2 Experimental method

The experimental set-up consists of a rigid NACA 0012 foil of chord length, $c_{R}=10~\text{cm}$ , pitched sinusoidally by a AC servo motor (Panasonic, Minas A4), as shown in figure 2(a). The motor gets its sinusoidal driving signal, $\unicode[STIX]{x1D703}=\unicode[STIX]{x1D703}_{max}\sin 2\unicode[STIX]{x03C0}ft$ , from a Stanford DG340 function generator. The experiments were done in a closed circuit water tunnel whose test section had a cross-section of $0.26~\text{m}\times 0.45~\text{m}$ with a length of 1 m. Flow speeds of up to $30~\text{cm}~\text{s}^{-1}$ could be achieved in the tunnel test section. The foil made of aluminium was accurately machined using a wire-electro discharge machining (EDM) technique and electroplated to avoid pitting by aluminium in water. The foil had a span of 30 cm. To promote two-dimensional flow over the foil, fixed end plates were placed 3 mm away from both the top and bottom ends of the foil.

Figure 2. (a) Schematic of the experimental arrangement showing the pitching foil in a water tunnel. The pitching oscillations are forced through a servo motor, with the load cell being in the rotating reference frame. PIV measurements are taken close to the central plane of the foil. (b) Schematic showing the oscillating pitching foil and the conventions used for positive values of pitching angle ( $\unicode[STIX]{x1D703}$ ), and the measured axial force ( $A$ ), normal force ( $N$ ) and moment ( $M$ ).

The experiments with the flexible trailing edge flap attached to the rigid foil were done by attaching a thin flexible flap of known flexural rigidity ( $EI$ ) and flap length ( $c_{F}$ ) to the trailing edge of the rigid foil. We shall refer to this combination as the flexible foil in this paper. The details of the different flaps including the measured flexural rigidities of the flaps used are given in table 1, these values being per unit span length. The flaps were attached to the rigid foil using thin tapes such that the boundary condition for the flap was close to a cantilever. Care was taken to ensure that the attaching procedure does not result in bumps or ridges. The flexural rigidity ( $EI$ ) of the attached flaps was varied over a large range, as shown in table 1, representing a variation of approximately 4 orders in magnitude. The flexible flap to total chord length ratio ( $c_{F}/c$ ) was varied from approximately 0.3 to 0.7.

The natural frequency of the flexible flap depends on the flexural rigidity ( $EI$ ), length ( $c_{F}$ ) and the inertia of the flap and the surrounding fluid (Dewey et al. Reference Dewey, Boschitsch, Moored, Stone and Smits2013; Quinn et al. Reference Quinn, Lauder and Smits2014; Paraz et al. Reference Paraz, Schouveiler and Eloy2016). The numerical values of natural frequencies of each of the flexible foils used in the present study were determined experimentally and are tabulated in table 1, which gives a complete list of all important flexible foil parameters used in the study. The natural frequencies, both first ( $f_{n1}$ ) and second mode ( $f_{n2}$ ), were determined by forced pitching oscillation experiments in still water, with the natural frequencies corresponding to the local peak in flexible flap tip amplitude. In some flap cases, $f_{n2}$ could not be determined as it was higher than the largest forcing frequencies that could be tested (2.5 Hz) due to the given motor torque limitations. The rigid foil oscillation amplitude for all tests was kept small and fixed by the parameter $2\unicode[STIX]{x1D6FF}_{R}/c_{F}=0.065$ , where $\unicode[STIX]{x1D6FF}_{R}$ is the amplitude of the rigid foil tip excursion. The corresponding mode shapes obtained for a flap in still water are shown in figure 3 in both the laboratory reference frame and the rigid foil reference frame, the latter being more representative of the actual deformation of the flaps (Michelin & Llewellyn Smith Reference Michelin and Llewellyn Smith2009). These two modes were clear with the flexible tip amplitude showing a sharp local peak at mode 1 and a more broadband peak at mode 2, as in Paraz et al. (Reference Paraz, Schouveiler and Eloy2016). It may be noted here that the ratio of second mode to first mode natural frequency ( $f_{n2}/f_{n1}$ ) found in our studies was different from the expected classical value of 6.3, with the difference likely due to the low mass ratios used in our experiments combined with the fact that the oscillating rigid foil in our configuration sets up a significant flow around the flexible flap, even in still water, as shown by Shinde & Arakeri (Reference Shinde and Arakeri2014). Another point to note here is that the first natural frequency ( $f_{n1}$ ) in our experiments scales well with $\sqrt{EI/(\unicode[STIX]{x1D70C}c_{F}^{5})}$ , as expected from simple dimensional analysis and consistent with the expressions in Paraz et al. (Reference Paraz, Schouveiler and Eloy2016) for low mass ratios, with the actual values in our case across different $EI$ and $c_{F}/c$ values being well represented by the expression: $f_{n1}=\unicode[STIX]{x1D6FC}\sqrt{EI/(\unicode[STIX]{x1D70C}c_{F}^{5})}$ , where $\unicode[STIX]{x1D6FC}=0.0211$ and $\unicode[STIX]{x1D70C}$ is the fluid density.

Figure 3. The mode shapes of the flexible flap in still water corresponding to (a) mode 1 and (b) mode 2 in both the laboratory reference frame (top) and the reference frame of the rigid pitching foil (bottom). The data shown correspond to the flexible flap with $EI=5.07\times 10^{-4}~\text{Nm}$ and $c_{F}/c=0.45$ obtained at the first mode natural frequency, $f_{n1}=0.3~\text{Hz}$ , and the second mode natural frequency, $f_{n2}=7.3f_{n1}$ . The rigid foil oscillation amplitude for this test was fixed by the parameter $2\unicode[STIX]{x1D6FF}_{R}/c_{F}=0.065$ , where $\unicode[STIX]{x1D6FF}_{R}$ is the amplitude of the rigid foil tip excursion, with this corresponding to $\unicode[STIX]{x1D703}_{max}=2\,^{\circ }$ for the $c_{F}/c=0.45$ case.

