UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE Progress Report No. 1 April - December 1951 THEORY AND DESIGN OF A VARIABLE MTACH NUMBER CORNER NOZZLE By J. S. Murphy Research Associate H. Buning Research Assistant WTM - 221 Project M-951 Air Research and Development Command U. S. oAir Force Contract AF - 33(038)-23070, E. 0. No. 460-31-14

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE TABLE OF CONTENTS Page I SunSmmary.................................., 1 II Introduction...... v..*.****..*..... 2 III Supersonic Portion of Nozzle2..................... 2 (a) Design Criteria...........,..,........... 2 (b) Initial Approximation of Nozzle Shape..,. 3 (c) Curvature of Nozzle Contours............. 6 (d) Nozzle Coordinates....................... 7 IV Transonic and Subsonic Portion of Nozzle,**..***......, 8 (a) Transonic Region.......... *........ 8 (b) Subsonic Region.'.............. 8 V Boundary Layer Computations.........*........ 9 VI Flexibility Criteria for Nozzle Contour........... 9 (a) Corrections for Viscous Effects*......... 9 (b) Corrections for Local Non-Uniformities... 10 VII Preliminary Experimental Results In The Subsonic and Transonic Regions of The Nozzle............... 13 PERSONNEL* ~~c~~*0,00.X.*..*..*0*. **.***.*..*.*..*....*. 15 REFERENCES........................................... 16 TABLE I......*...........*....,............... 17 TABLE II0*.*.......................................... 20 o^TsABL ___

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE WTM - 221 February 1952 Progress Report No. 1 THEORY AND DESIGN OF A VARIABLE MACH NUMBER CORNER NOZZLE I Summary Theoretical considerations are presented on the flow in an asymmetric variable Mach number nozzle in which change in Mach number is accomplished by translation of one contour relative to the other. Design criteria for a specific class of nozzles are set down and a procedure for designing a nozzle to cover any specified Mach number range is proposed. The procedure makes use of an iteration process with the method of characteristics. An approximate theory is included which indicates how the first step in the interation procedure- may be chosen so as to bring about rapid convergence to a solution. The theory also predicts the overall length of nozzles of the class under consideration as a function of the design Mach number range and expansion angle. The procedure is applied to the design of a nozzle which, in operation, is expected to cover the Mach number range 1.4 to 4.0. The length of the supersonic portion of this nozzle is approximately 8 test section heights. Considerations on the flow in the subsonic and transonic regions of this nozzle are included as well as the results of calculations of viscous effects for the case where the nozzle is operated with standard atmospheric stagnation conditions. The nozzle will be evaluated by experimental investigation of test section flow. In order to be able to correct for viscous effects for the entire Mach number range and to have some control over the uniformity of test section flow the nozzle will be constructed with flexible surfaces. The amount and types of flexibility required are discussed.

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE II Introduction Several types of asymmetric variable Mach number nozzles have been proposed. Allen (Ref. 1) investigated the flow in an asymmetric nozzle which he considered to be one-half of a two-dimensional plug-type nozzle proposed by A. Silberstein alt the NACA Cleveland Laboratory. With this type of nozzle, the test section Mach number is varied by translation of one nozzle contour relative to the other. Improvements in the method of obtaining the coordinates of the contour of such a nozzle were presented by Syvertson and Savin (Ref. 2), who suggested the use of an inclined and curved sonic line. Evvward and Wyatt (Ref. 3) investigated the flow in a nozzle whose contours were based on the Prandtl-Meyer theory for flow around a corner, Ferri, Burbank, and Byrne (Ref. 4) studied the flow in a nozzle similar to the type proposed by Allen with the exception that the sonic line was eliminated as a design parameter. This was accomplished by design of the subsonic inlet so that the sonic line was straight and perpendicular to the nozzle wall at the throat over the design Mach number range. Nozzles designed by the above procedures have been evaluated by experimental investigation of the test section flow, and were found to perform satisfactorally over a limited Mach number range (Ref. 2,4). At the time the present investigation was initiated, the upper limit of the Mach number range which had been obtained was approximately Mach 3. The investigation was undertaken to make a study of the general characteristics of variable Mach number asymmetric nozzles, and in particular to obtain a design which would cover the Mach number range 1.4 to 4.*O0 This report presents the progress made between April and October, 1951l III Supersonic Portion of the Nozzle. (a) Design Criteria. A preliminary investigation (Ref. 6) of the two design methods given in Ref. 2 and 4, showed that the overall length of a nozzle designed by either procedure for a given Mach number range and expansion angle is approximately the same for Mach numbers up to 3. The essential difference between the two design procedures is associated with the shape of the sonic line. The nozzles considered in this report are of the type proposed by Allen and use translation of the lower contour as means of changing the Mach number. They incorporate the following criteria some of which were suggested by A. Ferri. I. The sonic line should be straight and perpendicular to the nozzle -2

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE wall at the throat. 2. The nozzle contour should have no inflection points i.e., the second' derivative should not change sign in the supersonic flow region. 3. The variation of the first derivative of the' contour should be continuous and smooth over the nozzle length. 4. No compression wavelets should be introduced in the theoretical design of the contour. In order to cover a specified Mach number range (ML to Mi), two design Mach numbers must be chosen. Let M* and M2 represent the lower and upper design Mach numbers respectively. M3 and M2 satisfy the relations MzN> Ml: M2C Ml. Experimental results (Ref. 4) indicate that the interval (M1 to MN) can be made larger than the interval (Ml to M2). The respective values for' that nozzle are: M1 = 1.71; Mj = 1.27 M2 = 2.63; MN = 2.75 After a suitable choice of design Mach numbers is made, it is assumed that the lower design Mach number M1 is obtained by a simple wave flow in the nozzle. Therefore, all waves originating at the lower contour are cancelled upon striking the upper contour. This assumption fixes the expansion angle of the nozzle as that angle of turning- required to expand a Mach 1 flow to El- Let this angle be called Q,. Also, let the amount of turning required to expand a Mach- 1 flow to the upper design Mach number M2 in a simple wave flow be called Q2. Then with the nozzle in position to give M2,'(o2 - Qo) degrees of expansion are required by reflected waves to obtain M2. The assumption is made that the reflection of waves takes place along a straight wall on the upper surface. Both of the above assumptions were shown to be valid in Ref. 4* On the basis. of the above criteria a variable Mach number nozzle can be designed as discussed in the subsequent paragraphs. An approximate theory is developed which provides an initial estimate of the nozzle shape and predicts the overall length of a nozzle. (b) Initial Approximation of Nozzle Shape. The supersonic portion of the nozzle in the two design positions is shown schematically in Fig. 1. Let the distances along the upper and lower contour of the nozzle -3

