Three-dimensional instability of the flow over a forward-facing step

Lanzerstorfer, Daniel; Kuhlmann, Hendrik C.

doi:10.1017/jfm.2012.28

Cited by 35 publications

(58 citation statements)

References 16 publications

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“…is called the base flow U B and is linearly stable for the 2D backward-facing step with expansion ratio Γ = 0.5 at the chosen Re = 500, with its first unstable mode appearing in 3D for Re cr ∼ 748 (Barkley et al (2002) and Lanzerstorfer & Kuhlmann (2012)). The classical approach is to study the linear response to harmonic forcing around this stable base flow.…”

Section: Linear Transfer Functionmentioning

confidence: 99%

“…The model is specifically applied to the well known backward-facing step case study in 2D at Re = 500. It is globally stable since the first bifurcation is a 3D global instability at Re cr ∼ 748 (Barkley et al (2002) and Lanzerstorfer & Kuhlmann (2012)). First, a weakly nonlinear asymptotic expansion is developed around the stable base flow, showing the Reynolds stress as the key nonlinear term in the saturation process.…”

Section: Introductionmentioning

confidence: 99%

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Self-consistent model for the saturation mechanism of the response to harmonic forcing in the backward-facing step flow

Mantič-Lugo

Gallaire

2016

J. Fluid Mech.

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Certain flows denominated as amplifiers are characterised by their global linear stability while showing large linear amplifications to sustained perturbations. As the forcing amplitude increases, a strong saturation of the response appears when compared to the linear prediction. However, a predictive model that describes the saturation of the response to higher amplitudes of forcing in stable laminar flows is still missing.While an asymptotic analysis based on the weakly nonlinear theory shows qualitative agreement only for very small forcing amplitudes, the linear response to harmonic forcing around the DNS mean flow presents a good prediction of the saturation also at higher forcing amplitudes. These results suggest that the saturation process is governed by the Reynolds stress and thus motivate the introduction of a simple self-consistent model.The model consists of a decomposition of the full nonlinear Navier-Stokes equations in a mean flow equation together with a linear perturbation equation around the mean flow, which are coupled through the Reynolds stress. The full fluctuating response and the resulting Reynolds stress are approximated by the first harmonic calculated from the linear response to the forcing around the aforementioned mean flow. This closed set of coupled equations is solved in an iterative manner as partial nonlinearity is still preserved in the mean flow equation despite the assumed simplifications.The results show an accurate prediction of the response energy when compared to Direct Numerical Simulations (DNS). The approximated coupling is strong enough to retain the main nonlinear effects of the saturation process. Hence, a simple physical picture is formalised, wherein the response modifies the mean flow through the Reynolds stress in such a way that the correct response energy is attained.

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Section: Linear Transfer Functionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Self-consistent model for the saturation mechanism of the response to harmonic forcing in the backward-facing step flow

Mantič-Lugo

Gallaire

2016

J. Fluid Mech.

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“…These restrictions apply to the SC model on the backward-facing step flow at the studied Reynolds numbers since the flow presents mainly 2D dynamics [12] and it is stable up to Re cr ∼ 748 [27,28], where a steady 3D instability appears. Although the model can be extrapolated to 3D flows, one would have to cope with the natural growth and saturation of the static 3D instability mode and thereby generalize the proposed model.…”

Section: Discussionmentioning

confidence: 99%

“…The model is applied to a canonical amplifier flow, the incompressible backward-facing step flow, which is archetypical in fundamental studies of flow separation induced by abrupt changes of geometry. The flow is globally stable at the Reynolds numbers considered, Re = 500 and 700 [27,28]. The work presented herein is an extension to white noise forcing of the self-consistent model [19] previously introduced for the nonlinear saturation of the response to harmonic forcing.…”

Section: Introductionmentioning

confidence: 91%

“…For an expansion ratio = 0.5, the flow is globally stable at this Reynolds number, presenting mainly a 2D response to white noise [12] and thus supporting the choice of a 2D analysis. The threshold for the 3D global instability was found to be Re cr ∼ 748 [27,28]. Assuming a small amplitude of the forcing, the exact nonlinear response can be approximated linearly as…”

Section: B Linear Transfer Functionmentioning

confidence: 99%

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Saturation of the response to stochastic forcing in two-dimensional backward-facing step flow: A self-consistent approximation

Mantič-Lugo

Gallaire

2016

Phys. Rev. Fluids

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Selective noise amplifiers are characterized by large linear amplification to external perturbations in a particular frequency range despite their global linear stability. Applying a stochastic forcing with increasing amplitude, the response undergoes a strong nonlinear saturation when compared to the linear estimation. Building upon our previous work, we introduce a predictive model that describes this nonlinear dynamics, and we apply it to a canonical example of selective noise amplifiers: the backward-facing step flow. Rewriting conveniently the stochastic forcing and response in the frequency domain, the model consists in a mean flow equation coupled to the linear response to forcing at each frequency. This coupling is attained by the Reynolds stress, which is constructed by the integral in frequency of the independent responses. We generalize the model for a response to a white noise forcing δ-correlated in space and time restricting the flow dynamics to its most energetic patterns calculated from the optimal harmonic forcing and response of the flow. The model estimates accurately the response saturation when compared to direct numerical simulations, and it correctly approximates the structure of the response and the mean flow modification. It also shows that the response undergoes a selective process governed by the nonlinear gain, which promotes a response structure with an approximately single frequency and wavelength in the whole domain. These results suggest that the mean flow modification by the Reynolds stress is the key nonlinearity in the saturation process of the response to white noise.

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