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'I2 2 1i t7 3.c/.7 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1212 ON THE THEORY OF THE LAVAL NOZZLE* By S. V. Falkovich In the present paper, the motion of a gas in a planeparallel Laval nozzle in the neighborhood of the transition from subsonic to supersonic velocities is studied. This problem was first consid ered by Meyer (reference 1) who sought to obtain the velocity poten tial in the form of a power series in the coordinates x,y of the flow plane. The case of the nozzle with plane surface of transi tion from subsonic to supersonic velocities was further considered in a paper by S. A. Christianovich and his coworkers (reference 2). For computing the supersonic part adjoining the transition line, Christianovich expanded the angle of inclination of the velocity and a specific function of the modulus of the velocity in the power Series, using the velocity potential and the stream function as the unknown variables. In a recently published paper, F. I. Frankly (reference 3), applying the hodograph method of Chaplygin, under took a detailed investigation of the character of the flow near the line of transition from subsonic to supersonic velocities. From Sthe results of Tricomi's investigation on the theory of differ ential equations of the mixed elliptichyperbolic type, Frankl introduced as one of the independent variables in place of the modulus of the velocity, a certain specially chosen function of this modulus. He thereby succeeded in explaining the character of the flow at the point of intersection of the transition line and the axis of symmetry (center of the nozzle) and in studying the behavior of the stream function in the neighborhood of this point S by separating out the principal term having, together with its derivatives, the maximum value as compared with the corresponding corrections. This principal term is represented in Frankl's paper in' the form of a linear combination of two hypergeometric func tions. In order to find this linear combination, it is necessary to solve a number of boundary problems, which results in a complex analysis. In the investigation of the flow with which this paper is concerned, a second method is applied. This method is based on S the transformation of the equations of motion to a form that may be called canonical for the system of differential equations of *"K Teorii Sopla Lavala." Prikladnaya Matematika i Mekhanika. Vol. 10, no. 4, 1946, pp. 503512. NACA TM No. 1212 the mixed elliptichyperbolic type to which the system of equations of.the motion of an ideal compressible fluid refers. By studying the behavior of the integrals of this system in the neighborhood of the parabolic line, the principal term of the solution is easily separated out in the form of a polynomial of the third degree. As a result, the computation of the transitional part of the nozzle is considerably simplified. 1. Fundamental equations. The equations of the twodimensional, steady, nonvortical motion of an ideal gas in the absence of friction and heat conductivity have the form T y = =o (1.1) W2 + P (1.2) 2 X1 P X1 Po where u and v are the components of the velocity along the x and y axes, p is the density, p is the pressure, W = /u2 + v2 is the magnitude of the velocity, X = c/c PO and po are the density and pressure of the gas at rest. Equations (1.1) represent the condition of the absence of vor tices and the equation of continuity. Equation (1.2) is Bernoulli's equation for adiabatic motion for which Po P (1.3) For the velocity of sound a a2 = X (1.4) P From equations (1.2), (1.3), and (1.4), the following equation is derived: 1 P =po 2,2.) (1.5) ao/ NACA TM No. 1212 (a02 = Xp/p0 is the velocity of sound in the gas at rest) from which d PO W " T P a (1.6) From equation (1.1), it follows that there exist two functions: the velocity potential cp(x,y) and the stream function (x,y), which are determined by the equations dap = udx + vdy In place of the velocity components coordinates, setting u = W cos 0 and v the angle between the velocity vector and stituted. Equations (1.7) are solved for obtaining d =  ( vdx + udy) PO (1.7) u and v, the polar = W sin 8, where e is the x axis, are sub dx and dy, thus cos 0 P0 sin 8 d W pW dW dx = dcp d sin 0 PO cos dy dcp+ p (1.8) If x and y as well as W and 0 are considered as func tions of the variables y and *J, then dx and dy must be total differentials, so that the following equations must hold: S/cos 9\ o0 sin 6 Tp P S/sin )( oa cos e In carrying out the differentiation,in taking account of the fact that according to equation (1.5) in which p0/p depends only on the magnitude of the velocity W, and in making use of equation (1.6), the following equations are obtained: b0 cos 0 W 