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE be designated in the following manner: L; = length of curved portion of lower contour where expansion waves originate. Lu = length of curved portion of upper contour where expansion waves are cancelled, R = axial distance covered by the last simple wave in the nozzle in the Mach M2 position. P = axial distance covered by the last simple wave in the nozzle in Mach M1 position. Z = length of' straight portion of upper contour where waves are reflected. i.e., distance between points 0 and F (Fig. la) h = test section height W= h tan Q~ L" Mach angle corresponding to Ml P2 Mach angle corresponding to M2 s = distance along nozzle contour Therefore (1) hP R h =tan pi tan P Neglecting second order differences we may write for the nozzle in the two design positions-:e (2) Z +Lu=R +LzL W (3) Lu = P + Lt +W Subtracting (3) from (2) yields: (4) z =R-P = h( 1 This determines the position of point F relative to the throatF ~v -4

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE Now consider the nozzle in position to give M2, Fig. 2. The Mach number at the point F on the upper surface, MF, is determined as the Mach number obtained by an expansion of (Q! - 91) from Mach 1. The Mach number at point B on the lower surface is, t We consider the case where the las reflected wave leaves the upper surface at F, strikes the lower surface at B and is reflected from the lower wall as the last simple wave in the nozzle. Consider the wave crossing the nozzle from F to B (Fig. 2). It can be shown that to a fair degree of approximation, the curved wave can be replaced by a straight wave making the angle, (5) P3 = PF + + x' (OF = Mach Angle for MF) 2 with the direction OF and having the length i. The length ~ can be determined from the condition of continuity. Let the length of the last wave in the nozzle be 12. It is determined by the relation sin P2 i or (6) 2 = M2 The mass crossing the wave 12 per second is (7) Q = hM2p2c2 where P2 = density at M2 Cz = speed of sound at M2 The mass, crossing the wave 1 per second, to a sufficient approximation, is given by (8) cX + 3. )

06==44: tt~~~~~~~~~~~~~~~~~~~~~t U' Ps

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE where P -k = density at M = M2 - 24(MX2 MF) c* = speed of sound at M = ps = density at M = M2 - 3/(M2 - MF) CH M cAl/ = ~~%* at M =t Since by continuity Q = Q2 we have (9) i X = M p,/+ c,/+ + p3/4c/+ Hence the position of point B relative to point F is known from eqs. (5) and (9). Equation (6) defines the location of point A relative to B. Point A is located at the nozzle exit. The overall length, L, of the nozzle is then given by the equation: (10) L = Z + 1 cos q3 + M2 cos (91 - 2) Fig. 3 shows the variation of overall nozzle length as a function of expansion angle for various upper design Mach numbers, M2. Since expansion angle determines the lower design Mach number M1, the ordinate is shown as both expansion angle Qs and lower design Mach number M1. Since the locations of points F, B, and A are known relative to the throat, they serve as an initial approximation to the nozzle shape' The distribution of curvature along the upper surface between points F and A and along the lower surface between points 01 and B remains to be determined, (c) Curvature of Nozzle Contours, The next step in the design is to determine the distribution of curvature along the lower surface from OL to B and along the upper surface between points F and A, (see Fig. 2), such that the exit flow is uniform for the two design Mach numbers M1 and M2. This is accomplished by meaxns of an iteration procedure using the method of characteristics (Ref. 6). A curvature distribution is assumed for the lower surface between points 01 and B. Let this distribution be K1(s), Using K1(s) along the lower surface, a distribution of curvature is found along the upper surface between F and A which -6

e 6 1 g 9 z o 1 ttE EIfE g E E.IDISH NOID E, 2 OZ HLOE IZZON 1IO OI;oi -[ X 4e~~~~~~ttttc~'-tCS/ 0;- i t X W g Sl I II ow t w 4 W w g R W W m g i 4- W 1 111111111i1 11 WL w w m I 1w 011111111 GS m m E g wtXM~~~~~~~~~~-iM 11111111111 -11111111 m o~Z 9'/ t99 - jg/011} |[|W o4~~~~~~~~~~~~~~~~~~~~ / ~5Zo.~~~~~ Zg~ MZ I I'xl.+ 09 i I TI I I Itt- 06 i 71;) ~ l':dz3- ||0 5 14~~~~~~~~~~~~~i-i~~~~~~~~~~~i ozT:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ I9Z ('N $o -llllllll)~llllllllm~lll~lttlll~lllW~l t T I e I'g'[ = ~9'I "L f:it: 0 F;~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~FT IIl I(I~ltV Illllllllllll~lllll'lllliIIII d'Rq{!IXIIIIIII Illlulllllll~l l —llt k~ lll4l~ 1-i — 1111111 ooI'~:' Ll, _iii.I l I~ l~, a.Ino7 —-T~~~~~~~~~~~~s:-:-r ~~.i-r~~~~~' -H+ti oo'~Iti,[V 5,I O jrI~N pcU',0 qq iO -'.0 S IO -. I I A 0/'r X F~~~~~~~~~:tti-(t~~~~~~~Ci — r -:!~t!- -~ 1~~~~-t+ i —IL -14 NIL~ii I + ITr; |~~~~~~~~~T O Ol llll tlllllIU.!'L IIIIIIl ST U ITY 0 0 (11TH I D IIN'&EiWe Z S A,l., TI IL I I5 0 I |j |-E-le -| ||!.. I I -t-t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~IT " o f' I~~~~~~~~~~~~~~~~~~~~~ ii~ 11 -i~~~~~~~~~~~~~~~~~~~~~~~~~~~~11 31 OT__1~aMtff (s~viflriv AJ —----. ( i f, ( i f T I 1 1 # -4. I- --- ----- _ 1-f.i~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 1. i ------ -- 4Al..i ~-~ —i —r *~ -- -i~- ~ ot'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~bi- M~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~T I - (~~~~~a J~~~~ ~n),gr ~~~~~~~~~~Og~E ZI~~~~~~~~I'(ye T: 7:i- i~- I~~~~~JOJT I II II iT I I b ODIVA'60J SM'ZON OIUL'L IVERSY, DNTI TNU SAO — " ~~~~~~~~~~~~~~~~~~~f II~~~~~~~~~~~~~~~~~~~~~ll

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE gives uniform exit flow at M2. Let this distribution be Ku(s). With the nozzle in position to give M1, Ku(s) defines a new curvature distribution Ko(s) along the lower surface which gives uniform exit flow at M1. Ko(s) and K1(s) are compared and form the basis for a new estimate of the curvature of the lower surface, say K (s). Kj(s) defines a new curvature distribution for the upper surface, Ku(s) to give uniform MN flow. K,(s) defines a new lower surface curvature distribution Ks(s) to give uniform M1 flow. Ks(s) is compared to Ki(s) and adjustments made so that they are the same, say Klj(s). This procedure is repeated n times until the curvature distributions K.(x) along the lower surface and Knul() along the upper surface give uniform exit flow at both M1 and M2. Experience indicates that approximately 8 to 10 steps are necessary. Each of the above steps is carried out by means of the method of characteristics. Ref. 4 indicates that the use of a 1' wave approximation in the iteration procedure is sufficiently accurate. Experience at the University of Michigan'indicates that this is trueabut that the last few steps in this process should be made with a 1/2" wave approximation. Figs. 4 and 5 show the characteristic diagrams of flow in two nozzles calculated in this manner. The first has design Mach numbers 1.64 and 3060, and the second design Mach numbers 1.64 and 3.87, Both nozzles have an expansion angle of 16'. These nozzles are also shown in Fig. 3 as is the nozzle given in Refo 4. It can be seen from this figure that the agreement between the approximate theory and the characteristic solutions is such that the approximate theory can be used to determine the overall length of a nozzle of the class considered, Fig. 3 shows that for a given upper design Mach number M., overall nozzle length decreases as expansion angle increases, Consequently, since it is desirable to cover a given Mach number range with as short'a nozzle as possible, a design was initiated with design Mach numbers, M1 - 1845, M2 = 3.87 and @ = 22~, which in operation was expected to cover the Mach number range. to 4~0 with a saving in length of approximately 3/4h as compared with the M1 = 1.64, M2 = 3.87, Q1 = 16' nozzle, The results of the approximate theory determined the initial approximation used in the iteration procedure. However, the iteration procedure did not converge readily enough to allow completion of the design in the time available. The exact cause of the non-convergence to a solution is not known. (d) Nozzle Coordinates. Comparison of the two 16' nozzles show that the M2 = 3.60 nozzle is shorter than the M2 = 3.87 nozzle. However, it is believed that the M2 = 3,60 nozzle would not give as uniform a M = 4 flow as the M2 = 3.87 nozzle, because the interval (MN - M2) =.4 is probably too large. The M2 = 3~60 nozzle might be expected to give satisfactory flow at NM 3.80 and the M2 - 3.87 is expected to give satisfactory flow at M = 4.00. Checks of the MN 1.64, M2 = 3.87, @ - 16~' nozzle at Mach numbers M 2.37, 3.23, 3.60 and L.ol indicate that the test section flow is uniform within the approximations of the characteristic analysis at these intermediate Mach -7

