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.