Force measurements were done using a custom-built strain gauge based load cell. The load cell was mounted just above the foil and rotated along with the foil, as shown in figure 2(a). The load cell could hence measure the force normal to the foil ( $N$ ) and the axial force along the chord of the foil ( $A$ ), as illustrated schematically in figure 2(b). The instantaneous angular position of the foil ( $\unicode[STIX]{x1D703}$ ) was measured using a potentiometer, which could then be used to resolve the measured normal ( $N$ ) and chordwise ( $A$ ) forces into the streamwise (thrust/drag) and the cross-stream (lift) components. In the case of flexible foils, the normal and axial directions correspond to the normal and axial directions, respectively, of the rigid part of the foil. The natural frequency of the load cell with the foil in air was 14 Hz, while it was close to 6.5 Hz when the foil was immersed in water. The pitching frequencies used were in the range of 0.5–1.5 Hz, which is reasonably below the natural frequency of the system in water. It should be noted that this natural frequency is related to the stiffness of the load cell, and is different from the natural frequencies of the flexible flap given in table 1, which is related to the flexural rigidity of the flap. For rigid foil experiments, the pitching frequency range of 0.5–1.5 Hz was used along with a flow speed of $U=10~\text{cm}~\text{s}^{-1}$ to cover a reduced frequency range between 1.5 and 5, and another set of experiments was done at $U=5~\text{cm}~\text{s}^{-1}$ using the same frequency range to cover a reduced frequency range between 3 and 10. Data in the overlap region show that effects due to changes in Reynolds number caused due to the change in free-stream velocity between these two sets are not significant. This procedure was adopted to keep the oscillation frequencies ( $f$ ) well below the natural frequency of the load cell system in water (6.5 Hz). Independent measurements, not presented here, showed that at a fixed $k$ , Reynolds number variation between 4000 and 15 000 showed nearly no effects on the thrust generated. Forces due to foil inertia were measured by pitching the foil in air and these were subtracted from the measurements done in water to get the forces acting on the foil only from the fluid. Signals from the load cell were amplified 3000 times using a Vishay 2210 signal conditioning amplifier. These pre-conditioned signals were then acquired at 200 Hz into a PC using a 4-channel NI-cDAQ 9172 data acquisition card. Typically around 150 foil oscillations were acquired and used for analysis.

The measured normal ( $N$ ) and chordwise ( $A$ ) forces are non-dimensionalized as $C_{N}=N/(1/2\unicode[STIX]{x1D70C}U^{2}sc)$ and $C_{A}=A/(1/2\unicode[STIX]{x1D70C}U^{2}sc)$ , respectively, and the moment $M$ , is non-dimensionalized as $C_{M}=M/(1/2\unicode[STIX]{x1D70C}U^{2}sc^{2})$ , where $s$ is the span and $c$ is the total chord of the foil. In the case of flexible foils, the total chord is the sum of the rigid foil chord ( $c_{R}$ ) and the flexible flap length ( $c_{F}$ ). The thrust was calculated from resolving the normal and chordwise force coefficients in the free-stream direction using the position signal, $\unicode[STIX]{x1D703}(t)$ , as $C_{T}=-C_{N}\sin \unicode[STIX]{x1D703}+C_{A}\cos \unicode[STIX]{x1D703}$ . The obtained data were then cycle averaged to get the mean thrust coefficient ( $\overline{C_{T}}$ ). The cycle-averaged input power coefficient ( $\overline{C_{P}}$ ) is defined as $\overline{C_{P}}=\overline{P}/(1/2\unicode[STIX]{x1D70C}U^{3}sc)$ , where the input power ( $\overline{P}=1/\unicode[STIX]{x1D70F}\int M\dot{\unicode[STIX]{x1D703}}\,\text{d}t$ ) was obtained from the measured moment ( $M$ ) ( $\unicode[STIX]{x1D70F}=1/f$ is the time period of oscillation).

Flow field visualizations in the wake were done using PIV. For this, the flow was illuminated using a double pulsed 15 mJ/pulse Nd:YAG laser. Images were captured using a 10 bit Flow-sense PIV camera with a resolution of 1600 $\times$ 1200 pixels. A typical captured flow field area size was 280 mm $\times$ 220 mm. The laser and the camera were synchronised using a Stanford DG 535 digital delay generator. The time interval between the two captured PIV frames was chosen to be between 4 and 15 ms based on the flow speed ( $U$ ) and the oscillation frequency ( $f$ ) of the foil. In each of the foil cases, 30 PIV fields were obtained at each of 16 equally spaced phases. From these phase referenced PIV fields, the phase-averaged velocity fields were calculated. The processing of acquired images, averaging the flow field information, calculation of vorticity field and circulation of individual vortices were done using a MATLAB based PIV code, as described in Das, Govardhan & Arakeri (Reference Das, Govardhan and Arakeri2013), which is based on the code described in Govardhan & Williamson (Reference Govardhan and Williamson2000).

3 Rigid foil

In this section, we briefly present results from force measurements on a pitching rigid foil and compare cycle-averaged thrust coefficients and propulsive efficiency values with earlier studies in the literature.

