PMr^^i?.3y, SMLLEADOM y NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1236 A CLASS OF de LAVAL NOZZLES By S. V. Falkovich Translation 'Institut Mekhaniki Akademii Nauk Siuza, SSR" Prikladnaia Matematika i Mekhanika, Tom XI, 1947. Washington October 1949 DOCUMENTS DEPARTMENT -7 10 (51 o^ "^'^^ NATIOML ADVISORY COMMITTEE FOR AERONAUTICS TECMICAL MEMORANDUM 1236 A CLASS OF de LAVAL NOZZLES* By S. V. Fallcovich A study is meuie herein of the Irrotational adlabatlc motion of a gas in the transition from subsonic to supersonic velocities. A shape of the de Laval nozzle is given, which transforms a homoge- neous plane-parallel flow at large subsonic velocity into a super- sonic flow without any shock waves beyond the transition line from the subsonic to the supersonic regions of flow. The method of solution is based on integration near the transition line of the gas equations of motion in the form Investigated by S. A. Christiauaovich (reference 1). 1« Fundamental equations. - A plane, steady, irrotational, adlabatlc flow of a gaa is considered. In this case, the equations of motion, as is known, have the form W^ _K_ p K Pq 2 K-1 p " K-l P where p density of gas u,v components of velocity along i- and y-axes p pressure W absolute value of velocity K adlabatlc exponent Subscript denotes condition of gas at rest. (1.2) ♦"Institut Mekhaniki Akademii Nauk Siuza, SSR" Prlkladnala Matematika 1 Mekhanika, Tom XI, 1947. KACA TM 1236 From equations (l.l) there exiat two functions, the velocity potential cp (i,y) and the stream function \i'(x,y), determined by the equations dtp = u dx + V dy d^|/ = — (-v dx + u dy) (1.3) If u = W cos e and v = W sin 6, where 6 is the angle between the velocity vector and the x-axis, are substituted in equations (1.3) and are solved for dx and dy. cos_0 ^^ _ ^ 8i£_e ^^ ^y ^ si£_0 ^ ^ ^ COO a>lr (1.4) "^^ " w ""^ p w "' " w ■ P w The concepts x emd y and

a, s will be purely imaginary and K negative. .Setting s = i¥ and K = -K in equations (1.9) gives Thus, if in the solution ^6,b) and ^e,s) of equations (1.9) determining the flow in the subsonic region, s is set equal to is" euid the real parts of the expressions thus obtained are taken, the solution of equations (l.ll) determining the flow of the gas in the supersonic region near the transition line is obtained. 2, Investigation of equations (1.9) near the transonic line W «= a^^^ From equation (1.8), it follows that in the plane of the variables 6,b the upper half -plane s > will correspond to the region of subsonic velocity and the axis of the abscissa B = to the line of soimd velocity. It is therefore necessary to consider the behavior of the function K(s) for small values of the variable s . Equation (1.2) is represented in the form ^^iJCilla,^ (2.1) 2+(lt-l) M^ NACA TM 1236 Substituting this value of W In equation (1.8) gives t 8 = (l-t2)(h2.t2) V " K-1, (l-t")(h^-t^) Computing the integral gives 8 = - log /h-t\^ 1+t \h+tj 1-t If the equation is expanded in a power series In t. Then, 3 = ^ t^ . ^ t5 . ^ t^ . 3h^ 5h^ 7h^ (2.2) where 4. 1/3 3/3 5/3 t = a-j^s ' + ajs ' + ags ' + . , . 3| — T *1 " \/ 2 (2.3) h -1 Further, from equation (1.2) and the adiabatic condition after simple transformations, 1 p " p* V hV Substituting this value In expression (l.lO) gives for K(s) 1 (2.4) '^ = ^H"S .K-l (2.5) 6 KACA TM 1236 Substituting In equation (2.5) the series for t (equation 2.3) gives '\lK(r) ,,sl/' . .3=3/3 , ,^,5/3 .... (., = g ^ji^ (2.6) It follows from equation (2.6) that for a velocity near that of sound, that is, for small s, consideration may he restricted in series (2.6) to the first term hy setting 'y/lT « "bjB^f^, The fundamental equations (1.9) then assume the form 3^ - ^1^ Si- Sep _ -u _l/3 S\|/ S5- - ^1^ 5? (2.7) whence for the functions "^(6,8) and Cp(0,s) there is obtained (2.8) ^ + ^ + A. ^^ = ^2 V 2 38 ^ de OS S^ . S^cp 1 Sep _ Q ^2 -.2 3s OS Se Ss Having determined from equations (2.8) the velocity potential (0,s) and the stream function '^(G^b), by equations (l.S) the coordinates x and y are found in the flow plane. It is easily seen that equations (1.5) can be