- ---- --

BRL 1086 c . 1.A

k f l llllnllllu - - - ..IIIlIli. 11111111111111'"'~~Y Uhr "111111"

lit1 nnl~~~nn nnnn~~,. IIIIIIII~IIII IIIIIIIIII,. IIII~IIIIII 111111111111 llllllllllll~ ll11111111ll11 #IIIIIlIII1~ IIHIIRIIII

I ttl AllllllllU IIHIIIIIII~' llllllllllllll lllllllllll' IlllIl#IIlI ' [;.$I 11 11 11 11 11 11 II I1 !I 1 ' 11 11 11 11 11 11 11 II IIIII~" 11 11 11 I1 11 11

D 11 11 11 11 11 11 !!I! t! " ll llllllll llllll 11 ul!"" 11 11 11 11 11 11

I I:'FI ll11111111ll IIIIIIII~, llllllllllllll IIIIIIIII HHlllllll 1111 1111 IIIIIIIIIIH h IIIIIIIIHIIII IIIIHIIIIII IIIIIIIIIIII 111111 IIIHlllllllt Illl#IIIIIIIl 111111111111 llllllllllil 11

El!] . i IIIIIIh~~II~'" lRY RIIRll llllllllllllrl~ .Illlllll.,.,lrllllll

1 @

1111111111" ll I1 H H H ll ll ll k 'I H U II I1 '. d H H H Y ll ll l U ll ll il ll I1 ll iw gr;l .' ,

511 IANlIARY 1960 -. -. .-- [;$I 1'1 [,if1

[';I .. .. 181 k:C] ,

kf:j 1:Lj W . ,

fg] Ffi;]

[;g;l If1 . C

. ,: . ... I$] i'ra nnnadmnnt W G ~ U I L I I I ~ I ~ r VI nf t h o I I n r Arm11 IIIJ Prniod I I V J W V S hh IVU. , 5Rn?-O?-fK)l W W H

r r . - A _ - - - - . A , - - - ~ - ~ - + C + r n n e + m l ~ f i P A ~ P ~nIn. 11.814

[fj urananct: lv\arlayt:~~rc;~ 11 air ULIUI G LUUG /u*u

2 ; . ,. BALLISTIC RESEARCH LABORATORIES

snnmrr rnn n r r r n h n l h l ~ ~ ~ n ~ ~ C ) n\J(/ \IdDIARIC pQAn)FhrrC I A b U 3 r V K vrIrnrv\IlunI I U I ~ ur I Luvr n u ~ I \ ~ W I L I I I J

o r u I \In rI ID\rrn c u n p K WA\/FC DLII I I Y V b U l \ V ~ v J I I ~ W I \ r r n v LJ

LI-&L,-- f ' n m m h n w IYdll ldl l Ut:l Utfl

Joan M. Bartos

ABERDEEN PROVING GROUND, MARYLAND

B A L L I S T I C R E S E A R C H L A B O R A T O R I E S

REPORT NO, 1086

JANUARY 1960

TABLES FOR DETERMINClTION @ FLOW VARIABU GRADIENTS BEHIND CURVED SHOCK WAVES

Nathan Gerber

Joan M e Bartos

Department of t he Army Project No. 5BO3-03-001 Oriimnce Management Structure code - 5010.1l.814

(Ordnance Research and Development Project No. TB~-0108)

A B E R D E E N P R O V I N G G R O U N D , M A R Y L A N D

B A L L I S T I C R E S E A R C H L A B O R A T O R I E S

REPORT NO. 1086

~~erber/JMBartos/ebh Aberdeen Proving Ground, Md. January 1960

TABUS FOR DEZERMDUTION aF FLOW VARIABU G R A D m S BEHIND CURVED SHOCK WAVES

ABSTRACT

The shock wave r e l a t . i ~ n s r equations of is0energet. i~

two-dimensional or axisymmetric flows can be combined to yield

expressions for various useful quantities behind a shock wave if

the curvature is known in addition to the position and slope. Tables

are presented here of computations, for y = 1,4, of seven functions of Mach number and shock wave slope, from which the following are

easily computed: 1) streamline curvature, 2) velocity gradient along

streamline, 3) angle between streamline and density contour, and 4) angle between stredine and Mach number contour. In addition, the

Mach number and angle of inclination of streamline are listed. A

Mach number range from 1.1 to 10 is covered by these computations.

