Abstract
The method of matched asymptotic expansions is applied to the investigation of transitional separation bubbles. The problem-specific Reynolds number is assumed to be large and acts as the primary perturbation parameter. Four subsequent stages can be identified as playing key roles in the characterization of the incipient laminar–turbulent transition process: due to the action of an adverse pressure gradient, a classical laminar boundary layer is forced to separate marginally (I). Taking into account viscous–inviscid interaction then enables the description of localized, predominantly steady, reverse flow regions (II). However, certain conditions (e.g. imposed perturbations) may lead to a finite-time breakdown of the underlying reduced set of equations. The ensuing consideration of even shorter spatio-temporal scales results in the flow being governed by another triple-deck interaction. This model is capable of both resolving the finite-time singularity and reproducing the spike formation (III) that, as known from experimental observations and direct numerical simulations, sets in prior to vortex shedding at the rear of the bubble. Usually, the triple-deck stage again terminates in the form of a finite-time blow-up. The study of this event gives rise to a noninteracting Euler–Prandtl stage (IV) associated with unsteady separation, where the vortex wind-up and shedding process takes place. The focus of the present paper lies on the triple-deck stage III and is twofold: firstly, a comprehensive numerical investigation based on a Chebyshev collocation method is presented. Secondly, a composite asymptotic model for the regularization of the ill-posed Cauchy problem is developed.
Highlights
The present paper is concerned with the investigation of the early stages of laminar–turbulent transition induced by localized boundary layer separation
The present paper essentially is a revision of the fundamental work of Elliott and Smith [29]
A spectral collocation method based on Chebyshev polynomials has been developed for treating the second—shorter—interactive stage of marginally separated flows
Summary
The present paper is concerned with the investigation of the early stages of laminar–turbulent transition induced by localized boundary layer separation. The mentioned structures, of size comparable with the laminar boundary layer thickness, convey the early stage of the turbulent part of the flow and persist a certain distance downstream of the mean reattachment point until they disintegrate into small-scale vortices (the ‘cloudy’ pattern at the top right of Fig. 1) characteristic of the developing turbulence This laminar–turbulent transition scenario generated by a locally detached shear layer is not yet fully understood, but it is commonly accepted that the repetitive bursting of a LSB, i.e. the (largely) periodic vortex shedding at its rear has to play a key role insofar as it precedes and initiates rather than follows full transition, see e.g. As shown below in more detail, the successive passage of these distinct
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