The Performance of a Supersonic Airplane Citation Weldon, Thomas Franklin (1947) The Performance of a Supersonic Airplane. Engineer's thesis, California Institute of Technology. doi:10.7907/T271-MK09. https://resolver.caltech.edu/CaltechETD:etd-09142005-112442 Abstract This thesis is primarily a survey of the methods of calculating drag and performance of an airplane capable of supersonic flight. As a basis of presentation of the findings of this survey the drag and performance calculations are worked out for a single place airplane which has a rocket power plant, sweptback wings and other low drag features. These calculations reveal that this airplane should be able to attain a Mach number of 2 at an altitude of 42,000 feet. Item Type: Thesis (Engineer's thesis) Subject Keywords: Electrical Engineering Degree Grantor: California Institute of Technology Division: Engineering and Applied Science Major Option: Electrical Engineering Aeronautics Thesis Availability: Public (worldwide access) Research Advisor(s): Puckett, Allen E. Group: GALCIT Thesis Committee: Unknown, Unknown Defense Date: 1947 Record Number: CaltechETD:etd-09142005-112442 Persistent URL: https://resolver.caltech.edu/CaltechETD:etd-09142005-112442 DOI: 10.7907/T271-MK09 Default Usage Policy: No commercial reproduction, distribution, display or performance rights in this work are provided. ID Code: 3531 Collection: CaltechTHESIS Deposited By: Imported from ETD-db Deposited On: 14 Sep 2005 Last Modified: 16 Mar 2023 22:50 Thesis Files Preview PDF (Weldon_tf_1947.pdf) - Final Version See Usage Policy. 2MB Repository Staff Only: item control page

THE PERFORMANCE OF A SUPERSONIC ATRPLANE Thesis by Thomas F, Weldon llment of the Requirements for the Professional ge in Aeronautical Engineering Califernia Institute of Technology Pasadena, California =1LO47 = ACKNOWLEDGEMENT The author is deeply indebted to Mr. A. E. Puckett of GALCIT for his invaluable assistance, cooperation 4 and advice in the preparation of this paper. T. F. Weldon California Institute of Technology Pasadena, California May 23, 1947 fie TABLE OF CONTENTS: PAR? TITLE PAGE Acknowledgement i Table of Contents Li I Summary of Thesis 1 It Deseription of Airplane 2 Tir Drag Caleulations 6 IV Performance Calculations 38 Vv Table of Symbols 47 VI References 49 PART I SUMMARY OF THESIS: This thesis is primarily a survey of the methods of calculating drag and performance of an airplane capable of supersonic flight. As a basis of presentation of the findings of this survey the drag and performance calcule.tions are worked out for a single place aire plane which has a rocket power plant, sweptback wings and other low drag features, These calculations reveal that this airplane should be able to attain a Mach number of 2 at an altitude of 42,000 feet. D3 PART Ii DESCRIPTION OF THE ATRPLANE The only claim made for the airplane about to be described is that it is feasable. It was designed on the basis of minimum drag with due consideration of stability and control and overell practi-= cability. It is beyond the purpose of this thesis to delve deeply into the design details. Rather it is assumed that such an airplane is to be built and an attempt is made to compute its probable per formance. The fuselage is just large enough to contain a pilot and a reasonable quantity of fuel. Jt has a slenderness ratio of 11.5 and is conical at both ends, The optimum nose cone of semi-vertex angle of about 4900' was not used as the saving in drag was slight over the more pratical one of 7930', The optimum boattail cone of 7°15' was, however, installed, The wings are of NACA 0012 profile perpendicular to the leading edge with 60° sweepback at the leading edge and 45° sweepback at the trailing edge, With this amount of sweep the silerons can easly be enployed as elevons and thus eliminate the necessity of a horizontal stabalizer. The vertical fin has a NACA 0006 profile and a 60° sweepback at the leading edge. The lending gear consists of a main double wheel located between the two fuselage fuel tanks, a nose wheel forward of the pilot and two small “outrigger” wheels, all retractable. The power plant is a 7,000 pound regeneratively cooled acid= aniline rocket motor. A brief weight statement is as follows: Fuselage hull 900 pounds Wings 500 Fin 50 Ianding gears , 200 Fuel tanks 150 Rocket motor 200 Pumping unit complete 200 Plumbing 20 Cockpit accessories and instrumentation 50 Breathing oxygen equipment 30 Pilot 200 Ne fuel total weight 2500 pounds Fuel weight 2900 Starting weight 5400 pounds Reference (1) was used as a guide in the above weight estimatione Some essential data required for drag calculations is as follows: Wind span Wing plan area external of fuselage Average chord of wind parallel to line of flight Maximum wing loading Minimum wing loading Pin plan area Average chord of fin parallel to line of flight Fuselage frontal area Rocket nozzle exit diameter Fuselage wetted