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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]
