UFDC Home  Search all Groups  World Studies  Federal Depository Libraries of Florida & the Caribbean   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
N A( M1o
r * .1 /  NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1260 EXACT SOLUTIONS OF EQUATIONS OF GAS DYNAMICS* By I. A. Kiebel The equations of the twodimensional stationary dynamics are of the form 6u u U V + = dx by U  + V  = problem of gas 1 ap P 6y P y + 1 = 0 oy u a I 1 p + p = 0 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 = cp/c_, the ratio of specific heats. The equation of continuity permits the introduction of the stream function from the equations Pu =  ay pv = 3x The last of equations (1) gives 1 p = pS (M) *"Primer Tochnogo Reshenia Ploskoi Vikhrevoi Zadachi Gazovoi Dinamiki." Prikladnaya Matematika i Mekhanica. Vol. XI, 1947, pp. 193198. 7/pb y P1 v Ps NACA TM 1260 where 6 is a certain function only of *. The first two of equations (1) give the Bernoulli law u2+v2 2 K1 K K + K p K 1 = io (M) where io is a function of I, and the equation for the vorticity is xv uy ax ay " odi ( dF K1 \ S KK d4 K1 dI) For the solution of the vortex problem in which at least one of the derivatives dio/d* and d4/d* is different from zero, it is convenient to pass from the variables x and y to the vari ables x* = x and *. Equations (2) and (5) then assume the form (See, for example, reference 1.) ox* u ay 1 34 pu Op v 5T 5x* The problem reduces to u, v, p, p, and y of and (6). the determination of the five functions x* and \ from equations (3), (4), By using the last of equations (6), the function duced with the aid of the equations P a V _ X is intro (7) NACA TM 1260 The first two equations of equation (6), on the basis of equa tions (3) and (4), then assume the forms x i2 2K i o 1 84' L2i K1 1 ciL axa 2K 21 K K 1 K1 1 (x/, 1 2 x K K1 1 K 2 2 \x \a where the asterisk on x has been dropped. By differentiating equation (8) with respect to * tion (9) with respect to x, y can be eliminated and a equation for the function X obtained. and equa single In order to obtain an example of the exact solution of the system of equations, io is set equal to a constant: K1 X = H (') x 1 (10) where H is a certain function of W to be determined. Equa tions (8) and (9) then give K1 6y K+12 H' x (2 OX 2 SK1 ( 1 io 2 51 2o  t F 2K Kl K1 \K+1 /K1 \K +1 K 1 H) K + H'2 K1 K K1 2 2 2L K+1 (11) 1 2 +1 + H'2 x K 1 (12) NACA TM 1260 When equation (11) 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 H" K1 H) K + H" \K / 2 (13) B' (H2 +KH H")' = (K1) H" (H' 2+ H") Successive integration of the second of equations (13) gives H'2 + KH H" = ClH'1'1 K (14) H = C2 (C_H 1+) K From the last equation, the relation between be found with the aid of quadratures. It is more ever, to introduce a numbering of the streamlines the aid of H' and not 4. * and H can convenient, how directly with Thus, equations (14) give H as a function of H' and, using equations (13), permit finding t in terms of H' in the form KI C K1/ 2 W02 2K K+1 1K K 2M K+1 H i1 K C, H 1+ K (15) Substituting in equation (11) and integrating yields, for the streamlines, yK1 L1 1 2 dr CIH ,1x + dx + F (H') (16) K NACA TM 1260 Comparison with equation (12) shows that F = constant and without loss of generality can be taken as F 0. For the determination of u, v, p, and p the following relations are written: r `2 S1K 1+K u2 =2io C1H'"Kx K1 v = H'x K+1 K 2K Sc (CI +K 1+ K 1+K p = C2 (CxE' I 2K+1 2 2Ki 2 (c1H1+) K+1 K1 + K (17) (18) (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 L (20) H, =1K =1 where i K+1 1 2 K1 K1 L = (?21o) C1 (21) Equations (16) and (17) then assume the following simple form: 6 NACA TM 1260 1s P 2 1  m= m +1 i d1 . y m (22) x = Tm () J u 21o (1m ) (23) K1 2 I1 +1 1 il v =21 m =A/2i m If K= 1.4, the integral in equations (22) is evaluated and gives y 6 1 [1 + 2 ( m) + ( 1)2] (24) and the streamlines can therefore be easily constructed. Figure 1 shows the streaml.nes for q 1, 2, 3, 19. This motion possesses both supersonic and subsonic velocities, for the line of transition (shown dotted in fig. 1) is obtained if 2 K1 u + v 2 1Ki that is, 2 K1 2 K1 2 2 i+1 1 K+1 K1 