A derivation of the shock gradient functions is presented; and

several applications of the calculations are given, including a set

of tables for determining the slope of the sonic line.

- rectangular coordinates - natural coordinates normal and tangential respectively

to stre-ine

2 2 2 1/2 = (cos B + g sin p)

- speed of sound = (ap/&) LIZ

- shock gradient functions

- enthalpy - curvature of shock wave - curvature of streamline - inverse of pressure ratio across shock = P1/P

- Mach number = M1 sin f3

- pressure - velocity - ratio of stagnation densities across shock - entropy - temperature - angle between shock wave and x-axis - ratio of specific heats (= 1.4 for air)

- 0 for 2-dimensiona,l flow 1 for axisymmetric flow

- angle between stredine and l i ne ~f c ~ n s t c q t density

- angle between stredine and x-axis - angle between streamline and line of constant

*I - -I_ ------- 7. men numDer

- density - arc length along shock wave

I. INTRODUCTION

The problem of determiaing the gradients of the flow variables

behind 8 shock wave in ieoenergetic two-dheneional flow hae been considered by several investigators 1 2 3 . ~hese grsdiente are very

useful; for instance, the slope of the streemline at shock polar

points i n the magraph pl-me c a be obbined from them (the Bueemann 4

"hedgehog" ) . A particularly simple derivation wae indicated by Sternberg , wing natural coordinates. It is found that the flow vsriable gradients

are proportiond. to the ehock wave curvature. Tablee of coefficients for 2

etreemline c m t u r e have been computed by Thomas for a limited range of Mach number,

The same gradients can be comguted for axiaynnnetric f l o w . However, theyare now linear combinations of the two shock curvatures, Kg in the

meridiaPal plane and l/r in the azimuthal plane. h he two-dimensional in obtained saqly by the lIr term.) Beesues of Ws

the general usefulness of these gradient functions is restricted in that

one can no longer, as in two-dimensional flow, obtain certain quantities

(e.g., elope of sonic l ine) without having t o specify the vsluee of r and

Kg, Expressions for the slope of these contours (density, pressure, WBch 5 number) have been given by Wood and Gooderum .

FIGURE 1.1

hide from the results of Thomas cited above no calculations

of +&e & ~ ~ e - ~ n t i ~ ~ d Q u e n t q i m k i t i e e haw to the &ors1 howledge

been published. This report supplies such a set of calculations ("shock

gradient iunctions" ) which can be used to determine significant flow in-

formstion f m m physical .measurements. It is felt that these computations

can also be helpful in obtaining a clearer general picture of flow behind

a curved shock wave. These fU~tion.8 will find further application in the

study of f l o w of a reLaxing ges behind shock waves at large Mach numbers.

The particular quantities to be obtained here are (using the notation

of Fig. 1.1) : 1) bq/as, the velocity gradient along the streamline j 2) \A /\ - t rL, ,l,,, ,,,-I, ~ l o s (I 91, the cimvatiire of the etreamline; J) Q + 5, bu: asupc: -LC

of the isopycnal, or density contour; and 4) Q + k, the slope angle of the Mach number contour. It is assumed that the gas is ideal, and that the P7n-r J - n --4 - ~ r r - r - d 4 4 n a l ~ . . ~ r r a + ~ a a r n l . t m a a A A U W AD u - A auu r D c u u A u y r G arvq D W A G ~ A ~ L G O ~

