**Marquet O., Leontini J.S., Zhao J. & Thompson M.C (2022).**Hysteresis of two-dimensional flows around a NACA0012 airfoil at Re=5000 and linear analyses of their mean flow, accepted for publication in*International Journal of Heat and Fluid Flow*. The published version is temporally available here.

**Abstract: **Two-dimensional numerical simulations of the flow around a NACA0012 profile at Reynolds number 5000 show that unsteady periodic flows reach different saturated states when increasing or decreasing the angle of attack between 7° and 8° . Within this range, the lift signal shows co-existing periodic states and period-doubling, as the wake undergoes a substantial change in character from the standard von-Kármán vortex street. Results of experiments in a water channel also indicate a change of the flow topology but at slightly lower angles of attack 6°. A discussion of the discrepancy between numerical and experimental results is proposed in light of results about the three-dimensional transition of wake flows behind bluff bodies and airfoils. Finally, eigenvalue and resolvent analyses of time-averaged flows are used to investigate the two-dimensional transitions further. While a peak of energetic amplification is obtained at the frequency of a single periodic state, a double peak is observed for co-existing periodic states, the second one being at the frequency of the periodic state not used to compute the time-averaged flow. This behaviour also characterizes the resolvent analysis of the period-doubled states, although less pronounced.

**Pfister J.-L., Fabbiane N. & Marquet O.**Global stability and resolvent analyses for characterizing the attenuation of boundary-layer flow instabilities by visco-elastic patches, accepted for publication in*Journal of Fluid Mechanics.*

**Abstract:** The attenuation of two-dimensional, boundary-layer instabilities by a finite-length, visco-elastic patch is investigated by means of global linear stability theory. First, the modal stability properties of the coupled problem are assessed, revealing unstable fluid-elastic travelling-wave flutter modes. Second, the Tollmien-Schlichting instabilities over a rigid-wall are characterised via the analysis of the fluid resolvent operator in order to determine a baseline for the fluid-structural analysis. To investigate the effect of the elastic patch on the growth of these flow instabilities, we first consider the linear frequency response of the coupled fluid-elastic system to the dominant rigid-wall forcing modes. In the frequency range of Tollmien-Schlichting waves, the energetic flow amplification is clearly reduced. However, an amplification is observed for higher-frequencies, associated to travelling wave flutter. This increased complexity requires the analysis of the coupled fluid-structural resolvent operator; the optimal, coupled, resolvent modes confirm the attenuation of the Tollmien-Schichting instabilities, while also being able to capture the amplification at the higher frequencies. Finally, a decomposition of the fluid-structural response is proposed to reveal the wave cancellation mechanism responsible for the attenuation of the Tollmien-Schlichting waves. The visco-elastic patch, excited by the incoming rigid-wall wave, provokes a fluid-elastic wave that is out-of-phase with the former, thus reducing its amplitude.

**Moulin J. & Marquet O. (2021).**Flow-induced instabilities of springs-mounted plates in viscous flows: a global stability approach, Physics of Fluids**33**, 034133*.*

**Abstract: **The linear stability of a typical aeroelastic section, consisting in a rectangular plate mounted on flexion andtorsion springs, is revisited here for low-Reynolds-number incompressible flows. By performing global stability analyses of the coupled fluid-solid equations, we find four types of unstable modes related to different physicalinstabilities and classically investigated with separate flow models: coupled-mode flutter, single-mode flutterand static divergence at high reduced velocity U∗ and vortex-induced vibrations at low U∗. Neutral curves forthese modes are presented in the parameter space composed of the solid-to-fluid mass ratio and the reducedvelocity. Interestingly, the flutter mode is seen to restabilize for high reduced velocities thus leading to afinite extent flutter region, delimited by low-U∗ and high-U∗ boundaries. At a particular low mass ratio, bothboundaries merge such that no flutter instability is observed for lower mass ratios. The effect of the Reynolds number is then investigated, indicating that the high-U∗ restabilization strongly depends on viscosity. The global stability results are compared to a statically calibrated Theodorsen model: if both approaches convergein the high mass ratio limit, they significantly differ at lower mass ratios. In addition, the Theodorsen modelfails to predict the high-U∗ restabilization of the flutter mode.