area Total wetted area 16 86 6501 29e1 20 7207 243 460 feet square feet pounds pounds feet square foot square square feet per square foot per square foot feet feet feet ‘Lane z Airp The Fisure 1, 9 - a ee en SS ee Spe TO auTTruy |! dumg | Se iS eS Pt 3 . ot me FEE fb oe ae tee ee ee 2. al en acer | ate “ a a a ( CS _ wan * = — eee : \ ~ l _- 009 PART TIT DRAG CALCULATIONS Between M= 0 and M= 2 there are three regimes of speed in which the methods for calculating drag differ. They ares le Subsonic M= 0.0 to M¥ 0.8 2. Transonic M=0O,.8 to M=1.2 3. Supersonic M= 1.2 to M= 2.0 In the subsonic regime the equivalent flat plate area method is used for computing drag. This method is in agreement with both theory and experiment. At trans and supersonic speeds the pressure drag and the skin friction drag of each component of the airplane must be com= puted seperately. To obtain the various pressure drag forces we must lean heavily on experimental data since the theoretical methods of computing this kind of drag are still in the early stages of develop» ment. The technique necessarily employed to obtain this: drag is, therefore, the "brute force" method of searching the literature for wind tunnel tests on configurations similar to those on the airplane. Consequently, to obtain the complete drag picture the following drag calculations are necessary: A, Subsonic total drag. Be Transonic fuselage pressure drag. C. Transonic fuselage skin friction drag. D. Transonic wing pressure drage Be Transonic wing skin friction drag. FR. Transonic Pin pressure AYage G Transonic fin skin friction drag. H. Supersonic fuselage nose pressure drage I. Supersonic fuselage boattail pressure drage J. Supersonic fuselage skin friction drage Ke Supersonic wing pressure drage Le Supersonic wing skin friction drag. M. Supersonic fin pressure drage N. Supersonic fin skin friction drag. fhe drag calculations will be confined to the speed range M= Oez to M= 2.0 and to the altitude range sea, level to 60,000 feet. The calculations are first made assuming an “ambient jet", ieee, it is assumed that at all times the pressure at the blunt part of the boattail is the same as the local atmospheric pressure. The jet-on and the jet-off drag are then computed from the ambient jet drage There are two reasons for doing this: 1. The airplane is then com- pletely isolatedf rom its power plant during most of the calculations and alot of possible confusion is thereby avoided. 2. The nature of the drag data available is such that the jet-on drag is more easily obtained by this method of approach. The drag due to Lift is computed for the no fuel condition as this drag component has more significance during the glide and landing pore tion of the flight. The drag due to lift is so small at trans and supersonic speeds that it is disregarded. Throughout the calculations certain working data are continually required, They are: 1. The equivalent flat plate area of the airplane. (Subsonic) 2e A table of feet persecond vs Mach number and altitude. 3. A graph of unit length Reynolds number vs Mach number at various altitudes, 4, <A graph of skin friction coefficient vs Reynolds number. a A discussion of these items follows immediately in numerical order. The equivalent flat plate area of the airplane is estimated from ari "Aerodynamic Cleanliness" graph as shown in Figure 2. A comparison is made with a hypothetical P-51 having the same wetted area as the XXX. Such a P51 would have an equivalent flat plate area of 1.7 square feet. Since the XXX is even more "stream= lined" than the P#51 its equivalent flat plate area will probably be a little less than this. However, in the interests of conservativisn @ we will use this figure. The speed of sound varies with altitude, Consequently the speed corresponding to a certain Mach number varies with altitude. From the table in Figure 3, one can obtain the speed fora given Mach number and altitude. These speeds were computed? rom the NACA stand= ard atmosphere The Reynolds number of the various parts of the airplane is frequently considered when calculating skin friction. A plot of unit length Reynolds number vs Mach number and altitude is found in Pigure 4. To find the Reynolds number of a given body at a certain Mach number and altitude just multiply the length of the body in feet by the unit length Reynolds number fround on this graph. There are a number of different empirical formulae and graphs in the Literature expressing the relation between the smooth flat plate skin friction coefficient and the free stream Reynolds number in the s subsonic regime. They are all in fairly close agreement. According a > & 2 2 £. to Gharters, (3), there is also experimental evidence to show that these same relations hold true in the trans and supersonic regims. He recommends the graph which is reproduced here in Figure 5 and can also be found in reference (4). fowl > ad by the quotient: ‘erodynamic Z irs valent flat plate area and w= total wetted area, oP a sed © £33) er Figure 3. Altitude Sea Level 10,000' 20,000! 