Theoretical and experimental investigation of boundary layer control in low-density nozzles by wall suction and cooling

where t h e v a r i a b l e of i n t e g r a t i o n has been changed from y t o T (see Appendix A). It is shown i n Appendix A t h a t With t h i s s u b s t i t u t i o n , we have S u b s t i t u t i n g f o r 6, t h e expression 6, = 2MRe Go (see Appendix A) and

using t h e d e f i n i t i o n s of Re and Go given i n Appendix A, equation (29)

s i m p l i f i e s t o 0 0 To c a r r y o u t t h e i n d i c a t e d i n t e g r a t i o n , i t i s necessary t o assume a v e l o c i t y p r o f i l e . I n Reference 9, a v e l o c i t y p r o f i l e of t h e form u/ul = s i n is used and good c o r r e l a t i o n with experimentally obtained v a l u e s of 6 i s shown. An exact s o l u t i o n f o r the s u c t i o n case is given i n Reference 4, b u t the computations a r e laborious. Lew and

Romano suggest an exponential p r o f i l e of the form u / u 1 = 1 - e-"(l - TK)

(see Appendix A ) . This p r o f i l e showed good agreement with an e x a c t solu- t i o n f o r t h e two-dimensional asymptotic s u c t i o n case [4] a t Mach number 10 and To/Tl = 4. For t h e same conditions, i t over-estimated t h e boun- dary l a y e r t h i c k n e s s f o r the no-suction case (exact s o l u t i o n , Reference 20) by about 20%. S u b s t i t u t i n g t h e above v e l o c i t y p r o f i l e (equation (11) of Appendix A) and i n t e g r a t i n g , w e g e t where terms involving e-Te and e-2Te have been dropped s i n c e they a r e n e g l i g i b l e . h is determined by Lew and Romano's s o l u t i o n , equation (13), Appendix A.

where A l l q u a n t i t i e s needed t o e v a l u a t e 6 a r e known, and A is determined by an i t e r a t i o n of equation (32).

With A known, K may be evaluated, and -re determined by equation (11) of Appendix A f o r u/ul = . 9 9 9 , o r whatever v e l o c i t y r a t i o has been s e l e c t e d t o d e f i n e t h e edge of the boundary layer.

The second i n t e g r a l i n equation (27) may be w r i t t e n (following a development s i m i l a r t o t h a t f o r t h e f i r s t i n t e g r a l ) : 0 0 The expression f o r y as a f u n c t i o n of z i s obtained by i n t e g r a t i n g equation (6) of Appendix A , which may be w r i t t e n i n t h e form S u b s t i t u t i n g the exponential v e l o c i t y p r o f i l e f o r G/u1 and c a r r y i n g o u t t h e i n t e g r a t i o n , one o b t a i n s where - .

I f w e now replace t h e i n t e g r a l expressions i n equation (27) by e q u a t i o n s (31) and (33), and use equation (35) f o r y, we g e t - 1 + K ]

6* ( 1 - 2a

I n t h i s equation 6*, a, po, vo, py, To/T,, and cos w are a l l known q u a n t i t i e s . The term ye is given by equation (35) which becomes, after terms involving emTe are dropped (Te i s always of the order of 4 . 5 t o 8), + ( % - . l + v N 2 5 ) ( ~ e - 1 + K ) ] . 1 To I n t h e above set of equations, h and a l l t h i n g s determined by it (namely, K, ‘te, (u/u,), and Q) must b e considered as a f u n c t i o n of a They cannot be e v a l u a t e d a t t h e f l a t p l a t e x, which we could c a l l x’, p a r t i c u l a r nozzle x under consideration, s i n c e then e q u a t i o n (37) would b e o v e r s p e c i f i e d . It is noted t h a t h is determined by equation (32) and t h e x ' dependence e n t e r s as t h e Reynolds number. Thus, t o determine 6, we assume f o r the moment t h a t t h e boundary l a y e r i s on a f l a t p l a t e w i t h a known Mach number, To/Tl and 6'k, and v a r y t h e x' u n t i l equations (37) and (38) a r e simultaneously s a t i s f i e d . The term 6 i s , of course, t h e v a l u e of ye f o r which the equations a r e s a t i s f i e d ; One problem occurred i n the computation of 6 by t h e method o u t l i n e d herein. For s u f f i c i e n t l y high u n i t Reynolds numbers i n t h e s u c t i o n case, no s o l u t i o n f o r 6 could be obtained, s i n c e f o r t h e 6*, Mach number and o t h e r c o n d i t i o n s given (which a r e those occurring i n a nozzle, not a f l a t p l a t e ) , t h e boundary l a y e r h e i g h t on a f l a t p l a t e does n o t reach a high enough v a l u e a t any x' t o s a t i s f y t h e equations.

Suction boundary l a y e r s doanot continue t o grow i n d e f i n i t e l y , b u t reach a maximum h e i g h t a t the p o i n t where t h e s u c t i o n v e l o c i t y balances t h e r a t e of growth. This l i m i t a t i o n w a s encountered only once i n t h e course of t h i s study. The conditions f o r which no s o l u t i o n could b e reached occurred i n a 10" h a l f - a n g l e nozzle a t a Mach number of 7.66, a u n i t Reynolds number of 5200/inch, with a s u c t i o n v e l o c i t y r a t i o V O / U ~ of -.0305 (the minus s i g n i n d i c a t e s s u c t i o n ) . The w a l l - t o - f r e e s t r e a m temperature r a t i o , To/T,, was 1.88. The l i m i t a t i o n thus a p p a r e n t l y occurs only a t r e l a t i v e l y high u n i t Reynolds numbers and s u c t i o n veloc- i t y r a t i o s ( t h e open a r e a i n t h e above case was 60% of the w a l l area) even f o r highly cooled w a l l s . For most cryopumped wind tunnels, it i s considered t h a t the l i m i t i n g conditions would r a r e l y b e encountered.

IV. METHODS O F CALCULATION A. Displacement Thickness, 6" W e now proceed t o d i s c u s s t h e methods of solving equation (20).

Two b a s i c approaches a r e p o s s i b l e , depending on what information i s given.

I n both methods, it i s necessary t o i n t e g r a t e equation (20) numerically.

A l s o , an i t e r a t i v e procedure i s necessary i n both methods.

The f i r s t method a t t a c k s t h e problem of c a l c u l a t i n g the Mach number and uniform core r a d i u s with a given nozzle shape and p o r o s i t y d i s t r i b u t i o n . The procedure i s of t h e i t e r a t i v e type because t h e Mach number depends on t h e magnitude of t h e s u c t i o n , and i n t u r n the s u c t i o n mass flow depends on t h e Mach number d i s t r i b u t i o n . It is necessary t o assume hole diameter D as a f u n c t i o n of x. It is considered t h a t t h e h o l e diameters can b e s a f e l y made t o b e 20% of the expected boundary l a y e r thickness. The c a l c u l a t i o n s are n o t s e n s i t i v e t o t h e h o l e s i z e chosen, and t h u s conservative v a l u e s should be used. The nozzle is divided i n t o s e v e r a l increments of length, Ax. The c a l c u l a t i o n is n o t p a r t i c u l a r l y s e n s i t i v e t o t h e number of increments chosen. For Mach numbers of t h e order of 12, twenty increments u s u a l l y g i v e s an accuracy The c a l c u l a t i o n i s begun c o n s i s t e n t w i t h the accuracy of t h e theory.

a t t h e f i r s t x and proceeds down the nozzle, 6* and Mach number being c a l c u l a t e d a t each x . It i s necessary t o compute S only a t t h e last x.

