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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1147 GENERAL CHARACTERTSTICS3 OF THE FLOIl THHOUCH U03ZLE3 AT iEAR CRITICAL SPEEDS' .3y R. Sauer The characteristics of the position and form of the transition surface tirou.gh the critical velocity are computed for flow thrUC.. fl:t and round nozzles from subsonic to supersonic velocity. Corresponding con siderations were carried out for the flow about profiles in the vicinity of sonic ,velocity. I. INrROTDUCTIOi; 'rhile useful retho.ds of calculation exist for pure subsonic :ndl for pure sunersor.ic flows of comouressible gasez difficultie arise in th i' athematical treatment of mi:edi flows with 3ubtonic and sup.ersonic raies.. Such mixed flows exist: (1) in the tira!.sition from subsonic to sutpersonic speeds in Lavai nozz .s, (2) about rocfiles in a flow at high subsonic speed ,:. the apple. ar:nce of local sup,ronic ranges, (5) in froit of blunt bodies in a flow &t supersonic 'velocit' witih a locel subsonic regicn behind the comnorssioa shcci: in the vicinity of the stagnation point. The present report taes vui the firs; as the simplest of the three problems nar.;ed. It is a fact that for this case a rough view of flov conditions suffi cient for many problems cen be obtained by che methods of hydraulics, that is, the nozzle is considered a flow tube and the magnitude of the velocity is considered "Allgemeine Eigenschaftan der Stromung durch Disen in der Wfho der kritischen Geschwindigkeit," FB 1992, Zentrale fur wissenschaftliches Berichtswesen der Luftfahrtforschung des Generalluftzu&mnistere (ZWB) Berlin Adlorshof, Sept. 25, 1944. NACA TM No. 1147 constant in every crosssectional plane. The critical velocity (flow velocity: = sonic velocity) is then reached exactly in the minimum cross section of the Laval nozzle. In the strict twodimensional treatment, however, 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. Gbrtler (3), and Kl. Oswatitsch and H. Rothstein (4), expansions in power series for the determination of the critical curves are given. In what follows, general statements on the position and curvature of the critical curve for a given nozzle are obtained from such expansions in power series by breaking off the series after the first two terms. The investigations are concerned with 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 with curved axes. In this way, information is obtained on the variation of the critical curve for profile flows with local supersonic regions. II. POTENTIAL EQUATION The nozzle axis is selected as the xaxis 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.) With the Lhyothesis of non vortical flow and perfect flow of a perfect gas with constant specific heats c c ( = c /c ) the potential equation F (a2 2) + (2 2) 2 7 +.oa2v= 0 (1) NACA TM No. 1147 3 holds. In this u, v are the x and ycomponents of the flow velocity and a, the local sonic velocity, which is related to Li, v and the critical velocity a' through a2 = a a ku + v (2) S= 0 cha:acterizes CL.e flat or twodimensional flow, 0 = 1 the threedimensional flow through stream tubes .of of circular cros section; x, y are cartesian for a =.0, cylindrical coor'linates for a = 1i Substitutin. (2) i" equation (1) and introducing the di:arensriories velocity co.oponents l  a', V = /a (3) the .~'~ ential eauatio. (1) bco,.e; UU2 V) U '1 ;2 + 1l U2 V x: + + ..Luv r 1i _1 2+ v .=0 (,) k + 1 I:+ .v Since the rrezent investigations w.as litrited to the vicintty of the critical curve, U = 1 + u, V = ) in which u an.I v are small quantities, and equation (.) becomes S: + + T + S U k ( + u) v  (k 1) K 't + v  2(k u k + i (1 j j (6) S+1 + NACA TM No. 1147 As xO0, >0, u,. and v approach zero and also on the basis of (6) 6v/,y and,consequently, the quotient v/y approach zero, while the velocity rise 6u/bx along the axis ought not.vanish at the origin. Then if the small quantities u, v, 3 v/6y, and v/y are considered linear only, the following approximate relation is obtained from (6) I I (k + 1) u u + 2v 0 (7) (i Ys v o o_ (7) Since the nozzle flow is symmetrical with respect to the xaxis, 6u/6y also approaches zero for x>O, y>0. Consequently the term 2v 6u/6y may be ignored, by means of which (7) becomes further simplified to ("^ "l0^ 8 b u 6v v (k + 1) u 6 = 0o (8) dith a = 0, (8) was already applied in another con nection by Kl. Oswatitsch and K. Wieghardt (5). III. FLAT NOZZLES Equation (8) furnishes general relations, for a = 0, for the flow through flat nozzles in the vicinity of the critical curve. Like Kl. Oswatitsch and W. Rothstein (4), taking into account the symmetry around the xaxis, setting = f0(x) + y2f2(x) + y f (x) +... u = x= f'o(x) + y2f'2(x) + y4f'4(x) ... (9) v = __ = 2yf2(x) + 4y5f(x) Ir(' 6y 2 NACA TM No. 1147 in which the primes signify derivatives with respect to x, and, obtain from (8). (k + I) (f' + y2f' + () (f"0 =0 2!V = 2f2 + 12y2f ... + 2 (f + By arranging in order of powers of y coefficients of the individual powers system. of equations is obtained + 2 ^ 2 + y2f"2 +.. 22 A 2 3 4,.. 2yf4 +..1 and equating to zero this 2(1 + c) f = (k + 1) f' f" 2 0 0 (10) 2(6 + 2o*)f, = (k + 1) (p f" + fo f, ) ... 0 2 0 2 / through which the functions f2(x), f (x)... in order 4C can be expressed by the function f' (x) and its derivatives. The function f' = uox) which characterizes the velocity distribution along the nozzle axis, remains undeterrAined. Given u ( ) the flova is completely established through (10) and can be understood to be nozzle flow, if a pair of streamlines that are mirror imamres of one another with respect to the xaxis are assumed as nozzle walls. For further discussion, setting f'o = uo(x) = a x (11) 0 o from which k + 2x, 2 2 2 S (k + 1)2 5a f 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. (12(a)) NACA TM No. 1147 follows immediately from (10) with 0 = 0. According to (9) Sk + 1 2 2 u = ax+ 2 2y2 + ... 2 (15(a)) v = (k + 1) a2xy + (k + 1) a3y5 + . is obtained for the components of the flow velocity. The dots represent terms of higher order of y. If (11) is taken as the first term of a power series expansion for uo(x), the expressicnsfor f2 and f4 in (12(a)) are also the first terms of power series expansions. From (15(a)) the critical curve is obtained from the requirement 2 2 (1 + u) v = 1 from which in (13(a)) the parabola x k + 1 ayK2 (14(a)) is obtained, therefore, with u = 0 as a first approximation. It cuts the nozzle axis, normally, at the point x = y = 0 and its curvature there is 1/pK = (k + 1) a (15(a)) The vertices, therefore the points, of the streamlines, adjacent to the nozzle axis, are given by v = 0 and, according to (13(a)), lie on the curve Sk 1 ay2 (16(a)) Considering, now, the streamlines at a given vertex distance yS (fig. 1) from the nozzle wall, then E =k + 1 2 (17(a)) E = "xS "S(1 Ys NACA TM No. 1147 7 is the distance from the center point of. the narrowest cross section downstream to the intersection point of the naozzle axis and the critical curve. The junction point of the critical curve with the nozzle wall is upstream and is obtained, as a first approximation from (14(a)) with yK = yS, which yields S = a2 = 2y (18(a)) for the distance r from 1the junction point to the narrowest cross section. Finally the ca L.:ulIs.tion of the curvature at the vertex of the nozzle wall is to o' i..:dc. In moving outward from thi vertex. to the wall by an element of curve: dI = dx, the tangent turns through the angle (.d v 1 6v 1 .iv + u v = ds 1 + u 1 + + u oX Therefore the curvature at the vertex is 1 d 1 v PS s~ 1 + u oX and taking into account (15(a)) ani (15(a)) 1/pS = (k + 1) a2y = l/PKaS (19(a)) Up to now the velocity distributiDn uo(x) had been considered as Liven by equation (11) and the nozzle walls computed for it. Practically, tie reverse is the rrobl ir:., unely to asc'rtain for a particular nozzle, that is f'or assigned valu.S f y,, PS, the flo: in the norrowust cross sct: n, ther.fore the NACA TM No. 1147 values a, c, 1T, and pi. On the basis of the equations that have preceded it Ts possible to write down the solution