NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 
 
 No. 1147 
 
 GENERAL CHARACTERISTICS OF THE FLOW THROUGH 
 
 NOZZLES AT NEAR CRITICAL SPEEDS 
 
 By R. Sauer 
 
 Translation 
 
 "Allegemeine Eigenschaften der Stromung durch Diisen 
 in der Nahe der kritischen Geschwindigkeit" 
 
 Deutsche Luftfahrtfdrschung, Forschungsbericht Nr. 1992 
 
 NACA 
 
 UNIVERSITY OF FLORIDA 
 Washington DOCUMENTS DER^RTMENT 
 Tune 1947 ^ 20 MARSTON SCIENCE LIBRAFlY 
 
 RO. BOX 117011 
 GAINESViLLE, FL 2^m 1 -10^ 1 UEA 
 
?n fsKi ^^'^ 
 
 HATICNAL ADVISORY COMvIITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORAKDUM NO. 11^7 
 
 GENERAL CHARACTERISTICS OF THE PLOW THROUGH 
 
 NOZZLES AT NEAR CRITICAL SPEEDS"'' 
 
 By R, 'Sauer 
 
 SUMMARY ' 
 
 The characteristics of the position and form of 
 the transition surface bhrough the critical velocity 
 are computed for flow through flat and round nozzles 
 from subsonic to supersonic velocity. Corresponding con- 
 siderations were cari'ied out for the flov/ ahout profiles 
 in the vicinity of sonic velocity. 
 
 I. II^TRODuGTIOK 
 
 Vih,ile useful methods of calculation exist for pure 
 subsonic and for pure supersonic flov/s of comDresslble 
 gases difficulties arise in the mathematical treatment 
 of mixed flows with subsonic and supersonic r singes. Such 
 mixed flows exists (l) in the transition from subsonic to 
 supersonic speeds in Laval nozzles, (2) about Drcfiles 
 in a flow at high subsonic speed on the appearance of 
 local supersonic ranges, (5) in front of blunt bodies 
 in a flow at supersonic velocitjr with a local subsonic 
 region behind the compression shock In the vicinity of' 
 the stagnation poliat. , ' 
 
 The present report takes up the first as the 
 simplest of the three problems na.med'. it is a fact that 
 for this case a rough view of flow conditions suffi- 
 cient for many problems can be obtained by bhe methods 
 of hydraulics, that is, the xiozzle is considered a 
 flow tube and the m.agnitude of the velocity is considered 
 
 'Allgemeine Eigsnschaf ten der Stromung durch 
 Dusen in der Ni-ihe der kritischen Geschwlndigkeit, " 
 FB 1992, Zentrale fur wissenschaf tliches Berichtswesen 
 der Luftfahrtforschung des Generalluf tzeugmoisters (ZWB) 
 Berlin - Adlershof, Sept. 25, 1944. 
 
2 .. . „ NACA TM No. lli^T 
 
 constant in every cross-sectional plane. The critical 
 velocity (flow velocity- = sonic velocity) is_then reached 
 exactly in the minimum cross section of the Laval 
 nozzle. In the strict tvjo-dimensional treatment, 
 lxo\"evev , a curved line (critical curve) is obtained for 
 the passage of the velocity through the critical value 
 in the plane of the longitudinal section of the nozzle; 
 It begins at the nozzle wall in front of the minimum 
 cross section and cuts the nozzle axis downstream of 
 the minimum cross section. (See fig. 1.) 
 
 In various reports, especially those of Th. Meyer (1), 
 G, J, Taylor (2), H. Gdrtler (5), and Kl, Oswatitsch and 
 li. Rothstein (II), expansions in power series for the 
 determination of the critical curves are given. In 
 what follov.'s, general statements on the i:osition and 
 curvature of the critical curve for a given nozzle are 
 obtained from, such expansions m power series by 
 breaking off the series after the first two term.s , The 
 investigations are concerned v;ith flat and round nozzles 
 and provide sufficiently accurate results for practical 
 purposes for nozzle curvatures that are not too sharp. 
 
