# Bioinspired propeller propulsion : reconfiguration and intermittency

Tristan Aurégan

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Tristan Aurégan. Bioinspired propeller propulsion : reconfiguration and intermittency. Fluid mechanics [physics.class-ph]. Université Paris Cité, 2023. English. NNT : tel-04254782

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THÈSE DE DOCTORAT
D'UNIVERSITÉ PARIS CITÉ

Spécialité : Physique

École doctorale n°564: Physique en Île-de-France

réalisée

au Laboratoires de Physique et Mécanique des Milieux Hétérogènes,
et Matière et Systèmes Complexes

sous la direction de Benjamin THIRIA et Sylvain COURRECH DU PONT

présentée par

Tristan AURÉGAN

pour obtenir le grade de :

DOCTEUR D'UNIVERSITÉ PARIS CITÉ

Sujet de la thèse :

Propulsion bioinspirée par hélice : reconfiguration et intermittence

Bioinspired propeller propulsion : reconfiguration and intermittency

Présentée et soutenue publiquement le 18 septembre 2023

devant le jury composé de :

<table><tr><td>Pr. Camille Duprat</td><td>École polytechnique</td><td>Rapportrice</td></tr><tr><td>Pr. Médéric Argentina</td><td>Université Côte d'Azur</td><td>Rapporteur</td></tr><tr><td>Pr. Eva Kanso</td><td>University of Southern California</td><td>Examinatrice</td></tr><tr><td>Pr. Christophe Eloy</td><td>Centrale Marseille</td><td>Examinateur</td></tr><tr><td>Pr. Benjamin Thiria</td><td>Université de Paris</td><td>Directeur</td></tr><tr><td>Pr. Sylvain Courrech du Pont</td><td>Université de Paris</td><td>Directeur</td></tr><tr><td>MCf. Sophie Ramananarivo</td><td>École polytechnique</td><td>Membre invité</td></tr></table>
<figure><img src="image_1.png" /></figure>
Résumé

Cette thèse s'intéresse à la propulsion par hélice au travers de deux nouveaux concepts bioinspirés : la reconfiguration et l'intermittence, avec un objectif d'amélioration de l'efficacité de la propulsion. La reconfiguration consiste à rendre les pales souples afin qu'elles se déforment passivement sous l'action de l'écoulement. De récents travaux de Cognet et al. (2017) ont montré que dans le cas d'une éolienne, cette stratégie permet d'extraire de l'énergie dans une large gamme de vitesses de vent. La flexibilité permet à l'éolienne d'adopter passivement une forme optimale, ce qui est actuellement effectué de manière active à l'aide de moteurs.

Le second axe pour améliorer l'efficacité de la propulsion est l'intermittence : une stratégie de locomotion observée chez certaines espèces d'oiseaux et de poissons qui consiste à alterner des phases de propulsion actives (l'animal bat des ailes ou de la queue) et des phases passives (en vol plané ou simplement en se laissant glisser). Cette stratégie semble a priori néfaste pour l'efficacité de la propulsion, car les accélérations et décélérations sont souvent associées à un cout énergétique supplémentaire. Cependant, de précédentes études ont montré que pour certaines espèces, cette stratégie permet au contraire de réduire la consommation énergétique lorsqu'elle est associée à un changement de forme durant la phase passive qui permet de réduire la résistance exercée par le fluide environnant sur l'animal (Weihs, 1974; Li et al. (2021, 2023)). Cette thèse a donc un double objectif : explorer la possibilité de transposer les résultats obtenus sur les éoliennes aux hélices dans l'eau pour les rendre plus efficaces et discuter de l'applicabilité de la stratégie de propulsion intermittente aux hélices.

<figure><img src="image_2.png" /><figcaption>Figure 0.1: (a) Déformation de l'hélice entièrement flexible. Superposition d'images à plusieurs vitesses de rotation, l'écoulement vient du haut de l'image. (b) Déformation de l'hélice avec un bord d'attaque rigide. Superposition d'images à plusieurs vitesses de rotation, l'écoulement vient du haut de l'image. Le bout de la pale est éclairée par un laser. (c) Angle de flexion en bout de pale en fonction d'un nombre de Cauchy modifié par le coefficient de poussée. La couleur indique la vitesse de l'écoulement. Les symboles avec un contour noir correspondent aux pales très flexibles, les autres aux pales légèrement flexibles. La ligne de tendance en noir a une pente de 0.18 rad.</figcaption></figure>

Elle se compose essentiellement de travaux expérimentaux à échelle réduite dans un canal hydrody-
namique spécialement conçu à cet effet. Nous utilisons des hélices à géométrie simplifiée pour mettre
en valeur les mécanismes physiques importants, au détriment de l'efficacité. Dans un premier temps,
nous explorons le problème d'interaction fluide-structure d'une hélice flexible dans l'eau et mettons en avant les mécanismes clés de la déformation. Puis nous étudions l'efficacité de la propulsion intermittente à l'aide d'une hélice déformable.

Grâce à une première hélice avec des pales entièrement souples ressemblant à des rubans (Fig. 0.1, (a)), nous étudions un nouveau problème d'interaction fluide-structure. Les pales peuvent se plier et se tordre, ces deux rotations étant couplées par le forçage hydrodynamique (dans le cas de nos hélice flexible dans l'eau la force centrifuge est négligeable). Nous mesurons à la fois la déformation de flexion des pales et la force de poussée associée. Selon les conditions précises de l'expérience, d'importantes déformations vers l'amont ou vers l'aval peuvent être observées. Dans la limite des petites déformations, une loi d'échelle basée sur des arguments géométriques permet de prédire la direction et d'estimer l'amplitude de la flexion. La déformation de flexion en bout de pale $\theta_M$ pour les déformations modérées vérifie :

$$ \theta_M \sim C_Y^B (1 + 1/\lambda^2) (\gamma_0 - \arctan 1/\lambda), \quad (0.1) $$

avec $C_Y^B$ le nombre de Cauchy de flexion, un nombre sans dimension comparant l'importance relative de la force hydrodynamique par rapport à l'élasticité de la pale. L'angle $\gamma_0$ est l'angle de calage de la pale : l'angle entre le pied de pale et le plan de rotation. Enfin, $\lambda$ est la vitesse de rotation réduite (ou tip speed ratio en anglais, $\lambda = R\Omega/U_\infty$) un paramètre clé de l'étude des rotors.

L'absence de mesure précise de la torsion empêche de prédire exactement la déformation dans le cas des grandes déformations (la pale peut se plier de plus de 90 deg.). Pour dépasser cette limite, nous utilisons la poussée produite par le rotor (à travers le coefficient de poussée $C_T$) pour estimer le forçage sur les pales. L'angle de flexion en bout de pale vérifie alors :

$$ \theta_M \sim C_Y^B C_T \cos \gamma_0 \frac{\pi R^2}{N_b c R}, \quad (0.2) $$

Cette relation met en valeur un lien fort entre la déformation de flexion et la poussée produite par l'hélice : étant donné une géométrie d'hélice, mesurer la déformation permet de déduire la poussée (Fig. 0.1, (c)).

<figure><img src="image_3.png" /><figcaption>Figure 0.2: (a) Efficacité $\eta$ de l'hélice en fonction de la vitesse réduite $\lambda$. (b) Déformation en bout de pale $\varepsilon$ en fonction de son expression théorique $\varepsilon_1$. La couleur indique la flexibilité de la pale : le bleu correspond aux pales rigides et le gradient de jaune à rouge correspond à des pales de plus en plus flexibles. Les symboles correspondent à l'angle de calage $\gamma_0$.</figcaption></figure>

Dans un second temps, nous étudions une hélice dont le bord d'attaque est rigide : seule la déformation de torsion est autorisée (Fig. 0.1, (b)). Rendre cette hélice flexible permet de s'adapter passivement aux changements de conditions extérieures si la flexibilité est bien choisie. Cet effet
est par exemple visible dans la Fig. 0.2 (a) où l'on trace l'efficacité de l'hélice η en fonction de la vitesse réduite λ. La performance de l'hélice rigide servant de référence est tracée en bleu pour plusieurs angles de calage γ₀, représentés par différents symboles. Les losanges de jaune à rouge correspondent à des hélices de plus en plus flexibles avec un angle de calage de 40 deg. Dans le cas optimal (losanges rouge clair) l'hélice flexible est plus efficace que l'hélice rigide quelque soit son angle de calage. L'angle de torsion de l'hélice rigide s'adapte passivement, lui permettant de rester efficace dans un large gamme de vitesses réduites λ ∈ [3, 5.5].

Connaître l'angle de la pale par rapport au plan de rotation est ainsi crucial pour déduire l'efficacité de la propulsion. Une modélisation simple de la pale en poutres attachées par des ressorts de torsion au bord d'attaque permet d'obtenir l'équation régissant l'angle de torsion de la pale γ :

$$ \gamma(r) = \gamma_0 - C_Y^S u_p^2 (C_L(\alpha) \cos \alpha + C_D(\alpha) \sin \alpha) \delta_{AC}^{LE}, \quad (0.3) $$

avec $C_Y^S$ le nombre de Cauchy correspondant à la géométrie du problème, $u_p$ la norme locale de la vitesse et $\alpha$ l'angle local d'incidence sur la pale. $\delta_{AC}^{LE}$ est la distance entre le point d'application de la force hydrodynamique et le bord d'attaque, et $C_L$ et $C_D$ sont respectivement les coefficients de portance et traînée du profil d'aile utilisé. Un développement limité cette équation nous permet d'obtenir la valeur théorique $\varepsilon_1$ de la déformation en bout de pale $\varepsilon = \gamma_M - \gamma_0$, en loi d'échelle. Cette loi d'échelle est tracée sur la Fig. 0.2 (b) et décrit bien l'évolution de l'angle en bout de pale pour des déformations modérées.

<figure><img src="image_4.png" /><figcaption>Figure 0.3: (a) Vitesse de rotation dans la phase active $\Omega$ en fonction du rapport cyclique $\Delta$ pour plusieurs périodes (couleurs). Les losanges correspondent à l'hélice rigide et les cercles à l'hélice pliable. Les courbes en tirets et en pointillés sont obtenues à l'aide de l'Eq. (0.4) pour les hélices rigides et pliables respectivement. (b) Coefficient de puissance $C_P$ en fonction du rapport cyclique $\Delta$. L'hélice rigide est tracée séparément dans l'insert. La ligne grise horizontale correspond au cout associé au régime continu ($\Delta = 1$). Ces données sont obtenues avec une vitesse d'écoulement de 0.11 m/s.</figcaption></figure>

Une autre stratégie bioinspirée pour améliorer l'efficacité de la propulsion est l'intermittence et consiste à alterner les phases de propulsion actives et passives pour réduire le cout énergétique moyen. Cette stratégie est utilisée chez plusieurs espèces de poissons ou d'oiseaux. Nous construisons une expérience capable de reproduire cette stratégie pour un bateau à hélice : l'hélice est actionnée périodiquement et le bateau se déplace librement dans l'eau. 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. Nous avons donc fabriqué une hélice capable de s'ouvrir et de se fermer passivement lorsque le moteur se met en marche, à l'aide de charnières permettant aux pales de se refermer sur l'axe de rotation.

L'expérience consiste alors à actionner l'hélice périodiquement pendant une fraction $\Delta$ de la période du mouvement (fraction aussi appelée rapport cyclique). La vitesse de l'écoulement est fixée et la vitesse de rotation de l'hélice qui permet l'équilibre du système doit être trouvée manuellement pour réussir l'expérience. Cette vitesse de rotation dans la phase active de propulsion est tracée sur la Fig. 0.3 (a) et augmente lorsque le rapport cyclique diminue. Cette vitesse de rotation $\Omega$ peut être prédite grâce à l'équilibre des forces en moyenne sur le bateau :

$$ \Omega = \Omega_c \sqrt{\frac{C_{D,C}}{C_{D,B}} \left( \frac{1}{\Delta} - 1 \right) + 1}, \quad (0.4) $$

avec $\Omega_c$ la vitesse de rotation associée à la propulsion en continu à la même vitesse moyenne (à $\Delta = 1$). $C_{D,C}/C_{D,B}$ correspond au ratio entre le coefficient de traînée dans la phase passive et celui dans la phase active. Ces coefficients sont mesurés séparément et permettent de prédire la vitesse de rotation à imposer (courbes en tirets et pointillés de la Fig. 0.3 (a)).

Le coût de la propulsion est mesuré en calculant la puissance mécanique moyenne au cours d'un cycle à l'aide de capteurs de couple et de vitesse de rotation. Le coefficient de puissance $C_P$ est obtenu en comparant ce coût à la puissance typique dissipée par le frottement de l'eau à une vitesse donnée. En utilisant l'hélice pliable, le coût de la propulsion peut être réduit de 25 % comparé à la propulsion continue à la même vitesse (Fig. 0.3 (b)). Sur une large gamme de rapports cycliques et pour les périodes les plus longues ($T > 5$ s), la propulsion intermittente est plus efficace que la propulsion en continu (trait gris horizontal). Ce n'est cependant pas le cas de l'hélice rigide (en insert) : dans ce cas la traînée n'est pas suffisamment réduite dans la phase passive et la propulsion continue est la plus efficace.

<figure><img src="image_5.png" /><figcaption>Figure 0.4: (a) Amélioration relative du coût de la propulsion intermittente par rapport à la propulsion continue pour le cycle intermittent optimal, en fonction du contraste de traînée. Les lignes représentent le résultat du modèle et les symboles les points expérimentaux. (b) Rapport cyclique correspondant au cycle intermittent optimal.</figcaption></figure>

Afin d'obtenir le contraste de traînée à partir duquel l'intermittence devient plus efficace que la propulsion en continu, nous construisons un modèle capturant la dynamique du bateau au cours
du cycle. Ce modèle calcule la vitesse instantanée du bateau à partir des forces qui s'appliquent
sur celui-ci et permet d'en déduire le coût énergétique moyen. Tous les paramètres du bateau et de
l'hélice ont été mesurés au préalable et ce modèle est ainsi capable de prédire le coût énergétique
d'un cycle de propulsion intermittent, sans paramètre d'ajustement. Pour un contraste de traînée et
une vitesse moyenne donnés, nous calculons le cycle ayant le coût énergétique le plus faible. Sur
la Fig. 0.4 (a), nous traçons l'amélioration du coût de propulsion par rapport au cycle continu $\tau$.
Ce modèle capture correctement les deux séries d'expériences que nous avons réalisées avec les
pales pliables et prédit le résultat suivant : si la traînée n'est pas réduite d'au moins 65 % durant la
phase passive, le régime continu est le plus efficace. Pour des contrastes de traînée sous cette limite,
$\tau$ augmente rapidement et le rapport cyclique associé au cycle optimal $\Delta_{max}$ s'approche de zéro.

Les stratégies bioinspirées explorées dans cette thèse présentent un intérêt important pour l'amélioration
de l'efficacité de la propulsion maritime. Des hélices souples pouvant s'adapter passivement aux
variations de vitesse de rotation peuvent par exemple compenser les variations rapides de vitesse
dans le sillage de la coque d'un navire. L'intermittence possède un potentiel important de réduction
du coût de la propulsion. Elle nécessite en revanche de modifier l'ensemble du système hélice et
bateau pour que la traînée soit suffisamment réduite dans la phase passive.
Remerciements

Je souhaite adresser me remerciements aux membres du jury qui ont accepté de relire et corriger
mon manuscrit. La discussion scientifique autour des problématiques rencontrées pendant ma
these a été enrichissante et m’a permis de prendre du recul sur les résultats de ces trois années.
J’aimerais remercier mes directeurs de these pour leur accompagnement qui m’a permis d’apprendre
à avancer un projet de recherche dans la durée. Sylvain a toujours apporté un regard nouveau et des
propositions originales pour débloquer la situation. Benjamin a su garder une légèreté agréable dans
ses explications et m’a aidé à garder le bon cap au quotidien.

L'excellente ambiance dans le laboratoire a rendu mon experience de these particulièrement ap-
préciable et j'aimerais remercier tous ceux qui y contribuent : Fred, Claudette, Philippe et Damien
en particulier. J'ai passé de très bon moments scientifiques ou non avec nombre de personnes du
laboratoire aux Arènes, et meme si faire la liste complete ici serait trop long, ca a été un plaisir de
partager ces trois années avec vous tous qui faites du PMMH un endroit aussi spécial.

L'équipe BIOMIM a évidemment été centrale dans cette bonne ambiance générale et a fait que j'ai
toujours été content d'aller au laboratoire. Pouvoir faire une pause et trouver quelqu'un avec qui
discuter a tout moment a rendu le travail de thèse très agréable. En particulier, Roméo m'a appris
toutes les bonnes techniques de fabrication, me transformant de "touriste" en véritable doctorant.
Partager le quotidien de la thèse avec Baptiste et Gatien du début à la (presque) fin a été un privilège
grace a leur complicité, bonne humeur et disponibilité pour discuter science, échecs, Tour de France
ou colormap. Gauthier a été une source intarissable de rire et de bons moments, autour d'une manip
ou a la salle d'escalade. La course du mercredi midi et les débats avec Vincent ont été une pause
précieuse pendant la redaction, 10/10. Camille a ramené le BIO dans la salle BIOMIM, et avec, sa
gentillesse et de nombreux gâteaux qui manquaient depuis le depart de Roméo. Merci enfin à Alexis
pour son assistance sur mes problèmes numériques et sa positivité a toute épreuve.

Hors du laboratoire, merci aux amis qui m'ont soutenu pendant ma these : Gwendal, Goulven et surtout Jules avec qui, comparer nos experiences de thèse autour d'une partie de palet ou a l'escalade a permis de prendre du recul lorsque c'était nécessaire. Merci enfin à mes parents et ma sœur pour leurs conseils et leurs mots d'encouragement tout au long de ces trois années.
# Contents

<table><thead><tr><th></th><th></th></tr></thead><tbody><tr><td>1 Introduction</td><td>1</td></tr><tr><td>1.1 Context of the study</td><td>1</td></tr><tr><td>1.1.1 Reconfiguration</td><td>2</td></tr><tr><td>1.1.2 Intermittency in biolocomotion</td><td>4</td></tr><tr><td>1.1.3 Rotors</td><td>7</td></tr><tr><td>1.2 A blade element description of rotors</td><td>9</td></tr><tr><td>1.2.1 Blade element model</td><td>9</td></tr><tr><td>1.2.2 Flows around foils</td><td>12</td></tr><tr><td>1.2.3 Propeller performance</td><td>14</td></tr><tr><td>2 Experimental setup</td><td>23</td></tr><tr><td>2.1 Hydrodynamic channel</td><td>23</td></tr><tr><td>2.1.1 Description</td><td>23</td></tr><tr><td>2.1.2 Calibration</td><td>24</td></tr><tr><td>2.2 Boat hangar</td><td>28</td></tr><tr><td>2.2.1 Submarine</td><td>29</td></tr><tr><td>2.2.2 Semi-submerged boat</td><td>30</td></tr><tr><td>2.2.3 Surface boat</td><td>31</td></tr><tr><td>2.3 Materials</td><td>35</td></tr><tr><td>3 Reconfigurating propellers</td><td>39</td></tr><tr><td>3.1 Fully flexible propeller</td><td>39</td></tr><tr><td>3.1.1 Setup</td><td>40</td></tr><tr><td>3.1.2 Mathematical description</td><td>41</td></tr><tr><td>3.1.3 Effect of bending on thrust production</td><td>44</td></tr><tr><td>3.1.4 Measuring the efficiency</td><td>50</td></tr><tr><td>3.1.5 Numerical efficiency improvements</td><td>55</td></tr><tr><td>3.1.6 Concluding remarks</td><td>59</td></tr><tr><td>3.2 Chordwise-flexible propeller</td><td>60</td></tr><tr><td>3.2.1 Setup and minimal model</td><td>60</td></tr><tr><td>3.2.2 Passive twisting improves the efficiency</td><td>63</td></tr><tr><td>3.2.3 Evolution of the twist angle</td><td>70</td></tr><tr><td>3.3 Conclusion</td><td>72</td></tr><tr><td>4 Intermittent propulsion</td><td>75</td></tr><tr><td>4.1 Folding propeller</td><td>75</td></tr><tr><td>4.1.1 Setup and expected behavior</td><td>76</td></tr><tr><td>4.1.2 A geometrical coupling between the opening and pitch angles</td><td>77</td></tr><tr><td>4.1.3 A parametrization of the folding propeller</td><td>81</td></tr><tr><td>4.2 Intermittent locomotion experiment</td><td>85</td></tr></tbody></table>
CONTENTS

ix

<table><tbody><tr><td>4.2.1 Setup</td><td>85</td></tr><tr><td>4.2.2 Cycle-averaged results</td><td>88</td></tr><tr><td>4.2.3 Minimal drag reduction</td><td>93</td></tr><tr><td>4.3 Conclusion</td><td>100</td></tr><tr><td>Conclusion</td><td>103</td></tr><tr><td>Nomenclature</td><td>105</td></tr><tr><td>A Fully flexible propeller</td><td>107</td></tr><tr><td>B BEMT corrections</td><td>111</td></tr><tr><td>C Chordwise-flexible propeller data</td><td>117</td></tr><tr><td>D Image processing</td><td>123</td></tr><tr><td>D.1 Fully flexible propeller</td><td>123</td></tr><tr><td>D.2 Chordwise-flexible propeller</td><td>125</td></tr></tbody></table>
# Chapter 1

## Introduction

### 1.1 Context of the study

The globe is warming due to anthropogenic emissions of greenhouse gases (IPCC, 2022). In particular, maritime transport is responsible for about 3% of the annual emissions (IMO, 2020) and must therefore reduce its impact on the environment. Practical solutions are available, such as reducing traffic, slowing down ships or upgrading outdated propulsion systems to new standards. In this context, the search for new ideas on how to further improve the efficiency of transportation can be very useful in achieving climate goals.

