Hostname: page-component-cd9895bd7-hc48f Total loading time: 0 Render date: 2024-12-19T03:13:13.540Z Has data issue: false hasContentIssue false
  • English
  • Français

A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow

Published online by Cambridge University Press:  10 February 1997

Sanjay S. Joshi ,
Jason L. Speyer  and
John Kim
Sanjay S. Joshi
Affiliation:
School of Engineering and Applied Science, University of California, Los Angeles Los Angeles, CA 90024, USA
Jason L. Speyer
Affiliation:
School of Engineering and Applied Science, University of California, Los Angeles Los Angeles, CA 90024, USA
John Kim
Affiliation:
School of Engineering and Applied Science, University of California, Los Angeles Los Angeles, CA 90024, USA
Get access Rights & Permissions [Opens in a new window]

Extract

A systems theory framework is presented for the linear stabilization of two-dimensional laminar plane Poiseuille flow. The governing linearized Navier-Stokes equations are converted to control-theoretic models using a numerical discretization scheme. Fluid system poles, which are closely related to Orr-Sommerfeld eigenvalues, and fluid system zeros are computed using the control-theoretic models. It is shown that the location of system zeros, in addition to the well-studied system eigenvalues, are important in linear stability control. The location of system zeros determines the effect of feedback control on both stable and unstable eigenvalues. In addition, system zeros can be used to determine sensor locations that lead to simple feedback control schemes. Feedback controllers are designed that make a new fluid-actuator-sensorcontroller system linearly stable. Feedback control is shown to be robust to a wide range of Reynolds numbers. The systems theory concepts of modal controllability and observability are used to show that feedback control can lead to short periods of highamplitude transients that are unseen at the output. These transients may invalidate the linear model, stimulate nonlinear effects, and/or form a path of ‘bypass’ transition in a controlled system. Numerical simulations are presented to validate the stabilization of both single-wavenumber and multiple-wavenumber instabilities. Finally, it is shown that a controller designed upon linear theory also has a strong stabilizing effect on two-dimensional finite-amplitude disturbances. As a result, secondary instabilities due to infinitesimal three-dimensional disturbances in the presence of a finite-amplitude two-dimensional disturbance cease to exist.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 332 , February 1997 , pp. 157 - 184
Copyright
Copyright © Cambridge University Press 1997

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

Bayly, B. J., Orszag, S. A. & Herbert, T. 1988 Instability mechanisms in shear-flow transition. Ann. Rev. Fluid Mech. 20, 359–391.Google Scholar
Beringen, S. 1984 Active control of transition by periodic suction and blowing. Phys. Fluids 27, 1345–1348.Google Scholar
Bower, W. W., Kegelman, J. T. & Pal, A. 1987 A numerical study of two-dimensional instabilitywave control based on the Orr-Sommerfeld equation. Phys. Fluids 30, 998–1004.CrossRefGoogle Scholar
Bryson, A. E. JR. & Ho, T.-C. 1975 Applied Optimal Control. Taylor and Francis.Google Scholar
Burns, J. A. & Ou Y.-R. 1994 Feedback control of the driven cavity problem using LQR designs. Proc. 33rd Conf. on Decision and Control, December, 1994, pp. 289–294.Google Scholar
Butler, K. M. & Farrell, B. F. 1992 Three-dimensional optimal perturbations in viscous shear flows. Phys. Fluids A 4, 1637–1650.CrossRefGoogle Scholar
Choi, H., Moin, P. & Kim, J. 1994 Active turbulence control for drag reduction in wall bounded flows. J. Fluid Mech. 262, 75–110.Google Scholar
Drazin, P. G. & Reid, W. H. 1981 Hydrodynamic Stability. Cambridge University Press.Google Scholar
Farrell, B. F. 1988 Optimal perturbations in viscous shear flows. Phys. Fluids 31, 2093–2102.Google Scholar
Franklin, G., Powell, J. D. & Emami-Naeini, A. 1988 Feedback Control of Dynamic Systems. Addison-Wesley.Google Scholar
Grace, A., Laub, A. J., Little, J. N. & Thompson, C. M. 1992 Control System Toolbox: For use with MATLAB. The Math Works.Google Scholar
Gunzburger, M. D., Hou, L. & Svobodny, T. 1992 Boundary velocity control of incompressible flow with an application to viscous drag reduction. SIAM J. Control Optimization 30, 167–181.Google Scholar
Henningson, D. S. 1994 Bypass transition - Proceedings from a Mini-workshop. Tech. Rep. Royal Institute of Technology, Department of Mechanics, S-100,44, Stockholm, Sweden.Google Scholar
Joshi, S. S. 1996 A systems theory approach to the control of plane Poiseuille flow. PhD thesis, UCLA, Department of Electrical Engineering.Google Scholar
Joslin, R. D., Erlebacher, G. & Hussaini, M. Y. 1994 Active control of instabilities in laminar boundary layer flow- Part I: An overview. ICASE Rep. 94-97. NASA Langley Research Center, Hampton, VA.Google Scholar
Kailath, T. 1980 Linear Systems. Prentice-Hall.Google Scholar
Kim, J., Moin, P. & Moser, R. 1987 Turbulence statistics in fully developed channel flow at low Reynolds numbers. J. Fluid Mech. 177, 133–166.Google Scholar
Kreiss, H.-O. & Lorenz, J. 1989 Initial Boundary Value Problems and the Navier-Stokes Equations. Academic.Google Scholar
Ladyzhenskaya, O. A. 1969 The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach.Google Scholar
Nosenchuck, D. M. 1982 Passive and active control of boundary layer transition. PhD thesis, California Institute of Technology.Google Scholar
Ogata, K. 1990 Modern Control Engineering. Prentice-Hall.Google Scholar
Orr, W. M'F. 1907 The stability or instability of the steady motions of a perfect liquid and of a viscous liquid. Proc. R. Irish Acad. A 27, 9–68.Google Scholar
Orszag, S. A. 1971 Accurate solution of the Orr Sommerfeld stability equation. J. Fluid Mech. 50, 689–703.Google Scholar
Orszag, S. A. & Patera, A. T. 1983 Secondary instability of wall-bounded shear flows. J. Fluid Mech. 128, 347–385.Google Scholar
Reed, H. L. & Nayfeh, A. H. 1986 Numerical perturbation technique for stability of flat plate boundary layer with suction. AIAA J. 24.CrossRefGoogle Scholar
Schubauer, G. B. & Skramstad, H. K. 1947 Laminar boundary layer oscillations and stability of laminar flow. J. Aeronaut. Sci. 14, 68–78.Google Scholar
Sommerfeld, A. 1908 Ein Beitrag zur hydrodynamicischen Erklaerung der turbulenten Fluessigkeitsbewegungen. Proc. 4th Intl Congress of Mathematics, Vol. Ill, pp. 116–124.Google Scholar
Temam, R. 1984 Navier-Stokes Equations. North-Holland.Google Scholar
Trefethen, L. N., Trefethen, A. E., Reddy, S. C. & DRISCOLL, T. A. 1993 Hydrodynamic stability without eigenvalues. Science 261, 578–584.Google Scholar