1 linné flow centre kth mechanics isaac newton institute workshop 8-12 september, 2008 dan...

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Linné Flow CentreKTH Mechanics

Isaac Newton Institute Workshop 8-12 September, 2008

Dan Henningson

collaborators

Shervin Bagheri, Espen ÅkervikLuca Brandt, Peter Schmid

Input-output analysis, model reduction and control applied to the Blasius boundary layer - using balanced modes

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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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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, creating 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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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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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

• Input-output formulation ideal for analysis and design of feedback control systems – applied in Blasius flow

• Balanced modes Obtained from snapshots of forward and adjoint solutions 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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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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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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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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The forced problem: input-output

• Ginzburg-Landau example• Input-output for 2D Blasius configuration• Model reduction

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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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Feedback control

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

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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

1 linné flow centre kth mechanics isaac newton institute workshop 8-12 september, 2008 dan...

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