Optimal Control of Plane Poiseuille Flow Workshop on Flow Control, - - PowerPoint PPT Presentation

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

Controller Plane Poiseuille Flow + Flow Disturbance Lower Wall Upper Wall Actuation Sensing streamwise, x wall−normal, y

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

about base flow

Optimal Control of Plane Poiseuille Flow – p. 13/22

Implementation of Controller

FFT to compute measurements

from

States estimated using;-

• ✁

• ✁

• ✁

Control signals

• ✂

• ✂

computed using;-

(LQR)

• ✁

(LQG) Integration for

and

Inverse FFT for

Optimal Control of Plane Poiseuille Flow – p. 14/22

Open Loop Results

• ✝

1 2 3 4 5 1 2 3 4 5 6 7 8 x 10

−6

Time(s) Transient Energy Density/ρ Ucl

2

Non−linear Simulation Non−linear Simulation Estimate Linear Simulation Linear Simulation Estimate

Optimal Control of Plane Poiseuille Flow – p. 15/22

Open Loop Results

• ✝

1 2 3 4 5 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 Time(s) Transient Energy Density/ρ Ucl

2

Non−linear Simulation Non−linear Simulation Estimate Linear Simulation Linear Simulation Estimate

Optimal Control of Plane Poiseuille Flow – p. 16/22

Closed Loop Results - LQR

State Feedback,

1 2 3 4 5 0.002 0.004 0.006 0.008 0.01 0.012 Time(s) Transient Energy Density/ρ Ucl

2

Non−linear Simulation Linear Simulation

Optimal Control of Plane Poiseuille Flow – p. 17/22

Closed Loop Results - LQG

Output Feedback,

1 2 3 4 5 0.005 0.01 0.015 0.02 0.025 Time(s) Transient Energy Density/ρ Ucl

2

Non−linear Simulation Non−linear Simulation Estimate Linear Simulation Linear Simulation Estimate

Optimal Control of Plane Poiseuille Flow – p. 18/22

Closed Loop Results

Output Feedback,

1 2 3 4 5 −0.025 −0.02 −0.015 −0.01 −0.005 0.005 0.01 0.015 0.02 0.025 Time(s) Upper wall transpiration vel/Ucl Non−linear Simulation Linear Simulation

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

• ✂

rather than TS waves

• ✏

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