**Benetti Ramos L., Marquet O., Bergmann M. & Iollo A. (2021).**Fluid-solid Floquet stability analysis of self-propelled heaving foils,*in Journal of Fluid Mechanics, vol 910, 10 March 2021, A28.*

**Abstract:** We investigate the role of linear mechanisms in the emergence of nonlinear horizontal self-propelled states of a heaving foil in a quiescent fluid. Two states are analyzed: a periodic state of unidirectional motion and a quasi-periodic state of slow back & forth motion around a mean horizontal position. The states emergence is explained through a fluid-solid Floquet stability analysis of the non-propulsive symmetric base solution. Unlike a purely-hydrodynamic analysis, our analysis accurately determine the locomotion states onset. An unstable synchronous mode is found when the unidirectional propulsive solution is observed. The obtained mode has a propulsive character, featuring a mean horizontal velocity and an asymmetric flow that generates a horizontal force accelerating the foil. An unstable asynchronous mode, also featuring flow asymmetry and a non-zero velocity, is found when the back & forth state is observed. Its associated complex multiplier introduces a slow modulation of the flapping period, agreeing with the quasi-periodic nature of the back & forth regime. The temporal evolution of this perturbation shows how the horizontal force exerted by the flow is alternatively propulsive or resistive over a slow period. For both modes, an analysis of the velocity and force perturbation time-averaged over the flapping period is used to establish physical instability criteria. The behaviour for large solid-to-fluid density ratio of the modes is thus analyzed. The asynchronous fluid-solid mode converges towards the purely-hydrodynamic one, whereas the synchronous mode becomes marginally unstable in our analysis not converging to the purely-hydrodynamic analysis where it is never destabilised

**Pfister, J.-L. and Marquet O. (2020)**Fluid structure stability analyses and nonlinear dynamics of flexible splitter plates interacting with a circular cylinder flow.*Journal of Fluid Mechanics*, vol. 896.

**Abstract:** The dynamics of a hyperelastic splitter plate interacting with the laminar wake flow of a circular cylinder is investigated numerically at a Reynolds number of 80. By decreasing the plate’s stiffness, four regimes of flow-induced vibrations are identified: two regimes of periodic oscillation about a symmetric position, separated by a regime of periodic oscillation about asymmetric positions, and finally a regime of quasi-periodic oscillation occurring at very low stiffness and characterized by two fundamental (high and low) frequencies. A linear fully coupled fluid–solid analysis is then performed and reveals the destabilization of a steady symmetry-breaking mode, two high-frequency unsteady modes and one low-frequency unsteady mode, when varying the plate’s stiffness. These unstable eigenmodes explain the emergence of the nonlinear self-sustained oscillating states and provide a good prediction of the oscillation frequencies. A comparison with nonlinear calculations is provided to show the limits of the linear approach. Finally, two simplified analyses, based on the quiescent-fluid or quasi-static assumption, are proposed to further identify the linear mechanisms at play in the destabilization of the fully coupled modes. The quasi-static static analysis allows an understanding of the behaviour of the symmetry-breaking and low-frequency modes. The quiescent-fluid stability analysis provides a good prediction of the high-frequency vibrations, unlike the bending modes of the splitter plate in vacuum, as a result of the fluid added-mass correction. The emergence ofthe high-frequency periodic oscillations can thus be predicted based on a resonance condition between the frequencies of the hydrodynamic vortex-shedding mode and of the quiescent-fluid solid modes.