To begin t h e c a l c u l a t i o n , it i s necessary t o assume a v a l u e of t h e Mach number a t t h e f i r s t s t a t i o n considered.

I n t h e i n i t i a l p o r t i o n of the nozzle, where t h e boundary l a y e r i s t h i n , a good f i r s t guess is t h e Mach number which would occur w i t h no boundayy l a y e r and no suction.

F u r t h e r down t h e nozzle, it is b e t t e r t o add small Mach number increments t o t h e Mach number a t t h e previous s t a t i o n u n t i l agreement is obtained.

The c a l c u l a t i o n is summarized as follows:

For 5, assume a

Mach number M,, where n r e f e r s t o the n o z z l e l o c a t i o n .

Compute ply T1, p l , ul, from known s t a g n h t i o n conditions and i s e n t r o p i c equations.

2 L D 1 = Reynolds number based

ReD =L 4 5

on h o l e diameter and

( 1 ++) 2(7-1)

s o n i c conditions.

= where c ~ p c i s given by I J - ~ PlX - P

- - n +n- 1

= average p r e s s u r e over the i n t e r v a l Ax.

Pl avg . 2

i s r e a d from F i g u r e 5 a t proper v a l u e @'I@ = same as and

vo/ul = -.0165 A

M l @ = T, (1 + 9 M12 &) = a d i a b a t i c wall temperature.

Taw C

a t proper value of ~k

is taken from Figure 4

&

TO 4 a t proper value o f 6 (6fc/0)2dinc is taken from Figure T O Taw

- - 1

(W/0)2d = (6fc/f3)2d ( T o / T i ) +

i n c T l

L u1 Javg 2

cf +xQT1-

w ( x ) = 2 cos w u1 To cos w When xn-l = 0 , take G(x) W(x) = 3 (Cf/2 COS w), Note: 'fav and G(x) is of t h e order of 1.

s i n c e Cf = 2 cos w

86: = -

a cos a w [l - Jl -

r = a - 6* cos w

e c (Summation of s u c t i o n flow up t o xn divided AS/m* = by the t o t a l nozzle flow) 1 'TO A*

* This equation r e s u l t s from forming t h e r a t i o of the flow p e r u n i t

a r e a a t any nozzle s t a t i o n t o t h e flow per u n i t a r e a a t the t h r o a t .

M

I L

m

ec

-

y + A

' TT1 (1 4- M 2 ) 2(y - 1) A*

A where i s t h e mass flow i n t h e expans ion ,core a t t h e s t a t ion con- s idered. Thus, 2 1 I f M, assumed is w i t h i n a s p e c i f i e d t o l e r a n c e of t h e Mach number given by A/%, the c a l c u l a t i o n of 6 9 : and Mach number i s complete.

Otherwise, a new value of M, i s assumed and t h e c a l c u l a t i o n repeated.

The second method begins with a known Mach number d i s t r i b u t i o n and computes the r e q u i r e d w a l l r a d i u s and uniform c o r e s i z e . It i s necessary t o s p e c i f y e i t h e r t h e p o r o s i t y d i s t r i b u t i o n (open area t o t o t a l w a l l a r e a , even though t h e w a l l area i s n o t known) o r t h e s u c t i o n m a s s flow d i s t r i b u t i o n , fiS/fifc. Since t h e Mach number is known, a l l Mach number dependent i t e m s can immediately be computed. I f it i s t h e p o r o s i t y d i s t r i - (percent open a r e a ) which has been s p e c i f i e d , t h e c a l c u l a t i o n pro- b u t i o n ceeds i n exactly t h e same manner as t h e f i r s t method up t o t h e G(x) c a l c u l a t i o n . A v a l u e f o r t h e w a l l r a d i u s i s now assumed. Then 01, 61, and &s/fi* a r e evaluated as before; re, is obtained from t h e equation 6;k i s given by The v a l u e of r r cos w cos w This new The w a l l radius a may now be computed from a = rec + 6 * cos w.

v a l u e i s now used throughout t h e computation and t h e process repeated u n t i l t h e computed a a g r e e s with t h e assumed a t o w i t h i n a p r e s c r i b e d tolerance. I n a l l computations performed during t h i s study, t h i s pro- cedure always converged.

I f it is the s u c t i o n weight flow r a t i o s which a r e s p e c i f i e d , a w a l l r a d i u s i s assumed and vo/ul computed from t h e equation &-tS and &lo are the incremental s u c t i o n m a s s flow and w a l l area where f o r t h e Ax considered. el and 6, are computed as i n t h e f i r s t method; ret, Sfc, and a new a are computed a s above, and t h e same i t e r a t i v e pro- cedure followed. A f t e r a i s found, t h e r e q u i r e d @ is g i v e n by where @ ' / P i is t h e same as &/A* and is known from F i g u r e 5 .

B. Boundary Layer Thickness, 6 With 6* and Mach number known from either of t h e above two methods t h e t h i c k n e s s of t h e boundary l a y e r is computed by simultaneously s a t i s f y i n g equations (37) and (38). I n t h e s e equations 6*, a, PO, VO, p1, The computation proceeds a s follows: Tl/To, and cos w a r e a l l known, (1) Assume an x ' .

Find h by an i t e r a t i o n of equation (32). A t a b l e of (2) v a l u e s of t h i s function is presented i n Table 11. The assumed v a l u e of x ' i s used f o r x i n equation (32).

-7 e T~ is determined from (u/ul)e = 1 - e (1 - T ~ K ) .

(4) The v a l u e of (u/ul), is t h a t which d e f i n e s the boundary l a y e r thickness (usually taken a s .999, b u t i n t h i s pro- cedure it . i s a r b i t r a r y ) .

A v a l u e f o r ye can now be computed from equation (38) and (5) s u b s t i t u t e d i n t o equation (37). The i n t e g r a l i n t h e l a s t (37) can be evaluated e i t h e r numerically, term of equation g r a p h i c a l l y o r a n a l y t i c a l l y . u/ul i s given by -7

1 - e (1 - TK) and Q by equation (36). Because of the

l a r g e number of terms i n t h i s expression, a g r a p h i c a l o r numerical i n t e g r a t i o n procedure is recommended.

(6) I f equation (37) is not s a t i s f i e d w i t h the assumed X I , a l a r g e r v a l u e is assumed and t h e process repeated; 6 is t h e v a l u e of ye f o r which equation (37) is s a t i s f i e d .

In t h e case of no s u c t i o n , (vg = 0 ) , t h e term i s indeterminate s i n c e A = 0 when vo = 0. An expansion of equation (32) f o r t h e case when A + 0 shows t h a t can b e replaced by 1.2 x'

E

V. EXPERIMENTAL INVESTIGATION AND COMPARISON WITH THEORY A. Experimental Arrangement A n experimental program w a s devised t o provide a check on the t h e o r e t i c a l calculations.ik A sketch of t h e experimental l a y o u t is p r e - sented i n Figure 6 . During operation, t h e n i t r o g e n used as t h e t e s t gas is obtained by vaporizing l i q u i d nitrogen. This gas passes through t h e 1.5-K.W. h e a t e r t o t h e s t i l l i n g chamber and then i n t o t h e nozzle. P a r t of t h e flow passes through t h e nozzle pores t o cryopump c o i l s surround- ing t h e nozzle, while t h e rest continues through t h e nozzle i n t o t h e vane-type " d i f f u s e r " t h a t serves as a precooler. This gas passes between the vanes t o t h e remaining cryopump c o i l s . The cryopump i s cooled by gaseous helium from a 350-watt r e f r i g e r a t o r . A l i q u i d - n i t r o g e n cooled s h i e l d reduces t h e r a t i a t i o n load t o t h e cryopump. A roughing pump i s used f o r i n i t i a l tank evacuation, and a d i f f u s i o n pump i s used t o remove any noncondensible gases.

