Lourenco Et Al

7/24/2019 Lourenco Et Al.

http://slidepdf.com/reader/full/lourenco-et-al 1/8

Experiments in F luids 18 0995) 421-428 9 Springer-Verlag1995

O n t h e a c c u r a c y o f v e l o c i ty a n d v o r t i c i t y m e a s u r e m e n t s w i t h P V

L. Lourenco A. Krothapal l i

A b s t r a c t A n u m b e r o f n u m e r i c a l t e c h n i q u e s a im e d a t

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

appr oa ch in P a r t ic le I m age Ve loc im e t r y , P IV, a r e p r op osed

and inves t iga ted . I n th is appr oac h the ve loc i ty d i sp lacem ent )

i s foun d as the loca t ion of a peak in the cor r e la t ion m ap . Based

o n a n e x p e r i m e n t a l m o d e l t h e b e s t p e r f o r m i n g p e a k f i n d in g

appr oaches a r e s e lec ted am on g d i f f e r en t s tr a teg ie s . S econd , an

a lgor i thm i s p r op osed w hich m in im izes e r r o r s on the e s t im a tes

of vor t i c i ty us ing ve loc i ty d i s t r ibu t ions ob ta ined by m eans o f

P IV. The pr op osed m e thod s a r e expe r im en ta l ly va l ida ted

aga ins t a f low wi th know n pr op e r t i e s .

Introduct ion

O n e of the cha l lenges in e xpe r im e nta l f lu id dynam ics i s the

m ea sur em ent o f the v or t i c i ty f ie ld . Th is d i f f icu l ty a r i se s f r om

the f ac t tha t vor t i c i ty is a qu an t i ty de f ined in t e r m s o f the

ve loc i ty g r ad ien t s . Hence , the ve loc i ty m ea sur em ent m u s t be

m ad e s im ul taneous ly ove r s eve r al c lose ly spaced loca t ions

f r om w hich spa t ia l de r iva t ives can be eva lua ted us ing f in i te

d i f f e r ence s chem es . Am ong the cur r en t ly ava i lab le ve loc i ty

m ea sur em ent t echn iques , P a rt i c le Im age Ve loc im e t r y P IV) i s

the m o s t su i t ed f o r th i s k ind o f m ea sur em ent a s i t i s capab le o f

pr ov id in g the ve loc i ty vec tor ov e r a s e lec ted two- d im ens iona l

r eg ion o f the f low, wi th suf fic ien t accur acy and spa t ia l

r e so lu t ion . Th is cons t i tu te s a g r ea t a s se t f o r the s tudy o f

a va r ie ty o f flows tha t evo lve stochas t i ca lly bo th in t im e and

space , such a s uns teady sepa r a ted , t r ans i t iona l and tu r bu len t

f lows. This ar t ic le discusses the ac curac y that ca n be achie ved

in the m easur e m en t o f ve loc i ty and vor t i c i ty us ing P I V.

N a t u r e o f t h e p r o b l e m

Let us cons ide r tha t the in - p lane ve loc i ty com ponen ts , u(u, v),

o f a two- d im ens iona l r eg ion of a g iven f low f ie ld a r e m easu r ed

with PIV, in a Car tes ian gr id, a t un iform ly spaced intervals .

Received: 6 M ay ~994 Accepted: 4 December 994

L. Loure nc o , A . Krotha pa l l i

F lu id Me c ha nic s Re se a rc h La bora tory ,

F A M U / F S U C o l l e g e o f E n g i n e e r i n g ,

P.O. Box 2175,

Tallahassee , FL 32316-2175, USA

Correspondence to: L. Loure nc o

W o r k s u p p o r t e d b y N A S A A m e s R e s e a r ch C e n t e r

T h e m e a s u r e d v e l o c i ty c o m p o n e n t s a r e u * = u + e a n d

v * = v + e ,

wher e e denotes the ab so lu te e r r o r . One way to

ob ta in the ou t - of - p lane com po nen t o f the vor t i c i ty fi e ld

cons i s ts o f eva lua t ing the ve loc i ty de r iva t ives wi th a su i t ab le

f in it e d i ff e r enc ing schem e , such a s the s econd o r de r s chem e in

Eq. 1)

\(u~+~-- u :- +-A)g ~ R+ A -y

1)

wher e ~R s tands f o r h ighe r o r de r t e r m s .

Equa t ion 1) shows tha t the e r r o r in the m easur em ent o f the

vor t i c i ty depend s on two com pone nts o f d i ff e r en t na tur e . O n e

i s the t r unca t ion e r r o r a s soc ia ted wi th the f in it e d i f fe r ence

schem e used , o f o r de r Ax , Ay2), the o th e r due to the

unce r ta in ty , e, in the ve loc i ty m ea sur em ent and of o r de r

( l lAx , l lAy ) .

Wher eas the t r unca t ion e r r o r dec r eases f o r

sm a l le r g r ids, the oppo s i te happens to the e r r o r due to the

ve loc i ty unce r ta in ty . Th is s i tua t ion sugges ts tha t f o r a g iven

ve loc i ty m ea sur em ent e r r o r , e, the r e i s an op t im u m spac ing

gr id) for which th e tota l er ro r is minim ized. I t a lso shows tha t

m eaningf u l vor t i c i ty e s t im a tes can on ly be ob ta ined i f the e r r o r

in the ve loci ty me asure me nt , e , is kept small .