30,000" 40,000' 50,000! 60,000" Nach No. 0.2 223 FL/Sec 215 207 199 19h, 19, 19h, 0.3 335 323i As 9B 291 291. 291 Oh hb 431 hus 398 3¢8 388 388 0.5 558 538 51897 1,85 485 Les 0.6 669 646 620 596 562 582 582 0.7 Tl 75k OH_C“‘éi TO OO 0.8 | 892 | 861 827 195. TTT TTT 1D 0.9 00h (8 933 895 87h 87, Th 10 6 ‘077. 103799 97 ML OTL Lr eer 118, «140 1083 106s isd (tst*«i EB 2 339292 125k 1193 1165 1651165 lak 1562 sot UL 392s 89 16-1785 1723-1657 5901553 15531583 1.8 2008 1938 «86, 2.0 2231 2153 2072 ~—«-1998 1941 19h1 19h a fond 5 7 ~ 28 Figure 5, sequny spTousey 201 ot ot oot ea T00°0 : ~~ / i Z00°O a LS NI _ | #700°0 \ \ \ \ 900°O \ \g00"0 QUeTOTTJeeO uotyotdy uTpNS 14 DRAG CALCULATIONS A. SUBSONIC TOTAL DRAG Total drag = D = varasite drag + drag due to lift. (5) p= PFVA + GSP? y* ol aTRE p= PFV2 » _aw* ol PVT e J p=2.0rn18 oo V2 ZPKAI0% o- ¥% where ££ = 1.7 square feet W = 2500 pounds e = 0.7 (a conservative estimate ) b= 16 feet /? = 0.002378 slugs per cubic fect x 77 Using the formula just derived the following table of parasite . * drag forces is made: Altitude Sea Level 10,000’ 20,000' 30,000' 40,000’ 50,000' 60,000' Mech Nos O62 100 69 46 29 0.3 226 150 106 63-4 Oo4 401 282 184 114.0—~Otts«*TA 46 0.5 625 436 306 178115 73 45 0.6 904 620 412 256 168 107 65 O67 1225 845 565 364 226 142 87 0.8 1600 1095 730 475 296 184 114 * All tabulated drag froces are in pounds. Using the formula derived on the previous page the following table of induced drag forces is made: Altitude Sea Level 10,000' 20,000' 30,000' 40,000' 50,000' 60,000' Mach No. 0.2 184 277 4.43 632 0.5 84 120 216 281 450 O64 47 68 Lei 158 253 408 0.5 30 43 77 Lol 162 26h. 420 008 al 30 54 74 Liz L80 292 Oe? 15 22 33 be 84 134 2l6 0.8 9 14 20 31 50 81 130 DRAG CALCULATIONS Be TRANSONIC FUSELAGE PRESSURE DRAG The data used here is taken from reference (6) which concerns a body of revolution pointed at both ends and with a slenderness ratio of 6. A transcript of this data is as follows: Mach Noe 0.9 1.0 lel C 0.050 0.250 0.285 Dey To obtain the pressure drag coefficients we must subtract from the above coefficients the respective skin friction coefficients accord= ind to the equation: C = ¢ - CQ x Wetted area ARON ARE A ARE MERE wp TT frontal area A typical calculation proceeds in the following manner. Let us find the drag at sea level at M= 1. The unit length Reynolds number at sea level at M= 1l.is obtained from Figure 4. It is 7.0 x 10°, The length of the model was five feet. Therefore the Reynolds number of the model is 3.5 x 10". Entering the graph of Reynolds number vs Ce Model wetted area Model frontal area we find Ch of model = 0.0025. Now = 16. Therefore the Cy of model = 0.250 = 0.0025 x 16 = 0.210. We use this Dip same Cy for the fuselage of the XXX. ae Therefore D = 0.210 x $ x 0.002378 x (1116 Pe x 7.07 = 2190 pounds. By similar computation we obtain the following table of fuselage pressure drag forces: Altitude Sea Level 10,000' 20,000' 30,000' 40,000' 50,000' 60,000° Mach No. 0.9 84 58 33 17 9 5 3 1.0 2190 1510 1005s 6 42 391 248 149 lel 3110 2130 «= 1415 891 548 3.45 213 DRAG CALCULATION Ce TRANSONIC FUSELAGE SKIN: FRICTION DRAG This drag component is obtained directly by use of the information contained in Figures 3, 4, and 5. As an example let us find the drag at M= 1.1 at 20,000 feet. The unit length Reynolds number under these circumstances is 4.5 x 108, The length of the fuselage is 34.5 feet. Consequently the fuselage Reynolds number is 1.5 x 108 and the corres= ponding o is 0.0020. Therefore D = 0.0020 xs x 0.001267 x (1140)” x 243 = 401 pounds. dé In this way we obtain the following table of fuselage skin friction drag forces: t Altitude Sea Level 10,000' 20,000' 30,000’ 40,000' 50,000 60,000" Mach Noe 029 577 419 288 198 124 83 56 1.0 692 503 345 239 151 100 67 1.1 827 601 401 280 179 117 79 DRAG CALCULATIONS D. TRANSONIC WING PRESSURE DRAG The data used here is gathered from references (7), and (8)e From the former are transcribed the following values of Cy for a wing of 52° sweepback at the leading edge and a similar plan form to the wing of the XXX. Mach Nos 0.9 1.2 Cy 0202 0. 