One novel research axis which may yield such efficiency improvements is bio-inspiration: the process of observing natural systems in search of alternative ways to solve complex problems traditionally addressed by engineers. It can open unexplored paths and sometimes lead to breakthroughs in non-biological fields, with examples ranging from the shape of the nose of the Shinkansen (a bullet train in Japan), that mimics the beak of the kingfisher, to tape that imitates the paws of geckos (Ng et al., 2021).

We propose to study of marine propulsion by means of two bioinspired strategies: reconfiguration and intermittency. Reconfiguration is the introduction of flexibility into a usually rigid system and the tuning of this flexibility to adapt to changing external conditions. It has already been successfully applied to wind turbines in a previous study by our group (Cognet et al., 2017). In this study, the shape of the rotor blade can passively adapt to changing external conditions and adopt the most efficient shape. In the case of a wind turbine, shape change allows energy to be extracted over a wider range of wind speeds compared with a rigid blade wind turbine. With propellers, reconfiguration could improve the efficiency of propulsion in off-design conditions. Most ship propellers operate in the wake of the ship's hull (Regener et al., 2018). As a result, they experience constantly varying flow speeds throughout the rotation. Passively adjusting to compensate for varying flow speeds would also greatly improve energy efficiency.

The second strategy is intermittency: a locomotion type found in fish and birds that involves passive gliding between active strokes. Intermittency has been found to reduce the cost of locomotion compared to continuous motion (Weihs, 1974; Rayner et al., 2001; Li et al., 2023). In the case of propellers, intermittency corresponds to switching the motor turning the propeller on and off periodically. Intermittent propulsion in the case of marine propellers has never been explored in
previous studies because accelerations are often associated with increased power dissipation, which reduces efficiency. However, it has been shown in the case of fish and birds that the cost of motion can be reduced if there is a shape change between the phases of motion: either gliding or streamlining. This shape change reduces the resistance exerted by the fluid on the animal in the passive phase and is beneficial for efficiency. Therefore, a propeller that dynamically changes shape is needed to mimic this strategy, coupling the intermittency and reconfiguration strategies.

In the following, we recall previous works relevant to the study of shape-changing propellers and intermittent propulsion. We first make a summary of the study of bioinspired reconfigurating structures, then detail the literature on intermittent locomotion in fish. Finally, we review the most important results on flexible rotating structures, either from wind turbines or ship propellers.

### 1.1.1 Reconfiguration

Reconfiguration is the ability of a structure to change shape under the influence of a fluid loading. The deformation caused by the flow modifies the force applied on the structure. This force change may then be used for instance to protect the structure against wind gusts or passively adapt to flow changes. In his seminal work of 1969, Vogel observed that many plant leaves change shape when exposed to high velocity air flows (Fig. 1.1, (a)). This passive mechanism reduces the force exerted by the air on the leaf and therefore allows it to survive strong gusts. The force generated by a flow pushing on a bluff body is called the drag, and can take several forms depending on the characteristics of the flow and the body. These properties are summarized in the Reynolds number that compares the relative importance of the inertial and viscous effects in the fluid. For a flow with a typical velocity $U$ and a kinematic viscosity $\nu$, interacting with an object of size $L$, the Reynolds number reads:

$$Re = \frac{UL}{\nu}. \qquad (1.1)$$

In the cases where the Reynolds number is larger than 1000, the drag force $D$ exerted by a flow on a bluff body takes the form:

$$D = \frac{1}{2} \rho S U^2 C_D, \qquad (1.2)$$

where $\rho$ is the density of the fluid, and $S$ is the frontal area of the object. $C_D$ is a dimensionless number called the drag coefficient. For flows where $Re > 1000$ the drag coefficient depends only on the shape of the object and is a constant of the flow velocity (Hoerner, 1965). The drag force therefore increases with the square of the flow velocity.

In the case of tuliptree leaves (Fig. 1.1, (a)) however, Vogel (1989) showed that the force grows slower than this expected behavior. The leaves roll up upon themselves and reduce the surface exposed to the wind, reducing the drag force. Instead of growing like the flow velocity squared, the force increases with a smaller exponent: about 1 for leaves. The leaves are therefore said to reconfigure: their shape adapts to the intensity of the wind. Similar results were found with other types of leaves, algae, or even small trees. All of these species follow drag laws scaling with the velocity with an exponent smaller than 2. The exact value of the exponent however differs between systems (Vollsinger et al., 2005; Harder et al., 2004; de Langre, 2008). In extreme case it is even possible to find systems where the drag is independent of the flow speed (Barois and de Langre, 2013; Marzin et al., 2022).

Artificial and model reconfigurating structures are often used to describe shape change theoretically. In particular, the two dimensional problem of a flexible cylinder in a flowing soap film was studied by Alben et al. (2002, 2004). Like the leaf, a flexible cylinder clamped in its midpoint bends when subjected to the flow and reduces the drag force applied upon it (Fig. 1.1 (b)). The authors emphasized the
<figure><img src="image_6.png" /><figcaption>Figure 1.1: Example of reconfigurating systems. (a) Tuliptree leaf reconfigurating in high winds (from Vogel (2003)). (b) Shape of a flexible cylinder attached in the center in a soap film flow coming from the top, (c) same shape but the lengths have been dilated by $L_0$ (Extracted from Alben et al. (2002)). (d) Flexible plate in an air stream (Adapted from Gosselin and de Langre (2011)).</figcaption></figure>

presence of an intrinsic length scale of the fluid-structure interaction problem: $L_0 = (2E/\rho f U^2)^{1/3}$, where E is the Young modulus of the cylinder and f the film thickness. They measured the shape of the deformed cylinder and showed that it is self similar: when the lengths are streched by $L/L_0$, all the shapes superimpose on a universal quasi-parabolic shape (Fig. 1.1 (c)). The curvature of the cylinder due to the fluid drag is concentrated in a region of size $L_0$.

The plant experiment was revisited by Schouveiler and Boudaoud (2006), using artificial leaves made out of flexible discs with a radial cut from the center to the edge. These discs roll upon themselves similarly to the tuliptree leaf under high speed flows. They were able to describe the evolution of the drag coefficient with the flow velocity using potential flow theory. Gosselin and de Langre studied flexible plates (Gosselin et al. (2010), Fig. 1.1, (d)) and poroelastic balls (Gosselin and de Langre, 2011) in an airstream. They highlighted two important parameters: first, the Cauchy (or elastohydrodynamical) number comparing the magnitude of the fluid forcing to the elasticity of the material:

$$C_Y = \frac{\rho L^3 U^2}{2EI}, \qquad (1.3)$$

where I is the second moment of area, a geometric property of the object. Second, they introduced the reconfiguration number:

$$\mathcal{R} = \frac{D}{D_0}, \qquad (1.4)$$

where $D_0$ refers to the drag of the rigid object with the same geometry. The reconfiguration number quantifies the effect of the flexibility by comparing the drag of the object with the one of its undeformed counterpart. While the Cauchy number is small the reconfiguration number is equal to one: the effect of flexibility is negligible. For $C_Y \gtrsim 1$ the reconfiguration number decreases as the object adapts to reduce the drag force.

The force on the object in the numerator of the Cauchy number must be rescaled by the drag coefficient to correspond the actual loading on the object. Doing so defines a rescaled Cauchy number $\tilde{C}_Y = C_Y C_D$, where $C_D$ is the drag of the undeformed object. All effects related to the shape of the objects are taken into account in the drag coefficient and therefore, the reconfiguration number for very different objects collapses onto a single curve as a function of the scaled Cauchy number (Fig. 1.2), showing the universality of the reconfiguration of these structures.
<figure><img src="image_7.png" /><figcaption>Figure 1.2: Reconfiguration number $\mathcal{R}$ as a function of the rescaled Cauchy number $\tilde{C}_Y = C_Y C_D$ for different objects. Black circles correspond to flat plates in the work of Gosselin et al. (2010). Green lozenges correspond to cones rolling up in Schouveiler and Boudaoud (2006). Blue triangles are discs cut along several radii in Gosselin et al. (2010). Red squares are cylinders in 2D soap films from Alben et al. (2002). Black crosses are poroelastic balls from Gosselin and de Langre (2011). Figure extracted from Hassani (2016).</figcaption></figure>

1.1.2 Intermittency in biolocomotion

Structures deforming by the action of flows are not always passive: fluid-structure interaction problems are for instance encountered almost everywhere in animal locomotion. Most animal bodies are flexible, and therefore propulsion through a fluid creates shape changes similarly to the reconfiguration of leaves. In contrast to the previous cases, the forcing initially comes from the body rather than the flow. Examples span large scales: from flexible millimeter-sized cilia to the flukes of the largest cetaceans: passive cambering improves efficiency (Fish et al., 2006; Costello et al., 2023). Flexibility is a key element of locomotion, among other examples, insects have been shown to rely on wing bending to propel themselves efficiently through the air (Ramananarivo et al., 2011).

One aspect of particular interest in biolocomotion is intermittency: the ability to dynamically recon-
figure to save energy. Many species utilize an intermittent gait when moving instead of continuous
motion (Kramer and McLaughlin, 2001). This strategy at first glance seems energetically unfavorable:
accelerations often lead to an increased overall dissipation compared with motions at a constant
velocity. However, several species of fish and birds seem to prefer this gait precisely for energy effi-
ciency (Videler, 1981; Muller et al., 2000; Ribak et al., 2005; Wu et al., 2007; Xia et al., 2018; Coughlin
et al., 2022).

In birds, there are two distinct intermittent flight modes: either gliding or bounding (Rayner et al., 2001). Gliding corresponds to the bird gaining height by actively flapping its wings and gliding down with the wings open. Bounding corresponds to the same strategy but with the wings flexed in the passive phase. For most birds gliding is energetically advantageous at most speeds while bounding is only interesting at high speeds (Wu et al., 1975; Tobalske, 2010; Taylor et al., 2010). The ability to store potential energy in the active phase is key to the energy efficiency of this flight mode
(Paoletti and Mahadevan, 2014). This technique is however not accessible to propulsion at the surface of water, and therefore we focused on fish swimming.

<figure><img src="image_8.png" /><figcaption>Figure 1.3: (a) Categories of fish swimming kinematics (Adapted from Sfakiotakis et al. (1999); Vatanabe et al. (2008)) (b) Schematic of a swimming fish, numerically separating the propulsive active part (the tail) and the passive part (the body) (Adapted from Akoz and Moored (2018)).</figcaption></figure>

Fish swimming kinematics vary widely between species (Fig. 1.3, (a)): from anguiliform for long and slender fish to thunniform for fish with wide and short tails like tunas (Sfakiotakis et al., 1999; Blake, 2004; Webb et al., 2011; Lauder, 2015; Di Santo et al., 2021). Lighthill (1971) created the elongated body theory, a model capable of describing the force generation from all these different kinematics, using only the tail tip movement. Efforts were then made to find commonalities in aquatic locomotion: the swimming kinematics can be summed up in one parameter called the Strouhal number *St*, that compares the typical velocity of the tip of the tail *fA* where *f* is the frequency and *A* the amplitude of the tail beat, to the swimming velocity *U*: *St* = *fA/U*. For many high Reynolds number swimming and even flying animals, this number is in a narrow range between 0.2 and 0.4 (Triantafyllou et al., 1991, 1993; Eloy, 2012; Gazzola et al., 2014). In this regime, the wake of the animal has a very particular shape with vortices alternatively on either side of the body. These vortices are oriented such that they produce a backwards jet on average and are called a reverse Bénard-von Kármán vortex street.

Still, important discrepancies remain between swimmers, the two most notable being the relative size of the tail compared to the body, and the type of motion adopted by the tail. The first one is measured by the Lighthill number: $Li = S_w/S_pC_D$, where $S_w$ is the wetted surface of the fish, $S_p$ the surface of its propulsor, and $C_D$ is the drag coefficient of the whole fish (Eloy (2012); Akoz and Moored (2018), see Fig. 1.3 (b)). The second important distinction between types of locomotion is the tail actuation. Simplifying the motion of a fish tail to that of a 2-dimensional airfoil (Fig. 1.3 (b)), there are two degrees of freedom available: pitch and heave. Pitch consists of angular oscillations around the leading edge of the wing, while heave corresponds to side-to-side motion. Any other 2 dimensional motion of a fish tail can be modeled by a combination of the two. As a result, a large body of literature has focused on finding the thrust and efficiency of such motions (Triantafyllou et al., 1991; Floryan et al., 2017a, 2018; Moored and Quinn, 2019; Van Buren et al., 2019; Wu et al., 2020).

In fish, intermittent locomotion is also called burst and coast. The fish activates its tail for one or two
beats and then glides passively using the kinetic energy accumulated during the active phase (see Fig. 1.4). Weihs (1974) was the first to theoretically investigate the efficiency of this type of gait. He showed that intermittent locomotion could be more energy efficient than continuous swimming if the drag experienced by the fish was reduced during the gliding phase. Indeed, undulatory swimming increases the drag produced by the body of the fish during the burst phase by a factor of up to three times the drag at rest during the coast phase. The mechanism leading to a high drag contrast between the two phases of the cycle is achieved via a combination of a geometrical effect (the body is aligned with the flow during the coast phase) and a dynamical one (the skin friction is increased during the burst phase due to the motion of the body), that increases the drag while the tail is moving (Ehrenstein et al., 2014; Godoy-Diana and Thiria, 2018; Akoz and Moored, 2018). Weihs showed that this drag contrast is crucial for taking advantage of intermittent kinematics. In particular, his model shows that intermittent locomotion is the most efficient swimming strategy when the drag during the coast phase is half the drag during the burst phase.

<figure><img src="image_9.png" /><figcaption>Figure 1.4: (a) Chronophotography of *Hemigrammus rhodostomus* swimming in a channel, the flow is coming from the top and time is from left to right. Courtesy of B. Lafoux. (b) Schematic of the time dependent velocity of a fish swimming in a burst and coast gait.</figcaption></figure>

Some experimental works have successfully measured this drag contrast. First, (Videler, 1981) estimated that the drag of a swimming cod is about 3.3 times greater than that of a gliding fish. With koi carps and using a PIV setup to measure the fish's wake, Wu et al. (2007) found a fourfold increase in drag in the active phase. The same study also concludes that burst and coast allows the fish to reduce the cost of swimming by 45% compared to steady swimming at the same average speed. To our knowledge, this work is the only experimental estimate of the energy savings of intermittent propulsion in live fish.

Other authors have been able to simulate 3D flows around a body kinematics directly extracted from measurements of fish swimming (Chung, 2009; Xia et al., 2018; Ashraf et al., 2020). It is then possible to vary the parameters accessible to the fish such as tailbeat frequency and amplitude or the fraction of the period spent actively propelling (also called duty cycle). Li et al. (2021, 2023) have shown that fish use parameters that are very close to the optimal choice to minimize their energy expenditure. Burst and coast locomotion in fish therefore results in reductions in energetic cost of up to 26% compared to continuous motion at the same speed (Fig. 1.5 (b)).

Akoz and Moored (2018) performed numerical simulations of intermittently pitching airfoils (Fig. 1.3 (b)). They found that the cost of motion at a given speed can be significantly reduced (Fig. 1.5, (a)) even if the drag of the body is unchanged between the two phases (Akoz et al., 2019). By modeling the unsteady forces on the airfoil, they showed that there are both drag-producing and thrust-producing inviscid contributions when pitching the airfoil. They later added the heave contribution (Akoz et al., 2021) and showed that heave dominated motions do not benefit from intermittency. Heaving motion creates thrust through circulation around the airfoil, creating a small amount of drag compared with pitching motion, which is primarily based on added mass. Swimmers who benefit the most from intermittent locomotion therefore have small Lighthill numbers (i.e., large tails) and perform mostly
pitching motions.

Finally, Floryan et al. (2017b) have studied intermittent locomotion experimentally, using a pitching airfoil in water. The airfoil is fixed and intermittently actuated. To make the comparison with burst and coast the authors have to make two strong assumptions: (i) the flow remains quasi-steady in the accelerated case, such that no effect is lost by keeping the foil stationary and (ii) the forces generated by the airfoil are independent of the incoming flow velocity (shown in a previous study from the same authors: Floryan et al. (2017a)). They found also in this case that intermittent motion can be more efficient than continuous swimming at the same mean speed.

<figure><img src="image_10.png" /><figcaption>Figure 1.5: (a) Cost of transport for a pitching airfoil, relative to the cost of transport of continuous motion at the same speed, as a function of the duty cycle. Different curves and colors correspond to different dimensionless tail amplitudes. Note that the x-axis is inverted (Adapted from Akoz and Moored (2018)). (b) Cost of locomotion Ω* of a small fish as a function of mean speed for continuous and optimal intermittent motions (Adapted from Li et al. (2023)).</figcaption></figure>

### 1.1.3 Rotors

Our goal is to apply the concepts of reconfiguration and intermittency to underwater propulsion. Intermittent propulsion with a propeller is essentially unexplored in the literature. However, flexible rotors have recently attracted engineering interest due to new blades made of composite materials. These are very light and can be made into any shape, but tend to deform more than traditional alloy blades. Designers of wind turbines or propellers must therefore take these deformations into account (Marsh, 2004; Hansen, 2015; Young et al., 2016).

Using flexible materials in rotors is however not a new idea: Winston (1968) explored the possibility of using extremely flexible helicopter blades to improve weight and easier storage. The blades are so flexible that they bend under their own weight and are straightened when rotating by the centrifugal force. In these conditions, important couplings between the flexibility and the fluid loading occur, generating aeroelastic instabilities (Tang and Dowell, 1993). The blade can start oscillating along its torsion axis in a phenomenon called flutter (de Langre, 2006). The aerodynamic lift on the blade amplifies the oscillations with a potentially destructive result. Sicard and Sirohi (2012, 2016) studied these instabilities in detail for ribbon-like helicopter blades and found that they can be damped by modifying the location of the center of gravity. By placing small masses at the tip of the blades near their leading edge, the aerodynamic force acts to stabilize the blade.
<figure><img src="image_11.png" /><figcaption>Figure 1.6: (a) Shape of a composite marine propeller blade deforming under hydrodynamic load. LE and TE respectively refer to leading and trailing edges (Adapted from Young (2008)). (b) Wind turbine blade deforming and changing its effective angle with the flow (Extracted from Cognet et al. (2020)). (c) Dimensionless power<fnref n="1" /> $C_P$ extracted by a wind turbine for varying external conditions. Red diamonds correspond to a wind turbine with flexible blades while all the other symbols are rigid wind turbines with various blade pitch. $\Lambda$ is the tip speed ratio and $\theta_0$ is the pitch angle. The inset displays a section of the blade deforming under the action of three forces: the centrifugal force $f_c$, the aerodynamic lift force $f_L$, and the aerodynamic drag force $f_D$ (Adapted from Cognet (2017)).</figcaption></figure>

Flexibility in rotor blades can also be utilized to enhance performance. Typically, blades are designed for a specific operating condition; however, wind turbines operate across a broad spectrum of wind speeds. In order to maintain peak efficiency, many modern wind turbines incorporate motors to adjust the angle of the blades. Changing the shape of the blades to adapt to the exact conditions. Drawing inspiration from the reconfiguration seen in natural systems, Cognet et al. (2017) showed that it is possible to do this same shape change passively, using flexibility alone (Fig. 1.6, (b) and (c)). Two forces act on the wind turbine blade: the aerodynamic load and the centrifugal force. These two forces act in opposite directions (Fig. 1.6, inset of (c)). Balancing their relative importance allows the blade deformation to be controlled precisely. Cognet et al. (2020) therefore showed that given a wind turbine geometry and a site's wind speed distribution, it is possible to find material properties (density and flexibility) for the blades such that the efficiency of the rotor is improved. Recently Durán Venegas et al. (2019) have proposed a mathematical description for this problem, for a rotor in which both fluid forcing and centrifugal force bend and twist the blades as they rotate.

