ABE Ho. LlH29 
 
 ^ 
 
 NATIONAL ADVISORY COMMITTCE FOR AERONAUTICS 
 
 WARTIME REPORT 
 
 ORIGINALLY ISSUED 
 
 November iS^h as 
 Advance Restricted Report IAI29 
 
 ON THE FLOW OF A COMPRESSIBLE FLUID 
 BY TEE HODOGRAPH METffi)D 
 II - FDNDAMENTAL SET OF PARTICULAR FLOW SOLUTIONS 
 OF THE CHAPLYGIN DIFFERENTIAL EQUATION 
 By I. E. Garrlck and Carl Kaplan 
 
 * . Lemgley Memorial Aeronautical Laboratory 
 
 Langley Field, Va. 
 
 NACA 
 
 WASHINGTON 
 
 NACA WARTIME REPORTS are reprints of papers originally issued to provide rapid distribution of 
 advcince research results to an authorized group requiring them for the war effort. They were pre- 
 viously held under a security status but are now unclassified. Some of these reports were not tech- 
 nically edited. All have been reproduced without change in order to expedite general distribution. 
 
 L - 11^7 
 
 DOCUMENTS DEPARTMENT 
 
--7 < I ^iO i f 
 
 MCA ARR No. li|I29 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 ADVANCE RESTRICTED REPORT 
 
 ON THE FLOV. OF A COMPRESSIBLE FLUID 
 BY THE HODOGRAPK METHOD 
 II - FUl-JD/iMENTAL SET OF PARTICULAR FLOW SOLUTIONS 
 OF TRE CEAPLYGIN DIFFERENTIAL EQ:UATION 
 By I. E. Garrick and Carl Kaplan 
 
 SUIIMARY 
 
 The differential equation of Chaplygin's jet problem 
 is utilized to give a systematic development of particular 
 solutions of the hodograph flow equations, which extends 
 the trestment of Chaplygin into the supersonic range and 
 completes the set of particular solutions. 
 
 The particular solutions serve to place on s rea- 
 sonable basis the use of velocity correction formulas for 
 the com.parison of incompressible and compressible flows. 
 It is shown that the geometric-mean type of velocity 
 correction formula introduced in an earlier paper, part I, 
 has significance as an over-all type of approximation in 
 the subsonic range. 
 
 A brief review of general conditions limiting the 
 potential flow of an adiabetic compressible fluid is 
 given end application is made to the particular solutions, 
 yielding conditions for the existence of singular loci 
 in the supersonic range . 
 
 The combining of particular solutions in accordance 
 with prescribed boundary flow conditions is not treated 
 in the present paper. 
 
 INTRODUCTION 
 
 This paper presents a theoretical investigation that 
 may be regarded as a continuation of studies initiated 
 
2 NACA AHR No. iM-lZ') 
 
 In part I (reference 1). In psrt I an stterpt vy&s made 
 to unify t-ie results of Ghnplygin, von Karmsn and Tsien, 
 Temple and Ysrwood, and Prandtl and Glauert insofar as 
 their results v/ere concerned with velocity and pressure 
 correction factors for the correspondence of incompres- 
 sible and compressible flows. In addition, two new 
 velocity Gorrec"cion formulas were introduced that appeared 
 to nave a somewhat voider range of applicability thaii the 
 formulas of the afore-mentioned authors, i^'ost of the 
 results of part I were obtained with the use of two par- 
 ticular solutions of the hodograph equations. These tv.-o 
 basic solutions correspond to a vortex and a source in a 
 compressible fluid. 
 
 It was mentioned in part I thgt, in order to treat 
 the exact boundary problem, of uniform flow of a compres- 
 sible fluid past a prescribed body, a general set of 
 particular solutions of the hodograph equations had to be 
 obtained. Such c study is given in the present paper, 
 which incident,T:lly helps to clarify the nature of the 
 velocity correction factors of part I - in particular, 
 the one referred to as the "geometric-mean" type of 
 approximation. In addition, many interesting tjT^es of 
 flows are disclosed from a physical interpretation of 
 the particular solutions. A few such solutions have 
 already been obtained and discussed by Ringleb (refer- 
 ence 2 ) . 
 
 Several mathem.etical appro j^ches exist by m^eans of 
 which particular integrals of the hodograph equations 
 may be obtained. 'Two such approaches, mentioned in 
 part I, m^ay be attributed to Chapl^/gin (reference 5) and 
 Bers and Gelbert (reference Lj.) and are analogous to an 
 exponential and to a power-series approach, respectively. 
 Another method of defining particular Integrals is the 
 integral-operator method of Bergman (reference 5)* I^^ 
 the present paper the differential equation, first used 
 by Chaplygin in his treatment of jets (reference 5)> 
 provides the besls for the definition of a complete set 
 of psrticular solutions. 
 
 The scope of the present paper is lim.ited chiefly 
 to a systematic study of the fundamental solutions and 
 to the physical interpretation of some of the particular 
 flows represented by "them. The combining of particular 
 solutions to represent uniform flow past e prescribed 
 body is not treated herein. It is believed, however, 
 that the present stud^/ may serve as a basis for further 
 development and clarification of this important problem. 
 
NAG A ARR Fo . 1J4.I29 5 
 
 SYIvIBOLS 
 
 X, y rectangular coordinates in plane of flow 
 
 q magnitude of fluid velocity 
 
 9 angle included by velocity vector and posi- 
 
 tive direction of x-aiicis 
 
 p density of fluid 
 
 p pressure in fluid 
 
 a velocity of sound in fluid 
 
 M Kach number ( q/a ) 
 
 pr,, Pq, a^_, quantities referred to stagnation point q = 
 
 / velocity potential 
 
 Aj; stream function 
 
 Y ratio of specific heats (approx. I.I4. for air) 
 
 P = — (approx. 5/2 for air) 
 
 Y - 1 
 
 2[Ea^^ maximum fluid velocity (corresponding to 
 
 P = p = a = 0) 
 
 T dim.ensionless speed variable 
 
 / q"^ _ M- \ 
 
 T = 
 
 2pa^^2 2p + rr^ 
 
 T<, sonic vglue of t (t. = ; aporox. 
 
 ^ . ^ ^ ^ 2p + 1 
 
 l/'£ for air i 
 
 For p>0 (or 1<y'^"')7 the range of T is 
 < T < 1 
 
GEi:e:;AL PARTICI'Liia SCLUTICiyS OP^ THE FODOGRAPH E(o,UATIONS 
 
 Hodcgraph Equetions 
 
 The linear equations In the hodogrsph verifltles 
 end q, ■/.'hich relate the velocity potential >3 and the 
 streair; function i' for the steady twc— dime ns ions 1 flov/ 
 of a nonviscous co:-npre£3ible fluid, are 
 
 66 = ^-1^^^^ n 
 
 > (1) 
 
 ^^ _ ^ ,... ^^l 
 
 •^ 
 
 in which, for the adisb'^.tic equation of strte between 
 pressure and density, 
 
 -L 
 
 (1 - T) 
 
 &nd 
 
 d /P 
 
 uqy p( 
 
 ~ oq V '- J 
 
 1 - (^p + 1) T 
 q(l - T)P^^ 
 
 (See equations (21) pnd (i^) of reference 1.) 
 
 In the Incompres sibie c-ise t — ^3, equations (1) 
 can be expressed in the Cauchy-Riemann fortrx. Particular 
 solutions Q = '/J + '.ii can be e.^.pressed in this case as 
 any analytic function of the corr.plex variable 
 
 w = 9 + i lor q (2) 
 
NACA ARR No. rJ|I29 
 
 or as ?ny analytic function of the related exponential 
 function 
 
 e-^''^ = qe-^"e (5) 
 
 Thus, an infinite set of pprticular integrals of equa- 
 tions (1), in the incompressible case, referred to herein 
 as "the powers set," is w^. When k is a positive 
 integer, the particul?r solutions v&nish at the origin 
 (9 = 0, log q = 0) and, when k is a negstive integer, 
 the oarticular solutions are infinite at the origin. In 
 the case of nonintegral values of k, the or-igin is a 
 branch point of the functions w^ 
 
 fi^ 
 
 Anothez'' infinite set of particular integrals of 
 equations (1) in the incompressible cgse, referred to 
 herein as "the exponential set," is 
 
 (< 
 
 r ■-•■■• f = qke-i« 
 
 v/here, again, k can take on any value - integral, non- 
 integral, positive, or negative. 
 
