8/13/2019 Flow_control Blasius Eq Lam Turb
1/42
1
Linn Flow CentreKTH Mechanics Dan Henningson
collaborators
Onofrio Semeraro, Shervin Bagheri,
Luca Brandt
Model reduction for flow control:input-output analysis and flow control applied to transition
in the Blasius boundary layer on a flat plate
8/13/2019 Flow_control Blasius Eq Lam Turb
2/42
2
Linn Flow CentreKTH Mechanics
Flow on an Airplane Wing
Friction Drag on surface smaller for laminar thanturbulent flows
Delay the transition to turbulence to save fuel
8/13/2019 Flow_control Blasius Eq Lam Turb
3/42
5
Linn Flow CentreKTH Mechanics
Drag breakdown
G. Schrauf, AIAA 2008
8/13/2019 Flow_control Blasius Eq Lam Turb
4/42
6
Linn Flow CentreKTH Mechanics
Friction drag reduction
Possible area for Laminar Flow Control:
Laminar wings, tail, fin and nacelles -> 15% lower fuelconsumption
8/13/2019 Flow_control Blasius Eq Lam Turb
5/42
7
Linn Flow CentreKTH Mechanics
Transition control
Transition is caused by
breakdown of growing
disturbances inside the
boundary layer.
Prevent/delay transition by
suppressing the growth
of small perturbations.
8/13/2019 Flow_control Blasius Eq Lam Turb
6/42
8
Linn Flow CentreKTH Mechanics
exponentialgrowth of 2D TS
waves
turbulence
Low levels of free-streamturbulence (
8/13/2019 Flow_control Blasius Eq Lam Turb
7/42
9
Linn Flow CentreKTH Mechanics
Localized disturbance undergoingtransition in Blasius flow
Linear TS-wavepacket
Nonlinear wavepacket
Turbulent spot
8/13/2019 Flow_control Blasius Eq Lam Turb
8/4210
Linn Flow CentreKTH Mechanics
Control of transition in the Blasiusboundary layer
Aim is to extend the laminar region byeliminating boundary layer disturbances
Control based on linear theory sincetransitional disturbances usually smallinitially use linearized Navier-Stokes
Target control at typical disturbances foundin the boundary layer so called TS-waves
Use model reduction based on balancedtruncation to obtain low order controller
8/13/2019 Flow_control Blasius Eq Lam Turb
9/4211
Linn Flow CentreKTH Mechanics
Linearized Navier-Stokes for Blasius flow
Discrete formulation:state space form
Continuous formulation
8/13/2019 Flow_control Blasius Eq Lam Turb
10/42
12
Linn Flow CentreKTH Mechanics
The discretized linearizedNavier-Stokes equations
Disturbances around the laminar velocity i.e. U(x,y) + u(x,y,z,t)
The disturbance flow velocities are in the vector field u(x,y,z,t)
Horizontal directions (x,z) expanded in Fourier series
Normal direction yin Chebychev series
Discretization in time Runge-Kutta/Crank-Nicolson
Fringe region with volume force to make flow periodic inx
8/13/2019 Flow_control Blasius Eq Lam Turb
11/42
13
Linn Flow CentreKTH Mechanics
Dont store matrixA too large dimension
Matrix A very large for complex flows
Time-stepper technique:
Never store matricesSimulation code to steps velocity fields forward in time
Work only with series of snapshots
8/13/2019 Flow_control Blasius Eq Lam Turb
12/42
14
Linn Flow CentreKTH Mechanics
Input-output configuration for linearized N-S
8/13/2019 Flow_control Blasius Eq Lam Turb
13/42
15
Linn Flow CentreKTH Mechanics
Solution to the complete input-output problem
Initial value problem: flow stability
Forced problem: input-output analysis
8/13/2019 Flow_control Blasius Eq Lam Turb
14/42
16
Linn Flow CentreKTH Mechanics
Ginzburg-Landau example
Entire dynamics vs. input-output time signals
8/13/2019 Flow_control Blasius Eq Lam Turb
15/42
17
Linn Flow CentreKTH Mechanics
State-spaceformulation:
Solution:
Input-output behavior on the flat plate
w
B C
timetime
8/13/2019 Flow_control Blasius Eq Lam Turb
16/42
18
Linn Flow CentreKTH Mechanics
Traditional Model Reduction forLinear Systems
Obtain a set of functions and project the stateon this basis:
Insert into the plant to obtain reduced model:
T can be Balanced modesobtained by solving two Lyapunovequations related to controllability and observability Gramain.
Standard balanced truncation algorithm only computationallyfeasible for moderate-size systems (with less 10000 d.o.f)
m 5
8/13/2019 Flow_control Blasius Eq Lam Turb
17/42
19
Linn Flow CentreKTH Mechanics
Input-output operators
Past inputs to initial state: class ofinitial conditions possible to generatethrough chosen forcing
Initial state to future outputs:possible outputs from initialcondition
Past inputs to future outputs:
8/13/2019 Flow_control Blasius Eq Lam Turb
18/42
20
Linn Flow CentreKTH Mechanics
Most dangerous inputs, creating the largest outputs
Singular vectors of the Hankel operator balanced modes
Observability Gramian
Controllability Gramian
8/13/2019 Flow_control Blasius Eq Lam Turb
19/42
21
Linn Flow CentreKTH Mechanics
Controllability Gramian for GL-equation
Correlation of actuatorimpulse response in
forward solution
POD modes:
Ranks states most easilyinfluenced by input
Provides a means tomeasure controllability
Input
8/13/2019 Flow_control Blasius Eq Lam Turb
20/42
8/13/2019 Flow_control Blasius Eq Lam Turb
21/42
23
Linn Flow CentreKTH Mechanics
Controllability & Observability
Largest flow structures created by input forcing?
