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COPUE LEVELEI1,A COMPUTER PROGRAM FOR THE AERODYNAMIC

DESIGN OF AXISYMMETRIC AND PLANARONOZZLES FOR SUPERSONIC AND

WIND TUNNELSJ. C. SivellsARO, Inc., a Sverdrup Corporation Company

VON KARMAN GAS DYNAMICS FACILITYARNOLD ENGINEERING DEVELOPMENT CENTER

AIR FORCE SYSTEMS COMMAND.ARNOLD AIR FORCE STATION, TENNESSEE 37389

LUL December 1978

Final Report for Period December 1975 - October 1977

L Approved for public release; distribution unlimited.

Prepared for D DARNOLD ENGINEERING DEVELOPMENT CENTERIDOTR JAN 8 1979ARNOLD AIR FORCE STATION, TENNESSEE 37389

D

79 0BEST AVAIL.ABLE -COPY

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"4.

NOTICES

When U. S. Governiment drawings, specifications, or other data are used for any purpose otherthan a definitely related Government procurement operation, the Government thereby incurs noresponsibility nor any obligation whatsoever, and the fact that the Government may haveformulated, furnished, or in any way supplied the said drawings, specifications, or other data, isnot to be regarded by implication or otherwise, or in any manner licensing the holder or anyother person or corporation, or conveying any rights or permission to manufacture, use, or sellany patented invention that ma y in any way be related thereto.Qualified users may obtain copies of this report from the Defense Documentation Center.Reterences to named commerical products in this report are not to be considered in any senseas an indorsement of the product by the United States Air Force or the Government.

This report has been reviewed by the Information Office (01) and is releasable to the NationalTechnical Information Service (NTIS). At NTIS, it will be available to the general public,including foreign nations.

APPROVAL STATEMENTreport has been reviewed and approved.

Project Manager, Research DivisionDirectorate of Test Engineedng

Approved for publication:FO R TH E COMMANDER

ROBERT W. CROSSLEY, Lt Colonel, USAFActing Director of Test EngineeringDeputy for Operations

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UNCLASSIFIEDREPORT DOCUMENTATION PAGE BEFORE COMPLETING FORM

(6 W. COMPUTER.PROGRAM FOR THE AERODYNAMIC nal epaDc 7J. C./Sivellsl ARCO, Inc., a Sverdrup

Air Force Systems Command ogram Element 65807?ArnldAiFrcSatonTenese_338__

16. AGERSLIC NATET Ao hi. REpo(f dfeetfolotoln fi 1,SCRT LS,(fti eot

* IApproved for public release; distribution unlimited.

17, L3STRIOUTION STATEMENT (of tho abstract onteted ir Block 20, It different from Report)

IS. SUPPLEMENTARY NOTESAvailable in DDC.

19. KL Y WORDS (Contitnuo on reverse side It nce...ry and Identify by block "limber)wind tunnel design boundary layerstransonic nozzles* J supersonic nozzles* *..hypersonic nozzlesexhaust nozzle performance computer program

20. ISTYAA CT (Con linue on .nrer.o .1do I nteo eeary and Identify by block numbe r)A computer program is presented for the aerodyna-mic design ofaxisymmetric and planar nozzles for supersonic and hypersonic windtunnels. The program is the culmination of the effort expended atvarious times over a number of years to develop a method of de-signing a wind tunnel with an inviscid contour which has contin-uous curvature and which is corrected for the growth of theboundary layer in a manner such that uniform parallel flow can be

DD FOR 1473 EDITION OF I NOV 65 IS OBSOLET E~~~ ~UNCLASS 7 1 2 2

IT %j 01 0ow 1

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UNCLASSIFIED20, ABSTRACT (Continued)expected at th e nozzle ex-t. The continuous curvature isachieved through specification of a centerline distributionof -'elocity (o r Mach number) which has first and second deriva-tives that 1) are compatibl* -Lth a transonic solution nearthe throat and with radial flow near the inflection point and2. approach zero at the design Mach number. The boundary-layergrowtb ia calculated by solving a momentum integral equationby nuws'lcal iaitegration.

AILEVEJi, Wi,oil- D D C

JAN 8 1979IT ......................... .............

Slt. AVAIL

UNCLASSIFPTD--,,.

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PREFACEThe work reported herein was conducted by the Arnold Engineering

Development Center (AEDC), Air Force Systems Command (AFSC). The resultsof the research were obtained by ARO, Inc., AEDC Division (a SverdrupCorporation Company), operating contractor for the AEDC, AFSC, ArnoldAir Force Station, Tennessee, under AR O Project Numbers V33A-A8A andV32A-P1A. The Air Force project manager was Mr. Elton R. Thompson.The manuscript was submitted for publication on September 12, 1978.

Th e author wishes to acknowledge the assistance of Messrs. W. C.Moger and F. C. Loper, ARO, Inc., for providing thr. basic subroutinesfor smoothing and spline fitting, respectively, which were adapted foruse with the subject program. Mr. F. L. Shope, ARO, Inc., providedtechnical assistance in the preparation of this report. Prior to thepublication of this report, the author retired from ARO, Inc.

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CONTENTS

1.0 INTRODUCTION ................... ...................... 5*.0 TRANSONIC SOLUTION ........... .................. ... 103.0 CENTERLINE DISTRIBUTION .................. ......... 154 4.0 INVISCID CONTOUR ........... ................. .... 205.0 BOUNDARY-LAYER CORRECTION ..... ............. ..... 246.0 DESCRIPTION OF PROGRAM....... ................ ..... 347.0 SAMPLE NOZZLE DESIGN ........... .............. .... 388.0 SUMMARY .............................. ... 41

REFERENCES ............. ...................... .... 41

ILLUSTRATIONS

Figure1. A Foelsch-Type Nozzle with Radial Flow at the

Inflection Point .............. .................. 62. Nozzle with Radial Flow and a Transition Region

to Produce Continuous Curvature ............ 73. Nozzle Illustrating Design Method of Ref. 13 , , , 84. Nozzle Throat Region ............ ............... 95. Relationships Obtained from Cubic Distribution

of Velocity from Sonic Point to Point E forAxisymmetrin Nozzle ...... ................. .... 18

6. Limitations of Fourth-Degree Distribution ofMach Number from Eq. (39) ..... .............. ... 19

7. Characteristics Near Throat of Nozzle with R = 1 238. Variation of Wake Parameter, A, w.,ith

Reynolds Number (Incompressible) ... .......... ... 289. Variation of Skin-Friction Coefficient with

Reynolds Number (Incompressible) ... ........ .... 2910. Variation of Velocity Profile Exponent with

Reynolds Number Based on Boundary-LayerThickness ...... ........ .................. ... 303

h 4;

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TABLE

Page1. Input Cards for Sample Design ...... . . . . . . . 39

APPENDIXES

A. TRANSONIC EQUATIONS .................. . .. ... ... 450. CUBIC INTEGRATION FACTORS. .. . . . . . . . . . .. . 49C. INPUT DATA CARDS . ....... . . . . . . . . . . . . .52D. COMPUTER PROGRAM................ . . . . . . . . . . 61

: "NOMENCLATURE ....................... 13 9

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1.0 INTRODUCTION A

Supersonic and hypersonic wind tunnel nozzles can be placed in twogeneral categories, planar (also called two-dimensional) and axisymmetric.Early supersonic nozzles (circa 1940) were planar for many reasons: thestate of the art was new with regard to both the design and the fabrica-tion; the expansion of the air - the usual medium - was in one planeonly, thereby simplifying the calculations and requiring two contouredwalls for each test Mach number and two flat walls which could be usedfor all the Mach numbers; and the relatively low stagnation temperatureand pressure requirements did not create dimensional stability problemsin the throat region. Dimensional stability would in later years becomea primary factor in the development of axisymmetric nozzles.

Prandtl and Busemann, Ref. 1, aid the foundation for determiningthe inviscid nozzle contours by the method of characteristics. Foelach,Ref. 2, simplified the calculation of the contour by assuming that theflow in the region of the inflection point was radial, as if the flowcame from a theoretical source as illustrated in Fig. 1. Th e downstreamboundary of the radial flow is the right-running characteristic AC fromthe inflection point, A, to the point, C, on the axis of symmetry wherethe design Mach number is first reached. The flow properties along thischaracteristic can be readily calculated; and inasmuch as all left-running characteristics downstream of the radial flow region are straightlines in planar flow, the entire downstream contour can be determinedanalytically. Upstream of the inflection point, it was assumed that thesource flow could be produced by a contour which was a simple analyticcurve. In the Foelsch design the Mach number gradient on the axis isdiscontinuous at the juncture of the radial flow region and the begin-ning of the parallel flow region. This discontinuity produces a dis-continuity in curvature of the contour at the inflection point and atthe theoretical exit of the nozzle.

5

.- .

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UniformrS~Flow

Figure 1. A Foelsch-type nozzle with radial flowat the inflection point.As the state of the ar t progressed, it became desirable to cover a

range of Mach numbers without fabricating different nozzle blocks foreach Mach number. A limited range of Mach numbers could be covered byusing blocks with unsymmetrical contours which could be translatedrelative to each other to vary the mean Mach number in the test section.The widest range of Mach numbers with acceptably uniform flow in thetest section has been obtained in wind tunnels in which the contouredwalls consist of flexible plates supported by jacks which can be adjustedto vary the contour to suit each Mach number. Inasmuch as the curvatureof a plate so supported must be continuous, methods of calculatingcontours with continuous curvature were developed (Refs. 3, 4, and 5)by introducing a transition region, A B C J, downstream of the radialflow region (see Fig. 2). The shape of the wall between points A and Jwas controlled to give continuous curvature. The contours used for thevon K~rmgn Gas Dynamics Facility 40- by 40-in. Supersonic Wind Tunnel(A) at AEDC were obtained by the method of Ref. 5. Not only is acontinuous-curvature contour easier to match with a Jack-supported plate,but it also satisfies the potential flow criterion for zero vorticity,,

dq/dn - Kq (1)where q is the velocity measured along a streamline of curvature K and nis the distance normal to the streamline. Inasmuch as the inviscidcontour is a streamline, this criterion implies that the flow will bedisturbed where a contour has a discontinuity in curvature.