The above flow quantities are obtainable from expressions having the

following simplified forms:

n f a . a\ v a\ f~ 1-1 9 M13 PI 4\13 53 P I

tan 5 - - + cF6(7, Ml, (lIrf

4

~ ~ ( 7 , M1, B) Ks + E ~ ~ ( 7 , M1, B) (l/r) tan L - - n i (7, M ~ , 8) K~ + E ~ ~ ( 7 , xl, (l/rJ 7

where e is equal to zero for t w o - m i o n a l flow and one for axisymmetric P 1 rrT-. LAUW. q & s, m e +he nqmer velocity, respectively, in t h e

free stream; tan f3 is the slope of the shock wave; 7 is the ratio of specific

heats. The angle Q = Q (M~, j3) is known from the shock wave relations e . g = , ~ e f . 6 ) :

The gradient f'unctions Fly F2, ...., F are derived in detail in 7

SectionIf. These functions for the most part are long and complicated

expressions in M and which do not lend themselves to analytical treat- 1

ment or easy hand computation. It is, however, feasible to calculate them m on a high speed computing machine. Thus the functions F 1" F2, . o . . , E ~ ,

8 , and M have been programmed for computation by the EWAC in the Ballistic

Research Laboratories, and they can be obtained quickly for any particular

case of supersonic flowby specifying H 1 and 7 i n t i e computing machine input.

The simple form of the expressions in Eqs. (1.1) suggests the practi-

cability of a compilation of the coefficients of Ks and l/r which could be

employed as a labor saving means of determining the desired quantities.

To this end, the present paper provides a series of tables of shock wave

gradient functions versus f3 for 7 = 1.4, with M1 as parameter.

The angle f is of special interest to the authors, particularly in calculating a flow field (velocity, streamlines, etc . ) from given inter- ferometric density data (see Ref. 5, 7, and 8). The information resulting

from the measured and Ks reduces the uncertainty in density near a shock

wave which is inherent in the interferometric method.

Sample hand computations have verified that the formulas used in this

report yield the as form-fias given in the referenceso

11. DERIVATION @ GRADIENT FUNCTIONS

m L - I--- m e present derivation follows the one indicated i n Ref. 4. In

natural coordinates the equations of isoenergetic flow of an ideal non-

viscous gas are

(b)

(c)

where e = 0 for two-dimensionrcl, = 1 for axisymmetric flow. The coordinates s r; leFg+,hs d o n g the stredine dlF4 the =r-t-hcg-nd ?-ra:ect=

ory of the streanline, respectively; q is the speed of flow, and Q is the

angle between the streamline and the free stream direction (x axis); p is the

density, and p the pressure, T is the temperature; M is the Mach number

( = o/a, where a = (bp/&)112 = speed of sound).

The two essential equations to be used here are

(a)

(b)

obtained from Eqs . (2.1) and the definition of the velocity of sound. he foregoing equations are given in Ref. 9 and 10.) Entropy remains constant

along streamlines,

If a is arc length along the shock wave: it is seen from the relation

that at the shock wave (2.3)

a a ag a a a = ( 1 = K s = cos (B - Q) + sin (p - 0)

where and K m e the slope and clmv%me, respectively, of t h e s

shock wave @ig. 1.1). As the coordinates are defined, curvatures are

positive where curves are concave upward and negative where concave

By means of Eq. (2.3), Eqs. (2.2) are replaced by two simultaneous

linear algebraic equations for and &/as. The coefficients of these

equations are now converted into the desired form; namely, that of Eqso

(1.1). For this purpose use is made of the following definitions and

relations which can be found in many textbooks and reference books (the

authors employed Ref. 6 mainly) :

The subscript 1 always refers to free stream conditions.