 In the compressible case, the particular solutions 
 corresponding to the powers set wl'"- (that is, the par- 
 ticular solutions which reduce to w-^ in the incompres- 
 sible case T — >0) depend on whether the coefficient of 
 vr^ is real or imaginary - a consequence of the fact 
 that, in the compressible crse, and ^J/ do not 
 satisfy the same differential equation. For exat-p'le, 
 for k = 1, the two functions corresponding to w 
 and iv/, which have been developed in part I, are 
 
 W = 9 + iL 
 
 and 
 
 where 
 
 and 
 
 V = 1(9 + ih) 
 
 iVi 
 
 L = log q + f(T) 
 
 L = log q + g(T 
 
MAC A ARR Nc. l4l29 
 
 end f(T) and g(T) each vsnish for r = 0. (See 
 equetlons (26) and (27) of refer^ence 1.) 
 
 The doveloprr.ent of other functions corresponding to 
 tiie povver set vP'-'^ , for positive integral values of k, 
 follows according to the inethod of Bi^^rs and Gelbsrt. 
 (See expression (22) of reference 1.) Since the present 
 peper is chiefly concerned with the _ functions corre- 
 sponding to the exponential set e"-^-^^", the powers set 
 io not further dis cussed. 
 
 Chaplygin Differential Equstion 
 
 The Functions pj^ and Q^^ 
 
 Corresponding to the exponential sets in the incom- 
 pressible case 
 
 £nd 
 
 . -ikw k 
 
 le - q sxn 
 
 k9 + iq' cos k6 
 
 there appear in the compressible case functions designated, 
 respectively. 
 
 and 
 
 P|,-(q) cos ke - i Q|,_(q) sin k6 
 
 P^(q) sin ke + i Q:^{o) cos kG 
 
 where the functions F|^(q) and Q>(q) satisfy second- 
 order differential equations. These equations are easily 
 ob-cained by substituting in equations (1) the product- 
 type solutions 
 
 A. = ?,,(q) 3"^ (k9) 
 
 ii. t\. -ill 
 
 !in 
 
 *k = ^k'"^' =03 <-M' 
 
 (!+) 
 
 ^ 
 
NAG A ARR No. l4i29 
 
 In view of equstions (1) it is observed that 
 
 k PL.(q) = -^ q 
 P 
 
 ■>. 
 
 dPi., ( q ) 
 
 dq 
 
 -Kq 
 
 dq 
 d /P 
 
 > 
 
 ^(^)V.'. 
 
 (5) 
 
 The ruiictions Q,i^(q) satisfy the second-order differ- 
 ential equation 
 
 2 ^ 
 
 dq- 
 
 (l + M^^q ^ _ ^2(^^ _ j^2^ ^ ^ Q (g^ 
 
 dq 
 
 The functions Pp(q) can he obtained froin Qi^(q) by 
 means of the first of equations (5). Equation (6) may 
 be reduced to a standard type by introducing t as the 
 independent variable. put 
 
 Q.k(q) - q'" Yi,.(T) 
 
 (7) 
 
 Vt/here clearly Yi.(t) — >1 as t — ^0 ( inccrapressible 
 case). I'Vith tlio usa of the syrabolic relations 
 
 " dq dT 
 
 ,2 d 
 
 .2 d 
 
 2 
 
 dq^ dT^ 
 
 + 2t ii- 
 dT 
 
 and the relation 
 
 M' 
 
 2 _ 2pT 
 
 1 - T 
 
 the desired differential equation is 
 
 T(l-T) i+ Rk+l) - (k+ 1- °)t] -^+~pk(k+l) Y^ = (8) 
 
 dT 
 
MCA ARR No. 1I4.129 
 
 EQuatlon (8), which Is of the hyp ergoome trie type, was 
 first introduced by ChRplygin in his inerroir on gas jt-ts 
 (reference 3 ) . 
 
 The Functions Yv and Y_]^ 
 
 Ghpplygin treated the subsonic flow of s compres- 
 sible fluid through jets with straight-line boundaries, 
 for such problens the hodograph variables 9 and q 
 ore natural variables in the sense that the solid and 
 fluid boundaries are described by 9 = Constant and 
 q = Constant, respectively, and only the particular solu- 
 tions of equation (8) with positive characteristic 
 index k are needed. In the present paper a complete 
 ordered set of particular solutions of equation (6) is 
 obtained, v.'hich extends the results of Chaplygin into 
 the supersonic range and to negative values of the 
 index k. Two types of solutions of equation (8) for 
 nonintegral values of k are 
 
 Yi,(T) = ?(a5,, b^, k+1; t) (9) 
 
 ana 
 
 where 
 
 Sk + bj^ = k - p 
 
 and 
 
 ■(a, b, c; T) = 1 + ^T + 
 
 a(a + l)b(b + 1) ^2 ^ 
 2 1 c(c + 1) 
 
 It is now shown that only one of the solutions need be 
 used. For positive values of k, the requirement that 
 y^vi^) = 1 excludes the use of equation _( 10 ) . For nega- 
 tive values of the index, the solution Q,'_,/q)=q~ Y ^^(t) 
 obtained with the aid of equation (10) is,'* except for a 
 constant factor, equivalent to the solution Qj, (q,)''~q^" "^k^^^ 
 obtained with the aid of equation (9)« Thus 
 
NACA ARR No. rJ4l29 
 
 ^-k(q) = q~^ '^-k^''^ 
 
 - q-%^ F(e._y+k, b.,^+k, k+1; t) 
 = q-'-^rk p^^aj,,, b^,, k+ls t) 
 
 1 
 
 .2 pa "" 
 
 k 
 
 o / 
 
 Hence, only the solutions given by equption (9) Pi-e 
 needed for the determinftion of Qi-(q) and Q v^^^ 
 Then 
 
 Qv(q) - q^' Y,,(T) 
 
 k 
 
 q^ ?(a^., bj., k+1; t) , (11) 
 
 snd 
 
 Observe that both types of hypergeometrlc functions 
 appearing in equctions (9) snd (10) are utilized in the 
 expressions for Q^(q) end Q„k^^^' 
 
 The foregoing discussion has been limited to non- 
 integrel values of the index, positive or negative. When 
 the index is integrp-i and positive, equstions (9) and (11) 
 remain valid. When the index is integral snd negative, 
 however, equation (12) does not in general lead to a 
 meaningful solution end consequently another independent 
 solution is to be sought. The desired solution for 
 Y_i^(t) in such cases contpins s logarithmic term snd 
 again is subject to the condition tnst it reduce to unity 
 for T = ( Inc empress iblo case). The expression for 
 Q_ (q) is then given by 
 
 ^k^'^) = ^""' Y_i,(T) (15) 
 
10 
 
 MCA ARR No. li|I29 
 
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imOk ARR No. I4T2Q 
 
 11 
 
 Crse of 
 
 Y 
 
 - -1 
 
 Consider as an example the von Karman-Tslen treat- 
 ment oi compressible riow (reference f^) in which the 
 
 1 
 
 adiabatic index 
 
 -1 
 
 or 
 
 f- = - 
 
 2* 
 
 Then 
 
 k + 1 
 
 K 
 
 For a negative integral index, equation (15) ma^f appear 
 to be applicable, In which case the expression for 
 ^■^ Vr^''"' w^i-ild 'd3 a polynomial of degree k - 1. An 
 examination of equation (12) shovirs, hov.ever, that for this 
 esse no infinities rrise and that, vmsn the index is 
 negative, integral or nonintsgral. 
 
 Y_i,(T) - F(ijl, -|, 1-k; t) 
 
 The hypergeometrlc series represented by Y_i^ ( t ) con- 
 
 / 
 
 verges for values = |t| < 1. For the present case of 
 
 Y = -1 or p = --;, valuer of t co:ere spending to 
 
 positive values of M lie outside the range of conver- 
 gence. A closed expression for Y_i.(t) can be found, 
 
 however, for this case which, by analytic continuation, 
 is therefore valid for all values of t. Thus 
 
 ^-k<^) = "va 
 
 1 + (1 - T)^/' 
 
 Simll-L-rly, from equation (9)j when the index is positive. 
 
