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 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1295 
 
 GAS FLOW WITH STRAIGHT TRANSITION LINE 
 
 By L. V. Ovsiannikov 
 
 Translation 
 
 'Ob Odnom Gazovom Techenii s Priamoi Liniei Perekhoda. " 
 Prikladnaya Matematika i Mekhanika. Vol. XIII, 1949. 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 120 MARSTON SCIENCE UBRAm 
 RO. BOX 117011 
 
 GAINESVILLE. FL 32611-7011 USA 
 
 Washington 
 
 May 1951 
 

NATIONAL ADVISORY COMI-HTTEE FOR AERONAUTICS 
 
 TECHNICAL MElvlORANDUM 1295 
 
 GAS FLOW WITH STRAIGHT TRANSITION LINE* 
 
 By L. V. Ovsiannikov 
 
 On the basis of the solutions ohtained by S. A. Chaplygin 
 (reference l) , an investigation was made of the limiting case of 
 a gas flow when the constant pressure in the surrounding medium 
 is exactly equal to the critical pressure for the given initial 
 state of the gas; the results are presented herein. For a jet 
 flowing out of the opening in a vessel with plane walls, it is 
 sho'v/n that equalization of the flow in the jet is attained at a 
 finite distance from the start of the free jet, the line of transi- 
 tion being a straight line . 
 
 1. According to Chaplygin (reference l), every problem on the 
 determination of the subsonic flow that is satisfied by some condi- 
 tions reduces to the solution of the system 
 
 S9 
 
 2T ^ 
 P at 
 
 (1-T) 
 
 ^^"2aT(l-T)P^l^ 
 
 •^ 
 
 y 
 
 (1.1) 
 
 or the equivalent system 
 
 St 
 
 2T 50 
 
 ^t (l_T)P^'f 
 
 ST ^ _ 2aT(l-T)P+l 50 
 5<P a-T St 
 
 (1.2) 
 
 for the corresponding boundary conditions. 
 
 In these equations, \if = V(e,T) and <P = Sp(e,T) 
 function and the velocity potentia.., respectively^ t = "V /T' 
 
 are the stream 
 
 •2 Ar2 
 
 max^ 
 
 where 
 
 *"0b Odnom Gazovom Techenii s Priamoi Liniel Perekhoda," 
 Prikladnaya Matematika i Mekhanika. Vol. XIII, 1949, pp. 537-542. 
 
2 NACA TM 1295 
 
 V is the magnitude of the velocity vector and Vjj^g^ is the maximum 
 
 value of V that satisfies the given initial state of the gas; and 
 Q is the angle between the velocity vector and the x-axis . Further- 
 more, (3 = 1/(k-1) is the adlabatic power of the function of the 
 density on the temperature, where for brevity, 
 
 = 1 = ±^ 
 2P+1 K+l 
 
 so that the value T = a corresponds to the critical velocity 
 
 V K+1 
 
 er \l ,>■ T ^max 
 
 Chaplygin (reference l) gives the solution of the problem of 
 the flow of a gas jet out of a vessel with plane walls forming an 
 single of 180O into a medium with pressiire pq = constant in the form 
 
 00 
 
 Q 
 
 n=l 
 
 * '■ . d - ) - — sin 2n0 (1.3) 
 
 '•nO 
 
 T(i-T.)P (l-t)P 
 
 OB 
 
 1 + / i -^ x„ cos 2n0 
 ^ " ^nO J 
 
 2n0 (1.4) 
 
 where this solution satisfies the following boundary conditions 
 (fig. 1): 
 
 ij; = - 2 Q on ABC 
 
 \tr = I Q on A'B'C 
 
 In equations (1.3) and (1.4), Q is the relative quantity of 
 flow in the gas jet and C is an arbitrary constant depending on 
 the choice of the origin from which the values of cp are computed. 
 The magnitudes z^, z^q' ^^'^ ■% ^"^^ defined by the equations 
 
NACA TM 1295 
 
 ^ 
 
 "n 
 
 ^nO 
 
 
 
 y 
 
 (1.5) 
 
 (n = 1,2, . . .) 
 where yn('^) is that solution of the hypergeometric equation 
 t(1-t) Yn" +[2n + 1 (p-2n-l)T Jy^^' + pn (2n+l) y^ = 
 
 which is obtained for t = 0> Tq denotes the value of t cor- 
 responding to Pq. 
 
