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1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca Brandt, Peter Schmid Flow control applied to transitional flows: input-output analysis, model reduction and control

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2 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Outline Introduction with input-output configuration Matrix-free methods for flow stability using Navier-Stokes snapshots Edwards et al. (1994), … Global modes and transient growth Cossu & Chomaz (1997), … Input-output characteristics of Blasius BL and reduced order models based on balanced truncation Rowley (2005), … LQG feedback control based on reduced order model Conclusions

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3 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Message Need only snapshots from a Navier-Stokes solver (with adjoint) to perform stability analysis and control design for complex flows Main example Blasius BL, but many other more complex flows are now tractable …

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4 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Linearized Navier-Stokes for Blasius flow Discrete formulationContinuous formulation

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5 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Input-output configuration for linearized N-S

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6 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Solution to the complete input-output problem Initial value problem: flow stability Forced problem: input-output analysis

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7 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, The Initial Value Problem Asymptotic stability analysis: Global modes of the Blasius boundary layer Transient growth analysis: Optimal disturbances in Blasius flow

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8 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Dimension of discretized system Matrix A very large for complex 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

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9 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Krylov subspace with Arnoldi algorithm Krylov subspace created using NS-timestepper Orthogonal basis created with Gram-Schmidt Approximate eigenvalues from Hessenberg matrix H

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10 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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11 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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12 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Optimal disturbance growth Optimal growth from eigenvalues of Krylov sequence built by forward-adjoint iterations

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13 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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14 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Snapshots of optimal disturbance evolution Initial disturbance leans against the shear raised up by Orr- mechanism into propagating TS-wavepacket

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15 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, The forced problem: input-output Ginzburg-Landau example Input-output for 2D Blasius configuration Model reduction

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16 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Ginzburg-Landau example Entire dynamics vs. input-output time signals

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17 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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:

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18 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Most dangerous inputs and the largest outputs Eigenmodes of Hankel operator – balanced modes Observability Gramian Controllability Gramian

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19 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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20 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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21 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Balanced modes: eigenvalues of the Hankel operator Combine snapshots of direct and adjoint simulation Expand modes in snapshots to obtain smaller eigenvalue problem

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22 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Snapshots of direct and adjoint solution in Blasius flow Direct simulation:Adjoint simulation:

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23 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Balanced modes for Blasius flow adjoint forward

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24 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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25 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Model reduction Project dynamics on balanced modes using their biorthogonal adjoints Reduced representation of input-output relation, useful in control design

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26 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Impulse response DNS: n=10 5 ROM: m=50 Disturbance Sensor Actuator Objective Disturbance Objective

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27 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Frequency response DNS : n=10 5 ROM: m=80 m=50 m=2 From all inputs to all outputs

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28 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Feedback control LQG control design using reduced order model Blasius flow example

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29 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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 LL w z g (noise)

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30 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, LQG controller formulation with DNS Apply in Navier-Stokes simulation

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31 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Performance of controlled system Noise Sensor Actuator Objective controller

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32 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Performance of controlled system NoiseSensorActuatorObjective Control off Cheap Control Intermediate control Expensive Control

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33 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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 Outlook: incorporate realistic sensors and actuators in 3D problem and test controllers experimentally

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34 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 2008.

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35 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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)

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36 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Jet in cross-flow Horseshoe vortices Wake region Countair rotating vortex pair Shear layer vortices

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37 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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38 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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39 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Global eigenmodes Global eigenmodes computed using ARPACK Growth rate: 0.08 Strouhal number: 0.16 Perturbation energy Global mode energy 1st global mode time

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40 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Optimal sum of eigenmodes

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41 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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42 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, D Blasius optimals Streamwise vorticies create streaks for long times Optimals for short times utilizes Orr-mechanism

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43 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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44 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, A long shallow cavity Åkervik E., Hoepffner J., Ehrenstein U. & Henningson, D.S Optimal growth, model reduction and control in a separated boundary-layer flow using global eigenmodes. J. Fluid Mech. 579: 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

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45 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Global spectra Global eigenmodes found using Arnoldi method About 150 eigenvalues converged and 2 unstable

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46 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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47 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Maximum energy growth Eigenfunction expansion in selected modes Optimization of energy output

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48 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, 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

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49 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, LQG feedback control Estimator/ Controller Reduced model of real system/flow cost function

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50 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Riccati equations for control and estimation gains

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51 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Feedback control of cavity disturbances Project dynamics on least stable global modes Choose spatial location of control and measurements LQG control design

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52 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, September, Controller performance Least stable eigenvalues are rearranged Exponential growth turned into exponential decay Good performance in DNS using only 4 global modes

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