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Recovery of the inherent dynamics of noise-driven amplifier flows

Published online by Cambridge University Press:  16 May 2016

Juan Guzmán Iñigo ,
Denis Sipp  and
Peter J. Schmid
Juan Guzmán Iñigo*
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Denis Sipp
Affiliation:
ONERA-DAFE, 8 rue des Vertugadins, 92190 Meudon, France
Peter J. Schmid
Affiliation:
Department of Mathematics, Imperial College London, London SW7 2AZ, UK
*
†Email address for correspondence: juan.guzman.inigo@gmail.com
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Abstract

Unsteadiness in noise amplifier flows is driven and sustained by upstream environmental perturbations. A dynamic mode decomposition performed with snapshots taken in the statistically steady state extracts marginally stable dynamic modes, which mimic the sustained dynamics but miss the actual intrinsic stable behaviour of these flows. In this study, we present an alternative data-driven technique which attempts to identify and separate the intrinsic linear stable behaviour from the driving term. This technique uses a system-identification algorithm to extract a reduced state-space model of the flow from time-dependent input–output data. Such a model accurately predicts the values of the velocity field (output) from measurements of an upstream sensor that captures the effect of the incoming perturbations (input). The methodology is illustrated on a two-dimensional boundary layer subject to Tollmien–Schlichting instabilities, a canonical example of flow acting as a noise amplifier. The spectrum of the identified model compares well with the results reported in literature for the full-order system. Yet the comparison appears to be only qualitative, due to the poor robustness properties of eigenvalue spectra in noise-amplifier flows. We therefore advocate the use of the frequency response between the upstream sensor and the flow dynamics, which is revealed to be a robust quantity. The frequency response is validated against full-order computations and compares well with a local spatial stability analysis.

JFM classification

Boundary Layers: Boundary Layers Boundary Layers: Boundary layer stability
Type
Papers
Information
Journal of Fluid Mechanics , Volume 797 , 25 June 2016 , pp. 130 - 145
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Copyright
© 2016 Cambridge University Press 

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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.Google Scholar
Alizard, F. & Robinet, J.-C. 2007 Spatially convective global modes in a boundary layer. Phys. Fluids 19 (11), 114105.Google Scholar
Antoulas, A. C. 2005 Approximation of Large-Scale Dynamical Systems. SIAM.Google Scholar
Bagheri, S. 2013 Koopman-mode decomposition of the cylinder wake. J. Fluid Mech. 726, 596–623.Google Scholar
Bagheri, S. 2014 Effects of weak noise on oscillating flows: linking quality factor, Floquet modes, and Koopman spectrum. Phys. Fluids 26 (9), 094104.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.CrossRefGoogle Scholar
Ehrenstein, U. & Gallaire, F. 2005 On two-dimensional temporal modes in spatially evolving open flows: the flat-plate boundary layer. J. Fluid Mech. 536, 209–218.Google Scholar
Gaster, M. 1974 On the effects of boundary-layer growth on flow stability. J. Fluid Mech. 66, 465–480.CrossRefGoogle Scholar
Gautier, N. & Aider, J.-L. 2014 Feed-forward control of a perturbed backward-facing step flow. J. Fluid Mech. 759, 181–196.Google Scholar
Giannetti, F. & Luchini, P. 2007 Structural sensitivity of the first instability of the cylinder wake. J. Fluid Mech. 581, 167–197.CrossRefGoogle Scholar
Guzmán Iñigo, J., Sipp, D. & Schmid, P. J. 2014 A dynamic observer to capture and control perturbation energy in noise amplifiers. J. Fluid Mech. 758, 728–753.Google Scholar
Hervé, A., Sipp, D., Schmid, P. J. & Samuelides, M. 2012 A physics-based approach to flow control using system identification. J. Fluid Mech. 702, 26–58.CrossRefGoogle Scholar
Huerre, P. & Rossi, M. 1998 Hydrodynamic instabilities in open flows. In Hydrodynamics and Nonlinear Instabilities (ed. Godrèche, C. & Manneville, P.), pp. 81–294. Cambridge University Press.Google Scholar
Juillet, F., McKeon, B. J. & Schmid, P. J. 2014 Experimental control of natural perturbations in channel flow. J. Fluid Mech. 752, 296–309.CrossRefGoogle Scholar
Juillet, F., Schmid, P. J. & Huerre, P. 2013 Control of amplifier flows using subspace identification techniques. J. Fluid Mech. 725, 522–565.Google Scholar
Lumley, J. L. 1967 The structure of inhomogeneous turbulent flows. In Atmospheric Turbulence and Radio Wave Propagation, pp. 166–178. Nauka.Google Scholar
Qin, S. J. 2006 An overview of subspace identification. Comput. Chem. Engng 30 (10), 1502–1513.CrossRefGoogle Scholar
Rowley, C. W., Mezić, I., Bagheri, S., Schlatter, P. & Henningson, D. S. 2009 Spectral analysis of nonlinear flows. J. Fluid Mech. 641, 115–127.Google Scholar
Ruhe, A. 1984 Rational Krylov sequence methods for eigenvalue computation. Linear Algebr. Applics. 58, 391–405.Google Scholar
Schmid, P. J. 2010 Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 5–28.Google Scholar
Schmid, P. J. 2011 Application of the dynamic mode decomposition to experimental data. Exp. Fluids 50 (4), 1123–1130.Google Scholar
Schmid, P. J., Li, L., Juniper, M. P. & Pust, O. 2011 Applications of the dynamic mode decomposition. Theor. Comput. Fluid Dyn. 25, 249–259.CrossRefGoogle Scholar
Schmid, P. J. & Henningson, D. S. 2001 Stability and Transition in Shear Flows. Springer.Google Scholar
Seena, A. & Sung, H. J. 2011 Dynamic mode decomposition of turbulent cavity flows for self-sustained oscillations. Intl J. Heat Fluid Flow 32 (6), 1098–1110.CrossRefGoogle Scholar
Sipp, D. & Marquet, O. 2013 Characterization of noise amplifiers with global singular modes: the case of the leading-edge flat-plate boundary layer. Theor. Comput. Fluid Dyn. 27 (5), 617–635.CrossRefGoogle Scholar
Sipp, D., Marquet, O., Meliga, Ph. & Barbagallo, A. 2010 Dynamics and control of global instabilities in open-flows: a linearized approach. Appl. Mech. Rev. 63 (3), 030801.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Q. Appl. Maths 45, 561–571.CrossRefGoogle Scholar
Trefethen, L., Trefethen, A., Reddy, S. & Driscoll, T. 1993 Hydrodynamic stability without eigenvalues. Science 261 (5121), 578–584.CrossRefGoogle ScholarPubMed
Van Overschee, P. & De Moor, B. 1994 N4SID: subspace algorithms for the identification of combined deterministic-stochastic systems. Automatica 30 (1), 75–93.CrossRefGoogle Scholar