~~~~~ 1 2~~~~~~~~~~~2 3 4 5 8~~~~~~~~~~~~~~~~ 7 8 8 9~ 10 10 I 113 PO SITION FO M'36 12'1 13,~~~~~~~5 p /rS, ~~~~~~~~~~~~~~~~~~~~~~~ 15~~~~~~1 1 1 9 t iPOSITION FOR M=1.64 1 I CHARACTERISTIC NET FR = 6"4 I FIG. 4

2~~~~~~~~~~~~~~~~~~~~ dII~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~I 1' 2' 3' - -7~~~~~~~~~~~~I G. 5 1 12' / i' ~1 13 II~ 12~~1 147 P:C AOSISTION FOR M 3.87 0'r CHARACTERISTIG. NETS5FOR 9= 16 FIG. 5

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE nultbers. Because of these considerations the coordinates have been computed only for the M2 = 3.87, 9 = 16' nozzle. The procedure followed in this computation is indicated below. The results of the characteristic diagrams give the distribution of slope along the nozzle surface. This curve is faired and differentiated to obtain d2y which is also faired, and then the curves are integrated to obtain y =d(x) for the contour, which then has a smooth second derivative. The coordinates of the nozzle contours, comptted in this manner* are given in Tables I and II for the M2 = 3.87, M1 = 1.64, = 16' nozzle. IV Transonic and Subsonic Portion of the Nozzle. (a) Transonic region. Refs. 4, and 8 indicate that a sonic line which is straight and perpendicular to the contour can be obtained by proper shaping of the nozzle surface upstream of the throat. It is shown in Ref. 8 that a contour having the equation: 7Z (11) Z a (1 +.1924 x6) _Sonic' LlZe for the region upstream of the throat will give such a sonic line.. (See accompanying sketch for definition of symbols) In practical terms, this equation indicates that the curvature of the wall should be essentially zero for a distance of 1 to 1 1/2 throat heights upstream of the sonic line in order to satisfy the condition used as a starting point in the characteristic analysis of the supersonic flow. This criterion was followed in design of the transonic region of the nozzle. (b) Subsonic region. The flow in this region of the nozzle was analyzed by means of one-dimensional flow theory. However, for the low Mach number positions of the lower contour, this method is not practical. Therefore a few exploratory experiments have been made in the 8 x 13 inch Supersonic Tunnel of the University to check out this portion of the nozzle as well as the transonic region. The results of these experiments are included-in section VII of this report. The entire nozzle contours obtained by the preceding perfect fluid analysis are shown in Fig. 9. These contours are designated as y = yo(x) in the figure which shows the nozzle in the Mach 4.0 and 1.35 positions. -8

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE TV Boundary Layer Computations. A series of computations have been made which give boundary layer displacement thickness along the supersonic contour for various test section Mach numbers (Ref. 10), for standard atmosphere stagnation conditions. The Tucker (NACA) method was followed. The results of these computations are presented in Figs. 6 and 7 which show displacement thickness, 6*, for upper and lower supersonic contours respectively at Mach numbers 1.64, 2.37, 3.23, 3.60, 3.87, and 4*01. The abscissa is the distance along the contour measured downstream from the throat. From Fig. 7 it can be seen that a cross plot of the values of 6* at a given station for all Mach numbers between 1*6 and 4,01 can be. prepared. Fig. 8 is an example of such a cross-plot for stations 15, 12 5, 10, 7.5, and 5 inches downstream of the sonic line along the lower surface, which shows that the data presented in Figs. 6 and 7 sufficiently difines 6* at all points on the supersonic contour for all Mach numbers between 1.6 and 4.01. These data, however, do not take into consideration secondary flows. It is expected that these flows will alter the values of 6* given in Fig. 6 and 7. The extent of this alteration must be determined experimentally, at the present time. VI Flexibility Criteria for Nozzle Contour, The nozzle under investigation is to be used as an instrument to determine the degree of uniformity of the test section flow obtainable with asymmetric nozzles. Consequently, the surfaces will be flexible in order to correct for any flow hon-uniformities resulting from the use of the contours found by the perfect fluid analysis described in sections III and IV. There are two types of flexibility required: (1) large deviations from perfect fluid contour necessary to correct for viscous effects. (2) small deviations from perfect fluid contour necessary to correct for any local non-uniformities which may occur in the test section flow at any Mach number, between 1.4 and 4.0. (a) Corrections for viscous effects. Let the deviations given under (1) be considered as deviations from the theoretical contours Then we may write the equation of the mean contour in the following manner: (12) ym=yo(X ) +A(x) where yo is given in Tables I and II, and A (x) is a deviation of the type -9

VDEELOP1TE OF BOUNDARY LAYER DISPLACEIENT THICKNESS ALONG UPPER SURFACE FOR SEVEuRAL MACH NUMBERS 17- 7.7 ~~ ~i:'~., fi - TAi'' lii~~~~~~~~~~~~~~ O ~~~~ ~ ~ ~ ~ ~Y~~~~~lit X~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ L.~ C1l- C tttttmt~t~~H~!nI ln I~rrlr~lrltt+it-r~~rnI1II1I1 I Il~~lttrI13Hl I Jill 11 IHRfRK~-b~P r' -: —.-c'- tre~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ttl - I: I W~~~~~~~~~~~~~~~~~I.. ~~~~~~-r1 1I I I I JI ~lrrLnL I I I -_LT I I II I I IljIT~T-7~~~ LI II TFFI 144FF

DEVELOPMENT OF BOUNDARY LAYER DISPLACEI~ENT THICKNESS ALONG LOWER SURFACE FOR SEVERAL MACH HUMBEES. i t Hii ~l~fttttmftm tmmlm-+lt Itm+~ 3 I-I F lil.Owl I - ~ ~ ~ ~ ~, I~s S, DISTAN~CE FROM THROAT, INCHES7

t_., 4.d+4ii~-f~ c~i-b t 1- - -l-ttl~ t'i~-4t tiE>-r t ~~~~~VARIATION OF BOUNDARY LAYER DISPLACEMENT THICKNESS WITH TEST MACH NUMBER FOR SEVERAL AXIAL STATIONS ON LOWER BLOCK i' C,, H to ~r~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~A C.) H I E-4. PI ~ ~ ~ ~ ~ ~ ~ < Co~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~o H. --;-;-: +?ay Cc.L~L~~-C(- I -C~-I-U-~-CC -I-IC1 — tt- - - I lT A N -r-~~~~~~~~~~~~~~~~4 ~~~C;~' c-1 — C. ~~TET. C NIAE

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE (1). It follows from (12) that (13) d2ym = d2yo + d2 and dym = dyo + dd x2z (3 dxz dcx~ dx dx The boundary layer displacement thickness computations have been used to calculate d(x). In order to correct for viscous effects in a manner which makes (x) as small as possible, the following procedure has been used. The lower nozzle block is rotated about a point near the throat so that the point B is displaced normal to the original contour an amount 6*B, where 6-*B = displacement thickness at point B. Then other points on the contour are moved by flexing the surface so that each point has moved a net amount 6* from its original unrrotated position where 6* = boundary layer displacement thickness at the point under consideration. The movement of the contour by flexing is denoted by A(x). The greatest amount of flexing is required at M = 4.0 and the curve of A (x) fcrn this case is given in Table I, along with its first and second derivatives. The flexing of type (1) at Mach number ~ 1.6 is negligible, and that at any intermediate Mach number is less than AM 5 4* Consequently, the mean contour must assume the extreme shapes given by (l) (Yin:M yo(X) Ym yO(x) + A(x)M =4 and the shape at any intermediate Mach number lies somewhere between the extremes and is similar to one of the equations (14). A similar procedure has been used for the upper surface except that the rotation is such that the contour is displaced normal to its original position an amount 6* at the nozzle exit (point A). The values of nfor the upper surface are given in Table II. Fig. 9 shows the contour in its rotated M = 4 position and in its rotated and flexed M = 4 position. (b) Corrections for Local Non-Uniformities. Deviations of the type (2) are restricted to the region just downstream of the throat on the lower surface and to that just upstream of the exit on the upper surface. Small deviations from the mean, contour in these regions will give a control over the uniformity of test section flow at all Mach numbers. Analysis of the design contours with the method of characteristics at off-design Mach numbers indicates that the deviations of type (2) are quite small. If we characterize these deviations as A (x), the shape of the -10

