Control of Turbulent Boundary Layers:Success, Limitations & Issues

John Kim

Department of Mechanical & Aerospace EngineeringUniversity of California, Los Angeles

OutlinePart I. Some Comments on Boundary-Layer Control using the Lorentz Force

• Kim (Dresden, 1997), Berger et al. (POF, 2000)• Du and Karniadakis (Science, 2000), Du et al. (JFM 2002)• Breuer et al. (POF, 2004)

Part II. Analysis of Boundary-Layer Controllers: A Linear System Perspective

• Motivations• Linear Optimal Controllers• Analysis of Linear Systems

– Eigenvalue Analysis and Transient Growth– Singular Value Decomposition and “Optimal” Disturbances– Relevance to TBL?

• Beyond Canonical TBL• Limitations and Issues• Concluding Remarks

Part II

Analysis of Boundary-Layer Controllers:A Linear System Perspective

Motivations

• Several investigators have shown that linear mechanisms play an important role in wall-bounded shear flows:

– Near-wall turbulence structures in TBL are “optimal” disturbances of the linearized Navier-Stokes system (Farrell et al.)

– Transient growth (due to a linear mechanism) can cause by-pass transition at sub-critical Reynolds numbers (Henningson et al.)

– Near-wall turbulence could not be sustained without a certain linear mechanism (Kim and Lim)

• Successful applications of linear controllers to transitional and turbulent flows have been reported (UCLA,UCSD,KTH).

• The fact that a linear mechanism plays an important role in turbulent flows allows us to investigate the flow from a linear system perspective. We apply the SVD analysis in order to gain new insights into the mechanism by which these controllers are able to accomplish the viscous drag reduction in turbulent boundary layers.

Linear Optimal Controllers:Systems Control Theoretic Approach

• Linear optimal control theory synthesizes optimal control inputs to minimize (or maximize) a cost function.

• Does not require extensive intuitive understanding of the dynamics of the system to be controlled.

• Represent the system in state-space form, which consists of a state (x), control (u), measurement (z), and system matrices:

• Choose the control input (u) to minimize,

x Ax Bu

z Cx Du

* *

0

( )J xQx u Ru dtg¥

= +ò

State-Space Representation of N-S

0= +

os

wally yc sq

v L vdBv

L Ldt w w

é ùé ù é ùê úê ú ê úê úê ú ê úë û ë ûê úë û

u

x

Linear Quadratic Regulator (LQR)

• If the full internal state (x) is available, LQR synthesis provides a control law to minimize a quadratic cost function.

• An optimal control gain matrix, K, is obtained from the solution to the algebraic Riccati equation.

* *

0

( )J xQx u Ru dtg¥

= +ò

1 * 0AP PA Q PBR B Pg -+ + - =

1 *( )u R B P x

Kx

g -= -

= -

Different Cost Functions

LQR Control of Turbulent Channel

25 ~ 30 %

Control input~ 10 % of u

LQR Control of Turbulent Channel

LQR Summary

• Although the flow system is fully nonlinear, LQR successfully reduced all cost functions.

• For all these cases, there is a significant amount of mean drag reduction.

• To achieve mean drag reduction, it is important to eliminate the flow structures near the wall.

– The sources of turbulence has to be significantly reduced by control actuation.

– Cost function should take into account the near-wall activity.

• Various fine-tuning efforts can lead to further drag reduction (e.g., gain scheduling approach with evolving mean profiles led to laminarization of low Re flows).

SVD Analysis of Linear Systems (Lim and Kim, POF 2004)

• The linearized N-S equations for incompressible flows,

or

have a solution

• It can be written in terms of eigenvectors of A

where i and si denote eigenvalues and eigenvectors of A, respectively.

• In classical linear stability analysis,

Classical Linear Stability Analysis:Eigenvalue Decomposition

xx

Adtd

ysqc

os

y

v

LL

Lv

dtd

0

.oAte xx

i < 0 , stablei > 0 , unstable

.221121

Nt

Ntt Nececec sssx

• Physically, we are interested in the disturbance energy.

• Let

where and denote the energy- and 2-norm, respectively, and Q is a known operator defining the disturbance energy.

• A is not self-adjoint and its eigenvectors are NOT orthogonal to each other.

• Linear stability analysis predicts correctly the asymptotic behavior, but it ignores the transient behavior due to the non-orthogonality of the eigenvectors.

Eigenvalue Decomposition – contd.

2xx Q

E

2

E

Examples of parallel eigenvectors

• Example: Rec=5000, kx=0., kz=2.044 max(Ek/Eko) = 4897

“Optimal” disturbance

Transient Growth of “Optimal” Disturbance

Transient Growth of Optimal Disturbance

• We are interested in a disturbance that has the largest energy amplification, thus

• But this is the induced 2-norm of matrix, thus

• Recall that the 2-norm of a matrix corresponds to the largest singular value of the matrix.

