**Principal investigators:**

Ramiro Godoy-Diana

https://blog.espci.fr/ramiro/

**Research opportunities**

Contact us if you are looking for an undergraduate or MSc internship as well as for PhD or postdoctoral projects.

Welcome to the Biomimetics and Fluid-Structure Interaction (aka Swimming and Flying aka BIOMIM) group at PMMH. Our team has been working for a few years in the physics of bio-inspired locomotion at intermediate Reynolds numbers such as flapping flight and undulatory swimming. Among other projects, we have focused on simplified models of flapping foils in hydrodynamic tunnel experiments; mechanical models of flapping flyers with flexible wings in self-propelled configurations; experimental models of undulatory swimming. Closer to experimental biology, we have worked on collective fish swimming, insect flight, impulsive underwater manoeuvres such as a snake strike, among others. Beyond bio-locomotion, we have addressed a few problems of fluid-structure interaction applied to wind and wave energy conversion.

**Current members of the group:**

Roméo Antier (PhD 2018-2021)

Bill François (PhD 2017-2020)

Vincent Cognet (PostDoc 2018)

Clotilde Nové-Josserand (PhD 2015-2018)

*Past PhD students and postdocs:*

- Intesaaf Ashraf (PhD 2014-2017)
- Marion Segall (PhD student 2014-2017)
- Vincent Cognet (PhD student 2014-2017)
- Miguel Piñeirua (Postdoc 2014-2015)
- Sophie Ramananarivo (MSc Internhsip 2010, PhD student 2010-2013)
- Verónica Raspa (Postdoc 2010-2013)
- Catherine Marais (MSc internhsip 2007, PhD student 2007-2010)

### News

Check out our review papers in the December 2018 issue of Current Opinion in Insect Science, on the special section on Insect bio-inspired micro and nanotechnologies edited by Jérôme Casas.

Juan’s paper on forced wake just published in Phys. Rev. Fluids

Clotilde’s work on bio-inspired harvesters is now published in Bioinspiration and Biomimetics.

Our review on the role drag in animal locomotion has been published in Interface

Check out the last news on our PNAS paper in Hakai Magazine

Read new Miguel’s JFM and Intesaaf’s PNAS on fish swimming.

Check out Vincent’s work on bio-inspired wind turbines published in Proceedings of the Royal Society A and covered by Science.

Check out our mini-film (in French), part of the Nature=Futur series produced by "La Belle Société" on flapping wings and bio-inspired urban windmills.

### Publications

**On flapping flight mechanisms and their applications to wind and marine energy harvesting**

In this paper, we present a short review on some of significative results on insect flapping flight. In particular, we focus on the time varying shape mechanisms observed during the flapping cycle that are used by insects to enhance the production of aerodynamic force. We then discuss a few examples on how these mechanisms are adapted to energy harvesters in engineered applications.

B. Thiria Current Opinion in Insect Science 30, 39–45 (2018), doi.org/10.1016/j.cois.2018.09.001

**Insect and insect-inspired aerodynamics: unsteadiness, structural mechanics and flight control**

Flying insects impress by their versatility and have been a recurrent source of inspiration for engineering devices. A large body of literature has focused on various aspects of insect flight, with an essential part dedicated to the dynamics of flapping wings and their intrinsically unsteady aerodynamic mechanisms. Insect wings flex during flight and a better understanding of structural mechanics and aeroelasticity is emerging. Most recently, insights from solid and fluid mechanics have been integrated with physiological measurements from visual and mechanosensors in the context of flight control in steady airs and through turbulent conditions. We review the key recent advances concerning flight in unsteady environments and how the multi-body mechanics of the insect structure — wings and body — are at the core of the flight control question. The issues herein should be considered when applying bio-informed design principles to robotic flapping wings.