The nozzle schematic is presented i n Figure 7 . The nozzle i s The forward s e c t i o n a 13" half-angle cone f a b r i c a t e d i n two s e c t i o n s .

*

Experiments were conducted i n t h e Hyperaltitude F a c i l i t y of t h e Environmental Division of t h e U. S . Naval M i s s i l e Center, Point Mugu, California.

was machined from s o l i d copper and includes t h e c o n t r a c t i o n s e c t i o n , t h e t h r o a t , and the supersonic s e c t i o n t o a diameter of 2".

The 9/32" diameter t h r o a t is a c y l i n d e r about one diameter long w i t h t h e corners s l i g h t l y rounded.

The second s e c t i o n i s t h e porous s e c t i o n t h a t extends from t h e 2" diameter t o t h e 12" diameter. It w a s r o l l e d from a 3/16" s h e e t of copper a f t e r t h e pores had been d r i l l e d . The h o l e p a t t e r n and t h e r e s u l t - ing p o r o s i t y are a l s o given i n Figure 7. The p o r o s i t y d i s t r i b u t i o n is e s s e n t i a l l y l i n e a r w i t h a x i a l distance. The h o l e r a d i i w e r e s e l e c t e d t o keep t h e r a t i o of t h e h o l e r a d i u s t o l o c a l boundary l a y e r h e i g h t equal t o about -1; however, near t h e f r o n t of t h e porous s e c t i o n t h i s r a t i o approached -6.

The l i q u i d - n i t r o g e n cooling c o i l s w e r e s o l d e r e d on t h i s s e c t i o n The c o i l s d i d n o t cover any of t h e w i t h approximately 6'' between c o i l s .

pores.

A f t e r j o i n i n g , the two s e c t i o n s w e r e g i v e n t h e f i n a l machining s o the n o z z l e c o o r d i n a t e s are w i t h i n 1% of a t r u e 13" h a l f - a n g l e cone.

B. I n s t r u m e n t a t i o n T e s t s e c t i o n p i t o t pressure w a s measured on an Alphatron (NRC Model 520). A c a l i b r a t i o n made p r i o r t o t h e test showed t h a t i n

t h e range of most of t h e testing ( 1 0 0 ~ - 300p), t h e instrument was as

a c c u r a t e as t h e scale could be read. This corresponds t o from 5 5% t o -F 1.7%. Limited d a t a taken below 1 O O p r e q u i r e d c o r r e c t i o n s of less t h a n 10% t o achieve reading accuracy.

Stagnation pressure w a s measured using a Bourdon gage a t p r e s - s u r e s above 20 mm Hg and a McLeod gage f o r lower p r e s s u r e s . The accuracy of t h i s system was approximately i- 2%.

Tunnel s t a t i c p r e s s u r e was measured using a thermocouple gage was a v a i l a b l e , b u t designed f o r t h e law-micron range. No c a l i b r a t i o n t h e reading accuracy and the r e p e a t a b i l i t y were approximately 2 .2p i n t h e testing range.

C. Experimental R e s u l t s 1. Porous Wall Nozzle a range of stagna- Radial p i t o t probe surveys were made f o r t i o n temperatures and pressures. I n g e n e r a l a .312" O.D., .270" I . D .

p i t o t tube w a s used. Since t h e s e dimensions are t h e same o r d e r of magni- as t h e mean f r e e p a t h of freestream gas, tests were made on a l a r g e r tude probe (.875" O.D., .785" I . D . ) and a smaller probe (.125" O.D., .106" I . D . ) t o determine the e f f e c t of probe s i z e on the i n d i c a t e d p i t o t pressure. The r e s u l t s of these t e s t s , shown i n Figure 8, were used t o c o r r e c t the p i t o t probe surveys. Only t h e d a t a taken w i t h s t a g n a t i o n pressures l e s s than .5 p s i a r e q u i r e d c o r r e c t i o n s of 10% or more.

These c o r r e c t e d surveys are presented i n Figure 9. For t h e s e s t a g n a t i o n conditions, t h e nozzle developed Mach numbers ranging from 9.15 t o 11.0 a t f r e e s t r e a m Reynolds numbers ranging from 100/inch t o 500/inch. A t the lowest Reynolds number, t h e p i t o t p r e s s u r e r a t i o i s lower than would be expected. Despite t h e r e g i o n of uniform p i t o t p r e s s u r e (obtained with t h e .125" O.D. probe), it appears t h a t the boundary l a y e r has merged and the stream s t a g n a t i o n p r e s s u r e i s decreas- Since t h e p i t o t p r e s s u r e i s r a t i o e d t o t h e upstream s t a g n a t i o n ing.

p r e s s u r e , t h i s would account f o r t h e low r a t i o .

A s w i l l b e shown i n t h e next s e c t i o n , t h i s i s c o n s i s t e n t with t h e t h e o r e t i c a l c a l c u l a t i o n s .

A t t h e higher Reynolds number t h e p r o f i l e s show a n increasing p i t o t pressure a s t h e probe moves from t h e c e n t e r .

This i s t y p i c a l of low-density conical nozzles when they a r e operated a t higher d e n s i t i e s . The e x t e n t of t h i s higher d e n s i t y regime can be estimated from the r a t i o of boundary l a y e r h e i g h t t o t e s t s e c t i o n r a d i u s (8/a). This i s i l l u s t r a t e d i n Figure 10, where t h e r a t i o of t h e p i t o t pressure a t t h e edge of t h e core t o the c e n t e r l i n e p i t o t p r e s s u r e has been p l o t t e d versus t h e r a t i o (8/a) f o r a s e r i e s of c o n i c a l

nozzles. The d a t a covers t h e Mach number range 4.5 - 11.0 and cone h a l f -

angles from 10" t o 20". From t h i s p l o t i t can be seen t h a t a s (8/a) i s reduced below about .7, t h e p i t o t p r o f i l e v a r i a t i o n s exceed 5%. Thus, t o avoid transverse Mach number g r a d i e n t s , simple, conical nozzles must be operated a t low d e n s i t i e s i n order t h a t ( s / a ) exceeds .7. For higher d e n s i t y operation, it appears necessary t o u s e a smaller cone angle o r t o contour the nozzle walls.

2. Solid Wall Nozzle I n order t o o b t a i n a d i r e c t comparison of a solid-wall nozzle and a porous-wall nozzle, t h e e x t e r i o r of the nozzle w a s covered w i t h aluminum f o i l . While t h i s prevented any outflow, it d i d leave the i n t e r i o r nozzle w a l l rough due t o t h e closed pores; however, it was f e l t t h a t these closed pores would n o t p r e s e n t any g r e a t e r d i s t u r b a n c e s than they had when used as a porous w a l l .

The p i t o t p r o f i l e s obtained from t h i s solid-wall nozzle a r e presented i n Figure 11. The corresponding porous-wall d a t a a r e sets of d a t a , it i s c l e a r a l s o presented. From a comparison of t h e two t h a t t h e main e f f e c t of t h e porous w a l l was t o i n c r e a s e t h e Mach number while maintaining approximately t h e same boundary l a y e r h e i g h t .

Comparison of Theory and Experiment D.