9 = A GI1'

3 3 => 1 /2 where A = (cos- B + g- sin- B)-'

After differentiation for aq/&, a@/?@, and as/@ (noting that &/as = o), one obtains expression for F1, F2,FJ, and Fq given by the

following sequence of formulas I

( l / q , ) -L - a d a s = F, K- + E F,, (l/r) 3 s 4

The following re la t ions , obtained from the flow equations, a re used

to find ( and 5 :

where the subscript t denotes the stagnation value (apt/& = apt /& = 0) ,

J . / ( ~ - ij - ij In = p / n = ~ = i - - g rtl t' rtl (see R e f . 6)

By means of Eqs . (2.3) and (2.25), ap/as and &/an can be expressed

in terms of ap/bs, ap/an, and other quantities already determined in this

report. Substituting these expressions into Eqs . (2.1), one then derives

expressions for ap/as a d &/an; a d putting these into the relation for

the angle between the density contour and the streamline, namely,

tan f = - (ap/hs)/(hp/an) (2.26)

one obtains F K + E F~ (111)

3 tan ' = I? Ks + E F6 (1/r) where

5

- 4 cos p F5 - r + l

A positive value of f indicates a counterclockwise mtation from the

streamline to the density contour.

The angle 5 between the Mach number contour and the streamline is determined fmm

tan s = - (aM/as)/(aM/an) = - (d/as)/(d/an), where k has the same sign convention as 5 . since

Id2 = (~d)/(*), &?as and &?/an can be expressed in terms of the derivatives of p and q,

which in turn are expressiblein terms of previously determined quantities.

Conseqyently one obtains

F K~ + E F~ (l/rj tan 5 = , where

F? K + E F6 ( l I r ) f s

2 I ( A 3 7 1 m 2 / n ) + 2 A g h p 1 - 2

s i n p 1 J

111. DISCUSSION (X? C-TIONS

Tabulations of the functions Fly F2, . . . . , F 7' 0, sad My calculeted

f o r y = 1.4, are presented in Section 4 fo r values of M1 - rang* from 1.1

+,c lQ.Qe 'Fhp ~ m c t l n m are l i s t e d in order of decreashg f3, from f3 = 90'

t o a value somewhat greater than p = ~ i n - ~ ( l / ~ ~ ) . The l i s t e d values of Y andp were chosen so as t o permit the determination of the functions t o a

reesonsble degree of accuracy f o r any M1 and p by means of graphical inter-

polat ion.

Figure 2.1 shows an example of resul t s obtainable from these calculations. & L,,J - * +hfi Trarqcl+qr\n u L A I~ZULU L~~~ r u v w r rur .,,,,,,, in slope of the constant density l ines

along the shock wave of a supersonic sphere i n nitrogen (shorn a t the l e f t )

determined from shock wave coordinate measurements a,nd the appropriate shock

gradient functions. For reference purposes, the left hand f igure shows the

sonic point and "Crocco point" (defined below), and indicates the location

of typical values of rKa. It must be noted tha t the accuracy of resui t s - like those in Figure 3.1 is res t r ic ted by the l u t e d accuracy of and Ka &a

determined from experiment. (see Ref. 5 and 8 for discussion on deter- mination of B and KB .)

A point often referred to in two-dimensional flow studies (e.g., Ref.

3) is the "~rocco point", where the streamline curvature is zero. Figure

3.2 shows the geometrical conditions which must exist at the Crocco point in plane and axisymmetric flow for y = 1.4. It can be seen that in two-

dFmensional flow the Crocco point dways lies below the sonic point on a

shock wave of continuosuly decreasing slope. The two points approach

coincidence with increasing Mclch nurriber. On the other hand, in axisymmetric

flow the sonic point lies below the Crocco point for all values of M1 and

rKs shown. By continuity there would be values of these two parameters

for which the sonic point lies above the Crocco point.

The slope of the sonic line is a quantity of considerable interest. It

can be otained from the tables by evaluating the f3 and shock gradient

functions corresponding to M = 1 (e.g., by graphical interpolation); it is

found to be given by

where B 1, B2, and B are given in Table 3.1 for 7 = 1.4. It is seen that, 3

formally, rKs = - w corresponds to two-dimensional flow. This particular

case is plotted, in slightly different form, in Ref. 11.