 In Chaplygin's investigation, it was assumed that T -^To^cc. In 
 this case, equations (1.3) and (1.4) converge everywhere in the region 
 of flow and give the proposed solution of the prohlem. The case 
 where Tq = a is considered in the following development: 
 
 It is recalled that the fimctions Zn(T) and xn(T) for 
 O^T^a satisfy the following inequality of Chaplygin: 
 
 r(l-T)^^ 
 
 23 
 
 llo(i-'^o)'^^J 
 
 n 
 
 "nO 
 
 ^n 
 
 (1.6) 
 
 ^-d--^) nl/3 (l-T) "" Va(l-T 
 
 q ='\/2p2 (l+2|3) = constant 
 
 (1.7) 
 
 2. It shall he shown that the potential of the velocity cp given 
 hy equation (1.4) for T — >Tq 
 
 (a) increases without limit if TQ«=ca 
 
 (b) remains finite if T q = a 
 
WACA TM 1295 
 
 It shall "be assiimed that the limit T — ^Tq is effected hy 
 moving along some streamline. The following notation is introduced: 
 
 e =^ 
 
 (l-T)^P 
 
 '^o(i-'^o)^^ 
 
 P(0^'^j''"n) = > - — - Xn cos 2n9 
 
 n=l 
 
 It is readily seen that for O-^^- 
 
 ■ a, then -^ ^ "^ 1 < 
 
 For the potential cp at the points of the x-axis (9=0), the 
 following expression is obtained from equation (1,4): 
 
 ^»=c. 
 
 1 r dT 
 
 2J T(l- 
 
 T)P (l-T)PL 
 
 1 + P(0,TjTo) 
 
 (2.1) 
 
 On the basis of the inequalities of equations (1.6) and (1.7)^ 
 for the magnitude of P(0,TjTq), 
 
 Va.(l-T) /__ri^ 1-T / 
 
 n=l n=l 
 
 _1 
 
 nV3 
 
 / a-T 
 
 \ 1 Zn 
 
 a(l^) 
 
 / , ^ -no 
 n=l 
 
 
 (2.2) 
 
 Assertion (a) now follows in an obvious manner from the second 
 of inequalities (2.2) because z^/z^^q — >1 for t — >Tq and the sum 
 
 of the series with the general term z^/hz^q therefore increases 
 without limit; whereas the coefficient preceding the term, because 
 Tq ^ a, approaches a positive limit. 
 
 From assertion (b), it is seen that 
 
 n-1 
 
 i ^n = _ log(l-^) = - log 
 
 ■(1-t)2P 
 
 _ To(l-To)2ii_ 
 
 1 - 
 
MCA TM 1295 
 
 ty virtue of which, for Tq = a as T— »a in equation (2.2) on the 
 left, the first component approaches zero and the second component 
 approaches a finite magnitude equal to 
 
 The same reasoning, together vith the first of inequalities 
 (2.2) is repeated for P(9,Tja) for 9^0, so that the assertion 
 is completely proven. 
 
 3. From the previously proven houndness of the velocity potential 
 it follows that at a finite distance from the opening BB', the jet 
 is intersected Toy a certain line L along which t = a, so that the 
 velocity is equal to the velocity of sound. It will now he shown that 
 at all points of L, 9=0. 
 
