Base pressure at subsonic speeds in the presence of a supersonic jet

VLi:GTUIC20UV/iCUNDE Michiel 6e Ruyleweg 10 - DELH

"' okim

KluyvenA'eg 1 - , 3 DELFT

THE COLLEGE OF A E R O N A U T I C S

CRANFIELD

BASE PRESSURE AT SUBSONIC SPEEDS IN

THE PRESENCE OF A SUPERSONIC JET

by

T H E C O I, L JH G_ E . . __0 F_^_ . A - E _R , 0 K A U Ï I 0 _S O R A IT F I E L D

Base P r e s s u r e a t Subsonic Speeds i n t h e P r e s e n c e of a S u p e r s o n i c J e t

b y

-A. H, C r a v e n , M . S c , P h . D . , D.G.Ae.

SmiL/IARY

T h i s p a p e r p r e s e n t s t h e r e s u l t s of an o x p e r i n i e n t a l i n v e s t i g a t i o n i n t o t h e e f f e c t of s u p e r s o n i c j e t a upon t h e b a s e x^ressure of a blixff c y l i n d e r i n a u n i f o m s u b s o n i c f l o w . The r a t i o of j e t d i a o e t e r t o b a s e diame t e r v/as 0 . 1 8 7 5 .

J e t s t a g n a t i o n p r e s s u r e s g i v i n g s l i g i i t u n d e r - e x p a n s i o n of t h e j e t c a u s e an i n c r e a s e i n t h e b a s e p r e s s u r e b u t f o r l a r g o r j e t s t a g n a t i o n p r e s s u r e s t h e b a s e p r e s s u r e i s a g a i n r e d u c e d .

A siniple t h e o r y , b a s e d on a momentum i n t e g r a l , sho\TO t h e dependence of t h e b a s e d r a g upon t h e j e t and f r e e s t r e a m s p e e d s and upon t h e dir.iensions of t h e j e t and t h e b a s e .

The a u t h o r v/ishes t o aclanov/lcdge tlio perriiission g i v e n by tlie CoriïïAondant of tlie Royal A i r Force Tecüraiioal C o l l e g e t o undertaJ:e a t t h e C o l l e g e of A e r o n a u t i c s , G r a n f i e l d , t h e s t u d y r e p o r t e d i n t h i s p a p e r ,

Summary

L i s t of Symbols

I n t r o d u c t i o n 1 The dependence of the base drag c o e f f i c i e n t

on j e t conditions 2

Apparatus 8

The scope of the Tests 9

Test procedure 9

Results 10 Discussion 12 Conclusions 15 References 16 Appendix - A potential flow model for 17

the flow in the vicinity of a bluff body

C_^ b a s e d r a g c o e f f i c i e n t

T3 ^ p u'' ir{R3 - R i )

CO CO x5 d

uL, modified base drag coefficient defined in eqn. 15 C T jet momentum coefficient mVv/A- p u ^ S

«J 17 CO CO

Q

PL base pressure coefficient

d radius of mixing region in the plane of the velocity traverse

J jet stagnation pressure parameter "~ "•=- '^^=- • • • —

( , . ^ „ ^ ) =^ry-i)

D

1 distance of the plane of the velocity traverse from the base m rate of jet mass flow

M T nozzle design Mach Number P^ jet stagnation pressure p static pressure

pu base pressure

p^ jet static pressure at nozzle exit r radial distance from jet centre line r ' radius of a vortex ring

r r/r'

R^ base radius R_ jet radius

S base area '"'0^ " '^j ) u streanii-rfLse velociiy

u_ jet velocity at exit

V velocity normal to free stream direction

V, equivalent jet velocity (jet velocity attained in isentropic expeinsion from jet stagnation pressure to fi-^DO stream

x ' distance of vortex ring downsti-eara of base

P density

P _ jet density at nozzle exit d

r vortex strength

n

r/d

ê x/1

Si^f icoj

00 undist\irbed free streaia

1 free stream in plane of the base

2 free stream in plane of the velociiy traverse D at nozzle design conditions

1 . I n t r q d u c t i o n

I n the p a s t the majo]rity of vrork on base pressures has concentrated iipon tlie base i n simersonic flow. A comprehensive bibliography has been pi^pared by T/ilson (Ref. 1 ) , Very l i t t l e infornation i s a v a i l a b l e on base drag i n subsonic flow. The present author has considered tlie e f f e c t on the base drag of a subsonic j e t i n the choked and ttnchoked condition i s s u i n g from the base (Ref. 2^. The e f f e c t of j e t deflection (Ref. 3)

and of body incidence (Ref, 1+) liave also been studied, both v/ith a subsonic j e t i s s u i n g from the b a s e . I n each case the base diameter has been large conipared v/ith the j e t diameter.

I t has been sliown (Ref. 2) t h a t , for a subsonic j e t , the presence of a bubble ( o r a volxime of r e c i r c u l a t i n g f3.uid) extending from the base to some four body diameters dov/nstream, and considerable regions of reversed f l a T , cause s u b s t a n t i a l reductions i n the base pressure and increases i n the base drag. These e f f e c t s increase i n magnitude as the j e t stagnation pressure i n increased.

The p r e s e n t paper p r e s e n t s the r e s u l t s of an experimentail i n v e s t i g a t i o n i n t o the e f f e c t of siipersonic j e t s upon the base pressure on a bluff

cylinder i n uniform subsonic flo\7. The geometry of tlie b a s e , j e t diameter/ body diameter = 0,1875» i s not intended to be r e p r e s e n t a t i v e of current

a i r c r a f t p r a c t i c e , although i t i s f a i r l y r e p r e s e n t a t i v e of the base geanetiy of unguided r o c k e t s . The main xxjason for s e l e c t i n g t h i s geometry was a desire t o esqplore the flow i n the v i c i n i t y of the base and tlierefore the consequent advantages of using the above dimensions are obvious,

A simple theory, based on a momentum i n t e g r a l , i s derived v/hich shows the dependence of the base drag on the j e t and free stixjajn speeds and upon the dimensions of the j e t and body. I n order t o confirm the flow p a t t e r n downstiream of the base a simple p o t e n t i a l flov/ model has been talcen c o n s i s t i n g of a t o r o i d a l vortex to represent the bubble and a l i n e d i s t r i b u t i o n of sinks along the j e t c e n t r e l i n e to represent the e n t r a i n -ment e f f e c t of the j e t . I t i s shown t h a t t h i s model y i e l d s a pressiJre d i s t r i b u t i o n on the base sii'nilar to t h a t obtained by esqxiriment.

