Structure of Absolute Instability in 3-D Boundary Layers: Part 2. Application to Rotating-Disk Flow.

Itoh, Nobutake

doi:10.2322/tjsass.44.101

Cited by 11 publications

(4 citation statements)

References 7 publications

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“…6 as 258 and 50.6, respectively, which are slightly lower than the values R c ¼ 290, 282 and 284 for C-F mode given by Lingwood, 11) Itoh 12) and Pier, 13) and R c ¼ 69, 64 and 63 for the S-C mode by Faller,8) Balakumar and Malik 16) and Itoh,12) respectively, as well as the experimentally observed lower limits of C-F mode listed by Malik et al 17) However, the Orr-Sommerfeld equation gives R c ¼ 177 for the C-F mode, 15) which is much lower than the above non-parallel estimations. Since the present study restricts non-parallel terms only to those of the most fundamental activity in the multiple instabilities, it is reasonable that the equations proposed give a critical value midway between the parallel approach and the non-parallel attempts.…”

Section: Eigenvalue Problems For Multiple Instabilitiesmentioning

confidence: 53%

“…Thus, various forms of stability equation system have been proposed to determine the complex dispersion relation for Kármán's rotating-disk flow (i.e., Malik 10) and others [11][12][13] ), though none of them seems to be commonly acceptable in terms of reliability and convenience. It may be natural to expect that different instabilities should be described by different stability equations, as those for Tollmien-Schlichting instability and Görtler instability of two-dimensional boundary layers along a weakly concave surface.…”

Section: Introductionmentioning

confidence: 99%

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Stability Analysis of Rotating-Disk Flows with Uniform Suction

Itoh

2017

Trans. Japan Soc. Aero. S Sci.

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Instability of the flow on a rotating disk is governed by linearized disturbance equations of the partial differential with respect to the radial distance from the rotation axis and the normal distance from the disk surface. Applying uniform suction from the surface brings a small parameter associated with displacement thickness of the circumferential velocity profile into a dimensionless form of the equation system. Two kinds of series solutions expanded by the powers of this parameter are obtained to describe the cross-flow and centrifugal instabilities of the flow having a twisted velocity profile. The leading terms of the series solutions are determined from two eigenvalue problems of slightly different ordinary differential equations, and the superposition of those equations leads to an eigenvalue problem applicable to multiple-instability characteristics of such three-dimensional boundary layers.

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Section: Eigenvalue Problems For Multiple Instabilitiesmentioning

confidence: 53%

Section: Introductionmentioning

confidence: 99%

Stability Analysis of Rotating-Disk Flows with Uniform Suction

Itoh

2017

Trans. Japan Soc. Aero. S Sci.

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“…The critical Reynolds number she estimated was 507, under an assumption of parallel flow. However, Itoh [14,15] showed that under a slightly non-parallel-flow condition, the flow field became stable for absolute instability. Davies & Carpenter [16] in their numerical simulation showed that, in a non-parallel flow field, the velocity fluctuation consequently grew by the convective instability.…”

Section: Introductionmentioning

confidence: 99%

Relation Between Convective Instability and Global Instability on a Rotating Disk

Lee¹,

Nishio²,

Izawa³

et al. 2018

TOMEJ

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Background: The velocity fluctuations grow dominantly by convective instability form 32 spiral vortices which are stationary with respect to the disk. However, recent researches suggest that the global instability plays a role in the boundary layer transition. Objective: The study looks into the relation between convective instability and global instability. Method: A finite difference method is used to carry out numerical simulation. The full Navier-Stokes perturbation equations and the continuity equation solved by simulation code. Results: A disturbance is continuatively introduced to excite the convectively unstable mode, which successfully generates a flow field with 32 spiral and stationary vortices. Next, a short-duration artificial disturbance with an azimuthal wavenumber of 64 is introduced at Reynolds number of 530 in order to introduce a velocity fluctuation of the traveling mode, which is globally unstable. It is shown that the source of vibration for the globally unstable mode exists between Reynolds number of 560 and 670. Finally, the global and traveling wavenumber 64 component is excited in a flow field which is dominated by the convective and stationary wavenumber 32 component. It is shown that the wavenumber 64 component grows by the global instability even when the excitation is very weak. Conclusion: The results suggest that the reason why the globally unstable mode has not been observed in experiments is because the boundary layer transition caused by the convective instability takes place before the globally unstable mode can start to grow by itself.

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“…However, it was found that the results would be different if the target of the investigation was the global instability instead of the absolute instability. Itoh [14,15] used the theoretical method and employed the vicinity of the singularity point for the zero complex group velocity. He showed that the flow field under the slightly non-parallel-flow condition was absolutely stable, and therefore concluded that applying the parallel-flow assumption is not proper.…”

Section: Introductionmentioning

confidence: 99%

Growth of Spiral Vortices in a Globally Unstable Region on a Rotating Disk

Lee¹,

Nishio²,

Izawa³

et al. 2018

TOMEJ

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Background: Instabilities on a rotating disk ﬂow can be classiﬁed into two distinct groups, convective instability and global instability. Contrary to the convective instability, many characteristics of the global instability are left unknown despite of a lot of researches. Objective: The study investigates the characteristics of the globally unstable mode. Method: Numerical simulation is carried out by a finite difference method. The simulation code solves the full Navier-Stokes perturbation equations and the continuity equation. Results: Four cases with azimuthal domain sizes of 2π/10, 2π/70, 2π/80 and 2π/90 are compared. In all cases, a short-duration wall-normal random suction and blowing is introduced from the wall at the beginning of the computation. In the computation for the wider size 2π/10 domain, wavenumber components of 70, 80 and 90 are all found to co-exist in the ﬂow ﬁeld, with the wavenumber 80 component being much stronger than the other two components. The strength of the wavenumber 80 component is equivalent to the narrower domain case. For the other two wavenumber components of 70 and 90, the strengths in the wider domain case are much lower compared to the corresponding narrower domain cases. When the same wavenumber components are compared between the wider and narrower domain computations, no difference can be found, indicating that each wavenumber component grows by the global instability. Conclusion: The results imply that the amplitude saturation levels of wavenumber components 70 and 90 are suppressed by the wavenumber 80 component through the nonlinear effect.

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Structure of Absolute Instability in 3-D Boundary Layers: Part 2. Application to Rotating-Disk Flow.

Cited by 11 publications

References 7 publications

Stability Analysis of Rotating-Disk Flows with Uniform Suction

Stability Analysis of Rotating-Disk Flows with Uniform Suction

Relation Between Convective Instability and Global Instability on a Rotating Disk

Growth of Spiral Vortices in a Globally Unstable Region on a Rotating Disk

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