A linear state-space representation of plane Poiseuille flow for control design: a tutorial

A method for the incorporation of wall transpiration into a model of linearised plane Poiseuille flow is presented, with the aim of producing a state-space model suitable for the development of feedback control of transition to turbulence in channel flow. The system state is observed via wall shear-stress measurements and controlled by wall transpiration. The streamwise discretisation in the linearised model is by Fourier series and the wall-normal discretisation is by a Chebyshev polynomial basis, which is modified to conform to the control boundary conditions. This paper is intended as a tutorial on the addition of boundary control to a spectral model of a fluid continuum, to form a state-space model, as used in the emerging multidisciplinary field of flow control by means of Microelectrical Machines (MEMs). The ultimate aim of such flow control is the reduction of skin-friction drag on moving bodies.

[1]  The Stability or Instability of the Steady Motions of a Perfect Liquid and of a Viscous Liquid. Part I: A Perfect Liquid , 2008 .

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

[3]  Haim H. Bau,et al.  Feedback control to delay or advance linear loss of stability in planar Poiseuille flow , 1994, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[4]  Ole Morten Aamo.  Modelling and control of fluid flows and marine structures , 2002 .

[5]  Panagiotis D. Christofides,et al.  Drag reduction in transitional linearized channel flow using distributed control , 2002 .

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

[7]  George Papadakis,et al.  OPTIMAL CONTROL OF PLANE POISEUILLE FLOW , 2004 .

[8]  Wilhelm Heinrichs.  Improved condition number for spectral methods , 1989 .

[9]  L. Fox,et al.  Chebyshev polynomials in numerical analysis , 1970 .

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

[11]  R. Peyret.  Spectral Methods for Incompressible Viscous Flow , 2002 .

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

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

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

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

[16]  Jack Dongarra,et al.  Chebyshev tau-QZ algorithm methods for calculating spectra of hydrodynamic stability problems , 1995 .

[17]  D. Rempfer.  LOW-DIMENSIONAL MODELING AND NUMERICAL SIMULATION OF TRANSITION IN SIMPLE SHEAR FLOWS , 2003 .

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

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

[20]  Thomas R. Bewley.  Optimal and robust control and estimation of transition, convection, and turbulence , 1999 .

[21]  L. Råde,et al.  Mathematics handbook for science and engineering , 1995 .

[22]  Y S J O S H I,et al.  A systems theory approach to the feedback stabilization of infinitesimal and finite-amplitude disturbances in plane Poiseuille flow , 1997 .