0 o s e 0 sin 8 :2 w sin e + = + cos e " o sin 0 w PO sn e PO sin 8 ( cos sin T' w W ap 1 2 a bJI~~~ k ~/ a~P ac NACA TM No. 1212 By solving these equations for the derivatives 90/~qc and e P W = TC Pow F* oe PO a2W2 _ +p iP O= 0 This system of differential equations will be of the elliptical type if the magnitude of the velocity W is less than the velocity of sound and will be of the hyperbolical type for supersonic velocities. The new function iT is considered instead of velocity is related to W in the following manner (reference 3): W and (1.10) S= aW dW /3 W Equations (1.9) then assume the form + b 0 aJ+b(rl) b o i b( q = ,1 b =o0 (1.11) P V a a2 b(ij) =~2 a (1.12) as a result of (1.10), is a function of the variable T. Equations (1.11) are the fundamental equations for the inves tigation of twodimensional, nonvortical motion of a gas when the velocity of the flow passes from subsonic to supersonic velocity. In some cases,it is more convenient in these equations to sub stitute 8 and n as the independent variables and take cp and I as the required functions. After this transformation, equa tions (1.11) assume the form T + b T) = o i b(l) O = 0 2. Investigation of variable T. The variable T deter mined by equation (1.10) is considered in more detail. For (1.9) where NACA TM No. 1212 computing the integral entering this equation, the square of the velocity of sound is 2 k+1 2 k1 2 a =2 a 2 (2.1) In substituting the preceding equation in (1.10) (e1 ii r 1 h i2 /3 The integration results i The integration results in W 2 a X = h a* l+h h2X2  2 1h h V 2? 1 = By expanding equation (2.3) in a series T1 = (h(h21) /3 h2\2 l+0(l?2) From equation (2.3), it follows, that 1 > 0 for A< 1 i < 0 for X > 1, that is, in the plane of the variables e and T, the region lying in the upper halfplane will correspond to the region of subsonic velocities and the region lying in the lower halfplane will correspond to the supersonic velocities. The line of transition from subsonic to supersonic velocity will correspond to the line T = 0, that is, the axis of abscissas. From equa tion (1.10), the value of the velocity W = 0 in the plane e,r corresponds to an infinitely distant point. For A > 1, equa tion (2.3) assumes the form _= (2/3 (h arc g 2 tg F arc tg h 2 h I/ characteristics in the plane of the hodograph of the for twodimensional, nonvortical motion of the gas are epicycloids (fig. 1), the equations of which are (refer (2.3) (2.4) and The velocity known as ence 4). (2.5) NACA T3 No. 1212 S= C & h arc tg h2_ arc tg h Vh? Vh2 X2 Because for a point transformation characteristics go over into characteristics, the following equations of the character istics in the plane of the variables 8 and n are found by using equation (2.5): e = ( + C (2.6) from which it follows that the characteristics assume the form of semicubical parabolas with the cusps on the axis of abscissas (fig. 2). 3. Differential equations of motion of a gas in neighborhood of transition line. The flow in a Laval nozzle near the line of transition from the subsonic to the supersonic velocities is con sidered. This line is hereinafter designated the sound line. If a straight line perpendicular to the axis of symmetry of the nozzle is directed away from the axis, it will intersect the streamlines with constantly increasing curvatures and will there fore encounter particles of the gas having constantly increasing velocity. The sound line will therefore be a curve that is con vex toward the supersonic velocities1 with vertex on the axis of symmetry (fig. 3). The point of intersection of the sound line with the axis of symmetry is, according to Frankl, denoted as the center of the nozzle. In the plane of the variables cp and 4, the region of flow is transformed into a strip the width of which is determined by the amount of gas flowing through the nozzle (fig. 4). The point of origin of coordinates in the cp,i plane cor responds to the center of the nozzle in the flow plane. The determination of the flow reduces to finding two functions T=T(p,i) and 9=0()cp,) that satisfy equations (1.11). Because the flow is to be symmetrical with respect to the streamline * = 0, it is necessary that the required functions satisfy the conditions 1When the streamlines have points of zero curvature, the sound line will be a straight line perpendicular to the axis of symmetry; this case was considered by S. A. Christianovich (reference 2). NACA TM No. 1212 n(cp,4,) = T(cpq*) e(cp,I) = 0e(cp) I(o,o) = 0 (3.1) which are based on equations (2.2). The solution of equation (1.11), in the form of a power series in the variables cp and 4, takes into account equations (3.1) 2 2 3 2 S= 1cp + a2p + a3i i acp + a5c . 