the calculation of the turbulent boundary layer

8/11/2019 The Calculation of the Turbulent Boundary Layer

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i ~ "- " . . . . . ' CR a~r /~ IV . .

' " TH E CA L CU L AT I ON . O F T H E T U R B U L E N T. .B U U / q D A K I " L A Y P. ~ o : .

. . . . " " .

117. T h e P r o b l e m o f B o u n d a r y L a y e r C a l c u l a t i o n . i+

The f ina l goa l o f boundary layer theory i s to be ab le to p red ic t in q u a n f l ~ w a y th e de ve lo p m e nt o f t h e b o u n d a ry ] ~y er w h i ch fo ra mo n ~ 1 1 i n given environm ent . This problem ean, in the preach s ta te

.invem e p ro b le m ; i t i s ooneerned w it h t h e w s t h e b ehaVio ur',d ' u rbu len t bou ndary layers, w hi ch ha ve n o t 13een obmn-ved,-but w hichwill develo p wh en the pressure d istr ibutio n an d surfaee ..spu~hnem an ,

TI~ , chapter wi l l be devo ted to these Froblem, and wi l l g ive review.of" the kno wn m etheds des igned for approxim ate ealoula tion ~ f thebvlen t boundary layer.

17.1 B o u n d a r y l a y e r e q u a t i o n s . . "Appro:sdmate m ethods fo r boun dary lay er ~]o ula t io n f i rs t found

appl ication to lamin ar lay er The des ire fo r s imple methods arose , J nthis case , f rom ~he di~ cul t ies w hich are eonnected ~ t h the aolutios~ o fnon- l inear par t ia l d i fferentia] equat ions for bo und ary lay er flow. ~ et ral~formation of the equations to a ~ st em of a f inite number of ordin-ary diffe~ntia ] equations and ~he .~nu]taueous use of a family ofvelocity proK]es depending on one or more l~rA,~eters, offers the p o l i -bil i ty of ace,ompl i~h~g su~c~ent ly exa c t so lu t iom wi th a ~ uo ns b l e .expend i ture o f labour.

The par t icular forms o f the bou ndary layer equat ion being oonsideredfor such a trea tm en t m ay be divided into two ca~-gorie~ ~ theTe arethe in tegral forms, which can be der ived f rmn the boundary laycr]~q. (4 .9) in a qui te general m anner b y m ul t ip lying each .*~rm o f t h e

zs 1~

o f k n o w ] ed p , b e a ~ o n ly b y sem i~-m pirioa] m e t h o d ~ T h is f ~ t

b rin gs a b o u t s d u al ' d i ~ t i o n o f t h e p ro ble m . M y , a n ~ y m o fexp erhn ent~ da ta .is to be m ade in o rder to develop the requ i red em pi .r i ~ re la t iom . Thi . implies ths t for experimenta l b o u n d t~ la y e n ~somet imes ~ quan t i ti es which h~ve n o t been ~ am to .b e

c ~ cu ~ te d fro m th e b o u n ~ r y ~ y e r e q u s a . s ~ A o ~ U n g to C o ~ e ~ ,problem rosy be ~ lis d the dimot problem. Th e ~ problem-is the


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d is t an c e o f t h e ~ ~ " ~ , a n d t h e i n ~ o ve ~ y. I f t h e , b es t.

~- according to ~ q. (4.6) .is /ntroduced into Eq . (~.9) , th e fol lowing equ s-t ion i s ob ta ined . '

f d y f d , .- . .

o o1 o

o 1

f o r s clko gete l a y e r, f o r ~ n e e 9 - O o r y - 8. ~ _ e ~ u ~ o n f o r y - 0

repreN nt l whab is cal led the w al l . co ns t~ in t , w hi~ . i sa n - h n p o r t a u tfea tu re o f t he P o hlh au m n m e th o d f or c alc ula tio n o f ~ b o ~layerL The equa tions which have found &ppHcaMon in t h e k n o w nm ethock for turb ulent bo und ary layer ea loula~on are oompiled" inTab le & .

' TA B L E "e~ ,~ ,~ ~ o ~ ~

fq , ( l? . l)|

A someWha~ differen~ form of the energy i n t e ~ equat ion is obta inedb y i n t e g r a t i n g Eq . ( 8 . 1 0 )

l d I d ~ ~ "

@ " 0 8

0 Q "' . 0 . - a

of the con t inui ty equat ion. A deta i led developmen t o f tk ia eq~__tion h ~been g iven by Te te rv tn andl " . i n s a a n d ~ , h , . l ~ 9 . e d forma hawJb e e ndiscussed by Truekenbrodt tat . F or some com binations o f exponentu ,g iven inT ab le 4 , th i s equs t i0n can be in te rpre ted in t e rms o f phys ieldqu sn giU ee . ~.

In t he No ond ~ t , egow the b o un da ~ l aye r equs~ ion is a~ tua l ly ~ t i a f led


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I s 6 , L C . l ~ : a ' r x

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i : g . "

( 1 7 . 3 )

In. thi equ ion,.-hi w u used -or Z u tion zn ethod . r, t t bYthe present wri ters, t.7, 99, the wo rk do ne b y the m ean stre sses is re-p laced by the i n t e ~ o~ the to ta l r ~ . o f vi scous energy cUsaipa~ion E ,W here E isgiven b y Eq .( 7 . ) .

W hile fo r lam inar boundary layers &ppro~ ,-~ t~ methods a re usedonly to a l l e ria te the ms~em aCica l~m~ ut tie~ inth e i ~ t i a l ~eVerentiff ilequat ions , the 0 i t u ~ o n is ent i re ly di fferent in the turbu /ent case . Th ecironmatam, et h s t t he t u rbu l en t mo t ion i s no t acce~ ib l e t o & m s t h ~m f i ~ t r e a t m e n t ,r endemth e exa ct solutioffi impossible. Th e e x te ntto-which the g enend system o f Eqs . (2 .2) to (2 .4) i s to be - [mpl i fied de ,pends on ho w f a r i t appears poss ible to der ive , in an empir ical manne~;general :~l~.f ions fu rt l ie nn]m ow n q uanti t iee from available experim entawhich. in turn depends on how many det~ls of t l ie f low are experL-m enta l ly determ ined. I t is no t Surprising thr.', theT~iOUS nves t,~pLtomh~ ve very~diverging opinions w ith, regard t o the quest ion a a to ho w ~Lr

t h e equa tions should be simpli fi ed or how co m pl ica t~ the sys tem ro sybe . This s ta te o f :affai rs has r~ u / te d in g rea t var /e ty o f p roposedmethods . Very.crude -impl if ica tiona of the problem require o nly s im ple

. empirie.al relat io n- and give equation s wh ich e.an e u il y be solved, b u tthe ac tua l beh~viour o f the flow ia on ly insu~c ien t ly t aken in to aoeoun~V ery complicated methodJ , on the o ther hand, require , s c om pM en aiv ek no w le dg e o f tu r b ul en t q u an titie e, a n d t h e r el ia b il it y o f su c h m ~supers often from inoorreot h ypo theses on w hicht heempi r i ca lrelation-.s r e based . ~ o m the theo ret ica l po in t o f v i e w i t i s im por t an t t o knowto w hat degree o f ex~t~e~_ and re l iab i l ity the tu rb u len t bo und ary

aye r can be c a l ~ w i th p ree en t knowledge, w i thou t r ega rd tOthe t ime necessary for pract ica l ca lculs t ion. The subsequent review o fthe e x i s t i n gm ethods wi l l bem a d ef rom such s tandpo/n t . Al l equations wi l l be given here for s t : ic t ly two-dimensional mean

flow, s/ though i t i s no t d~m~ult to der ive the e q u s ~ n s fo r l a te ra l lyconvergent or d ivergent f low.and for a b od y o f revolut ion in symm ~tr i-eal flow. ~

18. D e s c r i p t i o n o f th e M e a n Ve l o ci W P r o f i l e s

One of the ea rl ies t exper im enta l o beer~ t ions , which has , aga in andagain., found surprisingly good confirm stion by various inve st igators, isthe a l ready m ent ioned poss ibi li ty of rep~r~enting the mean velocity


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ac-tu al b e ~ v i o u r i n t h e p r o ~ t y o f t h e w i l l. A ft e r t h e e x p e r im e n A l~ -e rif i~ t.io n o f t h e u n i ~ . l a w o f t h e w a ll , i t w a s a l o g i ~ l ~ , p t o spp ..y .I .he oonoep t o f s ing ]e -pa ramete r vd o c i ty p ro f il es to t h e ~ -eloci ty de fec t -profi lm , def in ed b y . . . " ..- . . ' . . . . :: '

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v e lo c i ty p ro ~ ies f ro m ~ e n t a b y L u d w ie g ~ d~a re pre~___nted i . t h i s m s= n = in F ig . IS~ . . Th e va lue o f de f ined byE q. (15.10) m a y b e o ons idered as t h e ~ g ,h ~p e 1~r~m etm r.I f t la ea e d e f e c t p ro f il es f o r m i n f a ~ , s o n e - p a ~ a n e t e r f a m i l y, t h e n t h eoons t a= t E o f t h e . a~ 3m p t ot ie r e l a t io n ( l l; . ~ ) is a f u n ~ a o n o f l . ~ l o n e /F u r t h e r m o r e , a n y h i g h e r m ~ l er , m o m e n t o f v elo c i'ty ~ u t i o n i sun iqu e ly r e l s ted , t o I , fo r exam ple the t ai ird o rd e r momem~ . ' . :

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re la tao~ ,a . , d m p l e ap p r o - ; - ~ a ti o n f o r . th e ~ l o e i ~ p ro f il e ] ~ s. b e e n p r o -po sed .by R ot t~es , ~7. wh ich w as inde pen den t ly a l sor~gge,~lb y R o man d l~)ber tamn~2. T h e t e nta tiv e a pp ro ~ m a tio n ~ o f t he l ~ -m i e l a w o f t h e w a ll ,E q . ( 11 . 4 0 ) , t o w h i c h a l i n e a r t e r m i s a d d e d ,

t , -~ , , , ~ -~ + o . .fo, o,~.~ ,= a, (~ .~ )

wh er e A denotes free.parameter.-The hi e~ eu of h e . ] ~ ~ layeri s de f ined by the c~nd i t ionU - U.o fo r s ~ . Th en E q . ( 1 8 . 7 ) ea n .ber e w r i t t e n i n t h e f o r m o f a d e f e ~ l a w -

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b ~ U . I + A- , . ( ~ s . ~ )


8/67

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W ith th e use of t~his V~lue the qu ~nt i tiee K , I t n d 1~ e~m befrom Eq . (18.8) ~ t fu nct ion of th e lum m~eter A: .

X - _ x + . a +/ I f K K -

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merits n the turbu len t m ixiug zoneof, ,: freeplr,.ue e t by LiepmainnLudL--u,t'er a.,t h e p l a t e b o u n d a r y l a y e ~ a n d b o u n d m 7 l ~ . v e r s i na d v e r s e :p r e es u r e g r a d ie n t s f r o m ' v a rio u s p s p e r s . A m ; - o r d e f e ~ w h i c h th i s v e l o -c i t y p r o fi le h a s i n e o m m o n w i t h t ~he t w o p r e v i o u s l y m e n t i o n e d r e p r e -sen ta~t ionm is . thAt the der iva t ive~ U / ~ i s no t ex ac t ly zero . a t y - &V e l o c i t y p r o f il es c z lc u l a t e d b y . t h i s m e t h o d f o r a s e p a r a t i n g tu r b u l e n t.b o u n d a r y l a y e r ~ p l o t t e d a s s o l id c u r v e s in F ig .. 1 5 . 1 .

