ACR No. 3L23

NATIONAL ADVISORY COMMITEE FOR AElONAmCS

December 1943 as Advance Confidential Report 3 L23

INVESTIGATION O F FLOW IN AN AXIALLY SYMMETmCAL

HEATED JET OF AZR

By Stanley Corrsin ' CaliforniaInstituteofTechnalogy

NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of advance research results to an authorized group requiring them for the war effort. They were pre- viously held under a security status but a re now unclassified. Some of these reports were not tech- nically edited. All have been reproduced without chmge in order to expedite general distribution.

WBTIONAL ADVISORY COMMITTEE FOR AERONAUTICS

ADVANCE C O N F I D E N T I A L R E P O R T

I N V E S T I G A T I O N OF BLOW I N AN AYIALLY SYMMBTRICAL

HEATED JET OF AIR

By S t a n l e y C o r r s i n

The f t r s t p h a s e of t h i s i n v e s t i g a t i o n was c o n c e r n e d ' p r i m a r i l y w i t h t h e d e t e r m i n a t i o n of a s u i t a b l e a z r o d y -

n&mi c s e t u p . When t h e f i n a l flow a r r a n g e m e n t was a c h i e v e d , a t t e n t i o n was t u r n e d t o t h e m e a s u r i n g equ ipmen t .

I n o r d e r t o p e r m i t a t h o r o u g h i n v e s t i g a t i o n of t h e mechanism of t h e s p r e a d of f r e e t u r b u l e n c e , a new f o u r - w i r e , t w o - a m p l i f i e r h o t - w i r e s e t was c o n s t r u c t e d . h u t o - m a t i c a l l y r e c o r d i n g mean v e l o c i t y and t e m p e r a t u r e i n s t r u - ments a l s o were maae and were u s e d i n t h e i n i t i a l s t a g e of t h e r e s e a r c h i n t h e f i n a l j e t s e t u p . The p r e l i m i n a r y r u n s c o n s i s t e d of measurergents of mean v e l o c i t y and t e m p e r a t u r e d i s t r i b u t 2 o n s a l o n g d i a m e t e r s of s e v e r a l s e c t i ons normal t o t h e a x i s of t h e h e a t e d a i r j e t i s s u i n g f rom a c o n v e r g e n t - s t r a i g h t n o z z l e w i t k a mouth d i a m e t e r of 3 i n c h e s . The s e c t i o n s r a n g e d f rom t h e n o z z l e mouth t o a n a x i a l d i s t a n c e of 12 d i a m e t e r s , o r 3 6 i a c h e s . P r e l i m i n a r y measurements a l s o were made of t h e a x i a l component of t u r b u l e n c e i n t h e c e n t e r r e g i o n of e a c h c r o s s s e c t i o n .

The p r e l i m i n a r y r e s u l t s d i s p l a y e d s e r i o u s asymmetry a t r e l a t i v e l y low a x i a l d i s t a n c e s . S i n c e t h e e f f e c t ap- p e a r e d t o Be due t o e x t e r n a l d i s t u r b a n c e s , t h e change t o a s m a l l e r j e t s i z e was i n d i c a t e d t o p e r m i t measurements a t g r e a t e r a x i a l d i s t a n c e s . A 1 1 t h e f i n a l measurements were c a r r i e d out by t h e a u t h o r i n t h e j e t i s s u i n g f rom a n o z z l e 1 i n c h i n mouth d i a m e t e r ,

In t h i s 1 - i n c h h e a t e d j e t , L a t e r a l d i a m e t r i c a l t r a v e r s e s of mean v e l o c i t g , t e m p e r a t u r e , and t h e a x i a l component of t h e t u r b u l e n c e were r u n a t s e v e r a l s t a t i o n e be tween t h e end of t h e p o t e n t i a l cone and a n a x i a l d i s - t a n c e of 40 d i a m e t e r s from t h e mouth. A t a n a x i a l d i s t a n c e of 20 d i a m e t a r s , measurements a l s o were made of t h e r a d i a l component of t h e t u r b u l e n c e , of t h e c o r r e l a - t i o n be tween a x i a l and r a d i a l components of t u r b u l e n c e a t t h e same p o i n t , and of t h e c o r r e l a t i o n between t h e a x i a l

t u r b u l e n t velocities a t equa l - speed p o i n t s 'on opposf t e s i d e s of t h e J e t a x i s .

~ s c i l l o ~ r a m s -were t a k e n bf t h e " a x i a l v e l o c i t y f l u e - I * t u a t i o n s a t s e v e r a l p o i n t s a l o n g t h e j e t r a d i u s a t 20

d i a m e t e r s . These o s c i l l o g r a m s t o g e t h e r w i t h t h e c o r r e l a - t i o n measurements i n d i c a t e t h a t c o m p l e t e l y t u r b u l e ~ t f l o w e x i s t s o n l y i n t h e c e n t e r r e g i o n of t h e j e t , fo rming a t u r b u l e n t c o r e . I t i s found t h a t t h e f l o w i n t h e ex-

. t r e m e l y ' l o w v e l o c i t y r e g i o n a t tkid edge i s a p p a r e n t l y l a m i n a r , c o r r e s p o n d i n g t o a s o r t of l a m i n a r c o l l a r ; w h i l e between t h e s e i s a n a n h u l a r " t r a n s i t i ~ n l ~ r e g i o n where t h e f l o w f l u c t u a t e s between t h e l a m i a a r and t h e t u r b u l e n t s t a t e s . , . , *.

i" . The t e m p e r a t u r e was found t o s p r e a d more r a p i d l y ' ; I

t h a n t h e v e l o c i t y , a s h a s been r e c o r d e d i q a l l p r e v i o u s l y p u b l i s h e d r e s u l t s . , .

The d i a m e t r i c a l d i s t r i b u t i o n s o f " mean v e l o c i t y and +

t e m p e r a t u r e a r e compared w i t h t h e r e ' s u l t s o f t h r e e the'o- . r i e s f o r f u l l y d e v e l o p e d t u r b u l e n t f l o w i n a f re 'e h e a t e d j e t . w i t h c o n s t a n t d e n s i t y and v i s c o s i t y . I t i s found . , t h a t s i n e a c h c a s e , e i t h e r t h e v e l o c i t y o r t h e t e m p e r a t u r e . d i s t ~ t b u t i o n ( d e p e n d i n g upon t h e s c a l e of a b s c i s s a s y can be &a%e t o g i v e a f a i r l y good check between t h e o r y and ex- p e r i & e n t i n t h e t u r b u l e n t j e t c o r e , bu t n e v e r b o t h v e l o c - i t y and t e m p e r a t u r e ' . s i m u l t a n e o u s l y .

An o s c i l l o g r a m a l s o ' w a s t a k e n of t h e r e g u l a r f l u c - t u a t i o n s e x i s t i n g i n t h e r e g i o n of t h e j e t b e f o r e t h e end of t h e p o t e n t i a l c o n e . T h i s t y p e of f l u c t u a t i o n was f i r s t obse rved in t h e 3 - i n c h Je t by M r . T h i e l e i n 1940.

The work done under t h i s c o n t r a c t f a l l s e s s e n t i a l l y i n t o two -pa r t s : The f i r s t p a r t was t h e d e s i g n and con- s t r u c t i o n of t h e e q u l p a e n t and t h e r u n n i n g of p r e l i m i n a r y t e s t s on t h e 3 - i n c h Jqt . , c a r r i e d hut by M r . C a r l T h i e l e i n 1940; t h e second p a r t , c o n s i s t i n g of t h e measurements i n t h e 1 - i n c h j e t and t h e p r e p a r a t i o n of t h i s f i n a l r e - p o r t , was c a r r i e d out by t h e a u t h o r .

> *

For a comple te i n v e s t i g a t i o n o f t h e g e n e r a l c h a r a c - t e r i s t i c s of a t u r b u l e n t f low, some r a t h e r c o m p l i c a t e d

measurements a r e necessa ry . Because of t h i s and because of t h e i n c r e a s i n g d i f f i c u l t y i n o b t a i n i n g e l e c t r i c a l s u p p l i e s , i t was d e c i d e d i n 1941 t h a t any new equipment needed i n t h e f u r t h e r i n v e e t i g a t i o n of t h e g e n e r a l t u r - b u l e n c e problem shou ld be c o n s t r u c t e d immedia te ly , A c c o r d i n g l y , a ve ry comple te hot-wire s e t was des igned and b u i l t i n t h e f a l l of 1941 , and was f i r s t !appl ied i'n t h i s r e s e a r c h p r o j e c t ,

I n a n approach t o t h e g e n e r a l problem of f r e e t u r - b u l e n c e , an : inves t i g a t f on of t h e f low i n a f r e e J e t i s of p a r t i c u l a r i n t e r e s t . U p t o t h e p r e s e n t t im'e, i t h a $ been assumed t h a t t h e f low i n t h e f r e e j e t i s s u i n g from a c i r c u l a r o r i f i c e i n t o a s t a t i o n a r y medium of t h e same d e n s i t y and v i s c o s i t y r e a c h e s a f u l l y developed c o n d i t i o n a t about 8 or 1 0 d i a m e t e r s from t h e o r i f i c e - t h a t i s , f u l l y developed i n t h e s e n s e of a t t a i n i n g a c o n f i g u r a t i o n mhich i s m a i n t a i n e d 1 ~ 9 t h s i m i l a r i t y bownstream. I n a n a x i a l l y symmetr ica l j e t i t h a s been found t h a t t h e "poten- t i a l c o n e , " i n mhich t h e v e l o c i t y i s e q u a l t o t h a t a t t h e n o z z l e mouth, h a s i t s apex about 4* d i a m e t e r s from t h e mouth , but t h a t t h e f i n a l ( d i m e n s i o n l e s s ) v e l o c i t y pro- f i l e i s not r eached u n t i l the ' d i s t a n c e of 8 .or 1 0 diam- 8% e r s p r e v i o u s l y mentioned.

Buden ( r e f e r e n c e 1 ) and Luethe ( r e f e r e n c e 2 ) have bothemade measurements i n j e t s w i t h a x i a l symmetry. Ruden measured v e l o c i t y and t e m p e r a t u r e d i s t r i b u t i o n s ou t t o -an a x i a l d i s t a n c e of about 15 d i a m e t e r s i n a h e a t e d j e t ; w h i l e Kuethe measured a s t a i l e d v e l o c i t y d i s - t r i b u t i o n s and made some t u r b u l e n c e measurements out t o 9 d i a m e t e r s i n a n u n h e a t e d j e t . Both i n v e s t i g a t o r s found t h a t s i m i l a r i t y of v e l o c i t y p r o f i l e s was reached b e f o r e 1 0 d i a m e t e r s , which l e d & t o t h e g e n e r a l b e l i e f t h a t comple te mechanica l s i m i l a r i t y I n a j e t mas reached t h e r e , I t was t h o u g h t , however, t h a t t h e t u r b u l e n c e d i s - t r i b u t i o n i n a r a d i a l d i r e c t i o n might s e r v e a s an i n d i c a - t i o n of a f u l l y developed s t a t e . F u r t h e r m o r e , i t w a s f e l t that t h e l i m i t of 1 5 d i a m e t e r s , mhfc'h appeared t o be t h e g r e a t e s t a x i a l d i s t a n c e i n c l u d e d i n any p r e v i o u s l y p u b l i s h e d r e s u l t s , might b e f n s u f f i c i e n t t o g i v e a com- p r e h e n s i v e p i c t u r e of t h e na tu re , o f t h e f low. The inves- t i g a t i o n program, t h e r e f o r e , was s e t up t o i n c l u d e c o m ~ l e t e measurements of a t l e a s t t h e a x i a l component of t u r b u l e n c e and s e c t i o n a l t r a v e r s e s out a s f a r a s 40 d i a m e t e r s from t h e nozg le mouth.