to the problem, immediately, namely n W t (k + 1)pSyS i PSYS PK = k + 1 (20(a)) n = 26 = (k + 1) 13 PS If the flow is in the positive direction of the xaxis, as in figure 1, then the subsonic region is to the left and the supersonic region to the right and, accord ingly, a> 0. In practice it is recommended that all distances be expressed, dimensionless, in terms of yS, and accordingly set yS = 1 throughout (20(a)). IV. ROUND NOZZLES Corresponding relations for round nozzles come from (9) and (10) with a = 1. retainingg (11) for the velocity distribution u (x) along the nozzle 0 axis, the following relations are obtained in place of (12(a)) through (20(a)): f2 = 1 2x f2 = u x, (12(b)) (k + 1)2 f = 64 a INACA TM No. 1147 i1 k + 1 2 ( + 1) .. v = +2 + 0 , k +1 2 xK = a4 K 1 x =  '" + 1 a k+1   +1  . J =. +1 i2 : "(;;= : ) = (': + 1) 7 T2 : c 1 k+1+ 1 p 2 a' .= y (i(1j)) (1Si:h) (1i~) NACA TM No. 1147 f  2 = (k + l)p y 1 PS s I 1 A 2(k + 1) I S To compare a flat nozzle with a round one of the same longitudinal section, formulas (20(a)) and (20(b)) are compared. (See fig. 2.) For the flat nozzle T = 2E, for the round nozzle T) =. Moreover E round 3  C flat .\2 ~ 1.06 rn ound 3 0.5 '0 flat \ 0 a round i = o.41 flat  The velocity increase is, therefore, around 40 percent larger for the round nozzle than for the flat nozzle. All of these results are fully confirmed by the nozzle flows calculated by Kl. Oswatitsch and "'. Rothstein, as the enclosed table on page 18 shows. PK I. S2pSyS k + 1 (2C(b)) YS 8 NACA TM No. 1147 1: V. C,(1 i.T:0 T FLO'iT'U ..'..: T..PC Th sutho2'.3 rC .s ilt e .. :, ..e r.,;".O w i .t. thr f o1'0 ' b'.b? ar,.:.' oxi * .[.l" 9 ,;.,";,." .. , .;,.'.?o: e t1 I? t nZ 'I in veloc t.; ct icr ,t ." ( " )) .: ( by he foe f low b eci .~'s r .=. .. ro. , Sden ity lon the ton ,'1 a:i in U.c vrc i.. o 'nLic narrov,/e L& cros s cto ... l ::c: r o"l'. "s e = . ij pe. = CO. t., ~ co,I t. '1 + 1 \ )"  1 + 1 221 :I. (1 + T) 1 . SI 1 (1 + u) 2uL 1  conas. (1 u.) Taking the velocit' d is.tri ';,1 i.n (].) 1 a ,.s.sf, for the ozze r ,.oz e w s .in bt.. ..1 t 1. ,'.rro...t cros2 section tlihe 1.'II ;.n '.. oj'c.i.r i n ..l c? o' a flat nozTlt 2/ ). 2= 1 7 1k u = c0,1 = V = i' .r L ' S. 2 = 1 + o: and. in the c.s of r'. rof ;'. n i ziz'. z. 1 ' u = c on:. = 7 V .= .1. Fi'otil thi.s, th curvature at the vc.'t.x 1. 1/p, = .* " and 1/p., = a2:' ., s + X respectively, NACA TM No. 1147 and in place of the first equations of (20(a)) and (20(b)) 1 a V 2pS my a 0  (21(a)) (21(b)) The comparison of (20(a),20(b)) and (21(a),21(b)) shows that the velocity increase a in the flowtube approxi mation for both flat and round nozzles is too large by a factor k + 1, that is for k = 1.! about 10 percent. The values for the flowtube aproximation are added to the table, pnge 18, in square brackets. VI. FLOV ABOUT PROFILES Through similar powerseries expansions the flow past a profile in the vicinity of transit through the critical velocity can be discussed. If a streamline adjacent to the profile is thought of a rigid wall, the flow ;* also be explained as flow through a nozzle with a curved axis. The origin of the coordinate system is located at that point of the profile at which the critical velocity is just reached, the xaxis downstream on the profile tangent and the yaxis normal to the profile, outward. (See fig. 3.) Since the flow is no longer symmetrical to the xaxis, instead of (8) the rather complicated equation (7) which must be specified with G = 0, by all means, is taken as a basis and in place of (9) S= f0(x) + yfl(x) + y2f2(x) + y3f (x) +... u = x= f'o(x) + yf' (x) + y22(x) + yf3(x) +... (22) v Q fl(x) + 2yf2(x) + 5y2fj x) +... 6y 2 2 NACA TM No. 1147 13 3,y putting (22) in (7) (k + l) (f + yf'+... (f0 + yf ..) = (2 + ,f2 +.. + yf2+...), + 2',, ) and by comparing coefficients k+l if + ____ 0 (23) f = ( + 1) (f f + r" f' + ( f + f' 0 0 1f while )nly the f unction f'o(x) haJ renmainel arbitrary in the n)zzle flow, here there are tv..o arbitrary functions f'0(x), fl(x). :.ith f'o(x) and fl(x) the remaining coe'fficient tf~nctio)ns f2(x), f(x),... in order may be c'omp.uted fron1 ..he systemi of equations (25), by which the flow 1i completely .established. Similar to (11) sot f0 = ax, fl = PX (24) and obtain fror (25) f2 = a + 7 k + 1 a. 4P 3 (+ 1 2 +p2x S6 3 2 and from (22) NACA TM No. 1147 u = ax Py + a + 2... v = x + k + 1)a2 + 2 'xy (25) + r (' + 1)1 2 22 x ! 