An asymptotic description of transitional separation bubbles - compressibility effects and acoustic radiation

01.10.2009 - 30.06.2014
Research funding project
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.

People

Project leader

Institute

Grant funds

  • FWF - Österr. Wissenschaftsfonds (National) Austrian Science Fund (FWF)

Research focus

  • Computational Fluid Dynamics: 25%
  • Mathematical and Algorithmic Foundations: 25%
  • Modeling and Simulation: 50%

Keywords

GermanEnglish
Auftreten von Singularitäten nach endlicher Zeit finite-time singularities
angepasste asymptotische Entwicklungenmatched asymptotic expansions
Dreierdeck-Grenzschichttheorietriple-deck boundary layer theory
laminare Ablöseblasenlaminar separation bubbles
laminar-turbulente Transitionlaminar-turbulent transition
nichtlineare Integrodifferentialgleichungennonlinear integro-differential equations

Publications