A l l t u r b u l e n t - j e t a n a l y s e s p u b l i s h e d up t o t h e p r e s e n t

tfme ape based upon t h e fuadamentab assumption t h a t tbe flew i s oamplete ly t u r b u l e n t e ~ r o s s t h e e n t f r s width 62 t h e J e t . The two t u r b u l e n t j e t G h e ~ f i s a ~ 8 8 8 widela se* eepted a t t h e p r e s e n t time a r e f r a n d t i l t s momentum t r a n s - f e r t heo ry ( r e f e ~ e n c e J ) , and t h e modified r a r t i c i t y t r a n s f e r t heo ry a9 Q. I. Taylor ( . reference 4 ) . In t h i s r e p o r t t h e r e a l s o a r e inc luded t h e r e s u l t s o f 8 t h f r d methad af a$tea@k en t h e geaer a 1 t u r b u l e a a e pralblea, ?re- sented bg P, Y, Ghou ( ( reference 5 ) , he p a r t i c u l a r appliear-Q;ioa of t h i s t heo ry t o a j e t haat been made by C. C , &in, cu r reGt ly a t t h e OALOfT, Thfs spproaab i s equ iva l en t t o the easnmptions suggested by T. van ~ i r m a n (ref ereace 6 ) The s p p l i a a t f on of t h e mamenbum transg'es a w w m ~ t f ons t o t h i s preb3.em was c a r r i e d out by V , TalLaiee (rtsfwaace 7). the @alauXa%ina of the moQiff ed r e r t f c f t r t ra~rPw prlnm c i p l e spp5ied t o ai j e t was made 'by Tomotite [ r s i e r e n c e ' 8 ) . and t h e r e s u l t s of t h e two theorfss were sumaarioe by L. Bawa~tb ( r e f e r e n c e 9), Qlm, the prablem of t- + e l & c i t y b i a t r i b u t i a a fg t h e &naula r rsg.lion are~nl i ) the p e t a n t g a l cone of an a x i a l l y sya l a s t r i oa l de t ha+ )st%% solved an t h e b a s i s af tba aomaatnm t e r assawtidm by A. L Kuetkae. S h ~ ~ t a g s af trfm has p~ec3udatt %he c a w paxiso&% of t h e 3-inch J e t aeasuremeats witb Kuetha's csalat i on ,

A c o m ~ r i s o n of t h e dfs%ributbons of mesa v e l a c i t p and t empera ture fn r free j e t t b e iast ial tempe~alr-wre of wkfeb i s s l i & h t X t d i f f e r e n t from t ka ) o f , the ress ivLng medium g rasenes en exeeXlent a p p q r t u a i t y f a r comparison of the r a t e s of diffusion a f t h e two q u a n t i t i e s , msmeqtanr and heat, without i n t roduc ing t h e complexity of r a ~ i a b l s a $ s e o s i t y and d e n s i t y ,

Ph i s i n v e s t test ion. ~ ~ n d u ~ t e d . a t t h e Calff ornin I n s t i t u t e of Technology, wats sponsored by, and conducted wi th fineacfal a s s i s t a n c e from, t h e S a t f o n a l Advisga# Committee f o r Aeronau&&tse 2be work was aarried out under the gene ra l s u p e r v i s i o n of D r . T, von ~hpm&a and br, C. 3. M i l l i w p t whose i n t e r e s t and adviice ere g r a t s f u l l y ackaowledged. P a r t i c u l a r thanks a r e d$e ts Mr. C a r l Th ie l e f a r h i s sp l end id job of Besigning and c o n a t r ~ c t i h g t h e equipm9nt end g e t t i n g f h s research under way and t a Pr, Hans L t epaan-a f a r h i s invaluable couasel aad adv iae througbouf t h e researeh, Theoretical d iscus- s i o n s on many aecas ions wi th MF* CI . 6 % Si.q were alrso of great asrisstance.

d d i a m e t e r of n o z z l e mouth

x a x i a l d i s t a n c e f rom mouth

r r a d f a l d i s t a n c e f rom j e t a x i s

U a x i a l component of mean v e l o c i t y ,* , 1 I -

. k a d i a l component of mean v e l o c i t y

W t a n g e n t i a l ( r o t a t i o n a l ) component of mean v e l o c i t y

U, = maximum v a l u e of U a t a s e c t i o n ( i . e. , on t h e a x i s )

U o maximum v a l u e of U i n t h e j e t ( i n p o t e n t i a l c o n e )

r o t h e v a l u e r a t tany s e c t i o n f o r a h i c h U = 1 U, 2

u a x i a l component of i n s t a n t a n e o u s v e l o c ' i t y f l u c t u a t i o n

v r a d i a l component of i n s t a n t a n e o u s v e l o c i t y f l u c t u a t i o n

w t a n g e n t i a l component of i n s t a n t a n e o u s v e l o c i t y f l u c t u a t i o n

1 . 2 d L. t u r b u l e n c e l e v e l s u ' u ' u P p r e s s u r e

P a i r d e n s i t y --------------_I_ - - - I _ - ' I . .

"Mean v a l u e s of v e l o c i t y a n d t e m p e r a t u r e r e f e r r e d t o i n symbols a r e t i m e means.

P a i r v i s c o s i t y coef f i o i e n t

v = E a i r k inemat ic v i s c o s i t y P

T shea r . ';

T . - mean tempera ture , measured above room tempera ture

T* maximum va lue of maan tempera ture a t a s e c t i o n ( i , e , , on t h e arets) - . " . s ,

To maximum va lue o f mean t e a p e r a t u r e i n t h e j e t ( a t t h e mouth)

t i n s t an t aneous telnp&ature f l u c t u a t %on

t t - 2 "mixing leng thJ t i n momentum t r a n s f e r t heo ry

X "mj.cso-scale" of t u rbu l ence -- 43 c o r r e z a t i o s between any' t a o f l u c t u e t i ng . q u a n t i t i e s ,

A and B . . -- = - c o r r e l a t i o n c o e f f i c i e n t , .

Ruv u ' v l

U U RU = -A5& c o r r e l a t i o n c o e f f i c i e n t u

ET t o t a l k i n a t i c energy c roga ing a s e c t i o n of t h e j e t i n u n i t t ime

EM k i n e t i c energy @ f mean P l o m c r o s s i n g s e c t i o n i n a n i t t ime

E' t u r b u l e n t k i n e t i c energy c r o s s l n g s e c t i o n i n u n i t t ime

Aerodynamic Betup

F i g u r e 1 8 8 a schemat ic diagram of t h e wind-tunnel and i n s t a l l e d j e t u n i t e . The f ovr-blade s tee l p r o p e l l e r i s d r i v e n by a t h r e e - ~ h a e e f n d u c t i o n motor r a t e d a t 1 5 horeepower and r u n from a v a r f a b l e f r equency g e n e r a t o r , I n a l l t h i s work, t h e u n i t was r u n a t on ly a amall f r a c - t i o n of r a t e d power,

The 6Q-ioot s q u a r e P s p r e s s u r e box@wwas a d a p t e d from t h e fo rmer G & C % I c a l i b r t a t i o n ' tunnel , The a i r i s h e a t e d by means of 16 ealrod h e a t i n g u n i t s mounted i n t h e p r imary c o n t r a c t i o n ( f i g . 2 ) , and t h e r e are two s c r e e n s mounted between t h e h e a t e r s and t h e f f n a l j e t c o n t r a c t i o n , which was a spun aluminum convergent nozz le . I n a n e f f o r t t o o b t a i n a uni form t e m p e r a t u r e d i s t r i b u t S o n a c s o s s t h e mouth of t h e n o z z l e , p a r t of t h e h e a t e d a 9 s wae;l d u c t e d a l o n g t h e o u t s i d e of t h e l a r g e c i r c u l a r p i p e leadPng t o t h e n o z z l e , so t h a t t h i s p i p e and t h e b e g i n n i n g of t h e n o z z l e c o n t r a c - t i o n were Bmmerged i n h e a t e d a i r . That % h i e scheme mas n o t c o m p l s t e l y s u c c e s s f u l c a n be s e e n from t h e t e m p e r a t u r e d i s t r i b u t i o n measured a t t h e mouth. I t d i d r e p r e s e n t , however, a d i s t i n c t improvement over t h e wooden n o z z l e f i r e t C r i e d , P o s s i b l y thira I n a c c u r a c y i n one of t h e tem- p e r a t u r e boundary c o n d i t i o n s had a n e f f e c t upon t h e tem- p e r a t u r e p r o f i l e a i n t h e r e g i o n of t h e j e t ups t ream from t h e end of t h e p o t e n t i a l c o n e , a s w i l l be d i s c u s s e d l a t e r .

A l l t e a t s on t h e 3-inch j e t were s u n a t a mouth ve- l o c i t y of 20.5 m e t e r s 8 e r second and a maximum t e m p e r a t u r e d i f f e r e n c e of about l 3 (2,

The p l a t i n u m h o t m i r e s used t o measure u 8 were e t c h e d Wol las t o n , 0. OQ025 i n c h fn d%arneter and approxi - ma te ly 2 m i l l i m e t e r s i n l e n g t h . Mean v e l o c i t y ,mas meas- u r e d ~ i t h a s m a l l t o t a l head t u b e , and t e m p e r a t u r e was measured w i t h a s m a l l copper-cons tantan thermocouple.

The a m p l i f i e r o u t p u t f o r t u r b u l e n c e r e a d i n g s n a s p u t t h r o u g h a therruocsuple i n t o a c r i t i c a l l y damped w a l l galvanornet e r w i t h a 3-second p e r i o d and a f u l l s c a l e de- f l e c t i o n 02 approx imate ly 20 c e n t i m e t e r s ,

330th mean speed and temperature were phot ographioal ly recorded by means o f an automat ic- t ravsra/ng arrangement. The total-head p ressu re l i n e w a s run i n t o a small copper bellows ~ h i c h t i l t e d R mirror, thereby d e f l e c t i n g a narrow l i g h t beam upon a uni f ormlf .zxoviag sheet of s e n s i t f sed paper.

The recording of temperature on the s e n s i t i z e d paps? u t i l i g a d d i r e c t l y t h e l i g h t beam r e f l e c t e d f r o m t he mOrror of th6 t a l l ga lvanmet e r .

Ehe traversing aas aceomplisherd by meaas of a screw- dr iven a a r r i a g e ranning h o r i e o n t a l l y along a s t e e l t r ack . The acrew was r o t a t e d by a r e v e r s i b l e e l e c t r i o m o t o r and, during t h e photographical ly recorded runs , t h e s e n ~ i t i aed papeo was mechanically connected t o t h e hot-wire c e r r i a g e s o t h a t t h e absc ipsas aP t h e recorded curves would be ez- a c t l y equal t o t h e t r u e l e t e r a 3 d is tances , 'Bigurea 4 and 5 show an over-all view of t h e t r a v e r s i n g u n i t and a close- up of t h e c a r r i a g e an& X B Q ~ O F ~ F h ~ t o g r ~ l p h s a r e of t he 1- ingh j e t i n s t a l f a t i ~ n , w h $ c l ~ waa Sdent ica l with t h a t f o r t h e 3-inch j e t u n i t , except f o r t h e @ a l l In the p lane of t h e n o s ~ l e mouth i n t h e s e p ic tq ree ,

, . Measuring Equipnent and Procedurs

This new 2iot-wire s e t iphotograpned i n fig. 6) con- s i s t s o f t h e fol lowing elements:

1, Four hot-wire hea t i ag c i r c u i t s and potent iometer

2, One bridge c i r c u i t vhich can be used with any of t h e four hea t ing c i y c u i t s

3, T m o compensated a m p l i f i e r s a i t h very nearly t h e same phase s h i f t and ga in charact e r i s t i c e

4, One ampl i f ie r i n which t h e outputs of t h e other two a m p l i f i e r s can be added

With t h i s instrument the. fol lowing flow p r o p e r t i e s can be measured: -

1. Mean apetad, U

2, The t h r e e p r i n c i p a l turbulence component 6 , t e v w '

U"' u

-.- -- 3. The c o r r e l a t i o n s a t a - p o i n t , uv and urn

4. The c o r r e l a t i o n s between t u r b u l e n t v e l o c i t y com- p o n e n t s a t two p o i n t s, i n f l o s s a i t h o r wi thou t a g r a d i e n t i n t h e mean v e l o c i t y

The b r i d g e c i r c u i t i s s o a r r a n g e d t h a t t h e hot-wire n t i m e c o n s t a n t " c a n be de te rmined by super imposing e q u a l a l t e r n a t ing-cur ren t v o l t a g e s a t two f r e q u e n c i e s upon t h e ba lanced d i r e e t - c u r r e n t b r i d g e c i r c u i t .

The g a i n of each a m p l i f i e r i s c o n s t a n t t o w i t h i n f3 p e r c e n t o8er a f r e q u e n c y r a n g e from 5 t o 9000 c y c l e s p e r aecona.