 Corresponding expansions in power series are 
 applied, in conclusion, to the flows through flat 
 nozzles v/lth curved axes. In this way, information is 
 obtained on the variation of the critical curve for 
 profile flov.'s v/lth local supersonic regions. 
 
 II. POTENTIAL SQUATIOIT 
 
 The nozzle axis is selected as the x-axis and the 
 origin of the coordinate system, is placed at the point 
 on the axis at which the critical velocity is first 
 reached, (See fig. 1.) V'ith the hypothesis of non- 
 voi'tical flow and perfect flov; of a perfect gas with 
 constant specific heats c , g„ (k = c /c ) the ootential 
 
 equation 
 
 ^ (a^ - u2) ^ ox (^^2 . r,2) . 2nv ^ +-aa2v= o (1) 
 
NACA TM No . 11 if? 
 
 holds. In this ii, v are the x- and y-components of the 
 
 floY;; velocity and a, the local sonic velocity, which 
 
 is related to n, v and the critical velocity a'"" through 
 
 a' 
 
 k + 1 ..p k - 1 
 
 £i 
 
 
 (2) 
 
 ^ = characterizes the flat or two-dimensional flow, 
 O = 1 the three-dimensional flov; through stream tubes -^f 
 of circular cross section; x, y are cartesian for = -0, 
 cylindrical coordinates for a = 1. 
 
 Substituting (2) in equation (1) and introducing 
 the ditiienslonless velocity components 
 
 'J " u/a^'', V = v/a"'"' 
 
 the potential equation (l) becomes; 
 
 \ 
 
 ^^ + 1 03^ 
 
 -^ ic + 1 r ^ y 
 
 1=0 
 
 (3) 
 
 (li) 
 
 Since the present investigation was llinited to the 
 vicinity of the critical curve. 
 
 U = 1 + u, V = V (5.) 
 
 in v;hich u and v are small quantities, and equation (k) 
 becomes 
 
 
 ^ P' k - 1 21 
 u + u^- + TTTT ^^ 
 
 ■dv! 2 
 
 X, 
 
 - 24# 
 
 (k - 1) 
 
 yjk + 1 Ik +'TJ 
 
 u 
 
 ^^-:-i u2 - v2 
 
 !^ 6u 
 k + 1 ^ 
 
 (1 + u ) V - a 
 
 2 
 k + 1 
 
 -^^^-Zi-Hf^--^ 
 
 V • 
 
 = 0" 
 
 ■ (^) 
 
NACA TM No. Il47 
 
 As X— >0, f — >0, vl, and v approach zero and also 
 oil the basis of (6) bv/bj and, consequently, the 
 quotient v/y approach zero, while the velocity 
 rise 6u/dx alone; the axis ought not, vanish at the 
 origin. Then if the small quantities u, v, '^•v/oy, 
 and v/y are considered linear only, the following 
 approxinate relation is obtained from (6) 
 
 (k + 1) u 
 
 du 
 
 C v 
 
 (Sx. " 5y- 
 
 + 2v r 
 
 - a 1= 
 
 (7) 
 
 Since the nozzle flow is sjnrranetrical with respect 
 to the X-axis, ou/Cy also approaches zero for x~>0, 
 y — 7>0. ConFequ3nt3.y .the term 2v 6u/oy may be ignored, 
 by means of v;hich (7) becomes further siinplified to 
 
 (8) 
 
 t/ith a = 0, (o) v.'as already applied in another con- 
 nection by Kl. Osvatitsch and K. Vv'leghardt (5). 
 
