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Bifurcation analysis and frequency prediction in shear-driven cavity flow

Published online by Cambridge University Press:  23 July 2019

Y. Bengana Open the ORCID record for Y. Bengana[Opens in a new window] ,
J.-Ch. Loiseau Open the ORCID record for J.-Ch. Loiseau[Opens in a new window] ,
J.-Ch. Robinet Open the ORCID record for J.-Ch. Robinet[Opens in a new window]  and
L. S. Tuckerman Open the ORCID record for L. S. Tuckerman[Opens in a new window]
Y. Bengana
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS; ESPCI Paris, PSL Research University; Sorbonne Université; Univ. Paris Diderot, 75005 Paris, France
J.-Ch. Loiseau
Affiliation:
Laboratoire DynFluid, Arts et Métiers ParisTech, 75013 Paris, France
J.-Ch. Robinet
Affiliation:
Laboratoire DynFluid, Arts et Métiers ParisTech, 75013 Paris, France
L. S. Tuckerman*
Affiliation:
Laboratoire de Physique et Mécanique des Milieux Hétérogènes (PMMH), CNRS; ESPCI Paris, PSL Research University; Sorbonne Université; Univ. Paris Diderot, 75005 Paris, France
*
†Email address for correspondence: laurette@pmmh.espci.fr
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Abstract

A comprehensive study of the two-dimensional incompressible shear-driven flow in an open square cavity is carried out. Two successive bifurcations lead to two limit cycles with different frequencies and different numbers of structures which propagate along the top of the cavity and circulate in its interior. A branch of quasi-periodic states produced by secondary Hopf bifurcations transfers the stability from one limit cycle to the other. A full analysis of this scenario is obtained by means of nonlinear simulations, linear stability analysis and Floquet analysis. We characterize the temporal behaviour of the limit cycles and quasi-periodic state via Fourier transforms and their spatial behaviour via the Hilbert transform. We address the relevance of linearization about the mean flow. Although here the nonlinear frequencies are not very far from those obtained by linearization about the base flow, the difference is substantially reduced when eigenvalues are obtained instead from linearization about the mean and in addition, the corresponding growth rate is small, a combination of properties called RZIF (real zero imaginary frequency). Moreover growth rates obtained by linearization about the mean of one limit cycle are correlated with relative stability to the other limit cycle. Finally, we show that the frequencies of the successive modes are separated by a constant increment.

JFM classification

Nonlinear Dynamical Systems: Bifurcation Boundary Layers: Free shear layers Vortex Flows: Vortex shedding
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 875 , 25 September 2019 , pp. 725 - 757
Copyright
© 2019 Cambridge University Press 

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Bengana et al. supplementary movie 1

Vertical velocity fluctuations for limit cycle $LC_2$

Download Bengana et al. supplementary movie 1(Video)
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Bengana et al. supplementary movie 2

Vertical velocity fluctuations for limit cycle $LC_3$

Download Bengana et al. supplementary movie 2(Video)
Video 243 KB