RESULT S OF THE PRELIMI R U P MEASUREMENT S

The measured r e s u l t s a r e p l o t t e d i n f i g u ' r e s 8 t o 16. F i g u r e 8 shows t h e d i s t r i b u t i o n of mean v e l o c i t y , mean t e m p e r a t u r e , and t u r b u l e n c e a l o n g t h e a x i s of t h e j e t . The v e l o c i t y i s p l o t t e d both u n c o r r e c t e d an8 approx imate ly c o r r e c t e d f o r t h e e f f e c t of t u r b u l e n c e upon t h e t o t a l h e a d t u b e r e a d i n g s , Assuming c o n s t a n t s t a t i c p r e s s u r e through- out t h e j e t , t h e a v e r a g e p r e s s u r e a t t h e mouth of t u b e %a a t u r b u l e n t f low is:

where U i s t h e a x i a l mean v e l o c i t y and u i s t h e in- s t a n t a n e o u s a x i a l v e l o c i t y f l u c t u a t i o n , Theref o r e ,

s i n c e

and t h e dynamic p r e e s u r e c o r r e s p a n d i n g . t o t h e mean ve- l o c i t y i s

" N e g l e c t i n g t h e e f f e c t of l a t o r a l f l u c t u a t f o n s .

Figuses 9 t o 16 g i v e t h e measured mean v e l o c i t y and t e m p e r a t u r e d i s t r i b u t i o n s a c r o s s s e c t i o n s normal t o t h e j e t a x i s . A l l of t h e s e v e l o c i t y and t e m p e r a t u r e d i s t r i - b u t i o n s n e r e c o n t i n u o u s l y r e c o r d e d p h o t o g r a p h i c a l l y , and t h e lotted r e s u l t s a r e c u r v e s f a i r e d t h r o u g h t h e photo- g r a p h i c d a t a . S ince t h i s c o n t i n u o u s r e c o r d i n g method t e n d s t o g i v e i n s t a n t a n e o u s f l u c t u a t i o n s i n t h e q u a n t i t y be ing measured ( u n l e s s d o n s i d e r a b l e .overdamping i s em- p l o y e d ) , t h e s c a t t e r of t h e p h o t o g r a p h i c c u r v e s i s appre- c i a b l y g r e a t e r t h a n t h a t of point-by-point measurements n h i c h i n v o l v e a p r o c e s s of men ta l t ime-averaging i n t h e r e c o r d i n g of t h e obse rved d a t a .

The d i s t r i b u t i o n s of t u r b u l e n c e l e v e l &long t h e a x i s and i n s e c t i o n s normal t o i t a r e a l s o g i v e n i n t h e s e f fg - u r e s , U n f o r t u n a t e l y , e i n c e t h e t r a n s v e r s e measurement a have been made i n only t h e c e n t r a l p a r t of t h e j e t , a comple te p i c t u r e cannot be g o t t e n from them.

DI SCU 591 ON OF RESULT S

Veloaf%y a.nd Temperature

Asymmetry i n t h e v e l o c i t y p r o f i l e s becomes a p p r e c i - a b l e a t a n a x i a l d i s t a n c e o f about 4 d i a m e t e r d , and ap- p e a r s even sooner i n t h e t e m p e r a t u r e c u r v e s , The n a t u r e of t h e v e l o ~ i t y asymmetry, a h i g h r e g i o n on t h e l e f t s i d e

U of t h e a x i s f o r - < 0 .4 , seems t o i n d i c a t e t h a t t h e Um

f l o w c o n d i t i o n a c t u a l l y e x i s t e d , due p e r h a p s t o e x t e r n a l d i s t u r b a n c e o r t o misa l i aement of t h e n o z z l e r e l a t i v e t o i t s c i r c u l a r duc t , On t h e o t h e r hand, t h e t e m p e r a t u r e asymmetry c o n g i s t s of a n a p p a r e n t change i n t h e eoro- o r d i n a t e l e v e l between o p p o s i t e sidea; of t h e j e t , and i t a p p e a r s morg l i k e l y t h a t t h i s i s due t o t h e method of measurement t h a n t h a t i t i s r e p r e s e n t a t i v e of a t r u e con- d i t i o n of & h e j e t .

One of t h e t h o o r e t i c a l boundary c o n d i t i o n s f o r t h i s j e t problem i s nlat p r e c i s e l y s a t i s f i e d : The t e m p e r a t u r e p r o f i l e a t t h e . : mouth d e v i a t e s a p p p c i a b l y f r o m a rec - tangurlar shape . . s p i t e of t h e d u c t i n g s y ~ t e r n f o r h e a t i n g t h e malls of t h e c h a n n e l l e a d i n g t o t h e n o ~ a l e , Sven s o , t h i s t e m p e r a t u r e p r o f i l e r e p r e s e n t e d a c ~ n s i d e r a b l e Sm- provement over t h a t e x i s t i n g % n the j e t emerging f rom t h e

vooden convergent n o z z l e w i t h n h i c h t h e s e exper iments were f i r s t t r i e d .

I t h a s not been de te rmined whether t h e n a t u r e of t h e d e v i a t i o n from t h i s p a r t i c u l a r boundary c o n d i t i o n h a s any a p p r e c i a b l e e f f e c t on t h e d i f f e r e n c e i n t h e r e l a t i o n of t h e v e l o c i t y and t e m p e r a t u r e p r o f i l e s i n t h e two r e g i o n s of t h e j e t s e p a r a t e d by t h e s e c t i o n a t a n a x i a l d i s t a n c e of about 7 d i a m e t e r s from t h e mouth. F o r x < 7 d , t h e

T U c u r v e of -- i s l o n e r t h a n -- i n t h e c e n t r a l p a r t of *m urn

t h e j e t , w h i l e f o r x 2 7 6 t h e t e m p e r a t u r e i s everymhere h i g h e r . I t i s p o s s i b l e t h a t t h e e x i s t e n c e of a d e f i n i t e t e m p e r a t u r e g r a d i e n t i n t h e main body of t h e j e t a t x = 0 , where t h e r e i s no c o r r e s p o n d i n g v e l o c i t y g r a d i e n t , h a s a c c e n t u a t e d t h e c o n d i t f on.

I t seems p r o b a b l e , however, t h a t t h i s change i n r e l - a t i v e shapes of t h e v e l o c i t y and t e m p e r a t u r e p r o f i l e s i s a t r u e c h a r a c t e r i s t i c of a h e a t e d j e t . I n t h e r e g i o n up- s t r e a m of t h e p o t e n t i a l cone t h e f low mag be approximated. by t h e two-dimensional c a s e of t h e s i n g l e mixing r e g i o n between a ~ e m i - i n f i n i t g moving body of h e a t e d a i r and a s e m i - i n f i n i t e body of s t a t i o n a r y c o o l e r a i r . I n t h i s c a s e ---------- i t seems e v i d e n t t h a t , i f t h e e f f e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t i s assumed g r e a t e r t h a n t h e e f f e c t i v e s h e a r e o e f f i c i e n t , t h e d i ~ e n s i o n l e s s p l o t s of t e m p e r a t u r e and v e l o c i t y a c r o s s a s e c t i o n normal t o t h e f low ( i . e . , c u r v e s of U/Um and T/T, a g a i n s t t h e same a b s c i s e a r / ro ) w & % f ~ s h e m t e m p e r a t u r e lower t h a n v e l o c i t y i n t h e r e g i o n of h i g h e r v a l u e s of t h e two v a r i a b l e s ; c o n v e r s e l y , t h e y w i l l show t e m p e r a t u r e h i g h e r t h a n v e l o c i t y i n t h e r e g i o n of lower v a l u e s . For t h e f u l l y develoged j e t w i t h a x i a l symmetry i t i s obvious t h a t t h e same assumpt ion of h e a t - t r a n s f e r c o e f f i c i e n t g r e a t e r than s h e a r c o e f f i c i e n t l e a d s t o d & m _ e ~ s i o n l q a t e m p e r a t u r e p r o f i l e s everymhere h i g h e r t h a n t h e c o r r e s p o n d i n g v e l o c i t y p r o f i l e s .

I t s h o u l d a l s o be no ted t h a t t h e s e v e l o c i t y and tern- p e r a t u r e p r o f i l e s check t h e measurement s of Ruden f a i r l y we l l ,

Turbu lence

The p r i n c i p a l i t em of i n t e r e s t connec ted w i t h t h e . t u r b n l e n c e measurements i s t h e ext remely r a p i d s p r e a d of

- - f 2 .-

b

$a rbu leaae f n t o t h e p a t e n t i d l cone. I t w 9 1 1 be noted %ha% t h e t u rbu lence l e v e l on t h e j e t a x i s beg ins t o r i a @ a t an a x i a l d i s t a n c e of only 1 diameter from t h e nozzle mouth; whi le t h e v e l o c i t y de fec t doas n o t ' r e a c h t h e a x i s unt i l about % diameters , T h i s r e s u l t was observed by Kuathe and ha$ been recogded i n t h e 1-inch 3 e t desc r ibed l a ter . I t sl s i g n i f i cance wi th r e spec t t o open-throat wind . t u n n e l s f s apparen t , .

The q u a n t i t y u1 /O l o s e s some of i t s u s e f u l n e s s f o r h i g h values8 so u i /u , h a s also been p l o t t e d f o r each s e c t i o n , The f u l l b e n e f i t of p l o t t i n g e i t h e r u l /Um a r u1/U, w i l l become Ovident i n t h e 1-inch j e t where . t u rbu lence has been.measured out t o t h e edge of t h e j e t .

11, FINAL MEASUREMENTS ON Q JET 1 - I N C H I N D I U E T E R

.*he experimant a 1 apparatus employed i n t h i s p a r t o f tba ma$k fa b a s i c a l l y t h e same a s t h a t used i n t h e pre- l%miqea.atjr- C&;&G&, afftLk hke. PeJLlowing modif ica t ions :

1. B & - i ~ ~ h ~g.>n~&$~rp, ipu c g a y ~ y g e n t noazle was sub- st itqt ad f a r $ha. 3-inch nii~eltJ. ' Tha P a t e r n a l flow- a r r e t n 8 w ~ n t .v.e~l t p ~ stiqfi), '

2. .A w a x 1 wag qerb up -in t h e p lane of thq 39% mouth, perpesct iaulas Co thg,. gixj8, dCn orde r t o seg?rr.#oc?e more p r e c i s e l y the boundary ( ? ~ n d % t l ~ n o assumed in $ 4 ~ ~ t & e o r i e s p A cheek r u n ' a a s made w $ * i o u b $ 3 1 ~ valP i n p$i l$b 'op, aqd a l though no appraclsrblre diffe~lpance wrs$'foun4'Sn tgs ' $ & o w c h a r a c t e r i e t lcti f a the f l fu l%y dtms lo~od! ! . yepion n! the j e t , i t waa dactdeb $ a .PUB qi'tk the wq53. In pq$ff;ipn, . 1 . I '

3. No p h o t o ~ r a g h 5 ~ ~ x s ~ ~ r d # i ~ ~ tsolrlniclve was ?@a$ f o r any of t h e c luaa t i t ca t iv~ weqsarelprent-a, @%nee S $ was 4BougbSF %'&a-t gome of tbes ~ s y m n a t r y far't;Pa @-$rich 3wrt %sagur@rp@n@cr might pave- been due t o pgrr~aaant kjet t.9 thq mnvlng p a r t e of t h e reoord%n@ q y e t ~ n ,

4. Nean ffipeed wae mgawupe6 with @ n h ~ g l a Fn* wire i n s t s a d o f ca t o t a l hsa4 tub@& Qqrs ohaek run vkap 4$@$1 wiBk a t o t a l head t u b e , and the qgpeamqnt wgs b a t i s l b q i i ~ y y ~ ,

8 .

- < f-

The bean vel6cSty measured by t h e h o t ' w i r e was fiT-Tp r a t h e r t h a n aimply V. Therefore t h e expe t i - menta l r e s u l t s have bean l g b e l e d ---- accordingly. , wi th t h e symbol UR t o s i g n i f y f i + V 2 . Over a l l but t h e edge of t h e j e t UR D, and i n t h e comparison between the- ory and exper imental r e s u l t s , U ( t h e o r e t i c a l ) i s compared w i t h UR ( e x p e r i m e n t a l ) , A s m i l l be p o i n t e d out l a t e r , t h e l a r g e discrepancy between experiment and t h e molst s a t i s f a c t o r y . t h e o r y ( f ig , 50) a t t h e edge of t h e j e t i s not due t o t h i s comparison of d i f f e r e n t ve- l o c i t y components,

I t ahould a l s o be remarked t h a t t h e t u rbu l ence com- ponent measured by t h e s i n g l e bot ? i r e i s t h e component i n t h e d i r e c t i o n of URd However, i t i s everywhere w r i t t e n merely a s u t , s inge i n t h e t u r b u l e n t c o r e of t h e j e t t h e measured component of t u rbu l ence i s probably equa l t o t h e t r u e a x i a l component ( u l ) w i t h i n t h e l i m i t s of accuracy of t h e measurements.