The results presented are for a foil pitched sinusoidally about the quarter chord with angular amplitude $\unicode[STIX]{x1D703}_{max}$ of $10\,^{\circ }$ , and with some results also for $\unicode[STIX]{x1D703}_{max}$ of $5\,^{\circ }$ . The reduced frequency, $k=\unicode[STIX]{x03C0}fc/U$ , is varied in the range of 1–10 corresponding to a Strouhal number ( $St_{R}=f(2\unicode[STIX]{x1D6FF}_{R})/U$ ) of between 0.1 and 1.1, this being based on the excursion of the rigid foil trailing edge ( $2\unicode[STIX]{x1D6FF}_{R}$ as illustrated in figure 20).

We shall first present the methodology used to obtain the individual contributions to thrust from the measured normal ( $N$ ) and chordwise or axial ( $A$ ) forces, as illustrated in figure 2(b). At any instant, the total thrust coefficient ( $C_{T}$ ) is obtained as the sum of the contributions from the normal ( $C_{TN}$ ) and axial forces ( $C_{TA}$ ) by resolving the normal force ( $C_{N}$ ) and the chordwise force ( $C_{A}$ ), respectively, along the free-stream direction using the measured position ( $\unicode[STIX]{x1D703}$ ) as given below:

We compute the cycle-averaged quantities directly from the above equation, to obtain mean total thrust ( $\overline{C_{T}}$ ) and the individual contributions from the normal force ( $\overline{C_{TN}}$ ) and the axial or chordwise forces ( $\overline{C_{TA}}$ ). It is however useful to see what the averaging process does in the present case of relatively small amplitude sinusoidal oscillations. For small pitching angles, $\sin \unicode[STIX]{x1D703}\approx \unicode[STIX]{x1D703}$ and $\cos \unicode[STIX]{x1D703}\approx 1$ , leading to the equation,

Further, substituting for $\unicode[STIX]{x1D703}(t)=\unicode[STIX]{x1D703}_{max}\sin 2\unicode[STIX]{x03C0}ft$ and then averaging the thrust over the oscillation cycle, we obtain an expression for the cycle-averaged thrust coefficient ( $\overline{C_{T}}$ ) as:

Figure 4. Variation of the measured normal force ( $C_{N}$ ) and mean axial force ( $\overline{C_{A}}$ ) with reduced frequency ( $k$ ) for the rigid foil ( $\unicode[STIX]{x1D703}_{max}=10\,^{\circ }$ ). In (a), the amplitude of the normal force and the mean axial force are shown indicating the large values of normal force compared to the axial force. In (b), the phase of the normal force with respect to the foil motions ( $\unicode[STIX]{x1D719}_{CN}$ ) is shown, which appears to saturate at around $-120\,^{\circ }$ at large $k$ values. ●, Present ( $\unicode[STIX]{x1D703}_{max}=10\,^{\circ }$ ); ♢, Mackowski & Williamson (Reference Mackowski and Williamson2015) ( $\unicode[STIX]{x1D703}_{max}=8\,^{\circ }$ ).

It can be seen from this equation that the mean thrust involves two parts. The first part from the normal force essentially depends on the amplitude of the first harmonic of $C_{N}$ , represented here as $[C_{N}]$ , and the phase difference $\unicode[STIX]{x1D719}_{CN}$ between the first harmonic of $C_{N}$ and the pitching angle $\unicode[STIX]{x1D703}$ , with positive values of $\unicode[STIX]{x1D719}_{CN}$ corresponding to this signal ( $[C_{N}]\sin (2\unicode[STIX]{x03C0}ft+\unicode[STIX]{x1D719}_{CN})$ ) lagging $\unicode[STIX]{x1D703}$ . The second contribution coming from the axial force is essentially just the time average of the chordwise force $C_{A}$ . As one might expect, it is the contribution from the normal force that is mainly responsible for thrust generation in a rigid foil, with the axial or chordwise contribution being small and negative.

The variation with the reduced frequency ( $k$ ) of the measured normal and axial forces on the pitching foil is shown in figure 4. In the case of the normal force, both the amplitude of the first harmonic of $C_{N}$ ( $[C_{N}]$ ) and its phase ( $\unicode[STIX]{x1D719}_{CN}$ ) are shown, while in the case of the axial force, only the time-averaged value ( $\overline{C_{A}}$ ) is shown, these being the most important for determining the cycle-averaged thrust coefficient ( $\overline{C_{T}}$ ), as seen in (3.4). The normal force amplitude ( $[C_{N}]$ ) is found to increase rapidly with reduced frequency ( $k$ ) reaching large amplitudes of approximately 40 at reduced frequencies of approximately 10. The time-averaged axial force ( $\overline{C_{A}}$ ) on the other hand remains very small, as one would expect, with magnitudes of approximately $-0.2$ at a reduced frequency of close to 10. The phase of the normal force ( $\unicode[STIX]{x1D719}_{CN}$ ), shown in figure 4(b), is found to vary from about $-70\,^{\circ }$ at low reduced frequency to an apparently asymptotic value of close to $-120\,^{\circ }$ at high reduced frequencies. This value of phase difference is consistent with the recent measurements of Mackowski & Williamson (Reference Mackowski and Williamson2015), whose data are also shown in figure 4(b) and who report similar values of approximately $-120\,^{\circ }$ at $\unicode[STIX]{x1D703}_{max}$ of $8\,^{\circ }$ .