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D

B C

Figure 2. Nozzle with radial flow and a transitionregion to produce continuous curvature.

The usual wind tunnel criterion concerning temperature is that theconstituents of the gas should not liquefy during the expansion processrequired to reach the test Mach number. For the usual pressure levelsinvolved, ambient stagnation temperatures can be used up to a Machnumber of about five. As the stagnation temperature is raised, dimen-sional stability becomes more difficult to maintain in a planar nozzle.Therefore, axisymmetric nozzles are used when elevated stagnation tem-peratures are involved. Axisymmetric nozzles have also been used forlow-density tunnels (Ref. 6) because their boundary-layer growth is moreuniform than that of planar nozzles, which inherently have transversepressure gradients on the flat walls. The obvious disadvantage ofaxisymmetric nozzles is that each one must be designed for a particularMach number. Moreover, disturbances created by imperfections in thecontour tend to be focused on the centerline.

Before the advent of high-speed digital computers, it was extremelytime consuming (Ref. 7) to calculate axisymmetric nozzle flow by themethod of characteristics (Ref. 8). Inasmuch as the assumption ofsource flow saved time in designing a planar nozzle, it was logical touse source flow as a starting point in the design of an axisymmetricnozzle. In Ref. 9, Foelsch develops an approximate method of convertingthe radial flow to uniform flow. Beckwith et al., Ref. 7, show thatFoelsch's approximations were quite inaccurate but utilized the idea of

7./.

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a region of radial flow followed immediately on the axis by uniformflow, as in Fig. 1. As in the case of planar flow, the discontinuity inMach number gradient on the axis produces a discontinuity in curvatureon the contour (Ref. 10). Such discontinuities have been eliminated bythe design methods of Refs. 10, 11, and 12; here, an axial distributionof Mach number (or velocity) between points B and C (Fig. 2) introducesa transition region between the radial and parallel flow regions, thusgradually reducing the gradient and/or second derivative to zero from theradial flow values at the beginning of the parallel flow. As shown inFig. 3, the upstream boundary of the radial flow region is a left-runningcharacteristic from the inflection point, G, to the axis at point E. Theflow angle is the same at points G and A. Both are shown to illustratea general nozzle design. As described in Ref. 12, the contour upstreamof the inflection point can be calculated for an axial distribution ofvelocity in the region between points I and E, which makes the transitionfrom sonic values to radial flow values. On the axis, the sonic valuesof first and second derivatives of velocity with respect to axial distancewere calculated by an adaptation of the transonic theory of Hall, Ref.13, or Kliegel and Levine, Ref. 14. The upstream limit of these cal-culations was the left-running characteristic from the sonic point onthe axis.

Inflection Reglon

M 6SI F -

Figure 3. Nozzle illustrating design method ofRef. 13,

8

-:.'*. ... U,,.J-

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This characteristic is also called a branch line. Between the theo-ret ical location of the throat and the intersection of the branch l inewith the contour was a region which was not calculated but which in-creased in size as the throat curvature increased. This gap in thecontour has been eliminated by the method described herein which util-izes a right-running characteristic originating at the throat as shownin Fig. 4 (where point I has been moved from the sonic line to thethroat characteristic). With this latest improvement upon the method ofRef. 12, contours can be designed which have throat radii of curvatureof the same order of magnitude as the throat radii although such anextreme curvature would not normally be recommended from other stand-points. A recent (1975) design of a Mach 6 nozzle utilized this methodwith a throat radius of curvature of about 5.5 times the throat radius.

Sonic Line Branch Line

YO! Throat Characteristic

Figure 4. Nozzle throat region.

After the design method was developed for axisymmetric nozzles, itwas adapted for planar nozzles having a prescribed centerline dis-tribution of Mach number (or velocity). This approach to such a designis considerably different from that of Ref. 5. The current designmethod is incorporated into the computer program included herein. As anoption in the program, a complete centerline Mach number distribution

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can be used which does not include a radial flow region. Parts oZ thecomputer program are subroutines for computing the boundary-layer correctionto the inviscid contour, for smoothing the contour, and for interpolat-ing points at even axial positions by means of a cubic spline fit of thecontour.

2.0 TRANSONIC SOLUTION

In many early nozzle designs, it was assumed that the flow at thethroat was uniform (M - 1) nd parallel. This assumption impliey thatthe wall curvature is zero and that the acceleration of the fl , Iszero (i.et, the acceleration starts from zero at the beginning .-.hecontraction, reaches a maximum in the contraction but is reduced to zeroagain at the throat, and must be increased again in the beglaning of thesupersonic contour and reduced to zero at the nozzle exit). A nozzle sodesigned therefore becomes considerably longer than one in which theflow reaches its maximum acceleration in the vicinity of the throat,where it is approximately proportional to the reciprocal of the squareroot of the radius of curvature. The above argument indicates thefallacy of some so-called "minimum length" nozzles, although somedesigners have combined a contraction having a relatively high throatcurvature with the supersonic section having zero throat curvature.

For a throat with a finite radius of curvature there have been manytransonic solutions. Hall, Ref. 13, developed a small perturbationtransonic solution for irrotational, perfect gas flow, in both two-dimensional and axisymmetric nozzles, by means of expansions in inversepowers of R, he ratio of the throat radius of curvature to the throathalf-height, or radius. His solution gives the normalized (with thevelocity at the sonic point) axial and normal velocity components in theform

(y,Z) + . . .. + + .Y. . . (2)R R2 R310

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2- + 12 Lv (y,z) v+(y,z) + v,(y,)V(L,( 1 2 +L. .( 3 )where

and x and y are coordinates normalized with the throat half-height orradius, yo. The value of a is zero for two-dimensional flow and one foraxisymmetric flow. Kliegel and Levine in Ref. 14 extended the applica-bility of Hall's axisymmetric solution to lower values of R essentially bymaking the substitution

- S-1 + 9-2 + S- . .(5)

where S - R + 1, into Eqs. (2) and (3). In the method used herein, thesame substitution is made in Eq. (4) fo r two-dimensional flow as well asfo r axisymmetric flow and therefore becomes a special case of the generaltransonic solution described in Ref. 15. Th e complete general equationsin terms of S are given in Appendix A.

At the throat, x - 0, y - yo' v 0, for planar flow,n= 1+ _1_ - (14y-7 5 ) + (274Y2 - 861y,+ 4464)3S 270S 2 170100 3

d u . -(32y 2 + 87y 561) +dx/y. S 540S2

and, for axisymmetric flow,

+ 1-4y -57) + (2364y 2 - 3915y + 14337) + .. . (8)4S 288S 2 82944S3

Sd__+__ = _ -(64y2 + I17y - 1026) + (9)dX/yu as 1152S 2

;!: 11

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where the derivatives are with respect to x nondimensionalized by thethroat half-height or radius, respectively, and

2A _____(10)

On the axis, y - 0, v - 0, for planar flow,u = 1 _i + Y-5 782y 2 + 3 5 07y - 7767 +6S 270, 2 2721 60S'

+ + 34y2+ 429y_ 12+].4Y" 4320S 2 1

,Y," 6 36S

(,&z' (2y 2 - 33y + 9)/72 + . (1)and, for axisymmetric flow,

S1 - _. + IOy-. 1L5 _ 2708y2 + 2079y + 2 1 1 5+4S 288S 2 82944S 2

+&1 - I + 92y2 + 180X 9 ++Y \ 8S 115282

()2 (.2~ + +

(--A)' (4Y2 - 57y + 27)/144 + (12)r,Because the sonic line is curved for finite values of R, the mass

flow through the throat is reduced by the factor CD (discharge coef-ficient), which is the rat 4 .o of accual mass flow to that which couldflow if R were infinite and the sonic line were straight. For planarflow,

CD I- [i - 4y24 + 3342 - 457y + 4353 + (]3)I12

i 12

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and, for axisymmetric flow,C[ 1 - y [1 - - .754y2 -757y +36U. + (14)96 S' 24S 28EIOS'

The flow which passes through the throat also passes through thesonic area of the source flow which is at a distance r from the source.In planar flow, 0(

! oroY/r = ,/C1 (16)

where the inflection angle, n, is in radians.

In axisymmetric flow,7y.ry~o CD =21 r 1 -cs (17)

or Iyo/r- 2 sin (1/2)/cD (18)

In the calculation of the throat characteristic used herein, thevalue at x - 0, y - yo Eq. (6), is the starting point. The half-height'~or radius, y0, is divided into 240 equally spaced values of y. Inasmuchas the characteristic is right running, it s slope at each point is

dy/dx tan( (19)

wheresin I1/M (20)

AlsoW = M + K-l M2) (21)y+1 Y+-

sin = v/W (22)and

drdo d-o '""a d (23)y

13 - -

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Sde dx/cos(o - t) - dy/sin(o - 1) (24)

'The term i is the Prandtl-Meyer angle in two-dimensional flow,yan. (2I - tan-1 (I2 -1) (25)Y-1 y+l

Equations (19) and (23) are the characteristic equations and are solvedby finite differences. If all values are known at point 1, the valuesat point 2 are found (y is known a t both points) by

S + 2(y-y) (26)x2 1 tan( l-A, + tan

A Y2 -Y)2 + (x2! - x1 )2 (27)

V0 = 01 + 01 - 0 2 + VI + - V 2 (28)At the starting point W is the value of u because v - 0. Values of v2Iare calculated at each point (x 2 , y2 ) from the transonic solution,and Eqs. (26) to (28) are iterated until convergence is reached. Forevaluating the term in brackets in Eq. (28), the ratio v/y is defined bythe transonic solution even on the axis where both v and y are zero.This fact eliminates thbi general problem ia axisym-etric characteristicssolutions of evaluating the indeterminate sin */y in Eq. (23) on theaxis of symmetry.

It may be noted that the value of W as calculated from the character,-istic value from Eq. (21) differs from the value (u2 + V 2)I12 calculatedfrom th e transoutic equations, but the difference decreases wT:Uh in-creasing R. Fo r the final point of the throat characteristic which 1,uson the axis, the value of d 3 u/dx3 from the transonic solutioa fo r th eaxial distribution is "corrected" to make u - W for the axisymmetriccase fur values of R less than 12 . The correction is abvut 16 perccntfor R I and decreases rapidly as R increases. This correction is made

14

F ' " i I i 'I ii i i i ln n. . " .

