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REPORT No. 886

EN I AC Computation of •a

Two-Dimensionai Supersonic Nozzles

NATHAN GERBER

DEPARTMENT OF THE ARMY PROJECT No 503-06-004 ORDNANCE RESEARCH AND DEVELOPMENT PROJECT No. TB3-0108

BALLISTIC RESEARCH LABORATORIES i.^ijirau.'. jn-g

ABERDEEN PROVING GROUND, MARYLAND

THIS REPORT HAS BEEN DELIMITED

AND CLEARED FOR PUBLIC RELEASE

UNDER DOD DIRECTIVE 5200,20 AND

NO RESTRICTIONS ARE IMPOSED UPON

ITS USE AND DISCLOSURE,

DISTRIBUTION STATEMENT A

APPROVED FOR PUBLIC RELEASE;

DISTRIBUTION UNLIMITED.

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BALLISTIC RESEARCH LABOR A i C it I 2 S

REPORT NO. 886

DECEMBER 19^3

I ENIAC CO 'MFJTATI ION OF TWO-DIMENSIONAL SUPERSONIC NOZZLES

\ Nathan Qerber

Department of the Amy Project No. ?03-06-O0U Ordnance Research and Development Project No. TE3-0108

i

ABERDEEN PROVING GROUND, MARYLAND

i

SYTBOLS

x, y rectangular position coordinates

q velocity vector

u, v rectangular velocity components

q, 9 polar coordinates of velocity

q speed of radial flow on initial circular arc

a local speed of sound

y ratio of specific heats (al.U for air)

M Mach number (»q/a) i

0 inclination of bounding ray of radial flow

•fc curvature of nozzle

Oi , p characteristic coordinates (a varies on the patching

characteristics)

j H=a2-u2 L=a2-v2

Ks -uv R= aCq^a2)1/2

X, n direction cosines of straight line (ex = const.) simple

wave characteristic

r length of CX - const, simple wave characteristic

p density

<£ , 7] x, y along characteristic which patches the transition

and simple wave regions

s arc length along transition section of nozzle

I A 8 parameter governing rate of growth of curvature of

transition section

I

BALLISTIC RESEARCH LABORATORIES

REPORT NO. 886

NGerber/lr Aberdeen Proving Ground, Md. December 1°$3

ENIAC COMPUTATION OF TWO-DIMENSIONAL SUPERSONIC NOZZLES

ABSTRACT

A procedure based on the method of characteristics is described in detail for computing two-dimensional supersonic nozzles with continuous curvature everywhere. The results of ENIAC calculations of a series of nozzles by this method are summarized briefly.

J

SECTION I

INTRODUCTION

This report will describe a method of determining two-dimensional supersonic nozzles on the basis of piano iaentropio flow of a perfect gas. The computational procedure is based on the method of characteristics (ref. [lD )•

Fig 1.1

The procedure commonly used for designing these nozzles (ref»|2j) is to patch a "simple wave" onto an assumed initial radial flow to obtain the "cancellation" region which leads to uniform flow. These several regions are indicated in Fig 1.1, which shows a representative two-dimensional nozzle. (Because of symmetry we need only refer to the upper half of the diagram.) The radial flow originates from a two-dimensional source at point 0, the ray OB being tangent to the contour TBJ at the "inflection" point B. BD and CJ are characteristics (or Mach lines)*. The problem is the determination of BJ•

Nozzles designed in the above manner xri.ll, in general, have discon- tinous wall curvature at points B and J (Fig 1.1). While this condition is permissible for fixed throat tunnels, it is unsatisfactory for flexible threat tunnels, the walls of which must have continuous curvature everywhere. The process used in the calculations described in this report avoids

* See, e.g., pp. 259 - 286 of ref. [3J for discussion of simple waves and Mach lines.

p*

this trouble by inserting between the radial flow and the simple

Fig 1.2

wave a transition section (EDFE in Fig 1.2) the wall curvature of which varies continuously from zero to the value at the beginning of the cancellation section. The transition section can be fitted in many ways. The particular method used here is described in Section II by eqn. (2.6) and its accompanying discussion.