 :il. FLAT KOZiZLES 
 
 -liquation (8) furnishes general relations, for a = 0, 
 for the flovir through flat nozzles in the vicinity of the 
 critical curve. Like Kl, Oswatitsch and W. Fiothstein ([|.), 
 taking into account the sjnnmetry around the x-axis, 
 setting 
 
 <9 
 
 f,(x) 
 
 + y' 
 
 h; 
 
 -f^(x) + y^-f, (x) + 
 ^ u. 
 
 u = ^= f'o(x) + y2f' (x) + y^f',,(x) . 
 
 ox 
 
 h' 
 
 V = ^-£L= 2yf rx) + ky5f, (x) ^A^':i'^'^'^''^-^- 
 
 ^y 
 
 J 
 
 Y is) 
 
MCA TM No. 1147 5 
 
 in which the primes signify derivatives with respect 
 to X, and, obtain from (o), 
 
 (k + 1) (f ' + y2f ' +A.) (f" + y2fi. +...<? ) 
 ^ '» t- K 
 
 - 2f2 + '^'^l^]y +•••• + 2 a (f^ + Sy'^fh +••! ) 
 
 By arranging in order of powers of y and equating 
 coefficients of the individual powers to zero this 
 system of equations is obtained 
 
 2(1 + a) f = (k + 1) f ' f" 
 2 
 
 (10) 
 2(6 + 2a*)f, = (k + 1) (f» f' + f" f t^) 
 
 through which the functions f (x), . f) (x)... in order 
 
 can be expressed by the function f ' (x) and its , 
 derivatives. The function 
 
 f'Q = u^(x).. .. 
 
 which characterizes the velocity distribution along the 
 nozzle axis, remains undetermined. Given u' (x) the 
 
 ' O' 
 
 flow is completely established through (10) and can be 
 understood to be nozzle flow, if a pair of streamlines 
 that are mirror images- of one another with respect to 
 the X-axis are assumed as nozzle walls. For further 
 discussion, setting 
 
 f ' = u (x) = a X (11) 
 
 u o 
 
 from which 
 
 - 2 " 
 
 ^ - k + 1 ^2^ - (k + l) „5 M o / « > ^ 
 fg = — 2 — 0. X, ^1 =^ %\ cc-',... (12(a)) 
 
 "Translator's note: The number 2 was omitted from the 
 German report. The correction here changes several 
 subsequent equations and several values in the table. 
 
6 MCA TM No. 1147 
 
 follows immediately from (10 ) with <7 = 0. According 
 to (9) 
 
 , k + 1 ? 2 . 
 u = ax + — p — ay + 
 
 V = (k + 1) a^xy + ^^"' t ' ^^ a5y3 ^ 
 
 a-'y-' + ., 
 
 (13(a)) 
 
 is obtained for tlie components of the flow velocity. 
 ITrie dots represent terms of higher order of y. If (11) 
 is taken as the first term of a power series expansion 
 for Uq(x), the expresslcns for fp and ft^ in (12(a)) 
 
 are also the first terms of pov/er series expansions. 
 
 Prom (15(-)) the critical ciAi've is obtained from the 
 requirement 
 
 (1 + u)" + v^ = 1 
 
 from which in (lj(a)) the parabola 
 
 iv 
 
 k + 1 
 
 2" "^K 
 
 (li|(a)) 
 
 is obtained, therefore, with u = as a first 
 approximation. It cuts the noazle axis, normally, at 
 the ooint x = y = and its curvature there is 
 
 l/p^ = (k + 1) a 
 
 (15(a)) 
 
 The vertices, therefore the points, of the streamlines, 
 adjacent to the noszle axis, are given by v = and, 
 accordin^j to (15(a)), lie on the curve 
 
 Xc 
 
 H^ -^-s^ 
 
 (16(a)) 
 
 Considering, nov;, the streamlines at a given vertex 
 distance y^ (fig. 1) from the noazle v;all, then 
 
 """S 
 
 k + 1 2 
 
 (17(a)) 
 
mCA TM No. 1147 ' '7 
 
 is the distance from the center point of- the narrowest 
 cross "section downstream to the intersection point of 
 the nozzle axis and the critical curve. 
 
