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Int. J. M ultiphase Flow Vol. 19, No. 2, pp. 347-367 , 1993 0301-9322/93 $6.00 + 0.00Printed in Gre at Britain. All rights reserved Copyrigh t © 1993 Pergam on Press Ltd

A L A G R A N G I A N M O D E L F O R S O L I D P A R T I C L E S I N

T U R B U L E N T F L O W S

Q. Q. Lu,t J. R. FONTAINE an d G . AUBERTINIns t i t u t N a t io na l de R eche rche e t de S6cur it 6, A v enue de B ourgogn e , 54501 V an doeu vre C 6dex, F ranc e

(Received 14 Marc h 1991; in revised fo rm 30 Novemb er 1992)

A bs t r ac t - -A pp ly ing t he t im e se r i e s ana lys i s i dea t o t he t em pora l and spa t i a l f l u id ve loc i t y co r r e l a t i ons ,a t h r ee -d im ens iona l L agran g i an m o de l fo r t he m ot io n o f pa r t i c le s i n tu rbu l en t fl ow s has been e s t ab l i shed .T h i s m o d e l h a s b e e n u s e d t o s i m u l a t e t h e e x p e r i m e n t s o f o t h e r w o r k e r s . T h e c o m p u t e d r e s u l t s a r ecom pared w i th t he ex pe r im en ta l da t a fo r t he pa r t i c l e d i spe r s ion , ve loc i t y co r r e l a t i ons and ve loc i t y decay .I n t h e c a s e w h e r e t h e m e a n t u r b u l e n t f lo w h a s o n e m a i n d i r e c t io n , t h i s m o d e l h a s b e e n e x t e n d e d t o i n c l u d ethe E u le r i an t em pora l ve loc i t y co r r e l a t i on ; a so -ca l led m ixed m ode l has bee n dev i sed . T h i s m ode l has beenused t o com p ute pa r t i c l e d i spe r s ion i n s t a t i ona ry , hom ogene ous , i so t rop i c and i ncom pres s ib l e t u rbu l ence .C om par i so n i s m ade w i th t he t he roe t i ca l l ong- t im e pa r t i c l e d i f fus ion coe ff ic ien ts fo r ca ses w here t hec ros s ing - t r a j ec to ry e f f ec t i s im por t an t o r un im por t an t . G ood ag reem en t i s ob t a ined .

K ey W ords : L agrang i an m ode l , pa r t i c l e s , t u rbu l ence

1 . I N T R O D U C T I O N

A n u m b e r o f c o m p u ta t i o n a l m e th o d s h a v e b e e n d e v e lo p e d t o d e s c r ib e t u r b u l e n t flo ws l ad e n w i th

par t ic les . These methods use e i the r Eu le r ian o r Lagrang ian ana lyses . In the Eu le r ian approach ,

b o th t h e f l uid c a rr i e r a n d t h e p a r t ic u l a t e p h a s e a r e a s s u m e d t o b e c o n t i n u o u s m e d ia a n d s et s o f

coup led d i f fe ren t ia l equa t io ns a re de r ived fo r each phase (e .g . Abb as e t a L 1980 ; Durs t e t a l . 1984;

E lg h o b a s h i e t a l . 1984) . In the Lagrang ian approach , pa r t ic les a re t rea ted ind iv idua l ly by so lv ing

th e d y n a m ic e q u a t i o n o f p a r ti c l e m o t io n a n d t h e b u lk p r o p e r t ie s o f t h e p a r t ic u l a t e p h a s e a r eob ta ined by averag ing over a s ta t i s t ica l ly s ign i f ican t number o f pa r t ic les .

T h i s p a p e r m o d e l s p a r ti c l e m o t io n i n t u r b u l e n c e b y a L a g r a n g i a n m e th o d . Go s m a n & Io a n n id e s

(19 81 ) p r o p o s e d a m o d e l t o a c c o u n t f o r t h e e f f e ct o f f l uid t u rb u l e n c e o n p a r ti c le s wh e n t h e m e a n

f lu id ve loc i ty i s kno wn and the f luc tua t ing pa r t i s ob ta ine d by sampl ing f rom a G auss ian p robab i l i ty

d is t r ibu t ion func t ion (p .d . f . ) whose va r iance i s p ropor t iona l to the loca l tu rbu len t k ine t ic energy .

T h e d y n a m ic e q u a t i o n o f p a r t i c l e m o t io n i s i n t e g r a t e d w i th t h e f l u id i n s t a n t a n e o u s v e lo c i t y

unc han ged un t i l a pa r t ic le -ed dy in te rac t ion t ime exp ires . Th is pa r t ic le -e ddy in te rac t ion t ime i sd e f i n ed t o b e t h e m in im u m o f t h e e d d y l if e tim e a n d t h e t r a n s i t t im e . In t h is m o d e l , t h e t u r b u l e n t

ins tan ta neou s ve loc i ty f ie ld is supposed to be un i fo rm wi th in the eddy . In o th e r words , th i s mod e l

does no t a ccou n t fo r the co n t inu i ty ef fec t (Csa nady 1963) which leads to a d i f fe rence be twee n the

par t ic le long i tud ina l an d t ransverse long- t ime d i f fus ion coef f ic ien ts (Cs anad y 1963 ; Reeks 1977; N i r& P ism en 1979). Shuen e t a l . (1983, 1985) used the s am e idea. Kall io & R eeks (1989) m odif ied th is

m o d e l b y d e t e r m in in g t h e e d d y l if e tim e w i th a n e x p o n e n t i a l p.d .f , d i s t r ib u t i o n h a v in g t h e m e a nequa l to the Lagrang ian in teg ra l t imesca le . Burnage & Moon (1990) advanced fu r the r in th i s

d i r e c ti o n . T h e i r m o d e l c o n t a in s b o th a r a n d o m t im e s c a l e a n d a r a n d o m l e ng th sc a le . T h e d y n a m ic

e q u a t i o n o f p a r t i c le m o t io n wa s i n t e g r a t e d w i th t h e f l u id i n s t a n t a n e o u s v e lo c i ty u n c h a n g e d u n t il

e i the r the in teg ra t ing t ime o r the d i s tance be twee n the pa r t ic le and a f lu id po in t i s g rea te r than the

c o r r e s p o n d in g r a n d o m s c a l e . T h e two r a n d o m s c a l e s a r e g iv e n b y r a n d o m s e l e c t i o n s f r o m two

P o i s s o n p r o c e s s e s wh o s e m e a n s a r e t h e t u r b u l e n t L a g r a n g i a n i n t e g r a l t im e s c a l e a n d t h e e d d y

lengthscale .O r m a n c e y & M a r t i n o n (19 84 ) i n t r o d u c e d a n o th e r w a y o f m o d e l i n g p a r t ic l e m o t io n i n t u r b u l e n t

f lows . They cons t ruc ted a sampl ing p rocess fo r the f luc tua t ing ve loc i ty o f a f lu id po in t dur ing af in ite in te rva l o f t ime . F or the f luc tua t ing ve loc i ty o f the f lu id a t the p a r t ic le pos i t ion how ever , they

? P r e s e n t a d d re s s : D e p a r t m e n t o f C h e m i c a l & N u c l e a r E n g i n e e ri n g , T h e U n i v e r s i ty o f C a l i fo r n i a , S a n t a B a r b a r a , C A 93106

U.S.A.

47

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348 Q, Q. LU et a l .

used ano ther r andom process which accoun t s fo r t he f lu id Eu le r i an long i tud ina l and t r ansver se

cor r e l a t ions . A s a r esu l t , t h is mo de l i nc ludes the con t inu i ty e f fec t on p a r t i c le d i sper s ion . Th ey

fo l lowed , s imul t aneous ly , a pa r t i c l e and a f lu id po in t un t i l t he d i s t ance be tween them exceeds

a ce r t a in g iven l eng thsea l e . Ber l emont et a l . (1990) pur sued th i s approach bu t expressed the

f luc tua t ing f lu id ve loc i ty a long the f lu id t r a j ec to ry as t he we igh ted sum of i t s pas t va lues p lus a

r and om var i ab le , a s sugges t ed by Par tha sa ra th y & Fae th (1990). These two m ode l s fo l lowed a flu id

po in t du r ing a f i n i t e t ime in t e rva l and used no i se t e rms in the t empora l and spa t i a l co r r e l a t ionre l a t ions tha t a r e mean-zero , Gauss i an va r i ab les .

The s t a r t ing po in t o f the p r esen t s tudy i s the t ime se r ies ana lys i s i dea (Box & Jenk ins 1976) . By

app ly ing i t t o bo th the f lu id Lagrang ian and the Eu le r i an spa t i a l co r r e l a t ions , t he f luc tua t ing f lu id

ve loc i ti e s a t two success ive pos i t i ons o f t he p a r t i c le a r e spec if ied . In t he p r esen t m ode l , t he f lu id

loca t ion a nd ve loc i ty change a t every t ime s t ep . The c o r r e l a t ion func t ions a r e speci fi ed on ly fo r

o n e t i m e s t e p o f c o m p u t a t i o n , A t , a n d f o r t h e d i s t a n c e d e v e l o p e d b e t w e e n t h e p a r ti c le a n d a f l ui d

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

mean-zero , Gauss i an va r i ab les .

Th i s pap er i s o rgan ized as fo l lows . Sec t ion 2 i s dev o ted to t he es t ab l i shm ent o f mode l 1 . The

cho ice o f t he co r r e l a t ion fun c t ions and the assoc ia t ed pa ram ete r s a r e d i scussed in subsec t ion 3 .1 .