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so that v,'alues of du/dx and d udx 2 can be calculated from the transonicsolution for later application. Th e correction is believed to be JuatifiedI1'much as the accuracy of the transonic solution -s imited, particularlyfor low values of R, because the series expression for u is truncatedafter the x term.

3.0 CEN'TERLMN DISTRIBUTIONIn the radial flow region, the distance r, measured from the sourne,

is related to the locAl. Mach number byLr)+ _ (_L + h.-y+ + M 2)qy (29) i r Y+1 ++

or)1+a W-(Y+' I W2-- (30)

First, second, and third derivatives of W-itM with respect to r/r canbe obtained as described in Ref. 12. Along the axis x - r when x ismeasured from the source. Inasmuch as all, coordinates ma.st be normalizedby the same factor, r,, the transonic equation in terms of A/yo and y/yocan be transormed by Eqs. (16) and (18), after which the distance froomthe source to the throat station must be taken I.-ntc account. Thislatter distance is generally unknoun until after the distance from pointI to point E is determined.

In radial flow, the term on the right-hand side of Eq. (23) can beevaluated dimply. Inasmuch as sin y/r and d& dr/cos v,H ik9! S2 1 tan fy rbut

. -tan(M 2 -I ) 2

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and, from Eq . (29) 'for a - 1,. dr (2- 4M

r 2(1 +2 : M2) Mi2Thus

,tnan sr = (M 2 -1) 2 dMr 2(1+-Z2 MN) M2I

From Eq. (25), do M(I +Y-1 M ) M2"therefore, Eq. (23), in radial flow, becomes

dot/ do4 - 2- dot (31)21which applies for characteristic AB or GF. Similarly, for the left-running chaivacteristic EG,

do - do - ad (32)Thc.:efore, 2

13 - A + -G (33)and G - OF, = (a + 1) (34)and, from the design values n and MB (and/or ), MA, M, M' WE, andthe necessary derivatives can be calculated.

Within the accuracy of Eqs. (11) and (12), the second derivative ofvelocity ratio at the sonic point is negative for values of R loss than11.767 for planar flow and 10.525 for axcisyrmetric flow. The secondderivative of Mach number at the sonic point is positive for all valuesof R. Inasmuch as the second derivative of either Wor M is negativefor source flow, it seems better to use a velocity distribution ratherthau a Mach number distribution between points I and E. On the otherhand, a Mach number distribution between points B and C is preferable

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because the velocity ratio approaches the constant value of [(Y +1)/Y _ 1)] 1/2 as the Mach number increases to infinity; therefore, thechange in velocity between points B and C becomes small relative to thechange in Mach number.

I.The velocities and their first an d second derivatives at points I

and E are used to determine the coefficients of the general fifth degreepolynomial

.,W C1 + C2 X + C3 X2 + C4 X3 + C'5X4 + C6X5 (35)where

X - (x - xl)/(XF - X[) (36)Similarly, the Mach numbers and their first and second derivatives atpoints B an d C are used to determine the coefficients of the polynomial

M= 1) + D2 X + D X2 + 0) X + D5 X4 + D6X" (37)where, in this case,

X .: (x - XI)/(X. - xv) (38)and the first and second derivatives at point C are usually set equal tozero.

In these equations, the lengths (xE-xI) and (xC-xB) must be specified,but can be determined by the conditions that C6 and D6 equal zero,thereby reducing the polynomials to fourth-degree ones. If the velocityat point E is determined by iteration, the third derivative at point Ior E can be included as a criterion for the fourth-degree polynomial;or, by setting C5 - 0, one can find a third-degree polynomial with a con-stant third derivative. In either case, the Mach number at point B isfound from Eqs. (33) and (34) after the value at point E is found. All"of these options are included in the program, but unless there are otherfactors involved, the preferred options are the cubic between points Iand E an d the quartic between points B an d C.

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* For the cubic distribution for axisymmctric flow, the Mach numberat point E is related to the radius ratio as shown in Fig. 5 for y

1.4 for various values of inflection angle. Cross plotted are linesof constant values of the ratio *E/n. Such values for most axisymmetricnozzles lie in the range covered in this figure, and inasmuch as *F/n

h + 4, values of MF can also be obtained.

2,8

2.41,4 "-2-2,2

1, 8 - 12.11.61,2 4

0 4 8 12 16 20 24Radius Raik (R)Figure 5. Relationships obtained from cubicdistribution of velocity from tonicpoint to point E for axisymmetricnozzle.

j18- - - - -' ,. -

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In determining the length of the segment between points B and C,using th e fourth-degree polynomial distribution, there is a minimumvalue of the Mach number at point B for the design Mach number at pointC. As given in Ref. 12 , M=; + 0.715 M" /M (39)

where the primes indicate derivatives with respect to r/rI. Thisrelationship is shown in Fig. 6. For an axisymmetric nozzle designedfo r a Mach number greater than about 3.4, the minimum Mach number at

20+ ~1 8

1614

,, 1210 Axisymmetric

Planar

0 I . , I0 2 4 6 8 10 12 14

Minimum Mlach Number at Point BFigure 6. Limitations of fourth-degree distribution

of Mach number from Eq. ?39).point B is about two-thirds of the d2sign Mach number. Using such avalue visually causes the length to be xcessive, and more realistic

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values of B are 75 to 80 percent of M It is important, however, asillustrated in Ref. 16, that the distance between points B and C besufficient to allow for accurate machining of the contour betweenpoints A and J, which lie on the characteristics through points B an d C,respectively.

4.0 INVISCID CONTOUR

The flow properties are determined at a desired number of pointsalong the key characteristics (i.e., the throat characteristic, TI , asdescribed earlier (a sub-multiple of 240 is used for subsequent calculations),the characteristics EG and AB bounding the radial flow region by Eqs.(33) an d (34) for equal increments in n, an d the final characteristic CDalong which the Mach number is constant an d the flow angle is zero).The flow properties are also determined at axial points from Eqs. (35)and (37). The network of characLeristics is then calculated in theregion TIEG starting at point E and progressing upstream and in theregion ABCD starting at point B and progressing downstream.

The equations for a right-running characteristic were given previously.dy/dx = tan(o - p,) (19)+ ,. d (23)

wherede - dx/cos( -It) dy/sin(r/ - p) (24)

For a left-running characteristic, the equations aredy/dx tan (0 + p) (40)

do- =b '!2s_0nL,uiJ d4 (41)ywhereAldo dx/cos (0 + p) dy/sin(0 + p) (42)

:, Alsodo - .~J- dM acot dW (43)(1 +1.i M2) M W2

[I 20A__ __

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Values of x, y, *,and M are known at the general point 1 on theright-running characteristic, C, and at the general point 2 on the left-running characteristic, C. The characteristics intersect at the generalpoint 3 where the values are calculated by numerical integration of Eqs.(23) and (41) along the respective characteristics.

S(si 6,mn A + , 02. n P~2 ) (44)where

A4 c " 2) sec 1 (45)and

Y3 Y2 tan~ tan (0 3 + jA) + 1. an (0 2 + A2 (46)3 222-3 011 + (3- 01) = =

4, (Min0mi&AMn +. sinl 6minp (47where (3- x1) sec. a (48)and

=tan a Itan ((3 13 +1L tan(~ (49)X -X 2

Adding, substracting, and rearranging gives03 (0 + - 0 + S6 + P2 + )(50)

203 -01 2 + 0 ,+ 02 + --(51)2

In planar flow, P1 p2 0 because a -0 and Eqs. (50) and (31)can be solved directly, 14 is obtained from *j by t~he inverse applica-14-1 tion of Eq. (25), and vNsin (1/M ). In xisymnietric flow, the equa-tions must be solved by iteration. A useful first approximation for P1and P2 is th e radial flow values, P1 -(p- 1)/2 and P2 )3 /2.

21

.- . ...

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At all points except on the axis in axisymmetric flow, Eqs. (44)and (47) are defined because Y2 and y, are nonzero. On the axis, theterms sin /Y2 and sin 0,/y, are indeterminate with the form zero/zero.These indeterminates can be evaluated by assuming that the generalpoints 1 and 2 on the axis are very close together and that U f P s 13and W OW W2 * W . Equation (41) can be written

Cot g L do + 'in 6 sin I dx (52)W y cos ( + -)and Eq. 23 can be written

cot A dW= -d9 + sin 0 idnp dx (53)W y cos( -)as 964 9S-Psino, OIA-.ts

and tan 3 = i__i..-- =. 33 2 1 3

In finite-difference form,co in t an A..X3 -x2

t a (W3 - W2) = 3 + . 3 3 2)

0 tan X sin 0. tan P (X3 - X2)Y3 Y3

-- 2 sin 03 tan M3 (x3 - x2)/y 3 (55)Similarly

S(WlW3) =3 + sin 0, tan /I3 (x1 - x3)/y 3 (56)Wa-* 2 sin 03 tan p 3 (xl - x 3)/y 3 (57)

Adding Eqs. (55) and (57) and rearranging,lia =i ~ _ w (58)y4O Y 2 W dx

and s"' 0'2 Min (M1-)Y (d' (59)

2 2. , ,. . - . , ,, . , ./ '. , . . . , 2 2

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AEDC-TR-78-63fo r use in Eq. (44) when point 2 is on the axis, and

HilIl Ht 410Yl 2W W \I (60)fo r use in Eq. (47) when point I is on the axis.

