^AC/f7Al-|Z3'i^fc f^M-LEADON 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1239 
 
 TWO-DIMENSIONAL MOTION OF A GAS AT LARGE 
 
 SUPERSONIC VELOCITIES 
 
 By S. V. Falkovich 
 
 Translation 
 
 "Institut Mekhaniki Akademii Nauk Soiuza, SSR" 
 Prikladnaia Matematika i Mekhanika, Tom XI, 1947 
 
 Washington 
 October 1949 
 
 ftOCUME^fTS DEPARTMENT 
 
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 NATIONAL ArVISOEY COMMITTEE FOE AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1239 
 
 TWO-DIMENSIONAL MOTION OF A GAS AT LAEGE SUPERSONIC VELOCITIES* 
 
 By S. V. Falkovich 
 
 A large number of papers have been devoted to the problem of 
 integration of equations of two-dimensional steady nonvortical 
 adiabatic motion of a gas. Most of these papers are based on the 
 application of the hodograph method of S. A. Chaplygin in which 
 the plane of the hodograph of the velocity is taken as the region 
 of variation of the independent variables in the equations of 
 motion; the equations become linear in this plane. The exact 
 integration of these equations is, however, obtained in the form 
 of infinite series containing hypergeometric functions. The 
 obtaining of such solutions and their investigation involve exten- 
 sive computations. As a result, methods have been developed for 
 the approximate integration of the equations of motion first 
 transformed to a linear form. S. A. Chaplygin in' reference 1 
 first pointed out such an approximate method applicable to flows 
 in which the Mach number does not exceed 0.4. 
 
 S. A. Christianovich (reference 2), in solving the problem 
 of the flow with circulation about a wing in a supersonic stream, 
 gave as a first approximation a generalization of the method of 
 Chaplygin to the case where the region of variation of the velocity 
 in the hodograph plane lies within a sufficiently narrow ring 
 entirely inside the circle \J <e^. At the same time and inde- 
 pendently an analogous method was proposed by Tsien and von Karman 
 (references 3 and 4). These methods are not applicable for 
 Mach numbers near unity. In the papers by F. I. Frankl (refer- 
 ence 5) and S. V. Falkovich (reference 6), approximate equations 
 of motion are given suitable for the investigation of flows in 
 passing through the velocity of sound. In his recently published 
 paper, S. A. Christianovich (reference 7) showed that for Mach 
 numbers 1.05 < M <3.5 the equations for the stream function and 
 the velocity potential may approximately be replaced by the equa- 
 tions of Darboux with integral coefficients and hence the general 
 integral obtained in finite form. 
 
 *"lnstitut Mekhaniki Akademii Nauk Souiza, SSR" Prikladnaia 
 Matematika i Mekhanika, Tom XI, 1947. 
 
MCA TM 1239 
 
 The equation of motion of a gas is investigated herein for 
 large supersonic velocities and a method is shown for the approxi- 
 mate integration, vhich gives sufficient accuracy for Mach numbers 
 greater than 4. 
 
 1. Fundamental equation and its transformation. - The equa- 
 tion determining the velocity potential of the two-dimensional 
 irrotational adiabatic motion of a gas has, as known, the form 
 
 (e2 - u2) § - 2uv-% . (a^ . ^2) ^ = (l.l) 
 Sx^ dx dy ^yZ 
 
 To this equation the transformation of Legendre is applied. 
 As the independent variables in place of x and y are taken 
 Scp/Sx = u and 5cp/Sy = v and in place of the velocity poten- 
 tial cp(x,y), the Legendre potential is introduced. 
 
 $ (u,v) = ux + vy - cp(x,y) (1.2) 
 
 Equation (l.l) becomes 
 
 (a2 - v2) ^ . 2UV -^ . (a^ - u^) ^ = (1.3) 
 
 If the function $(u,v) is determined, then by differen- 
 tiation of equation (1.2), the coordinates x and y of the 
 corresponding point in the flow plane are determined. 
 
 X = BVSu y = S$/Sv (1.4) 
 
 Pass in the hodograph plane u,v to polar coordinates and 
 set u = W cos 6 and v = W sin 0; equations (1.3) and (1.4) 
 then become 
 
 a^ - w2 S2$ a£$ a2 - W^ S^ _ ^ ,. ^, 
 
 a^w^ ae2"sw2' a^W ^^ = ^''"^ 
 
 _ S$ sin 5$ . „ 5$ cos 5$ /, ^. 
 
