of attack $\alpha = \gamma - \varphi \approx \gamma$ reduces to the twist of the blade $\gamma$. Therefore, if $\gamma_0 = 60$ deg, improving the thrust by reducing the twist is possible. This is what we see for the least flexible blades (yellow curve). The mechanism that leads to the thrust reduction of the more flexible blades is different to the one in the previous section. It is due to a reduction of the angle of attack to smaller values than the one producing maximum thrust. In this case, even the most flexible blade still produce positive thrust at high tip speed ratio. At the same time, this reduction of the angle of attack leads to a quick decrease of the torque as well (panel (b)).

We can appreciate here the effect of changing the tools used to measure the thrust and torque. The uncertainty between experiments (error bars) is limited and only concerns low tip speed ratios, a range that is not the primary focus of this section.

Using this particular pitch angle ($\gamma_0 = 60$ deg), we can see that all flexible blades improve the efficiency of the propeller (panel (c)). This is because of the angle of attack on the rigid blade. At high $\lambda$, the angle of attack at the tip is approximately $\alpha \approx \gamma$, which in the case of the rigid blades corresponds to the pitch angle. However, in Sec. 1.2.2, we found an optimal angle of attack for the lift to drag ratio $\alpha_{opt}$ of about 17 deg. Reducing the angle of attack through twist is therefore almost always beneficial at high tip speed ratios. This mechanism creates an optimal Cauchy number: the red curve (P3 blades) is above the most flexible dark red one (P4 blades). If the twist reduces the angle of attack beyond $\alpha_{opt}$ over most of the blade, then the efficiency will reduce again. This is visible in the last panel (d) where the most flexible dark red points are below 10 deg at high tip speed ratio. This mechanism is summed up in Fig. 3.23. We focus here on the tip of the blade because it is where most of the force is generated as the velocity norm grows like the square of the distance to the axis of rotation. However, the global shape of the blade is modified: the tip is more twisted than the root creating a spanwise torsion (bottom row of Fig. 3.23).

<figure><img src="image_52.png" /><figcaption>Figure 3.23: Schematic of the mechanism that leads to improved efficiency. (Top row) Evolution of the angle of attack with the material flexibility for one given blade section. (Bottom row) Evolution of the global shape of the blade.</figcaption></figure>

Finally, we can see the effect of the proposed correction of the tip twist angle $\gamma_M$. Markers with a dark edge color correspond to the data corrected by adding $\gamma_{corr}$: transforming the raw data measured on the side of the rotor to the mean angle over a whole rotation. The effect of this modification is
to decrease the tip deflection by a small amount (less than 5 deg) with a smaller contribution as $\lambda$ grows larger.

On the right column of Fig. 3.22, we plot the results obtained from solving Eq. 3.36 over the whole span of the blade, with the exact same parameters as the experiment. This very simple model produces very good predictions of the behavior of the rotor. In particular the deflection as a function of the tip speed ratio is very similar to the one obtained in the corrected experiments. Forces are well predicted as well, even if the torque is slightly smaller than in experiments: no losses due to flow curvature or finite blade size effects have been implemented in this model. The resulting efficiency is more important and more peaked than the one found in the experiments, yet the relative order between flexibilities is well respected. All the experimental data for the thrust, torque, and twist angle for other pitch angles can be found in App. C.

However, the model does not quantitatively agree with the experimental data. This means that our correction of the tip twist angle $\gamma_{corr}$ is an approximation of the real correction required. It still has the right direction and order of magnitude and the tip angle data in the following is the corrected one.

There are two metrics that can be used to compare the efficiency of different propellers. The first and most natural one is to compare the maximal efficiency: when the conditions are optimal, what is the best performance that a propeller can produce ? The second one in contrast, compares the resilience to a change of external conditions: how flat is the efficiency versus tip speed ratio curve near its optimum ? We will see in this section that these flexible blades can improve the efficiency on both fronts.

## Resilience to variations of tip speed ratio

The most important efficiency improvement is the resilience to changes of external conditions. This metric is the direct equivalent of the result obtained by Cognet et al. (2017). In Fig. 3.24 panel (a), we plot the efficiency as function of the pitch angle for all the blades with $\gamma_0 = 40$ deg, the optimal pitch angle for the flexible blades. To compare the optimal flexible blades with the rigid ones, we also plot the efficiency curves for every pitch angle for the rigid blades (in blue). Markers indicate the pitch angle, with 40 deg being indicated by diamonds.

The rigid efficiency curves all display a maximum for an intermediate tip speed ratio (except for $\gamma_0 = 80$ deg which remains very inefficient). The tip speed ratio corresponding to this optimum decreases when increasing the pitch angle: from $\lambda = 7$ at $\gamma_0 = 20$ deg, deg to $\lambda = 3$ at $\gamma_0 = 70$ showing the necessity to adapt the pitch angle to stay at an optimal efficiency. Conversely, this defines an optimal pitch angle for each tip speed ratio, that we denote as $\gamma_0^{opt}(\lambda)$ and is specific to this propeller geometry.

From $\lambda = 3$ to 5.5, the P3 blades (red) with $\gamma_0 = 40$ deg are as efficient or more than any rigid propeller. Their effective pitch angle is passively modified by the flow, maintaining a high efficiency for a wide range of tip speed ratios. The P3 blades are optimal as the less flexible P1 and P2 blades (yellow and orange) have an efficiency closer to their rigid counterpart while the more flexible P4 (dark red) have an overall reduced efficiency.

The efficiency of the propellers is closely linked to the twist angle $\gamma$ along the span as shown in the schematic of Fig. 3.23. We want to know what the evolution of the tip twist angle should be in order to have a blade that remains as efficient as any rigid one. We therefore plot in Fig. 3.24 panel (b) the tip twist angle for all the blades with $\gamma_0 = 40$ deg (yellow to red, blue for the rigid blades). In addition to that, we plot $\gamma_0^{opt}(\lambda)$ measured experimentally (in black): the pitch angle which results in
<figure><img src="image_53.png" /></figure>

Figure 3.24: Efficiency and tip twist as a function of the tip speed ratio. Left column corresponds to experimental results, right column to numerical computations. (a) Efficiency of the $\gamma_0 = 40$ deg flexible propellers (diamonds, yellow to red). (blue) Rigid propeller with various pitch angles (differentiated with markers). (b) Associated tip twist. (black curve) Optimal pitch angle of the rigid blades $\gamma_0^{\text{opt}}(\lambda)$. The tip twist is corrected by adding $\gamma_{\text{corr}}$. (c) Numerical efficiency with $\gamma_0 = 50$ deg for the flexible blades (yellow to red). (blue) Rigid propeller with various pitch angles. (black) Efficiency of an optimally adjusted rigid propeller. (d) Numerical tip twist for the flexible blades. (black) Optimal pitch angle $\gamma_0^{\text{opt}}(\lambda)$.

the largest efficiency of the rigid blades for each tip speed ratio. It is modified in steps because the rigid propeller experiments were carried out every 10 deg of pitch angle. To maximize efficiency, the tip twist should be close to the optimal pitch angle. Indeed, the P4 blades, which are far from $\gamma_0^{\text{opt}}(\lambda)$ are very inefficient. The slope of the tip twist $\gamma_M$ for the P3 blades matches approximately the curve $\gamma_0^{\text{opt}}(\lambda)$ in the range $\lambda \in [3, 5.5]$. The exact value is however slightly smaller than was expected using the rigid blade data.

In panels (c) and (d), we plot the numerical equivalents to the experimental data from panels (a) and (b). Panel (c) shows the efficiency with several flexible blades (yellow to red) with pitch angle $\gamma_0 = 50$ deg. The value of $\gamma_0$ plotted here is not the same as the experimental one because it numerically corresponds to higher efficiencies. However, the same discussion is possible with other pitch angle without changing the results. Added to that we plot several efficiency curves for rigid propellers, with various pitch angles (light blue) as well as the maximum efficiency reachable in black (the efficiency of a rigid propeller following $\gamma_0^{\text{opt}}(\lambda)$ or the envelope of the blue curves). The red curve with a bold line shows the optimal flexible blade for this pitch angle. It remains more efficient than any rigid propeller for $\lambda \in [1.95, 6]$.

Panel (d) displays the corresponding tip twist. The black line corresponds to $\gamma_0^{\text{opt}}(\lambda)$. Similarly to the experimental result, the most efficient flexible propeller has a tip twist curve that follows the same evolution as the optimal pitch angle, but a few degrees below it.
In contrast to the previous propeller studied, the effect of flexibility is to decrease the twist angle and reduce the angle of attack. As a consequence, the flexible propellers passively adjust to a change in external conditions: if $\lambda$ increases, the angle of attack increases, increasing the loading on the blade and decreasing the twist angle, that finally reduces the angle of attack and compensates for the increased $\lambda$. They have therefore an overall increased efficiency.

## Maximal efficiency

The flexible blades reach a higher maximum efficiency than the rigid ones. For a pitch angle of 40 deg the P3 blades are more efficient than any rigid ones at around $\lambda = 4$. We interpret this observation as being due to the spanwise distribution of twist along the blade (See Fig. 3.23, bottom row). In the previous analysis we only considered the tip angle because it corresponds to the location where most of the force is generated. However, the whole blade deforms in response to local constraints. For the same tip twist, the shape adopted by the rest of the blade can modify the resulting forces. This torsion effect is secondary compared with having the right angle near the tip.

In Fig. 3.25 (panel (a), green curve) we show the twist as a function of the spanwise position $s$ on the blade for $\gamma_0 = 40$ deg and $\lambda = 4$ (close to maximum efficiency)<fnref n="1" />. Close to the hub, there is almost no deflection: $\gamma \approx \gamma_0 = 40$ deg. The angle $\gamma$ decreases when going from the root ($s = 0.3$) to the tip ($s = 1$) of the blade. This trend follows the same evolution as the local flow angle $\varphi = \arctan(\cos \delta/(s\lambda))$ (blue curve) and therefore reduces the angle of attack fluctuations ($\alpha = \gamma - \varphi$) on the blade compared with a rigid one where $\gamma(s) = \gamma_0$. In panel (b), we show the global efficiency optimum obtained with the numerical model using an optimization algorithm searching the $(\gamma_0, \lambda, C_Y^S)$ parameter space: the curves for the twist and flow angle keep an almost constant difference. This difference is close to the optimal angle of attack that maximizes the lift to drag ratio: $\alpha_{opt} = 17$ deg.

<figure><img src="image_54.png" /><figcaption>Figure 3.25: Local flow angle $\varphi(s)$ (blue) and twist $\gamma(s)$ (green) as a function of the spanwise position $s$. (a) The twist corresponds to experimental data for the P3 blades at $\gamma_0 = 40$ deg and $\lambda = 4$. (b) The twist corresponds to the global efficiency optimum obtained with the numerical model ($\gamma_0 = 70$ deg, $\lambda = 2.1$, and $C_Y^S/\lambda^2 = 0.93$). In either case the flow angle is obtained via $\varphi = \arctan(\cos \delta/(s\lambda))$.</figcaption></figure>

The strategy for maximizing efficiency is the same as was noticed in Sec. 1.2.2: keep the angle of attack as close to $\alpha_{opt}$ as possible on most of the blade. The numerical solution keeps the angle of

<footnote n="1">¹We measured it at several spanwise positions by moving the laser sheet.</footnote>
attack $\alpha = \gamma - \varphi$ between 12 and 15 deg almost everywhere. Experimentally, we are not able to reproduce this solution: the P4 blades (dark red markers) should have more spanwise torsion but are almost never more efficient than the less flexible P3. Two factors might cause this discrepancy: first, the important chordwise curvature of the blades when highly deformed is detrimental for hydrodynamic efficiency. Flexible blades are less efficient than ones with a torsion spring as in the model. Second, we use a flexible plate instead of an assembly of independent beams. This might prevent us from reaching the 30 deg of spanwise torsion required for this optimal solution as the elastic energy associated with the spanwise torsion is no longer negligible and a model taking into account the plate geometry is required.

Still, the much more moderate evolution of the pitch angle in the optimal experiment (Fig. 3.25, panel (a)) allows the angle of attack to stay close to the one of maximal lift to drag ratio. At the tip, $\alpha \approx 20$ deg and at mid span it is about 15 deg, resulting in high lift to drag ratio over most of the blade (Fig. 1.8). This spanwise torsion allows the flexible blades to be more efficient than any rigid blades. It should be noted that we would most likely not see this improvement in maximum efficiency if the shape of the propeller at rest were twisted to maximize the performance of the rigid propeller.

### Effect of the pitch angle

<figure><img src="image_55.png" /><figcaption>Figure 3.26: Efficiency as a function of the tip speed ratio. Colors indicate blade flexibility (blue) rigid, (yellow to red) increasingly flexible blades. Markers indicate pitch angle. Each panel corresponds to a single pitch angle from 30 to 90 deg. For reference, the efficiency of the rigid blades with every pitch angle is plotted on each panel.</figcaption></figure>

Finally, we can compare how the efficiency curve of Fig. 3.24 is modified by changing the pitch angle. In Fig. 3.26, we plot the equivalent graph for all pitch angles tested. In each panel we plot
all the flexible curves with a particular pitch angle (yellow to red) and all the rigid curves in the background.

The benefits of using flexible blades are most visible at high pitch angle: when the efficiency associated with the rigid blades is extremely poor. The blade that most often achieves maximum efficiency is the P3 blade (light red markers), except for the largest pitch angle ($\gamma_0 \ge 80$ deg) where even more twist is required, and the most flexible P4 blades (dark red markers) are more efficient. For pitch angles of 30, 40, and 50 deg there are flexible blades which are more efficient than any rigid ones.

<figure><img src="image_56.png" /><figcaption>Figure 3.27: (red curves) Tip twist angle $\gamma_M$ as a function of the tip speed ratio for several pitch angles (markers). Data obtained with the P3 blades. (blue curve) Pitch angle $\gamma_0$ which maximizes the efficiency of the rigid blades as a function of $\lambda$. Dark edge colors correspond to data points with an associated efficiency larger than any rigid propeller at the same tip speed ratio. The tip twist has been corrected by adding $\gamma_{corr}$.</figcaption></figure>

In Fig. 3.27, we plot the tip twist for every pitch angle using the P3 blades. The black line corresponds to $\gamma_0^{opt}(\lambda)$ obtained experimentally. Using the P3 blades ($C_Y^S/\lambda^2 = 0.24$, red markers), the slope of the tip twist curve is similar to the one of the optimal pitch angle curve, indicating that they are well suited to improve the overall efficiency of the propeller. The data points with an efficiency larger than any rigid propeller have a dark edge color. The $\gamma_0 = 40$ deg (lozenges) curve is the most efficient in the range $3 < \lambda < 5.5$. The data points corresponding to high efficiency show that the tip twist should be slightly smaller than the optimal pitch angle for the rigid blades at a given $\lambda$. This result is due to the spanwise torsion of the blade. At high tip speed ratio $\lambda$ all the P3 blades except from the one with $\gamma_0 = 20$ deg most likely have a too important curvature to be efficient.

### 3.2.3 Evolution of the twist angle

We have seen that knowing the tip twist is a good indicator to predict the efficiency of a flexible blade. Therefore, in the following we will focus on predicting its evolution with the three parameters of the problem (pitch angle $\gamma_0$, tip speed ratio $\lambda$, and Cauchy number $C_Y^S$).