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE contour is given by the equation (15) y = ym(x) + A (x) where ym is determined by equation (14). It follows that (16) d\ *d2y dz dA The effect of deviations of the type A on the test section flow can be found by the following analysis (Ref. 11). If we assume the nonuniformities in the test section flow to be characterized by a two dimensional perturbation potential (17) p - (xyo) where xo = rect. cart. coord. in direction of tunnel axis Yl I it It II It i f YO m tt Then the non-uniformities in velocity are (18) u i_'P in xo direction v in yo direction, ayo and the non-uniformities in flow inclination are (19) + where U = test stream velocity (See Fig:, 10). Because of (18), we may consider the stagnation pressure Po to be a constant. Then the static pressure at a given point in the flow is related to Mach number at the point by (20) Q P — 11~ 2...... _..

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE and static pressure deviations are given by (21) Up M2 P 1~~ 2 M This pressure variation is related to the inclination deviations by the equation: (22) AP = + yM2 AO p'2f__ Now if a change in wall slope dA occurs at some point Xu, Xu', Xu" or X1, XI, XV, etc., along the supersoniyn ozzle surface as shown in Fig. 10, the change in flow inclination at a point Xo along the tunnel centerline is given by (23) + dA -dx and the change in pressure at X0 by (24h) ~y +yM2N dX where N indicates the number of times that a wave from Xu, Xut, or X1, Xi etc., reflects before it reaches the point X0. To obtain an order of magnitude on the values of dA which may be necessary in order to be able to correct for non-uniform fli in the test section, we consider a deviation in Mach number of.02. The change in pressure associated with this Mach number deviation is given by (25) dp= _*02M2 1 P becomes by use of (24) which becomes by use of (24).(26) 02yM2 1 yM2N i -12

A-L AO ~ ~ ~ ~ fR~~~~~~I CO I~~~~~~~~~~~~~~~~~~~~~~ Jr l1# - ozz- - - - - - - -------— ~~~ t~~~~~~ il;~~~~~~~~-t-t~~~~~~~~~~~~~~~~~~t-:-1-1 ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~I

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE s0 that (27).02 M.2-1 7X MN(lrM2) The values of d obtained by solution of (27) are shown in Fig. 11 as a function of Mach nuter for N = 1,5 The extreme values of the function A (x) which may be necessary for the upper and lower surfaces are given in Table III, where the + signs indicate that perturbations both above and below the mean contour are desirable. VII Preliminary Experimental Results In The Subsonic And Transonic Regions Of The Nozzle. As indicated in Section IV, a series of simple experiments have been made to check out the subsonic and transonic regions of the nozzle, In order to perform the tests economically, models of the nozzle contour up to the throat were constructed from wooden blocks. The blocks were fashioned in such a manner that they could be inserted in the existing 8" x 13" Supersonic Wind Tunnel, in the M = 3.87 and M - 1.40 positions relative to one another. Instrumentation for tests in these two positions was limited to schlieren photography, china-clay-film visualization studies, and a small amount of pressure measurement with a five-prong total head probe. The measurements obtained in this experimental work are reported fully in Ref. 9, but some of the more important results are reproduced here. Investigations were conducted with two different contours for the subsonic regions. The results with the original contours showed that the sonic line was straight and perpendicular to the nozzle wall a t both M = 3.87 and M = 1.40 positions. Evidence of this is shown in Fig. 12.' Fig. 12a is a schlieren photograph of the nozzle in the M = 3.87 position, while Fig. 12b was taken with the nozzle in the M = 1L40 position. The Mach waves just downstream of the throat are slightly inclined and straight. Comparison of the experimental Mach wave distribution just downstream of the throat with that predicted theoretically in Ref. 8 (see Fig. 11 of that Ref.).for the case of a straight sonic line leads to the conclusion that a straight sonic line exists in Fig. 12 even though it is, of course, invisible. Another result with the original contour was that the viscous flow in the neighborhood of the M = 3.87 throat was adequately predicted by theory. However, a separation occured approximately two inches upstream of the lower block leading edge in the M = 1.40 position. Fig. 13a shows the extent of the separation; while Fig. 13b shows that the separated flow reattached to the lower surface approximately 3 1/2" upstream of the M = 1.40 throat. The streaks visible near the lower block leading edge in Fig. 13a are the results of application of a small amount of China-clay-film on the -13

I.H. 4 -44-i +4- J- +f- ii f+ + f fil.1 I it 4CO 0............ C\1 ------- P9 0 ",02-4 IA, --------- 0 E-40 if - - - - - - - - - - - - - - - - - -....... I if E-4 11 lilt i f i l l E-4 11 fill 104 0 11 E —4 E-4 -<4 E-4 co CO 1.4 I S i l l f i l l 1 1 i i I I f i l l I I l l I i f I I t 1 1 1 f i t I I t 1 1.4-H i+ F4- 4 4J'r- T-4-:7-T 77; T r r trLI: 4 -Hfl+11-H f-i 4;;............. ir A 717 +fl 14'1+ 14-H f,

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE sidewall. They represent essentially the stream line directions in the boundary layer near the sidewall. The separation seemed to be due to tsodimensional effects associated with the asymmetry of the channel in the M = 1.40 position, As a result of the separation, a modification of the subsonic contours was considered necessary. The modification essentially decreased the curvature of the subsonic inlet; and consequently led to an inlet approximately one test section height longer than the original inlet, Since the modification did not extend into the transonic region of the nozzle, it did not affect the sonic line. The coordinates of the modified contour are given in Tables I and II. Fig. 14 is a photograph of the modified nozzle blocks; while Fig. 15 shows the method of attachment of the blocks to the top and bottom sidewalls of the 8"1 x 13" channel. The plug was attached to the block by 6 wood-screws, and was inserted in a hole in the tunnel wall. A bolt was inserted through the channel (which rested against the outside tumilel wall) and fitted the plug tightly in place. An indication of the fact that the modification eliminated the separation is shown in Fig. 16. Fig. 16 is a photograph of the Mach 1.4 throat region after the modification. It was made with a highly sensitive setting of the schlieren apparatus in order to bring out the flow details, The boundary layer on the lower surface appears to behave in a satisfactory mannimer, although the secondary or cross-flows in the sidewall boundary layer ap-pear to be rather large near the leading edge of the lower block. These preliminary experimental results thus indicate that with the modified inlet contours: (a) a straight sonic line is obtained as predicted by theory, and (b) viscous effects up to the throat should not materially influence the flow in the supersonic portion of the nozzle.