In starting the calculation of the network of characteristics inthe region TIEG, point E becomes point 1 and the first axis point up-steam of point E becomes point 2. The complete left-running character-istic approximately parallel to EG is calculated, and the point on thecontour is determined from mass flow considerations as described in Ref.17. The flow properties along this characteristic are then used tocalculate the next left-running characteristic, again starting on theaxis. This process is repeated until point I is reached, after whichthe starting point for each left-running characteristic is a point onthe throat characteristic as illustrated in Fig. 7. The process inregion ABCD is similar except that right-running characteristics arecalculated for each point on the contour.,4

UI

1.0

0.8 -Y0.6 "-0.4 - 0.2010 0.2 0.4 0.6 0.8 1,0 1.2 1,4 1,6 1. 8 2.0

xFigure 7. Characteristics near throat of nozzle

with R = 1.23

-. 'IL,

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5.0 BOUNDARY-LAYER CORRECTIONTo each ordinate of the inviscid contour must be added a correction

for the boundary-layer growth to obtain the viscid or physical contourof the nozzle. Except for very low stagnation pressures, the boundarylayer is assumed to be turbulent. Generally, the boundary-layer cor-rection will be made for on e design condition of stagnation pressure andtemperature although it is theoretically possible to reshape a flexible-plate type of planar nozzle to account for different boundary-layerthicknesses corresponding to different stagnation conditions. Thecorrection for a planar nozzle is usually applied to the contoured wallsonly, but the correction also allows for the growth of the boundarylayer on the parallel walls in order to maintain a constant Mach numberalong the test section centerline. Therefore, the correction applied isgreater than the displacement thickness on the contoured walls, and theflow in the test section is diverging in the longitudinal plane normalto the contoured walls. In the longitudinal plane normal to the parallelwalls, the flow is converging because of the boundary-layer growth;moreover, there is a tendency for the boundary layer to be thicker onthe wall centerline because of the transverse pressure geadients presenton the parallel walls. Although these physical effects make a truecorrection impossible for a planar nozzle, the calculations describedherein are made as if the cross section were circular, with the cir-cumference at each station equal to the periphery of the actual rec-tangular cross section.

The method of calculating the boundary-layer growth is based onobtaining a solution to the von Kfrmfn momentum equation written fo raxisymmetric flow.

0I M 2 -M 2 +H dM + I dr C (61)dx MEl + (y.-_) M2 /2 dx rw dx 2The term [(I/rw)(drw/dx)1 becomes an effective one fo r planar flow asjust described. For either type of nozzle, the inviscid value is used

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as a first approximation. The entire solution is iterated several timeswith ne w values of rw and drw/dx - tan 0w obtained each time by addingvectorially the displacement thickness to the inviscid contour.

The value of mcmentum thickness used in Eq. (61) is defined byPI -__d__(62)0___ P'I '( 1-- q )d (2

where z is measured normal to the wall.

Also 5i iO. (,_ dz (63)The quantities P* and 0 may be considered to be the displacement andmomentum thicknesses when the boundary-layer thickness is small withrespect to the radius, r . These values are related to tota2 valueswa* and 0, obtained from mass-defect and momentum-defect considerationsby 12

- (64)and -- . 62 (65)Because rw-6 coo Cw + y, where y is the inviscid radius, Eq. (64) mayw a~ wbe rearranged to give

4-- w I Y SC2 2 - y sec 9W (66)For the final correction, the value *asec w is added to the in-viscidradius in order that no correction be made to the longitudinal location.

The integrations of Eqs. (62) and (63) are performed numericallyusing Gauss' 16-point formula, with the assumption of the power-lawvelocity distribution

q/q, (67)

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pl/p " T T (68)where

T = T (Taw - Tw) q/q, + [Te - a - Tw) - Tw] (q/q.) 2 (69)which is Crocco's quadratic temperature distribution if a - 1. However,as shown in Ref. 12, a value of a - 0 gives a parabolic distributionwhich agrees better wit1' data obtained in hypersonic wind tunnels withwater-cooled walls. The same distribution is obtained if T - Taw'w &which is likely to be the case for planar, flexible-plate nozzles.Before using the Gaussian integration, one must replace the values of zand dz with 6(q/qdN and N6 (q/qe)N-1 d(q/qe), respectively, in order toavoid the infinite slope, dq/dz, when q and z equal zero.

The value of the compressible skin friction coefficient, Cf, inEq. (61) is assumed to be related to an incompressible value, Cfiby a factor Fc, introduced by Spalding and Chi, Ref. 18 ,

Fc Cf =Cr (70)and Cf is related to an incompressible Reynolds number, R, which isfIrelate to the compressible value, Re , by a factor F,c

FIa RO, RO (71)

The factor Fc, also used by van Driest, Ref. 19, is given byL I/(p/pe)2 d (q/qe)] (72)

which uses Eqs, (68) and (69), In Refs. 18 and 19, a value of a ' 1was implied, but Eq. (72) is used herein with a - 0 also, to givea "modified" value of V . The factor Fc may be considered to be the ~cratio of a reference temperature to the free-stream temperature. Thefactor FR , as used by van Driest, isV F68 = ,/JAW (73)

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The compressible momentum thickness, 6 upon which R is based isthe flat-plate valueC

li P% qq" 4

because the values of Fc and FR were developed to correlate flat-plate data.

The equation used herein fo r incompressible skin-friction coef-ficient is that of Ref, 20,

If (log H0 + 4.56]) (log t0o - 0.546)

This equation is believed to agree with experimental data slightlybetter than the von Kgrm~n-Schoenherr equation,

C (0,242)2, (fI (log 110 1.1696) (log lie + 0.3010) (76)at high Reynolds numbers. Also as shown in Ref. 20, Eq. (75) agreeswith the equation, Ref. 21, based on Coles' law of the wall and law ofthe wake,S~L (2/C1 )2 fn 15 + 0.5 Fn (C /2) + YC + 211 (77)if 1I aries as shown in Fig. 8 from about 0.41 at R6 - 400 to a maximumof 0.5885 at Re - 50,000 and then decreases to about 0.49 at R.7i107. in order for Eq. (76) to agree with Eq. (77), I1 ust continuallyincrease with increasing R0 as shown in Fig. 8. Th e data shown in Fig.8 were computed by Coles iniRef. 21 from Wieghardt's flat plate data,Ref. 22. A comparison of friction coefficients from Eqs. (75) and (76)is shown in Fig. 9 together with Wieghardt's values as recomputed byColes. The constants K and C are 0.41 and 5.0, respectively. Therelationship between e and 6 is obtained from the logarithmic velocityprofile by neglecting the laminar sublayer, representing the wake function2by a sine distribution, and integrating to obtain

____ (78)

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and_8* C= (~,. 2 179 11 + 1.5 n2) (Y9)

8 8 2K

,'"0.6 ' 0 0

0.50U From Eas. (76) and (77)

'-Iata Tabulated InRef. 21 ;Identifled as Wloghardt Flat Plate Flow0.3

0.2I~ l ll JI l lll ] I I ~ ll I I I I I f ll I I IJ I ll

103 104 105 106 107

Figure 8. Variation of wake parameter, n, withReynolds number (incompressible).The value of N in Eq. (67) is assumed to be a function of Reynolds

number based on the actual boundary thickness, not corrected by FRand is evaluated through th e use of the kinematic momentum thickneisq I- dz (80)

from whichOk/6 N/(N 2 + 3N + 2) (81)

or

N -a+ - 6\ + 1I (82)

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A EDC-TR -78-630 006 r I I [1T l 1 TrTiJ I l f l I ! I I 1 1 1 1

0,003

O.004 Wleqhardt's Flat Plate DataTabulated In Ref, 21t,0.003

0.002 Eq .(75)

0.L0 Eq . (76100 1 1 11 tl 1 1f11 ilj ,J I J iIlll ,I I I I ILill'

104 1 5 106 Ia7Re ,Figure 9. Variation of skin-friction coefficient withReynolds number (incompressible).

The value of 0k/ 6 is obtained from Eq. (79), where the value of nI sevaluated from Eqs. (75) and (77) with ek used instead of 0i. The re-sulting variation of N with Rd is shown in Fig. 10.

Two options contained in the program subroutine for the boundarylayer utilize Coles' law of corresponding stations (Ref. 23),

(83)if Cf /Cf . F is calculated from Eq. (72) for a - 0 or a 1, then onei coption gives

F 1 1 8 ~ ~/I~''~ ~(84)The second option divdes Eq. (83) into the two parts,

Cfl/( =f I' /T,, P11 (85)

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and= !S/!L.(86)

where pis evaluated at the temperature

'T', . + 1?.2(Cf/2)2 a(Ta - T) 305(Cf,/2) [a(Taw - Tw) + Tw T.](87)

10

.1 8N6

104 106R6Figure 10. Variation of velocity profile exponent with Reynoldsnumber based on boundary-layer thickness.

Still another option defines the incompress~ible skin-frictioncoefficient as

Cf 0,0888(log 118 + 4.6221) (log 118 1.4402) (88)

whereCTWAW (89)

and F is calculated from Eq. (72).

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Th e wall temperature in the above equations can be the adiabaticwall temperature or can be allowed to vary between a throat wall tem-perature, TWT, and a nozzle-exit wall temperature, TWD, both of whichare input to the program. Two options are available for the variationof wall temperature,

I- (A i/A*)'- I A/A* (90)

where m can be 1/2 or 1, A/A* is the area ratio corresponding to localMach number, and Ac/A* is the area ratio corresponding to the design Machnumber at the nozzle exit. Equation (90) is used in lieu of moreaccurate values and approximates the way the heat transfer decreases asthe Mach number increases from 1 at the throat to the design value atthe exit. For a water-cooled throat, the value of T can also becalculated by the program, T

T + 00 I('l')-1 )'I; T= (91)h. + 0where h is the airside heat-transfer coefficient at the throat asacalculated by Reynolds analogy from the throat skin-friction coef-ficient Sp p 2/ c2 (92)

with a constant specific heat based on the thermochemical BTU

T (y -. i) 777,64885, (93)and Q is an input which is a function of the properties of the throatmaterial, the cooling water, and the geometry and would be a constant ifthe properties were constant. The assumption is made that the bulktemperature of the water is 15*F less than T and that p2 /3 is thesquare of the recovery factor used to obtain Vhe adiabatic wall tempera-ture, T

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For "he integration of Eq. '61), th e values of x, y, dy/dx, M,and d&/dy .. : obtained from the iviscid contour at unevenly spacedpoints as a result of the characterio:t:cs solution. With the inputs ofstagnation pressure and temperature, ga, constant, and recovery factor,the unit Reynolds number and static an d adiabatic wall temperatures canbe calculated at the same points as functions of Mach number withSutherland's equation used for viscosity. With the inputs of T and_TWD, the wall temperatures can also be calculated as functions of Machnumber, although T may need to be obtained by interation if theToption to input a value of Q is exercised. Sutherlandts equation isalso used with wall temperatures to obtain the viscodities at the wall.For any static temperature below the Sutherland temperature, 198.72*R asused herein, the viscosity variation with temperature is assumed to belinear.