The p red ic t ed r esu l ts f o r pa r t i c le d i sper s ion , f o r t he dec ay o f fl uc tua t ing ve loc i ty and fo r t hev e l o c it y c o r r e l a ti o n a r e c o m p a r e d w i t h e x p e r im e n t a l d a t a o f S n y d e r & L u m l e y ( 1 97 1) a n d o f W e l ls

& S tock (1983) in subsec t ions 3 .2 and 3 .3 . In subsec t ion 3 .2 , a compar i son i s a l so made wi th the

theore t i ca l r e su l t s o f Ni r & P i smen (1979) fo r ve loc i ty co r r e l a t ions o f t he copp er pa r t i c le o f the

e x p e r i m e n t o f S n y d e r & L u m l e y . T h e s e n s it iv i ty o f m o d e l 1 t o t h e t i m e s t e p a n d a n u m e r ic a l

pa ra me te r i s exam ined in subsec t ion 3 .4 . In subsec t ion 3 .5 , an ex tens ion o f m ode l 1 i s i n t rodu ced

tha t i nco rpora t es t he Eu le r i an t em pora l ve loc i ty co r r e l a t ion . Th i s es t ab l i shes mod e l 2 ( the mixed

mode l ) . Th i s mode l i s app l i ed to pa r t i c l e d i sper s ion in a s t a t i onary , homogeneous , i so t rop ic and

incomp ress ib l e t u rbu len t f l ow. The p red ic t ed long- t ime p ar t i c l e d if fus ion coef f ic i en ts a r e com pare d

wi th the theore t i ca l r e su l ts o f Csan ady (1963) , Ni r & P i smen (1979) and P i smen & N i r (1978), b o th

in the p r esence and the absence o f t he c ross ing- t r a j ec to ry e f f ec t .

2. E S T A B L I S H M E N T O F TH E M E T H O D

2 . I . Pa r t i c l e m o t i o n

The p resen t s tudy neg lec t s t he Ba sse t t e rm a nd the t emp ora l de r iva t ives o f the f luc tua t ing f lu id

ve loc i ty a long the t r a j ec to r i es o f so l id an d f lu id pa r ti c l es. Such a s imp l i fi ca t ion h as been jus t if i ed

f o r l o w tu r b u l e n c e i n te n s it ie s a n d m o d e r a t e d e p a r t u r e f r o m h o m o g e n e i t y ( O r m a n c e y 1 9 8 4 ) . T h e

cor r espond ing der iva t ives o f t he f lu id mean ve loc i ty , however , a r e p r ese rved . By neg lec t ing the

in f luence o f s t reaml ine cu rva tu re and the in t e r ac t ion be tw een par ti c l es , t he mo t ion o f a spheri ca l

and r ig id pa r t i c l e i s p r esen ted by the fo l lowing equa t ions :

dV 3 O(V - U)

PP "~- = 4dp PrCD(V -- U)IV - UI - 0.5pf dt + (pp - Pr)g [1]

a n d

d X- - = V [ 2 ]d t

where pp and p f a r e t he pa r t i c l e and f lu id dens it i es , r e spec t ive ly , V and U a r e the ins t an tane ous

ve loc i ti e s o f pa r t i c l es a nd f lu id , r e spec t ive ly , dp i s t he pa r t i c l e d i am ete r and g i s t he g r av i t a t iona l

acce l e r a t ion . The coef f i c i en t CD i s i n t roduc ed fo r t he d r ag t e rm and i s g iven by

24C D ~-- (-~C)p[1 + 0 .1 5 (R e) °'687] for (Re)p < 200,

where (Re)p , t he pa r t i c l e Reyno lds number , i s de f ined as

IU -- VldpR e ) p =

V

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A LAGRANGIANMODEL FOR SOLIDPARTICLES IN TURBULENT FLOWS 349

In th i s s tudy , t he l i f t f o r ces , due to e i the r pa r t i c l e ro t a t ion (Rub inow & Kel l e r 1961) o r t o f l u id

ve loc i ty g r ad ien t (Saf fman 1965) , a r e no t i nc luded , bu t t he v i r tua l mass t e rm i s .

The e f f ec t o f f lu id tu rbu lence on the pa r t i c les i s cons ide red bu t t he mo di f i ca t ion o f t he tu rbu le n t

f ie ld d ue to t he p r esence o f pa r t ic l es i s i gnored , There fo re , a l l t he mea n da t a o f t he f lu id fi eld,

i n c lu d i n g t h e m e a n v e l o c it y , t h e m e a n t u r b u l e n t k i n e t ic e n e r g y a n d t h e m e a n t u r b u l e n t k i n et ic

energy d i ss ipa t ion r a t e , a r e r egarded as known a priori. T h e y c o u l d b e o b t a i n e d b y m e a s u r e m e n t s

o r b y a t u rbu len t mo de l . To kn ow the s t a t is t i c p roper t i e s o f pa r ti c l es , each par t ic l e i s f o l lowed a longi ts t r a j ec to ry by in t eg ra t ing [1 ] and [2] . To do th i s, i t is necessa ry to kn ow the ins t an tane ous ve loc i ty

o f t he f lu id a t t he loca t ion p o in t s o f the pa r t i c le . S ince the f lu id mea n v e loc i ty i s supp osed to be

known, i t i s on ly necessa ry to es t imate the f luc tua t ing f lu id ve loc i ty a t t he loca t ion po in t o f t he

par t ic le .

2 . 2 . Mode l 1

At the ins t an t t , t he pa r t i c l e and a f lu id pa r t i c l e s t a r t ou t f rom the same pos i t i on X~. Af t e r one

t ime s t ep o f com puta t ion , At , they a r r ive a t Xp and Xr , r e spec tive ly , and the d i s t ance b e twee n Xp

and Xf i s As , a s sho wn in figu re 1 . A r e l a tive coord ina t e sys t em O -O Sf ~ i s de f ined . The r e l a tive

coo rd ina t e sys t em O -O ~f ~ i s chosen such tha t i t s o r ig in i s l oca t ed a t Xr and the e ax i s passes

th rou gh the tw o pos i t i ons Xf and Xp . He re Xs , Xf and Xp a r e the pos i t i on vec to r s i n t he abso lu t ec o o r d i n a t e s y s t e m o - x y z .

Since , i n p rac t i ca l app l i ca t ions , f l ow f ie lds a r e o f t en non-ho m oge neo us an d no n- s t a t ionary ,

the f luc tua t ing ve loc i ty u~ i s no rmal i zed by the squa re ro o t o f i t s loca l va r i ance to l e ssen the

e f fe c ts o f r e f er e n c e ti m e a n d p o s i ti o n . T h e n o r m a l i z e d f l u c t u at in g c o m p o n e n t i n t h e / - d i r e c t i o n

by W~, i.e.

UiW, = ~u~u: (i = 1, 2, 3),

whe re the su bsc r ip t s 1 , 2 and 3 r ep resen t , r e spec t ive ly , t he d i rec t ions o f the r e l a t ive O , ~ and [~

axes . The quan t i t i e s wi th overbar s i nd ica t e t he ensemble average va lues .

In the r e l a t ive coord ina t e sys t em O-O,~ ,D , t he normal i zed f luc tua t ing ve loc i t i e s a t pos i t i ons X , ,

X / , Xp a r e ass um ed to hav e the fo l lowing co r r e l a t ion r e l a tions :

a n d

Wi (X f )Wj (X s ) = W ~ ( X s ) W y ( X s ) f b ( A t ) ( i , j = 1, 2, 3) [ ]

Wi(Xp)Wj(Xf) .~. Wi(Xf)Wj(Xf)gij(A8) ( i , j ---- 1, 2 , 3) . [ 4 ]

Rela t ion s [3] and [4 ] a r e ca l l ed , re spec t ive ly , t he L agrang ian au to -co r r e l a t ion and Eu le r i an spa t i a l

ve loc i ty co r r e l a t ion func t ions . I t shou ld be po in t ed tha t t he two r e l a t ions a r e pos tu l a t ed fo r one

t im e s t e p o f c a l c u la t io n , A t. T h r o u g h o u t t h is p a p e r , n o s u m m a t i o n c o n v e n t i o n i s u s ed .

O

Figure I. The locations of the particle and a fluid particle at the instants t and t + At.

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350 Q.Q . LU et al.

O n l y t h e i s o t r o p i c c a s e is c o n s i d e r e d . T h e R e y n o l d s s t r es s c o m p o n e n t s u~uj a r e , t h u s , ze r o f o r

i ~ j an d th e co r r e l a t i o n r e l a t i o n s [3] an d [4 ] b eco m e :

Wj(Xf)W~(Xs) = W I ( X s ) W i ( X , ) f L ( A t ) (i = 1, 2, 3) [5]

a n d

W i ( X p ) W i ( X f ) = Wi(Xf)Wi(Xf)gi i (A$) (i = 1, 2, 3). [6]

Th e id ea o f a t im e se r i e s an a ly s i s (B o x & J en k in s 1 9 7 6) i s u sed f o r t h e s to c h as t i c p r o cess o f th e

f lu id p o in t f r o m X s t o X f . Th u s ,

W~(Xf) = a, W,.(Xs) + ~, (i = 1, 2, 3), [7]

wh ere a j ( i = 1 , 2 , 3) a re c oef f ic ien ts ye t to be de te rm ined and ~t~ ( i = 1 , 2 , 3 ) a re mea n-ze ro , ran do m

v ar i ab l e s i n d e p en d en t o f W, .(Xs) b u t n o t n ecessa r i l y Gau ss i an , w h o se p r o p e r t i e s w i l l b e d i scu ssed

la t e r. T o d e t e r m in e th e co e f f ic i en t s a~ , t h e tw o s id es o f [7 ] a r e m u l t ip l i ed b y W~ (X ~) an d th e en sem b le

av e r ag e i s t ak en . Af t e r acco u n t in g f o r t h e i n d ep e n d en c e o f ~t; an d W~ (X ,) , t h e f o l lo w in g r e l a t i o n

i s o b ta in ed :

W~ (Xf)W~(X~ ) = a~ W,(X~ )W~(X~ ) (i = 1, 2, 3). [8]Subst i tu t ion of [5] in to [8] g ives

at = f ~ ( A t ) . [9]

S qu a r in g b o th s id es o f [7 ] an d t ak in g th e en sem b le av e r ag e , t h e v a r i an ce o f a~ i s d e t e r m in ed to b e

( ¢ , ) ~ = x / 1 - a ~ ( i = 1 , 2 , 3 ) .

