Hostname: page-component-cd9895bd7-gvvz8 Total loading time: 0 Render date: 2024-12-29T16:14:18.085Z Has data issue: false hasContentIssue false
  • English
  • Français

Stochastic dynamics and model reduction of amplifier flows: the backward facing step flow

Published online by Cambridge University Press:  19 February 2013

G. Dergham ,
D. Sipp  and
J.-Ch. Robinet
G. Dergham
Affiliation:
DynFluid Laboratory, Arts et Métiers ParisTech, 151 Boulevard de l’Hôpital, 75013 Paris, France ONERA, The French-Aerospace Lab, 8 rue des Vertugadins, 92190 Meudon, France
D. Sipp
Affiliation:
ONERA, The French-Aerospace Lab, 8 rue des Vertugadins, 92190 Meudon, France
J.-Ch. Robinet*
Affiliation:
DynFluid Laboratory, Arts et Métiers ParisTech, 151 Boulevard de l’Hôpital, 75013 Paris, France
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

Methods for investigating and approximating the linear dynamics of amplifier flows are examined in this paper. The procedures are derived for incompressible flow over a two-dimensional backward-facing step. First, the singular value decomposition of the resolvent is performed over a frequency range in order to identify the optimal and suboptimal harmonic forcing and responses of the flow. These forcing/responses are shown to be organized into two categories: the first accounting for the Orr and Kelvin–Helmholtz instabilities in the shear layer and the second for the advection and diffusion of perturbations in the free stream. Next, we investigate the dynamics of the flow when excited by a white in space and time noise. We compute the predominant patterns of the random flow which optimally account for the sustained variance, the empirical orthogonal functions (EOFs), as well as the predominant forcing structures which optimally contribute to the sustained variance, the stochastic optimals (SOs). The leading EOFs and SOs are expressed as a linear combination of the suboptimal forcing and responses of the flow and are related to particular instability mechanisms and/or frequency intervals. Finally, we use the leading EOFs, SOs and balanced modes (obtained from balanced truncation) to build low-order models of the flow dynamics. These models are shown to accurately recover the time propagator and resolvent of the original dynamical system. In other words, such models capture the entire flow response from any forcing and may be used in the design of efficient closed-loop controllers for amplifier flows.

JFM classification

Flow Control: Flow Control Instability: Instability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 719 , 25 March 2013 , pp. 406 - 430
Copyright
©2013 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
Alizard, F., Cherubini, S. & Robinet, J.-C. 2009 Sensitivity and optimal forcing response in separated boundary layer flows. Phys. Fluids 21, 064108.Google Scholar
Alizard, F. & Robinet, J.-C. 2007 Spatially convective global modes in a boundary layer. Phys. Fluids 19 (11), 114105.CrossRefGoogle Scholar
Bagheri, S., Brandt, L. & Henningson, D. S. 2009 Input–output analysis, model reduction and control of the flat-plate boundary layer. J. Fluid Mech. 620, 263–298.Google Scholar
Barbagallo, A., Dergham, G., Sipp, D., Schmid, P. J. & Robinet, J.-C. 2012 Closed-loop control of unsteadiness over a rounded backward-facing step. J. Fluid Mech. 703, 326–362.Google Scholar
Barbagallo, A., Sipp, D. & Schmid, P. J. 2011 Input–output measures for model reduction and closed-loop control: application to global modes. J. Fluid Mech. 685, 23–53.Google Scholar
Berkooz, G., Holmes, P. & Lumley, J. L. 1993 The proper orthogonal decomposition in the analysis of turbulent flows. Annu. Rev. Fluid Mech. 25, 539–575.Google Scholar
Blackburn, H. M., Barkley, D. & Sherwin, S. J. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.CrossRefGoogle Scholar
Brandt, L., Sipp, D., Pralits, J. & Marquet, O. 2011 Effect of base-flow variation in noise amplifiers: the flat-plate boundary layer. J. Fluid Mech. 687, 503–528.CrossRefGoogle Scholar
Chomaz, J.-M. 2005 Global stabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Cossu, C., Pujals, G. & Depardon, S. 2009 Optimal transient growth and very large-scale structures in turbulent boundary layers. J. Fluid Mech. 619, 79–94.Google Scholar
Crouch, J. D., Garbaruk, A. & Magidov, D. 2007 Predicting the onset of flow unsteadiness based on global instability. J. Comp. Phys. 224, 924–940.Google Scholar
Dergham, G., Sipp, D. & Robinet, J.-C. 2011 Accurate low dimensional models for deterministic fluid systems driven by uncertain forcing. Phys. Fluids 23 (9), 094101.Google Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows. J. Fluid Mech. 536, 209–218.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1993a Stochastic forcing of the linearized Navier–Stokes equations. Phys. Fluids A 5 (11), 2600–2609.CrossRefGoogle Scholar
Farrell, B. F. & Ioannou, P. J. 1993b Stochastic dynamics of baroclinic waves. J. Atmos. Sci. 50 (24), 4044–4057.Google Scholar
Farrell, B. F. & Ioannou, P. J. 1996 Generalized stability theory. Part I: autonomous operators. J. Atmos. Sci. 53 (14), 2025–2040.Google Scholar
Farrell, B. F. & Ioannou, P. J. 2001 Accurate low-dimensional approximation of the linear dynamics of fluid flow. J. Atmos. Sci. 58, 2771–2789.2.0.CO;2>CrossRefGoogle Scholar
Fontane, J., Brancher, P. & Fabre, D. 2008 Stochastic forcing of the Lamb–Oseen vortex. J. Fluid Mech. 613, 233–254.CrossRefGoogle Scholar
Grosch, C. E. & Salwen, H. 1978 The continuous spectrum of the Orr–Sommerfeld equation. Part 1. The spectrum and the eigenfunctions. J. Fluid Mech. 87, 33–54.Google Scholar
Hecht, F., Pironneau, O., Hyaric, A. Le & Ohtsuka, K. 2005 Freefem++, the book. UPMC-LJLL Press.Google Scholar
Laub, A. J., Heath, M. T., Paige, C. C. & Ward, R. C. 1987 Computation of system balancing transformations and other applications of simultaneous diagonalization algorithms. IEEE Trans. Autom. Control 32 (2), 115–122.Google Scholar
Marquet, O., Sipp, D., Chomaz, J.-M. & Jacquin, L. 2008 Amplifier and resonator dynamics of a low-Reynolds number recirculation bubble in a global framework. J. Fluid Mech. 605, 429–443.Google Scholar
Monokrousos, A., Åkervik, E., Brandt, L. & Henningson, D. S. 2010 Global three-dimensional optimal disturbances in the blasius boundary-layer flow using time steppers. J. Fluid Mech. 650, 181–214.Google Scholar
Moore, B. 1981 Principal component analysis in linear systems: controllability, observability, and model reduction. IEEE Trans. Autom. Control 26, 17–32.CrossRefGoogle Scholar
North, G. R. 1984 Empirical orthogonal functions and normal modes. J. Atmos. Sci. 41 (5), 879–887.2.0.CO;2>CrossRefGoogle Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurcation Chaos 15, 997–1013.Google Scholar
Sipp, D. & Marquet, O. 2012 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. doi:10.1007/s00162-012-0265-y.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, 030801.CrossRefGoogle Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & Driscoll, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar

Dergham et al. supplementary movie

Time-evolution of the streamwise component of the velocity field in the DNS (up) and the ROM (down) respectively. The middle figure represents the measurement me extracted from the DNS.

Download Dergham et al. supplementary movie(Video)
Video 8.4 MB