On the role of tip curvature on flapping plates

The two rigid curved plates with a small and a large tip radius of curvature, plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$, respectively, are compared with a rigid flat plate, plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. This comparison is made to establish a baseline regarding the effect of curvature on the generated forces and torques if a rigid flat plate could be actuated into a similar curved geometry. The curvature of plate $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ is selected to be similar to the maximum achievable curvature through dynamic actuation with plate $\newcommand{\PlateD}{{\rm AC1}} \PlateD$, while the curvature of plate $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ is selected as a semicircle to investigate the impact of curvatures larger than plate $\newcommand{\PlateD}{{\rm AC1}} \PlateD$ can realize. The instantaneous C_S, C_T, and $C_{\tau}$ as functions of the non-dimensional time $t^* = t/(2t_s)$ are shown in figure 4 for a typical cycle and the $\overline{C}_S$, $\overline{C}_T$, $\newcommand{\e}{{\rm e}} \eta_1$, and $\newcommand{\e}{{\rm e}} \eta_2$ per cycle are shown in figure 5 for plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$, $\newcommand{\PlateB}{{\rm RC1}} \PlateB$, and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$. The forward stroke, when a concave geometry is presented by the rigid curved plates towards the incoming flow, corresponds to 0  <  t^*  <  0.5 while the backward stroke, when a convex geometry is presented towards the incoming flow, corresponds to 0.5  <  t^*  <  1.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Comparison of the instantaneous C_S in figures 4(a)–(c) between the three plates for all sets of kinematics shows that increasing the curvature decreases the peak side force generated, even when accounting for the difference in the planform area with the denominator of C_S. From the data shown in figure 4, plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ decrease the peak side force by 5.3% and 30.8% on average during the forward stroke, respectively, and by 10.8% and 38.3% on average during the backward stroke, respectively, when compared with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. The results from the instantaneous $C_\tau$ in figures 4(g)–(i) corroborates the results from C_S. Comparison of the instantaneous C_T in figures 4(d)–(f) between the three plates for all sets of kinematics shows a decrease in the peak thrust generated by plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ compared to $\newcommand{\PlateA}{{\rm R0}} \PlateA$. The data shown in figure 4 indicates that plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ decrease the peak thrust by 9.8% and 25.9% on average during the forward stroke, respectively, and by 15.2% and 43.3% on average during the backward stroke, respectively, when compared with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. These results can be attributed to the reduced planform area of plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ creating a more streamlined geometry further discussed in section 3.4. A more streamlined body opposes the flow less, decreasing the side force, and pushes less fluid in the thrust direction, decreasing the generated thrust. Regarding the difference between the forward and the backward strokes, the greater decrease in side force is due to the lower drag coefficient of a convex geometry compared with a concave geometry [25], while the greater decrease in thrust is due to a convex geometry creating a significant amount of suction, highlighted in section 3.6.

Comparison of $\overline{C}_S$ and $\overline{C}_T$ between the three plates in figures 5(a) and (c), respectively, corroborates the previous conclusion from the instantaneous C_S and C_T that an increased curvature decreases all of the generated forces. It should be noted that because plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ present a different geometry towards the incoming flow during the forward and the backward strokes, a net side force is generated. The net side force is computed as $\overline{F}_x$ in contrast to the average side force computed as $\overline{\vert F_x\vert }$ in $\overline{C}_S$. The net side forces generated by plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ are 4.0% and 7.3% of the average side force generated by plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$, respectively, in the direction opposite that of the forward stroke. This would cause an AUV to turn if a static geometry is used as a propulsor. A reduction in all of the forces could increase the efficiency if the side force or the torque is reduced more than the thrust. However, as shown in figures 5(b) and (d), both measures of efficiency for plates $\newcommand{\PlateB}{{\rm RC1}} \PlateB$ and $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ do not differ significantly from those for plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$.

The Nitinol-actuated plate $\newcommand{\PlateD}{{\rm AC1}} \PlateD$ is used to determine the impact of a dynamically-actuated curvature. The actuation profiles used to contract the Nitinol wire are created by modifying the duty cycles of square waves with an identical frequency to that of the motion. Actuation profiles 1, 2, and 3 have duty cycles (DC) of 12.5%, $25\%$, and $50\%$, respectively, with phase angles $\theta_F$ of 67.5°, 0°, and 0°, respectively, for actuation during the forward stroke and phase angles $\theta_B$ of 247.5°, 180°, and 180°, respectively, for actuation during the backward stroke. These actuation profiles cause the wire to actuate and remained contracted during the forward stroke from a t^* of 0.1875 to 0.3125, 0 to 0.25, and 0 to 0.5 respectively and during the backward stroke from a t^* of 0.6875 to 0.8125, 0.5 to 0.75, and 0.5 to 1 respectively. Characteristics of the actuation profiles and the t^* intervals when the plates are contracted are summarized in tables 2 and 3. The parameters were selected to cover a variety of possible profiles to assess whether the decrease in forces could be triggered and maintained effectively and arbitrarily. It should be noted that within a single trial, actuation only occurs during either the forward or the backward stroke.