The observed variation in $\unicode[STIX]{x1D719}_{CN}$ has a direct influence on the cycle-averaged thrust coefficient ( $\overline{C_{T}}$ ), as seen from (3.4). At low reduced frequencies, $k\ll 1$ , the normal force ( $C_{N}$ ) may be expected to be in phase with the angular location ( $\unicode[STIX]{x1D703}$ ) implying that the phase $\unicode[STIX]{x1D719}_{CN}$ would be zero, which is consistent with the trend of the data in figure 4(b). At $k\approx 2$ , $\unicode[STIX]{x1D719}_{CN}\approx -90\,^{\circ }$ , which implies that the cosine term becomes nearly zero, and the normal forces in this case make no contribution to the mean thrust. Physically, the phase $\unicode[STIX]{x1D719}_{CN}$ is important as its value determines the fraction of the oscillation cycle when the normal force has a component in the thrust direction. For $\unicode[STIX]{x1D719}_{CN}=0\,^{\circ }$ , the normal force only has a drag component. For $\unicode[STIX]{x1D719}_{CN}=-90\,^{\circ }$ , the normal force has a thrust component for 50 % of the cycle and a drag component for the remaining 50 % of the cycle, with the cycle-averaged thrust being zero. For the extreme case of $\unicode[STIX]{x1D719}_{CN}=-180\,^{\circ }$ , the normal force will have a thrust component for the full oscillation cycle, but we see from our measurements that the phase never reaches such high values and instead saturates to $\unicode[STIX]{x1D719}_{CN}\approx -120\,^{\circ }$ at high reduced frequencies. The extreme case of $\unicode[STIX]{x1D719}_{CN}=-180\,^{\circ }$ is possible in pure potential flow, where only added mass forces due to the rotational accelerations are present, as suggested by Daniel (Reference Daniel1984).

Figure 5. Variation of the time-averaged thrust with reduced frequency for the rigid foil ( $\unicode[STIX]{x1D703}_{max}=10\,^{\circ }$ ). The contributions to thrust from both the normal force ( $\overline{C_{TN}}$ ) and the axial force ( $\overline{C_{TA}}$ ) are shown, in addition to the total thrust ( $\overline{C_{T}}$ ). The main contribution to thrust is from the normal force ( $\overline{C_{TN}}$ ) with small negative values from the axial ( $\overline{C_{TA}}$ ). ● (Solid line), total thrust ( $\overline{C_{T}}$ ); ○ (dashed line), thrust from the normal force ( $\overline{C_{TN}}$ ); ▫ (dashed-dot line), thrust from the axial force ( $\overline{C_{TA}}$ ).

In figure 5, we present the cycle-averaged total thrust ( $\overline{C_{T}}$ ) and the individual contributions from both normal ( $\overline{C_{TN}}$ ) and axial ( $\overline{C_{TA}}$ ) forces as a function of the reduced frequency. As one might expect, the normal force contribution to thrust is much larger than the axial force contribution, the ratio between the two being a factor of approximately 8 at a $k$ of around 10. The total thrust ( $\overline{C_{T}}$ ) is thus very close to the contribution from the normal force ( $\overline{C_{TN}}$ ), with the values being a little lower due to the negative contribution (or drag) from the axial force. It may be noted that in this rigid foil case, the axial force contribution is negative (or drag) at all reduced frequencies, which will be contrasted with the flexible trailing edge flap cases in the next section. We have compared our mean or cycle-averaged thrust measurements with other studies in the literature for pitching rigid foils with comparable $\unicode[STIX]{x1D703}_{max}$ values in figure 6(a). This includes values from two numerical simulations (Sarkar & Venkatraman Reference Sarkar and Venkatraman2006; Lu, Xie & Zhang Reference Lu, Xie and Zhang2013) at $\unicode[STIX]{x1D703}_{max}$ values of $5\,^{\circ }$ and $10\,^{\circ }$ , and the relatively recent measurements of Dewey et al. (Reference Dewey, Boschitsch, Moored, Stone and Smits2013) ( $\unicode[STIX]{x1D703}_{max}=14.3\,^{\circ }$ ) and Mackowski & Williamson (Reference Mackowski and Williamson2015) ( $\unicode[STIX]{x1D703}_{max}=8\,^{\circ }$ ). As seen in the figure, the data from the present experiments ( $\unicode[STIX]{x1D703}_{max}$ = $5\,^{\circ }$ , $10\,^{\circ }$ ) is broadly consistent with the results from these prior studies. Also shown as an inset in the figure is the same data plotted versus Strouhal number ( $St_{R}$ ) formed using the excursion of the rigid trailing edge ( $2\unicode[STIX]{x1D6FF}_{R}$ ). With this scaling, our data at $\unicode[STIX]{x1D703}_{max}$ of $5\,^{\circ }$ and $10\,^{\circ }$ collapse, as suggested by the measurements of Mackowski & Williamson (Reference Mackowski and Williamson2015) for $\unicode[STIX]{x1D703}_{max}<16\,^{\circ }$ . The data of Dewey et al. (Reference Dewey, Boschitsch, Moored, Stone and Smits2013) are however higher, and is likely related to the different configuration used in their study with differences both in the pitching axis location (in their case the leading edge) and the presence of a fixed fairing upstream of the leading edge of the panel in their case.

Figure 6. Comparison of the present (a) time-averaged thrust ( $\overline{C_{T}}$ ) and (b) propulsive efficiency ( $\unicode[STIX]{x1D702}$ ) measurement for a pitching rigid foil with data from the literature. The data are plotted with the reduced frequency ( $k$ ) in the main plot, and with the Strouhal number ( $St_{R}$ ) formed using the excursion of the trailing edge in the inset plots. ●, Present ( $\unicode[STIX]{x1D703}_{max}=10\,^{\circ }$ ); ○, present ( $\unicode[STIX]{x1D703}_{max}=5\,^{\circ }$ ); ▿, Dewey et al. (Reference Dewey, Boschitsch, Moored, Stone and Smits2013) ( $\unicode[STIX]{x1D703}_{max}=14.3\,^{\circ }$ ); ▫, Lu et al. (Reference Lu, Xie and Zhang2013) ( $\unicode[STIX]{x1D703}_{max}=5\,^{\circ },10\,^{\circ }$ ); ♢, Mackowski & Williamson (Reference Mackowski and Williamson2015) ( $\unicode[STIX]{x1D703}_{max}=8\,^{\circ }$ ); ▵, Sarkar & Venkatraman (Reference Sarkar and Venkatraman2006) ( $\unicode[STIX]{x1D703}_{max}=5\,^{\circ }$ ).

