Asymptotic Analysis of the Orr-Sommerfeld Problem for Boundary-Layer Flows

LAKIN, W. D.; Reid, W. H.

doi:10.1093/qjmam/35.1.69

Cited by 7 publications

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“…We determined the minimum critical Reynolds number Re,,, and the corresponding critical values of aCrit and cCrit. These were Recrit = 46095, a,,,, = 0.1592, cCrit = 0.1564, and compare very favourably with those found by Lakin & Reid (1982) We performed a numerical solution of the full system of differential equations and boundary conditions (6.27)-(6.33) by a similar method to that described in $5. Our results are displayed in table 2 for the model lead alloy data given in table 1 but with Sc = 5.…”

Section: )supporting

confidence: 71%

“…This confirms the above analysis that indicates that the shear flow mode of instability is unaffected by the presence of the freezing interface for high Schmidt numbers. The quantitative disparity between the asymptotic results and the numerical solution of the governing equations can be attributed to the neglect of the D3 operation in the approximations to the Orr-Sommerfeld equation, as was remarked by Lakin & Reid (1982).…”

Section: )mentioning

confidence: 97%

“…They developed approximations in the limit of large Reynolds number to the solutions of the Orr-Sommerfeld equation by heuristic arguments. Subsequently Lakin & Reid (1982) derived uniformly valid approximations to the Orr-Sommerfeld equation in this limit and so determined more accurate values for the critical Reynolds number. Numerical solutions of the Orr-Sommerfeld equation have been obtained by Ng & Reid (1980) and Mack (quoted in Drazin & Reid 1981).…”

Section: Introductionmentioning

confidence: 99%

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Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model

Forth

Wheeler

1989

J. Fluid Mech.

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In this paper we consider the effect of a model boundary-layer flow on the hydrodynamic and morphological stability of a simple model of the solidification of a binary alloy. We conduct a linear analysis and develop asymptotic solutions for large Schmidt number and large Reynolds number. We also present numerical solutions for data appropriate to a lead–tin alloy. We show that for modes parallel to the free-stream velocity the flow is responsible for the appearance of travelling waves and, for common values of the material parameters, may stabilize the morphological stability of the interface. However the morphological stability of modes perpendicular to the free-stream velocity is unaffected by the presence of the flow. The hydrodynamic stability of the boundary layer is very weakly affected by the presence of the interface, which we attribute to the large Schmidt numbers associated with real crystal growth situations.

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Section: )supporting

confidence: 71%

Section: )mentioning

confidence: 97%

Section: Introductionmentioning

confidence: 99%

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Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model

Forth

Wheeler

1989

J. Fluid Mech.

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“…On the other hand, high resolution is required in the thin active region of the perturbations, making it impractical to integrate much further from the interface than the outer limit of the active regions. The imposition of an artificial top or floor is known to yield spurious modes in the resulting eigenfunction spectrum (Lakin & Grosch 1982). Therefore we have truncated the velocity distribution in the air by choosing simple G .…”

Section: Model Of the Shear Flowmentioning

confidence: 99%

Instability waves on the air–sea interface

Wheless¹,

Csanady²

1993

J. Fluid Mech.

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We used a compound matrix method to integrate the Orr–Sommerfeld equation in an investigation of short instability waves (λ < 6 cm) on the coupled shear flow at the air–sea interface under suddenly imposed wind (a gust model). The method is robust and fast, so that the effects of external variables on growth rate could easily be explored. As expected from past theoretical studies, the growth rate proved sensitive to air and water viscosity, and to the curvature of the air velocity profile very close to the interface. Surface tension had less influence, growth rate increasing somewhat with decreasing surface tension. Maximum growth rate and minimum wave speed nearly coincided for some combinations of fluid properties, but not for others.The most important new finding is that, contrary to some past order of magnitude estimates made on theoretical grounds, the eigenfunctions at these short wavelengths are confined to a distance of the order of the viscous wave boundary-layer thickness from the interface. Correspondingly, the perturbation vorticity is high, the streamwise surface velocity perturbation in typical cases being five times the orbital velocity of free waves on an undisturbed water surface. The instability waves should therefore be thought of as fundamentally different flow structures from free waves: given their high vorticity, they are akin to incipient turbulent eddies. They may also be expected to break at a much lower steepness than free waves.

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Observations on the Role of Vorticity in the Stability Theory of Wall Bounded Flows

Herron

1991

Stud Appl Math

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The linearized equations for the evolution of disturbances to four wall bounded flows are treated. The flows are plane Couette flow and plane Poiseuille flow, Hagen-Poiseuille pipe flow, and the asymptotic suction profile. By looking at the vorticity it is proved simply that plane Couette flow and Hagen-Poiseuille flow are linearly stable. Further study is made of the structure of the disturbance equation by the introduction of a special vorticity adjoint.

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Asymptotic Analysis of the Orr-Sommerfeld Problem for Boundary-Layer Flows

Cited by 7 publications

References 0 publications

Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model

Hydrodynamic and morphological stability of the unidirectional solidification of a freezing binary alloy: a simple model

Instability waves on the air–sea interface

Observations on the Role of Vorticity in the Stability Theory of Wall Bounded Flows

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