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C o m p a r i s o n b e t w e e n t w o a n d t h r e e - d i m e n s i o n a l
P O D i n a t u r b u l e n t b o u n d a r y l a y e r u s i n g
m u l t i - p l a n e s t e r e o s c o p i c P I V
A l e x L i b e r z o n * , R o i G u r k a a n d G a d H e t s r o n i
T e l A v i v U n i v e r s i t y , B e n G u r i o n U n i v e r s i t y , T e c h n i o n
E T C 1 3 , S e p 1 2 - 1 5 , 2 0 1 1
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O u t l i n e
B a c k g r o u n d a n d m o t i v a t i o n
D e s c r i p t i o n o f t h e e x p e r i m e n t a l s e t u p
D e s c r i p t i o n o f P O D i n 2 D a n d i n 3 D
C o m p a r i s o n
S u m m a r y a n d c o n c l u s i o n s
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B a c k g r o u n d a n d m o t i v a t i o n
C o h e r e n t s t r u c t u r e s i n t u r b u l e n t o w s
O b j e c t i v e ( ! ) d e t e c t i o n a n d c h a r a c t e r i z a t i o n ( g e o m e t r i c ,
d y n a m i c , e t c . )
L u m l e y ( 1 9 7 0 ) i n t r o d u c t i o n t o P r o p e r O r t h o g o n a l
D e c o m p o s i t i o n
B r o a d u s e a n d p o t e n t i a l a b u s e o f P O D d e t e c t i n g c o h e r e n t
m o t i o n s
O n e o f t h e m a i n a s p e c t s i s d u e t o t h e 2 D c r o s s - s e c t i o n s a s
c o m p a r e d t o f u l l 3 D a n a l y s i s
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E x p e r i m e n t a l s e t u p
62
3
5
4
1
11 12109 13
8 7
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X P I V m e t h o d
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X P I V a l g o r i t h m
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X P I V a l g o r i t h m c o n t d .
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P r o p e r o r t h o g o n a l d e c o m p o s i t i o n
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P O D f o r t u r b u l e n c e
L u m l e y 1 9 7 0
H o l m e s , P . a n d L u m l e y , J . L . a n d B e r k o o z , G 1 9 9 6
S i r o v i c h s n a p s h o t s m e t h o d 1 9 8 7
R ( x , x )( x )d x = ( x ) ( 1 )
fk
( x ) =
K
n
=1
a
k
n n
( x ) ( 2 )
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S n a p s h o t s m e t h o d ( S i r o v i c h 1 9 8 7 )
F o r t h e t h r e e - d i m e n s i o n a l a n a l y s i s , t h e k
t h
s n a p s h o t i s d e n e d
b a s e d o n a s e t o f N
x
N
y
N
z
g r i d p o i n t s a n d e x p r e s s e d a s a
v e c t o r o f t h e l e n g t h n=
N
x
N
y
N
z
:
=
k . . . kn
T
( 3 )
C=
( )T ( )
( 4 )
C = ( 5 )
( n ) =M
j
=1
(
n
) j
j ( 6 )
H e r e (
n
) j
d e n o t e s t h e j
t h
e l e m e n t o f t h e e i g e n v e c t o r ,
c o r r e s p o n d i n g t o t h e j
t h
e i g e n v a l u e (n ) .
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O u r P O D a p p r o a c h
V o r t i c i t y i s u s e d i n s t e a d o f v e l o c i t y
L i n e r c o m b i n a t i o n o f t h e m o d e s i n t h e s p i r i t o f c h a r a c t e r i s t i c
e d d y ( L u m l e y 1 9 7 0 )
( x ) =K
n
=1
(n )(n )(( x )), K N ( 7 )
2 D v s 3 D : e a c h p l a n e i s a n a l y s e d s e p a r a t e l y o r a l l t h e p l a n e s
t o g e t h e r
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R e s u l t s a n d d i s c u s s i o n
0 50 100 1500
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
#modes
Cumu
lativeenergy
3D - PODSum of slice-POD
middle plane
lower plane
higher plane
F i g u r e : C u m u l a t i v e e n e r g y o f t h e P O D m o d e s u s i n g t h e 3 D - P O D ( s o l i d
l i n e ) a n d t h e s l i c e - P O D f o r s e p a r a t e p l a n e s a n d t h e i r s u m m a t i o n
( d a s h e d - d o t t e d l i n e ) .
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D i e r e n t r s t m o d e s
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D i e r e n t h i g h e r m o d e s
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D i e r e n t c o m b i n a t i o n o f m o d e s
( a ) ( b )
F i g u r e : C o m p a r i s o n o f t h e w a l l - n o r m a l e n s t r o p h y c o m p o n e n t
2
y
o b t a i n e d
u s i n g a 3 D P O D ( a ) v e r s u s t h e e l d o b t a i n e d t h r e e s i n g l e - p l a n e
s l i c e - P O D ( b ) .
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S u m m a r y
T h e m u l t i - p l a n e s t e r e o s c o p i c p a r t i c l e i m a g e v e l o c i m e t r y
m e a s u r e m e n t s y s t e m ( X P I V ) w a s u t i l i z e d t o m e a s u r e t h e
b o u n d a r y l a y e r i n a u m e . T h r e e - c o m p o n e n t v e l o c i t y v e c t o r
e l d s a t t h r e e p a r a l l e l s t r e a m w i s e - s p a n w i s e p l a n e s w i t h a
x e d d i s t a n c e b e t w e e n t h e m w e r e o b t a i n e d .
T h e p r o p o s e d a n a l y s i s e m p h a s i z e s t h e n e c e s s i t y o f t h e
t h r e e - d i m e n s i o n a l m e a s u r e m e n t s i n t u r b u l e n t b o u n d a r y l a y e r s .
W e c a l c u l a t e t h e P O D m o d e s o f t h e w a l l n o r m a l c o m p o n e n t o f
t h e v o r t i c i t y e l d s . T h i s p r o c e d u r e w a s p e r f o r m e d u s i n g t w o
m e t h o d s : A ) t w o - d i m e n s i o n a l s l i c e s o f t h e d a t a w e r e u s e d
s e p a r a t e l y i n a s o - c a l l e d s l i c e - P O D , a n d B ) t h e r e s u l t s a r e
u t i l i z e d a s a v o l u m e t r i c d a t a s e t t o o b t a i n t h e 3 D - P O D m o d e s .
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