Nonlinear modelling of vortex shedding control in cylinder wakes

Roussopoulos, Kimon; Monkewitz, Peter A.

doi:10.1016/0167-2789(96)00151-0

Cited by 50 publications

(39 citation statements)

References 14 publications

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“…Closed-loop control may be viewed as a special case of operator perturbation η 2 L specifically designed to restabilize the linear evolution operator L. For example, the proportional feedback control of the Global mode for the complex Ginzburg-Landau equation with varying coefficients similar to Equation 10 implemented by Roussopoulos & Monkewitz (1996) corresponds to the feedback loop discussed above with the actuator location at x 1 = 0 and the sensor location at x 2 = 1.5. In this case, ω − ω G (R) = −iη 2 Cψ * G (x 1 )φ G (x 2 ) ψ G |φ G −1 and for any given x 1 and x 2 , it is possible to restabilize the flow near the threshold by choosing the amplitude and phase of the complex gain C in a particular range.…”

Section: Sensitivity Of the Global Spectrum Effect Of Feedback And Cmentioning

confidence: 99%

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

Chomaz

2005

Annu. Rev. Fluid Mech.

607

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: Sensitivity Of the Global Spectrum Effect Of Feedback And Cmentioning

confidence: 99%

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

Chomaz

2005

Annu. Rev. Fluid Mech.

607

565

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“…consisted of the quick reduction, perhaps to zero, of the free-stream velocity, with a subsequent quick return to its initial value U 0 (corresponding to given Re > Re cr ). Schumm, Schumm et al, and Park, also employed several other control methods such as bleeding of fluid from the rear part of the cylinder, wake heating, or forced vertical vibrations of a cylinder with a small amplitude a 0 D. (All these operations at supercritical Re > Re cr strongly suppress vortex shedding; see, e.g., Monkewitz's surveys (1993Monkewitz's surveys ( , 1996 and the papers on wake control by Roussopoulos (1993); Schumm et al (1994); Park et al (1993Park et al ( , 1994; Park (1994); Roussopoulos and Monkewitz (1996); Gunzburger and Lee (1996), and Gillies (1998) containing many additional references). However, the abovementioned control methods are applicable only at supercritical Reynolds numbers and can provide no information about the values of coefficients of Eqs.…”

Section: Wake Flows: the Case Of A Circular-cylinder Wakementioning

confidence: 98%

“…A more complete two-dimensional Ginzburg-Landau equation for anoscillation amplitude A = A (x, y, t) dependent on two spatial coordinates was applied to wake flows by Park and Redekopp (1992) and Chiffaudel (1992), and Roussopoulos and Monkewitz (1996). Park and Redekopp considered the G-L equation of the form…”

Section: Wake Flows: the Case Of A Circular-cylinder Wakementioning

confidence: 99%

Stability to Finite Disturbances: Energy Method and Landau’s Equation

Yaglom

Frisch

2012

Fluid Mechanics and Its Applications

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The main part of Chap. 2 and the whole of Chap. 3 were devoted to topics of linear stability theory dealing with the evolution of very small flow disturbances satisfying the linearized fluid dynamics equations. In Chap. 2 it was shown that the classical normal-mode method of the linear theory of hydrodynamic stability often leads to results which strongly disagree with experimental data. It was also indicated there that these disagreements are apparently due to nonlinear effects, which make linearization of the equations of motion physically unjustified. In Chap. 3 it was explained that the necessity for consideration of the full nonlinear dynamic equations often follows from the fact that many solutions of the initial-value problems for linearized fluid dynamics equations grow considerably at small and moderate values of the time t even in the cases when the normal-mode analysis shows that these solutions decay asymptotically (i.e. at t → ∞).The nonlinear theory of hydrodynamic stability has achieved a high level of development. Although the theory is still far from being completed, it has elucidated many formerly mysterious properties of fluid flows which are interesting for physicists and important for engineers. There is now an enormous literature on this subject and only a small part of it, dealing with relatively simple flows of incompressible fluids, will be considered in this book. In the present Chapter two topics from the nonlinear stability theory will be discussed: the energy method of stability analysis (short introductory consideration of this method was included in Sect. 3.4 above) and Landau's approach to the weakly nonlinear stability theory which described the initial period of the nonlinear development of flow disturbances.

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“…In particular, the CGL model has proven to be accurate in determining global frequency criteria in both the hnear (Chomaz et al 1991) and nonlinear (Pier et al 1998) regimes. Moreover, because the CGL model roughly captures the streamwise structure of the system eigenfunctions and the variation of the complex frequency of these eigenmodes with Reynolds number, the CGL model has also allowed quantitative predictions of the effects of proportional feedback control on the actual flow system in several previous studies (Monkewitz 1989(Monkewitz , 1993Monkewitz et al 1991;Roussopoulos & Monkewitz 1996). In the present paper, we thus consider the CGL model of the flow exclusively.…”

Section: The Ginzburg-landau Model Of Weakly Nonparallel Flowsmentioning

confidence: 99%

“…The spatial position xt, which is generally taken to be complex, is found by analytic continuation of local dispersion relations (Hammond & Redekopp 1997), and characterizes the hydrodynamic resonance phenomenon (Chomaz et al 1991). The parabolic form used here is motivated by many previous studies which focused on the modelling of spatially developing flows (Chomaz et al 1987(Chomaz et al ,1990Huerre & Monkewitz 1990;Roussopoulos & Monkewitz 1996). Using this parabolic form, it was shown by Chomaz et al (1987) that local instability appears in a finite region in the system when Ho > 0, this local instability being everywhere convective if no < Ha = U'^^(i')/4\u\'^, and absolute in a portion of the unstable region if no > fJ'a-Signiflcantly, the localized control forcing term applied to (2.1) does not change these local instability properties of the system, though it can substantially alter its global dynamics.…”

Section: The Ginzburg-landau Model Of Weakly Nonparallel Flowsmentioning

confidence: 99%