A series of calculations described in Section IV were performed on the EN1AC to provide design data for the 13" x lf>" flexible nozzle wind tunnel (No, 1) now under construction for the Supersonic Wind Tunnels Branch of the Exterior Ballistics Laboratory , These shapes were selected from a three-parameter ( qQ, 9 . ^9) family of nozzles. The particular choices of parameters were guided by factors such as physical limitations on tunnel lengths and permissible plate curvatures, The object was to get a continuously varying family of nozzles from which could be obtaii«sd reasonable shapes of curves ("jack displacement" vs. Mach number) for the design of cams governing automatic settings of the jacks which hold the wall in place.

To obtain the final family of eighteen nozzles required about one hundred and ten trials at about forty-five minutes per case. Clearly computations on this scale would never have been undertaken by hand.

wSST*

L «—J

I SECTION II

EQUATIONS AND BOUNDARY CONDITIONS

The equations describing the two-dimensional flows are (ref. \l"\t p.12)

(2 .1) H 3u/dx + K (6u/3y + ov/5x) + L 6v/3y = 0 \

ov/dx - chi/5y = 0

2 2 where H = a -u .

2 2 K = -uv, L = a -v .

The flow field is divided into four regions (Fig 2.1): I — radial flew, II — general two-dimensional flow, III — simple wave, and IV — uniform fl~*r. These regions are separated by the characteristic curves

Fig. 2.1

BD, EFj and FJ; BB' is a circular arc.

The boundary conditions are

a) v«0 when ym0;

that is, the air flows along the center line of a cross-section of the nozale.

b) v/u « f (x) along a contour f (x), which is to be determined as part of the solution to the problem. This contour is sub- ject to the following conditions: f' (x) = tan 9Q, f" (x) = 0 at point B; f (x) • 0 at point Jj f (x) has continuous curvature-^ (x) everywhere between B and J»

u

f

We first establish the coordinate system. The origin is determined so that Xg • 0, yv. = 1 (Fig 2 .2).

+-X

Fig. 2.2

The initial information at our disposal is the radial flow, with speed q on BB". Since the streamlines coincide with the rays from point 0, 0 measures the inclination of both. The angle 8_ is divided into n, intervals of magnitute cT 9 = 9 /n., where n, is an integer.

Clearly, on equally spaced points on BB1

x(n) = esc 9Q cos (9Q - r& Q/n^) - cot 90

y(n) ••* esc 9 o sin (9Q - tB Jn^

u(n) - qQ cos (9Q - n^/n^

v(n) = qQ sin (eQ - I^/I^)

where n • 0, 1, 2,.-.,n . Accordingly, n • 0 for point B and increases along the curve up to nJ at point B'.

Vre designate c* as the variable etlong BD, the Mach line patching together the regions I and II. In the solution of problems by characteristics the choice of characteristic parameter along certain bounding Mach lines is arbitrary (see ref. [lj, p. 21). Thus we are able to

(2.2)

8

eK»J

f— n

choose Ot=n along DDj that is, (X will vary continuously with 9 fror; zero at B to TL, = C( at D, and at any point P (Fig. 2.2), where 9=^ (l-n/a,),CX

will be equal to n.

Since the hodograph of the plane characteristic BD is a Prandtl- Busemann epicycloid, the velocity distribution on BD is described by

9Q - 9 - f(M) - f(Mo), where

f(M) • cos"1 a/to) - v'(r+i)/(r-i) tan"1 (/M2-! /(r-i)/(m>),

r i / o / FT oH (2«"}) (see ref. [Uj, p. 26) and M(q) = \f2q7' |_(r-l)(l-q-)J > also MQ = M(qQ) .

To determine the curve BD, consider the streamline OP (Fig. 2.2)

oF/or' = yp/yp.