The l a t e r a l t u rbu l ence l e v e l i s w r i t t e n a s v'/UR a l though t h e v-meter was always s e t p a r a l l e l t o t h e j e t a x i s . However, t e s t s showed t h a t t h e e f f e c t of small d e v i a t i o n s i n al inement of t h e ins t rument from t h e meRn flow d i r e c t i o n upon t h e output mas n e g l i g i b l e r e l ~ t i v e t o exper imenta l s c a t t e r .

5, f h e l a t e r a l -- component of t u rbu l ence and t h e t u r - bu len t c o r r e l a t i o n u v , mere measured v i t h a b ip l ane X- t y p e meter composed of t w o p l a t i num v i r e s , each s e t a t an a n g l e of hbout 450 t o t h e mean f l ow , and ther-efore p e r p e n d i c u l a r t o each o ther . The n i r e s were e tched . Wollast on, 0.. 00025 inch i n diameter and approximately 4 m i l l i m e t e r s i n l eng th .

6 , A Dunont o s c i l l o s c o p e mi t h 5-inch phot bgraphic- type t ube and a General Radio high-speed camera were u sed t o r eco rd t h e osc i l lograms .

7. The s c a l e of u t t u r b u l e n c e on t h e j e t a x i s was measured by mounting a second hot-vi re c a r r i a g e on t h e s t e e l t r a v e r s i n g t r a c k ,

A 1 1 r uns v e r e made a t a noaz le v e l o c i t y of 10 meters pe r second, and a maximum tempera ture d i f f e r e n c e o f about 10° C. I n o rder t o check t h e assumption t h a t t h i s temper- a t u r e d i f f e r e n c e has no a p p r e c i a b l e e f f e c t upon t h e hydro- J V -'

dynamic j e t cha rac t e r f s t i e s , one t r a v e r s e n a s r epea t ed wi thout t h e a d d i t i o n of hea t . The UB/Um and u t /UR cu rves f o r t h e two c a s e s checked w i t h i n t h e exper imental s c a t t e r .

I n c l i n a t i o n of t h e Jet . a x i s due t o f r e e .convect ion was n e g l i g i b l e .

TEST S AND G E D R A L .PROCIIDURI!I

1, Mean v e l o c i t y . , ' U.R: mea'sured wi th a s i n g l e hot w i r e by e f t h e r cons t an t cu r r en t or cons t an t resistance method

2. Mean t empera tu re , T: peasured wi th a s n s l l copper- cona t an t an thermocoupls a c r o s s R w a l l galva- nometer

3 , A x i ~ l t u r b u l e n c e , u1 : measured wi th a s i n g l e hot ' wire i n t h e convent iona l manner

4. Rad ia l turbulence. , v ' z measured w i t h a b i p l a n e I - type meter composed of two hot w i r e s t h e ou tpu t s of which a r e s u b t r a c t e d b e f o r e , be ing pu t i n t o t h e a m p l i f i e r

5. Double c o r r e l a t i o n a t a p o i n t ( t u r b u l e n t s h e a r ) , - uv: measured wi th t h e same meter a s v t . The ou tpu t s of t h e two e i r e s a r e pu t through t h e a m p l i f i e r s e p a r a t e l y , and t h e c o r r e l a t i o n i s computed from t h e s e r e s u l t s ,

6; Double c o r r e l a t i o n of a x i a l t u rbu l ence a t tno ' equal-veloci ty p o i n t s on oppos i t e s i d e s of t h e ---- a x i s , u,u,: measured n i t h t w o s i n g l e ho t ~ i r e s . I n o rder t o ge t a measure of t h e s c a l e of t u r - ---- bulenee , t h e u,u, mas measured a s a f u n c t i o n of t h e d i s t a n c e between t h e mires.

'7, Osci l lograms of t h e fluctuations mere made, u s i n g t h e s i n g l e ho t v i r e . The a m p l i f i e r out- put na$ photographed f r o & t h e s c r een of an o s c i l l o s c o p e t h e smeep of which had been cu t out . The f i l m motion supp l i ed t h e con t inuous stveep.

U,nless s p e c i f i c a l l y desc r ibed o t h e r a i s e , a l l quant i - t i e s p l o t t e d i n t h i s r e p o r t a r e unco r rec t ed f o r t h e ef- f e c t s of t u rbu lence upon t h e measuring ins t ruments . The n a t u r e of t h e s e c o r r e c t i o n s m i l l be d i scussed l a t e r .

Mean Veloc i ty and Temperature

The measured v e l o c i t y and t emperature d i s t r i b u t f ons a long t h e j e t a x i s and a t each s t a t i o n a r e inc luded i n f i g u r e s 17 and 22, A s has been found by Ruden, t h e tem- p e r a t u r e spreads a t a g r e a t e r ang le t h a n t h e v e l o c i t y i n t h e r eg ion of x g r e a t e r t han 8 o r 1 0 diarnete'rs, I t w i l l be noted a l s o t h a t a t 5 d i ame te r s , t h e temperature d i s t r i b u t i o n i s narrower t h a n t h e v e l o c i t y i n t h e c e n t e r p a r t of t h e j e t , a s recorded and d i scussed i n t h e f i r s t p a r t of t h e r epo r t . No d e t a i l e d measurements were made i n t h e 1-inch j e t f o r x . l e s s t han 5 d iametere , but t h e tempera ture d i s t r i b u t i o n a t t h e nozz le mouth was checked a s being of t h e same n a t u r e a s t h a t i n t h e 8-inch J e t .

The a x i a l distributions show a c r o s s i n g , o r a t l e a s t a co fnc idence , of t h e v e l o c i t y and temperature curves a t l a r g e d i sbances from t h e nozz l e , i n s p i t e of t h e f a c t t h a t t h e i n i t i a l drop-off occurs much sooner f o r t h e tempera- t u r e t h a n f o r t h e v e l o c i t y ,

Turbulence

a) TurhUaacaL,~ual. - The measurer2 l o c a l t u rbu lence l e v e l ut /Ua aa w e l l a s t h e va lues of uv/Um computed from ut/UE and t h e f a i r e d v e l o c i t y p r o f i l e , a r e inc luded i n f i g u r e s 18 t o 22. Probably t h e q u a n t i t y u l / U ~ l o s e s i t s s i g n i f i c a n c e i n r eg ions of extremely low v e l o c i t y l i k e t h e edge s f t h e j e t . A b e t t e r comparison of t h e turbu- l ence d i s t r i b u t i o n s a t success ive s t a t 5 o a s can be gained from f i g u r e 23 mhere u8/Uo i s p l o t t e d a g a i n s t t h e r ad ius . Th i s c l e a r l y shows. t h e r a d i a l spread of t h e f l u c t u a t i n g energy ( r e a l l y , t h e square roo t of t h e energy) a s t h e flotp t r a v e l s i n t h e a x i a l d i r e c t i o n .

The only measurement of t h e r a d i a l t u rbu lence compo- nent v' i s compared i n f i g u r e 24 wi th t h e u 1 a t t h e same s t a t i o n .

b) Eqrrqlat$oa js%.JvmPmbg ame n o i n t . - The c g r ~ e l a t i o n and .I-

t i c n t a r e giveh i n f i g u r e 25. The c o e f f i c i e n t i s de f ined @Y

t h e t u r b u l e n t shear i e , of cou r se , -

T -p UV

and m i l l be disousged 4n t h e next see%idxi.

The tremsndoue s o a t t e s i n %, i n t h e ou te r annulus of t h e J e t , a s s e l l as tbs eharp decqease i n b o t ? Ruv -- and uv toward t h i s p e g % ~ & , eeem t o int3ic'at.e a d e v i a t i o n from t h e t r u e t u i b u l s n t s t a t e , a hypo thes i s which r e c e i v e s e s p e c i a l support frea the a s ~ S l l o g r a m s of t h e v e l o c i t y f l u c t u a t i o n s . .

6 ) ! & X X ~ & A ~ ~ ~ - ~ ~ ~ H B & % X ~ L ~ ~ ~ l ! a 4 3 : ~ & & & f & s r - e & j o i n t a , - F igu re 26 %s the one meaaured d i s t r i b u t i o n of - B, b ~ t v e e n p o i n t s symmetrical about t h e j e t ax i s . ' Th is c o r ~ e l a t i o n eos f f i c i e a t f a de f ined by

An important r e e u l t s h o ~ n by t h i s curve i s t h a t , t h e s e f as no whipping sf t h e j e t a$ an whole; whipping couyd r e s u l t from a general. i n s t a b i l i t y of t h e a y s t s ~ . Ef saeh a con- d i t S o n exf seed, t h s c o r r e l a t i o n would not go t o z6ro n i t h f nc reas ing Y, but , would i n c r e a s e nega t ive ly .

d ) O_s_c_f,1,L~~rams~ - The s e r i e s of 'osci l logsame' i n f i g - ure 27 i n d i c a t e e a d e f i n i t e d e v i a t i o n from t h e f u l l y tur- bu len t regime toward t h e edge of t h e j e t . A completely t u r b u l e n t co re exf sts out t o approximately r o , t h e r a d i u s a t which t h e mean v e l o c i t y i s one-half t h e maximum mean v e l o c i t y . a t t h e s e c t i o n , which i s , of c o u r s e , on t h e j e t a x i s , .-.

Figure 29 shows t h e type of r e g u l a r f l u c t u a t i o n s f i r s t observed i n a 3-inch j e t by T h i e l e in 1940. These a r e

o s c i l l o g r a m s of t h e u - f l u c t u a t i o n s i n t h e 1-inch j e t a t an a x i a l d i s t a n c e of 2 d i a m e t e r s . The bottom p a r t i s a 2 0 0 cycle-per-second t i m i n g wave, which a p ~ l i e s t o both f igures; 'Z?*aQd@28; f i g u r e 28a was r e c o r d e d on t h e a x i s i n t h e p o t e n t i a l cone and f i g u r e 28b was r e c o r d e d a t a r a d i u s e q u a l t o t h e r a d i u s of t h e n o z z l e mouth. T h i s i s approx imate ly t h e i n n e r edge of t h e f r e e boundary l a y e r . The l a y e r i s e v i d e n t l y t u r b u l e n t , o r t h e r e g u l a r waves would c o n t i n u e th rough i t .

THE ORDT I GAL ANALY SE S

V e l o c i t y

For a f r e e j e t of t h i s t y p e t h e u s u a l t u r b u l e n t boundary l a y e r a s sumpt ions a r e a p p l i e d t o t h e g e n e r a l Navier-St okes e q u a t i o n . The s i m p l i f i e d e q u a t i o n must be s o l v e d w i t h t h e boundary c o n d i t i o n s V = 0 and

= 0 at r = 0 (on t h e a x i s ) . A l s o t h e v e l o c i t i e s a r and t h e i r d e r i v a t i v e s must v a n i s h a s r becomes i n f i n i t e , S ince t h e a x i a l p r e s s u r e gradien ' t i s n e g l e c t e d , t h e r a t e of f low of a x i a l momentum a c r o s s a l l s e c t i o n s i s t h e same:

2n p r XJ2 r d r = c o n s t a n t b

Since v i s c o s i t y i s n e g l e c t e d ,

a ) lilome~turn trii.933 e r t h ~ s r y - The a s s u m ~ t i o n t h a t momentum i s t h e t r a n s p o r t a b l e q ~ a n t i t y w i t h a mechanism ana logous t o t h e k i n e t i c t h e o r y of g a s e s l e a d s t o

v h i c h g i v e s t h e e q u a t i o n of motion:

au P s a n a t l f u r t h e r s u g g e s t e d t h e aesurnption of v = 1 - ar

or, s i n c e -- a' 3.8 i n t r i n s i c a l l y n e g a t i v e i n 'an a x i a l l y Br

~ y m m a t r i c a l J e t , v = - 1 [E Thf s leads t o

T = p z 2 [E] (3$ and the e q u a t i o n of mot ion may be

To l lmien assumes s f m i l a r f t y a t d i f f e r e n t s e c t i o n s of t h e j e t and assumes t h a t 1 i s c o n s t a n t a c r o s s any s e c t i o n and p r o p o r t i o n a l t o t h e a a d i i ar0 o f ' t h e sec- t i o n s , Having found e x p e r i m e n t a l l y t h a t r - , h e assumes 1 = cw.

b ) Rodif i e d voxt iq&&y t r a n s f e r ' C h e o r ~ ~ - Taylor assumes t h a t v o r t i c i t y i s t h e t r h n s p o x t a b l e q u a n t i t y and i s c a r r i e d unchanged from one l a y e r of f l u i d t o a n o t h e r , T h i s , mfth t h e assumpt ion of i s o t r a p y o f t u r b u l e n c e , yf e l d s t h e e q u a t i o n o f motion

* -- Assuming m P t h G a n d t l t h a t zv = 1 I a ~ ' , Howarth and

T o n o t i k a have

and t h e y a l s o assume 1 = cx.

c ) A t h i r d approach t o t h e c a l c u l a t i o n of t h e ve loc- i t i e s i n t u r b u l e n t f low mas p r e s e n t e d by P. Y. Chou, The e q u a t i o n s a r e o b t a i n e d by c o n s t r u c t i n g c o r r e l a t i o n f u n c t i o n s from t h e e q u a t i o n s of t u r b u l e n t f l u c t u a t i o n s .

