## Periodic Reporting for period 3 - AEROFLEX (AEROelastic instabilities and control of FLEXible Structures)

Reporting period: 2018-07-01 to 2019-12-31

Structural stiffness is considered as a crucial aspect in the traditional design of many engineering systems, as aircrafts, bridges, wind mills or offshore drilling risers. The design paradigm is that the structural compliance should be minimized to avoid large deformations induced by hydrodynamic loads. Dynamic structural deformations induced by fluid loads were first ignored, unfortunately leading to dramatic consequence. The accident of the Tacoma Narrows suspension bridge is a famous example of disasters due to a dynamic fluid/structure interaction, known as the flutter phenomenon. Such phenomenon is also well known in aeronautics, where large-amplitude deformations of the wings may occur when the airplane’s speed is increased beyond a critical speed, thus limiting the flight envelope. Although the flutter manifests through large-amplitude structural-deformation, it is due to a linear instability arising from the interaction between structural and fluid dynamics, which both exhibit stable dynamics in the absence of coupling. Infinitesimal displacements of the structure create flow perturbations exerting hydrodynamic forces on the structure that counterbalance the restoring elastic forces, thus leading to an amplification of the structural displacement. If flutter could be controlled at cruise speeds, not only the flight envelope could be extended, but also lighter wings could be designed, thus improving the energetic efficiency of airplanes. In the offshore marine industry, very elongated drilling risers are used to extract oil from the bed sea. Large displacements of the risers are observed as a result of large-scale vortices shed in the wake flow behind the risers. The vortex-induced deformation of the risers is also due to a linear instability arising from the interaction between structural and fluid dynamics, but with a different physical origin: the resonance between two physical oscillators characterized by very close natural frequencies. In all of these examples, the large displacements or deformations of the structures are induced by flow-induced elastic-deformation instabilities and must be avoided, since they are currently limiting the capacities of products in various industrial branches such as aeronautics, marine industry and wind electricity production.

If suppressing flow-induced elastic instabilities is an ultimate goal in most today’s industrial applications, observation of nature strongly suggests that structural compliance can also be an advantage. A first striking example is the static deformation of plants. Under the action of water or wind stream, they tend to passively bend and twist, and thus reduce their hydrodynamic drag load when compared to the rigid-body case. A second striking example is the dynamic small-amplitude deformation of dolphin’s skin that helps them to reduce their skin-friction drag by delaying the laminar-turbulent transition process. A last example is the large-amplitude motion/deformation of appendices used by swimming or flying animals to generate propulsive and lift forces, respectively. For insects, the elastic deformation of their wing is passive but it induces instantaneous redistributions of the aerodynamics forces, which are believed to additionally benefit to the sustentation and propulsion. For fishes, such deformations are active since they use the bending of their backbones to flap their caudal fins and thus swim. Finally, thanks to the elastic property of their body, eels create wave-like movement of their whole body to generate thrust and thus swim. All of these examples highlight that the use of flexibility of structures to decrease their drag seems to be already effective in nature.

The objectives of the AEROFLEX are both physical and methodological. The two physical objectives are (i) the suppression of flow-induced elastic instabilities that are limiting the capacities of products in various industrial configurations and (ii) the use of elastic struc

If suppressing flow-induced elastic instabilities is an ultimate goal in most today’s industrial applications, observation of nature strongly suggests that structural compliance can also be an advantage. A first striking example is the static deformation of plants. Under the action of water or wind stream, they tend to passively bend and twist, and thus reduce their hydrodynamic drag load when compared to the rigid-body case. A second striking example is the dynamic small-amplitude deformation of dolphin’s skin that helps them to reduce their skin-friction drag by delaying the laminar-turbulent transition process. A last example is the large-amplitude motion/deformation of appendices used by swimming or flying animals to generate propulsive and lift forces, respectively. For insects, the elastic deformation of their wing is passive but it induces instantaneous redistributions of the aerodynamics forces, which are believed to additionally benefit to the sustentation and propulsion. For fishes, such deformations are active since they use the bending of their backbones to flap their caudal fins and thus swim. Finally, thanks to the elastic property of their body, eels create wave-like movement of their whole body to generate thrust and thus swim. All of these examples highlight that the use of flexibility of structures to decrease their drag seems to be already effective in nature.

The objectives of the AEROFLEX are both physical and methodological. The two physical objectives are (i) the suppression of flow-induced elastic instabilities that are limiting the capacities of products in various industrial configurations and (ii) the use of elastic struc

The AEROFLEX aims at controlling flow-induced structural instabilities, which are linear instabilities arising when a rigid or an elastic structure is immersed in a viscous incompressible flow. The work performed so far is described in the following three sections: the first section describes the mathematical and numerical modelling of flow-induced structural instabilities, the second section details results obtained for various fluid-structure configurations and the third section shows the stabilization of flow-induced structural instabilities using various physical strategies.