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~ - - 1 , , ~ ? - -t -o o -z o -6 ~-~ 14 ~ 2"4 a 4 ~ '8 ~ , 0

H

1 ~ o . 1 $ . 6 . . l t e l s t io n b e t w e e ~ v e l o c l ty - p r o f i le s h a p e p a r a m e t e r s B " m. r ' / O , .r.f m ~ ' / e . a n d l o c a l e k i n i ~ r ic ti o n e o e ~ e . i e n t t e z c o r d i n g t o F.x ps. ( I S . 1 0 )

a n d ( 1 8 .1 5 ) a f t e r R o t t a t %


12/67

.

A c o ~ x li ng t o t h e m ~ o n s a n d t h o s e o f S e e t i o n 11 t h e m e a n v e l o c it y

f ie ld inc lud ing the f low in the rsub laye~ , is. de te rm ined w he ne ve r the .k inema~io v iscos i ty v, th e surfsee r o u = h - ~ . ~ l~A,~-terized by . the sca le"krand tbz ee o f t he four pa ram ete r s ~ 'm; 1 ,,, ~ iand B a re kn ow n. Be eau mo f the re la t ions Eq . (18.14J and (18.15) the pa ram etem 8 and B m ay berep laced by tw o o the r bo und ary l aye r th ieknees pa ramete r s , e .g . ~* an d8. T h i s m e a n s , t h a t t h e v e l o c i ty p ro f il e is c o m p l e t e ly d e t e r m i n e d b yv, . ~ ~,o , 0 a n d t h e s h a p e p e ~ m e t e r H . A s a n e x a m p l e t h e r e la ~ o n b e -t w e e n t h e th /c kn ee a r ati o ~ H e - ~ * I 0 ,a n d loca l sk in f~ ie t ion eoeff le ien~c~ is repre sent ed in Fig. 18.6, This dis4Fsm is baae d on vM ues o f I ~nd.I z a e e o r d i n g t o E q s . ( l& 1 0 ) , w h i c h a r e n o t ~ e r y d i f fe r e n t f r o m t h r o e o fEq& (18.14). ..

W hen u , sppro aehes ze ro (i~e. lhn ~ -* 0 ), th e pa r am ete r A o r B inE q & ( l& 8 )' a n d (18.12) t e n d s t o i n fi n it y. F r o m t h e b o u n d a r y c o n d i t i o nU . , 0 f o r y - 0 w e h a v e .

, . ..' " .

l i ra ~,~v, , . 1 ~ d . l i ra ~,B- u , . .=. i .. ( l $ . l e ). . . . ~ - e ~17,, . vv~e ~ . 17,,

. , . .. ". - . ,- " . .

" U r n / 7 ~ / B a n d I / m U " . w ~ I S ) 1 8 . i 7 ). , ' U . . , - 0 U -

respect ively. FAther o f the se equ afio rm pro vide e h e _~__~peo f the p ro f i les t p o i n t o f s e pa r at io n o r r e - ~ t . " .

Whereaa the ~ / ~ ; l~ i ty l&wa fo r the wa l l f low and a l so fo r the ve loc i tydefec t o f the f i at p la te a n d equ i l ib r iu m .boundary layer s, a re based o nclear phys ica l ideas , the s /mflar i ty co ncep t m vn lve d in Eq . (18 .12) go esfa r bey on d the ] iml ta o f ~ ime.nsiona l - a,~lys i~ There fo re these ~ o n s

eLn be app l i ed to con d i t ions~ ut~ id e t h e m n p o f o ~ o n a o n ly w i t hsom e rese rvst ions . A specia l e su t io n i s in o rde r when th e b ou ndary l a ye ris s u b j e ct t o e x t r a o r d i n a r y c o n d i t i0 n s a s i n t h e n e i g h b o u r h o o d o f t ra n a i-t i o n L ~o m l a m i n a r t o t u r b u l e n t f lo w, n e a r s e p a ~ t i o n , r e - a t t a c h m e n tb e h i n d o b s t ~ l e s , a n d s u d d e n t r a d i t i o n f ro m s m o o t h to r o u g h s ur.~ ceor v ice ve r sa , Also h igh l eve l o f f r ee s tr eam tu rbu lence m ay hav eas t ro ng inf luenceo n t h e s h a p e o f t h e v e l o c i t y p ro f il e a s h a s e x p e r i m e n t a l lyb e e n sh o w n b y W i e g h s r d ~ U .

T h e e x ee ll en ~ c o n ~ n & t i o n f o r s l a rg e n u m b e r o f m e a s u r e m e n t s u

e .ddueed by Coles, ca l l s -d i rec~y for s .phyaica l e xplan at ion . Co les bug-gest,e d a n i n t ui t iv e , i n t e r p r e t a ti o n w h i c h m a y. h e l p t o u n d e r s t a n d t h ep h e n o m e n o n . A h y p o t h e t i e M v e l o c i t y p ro f il e co m p o s e d o f t h e l aw o f t h ew a l l a n d t h e l a w o f t h e w a k e is s h o w n i n F ig . 1 8 .7 . T h e d s a h e d l in erepresen t s the l aw o f the wMl a~cord lng to Eq . (11 .3 ). The dM h-po in t


13/67

":" "i " " . . . . . . . . .

~ ( 2 - w ) / ~ , a u d t h e in te r ce p t a t y m 0 o f t h e e q u i v a l en t w a k e p ro file ..t3~re~bre d i~e~) f rom the ve loc i ty i n t he ~ s t r eam by an a m o u ~~ / ~ . S in ce t h e ~ l e n t m o t io n in t h e o u ter p ~ o f a b e u n d a ~ l ~ y e ri s e~ec t ive ly, u n r m ~e te d and the p ro ee~ o f en tr aiu rmmt o f -nan -"t ~ b u l e n t f la id ~ k m p la o e b y p m o e ~ m v e r y ~ t o t h o se o b ee rv edin W akm and | e ~ , t he bounda ry l~ye r m ay be v i ewed a s a wa ke flow.,i n t o w h ic h a ~o lid t ~ , p k t e h p ls e~ d a t t h e c en tr al p l a u e , - ~ v e l 0 ( ~ "d e f e ~ o f ~ e w a k e b e in g 1 7 , ) - U - ~ u ~ ) / ~ a t ~ e c en tr e. A t ~ ) ~ o f t h e p l st ~ t h e b o u n d a ~ o o n d i~ o m o f v - -~ -~ in g v elo ~iW ~ d m o l e -o u ~ f ric tio n a r e t o b e ~ t ~ e d . T h e s e o o n d~ io n ~ i m p o s e an ~ d d i t ~ n ~ l

c on str a in t o n t h e fl ow, w h o s e e f e ~ k t o m o d if y t ~ e m e n u , v e i o ~ i ~d i sm~b ~ion a s show n by ~e .~o l id l ine i n F ig . 18.7 . Nea r t he p l a~ , wh ere

t H I. o .

I ~ / / "" " " " " " "

I ~ " " / / '

- - - " ' ( ~ ' } - - - - - ~ ' , ) ' ( ~ ) "/ t . . .

/

/ , / / . " .

- . - . . . . . i t . . J / .

. U

~ a . 1 8 .% l ~ p o ~ , h e f i o s ~ v e l . ~ t y p r o f i le o f u ) r b u l e n t b o l m d l ~ l S :n w .

- - - - - - v e l o o i ~ , c t i ~ . - i b u U o n ~ t o t h e l a w o f t h e

Eq. (n3) . v e l o c i t y d i s t r l b ~ t i o n o f w a k e ~ to w . .


14/67

1 6 6 3 . C . F, O T ' r.a .

" t he m e a n w a k e v e l o c i t y i s n e a r ly co n s t an t , t h e c o n s tr a in t p r o v i d e d b y. v i a c o ~ t y produces ~ flow pa t t e rn. a s d e m n ' ib e d b y t h e s i m i l ar i ty l ~w o fw a l l f lo w ~ "

Th=e .e=~i,,ome o the r e m p i ~ methodso~ ~ ~ ~:ve lo city ln'ofile.O f turbulent ; bou ndary ' Isyem,. w hic hw e . m e n t i o n h e r ef o r t h e ~ e o f c o m p l e t en e m . H ; _,d ;m o t o ~ t a ' o p o ~ l a o n e p a r a m e t e r. r e .

o f p ro f il m c o m p ~ e d o f t w o u n i ve m a l f im c ti o ns . Ve r y r e c e n tl y a n a n ~ I y ti c al r el& t io n c o n t ~ ; ~ t h r e e p a ra m e te r~ h a s b ee n g i v e n b y, .~md-

. ~ ~ 0 l . . - : . . . . . . -. .... . . .

1 9 . L o c a l S k i n F r i c t i o n C o a t t l c t ~ n t

~The e~l.lC, la t io n o f th e leea l akin f i - ict ion coeff icient d e/ ~ ed as " ' , ,

. ~ 1 , wcf - ( ] 9 . 1 ) p ~ . ,

- " . . :

' beco m es necessary m co nne ct io n w i th tw o di~erenb problem s. Th e ~h~st n e is t h e d e t e r ~ , , - ~ i o n o f t h e s k i n f zi ct io n in t h e f i ~ m e w o r k o f. ab o u n d a r y l a y e r p r e d i c ti n g c al o ul af io n . T h e s e co n d p r o b l em i s t h e d e t e r -m l n ~ t i o n . o f - ~ -f i ' i e t h m i na bo und ary l ayer, fo r w hich" the m ean vo le -

c i t y f ie ld is e x p e r i m e n t a l l y d et er m ln ,~ d , b u t n o d i r e s t m e ~ u r e m e n t o f ,' ~e w a l l sh ea r ing o t reas h , , - been ram, e . :.

1 9 . 1 S k i n f r i c t i o nf o r m u l a e ' : S e ~ w Of' he Sener~y"ob~,r~ed ' ~ that them w v e l o ~ t y p r o f i l .

fo rm near ly a on e pa ra m ete r fLm]ly, the loca l ak in f~ ict ion coemcien6c a n a p p r o , ~ , ~ t e l y 1)o c o r r e l a t e dw i t h lo c a l qu a n ti'd e s l i k e b o u n d a r yl a ye r th i rknesa , ve lo c i ty Um ou t s ide , the l ayer, v i~ ws i ty v, rous lmeas10arzmeters , a nd prof i le Shape param eter. A nu m be r o f empir ica l rely-

t i o n s h a v e b e e n p r o p o s e d fo r t h e - ~ n f r ic t io n o f t h e f i at p l a t e h e un cla zTlayer on sm oo th su r faces . The o lde ~ lmo w n re la tion is,. -' / 1 7 , o 0 ~ , - u 4 - '

.~ m

w h i c h w a s d e d u c e d f r o m m e a s u r e n ~ n t a o f t h e p r es su r e .dro p i n f l o wthro ug h & p ipe. A com par i son o f var ious fo rm ulae has been g iven by

Granvi l le~ . T h e e f fe ct o f t h e s h ap e p a ra m e t e r w a s d i ~ u t ~ U , Is7 f o r alo ng t im e u nti l e lee~re x p e r hn e n t al e v id e n ce w a s a c hi e ve d b y ~ h u b a u e ra n d K l e b a n o f l~ a an d Lu dw ieg and T~II,,~A,~,s7 th at th e s kin fr ict io nco e~c ien t dec reases wh en the shape pa ram ete r H increases .

Th e phys ica l bas is fo r th e ca lcu la tion o f t he wa l l shea r ing s tr e ss i sp r o v i d e d b y t h e p o s t u l a t i o n o f t h e u n iv e rs al la w f o r t h e m e a n v e l o c i t yd i s tr i b u ti o n n e a r t h e w a ll . I f a m e a n v e l o c it y p r o w l s c o m p o N d o f th e


15/67

. . . . .

o f the wan onoado pted , the loca l ~z in 'L- ic tion co ~ c i en t can be ~omputed f romE q .

