Master thesis
PhD thesis

JeanLou PFISTER, Instabilities and optimization of elastic structures interacting with laminar flows, University Paris Saclay, June 2019.
Abstract: Large static and dynamic deformations arise when elastic solids interact with viscous flows. They may accurately be captured by considering a strong numerical coupling between the Lagrangian solid dynamics and the Eulerian fluid dynamics, especially when large addedmass effects are at play. Besides running unsteady nonlinear simulations, linearised modal approaches are useful to identify hydroelastic instabilities at the origin of those vibrations. They can also be used to design passive control strategies aiming at attenuating or even suppressing the structural vibrations. The objectives of this thesis are to develop and apply methods, first to accurately describe the linear dynamics of strongly coupled fluidsolid systems, and then to optimize the shape or the elastic properties of the solid so as to control the linear dynamics. The first part of this thesis presents the theoretical and numerical methods developed to investigate the linear dynamics of fluidsolid perturbations around nonlinear steady states. The fluid dynamics is governed by the incompressible NavierStokes equations, while the solid is described by hyperelastic models. An Arbitrary Lagrangian Eulerian coupling is chosen, resulting in a conformal description of the fluidsolid interface in a timeindependent reference configuration. An exact linearisation of this formulation is derived, and two analyses of the resulting fully coupled, linearised fluidsolid operator are considered. An eigenvalue analysis allows to determine selfsustained fluidsolid instabilities responsible, for instance, for the vortexinduced vibrations of bluff bodies or the flutter of slender bodies. The resolvent analysis, i.e. a singular value analysis of the fluidsolid operator, allows to determine the linear response of the fluidsolid system to external forcings, such as gusts. The second part is devoted to the analysis and control of the vibrations of elastic plates attached downstream of a rigid circular cylinder, and immersed in a uniform incoming flow. First, complex eigenmodes, related to vortexinduced vibrations, are identified by means of the eigenvalue analysis. These modes become unstable when reducing the stiffness. A further decrease of stiffness yields to the destabilization of a real eigenmode, characteristic of a symmetrybreaking divergence instability. Nonlinear steady and unsteady simulations are performed to elucidate the nonlinear interactions between the unstable modes. Secondly, an adjointbased shape optimization of the rigid body supporting the elastic plate is proposed to control the unstable complex modes, aiming either at decreasing the growth rate or varying the frequency. A stabilization of the complex mode is achieved by a thinning of the rigid body. More exotic shapes are obtained when considering the variation of the frequency. A frequency decrease is achieved by Dshaped cylinders, while a frequency increase is obtained with Cshaped cylinders. The last part of the thesis is dedicated to the delay of laminar/turbulent transition in twodimensional boundarylayer flows thanks to viscoelastic, finitelength coatings. A resolvent analysis of the fluidsolid operator is used to quantify the attenuation of lowfrequency TollmienSchlichting instability waves when the stiffness of the coating is reduced. On the other hand, the eigenvalue analysis shows that highfrequency solidbased modes are destabilized when the solid viscous damping is too low. A gradientbased strategy to optimize the stiffness distribution of the coating with respect to the energy amplification of both instabilities is eventually proposed. The optimized coatings have an overall structure organized in layers aligned with the flow, with a much stronger anisotropy in both the streamwise and transverse directions close to the edges, and make it possible both to attenuate TollmienSchlichting waves and to limit the development of solidbased instabilities.

Johann MOULIN, On the flutter bifurcation in laminar flows: linear and nonlinear modal methods, University Paris Saclay, December 2020
Abstract: The flutter instability has been the focus of numerous works since the middle of the twentieth century, due to its critical application in aeronautics. Flutter is classically described as a linear instability using potential flow models, but viscous and nonlinear fluid effects may both crucially impact this aeroelastic phenomenon. The first part of this thesis is devoted to the development of theoretical and numerical methods for analyzing the linear and nonlinear dynamics of a typical aeroelastic section, i.e. a heaving and pitching springmounted plate, immersed in a twodimensional laminar flow modeled by the incompressible NavierStokes equations. A semianalytical weakly nonlinear analysis (WNL) is first developed in order to derive an amplitude equation for the flutter bifurcation. In order to bypass the inherent limitations of this method to weak nonlinearities, we then develop a harmonic balance type method, known as the Time Spectral Method (TSM), allowing to efficiently compute (possibly unstable) highlynonlinear periodic flutter solutions. The challenging task of solving the TSM equations, especially when large numbers of Fourier harmonics are considered, is tackled via a timeparallel NewtonKrylov approach in combination with a new, socalled blockcirculant preconditioner, for which the robustness with the number fo harmonics is numerically demonstrated. The second part of this thesis focuses on the physical investigation of the flutter bifurcation of the springmounted plate. We start by revisiting the linear stability problem using a NavierStokes fluid model allowing to highlight, in particular, the effect of viscosity. Comparisons to classical quasisteady and unsteady potential flow (Theodorsen model) theories are performed. Contrary to what happens in potential flows, the flutter instability is shown to restabilize at very high reduced velocities in viscous flows. We continue our route on the flutter bifurcation by investigating the effect of fluid nonlinearities. Low solidtofluid mass ratios and increasing Reynolds numbers foster subcritical bifurcations. The role of leadingedge shear layers is pointed out. For intermediate mass ratios, an unusual bifurcation scenario that combines a supercritical bifurcation and the existence of subcritical highamplitude flutter solutions is discovered. We conclude our study of the flutter bifurcation by investigating the appearance of lowfrequency amplitude modulations on top of a previously established periodic flutter solution. Using an original TSMbased Floquet stability analysis, we explain this behavior by the destabilization of the periodic solutions by a pair of complexconjugate Floquet modes. An analysis of the latter shows that the physical mechanism governing the instability borrows elements from the classical flutter instability arising on steady solutions. The last part of this thesis aims at initiating the extension of the different methods previously evoked to largescale threedimensional configurations. As a first step towards this longterm goal, we develop an opensource massively parallel tool, based on the FreeFEM library and its PETSc/SLEPc interface, able to compute the nonlinear steadystate flow and subsequently solve the linear stability eigenproblem, for threedimensional flows (the structure is fixed) possessing several tens of millions of degrees of freedom.

