We analyse the three-dimensional non-parallel instability mechanisms responsible for transition to turbulence in regions of recirculating steady laminar two-dimensional incompressible separation bubble ®ow in a twofold manner. First, we revisit the problem of Tollmien{Schlichting (TS)-like disturbances and we demonstrate, for the rst time for this type of ®ow, excellent agreement between the parabolized stability equation results and those of independently performed direct numerical simulations. Second, we perform a partial-derivative eigenvalue problem stability analysis by discretizing the two spatial directions on which the basic ®ow depends, precluding TS-like waves from entering the calculation domain. A new two-dimensional set of global ampli ed instability modes is thus discovered. In order to prove earlier topological conjectures about the ®ow structural changes occurring prior to the onset of bubble unsteadiness, we reconstruct the total ®ow eld by linear superposition of the steady two-dimensional basic ®ow and the new most-ampli ed global eigenmodes. In the parameter range investigated, the result is a bifurcation into a three-dimensional ®ow eld in which the separation line remains una¬ected while the primary reattachment line becomes three dimensional, in line with the analogous result of a multitude of experimental observations.

No abstract

A simple extension of the classic Görtler–Hämmerlin (1955) (GH) model, essential for three-dimensional linear instability analysis, is presented. The extended Görtler–Hämmerlin model classifies all three-dimensional disturbances in this flow by means of symmetric and antisymmetric polynomials of the chordwise coordinate. It results in one-dimensional linear eigenvalue problems, a temporal or spatial solution of which, presented herein, is demonstrated to recover results otherwise only accessible to the temporal or spatial partial-derivative eigenvalue problem (the former also solved here) or to spatial direct numerical simulation (DNS). From a numerical point of view, the significance of the extended GH model is that it delivers the three-dimensional linear instability characteristics of this flow, discovered by solution of the partial-derivative eigenvalue problem by Lin & Malik (1996a), at a negligible fraction of the computing effort required by either of the aforementioned alternative numerical methodologies. More significant, however, is the physical insight which the model offers into the stability of this technologically interesting flow. On the one hand, the dependence of three-dimensional linear disturbances on the chordwise spatial direction is unravelled analytically. On the other hand, numerical results obtained demonstrate that all linear three-dimensional instability modes possess the same (scaled) dependence on the wall-normal coordinate, that of the well-known GH mode. The latter result may explain why the three-dimensional linear modes have not been detected in past experiments; criteria for experimental identification of three-dimensional disturbances are discussed. Asymptotic analysis based on a multiple-scales method confirms the results of the extended GH model and provides an alternative algorithm for the recovery of three-dimensional linear instability characteristics, also based on solution of one-dimensional eigenvalue problems. Finally, the polynomial structure of individual three-dimensional extended GH eigenmodes is demonstrated using three-dimensional DNS, performed here under linear conditions.

A three-dimensional, incompressible boundary-layer flow is investigated theoretically with respect to primary and secondary instability. These investigations accompany a basic transition experiment, which is being performed at the DLR in Göttingen [B. Müller, in Laminar-Turbulent Transition, edited by D. Arnal and R. Michel (Springer-Verlag, Berlin, 1990), p. 489]. Primary stationary and secondary nonstationary disturbances are used to model the mean flow and the fluctuations of a measured (quasi-) saturation state. The analysis is based on a Falkner–Skan–Cooke approximation of the undisturbed flow. Good agreement of the secondary stability model with this experiment is obtained, especially concerning the spatial distribution of the rms fluctuation. A striking change of the vortex pattern due to secondary instability has not been observed.

No abstract