Table 2. Characteristics of the actuation profiles.

Profile	DC ($\%$)	$\theta_F$ $({\hspace{0pt}}^\circ)$	$\theta_B$ $({\hspace{0pt}}^\circ)$
1	12.5	67.5	247.5
2	25	0	180
3	50	0	180

The impact of actuation is assessed by evaluating the difference in the generated forces between actuated and unactuated cases. These instantaneous differences in C_S and C_T are denoted $\Delta C_S$ and $\Delta C_T$, respectively, in figure 6 for a typical cycle. The dotted horizontal lines in figure 6 correspond to the noise level of the transducer during the motion, computed from the maximum $\Delta C_S$ and $\Delta C_T$, between trials with identical kinematics when no actuation occurs. The solid line corresponds to actuation during the forward stroke, while the dashed line corresponds to actuation during the backward stroke. For a 60° stroke angle and a 1 s stroke time, convection cooling during the forward stroke, when the incoming water flows directly over the wire, prevents actuation. It should be noted that the solid horizontal line at zero is only meant to highlight the zero-line and does not correspond to any data. The results for the 0.381 mm thick polycarbonate plate with a 90° stroke angle and a 2 s stroke time using actuation profiles 1 and 2 are shown in figures 6(a)–(d), while the results for the 0.508 mm thick polycarbonate plate with a 60° stroke angle and a 1 s stroke time using actuation profiles 1 and 3 are shown in figures 6(e)–(h).

Zoom In Zoom Out Reset image size

Comparison of $\Delta C_S$ or $\Delta C_T$ in figure 6 across all actuation profiles shows that the forces are decreased whenever the plate is actuated, which corroborates the previous conclusion regarding the impact of curvature, with the added benefit that the reduction in forces can be triggered and maintained arbitrarily. This can be seen by comparing the duration of the decrease in forces between actuation profiles 1 and 3 in figures 6(e)–(h). Using actuation profile 1, the largest impact is achieved; the maximum reduction in C_S is by 12.8% and 9.0% while that in C_T is by 8.8% and 12.3% at a 90° stroke angle with a 2 s stroke time and a 60° stroke angle with a 1 s stroke time, respectively, when compared to the case without actuation. It should be noted that characterization of the Nitinol wire by actuating the wire on a motionless plate in quiescent fluid generates forces below the noise threshold. This implies that the decrease in forces is primarily due to a static change in curvature rather than a dynamic bending motion. These results show that dynamic chord-wise curvature control through Nitinol wire actuation can be triggered and maintained effectively to reduce the generated forces by approximately $10\%$. Furthermore, this decrease in the forces is comparable to the decrease in the peak amplitude of C_S and C_T from plate $\newcommand{\PlateB}{{\rm RC1}} \PlateB$, which had a similar chord-wise tip radius of curvature and β as plate $\newcommand{\PlateD}{{\rm AC1}} \PlateD$, meaning that if greater curvatures could be realized, the forces could potentially be decreased by approximately $31\%$, as seen with plate $\newcommand{\PlateC}{{\rm RC2}} \PlateC$. Characterization of the influence of flexibility will require future studies.

Plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ is used to isolate the effect of curvature as the only difference between plates $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ and $\newcommand{\PlateA}{{\rm R0}} \PlateA$ is that plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ has a tip radius of curvature of 23 mm and a β of 0.5. Plate $\newcommand{\PlateF}{{\rm RM2\mbox{-}F}} \PlateF$ is used to study the impact of spanwise flow, which has been shown to significantly affect the generation of tip vortices and thrust [10]. By attaching a fence to the tip of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ to create plate $\newcommand{\PlateF}{{\rm RM2\mbox{-}F}} \PlateF$, flow in the spanwise direction near the tip of the plate is impaired because here, during the forward stroke, the flow is redirected normal to the plate. The instantaneous C_S, C_T, and $C_{\tau}$ as functions of t^* are shown in figure 7 for a typical cycle and the $\overline{C}_S$, $\overline{C}_T$, $\newcommand{\e}{{\rm e}} \eta_1$, and $\newcommand{\e}{{\rm e}} \eta_2$ per cycle are shown in figure 8 for plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$, $\newcommand{\PlateE}{{\rm RM2}} \PlateE$, and $\newcommand{\PlateF}{{\rm RM2\mbox{-}F}} \PlateF$.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

