The triple-deck stage of marginal separation

J Eng Math. 2021;128(1):16. doi: 10.1007/s10665-021-10125-3. Epub 2021 Jun 1.

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.

Supplementary information: The online version contains supplementary material available at 10.1007/s10665-021-10125-3.

Keywords: Chebyshev collocation method; Finite-time blow-up; Interaction boundary layer theory; Laminar separation bubble; Laminar–turbulent transition; Unsteady separation.