Direct spatial resonance in the laminar boundary layer due to a rotating-disk

Türkyılmazoğlu, Mustafa; Gajjar, Jitesh S. B.

doi:10.1007/bf02703508

Cited by 17 publications

(16 citation statements)

References 48 publications

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“…Lingwood suggested that type I and type III were the coalescing branches at the origin of the type I absolute mode. Numerical investigations of the linear stability equations conducted by Turkyilmazoglu & Gajjar (2000) and of the linearized Navier-Stokes equations by Davies & Carpenter (2003) yielded results consistent with those of Lingwood when the base flow is parallel, confirming the existence of an absolute instability. However, Turkyilmazoglu & Gajjar (2000) argued that if the nonparallel effects were taken into account, the transition from a convective to an absolute regime might occur at a significantly higher Reynolds number, possibly above the experimental onset of turbulence.…”

Section: Introductionsupporting

confidence: 53%

“…Numerical investigations of the linear stability equations conducted by Turkyilmazoglu & Gajjar (2000) and of the linearized Navier-Stokes equations by Davies & Carpenter (2003) yielded results consistent with those of Lingwood when the base flow is parallel, confirming the existence of an absolute instability. However, Turkyilmazoglu & Gajjar (2000) argued that if the nonparallel effects were taken into account, the transition from a convective to an absolute regime might occur at a significantly higher Reynolds number, possibly above the experimental onset of turbulence. Moreover, they investigated the coalescence of type I and II modes (propagating in the same direction), and found that they are at the origin of a direct spatial resonance at a Reynolds number (445) significantly lower than the one associated with the absolute instability threshold (507).…”

Section: Introductionsupporting

confidence: 53%

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The elephant mode between two rotating disks

2008

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Spectral direct numerical simulations (DNS) are carried out for a source-sink flow in an annular cavity between two co-rotating disks. When the Reynolds number based on the forced inflow is increased, a self-sustained crossflow instability of finite amplitude is observed. We show that this nonlinear global mode is made up of a front located at the upstream boundary of the absolutely unstable domain, followed by a saturated spiral mode, and that its properties are in good agreement with results of the local stability theory. This structure is characteristic of the so-called elephant mode of Pier & Huerre (J. Fluid Mech. vol. 435, 2001, p. 145). The global bifurcation is subcritical since only large-amplitude initial perturbations are found to trigger the elephant mode. Small-amplitude perturbations induce a long-lasting transient growth but lead eventually to a damped linear global mode, showing that non-parallel effects counteract the absolute instability and restabilize the flow. A similar linear global stabilization due to non-parallel effects has been found in the case of the flow above a single rotating disk. For the single-disk geometry, the existence of an elephant mode would imply, together with results of Davies & Carpenter (2003) a subcritical global instability, which has not yet been demonstrated. Although the present geometry differs from the single-disk case, the existence of a subcritical global bifurcation is now established, allowing a precise analysis of the transition scenarios.

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Section: Introductionsupporting

confidence: 53%

Section: Introductionsupporting

confidence: 53%

The elephant mode between two rotating disks

2008

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“…In fact, as shown by Lingwood (1995) and Cooper & Carpenter (1997a), they do coalesce, but the result is an algebraically growing disturbance rather than an absolute instability. This phenomenon was investigated in I and also in a recent paper by Turkyilmazoglu & Gajjar (2000). Huerre & Monkewitz (1990) give an excellent review of the global instability of spatially developing flows, especially its relationship to absolute instability.…”

Section: Referred To As I)mentioning

confidence: 98%

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

2003

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A study is carried out of the linear global behaviour corresponding to the absolute instability of the rotating-disc boundary layer. It is based on direct numerical simulations of the complete linearized Navier–Stokes equations obtained with the novel velocity–vorticity method described in Davies & Carpenter (2001). As the equations are linear, they become separable with respect to the azimuthal coordinate, $\theta$. This permits us to simulate a single azimuthal mode. Impulse-like excitation is used throughout. This creates disturbances that take the form of wavepackets, initially containing a wide range of frequencies. When the real spatially inhomogeneous flow is approximated by a spatially homogeneous flow (the so-called parallel-flow approximation) the results ofthe simulations are fully in accordance with the theory of Lingwood (1995). If the flow parameters are such that her theory indicates convective behaviour the simulations clearly exhibit the same behaviour. And behaviour fully consistent with absolute instability is always found when the flow parameters lie within the theoretical absolutely unstable region. The numerical simulations of the actual inhomogeneous flow reproduce the behaviour seen in the experimental study of Lingwood (1996). In particular, there is close agreement between simulation and experiment for the ray paths traced out by the leading and trailing edges of the wavepackets. In absolutely unstable regions the short-term behaviour of the simulated disturbances exhibits strong temporal growth and upstream propagation. This is not sustained for longer times, however. The study suggests that convective behaviour eventually dominates at all the Reynolds numbers investigated, even for strongly absolutely unstable regions. Thus the absolute instability of the rotating-disc boundary layer does not produce a linear amplified global mode as observed in many other flows. Instead the absolute instability seems to be associated with transient temporal growth, much like an algebraically growing disturbance. There is no evidence of the absolute instability giving rise to a global oscillator. The maximum growth rates found for the simulated disturbances in the spatially inhomogeneous flow are determined by the convective components and are little different in the absolutely unstable cases from the purely convectively unstable ones. In addition to the study of the global behaviour for the usual rigid-walled rotating disc, we also investigated the effect of replacing an annular region of the disc surface with a compliant wall. It was found that the compliant annulus had the effect of suppressing the transient temporal growth in the inboard (i.e. upstream) absolutely unstable region. As time progressed the upstream influence of the compliant region became more extensive.

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“…for which I 6 is given by a complex contour integration as 9) and the correspondence of C 2 can be obtained from (3.13) as 10) where…”

Section: Linear Resultsmentioning

confidence: 99%

“…The absolute instability is a linear one studied first in [3] and [4]. These studies, together with later investigations in [5,6] and [7,8,9,10], have proven that the infinite growth of the unstable disturbances in the radial directions may be a reason for transition in rotating disk flow. On the other hand, the surface coating suggested in [11] or the suction through the wall as applied in [12] and [13] may effectively delay the occurrence of absolute instability.…”

Section: Introductionmentioning

confidence: 99%

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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Direct spatial resonance in the laminar boundary layer due to a rotating-disk

Cited by 17 publications

References 48 publications

The elephant mode between two rotating disks

The elephant mode between two rotating disks

Global behaviour corresponding to the absolute instability of the rotating-disc boundary layer

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

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