Flow control applied to transitional flows: input-output analysis, model reduction and control

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Linné Flow CentreKTH Mechanics

7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.

Dan Henningson

collaborators

Shervin Bagheri, Espen ÅkervikLuca Brandt, Peter Schmid

Flow control applied to transitional flows:input-output analysis, model reduction and control

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Outline• Introduction with input-output configuration

• Matrix-free methods for flow stability using Navier-Stokes snapshots

Edwards et al. (1994), …

• Global modes and transient growthCossu & 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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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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Linearized Navier-Stokes for Blasius flow

Discrete formulationContinuous formulation

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Input-output configuration for linearized N-S

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Solution to the complete input-output problem

• Initial value problem: flow stability

• Forced problem: input-output analysis

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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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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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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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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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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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Optimal disturbance growth

• Optimal growth from eigenvalues of

• Krylov sequence built by forward-adjoint iterations

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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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Snapshots of optimal disturbance evolution• Initial disturbance leans against the shear raised up by Orr-

mechanism into propagating TS-wavepacket

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

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Ginzburg-Landau example

• Entire dynamics vs. input-output time signals

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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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Most dangerous inputs and the largest outputs

• Eigenmodes of Hankel operator – balanced modes

• Observability Gramian

• Controllability Gramian

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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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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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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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Snapshots of direct and adjoint solution in Blasius flow

Direct simulation: Adjoint simulation:

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Balanced modes for Blasius flow

adjoint forward

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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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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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Impulse response

DNS: n=105

ROM: m=50

Disturbance Sensor

Actuator Objective

Disturbance Objective

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Frequency response

DNS: n=105

ROM: m=80 m=50 m=2

From all inputs to all outputs

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

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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

KLw z g(noise)

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LQG controller formulation with DNS

• Apply in Navier-Stokes simulation

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Performance of controlled system

Noise

Sensor

Actuator

Objective

controller

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Performance of controlled system

Noise Sensor Actuator Objective

Control offCheap ControlIntermediate controlExpensive Control

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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 flowReduced order models with balanced modes used in LQG controlController 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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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 propertiesKrylov-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 systemsBalanced truncation: Rowley (2005)Global modes for shallow cavity: Åkervik et al. (2007)Ginzburg-Landau: Bagheri et al. (2008)

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Jet in cross-flow

Horseshoe vortices

Wake region

Countair rotating vortex pair

Shear layer vortices

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Direct numerical simulations

• DNS: Fully spectral and parallelized

• Self-sustained global oscillations

• Probe 1– shear layer

• Probe 2 – separation region

2 Vortex identification criterion

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2

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Basic state and impulse response• Steady state computed using the SFD method (Åkervik et.al.)

• Energy growth of perturbation

Unsteady Time-averagedSteady-state

Steady state

Perturbation

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Global eigenmodes• Global eigenmodes computed using ARPACK

• Growth rate: 0.08

• Strouhal number: 0.16

Perturbation energyGlobal mode energy

1st global mode

time

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Optimal sum of eigenmodes

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

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Input-output analysis

• Inputs: Disturbances: roughness, free-stream turbulence, acoustic wavesActuation: 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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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

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

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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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Maximum energy growth

• Eigenfunction expansion in selected modes

• Optimization of energy output

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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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LQG feedback control

Estimator/ Controller

Reduced model of real system/flow

cost function

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Riccati equations for control and estimation gains

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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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Controller performance

• Least stable eigenvalues are rearranged

• Exponential growth turned into exponential decay

• Good performance in DNS using only 4 global modes

Flow control applied to transitional flows: input-output analysis, model reduction and control

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