}^hCf\fifh'l^('(^^m i^mLLtJkDON NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1260 EXACT SOLUTIONS OF EQUATIONS OF GAS DYNAMICS By I. A. Kiebel Translation << 1 'Primer Tochnogo Reshenia Ploskoi Vikhrevoi Zadachi Gazovoi Dinamiki." Prikladnaya Matematika i Mekhanica. Vol. XI, 1947. Washington June 1950 ; Documents department NATIONAL ADVISOEY COMMITTEE FOE AERONAUTICS TECHNICAL MMORANDUM 1260 EXACT SOLUTIONS OF EQUATIONS OF GAS DYNAMICS* By I. A. Kiebel The equations of the tvo -dimensional stationary problem of gas dynamics are of the form u Su Bu V = _ 1 P ap u ^x' ^§7 = _ 1 P Bp ay Spu hx ^ Spv Sy (1) where u and v are the components of the velocity along the coordinate axes x and y, respectively, p is the pressure, p the density, and K= c_/c^, the ratio of specific heats. The equation of continuity permits the introduction of the stream function from the equations Pu = — By pv hx (2) The last of equations (1) gives 1 K P = P* (>!/) (3) ♦"Primer Tochnogo Reshenia Ploskoi Vikhrevoi Zad.achi Gazovoi Dinamiki." Prikladnaya Matematika i Mekhanica. Vol. XI, 1947, pp. 193-198. NACA TM 1260 where ^ is a certain function only of ^. The first two of eq,uation3 (1) give the Bernoulli law (4) where i^ is a function of ^^ and the equation for the vorticity Is dx ■ Sy ■ ■ ^\d^i' "k-1 ^ dv|// ^^^ For the solution of the vortex problem in which at least one of the derivatives diQ/d>i/ and d^/d'4' is different from zero, it is convenient to pass from the variables x and y to the vari- ables T* = X £Lnd ^|/. Eq.uation3 (2) and (5) then assume the form (See, for example, reference 1.) ^ V hx* ~ u ^ Sy 1_ pu (6) Sp _ _ Sv hii hx* J The problem reduces to the determination of the five functions u, V, p, p, and y of x* and '^ from eq.uation3 (3), (4), and (6). By using the last of equations (6), the function X is intro- duced with the aid of the equations P = ax Sx* ax V = - h^ J (7) NACA TM 1260 The first two equations of equation (6), on the "basis of equa- tions (3) and (4), then assume the forms K-1 _ 1 -o - 1^ * te)' (8) ^_ , to 1 K K-1 2i 2K IbX K -1 * V Sxy 1 "2 (9) where the asterisk on x has been dropped. By differentiating equation (8) with respect to ^ and equa- tion (9) with respect to x, j can be eliminated and a single equation for the function X obtained. In order to obtain an example of the exact solution of the system of equations, i^ is set equal to a constant: X = - H (\^) X K-1 K+1 (10) where H is a certain function of 'I' to be determined. Equa- tions (8) and (9) then give ^ = H' X ox K-1 K+1 2i, 2K , /k-1 "^ ^ K-1 \K+1 K-1 Hi + H'^ -2 ^]-2 K+ll (11) |l.W-lH)'x^^^k- d\l/ Vk+1 / > ° K-1 2JL,(1z1h]' .h.2 K-1 Vk+1 1 K+1 (12) NACA TM 1260 When equation (ll) is differentiated with respect to '^ and equation (12) is differentiated with respect to x, terms with the same degree of x are collected and after simple trans - formations there is obtained _1 Wl J 2 ^ (13) H' (H'^+KHH")' = - (k-1) H" (H'^+KHH")^ Successive integration of the second of equations (13) gives H'^ + KH H" = CiH'-'-"'^ J5_ / (14) H = Co (C,-H'^+'*' '2^-1 From the last equation, the relation between "^ and H can "be found with the aid of quadratures. It is more convenient, how- ever, to introduce a numbering of the streamlines directly