 First, it will be observed that along any fixed streamline, 
 9 veiries monotonically, as is true of the boundary streamline, and 
 the trsmsformation (9,t) — >^(9,i|/) is a single sheet transformation 
 at every interior point of the flow region. Next, by fixing some 
 
 value ^' = W for which |ij/|< ± Q, the limit is approached as 
 
 T— .^a and 9q ~ lim 9 is set for t— ♦a and if = f; this lirndt 
 exists because of the finiteness of 9 ana the monoticity of its 
 variation. 
 
 Approaching this limit in equation (1.3) for i|f = t yields 
 
 . Z / n 
 
 ®o -0 - -^}= - ^ if 9o>o 
 
 ^ -^ . _ 00 _ ^ _ sin 2n0o = < 
 n=l 
 
 -«0 -(-1-^)= + !" 90>0 
 
 that is, in all cases for Qq / 0, \^\ = p- Q, which contradicts the 
 assumption that l^irl^-^ Q. Hence, the equality 9 = must hold. 
 
 It will now be sho-vm that the line L is straight: Along L, 
 9 = constant) this result is readily obtained if the previously con- 
 sidered transition to the limit is carried out in equation (1.4). 
 
NACA TM 1295 
 
 It is sufficient to note that every displacement in the plane xy 
 is connected with the corresponding displacement in the plane cpV hy 
 the relation 
 
 , cos 6 ,_ sin ,, /-z T \ 
 
 dx = -y- d,p - ^^^^-^ dt (3.1) 
 
 from which it follows that in a displacement along L in the 
 xy-plane, dx = 0, because in a displacement along L in the 
 cpvjr-plane, according to what was previously proven, 8 = and 
 dp = 0. The line L in the xy -plane is thus a straight line 
 perpendicular to the x-axis. 
 
 The equalization of the jet occurs along the line L. Behind 
 this line the jet hecomes uniform, flowing with constant velocity 
 everywhere equal to the velocity of sound. 
 
 The distance of the line L from the edge of the opening a^tlII 
 be computed. Along the boundary of the jet, ij; = constant and 
 T = a = constant, so that at the points of this boundary, equa- 
 tion (3.1) assumes the form 
 
 dx = S2±± p d9 
 Vcr S^ 
 
 Substituting the expression for (p (equation (1.4)) taken 
 for T = a yields 
 
 00 
 
 dx = -^ — 7-T7- ) 2xn(a) sin 2n0 cos d0 (3.2) 
 
 nV (l-a)P^ 
 
 n=l 
 Inasmuch as 
 
 2 sin 2n0 cos = sin (2n+l)0 + sin (2n-l)0 
 
 integrating equation (3.2) from x = x-g, 0= — «, to x = x^, = 
 yields 
 
 ^L ~ ^B - 
 
 nV^Jl-a.) 
 
 OP 
 
 Q \ 4n , V 
 
 1-a)!- Z_^4n^-1 
 
 ^^' • n=l 
 
NACA TM 1295 
 
 If h denotes the rndth of the jet where it is uniform, then 
 Q = V^j.(l-a)P h, so that the required distance is finally obtained 
 in the form 
 
 '-Za:i=<n(a) (3.3) 
 
 n=l 
 
 4. If any two streamlines of the obtained flow are taken as the 
 walls of a certain nozzle, the subsonic flow within this nozzle \t±11 
 be determined by equations (l.o) and (1.4). This flow becomes ^JJliform 
 at a certain distance and has a straight transition line. This fact 
 corresponds entirely to the result obtained by S. A. Christianovich 
 in his investigation (reference Z) , where a general device is given 
 for the construction of a Laval nozzle with straight transition line. 
 
 The first derivatives of 9,T v;ith respect to 9^1^ are 
 estimated near the line L inasmuch as the behavior determdnes the 
 possibility of continuing the obtained flow across L as a super- 
 sonic flow. The investigation of this complicated problem has been 
 started and, it is hoped, will be presented in a forthcoming report. 
 Remarks herein will be restricted to the following: 
 
 In the first place, from the previously noted properties of the 
 line L, it follows that the derivatives Ss/S^j/, S V^^^ ^^^ S0/5(;p 
 are equal to zero on L. 
 
