Bypass transition: an asymptotic approach

The research proposal entitled "Bypass transition: an asymptotic approach" is devoted to a theoretical description of the transition process from the laminar to the turbulent flow state in a generic aerodynamic setting, called laminar separation bubble. Whereas the former state is characterized by smooth, streamlined motion of fluid which induces comparatively low friction, the latter typically shows highly disordered movement and mixing of fluid particles, a fact which causes substantial drag forces onto immersed solid bodies like e.g. aircraft wings. From the viewpoint of practical efficiency - e.g. minimizing fuel consumption - it is worth striving for a sound knowledge of the conditions leading to transition and the mechanisms involved. Furthermore, a fundamental understanding of the transition process would contribute to an insight into the complex dynamics of turbulence. The approach chosen to tackle this long-standing issue is based on so-called perturbation theory. The therefore necessary assumptions reflect a situation that is actually encountered in many practical flows of air or water, namely, that the relevant flow velocity is sufficiently large and the viscosity of the considered fluid is very low. As a result of this asymptotic analysis, the rapid transition process, triggered by slightly perturbed localized reverse flow regions, is decomposed into a series of subsequent stages (with distinctive length and time scales), with each of them being governed by a simplified, but still nonlinear, set of equations. Most important, the respective equations reveal the essential physical mechanisms at work and their controlling parameters. The proposed approach differs significantly from more common research methods like direct numerical simulations based on the full and computationally expensive equations of fluid mechanics, considerations derived from stability theory and experimental techniques since it makes rigorous use of asymptotic methods which primarily yields qualitative, universally valid results rather than quantitative, strongly case-dependent data. In further consequence, this enables the systematic evaluation of flow control measures and their optimization towards either preventing or forcing transition in the most effective way. The main focus of the planned work is to investigate the formation of coherent vortical structures, characteristic of transitional flows. As it turns out, this process is predominantly inviscid, entirely self-induced and exhibits generic features known from the analysis of a setting referred to as unsteady separation. The proposed object of investigation is a logical extension of fundamental research initiated in the early nineteen-eighties in Russia and England and combines methods of applied and numerical mathematics.

Although it is a fundamental research question in classical physics, the turbulent flow state is still far from being fully understood. The present research project attempts to approach this complex problem by examining the turbulence in its origin, the so-called laminar-turbulent transition, in the case under consideration caused by locally separated flow (so-called laminar separation bubbles). This phenomenon typically occurs in the leading edge area of the suction side (usually the top) of modern (slender) airfoils in normal flight conditions and makes a decisive contribution to their performance (in the form of the lift to drag ratio). The selected mathematical solution approach uses perturbation methods, taking advantage of the fact that the effect of friction can be neglected in most of the flow field and only needs to be taken into account in the immediate wall region (boundary layer) of the body. It therefore differs significantly from the more common investigation methods such as direct numerical simulation of the underlying Navier-Stokes equations or flow measurement and visualization techniques. In particular, the incipient flow transition can be divided into several, successive stages using perturbation theory, each of which is described by characteristic spatio-temporal scales and simplified model equations, so-called similarity laws. The solutions to these equations require the use of suitable numerical methods and are characterized by the fact that they each form singularities in a finite time (so-called blow-up). On the one hand, this indicates the (local) break down of the respective model equations, but on the other hand it clearly defines the conditions for the spatio-temporal scales and the model of the subsequent stage. The present project makes significant contributions to stages S3 (Triple-Deck) and S4 (Euler-Prandtl). S3 describes the initial bursting of the separation bubble and S4 the subsequent formation of a coherent vortex structure. For the first time, numerical solutions of the triple-deck stage including the behavior in the vicinity of the blow-up could be calculated. Building on this, the construction of numerical solutions for the Euler and Prandtl regions was successful, but the (expected) occurrence of singularities could not be verified because the calculation methods were not yet sufficiently sophisticated. The calculation of blow-up solutions for stage S2 (marginal separation) using the method of matched asymptotic expansions in the extended case of a three-dimensional flow field was even possible beyond the blow-up time. The moving singularities that arise immediately afterwards can be interpreted as kernels of so-called lambda-vortex structures, which are known from the visualization of transitional separation bubbles. The regularization of these singularities based on the Navier-Stokes equations is the subject of a future research project.