polymers bounded shear flows v2

8/8/2019 Polymers Bounded Shear Flows v2

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D i l u t e p o l y m e r e e c t s o n t u r b u l e n c e

D i l u t e p o l y m e r e e c t s o n t u r b u l e n t b o u n d e d o w s

A l e x L i b e r z o n

T e l A v i v U n i v e r s i t y

N o v e m b e r 3 , 2 0 1 0


2/43


A c k n o w l e d g m e n t s

E T H Z u r i c h :

B . L t h i , M . G u a l a ( U M N ) , M . H o l z n e r ( M P I ) , U . R e i t e r ( P S I ) ,

W . K i n z e l b a c h

T u r b u l e n c e S t r u c t u r e L a b o r a t o r y ( w w w . e n g . t a u . a c . i l / e f d l ) , T e l

A v i v U n i v e r s i t y

M . K r e i z e r , D . R a t n e r , R . E l f a s s i , A . T s i n o b e r


3/43


I n t r o d u c t i o n

O u t l i n e o f t h e t a l k

B a c k g r o u n d a n d m o t i v a t i o n

E x p e r i m e n t s : m e t h o d s a n d f a c i l i t i e s

R e s u l t s

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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M a i n f o c u s i s t u r b u l e n c e

Turbulence

Polymers

Particles

Forcing

LagrangianEulerian


5/43



M o t i v a t i o n t o s t u d y p o l y m e r s i s b o t h b a s i c a n d p r a c t i c a l

D r a g r e d u c t i o n h a s b e e n s t u d i e d s i n c e

1 9 4 8 T o m s e e c t

B o d y o f l i t e r a t u r e i s h u g e , w i t h t h e

i m p o r t a n t c o n t r i b u t i o n s o f t h e

a u d i e n c e

T h e t u r b u l e n c e w h i c h o c c u r s i n t h e p r e s e n c e o f d r a g - r e d u c i n g

a d d i t i v e s i s d i e r e n t f r o m t h e t u r b u l e n c e w h i c h o c c u r s i n t h e

s o l v e n t a l o n e . I n d e e d , i n s o m e c a s e s o f v e r y d i l u t e p o l y m e r

s o l u t i o n s , t h e a n o m a l o u s ( i . e . l e s s d i s s i p a t i v e ) t u r b u l e n c e i s

p r o b a b l y t h e o n l y d e t e c t a b l e n o n - N e w t o n i a n e e c t . M c C o m b 1 9 9 0


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T h e r e i s d r a g r e d u c t i o n , b u t t h e r e i s m u c h m o r e


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P h e n o m e n o l o g y o f p o l y m e r e e c t s

F u c t u a t i n g a n d c o m p l e x s t r a i n

e l d i s n e c e s s a r y t o t u r n t h e

e e c t o n

R e a c t i o n b a c k c h a n g e s t h e e l d

o f s t r a i n , e . g . r e s i s t a n c e t o l a r g e

s t r a i n , s u p p r e s s i o n o f s t r o n g

e v e n t s , b u r s t s

A s s o c i a t i o n s o f p o l y m e r

m o l e c u l e s h a v e b e e n o b s e r v e d i n

d i l u t e s o l u t i o n s

T h e o w c o u l d b e c o n s i d e r e d

i n t e r m i t t e n t l y r h e o l o g i c a l


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E x p e r i m e n t s

T h r e e e x p e r i m e n t s , o p t i c a l m e t h o d s o f P T V / P I V

supportingframe

grid

field of view

glasstank

front view top view

lasersheet

linear motor

high-speedcamera

z

x

y

200 mm

4 mm

x

0 0.2 0.4 0.6

120

0

120

t(s)

grid velocity(mm/s)

FIG. 1. Schematic of the oscillating grid-stirred tank experimental setup. A

sample of the grid velocity in time is shown in the upper right corner.

