1. Control of Turbulent Boundary Layers: Success, Limitations & Issues John Kim Department of Mechanical & Aerospace Engineering University 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 II Analysis 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. State-Space Representation of N-S u 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. 25 ~ 30 % Control input ~ 10 % of u  LQR Control of Turbulent Channel

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 Re flows).

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

13. 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 s i denote eigenvalues and eigenvectors of A, respectively. In classical linear stability analysis, Classical Linear Stability Analysis: Eigenvalue Decomposition i < 0, stable i > 0, unstable

14. 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.

15. Examples of parallel eigenvectors Example: Re c =5000, k x =0., k z =2.044 max(E k /E ko ) = 4897 “Optimal” disturbance Transient Growth of “Optimal” Disturbance Transient Growth of Optimal Disturbance

16. 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)

17. 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. The right singular vector is the “optimal” disturbance The left singular vector shows the amplified “optimal” disturbance

18. Channel at a sub-critical Reynolds number: Re c =5000, k x =0, k z =2.044, max(E k /E ko ) =  max = 4897 Singular Value Decomposition – contd.

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 is interrupted by nonlinear activities before its potential maximum state can be reached. A turbulence time scale t g, 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).

21. SVD of Turbulent Channel Flow kxkx kzkz  max E/E 0 =32.5 for k x =0 and k z =6 ( z + =100) with t g + =80

22. “Optimal” Disturbance in Turbulent Channel Singular Values Evolution of Energy E/E 0 =32.5 for k x =0 and k z =6 ( z + =100)

23. “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 Re  =100, k x =0, k z =6.0 ( z + =100 ) z

24. SVD Analysis of Linear Systems with Control

25. 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)t Q -1 instead of Qe At Q -1. Linearized Navier-Stokes System with Control

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 a reference case against which other controllers to be compared. –About 30% drag reduction was achieved with y d + = 10-15. –Drag was increased significantly with y d + > 20.

27. 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)t Q -1 instead of Qe At Q -1. Linearized Navier-Stokes System with Control What is K for opposition control?

28. 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 y d, (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. ydyd KvKv KK

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  max for larger y d + is due to the increase of  max at k z =0.

31. 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 k x = 0 wavenumbers persists, implying that opposition control will continue to be effective at high Re in controlling streamwise vortices. For ( k x = 0, z +  100)

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

33. 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. 0

34. Virtual Flow Result Laminarization!

35. SVD of the Virtual Flow Non-normality of of the linear system is reduced without the linear coupling term, L c. 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 d x /dt=A x. Reduction of non-normality or large singular values can be used as a control objective in controller design. Singular values with and without L c for k x = 0 and k z = 6 Regular channel Virtual flow

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

37. 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)

38. SVD of Boundary-Layer Control Singular Values  No control LQR (  = 0.1 )  Opposition ( y d + =10)  Lc=0

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

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. Approach Overview Navier-Stokes equations State EstimatorControl Actuator Approximate Linear Model Measurement Pressure Vorticity Shear stress Actuation Blowing Suction 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.

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

44. 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 Blasius Boundary Layer Separation region Suction V(x) 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. Approximate Linear Model The ARX model (in discrete time) The system’s state-space representation can be constructed using the identified model: y: measurement, u: input signal N: model order, N k : measurement delay a T and b T matrices are determined by least- square estimate. Remarks –Time delay in ARX model are estimated using the convective velocity –Assumed zero feed-through –Long delay (large N k ) may lead to large system matrices (A, B, C, D) –Insufficient data length leads to inaccurate system identification

46. Identification Procedure Approximate Linear Model Navier-Stokes equations Measurement Pressure Vorticity Shear stress Actuation: Broad-band noise Phase 1: Record input-output data 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 Controller

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

51. 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.