I n t h e a p g l i c a t i o n of t h i s t h e o r y t o t h e j e t , L in h a s emploged, i n a d d i t i o n t o t h e boundary l a y e r approxi - rcat ions:

1. The assumpt ion of mechan ica l s i m i l a r i t y of mean q u a n t i t i e s extended t o i n c l u d e a l l t h e double (and t r i p l e ) c o r r e l a t i o n s

2. C e r t a i n c o n s i d e r a t i o n s based upon s i m i l a r i t y o f f l u c t u a t i o n s , and t h e m i c r o s a a l a of t u r b u l e n c e

3. The assumpt ion t h a t t h e t r i p l e c o r r e l a t i o n s of t h e t u r b u l e n t f l u c t u a t i o n n h i c h v a n i sh a t t h e c e n t e r o f t h e j e t a r e n e g l i g i b l e throughout t h e j e t

These assumpt ions r e s u l t i n a n e x p r e s s i o n f o r t h e s h e a r n h i c h i s i d e n t i c a l t o t h e f o r x s u g g e s t e d by von ~ g ~ r n k n i n 1937: . -

-+-

?As au = -- -- w ar

I n a d d i t i o n , ' L in assumes -

8 4, That v and X a r e c o n s t a n t . T h i s f i n a l

assumpt ion amounts t o a c o n s t a n t s h e e r coef- f i c i e n t , * , and t h e form of t h e s o l u t i o n i s t h e same a s t h a t f o r a l a m i n a r j e t :

*S ince t h e w r i t i n g of t h i s r e p o r t , c o p i e s of t a o new t h e o r e t i c a l p p p e r s on t h e problem of f r e e t u r b u l e n c e have been r e c e i v e d . Both a u t h o r s a d ~ r o c a t e t h e assumpt ion of a c o n s t a n t s h e a r c o e f f i c i e n t f o r t h e c a l c u l a t i o n of t h e f low i n a f r e e j e t . The p a p e r $ a r e :

P r a n d t 1, L. : - Bomerkung z u r ~ T h e o r i e d e r f r e i e a Turbulenz , Z t f . a , M , M . , v o l e 2 2 , Oct. 1942.

G z r t l e r , H,: Berechnung von Aufgaben d e r f r e i e n Turbulenz auf Grund e i n e s neuen Niiberungsansatzes.

where x I - ;

Tempera ture

"Tith t h e u s u a l boundary FTayer appraximat i o n s , t h e e q u a t i o n f o r t h e t e m p e r a t u r e T i s

The same g e n e r a l boan6axy c a n d i e i o n s app ly t o t h e temper- a t u r e as were a p p l i e d t o veloc0ty.

a ) Idomenturn t r + n s f e r - t h ~ ~ ~ . - S i n c e t h e above equa- t i o n i n T i s t h e aaqe t i8 t h e e q u a t i o n of mot ion ( i n l?) c o r r e ~ p o n d i n g t o t h e mornanturn t r a n s f e r t h e o r y , i t f o l l o w s t h a t t h e t e m p e r a t u r e d i s t r i b u t i o n i s t h e same a s t h e v e l o c i t y d i s t r i b u t i o n .

k ) Modif ied vor4~g~~-&~gasfer t h e o r y , - The p r e v i o u s l y o b t a i n e d s o l u t i o n s f o r U and V a r e p u t i n t o t h e tem- p e r a t u r e equation, a l o n g w i t h t h e assumpt ion t h a t

-- 1v = -ox 2

as before . 3 r

T h i s l e a d s t o a s o l u t i a n f o r T / T m a s a n e x p o n e n t i a l f u n c t i o n which convqrgae much more r a p i d l y t h a n t h e ve loc- i t y U/U,.

c ) I n s o l v i n g f o r t h e t e m p e r a t u r e d f s t r i b u t i o n , L i n u s e s a c o n s t a n t c o e f f i c i e n t o? t h e r m a l d i f f u s i o n , ana logous t o t h e u s e of a c o n s t a n t shea r o o e f f f c i e n t . T h i s l e a d s t o a power law r e l a t i o n s h i p between t h e v e l o c i t y and ternpera- t u r e d i s t r i b u t i o n ; which ie w r i t L e n .

where C a i s not g i v e n By t h e t h e o r y but must be d e t e r - mined e x p e r i m e n t a l l y ,

'bl &CU s 81 ON 0 9 RESULT 8

The Nature of t h e Flow

F i g u r e 27 i s a s e r i e s of o s c i l l o g r a m s of t h e axial v e l o c i t y f l u c t u a t i o n s a t v a r i o u s r a d i a l p o s i t i o n s on a s e c t i o n 2 0 d f a m e t e r s from t h e n o z a l e mouth. T h i s s e r i e s shows t h a t t h e f low i n a " t u r b u l e n t j e t " i s f u l l y t ~ r b u - l e n t only out t o approx imate ly r = rO. F o r r > ro , t h e r e e x i s t e f i r s t a n a n n u l a r " t r a n s i t i o n r e g f onn i n which t h e f low a l t e r n a t e s between t h e t u r b u l e n t and t h e l a m i n a r regimes. The f r a c t i o n of t h e t o t a l t ime d u r i n g which t h e t u r b u l e n t s t a t e p r e v a i l s a e c r e a s e s as t h e r a d i a l d i s t a n c e i s i n c r e a s e d . Then, n e a r t h e edge of t h e j e t and e x t e n d i n g t o g e r o v e l o c i t y i s what might b e te rmed t h e a n n u l a r " l a m i n a r c o l l a r : " Of c o u r s e , t h i s l aminar c o l l a r d i f f e r s f r o m t h e u s u a l concept of a lam- i n a r boundary l a y e r o r a l aminar a u b l a y e r i n a t u r b u l e n t boundary l a y e r next t o a m a l l ; t h e d i f f e r e n c e i s t h a t t h e r e e x i s t s a n a p p r e c i a b l e r a d i a l ( v ) component i n t h e mean v e l o c i t y . The d i s a p p e a r a n c e , however, of <?-corro- l a t i o n tamard t h e edge of t h e j e t can be t a k e n as a c l e a r i n d i c a t i o n t h a t t h e momentum t r a n s f e r h e r e i s e s s e n t i a l l y 1 ami n a r ,

The cause of t h e O e t o r i o r a t i o n o f t h o t u r b u l e n * f lom toward t h e e d g e of t h e j e t i s not immediately a p p a r e n t , s i n c e t h e r e i s no s o l i d w a l l t o damp out t h e v - f l u c t u a t i o n s a s i n t h e c o n v e n t i o n a l t u r b u l e n t boundary l a y e r .

The s i m i l a r i t y between t h i s e n t i r e f lom c o n d i t i o n and t h e h i s t o r y i n the mean flom d i r e c t i o n of t h e f l o n i n t h e boundary l a y e r a l o n g a w a l l i s s t r i k i n g . The p r i n c i p a l d i f f e r e n c e i s t h a t t h e sequence of f lom regimes e x t e n d s i n a d i r e c t i a n p e r p e n d i c u l a r t o t h e . mean f l o w , p h i l o i n t h e l a t t e r c a s e i t e x t e n d s i n t h e mean f low d i r e c t i o n . The o s c i l l o g r a m s of t h e j e t - t r a n s i t i o n r e g i o n a r e i d e n t i c a l w i t h some of t h e o s c i l l o g r a m s of t h e t r a n s i t i o n . f r o m a l a m i n a r t o a t u r b u l e l i t boundary l a y e r ,

A rcore s i g n i f i c a n t a s p e c t of t h e v e r i f i c a t i o n of t h i s f low p a t t e r n i n t h e t u r b u l e n t j e t i s t h a t t h i s may p r e s e n t a n i n s i g h t i n t o t h e n a t u r e of t h e f l o a . i n a f u l l y developed t u r b u l e n t boundary l a y e r next t o a s o l i d mal l . I t i s n e l l known t h a t a t h i n l a m i n a r s u b l a y e r e x i s t s immediately ad- j a c e n t t o t h e s o l i d boundary , and t h a t t h e r e i s no sha rp

p o i n t of d e m a r c a t i o n , but a narrow r e g i o n of t r a n s i t i o n , The f l o w i n t h i s regi:on.-i%"prob&rb$y~ pb t h e same n a t u r e a s t h e f l o w i n t h e a n n u l a r t r a n s i t i o n r e g i o n of t h e t u r b u l e n t j e t . ~ 3 : ~ *

I t w i l l be n o t e d t h a t t h e g e n e r a l l o c a t i o n of t h e t r a n s i t i o n . reg ion1 i n t h e ' j e t &B or: Less the,-same a s t h o , l o c a t i o n of the maximuns'.of .the u i /UR c u r v e s a t

a l l ' s e c t i o n s ( s e e f i g s . 18 t o kg). T h i s may mean Ithst a p a r t of t h e " t u r b u l e n c e " i s du$ no t t o t h e u s u a l Imrbu- l e n t v e l o c i t y f l u c t u a t i o n s , b & - t o a c t u a l d i f f e r e n c e s i n t h e l o c a l mean v e l o c i t y a t a pcrf.nt, a s t h e f l o w o s c i l l a t e s between t h e t u r b u l e n t and t h e Z ~ ~ l a i n a r s t a t e s . f h f g ex- p l a n a t i o n 'nap f i r s t g i v e n by Lf bpmann i n c o n n e c t i o n d i t h t h e ratudies r e p o r t e d i n r e f e rezk~e ;? 10 t o e x p l a i n l t k i ! h i g h a p p a r e n t t u r b u i e n c e l e v e l i n t h g - t r a r r s i t i o n ~ e g i o n ; p f - . a boundary Payer a d j a c e a t t o a s9 l i . d mall,_ I t . phoulb; '

however , b e r e c o r d e d t h a t no d e f i n i t e one-sihbd % u r b u l e n t b u r e t s on t h e o s c i l l o g r a m ~ were s a e n i n the case,of t h e J e t . ..,

B

Comparison betweeh Theory and Experiment $ .

a ) V e 1 o c i t s . - On f i g u r e & 2 9 , 30, and 31 are- ,&.ot ted t h e r e s u l t s o f t h e t h r e e t h e o r i e s b r i e f l y discus,.sh'ci and t h e e x p e r i m e n t a l l y observed p o i n t q . For p u r p o s e s ' of com- p a r i s o n , t h e t h e o r i e s and exper iment have a l l ..be.e-en matched a t r = r T h i s seems t o be a n e s p ~ c i a l l y ~ u i t ~ a h l e , method i n v iew of t h e f a c t t h a t t h i s r a d i u s i s approximaCely t h e boundary o f t h e c o m p l e t e l y t u r b u l e n t j e t c o r e .

