Minimizing transient energy growth in plane Poiseuille flow

The feedback control of laminar plane Poiseuille flow is considered. In common with many flows, the dynamics of plane Poiseuille flow is very non-normal. Consequently, small perturbations grow rapidly with a large transient that may trigger non-linearities and lead to turbulence, even though such perturbations would, in a linear flow, eventually decay. This sensitivity can be measured using the maximum transient energy growth. The linearized flow equations are discretized using spectral methods and then considered at one wave-number pair in order to obtain a model of the flow dynamics in a form suitable for advanced control design. State feedback controllers that minimize an upper bound on the maximum transient energy growth are obtained by the repeated solution of a set of linear matrix inequalities. The controllers are tested using a full Navier—Stokes solver, and the transient energy response magnitudes are significantly reduced compared with the uncontrolled case.

[1]  Eric Rogers,et al.  H∞ control of nonperiodic two-dimensional channel flow , 2004, IEEE Trans. Control. Syst. Technol..

[2]  Brian F. Farrell,et al.  Optimal excitation of perturbations in viscous shear flow , 1988 .

[3]  Elmar Plischke,et al.  Transient Effects of Linear Dynamical Systems , 2005 .

[4]  F. Wirth.  Transient Behavior and Robustness , 2004 .

[5]  George Papadakis,et al.  Minimisation of transient perturbation growth in linearised Lorenz equations , 2005 .

[6]  L. Trefethen,et al.  Spectra and pseudospectra : the behavior of nonnormal matrices and operators , 2005 .

[7]  S. C. Reddy,et al.  Energy growth in viscous channel flows , 1993, Journal of Fluid Mechanics.

[8]  P. Schmid,et al.  Stability and Transition in Shear Flows. By P. J. SCHMID & D. S. HENNINGSON. Springer, 2001. 556 pp. ISBN 0-387-98985-4. £ 59.50 or $79.95 , 2000, Journal of Fluid Mechanics.

[9]  E. Plischke.  Stabilization of Linear Systems with Prescribed Transient Bounds , 2004 .

[10]  James F. Whidborne,et al.  On minimizing maximum transient energy growth , 2005 .

[11]  Bassam Bamieh,et al.  Componentwise energy amplification in channel flows , 2005, Journal of Fluid Mechanics.

[12]  Vladimir A. Yakubovich,et al.  Linear Matrix Inequalities in System and Control Theory (S. Boyd, L. E. Ghaoui, E. Feron, and V. Balakrishnan) , 1995, SIAM Rev..

[13]  James F. Whidborne,et al.  On the Minimization of Maximum Transient Energy Growth , 2007, IEEE Transactions on Automatic Control.

[14]  Eric Rogers,et al.  Control of Nonperiodic Two-Dimensional Channel Flow , 2004 .

[15]  Anne E. Trefethen,et al.  Hydrodynamic Stability Without Eigenvalues , 1993, Science.

[16]  Karolos M. Grigoriadis,et al.  A unified algebraic approach to linear control design , 1998 .

[17]  John McKernan.  Control of plane poiseuille flow: a theoretical and computational investigation , 2006 .

[18]  George Papadakis,et al.  Linear quadratic control of plane Poiseuille flow–the transient behaviour , 2007, Int. J. Control.

[19]  P. Schmid.  Nonmodal Stability Theory , 2007 .

[20]  Dan S. Henningson,et al.  On stability of streamwise streaks and transition thresholds in plane channel flows , 1998, Journal of Fluid Mechanics.

[21]  S. Orszag.  Accurate solution of the Orr–Sommerfeld stability equation , 1971, Journal of Fluid Mechanics.

[22]  George Papadakis,et al.  A linear state-space representation of plane Poiseuille flow for control design: a tutorial , 2006, Int. J. Model. Identif. Control..

[23]  Thomas Bewley,et al.  Optimal and robust control and estimation of linear paths to transition , 1998, Journal of Fluid Mechanics.

[24]  Elmar Plischke,et al.  State Feedback Stabilization with Guaranteed Transient Bounds , 2002 .

[25]  Eric Rogers,et al.  Robust Control of Linearized Poiseuille Flow , 2002 .

[26]  W. R. Howard,et al.  Mathematical Systems Theory I: Modelling, State Space Analysis, Stability and Robustness , 2005 .

[27]  Kathryn M. Butler,et al.  Three‐dimensional optimal perturbations in viscous shear flow , 1992 .

[28]  Sheila E. Widnall,et al.  A flow-visualization study of transition in plane Poiseuille flow , 1982, Journal of Fluid Mechanics.

[29]  Sanjay S. Joshi,et al.  A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow , 1997, Journal of Fluid Mechanics.

[30]  John C. Doyle,et al.  Robust flow stability: theory, computations and experiments in near wall turbulence , 2004 .

[31]  Stephen P. Boyd,et al.  Linear Matrix Inequalities in Systems and Control Theory , 1994 .

[32]  George Papadakis,et al.  Large Eddy Simulation of Turbulent Flow in a Rushton Impeller Stirred Reactor with Sliding-Deforming Mesh Methodology , 2004 .

[33]  Bruce A. Francis,et al.  Feedback Control Theory , 1992 .

[34]  James F. Whidborne,et al.  MINIMIZATION OF MAXIMUM TRANSIENT ENERGY GROWTH BY OUTPUT FEEDBACK , 2005 .