**Pfister, J.-L. ,Carini, M. & Marquet, O. (2019)**, Linear stability analysis of strongly coupled fluid-structure problems with the Arbitrary-Lagrangian-Eulerian method.*Computer Methods in Applied Mechanics and Engineering, 355,*663-689

**Abstract**: The stability analysis of elastic structures strongly coupled to incompressible viscous flows is investigated in this paper,based on a linearization of the governing equations formulated with the Arbitrary-Lagrangian–Eulerian method. The exact linearized formulation, previously derived to solve the unsteady non-linear equations with implicit temporal schemes, is used here to determine the physical linear stability of steady states. Once discretized with a standard finite-element method based on Lagrange elements, the leading eigenvalues/eigenmodes of the linearized operator are computed for three configurations representative for classical fluid–structure interaction instabilities: the vortex-induced vibrations of an elastic plate clamped to the rear of a rigid cylinder, the flutter instability of a flag immersed in a channel flow and the vortex shedding behind a three-dimensional plate bent by the steady flow. The results are in good agreement with instability thresholds reported in the literature and obtained with time-marching simulations, at a much lower computational cost. To further decrease this computational cost, the equations governing the solid perturbations are projected onto a reduced basis of free-vibration modes. This projection allows to eliminate the extension perturbation, a non-physical variable introduced in the ALE formalism to propagate the infinitesimal displacement of the fluid–solid interface into the fluid domain.

**Computer codes** are available here: /erc-aeroflex/sites/default/files/2023-10/pfister_carini_marquet_cmame_2019.tar.gz

**Moulin, J., Jolivet, P., & Marquet, O. (2019).**Augmented Lagrangian preconditioner for large-scale hydrodynamic stability analysis.*Computer Methods in Applied Mechanics and Engineering*,*351*, 718-743.

**Abstract**: Hydrodynamic linear stability analysis of large-scale three-dimensional configurations is usually performed with a“time-stepping” approach, based on the adaptation of existing solvers for the unsteady incompressible Navier–Stokes equations. We propose instead to solve the nonlinear steady equations with the Newton method and to determine the largest growth-rate eigenmodes of the linearized equations using a shift-and-invert spectral transformation and a Krylov–Schur algorithm. The solution of the shifted linearized Navier–Stokes problem, which is the bottleneck of this approach, is computed via an iterative Krylov subspace solver preconditioned by the modified augmented Lagrangian (mAL) preconditioner (Benzi et al., 2011). The well-known efficiency of this preconditioned iterative strategy for solving the real linearized steady-state equations is assessed here for the complex shifted linearized equations. The effect of various numerical and physical parameters is investigated numerically on a two-dimensional flow configuration, confirming the reduced number of iterations over state-of-the-art steady-state and time-stepping-based preconditioners. A parallel implementation of the steady Navier–Stokes and eigenvalue solvers, developed in the FreeFEM language, suitably interfaced with the PETSc/SLEPc libraries, is described and made openly available to tackle three-dimensional flow configurations. Its application on a small-scale three-dimensional problem shows the good performance of this iterative approach over a direct LU factorization strategy, in regards of memory and computational time. On a large-scale three-dimensional problem with 75 million unknowns, a 80% parallel efficiency on 256 up to 2048 processes is obtained.

**Computer codes** are available here: https://github.com/prj-/moulin2019al

**Marquet O. & Lesshafft L. (2015),**Identifying the active flow regions that drive linear and nonlinear instabilities, arXiv:1508.07620 [physics.flu-dyn]

**Abstract: **A new framework for the analysis of unstable oscillator flows is explored. In linear settings, temporally growing perturbations in a non-parallel flow represent unstable eigenmodes of the linear flow operator. In nonlinear settings, self-sustained periodic oscillations of finite amplitude are commonly described as nonlinear global modes. In both cases the flow dynamics may be qualified as being endogenous, as opposed to the exogenous behaviour of amplifier flows driven by external forcing. This paper introduces the endogeneity concept, a specific definition of the sensitivity of the global frequency and growth rate with respect to variations of the flow operator. The endogeneity, defined both in linear and nonlinear settings, characterizes the contribution of localized flow regions to the global eigendynamics. It is calculated in a simple manner as the local point-wise inner product between the time derivative of the direct flow state and an adjoint mode. This study demonstrates for two canonical examples, the Ginzburg-Landau equation and the wake of a circular cylinder, how an analysis based on the endogeneity may be used for a physical discussion of the mechanisms that drive a global instability. The results are shown to be consistent with earlier 'wavemaker' definitions found in the literature, but the present formalism enables a more detailed discussion: a clear distinction is made between oscillation frequency and growth rate, and individual contributions from the various terms of the flow operator can be isolated and separately discussed

