Blow-up phenomena in localized boundary layer separation

Laminar-turbulent boundary layer transition is of great importance to researchers and engineers in their efforts to understand the complicated structure of turbulence and also to develop appropriate engineering models for the prediction of flow characteristics. Of special theoretical as well as practical interest is the accurate calculation of lift and drag forces acting on aerodynamic bodies which requires comprehensive knowledge on whether the flow is laminar or turbulent, attached or separated. The theory of boundary layer flows with special emphasis on the most important issues separation and transition when the Reynolds number is asymptotically large has been a field of active research since Prandtl first developed his - now classical - theory for laminar steady two-dimensional flows in 1904. Milestones are, among others, the discovery of the singular behaviour and breakdown of the classical boundary layer equations near a point of vanishing skin friction (separation point) by Goldstein 1948 and that of viscous-inviscid interaction independently made by Stewartson, Messiter and Neiland in the late 1960s which has generally become known as the triple deck theory. In conventional triple deck problems sudden changes of boundary conditions or singular behaviour of the imposed pressure gradient initiate the interaction mechanism. On the contrary, in cases of socalled marginal separation an increase of the smooth imposed adverse pressure gradient controlled by a characteristic parameter leads to the onset of the interaction process and localized separation. In the early 1980s, Ruban and Stewartson, Smith and Kaups independently formulated a rational description of the local interaction mechanism now commonly referred to as the theory of marginal separation. It serves as the starting point for the proposed work which deals with the investigation of the transition process in laminar separation bubbles. As is well known, the theory of marginal separation predicts an upper bound of the control parameter for the existence of strictly steady, i.e. unperturbed, flows. The incorporation of unsteady effects led to the conclusion that the onset of transition is associated either with exceeding the critical value of the control parameter or the presence of a sufficient perturbation level in case of below critical conditions. Within the framework of the existing theory, vortex shedding from the rear of the separation bubble manifests itself in the occurrence of a finite time singularity. Surprisingly, in that case recent findings strongly suggest the presence of a unique blow-up pattern, entirely independently of the previous history of the flow, i.e. the initial condition, the value of the controlling parameter, etc. The associated breakdown of the flow description implies the emergence of shorter scales and the subsequent evolution of the flow then is described by a fully nonlinear triple deck interaction which seems to suffer finite time breakdown as well. The tracking of this `breakdown cascade` is of particular interest and a main focus of the project since it reflects the successive genesis of shorter spatiotemporal scales which is a distinctive feature of the vortex generation process. Since three-dimensionality is believed to be an essential ingredient of the real transition process, we aim at the extension of the current theory with regard to three-dimensional effects. We expected to gain deeper insight into the questions concerning the uniqueness of the blow-up profile in case of three-dimensional flows and the possibility to continue solutions beyond the blow-up time. Furthermore, aspects of flow control and transition detection round off the scope of the project in view of its embedding into the research activities of the institution. Extensive use of perturbation techniques and novel numerical schemes characterize the present approach to a fundamental problem of fluid mechanics.

Viscous effects which cause friction drag of streamlined bodies (e.g. airfoils) are confined to a thin (boundary) layer adjacent to the solid body wall if the free stream flow velocity is sufficiently high and the fluid viscosity low. Flow conditions which lead to localized flow separation are of particular interest. In general, flow separation is known to destabilize the overall flow field (in extreme cases this may even lead to a breakdown of lift), localized separation bubbles typically trigger the laminar-turbulent transition process within the boundary layer flow. The investigation of this phenomenon is of great importance since the turbulent boundary layer flow downstream of the separation bubble is usually associated with an increase of drag (fuel consumption) in comparison with the corresponding conditions in strictly laminar flow. Furthermore, a fundamental knowledge of the transition process would contribute to an insight into the complex dynamics of turbulence. The current analysis is based on an analytical approach (singular perturbation theory, method of matched asymptotic expansions) and the thus gained equations require numerical treatment in most cases. Furthermore, due to inconvenient solution behavior (stiffness of equations) innovative numerical methods must be applied to solve these equations.The current work is a logic extension of fundamental research initiated in the early nineteen-eighties in Russia and England and essentially comprises: (i) an alternative, elegant derivation of the fundamental similarity laws which govern the unsteady, three-dimensional localized separation phenomenon. This approach enabled - besides the deduction of the well-known leading order equation - (ii) the derivation of higher order corrections which (iii) turned out to be crucial for an adequate asymptotic representation of the laminar-turbulent transition process. As is well-known, these model equations in general exhibit ill-posedness, i.e. small amendments of the initial conditions lead to fundamental different solution behavior. A detailed investigation of this fact (iv) led to a successful regularization of this problem, the corresponding modified equations may now be designated as well-posed. (v) The application and development of novel numerical techniques for the solution of these initial value problems, i.e. the computation of the spatiotemporal evolution of relevant flow characteristics, which in essence base on spectral methods for unbounded domains, round off the processed project scope.