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

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Dan Henningson

collaborators

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 …

Continuous formulation

Discrete formulation

- Initial value problem: flow stability
- Forced problem: input-output analysis

- Asymptotic stability analysis:
Global modes of the Blasius boundary layer

- Transient growth analysis:
Optimal disturbances in Blasius flow

- 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 created using NS-timestepper
- Orthogonal basis created with Gram-Schmidt
- Approximate eigenvalues from Hessenberg matrix H

- Least stable eigenmodes equivalent using time-stepper and matrix solver
- Least stable branch is a global representation of Tollmien-Schlichting (TS) modes

- 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 growth from eigenvalues of
- Krylov sequence built by forward-adjoint iterations

- Full adjoint iterations (black) sum of TS-branch modes only (magenta)
- Transient since disturbance propagates out of domain

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

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

- Entire dynamics vs. input-output time signals

- 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:

- Eigenmodes of Hankel operator – balanced modes

- Controllability Gramian

- Observability Gramian

- Input

- Correlation of actuator impulse response in forward solution
- POD modes:
- Ranks states most easily influenced by input
- Provides a means to measure controllability

- 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

- Combine snapshots of direct and adjoint simulation
- Expand modes in snapshots to obtain smaller eigenvalue problem

Direct simulation:

Adjoint simulation:

adjoint

forward

- Largest outputs possible to excite with chosen forcing
- Balanced modes diagonalize observability Gramian
- Adjoint balanced modes diagonalize controllability Gramian
- Ginzburg-Landau example revisited

- Project dynamics on balanced modes using their biorthogonal adjoints
- Reduced representation of input-output relation, useful in control design

Disturbance Sensor

Actuator Objective

Disturbance Objective

DNS: n=105

ROM: m=50

From all inputs to all outputs

DNS: n=105

ROM: m=80

m=50

m=2

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

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

- Apply in Navier-Stokes simulation

controller

Noise

Sensor

Actuator

Objective

Control off

Cheap Control

Intermediate control

Expensive Control

Noise

Sensor

Actuator

Objective

- 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)

Countair rotating vortex pair

Shear layer vortices

Horseshoe vortices

Wake region

- 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

- Steady state computed using the SFD method (Åkervik et.al.)
- Energy growth of perturbation

Steady state

Perturbation

- Global eigenmodes computed using ARPACK
- Growth rate: 0.08
- Strouhal number: 0.16

1st global mode

time

Perturbation energy

Global mode energy

- 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

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

- 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

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

- Forward and adjoint mode located in different regions
- implies non-orthogonal eigenfunctions/non-normal operator
- Flow is sensitive where adjoint is large

- Eigenfunction expansion in selected modes
- Optimization of energy output

- 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

cost function

Reduced model of real system/flow

Estimator/ Controller

- Project dynamics on least stable global modes
- Choose spatial location of control and measurements
- LQG control design

- Least stable eigenvalues are rearranged
- Exponential growth turned into exponential decay
- Good performance in DNS using only 4 global modes