We use the propulsive efficiency ( $\unicode[STIX]{x1D702}$ ), as in Anderson et al. (Reference Anderson, Streitlien, Barrett and Triantafyllou1998) and other studies, as the ratio of the useful power based on the cycle averaged thrust ( $\overline{T}U$ ) to the cycle-averaged input power ( $\overline{P}$ ), $\unicode[STIX]{x1D702}=\overline{T}U/\overline{P}=\overline{C_{T}}/\overline{C_{P}}$ , where the input power is obtained from integration of the instantaneously measured moment, as discussed in the experimental methods section. The propulsive efficiency values obtained from the present work are shown in figure 6(b). There have been very few prior direct experimental measurements of this, and we present in figure 6(b) data from the recent work of Dewey et al. (Reference Dewey, Boschitsch, Moored, Stone and Smits2013) and Mackowski & Williamson (Reference Mackowski and Williamson2015). For the rigid pitching foil, it is clear that the propulsive efficiencies are quite low at approximately 10 %, both from our experiments and those of Mackowski & Williamson (Reference Mackowski and Williamson2015), which are both for the same geometry. The efficiencies of Dewey et al. (Reference Dewey, Boschitsch, Moored, Stone and Smits2013) are higher, this being likely linked to the differences in geometrical configuration, as stated earlier. The same data are also plotted versus $St_{R}$ in the inset to the figure, and shows that the peak efficiencies are found in the $St_{R}$ range of 0.4–0.5 in both our case and that of Dewey et al. (Reference Dewey, Boschitsch, Moored, Stone and Smits2013).

4 Rigid foil with flexible trailing edge flap

We shall now proceed to investigate the thrust and efficiency characteristics of a flexible foil. As discussed earlier, the flexible foil is constructed by attaching a flexible flap of known length ( $c_{F}$ ) and flexural rigidity ( $EI$ ) to the rigid foil of chord length $c_{R}$ used for the study in the previous section. The new flexible foil thus created has a total chord length $c$ ( $c=c_{R}+c_{F}$ ). Throughout the rest of the paper, the chord $c$ for the flexible foil will imply the total chord, and all normalizations such as those for the reduced frequency ( $k=\unicode[STIX]{x03C0}fc/U$ ) and the force and power coefficients are done using this total chord ( $c$ ). We can thus view the present results as the effects of making a part of the foil flexible. This configuration of the flexible foil brings in two additional parameters, namely, the flexural rigidity of the (flexible) flap ( $EI$ ) and the ratio of the flexible flap length to the total chord length ( $c_{F}/c$ ). The flexural rigidity ( $EI$ ) may be normalized to yield a non-dimensional flexibility parameter, $R^{\ast }$ , along the lines of Michelin & Llewellyn Smith (Reference Michelin and Llewellyn Smith2009) and others, as given below:

It is important to note here that the flexibility parameter $R^{\ast }$ may be related to the ratio of the actual foil oscillation frequency to the first natural frequency of the flap ( $f/f_{n1}$ ), by using the relation: $f_{n1}=\unicode[STIX]{x1D6FC}\sqrt{EI/(\unicode[STIX]{x1D70C}c_{F}^{5})}$ discussed in § 2, as:

We shall use this expression to link $f/f_{n1}$ to $R^{\ast }$ throughout the rest of the paper to enable discussions on the possible effects of resonance.

There are thus three main parameters defining the flexible foil performance, namely, the reduced frequency ( $k$ ), the flexural rigidity ( $EI$ ) or the non-dimensional flexibility parameter, $R^{\ast }$ (or $f/f_{n1}$ ) and the ratio of the flexible flap length to the total chord length ( $c_{F}/c$ ). It may be noted that $R^{\ast }$ also contains the flap length $c_{F}$ . In general, the angular amplitude ( $\unicode[STIX]{x1D703}_{max}$ ) of the rigid foil, the location of the pitching point and $Re$ , are also parameters. However, in the present work, we keep the first two of these parameters fixed throughout the study at $10\,^{\circ }$ and the rigid foil quarter-chord point ( $c_{R}/4$ ), respectively, while as stated earlier, the flow is nearly independent of $Re$ over the range of $Re$ from 5000 to 15 000 studied. It may be noted that the Strouhal number for the flexible foil ( $St_{ft}=f(2\unicode[STIX]{x1D6FF}_{ft})/U$ ) may be defined based on the lateral excursion of the flexible flap tip ( $2\unicode[STIX]{x1D6FF}_{ft}$ as illustrated in figure 20), while noting that both $\unicode[STIX]{x1D6FF}_{ft}$ and $St_{ft}$ are not known a priori in these flexible flap cases.

Apart from the above parameters, another parameter related to the flexible flap is the ratio of the flap’s inertia to the fluid inertia. This is typically referred to as the mass ratio, $\unicode[STIX]{x1D707}$ , and is defined here along the lines of Thiria & Godoy-Diana (Reference Thiria and Godoy-Diana2010) as $\unicode[STIX]{x1D707}=(w/g)/(\unicode[STIX]{x1D70C}(2\unicode[STIX]{x1D6FF}_{ft}))$ , where $w$ is the weight per unit area of the flap, $\unicode[STIX]{x1D70C}$ is the fluid density and $2\unicode[STIX]{x1D6FF}_{ft}$ is the flap tip’s peak to peak traverse. In our cases, with plastic flap material in water, the mass ratio, $\unicode[STIX]{x1D707}$ , is found to be of the order of $10^{-3}$ for all cases, as shown in table 1. This indicates that the flap’s inertia is negligible compared to the fluid inertia, and therefore in our experiments, the flap’s mass is not an important parameter.

