Input-Output Analysis and Control Applied to Spatially Developing Shear Flows

1. Input-Output Analysis and Control Applied to Spatially Developing Shear Flows Dan Henningson collaborators Shervin Bagheri, Espen Åkervik, Antonios Monorkousos Luca Brandt, Philipp Schlatter Peter Schmid, Jerome Hoepffner

2. Message • Stability analysis and control design for complex flows can be performed using tools from systems theory and linear algebra based on snapshots of the linearized and adjoint Navier-Stokes equations • Main example Blasius, others: GL-equation, jet-in-cross-flow, shallow cavity

3. Outline • Matrix-free methods using Navier-Stokes time-stepper as linear evolution operator • The initial value problem and global modes/transient growth • Particular/forced solution and input-output characteristics • Reduced order models preserving input-output characteristics, balanced truncation • LQG feedback control based on reduced order model • Examples …

4. 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)

5. Input-output configuration for linearized N-S

6. Solution to the complete input-putput problem • Initial value problem: flow stability • Forced problem: input-output analysis

7. The Initial Value Problem

8. Dimension of discretized system • Matrix A very large for spatially developing flows • Use Navier-Stokes solver (DNS) to approximate the action of exponential matrix/evolution operator • Eigenvalues of evolution operator related to NS-spectrum

9. Krylov subspace with Arnoldi algorithm • Krylov subspace created using matrix or NS-timestepper • Orthogonal basis created with Gram-Schmidt • Approximate eigenvalues from Hessenberg matrix H

10. Global spectrum for Blasius flow • Time-stepper vs. matrix solver • TS-branch from both approaches agrees • Least stable TS-mode

11. Global spectrum vs. Orr-Sommerfeld modes • Global Tollmien-Schlichting branch temporally damped since instability is convective • Local spatial growth identical to global envelope • Shape functions identical

12. Optimal sum of eigenmodes

13. Optimal disturbance growth • Krylov sequence built by forward-adjoint iterations • Adjoint propagation operator solves adjoint equations backward

14. Evolution of optimal disturbance • Adjoint iterations: black • TS-branch only using sum of modes:magenta • Transient since disturbance propagates out of domain • Eigenfunction expansion ill-conditioned for large number of modes and large Reynoldsnumber

15. Snapshots of optimal disturbance evolution • Orr-mechanism: initial disturbance leans against the shear raised up into propagating TS-wavepacket

16. 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 packets grows due to local exponential growth, but globally represents transient disturbances since they propagate out of the domain

17. The forced problem: input-output

18. Input-output analysis • Inputs: • Rougness, free-stream turbulence, acoustic waves, blowing/suction etc. • Outputs: • Measurements of pressure, skin friction etc. • Perserve dynamics of input-output relationship in reduced order model for control design

19. Ginzburg-Landau example • Entire dynamics vs. input-output time signals

20. 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:

21. Most dangerous input • Eigenmodes of Hankel operator • Controllability Gramian • Obsevability Gramian

22. Input 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

23. 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

24. Snapshots of direct and adjoint solution Direct simulation: Adjoint simulation:

25. Balanced modes • Combine snapshots of direct and adjoint simulation (Rowley 2005) • Expand modes in snapshots to obtain smaller eigenvalue problem adjoint forward

26. Properties of balanced modes • Largest outputs possible to excite with chosen forcing • Balanced modes diagonalize observability Gramian • Adjoint balanced modes diagonalize controlability Gramian • Ginzburg-Landau example revisited

27. Model reduction • Project dynamics on balanced modes using their biorthogonal adjoints • Optimal representation of input-output relation, useful in control design

28. Impulse response Disturbance Sensor Actuator Objective Disturbance Objective DNS: n=105 ROM: m=50

29. Frequency response From all inputs to all outputs DNS: n=105 ROM: m=80 m=50 m=2

30. Optimal Feedback Control – LQG cost function g (noise) y f z w controller Find control signal f(t) based on the measurements y(t) such that the influence of external disturbances w(t) and g(t) on the output z(t) is minimized.  Solution: LQG/H2

31. Performance? Control design – 5 steps Reduced model controller • Construct plant: Flow, inputs, outputs • Construct reduced model from the plant using balanced truncation • Design controller using the reduced-model (LQG/H2) • Closed-loop: Connect sensor to actuator using the reduced controller • Run small controller online and evaluate closed-loop perfomance

32. LQG feedback control cost function Reduced model of real system/flow Estimator/ Controller

33. Riccati equations for control and estimation gains

34. LQG controller formulation with DNS • Apply in Navier-Stokes simulation

35. Performance of controlled system Noise Sensor Actuator Objective

36. Control off Cheap Control Intermediate control Expensive Control Performance of controlled system Noise Sensor Actuator Objective

37. Additional flows/effects • Jet in cross-flow • 3D Blasius optimals • Model reduction and control in shallow cavity

38. Jet in cross flow Countair rotating vortex pair Shear layer vortices Horseshoe vortices Wake region

39. Stability analysis – 4 steps • Simulate flow with DNS: Identify structures and regions • Compute baseflow: Steady-state solution • Compute impulse response of baseflow: • Compute global modes of baseflow:

40. Direct numerical simulations • DNS: Fully spectral and parallelized • Self-sustained global oscillations • Probe 1– shear layer • Probe 2 – separation region 1 2 Vortex identification criterion 2

41. Unsteady Time-averaged Steady-state Impulse response • Steady state computed using the SFD method (Åkervik et.al.) • Asymptotic energy growth of perturbation Steady state Perturbation

42. Global eigenmodes • Global eigenmodes computed using ARPACK • Growth rate: 0.08 • Strouhal number: 0.16 1st global mode time Perturbation energy Global mode energy

43. 3D Blasius optimals • Streamwise vorticies create streaks for long times • Optimals for short times utilizes Orr-mechanism

44. A long shallow cavity • 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 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.

45. Global spectra • Global eigenmodes found using Arnoldi method • About 150 eigenvalues converged and 2 unstable

46. 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

47. Maximum energy growth • Eigenfunction expansion in selected modes • Optimization of energy output

48. 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

49. Feedback control of cavity disturbances • Project dynamics on least stable global modes • Choose spatial location of control and measurements • LQG control design

50. Controller performance • Least stable eigenvalues are rearranged • Exponential growth turned into exponential decay • Good performance in DNS using only 4 global modes