The author v/ishes t o express h i s g r a t i t u d e to Mr, G. M. L i l l e y f o r h i s help and encouragement throughout the study, to Dr. R, Fo Sargent of the B r i s t o l - S i d d e l e y Aero-engine Company for providing the nozzle o r d i n a t e s , t o Mr, S, H. L i l l e y for the design and e r e c t i o n of the experimental

equipment, t o Mr, H. Stanton for making the nozzles and otliei- equipment and t o the laborator^"- a s s i s t a n t s of the Aerodynamics Department vdio were l a r g e l y responsible for talcing the esqperimental measurements.

2 . Thq^ .A?J9Q.^4?."^-°-^--i^J^-J<.-^.'j-J'-ë:^-Q, J ^ ^ - ° S ' ^ . I ! ^ 4 Q A S I J : " * L - Q I I A ^ i , ^ c o n d i t i o n s

Consider the flow i n t o and out of the element ADGEEP ( P i g , 1 ) . BC coincides vidth the base and EP i s s u f f i c i e n t l y f a r dOTmstreai:i for the element to include corapletely the bubble and any reversed flOT/.

Let the j o t radius (AB) = R^ the base radius (AC) = Rp. and the radiiis of the

mixing region ( P E ) = d a t a distance 1 dov/nstream of the b a s e ,

We v/rito 6 to represent the boundary l a y e r thickness on tlie body a t C. 2«1 • The_ "basic__eqmtiCTis

The equation of conservation of mass flov/ applied to tlie element y i e l d s

2'7rr p^ u^ dr + / 27rr pu dr = / 27rr pu dr + 27rd / pv dx

AB CD PE DE (1)

and from the conservation of momentum

/ 2 ï r r (p^. + Pj u?.)dr + / 2 7rr p^ dr + / 2 ï r r (p^ + pu^)dr AB BG CD 2 7rr ( p + p u ^ ) d r + Zvö. j p v u d x EB DE ( 2 ) form . -0 ^ J ^ J ^ J + 1 1

I f v/e neglect the j e t boundary l a y e r thickness ( l ) can be 'iTritten in the

d ^ - ( R g - . 6 ) ^ ] p^u + 2 P ^ u j ( R 3 * y ) ^ ^ dy • • o • • \ ^ J vdiere 77 = r / ^ ; g = x/^ = 2 d^ pu 77 d/7 + 2 d 1 p u / - ^ d S • 1 1 • p u o 0 1 1

and (2) becomes

R^ ( p j + Pj u p + [ d M E B + S ) ' ] (p, + Pi u^) + 2 P^u^j (Rg + y) -^^dy + 2/ r Pg dr P,u, - 2 d^j (p + pu^) 77 dr? + 2 d l P^uJ / - ^ dC •' J o u o ^ 1 ^ ^ R. (4) 2 , 2 . The_bas£^^^ag

I f nov/ T/e defiriQ the base drag c o e f f i c i e n t C_ by B 2 ^ 1 (Poo - PB^ r dr R,

"^ = 1

^ ^co < ^ ' - ( ï ^ - ^ j ) (5)

v/hich does not include the pressure i n t e g r a t e d over the area of the j e t a t the nozzle e x i t , we can replace p^ i n (4) from (5) t o give

\ ^ P.^1^^ - ^J ) = ^J ( P j -^^J ^ j ' ) + Poo (P^ - Rj )

+ f d ^ - (R^ +6)^1 (p^+ p^ u p + 2 P^ u ^ f (Rp+y) ' ^ ^ d y

- 2 # / (p + ptO ?7<3-7? - 2 d l p ^ u f / ^ ^ d S (6)

o

PA

and, eliminating R^ P ^ u^ betv/een (3) and ( 6 ) , v/e have «J J J

+ Peo (R^ - Rj) + 2 / (Rg. + y) Pu(u - Uj)dy o

+ 2d^ / (pu Uj - Pu^- p) T7dT? - 2 d l /pv(u - Uj)d^

• 0 o

Now v/ith constant s t a t i c pressure along FE

p 77 d?7 = 2 "

and (7) niay be rev/ritten i n the form

d=^(p,-pj

1

_ R J ( P T - P C O ) Poo - P / ^ + ^ / R B ) ' -^ P u" P u^' 1 1 u , + 2 -^-^- fl -^ P0.4. \ "^ - ( I ^ . ö ) ^ 4 R ^

4-

R; 2 ^2 'J R 3 - R 1 + y U T \ u_ _ J | Pu .n Pu ^ J o ^

ex

u. ^ - ^ ) d ^ P u \ u u "^CO CO \ CO CO ( 8 )

I t i s reasonable to take the speeds u^, u^ and the pressure p^ » Pg , Po outside the mixing region equal r e s p e c t i v e l y . Tlixis (8) reduces t o

PT •" P • ^ J -^co (^jv = ^ J ^co / _ _ ! _ - « „ i Pco ^ H/R^ " "" + 2 1 -U -

a' - " '

S^

V V E ^ - R J V

'

i^'

R: P U P.. U . 1 -_u_ u^ T7dr} 4 d l 2 2

i Tx

u / u CO J c O u _{d ?} (9) vdiere the boundary l a y e r thickness 6 has been assumed ver;^' small coirpared with Rp,.

