Hostname: page-component-745bb68f8f-lrblm Total loading time: 0 Render date: 2025-01-09T21:59:19.528Z Has data issue: false hasContentIssue false
  • English
  • Français

Input–output analysis, model reduction and control of the flat-plate boundary layer

Published online by Cambridge University Press:  10 February 2009

SHERVIN BAGHERI ,
LUCA BRANDT  and
DAN S. HENNINGSON
SHERVIN BAGHERI
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
LUCA BRANDT
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
DAN S. HENNINGSON*
Affiliation:
Linné Flow Centre, KTH Mechanics, SE-100 44 Stockholm, Sweden
*
†Email address for correspondence: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The dynamics and control of two-dimensional disturbances in the spatially evolving boundary layer on a flat plate are investigated from an input–output viewpoint. A set-up of spatially localized inputs (external disturbances and actuators) and outputs (objective functions and sensors) is introduced for the control design of convectively unstable flow configurations. From the linearized Navier–Stokes equations with the inputs and outputs, controllable, observable and balanced modes are extracted using the snapshot method. A balanced reduced-order model (ROM) is constructed and shown to capture the input–output behaviour of the linearized Navier–Stokes equations. This model is finally used to design a 2-feedback controller to suppress the growth of two-dimensional perturbations inside the boundary layer.

Type
Papers
Information
Journal of Fluid Mechanics , Volume 620 , 10 February 2009 , pp. 263 - 298
Copyright
Copyright © Cambridge University Press 2009