12 
 
 KACA ARR No. li|l29 
 
 )bservs th&t 
 
 ^k^^ 
 
 p(l_^k, |, i^k; t) 
 1 + (1 - T)l/2 
 
 -k 
 
 Qi,(q) = 
 
 Q_k(q) 
 
 ^ (1 - T)^/2 
 
 k 
 
 (li|) 
 
 This Identxty for the von Ksrraen-Tslen esse corresponds 
 
 ,k _ 1 
 
 to the identity q 
 
 Case of k = 1: 
 For k = 1, 
 
 ^■1 = 1 
 
 ,-k: 
 
 for the incompressible case. 
 
 bi - -P 
 
 C-, = <::: 
 
 Then, for the positive index, 
 
 Qi(q) 
 
 q Y^(t) 
 
 q P(l, -p, 2; T) 
 
 1 - (1 - T)P^^ 
 q i i 
 
 (P + 1)t 
 
 (15) 
 
 For the negetive integral index, it may appear at first 
 glance that equation (13) is needed; however, equation (12) 
 does yield a relevant and finite result and accordingly 
 is the equation to be used. Thus 
 
NACA A'^R No. L[^I29 
 
 13 
 
 Lim F/'av-k, bj^-k, l-k; t) = 1 + |. t ■? T^ + ^^ ^^ " -^^ 
 
 c->l ^ 2 2x2.' 2x3.' 
 
 2 X J4.' 
 
 = 1 + 
 
 i_i_r 
 
 2 p + 1 L 
 
 1 - (1 -t) 
 
 p+1 
 
 snd therefore 
 
 ^-l( 
 
 
 '.' I" 0-1- 1 
 
 _h_ I 1 _ (1 - T)^^^ 
 
 (16) 
 
 J 
 
 Case of k = 0: 
 
 The exceptional C£se of k - is directly treated 
 by means of equation (0). The differential equation for 
 Yq(t) or Q^, (t ) then is 
 
 a 
 d7 
 
 I dq^l 
 
 (1 _ T)P dTJ 
 
 = 
 
 The gereral solution of this equation can be written as 
 
 Qo(q) = 2c^ lo- q + C;l / L^^ ■ ""^^ ~ ^J T" ^ ^2 
 
ll^ MCA ARR No. l4l29 
 
 where C-i end Co are arbitrary constants of inte3r8- 
 
 tlon. The consttnts C-^ and C^ are determined by the 
 imposed condition that the expression for Q.q^'^^ reduce 
 
 in the incompressible case simply to log q. Then 
 
 r, _ 1 
 
 -1-2 
 
 = 
 
 end therefore 
 
 Qo(q) - log q + J ["""[(1 - T)P - l] ^ (17) 
 
 In a sirnilar manner, from the differential eouaticn for 
 fo' 
 
 _d_ 
 dT 
 
 t(1 - T)^"^^ dP^ 
 1 - (23 + 1)t dT 
 
 = 
 
 the expression for P^-, is obtained as 
 
 P (q) = log q + p 
 
 I - (2p + 1)t _ ^' 
 
 (1 - T) 
 
 (i+1 
 
 ^ (IS) 
 
 md P^(q) 
 
 It is rem^^rked thet the functions (^^(q) 
 are identical with the elementary functions L(q) 
 and L(q), respectively, introduced in part I (refer- 
 ence 1) and are associated vvith a vortex end a source 
 tTpe of flow. 
 
 The Functions 
 
 ^^k 
 
 and 
 
 'k 
 
 A linear homogeneous differential equation of 
 order n can, in general, be reduced to a differential 
 equation of order n - 1 by means of an exponential- 
 type substitution for the dependent variable. Chaplygin 
 made use of such a substitution to reduce the second- 
 order differential equations satisfied by P, and -X 
 
NACA ARR IIo. l!.;.I29 
 
 15 
 
 to first-order equations: of the Hiccati forr.i, in order 
 to study properties of the functions Pi, and Qi^ in the 
 subsonic range for only positive values of k. In the 
 present analysis the Rlcceti equations are pIso found 
 useful in order to extend the study of the functions P^ 
 arid Qj. to the supersonic rrnge for both positive and 
 negative values of the index k. 
 
 The second-order diff ei-ential equations for P^ 
 end 0^, v«ith t as the independent variable, are 
 
 _d_ 
 
 1 _ T)P-'l dP. 
 
 1 - (2(3 + 1)t dT 
 
 
 (1 - T)P 
 
 Pk = 
 
 and 
 
 A r J £> I _ l£ 1 - (2p + 1)T ^,,0 
 
 dT L(i _ ,)^ dT J J^ ^^^ _ ^.p+l ^'^ 
 
 The corresponding first-order Riccati equations are 
 obtained by substituting for Pj^ and Qj^ new dependent 
 variables Ri. and S-^_, respectivelyj as follows: 
 
 ^k = 
 
 n k 
 J Z7% 
 
 dT 
 
 or 
 
 R = 2t_ J_ ^ 
 "k k pj^ dT 
 
 2t d ^ 
 
 — — loe: Pi. 
 
 K 
 
 iT 
 
 (19) 
 
 md 
 
 ^V. = ^ 
 
 s 
 
 _k_ 
 2t 
 
 ^. 
 
 dT 
 
16 
 
 MAC A ARR No. LI+129 
 
 or 
 
 _ 2t 1 dr^ 
 
 
 K 
 
 k Qj^ dT 
 
 2t _d_ 
 k dT 
 
 lo 
 
 S %. 
 
 (20) 
 
 The equations satisfied by P^j^Ct") and S^^(t) are 
 
 ^-'^k , 1 + (2p + l)T 
 
 dT 
 and 
 
 l-(2p + l)T 1-T 
 
 Rk-^ 
 
 K 
 
 2t 
 
 R 
 
 2 1 - (2P + 1)t 
 
 1-T 
 
 k 
 
 = 
 
 (21) 
 
 dS 
 
 k 
 
 dT 
 
 1-T 
 
 k 2t 
 
 'k 
 
 1 - (2p + 1)t 
 
 X - T 
 
 = 
 
 (22) 
 
 Initial conditions for R|r(T) rnd S^Jt ) tre found by 
 exaninstion of the Incompressible case T — >0. In this 
 
 case Fv = Qt^ = q^ and, since 2t ^ = q ^, it follows 
 
 dT dq 
 from equations (19) and (20) that 
 
 Pu^,(0) = S,,(C) = 1 
 
 The following important relation exists between th^ 
 functions ■R't<(''') ^^d S-]^(t): 
 
 R^(.) S,(T) =l^i2£^ 
 
 1 - }!1 
 
 ,t2 
 
 (25) 
 
 Equation (23) cm be verified directly from the hodogreph 
 equations (1). It may be noted at this point that this 
 result 13 of significance in connection with the 
 geometric-mean type of velocity correction factor intro- 
 duced in part I and is discussed more fully in a later 
 section. 
 
NACA ARR No. 14X29 
 
 1? 
 
 Before the functions Ri^(t) 
 
 are treated, 
 
 .nd S^ ( T ) 
 certain generel observations can be made regarding the 
 functions I'lrd), Q^(t), ^^^(t), end S^(t). Chaplygin, 
 v;ho limited his investigations to the subsonic range and 
 to positive values of the index k, has shown thft Qi, 
 and consequently the other functions possess no roots 
 for anj value of the independent variable in the subsonic 
 range, with M = excluded. In the supersonic range 
 K > 1, J^ir^^ ^^^ Qi-(''") ^^^ g'^neral possess zeros. 
 Certain rel£tions obtained by means of equations (19), 
 (20), and (25) between P-^, Q^, R^, end S^ at the 
 
 zeros of P^. 
 
 K 
 
 and 
 
 Qk 
 
 are sunimaiazed as follows: 
 
 ^k 
 
 '% 
 
 dPj, 
 dT 
 
 dT 
 
 ^'k 
 
 \ 
 
 
 
 K^x or -niin 
 
 » • O 
 
 ^J 
 
 CO 
 
 
 
 Max or nln 
 
 
 
 
 
 r « • 
 
 
 
 » 
 
 as 
 
 It is rei.iarked thet the number of zeros of Q^, 
 a function of the index k, can be found from an 
 
 expression developed by Klein and Hurwitz (reference 7) 
 in connection with the zeros of the h^-pergeometrlc func- 
 tion. In general, the nuiaber of zeros increases with 
 the magnitude of tiie index k and is infinite for 
 k = ±^. 
 