With the same approximations for the speeds and pressxxrcs outside the lïiixing region, equation (3) can be v/ritten

^J "j ^J 2 2 d - R 3 ^-^P)P^ R^-Rj R^-^R. o 2 * 2 _J Pco-Uco p 2 2 R 3 - R . o

^ ds

Uoo

NOT/ the l a s t term of (9) i s

dl_ R^

^

-^co \ ^00 " 11c dS

(10)

and along DE (v/hich is outside the mixing region) u = u ^ , Thus v_ 2 1,2 J U J O R ^ - R u. _{dl /^J} - i - ^ I d? = - 1 = 1 ~ dg(ll) J o and, substituting from (10) and (11) into (9) , v/e obtain

\

4-Pu ^2 J P„ u„ J o R ~ •• R'' •' '00 "CO 1 - — ) T7 d r? ( 1 2 ) I f v/e define a t h r u s t c o e f f i c i e n t G by °T = J e t t h r u s t 2 / - 2 2 P o o < ' ^ ( ^ - ^ j ) then P j -• Pco + P J ^ J 1 2 2 P U CO CO R: 2 2 ^ •" ^ j (13) and (12) can be w r i t t e n

^^~- "^' ^ - ^ l

P u Pon U . 1 « ^ j n d n (14) f o r u^ » u^ . 2 , 3 . The c a s e when R^ *^*^ ^ ^^^ '^j •*•* '^c I f , i n e q u a t i o n ( 1 2 ) , we v / r i t e \ B 1 2 2 ''co 1i<„ R: ^ - R. (15)

we h a v e , s i n c e d = Rp. + 5 and 6 i s s m a l l compared v/ith R^,

^ - ' ^

PJ3

h

p u^ Rj _ co co -IJ

,_l,.

P U p u V u oo co V ~ U - 1 d77 Thus ^<^%. , ^z P T UT-^ UT-^ J S UT-^D ^ J ^ J B ^ 1 _ P^ u^ , ^D -^D ° u 1 -^ . D " j / " -R j PU u , D

^j

'j "i

u d n (16)

wheire ^J-^ and J_^ a r e t h e d e n s i t y and v e l o c i t y a t tlie n o z z l e e x i t u n d e r

design conditions, NOT; P _ U^ ^D ^D D T c ^/ V ^yst / v R T M. D +1

1 + ^ M l

2 -"J. D y+l 2 ( y ^ ) and

^3

P , T £ ^D "^D 'D (17) (18) \Then t h e e x i t v e l o c i t y i s s i i p e r s o n i c f o r , t h e n , M = IvL.

Thus, provided the Mach number of the j e t a t e x i t i s supersonic, v/e have by s u b s t i t u t i n g ( l 7 ) and (18) i n t o (16) M, Tlco \

21

Poo D /+1

, . ^ M J J ^ - ^ > i

r:

^ -Uco\ J-_D ^ / ^ 0 _R, 3 2 pu _{^ .} D Pj^J ^ - 1 u „ (19) dT?

Prom velocity traverses across the mixing region it has been found that for any partic;alar jet and streaja conditions the integrated second term in (19) is roughly independent of the position of the traverse (provided the position of the traverse is in the established wake) and tliat it is of the same order of magnitude as the first (constant) teim. Thus we dedioce that the integrand is nearly independent of u,/ but is a function of the jet

«J/11 CO conditions u.

Vu,

and P J/P. D D Hence u _\ ^J y+i_ (20)

NOT7 the jet static pressure at exit is not markedly different from the free

stream static pressure and R^ « R^. Hence 0_. v/ill be of the same order as Gj., Thus, defining a jet stagnation jjressure parameter J by

J = .^ . ->->^ P .. . , (21)

,,Yf^^y^'^)

we have for large jet velocities

u^ C = f (J) (22)

I t i s shOTvn i n P i g , 3 t h a t the r e s u l t s f o r the base drag c o e f f i c i e n t s , found i n the experiments described i n l a t e r s e c t i o n s of t h i s p a p e r , f o r d i f f e r e n t j e t d e a i g i Mach numbers collapse on the b a s i s of u ^ C_^ p l o t t e d a g a i n s t J f o r supersonic e x i t v e l o c i t i e s ,

2,4. ^ e case v/hen R, « Rp and u^ < u^

Although accurate experimental results are not available for the case when the jet velocity is less tlian the free stream speed, it is of interest

to consider the case for it represents the problem of "base bleed" associated with a small central jet.

•tto

From equations 12 and 13 i f d = R^, R_ » R_ and

< 1

77 dT7

If Cm ~ 0 it is seen that, because the integral is always positive, the base drag is reduced when PT U-r > 0 and u < u^^^

3, Apparatus'

3*1 • The jmnd tunnel and instrumentation

The t e s t s v/ere performed i n a s t r a i g h t through wind tunnel having a closed working s e c t i o n measuring 3 f t , square. The compressed a i r supply f o r the j e t was l e d along the centre l i n e of the tunnel t o the v/orking s e c t i o n i n a 32" ^ « diameter pipe which v/as threaded a t i t s downstream end t o a t t a c h the models. The siipply pipe v/as encased i n a duralumin sleeve 4 i n , i n diameter, the space betv/een the sleeve and the supply pipe being occupied by the pressure t u b e s . The surface p r e s s vires from the model v/ere read from a raulti-tuhe water manometer.

3 , 2 . The_jnodels

The models used i n these t e s t s were each r i g h t cylinders 4 i n , i n diameter and 12 i n , long turned from l i g h t a l l o y . The i n t e r n a l cavity of each model was machined to give smooth i n t e r n a l flOTv i n t o a convergent-divergent nozzle ^ i n . e x i t diameter, each model having a nozzle designed to give pairallel flOTv a t the j e t e x i t a t i t s design Maoh number. The design Mach numbers of the n o z z l e s , allowing for a nominal boundary l a y e r c o r r e c t i o n , v/ere 1,0, 1 , 2 , 1 , 4 , 1 . 6 , 1 ,8 and 2 , 0 , An i n t e r n a l gauze screen was f i t t e d betv/een the model and the supply pipe to eliminate non-uniformities

i n the compressed a i r fIOT/ from the sttpply pipe i n t o the model* s pressure c a v i t y ,

Polythene tubing for pressiore measurements v/as i n s e r t e d i n s l o t s along the base r a d i i and the models generators a t angular i n t e r v a l s of 22-2-and secured v/ith a r a l d i t e ,

4. The Scope of the Tests

The tests on each of the models covered a range of free stream speeds from 50 to 100 ft/sec. The actual "design" Mach numbers l^L. of the nozzles

D

tested v/ere 1,0, 1.23, 1.41, 1 .60, 1,82 and 1,98. Defining the jet stagnation pressure parameter J as in equation 21, i.e,

^ ^J ^D

Pco / \ Z±l

(..v^^Jaly:.)

where P = j e t s t a g i a t i o n pressure p^ = free stream s t a t i c pressure M^. = j e t design Mach number,

^D

s t i f f i c i e n t j e t stagnation pressure v/as a v a i l a b l e t o enable J to be v a r i e d i n the range 0 to 4 . 0 .