6 = bfpy 3 b2cpW + b3 3+ b4C9p+ .. (3.2) from which it follows that if the flow in the neighborhood of the origin of coordinates is considered, that is, if cp and are assumed to be small magnitudes, the following equations may be obtained from equation (3.2): I = 0(c) e = (cpW) = O0() e= 0() = 0(1) 0(cp) (3.3) With the use of equation (2.1) and the notations introduced in equations (2.2), equation (1.12) for the function b(T) may be reduced to the form 0o /h(112) b(n) = P \h ) P P 'V(h2^2)T In accordance with equation (2.4), the following equation is derived: 1 Po 1/3 k k1 1/3 b(0) ) (xl)3 (k)l) In taking account of the order of smallness of all terms entering equations (1.11), it is concluded that near the origin of coordinates the system of equations (1.11) may be replaced by the following equations: NACA TM No. 1212 Sb(o) 0o dcp By setting b(0)W =W, the final result is n e 0 two 0 p be T1 Cn 0 where, for simplicity, the bar over has been dropped. 4. Investigation of flow in neighborhood of center of nozzle.  It is evident that the functions 6 =A2p* A3 %3 3 T = Acp 2 2 where A is an arbitrary constant, are integral of the system of equations (3.4), and satisfy conditions (3.1). The significance of the constant A will be explained. From the second equation (4.1), ) = Acp along the axis of symmetry of the nozzle ( 0 = 0). Differentiation results in A = dr = d1 dW dx dp dW dx dp Furthermore, by using successively equations (1.10) and (2.1) S1 a2W2 _ha 1?_ dW aW 7 (h2)_2 Moreover, along the line 4 = 0 dx 1 6=p w dW =u dx ox Hence,for A the following relation is obtained: 17?2 ux 1th2 ) x )y =0 y=0o  S#0 T6 (3.4) (4.1) A( h A= W2 Wa (x 2l) a (Z0) 7=0 (4.2) NACA TM No. 1212 where to obtain the last result, equation (2.4) was used. The value of A is thus proportional to the value of the derivative of the velocity at the center of the nozzle. It is assumed that Zu/ox > 0 so that A will be a negative quantity. Along the sound line, T = 0. Hence, according to the second equation (4.1), the following equation of the sound line is derived: A= 2 (4.3) that is, in the plane cp,4', the sound line will be a parabola. From equation (3.4), the differential equation of the char acteristics has the form \dIJ  By substituting the value of i from equation (4.1) ()2 A22 Ap In the integration of this equation, set 2 Acp= 2 A* = x The equation then assumes the form (1222e xy 2 2 or 12y22y = y After separating the variables and integrating, the following equations of the characteristics are obtained: 2/3 1/3 2/3 1/3 x(y+l) (2y1) = C x(yl) (2y+l) = C L. NACA TM No. 1212 In order to obtain the characteristics passing through the origin of coordinates, set C = 0. Thus the variables qp and * become A.2 A(4) CP = 2 44) Hence, the characteristics passing through the origin of coor dinates in the cp,$ plane are parabolas tangent to each other at this point and tangent to the sound line (fig. 4). The origin of coordinates will therefore be a singular point of the integral of equations (3.4) determining the flow in the nozzle. In considering the character of this singularity, it is evi dent from figure 4 that the characteristics and the sound line divide the neighborhood of the center of the nozzle into six regions. It shall be investigated how the neighborhood of the center of the nozzle is transformed in the plane of the variables 0 and TI by the integral of equation (4.1). By eliminating from equation (4.1) the variable cp, the following cubical parabola is used in determining the stream function: A33 + 3An 30 = 0 (4.5) This equation has one real root if its discriminant 8 = 902/4 + T3 > 0 and three real roots if 5 < 0. Because the point (cp = 0, = 0) corresponds in equation (4.1) to the point (9 = 0, T = 0), the equations of the characteristics cor responding to equations (4.1) are in accordance with equation (2.6). S02 3 = 0 Thus regions I, II, and III of the plane are transformed into the same region of the plane 0,6T lying between the characteristics 3 Furthermore, the streamlines ( = & q correspond, as seen from equation (4.5), to the straight lines in the 0,,T plane. 2A 2 al4_ NACA TM No. 1212 The transformation of the neighborhood of the nozzle in the G,I plane will thus have the form of the folded surface shown in figure 5. The corresponding regions in figures 4 and 5 are denoted by the same numbers. In order to compute the streamlines in the flow plane, equa tions (1.8) are used in which dW is set equal to 0, after which they assume the form Scos dx = d cp dy sin W By substituting for magnitude of the velocity function of the variable dx = w cos (A A2q dc its value from equations (4.1), the is, according to equation (1.10), a Thus along the streamlines I = q dy= sin dy = sin (2Kq A2q) dp Integration results in r 1 x= = Cos 0 y= k 3  AZqcpdcp 0 W1m sin +3 A2 dcp (4.6) where H is the width