1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18...

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

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

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 …

Linearized Navier-Stokes for Blasius flow

Discrete formulationContinuous formulation

Input-output configuration for linearized N-S

Solution to the complete input-output problem

• Initial value problem: flow stability

• Forced problem: input-output analysis

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

• 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 created using NS-timestepper

• Orthogonal basis created with Gram-Schmidt

• Approximate eigenvalues from Hessenberg matrix H

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

• 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 growth from eigenvalues of

• Krylov sequence built by forward-adjoint iterations

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

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

The forced problem: input-output

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

Ginzburg-Landau example

• Entire dynamics vs. input-output time signals

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

• Eigenmodes of Hankel operator – balanced modes

• Observability Gramian

• Controllability Gramian

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

• 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

• 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

Direct simulation: Adjoint simulation:

Balanced modes for Blasius flow

adjoint forward

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

• Project dynamics on balanced modes using their biorthogonal adjoints

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

Impulse response

DNS: n=105

ROM: m=50

Disturbance Sensor

Actuator Objective

Disturbance Objective

Frequency response

DNS: n=105

ROM: m=80 m=50 m=2

From all inputs to all outputs

Feedback control

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

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)

LQG controller formulation with DNS

• Apply in Navier-Stokes simulation

Performance of controlled system

Sensor

Actuator

Objective

controller

Performance of controlled system

Noise Sensor Actuator Objective

Control offCheap ControlIntermediate controlExpensive Control

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

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)

Jet in cross-flow

Horseshoe vortices

Wake region

Countair rotating vortex pair

Shear layer vortices

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

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

• Energy growth of perturbation

Unsteady Time-averagedSteady-state

Steady state

Perturbation

Global eigenmodes

• Global eigenmodes computed using ARPACK

• Growth rate: 0.08

• Strouhal number: 0.16

Perturbation energyGlobal mode energy

1st global mode

Optimal sum of eigenmodes

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

• Streamwise vorticies create streaks for long times• Optimals for short times utilizes Orr-mechanism

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

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

Global spectra

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

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

• Eigenfunction expansion in selected modes

• Optimization of energy output

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

Estimator/ Controller

Reduced model of real system/flow

cost function

Riccati equations for control and estimation gains

Feedback control of cavity disturbances

• Project dynamics on least stable global modes

• Choose spatial location of control and measurements

• LQG control design

Controller performance

• Least stable eigenvalues are rearranged

• Exponential growth turned into exponential decay

• Good performance in DNS using only 4 global modes

1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18...

Documents

Transcript of 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18...

As origens da classificação de plantas de Carl von Linné no ensino ...

Mantle histology, histochemistry and ultrastructure of the ...archimer.ifremer.fr/doc/1992/publication-2770.pdf288 INTRODUCTION The black pearl oyster Pinctada margaritif'era (Linné,

Texto 12_As Origens Da Classificação de Plantas de Carl Von Linné No Ensino de Biologia

1 Linné Flow Centre KTH Mechanics Isaac Newton Institute Workshop 8-12 September, 2008 Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca.

Geometry Control of Precast Segmental Concrete Bridges · Geometry Control of Precast Segmental Concrete Bridges Brice F. Bender, PE Senior Consultant Special Concrete Bridges Henningson,

Summer Occurrence of Vespertìlio murinus LINNÉ …verlag.nhm-wien.ac.at/pdfs/088B089B_281291_Baggoe.pdfSummer Occurrence of Vespertilio murinus and Eptesicus serotinus 283 Having

Engineering mechanics powerpoint fluid mechanics

Fracture Mechanics: Linear Elastic Fracture Mechanics …people.unipi.it/static/ciro.santus/MaterialeDidattica/FM_Classes... · Fracture Mechanics: Linear Elastic Fracture Mechanics

Malacoshare - Linné C. 1766 - Sistema Naturae - Ed. XII T.1

Optimal Disturbances and Receptivity of 3D Boundary Layers · David Tempelmann Ardeshir Hanifi, Dan Henningson Ercoftac SIG 33 Meeting, 16-18th October 2008 The work described in

Earth’s New MOON · Relief map of Linné crater (2.2 km diameter) created from an Lunar Reconnaissance Orbiter Camera (LROC) stereo topographic model. Linné is a very young and

Claves para el Género Dracaena Vandelli ex Linné …Claves para el género Dracaena Vandelli ex Linné en la provincia de Valencia Bouteloua 5: 3-12 (VI-2008). ISSN 1988-4257 4 fa

… including a contribution by Hanifi FOI and Linné Flow Center Stockholm , Sweden

2fIIIIIIIIIII lllfll.lfllflfll IEEIIIEIIIIEI ... · F47,4-78.-C-,029 9. P RFORMNG ORGANIZATION NAME AND ADDRESS 10. PPOGPAM ELEMENT, PROJECT, TASK APEA A WORK UNIT NUMBERS Henningson,

EARLINET-ASOS Symposium Geneva Sept. 20, 2010 EARLINET – ASOS -NA6: Database construction and operation Holger Linné Max-Planck-Institut für Meteorologie.

On nomenclature and identity of Scarabaeus aeruginosus ... · the international Code of Zoological Nomenclature (iCZN 1999). Scarabaeus aeruginosus linné, 1767 (Rutelinae: Anticheirina),

What is Engineering Mechanics - Home - Dynamics Course · Engineering Mechanics Mechanics of rigid bodies Statics Dynamics Kinetics Kinematics Mechanics of deformable bodies Mechanics

Title Headline · 2014-02-06 · Title Headline. ublication number Highlighted Lunar Features Linné Crater With its bowl shape and pristine features, lunar scientists consider Linné,

LAFS Game Mechanics - Tactical Mechanics

Lower-branch travelling waves and transition to turbulence in pipe flow Dr Yohann Duguet, Linné Flow Centre, KTH, Stockholm, Sweden, formerly : School.