Theoretical c a l c u l a t i o n s were made f o r t h e various t e s t condi- t i o n s , and t h e comparison between the t h e o r e t i c a l and experimental Mach siumbers and boundary l a y e r h e i g h t s is presented i n Figures 1 2 and 13.

For d i r e c t comparison t h e t h e o r e t i c a l values of Reynolds number/ inch have been used a s the a b s c i s s a f o r both sets of data.

Figures 1 2 and 13 show t h a t , with t h e exception of t h e lowest Reynolds number/inch point, t h e theory p r e d i c t s the Mach number t o w i t h i n 5% and t h e boundary l a y e r h e i g h t t o within 10%. Since these c a l c u l a t i o n s w e r e made using a 145'R nozzle w a l l , while i n t h e experiment t h e t h r o a t block was a t a higher temperature (about 285OR), c a l c u l a t i o n s were a l s o made t o a s s e s s t h e e f f e c t of t h i s temperature d i s t r i b u t i o n . The r e s u l t s showed t h a t the temperature d i s t r i b u t i o n changed t h e Mach number by less than .5% and t h e boundary l a y e r height by less than 2%. The boundary l a y e r c a l c u l a t i o n on t h e lowest Reynolds number/inch p o i n t showed t h a t t h e boundary l a y e r s had merged. This agrees with t h e experimental r e s u l t s discussed i n t h e previous section.

Although t h e r a t i o of mean f r e e path t o pore r a d i u s v a r i e d from about . 1 t o 1 . 3 , t h e s u c t i o n mass flow was computed using f r e e molecular flow v a l u e s of &/15* f o r a l l conditions. It is estimated t h a t t h i s s i m p l i f i c a t i o n reduces t h e t o t a l suction mass flow by l e s s than 10%.

Since t h i s c a l c u l a t e d s u z t l o n mass f l o w v a r i e d f r m 3% t o 13% of t h e t h r o a t mass flow, the e r r o r is l e s s than 2% of t h e t h r o a t mass flow.

The experimental d a t a with comparable s t a g n a t i o n conditions w e r e a l s o used t o determine t h e suction mass flow by assuming t h a t t h e d i f f e r e n c e between t h e e x i t Mach numbers f o r the s o l i d wall and t h e porous w a l l was due t o t h e s u c t i o n mass flow reducing the e x i t mass flow. I n t h i s case t h e s u c t i o n mass flow is determined from

m out A/&) s o l i d wall

- = 1 - (

(A/&) porous wall .

&* The a r e a r a t i o s a r e determined from the e x i t 'Mach numbers. This pro- cedure.assumes that t h e e f f e c t s of small changes of Reynolds number, A comparison of t h e r e s u l t s v e l o c i t y p r o f i l e , and 6/a a r e negligible.

of t h e s e c a l c u l a t i o n s with t h e t h e o r e t i c a l values, presented i n Figure 14, appears t o j u s t i f y t h e above assumptions.

In a n attempt t o l e a r n more of t h e d e t a i l s of t h e flow coming from t h e pores, temperature and p i t o t p r e s s u r e s were measured a t the Due t o the probable nonequilibrium con- e x i t of a 5/8" diameter pore.

d i t i o n of t h i s e x i t i n g flow, and t h e unknown p i t o t probe e r r o r s , t h e r e s u l t s were only q u a l i t a t i v e . The measured temperature was about 250"R s t a t i c pressure. Since and t h e pressure was about t h r e e times tunnel t h e temperature and s t a g n a t i o n p r e s s u r e i n t h e boundary l a y e r reach these values a t d i s t a n c e s of the order of one mean f r e e p a t h i n t o t h e boundary layer, t h e s e v a l u e s seem reasonable, The measured e x i t tempera- t u r e being above the nozzle wall temperature suggests t h a t nozzle pores with small v a l u e s of L / D cannot be used t o precool t h e s u c t i o n flow t o w a l l temperature. For complete cooling, t h e pore l e n g t h and diameter both must be s i z e d r e l a t i v e t o the tunnel w a l l mean f r e e p a t h i f t h e tunnel boundary l a y e r temperature rises s i g n i f i c a n t l y i n d i s t a n c e s of t h e order of the mean f r e e path.

To check the v a l i d i t y of t h e theory a t h i g h e r Reynolds numbers, c a l c u l a t i o n s w e r e made on t h e cooled porous nozzle r e p o r t e d i n Refer- 8. This 10" h a l f - a n g l e nozzle had a .104" diameter t h r o a t , a ence 60%. The comparison of 1.9" diameter e x i t , and a maximum p o r o s i t y of t h e r e s u l t s of t h e c a l c u l a t i o n s and t h e experiments i s presented i n Figure 15. The t h e o r e t i c a l v a l u e s of Reynolds number/inch have been used as the a b s c i s s a . With t h e exceptions of t h e two h i g h e s t Reynolds numbers where t h e nozzle flow had strong t r a n s v e r s e g r a d i e n t s , t h e theory p r e d i c t s the Mach number t o w i t h i n 3%. S i m i l a r l y , w i t h t h e exception of a few p o i n t s , the boundary l a y e r h e i g h t s a r e i n good agree- ment. The s c a t t e r i n t h e boundary l a y e r p l o t s i s believed t o be due t o t h e d i f f i c u l t y i n d e f i n i n g the edge of t h e boundary l a y e r on t h e experi- mental p i t o t p r o f i l e s .

V I . COOLING AND SUCTION EFFECTS ON NOZZLE AND TEST CORE S I Z E FOR G I V E N MACH NUMBER DISTRIBUTIONS Machine c a l c u l a t i o n s w e r e c a r r i e d o u t using t h e second t h e o r e t i c a l method outlined i n Section I V t o determine the b e n e f i t s of w a l l s u c t i o n and cooling f o r a f i x e d Mach number. A Mach number d i s t r i b u t i o n was assumed and t h e r e s u l t i n g boundary l a y e r thickness, nozzle diameter , and uniform core thickness w e r e c a l c u l a t e d .

a t s u i t a b l e Mach number d i s t r i - A s i m p l e , r a p i d method of a r r i v i n g butions t o u s e i n the c a l c u l a t i o n s w a s supplied by t h e f i r s t t h e o r e t i c a l method, where a given nozzle shape i s s p e c i f i e d and a nozzle Mach number d i s t r i b u t i o n i s predicted. With a few t r i a l s using c o n i c a l w a l l shapes, it w a s p o s s i b l e t o p r e d i c t Mach number d i s t r i b u t i o n s w i t h e x i t Mach numbers close t o 3 , 6 , 9, and 12, t h e range which w a s of i n t e r e s t i n t h i s study. Thus, although t h e d i s t r i b u t i o n s used were n o t aimed a t a p a r t i c u l a r t e s t s e c t i o n flow f i e l d , they do r e f l e c t smooth expansions which would c l o s e l y approximate t h e flow i n a c t u a l nozzles of the same t e s t s e c t i o n Mach number, and a r e adequate f o r showing trends.

It was endeavored a t a l l Mach numbers t o p i c k a t h r o a t s i z e such t h a t t h e r e s u l t i n g uniform core was about s i x inches i n r a d i u s .

A n o z z l e l e n g t h of four f e e t w a s used i n a l l c a l c u l a t i o n s , and i n most cases r e s u l t e d i n reasonable nozzle w a l l angles.

F i g u r e 16 shows t h e nozzle wall r a d i i , boundary l a y e r thicknesses, and uniform core r a d i i which were c a l c u l a t e d f o r f o u r Mach numbers.

