THE UNIVERSITY OF MICHIGAN RESEARCH INSTITUTE ANN ARBOR, MICHIGAN Scientific Report No. ES-3 TABLE OF PRESSURE RATIOS Pi/Pc and Pc/Pa VS. MACH NUMBER FOR A 15~ RIGHT CIRCULAR SUPERSONIC CONE (7 = 1.405) Revised from an earlier report by N. W. Spencer The research reported in this document has been made possible through support and sponsorship extended by the Geophysics Research Division of the Air Force Cambridge Research Center, under Contract No. AF 19(604)-545. It is published for technical information only, and does not necessarily represent recommendations or conclusions of the sponsoring agency. UMRI Project 2597 UNITED STATES AIR FORCE, ARDC AIR FORCE CAMBRIDGE RESEARCH CENTER GEOPHYSICS RESEARCH DIRECTORATE CONTRACT NO. AF 19(604)-1948 LAURENCE G. HANSCOM FIELD BEDFORD, MASSACHUSETTS September 1958

ABSTRACT This report presents a table of pressure ratios to be found on a 15~ right circular cone tabulated versus the free-stream Mach number. The pressure ratios are given to five significant figures against 0.001 increments of the argument. The argument ranges from M1 = 1.048 to M2 = 3e895o Given are the ratio of total head pressure (Pi) to the cone-surface pressure (Pc) and the ratio of the conesurface pressure (Pc) to the free-stream static pressure (Pa). With these tables and two pressures on a right circular cone, it is possible to determine the freestream Mach number and the free-stream static pressure. By measuring the freestream velocity (or the velocity of the cone through the media if that is the situation) in addition, it is possible to compute free-stream static temperatures and densities. iii

INTRODUCTION Measurement of the ambient pressure and temperature of the atmosphere as a function of altitude is possible through a determination of the Mach number of the flow about a right circular cone moving at supersonic velocities.1'2 Local measurements of selected pressures, specifically the total head pressure, and a cone-wall pressure, permit evaluation of the Mach number. In performing cone-surface pressure measurements, the cone-surface pressure tap should be located at approximately Oe9 the axial distance from the leading edge. This has been shown as a result of wind-tunnel experiments conducted at the University, which have demonstrated that conical regime pressures3'4 are established on the cone within three diameters (based on the truncation section diameter) along the cone's surface. Having the total head pressure Pi and the cone-surface pressure P., it is possible to compute the free-stream Mach number and the free-stream static pressure, beginning with the following identity: P Pi Pi P Pa Pw - - x — x x - (1) Pc Ps Pa Pw Pc Since each of the right-hand factors is a function of the Mach number, the pressure ratio Pi/Pc is an index of the free-stream Mach numbero Each of the righthand factors may be expressed in terms of the Mach number as follows. ~Pi /~71 2' -_ = i + -M2, (2) Ps 2 PS 2 / where [ (y-1) M2 + 21/2 27 Ml - (y-l) Then the ratio of pressures across the oblique shock is Pw = 2y M sin2 Q (7-1) Pa 7+1 w (7+1) and 1

Ps _ 2y M2 - (y-1) Pa 72+1 (5) Lastly, Pw i I 1 - (u,/c) - (vW/c) 2' 1 - (/)2 - (vsjc) (6) Because of the complexity of the calculation of the parameters for the small increment in the Mach number desired, it was found more satisfactory to determine the parameters at larger intervals of the Mach number and depend on a polynomial fitted to these main values for the calculation of the intermediate values of the argument. The following table lists the principal points resulting from Eqs. (1) through (6), to which polynomials were fittedo Correspondingly, the relation between the free-stream static pressure Pa and the cone-surface pressure Pc for 0.001 increments of the Mach number were calculated from polynomials fitted to the identity Pc / -P Pa V PW/ PW i [Pa, (7) Ml ME Pc/Pw Pw/Pa Ps/Pa Pi/Ps Pi/PC 1.0484 0 9546 1 0773 1,0177 1.,158 1 7990 1.8313 1.0737 0.9329 1 0774 1,0151 11785 1.7560 lo8959 1,1144 0o9008 1.0797 1o0105 1 2825 16940 1.9913 1.1902 0.8484 1o0823 1 0095 1l4866 1 6025 2 1804 1.3577 0o7586 1o0968 1 0103 1.9853 1o4659 2.6263 1,5462 0,6861 1 1111 1 0147 2.6225 15700 3.1868 1 7610 0o6236 1 1289 1,0214 354513 1,2992 3.8886 2,0108 0o5741 1>1487 1o0324 4,5505 1.2503 4~7975 2o3105 0O5322 1 1699 1o0508 6.0615 1>2128 5.9801 2.6847 0.4968 1,1914 1,0822 8.2422 1,1836 7y5664 3,1795 0o4659 1o2105 1.1386 11 6274 1,1605 9o7900 3 8946 0.4379 1 2216 1 2489 175292 11407 13,1055 Figures 1 and 2 are qualitative representations of the functions over their entire tabulated range. Four equations represent Pi/Pc over its entire range of Mach numbers, while five equations were necessary to represent Pc/Pa for a slightly shorter range, The choice of a smaller MI range for PC/Pa was dictated by the lack of knowledge concerning the behavior of the PC/Pa function in the transonic 2

