Optimal and robust control of small disturbances in a channel flow with a normal magnetic field

Active closed-loop control of subcritical and supercritical instabilities amplified in a channel flow submitted to a constant normal magnetic field is investigated. Control is carried out at both the upper and lower walls by blowing and suction (velocity control) or by a perturbation of the normal magnetic field (magnetic control). Even if a velocity control is more efficient than a magnetic one, we found that magnetic control succeeds in stabilizing supercritical instabilities. Development of new actuators using magnetic field for flow control may thus be promising. Closed-loop control modifies the optimal perturbation but does not destruct the lift-up effect.

[1]  T. Alboussière,et al.  On the stability of the Hartmann layer , 1999 .

[2]  P. Moresco,et al.  Experimental study of the instability of the Hartmann layer , 2004, Journal of Fluid Mechanics.

[3]  Miroslav Krstic,et al.  Inverse optimal boundary control for mixing in magnetohydrodynamic channel flows , 2003, 42nd IEEE International Conference on Decision and Control (IEEE Cat. No.03CH37475).

[4]  Frank L. Lewis,et al.  Optimal Control , 1986 .

[5]  Chih-Ming Ho,et al.  MICRO-ELECTRO-MECHANICAL-SYSTEMS (MEMS) AND FLUID FLOWS , 1998 .

[6]  Thomas R. Bewley,et al.  A noncausal framework for model-based feedback control of spatially developing perturbations in boundary-layer flow systems. Part II: numerical simulations using state feedback , 2004, Syst. Control. Lett..

[7]  Liang Shao,et al.  A new dynamic subgrid eddy viscosity model with application to turbulent channel flow , 2004 .

[8]  M. Landahl.  A note on an algebraic instability of inviscid parallel shear flows , 1980, Journal of Fluid Mechanics.

[9]  Paolo Luchini,et al.  Reynolds-number-independent instability of the boundary layer over a flat surface , 1996, Journal of Fluid Mechanics.

[10]  E. Reshotko,et al.  Spatial theory of optimal disturbances in boundary layers , 2001 .

[11]  P. Luchini.  Reynolds-number-independent instability of the boundary layer over a flat surface: optimal perturbations , 2000, Journal of Fluid Mechanics.

[12]  Paolo Luchini,et al.  Algebraic growth in boundary layers: optimal control by blowing and suction at the wall , 2000 .

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

[14]  Fernando Paganini,et al.  Distributed control of spatially invariant systems , 2002, IEEE Trans. Autom. Control..

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

[16]  Fernando Paganini,et al.  A Course in Robust Control Theory , 2000 .

[17]  Jan O. Pralits,et al.  Optimization of steady suction for disturbance control on infinite swept wings , 2003 .

[18]  Dan S. Henningson,et al.  Linear optimal control applied to instabilities in spatially developing boundary layers , 2002, Journal of Fluid Mechanics.

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

[20]  D. Gérard-Varet.  Amplification of small perturbations in a Hartmann layer , 2002 .

[21]  Lennart Löfdahl,et al.  MEMS applications in turbulence and flow control , 1999 .

[22]  David J. N. Limebeer,et al.  Linear Robust Control , 1994 .

[23]  G. Gerbeth,et al.  Experimental study of non-normal nonlinear transition to turbulence in a rotating magnetic field driven flow , 2003 .

[24]  Alessandro Bottaro,et al.  Optimal control of Tollmien–Schlichting waves in a developing boundary layer , 2001 .

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

[26]  Dan S. Henningson,et al.  Linear feedback control and estimation of transition in plane channel flow , 2003, Journal of Fluid Mechanics.

[27]  A. Schaft.  G.E. Dullerud and F. Paganini - A course in robust control theory: a convex approach. Berlin: Springer-Verlag, 2000 (Text in Applied Mathematics ; 36) , 2001 .

[28]  George S. Dulikravich,et al.  Magnetofluiddynamics in Channels and Containers , 2001 .

[29]  L. Gustavsson.  Energy growth of three-dimensional disturbances in plane Poiseuille flow , 1981, Journal of Fluid Mechanics.

[30]  A. Bottaro,et al.  Optimal and robust control of streaks in pipe flow , 2005, Journal of Fluid Mechanics.

[31]  Thomas Boeck,et al.  Numerical study of the instability of the Hartmann layer , 2004, Journal of Fluid Mechanics.

[32]  D. Henningson,et al.  Optimal disturbances and bypass transition in boundary layers , 1999 .

[33]  Dan S. Henningson,et al.  Adjoint-based optimization of steady suction for disturbance control in incompressible flows , 2002, Journal of Fluid Mechanics.

[34]  Satish C. Reddy,et al.  A MATLAB differentiation matrix suite , 2000, TOMS.

[35]  Thomas Bewley,et al.  Flow control: new challenges for a new Renaissance , 2001 .

[36]  P. Moresco,et al.  Weakly nonlinear stability of Hartmann boundary layers , 2003 .

[37]  Christophe Airiau,et al.  On the amplification of small disturbances in a channel flow with a normal magnetic field , 2004 .

[38]  Alessandro Bottaro,et al.  Optimal perturbations for boundary layers subject to stream-wise pressure gradient , 2000 .