Figure 3.3 shows plots of the variation of the angle between

the streamline and Mach number contour at the sonic point. For two-dimensional

flow 5 is negative, indicating that the sonic line has a smaller slope than the streamline. For axisymmetric flow the figure indicates that the sonic

line can have a greater slope than the streamline. st he streamline curvature must be positive in this case. This can be seen from Eqs. (2.lb) and (2.1~)

and the fact that dq/do ( 0 and dM/du ) 0 along a shock wave with continuously

decreasing slope.) If rKs = -.2 at the sonic point, the sonic line and

streamline are nearly tangent to each other over a very wide range of Mach

number.

---..... Sonic point

Figure 3.2. Conditions for zero c u r v o t ~ r e of streamlrfie

id8 idss0 j a t shock wave i c r o c c o ~ o i n t j .

(NOTE: FOR THE r K c O CURVE, 8 +(= 8 )

80 1 I 1 1 1

I: I I J- - - - t - - -q- - - - I I i I - - - i R 0 -e

- 0 0 f I'

! A I \\\ I I I I i I i I I I - 1 -

--a . I I I I

rK,= O w v I ,. S- = -."& - na

r K s s - . 0 5

I.' I I

,T 1 1 rKes-.2 .I

I tK,=-.5

I 1 I r-I r K , z = a

-2Ci-i I I i / ' v= I/ n I I I I I 1 I

,*I - 1 I I I I

1

I S POSITIVE F O R A COUNTERCLOCKWISE I I I I / O ~ T A T I ~ \ M F P ~ U TuF 9 T R C A M I INE 10 THE I I I I / n~ 1 lvl- r \ v r ~ w rn I. w . - .--.-- -.---

- 6 0 1 ~ 71/,~wC LINE. E

II 1 Y ! I 1 I !

I I I II I A

I -80 r CORRESPONDS TO TWO:

I I / I ( m w u s , o N * L FLOW. 1 I I' / I I I

i i I I I -100 5 I 5

1

I 2 3 4 5 6 7

"": -.A,. A - r r r A ~ h p t \ A , C C l 1 e T m C I M C

FIGURE 3.3. V A R i A l IVN ur R N ~ L E oc I n c c l ~ 9 1 n r r r ~ w ~ c ~ u . ~

AND SONIC LINE AT SHOCK WAVE.

18

me vsl..aa -9 li errs a l - A ..lrrrA.l 4 - P-l-A-t-rr +LA C . ~ A I I I A* &LA A U G V C Y - 0 UL A.1 CLIC OIPV WCIW AU A ALLCLLU& bALC OIUYeD U I

characteristics at the shock wave in the supersonic portion of the

flow field.

For quick reference, Eqs. (1.1) are resteted here.

TABU 3.1

Sonic Point Table ( 7 = 1.4)

The authors wish to express their appreciation to Dr. Rapnd Sedney

for his helpful advice and to Mr. P a h e r Schlegel for programming and

performing the WAC calculations.

1. Schaefer, M., "The Relation Between W a l l Curvature and Shock Front Curvature i n Tvo-Dimensional Gas FW' Tech Report No. F-TS-1202-u, A i r Materiel Comnnand, U. 9. A i r Force, 1949.

2. Thomas, T. Y., "Calculation of the Curvature of Attached Shock waves" Jour . Math. and Phys . , 3~ 279 (1948). (Also see Jour . Math. and Phys . , 26; 62 (1947) .) -

3. Sears, W. R e , Editor, " ~ i g h Speed Aerodynamics and J e t FYopulsionii , Vol V I 678). Princeton University Press, Princeton, New Jersey (1954)

5. Wood, G. P., and Goodem, P. B., " ~ e t h o d of Determining I n i t i a l Tangents of Contours of Flow Variables Behind a Curved, Axially Symmetric Shock waveii. NACA TN 2411, ( ~ u l ~ 1951).

