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'7 / C, 7 C' ' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDIM 1236 A CLASS OF de LAVAL NOZZLES* By S. V. Falkovich A study is made herein of the irrotational adiabatic 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 planeparallel 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. Christianovich (reference 1). 1. Fundamental equations. A plane, steady, irrotational, adiabatic flow of a gas is considered. In this case, the equations of motion, as is known, have the form OPU ov 0 + = 2 p av au K1 PO 0 where p density of gas u,v components of velocity along x and yaxes p pressure W absolute value of velocity K adiabatic exponent Subscript 0 denotes condition of gas at rest. (1.1) (1.2) *"Institut Mekhaniki Akademli Nauk Siuza, SSR" Prikladnala Matematika I Mekhanika, Tom XI, 1947. NACA TM 1236 From equations (1.1) there exist two functions, the velocity potential cp(x,y) and the stream function 4(x,y), determined by the equations dP = u dx + v dy If u = W cos 8 and d* = p (v dx + u dy) 0 v = W sin 8, whe2 between the velocity vector and the xaxis, equations (1.3) and are solved for dx and Sco dP! sin d w P W sin W (1.3) ae 6 is the angle are substituted in dy, 0 cos 0 p w (1.4) The concepts x and y and variables W and 0; then 4p and are functions of the dq = dW + d dI = dW 4 aw a* d ae If these expressions are substituted tions (1.4), dx = C aP dW + \w VT vW aw dy = sin e a \w aw p cos 0 PO v dW aw/ for d0P and d4I in equa cos acP O sin 0 8 \ w p W 80/ (sin 0 w aq' ae cos 0 W aeKj (1.5) In order for dx and dy, determined from equations (1.5), to be total differentials, it is necessary and sufficient that the fol lowing qualities hold: \ NACA TM 1236 6P_ P0 sin a\ /COB arp 0 sin e wp w w \w p W 6C PO cos e 6a w + p W X) S/ine a P cos e = \ w e p W awN (1.6) aw 5e 89 From equation (1.2) and the condition of adiabatic flow, d /PO\ PO W W C P a2 where a velocity of sound If the differentiation In equations (1.6) is carried out, 6cP PO (1.7) = (1M2)1 W i w Te where M = W/a is the Mach number. In equations (1.7), as the independent variable, in place of the velocity W a new variable a (reference 1) is introduced, which is connected with W by the relation  a Jv W2 dW aW Equations (1.7) then assume the form  = ao where K(s) = 2 (1M2) inasmuch as equation (1.8) is a function of s. a (cos s  (ine ae W w a U a0' (1.9) (1.10) (1.8) NACA TM 1236 The system of equations (1.9) in the regions where the flow velocity is subsonic, W < a, is of the elliptic types Hence, any solution qC= cp(e,s) and = *(e,s) of equations (1.9) rep resents two functions analytic in the variables s and e up to the line of transition to the region of supersonic velocities. If the Jacobian of the transformation of the region in the plane e,8 on the plane qP,% does not become zero over a certain segment of the transition line and therefore also in a certain region on the supersonic side, the functions ((e,s) and 1(6,s) may be analytically continued across this segment into the supersonic region of the flow. Hence, the flow in the subsonic region determines the flow in the supersonic region near the transition line. Equations (1.9) retain their meaning also for supersonic velocities. For, if W > a, s will be purely imaginary and K negative. .Setting s = ii and K = K in equations (1.9) gives Thus, if in the solution P(e,s) and L(K,s) of equations (1.9) determining the flow in the subsonic region, s is set equal to iT and the real parts of the expressions thus obtained are taken, the solution of equations (1.11) 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 V = aE. From equation (1.8), it follows that in the plane of the variables 6,s the upper halfplane s > 0 will correspond to the region of subsonic velocity and the axis of the abscissa s = 0 to the line of sound velocity. It is therefore necessary to consider the behavior of the function K(s) for small values of the variable a. Equation (1.2) is represented in the form W2 (+l)a*2 