, , ,

(is .s) to give . . . . . . . . . . . . : : . . . . :

~ n o e , w i t h t ~ e ~ . p p o s i t i o ~ , ~ is a fu nc tio no f' th e a h a ~ p a r a m e teI a ~ z r d i n g t o i ~ . 18.3, n d t h e m o m e n t u m t h i c k n e ~ e is r el at ed o.t h e ~ , m e n t t ~ e m ~ * ~ y E q . (XS.;~). h ~ ; o ~ ~ ~ o .c oe ff ic ie nt c a n o o n v e m e n t l y b e e x p r e m m d ,,, a f u n c t i o n o f R e y n o l d ~num ber U~8/~ , s lmpe Im ~m ete r H and r a t io o f r0ughne~ ~ .a le t o m e-m entu m ~] t i~m e~ ~v~e, in the form : . - .

- - , . : .

~ - ~ ( s , t , / s ) . (z~,4)

cient f romt h e pro pe rt im o f. th e ve locity profile, wh ich const~tutee t h e~ ; " advan tage o f d iv id ing t he under ly ing ve loc it y p ro~ e i n to twoInufiles t t w o d iffenmt k y e r a . : ' .

F c ~ m m o o t h m f r f a ~ ; Eq. (19.3)~ u b e r e wr it te n w i t h ~ he m e : o fEq. (18.15) ~og ive . . . . "

.. ~ - 11 1 - n - - --. '- . 1 - + . (19. .5)

- K % )* i ~ - ,

Eq ; ( lg .S) w~th the a id of E qL (18.10), ( lS. l l} , an d (IS:lb '} is ~ inF ig . 19.1. T h e ~ o y o f t h i s re ~ttio n relies o n t h e em p irica l ~ "x and 0(0) o f the l aw o f the wal l. I n o rder to & void the uno er ta tn t i~which m ay ~eventu&lly be ~uJed b y inacour~ . ie l in these c o n ~ u ts ,L u dw i eg a n d T ~ , , ~ . e 7 d e pa rt ed fr om t h e ~ e n t & l .fzio tio n l aw

of the con s tan t p reemlre l ayer. These au thors der ived on th e ba rn o fthe i r exp er im en~ in com bina tion w i th the su~-position of s imul tancowezis tenoe o f a un iversal law o f the wal l and l ingle t in--m eter velo~Tp r o f i l m t h e approx ima te fo rm u la

c , - o.24s .1 0 ~ - o . , = ~ n ( U . 8 / , ) - o . m , ( 1 9 . 6 )w b ~ ~e lds fo r U.~8/v> 1 0 0 0 a n d H < 9. e m e ~ ; , ~ . y t h e sa m e r e s u l ~. E q . ( 19 . 5) .T h e r ec en t m e a m z m m e n t ~ m a d e b y S m i t h a n d Wa l k e e zzin a ddi t ion to Schul tz-Grunow'e n s data g iveg o o d m r p l x ) r t t o t h e f z i o -t io n I sw o f ]~q. (13.13), aa seen fzom F ig. 13.5. Since ~.q. (19.$) sgreesform ally w ith E q. (13.13), i t ia to be expeeted th s t Eq . (19.~) wil l yie ldcorrect resutte for the same range o f Reyno lds n-m bera .

T h e typic al belmviour o f turb ulen t flow on rough surfaces , as dee-cribed in Section 11 is that for very small roughness the value ofC' ;,,


16/67

1 6 8 : . ". ~ J . . G . ~ . " "

constant:such th s t ~ f r o m Eq. (19.4) is independent of the r~ o/ ~/ 8.The s u r t ' ~ . i shydmu~e.~ly ~anoo~ in t h i s l~m~t, ng ~ With inca~u-

ing heights of roughness the ratio. ~/8 ~n~ more and more influencerelative to the Reynolds number/7.~/v, until finally the skin frictioncoe~cient is-s function of ~/~ and ~v only. This.is the other llm~tingcase of &completely rough surface.The value of ~ obeys the asymptotioform given by Eq. (II.4~). ll 't hi s relation is introduced in Eq, (19.3),the local skin friction coefficient can be c s l ~ from " ...

" l ln c , , . , ,

The result is given in Fig. 19~ for 1 T ~ l s e ' s sand ~ rou~mesLThis d i a ~ m has s general siguificanco sinco, in the completely roughregime, any t ype of roushne~ can beexprewed by an equivalent sand

rou4~hneea, as.outlined in Sect/on 11, . .. ..+ ,

, o , I tt . '

o . I

0 , ~ ~ 2 ~ - ~ ' ~ . ~ - - . ~ , . s

0- ,

1 S ~ * O6 I 0

~"~o.19.1. Local ~in fl-lethm coemclent ot turbulent ~u nd~ F l~ye~profiles o~ smooth sur/'scu from .Eqs. (19-6), (18.10}, (IS.II), and

( 1 8 . 1 . 5 ) . ,r - , 0 - 4 ~ C ( O ) - , 6 - 2 .

For the transient re ~ e between hydrsulically smooth, and com-pletely ro.gh surfaces the repreeent~tio~ is more complex,.sinco seoord-ing to Eq. (19.4) c~ ia s function of three independent v&ri~bles, whichcannot be given explicitly. Apart from this dit~cnlty there is not s un/que


17/67

. . - . . . .

. , . . . ' , .

ion,. i f . for b the. grain ~ of the eq ub ~lent sand ~ roug hu e~:ki n t r o d u a ~ T h is is d e a r ly d e m o n s t n t e d b ythe exl~en'~d result~ "oom pi led in l~i~; 11 .14 . The d ev ia t ions f rom t he s t ra igh t l ines t en d to :di fferent d i r e c t i o n s f or t h e v a ~ o u s t y p e s o f r o u ~ h n e ~ . M o s tl y, i tt o b e j ~ to. u s u m e fo r b o ~ la ye r ~a/1 u atiom~ .c o m p l e t el y r o u g h o r a h y d ra u l ic a ll y s m o o t h m a r fa ce d e l ~ . ~ i ~ e o n

- . . .

" ' I . " I : I : I ' t , , ~ t < , " - "

I .~ ". I ~ , / ~ ' * . ~ ~

.

1 , 0 I ; 0"

l i ~ ' ~ 1 0 " ; , l J l O ' i . I ~ D "~ . I

l ~ n .,

~a. I0.~. Loe~ ,kin ~o'.~oB eoem~ent of ~bulmt ~ ~prollk, s on eomplewly rough ~u~ra~ wit~ mud ~- t~3pe of ~

ca lcula ted . f rom Eqe, (19 .7) , (15 ,10) , (15 .11) and ( lg .15) . .

, ( - 0 ~ 4 , C , - , 8 " 4 . "F o r ~ . O l , < ( U . ~ l , ) , t h e ~ i s o b e o o n ~ d m X ] - . b e b ~

w hic h of the two -~ases g ives a h igher va lu e for th e ~ f r ic tion c0ef l i-e i en t . F or th i s r eason , in ~'2g . 19 .2 d o t ted cnrves a re d rawn in fo r con-s t ~ nt R e y n o l d s h u m o r s ( U ~ / v ) s , f o r ~ h i c h ~ q . ( 19 .5 ) f o r m n o ot h m n~face g ives the sam e va]ue o f "skin f r ic t ion e oe ~c ien t as ~ l . (19 .7 ). ~Themt h e a c t u a l R e y n o l d s n u m b e rU ~ / v i so w e r t h a n ( U ~ / v ) , t h e s u z f a ~i s t o b e c o n si d er e d a s b e in g h y d r a u l i ~11 y sm o o t l ~ ~ e s e e u r v ~ a r eo b t a i n e d by equat ing the r ight-hand s ides of Fxls . (19.b~ and (19.Y~T h i s g i r e s

~ " " ~ , ~ ~ .e / , C X l ~ ( , , [ C , -0 ( o ) ] ) . ( z g . s )


18/67

17.0 J'. C. R ~ TA -

19 -1 D e t e r m i n a t i o n o fs k i n f r i c t io n c o e f f ic i e nt f o r . e x p e r i m e n t a l b o u n d a r y l a y e r s

T h e com put s t i on o f t he l oca l sk in fi~ction i'o r ~ n t a l ve loc i typro~des is s prob lemo f grest pract ical s i g n i f i c a n c e , s i n c e t h e e x p e r i -m e n t a l determ;,~*~ion o f th ew a l l s h e a r s tr e ea i s aa l l .c u l t t a ~ w h i c hrequires the use oC special devicm . Su ch d eviom are ver y o t~ n e~thernot & vzil~ble or not appUcsblec T h e m etho d o f calcuIs t ing th e 'veal]s h ~ s tr es s f r o m N e w t o n ' s r el st d on

1 ~,O'i (19 .9 )

f . n . i nm o s t c as e e a in e e it k . a l m o ni m p o ~ b l e t o m a k e r a lis ble v e lo c it ym e u u z e m e n t l c lo s e e n o u g h t o t h e w i l l t o b e t b l e to" d e t er m i n e f ~ e

~ o c i t y g r a d ie n ~ a U l ~ . w i t h ~ e n t ~ ~ t t h e . ~ ] L A p ~~ o m t hk ~ t h e m e t h o d ~ n o~ s p p li ea b l e i n t h e e m o f r o u g h

. 0 . 0 1 2 5 . ."

/ , ,' ' . " 0 . 0 1 4 0

) /. . . .

0 . 0 0 I , , "

" I' " '

O . O 0 : L q ~ ' . . e , ,d , ~- w u . _ - .

: I 1 . s 4 S

~ m

F z e . 19.3. C41eubsted oca l akin~ie t icm eoegk ,h m t o r s t u r b u l e n t h o u r i.~ r y Lsye~ n r h ~ p ro.u rn ~ e r w i e s ~ V ' .

~ u t h e t ~ U m e ~ o m d m om e uu ~m i n te e m l e q u a l;o no ~ r ~ . 1 9 .1 .

F o r s l o n s t i m e i t w a s n o t p o s s ib l e a t a l l t o m e a s u r e t h e l o c a l, I t ; ,1 f r i c -t i o n i n 6 t u r b u l e n t b o u n d a r y l ~ y e r w i t h s r b i t r a r ~ p r e ss u r e d is tn ~ b u t io n .

T h e o n l y p o s s ib i l it y t h e n w a s t o c s l c u l s t e t h e s i g n f i- ic t io n c o e ~ e i e n t

f ro m y o n I ~ r m ~ ' s m o m e n t u m in te g r al e q u a t i on ( 4 .11) , o r N o . 1 o fTa b l e 5 . T h i s m e t h o d , i f c o z e f u l l y c a rr ie d o u t , g i v e s q u i t e r ea s o n a b l er e s u l ts f o r b o u n d a r y l a y e rs i n c o n s t a n t o r f a l l in g p r e ss u r e . H o w e v e r,w i t h b o u n d a r y ~ y er s i n r is in g p r e ss u r e t h e d i ~ c u l t y i s t h a t t h e t w o


19/67

. . . . " ., ". ' . " . : ~ .

o p p o s i t e ~ s o t h ~ r 4 i s t o b ee.~lculated a s s sm al l difl'erenoe between"t w o l a rg e va lu e s, w h i c h i n t h e i r t u r n m u s t b e c o m p u t e d f r o m m e a s u r e dquan t i t i e s by num er ica l o r g raph ica l d i ffe ren ti a tion . C o nse qu en t ly th eresu l ts o f th i s m e th o d a re ex t rem ely sens it ive to smal l -~Lv0idsb le

e r r o r s i n t h e m e a s u r e m e n t s a n d e a le u la ti o n a. I n a d d it io n , a n y S m alld e v i s t i o n f r o m t w o - d b , * m l o n a l i t y o f t h e m e a n flo w, m p e ci aH y l a te r a l

c o n v e rg en o e o r d i v e /g e n e e o ft h e ~ e t m - l i n m , p r 0d u e w c o n d d em b l eer ro r s . Th i s i s dem on s t ra ted in F ig . 19.$ by com par ing , fo r.& bound sa7 ayer i~ r i sing p ressure , th e loea l sk i n f r i c tion coeff ic ien ts ~ d ~

f ro m t h e t w o ~ i m e n si o n s] m o m e n tu m i nte gr al e q u t t i o n w i t h t h o ~. . . . ' . . . . . .