with the aid of H' and not '^. Thus, equations (14) give H as a function of H' and, using equations (13), permit finding ^ in terms of H' in the form l-K 2n Substituting in equation (ll) and integrating yields, for the streamlines. K-1 _2 »«^ _1 y = I H-x*^""^ [21^ - CiH'^-'^x "^^^ ^ dx + F (E-) (16) NACA TM. 1260 Comparison with equation (12) shows that F = constant and without loss of generality can he taken as F s 0. For the determination of u, v, p, and p the following relations are written : u^ = 21^ - C]^E'^"'^x 1-K 1+K ^ V = H'x K-1 K+1 y (17) p = Cg fci-H K 2k l+,^\ 1+K 1+K (18) 2K+1 2 H' X (19) For simplification it is convenient to introduce the nondimen- sional variables x and y and in place of H' introduce t\ with the aid of the following equations; X = Lx y = Ly (20) H'-'-'*'^ = C 1 "2 where L = (2i_) 1 K+1 J^ 2 K-1 K-1 (21) Equations (16) ajid (17) then assume the following simple form: NACA TM 1260 m K-1 K+1 _1 l] dm ^ X = Tim (y) -2 ^-4 u2= 2i„ (1-m *'^^) ~\ V"oi K-1 2 1 K+1 /TT" 1 Itfl m = A/2i- — m (22) (23) gives If K = 1.4, the integral in equations (22) is evaluated and y = 6 A^/m - 1 1 + I ( ^f^-1) + I ( .^-1)2 (24) and the streamlines can therefore he easily constructed. Figure 1 shows the streaml-nes for t^ ± 1, 2, 3, . . . , 19. This motion possesses hoth supersonic and subsonic velocities, for the line of transition (shown dotted in fig. 1) is obtained if 2 2 o 1^-1 • u + V = 2 r- i_ K+1 ° that is. or "^• 1 - m 2±l .2 !i^ K+1 1 K+1 K-l + — m K+1 o K-1 ^ K+1 . _ 1 V n / J (25) NACA TM 1260 It is possible, without difficulty, to construct the character- istics in the x,y- and u,v-planes. Instead of the equation of the epicycloid in the u,v-plane, in the case considered a more complicated equation arises. In the vortlclty problem, along the characteristic there occurs ctg a sin a cos a , , , dp T — - — dw = ± d log ^ (26) w K-1 vhere a is the Mach angle, w the magnitude of the velocity, and P the angle of Inclination of the velocity to the x-axis. Equa- tions (23) yield _ 21^-u^ 21.-w^ cos^ p ■ n = ° = —2 (27) 1^ v^ sln^ p But ^ depends only on H', that is, on t^ . According to equation (15), the following is obtained: ^ = 1^ (=^ #""=1 T <^^-'^ , .ill p K-1 2K «ll/n_ «-1\k ^1 /„2 „, 2 ,.-7TT /o, „2. ir(^2^)^ .„ / , „ (v-Bln-^p)K.l (2i,-w^)K.l (28) 2iQ-w cos" p Substituting this value of ■d in the right side of equa- tion (26) yields the differential equation for the characteristics In the PjW-plane. Finally, the question arises whether it is possible to obtain such vortex motion by transition through a surface of strong dis- continuity of some other kind but with Irrotatlonal motion. This NACA TM 1?60 problem may "be answered in the affirmative. For on a surface of discontinuity there must^ among others, be satisfied the relations K-1 2 K 0_ K+1 K+1 n fl ^ "\ (29) _> where ^+) "+j and 0^ are the pressure, density, and velocity of propagation of the surface of discontinuity on one side of this surface and p_ emd p_ are the pressure and density on the other side. The magnitudes p^ and p^. may be taken from the vortex motion. The magnitude 6^ is found in terms of the elements des- cribing the motion and in terms of the inclination of the surface of discontinuity. Finally, p_ and p_ can be