Unce r ta in t i e s in the ve loc i ty m ea sur em ent inc lude e r r o r s

in t r odu ced dur ing the r ecor d ing of the m ul t ip le exposur e

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

the s cene be ing r ecor d ed by the cam er a l ens , l im i ted l ens and

f ilm r eso lu t ion , th r ee - d im en s iona l e ff ec ts Mey nar t and

Lour enco 1984) , b ia s in t r oduc ed by la r ge ve loc i ty g r ad ien t s

Adr ian 1988, Keane and Adr ian 1989) , and the inaccur ac ie s

due to the p r oces s ing a lgor i thm s . I n th i s a r ti c le, s chem es

a i m e d a t t h e m i n i m i z a t i o n o f t h e e r r o r c o n t r i b u t i o n d u e t o t h e

pr oces s ing a lgor i thm a r e d i s cussed . I n add i t ion , an ad ap t ive

m esh schem e f or the eva lua t ion of the ve loc i ty de r iva t ives is

p r oposed . Th is s chem e i s des igned to m in im ize e f f ec t o f the

loca l ve loc i ty m e asur em ent e r r o r s on the ve loc i ty de r iva t ives

and hence the vo r t i c i ty e s t im a te .

3

T h e P l Y te c h n iq u e

The m easur ing pr inc ip le o f P a r t i c le I m aging Ve loc im e t r y

cons i s ts o f ob ta in ing the ve loc i ty o f a f lu id by m easur ing the

ve loc i ty o f t r ace r pa r t ic le s in suspen s ion . Th is i s acco m pl i shed

by i l lum ina t ing a p lana r r eg ion of the f low wi th h igh in tens i ty

s t r obe i l lum ina t ion , usua l ly a pu l sed l a se r, and r ecor d ing the

42



t race r pos i t ion a s a func t ion o f t im e in doub ly o r m ul t ip le

exposu re pho tograph s . The ve loc i ty i s g iven by the ra t io

be tween cor re sponding t race r im ages and the t im e be tween

s t robe pu l ses. W ha t d i f fe ren t i a te s PIV f rom o the r w hole f i e ld

ve loc i ty m ea surem ent t echn iques , such a s P a r t i c le T rack ing , i s

tha t in idea l ope ra t ing condi t ions the t race r co ncen t ra t ion is

h igh eno ugh such tha t a l a rge num ber o f pa r t ic le s ex i st wi th in

a spec i f i ed m easur ing vo lum e (Lourenco and Kro thapa l l i

1987) . The m easur ing v o lum e i s a sm a l l reg ion wi th in the f lu id ,

where the f low i s cons ide red to be suf f i cien t ly un i fo rm . The

local ve loc i ty is t aken a s the ave rage ve loc i ty , o r g roup

ve loc i ty , o f the pa r t i c le s recor ded wi th in the m easur ing

volum e .

There a re s eve ra l m e thod s ava i l ab le to con ver t the da ta

con ta in ed in the pho tog raphs , in to ve loc i ty da ta (M eynar t 1983 ,

Mey nar t and L ourenco 1984, Adr ian an d Yao 1984) . How ever ,

t h e m o s t c o m m o n l y u s ed t e c h n i q u e s u s e t h e a u t o c o r r e l a ti o n

approach . In th i s approach the two d im ens iona l au to-

cor re la t ion o f sm a l l in te r roga t ion reg ions i s ob ta ined .

T h e average disp lacem ent wi th in the in te r roga t ion reg ion

cor responds to the coord ina te s where the au tocor re la t ion

func t ion has a m axim um va lue , exc lud ing the s e l f-cor -

re la t ion peak . The au tocor re la t ion func t ion i s e f f ic ien t ly

eva lua ted us ing two 2D F our ie r t rans form a t ions . To pe r form

these t rans form s d i f fe ren t s chem es a re used tha t em ploy

opt ica l p roces sors , d ig i ta l p roces sors o r a com bina t ion the reof .

The c om m o n fea tu re o f any of these : .p rocessors i s th a t the

au tocor re la t ion is ob ta ined in a d ig i t a l a r ray fo rm a t . In

th i s two-d im ens iona l a r ray , the loca t ion o f the m axim um va lue

in the a r ray , exc lud ing the s e l f cor re la t ion peak , i s p ropo r t iona l

to the loca l d i sp lacem ent .

The pr im a ry ob jec t ive o f th i s s tudy i s to eva lua te d i f fe ren t

s t ra teg ie s tha t can b e used to a ccura te ly de te rm ine the pos i t ion

of the cor re la t ion peak w i th sub-gr id re so lu t ion . Th i s i s

ach ieved us ing an expe r im enta l approach , and ca r r i ed ou t

us ing a series of

calibration experiments .

P e a k f in d i n g a l g o r i th m s

The d i sp lacem ent ~(A x, Ay) i s g iven by the pea k pos i t ion o f the

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

A(x ,y )

of the pa r t i c le

im age pa i rs . Assum ing two d im en s iona l m ot ion , the ensem ble

of pa r t i c le im age double t s i s we l l represen ted (Lourenco e t a l.