W - csc 9 o

Thus jfp~ OT = (yp/yp.) csc 9Q.

P and P' may be taken as two points on a two-dimensional nozzle sur- face. Then (ref. [£], p. 3U)

7p/7p. - (ApA*) / (Ap'A*)

A/A* is the ratio of the cross-sectional area at any point in a two-dimensional nozzle to that at the throat of the nozzle (according to one- dimensional theory, M»l at the throat) and is a function of q.

k/k = a/to) {2+(i-l)M?}/(>+l)] 0-+l)/(2V"2)

Since qpi ("q0) and qp (cO arc known, /p (or) can be determined.

Let 0 - 9Q(1 - n/i^) - 0o(l -or/cx _,) * 9 (a) at point P.

Then ^ x(o(5 = fp cos (1 -or,^) eo - cot 9(

7(<y) = JTP sin (i -«AX) e0

u(«) = q(oc) cos (1 -oc/fe^) 90

v(oc) - q(6X) sin (1 - O^) 9Q

(2 JO

9

r

.

Consequently x(of), y(cX), u(cx), v(or) are determined at c* +1 points on BD,

We transform to the characteristic (Oi,B) plans to determine the flow in region II. The characteristic equations are (ref. [ij )5

-^ Cy^ =qy-/^, y^ Sdy/sp, etc.) (K-R) 7<x- Lxa = 0

H yp - (K-R)xp - 0

Kuw+ (K-R)vo< = 0 >•

(K-R)uR + Lv - 0 P P V

(2.5)

where F ayK^ -- HL. Fig. 2.3 shows the map of region II in the characteristic plane. The two bounding streamlines BE and DF map into the lines B««

*- oc

Fig. 2.3

and B-o<-ot respectively (dB/d« = 1 on them).

We arbitrarily demand that the inclination of the nozzle on BE be of the form

9=9 .-ex" A9 o (2.6)

where A0 >0 is a prescribed constant. The curvature on BE is

K± = dB/ds = - 2ex AQ/s^ •

At B, or• 0, so-^\ • 0. Thus the condition of zero curvature of the nozzle l 2

at point B is satisfied. More generally 6 = 9 -ex g(cx), where g(o) is an arbitrary function of CX , satisfies this condition.

Since

ds/dpr= (3x/30f) Vl + (dy/dx)2 - (9x/a«)/l + tan2 (9 -CX2A9),

10

r the curvature is

^ . - 2 Ot Ae/["(dx/d<*) /l + ten2 (Oo-w2 A9) 1 (2.7)

The flow in region III is a simple wave; that is, the velocity is constant along the characteristics ex = constant, which are straight lines Consequently the points on EJ, the remainder of the nozale, are simply functions of CX . In particular, 9. •' 8 pC.) » 0, and q^ is the speed of the final uniform flow.

Fig. 2.U

We wish to locate a point, say T, on EJ (Fig. 2«]j.) which corresponds to the point G on EF. Let x • 4 , y = *) on EF. Then XQ - £ (<*), yQ = Y) (<*),

The slope of GT is

m(«) = [(§y/op) / (9x/dp) |Q =• (K-R) /"H (from eqn. (2.5) ).

The unit vector tangent to GT i3 ( A,H), where

X - - H//H2 + (K-R)2 , n = - (K-R) /? + (K-R)2. (2.8)

Then on EJ

x(ot) = £ (a) + \ (or) r (of) - Xj

y(oc) - r) (a) + ^ (oc) r (Of) « yT r (or) = GT

u(ot)

v(0f)

*T

s Vr,

(2.9)

".\Te find r(°0 essentially from the equation of continuity by equating

11

.

the mass per second of air flowing across GT to the mass per second across EG.

(^,T - mass/sec. across GT = p(oc) lu(o<) n (or) - v(of) $Oc)Jr(o<).

