Flow_control Blasius Eq Lam Turb

8/13/2019 Flow_control Blasius Eq Lam Turb

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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


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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


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Drag breakdown

G. Schrauf, AIAA 2008


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Friction drag reduction

Possible area for Laminar Flow Control:

Laminar wings, tail, fin and nacelles -> 15% lower fuelconsumption


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Transition control

Transition is caused by

breakdown of growing

disturbances inside the

boundary layer.

Prevent/delay transition by

suppressing the growth

of small perturbations.


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exponentialgrowth of 2D TS

waves

turbulence

Low levels of free-streamturbulence (


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Localized disturbance undergoingtransition in Blasius flow

Linear TS-wavepacket

Nonlinear wavepacket

Turbulent spot


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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


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Linearized Navier-Stokes for Blasius flow

Discrete formulation:state space form

Continuous formulation


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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


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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


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Input-output configuration for linearized N-S


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Solution to the complete input-output problem

Initial value problem: flow stability

Forced problem: input-output analysis


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Ginzburg-Landau example

Entire dynamics vs. input-output time signals


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State-spaceformulation:

Solution:

Input-output behavior on the flat plate

w

B C

timetime


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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


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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:


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Most dangerous inputs, creating the largest outputs

Singular vectors of the Hankel operator balanced modes

Observability Gramian

Controllability Gramian


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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


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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


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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


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Properties of balanced modes

Balanced modes diagonalize observability Gramian

Adjoint balanced modes diagonalize controllability Gramian

Ginzburg-Landau example revisited


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Input-Output system for 3D Blasius


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Disturbance, TS wavepacket B1


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Actuators and Sensors- B2 and C2

Actuators represented by localized volume forcing close to the wallSensors flow variables weighted with localized Gaussian function


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Objective function C1

POD1 - TS

Projection of flow on first 10 POD modes of wavepacket,targeting the 10 most energetic structures


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Snapshots of direct and adjoint solution in Blasius flow

Direct simulation: Adjoint simulation:


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Balanced and adjoint balanced modes

Leading balanced mode: wave-packet structure mainly located

downstream

Leading adjoint balanced mode: tilted structure located upstream


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Model reduction

Project dynamics on balanced modes using theirbiorthogonal adjoints

Reduced representation of input-output relation,useful in control design


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Performance of the reduced order model

Disturbance Sensor

Actuator Output

Disturbance Output

System degrees offreedom: n>107

Reduced ordermodel: r=60


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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)


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LQG controller formulation with DNS

Apply in Navier-Stokes simulation


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TS-packet evolution with and without control


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Sensor and actuator signals for controlled flow on thecenterline

Wave-packet in sensor and resulting control signal

Input output signals


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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.


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Parametric analysis - actuators

7 actuators, 7 sensors, 1 compensator/controller

Reference case: full setup, 9 actuators, 9 sensors


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Input stochastic forcing


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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

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Flow_control Blasius Eq Lam Turb