The thickness of the t u r b u l e n t boundary l a y e r on the side of the body a t the base section i s 0,6" approximately from which we deduce t h a t the e f f e c t i v e length of the body i s 2.3 f t . Based on t h i s length and a

t'unnel speed of 100 f t / s e c , , the Reynolds number of these t e s t s v/as 1 ,5 x 10*, 5, Test procedujre

The ordinary pressia:^ p l o t t i n g techniques v/ere used in these t e s t s . '• " P r e s s u r e measurements vrere taken a t tunnel speeds of 50, 55> 60, 65, 70,

80, 90, 100 and 120 f t / s e c , for each of twelve j e t stagnation p r e s s u r e s , Total head and s t a t i c pressure t r a v e r s e s were made i n a diametral plane across the j e t and mixing region a t two, t h r e e , f o u r , s i x and e i g h t body diameters dOT/nstream of the b a s e . The t o t a l head lïioas'urements were made using a Conrad yawmeter f ran v/hich the flow d i r e c t i o n a t any s t a t i o n ' was determined. The s t a t i c tube v/as aligned i n t h i s d i r e c t i o n v/hen making

s t a t i c pressure measiorements. The t o t a l head and s t a t i c tubes were c a l i b r a t e d a t lov/ speeds; no f u r t h e r corrections being applied to tlie readings,.

6, Results

6,1 , Presentation .^_r;e suits

The analysis of section 2 shov/s that the product of base drag coefficient and free stream speed is dependent only on jet conditioiis for the supersonic jet, "ÏÏQ may infer that the product of the base pressijre coefficient and free stream speed also depends only on jet conditions and is independent of free stream speed. Thus the press\are distribution on the base is presented in

the form of graphs of C n^ against r/R for given values of the jet stagnation parameter J (Figs, 2 b - d) .

The base pressures have been i n t e g r a t e d over the base to detennine the base drag c o e f f i c i e n t C_^ r e f e r r e d t o base area f i . e , ^''(R^ - RT)~] and free stream speed. C_^ v^ i s p l o t t e d a g a i n s t j e t s t a g n a t i o n pressure J i n F i g , 3 .

The foregoing method of p r e s e n t a t i o n breaks dOT/n f o r l-cm values of j e t stagnation p r e s s u r e . I t i s knOT/n (Ref, z) tliat the base i^ressure and base drag c o e f f i c i e n t s i n subsonic flow v/ith no j e t are constant and independent of free stream speed. I t i s to be expected t h a t any Cp, v^^ curve v/ould

B

brealc up i n t o s e v e r a l branches (one f o r each tunnel speed) f o r values of J l e s s than J ^ . Hov/ever f o r such j e t s t a g n a t i o n pressi:ires i t i s knavn t h a t the flow from any of the nozzles i s subsonic and i t has been sliOTvn (Ref, 2) t h a t , for subsonic j e t flov/, the base pressTjre and base drag c o e f f i c i e n t s are properly presented i n terms of the j e t momentum c o e f f i c i e n t C^. defined by

d m V, C ^ J -1 2 _ è p u S •^ CD CO

vAiere m i s the r a t e of mass flOTV from the j e t and v, i s the equivalent j e t v e l o c i i y ( i . e , the v e l o c i t y which the j e t would a t t a i n i n i s e n t r o p i c

expansion from i t s stagnation x-^ressure to free stream s t a t i c pressi^re* • P i g s . 5 a. - f shOTV c l e a r l y t h i s dependence and the p o i n t a t v/hich t h i s p r e s e n t a t i o n a l s o breaks dOT/n, The r a d i a l pressvire d i s t r i b u t i o n for the subsonic j e t i s shOT/n i n F i g . 2a i n terms of G^.,

6.2, The base pressiJre ^distribution (Pig, 2)

For any given jet conditions the base pressure distribution follows the same general pattern. From tlie edge of the jet the base pressxjre falls v/ith incr-ease of radial position to a minimum at 0,7 Rp approximately, after v/hich it rises steadily to the circumference of the base. The position of the minimum base pressure moves outv/ards from 0.68 Rp at the low values of jet stagnation pressxare to 0,73 Rp at the hi^est available pressures. A variation from the general pattern of the pressiore distribution was noticed near the jet exit for values of J less than J,^, In tlie region from the jet exit ( /EJ_ = 0,1875) to some point close to r/El^ = 0,3 the base pressure rises slightly before conforming to the general reduction noted previously, Furthermore the position of maximum base pressure moves in\Tards as the jet

stagnation pressure is increased iip to J_^. IWien the nozzle is at its design condition the pressure variation in tlie region 0,1875 < ^/RQ "^ 0.3 is

negligible. j At any radial position the base pressure falls as tlie jet stagnation

pressure rises to its design value for any nozzle. As P^ increases further d

I n the range J_ < J < 3.0 the base pressure r i s e s only t o f a l l again f o r J > 3 . 0 ,

6 . 3 . ,The ba.se dra.g ( F i g . 5)

The base drag coefficient increases v/ith jet stagnation pressure for J loss than J,

'D' For J between J_, and 3.0 increase of jet stagnation pressure causes a substantial reduction of base drag coefficient; but for J greater than 3.0 the base drag coefficient again increases,

6.4.

d i s t r i b u t i o n i n the mixine

R e l i a b l e v e l o c i t y t r a v e r s e s were obtainable only a t d i s t a n c e s g r e a t e r than four body diameters dOT/nstream of the b a s e . I n a l l cases the v e l o c i t y

d i s t r i b u t i o n v/as of the form shown i n F i g . 1 . The d i s t r i b u t i o n s , vtfhen i n t e g r a t e d according t o equations 12 and 16 of paragraph 2 , gave valines of the base

drag c o e f f i c i e n t shown i n F i g . 4 .

Ejcanrples of the v e l o c i t y d i s t r i b u t i o n s ' ejte given i n P i g s , 11 and 12, The readings from which tliese d i s t r i b u t i o n s were o b t a i l e d were taken i n r e g i o n s v/here the j e t flOTv had become subsonic.