**SUBMITTED:**

**IN PREPARATION:**

**Moulin J. & Marquet O.**Low-frequency amplitude modulation of flutter oscillations: a Floquet analysis based on the Time Spectral Method, in preparation for Journal of Fluid Mechanics.

**Abstract:** The flutter oscillations of a thin plate mounted on a system of bending/torsional springs and immersed in a laminar incompressible flow are investigated numerically for the low Reynolds number $\textit{Re}=500$ and the large solid-to-fluid mass ratio $\mtildebis=1000$. Time-marching simulations of the fluid-solid interaction first show that, when increasing the reduced velocity above the critical value $\Uscrit$, periodic oscillations of the plate are first observed, resulting from a \textit{primary} flutter instability of the steeady solution. For these low-frequency oscillations ($\omega_{0}\sim 0.15$), the flow is quasi-steady and remains attached to the plate during (almost) the whole period. For slightly larger reduced velocity, a very-low-frequency modulation of the pitching angle is then observed, associated to stronger flow separation occurring when the pitching angle oscillates between larger values. Drawing a Poincaré map clearly indicates that the quasi-periodic solution is a torus attractor. To explain the emergence of the quasi-periodic solutions, a secondary instability analysis is then achieved, based on a Floquet analysis that fully relies on the Time Spectral Method, not only to compute the unstable periodic limit cycle oscillations, but also to determine the leading Floquet modes. We thus show that an asynchronous Floquet mode gets unstable for values of the reduced velocity where quasi-periodic solutions are observed. Their very-low frequency $\omega$ is well predicted by the Floquet analysis, especially slightly above the secondary critical velocity. An analysis of the pitching and heaving component of the complex Floquet mode shows that they are almost in phase for their real part, but out-of-phase for the imaginary part. A spectral analysis of these Floquet components further reveals that the pitching angle predominantly oscillates at a slightly higher frequency $\omega_0+\omega$) than the heaving displacement which oscillates (predominantly) at $\omega_0-\omega$, in agreement with results of nonlinear simulations. This explains that the phase difference between the pitching and heaving signal continuously drifts during the very-slow oscillation. The recontruction of this oscillating components allows to better understand the physical origin of the very-low-frequency modulation. When the pitching (resp. heaving) motion precedes the heaving (resp. pitching) motion, energy is extracted from (resp. transmitted to) the flow and the plate exhibits a flutter (resp. anti-flutter) motion.

**Moulin J. & Marquet O.**Hard and soft flutter of thin plates in laminar incompressible flows: weakly and strongly nonlinear analyses, in preparation for Journal of Fluid Mechanics.