The Mach number d i s t r i b u t i o n s and t h r o a t s i z e s t h a t were used a r e given i n Table I. Nitrogen w a s assumed as t h e nozzle f l u i d . I n a l l c a s e s a w a l l temperature of 180"R was used, corresponding roughly t o t h a t f o r a w a l l cooled by l i q u i d nitrogen. Results a r e given f o r the no-suction case and f o r t h e case where 22% of the nozzle flow is removed v i a w a l l suction. A u n i t Reynolds number range up t o about 300 p e r inch was s e l e c t e d , s i n c e f o r the nozzle s i z e chosen t h i s corresponds approxi- mately t o t h e maximum weight flow which can be accepted by a one- k i l o w a t t cryopump.

It is r e a d i l y apparent from the f i g u r e t h a t l a r g e r e d u c t i o n s i n t h e boundary l a y e r thickness occur w i t h 22% s u c t i o n flow. For t h e Mach number 1 2 nozzle, 6 i s reduced about four inches a t a u n i t Reynolds numbsr of 160/inch, corresponding t o a flow rate of 2 grams/sec.

The uniform c o r e r a d i i , however, a r e remarkabley c l o s e a t a l l Reynolds numbers, so t h a t the n e t e f f e c t of s u c t i o n i n a l l cases shown i n F i g u r e 16 i s t o reduce t h e nozzle diameter.

What may b e of more importance, however, i s the b e n e f i c i a l e f f e c t of precooling i n t h e s u c t i o n case. It is d i f f i c u l t t o e x a c t l y assess t h e b e n e f i t s of t h e precooling on the nozzle s i z e and c o r e r a d i i , s i n c e t h i s is considerably influenced by t h e n a t u r e of t h e o v e r a l l enthalpy removal system, i n p a r t i c u l a r , whether a d i f f u s e r and precooler a r e used downstream of t h e nozzle and how e f f e c t i v e they are. I n t h i s r e g a r d it i s probable t h a t t h e reduced boundary l a y e r i n t h e s u c t i o n case would a i d a t t e m p t s t o d i f f u s e and precool the nozzle flow. A c a l c u l a t i o n has been made using simple assumptions about t h e system t o provide an e s t i m a t e of t h e p o s s i b l e i n c r e a s e i n c o r e s i z e due t o t h e u s e of s u c t i o n and precooling. I f it is assumed i n t h e no-suction case t h a t a l l of t h e enthalpy of the incoming flow is removed by the cryopump, and, f o r t h e s u c t i o n case, 22% of t h e flow goes through t h e galls where t h i s s u c t i o n flow is a l l precooled t o 180'R ( l i q u i d - then f o r a s t a g n a t i o n temperature of 1460"R @bch 12) n i t r o g e n c o o l a n t ) , we have

- 235 cal/gram

= 1.16. -

7 8 (235 cal/gram) + .22 (90 cal/gram)

Thus t h e t e s t s e c t i o n a r e a could b e increased by 16% w i t h a r e s u l t i n g 7.5% increase i n t h e uniform core s i z e . Somewhat h i g h e r s u c t i o n rates would r a i s e t h i s number, b u t it i s doubtful t h a t s u c t i o n flows exceed- ing 30% of t h e t h r o a t flow are p r a c t i c a l , e s p e c i a l l y i f it i s intended t o cool t h e s u c t i o n flow t o t h e temperature of the w a l l .

Figure 17 gives t h e r e s u l t s of c a l c u l a t i o n s on t h e e f f e c t of v a r i o u s degrees of w a l l cooling on a , 6 and ruc f o r Mach number 12 and a u n i t Reynolds number of 800/inch. This corresponds t o a flow of t e n grams per second f o r t h e nozzle s i z e shown. The high Reynolds number w a s used i n t h i s p l o t t o keep t h e w a l l mean f r e e p a t h t o reasonable v a l u e s i n the a d i a b a t i c wall case. Even f o r t h i s Reynolds number, however, t h e adia- b a t i c w a l l mean f r e e p a t h i s about 25% of t h e boundary l a y e r thickness.

Results are given f o r t h e no-suction case and f o r a s u c t i o n case w i t h gmaX = .66. Again, l a r g e decreases i n 6 are p o s s i b l e w i t h t h e u s e of cold w a l l s , b u t t h e uniform core r a d i i a r e very c l o s e i n t h e s u c t i o n and no-suction cases f o r any w a l l temperature. A perhaps s u r p r i s i n g f e a t u r e of t h e p l o t i s the i n c r e a s e i n uniform core r a d i i as t h e a d i a b a t i c w a l l temperature (no cooling) i s approached. Of course, without cooling t h e nozzle r a d i i g e t very l a r g e , and f o r t h e p a r t i c u l a r case shown h e r e , t h i s would r e s u l t i n a very l a r g e w a l l divergence even i f s u c t i o n w e r e used.

Moreover, it i s u n c e r t a i n how s u c c e s s f u l l y such a t h i c k boundary l a y e r could be cooled a f t e r leaving t h e nozzle, and t h e r e f o r e a h o t w a l l may s t i l l b e i n f e r i o r t o a cold w a l l w i t h t h e r e s u l t i n g i n c r e a s e i n a c c e p t a b l e weight flow. Obviously, r e s e a r c h i s r e q u i r e d regarding d i f f u s i o n and cooling of flows with t h i c k boundary l a y e r s before meaningful conclusions can be made concerning t h i s p o i n t .

Figure 18 shows, f o r a t y p i c a l nozzle, t h e maximum r a t i o s of open area t o t o t a l w a l l a r e a which are r e q u i r e d t o achieve v a r i o u s s u c t i o n flow r a t i o s . A l i n e a r s u c t i o n rate has been used i n a l l cases shown; i.e,, s t a r t i n g a t .4 f e e t down t h e nozzle, t h e s u c t i o n flow i s l i n e a r l y increased and r e s u l t s i n a t o t a l s u c t i o n flow of e i t h e r 11% o r 22% of t h e t h r o a t mass flow. This a l s o r e s u l t s i n a n e a r l y l i n e a r @, the maxi- mum @ i n a l l cases occurring a t t h e nozzle exit. P h y s i c a l l y impossible @ I s are indicated f o r t h e 22% s u c t i o n r a t i o of Mach number 1 2 . This case w a s used i n Figure 16 f o r comparison purposes, b u t should n o t a f f e c t t h e v a l i d i t y of t h e i n d i c a t e d trends.

VII, CONCLUDING REMARKS Based on a comparison of t h e r e s u l t s of t h e o r e t i c a l and experi- mental i n v e s t i g a t i o n s , it i s concluded t h a t t h i s theory p r e d i c t s the Mach numbers t o about 5% and the boundary l a y e r h e i g h t t o about 10% when t h e theory i s applied t o l i q u i d - n i t r o g e n cooled low d e n s i t y nozzles. The i n v e s t i g a t i o n covered t h e range of Reynolds numbers from t h e range of Mach numbers form 7.5 t o 11.

