Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-26T19:50:41.888Z Has data issue: false hasContentIssue false
  • English
  • Français

Sensitivity analysis and passive control of the secondary instability in the wake of a cylinder

Published online by Cambridge University Press:  01 February 2019

F. Giannetti Open the ORCID record for F. Giannetti [Opens in a new window] ,
S. Camarri Open the ORCID record for S. Camarri [Opens in a new window]  and
V. Citro Open the ORCID record for V. Citro [Opens in a new window]
F. Giannetti
Affiliation:
DIIN, Università di Salerno, 84084 Fisciano (SA), Italy
S. Camarri*
Affiliation:
Dipartimento di Ingegneria Civile ed Industriale, Università di Pisa, 56126 Pisa, Italy
V. Citro
Affiliation:
DIIN, Università di Salerno, 84084 Fisciano (SA), Italy
*
†Email address for correspondence: s.camarri@ing.unipi.it
Get access Rights & Permissions [Opens in a new window]

Abstract

The stability properties of selected flow configurations, usually denoted as base flows, can be significantly altered by small modifications of the flow, which can be caused, for instance, by a non-intrusive passive control. This aspect is amply demonstrated in the literature by ad hoc sensitivity studies which, however, focus on configurations characterised by a steady base flow. Nevertheless, several flow configurations of interest are characterised by a time-periodic base flow. To this purpose, we propose here an original theoretical framework suitable to quantify the effects of base-flow variations in the stability properties of saturated time-periodic limit cycles. In particular, starting from a Floquet analysis of the linearised Navier–Stokes equations and using adjoint methods, it is possible to estimate the variation of a selected Floquet exponent caused by a generic structural perturbation of the base-flow equations. This link is expressed concisely using the adjoint operators coming from the analysis, and the final result, when applied to spatially localised disturbances, is used to build spatial sensitivity and control maps. These maps identify the regions of the flow where the placement of a infinitesimal small object produces the largest effect on the Floquet exponent and may also provide a quantification of this effect. Such analysis brings useful insights both for passive control strategies and for further characterising the investigated instability. As an example of application, the proposed analysis is applied here to the three-dimensional flow instabilities in the wake past a circular cylinder. This is a classical problem which has been widely studied in the literature. Nevertheless, by applying the proposed analysis we derive original results comprising a further characterisation of the instability and related control maps. We finally show that the control maps obtained here are in very good agreement with control experiments documented in the literature.

JFM classification

Flow Control: Instability control Wakes/Jets: Wakes
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 864 , 10 April 2019 , pp. 45 - 72
Check for updates
Copyright
© 2019 Cambridge University Press 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Barkley, D. & Henderson, R. D. 1996 Three-dimensional Floquet stability analysis of the wake of a circular cylinder. J. Fluid Mech. 322, 215–241.Google Scholar
Bottaro, A., Corbett, P. & Luchini, P. 2003 The effect of base flow variation on flow stability. J. Fluid Mech. 476, 293–302.Google Scholar
Brandt, L., Sipp, D., Pralits, J. O. & Marquet, O. 2011 Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.Google Scholar
Camarri, S. 2015 Flow control design inspired by linear stability analysis. Acta Mechanica 226 (4), 979–1010.Google Scholar
Camarri, S., Fallenius, B. & Fransson, J. 2013 Stability analysis of experimental flow fields behind a porous cylinder for the investigation of the large-scale wake vortices. J. Fluid Mech. 715, 499–536.Google Scholar
Davis, T. A. 2004 Algorithm 832: UMFPACK, an unsymmetric-pattern multifrontal method. ACM Trans. Math. Softw. 30 (2), 196–199.Google Scholar
Drazin, P. G. 2002 Introduction to Hydrodynamic Stability. Cambridge University Press.Google Scholar
Gavarini, I., Bottaro, A. & Nieuwstadt, F. T. M. 2004 The initial stage of transition in pipe flow: role of optimal base-flow distortions. J. Fluid Mech. 517, 131–165.Google Scholar
Giannetti, F., Camarri, S. & Luchini, P. 2010 Structural sensitivity of the secondary instability in the wake of a circular cylinder. J. Fluid Mech. 651, 319–337.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the cylinder first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1987 Fluid Mechanics, 2nd edn., Course of Theoretical Physics, vol. 6. Butterworth-Heinemann.Google Scholar
Luchini, P. & Bottaro, A. 2014 Adjoint equations in stability analysis. Annu. Rev. Fluid Mech. 46, 493–517.10.1146/annurev-fluid-010313-141253Google Scholar
Luchini, P., Giannetti, F. & Pralits, J. 2009 Structural sensitivity of the finite-amplitude vortex shedding behind a circular cylinder. In IUTAM Symposium on Unsteady Separated Flows and their Control, IUTAM Bookseries, pp. 151–160. Springer.Google Scholar
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2014 Self-consistent mean flow description of the nonlinear saturation of the vortex shedding in the cylinder wake. Phys. Rev. Lett. 113, 084501.Google Scholar
Marquet, O., Sipp, D. & Jacquin, L. 2008 Sensitivity analysis and passive control of cylinder flow. J. Fluid Mech. 615, 221–252.Google Scholar
Meliga, P. 2017 Computing the sensitivity of drag and lift in flow past a circular cylinder: time-stepping versus self-consistent analysis. Phys. Rev. Fluids 2 (7), 1–25.Google Scholar
Meliga, P., Boujo, E. & Gallaire, F. 2016 A self-consistent formulation for the sensitivity analysis of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 800, 327–357.Google Scholar
Meliga, P., Boujo, E., Pujals, G. & Gallaire, F. 2014 Sensitivity of aerodynamic forces in laminar and turbulent flow past a square cylinder. Phys. Fluids 26 (10), 104101.Google Scholar
Noack, B. R., König, M. & Eckelmann, H. 1993 Three-dimensional stability analysis of the periodic flow around a circular cylinder. Phys. Fluids A 5 (6), 1279–1281.Google Scholar
Noak, B. R. & Eckelmann, H. 1994 A global stability analysis of the steady and periodic cylinder wake. J. Fluid Mech. 270, 297–330.Google Scholar
Rai, M. M. & Moin, P. 1991 Direct simulations of turbulent flow using finite-difference schemes. J. Comput. Phys. 96, 15–53.Google Scholar
Sipp, D., Marquet, O., Meliga, P. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.Google Scholar
Williamson, C. H. K. 1988 The existence of two stages in the transition to three-dimensionality of a cylinder wake. Phys. Fluids 31, 3165–3168.Google Scholar
Zhang, H. Q., Fey, U. F. & Noack, B. R. 1995 On the transition of the cylinder wake. Phys. Fluids 7 (4), 779–794.Google Scholar