represented in the form ■^ dx = i W dy =i (cos e p \ OS sq> sin ^Us -H (cos ^- ^ sin 6 ^Ue S^' Se sqp Se. ,sin -r^ + -— COS e -— Ids + \sin e ^^ + -^ cos e -^^ Ide V Ss P Ss / \ Se p Sey""'_ (2.9) If, however, we pass from the exact equations (1.9) to the approxi- mate equations (2.7), the expressions (2.9) cease to be exact dif- ferentials. Hence, simultaneously with the passing from equa- tions (1.9) to equations (2.7) it is necessary to introduce NACA IN 1236 Instead of the relation (1.8) between s and W a new relation such that the expressions (2.9) remain exact differentials. In order to obtain this relation In equations (2.9) are substituted the values of the derivatives of the function Cp from equations (2.7) after which the condition that dx from equations (2.9) is a total differential assumes the form fb. s^/^cos e ^ . !2 sin e , ^\. a Ls^/^ose^^^slne^V Carrying out the differentiation and making use of the first of equations (2.8) satisfies this condition if the following equations are satisfied (the expression for dy likewise then becomes a toteJ. differential) : fo , 1/3 d PW 1^ ds V W 1/3 W 2 3/2 Setting s = =• T) gives 12 PW ^1/3 V _ J. fO W ' dri PW J:_ ^ 4 u (2.10) Thus, the equation determining l/W is obtained, namely, (2.11) The functions satisfying this equation are called Airy functions. Tables of these functions have been computed by V. A. Fock (reference 5). Thus J = Cj^ u(ti) + Cg v(ii) (2.12) NACA TM 1236 where u(ii) emd t(t)) eire two linearly independent tabulated integrals of the equation (2.11). The constants of integration are determined from the conditions \=o ^* "^^ \^/ti=0 V2y l^iP^a^^ a» \ " \l h2-l 2h where the latter relation is obtained from equations (2.10) and (2.6). After computation, where from reference 5, u(0) + iv(0) = .^^ exp ^ 3^/^r(2/3) u'(0) + iv'(O) =-f# «^p4^ 3'/'r(4/3) From the first of equations (2.10) and (2.12), Pq ( \1/7> C u'(ti) + C v'(ti) The expressions (2.12) and (2.14) thus found for l/W and Pq/p must be substituted in equations (2.9). 5. Determination of flow in feed part of de Laval nozzle. - The shape of the de Laval nozzle was determined such that the distribution of the velocity over the section tended, with increasing distance from the critical section upstream of the flow, to a uni- form flow with a certain subsonic velocity Wq near the velocity NACA TM 1236 of sound. It was necessary that the walls of the nozzle for x-> - » have a horizontal asymptote j = ±H (fig. 1), the mag- nitude H being determined from the amount of gas flow and the velocity Wq. In the plane of the variables 9,8, there corresponds to the Infinitely distant section in which the velocity is constant and equal to Wq, the point A with coordinates 0=0, s = Sq. To the lines of flow there correspond in the 9, a plane a bundle of curves issuing from the point A. In order to obtain a solution of equations (2.8) having the stated properties, in the upper half- plane of 0,8 bipolar coordinates are introduced (fig. 2). 2 2 +(s+s^) 2s„0 a=log^|— ^^— P = arc tg ^ (3.1) 02+(s-Sq)2 ©^s'^-Sq'^ Thus the lines a = constant constitute a family of circles with centers on the s-axis and the lines 3 = constant constitute a family of circles with centers on the 0-axis passing through the point A (fig. 2). The equations of the families of these circles are, respectively, 2 2 02+(3+Sn cth a)2 = -^ (0-a„ ctg 3)^ + s^ = ^0 ^ sh^a ^ sin2 p The first of equations (2.8) transformed into bipolar coordinates has the form 2 2 3 sh a(ch a+cos 3) (l+ch a cos P)-r— + sh a sin 3 -r— oa dp ;].. (3.2) l/6 \l/= (ch a + cos p) ' X(a,p) (3.3) Equation (3.2) is reduced to the form SSc o?X cth g Sy 1 ri t-Ti A\ dx^ S"p in which the variables are separable. 