150/inch t o 3000/inch and When t h e theory was used t o determine t h e e f f e c t s of w a l l p o r o s i t y and cooling on t h e r a d i u s of t h e uniform core, it w a s found that, f o r a f i x e d t h r o a t mass flow and f i x e d Mach number d i s t r i b u t i o n , t h i s r a d i u s w a s v i r t u a l l y unchanged. Since the cooling and w a l l s u c t i o n both served t o reduce t h e boundary l a y e r height, t h i s means t h a t the cooling reduced 6*/8 enough t o keep t h e boundary layer mass flow c o n s t a n t , while t h e w a l l flow a t a r a t e t h a t kept c o n s t a n t t h e sum of the s u c t i o n removed mass s u c t i o n mass flow and t h e boundary layer mass flow. While t h i s showed t h e r e was no d i r e c t b e n e f i t from cooled porous w a l l s , t h e r e were important secondary e f f e c t s . One of t h e s e was t h e precooling of t h e s u c t i o n flaw, f o r a f i x e d which would a l l o w a higher mass flow (and l a r g e r nozzle) cryopump s i z e . Another advantage was t h a t the smaller boundary l a y e r meant a p h y s i c a l l y smaller and more manageable nozzle. Further, t h e smaller boundary l a y e r might r e s u l t i n d i f f u s e r p r e s s u r e recovery s u f - f i c i e n t t o a l l o w conventional mechanical pumps t o handle a s i g n i f i c a n t , f r a c t i o n of t h e flow, p e r m i t t i n g a l a r g e r t o t a l mass flow and a l a r g e r nozzle. Since t h e s e secondary advantages depend upon t h e o v e r a l l system, it is concluded t h a t t h e m e r i t s of a cooled porous w a l l must be decided from a n a n a l y s i s of t h e complete wind tunnel system.

1s 6Nin.

FI'W,#RE 1 EFFECT OF C O O L I N G A N D S U C T I O N O N FLAT PLATE.

BOUNDARY LAYER HEIGHT ON A X^I in.

F I G U R E 2 E F F E C T OF U N I T R E Y N O L D S NUMBER ON BOUNDARY LAYER HEIGHT ON A FLAT PLATE.

2 P

1.0 0 1.0 2.0 FIGURE 4. S K I N F R I C T I O N C O E F F I C I E N T A N D ( S * / Q ) FROM Zdi nc RESULlS OF I O L I L C H (REFERENCE 1 ) ) 1.0 .8 .6 .2 6.8

I I

001 1 1 .o 10 1 o2

FIGURE 5, E F F E C T OF R E Y N O L D S NUMBER A N D T U B E L / D O N M A S S FLO,W.RATIO ii Y n c c L p.

n

s

p.

I , c

A

I

I

I

I

ry W L

I

SIZE O F HOLES

I

I

0-0-I 0 .5 1.0 1.5 2.0 x feet

I . 4

I .2 I .o .8 . I F I G U R E 8 8 E F F E C T OF K N U D S E N N U M B E R ON M E A S U R E D P I T O T PRESSURE.

‘FIG09 8 V A R I A T I O N O F P I T O T P R E S S U R E D I S T R I B U T I O N

W I T H R E Y N O L D S N U M B E R .

1.4 P ’ Te P i B 1.2 1.0

0 . 2 04 .6 .% 1 .o

Q FIGURE 10, V A R I A T I O N O F P I T O T P R E S S U R E R A T I O W I T H

--

Seltd Wall Porous Wa I I 2 1 0 1 2 3 + 6 r N in.

FIGURE 1, COMPARISON OF PITOT PRESSURE DISTRIBUTION, SOLID W A L L A N D POROUS W A L L Mach 100 200 400 600 800 lo00

Re - (l/in.)

F I G U R E 12, C O M P A R I S O N OF T H E O R E T I C A L A N D E X P E R I M E N T A L M A C H NO.

4.

6 (in.)

(" = -999)

"1 2.

1.

Re (l/in.)

F I G U R E 13, C O M P A R I S O N OF T H E O R E T I C A L A N D E X P E R I M E N T A L I b O U N D A R Y L A Y E R H E I G H T

(.I)

I I I 1 0 I F I G U R E 14, C N T A L Mach No.

.4 .4 .2 .l 100 1000 loo00 RO N (l/ik) F I G U R E IS, C O M P A R I S O N W I T H E X P E R I M E N T A L D A T A OF R E F E R E N C E 8 IO t

3 V

C I

-

Y (0 1 0 IO 9 9 e, S 5 4 Q 3 3 2.0 3.0 FIGURE 16, E F F E C T O F U N I T REYNOLDS NUMBER ON a, 6 , a n d r U c A T FOUR M A C H NUMBERS ( NorzleLength=4foet, Wall Tompmtum= 180%) M A C H NUMBER : 12 N O Z Z L E L E N B T H I 4 F E E T THROAT RADIUS : .0234 F T .

R o / i n . : 8 0 0

; * I 10piurr / t.C

S T A C B N A T I O N TEMPERATURE : 1460'R FIGURE 1 7 , EFFkCT OF R A T I O OF WALL TEMPERATURE TO F R E E S T R E A M TEMPORATWRE O N a 8 6 8 a n d r u c CONDITIONS: 1.4 1.0 .8 *nmx -6 .4 .2

-

1 1 FIGURE 18, MAXIMUM OPEN A R E A / T O T A L A R E A R E Q U I R E D TO G I V E VARIOUS S U C T I O N F L O W RATES.

TABLE I MACH NUMBER DISTRIBUTIONS FOR FOUR-FOOT NOZZLES C (feet) MACH NUMBER 2 . 2 1 3 . 2 1 . 2 1 . 4 3 3.77 2.75 4 . 1 3 4 . 8 9 .4 1 . 6 4 .6 1 . 8 0 3 . 1 6 4 . 7 9 5 . 7 0 2 . 0 7 3 . 8 0 5 . 7 5 6 . 9 4 1 . 0 2 . 1 8 4 . 0 5 6 . 1 2 7 . 4 5 1 . 2 2.29 4 . 2 8 6 . 4 5 7 . 9 1 1.4 2.39 4.49 6 . 7 4 8 . 3 4 1 . 6 2 . 4 8 4 . 6 8 7 . 0 1 8 . 7 3 1 . 8 2 . 5 6 4 . 8 5 7 . 2 5 9 . 1 0 2.0 7 . 4 8 2.2 2.65 5 . 0 2 9 . 4 5 2.72 5.17 7 . 6 9 9 . 7 8 2 . 4 2 . 8 0 5 . 3 1 7 . 8 8 1 0 . 0 9 2.6 5 . 4 4 8 . 0 6 1 0 . 3 9 2.87 2.8 5.57 8 . 2 2 1 0 . 6 7 2 . 9 4 3 . 0 8 . 3 9 1 0 . 9 5 3.01 5 . 6 9 3 . 2 5 . 8 1 8 . 5 4 1 1 . 2 2 3.07 3 . 4 5 . 9 2 8 . 7 0 11.47 3 . 1 4 3.6 8 . 8 2 1 1 . 7 2 3 . 2 0 6 . 0 2 3.8 8 . 9 5 1 1 . 9 6 3.25 6 . 1 2 4 . 0 .08 .0351 .0234 rhroa t .284 R a d i i ( f e e t ) TABLE I1 ~~ -A f (A) -A

f (N

0 0 .26 .6816

. 01 .00559€

.27 ,7594 .02 .00228C .28 .8446 .03 .005225 .29 .9381 .009464 -04 .30 1.0409 .05 .01507 .31 1.1540 .06 .02213 .32 1.2787 .07 .0307 2 .33 1.4165 .04094 .08 .34 1.5692 .09 .05290 .35 1.7390 .10 .06669 .36 1.9283 .11 .08 244 .37 2.1405 .12 .lo03 .38 2.3795 .13 .1204 .39 2.6504 .1428 .14 .40 2.9597 .15 .1678 .41 3.3159 .16 .1956 .42 3.7309 .17 .2262 .43 4.2212

.18 .2601 . .44 4.8107

.19 .2974 .45 5.5373 .20 .3384 .46 6.4640 .21 .3834 .47 7.7090 .22 .4327 .48 9.5385 .23 .4867 ' .49 12.8017 .24 .5458 24.1271 .499 .25 .6106 ,4999 35.6081

APPENDIX A>k COMPRESSIBLE LAMINAR BOUNDARY LAYER ON A FLAT PLATE W I T H UNIFORM SUCTION AND WALL COOLING Consider t h e compressible flow over a f l a t p l a t e having a uniform s u c t i o n v e l o c i t y vo and constant w a l l temperature To.

