The effects of freestream pressure gradient on a corner boundary layer

Dhanak, Manhar R.; Duck, Peter W.

doi:10.1098/rspa.1997.0097

Cited by 34 publications

(45 citation statements)

References 20 publications

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“…In seeking solutions to corner boundary-layer problems, Ridha (1992) and Dhanak & Duck (1997) found that if a cross-flow component of flow is permitted, which grows linearly in the cross-flow direction (the other two velocity components are independent of this distance), then two distinct solution branches are obtained. A corollary which arises from this work is that the (two-dimensional) Blasius solution has a three-dimensional alternative, which possesses a jet-like cross-flow velocity component.…”

Section: Introductionmentioning

confidence: 99%

Continua of states in boundary-layer flows

2002

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We consider a class of three-dimensional boundary-layer flows, which may be viewed as an extension of the Falkner-Skan similarity form, to include a cross-flow velocity component, about a plane of symmetry. In general, this provides a range of threedimensional boundary-layer solutions, parameterized by a Falkner-Skan similarity parameter, n, together with a further parameter, Ψ ∞ , which is associated with a cross-flow velocity component in the external flow. In this work two particular cases are of special interest: for n = 0 the similarity equations possess a family of solutions related to the Blasius boundary layer; for n = 1 the similarity solution provides an exact reduction of the Navier-Stokes equations corresponding to the flow near a saddle point of attachment. It is known from the work of Davey (1961) that in this latter class of flow, a continuum of solutions can be found. The continuum arises (in general) because it is possible to find states with an algebraic, rather than exponential, behaviour in the far field. In this work we provide a detailed overview of the continuum states, and show that a discrete infinity of 'exponential modes' are smoothly embedded within the 'algebraic modes' of the continuum. At a critical value of the cross-flow, these exponential modes appear as a cascade of eigensolutions to the far-field equations, which arise in a manner analogous to the energy eigenstates found in quantum mechanical problems described by the Schrödinger equation.The presence of a discrete infinity of exponential modes is shown to be a generic property of the similarity equations derived for a general n. Furthermore, we show that there may also exist non-uniqueness of the continuum; that is, more than one continuum of states can exist, that are isolated for fixed n and Ψ ∞ , but which are connected through an unfolded transcritical bifurcation at a critical value of the cross-flow parameter, Ψ ∞ .The multiplicity of states raises the question of solution selection, which is addressed using two stability analyses that assume the same basic symmetry properties as the base flow. In one case we consider a steady, algebraic form in the 'streamwise' direction, whilst in the other a temporal form is assumed. In both cases it is possible to extend the analysis to consider a continuous spectrum of disturbances that decay algebraically in the wall-normal direction. We note some obvious parallels that exist between such stability analyses and the approach to the continua of states described earlier in the paper.We also discuss the appearance of analogous non-unique states to the Falkner-Skan equation in the presence of an adverse pressure gradient (i.e. n < 0) in an appendix. † Manchester Centre for Nonlinear Dynamics.‡ Current address:

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Section: Introductionmentioning

confidence: 99%

Continua of states in boundary-layer flows

2002

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“…Despite being somewhat less intuitive than the primitive variable formulation, this form of (1) is well known in relation to corner boundary-layer flows, see for example the formulation of Dhanak and Duck [15] or Pal and Rubin [16]. We will follow the formulation of [15] further by combining (5a) and (5b) to obtain expressions for the Laplacian of both Φ and Ψ given by…”

Section: Formulationmentioning

confidence: 99%

“…where φ = O(1) in order to match with the spanwise varying displacement induced by the parabolic solution (15) and ζ 0η + δ F is the (outer) η → ∞ limit of the (inner) Falkner-Skan solution. Rewriting (5a) and (5b) in terms of ζ andη, and since both U and Θ remain o(ζ −2 0 ), we find that φ is determined by the harmonic problem,…”

Section: Asymptotic Description For ζ 0 → ∞: the Macro-scale Slot Limitmentioning

confidence: 99%

Micro-slot injection into a boundary layer driven by a favourable pressure gradient