05 To get the C, of the model we use the formula : C, = C, = 20, « Dy Ds D F The applicable Reynolds number of the model wing were: 500,000 at M= 0.9 — 850,000 at M= 1.2. This results in the following skin friction coefficients: At M = 0.9 Ch = 0.0018 At M= 1.2 Ce = 0.0022 Whenee we arrive at the following model pressure drag coefficients: At M*= 0.9 At M* 1.2 No correction factor is made for the difference in sweepback between the model and the XXX, (only 8°), but a correction factor is intro= duced to account for the difference in thickness ratio. ‘The airfoil described in reference (7) has a 12% thickness perpendicular to its maximum thickness line which has a sweep of 45°, The airfoil on the XXX has a 12% thickness perpendicular to the leading edge which has ‘ 0 V 7 a sweep of 60°, Therefore the ratio of the thickness is fosine 60 _ Z, cosine 45° = 2 The pressure drag coefficient at these velocities varies as the square of the thickness ratio. Consequently we have as the applicable pressure drag coefficients of the wing of the XXX+s At M = 0.9 C = 06.0082 At M = 1.2 C. = 0,0228 These two figures are incomplete for our purposes, however, in thet they fail to shor in what mamer Cy goes from 0.0082 at M= 0.9 to B 0,0228 at M= 1.2. From reference (8) we find that for a wing of 45° sweepback, and a fortiori for a wing of larger sweep, the Cy curve is almost linear in the transonic region. Thus by linearly filling in the missing points we have: Mach No. 0.9 1.0 lel Le2 Cy 0.0082 0.0131 0.0180 0.0228 p Using these drag coefficients we construct the following table of wing pressure drag forces: Altitude Sea Level 10,000 20,000! 30,000! 40,000% 50,000% 60,000? 0.9 907 581 388 252 157 92 60 1.0 1800 1143 765 440 310 192 119 1.1 e775 1910 1274 807 512 518 197 DRAG CALCULATIONS E, PRANSONIC WING SKIN FRICTION DRAG This drag component is derived in the same manner as in Drag Calevlations C. We use as the characteristic length the average chord of the winge The tabulated results are as follows: Altitude Sea Level 10,000f 20,000! 30,000 40,000! 50,000" 60,000! Mach Noe 0.9 477 354 242 159 116 88 35 1.0 590 437 298 196 125 82 44: lel 707 507 352 224 148 99 53 DRAG CALCULATIONS F. TRANSONIC FIN PRESSURE DRAG The same data is used here as was used to obtain the transonic wing pressure drag. Since the thickness ratio of the fin is one half that of the wing the correction factor 0,25 will have to be introduced. This results in the following: “Mach Nos 0.9 1.0 lel 1.2 Cy 0.00205 0.00327 0. 00450 0200570 P Using these drag coefficients we construct the following table of fin pressure drag forces: Altitude Sea Level 10,000 20,000! 30,000! 40,000! 50,000* 60,000! Mach Noe O29 49 54 22 14 9 5 3 1.0 96 65 44 26 17 10 6 lel 161 Lil 73 46 30 19 il DRAG CALCULATIONS G. TRANSONIC FIN SKIN FRICTION DRAG This drag component is computed in exactly the same way as in Drag Calculations E. The tabulated results are: Altitude Sea Level 10,000! 20,000! 30,000! 40,000 50,000 60,000! Mach Noe 0.9 110 82 56 37 a7 15 8 1.0 137 102 70 46 30 Lg LO lel 165 118 82 52 35 23 le DRAG CALCULATIONS He SUPERSONIC NOSE PRESSURE DRAG The preblem of pressure drag due to supersonic flow past a cone at zero angle of attack is the subject of references (9), (10), (11), and (12). Reference (13) discusses the merits of each of the above. The method that results in the highest drag for the conditions of our problem is the simple application of Karman's approximate formula, c =26* a . a 3a) THp where @ = semi-vertex angle, and @ = / Wf = 2d For a sample calculation let us find the nose pressure drag at M= 1.8 at an altitude of 30,000t, c, = 2(0.1309)2-%, EE = 0.0806 Pimp Q1309 Kf. 8)°—/ : ont where 0.1309 radians = 7930'. D = 0.0806 x = x 0.000889 x (1789)* x 7.07 = 850 pounds By Similar computation the following table is obtained: Altitude Sea Level 10,000 20,000! 30,000! 40,000' 56,000 60,0008 Mach No. L.e2 | 1624 1135 768 487 299 188 L117 1.4 1955 1345 900 580 356 225 140 1.6 2520 1595 1065 684 432 268 166 1.8 2730 1880 1355 800 4,97 314 195 229 3150 2170 1450 936 584 362 225 DRAG CALCULATIONS. I. SUPERSONIC BOATTAIL PRESSURE DRAG Reference (3) is used as a guide for finding this drag component. The recommended technique is to first compute the "square base" drag and. then subtract the reduction, is drag due to boattailing. The square base drag is obtained from a graph of the ratio of square base pressure to atmospheric pressure vs Mach number. The reduction in drag due to boattailing is also obtained from a graph, namely Ok vs Mach number. Both these graphs were constructed from empirical datae A transeript of the pertinant information is as follows: Sq. Base Press. 177% 72% 67% 62% 57% AK 0.0570 0.0490 0.0425 0.0375 0.0340 The aguare base area of the fuselage is 7.07 square feet, but with the ambient jet assumption the effective square base area is 6.285 square fete For a sample calculation we will find the boattail pressure drag at M= 1.2 at sea level. The square base drag under these conditions is D= (L = 0.77) x 6.285 x 2116 = 3050 pounds. The reduction in drag by use of boattailing under these conditions ao % is D = 0,057 xo x 6.285 x4 x 0.002378 x (1339)” = 1940 pounds. Therefore, the boattail drag = 3050 = 1940 = 1110 poundse By similar computation the following table is obtained: Altitude Sea Level 10,000! 20,000! 30,000" 40,000! 50,000" 60,000! Mach No. 162 1110 785 539 330 204 129 80 1.4 1490 1006 710 424 267 166 104 1.6 “1890 1270 892 531 555 208 133 1.8 2220 1530 1070 644 399 250 155 260 2550 1760 1290 756 258 284 176 DRAG CALCULATION J, SUPERSONIC FUSELAGE SKIN FRICTION DRAG The same technique is used here as in the previous skin friction computations. Altitude Sea Level 10,000* 20,000 30,000! 