1. Mathematical and numerical modelling of flow-induced structural instabilities:

The mathematical description of flow-induced structural instabilities is revisited by linearizing the non-linear equations governing the dynamics of elastic material modelled by the Saint-Venant Kirchhoff law and interacting with a viscous Newtonian fluid, modelled by the incompressible Navier-Stokes equations when considering laminar flow regimes. To cope with the coupling between the Eulerian and Lagrangian descriptions of the fluid and solid dynamics respectively, two different formulation are considered: the Fictitious Domain formulation and the Arbitrary Lagrangian formulation. In addition to the modal analysis classically considered when analyzing flow-induced structural instabilities of bluff-bodies, a original resolvent analysis is also developed to describe the interaction of spatially-growing flow instabilities interacting with a elastic structure. Finally, numerical methods and tools have been specificaly developed to compute the steady-state solution and compute the eigenmodes of insterest in large-scale problems.

Fictitious Domain versus Arbitrary Lagrangian Eulerian formulations:

First, a Fictitious Domain formulation of elastic structures immersed in incompressible flows is considered to obtain the linear equations governing the fluid-structure perturbation around a steady state. In that formulation, the spatial domain occupied by the solid is filled in with a fictitious fluid so that the governing fluid equations are written in a spatial domain independent of time. The presence of the solid is taken into account in the fictitious fluid equations by introducing distributed forcing term, determined so as to respect the equality between the fluid and the solid velocities. In the Fictitious Domain formulation, the fluid and structure boundaries do not necessarily conform. Secondly, Arbitrary Lagrangian Eulerian (ALE) formulation of the fluid-structure problems allows for a conformal description of the fluid-structure interface. Indeed, the solid displacement at the interface is artificially propagated into the spatial domain occupied by the fluid, so as to deform it adequately. The linearization of the ALE formulation has been handled by considering two decompositions of the fluid variables, yielding to two different linear formulations of the problem. The first linear formulation governs the perturbation of the Lagrangian fluid variables. It yields to the linearized ALE equations that include additional terms in the governing fluid equations, proportional to the extension displacement, which is a non-physical variable introduced in the formalism to deform the fluid domain. In the Lagrangian perturbation approach, the fluid equations are more complex than in the purely hydrodynamic case, and the size of the computational fluid-structure problem is larger as it involves an additional variable. The second linear formulation governs the perturbation of the Eulerian fluid variables and is attractive at first glance, as it is independent from the (non-physical) extension displacement, thus reducing the complexity of the linearized fluid equations. A comparison of numerical results obtained with the linearized Ficitious Domain formulation and the two linearized Arbitrary Eulerian Lagrangian (for the Lagrangian and Eulerian perturbations) has been per

1. Mathematical and numerical modelling of flow-induced structural instabilities:

The mathematical description of flow-induced structural instabilities is revisited by linearizing the non-linear equations governing the dynamics of elastic material modelled by the Saint-Venant Kirchhoff law and interacting with a viscous Newtonian fluid, modelled by the incompressible Navier-Stokes equations when considering laminar flow regimes. To cope with the coupling between the Eulerian and Lagrangian descriptions of the fluid and solid dynamics respectively, two different formulation are considered: the Fictitious Domain formulation and the Arbitrary Lagrangian formulation. In addition to the modal analysis classically considered when analyzing flow-induced structural instabilities of bluff-bodies, a original resolvent analysis is also developed to describe the interaction of spatially-growing flow instabilities interacting with a elastic structure. Finally, numerical methods and tools have been specificaly developed to compute the steady-state solution and compute the eigenmodes of insterest in large-scale problems.

Fictitious Domain versus Arbitrary Lagrangian Eulerian formulations:

First, a Fictitious Domain formulation of elastic structures immersed in incompressible flows is considered to obtain the linear equations governing the fluid-structure perturbation around a steady state. In that formulation, the spatial domain occupied by the solid is filled in with a fictitious fluid so that the governing fluid equations are written in a spatial domain independent of time. The presence of the solid is taken into account in the fictitious fluid equations by introducing distributed forcing term, determined so as to respect the equality between the fluid and the solid velocities. In the Fictitious Domain formulation, the fluid and structure boundaries do not necessarily conform. Secondly, Arbitrary Lagrangian Eulerian (ALE) formulation of the fluid-structure problems allows for a conformal description of the fluid-structure interface. Indeed, the solid displacement at the interface is artificially propagated into the spatial domain occupied by the fluid, so as to deform it adequately. The linearization of the ALE formulation has been handled by considering two decompositions of the fluid variables, yielding to two different linear formulations of the problem. The first linear formulation governs the perturbation of the Lagrangian fluid variables. It yields to the linearized ALE equations that include additional terms in the governing fluid equations, proportional to the extension displacement, which is a non-physical variable introduced in the formalism to deform the fluid domain. In the Lagrangian perturbation approach, the fluid equations are more complex than in the purely hydrodynamic case, and the size of the computational fluid-structure problem is larger as it involves an additional variable. The second linear formulation governs the perturbation of the Eulerian fluid variables and is attractive at first glance, as it is independent from the (non-physical) extension displacement, thus reducing the complexity of the linearized fluid equations. A comparison of numerical results obtained with the linearized Ficitious Domain formulation and the two linearized Arbitrary Eulerian Lagrangian (for the Lagrangian and Eulerian perturbations) has been per