**Abstract:** We investigate numerically the role of incompressible flow nonlinearities on the periodic flutter of a thin plate mounted on a system of bending/torsion linear springs located at its center of mass. The steady flow solution gets unstable to linear flutter eigenmodes at a critical reduced velocity. Close to that threshold, limit cycle oscillations of the plate appear for lower and upper reduced velocity, depending on the nature of the bifurcation. A weakly nonlinear analysis is first developed to compute the coefficients of the cubic amplitude equations that determine the subcritical or supercritical nature of the bifurcation close to the threshold. A parametric investigation of the solid-to-fluid mass ratio and Reynolds number shows that the bifurcation is supercritical (soft flutter) at low Reynolds numbers $\textit{Re}<90$ independently of the mass ratio, and gets subcritical for intermediate Reynolds number $90 < \textit{Re} < 2000$ and low mass ratio $\mtildebis < 100$. For larger values of the Reynolds number $2000 < \textit{Re} < \textit{Re}_{w}$, that remains bounded by the critical Reynolds number for the onset of vortex-shedding, the bifurcation is subcritical (hard flutter) independently of the mass ratio. The bifurcation scenarii are further investigated at the Reynolds number $\textit{Re}=500$ with a Time Spectral Method allowing to compute accurately periodic solutions with large amplitude oscillations. The transition from a supercritical to a subcritical bifurcation when decreasing the mass ratio is thus scrutinized, thus revealing a double-fold bifurcation scenario at intermediate mass ratio. The bifurcation is supercritical, as shown by the weakly nonlinear analysis, but a fold bifurcation of limit cycle solutions occurs slightly above the critical reduced velocity, leading to (unstable) limit cycle oscillations at lower reduced velocity. The second fold of periodic solutions then lead to the branch of large amplitude oscillations that are observed in time marching simulations. The double fold bifurcation is finally discussed in light of experimental results by Amandolese et al. (2016)

**Benetti Ramos L., Marquet O. & Bergmann M.**Transition between diffusion- and pressure-driven thrust of self-propelled heaving foils, in preparation for Journal of Fluid Mechanics.

**Abstract:** Flapping propulsion is a locomotion strategy adopted by living organisms whose thrust origin is normally associated to fluid acceleration rather than viscous friction. Studies show, however, that the latter diffusive forces might still play an important role on its thrust generation. In this work, we address this issue studying the diffusion and pressure time-averaged contributions to the thrust generated by a horizontally self-propelled heaving foil immersed in a quiescent fluid. Using numerical simulations we show that while increasing the flapping frequency or amplitude this system transition between two distinct thrust regimes. In the first one the time-averaged thrust is driven by viscous diffusion, with forces generated by the asymmetric shear on the foil lateral surface whereas in the second one the thrust is driven by the trailing edge pressure increase, a consequence of the fluid acceleration behind the foil. We finally study the effect of flapping amplitude and thickness-to-chord aspect ratio over these thrust regimes, highlighting that the diffusion-driven thrust regime is enhanced for smaller aspect ratios and that the transition between both regimes takes place for a constant Stokes number $\beta_A=fAc/\nu \approx 10$ based on $A$ and $f$ the flapping amplitude and frequency, $c$ the foil chord and $\nu$ the fluid viscosity.

**Benetti Ramos L., Marquet O., Bergmann M. & Iollo A.**Local and global bifurcations of a self-propelled heaving foil at high flapping frequencies, in preparation for Journal of Fluid Mechanics.

**Abstract:** When a foil is heaved along its symmetry axis and starts propelling in the orthogonal direction due to its interaction with the induced flow, complex self-propelled states may appear, as for instance a slow non-coherent back and forth motion of the foil at intermediate flapping frequency. We focus here on self-propelled states appearing for higher flapping frequencies, and, using time marching simulations, we report the existence of a new quasi-periodic self-propelled state when slightly increasing the frequency. Its propulsive wake does not only oscillate at the flapping frequency, but also slowly deviates upward and downward. When further increasing the frequency, the quasi-periodic oscillation disappears and a periodic and symmetric propulsive state is first obtained, followed by a permanently deviated propulsive state. To understand the emergence of these states, a time spectral method coupled to a pseudo arc-length continuation method is first used to follow the branch of periodic and symmetric propulsive solutions. It shows that, at high flapping frequency where the deviated propulsive solution is observed, this branch still exists, while a saddle-node bifurcation of periodic state occurs at lower frequency. The linear stability of these states is then investigated by performing a fluid-solid Floquet analysis. It reveals the existence of synchronous and asyncrhonous Floquet modes, both related to displacement of the wake vortices, that get unstable precisely when the periodic and quasi-periodic propulsive solutions are observed, respectively. If the transition from the symmetric propulsive solution to these two regimes is local in the sense of bifurcation analysis, the transition between the back \& forth and quasi-periodic propulsive turns out to be global. By analyzing the evolution of this dynamical system through its phase space representation, we finally show that a collision between the two regimes occurs as it approaches the saddle-node bifurcation of the periodic symmetric branch.