40,000° 50,000t 60,000! Mach No. 1.2 938 658 460 307 209 137° 93 1.4 1337 897 615 418 287 182 125 1.6 _ 1700 1145 780 530 359 2e9 158 1.8 2150 1440 990 673 445 294: Los 20 2580 1775 1215 839 533 363 246 DRAG CALCULATION Ke SUPERSONIC WING PRESSURE DRAG Due to the lack of applicable data in the literature, this drag component must be estimated by an extrapoletion from the transonic m of pressure drag. The basis of this extrapolation is the fact that the pressure drag coefficient of a highly swept wing increases moderately in the region M= 0.9 to M= 1.2 and then remains fairly constant until the Mach wave crosses the maximum thickness line or the leading edge, whichever occurs first, At this point the drag wefficient increases about two and a half times in value. (Reference 14), Consequently the pressure drag coefficient will remain at about 0.0228 from M= 1.2 to M= 1.8, and then at M= 2.0 it will be about 0.0540, With this infor= mation we can construct the table of supersonic wind pressure drag forces. Altitude Sea Level 10,000% 20,000! 30,000! 40,000' 50,000" 60,000! Mach Noe Le2 4160 2955 2016 1251 762 464 280 Led 5625 3870 2580 1613 1020 645 400 1.6 7110 — 4882 3240 2100 1265 815 506 1.8 9260 §300 4231 2709 1705 L060 655 200 27400 18850 12650 8220 5090 5160 1960 DRAG CALCULATIONS L. SUPERSONIC WING SKIN FRICTION DRAG Altitude Sea Level 10,000" 20,000 30,000 40,000' 50,000* 60,000! Mach Noe 1.2 804. 605 426 262 178 118 63 Let L115 788 526 355 240 151 93 1.6 1435 1003 670 462 314 202 128 1.8 1815 1245 867 584 381 289 157 200 2195 1540 1070 730 4.76 304 L96 DRAG CALCULATIONS M. SUPERSONIC FIN PRESSURE DRAG The same data is used here as was used to obtain the supersonic wing pressure drage To account for the difference in thickness ratio the applicable pressure drag coefficients are multiplied by 0.25. Altitude Sea Level 10,000 20,000! 30,000 40,000 50,000* 60,000* Mach Noe le2 242 172 107 73 44. 27 16 1.4 327 225 150 93 59 58 26 1.6 414 285 189 122 74. 47 29 1.8 535 387 246 158 99 62 38 220 1590 1095 735 276 296 183 114 DRAG CALCUIATIONS: N. SUPERSONIC FIN SKIN FRICTION DRAG © Altitude Sea Level 10,000! 20,000! 30,000' 40,000 50,000' 60,000! Mach Noe 1.2 183 161 98 61 41 28 15 1.4 258 182 122 82 55 35 ee 1.6 354 253 156 107 73 47 30 1.8 423 290 202 136 89 56 37 260 510 358 249 L170 110 71 46 NE DRAG SUMMATION The following table gives the total drag on the airplane for straight and level flight at the various altitudes and Mach numbers, The drag due to lift is neglected after M= 0.8, and the “ambient jet" assumption is maintained throughout. Altitude Sea Level 10,000 20,000 30,000* 40,000 50,000' 60,000! Mach Noe 0.8 284 346 489 0.3 310 270 322 544 0.4 448 350 305 272 327 0.5 655 479 383 279 277 354 0.6 925 650 466 330 280 287 357 Oo? 1240 867 598 416 310 276 307 0e8 1612 Lilz 756 514 359 286 279 0.9 2204 1528 1029 777 | 442 269 165 1.0 5505 3760 2525 1589 924 646 395 lel 7745 5292 3597 2300 1452 921 567 1.2 9061 6451 4416 2771 1737 1091 664 1.4 12107 8207 5603 3565 2284 1440 907 1.6 15203 10413 6982 4536 2840 1816 L150 1.8 19133 18152 8961 5704 8715 2275 1436 260 39975 27548 18661 12127 7540 4727 2963 Drag Coefficient Figure 6. This figure presents in graphical form a plot of the total ambient jet fusalage drag coefficient and the total wing drag coefficient against Mach number at sea level, ‘The characteristic area in both cases in the wing plan area, Mach Number 0.06 0.05 =f L / 0,04 | _ j { 0.03 |. es — _. Ww 0.02 oo Easalage | ee 0,01 |---- ! 0.00 : a : 0.8 1.0 Le2 14 1.6 1.8 2.0 igure 7, ‘This is a graph of the ambient jet drag ceofficient of the total airplane vs Hach number. 0.07 0.06 0.05 060k Drag Coefficient 0.01 0.00 0.03 - 0.8 1.0 “ach Number 1.6 ou JET = OFF DRAG Jet-eff drag is equal to the "ambient jet" drag plus a certain amount of suction drag at the square base part of the boattail. To find this suetion drag we have merely to know the base pressure as & percentage of ambient pressure for various speeds. This information is partly given in Drag Calculations I, It is found that in the region between M= 1.2 and M= 2,0 the base pressure varies linearly, and that the point M= 0, 8 = 1 is almost on an extention of this line. By estimating the curve between M= 0 and M¥ 1.2 we arrive at the comne= plete required datas VMaeh Noe Mach Noe O20 1.000 0.90 — 0,845 One Oe 980 1.0 Oe 820 0.4 06 970 lel Oe 795 Oe 4 0, 950 1.2 Oe? 70 Oe5 Oe 930 let 06720 O68 0,910 1G 06670 Oe7 0.890 128 0.620 0.8 0.870 2209 0.570 using this information we caleulate the following table of differences between ambient jet drag and jet-off drag. THE DIFFERENCE BETWEEN AMBIENT JET DRAG AND JET OFF DRAG wT Altitude ea Level 10,000 20,000' 30,000' 40,000" 50,000" 60,000! Mach Noe O28 42 29 19 12 8 5 3 063 63 44 29 19 12 7 4 0.4 106 73 49 $l 20 L2 7 0.5 148 102 68 44 ev L? li 0.6 190 131 87 56 35 22 | 14 Oe? 252 159 106 69 43 Pate 17 0.8 275 189 126 82 51 52 20 009 Set 225 155 97 61 38 23 1.9 380 261 174. 