. . . , . - . ; . .

" I -O . , "

. UeY .. " ".

e x l ~ n ~ . ~ v e l ~ . ' ~ I ~ a t ' ~ ~ t ~ , ' , . ..o

o ~ e d f r o m F i g . 19,1; T h e r e su l ts f r o m" the m o m e n t u m e q u at io ns i m u l s t e ~ s u d d e n r is e o f t h e - ~ - f r ic ti o n co e m o i e nt , w h i c h d o e s n o tactua l ly ooour. In fac t i '. i s s lm o st impo ss ib le to es tLbl ish s eo m ple te lytwo~f imen=iona l m ean f low w i th ]Lbom tory exper iments . T he secondary

flo w in s b o ~ d a r y ] sy e r w in d t - , . ~ ] h u b e en ~ t ~ d b y T ill-ma~rt14~. Therefore,t ] ~ m e t h o d o f c a lc u la t in g t h e s k i n f ri ct io n f r o mm e a s u r e d v e l o c i t y d i s t r i b u t i o n s h a s n o t p r o v e d t o b e a d e q u a t e f o rt u r b u l e n t b o u n d a r y k y e r s i n a d ~ s e p re m m r e g r ad i en t s. T h e s im p l e st

m an n er h o f Course to ee t im ste c4 fro m t h e z eint ions Eq s. (19.5), (19.6)o r (19.7 ). A no ther m e tho d supposee on ly the - - i ve m a] inw of the wa l l ,a n d i s n o t b o u n d t o t h e v a l i d i ty o f a u n i v e r sa l o n e p a r a m e t e r d e f e ctlaw. Th e m easure d veloci ty prof i le i s p lo t ted in a sem~log~ri~m~e dis -g ram. Th e f r i c t ion ve loc i ty, f rom w hich the loca l sk in f r ic t ion eo e~c ien t


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de ;--d or the- by th e re la t /on

- ( : 9 . : 0 )j f

' T h i s m e t ho d i s app l i csb le to sm oo th as weLl a s rough su rfaces , bu tt h e r e s ul f r el ie s o n t h e a c cu r ac y w i t h w h i c h th e a i o p e c a n b e d e t e r m l - o df ro m ' t h e m e a s u r e m e n t s a n d o f t h e c o n s t a n t x . T h e u ~ e e r t a ~ t i e s a r ered uce d ~ the m etho d p ropo sed by Clsnse rD i s app l ied U i l lus t ra ted i nFig . 19.4 . Th e law of th e waU aeeordin" to l r~ : 11 .1 i s replo t t sd as a

un ive r sa l f ami ly wi th va rio tm pam m etem ~ t , T he ac tua l u , i s de te rm inedby se lec ting the appropr ia te mem ber o f the fami ly which best fi ts them e a s u re m e n t s. T h i s m e t h o d r eq u ir es t h e s u rf ac e to b e s m o o t h o r t h ee~ 'ect ive mug]mesa sca le to be k no w ~ A ~m pl i~ea t ion o f C lause r'a e tho d i s suggee ted by Br ads haw 7 accord ing to w hich s cu rve U(.V) .i s t o b e p l o t t e d f ro m t h e l aw o f t h e w a ll a ~ - m ; n ga f l r ~ v a l u e o fy t t /~

- a n d vsxTing u , . The in te r sec tion o f th i s cu rve wi th the exper : .men ta lcurve/7( .V) de term ines the e~,tusl va lue o f ~,, a n d thus ~ .

. . . . : , .

2 0 ' D e t e r m i n a t i o n o f S h e m r i n ~ S t r e ss D i s t r i b u t i o nT h e in t er e st i n t im e h e a ~ s t r e s s d i st ri b ut io n o f t u r b ul e n t b o u n d a r y

layers has .i ts o r ig in in the fac t t h a to n the r /gh t -hand a ides o f a l l b o un -dary l aye r equa t ions o f : Tab le 5 , excep t fo r the. m o m ent um in tegra le q u a t io n , t h e m o c c ur s ~ t e r m c o n t a in i n g t h e~ g st~',,,~,~.~____..Henee..k n o w l e d g e i s r e q u i r e dof the shea r ing ~ d i s t ribu t ion as we l l a s them e ~ v e l o e i ty p r o ~ le a n d t h e l o c al ~ i n f ri ct io n e 0 e m e l e n t; o t h e r w i set h e e q u a ti o n s c a n n o t b e u s e d f o r n u m e r i c a l e a le u la ti o n o f t h e b o u n d a r y,l ay er . T h i s p r o b l e m d o e s n o t e x i s t f o r l a m i n a r b o u n d a r y l ay e r s , b e -causeh e r e t h e shearing s t r eu ~ be ca leu lz ted f i' om N ew ton ' s r e l at ion .Oncet h a ve loc i ty p ro fde is am um ed , the shea r s tr e ss i s de te rm ined by

8 / 7t . 2 o . l )

I f fo r ins tance , ~he bo und ary la yer prof iles are assum ed to be ~ on epara m eter fami ly, the n th e shear ing s tress prof iles a lso form a o ne para- -m e te r f ami ly. The sea rch fo r an adequ a te r e l a tion be tw een shea r ings tr ee s d i st r ib u t io n a n d m e a n v e l o c i ty p ro f il e a n d o t h e r h a r ~ t ~ i n gp a r a m e t er s i s t h e c e n tr a l p ro b l e m i n t h e a p p r ~ ; ~ - ~ t e e a le u la t io n o fgenera l turbulent boundary layers , for which a sa t i s fac tory so lu t ionh a s n o t y e t b ~ e n f o u n d . T h e d if f i cu l t y s t h a t t h e s h e a r i n g s t re s s dis t r i -b u t i o n m u s t b e k n o w n w i t h a r e m a r k a b l e d e g r e e o f a e c u r s e y , a s w i l lb e s h o w n l a t e r n t h i s c h a p te r .


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. . - - - , - , - - _ _ , : . / , , , ~ . , ~ , . a B o ~ . , ~ , . . . , r ~ , . _ . . ' ' . .in ~ . l=Z .

,"~ ~ , ~ , . . ~ ~ . . .

20~ .1 T h e d t . r e ~ I p r ' o b l ~ . } . i ' : ~ i . i : : .' . i..

- O : I . ~ ' / r ~ - I

: ~ . ~ - l e ~ - d e l~. s . e = , - l ~ - o

~ / . - - , ~ : 4 . ~ ' " - 0

a h ea ri ng s t.r ea d i s ~ u u t i o n r eq u ir es , O f e o m ~ , t h e d a t a o f n m n ~ .p r o b a b ly a g r e a t n u m b e r - o f t u r b u l e n t b c u n d a r y la y er s,f o r w h i ~ t h em e a n v e l o c i t y f ie ld .i s e ~ t a l l y k n o w n ; T h e e x p er im e n t a l d e t e r - . ; ~ - t i o n o f t h e s h e a ri ng s t re m i s p o s d b l e b y u s i n g t h e h o t w i re t e e h -nique~ B u t such ~ e n t a a re , : ;~ ou lt t o m ak~ and. th e n e o m a r ye ~ m e n t i s n o t alw a ys a va ila bl e. 0 ~ l y a f ew e g ~ n p l m a m f o u n d h r t h e l i te r a t u re a n d t h e s e h s v e b e e n m e n t i o n e d a bo v e . T h e r ef o r e g r e a t ~e ~ ex i s t , i n oompu t~ug t h e . hea .- iu ~ f rom the e~ 't~ex~ .m~i] lyd e t ~ i n e d m e ~ n v e l o c i ~ dism ~o ut~o n w i t h th e m e e f t h e b o u n d s r ylayer F~ . (4.9). Th is Fzo blem ~ be ca l led th e d i rec t prob lem. Suc h .c a l c u l a t i o n s w e r e m a d e m a n y y e a n a g o b yG z ~ r . h w i t ~ ,] ~ h u l t s - -

.O rU no w US a n d o t h e r s, T h e y ~ v e r y t i m e - e o m u m i n g a n d t h e M o c r a o yi s o ~ e n p o o r ~ o f t h e n e e d t o d i ~ e re n t ia t e e x p e rh n e n ta l f g n e t i m ~A s impl if i ca t ion Of th e p rob lem i s thus ve ry dee i rab ~

.Very p ro bab ly th e f ir s t a t t em pt to d e r i ve .an auLly ti cS] o r t h e ~ . h e a r i ~ . t i n = d iz t rib u t io n w a sm a d e b yF ~ .

~ F o ~ , .the Poh lhausem m eth o d fo r ~ ow, . he . tr ied to t l t Lp o l y n o m i n s l fo r th e shear in g s / ram prof i le , .:

. . . " '.." . ~ .

- . . . - - - ~ + ~ . / ~ ) . , f o ~ o < ~ < ~ (~0 .~1 q ' ~ ' ~ - I - . ' . .

w hich m, i sf ie s a num~)er o f bo und ary o o nd i t io ns" ~ t th e wa i l andt h e o u t e r e d g e o f t h e l ~y er. T h e f o l lo w i n g b o u n d a r y c o n d i ti o n s ~ - e i m p o ~ d

b y d e ~ n m o ~

f rom the boundary Isyer F.Xl . - (~ '5) , .f ro m t h e d e ri va ti v e o f t h e b o u n d a r y k y w E q. (4~) a nd co nt in ui ty Eq . (~.~;),f r o m t h e d e f in i ti o n o f t h e b o u n d a r y l a y e r

th ic kn es s ~, -.~ . ~ 1 ~ - 0 f r o m t h e c o n t i n u i ty o f t h e d e r i v a ti v e o f

t h e t o t a l h e a d o n t h e s tr e am l i n es a t t h e: " ou te r edge o f the l aye r.

F ro m th e co nd i t io ns 2 to ~ the fwst four eoe fl lc ien t~ o f Eq . (20 .~)ea~be evaluated . .

A com par ison o f she a r~ g s t re ss d i s tr ibu t ions ca leu |~ ted in th i s waywi th m easurem ents m ad e . by Schub&uer and I~ebano f l -~ d i sp layed


22/67

very d i sap l~ in t ing resu l ts . I t i s no ta;mcult to lhow whyt he se a u u m p -t ~ o n s . l e s d o = u e h p o o r a g r e e m e n tw ith exper im enta l zesult4 . Th e m eanveloci ty ' in tu rb ule nt bounck=3r . layer= increamm eve n w ithin th e sub-.l=yer to high. v a l u e s . k ~ . h a r e no longer ne~l~ible in ~ bo m zd ~il s y er e q u a ti o m C o n s eq u e nt ly t h e co n d i ti o n a t y - 0 a / o n e i s n o t r e -spona ib le fo r the shea r s t r eu nea r th e wa ll . Th i s fo l lows fzom F i ~ Uo12 ,i n w h i c h t h e s he a zi ng m d i s tr i bu t i o n s n e a r t h e w a ll w ~ c a I ~ .w i t h t h e a m m m p t io n o f a c o n l t a n t s k i n f r io t i o n ~ e n t . A s i m p l e b u tm o r e g e n er a ll y v a l i d e e t i m a t e c a n b e m a d e f r o m t h e r e l a t io n E q . ( l l . ~ q )g ivenb y C o l e ~ =, f h e var ia t ion o f ak in f r i ct ion ooe~c~en t i s t aken in toa~ o un t . Dif l ' e rent i st ion o f ]~q. (19 ,3) w i th reapeet to = ~

r o m ~ l . ( 1 8 . 1 5 ) t h e v a l u e o f f c a n .be expressed as = fu nct io n o fB = ~ / e z m d ~ , i = t h e . f o r m " "

. . , o ;

the d i~ere nt ia t ion o f wh ich gives .. . . . . . . .