Williams

Hewitt

2017

J Eng Math

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We investigate the effects of injection through a streamwise-aligned 'micro-slot' into a laminar boundary layer driven by a favourable pressure gradient of power-law type. The injection slot exists at all downstream locations, and is 'micro' in the sense that it has a finite spanwise width that is a fixed ratio of the local boundary-layer thickness. This approach is motivated by recent studies of micro-jets (of small spanwise and streamwise extents), which have indicated that for short spanwise scales, injection does not necessarily lead directly to separation. Injection in the absence of a free-stream pressure gradient has recently been analysed by Hewitt et al. (J Fluid Mech 822:617-639, 2017), and here we show that boundary layers in a favourable pressure gradient behave qualitatively differently. We present three-dimensional boundary-layer solutions affected by slot injection and contrast these with the corresponding zero pressure-gradient states. In the absence of a pressure gradient, injection results in low-speed streamwise-aligned streaks, where the amplitude and spanwise width of the injection determine the geometry of the streaks as one of the three possible types. The introduction of a favourable pressure gradient greatly reduces the spanwise extent of injection-driven streaks and removes the delineation between the three distinct flow regimes found in the zero pressure-gradient case. We present an asymptotic description in the limit of a large injection-slot width, thereby approaching the macro-slot limit from the micro-slot formulation. This description shows that not all injection rates and pressure gradients recover the expected Falkner-Skan solution at the centreline of the injection slot in the macro-slot limit. We explain this disagreement in terms of local spatial (cross flow) eigenmodes that are associated with a cross-flow collisional process at the centre of the injection slot.

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“…This approach assumes that spanwise lengthscales are comparable to the transverse boundarylayer thickness, ensuring that diffusion in the cross section is retained in both directions, but the longer streamwise lengthscale leads to neglect of streamwise diffusion. Similar formulations have previously been employed in high Reynolds number descriptions of (for example) corner boundary regions (Dhanak & Duck 1997), wakes behind elongated roughness elements (Goldstein et al 2016), flow near small-scale surface gaps (Hewitt & Duck 2014), the influence of upstream vorticity on transition (Wundrow & Goldstein 2001) and the generation of laminar streaks by freestream vorticity (Ricco & Dilib 2010). The formulation of van Dommelen & Yapalparvi (2014) is also equivalent if one considers the zero-curvature limit of their equations, although in their case the short-scale spanwise forcing is also assumed to be periodic, which simplifies the far-field behaviour compared to the (finite-spanwise extent) our problem.…”

Section: Introductionmentioning

confidence: 99%

Injection into boundary layers: solutions beyond the classical form

2017

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This theoretical and numerical study presents three-dimensional boundary-layer solutions for laminar incompressible flow adjacent to a semi-infinite flat plate, subject to a uniform free-stream speed and injection through the plate surface. The novelty in this case arises from a fully three-dimensional formulation, which also allows for slot injection over a spanwise lengthscale comparable to the boundary-layer thickness. This approach retains viscous effects in both the spanwise and transverse directions, and effectively results in a parabolised Navier-Stokes system (sometimes referred to as the 'boundaryregion equations'). Any injection profile can be described in this approach, but we restrict attention to three-dimensional states driven by a finite-width slot aligned with the flow direction and self-similar in their downstream development. The classical two-dimensional states are known to only exist up to a critical ('blow off') injection amplitude, but the three-dimensional solutions here appear possible for any injection velocity. These new states take the form of low-speed streamwise-aligned streaks whose geometry depends on the amplitude of injection and the spanwise width of the injection slot; intriguingly, although very low wall shear is typically obtained, streamwise flow reversal is not observed, however hard the blowing. Asymptotic descriptions are provided in the limit of increasing slot width and fixed injection velocity, which allows for classification of the solutions according to two bounding injection rates.

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The effects of freestream pressure gradient on a corner boundary layer

Cited by 34 publications

References 20 publications

Continua of states in boundary-layer flows

Continua of states in boundary-layer flows

Micro-slot injection into a boundary layer driven by a favourable pressure gradient

Injection into boundary layers: solutions beyond the classical form

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