R. Bomphrey and R. Godoy-Diana Current Opinion in Insect Science 30, 26–32 (2018), doi:10.1016/j.cois.2018.08.003

**Forced wakes far from threshold: Stuart-Landau equation applied to experimental data**

In this Rapid Communication, we study with the Stuart-Landau (SL) amplitude equation, a wake flow control scenario using experimental data from a cylinder wake forced by plasma actuators. Given the formal framework recently discussed by Gallaire et al. (Fluid Dyn. Res. 48, 061401, 2016) on pushing amplitude equations far from threshold, we analyze experimental data of a forced wake in order to test the SL reduced order model. Linear stability theory and global mode concepts are used to determine the SL parameters. The extension to forced wakes of the SL model had been proposed by Thira and Wesfreid (J. Fluid Mech. 579, 137, 2007) in the context of their study on stability properties, but its employment still remained an open question. Here, we show that a forced wake at a Reynolds number far from the first threshold can also attain the critical behavior described by the SL model.

S. Boury, B. Thiria, R. Godoy-Diana, G. Artana, J. E. Wesfreid and J. D’Adamo;

Phys. Rev. Fluids., 3, 091901(R), (2018)

**Surface wave energy absorption by a partially submerged bio-inspired canopy**

Aquatic plants are known to protect coastlines and riverbeds from erosion by damping waves and fluid flow. These flexible structures absorb the fluid-borne energy of an incoming fluid by deforming mechanically. In this paper we focus on the mechanisms involved in these fluid-elasticity interactions, as an ef cient energy harvesting system, using an experimental canopy model in a wave tank. We study an array of partially-submerged flexible structures that are subjected to the action of a surface wave field, investigating in particular the role of spacing between the elements of the array on the ability of our system to absorb energy from the flow. The energy absorption potential of the canopy model is examined using global wave height measurements for the wave eld and local measurements of the elastic energy based on the kinematics of each element of the canopy. We study different canopy arrays and show in particular that exibility improves wave damping by around 40%, for which half is potentially harvestable.

C. Nové-Josserand, F. Castro Hebrero, L.-M. Petit, W. M. Megill, R. Godoy-Diana and B. Thiria;

Bioinspir. Biomim., 13, 036006, (2018)

**On the diverse roles of fluid dynamic drag in animal swimming and flying**

Questions of energy dissipation or friction appear immediately when addressing the problem of a body moving in a fluid. For the most simple problems, involving a constant steady propulsive force on the body, a straightforward relation can be established balancing this driving force with a skin friction or form drag, depending on the Reynolds number and body geometry. This elementary relation closes the full dynamical problem and sets, for instance, average cruising velocity or energy cost. In the case of finite-sized and time-deformable bodies though, such as flapping flyers or undulatory swimmers, the comprehension of driving/dissipation interactions is not straightforward. The intrinsic unsteadiness of the flapping and deforming animal bodies complicates the usual application of classical fluid dynamic forces balance. One of the complications is because the shape of the body is indeed changing in time, accelerating and decelerating perpetually, but also because the role of drag (more specifically the role of the local drag) has two different facets, contributing at the same time to global dissipation and to driving forces. This causes situations where a strong drag is not necessarily equivalent to inefficient systems. A lot of living systems are precisely using strong sources of drag to optimize their performance. In addition to revisiting classical results under the light of recent research on these questions, we discuss in this review the crucial role of drag from another point of view that concerns the fluid–structure interaction problem of animal locomotion. We consider, in particular, the dynamic subtleties brought by the quadratic drag that resists transverse motions of a flexible body or appendage performing complex kinematics, such as the phase dynamics of a flexible flapping wing, the propagative nature of the bending wave in undulatory swimmers, or the surprising relevance of drag-based resistive thrust in inertial swimmers.

R. Godoy-Diana and B. Thiria;

Proc. R. Soc. Interface, 15, 20170715, (2018)

**Modelling of an actuated elastic swimmer**

We study the force production dynamics of undulating elastic plates as a model for fish-like inertial swimmers. Using a beam model coupled with Lighthill’s large-amplitude elongated-body theory, we explore different localised actuations at one extremity of the plate (heaving, pitching and a combination of both) in order to quantify the reactive and resistive contributions to the thrust. The latter has the form of a quadratic drag in large Reynolds number swimmers and has recently been pointed out as a crucial element in the thrust force balance. We validate the output of a weakly nonlinear solution to the fluid–structure model using thrust force measurements from an experiment with flexible plates subjected to the three different actuation types. The model is subsequently used in a self-propelled configuration – with a skin friction model that balances thrust to produce a constant cruising speed – to map the reactive versus resistive thrust production in a parameter space defined by the aspect ratio and the actuation frequency. We show that this balance is modified as the frequency of excitation changes and the response of the elastic plate shifts between different resonant modes, the pure heaving case being the most sensitive to the modal response with drastic changes in the reactive/resistive contribution ratio along the frequency axis. We analyse also the role of the phase lag between the heaving and pitching components in the case of combined actuation, showing in particular a non-trivial effect on the propulsive efficiency.