Singular Value Decomposition (SVD)

2

2

1

0

2

2

000

sup

supsupsup

o

oAt

o

oAt

Eo

EoAt

Eo

E

QQe

Q

Qee

o

ooo

w

w

x

x

x

x

x

x

w

xxx

2

1

00sup

QQeAt

E

E

o x

x

x

N

NNA

2

1

2121

~uuuvvv

• Therefore, if we let , the “optimal” disturbance we are interested in corresponds to the first right singular vector of and its amplification factor corresponds to the largest singular value of , i.e.,

or

Singular Value Decomposition – contd.

1~ QQeA At

A~A~

The right singular vector is the “optimal” disturbance

The left singular vector shows the amplified “optimal” disturbance

VAUUVAVUA HH ~~~

Channel at a sub-critical Reynolds number:Rec=5000, kx=0, kz=2.044, max(Ek/Eko) = max= 4897

Singular Value Decomposition – contd.

Any relevance to turbulent flows, which are known to be highly nonlinear phenomena?

• Henningson et al. have shown that this linear mechanism could lead to sub-critical transition.

• Farrell et al. attribute this mechanism responsible for the near-wall turbulence structures.

• Various control schemes investigated by the UCLA group suggest that a linear mechanism(s) is playing a key role in TBL.

• It has been shown that linear optimal controllers (LQR/LQG) work surprisingly well in TBL, suggesting that the wall-layer dynamics can be approximated by a linear model.

• Can we use the SVD to gain insights into different controllers we have used?

“Optimal” Disturbance in Turbulent Channel

• In contrast to the optimal disturbance in a laminar flow, the transient growth of “optimal” disturbances is interrupted by nonlinear activities before its potential maximum state can be reached.

• A turbulence time scale tg , during which an “optimal” disturbance can grow according to the linear mechanism, must be included in the analysis. Turbulent eddy turnover time in the wall layer is considered here (Butler and Farrell, 1993).

• The “optimal disturbance” in turbulent flows is the disturbance that will have the largest transient growth within the eddy turnover time.

• Find the the largest singular value (max) attainable within the eddy turnover time by all possible wavenumbers (ie all possible eddy sizes).

SVD of Turbulent Channel Flow

kx

kz

max

E/E0=32.5 for kx=0 and kz=6 (z+=100) with tg

+=80

“Optimal” Disturbance in Turbulent Channel

Singular ValuesEvolution of Energy

E/E0=32.5 for kx=0 and kz=6 (z+=100)

“Optimal” Disturbance in Turbulent Channel

• The optimal disturbance is found to be similar to the streamwise vortices and high-and low-speed streaks in TBL .

• The length scale of the optimal disturbance for a uncontrolled flow is universal for wide range of Re (z

+ 100).

y

0 0.5 1 1.5 2 2.5 3 3.5 4-1

-0.8

-0.6

-0.4

-0.2

0Re=100, kx=0, kz=6.0 (z

+=100)

z

SVD Analysis of Linear Systems with Control

• The linearized N-S equations for incompressible flows,

or

• With control,

• With a linear feedback control, u = - Kx,

• Need to perform SVD analysis of Qe(A-BK)tQ-1 instead of QeAtQ-1.

Linearized Navier-Stokes System with Control

uxx

BAdtd

xx

Adtd

ysqc

os

y

v

LL

Lv

dtd

0

xxxx

)( BKABKAdtd

Opposition Control

• One of the first successful active feedback flow controls for drag reduction in TBL (Choi et al, 1994).

– Notwithstanding its implementation problem in practice, it has been used as a reference case against which other controllers to be compared.

– About 30% drag reduction was achieved with yd+ = 10-15.

– Drag was increased significantly with yd+ > 20.

• The linearized N-S equations for incompressible flows,

or

• With control,

• With a linear feedback control, u = - Kx,

• Need to perform SVD analysis of Qe(A-BK)tQ-1 instead of QeAtQ-1.

Linearized Navier-Stokes System with Control

uxx

BAdtd

xx

Adtd

ysqc

os

y

v

LL

Lv

dtd

0

xxxx

)( BKABKAdtd

What is K for opposition control?

Opposition Control in State-Space Form

• Using the collocation matrix representation (Bewley and Liu, 1998), the control gain matrix K for opposition control can be expressed as

• Depending on the yd , (A-BK) will have different system dynamics.

– Unlike the linear optimal controllers, there’s no guarantee that the opposition-controlled system will be stable.

