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Experim ents in Fluids 14, 133 139 (1993)
x p e r i m e n t s i n F l u i d s
9 Springer-Verlag 1993
Study o f vortex breakdow n by particle trackin g velocimetry PT V )
P a r t 2 : S p i r a l t y p e v o r t e x b r e a k d o w n
C B r i i c k e r
Aerod ynam isches Ins t i tu t , RW TH Aachen , Wiil lnerst r . zw 5 and 7 , D-5100 Aachen, F .R. Ge rma ny
Abstract
The spi ral -type breakd ow n of a s lender vor tex was quan-
ti tatively investigated with PTV . Multiexposed pictures of the i l lu-
minated m er id ional midplane were processed to obta in the ins tanta-
neou s 2-D velocity field and v ortieity distribution. Th e periodic
chan ge of flow patterns with respect to t ime is clearly show n in a
time-series o f pictures. T he 2-D velocity fields con taine d a stag na-
t ion point in the midp lane located outside of the center l ine . A ddi-
tionally i t was observed, that this poin t rotates ar ou nd the centerline
in the same way as the outer f low. A com par ison wi th m easurements
of a "bubble"- type break dow n indica tes a s t rong s imi lari ty to the
spiral-type breakdown. The results reveal that the slope, winding
and d iam eter of the spiral vo rtex-c ore determine the different ob-
servable forms. The first part of the deflected vo rtex-co re nea r the
breakdown point causes an asymmetr ic backf low due to induct ion ,
the s t rength of which d epends on the s lope of the def lec ted v or tex-
core. This is responsible for the radial distance between stagn ation
point and center line . In case of the observed bubble- type brea kdow n
the spiral is compressed which results in a stable stagnation point at
the centerline.
1 I n t r o d u c t i o n
T h e n e a r l y a x i s y m m e t r i c b u b b l e a n d n o n - a x i s y m m e t r i c sp i-
r a l - t y p e b r e a k d o w n o f s l e n d e r v o r t i ce s a r e th e t w o p r e d o m -
i n a n t f o r m s a n d w e r e f i r s t r e p o r t e d a n d c l a s s i f i e d b y L a m -
b o u r n e a n d B r y e r (1 96 1 ). S u b s e q u e n t l y a l o t o f t h e o r e t i c a l
a n d e x p e r i m e n t a l i n v e s t i g a t i o n s h a v e b e e n c a r r ie d o u t a s
r e v i e w e d b y S a r p k a y a ( 1 9 7 1 ) , H a l l ( 1 9 7 2 ) , L e i b o v i c h ( 1 9 7 8 )
a n d E s c u d i e r (1 98 8 ). N u m e r i c a l s i m u l a t i o n s o f t h e t h r e e - d i -
m e n s i o n a l t i m e - d e p e n d e n t f lo w in t h e b r e a k d o w n r e g io n
w e r e p r e s e n t e d b y , e.g ., K r a u s e ( 1 9 90 ), B r e u e r ( 1 9 9 1) a n d
S p a l l e t a l. (1 99 0 ). I n e x p e r i m e n t a l r e s e a r c h f i rs t q u a n t i t a t i v e
r e s u lt s o f t h e b u b b l e - t y p e b r e a k d o w n w e r e o b t a i n e d w i t h
L a s e r D o p p l e r A n e m o m e t r y ( L D A ) b y F a l e r a n d L e i b o v i c h
( 19 78 ), E s c u d i e r a n d Z e h n d e r ( 19 82 ). N a k a m u r a a n d U s h i d a
( 1 9 8 7 ) p r e s e n t e d f i r s t q u a n t i t a t i v e r e s u l t s c o n c e r n i n g t h e s p i -
r a l - t y p e b r e a k d o w n . R e c e n t l y , B r i i c k e r a n d A l t h a u s ( 1 9 9 2 )
p u b l i s h e d q u a s i t h r e e - d i m e n s i o n a l i n s t a n t a n e o u s f l o w f ie ld
m e a s u r e m e n t s o f t h e b u b b l e - ty p e b r e a k d o w n w i th c o m b i n e d
P T V a n d l ig h t s h e e t s c a n n i n g . T h e r e s u l ts c l e a r l y in d i c a t e d
a s in g l e v o r t e x - r i n g a t t h e d o w n s t r e a m p a r t o f t h e b u b b l e
a n d a s t a g n a t i o n p o i n t a t t h e c e n t e r l i n e . T h e l o w p e r i o d i c
f l u c t u a ti o n s i n s id e t h e b r e a k d o w n r e g i o n m a y b e e x p la i n e d
b y t h e g y r a t i n g b e h a v i o r o f th e s l i g h tl y t i lt e d v o r t e x - r i n g a s
f i r s t d e s c r i b e d b y S a r p k a y a ( 1 9 7 1 ) .
I n t hi s w o r k t h e n o n - a x i s y m m e t r i c s p i r a l- t y p e v o r t e x
b r e a k d o w n h a s b e e n s t u d i e d w i t h P T V . V i s u a l i z i n g t h e v o r -
t e x a x i s w i t h a d y e f i l a m e n t , " t h e s p i r a l f o r m i s m a r k e d b y a
k i n k i n t h e f i la m e n t , f o ll o w e d b y a c o r k s c r e w - s h a p e d t w i s t-
i n g o f th e d y e " ( L e i b o v i c h 1 9 78 ). T h e s e n s e o f t h e s p i r a l ' s
w i n d i n g i s o p p o s i t e t o t h e s e n s e o f b a si c f l o w r o t a t io n . T h e
u p s t r e a m p a r t o f t h e b r e a k d o w n r e g i o n p e r i o d i c a l ly r o t a t e s
a r o u n d t h e t u b e a x i s w i t h a r e g u l a r f r e q u e n c y , a s m e a s u r e d
b y N a k a m u r a a n d U s h i d a ( 19 87 ). " T h e s p i ra l f o r m in p a r ti c -
u l a r r e v ea l s t h e su d d e n e s s o f b r e a k d o w n , a n d s u g g e s ts t h e
o c c u r r e n c e o f a s t a g n a t i o n p o i n t a t t h e v o r t e x a x i s " (E s c u d -
i e r 19 8 8) . B e c a u s e t h e r e i s n o e x p e r i m e n t a l i n f o r m a t i o n
a b o u t t h e ti m e - d e p e n d e n t i n t e r n a l f lo w in th e b r e a k d o w n
r e g i o n u p t o n o w , t i m e - s e r i e s o f t h e m e r i d i o n a l f l o w f ie l d o n
t h e t u b e a x i s w e r e t a k e n w i t h t h e m u l t i e x p o s u r e t e c h n i q u e
a n d a r e e v a l u a t e d w i t h P T V .