Luis BENETTI RAMOS, Selfpropulsion and fluidmediated interaction of flapping wings in viscous flows, University of Bordeaux, December 2020.
Abstract: A common locomotion strategy exploited by aquatic and flying animals and more recently in the innovative conception of engineering devices is the flapping motion of appendages such as wings, fins or tails. This locomotion strategy appears with the increase of fluid inertia and nonlinearities, that result in a transition of the flow dynamics where timereciprocal motions allow to achieve locomotion and collective dynamics, such as bird flocks and fish schools, become possible through fluidmediated interactions. In this thesis we study the emergence of flapping propulsion and the role of hydrodynamic interactions in collective dynamics of flapping wings through linear and nonlinear analysis of the coupled fluid/ selfpropelled wing system. The first part of this thesis is consecrated to study the horizontal selfpropulsion of a symmetric heaving foil in a twodimensional quiescent fluid. The problem is investigated numerically based on the resolution of the NavierStoles equations, written in a noninertial frame of reference that follows the foil centre of gravity, coupled to the foil horizontal acceleration. At first, we investigate the emergence of selfpropelled regimes through unsteady nonlinear simulations, adopting a fixed density ratio and flapping amplitude while varying the flapping frequency. At low flapping frequencies, two selfpropelled states are analysed: a periodic state of unidirectional propulsion and a quasiperiodic state of slow back and forth motion around a fixed point. These states emergence is explained through a fluidsolid Floquet stability analysis of nonpropulsive symmetric baseflows. Unlike purely hydrodynamic stability analyses, usually employed in the literature, the proposed analysis accurately determines the locomotion states onset. In addition, it highlights linear mechanisms responsible for the emergence of unidirectional propulsion and the slow direction switching of back and forth motion. A timeaveraged analysis of the modes horizontal force and velocity allows to establish a physical instability criterion for selfpropelled foils. We thus extend this analysis to higher flapping frequencies, where three regimes, quasiperiodic, reversed VonKármán wake and deviated wake propulsion, are obtained. We show that these regimes cannot be explained, as previously, by a fluidsolid Floquet stability of the symmetric nonpropulsive baseflow. These states emergence is thus explained by a nonlinear bifurcation. Using the timespectral method coupled to a pseudo arclength continuation, we reveal that quasiperiodic and deviated wake propulsion appear as instabilities of the reversed VonKármán wake propulsive solutions branch. We equally show that the transition between quasiperiodic propulsion and back and forth is a global bifurcation. The study of the selfpropulsion of the symmetric heaving foil in a quiescent fluid is concluded by a physical analysis of its thrust generation. Decomposing the thrust force into its diffusive and pressure contributions we reveal that for an increasing flapping frequency it exists a transition between diffusion and pressuredriven thrust regimes, the first regime being characterized by no vortex shedding and an asymmetric viscous shear alongside the lateral wall of the foil and the second one by vortex shedding and its resultant trailing edge pressure increase. For large flapping amplitudes, we show that the transition between these regimes is discontinuous, giving rise to the back and forth motion previously studied. The second part of this thesis is dedicated to the collective interactions of an infinite array of heaving wings confined in a channel. Since the channel configuration no longer allow to adopt a noninertial frame of reference to take into account the heaving motion, we adopt a fictitious domain formulation with distributed Lagrange multipliers. To understand the impact of the collective interaction on the array locomotion we have explored the effect of varying a fixed gap between wings and the flapping frequency, while maintaining the wings density ratio, the flapping amplitude and the channel height fixed. For large gaps the interaction effect is barely felt, and the array exhibit the same velocity as a single wing. As the gap is progressively decreased, the array velocity passes through discontinuous branches of stable solutions where it can become superior or inferior than a single wing velocity. We remark, however, that the power input to heave the wing is always inferior when collective interactions are at play. are observe that certain gaps present two coexisting solutions. Their emergence is studied through unsteady simulations with an imposed horizontal velocity to the array. Studying the timeaveraged horizontal force acting on the array wings, we highlight the existence of three rather than two equilibria of the system. The emergence of only two stable selfpropelled states is thus explained by the behaviour of the timeaveraged hydrodynamic force acting on the array that assumes a stabilizing character over two solutions and a destabilizing one in the remaining case.
Master thesis
Pauline Bonnet, Etude d’instabilités aéroélastiques par une méthode aux frontières immergées – cas de l’écoulement confiné autour d’une plaque flexible, Projet de Fin d’Etude, ENSTA ParisTech Université Paris Saclay, MarsAout 2017.
Georg Lopez Fawaz, Linear fluidstructure stability analysis of a flexible foil, Master’s thesis, Ecole Polytechnique, Université ParisSaclay, 28 August 2017.
Johann Moulin, Contrôle d’un écoulement de culot par interaction fluidestructure, Stage de fin d’Etudes, CentraleSupélec, MaiNovembre 2016.
JeanLou Pfister, Quasilinear modelization of the laminar vortexshedding behind twodimensional bluffbodies, defense of the Master's thesis, Ecole Polytechnique, Université ParisSaclay.