The nonlinear dynamics of singularities in boundary layer flow separation

Abstract

The phenomenon of laminar-turbulent transition in wall-bounded shear flows is not yet completely understood. We focus on the high Reynolds number asymptotic description of transition in separated boundary layers, which may be depicted through a succession of singular perturbation problems. Each significant change of the dominant balance between nonlinear (inertia), pressure and friction forces results in the breakdown of the current asymptotic flow stage through a finite-time (point) blow-up event, and initiates the subsequent (regularisation) stage in close proximity around the singularity. We consider two scenarios, that respectively serve the purpose to study the influence of time-dependence and three-dimensionality on the incipient transition process. We begin with the classical planar boundary layer stage and numerically study its unsteady suction-induced breakdown within a laminar channel flow. A suitable combination of finite differences and Chebyshev collocation, enables us to compare the emerging reversed flow area with its asymptotic representation of the Van Dommelen–Shen finite-time singularity structure. In the main part of the thesis, we switch to the quasi-steady limit of a laminar separation bubble, within a planar marginally separated boundary layer. We impose weak three-dimensional and time-dependent disturbances, that eventually cause a (repeated) bursting of the bubble and the creation of turbulence in its wake, a process which may happen commonly on the leading-edge suction-side of an airfoil during flight. Here, the initial evolution and amplification of the generated flow disturbances are related to blow-up solutions of a forced Fisher-KPP equation. This reaction-diffusion equation even allows an extension beyond the blow-up time, where the solution consists of two moving singularities that repel each other in spanwise direction. Using matched asymptotic expansions, we closely resolve their creation and initial motion in a succession of asymptotic sublayers and ultimately unveil their hidden self-similar structure to be a singular travelling wave with a time-dependent speed. The locations of the singularities are closely connected to the positions of large vorticity inside the boundary layer flow, enabling us a first glimpse of the creation of three-dimensional coherent vortical structures (lambda or hairpin vortices) in a new asymptotic stage of the laminar-turbulent transition process.