher mode contributions increase as effective flexibility increases. (b) Plot of the dimensionless thrust, $\bar{T}_{avg}$ , as a function of the effective flexibility, $\unicode[STIX]{x1D6F1}$ . The effective flexibilities corresponding to the analytic prediction of the first five harmonic response frequencies (Van Eysden & Sader Reference Van Eysden and Sader2007) for the five panels are indicated by symbols on the horizontal axis, from which emanate dashed vertical lines whose colour corresponds to the five modes in panel (a).

We also use the tethered panel framework to validate our simulation results by comparing predictions of peaks in thrust with analytical approximations from a linearized theory. Van Eysden & Sader (Reference Van Eysden and Sader2006, Reference Van Eysden and Sader2007, Reference Van Eysden and Sader2009) developed a theory for the frequency response of a cantilever beam with rectangular cross-section immersed in a viscous fluid. In that study the fundamental frequencies are a function of panel geometry, $EI$ and Reynolds number, and are based on the exact solution of the linearized Navier–Stokes equations produced by a zero-thickness, infinitely long oscillating blade. It is noted in Van Eysden & Sader (Reference Van Eysden and Sader2007) that these approximations to harmonic response frequencies become less accurate as the mode increases due to the heightened three-dimensional aspects of the cantilever–fluid system. As a way of validating our method, we compare the frequencies at which the tethered panels of our simulations achieve local peaks in thrust to the values of harmonic response frequencies corresponding to different modes predicted by Van Eysden & Sader (Reference Van Eysden and Sader2007). For each of our five panels with fixed geometry and different bending moduli $EI$ , we compute the harmonic response frequencies $\unicode[STIX]{x1D714}_{n}$ associated with mode $n$ using the linear theory. For that $\unicode[STIX]{x1D714}_{n}$ and $EI$ , we associate an effective flexibility $\unicode[STIX]{x1D6F1}$ of that panel waved at the harmonic response frequency. These effective flexibilities (for five panels and the first five modes) are indicated on the horizontal axis of figure 18(b), and vertical lines are drawn emanating from these values. For each mode $n$ , the theoretical values of effective flexibility are essentially equal for the different bending moduli. For the smallest values of $\unicode[STIX]{x1D6F1}$ that correspond to stiff panels or panels heaved at low frequencies, restrictive computational costs do not allow simulations that resolve the first mode, so we focus on modes 2–5. However, the effective flexibility corresponding to the theoretical harmonic response frequency of the second mode for $EI=1\times 10^{-7}$ and $5\times 10^{-8}~\text{N}~\text{m}^{2}$ differs from the simulation value of $\unicode[STIX]{x1D6F1}$ at which peaks occur by less than 0.3 %, and for the third mode by less than 10 %. Larger differences are associated with higher modes, as expected. For stiffer panels, $EI=1\times 10^{-6}$ and $5\times 10^{7}~\text{N}~\text{m}^{2}$ , the computed effective flexibilities for mode 2 are approximately 22 % lower than those predicted by the analytic model of Van Eysden & Sader (Reference Van Eysden and Sader2007). The least stiff panel, $EI=1\times 10^{-8}~\text{N}~\text{m}^{2}$ , corresponds to a relative difference of 30 % in effective flexibilities for mode 2. The agreement between the harmonic response frequencies for modes 2 and 3 estimated by two-dimensional linear theory and those computed using the full three-dimensional Navier–Stokes system serves as a validation of the simulations.

Figure 19 show scatter plots of the thrust of a tethered heaving panel and the speed of its untethered counterpart ( $\bar{T}_{avg}$ , $\bar{V}_{avg}$ ), for the five panels. We find that for tethered panels that produce positive thrust, $\bar{T}_{avg}$ , increasing thrust generally corresponds to increasing free-swimming speed $\bar{V}_{avg}$ . To further examine the differences between tethered and untethered panels, we examined the modal contributions of the tethered panels (figure 18 a). We find that the amplitude of modal contributions of the panels have peaks at the same $\unicode[STIX]{x1D6F1}$ value, although the amplitude of the modal contribution varies.

Figure 19.

Plot of the maximum modal contribution of each mode with respect to the heaving panels’ effective flexibility. Higher mode contributions increase as effective flexibility increases.