6. Ames Resesrch Staff : "~quat ions, Tables, and Charts for Compressible F W NACA R-1135 (1953).

7. Bennett, F. D., Carter, W. C ., aad Bergdolt, V. Em, "Interfernmetric Analyeis of Airflow About Projectiles i n Free ~ l i ~ h t " . Jour. Apple Phys. a 453, (1952). Aberdeen Proving Ground: BRL R-797, ( k c h 1952).

8. Sedney, R., Gerb er , N., and Bartos, J. Me, "~umerical Determination of S t r e m i n e s f l.om Density Data'' . Aberdeen Proving Ground: BRL R-1073 ( ~ p r i l 1959) l

9. Lie-, H. W., and Roshko, A., "Elements of ~asd~namics". John Wiley and Sons, New York (1957).

10. Lin, C. C., and Rubimv, S. I., "On the low Behind Curved shocks" . JW. w h . and P-w., 105 (1948).

11. Hayes, W e Dm, and Probstein, Re F . , " ~ ~ p e r s o n l c Flow ~heory" (P. 207). Academic Press, New York (1959).

IV - TABUS OF SHOCK GRADIENT 3FIJNCTIONS

00.00 00 .86 01 .72 02 56 03 .39 04.20 0 4 . 99 05.75 06 .48 07.17 0 7 0 8 2 nn 'nn V 7 . V V

10.00 1 0 . 8 1 11 .42 11 .84 12 .06 11 .16 09 .26 06 .44 . .

' l i ' B Z' j+ ti' w ?3 '1 '2 ' 6 ' 7

n n n n n ~ U U U U U 0 0 9 6 8 4 n n n n h ouvuuu Oo0030 0 0 0 0 0 0 (jogc(jfi O08G6 009553 -001675 000928 -eOO154 0 ~ 1 7 7 3

Ti' v. "4 3'

5 '6 F P- F 1 ' 2 5 7

nnnnn 0 U V \ J \ I U

3 0 9 5 : .?364-3 0 6 7642 .I2377 17288

0 2 1 9 2 7 7 r A ? ?

0 L O L J L J L .

2 9 3 6 6 .?I986

Y 3 ' ; " C . A

355,;tj

.3,3254 3 5 4 9 4 - r ~ n n

0 3 3 Y b O

3 2 3 1 3 3 7 3 c r ,

b L 1 i u w

,22589 , 2.2483 ,14360 .ll960

A T - r o b I L O O

? & I 7 i . U - / - r l A.

,03995 .02812

nnnnn n nnnn e w u v u Y L J P U U V U

- 0 0 0 1 6 7 0 0 1 9 2 3 - . 0 0 6 5 0 003761 - 0 0 1 3 9 4 0 0 5 4 4 4 - 6 0 2 3 2 7 0 6 6 9 2 8 -003371 O a 8 1 9 4 - 0 0 4 4 5 3 0,9248 - a 0 5 5 2 0 1.0110 -0c6521 1 0 c 8 e 8 - , 0 7 4 5 2 l e 1 3 7 4 - 3 O E 3 0 1 1 . 1 8 3 4 - 0 9 9 5 9 1 1.2535 - 0 1 0 7 2 6 1 0 3 0 5 7 -011457 1,3fi-90

n , , f i r - 0 1 1 y - 3 1 0 3 u u 3

m e 1 2 2 3 1 l o 4 2 6 6 - , I 2 3 2 3 1,5222 - = l l 8 I f i 1.6176 - 0 10942 1 0 7 0 6 5 - a 0 9 8 3 7 1 0 7 8 0 4 - 0 0 8 5 9 8 1 0 8 2 8 7 - - - , 0 7 3 0 0 l , a 3 u l