2 E x p e r i m e n t a l a r r a n g e m e n t
T h e s p i r a l - t y p e b r e a k d o w n w a s i n v e s t i g a t e d i n t h e s a m e
v e r t i c a l l o w s p e e d w a t e r - c h a n n e l a s u s e d b y B r t i c k e r a n d
A l t h a u s ( 1 9 9 2 ) . E v e n t h e i n f l o w c o n d i t i o n s w e r e t h e s a m e ,
s h o w n i n th e i r w o r k i n F i g . 2 ( se e B r / i c k e r a n d A l t h a u s 1 9 92 ).
T h e s e c o n d i t i o n s l e d t o a f lo w s t a g e w h e r e a n e a r l y p e r i o d i c
c h a n g e f r o m s p i r a l t o b u b b l e - t y p e b r e a k d o w n a n d v i c e v e r s a
w a s o b s e r v a b l e . T h i s o c c u r r e d a l s o i n t h e e x p e r i m e n t s o f
s e v e r a l a u t h o r s , e . g . L e i b o v i c h ( 1 9 8 8 ) . T h e v o r t e x a x i s w a s
v i s u a l i z e d w i t h f l u o r e s c e i n d y e . I n a d d i t i o n , s m a l l t r a c e r p a r -
t i cl e s ( m e t a l l ic c o a t e d p a r t i c l e s T S I 1 0 0 8 7 ) w e r e a d d e d t o t h e
f l o w u p s t r e a m o f th e s w i r l -g e n e r a t i n g g u i d e v a n e s .
3 R e s u l t s a n d d i s c u s si o n
A " c u t " t h r o u g h t h e b r e a k d o w n r e g i o n i n t h e m e r i d i o n a l
m i d p l a n e i s s h o w n i n F i g . 1 ( t he p i c t u r e b e l o n g s t o t h e t i m e -
s e r ie s i n F i g . 3 w h e r e i t s e v a l u a t i o n i s s h o w n i n F i g . 3 a ).
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134
Fig. 1. Multiexposed picture o f the particles an d fluorescein in a
light-sheet in the midplanc of the channel; the picture shows the
spiral -type breakdown; the flow is coming from left; the dark
vertical stripe o n the left side represents the do wns tream edge of the
diffusor; the 4 lightened marks in the picture corners are used for
calibration
Fig. 2. Gross structure of the instantaneous flow field in Fig. 1;
recon struct ed from the intersection lines of the streamsurfaces with
the illuminated midplane and fro m the particle p aths in Fig. I. The
scale D is used as a characteristic diameter for normalizing the
coordinates
In the sp i ra l- type case the b reak dow n po in t i s de f ined a s
the loca t ion where the vor tex ax i s , m arked wi th f luoresce in
dye , depa r t s f rom the cen te r l ine . In F ig . 1 , the b reakdown
takes p lace a t the downs t r eam end o f the t ranspa ren t d i ffu -
sor . The s t reaky pa t t e rn o f the pa r t i cle ' s d i s t r ibu t ion i s due
to the intersect ion of the rol led up s treamsurface s with the
l ight-sheet. A reco nstr uct i on of the gross f low fie ld is show n
in F ig . 2 , ob ta ined f rom the pa r t i c le pa ths and s t reaks in
Fig. 1.
The l ines give in fo rm a t ion a bou t the ins tan tan eous d i rec -
t ional f ie ld of the projected veloci ty vectors in the i l luminat-
ed midplane. On the left s ide of F ig. 2 the divergin g in fo w is
shown. The f luorescein f i lament is s t ra ight and c losely sur-
roun ded by s ligh t ly d ive rg ing f low. Even downs t r eam of the
breakdown po in t , the f luoresce in f i l am ent i s s t i l l em bedded
in the s t reaky pa t t e r n o f the pa r t i c le 's d i s t r ibu t ion . The pa ths
of pa r t i c le s nea r b re akdo wn show s t rong ly dec reased ax ia l
veloci t ies , whereas the radia l veloci t ies are responsible for
the sudden f luoresce in d i sp lacem ent f rom the tube ax i s .
The re fore , the b rea kdo wn po in t , a s de f ined above , is no t a
s tagna t ion po in t where a l l com pon en ts o f the ve loc i ty a re
zero. In fact , F ig. 1 and Fig. 2 show the exis tence o f a s tagna-
t ion po in t in the m idp lane bu t
p rt
f rom the cen te r l ine and
oppos i t e to the curved d i sp laced f luoresce in . At the s t agna -
t ion po in t the appro ach in g f low s t rong ly sp reads ou t , a f fec t-
ed by the reversed f low region.