In the specific case of marine propellers, an extensive body of literature has focused on the effects of flexible composite propellers on hydrodynamic efficiency (Marsh, 2004; Young et al., 2016; Hussain et al., 2021). The first simulations and experiments of Young (2008) showed the potential of composite propellers flexing to alter the performance under normal or cavitating conditions and dampen the vibrations of the structure (Fig. 1.6, (a)). This work highlighted the importance of the fiber layup direction in controlling the mechanical response of the blades. The resulting anisotropy may be used to improve performance, as shown by Motley et al. (2009). A coupling between the bending and twisting deformations of the blade can make the blade passively adapt to variations of external conditions. In particular, they showed that propellers can passively adapt to remain efficient when operating in the spatially varying wake of the hull of a ship.

<footnote n="1">¹The power coefficient $C_P$ represents the fraction of the power available in the incoming wing extracted by the wind turbine. With $Q$ and $\Omega$ the torque and angular velocity of the rotor respectively, the power coefficient is $C_P = Q\Omega/(1/2\rho\pi R^2 U_\infty^3)$. $\rho$ is the fluid density, $R$ is the radius of the turbine and $U_\infty$ the wind velocity.</footnote>
During the course of this thesis, Eldemerdash and Leweke (2021) studied experimentally a model scale propeller with highly flexible blades. Using flow field visualization and blade tracking they observed large deformations up or downstream depending on the angle of the blades and the angular velocity. Under braking conditions, when the rotor produces negative thrust, large amplitude fluctuations of the blade shape are observed. They are interpreted as the blade rotating in the vortex generated by its own wake, a phenomenon called the Vortex Ring State in helicopters. Building upon the model created by Durán Venegas et al. (2019), they are able to accurately predict the small deformations of the rotor (Eldemerdash, 2023).

# 1.2 A blade element description of rotors

Rotors are particular systems due to their screw geometry. The description of their behavior requires a careful analysis of the local flow velocity as well as the geometrical properties of the rotor. An approach to describe the local and global loads on a rotor is described in detail below. Models relying on this approach are used throughout the thesis, adding each time a different layer of complexity: folding blades, flexibility, or curvature effects. We explain here the general technique allowing the estimation of these loadings, and derive a few classical results on propellers.

### 1.2.1 Blade element model

A classical tool used in the analysis of rotating systems (boat or plane propellers, wind turbine etc.) is the blade element model. It consists in dividing the rotor in small spanwise cuts along each blade and analyzing them separately. Sections along the same blade are supposed independent of each other and multiple blade interactions are neglected. The technique detailed here applies to rotors subjected to flows along their axis of rotation (i.e. propellers or turbines) but the specific details could be modified to work for rotors with flows perpendicular to the rotation (for instance, helicopter rotors). This division of the global problem can be thought of as dividing the incoming flow into concentric streamtubes that pass through the rotor and only interact with the blades at a single radius, not influencing the adjacent tubes (See Fig. 1.7). The approach here is dedicated to steady flows, if there are important variations faster than a typical rotation of the propeller, the analysis should be modified.

Dividing the problem in such a way allows us to transform the 3D problem of the flow around a rotating structure into a series of 2D problems around an airfoil. This transformation allows a very fast and simple estimation of the forces involved at the expense of several strong hypotheses:

* Radial flows are neglected: they are typically smaller than the axial or orthoradial flows and do not generate circulation around the blades. The radial flow therefore do not contribute to the lift force which is the main source of thrust. Near the tip or root, vortices are generated due to finite size effects and therefore important radial flows are present, they are neglected here.

* The rotor self-induction is not considered: when analyzing a streamtube passing through a propeller, the exact axial velocity at the location of the rotor is not the one far upstream. The flow is slightly accelerated before reaching the rotor. In the same way its angular velocity is not zero due to the rotor itself. This effect can be taken into account using the Blade Element Momentum Theory which is extremely common in the engineering of wind turbines (Glauert, 1983; Sørensen, 2011; Masters et al., 2011). However, this tool often requires semi-empirical corrections to be quantitative (Snel et al., 1994; Corrigan and Schillings, 1994; Mario, 2012;
MacNeill and Verstraete, 2017) which may be problem dependent. Our goal is to build a minimal model describing the behavior of a rotor in general, so we neglect these effects.

* Flow curvature effects are neglected: when interacting with a section of the blades, the flow is considered to be 2D and planar. In reality, the flow is along a tube. These curvature effect are all the more important when the blades occupy a large angular sector. A geometrical consequence of this effect is analyzed in Sec. 4.1 where it is particularly important.

* The problem is supposed axisymmetric: there are no variations of flow conditions during the rotation (for instance due to the presence of a hull). An example on the modifications required to take into account spatially varying flow conditions is detailed in Sec. 2.2.3

<figure><img src="image_12.png" /><figcaption>Figure 1.7: (left) Schematic of a streamtube interacting with a rotor. (right) Definition of the angles on a 2D profile interacting with the streamtube.</figcaption></figure>

The definitions and schematic of the model are shown in Fig. 1.7: the incoming axial flow has a velocity $U_\infty$, and the propeller is rotating with an angular velocity $\Omega$. The streamtube interacts with the blade at a radius $r$. Looking at a 2D section in particular (right figure, the $x$ direction corresponds to the axis of rotation and the $y$ axis to the plane of rotation), the blade has a chord length $c(r)$ and a local angle with respect to the plane of rotation $\gamma(r)$. At the root of the blade, this angle is $\gamma_0$: the pitch angle. In typical plane or ship propellers, the blades are twisted along the span, such that $\gamma(r) \neq \gamma_0$ in general. The pitch angle is a key property of the rotor and can be adjusted during the rotation in the case of wind turbines Hansen (2015) or plane propellers Glauert (1983). The incoming flow has an $x$ component of norm $U_\infty$ and a $y$ component of magnitude $r\Omega$, such that the local velocity is $U_p = U_\infty e_x + r\Omega e_y$. A key parameter of the study of rotating structures is tip speed ratio or reduced velocity $\lambda = R\Omega/U_\infty$, such that the angle $\varphi$ between the flow and the plane of rotation reads:

$$ \tan \varphi = \frac{U_{\infty}}{r\Omega} = \frac{R}{r\lambda}. \qquad (1.5) $$

The angle of attack of the airfoil at this particular location is thus:

$$ \alpha(r) = \gamma(r) - \varphi(r) = \gamma(r) - \arctan \frac{R}{r\lambda}. \qquad (1.6) $$

This angle of attack on a 2D airfoil generates a force due to the circulation of the fluid around the profile. This infinitesimal force is usually divided into a component in the direction of the incoming flow: the drag $dD$, and its orthogonal counterpart: the lift $dL$.
These two forces are proportional to the square of the local velocity $|U_p|^2$ and read:

$$dL = \frac{1}{2} \rho c(r) dr |U_p|^2 C_L, \quad dD = \frac{1}{2} \rho c(r) dr |U_p|^2 C_D \quad (1.7)$$

where $\rho$ is the fluid density and $dr$ is the infinitesimal width of the section of the blade. The lift $C_L$ and drag $C_D$ coefficients are dimensionless functions that relate the shape of the airfoil and the angle of attack to the forces produced. Their precise modeling is discussed in the following section (Sec. 1.2.2).

In the case of rotors, the quantities of interest are the thrust $F_{\text{thrust}}$ (the force along the axis of rotation) and the torque $Q$ (the moment required to spin the propeller). By projecting the lift and drag force along the $x$ and $y$, we obtain:

$$dF_{\text{thrust}} = dL \cos \varphi - dD \sin \varphi, \quad (1.8)$$

$$dQ = r(dL \sin \varphi + dD \cos \varphi). \quad (1.9)$$

These forces are usually made dimensionless using the angular velocity of the rotor such that we can define the thrust and torque coefficients as:

$$C_T = \frac{F_{\text{thrust}}}{1/2 \rho \pi R^2 (R\Omega)^2}, \quad C_Q = \frac{Q}{1/2 \rho \pi R^2 (R\Omega)^2 R}. \quad (1.10)$$

The reference surface used here is the area swept by the rotor as it spins<fnref n="1" />: $\pi R^2$. Combining Eqs. (1.7)-(1.10), we obtain for the infinitesimal thrust and torque coefficients:

$$dC_T = \frac{c(r)dr}{\pi R^2} \left( \left(\frac{r}{R}\right)^2 + \frac{1}{\lambda^2} \right) \left( C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi \right), \quad (1.11)$$

$$dC_Q = \frac{r c(r) dr}{R \pi R^2} \left( \left( \frac{r}{R} \right)^2 + \frac{1}{\lambda^2} \right) \left( C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi \right). \quad (1.12)$$

The full thrust and torque coefficients are obtained by integrating $dC_T$ and $dC_Q$ over the span of the blades (from root to tip) and multiplying by the number of blades $N_b$. We use a dimensionless abscissa $s = r/R$ from $s_{\min} = r_{\min}/R$ to 1 and the coefficients read:

$$C_T = \int_{s_{\min}}^{1} \sigma(s) \left( s^2 + \frac{1}{\lambda^2} \right) \left( C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi \right) ds, \quad (1.13)$$

$$C_Q = \int_{s_{\min}}^{1} \sigma(s) s \left( s^2 + \frac{1}{\lambda^2} \right) \left( C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi \right) ds, \quad (1.14)$$

where $\sigma(s) = N_b c(s)/\pi R$ is called the solidity of the rotor. These expressions can be further simplified by noticing that

$$\cos \varphi = \frac{1}{\sqrt{1 + 1/(s\lambda)^2}}, \quad \sin \varphi = \frac{1/s\lambda}{\sqrt{1 + 1/(s\lambda)^2}}, \quad (1.15)$$

which finally gives

$$C_T = \int_{s_{\min}}^{1} \frac{s}{\lambda} \sigma(s) \sqrt{1 + (s\lambda)^2} \left( C_L(\alpha) - \frac{C_D(\alpha)}{s\lambda} \right) ds, \quad (1.16)$$

$$C_Q = \int_{s_{\min}}^{1} \frac{s^2}{\lambda} \sigma(s) \sqrt{1 + (s\lambda)^2} \left( \frac{C_L(\alpha)}{s\lambda} + C_D(\alpha) \right) ds. \quad (1.17)$$

<footnote n="1">¹In the engineering literature on marine propellers a slightly different scaling convention can be found. Instead of the tip speed ratio, the advance ratio $J = U_{\infty}/nD$ is used where $n$ is the angular velocity of the rotor in rotations per second $n = \Omega/2\pi$ and $D = 2R$ the diameter. The thrust and torque coefficients are consequently written as $K_T = F_{\text{thrust}}/(\rho n^2 D^4)$ and $K_Q = Q/(\rho n^2 D^5)$. One can convert between the two systems with $\lambda = \pi/J, C_T = 8/\pi^3 K_T$ and $C_Q = 16/\pi^3 K_Q$.</footnote>
The regions of the blades that contribute most to the force generation are the regions near the tip (where $s\lambda \gg 1$) because the norm of the velocity is more important there. The thrust is mostly created by the lift on the blades: the drag coefficient is typically smaller than the lift and its contribution is scaled down by $s\lambda$. On the contrary, the torque is mostly created by the drag on the blade sections.

### 1.2.2 Flows around foils

As shown in the previous section, knowing the 2D flow and resulting forces around an airfoil for a given Reynolds number and angle of attack is key to estimating the performance of a rotor. This problem has an extensive body of literature around it as it is of prime interest for many engineering applications such as plane wings, sails and rotors. The lift and drag coefficients, a priori depend only on two parameters (except from the airfoil shape itself): the angle of attack $\alpha$ and the chord-based Reynolds number: $Re = Uc/\nu$, with c the chord. The coefficients can either be directly measured through experiments keeping the angle of attack and Reynolds numbers constant, through analytical modelling or through simulations.

In high Reynolds number flows, typically from 1000 onwards, a simplifying assumption can be made: the lift and drag coefficients have a weak dependence on the Reynolds number (compared with the angle of attack) (Lissaman, 1983). This allows the possibility of making experiments at scale without conserving the Reynolds number (which is difficult since the product $Uc$ must be kept constant if the fluid is unchanged). The Reynolds number mostly influences the boundary layer around the airfoil: depending on $Re$, it can either be turbulent or laminar, attached to the profile or detached. In particular, for relatively low Reynolds numbers, the classical stall phenomenon (during which the boundary layer separates from the profile and the lift force suddenly drops) can disappear in the force measurements (Alam et al., 2010; Mateescu and Abdo, 2010).

In the experiments described in the manuscript, the Reynolds number is defined using the local velocity norm in a given blade section:

$$Re = \frac{c(r)\sqrt{U_{\infty} + r\Omega}}{\nu}. \qquad (1.18)$$

Thus, its exact value varies along the span of the blade and with the angular velocity of the rotor. The typical range explored in experiments, from slow velocities at the blade root to high velocities at the tip is $Re \in [750, 10000]$. In this range of Reynolds number the coefficients mostly depend on the angle of attack. Therefore, scaling the system — changing only the Reynolds number by increasing the flow velocity or size — will have only a small effect on the performance (Sunada et al., 1997).

Several properties are expected of the lift and drag coefficients as functions of the angle of attack. If the airfoil profile is symmetric, zero angle of attack results in zero lift. Similarly, if the profile is fore-aft symmetric the lift should be zero for $\alpha = \pi/2$. Added to that the function $C_L(\alpha)$ should be antisymmetric since a negative angle of attack results in negative forces. Finally, the lift force at small angle of attack should be linear with $\alpha$ according to thin airfoil theory (Kaushik, 2019). In contrast, the drag coefficient should be symmetric, with a minimum in zero and maximal when perpendicular to the flow.

For airfoil with a thin flat shape, which we will often use in the following, a very common and simple model for the lift and drag coefficients that verifies the properties above can be written as follows (Cognet et al., 2017; Timmer, 2010):

$$C_L(\alpha) = C_L^1 \sin \alpha \cos \alpha, \quad C_D(\alpha) = C_D^0 + C_D^1 \sin^2 \alpha, \qquad (1.19)$$

with $C_L^1$, $C_D^0$, and $C_D^1$ constants that depend on the airfoil shape. This law uses a single coefficient to
describe the small angle of attack lift slope as well as the maximal lift reached which can be difficult to fit onto experimental data (usually $C_L^1 \approx 2$, $C_D^0 \approx 0.1$ and $C_D^1 \approx 2$).

Recently Li et al. (2022) studying small scale gliders, proposed a simple law able to better describe the force produced over the whole range of angles of attack. It matches the limiting cases of small and high angle of attacks using a smooth function

$$f(\alpha) = \frac{1}{2} \left( 1 - \tanh \left( \frac{\alpha - \alpha_0}{\delta} \right) \right),$$

where $\alpha_0$ is the stall angle and $\delta$ represents the smoothness of the stall. The lift and drag then read:

$$C_L(\alpha) = f(\alpha) C_L^1 \sin \alpha + (1 - f(\alpha)) C_L^2 \sin 2\alpha, \qquad (1.20)$$

$$C_D(\alpha) = f(\alpha) (C_D^0 + C_D^1 \sin^2 \alpha) + (1 - f(\alpha)) C_D^{\pi/2} \sin^2 \alpha. \qquad (1.21)$$

This model thus has 5 coefficients of fit, which allows it to have a richer phenomenology.

<figure><img src="image_13.png" /><figcaption>Figure 1.8: Lift and drag coefficients as a function of the angle of attack for a flat plate at Reynolds number 4500. Lines show fits of the data for two models.</figcaption></figure>

We measured the lift and drag forces generated by a wing profile using a modified version of our setup (similar to the one described in Sec. 2.2.1). The wing profile has a similar shape to the one used with the propellers but is scaled up by a factor 2 in order to have a similar Reynolds number ($Re = 4500$). Fig. 1.8 displays the lift and drag coefficients associated. For each of the two models, we adjusted the coefficients to fit this data. We find for the flat plate model: $C_L(\alpha) = 1.93 \sin \alpha \cos \alpha$, and $C_D(\alpha) = 0.10 + 1.6 \sin^2 \alpha$. This model fits well the data, except for small angles of attack where we see that the lift increases more rapidly. The model developed by Li et al. (2022) takes into account this effect. We find the following coefficients: $a_0 = 19$ deg, $\delta = 4$ deg and $C_L^1 = 2.4$, $C_L^2 = 0.95$, $C_D^0 = 0.1$, $C_D^1 = 2$, $C_D^{\pi/2} = 1.6$. While the agreement with the experimental lift at small angle of attack is improved, the drag is not significantly modified by using this formula.

In the rest of this manuscript, we mostly use the flat plate model for its simplicity. More complex laws such as the one from Li et al. (2022) could easily be used instead to more accurately model stall on the blades. However, it is important to keep in mind that the stall phenomenon is strongly Reynolds number dependent, especially in the lower limit of $Re$ studied here (Lissaman, 1983; Mateescu and Abdo, 2010; Alam et al., 2010). Corrections should therefore depend on the local Reynolds number on the blade, which adds complexity without substantially improving the accuracy of the models.
### 1.2.3 Propeller performance

<figure><img src="image_14.png" /><figcaption>Figure 1.9: Performance curves of a propeller as a function of the tip speed ratio. From left to right: thrust coefficient, torque coefficient and efficiency. Colors indicate the pitch angle $\gamma_0$.</figcaption></figure>

Once the propeller geometry and lift and drag coefficients are known, we can compute the thrust and torque using Eqs. (1.16) for any tip speed ratio. For this example we use the same shape (number of blades, chord, radius, etc.) as the propeller described in Sec 3.2, but this choice does not qualitatively change the results presented here. We introduce a parameter measuring the hydrodynamic efficiency of the propeller:

$$ \eta = \frac{F_{\text{thrust}}U_{\infty}}{Q\Omega} = \frac{C_T}{\lambda C_Q}. \qquad (1.22) $$

It compares the useful power $F_{\text{thrust}}U_{\infty}$ to the total mechanical power invested $Q\Omega$ and is the figure that engineers optimize after matching thrust requirements. We show the evolution of the thrust, torque, and efficiency as a function of the tip speed ratio in Fig. 1.9. The colors show the effect of changing the pitch angle of the blades (i.e., setting $\gamma(s) = \gamma_0$).

Looking at the thrust and torque curves at a given pitch angle, we can define three regions. At very low tip speed ratio, $C_T < 0$ and $C_Q < 0$: the flow creates a torque that spins the propeller and creates a backwards thrust. This region corresponds to wind turbines. The torque coefficient always becomes positive first defining a region where $C_T < 0$ and $C_Q > 0$. Spinning the propeller too slow compared with the flow speed creates a braking force. Finally, the thrust becomes positive, and we have $C_T > 0$ and $C_Q > 0$, energy is spent to create a forward pointing thrust force. The $\lambda$ threshold between these regions get smaller as we increase the pitch angle $\gamma_0$. The onset of positive thrust can be approximated by stating that the angle of attack near the tip must be positive, giving

$$ \lambda > \frac{1}{\tan \gamma_0}, \qquad (1.23) $$

which indeed decreases as $\gamma_0$ approaches $\pi/2$.

The corresponding efficiency is well-defined only for positive torque. For each pitch angle it reaches a maximum for a tip speed ratio slightly greater than the onset of positive thrust and then declines. For each tip speed ratio, there is one pitch angle which maximizes the efficiency of the rotor. This optimal pitch angle $\gamma_0^{\text{opt}}(\lambda)$ is a decreasing function of the tip speed ratio: high pitch angles are only
efficient at small tip speed ratios. As a consequence, to remain as efficient as possible throughout the range of tip speed ratio, the pitch angle $γ_0$ needs to be modified accordingly. In practice, aircraft propellers and wind turbines use motors to actively adapt the pitch angle to the conditions. This technique is rarely found in marine propellers because of its complexity in the marine environment but would add significant efficiency improvements.