. d Z 1 d U / d = ~ H ' t d ~ ( c , / ~ ) " "-. - - - , , - - . . ( s o . ~ d=" m ~ ( = ~ t s ) c ~ s d =

" ~ d / 7 . . . a ( = , i = ) ." ( = ~ = = ~ + - " d = ; ( s o . e )

i

~ t r o d u a n S : E q . ( = 0 . 3 ) = r i d ( S O . S ) s i ' , '= . . .

I d = , . I d O ' , ,~ , , d = / 7 , , d s

d E ' ~ / ~ c :+ / ( ~ ) ~ ( 1 / ~ ) d l n ( / 7 . ~ ' / , ) / d x + d O / d , z ]

t

- u - ( ~ - x ~ / d Z . . . ( s o . ~

= p p r o ~ y e q u a l t o , , ~ t y , M s ee n f r o m F ig . 1 S.3 . ~ t h t h e s e s @ -po sition s E q. ( .~0.?) re du cm to .. "

I d = r 1 d U = , I d H . .=~d = = u . d = U d = " ( ~ 0 . s )

Th en w i th th e use o f lZ,q . (11 .36) the shear ing s t ress d is t r ibut ion in th e


23/67

- - . . +-- . . ,. ,

~ n j u s t o u t . d e th e . u b k y e r can b e e k p l e m ~ ~ : ? ' "

. c ~ . r , O , i , ' - 0:. ( ~ ' a S O ) ] - = ' - - , . , , , ,

the ft.-s t, t h ~ b o u n d ~ 0ondit , o , ', , mentioned ~,bove, Eq. (~0.9)re s ~t~ remmrka~,blydiffenm t, from thos e of th e ,dmple. re. l~tion eve n for

w ~ d i s~ o e . . The Eq . (~0 .9 )m,.y ~ ~ su a id . t0ez t~p o l ., t, e , m .e ,u U~ ahea ,~g d is t r ibu t ion to ~ , - 0 , in order to f ind th e .v l iue ot h e w a l l .s h e a r s t z e ~ ' " . ~ ~ - , '

T h o o ~ l o . l ~ o n o f t h e e u t a e ~ h ~ ~ d ism "ou ~io= b m u o h,dmpli~ed by i~ o ~ uc in ~ s me~u ve loc i ty p ro~ le d e p e n d ~ o . i

paramete r, a s g iven in SeCt ion 18 , i n to the boundaz7 ls y e r ]~,q.(4.~). Th e deriva.t, ve o f th e bounclsz~ layer th id m e u is ~ b 7m e a n s o f t he m o m e n tu m in te g ra l e q u s t i on . T h e m m s f m m ~ o n o f t h e

. , . : . . ~--~-f (~/~;~) . . ~ o . ~ o .h ~ b e e n t re a te d b y T e t e : ~ .r od L i nx o . T h e ~ e a r ~ ~ -f o r ~ e ,h v elo c ity p r o m m e au b e r e - - t e d ~ ~ m m o f ~ tm m mw hiah ~ I ~ I ~ i o n ~ l t o th e ~ h ear ~ , t h e ~ S n K l i m t , ~ 1t h e r r a 4 i m t o f t h e d ~ a ~ ~ ~ v e ] y . T h is z ~ d o . r e~c~

w h e r e g . y l O . T h e e f fe c t o f h e P,eynolck norm al ~ k n e g l e e t ~Th e f ~ o , , , o f E q . (1 0.11) o sn r e a d ~ b e e o m p u t ~ it" th e t o n ~ o ~ Io f ~Cq.( lo . lo ) ~. , r ~ :

~(~:. ~ - l + ] , d [ ' - / ( C ) ] , d ~ .

~ c . ~ - ~ ; - , - . ~ / . , " , t~ - ~ - + , ) s C ) l S d ~ . " ~ o.i,)O ' r , , , ..

0 t


24/67

176 ' - J; C. ROTT~

~ t h e . p ~ a l + c a . Of po wer ~ , , p r o m . + , m o r d ~ t o ~ . ( m ., ) th e f- .,o -t io n s r e d u m t o -

++ H - I / x O t+ ~ < + . ~ - , - + + + + , > t ~ _ j +.+- - - - - -

+ + " , , 0 , + ,i ~ 2 O - s

. . . . . j . " . . . . .

Th e func t ions a re p lo t t ed in F igs. 20 .1 t o 20~3. A few examples ha veb e e n c a l c u l a t e d a n d c o m p a r e d . w it h e x p e ri m e n t a l r e s u lt s f z 0m S c h u -

bau e r a nd Klebanof l'L~ ; these a re show n in Eg . .20 .4 , In a U th ree e, ees

. . . .

, . , . . .b I I I ,~

0.4~

' " ,

t~ ffib .,t-, bou nd ,~ L , rm h . v ~ pow~ ~wp e ~ , ~ F~. (2au}.

t h e c a l c u la t e d c u r v e i s r e m a r k a b l y l o w e r t h a n t h e f a ir e d ~ .urve t h r o u g ht h e m e a s u r e d p o i n t s ; i ~ m u s t b e a ~ tm l tt ed h o w e v e r, t h s t t h e s c a t t e r o ft h e m e M u r e m o n t s is c o n s id e ra b le a n d t h e v a l ue s a re , O n t h e & v e m p ,too h igh . l~'ear the wal l the exp er im enta l shear s t resses a re ab ou t 4 0 %too h igh* . Under these c i r cums tances the ag reemen t be tween exper i -m en t and ca lcu lL t ion appea r s to be q u i t e sa ti s f sc to ry. I~ m ay be no t i cedw i t h r e g a r d t o t h e e r ro n e o u s re s u lt s o f th e w a l l sh e a r s t r e ~ c a l c u la t e df r o m t h e m o m e n t u m i n te ~ - al e q u a t io n t h a t t h e eft'o ct o f l a t e ra l c o n -vergen ce or d iverge nce of th e m ean s t reaml ines is e; iminA'.~d to a f i r s tor de r in ~ .qs . (20.11) to (20.13). Th us the ca lcu la t ion o f th e shear ings t re s s d i s t r ibu t ion o n th e basi s o f Eq . (20.11) m ay becom e a use fu l too l

* Av c o r d i n g t o a p r z ~ - a ~ o v a m u r ~ c a t i o n o f D r.. G . B . S c h u t m u e r t o t h e w r i t e r .


25/67

- . . ~ . . . . ,~ . . .

f u r t h e r e x a m i n a t i o n i s v e r y.d e s ir a b l e b e fo r e t h i s m e t h o d e a ~ g e n e ra l l y 'i ~ . r ~ . ~ d ~ L ~ l l y i t ~ l S b e n ot ic ed t l ~ t h e ~ l e u l ~ t ~ l ~ l ~ h ~ l [

p t ,,of i]e , d o no t u ,t&r,~y t~e: bou ndax T .ooi.~dit ion ~ q . ~ , , 0 wb . ~. . . .

.~k. 4

~ k " " I '

I / "

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. . . i~ 1 1 .~ i .. .i . i

~ . . " ~ . ~ , . , . , -~ -~ . - I ~ l , Q . 1 3

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~ , ~ .

t SI - I /

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. \

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m m m m m m

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t .m 'b u l e n t b o u n ~ A ~ I s . T i n h a v i n g p o w e r ] a ~ l ~ o 1 e 6 , f r o m iP .,c . (~ 1 0 ,1 1

0 i ~ 3 4 $ $ ? I Io l i 12Y

t


26/67

- s ; ~ ~ o u o . . ~ o m ~xo ~ u ~ a m ~ v ~ ~ m . . ~ h o . . ~ ~ , a ~d o n o t m n , , i~ f 3 v 'P,~ :le b o u n d a r yc o n d it io n ~ / 7 1 ~ - - 0 a t ~ / - - & ~ envea b e t ~ ~ ~ . i ~ e m s ~ ~ . t ~ is

i f a , m e e , n " r e l o ~ ~ z m ~ , l ooomlmeed O ft h o u n i v e x . ~ , Is w o ft h e w ' s l l a m dth ~ ~ ~ m ~ , ~ , x . ~ . , ~ , a ' ~ a . . m ~ , e a ~ 'o ~~ ~ o ~ , ~ ~ . .

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. d v ~ o X m m m S m U . u ~a d c u l , , te d ~ ~ (~ 0 , 1 X) a n d ( ~ 0 . 1 3 )

- , , - - , , - - e A L k u d s t e d ~ F,qan , ( 2 0 ~ ) m d ( ~ 1 8 )

a; m, 17"5 ft H m1 . 3 11

" , , 2 5 " 0 t ~ " ,2 " ~ 1

x~-quired re la t ion can be d er ived by" in tegrat ing th e non-dim ensionale q u a t i o n o f t u r b u l e n t b o u n d a r y l a y e r s, E q . (1 5 .8 ) , i f~ P l a ~ - ( ~ P I ~ I )d X l d g i s i n tr o d u c e d a ~ d t h e v a l u e I , d e f in e d b y E q . ( 15 . 1 0 ), i s th e .e h a m e t e t ~ n g p a r a m e t er . T h is r e la t io n w i l l n o w b e g lu ten . I t i s

1. ~ Z , = - - - - = * X [ ' l ,l , ~ / ( C r / 2 ) ] r ; % [ , I ,I , V ' ( C t p - ) ] - -


27/67

~ . . . . . . . . - , . " . . . ' -

r u n . i o n s o f t h .k e q u a t io n a n , d e r m e d - M . . . . . " "" " " " " B ' - " * :"" " "" " . . . . . ' - . " , " ." . ' " . , , . ' . " / . " ." . . , . "" . . . . , "

" ~ " "e . , : " : i " " " : o - " . " ' :

" . . . : " ' I I " ; . - . . ' ~ . " " . " ' . - - "

0 O " ~- . . . , , . ,. , , , , ,,m , , ,

. . . . . . . V ' ( c . . . . " ' : . . . . " " t " , " ~.

~ ' - c ; - : . . . -

" " O .."

: " i , "~ ' + ~ ~ i - ~ ~ , . ~. . . . i . e . S . - , , . . ~ - . . . :

. . . .

' ' ~ . ~ ( , , , / , ) z ~" '. , . ,

- : . . . . . . . " " ' . ( ~ o . ~ ) -

'~ 'h em = (U - -- 17 )]1 ,, i s , , .f u n c t i o n o f 9 a .nd I . W i t h t h e u m e s i m p l ~ c a t i o ~ w h i c h l e d t o E q ~ (2 0 , 8 ), E q . ( 1 ~. 9 ) g i v e s. . . . - '

" d h ~ l c d 2 ) : . : ( 2 0 . 1 e l . c ~ = ~ " .

. . . . .

Fa:l .(18 .12) . s i n g Co]e ' s wa~e .f~motio=.

