Hostname: page-component-cd9895bd7-gxg78 Total loading time: 0 Render date: 2024-12-28T21:08:34.708Z Has data issue: false hasContentIssue false
  • English
  • Français

Harmonics generation and the mechanics of saturation in flow over an open cavity: a second-order self-consistent description

Published online by Cambridge University Press:  04 August 2017

P. Meliga Open the ORCID record for P. Meliga [Opens in a new window]
P. Meliga*
Affiliation:
Aix Marseille Univ., CNRS, Centrale Marseille, M2P2, Marseille, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The flow over an open cavity is an example of supercritical Hopf bifurcation leading to periodic limit-cycle oscillations. One of its distinctive features is the existence of strong higher harmonics, which results in the time-averaged mean flow being strongly linearly unstable. For this class of flows, a simplified formalism capable of unravelling how exactly the instability grows and saturates is lacking. This study builds on previous work by Mantič-Lugo et al. (Phys. Rev. Lett., vol. 113, 2014, 084501) to fill in the gap using a parametrized approximation of an instantaneous, phase-averaged mean flow, coupled in a quasi-static manner to multiple linear harmonic disturbances interacting nonlinearly with one another and feeding back on the mean flow via their Reynolds stresses. This provides a self-consistent modelling of the mean flow–fluctuation interaction, in the sense that all perturbation structures are those whose Reynolds stresses force the mean flow in such a way that the mean flow generates exactly the aforementioned perturbations. The first harmonic is sought as the superposition of two components, a linear component generated by the instability and aligned along the leading eigenmode of the mean flow, and a nonlinear orthogonal component generated by the higher harmonics, which progressively distorts the linear growth rate and eigenfrequency of the eigenmode. Saturation occurs when the growth rate of the first harmonic is zero, at which point the stabilizing effect of the second harmonic balances exactly the linear instability of the eigenmode. The model does not require any input from numerical or experimental data, and accurately predicts the transient development and the saturation of the instability, as established from comparison to time and phase averages of direct numerical simulation data.

JFM classification

Instability: Instability Nonlinear Dynamical Systems: Nonlinear Dynamical Systems Wakes/Jets: Separated flows
Type
Papers
Information
Journal of Fluid Mechanics , Volume 826 , 10 September 2017 , pp. 503 - 521
Check for updates
Copyright
© 2017 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