The integration of Eq. (61) is started at the throat where it isassumed that dOidx w 0 in order to obtain a value of e. Iteration isinvolved at each point because C is a function of Reynolds number basedupon 8, and the relations e/6 and 8*/8 depend upon the value of N,which is a function of Reynolds number based upon 6. After all itera-tions converge within specified tolerances, the value of P is calculatedafrom the value of 6*, and the values of B and dO/dx are used in thecalculation at subsequent points. The values of dO/dx are integratednumerically to obtain the increment in 0 o be added to a previouslydetermined value of 8. The trapezoidal rule is used to determine thesecond point, the parabolic rule for the third point, and cubic integra-tion for the fourth and subsequent pointa.

For convenience, Eq. (61) ma y be written 8' + 8P - Q. The generalintegration for the nth point is

On =On3 + Gu_31 n 3 + Gn-2 0 n-2 +IGn 1+ Gn 0. (94)

32

S.... . . : ': : ..... .-- " -- ' ... .. -.. . . . .." + + + "+" , i ............. ....... ..........

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where th' C's are functiw~s of the spacings s, t, and u between thepoints and are given in Appendix B. Except tor e and ef, the othern fnvalues in Eq. (94) are known from previous calculations. Inasmuch as

0 Q P 0 (95)Eq. (92) can be rearranged to g(veon. (011_,1 111 0.,, 31. "_G t +G; 0' 4.- Q.(6(0 + Gri )

After convergence of the iterations, Eq. (95) is used to obtain de/dx.Inasmuch as Eq. (94) depends upon the knowledge ot en-3, the value ofa2 s calculated by

on-2 = On-. 3 + "' n - + + "'n-1 0,1-1 + p, lon (97)which becomes the Gn 3 for the next point to be calculated. Th e valuesof the Fls are also given in Appendix B. The values of 02 and e3 obtainedfrom Eq. (95) are used in the calculation of P* and 6P instead of theainitial values obtained by the trapezoidal or parabolic integration.

The success of the above type of integration depends upon thespacing of the points. The values of the increments s, t, and u mustbe of the same order of magnitude, although t is usually larger than sand smaller than u if the parameters itnvolved in the characteristicssolution are selected with care.

After the values of 6* sec w are calculated, the values ofa wd(6* sec 0 )/dx are obtained by parabolic differentiation and added toathe inviscid values of dy/dx to obtain dr /dx. This procedure is believedwto be more accurate than differentiating the value (* sac w + y).a wbecause dy/dx is obtained directly from the characteristics solution andnot by differentiating y with respect to x.

In general, the boundary-layer correction at the throat will havea gradient such that the viscid throat will be slightly upstream of the

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AEOC-TR-78,63inviscid throat. This displacement and the value of the viscid curve-ture at the throat are calculated using the assumption that both theinviscid throat and the boundary-layer correction are parabolic inshape.

6.0 DESCRIPTION OF PROGRAM

The computer program is written in Fortran 1V for use with the IBM370/165 Computer. The program consists of a main section, three functions,and 16 subroutines arranged so that the program can be overla4 ,d toconserve computer storage. The four overlays consist of AXIAL, CONIC,SORCE, and TORIC; PERFC; BOUND and HEAT; SPLIND and XYZ. The inputdata cards are described in Appendix C, and a listing of the program isgiven in Appendix D.Program MAIN. MAIN calls for the various overlays. The title card isread in with the designation as to whether the nozzle is planar oraxisymmetric. A card defining the gas properties and a few pertinentdimensions is then read in. The first subroutine called is AXIAL, inwhich the upstream axial distribution is defined. PERVC Is called tocalculate the upstream contour. AXIAL is recalled to define the downstreamdistribution, and PERFC is recalled to calculate the downstream contour.BOUND is called to calculate the boundary-layer growth. SPLIND iscalled to determine the coefficients of cubic equations to fit theunevenly spaced points along the contour, and XY Z uses these coeffici-ents to obtain ordinates at evenly spaced points along the axis or, inthe case of the planar nozzle, at discrete points along the surface ofthe flexible plate at which the supporting jacks are located.Subroutine AXIAL. In this subroutine, cards are read in with theparameters used to define the axial distributions of velocity and/orMach ,Lumber and with integers which define the number and spacing of thepoints on the axis and on the key characteristics and the sequence of

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subsequent calculations. If the throat characteristic is called for,the upstream end of the upstream distribution starts at the intersectionof the throat characteristic and the axis. An option can be exercisedto not use the throat characteristic and thereby start the distributionat the point where M - 1. This option would normally be used for anozzle with a large throat radius of curvature, e.g. a planar nozzle, orif it were desired to repeat a calculation as in Ref. 13. Another optionis to avoid a radial flow section altogether by using a polynomial dis-tribution from the throat to the beginning of the test cone or rhombus.Other options will be described in Appendix C when the input cards arediscussed.

Subroutine BOUND. This subroutine is used to calculate the turbulentboundary-layer correction to the inviscid contour. The stagnationconditions are input, as are the parameters to describe the wall tem-perature distribution, the temperature distribution in the boundarylayer, and the factors relating the compressible skin-friction coefficientsto incompressible values.

Subroutine CONIC. This subroutine is used within AXIAL to give thederivatives of Mach number with respect to r/r 1 in radial flow from Eq.(29).

Function CUBIC. This subroutine is used to obtain the smallest positiveroot of a cubic equation.Function FMV. This subroutine determines the Mach number for a givenPrandtl-Meyer angle.

Subroutine FVDGE, This subroutine is used within PERFC in conjunctionwith NE O Lo smooth the inviscid coordinates as desired.

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Subroutine HEMT. This subroutine is a dummy called by BOUND but isincluded so that with a more elaborate subroutine a heat balance can bemade to determine the wall temperature if the material conductivity isspecified and the cooling water passage geometry and quantity of floware specified.

Subroutine NEO, This subroutine is used with PERFC in conjunction withFVDGE to smooth the inviscid coordinates as desired by modifying theordinate such that the second derivative is more nearly linear aftersmoothing than beforehand,

Subroutine OFELD. This subroutine is used within PERFC to calculate theproperties at the intersection of a left- and a right-running char-acteristic.

Subroutine OREZ. This subroutine is used to make all values of an arrayequal to zero prior to a new calculation.

Subroutine PERFC. In this subroutine, the properties along the keycharacteristics are first calculated to go with those along the axis.Th e intermediate characteristics are then calculated and the contourpoints obtained by integrating the mass flow crossing each character-istic. If desired, certain designated intermediate characteristics maybe printed out. If smoothing of the ordinates is desired, the inputsassociated with the smoothing are read and the smoothing applied.Inasmuch as the wall angle is interpolated from mass-flow considera-tions, independently of the coordinates, the wall slopes are integratedfrom the inflection point toward the throat for comparison with the"interpolated ordinates. Parabolic integration is used for this purposeas well as for the mass flow. Also calculated for comparisorn are theordinates of a parabola and a hyperbola which have the same radiusratio, R, inasmuch as the transonic solution should be equally applic-able to these shapes for the number of terms retained in the series,

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Eqs. (2) and (3). Finally, the scale factor, the value of rI in inches,is applied to obtain the inviscid coordinates in inches, and the abscis-sas are also shifted as desired.

Subroutine PLATE. This subroutine is also a dummy to allow additionalcalculations to be made for a flexible plate contour after the coordinatesat each Jack location have been interpolated by SPLIND and XYZU

Subroutine SCOND. This subroutine is used in BOUND, NEO, and PERFC forparabolic differentiation of coordinates to obtain the slopes, or ofslopes and abscissas to obtain second derivatives. Three points at atime are used to establish the parabola, and the slope is obtained atthe center point. The slopes at the first and last point are also obtained,but with less accuracy.

Subroutine SORCE. This subroutine is used within AXIAL to give thederivatives of velocity ratio, W, with respect to r/rI in radial flowfrom Eq. (30).

Subroutine SPLIND. This subroutine computes the coefficients of cubicequations that fit the unevenly spaced points obtained from the char-acteristics solution. The initial and final slopes are used togetherwith the coordinates to determine the cubic coefficients.

Function TORIC. If the velocity gradient is known at the axial pointwhere M - 1, this function gives the value of radius ratio, R, whichwould produce such a gradient from the transonic theory used. Thisfunction is used in AXIAL if the option is exercised of specifying theMach number at point F but not specifying the value of R. It is alsoused to determine the value of R for calculating streamlines other thanthe contour itself.

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Subroutine TRANS. This subroutine calculates the throat characteristicfrom the transonic theory. In AXIAL, at the point where the throatcharacteristic intersects the axis, the derivatives of velocity andMach number are used to determine the coefficients of the polynomialdescribing the axial distribution. In PERFC, the flow properties alongthis key characteristic are used at the number of points specified asone plus a submultiple of 240.

Subroutine TWIXT. This subroutine is used in PERFC and BOUND to inter-polate the ordinate and other properties at a specified point. A four-point Lagrangian interpolation is used with two points on either side ofthe specified point.

Subroutine XYZ. This subroutine uses the cubic coefficients obtained inSPLIND for calculating the ordinate, slope, and second derivative atspecified values of the abscissa read as inputs in the MAIN section ofthe program. The points may be at even intervals in the abscissa or atarbitrary uneven intervals. The points may be the same points as thoseinput to SPLIND if a comparison is desired between the derivatives sodetermined and those obtained el8ewhere in the program.