 A further observation of interest con be made in con- 
 nection with equation (2^). Chsplygin has shown that, 
 for positive finite values of k (and the same is true 
 for negative finite values of k), 
 are not zero for the sonic value t 
 Prom equation (25) then, it follows 
 R.-(t) = for M = 1. 
 
 th 
 tnat 
 
 functions S^, (t ) 
 or M = 1. 
 the functions 
 
 In view of the relation between the functions 
 
 and 
 
 given by equation (25), only Sj, 
 
 'K 
 
 n: 
 
 ;a De ais- 
 
 cussed. The Riccatl equation (22) nay be used to discuss 
 certain properties of the fu,iction S^^ but in general, 
 for numerical cvalurtion, the original definition (equa- 
 tion (2J)) in terms of the function Q, m.ey be used 
 directly J 
 
18 
 
 NACA ARR No. l1+I29 
 
 2t_ J_ £Qk 
 
 k Qj^ dT 
 
 Sv = — — 
 
 or 
 
 ^k 
 
 . 1.2ld^ 
 
 ^k 
 
 dT 
 
 In general, the functions S^ are expressible in Infi- 
 nite series. For several values of k, however, 2^ 
 can be expressed in closed forms. For k = and 
 k = ±==, S]^ may be obtained by a limiting process from 
 equation (20); however, for these special cases the 
 Riccati equation (equation (22)) yields the results 
 directly. Thus 
 
 = (1 - T) 
 
 {2k) 
 
 end 
 
 >±00 
 
 1 - (2p + 1)t 
 
 1 - T 
 
 = (l-M^)^/^ ' 
 
 1/2 
 
 (25) 
 
 The cases k = 1 and k = -1 may also be expressed in 
 closed form. vVith the aid of the equations (15) and (l6) 
 for Q-| and Q_t, equation (20) yields 
 
 S3_ = 1 - 2 
 
 1 - (1 + p.T)(l - T) 
 1 - (l-T)f^l 
 
 (26) 
 
 and 
 
 '-1 
 
 1 - 
 
 Pt(1 - T)P 
 
 2 e + 1'- -• 
 
 (2?) 
 
 In order to illustrate the behavior of some of the 
 functions thus far introduced, a number of tables and 
 figures are given. All the calculations heve been 
 
NAG A Ax^^ r^o. li;l29 19 
 
 performed with the adlf-batic index y ~ l.i+. Table 1 
 givss V&1U2S cf Y^ as a function or IV; or t for 
 
 sevex^al positive and negative values of the index k. 
 riguro 1 s^iovis the Yj. functions plotted against M. 
 Values of the 3]^^ and R) functions ere given in 
 
 tables 2 and 3 and are plotted against M in figures 
 and 5 . 
 
 The r'unctions f^^"^' ^^^ S' '"""^ 
 
 In the incompressible case, the sets of functions Q|^ 
 end I'lr can be red\;ced to a single function log q by 
 
 means of a simple operftor r- log. Thus 
 
 and 
 
 1-1 k -I 
 
 ~ log q^ = leg q 
 
 This sa'iie operation applied to the functions Q,-,, and P^ 
 
 in the coiiipresyible case serves to define t'-vo useful sets 
 cf functions leg q + f^(T) and log q 'r g^(T), respec- 
 tively. Thus 
 
 ,- log Q.V = log q + f^(T) (23) 
 
 k 
 
 and 
 
 i log P.^ := log q + g^,.(T) (29) 
 
 k 
 
 [<. 
 
 From equation (7), i^amely. 
 
 it foliov-i; that 
 
 'k 
 
 f^J■r) = ,^ log Yi,(T) (50 
 
20 FACA ARR No. ri;l29 
 
 From equation (5) for P^ and equation (20), which 
 
 defines S^^, 
 
 It follows that 
 
 ^ (1 - T)P ^ ^ 
 
 1 , 1\(t) S,^(t) 
 
 gk(T) =^10 
 
 (l-T)P 
 
 1 . s, 
 
 = fj^(T) + ^ log ^^—- (51) 
 
 ^ (1 - T)^ 
 
 For example, for k = 1 and k = -1 and with the use of 
 equations (15) and ( l6 ) , 
 
 1 - (1 _ T)P+1 
 r (t) = log i Li li (32) 
 
 (P + 1)T 
 (1 - T)P[l + (2p. + i)t] - 1 
 
 Si(t) - log (55) 
 
 (P + 1)T(1 - T)P 
 
 ■ (t) = - log il + i — L [l - (1 - T)^"" 
 
 ■■^ I 2 p + 1 L 
 
 H 
 
 ^J 
 
 (51;) 
 
 g i(T) =- log (5P^i) - B(i. T)(l. T)P ^^^^ 
 
 2({i + 1)(1 - T)f' 
 
 For k = and k = ±°°, equations (50) and (51) require 
 a limiting process for their evaluation. Alternate forms 
 for fi<;(T) end §1^(1") may be obtained, hov;ever, b:/ 
 mear:i3 of equations ( I9 ) and (20) defining R^(t) and 
 Sit(T), which yield tlie results for k = and k = ±°° 
 directly. Thus 
 
MCA A..^".^ ¥o. lI|.I29 
 
 21 
 
 ma 
 
 ^k(^) 
 
 e\' ^ T 
 
 Jo 
 
 [Sk( 
 
 r - 
 
 rRi,(T) 
 
 ^j 
 
 T 
 
 1 £1 
 
 J T 
 
 (56) 
 
 (37) 
 
 where F.^^(t) end S^^(t) are relf;ted scco.-^dine to equa- 
 tion (23). Than 
 
 i-o(T) - 
 
 1 i 
 
 r T 
 
 (1 - T)P - Ij ^ 
 
 (38) 
 
 ffe(-^) 
 
 1 
 o 
 
 PT r 
 
 2P + 
 
 '-J c- 
 
 T)^ 
 
 ,■•1 
 
 1 i^ 
 
 (35) 
 
 and 
 
 ^fx)(T) = e±oo't) = - 
 
 2 J, 
 
 l££. 
 
 ^- 1)t 
 
 1 - T 
 
 - 1,-^ 
 
 dT 
 
 J ^ 
 
 (1^0) 
 
 It is v^orthy of special notice tLrt the functions f^d), 
 Eq^"^ ) ? p-^-i^ C.^ (O Si'''^ identical with the functions 
 ^(t)> g(T)^ S'-i'"'- h(-"), respec t.i.vel'/^ vv'hich formed the 
 bpsis ^i part I (leferencc i). In addition, the expres- 
 sion3 I03 q + fo(T), log q + ^^^(t), _^end log q+f^.^(T) 
 
 are ldfen"Gioal with the functions L, h, an6 H, 
 respectively/, which .leie introduced in part I. 
 
 A nuino'^r of functions fi, and 
 
 k 
 
 calculated, 
 
 6k 
 
 have been 
 
 .th 
 
 .'j^, for several positive and 
 
 negative values of the index k, and the values are 
 given in tables i;. end 5 and plotted in figures J4. and 5* 
 
22 MCA ARR No. li|l29 
 
 The opportunity is teken here to note that, for the 
 von Knrman-Tsien esse (y = -1 or p = — ), 
 
 f, = g, = - leg L^ii^.1)^ 
 
 = iog ^V^. 
 
 1 + "A - 
 
 end that the sets of functions P, and Q^, as in the 
 incoinpressible case, are reduced to a single function by 
 the operator — log; namely (compere equation (llj.)), 
 
 log q + log 
 
 l^/l -M" 
 
 1 + ^1 - llf 
 
 In fact, the complex flow potential ^ + i']f can be 
 expressed as an anrlytic function of a single complex 
 
 / ~2 
 
 variable 9 + i log q — — ■ Tsien has made use 
 
 1 + yi - M^ 
 
 of this complex variable in his hodograph treatment of 
 the compressible flow past an elliptic cylinder (refer- 
 ence 3) . 
 