 In the second place, the derivative St/S(P is evaluated at the 
 points of the x-axis. It is sufficient to consider the derivative 
 dCij/^T, inasm.uch as, on the x-axis, because 9 = and i|r = constant, 
 
 St 1 
 
 Sep Scp/ST 
 
 (4.1) 
 
 Substituting equation (1.3) for ijr in the second of equa- 
 tions (l.l) for T = a and setting z^^^^ = Zjj(a) yield 
 
 Sep Q(a-T) 
 
 ^"^ 2naT(l-T)3+l 
 
 L f ^na 
 
 1 + 2 > cos 2n9 
 
 n^ 
 
8 
 
 NACA TM 1295 
 
 or, on the x-axis where 0=0 
 
 Q(a-T) 
 
 Sep 
 
 S^ iTaT(l-T)e+l 
 
 _ oo - 
 
 2 / / ZncL 
 
 n=l 
 
 (4.2) 
 
 For estimating the value of the expression in brackets, which 
 is denoted by S, the last of relations (1.5) and the inequality of 
 Chaplygin (equation (l.V)) yield 
 
 Ma(l-T) 1-T ^dT ° ''' '^-^ Ya(l-T) 
 
 dT " ' T 'ycoCl-r) 
 Integrating equation (4.3) from t^t^^^o to T = cc yields 
 
 Pa 
 
 (4.3) 
 
 ■ex 
 
 \CL 
 
 n 
 
 I I ^-^ dT + an2/3 ^>.log ^ n J: / ^""^ 
 
 ^lo.{l-T) ' 1-T Z^ ^V^-^l--^^ 
 
 dT 
 
 Raising the upper and lowering the lower limits smd carrying 
 out the integration give 
 
 2n 
 
 (a-T)3/2 ^ qnV5(a_T) ^^^^ z^a 2r^ (a-T)3/2 
 
 If a-T = z(0'^2^a-T]_), where z — »»C as T — »a and 
 
 2 
 
 (4.4) 
 
 r 
 
 3T2^(Ja(l-a) 
 
 -5>0 
 
 --L 1-a 
 
 oaV a 
 equation (4.4) assumes the forni 
 
KAGA TI'.l 1295 
 
 Then the inequalities 
 
 exp(-rnz-V2-q-.n2/3s)'^ £^^exp{-bnz^/^) (-^-S) 
 
 ^no. 
 
 are obtained. 
 
 Because of equation (4.5)^ the following inequalities are 
 obtained for 3: 
 
 S-. = y^ exp(-rnz-^/2-q n2/3z)-«;CS<C- + / exp(-6nz3/2) = sg 
 
 2 T 
 n=l n-1 
 
 Tlie sum So is readily computedj 
 
 1^32 = i + i ^i + -1— (4.7) 
 
 In order to estimate the svaa Sj.} 
 u^ = exp(-q-i^n2/3z) 
 
 v^ = exp(-rnz3/2) 
 
 k=C 
 
 Ojp^ = y "'^^n ~ / ^-^ -q-|_n2/3z-)'nz3/^ 
 n=0 n=:C 
 
 Applying the trajQsformation of Abel to 0^-, yields 
 
 m-1 
 
 %.. =XZ (%-%+l) Sn' + ^^mSm' (^-8) 
 
 n=0 
 
NACA TM 1295 
 For in-2/3 a3sz^(ti]+l)-2/3 o), where w = r"^/^(lo3 2)2/3^ 
 
 ^n 
 
 
 k=0 
 
 The differences Urj-Un+l are evaluated Toy the formulax of finite 
 increments . Therefore 
 
 " 'S.+l "" e:<5 |^-q-Ln2/3zJ - exp [- q.]^(n+l)2/t' zj 
 
 Replacing ^ by 1 yields the inequality / 
 
 ^. - ^+1>^^ (n+l)-l/? exp[-qi(n+l)2/3 zJ (4.10) 
 
