Linear and non-linear dynamics of non-autonomous flows

Abstract: Fluid flows subject to time-dependent external forces or boundary conditions are ubiquitous in biology and technical applications. Whether one considers birds flying by flapping their wings or the gust response of wind turbines, the flow is non-autonomous. This thesis investigates the influence of external time-dependence on the non-linear evolution of fluid flows, as well as on the linear response to small disturbances that determines their stability.

For the analysis of the time-periodic pulsatile flow through toroidal pipes, an iterative fixed-point solver in frequency space is developed and validated to obtain the baseflows. The method is used to explore the effect of pulsations on the flow through tori with relevant curvatures. Using the Floquet framework, the linear stability of the flow close to criticality is investigated, revealing strong sensitivity to pulsations that are mostly stabilising.

Considering the local stability of pulsating plane Poiseuille flow, the eigen- pairs of the linear operator are tracked over time producing subharmonic eigenvalue orbits. Their appearance is traced to spectral degeneracies of the op- erator, leading to the transition of the harmonic disturbance between eigenvalue trajectories involving non-modal growth bursts. The same flow case is then used to assess the potential of the optimally time-dependent (OTD) framework for transient linear stability analysis of flows with arbitrary time-dependence using a localised linear/non-linear implementation aimed at open flows.

This framework is then used to track the linear stability of laminar separation bubbles on pitching wing sections. On a natural laminar flow airfoil, the global mode corresponding to an absolute local instability is identified at the rear of the bubble, causing its breakdown to turbulence. In the case of an airfoil undergoing dynamic stall, the OTD modes reveal the main instability on the shear layer of the bubble as well as growth bursts correlated with vortex shedding.

The influence of low-amplitude free-stream disturbances on the onset of dynamic stall is investigated and the onset of intermittent vortex shedding during the bubble bursting is documented. The repeated appearance of the phenomenon in a set of flow realisations confirmed its statistical relevance. The Proper Orthogonal Decomposition framework is extended to include time. This allows for the objective extraction of transient structures from data.