– More importantly, the so-called “optimal” disturbance will have different transient growth.

• Perform the SVD using (A-BK) instead of A, and examine max .

yd

0 1 0 0 0 0

0 0 1 0 0 0K

é ùê ú= ê úê úë û

L L L L L

L L L L L

Kv K

dwall y yv v +== -

SVD of Opposition Control

SVD of Opposition Control – contd.

• The length scales corresponding to the “optimal” disturbance with control are fairly universal.

• Increase of the largest max for larger yd+ is due to the increase of max at kz=0.

No control yd+ = opt yd+ >> optE/Eo 38.1 21 167length scale (inf, 105+) (630+, 125+) (420+, inf)E/Eo 30.7 19.2 113length scale (inf, 108+) (646+, 126+) (412+, inf)E/Eo 27.5 17.7 99.2length scale (inf, 108+) (620+, 118+) (413+, inf)E/Eo 27.8 17.1 53.9length scale (inf, 104+) (618+, 116+) (371+, inf)

590

Re_tau

100

180

395

SVD of Opposition Control at High Re

• An approximate estimation of max using the Reynolds-Tiederman profile for turbulent mean flows at high Re:

– Optimal range of the detection-plane location appears to exist.

– Reduction for kx = 0 wavenumbers persists, implying that opposition control will continue to be effective at high Re in controlling streamwise vortices.

Re_tau No control yd+ = 101000 29.5 18.72000 27.8 18.65000 27.3 17.8

For (kx = 0, z+ 100)

dydU

ikL

ReUikL

RedyUd

ikUikL

where

N

Νv

LL

Lv

dtd

zc

xsq

xxos

v

ysqc

os

y y

1

1

0

22

21

• Navier-Stokes Equations:

SVD of KL’s Virtual Flow

• Modified Navier-Stokes Equations:

SVD of KL’s Virtual Flow – contd.

• The operator in the modified system is closer to normal or self-adjoint.

• Provides insights into the role of the linear mechanism in TBL.• Provides guidelines for controller design in TBL.

yN

Νv

LL

Lv

dtd v

ysqc

os

y 0

0

Virtual Flow Result

Laminarization!

SVD of the Virtual Flow

• Non-normality of of the linear system is reduced without the linear coupling term, Lc.

• Reduction of non-normality led to reduction of large singular values. Conversely, large singular values were due to non-normality of the linear operator A in dx/dt=Ax.

• Reduction of non-normality or large singular values can be used as a control objective in controller design.

Singular values with and without Lc for kx = 0 and kz = 6

Regular channel

Virtual flow

Self-Sustaining Mechanism of Near-Wall Turbulence

Streamwise Vortices

StreaksStreamwise-varying modes, kx~=0

Nonlinear LcV

Streak instability

• If the full internal state (x) is available, LQR synthesis provides a control law to minimize a quadratic cost function.

• An optimal control gain matrix, K, is obtained from the solution to the algebraic Riccati equation.

• Here, Q is chosen to minimize disturbance energy.

SVD of Linear Quadratic Regulator (LQR)

0

γ dtRQJ ** uuxx

x

xu

K

PBR

PBPBRQPAAP

)(γ

γ

*

*

1

1 0

SVD of Boundary-Layer Control

Singular Values

No control 32.85Opposition y^+=5 27.27

y^+=10 17.03y^+=20 18.36

LQR 26.3 15.95 15.33

Lc=0 0.5424

No control LQR (= 0.1 )

Opposition (yd+ =10) Lc=0

Turbulent Channel Flow Control Result

Drag

t+

• Opposition control (yd+ = 10 ) and LQR control ( = 0.1 ) produced

similar drag reduction.

Virtual flow

LQROpposition control

No control

Summary of SVD Analysis

• The SVD provided new insights into opposition control and other linear controls regarding their capability of attenuating the transient growth of disturbances in turbulent boundary layers.

• The SVD of opposition control indicated existence of an optimal range of the detection plane. It also showed that opposition control using detection planes too far away from the wall could enhance the growth of certain disturbances, consistent with observations in DNS/LES.

• Trends observed through the SVD in turbulent channel flow were similar to those observed in DNS or LES.

• The SVD can provide useful guidelines for control of turbulent boundary layers.

Further details in Lim and Kim (POF,2004)

Beyond Canonical BL:Control of Separated Flow over an Airfoil

• System matrices are not known• Use the system identification theory to model the system, and then apply linear control

theories

No control

Control with single

frequency

Approach Overview

Navier-Stokes equations

State EstimatorControl Actuator

Approximate Linear Model Measurement

PressureVorticityShear stress

Actuation

BlowingSuction

LQG (Linear Quadratc Gaussian) compensator

Numerical simulation of separated flows

Separated boundary-layer flows are considered for preliminary controller design and testing, the ultimate goal being high angle-of-attack airfoil flows.