Figure 7. Flexible flap deflection at 16 phases during the oscillation cycle in both the laboratory reference frame (top) and the rigid foil reference frame (bottom). The data shown correspond to four flexural rigidities ( $EI$ ) of the flexible flap, (a) $1.10\times 10^{-2}~\text{Nm}$ ( $R^{\ast }=4.3$ ), (b) $5.07\times 10^{-4}~\text{Nm}$ ( $R^{\ast }=0.20$ ), (c) $5.53\times 10^{-5}~\text{Nm}$ ( $R^{\ast }=0.022$ ), (d) $1.04\times 10^{-5}~\text{Nm}$ ( $R^{\ast }=0.004$ ). In all cases, the ratio of the flexible flap to total chord ratio ( $c_{F}/c$ ) is 0.45 and the reduced frequency $k$ is close to 6.

The flexible flap deforms substantially during the oscillation cycle in all cases. The deformation is affected by the three parameters namely $k$ , $EI$ (or $R^{\ast }$ or $f/f_{n1}$ ) and $c_{F}/c$ and is a result of an interplay between fluid forces and the elastic rigidity of the flexible flap. Figure 7 shows example flap deflection profiles for four flexural rigidity ( $EI$ ) cases in a free stream, starting from the highest $EI$ case in (a) to the lowest $EI$ case in (d), with $k$ and $c_{F}/c$ being fixed at 6 and 0.45, respectively. For each $EI$ case, the flap deflection profile at 16 phases of the oscillation cycle are shown in both the laboratory reference frame on the top, and in the reference frame of the rigid foil on the bottom. In the laboratory reference frame, the flap deformations are obtained from direct imaging of the flaps, with the left end of the flap shown corresponding to the trailing edge of the rigid foil. We can see considerable deflection or bending of the flaps in all cases with the emergence of increasingly more complex mode shapes as $EI$ is reduced. This is especially clear near the flap tips where the least stiff flap exhibits very large curvatures, as compared to the more gradual variations in the stiffest flap case. An alternate method to look at the mode shapes, as done by Michelin & Llewellyn Smith (Reference Michelin and Llewellyn Smith2009), would be in the reference frame of the rigid foil. This, in our case, involves rotation of the measured deflection profile in the laboratory reference frame about the pitching axis of the rigid foil. The resulting mode shapes that are now more representative of the actual deformation of the flaps are also shown in figure 7, and one can see changes in the mode shapes more easily. It is clear that the highest $EI$ case shown in (a) corresponds to a simple mode 1 type bending, as seen in still water tests in figure 3, with higher and more complex modes visible as the flexural rigidity $EI$ is reduced.

We shall now present results from force measurements on flexible foils in three sections each corresponding to variations of one of the three main parameters, namely, reduced frequency ( $k$ ), flexural rigidity ( $EI$ ) and the length of the flexible flap ( $c_{F}$ ). In each of these studies, the value of one of these parameters is varied, while the other two are kept constant to help understand the effect of each of these parameters. An overview of all the different flexible foil experiments conducted is given in table 1. It should be noted that in all cases presented here, the force and moment coefficients are normalized with the total chord ( $c$ ).

5 Wake measurements

In this section, we shall present results from PIV measurements in the wake of the flexible oscillating foil. The results shown are based on PIV measurements for 6 different flexible foils with $R^{\ast }$ values ranging from 0.004 to 4.3 (4 cases with $c_{F}/c=0.45$ and one each with $c_{F}/c=0.29$ and $c_{F}/c=0.62$ ), besides the rigid foil case. We begin by showing the measured wake vorticity fields from PIV for the rigid case and a few flexible cases with decreasing values of the non-dimensional flexural rigidity ( $R^{\ast }$ ) in figure 17. In all the cases, the reduced frequency was held constant at $k\approx 6$ , and the vorticity field shown corresponds to the phase when the rigid foil is at the mean angular position ( $0\,^{\circ }$ ) and the trailing edge of the rigid part of the foil is moving upwards. In the rigid foil case in (a), we can see a clear deflected wake with the vortices moving along a line that is inclined upwards in the image shown. It should be noted that this asymmetry is despite the fact that the foil oscillates symmetrically about the free-stream direction. This type of deflected wake has also been seen by Cleaver, Wang & Gursul (Reference Cleaver, Wang and Gursul2012) for a rigid heaving foil, and is consistent with the frequency amplitude phase chart for the wake formation from a rigid pitching foil presented in Godoy-Diana et al. (Reference Godoy-Diana, Aider and Wesfreid2008). As shown in their map, the thrust producing wake of a purely pitching rigid foil consists of either symmetric or asymmetric (deflected) reverse Kármán vortex street.