113 70 44 a? lei 433 298 199 128 80 50 SL 1.2 436 334 223 144 90 56 35 Le 591 400 271 175 LO9 68 42 1.6 696 478 319 206 L129 80 5O 1.8 802 551 368 258 L148 92 57 200 908 624 416 269 L68 104 65 ov JET = OFF DRAG FOR STRAIGHT AND LEVEL PLIGHT Altitude Sea Level 10,000" 20,000! 30,000 40,000! 50,000! 60,090! Vach No. O62 326 375 508 873 0.3 375 314 S51 363 504: 0.4 554 423 354 303 547 466 0.5 808 BSL 451 - 323 3504 551 276 O25 LLi5 781 553 386 OLS 309 B74 O67 1472 1026 704 485 353 50S Bat 0.68 1837 L501 382 596 410 318 299 O06? 2531 L753 Lis4 874 503 307 188 1.0 5885 4021 2699 L702 294 S90 422 lel SL78 5520 S796 2428 L532 O71 598 isd 8547 6985 4639 2915 L827 L147 699 1.4 12698 38507 5874 8740 2593 L508 949 166 15899 10891 700L 4742 2969 1896 1200 1.8 19935 13703 9529 5942 3863 2387 1493 209 40883 28172 17077 12396 7728 4831 5028 ive {~ JET = ON DRAG The difference between ambient Jet drag and jeteon drag is caused by an added velocity at the rear and induced by the very rapid velocity of the exhaust. The only available data on this type of drag is the result of tests on a model V-2. (Reference 15). At M= 0.2 the jet-on parasite drag was twice the ambient jet parasite drag. This ratio de- ereased linearly to one at M= 0.9. In other words, at M= 0.9 the jet» on drag is the same as the ambient jet drag. For Mach numbers greater than 0.9 the ambient jet drag is greater than the jet-on drag and the difference increases linearly until at M= 2.0 the jeteon drag is 83% of the ambient jet drag. In tabular form we have: At M= 0.2 the jet-on parasite drag = 14/7 the ambient jet parasite drape At M= 0.3 1 tt ott to nm = 13/7 it n " " tt Ab M=0,4 " t" tt t to o= 4p 7 tt tt tt tt ft At M@=0.5 " tt otf tt " = 41 /? tt tt " t " At M=0.6 " ott " " = 40 7 " tt tt tt tt At M=0O.7 " es oof tt ! ox 9/7 " tt tt it 7 At M=0,8 ™ wm oft tt to ox 8/7 tt tt t w " At M= 0.9 tt tt ft t tom 7 17 tt ft tt Yt wt AG M=1.0 " " " " = 98.9% the ambient jet parasite drag. At M= 1,1 % tof " fos 97.8% " tr n tt tt At M=2.2 7 ft oon " Hos 96.7% tt tt tt tt tt Ab M= 1,4 i it to= 94, 5% tt tt tt tf t Ab M=1.5 ” "off " too ge, 3%, tt tt tt tt u AG M=1.8 " " " "= 91.1% " " " " " At M=2,0 " t ff ft to — 88.0% tt tt t " tt Using this information we construct the following table of jet-on drag forces for straight and level flight. JET = ON DRAG FOR STRAIGHT AND LEVEL FLIGHT Altitude Sea Level 10,000' 20,000! 30,000 40,000' 50,000' 60,0001 Mach No. 02 304 415-835 0.3 504 398 413 398 0.4 782 540 436 382 383 0.5 1000 738 557 381 342 381 0.6 1321 905 634 386 340 330 384 0.7 1585 1122 758 519 386 317 328 0.8 1834 1264 855 561 388 291 260 0.9 2204 1528 1029 777 44.2 269 165 1.0 5400 3680 2470 1565 905 633 387 1el 7500 5130 3482 2230 1408 893 B50 1.2 8700 6200 4220 2660 1665 1050 637 1.4 11390 7700 5270 3350 2150 1353 852 1.6 13990 9570 6420 4165 2280 1670 1060 1.8 17240 11830 8060 5140 2280 1670 1060 209 35080 242380 16440 10686 6640 4170 2640 oe j PART IV 3 b PERFORMANCE CALCULATIONS Before actually investigating the performance of the XXX a brief diseussion of its longitudinal stability and control seems appropriate, Since there is no horizontal stabalizer it is important that the center of Lift and center of gravity remain fairly close together for all operating speeds and weights, It is reasonable to assume that the subsonic lift is at the quarter chord and that the centroid of this lift is at one third the semiespan, This results in a center of lift at station 18, where station here means distance from the nose in feet. The airplane is designed so that its C.eGe is fixed at station 17.9%. Generally speaking, the airplane is then stable in the subsonic regime. In the supersonic regime for the range of Mach numbers considered here, the flow over the wing is almost conical and the center of pres sure may be readily computed since the pressure is constant along any radial line through the root of the leading edge. Thus for the wing of the XXX the center of pressure is at station 19.2, indicating even more stability than for subsonic flight. However, in the supersonic case the lift of the fuselage can play an important part since its lift usually may be considered as being concentrated at its nose, and therefore the complete stability is given by the equation JSC. = 7. Aer — tur Saw TK ex o/ « where is the distance f rom the nose of the C.G. and ur is the f ACL‘ YS XK mately 2 per radian based on fuselage frontal area or 0.164 based on Mew SH distance Prom the C.G. to the center of lift, will be approxi- wing area. varies with Mach number and for an example at M= 1.5 it is 3.1 per radian. Thus at M = 1,53 dc ie = 17.9 x 0.164 = (19.2 = 17.9) x 3.1 = = 1.1 per radian. For straight and level flight the XXX will have to maintain an angle of attach given by: L= «oa x rn ‘fp 5 Assume for an example that M= 1.5, L = 3,000 pounds, and that p + 628 1b./sqe ft. corresponding to 30,000 