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150 J O U R N A L O F T H E A E R O N A U T I C A L S C I E N C E S F E B R U A R Y , 1 9 5 7

is wel l known tha t , for two-dimen sional , tur bul ent m ixing between a je t and a dead region, the mixing zone grows l inear ly wi tha d ivergence angle of abou t 15 ( th is should be tak en only as a ru leof thu m b) . Now , if we ca lcula te the f ree-s t reaml ine pa t te rn fora p la te a i r fo i l wi th separa t ion , we f ind the dead-water region tobe rough ly wedge shaped wi th i t s edge a t the point of sepa ra t ion .I f the wedge angle i s grea ter than the angle of turbulent mixing,sepa ra t ion wi l l be unaffec ted . I f i t i s less , pres um ably the mixingwi l l kni t the f ree-s t ream f low back to the sur face . An an alyt ica lsolution, estimating the losses on an airfoil with frictionless surface but wi th th is k ind of separa t ion and rea t tachment , would bemo st welcome. This would , und oub tedly , be d i ff icul t to obta in .However , in the case of the channel , even the crudes t of inves t i ga t ions y ie lds much useful informat ion.

Tak i ng accoun t of t he above - men t i oned phenom enon o f r e a t ta chm ent , le t us re tu rn again to the d if fuser. Le t us say th a ts epa r a t i on ha s t aken p l ace and t ha t t he dead - wa t e r r eg i on s a t i s f ies the condi t ion of con s tan t pressu re . Some dis tance pas t thesepara t ion poin t the s t ream l ines wi l l have become para l le l . Th eangle bounding the dead region wi l l be the angle of d ivergence ofth e diffuser (see Fig . 2).

I f th is angle i s grea ter than the angle of mixing, rea t tachmentwill no t tak e plac e. If i t is less, sep ara tion will no t tak e place.Whether or not th is las t conclus ion i s t rue , the angle of separa t ion undoubtedly inf luences the behavior of the boundary layer ;and , whe r e s epa r a t i on ha s no t t aken p l ace , t he virtual angle ofseparation offers to be a facto r of con side rabl e influence. Byvirtual angle of separation i s meant the angle a t which the separa ted f low would leave the sur face if separa t ion took place a t agiven poin t . Any explanat io n of the behavior of the bo un dar ylayer must be in te rms of fac tors wi th in the boundary layer or int he i mm ed i a t e^ ad j acen t po r t i ons of t he f ree s t r e am. Conf in i ngsurfaces some dis tance away cannot in themselves inf luence thebou nda r y laye r . The y do , howeve r , de t e r mi ne t he v i r t ua l ang l eof separa t ion which i s a fac tor wi th in the boundary layer itself.This angle i s a point funct ion , var iable a long the sur face . I t sinf luence should be of impor tance even af ter ac tual separa t ionhas occurred , especia l ly when the d imensions of the separa tedregion are smal l .From these observat ions i t appears tha t , in any s tudy of theboundary layer in a channel , one should take cognizance of th isangle as wel l as the sur face veloci ty d is t r ibut ion and aspectra t io ( th is las t fac tor influences secon dary f low) . Conv erse ly ,i f two surfaces have the same pat tern of ve loci ty var ia t ion andof v i r tua l -separa t ion-angle var ia t ion , one would expect s imi larbounda r y - l aye r behav i o r , i r r e s pec t i ve o f t he ac t ua l geome t r y .This las t s ta tement can be taken as a speci f ica t ion of the condi t ions for quas i -s imi l i tude . Once i t becomes es tabl i shed tha t th isangle does truly account for the influence of confining walls, i tshould be poss ib le to genera l ize exper imenta l measurementsf rom a few typica l channels to a var ie ty of o the rs . For exa mple ,i t should be possible to predict , confidently, the behavior of amixed-f low impel ler f rom s tudies of a s ta t ionary channel modeledupon t he f ac to r s men t i oned abo ve .