interval. The lowest value for Ml is the relative minima for the function before it becomes double-valued, Information concerning Eqo (7) in this branch of the double-valued function is not sufficient to define the locus to the limits prescribed by the tabulation interval of the argument. CONCLUSION This report is a revised, corrected, and expanded version of an earlier report,* issued under Contract No AF 19(604)-545. It presents data that are being used in evaluating atmospheric structure, and which may be useful to others who may need this information relative to pressure studies on supersonic nose cones. The following tabulation covers a range of Mach numbers from 1.048 to 5.500. REFERENCES 1o The Rocket Panel, Phys, Rev,, 88, 1027 (1952). 2. Sicinski, HL S., Spencer, No Wo, and Dow, W. G., "Rocket Measurements of the Upper Air," J. Appl, Phys., Dec., 1953. 3. Lighthill, M. Jo, London Philosophical Mag., 7th series, 40, 214 (1949). 4. Giese and Bergdolt, "Interferometric Studies of Supersonic Flows About Truncated Cones," Ballistic Research Laboratories Report No. 830o *Scientific Report No. CS-1, September, 19530 3

FIG. I...... -FREE STREAM MACH NUMBER VERSUS...._. RATIO OF TOTAL HEAD PRESSURE TO CONE SURFACE PRESSURE,,,,......, _ _.i.., L, I I.. I I I I. I. A i A I 4.000 —. —----- -- -- -- -- - -- -- 3.800 -. —- -. _ _c__nn__H p —----------- 3.40( 3,20( 3.00( Ml 3,80( 2,60( 2.40 2.20( 2.00 1.80 1.60 1.40 1.20 i.00 v 1 1 / o / ____/ CS ------------- - r\ ^/ — --- --- --- --- -—. —-. —-. —-............. v I (.1. 1.1 1 1 i i i Wm r 1.800 3.800 5SOO.800 7.800 Pa 9.800 11.800 13.800 4

FIG. 2....____ FREE STREAM MACH NUMBER VERSUS RATIO OF CONE SURFACE PRESSURE TO STATIC PRESSURE 4.000 3,800.....-............. 3.600 ---- 3.400 ---- 3.200.... 3.0002.800- --- / 2.600 2.000_ / 2.200 2.000 ____ ___. Ml PPc BS 7 1.800 e_ -. Pa 1.600 _ 1.400 1.200 __ 1.000 1.100 1.200 1.300 1.400 1.500 1.6( PC Po 5

SURFACE NOTATION Pw C M\e s P Pa = Static pressure ahead of shock surface M1 = Mach number M2 = Mach number behind normal shock P = Static pressure behind normal shock Pi = Total pressure behind normal shock Pw =- Static pressure at conical shock surface PC =Surface pressure on cone y = Ratio of specific heats Cp/Cv = 1.405 9w = Shock wave angle vw = Radial velocity behind shock uw = Tangential velocity behind shock v5 = Radial particle velocity at cone surface us = Tangential particle velocity at cone surface Qs = Semiapex angle of cone 7.5~ 6