I t i s ev iden t f rom the def in i t ion of a~ tha t wh en Xf ap pro ac he s X~, a~ --* 1 and thus (a, )~ - -* 0 . In fac t ,

i f ~t~ i s r ep r e sen ted b y a G au s s i an wh i t e n o i se , f o r h o m o g e n eo u s tu r b u len t f l o ws , [7 ] w i l l r ed u ce

t o t h e e q u a t i o n u s e d b y P a r t h a s a r a t h y & F a e t h ( 1 99 0 ). F u r t h e r m o r e i f a~ i s r e p r e s e n t e d b y a n

ex p o n en t i a l f u n c t io n , [7 ] w i l l b e i d en t i ca l t o t h e equ a t io n o f K a p lan & D in a h (1 9 88) an d i f A t is

v e r y sm a l l co m p ar ed to t h e Lag r an g ian in t eg r a l t im esca l e , [7 ] w i l l b e equ iv a l en t t o t h e Lan g e v inequ a t io n (W ax 1 9 54 ; D u r b in 1 9 80 ; S aw f o r d & H u n t 1 9 86 ) . F o r t h e c lo se s im i l a r i t y o f [7 ] w i th t h e

L a n g e v i n e q u a t i o n , t h e s iz e o f th e c o m p u t a t i o n t i m e s t e p A t m a y b e s u b j e c t e d t o c e rt a i n l i m i t a t io n s

( see D u r b in 1 9 80 ) .

I f t h e co n cep t o f a tim e se r i e s an a ly s i s i s ex t en d ed to r ep r e sen t t h e e f f ec t o f sp a t i a l d i sp l a cem en t

o n th e f l u c tu a t in g v e lo c i t ie s o f t h e f l u id a t t h e p o in t s X f an d X p ,

W~(Xp) = b, W ,(Xf ) + fl, (i = 1, 2, 3), [10]

w h e r e b~(i = 1 , 2 , 3 ) an d f l; ( i = 1 , 2 , 3 ) has the sam e sense as a~ and a~ in [7]. By the pro ce du re

a l r ead y ad o p ted f o r [7 ] , i t i s f r o m [1 0 ] t h a t

b i = g i i ( A s ) . [ l l ]

In t r od uci ng [7] in to [10] to e l iminate W~(Xf) ,

W,(X p) = a,b,W~(X~) + b,~t ,+ fl, (i = l, 2, 3). [12]

D e n o t in g th e t e r m s u n d e r l in ed in [12 ] b y ~ ,~ , [ 12 ] can b e r ewr i t t en a s

W ~ ( X p ) = a , b , W ~ ( X ~ ) + ~ , ( i = 1 ,2 ,3) . [13]

H er e ~b~ ( i = l , 2 , 3 ) a r e a ssu m e d to b e m ea n - ze r o , Ga u ss i a n r an d o m v a r i ab l e s . T o co m p le t e [1 3] ,

t h e s t an d a r d d ev ia t io n s ( tr ~)~ o f ~b~ a r e n eed ed . B y squ a r in g b o th s id es o f [13 ] an d t ak in g th e

en sem b le av e r ag e , a s su m in g th e in d e p en d e n ce o f W, .(X, ) an d ~O~, t h e f o l lo win g i s o b ta in ed :

(a~,),- = ~/ 1 2 2- a , b , i = 1 , 2 , 3 ) . [14]Eq u a t io n [13 ] t o g e th e r w i th [14 ] i s c a l l ed m o d e l 1. Th e m e th o d e s t ab l i sh ed ab o v e i s b a se d o n th e

f lu i d L a g r a n g i a n t e m p o r a l v e l o c it y c o r r e l a t io n o f th e s t o c h a s t ic p r o c e s s r e p r e s e n ti n g t h e m o t i o n

o f t h e f lu id p a r ti c l e f r o m X , t o Xr . I t w i l l b e ca l led th e L ag r an g ian t em p o r a l co n s t r u c t io n .

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A LAGRANGIAN MODEL FOR SOLID PARTICLES IN TURBULENT FLOWS 351

In the p rese n t s tudy , [13] i s used as the m ode l equa t ion . I t rep resen ts the f luc tua t ing ve loc i tie s

o f the f lu id a t successive loca t ions o f the pa r t ic le and inc ludes bo th the t ime a nd space e f fec ts o f

the tu rbu le n t f ie ld v ia the coef f ic ien ts a i and b~. Th is m ode l , in co n t ras t to the con ven t io na l t ime

ser ies m etho d , ha s no res t r ic t ion tha t u~ and f li a re m ean-ze ro , G auss ian va r iab les . Ho wev er , ther e q u i r e m e n t t h a t t h e a p p r o p r i a t e l i n e ar c o m b in a t i o n h a s t h e s e p r o p e r ty , d o e s p l a c e t h e r e s tr i c ti o n

on the p roper t ie s o f ~ ; and fl~ .

I t i s no te d th a t th e coord ina te sys tem emp loyed in f igure 1 i s on ly es tab l ished fo r one t im e s tepa n d i t s h o u ld b e c h a n g e d c o n t i n u o u s ly i n t h e c o u r s e o f c o m p u ta t i o n .

2.3. The calculation procedures

(1) At t = 0 , the part ic le posi t ion and veloci ty are given. The ini t ia l f luctuat ing f lu id veloci tyco m po ne nts ui ( i = 1 , 2 , 3) a t th e part ic le positi__on are o btain ed fro m th e G aus sian variables

sa t i s fy ing the p .d . f , wi th the loca l va r iances u 2 . The ins tan tan eous f lu id ve loc i ty can thenb e o b t a in e d b y a d d in g t h e k n o w n m e a n v e lo c i ty a n d t h e f l u c tu a t in g p a r t .

(2 ) From the ins tan taneous f lu id ve loc i ty found above , the f lu id po in t pos i t ion a t t ime At , Xf

c a n b e c a l c u l a te d b y t h e E u l e r - Ca u c h y m e th o d ( r e fe r r in g t o f i gu r e 1 ). U s in g t h e g iv e n

par t ic le in i t ia l ve loc i ty and the ins tan taneou s f lu id ve loc i ty jus t ob ta ined , th rou gh [1 ] and

[2] , the pos i t ion o f the pa r t ic le a t the ins tan t At , Xv , i s ob ta ine d by the R un ge -K ut t a m etho d .Disp lacement As can then be ca lcu la ted (see f igure 1 ) .

(3) Es tab l i sh the re la t ive coord in a te sys tem O -O.=f~ . Fr om [9 ] and [11] , the coef f icien ts a t andb~ in [13] a re es t ima ted . Th e ran do m te rms ~ hav ing the s tand ard dev ia t ions g iven by [14]

a r e g e n e r a t e d b y c o m p u te r w i th t h e a id o f t h e s o f twa r e GAS D E V (Ve t t e r l i n g et al. 1988).The f luc tua t ing ve loc ity o f the f lu id po in t a t Xp ( the new par t ic le pos i t ion com pute d a t s tep 2 )

c a n b e f o u n d f r o m [1 3 ] .

(4) Le t Xp be the s ta r t ing po in t , i . e. X , fo r the nex t t ime s tep and re pea t s teps 2 and 3 un t i lc o m p l e t io n o f t h e c o m p u t a t io n .

3. V E R I F I C A T I O N O F M O D E L 1

3.1. Choice of the correlation functions and the associated parameters

In w h a t f o ll o ws , th e f o l l o win g f o r m s o f t h e t e m p o r a l a n d s p a ti a l c o r r e la t i o n f u n c t i o n s p r o p o s e d

by Frenk ie l (1948) a re adop ted :

W~ (Xf)W,(X,) -- W,(Xs)W ~(X,) exp (i = 1, 2, 3) [15]

a n d

Wi(Xp)W t(Xr) = W~(Xf)W~(Xf)exp cos (i = 1, 2, 3). [161

where AI i s the long i tud ina l l eng thsca le and A 2 a n d A 3 a r e the t ransv erse leng thsca les . Of course ,

o th e r fo rm s o f the co r re la t io n func t ion s can be used to rep lace [16], e .g . expon en t ia l func t ions . In

th e p r e s e n t s t u d y , a l l t h e s a m p le c o m p u ta t i o n s a r e c o n d u c t e d f o r i s o t r o p i c a n d i n c o m p r e s s ib l etu rbu lence . T here fo re , the Lag rang ian in teg ra l t imesca les eL and Eu le r ian leng thsca les Ai in [15]and [16] can be es t ima ted by

a n d

U 2

TL= ~2 = z3 = C e , - - , [17]

L 2

AI = Cc2"r ~ [18]

A 3 -- A2 -- Ce3A I, [19]

I JMF 19 /2 H

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352 Q . Q . L U et al.

w h e r e u = ~ (u ~ + u z + u ] ) a r e 8 i s th e t u r b u l e n t k i n e t i c e n e r g y d i s s i p a t i o n r a t e . R e l a t i o n s [ 1 7 ] a n d

[18] ca n be fo un d in Hinz e (1975) . A cco rd in g to Hin z e (1975, p . 398) , Ce l a nd Ce2 a re no t

i n d e p e n d e n t b u t s a t is f y th e f o l l o w i n g r e la t io n :

0.588C e 2 = - - [2 0]

Ce~

A g a i n , b a s e d o n t h e d i s c u s s i o n o f t h e t h e o r y o f i s o t r o p i c , i n c o m p r e s s i b l e t u r b u l e n c e b y H i n z e ( 19 75 ,

p . 4 2 6) , C e , = 0 .2 3 5 a n d C e3 = 0 . 5 . F o r c o n v e n i e n c e , t h e v a l u e s o f t h e a s s o c i a t e d p a r a m e t e r s u s e d

t h r o u g h o u t t h e p r e s e n t s t u d y a r e l is t e d i n t a b l e 1 .

