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A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake

Published online by Cambridge University Press:  07 July 2016

P. Meliga ,
E. Boujo  and
F. Gallaire
P. Meliga*
Affiliation:
Aix-Marseille Université, CNRS, Ecole Centrale Marseille, Laboratoire M2P2, Marseille, France
E. Boujo
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
F. Gallaire
Affiliation:
LFMI, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland
*
†Email address for correspondence: philippe.meliga@l3m.univ-mrs.fr
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Abstract

We use the adjoint method to compute sensitivity maps for the limit-cycle frequency and amplitude of the Bénard–von Kármán vortex street in the wake of a circular cylinder. The sensitivity analysis is performed in the frame of the semi-linear self-consistent model recently introduced by Mantič et al. (Phys. Rev. Lett., vol. 113, 2014, 084501), which allows us to describe accurately the effect of the control on the mean flow, but also on the finite-amplitude fluctuation that couples back nonlinearly onto the mean flow via the formation of Reynolds stress. The sensitivity is computed with respect to arbitrary steady and synchronous time-harmonic body forces. For a small amplitude of the control, the theoretical variations of the limit-cycle frequency predict well those of the controlled flow, as obtained from either self-consistent modelling or direct numerical simulation of the Navier–Stokes equations. This is not the case if the variations are computed in the simpler mean flow approach overlooking the coupling between the mean and fluctuating components of the flow perturbation induced by the control. The variations of the limit-cycle amplitude (that falls out the scope of the mean flow approach) are also correctly predicted, meaning that the approach can serve as a relevant and systematic guideline to control strongly unstable flows exhibiting non-small, finite amplitudes of oscillation. As an illustration, we apply the method to control by means of a small secondary control cylinder and discuss the obtained results in the light of the seminal experiments of Strykowski & Sreenivasan (J. Fluid Mech., vol. 218, 1990, pp. 71–107).

JFM classification

Flow Control: Instability control Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Wakes/Jets: Wakes
Type
Papers
Information
Journal of Fluid Mechanics , Volume 800 , 10 August 2016 , pp. 327 - 357
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Copyright
© 2016 Cambridge University Press 

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