A p p r o x i m a t e l y a t t h e m a x i m u m d i s p l a c e m e n t f r o m t h e
centerl ine , the f luorescein f i lament leaves the i l luminated
m idp lane and reappea rs a t the uppe r pa r t o f F ig. 1 . The
part ic le paths reveal several vort ices which can be interpret-
ed as the t race of the vortex core , rol led u p into a spira l .
However, as an unexpected resul t the f luorescein f i lament
does no t represen t the cen te r o f the vor tex core in the l igh t
sheet . This wil l be discussed la ter in deta i l . In order to get
m ore in fo rm a t ion conce rn ing the o f ten m ent ioned pe r iod ic
f low cha rac te ri s t i cs , the t em p ora l deve lopm ent o f the f low in
the m er id iona l m idp lane was recorded wi th a m ul t i exposure
techn ique (F ig. 3 a i) . The fram e frequenc y was 1.5 fram es/s
with a l ight pulse frequency of 12.5 Hz.
The da ta p roces s ing o f the m ul t i exposed p ic tu re s was
carried out as described by Brt icker and Althaus (1992) for
the bubb le - type b reakdown. The x -ax i s was a l igned wi th the
vortex-ax is of the inflow, with i ts origin a t the axia l bre ak-
down loca t ion . Al l axes have been nor m a l ized wi th a cha rac -
te r is t ic d iam e te r D of the f low; fo r sp i ra l - type b reak dow n the
d iam e te r o f m ax im u m f low d i l a ta t ion in the b reakdo wn re -
gion, defined as sho wn in F ig. 2 , was ch osen. This is com -
pa rab le wi th the def in i t ion o f D fo r bubb le - type b reak dow n
as the m axim um bubble d iam e te r (Brf icke r and Al thaus
1992). Here , D approximately equals the tube radius .
The f low in the b reakdown reg ion pe r iod ica l ly ro ta te s
aro un d the ccnterl ine; one perio d is show n in F ig. 3 a i . The
general f low s tructu re in F ig. 3i is s imilar t o F ig. 3 a which
indicates the beg inning of a new period. At the semipe riod in
Fig. 3e the f low fie ld looks l ike s imply turned aro un d the
cen te r linc a bou t 180- in com p ar i so n to F ig . 3 a . The dura -
t ion T of one pe riod can b e es t imated from Fig. 3 and gives
T = 5.3 sec.
The spa t i a l d i s t r ibu t ion o f the ca lcu la ted vo r t i c i ty com -
po nen t a lso shows perio dical beh avior. F irs t in F ig. 3 a , there
is a c lockwise rota t ing vortex above the centerl ine with i ts
m ax im u m nega t ive vor t i c ity a t the cen te r o f ro ta t ion . On the
o the r s ide o f the cen te r l ine , the re a re two co un te rc lockwise
rota t ing vort ices with pos i t ive vort ic i ty on the left and r ight
s ide o f the uppe r vor tex . These vor tex cen te rs m ar k the t race
of the sp i ra l fo rm ed vor tex-core . The g ros s vor tex d i s t r ibu-
t ion s lowly m oves dow ns t ream , so tha t in F ig . 3 b the lower
r igh t vor tex has a l ready l e f t the m easur ing p lane . Nea r the
breakdown po in t a new nega t ive vor t i c i ty f i e ld beg ins to
grow above the cen te r l ine , con t inuous ly inc reas ing in s i ze
and s lowly m ovin g fu r the r d own s t ream (F ig . 3 b i). The
same process takes place below the centerl ine with a reversed
s ign of vort ic i ty (F ig. 3 f- i ). S imilar ob serv at ion s w ere mad e
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1 3 5
- I 9 I , I
b
9 i . .. . _ _ % _ __ i . . . .
o., : : : : : : : : : : : : : : : : : : : : :
. / / l l x ~
1 1 1 i -
- ~ - . ~ r . . . . . . . ~ . . . . . .
o
I t/ " " . . . . . . x ~ l ~ l | l . . . . .
* t ' ~ \ \ l ~ l ~ ' , ~ l t l t t l l l l l t , , . .
~ N N X ~ X \ k ' . . / I / / / / I / / / / . . . . . .
-0 .5
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1 , , . . . . .
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- -. ~ .~ t - ' '
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. < . . . , , , , ~ . ~ . . r _ _ . ~ . , . . . . . . . .
~ ~
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0
~ s l t / ' . , ~ l l T / 7 l l t t / / .
~ . , t t t t t , , . . , b b ~ , . . . . . . : ~ . . . . . . . . . . . . . . . . . , . . . .
~ . ~ - ' , ~ . . ~ : : ~ - - . r . . . . ~ ' t : : . . . . . .