I n e a s e s o f i s o t ro p i c o r w e a k l y a n i s o t r o p i c t u r b u l e n c e , t h e f o l l o w i n g re l a t io n s h o l d b e t w e e n t h e

f l u c t u a t in g f lu i d v e l o c it y v a r ia n c e s i n t h e a b s o l u t e c o o r d i n a t e s y s t e m o - x y z a n d i n t h e r e l a t i v e

c o o r d i n a t e s y s t em O - O , F f l :

U l 2 m 2 2 2 2 2 2l ~ u x + m l u y + n , u ~

U2. .~_ 2 2 2 2 2 212U x + m 2u~ + n2uz

U 2 ~ 2 2 2 2 2 2 .13Ux q- m 3 u y - k n 3 u z ,

[21]

[ 2 2 ]

[23]

wh ere 1 ,, m m, n l ; 12, m 2 , n2 an d / 3 , m 3 , n3 a re the d i r ec t io n cos ines o f the O, ~ an d f l a xes r e la t ive

to the x , y and z axes , r espec t ive ly .

3.2. Simulat ion o f the experiment o f Snyde r & Lum ley (1971)

O n e o f t h e m o s t c o m p r e h e n s i v e e x p e r i m e n t s o n p a r t ic l e m o t i o n i n t u r b u l e n t f lo w s i s t h a t m a d e b y

S n y d e r & L u m l e y (1 97 3) . W i t h t h e u s e o f a g r i d s y s t e m , t h e y p r o d u c e d a n e a r l y i s o t ro p i c d e c a y i n g

t u r b u l e n t a i r f l o w ( a i r d e n s i t y = 1 .2 05 x 1 0 - 3 g / c m 3 an d k in em at ic v i s co s i ty = 14 .937 x 10 -2 cm 2/s ).

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

e x p e r i e n c e b o t h i n e r t i a a n d c r o s s i n g - t r a j e c t o r y e f f e c t s . T h e i r d i a m e t e r s , d e n s i t i e s a n d k i n e m a t i c

v i s c o s i ty a r e g i v e n i n t a b l e 2 . T h e p a r t i c le s w e r e i n j e c t e d i n t o t h e t u r b u l e n t f l o w a t x / M = 20 . The

m e a s u r e m e n t w a s c a r r ie d a t o r b e y o n d x / M = 6 8 , w h e r e x r e p r e s e n t s t h e d i s t a n c e f r o m t h e g r i da n d t h e g r i d s p a c i n g M = 2 .5 4 c m .

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

d a t a a r e g i v e n b y

U x = 655 (cm /s), U , = 0, U~ = 0, [24]

-5 - (Ux)2 [25]

42.4 - 16

- 5 u x ) ~ [2 61

u Y = x )3 9 . 4 ~ - - 1 2

nd

2 ~ 2u~ - u,.. [27]

F r o m T a y l o r ' s f r o z e n h y p o t h e s i s a n d t h e r e l a t i o n d k / d t = - 8 , w h e r e k, t h e t u r b u l e n t k i n e t ic

e n e r g y , i s d e f i n e d a s

k -- ½(u~ + u~ + u2), [28]

Table 2 . Parameters of the par t ic les used in the exper imento f S nyde r & Luml ey 0971)

Table I . Values of the associa ted coeff ic ients and the t ime Ho l low Corns tep At glass pol len Glass Cop per

Ce I Ce, Ce3 At (s) Diam eter ~ m ) 46.5 87.0 87.0 46.5

0.235 2.5 0.5 0.001 De nsity (g/cm 3) 0.26 1.00 2.50 8.90

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A L A G R A N G I A N M O D E L F O R S O L I D P A R T IC L E S I N T U R B U L E N T F L O W S 353

t

6

5

4

3

2

1

Hol low glassCorn pollen .... /

Glass ..... o /C o p l ~ T

Hol low glass 0 /

Corn pollen +Glass [ ] =C opper × y . . . . . ~ . . . . . . . .

+ . . . - . . . . . ...2...'..'..2 ..2 :~'222222. . . . . . . . . . . . . .

0 0.1 0.2 0.3 0.4 0.5Time (s)

Figure 2a. Pred icted and experimentalparticle transverse dispersions.

t h e t u r b u l e n t k i n e t ic e n e r g y d i s s i p t io n r t e i s o b t i n e d s

e = ~ 4 2 . 4 : X 1 6) 2 ' x 2 [29]- 3 9 . 4 \ ~ - 12

In a l l the fo l lowing com par i son s , the l ine s r epre sent the com pu ted re su l ts , whi le the sym bols

a re expe r imenta l da ta . F igure 2a compares the pred ic ted and expe r imenta l t r ansve r se pa r t i c l e

d i spe r s ions . To examine the ove ra l l e f fec t o f grav i ty on pa r t i c l e d i spe r s ion , f igure 2b shows the

d i s p l a c e m e n t o f pa r ti c le s i n t he g r a v i ty d i r e c t ion ( t he x - d i r e c t i on i n t he s t udy ) . T he c om p a r i s on

in f igure 2a i s in a f a i r agreement , a l tho ug h apprec iab le d i f f e rences a re obse rved . A la rge r d i ff e rencei s no t ic e d f o r he a vy pa r t ic l es . T he c o m pa r i s on s ugge s ts t ha t t he p r e s e n t m ode l w o r ks f o r t he ho l l ow

g l as s par t ic l e, f o r w h i c h s om e L a g r a ng i a n a pp r oa c he s f ai l ( O r m a nc e y & M a r t i no n 198 4; B e r l e m on t

e t a l . 1990) . As pred ic ted by prev ious s tud ie s (e .g . Reeks 1977) , f igure 2b impl ie s tha t pa r t i c l e sd i spe r se quick ly in the d i rec t ion of grav i ty due to the cont inu i ty e f fec t .

F i gu r e 3a c om pa r e s t he num e r i c a l a nd e xpe r i m e n t a l r e s u l t s on t he de c a y o f t he t r a n s ve r s e

ve loc i ty f luc tua t ion of the pa r t i c l e s . He re the pa r t i c l e f luc tua t ing ve loc i ty u i s norma l ized by the

squa re o f the longi tud in a l mean ve loc i ty U [he re 655 (cm/s )] . Q ua l i t a t ive agreemen t i s no ted bu t

~ 4

2

00

H o l l o w glassCorn pollen ....

Gla ss .....

0.I 0.2 0.3 0.4 0.5Ti me (s)

Figure 2b. Predicted particle longitudinaldispersions.

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3 5 4 Q . Q . L U e t a L

35

30F l u i d

H o l l o w glass . . . . .

C o r n p o l l e n . . . .

G l a s s . . . ..

2 5 Copper ......... xH o l l o w glass O

Corn pollen + ra2 0 G l a s s [ ]

. Copper X Xx x ..........................%--.--.' . .

~ 1 5 ..................... '." -'.".' . . . . . . . . . . ....... ....:.:.:.:~"7 i - . - ~- . . . . . ~

; . - -=,. .. .. .. .. .. .. .. .. .. : ' : ' ~ ' : ' 1 3 ' , 0 . . . . . . . . .1 0

r : _ , o +_ _ ~ _ . . . . . 6 . . . . . _ . ~ ~ '

6 . . . ¢ + - - ¢ - - - ' ~ - . •s l , l , - - $' ' ~ - - ~ * - ' - ' ~ - - -

0 I I I I0 0.1 0.2 0.3 0.4 0.5

Tun e (s)

Figure 3a. Predicted and experimental transverse fluctuating particle velocitydecays.

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

s a m p l i n g r a t e u s e d i n t h e e x p e r i m e n t . F i g u r e 3 ( b ) g i v e s t h e p r e d i c t e d l o n g i t u d i n a l f l u c t u a t i n g

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

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

o f o t h e r i n v e s t i g a t o r s ( e .g . R e e k s 1 97 7) .

F o r t h e r e a s o n s g i v e n b y N i r & P i s m e n ( 19 79 ), n o a t t e m p t w a s m a d e i n t h i s p a p e r t o c o m p a r e

t h e p r e d i c t e d r e s u l t s w i t h S n y d e r & L u m l e y ' s (1 9 71 ) p a r t i c l e v e l o c i t y c o r r e l a t i o n s . F i g u r e 4 , i n s t e a d ,

c o m p a r e s t h e c o m p u t e d L a g r a n g l a n c o r r e l a t i o n s o f t h e c o p p e r p a r t i c le t r a n sv e r s e v e l o ci t y a l o n g

w i t h t h e t h e o r e t i c a l r e s u l t s o f N i r & P i s m e n ( 19 7 9) . R e a s o n a b l y g o o d a g r e e m e n t i s o b s e r v e d i n

f i g u r e 4 , w h e r e x / M r e p re s e n t s t h e l o c a t i o n w h e r e t h e c a l c u l a t i o n w a s m a d e o f t h e c o r r e l a t i o nc o e f f ic i e n t d e f i n e d a s b e l o w :

R L ( A t ) = ui(At)u,(O ) (i = x ,y ). [30]

u/Z(0)

T h e Lagran~an c o r r e l a t i o n t i m e s c a l e s a re d e f i n e d a s

I ~R , , ( ~o )d t p ( i =x , y ) . [31]~ii -~" L

do

3 5

30

2 5

4545

5

1o

Fluid- Ho l low lass . . . . .

Corn po ll en . . . .Glass . . . . .

- Coppo r .........

m

I I I I

0 0 . I 0 . 2 0 . 3 0 . 4

T i m e ( s)

F i g u r e 3 b . P r e d i c t e d l o n g i t u d i n a l f l u c t u a t i n g pa r t ic . le e l o c i t y d e c a y s .