We derive a law for small deformations by developing at the first order Eq. (3.36). The tip deformation $\epsilon = \gamma_M - \gamma_0$ is small, and the angle of attack reads:

$$ \alpha \approx \epsilon + \alpha_0, \qquad (3.39) $$
where $\alpha_0 = \gamma_0 - \arctan \cos \delta/\lambda$ is the angle of attack on the undeformed blade. We then develop the expression in the right hand side of Eq. (3.36):

$$C_L(\alpha) \cos \alpha \approx C_L(\alpha_0) \cos \alpha_0 + \varepsilon \left( -C_L(\alpha_0) \sin \alpha_0 + \frac{\partial C_L}{\partial \alpha} \Big|_{\alpha_0} \cos \alpha_0 \right) \quad (3.40)$$

$$C_D(\alpha) \sin \alpha \approx C_D(\alpha_0) \sin \alpha_0 + \varepsilon \left( C_D(\alpha_0) \cos \alpha_0 + \frac{\partial C_D}{\partial \alpha} \Big|_{\alpha_0} \sin \alpha_0 \right). \quad (3.41)$$

We therefore obtain the approximated law for the deformation $\varepsilon$:

$$\varepsilon = -C_Y^S u_p^2 \delta_{AC}^{LE} \left( C_N(\alpha_0) + \varepsilon \frac{\partial C_N}{\partial \alpha} \Big|_{\alpha_0} \right), \quad (3.42)$$

where $C_N(\alpha) = C_L(\alpha) \cos \alpha + C_D(\alpha) \sin \alpha$ is the normal force coefficient and

$$\frac{\partial C_N}{\partial \alpha} \Big|_{\alpha_0} = -C_L(\alpha_0) \sin \alpha_0 + \frac{\partial C_L}{\partial \alpha} \Big|_{\alpha_0} \cos \alpha_0 + C_D(\alpha_0) \cos \alpha_0 + \frac{\partial C_D}{\partial \alpha} \Big|_{\alpha_0} \sin \alpha_0. \quad (3.43)$$

Finally, we use $u_p^2 = 1 + \cos^2(\delta)/\lambda^2$ and obtain the following relationship for the small twist deformation:

$$\varepsilon = - \frac{C_Y^S (1 + \cos^2(\delta)/\lambda^2) \delta_{AC}^{LE} C_N(\alpha_0)}{1 + C_Y^S (1 + \cos^2(\delta)/\lambda^2) \delta_{AC}^{LE} \partial C_N / \partial \alpha \Big|_{\alpha_0}} \equiv \varepsilon_1. \quad (3.44)$$

<figure><img src="image_57.png" /><figcaption>Figure 3.28: Twist deformation $\varepsilon$ as a function of its first order linearized value $\varepsilon_1$. (Black line) Linear fit for the moderate deformations (< 30 deg) of slope 0.65. (Inset) Deformation as a function of the Cauchy number only. The tip twist has been corrected by adding $\gamma_{corr}$.</figcaption></figure>

We use the flat plate lift and drag coefficients obtained in Sec. 1.2.2 to compute the first order linearized deformation $\varepsilon_1$. Using these functions, we verified that none the terms in Eq. (3.43) are negligible and that their total is of order one, meaning that first order development is required to capture the deformation of the blades. We plot the result in Fig. 3.28, and the deformation indeed follows a linear trend with its theoretical value for small deformations. The slope of this trend is 0.65 in radians, and it holds up to a deformation of about 40 deg. Beyond this value, for very flexible and high pitch blades, the slope decreases, but the data remains relatively collapsed. This scaling of the deformation is robust to changes of the exact coefficients used to compute the normal force $C_N$. We have used the ones fitted onto the lift and drag experiment in Sec. 1.2.2 but found that variations of 20% in these coefficients still collapses the data, although with a different slope.
The scatter of the data in the linear part is still relatively important (most of the points are within 5 degrees of the black line). Knowing the exact lift and drag coefficients (corrected for small Reynolds number or rotational effects) could help improve the accuracy of the formula. One other source may also be the uncertainty on the exact pitch angle manually set at the beginning of the experiment. Finally, taking into account the chordwise curvature may also improve the results, although significant curvatures are only obtained for large deformations.

## 3.3 Conclusion

In this chapter, we have introduced new bioinspired flexible propellers and studied their deformation and efficiency. The first part deals with the canonical problem of a fully flexible propeller with slender ribbon-like blades. This propeller exhibits extreme bending deformations towards the incoming flow as well as moderate twisting deformations. Under the assumption of small deformations, a scaling law based on geometric arguments allows us to predict the direction and estimate the amplitude of the bending. In the general case, we establish a strong relationship between the bending deformation and the thrust produced by the propeller: given a propeller geometry, the measurement of the deformation allows the deduction of the thrust.

This propeller presents an interesting fluid-structure interaction problem: the two directions of deformation are coupled by the hydrodynamic forces, and the loading is mainly due to the lift force instead of the drag. However, it is very inefficient. Using a combination of experiments and analytical results, we show that bending is always detrimental to the efficiency. Torsion can improve the efficiency of a given propeller by reducing the gap between the tip twist angle and the most efficient pitch angle. However, this occurs in a limited range of tip speed ratios, and we show that a rigid propeller with the optimal pitch angle $\gamma_0^{\text{opt}}(\lambda)$ will always be more efficient.

In a second part, we used propellers inspired from the works of Cognet et al. (2017, 2020) that are able to passively adapt to a change of external conditions. This propeller is flexible only in the chordwise direction, and the pivot point for its twist deformation is the leading edge. The difference between this problem and that of a flexible blade wind turbine is twofold: first, centrifugal force is always negligible in our problem, and therefore the material flexibility is the only adjustable property (its density is not relevant in this problem). Second, in order to remain as efficient as the best rigid blade over the entire range of tip speed ratios, the flexible wind turbine only needs to twist by 8 to 10 degrees. This is very different from the 40 degrees that the optimal flexible propeller must follow, which results in significant curvature effects that limit efficiency.

In experiments our best flexible blades can passively adjust to a change in tip speed ratio. In the range $3 < \lambda < 5.5$, we have found a pitch angle $\gamma_0$ and flexibility $C_Y^S/\lambda^2$ such that the flexible blades remain at least as efficient as the rigid ones following the optimal pitch angle $\gamma_0^{\text{opt}}(\lambda)$. This range corresponds to a 20 deg variation of $\gamma_0^{\text{opt}}(\lambda)$. We numerically found an even more efficient flexible propeller with a deformation that perfectly matches the spanwise variations of the local velocity. This solution is not accessible with the design used here because of the real blades have chordwise curvature and cannot match the extreme spanwise torsion required. One perspective of this work could therefore be to concentrate the flexibility near the leading edge, avoiding curvature effects and getting closer to the model of the torsion spring. Another possibility is to tune the chordwise flexibility along the span. This may allow the flexible blades to have a local angle of attack close to $\alpha_{\text{opt}}$ on all blade sections.

We used the tip twist angle as the unique information about the deformation of the blade. We show that comparing this tip twist angle to the optimal pitch angle $\gamma_0^{\text{opt}}(\lambda)$ in the rigid case, we can find
the most efficient pitch angle for the flexible blades. The tip twist follows a complex evolution with all the parameters ($\gamma_0$, $C_Y^S$, $\lambda$); using a linear development of the governing equation in the small deformation limit we are able to produce a law that collapses all the tip twist angle data. This law requires to know the airfoil characteristics used in the propeller (lift and drag versus angle of attack), but is not sensitive to exact parameters.

Finally, anticipating slightly onto the next chapter about intermittent locomotion, we will see that an important criterion is the drag of the propeller when not rotating. Because the propellers used in the second part are able to adapt to drastic modifications of external conditions, they might be suitable for this type of application. We measured the drag of the non-rotating propeller alone at various pitch angles with the rigid blades (Fig. 3.29) and as expected it reduces when the blades approach a flag configuration ($\gamma_0$ = 90 deg). A flexible blade would experience a similar drag as the deformation without rotation is small ($C_Y^S/\lambda^2 \ll C_Y^S$). Picking purposefully a very high pitch angle to minimize the drag while still maintaining a high efficiency might be a good strategy for intermittent locomotion. For instance, using the P3 blades with a pitch angle of 70 deg reduces the drag by 65% but retains 80% of the efficiency.

<figure><img src="image_58.png" /><figcaption>Figure 3.29: Drag coefficient of the rigid propeller as a function of the pitch angle. Obtained with $U_\infty$ = 0.15 m/s.</figcaption></figure>

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# Intermittent propulsion

In this chapter, we aim to build an experiment that applies the concept of burst and coast locomotion usually found in fish swimming to a new setting: propulsion with a classical boat propeller. The earlier studies (Weihs, 1974; Wu et al., 2007; Floryan et al., 2017a) have shown that the drag force experienced during the coasting phase of the motion is a key parameter of the problem. Fish experience up to four times less drag when gliding than when actively swimming. We mimic the passive drag reduction experienced by fish using a shape-changing propeller.

We will first describe the propellers used, and the mechanism of shape change in a static situation (the boat is fixed in the channel). We then measure the thrust and torque produced by this propeller in order to parametrize its behavior. We can then describe the complete setup of the intermittent locomotion experiment and data processing. Using cycle-averaged results, we demonstrate that efficient intermittent locomotion can be achieved with this shape-changing propeller. Finally, a model of the unsteady motion of the boat helps us find the conditions under which burst and coast locomotion is efficient.

## 4.1 Folding propeller

<figure><img src="image_59.png" /><figcaption>Figure 4.1: Folding propeller</figcaption></figure>
The propeller used in the intermittent locomotion experiment has one main objective: to minimize its drag when not in use. Given the binary nature of the requirements — either functioning like a normal propeller or minimizing the drag without any regards for performance — an inelastic system is sufficient. Elasticity is used in the other propellers to set the amplitude of the response to a given loading. Here we want the propeller to start functioning normally as soon as a threshold in angular velocity is met.

We therefore use a propeller with rigid blades, mounted on hinges, allowing it to fold downstream. This propeller is completely passive: the blades react to any force applied on them by freely rotating around their pivot. This type of propeller is not completely new as commercial versions of this idea — with the blades either folding or feathering — are available<fnref n="1" />. These propellers are used to reduce the drag of sailing boats. These propellers are also used when designing missions where efficiency and range are the primary concerns: Chen et al. (2013) uses a similar design to build an underwater glider, retracting the propeller to increase glide efficiency in the descent. Yet to our knowledge no analysis of how these propellers open exist.

When idle, the flow pushes back on each blade, forcing it to fold along the central axis of the propeller and reducing its frontal area. When the propeller is spinning the blade experiences the centrifugal force, and the hydrodynamic lift and drag on the blade, both of which tend to open the blade as the angular velocity is increased. We will first measure and describe how this propeller opens and then parametrize its performance for use in the following section.

### 4.1.1 Setup and expected behavior

The folding propeller is composed of three flat, 3D printed blades. These blades have a maximum radius of $R = 45$ mm, a minimum radius of $R_{hub} = 6$ mm and a thickness of $e = 1.5$ mm. Because they need to produce enough thrust to overcome the boat's drag while remaining of a reasonable diameter, their shape is closer to marine propellers than the ones we used in the previous chapter. The chord increases linearly from 9 mm at the root and 44 mm at the tip. These blades are attached to a hinge, and are tilted to impose the pitch angle (Fig. 4.4). This hinge is attached to the hub and allows the blades to bend downstream but prevents them from extending upstream. When extended, the blades are perpendicular to the axis of rotation.

The boat is static in the channel, and we impose the incoming flow velocity $U_∞$. Then we increase the angular velocity, and for each value we record the shape of the propeller using a camera and measure the angle $θ_e$ between the blades and the axis of rotation (defined by $θ_e = 0$ deg when the blades are folded and $θ_e = 90$ deg when fully extended, see Fig. 4.4). The tilt of the rotation axis with respect to the incoming flow introduces a small periodic variation of the angle $θ_e$ over the course of one revolution. We always measure the opening angle when the blade crosses the symmetry axis of the setup (above the shaft) to negate the effect of the tilt. We repeated the experiments for three flow speeds (0.12, 0.15 an 0.18 m/s) and three pitch angles (15, 30 and 45 deg.). To identify any blade-blade interaction effects, we also repeated one experiment (30 deg, 0.18 m/s) with a single blade.

The expected behavior of such a propeller when the flow is increased is the following. When a blade is tilted with respect to the incoming flow by an angle $θ_e$, the velocity components on a blade section located at a radius $r$ from the axis of rotation are modified. The flow velocity is decomposed in the spanwise direction and the component normal to the blade. Any spanwise force does not enter into the moment balance on the hinge, and we therefore neglect any contribution from this component. The remaining component, normal to the blade surface is $U_∞ \cos δ \sin θ_e$. Simultaneously, the distance to the axis of rotation is $r \sin θ_e$ and thus the angular velocity is $rΩ \sin θ_e$. In particular,

<footnote n="1">¹https://gori-propeller.com/ or https://www.darglow.co.uk/featherstream/</footnote>
<figure><img src="image_60.png" /><figcaption>Figure 4.2: Opening angle $\theta_e$ as a function of the tip speed ratio for several mean speeds (color). (green edge color) Experiments with a single blade instead of three. Smaller dots correspond to individual realizations, larger ones to the mean.</figcaption></figure>

we see that the local speed ratio $r\Omega/(U_\infty \cos \delta)$ (and therefore the angle of attack as well) remains the same whichever the opening angle $\theta_e$.

The hydrodynamic moment on the hinge results from the integration of the lift and drag on each blade section. The remark above means that the hydrodynamic moment reads $M_{hydro}(\theta_e) = \sin^2 \theta_e M_{hydro}(\theta_e = 90^\circ)$. In particular its sign is fully dictated by the tip speed ratio $\lambda = R\Omega/U_\infty$ and does not depend on $\theta_e$. Because our system is inelastic, and the centrifugal force is small, the equation setting the angle of the blades is $M_{hydro} = 0$. We therefore expect that the blades should be fully closed for low tip speed ratios (when $M_{hydro} < 0$) and switch to completely opened for higher $\lambda$.

Fig. 4.2 shows a very different behavior: the blades progressively open for $0 < \lambda < 3$. At a given tip speed ratio, the magnitude of the flow velocity does not change the opening angle $\theta_e$. This discrepancy with the expected behavior is not a blade-blade interaction effect as even the propeller with a single blade (dots with green edge color) shows a similar behavior. We cannot see an angular velocity threshold at which the blades start to open because the motor we use cannot rotate slow enough.

The exact tip speed ratio obtained when self propelling depends on the relative proportions of the boat compared to the propeller. In our case it is always over 3.5. This propeller is therefore suitable for our original goal which was to have a rotor that changes shape between the active and passive phases. The mechanical properties (thrust, drag at rest or opened) of this rotor are parametrized in Sec. 4.1.3.

The following section proposes an explanation for the observation that the propeller opens progressively instead of switching between fully opened or fully closed states.

### 4.1.2 A geometrical coupling between the opening and pitch angles

The two forces that act on the blade are the hydrodynamic force and the centrifugal force. The ratio between them reads:

$$ \frac{F_{\text{centrifuge}}}{F_{\text{hydro}}} \sim \frac{\rho_b R c e R \Omega^2}{\rho R c ((R\Omega)^2 + U_\infty^2 \cos^2 \delta)} \sim \frac{\rho_b e}{\rho R} \frac{1}{1 + \cos^2 \delta / \lambda^2} \approx 10^{-2}. \quad (4.1) $$
<figure><img src="image_61.png" /><figcaption>Figure 4.3: Schematic of the decomposition in ring elements.</figcaption></figure>

Where $\rho$ and $\rho_b$ are the densities of the fluid and the blade respectively, $c$ is the typical chord and $e$ the thickness. This ratio is always small in our case, and as a result, we will neglect the centrifugal force in the following. This result can also be obtained from observing that the opening angle only depends on the pitch angle: if the centrifugal force played a role, the force balance would differ when changing the flow velocity.

The expected behavior that we previously derived is based on the analysis in section 1.2, where we separated the incoming flow into concentric rings that interact with a blade section but do not influence each other. When the blades are thin and perpendicular to the flow (as in the previous sections), this division is equivalent to discretizing the span of the blade. Here, however, because the blades are tilted and occupy a significant angular sector (about half of the frontal area is filled by the blades), this is not the case. As can be seen in Fig. 4.3, a given flow tube cuts the blade at an angle. This cut has a non-trivial shape which depends on the opening angle and the spanwise position, and results in a modified effective pitch angle and chord length.