 Velocity Correction Factor 
 
 The solution of the problem of an exact corre- 
 spondence betv;een the flow past a prescribed body in an 
 incompressible fluid and the flow past the same body in 
 a compressible fluid is a difficult matter. This 
 problem can be solved exactly for certain types of flow 
 patterns (not past closed shapes), such ss flov/s inside 
 or outside angles or channels, and for certain flow 
 singularities such ps s vortex, source, and doublet - 
 tjrpes of flow which can be associated v/ith the particular 
 solutions Q, . Some of these types of flow are illus- 
 trated by examples in the following section. Combining 
 
NACA kRR Fo. li^-I^O 23 
 
 p&rticular solutions to repi'esent uniform flow pest a 
 prescribed body is e corriplic£.ted process, since the treat- 
 ment of infinite series in the x'urictions Qj,. for both 
 
 positive and negative vflueG of k is involved. Fux'-ther- 
 raore, the process of returning to the physical-plane 
 vrrlables from the hodogrsph-plane variables hinges on 
 nonelementary parts of dif I'erential geom.etry. Certain 
 types of jet problem.s can be properly treated in the sub- 
 sonic renge by series in Ch. with k positive, as was 
 shov/n by Chaplygin (reference y). Thus, it appears that 
 much vi/ork remains to be done in order to render feasible 
 exact and practical solutions for uniform flow past pre- 
 scribed bodies in a compressible fluid. Becsuse of the 
 difficulty tnd com-plexity of the general oroblei.i of flew 
 in a compressible fluid, attempts have been m^ade by a 
 nujTiber of Investigators to obtain results by means of 
 velocity correction formulas that serve to place in 
 correspondence velocities in an incompressible and in a 
 compressible fluid. 
 
 In part I the velocity correction factor v;as dis- 
 cussed with particular reference to the tv/o functions L 
 and L ( Q„ and P^ cC the present paper) associabed 
 with a vortex and source type of flow, respectively. The 
 main justification for tho results of part I was the 
 yielding and the unifying of the results of Chaplygin, 
 von Ka'rma'n and Tsien, Temple and Yarwood, and Prandtl 
 and Glauert. The knov;ledge of the infinite set of func- 
 tions Pi,- and Q,-^ discussed in the present paper can 
 now servo to establish further on a reasonable basis the 
 concept of a velocity correction foi^'mula. 
 
 In order that a single velocit7.'" correction factor 
 be feasible, even for a flow associated v/ith a particular 
 solu.tion, it is necessary that F^^; :; Q]^. Consider, for 
 
 example, the functions Q^ and P^^ insofp^r as the first 
 po.ver of the variable t is concerned. It can be shown 
 easily that 
 
 - log Q, = log q + f, (t) 
 
 log q - ^pT 
 
 c 
 
2L 
 
 MCA ARR No. li|l29 
 
 and 
 
 - log P, = log q + g, (t) 
 
 =; log q - -pT 
 
 Thus, to the first power of t and independent of k, 
 
 rv(T) = gv(T) 
 
 K' 
 
 .4pr 
 
 Then 
 
 Pk = Qi 
 
 r \qe / 
 
 The nature cf the correspondence betv/een the incompressiblo 
 flow ar/i the compressible flow is such that 
 
 ^i = ^c 
 ^^i = ^^c 
 
 (U) 
 
 Without going into any details here of the field point 
 correspondence or of the boundary distortion, the veloci- 
 ties in the incompressible and compressible cases may be 
 placed in correspondence a? follows: 
 
 (log q;.j_ = (log q - JP'^X 
 
 or 
 
 4pt 
 
 qi - %e 
 
 (ii2) 
 
NACA ARR Fo. li+KQ 
 
 25 
 
 Tbls resul-c implies chf t the ooriplex variable 
 
 S + 1 flog q - — pT ) in the coinpi'ess.ible case correnoonds 
 
 to the complex vorlsble 9+1 log q in the incompres- 
 sible c&so. EquC-tlon {l\.2) reprosents the approximation 
 of Temple and Yarwood discussed in part I. 
 
 Consider nov- the functions 0- 
 
 k 
 
 :nG 
 
 insofar £s 
 
 lerge values of the index k are concerned. It is 
 recalled thct, as the index k — > ±t», 
 
 Rk --> Sk 
 
 v^ - 
 
 .2V/^ 
 
 end that 
 
 whe re 
 
 - log Q, — > - log P, 
 k k; k k 
 
 — > log q + h(T) 
 
 h(T) 
 
 ,(t) - 
 
 SiJ-"") 
 
 Then, es k 
 
 --., +r 
 
 ^k = \ 
 
 
 The function h(T) Is expressed in integral form, in 
 equation (l+O) and has been evaluated and tabulated in 
 part I. (See also table li. and fig. 1+ of the present 
 paper-,) The correspondence of velocities in the incom- 
 pressible and the com^pressible case is given by 
 
 q. = q 3^(^) 
 
 ^1 ^G 
 
 (13) 
 
26 NACA ARR No. ll\.l2^ 
 
 Squaticn ([(.J) constitutes the geometric -me an velocity 
 correction formula introduced in part I and is limited 
 to the subsonic range ^ M ^ 1. It is observed that, 
 for positive values of k, h(T) lies between fv(f) 
 
 ond Si^(''") ii"' magnitude. Moreover, the deviation of 
 
 g-^lT; from e ^ and e"^*^ is quite small in the 
 subsonic ran^e. (See table 6.) 
 
 The foregoing remarks, together with the fact that 
 the geometric-mean type of approximation contains the 
 results of Chaplygin, von Xa'rman and Tsien, Temple and 
 Yarviiood, and in the limiting case of small disturbances 
 to the main flow the exact Prandtl-Glauert rule, lead 
 to the suggestion that it may be adopted as en over-all 
 type of approximation in the subsonic range. 
 
 f'low Patterns Cori-^'e spending to the particular Solutions 
 
 Before the flow patterns corresponding to the par- 
 ticular solutions 0-y^ and 'it^ given by equations {l^} 
 
 for tiie com.pressible case are discussed, it is instruc- 
 tive to examine the incompressible case. Consider the 
 complex velocity potential 
 
 Q = ^ + i\lf 
 
 where TJ and n are constants and z = x + iy. It is 
 well known thpt, if n = — where a is an angle between 
 
 and 2tt, equation (l+Lj.) represents the flow in a shrrp 
 angle. For example, the flov/ inside a right angle is 
 obLcinea with n = 2. and the flow outside a right angle 
 
 2 
 is obtained vi/ith n = — . Again, the value n = 1 or 
 
 5 1 
 
 a = TT corresponds to a uniform flow 8nd the value n = -p 
 
 or a = 2t! corresponds to the flow around s semi- 
 iiifinlte line. Clearly, all the angle flov/s are obtained 
 with values of n between l/2 and co. Other types of 
 flows are given by other values of n. For example, 
 n = -1 corresponds to a doublet and the remaining 
 
FACA ARR iJo. iJ+lSo 27 
 
 negative integers pre sssoolpted v-ith singularities of 
 higher order than the doublet. in addition to the flows 
 described by the powers z'^, thei-e are the two funda- 
 mental flows, the source and the vortex, associated with 
 tliO fuiiction loq, z. If, now, it is desired to obtain 
 generalizations for the conipressible crse of the fore- 
 going particular I'lo.vs, the procedure is first to express 
 >/ or \|/ for tne incorupressible flow as a function of 
 the hodograph variables q and 6 and then to replace 
 q^' by P,, cr Q^, respectively. Several exa:nples will 
 
 best illustrate this pi'ccedure: 
 
 (1) Consider the compressible generalization of the 
 angle flows. By means of the relation 
 
 dD, _ - iw; 
 d"^ ~ 
 
 where w = 9 + i log q, the hodograph co.-nplex variable w 
 is introduced as independent variable in place of z. 
 Prom equation {hl\.) 
 
 dQ ^, n-1 
 
 — = nUz 
 
 az 
 
 g-iw 
 
 Hence 
 
 1 
 
 . = _!__ (e-iw)-l 
 
 (Un) 
 
 and 
 
 1 
 n~ 
 
 n 
 
 Q = 
 
 U 
 
 TT^ (e-^'O' 
 
28 NACA ARH Fo . li|l29 
 
 Then 
 
 n 
 
 TT " 
 
 u 
 
 q^'^- 
 
 -1 
 
 sin 
 
 n 
 
 n-1 
 
 9 
 
 n 
 
 n-1 
 
 (Un) 
 
 
 
 
 
 
 ir — Y is replaced by k, the compressible genersli- 
 zrtion of the sn^le flov/s is given by 
 
 ^ k-1'' 
 
 The inside angle flows are given by values of k in the 
 range 1 < k < <» and the outside sngle flows, by values 
 of k in the range 1 ^ -k < «». For example, k = 2 
 for the flov; inside a right angle, and k = -2 for the 
 flew outside a right angle. Other types of flow are 
 given by values of k in the range -1 < k £ 1. 
 