feet altitude. Then @& = 0.62 desreesS. To determine the necessary elevon diflection we use the expressions Cy =O = «to —, Che fe where A, is the distance from the C.G. to the center of the elevon and is here equal to 5.5 feet. Then 0 = 0,62 x 0.01745 x (-1e1)- 5.50, x a a 86 whence C. = 0,021 , e Using the two dimentional dGe = 7 __ = 326, we haves Jf Xe 74> -/ ASXKe die Xe X 96 = ~ 0,021 He = - 0,0058 radians = 0.38 degreese AB OoCES af bo A&lfCeare a fees We< the ingtartancous weirht oy a << toe acecler BD tho Instentencous drag io hesdeorbal aoues thrust. ee a“ 4 Before serbinul m for our digcussl sion of 3. ao THRUST AVAILABLE We use the simplified expression for thrust: ze “ta * Po * te where Con = Cop X x Ne Both and y are usually equal to about 0695s The basic constants of the nitric acid and aniline rocket motor ares Py = 800 pounds per square inch and = 1.226 With this information and Pigure 9, which is copied from Reference (16) we can perform all the rocket computations necessary for our pur= poses here. 2 Arbitrarily selecting 25,000' as the altitude for opbimum nozzle expansion we then have: c 800 x 144 — = 55 e Bo 2116 x 0.3709 ° Entering the graph in Figure 9 with this number we arrive at an PH t. & of 7.4 and a Cot of 1.565 at 25,000'. Consequently the thrust at t 25,000! is F = 1.565 x 0.95 x 0.95 x 300 x 15.3 = 6,470 pounds. For other altitudes we have a different Po and consequently a Po different. C. and a different thrust. Altitude Po Co. F = ft Po See 20045 1684 «5, 540 pounds 10,000! 29275 1.45 6,000 pounds 37 (continued ) Altitude Pe Cop F Po 20,0008 44,60 1.53 6,320 pounds 30,000! 68.90 1.59 6,570 pounds 40,000! 110.50 = 1268 6,740 pounds 50,000! 178,00 1.66 6,860 pounds 60,000! 287.50 1.68 6,950 pounds THE INSTANTANEOUS WEIGHT The actual specific impulse of the acideaniline combination is 200 lb. sec./lb., this gives a duration of 89.1 seconds and a mass fuel flow of 32.6 pounds per second. Therefore the instantaneous weight of the I is: 5400 = 36.6 x T 0 v 89.1 seconds SOME SPECIFIC PERFORMANCE PROBLEM: To find the maximum Mach number curve of the XXX we use only two terms of the performance equations D=F . These two terms are plotted independently against altitude and Mach number in Figure 10, and a cross plot yields the maximum Mach number curve in Figure ll. To find the gliding characteristics of the XXX we again use only two terms of the performance equation: Dtw x sin = Oe The gliding performance is presented graphically in Figures 12 and 13. We will now find the maximum altitude attainable by taking off, making a 20 second turn to the vertical direction and continuing the climb straight ue The sea level thrust is 5540 pounds and the starting weight is 54006 pounds, a thrust loading of a little over one. We estimate 200 mph or 293 Pte /sece as a safe take-off speed and also estimate that the drag forces during the take-off will be such that the actual "ett, Takemoff speed will then be attained acceleration will be one in 9.1 seconds after a run of 1535 feet at the expense of 297 pounds of fuel. The upward turn after takeeoff will be considered in five segments of four seconds each, Each segment or leg has an average climb angle, average drag, average thrust, and consequently an average acceleration, The technique is to estimate a velocity at the end of a segment and from this, calculate the altitude at the end of the segment, the average drag, the average thrust and the average acceleration. Knowing the average acceleration one can then verify the estimated end velocity. Thus each step is essentially a cut and dry process. What we are acetually doing is a kind of a numerical integration of the original differential equation. ag *4e GTe*9 "45 go2*e °42 ogE*2 °4F OL0°T °43 082 *aTV pug 8/5 3/5 s/JZ s/J s/J i) ™ LK) gts 00S 99% pug “SAT 08g “sat O69¢ "ST Geog *saT gesg "Sal SPSS qd *Ay *SQT QOOT *sqT OOTT *SQT SZ0T "Sat OSS "SGT OGL qd °Ay °SaT ObbP *SqT Of£TY *SaT OOS *sqT Gee? *saT Led (rhuts m) ay FOABY OM WIOT TeTNGe, UT pequeserg *SaT 9TSP “SAT LvOF *saT LLLP *saT L067 °SQZ gf0G Mo tay 18 929 of? oA8 S T°62 Ss T°Ge S T°Te § T°4T S T°St *h Puy tas, 3e7T aU ime. in 6 ing : g 60 seconds of burn : 2 NA. LTT, if re. der the r Thus we arrive at the vertical direction 29.1 seconds after msi starting, at. 6,315 feet, at a speed of 523 ft./sec. at the expense We shall co of $50 pounds of fuel. "a3 00G°SG S/F OTST yS/F O°TS “SAT 0069 "SUT COLT “sat g6gz "a3 000*OS s/J Off S/F G°9Z *S4T 0089 “SAT OOBT °SAT Boz "$5 OLb°2h S/F FAIL 48/F B°0Z “SAT 0G99 "SAT OOBT “SAT ge6e *45 00e*ge s/F OGOT 28/8 ¥°OS SAT 0099 *S4T OOFT *Sa4T TTS "25 oge°6e s8/F 926 2/8 G°6l *SaT 0879 °SAT COOT *SaT 92g¢ "35 009°FZ S/F TI8 28/F GOT *S4T 0OP9 “SAT OOOT “SAT OLgE °4 o00f0! 