Dynamic stall due to unsteady marginal separation

Elliott, J. W.; Smith, F. T.

doi:10.1017/s0022112087001629

Cited by 29 publications

(36 citation statements)

References 6 publications

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“…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).…”

Section: Mathematical Basismentioning

confidence: 99%

Entrainment and boundary-layer separation: a modeling history

Veldman

2017

J Eng Math

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For many decades since Prandtl's presentation of the boundary-layer equations, all attempts to calculate separated boundary layers ended up in a break-down of the numerical algorithm. It was not until the late 1970s that the first successful calculations were reported. The paper describes the history and philosophy of the steps that led to understanding where the break-down originates and how to circumvent it. Especially, the role of Head's entrainment will be highlighted.

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Section: Mathematical Basismentioning

confidence: 99%

Entrainment and boundary-layer separation: a modeling history

Veldman

2017

J Eng Math

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“…1(a)) of sufficient magnitude or super-critical values of α lead to a finite-time breakdown of the local flow description at X s . The introduction of shorter scales then results in the flow being governed by another viscous-inviscid tripledeck interaction, [4]. The theory obtained from this approach is capable of both resolving the mentioned singular behaviour and reproducing the spike formation (iii) that, as known from experimental observations, sets in prior to vortex shedding at the rear of the bubble.…”

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confidence: 96%

“…1(a). Unless countermeasures are adopted, the triple-deck stage again terminates in form of a finite-time blow-up at a certain location x s , [2,4,6]. The study of this event gives rise to a non-interacting Euler-Prandtl stage (iv) comprising an outer inviscid rotational region, where, in essence, the vortex wind-up and shedding process takes place, and a viscous wall-layer.…”

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confidence: 99%

Interaction boundary layer theory applied to a benchmark flow problem

Braun

2017

Proc Appl Math and Mech

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Consider planar, incompressible laminar high Reynolds number (Re) flow between two parallel plates: fluid is removed through the permeable upper wall in order to generate adverse pressure gradient conditions in mean flow direction. As a consequence of relatively weak suction, the boundary layer at the impermeable lower wall is forced to separate locally. The corresponding reverse flow region (separation bubble) is known to react very sensitively to perturbations of any kind and typically undergoes rapid transition to the turbulent flow state. The present paper briefly summarizes the current status of the corresponding asymptotic flow description in the limit as Re → ∞.

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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…”

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confidence: 99%

“…2-4 (for E 1 = E 2 = 1). Current efforts aim at an improved spatial resolution and the confirmation of a possible blow-up scenario of (1) predicted in [3], which has been investigated numerically in [5]. …”

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confidence: 99%

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

Braun

Scheichl

2016

Proc Appl Math and Mech

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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...

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Dynamic stall due to unsteady marginal separation

Cited by 29 publications

References 6 publications

Entrainment and boundary-layer separation: a modeling history

Entrainment and boundary-layer separation: a modeling history

Interaction boundary layer theory applied to a benchmark flow problem

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

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