In the following, we assume that the flow is arranged in concentric cylinders that do not interact with each other. The goal of the following calculations is to compute the shape of the intersection of each cylinder with the blade at a given opening angle $\theta_e$. In particular, we want to obtain the effective pitch angle and chord length to derive the hydrodynamic moment on the hinge in a second part. The mathematical modeling is inspired from the works of Marzin et al. (2022) on origami folding: we parametrize the position of the blade by two vectors $w_0$ and $w_1$, attached respectively to the trailing and leading edges of the blade (green arrows in Fig. 4.3).

The axis $e_a$ is defined such that when the blade is fully extended, $w_0 = e_a$. In the general case the blade is rotated about $e_b$ by an angle $\frac{\pi}{2} - \theta_e$ such that

$$w_0(\theta_e) = \sin \theta_e e_a - \cos \theta_e e_c. \qquad (4.2)$$

The second base vector is obtained when fully extended by rotating first the vector $e_c$ by the pitch angle $\gamma_0$ around $e_a$ and then rotating $e_a$ around the new $e_c$ direction:

$$\mathbf{w}_1(\theta_e = \frac{\pi}{2}) = \cos \beta_{\max} \mathbf{e}_a + \sin \beta_{\max} (\cos \gamma_0 \mathbf{e}_b - \sin \gamma_0 \mathbf{e}_c), \qquad (4.3)$$

where $\beta_{\max}$ is the angle between $w_0$ and $w_1$ (57 deg here). The general case is obtained by multiplying
the above result by the rotation matrix of angle $\frac{\pi}{2} - \theta_e$ around $\mathbf{e}_b$ and we obtain:

$$ \mathbf{w}_1(\theta_e) = \begin{pmatrix} \cos \beta_{\max} \sin \theta_e + \cos \theta_e \sin \gamma_0 \sin \beta_{\max} \\ \sin \beta_{\max} \sin \gamma_0 \\ -\cos \theta_e \cos \beta_{\max} + \sin \theta_e \sin \gamma_0 \sin \beta_{\max} \end{pmatrix}_{(\mathbf{e}_a, \mathbf{e}_b, \mathbf{e}_c)} \quad (4.4) $$

Let M be a point on the blade, C the point where the (virtual) leading and trailing edges meet, B the intersection of the trailing edge and the hinge and A an arbitrary fixed point on the axis of rotation (see Fig. 4.3 right). The vector CM reads

$$ \mathbf{CM} = A_0 \mathbf{w}_0 + A_1 \mathbf{w}_1 \quad (4.5) $$

with $A_0$ and $A_1$ two constants. We parametrize the blades using the angle $\beta$ between CM and $\mathbf{w}_0$ and the distance $r$ from M to the axis of rotation. By imposing $r$, we ensure that we compute the shape of the blade at a constant radius i.e., the intersection with a cylinder. The point M verifies:

$$ \mathbf{CM} \cdot \mathbf{w}_0 = \|\mathbf{CM}\| \cos \beta \quad (4.6) $$

$$ (\mathbf{AM} \cdot \mathbf{e}_a)^2 + (\mathbf{AM} \cdot \mathbf{e}_b)^2 = r^2 \quad (4.7) $$

$$ \|\mathbf{BM}\| \le R - r_{\min}. \quad (4.8) $$

The first equation gives a relationship between $A_0$ and $A_1$:

$$ (A_0 + A_1 \cos \beta_{\max})^2 = (A_0^2 + A_0 A_1 \cos \beta_{\max} + A_1^2) \cos^2 \beta, \quad (4.9) $$

which simplifies into:

$$ A_0 = A_1 \left( -\cos \beta_{\max} + \frac{\sin \beta_{\max}}{\tan \beta} \right). \quad (4.10) $$

The second equation closes the system:

$$ (\mathbf{AB} \cdot \mathbf{e}_a + (A_0 - \|\mathbf{BC}\|) \sin \theta_e + A_1 (\cos \beta_{\max} \sin \theta_e + \cos \theta_e \sin \gamma_0 \sin \beta_{\max}))^2 + (A_1 \sin \beta_{\max} \cos \gamma_0)^2 = r^2. \quad (4.11) $$

By replacing $A_0$ we can solve analytically the system and get the shape of the intersection between the blade and a cylinder. The third equation ensures that the radius of the blade is never exceeded.

Fig. 4.4 shows examples of the shapes generated by this method. We can see that both the chord length and pitch angle are modified by the angle $\theta_e$. Panel (b) shows the local angle with the horizontal:

$$ \gamma(s, \theta_e, r) = \arctan \frac{\partial \mathbf{CM} \cdot \mathbf{e}_c}{\partial s}, \quad (4.12) $$

where $s = r\beta$ is the curvilinear abscissa along a cut. We can see that as the opening angle $\theta_e$ decreases, the mean pitch angle of the blade increases from the prescribed 30 degrees up to almost 90 degrees when the blades are completely folded.

In the following we model the complex shapes obtained previously by an effective chord and pitch angle that depend on the radial position $r$ and the opening angle $\theta_e$:

$$ c(\theta_e, r) = \int_0^{\beta_{\max}} \left\| \frac{\partial \mathbf{CM}}{\partial \beta} \right\| d\beta \quad (4.13) $$

$$ \gamma(\theta_e, r) = \underset{0 \le s \le s_{\max}}{\text{mean}} \gamma(s, \theta_e, r) \quad (4.14) $$
<figure><img src="image_62.png" /><figcaption>Figure 4.4: Geometry of the intersection between a ring and the blade for different opening angles (shades of green). Computed at $r = R/2$. (a) height as a function of the curvilinear abscissa (b) local angle with respect to the horizontal (defined in Eq. (4.12))</figcaption></figure>

We can now compute the local lift and drag contributions. The local velocity on a blade cut has a vertical component $U_\infty \cos \delta$ and horizontal component $r\Omega$ (see Fig. 4.3). We can define the local angle of attack:

$$ \alpha = \gamma(\theta_e, r) - \varphi(U_\infty, \Omega), \quad (4.15) $$

where $\varphi = \arctan U_\infty \cos \delta/(r\Omega)$.

The local velocity and angle of attack create local lift and drag forces (respectively $dL$ and $dD$), which in turn create a moment on the hinge with a moment arm $(r - r_{\min}) / \sin \theta_e$:

$$ M_{\text{hydro}} = \int_{r_{\min}}^{R} \frac{r - r_{\min}}{\sin \theta_e} (dL \cos \varphi - dD \sin \varphi). \quad (4.16) $$

We write the infinitesimal lift and drag forces in terms of lift and drag coefficients and obtain:

$$ M_{\text{hydro}} = \int_{r_{\min}}^{R} \frac{1}{2} \rho c(\theta_e, r) \frac{r - r_{\min}}{\sin \theta_e} \left( (U_{\infty} \cos \delta)^2 + (r\Omega)^2 \right) (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) \, dr. \quad (4.17) $$

Fig. 4.5 the experimental data and the integration from the model for three different pitch angles (colors) and three different flow velocities (markers). For all three pitch angles, the model captures with reasonable accuracy the progressive opening of the blades as a function of the tip speed ratio. The smaller the pitch angle, the higher the tip speed ratio required to extend the blades. It should be noted that the particular behavior of these propellers is a consequence of the fact that we placed the hinge at the trailing edge of the blade. Placing it at the leading edge would modify how they open, but most importantly would prevent the propeller from closing completely because the blades would obstruct each other.

The model always overestimates the forces and thus the opening angle of the propeller. This discrepancy might be due to the shape of the intersections shown in Fig. 4.4 being curved and therefore modifying the sectional lift and drag coefficients. Alternatively, the flow curvature might also modify the force balance. The blades occupy a large angular sector and therefore the lift and drag measured in a flat 2D setting might not apply directly here due to the curvature of the intersection between the cylinders and the blades. Semi-empirical correction for flow curvature effects in blade element model exist (Corrigan and Schillings, 1994), but we did not implement them.
<figure><img src="image_63.png" /><figcaption>Figure 4.5: Opening angle as a function of the tip speed ratio. Color indicates pitch angle. Markers differentiate flow speed. Dark edge color shows experiments ran with one blade instead of three. The lines are from the blade element model described above.</figcaption></figure>

We have found a numerical model that predicts the opening behavior of a foldable propeller. The coupling between the opening angle and the effective pitch angle (due to the large angular extent of the blades) makes them progressively open instead of all at once when reaching a threshold. The pitch angle is essentially 90 deg when the propeller is folded and progressively decreases back to its prescribed value when the blades open.

### 4.1.3 A parametrization of the folding propeller

In the next section, we will build a model of the unsteady motion dynamics. It requires information about the boat and propeller system such as drag, thrust, or torque. Therefore, in this section we make preliminary measurements of these quantities and parametrize their evolution to include them in the model. We use the bio-inspired propeller described in the previous section, which is able to change shape between the active and passive phases, thus mimicking the drag reduction observed in fish. For comparison, we also use a rigid propeller that has the same shape as the folding propeller when extended.

#### Thrust and torque

We evaluate the performance of both propellers by holding the boat in a fixed position with a traction/compression force sensor, thus measuring the axial force $F_a$: the sum of drag and thrust. Unlike all the propeller performance curves presented previously, we do not remove the boat's hull drag here. Indeed, we will be interested in the force balance when the boat is free to move, and therefore we need to take the hull drag into account. We define an axial force coefficient $C_a$ in a similar manner as for the thrust:

$$C_a = \frac{F_a}{1/2\rho\pi R^2 (R\Omega)^2} \qquad (4.18)$$

The axial force and torque coefficients, are plotted in Fig. 4.6 as a function of the tip speed ratio, for the rigid and folding propeller, as well as a typical propeller curve extracted from Boswell (1971) (the exact propeller used has the reference 4381). For the latter, we have added the hull drag, such that
<figure><img src="image_64.png" /><figcaption>Figure 4.6: (•) Axial force $C_a$ and (♦) torque $C_Q$ coefficients as a function of the tip speed ratio $\lambda$ for several propellers. (green) Folding, (blue) rigid, and (orange) propeller 4381 from Boswell (1971). The full lines are the fits the axial force and the dashed those of the torque coefficient, using Eq. 4.19 and 4.20. Experiments with $U_\infty = 0.15$ m/s.</figcaption></figure>

the curves are comparable. We see that all propellers follow the same general trend, although the propeller from the literature is obviously much more efficient (lower torque and higher thrust) than the very simple propellers we use. There is a discrepancy between the folding and rigid propellers, even though they should have the same shape when $\lambda > 3.5$ and the folding propeller is fully opened. Because the folding propeller produces a higher thrust associated with an increased torque, we interpret this fact as the result of the play in the hinges connecting the blades to the hub. This play allows the folding propeller to passively increase its pitch angle like the fully flexible propeller of Sec. 3.1. This discrepancy will not impact the results of this section as we are interested in the comparison with continuous motion for both propellers separately.

We parametrize the evolution of these coefficients with the tip speed ratio to model the unsteady dynamics of the boat. We are interested in the range $4 < \lambda < 8$ because it is the one explored in the experiments. The axial force can be divided into two components: the drag and the thrust. Our goal is to separate these two contributions by fitting the data in Fig. 4.6. The drag scales with the flow velocity squared and is negative, while the thrust scales with $(R\Omega)^2$ and is positive. By assuming that the coefficients of proportionality in both these scalings are constants (of $\lambda$ and $U_\infty$) we can write the axial force coefficient as:

$$C_a = a_T + \frac{c_T}{\lambda^2}, \qquad (4.19)$$

where $a_T$ and $c_T$ are two fitting parameters, representing the thrust and drag respectively. The coefficient $c_T$ represents the drag of the hull and of the rotating propeller (which may be different from the one of the non-rotating propeller). Similar separations have to be made in the case of fish swimming where the motion of the tail both produces thrust and increases the drag (Wu et al., 2007; Floryan et al., 2017a; Akoz and Moored, 2018).

Similarly, we model the torque as the sum of two contributions First, the incoming flow creates a torque that would spin the propeller is it was not attached — like a wind turbine — scaling as $U_\infty^2$ and negative. Second the resistive torque due to the rotation of the propeller, positive and scaling as $(R\Omega)^2$. We thus also have:

$$C_Q = a_Q + \frac{c_Q}{\lambda^2} \qquad (4.20)$$

with $a_Q$ and $c_Q$ two more fitting parameters. It should be noted that with the definitions above $a_T$
and $a_Q$ are positive, while $c_T$ and $c_Q$ are negative. We fit these coefficients to the curves in Fig. 4.6 (lines). This simple model describes well the evolution of both coefficients for $\lambda > 3.5$ which is the regime which we are interested in. We sum up the exact values of the fitting coefficients in Table 4.1.

<table><caption>Table 4.1: Fitting coefficients for the thrust and torque, for all propellers tested</caption><thead><tr><th>Propeller</th><th>a<sub>T</sub></th><th>c<sub>T</sub></th><th>a<sub>Q</sub></th><th>c<sub>Q</sub></th></tr></thead><tbody><tr><td>Rigid</td><td>+0.1111</td><td>-1.6065</td><td>+0.0545</td><td>-0.4196</td></tr><tr><td>Folding</td><td>+0.1268</td><td>-1.3777</td><td>+0.0689</td><td>-0.4082</td></tr><tr><td>4381</td><td>+0.1452</td><td>-1.6259</td><td>+0.0492</td><td>-0.4066</td></tr></tbody></table>

<figure><img src="image_65.png" /><figcaption>Figure 4.7: (left) Thrust and (right) torque coefficients for two different mean speeds, for both propellers.</figcaption></figure>

In the intermittent locomotion experiment, the effective flow speed is constantly changing due to the acceleration of the boat. We thus repeated the force sensing experiment for another mean flow speed (0.11 m/s compared with 0.15 m/s previously). We show the result in Fig. 4.7 and we can see that once made dimensionless according to 1.10 and as a function of the tip speed ratio, both $C_a$ and $C_Q$ are independent of the flow speed (i.e., of the Reynold number). This means that the fitting coefficients found in Tab. 4.1 are indeed independent of the flow speed.

## Drag

Finally, we also measured the drag of the boat in various configurations while the propeller is not spinning, representing the coasting phase. For several flow velocities we measured the drag of the boat without any propeller attached and with the rigid or folding propeller — which in this case is completely folded. We also measured the drag associated with using the rigid propeller, but letting it spin freely. To do so, we disconnect it from the motor because of the gearbox which prevents it from rotating if we keep it engaged.

We plot the drag coefficient $C_D = D/(1/2\rho S_{\text{boat}}U_{\infty}^2)$ in Fig. 4.8. For all configurations except for the disengaged propeller, the drag coefficient can be considered constant in our range of interest ($0.1 \text{ m/s} < U_{\infty} < 0.2 \text{ m/s}$). The presence of the rigid propeller add a substantial drag to the system, and therefore using the folding propeller is very interesting because it creates an important
<figure><img src="image_66.png" /><figcaption>Figure 4.8: Drag coefficient as a function of the mean flow speed. The text corresponds to the value averaged for large mean velocities ($U_\infty > 0.1$ m/s).</figcaption></figure>

drag contrast between active and passive phases. The drag in the active phase is computed using $C_{D,Burst} = -c_T S_{boat} / \pi R^2$ and the drag in the passive phase is the one directly measured in Fig. 4.8. This results in a drag contrast $C_{D,Burst}/C_{D,Coast}$ of about 0.2 for the folding propeller, similar to the ones encountered in fish swimming (Weihns, 1974; Wu et al., 2007). Using this model, we also find that the drag coefficient of the rigid propeller is increased when it is spinning. This results in a moderate drag contrast of about 0.65.

