In the context of linear stability analysis, considering unsteady base flows is notoriously difficult. A generalisation of modal linear stability analysis, allowing for arbitrarily unsteady base flows over a finite time, is therefore required. The recently developed optimally time-dependent (OTD) modes form a projection basis for the tangent space. They capture the leading amplification directions in state space under the constraint that they form an orthonormal basis at all times. The present numerical study illustrates the possibility to describe a complex flow case using the leading OTD modes. The flow under investigation is an unsteady case of the Blasius boundary layer, featuring streamwise streaks of finite length and relevant to bypass transition. It corresponds to the state space trajectory initiated by the minimal seed; such a trajectory is unsteady, free from any spatial symmetry and shadows the laminar-turbulent separatrix for a finite time only. The finite-time instability of this unsteady base flow is investigated using the 8 leading OTD modes. The analysis includes the computation of finite-time Lyapunov exponents as well as instantaneous eigenvalues, and of the associated flow structures. The reconstructed instantaneous eigenmodes are all of outer type. They map unambiguously the spatial regions of largest instantaneous growth. Other flow structures, previously reported as secondary, are identified with this method as relevant to streak switching and to streamwise vortical ejections. The dynamics inside the tangent space features both modal and non-modal amplification. Non-normality within the reduced tangent subspace, quantified by the instantaneous numerical abscissa, emerges only as the unsteadiness of the base flow is reduced.

This thesis focuses on numerical studies of subcritical transition to turbulence in shear flows. The thesis employs a framework based on nonlinear dynamics in the subsequent studies. The geometrical approach to subcritical transition pivots the concepts of edge manifold and edge state. Such concepts are explored in detail in the Blasius boundary layer. The identified edge trajectory is chaotic and presents a couple of high- and low-speed streaks akin to those identified in other shear flows. For long enough times the linear instability of the Blasiusboundary layer coexists with the bypass transition scenario. The edge is thus reinterpreted as a manifold separating both routes. On the edge manifold of the Blasius boundary layer, the fully localised minimal seed is identified. The minimal seed experiences a sequence of linear mechanisms: the Orr mechanism followed by the lift-up. The resulting perturbation approaches the same region in state space as identified from arbitrary perturbations.These insights from the edge trajectory identified in the Blasius boundary layer inspired a low-dimensional model. The model illustrates the e↵ect of the laminar attractor becoming linearly unstable and it agrees qualitatively withother recent studies in the literature.The edge has been identified as a hyperbolic Lagrangian coherent structure of infinite dimension. We show how two Lagrangian diagnostics can be used to locate the edge directly in state space. This allows us to revisit edge tracking as a method optimising a Lagrangian diagnostic instead of a binary algorithm.The two last studies of the thesis focus on the optimally time-dependent(OTD) modes as a basis for the linearised dynamics about a base flow with arbitrary time-dependence. The OTD modes are explored for a periodic flow in pulsating plane Poiseuille flow. The resulting OTD modes can be linked to thespectrum of the Orr-Sommerfeld operator. The results revealed perturbations which span more than one period of the base flow. Finally, the OTD frameworkis used on the edge trajectory starting from the minimal seed in the Blasiusboundary layer.

Time-dependent flows are notoriously challenging for classical linear stability analysis. Most progress in understanding the linear stability of these flows has been made for time-periodic flows via Floquet theory focusing on time-asymptotic stability. However, little attention has been given to the transient intracyclic linear stability of periodic flows since no general tools exist for its analysis. In this work, we explore the potential of using the recent framework of the optimally time-dependent (OTD) modes (Babaee & Sapsis, Proc. R. Soc. Lond. A, vol. 472, 2016, 20150779) to extract information about both the transient and the time-asymptotic linear stability of pulsating Poiseuille flow. The analysis of the instantaneous OTD modes in the limit cycle leads to the identification of the dominant instability mechanism of pulsating Poiseuille flow by comparing them with the spectrum and the eigenmodes of the Orr-Sommerfeld operator. In accordance with evidence from recent direct numerical simulations, it is found that structures akin to Tollmien-Schlichting waves are the dominant feature over a large range of pulsation amplitudes and frequencies but that for low pulsation frequencies these modes disappear during the damping phase of the pulsation cycle as the pulsation amplitude is increased beyond a threshold value. The maximum achievable non-normal growth rate during the limit cycle was found to be nearly identical to that in plane Poiseuille flow. The existence of subharmonic perturbation cycles compared with the base flow pulsation is documented for the first time in pulsating Poiseuille flow.