Figure 17. Phase-averaged vorticity field in the wake of an oscillating flexible foil. The cases shown correspond to non-dimensional rigidity parameter $R^{\ast }$ of (a) infinite (rigid), (b) 4.3, (c) 0.198, (d) 0.025. The ratio of flexible flap length to total chord ( $c_{F}/c$ ) in (b,c) is 0.45, while it is 0.62 in (d). In all cases, the rigid part of the foil is at its mean position with the trailing edge moving upwards and the reduced frequency, $k\approx 6$ . Solid and dashed line contours represent positive and negative vorticity, respectively. Vorticity contour levels shown are $\unicode[STIX]{x1D714}c/U=\pm 10,\pm 20,\pm 30,\ldots .$

In contrast to the deflected or asymmetric wake seen in the rigid case in figure 17(a), all the flexible cases (figure 17 b–d) show a symmetric (non-deflected) wake. Similar observations of flexible foils inducing symmetry in the wake are reported in the recent work of Marais et al. (Reference Marais, Thiria, Wesfreid and Godoy-Diana2012) for flexible foils of particular stiffness, and also by Shinde & Arakeri (Reference Shinde and Arakeri2014) for a pitching flexible foil in the absence of a free-stream velocity. The non-dimensional vorticity contours ( $\unicode[STIX]{x1D714}c/U$ ) are the same in all the cases shown, and hence, the figure directly gives a sense of the changes in the wake vortices caused by the decreasing flexural rigidity of the foil from (b–d). One can, for example, clearly see reductions in size of the vortices, lateral spacing between vortices and the circulation strength of the vortices as the rigidity parameter $R^{\ast }$ decreases. We see two single vortices being shed per oscillation cycle (2S formation) in all cases, except the case (d), where we see the clear formation of two pairs of vortices per oscillation cycle (2P formation). It may be noted here that both vortices of the pair in the 2P mode seen here are formed at the trailing edge of the flexible foil. There is no leading edge separation or vortex formation, as seen for example in Heathcote & Gursul (Reference Heathcote and Gursul2007). This is due to the fact that leading edge separation is determined by the local angle of attack at the leading edge, which in our case is part of the rigid foil that always pitches at the relatively small amplitude of $\pm 10\,^{\circ }$ . This type of 2P mode with both vortices formed at the trailing edge has been reported in pitching rigid foil cases by Koochesfahani (Reference Koochesfahani1989) and by Mackowski & Williamson (Reference Mackowski and Williamson2015), with no leading edge separation reported in either of the two cases. From the vorticity plots of Mackowski & Williamson (Reference Mackowski and Williamson2015) one can see that the trailing edge vorticity in the 2P case is shed as an elongated thick vortex, which then splits into two resulting in the formation of two like-signed vortices. One of these vortices then pairs with a split opposite-signed vortex from the previous half-cycle forming a counter-rotating vortex pair or the 2P vortex pattern in a manner similar to the observations and discussions of Williamson & Roshko (Reference Williamson and Roshko1988) for an oscillating cylinder. This type of vorticity splitting leading to the formation of the 2P vortex wake may be attributed to the increased strain rates in the central region of the elongated vortex, as discussed by Govardhan & Williamson (Reference Govardhan and Williamson2000) for an oscillating cylinder. In our case for the flexible foil composed of the rigid leading edge, we do not see any leading edge vortex, and the time sequence of vorticity plots suggest that the formation of the 2P vortex pattern is due to the splitting of the shed vortex from the trailing edge as in Mackowski & Williamson (Reference Mackowski and Williamson2015).

Figure 18. Variation in (a) lateral distance between the centre of the vortices normalized by the total chord ( $b/c$ ), (b) non-dimensional circulation ( $\unicode[STIX]{x1D6E4}/Uc$ ) and (c) the wake Strouhal number ( $St_{w}=fb/U$ ) with $R^{\ast }$ . Also shown in (c) for comparison are the corresponding values of the Strouhal number based on flap tip excursion ( $St_{ft}$ ) in open symbols (dashed line). Data are shown from the $EI$ variation study (circle) and the $c_{F}/c$ study (square). In all cases, $k\approx 6$ .

We shall now present wake parameters, the lateral spacing between the vortices ( $b$ ), the normalized circulation ( $\unicode[STIX]{x1D6E4}/Uc$ ) and the wake Strouhal number ( $St_{w}=fb/U$ ), all measured from phase-averaged vorticity fields, as shown for example in figure 17. The lateral vortex spacing ratio ( $b/c$ ) is the ratio of the lateral spacing ( $b$ ) to the total chord and is shown in figure 18(a) plotted versus the non-dimensional rigidity parameter $R^{\ast }$ , this being shown only for the 2S vortex formation cases, as the definition of ( $b/c$ ) is not clear for the 2P formation case. The data from the $EI$ variation study and the $c_{F}/c$ study are both shown in the plot and collapse well on to a single curve. At low $R^{\ast }$ values, the spacing ratio is found to be approximately 0.2, and is thereafter found to increase continually with $R^{\ast }$ reaching a value of approximately 0.32 at $R^{\ast }=4.3$ close to the peak thrust case, this being higher than the value of 0.29 for the rigid case. The continuous increase in lateral spacing ( $b/c$ ) seen here correlates well with the continuous increase in thrust coefficient seen earlier at between low $R^{\ast }$ values and $R^{\ast }=4.3$ . This is consistent with the Kármán formula for drag/thrust that is applicable to the 2S vortex formation mode, which indicates that the lateral spacing is important in deciding the mean drag/thrust, as discussed and observed by Michelin & Llewellyn Smith (Reference Michelin and Llewellyn Smith2009). The other important factor in the Kármán formula for drag/thrust is of course the circulation of the shed vortices, which we calculate as the vorticity integrated over a low vorticity contour level ( $\unicode[STIX]{x1D714}c/U=\pm 5$ ) surrounding the shed vortex. In the 2P case (figure 17 d), the circulation shown is the sum of the two same-signed vortices shed per cycle. The normalized circulation ( $\unicode[STIX]{x1D6E4}/Uc$ ) in figure 18(b) also increases continuously with $R^{\ast }$ starting from low $R^{\ast }$ values, where ( $\unicode[STIX]{x1D6E4}/Uc$ ) is approximately 0.5, and reaching a value of approximately 2 at $R^{\ast }=4.3$ close to the peak thrust case, this being almost similar to the value of approximately 2.2 for the rigid case. The Kármán formula applicable for the 2S mode suggests that the maximum thrust (or drag) would occur when the product of the lateral spacing ( $b/c$ ) and the circulation ( $\unicode[STIX]{x1D6E4}/Uc$ ) are maximized. In the present case, it is clear from the data in figure 18 that this would occur at large $R^{\ast }$ , beyond the value of 4.3, although we cannot determine an actual peak $R^{\ast }$ value as both ( $b/c$ ) and ( $\unicode[STIX]{x1D6E4}/Uc$ ) show a continually increasing trend until $R^{\ast }=4.3$ beyond which we do not have PIV data. This is however consistent with the peak thrust coefficient measured with the load cell, which shows that the peak thrust occurs at $R^{\ast }\approx 8$ .