To build an efficient propeller, engineers want to optimize the thrust to torque ratio. Using the associated definitions in Eq. (1.16), we can compute the local efficiency for each blade section (for a given $r$ and $λ$). One way to maximize the global efficiency is to maximize the infinitesimal contribution from each section by changing the local pitch angle: for each section we compute $γ(s)$ such that $dC_T/(λdC_Q)$ is maximized. We therefore create an optimal spanwise distribution of pitch angle that generates a very efficient propeller.

In Fig. 1.10, we show this distribution along with the local flow angle $φ$ (computed for $λ = 4$) as a function of the spanwise position. We see that both follow the same trend keeping an almost constant difference between them. This behavior can be rationalized using the following argument: maximizing the thrust can be approximated as maximizing the lift on each section, while minimizing the torque corresponds to minimizing the drag. Thus thrust to torque ratio is maximal at the same time as the lift to drag ratio on a particular blade section. For the flat plate model used here, the lift to drag ratio is shown in Fig. 1.10 (right) and we can see that it reaches a maximum between 15 and 20 degrees, about the same as the difference between the optimal propeller twist and the local flow angle. For the flat plate the optimal angle of attack is $α_{opt} = 17$ deg. A well functioning propeller needs to have a local angle of attack close to maximum lift to drag ratio on most of the blade, achieved by having important twist between root and tip.

<figure><img src="image_15.png" /><figcaption>Figure 1.10: (left) (blue) Local flow angle $φ$ (See. Fig. 1.7) and (green) optimal twist as a function of the dimensionless spanwise coordinate $s$. Computed at a tip speed ratio $λ = 4$. (Right) Lift to drag ratio $C_L/C_D$ as a function of the angle of attack for the flat plate model used.</figcaption></figure>

The dimensionless parameters involved in the problem of a flat bladed propeller are the geometrical properties: $c/R$, $r_{min}/R$, $γ_0$ and the hydrodynamical parameters $λ$ and $Re$. Changing the flow velocity at a constant tip speed ratio only influences the resulting coefficients (thrust or torque) through the Reynolds number. Its influence is limited to modifying the lift and drag coefficients, which are expected to vary only by a small amount. Modifying the flow speed at a constant tip speed ratio therefore does not affect the thrust and torque coefficients.

The blade element model is a useful tool to model the experimental observations and will be used several times in the thesis. Its simplicity allows us to highlight the important mechanisms at play.
## Thesis outline

Our goal is twofold: first, to study the deformation of a flexible propeller in a flow to investigate whether the propulsive efficiency can be improved using a technique similar to that of Cognet et al. (2017) in the case of a wind turbine. A propeller that can passively deform and adapt to a change in tip speed ratio would greatly improve efficiency when operating in the wake of a ship's hull. Second, we want to explore the possibility of using intermittency to reduce propulsion energy consumption. The results cited previously show that the drag of the whole system (boat and propeller) must be significantly reduced when gliding in order for this technique to be more efficient than continuous motion. Therefore, we also need to design a shape changing propeller to achieve this drag reduction.

Chapter two presents the setup and tools used to address these questions. We will describe the free surface water channel that was built specifically for this project and the systems that were used to hold the propellers and sensors.

Chapter three presents our results on shape changing propellers. We discuss the deformations and forces experienced by two different propellers. The first is built to improve our understanding of the fluid-structure interaction problem of a flexible rotor in water: it has slender ribbon-like blades that experience large amplitude bending and torsion. We will show how the direction and magnitude of the deformations can be understood through a geometric argument and highlight a strong link between loads and deformations. The second propeller studied is focused on improving the efficiency: its design is based on the work of Cognet (2017). We will show that using this propeller with the right flexibility can improve the resilience to a change in the tip speed ratio. We will predict its torsional deformation using a simple scaling law.

Finally, chapter four addresses the new problem of intermittent propulsion with a propeller. In a first part, we describe the shape-changing propeller used to reduce drag during gliding. Using a modified blade element model, we are able to explain its reconfiguration. We will then use it dynamically in a novel experimental setup capable of reproducing the time-dependent burst and coast motion of fish. Using this setup and the shape changing propeller, we will show that it is possible to reduce the cost of motion by using intermittent propulsion. A time-dependent model of the motion of the boat will help us identify the constraints on the boat and propeller system that make intermittent motion more efficient than continuous motion.

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Chapter 2

Experimental setup

This section describes the experimental tools used in during the thesis. First the hydrodynamic channel in which all the experiments were performed, then the different model scale boats used to house the sensors and their instrumentation. Finally we describe the material properties of the flexible sheets we use to build elastic propeller blades.

**2.1 Hydrodynamic channel**

**2.1.1 Description**

We show a simplified sketch of the channel in Fig. 2.1, the goal of the channel is to accommodate two types of experiments: first the study of any type of rotor by producing a steady, uniform flow at any desired velocity. Second the study of the intermittent motion: the device carrying the rotor must be allowed to move freely along the axis of the channel. Thus, the channel must be long enough to accommodate this motion while maintaining uniform conditions.

<figure><img src="image_16.png" /><figcaption>Figure 2.1: Simplified sketch of the free surface canal, (right) zoom on the convergent section.</figcaption></figure>
As a consequence, the channel is built with a very long test section (30 cm x 20 cm x 2 m) and with a free surface. We chose the frontal area and rotor size such that the blockage in the channel stays limited to not qualitatively change the flows studied (Whelan et al., 2009) (in practice always below 15%). Around this section, we built a return channel designed to keep the test conditions constant. A tank, which acts as a buffer and holds most of the water in the whole setup, is located at the end of the channel. This tank is connected to the pump that drives the flow (not visible in Fig. 2.1) with a PVC pipe. The flow generated by the pump is expelled upwards into a convergent section that ensures steady flow in the channel.

The pump (Pentair Whisperflo) is equipped with a variable frequency drive which allows us to adjust its rotational speed. The pump is designed to produce important flow rates with little added pressure, which is ideal for our application because the pressure drop in the entire channel is very small. When the entire chanell is mounted we measured fluxes from 2 to 10 l/s (i.e., flows from 5 to 25 cm/s). The pump has two drive modes: constant angular velocity or constant flux. The pump does not have a flux measuring sensor and uses a factory calibration to control the rotation rate using the electrical power drawn. The two modes produce a constant flow rate, but we will see below (Fig. 2.3) that the constant RPM mode allows us to obtain smaller velocities and is used throughout the thesis.

To transform the high speed, fluctuating flow generated by the pump into a steady, uniform flow in the test section, we use a convergent section that has three main roles:

* transform the fast flow, concentrated in a pipe into a flux source homogenous over the whole width and height of the channel,

* mitigate all the fluctuations that might be present in the initial source,

* uniformly accelerate the flow such that it can be used in the test section.

For the first objective, we use a T-shaped junction that is perforated with multiple holes so that the fast flow from the pump becomes multiple sources across the width of the channel (Fig. 2.1, right). These holes create localized jets that are directed away from the test section, towards a wall. The jets therefore dissipate most of their energy, and we obtain in a diffuse, near-zero velocity flow source. We then uniformly accelerate the flow by using a 4 to 1 reduction in width over 50 cm. To avoid boundary layer separation of the flow over the live edges, we added smooth 3D printed parts to hide these edges. Finally, to dampen the remaining fluctuations in the channel, we used two sets of honeycombs at the entrance and exit of the converging section. They are made of cylindrical straws aligned in a square pattern. A boundary layer develops inside each straw, forcing the flow to align with it. As a result, flow structures larger than the size of the straw disappear (Lumley and McMahon, 1967).

### 2.1.2 Calibration

To calibrate the flow speed in the channel and quantify the magnitude of the fluctuations, we have used two different tools: Particle Image Velocimetry (PIV) and Constant Temperature Anemometry (CTA).

The PIV technique consists of seeding the channel with polyamide particles that act as passive tracers: we illuminate them with a laser sheet placed vertically in the center of the channel and record their movement in this plane with a fast camera. The size of the particles is 100 µm so that they don't settle too fast at the bottom of the channel and still provide good contrast for the camera. We then process the images to extract the 2D velocity field in the illuminated plane. The algorithm compares
<figure><img src="image_17.png" /><figcaption>Figure 2.2: Example of flow field obtained using PIV. (left) Instantaneous snapshot of the reconstructed flow (superimposed with the raw image). Colors show the velocity norm while streamlines show its direction. (right) Time and space averaged vertical velocity profile</figcaption></figure>

two successive images to extract the field in the following way: the first image is sliced into small windows (in our case a square with a side of 61 pixels, out of a 1216x1200 image), and then the algorithm looks in the second image where the correlation with the small window is best. This gives the best estimate of the displacement of the particles in the window and thus an average velocity vector for that window (Tropea et al., 2007). This operation is then repeated for every window, for every frame, and we obtain a time dependent, 2D velocity field. It should be noted that any out-of-plane velocity cannot be detected using this method.

In Fig. 2.2 (left) we show an example of the instantaneous field in the channel. We see that the flow speed is very homogeneous in speed (color) except at the very bottom with only minor disturbances in the vertical direction. We also plot the time and *x* averaged (over 10 s and 0.2 m) velocity profile on the right, and we can confirm that the flow is homogeneous in the vertical direction in the middle of channel. At the bottom we see smaller velocities, indicating the presence of a boundary layer. We can measure the size of this boundary layer in the middle of our test section and see that 95% of the mean flow speed is always reached about 2 cm above the bottom wall. Even we did not measure it directly, it is most likely that similar boundary layers develop on the side walls.

We show the mean flow speed in the channel for both pump setting modes in Fig. 2.3. Both modes have a linear relationship with the flow speed. However, the constant flux mode (blue dots) is no longer linear for small fluxes and cannot produce mean velocities below 0.1 m/s. The constant RPM mode stays linear with the flow rate down to 0.05 m/s and since both modes produce similar fluctuations around the mean speed, we use the constant RPM mode in the following.

It should be noted that one advantage of the constant flux mode overlooked here is the feedback loop within the pump. When making important changes in the RPM setting the pump suddenly accelerates which produces back and forth oscillations in the channel. On the contrary, the constant
<figure><img src="image_18.png" /><figcaption>Figure 2.3: Mean flow speed in the channel as a function of the pump angular velocity and indicated flow rate. Error bars indicate the standard deviation of the flow speed. (black line) Best fit of the constant RPM data: $U$ [m/s] = $9.16 \times 10^{-5}\Omega_{\text{pump}}[\text{RPM}] + 2.57 \times 10^{-3}[\text{m/s}]$.</figcaption></figure>

flow rate mode progressively accelerates and directly settles on the right velocity. In the following we thus take care of waiting for the oscillations to dampen before starting an experiment after changing the flow velocity.

In order to make experiments as close to an open water situation as possible, the intensity of the fluctuations in the incoming flow must be kept to a minimum. Indeed, when a boat is moving through initially still water, the incoming flow on the propeller is only due to the boat speed and thus not fluctuating at all. In the experiments we change the frame of reference by imposing a constant velocity, and the fluctuations must also be minimal to imitate the open water situation. The quantity that measures the amount of fluctuations in the flow is the turbulence intensity and is defined as:

$$ \tau_U = \frac{\sqrt{\frac{1}{3}(u_x'^2 + u_y'^2 + u_z'^2)}}{\bar{U}}, \qquad (2.1) $$

where $\bar{U}$ is the mean flow speed (which is only horizontal in our case) and $u_i'$ is the fluctuation in the $i$ direction.

Two problems arise when trying to measure the turbulence intensity in the channel using PIV: First, the velocity component out of the laser plane is not accessible. Although its mean contribution is expected to be zero, its standard deviation has no reason to be. The second problem is that PIV has a finite resolution and if there are smaller flow structures, they won't be taken into account in the standard deviation. In our case, the size of the interrogation window is about 1 cm. If the incoming flow is indeed turbulent, the classical energy cascade results in important contribution from structures which can be as small as the Kolmogorov scale (Frisch, 1995). This scale is defined as

$$ \eta_K = \left( \frac{\nu^3}{\epsilon} \right)^{1/4}, \qquad (2.2) $$

where $\nu$ is the kinematic viscosity of water and $\epsilon$ the turbulent kinetic energy of the incoming flow: $\epsilon = 1/2(u_x'^2 + u_y'^2 + u_z'^2)$. If the turbulence intensity in the channel is about 5% at a velocity of 0.1 m/s, this gives a minimal scale of 0.4 mm. The PIV alone is thus not able to correctly measure it.

The first problem can be solved by approximating the fluctuations in the transverse direction to be the same as the ones in the vertical direction: both directions perpendicular to the main flow should
have fluctuations of the same order of magnitude. To overcome the second issue however, we need to use a different measuring method.

<figure><img src="image_19.png" /><figcaption>Figure 2.4: (a) Calibration of the hot-wire probe using the PIV data for both pump modes (blue) constant pump flux, (orange) constant RPM. (Black line) Best fit using the King's law: $\bar{U} = 1.82 \times 10^{-3}(V_{CTA} - 1.60)^4$ with $V_{CTA}$ in V and $\bar{U}$ in m/s. (b) Power spectral density (normalized by the mean speed) of the hot-wire measurements. Colors correspond to different pump angular velocity. (inset) Same PSD but the frequency axis is normalized by the angular velocity of the pump.</figcaption></figure>

We use hot-wire anemometry to better measure the intensity of the fluctuations in the channel. It allows for point wise measurements of the flow speed with very high temporal resolution. Its principle is based on measuring the temperature of a tiny wire exposed to the flow. It is heated by a current flowing through it and cooled by the flow carrying away heat energy. In our case we use Constant Temperature Anemometry where the temperature of the wire is held constant, and thus its resistance as well through a feedback loop. This means that a change in flow speed is measured through a change in the current flow through the wire. We use the MiniCTA circuit from Dantec Dynamics to control the hot-wire probe: it has a bandwidth of above 10 kHz and can be used in water applications. The probe is specially designed to work in conducting liquids and is made out of a 70 µm quartz fiber, 3 mm long covered by a nickel film with a thickness of 0.1 µm. The probe is straight and mainly sensitive to flows which are perpendicular to the axis of the fiber. We place it horizontally, perpendicular to the incoming flow, in the middle of the channel. The relationship between the sensor resistance $R_w$, the current flowing through it $I_w$ and the flow speed $U$ is known to follow Kings's law for incompressible flows (Tropea et al., 2007):

$$ \frac{R_w I_w^2}{R_w - R_a} = A + B U^{0.5}, \qquad (2.3) $$

where $R_a$ is the resistance of the unheated sensor and $A$ and $B$ are constants which depend on the wire and fluid properties (e.g., fluid temperature). We thus monitored the fluid temperature (mainly controlled by the thermostat of the lab room) with a thermometer when using this sensor. The output from the MiniCTA is a voltage $V_{CTA}$ proportional to $I_w$ and thus we expect the relationship between $U$ and $V_{CTA}$ to be of the form $U = a_1 + a_2 V_{CTA}^4$ where $a_1$ and $a_2$ are constants.

Fig. 2.4 (a) shows how the hot-wire is calibrated using the PIV measurements with both pump modes. On the right we show the power spectral density of the signal. We see a rapid decrease of the energy density as the frequency increases: above 100 Hz (the time resolution of the PIV) the most energetic
mode remaining beyond this value is 10³ less important than the mean flow. We can also notice that the energy is spread over all frequencies below 10 Hz. The high frequency peaks have a relatively low energy associated. The peak at 25 Hz present in all measurements most likely has an electrical origin since it is not visible in the PIV. The peaks from 100 Hz onwards seem to be noise created by the pump, indeed when plotting the spectrum against the reduced frequency (f/Ωpump) in the inset, we can see a peak for all curves at a value of 6, the number of blades of the pump's propeller.

The hot-wire measurement was performed before the honeycombs were installed, and the PIV was performed before and after. Although the exact turbulence intensity is not available for the final version of the channel, comparing the hot-wire and PIV results is still useful to get the order of magnitude of the fluctuations. Fig. 2.5 shows the intensity of the fluctuations in the channel for both configurations, for both pump modes. First, the hot-wire and PIV measurements are of the same order of magnitude for the channel without honeycombs (hollow bullets and triangles). The similarity of both measurement methods means that the structures we miss with PIV due to too coarse spatial or temporal resolution have low energy associated with them. Then, the effectiveness of the honeycombs in reducing the fluctuations in the channel is easily verified: from 15 to 20%, the turbulent intensity is reduced to below 5 % for all flow velocities. Since we do not study the small details of the flow, this value is satisfactory.

<figure><img src="image_20.png" /><figcaption>Figure 2.5: Fluctuation intensity in the channel as a function of the pump setting. The color indicates the pump mode as in Fig. 2.3. Hollow markers correspond to the channel without the honeycombs installed, while the full ones are with both of them installed. Bullets represent PIV measurements while triangles show hot-wire data.</figcaption></figure>

To avoid depending on the calibration of the pump, we finally added a flow meter (GPI TM200-N-P) in the return channel of the experiments. The magnitude of the flux is encoded by the angular velocity of a small impeller placed in the pipe. We can simply measure the water height in the channel to obtain the mean velocity.

## 2.2 Boat hangar

This section describes the different "boats" that were used during this thesis. Their aim is to hold all the required sensors and protect them from the surrounding water. We use different iterations
throughout the thesis, either because the previous one had limitations or because new constraints required important changes. We detail here their inner workings as well as the trade-offs that each design implies.

They are presented here chronologically. The first version is used in Sec. 3.1, the second one is used in Sec 3.1.4. The final boat is used in Secs. 3.2 and 4.

### 2.2.1 Submarine

<figure><img src="image_21.png" /><figcaption>Figure 2.6: First design of the hull</figcaption></figure>

The first boat (Fig. 2.6) has a very simple design that allowed us to spin a propeller in the flow at a desired angular velocity and record the force. A small electric motor (Portescap 22N78 311P), held in place by a metal rod, turns a shaft to which the blades are attached. The shaft is oriented upstream so that the incoming water flow is undisturbed when it reaches the blades. The motor has a 5 to 1 gearbox attached to it to be able to spin the blades through water without stalling. We measure the angular velocity of the propeller by placing a small magnet on the shaft and a Hall effect sensor in front of it. Each time the magnet passes in front of the sensor, the output goes from 0 to 5V. By measuring the frequency of the output signal, we can measure the angular velocity.

The motor and encoder are protected from the water by a 3D-printed hull, which is assembled in two parts, joined by screws and an O-ring to make the seam watertight. This hull is designed to minimize its impact on the surrounding flow: it adopts an airfoil profile in thin sections and is as thin as possible around the motor. The shaft is held in place by a bearing, and the seal is made by putting grease between the hull and the shaft. We attached the entire boat to a bending force sensor (CZL616C, measuring up to ±7.65 N) placed vertically and parallel to the incoming flow. This sensor thus measures the total axial force (thrust + drag) during the experiment and is attached at the other end to a fixed point in the laboratory.

This boat was used to measure the thrust and deformation of a fully flexible propeller (Sec. 3.1). We tried to measure the power required to turn the propeller using the electrical power consumed by the engine. However, two problems arose: (i) there is a lot of dissipation in the system, which is difficult to quantify (Joule effect in the motor, losses in the gearbox or in the bearing), and (ii) the efficiency of the motor itself: its conversion rate between electrical and mechanical power is not a constant over the range of angular speeds (as shown in the manufacturer's documentation). Overcoming these
difficulties would have added too much uncertainty to the power measurement, so we decided to
use a dedicated torque sensor in the second version.

2.2.2 Semi-submerged boat

<figure><img src="image_22.png" /><figcaption>Figure 2.7: Second design of the hull, adding a torque sensor</figcaption></figure>

This second design (Fig. 2.7) is an improvement of the first. The main idea remains the same: a shaft facing the incoming flow, upstream of the hull, is rotated by an electric motor. The main difference is the addition of a torque sensor (Burster 8661 4050 V0000, measuring up to ±0.05 N.m) between the motor and the propeller. It can measure the torque required to turn the propeller while the shaft is rotating. This measurement is done by strain gauges placed on the rotating shaft, which deform in response to the torque. The signal measured by these gauges is then transmitted without contact to a stationary receiver.

The new electric motor (Micromotors PS150-24-25) is more powerful and has an integrated 25-fold gear reduction. Placing the motor and torque sensor in line would mean a hull more than 20 cm long. To avoid too much hull disturbance in flow, we instead used the vertical space above the torque sensor and a pulley and belt system for the transmission. Finally instead of the Hall effect sensor, we used an off the shelf encoder instead (RLS RE22). The force sensor is the same as with the previous boat (CZL616C).