1994) by

D(x, y) = I(x, y ) | [5 (x, y) + 6 (x + Ax, y + Ay) ] (2)

w h e r e

I(x,y)

represen t s the record ed im ages o f the pa r t i c le s ,

which i s we ll approx im a ted by a Gauss ian func t ion (G oodm an,

1968), 6(x,y) i s the Di rac de l t a func t ion cen te red a t (x,y) a n d

| i s the convo lu t ion ope ra to r . The spec t rum , i .e . the v isua l

Young ' s f r inges, is ob ta ined by t ak ing the F our ie r t rans form of

Eq . (2 ) and m ul t ip ly ing wi th i t s com plex con juga te ,

ID *(ro . , coy)12= 2

I*(09,,, D r ) 2

{1 + cos

[2~(Axoox+Ayc@)] } (3)

cox and coy are the spat ia l freque ncies a long the x a n dy axis, an d

I* (co,, coy) and D*(cox, coy) are the Fou rier t ran sfor ms

o f I (x ,y )

a n d

D(x,y) .

Final ly, the auto cor rela t io n is obtained as the

F our ie r t rans form of Eq . 3 ) ,

A (x ,y ) = j j ID*(09x, o&)

12exp [ - 21rj (xcG +yc o ) ]

dcoxdoay

= 2 H ( x , y ) + H ( x + A x , y + A y ) + H ( x - - A x , y - A y)

4 )

w h e r e j = x / ~ a n d

H (x ,y ) = I~ [/*(co, a~)12 ex p [ -- 2~zj (xcox+ycoy) ]

dcoxdcoy

is the

F our ie r t rans form of the in tens i ty ha lo

I - / ~ [ .

Equa t ion (4 ) shows th a t the au tocor re la t ion A (x , y ) can be

represen ted a s the sum of s eve ra l cop ies o f the fun c t ion H(x , y) ,

i .e . the ave rage im age in tens i ty d i s t ribu t ion . We nam e the

func t ion 2 H(x , y) ce n te red a t the o r ig in o f the cor re la t ion m ap ,

the D. C . peak , wh ereas

H(x, + Ax , y+

Ay), or a l ternat ively

H ( x - - A x , y - - A y ) , is des ignated as the data peak.

Because the au toco rrela t ion is repre sent ed digi ta l ly, the

m ax im u m d i sp lacem ent i s in i t i a lly ob ta ined by the loca t ion in

the d i s c re te cor re la t ion a r ray wi th the m axim um va lue . F or

im pro ved accuracy , i .e . in o rde r to re so lve the peak

coord ina te s wi th subgr id accuracy , th i s e s t im a te needs to be

fur the r re f ined . Three d i f fe ren t s t ra teg ie s a re inves tiga ted :

a . The peak pos i t ion i s approx im a ted by the cen t ro id o f the

au tocor re la t ion func t ion in the v ic in i ty o f the d a ta peak ,

des igna ted a s H(x + Ax, y + Ay).

b . The d ig i t a l au toco r re la t ion in the v ic in i ty o f the peak

i s a p p r o x i m a t e d b y a k n o w n c o n t i n u o u s f u n c t i o n . T h e

coe f f ic ien t s o f the f i tt ing func t ion a re fo und by l eas t squa res.

The re f ined peak p os i t ion i s t aken wh ere the f i t t ing func t ion

has m axim u m , i . e. i t s de r iva t ives a re equa l to ze ro . A pa rabo l ic

and a Gauss ian f i t a re em ployed .

c . T he d ig i ta l au tocor re la t ion a roun d the pea k loca t ion is

com puted in a re f ined gr id us ing an in te rpo la t io n s chem e . The

W hi t t ake r ' s s ignal recons t ru c t ion t echn ique i s used .

The re la t ive accuracy g iven by these d i f fe ren t approaches

were inves t iga ted us ing

calibration

disp lacem ents . The

calibration

disp lacem ent f ie lds a re ob ta ined b y record ing

speck le pa t t e rns un dergo in g p lane , un i fo rm ro ta t io n and t rans -

la t ion . To ensure a h igh degree o f repea tab i l ity , the ca l ib ra ted

d i sp lacem ents were in t roduced v ia p rec i s ion t rans la t ion and

rota t ion s tages (Klinger Scient if ic mo dels MRL8.25 and TR 80),

which pro v ided accura te d i sp lacem ents w i th in 0 .1 ~ tm and 0 . 01

degrees re spec t ive ly . The speck le pa t t e rns were gen e ra ted by

doub ly expos ing an a lum inum sur face , wi th f ine g ra in , to the

coher en t i l lum ina t ion o f a HeN e la se r . The d i sp lacem ent f i e lds

a re recorde d on to ph o tograp h ic f ilm , Kodak 2415 Technica l

P an , by m eans o f a 35 m m Nikon F -3 cam era f i tt ed wi th

a 105 mm M acro lens . The lens was set a t a magnificat ion o f 0.5

and F of 2 . 8 . Th i s l ens was chosen because o f i ts op t im a l

pe r form ance , i. e. m in im u m d i s to r t ion , fo r p lane f i eld

record ing . To fu r th e r reduce the e f fec t o f im age abe r ra t ions

on ly the cen t ra l 1 /2 o f the reco rd ing was used fo r the

proces s ing . The f i lm re so lu t ion i s 320 l ine pa i r s /m m and i s

suffic ient to resolve, a l ias free diffr act ion l imited speckle

images w ith averag e s ize of 6.5 ~tm. F igure 1 is an exam ple of

one of such record ings , cor re spon ding to 0 .5 degrees ro ta t ion

and 100 pm t rans la t ion .

