1. 1 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Dan Henningson collaborators Shervin Bagheri, Espen Åkervik, Antonios Monokrousos Luca Brandt, Philipp Schlatter Peter Schmid, Jerome Hoepffner Stability, Input-Output Analysis and Control Applied to Spatially Developing Shear Flows

2. 2 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Message Need only snapshots from a Navier-Stokes solver (with adjoint) to perform stability analysis and control design for complex flows Main example Blasius, others: GL-equation, jet in cross-flow

3. 3 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Outline Introduction with input-output configuration Matrix-free methods using Navier-Stokes snapshots The initial value problem, global modes and transient growth Particular or forced solution and input-output characteristics Reduced order models preserving input-output characteristics, balanced truncation LQG feedback control based on reduced order model Conclusions

4. 4 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Background Global modes and transient growth Ginzburg-Landau: Cossu & Chomaz (1997); Chomaz (2005) Waterfall problem: Schmid & Henningson (2002) Blasius boundary layer, Ehrenstein & Gallaire (2005); Åkervik et al. (2008) Recirculation bubble: Åkervik et al. (2007); Marquet et al. (2008) Matrix-free methods for stability properties Krylov-Arnoldi method: Edwards et al. (1994) Stability backward facing step: Barkley et al. (2002) Optimal growth for backward step and pulsatile flow: Barkley et al. (2008) Model reduction and feedback control of fluid systems Balanced truncation: Rowley (2005) Global modes for shallow cavity: Åkervik et al. (2007) Ginzburg-Landau: Bagheri et al. (2008) Invited session on Global Instability and Control of Real Flows, Wednesday 8-12, Evergreen 4

5. 5 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Linearized Navier-Stokes for Blasius flow Discrete formulationContinuous formulation

6. 6 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Input-output configuration for linearized N-S

7. 7 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Solution to the complete input-output problem Initial value problem: flow stability Forced problem: input-output analysis

8. 8 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. The Initial Value Problem I Asymptotic stability analysis Global modes of the Blasius boundary layer Jet in cross-flow

9. 9 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Dimension of discretized system Matrix A very large for spatially developing flows Consider eigenvalues of the matrix exponential, related to eigenvalues of A Use Navier-Stokes solver (DNS) to approximate the action of matrix exponential or evolution operator

10. 10 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Krylov subspace with Arnoldi algorithm Krylov subspace created using NS-timestepper Orthogonal basis created with Gram-Schmidt Approximate eigenvalues from Hessenberg matrix H

11. 11 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Global spectrum for Blasius flow Least stable eigenmodes equivalent using time-stepper and matrix solver Least stable branch is a global representation of Tollmien- Schlichting (TS) modes

12. 12 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Global TS-waves Streamwise velocity of least damped TS-mode Envelope of global TS-mode identical to local spatial growth Shape functions of local and global modes identical

13. 13 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Jet in cross-flow Horseshoe vortices Wake region Countair rotating vortex pair Shear layer vortices

14. 14 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Direct numerical simulations DNS: Fully spectral and parallelized Self-sustained global oscillations Probe 1– shear layer Probe 2 – separation region 2 Vortex identification criterion 1 2

15. 15 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Basic state and impulse response Steady state computed using the SFD method (Åkervik et.al.) Energy growth of perturbation Unsteady Time-averaged Steady-state Steady state Perturbation

16. 16 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Global eigenmodes Global eigenmodes computed using ARPACK Growth rate: 0.08 Strouhal number: 0.16 Perturbation energy Global mode energy 1st global mode time

17. 17 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. The Initial Value Problem II Transient growth analysis Optimal disturbances in Blasius flow

18. 18 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Optimal disturbance growth Optimal growth from eigenvalues of Krylov sequence built by forward-adjoint iterations

19. 19 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Evolution of optimal disturbance in Blasius flow Full adjoint iterations (black) sum of TS-branch modes only (magenta) Transient since disturbance propagates out of domain

20. 20 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Snapshots of optimal disturbance evolution Initial disturbance leans against the shear raised up by Orr- mechanism into propagating TS-wavepacket

21. 21 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. The forced problem: input-output Ginzburg-Landau example Input-output for 2D Blasius configuration Model reduction

22. 22 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Input-output analysis Inputs: Disturbances: roughness, free-stream turbulence, acoustic waves Actuation: blowing/suction, wall motion, forcing Outputs: Measurements of pressure, skin friction etc. Aim: preserve dynamics of input-output relationship in reduced order model used for control design

23. 23 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Ginzburg-Landau example Entire dynamics vs. input-output time signals