Comparison of the instantaneous C_S in figures 7(a)–(c) between plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$ and $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ shows that curvature decreases the peak side force generated, similar to the conclusion from section 3.1. However, adding a fence with plate $\newcommand{\PlateF}{{\rm RM2\mbox{-}F}} \PlateF$ causes the side force generated during the accelerating portion of the forward stroke to approach that generated by the baseline flat plate. The results from the instantaneous $C_\tau$ in figures 7(g)–(i) corroborate the results from C_S. Comparison of the instantaneous C_T in figures 7(d)–(f) between the plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$ and $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ shows that curvature significantly decreases the thrust generated during the backward stroke, similar to the previous conclusion from section 3.1, but, interestingly, increases the thrust generated during the forward stroke; using the data in figure 7, plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ increases the peak thrust generated by 22.4% on average during the forward stroke but decreases the peak thrust by 26.4% on average during the backward stroke compared to plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. However, the addition of a fence with plate $\newcommand{\PlateF}{{\rm RM2\mbox{-}F}} \PlateF$ negates the benefit of presenting a concave geometry into the flow, causing the thrust during the forward stroke to decrease even below that of the rigid flat plate.

Comparison of the overall forces generated and the efficiency per cycle in figure 8 corroborate the previous result that plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ decreases the side force generated and increases the thrust generated, which may not be evident as plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ increases the thrust during the forward stroke but decreases the thrust during the backward stroke. Furthermore, the addition of a fence decreases the performance of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$. As expected, because plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ decreases the side force and increases the thrust, the efficiencies $\newcommand{\e}{{\rm e}} \eta_1$ and $\newcommand{\e}{{\rm e}} \eta_2$ are increased by 34.3% and 36.4%, respectively. Additionally, the net side force generated is only 3.3% of the average side force of the baseline flat plate, meaning that replacing a flat plate propulsor with a curved plate of an identical planform area may be viable.

The impact of the planform area can be isolated by comparing the results from plates $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ and $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ because they have an identical β but different planform areas. Comparison of the instantaneous C_S between the two plates in figures 4(a)–(c) and 7(a)–(c), respectively, shows that curvature alone (comparing plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ to $\newcommand{\PlateA}{{\rm R0}} \PlateA$) has a minimal effect during the forward stroke and a small effect during the backward stroke. Therefore, the result that plate $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ decreases the peak C_S and the overall $\overline{C}_S$ per cycle more than plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ decreases those when both are compared against plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$ must be accounted for by plate $\newcommand{\PlateC}{{\rm RC2}} \PlateC$'s reduced planform area (due to its smaller chord near the tip of the plate). This provides further evidence that decreasing the planform area creates a more streamlined geometry which reduces the side force, previously suggested in section 3.1.

Comparison of the instantaneous C_T between plates $\newcommand{\PlateC}{{\rm RC2}} \PlateC$ and $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ in figures 4(d)–(f) and 7(d)–(f), respectively, shows that the benefit of presenting a concave geometry towards the incoming flow is negated if the planform area is reduced. These results show that modifying the planform area by changing the chord near the tip of the plate can either significantly increase or decrease the generated thrust during the forward stroke. Furthermore, this highlights the required interaction between the planform area and the curvature to increase the thrust generated during the forward stroke which is illustrated in section 3.6.

Motivated by the results from plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$, plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ is designed to passively snap-buckle and present a concave geometry into the flow during both the forward and the backward strokes. The instantaneous C_S, C_T, and $C_{\tau}$ as functions of t^* are shown in figure 9 for a typical cycle and the $\overline{C}_S$, $\overline{C}_T$, $\newcommand{\e}{{\rm e}} \eta_1$, and $\newcommand{\e}{{\rm e}} \eta_2$ per cycle are shown in figure 10 for plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$, $\newcommand{\PlateE}{{\rm RM2}} \PlateE$, and $\newcommand{\PlateG}{{\rm AM2}} \PlateG$. Plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ is tested on a smaller subset of the kinematics as only large angular velocities induce snap-buckling.