Pn i n s p e c t i n g t h e s e f i g u r e s , i t must be remembered t h a t t h e e x p e r i m e n t a l p o i n t s have not boen c o r r e c t e & : f o r t h e e f f e c t of t u r b u l e n c e upon t h e ,hot w i r e : . T h i s c o r r e c - t i o n would a p p r e c i a b l y d e c r e a s e t h e measuled v e l o c i t y everywhere but n e a r t h e axf s . Q u a l i t a t i v e l y ; such, a :cor - r e c t i o n , fo l lowed by p r o p e r r e f i t t i n g of t h e c e x p e r i u a n t a l p o i n t s a t r r would p r i m a r i l y widen t h e p r o f i l e peak . S i n c e t h e af' c o r r e c t i n g hot-w'irc measure- ments f o r t h e e f f e c t of v e r y h igh ' t u r b u l e n c e l e v e l s (u?/U a%ove abou t 28 p e r c e n t ) - , i s s t i l l u n d e r c o n s i d e r a - t i o n ' a t t h e GALCIT, the r e s u l t s a r e p r e s e n t e d a s obse rved ai.1.d w i t h o u t c o r r e c t i o n . The q u e s t i o n o f turbulen.&' co r - r ' e c t i o n s i s t akpn up b r i e f l y i n append ix 11,.

i

I t shou ld be rnentqoned a l s o t h a t t h e mean s p e e d s a t s t a t i o n s 1 0 , 2 0 , and 40 -re measured by t h e c o n s t a n t - c u r r e n t method. I n t h i s method, t h e v a l u e s a t ex t remely low speeds a r e t o o h i g h because of c o n v a c t i o n c o o l i n g , ahen s u f f i c i e n t c u r r e n t i s u s e d t o g i v e r e a s o n a b l e s e n s i - t i v i t y a t t h e b i g h e r speeds . The c o n s t a n t - r e e i s t a n c e ( and, t h e r e f o r e , c o n s t a n t - s e n s i t i v i t y ) method i s much more s a t i s f a c t o r y when a v i d e v e l o c i t y r a n g e i s t o be c o v e r e d , and rpas employed a t s t a t i o n s 5 and 30, The ob- s e r v a t i o n s by t h e two methods d i v e r g e p e r c e p t i b l y only at t h e extreme edge of t h e j e t , I n f i g u r e 30 t h e s c a t t e r everywhere but a t t h e extreme edges i s o r d i n a r y exper i - ment a1 s c a t t e r ,

A s ment ioned b e f o r e , s i m i l a r i t y i n t h e v e l o c i t y p r o f i l e i s reached a t about x = 1 0 d i a m e t e r s ,

S ince t h e t h r e e t h e o r e t i c a l a n a l y s e s assume a t u r - b u l e n t s t a t e a c r o s s t h e f u l l w i d t h of t h e j e t , t h e d e g r e e of agreement between t h e o r y and experiment i n t h e r e g i o n f o r r > r o i s e v i d e n t l y of secondary impor tance . In- s p e c t i o n of t h e ( t h e o r e t i a a l l y ) s i g n i f i c a n t r e g i o n of r < xo i n d i c a t e s a s a t i s f a c t o r y check be tneen t h e oh,. s e r v e d p o i n t s and t h e curve 'based upon a c o n s t a n t s h e a r c o e f f i c i e n t . I t i s obvious t h a t t h e l a r g e d i s c r e p a n c y i n t h e o u t e r p a r t of t h e j e t b e t ~ e e n experiment and con- s t a n t s h e a r c o e f f i c i e n t t h e o r y cannot be due t o t h e method of v e l o c i t y measurement (hot-.-wire measures UR i n s t e a d of U ) ; c o r r e c t i n g , the e x p e r i m e n t a l p o i n t cr t o be U in- s t g a d of UR mouzd l n c r e a s e t h e d i sc repancy .

The p h y s i c a l r e q u i r e m e n t s of c o n s t a n t flux of momen- tum a c r o s s a l l s e c t i o n s of the j e t ~ l u s t h e u s u a l assump- t i o n of v e l o c i 0 y - p r o f i l e s i m i l a r i $ g l e a d t o t h e requ i rement of a h y p e r b o l i c d e c r e a s e i n t h e mean speed a l o n g the j e t a x i s . A p l o t of t h e a x i a l v e l o c i t y d i s t r i b u t i o n on loga- r i t h m i c c r o s s - s e c t i o n p a p e r ( f i g . 35) shoms t h a t t h e de- c r e a s e approx imates a n e a r l y h y p e r b o l i c power l a w from 1 0 t o 30 d i a m e t e r s , but d e v i a t e s markedly a t g r e a t e r d i s t a n c e s . T h i s d e v i a t i o n may i n d i c a t e e i t h e r t h a t t h e v e l o c i t y de- c r e a s e does not f o l l o a a s imple pbver l av o r t h a t e x t e r n a l d i s t u r b a n c e s have begun t o be f e l t . A check run of t h e a x i a l v e l o c i t y d i s t r i b u t i o n f o p U o more t h a n t w i c e as l a r g e a s u s e d i n a l l t h e ~ e measurements gave a n i d e n t i c a l d e v i a t i o n from t h e h y p e r b o l i c d i s t r i b u t i o n , From t h e ex- p e r i m e n t a l s e t u p i t a p p e a r s u n l i k e l y t h a t e x t e r n a l

d i s t u r b a n c e s would 'be f e l t b e f o r e ' x = 50 d i a m e ~ m s , Pf ;' . , t h e n , i t i s assumed t h k t e x t e r n a l e f f e c t s a r e n e g % i g i b l e , . th.1-s q e v i a t i o a seems t o deny t h e s x i s t e n e e .sf iexs6%' s t m i - , : l a r i t y , s i n c e t b e c o n d t a n t f l u x sf gomen$um . i t3 a s impl* phys i u a l r equ i rement , * The on% y apimaf n i n g pas s i b i l'it y 9%-' ' t h a t t h e a x i a l p r e s s u r e g s a k i e a t c a ~ a a o t be enggleet e8,

The s i m i l s r i t y p r o p e r i y , however, of t h e j e t a $ , ~ e $ n . r e s e a t e d by a l f a e a r i n c r e a s e i n t h e width of t h e v e l o e i f y . , p r o f i l e s w i t h downstream dis t t4nee i s w e l l checked out t o x = 40 diametere . : in f i g u r e 36, S u r p r i s i n g l y , ,.the l i n e '

p a s s e s veag nearly t h r o u g h t h e n o z z l e mouth when extrapom : l a t e d t o z e r o . je t w i d t h ,

A l l t h e t b a d r e t i e + l a r la lyses assume mixing l e n g t h i n c r e a s i n g l i n e a r l y w i t h axial d i s t a n c e : 1 = cx, A c e l - c u l a t i . 0 8 of ,o a t x 5 20 diamet e r s , based on t h e momeaa- turn ,$.ransf e s th.ectgPy l e a d s t o , .c p. q,0166. ToPllmien ~ i ~ e s a va1u.w of- 0 0x5 8:,

b ) ! F e m p s ~ a t . ~ r e , ~ - S ince L i n l n a p a l p s i s does not ex- p l i c i t l y g i v e the t e m p e r a t u r e g r s f i i e as n f u n c t i ~ n o f t h e v e l o i i t y p r o f i l e . but l e a v e s one o o n s t a p t , t o be de te r - . ' mined by exper iment , olrnly t h e r e s u 3 . t ~ of t h e o t h e r two t h e o r i e s have b e e n p l o t t e d i n f i g u ~ e s 32, 33, and 34 f o r comparison w l t h ' t h e exper imenta l r d s u l t s. - s .- . P

I n s p i t e of, t h e c o n a f d e r a b l e s c a t t e r , i.t i s h i d e d t t h a t t h e o r e t i c a l and 'experimentczl t empernture d i s t r i b u - t i o n e do not a g r e e i f t h e v e l o c i t y d i s t r i b u t i o n s .a re matched st r = r I f , on t h e o t h e r hand, t h e tempera- t u r e s were f i t t e d a t t h e r a d i u s where T = 1 / 2 ~i. t h e n t h e e x p e r i m e n t ~ l ' a n d t h e o r e t i c a l velocity p r o f f l e s would

' e x h i b f t R wide d ive rgence : w h i l e t h e t e m p e r a t u r e s would cheek r e a s o n a b l y w e l l , , i

, ?

S i n c e t h e r e i e no method of! d e t e r m i n i n k whiesh f i t - '

tang proce$iure f s more nearby c o y r e a t , ' t h e ,only p i ~ m i s s i - "$6 coael'uerian is t h a t aone of t h e ei%eVtlng ' -'theories g i v e s a s & Q P ~ % a c t o ~ y q u a n t i t a t lvas s e l ' s k i o n s h i p b$'ew,e,ea t h e spread 00 ves loai ty and t h e apttsa8~ of tempera ture , i n R turbu%enl h e a t e d g e t , . *

I t woulb seem t h a t a b s t t e ~ underqkandf.ng of t h e " 8 # % e r e l a t i i o n between the mechanie~mr~ of t h e d i i f fue ioq 0% v e l n e i t p and t e m p w a t u r e e b u l d be g a t $ e a by meesure- meats of t h e teapera$un°e f l u c t u a t 40x1 4%" $tag,' sad

---- p e r h a p s t It , c o r r e l a t i o n s ) a n 8 by t h e comparison of t h e s e measurement s w i t h t h e eor resgondrng k i n e m a t i c q u a n t i t i e s .

I t s h o u l a be n o t i c e d ( f f g , 35) t h a t t h e ax$&& tam- p e r a t u r e d i s t r i b u t i o n i s a>prox imate ly h y p e r b o l i c i n form and t h a t t h e t e m p e r a t u r e je t -width ( f i g . 3 6 ) fn- c r e a s e s v e r y n e a r l y l i n e a r l y w i t h a x i a l d i e t a n c e . . A g l a n c e a t f i g u r e 22 o r 34 w i l l show t h a t t h e p o i n t a t 48 d i a m e t e r s i s ex t remely u n r e l i a b l e .

Turbn lence

The q u a n t i t y u l / U R c a n s c a r c e l y be u s e f u l i n a f low where i n s t a n t a n e o u s v a l u e s o f u a r e o f t e n g r e a t e r t h a n U . T h i s i s c e r t a i n l y t h e c a s e a t t h e edge of t h e j e t , and t h e r e f o r e i t was f e l t t h a t a d i m e n s i o n l e s s q u a n t i t y u4/U, p s o p o r t i o n a l t o u l , p l o t t e d a s a func- t i o n of r a d i u s , would g i v e a more u s e f u l u i c t u r e o f t h e v e l o c i t y f l u c t u a t i o x ~ s i n t h e j e t . F u r t h e r , when t h e t u r b u l e n c e d i s t r i b u t i o n s a t v a r i o u s a x i a l p o s i t i o n s a r e t o be compared, t h e q u a n t i t y u t / U o i s p l o t t e d ' ( f . ig . 23) .

I t c a n be s e e n from f i g u r e 23 t h a t t h e l o c a l minimum a t t h e c e n t e r of t h e t u r b u l e n c e d i s t r i b u t i o a s .does not d i s a p p e a r b e f o r e a n a x i a l d i s t a n c e of a b o ~ t 20 d i a m e t e r s . Thus , a l t h o u g h s i m i l a r i t y of v e l o c i t y p r o f i l e s i s e f f e c - t i v e l y reached a t 8 o r 1 0 d i a m e t e r s , r e a l k i n e n a t i c s i m i - * I a r i t y i s not r eached u n t i l about 2 0 d i a m e t e r s ,

I t i s a l s o i n t e r e s t i n g t o n o t e t h a t t h e l o c a l maxi- mums in t h e c u r v e s f o r x 2 3 d i a m e t e r s remain a t about t h e same r a d i u s independent o f axial p o s i t i o n and t h a t t h i s r a d i u s i s approx imate ly t h e nozzl'e mouth ra .dius. Of c o u r s e , t h e r a t h e r a r b i t r a r y c a r v e f a i r i n g d e t e r m i n e s t h e e x a c t l o c a t i o n of t h e . y e a k s . However, t h i s c l a t t e r i s re- l a t e d t o t h e problem of t h e e v o l u t i o n f r o m a r e c t a n g u l a r v e l o c i t y p r o f i l e w i t h v e r y low t u r b u l e n c e t o t h e f u l l y developed t u r b u l e n t j e t , which st ill awai t s a n adequat e b a s i c e x p e r i m e n t a l i n v e s t i g a t i o n ,

The a x i a l d i s t r i b u t i o n of u l / U o i s p l o t t e d on f i g - u r e 3 5 , and a l t h o u g h a s t r a i g h t l i n e h a s been drawn ten- t a t i v e l y th rough t h r e e p o i n t s , t h e p o i n t s do not i n d i c a t e a s imyle power la^.