 X = cos ^ - -^ ^ y = sm ^ + -^ ^ (1.6) 
 
 If only supersonic velocities are considered, it is more con- 
 venient in place of the independent variable W to introduce in 
 equation (1.5) the new variable 
 
KACA TM 1239 
 
 z = 1//v/m^ - 1 
 
 (1.7) 
 
 Thus z will be small for large Mach numbers and vill increase 
 indefinitely as M-^1. 
 
 In order to carry out this transformation, from the equation 
 of Bernoulli the expression of W is found in terms of z. Thus 
 
 wf _a^ ^ h^a» 
 2 "^ K-1 2 
 
 {>^'-n-') 
 
 or 
 
 1 1 
 
 — + 
 
 v,2 2 
 
 ^ (k-1) M^ 2W^ 
 Tf M^ is replaced by z, from equation (1.7) 
 
 2 h\2(i^^2) 
 
 W^ = 
 
 T v,2 2 
 1+h z 
 
 (1.8) 
 
 By making use of this expression, pass in equations (1.5) and 
 (1.6) from the variable W to the variable z; then 
 
 h^ 
 
 2_^^2 
 
 (h^-1) 
 
 ^ + 5h^z^ + 2h^z^ + h^ - 2 5$ ^ Q 
 2 „ psp ,, ,? ?^? ^.2 ■" (1^22) (l+h2z2)z Bz 
 
 Sz^ (l+z2)2 (l+h2z2)2 S0^ 
 
 (1.9) 
 
 1 
 
 2^ '■> .^^2„2n 3^ . ^ 5$ 
 
 (1+z^) (l+h-^z^) , .^ . o -' 
 
 ■^ ^^r^ ^ cos 9 T— + sm T— 
 
 (h2-l)z Sz he 
 
 1 
 
 y = - w 
 
 (Uz2 ) (l+h2z2) . . 5$ - S$ 
 -^ ^-5-i ^ sm r^ cos 6 r- 
 
 (h2-i)z az he 
 
 (1.10) 
 
 The characteristics of equation (1.9) may be taken in the form 
 
MCA TM 1239 
 
 , = e- / _(l£ii)i5 , = a. / (^^ild^ 
 
 (l+z2) (Uh2z2) /_ {l+z2) (l+h2z2) 
 
 ''^'^^ '' ^■^■^" " > JO 
 
 The line of maximum velocity z = in the hodograph plane 
 vill correspond in the plane of the characteristics Xu to the 
 line \i - X = 0. If the integration is carried out, 
 
 X = e - (h arc tg hz - arc tg z) \i = -t- {h arc tg hz - arc tg z) 
 
 (1.11) 
 
 2. Investigation of equation (1.9). - If equation (1.9) is 
 referred to the characteristic coordinates Xn, 
 
 ^2$ h2 - 2 - 2z2 - h^z^ /a$ b^\^ Q /g ^, 
 
 BX Sn " 4(h2 - l)z \^l^ ^ 
 
 From equations (l.ll), 
 
 [i - X = 2(h arc tg hz - arc tg z) (2.2) 
 
 From equation (2.2) it follows that the coefficient of the 
 equation (2.1) 
 
 L(, . X) . ^-.-l^^£^ (2.3) 
 
 4(h^ - l)z 
 
 is a function of the difference |j. - X. Equations (2.2) and (2,3) 
 give this dependence in parametric form. 
 
 From equation (2.3) it follows that the function L(n - X) 
 
 is negative for z2 > (h^ - 2)/h and positive for 
 
 z2 < (h2 - 2)/h2. For z^ = (h^ - 2)/h^ the function L(u - X) 
 becomes zero. According to equation (1.7) this function corre- 
 sponds to the Mach number 
 
 M = 2/ v^3^ = 1.565 
 
 S. A. Christianovich showed (reference 8) that for a given 
 value of the Mach number there is a change in the direction of 
 curvatiire of the characteristics in the flow plane. The graph of 
 the function L{\i - X) is shown in figure 1. 
 