. o . - ~ < < < < < < < < ~ - - . ~ - . -0.5
0 . 5 ~
. . . . . . . ~ : / / ~ - ' - - - - - - . - ~ - - - . - - ~ -
~ ' , ' > ~
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~ . . . . . . . .. .. . : ~ ' ~ - - ~ . . . . ' : : ~ : ~
- , -. ~ \~ * * ' 5 : / I / I I / i ~ i l
- 0 . 5 ~ , ~ z < ~ x . ~ ~ - - ~ - - -
0 0 .5 1 0 0 .5 1
0 . 5
o
- 0 . 5
: : : : : : : : : : : : : : : : : : : : : : : : :
-
, ~ . . .' I I / /, : / l l / / / : ~ . ~ l l ~ l y L ~ - : -
~ J / l l l l l / : [ , l l l l l . / : : / : /
0 0 . 5 I
0 0 . 5
x / D x / D
F i g . 3 a - i . T i m e - s e r i e s o f t h e m i d p l a n e c r o s s s e c t i o n o f t h e s p i r a l-
t y p e b r e a k d o w n ; t h e t h i c k e r l i n e s s h o w t h e f l u o r e s c e i n c o n t o u r s i n
t h e p i c t u r e s ; t h e d i a m e t e r D i s t h e m a x i m u m d i l a t a t i o n o f t h e b r e a k -
d o w n r e g i o n a n d n e a r l y e q u a l s t h e t u b e r a d i u s ; e a c h f i g u r e i s d i v i d e d
i n t o 2 p a r ts : I 2 - D v e l o c i ty f i e ld (m a x i m u m - v e l o c i t y 2 c m / s ) ; I I
c o n t o u r s o f c o n s t a n t z - v o r t i c i t y ; c o n t o u r l e v e l s l ie i n t h e r a n g e f r o m
m i n i m u m ( c r o s s ) t o m a x i m u m ( c ir c l e ) i n 1 3 i n t e r v a l s ; a - 1 . 6 2 / s
- 1 . 4 4 / s ; b - l . 5 3 / s - l . 3 8 / s ; c - 1 . 4 6 / s - 1 . 7 6 / s ; d - 1 . 5 9 / s
- 1 . 9 8 / s ; e - 1 . 4 6 / s - 1 . 5 5 / s ; f - 1 . 4 8 / s - 1 . 7 2 / s ; g - 1 . 5 4 / s - 1 . 4 / s ;
h - 1 . 8 2 / s - l . 1 6 / s ; i - 1 . 8 6 / s - l . 4 8 / s
x / D
x / D
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1 3 7
_-:-
0
-0 .5 ~ . . , , . , .~_
[ : : - - - - - - -=- - - - -~-~- : - - 2_-77 _ , ,
0 . 5
- 0 . 5
b
0 . 5 - - ~
o.~~ ~ ~ ~
0 0 .5 1
x/D
x
~ l l r
x \ I
, - : : : ~ - : : - : ~ - - - - : i ~ : - : - - - - - - - ~
: Z --.?L
d 0 5 : 1 , :
. . . . i - - : : -~ . - ; :~ . i : . ~ -
~ : . . . . . . . . -'-.- - - 9 . ~ . : . ~ 9
.o
0 0 .5 1 0 0 .5 1 0 0 .5 1
x / D x / D x / D
Fig. 7 a- f . Time-series of the midplane cross section of the unsta ble bu bble ; the thicker l ines show the fluorescein con tours in the pictures;
the bubble diameter D nearly equals the tub e radius; each figure is divided in 2 pa rts: I 2-D velocity field (maximum -velocity 2 cm /s);
II contou rs of constan t z-vortici ty; c onto ur levels l ie in the range from m inimum (cross) to maxim um (circle) in 13 intervals; a -1 .7 1/ s
- 1 .6 9 /s ; b -1 . 6 1 / s - l . 7 1 / s ; e - l . 8 4 / s - l . 6 8 / s ; d - l . 8 4 / s - t . 6 t / s ; e - l . 9 7 / s - l . 2 5 / s ; f - 2 . 0 / s - l . 1 6 / s
t h e f ir s t c u r v a t u r e o f t h e d y e f i l a m e n t o n t h e o t h e r s i d e o f t h e
c e n t e rl in e . T h e v e l o c i t y f ie l d s i n F i g . 3 a r e m a i n l y d e t e r m i n e d
b y t h e i n d u c i n g e f fe c t o f t h e s p i ra l . T h e p o s i t i o n o f t h e s t a g -
n a t i o n p o i n t d e p e n d s o n t h e d i re c t i o n o f i n d u c e d b a c k f l o w .
P r i m ar i l y , t he s l op e o f the f i r s t cu r v e o f the sp i r a l i n f l uences
t h e r a d i a l d i s t a n c e b e t w e e n s t a g n a t i o n p o i n t a n d c e n t e r l i n e .
A p e r s p e c t i v e 3 - D v i e w o f t h e p o s i t i o n o f t h e s t a g n a t i o n
po i n t i n t he f l ow f i e ld is g i ven in F i g . 5 . Based up on qua l i t a -
t i ve ob se r v a t i on s S t auf en b i e l e t a l. (1987) a l so a r gu ed t h a t
t h e s t a g n a t i o n p o i n t i s l o c a t e d o u t o f t h e v o r t e x c e n t e r. T h i s
is a n i m p o r t a n t d i ff e re n c e f r o m b u b b l e - t y p e b r e a k d o w n ,
w h e r e t h e s t a g n a t i o n p o i n t i s l o c a t e d o n t h e c e n t e r l in e a s
d i scus sed by Br i i cke r and Al t haus ( 1992) .
W i t h i n t e n t t o f u r t h e r s t u d y t h i s i n t e r e s t i n g d i f f e r e n c e
b e t w e e n b o t h m o d e s , a ti m e - se r ie s o f a n u n s t a b l e b u b b l e -
t y p e b r e a k d o w n w a s r e c o r d e d w i t h t h e m u l t i e x p o s u r e t ec h -
n i q u e . T h i s m o d e i s c h a r a c t e r i z e d b y t h e t y p i c a l b u b b l e - t y p e
a p p e a r a n c e , w h e r e a s d i s t i n c t p e r i o d i c m o t i o n s a r e o b s e r v -
a b l e i n s i d e t h e b u b b l e . H o w e v e r , t h i s b u b b l e i s m o r e s t a b l e
i n a x ia l p o s i t i o n t h a n t h e a x i s y m m e t r i c f o r m . F i n e q u a l i t a -
t iv e s tu d i es o f th i s b r e a k d o w n m o d e w e r e m a d e b y S t o j an o f f
( 19 9 1) i n a f lo w c h a n n e l s i m i l a r to S a r p k a y a ' s . T h e a p p e a r -
a n c e o f t h e u n s t a b l e m o d e s e e m s t o b e a n i n t e r m e d i a t e s t a g e
b e t w e e n a x i s y m m e t r i c b u b b l e - a n d a s y m m e t r i c s p i ra l - ty p e
b r e a k d o w n . A m u l t ie x p o s e d p i c t u r e o f th e m e r i d i o n a l m i d -
p l a n e i s s h o w n i n F i g. 6 ( th e f lo w c o n d i t i o n s a r e t h e s a m e a s
in Fig. 3).
