1 linné flow centre kth mechanics julian d. cole lecture, aiaa theoretical fluid mechanics...
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Dan Henningson
collaborators
Shervin Bagheri, Espen Åkervik, Antonios MonokrousosLuca Brandt, Philipp Schlatter
Peter Schmid, Jerome Hoepffner
Stability, Input-Output Analysis and Control Applied to Spatially Developing Shear Flows
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Background
• Global modes and transient growthGinzburg-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)
• Invited session on Global Instability and Control of Real Flows, Wednesday 8-12, Evergreen 4
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Linearized Navier-Stokes for Blasius flow
Discrete formulationContinuous formulation
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Input-output configuration for linearized N-S
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Solution to the complete input-output problem
• Initial value problem: flow stability
• Forced problem: input-output analysis
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
The Initial Value Problem I
• Asymptotic stability analysis
• Global modes of the Blasius boundary layer• Jet in cross-flow
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Dimension of discretized system
• Matrix A very large for 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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Jet in cross-flow
Horseshoe vortices
Wake region
Countair rotating vortex pair
Shear layer vortices
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
The Initial Value Problem II
• Transient growth analysis
• Optimal disturbances in Blasius flow
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Optimal disturbance growth
• Optimal growth from eigenvalues of
• Krylov sequence built by forward-adjoint iterations
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Snapshots of optimal disturbance evolution
• Initial disturbance leans against the shear raised up by Orr-mechanism into propagating TS-wavepacket
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
The forced problem: input-output
• Ginzburg-Landau example• Input-output for 2D Blasius configuration• Model reduction
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Ginzburg-Landau example
• Entire dynamics vs. input-output time signals
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Most dangerous inputs and the largest outputs
• Eigenmodes of Hankel operator – balanced modes
• Observability Gramian
• Controllability Gramian
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Snapshots of direct and adjoint solution in Blasius flow
Direct simulation: Adjoint simulation:
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Balanced modes for Blasius flow
adjoint forward
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Impulse response
DNS: n=105
ROM: m=50
Disturbance Sensor
Actuator Objective
Disturbance Objective
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Frequency response
DNS: n=105
ROM: m=80 m=50 m=2
From all inputs to all outputs
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Feedback control
• LQG control design using reduced order model• Blasius flow example
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
LQG controller formulation with DNS
• Apply in Navier-Stokes simulation
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Performance of controlled system
Noise
Sensor
Actuator
Objective
controller
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Performance of controlled system
Noise Sensor Actuator Objective
Control offCheap ControlIntermediate controlExpensive Control
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Optimal sum of eigenmodes
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
3D Blasius optimals
• Streamwise vorticies create streaks for long times• Optimals for short times utilizes Orr-mechanism
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Global spectra
• Global eigenmodes found using Arnoldi method• About 150 eigenvalues converged and 2 unstable
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Maximum energy growth
• Eigenfunction expansion in selected modes
• Optimization of energy output
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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
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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Linné Flow CentreKTH Mechanics
Julian D. Cole lecture, AIAA Theoretical Fluid Mechanics Conference, Seattle, WA, June 24, 2008.
Controller performance
• Least stable eigenvalues are rearranged
• Exponential growth turned into exponential decay
• Good performance in DNS using only 4 global modes