MCA TM 1239 
 
 If the right side of equation (2.2) is expanded in a series 
 in powers of z, 
 
 \L^ = (h^ - 1) z - ^i-^ z^ + . . . (2.4) 
 
 from which 
 
 u. - A h^ + 1 , ^,3 
 
 z = -f^ — - + — — — — {[i - xr + . . . 
 
 2(h2 - 1) 24(h2 - 1)3 
 If this series is substituted in equation (2.3), 
 
 w .N h^ - 2 (h^ -t- 1) (h^ - 2) + 6 , ,v 
 
 ^^^ - ^) = pTt; xT 79 T? (u - A) + . . . 
 
 ^[^^■ - Aj 24(h2 - 1)2 
 
 Set h^ = 6. Then 
 
 L(n - A) = — — - 0.057 (^ - A) + . . . 
 
 For M > 4, assume 
 
 2 
 
 L(n - A) = 
 
 U - A 
 Equation (2.1) then becomes 
 
 ^2$ _ 2 /S$_ S$\_ . ,- _. 
 
 Equation (2.5) is the equation of Darboux with integral 
 coefficient. The general integral of this equation has the form 
 
 (Dfx ^) ^^ ^(^) - ^^^^ - ^'(^) -^ ^'(^^) p X(A) - Y(u) 
 
 ^^'^^ ^^ "SAB^ A -.a - (x.^,)2 ■ (A- ^)3 
 
 (2.6) 
 
 where X(A) and Y(i-i) are arbitrary functions of their arguments, 
 The expression (2.6) is the asjTnptotic integral of the exact 
 equation (2.1) for z-^0, that is, for M-^oo. 
 
MCA TM 1239 
 
 Equations (l.lO) for the coordinates x and y, after 
 transformation to the characteristics X \i, becomes on the 
 hasis of equation (l.ll) 
 
 W 
 
 y = -^w 
 
 ^S«5 ^ S$V.„ X + n ^ 1 /S$ M,^„ X + U 
 
 ^^ ^ S$\ X + u 1 f 5$ S$\ , X + u 
 
 > (2.7) 
 
 -/ 
 
 From equation (2.2) for small values of z, 
 
 r, = U - X ^ H - X 
 
 " ~ 2(h2 - 1) " 10 
 
 If this value of z is substituted in equation (2,7) and the 
 arbitrary functions $(X, n) eliminated from equation (2.7), 
 vith the aid of equation (2.6) the final solution is obtained. 
 
 X = - - 
 
 10 
 
 ^(vn)2 
 
 ' x" - r 
 v(^-n)^ 
 
 x« 
 
 (^-^)^ 
 
 sin 
 
 + P- 
 
 (X-^)4 (^.^)5J 2 
 
 y = + i 
 
 "^ w 
 
 \2 /^ ..\3 / 2 
 
 .(A-U)' 
 
 (X-u)^ 
 
 10 (^1^ . 6 ^li^ ^ 12 ^^^ sinH^ 
 V(X-n)3 (X-n)4 (x.^,)5y 2 
 
 Consider the equations for the velocity potential cp(W,9) 
 and the stream function •^{V,6): 
 
 S0 
 
 Sw 
 
 ^W 
 
 w ^ 
 
NACA TM 1239 
 
 If the variable z is introduced in place of the modulus of the 
 velocity W according to equation (1.7) and equation (1.8) used, 
 these eqviations are transformed to the form 
 
 K K K+1 >\ 
 
 ^=-(l+hz) (h-l) (l+z)z ^ 
 
 2-K K+1 
 
 ^cp /, .2 lvK+1 ,, 2n-1 /^2 ,v-l 1-K ^\\l 
 
 >(2.8) 
 
 dop /. .2 lvK+1 ,- 2%-l /,2 ^v 
 ^=-(l + hz) (1+z) (h-1) z ^ 
 
 y 
 
 For small z, equations (2.8) may he replaced approximately hy 
 the equations 
 
 K K+1 
 
 |?=- (h2 -l)l-^z^-"|^ 
 do dz 
 
 2-K _5fl 
 ^ _ ,^2 , \K-1 _1-K S4' 
 3z 
 
 /v2 , \^--i- J 
 = - (h - 1) z ^ 
 
 If the velocity potential op is eliminated, the equation for the 
 stream function \1/ is 
 
 i2 _ 1)2 ^ _ ^ii' ^ ^ ^ ^ 
 
 (h^-l)^^-^.i^f^^ = (2.9) 
 
