The triple-deck stage of marginal separation

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.

The classical unsteady boundary layer: A numerical study

“…There is some kind of freedom in the definition of the auxiliary function δ, we choose $$$$\begin{align} \delta (x,t) := \frac{1}{u_w}\sqrt {2 \int _0^x u_w(x^{\prime },t)\, dx^{\prime }}, \end{align}$$$$where the integral accounts for the growth of the boundary layer to some extent. In transformation (), we also separated a similarity solution $$\tilde{f}(\eta )$$, which is known as the Blasius or flat plate solution [2], $$\tilde{f}$$ already solves the BL equation in the limit of zero suction $$\alpha = 0$$ (i.e., for $$P_x=0$$, $$u_w=1$$).…”

“…where the integral accounts for the growth of the boundary layer to some extent. In transformation (5), we also separated a similarity solution f(𝜂), which is known as the Blasius or flat plate solution [2], f already solves the BL equation in the limit of zero suction 𝛼 = 0 (i.e., for 𝑃 𝑥 = 0, 𝑢 𝑤 = 1).…”

“…In the limit of $$Re\rightarrow \infty$$, the friction term in the Navier–Stokes equation becomes only relevant at certain flow areas; these are especially the areas near rigid surfaces, denoting a so‐called singular perturbation problem. We imagine that the whole process of laminar–turbulent transition can be described through a cascade of such problems [1, 2].…”

“…Usually, such a breakdown can be regularised via the introduction of new $$Re$$‐dependent scales and the next singular perturbation problem. While follow‐up stages have been extensively studied for a stationary starting point (i.e., the steady BL theory) [2, 4], there are only few studies on the fully time‐dependent scenario [5–7].…”

In memoriam: Alfred Kluwick (1942–2022)

“…In this way, the systematic analysis of so-called marginally separated flows succeeds, in the case of turbulent flows even largely without any turbulence modeling [8]. Recent developments in this area, initiated by Alfred Kluwick many years ago, concern the description of the laminar-turbulent transition process triggered by laminar separation bubbles solely on the basis of asymptotic methods and associated numerical techniques [1]. The recognition of Alfred Kluwick's expertise led to three memberships in committees of the International Union of Theoretical and Applied Mechanics (IUTAM), which is the world's most important scientific umbrella organization for mechanics: a 4-year membership of the 8-member Executive Committee (Bureau), a 9-year membership of the Congress Committee and a membership of the General Assembly.…”