In o rde r to m in im ize sys tem a t ic e r ro rs in t roduced by such

e f fec ts a s f i lm shr inkage , we op ted to co m p are how we l l the

d i f fe ren t a lgor i thm s t rack the change o f d i sp lacem ent wi th



42

posi t ion as imposed by the rota t ion. Thus the displacements 250

were measured a long an axis wi th the origin a t the center of

rota t ion, to, and compared with actual displacement values

given by the distance fro m the axis, r - r0, t imes the magnitu de -~ 200

of the impo sed ro ta t ion, r~. The t rans la t ion was primar i ly used

as a mean s of increas ing the dynam ic range o f the technique,

and allowed us to mea sure small displac emen ts in the vicinity ~ 150

of center of rotation. In the data tha t is prese nted in this article

this bias effect is rem ove d after processing. ~ - 100

In this s tudy the autocorrela t ion was obtained us ing

a comb inat ion of opt ical and digi ta l processors . In this

appro ach a col l imated laser beam w ith 300 ~tm in diam eter

illuminates the film negatives to pro duc e the interferenc e ~; 50

Young's fr inge pat tern in the focal plane o f a converging lens.

The Young's fringe pattern is digitized in the 256 x 256 x 8 bit

pixel format , and a second Fourier t ransfo rm is com puted 0

digitally to yield the autocorrelation. Accordingly, the

displacement which has ben recorded in the region i l luminated

by the laser beam is propo rt ional to the locat ion o f highes t

ampli tude p eak of the digi tal autoco rrela t ion excluding that a t

the origin, herein defined as the D.C. peak. This position is

found, wi th sub-grid accuracy, us ing the various methods

which are described in the following.

4 1

Centroid method

Equat ion (5) represents the average displacement computed

using the centroid method,

g Ax, Ay) = ~ ~ x, y) H x + Ax, y + Ay)

M ~ H x + Ax ,y+ Ay)

5)

where M is the magnif icat ion factor . The l imits of integrat ion

are variable , and set where the s ign o f the derivat ive of

H x + Ax,y +

dy) changes. Figure 2 is a typical result showing

the re la t ionship b etween the actual displacements , defined as

( r - ro) ' r~ , and the measured d i sp lacement ob ta ined a s the

locat ion in the autocorrela t ion m ap. This resul t shows that for

displacements larger than a l imi t value, good agreem ent is

obtained between measu red and actual displacement data. The

discrepancies observed in the low range o f displacements can

be accounted for by the contamina t ion of the da ta peak by the

D.C. peak. For small displacements, Ax~O and dy~O, making

Fig. 1. Calibration displacement field

f

100 150

Actual disptocemen{ ~m)

200 250

Fig. 2. Actu al vs. Measured Displacements (Centroid Schem e)

the D.C. pedestal, H x,y) to become superposed to the data

peak. Thus the centroid metho d should o nly be used in

s i tuat ions when the displacement is such that the funct ion

H x + dx, y + dy) i s c learly separated from H x, y).

4 2

Curve fitting

Figure 3 shows the actual displacement versus the measured

displacement, when a parabolic fit in the least squares sense is

used. This interpolator uses a 3 x 3 computational cell ,

centered a t the e lement wi th m axim um value. Figure 3 shows

that this interpolator can produce large errors as i t tends to

bias the resul ts towards displacements that correspond to

discrete integer frequencies, as shown by the ladder like

behavio ur in Fig. 3. This lack of perform ance is not surpris ing

and is accounted for by the fact that the shape of the

correla t ion peak, H x + Ax, y + Ay), i s poor ly approxima ted by

a parabol ic shape. In a preceding paragr aph i t has been shown

that the image intens i ty dis tr ibut ion, and the corre la t ion peak

are well approxim ated by a Gauss ian. The Gauss ian f i t assumes

that the co rre la t ion funct ion in the vic ini ty of the data peak is



250

- ~2 0 0

=t.

150

~ 1

o

~ 5O

I

0 200

I P

50 100 150

Actual d isplacement ( /~m]

250

Fig. 3. Actual vs. Measured Displacements Parabolic Fit)

250, , , ,

200

E

E

150

r~

.~_

100

( / 1

a

~

5 o

0 0 100 150 200 250

Actua l d isplacement ( izm)

Pig. 4. A ctual vs. Measured Displacements Gaussian Fit)

c l o se l y a p p r o x i m a t e d b y

H x ,y )= C e x p [ x - d x ) 2 y -A y ) 2 q

~ 2 ~ 6 )

whe r e C a nd ~ a r e t he f i t t i ng c ons t a n t s . The Ga us s i a n f i t i s

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

l i ne a r i z i ng ) o f Eq . 6 ).

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

i n F i g . 4 . I n c on t r a s t t o t he p r e v i ous r e s u l t s t h i s f i gu r e s hows

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

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

Ga u s s i a n i n t e r po l a t o r a r e i t s e f fi c i e nc y , a s i t r e qu i r e s ve r y

f e w c o m p u t a t i o n s , a n d a c c u r a c y . H o w e v e r i t i s e x p e c t e d

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

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

a p p r o x i m a t e d b y a G a u s s i a n s h a p e . T h i s c a n b e th e c a s e w h e n

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

25

- - 200

E

=1.

E 150

u l

100

5 0 1

i . . . .

50 100 150 200 250

Actua l d }splecement (pro)

Fig. 5. Actual vs. Measured Displacements Parabolic Fit with

Padding)

250

-~ 200

=1.