The first boat had a tendency to leak water through the gap between the rotating shaft and the hull,
even in the presence of grease. To better protect the sensors, we used a stern tube (or "tube etambot"
in French) from a model boat company. It is a long shaft and tube with a small tolerance between the
two that can be filled with grease and provides a much better seal against water. This boat was used
to measure the thrust and torque of the fully flexible propeller (Sec. 3.1.4).
<figure><img src="image_23.png" /><figcaption>Figure 2.8: Schematic of the final boat, with all the sensors out of the water.</figcaption></figure>

### 2.2.3 Surface boat

#### Description

The final design is different from the previous ones and results from the difficulties in precisely measuring the torque at this small scale. The friction is disproportionally important when reducing the size compared to a full scale boat. In physically interesting regimes, the parasitic torque can be as important as the hydrodynamic torque. This parasitic torque, which we measured before experiments also varies with the precise operating conditions (temperature of the grease, amount of grease / water in the tube, etc.), making it difficult to use the results. We eliminated this problem of sealing the sensors from water by tilting the shaft such that the sensors are out of the water channel, but the propeller still is in the middle of the channel. The compromise is that the propeller is now tilted by 20 degrees: the effects of this tilt are analyzed in the following section.

<figure><img src="image_24.png" /><figcaption>Figure 2.9: Torque with no propeller for the second and third boats.</figcaption></figure>

The rest of this boat is very similar to the previous one, but all inline (Fig. 2.8). The DC motor (Micromotors PS150-24-25) is at the end of the shaft to control the rotation, and we measure the angular velocity with the Hall effect sensor + magnet similarly to the first boat. It is then attached to the torque sensor (Burster 8661 4050 V0000) and the final part of the shaft is held in place by two stainless steel ball bearings — one of which is submerged — that have very small friction (see Fig. 2.9).

This boat is used for the intermittent locomotion experiment, so it must be able to glide back and forth in the channel. The hull is attached to a carriage on a linear air bearing. This guide rail is a 75mm x 99mm bar and a matching carriage machined to tight tolerances. Pressurized air is then introduced by a tube directly into the moving carriage, and is vented in the gap between it and the rail. The air stream creates a film that levitates the carriage. As a result, the only friction experienced
during motion is the shear in the lubricating air film, which is much smaller than any force in the problem studied.

The final sensor in the system is a position sensor (Baumer U500.DA0.2-UAMJ.72F) mounted on the carriage. It sends ultrasonic pulses to a screen at one end of the channel and measures the time it takes for the pulses to return to the sensor. It produces an analog signal proportional to the distance to the screen and can thus track the position of the boat in the channel in a time-resolved manner. This sensor is sensitive to the speed of sound in the room, which itself scales as the square root of the temperature. So we were careful not to have rapid temperature changes.

While the friction on the linear air bearing is negligible, the force from the cable supplying air to the carriage — and the smaller electrical cables — is not. This cable is a plastic tube that we mounted as high above the rail as possible to minimize the return force it produces. To characterize the force it generates, we released the carriage alone with zero initial speed from various positions in the channel and measured the acceleration from the position signal immediately after release. We show the norm of the corresponding force (simply dividing by the mass of the carriage) in Fig. 2.10 as a function of the initial position on the rail. The force has a flat profile in the center and then increases sharply, starting at 10 cm from the center. While the amplitude of this force is small (10 mN compared to 100 mN for typical drag), since we want the boat to be stationary on average in the channel, the hydrodynamic thrust and drag will cancel each other out, leaving this

force as the only net contribution. Therefore, in the Burst and Coast experiments, special care was taken to ensure that the average position of the boat was in the location where the spring force was minimal (within ± 2 cm).

**Force sensing experiments:** In order to measure properties of the rotor (Sec. 3.2 and 4.1) we needed to perform experiments with the boat fixed but measuring the force in the direction of the flow. We attached a traction / compression force sensor (FUTEK LSB210) between the carriage and a fixed point in the laboratory. The linkage between the force sensor and the carriage is a ball joint such that, when the pressurized air is turned on, the carriage can freely levitate up or down and transmits all the longitudinal force to the sensor. This design has the added advantage that, unlike previous designs, the highly sensitive force sensor does not carry the weight of the boat below it. The linear air bearing levitates the boat and propeller, but transmits all the longitudinal force to the sensor. We noticed that the combination of the new sensor and this design resulted in a much smaller drift — and resulting uncertainty — of the force measurement.

**Control and acquisition** In this case, like the previous ones, the control and acquisition of the signals is handled by a National Instruments card (NI USB-6221). This allows the automation of the experiments: any desired signal can be programmed in the computer and sent as an analog voltage to the DC motor (with a sampling frequency of 4096 Hz). The card is able to produce the right voltage, but the associated power is very low and not able to directly drive the motor. We thus amplify the analog signal with a high power, low frequency amplifier with a gain of 3.6 (KEPCO BOP 36-6M). At the same time, the device records the multiple signals coming from the experiment, which can be divided in two categories: analog and digital. Analog signals (force, torque, intensity, and position) are directly measured and later converted into the associated physical quantity using the sensor calibration. Digital signals (angular velocity and flow rate) encode their value in the time interval between two consecutive pulses from 0 to 5V.

<figure><img src="image_25.png" /><figcaption>Figure 2.10: Force due to the cables attached to the carriage.</figcaption></figure>
To ensure synchronicity between all signals they are acquired as analog signals, and a rising edge detection algorithm later converts them into their associated physical value in software. We sample at 32 768 Hz to ensure that this algorithm functions correctly. This sample rate is way too fast for all analog signals and we therefore downsample all signals by a factor 32 once the digital to analog conversion is done. Our effective acquisition frequency is 1024 Hz. The force signal is pre-amplified by a signal conditioner (SCAIME CPJ) which is designed specifically to amplify the signal coming from analog force sensors, while minimizing noise. The torque sensor does not require the same treatment as an amplification is built into the sensor.

### Tilted propeller

As shown in Fig. 2.8 the shaft and the propeller are at an angle to the incoming flow, breaking the symmetry of the system. Indeed, on one side of the propeller the angular velocity and the flow velocity combine in such a way that the blade experiences faster flow with a larger angle of attack. On the other side the opposite effect occurs, creating a variation of lift and drag forces with a period identical to the rotation rate. The net effect of this tilt besides periodically varying the thrust is to create a yawing moment (a rotation to the left of the channel in our case). This phenomenon occurs when a propeller-driven aircraft tilts up to generate more lift when landing and is known as p-factor or asymmetric blade effect (Rostami et al., 2022).

<figure><img src="image_26.png" /><figcaption>Figure 2.11: Force normal to the rotor (along the shaft direction) as a function of the angular position during the rotation (as seen from the front of the boat) on a single blade. The force is normalized by $1/2\rho\pi R^2(R\Omega)^2$. Color indicates the tilt angle. The three polar plots correspond to increasing tip speed ratios $\lambda$ from left to right.</figcaption></figure>

This section quantifies the effect on thrust, torque and eventual side loads due to this tilt $\delta$ of the shaft which is 20 degrees in our case. The local velocity is the combination of the flow velocity and the angular velocity. In the rotating frame aligned with the shaft ($e_r, e_\theta, e_z$) (see Fig. 2.8), the total flow velocity can be projected as:

$$ U_p = \begin{pmatrix} U_\infty \sin \delta \cos \Omega t \\ r\Omega - U_\infty \sin \delta \sin \Omega t \\ U_\infty \cos \delta \end{pmatrix}_{(e_r, e_\theta, e_z)} , \qquad (2.4) $$

where $\Omega$ is the angular velocity and $t$ the time. The tilt of the shaft therefore creates a varying velocity during the rotation.
A radial velocity exists along the blades, for instance when $\Omega t = 0 \mod \pi$ (i.e., when the blade is aligned with the flow, above or below the shaft). This radial velocity is small when the tilt angle $\delta$ also is, and does not interact with the blades to create lift forces. Its effect will thus be neglected in the analysis below.

For a blade element at a radius $r$, the norm of the incoming velocity normal to the blade is:

$$ \| \mathbf{U}_p \times \mathbf{e}_r \|^2 = (U_\infty \cos \delta)^2 + (r\Omega - U_\infty \sin \delta \sin \Omega t)^2, \quad (2.5) $$

and the angle between the local incoming flow and the plane of rotation verifies:

$$ \tan \varphi = \frac{\cos \delta}{r\Omega/U_{\infty} - \sin \delta \sin \Omega t}. \quad (2.6) $$

The quantity $r\Omega/U_\infty = \lambda r/R$ is the local speed ratio. If it is large compared to 1 everywhere on the blade (fast rotating propeller) then the only effect of the tilt of the shaft is the reduction of the incoming velocity by $\cos \delta$. To analyze further the effect of the tilt, we compute the macroscopic properties (thrust, torque, and efficiency) of a flat bladed propeller with a pitch of 30 degrees. This analysis is done similarly to the one described in 1.2, adding in the position dependent velocity.

Fig. 2.11 shows the force normal to the rotor (parallel to the shaft) as a function of the position during the rotation for several tip speed ratios and tilt angles, computed for a single blade. When the blade is around 270 degrees (the propeller is rotating clockwise, meaning that it is going down and towards the incoming flow) the force is more important. This effect is increasingly important as $\delta$ is increased and $\lambda$ is decreased. In this location the angular velocity and the incoming flow have an angle between them tighter than 90 degrees. The combination of the velocities results in (i) an increased local velocity norm and (ii) a smaller flow angle with the plane of rotation (or an increased angle of attack on the blade). Both modifications tend to increase the generated force and therefore much more thrust is produced on the right side of the rotor. As expected this effect is more important for smaller $\lambda$, since the fluctuation of $U_\infty$ has more relative importance compared to the angular velocity. In extreme cases the angle of attack could even change direction on the left side of the rotor, but it is never the case for the parameters relevant to the problem here. The effect is important with a single blade as presented here but with three (like is the case in Sec. 3.2 and 4.1), the angular position dependence of the total force is averaged: when one blade is in the position of high thrust (270 deg) both the others are on the opposite side where less force is produced. As a result, the fluctuation of the force during one cycle due to the tilt is about 1% with 20 deg of tilt.

We can then compute the thrust and torque of the rotor for a given $\lambda$ and $\delta$. The torque is unchanged from the formula in 1.2 (Eq. (1.16)), however the thrust is directed up instead of forward and has to be projected horizontally (by multiplying by $\cos \delta$). In Fig. 2.12, we show the thrust, torque, and efficiency of a three bladed rotor with for several tip speed ratios as a function of the tilt angle. The net effect on thrust depends on the tip speed ratio: for small values of $\lambda$ the accelerated flow on one half of the rotation more than makes up for the velocity deficit on the other side and the subsequent projection. As a result the thrust is slightly increased (light red curves, panel (a)). At higher tip speed ratios, the angular dependence effect is smaller and thus the only remaining effect of the tilt is the projection of the thrust force in the direction of motion (i.e., for $\lambda \gg 1$, $C_T(\delta) \sim \cos \delta$, visible in the dashed black line). For the torque the reasoning is similar, but the torque is always more important than the zero tilt value. At high tip speed ratio, the discrepancy reduces.

Finally, the efficiency ($\eta = C_T/\lambda C_Q$) is always reduced. At high tip speed ratio, we get the same scaling as the thrust: the tilt reduces the efficiency by a factor $\cos \delta$.

The effect of the tilt (in our case $\delta = 20$ deg.) is twofold: first, it reduces the performance of the rotor used by about 10% the normal operating conditions. This effect is not very important for our study
<figure><img src="image_27.png" /><figcaption>Figure 2.12: Relative variation of the dimensionless (a) thrust, (b) torque and (c) efficiency with the tilt angle. Colors indicate the tip speed ratio. (dashed line) cosine of the tilt angle</figcaption></figure>

since we are not interested in the absolute values of thrust and efficiency but rather in their variations with respect to a reference case. Second, it changes the local conditions experienced by the rotor, which can have an impact on deformation throughout the cycle.

## 2.3 Materials

All the flexible blades used in the following are made out of flexible plastic sheets. This section describes how we separately measured their flexibility to model the deformations observed. The blades used are cut out of sheets of different plastics: either polyethylene terephthalate (PET or Mylar) or polypropylene (PP) of different thicknesses. The main parameter of interest is the Young's modulus $E$ of the material, which controls the amplitude of the deformations. In practice, we often need to compute the bending modulus of the studied blade: $EI$ where $I$ is the quadratic moment of area equal to $ce^3/12$, with $c$ is the width of a section of the blade in a direction perpendicular to the bending direction and $e$ the corresponding thickness.

To perform this measurement, we used a precise materials testing machine (INSTRON 5865): it is able to record force / displacement curves with forces in the $10^{-4}$ N range and displacements of the order of 0.1 mm. We measured the bending modulus of the blades used in Sec. 3.1 by a bending test: the 40 mm blade is held horizontal, and clamped at one end. The force sensor pushes down on the tip of the blade (the blades have a negligible deflection under their own weight). In the small deflection limit, the relation between the deflection $\delta_B$ and the force applied $F_B$ reads (Marigo, 2014):

$$ \delta_B = \frac{F_B L_B^3}{3EI}, \qquad (2.7) $$

with $L_B$ being the length of the blade ($40 \pm 0.5$ mm in this case). The width of the blade is $5.8 \pm 0.1$ mm.

We show the result of the test for the four different materials used in Sec. 3.1 in Fig. 2.13. The curves stay linear throughout the range of deformation, indicating that we are still in within the regime of small deformation ($\delta_B/L_B < 10\%$). We measure the slope of this linear trend to obtain the bending modulus.
<figure><img src="image_28.png" /><figcaption>Figure 2.13: Force displacement curve for the four different blades (colors). Black lines are best fits for the slope of each blade.</figcaption></figure>

To model the deformation of the slender blades, we also need to compute the twisting modulus of the material *GJ*. *G* is the shear modulus of the material, obtained from the Young's modulus and the Poisson ratio *ν*<sub>el</sub> (assumed to be 0.43 both for PET and PP), *G* = *E*/(2(1 + *ν*<sub>el</sub>)). *J* is the torsional moment of area, which in the case of a rectangular cross-section and *e* ≪ *c* reads *J* = *c*e<sup>3</sup>/3 (Audoly and Neukirch, 2021; Audoly and Pomeau, 2010). The blades are numbered from least to most flexible and their properties are summed up in table 2.1. The blades F1 and F4 are used in Sec. 3.1 and in Sec. 3.1.4 the four materials described here are used, only the chord is modified: from 5.8 mm it is increased to 18 mm. The Young's modulus stays the same but the bending and torsion moduli are multiplied by about 3.

<table><caption>Table 2.1: Relevant physical constants of the blades used in the experiments of Sec. 3.1.</caption><thead><tr><th>Blade type</th><th>Thickness <i>e</i> (μm)</th><th>Young's modulus (GPa)</th><th>Bending modulus (N.m<sup>2</sup>)</th><th>Torsion modulus (N.m<sup>2</sup>)</th></tr></thead><tbody><tr><td>Rigid (Al)</td><td>460 ± 10</td><td>∞</td><td>∞</td><td>∞</td></tr><tr><td>F1 (Mylar)</td><td>330 ± 10</td><td>3.9 ± 0.2</td><td>(69.4 ± 8.1) 10<sup>-6</sup></td><td>(96.9 ± 11.3) 10<sup>-6</sup></td></tr><tr><td>F2 (Mylar)</td><td>270 ± 10</td><td>4.4 ± 0.5</td><td>(41.7 ± 10.1) 10<sup>-6</sup></td><td>(58.3 ± 14.1) 10<sup>-6</sup></td></tr><tr><td>F3 (Mylar)</td><td>200 ± 10</td><td>4.3 ± 0.7</td><td>(16.5 ± 4.69) 10<sup>-6</sup></td><td>(23.1 ± 6.56) 10<sup>-6</sup></td></tr><tr><td>F4 (Mylar)</td><td>110 ± 10</td><td>4.0 ± 0.5</td><td>(3.67 ± 0.64) 10<sup>-6</sup></td><td>(5.25 ± 0.91) 10<sup>-6</sup></td></tr></tbody></table>

In Sec. 3.2, we used slightly different blades only allowed one direction of bending. We model this bending as a pointwise torsional spring along the axis where the blade is clamped. To measure the spring constant we repeated the same bending test but along the chordwise direction and directly measured an effective spring constant. Doing so minimizes the uncertainty due to the width of the patch glued to the rigid rod. In this new version the experiment the width *W*<sub>B</sub> is 30 ± 0.5 mm, and the length 12 ± 0.5 mm. The spring constant per unit width *C*<sub>B</sub> that we measure is linked to the force and deformation through:

$$ \frac{F_B L_B}{W_B} = C_B \arctan \frac{\delta_B}{L_B}. \qquad (2.8) $$

The resulting curves are plotted in Fig. 2.14, and the associated spring constants are displayed in table 2.2.
<figure><img src="image_29.png" /><figcaption>Figure 2.14: Moment due to the bending of the plastic sheet as a function of the bending angle arctan $\delta_B/L_B$</figcaption></figure>

<table><caption>Table 2.2: Relevant physical constants of the blades used in the experiments of Sec. 3.2.</caption><thead><tr><td>Blade type</td><td>Thickness (μm)</td><td>Spring constant C<sub>B</sub> (N)</td></tr></thead><tbody><tr><td>Rigid</td><td>535 ± 10</td><td>∞</td></tr><tr><td>P1 (Mylar)</td><td>108 ± 10</td><td>(16.0 ± 1.8) 10<sup>-2</sup></td></tr><tr><td>P2 (PP)</td><td>100 ± 10</td><td>(5.11 ± 0.75) 10<sup>-2</sup></td></tr><tr><td>P3 (PP)</td><td>60 ± 10</td><td>(0.871 ± 0.12) 10<sup>-2</sup></td></tr><tr><td>P4 (PP)</td><td>35 ± 10</td><td>(0.232 ± 0.046) 10<sup>-2</sup></td></tr></tbody></table>

Bibliography

AUDOLY, B. and NEUKIRCH, S. (2021). A one-dimensional model for elastic ribbons: A little stretching makes a big difference. *Journal of the Mechanics and Physics of Solids*, vol. 153:p. 104457 36

AUDOLY, B. and POMEAU, Y. (2010). *Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells*. Oxford University Press 36

FRISCH, U. (1995). Turbulence: The Legacy of A. N. Kolmogorov 26

LUMLEY, J. L. and MCMAHON, J. F. (1967). Reducing Water Tunnel Turbulence by Means of a Honeycomb. *Journal of Basic Engineering*, vol. 89(4):pp. 764–770 24

MARIGO, J.-J. (2014). *Mécanique des Milieux Continus I. École polytechnique* 35

ROSTAMI, M., CHUNG, J., and NEUFELD, D. (2022). Vertical tail sizing of propeller-driven aircraft considering the asymmetric blade effect. *Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering*, vol. 236(6):pp. 1184–1195 33

TROPEA, C., YARIN, A. L., and FOSS, J. F., editors (2007). *Springer Handbook of Experimental Fluid Mechanics*. Springer, Berlin, Heidelberg 25, 27

WHELAN, J. I., GRAHAM, J. M. R., and PEIRÓ, J. (2009). A free-surface and blockage correction for tidal turbines. *Journal of Fluid Mechanics*, p. 12 24
Chapter 3

Reconfigurating propellers

This chapter is dedicated to the study of shape-changing propellers through two designs. The first one has fully flexible blades, made out of plastic sheets, that can bend and twist along their span. Its purpose is to study the main drivers of deformation in a canonical case study and the effects of the deformation on the performance of the propeller.

In a second step, we analyze the performance of a propeller inspired from the work of Cognet (2017). This propeller is designed to passively adopt the shape which optimizes its efficiency. To do so, its bending deformation is not authorized and the twisting direction is reversed compared to the first propeller.

**3.1 Fully flexible propeller**

<figure><img src="image_30.png" /><figcaption>Figure 3.1: Fully flexible propeller</figcaption></figure>

We experimentally study the canonical problem of a propeller with highly flexible blades where the main driver of deformation is the flow. The goal of this work is to get a better understanding of the
fluid forcing on the blades and its effect on the propeller thrust. The blades are made out of a flexible material and have a flat, rectangular shape tilted by an angle $γ_0$ (Fig. 3.1).

Eldemerdash and Leweke (2021a) have recently studied a very similar system of a rotor made out of slender, flexible plastic blades in water. They measured the flow field around the blades as well as the deformations and shown that important bending (greater than 90 deg in some cases) can be observed in forward motion as well as large amplitude oscillations in backward motion. In their study, the thrust is estimated for some parameters using the flow field but no direct measurement was made. In the present study, we directly measure the thrust produced by such a propeller and how it is modified by the blade deformation. We are interested in the propulsive performance of such a propeller in water. Thus, there are two key differences between this work and that of Eldemerdash and Leweke (2021a): we study only forward motion, and we use a wider range of pitch angles. These changes correspond to the parameters typically found in propellers of boat or airplanes moving forward instead of helicopters or small drones hovering. We use the mathematical description of the system based on the work of Durán Venegas et al. (2019) to show how the thrust of a flexible rotor is modified by the addition of flexibility. Then, we can understand the small deformations through the modification of the classical Cauchy number and link the bending of the blades with the thrust produced by the rotor. Using a second series of experiments and a numerical solution of the system, we show how the efficiency of such a rotor is modified.

