Crossflow instability in a spinning disk boundary layer

Kohama, Yasuaki; Suda, Kenta

doi:10.2514/3.11344

Cited by 16 publications

(6 citation statements)

References 12 publications

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“…These modes are the ones naturally observed in the experiments of [12], [17] and [30,31]. However, as first detected in the experiment of [19], there exist lower branch modes corresponding to a lower Reynolds number.…”

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confidence: 69%

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

Türkyılmazoğlu

2007

Quart. Appl. Math.

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Abstract. In this paper a theoretical study is undertaken to investigate the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible Von Karman's boundary layer flow due to a rotating disk. Particular attention is given to the short-wavelength non-linear non-stationary crossflow vortex modes at sufficiently high Reynolds numbers with reasonably small scaled frequencies. [497][498][499][500][501][502][503][504][505][506][507][508][509][510][511][512][513] for the stationary linear and non-linear modes, it is revealed here that the nonstationary modes with sufficiently long time scale can also be described by an asymptotic expansion procedure based on the triple-deck theory. Making use of this approach, which takes into account the non-linear and non-parallel effects, the asymptotic structure of the non-stationary modes is shown to be adjusted by a balance between viscous and Coriolis forces, and resulted from the fact of vanishing shear stress at the disk surface. As a consequence of the matching of the solutions in adjacent regions it is found that in the linear case the wavenumber and the orientation of the lower branch modes are governed by an eigenrelation, which is akin to the one obtained previously in [Proc. Roy. Soc. London Ser. A 406 (1986), 93-106] for the stationary modes. The asymptotic theory shows that the non-parallelism has a destabilizing effect. A Landau-type equation for the modulated vortex amplitude with coefficients that are often difficult to get from finite Reynolds number computations has also been obtained from a weakly non-linear analysis in the limit of infinitely large Reynolds numbers. The non-linearity has also been found to be destabilizing for both positive and negative frequency waves, though finite amplitude growth of a disturbance having positive frequency close to the neutral location is more effective.

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confidence: 69%

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

Türkyılmazoğlu

2007

Quart. Appl. Math.

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“…It was evident in the hot-wire signals of Wilkinson & Malik (1983. It has been studied experimentally in detail by Kohama (1984Kohama ( , 1987 and Kohama & Suda (1993) and theoretically by Balachandar, Streett & Malik (1992) by means of a Floquet analysis. More recently Corke & Knasiak (1996) have shown that a triad coupling between pairs of travelling eigenmodes and a stationary mode could be responsible for the final breakdown to turbulence in their experimental study.…”

Section: Introductionmentioning

confidence: 99%

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 1. Type I and II instabilities

Cooper

Carpenter

1997

J. Fluid Mech.

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A theoretical study into the effects of wall compliance on the stability of the rotating-disc boundary layer is described. A single-layer viscoelastic wall model is coupled to a sixth-order system of fluid stability equations which take into account the effects of viscosity, Coriolis acceleration, and streamline curvature. The coupled system of equations is integrated numerically by a spectral Chebyshev-tau technique.Travelling and stationary modes are studied and wall compliance is found to greatly increase the complexity of the eigenmode spectrum. It is effective in stabilizing the inviscid Type I (or cross-flow) instability. The effect on the viscous (Type II) eigenmode is more complex and can be strongly destabilizing. An analysis of the energy flux indicates that this destabilization arises as a result of a large degree of energy production by viscous stresses at the wall/flow interface.The Type I and II instabilities are shown to be negative and positive energy waves respectively. The co-existence of eigenmodes of opposite energy type indicates the possibility of modal interaction and coalescence. It is found that, compared with the rigid disc, wall compliance promotes the interaction and coalescence of the Type I and II eigenmodes. There is an associated strong instability which appears to be characterized by marked horizontal motion of the compliant surface. Modal coalescence is interpreted physically as producing local algebraic growth which could advance the onset of nonlinear effects.

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“…Making use of the asymptotic triple deck theory, the linear and non-linear evolution of upper branch modes of the rotating disk boundary layer flow, as far as the orientation of the non-stationary waves are concerned, were examined in [32] and [33]. These modes are the ones naturally observed in the experiments of [14], [20] and [34,35]. However, as first detected in the experiment of [23], there exists lower branch modes corresponding to a lower Reynolds number.…”

Section: Suction/blowing Influences On the Finite Amplitude Disturbanmentioning

confidence: 97%

Influence of suction/blowing on the finite amplitude disturbances having non-stationary modes of a compressible boundary layer flow

Türkyılmazoğlu

2008

Quart. Appl. Math.

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Abstract. In this paper, a weakly non-linear stability analysis is pursued to explore the effects of suction and blowing on a compressible mode of instability of the threedimensional boundary layer flow induced by a rotating disk. The main thrust of the research is to extend the stationary work of Seddougui and Bassom (1996) to cover for the non-stationary mode so that it is targeted to determine whether the major role in finite amplitude destabilization of the boundary layer is played by the stationary mode of Seddougui and Bassom (1996) or the non-stationary mode as calculated from the present study. Within this perspective, the basic compressible flow obtained in the large Reynolds number limit, together with a suction parameter entering into the wall with normal velocity at the wall, is perturbed by disturbances which are constituted as the non-linear interaction of fundamental modes and harmonics. The effects of non-linearity are then explored by deriving a finite amplitude equation governing the evolution of the non-linear lower branch modes and also allowing the finite amplitude growth of a disturbance close to the neutral location determined from a prior stability analysis. Although the form of the amplitude equation is not at all surprising (given a similar result for the stationary vortices in Seddougui and Bassom (1996)), the dependence of the coefficients of the Landau-type modulated vortex amplitude equation on the frequency parameter is established here. A close examination of the coefficients of this evolution equation indicates strongly that the non-linearity has a destabilizing effect for both suction and blowing through the surface of the disk, the effect of which is much more stronger for a suction. Also the impact of non-linearity is higher for the non-stationary compressible modes than for the stationary waves of Seddougui (1990) and Seddougui and Bassom (1996). Moreover, in the case of suction, there occurs a regime of non-stationary modes that cover not only the positive frequency waves, but also waves having negative frequencies, and these modes are always unstable no matter whether the wall is insulated or isothermal. The solution of the asymptotic amplitude equation further demonstrates that as the local Mach number increases, compressibility has the influence of stabilization by requiring smaller initial amplitude of the disturbance for the laminar rotating disk boundary layer flow to become unstable whenever fluid injection is applied. Unlike this, suction makes the underlying flow more convectively unstable as far as the compressibility is concerned, particulary for the modes generated as a consequence of an isothermal wall. Both for the suction and injection cases, disturbances having positive frequency are always shown to cause an instantaneous non-linear amplification prior to the negative frequency waves. Introduction.In the present study, we consider the compressible boundary layer flow on a rotating disk and its non-linear instability owing to the existence of threedimensional ...

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Crossflow instability in a spinning disk boundary layer

Cited by 16 publications

References 12 publications

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

Non-linear and non-stationary modes of the lower branch of the incompressible boundary layer flow due to a rotating disk

The stability of rotating-disc boundary-layer flow over a compliant wall. Part 1. Type I and II instabilities

Influence of suction/blowing on the finite amplitude disturbances having non-stationary modes of a compressible boundary layer flow

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