On the origins of unsteadiness and three-dimensionality in a laminar separation bubble

Abstract

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.

A pressure-gradient mechanism for vortex shedding in constricted channels

“…consequences of a sustained perturbation immediately downstream from the constriction on the resulting flow behavior. Further, our results may explain the topological flow changes that researchers such as Theofilis et al 22 suggest are present at the onset of unsteady flow behavior. Our findings may lead to novel methods for suppressing or controlling the vortex shedding phenomenon.…”

Structural changes of laminar separation bubbles induced by global linear instability

“…In agreement with boundarylayer theory, the wall-normal velocity components are an order-of-magnitude smaller than the horizontal ones, and are not shown in the figures. As shown in Theofilis et al (2000), the spatial structure of the direct eigenfunctions is centred inside the recirculation region and extends downstream, while the new result here concerns the adjoint eigenfunctions, which extend upstream. Furthermore, the maximum amplitude in the direct eigenfunctions is attained in the streamwise velocity component u, while that in the adjoint eigenvector is attained in its spanwise component w.…”

“…Inverse formulation of the non-similar boundary-layer problem A non-similar inverse formulation of the boundary-layer equations on a flat plate was used to obtain the basic flow. Two-dimensional DNS of the Navier-Stokes equations used in earlier work (Briley 1971;Theofilis et al 2000), in which a linear deceleration of the far-field streamwise velocity was imposed in order to force the appearance of the LSB, were dismissed in the present context. In choosing the current nonsimilar boundary-layer formulation, the appearance of the bubble shedding as the peak reversed flow increases is circumvented in the computation of the basic flow and may be subsequently recovered as one (or more) global eigenmodes of this basic flow.…”

“…The objective of the present work is to investigate the topological changes exerted by the global mode over the LSB. Unlike Theofilis et al (2000) where an analogous discussion was qualitative and based on theoretical arguments presented in earlier works (Dallmann et al 1997), the reconstructions of the flow field considered here are quantitative, and are based on linear superposition of the eigenmodes obtained from the present global linear instability analysis upon the respective basic state.…”

“…Independent investigations using linear instability theory and/or direct numerical simulations (DNS) (Rist & Maucher 2002;Fasel & Postl 2004) agree reasonably well with this threshold for the onset of absolute instability of wave-like disturbances of the LSB. However, it was Theofilis (2000) and Theofilis, Hein & Dallmann (2000) who first unequivocally demonstrated the self-excitation potential of LSB flows, by solving the partial-derivative-based eigenvalue problem pertinent to the (twodimensional DNS obtained) separated boundary layer ensuing Howarth's freestream linear deceleration on a flat plate (Briley 1971), without the need to invoke the assumption of weak non-parallelism of the basic state; at about the same time, an analogous conclusion was independently reached by Barkley, Gomes & Henderson (2002) in the LSB formed in the laminar backward-facing step flow.…”