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE PERSONNEL The following persons have been connected with the work included in this report: J* S. Miurphy..................................oFull Time H. Buningo*..................................Part Time D o V. Black................................*..Part Time C. WO Donnar Part Time RC P. OSchulz. X........ *.......... *......Part Time R PTalbot. **........... *...........Part Time R. T. Wagner...............................Part Time In addition H. P. Liepman, Director, Supersonic Wind Tunnel and Professors A. M. Kuethe and M. V. Morkovin of the Aeronautical Engineering Department have made valuable suggestions. Discussions with personnel at the Langley Field and Ames Aeronautical Laboratories, NACA, were quite helpful during the initial phases of the work.

FIG. 12a TRANSONIC REGION, M=3.87 POSITION (ORIGINAL CONTOURS) FIG. 12b TRANSONIC REGION, M=1.40 POSITION (ORIGINAL CONTOURS)

FIG. 13a SUBSONIC REGION, M 1.40 POSITION (ORIGINAL CONTOURS) FIG. 13b TRANSONIC REGION, M=1.40 POSITION (ORIGINAL CONTOURS)

0U

..........~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:.... ii;:::":iii::ii::::ili: i::i:::iii~~~~~~~~~~~~~~~~ii~~i~~ii~~i~~ii::i~~~i~~ii~~isii~~~iii iiiii::i~:::::~ i:::i:::i:;i:~i-~i ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ iiiiiiiii!~!?i iiiiyi:;!? 1iii~ 11?'!~;i~;ii!!!ii.?~.~:,:.............:;~3i'!!~iz~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iiii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iii!~Bil~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:~:::::si-ii~s-:`::i iiiii:i~!i!Siiiiii'' e: i;!B!i~llj;i ii~i;i; i ~::i~i~i ~:.i iii:iiiiii ii;:-::ii::ii~i~: Eilii9ibiii s:i~i:::-i i.,lki;:-':i:-:::i:i ii!Bi~ii;ii~iiii i-iiiB;!iiiLi-:::i~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~........'.... iii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~iiii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~il ~~~~~ ~ -ii~::;:~:;i~~i:i;:i!.;i;~;i!i:i!ii!!!iB;i!~ xm:............~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~::.:...:::~:::.:::::::::::.:::'~~ X.:................~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.:::::..,:::::i:~-::~::I:...:.-:i~rii:i:.........................:.........................~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:.:.i:.:::.i::'::: i::.:::':: ~ii::i~riri~iii::::.i:: 1ii i;111 ~i.:!i:i.i';! i.i~i?:ill: i: i: iiiii~i:~::i~i:......... i:::l'i~ii:::.ljii~ii~ii~iiF G 15 X:~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~:::::::::~:.::: i::::ii~i~:i::;::::~::::::::~: iiil:l::i:;:.i:il:::::::::::- ii:::::::::l::::::::::i~~~~~~~~~~i::i~~~l~ii::::::':''':'::::':::;'::::'::':::'':::::::iii~~~~~~~~~~~~~~ili'i'iiiiili'-iii:-liitatio:...:.::_::i:::::::i:.ii::::.~:i:~i:~:i.::i:::::::::.:::::~::i::::::'::::::::::::~::iii~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i:~:::::::'~'':'::iilil~~~~~~~~~~~~ii':i:-i~~~~~ii::ii'lii'i~~~~~~~i::::i::::::::-:::: —-::::':::~~............................j::::::::i:i:iii~:::ii: -~:::j~~~i,ll::: i i::i~~i:: i~~ii:::::::::i::ii::: ii~~~~~~i:-i::::::::.:::::::::~::::::::::::::.:::::i:::i~~~~~~~~~~~~~ii:~i::i::i~~~i~~i i':.: ~::::::;::::::'::::::::::':::i::-:i~~~~~~~~~~~i' ii:.:~.::::::~::-::::li~: i~~~~~~~~~i~i::::: i:.:::::.:::::::::i:::.:::::::::: ~..~-~~~~~~~~~x:'':'::::::::' "::':::'~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.....::::::::-:::::::-:::; ~~::::::~:i I~~~~~~~~~~~~~~~jij~~~iijil: i~~~~i':':::':'.~::::::.:::.:':.::::::.:.:::.:.~::~'i:::::: I~~~~~~~~~~~~~~~~~~~~~~~~~~~~~l.:ii~~~ii~~i::.:ii:.i~~~~~~i::':.:::::....................:' ~:::`:'::::::::~i::::i~~~ii':,::ij~~i~i::i":::::~: i::i~~i':':::r::::::: i: i::..:.....: i::.:::::::::::lilii:.liiilil~~~~~~ii: ii~il~iiiil li,,i:::i':iili:':.:...................... ii: i:::::i::::::::'''~'i::''P~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~i'i''it~~~~~~~~~iiii iii:;::,.:::....-.- i: ~.:::ii:::::::: ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~~~~~.:.......... i:: ~~~.:..::.:::ii-~i~-i-~~:~:-::::::::.:::.:~~~ ~~ ~~ ~~ ~~ ~~ ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~......::~::..:-.::::iiiiiiil:ii-iiiiFIG. 15ii

'7 7x....................................................................................................................................................... X A-:-,........................................................................................................................................................................................................................................................................................................................................................