7» Msoiossion

7 . 1 . Acct3ra.cy of the r e s u l t s

The j e t stagnation pressure was maintained, by continual adjustment of the o o n t r o l v a l v e , to an accuracy b e t t e r tlian 2,dfo during any t e s t . The tunnel speed could be kept constant t o within ^% aaid the surface p r e s s u r e s measured t o an accuracy of 0,02 i n , of water. Hence the o v e r a l l e r r o r i n

the pressiore c o e f f i c i e n t s and i n the base drag c o e f f i c i e n t s i s considered t o be l e e s than 5^, This i s borne out by exandnation of P i g , 3 i n which maximum and minimum values are shov/n,

No account has been talcen of tunnel interference e f f e c t s . Any e r r o r s from t h i s cause are expected to be small since the j e t v/as aligned along the tunnel c e n t r e - l i n e and the tunnel speed was adjusted t o i t s prescribed value as the j e t stagnation pressure v/as a l t e r e d and before any pressure readings were talcen,

7 . 2 . _TIie flow ^in the base region

The flOTT i n the base region, for zero j e t v e l o c i t y , c o n s i s t s of a large volume of slowly r e c i r c u l a t i n g flow (bubble o r , i f i d e a l i s e d , a t o r o i d a l vortex) , Only a t some tliree to four body diameters dOT^/nstreara of the base ( P i g . 6a) i s the v/ake w e l l e s t a b l i s h e d . The form of tlie base pressxire d i s t r i b u t i o n s and the regions of very slow and reversed flow i n d i c a t e d by the v e l o c i t y t r a v e r s e s shw/ t l i a t , i n the presence of a j e t , the bubble s t i l l e x i s t s and from i t s i n s i d e edge flOTv i s entrained i n t o the j e t .

When the j e t stagnation pressure i s l e s s than the n o z z l e ' s design pressure the j e t flOTv i s subsonic everyv/here i n the nozzle or passes through a normal shock i n the divergent portion and i s subsonic a t e x i t . For the subsonic j e t the maximum base pressure occurs near r/Rp = 0,3 ( P i g , 2a) and i n d i c a t e s the existence of an attachment l i n e t h e r e . The a i r impinging tqpon t h i s l i n e comes from the boundary l a y e r on the body ( P i g , 6b) and thus has a stagnation pressure much l e s s than t h a t of the free stream. The f a l l i n g pressure g r a d i e n t inv/ards i s i n d i c a t i v e of flOTV tOT/ards the j e t along the i n n e r p o r t i o n of the base r a d i i f o r the IOTT j e t speed ( P i g . 6 b ) . As the j e t v e l o c i i y i s increased the bubble decreases i n length and consequently the base pressure becomes more n e g a t i v e , Entrainment i n t o the j e t increases with increase of the j e t v e l o c i i y . I n other words the vortex g e t s stronger v/ith increase i n j e t v e l o c i t y . I t i s also noted ( P i g , 2a) t h a t the attachment l i n e moves inwards tov/ards the edge of the j e t as the j e t speed i n c r e a s e s ,

These trends continue u n t i l tlie j e t reaches sonic v e l o c i t y ( f o r the

convergent nozzle) or i t s design Mach number (for convergent-divergent n o z z l e s ) , For t h i s condition the bubble i s sliortest and tlie attachment l i n e has moved t o the edge of tlie j e t ( F i g . 6c) , The flOTv on the inner portions of the base i s outv/ards and very SIOT/,

I n a s l i g h t l y under-expanded j e t ( i , e , the j e t stagnation pressiare s l a g h t l y g r e a t e r than tlie design value) the length of the psexxLo-laminar mixing region o r i g i n a t i n g a t the j e t e x i t i s g r e a t e r tlian t h a t f o r a j e t a t i t s drr.ign oondj.tion. This means t h a t the strong extrainment occvirs a t some dj-etance from tlie base ( F i g , 6 d ) , Hence the length of the bubble must increase causing the base pressure t o increase ( i , e , OU i s reduced s l i g h t l y ) ,

For a more highly under-expanded j e t the e x t r a expansion of the j e t as i t leaves the nozzle r e s u l t s i n a smaller bubble ( F i g , 6 e ) , The vortex s t r e n g t h must i n c r e a s e causing the base pressure t o be more n e g a t i v e . For large under-expansions the j e t displaoement e f f e c t tends t o f i l l up the region downstream of the base and f o r s t i l l l a r g e r j e t stagnation p r e s s u r e s , the j e t gives a ooiif)ressive e f f e c t on the e x t e m a l flow ( P i g , 6 f ) , Thus i t i s

conjectured t h a t the base pressure reaches a maximum suction for c e r t a i n j e t oonditions and then must increase v/ith f u r t h e r increase of j e t stagnation p r e s s u r e . This second maximum i n , and subsequent reduction of, base suction v/as outside the range of j e t stagnation pressure a v a i l a b l e i n the experiment,

The s t r u c t u r e of the axi-sjTnmetric r e c i r c u l a t i n g flOTV p o s t u l a t e d here has s i m i l a r p r o p e r t i e s to those of the two-dimensional laminar separation bubble described by Eurravs and Nev/man (Ref, 5) ,

7 . 3 . Tho_^base pre^sure_ and ti;g^ base drag,

Previously (Ref. 2) i t has been shOTvn t h a t , for subsonic j e t s , the base pressure d i s t r i b u t i o n and the base drag c o e f f i c i e n t are indepiendent of free

stream speed wiien p l o t t e d a g a i n s t the j e t momentum c o e f f i c i e n t C^ defined by

0

mv^

' * p u : s

variations vsdtli forward speed only being apparent for jet stagnation presstxres approaching that at which the nozzles choked. For a supersonic nozzle, 0

J is again the controlling parameter provided that the jet stagnation pressure is not sufficiently large to choke the nozzle (Pig. 5 ) , As the nozzle design pressi:ire is approached variations with, forward speed are again apparent

but, for jet stagnation pressxires exceeding twice the design pressure, the dependence of base drag coefficient on jet momentum coefficient is not affected by free stream speed. Comparison of Figs, 5a — f show that the

presentation of 0 against G_ still leaves a dependence upon the nozzle design Maoh number.

The theory of paragraph 2 shavs that, for a base diameter considerably larger than the jet diameter, the product of the base drag coefficient and free stream speed is dependent only iipon conditions in tho jet. As b^se drag is obtained by integration of the base pressure, v/e v/ould expect from the theory that the product of base pressure and free stream speed is also dependent only upon jet conditions.