0.5

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0.8

i 0 . 6

.4

0.2

~~ ' ~ , ~ E lv

\ \ '~ x / M = 7 3 . . . . .\ \ , 1 2 3 x / M = 1 7 1 . . . .~ \ • x / M = 4 1 OX \ ' , , x /M = 7 3 +

\ ' \+ " , , tTt Q

I I i ~ t

40 80 120 160 200 240 280Tune m s)

F i gur e 4 . C ompar i son o f t he numer i ca l and t heor e ti ca l (N i r & P i smen 1979) Lagr ang i an co r r e l a t ions o fthe copper par t ic le t ransverse veloci ty .

As we know, inc reas ing pa r t ic le ine r t ia wi l l reduce the pa r t ic le f luc tua t ing ve loc i ty . On the o the r

hand , th i s wi l l a l so lead to an inc rease in the pa r t ic le co r re la t ion . For examin ing the la t te r , we

ca lcu la te the lon g i tud ina l and t ransverse Lag rang ian cor re la t ion coef f ic ien ts o f the pa r t ic le

f luc tua t ing ve loc i ty . S ince the pa r t ic le d i spers ions in the exp er ime n t a re sym me tr ic re la t ive to thex a x is , o n ly t h e v e lo c it y c o r r e l a ti o n s f o r th e x - a n d y - c o m p o n e n t s a r e c o m p u te d . T h e c o m p u ta t i o n

is pe r f o rm ed a t th ree sec t ions , x / M = 41 , 73 and 171. The in teg ra l t imesca les o f these co r re la t ion

a r e d e n o t e d b y T ~ i = x , y ) a n d t h e i r c o m p u te d v a lu e s a r e t a b u l a te d i n t a b l e s 3 - 6 . I t c a n b e s e e n

f rom these tab les tha t th e pa r t ic le ve loc i ty co r re la tes mo re in the d i rec t ion o f g rav i ty than in thed i rec t ion no rm al to i t. A s a resu l t , the in teg ra l t imesca le i s l a rge r in the g rav i ty d i rec t ion than i t s

coun te rpar t in the normal d i rec t ion . In tab les 3 -6 , the t imesca le ~L i s computed by [17] .

3.3. Simulation o f the experiment o f W ells d Stock (1983)

Wells & Stock (1983) used an iden t ica l g r id sys tem to Sn yder & Lum ley ' s (1971) to p ro duc e a

tu r b u l e n t a i r f l o w , b u t t h e m a in d i r e c t i o n o f t h e f l o w wa s h o r i z o n t a l . T h e m e a n d a t a f o r t h etu rbu len t f ie ld a re

Ux = 655 (cm/s) , Uy = O, U~= O, [321

1

--7 _ U~)2 [33]

u x - 5 3 . 2 2 4 ( M - 7 .0 53 ) '

Table 3 . Ho l low glass par t ic le Table 4 . Co rn Pol len par t ic le

X X

T~., m s) ~ , m s) ~,~ m s) M T~., m s) ~. , . m s) ~,~ m s)

41 14.93 15.83 16.68 41 29.90 29 .88 16.6873 36.82 38.82 36.21 73 42.60 39.55 36.21

171 95.39 96.70 95.79 171 80.62 64.62 95.79

Tabl e 5 . G l a s s pan i c l e Tab l e 6 . C opp e r

X X

M ~. , (m s) ~P>, (m s) ~L (m s) M ~ ., (ms) P,,. m s) ~ (m s)

41 41.07 50.96 16.68 41 53.86 56.23 16.6873 60.69 61.55 36.21 73 63.13 62 .95 36.21

171 85.11 72.30 , 95.79 171 90.13 77.25 95.79

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356 Q . Q . LU et al.

2Uy =

(ux ) 2

(6 5 4 6 _ ; , _ 8 . 8 6 7a n d

u 2 2. = U y .

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 d i s s i p a t i o n r a t e i s

l /

[ 3 4 ]

[35]

2 ] t3 6 jt ( x ' - 256.546 ~ -- 8.867

H e r e e i s o b t a i n e d i n t h e s a m e w a y a s t h e e x p e r i m e n t o f S n y d e r & L u m l e y ( 19 71 ).

I n t h i s e x p e r i m e n t , 5 a n d 5 7 ~ t m g l a s s p a r t i c l e s w e r e c h a r g e d b e f o r e t h e g r i d a n d a n a d j u s t a b l e ,

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

o n t h e p a r t i c l e s a n d , t h e r e f o r e , t h e p a r t i c l e t e r m i n a l v e l o c i t y V t. T h e d e n s i t ie s o f t h e 5 a n d 5 7 ~ tm

par t i c l es a r e 2 . 475 (g /cm 3) an d 2 . 420 (g /cm 3), r espec t ive ly . In th i s s im ula t io n , g rav i ty i s in the

n e g a t i v e d i r e c t i o n o f t h e y a x i s a n d t h e m a i n f l o w d i r e c t i o n c o i n c i d e s w i t h t h e x a x i s .F i g u r e s 5 a n d 6 g i v e t h e p r e d i c t e d a n d m e a s u r e d t r a n s v e r s e d i s p e r s i o n s f o r 5 a n d 5 7 / ~ m p a r t i c le s ,

r e s p e c ti v e l y . R e a s o n a b l e a g r e e m e n t i s o b s e r v e d w h e n t h e p a r t i c l e t e r m i n a l v e l o c i t y l it i s re l a t i v e ly

s m a l l . A s V~ b e c o m e s l a r g e r , t h e c o m p u t e d a n d e x p e r i m e n t a l d a t a d i s a g r e e c o n s i d e r a b l y . I n t h e

e x p e r i m e n t , t h e t e n d e n c y f o r t h e 5 7 / ~ m p a r t i c l e s t o d i s p e r s e f a s t e r t h a n t h e 5 / ~ m p a r t i c l e s w h e n

V t = 0 w a s r e p o r t e d . T h i s c a n n o t b e n o t e d i n t h e p r e d i c t i o n d a t a .

F i g u r e 7 a a n d 7 b c o m p a r e t h e p r e d ic t e d a n d e x p e r i m e n t a l d a t a o f 5 / ~ m p a r ti c l e f l u c t u a t in g

v e l o c i t y d e c a y i n t h e l o n g i t u d i n a l a n d t r a n s v e r s e d i r e c t i o n s , i . e . 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 ti v e l y . F i g u r e s 8 a a n d 8 b s h o w t h e r e s u l t s f o r t h e 5 7 / ~ m p a r t i c l e s . I t is se e n t h a t t h e p r e d i c t e d

r e s u l t s a g r e e f a i r l y w e ll w i t h t h e e x p e r i m e n t a l d a t a w h e n V t i s n o t t o o l a r g e . A s V t g e t s l a rg e r , t h e r e

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

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

2.5

2

t

1.5

1

0.5

I I I I

/ s ss °

~ p a~ Soo °s° s •

I ~ ° ' ° S s s S

s •Vt=0.00Vt~5.86 . . . . .

Vt=13.31 . . . .Vt=23.65 . . . .

Vt=0.00 oVt=5.86 +

Vt=13.31 r~Vt=23.65 x

I I I I0 20 40 60 80 100

x/M

Figure 5. P redicted and experimental transverse dispersions of 5/~m particles.

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A L A G R A N G I A N M O D E L F O R S O L ID P A R TI C L E S IN T U R B U L E N T F L O W S 357

3

2.5

2

.~ 1.5

1

0.5

s

s J

ss S

s~ S

ss S ° . °

1 -I- *"

Vt=O O0 . . . . • / :Vt=25.8 . . . . . . / + ° o S

° ° s .Vt=54.5 .. . . 9 / * * ° ~ [~,o °o . ° °

Vt=81.5 .. .. . / ,,' ,,- .. -V t f f i 1 2 1 . 6 ~ o / + , . - , -° • .

• ° ° °

V t = O O 0 o / , , , . ,'J~,

Vt=25.8 + / + ' . ' /...* ° ° ,o -w 5 4 5 . -

Vt----121.6 ,~ ÷

0 I I I I

0 20 40 60 80 1 0 0

x M

F i g u r e 6 P r e d i c t e d a n d e x p e r i m e n t a l t r a n s v e r se d i sp e r s i o n s o f t h e 57/~m p a r t i c l e s

Fi gu r e 9a a n d 9 b sh ow th e p r e d i c te d Lagran g ian p ar ti cl e v e l oc i ty c or r e la t i on s i n th e lon g i tu d i n a l

d i r e c t ion . A c c or d i n g to th e exp e r i m e n ta l i n for m at i on o n th e r a t i o o f th e Eu l er ian an d Lagr ang ian

t i m e sc a l e s w e c an d e d u c e fr om th e e xp e r i m e n ta l d a ta th e Lagr an gi an p ar t i c le ve l oc ity c or r e l a t ion

in the long i tudina l d irect ion for the case o f Vt = 0 . Th e der ived results are a l so p lotted in f igures

9 a a n d 9 b .

4

12

10

4t4t

6

F l u i dVt=0.00 .. . .Vt=5.86 .. . . .

V~13.31 . . . .

Vt=23.65 .. . . .Vt=0.00 O

Vt=5.86 +Vt=13.31 r~

Vt=23.65 ×

i I I

2 O 4 0 6 0 8 0

x M

0 0 1 0 0

F i g u r e 7a. Predicted a n d e x p e r i m e n t a l l o n g i t u d i n a l v e l o c i t y d e c a y s o f t h e 5 pm particles.

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5 8

14

Q . Q . L U e t a l .

I I

12

10

8

4t4t

6t

F l u i d - -Vt--0.00 ....

Vt=5.86 - .. ..Vt=13.31 ....

Vt=-23.65 .....

/ I I I I00 20 40 60 80 100

x/M

Figure 7b Predicted transverse velocity decays of the 5/Jm particles

T a b l e s 7 a n d 8 s u m m a r i z e t h e p r e d i c t e d v a l u e s o f z ~ f o r th e 5 a n d 5 7 m p a r t i c le s , r e s p e c t iv e l y .