T h e b u b b l e s h a p e i s c le a r l y s h o w n b y t h e f lu o r e s c e i n
c o n t o u r . H o w e v e r , a s a n u n e x p e c t e d r e s u l t , t h e p a r t ic l e p a t h -
l i n e s m a r k t h r e e s e p a r a t e d v o r t i c e s . U n d e r t h e a s s u m p t i o n
t h a t t h e v o r t i c e s m a r k t h e t r a c e o f a s p i r a l f o r m e d v o r t e x
c o r e , Fi g . 6 r a t h e r s h o w s a s p i r a l - t y p e b r e a k d o w n t h a n a
b u b b l e - t y p e o n e . T h e c o m p l e t e t i m e - s e r i e s i s g i v e n i n F i g .
7 a - f , w h e r e F i g . 7 c r e p r e s e n t s t h e e v a l u a t i o n o f F ig . 6.
C o m p a r i n g t h e v o r t i c i ty c o n t o u r s o f F i g . 7 a a n d F i g . 3 b
o b v i o u s l y y i e ld s s i m i l a r f e a tu r e s . M o r e o v e r , t h e s a m e t e m p o -
8/17/2019 Vortex Breakdown Ptv
6/7
138
r a l d e v e l o p m e n t c a n b e o b s e r v e d ; i n t h e s a m e w a y a s i n
F i g . 3 , a c l o c k w i s e r o t a t i n g v o r t e x i n t h e u p p e r p a r t o f
F i g . 7 a d e v e l o p s n e a r t h e b r e a k d o w n p o i n t , c o n t i n u o u s l y
i n c r e a s i n g i n s iz e, s lo w l y m o v i n g d o w n s t r e a m a n d f i n a l l y
r e a c h i n g t h e a x ia l p o s i t i o n o f m a x i m u m b u b b l e d i a m e t e r
( F ig . 7 f ). I n t h e m e a n t i m e , t h e d o w n s t r e a m v o r t e x m o v e s o u t
o f t h e m e a s u r i n g p l a n e ( F i g . 7 d ) w h i c h i s a l s o o b s e r v e d i n
F i g . 3 d - f c o n c e r n i n g t h e s p i r al - t y p e b r e a k d o w n . T h e
s e m i p e r i o d i s r e a c h e d a t F i g . 7 d ; t h e s p a t i a l d i s t r i b u t i o n o f
t h e v o r t i c it y c o m p o n e n t f ig u r es v o r t i c es o f o p p o s i t e s ig n
a l t e r n a t i n g a r o u n d t h e c e n t e r l i n e . I n c o m p a r i s o n t o F i g . 7 a
t h e s t r u c t u r e i s s i m p l y t u r n e d a r o u n d t h e c e n t e r l i n e a b o u t
1 8 0 ~ B o t h c o u n t e r c l o c k w i s e r o t a t i n g v o r t ic e s a r e n o t c l e a r l y
s e p a r a t e d ; h o w e v e r , t w o l o c a l m a x i m a o f v o r t i c it y a r e d i s ti n -
g u i s h a b l e . T h e n e x t p e r i o d b e g i n s o n e t i m e - s t e p l a t e r t h a n
F i g . 7 f , wh e r e a l r e a d y a s m a l l pos i t i ve v o r t i c i t y f i e ld a r i s e s a t
t h e b u b b l e f r o n t a b o v e t h e c e n t e r l i n e . A t t h e o t h e r s i d e t h e
g r o w t h o f v o r t i c i t y is se e n i n F i g . 7 c - d s t a r t i n g a t t h e t o p o f
t h e b u b b l e . S u b s e q u e n t l y t h e v o r t i c i t y r e g i o n i s e l o n g a t e d
a l o n g t h e b u b b l e c o n t o u r a n d c o n t i n u o u s l y i n c r e a s e s i n
s t r e ng t h .
T h e m o v e m e n t o f th e f l u or e sc e in c o n t o u r s a t t h e d o w n -
s t r e a m p a r t o f t h e b u b b l e s e e m s t o b e p e r i o d i c al , t o o . T h e
f l u o re s c e i n ta i ls a t t h e e n d o f t h e b u b b l e , e n v e l o p i n g t h e v o r -
t i c e s , a r e s we p t a wa y i n t he s a me wa y a s t he vo r t i c e s i t s e l f .
