Stability of Nonrotationally Symmetric Disturbances for Inviscid Flow between Rotating Cylinders

Krueger, E. R.; DiPrima, R. C.

doi:10.1063/1.1706531

Cited by 9 publications

(4 citation statements)

References 5 publications

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“…Therefore, the boundary conditions are ψ 1 (R 1 )=ψ 1 (R 2 ) = 0 imposing the condition k It can be checked that this particular condition agrees with the results of Krueger & DiPrima (1962) for µ → 1. However, we are interested here in the case k ≫ 1 for whichr 1 <R 2 .…”

Section: Taylor-couette Flowsupporting

confidence: 68%

“…The WKB approximation (7.2) is then valid throughout the interval R 1 < r < R 2 . Therefore, the boundary conditions are ψ 1 (R 1 ) = ψ 1 (R 2 ) = 0 imposing the condition k It can be checked that this particular condition agrees with the results of Krueger & DiPrima (1962) for µ → 1.…”

Section: Taylor-couette Flowsupporting

confidence: 63%

“…For example, Smyth & McWilliams (1998) and Gallaire & Chomaz (2003) have investigated numerically the stability of the Carton & McWilliams (1989) vortex profiles and shown that only the first azimuthal modes are unstable with growth rates smaller than for the axisymmetric mode. Regarding the Couette flow in the inviscid limit, Krueger & DiPrima (1962) and Bisshopp (1963) have shown, in some limiting cases (narrow gap, small azimuthal wavenumber), that the axisymmetric mode is always more unstable than non-axisymmetric modes for a given axial wavenumber. The goal of the present paper is to extend the Rayleigh criterion to nonaxisymmetric perturbations following an approach similar to the recent analysis of Kelvin waves by Le Dizès & Lacaze (2005).…”

Section: Introductionmentioning

confidence: 99%

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Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

2005

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The well-known Rayleigh criterion is a necessary and sufficient condition for inviscid centrifugal instability of axisymmetric perturbations. We have generalized this criterion to disturbances of any azimuthal wavenumber m by means of large-axialwavenumber WKB asymptotics. A sufficient condition for a free axisymmetric vortex with angular velocity Ω(r) to be unstable to a three-dimensional perturbation of azimuthal wavenumber m is that the real part of the growth rateis positive at the complex radius r = r 0 where ∂σ(r)/∂r =0, i.e.where φ =(1/r 3 )∂r 4 Ω 2 /∂r is the Rayleigh discriminant, provided that some ap o s t e r i o r ichecks are satisfied. The application of this new criterion to various classes of vortex profiles shows that the growth rate of non-axisymmetric disturbances decreases as m increases until a cutoff is reached. The criterion is in excellent agreement with numerical stability analyses of the Carton & McWilliams (1989) vortices and allows one to analyse the competition between the centrifugal instability and the shear instability. The generalized criterion is also valid for a vertical vortex in a stably stratified and rotating fluid, except that φ becomes φ =(1/r 3 )∂r 4 (where Ω b is the background rotation about the vertical axis. The stratification is found to have no effect. For the Taylor-Couette flow between two coaxial cylinders, the same criterion applies except that r 0 is real and equal to the inner cylinder radius. In sharp contrast, the maximum growth rate of non-axisymmetric disturbances is then independent of m.

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Section: Taylor-couette Flowsupporting

confidence: 68%

Section: Taylor-couette Flowsupporting

confidence: 63%

Section: Introductionmentioning

confidence: 99%

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Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

2005

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“…Rudraiah (1970) carried out the stability analysis of axial flow of heterogeneous, incompressible and :electrically conducting fluid between two fixed co-axial cylinders. DiPrima (1961), Krueger and DiPrima (1962) and Agrawal (1970) have gone further into the analysis by allowing asymmetric perturbations. The onset of thermal instability in a static horizontal layer of homogeneous fluid kept under a uniform temperature gradient has been investigated by many authors.…”

Section: Introductionmentioning

confidence: 99%

Thermal instability of Couette flow

Agrawal

Saini

1974

J. Aust. Math. Soc.

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SummaryThe present paper investigates the thermal instability of a non-homogeneous fluid rotating between two co-axial cylinders when the inner cylinder is being heated uniformly. The conditions are established under which the oscillatory and non-oscillatory modes exist and further it has been shown that the oscillatory modes are amplified due to the adverse temperature gradient. In the case of non-oscillatory modes, sufficient conditions for stability and instability are obtained.

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Bibliography

1998

Theoretical Fluid Dynamics

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Stability of Nonrotationally Symmetric Disturbances for Inviscid Flow between Rotating Cylinders

Cited by 9 publications

References 5 publications

Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

Generalized Rayleigh criterion for non-axisymmetric centrifugal instabilities

Thermal instability of Couette flow

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