H e r e t h e s u b s c r i p t x x s t il l d e n o t e s t h e l o n g i t u d i n a l d i r e c t i o n ( t h e h o r i z o n t a l d i r e c t i o n ) . F o r t h e c a s e

o f V , = 0 i t c a n b e d e d u c e d f r o m t h e e x p e r i m e n t t h a t ~ $ x = 1 9 . 6 a n d ¢ P x = 3 7 . 6 (m s ) f o r t h e 5 a n d

5 7 / ~ m p a r t i c le s , r e s p e c t i v e ly . T h e v a l u e s p r e d i c t e d f o r t h is c a s e a r e 2 5 . 0 3 a n d 3 8 . 3 5 ( m s ) fo r t h e5 a n d 5 7 m p a r t i c le s , r es p e c t i v e ly .

4

12

10

6t

Fluid

Vt=0.00 .. . .

Vt=25.8 .. . . .Vt=54.5Vt--81.5

Vt=121.6Vt=0.00Vt=25.8Vt=54.5Vt=81.5

Vt=121.6

oo *' ,° o*

oO S° °° °° .~ l

7 . . . - . s . ' ° . ~ -

o

B

0 I I I I

0 20 40 60 80 100x/M

Figure 8a Predicted and experimental longitudinal velocity decays o f the 57/~m particles

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359

12

10

4

8

~t.g

0 20 100

Flu id

Vt=O.O0 . . . .V t=25 .8 . . . . .V t=54 .5 . . . .V t=81 .5 . . . . .

Vt=121.6

4 0 6 O 8 O

x / M

A LAGRA NGIAN MOD EL FOR SOLID PARTICLES IN TURB ULENT FLOWS

14

Figu re 8b. Predicted transverse velocity decays of the 5 7/~m particles.

3.4. Sensi t iv i ty o f mo del I to the time s tep A t

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

o f A t . F o r t h is p u r p o s e , t h e e x p e r i m e n t a l s y s t e m o f S n y d e r & L u m l e y ( 19 71 ) w a s u s e d t o c a l c u l a te

t h e l o n g i t u d i n a l a n d t r a n s v e r s e p a r t i c l e d i f fu s i o n c o ef f ic i e n ts , D1~ a n d D 22 . T h e c o m p u t e d r e s u l t s

I I I I

0 8

.6

i 0 . 4

0.2

Vt=0.00Vt=-5 .86 . . . . .

V t=13 .31 . . . .V ~ 2 3 . 6 5 . . . . .

Vt=0.00 o

~s q : - : : - - - - . . . . . . . . . . .

0 20 4O 6O 80 100Time (ms)

Figu re 9a. Predicted Lag rangia n longi tudina l velocity correla tions for the 5/~m particles as well as theexperimental c orrela tion in the case o f Vt = 0.0.

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360 Q . Q . L U e t a l .

I I I I

0 .8

0 . 6

¢J

o=

0 . 4

¢.)

0 .2

¢ .

O

,,,...,% ,

o

o

Vt=O.O0V t = 2 5 . 8 . . . . .V t = 5 4 . 5 . . . .

V t = 8 1 . 5 . . . . .V t = 1 2 1 . 6 . . . .

Vt--O.O0 o

O

0 20 40 60 80 100 120 140T i m e ( m s )

F i g u r e 9 b . P r e d i c t e d L a g r a n g i a n l o n g i t u d i n a l v e l o c i t y c o r r e l a t i o n s f o r t h e 5 7 / z m p a r t i c le s a s we l l a s t h eexpe r ime n ta l co r r e l a t io n in the case o f l i t = 0 .0 .

are p resen ted in tab le 9 . The tab le a l so shows tha t the resu l t s a re s l igh t ly dependen t on the

t im e s te p A t . O u r e x p e r ie n c e w i th t h e c o m p u ta t i o n s h o ws t h a t i n o r d e r t o e n s u r e g o o d p r e d i c t io n s ,the fo l lowing cond i t ion may be regarded as a conse rva t ive c r i te r ia fo r the t ime s tep At in

mode l 1 :

• / L ~ A I A 2 /.A t = m m [ f , , . [37]

3.5. Ex tension o f mo del I to inc lude the f lu id Eulerian tempo ral corre la t ions

As sh ow n by the a bove p red ic ted resu l t s , m ode l 1 takes in to acc oun t the c ross ing- t ra jec to ry e f fec t

and the con t inu i ty e f fec t on the heav y par t ic le d i spers ion , ve loc i ty decay and ve loc i ty co r re la t ion

in t h e d ir e c t io n s p a r al l el a n d n o r m a l t o t h e d i r e c t io n o f g r a v i ty . H o w e v e r , in t h e a b s e n c e o f t h e

c ross ing- t ra jec to ry e f fec t, th i s mod e l cann o t g ive the p red ic t ion tha t the long- t ime par t ic le d i f fus ioncoef f ic ien t i s g rea te r than f lu id ' s. R eeks (1977) show ed tha t th i s can o ccur on ly whe n the f lu id

Eule r ian in teg ra l t imesca le i s l a rge r than the Lag rang ian o ne . In s ta t ionary , hom ogene ous , i so trop icand incomp ress ib le tu rbu lence , P isme n & N ir (1978) d i rec t ly re la ted the long- t ime par t ic le d i f fus ion

coef f ic ien t to the in teg ra l t imesca le o f the f lu id Eu le r ian tem pora l ve loc i ty co r re la t ion . T h is sugges tstha t in o rd er to p red ic t tha t , in the absence o f the c ross ing- t ra jec to ry e f fect , pa r t ic les d i sperse more

than the f lu id , the inc lus ion o f the f lu id Eu le r ian tem pora l ve loc i ty co r re la t ion i s necessa ry . In the

fo l lowing , mode l 1 i s ex tended to inc lude th i s co r re la t ion .

T a b l e 8 . 5 7 / ~ m P a r t i c l e d a t a

Tab le 7 . 5 / z m Pa r t i c l e da t a V, (cm/s ) z~ .~ (ms) zP (ms)y r

V, (cm/s ) T~ (ms) z~ (ms) 0 .00 38 .35 38 .67

0.00 25.03 22.60 13.5 37.07 37.042.73 20.48 23.29 25.8 35.84 36.78

5.86 21.04 23.86 39.7 32.56 34.6513.31 19.26 19.12 54.5 29.47 33.4017.06 15.70 18.67 81.2 28.0 6 28.9820.91 14.88 18.12 108.0 24.3 4 26.4 023.65 15.01 19.36 121.6 24.0 2 25.44

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T a b l e 9 . D e p e n d e n c e o f D u a n d D , , o n t h e t i m e s t e p A t

H o l l o w C o r ng l a ss p o l l en G l a s s C o p p e r

Du [At = 0.01 (s)] 5.42 3.65 2.88 2.67D22 [At =0 .0 1 (s)] 6.18 3.02 1.88 1.77D u [At =0 .00 1 (s) l 5 .58 4.09 3.03 2.79D22 [At = 0.001 (s)] 5.01 2.94 2.16 1.75D u [At = 0.0001 (s)] 5.66 3.58 2.93 2.91D22 [At -- 0.0001 (s)] 5.67 3.26 2.08 2.06

I f tu rbu len t f low has one p r inc ipa l mean f low d i rec t ion , a s i s o f ten the case in the exper imen ts

o f Sny der & Lu m ley (1971) and Wel ls & Stock (1983) and the theore t ica l inves t iga t ions o f Reeks

(1977) , P ismen & N ir (1978) and N ir & P ism en (1979), the re i s ano the r met hod o f mo de l ing the

f luc tua t ing f lu id ve loc it ie s a t the pos i t ions X , and Xp , de f ined in f igure 1 . Le t the m ean f low d i rec t ion

b e t h e x - d i r e c t i o n a n d t h e m e a n v e lo ci t y b e U r n. T h e r e la t iv e c o o r d in a t e s y s t e m O ' - O ' ~ ' f l ' , a s

show n in f igure 10, i s e s tab l ished . The o r io n o f the coord in a te sys tem i s a t Xo, and the 0 ' ax is

passes th ro ugh the two pos i t ions Xo, and Xp . In f igure 10, Xo, i s de f ined as X , + U , At i , w here i

i s the un i t vec to r o f the abso lu te x ax is . As in the d i scuss ion in subsec t ion 2 .2 , the normal ized

f luc tua t ing ve loc i tie s o f the f lu id a t pos i t ions Xs , Xo, and Xp have the fo l lowing cor re la t ion re la t ions :

W~(Xo.)Wj(Xs) -- Wi(X~)W j(X,) F~ (A t) ( i , j = 1, 2, 3) [38]

a n d

W i(Xp)W :(Xo.) = W~(Xo,Wj(Xo,) G~j(As,) ( i , j = 1, 2, 3). [39]

I t i s c lea r tha t [38] i s the E u le r ian tem pora l ve loc i ty co r re la t ion func t ion in a f ram e o f re fe rence

m o v in g w i th t h e v e lo c it y U r n, b e c a u s e, s e en f r o m th e c o o r d in a t e s y s t e m O ' - @ ' £ ' f f , X , a n d X o,

occupy the same space po in t , a s seen in theore t ica l works o f Reeks (1977) , P ismen & Nir (1978)and N ir & P isme n (1979) . For th i s reason , th i s m e tho d is ca l led the Eu le r ian tem pora l cons t ruc t ion .

T h e f u n c t io n Gij i s the same as gu in [4].

For reasons o f s impl ic i ty , the normal ized f luc tua t ing f lu id ve loc i ty in the coord ina te sys temO' -O ' [~] ' f l ' i s s t i l l deno ted by Wi . S imi la r ly , fo r i so t rop ic tu rbu len t f low cases , the fo l lowingcor re la t ion re la t ions ho ld :

W , X o . ) W i X , ) = W i X , ) W i X , ) F ~ A t ) i = I , 2, 3) [ 4 0 ]

a n d

W ,(Xp)W i(Xo,) = Wi(Xo.)W,.(Xo.)Gi l (As , ) (i ffi 1, 2, 3). [411

Fo r the f luc tua t ing f lu id ve loc it ie s a t the pos i t ions Xs , X o, and Xp , i t i s pos tu la ted , re spec t ive ly , tha t

W i ( X o . ) = c i W i ( X , ) + X , ( i = 1,2,3) [42]

a n d,(Xp) = d l W ,(Xo.) + 6i (i = 1, 2, 3), [43]

l I ~ t

F igure 10 . T he coord ina t e sys t em es t ab l i shed fo r m ode l 2 .