A l t h o u g h t h e f l u o r e s c e i n i n d i c a t e s a b u b b l e s h a p e , t h e
s i m i l a ri t ie s o f F i g . 3 a n d F i g . 7 l e a d u s t o t h e c o n c l u s i o n t h a t
a s p i r a l f o r m e d v o r t e x - c o r e p r e c e s s e s a r o u n d t h e c e n t e r l i n e
i m p l y i n g a s p i r a l- t y p e b r e a k d o w n . H o w e v e r , s o m e i m p o r -
t a n t d i f f er e n ce s c a n b e p o i n t e d o u t : d u r i n g t h e w h o l e p e r i o d
i n F i g . 7 t h e v e l o c i t y f i e l d i n d i c a t e s t h e s t a g n a t i o n p o i n t on
the centerline
a t th e t o p o f t h e b u b b l e . M o r e o v e r , t h e v o r t i c e s
i n F i g . 7 a re e l o n g a t e d s t r e am w i s e w h e r e a s t h e y a r e m o r e
s y m m e t r i c i n F i g . 3 . T h i s m a y b e a n i n d i c a t i o n f o r a m o r e
i n t e n s e s t r a i n - r a t e a l o n g t h e b u b b l e s h a p e . T h e s p i r a l d i a m e -
t e r , m e a s u r e d a s t h e m e a n v a l u e o f r a d i a l d i s t a n c e b e t w e e n
t he vo r t e x c e n t e r s , i s y D = 0 . 65 i n F i g . 7 a nd y D = 0 .5 in
F i g . 3 . A d d i t i o n a l l y , t h e s l o p e o f t h e s p i r a l s a s d e f i n e d i n
F i g . 4 is d i f fe r e n t ; in F i g . 7 it a m o u n t s t o 6 0 ~ m e a s u r e d t o t h e
c e n t e r l i n e i n c o m p a r i s o n w i t h 5 0 ~ i n F i g . 3 . C o n s i d e r i n g t h a t
t h e u n s t a b l e b u b b l e - t y p e b r e a k d o w n i s a n i n t e r m e d i a t e
s t a g e b e t w e e n t h e s p i r a l - a n d a x i s y m m e t r i c b u b b l e - t y p e
b r e a k d o w n , t h e c h a n g e o f m o d e s m i g h t b e t h e r e su l t f r o m
c h a n g e s o f th e s p i r a l ' s w i n d i n g a n d s l o p e w h i c h g i v e s u s a
n e w w o r k i n g h y p o t h e s i s f o r t h e r e a s o n o f d i ff e re n t m o d e s o f
v o r t e x b r e a k d o w n . T h i s h y p o t h e s i s i s c o n f i r m e d b y q u a l i t a -
t iv e o b s e r v a t i o n s i n s o m e v i d e o - r e c o r d i n g s ( se e A l t h a u s a n d
K r a u s e ( 19 9 0) ), t o o . A s a c o n s e q u e n c e o f th e h y p o t h e s i s ,
F i g . 6 a n d F i g . 7 s u g g e s t t h a t t h e b a s i c b r e a k d o w n t y p e i s a
s p i ra l t h a t m a y u n d e r g o c h a n g e s i n w i n d i n g a n d s l o p e w h i c h
l e a d t o d i f f e r e n t v i s u a l a p p e a r a n c e s , a d d i t i o n a l l y d e p e n d i n g
o n t h e k i n d o f f l o w v i s u a l i z a t io n .
W h y d o e s t h e f l u o r es c e i n c o n t o u r i n F i g . 6 a p p e a r a s a
b u b b l e s h a p e a l t h o u g h t h e i n t e r n a l f l o w i n d i c a t e s a s p i r a l
c o r e ? T h i s c a n b e e x p l a i n e d i n t h e f o l l o w i n g w a y : I f th e
s p i r a l i s c o m p r e s s e d , t h e i n d u c e d b a c k f l o w b e c o m e s m o r e
a x i s y m m e t r i c . T h is l e a d s t o a n a p p r o a c h o f t h e r o t a t i n g
s t a g n a t i o n p o i n t t o t h e c e n t e r l i n e . A t a c h a r a c t e r i s t i c s l o p e
o f t h e s p i r a l th e s t a g n a t i o n p o i n t i s l o c a t e d n e a r t h e c e n t e r -
l in e s o t h a t t h e f l u o r e s c e i n f i l a m e n t i s n o l o n g e r d e f l e c t e d a s
a w h o l e b u t s p r e a d s a w a y a t t h e s t a g n a t i o n p o i n t a n d f i l l s u p
t h e l a y e r b e t w e e n o u t e r f l o w a n d i n d u c e d b a c k f l o w . T h i s
r e g i o n i s c o n t i n u o u s l y f i l l e d u p w i t h f l u o r e s c e i n f r o m t h e
c e n t e r l i n e a n d f i n a l l y a b u b b l e - l i k e s h a p e i s v i s u a l i z e d . F u r -
t h e r c o m p r e s s i o n o f th e s p i r a l m a y r e s ul t i n a n e a r l y a x i s y m -
m e t r i c b u b b l e a s s h o w n i n B r i i c k e r a n d A l t h a u s ( 1 9 9 2 ) . T h e
a z i m u t h a l v o r t i c i t y i s t h e n c o n c e n t r a t e d i n a v o r t e x - r i n g l i k e
s t r u c t u r e w h e r e i t i s n o l o n g e r p o s s i b l e t o e x p e r i m e n t a l l y
o b s e r v e c l e a r l y s e p a r a t e d v o r t i c e s . F o l l o w i n g t h e h y p o t h e s i s ,
t h e t o t a l v o r t i c i t y i s m o r e c o n c e n t r a t e d i n a z i m u t h a l d i r e c -
t i o n i n c o m p a r i s o n t o a n e a r l i er s t a g e o f t h e p r o c e s s d u e t o
t h e c o m p r e s s i o n . T h i s l e a d s t o a s t r o n g e r i n d u c e d b a c k f l o w ,
m a y c h a n g e t h e f or c e s p r o v i d i n g e q u i l i b ri u m a n d h e n c e r e -
s u lt s in u p s t r e a m m o v e m e n t o f t h e b u b b l e , a s o f t en o b s e r v e d .