connected by the relation p_ = ■diP_, where is a constant (up to a certain degree of arbitrariness) magnitude. (At the left of the surface of discontinuity the motion is irrotatlonal. ) Inasmuch as (30) then where Inasmuch as GTi^ (ti2-1) il ^ i-1 2 1 K K 2 M = ^e+ 2K K+1 "A (31) (32) 6 = V COB 6 + u sin & + + + NAG A m 1260 where 5 is the angle between the normal to the surface of dis- continuity and the x-axis (the normal is directed toward the "positive" region); M may be expressed in terms of known magni- tudes and 6. By expressing tan 6 in terms of the derivative of x with respect to t| along the surface of discontinuity and using equations (22), (18), and (19), the differential equation for determining the surface of discontinuity (K = 1.4) is obtained after simple transformations: dm dTl m m-1 1+n \l \[^ 1 ± (33) where M is expressed in terms of i^ from the transcendental equation (31) in which the constant G is, to a great extent, arbi- trary. The velocity in the "negative" region will be determined first on the surface of discontinuity and then extended on the negative region by the usual graphical method of Busemann, A model of the motion about a contour having an angle is obtained. The surface of discontinuity extends out from the angle and on passing through the angle the motion reconverts to the rotational motion herein considered. A solution analogous to that developed can also be obtained for the problem with axial symmetry. Taking for the independent variable the distance r* = r from the axis of symmetry z and the stream function \|/ yields the relations br* (34) r*pvi, J where v^, and v, are the velocity components. As before, 1 K i{^)p 10 NACA TM 1260 Bernoulli's lav will have the form K-1 1/2 2\ Ki = i. The equations of Euler give dr* ~ ~SY The function X(r*,\l') can therefore be introduced from the conditions ^z = - Sl r* ar* J (35) Equations (34) and (35) now permit writing (the asterisks on r are dropped) Bz ^ _ ax i_ bii ~ S^ v^ K+l or Vor/ V. 1 axl 1 (36) where 2i. L i!l^^(^^)fiax 1^-1 \ r dr , K-1 By eliminating z from equations (36), a single equation for the determination of the function X is ohtained. Particular NACA TM 1260 11 solutions, analogous to the solutions in the first case, can he constructed hy seeking X in the form X= - r '^"'^ E(^) Translated by S. Reiss National Advisory Committee for Aeronautics . REFERENCE 1. Kochin, N. E., Kiebel, I. A., and Rose, N. V.: Theoretical Hydromechanics. Pt. II, 1941, p. 80. NACA TM 1260 13 12 11 10 9 8 7 6 5 4 3 2 1 J / / / y y ,y y y y y^ d / y / / ^ y y y^ y^ y / A y y y" y / / y y y ^ y x Supersonic zone ^ / / / ^ y y y" y y' ^ ^ / 1.2 1.4 1.6 1.8 y ^> ^ ^ ^ / f y X ^^ ^ ^ "^ - ^^ / / y y ,^ ^ ^ ^ -^ / y / /* / ^ ^ ^ ^ ^ -^ _,-*« / r / / y X > ^ ^ ^ .^ ::^ ^ § •^ / / / / y y. r^ ^ P ■^ / / / 7 / y y > < |^Su"bsonic~ zone f 7 ^ / ( f / / / / 7 'A y ^ y y 'A 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 1. NACA-Langley - 6-27-50 - gso to o ■H 0) W •H cd ra O (u On CO \ O u 0) \ W > < \ § ID U \ -H < < tJ a Z / Cm O o -p u ■p CO CO :-{ ^ d OJ 5 o .^ •r-t o; o +J •H VO • 3 y; OJ K (3 O +J o 2 4-> 0) CO -H CO H a OJ rO /3 o ^ < •H ■H (D o 10 -P •H VO CO 3 y: OJ (D H rH O ^H O • ITA gl CO < S CTn Eh .H O CJ cd 1— 1 1 ^ pp gl P4 -p u t; th