Por j e t s t a g i a t i o n pressures approaching and exceeding the design pressiore i t i s shown i n P i g . 2 t h a t , v/hatever nozzle v/as used, the product

c i}^ i s dependent only upon the j e t stagnation pressure parameter J and not e x p l i c i t l y iipon the j e t stagnation pressure or the design Mach number of the n o z z l e . I n P i g , 3 the dependence of G^ x^ upon J i s shov/n. In t h i s figure the maximum and minimum values f o r G^ Uco are shw/n f o r the d i f f e r e n t nozzles a t various values of J and i t i s c l e a r t h a t the r e s u l t i n g curve i s independent of the j e t Mach number e x p l i c i t l y provided the j e t s t a g n a t i o n pressure i s g r e a t e r than tlie design v a l u e ,

The subsonic base drag f o r a base -v/ithout j e t i s a constant independent of free stream speed. I t i s therefore not siorprising t h a t there i s considerable s c a t t e r i n the values of CL u^^ f o r J l e s s than J.^^. The departure of the

B

experimental r e s u l t s from the t h e o r e t i c a l p r e d i c t i o n for IOT/ j e t stagnation pressures i s a t t r i b u t e d t o imperfect establishment of tho e n t r a i n e d flOTV dOT/nstream of the base for such j e t c o n d i t i o n s ,

For values of J g r e a t e r than J_. the j e t i s imder-expandcd and, as i t leaves the nozzle considerable expansion occurs. Pur"üior the mixing region near the j e t e x i t i s probably pseudo-laminar and s i g n i f i c a n t entrainment occurs only davnstream of the compression region i n the j e t . The e f f e c t of t h i s expansion on the base pressure has been discussed i n the previous s e c t i o n , I t i s stmimarised i n P i g , 7 ,

7 , 4 . Comparison between theory and es^eriment ( F i g . 4)

The base drag c a l c u l a t e d from tiie v e l o c i i y t r a v e r s e s and equation 12

of section 2 , shOTVs reasonable agreement vdth tliat obtained by d i r e c t i n t e g r a t i o n of the base pressure d i s t r i b u t i o n for the same j e t c o n d i t i o n s . The theory

can be s i m p l i f i e d i n the case when R-j. « Rp and u.^ » i:^^ giving a value of base drag which underestimates the experimental value only s l i g l i t l y . The

7»5. A potential flOTV model to re'present the mixing region

In an attempt to show that the flow in the mixing region postulated in previous reactions is consistent with the experimental results a simple

potential flow model has been considered. The model, which is described more cornpletely in the appendix to this paper, represents only the flow outside the jet itself. The recirculating flow is represented by a vortex ring of strength P and radius r' at a distance x dOT/nstream of tlie base. The entrainment effect of the jet is represented by a distribution of sinks along the jet centre line. By setting up the appropriate image system the base automatically becomes a streamline, and the radial velocity on the base can be calcrulated,

The pressure distributions calculated from the radial velocity show the same general trends as the experimental distributions but there ai.'e some wide differences in the magtitiudes of the pressure coefficients involved, The major source of error lies in tlie use of free stream static pressxjre and speed as reference conditions. Since the recirc\ilating and entrained flavrs

near tlie base oooo mainly from the boimdaiy layer of tlie body it is necessary to use as reference conditions some lavfer valtie of speed as a reference. In the appendix a speed equal to tlie average speed in the boundary layer v/as used v/ith some improvement to the correlation between experimental and theoretical valties (Pigs, 9 and 10),

The model fails i:i that it is necessary to use expcrinientally determined boiDidary conditions to determine the values of the vortex strengtii and position, A mnach more sophisticated model is necessary if one v/ishes to predict base

pressures by purely theoretical means. The model postulated here does not malce the radial velocii^y zero at tlie outer edge of the base. Even so the pressure predicted tliere is not seriously in error and the movement of the centre of the recirculating flOT7 and its increase in velociiy is also predicted to a certain extent, Tho results obtained from this model are sufficient to shOTV that the aotijial flOT/ in the base region is as described in previous sections of this paper,

8, Gonolusiqns

1 , When the supersonic nozzle is not choked, the base pressure falls (and base drag increases) v/ith increase of jet stagnation pressure,

2, For choked nozzles (designed for Mach numbers in the range from 1 to 2) an increase of jet stagnation pressure beyond tlie design pressure

(j_^ < J < 3»0) causes an increase in base pressure. Further increase of jet stagnation pressure (j > 3.0) again causes a reduction in base pressure.

3. The f IOT/ downstream of a large base consists of a toroidal bubble covering most of the base area. The flOTv entrained into the jet near üie nozzle exit comes from the edge of this recirculating region. The normal flow associated vdth jet mixing is established only some three to fovir body diameter downstream of the base. The size of the bubble is reduced with increase of jet stagnation pressure,

4» A simple momenium a n a l y s i s and an elementary p o t e n t i a l flow model give r e s u l t s c o n s i s t e n t vnLiii the p o s t u l a t e d f IOT/ i n the mixing r e g i o n ,

9 , References 1 . 2. 3. 4. 5,

6.

Wilson, C W . J . Craven, A.H. Graven, A.H, Graven, A,H, BurrOTTS, P . M . , Nev/man, B.G. Graven, A.H.

Base Pressure and Jet Effects on Afterbody Base and Adjacent Sui-fa^es.

R.A.E, Library Bibliography 196, February 1958, Interference of a rearward facing jet on the flow over three represenbative afterbody

shapes.

College of Aeronautics Note No, 60, April 1957. The e f f e c t of j e t deflection on tlie i n t e r f e r e n c e of a rearv/ard facing j e t , '

College of Aeronautics Note No. 70, October 1957. Effect of body incidence on t\vo afterbodies witli a rearward facing j e t ,

College of Aeronautics Note No.92, December 1958. The a p p l i c a t i o n of suction t o a two-dimensional laminar separation bubble,

M i s s i s s i p p i State University Department of Aerophysics Research Report 27, October 1959. The e f f e c t of density on j e t flow a t subsonic speeds.