From the wake dynamics point of view, the most relevant length scale for a Strouhal number would be the wake width (Triantafyllou, Triantafyllou & Yue Reference Triantafyllou, Triantafyllou and Yue2000). With this in mind, we calculate a wake Strouhal number ( $St_{w}$ ) based on the lateral spacing of the vortices in the wake ( $b$ ) for all the flexible foil cases, and is shown in figure 18(c). The figure shows that $St_{w}\approx 0.35$ at $R^{\ast }=0.02$ close to the peak efficiency case, while $St_{w}\approx 0.55$ at $R^{\ast }=4.3$ close to the peak thrust case. The $St_{w}$ of about 0.35 that we find close to the peak efficiency condition is consistent with the range of Strouhal numbers (0.20–0.40) suggested by the experiments of Anderson et al. (Reference Anderson, Streitlien, Barrett and Triantafyllou1998) for high propulsive efficiency of a foil in combined pitch and heave. Also, shown for comparison in the plot is the Strouhal number based on the flexible foil flap tip excursion ( $St_{ft}$ ), which was discussed in the previous section. At low values of $R^{\ast }$ corresponding to highly flexible flaps, one can see that $St_{ft}$ values are significantly larger than $St_{w}$ values. This is related to the fact that in these cases, the highly flexible flap, although having large flap tip excursions (2 $\unicode[STIX]{x1D6FF}_{ft}$ ), is not effective in deciding the lateral spacing of the vortices ( $b$ ) that is relatively much smaller. Hence, $St_{w}$ values based on $b$ are much smaller than $St_{ft}$ values based on the larger $\unicode[STIX]{x1D6FF}_{ft}$ . As $R^{\ast }$ increases from these low values, the flap gets stiffer, leading to smaller values of $\unicode[STIX]{x1D6FF}_{ft}$ , which is also more effective in deciding the lateral vortex spacing ( $b$ ). As a consequence $\unicode[STIX]{x1D6FF}_{ft}$ becomes smaller and closer to $b$ , which in turn makes $St_{ft}$ values smaller and closer to $St_{w}$ . Apart from this trend, the $St_{ft}$ values again increase between $R^{\ast }$ values of 1 and 4, which is related to the increase in $\unicode[STIX]{x1D6FF}_{ft}$ values as shown in figure 20 that is likely related to the resonant mode 1 condition of the flexible flap as discussed in the next section. The comparison shown here between the two Strouhal numbers, $St_{w}$ and $St_{ft}$ , indicates that in the flexible foil cases one has to be cautious about the use of the Strouhal number based on the flexible foil tip excursion ( $St_{ft}$ ) as a surrogate for $St_{w}$ .

Figure 19. Phase-averaged vorticity field corresponding to (a) large propulsive efficiency ( $R^{\ast }=0.022$ ) and (b) large thrust coefficient ( $R^{\ast }=4.3$ ). In each case, a time sequence of two vorticity fields is shown corresponding to the upward motion of the flexible flap tip, the fields being separated by a time interval of $\unicode[STIX]{x1D70F}/8$ , where $\unicode[STIX]{x1D70F}$ is the oscillation period. In both cases, $c_{F}/c=0.45$ and $k\approx 6$ . Solid and dashed line contours represent positive and negative vorticity, respectively. Vorticity contour levels shown are $\unicode[STIX]{x1D714}c/U=\pm 10,\pm 20,\pm 30,\ldots .$

We have thus far given an overview of the wake measurements over a range of flexural rigidities and flexible flap lengths. We now focus on two cases of interest, namely, the flexural rigidity ( $R^{\ast }$ ) corresponding to values which are close to the maximum efficiency and maximum thrust cases. We present in figure 19(a,b), the phase-averaged wake vorticity fields for these two cases, at two phases in the oscillation cycle, as the flexible flap tip moves upwards with the two fields being separated by $\unicode[STIX]{x1D70F}/8$ in each case ( $\unicode[STIX]{x1D70F}$ is the oscillation period). One can see the significant differences in the two time sequences in figure 19(a,b). Apart from the already discussed points related to the circulation of the vortices and their lateral spacing, one can also see differences in the shedding of vortices from the flap tip. In both figures, the arrangement of individual phases has been done in precisely the same manner based on the location of the flexible flap tip. This is done to accentuate differences between the two cases, as the motions of the flap tip are likely to play an important role in vortex formation. Comparing vertically across the individual vorticity fields of the two sequences, one can see that a major difference between the two cases is the continuous shedding of vorticity from the more rigid flap (high thrust case), as opposed to the more flexible flap (high efficiency case). In the larger thrust case, for example on the right-hand side of figure 19(b), substantial vorticity can be seen coming from the flap tip, which will eventually roll up into the vortex. On the other hand, on the right-hand side of figure 19(a), there is almost no vorticity being shed from the flap tip at the same flap tip location, and the formed vortex is disconnected from the flap tip. Another obvious difference between the two cases is the shape of the flexible flap at any given instant. We shall discuss this in more detail in the next section.