Two reservat ions should enter in to the evaluat ion of theideas presente d abov e. Fi rs t , the val id i ty of v i r tu a l separa t io nin expla in ing boundary- layer behavior need not necessar i ly depend upon the mech anism s proposed here . Secondly , when oneat tempts to compute the angle of separa t ion , he may f ind ,say , tha t a t the separa t io n poin t i t i s in i t ia l ly zero , gradual lygrow ing to a finite valu e. In an effort to byp ass such local effects,the author has had some success in res t r ic t ing the analys is toovera l l mass movements of the f lu id ra ther than to local condi t ions . For example , in deal ing wi th the veloci ty reco very on thesurface of a vane in a turning cascade , the width of the separa tedzone in the plane of trail ing edges and the distance of this l inef r om t he s epa r a t i on po i n t we r e u s ed t o compu t e t he ang l e .Th i s p r ocedu r e r educes t he v i r t ua l - s epa r a t i on concep t t o no t h i ngmor e tha n a yard s t ick for comp ar ison purposes . As such, however , i t s t i l l promises to be useful in the s tudy of nar row turbo-mach i ne pa s s ages .

One cannot but fee l tha t there i s some more exact conceptly ing behin d th a t which has been d iscussed here . Now tha t theneed ha s been made ap pa r en t i t may s oon be f o r t hcomi ng .

-+ -L o c a t i o n of M a c h D is c s a n d D i a m o n d sS u p e r s o n i c A i r J e t s f in

D o n al d E. W i l c o x / A l e x a n d e r W e i r , J r . / * J . A . N i c h o l l s , * * * an dR o s e r D u n l a p * * * *

Aircraft Propulsion Laboratory, University of Michigan ,An n Arbor, Mich.

Sep tember 17 , 1956SYMBOLS

A t = t h r o a t a r e a of n o z z l eA e e x i t a r e a o f n o z z l eD d d i a m e t e r o f M a c h d i s cD e = e x i t d i a m e t e r o f n o z z l eM ae = e x i t M a c h N u m b e r c o m p u t e d f ro m E q . ( !)P a = a t m o s p h e r i c p r e s s u reP e = e x h a u s t p r e s s u r eP o = u p s t r e a m s t a g n a t i o n p r e s s u r ex = d i s t a n c e f r o m n o z z l e e x i t t o o b l i q u e s h o c k - w a v e i n t e r s e c t i o n p o i n t

or d i s t a n c e f r o m n o z z l e e x i t t o M a c h d i s c lo c a t i o n7 = r a t i o o f h e a t c a p a c i t i e s , a s s u m e d t o b e 1 . 4

INTRODUCTION

P T ^ H E COMPLICATED shock and rarefac t ion pat terns exis t ing in-- supersonic je t s have long been of fundam enta l in teres t in

the s tud y of fluid dyn amic s . I t i s wel l know n tha t an over-expanded nozzle (exi t pressure less than receiver pressure) leadsto obl ique shock waves a t the exi t of the nozzle which in tersec tand give the fami l iar "d ia mo nd " pat ter n . I f the degree of over-expans ion i s grea t enough, th is pa t te rn i s modif ied so as to te r minate the obl ique shocks wi th a normal shockor the so-called M ac h disc config uration . On th e oth er han d, a sufficientlyunderexpanded nozzle wi l l lead to a s imi lar Mach disc because ofth e focus ing of compress ion wave s f rom the je t bou nd ary . Atext reme degrees of underexpans ion, the obl ique shocks and the i rref lec t ion f rom the Mach disc are markedly curved, and theconf igura t ion i s appr opr ia te ly term ed a shock bot t le .

N u m e r o u s i n v e s t i g a t o r s 1 - 4 have devoted ef for t to exper i men t a l and ana l y t i c a l s t ud i e s of t he s e phenom ena . Howeve r ,most of these inves t iga t ions have been res t r ic ted to sonic nozzlesand re la t ive ly low-pressure ra t ios ( s tagnat ion pressure to rece iver pressure ) . Kelbe r and Jarv is 5 have u t i l i z ed t he me t hodof character i s t ics to analyze the f low f rom a nozzle opera t ing

f T h i s r e s e a r c h w a s s p o n s o r e d b y t h e O f fi ce of O r d n a n c e R e s e a r c h , U . S .A r m y . D i s c u s s i o n s of t h i s e x p e r i m e n t a l p r o g r a m w i t h P r o f s . T . C . A d a m -son, R . B . M o r r i s o n , a n d H . E . B a i l e y o f t h e A e r o n a u t i c a l E n g i n e e r i n gD e p a r t m e n t a r e g r a te f u l ly a ck n o w l e d g e d .