### 3.1.1 Setup

<figure><img src="image_31.png" /><figcaption>Figure 3.2: Typical deformation of the blades when the rotational velocity is increased (superposition of approximately 30 phase synchronous frames). Blades F4 with 20 deg pitch angle subjected to 0.08 m/s flow, with increasing rotational velocity ranging from 0 to 30 rad/s (left to right).</figcaption></figure>

We first conducted experiments utilizing the initial boat design (Sec. 2.2.1) to investigate propeller thrust and deformation. The experiments involved three types of blades, all having a rectangular shape with specific dimensions: an external radius of 46 mm, an internal radius of 4 mm (referred to as $R_{hub}$), and a chord length of 6 mm. However, the blades differ in terms of thickness and material composition. For the purpose of reference, we employed rigid blades made from thin aluminum sheets. The two sets of flexible blades are crafted from mylar sheets, each having a distinct thickness. The physical characteristics of the blades are provided in Table 2.1. We used the least and most flexible blades defined in that table: F1 and F4.

To secure the blades onto the rotating shaft, we utilized 3D printed components featuring tilted slits. These slits allowed us to set the pitch angle relative to the plane of rotation, as illustrated in Fig. 3.1. Our experiments were carried out with pitch values of 5, 10, 20, and 30 degrees. The blades were attached to a shaft with a diameter of 4 mm and a length of 75 mm positioned in front of the hull.
We use a high speed camera (Phantom Miro M120) to directly measure the deflection of the blades under load. It is synchronized with the rotation of the blades using a Hall effect sensor. A frame is shot each time the blades are perpendicular to the line of sight of the camera (see Fig. 3.2). We compute the position of the centerline of the blades and correct for the out of plane bending of the blades, and therefore we can track the flexion angle. The torsion angle can also be estimated by measuring the apparent width of a section of the blade. However, this method has low precision and the torsion angles are small so that conclusive results cannot always be obtained.

In Fig. 3.3, we show a sketch of the required correction to obtain the bending angle. In general, the blade bends in 3 dimensions. However Eldemerdash and Leweke (2021a) showed that the twist of the blade is small (of the order of a few degrees compared to the pitch of the blades which is of tens of degrees). As a consequence, we can consider that the blade centerline is in a plane. This means that the vector $d_1(s)$ defined locally in Fig. 3.4 is actually of constant direction. The plane containing the centerline is obtained by rotating the vertical plane of Fig. 3.3 around the vector $e_z$ by the pitch angle of the blade: $\gamma_0$. The camera we used to measure the bending angle is placed 2 m away from the rotor and can thus be considered at infinity, capturing the shape of the blade as projected onto the vertical plane. The distances measured along the $x$ axis are underestimated if not corrected. In order to get the actual shape of the blade in the plane of bending we correct the distances by a factor $1/\cos \gamma_0$.

For each set of parameters we capture at least 30 images, extract data separately and use the mean of the results as the bending angle. Similarly, we average values of rotational velocity and force measurements over 20 s, after a 10 s settling period when the electric motor is switched on. Technical details about the image processing are given in App. D.1.

<figure><img src="image_32.png" /><figcaption>Figure 3.3: Schematic of the projection occurring when photographing the deformed blades.</figcaption></figure>

### 3.1.2 Mathematical description

We obtain the equations describing the deformation of the blades similarly as in Durán Venegas et al. (2019) but with a few key differences. First the blades are submerged in water, modifying the balance of the forces acting on the blades. We also use important pitch angles (up to 30 degrees) and so small angles limits used in this article are not valid here. Finally, our blade shape is rectangular and flat, simplifying the constitutive relation.

Using Kirchhoff relations for a steady state in the local frame ($d_1, d_2, d_3$) described in Fig. 3.4, we have:

$$ \frac{\partial T_{el}}{\partial s} + f_{el} = 0, \\ \frac{\partial M_{el}}{\partial s} + d_3 \times T_{el} + m_{el} = 0, \qquad (3.1) $$

where $f_{el}$ and $m_{el}$ are the external forces and moments per unit of length respectively. $T_{el}$ and $M_{el}$ are the internal forces and moments and $d_3$ the vector tangent to the blade centerline (defined along
the centers of mass of each section). We define the Darboux vector:

$$ \Omega_{el} = \kappa_1(s) d_1(s) + \kappa_2(s) d_2(s) + \tau(s) d_3(s), \qquad (3.2) $$

with $\kappa_1$, $\kappa_2$ and $\tau$ the local curvatures in the local frame. Then the vectors $d_i$ verify:

$$ \frac{\partial d_i}{\partial s} = \Omega_{el} \times d_i, \qquad (3.3) $$

for $i = 1, 2, 3$. The geometry of the blades is such that they can be modeled as slender ribbons (Audoly and Pomeau, 2010), giving $\kappa_2 = 0$. The curvatures can be expressed via the angles $\theta$ and $\gamma$ (see Fig. 3.4):

$$ \frac{\partial \theta}{\partial s} = \kappa_1, \quad \frac{\partial \gamma}{\partial s} = \tau. \qquad (3.4) $$

The constitutive law for inextensible ribbons is:

$$ M_{el} = EI\kappa_1 d_1 + \frac{EJ}{2(1 + \nu_{el})} \tau d_3, \qquad (3.5) $$

where $\nu_{el}$ is the Poisson ratio, $E$ is the Young modulus, $I$ is the second moment of area in the $d_1$ direction and $J$ the torsion modulus of the blade. The complete elastic energy for inextensible ribbons contains non quadratic and coupled terms between the bending and torsion curvatures (see Audoly and Neukirch (2021)). However here, given that our blades are slender and thin and that twisting curvature remains small the simpler relation (3.5) remains valid.

<figure><img src="image_33.png" /><figcaption>Figure 3.4: (left) Axes and angles definition on the blade. (right) Angles and definitions on a cut (orange line) of the blade</figcaption></figure>

## Bending equation

The external forces acting on a blade are both the hydrodynamic forces and the centrifugal force. We separate the fluid force in lift and drag coefficients acting on each section of the blade normal to its span (see orange cut in Fig. 3.4). In this frame, the force normal to the surface of the blade (per unit span) reads:

$$ f_{hydro} = \frac{1}{2} \rho U_p^2 c (C_L(\alpha) \cos \alpha - C_D(\alpha) \sin \alpha), \qquad (3.6) $$

where $\rho$ is the fluid density, $U_p$ the local apparent velocity (see Fig 3.4), $c$ the chord of the blade, $C_L$ and $C_D$ are respectively the profile lift and drag coefficients and $\alpha$ is the local angle of attack. The centrifugal force tends to align the blade with the plane of rotation. It has in general a component in the $d_2$ and $d_3$ directions, however external forces in the $d_3$ direction do not contribute to the bending
and twisting motions as can be seen in the second part of (3.1). The centrifugal force per unit span projected in the $d_2$ direction reads:

$$f_{\text{centrifuge}} = A_b \rho_b \tilde{r} \Omega^2 e_r \cdot d_2, \qquad (3.7)$$

where $A_b$ is the cross-sectional area of the blade, $\rho_b$ its density and $\Omega$ the rotation speed. The distance to the rotation axis is denoted as $\tilde{r}$. The vector $e_r$ is the unit vector aligned with the direction of the centrifugal force: it is pointed from the axis of rotation outwards. The distance to the axis of rotation $\tilde{r}$ and the projection $e_r \cdot d_2$ can be computed analytically in the small twist limit (it is done in App. A) but their exact values are not necessary for the discussion here.

We combine the Kirchhoff equations (3.1) and the constitutive law (3.5). By projecting Eq. (3.1) along $d_1$ we get:

$$d_1 \cdot \frac{\partial M_{\text{el}}}{\partial s} + T_{\text{el},2} + m_{\text{el},1} = 0, \qquad (3.8)$$

where we denote the $i^{th}$ component in the local frame by the subscript i. Taking the derivative of the expression above, we need to compute the derivative of $d_1 \cdot \partial M_{\text{el}}/\partial s$ and using Eq. 3.2 and $\kappa_2 = 0$, we get:

$$\frac{\partial}{\partial s} \left( d_1 \cdot \frac{\partial M_{\text{el}}}{\partial s} \right) = \frac{\partial^2 M_{\text{el},1}}{\partial s^2} - \frac{\partial \tau M_{\text{el},2}}{\partial s}. \qquad (3.9)$$

Once again, since there is no bending in the $d_2$ direction, second term is zero. Using the constitutive law (3.5), we finally obtain the equation for $\theta$:

$$EI \frac{\partial^3 \theta}{\partial s^3} = - \frac{\partial T_{\text{el},2}}{\partial s} = f_{\text{el},2}. \qquad (3.10)$$

Using the expressions of the external forces projected in the direction normal to blade obtained in (3.6) and (3.7) we obtain the equation describing the bending of the blade during the rotation:

$$EI \frac{\partial^3 \theta}{\partial s^3} = \frac{1}{2} \rho U_p^2 c (C_L(\alpha) \cos \alpha - C_D(\alpha) \sin \alpha) - A_b \rho_b \tilde{r} \Omega^2 e_r \cdot d_2. \qquad (3.11)$$

## Twisting equation

The hydrodynamic center of a typical airfoil is located around 1/4 of the chord and since the blades have a homogeneous and rectangular profile, their center of mass is located at mid-chord. Thus, the blades tend to twist and increase their pitch angle when they are generating lift due to the hydrodynamic moment. The centrifugal force tends to resist this twist and brings the blade towards the plane of rotation. To obtain an equation on the torsion curvature $\gamma$, we project the Kirchhoff equation along $d_3$:

$$d_3 \cdot \frac{\partial M_{\text{el}}}{\partial s} + m_{\text{el},3} = 0. \qquad (3.12)$$

Similarly to the bending equation, we develop the first term to get:

$$d_3 \cdot \frac{\partial M_{\text{el}}}{\partial s} = \frac{\partial M_{\text{el},3}}{\partial s} - \kappa_1 M_{\text{el},2}. \qquad (3.13)$$

And since $M_{\text{el},2} = 0$ the equation describing the twist reads:

$$\frac{EJ}{2(1 + \nu)} \frac{\partial^2 \gamma}{\partial s^2} = -m_{\text{el},3} \qquad (3.14)$$
The moment along the span due to the hydrodynamic forcing is simply its projection along $d_2$ times the distance from the point of application of this forcing to the center of the chord. We will denote this distance $\bar{\delta}_{cm}^{ac}$ (equal to $c/4$ in general). The centrifugal restoring moment is obtained in App. A and reads:

$$dM_{\text{centrifuge}} = -\frac{\rho_b e \Omega^2 c^3}{24} \sin 2\gamma \cos \theta. \qquad (3.15)$$

This moment is along the direction $d_3$.

Finally, we obtain for the equation describing the twist:

$$\frac{EJ}{2(1 + \nu)} \frac{\partial^2 \gamma}{\partial s^2} = -\frac{1}{2} \rho_b c^2 U_p^2 \left( (C_L(\alpha) \cos \alpha + C_D(\alpha) \sin \alpha) \bar{\delta}_{cm}^{ac} \right) + \frac{\rho_b e c^3 \Omega^2}{24} \sin 2\gamma \cos \theta. \qquad (3.16)$$

### Dimensionless system

In our problem the typical length scale is $R$ and the typical velocity $R\Omega$. We use these quantities to make the equations of the deformation dimensionless and two Cauchy (or elastohydrodynamical) numbers appear: one for the bending of the blades and one for the twist. They compare the intensity of the hydrodynamic forces to the elastic moduli and read:

$$C_Y^B = \frac{\rho S_b (R\Omega)^2 R^2}{2EI}, \qquad C_Y^T = \frac{\rho S_b (R\Omega)^2 Rc(1 + \nu)}{EI}, \qquad (3.17)$$

where $S_b = Rc$ is the planar surface of a blade. The system of elastic equations can thus be written:

$$\frac{1}{C_Y^B} \frac{\partial^3 \theta}{\partial s^3} = u_p^2 (C_L(\alpha) \cos \alpha + C_D(\alpha) \sin \alpha) - 2d \frac{A_b}{S_b} \frac{\bar{r}}{R} e_r \cdot d_2, \qquad (3.18)$$

$$\frac{1}{C_Y^T} \frac{\partial^2 \gamma}{\partial s^2} = -u_p^2 \left[ (C_L(\alpha) \cos \alpha + C_D(\alpha) \sin \alpha) \delta_{cm}^{ac} \right] + \frac{1}{24} \frac{A_b}{R^2} \sin 2\gamma \cos \theta, \qquad (3.19)$$

where $u_p = U_p/R\Omega$, $\delta_{cm}^{ac} = \bar{\delta}_{cm}^{ac}/c$, and $d = \rho_b/\rho$ is the density of the blade. Note that we refer to the dimensionless curvilinear abscissa as $s$ for convenience. It is obtained by dividing the dimensional curvilinear abscissa by the span $R$. The dimensionless system lets us compare the order of magnitude of the different terms:

$$2d \frac{A_b}{S_b} \approx 10^{-2} \quad \text{and} \quad \frac{A_b}{24R^2} \approx 10^{-5} \quad \text{while} \quad u_p^2 \approx 1 + \frac{1}{\lambda^2} \approx 1. \qquad (3.20)$$

The centrifugal force and moment are orders of magnitude smaller than the hydrodynamic contribution except for very large deformations (i.e., large Cauchy numbers). In the following we will neglect the centrifugal force and focus on the coupling between the elasticity and the fluid loading.

### 3.1.3 Effect of bending on thrust production

We perform the experiments at a fixed flow speed $U_\infty$ (0.05 to 0.15 m/s) and pitch angle $\gamma_0$, and increasing the angular velocity from 0 to 35 rad/s. Fig. 3.5 shows the results of the experiments for $U_\infty = 0.15$ m/s and $\gamma_0 = 20$ deg. We plot two quantities of interest: the thrust produced by the rotor and the tip bending angle $\theta_M = \theta(s = R)$. We can see that both flexible blades exhibit significant bending deformations at large tip speed ratios. The F1 blades have similar thrust than their rigid counterparts while the F4 blades bend almost completely along this axis of rotation ($\theta_M > \pi/2$, see
<figure><img src="image_34.png" /></figure>

Figure 3.5: Thrust force generated by the rotor (left) and deflection of the tip of the blade (right) as a function of the rotation rate. Blue points correspond to the rigid blades, orange to the lightly flexible ones (F1 in Tab. 2.1) and red to the very flexible ones (F4 in Tab. 2.1). The error bars indicate the standard deviation of the measurement. The standard deviation of the tip bending angle is smaller than the marker size. All curves have the following parameters: $U_\infty = 0.15$ m/s, $\gamma_0 = 20$ deg. The F4 rotor exhibit two sets of points for the bending angle $\theta_M$ at a single angular velocity $\Omega$ because the two different blades composing the rotor bend differently.

Fig. 3.2) causing the production of negligible thrust. Note that at zero angular velocity, the upstream flow on its own is not strong enough to bend the F1 blades but does slightly bend the F4 blades backwards. These most flexible blades also exhibit sometimes different behaviors for the two blades with the same external conditions (see Fig 3.5 (right), red markers at $10$ rad/s < $\Omega$ < $15$ rad/s). We believe that this discrepancy is most likely due to small experimental deviations on the exact clamp angle of the blades. A small pitch difference becomes significant when the loading on the blades varies rapidly with the angle of attack (i.e., when the angle of attack is close to zero). The different behavior between the two blades is therefore only visible close to the angular velocity where sign of the bending changes (Eldemerdash and Leweke (2021a) have also noticed a similar behavior in their experiments).

The thrust coefficient (defined in Eq. (1.10)) of the three rotors are shown in Fig. 3.6 for all pitch angles tested and a fluid velocity of 0.05 m/s. The most flexible blades (red markers) start at very low tip speed ratios with a performance similar to the rigid ones but very quickly significantly deviate from the reference behavior. At high tip speed ratios, whichever the pitch angle, F4 blades do not produce any thrust. For the intermediate blades F1 (orange markers) the thrust coefficient only decreases at very high tip speed ratios and for high pitch angles. Otherwise, we observe a similar performance for F1 blades and the rigid ones (even though the associated deformation may reach $\theta_M \approx 30$ deg) with a slight increase in thrust production for intermediate tip speed ratios.

We now consider the effect of the fluid loading on the rotor shape to link it with the observations on the thrust production above. This deformation is characterized by two functions along the blade: the bending angle $\theta(s)$ and the twist angle $\gamma(s)$. The twist is small in experiments (described in more extensive details in Eldemerdash and Leweke (2021a)) and so we will focus on the bending angle. Fig. 3.7 shows the tip bending angle for flexible blades as a function of the bending Cauchy number. As expected, for $C_Y^B$ lower than one (corresponding to slow angular velocities and stiffer blades), the deformations are small. For larger angular velocities (larger $C_Y^B$), flexible blades exhibit widely
<figure><img src="image_35.png" /><figcaption>Figure 3.6: Thrust coefficient as a function of the tip speed ratio for all rotors tested at $U_\infty$ = 0.05 m/s. Markers indicate the pitch angle, color indicates the blade type (blue: rigid blades, orange: F1 flexible blades, red: F4 flexible blades).</figcaption></figure>

<figure><img src="image_36.png" /><figcaption>Figure 3.7: Tip bending angle $\theta_M$ as a function of the bending Cauchy number $C_Y^B$ (defined in (3.17)) for all flexible blades tested at $U_\infty$ = 0.05 m/s in log scale. Markers indicate the pitch angle, color indicates the blade type (orange: F1 flexible blades, red: F4 flexible blades).</figcaption></figure>
different behavior depending on the pitch angle: the bending deformation is a fast-growing function of the pitch angle. The blades with pitch $\gamma_0 = 5$ deg bend backwards in almost every flow condition (also for larger incoming velocity $U_\infty$, not shown in Fig. 3.7): it corresponds to a negative thrust produced by the rotor and a negative local lift on the blades causing them to bend downstream. The blades with a pitch angle of 10 degrees, however, bend backwards for small angular velocities but quickly jump to positive bending when the threshold between positive and negative thrust is reached (red diamonds, around $C_Y^B = 30$). It corresponds to the same threshold as the one described for the thrust: $\lambda > 1/\tan \gamma_0$, obtained in Sec 1.2. Similar behaviors can be observed with the F1 blades but with a much smaller amplitude. Changing the pitch angle by a few degrees can modify the resulting deformation from slightly backwards to 15 or 20 degrees forwards. In the range of parameters we explore, backwards bending is always much smaller in amplitude than forwards bending. This is because backwards bending can only occur for small tip speed ratios, which corresponds to relatively small Cauchy numbers in our case.

<figure><img src="image_37.png" /><figcaption>Figure 3.8: Tip bending angle $\theta_M$ as a function of a modified Cauchy number scaled by the tip velocity and angle of attack for the F1 blade.</figcaption></figure>

In the following, we aim at understanding the different bending angle behavior depending on the pitch angle using a scaling law and geometric arguments. Because it is constructed using small deformation assumptions this scaling law will *a priori* be applicable only to small or moderate Cauchy numbers but gives insight into which physical phenomena drives the deformations. Deriving a scaling law for $\theta_M$ from equation (3.18) we obtain:

$$ \theta_M \sim C_Y^B \left(1 + \frac{1}{\lambda^2}\right) \left(C_L(\alpha_M) \cos(\alpha_M) + C_D(\alpha_M) \sin(\alpha_M)\right), \quad (3.21) $$

where $\alpha_M$ is the angle of attack at the tip of the blade, where most of the hydrodynamic force is generated. We approximate the local apparent velocity at the tip by the combination of both the incoming flow velocity and angular motion (thus neglecting any induced velocity from the rotation). As a first approximation, using the zero twist limit, the angle of attack at the tip becomes

$$ \alpha_M \approx \gamma_0 - \arctan \frac{1}{\lambda}. \quad (3.22) $$

In typical operating conditions, the tip angle of attack is small so that (i) the contribution from the lift coefficient is greater than the one from the drag coefficient, and thus we can neglect the second term in (3.21). (ii) we can use the linear aerodynamics limit where the lift coefficient is proportional
to the angle of attack, and we get:

$$ \theta_M \sim C_Y^B \left(1 + \frac{1}{\lambda^2}\right) \left(\gamma_0 - \arctan \frac{1}{\lambda}\right) \qquad (3.23) $$

In this scaling law, the Cauchy number is rescaled by the lift coefficient of the equivalent rigid rotor. This strategy is similar to the strategy employed in Gosselin and de Langre (2011) where the authors compare the reconfiguration of very different systems by rescaling the Cauchy number by the rigid drag coefficient of the system studied.