UNIVERSITY OF MICHIGAN SUPERSONIC WIND TUNNEL ENGINEERING RESEARCH INSTITUTE REFERENCES 1. Allen, Ho Julian; "The Asymmetric Adjustable Supersonic Nozzle for Wind-Tunnel Application". NACA RM No. A8E17, July 23, 1948. 2, Syvertson, Co A., and Savin, R, C. "The Design of Variable Mach Number Asymmetric Supersonic Nozzles by Two Proceedures Employing Inclined and Curved Sonic Line&"' NACA RM No. A51A19, April 11, 1951. 3, Envard, John C., and Wyatt, D. D. "Investigation of a Variable Mach Number Supersonic Tunnel with Non-Intersecting Characteristics", NACA RM No. E8J13, November 15, 1948. 4. Burbank, P. B., and Byrne, R. W. "The Aerodynamic Design and Calibration of an Asymmetric Variable Mach Number Nozzle with a, Sliding Block for the Mach Number Range 1.27 to 2.75". NACA RM No. L50L15, arch 15, 1951* 5, Murphy, James S. "Description of Method Used to Determine Contour of Variable Mach Number Nozzle", Univ. of Mich. WTM-217, October 10, 1951. 6. Schulze, R. P., and Talbot, L. "Design of Variable Mach Number Supersonic Nozzles Employing the Method of Syvertson and Savin", Univ. of Mich. JTM-211, August, 1951. 7. Puckett, A. E. "Supersonic Nozzle Design", Transactions of the ASME, December, 1946. 8, Gortler, H. "On the Transition from Subsonic to Supersonic Velocities in Channels"o Zeit. Ang, Math. und Mech,, Dez, 1939, Translated by H. Buning at Univo of Mich. WTM-214, September, 1951. 9. Murphy, J, S. "Experiments on the Subsonic and Transonic Regions of the 4 x 4 Inch Variable Mach Number Nozzle"o 10. Buning, H, "Boundary Layer Thickness Computations". Univ. of Mich, WTM-216, September, 1951o 11. Puckett, Allen E. "Design and Operation of a 12" Supersonic Wind Tunnel", Preprint No. 160, Inst. of the Aero. Sciences, July 16, 1948. -16

TABLE I COXRDZNATES OF LOWER CONT0QU. " y,.....8 | l... dy,, |:/ dX I, r, 6,%.~..0oo 2.178 -3.75 2. 248 -3.50 2.*292 -3.25 2.328 -3.00 2*353 72.75 2.371 -2.50 2.386 -2.25 2.39~ -2.00 2.395.1.75 2.38 -1.50 2.388 -1.25 2.378 -1.00 2.361 -.7 2.338 -.50 2.312 -.25 2.280.00 2.2 -.:,O50 l e~.25 2.207 -.167.50 2.163 -.183.75 -2.115 -.199 *.00 1 2.0 64 -.214! 1.25 2.009 -228 1. 50..L950 o -.241 7145,|1.888 -.253 2.00 i,824 4.264 2.25;,1.756 -.274 2.50,.1.687 -.283 2, 75 i.o.61$ -.290 3*00.1542 -.297 3.25 1.467 -.302 3.50 1.391 -.306 3.75 i.314 -.308 4oo00 1.237.309 4.25.1..59 -.308 4.50 o il.o83 -.306 4.75 1.006 6 -.303 5.000..931. -.298 5.25.858 -.292 5.50.785 f -.285 5.75 I.75 1 -.276 6.00..64.7 -.2665 6.25. 582 -.2555 6.50..520.244 6. 75.60 -.231 7.00, i -.217 8.25.352. -.203 7.50.303 i -.88 7.75 258 I -.172 8.00.216 -.157 8.25 *1791 1135g, i _....., -17

-ib. COMRDlnATES OF LOL COOU (CoaT.) Y" | " |Y/x "x Idx/x 8.25.179 -.41,5 8.50.6 —.1260 8.75.16 -.3.105 9.00.090 -.0955 9.25.068 -.0815 9o50.050 -.0675 9.75.035.0260 -.054.00000 -.0o034 10.00.023.o258 -.0415 -.00:116 -.00292 10.25.o03.0255 -.030 -.00212 -.00257 10.50.0075.o0249 -.020 -.00300 -.00229 10.75.0035.0242 -.012 -.00374 -.oo002 11o00.0010.0233 -.006 -.00o41 -.0018 l,25.0000.0243 -.oo002 -.oo00501 -.00168 1.*50.0216.000 -.00552 -.00150 11*..o0198 -.00599 -.00136 12*.oo 0182 -.00oo637 -.00o 12.25.0167 -.00670 -.00108 12.50.0150 -.00698 -.00095 12. 75.0132 -.00721 -.00080 13.00.0114.00139 -.00068 13*25.0095 -.00753 -.00056 13.50o.0076 -.o0763 -.00042ooo 13.75.0057 -.00771 -.00031 soo00.oo38 -.00778 -.00021 14 25.0019.o00780 -.0001ooo1 14.50.0000.0000.0000 -.00782.0000 -.00002 14. 75.0004 -.0021.0012 -.00781.0097.00006 15.00.0015 -.oo4.00oo47 -.00779.0173.0003 15.25.0039 -.0059.0095 -.00774.0225.00020 15.50.0079 -.0078.0157 _ -.00769.0266.00028 15.75.0136 -.0098.0228 -.00760 ~0293.00034 16.00.0212 -.0117.0302 -.00750.0291.ooolo 16.25.0304 -.0135.0373 v.o00739.0279 000o45 16.50 o0415 -. 0154.0442 -.00725 0269.00051 16*75.0545 -,0172.051 -.00701 1.0260.00055 17.o00.0608 -.189.057 -.00oo695.0252 0060 17.25.084 1 -.0207.0637 -.00679.o244.o00064 17.50.102 -.o224.0697 -.00660.0263 00069 17.75.121, -.0240.0754 -.006140.0229.00072 18.00.J41 -.0256.0810 -.00620.0223.00077 I 18.25.163 -.0270.086% -.00599.0218.00080 18.50.186 -.0285.0920 -.o00577.0214 00083 18, 75.210 -.0299.0972 -.00554.0210.00087 19.00.236 -.0313.1023 -.00530.0208.00090 19.25.262 -.0326.107 -.00507.0206 19.50.291 -.0338.1128 -.0481.0204.00098 19.75.320 -.0350.1180 _-.00456.0202.00101 20.00.351 -.0361.1230 -0429.0201 00103 20.25.383 -.0372.1280 -.00402.0200 00108 20.50.416 -.0382.1330 -.00375,.0198.000o. 20.75 40 -091.3 j-.0034.019 j.001 20.75 |.450 1 -.0391.380 -.003l8.0197.00111