* R e s e a r c h A s s i s t a n t , E n g i n e e r i n g R e s e a r c h I n s t i t u t e .* * L e c t u r e r in C h e m i c a l E n g i n e e r i n g a n d A s s o c i a t e R e s e a r c h E n g i n e e r ,

E n g i n e e r i n g R e s e a r c h I n s t i t u t e .* * * I n s t r u c t o r in A e r o n a u t i c a l E n g i n e e r i n g .* * # * A s s i s t a n t in R e s e a r c h , E n g i n e e r i n g R e s e a r c h I n s t i t u t e .

Wm^ My--p it tqgfaotavL apr""' 1

f f tOCK iOTTLf

. . . , * I *ACH DISC

F I G . 1. Schl ieren pho tog raph of supersonic a i r je t .

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R E A D E R S ' F O R U M

under ex tremely h igh pressures . Such a techn i que fa i ls to ind ica te the p resence o f a Mach d isc so tha t the phenomenolog ica lconsid eratio ns for locat ing th e disc are in question. Weir6- 7 h a sreported in form ation on the va r ia t ion of d isc loca t ion with p res sure ra t io fo r a i r f low th rough two-d imens iona l s l i ts .

The in f luen t ia l pa ramete rs in de te rmin ing the Mach d isc location in a supersonic jet would appear to be the ratio of drivingpressure to rece iver p ressure , the nozz le ex i t Mach Number, andthe nozz le d ivergence ang le . In form ation of th is typ e is meagerand l imited in the range of va r ia t i on . Accord ing ly , the in i t ia lport ion of th is resea rch con trac t was devoted to a sys tematicexpe rim ental investigat ion of the effect of thes e vari able s . The seresu l ts have been used as an a id to the theore t ica l s tudy whichis s till in progre ss . Thi s pap er is dev oted solely to the exp erimenta l resu l ts .

E X P E R I M E N T A L A P P A R A T U S AN D P R O C E D U R EAir was obtained from a 2,000-psia s torage system with a total

volum e of 170 cu.ft. Th e air used thus had a very low absolu te

P / p 41 .7o a

P / P ^ s IT S

hum id i ty owing to the h igh s to rage p ressure . The a i r leav ingthe s to rage sys tem passed th rough two dome-contro l va lves inpara l le l , th rough the te s t nozz le , and then d ischarged in to thea tmo sphere . Th e pressure ups t ream of the te s t nozz le wasmain ta ined cons tan t by the dome-contro l va lves , which wereopera ted by ad jus t ing n i t rogen pressure above the d iaphragm.Any pressure desired from 0 to 600 psig could be maintainedby th is sys tem.

In o rder to inves t iga te quan t i ta t ive ly the in f luence o f nozz legeometry , four ax ia l ly symmetr ic nozz les with rounded in le ts ,throat diameters of 0.250 in. , and conical diverging angles of 10,15 , 20, and 30 (total angle) were designed to be connected tothe 1 -in. p ipe down s tream of the con tro l va lves . The M achNumber a t the ex i t was 3 .5 in a l l cases , a s computed by theisentropic flow relation

-| (7 + l ) / 2 ( 7 - D^ = M a [ IT (T + l ) / 2[ ( 7 - D / 2 ] M i, (1)Schlieren photographs were obtained of the flow issuing from

these nozz les a t ups tream s tagna t ion pressures f rom 55 to 615psia in 50-psi incre men ts . A conv entio nal schlieren syste m witha 6-in. field of view was used, the pictures being recorded on 8-by 10-in. film.

In o rder to inves t iga te the in f luence o f the ex i t Mach Numberon the flow, th e abov e-m entio ned nozzles were modified. Aftereach series of runs at one Mach Number, the nozzle was shortened un t i l the ex i t d iamete r agreed with tha t g iven by the a reara t io r e q u i re d for th e n e x t Ma c h N u mb e r . T h e Ma c h N u m b e rstest ed w ere 3.5, 3.0, 2.5, and 2.0.

Moe ^ 2 , 0

m%^23

^^^^^^ M aB~Z,0

m^^gmi mmm__F I G . 2 . Effec t of p ressure ra t i o on f low pa t te r n . Ex i t M achNu mb er = 2 .0 . D ivergence ang le = 10 . F I GMa e - 3,5

3 . Effec t o f ex i t Mach Number on f low pa t te rn ,r a t io = 4 1 . 7 . D iv e rg e n ce a n g le = 1 0 . P re s s u re