7.0 SAMPLE NOZZLE DESIGN

Th e design of a Mach 4 axisymmetric nozzle is selected to illus-trate use of the computer drogram. The input cards for the sampledesign are given in Table 1. An axiiymmetric nozzle is specified byleaving JD blank (JD - 0) on Card 1. Leaving SFOA blank ou Card 2specifies that the upstream axial velocity distribution is not a fifth-degree polynomial. Leaving FMACH blank cn Card 3 specifies that thevalue of FMACH will be computed by the program, and leaving IX blank onCard 4 specifies a cubic distribution. The computed value of FMACH is3.0821543, which is greater than the value of BMACH specified on Card 3;

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therefore, BMACH also becomes 3.0821543. The negative value of SFmeans that the inviscid exit radius of the nozzle is 12.25 in. Thevalue of PP means that the inflection point will be 60 in. downstream ofan arbitrary point. Leaving XC blank specifies the downstream axialdistribution will be a fourth-degree polynomial, and the positive valueof IN on Card 4 specifies a Mach number distribution. The values of MT,NT, MD, ND, NF, and LR determine the number of points on the key char-acteristics and are all odd numbers because each includes both endpoints of each distribution which is divided into an even number ofincrements. The negative value of NF specifies the contour points to besmoothed according to Card 5, and the negative value of LR specifiesthat the transonic distribution be printed as the first page of thesample output. The NX value of 13 specifies the spacing of the axialpoints between points I and E to be close together near Point I with thelast increment about 3.17 times as large as the first increment,(201.3 191.3). The JC value of 10 specifies that every 10th left-running characteristic will be printed for the upstream contour togetherwith the right-running characteristic through Point E. The smoothingincegers on Card 5 are used to control the emoothing subroutine.

Table I. Input Cards fo r Sample DesignCARD IITLE JUMACH4CARD 2

GA M AN ZO RO VISC VISM SFOA XAL144 1716s563 1. 0,896 2.26968E-8 198,7a 10000CARO 38.67 60 3. 4. -12.25 60.FTAD RC FMACH BMACH CNC SF PP MC

CARD 4MT NT Ix IN IG MO NO NF MP MG dA Jx JC IT LR NX41 21 10 41 49 -61 1 10 -21 13

CARD 9NOUP NPCT NODO5S 85 S,)CARO 6PPQ TO TwT IWAT GFUN ALPH IHT IR I1 LV200. 16J8* 900. 540. .36 1 5CARD 7

XSt XLOW XENO XINC Hi XMID XINC2 CN1000. 46. 172. 2.

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For the boundary-layer calculations fo r stagnation conditions of200 psia and 1638R, the value of QFUN of 0.38 overrides the specifiedthroat temperature of 900R and produces the throat temperature of 866Ras indicated on the output. Leaving ALPH blank causes the temperaturedistribution in the boundary to be parabolic for both the calculation ofthe boundary-layer parameters and the calculation of the referencetemperature. Leaving IHT blank causes the longitudinal distribution ofwall temperature to vary as a square-root function of the area ratio

* corresponding to the local Mach number; m a 1/2 In Eq. (90). ILeavi.ng IR:' blank causes the transformation from incompressible to compressible

values of skin friction coefficient to be calculated using a modifiedSpalding-Chi reference temperature and a Van Driest reference Reyuoldsnumber. Specifying ID - I takes into account that the boundary-layerthickness is not negligible relative to the radius of the inviscid core,and its positive value causes the boundary-layer calculations to beprinted for the first and last iteration; the number of iterations isspecified by the absolute value of LV (LV m 5 fc r the example).

For the final coordinates, interpolated at even intervals, speci-fying XST - 1,000 (the same value as XBL on Card 2) keeps the X-coordinates consistent with the location of the inviscid inflectionpoint at 60 in. downstream of an arbitrary point.

The main parameters selected for the sample problem were the inflec-tion angle, the curvature ratio, and the Mach number at the point B.The selected values of 8.67 deg, 6, and 3.0821543 (computed), respectively,are not necessarily optimum but result in a nozzle with an upstreamlength of about 14 in. from the throat to the inflection point, alength of about 31 in. from the inflection point to point 3 (see Fig. 3),and nearly 120 in. from the inflection point to the theoretical endof the nozzle. Such dcwnstream lengths are probably conservative andcould be reduced to some degree although experience with Mach 4 axisym-metric nozzles is very limited.

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The number of points used on the key characteristics should be con-sistent with the number of points used in the axial distributions inorder that the individual nets in the characteristics network should notbecome too elongated (e.g., see Fig. 7). The spacing of the points onthe final contour should also progress in an orderly manner. Severaltrials may be necessary to optimize the various inputs to the program.

8.0 SUMMARY

A method and computer program have been presented for the aero-dynamic design of planar an d axisymmetric supersonic wind tunnel noz-zles., The method uses the well-known analytical solution for radialsource flow and connects this radial flow region to the throat and testsection regions via the method of characteristics. Continuous curvatureover the entire contour is attained by specifying polynomial distribut-ions of the centerline velocity or Mach number and matching variousderivatives of these polynomials at the extremities of the radial flowregion, the test section, an d a throit characteristic. The inviscidcontour is obtained by initiating characteristics outward from thecenterline and then integrating the mass flux along these character-istics to compute the inviscid nozzle boundary. The final wall contouris then obtained by adding to the inviscid coordinates a boundary-layer correction based on displacement thickness computed by integratingthe von KArmAn momentum equation. To illustrate the method, a sampledesign calculation was presented along with the associated input andoutput data. A listing of the computer program and an input descrip-tion are included.

REFERENCES

1. Prandtl, L., and Busemann, A. "Nahrungsverfahren zur zeichnerischenErmittlung von ebenen Stromungen mit uberschall Geschwindigkeit."Stodola Festschrift. Zurich: Orell Susli, 1921.

41

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rAEDC-TR-78-632. Foel2ch, K. "A New Method of Designing Two Dimensional Laval

Nozzles for a Parallel and Uniform Jet." Report NA-46-235-1,North American Aviation, Inc., Downey, California, March 1946.

3. Riise, Harold N, "Flexible-Plate Nozzle Lesign for Two,.DimensionalSupersonic Wind Tunnels." Je t Propulsion Laboratory Report"No. 20-74, California Institute of Technology, June 1954.

4. Kenney, J. T. and Webb, L. M. "A Summary of the Techniques ofVariable Mach Number Supersonic Wind Tunnel Nozzle Design."AGARDograph 3, October 1954.

5. Sivells, J. C. "Analytic Determinatiton of Two-Dimensional Super-sonic Nozzle Contours Having Continuous Curvature."AEDC-TR-56-11 (AD-88606), July 1956.

6. Owen, J. M. and Sherman, F. S. Fluid Flow and Heat Transfer atLow Pressures and Temperatures: "Design and Testing of aMach 4 Axially Symmetric Nozzle for Rarefied Gas Flows."Rept. HE-150-104, July 1952, University of California,Institute of Engineering Research, Berkeley, California.

7. Beckwith, I. E., Ridyard, H. W., and Cromer, N. "The AerodynamicDesign of High Mach Number Nozzles Utilizing Axisymmetric Flowwith Application to a Nozzle of Square Test Section."NACA TN 2711, June 1952.

8. Cronvich, L. L. "A Numerical-Graphical Method of Characteristicsfor Axially Symmetric Isentropic Flow." Journal of the Aero-nautical. Sciences, Vol. 15, No. 3, March 1948, pp. 155-162.

9. Foelach, K. "The Analytical Design of an Axially SymmetricLaval Nozzle for a Parallel and Uniform Jet." Journal ofthe Aeronautical Sciences, Vol. 16, No. 3, March 1949, pp .161-166, 188.

42"i' ' 4

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10. Yu , Y. N. "A Summary of Design Techniques fo r AxisymmetricHypersonic Wind Tunnels." AGARDograph 35 , November 1958.

11. Cresci, R. J. "Tabulation of Coordinates for Hypersonic Axisym-nmetric Nozzles Part I - Analysis and Coordinates for TestSection Mach Numbers of 8, 12, and 20." WADD-TN-58-300,Wright Air Development Center, Dayton, Ohio, October 1958.

12. Sivells, J. C. "Aerodynamic Design of Axisymmetric HypersonicWind-Tunnel Nozzles." Journal of Spacecraft and Rockets,Vol. 7, No. 11, Nov. 1970, pp, 1292-1299.

13. Hall, I. M. "Transonic Flow in Two-Dimensional and Axially-* Symmetric Nozzles." The Quarterly Journal of Mechanicsand Applied Mathematics, Vol. 15, Pt. 4, November 1962,

pp. 487-508.

14. Kliegel, J. R. and Levine, J. N. "Transonic Flow in SmallThroat Radius of Curvature Nozzles." AIAA Journal, Vol. 7,No. 7, July 1969, pp. 1375-1378.

15. May, R. J., Thompson, H. D., and Hoffman, J. D. "Comparisonof Transonic Flow Solutions in C-D Nozzles." AFAPL-TR-74-110, October 1974.

16. Edenfield, E. E. "Contoured Nozzle Design and Evaluation forHotshot Wind Tunnels." AIAA Paper 68-369, San Francisco,California, April 1968.

17. Moger, W. C. and Ramsay, D. B. "Supersonic Axisymmetric NozzleDesign by Mass Flow Techniques Utilizing a Digital Computer."AEDC-TDR-64-110 (AD-601589), June 1964.

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18. Spalding, D. B. and Chi, S. W. "The Drag of a Compressible TurbulentBoundary Layer on a Smooth Flat Plate With and Without HeatTransfer." Journal of Fluid Mechanics, Vol. 18 , Part 1,January 1964, pp. 117-143.

19. Van Driest, E. R. "The Problem of Aerodynamic Heating."Aeronautical Engineering Review, Vol. 15, No. 10, October1956, pp. 26-41.

20. Sivells, J. C. "Calculation of the Boundary-Layer Growth in aLudwieg Tube." AEDC-TR-75-118 (AD-A018630), December 1975.

21. Coles, D. E. "The Young Person's Guide to the Data." ProceedingsAFOSR-IFP-Stanford 1968 Conference on Turbulent Boundary LayerPrediction. Vol. II, Edited by D. E. Coles and E. A. Hirst.

22 . Wieghardt, K. and Tillmann, W. Zur Turbulenten Reibungsschichtbe i Druckanstieg. Z.W.B., K.W.I., U&M6617, 1944, translatedas "On the Turbulent Friction Layer fo r Rising Pressure."NACA-TM-1314, 1951.

23 . Coles, D. E. "The Turbulent Boundary Layer in a CompressibleFluid." RAND Corporation Report R-403-PR, September 1962.