 The case k = 1 or n = ±«> is exceptional end, in 
 fact, corresponds to the incompressible flow 
 
 C = e°^ (U6) 
 
 where c is a constant. 
 
 (2) Consider the compressible generalization of the 
 doublet. The complex velocity potential for the incom- 
 pressible doublet at the origin is 
 
 z 
 
 The ref lected-velocity vector is 
 
 dO _ _ _1_ 
 
 -iw 
 = e 
 
NAG A Al-IR No. i4K9 29 
 
 Eencs 
 
 1 . 
 
 z = ie^ 
 
 and 
 
 1 . 
 
 - 3- iw 
 
 n = -ie 2 
 The 3trep":n furcfciori foi' tns incoxapress.'.ble doublet Is 
 
 -V,, 
 
 1 
 
 \1/ = - q cos :rf 
 
 The comprsssible generalizBtion of the doublet Is 
 therefore 
 
 ^1/2 ^ ~ ^1/2 ^°^ 2^ 
 
 (3) COiiSider the compressible generalization of the 
 source. The complex velocity potential for a unit source 
 at tne origxn is 
 
 Q 
 
 .oe z 
 
 The ref lected-V'.jloc ity vector is 
 
 dQ _ 1^ 
 dz ^ 
 
 = e"^'^ 
 
 Hence 
 
 z = eiw 
 
 and 
 
 Q, - Iw 
 
50 NACA ARR No. l)-\.l23 
 
 The velocity potential for the Incompressible source is 
 
 ^ = - log q 
 
 Thi; coirpresEible gener&lizstion of the source is then 
 given by 
 
 f- o •" o 
 
 (i|.) Consider the compressible generalization o^" a 
 point vortez. The complex velocity potential for a 
 vortex of unit strength at the origin is 
 
 D = - 1 log z 
 
 The ref lected-velocity vector is 
 
 dQ ^ _ X 
 dz ' z 
 
 _ ^-iw 
 
 •:ence 
 
 z = -ie^^ 
 
 and, except for en additive constant, 
 
 Q = w 
 
 The streejni function for the incompressible vortex is 
 
 '.j; = log q 
 
 The compressible generalization of the vortex is then 
 given by 
 
 o ^-<o 
 
MCA /IRR No. l4l29 
 
 31 
 
 Trpnsf ormatlon from the Hodograph 
 
 to the Physicpl Variables 
 
 Given the velocity potential {? and the stream 
 x'^iuxtlon ^ in terms of the hodo^rrph variables G 
 and q, it is possible to er.pres.-: the coordinates x 
 and y of the physical plane in teriiis of Q pnd o_ . 
 
 Prom the basic fiov; equations 
 
 6'^ _ Po 6^!/ 
 
 ox 
 
 -^^ 
 
 y 
 
 Pi. - ^2. kll 
 dj p r^ X 
 
 It follov.s (see equation (6) of reference 1) that 
 
 1 19^ w . , Po ^,\ 
 ds = - e ( dv? + 1 -i^ d;;; ) 
 
 H V p / 
 
 The real snc. imaginary parts of this equation yield 
 
 dx = i 
 
 q 
 
 fM cos e - ^ ^ sin 9) dq 
 _\dq. P oq / 
 
 + 
 
 ^^^ 
 
 (^ eos B -"-^^^ sin e) 
 Vc6 D 06 '^ 
 
 1 
 
 dy = 
 
 q 
 
 
 o. 
 
 d-i! 
 
 d9 
 
 in G + ~- ?-- 00s b j dq 
 
 
 
 d.'/ 
 
 Vc'a 
 
 P Cq 
 
 sin 9 + -2- ^4^ cos ej de 
 
 P 69 
 
 J 
 
 > 
 
 (4?) 
 
 Equations (1|7) relate the d:^f f erential line elements in 
 the physical x, y plane and tno hodograph 9, q plane. 
 
32 
 
 MAC A ARR No. 14 12 9 
 
 vVhen ezprosslons for ^ end \1/ as functions of 9 
 snd q are known for a given flow, the integrals of 
 equations ihl ) sre the equations of transformation of 
 Lhe 6, q to the x, y coordinates. It may be remarked 
 L.iiat the hodograph flovj equations (1) are the integra- 
 biiity conditions for the differential equations (Ij.?)' 
 The I'ight-hand sides of equations (i|7) sre therefore 
 perfect differentials. 
 
 Consider one set of particular solutions from equa- 
 tions (li) 
 
 ^ = Pk(ci) -OS k9 
 '^ = - Qi^(q) sin ke 
 
 v.'here k = ±1 and k = are excluded. By the use of 
 equations (5), it c ?n easily be verified that 
 
 ^k 
 
 V 
 
 k 
 
 K 
 
 
 -> 
 
 pq / 
 
 k 
 
 
 cosU + l)i 
 
 ;os(k - 1)9 
 k - 1 
 
 sin(k +1)8 
 k + 1 
 
 + Constant 
 
 > 
 
 
 sin(k - 1)9 
 k - 1 
 
 + Constant 
 
 (1|8) 
 
 The equations of trensf ormet ion corresponding to the 
 other set of particular solutions from equations (i4.) are 
 obtained by replacing in equations (i+S) the cosine by 
 the sine end the sine by the negative cosine. 
 
 The excluded ceses k = and 1 
 
 ic 
 
 :i 
 
 rre now 
 
 trer-ted. For k = 0, one set of particular solutions 
 corresponds tc p source snd is 
 
MCA ARR No. lJil29 
 
 ^ - ? 
 
 ^o = 3 
 
 Equations {\\'{) then yield 
 
 X - -^- cos 
 
 pq 
 
 Po 
 
 V = — sin 6 
 
 pq 
 
 The other set of particulfr solutions corresponds to o 
 
 vortex snd is 
 
 Equations {UD then yield 
 
 1 • c 
 
 X ~ — sin c 
 
 4 
 
 — COS 9 
 
 q 
 
 fPcr k = 1 with 
 
 ;Z^^ = p, COS 9 
 
 ^'l ~ ~ "^"l ^^^ ^ 
 
 equations iUl) yield 
 
 1 /P^ p.. \ 1 r/l ^P] Pn d^^ 
 
 ■^ ' -^ - '-^ O. cos 29 + 4 f ^ — ^ + -^ — i 
 
 1 1; V q pq -1/ -J\^ dq pq dq/ 
 
3k 
 
 MCA ARR No. 1^129 
 
 _l^Pl Po 
 
 
 - 1 /^^ ^- PQ^A 
 
 2 V^ P^ / 
 
 Vilth the use of equations (Ip) and (5), 
 
 ^1 ~ 2 
 
 1 - 
 
 1 - (1 - t) 
 
 P+1 
 
 (P + 1)t(1 - T)P 
 
 cos 26 + log q 
 
 e , , 1 2p + 1 
 — t — g ( T ) i 
 
 P + 1 2 p + 1 
 
 l 
 
 (1 - T)P 
 
 - 1 
 
 > (1|9) 
 
 yi - - 
 
 1 - (1 - T)P^i 
 
 (P + l)T(i - T) 
 
 sin 2e - 9 
 
 y 
 
 where g(T) = g^^(T) by equction (59) and is evaluated 
 in pai'^t I (reference 1). 
 
 For k = 1 with 
 
 ^1 ~ ^1 ^'^ ^ 
 
 ^'l ~ ""^'l °°^ ^ 
 
 -^1 = 
 
 ^•1 
 
 sin 29 + e 
 
 (P + 1)t(1 - T)P 
 
 cos 29 + log q 
 
 !> (50) 
 
 P f y 1 2p + 1 
 --— g(T) - - "^ 
 
 P >- 1 2 p + 1 1^(1 _ ^)P 
 
 - 1 
 
NAG A ARR ]\[o . hklZ') 
 
 53 
 
 For k = -1 v/ith 
 
 >2C_^ =^ P_-. cos G 
 
 ^\i ^ = Q 1 sin 6 
 
 -X -1 
 
 equstioiis (Il7) yield 
 
 X 
 
 
 U V q pi / 
 
 I cos ^9 + - I - — -i 
 
 2 J \q ciq 
 
 pq ciq J 
 
 dq 
 
 V 
 
 1 ,^p-^ , ^o^--l^ . 
 