**Freely spinning propeller** Interestingly, letting the propeller spin due to the hydrodynamic torque applied by the flow reduces the drag compared to forcing it to stop, but does not result in a constant drag coefficient. A simple explanation for this phenomenon is that the torque generated by the flow tends to make the blades align with the relative flow they encounter. If the angle of attack on a particular section of the blade is positive, the torque slows the propeller, decreasing the angle of attack and if it is negative the propeller speeds up. Having the blades near zero angle of attack is then associated with a reduced axial force. The details are obviously more complex due to the different angle of attack at different spanwise positions, but the net effect is to reduce the drag. We do not explore this effect further but, instead of using folding propellers like we do in the following, one could instead use a clutch to disengage the propeller in the coast phase.
## 4.2 Intermittent locomotion experiment

### 4.2.1 Setup

<figure><img src="image_67.png" /><figcaption>Figure 4.9: (Left) Position of the boat in the open water case (or in the fluid reference frame) (pink) and in the lab reference frame (cyan). Position is made dimensionless by the distance traveled during one cycle. Experimental signals obtained with the rigid propeller, $U_\infty = 0.18$ m/s, $T = 10$ s and $\Delta = 0.2$. (Right) Examples of trajectories in the Lab reference frame for different duty cycles (from white to red: 0.2, 0.5, and 0.8). Curves obtained with the rigid propeller, with $U_\infty = 0.11$ m/s and $T = 5$ s. The instant $t/T = 0$ corresponds to the moment from which we consider the boat to be stationary on average.</figcaption></figure>

The experiments we perform aim to reproduce the problem of a boat in open water cruising at an average velocity $U_\infty$ while using a burst and coast strategy. Because we are limited by the length of the channel, we change the frame of reference of the study to one that moves at the mean velocity of the boat (hereafter referred to as “lab frame”). This change of frame of reference transforms the accelerated and decelerated motion (while always moving forward) into a back and forth motion of finite amplitude in our channel (see Fig. 4.9 (left)). This transformation is a Galilean change of frame of reference: one is moving with a constant velocity $U_\infty$ with respect to the other. As a result, the dynamics of the problem is unchanged.

The propeller is cyclically switched on and off, and we call the duty cycle ($\Delta$) the ratio of the time spent in the active phase to the period $T$. During the active phase, the propeller rotates at a fixed angular velocity $\Omega$. We release the boat from any position close to the center and let it evolve freely. An experiment is considered successful if the average position of the boat during ten consecutive cycles remains less than 2 cm away from the center of the channel (Fig. 4.9 (right) shows examples of successful experiments for different duty cycles). If it is not the case, we adjust the angular velocity until we reach an equilibrium. Once we have successfully performed one experiment, we can change the duty cycle or the flow velocity and repeat the process. In practice, we saw that the tube providing pressurized air to the linear bearing carriage helps the boat stay just in front or behind the center (discussed in Sec 2.2.3). We therefore tried to have a mean position as close to the center as possible. We repeated several runs for the same parameters and found on average a fluctuation of about 1% of the angular velocity and 3.5% for the average power during a cycle.

We chose the periods such that the boat does not lose a significant portion of its momentum during
the coast phase (corresponding to $U(t) = -U_\infty$ in our case). This time can be found by balancing the inertia of the boat with the dissipation due to drag:

$$m^* \frac{U_\infty}{(1 - \Delta)t_{\max}} \sim \frac{1}{2} \rho S_{\text{boat}} U_\infty^2. \qquad (4.21)$$

We find for the maximum period

$$t_{\max} \sim \frac{m^*}{1/2 \rho S_{\text{boat}} U_{\infty}}, \qquad (4.22)$$

where $m^*$ takes into account the total mass of the system and the added mass due to the acceleration of the surrounding fluid. The typical order of magnitude of $t_{\max}$ in experiments is 10 s. We indeed find for the smallest duty cycle and for the rigid propeller that a substantial amount of the boat inertia has been dissipated in the coast phase. In practice, this limit corresponds to the boat moving at the same velocity as the fluid, and thus having very important position fluctuations in the channel, making reaching a stationary equilibrium more difficult.

A special case is the one of continuous motion, where we manually look for the right angular velocity such that, when releasing the boat right in the middle of the channel, it stays in position for about 30 s. In practice a small drift is always visible, and we repeated this process several times to ensure finding the right value.

<figure><img src="image_68.png" /><figcaption>Figure 4.10: (a) to (c), Raw signals of position, angular velocity and torque, measured with the rigid propeller, $U_\infty = 0.11$ m/s and $T = 5$ s. Panels (d) to (e) show their phase averaged counterparts. Colors identify the duty cycle: from white to red 0.2, 0.5, and 0.8.</figcaption></figure>

In Fig. 4.10 we show the instantaneous signals of position $x(t)$, angular velocity $\omega(t)$ ($\omega(t) = \Omega$ in the burst phase and 0 in the coast phase), and torque $Q(t)$ (frames (a) to (c), respectively). These signals are from successful experiments as the mean position stays close to $x = 0$. The boat starts each burst with backwards momentum and needs to end it with enough forwards momentum to keep a stationary position. The associated instantaneous torque is very noisy, and so we perform a phase averaging operation on the 10 periods that we measure to reduce the noise. By precisely detecting the beginning of each period, we are able to match each instant of one period with all the corresponding ones in the other repetitions. This phase average operation results in a drastic
reduction of the noise. We plot the result of this operation on panels (d) to (f). We can now see that the torque starts with an initially high value and decreases throughout the cycle. Some of this effect may be due to the rotational inertia of the propeller and surrounding water.

<figure><img src="image_69.png" /><figcaption>Figure 4.11: Instantaneous boat speed in the lab reference frame for the rigid propeller (solid line) and for the folding propeller (dashed line). Colors identify the duty cycle: from white to red 0.35, 0.6, and 0.85. Curves obtained with $U_\infty = 0.18$ m/s and $T = 10$ s.</figcaption></figure>

From the position signal that we record, we are able to deduce the instantaneous velocity of the boat in the lab reference frame $U(t)$ by using a robust differentiation method (Savitzky and Golay, 1964). In Fig. 4.11, we show the result of this operation for both types of propeller. The amplitude of the velocity variations is much more important for the rigid propeller than for the folding one. Indeed, the drag being reduced on the boat during the coast phase means that there is less of a need to accelerate in the burst phase. This is one of the key aspects of efficient burst and coast locomotion as we will see in the following section. Similarly, for smaller duty cycles, the variations get increasingly important.

The parameter which we want to compute to assess the efficiency of burst and coast locomotion is the power requirement of each propulsion style. In our case it is the mechanical power $P(t) = Q(t)\omega(t)$. We plot this power in Fig. 4.12 panel (a), relative to the typical power dissipated when moving through a fluid at a velocity $U_\infty$. This power comes from the product of drag scaling like $1/2\rho S_{\text{boat}}U_\infty^2$ and the mean velocity $U_\infty$ such that:

$$P_0 = \frac{1}{2} \rho S_{\text{boat}} U_\infty^3. \qquad (4.23)$$

Both the instantaneous torque and angular velocity increase in the burst phase when reducing the duty cycle, resulting in an almost five-fold multiplication of the instantaneous power requirement for $\Delta = 0.25$. However, whether a cycle is efficient requires comparing the energy expended per cycle and this discussion is the topic of the following section.

Other parameters of interest can be obtained from the experiment such as the instantaneous tip speed ratio $\lambda(t) = R\omega(t)/(U_\infty + U(t))$ (Fig. 4.12 panel (b)). At the beginning of the burst phase, the total velocity is small, and thus the tip speed ratio is high and decreases during the active phase as the boat accelerates. At constant angular velocity, the torque is a growing function of $\lambda$ (See Sec. 1.2). The decreasing trend in the torque seen in Fig. 4.10 therefore might be explained by the evolution of $\lambda(t)$.
<figure><img src="image_70.png" /><figcaption>Figure 4.12: Instantaneous examples of (a) the dimensionless power, (b) tip speed ratio and (c) boat velocity. The characteristic power is $P_0 = 1/2\rho S_{\text{boat}}U_{\infty}^3$. Curves obtained with the folding propeller, $U_{\infty} = 0.18 \text{ m/s}$ and $T = 10 \text{ s}$. Colors identify the duty cycle: from white to red 0.25, 0.5, and 0.75.</figcaption></figure>

### 4.2.2 Cycle-averaged results

While the phase averaged signals of the previous sections give insights into the physical mechanisms at play in the problem, comparing intermittent and continuous locomotion requires quantities representing the whole cycle.

**The force balance sets the angular velocity**

The first quantity of interest is the angular velocity during the burst phase $\Omega$ (Fig. 4.13). It increases when decreasing the duty cycle: staying at a given mean velocity implies that drag during the whole cycle can be approximated as being the same for all duty cycles. Therefore, the total thrust required to counteract the drag is also similar, but needs to be generated within shorter time intervals ($\Delta \times T$) as the duty cycle decreases. As a result, the instantaneous thrust during the burst phase — and the angular velocity needed to produce it — increase with decreasing duty cycle. Another striking result is visible in Fig. 4.13: the period of the motion has no influence on the angular velocity, except for the shortest period at the lowest speed (light blue dots on panel (a)). Changing the mean velocity shifts all the curves globally: the tip speed ratio $\lambda = R\Omega/U_{\infty}$ of continuous motion is a constant. This result is derived from the propeller parametrization from the previous section: during continuous locomotion the only force acting on the boat is the axial force (the sum of the thrust and drag). From Eq. 4.19, we obtain the tip speed ratio for continuous motion:

$$ \lambda_c = \sqrt{-\frac{c_T}{a_T}}. \qquad (4.24) $$

We can then build a scaling law for the angular velocity dependence on the duty cycle. The typical scaling of the thrust produced by a rotor is

$$ F_{\text{thrust}}(t) \sim \frac{1}{2} \rho \pi R^2 (R\omega(t))^2, \qquad (4.25) $$
<figure><img src="image_71.png" /><figcaption>Figure 4.13: Angular velocity in the burst phase as a function of the duty cycle for three different periods and flow speeds. (♦) Rigid propeller, (●) folding propeller. (dashed line) scaling from Eq. (4.28) (using the rigid propeller parametrization). (dot dashed line) scaling $\Omega \sim \Delta^{-1/3}$ with the angular velocity of the continuous motion (folding propeller) as a reference value. (dotted line) scaling from Eq. (4.28) (using the folding propeller parametrization). Panels correspond to different mean velocities.</figcaption></figure>

while the drag on the whole boat scales as:

$$D \sim \frac{1}{2} \rho S_{\text{boat}} (U_{\infty} + U(t))^2. \qquad (4.26)$$

The balance of the forces on the boat during one cycle is $\langle F_x \rangle = \langle D \rangle$, where $\langle \cdot \rangle$ denotes the average over a cycle. The period of the motion does not affect the angular velocity required and therefore the force balance. As a consequence, we assume that the velocity variations around the mean are negligible (Fig. 4.11 shows that at most $(\delta U/U_{\infty})^2 \lesssim 1/4$), such that (i) we can approximate the average drag by the drag at the average velocity and (ii) the thrust coefficient is constant throughout the burst phase. The thrust is active only during the burst phase and thus and thus the balance reads:

$$\frac{1}{2} \rho \pi R^2 (R\Omega)^2 \Delta \sim \frac{1}{2} \rho S_{\text{boat}} U_{\infty}^2 \langle C_D \rangle. \qquad (4.27)$$

The drag coefficient is not a constant throughout the cycle. For the folding propeller it changes drastically when the propeller closes. The rigid propeller drag coefficient is also modified between the phases of motion according to the parametrization from Sec. 4.1.3. The average drag coefficient is therefore: $\langle C_D \rangle = C_{D,B}\Delta + C_{D,C}(1 - \Delta)$. We use here $C_{D,B}$ and $C_{D,C}$ which are the drag coefficients of the burst and coast phases, respectively. We directly measured the drag coefficient in the coast phase, and we model the drag coefficient in the burst phase using the fit of the thrust (Fig. 4.6 and Tab. 4.1): $C_{D,B} = -\pi R^2 / S_{\text{boat}}c_T$. The scaling law for the evolution of the angular velocity therefore reads:

$$\Omega \sim \Omega_c \sqrt{\frac{C_{D,C}}{C_{D,B}} \left( \frac{1}{\Delta} - 1 \right) + 1}, \qquad (4.28)$$

with $\Omega_c = \lambda_c U_{\infty}/R$ the angular velocity of continuous motion. The drag contrast $C_{D,C}/C_{D,B}$ involved in this equation is equal to 0.20 for the folding propeller and 0.65 for the rigid one. We plot this scaling in Fig. 4.13 (dotted line and dashed line), the agreement with the data is good, especially
<figure><img src="image_72.png" /><figcaption>Figure 4.14: Power coefficient as a function of the duty cycle for the folding propeller for different velocities and different periods: from light to dark blue $T = 2.5, 5$ and $10$ s. (light gray line) continuous power coefficient (corresponding to the rightmost point). Insets show the same graph but for the rigid propeller (Note the different vertical scale).</figcaption></figure>

for the longer periods. This result validates the idea that velocity fluctuations play a minor role in setting the angular velocity. This model does not take into account period dependent effects and therefore cannot explain the difference observed for shorter periods.

## Efficiency of burst and coast locomotion

To compare the efficiency of different styles of locomotion we introduce the power coefficient $C_P$ defined as

$$C_P = \frac{\langle P(t) \rangle}{P_0} = \frac{\langle Q(t) \omega(t) \rangle}{1/2 \rho S_{\text{boat}} U_{\infty}^3} \quad (4.29)$$

Similarly to the thrust, the torque scales like the angular velocity squared (Eq. (3.24)). The torque is also modified when changing the tip speed ratio as the torque coefficient is not a constant (Eq. (4.20)). If this variation is small, at a given $U_\infty$ and $T$, and neglecting velocity variations, this means that the power coefficient scales like:

$$C_P \propto \Omega^3 \Delta. \quad (4.30)$$

The power coefficient must be increasing for increasing duty cycle intermittent motion to be more efficient than continuous motion. A necessary criterion can therefore be written as:

$$\Omega(\Delta) < \Omega_c \Delta^{-1/3}. \quad (4.31)$$

This limit is plotted in Fig. 4.13 (dot dashed line) and we can already see, as expected, that intermittent locomotion with the rigid propeller has no chance of being efficient as the scaling from Eq. (4.28) with a drag contrast of 0.65 grows faster than $\Delta^{-1/3}$. The variation of the torque coefficient further reduces this limit as at low duty cycles, the propeller experiences a higher tip speed ratio in the burst phase and higher tip speed ratios are associated with an increased torque coefficient (Fig. 4.6).

From our measurements of instantaneous torque and angular velocity, we compute the power coefficient directly. We plot it in Fig. 4.14 for the folding propeller for the three different flow velocities
(panels (a) to (c)) and for the three different periods (color). For each flow velocity there is a set of parameters for which the intermittent motion is more efficient than the continuous motion (gray line). This energy saving reaches a maximum of 25% (for the slowest mean flow) and a reduction of 15% can be achieved at any flow velocity. For all tested velocities, the duty cycle that optimizes energy savings is about 0.3. The insets show the corresponding power for the rigid propeller. Consistent with the prediction from condition (4.31), burst and coast locomotion is always less efficient than continuous locomotion. In fact, it increases the cost by more than a factor of three.

The general trend in all cases tested is that longer periods are beneficial to efficiency. This result may be due to the rotational inertia of the propeller and the surrounding water that the motor must overcome at the beginning of each burst phase, but does not recover at the end of the burst. This constant cost is comparatively less important when using a longer period.

One interesting note is that the total power produced by the motor benefits even more from intermittency than the purely hydrodynamic analysis above (See Fig. 4.15). In all the results presented, we use the hydrodynamic torque, which is obtained by subtracting the parasitic torque due to friction in the bearings from the total torque. This parasitic torque has an almost constant value over the angular velocity range and therefore shorter bursts are associated with less power dissipated in the bearings. In a real situation, the cost would be measured on the total power, and so this additional reduction would be beneficial: a 15% energy saving from the hydrodynamic contribution alone becomes more than 20% savings with the added friction.

