Flow control applied to transitional flows input output analysis model reduction and control
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Flow control applied to transitional flows: input-output analysis, model reduction and control. Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca Brandt, Peter Schmid. Outline. Introduction with input-output configuration

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

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

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

Dan Henningson


Shervin Bagheri, Espen Åkervik

Luca Brandt, Peter Schmid



  • 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



  • 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 …

Linearized navier stokes for blasius flow

Linearized Navier-Stokes for Blasius flow

Continuous formulation

Discrete formulation

Input output configuration for linearized n s

Input-output configuration for linearized N-S

Solution to the complete input output problem

Solution to the complete input-output problem

  • Initial value problem: flow stability

  • Forced problem: input-output analysis

The initial value problem

The Initial Value Problem

  • Asymptotic stability analysis:

    Global modes of the Blasius boundary layer

  • Transient growth analysis:

    Optimal disturbances in Blasius flow

Dimension of discretized system

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

Krylov subspace with arnoldi algorithm

Krylov subspace with Arnoldi algorithm

  • Krylov subspace created using NS-timestepper

  • Orthogonal basis created with Gram-Schmidt

  • Approximate eigenvalues from Hessenberg matrix H

Global spectrum for blasius flow

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

Global ts waves

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

Optimal disturbance growth

Optimal disturbance growth

  • Optimal growth from eigenvalues of

  • Krylov sequence built by forward-adjoint iterations

Evolution of optimal disturbance in blasius flow

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

Snapshots of optimal disturbance evolution

Snapshots of optimal disturbance evolution

  • Initial disturbance leans against the shear raised up by Orr-mechanism into propagating TS-wavepacket

The forced problem input output

The forced problem: input-output

  • Ginzburg-Landau example

  • Input-output for 2D Blasius configuration

  • Model reduction

Ginzburg landau example

Ginzburg-Landau example

  • Entire dynamics vs. input-output time signals

Input output operators

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:

Most dangerous inputs and the largest outputs

Most dangerous inputs and the largest outputs

  • Eigenmodes of Hankel operator – balanced modes

  • Controllability Gramian

  • Observability Gramian

Controllability gramian for gl equation

  • 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

Observability gramian for gl equation

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

Balanced modes eigenvalues of the hankel operator

Balanced modes: eigenvalues of the Hankel operator

  • Combine snapshots of direct and adjoint simulation

  • Expand modes in snapshots to obtain smaller eigenvalue problem

Snapshots of direct and adjoint solution in blasius flow

Snapshots of direct and adjoint solution in Blasius flow

Direct simulation:

Adjoint simulation:

Balanced modes for blasius flow

Balanced modes for Blasius flow



Properties of balanced modes

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

Model reduction

Model reduction

  • Project dynamics on balanced modes using their biorthogonal adjoints

  • Reduced representation of input-output relation, useful in control design

Impulse response

Impulse response

Disturbance Sensor

Actuator Objective

Disturbance Objective

DNS: n=105

ROM: m=50

Frequency response

Frequency response

From all inputs to all outputs

DNS: n=105

ROM: m=80



Feedback control

Feedback control

  • LQG control design using reduced order model

  • Blasius flow example

Optimal feedback control lqg

Optimal Feedback Control – LQG

cost function








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

Lqg controller formulation with dns

LQG controller formulation with DNS

  • Apply in Navier-Stokes simulation

Performance of controlled system

Performance of controlled system






Performance of controlled system1

Control off

Cheap Control

Intermediate control

Expensive Control

Performance of controlled system







  • 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



  • 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)

Jet in cross flow

Jet in cross-flow

Countair rotating vortex pair

Shear layer vortices

Horseshoe vortices

Wake region

Direct numerical simulations

Direct numerical simulations

  • DNS: Fully spectral and parallelized

  • Self-sustained global oscillations

  • Probe 1– shear layer

  • Probe 2 – separation region


2 Vortex

identification criterion


Basic state and impulse response




Basic state and impulse response

  • Steady state computed using the SFD method (Åkervik et.al.)

  • Energy growth of perturbation

Steady state


Global eigenmodes

Global eigenmodes

  • Global eigenmodes computed using ARPACK

  • Growth rate: 0.08

  • Strouhal number: 0.16

1st global mode


Perturbation energy

Global mode energy

Optimal sum of eigenmodes

Optimal sum of eigenmodes

Global view of tollmien schlichting waves

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

3d blasius optimals

3D Blasius optimals

  • Streamwise vorticies create streaks for long times

  • Optimals for short times utilizes Orr-mechanism

Input output analysis

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

A long shallow cavity

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.

Global spectra

Global spectra

  • Global eigenmodes found using Arnoldi method

  • About 150 eigenvalues converged and 2 unstable

Most unstable mode

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

Maximum energy growth

Maximum energy growth

  • Eigenfunction expansion in selected modes

  • Optimization of energy output

Development of wavepacket

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

Lqg feedback control

LQG feedback control

cost function

Reduced model of real system/flow

Estimator/ Controller

Riccati equations for control and estimation gains

Riccati equations for control and estimation gains

Feedback control of cavity disturbances

Feedback control of cavity disturbances

  • Project dynamics on least stable global modes

  • Choose spatial location of control and measurements

  • LQG control design

Controller performance

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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