## PhD thesis

**Jean-Lou 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 added-mass effects are at play. Besides running unsteady non-linear simulations, linearised modal approaches are useful to identify hydro-elastic 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 fluid-solid 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 fluid-solid perturbations around non-linear steady states. The fluid dynamics is governed by the incompressible Navier-Stokes equations, while the solid is described by hyperelastic models. An Arbitrary Lagrangian Eulerian coupling is chosen, resulting in a conformal description of the fluid-solid interface in a time-independent reference configuration. An exact linearisation of this formulation is derived, and two analyses of the resulting fully coupled, linearised fluid-solid operator are considered. An eigenvalue analysis allows to determine self-sustained fluid-solid instabilities responsible, for instance, for the vortex-induced vibrations of bluff bodies or the flutter of slender bodies. The resolvent analysis, i.e. a singular value analysis of the fluid-solid operator, allows to determine the linear response of the fluid-solid 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 vortex-induced 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 symmetry-breaking divergence instability. Non-linear steady and unsteady simulations are performed to elucidate the non-linear interactions between the unstable modes. Secondly, an adjoint-based 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 D-shaped cylinders, while a frequency increase is obtained with C-shaped cylinders. The last part of the thesis is dedicated to the delay of laminar/turbulent transition in twodimensional boundary-layer flows thanks to visco-elastic, finite-length coatings. A resolvent analysis of the fluid-solid operator is used to quantify the attenuation of low-frequency Tollmien-Schlichting instability waves when the stiffness of the coating is reduced. On the other hand, the eigenvalue analysis shows that high-frequency solid-based modes are destabilized when the solid viscous damping is too low. A gradient-based 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 Tollmien-Schlichting waves and to limit the development of solid-based 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 spring-mounted plate, immersed in a two-dimensional laminar flow modeled by the incompressible Navier-Stokes equations. A semi-analytical 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) highly-nonlinear periodic flutter solutions. The challenging task of solving the TSM equations, especially when large numbers of Fourier harmonics are considered, is tackled via a time-parallel Newton-Krylov approach in combination with a new, so-called block-circulant 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 spring-mounted plate. We start by revisiting the linear stability problem using a Navier-Stokes fluid model allowing to highlight, in particular, the effect of viscosity. Comparisons to classical quasi-steady and unsteady potential flow (Theodorsen model) theories are performed. Contrary to what happens in potential flows, the flutter instability is shown to re-stabilize at very high reduced velocities in viscous flows. We continue our route on the flutter bifurcation by investigating the effect of fluid nonlinearities. Low solid-to-fluid mass ratios and increasing Reynolds numbers foster subcritical bifurcations. The role of leading-edge shear layers is pointed out. For intermediate mass ratios, an unusual bifurcation scenario that combines a supercritical bifurcation and the existence of subcritical high-amplitude flutter solutions is discovered. We conclude our study of the flutter bifurcation by investigating the appearance of low-frequency amplitude modulations on top of a previously established periodic flutter solution. Using an original TSM-based Floquet stability analysis, we explain this behavior by the destabilization of the periodic solutions by a pair of complex-conjugate 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 large-scale three-dimensional configurations. As a first step towards this long-term goal, we develop an open-source massively parallel tool, based on the FreeFEM library and its PETSc/SLEPc interface, able to compute the nonlinear steady-state flow and subsequently solve the linear stability eigenproblem, for three-dimensional flows (the structure is fixed) possessing several tens of millions of degrees of freedom.

**Luis BENETTI RAMOS,**Self-propulsion and fluid-mediated 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 time-reciprocal motions allow to achieve locomotion and collective dynamics, such as bird flocks and fish schools, become possible through fluid-mediated 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/ self-propelled wing system. The first part of this thesis is consecrated to study the horizontal self-propulsion of a symmetric heaving foil in a two-dimensional quiescent fluid. The problem is investigated numerically based on the resolution of the Navier-Stoles equations, written in a non-inertial frame of reference that follows the foil centre of gravity, coupled to the foil horizontal acceleration. At first, we investigate the emergence of self-propelled regimes through unsteady nonlinear simulations, adopting a fixed density ratio and flapping amplitude while varying the flapping frequency. At low flapping frequencies, two self-propelled states are analysed: a periodic state of unidirectional propulsion and a quasi-periodic state of slow back and forth motion around a fixed point. These states emergence is explained through a fluid-solid Floquet stability analysis of non-propulsive symmetric base-flows. 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 time-averaged analysis of the modes horizontal force and velocity allows to establish a physical instability criterion for self-propelled foils. We thus extend this analysis to higher flapping frequencies, where three regimes, quasi-periodic, reversed Von-Kármán wake and deviated wake propulsion, are obtained. We show that these regimes cannot be explained, as previously, by a fluid-solid Floquet stability of the symmetric non-propulsive base-flow. These states emergence is thus explained by a nonlinear bifurcation. Using the time-spectral method coupled to a pseudo arc-length continuation, we reveal that quasi-periodic and deviated wake propulsion appear as instabilities of the reversed Von-Kármán wake propulsive solutions branch. We equally show that the transition between quasi-periodic propulsion and back and forth is a global bifurcation. The study of the self-propulsion 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 pressure-driven 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 non-inertial 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 time-averaged 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 self-propelled states is thus explained by the behaviour of the time-averaged 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, Mars-Aout 2017. **Georg Lopez Fawaz**, Linear fluid-structure stability analysis of a flexible foil, Master’s thesis, Ecole Polytechnique, Université Paris-Saclay, 28 August 2017. **Johann Moulin**, Contrôle d’un écoulement de culot par interaction fluide-structure, Stage de fin d’Etudes, CentraleSupélec, Mai-Novembre 2016.

**Jean-Lou Pfister, **Quasi-linear modelization of the laminar vortex-shedding behind two-dimensional bluff-bodies, defense of the Master's thesis, Ecole Polytechnique, Université Paris-Saclay.