Periodic Reporting for period 4 - AEROFLEX (AEROelastic instabilities and control of FLEXible Structures) Reporting period: 2020-01-01 to 2020-06-30 Summary of the context and overall objectives of the project The design paradigm of many engineering systems (aircrafts, bridges, wind mills or offshore drilling risers) 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. 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 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 second example is the large-amplitude motion/deformation of appendices used by swimming animal (fishes and eels) or flying animals (insect or birds) to generate propulsive and lift forces, respectively. The flexibility of structures to increase the propulsive forces is 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 structures to reduce the skin-friction drag of blunt bodies and potentially generate mean propulsive forces at a low energetic cost. To achieve these physical objectives, innovative methodologies are developed. General mathematical formulations and numerical methods are developed to (i) determine flow-induced elastic instabilities in steady or oscillations configurations that are encountered in industrial or natural configurations, and (ii) control (suppress or use) these flow-induced instabilities with adjoint-based optimization of the structure. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far The work performed during this project concern the theoretical and numerical methods developed to investigate flow-induced elastic instabilities around steady or oscillating configurations and the passive or active control of flow-induced instability by shape-optimization or material-optimization. The mathematical description of flow-induced elastic instabilities is revisited by linearizing the non-linear equations governing the dynamics of elastic material and interacting with a incompressible flows. The Arbitrary Lagrangian formulation is preferred over lmmersed Boundary Methods as it allows a conformal description of the fluid-structure interface, which is a crucial ingredient for accurate linezrized equations. In addition to the global stability analysis, a resolvent analysis is also developed to describe the interaction of spatially-growing flow instabilities interacting with elastic structures. Finally, numerical methods and tools, based on an open-source finite-element library, are developed in a fully parallel framework, to compute steady-state solutions, eigenmodes and resolvent modes in large-scale fluid-structure problems. As a result of those primary flow-induced instability, periodic oscillations of structures may arise, as for the aeroelastic flutter limit cycle oscillations. They may alos be induced by an external forcing, as in the case of flapping bodies. In obth cases, theoretical and numerical methods are dveloped to analyze the stability of those periodic solutions. The growth or decay of perturbations is analyzed based on Floquet theory, by accounting for the coupling between the fluid and solid perturbations. An unconventional Time-Spectral-based method is developed for computing the leading Floquet modes. This time-spectral method is also used to determine unstable periodic solutions that cannot be computed with classical unsteady simulations. The challenging task of solving the Time Spectral equations is tackled via a time-parallel Newton-Krylov approach and an efficient parallel block-circulant preconditioner developed during the project. Finally, the control of flow-induced instabilities is investigated in the framework of adjoint-based optimization where the objective function is either the eigenvalue (growth/frequency) of fluid-solid eigenmodes or the energetic gain of resolvent modes. We focused on solid actuations and consider the control of fluid-solid eigenmodes of a flexible splitter plate by (i) optimizing the shape of the rigid cylinder to which it is attached or (ii) by modifying the material properties of the splitter plate with shunted piezo-electric patches. Then we address the attenuation of Tollmien-Schlichting developing in a boundary layer flow by (iii) optimizing the material properties of the visco-elastic patch inserted in the wall or (iv) actuating the vibration modes to optimally minimize the flow perturbation that are maximized by the inflow excitation. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) The main achievement of the AEROFLEX project are (or about to be) published in peer-reviewed journals, and the manuscripts can be found on the web-site of the project, at the URL:https://w3.onera.fr/erc-aeroflex/journal-papers. They concern:- Linear stability analysis of hyperelastic structure strongly coupled to an incompressible flows with the Arbitrary-Lagrangian–Eulerian method - Global stability analysis and control of flexible splitter plates interacting with a circular cylinder’s wake flow. - Resolvent analysis and optimization of visco-elastic patchs for the attenuation of boundary-layer flow instabilities. - The global stability analysis of a springs-mounted plate revealing the viscous effect on the flutter restabilization- The role of flow nonlinearities on the nature of the flutter bifurcation (hard versus soft flutter) depending on the mass ratio and Reynolds number.- The self-propulsion of flapping foils at low and large flapping frequency: Oscillation of flexible splitter plates observed in the soap film experiment when varying its length Sensitivity of TWF resolvent modes to stiffness, damping and mass distribution Shape-optimization of the cylinder aiming at increasing the oscillation frequency of the plate Nature of the flutter bifurcation as a function of the Reynolds number and solid-to-fluid mass-ratio Eulerian- and Lagrangian-based decomposition for the fluid-structure stability analysis in ALE Effect of viscosity on the flutter instability Optimal control of TS resolvent modes by feedforward wave cancellation through wall actuation Flow-induced modes of a plate mounted on a system of bending and torsional springs Stiffness-optimization of the patch for minimizing the amplification of TS and TWF instabilities Decoupled and coupled eigenvalue analyses showing the destabilization of the vibration solid modes Unstable eigenmodes of a flexible splitter plate as a function of stiffness and Reynolds number Sensitivity of TS resolvent modes to stiffness, damping and mass distribution Decomposition of fluid-solid perturbations revealing the wave cancellation mechanism of TS waves Fluid-structure stability analysis of a hyper-elastic three-dimensional plate bend by a steady flow Resolvent analysis of a viscoelastic patch interacting with a two-dimensional boundary-layer flow