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AEDC-TH-78-63APPENDIX ATRANSONIC EQUATIONS

When Eq. (5) is substituted into Eqs. (2),(3) and (4), Eq. (2)can be written as:

1 GR GSu i3-- cy)S -S2 S3 "S T+ (i -aF + GT +Gv'' x,

8S 2 /+2X2 -2 V + )- + 33 K ++,-/-(1 - .- ...2 S 3

y2U Y2 U y6 U y4 +U 2y2+ 2+ 4 4 ... U2 + 63 . 43 + 232S _2__3

S y42(2+ UxP2 Y U0O )

222+ 3 - (1 0 - 30)y\+2 \ 4S + A i

where the coefficients are written in the terminology of the programand x and y are normalized with respect to yo. For planar flow,

GR - (15 - y)/270 (A-2)

US (782 2 + 3507 y + 7767)/272160 (A-3)GT - (134 y2 + 42 9 y + 123)/4320 (A-4)av - 5 y/18 (A-5)GK - (2y 2 - 33 y + 9)/24 (A-6)U4 2 - (y + 6)118 (A-7)U2 2 - y/9 (A-8)

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63 = (362 y2 + 1449 ,y 3177)/12960 (A-9)U43 y43 ' (194 y + 549 y - 63)/2592 (A-10)

U 3 (854 yL + 807 y + 279)/12960 (A-11)" Up2 (26 y + 27 y + 237)/288 (A-12)

2U0 = (26 y + 51 y - 27)/144 (A-13)For axisymmetric flow,

GR - (15 - 10 y)/288 (A-14)GS - (2708 y 2 + 2079 y + 2115)/82944 (A-15)GT w (92 y2 + 18 0 y - 9)/1152 (A-16)GV - (y + 0)/8 (A-17)GK - (4 y2 - 57 y + 27)/48 (A-18)U42 (2 y + 9)/24 (A-19)U2 2 (4 y + 3)/24 (A-20)U6 3 - (556 y + 1737 y + 3069)/10368 (A-21)U4 3 -(388 y2 + 777 y + 153)/2304 (A-22)

U Y2U2 3 (304 y + 255 y - 54)/1728 (A-23)P2 - (52 + 51 y + 327)/384 (A-24)

Up0 - (52 y2 + 75 y - 9)/192 (A-25)The first part of Eq. (A-i), which is independent of y, can be recognizedas Eq. (11) for planar flow or Eq. (12) for axisymmetric flow inasmuchas x and y are normalized here with the value of yo.

46

own'!--- ... :- iglo o "

i:- ,, _ T3:I r"l~:IIrI ]1'

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In a similar manner, Eq. (3) can be written as4 2

2-) 42 V22 +V 0 2S 2( 3 )s + s2

6 v y4 +V 2+ V6 3 - 4 3 +v 2 3 y -V 0 33'2+ x I (2y +- 2 - 3+)y 2y 1 .5q(9 - 3o)S

6U6 3 y4 -4 U4 3 + 2 U 3 +,S2( 4 U 2 'X12x22 Up PO+

2 S -2

+ x3 "4 y .. )/ (A-26)

For planar flow,

V4 2 - (22 y + 75)/360 (A-27)

V22 - (10 y + 15)/108 (A-28)V0 2 - (34 y - 75)/1080 (A-29)V6 3 - (6574 y + 26481 y + 40C59)/181440 (A..30)

"V4 3 - (2254 y2 + 6153 y + 2979)/25920 (A-,31)

2V2 3 - (5026 y2 + 7551 y 4923)/77760 (A-32)

-0 3 (7570 y + 3087 Y + 23157)/544320 (A-33)

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III: t A EDC-TR.78.03L,

, Fo r axisymmetric flow,

v4 2 " (Y 3)/9

v,2 (20 y + 27)/96 (A-35)V0 2 " (28 y - 15)/288 (A-36)V6 3 - (6836 y2 + 23031 Y + 30627)/82944 (A-37)V4 3 " (3380 y2 + 7551 Y + 3771)/13824 (A-38)V2 3 - (3424 y2 + 4071 y 972)/13824 (A-39)"0O3 - (7100 Y2 + 2151 Y + 2169)/82944 (A-40)

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* I ,

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APPENDIXCUBIC INTEGRATION FACTORSIf a curve through four points with ordinates a, b, c, and d,spaced at uneven increments in abscissa, s, t, and u, is defined

by a cubic equation, th e area under each section of the curve canbe found in the following manner:Area b F a+ b-+ Fc +dd (B-i)a-b as Fbs cs FdsAreab-c at a + Fbt b + F c + d (B-2)F Aresc-d "F a + Fbu b + F c +F d (B-3)- u u cu Fdu -3li

ota G a Gb + G c + Gd d (B-4)

where2S(s + 4t (+-5)

",2 Is + 4t + 2uiij a +SFbs +--)(B-6)bs 2 12 t(t + u)a3(s + 2t (2B)cs T2tu(a + t) 's- 7)

I:: s_(L+2t,)Fds 12 (s + t + u)(t + u)u (B-B)t 3t+2u (-9)t 12s(s + t)(s + t + u)F - 2+ t + 2u

(t 2 12s(t + u)t 2 (2s + t -2B1)Fct 2-u(s + t) (B-11)

49

_ _....__ _ _ _ _ _ __._ _.-- - .,.

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t 3 (2s + t) (B-12)Fdt -- 12u(t + u)(s + t + U)

F u3(2t + u)F = 32 )(B-13)au 12s(s + M)s + t + U)3iu3(2s + 2t +_y (B14

rbu - 12st(t + u) (B-14)u 2s + 4t + u) (-5

cu 2 12t(s + t)

.. u2 2s + 4t + 3u) (B-16)du 2 12(t + u)(s + t + u)

G -F + Fat + F (B-17)

Gb bs + Fbt + Fbu (1-18)

Gc F + F u (B-19)

G -F + F + F (B-20)d ds dt du

If all increments are equal, thens t - u h (B-21)

Fds -F t Fdt -Fau -h/24 (B-22)

F -F-5h/24 (B-23)Co bu

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F = F - 19h/24 (B-24)

Fas Fd u -9h/24 (B-25)

F t Fct - 13h/24 (B-26)Gt 3h/

d 3h/8 (B-27)

G - c 9h/8 (B-28)The values of G's in Eq. (96) correspond to those in Eq. (B-4).The value of F's in Eq. (97) correspond to those in Eq. (B-i).

*5 1

22$WA,,j'""

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APPENDIX CINPUT DATA CARDS

Input ColumnsCard 1ITLE 2-12 TitleJD 14-15 Blank (0) for axisymmetric contour,-1 for planar.Lard 2GAM 1-10 Specific heat ratio.AR 11-20 Gas constant, ft2/sec2 R.ZO 21-30 Compressibility factor for an axisym-metric nozzle, constant for entire

contour. Or, for a planar nozzle, ZOis half the distance (in.) between theparallel walls, and the compressibilityfactor is one.

RO 31-40 Turbulent boundary-layer recovery factor.VISC 41-50 Constant in viscosity law.V1SM 51-60 Constant in viscosity law. If VISM isequal to or less than one,

VISC* T lb-sec/ft 2If VISM is greater than one,1.5ISC*T 2"11 VI- T- lb-sec/ft2 . If"T + VISM

T is greater than VISM,VISC* T ; T

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on the characteristic diagram. If nega-tive, absolute value is distance from th ethroat to Point G. If Blank, 3rd- or 4th-deg distribution iu used depending on valueof IX on Card 4.

XBL 71-80 Station (in.) where interpolation isdesired (e.g., the end of a truncatednozzle). If XBL1000., the spline fitsubroutines are used to obtain values atincrements evenly spaced in length.

Card 3ETAD 1-10 Inflection angle in degrees if radial flow

region is desired. Two characteristicsolutions are obtained, one upstream andone downstream of Point A. If ETAD = 60.,the entire centerline velocity distributionis specified and only one solution isobtained and the inflection point must beinterpolated. If ETAD - 60., IQ 1, IX - 0,on Card 4.

RC 11-20 Ratio of throat radius of curvature tothroat radius. Must be given if ETAD - 60.or FMACH - 0. If FMACH is given, RC iscalculated. If LR - u, IX = 0 givws third-deg equation betweev Mach 1 and EMACH,matching first and second derivations ateach end. If LR 0 0, the value of RC foundfo r LR = 0 is used with given value of FMACHto define a fourth-deg equation. If IX = 1and FMACH is given, RC is calculated todefine a fourth-deg equation. If LR 0,a new value of FMACH is found, compatiblewith the value of RC calculated fo r LR = 0.

FMACH 21-30 Mach number at Point F if ETAD # 60. Nega-.tive value specifies Prandtl-Meyee angleat Point F as IFMACHI *ETAD (usually around-7). If PMACH and RC are given, IX = 0and 4th-deg distribution is used. IfFMACH - 0 and IX = 0, a 3rd-deg distribu-tion is used. If FMACH = 0, and IX = i,a 4th-deg distribution is used. FMACH iscalculated if not given. If ETAD = 60.,Point F is not defined.

53

. . . .

'' ' ' .... .. ' ," P , --- "-- --'; ' " .. ... " . .............. .. ........ .......

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BMACH 31-40 Mach No. at Point B if ETAD 60.(MC 41-50 Absolute value is design Mach No. at PointC. If ETAD 0 60, positive CMC giyes d 2M/dx 2-0, and negative CMC gives d2 M/dx' 0 0. If

ETAD - 60., CMC ts positive.SF 51-60 Scale factor by which nondimension coordi-nates are multiplied to give dimensionsin inches. If SF - 0, nozzle will have an

inviscid throat radius (or half-height) ofI in. If negative, nozzle will have aninviscid exit radius (or half-height) ofISF1 in.

PP 61-70 Station (in.) at Point A. PP - 0 givescoordinates relative to geometric throat.Negative PP gives coordinates relative tosource or radial flow (ETAD 0 60.).XC 71-80 Nondimensional distance from source toPoint C. XC 1. requires centerline Mach

No. distribution from Point B to Point Cto be read in as input data on Unit 9.Otherwise, positive XC gives 5th.-dog dis-tribution if CMC positive and 4th-deg if CMCnegative. XC - 0 gives 4th-deg dietributionif CMC positive and 3rd-deg if CMC negative.Negative XC and IN gives 3rd-deg distributionwith d2W/dx 2 not matching source flow atPoint B. If ETAD - 60. and XC > 1, XC is ratioof length, from throat to Point C, to throatheight. Negative XC gives 3rd-deg distribu-tion in M; XC - 0 gives 4th-deg distribution;XC > 1 gives 5th-deg distribution. XC - 1.requires centerline Mach No. distribution tobe read in as input data on Unit 9.