M42 (2.1) 2+(61) M2 NACA TM 1236 Substituting this value of W in equation (1.8) gives (h2l)t2 dt (t2)(h2t2) h2 2 +1) K1i Computing the integral gives 1 5 = log 2 [/(h l (2.2) If the equation is expanded in a power series in t, h 21 3 3h2 h41 t5 +5 5h4 h61 t7 7h6 t ala1/3 + a3e/3 + age5/3 + . (2.3) S3h al h 1 Further, from equation (1.2) and the adiabatic simple transformations, PO0 P 1 P 0. ( condition after 1 21 h2 K  Substituting this value in expression (1.10) gives for K(s) 1 (2.4) (2.5) = rt 0 I Then, where + . (t = 12 , NACA TM 1236 Substituting in equation (2.5) the series for t (equation 2.3) gives bsl/ b 81/ 3 + b953s/3 + 8. +(b = f h ) 1 (a) 13 5 5/5 1 1 (2.6) It follows from equation (2.6) that for a velocity near that of sound, that is, for small s, consideration may be restricted in series (2.6) to the first term by setting 4TK blsl/3. The fundamental equations (1.9) then assume the form = b~1/s a C b 81/3 s (2.7) whence for the functions *(6,s) and p(e,s) there is obtained + +0 L =a 0O (2.8) 2 2 39 a ae2 2 39 se Having determined from equations (2.8) the velocity potential q(9,s) and the stream function 4(0,s), by equations (1.5) 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 dz = 1 sc e o 6 sin e ds + cos sin d w as P as C ae P Lae j SP 9 de be (si a ] dy = sin + cos e ds + in e +  cose 6 de as P a ae P ae/ (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 TM 1236 instead of the relation (1.8) between a 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 P from equations (2.7) after which the condition that dx from equations (2.9) is a total differential assumes the form a 6 1/3co e 0 a ble rcos e sin 1 *e 6_\1e 0 p s = le /cos 9 c+ in 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 total differential): P 1/3 d 1/3 .1 d PW 1 d \W da 2 3/2 Setting = 3/2 gives P= ()1/ b b) 1 d 1/3b d P (2.10) Thus, the equation determining 1/W is obtained, namely, d2 1 0 (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 = C1 u(j) + C2 v(I) (2.12) NACA TM 1236 where u(Ti) and v(rT) are two linearly independent tabulated integral of the equation (2.11). The constants of integration are determined from the conditions (ii) 1 0 =j* (lda =(j)1/3 2( PO a7 blp*a* 4 where the latter relation is obtained from equations (2.6). After computation, 1 = [yv(O) v'(O) a* (2.3) ld (2.10) and 2 [u',(o) yu(0)] (2.13) where from reference 5, u(0) + iv(0) = u'(0) + iv'(0) = 2 Ahf in exp  34/3 (2/3) 2 41e in 4/3r(4/3)exp 4/3 exp 3 r(4/3) From the first of equations (2.10) and (2.12), (21/3 PO P c u'(T() + C v' () bl + C2 ClU(r) + C2v(?) The expressions (2.12) and (2.14) thus found for must be substituted in equations (2.9). (2.14) 1/W and P/P 3. 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 WO near the velocity NACA TM 1236 of sound. It was necessary that the x) have a horizontal asymptote nitude H being determined from the velocity W0. walls of the nozzle for y =B*H (fig. 1), the mag amount of gas flow and the In the plane of the variables 8,s, there corresponds to the infinitely distant section in which the velocity is constant and equal to WO, the point A with coordinates 8 = 0, a = s0. To the lines of flow there correspond in the e,s 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 e,s bipolar coordinates are introduced (fig. 2). 2 2 9 +(e+so) = o 82+( 0o)2 Thus the lines with centers on the a family of circles point A (fig. 2). are, respectively, 2s08 S= are tg e22 2 +280 2 (3.1) a = constant constitute a family of circles aaxis and the lines 0 = constant constitute with centers on the 8axis passing through the The equations of the families of these circles e2+(s+s0 cth a)2 0= 2 sho. (es0 ctg p)2 + s2 802 sin2 P The first of equations (2.8) transformed into bipolar coordinates has the form 2 _+ 1 (l+ch a cos P)L + sh a sin = 2 2 3 sh a(ch a+cos B) B + (3.2 W= (ch a + cos p) /6 (ap) (3.3 Equation (3.2) is reduced to the form 2% 6M.2 S. cth aC 1 C)2 3 i! + 6 X 0 ) (3.4) in which the variables are separable. NACA TM 1236 In seeking a solution of