 Tlie junction point of the critical curve v/ith the 
 nozzle wall is upstream and is obtained, as a first 
 approximation from (ll|(a)) with jj. = y^, which yields 
 
 . ^' = -6v- -si = ^H^ ""^^^ = 2' (18(a)) 
 
 for the distance v, from the junction point to the 
 narrowest cross section. ' 
 
 Finally the calculation of the curvature at the vertex 
 of the nozzle wall is to "be made. In moving outward 
 from the vertex to the wall by an elem^ent of 
 curve ds = dx, the tangent turns through the angle 
 
 ■ , , , d V 1 6v ^ 1 cv , 
 
 '^'^ = r-nr = mi ^ ^^ = nnr ^ ^^ ^ ; 
 
 Therefore the curvature -at the vertex is 
 
 1 _ d^ _ 1, ov . 
 Pq ~ ds 1 + u ox 
 
 and taking into account (13(a)) and (15(a)) 
 
 1/p^ = (k + 1) a^y^ = -/Pk^^^s (19(a)) 
 
 Up to nov/ the velocity distribution Uq(x) had been 
 considered as given by equation (11) and the nozzle 
 walls computed for it. Practically, the^rever^e is 
 the problem, namely to ascertain for a particular 
 nozzle/ that is for assigned values of y^., pg, the 
 flov/ in the narrov/est cross section, therefore the ' . 
 
8 
 
 MCA TM No. 1147 
 
 values a, c, y], and p, # On the basis of the equations 
 that have preceded it ±s possible to write down the 
 solution to the problem, irmnediately, namely 
 
 a 
 
 Pk = 
 
 TFT- 
 
 1/ PoYc 
 
 o S 
 
 I' k + 1 
 
 ^1 = Z€ = -^ |/(k + 1) 
 
 ys 
 
 (20(a)) 
 
 If the flow is in the positive direction of the x-axis, 
 as in figure 1, then the subsonic region is to the left 
 and the supersonic region to the right and, accord- 
 ingly, ci> 0. 
 
 In practice it is recomruended that all distances be 
 expressed, dimensionless, in tarnis of 
 
 J Q > 
 
 o 
 
 and 
 
 accordingly'- set y = 1 throughout (20(a)). 
 
 IV. ROUND NOZZLES 
 
 Corresponding relations for round nozzles come 
 from (9) and (10 ) with a = 1. .tietaining (11) for 
 the velocity distribution u (xj along the nozzle 
 
 axis, the follov/ing relations are obtained in place 
 of (12(a)) through''(20(a)): 
 
 ^2 = 
 
 k + 1 2 
 — Tl — a'^x. 
 
 % 
 
 (k + 1) ^ X 
 
 - — rrr, — a-' 
 
 F 
 
 (12(b)) 
 
MCA TM No. 1147 
 
 
 u 
 
 k + 1 p n 
 
 ax + — r— '- a'^y'- +. 
 
 (1: + lY 
 
 X 2. 
 
 k + 1 2 ■ \^- ' -^ ' ^-j ."J , 
 
 (13(0)) 
 
 k + 1 
 
 ^K = 
 
 k 
 
 a: 
 
 -K 
 
 (IIlCd)) 
 
 1 _ k + 1 
 
 'J 
 
 a 
 
 :i5(b)) 
 
 k + 1 2 
 
 (16(1:)) 
 
 k + 1 p 
 
 (17(b)) 
 
 
 1) ay./ ^ e (18(h)) 
 
 
 :i9) 
 
10 
 
 MCA TM No. 11^7 
 
 a = 
 
 iMk.Dp^y^ 
 
 Pk = 
 
 £ 
 
 S 2Ps2 
 
 ■! k + 
 
 8 
 
 
 
 
 
 = 
 
 1 
 \! 2(k 
 
 + 
 
 1 
 
 1 
 
 (2C(b)) 
 
 To coiupare a flat nozzle v/itli a round one of the same 
 longitudinal section, forniulas (20(a)) and (20(b)) 
 are compared. (See fig. 2.) 
 
 For the flat nozzle T] = 2€, for the roujad nozzle rj 
 
 = €. 
 
 Moreover 
 
 LIJ1}^= iv2 = 1.06 
 
 G flat U- ■ 
 
 n 
 
 "0 flat 
 
 ound 5 I 
 = - \! 
 