M. Piñeirua, B. Thiria and R. Godoy-Diana;

*J. Fluid Mech.* **829**: 731—750 (2017).

**Simple phalanx pattern leads to energy saving in cohesive fish schooling**

Fish school structures are firstly based on social life or prey–predator interactions, but another idea has often been raised

by hydrodynamicists, claiming that fish could take advantage of schooling behavior from a locomotion efficiency perspective. By using a controlled swimming experiment with real schools, the present work shows that fish swimming together effectively need a less demanding stroke rate to sustain high swimming velocities, using, however, a different collective strategy compared with the usually suggested diamond pattern predicted by vortex-based interactions. The observed strategy, simply consisting of synchronized side-by-side swimming with nearest neighbors, finally, appears to be a lot more convenient for reaching an energy-saving regime.

I. Ashraf, H. Bradshaw, J. Halloy, T.-T. Ha, R. Godoy-Diana, B. Thiria; Proc. Natl. Acad. Sci. (PNAS), 114 (36), (2017)

**Bioinspired turbine blades offer new perspectives for wind energy**

Wind energy is becoming a significant alternative solution for future energy production. Modern turbines now benefit from engineering expertise, and a large variety of different models exists, depending on the context and needs. However, classical wind turbines are designed to operate within a narrow zone centred around their optimal working point. This limitation prevents the use of sites with variable wind to harvest energy, involving significant energetic and economic losses. Here, we present a new type of bioinspired wind turbine using elastic blades, which passively deform through the air loading and centrifugal effects. This work is inspired from recent studies on insect flight and plant reconfiguration, which show the ability of elastic wings or leaves to adapt to the wind conditions and thereby to optimize performance. We show that in the context of energy production, the reconfiguration of the elastic blades significantly extends the range of operating regimes using only passive, non-consuming mechanisms. The versatility of the new turbine model leads to a large increase of the converted energy rate, up to 35%. The fluid/elasticity mechanisms involved for the reconfiguration capability of the new blades are analysed in detail, using experimental observations and modelling.

V. Cognet, S. Courrech du Pont, B. Thiria; Proc. Roy. Soc. A, 473, (2017)

**Pushing amplitude equations far from threshold: application to the supercritical Hopf bifurcation in the cylinder wake**

The purpose of this review article is to push amplitude equations as far as

possible from threshold. We focus on the Stuart–Landau amplitude equation

describing the supercritical Hopf bifurcation of the flow in the wake of a

cylinder for critical Reynolds number Rc 46. After having reviewed

Stuartʼs weakly nonlinear multiple-scale expansion method, we first demon-

strate the crucial importance of the choice of the critical parameter for the wake behind a cylinder. Although Sipp and Lebedev (2007 J. Fluid Mech 593 333–58) correctly identified the adequate bifurcation parameter, they have plotted their results adding an additional linearization, which amounts to using e’ as approximation to e. We then illustrate the risks of calculating ‘running’ Landau constants by projection formulas at arbitrary values of the control parameter. For the cylinder wake case, this scheme breaks down and diverges close to Re 100. We propose an interpretation based on the progressive loss of the non-resonant compatibility condition, which is the cornerstone of Stuartʼs multiple-scale expansion method. We then briefly review a self-con- sistent model recently introduced in the literature and demonstrate a link between its properties and the above-mentioned failure.