. 2 0 . 2 T h e i n v e r s ep r o b l , ~ m ; .T h e p r e ce d in g e q u ~ i o n s c a n n o t s er v e t h e p u r p o s e o f r e d u e h ~ t h e

b o u n d a r y l a y er e q u a t io n s o f Ta b l e 8 t o a s o l v a b le s .v st em o fe q u L t l o m , ,s i n c e t h e y o n l y m i k e a s t a t e m e n t a b o u ~ t h e s h e a.- in g s t re s s w h i o h i sr e q u ir e d b y t h e e q u i l ib r iu m o f fo r c es , i f a c e rt a in m e a n v e l o c i ty d i s t ~

b u f i o n i s b~iven . ~ rJ~a t i s r equ i red i s th e answ er t o th e qu es t io n , ' rWha ti s t h e a h e a r in g st re ss i f t h e d l s ~ b u t i o n o f m e a n v e l o c i t y h a s a w e l l .de f ined fo rm w i th r e spec t to th e =- and ~ /.eo-ordLna test ." A re la t ion i st h u s t o b e f o u n d w h i c h p r o v i d e s a n e q u i v a l e n t t o N e w t o n ' s r e l a t i o n ,E q . ( 2 0 .1 ) fo r la , ~ ; - * - f lo w.


28/67

i s o ' , l . c .. . , . .

T h e a p p l i e a t io n o f - s u c h h y p o t h e t i c a lr e l a t i o n s a sPrand t1 ' s -mix ingl e n g th t h e o r e m a n d y o n K ~ m ~ a~ 's d m i i a n 't y h y l m t h e a k d o n o t l ea d

d i rec tly to r e su l ts w h/c h a re quan t i t a t ive ly use fu l, bu t i t fo llows qua l/ota t ive ty tha t the d is t r ibut io n o f app aren t shear Stress - ph-~o u t s id e t h esubl~yer i s 8~ ie t ly re ls te dl to th e m ea n v~.locit prof ile, jus t as th e shea rs t ress o f ]am inar hLyers i s re ht ted to th e ~ e lo c i ty pro~le~ Th is suggeetat h a t t h e s h e ar s tr es s d i s t ri b u t i o n m i g h t b e e x p ~ b y

. . . ,

. . . . 5 " a

A n a t t e m p t t 0 ~ t h e t u r b u l e n t ~ s t re m d i s tr i b u ti o n w i t h t li o se u s u m p t i o n s h a s b e e n m a d e b y t h e p r e s e n t. w ~ i ~ o . Fediaevaky ' s

, power ser ies Eq. 2 0 . 2 )a n d t h e m e a n v e l o c i t y p r o f i l e a c c o r d i n g t o.Eq . ( IS .7 ) a re t ak en aa basi s, an d , ~ o f. the bou ndary co nd i t io na l ~ - dP,,Id= ., & l i n e ~ . r e l a t i o n b e t w e e n t h e f ir st e o e ~ e / e n t o f E q .(20.2) and th e Shape.p.aram ete o f Eq . (15.7) i s in t ro duced a s . "

: a x - 2 - A t ) , . ( 2 o . l s )

Where : i s ~ memp ir ica l co ns ta a t . A o (~ 0-45) i s th e ~ 'a lue o f A for th el ~ t pl at e. pr o fi le . T h e b o ~ m d ar y c o n d it io n ~ . l ~ z . 0 : a t . y m 0 isd ropped . A f i r s t e s t ima te o f the co ns tan t: c a n be m a d e b y m e a n s o fPran d t l ' s mix ing l eng th . re la t ion , Eq- (5.3 ). In . c0nnec t ion w i th yonKiamz/m ' s hy po the sk o f s imi la ri ty, a va lue o f a - 4 is ob ta ined , an di f the mix ing l eng th isa ~ u m e d t o be. l - 0 .4y th e resul t i s ~ - ~ Th isappro ach to the p ro b lem was o r ig ina l ly des igne tt in o rde r to f ind ana m f ly t ie a l r e la t io n b e t w e e n t h e s h a p e p a r a m e t e r o f t h e m e a n v e l o c i typrofi le and the in tegra l o f the tu rbu len t ene rgy d i s sipa tion . F ro m theinve st igat io n o.f f i f teen bounda1-y layers in adv erse pre ssure gra die nts

i t w a s f o u n d t h a t , o n t h e a v e ra g e, t h e e n e rg y i nt eg r al e q u a t io n E q .(17.9-) w as satisf ied best , i f s va lue o f = - 2 .57 w as cho sen. Tw o shea r-ing s t ress profi les are ea leula ted f ro m these re la t ions and .p lo t ted in Fig .20.4 , Fo r H - 1 .6 agreem ent w i th the shear s t r e~ d i s t r ibu t ion ca len-la ted f rom Eq . (20 .11) i s goo d, for H - 2 .22 the agre em ent in po or.

Ap ar t f rom t he spec ia l a ssump t ions under ly ing these ca lcu la t ions th eques t ion o f fundam enta l in te res t is how fa r th e no t ion t lu tt th e shea rs t ress prof iles are re la ted to the m ean veloci ty prof iles a lone can ac tual lyb e ~ fo r approx im ate bo und ary l aye r ca lcu la tion . F ro m the resu lt so f t h e d i re c t p r o b le m , t h i s u m ~ m p t i o n m a y a p p e ~ t o b e in c o m p a t i b lew i th the re la t ions Eqs . (20.11), an d (20 .14). .This argu m ent i s how eve rnot~ qui te coneluaive. S ince th e funct ions fz an d ~ , fo r the sam e pa rs-m ete r H , a re approx im ate ly s imi la r, i t i s conceivable tha t , fo r af i x e d


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- . - ' . . . . . : ~ "" " ' " ' -" . / . : : ~" i~ ' ." " '" " . .

always in a eertain combination, such that the shearing stre~ diltribu-:tion remains msentially unchanged. The argument is not fa~, un]em it "is raised also for -midst boundary layer calculation methods. Any

.objections to.the .ammmption in questio n must. there ~e r e l y upon.-other argument~ ~. '

It is very probable that the Re3molds shear h affected by upstreamconditions. The reason for this behaviour, which has already been dis-cussed in Section 15,2.k that the eddies are n~ p t downstream withthe mean flow. as is seen from "space time correlation measurement&

~'rom the balance of the turbulent energy for the fiat phLte boundarylayer i t is concluded that the convection terms arc negligibly mnallnearly everywhere across the layer except for the outer 15~/o. An/nflu-enee Of tbe u p ~ history on the shear stress is therefore to be ex-

.p e c ed for the outer part of the layer. Although it is generallytha t this effect must t ake pla~e in the development of th e turbulentboundary lsyor, no e=Perimenta proof is available to .demonstrate itquantit atively.. . -

T h e o n l y k n o w n a t te m p t t o m a k e u l e o f t h e p r e v iO ~ ~ i n sqm mt~ t~ tive ma n n e r h u b ee n m a de b y ~ a n d ~ m . l~ e d ~ -e~ky 'm boundary cond it ion (5) i s a lt ered to ~ 1 ~ - - T . ~ , i t y - S ,in order to ob ta in s c o n s ~ t slope at ~ he outer edge of the boundarylay~ for Iz:mitions downstream from z - z,, the initial position of th e d veree pressu re 8nLdient. ~ ~ t m Lkes t he sh ea ring s t n md~;bution depend on conditions upstream. Using a.mo~i~wd.poly-no~ - I expression for and incorpomt/ng the2~i sed boundsry eoadi-. ion (b'), together with the other four ~ conditions, ~ sadRobertemn.derived h e following ~ m :

. /~. - ~ l - ~ v /~ )] - ( , + ~ ) [ 1 - ~ v /8 )F + ( , + 1 ) t~ " ( W ~ ) ~ s, ' (~O .x t)w h m

.,8-1,e "~ ~ ~(de./dz)

- - - , ~ - . ( S O . 2 0 )q'm ~)~ q'w

A eoraparison between this relation, which is to be e]aesified as belong-ing partly to the direct problem and partly to t heinveme problem, andthe exper~nental resulte of ~.J~ubsuer a~d Kleb~noffahows s som~'h~tbet ter agreement than Fediaevsk-y's relstions, but it contains the short-eo,-ings i~herent in the boundary conditions at y -- O.


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2 0 . 3 T h e ~te~ ' a l o ft h e e n e r g y, d t s ~ t m t t o n "T h e i n t e ~ - t l o f t h e e n e q y d ~ W l S t io n ~ t h e b o ~ d a r y fr y er

r,h i ~ w h i ~ o ~ u r s in th e ene ,..,gy int~.,,~za~ F.,q . (17 ~) , ~ .c l ce c ly re-k t e d t o . th e sh ea r ing s t r e s s d ist,r ibu '~ion. Th e in teg ra l o f ~ e mrbu len '~e n erg y d ; ,, ~p e .t io n i s o b t a i n e d i n te rm - ' o f t h e w o r k d o n e b y t h e m e e a~ o w o n t h e R e y n o l d s ~ ~ a d ~ h e ~ o f t,he tu rbu . l en t ~ex"87 f lu x

Q .

o

f-,, d y - . g ~ - - - - d y - (u - v ' ) ~ . ~ d y - U ~ / t d ~ . (2 0 .2 1)Thenb y E q . ( 7 ,1 ) ~ d ( 4.6 ), t h e i n t e ~ s l o f t h e t ~ t s l e n e r ~ d I ~ / p s t i o nb e c o m m

E ' d ~ - a +- - . - | ~ - ~ _ d ~ - p ~ . o 7

O @

T h e I~.-~ t w o t e r m . o n t h e r / s h ~ h a n d J /d e . r e m e r ely c o ~ n t e n D .T h e m a i n c o n t~ - ib u ~ m t o t h e i n ~ s l o f t h e d i m i p s t io n t h u s co m e cf r o m t h e w i n k o f t h e s h e a r s t re w, ~ n d it x n ~y b e c o n c l u d e d t h a t b o t henergy dL~ipat ion snd shear s tress ob ey easent/a] ly the sam e lsw s.

I t w a s p o i n t e d o u t i n S e c ti o n 1 3.4 t h a t m o s t o f t h e e n e rg y k d i ~ .p a r e d in t h e t h i n l aye r ne a r t h e w s l l ,w h e r e t h e u n i v e r s a l l s w i s v a l id .A n a t t e m p t t o d e r i v e r e l s t i o n f o r t h e d i ss i p a ti o n i n te g r a l , t h e r e fo r e ,m a k e s u ~ o f th e . ~ n a . - i t y : e k t i o n 8 o f t h e w a l l 6 o ~ . S i nc e ~c tks ips t ion cont r ibu tes un ly wi~h/n the 8ub]syer re ,o n , the d i~p & t ioni n t e ~ c a n b e w x :tten a s

(2o.2~).

E d y - E d y + , d y . ( 2 o , ~ )

0 0 y

w h e r e y ' d e n o t e s t h e d i s t a n c e o f a n a r b i t ra r y p o i n t o u t e i d e t h e s u b -


31/67


32/67


33/67

. , . . ' , . , . .

resulted in a relat io n between. F and th ep a x a m e t ~ J o f E q . (I S.V )o f."'--,,.,~,oo z -~ " ~ - ~ : " - . '

~ . - ~ . . + 1 . . ~ + 1 . ~ . ~ (s o=s )

m e d is ~p atio n i n ~ ~ n b e ~ in term s o f a n o n ~ , , , ~d i , u i p a t i m ~eo e~ eien t z~ b y . ,. ; .. : " : ... . ..

. , , : . . . . , . ,-

. . ,

g D . ' t -: . .

- . . . ,

a n d ' s oE q . ( S 0 . = 6 ) ~ . '- ~ : : - . . . . .

UsingEq, . (S0~8), (18.10) and (18.15),. he. d J ~ l ~ t i o n~ e i m t e s n .be r e l ~ t ~ t e d u a f u n c t i o n o f the aha])epazamete~. H and, skin fxiotiozz

...eoe~fieientz; u s h o w n ' ; , " . : F L I ~ .~o,e . ' . . .

- . - ' - . . . . - . . .

