A Nonlinear Investigation of the Stationary Modes of Instability of the Three-Dimensional Compressible Boundary Layer Due to a Rotating Disc

Seddougui, Sharon O.

doi:10.1093/qjmam/43.4.467

Cited by 15 publications

(67 citation statements)

References 9 publications

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“…All viscous effects are therefore contained in the lower deck, which has to satisfy the no-slip condition on the surface of the cone. The analysis that follows is consistent with that of Seddougui (1990) who considered the related problem of the compressible rotating-disk boundary layer (i.e. ψ = 90 • ).…”

Section: Of 21supporting

confidence: 80%

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Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

et al. 2016

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We investigate the effects of compressibility and wall cooling on the stationary, viscous (Type II) instability mode within the three-dimensional boundary layer over rotating cones with half-angle greater than 40 • . The stationary mode is characterised by zero shear stress at the wall and a triple-deck solution is presented in the isothermal case. Asymptotic solutions are obtained which describe the structure of the wavenumber and the orientation of this mode as a function of local Mach number. It is found that a stationary mode is possible only over a finite range of local Mach number. Our conclusions are entirely consistent with the results of Seddougui [Q. J. Mech. Appl. Math. 43]. It is suggested that wall cooling has a significant stabilising effect, while reducing the half-angle is marginally destabilising. Solutions are presented for air.

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Section: Of 21supporting

confidence: 80%

“…The analysis follows the closely-related rotating-disk studies of Hall (1986) and Seddougui (1990). However, unlike those papers, we are particularly interested in the use of wall cooling as a potential flow control mechanism and do not consider the adiabatic (i.e.…”

Section: Introductionmentioning

confidence: 99%

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

et al. 2016

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“…With suction the variations in the coefficients of amplitude equation (2.14) are predominantly due to the changes in the wall shears; these are in turn due to the changing boundary condition (2.4c) for the basic flow. Preliminary consideration of the compressible version of the present problem also suggests that the modifications required in order to account for suction in Seddougui's (1990) …”

Section: Discussionmentioning

confidence: 98%

“…Seddougui (1990) considered the effects of compressibility on the subcritical mode of M and found that for both an adiabatic wall and an isothermal wall the stationary mode is only possible over a finite range of Mach numbers. The analysis conducted by Seddougui (1990) is naturally more involved than that of M but we have seen here that, at least for the incompressible flow problem, the inclusion of suction requires only relatively minor changes to the nonlinear analysis. With suction the variations in the coefficients of amplitude equation (2.14) are predominantly due to the changes in the wall shears; these are in turn due to the changing boundary condition (2.4c) for the basic flow.…”

Section: Discussionmentioning

confidence: 99%

The effects of suction on the nonlinear stability of the three-dimensional boundary layer above a rotating disc

Bassom

Seddougui

1992

Proc. R. Soc. Lond. A

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ASA 92-00673 AbstractThere exist two types of stationary instability of the flow over a rotating disc corresponding to the upper-branch, inviscid mode and the lower-branch mode, which has a triple deck structure, of the neutral stability curve. A theoretical investigation of the linear problem and an account of the weakly nonlinear properties of the lower branch-modes have been undertaken by Hall (1986) and MacKerrell (1987) respectively. Motivated by recent reports of experimental sightings of the lower-branch mode and an examination of the role of suction on the linear stability properties of the flow here we investigate the effects of suction on the nonlinear disturbance described by MacKerrell (1987). The additional analysis required in order to incorporate suction is relatively straightforward and enables us to derive an amplitude equation which describes the evolution of the mode. For each value of the suction a threshold value of the disturbance amplitude is obtained; modes of size greater than this threshold grow without limit as they develop away from the point of neutral stability.

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“…This certainly requires the multiple layer analysis of [9] and [40] (see also [41]), which seems most appropriate for the neutral waves found in this work. Finally, it was shown in [42] that the non-linearity is also destabilizing in the compressible rotating-disk boundary layer flow as far as the stationary modes are concerned. The effects of wall insulation and isothermal wall were also found to be destabilizing, indicating the greatest likelihood of instability through highly cooled walls.…”

Section: Discussionmentioning

confidence: 99%

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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A Nonlinear Investigation of the Stationary Modes of Instability of the Three-Dimensional Compressible Boundary Layer Due to a Rotating Disc

Cited by 15 publications

References 9 publications

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

Viscous modes within the compressible boundary-layer flow due to a broad rotating cone

The effects of suction on the nonlinear stability of the three-dimensional boundary layer above a rotating disc

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

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