8/F BTL s/ZJ T°Zl SAT OG29 “*SaT OSOT "SAT g9OZe "43 OOP*9T 8/5 629 8/3 8°8 “SAT OOT9 *SaT OGOT “sat T96E *a3 oge'ZI 8/5 99g 8/F 6°9 —*84T OS09 *SqT OOOT °*SaT 49TF "a5 00556 «S/F LEG = S/F OY = *SAT COGS «SAT OOOT “SAT eae% the same manner, only in intervals of six seconds each instead of four seconds. *aTy Pua A pus e ‘Ay a *ay aq *ay Meany irs) 6 ci t for ¥ = 1.2 an fficien Ideal Thrust Coe | } , a t i : 1 3 t \i \ \ ij im " i b ~ ca ~ ! cen ™—_ ON a) Dp a rc a 1.6 Lod 0.8 0.6 4 Fy quepoTyyeoy qsnayy Teepl 10 tO w ——— 2,.0,000! £ oo er ie) spimo,j g9 SpUue Sno, ' 1.0 50,000! 40,000" 20,0008 ee 10,0008 bee a,c 1.0 1.2 oA 1.6 1.8 29 2 ia Pirure Drag Jet~off ‘ngle Thrust vs ide 1 re iy Glide angle thrust = 2500° x sin WV’ } | so00f09 P| | | | ! ~ | | | | N\ | { i | \ | | | | | | | | \ | | \ \ | | | | | | | | OO | | \ | | | , | | | | | spuno, Jo spuvsnoul, UT qsnstyg, eTSUy epETy ae) “yy e 1.4 1.2 1.0 0.€ 0.6 ach Number oy rod ber m ate al “Or ae) ak exina, Eoay qeo JU spuesnoyl ul epnyTay. 1.6 oe 4 % me 1. 1.C Oo QO, oO Veit Thus at end of burning we arrive at the height of 58,500 feet “ata speed of 1516 ft. /sece Further altitude can be reached by allowing the airplane to coast to a stop in the vertical directions Ve will assume that the effect of drag is to increase the eravity effect by 50%. An additional height of 23,800 feet or a total height of 82,500 feet can then be attained. “E é PART V TABLE OF SYMBOLS: acceleration in feet ver second per second /w-1 angle of attack angle of deflection of elevon aspect ratio wing span gravity center of drag coefficient based on frontal area pressure drag coefficient based on frontal area LF nose pressure drag coef: tole ey °ielent based on frombal area pressure drag coefficient skin friction coefficient + <3 Q o ry im iets 3 hte c 94 3 te actual thrus theoretical thrust coefficient moment coefficient lift coefficient of wing lift coefficient of eleven ft coefficient of fuselage 1L +5 pode airplane efficiency factor 2 & nogzle efficiency factor rp o ve ces @ thrus ares. nogszle exit area nogzle throat area ratio of specific heats us ial 3 SY 02 + 5x Cas a drag coefficient found in ballistic 1] a nozzle efficiency factor distance from C.G. to nose distance from C.G. to elevon distance from C.G. to center of Lift Mach number a fighter type military aircraft rocket chamber pressure ambient pressure density of air in slugs per cubic foot semi~vertex angle of nose cone density ratio wing plan area velocity in feet per second wetted area weight of airplane the airplane described in Part II climb angle PART VI REFERENCES alee 1. UDET and ERNS?; “Proposal Specifications for a Research Airplane for the Investigation of Aerodynamic Phenomena in the Transonic and Supersonic Range“; C.G.D. No, 121; Translated by Bell Aircraft Corp., Buffalo, N.Y. (Originally published in German) 2. NACA TR 21A and NACA TR 518; “Properties of the Standard Atmos» phere", 3, CHARTERS, ALEX C.: “Some Ballistic Contributions to Aerodynamics"; Paper presented at the symposium on aerodynamics held December 6 and 7, 1945 at Applied Physics Leboratory, The Johns Hopkins University, Silver Spring, Maryland, | 4, VENNARD, JOHN K,z “Elementary Fluid Mechanics"; John Wiley & Sons Ine., New York, 1940, 5. OSTWALD, W. Be: “General Formulas and Charts for the Calculation of Airplane Performance"; NACA Technical Report 408; (1932). 6. BAILEY, F. Je, Jr., MATHEWS, C. W., and THOMPSON, J. Ro: “Drag Measurements at Transonic Speeds of a Freely Felling Body"; NACA ACR No, L5HO3; 1945, 7, LUDWIG, H.: “Sweptback Wings at High Velocities“; C.G.D. No. 19; Translated from the German by N, S, Medvedeff, Goodyear Aircraft Corp. in Arkon, Ohio. Date of original publication December 8, 1940, 8, MATHEWS, C. We, and THOMPSON, J. Rw: "Comparative Drag Measurements at Transonic Speeds of Rectangular and Sweptback NACA 65-009 Airfoils Mounted on a Freely Falling Body“; NACA ACR No. L5G30; 1945, 9. VON KARMAN, THEODORE: “The Problem of Resistance on Compressible Fluids"; Reale Academea D'Italia; 1936. 10, VON KARMAN, THEODORE and MOORE, N. D.z “Resistance of Slender Bodies Moving with Supersonic Velocities with Special Reference to Projectiles"; Trans, A.S.M.E. Applied Mechanics, page 303; 1932. 11, TAYLOR, G. I. and MACCOLL, J. We: “The Air Pressure of a Cone | Moving at High Speeds"; Proc. Roy. Soc.; Vol. 139, page 278-311; 1933. 12, SAUER, R.: “Theoretesche Hinfuhrung in die Gasdynamik”; page 49, Springer, Berlin; 1943, 13, STEWART, H. J.: “The Linearized Theory of Supersonic Flow", Paper presented at the symposium on aerodynamics held December 6 and 7, 1945, at the Applied Physics Laboratory;: The Johns Hopkins Univ-= ersity, Silver Spring, Maryland. 14, PUCKETT, &. E. and STEWART, H. d.: “Aerodynamic Performance of Delta Wings at Supersonic Speeds“; To be published in Journal of Aeronautical Sciences, 15, ERDMAN; "“Widerstandbeiwerte fur das A4VIP mit Berucksichtung des Strahlund Reibungseinflusses fur Unter = und Ubershellgesch= windigkeiten, Untersuchungen der Strehlexpansion™; HAP No, 66/105; 1934, 16. “Jet Propulsion"; A reference text prepared by GALCIT for the fir Technical Service Command; 1946,