Asymptotic desription of transitional separation bubbles

A systematic asymptotic analysis of Prandtl`s classical boundary layer equations has shown that they inevitably lose their validity near a point of vanishing skin friction. Starting in the late 1960s, however, a number of authors have developed strategies to overcome this deficiency by exploring the idea that inviscid and viscous regions are allowed to interact already to leading rather than higher order. It was thus found that there exist two different routes leading to separation of a laminar boundary layer under steady flow conditions. Firstly, a firmly attached boundary layer may be forced to separate due to the presence of a large adverse pressure gradient acting over a rather short distance. This approach led to what has become known as the triple-deck theory. In the second strategy, again the interaction region splits into three layers with different physical properties. But in this case, called marginal separation, an adverse pressure gradient that is imposed over a notably longer distance and is controlled by a characteristic parameter G gives rise to the interaction process and, consequently, localized separation. The concept has already been extended to three-dimensional as well as unsteady and, recently, also compressible flow conditions. The results obtained from this powerful generalized theory serve as the starting point for the proposed research, which will be devoted to an asymptotic analysis of the laminar-turbulent transition process caused by the bursting of separation bubbles. Within the framework of marginal separation theory, the flow in the neighborhood of the bubble is governed by an integro-differential equation, whose steady two-dimensional solutions exist up to a critical value G c only. Furthermore, it depends, also parametrically, on the local Mach number. Previous investigations showed that a substantial change of the flow field, manifesting itself in the occurrence of a finite-time singularity, has to take place as either G becomes larger than G c or the flow is forced to deviate from its steady state by the activity of sufficiently large external disturbances. Moreover, it turned out that for each sub-critical value of G, there exists a threshold value for the Mach number the exceeding of which as well triggers the blow-up event. The singular behavior of the solutions invalidates the theoretical basis underlying the concept of marginal separation, but at the same time heralds the onset of transition. That is to say, the finite-time singularities can be interpreted as representing vortical structures qualitatively similar to those emerging in direct numerical simulations of transitional separation bubbles. Surprisingly, a recent study revealed the existence of a unique blow-up profile developing entirely independently of the previous history of the flow, i.e. the type of forcing, the imposed initial conditions and the value of the controlling parameter. The breakdown of the marginal separation theory and thus the necessity to introduce smaller scales in time and space result in the subsequent evolution of the flow being described by a fully nonlinear triple-deck interaction. The aims of the study are primarily twofold: (i) a detailed investigation of this problem, with special emphasis placed on the incorporation of compressibility effects and on the calculation of the solution that represents the continuation of the unique blow-up structure obtained from marginal separation theory, and (ii) exploring the mechanism through which the fluid motion inside the boundary layer is transformed into aerodynamic sound. The latter investigation is intended to enable the detection of the distinctive sound pattern reflecting the status of the boundary layer with respect to transition. Furthermore, it is planned to generalize the theory so that three- dimensional effects are included and, if in turn a finite-time blow-up occurs, to investigate the next stage of the bursting phenomenon.

The viscous effects which cause the friction drag of streamlined bodies, such as air-foils, 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 is sufficiently low. In this connection, especially flow conditions which lead to localized flow separation are of particular interest. Flow separation is known to destabilize the overall flow field, in extreme cases, this may even lead to a breakdown of lift. On the other hand, localized separation bubbles typically trigger the laminar-turbulent transition process within the boundary layer flow. The analysis of this phenomenon is of great importance since in comparison with the conditions in strictly laminar flow, the turbulent boundary layer flow downstream of the separation bubble is usually associated with an increase of drag and, by that, an enhanced fuel consumption of a motorized aircraft. Furthermore, a fundamental understanding of the transition process would contribute to an insight into the complex dynamics of turbulence as well. The investigations presented here are based on an analytical approach applying the so-called singular perturbation theory, in particular the method of matched asymptotic expansions. In general, the thus gained equations require a numerical treatment. In our case, the difficulties resulting from the unpleasant features of the equations, in particular their stiffness, had to be circumvented by the employment of innovative computational schemes.The current work is a logical extension of the fundamental research initiated in the early nineteen- eighties in Russia and England, which started with the investigation of small separation bubbles in a steady flow, i.e. in a flow where the properties do not change over time. Based on these findings, many following studies then have dealt with the identification and investigation of the unsteady mechanisms that could be responsible for the transitions process from laminar to turbulent flow. The main difficulties encountered in this endeavour are owing to the fact that the transitional flow passes a series of stages, with each of them being characterized by the occurrence of singularities in the solutions to the governing equations. These singularities then in turn trigger the development of a subsequent regime. In the present project, an intermediate stage was investigated which follows two stages that have already been studied in detail. As is also known from these earlier investigations, the model equations to be solved in this cascade, in most cases, exhibit ill-posedness, that is to say that small changes in the initial conditions lead to a fundamental different solution behaviour. A thorough analysis of this fact led to a successful regularization of the equations governing this intermediate stage such that the now modified problem can be regarded as well-posed. The development and application of novel numerical techniques for solving the initial value problems arising in the series of regimes, i.e. for the computation of the spatio-temporal evolution of the relevant flow characteristics, constitute a further cornerstone of the project. By means of these schemes, which are primarily based on spectral collocation methods for unbounded domains, the analysed intermediate stage turned out to again feature singular solutions. On the other hand, potential candidates for the regimes to follow were identified.