I f the s u c t i o n v e l o c i t y i s small compared t o t h e f r e e stream v e l o c i t y u, t h e P r a n d t l boundary l a y e r equations can be used t o d e s c r i b e t h e flow.

Momentum Equation Continuity Equation

a p u + a P v = *

ax ay

Energy Equation I f the P r a n d t l number i s assumed t o b e one, and t h e w a l l tempera- t u r e constant, the energy equation can be replaced by t h e following r e l a t i o n s h i p : * This appendix is a synopsis of the work of Lew and Romano, Reference 6 .

A p r i o r work by Lew [23] concerning s u c t i o n on i n s u l a t e d w a l l s g r e a t 1Y f a c i l i t a t e s the understanding of Reference 6 .

Equation of S t a t e p = pRT.

The c o n t i n u i t y equation can be i n t e g r a t e d t o g i v e I f t h i s expression is s u b s t i t u t e d i n t o t h e momentum equation, t h e Von Karman momentum i n t e g r a l may be obtained i n t h e form The x and y v a r i a b l e s a r e now transformed a s follows: Let and dx = L R e ds where L i s t h e p l a t e c h a r a c t e r i s t i c l e n g t h and, R e = PO Equation (3) can be w r i t t e n in t h e form where

- U

- U , = u = and u1 - 0 Combining equations (6) and (8), we have S u b s t i t u t i n g equations (7) and (9) i n t o ( 5 ) , and r e a r r a n g i n g , one can o b t a i n the transformed momentum i n t e g r a l where 6t is a measure of the boundary l a y e r thickness and Reference 6 suggests both an exponential and a q u a r t i c p r o f i l e f o r sub- s t i t u t i o n i n t o the i n t e g r a l r e l a t i o n s h i p . However, t h e q u a r t i c p r o f i l e l e a d s t o impossible p r o f i l e shapes f o r l a r g e Reynolds numbers, and the The exponential v e l o c i t y pro- exponential p r o f i l e has been s e l e c t e d .

f i l e i s U -a - = l - e ( 1 - a K ) U 1 where T = t / 8 t and K must be determined by t h e boundary conditions.

The boundary conditions which the v e l o c i t y p r o f i l e must s a t i s f y a r e

- - a; a2 G

a +m: u + U l , - dT + 0 ,

a . r Z + O

The f o u r t h boundary condition was obtained by e v a l u a t i n g the momentum equation a t the w a l l f o r the case when p i s p r o p o r t i o n a l t o ( I f p i s n o t assuped proportional t o T , the c a l c u l a t i o n s become T .

c o n s i d e r a b l y more complex).

Equation (11) s a t i s f i e s the f i r s t t h r e e boundary conditions and t h e f o u r t h g i v e s

1 1 1 + 2A)

K = - -

2 (1 + A)

where h h a s been s u b s t i t u t e d f o r the combination of v a r i a b l e s I n s e r t i o n of the v e l o c i t y p r o f i l e i n t o t h e momentum i n t e g r a l r e l a t i o n l e a d s t o t h e d i f f e r e n t i a l equation - = d h e ds U 1 y i e l d This equation can be solved by i n t e g r a t i o n t o where and is a known q u a n t i t y .

Equation (13) thus i s used t o o b t a i n h by a t r i a l and e r r o r pro- Quantities such as Cf or 6, which can b e expressed i n terms of cess.

Tl/To, and the Reynolds number, can then be determined.

A, vo/ul,

APPENDIX B SIMILARITY BETWEEN COMPRESSIBLE AND INCOMPRESSIBLE EQUATIONS Examination of t h e transformed momentum i n t e g r a l , equation (10) of Appendix A, has shown t h a t i t has t h e same form and boundary conditions as t h a t f o r incompressible flow w i t h s u c t i o n . The Karman momentum i n t e - g r a l f o r f l a t p l a t e flow w i t h P r a n d t l number of one and having a s u c t i o n v e l o c i t y vo a t the w a l l i s The boundary conditions which must be s a t i s f i e d by the v e l o c i t y p r o f i l e which i s used i n equation (1) f o r incompressible flow a r e The second boundary condition i s obtained by e v a l u a t i n g , a t t h e w a l l , the momentum equation The transformed compressible momentum i n t e g r a l is

- -

A f t ( ; G1 - c2) d t - R e u1 vo = (E)

(3) ds where t i s a normal d i s t a n c e parameter i s a measure of t h e boundary l a y e r t h i c k n e s s 6,

- U Po- L

u = Re = L = c h i i r a c t e r i s t i c l e n g t h of p l a t e .

(3) may be w r i t t e n as: Equation L e t t i n g Jr = L t and t h e upper l i m i t be m, we have This equation has been transformed in y only.

The boundary conditions t o be s a t i s f i e d a r e

*

Jr = 0: u = 0, v o ($)o = E ($)o

a U a Z u

J r r ~ : u - 3 u - + o , -

1, a$ a@ -+

*

This boundary condition was derived using the v i s c o s i t y r e l a t i o n Fortunately, f o r the wall t e m p r a t u r e s of i n t e r e s t i n t h i s p/po = T/To.

those near t h e temperature of l i q u i d n i t r o g e n ) , this simple p r o j e c t (i.e., r e l a t i o n e x a c t l y s a t i s f i e s the accurate Sutherland r e l a t i o n i n t h e v i c i n i t y of t h e w a l l . The S u t k r l a n d law may be w r i t t e n (Footnote continued a t bottom of next page) 5 1 Thus, t h e momentum i n t e g r a l and boundary c o n d i t i o n s have t h e same forms i n the transformed compressible a s i n t h e incompressible case.

I n both cases, t h e boundary l a y e r form f a c t o r € ? C / O i s a f u n c t i o n of 5 , t h e nondimensional d i s t a n c e along t h e p l a t e , which has previously been defined. Because of t h e s i m i l a r i t y of t h e y-transformed compressible equations and the incompressible equations, 5 evaluated a t the w a l l i s equivalent i n t h e two cases. A s shown on page 11 of t h i s r e p o r t , t h e incompressible and compressible 5 ' s a r e then r e l a t e d by t h e equation P O where s i s a constant depending on t h e gas considered, and t h e s u b s c r i p t o r e f e r s t o w a l l values. The term i s of course 1 a t t h e w a l l , and t h e r a t e of change of t h i s term a t the wall i s 1 T O + S

$ [E ~ O + + s s ) ] O = [- 2 T + s

= [ * - * I ( $ ) 0 = O

f o r s = To = 100°K f o r nitrogen.

REFERENCES Lee, John D., "Axisymmetric Nozzles f o r Hypersonic Flows," 1.

WADC TN 59-228, The Ohio S t a t e University Research Foundation, June 1959.

2. Hurlbut, F. C . , and D. E. Beck, "New S t u d i e s of Molecular S c a t t e r - ing a t the S o l i d Surface," University of C a l i f o r n i a (Berkely) Technical Report HE 150-166, August 1959.

D r i e s t , E. ,R., " I n v e s t i g a t i o n of L a m i n a r Boundary Layer i n Van 3.