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

REFERENCES

Ahuja, S., Rowley, C. W., Kevrekidis, I. G. & Wei, M. 2007 Low-dimensional models for control of leading-edge vortices: equilibria and linearized models. AIAA Paper 2008-553, 46th AIAA Aeros. Sci. Meeting Exhibit.CrossRefGoogle Scholar
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 27, 501–513.CrossRefGoogle Scholar
Åkervik, E., Hœpffner, J., Ehrenstein, U. & Henningson, D. S. 2007 Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid. Mech. 579, 305–314.CrossRefGoogle Scholar
Anderson, B. & Moore, J. 1990 Optimal control: Linear Quadratic Methods. Prentice Hall.Google Scholar
Bagheri, S., Hœpffner, J., Schmid, P. & Henningson, D. 2009 Input-output analysis and control design applied to a linear model of spatially developing flows. Appl. Mech. Rev. In press.CrossRefGoogle Scholar
Barkley, D., Gomes, M. G. & Henderson, R. D. 2002 Three-dimensional instability in flow over a backward-facing step. J. Fluid Mech. 473, 167–190.CrossRefGoogle Scholar
Bewley, T., Temam, R. & Ziane, M. 2000 A general framework for robust control in fluid mechanics. Physica D 138, 360–392.Google Scholar
Blackburn, A. M., Barkley, D. & Sherwin, S. 2008 Convective instability and transient growth in flow over a backward-facing step. J. Fluid Mech. 603, 271–304.CrossRefGoogle Scholar
Butler, K. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flow. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Chevalier, M., Hœpffner, J., Åkervik, E. & Henningson, D. S. 2007 a Linear feedback control and estimation applied to instabilities in spatially developing boundary layers. J. Fluid Mech. 588, 163–187.CrossRefGoogle Scholar
Chevalier, M., Schlatter, P., Lundbladh, A. & D. S., H. 2007 b A pseudo spectral solver for incompressible boundary layer flows. Tech Rep. 7. Trita-Mek, KTH Mechanics, Stockholm, Sweden.Google Scholar
Chomaz, J. M. 2005 Global instabilities in spatially developing flows: non-normality and nonlinearity. Annu. Rev. Fluid Mech. 37, 357–392.CrossRefGoogle Scholar
Chorin, A. & Marsden, J. 1990 A Mathematical Introduction to Fluid Mechanics. Springer.CrossRefGoogle Scholar
Curtain, R. & Zwart, H. 1995 An Introduction to Infinite-Dimensional Linear Systems Theory. Springer.CrossRefGoogle Scholar
Dullerud, E. & Paganini, F. 1999 A Course in Robust Control Theory: A Convex Approach. Springer.Google 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.CrossRefGoogle Scholar
Gillies, E. A. 1998 Low-dimensional control of the circular cylinder wake. J. Fluid Mech. 371, 157–178.CrossRefGoogle Scholar
Hanifi, A., Schmid, P. & Henningson, D. 1996 Transient growth in compressible boundary layer flow. Phys. Fluids 8 (3), 826–837.CrossRefGoogle Scholar
Högberg, M., Bewley, T. R. & Henningson, D. S. 2003 Linear feedback control and estimation of transition in plane channel flow. J. Fluid Mech. 481, 149–175.CrossRefGoogle Scholar
Ilak, M. & Rowley, C. W. 2008 Modeling of transitional channel flow using balanced proper orthogonal decomposition. Phys. Fluids 20, 034103.CrossRefGoogle Scholar
Kailath, T. 1980 Linear Systems. Prentice-Hall.Google Scholar
Kim, J. & Bewley, T. R. 2007 A linear systems approach to flow control. Annu. Rev. Fluid Mech. 39, 383–417.CrossRefGoogle Scholar
Kreiss, G., Lundbladh, A. & Henningson, D. 1993 Bounds for threshold amplitudes in subcritical shear flows. J. Fluid Mech. 1994, 175–198.Google Scholar
Lauga, E. & Bewley, T. R. 2003 The decay of stabilizability with reynolds number in a linear model of spatially developing flows. Proc. R. Soc. Lond. A 459, 2077–2095.CrossRefGoogle Scholar
Lewis, F. L. & Syrmos, L. V. 1995 Optimal Control. John Wiley.Google Scholar
Moore, B. 1981 Principal component analysis in linear systems: Controllability, observability, and model reduction. IEEE Trans. Auto. Control 26 (1), 17–32.CrossRefGoogle Scholar
Noack, B., 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
Nordström, J., Nordin, N. & Henningson, D. 1999 The fringe region technique and the fourier method used in the direct numerical simulation of spatially evolving viscous flows. SIAM J. Sci. Comput. 20 (4), 1365–1393.CrossRefGoogle Scholar
Pazy, A. 1983 Semigroup of Linear Operators and Applications to Partial Differential Equations, 2nd ed.Springer.CrossRefGoogle Scholar
Rempfer, D. & Fasel, H. 1994 Evolution of three-dimensional coheren structures in a flat-plate boundary layer. J. Fluid. Mech. 260, 351–375.CrossRefGoogle Scholar
Rowley, C. W. 2005 Model reduction for fluids using balanced proper orthogonal decomposition. Intl J. Bifurc. Chaos 15 (3), 997–1013.CrossRefGoogle Scholar
Siegel, S. G., Seidel, J., Fagley, C., Luchtenburg, D. M., Cohen, K. & McLaughlin, T. 2008 Low dimensional modelling of a transient cylinder wake using double proper orthogonal decomposition. J. Fluid Mech. 610, 1–42.CrossRefGoogle Scholar
Sirovich, L. 1987 Turbulence and the dynamics of coherent structures. Part I–III. Quart. Appl. Maths 45, 561–590.CrossRefGoogle Scholar
Skogestad, S. & Postlethwaite, I. 2005 Multivariable Feedback Control: Analysis to Design, 2nd edn.Wiley.Google Scholar
Trefethen, L. & Embree, M. 2005 Spectra and Pseudospectra: The Behavior of Nonnormal Matrices and Operators. Princeton University Press.CrossRefGoogle Scholar
Willcox, K. & Peraire, J. 2002 Balanced model reduction via the proper orthogonal decomposition. AIAA J. 40 (11), 2323–2330.CrossRefGoogle Scholar
Zhou, K., Doyle, J. & Glover, K. 2002 Robust and Optimal Control. Prentice Hall.Google Scholar
Zhou, K., Salomon, G. & Wu, E. 1999 Balanced realization and model reduction for unstable systems. Intl J. Robust Nonlinear Control 9, 183–198.3.0.CO;2-E>CrossRefGoogle Scholar