LEAOON /\)/\CA-TrO'pl^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1212 ON THE THEORY OF THE LAVAL NOZZLE By S. V. Falkovich Translation 'K Teorii Sopla Lavala." Prikladnaya Matematika i Mekhanika. Vol. 10, no. 4, 1946. Washington . ,^, ^^.,ty QF FLORIDA April 1949^^' rSOP-HTMEMT^^ P.O. Boa GA- ■' ' .U -7011 US/. Ja'^ ro:x ^1^ 3>ro^x:^y NATIONAL AD7IS0EY COMMITTEE FOE 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 plane-parallel 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 eind 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. Frankl (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 the results of Tricoml's investigation on the theory of differ- ential equations of the mixed elliptic-hyperbolic 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 syimnetry (center of the nozzle) and in studying the behavior of the stream function in the neighborhood of this point 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 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. 503-512. NACA TM No. 1212 the mixed elliptic-hyperbolic type to which the system of equations of .the motion of an ideal compressible fluid refers. By studj-'ing 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 sin^jlified. 1. Fimdamental equations. - The equations of the two-dimensional, steady, nonvortical motion of an ideal gas in the absence of friction and heat conductivity have the form where u and v are the components of the velocity along the and y eixes, p is the density, p is the pressure, V7 = ^u^ + v2 is the magnitude of the velocity, X = %/'^y' % ^^^ ^0 ^^® the density and pressure of the gas at rest. Equations (l.l) 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 f- = (^] (1.3) For the velocity of sound a a2=X^ (1.4) From equations (1.2), (1.3), and (1.4), the following equation is derived: (1.5) NACA TM No. 1212 (eq = XPq/p^ is the velocity of soimd in the gas at rest) from which ' \^^ * (1.6) P a2 From equation (l.l), it follows that there exist two fimctions; the velocity potential cp(x,y) and the stream function >i'(x,y), which are determined by the equations dcp = udx + vdy d^/ = — (- vdx + udy) (l*^) PQ In place of the velocity con5>onents u and v, the polar coordinates, setting u = W cos and v = W sin 0, where is the angle between the velocity vector and the x axis, are sub- stituted. Equations (1.7) are solved for dx and dy, thus obtaining f cos , ^0 sin dx = — ^^ dcp - — — i^ d\|/ , sin , PQ cos , I ^1 n^ If X and y as well as W and are considered as func- tions of the variables c,") and \}/, then dx and dy must be total differentials, so that the following equations must hold: In carrying out the differentiation, in taking account of the fact that according to equation (1.5) in which Pq/p depends only on the magnitude of the velocity W, and in making use of equation (1.6), the following equations are obtained: . _ 'be cos Sw cos ^ - ^i2_£ M = - ^ sin ^ 4 NACA TM No. 1212 By solving these equations for the derivatives S6/Bcp and 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 fxinction t) is considered instead of velocity W and is related to W in the following manner (reference 3): aW \2/3 dW (1.10) Equations (1.9) then assiime the form |l^t()|£.o n|ll--f^|£ = o (1.11) S\j/ ^ '^ dcp ocp b(Ti) d^ where b(^)=-^^^^^ (1.12) as a result of (l.lO), is a function of the variable r|. Equations (l.ll) are the fundamental equations for the inves- tigation of two-dimensional, 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 6 and T) as the independent variables and take cp and \J/ as the required functions. After this transformation, equa- tions (l.ll) assime the form M = ^ - .U.) S = ^ + b(Tl) ^ = ^- Tlb(Tl) ^ 2. Investigation of variable r], - The variable t] deter- mined by^quation(l7To)is~consTd^ 5.n more detail. For NACA TM No. 1212 computing the integral entering this equation, the square of the velocity of sound is 2 _ k+1 2 a - -g- a ^^ 2 ^ (2.1) In suhstituting the preceding equation in (l.lO) 1 T = i-^^.^'-n <-' The integration results in r T) =\l ^ 1 + h 1 - h By expanding equation (2.3) in a series 1-A^ h2-x2 A 2/3 y (2.3) Mh^-i) v2/3 1-A^ h - A •- -1 2/3 1+0(1-X^) (2.4) From equation (2.3), it follows, that t| > for A< 1 and T] < for X > 1, that is, in the plane of the variables and Tj, the region lying in the upper half -plane will correspond to the region of subsonic velocities and the region lying in the lower half -plane 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 (l.lO), the value of the velocity W = in the plane 0,ti corresponds to an infinitely distant point, tion (2.3) assumes the form For X > 1, equa- ^=-l2 2/3 h arc tg 2 - arc tg h (2.5) The characteristics in the plane of the hodograph of the velocity for two-dimensional, nonvortical motion of the gas are known as epicycloids (fig. 1), the equations of which are (refer- ence 4) . NACA TM No. 1212 = C ± h arc tg Because for a point transformation characteristics go over into characteristics, the following equations of the character- istics in the plane of the variables 6 and t] are found by using equation (2.5): = ±1 U] + 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). 