Dan Henningson

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Flow control applied to transitional
flowsinput-output analysis, model reduction and
control

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

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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 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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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 formulation
Continuous 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

• Controllability Gramian

• Observability 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

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Observability Gramian for GL-equation
Output

• Correlation of sensor impulse response in adjoint
  solution
• Adjoint POD modes
• Ranks states most easily sensed by output
• Provides a means to measure observability

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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
Disturbance Sensor
Actuator Objective
Disturbance Objective
DNS n105 ROM m50
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Frequency response
From all inputs to all outputs
DNS n105 ROM m80 m50 m2
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Feedback control

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

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Optimal Feedback Control LQG
cost function
g (noise)
Ly
fKk
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
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LQG controller formulation with DNS

• Apply in Navier-Stokes simulation

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Performance of controlled system
controller
Noise
Sensor
Actuator
Objective
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Performance of controlled system
Noise
Sensor
Actuator
Objective
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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 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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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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Jet in cross-flow
Countair rotating vortex pair
Shear layer vortices
Horseshoe vortices
Wake region
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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
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Basic state and impulse response

• Steady state computed using the SFD method
  (Åkervik et.al.)
• Energy growth of perturbation

Steady state
Perturbation
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Global eigenmodes

• Global eigenmodes computed using ARPACK
• Growth rate 0.08
• Strouhal number 0.16

1st global mode
time
Perturbation energy Global mode energy
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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 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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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.
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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-norma
  l 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 y10 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
cost function
Reduced model of real system/flow
Estimator/ Controller
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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