150

N

~ 100

~ 50

i , ,

I I I I

0 100 150 200 250

Actual d isptacement (Fm)

Pig. 6. A ctual vs. Measured Displacements Gaussian Fit with

Padding)

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

t i me s ge ne r a t e s t r e a ks .

4.3

Padding the FFT S

A n a l t e r n a t i v e w a y o f i n c r e a s i n g t h e r e s o l u t i o n o f th e

~ x , y )

c o r r e l a t i o n a x e s i s p a d d i n g t h e d i g i t i z e d f r i n g e p a t t e r n w i t h

z e r o s S e a r n s a n d H u s h , 1 9 9 0) . F o r e x a m p l e i f t h e f r i n g e

pa t t e r n i s o r i g i na l l y ob t a i ne d i n a 128 x 128 f o r ma t , t he n i t is

a p p e n d e d w i t h z e r o s t o b e c o m e 2 5 6 x 25 6 i n s iz e. A s a r e s u lt

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

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

t h e s e c o n d 2 D F F T o p e r a t i o n b y a f a c t o r o f f o u r . U s e o f p a d -

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

r e m a r k a b l e i m p r o v e m e n t s i n a c c u r a c y F i g s. 5 a n d 6 ). T h e

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



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

d a t a to t h e tr u e m a x i m u m o f t h e c o n t i n u o u s a u t o c o r r e l a t io n .

250

4 .4

W h i t t a k e r ' s r e c o n s t r u c t i o n ~. 200

B e c a u s e th e Z e r o P a d d i n g t e c h n i q u e i s c o m p u t a t i o n a l l y

d e m a n d i n g it i s d e s i r a b l e t o e m p l o y a l t e r n a t i v e i n t e r p o l a t i n g ~ 1 50

s c h e m e s t h a t o f f e r c o m p a r a b l e a c c u r a c y w h i l e r e m a i n i n g 8

c o m p u t a t i o n a l l y e f f i ci e nt . T h e s i g n a l r e c o n s t r u c t io n t e c h n i q u e

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

100

i s t h e s c h e m e c h o s e n i n t h i s in v e s t i g a t i o n ( S t e a r n s a n d H u s h ,

1 9 90 ). T h e a m p l i t u d e o f t h e a u t o c o r r e l a t i o n a t a n y s u b p i x e l g

l o c a t io n i s g i v e n b y t w o - d i m e n s i o n a l r e c o n s t r u c t i o n a s : ~ 5 0

i 2 / 2

H x , y ) = ~ , ~ , H k 2 . , 1 2 y )

k = i 2 l = j 2

o

~ x - k 2 x ) ~ y - - ~ y )

w h e r e 2 ~ a n d 2 y a r e t h e s a m p l i n g i n t e r v a l s , c o n s i d e r e d

3

i d e n t i c a l, i n t h e x a n d y d i r e c t i o n s r e s p e c t i v e l y ; i a n d j a r e

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

v a l u e . A s s h o w n i n E q . ( 7) t h e s u m m a t i o n i s c a r r i e d o v e r 2

a l o c a l i z e d 5 x 5 c e ll r e g i o n . T h e a p p l i c a t i o n o f t h i s s c h e m e i s

i l l u s t r a te d i n F i g . 7 . T h e a p p r o x i m a t e l o c a t i o n o f t h e x 1

a n d y c o m p o n e n t s o f th e d i s p l a c e m e n t i s f ir s t g i v e n b y t h e

d i s c r e te c o o r d i n a t e s o f t h e m a x i m u m i n t e n s i ty o f t h e

a u t o c o r r e l a t i o n ( i, j ) . T h e v a l u e o f t h e a u t o c o r r e l a t i o n i s ~ 0

e v a l u a t e d i n a n e w g r i d , w i t h h a l f t h e s p a c i n g , a t e i g h t t~

i n t e r m e d i a t e l o c a t io n s ( 1 t h r o u g h 8 ) . A n e w m a x i m u m -1

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

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

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

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

p r o c e d u r e i s r e p e a t e d u n t i l t h e d e s i r e d r e s o l u t i o n i s a c h i e v e d .

T y p i c a ll y , s ix o f t h e s e c y c l e s a r e p e r f o r m e d t o a c h i e v e

a n o m i n a l a c c u r a c y o f 1 / 64 th o f a p i x e l.

F i g u r e 8 s h o w s a t y p i c a l r e s u l t w h e n t h e W h i t t a k e r

i n t e r p o l a t o r i s u s e d . D e t a i l e d c o m p a r i s o n s s h o w t h a t t h i s

I / I i I

50 100 150 200 250

Actuot d isptocement Fm)

Fig . 8 . Actual vs . Measured Di sp l acem ent s (Wh i t t aker s i n te rpo l a t i on)

9 : . j

. t . .

9 o 9 ~ o e ~ 1 7 6 1 7 69 - o 9

" o 9 9 _ - 9 9 o O r -

9 - - - o ' 9 1 4 99 1 4 9 - t o 9 9

# o 9 ~ 9 9

9 9 9

9 " t

i P

5O

100 150 200

Actuo[ disp[ocement ,u,m)

250

Fig . 9 . Error vs . pos i t ion

4 2

F i rs t su b g r id

Second sub g r id

Orig inal . gr id

2 3

i . j 4

6 5

F i g . 7 . G r i d f o r t h e W h i t t a k e r s i n t e r p o l a t o r

s c h e m e r e t a i n s t h e s a m e a c c u r a c y o b t a i n e d w i t h t h e G a u s s i a n

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

c o m p u t a t i o n t i m e . F i g u r e 9 s h o w s t h e m e a s u r e m e n t e r r o r a s a

f u n c t i o n o f t h e p o s i t i o n a l o n g t h e i n t e r r o g a t i n g a x i s . I n t h i s

s e r ie s o f e x p e r i m e n t s t h e e r r o r s a r e b o u n d e d a n d o f t h e o r d e r

o f _ 1 p m .