5. Differential equations of motion of a gas in neighborhood of transition line . - The flow in a Laval nozzle near the line of treinsition 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 synnnetry 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 velocities^ 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 Frank 1, denoted as the center of the nozzle. In the plane of the variables cp and vj/ , 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,\l' plane cor- responds to the center of the nozzle in the flow plane. The determination of the flow reduces to finding two functions Ti=Tj(q3,\l/) and 0=0(cp,\i/) that satisfy equations (l.ll). Because the flow is to be symmetrical with respect to the streamline \i/ = 0, it is necessary that the required fiinctions satisfy the conditions When the streamlines have points of zero curvature, the sovmd 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 7 Ti(cp,\J/) = Ti(cp-\i/) e(cp,\^) = - 0(cp-\l/) Tj(0,0) = (3.1) which are based on equations (2.2). The solution of equation (l.ll), in the form of a power series in the variables cp and "^ , takes into accoimt equations (3.1) 2 2 3 2 T] = a^cp + a2Cp + a.^\\i -t- a>cp + ac-cpvj; + . . . = b^cpvl/ i- bgCp^ + b3^1/^+ b^cp^+ . . . (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 ^i' are assumed to be small magnitudes, the following equations may be obtained from equation (3.2): 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 In accordance with equation (2.4), the following equation is derived: In tEiking account of the order of smallness of all terms entering equations (l.ll), it is concluded that near the origin of coordinates the system of equations (l.ll) may be replaced by the following equations: 8 NACA TM No. 1212 By setting b(0)\tf =\J;, the final result is where, for simplicity, the bar over ii has been dropped. 4. Investigation of flow in neighborhood of center of nozzle. • It is evident that the functions 3 2 e = A^^>ii- ^y\i^ Ti=A(p-^\^2 (4.1) where A is an arbitrary constant, are integrals 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), t) = Acp along the axis of symmetry of the nozzle (\|; = 0). Differentiation results in A = ^ = d]! dW dx dqp dW dx dcp Furthermore, by using successively equations (l.lO) and (2,1) 2_y2 ha^ / l-X^ dw = - aw\/ Ti - - X Vn(h'-x2) Moreover, along the line \V = dx _ 1 dW ^ Su dcp ~ W dx ^ Hence, for A the following relation is obtained: y=o y=o 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 Su/Bx > so that A will be a negative qusmtity. Along the sound line, t^ = 0. Hence, according to the second equation (4.1), the following eqviation of the sound line is derived: cp = P \i/ (4.3) that is, in the plane cp,\i/, the sound line will be a parabola. From equation (3.4), the differential equation of the char- acteristics has the form By substituting the value of t\ from equation (4.1) dmY _ a2\1/2 dV = -2- - Acp In the integration of this equation, set Acp = |- - x2y2 A\}/ = X The eqiiation then assxunes the form l-2y2-2xy g) = 7^ or l-2y2-2xy g = ± 7 After separating the variables and integrating, the following equations of the characteristics are obtained: x(y.l)^/^ (2y-l)^/^ = C x(y-l)^/^ (2y.l)^/^ = C 10 NACA TM No. 1212 In order to obtain the characteristics passing through the origin of coordinates, set C = 0. Thus the variables ro and \j/ become A\i/2 A\l/2 qp = - -g- qp = -4- (^•^) Hence, the characteristics passing through the origin of coor- dinates in the cp,\i; plane are parabolas tangent to each other at this point and tangent to the so\xnd line (fig. 4) . The origin of coordinates will therefore be a singular point of the integrals 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 6 and T] by the integrals of equation (4.1). By eliminating from equation (4.1) the variable cp, the following cubical parabola is used in determining the stream function: A^\|;^ + 3ATi\^ - 30 = (4.5) This equation has one real root if its discriminant 6 = 99 /4 + T] > and three real roots if 6 < 0. Because the point (cp=0, \i/=0) corresponds in equation (4.1) to the point (0 = 0, T] = 0), the equations of the characteristics cor- responding to equations (4.1) are in accordance with equation (2.6). I 02 , Ti3 = Thus regions I, II, and III of the plane are transformed