n r n n ~ - ~ U O U U O 1 0 7 6 9 3 = , 3 4 7 7 0 I O U J - I U 1 L C ~ L

- 0 0 3 6 3 9 1 0 3 7 8 0 - r 0 2 6 5 3 0 , 8 6 6 4

NO. of Copies

NO. of Organization Copies -

Director 1 Iu'ational, Aeronautics and Space Administration

T m n m l nv Raaasnnh IIA-+A- r r = o c - bu uCUbCL

W 1 e y Fierl , W r g i a g Attn: . G. P, W o o d - 1

Director National Aeronautics and Space Administration

1cnn w H L - - - - L -T T T

1 I;)CU n DxreeT;, n. w. Washington 25, D. C. Attnr Diviainn of Rese-ch

Informat ion I

Army Research Off ice Arlington H a l l Station -- - Ariington, vrrginia 1 A &A-

a b b L 1 i Lt. Cole $0 e

Commending General Army Ballistic Missile Agency 1 Reds tone Arsenal, Alabama Attn: Technical Library

Commander A-tr Rnnlrcl+ A Il..-la~a r u v r b u b n x z u cluu ULLIUCU

Missile -Agency - I

Redstone &se~al, Alabama Attn: Technical Library

o m - O T L

com&iw (yenerd

A m t r nr.Anam,a ~ -1ao .11 - f i r r - n - ~ - u a + J W r - A b G A . A A . O O A I C UU-U

Redatone Arsenal I Alabama Attn: Major D. H. Steininger

Commanding General White Sands Missile Range 1 Nnw Mnv-Inn w A . 4 G A A c . V

Attnt flRTTR.q-Ts -TE

Zlos Alanaos Scientific Laboratory

P. 0. BOX 1663 Las m-s, New &Aoo

A h D, Tnn -- -- , I.-. 30 Memorial Drive Cambridge 42, Massachusetts

Applied Physics Laboratory &2i Georgia Avenue Silver Spring, Maryland

AVCO Research aborgtcry Wilmington, Massachusetts Attn: Dr. Mac C. Adams

Boeing Airplane Company Seattle 14, Washington A&&-. abbLLi ]jr. 3. w e r ~

C O W ' Division of General Dyaamics Corporation

Ordnance Aerophysics Lab. Daingerfield, Texas

CO*AE. Division of General Dynmcr ! Corporation

Sasl Diego Division San Diego, Calif ornla ~ttn: R. D. Unneii

Cornell Aeronautical Lab., Inc. 4453 Gemgee Street Buffalo 5, New York Attn: Dr. F. .K. Moore

Cornell University G r & u t e School of

Aa-rrmrrr .+- t - 1 7 ----a -- G A A r u 4 & A 1 1 c c z 4 - I 4 5

IthA-c&, New Yerk Attn: Prof. W. R. Sears

DISTRlBVl!ION LIST

Chief of Ordnance Degart~nt of the Army Washington 25, D. C. Attn: - m- See

C o m d i n a Officer Dianond Ordnance Fuze Lab. Washington 25, D. C. Attn: O m L - 012 Office of Technicd

a . . ---3 -- DervlceE

n--ar*+mar?+ cf 'C~mmerce us- -LA-

Washington 25, D. C.

Dlrector Armed Services Technical Information Plgency

Arlington Hall Station A --1 r , ,A-, 7 V 4 Wmq m-1 ArLrLl& W L L , v r l B-A-

A t t n : T E R

British Joint Services Mission

1800 K Street, N. W e Washington 6, D. C. Attn: Reports OPficer

Cnn&sn Staff 2450 Massachusetts Avenue Washington 8, D. C.