T:60 T T60 T -" 60 T - T60 T 0S6 T g 6 T Og6'[: T6 T 980 7 980 T: 9980 T 5:5'08"0...... T....' g0o T ~80 T 860 T 0 T6 0 90 T -- 6 Q T- -" 8 06 TS ": 6 2. 0" T 2 6 0 T 906 T 8LO T -— g-6-o....]: ---- r6~ —"[:'zL-T 8 60 T T06 T 9LO T 360 T 968 T LO T........ -'. _...................................-......'..... 36.0 T_____!A 9__T_______ j 0 T69 L 0 g 6,0 T T68 T t 0 T 360- T 9 9 9 -T LO T Z-~60......- -I-SgIC T_5 _-i:' —~~II~ —I-~- -l:t~TDlO T~C~60 T 998 T OLO T ~60 T T8 8 T 990 T "~-6-OCT ——'lJ 8~Z'I~~ L 9' —— L-"9 C60 T 9,8. T 990 T C- 6 0"8 T"-0 ~ ~.....................- "-... -. 9 90.... g60 T TL8 1: 90 T __~ 6 0 T _ __ _ __ ^:^8 "I: t^90 T_0'I *^ 50~s~~o ~ -Te —~~~~-~~-~-~ ~ -~- ~ ~ - -------- 9 0 I -_ 6 0 T 998 T g90 T............ -o..... —----— ~9-ff —._........................i:t60 T T98 T 090 T...._ ^ _.... _ _'-........T,........................0 ~".... ^..............................6"O...'1... g60 T 998 T 80 T'-' —5-6^-'- --- -.~....-E-' —----—.-.^-^^ — __ ~60 T Og8 T 9S T --- -"55 Q~` —-— ~~-~ — " —- ~-~- --- Q7~~g- ~7~-^-~-~-~-'~ -- --- -___-__^^.__ _ 60 T 9 8 T t, S;O T — ^^OT~~~'-TCI- ----— " - -E^Q —3- ------- _-_^_^_^.-^ _ 960 T Ot8 T Z90O T 9-6 --- T. —- 0... —- --—.g-.....T.960 T O z8 T OS0 T.-L.- 60'T 460 T 8yf@' T O0 8 T 6 0O.T. 8!z0 T TzI

Pi PC PC Pa 1 089 1 932 1 091 -— 1_. Q9. Q.__________.193__Q-9___L2__ 1 091 1 937 1 091.,1 9__.__...__ 4 _90 1n o i... 1 093 1 942 1 091 I —..0 9 4 _.__..__.. ____1. 4_.. ~... - _ 1....._. 9__ 1 095 1 947 1 091 -_1__0. 9.6 1........_._.___. __.1 4 - -.......-1.0-9.1__ 1 097 1 951 1 091 09 9 —______ 1.. 1-__9.5 — 1 099 1 956 1 091.-..._. 1,00 0.I......__....... 9.5.9.__ ____ 1...._ 0_9 _._. 1 101 1' 961 1 091.LL....0_O.______..1-9 63____ 091_ 1 103 1 966 1 091 __1....d...4.........__ _..___........968.................__._m.9I.__. 1 105 1 971 1 091 _..a... _ 0 _._ ____1.......7... O 1 107 1 975 1 091 ___!_.J8 J_ 1 8 9 7 8_9 __ _ 1_.0931. 1 109 1 980 1 091 1_ 110 _ 1__ 982 _ 1_091 1 111 i 985 1 091 _ _2______... 98 7__.______ 9 1 113 1 989 1 091 11__1..4. __.1 992 _ ____ _ 091 1 115 1 994 1 091 ___1 6__ 1 997 _ 1091 1 117 1 999 1 091 4___;:.1_8^ _~..... 20 0 0 2___0_____ 1 119 2 004 1 091 _ 1.120 2 00.6 1 091_ 1 121 2 009 1 091 1 123 2 014 1 091 1 __1_2_4......................_ 2.....016_........1.,. _._..__9,,..9. 0 __ 1 125 2 018 1 091 _ 1 26 2..._126 __ ___ _ 2 021 . 09 __. 1 127 2 023 1 091 _-A-_2..8__.........____0_...... 0 9,__ __ 1 129 2 028 1 091,.1_ 13 0 _..... 13 __ lfl9 1 __.2 _. 3 _____ _ i 13;1 2 033 1 091 — 2___;___. o5_...1. 91 I 133 2 038 1 091

