Marginal Separation

Abstract

An analytical and numerical study is presented on the response of high Reynolds number flow when, starting from a fully attached state, it is forced to gradually approach separation as a certain critical parameter is increased. The end of attached flow near a rounded leading edge is the particular problem considered, although the application of the theory is much wider. The critical parameter is the angle of incidence. The Goldstein singularity first appears only weakly, and then the attached-flow concept can even admit a small local separation bubble. As the singularity increases in strength, however, a solution of this type suddenly ceases to exist and a dramatic transition to another, separated form of motion seems to be implied. Nonuniqueness of the attached-flow solution is also found.

A resolution of Stewartson’s quarter-infinite plate problem

“…In our asymptotic descriptions, we have allowed for discontinuous ζ gradients; 1 however, we cannot rule out that essential singularities have some role to play in smoothing such discontinuities over sufficiently small ζ scales. It should be noted that instances of discontinuous derivatives have been found in the past in boundary-layer flows, for example in marginal separation [9,16]. Interestingly, Ruban and Vonatsos [10] recently presented (unsteady) boundary-layer results that go further than merely discontinuous derivatives, presenting evidence of discontinuous solutions.…”

“…As we shall see, the solution with λ = 0 has a discontinuity in ζ derivatives at a critical location (ζ =ζ 0 ) off the plate, whilst the λ = 0 state does not. For the solution with λ = 0, it is only the leading-order velocity component (u) that approaches its freestream value; as ζ → −∞ there is a displacement effect on the crossflow (w) velocity component connected to (16).…”

Entrainment and boundary-layer separation: a modeling history

“…Also, the strong interaction between boundary layer and local inviscid flow region in the asymptotic triple-deck theory, with its reversal of hierarchy, points in this direction; see, e.g., [1,2,23,47,48]. The latter asymptotic theory can be extended to describe boundary-layer flow with marginal separation [9,49], until the separation bubble becomes unsteady and vortex-shedding sets in [50] (in terms of Fig. 11, there is no intersection anymore of the inviscid flow relation and the boundary-layer relation).…”

Concerning marginal singularities in the boundary-layer flow on a downstream-moving surface

“…It is widely recognized now that practical implications of the singular termination of the flow at separation in this and in many other examples of the Prandtl formulation with a prescribed pressure depend on the particular physical situation and can be dramatic in view of the often inevitable global separation of the boundary layer in such cases, with immediate impact on the stability of the flow, drag, performance of the airfoil, etc; for a discussion and further references see e.g. Stewartson (1974), Smith (1982~) and Sychev et al (1987).…”