**Pfister J.-L. & Marquet O.**Shape optimization of a rigid cylinder for controlling flow-induced instabilities of a flexible splitter plate.

**Abstract:** The oscillations of a flexible splitter plate interacting with the wake flow behind a rigid cylinder result from the destabilization of fluid-elastic global modes (Pfister & Marquet, JFM2020). An adjoint-based shape-optimization of the rigid cylinder is developed here, aiming at varying the eigenvalue associated to such fluid-elastic global mode. Two components are identified in the shape sensitivity function. The perturbative component accounts for the shape deformation in the unsteady equations governing the linear perturbation that develop around the steady base-flow. The base-flow component accounts for the shape deformation in the steady flow equations, that induces base-flow modification in the linear perturbation equations. The shape sensitivity functions, related to the growth rate and frequency of the unstable fluid-elastic global mode, are first discussed. For the frequency, the shape sensitivity is strongly dominated by the base-flow component and indicates that the rigid cylinder should be slendered to increase the oscillating frequency of the flexible plate. For the growth rate, the two components are of similar amplitude but of opposite trends, resulting in bell shapes to stabilize the fluid-elastic mode. Using this shape sensitivity functions, optimizations of the rigid cylinder's shape are then performed to control the flexible plate oscillation. When targeting independently a prescribed growth rate or frequency, very similar results are obtained. Slendering the rigid cylinder tends to stabilize the fluid-solid eigenmode and increase its frequency. With the objective of controlling the oscillation frequency of the flexible plate, we perform a shape optimization for an objective function minimizing the gap of both growth rate and frequency to target values. The shape of the rigid bodies leading to higher and lower frequency oscillations of the flexible plate are finally discussed and results are compared to time-marching simulations.

**Pfister J.-L., Allandrieu R., Marquet O. & Couliou M.**Symmetry breaking of flexible splitter plates: experiments and quasi-steady stability analysis, in preparation for Journal of Fluid Mechanics.

**Abstract:** The dynamics of elastic splitter plates interacting with the wake flow of a circular cylinder is investigated experimentally at the Reynolds number 350. The deviation of the plate and the oscillation frequency are discussed for a large panel of splitter plate length. By decreasing the plate’s length, three regimes of flow-induced vibrations are identified: two regimes of periodic oscillation about a symmetric position, separated by a regime of periodic oscillation about devaited positions. For splitter plate under a critical length, we observe a new re stabilisation of the plate with a zero mean deviation. A quasi-steady analysis is used to identify the linear mechanisms at play in the destabilization. The quasi-static static analysis allows a novel explanation the different symmetry breakings of flexible splitter plate. This approach proved itself to be particularly relevant for the static analysis of the phenomenon, and predicts the stationary instability affecting the average position of the filament with precision. Unlike existing models, it provides a good prediction of the short plate deviation.

**Leclercq T. & Marquet O.**Optimal control of ampifiers flows by feedforward wave cancellation, in preparation.

**Abstract:** We derive the formulation for the problem of the optimal control of spatially-developping amplifier fows and investigate its performances and characteristics on the example of the two-dimensional instability of the Blasius boundary layer. Our optimization is performed based on an input-output representation of the system in the frequency domain, and it is designed to minimize the optimal gain of the global resolvant operator by means of actuation of the flow at the wall boundary. Our formulation of the problem provides a feedforward optimal control law that inputs the incident exogenous perturbation at the upstream boundary of the computational domain, instead of a feedback law that inputs the flow state. As such, our strategy does not target the properties (eigenvalues, non-normality) of the linearized Navier-Stokes operator as feedback loops would. It does on the other hand affect the resolvant operator, by tuning the forcing terms at the wall boundary in order to produce an additional flow disturbance that best cancel out the effect of the incident perturbation. In this sense, our controller truly performs the optimal wave cancellation.