Ml P1i. PC Pc Pa 1 —134 2 040 i 1091..3..5.........2 043 ---—.........91 1 136 2 045 1 091 _... 1.7.............0... 8.............. _9 _1_ 1 138 2 050 1 091 _IJ9_Z.0I____QA_....__.,D5_2. _...0..91 1 4 0 2 055 1 0 - 91 1 141 2 057 1 091 1 142 2 060 1 092 114_____i____.....6....X___1 02, 1 144 2 065 1 092 1 146 2 069 1 092......1 14................. 2 072....4 1 092... 1 148 2 074 1 092 4 A__J_1.._.4 _29_____,_ 2......07 2.092.___1 1 150 2 079 1 092... 0.... 1........ 1.5. 1.__...0.......... 2. 08.2............ 1 9.......2 _ 1 152 2 084 1 092._5. 2-0&7.......____5 __J__09._._77_____. 2 1 154 2 089 1 092 -. 1.. 15... 55................. 2.92... _ 9.2..................._t..... 0_92___ 1 156 2 094 1 092 __1.7....... 1.. _........ _ _...... 0 9.__..........._....... 1._....__9_2 ~... 1 158 2 099 1 092 _1_ZJ_.~-_9_ _ _ __i.O.l____.2... 1 160 2 10 4 1 092 1 161 2 106 1 092 I 162 2 109 1 092 I ~,j.6 3 ~2 1.111_.________2 1 164 2 114 1 093 1 165 2116 1 _ 093 1 166 2 119 1 093 1 167 2 121 1__ 093 1 168 2 123 1 093 1_69....2....1 9 1 170 2 128 1 093 1171 _ 2 313_ 1 093_ 1 172 2 133 1 093 ___LU_7J_._.7..3__._ __ 31 9.3 1 174 2 138 1 093.1....1..7_5 14....._.__........21........1.....___._.....__. 1._.a93.. — 1 176 2 143 1 093 _L___1 7_7............. _2-._....... 146..................___...1_31 178 2 148 1 093 1 179______2 15 -I_1. n 09 3 I I 8 2 153 118 2 93 —093...

P1i PC Pc Pa 1 181 2 156 1 094.. 1 182....... 2 158.1.09.4. 1 183 2 161 1 094....-.1....._..8..4....................... 2 163.......__._.................__....1.......0._94_.4.... 1 185 2 166 1 094 __1i.8.6..2___ _26..8_____. __A 1 187 2 171 1 094 1.. 1...... 2 1... 7... 1 9.......4__ 1 189 2 176 1 094 ______1.90..... ___1__ i__9_.__21-7-8__1 9 1 191 2 181 1 094.1 19 2. 182.................... i 1 _... 0. 9-4_ 1 19.3 2 186 1 094.......4........ 2... 188........ _. 9A __ 1 195 2 191 1 094 L__1 1i9.6- 9._____...____.9. 0__A__ 1 197 2 196.1 1.9_ 9..... 9 2. 19 I 199 2 201 1 095.......... —..... 9. 5 1 095 _L _1______2_OQ_.......__a....8_0._ __ 09 5 1 201 2 206 1 095 1202 2 208..1095 1 203 2 211 1 095 1 204 2 214 095 1 205 2 216 1 095 I 2,_ 2 2_ 9.C 5. _1_-20_ Q~_j.?__________2.._____ ____..~^__ 1 207 2 221 1 095 1 208 2 224 1 09..... 1 209 2 226 1 095 21.2229._.._._...____1 Q95 1 211 2 231 1 095 1. 212 2 234 096 1 213 2 237 1 096 1.....214 2 239 _1 0._6... 1 215 2 242 1 096 1 216 2 2___...4_44 i________.xL9-i 1 217 2 247 1 096 2 218 2 249 2.4. 1 096 1 219 2 252 1 096 _._2 2 CL-5.....2_5__.5_J.... n__9 6 09_6.. 1 221 2 257 1 096 1.....2 22.... 22... 60........_.....1... 09..6....... 1 223 2 262 1 096 1 224 2 265.._ 9 6 1 225 2 267 1 096 t2 26 ___.___ 226 _n_i%._.______& 096 1 227 2 273 1 097