S z . T h e " Z " o ~ u . t a ~ t B ~ d ~ ? L a y e r o n a Zq m z P l a t o . ~ - o

: Z Z . l I n t e ~ z ~ t i o a o f th em o , - e n ~ e q ~ t i ~ z ~ ' .. ' ".

sk in f ~ o t i o n ~ d t h e t h i c k n e n o f t h ek y e ~ a s , v f u u o t i o nof t he k n l O h .= . T h k ~ b ea~omp~;-~,,.,~ w ith th e aid of th e r e l ~ for.

t h e 1o~1 ~ g iv m in ~ . i o n IS -- ,d t ~ m o m m t u m i n ~ leq ua t io n o f Tab le 5. S inoe the ve loci~3 U , out~ ide the ky e r k o o n ~

. ., . . , . .- . _ , , ~ - ~ _ ~d #. . ~ ' ~ ~ , .~ 1 z z , , ~ ' . . - (sz.z)

~he~e the la ~ term o~ the right-hand aide o f this equation k unua]lyneglected. According to the ~ u ~ z i t y re la t ions of Seefion 15 thet , t ios I an dE t ' ,-ve fixed v~u ee so th att h e l o c a l a kin f r ic t io n e o e ~ t ~c a n b e e x p r e s s e d ~ a ~ u n ~ t i o n o f U ~ , e ,~ , a n d kr. I f ~ i ~ z er o o r o o n ~ m tE q . ( ~1 .1 ) c a n b e in t e ~ - s t e d t o ~ e

f ~d~= " ~ e . e , , , ~ , ) :" - - C - z = ) :O n ~ bas i s , t h e ea~ca la t lon hasbeen earned ou t forth e lawyer on

smo oth surface by ColeslS. Unfortunately, c~ ca nn o tbe writ ten ex pli-citl y in ter m s o f 8, Th is m akcs direct integ ratio n A~W~cult.Fo r thist e a -sou a new vL, ' i ab lo of in tegrat ion~ - ~ ( 2 / ~ ) is s u b s t i t u t ~ .Then

s3


34/67

. . . : , . . . . . ,

186 J . C . t t 0 v ~

ca n be ia rstai lsivi .

. . . . . f - . fTh e m o m en tum t~h/ckness can no w be expressed in t e s t s o f {~,

w hich is ob ta ine d firom Eqs . (13;13) and (13 .15) , Th e m agni tu des x , X ,s ac l I s r e u a i v e . , ~ c o n s t ~ t s a n d t h e v s lu e o fC ( ~,,] ~/v ) - C ( U , , ~ [ ~ )i s s funo t ion , o f the su r f sm ro ug hn e~ d i s tr /bu t ion .

T h e p r o p e r c h o i ce o f t h e T ~ , ,~ o f / n t e g r ~ i o a i n E q . ff-1.3) r e q u ir e sc o n s i d e n ~ - - T h e u su al ~ on di~ io n i s ~ t t h e b o u n d a z y l a y e rth io k n e u ;- .zero. t the leading edge , Le . 0 - 0 a t = = 0 . Ho wev er, ~ho

~- s u b je ~ , t o ~ r y d i f f e re n t o o u d i t i o n s n e a r t h e fro n~ o f t h e . p l s t a .l a m o ~ e ase s t h e b o u ~ d a ~ l s y e r / J ~ f o r s ~ l e n g t h f r o mt h e l e a d in g e d g e . . h o ~ h ~ o ases, s e p s r ~ i o n ~ t t h e l e a di n g e d ge o r s tan ob s ta~e and sub sequen t r ea tt achmen~ of the f low pro duce s s lwla-t i v e ly t h i c k . tu r b u l e n t l ~y e r n e a r t h e l e a d in g ed g e, s u c h t h ~ t h e b o u n -

d a r / k y e r is turbulen~ s l m o ~ f ro m th e . le a d in g edge d o ~ T he~ t y c o nc ep t in tr o du ce d i n .Sectio n 1 3a as um e s th s~ t h e e ~e ct o fco nd it io ns n e a r t h e lead ing ed g e o n t h e lo cal flo w ~ c s f a r. d ~ w il l van ish . Tn o ther,w ordm , d o w nst rea m o fs l ~ ' ~ l a rpo~/ t Jon s ' - t~b . bo undary I sye r wi l l deve lopu l=edi~cedb y E q L .(~1.3 ) sa d (21.6 ) ~ v e l y i ~ t , he in i ti a l oond i~ /on i s p ro per ly chosen ;I n ord er to be &ble to a~eoun~ oo nven/ent ]y fo r th e d / fl 'e re=t bo un da ry .cond i t ions , i~ h ~ beeom e s t~ud sn i p rac t i ce to cons ide r an idea l tu~bu-len~ b o u n d a ry k y e r o r i ~ a ~ w i t h zc~o m o m e n t u m t h i ck m m , ~ - O,a~ s p o in t ~ - zo . Th i s po i : , ca l led the v i r tu a l o r ig in o f th e hy er , wi l l

genem1y no t coincide w i th th e leading ed ge .o f the p la~e . T he idea lbo unda ry I sye r n ~ y be ea lcu l s t ed f rom E q . (~1.3 ) and (~1.~) a l tho ughn e a r ~ - ~ t h e c o n d i t i o n ~ a r e b e y o n d t h el ~ m l t ~ w i ~ h ~ - which , th eund er y ins Msumpt ion~ a re va l id .

T h e c o n d it io n s s t t h e v i r tu a l o r ig in a re ~ - s c t e r i z e d b y sva lue o f ~ - -~o , wh;ch ; - ob ta ined f rom Eq . (~1.4 ), I m t t ~ ~ = 0 , th s tis

~o - I fo r = - ~o. (~1.~)Wi~h th is in i t ia l co ndi t io n ~q . (~1.3) ~e.kes th e fo ~n

~fE q. (21.4) i s in t ro duce d in to Eq . (21 .6), th e in t~ .gmtion can be ~ r r i e do ut ans ly~i~d/y in spec-;x oaaes . Th us s re ls t io n i s ee t~b lkhe d


35/67

. . . . . . . . . . . . .

. . . . " " I ~ . ~. ~ ' ~ , = ~ ~ / ~ , iS ~ " . : ~sT

T h e c o r r es p o n d in g m o m e n t u m , t h i e lm e s s i s t h e n r e a d i ly c a l c u l a t e d f r o m' E q . { 2 1.4 ). I t m & y b e m e n t io n e d f o r t h e s a k e o f e o m p l e t e m e l t lm ta c c o r d i n g t o t h e i n i t ia l c o n d i t io n ~ q . ( 2 1 . b ') t h e d i s p l a c e m e n t th i r _ k n e li s n o t e x a c t l y .z er o s t = - = ~ F r o m E q . ( 13 .1 3 ) i t i s f o u n d t h a t ..-

~ o e p e ola ] o==e e w f l] b e t r e a ~ , i n m o r o d e t a i l ~

. " io -

~ L _ ~

~ , i o .

I " '

t O s

B ' .

%

"" I

"I

b

, , i ( i+ , ) ] =_ . . I k _ ( _ , + , ) . +

on s smooth surr~os . Loes l sk in f r i ~ ~ ~ sooow U ~ to ~ i -

(s 1.8 ) 8 r id m ~ o o f m = = e n ~ ~ e ~ t o ~ 1 . { = I. @ t od ts m m a~ R - m . .- ~ o r l ~ T n o ] d s n u m b e r U . ( ~ - m , ) / , . .= - 0 .4 , X + O (0 ) - 3 .8 , . I - e -~ -

2 1.2 B o u n d a r y l a y e r o n s m o o t h s u r f a c e '

T h e - ~m p t es t a n d = ~ o t h e m o s t i m p o r t z n t e a se is t h e b o u n d a r y l a y e ro n s s m o o t h s u rfa ce . T h e ~s ) u e o f 0 i s o o n ~ m t . T h e n b y s o l v i ~ E q .(9-1.6 ) and re in t rodu- .~ng ~ / (2 /c~) fo r ~ , th e f r i c t ion l aw o f th e tu rbu len tf ia t p l s t e b o u n d a r y l a y e r o n s s m o o t h ~ b e c o m e e


36/67

1 8 8 " . ~ ~ : "q ~ o , .: : . . . 7

r o s y w e l l b e n e g le c t e d . T h e l o c ~ s k i n f ii c t~ o n c m ~ t ic i e n t , ~n d t h e o o x z ~ -

p o n d i n g . m o m e n t u m - t ~ c k n e m f ro m E q . ( 2 L Q i s sh o 'w ~ i n F i g .2 1 . L

' - . . . , . . .

2 4 0 C ' '

=o ~ JI . , / / j , ~ "

I

R ~ , Ir P O C

P ~ of Z ~

.

ct s,O "c,,,Jo3

\I \

W i l I I

Fxa . 21 .2 . Deve lopmm st o f the bou ndm 7 l sym. oa ~(a ) T h l e k n m(b ) ~ ~ ri at ~n a ~ u a m .

L ~ ~ e ( t pT ~ pos i t ion of t z 'm ~t /on"V'.O. V 'ir tusl o~ g ln o f turb ulent ls ye rP ~ e - U ~ / , , R o S - U , a ' / ,

~ b s t l o m fo rt h o ~ ~ y ~ l ,P,o # - 0 -t~ 64 V P , ~ , P ~ e . - 1 . 7 2 ~ / l ~ , . c t - 0 . e S 4 1 V ~ , . a , -

vs luee accord ing t o c ,, /cu l st ion . ( ud dm tnms/ t lou)- - . - - . . - - - me lm v s l u e , , o f ~ t u t l bounda ry 1 8 ~ . ( tz 'u a ition p rooem.ex tend s to s f in/re region).


37/67

. . . . . . . . : . , . ~ . : . ' . . . . .

. . " ,,

F r o = t ~ a ~ the ~ o o t h ~ p ~ b oo n e. 3 ~ y ~ n ~ m l y ~ d ~

t e rmined , i f thei n i t i a leo nditio ns are d e f in e d , . : . . : ~ order~ m u s t r ~ e t h e m e t n i n g o f t~ ~ , - ~ u a lorigin ofthe turbd le=Sl ay e r, th e d e ve lo p m e n t o f th e b o u n d a r y l ay e r o n a s m o o t h . ~ t p l t t e ,

h e r e t h e f low is lam inar fora e e s t a i n l e ~ , t h f rom t h e l e a d i n g edge,.ia.show n in Fig. 2 1~ , .The ealcul~tidn is usual ly baaed on th e a m um p6o n-t l ~ t t h e flow c h a ~ m sudden ly f romth e l a n ~ t ot h e fu lly tu rbu len t :s t a t e a t l x~ i t ion T. Th e v i r tua l or ig in i s chosen su ~ tha t t he l am im ~and . tu rbu len t l aye r ]~ve the same m om entum th ickness a r T . ~d i s - .plaoementt,h i ~ m e m a n d I o o a l. ld . . fr ic t ion shOW sudden ~ ,m gma t

r a n ~ t io n . I n a n e ~ -tu a lflow, t h e c h a n . e ef rom l amina r t o t u rbu len tf l0w ~ no t t ake p laee suddenly. The turbulen t Oecum fnut in smal l "sp ot. , w hich m o ve d o ~ gro wing in ~ a nd ~rrmber u n ti lt h e . -suremce is entirely covered w ith a tur bu len t kye~ . Th is process extends"ov er .a f ini te region. All m ean value= o f f low qua nti t ies Var~g r a d ~ _ ~ yt h ro u g h t h e r e gio n o f m ~ i ~ i o n =8 i n d i c t e d i n 1~ 8.S ~ . .

Somet imest h e t u r b u l e n t b o u n d a r y l ~ v e ~ t r an s i t i on = i n i t is ~ ed byo b st a cl e s o r b y l a m i n a rf l O W s e p a r a t io n n e a r t h e' l ead in g .edge. T h isi s E e ne r a l l y a ~ o m p a n i e d b ya t h i e k e n ~ - g o f t h e ' l ~ 3 e r. I t i s t h e nlm~sible for th e vir t~ d ~ o f the .turbulent layer to be lee~ted uP- i~ dze am o f ~ ] e a d i ~ e d Ca .

o " " t , . .. . .