<figure><img src="image_73.png" /><figcaption>Figure 4.15: Power coefficient as a function of the duty cycle before (blue) and after (green) subtracting the power dissipated by mechanical friction. (Obtained with $U_{\infty}$ = 0.18 m/s and $T$ = 10 s.)</figcaption></figure>

## A side note on energy per distance

In all the above considerations, we compared cycles keeping the mean velocity constant. Not changing the end use (getting from point A to point B in a fixed time) is the natural choice for comparing locomotion styles. However, some authors, especially in the animal locomotion literature, compare cycles with a fixed actuation on a per-distance basis (Blake, 1983; Floryan et al., 2017b). This makes sense for animal locomotion, as physiological constraints may prevent access to the most efficient gait. However, energy savings are then dominated by reductions in average speed, as power scales with the cube of speed. In animals, metabolic rate (a constant energy expenditure over time) can partially compensate for this effect, but it is not present in our system.

To investigate what would happen in a system where the angular velocity is fixed, but the mean velocity allowed to change we follow the steps proposed in Floryan et al. (2017b) for propulsion with a pitching airfoil. We compare the energy expenditure per distance traveled with the same angular velocity — for fish swimming the tail beat frequency and amplitude is fixed. The energy ratio between intermittent and continuous motion, travelling a distance $d$, reads:

$$ \zeta = \frac{P(\Delta)d/U_{\infty}(\Delta)}{P_c d/U_c} = \frac{P(\Delta)U_c}{P_c U(\Delta)}, \quad (4.32) $$

where $U_c$ and $P_c$ are the velocity and power associated with the continuous cycle. We did not perform constant actuation experiments and thus $U_c$ and $P_c$ are computed from the static thrust and torque measurements that we performed. Using the data obtained when the boat was fixed in Sec. 4.1.3, for
a given angular velocity we have:

$$U_c(\Omega) = \frac{R\Omega}{\lambda_c}, \quad P_c(\Omega) = \frac{1}{2}\rho\pi R^2 (R\Omega)^3 C_Q \left(\frac{R\Omega}{U_c}\right). \qquad (4.33)$$

For a given period, mean flow velocity $U_\infty$, and duty cycle $\Delta$, we measure the angular velocity and power required to sustain intermittent motion. We then calculate the average velocity and power required to sustain continuous motion at the same angular velocity using the equations above.

<figure><img src="image_74.png" /><figcaption>Figure 4.16: Energy ratio between intermittent and continuous motion per distance travelled. Colors and symbols are identical to Fig. 4.14</figcaption></figure>

We plot the ratio $\zeta$ in Fig. 4.16 and, as expected with this metric, intermittent locomotion is almost always advantageous compared with continuous locomotion. In the previous sections, coasting did not cost any energy, but required more thrust to be generated in the burst phase to maintain a constant average speed. Here, coasting is truly "free" because we have removed the mean velocity constraint.

## Amplitude of velocity variations

An important parameter of intermittent locomotion is the amplitude of the variations in velocity (Weihs, 1974, 1981). Indeed, the main source of dissipation added when using burst and coast, is the quadratic dependence of the drag with the velocity at high Reynolds numbers. The effect of this quadratic dependence is that the average drag when using intermittent locomotion is greater than the drag at the average velocity:

$$\langle D \rangle \sim \frac{1}{2} \rho S_{\text{boat}} \langle (U_{\infty} + U(t))^2 \rangle = \frac{1}{2} \rho S_{\text{boat}} (U_{\infty}^2 + \langle U(t) \rangle^2) > \frac{1}{2} \rho S_{\text{boat}} U_{\infty}^2. \qquad (4.34)$$

This variation in velocity can be estimated using the coast phase dynamics: the only force acting on the boat is the drag and the equation of motion reads

$$m^* \frac{dU}{dt} = -\frac{1}{2} \rho S_{\text{boat}} C_D (U_{\infty} + U(t))^2. \qquad (4.35)$$
If the variations in velocity are small with respect to the mean speed, we can rewrite this equation as a scaling law:

$$m^* \frac{U_{\max} - U_{\min}}{(1 - \Delta)T} \sim \frac{1}{2} \rho S_{\text{boat}} C_D U_{\infty}^2. \qquad (4.36)$$

Finally, the amplitude of the velocity variations scale like

$$A_u = (1 - \Delta)T \frac{1/2 \rho S_{\text{boat}} C_D U_{\infty}^2}{m^*}. \qquad (4.37)$$

We plot the standard deviation of the velocity in Fig. 4.17 as a function of this scaling for both propellers using the drag coefficients measured in 4.1.3. We can see that it describes very well the scaling of the standard deviation of the velocity. The coefficient of this scaling law is 0.18.

Variations in velocity are also associated with a peak of tip speed ratio at the beginning of the cycle (See Fig. 4.12) which is also detrimental to performance as high tip speed ratios are linked to high torque coefficients. Everything else equal, minimizing the velocity variations seems beneficial for efficiency. However, we see in Fig. 4.14 that longer periods (associated with important amplitude) are generally more efficient than shorter ones. Added to that the scaling laws plotted in Fig. 4.13 do not take this effect into account and still describe well the evolution of the angular velocity $\Omega$ with the duty cycle $\Delta$. These clues show that while high velocity variations add dissipation, the magnitude of this effect is small in the experiments compared with a change in the drag contrast or the period.

In the following we clarify the influence of the amplitude of the velocity variations on the performance using a model that describes the dynamic behavior or the boat during the cycle.

<figure><img src="image_75.png" /><figcaption>Figure 4.17: Standard deviation of the boat velocity as a function of the theoretical amplitude defined in 4.37. (green line) linear least square regression of the data with a slope of 0.18</figcaption></figure>

### 4.2.3 Minimal drag reduction

In this section, we construct a model which is able to reproduce the dynamic motion of the intermittent locomotion experiment. The goal of this model is twofold: (i) to predict the exact parameters for which propulsion is most efficient (period, duty cycle, etc.) and (ii) to identify the region in the parameter space where burst and coast is more efficient than continuous motion. Because we have seen in Fig. 4.14 that long periods most efficient, we will in the following neglect any unsteady dissipative effect.
The main force acting on the boat throughout the cycle is the drag force. In a general unsteady setting, this force on a blunt object can be divided into three terms (Rivero et al., 1991):

$$D = \underbrace{\frac{1}{2} \rho S_{\text{boat}} (U_{\infty}^2 + U^2(t)) C_{D,C}}_{\text{Quasi-steady}} + \underbrace{m_a \frac{dU}{dt}}_{\text{Added mass}} + \underbrace{F_H(t)}_{\text{History}} \quad (4.38)$$

The first one is the quasi-steady drag, which is the same as the drag with a steady flow, with the time dependent velocity. The added mass is a potential correction, proportional to the acceleration which represents the inertia of the water the boat has to accelerate with it. This term is not dissipative as it simply adds more inertia to the motion: it slows the boat on acceleration but helps conserve momentum when slowing down. The last term is the history force, which contains all the remaining phenomena due to the acceleration. In the small Reynold number limit, this force takes the form of an integral over the whole history of the acceleration of the object (also called the Basset force). This remaining force might also contain unsteady wave drag (Dode et al., 2022) or shape specific vortex shedding along the live edge of the boat. Our goal is to construct a minimal model capable of describing the experimental observations of long period dynamics, and we will therefore neglect these contributions.

We divide the model according to the two phases of motion, assuming an instantaneous transition between the two.

**In the coast phase** the only force acting on the boat is the drag. The equation for the total velocity of the boat in this phase ($U_1$) reads:

$$m^* \frac{dU_1}{dt} = -\frac{1}{2} \rho S_{\text{boat}} U_1^2(t) C_{D,C}, \quad (4.39)$$

where $C_{D,C}$ is the drag coefficient during the coast phase, and $m^* = m_{\text{boat}} + m_a$ is the total inertia of the system taking into account the added mass.

**In the burst phase** the boat propels forward, and the drag is still resisting the motion. We have seen in Sec. 4.1.3 that the combination of these forces can be modelled in the as the sum of a term proportional to $(R\Omega)^2$ and a term proportional to $U_2^2$, with $U_2$ the total velocity in the burst phase. The equation of motion is:

$$m^* \frac{dU_2}{dt} = \frac{1}{2} \rho \pi R^2 (a_T (R\Omega)^2 + c_T U_2^2(t)), \quad (4.40)$$

where $a_T$ and $c_T$ are two coefficients as defined in Eq. (4.19) and Tab. 4.1.

Finally, the boat speed being periodic and continuous gives the boundary conditions:

$$U_2(\Delta T) = U_1(\Delta T), \quad U_1(T) = U_1(0). \quad (4.41)$$

We make this system dimensionless using the period $T$ as the reference time and $R\Omega$ as the reference velocity. We denote the dimensionless velocities with a lowercase $u$ and the dot ($\dot{u}$) denotes a dimensionless time derivative. The system now reads:

$$\dot{u}_2 = M (a_T + c_T u_2^2), \quad (4.42)$$

$$\dot{u}_1 = -M \frac{S_{\text{boat}}}{\pi R^2} C_{D,C} u_1^2, \quad (4.43)$$

$$u_2(\Delta t) = u_1(\Delta t), \quad u_1(1) = u_2(0). \quad (4.44)$$
<figure><img src="image_76.png" /><figcaption>Figure 4.18: Power coefficient, relative to the continuous motion for (a) the folding propeller and (b) the rigid propeller ($T \ge 5$ s). (Full line) Result from the model with the same parameters, (dashed line) solution from Eq. 4.52.</figcaption></figure>

The inertial number $M$ is a dimensionless parameter that compares the fluid inertia to the one of the boat:

$$M = T\Omega \frac{\rho\pi R^3}{2(m + m_a)} \qquad (4.45)$$

The equations of motion integrate into:

$$ \begin{aligned} u_2(t) &= \sqrt{\frac{-a_T}{c_T}} \tanh \left( \sqrt{-a_T c_T} (B + Mt) \right), \\ u_1(t) &= \left( A + M \frac{S_{\text{boat}}}{\pi R^2} C_{D,C} (t - \Delta) \right)^{-1}, \end{aligned} \qquad (4.46) $$

where $A$ and $B$ are integration constants and the dimensionless time is also referred to as $t$ for convenience. It should be noted that $\sqrt{-a_T c_T}$ is well-defined as $a_T$ and $c_T$ have opposite signs. The boundary conditions close the system, giving two equations for $A$ and $B$:

$$ \begin{aligned} \sqrt{\frac{-a_T}{c_T}} \tanh \left( \sqrt{-a_T c_T} (B + M\Delta) \right) &= A^{-1} \\ \sqrt{\frac{-a_T}{c_T}} \tanh \left( \sqrt{-a_T c_T} B \right) &= \left( A + M \frac{S_{\text{boat}}}{\pi R^2} C_{D,C} (1 - \Delta) \right)^{-1} \end{aligned} \qquad (4.47) $$

We numerically solve the system for the parameters $A$ and $B$ using a root finding algorithm. Once the time dependent velocity $u(t)$ is known, we can compute all the other parameters of interest in the system: mean velocity $\langle u(t) \rangle$, time dependent tip speed ratio in the burst phase $\lambda(t) = 1/u_2(t)$, and finally the power coefficient using Eqs. (4.20), (4.29) and the table 4.1, giving:

$$ \begin{aligned} C_P &= \frac{\langle Q(t)\omega(t) \rangle}{1/2\rho S_{\text{boat}} U_{\infty}^3} = \frac{\pi R^2}{S_{\text{boat}}} \frac{1}{\langle u(t) \rangle^3} \int_0^{\Delta} C_Q(\lambda(t)) \, dt \\ &= \frac{\pi R^2}{S_{\text{boat}}} \frac{1}{\langle u(t) \rangle^3} \int_0^{\Delta} \left( a_Q + \frac{c_Q}{\lambda^2(t)} \right) \, dt. \end{aligned} \qquad (4.48) $$
The parameters $a_Q$ and $c_Q$ are measured using the static experiments, and there are therefore no adjustable parameters in the model.

We plot the relative modification of power consumption compared to continuous motion in Fig. 4.18 for the model above (full line) and the experiments with a long period. The model is able to reproduce the power requirement for both propellers fairly well. In particular the duty cycle corresponding to optimal efficiency matches the experimental data. It tends to underpredict the power at low duty cycles, most likely because we have neglected unsteady dissipative effects.

Our model matches well with experiments, and in the following we use it to analyze the effect of changing the drag contrast $C_{D,C}/C_{D,B}$ and the inertial number $M$. The performance of intermittent locomotion is quantified by the maximal efficiency improvement:

$$ \tau = - \min_{\Delta} \frac{C_P(\Delta) - C_P(1)}{C_P(1)}, \qquad (4.49) $$

and the associated duty cycle:

$$ \Delta_{\max} = \underset{\Delta}{\operatorname{argmin}} \frac{C_P(\Delta) - C_P(1)}{C_P(1)}. \qquad (4.50) $$

In addition to varying numerically the drag contrast, we also performed experiments with an increased boat draft (we increase the volume of the hull actually in water by changing the water depth). Doing this increases the hull drag which is present both in the active and passive phases and thus decreases the contrast (going from $C_{D,C}/C_{D,B} = 0.20$ to 0.26).

<figure><img src="image_77.png" /><figcaption>Figure 4.19: (a) Optimal efficiency improvement as a function of the drag contrast between the active and passive phases. Error bars correspond to uncertainty in the data: estimated by repeating experiments (b) Corresponding optimal duty cycle, as a function of the drag contrast. Error bars correspond to the width of the range where the power coefficient is less than 5% away from the optimum. Markers correspond to experimental points, where water depth is (•) 190 mm and (♦) 200 mm. Full lines correspond the model.</figcaption></figure>

We plot these two quantities for the model and the experimental data in Fig. 4.19 and we once again see a good agreement between both series of experimental points and the model. The model highlights a key result of the intermittent locomotion system: if $C_{D,C}/C_{D,B} \gtrsim 0.35$, continuous locomotion is the most efficient propulsion style. In other words, for burst and coast locomotion
to be more efficient than continuous locomotion, the total drag must be reduced by at least 65% in the coast phase. Below this threshold, the efficiency improvements increase monotonically when $C_{D,C}/C_{D,B}$ is reduced. At the same time, the duty cycle for which this optimum occurs is reduced and approaches zero.

### Velocity variations

<figure><img src="image_78.png" /><figcaption>Figure 4.20: (a) Instantaneous velocity as a function of time for (red) experimental data, (green) model with different masses ($U_\infty = 0.15$ m/s, $T = 10$ s, $\Delta = 0.5$, rigid propeller). (b) and (c) associated angular velocity (compared to the angular velocity of continuous motion) and power coefficient respectively, as a function of the inertial number $M$. (gray line) Experimental value of $M$.</figcaption></figure>

The model predicts that slower average speeds are benefit more from intermittent locomotion because the associated velocity variations are smaller, reducing the average drag and the variations of the tip speed ratio. The amplitude of the motion appears to play an important role in determining the efficiency of a given cycle, albeit secondary to drag reduction. This amplitude is modified by the period and the flow velocity, but also by the inertia of the system, visible in the inertial number $M$.

Fig. 4.20 shows the instantaneous velocity for experimental data ($M_{exp} \approx 8$), and numerical results with different $M$ keeping the mean velocity constant (the angular velocity in the burst phase therefore varies). The resulting velocity profiles are widely different: from high accelerations and important velocity variations at low inertia (high $M$) to almost flat profiles at high inertia (low $M$). The associated angular velocity in panel (b) grows monotonically with $M$ (i.e., when the relative inertia of the boat is reduced). This indicates that the required thrust to sustain the motion also grows: the average drag gets increasingly important for small inertia. The power coefficient also increases through two mechanisms: (i) at a constant mean velocity, the power coefficient is proportional to $\Omega^3$ and (ii) when there are important velocity variations, the tip speed ratio peaks at the beginning of the cycle (See Fig. 4.12) and high tip speed ratios are associated with high torque. A similar result was also obtained by Li et al. (2023) in fish, were they found that important velocity variations are associated with poor energy efficiency. Optimized burst and coast strategies in fish involve moderate velocity variations and small periods.