into the same region of the plane 0,ti lying between the characteristics Furthermore, the streamlines \)/ = ± q correspond, as seen from equation (4.5), to the straight lines in the 0,t] plane. NACA TM No. 1212 11 The transfomiation of the neighborhood of the nozzle in the 0,11 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 d\l/ is set equal to 0, after which they assume the form , cos , , sin j„ dx = — ^j— dcp dy = — ^^ dcp By substituting for d its value from equations (4.1), the magnitude of the velocity W is, according to equation (l.lO), a function of the variable t) . Thus along the streamlines \J/ = ± q Integration results in X / WRT '^^^ VV^ - A^qcpjdcp y = * 1 J wfe ^^^ (^ * A^q-^dcp-e Ej (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 The computation of the integrals in equation (4.6) reduces, evidently, to the computation of the two integrals of the type 12 MCA TM No. 1212 with the aid of which x and y are expressed as follows: aV r . A^ ^ /; . aV , A^3 X = I-j_ cos — ^ + Ig sin J* y = ± II2 ^^° J* - I]^ cos — r^ 3 ■ -^ "— 3 ^ y- and n-N /-N \ n *- y=yjx=+o (ou/dx; _Q are positive. From equation (4.1), it is evident that the magnitude A will have the value A = A]_ in the regions VI, V, and IV (fig. 4) and the value A = A„ in the region III where, according to equa- tion (4.2), Ai < and Ag < 0. From equations (4.1), it is concluded that in the regions VI, V, and IV 2 •'^l^ 3 ^1^ ? and for region III ^-3 k2 0=A2V-^^^ n = A2CP- -|-\^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 cp = -4— cp= 2~ (5-3) Substituting the first of these equations in equations (5.1) and the second in equations (5.2) NACA TM No. 1212 13 ® - " 12 AiV ^ = - 4 e=^fA,V Tj = - A2\l^ rp = + Ai\l/2 (5.4) A2\|/2 CD = 5- (5.5) In order that the flow in the nozzle has no discontinuities, it is necessary to determine 9 and ti 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 where f emd 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/\l/2 2f-2tf'-g' = ff'+3g-2tg' = (5.7) By the elimination of G' g = i [4tf - (f + 4t2) f] (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 '^ - 2tf ' + 2f = (5.9) From equations (5.6), (5.4), and (5.5), it follows that the boundary conditions for the function f will be An ^ At Ao f = - -L. f or t = — f = - Ap for t = - -^ (5.10) 44 '•2 In order to integrate equation (5.9), it is written in the form 14 NACA TM No. 1212 f +2t ' 2tf '-f = (The solutions 2tf' - f = 0, that is, f = c ^/t" which do not satisfy equations (5.10) are eliminated.) In carrying out the quadrature f ' f2t 1 ~, 1 f ^ ^^1^ 2tf '-f 2c3^ ' 2(t-ci) t-Cj^ that is, the integration of the linear equation results in f = 4c^t - Sc^^ -I- Cg yE^ (5.11) The boundary conditions, equations (5.10), which the obtained solution equation (5.11) must satisfy, have the form: f = f j^ for t = t, and f = f 2 for t = tp where it is easily seen from equation (5.10) that the points (t-,, f , ) and (to, fg) lie on the 2 parabola f = -4t eind that t^ < Eind tg > 0. Hence, in order to satisfy the bo\mdary conditions, it is necessary from the family of parabolas equation (5.11) to choose the parabola passing through {t-^, f-^) and (tg, fg)- Upon satisfying these conditions tj^^^tj^tg+tg^ 2 16(t3_-t2)^ (t3^+2t2)^ (2t3^ + tg)^ It is necessary that along a streamline the velocity in the flow direction should increase monotonically, that is, that r\ should decrease monotonically. Because t^ = f\jr according to equation (5.6), f ' < must be in the range t]_ < t when in accordance with equations (5.10), the following condition is obtained Ai< A2<^ for which a flow without discontinuity is possible. Translated by S. Reiss National Advisory Committee for Aeronautics. NACA TM No. 1212 15 REFERENCES ■ " .. 1. Meyer: Uber zweidimensionale Bevegungsvorgange in einem Gas das mit Uberschallgeschwindigkeit stromt. Forschungshef te, Nr. 62, 1908. 2. Levin, Astrov, Pavlov, and Christianovich: On the Computation of Laval Nozzles. Prikladnaya Matematika i Mekhanika, vol. VII, no. 'l, 1943. 3. Frankl, F. I.: On the Theory of Laval Nozzles. Izvestia Akademii Na\ik SSSR, Ser. Matematika, vol. IX, 1945. 4. Kochin, Kibel, and Eose: Theoretical Hydrodynamics. 1941. 16 NACA TM No, 1212 Figure 1. Figure 2. Figure 3. Figure 4. Figure 5. CO o •H o •H > O ;^ Id PP c\j O I •H CO ca g -p o 05 U +> CO 5 ON H L d o o o +J 0) -H 0) CO •H . =1 u H o •H x) ttJ -p to i-H 0} >H -H d d d -p ■p a d) o o o r^ d W -H XI •H •H •H 1 ■P +J +> m (1) iH cd 3 O o (0 0) Xi (0 +J -P 0) r-l ■p •p ■p (0 0) > 0) d H C X> d O •H -d ^ _: W -P t-^ O H CO O g > d ,o d O 0} CO d -P •H to M O •H 3 >^ o ed -H d D^ H •H : P< 0) d bQ-P 0) -H o (0 0) CO 3 >> ^^ CO a] CO M O od H CO (d Id ^1 CO •H CO o ® -d 0) 4) •H CO 3 iH ^ d ^ J3 s ,d «H -P