5

E x p e r i m e n t a l v e r i fi c a t io n : c o n t r o l e x p e r i m e n t

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

s e l ec t e d o n t h e b a s is o f it s im p r o v e d p e r f o r m a n c e v e r s u s t h e

o t h e r t e c h n i q u e s . T o f u r t h e r e v a l u a t e it s a c c u r a c y u n d e r r e a l

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

o u t . T h e e x p e r i m e n t c o n s i s t e d o f t h e m e a s u r e m e n t o f th e

v e l o c i ty a n d v o r t i c i ty d i s t r i b u t io n o f a t w o - d i m e n s i o n a l ,

i n c o m p r e s s i b l e , l a m i n a r w a l l - je t is s u i n g f r o m a r e c t a n g u l a r



Fig. 10. Doubly-exposed photograph of

plane containing the jet s mid axis

channel into st i l l air . This test flow is chosen because i t

contains two regions with large velocity gradients, the

free-shear layer and the boundary layer , and a wide veloci ty

dynamic range, f rom zero near the wall , up to a maxim um near

the jet axis. Additionally, a numer ical simulation of the ftow

using the Boundary Layer approximations is available. This

numerical solution is second order accurate in both the

x (streamwise) and y (cross-strea m) directions and is used as

a means to test and validate the accuracy of the PIV results.

Result s f rom the numerical code com pare ext remely well wi th

Glauert s similari ty s olution for fully developed flow.

In this experiment the jet width is 5 mm, has an aspect rat io

of twenty and the wall is one hundred widths long.

Measurements are carr ied out wi th in twenty widths f rom the

exit , where the flow can be considered two-dimensional. The

f low Reynolds numbe r base d on the jet width and the average

mass flow velocity is 1,300. For these conditions the exit

velocity profi le, at the end o f the channel, has a pa rabolic

shape. For the PIV measurement , the pr imary ai r jet and the

ambient air are seeded with small oil smoke part icles, 1-5 ~tm

in diameter. A single double-pulsed, frequency doubled

Nd-Yag laser (Lumonics HY-400), provides the i l lumination

sheet . The t ime separation between the double l ight pulses is

adjusted according to the m axim um f low veloci ty (Lourenco

and Krothapalli, 1987) and set at 18.9 ~tseconds. The doubly

exposed f rames are captured by means of a Nikon 35 mm

camera equipped with a 50 mm macro lens. A velocity bias

device, consist ing of a scanning mirror (Lourenco et al . 1986),

is used to accommodate the large velocity range within the

flOW

Figure 10 is a typical doubly e xpose d photo gra ph of the

central plane of wall-jet , covering the region from the jet exit

up to 3.3 widths downstream. This photograph clearly shows

the two fluid streams that form the jet : the primary jet , heavily

seeded and the l ightly seeded ambient st i l l air . The part icle

image doublets are also visible. The image displacement

on these f rames ranges f rom a minimum (corresponding

to the velocity bias magnitude), in the near wall-region, to

a maximum near the jet axis. The clear, riple free, interface

between the pr imary and ent rained f lu id depicted in the

photograph clear ly demonst rates that , in the in i t ial region of

jet , the amplitud e of flow instabil it ies can b e neglected. Thus

comparison wi th the numerical so lu t ion for laminar f low is

appropr iate.

Figure 11 is a typical comparison, showing good agreement,

between the measured velocity distribution, at a t ime instant ,

using convent ional photographic PIV and the numerical

1.0

0.8

O.6

0.4

0.2

0

]

o PIV

Numeric.ol

P

I 2 3

Y / Y ~ •

Fig. 11. Wall-jet velocity profile at x/h = 3

simulat ion at a downst ream locat ion corresponding to three

widths. In this figure the velocity is normalized with the

maximum velocity, and the abscissa with the location

corresponding to the max imu m velocity . This compar ison was

carr ied at downst ream locat ions where the f low remained

two-dimensional. Similar levels of agreement were obtained.

As i t wil l be discussed in the next section, the real challenge

remains the evaluation of the vortici ty from the velocity data

wi th minimal er ror .

Computation of vorticity: adaptive scheme

The normal ized vor t ici ty d is t r ibut ion corresponding to the

velocity profi le in Fig. 11 was first evaluated using the second

order accurate cent ral -d i f ference scheme. In th is scheme the

t runcat ion error has a cont r ibut ion of order

Ax 2, Ay2).

Therefore i t is expected that a smaller spacing will produce

better est im ates of the vortici ty distribution. This simplist ic

analysis does not account for the contribution of error

affecting the veloci ty measurem ent d iscussed in the

introduction. Figure 12 presents a comparison between the

vor t ici ty d is t r ibut ion evaluated by means of the second order

scheme and that computed using the numerical s imulat ion .

Although the general trend of the profi le is found, i t is clear

that individual est imates of the vortici ty can be affected by

large error.