Minimizing the velocity variations is therefore beneficial for the efficiency. This can be done by adding mass to the boat or shortening the period. Experimentally, we have seen that unsteady dissipative effects in practice limit the minimal possible period. In the experiments we have $(0.9 \lesssim M \lesssim 12)$ and
in the numerical model we use a total mass $m^* = 3$ kg. We repeated a series of experiments with the folding propeller, adding a 4 kg weight to the boat ($U_\infty = 0.15$ m/s, $T = 5$ s). We found a very small 2% average decrease of the power coefficient throughout the range duty cycle. This reduction is smaller than the uncertainty of our experiments and highlights the diminishing returns when $M$ approaches 0. This shows that in the case of our experiments, increasing the period is beneficial as it reduces the unsteady dissipative effects more than it increases the cost because of the amplitude of the velocity variations. However, we expect that there should be a limit period where the opposite occurs and the cost increases again with period. The velocity variations should begin to dominate when a significant amount of momentum is lost during the coast phase. This corresponds to the limit period $t_{max}$ (defined in equation (4.22)).

## Quasi-steady limit

In the extreme case of zero velocity variations in the active phase ($M \to 0$), the expression for the power coefficient can be further simplified:

$$C_P = \frac{\pi R^2}{S_{\text{boat}}} \left( \frac{R\Omega}{\langle U_{\infty} + U(t) \rangle} \right)^3 \int_0^{\Delta} C_Q(\lambda(t)) \, dt \approx \frac{\pi R^2}{S_{\text{boat}}} \left( \frac{R\Omega}{U_{\infty}} \right)^3 \Delta C_Q \left( \frac{R\Omega}{U_{\infty}} \right). \quad (4.51)$$

Using Eq. 4.28, we can replace $\Omega$ and obtain:

$$C_P = \Delta \frac{\pi R^2}{S_{\text{boat}}} \left( \frac{R\Omega_c}{U_{\infty}} \right)^3 \left( \frac{C_{D,C}}{C_{D,B}} \left( \frac{1}{\Delta} - 1 \right) + 1 \right)^{3/2} \left[ a_Q + \frac{c_Q (U_{\infty}/R\Omega_c)^2}{C_{D,C}/C_{D,B} (1/\Delta - 1) + 1} \right], \quad (4.52)$$

where $R\Omega_c/U_\infty = \lambda_c = \sqrt{-c_T/a_T}$ is the tip speed ratio of continuous motion. This law does not depend on the inertial number $M$ anymore: velocity variations have no influence on the power coefficient in this limit (nor do the period, or inertia). The flow velocity only enters the equation through the continuous tip speed ratio and therefore this law does not depend on the absolute velocity anymore. The dashed line in Fig. 4.18 displays this asymptotic limit, and we can see that even though it is very simplified, it predicts relatively well the evolution of the power coefficient. It obviously underpredicts the power requirement as we have neglected all the dissipation sources mentioned above.

This law is very similar to the one obtained by Floryan et al. (2017b), which in the case of intermittent propulsion with a pitching airfoil found $P \propto \Delta(C_{D,C}/C_{D,B} (1/\Delta - 1) + 1)^{3/2}$ in our notations (the paper uses a slightly different convention). Their law is equivalent to having a constant torque coefficient throughout the range of tip speed ratio for us (i.e., $c_Q = 0$). This difference comes from the fact that the power coefficient in fish swimming scales like the Strouhal number cubed ($St = fA/U_\infty$ where $f$ and $A$ are respectively the tail beat frequency and amplitude), without a negative offset like in our case. In this simpler case, the onset of efficient burst and coast locomotion is equivalent to threshold of $\Omega(\Delta) < \Omega_c\Delta^{-1/3}$, which gives:

$$\frac{C_{D,C}}{C_{D,B}} \left( \frac{1}{\Delta} - 1 \right) + 1 < \Delta^{-2/3}. \quad (4.53)$$

By differentiating the previous relation with respect to $\Delta$, we obtain that in this case burst and coast is more efficient than continuous motion only if:

$$\frac{C_{D,C}}{C_{D,B}} < \frac{2}{3}. \quad (4.54)$$
<figure><img src="image_79.png" /><figcaption>Figure 4.21: (left) Efficiency improvement for different propeller models as a function of the drag contrast. (dashed line) Efficiency improvement for the modified cycle with constant $\lambda$. (dotted line) Efficiency improvement with the analytical model, using the folding propeller. (right) Efficiency improvements changing the parameter $M$, (dotted line) Efficiency improvement with the analytical model</figcaption></figure>

This threshold is much easier to reach than the one we found for propellers ($C_{D,C}/C_{D,B} < 35\%$), indicating that the burst and coast strategy is more suited to fish-like locomotion than our propeller based system.

Unfortunately, the more complex law 4.52 cannot give such a simple analytical threshold for efficient intermittent locomotion. In the following paragraphs we describe how this threshold is modified by the change of some parameters of our system. We plot the optimal efficiency improvement $\tau$ as a function of the drag reduction computed from the full model in Fig. 4.21 (right) for several inertial numbers $M$. We see as expected that the smaller $M$ is (the higher the relative boat inertia), the more efficient burst and coast becomes. The values eventually converge as $M \to 0$ towards the scaling of Eq. 4.52 (dotted line), indicating that the analytical description above corresponds to the natural limit of infinite inertia. The zero velocity variation limit is equivalent to performing the experiments with a fixed boat instead of allowing it to move back and forth. Doing so results in an underprediction of the cost of the locomotion, even if unsteady dissipation is neglected. For all these curves, the threshold where burst and coast becomes more efficient than continuous locomotion ($C_{D,C}/C_{D,B} < 35\%$) is essentially unchanged.

In all the previous analysis of the efficiency improvement $\tau$ we have used the fitting coefficients from the folding propeller, but we can also use the slightly different ones from the rigid propeller or the propeller 4381. Doing so at fixed $M$ and flow velocity, gives the curves on the left panel of Fig. 4.21 (full lines). The folding and rigid propellers curves superimpose while the more efficient propeller 4381 gives a slightly lower value at $C_{D,C}/C_{D,B} < 33\%$ (keeping in mind that we have had to manually add the hull drag to the fitting coefficients here). Still, the order of magnitude remains similar which consolidates the result, especially given that the propeller from the literature has a maximal efficiency of about 0.7, compared to 0.25 for ours.
## Constant tip speed ratio cycle

Finally, another actuation type could be considered: instead of using a constant angular velocity burst phase, we can use a more progressive constant tip speed ratio burst phase. The total velocity is small at the beginning of the burst phase, and so the angular velocity as well. Both increase at the same time during the burst phase. This is more efficient than the constant $\Omega$ cycle as the acceleration is more progressive and the torque coefficient remains constant. However, it is more difficult to perform experimentally as it would be very sensitive external variations, or we would have to use a control algorithm to keep the tip speed ratio constant. We did not perform this type of experiment, but we can analyze numerically what the change would be.

Writing the system of equations Eq. (4.39) and (4.40) in this new setting gives:

$$ 
\begin{aligned}
\dot{u}_1 &= -\frac{1/2\rho S_{\text{boat}} U_{\infty} C_{D,C}}{m^*} u_1^2(t), \\
\dot{u}_2 &= \frac{1/2\rho\pi R^2 U_{\infty}}{m^*} T(a_T \lambda^2 + c_T) u_2^2(t),
\end{aligned}
\qquad (4.55) $$

where we have used $U_{\infty}$ as the reference velocity instead of $R\Omega$. By imposing the boundary conditions and $\langle u(t) \rangle = 1$, we obtain the tip speed ratio $\lambda$ that is solution to the system. We plot optimal efficiency improvement using this cycle in Fig. 4.21 (left, dashed line) for the same parameters as the folding propeller. The constant $\lambda$ cycle is indeed slightly more efficient than the one with constant $\Omega$ for all drag reductions. However, this improvement is very small: the high inertia limit is still more efficient than this cycle. One benefit of this cycle not visible here is that we have seen experimentally that shorter periods are less efficient, maybe because of the angular inertia that the motor has to overcome. Using the constant $\lambda$ cycle might reduce this effect and thus make the cycle more efficient.

## 4.3 Conclusion

This chapter presents how intermittent locomotion — an idea inspired from studying fish swimming — might apply to the movement of a boat with a propeller. Using a small scale experiment able to reproduce all the unsteady effects of an open water test we describe the behavior of such a system. We measure the time dependent torque and angular velocity, allowing us to compute the cycle averaged energy cost of each propulsion strategy.

Comparing intermittent and continuous locomotion at the same mean velocity, we show experimentally that the burst and coast strategy can be more efficient than continuous locomotion. Using a bioinspired propeller able to dynamically reconfigure between the active and passive phases — reducing the drag of the system by 80% — the cost of sustaining the motion can be reduced by about 15% at any speed and up to 25% for the most efficient cycle. The longest periods avoid unsteady dissipative effects and are therefore the most efficient.

We develop an analytical model reproducing the main forces acting on the boat during the cycle, based on the scaling of the thrust and torque coefficients of the propeller. This model agrees well with the experiments with no fitting parameters once the important properties (drag, thrust, and torque) of the boat are measured. We highlight the importance of the drag contrast between the active and passive phases: the drag must be reduced by 65% in the coast phase in order for intermittent locomotion to have an energetic interest. Finally, we show that inertia plays a secondary role in the problem, but that high inertia, with small velocity variations is the most efficient.

An interesting perspective of this work is the precise role of the period of the motion. Experimentally, we see that longer periods are increasingly efficient. This trend should eventually be reversed, since
the model shows that increasing the period (or *M*) leads to higher velocity variations, which increase the cost of a given cycle. At some point, this effect should begin to dominate, as velocity saturates when the period is too long. We therefore expect that there should be an optimal period in between. Finding this optimum requires modeling the unsteady dissipation that occurs in the short period experiments. Another aspect not explored here, but relevant to the problem, is the accurate modeling of unsteady wave drag. Dode et al. (2022) recently found that the mean drag on a surface-piercing towed hull can be reduced if its velocity oscillates around a mean value. Therefore, setting the period accordingly can further reduce the energy required for propulsion. It should be noted that this effect occurs for Froude numbers and periods that are far from those we use, so our setup does not benefit (or suffer) from this effect.

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Conclusion

This thesis explored the possibility of using new bio-inspired strategies to improve the energy
efficiency of propulsion in water. It consists mainly of experimental work carried out at model scale
in a water channel built for this purpose. We focused on two axes: reconfiguration and intermittency.

We built and analyzed the behavior of flexible propellers that can passively change their shape. The first one, a fully flexible propeller with ribbon-like flexible blades, proved to be a very interesting fluid-structure interaction problem. It has two deformation directions: bending and spanwise twisting, which are coupled by the hydrodynamic loading. Using a simple geometric argument, we were able to predict the direction and approximate magnitude of the moderate bending deformation. We showed that for any given deformation, there is a strong relationship between the bending angle and the thrust generated by the rotor, allowing the measurement of only one of these quantities.

However, this propeller design is very inefficient because of the bending, and therefore we explored
another design with a rigid leading edge. In this case, the twisting deformation acts to reduce the
angle of attack of the blades. By choosing the right material properties, we have shown that these
blades can be more efficient than any rigid equivalent and passively adapt their pitch angle to
varying conditions. The blades also experience spanwise torsion and as a result adopt a shape that
improves their maximal efficiency. The tip twist is a key parameter to track the efficiency of the rotor,
and we were able to predict its value using a linear development of the constitutive equation. For
both flexible propellers, we built simple analytical models able to reproduce all the phenomenology
observed in experiments.

We then studied intermittent locomotion by building a setup that can reproduce the back-and-forth
motion of a fish swimming freely in a channel. We mimic the passive drag reduction that fish
experience when gliding by using a shape-changing propeller. This propeller is inelastic, because
we only need it to be in a fully open or fully closed position. We are able to describe its progressive
opening by introducing a coupling between the opening angle and the pitch angle of the propeller.

Using this propeller, we have shown that cyclically turning the electric motor on and off can be
more efficient than driving continuously at the same speed. This intermittent strategy leads to a
reduction in propulsion cost of between 15 and 25%, depending on the average velocity. This is
possible because the drag in the passive phase is sufficiently low. This drag reduction lowers the
angular velocity required to propel the boat and thus the cost of the entire cycle. We were able to
develop scaling laws that capture this effect and predict the required angular velocity. By modeling
the time-dependent dynamics, we obtained the minimal drag reduction for intermittent locomotion
to be efficient: drag must be reduced by at least 65%. The same model shows that velocity fluctuations
play a minor role compared to other sources of dissipation. Experimentally, it is best to approach a
quasi-steady regime.
Nomenclature

<table>
  <tr>
    <td>(<b>d</b><sub>1</sub>, <b>d</b><sub>2</sub>, <b>d</b><sub>3</sub>)</td>
    <td>Local frame along the centerline of the elastic blade</td>
  </tr>
  <tr>
    <td>α</td>
    <td>Angle of attack</td>
  </tr>
  <tr>
    <td>α<sub>opt</sub></td>
    <td>Angle of attack that maximizes the lift to drag ratio</td>
  </tr>
  <tr>
    <td>δ&#773;<sup>LE</sup><sub>AC</sub></td>
    <td>Distance between the aerodynamic center and the leading edge</td>
  </tr>
  <tr>
    <td>δ&#773;<sup>ac</sup><sub>cm</sub></td>
    <td>Distance between the aerodynamic center and the center of mass of a section</td>
  </tr>
  <tr>
    <td>Δ</td>
    <td>Duty cycle</td>
  </tr>
  <tr>
    <td>δ</td>
    <td>Tilt angle of the shaft</td>
  </tr>
  <tr>
    <td>Δ<sub>max</sub></td>
    <td>Duty cycle corresponding to maximal efficiency improvement</td>
  </tr>
  <tr>
    <td>η</td>
    <td>Efficiency η = C<sub>T</sub>/(λC<sub>Q</sub>)</td>
  </tr>
  <tr>
    <td>γ</td>
    <td>Angle between the blade and the plane of rotation</td>
  </tr>
  <tr>
    <td>γ<sub>0</sub></td>
    <td>Pitch angle</td>
  </tr>
  <tr>
    <td>λ</td>
    <td>Tip speed ratio</td>
  </tr>
  <tr>
    <td>Ω<sub>el</sub></td>
    <td>Darboux vector</td>
  </tr>
  <tr>
    <td><b>f</b><sub>el</sub></td>
    <td>External force of the elastic blade</td>
  </tr>
  <tr>
    <td><b>M</b><sub>el</sub></td>
    <td>Internal moment of the elastic blade</td>
  </tr>
  <tr>
    <td><b>m</b><sub>el</sub></td>
    <td>External moment of the elastic blade</td>
  </tr>
  <tr>
    <td><b>T</b><sub>el</sub></td>
    <td>Internal force of the elastic blade</td>
  </tr>
  <tr>
    <td><b>U</b><sub>p</sub></td>
    <td>Local, relative flow velocity</td>
  </tr>
  <tr>
    <td>ν<sub>el</sub></td>
    <td>Poisson's ratio</td>
  </tr>
  <tr>
    <td>Ω</td>
    <td>Angular velocity</td>
  </tr>
  <tr>
    <td>ω</td>
    <td>Time dependent angular velocity</td>
  </tr>
  <tr>
    <td>Ω<sub>c</sub></td>
    <td>Angular velocity of the continuous motion</td>
  </tr>
  <tr>
    <td>ρ</td>
    <td>Fluid density</td>
  </tr>
  <tr>
    <td>ρ<sub>b</sub></td>
    <td>Blade density</td>
  </tr>
  <tr>
    <td>σ</td>
    <td>Solidity σ = N<sub>b</sub>c/(πR)</td>
  </tr>
  <tr>
    <td>τ</td>
    <td>Maximal efficiency improvement compared with continuous motion</td>
  </tr>
  <tr>
    <td>θ</td>
    <td>Bending angle</td>
  </tr>
  <tr>
    <td>θ<sub>e</sub></td>
    <td>Opening angle</td>
  </tr>
  <tr>
    <td>ε</td>
    <td>Twisting deformation, ε = γ<sub>0</sub> − γ</td>
  </tr>
  <tr>
    <td>φ</td>
    <td>Local flow angle</td>
  </tr>
</table>
$\zeta$ Power expanded per unit distance, relative to continuous motion