R o t a t i n g d i s k s O s c i l l a t i n g g r i d L i d - d r i v e n c a v i t y

W h y t h e s e ? b e c a u s e w e n e e d a ) s m a l l s c a l e d e r i v a t i v e s o f v e l o c i t y -

d i s s i p a t i o n , s t r e t c h i n g , e t c . b ) o p t i c a l a c c e s s a n d n o t v e r y f a s t o w s t o

g e t P T V w e l l , c ) a k n o w n e n e r g y i n p u t , d ) h i g h s h e a r s o m e w h e r e i n t h e

o w .

L i b e r z o n e t a l ( 2 0 0 5 , 2 0 0 6 , 2 0 0 9 )


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3 D - P T V

(a) (b)


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P T V a l g o r i t h m

T h e i m p o r t a n t t h i n g i s t h a t w e m e a s u r e d i r e c t l y t h e f u l l

g r a d i e n t t e n s o r a l o n g t h e p a r t i c l e t r a j e c t o r i e s :

u

i

/x

j

a n d i t s e v o l u t i o n i n t i m e .


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R o t a t i n g d i s k s

C o u n t e r - r o t a t i n g d i s k s w / o b a e s , q u a s i - i s o t r o p i c o w i n

t h e c e n t e r


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V e l o c i t y d e r i v a t i v e s a n d h i g h e r m o m e n t s : v o r t i c i t y , s t r a i n

a n d t h e i r p r o d u c t i o n t e r m s , i

j

s

i j

, si j

s

j k

s

k i


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S t r e t c h i n g r a t e s , e i g e n v a l u e s , a l i g n m e n t s


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S t r e t c h i n g r e l a t e d q u a n t i t i e s - C a u c h y - G r e e n t e n s o r

e i g e n v a l u e s

i

(t

) =B

i j

(t

)(0

), d B

i j

/d t

= (u

i

/x

k

)B

k j

,B

i j

(0

) = i j

,

W

i j = B i k B k j


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S t r e t c h i n g d y n a m i c s o f i n n i t e s i m a l m a t e r i a l l i n e s t h r o u g h a

s i n g l e t e n s o r

i

(t

) =B

i j

(t

) j

(0

),d B

i j

/d t

= (u

i

/x

k

)B

k j

B

i j

(0

) = i j

i

j

s

i j

= Bi k

B

j m

s

i j

k

( 0 )m

( 0 ) Tk m

( t )k

( 0 )m

( 0 )

T

k m

( t )(0 )m

( 0 ) = 2 ( 0 )T

i

c o s

2 (( 0 ), i

)

i

j

s

i j

= Ti

c o s 2 ( 0 , i

) = 13

2 ( 0 )T1

+ T2

+ T3

1 . t r a c e t r

(T)i s p o s i t i v e o n a v e r a g e

2 . e m p i r i c a l l y f o u n d t h a t o n e e i g e n v a l u e i s t h r e e o r d e r s o f

m a g n i t u d e l a r g e r t h a n o t h e r s

3 . i t w a s s h o w n t o b e s t r o n g l y r e d u c e d i n d i l u t e p o l y m e r s o w


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S t r o n g r e d u c t i o n o f t h e s t r e t c h i n g e i g e n v a l u e i n p o l y m e r s

FIG. 5. PDF of the first eigenvalue 1

of the Tmatrix for water solid lines

and polymer solution dashed lines for different time moments.



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I n t e r m e d i a t e s u m m a r y

P r o b i n g d i r e c t l y t h e s t r e t c h i n g o f m o l e c u l e s b y f o l l o w i n g

r a t e - o f - s t r a i n t e n s o r e v o l u t i o n a l o n g t r a j e c t o r i e s .

s t r e t c h i n g i s a e c t e d m o r e t h a n t h e o w q u a n t i t i e s t h e m s e l v e s ,

a l o n g w i t h e i g e n v a l u e s o f t h e C a u c h y - G r e e n t e n s o r b e c a u s e o f

t h e h i s t o r y e e c t

t h e c h a n g e s a r e a t t h e s m a l l e s t s c a l e s , a s s h o w n b y p r o d u c t i o n

t e r m s o f v o r t i c i t y a n d s t r a i n

t h e c h a n g e s a r e n o t q u a n t i t a t i v e b u t a l s o q u a l i t a t i v e - c h a n g e

o f a l i g n m e n t s



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D o e s t h e w a y o f t h e e n e r g y i n p u t m a t t e r ?