 ^ 0.5; 
 
 0^ round _ J ^ ^ . • T 
 
 — = \ ii ~ J. . llI 
 
 a flat ^' 
 
 The velocity increase is, therefore, aroirad ij.0 percent 
 larger for the round nozzle than for t'ne flat nozzle. 
 All of ■Ghese results are fully confirmed by the nozzle 
 flows calculated by Kl. Oswatitsch and '.«■ , Rothstein, as 
 the enclosed table on page i3 shov/s . 
 
NACA TM No. 11^7 H 
 
 V. COMPARISON! .'ITH 'HT. FLOW-TU.B:; .^FuOllTl'AHIO^ 
 
 The author's result:" are nov! to be coii"-)?Ared with 
 the flow-tube approxl^nation bv coi«p;\tin;;^ ths iucrea^'e 
 in velocity a by formulas (20(a)') end (2C(b)), a^id 
 hj the flow- tube ■cliecr^'-. For' this yurpo?e the xloj 
 deneitj- along the nos?:le axis in the vicinity of the 
 narrowest crosf section 5 3 expanded, as follov;s 
 
 9 zr -£.- U 
 
 •'V- + 1 -^x ^"TT 
 = const. II j™^^ - U^)- ^ 
 
 \ K - 1 / 
 
 .1 
 
 = const. (1 + u)ir~— -4 - 1 " 2u - u'^V""''' 
 
 ~ const, (1 + u)'Tt — =— i" - 2u'!"" 
 
 .k - 1 
 
 / 
 
 ~ const. (1 - U-) 
 
 Taking the velocity distribution (11 ) s.f- a basis, for 
 the nozzle walls in the neighV-orho'jd of ths narrowect 
 cross section the following is obtained in the case of 
 a flat nozGle 
 
 yil - \\-] - const. =^ " .. , ".;■ = y,\l - u .■ ~ ^'V, 11 + a/'x"/ 
 
 and in the case of a roiind nozzle 
 
 1 _ u-; = y^l + ? X / 
 From this , the curvature at the vertex is 
 
 and 
 
 1/p, = a^yg 
 
 respectively, 
 
12 
 
 WACA TM No. 1147 
 
 and in place of the first eaue.tio.ns of (20(a)) 
 and (20(b)) 
 
 a = u^ 
 
 i J. 
 
 
 (21(a)) 
 
 a 
 
 \ P.y. 
 
 (21(b)) 
 
 O '■'J 
 
 Ttie Goniparison of (20(a) ,20 (b) ) and (21 (a) ,21 (b) ) shows 
 that the velocity increase G in the flo\v-tube approxi- 
 mation for both rist and round noi^zles is too large hy 
 
 a factor vji£_±-i, that is for k = l,k, about 10 oercent. 
 
 V 2 
 llie values for the flow- tube approxlrriation are added 
 to the table, page iG, in square brackets. 
 
 VI. PLOW ABOUT PROFILES 
 
 ITarough slmilai- power-series expansions the flow 
 past a profile in the vicinity of transit through the 
 critical velocity can be discussed. If a stree^iiline 
 ad^jacent to the profile is -chousht of a rigid v/all, 
 the flow may also be explained as flow throurh a 
 nozsls with a curved axis. u:ne origin of the coordinate 
 system is located at that point of the profile at 
 which the critical velocity is just reached, the 
 X-axis downstream on the profile tangent and the y-axis 
 normal to the profile, outward, (See fig. 3») 
 
 Since the flow is no longer symmetrical to the x-axis, 
 instead of (8) the rather complicated equation (7) 
 which m.ust be specified with cr = 0, by all means, is 
 taken as a basis and in olace of (9) 
 
 cp = 
 
 u = 
 
 v = 
 
 fpCx) + yf^(x) + y2f^(x) + y?f..(x) +... 
 
 mo 
 Ox 
 
 p 
 
 .rO. 
 