Zoom In Zoom Out Reset image size

Zoom In Zoom Out Reset image size

From figures 9(d)–(f), the significant increase in thrust during both strokes is apparent as plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ increases the peak thrust by $121\%$ on average compared with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. It should be noted that the flat region immediately preceding the peak in C_T corresponds to the period of time when snap-buckling occurs; during this transition from one curved geometry to another, the thin polycarbonate sheet is completely passive to the flow, so no additional force is generated. However, snap-buckling is an abrupt process causing the peaks in C_S shown in figures 9(a)–(c). Plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ increases the peak side force by 47.6% on average compared with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. The results from the instantaneous $C_\tau$ in figures 9(g)–(i) corroborate the results from C_S. Overall, plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ increases $\overline{C}_T$ per cycle by 34.8% but causes $\overline{C}_S$ to approach that generated by the baseline flat plate, which increases $\newcommand{\e}{{\rm e}} \eta_1$ and $\newcommand{\e}{{\rm e}} \eta_2$ by 30.1% and 23.6%, respectively, compared with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. These results show that using a snap-buckling design can successfully take advantage of an increased thrust during both the forward and the backward strokes with the added benefit, compared with a rigid curved design, that no net side force should be generated as the geometry presented into the flow is identical during both strokes.

Dye visualization is used to provide qualitative insight into the flow structures generated by plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$ and $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ to investigate possible sources for plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$'s significantly increased thrust and efficiency compared to that of plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. Plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ is effectively rigid during each stroke and therefore no dye visualization is provided for the first forward stroke as the generated flow field would be nearly identical to that of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$. The first forward stroke is investigated to highlight the effect of the plate's geometry without the influence of a previously generated flow field. The locations of the dye injection points are concentrated near the tip of the plates because the flow is similar on the pressure side of the two plates at locations closer to the pitching axis. Although not visualized, flow at locations closer to the pitching axis simply move in the chord-wise direction away from the centreline towards the edge of the plate. This flow rolls up into an edge vortex that tends to follow the trajectory of the pitching plate as expected from DDPIV studies of a flat plate by Kim and Gharib [10]. This behavior is also exhibited by plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ because the curvature at locations closer to the pitching axis is small compared to that near the tip of the plate.

Representative snapshots of the generated 3D flow field near the tip of plates $\newcommand{\PlateA}{{\rm R0}} \PlateA$ and $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ pitching with a 60° stroke angle and a 1 s stroke time are shown in figure 11. The flow behaves in a similar manner as that from other kinematic sets. From the dye visualization of plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$ in figures 11(a)–(c), the dye originating from the two injection points closest to the tip shows a developing tip vortex which tends to follow the trajectory of the pitching plate with minimal outwards motion. The dye originating from the other two injection points shows a chord-wise flow away from the centreline, taking the most direct path towards the edge of the plate, as expected from studies by Kim and Gharib [10]. From the dye visualization of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ in figures 11(d)–(f), the dye originating from the two injection points closest to the tip shows a smaller developing tip vortex which almost immediately sheds by $t^*\sim$ 0.25 and appears to transition into a shape similar to a vortex ring by $t^*\sim$ 0.33. The dye originating from the other two injection points shows a flow moving towards the tip of the plate instead of the edge of the plate as seen with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$. Additionally, the influence of the geometry moves this flow towards the centreline of the plate to eventually supplement the development of the tip vortex. Compared to plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$, the geometry of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ appears to cause the tip vortex to shed earlier and travel further downstream. Furthermore, this geometry appears to 'channel' the flow near the tip of the plate, causing the dye to move towards the tip rather than the edge of the plate. The early separation of the tip vortex of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ may have been influenced by the increased spanwise flow near the tip, due to the geometry of the plate, or the induced velocity of the edge vortices which attach at an angle on the suction side of the plate.

Zoom In Zoom Out Reset image size

DPIV is used to provide quantitative preliminary insight into the mechanism that increases the thrust and the efficiency of plates $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ and $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ compared with those from plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$ at $\phi =$ 60° and t_s  =  1 s. The flow behaves in a similar manner as that from other kinematic sets. This 2D visualization technique involves high speed imaging of the illuminated particles following the flow. The laser sheet is illuminated along the mid-plane of the plate, indicated by the dashed line in figure 3, where 3D effects are minimal. Evolution of the wake in time is illustrated in figure 12 by first phase averaging the flow field based on the pitching frequency of the plate after reaching steady state, then selecting a single line along the x-direction at a fixed y-location, and finally plotting the velocity field along this line at every instance in time. The line in the x-direction at a fixed y-location is taken at a distance of 1.17 times the span from the pitching location of the plate. The orientation of the pitching motion in figure 12 is identical to that shown in figure 1(b) and, relative to this orientation, flow in the positive x-direction corresponds to a positive U^*, while flow in the negative y-direction corresponds to a positive V^*. The corresponding phase averaged force data for the generated flow field is shown in figures 12(a) and (b); the positive direction of the forces is denoted by the axes in figure 1(b). The non-dimensional component of velocity in the direction of C_S and C_T, $U^* = U/U_{\rm tip}$ and $V^* = V/U_{\rm tip}$, respectively, along a line as functions of t^* and x^*  =  x/L, where $L=s(\sin{30^o})$ is the excursion amplitude of the plate's tip, are shown in figures 12(c)–(h). From figure 12, the U^* velocity field generated by all of the plates appears similar, which is reasonable, as C_S is similar between all three plates. Additionally, no obvious event in the flow history is linked with the peak in C_S for plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$, which implies that these peaks are primarily due to an abrupt snap-buckling motion rather than a flow phenomenon.

