On a similarity solution in the theory of unsteady marginal separation

Kluwick

2011

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If the angle of attack α of a slender airfoil reaches a critical value α s flow separation is known to occur at the upper surface. Further increase of α initially leads to the formation of a short laminar separation bubble which has an extremely weak influence on the external flow field -a phenomenon known as marginal separation -but then rather rapidly causes a severe change of the flow behaviour, leading to leading edge stall. According to the asymptotic theory of marginal separation holding in the limit of large Reynolds numbers Re, the flow in the neighbourhood of the separation bubble is governed by an integrodifferential equation. This so-called interaction equation contains a single controlling parameter Γ which relates the angle of attack to the Reynolds number, with a value Γ s corresponding to α s . Some recent results concerning higher order corrections to this theory and their effect on the stability of steady solutions will be presented. Problem formulation and perturbation analysisAs a starting point, consider the planar, subsonic, incompressible, steady and laminar high Reynolds number flow around the leading edge of a thin airfoil at a small angle of attack α. As required by the theory of marginal separation, see [1, 2], we concentrate on the limiting case α ≃ α 0 , where the classical boundary layer calculation yields a marginal separation singularity at x = x 0 on the suction side and the interaction between the local thickening of the boundary layer and the feedback effect of the induced pressure in the outer inviscid flow field can no longer be neglected. First, an investigation of higher order steady corrections will be presented, which was carried out in order to obtain more accurate results as far as the comparison with future DNS-calculations is concerned. Secondly, higher order unsteady effects resulting from forcing disturbances (e.g. surface mounted obstacles and/or suction/blowing distributions), cf. [3], will be studied as well.For the scalings applied to the field quantities and the nondimensional coordinates defining the three layers in the interaction zone, the reader is referred to [3] and the references therein. Again, the Strouhal number Sr =L/(ũ ∞Θ ) setting the time scale is taken to be of O(Re −1/20 ). As it turns out, the incorporation of higher order effects into the three layers of the interaction region requires an asymptotic expansion of the basic equations in ascending powers of Re −1/20 , with Re =ũ ∞L /ν ∞ → ∞. To leading order, the solvability condition (fundamental equation) given e.g. in [3] for the scaled wall shear A(X, T ) ∝ Re 7/10τ w /(ρũ 2 ∞ ) is recovered:

Section: Third Order Unsteady Effectsmentioning

confidence: 99%

“…To leading order, the solvability condition (fundamental equation) given e.g. in [3] for the scaled wall shear A(X, T ) ∝ Re 7/10τ w /(ρũ 2 ∞ ) is recovered: …”

mentioning

confidence: 99%

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confidence: 99%

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On higher order effects in marginally separated flows

Kluwick

2011

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“…The paper will focus on the numerical treatment of the initial phase of the latter stage. Based on the investigations regarding finite time blow-up solutions of the well-known fundamental equation of marginal separation, [4,6], we want to address the corresponding formulation of the subsequent triple deck stage, [2,3], numerically. In suitably nondimensionalized and scaled form the spatio-temporal evolution of the stream function ψ(x, y, t) for planar incompressible flow is given by a boundary layer type equation with both, prescribed adverse, and induced pressure gradient…”

mentioning

confidence: 99%

“…Finally, the time derivative is replaced by a backward second order accurate finite differencing formula with the possibility of adaptive time stepping on a mapped domain, [4]. The resulting dense system of nonlinear equations is solved using a modified Powell hybrid method.…”

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confidence: 99%

On the initial phase of the triple deck stage in marginally separated flows

2016

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The method of matched asymptotic expansions is used to investigate marginally separated boundary layer flows (laminar or alternatively transitional separation bubbles) at high Reynolds numbers. Typical examples include, among others, the flow past slender airfoils at small to moderate angels of attack and channel flows with suction. As is well-known, classical (hierarchical) boundary layer computations usually break down under the action of an adverse pressure gradient on the flow, a scenario associated with the appearance of the Goldstein separation singularity. If, however, the parameter controlling the strength of the pressure gradient (the angle of attack or the relative suction rate in the examples mentioned above) is adjusted accordingly, the application of a local viscous-inviscid interaction strategy is capable of describing localized boundary layer separation. Moreover, taking into account unsteady effects and flow control devices allows the investigation of the conditions leading to forced or self-sustained vortex generation and the subsequent evolution process culminating in bubble bursting. Within the asymptotic formulation of this stage bubble bursting is associated with the formation of finite time singularities in the solution of the underlying equations and a corresponding break down. The distinct blow-up structure gives rise to a fully non-linear triple deck interaction stage featuring shorter spatio-temporal scales characteristic of the successive vortex evolution process. The paper will focus on the numerical treatment of the initial phase of the latter stage. In suitably nondimensionalized and scaled form the spatio-temporal evolution of the stream function ψ(x, y, t) for planar incompressible flow is given by a boundary layer type equation with both, prescribed adverse, and induced pressure gradientsupplemented with the interaction law which relates the induced pressure P(x, t) and the displacement function A(x, t). Here x, y, and t denote the streamwise and wall normal coordinates and the time. Equations (1) are subject to the no slip boundary conditions ψ = ∂ψ/∂y = 0 at the solid wall y = 0, the far-field behaviors ψ ∼ (y + A) 3 /6 + · · · as y → ∞, and ψ ∼ y 3 /6 , A, P → 0 as x → ±∞. The connection to the triggering blow-up event in the preceding marginal separation stage is established via the matching, or equivalently, initial conditionwithÂ ∼Â 1 (x) + |t|2 (x) + · · · andψ ∼ψ 2 + |t| (2) into (1) recovers the blow-up structure of the preceding stage in form of the linearized boundary layer equationswith vanishing leading order right hand side b 1 = 0. Application of the adjoint operator approach to (3) and making use of Fredholm's alternative then leads to equations (solvability conditions, not repeated here) determining the unique blowup profileÂ 1 (x), the eigenfunctionsê 1 (x) = E 1Â ′ 1 ,ê 2 = E 2 (Â 1 + 2 3xÂ ′ 1 ) (the arbitrary amplitudes E 1 , E 2 enable the embedding into a global solution), the perturbation stream functionψ 2 (x,ŷ) and the next order correction of the displ...

Asymptotic description of incipient separation bubble bursting

Kluwick

2012

The appearance of short laminar separation bubbles in high Reynolds number (Re) wall bounded flows due to appropriate adverse pressure gradient conditions is usually associated with minor effects on global flow properties (e.g. lift force). However, localized reverse flow regions are known to react very sensitively to perturbations and in further consequence may trigger the laminar‐turbulent transition process or even cause global separation. The present investigation of marginally separated boundary layer flows is based on an asymptotic approach Re → ∞. Special emphasis is placed on solutions of the corresponding model equations which blow up within finite time indicating the ejection of a vortical structure and the emergence of shorter spatio‐temporal scales reminiscent of the early transition scenario (‘ bubble bursting’ ). Within the framework of marginal separation theory, an alternative adjoint operator method is used to formulate evolution equations governing the viscous‐inviscid interaction process in leading and higher order correction required for the study of later stages of the flow development. Their blow up structure specifies the initial condition of and the match to the subsequent triple deck stage. (© 2012 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)