* As gent ioned p r e v i o u s l y , t h ? curves of f i g u r e 2 3

g i v e 'a c l e a r p i c t u r e of t h e spread of t h e t u r b u l e n t k i n e t i c energy. Of c o u r s e , t h e r e i s a cont inuous gener- a t i o n ' of t u r b u l e n t k sne t i c $energy from t h e k i n s t i c energy of t h e mean flow, A q u a n t i t a t i v e p i c t u r e of t h e k i n e t i c energy h i s t o r y of t h e j e t i s g o t t e n by i n t e g r a t i n g t h e aquares of t h e v e l o c i t y p r o f i l e s and t h e squares of t h e curves on f i g u r e 23, The t u r b u l e n t k i n e t i c energy i n t h e l a t e r a l f l u c t u a t i o n e ( v ) $ 8 g o t t e n f r o = t h e v ' d is- t r i b u t i o n measured a t x = 20 diameter^. I t has t hen been assumed t h a t a t a l l t h e o t h e r s t a t i o n s t h e v-energy i s i n t h e same r a t i o t o t h e u-energy a s a t 20 diameters . A l s o , t h e t o t a l w-energy at a s e c t i o n i s t aken a s equal t o t h e v-energy f o r t h e purpose o f t h i s ca l cu l a t i on :

The r e s u l t s a r e p l o t t e d on f i g v x e 37. The t o t a l k i n e t i c energy ET and t h e k f n e t i c energy of t h e mean flow EM . c r o s s i n g a s e c t i o n i n u n i t t ime a r e p l o t t e d a g a i n s t a x i a l p o s i t ion, The d i f f e r e n c e between t h e s e two curvaa i s t h e k f n e t i c energy af t h e t u r b u l e n t f luc- t u a t i o n s El. The t h i r d curve g i v e s t h e f r a c t i o n of t h e t o t a l k i n e t i c energy t h a t c o n s i s t s of t u r b u l e n t energy , a t a s e c t i o n , T h i s Curve seems t o approach a cons tan t va lue approximately where t h e J e t ach ieves complete s i m i l a r i t y , A t l a r g e a x i a l d i i t a n c e s , however, i t ap- p a r e n t l y beg ins t o drop o f f . I t i s p o s s f b l e t h a t t h i s i s merely exper imental s c a t t e r . The d e v i a t i o n a t x = 40 d iameters i s a l s o c l e a r l y v i s i b l e on f i g u r e 38.mhich i s

% a loga r i t hmic p l o t of t h e t o t a l , t h e mean f lom,and t h e t u r b u l e n t k i n e t i c ene rg i e s ,

Shear -...

The t u r b u l e n t . shea r i s p r o p ~ r t i o a a l t o t h e uv cor- r e l a t i o n a t a p o i n t , which i s p l o t t e d i n dimensionless form a t t h e b o t t o m of f i g u r e 25. Fron t h i s i t is seen t h a t t h e t u r b u l e n t shear drops of f '?nore r a p i d l y a long a r a d i a l l i n g than doe$ t h e mean v e l o c i t y . The r a t i o of t u r b u l e n t shear t o laminar shear i s of more p ra . c t i ca1 s i g n i f i c a n c e , and t h i s i s p l o t t e d on f i g u r e 39. I n t h e

. computat i on of t h e laminar shear -t.he v e l o c i t y p r o f i l e s lopes have been determined g r a p h i c a l l y so t h a t a s i z a b l e p robab le e r r o r has been in t roduced , p a r t i c u l a r l y near t h e c e n t e r of t h e jet. 1n f a c t , i t i s p o s s i b l e t h a t t h e max- imum l o c a t e d off t h e - a x i s i s merely due t o s c a t t e r . .

From f i g u r e 39 st can B o s e e n t h a t i n t h e r a d i a l d i r e c t i o n t h e t u r b u l e n t and l a m i n a r s h e ~ . ~ s r e a c h t h e same o r d e r of mngnituhe b e f o r e t h e meeLaa v e l o c i t y r e a c h e s z e r o , Th i s i s one of t h e i n d i o a t i o n s t h a t t h e f low i s not comple te ly t u r b u l e n t out t o t h e edge of t h e j e t The c o r r e l a t i o n c o e f f f c i e n t . RUT, p l o t t e d on f i g u r e 25, renches a maximum v a l u e of about 0 ,428 Wattendorf !s maximum of 0.33 and R e i c h a r d t b mmximum o f 0.24, b o t h computed by .- Vo-g-~Armhms ( r e f e r e n c e -- X I ) f o r p i p e f l o w , a r e valu_e,~r ~f uv/u2 i n s t e a d of u v / u f v * . The maximum v a l u e of u v / u 2 at t h i s Cross , s e c t i o n i n t h e j e t i s 0 ~ 3 4 ,

The extreme s c a t t e r nea r t h e edge of t h e j e t appar- e n t l y i s due t o t h e f a c t t h a t i n t h i s r e g i o n one f a c t o r i n t h e c a l c u P n t i o n of RUT i s a s m a l l d i f f e r e n c e between l a r g e q u a n t i t i e s each of which a l r e a d y has t h e normal amount of exper imenta l s c a t t e r . This e x c e s s i v e s c a t t e r may b e t a k e n a l s o a s a n i n d i c a t i o n t h ~ t t h e f low i n t h i s r e g i o n i s not c o n t i n u o u s l y t u r b u l e n t , The d o t t e d p a r t of t h e c u r v e h a s no s i g n i f i c a n c e and i s p u t i n o n l y f o r comple teness .

M r , L in has g i v e n a method of c a l c u l a t i n g t h e t o t a l s h e a r a t a s e c t i o n from t h e v e l o c i t y p r o f i l e . The method i s p r e s e n t e d i n appendix I , and a comparison between t h e t o t a l s h e a r computed i n t h i s way and t h e e x p e r i m e n t a l l y de te rmined shea r i s g i v e n fn f i g u r e 40. For t h e l a t t e r , o n l y t h e t u r b u l e n t s h e a r need b e p l o t t e d , s i n c e t h e l a m - i n a r component i s n e g l i g i b l e i n t h e range f o r whfch t h e p o i n t s a r e g i v e n . The curves a g r e e v e r y w e l l i n shape , a l t h o u g h t h e r e a s o n f o r t h e d i s c r e p a n c y i n magnitude toward t h e edge of t h e j e t i s n o t e v i d e n t .

I t i s wor th remarking h e r e t h a t t h e r a d i a l mixing- Pkngth B i s t r i b u t i o n , cornput ed from t h e measured t u r b u l e n t s h e a r by t h e assumpt ions of t h e momentum t r n n s f e r t h e o r y , shows a maximum d e v i a t i o n of o n l y f 2 5 p e r c e n t from t h e a v e r a g e v ~ l u e f o r t h e range 0,2 r o c r c 1,8 ro. The e u r v e i s p l o t t e d i n f i g u r e 41 , As i n t h e ca.se of f i g u r e 39? t h e a p p a r e n t l y complex n a t u r e of t h e c u r v e , w i t h two extremes on one s i d e o f t h e a x i s , may be due t o t h e method of o b t a i n i n g t h e r e s u l t s , which i n c l u d e s a graph- i c a l d e t e r m i n a t i o n of t h e v e l o c f t y p r o f i l e s l o p e s .

Scale . of Turbulence . --- -

The measurement of t h e u,u,/uD c o r r e l a t izon--coef- f i c i e , n t c u r v e , s y m ~ l e t r i c a l about t h e j e t a x i s a t 2:0 diameters ( f i g . 26), served a double purpose, F i r s t , i t gave t h e s c a l e of t u rbu lence a t a "po in t r t i n t h e j e t . Second, it showed that ' t h e r e e x i s t e d no i n s t a b i l i t y of t h e j e t a s a whole, such a s mould cause whipping and t h e r e f o r e a n aitdfttoh3L nega t ive c o r r s l a t ion between p o i n t s on oppos i t e s i d e s of t h e j e t .

C ONCLU SI ONS

1. I n a f u l l y developed t u r b u l e n t j e t wi th a x i a l symmetry, a completely t u r b u l e n t f l o a e x i s t s only fn t h e co re r e g i o n ont t o t h e r a d i u s a t whioh t h e v e l o c i t y i s about one-half t h e maximum v e l o c i t y a t t h e c r o s s s ec t ion , Outside of t h i s c o r e 18 a nfde annula r t r a n s i t i o n r e g i o n , and from t h e out s i d e of t h a t t o t h e edge of t h e j e t t h e P l o w i s in t h e n a t u r e of a laminar c o l l a r .

2, I n a t u r b u l e n t hea ted j e t t h e temperature d i f f u s e s more r a p i d l y t h a n t h e v e l o c i t y , a s a l s o h a s been recorded i n a l l p r e v i o u s l y pub l i shed r e s u l t s ,

3. None of t h e e x i s t i n g tu rbu lence t h e o r i e s g i v e s a s a t i s f g c t ory q u a n t i t a t i v e r e l a t i o n s h i p between t h e spread of v e l o c i t y and t h e spread of temperature .

4, If t h e o r e t i c a l and exper imental v e l o c i t y p r o f i l e s a r e matched a t t h e r a d i u s where t h e v e l o c i t y has one-half i t s maximum va lue a t t h e s e c t i o n , t h e t h e o r e t i c a l curve ' r e s u l t i n g from t h e aseu@pt ion of a cons tan t e f f e c t i v e shear c o e f f i c i e n t gives t h e bes t agreement n i t h experiment Over t h e f u l l y t u r b u l e n t core of t h e j e t , nhjch i s t h e only r e g i o n where t u rbu lence theo r i e f t may be expec'ted t o &PP IY *

5. Reasonably good s i m i l a r i t y of v e l o c i t y and turhu- l ence p r o f i l e s i n an a x i a l l y ~ y m m e t r i c a l j e t i s not reached u n t i l an a x i a l d i s t a n c e of a t l e a s t 2 0 d iameters a h e r e t h e p r o f i l e of tuFbulence u f /Um l o s e s i t s l o c a l minimum on t h e j e t a x i s ,

The non-hyperbalic na tu re of t h e a x i a l v e l o c i t y dis- t r i b u t i o n , hornever, i n d i c a t e s t h e p o s s i b i l i t y t h a t gven then t h e s i m i l a r i t y i s only approximate.

6 , I n a h e a t e d a x i a l l y symmetr ica l j e t , t h e dimen- s i o n l e s s t e m p e r a t u r e d i s t r i b a t i o n a c r o s s a s e c t i on nea r t h e o r i f i c e i s narrower a t t h e t o p and wider a t t h e bot- to rn , than t h e c o r r e s p o n d i n g d i m e n $ l o n l e s s v e l o c i t y d i s t r i b u t i o n . A t l a r g e a x i a l d i s t a n c e s , t h e t e m p e r a t u r e p r o f i l e i s everywhere wider t h a n t h e v e l o c i t y . The cbm- mon boundary of t h e s e two r e g i o n s i s a.t Bn & z i a l d i s t a n c e of about 7 srif i c e d i a m e t e r s l

T h i s checks t h e known r e s u l t t h a t t h e e f f e c t i v e h e a t - t r a n s f e r c o e f f i c i e n t i s g r e a t e r t h a n t h e e f f e c t i v e - s h e a r c o e f f i c i e n t .

O a l i f o r n i a I n s t i t u t e of Technology, Pasadena , C a l i f . , June 1945.

I

CBZCULATI ON OF SHEAR D I STRIBUT I ON FBOM MEL OCITY PRO?i'lLE

Consider t h e "boundary-layer f ormfl of t h e Navier- S tokes e q u a t i o n i n c y l r n d r i c a l c o o r d i n a t e s :

u E + v ? L Z = L . ; ) ( P T ) (1) 3 x ar PX ar

The e q u a t i o n i s t r a n s f o r m e d i n t h e c o n v e n t i o n a l f a s h i o n , by assuming s i m i l a r i t y , changing t o a n independ- 4 2 -

e n t v a r i a b l e = , and t a k i n g a s t ream f u n c t i o n i n t h e X

f o r m $I = A x f f ( q ) , w h e r ~ A i s a a o n s t a n t . The v e l o c i t y components a r e t h e n

Next, a n ,expressSonA for t h e . s h s a r i s assumed: s .-

< , I

S u b s t i t u t i n g e q u a t i o n s ( a ) . : &ad (3) i n t o (I) and s i m p l i f y i n g t h e r e s u l t i n g e q u a t i o n g i v e s

and now n = -1 i s chosen s o t h a t x d i s a p p e a r s from t h e e q u a t i o n , and t h e s i m i l a r i t y r e q u i r e n e n t i s s a t . i s f ied, Combining t h e two t e r m s on t h e l e f t s i d e , and m u l t i p l y i n g th rough t h e s q u a t i o n by q!

which immedia te ly i n t e g r a t e s a s

where t h e c o n s t a n t of i n t e g r a t i o n i s ze ro ,

T h i s g i v e s t h e shear i n s i m p l e s t form:

a l t h o u g h f o r c a l c u l a t i o n p u r p o s e s , a more conven ien t form i s

31

APPBNDTX II

FOR IHE EFFEC'P OF TURBULZBCE

If U i s d e f i n e d a s t h e t r u e mean v e l o c i t y ( n e g l e c t - i n g V a n d W ) , u, v , and w a s t h e i n s t a n t a n e o u s componeiits of t h e f l u c t u a t i o n v e l o c i t y a n d q a s t h e t o t a l v e l o c i t y a t a n y i n s b a n t , t h e n

I n u s i n g t h e h o t w i r e as a mean-speed measur ing d e v i c e i t i s a e s i r a b l e t o know U, b u t t h e h o t w i r e a c t u a l l y - measures q , a n d when t h e t u r b u k e n c e l e v e l i s s u f f i - c i e n t l y h i g h some c o r r e c t i o n aust be a p p l i e d .