 T- ( — ^ + -^ — — i Sin 
 
 ^ \ q pq / 
 
 !9 - - 
 
 2 V q oq '-V 
 
 With the use of ecuf.tlorxS (l6) and (';>), 
 1 R f? + ? 1 __J 
 
 ^^-^ 4q2 ! p.-l (1-T)^^ Ml 
 
 1 + |?t) 
 
 COS 2P 
 
 + -J-_ lo, q + 5L_-^_J ^(T) 
 
 + 
 
 3p + 2. 2[; + 1 
 
 8({3 + 1) p,e/- 
 
 (1 - T) 
 
 - 1 
 
 _i 
 
 > 
 
 (5i) 
 
 7 
 
 -1 
 
 J- 
 
 5p + 2 1 
 
 p 
 
 (1+ PT) 
 
 ^q- 
 
 P-1 (1-T)'-^ 
 
 p + 1 
 
 
 >in 2& 
 
 + 
 
 i|a. 
 
 ^ 
 
56 
 
 NACA ARR No. li+l29 
 
 For 
 
 v;lth 
 
 /^_1 "^ - P_]_ sin 
 \|/_-L = Q_-j_ cos 9 
 
 1 I ^ B 
 
 ^•-1 
 
 p + 2 1_ 
 
 l,q- [_p + l (1 _T)(^ [3 + 1 
 
 1 r. 
 
 (1+ P) 
 
 sin 29 
 
 ^^=*o 
 
 -1 
 
 h^' 
 
 3P+2 1 
 
 p + 1 ( 1 - T ) ^-^ p + 1 
 
 (1 + I3T) 
 
 COS 29 
 
 > ij2) 
 
 + _J_^ log q + ^P "• ^ ^ g(T) 
 
 1 c. 
 
 + 
 
 aa^ i|(p + l)a^ 
 
 3p + 2 2p + 1 
 
 (P + 1) p£ 
 
 (1 - T) 
 
 -1 
 
 J 
 
 y 
 
 Ringleb (reference 2) gives an example of the flov; 
 of a coj:ipressible fluid tround a serri-lnf inite line. An 
 
 exeir.ination of Ringleb' s stream function ^ - n ^^^'^ ® 
 
 skovvs that it is a lineer corabination of \!/^ end '^'•'.i* 
 til at is , 
 
 it 
 
 - ^^ -H Q_^ ) sin 9 
 
NAG A ARR Fo. lIiI29 
 
 57 
 
 In fact, all the exter'nsl angle flows (l = -k < =° j are 
 ncnunique ; for, in view of the discussion preceding 
 equation (11), a general form of \['_i. is 
 
 ^' , = q' 
 
 k 
 
 E'in k9 
 
 wher'.^ A is en rrbitrary constant. 
 
 Obi-ervations on Liiriit Lines 
 
 In the present soctlon there are reviewed briefly 
 certain conditions, discussed by Tollmien (reference 9) 
 and Ringleb (reference 10), witli regard to possible llmi' 
 tat ions on the potential flew of en adiabatic compres- 
 sible fluid. 
 
 Consider the fsjiiily of streajrlines 
 
 \l/(6.'^) ~ Constant 
 
 Then, alon^; a sere ami ine 
 
 d4/ = 1;^ dG + ^^i dq - 
 c-3 oc 
 
 and, frora equations {h.J) , the line elements along a 
 streaiiiliae are 
 
 dx = -il 
 
 foi^^ _ fd±\^\ 2 
 
 ;os 
 
 
 V (53) 
 
 ,'->..,, s 2 
 
 a2 j sin e 
 
 
 dq 
 
 p [_q 
 
 bif/ cQ 
 
 J 
 
 Singular points along a strea^iiline are characterized by 
 the vanishing of tnr comrion factor of equations (53): 
 
58 
 
 NACA kRR No. ri|l29 
 
 2 
 
 (MT - W - 'XW- - ° 
 
 (stagnation points at v.hich 
 vortex for v:hich ^^ = 
 
 and 
 
 d-ir 
 
 (5U) 
 
 6 -J/ 
 ^9 
 
 5T -— 6q 
 and the source for which 
 
 vanish, the 
 
 oq 
 
 are excluded fron this discussion. ) Observe now from 
 equ.9tlons (1^7) that the Jacobian of the transformation 
 from the hodograph variables 9 and q to the physical- 
 
 plane variables 
 
 X 
 
 and 7 is given by 
 
 
 e 6 
 
 ^ 
 
 ^q 
 
 on 0./ 
 
 -u'-ff ^-w-^m 
 
 L_ 
 
 (55) 
 
 Thus, the vanishing of the Jacobian is equivalent to the 
 condition for the existence of a singular locus for the 
 family of streamlines 
 
 ^(6,q) = Constant 
 
 This singular locus consists of points at which the 
 streamlines undergo an abrupt change of curvature and 
 
 m.eans, physically, that the acceleration q -~ 
 fluid particle is infinite st such points. 
 
 of a 
 
 Both Ringleb and Tollmien have sh 
 sinf^ular locus for the screaniines is 
 of the Mach lines in the plane of flow 
 are related to the streamlines in such 
 component of the fliiid velocity normal 
 equal co the local velocity of sound. 
 are identical v/ith the so-called chara 
 of the second-order partial differenti 
 for and iJ; and are the integral c 
 n:^,ry differential equation 
 
 own that the 
 
 also the envelope 
 The Mach lines 
 a way that the 
 to a llach line is 
 The Mach lines 
 
 cteristic curves 
 
 al equations 
 
 urves of the ordi- 
 
IIACA ARR No. Li+I29 
 
 59 
 
 ^-2 
 
 ~\^r - 1) clq- = 
 
 or 
 
 d0 := ± |VM^ - 1 dq 
 
 (56) 
 
 The real solutions of this dif f e^^entlal equation inter- 
 preted in the physical x, y plane yield the Mach lines 
 for a given flow. The solution of equation (56) is 
 
 e-e^ = ± 
 
 tai 
 
 ./T 
 
 n~\jT:: Jl^- - 1) - tan"\42 - 1 
 
 (57) 
 
 J 
 
 where t - j"-— ~- and where 9, assuries the values 
 s ^ + 1 
 
 of 3 rlont-: th=; I', - 1 line fc:' b given flow. 
 It is recalled thfit tl'io fanotion 
 
 T-T 
 
 = log q + h(T) 
 
 introduced in part I (refex^^nce 1; in connection v/ith 
 the geon-etric-msan typr of velocity correction formula, 
 is a solution of the dif f ei^ortlal equation 
 
 / 
 
 w- 
 
 dll - ± -■ — dq 
 
 in the £ub3onic rrnge. Observe that a continuation of 
 the function B Into the supersonic rpnge is given by 
 equation ( jf- ) as 
 
 d9 
 
 + /^^^^l 
 
 dq 
 
 In the supersonic range, t>.e function E = 9 - 9^ can 
 thus be Interpr'^tc'd as the hodograph of the Mach lines 
 for a givi-n flo'.v. 
 
liO 
 
 FACA ARR No. l4l29 
 
 The differential line elements dx and dy for the 
 IJach l.inss in the physical plane are now given. Fro-i 
 equations (-'.^7) ^^^ (5^), the line elements along a Ivlach 
 line for a given flov/ are 
 
 dx 
 
 pq \ 
 
 (t.J^i 
 
 cos 
 
 
 (58) 
 
 Singular points rlong a Mach line are characterized by 
 the vanishing of the common factor of equations (58) 
 
 c q q ^ ^' 
 
 56- 
 
 
 
 (59) 
 
 Equation ('39) represents in the plane of flow tv;o pos- 
 sible singular loci or "limit lines" for the two families 
 of iiach lines associated v;lth the plus and minus signs 
 in equation (56). Clerrly, the two singular loci cannot 
 occur simultaneously since the two conditions cannot be 
 .satisfied simxultsneously . Observe that equation (59) is 
 equivalent to the vanishing of the Jacobian given in 
 equations (55 )• Thus, the vanishing of the Jacobian is 
 not only the condition for the existence of a singular 
 (cusp) locus for the streamlines but also the condition 
 for the existence of a limit line (envelope) for the 
 jMach lines. 
 