0

2

- I

J t

9

~

~ x ~ ~ 9 P I V

I

Y/Ymo~

Fig. 12. Wal l -jet vort ici ty dist ribu t ion ( central -d i fferenc e sche me )

A d i f f e r e n c in g s c h e m e a i m e d a t th e m i n i m i z a t i o n o f t h e t o t a l

e r r o r a f f e c ti n g th e v o r t i c i t y e s t im a t e i s p r o p o s e d . T h i s s c h e m e

i s b a s e d o n t h e R i c h a r d s o n s e x t r a p o l a t i o n p r i n c i p l e . I n t h is

s c h e m e t h e d e r i v a t i v e s o f th e v e l o c i t y a re f i r st c o m p u t e d i n

c o a r s e g r id s , i n o r d e r t o k e e p t h e e r r o r c o n t r i b u t i o n d u e t o t h e

e x p e r i m e n t a l e r r o r , e l l, r e la t iv e l y s m a ll w h e n c o m p a r e d t o t h e

t r u n c a t i o n e r r o r e r . T h e g r i d i s t h e n f u r t h e r r e f i n e d u n t i l t h e

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

t h a t o f t h e t r u n c a t i o n e r r o r . T h e e x p e r i m e n t a l a n d t h e

t r u n c a t i o n e r r o r s a r e e x p r e s s e d a s f u n c t io n s o f h, t h e g r i d

s p a c i n g :

8 . = ~ 8 )

eV = aoh 2 + ai h4 ~- a2 h6 + ' ( 9 )

T h e a p p l i c a t i o n o f R i c h a r d s o n s e x t r a p o l a t i o n i n c r e a s e s t h e

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

a d e r i v a t iv e e s t i m a t e , b y c o m b i n i n g v a l u e s o f d e r i v a ti v e s

e v a l u a t e d i n t w o d i f f e r e n t g r i d s a s f o l l o w s .

C o n s i d e r t h a t • i s t h e d i v i d e d c e n t r a l d i f f e r e n c e o f t h e

f u n c t i o n u ( v e l o c i ty c o m p o n e n t ) o b t a i n e d w i t h a la r g e d a t a

s p a c i n g e q u a l t o 8 h , a n d f4 t h e d i v i d e d c e n t r a l d i f f e r e n c e

o b t a i n e d f o r a s p a c i n g e q u a l t o 4 h :

f a = u J + 8 - - u J - a - f e x a c t + - ~

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

a n o t h e r a p p r o x i m a t i o n o f t h e d e r iv a t i v e w i t h f o u r t h o r d e r

t r u n c a t i o n e r r o r u s i n g g r i d s p a c i n g 2 h a n d 4 h . T h is e s t i m a t e i s

d e n o t e d a s ~ * :

f 2* 4 ~ - f 4 7 e

- - 3 q -1 2-h + 6 4 a l h 4 + ' ( 1 3 )

N e x t t h e e s t i m a t e s f 4* a n d J ~* a r e u s e d t o e l i m i n a t e t h e f o u r t h

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

f 2* * 1 6 f 2 * - - f 4 * 4 9 e - - - 6

= ]-~ + 7 ~ + 5 , 3 7 6 a 2 n + . ..

( 1 4 )

T h i s s c h e m e c a n b e a p p l i e d i n s u c c e s s i o n t o e l i m i n a t e

t h e h i g h e r o r d e r t r u n c a t i o n e r r o r s a s s h o w n i n T a b l e 1 .

H o w e v e r i t i s i m p o r t a n t t o n o t e t h a t t h e e r r o r d u e t o t h e

u n c e r t a i n t y i n t h e m e a s u r e m e n t o f th e v e l o c i ty r e m a i n s

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

i n v o l v in g t h e s a m e g r i d s p a c i n g . F o r e x a m p l e t h e e x p e r i m e n t a l

u n c e r t a i n t y a f f e c t in g t h e f a n d f * e s t i m a t e s a r e t h e s a m e .

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

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

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

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

t h o s e w i t h s a m e e x p e r i m e n t a l e r r o r , f o r e x a m p l e t h e

d i f f e r e n c e s f o r m e d w i t h 3 ~ -f 4 a n d ~ - f 4 * . T h e g r i d i s f u r t h e r

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

t h e s a m e e x p e r i m e n t a l e r r o r , e . g ., j ~ -) ~ * , i s s m a l l e r t h a n t h e

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

e .g . , ) ~* -f 2* . I n t h is c a s e t h e n e w d e r i v a t i v e e s t i m a t e b e c o m i n g

j ~ * * a n d s o o n .

V e r y g o o d a g r e e m e n t b e t w e e n t h e v o r t i c it y d i s tr i b u t i o n

o b t a i n e d f r o m t h e d i f f e r e n ti a t io n o f th e P I V d a t a , u s i n g t h e

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

shown in F ig . 13 .

F u r t h e r i n c r e a s e i n a c c u r a c y i n t h e e v a l u a t i o n o f th e

d e r i v a t i v e c a n b e a c h i e v e d i f t h e d e r i v a t i v e e s t i m a t e s j ~ , 3~ a n d

)~ a r e c o m p u t e d u s i n g t h e l e as t s q u a r e s s e c o n d o r d e r

p o l y n o m i a l a p p r o x i m a t i o n t o t h e d a t a s e t { u~ }. I n t h is m a n n e r

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

i n c lu d e s i n f o r m a t i o n f r o m a ll t h e p o i n t s c o n t a i n e d b e t w e e n

i + 8 a n d i - 8 , i + 4 a n d i - 4 a n d i + 2 a n d i - 2 r e sp e c ti v el y . I n

t h i s f a s h i o n t h e c o n t r i b u t i o n o f t h e e r r o r e is d i m i n i s h e d , a s

m o r e d a t a p o i n t s a r e c o n s i d e r e d t o a p p r o x i m a t e t h e f u n c t i o n .