(yji - v$ is the scalar product of the vector q = (u,v) and the vector (li, - \), a unit vector normal to the line GT. It is also the component of the velocity normal to the characteristic GT, which is equal to a(©r), the speed of sound *

From ref. [l] , p. 12, p = p (1 - q ) '^ " ' (p being constant 2 2 - for the flows considered here; h«l), and a = (V-l)(l-q~)/2. Hence

QOT - P0 (1 - q2)l/()'-1) a(or) r (or).

A unit vector, normal to the curve EF is (-Y) //5»occ+ 70s ,

ôf/j/â + y)oc )j and QgQ s mass/sec. across EG « \ p(v^w- ul^)/

FTTll. /~""2 2

Sor 4» *)« J so

QSG " Po I! Cl-q2)l/°r-1) (v^- uwor) do,. E

Equating QgG and GjT, we get (taking X = I.I4.)

r (a) = jJÛ-q2)2'* <•**- uTJbr) dor] /[a(l-q2)2 •*]

According to eqn. (2.5)^^= L/(K-R) ^ - (K+R)£w /H j so

finally p ex

r(or) =| 4»E (l-q2)2-5[v - (K+R)u/fc]£<xdcrj /j*(«) [l-q2 (<*)] * **/

Along EJ dy/dx = v(o() /u(oi), d2y/dx2 - [d(v/u)/d°J /(dx/d<*). The curvature is _ ,_

y^[d2y/dx2 / [l+(dy/dx)2J = (u/q)3[d(v/u)/dcvl /(dx/dor) (2.11)

At point E, where r = 0, we note that (by eqn. (2,9))

(dx/d«)]E = (d^/doO] E + X (dr/dor)] £ (2<12)

12

i

By eqn. (2.1L) (

. (2.10.) . , "1 , 2\l/(Y-l)f , 1 / "1 1 1~?)W**) a dr/doj E

=(l-q } " Yd* /dD<_ E-*9 H E)

This leads to the relation

dr/dttj E - j- q2>- / (a2 ~u2)| d£/d»JE ,

which upon substitution into eqn. (2.12) leads to

dx/dO(J B = - j 2\u \/q2 - a2 / (a2 - u2) J d^ / da J £

By eqn. (2.5), X du/da + u-dv/da =• 0, and 2 / Jr 9" p d(v/u) / do( • (uvor - vu )/u = - (\/q- - a / jjur) tt0

(2 .13)

Substituting this relation and eqn. (2.13) into eqn. (2.11) we finally obtain

TC& - i (a - u ) /(2qA|i) (uof/^Jg (2.1U)

Actually, the length of the transition section is not known a priori. In the computation at each point on BJ (Fig. 2.1) both 4C (eqn.. (2.7)) and 1(v (eqn. (2.1U)) are computed, When a point E where H^ •"£- is found,

generally b~r interpolation, the characteristic EF is constructed and the cancellation section thus started. To achieve a desired design Mach number at the uniform flow region ssveral trials must generally be made with various initial qQ's on BB'.

13

I

SECTION III

COMPUTATIONAL PROCEDURES

The initial data for determining the flow or. BD (Fig. 2,1) are available on IBM output card3 as a result of a previous ENIAC computation of Frandtl-Meyer flow. (See eqn. (2 .3).) A/A* and q have been tabulated at .0$ degree intervals of co = 9 + const. The input deck for a computation is extracted from this main deck by selecting first the card with q • q (on which co is, say, co0), then those cards with co » co +<P<); co • 00 + 2<T9,..., co = co + n-xfQ - con + 9n. Sqns . (2 .U)

then then yield x,y,u,v at CL+1 equally spaced joints on BD (8=0) in the characteristic plane ( Fig. 2.3).