 ?z 
 
 ae2 Sz^ ^ ^2 
 
 the characteristics of which have the form 
 
 X = - (h2 - l)z u = e + (h^ - l)z 
 
 If eqviation (2.9) is referred to the characteristics, it is trans- 
 formed into the form 
 
 h^^ ^ h2 [Mi SuX _ 
 
 ^rs;: ■" 2(x - u) l^x " ^^ = ° 
 
 Setting h2 = 6 gives 
 
 5fl^Tf,(|-M|) = ° (-0, 
 
NACA TM 1239 
 
 The equation of Darboxuc with integral coefficient, which is anal- 
 ogous to equation. (2.5), is integrated in finite form. 
 
 5. Criterions of similarity. - By examining equation (1.9) 
 for Mach nunibers near 1 (z-^oo) and also for large Mach numbers 
 (z— ^0), certain criterions of similarity can be established 
 that may be useful in evaluating experimental data obtained in 
 wind-tunnel tests. For large values of z, equation (1.9) can 
 be replaced by the approximate equation 
 
 S2$ (h2 - 1)2 ^2$ 3 3^ 
 
 and for small values of z by the approximate equation 
 
 |!|. (•h2-l)2£|,^_:^|^.0 (3.2) 
 Sz2 ^02 z dz 
 
 A thin slightly cambered airfoil at small angle of attack is 
 now considered in a plane-parallel nonvertical gas flow with Mach 
 number Mq at a large distance from the airfoil. The profile 
 
 chord is denoted by I and its maxijmam thickness is denoted by 5. 
 If it is assumed that Mq is near lonity, in equation (3.1) 
 
 = 0*6/2 z = z»Zq (zq = I/ZMq^ - 1^ 
 and equation (3.1) becomes 
 
 _d^ h^ - 1 ^2$ 3 ^d, 
 dz 
 
 2 6-^ h - 1 6"^ 3 d<P _ ^ ,„ _. 
 
 where 
 
 Ki = I y(M2 - 1)3 (3.4) 
 
 The number K-^ may be called the criterion of similarity in 
 
 the sense that two flows having different Mach numbers about two 
 airfoils of different thicknesses 5 and different chords Z but 
 for which the values of K-^ are the same will be determined by 
 
 the same "nondimensional" equation (3.3); hence, for the drag 
 coefficient C^^ 
 
 C^ = f (%) 
 
NACA TM 1239 
 
 In em entirely analogous manner starting from equation (3.2), 
 a second criterion of similarity valid for large Mach numbers is 
 
 ■^■1^ 
 
 1 s I M^ 
 I 
 
 This criterion of similarity has recently been obtained by Tsien 
 by a different considerably more complicated method. 
 
 Translated by Samuel Eeiss 
 National Advisory Committee 
 for Aeronautics 
 
 EEFEEENCES 
 
 1. Chaplygin, S. A.: Gas Jets. NACA TM 1063, 1944. 
 
 2. Christianovich, S. A.: Flow of a Gas about a Body at High 
 
 Subsonic Velocities, Eep. No. 481, CAHI. 
 
 3. Tsien, Hsue-Shen: Two-Dimensional Subsonic Flow of Com- 
 
 pressible Fluids. Jour. Aero. Sci., vol. 6, no. 10, 
 Aug. 1939, pp. 399-407. 
 
 4. von Karman, Th.: Compressibility Effects in Aerodynamics. 
 
 Jour. Aero. Sci., vol. 8, no. 9, July 1941, pp. 337-356. 
 
 5. Franklj F. I.: On the Theory of the Laval Nozzle. Izvestia 
 
 Akademii Nauk SSSE, Ser. Matematicheskaya, vol. IX, 1945. 
 
 6. Falkovich, S. V.: On the Theory of the Laval Nozzle. NACA 
 
 TM 1212, 1949. 
 
 7. Christianovich, S. A.: Approximate Integration of Equations 
 
 of the Supersonic Gas Flow. Trans. File X-121, Ballistic 
 Res. Labs. (Aberdeen Proving Ground, Md.), March 11, 1948. 
 
 8. Christianovich, S. A.: On Supersonic Gas Flows. Eep. No. 
 
 543, CAHI, 1941. 
 
10 
 
 NACA TM 1239 
 
 -1 
 
 1 2 3 A ' 
 
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