Dissipative dynamical systems characterized by two basins of attraction are found in many physical systems, notably in hydrodynamics where laminar and turbulent regimes can coexist. The state space of such systems is structured around a dividing manifold called the edge, which separates trajectories attracted by the laminar state from those reaching the turbulent state. We apply here concepts and tools from Lagrangian data analysis to investigate this edge manifold. This approach is carried out in the state space of autonomous arbitrarily high-dimensional dissipative systems, in which the edge manifold is reinterpreted as a Lagrangian coherent structure (LCS). Two different diagnostics, finite-time Lyapunov exponents and Lagrangian descriptors, are used and compared with respect to their ability to identify the edge and their scalability. Their properties are illustrated on several low-order models of subcritical transition of increasing dimension and complexity, as well on well-resolved simulations of the Navier-Stokes equations in the case of plane Couette flow. They allow for a mapping of the global structure of both the state space and the edge manifold based on quantitative information. Both diagnostics can also be used to generate efficient bisection algorithms to approach asymptotic edge states, which outperform classical edge tracking.

The transition to turbulence in many shear flows proceeds along two competing routes, one linked with finite-amplitude disturbances and the other one originating from a linear instability, as in, e.g., boundary layer flows. The dynamical systems concept of an edge manifold has been suggested in the subcritical case to explain the partition of the state space of the system. This investigation is devoted to the evolution of the edge manifold when linear stability is added in such subcritical systems, a situation poorly studied despite its prevalence in realistic fluid flows. In particular, the fate of the edge state as a mediator of transition is unclear. A deterministic three-dimensional model is suggested, parametrized by the linear instability growth rate. The edge manifold evolves topologically, via a global saddle-loop bifurcation of the underlying invariant sets, from the separatrix between two attraction basins to the mediator between two transition routes. For larger instability rates, the stable manifold of the saddle point increases in codimension from 1 to 2 after an additional local pitchfork node bifurcation, causing the collapse of the edge manifold. As the growth rate is increased, three different regimes of this model are identified, each one associated with a flow case from the recent hydrodynamic literature. A simple nonautonomous generalization of the model is also suggested in order to capture the complexity of spatially developing flows.

Subcritical transition to turbulence in spatially developing boundary layer flows can be triggered efficiently by finite amplitude perturbations. In this Rapid Communication, we employ adjoint-based optimization to identify optimal initial perturbations in the Blasius boundary layer, culminating in the computation of the subcritical transition critical energy threshold and the associated fully localized critical optimum in a spatially extended configuration, the so called minimal seed. By dynamically rescaling the variables with the local boundary layer thickness, we show that the identified edge trajectory approaches the same attracting phase space region as previously reported edge trajectories, and reaches the region more efficiently.

Recent progress in understanding subcritical transition to turbulence is based on the concept of the edge, the manifold separating the basins of attraction of the laminar and the turbulent state. Originally developed in numerical studies of parallel shear flows with a linearly stable base flow, this concept is adapted here to the case of a spatially developing Blasius boundary layer. Longer time horizons fundamentally change the nature of the problem due to the loss of stability of the base flow due to Tollmien-Schlichting (TS) waves. We demonstrate, using a moving box technique, that efficient long-time tracking of edge trajectories is possible for the parameter range relevant to bypass transition, even if the asymptotic state itself remains out of reach. The flow along the edge trajectory features streak switching observed for the first time in the Blasius boundary layer. At long enough times, TS waves co-exist with the coherent structure characteristic of edge trajectories. In this situation we suggest a reinterpretation of the edge as a manifold dividing the state space between the two main types of boundary layer transition, i.e. bypass transition and classical transition.