The mathematical description of fluid-structure instabilities has been revisited without the classical assumptions made in traditional aeroelasticity. The general approach developed in the project is to consider the non-linear equations governing the full coupled fluid-structure dynamics, using various models for the fluid and the structure, and to linearize them around steady solutions. Although such approach is well established when investigating hydrodynamics instabilities, it is much less standard when considering hydro/aero-elastic instabilities. Indeed, the starting point of a classical flutter analysis is rather the structural equations where the fluid forces act through excitation terms. In the classical approach, the fluid variables are eliminating ab-initio from the mathematical problem and the aerodynamic effect is reduced to an excitation term in the governing equations. This excitation can be either modelled or estimated with computational fluid dynamics by determining the linear flow response to imposed structural displacements of the structure, given by the free-vibration eigenmodes of the structure. This traditional approach suffers from at least two well-known drawbacks. First, it neglects the fluid growth/damping when determining the linear flow response, so that results of this approach are only valid for the critical parameters at which the growth rate vanishes. Secondly, in case of elastic structures, it neglects the deformation of the fluid/solid boundary when evaluating the aerodynamics stresses of the linear flow response. The mathematical modelling that consists in considering ab-initio the fully coupled fluid-structure problem overcomes these two issues. The two linearized Arbitrary Lagrangian Eulerian (ALE) formulations, governing the Lagrangian and Eulerian perturbations, had been derived in previous studies, but their spatial discretization had never been tested and compared on relevant fluid-structure configurations. I the project, the incompressible Navier-Stokes equations and the linear elasticitiy equations governing the fluid-and structure dynamics have been both discretized with a finite element approach and using advanced parallel preconditionners for solving efficiently the large-scale linear equations arising when computing steady-state solutions and eigenvalue problems with a shift-and-invert strategy. A numerical comparison of these formulations has been performed on a model problem, an elastic splitter plate clamped on a rigid cylinder, that exhibits unstable eigenmodes of various physical origin when varying the Reynolds number and plate's stiffness. The Eulerian perturbation formulation yields unconverged numerical results , unlike the Lagrangian perturbation formulation. For that reason, the latter formulation is favored in the project.

Based on these results, control strategies aiming at stabilizating the unstable eigenmodes have been designed, either by optimizing the shape of the rigid support, or by introducing shunted piezoelectric patches on the splitter plate to change its local stiffness. In the first case, the minimisation algorithm is based on the knowledge of adjoint steady states and fluid-structure eigenmodes. These adjoint fluid-structure eigenmodes have also been used to determine the physical origin of fluid-structure instabilities, using a specific overlapping of the direct and adjoint eigenmodes. By performing an asymptotic developement of the eigenvalue problem close to the neutral curves, we have shown how to identifiy the added-stiffness mechanism at the origin of an unstable static divergence eigenmode. Vortex-induced vibrations, coupled mode flutter, static divergence or stall flutter are flow-induced instabilities that arise spontaneously when coupling a fluid and a structure. These self-sustained flow-induced structural instabilities are investigated by examining the long-term temporal behavior of the linear perturbation. This temporal behaviour is convenie

Based on these results, control strategies aiming at stabilizating the unstable eigenmodes have been designed, either by optimizing the shape of the rigid support, or by introducing shunted piezoelectric patches on the splitter plate to change its local stiffness. In the first case, the minimisation algorithm is based on the knowledge of adjoint steady states and fluid-structure eigenmodes. These adjoint fluid-structure eigenmodes have also been used to determine the physical origin of fluid-structure instabilities, using a specific overlapping of the direct and adjoint eigenmodes. By performing an asymptotic developement of the eigenvalue problem close to the neutral curves, we have shown how to identifiy the added-stiffness mechanism at the origin of an unstable static divergence eigenmode. Vortex-induced vibrations, coupled mode flutter, static divergence or stall flutter are flow-induced instabilities that arise spontaneously when coupling a fluid and a structure. These self-sustained flow-induced structural instabilities are investigated by examining the long-term temporal behavior of the linear perturbation. This temporal behaviour is convenie