For region II there are three computational routines: process for points on

the stream

>-<*

Fig. 3.1

DF (Fig. 3.1), the nozzle process for points on BE, and the general process for all other points* The second order iterative method described in detail in ref. [l] is used to solve the differential

1U

MSSXMO«V**-»lltf**^l",H1«l»Ytt*V•*>9V.,VWT*0 '-***

"

s

equations numerically' . The partial derivatxves are replaced by differences, so that far a po:int F (Fig. 3.2), with information known

M V • •

• •

u

-»- ex

j?ig. .5.2

at points L and U, eqns. (2.$) become

(K-R)^(yp^rL) - L^Cxp-^) = 0

V7?^ - <K-R VW = ° Vi(VV + <K-RWVL> = ° <K-R VW + VW = °

^

>• (3.1)

The notation H^, means that H is evaluated first at Q, then in the

second iteration at M. HQ=(Hu+HL)/2j HM«(Hp+Hj^j H^Hp+Hy)/^, where

Hp is computed from u_, v_ determined in the first iteration. This

notation applies similarly to all other coefficients.

The solution (general process) to eqns. (3.1) is

XP=(K"R V1H^ >rU - (K-R>0N *u}/D "

(K-R)^{(K-R)^yL-L^xL)/D V l

<*•*>«*, {<K-*)(to »u + % vu}/^

(3.2)

w It should be remarked that the ENIAC coding used was actually adapted (to save coding time) from existing axisymmetric flow coding by suppressing a*-v/y terms.

1$

L

where

A= Em L<M - (K'R)0 (K"R)^ Similarly, the stream process computation (DF in Fig. 3*1) for

point F in Fig, 3»3 is

I IA P

+~o<

Fig. 3.3

(3.3) vD = 0

The equations used for the ncasle process (BE in Fig, 3A) are

Hyrt - (K-R) x0 « 0 P r

(K-R)Ug + L v - 0

dy/dx = tan (9-c<2A9)

v - u tan (er -orAS) o

P

>-

D/

Fig. 3J4

16

-*-cx

(3.U)

,. S—•

(3.5)

j.n difference form they are (see Fig. 3.U)

(H/(K-R)} ^ Gv-yJ - (Xp-Xy) - 0

(-Op - V • (L/(K-R)j ^(Tp-r0) - 0

(7p - 7A) A'xp-xA) - tan M (90 -cx-2A9) * 0

up tan (9Q -CX|A9 ) - vp = 0 _

where tan-M (9Q -u2A0 ) = (1/2) [tan(9o-o:2£9) + tan (9Q -OTpA9 )j •

The solution to eqns. (3.5), which locates the nozzle points, is

xp = (1/A) {(_H/(K-R)J r/Q 7u - Xyj- +

(1/A)|H/(K-R)] ^ [-yA+xA tanM(9o -Qr2A9)jj

yp = (1/A) ,T-yA + xA tanM(9Q -c*2A9) - -

(1/A) f- [H/(K-R)] ^ y^xjl tanM(9o-a2A9)

up - Cl/B)|[L/(K-a)] ^ V^}

v = (i./B) J [L/(K-R;t^ vTT+uTr\ tan (9Q -«2 A9 )

>•

(3.6)

A =iH/(K«H)^T/n tan M (9n -oCA9 ) - 1 ^Q M v o

B = |L/(K-R)| ..p tan (9Q -<*2A9 ) + 1

The flow is computed at Of., + 1 equally spaced points on 6 • 1, then

on P • 2, p = 3, and so on.

The partial derivatives in. the curvature expression eqns. (2.7) and (2.1U) are approximated by linear difference quotients. Thus (see Fig. 3.U)

(8x/SOf)p ^ (Xp,~xp)/(*pt-wp); (Su/^)p = (up,-up)/(a p,-°p) .

17

r,

•

U

For every nozzle point iC and ^£ = (a^-u )u I / 2qAp. Xcx\

are computed, and cT—^"^ ^s recorded, cfvaries monotonically along

the nozzle; and when it changes sign at some point G (Fig. 3.1), wc know that the point 2, where the transition region curvature equals that for a simple wave, has been reached and passed. Using the information at G and the two preceding nozzle points, we determine by three point interpolation the variables for<T= 0; namely, xE, y~, tu, v~, «C, $v.