W e c o m p a r e r e s u l t s o b t a i n e d w i t h b u e s a n d s m o o t h d i s k s



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R - Q m a p s i n d i l u t e p o l y m e r s



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T K E c h a n g e s i n b a e s / s m o o t h d i s k s c a s e s

FIG. 5. PDF of the turbulent kinetic energy, u2, for the four flow cases:

average values are 12, 7.5, 8.6, and 8.8105 m2 s2.



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T K E p r o d u c t i o n a n d t e r m s



22/43



c o s (

u

i

u

j

S

i j

) - d i m e n s i o n l e s s q u a n t i t y , r e l a t e d t o t h e s t r u c t u r e o f t u r b u l e n c e


23/43



24/43


O s c i l l a t i n g g r i d

O s c i l l a t i n g g r i d a p p a r a t u s - s h e a r l e s s t u r b u l e n c e + i n t e r f a c e

+ e n t r a i n m e n t

supporting

frame

grid

field of view

glass

tank

front view top view

laser

sheet

linear motor

high-speed

camera

z

x

y

200 mm

4 mm

x

0 0.2 0.4 0.6

120

0

120

t(s)

grid velocity

(mm/s)



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M o s t s i g n i c a n t c h a n g e - t h e i n t e r f a c e s h a p e

x(mm)

y(mm

)

0 100 200

100

50

0

0 100 200100

50

0

2

1

0

1

2

2

1

0

1

2

x(mm)

y(mm

)

x

y

(x t),

ua

w

p

) b) -1(s )

-1(s )



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F r o n t p r o p a g a t i o n a n d e n t r a i n m e n t c h a n g e d

s o m e t h i n g i s d i e r e n t i n t h i s t u r b u l e n c e

0 2 4 6 8 10 12 140

2

4

6

t (s)

H

(cm) 0 20 40

0

100

200

C (ppmw)

K

(mm

2s

1

)

water 25 ppm50 ppm

0.5 1 2 3 4 5 10

100

t (s)

H

(cm)

25 ppm

50 ppm

water

t1/2

H=

K t , K - e e c t i v e g r i d a c t i o n



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v e l o c i t y a n d l e n g t h s c a l e s

100

10110

0

101

t [s]

v

[mm/s]

100

101

100

101

t [s]

u

mm/s]

100

101

101

102

t [s]

L

[mm]

water25 ppm50 ppmt1/2



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A n a l y t i c a l e x e r c i s e t o e x p l a i n t h e m o d i e d s h a p e a n d

e n t r a i n m e n t c o e c i e n t

u s i n g P h i l l i p s 1 9 7 2 m e t h o d a n d m u l t i - f r a c t a l d e s c r i p t i o n o f t h e i n t e r f a c e g e o m e t r y

e n e r g y u x v 3r m s

/L , e n e r g y d i s s i p a t i o n = 2 si j

s

i j

, v i s c o u s

d i s s i p a t i o n v

= , p o l y m e r d i s s i p a t i o n p

= ( 1

)

v

e

A

L

= u A , u = 1 /4 1 /4 C 1 /4 v 1 /4

r m s

L

1 /4

,

A AL

L

2

d A

L

L

1

/3 1 /2

1 / 4

A

L

L

1 /3C

1 /1 2v

1 /4r m s

L

1 /1 2 1 /4

v

e

= vr m s

, C = C , 1 /3 C 1 /3 - e n t r a i n m e n t c o n s t a n t : w a t e r - w 0 . 8 p o l y m e r s - p 0 . 7 0 . 7 2

7 0 % o f e n e r g y i n p u t i s d i s s i p a t e d b y v i s c o u s e d d i e s

