![Page 1: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/1.jpg)
1
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
![Page 2: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/2.jpg)
2
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
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
![Page 3: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/3.jpg)
3
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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 BL, but many other more complex flows are now tractable …
![Page 4: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/4.jpg)
4
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Linearized Navier-Stokes for Blasius flow
Discrete formulationContinuous formulation
![Page 5: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/5.jpg)
5
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Input-output configuration for linearized N-S
![Page 6: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/6.jpg)
6
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Solution to the complete input-output problem
• Initial value problem: flow stability
• Forced problem: input-output analysis
![Page 7: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/7.jpg)
7
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
The Initial Value Problem
• Asymptotic stability analysis:Global modes of the Blasius boundary layer
• Transient growth analysis:Optimal disturbances in Blasius flow
![Page 8: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/8.jpg)
8
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
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
![Page 9: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/9.jpg)
9
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Krylov subspace with Arnoldi algorithm
• Krylov subspace created using NS-timestepper
• Orthogonal basis created with Gram-Schmidt
• Approximate eigenvalues from Hessenberg matrix H
![Page 10: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/10.jpg)
10
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 11: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/11.jpg)
11
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 12: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/12.jpg)
12
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Optimal disturbance growth
• Optimal growth from eigenvalues of
• Krylov sequence built by forward-adjoint iterations
![Page 13: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/13.jpg)
13
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 14: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/14.jpg)
14
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Snapshots of optimal disturbance evolution
• Initial disturbance leans against the shear raised up by Orr-mechanism into propagating TS-wavepacket
![Page 15: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/15.jpg)
15
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
The forced problem: input-output
• Ginzburg-Landau example• Input-output for 2D Blasius configuration• Model reduction
![Page 16: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/16.jpg)
16
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Ginzburg-Landau example
• Entire dynamics vs. input-output time signals
![Page 17: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/17.jpg)
17
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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:
![Page 18: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/18.jpg)
18
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Most dangerous inputs and the largest outputs
• Eigenmodes of Hankel operator – balanced modes
• Observability Gramian
• Controllability Gramian
![Page 19: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/19.jpg)
19
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 20: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/20.jpg)
20
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 21: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/21.jpg)
21
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Balanced modes: eigenvalues of the Hankel operator
• Combine snapshots of direct and adjoint simulation
• Expand modes in snapshots to obtain smaller eigenvalue problem
![Page 22: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/22.jpg)
22
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Snapshots of direct and adjoint solution in Blasius flow
Direct simulation: Adjoint simulation:
![Page 23: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/23.jpg)
23
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Balanced modes for Blasius flow
adjoint forward
![Page 24: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/24.jpg)
24
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 25: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/25.jpg)
25
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Model reduction
• Project dynamics on balanced modes using their biorthogonal adjoints
• Reduced representation of input-output relation, useful in control design
![Page 26: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/26.jpg)
26
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Impulse response
DNS: n=105
ROM: m=50
Disturbance Sensor
Actuator Objective
Disturbance Objective
![Page 27: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/27.jpg)
27
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Frequency response
DNS: n=105
ROM: m=80 m=50 m=2
From all inputs to all outputs
![Page 28: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/28.jpg)
28
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Feedback control
• LQG control design using reduced order model• Blasius flow example
![Page 29: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/29.jpg)
29
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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)
![Page 30: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/30.jpg)
30
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
LQG controller formulation with DNS
• Apply in Navier-Stokes simulation
![Page 31: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/31.jpg)
31
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Performance of controlled system
Noise
Sensor
Actuator
Objective
controller
![Page 32: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/32.jpg)
32
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Performance of controlled system
Noise Sensor Actuator Objective
Control offCheap ControlIntermediate controlExpensive Control
![Page 33: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/33.jpg)
33
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
• Outlook: incorporate realistic sensors and actuators in 3D problem and test controllers experimentally
![Page 34: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/34.jpg)
34
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
![Page 35: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/35.jpg)
35
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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)
![Page 36: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/36.jpg)
36
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Jet in cross-flow
Horseshoe vortices
Wake region
Countair rotating vortex pair
Shear layer vortices
![Page 37: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/37.jpg)
37
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 38: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/38.jpg)
38
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 39: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/39.jpg)
39
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Global eigenmodes
• Global eigenmodes computed using ARPACK
• Growth rate: 0.08
• Strouhal number: 0.16
Perturbation energyGlobal mode energy
1st global mode
time
![Page 40: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/40.jpg)
40
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Optimal sum of eigenmodes
![Page 41: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/41.jpg)
41
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 42: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/42.jpg)
42
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
3D Blasius optimals
• Streamwise vorticies create streaks for long times• Optimals for short times utilizes Orr-mechanism
![Page 43: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/43.jpg)
43
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 44: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/44.jpg)
44
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 45: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/45.jpg)
45
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Global spectra
• Global eigenmodes found using Arnoldi method• About 150 eigenvalues converged and 2 unstable
![Page 46: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/46.jpg)
46
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 47: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/47.jpg)
47
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Maximum energy growth
• Eigenfunction expansion in selected modes
• Optimization of energy output
![Page 48: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/48.jpg)
48
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 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
![Page 49: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/49.jpg)
49
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
LQG feedback control
Estimator/ Controller
Reduced model of real system/flow
cost function
![Page 50: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/50.jpg)
50
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Riccati equations for control and estimation gains
![Page 51: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/51.jpg)
51
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Feedback control of cavity disturbances
• Project dynamics on least stable global modes
• Choose spatial location of control and measurements
• LQG control design
![Page 52: 1 Linné Flow Centre KTH Mechanics 7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008. Dan Henningson collaborators Shervin Bagheri,](https://reader036.vdocuments.net/reader036/viewer/2022062320/56649c815503460f94938daf/html5/thumbnails/52.jpg)
52
Linné Flow CentreKTH Mechanics
7th European Fluid Mechanics Conference, Manchester, 15-18 September, 2008.
Controller performance
• Least stable eigenvalues are rearranged
• Exponential growth turned into exponential decay
• Good performance in DNS using only 4 global modes