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362 Q.Q. LU et a l .

w he re c~, ; t; , d~ an d 6~ (i = 1, 2 , 3) ha ve the sa m e sens e as a~, ~t~, b~ an d fl~ in [7] an d [10]. B y s imi lar

p r o c e d u r e s ,

c i = F E(A t), [44]

= G . ( a s , ) [451

a n d

W~(Xp) = Gd~ W~ (Xs) + ~, (i = 1, 2, 3), [461

wh e r e ~i = c i~ + 6 ~ ( i = 1 , 2 , 3 ) h av e th e sam e m ea n in g a s ¢~ . Th e i r s t an d a r d d ev ia t io n s ( ac )~ a r e

- c , d , i = 1 , 2 , 3 ) . [ 4 7 ]

S in ce [1 3 ] l ead s t o t h e co r r ec t l im i t in g r e su l t f o r f l u id d i sp e r s io n , t h e f o l lo win g r e l a t i o n s a r e

p r o p o s e d i n m o d e l 2 :

~a~b, W,.(X,) + qJ, ( i = 1, 2 , 3) if as l ~ R a s [48]W~(Xv) = )c~di W,.(Xs) + ( , ( i = 1, 2 , 3) if as , < R A s "

Eq u a t i o n [48 ] i s a l so ca l l ed th e m ix ed m o d e l i n l a t e r d iscu ss io n s . H e r e R i s an ad ju s t ab l e p a r am e te r .

W h en R = 0 , m o d e l 2 red u ce s t o m o d e l 1 . I f R = 0 % th e p r e sen t m o d e l w i ll b e s im i l a r t o t h a t u se db y Reek s (1 9 7 7 ) an d P i sm en & Ni r (1 9 7 8) . As lo n g a s R # 0 % th e r e su l t i n g sch em e wi l l l e ad to

th e th eo r e t i ca l r e su l t o f Tay l o r (1 9 2 1 ) f o r t h e f l u id d i sp e r s io n co e f f i c i en t. I n t h e s im u la t io n o f t h i s

su b sec t io n , s im p ly R = 1. I t sh o u ld a l so b e n o te d th a t t h e two a l t e r n a t iv e eq u a t io n s i n [48 ] a r e n o t

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

B ef o r e v a l id a t in g m o d e l 2 , so m e in s ig h t i n to i t s p h y s i ca l b a s i s sh o u ld b e g iv en . Two l im i t in g

s i tu a t io n s can b e co n s id e r ed . O n e i s i n t h e ab sen ce o f t h e c r o ss in g - t r a j ec to r y e f f ec t ; t h e o th e r i s t h e

case wh e r e th e p a r t i c l e f r ee - f a l l v e lo c i ty i s m u ch l a r g e r t h an th e tu r b u len ce in t en s i ty .

In t h e f i r s t c a se , wh en p a r t i c l e i n e r t i a i s n eg lig ib le , su ch a s f o r a f l u id p o in t , p a r t i c l e s ca n r e sp o n d

to a l l th e t u r b u len t f l u c tu a t io n s ; t h u s A s -- . 0 an d As , t> R a s a lway s h o ld s . As a r e su l t , t h e f i r s t

a l t e r n a t iv e in [4 8 ] co n t r o l s t h e p a r t i c l e m o t io n . In o th e r wo r d s , p a r t i c l e m o t io n i s g o v e r n ed b y i t s

au to - co r r e l a t i o n . Th i s ag r ees wi th p h y s i ca l i n tu i t i o n . H o we v e r , i f p a r t i c l e i n e r t i a is so la r g e th a t

i t r e m a i n s i m m o b i l e r e l at i v e t o t h e c o o r d i n a t e s y s t e m m o v i n g w i t h t h e v e l o c i ty U m , t h e n t h e v e l o c i t y

co r r e l a t i o n o f t h e f l u id a t t h e p a r t i c l e p o s i t i o n i s t h e Eu le r i an t em p o r a l v e lo c i ty co r r e l a t i o n in t h e

s a m e f r a m e o f r e f er e n c e, a s h a s b e e n a l r e a d y p o i n t e d o u t b y P i s m e n & N i r ( 1 9 78 ) . I n t hi s c as e ,

As t --* 0 in m o d e l 2 . I n f ac t , wh en a s t ~ 0 , o n ly th e seco n d a l t e r n a t iv e in [4 8] is u sed an d c~d~ -- . c~.

An o th e r l im i t in g ca se i s wh e n th e p a r t i c l e f r ee - f a l l v e lo c i ty Vt i s m u ch l a r g e r t h an th e tu r b u le n ce

in t en s i ty . Le t V t b e d i r ec t ed in th e n eg a t iv e d i r ec t io n o f t h e ab s o lu t e z - ax i s . Th en th e v ec to r X p - X f

(see f igure 1) and Xp - Xo, can b e exp resse d as x ' i + y ' j + (z" - VtA t)k an d x " i + y " j + ( z " - V, At)k ,

r e sp ec t iv e ly , wh e r e i , j an d k a r e t h e u n i t v ec to r s o f t h e x , y an d z ax es . W h en Vt i s l a r g e en o u g h

s o t h a t x ' , y ' a n d z ' ( o r x " , y " a n d z " ) c a n b e n e g l ec t e d , e i t h e r x ' i + y ' j + ( z ' - V t A t ) k o r

x " i + y " j + ( z " - V t A t ) k c a n b e a p p r o x i m a t e d b y - V t A t k i n b o t h a l t e rn a t i v e s in [4 8]. C o n s e -

q u e n t l y , t h e t e m p o r a l ( e it h e r L a g r a n g i a n o r E u l e r i a n ) e f f e c t o f t h e t u r b u l e n c e c a n b e n e g l e c t e d i n

c o m p a r i s o n t o t h e s p a t ia l e f f e c t a n d a~b~ a n d c~d~ a n d c~d~ of [48] are e qua l to b~ an d d ; . S ince g ,

a n d G ~ a r e t he s a m e a n d t h e y a re a p p r o x i m a t e l y e q u a l t o g ~ ( V t A t ) , t h e v e l o c i ty c o r r e l a t io n o f t h e

f lu i d a t t h e p a r t ic l e p o s i t i o n n o w r e d u c e s t o t h e E u l e r i a n s p a t i a l c o r r e l a t i o n i n t h e c o o r d i n a t e s y s t e m

m o v in g wi th U m in b o th a l t e r n a t iv e s i n [4 8] , r eg a r d l e ss o f th e p a r t i c l e i n e r t ia . Th a t i s, wh en Vt i s

s ig n i fi can t en o u g h , m o d e l s 1 an d 2 a r e t h e sam e .

T o v e r if y m o d e l 2 , a s im u l a t i o n f o r p ar t ic l e d i s p e r s io n i n a s ta t i o n a r y , h o m o g e n e o u s , i s o t r o p i c

a n d i n c o m p r e s s i b l e t u r b u l e n t f l ow is p e r f o r m e d . T h i s p e r m i t s a c o m p a r i s o n w i t h t h e e x is t in g

t h e o r e ti c a l w o r k s o f C s a n a d y ( 1 96 3 ), P i s m e n & N i r ( 1 9 78 ) a n d N i r & P i s m e n ( 19 7 9 ). I n s t a t i o n a r y ,

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

integral timescale and th___eLag r a n g ian in t eg r a l t im esca l e a r e i n d ep en d en t o f t h e re f e r en ce d ir ec t io n .

Th ey a r e d en o ted b y u 2, z E an d t L . Th e co r r e sp o n d in g co r r e l a t i o n s t ak e th e f o l lo w in g f o r m s :

W~(Xo,) W ~(XJ = W , ( X D W ~ ( X , ) e x p ( - A t / t ~ ) (i = 1, 2, 3), [49]

a n d

W~(X~)Wf(Xo.) = W ~(Xo ,)W ~(X o,)exp (- -As t /2A ,)cos(a s , /2A ,) ( i = 1 , 2 , 3 ). [50]

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364 Q . Q . LU et a l .

70

~o

60 Ruid / / ~TI~ 50 (ms) . . . .

50

4O

30

o " :" =" " : ; : " " " " I

0 1 2 3 4Time (s)

Figure la. Predicted longitudinal particle dispersions in the presence of the crossing-trajectory effect.

t han in t he norma l d i r ec t ion . Ho we ver , i t shou ld be po in t ed ou t t ha t i n t he absen ce o f Vt [59] and

[60] are no long er val id , exc ept in the l imi t of ze ro pa r t ic le iner t ia . Form__ulas [59] and [60] are

cons i s t en t wi th the wo rks o f Reek s (1977) an d P i smen & N i t (1978) as u 2 / V 2 i s smal l enough .

S u b s t i t u t i n g t h e p r e s e n t v a l u e s o f A , a n d Co2 i n t o [ 5 9 ] a n d [ 6 0 ], D , a n d D22 are o b t a i n e d . I n

the ca l cu la t ion , g r av i ty i s d i r ec t ed in t he oppos i t e d i r ec t ion to t he x ax i s . Here the subsc r ip t s

11 and 22 r ep resen t t he d i r ec t ions pa ra l le l and perpen d icu la r t o t he g r av i t a t iona l f o r ce . F igures 1 a

and l i b r ep resen t t he p r ed ic t ed d i sper s ions fo r pa r t i c l es wi th d i f f e r en t t ime cons t an t s Tp .