Pi PC PC Pa 1 ^22_-~8 2 275_.^ - __- ~ 1 097 1 230 2 280 1 097 1 232 2 286 1 097 2_....._3__3 2 2__28.8_..._.... __1 097 1 234 2 2 9 1 1 097. 235 _. 293. 1 09 7 1 236 2 296 1 097 J237__; _ _2_ ____2_.9,.-?_1.....09_77 1 238 2 301 1 097 _ ___.2 _3 304.._.7_____ __92_..__-._4................. I..__.097 1 240 2 306 1 0_98 1 242 2 312' 1 098 1 243..............2 31.....4 0 9...8.... 0 8__ 1 244 2 317 1 098 1.... 2.45 2 3.9..2.28_0................0 9.8__ 1 246 2 322 1 098.__,__...2.Z..2-______1 09..O_8 1 248 2 327 1 098..... _....... 2_...4.9................,..... _.....3.._.............._..............3.._...._..__ _.Q-9 8__..... 1?25J _____5......._.. _.____ 0982__ 1 252 2 338 1 098 _1 253 2 34 __ _34 1_099_ 1 254 2 343 1 099 1 255 2 _ 346 1 099 1 256 2 348 I 099._ 2 5 7 2_ 3 5..____d _IQ99 1 258 2 354 1 099 1 260 23 599 1 0 9 9 1 2 6 2_ __3____6_2 I ____ 1 262 2 364 1 099 1 2 63 2 367 099 1 264 2 370 1 099 1. 265 2 37.....2..__9... 1 266 2 375 1 100 1_2 7_ U_____1_672.___2-ZB _7_:l....0 1 268 2 380 1 100 _ 1. 26..9....1. _3................... 2.....................I........._.. 1 270 2 386 1 100 Lia_.................. _ _......... _.... 1 272 2 391 1 100

M1 Pi P, Pc Pa 1 273 2 394 1 100 — 1 — 2-74 _ ---- 3 96 --- _ _ _1-0 - 1 275 2 399 1 100..1 27 6........... _.. 4 2._ 1 _ 0 1 277 2 404 1 100 -__Ja —— 277__-... _ _ _ 2- 4 —-01- 1. 01... - 1 279 2 410 1 101 1 —-- 8 0 _- 2 --— 41 2 - --- 0-1 —1 281 2 415 1 101 -1 -282-8 —-- -2 —-4 —-- 4-. —-1 l-0O — 1 283 2 420 1 101 1. 2.84 _..2. 2_3 _. 1.__.. __ - 1. 010 1 285 2 426 1 101 _.l_28I 2 8.4 9...__ 1___Qn__ 1 287 2 431 1 101 1 2-__.__.__ 4A34_1 0. 1 289 2 437 1 102 1-_. 2 90.......3 9.. 1 -1 -0Q1 291 2 442 1 102 1 293 2 447 1 102 1 2 94 _ 2 45Q ___5._.Q._ 1 295 2 453 1 02 _196 2 456 ___ 1 1_0 2 1 297 2 458 1 02 ___1 299 2 46__ _ 1 102a 1 299 2 464 1 102 1 300 1 301._:.,__..._20. _...... 1 303 1 304 1 305 1 306 1 307 _.J1. 8___0_8 1 309 2466.. 1 20_ 2 469 1 103.?_-3 4.72 _____ 1 2 475 1 103 2 477 1 13_ 2 480 1 103 2. 483......... 4a,..1 3 __. 2 486 1 103 _._2 4__ __a... __. 1.0 3 2 49 1103 1 310 1 311.1 3 1_ 1 1 313 1 314 1 315 1 316 1 317 13 19 1 319 2 4 9 4 2 496 _._....2.. 4_9_ 2 502 2.. 505 2 507... 2 510 2 513 ___. _2....-12 518 -- 1 1031 103 1 104 - -1-1 04 1 104 1 1-. 10-4... — 1 104 I 1. An 1 104

