Instability of flows in spatially developing media

Hunt, Rosanna; Crighton, D. G.

doi:10.1098/rspa.1991.0132

Cited by 45 publications

(24 citation statements)

References 8 publications

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“…The optimal initial perturbation for which the energy of the response is the largest after a time t may be computed by using the direct and the adjoint operator. When the non-normality is large, the globally stable flow sustains, close to the global instability threshold, extremely large transient growth of the initial perturbation energy, as demonstrated by Cossu & Chomaz (1997) on the basis of the results of Hunt & Crighton (1991) for the Ginzburg-Landau equation (Equations 6 and 10). Figure 6a illustrates the gain in energy over all initial perturbations and time intervals.…”

Section: Response To Forcing Amplifier Behavior and Transient Growthmentioning

confidence: 91%

GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

Chomaz

2005

Annu. Rev. Fluid Mech.

616

565

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Key Words absolute/convective instabilities, transient growth, fronts, Global modes, passive and active control ■ Abstract The objective of this review is to critically assess the different approaches developed in recent years to understand the dynamics of open flows such as mixing layers, jets, wakes, separation bubbles, boundary layers, and so on. These complex flows develop in extended domains in which fluid particles are continuously advected downstream. They behave either as noise amplifiers or as oscillators, both of which exhibit strong nonlinearities (Huerre & Monkewitz 1990). The local approach is inherently weakly nonparallel and it assumes that the basic flow varies on a long length scale compared to the wavelength of the instability waves. The dynamics of the flow is then considered as a superposition of linear or nonlinear instability waves that, at leading order, behave at each streamwise station as if the flow were homogeneous in the streamwise direction. In the fully global context, the basic flow and the instabilities do not have to be characterized by widely separated length scales, and the dynamics is then viewed as the result of the interactions between Global modes living in the entire physical domain with the streamwise direction as an eigendirection. This second approach is more and more resorted to as a result of increased computational capability. The earlier review of Huerre & Monkewitz (1990) emphasized how local linear theory can account for the noise amplifier behavior as well as for the onset of a Global mode. The present survey first adopts the opposite point of view by demonstrating how fully global theory accounts for the noise amplifier behavior of open flows. From such a perspective, there is strong emphasis on the very peculiar nonorthogonality of linear Global modes, which in turn allows a novel interpretation of recent numerical simulations and experimental observations. The nonorthogonality of linear Global modes also imposes severe constraints on the extension of linear global theory to the fully nonlinear régime. When the flow is weakly nonparallel, this limitation is so severe that the linear Global mode theory is of little help. It is then much more appropriate to develop a fully nonlinear formulation involving the presence of a front separating the base state region from the bifurcated state region.

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Section: Response To Forcing Amplifier Behavior and Transient Growthmentioning

confidence: 91%

GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

Chomaz

2005

Annu. Rev. Fluid Mech.

616

565

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“…Hunt and Crighton [15,16] obtained analytic solutions of the linearized complex Ginzburg-Landau equation for cases where the multipliers U and μ were given a prescribed linear or quadratic dependence on the spatial co-ordinate r. They determined the Green's function G(r, t) representing the response to an impulsive forcing of the form δ(r)δ(t), applied to the right-hand side of Eq. 7.…”

Section: Impulse Solutions Of the Linearized Ginzburg-landau Equationmentioning

confidence: 99%

Global stability of the rotating-disk boundary layer

2006

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The global stability of the von Kármán boundary layer on the rotating disk is reviewed. For the genuine, radially inhomogeneous base flow, linearized numerical simulations indicate that convectively propagating forms of disturbance are predominant at all radii. The presence of absolute instability does not lead to the formation of any unstable linear global mode, even though the temporal growth rate of the absolute instability increases along the radial direction. Analogous behaviour can be found in the impulse solutions of a model amplitude equation, namely the linearized complex Ginzburg-Landau equation. These solutions show that, depending on the precise balance between spatial variations in the temporal growth rate and the corresponding shifts in the temporal frequency, globally stable behaviour can be obtained even in the presence of a strengthening absolute instability. The radial dependency of the absolute temporal frequency is sufficient to detune the disturbance oscillations at different radial positions, thus overcoming the radially increasing absolute growth, thereby giving rise to a stable global response. The origin of this form of behaviour can be traced to the fact that the cylindrical geometry of the rotating-disk flow dictates a choice of a globally valid time non-dimensionalization that, when properly employed, leads to a significant radial variation in the frequency for the absolute instability.

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“…These principles have been rigorously proved by Chomaz, Huerre & Redekopp (1991) for a class of Ginzburg-Landau equations within the WKBJ framework. Hunt & Crighton (1991) also found that the frequencyselection principle is exact in their study of the global instability of the linearized Ginzburg-Landau equation.…”

Section: Referred To As I)mentioning

confidence: 99%

“…This corresponds qualitatively, more or less, with the rotating-disc characteristics illustrated schematically in figure 2(a). Hunt & Crighton (1991) did not consider this case with complex µ explicitly. But, as Monkewitz explains, the generic behaviour referred to above can be deduced fairly easily from the exact Green's function (equation 31 of Hunt & Crighton).…”

Section: Referred To As I)mentioning

confidence: 99%

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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Instability of flows in spatially developing media

Cited by 45 publications

References 8 publications

GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

GLOBAL INSTABILITIES IN SPATIALLY DEVELOPING FLOWS: Non-Normality and Nonlinearity

Global stability of the rotating-disk boundary layer

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

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