The characteristic ,8-pp (EF in Fig. 3.1), on which x=*£(or), y=>? (a)

is then computed in the same manner as the preceding p = const, characteristics. The Mach number- at F ic the final exit Mach numbers

The remainder of the flow is the cancellation region and is computed fromeqns. (2.9) and (2.10).

In evaluating the integrand in eqn. (2.10)

1 1 ( -^<*

L PR •

Fie. 3.5

we approximate the derivatives / <x(*0 by central differences (i.e., IrrC")^ f^^+l) - Q («-l)l /[2<y]) for all points but the first

two and the last two. For the first two points S^/d^ p=l(^Tj**£s<)A0fD**%)J

(see Fig. 3.5), and for the last two points oe/o~iP ~ ,(£r-S.)/(»,, —o«T )j . ' -* X *— X Jj * i-J <•!

We approximate the integral in eqn. (2.10), which we write as pot

I (ex) •- J J (o») d» , by Simpson's formula for all points but- the first E

two and the last one: namely,

T (c<) = T (or -2) + | [j (ex -2) + UJ (« -l) + J (Of)] ftl .

T («-Org) « 0 for the first point. Tp « TL + [(JL + Jp) (orp -<XL)/2~\

(see Fig. j).5>) for the second and last points.

Finally, for the wind tunnel designer's convenience, the coordinates of the nozzle points were transformed so that the exit point F has the coordinates Xp = 0, Yp • E, an assigned constant. This calculation places

all the nozzles on a scale based on a fixed height of the wind tunnel test section. The transformation was accomplished by

X-(E^)(^-x)L

Y - (E/yv) y J . (3*7}

18

"5

SECTION IV

SUMMARY CF COMPUTATIONS

Fig. l» „1 indicates the flow regions and Kach lines of a typical nozzle calculation.

The nozzles computed for the Hind Tunnels Branch are summarized graphically in Fig, U.2. Contours were computed at intervals of .25 for exit Kach numbers ranging from 1.25 to 5.5. The actual numerical computations arc available in the files of the Data Reduction Unit of the Supersonic Wind Tunnels Branch, Exterior Ballistics Laboratory.

The designs for the remaining sections of the wind tunnel walls ("subsonic" and ''expansion" sections - ST and TB in Fig. 1.1) were obtained empirically after the ENIAC data became available. Boundary layer corrections were included in all the jack settings.

i

These calculations were suggested by J. Sterriberg and were performed under the direction of J„ H. Giese. The author wishes to express his acknowledgements to A „ J. Fine and J , T, Lcerwald of the Airflow Branch; to H. Malloy of the Hind Tunnels Branch; and to P., Freshour, M« Gramberger, H. Mark, and W, Grant of the ENIAC Branch of the Computing Laboratory for their contributions to the calculations.

%cdbfua,n-' qfjjUvtS)

NATHAN GERBER

19

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r REFERENCES

1. Clippinger, R. F. and Gerber, N. : Supersonic Plow over Bodies of Resolution (with Special Reference to Hi»h Speed Computing). tsRL Report 719. May 1?50.

2 , Foelsch, Kuno; A New Method of Desi^ninp; ?--:o TJincnsional Laval Noszlos for a Farallel and Uniform Jet. Perth American Aviation Report NA-I4.6- 235. Parch 19hb.

3. Courant, R. and Friedricks, K. 0. : Supersonic Flow and Shockwaves. Interscience Publishers, 19U8,

U. F^rri, Antonio: Aerodynamics of Supersonic Flows. The Mac Millan Co., (New York), 19U9.

5. Liepmann, H. W. and i'uekett, A. E.: Introduction to Aerodynamics of a Compressible Fluid. John Wiley and Sons Inc., 19U7.

22

r gieo*>

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25