10 MCA TM 1236 In seeking a solution of equation (3.4) of the form X= X(a) Y(p), the ordinary equations for X(a) and Y(p) are obtained : ^ . n^Y = dp where n is an arbitrary constant. If n is set equal to 0, d^Y ^ Q dfx ^ cth g dX ^ J^ ^^ ^ q (3.5) 2 2 3 da 36 dp da The first of equations (3.5) gives Y = C, + CgP and the second, by the su equation 2 by the substitution t = ch a, reduces to the hyp ergeome trie td-t) ^ ^ (l - It] ^-_L.X = (3.6) 2 \2 6 / dt 144 the general integral of which can be represented in the form ^ = ^3* ^ \i2' H' ^' "tj + ^^4^ Vl2' 12' 2' * If C-]^ = C^ = and considering equation (3.3), the required solu- tion of equation (3.2) is in the form 1/6 Returning to the initial variables and s according to equations (3.1), after simple transformations NACA TM 1236 11 (3.8) This solution corresponds to the flow of gas in a nozzle having the shape shown in figure 1. Such flow cannot, however, be continued into the region of supersonic velocities. In order that a certain subsonic flow having a straight streamline may be continued into the supersonic region, it is necessary and sufficient, as has been shown by F. I. Frankl (reference 2) (see also S. A. Chrlstianovich, reference 3, ch. V.), that the stream function ^(e,s) have on the transition line the form Me,o) = A^e ' + A30 + A^e'^ + . . . (3.9) The solution (3.8) does not, however, satisfy this condition inasmuch as on the transition line (s = 0) it has the form In order to continue the flow with the stream function of equa- tion (3.8) into the supersonic region, it is necessary to add to equation (3.8) the solution of the first of equations (2.8) satis- 1/3 fying the condition ^-,(^,0) = 9 . Such a solution, as is shown in reference 4, has the form A'^^^^ + 3Ati\;/^ - 30 = 1/3 whence setting A = 3 , we obtain (3.10) ^ye -i\je^ + 12 MCA TM 1236 Hence, in order to construct the flow in the de Laval nozzle having the shape shown in figure 1, it is sufficient for the stream function \|/(0,s) to assume the form Me,B) = Aq\1/q + A^xl/^ + A3\^3 i% = G) (3.11) where Aq, Ai, and A, are arbitrary constants and the functions ^Q and ^j/-, are given by equations (3.8) and (3.10). Equation (3.11) for ^(0,s) is obtained if in the expansion (3.9) the first two terms are retained. With the values of the constants Aq, A^, and A_, a nozzle may be constructed sufficiently near the given nozzle of the shape under consideration. The equations determining the functions ^q and ^^ in the supersonic region will be determined. In the expression (3.8) for 3=0, the argument of the hypergeometric function attains the value unity and for the supersonic velocities, that is, for imagi- nary s, although remaining real, becomes greater than unity. The formula giving the analytic continuation of the hyper- geometric series (reference 6) is used F (a,b,c,t) = r(c) r(c-a- bl p (a,b,a+b-c+l, 1-t) + r(c-a) r(c-b) r(c) r(a^b-c) (i.^)C.a-b J. (^_^^ ^_^^ ^_^_^^^^ ^_^^ r(a) r(b) which in the case considered has the form ^ (l2' ^' ^' *) = r (11/12)^5/12) ^ \p' TZ' I' ^"7 ^ ^. 1-t' r(-l/3) (. t^^/^T,/^ 11 ^ r(i/i2)r(7/i2)^^'^^ F(— , — , -, £ind the characteristic coordinates X= 6 - is and |a = + is are determined. After computations, the following equations are obtained : ^ NACA TM 1236 13 ^i'O^^'t^)^ r(i/5) V (±±1 r(ll/12) r(5/12) Vl2'l2'3' 8o^(m-?0' r(-l/3) faoi^-'K) I r(7, \2/3 r(l/12) h7/12)U^^32 Vl2'l2'3' Al/3 / 3, arc tg 28q(X+h) Xn-s (3.12) ^'liK^)={^ (\jf^^^ +\j^-4^J (3.13) These expressions determine the flow in the regions 1 and 2 between the transition line and the characteristic passing through the center of the nozzle and directed upstream of the flow (fig. 1). The fur- ther computation of the supersonic part of the nozzle can be carried out by the method of characteristics. Translated by Samuel Reiss National Advisory Committee for Aeronautics. REFERENCES 1. Christlanovlch, S. A.: Flow of a Gas about a Body for Large Subsonic Velocities. Rep. No. 481, CAHI, 1940. 2. Frankl, F. I.: On the Theory of the Laval Nozzle. Izvestia Akademll Nauk SSSR, Ser. Matematicheskaya, vol. IX, 1945. 3. Christ ianovich, S. A.: On Supersonic Gas Flows. Rep. No. 543, CAHI, 1941. 4. Falkovich, S. V. : On the Theory of the Laval Nozzle. NACA TI^ 1212, 1949. 5. Fock, V. A.: Tables of Airy Functions. Nil, No. 108, NKEP, 1946. 6. Whittaker, E. T., and Watson, G. N. : Modern Analysis. The Macmlllan Co. (New York), 1943. 14 NACA TM 1236 Figure 1, NACA TM 1236 15 a = constant 3 = constant Figure 2. 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