F. Gallaire, E. Boujo, V. Mantic-Lugo, C. Arriatia, B. Thiria, P. Meliga ; Fluid Dyn. Res, 48, 061401, (2016)

**Synchronization and collective swimming patterns in fish (Hemigrammus bleheri)**

In this work, we address the case of red nose tetra fish Hemigrammus bleheri swimming in groups in a uniform flow, giving special attention to the basic interactions and cooperative swimming of a single pair of fish. We first bring evidence of synchronization of the two fish, where the swimming modes are dominated by ‘out-phase’ and ‘in-phase’ configurations. We show that the transition to this synchronization state is correlated with the swimming speed (i.e. the flow rate), and thus with the magnitude of the hydro- dynamic pressure generated by the fish body during each swimming cycle. From a careful spatio-temporal analysis corresponding to those synchronized modes, we characterize the distances between the two individuals in a pair in the basic schooling pattern. We test the conclusions of the analysis of fish pairs with a second set of experiments using groups of three fish. By identifying the typical spatial configurations, we explain how the nearest neighbour interactions constitute the building blocks of collective fish swimming.

I. Ashraf, R. Godoy-Diana, J. Halloy, B. Collignon, B. Thiria ; Proc. R. Soc. Interface, 13, 20160734, (2016)

**Does aquatic foraging impact head shape evolution in snakes ?**

Evolutionary trajectories are often biased by developmental and historical factors. However, environmental factors can also impose constraints on the evolutionary trajectories of organisms leading to convergence of morphology in similar ecological contexts. The physical properties of water impose strong constraints on aquatic feeding animals by generating pressure waves that can alert prey and potentially push them away from the mouth. These hydrodynamic constraints have resulted in the independent evolution of suction feeding in most groups of secondarily aquatic tetrapods. Despite the fact that snakes cannot use suction, they have invaded the aquatic milieu many times independently. Here, we test whether the aquatic environment has constrained head shape evolution in snakes and whether shape converges on that predicted by biomechanical models. To do so, we used three-dimensional geometric morphometrics and comparative, phylogenetically informed analyses on a large sample of aquatic snake species. Our results show that aquatic snakes partially conform to our predictions and have a narrower anterior part of the head and dorsally positioned eyes and nostrils. This morphology is observed, irrespective of the phylogenetic relationships among species, suggesting that the aquatic environment does indeed drive the evolution of head shape in snakes, thus biasing the evolutionary trajectory of this group of animals.

M. Segall, R. Cornette, A.-C. Fabre, R. Godoy-Diana, A. Herrel ; Proc. R. Soc. B, 283, 20161645, (2016)

Our last poster in collaboration with Francisco Huera-Huarte from Universitat Rovira i Virgili presented at the APS Gallery of Fluid Motion in the DFD 2015 Meeting.

Volumetric velocimetry of wake vortices produced by an undulatory swimmer

**Resistive thrust production can be as crucial as added mass mechanisms for inertial undulatory swimmers**

In this Rapid Communication, we address a crucial point regarding the description of moderate to high Reynolds numbers aquatic swimmers. For decades, swimming animals have been classified in two different families of propulsive mechanisms based on the Reynolds number: the resistive swimmers, using local friction to produce the necessary thrust force for locomotion at low Reynolds number, and the reactive swimmers, lying in the high Reynolds range, and using added mass acceleration (described by perfect fluid theory). However, inertial swimmers are also systems that dissipate energy, due to their finite size, therefore involving strong resistive contributions, even for high Reynolds numbers. Using a complete model for the hydrodynamic forces, involving both reactive and resistive contributions, we revisit here the physical mechanisms responsible for the thrust production of such swimmers. We show, for instance, that the resistive part of the force balance is as crucial as added mass effects in the modeling of the thrust force, especially for elongated species. The conclusions brought by this work may have significant contributions to the understanding of complex swimming mechanisms, especially for the future design of artificial swimmers.

M. Piñeirua, R. Godoy-Diana, and B. Thiria

*PRE* **91**: 021001(R) (2015).

**Four-winged flapping flyer in forward flight**

We study experimentally a four-winged flapping flyer with chord-wise flexible wings in a self-propelled setup. For a given physical configuration of the flyer (i.e. fixed distance between the forewing and hindwing pairs and fixed wing flexibility), we explore the kinematic parameter space constituted by the flapping frequency and the forewing-hindwing phase lag.

Cruising speed and consumed elec- tric power measurements are performed for each point in the ( f, φ) parameter space and allow us to discuss the problem of performance and efficiency in four-winged flapping flight. We show that different phase-lags are needed for the system to be opti- mised for fastest flight or lowest energy consumption. A conjecture of the underlying mechanism is proposed in terms of the coupled dynamics of the forewing-hindwing phase lag and the deformation kinematics of the flexible wings.