Åkervik, E., Ehrenstein, U., Gallaire, F. & Henningson, D. S. 2008 Global two-dimensional stability measures of the flat plate boundary-layer flow. Eur. J. Mech. (B/Fluids) 27 (5), 501–513.CrossRefGoogle Scholar
Barbagallo, A., Sipp, D. & Schmid, P. J. 2009 Closed-loop control of an open cavity flow using reduced-order models. J. Fluid Mech. 641, 1–50.CrossRefGoogle Scholar
Barkley, D. 2006 Linear analysis of the cylinder wake mean flow. Europhys. Lett. 75, 750–756.CrossRefGoogle Scholar
Beaume, C., Chini, G. P., Julien, K. & Knobloch, E. 2015 Reduced description of exact coherent states in parallel shear flows. Phys. Rev. E 91, 1–18.CrossRefGoogle ScholarPubMed
Chomaz, J.-M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Dušek, J., Le Gal, P. & Fraunié, P. 1994 A numerical and theoretical study of the first Hopf bifurcation in a cylinder wake. J. Fluid Mech. 264, 59–80.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 2012 Dynamics of streamwise rolls and streaks in turbulent wall-bounded shear flow. J. Fluid Mech. 708, 149–196.CrossRefGoogle Scholar
Farrell, B. F., Ioannou, P. J., Jiménez, J., Constantinou, N. C., Lozano-Durán, A. & Nikolaidis, M.-A. 2016 A statistical state dynamics-based study of the structure and mechanism of large-scale motions in plane Poiseuille flow. J. Fluid Mech. 809, 1–26.CrossRefGoogle Scholar
Fornberg, B. 1980 A numerical study of steady viscous flow past a circular cylinder. J. Fluid Mech. 98, 819–855.CrossRefGoogle Scholar
Hammond, D. A. & Redekopp, L. G. 1997 Global dynamics of symmetric and asymmetric wakes. J. Fluid Mech. 331, 231–260.CrossRefGoogle Scholar
Malkus, W. V. R. 1956 Outline of a theory of turbulent shear flow. J. Fluid Mech. 1, 521–539.CrossRefGoogle 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.CrossRefGoogle ScholarPubMed
Mantič-Lugo, V., Arratia, C. & Gallaire, F. 2015 A self-consistent model for the saturation dynamics of the vortex shedding around the mean flow in the unstable cylinder wake. Phys. Fluids 27, 074103.CrossRefGoogle Scholar
Mantič-Lugo, V. & Gallaire, F. 2016 Self-consistent model for the saturation mechanism of the response to harmonic forcing in the backward-facing step flow. J. Fluid Mech. 793, 777–797.CrossRefGoogle Scholar
Maurel, A., Pagneux, V. & Wesfreid, J. E. 1995 Mean-flow correction as non-linear saturation mechanism. Europhys. Lett. 32, 217–222.CrossRefGoogle Scholar
Meliga, P., Boujo, E. & Gallaire, F. 2016a A self-consistent formulation for the sensitivity analysis of finite amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 800, 327–357.CrossRefGoogle 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, 104101.CrossRefGoogle Scholar
Meliga, P., Cadot, O. & Serre, E. 2016b Experimental and theoretical sensitivity analysis of turbulent flow past a square cylinder. Flow Turbul. Combust. 97, 987–1015.CrossRefGoogle Scholar
Meliga, P., Chomaz, J.-M. & Sipp, D. 2009a Global mode interaction and pattern selection in the wake of a disk: a weakly nonlinear expansion. J. Fluid Mech. 633, 159–189.CrossRefGoogle Scholar
Meliga, P., Pujals, G. & Serre, E. 2012 Sensitivity of 2-D turbulent flow past a D-shaped cylinder using global stability. Phys. Fluids 24, 061701.CrossRefGoogle Scholar
Meliga, P., Sipp, D. & Chomaz, J.-M. 2009b Elephant modes and low frequency unsteadiness in a high Reynolds number, transonic afterbody wake. Phys. Fluids 21, 054105.CrossRefGoogle Scholar
Noack, B. R., Afanasiev, K., Morzynski, M., Tadmor, G. & Thiele, F. 2003 A hierarchy of low-dimensional models for the transient and post-transient cylinder wake. J. Fluid Mech. 497, 335–363.CrossRefGoogle Scholar
Pier, B. 2002 On the frequency selection of finite-amplitude vortex shedding in the cylinder wake. J. Fluid Mech. 458, 407–417.CrossRefGoogle Scholar
Pralits, J. O., Bottaro, A. & Cherubini, S. 2015 Weakly nonlinear optimal perturbations. J. Fluid Mech. 785, 135–151.CrossRefGoogle Scholar
Schmid, P. J. 2007 Nonmodal stability theory. Annu. Rev. Fluid Mech. 39, 129–162.CrossRefGoogle Scholar
Sipp, D. & Lebedev, A. 2007 Global stability of base and mean flows: a general approach and its applications to cylinder and open cavity flows. J. Fluid Mech. 593, 333–358.CrossRefGoogle Scholar
Stuart, J. T. 1971 Non-linear stability theory. Annu. Rev. Fluid Mech. 3, 347–370.CrossRefGoogle Scholar
Turton, S. E., Tuckerman, L. S. & Barkley, D. 2015 Prediction of frequencies in thermosolutal convection from mean flows. Phys. Rev. E 91, 043009.Google ScholarPubMed
Zielinska, B. J. A., Goujon-Durand, S., Dušek, J. & Wesfreid, J. E. 1997 Strongly nonlinear effect in unstable wakes. Phys. Rev. Lett. 79, 3893–3896.CrossRefGoogle Scholar