Zoom In Zoom Out Reset image size

Comparison of the V^* velocity field between the three plates illustrates a possible cause of the improved thrust performance of plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$. Compared with plate $\newcommand{\PlateA}{{\rm R0}} \PlateA$, plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ generates a V^* velocity field of a significantly greater magnitude. From a simple momentum argument, the greater the momentum imparted to the flow, the greater the thrust generated. The large thrust peaks of plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ are a result of the dark red regions in figure 12(h). Within these regions, the flow is almost impulsively 'channeled' outwards from the plate in the spanwise direction immediately following snap-buckling, illustrated by the abrupt change in color around a t^* of 0.2 and 0.7; the peak in C_T appears in the force measurements before the corresponding event in the flow because the recently-imparted momentum into the flow has to convect to the line where the velocity field is sampled.

A similar 'channeling' effect is present during the forward stroke of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ in figure 12(f). Immediately following a change in direction, when a concave geometry is presented into the flow around t^*  =  0.2, a red concentrated region appears, but with a smaller magnitude than that from plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ because no abrupt snap-buckling occurs. However, when a convex geometry is presented into the flow during the backward stroke, the V^* velocity approaches and even decreases below zero, which explains the poor thrust performance of the rigid curved geometries during the backward stroke, as described in sections 3.1 and 3.3. It should be noted that the velocity field interestingly resembles a pulsing flow, as most of the thrust is generated during the forward stroke followed by a 'recovery' phase during the backward stroke. Overall, these results suggest that the peak in C_S for plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ is caused by the abrupt snap-buckling motion and that the increase in thrust during the forward stroke of plate $\newcommand{\PlateE}{{\rm RM2}} \PlateE$ and both strokes of plate $\newcommand{\PlateG}{{\rm AM2}} \PlateG$ may have been caused by a 'channeling' effect where the incoming flow is redirected outwards from the plate in the spanwise direction. A definitive answer regarding the underlying mechanisms requires future studies with a quantitative 3D flow visualization technique.

The effect of chord-wise tip curvature on the hydrodynamic forces was tested with flapping plates of different geometries. The first case study involved the impact of physically bending the two corners of a plate towards each other. From the baseline study using rigid curved geometries, the results suggested that increasing the curvature decreases the hydrodynamic forces and torques with a minimal loss in efficiency. This concept was explored using a plate with a dynamically-actuated chord-wise tip curvature using a Nitinol wire, which corroborated the result from the baseline study with the added benefit that the amplitude and the duration of the decrease in forces could be arbitrarily modulated.

The second case study involved isolating the impact of chord-wise tip curvature by using plates with a similar planform area to that of the baseline flat plate. Similar to the previous case study, increasing the curvature decreased the generated forces and torques, with the exception that the thrust generated during the forward stroke increased. This benefit was leveraged using a snap-buckling plate to present a concave geometry into the flow during both the forward and the backward strokes, which significantly increased the overall thrust generated per cycle. Investigation through dye visualization and DPIV suggested that this increase in thrust may have been due to a 'channeling' effect, where the incoming fluid is redirected outwards from the tip of the plate rather than moving around it as a result of the geometry of the plate, imparting more momentum to the flow in the thrust direction. Future studies using a quantitative 3D flow visualization technique will be necessary to provide a more definitive answer regarding the underlying mechanisms.

These results suggest that two new mechanisms could potentially be used to increase the maneuverability or the efficiency of AUVs. If the goal is improved maneuverability, implementing actuated chord-wise tip curvature shows promise to assist in braking and turning without needing to modify the trajectory. If the goal is improved efficiency, replacing a rigid flat plate propulsor with a snap-buckling plate of a similar planform area shows promise to achieve this goal, provided the angular velocity of the plate is sufficient to snap-buckle the material.