?he r e l a t i o n s h i p between a n d U may b e o b t a i n e d by w r i t i n g

---------+-------------------- ..................... F = U ~ l + 2 9 + ( 7 VY+ (a)" ( 2 )

I

o r , by l e t t i n g

and oxpanding f h e r a d i c a l out t o t e rms i n <-$y

- s i n c e u = 0.

This e x p a n s i o n converges o n l y f o r

If e q u a t i o n ( 3 ) i s r e w r i t t e n i n t h e f o ~ m

a n expans i o n convergent f o r

can b e ob ta ined .

It must be xemembered t h a t U/U i s a f l u c t u a t i n g q u a n t i t y w i t h a d i s t r i b u t i o n p r o b a b l y s i m i l a r t o t h e Gauss ian and symmet r i ca l about z e r o * This means t h a t t h e convergence r e q u i r e m e n t s must be met by t h e main body of . t h e d i s t r i b u t i o n , i f not by t h e edges.

The p r i n c i p a l d i f f i c u 1 t . y w i t h such a t h e o r e t i c a l cor- I--*..--ro

r e c t i o n i s t h a t t h e v a l u e s of a r e unknown f o r t h e

' I i n t e g e r In) > 2 , There fo re , t h e expansion cannot be computed t o a n y d e s i r e d accuracy . I f a l l powers of U/U h i g h e r t h a n t h e second a r e o m i t t e d , t h e expsnsi.ons of e q u a t i o n s ( 3 ) and ( 5 ) g i v e t h e seme r e s u l t , I f , t h e n , i t i s assumed t h a t a = 2

which i s p l o t t e d i n f i g u r e 42. This must be considered a s a q u a l i t a t i v e r a t h e r than a q u a n t i t a t i v e correc t ion .

A s92nilar a t t a c k on t h e problem sf c o r r e c t f a g hot- wire turbulence-level readings f o r t h e e f f e c t of t h e tur- bulence upon t h e inatruinept meets wi th even g r e a t e r d i f f i c u l t iea . Thus, f t would appear t h a t at% esperimental approach t o both problerae would be more f r u i t f u l .

Up t q t h e present tfme, no measurements have been made a t t h e GALCIT t o determine t h e e f f e c t of turbulencd upon mean-erpeed readings , but aoma greliminapy measure- ments have been carries out on t h e e f f e c t of turbulence upon hot-mtre readings of t h e turbulence.

The prooeiuiture wae t o v i b r a t e a bot a i r e sinusoida' l ly i n t h e flow d b r a c t i o ~ of a r e l a t i v e l y low-turbulence a i r stream. The v i b r a t i n g device was t h e GAL019 v i b r a t o r , cons t ruc ted by F, Kaoblock and C. Thie le in 1837 f o r t h e purpose of determining hot-wire time-constant s t o permit proper, conpensation f o r thermal. lag, The hot-wire c i r c u i t was proper ly compensated, and t h e t r u e turbulence l e v e l determined ba measurement of t h e amplttude gnd frequency of t h e ~ a c i l l a t i o n and the speed of t h e a i r atroam.

A @omparison between the t r u e and the measured tur- b u l e ~ c a l e v e l s f s given i n f i g u r e 43 f o r t h e same bot mire and t h r e e d i f f e r e n t speeds. I t i s s u r p r i s i n g t o no t i ce t h a t the ourve i s concave upward f o r a range be- f o r e t h e main drop-off begins. The apprec iable dev ia t ion of t h e p o i n t s a t t h e higheert speed i n d i c a t e s t h a t the curve may be dependsnt upon bhe s e n s i t i v i t y o r the tempex- attare of t h e hot mire; and, of course , t h e app l i ed eom- pansa t ion f o r tAerma1 l a g waB c o r r e c t only f o r r eba t ive ly lorn h r h L a s s o a Xevals.

I t i s hoped t h a t t e s t s may be made in thc aear f u t u r e t o determine t h e e f fec t a of high turbulence upon the read- Aags af both mean v e l o c i t y and t u r b u l e m e lgve l with t h e hot-wire anemornet e r .

1, Ryden, P* P Turbu len t e Aurbr ef tungsvorg&nge i m . j s r e i s t r a h l . Die Naturwissenshaf t e n , v o l . 21,

nose 21/23, Ma$ 1933s PPe 375-3780

2 e Kueths , Arnold M. r I n v e s t i g a t i o n s of t h e Turbu len t Mixing Re i o n s Formed b y J e t s , J O U P ~ Appl. MechanPcs (n .s ,u .a , f , ~ + t . 19351 pp. trp-95.

3. P r a n d t l , La: The Mechanics of Viscous P l u i d s , Vole 1x1, d i v e G9 of Aerody,namic Theory, W , 3. h r e n d , ed., J u l i u s s p r i n g e r ,CBerl in) , 1935.

L

T a y l o r , &, I , : The Trarisport of VorOEaiCy an& Beat t h r o u g h a F l u i d i n ' T u r b u l e n t Motibn, P r o e * 'Roy, Soel, Zondon, s e r , 8 , v o l e 135, 1932, pp, 685-702

Ohou, R, F o g On a n l x t e n e i a n 00 ~ e ~ g 1 d s ' Method of B ind ing Apparent S t r e s s and t h e X a t u r a of Turbulence , Chinese Your. of P h y s i c s , v o l e .4, J a n , 1940;-

von Xbrmtin, Th,: The Fundam@nts ls of t h e S t a t i s t i c a l Theory of Turbulencee J o u r e Aeron, Sci . , v o l , 4 , no, 4, Feb, 1937, pp, l j l - l ? g a

e .

d o l l m i e p , W ~ l t e r 8 Bereohnung ~ u r b u l f f n t er Ausbre i tungs- vorgnnge. Z . f e a e M * M y . ~ o l r 6, Deo. 1 9 ~ 6 ~ pp. 46g3 478, : ,

. ,.' . Tomot i k a , S, r A p p l i c a t i o n o f . t h e Modif ied Vor.tPci.t.y

Z r a n s p o r t Theory t o t h e Tur'$ulent Spread ing ' o P s: J e t of Airo Proc* Boye Soco London, v o l Q 1.65~. ',., Maroh 18 , 1938, pp, 65-72. : I .,

I

Howarth, La: Cpacerning t h e ~ e l c i j ' h ~ and fernpesa tu te D i s t r i b u t f o n s i n P l a n e and A x i d l y Symmetrical J e t s , Proc . Cambtridge P h i l . Soc.. v o l . 34. <1?38r

. r

Liepmann, Hans W. : I n v e s t i g a t i o n s on Larni'nar Boundary- Layer S t a b i l i t y and T ~ a n s i t i o n on Curved Boundar ies ; RAGA ACE 80, jH30, Aug. 1943, ( g i g , 20)

11. von KBrntin, Th,: Some Aspec t s of t h e Turbulence Problem, P ~ o c ~ of t h e F o u r t h I n t , Congr, f o r Appl, Mechanics, The Univ* P ~ e s s (Cambridge) 1935,

NACA F

ig. 1

NACA

Figure 2 . - The

Beating uni t mounted i n the main tunnel contraction,

Figure 3 . - View

looking upstEeam at mctor- propeller uni t from inside the psessrare box.

F % p r e 4,- J e t

o r i f i ce and traversing uni t from domst-ream end of j e t .

Figwe 5.- Close-up

of % ; F & c ~ , drxve mechanigsm md hot-wire carriage,

Figme 6.- The 4-wire

2-eugpllfier mite AUX- i l l a r y agaplif ier ie in a aepasate case, not in the pistme,

F i w e 7 . - A %-wire

uv-meter o r v-aeter mounted on the carsiwe The carriage takeE3 ordinary u-metere with no external lea

THREE- INCH HEATED J E T

A r i d D/kfribufions A/ony CeoScr L ~ n e

Va loc;ty (uncorrected) - (correcfed ) ------

T ~ * l p e m f ~ ( r e --

.. P i g e . 8 , S u- = m- v&* on ax#; t & = ** .* et mouth Z - cnrn kmpemkrre on 6 T a .' ** *t mouth

Figure 8 . - ox,b/ d;s.tonc= ,ir dlhmcfem a (1 block = 13 divioione on 1/40H ~ngr. scale. Hold on slight angle.)

THREE -/NCH HEATED JET ~ a t e d ' ~ r o v e r s e 4f /y = 0 DiOmeters

Figure 9. radius: r - cm. I l .h lnnlr - In A 4 . r i o i n n o nn 1 1 1 2 H Arrh ern1 e)

I A C A Figs. 10,ll

THREE-INCH NEA TED JE T La*-/ Foreme et X = / Diameter

V e h i Z y .- -

md;ist rd em.

(1 block = 10/32")

THREE-/NCH HEATED JET Lo tero/ Tmverse ot X = 2 Diameters

ZT -meon veloc;

u- = I*

9' on ww&

u i. a ; o I <h.htctuojjon re/eci+ rn TI mean trmpershre

z P 8 1 or o n a u k

*dies : F d cm Figure 11.

NACA

THREE - /NC/ i HEATED JET L o t e m / Tjeverse o f X* 4 D h m e k r ~

r ip . ia,rs U m e w ra/.ci/y Urn n - 8 . a d I2 . w;.d ~/"c& vo/.ci& us = Is T = mu, t6mpmtUrr & b " m PUS

Figure 12.

THREE- INCH HEATED JET f.oferu/ Truverse c f X a 6 D/amr+ers

Figure 13. radl#s: r -.

THREE-INCH HEATED JET Lo tern l Troveme at . x a 8 Diameters

Figure 14. mdrus: rm em.

(1 block = 10/32")

THREE-JNW HEATED J E T L ~ j e r a / Traverse at X = / O Diome f e r s

u = mew ve/a~.v a= " " on-

Figure 15. mdrits . P d on

U P vJ4pY FigS. l6 8 l''

THQEE-/NCH HEATED JET K 5 * o w e x i s u e x / * / i? /~ / .u&on I Y C / O U ' ~ 1 a tend Traverse of X = /2 Diomelhet-s. ( I 1 = ?B

Figure 16.

Figure 17. (1 block = 10/4UN")

NACA

ONE-/NcU HEA TED J E T L ~ f e r a f Jraverse Q? X = 5 D ~ o m e f e r s

Figs. 18,19

Figure 18 p- cm. (1 h l 3 c k = 19/37")

ONE-/NCH HEATED JET L a t eroC G-averse ot X = I0 D/bmeters

Figure 19. p - cm.

r- cm. Fi11;11re 20. (1 block = 10/32*)

ONE-INCH. HEATED JET Loferal Troverse a t X = 30 Diameters

ONE- /NCH HEATED JET L o t e r o l T-verse at X -- 4 0 Diamet-ers

Figs. 22,23

OM-/NCH HEATED JET D/>fribuf/bns of Turbulent Velocifies

ONE-/NCH HEATED JET CompoT"r~ o fc lx io / o o d Rodis/ Turbu/encr

2 = 2 0 4%.

I p - cm.

Figure 24.

(1 block = 10/3aV')

NA

CA

Figure 29. (1 block

10/3zn)

( a ) P = 0 (on axis)

( c ) r = 7 . 5 cm= 1.79 r,

(d) r = 9.5 o m = 3-26 ro

(e) P = 14*5 cm = 3,45 r,

F i p e 27.- One-inch heated je t , x = 20 diaaneters.

( a ) r = 0 (on axis )

(b) s = 'Jro27 e m = d/B

( c ) 'Phi= wave? 200 c.pa ( for figs, a7 and 28)

Figure 88.- Qne-iacb bnea%ed j e t , x = 2 d8meterbs.

NACA

Figure 29. (1 block 10/3zn)

(1 black = 10132")

ONE-/NCIY HEATED JET Lotem/ 72i;mperofurcr Dljfr/>u tton Compomd 4 t h TA~oF~

X 40 die.

* (1 block = 131b2'')

NACA

ONE-/NCH HEATED JET Speed of Man i/040city and Z m p e r e f ~ r e

Figs. 36,37

ONE-INCH HEATED JET

0 5 10 20 25 SO 35 90

Figure 37. 24 ( 1 b l x k = 1@3/40 ' ' ~ )

Pigs. 38,39

WE-//VCH HEATED JET

Figs. 40,41

Figure 40.

Figs. 42,43

GALCIT HOT- WIRE VIBRATOR Zst of Hot-wire Revonre to h Irurbu/enre Levels

8

0: !o I L !o L 1 !o I 9L L /lo L I l!O L I Figure 43 .