Chief, Bureau of Naval Weapons &partme-' -0 LL- ?a '"' n UI ~ U C r1avy + , D i ;

Commander U. S. Naval Weapons Laboratory Dahlgren, Virginia

NO. of Copies Organization - - -

2 Commander \TI--- 1 fkr7.~1mm/.f i T . m h r \ . ~ ~ l . f . n y ~ L ' I a v w V l WroUbb -uv.lu---J

mite O&k Silver Spring, Maryland Attn: Library

Commander Naval &dnance Test Statioii China -e, Cd;forris

Attn: Technicd Library

Superintendent Naval Postgraduste School Monterey, California

Commander A 3 -- TT-3 " 4 +.r nu- UULVCL-DL UJ = A i r Force Base Alabama Attn: Air University Library

Commander Air Research a d b v e l ~ p ~ c t Command

A - d r r l r r . r a finre@ R u e r S I L u - G w D & V* -- -- - W~shtggton 25, D. C

1 Director National Aeronautics and Space Administration

k w i s Fl ight P r o ~ d ~ i ~ Z l T ~ K L U W A n h r \ w m + n r l r u UWL J

Cleveland Airport Cleveland, Ohio

1 Director National Aeronautics and Space Administration

A,,- D - n a m w n h p m n t . ~ ~ AJIlCD n vrrr.vu-

gcffett F i e l d , California

DISTRIBUTION LIST

No. of Copies

No. of Copies Organization Organization

California Institute of Technology

Gi-genheb A e r ~ p ~ u + , i ~ a l h h . 1500 Normandy Drive Pasadena 4, California Attn: Prof. L. Lees 1

Prof. H. W. Liepmann

Jet Propulsion Laboratory California Institute of

rn,,L--7 ..--- A c L L U L U Lu&y

Pasadena 3, California

Lockheed Aircraft Corporation Research and Development Laboratory

Missile Systems Division Palo &to, California Attn: Dr. D. Bershader

1 . Duke University Box CM, Duke Station Durhan, North Carolina Attn: Dr. R. J. Duffin

Glenn L. Martin Company Baltimore, Maryland Attn: S. C. Traugott

Douglas Aircraft Company 3000 Ocean Park Blvd. Santa Monica, California A++- e . &. II. buskin

Mr. R. J. Hakkinen Space Technology Laboratories, Inc .

P, 0. Box 95002 Los 45, California Firestone Tire and Rubber

company Defense Research Division 1 &on 17, Ohio

Ramo -Wooldridge Thoxpson 57760 Arbor Vitae Street Los Angeles '45, California

General E lec t r ic Company Aeronautics and Ordnance 1 Systems Division

1 River Road Schenectady 5, New York Attn: Dr. H. T. Nagamatsu

Dr. D. R. White

University of Maryland Institute for Ffluid Dynamics and Applied Mathematics

College Park, Maxyland Attn: Mr. S. I. Pai

University of Southern California Engineering Center

I;os Angeles 7, California

General Electric Company MSVD 3198 Chestnut Street Philadelphia, Pennsylvania Attnt Dr. F. G. Gravalos Princeton University

Aeronautical Department Forrestai Research Center Princeton, New Jersey Attn: Prof. W. Hayes

Prof. J. Bogdonoff

North American Aviation, Inc. 12214 Lakewood Blvd. Downey, California Attn: Aerophysics Laboratory

DISTRIBUTION LIST

No. of Copies 7

Organization No* of Copies

1 Brown University 1 Graduate Div. of Applied Mathematics

Providence 12, Rhode Island Attn: Dr. R. Probstein

Rensselaer Polytechnic Institute 2 Troy, New York Attn: Prof. T. Y, Li

Prof. G. H. Haadelman

W t e d Aircraft Corporation Research Department 1 East Hartford 8, Connecticut

University of Illinois Aeronautical Institute U r b a n a , Illinois

Standford University Palo Alto, California Attn: Dr. M. D. Van Dyke

Organization

Dr. L. H. Thanas Watson Scienttfic Computing Iaboratory

612 West 116th Street New York 27, New York

Professor G. F. Carrier H, W. Ehmons

Division of Applied Science Harvard University Cambridge 38, hkssachusetts

Professor F, H. Clauser, Jr. Johns Hopkins University Department of Aeronautics Baltimore 18, Wryland

Dr. A. E. Puckett Hughes Aircraft Company Florence Avenue at Teal St. Culver City, California