In order to establish the effect, on the base presstire distribution, of the viscous and tvirbulent mi.xing processes in the separated flOTV region downstream of the base it is instructive to consider the corresponding potential flow model. Only the region outside the jet will be considered, The inflow into the jet is represented by a continuous distribution of sinks of strength q per imit length on the axis of the jet. The circulating flOTv in the base region is represented by a vortex ring of strength P and radius r' placed at x' downstream of the base (Pig. 8;. To satisfy the condition of no flow normal to the base an image vortex ring of strength "P is taken at -x^ and a continuous distribution of sinks is placed on the reflection of the jet centre line. The external flow is represented by a surface distribution of sources of strength p u r do per unit length in the place x = 0 and in the region Rp < r < co. This source distribution is assumed to give no flow radially in the plane x = 0. It should be noted that this model only attempts to represent the flow on and dOT/nstream of the base,

By virtue of the radial symmetry, the radial f IOT/ along the base x = 0 is given by v(r) = x^ r^ r 2 w zir cos e m [x^^ + r^ + r'^- 2 r r^cose/^^ ^"^ ° ( T T T ) ^ ^

7rr'J(|)^+(l + f ) ^ L

K(k) - / 1 + , f^ ffi(k)

V (^/xO^+(f/-i)V

23rr (A.I)

where K ( k ) , E(k) are respectively complete elliptic integrals of the first and second kind

4r/^

'X/ / \2 T/ , N 2 (Vr')%(l -.Vr')

and q has been taken constant, or writing x' /r' = x and r/r = r

/ V _ _ r _ X

wr' r T

|j?+(l + ? ) ' ' ]

wiiere k = 4 ,^-— ? + (1 + r)2

K(k) - r 1 + --^^^^»~ "j E(k)

l P+(r-l)^J

2 7 / r (A.2)

I t i s now necessary t o determine values for the vortex strength T and i t s p o s i t i o n ( x , r*') from bo-undaiy c o n d i t i o n s . We note tliat the base pressure i s a minimum a t T/R^ = 0,7 approximately from v/hich v/e dediice t h a t the r a d i u s r ' of the v o r t e x r i n g i s given by r'/Eip = 0 , 7 .

Case 1• The subsonic j e t

From the experimental r e s u l t s i t can be seen t h a t v = 0 when r/Eip = 0,3 ; i , o , r = 0,417 Rg. Thus (A,2) becones

r X

1.558 [x2+ 2,008]2

K(k) - 1 +

-M3k_

3?+ 0,340 E(k) 1,882 = 0 (A.3) with k = 1,668 x^ + 2,008

The pressiore c o e f f i c i e n t be.sed on free stream conditions and measiJred a t r = r shows t h a t y(r) ^ c o for U T / = 6 and ( A , 2 ) becomes r X 1

's2+ h. Y

witli k = -^—^

S^ + 4

K -= 1,25

) ' • [ ' ' ' + r-2lE(^) - 2 = ^•'^^^ "^"^ ^^-^^

We may approximate to the strength of the sink distribution rejxpesenting the jet by applying the resxolts of Ref, 6, Por ^ / „ = 6 we find

' 0 0

<1

where m- (=:wp_ XL. BI) i s the mass flow i n the j e t

i , e . q « - ^ Uj Rj (A,5)

Substituting from (A,5) into (A,4) and (A,3) and solving we find

X = 3.10

r = 44.4

and t h u s , f o r a subsonic j e t (u (A,2) may be v/ritten

Vu„

(A.6)

= 6 ) , the radial velocity distribution

62,6 [9.60 +(1 + r)''j K(k) « r 1 + 2 r 9.60 +{r -1)^ J

"lE(k)

0,025 Rg (A.7) and k =

A_?

9.60 + (1 + r) -\ 2

On the assumption that the pressure at the point r/Rp = 0,3 vdiere v = 0 is the stagnation pressure of the free stream, the radial pressure

distribution has been calculated for the case u,/ = 6 and is shOTvn as curve A-A in Pig. 9 and compared vdth an experimentally determined pressure distribution C-0. The main discrepancy lies near the attachment point v/here v/e have assumed that full free stream stagnation pressure is reached. HOTvever the flow attaching at r/Elp = 0,3 comes from the boundary layer on

the body and is thus at a much lower total pressure than free stream. If

now we talce the average speed in the boundaiy layer at the end of the body ; as the reference speed and the static pressure as measured at tlie end of the body as a reference pressure and calculate the radial pressure distribution we obtain the curve shown as B-B in Pig, 9,

Oase 2, The Sonic Jet

In tills case the attachment point has moved to the edge of the jet and ly take u / = 1 0 ,

' 00

sr'jik s t r e n g t h i s g i v e n b y

vre may t a k e u_ / = 1 0 , Prom Ref, 6 i t i s s e e n t h a t t h e a p p r o p r i a t e

"^ ~ 10 With t h e boundary c o n d i t i o n s ( i ) V = 0 vAien X/B^ = 0.1875 ( i i ) C = 1 , 0 ( i , e , ~ = 1,4) vAien r/ÏL. = 0 . 7 0 P U» D WB f i n d t h a t (A,2) y i e l d s X = 2 , 8 6

r =

48.6

S u b s t i t u t i o n of these values i n (A,2) allOT/s the r a d i a l pressure d i s t r i b u t i o n to be calculai^ed as b e f o r e . Conrparison of the t h e o r e t i c a l and experimentally determined pressure d i s t r i b u t i o n s are given in P i g , 10

Case 3 . The supersonic j e t

The experimental r e s u l t s suggest t h a t the attachment p o i n t does not move once the j e t has become supersonic. Thus the only v a r i a t i o n s i n P and X come from v a r i a t i o n s i n the sink s t r e n g t h q v/hich i s dependent upon the speed r a t i o u^ / , However v/e may i n f e r from Ref, 6 t h a t an increase i n "O./ above 10 has l i t t l e e f f e c t on q and hence v/e may deduce t h a t the r a d i a l pressure d i s t r i b u t i o n viiich the p o t e n t i a l flOT/ model p r e d i c t s f o r the supersonic j e t w i l l vary l i t t l e from ü i a t found for the sonic j e t .

-I-O Cp - 0 4 0 4 V rt. ^ ^ ^ ^ 9 n

M

J^

" ^ « r\ / / / ' / / /

7

A e\ / / / / /

7/

/ f • A ^ -*— / ' / ^ £. n ^ "^ \

N ,

S ^ rt

s,

N\

\ > V. ^

-\1

\ \ ^ C j I2'S -lOO -»o - 6 0 CpUco t p - r - 4 0 - J O 0 _{* o} M j ^ 123 J i 1 2 6 Mj " |'4I J o " l < 0 -• -• ^ 1 O

I,

/ / 3 O 4 Q 'S Q ^ ^ ^ ^ j ^ - l » 2 Ji,-l-60 > h& O 7 O _{'( •} ^ '< h(

FIG. 2a. JET SUBSONIC FIG. 2b. JET AT DESIGN CXDNDITIONS.

