Multi-Instability Analysis of Swept-Wing Boundary Layers Part 1. A Nonparallel Model of Stability Equations

Itoh, Nobutake; Atobe, Takashi

doi:10.2322/tjsass.45.195

Cited by 4 publications

(7 citation statements)

References 14 publications

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“…Slightly different neutral curves will be obtained from different theoretical models. In fact, the new model equation 8 predicts a lower critical value of C-F instability than that of S-C instability, in contrast to the theoretical prediction explained earlier. The other possibility for the inconsistency with the theoretical prediction is that initial amplitudes created near the source point are not the same for all wave-number components.…”

Section: Discussioncontrasting

confidence: 79%

Experiments on Streamline-Curvature Instability in Boundary Layers on a Yawed Cylinder

2005

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Instability of the three-dimensional boundary layer on a yawed circular cylinder placed in a uniform flow is investigated experimentally by introducing acoustic disturbances from a point near the attachment line. To exemplify the flow dominated by streamline-curvature instability rather than crossflow instability, which has been often observed in many swept-wing flows, is the aim here. In upstream regions of the disturbance wedge originating from the point source, both streamline-curvature and crossflow disturbances are superposed on each other and yield complicated amplitude distributions. A newly proposed method enables the decomposition of the distorted amplitude distribution into contributions from the two instability modes. Detailed observations, however, show that the crossflow mode decays with the distance from the source much faster than the streamline-curvature mode and allows the latter to be dominant in a region further downstream. A fundamental characteristic of the streamlinecurvature instability wave is confirmed by examining its phase distribution in the spanwise and normal directions. Wave numbers and spatial growth rates are in good agreement with theoretical predictions. NomenclatureA = spanwise amplitude distribution A = spanwise total amplitude distribution a = total wave function C = inclination of phase D = cylinder diameter f = dimensional frequency L = streamwise length l s = surface length m = Falkner-Skan parameter N = spatial growth rate of disturbance Q = streamwise (parallel to the local external streamline) local velocity q = fluctuation of streamwise local velocity Re Q = uniform-flow Reynolds number Re δ = local Reynolds number t = time U = chordwise local velocity u = fluctuation of chordwise local velocity V = spanwise local velocity X = nondimensional chordwise (normal to the attachment line) distance from the attachment line along the surface x = dimensional chordwise distance Y = nondimensional spanwise (parallel to the attachment line) distance from the disturbance source Y 0 = nondimensional spanwise location where amplitude becomes maximum Z = nondimensional radial (normal to the surface) distance from the surface α r = chordwise wave number of disturbance β r = spanwise wave number of disturbance Presented as Paper 99-0814 at Science and Engineering, Utsunomiya. ϕ = phase difference between crossflow and streamline-curvature modes δ = boundary-layer thickness δ 99 = boundary-layer thickness where streamwise local mean velocity Q is 99% of edge velocity Q e = sweep angle λ = wave length ν = kinematic viscosity σ = standard deviation of amplitude distribution ϕ = spanwise phase of disturbance ϕ 0 = offset of phasē ϕ = spanwise total phase distribution = phase modulation ω = angular frequencŷ ω = nondimensional frequency Subscripts CF = crossflow mode e = local edge SC = streamline-curvature mode ω = component with angular frequency ω ∞ = uniform flow

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Section: Discussioncontrasting

confidence: 79%

Experiments on Streamline-Curvature Instability in Boundary Layers on a Yawed Cylinder

2005

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“…It should be recalled here that the familiar G ortler equations based on the parallel-ow assumption gave an unreasonal critical point located at inÿnitesimally small value of ÿ. In contrast, we can expect that the neutral curve obtained from the present equation system possesses a critical point at a ÿnite value of ÿ and its left-hand branch goes up to inÿnity as ÿ tends to zero, because the disturbance equations (4.3) include all terms of the model equations given by Itoh (1995), who added only the nonparallel term W to the G ortler equations and thereby succeeded in getting a neutral curve of the expected shape. Putting Ä = G 2 in Eqs.…”

Section: Eigenvalue Problemmentioning

confidence: 80%

“…The natural deduction that ÿ 2 G 2 will approach a nonzero constant as ÿ decreases to zero indicates sharp rising of G along a hyperbolic asymptote of our neutral curve as ÿ → 0. This fact suggests the presence of a critical point within a ÿnite range of the spanwise wavenumber, unlike the parallel-ow case of G ortler's eigenvalue problem, whose dependence on G through the term ÿ 2 G 2 is destroyed in the boundary conditions at the limit of small ÿ, as discussed in more detail by Itoh (1995).…”

Section: Eigenvalue Problemmentioning

confidence: 98%

A non-parallel theory for Görtler instability of Falkner–Skan boundary layers

Itoh¹

2001

Fluid Dyn. Res.