Compressible F l u i d s Using the Crocco Method," NACA Technical Note 2597, January 1952.

Lew, H. G. and J. B. Fanucci, "On Laminar Compressible Boundary 4.

Layer over a F l a t P l a t e with Suction o r I n j e c t i o n , " J o u r n a l of the Aeronautical Sciences , pp. 589-597 , Volume 22, Number 9 , September 1955.

5. S t a l d e r , J. R., "The U s e of Low Density Wind Tunnels i n Aero- d y n a m i c Research," Rarefied G a s Dynamics, Proceedings of the F i r s t I n t e r n a t i o n a l Symposium, h e l d a t N i c e , Pergamon P r e s s , 1960.

6 . Lew,H, G. and F. B m n o , "The Compressible Lsiainar Boundary Tdyer . over a F l a t P l a t e w i t h Uniform Suction and H e a t Transfer a t t h e w a l l , " PIBAL Report No. 132, September 1948.

7, Enkenhus, K. R., "The Design, Instrumentation and Operation of t h e UTIA Low Density Wind Tunnel," U T I A Report No. 44, I n s t i t u t e of Aerophysics, U n i v e r s i t y of Toronto, June 1957.

8 . Rogers, Kenneth W., "Preliminary Experiments on a Low Density Hyper- s o n i c Wind Tunnel Using a Cooled Proous Nozzle and Diffuser," USCEC U n i v e r s i t y of Southern C a l i f o r n i a Engineering Center, Report 65-48, November 1962.

9. Durand, J. A. and J. L. P o t t e r , "Calculation of Thicknesses of i n Axisymmetric Nozzles w i t h Low Density, Laminar Boundary Layers Hypervelocity Flows," AEDC-TN-61-146, December 1961.

10. Johnson, Arlo F., "A Method of Calculating Boundary Layer Thickness i n Axisymmetric Nozzles with Laminar Hypersonic Flow," Sandia Corp.

Report SC-4370 (RR), October 1959.

11. Maslach, G. J. and F. S . Sherman, "Design and Testing of a n Axisym- m e t r i c Hypersonic Nozzle f o r a Low Density Wind Tunnel," WADC-TR- 56-341, U n i v e r s i t y of California Report 150-134, August 1956.

(AD 97178).

5 3 REFERENCES (Cont' d) 12. Probstein, Ronald F. and David E l l i o t t , "The Transverse Curvature E f f e c t i n Compressible A x i a l l y Symmetric Laminar Boundary Layer Flow," Journal of t h e Aeronautical Sciences, Volume 23, No. 3 , pp. 208-222, March 1956.

13. I g l i s c h , R., "Exact Calculation of Laminar Boundary Layer i n Longi- t u d i n a l Flow over a F l a t P l a t e w i t h Homogeneous Suction," NACA TM 1205, 1949.

14. S i v e l l s , J. C. and R. G. Payne, "A Method of Calculating Turbulent- Boundary-Layer Growth a t Hypersonic Mach Numbers ," AEDC-TR-59-3, March 1959.

15. Cohen, Clarence B. and E l i Reshotko, "The Compressible Laminar Boundary Layer w i t h Heat Transfer and A r b i t r a r y P r e s s u r e Gradient," NACA-TR-1294, 1956.

Loeb, Leonard B . , "The K i n e t i c Theory of Gases," Third E d i t i o n , 16.

Dover P u b l i c a t i o n s I n c . , 1961.

1 7 . Liepmann, H. W . , "A Study of E f f u s i v e Flow," Reprinted from Aeronautics and A s t r o n a u t i c s , GALCIT P u b l i c a t i o n No. 486, 1960.

18. Brown, DiNardo, Cheng, and Sherwood, "Flows of Gases i n Pipes a t Low Pressures," Journal of Applied Physics, October 1946.

19. Shapiro, A. H . , "Compressible F l u i d Flow," Volume 1, The Ronald Press Company, 1953.

20. Cohen, Clarence B. and E l i Reshotko, "Similar S o l u t i o n s f o r the Compressible Laminar Boundary Layer w i t h Heat Transfer and Pres- s u r e Gradient," NACA-TR-1293, 1956.

.

21. Chuan, Raymond L., "Research on Rarefied Gasdynamics and Plasma- dynamics," USCEC Report 83-101, U n i v e r s i t y of Southern C a l i f o r n i a Engineering Center, September 1962.

22. Kosterin, S. I., N . I. Yschenk, N . T. Belova, and B. D. Kamaev, " I n v e s t i g a t i o n of t h e E f f e c t of Rarefied Supersonic Flow on the

T o t a l Pressure Readings of Impact Probes," Engineering - Physical

Journal, Vol. V, I s s u e 12, December 1962. (Russian).

Lew, H. G., "On t h e Compressible Boundary Layer over a F l a t P l a t e 23.

w i t h Uniform Suction," Reissner Anniversary Volume, Contributions t o Applied Mechanics, Edited by the Department of Aeronautical Engineering and Applied Mechanics of t h e Polytechnic I n s t i t u t e of Brooklyn, Published by J. W. Edwards, Ann Arbor, Michigan, 1949.

NASA TM X-53008 APPROVAL February 18, 1964 THEORETICAL AND EXPERIMENTAL INVESTIGATION OF BOUNDARY LAYER CONTROL I N LOW-DENSITY NOZZLES BY WALL SUCTION AND COOLING

*

M. R. B o t t o r f f and K. W. Rogers The information i n t h i s r e p o r t has been reviewed f o r s e c u r i t y c l a s s i f i c a t i o n . R e v i e w of any information concerning Department of Defense o r Atomic Energy Commission programs has been made by t h e MSFC S e c u r i t y C l a s s i f i c a t i o n Officer. This r e p o r t , i n its e n t i r e t y , has been determined t o be u n c l a s s i f i e d .

T h i s r e p o r t has a l s o been reviewd and approved for t e c h n i c a l accuracy.

Chief, Aerodynamics D i v i s i o n i 7 / .

L ’ & c ; -7. , A & E. D. Geissler D i r e c t o r , Aero-As trodynamics Laboratory

*

Engineering Center, U n i v e r s i t y of Southern C a l i f o r n i a DISTRIBUTION INTERNAL EXTERNAL D I R Ames Research Center .

NASA R-DIR M o f f e t t F i e l d , C a l i f o r n i a R- P&VE-D I R Manned Spacecraft Center NASA R-QUAL-DIR Houston 1, Texas R-RP-DIR AED C Tu1lahoma , Tenn.

R- TE ST-D I R Celestial Research Corporation R-AERO 1015 Fremont Ave.

D r . G e i s s l e r South Pasadena, C a l i f o r n i a M r . Murphree Attn: D r . R. Chum M r . D a h M. R. B o t t o r f f M r . Holderer K. W. Rogers M r . W. W. Vaughan M r . Scoggins S c i e n t i f i c and Tech. Info. F a c i l i t y ( 2 5 ) M r . Ballance Attn: NASA Representative (S-AK/RKT) M r . Felk P. 0. Box 5700 M r . C l a r k Bethesda, Maryland M r . Baur Langley Research Center (NASA) MS- IP Langley A i r Force Base, V i r g i n i a MS-IPL (8) L e w i s Research Center 21000 Brookpark Road MS-H Cleveland 3 5 , Ohio cc-P Goddard Space F l i g h t Center (NASA) HME -P Greenbelt, Maryland Jet Propulsion Laboratory 4800 Oak Grove Drive Pasadena 3 , C a l i f o r n i a