- 1 0 0 - • 0 - t o CpU., lp.1. - 2 0 C J ('•0> ~ J 0 5 -^W^

y

il

i

_f

/ ^

7

_(!' ' — ^ ^

r

^ \ , ^

1

-s,

§?;

^ : V j l'80 2'60 3 0

FKJ. 2c. JET UNDER-EXPANDED J o < J « 3 - 0 FIG. 2d. JET UNDER-EXPANDED J > 3 - 0 FIG. 2 RADIAL PRESSURE DISTRIBUTIONS.

• M ^ -+ X V A 0 •o ' 2 3 ' 4 1 • 6 0 ' 8 2 • 9 6 4 0 "=o,Ua. f.p.s. 2 0 lO

1

rK

9 » ' O

S:

- ^

k

N

• o ^ • ^ o ..

V

,« •

^ ^ — FROM INTEGRATION OF BASE PRESSURES.

* FROM VELOCITY TRAVERSES IN W K E . Mjj, =-1 BY EON. 2-12 K FROM VELOCITY TRAVERSES N WW<E. M ^ - 1 BY EON. 2-16 ~ O FROM VELOCITY TRAVERSES IN WAKE. Mj^ - 196 BIT BON. 2'I2 ~"

S FROM VELOCITY TRAVERSES IN WAKE. M j ^ - I ' 9 e BY EON. 2 16

I'O 2 0 3 0 JET STAGNATION PRESSURE M R A M E T E R .

I'O 2 0 S O JET STAGNATION PRESSURE PARAMETER.

FIG a MKRIATION OF BASE DRAG COEFFICIENT WITH JET STAGNATION PRESSURE AND NOZZLE

DESIGN MACH NUMBER.

FIG 4. COMPARISON OF BASE DRAG COEFFICIENTS OBTAINED FROM THE BASE PRESSURES

OS Ol

k

r \ f \

/ \ 1

I v o r

/ ^

/

"X"--»'

\ \ ,

K y

y

r

V^

_is*^ y* <

y'

40 AC JET MOMENTUM COEFFICENT C j

F I G 5 a . S O N I C J E T ao 100 O-B 1 °' 2 ^ > : V^ \ ^ ^^^<^ \ ''

Y^

%u„-soi^«. \ >

V -^

r ^ »'' 20 40 • o t o o JET MOMENTUM COEFFiaENT C j

F I G . 5 b . M j = 1 - 2 3

' J D

-JET MOMENTUM COEFFICIENT Cj

FIG. 5c. M , = 1 - 4 1 y-( j g

1

/ / * •

S

is / y ^ «'^ — ^••^ ^

,X >'

V

*>^ *~"^v' L* \ _ > ^ -SO I p. I. V X > » ' ^^"^ ^---^ r

JET HOtCr^TlM COCFFIOENT C} JET MOMENTUM COEFFICIENT Cj JET MOMENTUM C0EFF1OCNT C,

FIG. 5e. M j ^ = l - 8 2 FIG. St M j p - 1-98

FIG. 6a. NO JET

DGE OF POTENTIAL CORE.

FIG. 6b. JET VELOCITY I SMALL AND SUBSONIC.

OUTER EDGE OF JET MIXING REGION.

OUTER EDGE OF JET MIXING REGION.

EDGE OF POTENTIAL CORE.

FIG. 6c. JET VELOCITY SONIC OR DESIGN.

•-JDUTER EDGE OF JET MIXING REGION

PSEUDO-LAMINAR MIXING REGION.

FIG. 6«. JET MUCH UNDER-EXPANDED AT EXIT

PSEUDO-LAMINAR MIXING REGION.

FIG. 6d. JET VELOCITY SLIGHTLY GREATER THAN DESIGN.

OUTER EDGE OF MIXING REGION.

PSEUDO - LAMINAR MIXING REGON.

R G 6f. COMPRESSIVE EFFECT OF SEVERELY UNDER-EXPANDED JET

FIG. 6. FLOW IN MIXING REGION. [DIAGRAMMATIC.]

SUBSONIC JET DECREASING

BUBBLE LENGTH. FURTHER EXPANSION OF JET CAUSING SHORTEMNG OF BUBBLE.

LARGE EXPANSION OF JET COMPRESSIVE EFFECT ON EXTERNAL FLOW

FIG 7 EFFECT OF VARIATIONS IN FLOW CHARACTERISTICS' ON THE BASE DRAG.

r

•:{*

X-K-X-X-X-X-X-X-X-H—ll N II II I W » W

aööt.

« & M C C tTMCNBTH

Cf 0-6 0'4 0'2 o O-a 0'4 0-6 0-8 I-O c o B A 2 0 '

J

j

.V

Vi

/

1

1

c:

r

1

6 O ^ ^ ^ " \ ' -0

A — A CALCULATED USING FREE STREAM STAGNATK3N PRESSURE. B — B CALCULATED USING MODIFIED

STAGNATION PRESSURE. C—C EXPERIMENTAL. - O ' S ( o i

y

s V

J

;J

,

J

/

I

1

' \ 4 / 0 / /

f

\

V

_\ 6 OS V R _ I O

FIG 9. COMPARISON BETWEEN POTENTIAL FLOW MODEL AND EXPERIMENT

SUBSONIC JET _{" f U e o - '}

R G I O COMPARISON BETWEEN POTENTIAL FLOW MODEL AND EXPERIMENT

SONIC JET ^J/uoo= 10

' / u .

c

1^

^ i

^

"S

JET AT DESIGN CONDITIONS

ET DIAMETER i r EXIT ^ ^ - > , 0 5 1' J - I ' 7 S 0 r I-!

JET UNOER EXPANDED J - 2 3 5

FIG. 11. TYPICAL VELOCITY PROFILES. SONIC JET

JET UNOER EXPANDED J - 2 - 9

FIG 12 TYPICAL VELOCITY PROFILES. JET DESIGN MACH N U M B E R - 1 - 9 8