$A_b$ Cross sectional area of a blade

$a_Q, c_Q$ Fitting coefficients for the torque

$a_T, c_T$ Fitting coefficients for the axial force

$C$ Torsion spring constant

$c$ Chord length

$C_a$ Axial force coefficient

$C_D$ Drag coefficient

$C_L$ Lift coefficient

$C_N$ Normal force coefficient

$C_P$ Power coefficient

$C_Q$ Torque coefficient, $C_T = Q/(1/2\rho\pi R^2 (R\Omega)^2 R)$

$C_T$ Thrust coefficient, $C_T = F_{\text{thrust}}/(1/2\rho\pi R^2 (R\Omega)^2)$

$C_Y^B$ Bending Cauchy number

$C_Y^S$ Twisting Cauchy number for the semi-flexible propeller

$C_Y^T$ Twisting Cauchy number

$C_{D,B}$ Burst phase drag coefficient

$C_{D,C}$ Coast phase drag coefficient

$D$ Drag force

$d$ Blade relative density, $d = \rho_b/\rho$

$E$ Young's modulus

$e$ Thickness

$F_{\text{thrust}}$ Thrust force

$I$ Quadratic moment of area

$J$ Torsion modulus

$L$ Lift force

$M$ Dimensionless quantity comparing the fluid versus boat inertia

$m$ Boat mass

$m^*$ Total mass, $m^* = m + m_a$

$m_a$ Added mass

$N_b$ Number of blades

$P_c$ Typical power dissipated by viscous drag

$Q$ Torque

$R$ Radius (maximum)

$r$ Radius (local)

$r_{\text{min}}$ Radius (minimum)

$S_{\text{boat}}$ Reference surface of the boat

$S_b$ Planar surface area of a blade

$T$ Period

$U$ Boat velocity

$U_\infty$ Incoming flow velocity

$u_p$ Dimensionless velocity magnitude, $u_p = U_p/(R\Omega)$

$U_t$ Orthoradial velocity

$U_x$ Axial velocity
Fully flexible propeller

<figure><img src="image_80.png" /><figcaption>Figure A.1: Left, view of the bent blade in the ($e_y, e_z$) plane (as seen from the incoming flow). Right, view obtained by rotating along $e_y$ by $\gamma_0$ so as to see the blade from the side. Note that $\tilde{r}$ is coming from the back of the schematic here.</figcaption></figure>

This appendix details the computation required to obtain the equation for the deformation of the fully flexible propeller in Sec. 3.1. In particular, in the limit of small twist we first obtain an expression for $\tilde{r}$ — the distance between a bend blade element and the axis of rotation. Then, we compute the projection $e_r \cdot d_2$ required to get the centrifugal force and finally reach the expression for the moment due the centrifugal force.

**Distance to the axis of rotation** We compute the distance $\tilde{r}$ to the axis of rotation, which in the case of no pitch and pure bending, reads:

$$ \tilde{r} = r_{\min} + \int_{r_{\min}}^{R} \cos \theta dr. \qquad (A.1) $$

This formula does not hold in our case due to the finite pitch of the blades and the twist. The exact
formula requires integrating both bending and twisting angles along the span and adds a lot of unnecessary complexity. Indeed, in our system the twist is small compared to the pitch of the blades (seen in our and in Eldemerdash and Leweke (2021) experiments).

In the general case, $\tilde{r}(s)$ is the combination of its projection along the $z$ and $y$ directions: $\tilde{r}^2(s) = r'(s)^2 + y^2(s)$. Under the hypothesis that the twist is small, the $z$ projection corresponds to $\tilde{r}(s)$ in the zero pitch case (i.e., looking at the blade from the side as in Fig. A.1 is the same as if the blade was flat.). The projection in the $y$ direction corresponds to the displacement in the $x$ direction in the case of zero pitch, then rotated around $e_z$ by an angle $\gamma_0$ and finally projected onto $e_y$. In other words: $y(s) = \int_{r_{\min}}^{R} \sin \theta dr \times \sin \gamma_0$. Finally we obtain:

$$ \tilde{r}^2 = \left( r_{\min} + \int_{r_{\min}}^{R} \cos \theta dr \right)^2 + \left( \sin \gamma_0 \int_{r_{\min}}^{R} \sin \theta dr \right)^2 . \qquad (A.2) $$

We compute the scalar product $e_r \cdot d_2$ by expressing both vectors in the frame of reference ($e'_x, e'_y, e'_z$), obtained by rotating the frame ($e_x, e_y, e_z$) around $e_z$ by the pitch angle $\gamma_0$. We have:

$$ e_r = \cos \psi e_y + \sin \psi e_z, \qquad (A.3) $$

$$ d_2 = \sin \theta e'_z - \cos \theta e'_x. \qquad (A.4) $$

By writing $e'_x$ in terms of $e_x$ and $e_y$, we can finally write:

$$ e_r \cdot d_2 = \sin \psi(s) \sin \theta(s) - \cos \psi(s) \sin \gamma_0 \cos \theta(s). \qquad (A.5) $$

The angle $\psi$ is obtained using the values of $r'(s)$ and $z(s)$:

$$ \psi = \frac{r_{\min} + \int_{r_{\min}}^{R} \cos \theta dr}{\sin \gamma_0 \int_{r_{\min}}^{R} \sin \theta dr}. \qquad (A.6) $$

<figure><img src="image_81.png" /><figcaption>Figure A.2: Creation of a restoring moment due to the misalignment of the centrifugal forces.</figcaption></figure>

**Moment of the centrifugal force** The centrifugal force creates a moment which tends to twist the blade towards the plane of rotation due to the misalignment of the centrifugal forces in a section (Fig. A.2). We compute its analytical expression also using the hypothesis that the twist is small compared to the pitch of the blades. In a small blade element at a radius $r$ and chordwise coordinate $u$, the magnitude of the centrifugal force $d^2 F_c$ reads:

$$ d^2 F_c = \rho_b e \Omega^2 \sqrt{\tilde{r}^2 + u^2 \cos^2 \gamma_0} du dr. \qquad (A.7) $$
This force is directed mostly radially but also slightly away from the center of rotation because of the offset $u$. The projection of this force in the orthoradial plane corresponds to a multiplication by $\sin \delta_{mc}$, where $\delta_{mc}$ verifies:

$$ \tan \delta_{mc} = \frac{u \cos \gamma_0}{\tilde{r}}. \qquad (A.8) $$

Finally, taking into account the bending of the local section of the blade, the infinitesimal centrifugal force reads:

$$ d^2 F_c = \rho_b e \Omega^2 u \cos \gamma_0 \cos \theta du dr. \qquad (A.9) $$

The moment along $d_3$ created by this distribution of force is:

$$ dM_{centrifuge} = 2 \int_{0}^{c/2} u \sin \gamma_0 d^2 F_c(u). \qquad (A.10) $$

After computing the integral (the result would differ if the blade had a varying thickness or chord), we find:

$$ dM_{centrifuge} = \rho_b e \frac{c^3}{24} \Omega^2 \cos \theta \sin \gamma_0 dr \qquad (A.11) $$
BEMT corrections

This section efforts made to improve the quantitative agreement between the blade element model and the experiments. In particular it describes the widely used blade element momentum theory (BEMT) originally developed by Glauert (1983). This model takes into account the induced velocities due to the rotation of the rotor and estimates them via the conservation of momentum. It is widely used as a preliminary step in the engineering of rotating structures, wind turbines of propellers (Sørensen, 2011; Masters et al., 2011; MacNeill and Verstraete, 2017; Jiménez et al., 2022; Mohd Zawawi et al., 2014). Many corrections exist, taking into account different limiting aspects: compressibility Jiménez et al. (2022), low Reynolds aerodynamics MacNeill and Verstraete (2017), or flow curvature Snel et al. (1994); Corrigan and Schillings (1994).

In the following we derive the BEMT model and compare various implementations with the exper-
imental thrust and torque of the chordwise-flexible propeller (any other data set would lead to a
similar discussion). We do not couple the BEMT with a model for flexibility in the example, so the
blades are considered rigid. We do not explicitly include the effects of the tilt of shaft in the following,
but it can be readily adapted, either by including a position dependent local velocity (See for instance
Mario (2012)) or its average effect: reducing the thrust by a factor cos δ.

<figure><img src="image_82.png" /><figcaption>Figure B.1: Schematic of the actuator disk model (left) and definition of the induced velocities in red and associated angles (right)</figcaption></figure>
## Blade Element Theory

We consider a section of the blade, located at the curvilinear position $s$ along the span of the blade. The axial and orthoradial velocities are modified by the induced velocities $U_a$ and $U_r$ respectively (see Fig. B.1). The axial velocity $U_a$ is due to the fact that the flow is already slightly accelerated before reaching the rotor and the orthoradial velocity is due to the fluid being accelerated into rotation by the blades. The total axial velocity is therefore $U_x + U_a$, where $U_x$ is $U_\infty$ in the case of a flat rotor aligned with the flow, but in the case of a tilted rotor $U_x \approx U_\infty \cos \delta$. The total angular velocity is $r\Omega - U_r$. With the choice of directions for $U_a$ and $U_r$, both of these induced velocities should be positive when the propeller is behaving normally (producing positive thrust and experiencing positive torque).

The total (local) velocity $U_p$ and local flow angle $\varphi$ read:

$$U_p^2 = (U_x + U_a)^2 + (r\Omega - U_r)^2, \quad \varphi = \tan^{-1} \frac{U_x + U_a}{r\Omega - U_r} \qquad (B.1)$$

We then use the same modelling as in Sec. 1.2, discretizing the flow in concentric annuli, interacting with a section of the blades. The thrust and torque therefore read:

$$dF_{\text{thrust}} = \frac{1}{2} \rho N_b c U_p^2 (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) \, dr, \qquad (B.2) \\ dQ = \frac{1}{2} \rho N_b r c U_p^2 (C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi) \, dr.$$

To solve the system and find $U_a$ and $U_r$, we will find another way of expressing $dF_{\text{thrust}}$ and $dQ$.

## Momentum Theory

We consider the propeller as an actuator disk through which the velocity is continuous but where there is a pressure jump $\delta p$ (Fig. B.1, left). By writing Bernoulli's equation from infinitely upstream to the disk and from the disk to infinitely downstream, we get the system:

$$\begin{aligned} p_{\infty} + \frac{1}{2} \rho U_x^2 &= p' + \frac{1}{2} \rho U_1^2, \\ p' + \delta p + \frac{1}{2} \rho U_1^2 &= p_{\infty} + \frac{1}{2} \rho U_2^2. \end{aligned} \qquad (B.3)$$

It should be noted that the angular velocity imparted to the flow in the wake of the propeller is expected to be small, such that it does not enter into the balance above. We thus have the relation,

$$\delta p = \frac{1}{2} \rho (U_2^2 - U_x^2). \qquad (B.4)$$

The thrust produced by the propeller is the integration of the pressure around the actuator disk. In our simplified case this reduces to the change in pressure times the area of the rotor:

$$F_{\text{thrust}} = \pi R^2 \Delta p = \tilde{A} \frac{1}{2} \rho (U_2^2 - U_x^2). \qquad (B.5)$$

The thrust can also be obtained through the increase of momentum of the flow as it passes through the disk. It can be expressed as the mass flow through the disk $\rho\pi R^2 U_1$ times the increase in velocity $U_2 - U_x$:

$$F_x = \pi R^2 \rho U_1 (U_2 - U_x) \qquad (B.6)$$
Combining these two expressions we get that the velocity at the disk is the mean of the velocities up and downstream:

$$U_1 = \frac{U_2 + U_x}{2}. \qquad (B.7)$$

To compute the local thrust and torque, we need to apply this reasoning not to the whole flow around the rotor, but to the previously defined streamtubes of the flow. The conclusions stay the same, only they need to be applied locally instead of to the whole disk. For the tube crossing the blade associated with the spanwise coordinate s, we have $U_1(s) = U_x(s) + U_a(s)$ (or $U_2(s) = U_x(s) + 2U_a(s)$). The thrust contribution of an annulus of radius r and thickness dr is obtained by writing the increase in momentum through the annular portion of the rotor:

$$dF_{\text{thrust}}(r) = 4\pi r \rho (U_x + U_a) U_a dr. \qquad (B.8)$$

The conservation of angular momentum similarly gives $\Omega_1 = \Omega_2/2$. The torque contribution from the annulus is obtained by considering the angular momentum increase ($2U_r$), multiplied by the axial flow rate ($2\pi r dr \rho(U_x + U_a)$), and the moment arm r, giving:

$$dQ(r) = 4\pi r^2 \rho (U_x + U_a) U_r dr. \qquad (B.9)$$

Combining equations (B.2), (B.8), and (B.9) we close the system for $U_a$ and $U_r$:

$$\frac{1}{2} \rho N_b c U_p^2 (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) = 4\pi r \rho (U_x + U_a) U_a, \qquad (B.10)$$

$$\frac{1}{2} \rho N_b r c U_p^2 (C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi) = 4\pi r^2 \rho (U_x + U_a) U_r. \qquad (B.11)$$

The typical length scale of the problem is R and the typical velocity is $R\Omega$. Thus, we denote the dimensionless velocities with a lowercase u and the solidity of the propeller: $\sigma = cN_b/\pi r$. We obtain the dimensionless set of equations:

$$\frac{\sigma}{8} u_p^2 (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) - (u_x + u_a) u_a = 0, \qquad (B.12)$$

$$\frac{\sigma}{8} u_p^2 (C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi) - (u_x + u_a) u_r = 0 \qquad (B.13)$$

with,

$$\tan \varphi = \frac{u_x + u_a}{r/R - u_r}, \quad u_p^2 = (u_x + u_a)^2 + \left(\frac{r}{R} - u_r\right)^2, \quad \text{and} \quad \alpha = \gamma - \varphi. \qquad (B.14)$$

Using a root finding algorithm, we can find $u_a$ and $u_r$ for any given blade section. This result finally allows us to compute the thrust and torque coefficients:

$$C_T = \int_{s_{\min}}^{1} \frac{N_b c}{\pi R} u_p^2 (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) ds \qquad (B.15)$$

$$C_Q = \int_{s_{\min}}^{1} \frac{N_b c}{\pi R} \frac{r}{R} u_p^2 (C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi) ds. \qquad (B.16)$$

In Fig. B.2 (resp. B.3) we show the thrust coefficient (resp. torque coefficient) from several models, compared with the experimental data from Sec. 3.2. Four of the models (blue, green, purple, and black) utilize the blade element defined in the beginning of the thesis (Sec 1.2) where the axial and orthoradial velocities are assumed to be the flow velocity and the angular velocity respectively,
<figure><img src="image_83.png" /><figcaption>Figure B.2: Thrust coefficient versus tip speed ratio for various pitch angles $\gamma_0$. (●) Experimental data, (—) Blade element and flat plate model, (---) Blade element and glider model, (- - -) Blade element, angular position dependence and flat plate model, (--- -) BEMT and flat plate model, (--- - -) BEMT, angular position dependence and flat plate model.</figcaption></figure>

without self induction. The two remaining (red and orange) use the Blade Element Momentum Theory presented in this section. Without going into the details, it can be seen that the BEMT consistently underpredicts the thrust and torque. In contrast, the blade element model reaches the right order of magnitude for the thrust and is closer to the experimental torque. Forces are reduced when using the BEMT because the angle of attack is reduced when both induced velocities are positive. This is surprising as there is a large body of literature showing that the BEMT is an efficient tool to estimate the forces produced by a rotor, even in cases similar to ours (Eldemerdash, 2023; Jiménez et al., 2022; MacNeill and Verstraete, 2017; Durán Venegas et al., 2019). We identified several possible modifications to the models to help the quantitative agreement such as the effect of the tilt of the propeller, and the uncertainty on the sectional lift and drag coefficients.