Tab les 10 and 11 g ive the p r ed ic t ed an d theore t i ca l ( com puted f rom [59] and [60]) l ong- t ime par t i c le

d i f fus ion coef f i c i en t s , r e spec t ive ly . I t i s obse rved f rom the two t ab les t ha t wi th inc reas ing Tp , t he

p r e d i c t e d v a l u e s f o r D . , D22 a n d D 2 2 ~ . g r a d u a l l y a p p r o a c h t h e th e o r e ti c a l re s u lt s g i v en b y [ 59 ]and [60]. Th i s m eans tha t a s l ong a s x /u 2 i s smal l enou gh , t he p r ed ic t ed long- t ime par t i c le d i f fus ion

coef f iden t s D . and D22 a r e i nver se ly p rop or t ion a l t o Vt and due to t he con t inu i ty e f fec t , D22/DIt ends asympto t i ca l ly t o 0 .5 .

A no th er i n t e res t ing case fo r t he ca l cu la t ion o f pa r ti c l e d i sper s ion i s wh en the e f f ec t o f

~oss ing- t r a j ec to r i es i s absen t . T h i s a l l ows an exam ina t ion o f the e f f ec t o f pa r ti c l e i ne rt i a . F igure 12

g ives co m pu ted par t i c l e d i sper s ions fo r t h i s case . S ince the in t eg ra l t imesca le o f t he Eu le r i an

tem pora l ve loc i ty co r r e l a t ion i s g r ea t e r t han the Lagrang ian one in t he p r esen t work , t he long- t ime

70

6

40

20

10

0

Fluid / ,ffi 5 0 m s )

I

2 3 4 5

Time s)

Figure I lb. Predicted transverse particle dispersions in the presence of the crossing-trajectoryeffect.

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A L A G R A N G I A N M O D E L F O R S O L I D P A R T I C L E S IN T U R B U L E N T FL O W S 365

Table 10. Predicted long-time particle diffusion coefficientsTable 11. Theoretical long-time particle diffusion co-

efficients

Tp (ms) 50 150 Tp (ms) 50 150D n (em2/s) 2.51 1.16 D u (em2/s) 3.65 1.46/)22 (em2/s) 1.63 0.66 D22 (em2/s) 2.11 0.74D22/Du 0.65 0.57 D22/Du 0.58 0.51

p a r t i c l e d i f f u s io n co e f f ic i en t i n c r ease s wi th t h e p a r t i c l e t im e co n s t an t ; i t is l a r g e r t h an th a t o f t h e

f lu id , a s ex p la in ed b y Reek s (1 9 7 7 ) . Th i s i s sh o wn c l ea r ly i n f i g u r e 1 2 . Th e p r ed ic t ed lo n g - t im e

p a r t i c l e d i f f u s io n co e f f ic i en t s f o r t h e 1 6 0 0 (m s) p a r t i c l e an d f lu id a r e , r e sp ec t iv e ly , 9 .1 a n d 6 .0

(cm2/s) . Thei r theore t ica l counterpar ts a re 8 .59 and 6 .272 (crn2 /s) , g iven by [51] and [52] ,

r e sp ec t iv e ly . Th e two a r e i n a fa i r ag r eem en t . Th e r e s t r i c t i o n o n th e t im es t ep o f m o d e l 2 is

• / L E Al A2'~A t m , n ~ z , , z , , ~ t t , V t j [6 2]

4 . D I S C U S S I O N A N D C O N C L U S I O N S

B ase d o n th e id ea o f a t im e se ri e s an a ly s i s (B o x & J en k in s 1 9 7 6) , tw o Lag r a n g ian m o d e l s , t h a t

i n c l u d e t h e e f fe c ts o f t h e t e m p o r a l a n d s p a t i a l v a r i a ti o n s o f t h e t u r b u l e n c e o n p a r ti c le s , h a v e b e e n

es t ab l i sh ed • M o d e l 1 i s u sed to s im u la t e p a r t i c l e d i sp e r s io n in t h e two g r id - g en e r a t ed d ecay in g

i s o t r o p i c t u r b u l e n t a i r f l o w s s t u d i e d b y S n y d e r & L u m l e y ( 1 9 7 1 ) a n d W e l l s & S t o c k ( 1 9 8 3 ) .

C o m p a r i s o n i s m a d e o f th e p r e d i c te d a n d m e a s u r e d p a r t ic l e d i s p e r s io n a n d v e l o c i t y d e c a y i n th e

t r a n s v e r s e d i r e c ti o n . C o m p u t e d p a r t ic l e d i s p e rs i o n a n d v e l o c i t y d e c a y a r e a l s o p r e s e n te d f o r t h e

l o n g i t u d i n a l d i r e c ti o n . T h e c r o s s i n g - t r a j e c t o r y a n d c o n t i n u i t y e f f e ct s c a n b e o b s e r v e d c l e a rl y o n t h e

p a r t ic l e d i s p e r s i o n a n d v e l o c it y d e c a y . T h e n u m e r i c a l r e s u l t s a re a l s o c o m p a r e d w i t h t h e t h e o r e t i ca l

r e s u lt s o f N i r & P i s m e n ( 1 97 9 ) f o r t h e c o p p e r p a r t i c le v e l o c i t y a u t o - c o r r e la t i o n o f th e e x p e r i m e n t

o f S n y d e r & L u m l e y ( 1 9 71 ) . I n a d d i t io n , i n o r d e r t o e x a m i n e t h e c o m b i n e d e f fe c t s o f in e r ti a ,

co n t in u i ty a n d c r o ss in g - t r a j ec to r i e s , t h e i n t eg r a l t im esca l e s o f p a r t i c l e au to - co r r e l a t i o n s a r e g iv en

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

o t h e r s a r e n o t . F o r e x a m p l e , m o d e l 1 u n d e r p r e d i c t s t h e 5 7 / ~ m p a r t ic l e d i s p e r s io n s f o r t h e

e x p e r i m e n t o f W e l ls & S t o c k ( 1 98 3 ) a n d i t c a n n o t s h o w t h a t 5 7 / t m p a r t ic l e s d i s pe r s e f a s t e r t h a n

5 / ~ m p a r t i c l es w h e n V t = 0 . T h e s e n s i t iv i ty o f m o d e l 1 t o t h e t i m e s t e p A t o f t h e c o m p u t a t i o n w a s

t e s t ed ; t h e r e su l t s i n d i ca t e t h a t p a r t i c l e d i sp e r s io n d ep en d s o n ly s l i g h t ly o n At .

I n t h e c a s e w h e r e t h e t u r b u l e n t m e a n f l o w h a s o n e m a i n d i r e c t io n , m o d e l 1 h a s b e e n e x t e n d e d

t o i n c lu d e t h e E u l e r i a n t e m p o r a l v e l o c i ty c o r r e l a t io n a n d t h e s o -c a l le d m i x e d m o d e l ( m o d e l 2 ) h a s

140 , aI

120

~, 100

6O

2O

fO. . -

FluidTp= 800 (ms) . . . .

Tp=1600 (ms) . . . . .° o - ° °

. °°o

2 3 4

Tm ~ e (s)0 5

Figure 12. P redicted particle dispersions in the absence of the crossing-trajectory effect.

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366 Q.Q. LU et al.

been dev ised . The m ixed mode l i s then app l ied to pa r t ic le d i spers ion in a s ta t ionary , h om ogen eous ,

i so t rop ic and incompress ib le tu rbu len t f low. The numer ica l re su l t s show tha t mode l 2 i s

capab le o f p red ic t ing , in the absence o f the c ross ing- t ra jec to ry e f fec t , tha t the long- t ime par t ic led i f fus ion coef f ic ien t i s l a rge r than tha t o f th e f lu id i f the E u le r ian in teg ra l t imesca le o f the ve loc i ty

cor re la t ion i s l a rge r than the Lagrang ian one , a s po in ted ou t by Reeks (1977) . The long-

t ime par t ic le and f lu id d i f fus ion coeff ic ien ts com pute d by m ode l 2 ag ree wi th the theore t ica l

resul ts of Pisme n & Ni r (1978) an d T ay lor (1921), respect ively . I t is a lso verif ied th at w hen th epar t ic le d r i f t is dom inan t , the p red ic ted long- t ime long i tud ina l an d t ransverse pa r t ic le coeff ic ien ts

g r a d u a l l y a p p r o a c h t h e two f o r m u la s o f Cs a n a d y (1 9 8 3 ); i .e . t h e l o n g - tim e p a r t ic l e d i f fu s ion

coef f ic ien t in the g rav i ty d i rec t ion i s twice the t ransverse on e an d tha t bo th d i f fus ion coeff ic ien ts

a re inverse ly p rop or t ion a l to the pa r t ic le f ree- fa ll ve loc ity . Th is impl ies tha t mo de l 2 takes

in to acco un t ine r t ia , con t inu i ty a nd c ross ing- t ra jec to ry e ffec ts . H owev er , i f Vt i s smal l, the long-

t ime par t ic le d i f fus ion coef f ic ien t g iven by the p resen t s tud y d isagrees some wh at wi th C sanad y ' s

resul t .

F u r th e r m o r e , c a l c u l a t i o n s , n o t p r e s e n t e d h e r e , s h o w th a t , e x c e p t f o r t h e c a s e wh e r e t h e

c ross ing- t ra jec to ry e f fec t i s absen t and the pa r t ic le t im e con s tan t i s la rge , the d i f fe rences be tween

the resu l t s a ri s ing f rom mo de ls 1 and 2 a re ins ign i fican t.

Al l the ca lcu la t ions p resen ted abo ve w ere ob ta ined by averag ing ove r 5000 par t ic les t ra jec to r ieson a S U N SP A RC 1 s ta t ion . Fo r a typ ica l run , the pa r t ic le d i spers ions show n in f igures 2a and

2b , requ i red abou t 1800 C PU seconds .

Acknowledgements--Thanks are g iven to Professor H. Bu rnag e (Institute de M 6canique des Flu ides deStrasbourg, F rance) an d Professo r T. J. Hanratty, as w ell as to the five othe r referees, for their helpful andilluminating comments w hich have led to the significant improvement o f th is study.

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