**Centrifugal instability of Stokes layers in crossflow**

A circular cylinder oscillating in a viscous fluid produces an axisymmetric Stokes layer, a fundamental flow susceptible to centrifugal instabilities. In the present work we study such problem in the wake flow around a circular cylinder at Re = 100 performing rotary oscillations. For a forcing frequency ff and amplitude Ut, the non-dimensional control parameters f+ = ff /fn and A = Ut/U∞ are related to the flow properties, the inflow velocity U∞ and the natural vortex shedding frequency fn. In a previous work (D’Adamo et al., Phys. Rev. E. 84, 056308, 2011), we identified experimentally a zone in the parameter space with forcing at frequencies lower than the natural vortex shedding frequency, where the flow exhibited some turbulence features such as a continuous spectrum for the velocity components, and multiple vortex splitting interactions in the wake. We show here using numerical simulations that these observations result from a 3D centrifugal instability.

J. D’Adamo, R. Godoy-Diana & J. E. Wesfreid

*Proceedings of the Royal Society A* **471**: 20150011 (2015).

**Ratchet pump**

We investigate a mechanism that effectively transports fluids using vibrational motion imposed onto fluid boundary with anisotropy. In our experiment, two asymmetric, sawtooth-like structures are placed facing each other and form a corrugated fluid channel. This channel is then forced to open and close periodically. Under reciprocal motion, fluid fills in the gap during the expansion phase of the channel and is then forced out during contraction. Since the fluid experiences different impedances when flowing in different directions, the stagnation point that separates flows of two directions changes within each driving period. As a result, fluid is transported unidirectionally.

**Large-amplitude undulatory swimming near a wall**

We study experimentally the propulsive dynamics of flexible undulating foils in a self-propelled swimming configuration near a wall. Measurements of swimming speed and propulsive force are performed, together with full recordings of the elastic wave kinematics and particle image velocimetry. We show that the presence of the wall can enhance the cruising velocity in some cases up to 15%. The physical mechanism responsible for this augmentation is first discussed qualitatively by studying the vorticity dynamics in the wake. A quantitative picture of the problem is then established using a POD analysis of the velocity fields in the wake, showing that the first POD mode is a good indicator of the kinetic energy in the propulsive jet.

**Drag in undulatory swimmers**

During cruising, the thrust produced by a self-propelled swimmer is balanced by a global drag force. For a given object shape, this drag can involve skin friction or form drag, both being well-documented mechanisms. However, for swimmers whose shape is changing in time, the question of drag is not yet clearly established. We address this problem by investigating experimentally the swimming dynamics of undulating thin flexible foils. Measurements of the propulsive performance together with full recording of the elastic wave kinematics are used to discuss the general problem of drag in undulatory swimming. We show that a major part of the total drag comes from the trailing longitudinal vortices that roll-up on the lateral edges of the foils. This result gives a comparative advantage to swimming foils of larger span thus bringing new insight to the role of aspect ratio for undulatory swimmers.

**Propagating waves in bounded elastic media**

Confined geometries usually involve reflected waves interacting together to form a spatially stationary pattern. Our recent study on the locomotion of a self-propelled elastic swimmer on a free surface [Ramananarivo et al. 2013], however, has shown that propagating wave kinematics can naturally emerge in a forced elastic rod, even with boundary conditions involving significant reflections. This particular behavior is observed only in the presence of strong damping. Based on those observations, this study aims at giving a quantitative description of the mechanism involved to prevent the built up of standing waves and generate traveling solutions. The question is discussed here in the framework of hand-made artificial swimmers as an example of practical application but we believe that its potential is beyond this scope.

**Dynamics of a swimming elastic filament on a free surface**

Many living organisms use body undulations to propulse themselves through fluids: they achieve net forward motion by propagating wave of curvature down their deformable body. In inertial regimes, the anguilliform swimming mechanism has first been addressed in pioneer study by Lighthill [J. Fluid Mech., vol 9, 305-317, 1960]. In his so called reactive model, Lighthill considered the inertial momentum redistribution caused by the undulations within the fluid and he showed that the thrust force generated through this process could be estimated from the kinematic of the tail of the swimmer alone. A vast amount of theoretical and numerical works has followed, providing the basis for a broad spectrum of applications, especially in robotic and engineering.