equation (3.4) of the form X= X(a) Y(B), the ordinary equations for X(a) and Y(P) are obtained: d2Y 2 d2y + n2Y = 0 2 do d2X cth a dX + d + 3 d+ 36 da2 3 a. 36 n2)X = 0 where n is an arbitrary constant. If n is set equal to d2X cth a 2 3 da dX 1+ x= da 31= do. 36 The first of equations (3.5) gives Y = C1 + C20 and the second, by the substitution t = ch a, reduces to the hypergeometric equation (3.6) t(lt) 1d t) dX 1 X = 0 2 2 6 dt 144 dt the general integral of which can be represented in the form X t1/12 F 7 1 1 i 12' i, + C4F ( 72 1 , If C1 = C4 = 0 and considering equation (3.3), the required solu tion of equation (3.2) is in the form 1/6 S ch a. + cos 60 = ch a F 1 1 7F 12 12 mh2 V" i" ch" a/ Returning to the initial variables e and s according to equations (3.1), after simple transformations d2Y 0 2 do (3.5) (3.7) NACA TM 1236 S/ 2800 6 /, e2 +(sF 0 2+(s+so)22 "; 2 1 2 l\2 / 12 l (e2.82+802)2 2s0e tan'l 290 02tG2_B2 (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. Frankly (reference 2) (see also S. A. Christianovich, reference 3, ch. V.), that the stream function 1(8,s) have on the transition line the form *(e,o) = A1e/3 + A9 3/3 + Aso5/3 c A5B  . (3.9) The solution (3.8) does not, however, satisfy this condition inasmuch as on the transition line (a = 0) it has the form (eo0) = ( 1/6 F 1,, 1 t n 21 so 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 fying the condition *i(e,0) = e /3. Such a solution, as is shown in reference 4, has the form 2 = s 2/3 A3*5 + 3AT1 3 = 0 whence setting A = 3/3, we obtain whence setting A = 3 we obtain (1 / ( 6 2 + 2 + (2) (3.10) NACA 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 4(e,s) to assume the form (e0,s) = AO 0 + A11 + A33 (13 = 0) (3.11) where AO, Al, and A3 are arbitrary constants and the functions 0 and *1 are given by equations (3.8) and (3.10). Equation (3.11) for *(e,s) is obtained if in the expansion (3.9) the first two terms are retained. With the values of the constants A0, A1, and A., a nozzle may be constructed sufficiently near the given nozzle of the shape under consideration. The equations determining the functions 40 and *1 in the supersonic region will be determined. In the expression (3.8) for a = 0, the argument of the hypergeometric function attains the value unity and for the supersonic velocities, that is, for imagi nary a, 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(cab) F (a,b,a+bc+l, lt) + r(ca) r(cb) r(c) r(a+bc) (lt)cab F (cb, ca, cab+l, 1t) r(a) r(b) which in the case considered has the form 1 7 ( 1 7 2 , S 12' = (11/12) (5/12) 2 12' 3 r(l/3) 1/3 /5 11 4 t) rpl7) y 71zy(lt) F , , 1t (1/12 /12) 12 3 and the characteristic coordinates e= 9 is and p = e + is are determined. After computations, the following equations are obtained: NACA TM 1236 / 2 /6 2sP7o / "4 s~ r(1/3) (11/12) r(5/12) r ri/3 r so(WX) 2/3 F r r(/1) XV92 ) r(1/12) 7/12) 2 Ap4e0 / 5 11 4 12 l2 S 2 (A1+82) 2 are tg 2s(+) 2 o98 l (=,7) 1(i /3 (3qTx+,g+ q4XF) (3.12) (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. Christianovich, S. A.: Subsonic Velocities. Flow of a Gas about a Body for Large Rep. No. 481, CAHI, 1940. 2. Frankl, F. I.: On the Theory of the Laval Nozzle. Izvestia Akademii Nauk SSSR, Ser. Matematicheskaya, vol. IX, 1945. 3. Christianovich, S. A.: On Supersonic Gas Flows. Rep. No. 543, CAHI, 1941. 4. Falkovich, S. V.: On the Theory of the Laval Nozzle. NACA TM 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 Macmillan Co. (New York), 1943. 1 7 2 12' 12'3 ( 0 o ) 2 (P )2 ~o ( NACA TM 1256 Figure 1. NACA TM 1236 0 = constant o Figure 2. MACALangley 10549 950 *+ <( 0 a .1 mi  > 0ID 7 U 0 O , P 0. 1 'I I oCI S4 0 * 0 i 0 O 00 + o _o 4 o a) 10 .034m> > m c..0" r r. ) 3 ) 0 C0 S0r rl 4 N ie 44 on rl Oo 0 4 0r4 0 C 1 ONOO 14 r 4 0 0 4 0P00m0 o' 4+a M gi r4 14 ILId. m 1 *1 0 q1 4o CD a) o 0 Sq 0 4 00004 rd WCD 0 34 0 d 03 00 CD 0 4d 34 1e a< go+o i id E3 o 34P 00 0r 00 o 0 a "4P 0"4.0 0 Wri 0. 01 01 a tm 00 ID O Cr g>@m~ee UNIVERSITY OF FLORIDA II I 1 I 11111111111 I I I 31262 081058264 