Flow control applied to transitional flows input output analysis model reduction and control
Sponsored Links
This presentation is the property of its rightful owner.
1 / 52

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


  • 93 Views
  • Uploaded on
  • Presentation posted in: General

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

Download Presentation

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

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


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

  • 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


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

Continuous formulation

Discrete 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

  • Controllability Gramian

  • Observability Gramian


  • 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

  • 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

Disturbance Sensor

Actuator Objective

Disturbance Objective

DNS: n=105

ROM: m=50


Frequency response

From all inputs to all outputs

DNS: n=105

ROM: m=80

m=50

m=2


Feedback control

  • LQG control design using reduced order model

  • Blasius flow example


Optimal Feedback Control – LQG

cost function

g

(noise)

Ly

f=Kk

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


LQG controller formulation with DNS

  • Apply in Navier-Stokes simulation


Performance of controlled system

controller

Noise

Sensor

Actuator

Objective


Control off

Cheap Control

Intermediate control

Expensive Control

Performance of controlled system

Noise

Sensor

Actuator

Objective


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


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)


Jet in cross-flow

Countair rotating vortex pair

Shear layer vortices

Horseshoe vortices

Wake region


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


Unsteady

Time-averaged

Steady-state

Basic state and impulse response

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

  • Energy growth of perturbation

Steady state

Perturbation


Global eigenmodes

  • Global eigenmodes computed using ARPACK

  • Growth rate: 0.08

  • Strouhal number: 0.16

1st global mode

time

Perturbation energy

Global mode energy


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 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

  • 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 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

cost function

Reduced model of real system/flow

Estimator/ Controller


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


  • Login