Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca Brandt, Peter Schmid

1. Input-output analysis, model reduction and control applied to the Blasius boundary layer - using balanced modes Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca Brandt, Peter Schmid

2. Linearized Navier-Stokes for Blasius flow Continuous formulation Discrete formulation

3. Input-output configuration for linearized N-S

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

5. 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:

6. Most dangerous inputs, creating the largest outputs • Eigenmodes of Hankel operator – balanced modes • Controllability Gramian • Observability Gramian

7. 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

8. 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

9. Balanced modes: eigenvalues of the Hankel operator • Combine snapshots of direct and adjoint simulation • Expand modes in snapshots to obtain smaller eigenvalue problem

10. Snapshots of direct and adjoint solution in Blasius flow Direct simulation: Adjoint simulation:

11. Balanced modes for Blasius flow adjoint forward

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

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

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

15. Optimal Feedback Control – LQG cost function g (noise) Ly f=Kk z w controller Find an optimal control signal f(t) based on the measurements y(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

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

17. Performance of controlled system controller Noise Sensor Actuator Objective

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

19. Conclusions • Input-output formulation ideal for analysis and design of feedback control systems • Balanced modes • Obtained from snapshots of forward and adjoint solutions • 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

21. 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

22. 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

23. 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

24. The forced problem: input-output • Ginzburg-Landau example • Input-output for 2D Blasius configuration • Model reduction

25. 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

26. Feedback control • LQG control design using reduced order model • Blasius flow example

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

28. Riccati equations for control and estimation gains