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P1 PC Pc Pa 1i — 367 2..6...53 1- " 109 9. 36 _ 2....._........__.. 65.6 - - 1 1_..1,- -9 — 1 369 2 659 1 109 t 3. ___1__.... 2,6__._l. 1...0 _ 1 371 2 664 1 109 1......372 2 6.6..........-........2.....6 _6. _ -..........., -0-_____ 1 373 2 670 1 109.-1. 3:7..4... 1......1. 0__ 2_....3-....._..........9...... 0-0 9__ 1 375 2 676 1 1.09.-_J.7-__76__._ _____ -_2-7-629. 1 —1-0 — 1 377 2 682 1 110 1 378...68..4..... -12_._3_7....... 8_..... 84.................... 8.4..................1_ 10~_ 1 379 2 687 1 110 _~;8 -10_.._ __ __6-_6_....9Q _____ 10 1 381 2 693 1 110 -.._ 1 _38. 2 2.6......9 6.............2...66...__ _1_1 0_-. 1 383 2 699 1 110.-.......:..._..384.........702.............1..1.. 1.- -.-.7. 1 385 2 705 1 110 1__....8_6.7......0___ __,_2__7...0__8 1 1_ 1 387 2 710 1 111 1 3889 _ 2 713 __ 1 111 1 389 2 716 1 111 1_390 2 _719 1_111. 1 391 2 722 1 111 1 392 2 725 1 __ 1 393 2 728 1 111. 394..2 73............ 1 395 2 734 1 111 __1 396_...2_.2 _7 3._ 511 7_W__ 1 397 2 739 1 112 1 398 2 742. 1 11... 1 399 2 745 1 112 ______4 02Q......___ ____.Y_.4_......... _ 11..2.. 1 401 2 751 1112 1. 40. 2,2 75.. 4..._........1.2.... 1 403 2 757 1 112 1.404 __,_2 760 _ 112 1 405 2 763 1 112.1 4.4QA _.... ____ ~_ __2___7-6-jl-.______ 1 1 2 1 407 2 769 1 113 1 408 2 771........1 13. 1 409 2 774 1 113 __1_. 41..4......2.._ 78.07-. _1. 1. 1 411 2 780 1 113

Pi PC Pc Pa 1 412 2 783 1 113.-. -1 4 13.......... 2 786 —.- -.-. —- - -.-: — -L -— _ 1 -- 1 414 2 789 1 113.__.1. _4_t.5 _........792 - -.......____ 1 416 2 795 1 113 1417. _ _. __ 2-_Z -fi _________ 1 1 1 4 1 418 2 801 1114. 4..9.... _............._............ _ __41__ 4___ 1 420 2 807 1114.2.2:X..0A.___ __ 281 1 422 2 813 1 114._....... 4 2.. _......_3_........ _..._............ -8 1... -........8_.... ____.4.. 1 424 2 819 1 114.1 _ 2 5......2.8 2 2 _....1.1 14 1 426 2 825 1 114 _ __ 42.___ _..........__ — 1 5 1 428 2 830 1 115 1 429. 2 83 1 115. 1 430 2 836 1 115 _141._ 3 1 _._ 2 _8.________ 1 5 1 432 2 842 115.......4. 3............ 8 4 S _ 1 1.-.1 1 434 2 848 1 115.1 43.5 2. 851.... 1 1.. 5- -.. 1 436 2 854 1 115 4 _1_Au- 2.___L.2..___8_7 8_-5_- _.1 16 1 438 2 860 1 116...1... 4 9............... 2..86. 1._. 1 _. l 6 1 440 2 866 1 116 14.4.____44. 2 86_9..______. 1 16_ 1 442 2 872 1116 1 4 43 2 8 7 5_1 6 1 444 2 878 1 116 1. 4 5.......... 4... 18__.... 1 446 2 884 1 117 1 4 47.. — 887 a. 1__2_____ 1 448 2 890 1 117 1... 44..9.. 2.... 8.9 3.... 1..117 1 450 2 896 1 117 1 452 2 902 1 117.1. 4 53.... 2. 9 1... 1.7 - 1 454 2 908 1 117 1.4 5 5... 2.... 9 1.1 _.. I.-. 117..... 1 456 2 914 1 118 4.. 57 L__ __W._2LZ 11_f 1 458 2 920 1 118

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