Flow control applied to transitional flows inputoutput analysis

Flow control applied to transitional flows: input-output analysis, model reduction and control Linné Flow Centre KTH Mechanics Dan Henningson collaborators Shervin Bagheri, Espen Åkervik Luca Brandt, Peter Schmid 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 1

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), … Linné Flow Centre KTH Mechanics • 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 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 2

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 … Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 3

Linearized Navier-Stokes for Blasius flow Continuous formulation Discrete formulation Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 4

Input-output configuration for linearized N-S Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 5

Solution to the complete input-output problem Linné Flow Centre KTH Mechanics • Initial value problem: flow stability • Forced problem: input-output analysis 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 6

The Initial Value Problem Linné Flow Centre KTH Mechanics • Asymptotic stability analysis: Global modes of the Blasius boundary layer • Transient growth analysis: Optimal disturbances in Blasius flow 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 7

Dimension of discretized system Linné Flow Centre KTH Mechanics • 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 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 8

Krylov subspace with Arnoldi algorithm • Krylov subspace created using NS-timestepper • Orthogonal basis created with Gram-Schmidt • Approximate eigenvalues from Hessenberg matrix H Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 9

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 Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 10

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 Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 11

Optimal disturbance growth • Optimal growth from eigenvalues of • Krylov sequence built by forward-adjoint iterations Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 12

Evolution of optimal disturbance in Blasius flow • Full adjoint iterations (black) of TS-branch modes only (magenta) • Transient since disturbance propagates out of domain sum Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 13

Snapshots of optimal disturbance evolution • Initial disturbance leans against the shear raised up by Orrmechanism into propagating TS-wavepacket Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 14

The forced problem: input-output • • • Ginzburg-Landau example Input-output for 2 D Blasius configuration Model reduction Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 15

Ginzburg-Landau example • Entire dynamics vs. input-output time signals Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 16

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: Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 17

Most dangerous inputs and the largest outputs Linné Flow Centre KTH Mechanics • Eigenmodes of Hankel operator – balanced modes • Controllability Gramian • Observability Gramian 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 18

Controllability Gramian for GL-equation Linné Flow Centre KTH Mechanics • Correlation of actuator impulse response in forward solution • POD modes: • Ranks states most easily influenced by input • Provides a means to measure controllability Input 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 19

Observability Gramian for GL-equation Output Linné Flow Centre KTH Mechanics • Correlation of sensor impulse response in adjoint solution • Adjoint POD modes: • Ranks states most easily sensed by output • Provides a means to measure observability 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 20

Balanced modes: eigenvalues of the Hankel operator • Combine snapshots of direct and adjoint simulation • Expand modes in snapshots to obtain smaller eigenvalue problem Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 21

Snapshots of direct and adjoint solution in Blasius flow Direct simulation: Adjoint simulation: Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 22

Balanced modes for Blasius flow adjoint forward Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 23

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 Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 24

Model reduction • Project dynamics on balanced modes using their biorthogonal adjoints • Reduced representation of input-output relation, useful in control design Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 25

Impulse response Disturbance Actuator Sensor Objective Linné Flow Centre KTH Mechanics Disturbance Objective DNS: n=105 ROM: m=50 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 26

Frequency response From all inputs to all outputs Linné Flow Centre KTH Mechanics DNS: n=105 ROM: m=80 m=50 m=2 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 27

Feedback control • • LQG control design using reduced order model Blasius flow example Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 28

Optimal Feedback Control – LQG cost function w Linné Flow Centre KTH Mechanics g (noise) Ly f=Kk z 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/H 2 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 29

LQG controller formulation with DNS • Apply in Navier-Stokes simulation Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 30

Performance of controlled system controller Noise Linné Flow Centre KTH Mechanics Sensor Actuator Objective 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 31

Performance of controlled system Linné Flow Centre KTH Mechanics Noise Sensor Actuator Objective Control off Cheap Control Intermediate control Expensive Control 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 32

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 Linné Flow Centre KTH Mechanics • Balanced modes give low order models preserving inputoutput 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 3 D problem and test controllers experimentally 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 33

Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 34

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) Linné Flow Centre KTH Mechanics • 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) 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 35

Jet in cross-flow Countair rotating vortex pair Shear layer vortices Horseshoe vortices Linné Flow Centre KTH Mechanics Wake region 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 36

Direct numerical simulations • DNS: Fully spectral and parallelized • Self-sustained global oscillations • Probe 1– shear layer • Probe 2 – separation region Linné Flow Centre KTH Mechanics 1 2 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 2 Vortex identification criterion 37

Basic state and impulse response • Steady state computed using the SFD method (Åkervik et. al. ) Steady state Unsteady Time-averaged Steady-state Linné Flow Centre KTH Mechanics • Energy growth of perturbation Perturbation 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 38

Global eigenmodes • Global eigenmodes computed using ARPACK • Growth rate: 0. 08 • Strouhal number: 0. 16 1 st global mode Linné Flow Centre KTH Mechanics time Perturbation energy Global mode energy 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 39

Optimal sum of eigenmodes Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 40

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 Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 41

3 D Blasius optimals • • Streamwise vorticies create streaks for long times Optimals for short times utilizes Orr-mechanism Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 42

Input-output analysis Linné Flow Centre KTH Mechanics • 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 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 43

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 Linné Flow Centre KTH Mechanics 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. 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 44

Global spectra • Global eigenmodes found using Arnoldi method • About 150 eigenvalues converged and 2 unstable Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 45

Most unstable mode Linné Flow Centre KTH Mechanics • Forward and adjoint mode located in different regions implies non-orthogonal eigenfunctions/non-normal operator • Flow is sensitive where adjoint is large 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 46

Maximum energy growth • Eigenfunction expansion in selected modes • Optimization of energy output Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 47

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 Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 48

LQG feedback control cost function Linné Flow Centre KTH Mechanics Reduced model of real system/flow Estimator/ Controller 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 49

Riccati equations for control and estimation gains Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 50

Feedback control of cavity disturbances • Project dynamics on least stable global modes • Choose spatial location of control and measurements • LQG control design Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 51

Controller performance • Least stable eigenvalues are rearranged • Exponential growth turned into exponential decay • Good performance in DNS using only 4 global modes Linné Flow Centre KTH Mechanics 7 th European Fluid Mechanics Conference, Manchester, 15 -18 September, 2008. 52