3 0 % o f e n e r g y i n p u t i s d i s s i p a t e d b y p o l y m e r s



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T h e r a t i o o f v i s c o u s d i s s i p a t i o n t o e n e r g y i n j e c t i o n a s af u n c t i o n o f D e b o r a h n u m b e r D e = r /

FIG. 5. Color online Dependence of =/i on the Deborah number.

marks the numerical results Ref. 6, a triangle marks the result of Ouelletteet al. Ref. 34; see also Ref. 9, + is for Ref. 18, and the circle denotes the



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O s c i l l a t i n g g r i d s u m m a r y

p r o p a g a t i o n o f t u r b u l e n t f r o n t c h a n g e d d u e t o t h e s h a p e o f t h e

f r o n t t h a t c h a n g e d s u b s t a n t i a l l y

p o l y m e r r e d u c e d e n t r a i n m e n t c o n s t a n t ,

w e c a n e s t i m a t e t h a t p o l y m e r s d i s s i p a t e a b o u t 3 0 % o f t h e

e n e r g y ( i n r e s p e c t t o u

3 /L ) .

t o b e c o n t i n u e d w i t h j e t s : p o l y m e r i n p o l y m e r , p o l y m e r i n

w a t e r , e t c .




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L i d d r i v e n c a v i t y

L i d d r i v e n c a v i t y o w




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M e a n o w d o e s n ' t k n o w m u c h a b o u t p o l y m e r s

Figure 2: Mean flow field, arrows denote the two-dimensional velocity field U, V andbackground color emphasizes the velocity magnitude, normalized with the belt speed, Ub.

(color on line)




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T u r b u l e n t i n t e n s i t y , a u t o c o r r e l a t i o n a n d s p e c t r u m : w e a k e r

t u r b u l e n c e , l a r g e r s c a l e s , r e d u c e d s m a l l s c a l e s

Figure 3: (a) PDF of turbulent intensity , u2 +v20.5/U2+V20.5 (b) velocity autocorre-lation function, (r) = u(x)u(x + r)/u2 and (c) spectrum of the fluctuating velocity,E(k) of water (circles) and polymers (squares).




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P O D ( 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 ) s t r e n g t h e n s t h e

s m a l l s c a l e d e p l e t i o n h y p o t h e s i s

P a r t I . D i s t r i b u t i o n o f e n e r g y a m o n g t h e s c a l e s / b a s i s f u n c t i o n s




35/43


P O D m o d e s

P a r t I I . D e p l e t i o n o f s m a l l s c a l e s , t r a n s f e r o f e n e r g y u p s c a l e , o r g a n i z a t i o n o f p a t t e r n s




36/43


R e y n o l d s s t r e s s - n o t o n l y t h e a m p l i t u d e , b u t a l s o t h e

d i s t r i b u t i o n i m p o r t a n t

Figure 6: Reynolds stress distributions for water (a) and dilute polymer solution (b) flows,respectively. Color level is of the Reynolds stress magnitude, u v, [mm/s]2. The arrows




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P D F o f t h e R e y n o l d s s t r e s s e s a n d T K E p r o d u c t i o n s h o w

u s u a l d e c r e a s e

Figure 7: Probability density functions of (a) Reynolds stresses, uv, (b) turbulentkinetic energy production term, uvSuv. Both quantities are normalized in respect tothe lid velocity, Ub.




38/43


J o i n t P D F o f t u r b u l e n t v e l o c i t y c o m p o n e n t s s h o w

d e c o r r e l a t i o n o f v e l o c i t y c o m p o n e n t s

Figure 8: Joint PDF of the horizontal (u) and the vertical (v) components of turbulent




39/43


J o i n t s t a t i s t i c s o f R e y n o l d s s t r e s s a n d m e a n s t r a i n s h o w

d i e r e n t b e h a v i o u r

Figure 9: Scatter plots and the regression lines of the Reynolds stresses, uv versus themean strain field, Suv. Each quantity is normalized with the respective r.m.s., emphasizingthe distribution and the correlation of water (circles), fresh polymer solution (squares).