We present here a swimmer able to propulse itself at the surface of a water tank. The set up consists in a flexible filament forced to oscillate by imposing an harmonic motion to one of its extremities (using magnetic interactions). We fully characterize its dynamics, with the objective to bring a better understanding of fluid-solid interactions in undulatory propulsion. The characteristics of the propagating wave are crucial in determining the swimming performance. Modeling the filament as a forced beam under fluid loading, we pinpoint the different elements that can account for the observed kinematic. In particular, we show that in our Reynolds number regime, a quadratic fluid dissipation term is needed to propagate passively a wave in a finite elastic beam. The order of magnitude of this term is estimated by comparing the theoretical predictions of the model against the experimental data. When injecting the prescribed body deformations into Lighthill’s model, we show that this reactive theory gives good predictions of the performance of the swimmer (its forward speed).

31st Annual Gallery of Fluid Motion (Pittsburgh, Pennsylvania, USA, 2013)

**Force balance in the take-off of a butterfly**

Up to now, the take-off stage remains an elusive phase of insect flight relatively poorly explored compared to other maneuvers. An overall assessment of the different mechanisms involved in the force production during take-off has never been explored. Focusing on the first downstroke, we have addressed this problem from a force balance perspective in butterflies taking-off from the ground. In order to determine if the sole aerodynamic wing force could explain the observed motion of the insect, we have firstly compared a simple analytical model of the wings force to the acceleration of the insect’s center of mass estimated from video tracking of the wing and body motions. Secondly, the wing kinematics has also been used for numerical simulations of the aerodynamic flow field. Similar wing aerodynamic forces were obtained by the two methods. Both are however not sufficient, nor is the inclusion of the ground effect, to predict faithfully the body acceleration. We have to resort to the legs forces to obtain a fitting model. We show that the median and hind legs display an active extension responsible for the initiation of the upward motion of the insect’s body, occurring before the onset of the wing downstroke. We estimate that legs generate, at various times, an upward force which can be much larger than all other forces applied to the insect’s body. The relative timing of leg and wing forces explain the large variability of trajectories observed during the maneuvers.

**Topology and symmetry properties of biomimetic propulsive wakes**

It is known that the wake pattern observed in a cross-section behind swimming or flying animals is typically characterized by the presence of periodical vortex shed- ding. However, depending on species, propulsive wakes can differ according to their spatial ordering: symmetric (squid-like) or asymmetric (fish-like), with respect to the motion axis. We conducted a very precise experiment to analyse the role of the wake topology in the generation of propulsion. Self-propulsion is achieved by the flapping motion of two identical pitching rigid foils, separated by a distance d. By keeping the momentum input unchanged, we compared both symmetric and asymmetric flapping modes. For the entire explored range of parameters, the symmetric squid-like mode proves to be more efficient for thrust generation than the fish-like asymmetrical one. We show here that this difference is due to a pressure effect related to the ability of each wake to produce, or not, significant mixing in the near wake region.

**Wing compliance in self-propelled flapping flyers**

Saving energy and enhancing performance are secular preoccupations shared by both nature and human beings. In animal locomotion, flapping flyers or swimmers rely on the flexibility of their wings or body to passively increase their efficiency using an appropriate cycle of storing and releasing elastic energy. Despite the convergence of many observations pointing out this feature, the underlying mechanisms explaining how the elastic nature of the wings is related to propulsive efficiency remain unclear. Here we use an experiment with a self-propelled simplified insect model allowing to show how wing compliance governs the performance of flapping flyers. Reducing the description of the flapping wing to a forced oscillator model, we pinpoint different nonlinear effects that can account for the observed behavior ---in particular a set of cubic nonlinearities coming from the clamped-free beam equation used to model the wing and a quadratic damping term representing the fluid drag associated to the fast flapping motion. In contrast to what has been repeatedly suggested in the literature, we show that flapping flyers optimize their performance not by especially looking for resonance to achieve larger flapping amplitudes with less effort, but by tuning the temporal evolution of the wing shape (i.e., the phase dynamics in the oscillator model) to optimize the aerodynamics.