This scaling law is plotted in Fig. 3.8 for the F1 blades which experience moderate deformations. The rescaling gives a reasonable collapse onto a single line of slope approximately 0.15 rad. This indicates that the problem of knowing a priori the deformed shape of the blades can be simplified to a geometrical problem that consists in comparing two angles: the pitch angle $γ_0$ and the apparent flow angle at the tip $arctan 1/λ$. In particular, the positive or negative bending of the blades is completely set by the sign of the modified Cauchy number. This result can be linked to an observation made by Eldemerdash and Leweke (2021b): when the bending angle of the blade changes sign it is associated with a change of sign of the vorticity created at the blade tip. This sign is itself directly correlated to the local angle of attack at the tip. Our approximation of the angle of attack (3.22) is therefore valid near zero.

Still, the dispersion of the points is fairly important, in particular at large bending angles. It is due to the multiple hypothesis made to obtain this law. The most important ones being the assumptions that there is no twist between the root and the tip of the blade (while the twist is small, since the lift coefficient is proportional to the angle of attack any deviation has a strong influence on the bending of the blades) and using only the tip values instead of the integral over the blades. The twist is visible in the experiments but difficult to measure accurately. We can still see from Eq. (3.19) that the forcing term for the twist is proportional to the forcing term for the bending angle. This identical scaling means that large deformations in the bending direction corresponds to important twist as well. It also explains why the linear dependency of the bending angle with respect to the modified Cauchy number shown in Fig. 3.8 for the F1 blades does not hold anymore for the large deformations experienced by the F4 blades.

In order to incorporate the effects of a small but non zero twist into a predictive scaling law, Eq. (3.22) has to be modified to include twist effects (e.g., a contribution proportional to the twisting Cauchy number). One other possible improvement to better estimate the effective angle of attack on the blades would be to couple the bending and twisting equations to a rotor model that computes the flow around the propeller (e.g., the classical Blade Element Momentum Theory (Glauert, 1983) or a more complete wake model like in Durán Venegas et al. (2019)).

In the following we derive a scaling law that allows to collapse the tip bending angle onto a master curve without explicitly taking the twist into account. In order to do so, we rewrite Eq. (3.18) using the theoretical thrust and torque coefficients:

$$ C_T = \frac{N_b}{1/2\rho\pi R^2 (R\Omega)^2} \int_{R_{hub}}^{R} \frac{1}{2} \rho c U_p^2 (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) \cos \theta dr, \qquad (3.24) $$

$$ C_Q = \frac{N_b}{1/2\rho\pi R^2 R(R\Omega)^2} \int_{R_{hub}}^{R} \frac{1}{2} \rho c U_p^2 (C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi) \tilde{r} dr, $$

where $N_b$ is the number of blades. Using the values at the tip of the blade (subscript M), the scaling
of these two coefficients is:

$$ 
\begin{align} 
C_T &\sim \frac{N_b c R}{\pi R^2} U_p^2 (C_L(\alpha_M) \cos \varphi_M - C_D(\alpha_M) \sin \varphi_M) \cos \theta_M, \tag{3.25} \\ 
C_Q &\sim \frac{N_b c R}{\pi R^2} U_p^2 (C_L(\alpha_M) \sin \varphi_M + C_D(\alpha_M) \cos \varphi_M) \cos \theta_M. \nonumber 
\end{align} 
$$

By replacing $\alpha$ by $\gamma - \varphi$ in the sines and cosines of the bending equation (Eq. (3.18)) and neglecting the centrifugal force we get:

$$ \frac{1}{C_Y^B} \frac{\partial^3 \theta}{\partial s^3} = U_p^2 \left[ \cos \gamma (C_L(\alpha) \cos \varphi - C_D(\alpha) \sin \varphi) + \sin \gamma (C_L(\alpha) \sin \varphi + C_D(\alpha) \cos \varphi) \right]. \quad (3.26) $$

Rewriting this equation as a scaling law, using the values at the tip and considering the fact that the twist is small compared to the pitch angle, we can substitute the lift and drag coefficients by the thrust and torque:

$$ \theta_M \sim \frac{C_Y^B}{\cos \theta_M} \frac{\pi R^2}{N_b c R} (C_T \cos \gamma_0 + C_Q \sin \gamma_0), \quad (3.27) $$

We are interested in describing the small to moderate deformations of the rotor which means that we can make two simplifications to this relationship. First the cosine on the right hand side can be linearized to 1 in the small angles limit. Next, in normal propeller operating conditions, the thrust coefficient produced by the rotor is larger than the torque coefficient. The previous relationship can thus be simplified to:

$$ \theta_M \sim C_Y^B C_T \cos \gamma_0 \frac{\pi R^2}{N_b c R}, \quad (3.28) $$

where we measure all the terms experimentally. This scaling law gives, for small deformations, a linear relationship between $\theta_M$ and integrated, measurable parameters of the rotor.

<figure><img src="image_38.png" /><figcaption>Figure 3.9: Flexion angle as a function of the modified Cauchy number scaled by the thrust coefficient. Colors indicate the flow velocity, markers show the blade pitch angle and markers with a dark edge color are from F4 blades while the others are from the F1 blades. The trend line in black has a slope of 0.18 rad.</figcaption></figure>

This new scaling relationship is plotted in Fig. 3.9. For the F4 blades, in the cases where the two blades exhibit different behaviors we use the mean bending angle between the two blades since the thrust produced by the rotor is the result of the integration of the lift on both blades. All the data for small
bending angles (positive or negative) collapse on the same line. This region of linear relationship strikingly extends from -30 degrees to 40 degrees of bending, way further than the predictive scaling law (3.23). In this previous relationship we neglected any feedback from the deformation onto the loading itself as well as any twist along the blades. The collapse of all data on this curve but not in Fig. 3.8 shows that these assumptions are not true for moderate bending angles. However, the fact that our new scaling (3.28) works demonstrates that these non linearities are all taken into account in the thrust coefficient (or are small enough to not be visible).

We can then note that beyond the limits of the linearity of the scaling law, all the data still collapse on a single curve. This curve is not linear and features more dispersion than the linear part. Still it means that for all external conditions tested, the scaling law makes a link between the thrust produced by the rotor and the bending angle. For a given propeller geometry, for all (positive) pitch angle values, measuring one or the other is equivalent. For instance measuring only the bending deformation gives without ambiguity the thrust produced by the rotor if its bending Cauchy number and pitch angle are known.

### 3.1.4 Measuring the efficiency

<figure><img src="image_39.png" /><figcaption>Figure 3.10: Fully flexible propeller, with wider blades</figcaption></figure>

Cognet et al. (2017) have shown that adding flexibility to rotating systems can improve the performance of a wind turbine. In this case the figure of merit is the power coefficient: the power extracted compared to the total kinetic power density of the incoming wind. In the case of propellers, the performance is measured by the efficiency: $\eta = F_{\text{thrust}}U_{\infty}/Q\Omega$, the ratio of useful power to total expanded power. This section describes the modified setup used to measure the efficiency of the fully flexible propeller and the challenges encountered.

In Sec. 1.2, we showed that an efficient propeller needs to have a local angle of attack $\alpha$ close to the one maximizing the lift to drag ratio on every blade section ($\alpha_{\text{opt}}$ is about 15 to 20 deg.). With our flat and rigid propellers, it is not possible as the local flow angle $\varphi$ changes throughout the span. However, the local flow velocity and angle of attack are maximal near the tip, resulting in most of the force being generated there. A rigid propeller is efficient if the angle of attack near the tip is close to the optimal one.

When the tip speed ratio changes it is therefore necessary to modify the pitch angle $\gamma_0$ in order to compensate the change of the local flow angle $\varphi$. At the tip, this effect reads $\gamma_0 - \arctan(1/\lambda) \approx \alpha_{\text{opt}}$.
For each tip speed ratio there is therefore a pitch angle that maximizes the efficiency, defining a curve $\gamma_0^{\text{opt}}(\lambda)$.

Wind turbine and aircraft propellers use motors to actively change the pitch angle of the blades and maximize efficiency. In the following, we investigate how flexibility changes the twist angle distribution, allowing for a passive modification of the angle of attack.

### Setup

We repeated the experiments of the previous section with the aim of measuring the propulsive efficiency. There are a few key differences with the experiments of the previous section. First we added a torque sensor, which is required to record the mechanical power produced by the motor. To accommodate this new, larger sensor, we used a revised version of the hull (described in Sec. 2.2.2) which itself is larger. Finally, we used a long tube filled with grease to allow the shaft to come out of the hull while sealing the torque sensor from water.

Several issues arose when first attempting to perform the experiments directly with the same protocol as with the previous boat. First, the same force sensor became more unpredictable than previously. The larger volume of the new version of the hull creates an upwards force due to the large volume of air it encloses. This vertical load is taken up by the bending force sensor. Given its direction, it should not affect the horizontal force measurement, but we found that the zero force measurement often showed a slow drift which we interpret as being due to the two roles of the sensor: holding the boat in position (in both horizontal directions and in the vertical direction) as well as measuring the force in the direction of the flow. Measuring the reference force before every set of experiments (i.e., every 30 mins) did not seem to be sufficient to compensate for this effect.

Second, the shaft being surrounded by thick grease creates an important resistive torque (about 2 mN.m) which is about the same order of magnitude as the hydrodynamic torque from the propeller (Fig. 3.11). We measured this parasitic torque before every set of experiment to subtract it from the total torque measurement. However, we also noticed it had a tendency to vary as well, likely caused by the temperature increase of the grease or leakage of grease into the water.

Both the thrust and the torque are negative at very low tip speed ratios and positive in the opposite limit. Measurement precision is therefore of paramount importance in the region where the values are close to zero. As a result, we modified the setup in two ways: first, we modified the blades by multiplying their chord by three. This modification, according to Eq. (3.24) should at first order multiply the hydrodynamic thrust and torque by the same amount — the thrust and torque coefficients reflect this change as they are normalized by the area swept by the rotor instead of the blade surface area. Increasing the magnitude of the forces makes the fluctuations of the reference value smaller in comparison.

We also found that we obtained better repeatability by sweeping the angular velocity instead of using fixed values. We ramp up the angular velocity of the rotor very slowly to keep a steady system (50 s for a full cycle). We repeat angular accelerations and decelerations about 10 times, measuring the force reference every time the rotor has stopped. Then, we can suppress the force drift and match the data points with the same angular velocity. It should be noted that we cannot measure the torque drift with this technique as the torque measured when the rotor has stopped depends on residual solid friction on the shaft and thus varies every time. With these two modifications, we obtain typical torque measurements like the one in Fig. 3.11, where we plot the total torque (with the friction) and the reference torque for several repetitions of the same experiment (rigid blades, $\gamma_0 = 30$ deg, $U_\infty = 8.5$ cm/s). The repeatability of the torque measurement is good but the exact value at which the hydrodynamic torque is zero is difficult to estimate, due to fluctuations of the reference torque
(full lines). For this reason, we repeated every set of experiment at least thrice, at the expense of testing several mean speeds. At constant tip speed ratio $λ$, the effect of changing the mean speed is to modify the Reynolds and Cauchy numbers. The effect of the former is expected to be small (Sec. 1.2) and we used five different blade materials to explore a wide range of Cauchy numbers.

<figure><img src="image_40.png" /><figcaption>Figure 3.11: Examples of total torque measurements (dots) and the associated reference (full line) as a function of the angular velocity. Colors represent different repetitions. Rigid blades, $γ_0 = 30$ deg, $U_∞ = 8.5$ cm/s</figcaption></figure>

The blades are slightly modified from the previous section (Fig. 3.10). They are three times as wide and thus cannot be attached in the same manner. The blades are glued to a glass fiber T shaped junction (over about 1 mm), held in place by a 3D printed hub with tilted slits. We used a rigid glass fiber propeller for reference and the four materials F1 to F4 described in Sec 2.3 — the values for the bending and torsion moduli should be multiplied by 3 due to the shape change. We also use increased pitch angles (20, 30, and 45 deg) since those are closer to the maximum efficiency for rigid propellers. Because we saw that the bending angle is linked to the thrust produced by the rotor in the previous section, we did not record the deformation in this configuration.

Data

In Fig. 3.12, we show the thrust coefficient $C_T$ as a function of the tip speed ratio, for all the parameters tested (all blades and pitch angles). Since we use only one mean speed, each blade is characterized by a unique number obtained by taking the ratio between the Cauchy number $C_Y^B$ and the tip speed ratio $λ$ squared:

$$C_Y^B/λ^2 = \frac{ρS_bU_∞^2R^2}{2EI} \qquad (3.29)$$

This value ranges from about 0.01 for the least flexible blade to 1 for the most flexible and measures the magnitude of the blades bending due to the incoming flow alone. A similar number could be obtained with the twisting Cauchy number $C_Y^T$. From yellow to red, the blades are increasingly flexible and blue indicates the rigid propeller.

The thrust coefficient of the rigid propeller follows the same general trend as the one found in the previous section and its magnitude is a bit less than three times as big. Like in the previous case, the least flexible blades (yellow) follow almost exactly the behavior of rigid propeller while the most flexible ones quickly decrease to zero as the tip speed ratio $λ$ is increased. For smaller pitch angles, almost all blades exhibit a range of tip speed ratio where they produce more thrust than the rigid ones. The blades twisting results in an increased effective pitch angle $γ(s)$ near the tip of the blade. If the pitch angle $γ_0$ of the blade is too small compared to the one corresponding to maximal thrust, this twisting is beneficial. The thrust increase can only occur if the bending is limited as the bending deformation only reduces the thrust (Eq. (3.25)). At high pitch angles, increasing further the pitch begins to reduce the thrust. Thus, there does not seem to be any significant increase in thrust from using flexible blades at 45 deg.

The measurement of the torque coefficient has proven more challenging than that of the thrust coefficient (see Fig. 3.13). At low tip speed ratios, a significant relative uncertainty arises when comparing the reference torque to the hydrodynamic torque. Furthermore, the subsequent division $(RΩ)^2$, especially when the angular velocity is small, results in a notable variation or discrepancy
<figure><img src="image_41.png" /><figcaption>Figure 3.12: Thrust coefficient $C_T$ as a function of the tip speed ratio $\lambda$, for three different pitch angles $\gamma_0$. Color indicates the flexibility of blade (see Eq. 3.29), with blue corresponding to fully rigid propellers. Dots correspond to individual realizations, the lines are the mean values.</figcaption></figure>

between repeated measurements. Values either approach $-\infty$ as expected or exhibiting unexpected fluctuations that may lead to disproportionately large values. As a result, values below a tip speed ratio of about 5 to 8 depending on the pitch angle are difficult to interpret.

<figure><img src="image_42.png" /><figcaption>Figure 3.13: Torque coefficient $C_Q$ as a function of the tip speed ratio $\lambda$, for three different pitch angles $\gamma_0$. Color and symbols are identical to Fig. 3.12.</figcaption></figure>

At high tip speed ratio however, we clearly see a reduction of the torque coefficient for the most flexible blades. This reduction is due to the important bending of the blades: at 45 deg, the F4 blades (red) reach a 90 deg bending for high tip speed ratios. The reduction of the moment arm between the furthest blade sections and the axis of rotation means that the torque increases slower than $\Omega^2$ and thus the torque coefficient $C_Q$ is reduced (the F4 blades even reach a torque plateau for high tip speed ratios). The thrust increase visible for the F4 blades at $\gamma_0 = 30$ deg and $\lambda \approx 5$, is associated with an increase of the torque coefficient. To know whether this simultaneous increase of the thrust and
torque is beneficial, we need to compute the efficiency. However, the large uncertainty on the torque coupled with its inclusion in the denominator of the efficiency greatly hinders the interpretability and readability of the experimental data, making it challenging to extract meaningful information.

To still get an idea of the evolution of the efficiency when using flexible blades, we plot in Fig. 3.14 a modified version of the efficiency where we do not remove the parasitic torque from the measurement:

$$ \eta = \frac{F_{\text{thrust}}U_{\infty}}{Q\Omega}, \qquad (3.30) $$

where $Q = Q_{\text{hydro}} + Q_{\text{friction}}$ is the total torque measured, without removing the friction from the grease. Using the total torque regularizes the relative uncertainty issues mentioned above as the torque is now always positive and never approaches zero. However, it raises several issues: (i) the magnitude of the efficiency is reduced, (ii) the shape of all curves is modified because the friction torque $Q_{\text{friction}}$ is an affine function of the angular velocity (Fig. 3.11) and (iii) any fluctuation of the friction torque might perturb the final result. The curves of Fig. 3.14 are thus presented to get a general idea of the behavior of the flexible blades, but most likely cannot be used quantitatively.

The maximal efficiency of the rigid propeller seems to be reached with 30 deg of pitch angle at about $\lambda = 10$. For the very flexible propellers, both the thrust and torque are reduced at high tip speed ratios and result here in the efficiency plummeting. This efficiency reduction shows that the bending deformation is always detrimental for efficiency as could be expected.

For the rigid propeller, the efficiency is larger with $\gamma_0 = 30$ deg than 20 deg, even at $\lambda = 15$ (the blue curve of the middle panel is always above the one in the left panel). This gap shows that setting a pitch angle of 20 deg is too small for all conditions tested here. Blade twist is therefore required to increase the angle of attack to its optimal value $\alpha_{\text{opt}}$. We see indeed that the flexible blades seem to have a better efficiency than the rigid ones for $\gamma_0 = 20$ deg. The range where it is the case is at moderate $\lambda$ for very flexible blades and at high $\lambda$ for the less flexible F1 and F2 blades.

For larger pitch angles, the twist further increases the pitch, beyond the most efficient angle and results in a decrease of the efficiency. The evolution of the twist angle $\gamma$ with the tip speed ratio and its effect on the efficiency is the subject of the following section, analyzing numerically the effect of elasticity in the system.

<figure><img src="image_43.png" /><figcaption>Figure 3.14: Modified efficiency $\eta$ as a function of the tip speed ratio $\lambda$, for three different pitch angles $\gamma_0$. Color and symbols are identical to Fig. 3.12</figcaption></figure>
### 3.1.5 Numerical efficiency improvements

We solve numerically equations (3.18) and (3.19) for the bending and twist angles. The aim of this model is to explore the phenomenology of blade behaviors that we can encounter with these fully flexible blades and shed light onto the potential efficiency improvement that this type of design could bring. It does not however, aim at being a quantitative predictor of the exact performance of our propeller, but rather a simple toy model. Numerous corrections to this type of model exist, the most common of which is the blade element momentum theory, widely used in engineering and modelling of wind turbines and propellers (Sørensen (2011); Glauert (1983); Masters et al. (2011); Snel et al. (1994)).

We employ the same methodology as in Sec 1.2: dividing the blade into spanwise sections. We solve the system using finite differences and a forward Euler discretization. The system is non-linear through the velocity norm and angle: in the plane normal to $d_3$ at the curvilinear abscissa $s$, the axial velocity is $U_x = U_\infty \cos \theta$ and the orthoradial velocity is $U_t = \bar{r}\Omega$, with $\bar{r}$ the distance to the axis of rotation, computed in the small twist limit in Eq. (A.2). Therefore, the dimensionless velocity norm reads:

$$u_p^2 = \frac{\cos^2 \theta}{\lambda^2} + \left(\frac{\bar{r}}{R}\right)^2, \qquad (3.31)$$

and the local angle of attack couples the bending and twist together:

$$\alpha = \gamma - \arctan \frac{R \cos \theta}{\lambda \bar{r}}. \qquad (3.32)$$

The boundary conditions for $\theta$ and $\gamma$ also need to be added to solve the system. The blade is clamped at the hub and has no torque or force at the tip:

$$\theta|_{s=0} = 0, \quad \left.\frac{\partial\theta}{\partial s}\right|_{s=1} = 0, \quad \left.\frac{\partial^2\theta}{\partial s^2}\right|_{s=1} = 0, \quad \text{and} \quad \gamma|_{s=0} = \gamma_0, \quad \left.\frac{\partial\gamma}{\partial s}\right|_{s=1} = 0. \qquad (3.33)$$

We keep the geometry of the blades a constant, as a result the ratio between the twisting and bending Cauchy numbers stays the same, equal to the of the blades.