 = f^Qix) + 3^f '.^(x) + y^f '^(x) + y-'f 'j(x) +... V (22) 
 
 ( 
 
 ^ = f. (x) -1- 2yf (x) 4- 3y2f,(x) -r 
 oy J- «2 'j 
 
 J 
 
NACA TM No. 11^7 
 
 13 
 
 3y pu-bting (22) in (?) 
 
 and by comparing coefficients 
 
 f^ = 
 
 ^'o ^'o ^ ^1 ^'l 
 
 .9 
 
 = (k + 1) (f -o^"! "■ ^V'O "■ ^^(V2 ^ ^'/2) 
 
 > (23) 
 
 while only the function f'gCx) had remained arbitrary 
 in the nozzle flow, here there are tv/o arbitrary 
 functions f'g^^^ ^l^'"^)* "i^h f'o(x) and f;j_(x) 
 the remaining coefficient functions f-^^ix), fz (>:),..• 
 
 in order may be com.puted frora the system of equations (25), 
 by vrtiich the flow is completely established. 
 
 Similar to (11) set 
 
 f '^^ = ax, fi = -Px 
 
 izk) 
 
 and obtain from. (25) 
 
 ^5 = 
 
 f2 = 
 
 k + 1 
 
 = (~~ a2 . p2)x 
 
 ap 
 
 . kP (k_|J-. a2 + p^y 
 
 and from (22) 
 
14 
 
 NACA TM No. 114? 
 
 u = ax - py + V 2 " ^^ ^ ^ '' ^ +••• 
 
 N- 
 
 V = - px + 
 
 n 
 
 (k + Da^ + Sp^j- ^xy 
 
 (k + 1) 2 2j '^ 2^ 
 
 (25) 
 
 Here a signifies tiie velocity Increase -r- alon£:, the profile 
 
 ox 
 
 on transit through the critical velocityj it is therefore 
 
 positive at the starting point b. and negative at the 
 
 terminal point B (fig. .a) of a local supersonic region. 
 
 The profile curvature is given v/ith 3 > Q at the 
 point A or B as 
 
 Pp 
 
 V5x/x=ry=0 ' 
 
 (26) 
 
 For the slope of the critical curve v/lth respect to 
 the profile tangent with 
 
 (1 + u)'^ + v^ = 2u - 
 
 frorn (2S) the relation 
 
 tan -^ = I ~ I = ap 
 
 (2?) 
 
 is obtained. 
 
 ""'Translator's Note: This 2 was omitted through 
 error in the original German report. The correction 
 applied here changes values of the constant in the 
 succeeding equations and some values in the table. 
 
NACA TM No. 114? 
 
 15 
 
 For the slope of the streamline at a distance y,. 
 froir. tiae profile '^ 
 
 / 1 ^v\ ~ 
 
 y 1 - py 
 
 X = 
 
 p - 
 
 s L 
 
 (k + 1 
 
 -'3 
 
 ~ p - [(k + 1) a2 + p2J y^ 
 
 is obtained and t&king (26) into consideration 
 
 1 ^ P l1 
 
 = 4_ > i (k + 1 ) a'^^ + -^ I y 
 
 o ?i ^ 
 
 Ps Pp 
 
 ps " pp ■" ^s r ■*" ^ 
 
 k + 1) a2pp2J 
 
 (28) 
 
 according to (27) the critical curve at A (a > 0) 
 and B (a < 0) is steeper the larger the v?'locity 
 increase or decrease and the less the profile is 
 curved. By (281 in the limiting case of a = the 
 circle of curvattue at S is concentric ivith the 
 circle of curvatiire of the profile and, as is already 
 kno^vn from the f lovr-tube approximation, becomes still 
 flatter 'Afith increasing a. 
 
 VII. SUMMARY 
 
 Approximation formulas were developed for the 
 position and curvature of the critical curve for 
 transition through the critical velocity in the 
 neighborhood of the nai"rov/est cross section of flat 
 and round Laval nozzles. In comparison M'ith 
 
16 MCA "Bl Wo . ll^J-7 
 
 noszle flows calculated by Kl, Oswatitsch anc^ 
 W, Rothstain (li.) they showed satlsf ftctoTA?- arreev.ient. 
 In addition, corresponding approximation formulae- were 
 dediiced for the flo?/ at profile? vith local super- 
 sonic regions. 
 