i m +m  2 K+1 S(25) or 2 K K+1 1 m K+1 2, NACA TM 1260 It is possible, without difficulty, to construct the character istics in the x,y and u,vplanes. Instead of the equation of the epicycloid in the u,vplane, in the case considered a more complicated equation arises. In the vorticity problem, along the characteristic there occurs ctg a sin a cos a d F dw = d log v Ki (26) where a is the Mach angle, v the magnitude 0 the angle of inclination of the velocity to tions (23) yield 2 21ou 2iow2 cos2 w2 sin2 B of the velocity, and the xaxis. Equa (27) But 0 depends only on H', that is, on equation (15), the following is obtained: K 1 According to 2K  ( 21)K+l TI C K1 2K 2 2 (v2 sn' 0) K'+l (2zio.W)v2 21w cos O (28) Substituting this value of 0 in the right side of equa tion (26) yields the differential equation for the characteristics in the O,wplane. 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 irrotational motion. This 1K K1 E~ 1K K1 Il r' h K (2 K+1 8 NACA TM 1260 problem may be answered in the affirmative. For on a surface of discontinuity there must, among others, be satisfied the relations Ki 2 K P  + ++ +1 +e+ (29) p1 P. 2 ( K+l K P+ 9 P+ P +06;+P) where p., p0, and 0+ are t] of propagation of the surface o: surface and p_ and p_ are t] side. The magnitudes p+ and motion. The magnitude e9 is cribing the motion and in terms of discontinuity. Finally, p_ relation p_ = diP _ where degree of arbitrariness) magnitl discontinuity the motion is irr he pressure, density, and velocity f discontinuity on one side of this he pressure and density on the other oP may be taken from the vortex found in terms of the elements des of the inclination of the surface and p can be connected by the t1 is a constant (up to a certain ude. (At the left of the surface of otational.) Inasmuch as 1 P". K + 19 P+) 0 b (30) then 1 Kl  = GTC (n21) Cl 2 C = Kl GZK " 2 M= p, I1 2(Al Inasmuch as  0 = u cos 6 + V sin 6 + + + where 2K K+1 (31) (32) NACA M4 1260 where 6 is the angle between the normal to the surface of dis continuity and the xaxis (the normal is directed toward the "positive" region); M may be expressed in terms of known magni tudes and 6. By expressing tan 8 in terms of the derivative of x with respect to r 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: m A .A/ 71+1e2 ) S3 M6 (1q21) (33) where M is expressed in terms of 12 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 z Vz r (34) z_ 1 or* r*pvr where vr and vz are the velocity components. As before, 1 NACA TM 1260 Bernoulli's law will have the form K1 1 ( 2zV 2) + 2 Kp 2 ('r ^z K1 p = i The equations of Euler give ovr* r*p z' rwp are at The function X(r*,W) conditions can therefore be introduced from the v = ax i W p r r* or* (35) Equations (34) and (35) now permit writing (the are dropped) asterisks on r Oz aX I 0* lJ vr (36) 1 K+1  r r/ Vr= [2io  2 1 K1 ic 1 , 2 2 kcw/ By eliminating z from equations (36), a single equation for the determination of the function X is obtained. Particular where NACA TM 1260 solutions, analogous to the solutions in the first case, can be constructed by seeking X in the form 2K1 2 X= r + H() Translated by S. Reiss National Advisory Committee for Aeronautics. REFERENCE 1. Kochin, N. E., Kiebel, I. A., and Rose, N. V.: Hydromechanics. Pt. II, 1941, p. 80. Theoretical NACA TM 1260 13 12  11  0 . 10 / Supersonic 1.2 1.4 1.6 1.8 800 7 00  2 00 Subsonic 1"0 '0 zone 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Figure 1. NACALangley 62750 50 U2 *4 In C1 0 o H . , 0 o UN ' c G \  L< Rc C (U 0 fl *r ,0 r 0 71C,, 40 b 0 4) 0 r 1'4 0 Cd \ s O z O 4 Xd 0 0 C .. L , 4 J 0 0 1 M S0 Ho S m O ,, u , 0 ( O 0 q4 0 m wo i cd .I u +3 pq Z Sm m 0 3 g + )  N ar) 4) C 4.) cd r01 0 x m ( 0 4 j a) 0 0 d S+ o +3 rd+ m m m H a 'd I 0 0 2 4> a W W IP 0 0 W 0 hH. ) d 0U' d mO CO U H 0 d Qj ( P 00 2 u rn tH 0 ro 0 ,0 i 3 0O 'hOc 02 0 CS 03 0 *0 l 0S cu C ,0 0 u 4 0 + 4 0 C k3 a 40 0 y j r; ra . O 4 d , c C 02 F i H .2U '0 0 O 3 00+k+0 SFQ Hl ' 4.P 0 0o a WWOJI B c ,o D r H 01 P, D U+j *k r0 0 + p ? CU IP4 0 4) BH 0 rdI ) a w SO'd m o P oErl D C m ao c r 3* c:c d (U0 ( : o i a p o f UNIVERSITY OF FLORIDA 1262 08 05 029 I 