Conc lus ions
W i t h p a r t i c l e t r a c k i n g v e l o c i m e t r y ( P T V ) f i rs t q u a n t i t a t i v e
r e s u l ts o f t h e 2 - D f l o w f ie ld w i t h i n t h e b r e a k d o w n r e g i o n o f
t h e s o c a l le d s p i r a l -t y p e m o d e w e r e o b ta i n e d . O n e o f t h e
r e s u l t s i s t h a t t h e s t a g n a t i o n p o i n t i s n o t l o c a t e d o n t h e
c e n t e r l i n e a s o f t e n s u p p o s e d b u t r o t a t e s a r o u n d i t. I f t h e
m a r k e r i s n o t e x a c t l y a l i g n e d w i t h t h e v o r t e x a x i s , i t w i l l
a d d i t i o n a l l y r o t a t e a r o u n d t h e s p ir a l fo r m e d v o r t e x - a x i s a n d
m a y p r o d u c e a d i s t u r b e d p a t t e r n a fe w t u rn s l a t er . I n v e s t ig a -
t i o n s o f a n u n s t a b l e b u b b l e - t y p e m o d e y i e l d a s p i r a l
f o r m e d v o r t e x - c o r e i n s id e t h e b u b b l e , w h i c h i n d i c a te s t h a t
t h is m o d e a c t u a l l y i s a s p i r a l - t y p e m o d e . O n e c a n e x p l a in
t h e d i f f e re n c e o f b o t h v i s u a l i z e d f o r m s b y c h a n g e s o f s p i r a l 's
s l o p e a n d w i n d i n g . I n c a s e o f t h e u n s t a b l e b u b b l e - t y p e m o d e
t h e c o m p r e s s e d s p i r a l i n d u c e s a m o r e a x i s y m m e t r i c b a c k -
f l o w a n d t h e r e f o r e t h e s t a g n a t i o n p o i n t i s l o c a t e d c l o s e r t o
t h e c e n te r l i n e . I f t h e s t a g n a t i o n p o i n t a p p r o a c h e s t h e c e n t e r -
l i n e , t h e f l u o r e s c e i n c o n t i n u o u s l y f e e d s t h e l a y e r b e t w e e n
o u t e r f l o w a n d i n d u c e d b a c k f l o w a n d t h i s f i n a ll y l e a d s t o t h e
b u b b l e - l i k e a p p e a r a n c e . O v e r a l l , th e i n d u c t i o n e f fe c t p l a y s
a n i m p o r t a n t r o l e fo r th i s k i n d o f v o r t e x d y n a m i c s . I n f u tu r e
w o r k t h e r e a s o n s f o r t h e c h a n g e o f t h e s p i r a l f o r m w i ll b e
i nve s t i ga t e d .
References
Althaus, W.; Krause, E. 19 90: Flow v isualization of flows with con -
centra ted vor tic ity . P rogress Rep or t for the Com miss ion of the
Eur ope a n Communi t i e s
Breuer, M. 1991: Num erische L6s ung der Navier-Stokes-Gleichun-
gen ffir dreidimensionale inkompressible instationfire Str6m un-
gen zur Simulation des Wirbelaufplatzens. Dissertation RW TH
Aachen
Brf lcker, Ch. ; Al thaus, W. 1992: Study o f vortex brea kdow n by
particle tra ckig velocimetry {PT V). Pa rt l : Bubble-type vortex
break dow n. Exp . Fluids 13, 339 349
8/17/2019 Vortex Breakdown Ptv
7/7
Escud ier , M. P.; Zehn der, N , 198 2: Vortex-f low regimes. J . Flu id
Mech . 115, 105 121
E s c u d ie r , M . 1 9 8 8 : V o r t ex b r e a k d o w n : O b s e r v a t i o n s a n d e x p l a n a -
t ions . P rog . Aerospace . Sc . 25 , 189-229
Fa le r , J. H . ; Le ibov ich , S . 1978 : An exp er imen ta l ma p o f the in te rna l
s t ruc tu re o f a vor tex b reakdo wn. J . F lu id Mech . 86 , 313-3 35
Hal l , M. G . 1972 : Vor tex b reakdown. Ann . Rev . F lu id Mech . 4 ,
195 217
K r a u s e , E . 1 9 9 0 : V o r t ex b r e a k d o w n : P h y s i c a l is s u e a n d c o m p u t a -
t iona l s imula t ion . 3 . In t . Cong . o f F lu id Mech . , Ca i ro , Egyp t
Lam bourne , N . C . ; Bryer , D . W. 1961: The burs t ing o f l ead ing-edge
vor t i ces some obse rva t ions and d i scuss ion o f the phenom enon .
A e r o n a u t i c a l R e s e a rc h C o u n c i l, R e p o r t s M e m o r a n d a R . M .
No. 3282 , pp . 1 36
Le ibov ich , S . 1978 : The s t ruc tu re o f vor tex b reakdo wn. Ann . Rev .
Flu id Mech. t0 , 221 246
Na kam ura , Y .; Uch ida , S . 1987 : Severa l appro aches to the s tudy o f
vor tex b reakdo wn. 2 . In t . Co l l . on vor t i ca l f lows , Apr i l 6 . 7 .
1987 , BBC Research Cen te r , Swi tze r land
139
Sarpkaya , T . 1971 : On s ta t ionary and t rave l ing vor tex b reakdown.
J . F lu id Mech . 45 , 545-559
Sol ignac , J . L . 1990 : Ai r flow v i sua l iza t ion to the s tud y o f vor tex
breakdow n. In : F low V isua l iza t ion V , R . Rezn icek , ed . , pp . 490
496 , Wash ing ton : Hemisphere
Spall , R. E.; Gatski , T. B.; Ash, R. L. 1990: The structure and
dynam ics o f bubb le - type vor tex b reakdo wn. P roc . R . Soc . Lond .
A 429, 613- 637
Staufenb ie l , R . ; He lming , Th . 1985 : Exper iments on the b re akdo wn
of vort ice s in pressure f ields. 1 . Int . Coll . on vorte x brea kdo wn ,
Feb. 11. 12. 1985, Aac hen
Staufenb ie l, R . ; He lming , Th . ; V i t t ing , Th . 1987 : Con t ro l l ed b rea k-
dow n o f t ip vort ices. 2. Int . Coll . on v ort ica l f lows, Ap ri l 6. 7 .