 Translated hj Dave Feingoid 
 National Advisory Conirniittee 
 for Aeronautics 
 
NACA TM No. 114? ^'^ 
 
 VIII. REFERENCES 
 
 1. Meyer^Th.: Uber zveidlmensionale Bevegiingsvorgange 
 eines Gases, das mit Uberschallgeschwlndigkeit 
 streJmt, Diss. Gottlngen 1907. 
 
 5. Taylor^ G. J.: The flov of air at high speeds past 
 curved surfaces, Aeron. Res. Comm. Rep. 
 a. Mem. N. 1381, London 1930. 
 
 3. Gortler,H.: Zxim tTbergang von Unterschall-zu 
 tfberschallgeschwindigkeiten in Diisen, 
 Z.angew.Math.Mech.19 (1939), pp. 325 -337. 
 
 {^) 4. Oswatitsch, Kl. and Rothstein, ¥. : Das Stroraungsfeld 
 in einer Lavaldiise, Jahrbuch 19^2 der deutschen 
 Luftfahrtforschung, I 91-102. 
 
 5. Oswatitsch, Kl. and Wieghardt, K. : Theoretische 
 
 Untersuchungen Uber stationare PotentialstrSmungen 
 ■bei hohen Geschwindigkeiten (noch nlcht 
 verbffentlicht) . 
 
 L^) 
 
 5^_.^ 1AI^^ /14. F-Tt^- /U^/i'A/J),f'^^ 
 
•MCA TM No: 1147 
 18 TABLE 
 
 COMPARISON OF APPROXIMATION PORIvRJLAS (20(a)) AND (20(b)) 
 
 WITH THE NOZZLE FLQ^S COMPUTED SI V.L. OSvVATITSCH . 
 
 ■; -i- '■ ■■'and W. HOTIISTEIN 
 
 " 1 
 
 .• 
 
 10 
 
 Pg/Vg = 5 
 
 10/3 
 
 a flatp ■ 
 
 0.20 (0.20)' 
 
 , 0.2>- (0.27) 
 
 0.35 (0.32) 
 
 'a fi^r 
 
 I Tube _ , 
 c round 
 
 &..22]' 
 
 ■'O.29 (b-..28) 
 
 ]0.32J 
 
 0.I4.1 (■6.'.57> 
 
 lc.39] 
 
 •.. 0.30 (0.[:2) 
 
 i 
 1 
 
 a rouii3l 
 _ Tubs J 
 
 
 
 c.5'2' 
 
 t^^l' 
 
 ■• '-■^■^■.53! 
 
 e flat 
 
 C.08 (O.Oo) 
 
 0,12 (0.12) 
 
 o.iU (0.16) 
 
 e round 
 
 0.09 (0.10) 
 
 0.12 (0.li|) 
 
 o.ili (0.18) 
 
 j 
 
 T] flat 
 
 0.16 (0.16) 
 
 t).25 (0.23) 
 
 i 
 
 0.20 (C.23) ! 
 
 T] roiuid 
 
 1 
 
 0.09 (o.cS) 
 , 
 
 0.12 (0.12) 
 
 O.lii (0.ll|) 
 
 The unbracketed nurabsrs have been computed with approxi- 
 mation formulas (20(a)) and (20(b)), the niJinbers in 
 curved brackets taken from figures 7 and 10 of the 
 paper by Kl. Oswatitsch ajnd Vv'. Roth.stt;in Hi), 'I^ae 
 square bracketed numbers are from formulas (21(a)) 
 and (21(b)) for the flov.'-tube approximation. 
 
NACA TM No. 1147 
 
 Figs. 1,2 
 
 Figure 1. Illustration of terms. 
 
 Figure 2. Comparison of a flat and a round nozzle. 
 
Figs. 3,4 
 
 NACA TM No. 1147 
 
 Figxire 3. Profile flow at transit through the critical velocity. 
 
 / 
 
 / 
 
 \ 
 
 \ 
 
 / \ 
 
 i supersonic ^ 
 
 / \ 
 
 Figure 4. Profile with supersonic region. 
 
UNIVERSITY OF FLORIDA 
 
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