All the curves in Fig. B.2 and B.3 incorporate the mean effect of the tilt: a reduction of the incoming velocity by $\cos \delta$ and a reduction of the thrust by the same amount because of the projection of the force normal to the rotor. We tested the effect of including the angular position dependent velocity (like in Sec. 2.2.3). For the simpler blade element model, the result is plotted in purple (- - -) to be compared with the reference case of a full blue line. Explicitly including the tilt result in an increase of the thrust produced, which reduces at high tip speed ratio. For the BEMT, this corresponds to dividing the flow a second time in annular tubes (Mario, 2012). The same effect is visible: the dotted-red line includes the full tilt effect and the dotted orange line does not. The tilt is not sufficient to explain the discrepancy between the BEMT and the experiments. Another clue that the issue is not the tilt is that using the data from boats 1 and 2 where the shaft is aligned with the flow leads to similar results.

Often times, corrections have to be added to the lift and drag coefficients to make BEMT quantitative
(Corrigan and Schillings, 1994; Snel et al., 1994; Masters et al., 2011; Jiménez et al., 2022). These semi-empirical corrections can model the effect of flow curvature, compressibility, stall, or tip losses. However, most of these reduce the thrust obtained in the model and thus cannot explain the discrepancy. We still tried to implement corrections for flow curvature and tip losses and verified that their effect does not explain the discrepancy between BEMT and experimental data (not shown). Some corrections incorporate several fitting parameters which may be problem dependent. Instead of fitting our model to the data, we decided to have a simple model which qualitatively predicts the thrust and torque. The blade element model is simpler and qualitatively fits better our experimental data. We therefore decided to use it instead of the full BEMT. Similarly, using the more complex 2D model for lift and drag from Li et al. (2022) does not improve the accuracy, and we therefore use the flat plate model.

<figure><img src="image_84.png" /><figcaption>Figure B.3: Torque coefficient versus tip speed ratio for various pitch angles $\gamma_0$. (●) Experimental data, (—) Blade element and flat plate model, (---) Blade element and glider model, (---) Blade element, angular position dependence and flat plate model, (------) BEMT and flat plate model, (---) BEMT, angular position dependence and flat plate model.</figcaption></figure>

Bibliography

CORRIGAN, J. J. and SCHILLINGS, J. J. (1994). Empirical model for stall delay due to rotation. In *American Helicopter Society Aeromechanics Specialists Conference*. San Francisco, CA 111, 115

DURÁN VENEGAS, E., LE DIZÈS, S., and ELOY, C. (2019). A strongly-coupled model for flexible rotors. *Journal of Fluids and Structures*, vol. 89:pp. 219–231 114

ELDEMERDASH, A. (2023). *Fluid-structure interaction of strongly deforming flexible rotors*. Ph.D. thesis, Aix Marseille Université 114
ELDEMERDASH, A. S. and LEWEKE, T. (2021). Deformation and wake of a flexible rotor in water. *Journal of Physics: Conference Series*, vol. 1934(1):p. 012006 108

GLAUERT, H. (1983). *The Elements of Aerofoil and Airscrew Theory*. Cambridge University Press 111

JIMÉNEZ, J., HOYOS, J., ECHAVARRIA MARTÍNEZ, C., and ALVARADO, P. (2022). Exhaustive Analysis on Aircraft Propeller Performance through a BEMT Tool *. *Journal of Aeronautics, Astronautics and Aviation, Series A*, vol. 54:pp. 13–24 111, 114, 115

LI, H., GOODWILL, T., JANE WANG, Z., and RISTROPH, L. (2022). Centre of mass location, flight modes, stability and dynamic modelling of gliders. *Journal of Fluid Mechanics*, vol. 937:p. A6 115

MACNEILL, R. and VERSTRAETE, D. (2017). Blade element momentum theory extended to model low Reynolds number propeller performance. *The Aeronautical Journal*, vol. 121(1240):pp. 835–857 111, 114

MARIO, H. (2012). Aerodynamic Propeller Model for Load Analysis. MSc Thesis 111, 114

MASTERS, I., CHAPMAN, J. C., WILLIS, M. R., and ORME, J. A. C. (2011). A robust blade element momentum theory model for tidal stream turbines including tip and hub loss corrections. *Journal of Marine Engineering & Technology*, vol. 10(1):pp. 25–35 111, 115

MOHD ZAWAWI, F., LV, P., PROTHIN, S., MORLIER, J., MOSCHETTA, J.-M., and BÉNARD, E. (2014). Study of a flexible UAV proprotor. *International Journal of Engineering Systems Modelling and Simulation*, vol. 6(3/4):pp. 149–161 111

SNEL, H., HOUWINK, R., and BOSSCHERS, J. (1994). Sectional prediction of lift coefficients on rotating wind turbine blades in stall. Tech. Rep., Netherlands: N. p. 111, 115

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Chordwise-flexible propeller data

We present here all the experimental data obtained with the chordwise-flexible propeller. The plots are presented similarly to Fig. 3.26, but show the thrust (Fig. C.1), torque (Fig. C.2), and twist angle (Fig. C.3). Colors indicate flexibility from yellow to red, with blue being assigned to the fully rigid blades. Each panel shows the data for a particular pitch angle, but the rigid blade data is plotted on each panel to facilitate comparisons (except for the twist angle).

Fig. C.4 displays the results of the numerical model, exploring the ($\gamma_0$, $\lambda$) plane. The top row corresponds to a rigid propeller and the bottom row to a flexible one ($C_Y^S/\lambda^2 = 0.24$). Comparing panels (a) and (d) shows how most of the time flexibility reduces the thrust, except for large pitch angles ($\gamma_0 > 45$ deg) where reducing the angle of attack to 45 deg is beneficial. At the same time, panels (b) and (c) show that the torque is always reduced resulting in an improved efficiency ((c) and (d)). The narrow range of efficient propulsion close to the positive thrust onset is widened to most of the domain explored. The tip twist (g) reduces when $\gamma_0$ or $\lambda$ increase and is positive in the bottom left region of the plot (low $\lambda$ and $\gamma_0$, when the angle of attack is negative on most of the blade).
<figure><img src="image_85.png" /><figcaption>Figure C.1: Thrust coefficient as a function of the tip speed ratio. Colors indicate blade flexibility (blue) rigid, (yellow to red) increasingly flexible blades. Markers indicate pitch angle. Each panel corresponds to a pitch angle from 30 to 90 deg. For reference, the thrust of the rigid blades with every pitch angle is plotted on each panel.</figcaption></figure>
<figure><img src="image_86.png" /><figcaption>Figure C.2: Torque coefficient as a function of the tip speed ratio. Colors indicate blade flexibility (blue) rigid, (yellow to red) increasingly flexible blades. Markers indicate pitch angle. Each panel corresponds to a pitch angle from 30 to 90 deg. For reference, the torque of the rigid blades with every pitch angle is plotted on each panel.</figcaption></figure>
<figure><img src="image_87.png" /><figcaption>Figure C.3: Twist angle as a function of the tip speed ratio. Colors indicate blade flexibility (blue) rigid, (yellow to red) increasingly flexible blades. Markers indicate pitch angle. Each panel corresponds to a pitch angle from 30 to 90 deg. Dark edge colors correspond to the correction proposed in Fig. 3.21</figcaption></figure>
<figure><img src="image_88.png" /><figcaption>Figure C.4: Propeller performance from the numerical model. Top row corresponds to a rigid propeller, bottom one to a flexible propeller ($C_Y^S/\lambda^2 = 0.24$). Panels (a) and (d) show the thrust coefficient $C_T$, (b) and (e) show the torque coefficient $C_Q$, (c) and (f) show the efficiency $\eta$ and panel (g) shows the tip deflection $\gamma - \gamma_0$.</figcaption></figure>
Image processing

**D.1 Fully flexible propeller**

We detail here the processing required to go from the raw images of the rotor in Sec. 3.1 to the bending angle data. The steps are illustrated in Fig. D.1. We take 30 pictures (panel (a)) in the same operating conditions and each one of them is treated separately. The bending angle is the mean of the resulting data after removing the outliers with a median filter. The initial frame contains both blades and by symmetry, we can process the second blade in the same way as the first one.

**Crop and contrast boost:** we select only one blade (panel (b)) by cropping out the region of interest. In extreme cases, the tip of the blade can be above the hub of the rotor, in which case we hide the latter by making it fully white. We then enhance the contrast by an affine function of the pixel data.

**Binarization:** We threshold the image to make it binary (panel (c)). After filling every hole in the binary image, we label each non-zero connected region of the image and keep only the largest one, eliminating any noise or trace of particle in the flow.

**Edge detection:** We detect the edge of the blade (panel (d)) and compute a curvilinear abscissa *s* along the edge. To do so, we order the points along the edge, starting with the bottom left.

**Tip detection:** To detect the centerline, we first need to separate the leading and trailing edges and in order to do so we first need to detect the tip of the blade. We therefore compute the norm of the gradient along this edge, measuring abrupt changes in direction (panel (e)). By using a peak detection algorithm, we detect the two tips (green and orange dots) of the blade.

**Alignment of the edges:** We separate the leading and trailing edges, and we want to compute the coordinates of the centerline $x_c(s) = (x_{LE}(s) + x_{TE}(s))/2$ where LE and TE refer respectively to the leading and trailing edges (the same can be written for the *z* coordinate). We need a common parametrization of both edges in order to compute the right quantity (because of the pixelization of the edges, they might not have the exact same length). We therefore compute the length of each edge in order to align them properly (visible in the black bars in panel (f)).

**Centerline projection:** Because the blade is tilted towards (or away from) the camera we need to correct the *x* coordinate of the centerline (cf. Fig. 3.3) by dividing by $\cos \gamma_0$. The uncorrected centerline is the blue line in panel (g) and the corrected one is the green dashed one (here $\gamma_0 = 20$ deg).

**Bending angle:** We finally compute $\theta(s) = \arctan \frac{\partial x}{\partial z}$ and obtain the blue curve in (h). To
<figure><img src="image_89.png" /><figcaption>Figure D.1: Schematic of the processing to get from an image of the rotor to the spanwise distribution of bending angle. Details of each step are given in the text.</figcaption></figure>
smooth out any remaining errors (such as the deflection near the tip), we fit $\theta(s)$ with a third degree polynomial.

## D.2 Chordwise-flexible propeller

This section describes the processing used to extract the tip twist angle data from the series of pictures of the spinning rotor in Sec. 3.2. The camera is tilted in order to compensate the angle of the shaft. In contrast to the post-treatment of the fully flexible propeller, we trigger the camera 25 times per rotation not only when the blades are in the right position. We therefore first have to extract the images where the blades are perpendicular to the camera.

**Reslice:** We select a vertical slice of the image that the lit part of the blade cuts through when passing in front of the camera (panel (a), vertical red line) and plot the pixel data of each frame as a function of the vertical coordinate (white lines of panel (b) time is going upwards).

**Detection of front facing images:** Using a peak finding algorithm, we detect the frames where a blade is close to the horizontal line indicating the perfectly aligned position (blue line). Note that because 25 is not divisible by 3, the three blades are not photographed exactly in the same position: in the present case two are slightly below the middle and one is above. The maximal alignment error occurring because of this technique is $360 \text{ deg}/25 = 14.4 \text{ deg}$ up or downwards, which has little impact on the detected shape. From panel (c) onwards, we only use the front facing images.

**Binarization:** We crop the image and make it binary to prepare it for the next step. Because the background contains the nut used to fasten the blade as well as the other blades, the binarization is not as straightforward as a value threshold. Instead we use a combination of three thresholds: a pixel has a value of 1 in the binary image if it is brighter than the pixels in a wide region around it, the region close to it is sharper than most of the image, and it is bright enough.

**Hough transform:** Despite the efforts to detect only the blade in the binarization step, noise can be present or not all the blade may be detected. Instead of extracting the full shape of the lit part of the blade we use a Hough transform to extract its direction only. Its principle is to transform an image from regular 2D space to a position direction space. Each point in this space represents a line, defined by its slope (the angle we are looking for) and a distance from a reference point (here the top left corner of the image). The value of the Hough transform in this points simply is the number of pixel along the line with a value of 1. The result of this operation is plotted in panel (e), along with the maximum value at each given angle on top of it, since we are not interested in one particular location. The tip twist corresponds to the maximum of the curve. The result is shown in panel (f): the line captures the straight portion of the blade, near the trailing edge.
<figure><img src="image_90.png" /><figcaption>Figure D.2: Schematic of the processing to get from a sequence of images of the rotor to the tip twist angle. Details of each step are given in the text.</figcaption></figure>
Résumé

Cette thèse expérimentale s'intéresse à la propulsion par hélice au travers de deux nouveaux concepts bioinspirés : la reconfiguration et l'intermittence. La reconfiguration consiste à rendre les structures déformables afin qu'elles s'adaptent aux forces qui s'exercent sur elles. De précédents résultats dans le cas des éoliennes ont montré que cette déformation peut être exploitée pour s'adapter passivement aux changements de conditions extérieures si la flexibilité du matériau est bien choisie. Cette capacité à changer de forme peut aussi être utilisée de manière dynamique pour se propulser de manière intermittente. Cette stratégie est adoptée par de nombreuses espèces de poissons ou d'oiseaux pour diminuer l'effort lié à la locomotion.

Les travaux présentés ici s'intéressent donc dans un premier temps au problème d'interaction fluide-structure constitué d'une hélice aux pales souples dans un écoulement d'eau. Nous mettons en évidence les lois régissant la déformation des pales lors de la rotation. L'efficacité de la propulsion est modifiée par ces déformations et une hélice à pales flexibles peut ainsi être rendue plus résiliente aux variations de vitesse que sa version rigide.

Dans un second temps, nous étudions la propulsion intermittente qui consiste à alterner les phases de propulsion actives et passives pour réduire le coût énergétique : dans notre cas, cela signifie actionner périodiquement l'hélice. Pour que la propulsion intermittente soit efficace il est crucial que la force de traînée s'exerçant sur le bateau soit réduite dans la phase passive. En utilisant une hélice capable de s'ouvrir et de se fermer passivement, nous montrons que le coût de la propulsion peut être réduit de 25% comparé à la propulsion continue à la même vitesse.

Abstract

This experimental thesis focuses on propeller propulsion through two new bio-inspired concepts: reconfiguration and intermittency. Reconfiguration involves making structures flexible to adapt to flow conditions. Previous results have shown that this deformation can be exploited to passively adapt to changing external conditions if the flexibility is appropriately chosen. This ability to change shape can also be exploited dynamically through intermittent propulsion, a strategy adopted by many fish and bird species to reduce the cost of locomotion.

The work presented here initially addresses the fluid-structure interaction problem of a propeller with flexible blades in a water flow. We highlight the laws governing the blade deformation during the rotation. These deformations modify the efficiency of propulsion, and a propeller with flexible blades can be made more resilient to velocity variations compared to its rigid version.

Next, we investigate intermittent propulsion, which involves alternating active and passive propulsion phases to reduce energy costs. In our case, this means periodically rotating the propeller. For intermittent propulsion to be effective, it is crucial to reduce the drag force acting on the boat during the passive phase. By using a propeller that can passively open and close, we demonstrate that propulsion costs can be reduced by 25% compared to continuous propulsion at the same speed.

Keywords

Bio-inspiration, Fluid-structure interactions, Energy
Bio-inspiration, Interaction fluide-structure, Énergie