of the nozzle at the critical section. In equations (4.6), set according to equations (4.1) A2 2 T = Acp q 2 The computation of the integrals in equation (4.6) reduces, evidently, to the computation of the two integrals of the type C2 12 = sin A2qp W(Tl) d Il=1 0 cos A2q d wnrr 12 NACA TM No. 1212 with the aid of which x and y are expressed as follows: 33 33 A 333 x = Il cos A  + Ia sin A 3 y = 2 sin "A I1 cos A, (4.7) 5. Nozzle with surface of weak discontinuity. The case in which weak discontinuities are formed along the Mach lines issuing from the center of the nozzle is here considered. For this inves tigation, it is necessary and sufficient that the derivative (ou/ax)y=0 possess a discontinuity at the center of the nozzle (reference 3). It is assumed that both values [(uox) x and [(ou/ox)y=O are positive. From equation (4.1), it is evident that the magnitude A will have the value A = A1 in the regions VI, V, and IV (fig. 4) and the value A = A2 in the region III where, according td equa tion (4.2), A1 < 0 and A2 < 0. From equations (4.1), it is concluded that in the regions VI, V, and IV 2 A13 3 A2 2 eA = A, = Al 2 (5.1) and for region III SA23 3 A22 2 9 = A222 =2 (5.2) According to equations (4.4), the equations of the character istics separating the regions IV and V from regions I and II and the equations of the characteristics separating regions I and II from region III have the forms Al 2 A2\2 P= 4 CP= (5.3) Substituting the first of these equations in equations (5.1) and the second in equations (5.2) NACA TM No. 1212 A1l33 S8= 12 e =+ A213 *~ Tr TI =  A1W2 rp = + A2 2 In order that the flow in the nozzle has no discontinuities, it is necessary to determine 8 and q in regions I and II from equations (3.4) in such a manner that the characteristics condi tions, equations (5.4) and (5.5), are satisfied. In order to integrate the system, equations (3.4), set 1 = (f 92 0e= (5.6) where f and g are functions to be determined. For this substitution, equations (3.4) are transformed into a system of ordinary differential equations with the independent variable t = cp/2 2f2tf'g' = 0 ff'r3g2tg' = 0 (5.7) By the elimination of G' g = [4tf (f + 4t2) f'] 3 (5.8) By differentiating equation (5.8) and substituting the result in the first equation (5.7), a differential equation of the second order for determining f is obtained (4t2 f) f" f'2 2tf' + 2f = 0 (5.9) From equations (5.6), (5.4), and (5.5), it follows that the boundary conditions for the function f will be AI2 f = for 4 t A1 4 2 f = A for t = (5.10) In order to integrate equation (5.9), it is written in the form (5.4) (5.5) NACA TM No. 1212 f'+t = 0 2tf'f (The solutions 2tf' f = 0, that is, f = cq/ which do not satisfy equations (5.10) are eliminated.) In carrying out the quadrature f'+2t 1 or1 2 or f f 2tf'f 2c1 2(tcl) tcl that is, the integration of the linear equation results in f = 4Clt 8c12 + C2 1 (5.11) The boundary conditions, equations (5.10), which the obtained solution equation (5.11) must satisfy, have the form: f = fl for t = t1 and f = f2 for t = t2 where it is easily seen from equation (5.10) that the points (tl, fl) and (t2, f2) lie on the parabola f = 4t2 and that t < 0 and t2 > 0. Hence, in order to satisfy the boundary conditions, it is necessary from the family of parabolas equation (5.11) to choose the parabola passing through (tl, fl) and (t2, f2). Upon satisfying these conditions tl2_tlt2t22 2 16(tlt2)2 (tl+2t2)2 (2t t2)2 cl = 3(t+ t2) 2 = 27(tlt2)3 It is necessary that along a streamline the velocity in the flow direction should increase monotonically, tat is,that T should decrease monotonically. Because T = f according to equation (5.6), f' < 0 must be in the range tI < t condition is possible only for 2t1 + t2 < 0 and tl + 2t2 > 0 when in accordance with equations (5.10), the following condition is obtained Al Al I A2  for which a flow without discontinuity is possible. Translated by S. Reiss National Advisory Committee for Aeronautics. NACA TM No. 1212 15 REFERENCES I1 1. Meyer: Uber zweidimensionale Bevegungsvorgange in einem Gas das mit bberschallgeschwindigkeit stromt. Forschungshefte, Nr. 62, 1908. 2. Levin, Astrov, Pavlov, and Christianovich: On the Computation of Laval Nozzles. Prikladnaya Matematika i Mekhanika, vol. VII, no.'1, 1943. 3. Frankl, F. I.: On the Theory of Laval Nozzles. Izvestia Akademii Nauk SSSR, Ser. Matematika, vol. IX, 1945. 4. Kochin, Kibel, and Rose: Theoretical Hydrodynamics. 1941. NACA TM No. 1212 Figure 2. I / Figure 4. Figure 4. Figure 1. Figure 3. Figure 5. 0 m o m0 I 0 SI .I Oo O r 0* A o to 2 . o *5 o m 0 a ! < g I 0 o hmi; 43S 9 Q to 11 0l 04 0 0 Ol r4 0 1 0 M14M P 943 + a a 0> 0 1 oD flJ 6 4> + 4 5 0 0a gokno 0ri 4P 0W R 0 0 u S rd aEq4 0 0i I d 5 5 ai UNVEROEY OF FLORt 2iLUEFJI1 