Mathematical Models• Plant dynamics – Navier-Stokes equations

2

q 0q 1(qq) q

Rep

t

• State-space representation of dynamic system

xAx Bu

y Cx Du

d

dt

• State estimator

xAx Bu L(z z)

y Cx Du

d

dt

• Cost function to be minimized

* *

0(x Qx u Ru)J dt

Variablesq : velocity vector

A, B, C, D: system matrices

x: state

x: estimated state

y: measurement

u: control actuation

K: control gain matrix

L: Kalman filter

• Control law produced by the LQG (Linear Quadratic Gaussian) synthesis

u Kx

Separation on a Flat Plate• DNS of boundary layer separation caused by suction on the opposite boundary

• A simplified model for leading edge separation of an airfoil

BlasiusBoundaryLayer

Separation region

Suction V(x)

• Transition takes place abruptly around x=3.5 due to strong inviscid instability

5

0 0

Re 10

1( ) 0.3

Y Z

X

L L

Z Y

S V x dxdzU L L

Total vorticity on a spanwise plane

Streamwise vorticity on ahorizontal plane

Approximate Linear Model

• The ARX model (in discrete time)

T T

1 1

y( ) a y( ) b u( )N N

Kq q

t t q t q N

• The system’s state-space representation can be constructed using the identified model:

y: measurement, u: input signal

N: model order, Nk: measurement delay

aT and bT matrices are determined by least-square estimate.

xAx Bu

y Cx Du

d

dt

• Remarks– Time delay in ARX model are estimated using the convective velocity– Assumed zero feed-through

– Long delay (large Nk) may lead to large system matrices (A, B, C, D)

– Insufficient data length leads to inaccurate system identification

Identification Procedure

Approximate Linear Model

Navier-Stokes equationsMeasurement

PressureVorticityShear stress

Actuation:Broad-band noise

Phase 1: Record input-output data

Phase 2: Construct approximate linear model using selected model structure

Least-squareestimate

Input-ouptutdata

Phase 3: Perform the LQG (Linear Quadratic Gaussian) synthesis and form the feedback loop

Navier-Stokes equations

Controller

Estimator Performance

• For attached or mildly separated flows, the state estimator (blue line) is able to follow the outputs of the ARX model (black line) and Navier-Stokes simulations (red line)

• Challenges for massively separated flows– Separation bubble intermittently bursts or completely breaks down– Signals from the large amplitude, low-frequency oscillations can

contaminate identification results– Improved signal processing techniques are under development

Preliminary Results

• Time average of 500 fields

• The separation bubble boundary (blue line) is defined by the zero contour of the streamwise velocity

• Figures are magnified in the vertical direction for clarity

2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

2 2.5 3 3.5 4 4.50

0.1

0.2

0.3

Controller OFF

Controller ON

Limitations/Issues

• Except for few cases, most examples shown here result in 20-30% reduction of the mean viscous drag in spite of much larger reduction in the cost function

• Choice of cost function to yield the optimal result (drag in the present example) is not clear

• LQG/LTR– Control objective (e.g. reduction of disturbances) have been met,

but only about 15-20% reduction of the mean viscous drag– Estimation significantly affects the overall performance– Effect of control is confined very close to the wall– Cost function that allow the effect of control to penetrate further

into the flow field?– Effect of the base flow profile?– Other nonlinear effects?

Limitations/Issues (contd.)• Model reduction:

– Essential for TBL control– Currently based on observability/controllability– Contributions to the cost function should be included

• Decentralized control• Complex flows for which we don’t have the system matrices:

– How robust is the system identification approach?– Is a linear model still applicable?

• Some observed numerical issues:– System matrices are ill conditioned (high condition numbers).– Some under-resolved modes (due to a finite-dimension representation of

the infinite-dimension system) are very controllable and/or observable, and may (inadvertently) affect the controller design and its performance

• Laboratory validation:– Actuators, sensors, frequency response, etc.

• Many more outstanding issues

Concluding Remarks

• Applications of modern control theories, which had not been widely embraced by the fluids community, to turbulence control turned out to be very promising.

• In some nonlinear flows (e.g., TBL), linear mechanisms play an important role, and much can be accomplished by utilizing linear control theories.

• In spite of some promising demonstrations, many issues – regarding both control theories and numerical implementation – need to be resolved before such an approach can be used in designing a practical controller.

• Further collaborations between control theoreticians and fluid dynamicists will result in much progress in flow control, particularly turbulence control.