-19COORDINATES OF LOWER CONTOUR (CONT.) X (. y. A _____ - - /-,~ &A,............ 20o.75 o -.0391.38 -.z*00348.*0197.ooO 21.00.486 -.0399.1428 -,00320.0195.00113 21.25.524 -.0407.1475 -.00291.0194.00115 21.50.561 -.o414.1524 -.00261.0192.00118 21.75.600.0.419.1570 -.00231.0190. 00119 22.00.641 -.0425.1618 -.00201.0188.00120 22.25 ~.683 -.049.1664 -.00171.0185.00121 22.50.726 -.0433.1710 -.001,0.0183.00122 22.75.769 -.0436.1755 -.00310 0180.00123 23.00.814 -.0438.1800 -.00079.0176.00124 23.25.860 -.o440.1843 -.o00047.0172.00126 23.50.908 -.o1J.1887 -.0001016.00128 23.75 *956 -.041.1929.00016.0164.00129 24*00 1.o00 -.o4.1968.00047.0159.00o30 24.25 1.055 -.0438.2008.00079.0154.00330.21450. 1. 106 -.o436.2044.00110.og9.OOU3o 24.75 1.158.o0434.2080.o00141.o5.00o31 25*00 1.211 -.0429.2116.00174.01o1.00132 25.25 1.265 -.0425.2349.00205.036 o132 25.50 1.320 -.0418.a2180.00238.0133.0013325*75 1.375 -.0413.2213.00270.0130 o003,4 26.00 1.431.0405.2246,.00302.0127 OO135 26.25 1.488 -.0397.2277 i.00335.0124.0oo3 26.50 1.545 -.0388.2308 i.00367.0123.00136 26.75 -1.604 -.0378.2338.004o0.0122.00137 27.00 1.664 -.0368.2368.00431.0121.oo37 27*25 1.724 -.0357.2397 00oo466 0120.00137 27.50 1.784 -.0345.2428.o00498.0120 0017 27.75 1.846 -.0332.2456.00531.0120.00137 28.00 1.908.o0318.2485.oo565.0120.o0038 28.25 1.970 -.0305.2515.oo598.0120.0038 28.50 2,034 -.0289.2545.00630.0120.00138 28.75 2.098 -.0273.2575.00664.0120.00oo38 29.00 2.164 -.0256.2605.00oo697 o0120.00339 29.25 2.229 -.0238.2633.00730.0120.00139 29.50 2.296 - 0219.2664.00763.0120.00o39 29.75 2.364 -. 0200.2695.00798.0120.00139 30.0o 2.432 -.0179.2723.00831.0120.00139 30.25 2.500 -.0158.2752.00866.0117.00339 30.50 2.570 -.0136.2781.00899.0112. 00138 30.75 2.570 -.01336.2781.00899.0112.00138 30.75 2.640 -.0113.2810.00932.0100.00330 31.00 2. 711 -.0088.2832.00969.0083.00123 31.25 2.782 -.0064.2850.01000.0060.00115 31.50 2.854 -.0038.2861.01026.0030 *00102 31*75 2,925 -.0013.28675.01045.0005.0oo89 32.00.003.*01060.0000.00074 32.25 *.0040.01069.00060 32.50 J cm.0067.01072.00042 32.75 ~ i ".0094.00028 33.00 1.0120 I00014 33.25.0146 N I.00003 33.50.0174.00 595 "'*oI1i~I.Tr

my0X" d__ YA/'A kt A/cI xL -3.83 8.982, -3.75 8.960 -3.50 8.890 -3.25 8.818 -3.o00 8.746 -2075 8.673 2.50 8.600 -2.25 8.528..20O 8.450 -.75 8.359 -1.50 8.250 -1.25 8.122,1.00oo 7.963.75 7.757 -.5o 0.561 -.25 7.313.oo00 7.07 00.25 6.765 00.50 6.473. 00.75 6.162 1.00 5.846 1.25 55o22 1.50 5.200 1.'75 I.87o 2.00 4.545 2.25 4.220 2.50 3.913 2.75 3.600 3o00 3.3oo00 3.25 3.010 3.50 2.723 3.75 2.450 4.00,oo 2.182 4.25 1.936 4*.5o 1.694.915 4.75 1.478.818 5.00 1.286.730 5.2 ].:L.650 5.50.960.580o 5.75.83.520 6.00.?701.401 6.25.593.408 6.50.097.360 6.75.4313 -.023.318.00000 -.00600 7.00.340 -.0241.273 -.0031 -.00500 7.25.276 -.0236.235 -.0025 -.00J15 7.50.222 -.0228.200 00003 -.000348 7.75.175 -.02:9.170.00404 -.00292 8.00.136 -.0208.145 -.0045O8 -.00246 8.25 1.03 -.0197.120 -.00506 -.00209 8.950.076 -.0183.098.oo545 -.oo00175 8.75 050.078 1.00578 -00147 9.00.0375 -.0154.060 -.00608 -.00123!~ ~ ~ ~~~,oo ~02

-21COORDINATES OF UPPER CONTOUR 9II A-AZ.c2A1 9.00.0375 -.0154.060 -.00608 -.00123 9.25.0245 -.0139 -.o045 -.00632 -.00100 9.50.0150 -.0123 -.030 -.0oo0653 -.00080 9.75.00o82 -.0107 -.020 -.o00670 -.00063 loo10.00.0040 -.0089 -.012 -.00685 -.00046 10.25.0015.00o72 -.008 -.00696 -.ooo31 10.50.0005 -.0054 -.003 -.00705 -.ooo18.0075.0000 -.0037.OOO o.00712 605 1l.00 -.0018 -.0oo0716.00006 11.25.0000 -.0071900016.11.50.0017 -.00720.00025 11.75.0036 -.00717.00035 12.00o.oo0054 -.oo00714.00042 12.25.0072 -.00708.00050 12.50.0089 -.00702.00055 12.75.0107 -.00694 00061 13.00oo.0124 -.oo00685.ooo66 13.25.0140 -.00673.00071 13.5o.o0157 -.oo661.oWo75 13.75.0173 -.00648.000oO79 14.00.0189 -.00633.00082 14.25.0206 -.00617.00085 14.50.0221 -.00600.00087 14.75.0236.00583.00090 15.00.0250 -.00564.00092 15.25.0264 -.0o0543.00093 15.50.0277 -.00520.0ooo095 15.75.0290 -.00495.000955 16.00.0302 -.00oo470.00096 16.25.0313 -.00445.00096 16.50.0324 -.00420 I.00096 16. 75.0334 -.00395.00096 17.00 0343 -.00372.000955 17025.0353.00348.00095 17.50.0362 i -.00324.00094 i 17.75.0368 -.00300.00092 18.00.0000 0376 o.000.00279.00090 18.25.0383 — 00258 o00087 18.50.0383 -.00236.00085 18.75.0394 -.00218.00082 19.00.0399 1 -.00200.78 19.25.0404 i -.00182.oo00075 ~~19.~50 i.0408 -.00165.00070 19.75.0)42 -.00150.00065 20.0O.0417 -00135.00061 20.25.0420 -.00123.00056 20.50.0423 o.00110.00051.0423~~~~~~M -.OOllO 20.75.0426 -.00100.00046 21.00.0428 -.00090.00041 21.25.0430 -.00oo083 I 0.00036 ~21.50.0000.0432.0000 -.00075.0000 21.75.0000 034.0000 -.00070 -.0026,.00026 22.00 -.0001.0436 -.0012.0066 -.0064 123 -1.t me.... _-.... _ _........

