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Milan Milovanovic, Marek Gayer, Joris Michielsen and Ole Morten Aamo

48th IEEE Conference on Decision and Control, CDC 2009, Shanghai, China

Model-based Stabilization of Vortex Shedding with CFD Verification

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Von Kármán vortex street:

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for , with boundary conditions

and

A complex Ginzburg-Landau equation:

),0( dxx

)(),0( tutA 0),( txA d

2

1 2 32( ) ( )

A A Aa a x a x A

t x x

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With the control input:

A complex Ginzburg-Landau equation becomes:

for with boundary conditions:

( ) ( )

( ) ( )t R xx R I xx I

t I xx I R xx R

a b x a b x

a b x a b x

0,1x

(0, ) 0, (0, ) 0

(1, ) , (1, ) ( )R I

t t

t u t u t

1

1 ,10

1

,1 10

[ ( ) ( , ) ( ) ( , )] ,

[ ( ) ( , ) ( ) ( , )]

R c

I c

u k y y t k y y t dy

u k y y t k y y t dy

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Looking for:

to transform sistem into:

for with boundary conditions:

0

0

( , ) ( , ) [ ( , ) ( , ) ( , ) ( , )]

( , ) ( , ) [ ( , ) ( , ) ( , ) ( , )]

x

c

x

c

x t x t k x y x t k x y y t dy

x t x t k x y x t k x y y t dy

( ) ( )

( ) ( )t R xx R I xx I

t I xx I R xx R

a f x a f x

a f x a f x

0,1x

(0, ) (0, ) (1, ) (1, ) 0.t t t t

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The pair of kernels and satisfy the PDEs:

( , )k x y ( , )ck x y

, ,

( , ) ( , ) ,

( , ) ( , )

xx yy c c

c xx c yy c c

k k x y k x y k

k k x y k x y k

for with boundary conditions:

( , ) , : 0 1 ,x y x y y x

0

0

1( , ) ( , ) , ( ,0) 0,

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( , ) ( , ) , ( ,0) 0,2

x

x

c c c

k x x d k x

k x x d k x

where

2 2

2 2

( ) ( ) ( ) ( )( , ) ,

( ) ( ) ( ) ( )( , ) .

R R R I I I

R I

R I I I R R

R I

a b y f x a b y f xx y

a a

a b y f x a b y f xx y

a a

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Simulation setup

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Measurement curves:

Suction/blowing actuation slots:

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GL parameters:

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The residue between real CFD data and GL simulations:

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Controller kernels:

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The state feedback controller:

1 ,10

201

1( ) ( , )

1exp ( )

2

dx d dc

d d d

x

x x x xu t k ik A x t

x x x

a d dxa

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Transverse velocity:

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The control input:

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Stabilization of fully developed vortex shedding at Re=60:

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Conclusion:• Vortex shedding is successfully removed by the

control at Reynolds number Re=60.• The link is established between controllers previously

designed for the Ginzburg-Landau model of vortex shedding and the actual flow dynamics.

Additional effort:• Ongoing work: trying for Re>60• Future work: involve CFD verification of the output

feedback problem, which was solved for the GL equation in [Aamo2007]