**The wake of a flapping foil**

The vortex streets produced by a flapping foil of span-to-chord aspect ratio of 4:1 are studied in a hydrodynamic tunnel experiment. In particular, the mechanisms giving rise to the symmetry breaking of the reverse Bénard-von Kármán vortex street that characterizes fish-like swimming and forward flapping flight are examined. Two-dimensional particle image velocimetry measurements in the mid-plane perpendicular to the span axis of the foil are used to characterize the different flow regimes. The deflection angle of the mean jet flow with respect to the horizontal observed in the average velocity field is used as a measure of the asymmetry of the vortex street. Time series of the vorticity field are used to calculate the advection velocity of the vortices with respect to the free stream, defined as the phase velocity, as well as the circulation of each vortex and the spacing between consecutive vortices in the near wake. The observation that the symmetry breaking results from the formation of a dipolar structure from each couple of counter-rotating vortices shed on each flapping period serves as starting point to build a model for the symmetry breaking threshold. A symmetry breaking criterion based on the relation between the phase velocity of the vortex street and an idealized self-advection velocity of two consecutive counter-rotating vortices in the near wake is established. The predicted threshold for symmetry breaking accounts well for the deflected wake regimes observed in the present experiments and may be useful to explain other experimental and numerical observations of similar deflected propulsive vortex streets reported in the literature.

**PhD theses**

**HDR**

B. Thiria: Sillages et interactions fluide/structure: Contrôle, biomimetisme et conversion d’énergie. *Habilitation à diriger des recherches*, Université Paris Diderot, 2014. (Soutenue le 4/12/2014)

*Short term internship students*

- Nicolas Cattaud (Université Cergy-Pontoise L3, 2018)
- Alexandre (Université Paris Diderot L3, 2018)
- Sarah (PSL, 2018)
- Bill François (ENS 4th year internship, 2017)
- Matthieu Baron (ENS Cachan L3, 2017)
- Baptiste Ferrero (ESPCI Internship, 2017)
- Martyna Góral (M2 Fluid Mechanics, UPMC/Ecole Polytechnique, 2017)
- Luce-Marie Petit (M2 Biomechanical engineering, Université Paris-Saclay, 2017)
- Hanaé Bradshaw (L3, FdV, CRI 2016)
- Pedro Miguel Fernandes (ENSTA 2A, 2016)
- Florian Benoit (ESPCI 3A, 2016)
- Tùng Hà Thanh (ESPCI 3A, 2016)
- Clément Haeck (ENS Cachan L3, 2016)
- Thierry Ksstentini (UPMC L3, 2016)
- Gabriel Victorino Cardoso (Ecole Polytechnique 3A, 2016)
- Jérôme Hardoüin (MSc intern, Exole Polytechnique 2015)
- Keziah Reynoso (MSc intern 2014-2015)
- Lucien Causse (ENS Cachan Internship 2014 and 2015)
- Arnaud Choux (ENS Cachan Internship 2014)
- Olivier Spitz (ESPCI Internship 2014)
- Jérôme Vacher (ESPCI Internship 2014)
- Prasann Jain (IIT Kharagpur, India, Internship 2013)
- Karanvir Singh (IIT Kanpur, India, Internship 2013)
- Antoine Gaillard (ENS Internship 2013)
- Mariana Centeno (UNAM, Mexico, MSc Internship 2012)
- Maxime Dana (ENSTA Internship 2012)
- Cécile Gaubert (MSc Internship 2011)
- Alexis Weinreb (ESPCI Internship 2011)
- Olivia Gann (ESPCI Internship 2011)

*Visitors*

- Alexander Alexeev (Georgia Tech, 2018)
- Juan D’Adamo (LFD, FIUBA PhD student, 2018)
- Gen Li (Chiba University, 2017)
- Federico Castro-Hebrero (LFD, FIUBA PhD student, 2016)
- Francisco Huera-Huarte (URV, Tarragona, 2015)
- Rafael Fernández Prast (PhD visiting student, 2013)
- Geoffrey Spedding (USC, 2013)
- Roberto Zenit (UNAM, 2013)

*Past and present collaborators in PMMH*

- José Eduardo Wesfreid
- Daniel Pradal
- Jean-Luc Aider