24. 24 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Input-output operators Past inputs to initial state: class of initial conditions possible to generate through chosen forcing Initial state to future outputs: possible outputs from initial condition Past inputs to future outputs:

25. 25 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Most dangerous inputs and the largest outputs Eigenmodes of Hankel operator – balanced modes Observability Gramian Controllability Gramian

26. 26 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Controllability Gramian for GL-equation Correlation of actuator impulse response in forward solution POD modes: Ranks states most easily influenced by input Provides a means to measure controllability Input

27. 27 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Observability Gramian for GL-equation Correlation of sensor impulse response in adjoint solution Adjoint POD modes: Ranks states most easily sensed by output Provides a means to measure observability Output

28. 28 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Balanced modes: eigenvalues of the Hankel operator Combine snapshots of direct and adjoint simulation Expand modes in snapshots to obtain smaller eigenvalue problem

29. 29 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Snapshots of direct and adjoint solution in Blasius flow Direct simulation:Adjoint simulation:

30. 30 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Balanced modes for Blasius flow adjoint forward

31. 31 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Properties of balanced modes Largest outputs possible to excite with chosen forcing Balanced modes diagonalize observability Gramian Adjoint balanced modes diagonalize controllability Gramian Ginzburg-Landau example revisited

32. 32 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Model reduction Project dynamics on balanced modes using their biorthogonal adjoints Reduced representation of input-output relation, useful in control design

33. 33 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Impulse response DNS: n=10 5 ROM: m=50 Disturbance Sensor Actuator Objective Disturbance Objective

34. 34 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Frequency response DNS : n=10 5 ROM: m=80 m=50 m=2 From all inputs to all outputs

35. 35 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Feedback control LQG control design using reduced order model Blasius flow example

36. 36 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Optimal Feedback Control – LQG controller Find an optimal control signal   (t) based on the measurements (t) such that in the presence of external disturbances w(t) and measurement noise g(t) the output z(t) is minimized.  Solution: LQG/H2 cost function K LL w z g (noise)

37. 37 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. LQG feedback control Estimator/ Controller Reduced model of real system/flow cost function

38. 38 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Riccati equations for control and estimation gains

39. 39 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. LQG controller formulation with DNS Apply in Navier-Stokes simulation

40. 40 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Performance of controlled system Noise Sensor Actuator Objective controller

41. 41 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Performance of controlled system NoiseSensorActuatorObjective Control off Cheap Control Intermediate control Expensive Control

42. 42 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Conclusions Complex stability/control problems solved using Krylov/Arnoldi methods based on snapshots of forward and adjoint Navier-Stokes solutions Optimal disturbance evolution brought out Orr-mechanism and propagating TS-wave packet automatically Balanced modes give low order models preserving input- output relationship between sensors and actuators Feedback control of Blasius flow Reduced order models with balanced modes used in LQG control Controller based on small number of modes works well in DNS

43. 43 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Optimal sum of eigenmodes

44. 44 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Global view of Tollmien-Schlichting waves Global temporal growth rate damped and depends on length of domain and boundary conditions Single global mode captures local spatial instability Sum of damped global modes represents convectively unstable disturbances TS-wave packet grows due to local exponential growth, but globally represents a transient disturbance since it propagates out of the domain

45. 45 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. 3D Blasius optimals Streamwise vorticies create streaks for long times Optimals for short times utilizes Orr-mechanism

46. 46 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. A long shallow cavity Åkervik E., Hoepffner J., Ehrenstein U. & Henningson, D.S. 2007. Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579: 305-314. Basic flow from DNS with SFD: Åkervik et al., Phys. Fluids 18, 2006 Strong shear layer at cavity top and recirculation at the downstream end of the cavity

47. 47 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Global spectra Global eigenmodes found using Arnoldi method About 150 eigenvalues converged and 2 unstable

48. 48 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Most unstable mode Forward and adjoint mode located in different regions implies non-orthogonal eigenfunctions/non-normal operator Flow is sensitive where adjoint is large

49. 49 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Maximum energy growth Eigenfunction expansion in selected modes Optimization of energy output

50. 50 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Development of wavepacket x-t diagrams of pressure at y=10 using eigenmode expansion Wavepacket generates pressure pulse when reaching downstream lip Pressure pulse triggers another wavepacket at upstream lip

51. 51 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Feedback control of cavity disturbances Project dynamics on least stable global modes Choose spatial location of control and measurements LQG control design

52. 52 Linné Flow Centre KTH Mechanics Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008. Controller performance Least stable eigenvalues are rearranged Exponential growth turned into exponential decay Good performance in DNS using only 4 global modes