1987 , BBC Research Cen te r , Swi tze r land
Sto janof f, C . 1991 : P r iva te com mun ica t ion . RW TH Aachen
Received June 25, 1992
echn ica l no tes
e w c o n d u c t i v i t y p r o b e s f o r d i s p e r s i v i t y m e a s u r e m e n t s i n p o r o u s m e d i a
J . P i q u e m a l , J . - E B o u r r e l , G . L a v i l le
lns t i tu t de M6can ique des F lu ides , Uni t6 Associ~e au C .N .R.S . , Avenue du Professeur Cam i l le Sou la 314 00 Toulouse (France)
A b st r ac t . The n ew e lec t r i c p robes d esc r ibed in th i s no te a re fabr ica t -
ed us ing p r in ted c i rcu i t t echn iques . They o f fe r va r ious advan tages
over the ex i s t ing mod e ls : h igh min ia tu r iza t ion , smal l d i s tu rbance o f
the poro us m edium, p rec i se loca t ion , low cos t and easy fabr ica t ion .
1 P r o b l e m s t a t e m e n t
P r o b e s p r e s e n t e d h e r e w e r e d e v e l o p e d f o r d i s p e r s i o n s t u d i e s
i n h e t e r o g e n e o u s u n c o n s o l i d a t e d p o r o u s m e d i a c o m p o s e d o f
2 5 0 ~ tm s a n d , i n c l u d i n g i m p e r v i o u s s p h e r e s w i t h l a r g e d i -
a m e t e r s ( fe w cm s ) . E x p e r i m e n t s a r e c a r r i e d o u t i n a c i r c u l a r
c o l u m n i n w h i c h a s a t u r a t i n g l i q u i d A is d i s p l a c e d b y a
m i s c i b l e l i q u i d B . D u r i n g d i s p l a c e m e n t a m i x i n g z o n e g e n e r -
a t e d b y d i s p e r s i o n a p p e a r s b e t w e e n t h e t w o l i q u i d s . L i q u i d
B i s i n j e c t e d a t t h e c o l u m n i n l e t i n t h e f o r m o f a s t e p f u n c t i o n
a n d t h e l o n g i t u d i n a l c o e f f i ci e n t o f d i s p e r s i o n is d e t e r m i n e d
b y m e a s u r i n g t h e c o n c e n t r a t i o n C o f fl u i d A i n fl u i d B i n t h e
m i x i n g z o n e i n a s e c t i o n n e a r t h e c o l u m n o u t l e t ( s ee f o r
i n s t a n c e B e a r 1 96 8). E x p e r i m e n t s u s i n g l i q u i d s m a r k e d b y
s a l i n e t r a c e r s a r e m o s t c o m m o n l y p e r f o r m e d . W e h a v e u s e d
t h i s t y p e o f e x p e r i m e n t .
I n o u r p r o b l e m t h e p o r o u s m e d i u m i s h e t e r o g e n e o u s a n d
s o i t i s n e c e s s a r y t o m e a s u r e t h e a v e r a g e d c o n c e n t r a t i o n i n
a g i v e n s e c t io n . I n a c y l i n d r i c a l c i r c u l a r c o l u m n ( r a d i u s R ,
c o o r d i n a t e s z , r , 0 ), o w i n g t o t h e h e t e r o g e n e i t y , t h e c o n c e n -
t r a t i o n i s : C = C (z , r , 0 , t ). I n a g i v e n s e c t i o n a n d a t t i m e t ,
C ~. t = C (r , 0 ), t h e a v e r a g e d v a l u e o f C z . t o v e r a s e c t i o n i s :
1 R
C z , t - R 2 _ ~ gC z , ( r , O ) r d r d O .
(1)
E x p e r i m e n t a l v e r i f i c a t i o n o f E q . 1 c a n b e e f f e c te d u s i n g a
v e r y h ig h n u m b e r o f p o i n t p r o b e s l o c a t e d i n t h e s e c ti o n .
H o w e v e r , t h e s e p r o b e s w o u l d i n t e r a c t a n d g r e a t l y d i s t u r b
t h e p o r o u s m e d i u m r e s u l t i n g i n l a r g e e x p e r i m e n t a l e r r o r s .
H e n c e w e m u s t u s e a p r o b e t h a t w i l l m e a s u r e a v e r a g e d
c o n c e n t r a t i o n d i r e c t ly . I n o r d e r t o s i m p l i f y t h e p r o b l e m , w e
c a n r e a s o n a b l y a s s u m e t h a t t h e c o n c e n t r a t i o n a v e r a g e d o v e r
a s e c t i o n i s i n d e p e n d e n t o f 0 a n d s o w e d e t e r m i n e :
I R
C = , t =
R ~ R C ~ . t ( r ) r d r .
(2)
T h e c o n c e n t r a t i o n i s o b t a i n e d b y m e a s u r i n g t h e e l ec t r i c a l
c o n d u c t i v i t y o f t h e l iq u i d b y m e a n s o f a p r o b e c o m p r i s i n g
t w o e l e c t r o d e s . C ~ ., ( r) a n d t h e l i q u i d c o n d u c t i v i t y 7 z. t ( r) a r e
r e l a t e d b y C z , t ( r ) = g ( r ) 7 ( r , t ), g (r ) b e i n g t h e g e o m e t r i c a l
c h a r a c t e r i s t i c o f t h e p r o b e . I n o r d e r t h a t t h e p r o b e c a n y i e l d
r e l a t i o n 2 , g ( r) m u s t b e i n d e p e n d e n t o f r.
C (r ) i s a p o i n t c o n c e n t r a t i o n v a l u e . C o n d u c t i v i t y i s m e a -
s u r e d b y m e a n s o f a n e l e c t r i c c u r r e n t a c r o s s t h e e l e c t r o d e s .