21.~ B o u n d a r y. l a y e r o n u n i f o r m l y ro u p,h . m n ~ t e ~ -. I f t he su r f ~e i s cove redu n i f o r ml y w i t h r o n g h n e m m o f m e h =m e e a

, ~ t , t ~ t t he e 0 n ~ s o n o f~ ] ~ d y r on g h .u r fa ee ~- ~ v e ~ t h e v ~ u e 'of o ( ,, .J~ /, ) obeys t he ~ = a =. . . : . .

g i v e n b y E q . ( l l . 4 ~ ) , ~ n a~ 1 .{ $ 1 A ) t h e nt a k e s t h e | o ~ m

(SLg)

(=uo)

which k i n d ~ d e n t o f v i seos i ty : In t rod uc t ion in F_~ . (21.6} and in te -g r a t i o n gives

(SLn)


38/67

J'; C .. R o ' t ~ -

" ' v ~ . #. p ~ I

- . . . . - '

q

; = : t h e l ogaz i th , ,n ; ' ,= - in tegml , w h l c h i s f o u n d i n J ' ~ . a u d " l ~ J ~ d e ' =T - h i m o f F u n e t i o o s f o r / m ' t a , o ~ , F o~ ' v ' s .] u ~ , c . ' V ' ( ~ ) > , I t h e fo llo w ' in Sm n ~ , . . ~ Ye r 1 [ ~ ~ e d ~ ~ b e u s e d :

. = q ~ p : . . . . .

. . . , - . - .

( = L I ~. . .

o ~ . - - ; r ~ 7 , , ,u g h ~ ~

= - ~ , - - - , + ~ 1 " ~ t~ " -q+ ~ z - - + = ~ . , . . 1 . .. . : x L ~ /~ , = / 2xa xa~; = / . 7 j

( ~ I ] s ) :. ' ; . . . --

h , 0 4 ,. ' . . " " .

JO" ' -'

- - , < . . . . . . , . . . "

. . :. .

a=lO 1 , ~ , ,

t . - ~IV

a u a / f o m d 7 r o u g h r ~ r [ ~ L o c al s k~= f r b t io n e o e ~ t " ~s e o o ~ ; , , ~t o E q . ( 2 1. 13 ) s a d r s t l o o f m o m e n t u m t h ic ] m e m ~ m x x ~ m d ~ t o ~ q .( = 1 . 1 0 ) t o d J B t e , m ~ ~ . - - ~ c ut & f u J B ~ o n o f n ~ t .i o ( = - - = ~ ) / / r ~ ~ ,k t h e ===de

ot thee q u / v s l ~ t ~ [ ~ ~ , ~ .= -- 0"4 , X + C ' r = ~ ' I , Z - - 6"2-


39/67

. , ; ; : ; - : i . ; ' '-

~ t ~ = ~ t ~ ~ . ~ ~ t , is o t ~ e , ~ o - ~ a ~ , 7 ~ , ~ ' .

ig . ~ I .~ - ~ w s t h e I ~ ~ c ~ o ~ ~ ~ t ~ , , , ~ ~ . . .~ e o ~ t o * .~ , s w, ~ ' . - ;.. . . . .

.|

-. , . . ."

-I "-~" I I _ . . . . . . . . . -,".i.~ - - . '

, , , ~ f ~ ~

i s d e ~ . ~ @ - . - ." D r ( z ) .

- . .. . . o ~ , - . . e , N ) ~ . s ~ z 6 . , . . ~ ( 2 z , z , ~ )

w here 2 )~(=) i s t he f r i c~on drag o f one s ide o f th e p l~t~ o f ~ Z~ dw idth b . Th is to ta l sk in f r io t ion ~ t ~ i o i ~ i s obta ined f rom ]~q. (21.1)b y i n ~ . . .

w here t e turb ulent f luc tu a t ion term is aga in suppressed . This:~I~Uo~ap plies to b ot h A,~inar and tu rb u lent bou nda ry lsy er ~ T he 'v'a=isL'llonof the to ta l .~ n f ric tior, coe ~cie nt aoc~rding~ ~ , ( ~1 .I ~) is p ~ e din F ig . 2 1 .2 for th e exsm p}e repreeente&

6 f / S. ; I

| I

. . . . . . ; . . . ~ . . . . . : " ~ . . . . . ; ' . ' . . . . . ' .

. o . . .

o . . . . . . o . . . . .

~ , = . . = ~ . , . T ~ , 1 i . ~ ~ : ~ ~ . i ~ ~ Q,~ . ., ~ . ~ , , ~ . ~ . v ~ , . , ~ i ~ u = i l p o i n t . . : "

- .. . ~ , , ~ ~

~ ~ ~ ~ ~ . " .:. ..

. . . . . ( l ) ~ t ~ , ~ , = . - ~ , o , - . , . .

1 . . ( s ) ~ ~ . . . . . -.- . . o .

2 1 .4 T h e t o t a l ' s k i n h - l e t i o n c o d ~ d ~ " , "


40/67

z g s J .' C . :

' T h e t o ta l . ~ - ~ c t i o n o e~ ie ie nt o f t h e sm o o t h ~ t : p l s t e w i t hd i ~ ' e r -en t pos i t ions o f t r ans i t ion f i~m. lAm~-ar to tu rbu len t f low can be repre -

" sen ted tog e the r w i th t he "sk in ~ ic t ion coeff ic ien t o f th e roug h p la te o n as in gle .d ia gr am , a s 'g iv en in Fig . 21 .4 . T he v i r tu a l ori~,m., o f the ro ug hp la te - is a s smned to co inc ide wi th ih e l ead ing edge (ze . - 0 ) .. "

~ o a t t e m pt ~ m ad e he re ..to t~ r i ew th e o lde r f~ ict~on h tws . I t m ayon ly be no ted th a t the a fo rement ioned sk in f r ic t ion l aw has i t s p redeces-so r in P rand t l -Sch l i ch ting 'aes f r ic t ion l aw o~ sm o oth and rough p la tes .Th i s l aw i s baaed o n a : logan ' thmie Ve loci ty d i J t r ibu t ion th ro ugh ou t th elayer, in o th e r wo rda th e dev ia t ion o f tb e de fec t l aw f ro m the logar iehm~el a w a s s e en i n F i g, 1 3..* i s s u p p ~ I n c o n s e q u e n c e t h e . ~ - f r ic t io n c o e ~ c i e n t s o f P m n d t l - S e h I i c h t i ~ ' l fi-ic tio n l a w a r e a ll to o h i ~ - b y a n.a m o u n f; o f S to10%. . " . . :

: : , ' , . - . . f ' , , . . .

. . ' . . . . . . . .

2 2 . T h e S h a p e P a r a m e t e r ' E q u a t i o n ' .

In th e case o f t I~e ,generaJ bo unda ry l aye r the ~ lo c i ty -p ro f i l e is n o t. Only a f l ' ec t~ b y the loca l Reyno lds num ber and su raco roughness d i s . tn 'bu tion , bu t a im by the ex te rna l p ressure g rad ien t . I f th e ca lcu la t ioni s b a se d o n m e a n v e l o c i ty p r o f il e s d e p e n d in go n a s ~ . l e f re e s h ~ pe .p a m -

m ete r i ike thoN. desc r ibed in Sec t ion 18, then tw o unkno wn-fun e t ionso c c u r i n t h e c a le n la ti o n . T he s e a r e th e . b o u n d a r y l a y e r t h ic k n e es a n dth e ah~pe, param eter. W he n these tw o funetiorm, are de term ined, a l lo th er qua nti t ie s o f inte rest can easil" be.ca lcu]a , e df rom th e re l st ionag iven in the p reced ing sec t ions .

I n n t o ~ c a se s t h e ~ o u n d a r y l a y ~ t h i e k n e ~ i s c a lc u la t ed f r o m . th em o m e n t u m i n te g r a l e q u a ti o u . T h e d e te r m ~ -A t io n o f t h e a h~ pe p a r a e

m e t e r requi res s se con d equat ion , th e so-ca l led s tutpe .paZameter equ a-t i o n o rs n , r i I i e . r y e q u a ti o n , B e ca u se o f t h e i n c o m p l e t e s t a t e o f k n o w e d g eon tu rbu len t fl0w the shape pa ram ete r equ a t ion re li e s en t ' tr e lyor.pa r t lyon empi r i ca l obse rvat ions .

T h e p ro b l e m h ~ s t i m u h t t ~ m a n y in v e s ti g a to r s a n d a c o n s id e ra b len u m b e r o f di ff e re n t f o r m s o f t h e sh a p e p a r a m e t e r e q u a t i o n l m b e e np r o p o se d . To b e g in w i t h , t h e r e i s m o r e t h a n o n e e h o i co f o r th e e h ~ t c t e r -i z ing shape p a ram ete r i tse lL I f th e Gbape pa ram ete r i s genera lly den o tedb y P, n o m a t t e r h o w i t i s d ef in ed , a ll k n o w n s h a p e p a r a m e t e r e q u a ti o n se~n be rea r ranged to f it i n to the scheme

" , i= - _ , v ,

wh ere L assumes the Value Z or O. Th e symbo ls ~ and ~ den o te rune-t ions o f 1 ' . and - - in t t .e ca e o f roug h sur fae es - -k , / e . T h e d a t ~ f o r a n u m b e r o f m e t h o d a a r e c o l le c te d i n Ta b l e 6. T h e m e t h o d


41/67

. . . . ; - . . - . . . .. . . . . . . . " . . . . .

. . . . . : . * " : . . . . - . ' : . , . .- * - . . . / - - . . .

, . . ,

p o po b y a pp n t o b o t h - - o o t h r o S h surfsoes. A|] other methodsof Table 6 are o o~ned to smooth surfaces. "-Bud's met.hod neglects any history efl'e~ on the development of the

pr o~e shape. I t is demonstrated in Fig. I~.3 that ~ , supposit3aa is.insumcient~ The. other methods ate based on the ide~ that a ri dde n ~'change in the exter~__~ ~ will produee a change m alP/d= ra therthan in P itself. Although some of the earlier methods, m ~ l l y t]ao~of Buds and GmsehwiU~,.seem now to be m~ y of ~c al tnfere~the ~iversity O ~ is o o n f ~ g . Thls diversity inevitably al~eS~sinoe a multipl icity of obviously plausible hypo the~ offer themmlw~.

" Indeed, the development hasno t ye t come to a u t i s f s o t ~ oonelusionup to the pre~ent day. This state of aft'airs makes it appear aseful todi sc t~ the theoretival an d emp id ~l barn from a general point of view.

Some .a~q~ts conoerning the variation of the shape parameter haVealready been di~-u~ed in Section 14. This shows that it k im po~e le "~o lay down a plain principle for the shape pammeterequatien, which"takes properly into account the ~ynamical and non-linear eharsoterof the turbulent motion. The pre~n t disenssion'will therofcn.-startfrom an equation of the .type . . . . . .

. ' . . : . .- -. . *.. . . .

.. . . . . : . . . ' . .

It is a~umed t ha t the relation b e t w m the fiVe'VL'iLbles S notafl'eeted by the upstream history of the boundary layer. If the oeneept

. of the law of th~ wall and defect law k introduced, Re wand k v /8 e ~ Bereplaeed by ~.and p. An a l ~ version of .Eq. (.~.e.) is the-

, e d U ' p , 0 . . 1 ~ 1

. 0 U . d = . " "

which il equiva lent to ~/ s. (16.11) or (15A$). ~ e shapeequation c=n theoreti~lly be 01~ i ~ d by oomb/ning

the calculation of the turbulent boundary layer

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