6u Here a signifies tile velocity increase T along the profile on transit through the critical velocity; it is therefore positive at the starting point A and negative at the terminal point B (fig. 4) of a local supersonic region. ITe profile curvature is given with p > O at the point A or B as 1  P... p x=y=0 (26) For the slope of the critical curve with respect to the profile tangent with (1 + u)2 + v2 2u = 0 from (25) the relation (27) tan ~ p x P I ~ ___~~I 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. 1147 15 For the slope of the streamline at a distance y fror the profile I 1 1 iv ~ r ,2L 2 y 1k + 1_1 a2 + 2P yI S sL J Z [(k + 1) a2 + P2] y is obtained and taking (26) into consideration S i(k + 1) a2+ + y PS PD I PP 2 SL (28) I p: P + Y 1 + (k + 1) a2 21 PS P P according to (27) thc critic tl curve at A (a > 0) srd B (a < 0) is steeper the larger the velocity increase or decre.c. anud the less the profile is curved?. By (231 in the limiting case of a = 0 the circle of curva'tur.e at S is concentric with the circle of curvatu'e of' the profilee and, as is already known from the flowtube approximation, becomes still flatter with increasing a. VII. 3SU1J.ATcY Approximation formulas were developed for the position and curvature of the critical curve for transition through the critical velocity in the neighborhood of the narrowest cross section of flat and round Laval nozzles. In comparison with 16 NACA TM.No. 1147 nozzle flows calculated by Kl. Oswatitsch and W. Rothstein (it) they showed satisfactory agreement. In addition, corresponding approximation formulas were deduced for the flow at profiles with local super sonic regions. Translated by Dave Feingold National Advisory Committee for Aeronautics NACA TM No. 1147 VIII. REFERENCES 1. Meyer, Th.: Uber zweidimensionale Bewegungsvorgdnge eines Gases, das mit Uberschallgeschwindigkeit strLmt, Diss. Gbttingen 1907. 2. Taylor,G.J.: The flow of air at high speeds past curved surfaces, Aeron. Res. Comm. Rep. a.Mem.N.1381, London 1930. 3. G'rtlerH.: Zum tjbergang von Unterschallzu Uberschallgeschwindigkeiten in Disen, Z.angew.Math.Mech.19 (1939), pp.325337. 4. Oswatitsch,Kl. and Rothstein,V.: Das Stromungsfeld in einer Lavaldise, Jahrbuch 1942 der deutschen Luftfahrtforschung, I 91102. 5. OswatitschKl. and Vieghardt, K.: Theoretische Untersuchungen iber stationare Potentialstr6mungen bei hohen Geschwindigkeiten (noch nicht ver6ffentlicht). A6 N 7? ... .5,D ,1 18 TABLE ACA TM No. 1147 18 TABLE COMPARISON OF APPROXIMATION FORMULA. (20(a)) AiD (20(h)) WITH THE NOZZLE FLOWS COA;';'UT. 37 EL. COS,'ATIYTCH AND W. r 'OT '..1 . 0 r 10 i 10/3 a flat I 0.20 (0.2"') in fl.. t1 I !O. 2 L Iute _ a ru.mCd G.29 (C.28) Ia ro ui I i' " Tubj I '.3l C flat c.o8 (0.03) E round 0.09 (0.10) fI flat 0.16 (0.16) rn round 0.09 (O.CS) xI .; 0. ; ( .,7) 0.o 5 ((. 41 (0.37) V c.530 (o. I..  1_  4  0.12 (0.12) 0.12 (0.12) ).25 (0.25) 0.12 (0.12) 32) w The unbracketed numbers have been computed with approxi mation formulas (20(a)) and (20(b)), the numbers in curved brackets taken from figures 7 anrd 10 of thJ paper by Kl. Oswatitsch and W. Rothstoin ()4). The square bracketed numbers are from formulas (21(a)) and (21(b)) for the flowtube approxima.tion. 0.lL (0..16) o.ilL. (0.i38) 0.28 (0.25) 0.14 (0.14) ___ ~ _I~ _I ~ I NACA TM No. 1147  criLical curve Figure 1. Illustration of terms. flat nozzle ....... round nozzle Figure 2. Comparison of a flat and a round nozzle. Figs. 1,2 Figs. 3,4  critical curve Figure 3. Profile flow at transit through the critical velocity. Figure 4. Profile with supersonic region. NACA TM No. 1147 UNIVtH5I Y UP 2LUHIUA 3 1262 08106 299 3 
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