COORDINAteS OF UPPER CONTOUR ~x | ~~~~YQe LV / d1Yo/, |~ d^/AX l 22.00 -.0001.0o436 -.0012 -.00066 -.0064.00023 22.25.0006.0o437 -.0031 -.00064 0093.00018 22.50 -.0017.0439 -.00o58 -.oo00063 -.0:%16.00015 22.75 -.0036.O441 -.0090 -.00062 -.0135.00012 23.00.oo0063.o442 -.0126 -.00062 -.0152.00008 23.25 -.Q099.o444 -.0165 -.00062 -.0165.00006 23.50.0145.o446 -.0208.00oo62 -.0173.00005 23.75 -.0203.0448 -.0252 -o00062 O-.0176.00003 24.00 -.0271.0~449 -.0296 -.00062 -.0177.00002 24.25 -.0351.0451 -.0340 -.00061 -.0176.00001 24.50 -.0o44.o52 -.0384.00060 -.0175.00001 24.75 -.0543.0454.042 -.00060 -.0173.oooo00001 25.00 -.0655.o456 -.0470 -.00059 -.0170.000025 25.25 -.0778.0457 -.0513 -.00058 -.0168.oooO4 25.50 -.0911.0458.o0554 -.00057 -.0166.0000ooo5 25.75 -.105.0459 -.0596 1 -.00055 -.0165.00006 26.00 -.121.o460 -.O637 -.00054 -.0163.0007 26.25 -.137.0461 -.0678 -.00051 -.0163.ooo10 26.50 -.055,0462 -.0718 -.00049 -.0161.00011 26.75 -.173.o64 -.0758 -.ooo47 -.0160.00013 27.00 -.192.0465 -.0797 -.0oo44 -.0160.00015 27.25 -. 213.o466 -.0837 -.ooo10o -.0160.00017 27.50 -.235.0467 -.0877 -,ooo35 -.0161.00019 27.75 -.257.0467 -.0917 -.oo00031 -.0162.00021 28.00 -,280.0468 -.0956 -.00030 -.0163.00023 28.25 -.304 -.o0468 -.0996 -.000o 0 -.01614.00025 28.50 -.330 -.01469 -.1037 -.00015 -.0164.00027 28.75 -.356 -.0469 -.1076 -.00008 -.0165.ooo29 29.00 -.3814,.0169 -.1117 -.00001 -.0165.00031 29.25 -.4:12 -.01470 -.1158.0oooo5 -.016.00033 29.50 -.442 -.0470 -.198.ooo0 -.o0165.oo35 29.75 -.472 -.0470 1239 00021 -.0164 00036 30.00 -.503 -.o469 -.1280,00030 -.0163.00037 3025 5 -.36 =.0146.1321.00039 -.0162.00038 30.50 -.569 -.0468 -.1361.00047 -.060.00040 30.75 -.604 -,0467 -.1402.00056..0158.000041 31.o0 -.640..0o466.1440.ooo0065.05 00042 31.25 j.677.046o4.1480.900075.oo0 oc 31.50 j -.714 -.0462 -.117.ooo8 -.9.ooo00044 31.75. -.752.01460 -.1553.o 0095 1.o116,000145 32.00 1.792 M.0147 -.189.0010% -0OL. 000146 32.25 -.832 -.045 -.1623,0oo011.ouo.ooo47 32.50.o872 -.1452 -.1659.o12 -.o137.0148 32.75 -.914 -.0449 -.1692.0013 -.o0135.00050 33.00 -.957 "-.0446 -.1726.00o146 -.0132.o00051 332!-101 -.4..1758.0157 o.0129 -,0052 33.50 j-1.045 -.048 -.1790.00168 -.0126.00053 33.7, 1 -1.090 -.0434 -.1821.00180 -.0124.00054 34,o00 1.136 -.0430. -.185.0190 -.0121.000545 34*. 25 -1.183 -o.01425 01881.00202 -,o 8.o00055 34.50 -1.230 -.0420 -.1911.00214,-.0116.0055 34.75 -1.279.1 0415 -.1939.00225.-.01114, 00055 35.00 -1.327 -.0o410 -.1967.00237 -.0111.00056 L;ss=L _........ w..........................................................l a.......t

-23OO000DNATES CO CPPER ONTOlOR _ _ _ Yo". All ('/,X 35.00 -1.327.0410 -.1967.00237 -.01:L.00056 35.25 -1.377 -o0404 -.1993.00250 -.0109.00056 35.50 -1i427 -.0398 -.2021.00261 -.o0107.00056 35.75 -1.478 -.0392 -. 2050.00272 -.0106,00056 36.00 -1.529 - -.0305 -.2076.00285 -.0105.00056 36. 25 -1.581 -.0378 -.2102.00298 -.0104.00056 36.50 -1.634 -.0371 -.2128.00310 -.0104.00056 36.75 -1. 688 -.0363 -. 2156.00232 -.0104.0005537.00 -1.742 -.0356 -.2182.00336 -.0103.00055 37.25 -1.797 -.0347 -.2208.00352 -.0103.00055 37.50 -1.853 -.0339 -.2234.00362 -.0103.00055. 37.75 -1.909 -.0330 -.2261.00375 -1.0102.OOD55 38*00 -1.966 -.0321 -.2286.00388 -.0102.00055 38*25 -2.023 -.0311 -.2312.00400 -.0102.00055 38.50 -2.082 -.0301 -.2337.90415 -.0101.000545 38.75 -2.o140 -.0291 -.2362.00427 -.0101.00054 39.00 -2.199 -.0280 -.2387.00o411 -.0100.00053 39.25 -2.259 -.0268 -.2413.00455 -.0100.000525 39.50 -2.320 -.0258 -.2438.00470 -.01ou.00052 39.75 -2.381 -.0246 -.2463.0oo0483 -.100.00051 40.00 -2.443 -.0235 -.2489.00oo497 -.01o).000505 40.25 -2.506 -.0222 -.2513 -.0051 -.o0100.00050 40.50 -2.569 -.0210 -.2540.00525 -.0100.00049 40. 75 -2.633 -.0197 -.2565.oo00537 -.100.00047 41.00 -2.697 -.0183 -.2589.00550 -.00oo.00045 41.25 -2.762 -.0168 -. 26i4.00563 -.0100.00042 $ 41.50 -2.820 -0155 -.2640.00574 -.o, 00.0004 41.75 -2.894 -.0142 6 -.2667.00582 -.o 0100 00035 42.00 -2.961 -.0127 - _.2692.00590 -.0100.00031 42.25 -3.029 -.0113 -.2718.00595 -.0100ooo.00026 42.50 -3.097 -.oo98 -.2742.o598 -.0100.00021 42.75 -3.166 -.0oo84 -.2768.00600 -.000ooo 00017 - 43.oo -3.235 -.0069 -.2790.00600o -.0096.00013 43.25 -3.306 -.0oo54 -. 2816.00600 -.0088.00009 43050 -3.376 -.0040 -.2837.00600 -.0075.00005 4375 -3.447.oo0024 -.2853.00600 -.oo0054.002 44.00 1 -3.519 -.o009 -.2364.00600 -.0029.o0000 _44,125 -30.554 -.0000 -.28675.00600.0 0000.00

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