Card4MT 1-5 Number of points on charact',ristis EG if

ETAD ' 60. or CD if ETAD - 60. Maximumvalue about 125. Use odd nuwjer. A zero ornegative value stops calculation after tonter-line distribution is calculated if NT positive.

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=l . . .. .. i IIII .ll N. . . . . . f , ,

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NT 6-10 Number of points on axis IE. Maximum valueis 149-LR. Use odd number. A zero or nega-tive value stops calculation before center-line distribution is calculated but afterparameters and coefficients of distributionare calculated.

IX 11-15 Determines if third derivative of velocitydistribution is matched. IX - 1 matchesthird derivative with transonic solution.IX w -1 matches third derivative with sourceflow value. IX - 0 does not match third deriva-tive but gives constant third derivative ifRC - 0 or FMACH - 0.

IN 16-20 Determines type of distribution from Point Bto Point C, positive for Mach No . distribution,negative for velocity distribution. IN - 0 forthroat only. If XC is greater than 1., thedownstream value of the second derivative atPoint B is 0,1* JINJ times the radial flowvalue. Similarly, If ETAD - 60., the secondderivative at Point I is 0.1i*IN times the"transonic value.

IQ 21-25 Zero for a complete contour if ETAD 0 60., 1 orthroat only or if ETAD - 60., -1 or downstreamonly.MD 26-30 Number of points on characteris'ic AB. Maxi-mu m value about 125. Use odd number. A zeroor negative value stops calculation similarly

to MW.

ND 31-35 Number of points on axis BC. M~aximum value is150. A zero or negative value acts like NT.NY 36-40 Absolute value is number of points on character-istic CD for ETAD , 60. Maximum value is 14 9or 200 - ND - MP - jMQj - number of points onupstreem contour. Negative value calls forsmoothing subroutine.MP 41-45 Number of points on conical section GA ifFMACH 0 BMACH. Use lialue to give desired in-

crements in contour -, usually not known forinitial calculation.

55*--............. . . . . . .

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* I MQ 46-50 Number of points downstream of Point D ifparallel inviscid contour desired. A nega-tive value can be used to eliminate theinviscid printout.rB 51-55 Positive number if boundary-layer calcula-

tion ir desired before spline fit. Nega-tive number transfers control of program toJX. Absolute values greater than on e areused to approximately halve the number ofpoints on the upstream contour even thoughLR + NT - I points are calculated from char-*acteristic network if LR > 2, or (NT + 1)points if LR a 0.JX 56-60 Positive number calls for calculation of stream-lines, zero calls for repeat of inviscid calcula-tions requiring new cards 3 and 4, or, ifXB L 1000., for spline fit after inviscid calcu-lation, negative number calls for repeat of cal-culations requiring new cards 1, 2, 3, and 4.JC 61-65 If not zero, calls for printout of intermediatecharacteristics within upstream contour if JCis positive and downstream contour if JC isnegative. Characteristics are (NT - 1)/JC or

(N D - 1)/(-JC). Opposite running characteristicthrough Point E (or B) is also printed.IT 66-70 Number of points at which spline fit is desiredif points are not evenly spaced, such as Jacklocations for a flexible plate. Used only fora planar nozzle, inasmuch as a nonzero valuecalculates distance along curved plate surface.Positive value of IT tequires additional cards

to be read in (8 points per card) after boundarylayer is calculated.LR 71-75 Absolute value is number of points on throatcharacteristic used in characteristics solution.Negative values give printout of transonic solu-

tion. LR - 0 gives M - 1 at Point I.NX 76-80 Number from 10 to 20 determines spacing of pcintson axis for upstream contour. NX w 10 giveslinear spacing. NX > 10 gives closer spacing ofpoints at upstream end than at downstream end.

NX - 0 same as NX 20. Ratio of downstream

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increment to upstream increment is (NT - 1 )NX/10 --NX/1O(NT - 2) . Optimum values, usually 13 to 15,

determined by trial and error for specific con-tour desired. Negative NX used with negative LRlimits printout to transonic solution.

NOTE: A zero value of MT , NT, MD, or ND will allow a repeat of cal-culations for parameters specified by new cards Nos. 3 and 4.A negative value will allow a repeat of calculations for newcards Nos. 1, 2, 3, and 4.

Card 5NOUP 1-5 If smoothing is desired, negative NF. Number oftimes upstream contour is smoothed.NPCT 6-10 Smoothing factor in percent. Smoothing factor- NPCT/100.NODO 11-15 Number of times doimstream contour is smoothed.Card 5 If boundary-layer calculation is desired usingorinviscd points calculated from characteristicsor solution. (N o smoothing).Card 6 If boundary-layer calculation is desired usingevenly spaced points interpolated from splineor fit of points from characteristics solution.Card 7 If boundary-layer calculation is desired usingevenly spaced points interpolated from spline

fit of smoothed points.4 PPQ 1-10 Stagnation pressure (psia).

TO 11-20 Stagnation temperature, Rankine.TW T 21-30 Throat wall temperature, Rankine, if QFUN - 0.If TW T - 0, the wall temperature is assumed to

be the adiabatic value.TWAT 31-40 Wall temperature, Rankine, at Point D. Fo r

water-cooled wall, the bulk water temperaturei.s assumed to be 150 lower than specifiedTWAT. The cooled wall temperature distribu-tiou is assumed to be

57

t=,.. ........ . !. j. "u' ..........

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(TWT CT) "' A* 1)TWTWAT +where A/A* is the area ratio corresponding tolocal value of Mach number and Ac refevs toPoint C.For negative IHT

TW - TWAT + ,TWT-IA x A A*

QFUN 41-50 Heat-transfer function at the throat.QFUN - ha(Taw - TWT)TWT - TWAT + 15

where ha has dimensions of BTU/sec/sq ft/R andis obtained by Reynolds analogy from the skin-friction coefficient. If QFUth is specified,input value of TW T is ignored and TW T is calcu-lated from QFUN.

ALPH 51-60 Parameter specifying temperature distribution inboundary layer. ALPH - I. uses quadratic dis-tribution both in the calculazion of the refer-ence temperature TP and the calculation ofboundary-layer shape parameters. ALPH w 0 usasparabolic distribution in boti. calculatiovs.ALPH -u 1. uses quadratic distribution fo r TPand parabolic in the calculation of boundary-layer shape parameters. Within bcundary layer,T - Tw + a(Taw- TW ) (U/11)

+ Tes- (Taw- Tw) - Twj (U/Ue) 2where a 1 for quadratic dist.

a -0 for parabolic dist.IHT 61-65 Integer which determines temperature distribu-tion (see TWAT). If nonzero, IRiT determineshow often subroutine HFAT is called. An absolutevalue of IHT greater than KO , the number of points"on the upstream contour, will prevent HEAT frombeing called but will allow the choice of tempera-

ture distribution to be made.

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NOTE: HEAT is a special purpose subroutine fo r determ.ningheat-transfer values fo r the upstream contour. Thesubroutine HEAT incorporated in this program is adummy.

66-70 Integer, parameter specifying transformationfrom incompressible to compressible values.If YR w 2, Coles' transformation is used fo rCf and Re If IR - 1, TP is calculated bya modification of the Spalding-Chi (Van Driest)method. If IR - 0, th e Van Driest value ofRe6 is used, but if IR- -1,Colas' lav: ofcorresponding stations is used.

Cf Cf * TE/TP, Re - RD*Ree,i fID 71-75 Integer. If ID - 1, axisyminetric effectsare included in momentum equation and in cal-

culation of boundary-layer parameters (6 nornegligible relative to coordinate normal toaxis). If ID - 0, hese effects are omitted.Negative ID suppresses the printout of theboundary-layev calculations.LV 76-80 Integer. Absolute value, usually 5, deter-mines number of rimes boundary-layer solutionis iLerated so that radxus terms in momentumequation refer to viscid radius instead ofinviscid radius. Value of 0 or absolute val-eof 1 uses inviscid radius, Positive LV repeatsboundary-layer calculations for new set ofparameters on a new card if XBL j 1000.Card 5 If streariflines are desired, JX positive, (No

smnoo thing.)METAD 1-10 Inflection angle in degrees for streamlinedesired if ETAD 0 60. for Card 3. If ETkD -60. on Card 3, use ETAD - 60 on this card.* 11-20 Fract ion of r'.ottour desir.-ed if ETAD - 60.Otherwise, QON lTAD (in Card 5 divided byETAD oa Card 3.X,1 21-30 Value to updite JX for subr.eq4oent calcula-

tion, JX XJ.

,, 2*.;. a a'11d598Y '

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Card 5 If SPLIND used after inviscid calculation* (JX zero or negative and JB zero or nega-or tive). (N o smoothing.)Card 6 If SPLIND usad after viscid contouir (,7B posi-tive and LV zero or negative). No v'.oothingof inviscid contour. Or, if inviscid contouror is smoothed before SPLIND is used.Card 7 If inviscid contour is smoothed, boundary layeris added and SPLINE Ls desired.XS T 1-10 Station (in,) for throat value of X. IfXS T a 1000., program uses value previously

determined by specifying PP on Card 3. Other-Wise, value of XST is used to shift contourpoints by desired increments for arbitrmryStation 0.XLOW 11-20 Starting value for interpolation. Second valueof interpolated X a XLOW + XINC.XEND 21-30 End value for interpolation. If zeto, SPLINDis used to calculate slcpe and d2y/dx2 at samepoints as previously defined.XINC 31-40 Increment in X for interpolation. If zero, andBJ > 10, contour is divided into BJ increments.BJ 41-50 Value to update JB for subsequent calculation.JB - BJ. If negative and XEND - 0, interpola-tion is made at diecreate points read in on sub-sequent cards similar to case when IT > 0.XMID 51-60 Intermediate value for interpolaticn. Distance

(XMID-XLOW) is divided into increments definedby XINC, and distance (XEND-XMID) is dividedinto increments defined by XINC2.XINC2 61-70 Inctements in X between XMID and XEND if differ-ent than XINC.CN /1-80 Number of copies desired of final tabulation ofcoordinates if more than on e copy is desired.

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