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Centrifugal instability of two-dimensional similar boundary layers along a concave wall is governed by partial di erential equations with respect to the normal-to-wall coordinate in a nondimensional form and the Reynolds number based on local velocity of external stream and a boundary-layer thickness. In a particular case of the stagnation-point ow with the Falkner-Skan parameter m = 1, however, the exact equations admit a series solution expanded in inverse powers of the Reynolds number and its coe cients can be obtained by solving a sequence of ordinary di erential systems. Of particular importance is that the leading term is determined from an eigenvalue problem more involved than G ortler's parallel-ow approximation. Numerical evaluation of the series solution thus obtained shades light on fundamental e ects of the boundary-layer nonparallelism and ÿnite Reynolds numbers on theoretical prediction of G ortler instability.

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“…The experimental results show good agreement with Itoh's prediction (Itoh 1996a) using linear stability theory for swept Hiemenz flow but not with the solution of the improved model equation (Itoh and Atobe 2002) which takes into account the variation of the Falkner-Skan-Cooke parameter, m, and the wall curvature.…”

Section: Discussionmentioning

confidence: 58%

“…The dotted line is the theoretical result (Itoh 1996a) for swept Hiemenz flow modeled by an equation composed of the Orr-Sommerfeld (O-S) equation plus additional terms relating to the non-parallelism and curvature of the external streamline. The broken line (Itoh and Atobe 2002) is the solution for the yawed cylinder boundary layer using a recently improved model equation which takes into account the variation of the F-S-C parameter, m, and the wall curvature. The difference between these theoretical results becomes larger as the attachment line is approached, that is smaller values of 1/γ .…”

Section: Critical Frequency and Critical Wavenumbermentioning

confidence: 99%

Experimental investigation of the flow instability near the attachment-line boundary layer on a yawed cylinder

Nishizawa¹,

Tokugawa

Takagi

2009

Fluid Dyn. Res.

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The behavior of small disturbances in a 3-D laminar boundary layer on a yawed cylinder was experimentally investigated. This setup simulates the flow around the leading edge of swept wings. Since multiple instability modes appear near the attachment-line region, a point-source disturbance was artificially introduced to separate these modes. Amplitude and phase distributions of the disturbances originating from the point source were measured using a hotwire probe near the attachment-line flow to test existing theoretical predictions. Hotwire measurements show that two instability modes definitely coexist and overlap in the middle portion of the wedge-shaped region developing downstream of the point source. Decomposition by 2-D fast Fourier transform (FFT) analysis enables us to separate one oblique wave from the other. One of the oblique waves belongs to the cross-flow instability mode, which travels to the attachment line and grows even at Reynolds numbers that are slightly lower than the critical Reynolds number for the attachment-line instability. The origin of the other mode is not identifiable, because it has peculiar characteristics different from both the streamline-curvature instability mode and the crossflow instability mode. This mode decays in the downstream direction for all frequencies examined. By investigating the spatial characteristics of the small disturbance, the critical Reynolds number for cross-flow instability was successfully determined in the off-attachment-line region. The value, R c = 543, was lower than the critical Reynolds number of R c = 583 for the attachment-line flow. Furthermore, the critical frequency and wavenumber were in good agreement with existing predictions based on linear stability theory.

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Multi-Instability Analysis of Swept-Wing Boundary Layers Part 1. A Nonparallel Model of Stability Equations

Cited by 4 publications

References 14 publications

Experiments on Streamline-Curvature Instability in Boundary Layers on a Yawed Cylinder

Experiments on Streamline-Curvature Instability in Boundary Layers on a Yawed Cylinder

A non-parallel theory for Görtler instability of Falkner–Skan boundary layers

Experimental investigation of the flow instability near the attachment-line boundary layer on a yawed cylinder

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