Control of Turbulent Boundary Layers: Success, Limitations & Issues

1. Control of Turbulent Boundary Layers:Success, Limitations & Issues John Kim Department of Mechanical & Aerospace EngineeringUniversity of California, Los Angeles

2. Outline Part 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

3. Part IIAnalysis of Boundary-Layer Controllers:A Linear System Perspective

4. 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.

5. 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,

6. u State-Space Representation of N-S x

7. 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.

8. Different Cost Functions

9. LQR Control of Turbulent Channel

10. LQR Control of Turbulent Channel 25 ~ 30 % Control input~ 10 % of ut

11. 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 Reflows).

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

13. Classical Linear Stability Analysis:Eigenvalue Decomposition • The linearized N-S equations for incompressible flows, or have a solution • It can be written in terms of eigenvectors of A where li and si denote eigenvalues and eigenvectors of A, respectively. • In classical linear stability analysis, li < 0 , stable li > 0 , unstable

14. Eigenvalue Decomposition – contd. • 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.

15. “Optimal” disturbance Transient Growth of Optimal Disturbance Examples of parallel eigenvectors Transient Growth of “Optimal” Disturbance • Example:Rec=5000, kx=0., kz=2.044 • max(Ek/Eko) = 4897

16. Singular Value Decomposition (SVD) • 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.

17. The right singular vector is the “optimal” disturbance The left singular vector shows the amplified “optimal” disturbance Singular Value Decomposition – contd. • 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

18. Singular Value Decomposition – contd. Channel at a sub-critical Reynolds number: Rec=5000, kx=0, kz=2.044, max(Ek/Eko) = smax= 4897

19. 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?

20. “Optimal” Disturbance in Turbulent Channel • In contrast to the optimal disturbance in a laminar flow, the transient growth of “optimal” disturbances isinterrupted 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 (smax) attainable within the eddy turnover time by all possible wavenumbers (ie all possible eddy sizes).

21. E/E0=32.5 for kx=0 and kz=6 (lz+=100) with tg+=80 SVD of Turbulent Channel Flow smax kz kx

22. “Optimal” Disturbance in Turbulent Channel E/E0=32.5 for kx=0 and kz=6 (lz+=100) Singular Values Evolution of Energy

23. Ret=100, kx=0, kz=6.0 (lz+=100) z “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 (lz+ 100). y

24. SVD Analysis of Linear Systems with Control

25. Linearized Navier-Stokes System 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.

26. 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 areference 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.

27. What is K for opposition control? Linearized Navier-Stokes System 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.

28. Kv Kw yd 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 smax .

29. SVD of Opposition Control

30. SVD of Opposition Control – contd. • The length scales corresponding to the “optimal” disturbance with control are fairly universal. • Increase of the largest smax for larger yd+ is due to the increase of smax at kz=0.

31. SVD of Opposition Control at High Re For (kx = 0, lz+ 100) • An approximate estimation of smax 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.

32. SVD of KL’s Virtual Flow • Navier-Stokes Equations:

33. 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. 0 SVD of KL’s Virtual Flow – contd. • Modified Navier-Stokes Equations:

34. Virtual Flow Result Laminarization!

35. SVD of the Virtual Flow Singular values with and withoutLc for kx = 0 and kz = 6 Regular channel • 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. Virtual flow

36. Self-Sustaining Mechanism of Near-Wall Turbulence Streamwise Vortices Nonlinear LcV Streamwise-varying modes, kx~=0 Streaks Streak instability

37. SVD of 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. • Here, Q is chosen to minimize disturbance energy.

38. SVD of Boundary-Layer Control Singular Values No control  LQR (g = 0.1 )  Opposition (yd+=10) Lc=0

39. Turbulent Channel Flow Control Result No control Drag Opposition control LQR Virtual flow t+ • Opposition control (yd+ = 10 ) and LQR control (g = 0.1 ) produced similar drag reduction.

40. 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)

41. 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

42. Navier-Stokes equations State Estimator Control Actuator Approach Overview Numerical simulation of separated flows Actuation Measurement Approximate Linear Model Blowing Suction Pressure Vorticity Shear stress LQG (Linear Quadratc Gaussian) compensator Separated boundary-layer flows are considered for preliminary controller design and testing, the ultimate goal being high angle-of-attack airfoil flows.

43. Mathematical Models Variables • Plant dynamics – Navier-Stokes equations • State-space representation of dynamic system • State estimator • Cost function to be minimized • Control law produced by the LQG (Linear Quadratic Gaussian) synthesis

44. Suction V(x) Blasius Boundary Layer Separation region 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 • Transition takes place abruptly around x=3.5 due to strong inviscid instability Total vorticity on a spanwise plane Streamwise vorticity on a horizontal plane

45. y: measurement, u: input signal N: model order, Nk: measurement delay aT and bT matrices are determined by least-square estimate. Approximate Linear Model • The ARX model (in discrete time) • The system’s state-space representation can be constructed using the identified model: • 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

46. Approximate Linear Model Controller Identification Procedure Phase 1: Record input-output data Actuation: Broad-band noise Measurement Navier-Stokes equations Pressure Vorticity Shear stress Phase 2: Construct approximate linear model using selected model structure Least-square estimate Input-ouptut data Phase 3: Perform the LQG (Linear Quadratic Gaussian) synthesis and form the feedback loop Navier-Stokes equations

47. 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

48. 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 Controller OFF Controller ON

49. 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?

50. 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