Optimal Control of Plane Poiseuille Flow Workshop on Flow Control, Poitiers 11-14th Oct 04 J. Mckernan, J.F.Whidborne, G.Papadakis Cranfield University, U.K. King s College, London, U.K. Optimal Control of Plane Poiseuille Flow p. 1/22
Content Background Schmid and Henningson s spectral model (2001) Linear state-space model (modified Bewley 1998, Hogberg 2003) State feedback estimator and controller (Joshi 1995, ) Initial conditions (Butler and Farrell 1992) Controller implementation in full FV Navier-Stokes solver Results and Conclusions Optimal Control of Plane Poiseuille Flow p. 2/22
Background Theoretical, rather than experimental approach Laminar Poiseuille (closed channel) flow Simple base flow (constant), geometry, boundary conditions Linearly unstable, Transition Here, 2D (streamwise, wall-normal) Synthesize linear optimal controllers Minimise Transient Energy Growth (Time integral) Test controllers in full Navier-Stokes Solver Optimal Control of Plane Poiseuille Flow p. 3/22
Control of Poiseuille Flow Upper Wall wall normal, y Plane Poiseuille Flow + Flow Disturbance streamwise, x Lower Wall Actuation Sensing Controller Optimal Control of Plane Poiseuille Flow p. 4/22
Spectral Model Schmid and Henningson 2001 Linearised Navier-Stokes equations Velocity-vorticity formulation Spectral discretisation Chebyshev in wall-normal direction Fourier in streamwise and spanwise directions One wave number pair (here ) Homogeneous wall boundary conditions Optimal Control of Plane Poiseuille Flow p. 5/22
State-Space Model Form;- Input : rate of change of wall-normal velocities at walls ( now inhomogeneous ) Output : Wall-shear stress measurements States : Coeffs of Novel Chebyshev Recombinations Wall-normal velocities at walls Optimal Control of Plane Poiseuille Flow p. 6/22
% ' State-Space Model Navier-Stokes equations;- Linear state-space form;- $$ " ) ( ( " &" $$ " $$! #" Optimal Control of Plane Poiseuille Flow p. 7/22
Optimal State Feedback Given the real system;- Feedback control signal to minimize;- Given by where from ARE Optimal Control of Plane Poiseuille Flow p. 8/22
Weighting Matrices Choose to form energy density (Bewley 1998);- Curtis-Clenshaw quadrature for integration Choose Max energy vs plotted from linear simulations Choose Optimal Control of Plane Poiseuille Flow p. 9/22
Optimal State Estimation Given the real estimator;- The optimal to minimize;- Is given by where from ARE Optimal Control of Plane Poiseuille Flow p. 10/22
Weighting Matrices represents covariance of process noise Choose is physical distance between states, represents covariance of measurement noise Choose Fastest pole vs plotted from linear simulations Choose, for fastest estimator pole plant Optimal Control of Plane Poiseuille Flow p. 11/22
Initial Conditions Worst not unstable eigenvector Non-orthogonal system matrix (Trefethen 1993) Variational method used (Butler and Farrell 1992, Bewley 1998) States, Energy Transient Energy Growth from Hence (TS waves) Optimal Control of Plane Poiseuille Flow p. 12/22
Non-Linear Simulations Finite-volume full Navier-Stokes solver Second order in space (CDS), implicit first order in time PISO algorithm, Collocated grid ( ) BCs: Streamwise - cyclic Walls - transpiration Spanwise - symmetric Code modified to solve for the perturbation flow about base Optimal Control of Plane Poiseuille Flow p. 13/22
Implementation of Controller FFT to compute measurements States estimated using;- from Control signals computed using;- (LQR) (LQG) Integration for and Inverse FFT for Optimal Control of Plane Poiseuille Flow p. 14/22
Open Loop Results 2 Transient Energy Density/ρ U cl 8 x 10 6 7 6 5 4 3 2 Non linear Simulation Non linear Simulation Estimate Linear Simulation Linear Simulation Estimate 1 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow p. 15/22
Open Loop Results 2 Transient Energy Density/ρ U cl 0.08 0.07 0.06 0.05 0.04 0.03 0.02 Non linear Simulation Non linear Simulation Estimate Linear Simulation Linear Simulation Estimate 0.01 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow p. 16/22
Closed Loop Results - LQR State Feedback, 0.012 0.01 Non linear Simulation Linear Simulation 2 Transient Energy Density/ρ U cl 0.008 0.006 0.004 0.002 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow p. 17/22
Closed Loop Results - LQG Output Feedback, 2 Transient Energy Density/ρ U cl 0.025 0.02 0.015 0.01 0.005 Non linear Simulation Non linear Simulation Estimate Linear Simulation Linear Simulation Estimate 0 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow p. 18/22
Closed Loop Results Output Feedback, 0.025 0.02 Non linear Simulation Linear Simulation Upper wall transpiration vel/u cl 0.015 0.01 0.005 0 0.005 0.01 0.015 0.02 0.025 0 1 2 3 4 5 Time(s) Optimal Control of Plane Poiseuille Flow p. 19/22
Conclusions Optimal controllers for 2D Poiseuille Flow synthesized. Controller implemented in FV CFD code. Small Perturbations Spectral Linear and FV CFD results identical. Large Perturbations Open loop - CFD simulation saturates, but still unstable. Closed Loop - simulations stabilised, CFD needs longer than linear. Optimal Control of Plane Poiseuille Flow p. 20/22
Future Targets Investigate control of linearly stable but worst perturbation CFD simulations - Streamwise Vortices than TS waves rather Actuation by wall-normal and tangential transpiration More control degrees of freedom LMI Controllers Minimise upper bound on max transient energy growth Optimal Control of Plane Poiseuille Flow p. 21/22
The End Thank you. Optimal Control of Plane Poiseuille Flow p. 22/22