Respective correlation values are 0.12, -0.02, respectively.




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M i x e d t y p e q u a n t i t i e s t r a n s f e r i n f o r m a t i o n u p - s c a l e

u v/

y

1

2

/x

v

2 u

2

=

u

x

=

y

w

z

v

t o p p a n e l ( a , b ) :

z

v

b o t t o m p a n e l ( c , d ) :

u v/

y




41/43


L i d - d r i v e n c a v i t y r e s u l t s s u m m a r y

t h e e e c t s i n t h e o r d e r o f i n c r e a s i n g s p a t i a l s c a l e s : t h e

m i x e d - t y p e r e l a t i o n s t h a t r e p r e s e n t c o r r e l a t i o n s b e t w e e n s m a l l

s c a l e s ( v e l o c i t y d e r i v a t i v e s ) a n d l a r g e s c a l e s ( v e l o c i t y ) , s u c h

a s a r e m o d i e d c o n s i d e r a b l y .

S i m i l a r t o t h o s e o f s t r o n g l y r e d u c e d d e r i v a t i v e s o f t h e R e y n o l d

s t r e s s e s ,

/

y a n d a l s o t o d e c r e a s e d R e y n o l d s s t r e s s e s

M o r e o v e r , s t r o n g l y r e d u c e d c o r r e l a t i o n b e t w e e n = a n d S

u v

M o d i c a t i o n o f t h e t u r b u l e n t k i n e t i c e n e r g y p r o d u c t i o n ,

= S

u v


S u m m a r y


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S u m m a r y

P o l y m e r s n e e d L a g r a n g i a n a p p r o a c h a n d s m a l l s c a l e d e r i v a t i v e s

M a t e r i a l l i n e s a r e m o s t s e n s i t i v e s e n s o r s - h i s t o r y e e c t i s

i m p o r t a n t i n p o l y m e r s ( C a u c h y - G r e e n t e n s o r d y n a m i c s )

F l o w c h a n g e s a r e c o m p l e x : a l i g n m e n t s , i n t e r m i t t e n t

r h e o l o g i c a l e e c t s , e t c . P r o b a b l y c a n n o t b e e x p l a i n e d u s i n g

s i m p l e p a r a m e t e r s . M a y b e c a n p r e d i c t g r o s s d r a g r e d u c t i o n .

T u r b u l e n t e n t r a i n m e n t i s c h a n g e d b y r e d u c i n g t h e c u r v a t u r e o f

t h e i n t e r f a c e . N o t c l e a r h o w , b u t l i n k e d t o t h e s t r o n g g r a d i e n t s

a t t h e i n t e r f a c e a n d p r o b a b l y a l i g n m e n t s o f m o l e c u l e s . o n e c a n

e s t i m a t e t h e a m o u n t o f d i s s i p a t i o n b y p o l y m e r s a t 3 0 %


S u m m a r y


43/43

C o n c l u s i o n s

A l t h o u g h t h e s m a l l e s t s c a l e o f t h e p o l y m e r a c t i o n i s n o t c l e a r y e t ,

t h e c h a i n o f a c t i o n a p p e a r s a s : p r o d u c t i o n o f v e l o c i t y d e r i v a t i v e s

d u e t o m i s a l i g n m e n t s w i t h t h e e i g e n f r a m e o f s t r a i n e l d , v e l o c i t y

d e r i v a t i v e s , m i x e d - t y p e q u a n t i t i e s u , u v /y , d e c o r r e l a t i o n o f v e l o c i t y c o m p o n e n t s , r e d u c t i o n o f m o m e n t u m t r a n s f e r ,

m i s a l i g n m e n t w i t h t h e m e a n - r a t e - o f - s t r a i n , S

u v

, h i n t e d t u r b u l e n t

k i n e t i c e n e r g y p r o d u c t i o n a n d c o n s e q u e n t d r a g r e d u c t i o n ( l e s s

e n e r g y i s s p e n t o n t u r b u l e n c e p r o d u c t i o n ) .

polymers bounded shear flows v2

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