Linear quadratic control of plane Poiseuille flow–the transient behaviour

This paper describes the design of optimal linear quadratic controllers for single wavenumber-pair periodic 2-D disturbances in plane Poiseuille flow, and subsequent verification using a finite-volume full Navier–Stokes solver, at both linear and non-linear levels of initial conditions selected to produce the largest linear transient energy growth. For linear magnitude initial conditions, open and closed-loop finite-volume solver results agree well with a linear simulation. Transient energy growth is an important performance measure in fluid flow problems. The controllers reduce the transient energy growth, and the non-linear effects are generally seen to keep energy levels below the scaled linear values, although they do cause instability in one simulation. Comparatively large local quantities of transpiration fluid are required. The modes responsible for the transient energy growth are identified. Modes are shown not to become significantly more orthogonal by the application of control. The synthesis of state estimators is shown to require higher levels of discretiation than the synthesis of state-feedback controllers. A simple tuning of the estimator weights is presented with improved convergence over uniform weights from zero initial estimates. Nomenclature Greek symbols α  streamwise (x) wave number, cycles per 2π distance β  spanwise (z) wave number, cycles per 2π distance ε(t)  synchronic transient energy bound at time t  synchronic error energy bound at time t ζ  eigenvalue in synchronic transient energy bound eigensystem η(x,y,z,t)  wall-normal vorticity perturbation  η Fourier coefficient at wavenumber pair α,β θ  diachronic transient energy bound θ Error  diachronic error energy bound θ Est  estimated energy bound  ith eigenvalue  diagonal eigenvalue matrix μ  dynamic viscosity ρ  fluid density  ith singular value of A (A)  spectral norm or largest singular value of A  modal amplitude vector, χ0  initial χ, at time t = 0 Ψ  matrix of right eigenvectors ψ i  ith right eigenvector  frequency Roman symbols  system matrix  input matrix  output matrix  multiplying co-efficient for nth Chebyshev polynomial c i  amplitude of mode i E(t)  transient energy, , at time t E Est(t)  estimated transient energy, , at time t E Error (t)  error energy , at time t E 0  E of worst open-loop perturbation of , at t = 0 (2.26 × 10− 9) E pair,bound   upper bound on mode pair energy growth h  channel wall separation  identity matrix j    state feedback gain matrix  estimator gain matrix N  highest Chebyshev polynomial degree used, final collocation point index P  pressure P b  steady base flow pressure p  pressure perturbation  state variable weighting (energy) matrix  control weighting matrix R  Reynolds number r  control weight multiplier s  measurement noise weight multiplier  invertible matrix for conversion between state variables and , excludes next-to-wall velocities and wall vorticities t  time x,y,z  streamwise, wall-normal and spanwise co-ordinates  flow velocity vector  steady base flow velocity U cl  Ub at centreline  velocity perturbation vector  u,v,w Fourier coefficients at wavenumber pair α,β  control vector  measurement noise power spectral density  process noise power spectral density  state variable vector  state estimates vector  estimate error vector,   which generates θ   which generates θ Error   transformed to values at collocation points  state variables transformed to , thus  measurement vector y n  y at nth Chebyshev–Gauss–Lobatto collocation point