 Combining equations (4.9), (4.I0), and ('l.ll) yields 
 m-1 m 
 
 ^m>/ 3 qi"(n+l)2/3 exp [-qT_(n+l)'V3 -J - ^ ) qin2/3 3 exp [-q-Ln2/3 zj 
 
 n=0 n=l 
 
 The function . 
 
 f(x,y) - x'y2/3 exp(-xy2/3) (4.II) 
 
 is considered for y^O, x5»-0. 'Tne derivative \rith respect to y 
 varies as 
 
 Of 2 -1/3 , 2/3w^ 2/3 N^ 
 -— = - xy ' exp(-xy ' )(l-xy ' )^ 
 
 oy -^ 
 
 r 
 
 >0 for 0^y-«= x"-^/'^ 
 
 ^ ^ -3/2 
 
 ^ for y = x ' 
 
 3/2 
 
 =^ for y >• x 
 
 For fixed x>C, the function f(x,y) increases at first from 
 
 to e-1, then decreases and, for y — »od, approaches zero. Hence, 
 for any x>.c, the inequality 
 
MCA TM 1295 
 
 11 
 
 1 +-/_- T^/^ X expC-nV^x); 
 
 n=l 
 
 is obtained. 
 
 m 
 
 xy^/^ exp(-xy2/-^) dy = g 
 
 (4.12) 
 
 The value of the integral g is obtained by applying the 
 substitution xy^/^ = t. Then 
 
 3/2 
 
 =/2 e-^ dt fX-m2/3x) (4.13) 
 
 Substituting equation (4.13) in eauation (4.12), setting 
 X = a-^z, and noting that m^/^z^-jj yield 
 
 q, nS/i z exp(-qTn2/3z)> — 5 — ,_ 
 
 2qi3/2 
 
 n=l 
 hence 
 
 q.icB 
 
 :2/3 e-t dt 
 
 :-3/2 _ 1 
 
 2a. "/^ 
 
 pq.-i"j 
 
 oo 
 
 .2/3 
 
 e~^ dt 
 
 r3/2 (4.14) 
 
 On the basis of inequalities (4.6), (4.7), and (4.14), it inay 
 be concluded that there are two positive constants 5i and n such 
 that the inequalities 
 
 >lz3/2 
 
 
 /2 
 
 (4.15) 
 
 hold over the entire interval 0-=rz^a - T-j^. Returning to the 
 variable t and coKparing equation (4.2) with equation (4.15) gives 
 the following result: Two positive constants 62 and T2(o2'*^'2^ 
 exist such that for any T in the interval Tt^.^Tc^'^^ 
 
 6'^ 
 
 SgV^ ^"^ T2V^ 
 
 (4.16) 
 
12 MCA TO 1295 
 
 From equations (4.1) and (4.15) there follows finally 
 
 &2//H^>|l>r2/\/(^ (4.17) 
 
 which is proven for the points of the x-axis . 
 
 In a similar manner, it may he shown that on the x-axis the 
 second derivative ^^/h^> remains finite as T — *a.. 
 
 Translated by S. Reiss 
 National Advisory Committee 
 for Aeronautics 
 
 REFERENCES 
 
 1. Chaplygin, S. A.: Gas Jets. NACA TM 1063. 1944. 
 
 2. Astrov, Levin, Pavlov, and Christianovich: On the Computation of 
 
 Laval Nozzles. Prik. Mat. i Mek., vol. VII, no. 1, 1943. 
 
NACA TM 1295 
 
 13 
 
 NACA-Langley - 5-25-51 - 1000 
 

 
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UNIVERSITY OF FLORipA 
 
 3 1262 08106 669 7 
 
 UNIVERSITY OF aORlDA 
 DOCUMENTS DEPARTMENT 
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 P.O. BOX 117011 
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