In June 2019, Jean-Lou Pfister defended his PhD thesis, entitled "Instabilities and optimization of elastic structures interacting with laminar flows". The manuscript written in english is available by clicking on the link. The full abstract of this thesis is written below.
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.