Which flow structures generate largest output energy?
Strong
controllability
Strong
observability
Large
response
to inputInput
Output
Give rise to
large output
8/13/2019 Flow_control Blasius Eq Lam Turb
22/42
24
Linn Flow CentreKTH Mechanics
Balanced modes:singular vectors of the Hankel operator
Combine snapshots of direct and adjoint simulation
Expand modes in snapshots to obtain smaller eigenvalue problem
Balanced modes
8/13/2019 Flow_control Blasius Eq Lam Turb
23/42
25
Linn Flow CentreKTH Mechanics
Properties of balanced modes
Balanced modes diagonalize observability Gramian
Adjoint balanced modes diagonalize controllability Gramian
Ginzburg-Landau example revisited
8/13/2019 Flow_control Blasius Eq Lam Turb
24/42
26
Linn Flow CentreKTH Mechanics
Input-Output system for 3D Blasius
8/13/2019 Flow_control Blasius Eq Lam Turb
25/42
27
Linn Flow CentreKTH Mechanics
Disturbance, TS wavepacket B1
8/13/2019 Flow_control Blasius Eq Lam Turb
26/42
28
Linn Flow CentreKTH Mechanics
Actuators and Sensors- B2 and C2
Actuators represented by localized volume forcing close to the wallSensors flow variables weighted with localized Gaussian function
8/13/2019 Flow_control Blasius Eq Lam Turb
27/42
29
Linn Flow CentreKTH Mechanics
Objective function C1
POD1 - TS
Projection of flow on first 10 POD modes of wavepacket,targeting the 10 most energetic structures
8/13/2019 Flow_control Blasius Eq Lam Turb
28/42
30
Linn Flow CentreKTH Mechanics
Snapshots of direct and adjoint solution in Blasius flow
Direct simulation: Adjoint simulation:
8/13/2019 Flow_control Blasius Eq Lam Turb
29/42
31
Linn Flow CentreKTH Mechanics
Balanced and adjoint balanced modes
Leading balanced mode: wave-packet structure mainly located
downstream
Leading adjoint balanced mode: tilted structure located upstream
8/13/2019 Flow_control Blasius Eq Lam Turb
30/42
32
Linn Flow CentreKTH Mechanics
Model reduction
Project dynamics on balanced modes using theirbiorthogonal adjoints
Reduced representation of input-output relation,useful in control design
8/13/2019 Flow_control Blasius Eq Lam Turb
31/42
33
Linn Flow CentreKTH Mechanics
Performance of the reduced order model
Disturbance Sensor
Actuator Output
Disturbance Output
System degrees offreedom: n>107
Reduced ordermodel: r=60
8/13/2019 Flow_control Blasius Eq Lam Turb
32/42
34
Linn Flow CentreKTH Mechanics
Optimal Feedback Control LQG
controller
Find an optimal control signal (t) based on themeasurements (t) such that in the presence ofexternal disturbances w(t) and measurement noise g(t)the outputz(t) is minimized.
Solution: LQG/H2
cost function
=KLw zg
(noise)
8/13/2019 Flow_control Blasius Eq Lam Turb
33/42
35
Linn Flow CentreKTH Mechanics
LQG controller formulation with DNS
Apply in Navier-Stokes simulation
8/13/2019 Flow_control Blasius Eq Lam Turb
34/42
37
Linn Flow CentreKTH Mechanics
TS-packet evolution with and without control
8/13/2019 Flow_control Blasius Eq Lam Turb
35/42
38
Linn Flow CentreKTH Mechanics
Sensor and actuator signals for controlled flow on thecenterline
Wave-packet in sensor and resulting control signal
Input output signals
8/13/2019 Flow_control Blasius Eq Lam Turb
36/42
8/13/2019 Flow_control Blasius Eq Lam Turb
37/42
40
Linn Flow CentreKTH Mechanics
Energy of the controlled and non-controlled cases
Three cases considered: cheap controller (l=100),intermediate controller (l=250), expensive controller (l=500).
9 actuators, 9 sensors, 1 compensator/controller.
8/13/2019 Flow_control Blasius Eq Lam Turb
38/42
41
Linn Flow CentreKTH Mechanics
Parametric analysis - actuators
7 actuators, 7 sensors, 1 compensator/controller
Reference case: full setup, 9 actuators, 9 sensors
8/13/2019 Flow_control Blasius Eq Lam Turb
39/42
42
Linn Flow CentreKTH Mechanics
Parametric analysis - actuators
5 actuators, 5 sensors, 1 compensator/controller
Reference case: full setup, 9 actuators, 9 sensors
8/13/2019 Flow_control Blasius Eq Lam Turb
40/42
43
Linn Flow CentreKTH Mechanics
Parametric analysis - actuators
3 actuators, 3 sensors, 1 compensator/controller
Reference case: full setup, 9 actuators, 9 sensors
8/13/2019 Flow_control Blasius Eq Lam Turb
41/42
44
Linn Flow CentreKTH Mechanics
Input stochastic forcing
8/13/2019 Flow_control Blasius Eq Lam Turb
42/42
Linn Flow CentreKTH Mechanics
Conclusions
Input-output formulation ideal for analysis and design offeedback control systems applied in Blasius flow, a modelof an airplane wing
Balanced modes
Obtained from snapshots of forward and adjoint solutionsGive low order models preserving input-output relationshipbetween 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 3Dproblem and test controllers experimentally