F i g u r e 1 4 p r e s e n t s t h e v o r t i c i t y d i s t r i b u t i o n w h e n t h i s

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

4 2

+ 4 , 0 9 6 a l h 4 q - . . . ( 1 0 )

~ = u j 4 - u j - 4

8h fexact ~ ~ ~ 1 6 a 0 h 2

+ 2 5 6 a l h 4 + - . . ( 1 1 ) h 2 h 4

w h e r e f ex ac , d e n o t e s t h e e x a c t v a l u e f o r t h e d e r i v a t i v e . T h e

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

y i e l d a b e t t e r e s t i m a t e o f t h e d e r i v a t i v e a s : 8 h

4h )~ A*

f4* 4f 4- - f s 7s 2h ]~ j~*

= 3 + 2 - - ~ + 1 , 0 2 4 a l h 4 + . . . ( 1 2 ) h f f *

Table 1. R ichardso n s ex t rapol a t i on

h h a

E r r o r d u e t o

e x p e r i m e n t a l

uncer t a in ty

e/8h

e/4h

f2** e/2h

f ** f *** e/h



0

1

j . j

o PIV

-o~ -- Numerical

simulation

i i i

1 2

Y/Ymax

Fig. 13. Wall-jet vorticity distribution (adaptive scheme)

0

-1

o

P V

- - Numerical

simulation

1 2 3

Y Y m a x

Fig. 14. Wall-jet vort icity distribution (adaptive scheme a nd least

squares)

7

onclusions

A d e t a i le d s t u d y o n t h e a c c u r a c y o f d i f f e re n t p e a k i n t e r p o l a t i n g

a lg o r i th m s w a s c a r r i e d o u t . B a s e d o n th e f in d in g s o f t h i s s tu d y

t w o a p p r o a c h e s w e r e r e c o m m e n d e d . T h e G a u s s i a n p e a k

i n t e r p o l a t o r a n d t h e W h i t t a k e r s i n t e r p o l a t o r . T h e s e le c t io n o f

th e s e a lg o r i th m s w a s b a s e d o n th e a n a ly s i s o f th e s ig n a l b e in g

in te rp o la t e d a s w e l l a s o n s o m e e x p e r im e n ta l m o d e l s . T h e

p e r f o r m a n c e o f th e s e a p p r o a c h e s w a s f u r t h e r v e r if i e d b y

c o m p a r in g a w e l l e s t a b l i s h e d f lo w f i eld v e lo c i ty d i s t r ib u t io n s ,

in a l a m in a r w a l l - j et , w i th d a ta o b t a in e d b y m e a n s o f P IV .

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

d e r iv a t iv e s , a n d h e n c e th e v o r t i c i ty , w a s p ro p o s e d . I t i s d e m o n -

s t r a t e d th a t t h i s s c h e m e p ro v id e s b e t t e r e s t im a te s o f t h e

v o r t i c i ty d i s t r ib u t io n fo r t h e w a l l - j e t . T h e s c h e m e c a n b e

a p p l i e d to o th e r m e th o d s o f c o m p u t in g th e v o r t i c i ty , e .g ., t h e

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

e s t im a te s a f f e c t e d b y t ru n c a t io n e r ro r s t h a t a re e x p re s s e d b y

Eq. (9).

R e f e r e n c e s

Ad rian R] (1988) Statistical properties of particle im age velocimetry

measurements in turbulent flow. In: Laser Anem ome try in Fluid

Mechanics, Vol. III, Springer Verlag, 115-129

Ad rian R ]; Yao CS (1984) Development of pulsed laser velocimetry

(PLV) for measure ment o f turbulent flow. In: Proc Symp Turbl

Rolla: Univ. Missouri, 170-186.

Goo dm an ]W (1968) Introdu ction to Fourier Optics. New York:

McG raw-Hill. 287

Keane RD; Ad rian RJ (1989) Optimization of particle image

velocimeters. In: Optical meth ods in flow and particle diagnostics.

LIA Proc. Vo l. 68, Orlando: Laser Inst Am er 141-161

Lourenco L; Gogineni SP; LaSalle RT (1994) On-line particle image

velocimeter: an integrated approach. Appl Opt 33:2465-2470

Lourenco L; Krothapalli A (1987) The role of pho togra phic

parameters in laser speckle or particle image displacement

velocimetry. Exp Fluids 5:2 9-3 2

Lou renco L; Krothapalli A; Riethmuller ML; Buchlin JM (1986)

A non-invasive experimental technique for the measurem ent of

unsteady velocity and vorticity fields. AGARD-CP-413,231-239

Meynart R; Lourenco L (1984) Laser Speckle Velocimetry. von

Karman Institute Lecture Series. Brussels, Belgium

Meynart R (1983) Speckle velocimetry st udy of vorte x pairing in a low

RE unexcited jet, Phys Fluids 26:2074-2079

Stearns SD; Hush D (1990) Digital Signal Analysis. Second edition.

Prentice Hall. 75-84

Documents

Lourenco Et Al