In Fig. 3.15 we show the result of this computation versus the thrust $C_T$ and bending angle $\theta_M$ data from the blades presented in Sec. 3.1 (the long and slender blades for which the ribbon model best applies). The qualitative agreement between experiments and the numerically integrated model is quite good given its simplicity. The phenomenology is well reproduced: see the sign of the bending angle as a function of the tip speed ratio and the pitch angle, and in particular the very quick sign change at about $\lambda = 4$ for the blades with a pitch angle of 20 degrees (red squares). The thrust variations are also captured, in particular the zero thrust limit at high $\lambda$ for the F4 blades, or the slight overshoot and then decrease seen in the less flexible F1 blades.

The agreement unfortunately is not quantitative (see the different vertical scales) which was to be expected given the important assumptions regarding the flow around the blades (no induced velocities in particular). To a lesser extent, the elastic model used in not perfectly accurate for inextensible ribbons. We implemented the more complete model from Audoly and Neukirch (2021), and the difference was very small, even in the extreme bending cases. The plateau seen for the F4 blades with $\gamma_0 = 30$ deg is a result of a numerical threshold imposed to remove non-physical solutions: the blades fold further than 180 deg and start producing negative thrust.

We explored the parameter space (pitch angle $\gamma_0$ and tip speed ratio $\lambda$) for both rigid and flexible blades using this model (see Fig. 3.16). On the top row (rigid propeller), we find the classical behavior of a propeller. For all pitch angles, the thrust coefficient (panel (a)) is a growing function of the tip speed ratio and the threshold at which it becomes positive defines the region where the rotor is
<figure><img src="image_44.png" /><figcaption>Figure 3.15: Comparison between experimental data (left) and the numerical model (right). (Blue) rigid blades, (orange) F1 blades (red) F4 blades. (top row) Thrust coefficient $C_T$, (bottom row) tip bending angle $\theta_M$. Experiments are with the long and slender blades, $U_\infty = 0.15$ m/s.</figcaption></figure>

acting as a propeller. Maximum thrust occurs for very high tip speed ratios around 50 deg. In the region of thrust production, the torque coefficient (panel (b)) is effectively independent of the tip speed ratio (this is specific to the geometry we use here) and grows rapidly with the pitch angle with a maximum at 90 deg (not shown). The resulting efficiency (panel (c)) is therefore maximal very close to the delimitation of positive thrust and then decreases when increasing either the tip speed ratio $\lambda$ or the pitch angle $\gamma_0$. The maximum efficiency is reached for $\gamma_0 = 47$ deg and $\lambda = 2.6$. For a given tip speed ratio (i.e., at a given operating condition) the pitch angle that maximizes the efficiency $\gamma_0^{\text{opt}}(\lambda)$ is shown in red: it is a decreasing function of the tip speed ratio. It approximately corresponds to keeping a constant angle of attack at the tip of the blade. This angle is given by the gap between the red (most efficient pitch) and white (flow angle at the tip $\varphi(r=R)$) curves and is approximately 18 deg, very close to $\alpha_{\text{opt}} = 17$ deg.

Comparing these reference results to the behavior of a flexible blade (middle and bottom rows, at $\lambda = 4$, $C_Y^B = 1.2$) we highlight a few differences. At high tip speed ratios and especially for high pitch angles, the bending is extremely important ($\gtrsim 90$ deg, panel (h)). Both the thrust coefficient (panel (d)) and the efficiency (panel (f)) are therefore close to zero in this region.

In the region where the propeller produces negative thrust, the angle of attack is negative on most blade sections and therefore there is slight negative bending (panel (h)) and twist (panel (g)) (not visible because of the scale of the positive deformation). The limit between positive and negative bending (black line) is very similar to the limit of positive thrust (the same is true for the twist). This observation is coherent with the scaling law 3.28, and shows that the twist would follow a similar law. Finally, we see the small but finite amplitude of the twist ($\gamma_M - \gamma_0 \lesssim 15$ deg), that modifies the performance of the rotor.

Looking closer at the efficiency of the folding propeller, we plot in Fig. 3.17 (a) the efficiency as a function of the tip speed ratio only. In blue, we plot the curves obtained with the rigid propeller for
<figure><img src="image_45.png" /><figcaption>Figure 3.16: Maps of the results produced by the model as a function of the tip speed ratio $\lambda$ and pitch angle $\gamma_0$. Top row corresponds to rigid blades, and the bottom two to flexible ones (about as flexible as the F2 blades). Panels (a) and (d) show the thrust coefficient $C_T$, panels (b) and (e) show the torque coefficient ($C_Q$) and panels (c) and (f) show the efficiency $\eta$. Panels (g) and (h) show the tip twist $\gamma_M - \gamma_0$ and bending angles $\theta_M$ respectively, with regularly spaced contours (the black lines highlights zero deformation).</figcaption></figure>
various pitch angles and in black the maximal efficiency over all pitch angles (i.e., the envelope of the blue curves or the efficiency that can be obtained if the pitch angle is modified to follow $\gamma_0^{\text{opt}}(\lambda)$). On top of it, we plot the efficiency for flexible propellers with varying flexibility (yellow to red when increasing $C_Y^B/\lambda^2$) and a pitch angle of 30 deg (any other pitch angle would lead to a similar discussion).

<figure><img src="image_46.png" /><figcaption>Figure 3.17: (a) Efficiency $\eta$ as a function of the tip speed ratio $\lambda$. (light blue) Curves for different pitch angles with the rigid propeller. (black) Maximal efficiency for each $\lambda$. (yellow to red) Flexible propeller with varying Cauchy number. (b) Twist angle $\gamma$ as a function of the tip speed ratio. (black line) Pitch angle giving the optimal efficiency for the rigid propeller (red line from Fig. 3.16). (yellow to red) Tip twist of the flexible propeller.</figcaption></figure>

Flexible blades never exceed the envelope of the rigid blades: whichever the tip speed ratio, using the rigid propeller with the right pitch angle $\gamma_0^{\text{opt}}(\lambda)$ results in a better efficiency. The inefficacy of the flexible blades is due to the fact that (i) bending is always detrimental for efficiency, reducing the surface swept by the blades and (ii) the spanwise torsion of the blade is in the wrong direction. Efficient propulsion requires that the local twist angle $\gamma$ follows the evolution of the local flow angle $\varphi$: decreasing from root to tip. However, the twist in the case of these blades always increases $\gamma$ going from the root to the tip, further reducing the efficiency.

The flexible blades can be more efficient than a rigid propeller with the same pitch angle. At tip speed ratios smaller than the one reaching maximum efficiency ($3 < \lambda < 5$ here), the flexible blades are more efficient. The bending is limited in this region close to the zero thrust limit, reducing its detrimental effect on efficiency. In Fig. 3.17 (b) we plot the optimal pitch angle of the rigid blades $\gamma_0^{\text{opt}}(\lambda)$ (black curve) and the tip twist $\gamma(r=R)$ of the flexible ones with $\gamma_0 = 30$ deg (yellow to red). When $3 < \lambda < 5$, the pitch angle is smaller than the one required for maximal efficiency. The flexible blades twisting reduces this discrepancy. At a given tip speed ratio, increasing the twisting Cauchy number increases the efficiency up to a point where the sections of the blades near the tip have a pitch angle larger than $\alpha_{\text{opt}}$ and therefore generate drag, decreasing the efficiency. These results are coherent with the experimental observations in Fig. 3.14: the rigid blades with $\gamma_0 = 20$ deg have a
pitch angle too small for the tip speed ratios considered and thus all the flexible blades are more efficient. The most flexible ones are more efficient at lower $\lambda$. For more important pitch angles, any efficiency improvement occur at tip speed ratios $\lambda$ smaller than the one where the rigid blades reach their maximum.

### 3.1.6 Concluding remarks

These first results, intended at characterizing in broad lines the phenomenology encountered when dealing with highly flexible propellers, allowed us to obtain the following main results. The centrifugal force is always negligible in the problem, the complexity of the deformations observed comes from a coupling of the bending and twisting deformations through the hydrodynamic lift force. Even though classical reconfigurating systems (Gosselin and de Langre, 2011; Alben et al., 2002; Schouveiler and Boudaoud, 2006; Marzin et al., 2022) are driven by the drag force, we found that the techniques used to predict deformations (rescaling the Cauchy number by a quantity describing the initial loading on the system) can be directly transposed to this system for moderate deformations. When the bending is extreme, we found that we can still link two macroscopic quantities of the rotor (tip bending angle and thrust), although not in a predictive manner.

Using a simple toy model coupling an inextensible ribbon model for the blade elasticity and a blade element model for the hydrodynamic loading, we show that we can qualitatively predict the deformation of the blades. The novel fluid-structure interaction problem is the main interest of these blades. The bending deformation always reduces the thrust and efficiency compared with a rigid propeller. The twisting deformation only ever increases the pitch angle and thus is interesting for efficiency only in the range where the prescribed pitch angle $\gamma_0$ is too small. This effect can be reversed if the center of mass (or pivot) of the blade is placed in front of the hydrodynamic center. This is the topic of the following section.

Finally, these first experiments allowed us to identify the important parameters and challenges of measuring the deformations and performance of these types of model scale propellers. The fact that the region of interest in terms of the performance of the blade studied overlaps with the region where the uncertainty is maximal resulted in important challenges in obtaining repeatable results. In particular, the difficulties in accurately measuring the torque — and to a lesser extent the force — led us to design the third type of boat that we use in the remainder of the thesis, which completely eliminates these problems.
## 3.2 Chordwise-flexible propeller

<figure><img src="image_47.png" /><figcaption>Figure 3.18: Semi-flexible propeller</figcaption></figure>

This section describes a similar experiment to the one in the previous section but with a key difference: the pivot around which the blade rotates is the leading edge of the blade instead of its centerline. This design reverses the sign of the hydrodynamic twisting moment, reducing the local pitch angle instead of increasing it. The rigid leading edge also stops the bending of blades.

This concept is very similar to the one used by Cognet et al. (2017) in the case of wind turbines, with a rigid leading edge and a flexible blade. In this case, the hydrodynamic force increases the pitch angle while the centrifugal force decreases it. By balancing the two forces, the authors are able to follow the twist angle that maximizes the power extracted as a function of the tip speed ratio, thus extending the range of efficiency of the wind turbine. We investigate the possibility of obtaining a similar efficiency improvement in the case of propellers in water, where the force balance is modified in two ways. First, the centrifugal force is negligible for the same reasons as mentioned in the previous section. In addition, the lift force driving the deformation has an opposite sign between the case of a wind turbine and a propeller.

We first describe how the setup has been modified from the previous section, solving the main precision issues, and how we model the flexible propeller. We then analyze the effect of the deformation on thrust, torque, and efficiency in a particular case before describing how this flexible propeller can be used to improve efficiency in terms of magnitude and resilience. Finally, a first-order development of the governing equation allows us to compute a law that predicts the tip deformation of the propeller.

### 3.2.1 Setup and minimal model

We use blades with a rigid leading edge made of a small threaded shaft with flexible Mylar blades glued to it (Fig. 3.18). We sanded the threaded shaft in order to have a flat spot to glue the blades to. The blades and shafts are then threaded to the hub and fastened using nuts. This linkage allows us to change the pitch angle by varying the thread length. For reference, we also use rigid fiberglass blades made in the same way.

We used the third boat described in section 2.2.3, that allows us to measure simultaneously the thrust, torque and angular velocity of the rotor. We perform experiments by setting a flow velocity $U_\infty$ and
an angular velocity Ω and recording the signals of thrust and torque for 20 s. The values presented below are the mean over the last 10 s. Each experiment was repeated at least twice and the errorbars in the following represent the standard deviation between repeated measurements. Similarly to the previous case we do not vary the flow speed across experiments, instead focusing on the angular velocity and blade material. We use a flow velocity of $U_∞ = 0.15$ m/s, the blades P1 to P4 defined in Sec. 2.3 and angular velocities ranging from 4 to 22 rad/s. We used pitch angles $γ_0$ from 30 to 80 deg with all the blades. We also tested the rigid blade at 20 deg and the flexible ones at 90 deg (when not rotating the blades are aligned with the flow).

At the same time, we use a high-speed camera to capture the shape of the blade at the tip. This camera is synchronized with the Hall-effect sensor on the boat and therefore takes 25 pictures per rotation of the propeller. The blade is illuminated by a laser sheet that illuminates the entire chord at a given radius. We always place it a 1 to 2 mm from the tip, corresponding to 90% of the maximum radius. We then extract the images where the blade is aligned towards the camera. Doing so means that we record the shape of the blade always in the same position in the rotation, where the incoming flow is modified due to the shaft tilt (it corresponds to $Ωt = 3π/2 \mod 2π$ in the analysis from Sec. 2.2.3).

We employ a simpler modeling of the elasticity of the blade than in the previous section. Instead of solving for the shape of the blade as one single plate, we divide it into independent spanwise beams clamped at the leading edge (Fig. 3.20, (a)). This technique has been shown by Cognet (2017) to converge to the same solution when the number of beams is increased in the case of wind turbines. In doing so, we neglect the elastic energy associated with the torsion due to the different bending between root and

tip. In experiments, it is small compared to the bending itself (Fig. 3.19). We make one further assumption: instead of describing the curvature of each beam along the chord, we reduce the curvature to its average deflection by modelling each beam as rigid but attached to the leading edge by a torsion spring (Fig. 3.20, (b)). Doing so, we neglect the effect of the chordwise curvature of the blade. The effect of this curvature is to modify the sectional aerodynamic coefficients $C_L$ and $C_D$ in a non-trivial way. We measured the effective spring constant with a bending test in Sec. 2.3.

Experimentally, we extracted the tip twist angle $γ_M$ from the images of the deformed blade (Fig. 3.19). We used a Hough transform (Duda and Hart, 1972) to detect the main direction of the lit shape in the images. In practice, we measure the angle of the straight portion near the trailing edge (see App. D.2).

**Mathematical description** For a blade section a the distance $r$ from the axis of rotation, we write the conservation of angular momentum at the leading edge of the blade:

$$0 = C(γ(r) - γ_0) + \frac{1}{2}ρcU_p^2(C_L(α)\cos α + C_D(α)\sin α)\bar{\delta}_{AC}^{LE}, \quad (3.34)$$

where $C$ is the per unit width spring constant, $γ_0$ the pitch angle, $U_p$ the local flow velocity, and $\bar{\delta}_{AC}^{LE}$ the distance between the aerodynamic center and the leading edge. This distance is modeled as being equal to $c/4$ in this whole section (Li et al., 2022). Using $RΩ$ as a reference velocity and $c$ as the

<figure><img src="image_48.png" /><figcaption>Figure 3.19: Example of the deformation of the blades as the angular velocity is increased. Obtained with $U_∞ = 0.15$ m/s, $γ_0 = 30$ deg and the P3 blades.</figcaption></figure>
<figure><img src="image_49.png" /><figcaption>Figure 3.20: (a) Schematic of a blade twisting and its division into spanwise beams. (b) Definition of the angles on a single beam</figcaption></figure>

reference length, we obtain a new Cauchy number:

$$C_Y^S = \frac{1/2 \rho c^2 (R\Omega)^2}{C}, \qquad (3.35)$$

such that the balance of torques read:

$$\gamma(r) = \gamma_0 - C_Y^S u_p^2 (C_L(\alpha) \cos \alpha + C_D(\alpha) \sin \alpha) \delta_{AC}^{LE}, \qquad (3.36)$$

where $\delta_{AC}^{LE} = \bar{\delta}_{AC}^{LE}/c = 1/4$ and $u_p = U_p/R\Omega$. This equation is non-linear through the feedback from the hydrodynamic forcing: the angle of attack is $\alpha = \gamma - \varphi$.

**Correcting for the tilt of the shaft** The shaft is tilted and therefore the loading on the blades varies throughout the rotation. Using the model above, we estimate the resulting variation of the deformation. From the results obtained in Sec. 2.2.3 we have the angular position dependent angle of attack and velocity norm:

$$\alpha = \gamma - \arctan \frac{\cos \delta}{s\lambda - \sin \delta \sin \Omega t}, \quad \text{and} \quad u_p^2 = \frac{\cos^2 \delta}{\lambda^2} + \left( s - \frac{\sin \Omega t \sin \delta}{\lambda} \right)^2, \qquad (3.37)$$

where $s = r/R$, $\Omega t$ is the angular position in the plane of rotation, and $\lambda = R\Omega/U_\infty$ the tip speed ratio. We solve Eq. (3.36) with the position dependent angular velocities at the tip of the blade and extract several quantities of interest. First, the average twist angle over a rotation $\bar{\gamma}_M$. Second, $\gamma_M(\Omega t = 0 \mod \pi)$ the twist angle in the symmetric position (above or below the shaft). And third $\gamma_M(\Omega t = 3\pi/2 \mod 2\pi)$ the twist in the direction of the measurement. All of these are computed at the tip of the blade, where the deflections are the largest. We explored the parameter space of pitch angle $\gamma_0$, tip speed ratio $\lambda$ and Cauchy number $C_Y^S$.

In Fig. 3.21 (left) we show that the average twist during a rotation and the twist in the symmetric configuration are essentially identical. The equality between averaged and symmetric twist angles shows that it is sufficient to perform computations assuming $U_x = U_\infty \cos \delta$ and $U_t = r\Omega$ instead of computing the deflection throughout the rotation. We use this result in subsequent numerical computations.

In Fig. 3.21 (right) we show the difference between the angle measured when $\Omega t = 3\pi/2 \mod 2\pi$ and the average angle during the cycle. We vary $\gamma_0$ and $\lambda$ for a Cauchy number of about 1. The
<figure><img src="image_50.png" /><figcaption>Figure 3.21: (Left) Tip twist angle in the symmetric position versus mean twist angle. The numerical data is computed for $10 \deg < \gamma_0 < 90 \deg$, $10^{-2} < C_Y^S < 30$ and $1 < \lambda < 8$. (right) Difference between the tip twist angle in the measurement position and the mean twist as a function of the pitch angle $\gamma_0$ and the tip speed ratio $\lambda$.</figcaption></figure>

difference is a few degrees and is most important for small tip speed ratios. At the location of the measurement the blade is experiencing a larger velocity norm and a larger angle of attack. This larger angle of attack stems from the fact that the angle between the angular velocity and the flow velocity is narrower than 90 deg, and explains why the deflection measured is always larger than the mean value throughout the cycle. We therefore compute a theoretical correction $\gamma_{corr} = \hat{\gamma}_M - \gamma_M (\Omega t = 3\pi/2 \mod 2\pi)$, a function of $\gamma_0$, $\lambda$ and $C_Y^S$. In the following, we apply this correction to our experimental result in order to compensate for the tilt of the shaft.

## 3.2.2 Passive twisting improves the efficiency

### Typical experiment

We plot a typical example of the all the data that we measure in an experiment in Fig. 3.22. The left side (panels (a) to (d)) is experimental, the right side shows the corresponding numerical model. All data correspond to a pitch angle of $\gamma_0 = 60 \deg$. Similarly to the previous section, the different blades correspond to different colors according to the Cauchy number rescaled by the tip speed ratio squared:

$$C_Y^S / \lambda^2 = \frac{1/2 \rho c^2 U_\infty^2}{C} \qquad (3.38)$$

The P1 to P4 blades correspond to a value of $C_Y^S / \lambda^2$ respectively of $1.5 \times 10^{-2}$, $4.7 \times 10^{-2}$, $2.4 \times 10^{-1}$, and 1.04.

Blade flexibility most often reduces the thrust and torque coefficients. On panel (a) we see that all but the least flexible blade produce less thrust than the rigid one. This is because the effect of the flexibility is here to reduce the pitch angle on the blade (panel (d)). Most often this results in a reduction of lift and therefore of the thrust produced. However, the maximum lift a flat plate can produce occurs when the angle of attack is about 45 deg (Sec 1.2.2). At high tip speed ratio, the angle
<figure><img src="image_51.png" /><figcaption>Figure 3.22: Propeller performance as a function of the tip speed ratio $\lambda$ with a pitch angle of $\gamma_0 = 60$ deg. Panels (a) to (d) are experimental while (e) to (h) are numerical. Row 1 displays the thrust coefficient $C_T$, row 2 displays the torque coefficient $C_Q$, row 3 displays the efficiency $\eta$, and row 4 displays the tip twist angle $\gamma_M$. Left and right columns use the same vertical scales. Markers with dark edge color on panel (d) correspond to data that has been modified by adding $\gamma_{corr}$.</figcaption></figure>