Upper branch nonstationary modes of the boundary layer due to a rotating disk

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.

Section: Linear Results (δE Terms)mentioning

confidence: 99%

“…Making use of the asymptotic triple-deck theory, the linear and non-linear evolution of the upper branch modes of the rotating-disk boundary layer flow, as far as the orientation of the non-stationary waves is concerned, were examined in [28] and [29]. These modes are the ones naturally observed in the experiments of [12], [17] and [30,31].…”

mentioning

confidence: 99%

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

2007

Quart. Appl. Math.

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“…It was shown that the lower branch corresponds to an effective velocity profile having zero shear stress at the surface of the disk. 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].…”

Section: Suction/blowing Influences On the Finite Amplitude Disturbanmentioning

confidence: 99%

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

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 ...

“…An intriguing point which deserves attention is that, unlike the other boundary layer flows, like the Blasius flow (see [26]), here the stability quantities found in equations (4.1-4.2), and as a result, the structure of the lower branch modes arise from a balance between the viscous forces and Coriolis effects. The eigenrelation determining the upper branch modes on the other hand arises from a balance between various jumps across the critical layers and Stokes layer shift, see [29]. In figures 3(a-d), a comparison has been made between the computed numerical results of parallel flow approximation (see [7]) and the asymptotic predictions as obtained from equations (4.1-4.2) at a suctions = 1 and blowings = −1.…”

Section: Turkyilmazoglu Zampmentioning

confidence: 99%

“…Both the upper branch and lower branch stationary neutral modes and their asymptotic structures were obtained within the framework of asymptotic expansion at large Reynolds numbers. Making use of the asymptotic triple deck theory, the linear and non-linear evolution of the upper branch modes of the rotating-disk boundary layer flow, as far as the orientation of the non-stationary waves is concerned, were examined in [28] and [29]. These modes are the ones naturally observed in the experiments of [12], [17] and [30,31].…”

Section: Introductionmentioning

confidence: 99%

Effects of suction/blowing on the lower branch modes of the incompressible boundary layer flow due to a rotating-disk

2007

Z. angew. Math. Phys.

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In this study a theoretical approach is pursued to investigate the effects of suction and blowing on the structure of the lower branch neutral stability modes of three-dimensional small disturbances imposed on the incompressible von Karman's boundary layer flow induced by a rotating-disk. Particular interest is placed upon the short-wavelength, non-linear and nonstationary crossflow vortex modes developing within the presence of suction/blowing at sufficiently high Reynolds numbers with reasonably small scaled frequencies. Following closely the asymptotic framework introduced in [1], the role of suction on the non-linear disturbances of the lower branch described first in [2] for the stationary modes only, is extended in order to obtain an understanding of the behavior of non-stationary perturbations. The analysis using the rational asymptotic technique based on the triple-deck theory enables us to derive initially an eigenrelation which describes the evolution of linear modes. The asymptotic linear modes calculated at high Reynolds number limit are found to be destabilizing as far as the non-parallelism accounted by the approach is concerned, and they compare fairly well with the numerical results generated directly by solving the linearized system with the usual parallel flow approximation. An amplitude equation is derived next to account for the effects of non-linearity. Even though the form of this equation is the same as that of found in [2] for no suction, it is under the strong influence of suction and blowing. This amplitude equation is shown to be adjusted by a balance between viscous and Coriolis forces, and it describes the evolution of not only the stationary but also the non-stationary modes for both suction and injection applied at the disk surface. A close investigation of the amplitude equation shows that the non-linearity is highly 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 at destabilization of the flow under consideration. Finally, a smaller initial amplitude of a disturbance is found to be sufficient for the non-linear amplification of the modes in the case of suction, whereas a larger amplitude is required if injection is active on the surface of the disk.