 The existence of a singular locus may be looked upon 
 as being equivalent to the vanishing along a curve of 
 
 the Jacobian J 
 
 \Q,qJ 
 
 of the transformation from the 
 
 H to the physical- 
 
 hodograph-plene variables 6 and q 
 
 plane variables x and y. it is remarked that 
 
 singular solutions exist for which the Jacobian j( ' ' 
 
 of the transformation from the physical -plane variables x 
 and y to the hodograph-plane variables 6 and q 
 vanishes identically in a region of the physical plane. 
 In this case, as Tollm.ien pointed out, 9 and q are 
 
NACA A:^R No. LliT29 
 
 J|l 
 
 no longer independent variables and the flov/ cannot be 
 described in the hcdogrcph plf^ne. Examples of these 
 "Liissed flows" are the solutions of Meyer (reference 11) 
 for supersonic flow inside and outside sharp an^^ies . 
 
 It is of special interest to apply the condition 
 for the vanishing of the Jscobian to the particular 
 solutions j^^v a-id ^J'^ treated in the early part of 
 tl'iis paper. The expression for the Jacobian for a 
 particular solution 
 
 0", = P, cos k9 
 '^k k 
 
 ^k = -\ ^^^ ^^^ 
 
 is, with the use of equations (p), for k ^ , 
 VcG oq " 6q 69/ 
 
 J = 
 
 pq 
 
 k' 
 
 2 
 
 q-^^ 
 
 P^- sin^ke + (^) (l - ir) Q^^ ccs^k9 
 
 (60) 
 
 Clearly, this expression for J is positive in the sub- 
 sonic range M < 1. At the sonic value M = 1, P^ ^^ ^ 
 (see table following equation (2^)) and J is again 
 positive. At tne first zero of P^. in the supersonic 
 range M > 1, Q,. ^ 0; hence, J 'is negative. The 
 
 values of :vl, for all the pairs of values 6, 1,1 for 
 which the Jaccbian J vanishes, therefore lie between 
 
 II - 1 
 
 •nd the value of M at the first zero of P 
 
 k 
 
 (or S\ ) in the supersonic range 
 3y means of the relation 
 
 
 
 Pv 
 
 f ^'k^k 
 
4^ 
 
 the vr::ni3hing of the Jacoblan yields 
 
 NACA ARR No. li|l29 
 
 cot ke = + --=JL= (6i) 
 
 V^ - 1 
 
 Equation (51) is the relation for pairs of values 9, M, 
 which interpreted in the physical x, y plane constitute 
 the lir.iit line for the particular flow j^C,, t|.v. The 
 
 values of M thpt satisfy equation (bl) accordingly lie 
 between M = 1 and the value of M at the first zero 
 of Si- in the supersonic range. 
 
 This paper is closed v/ith the following remarks on 
 limiting values of IvI in connection with the use of 
 velocity correction formulas. The limiting local values 
 of M in the case of uniform flow past a prescribed 
 boundary, in general, depend on shape parameters. The 
 use of a velocity correction form/ala, however, yields a 
 constant limiting value of M that depends only on the 
 particular correction formula used. The gecmetric-mean 
 correction formula yields the value M = 1; the approxi- 
 mation of Temple and Yarwood yields I] = 1.55; snd the 
 arithmetic -mean correction formula given in part I 
 (reference 1), which is based on a linear combination of 
 a source (lim:iting value M = 1) and a vortex (limiting 
 value ■;; = <») or a spiral flow, yields the value M = I.I5. 
 
 Langley Memorial Aeronautical Laboratory 
 
 National Advisory Committee for Aeronautics • 
 Langley Field, Va. 
 
NACA ARR No. L'+I29 1^3 
 
 REFERENCES 
 
 1. Garrlck, I. S., and Kaolan, Carl: On the Plow of a 
 
 Compressible Fluid by the Hodograph Method. 
 I - Unification and Extension of Fresent-Day 
 Results. NACA ACR No. Ll4.C2i+, 19i^Ij-. (Classification 
 changed to Restricted, Got, 19)411..) 
 
 2. Rlngleb, Friedrich; Exakte Lbsungen der Differential- 
 
 gleichungen eiuer adiabatischen Gasstromung. 
 Z.f.a.M.M., Bd. 20, Reft l\., Aug. 19)40, pp. I85-I98 
 (available as British Air Ministry Translation 
 No. 1609). 
 
 5. Chaplygln, S. A.: On Gas Jets, (Text in Russian.) 
 
 Scl. Ann., l\"osoov; tTiperial Univ., Math.-Phys. Sec, 
 vol. 21, looLi., op. 1-121 (available as NACA TM 
 Fo. 1065, 19)4)4). 
 
 k-. B^rs, Lipman, and Gelbart, Abe: On a Class of Dif- 
 ferential Equations in Mechanics of Continua. 
 Quarterly Arpl. Math., vol, I, no. 2, Julv 19^3, 
 pp. 168-188".' 
 
 5. Bergman, Stefan; The Hodograph Alethod 3n the Theory 
 
 of Compressible Fluid. Advanced Instruction and 
 Research in Mechanics, Brov;n Univ., Sxjmmer 19'-l-2. 
 
 6. von Karman, Th. : Compressibility Effects in Aero- 
 
 drnaruics. Jour. Aero. Sol., vol. 3, no. 9* 
 July 19/4I, ^r. 557-35^. 
 
 7. Hurwitz, Adolf: Mathemati sche Werke . Bd. 1 - 
 
 Funktionentheorie . Emil Birkhanser '■ Cie. 
 (Basel), 1032, r)p. 311^-320 and 652-6^14. 
 
 8. Tsien, Hsue-Slien: Two-DiTiensional Subsonic Flow of 
 
 Compressible Fluids. Jour. Aerc. Sci., vol. 6, 
 no. 10, Aug. 1959, u^- 399-^1-07. 
 
 9. Tollmlen, W.: Grenzlinien adiabatischor Potential- 
 
 strbm-urgen. Z.f.a.M.M., 3d. 21, Heft 3, -Tune I9I4I, 
 pp. 1I4O-I52. 
 
 10. Ringleb, Friedrich: Uber die ^Dlfferentialgleichungen 
 einer adiabatischen Gasstromung und den Stroraiongs- 
 stoss. Deutsche Mathem.atik, Jahrg. 5, Heft 5> 
 I9I4O, pp. 377-38I4. 
 
y.; NACA ARR No. 1J+129 
 
 11. Meyer^ Th. : ITber z^veldimenslonale Bewegungsvorgange 
 
 in eineiTi Gss, das mit Uberschallgeschwlndigkeit 
 strbiut. . Porschungsarbelten a. d. Geb. d. 
 Ingsnieurwesens, Ksft 62, VDI-Verlag G.m.b.K. 
 (.Berlin), 1903, pp. 3I-67. 
 
 12. Kraft, Hans, and Dibble, Charles G.: Some Two- 
 
 Dl"'!ensional Adiabatic Compressible plow patterns. 
 Jour. Aero. Sci., vol. 11, no, li, Oct. 19^4-^ 
 PP. 283-298. 
 
 This reference, which has appeared since the present 
 
 paper was submitted for publication, contains 
 
 graphical flow natterns associated v»fith the 
 
 ' 1 '^ 
 
 particular solutions of index k = i— , ±^, and ±2, 
 
NACA ARR No. L4I29 
 
 45 
 
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 C^OCOrHrHOcONi) KNQn 
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 O V Cr- CTnOnOn on on CTnCO CO 
 
 OnC-00 i/nsO i/nO OCOnO 
 O O onCOnT - ■ - 
 
 NO ONrH_;tp _ . 
 
 NO LTMTvd" KNKNCM CM H O 
 
 CO CO CO CO CO CO CO CO CO 00 
 
 tC\r-i OnCOCO ONLTNtfNirNKN 
 00 O rH KN iT'p-rH iT-CO t^ 
 CM t^rH iTNCJNKNirNNO QnO 
 O OnCTnCO C~-t"-NO LTnlCNKN 
 
 CO f-c^p^p-p-c^r-r-^- 
 
 CM QNiTNOJ ONC^tJNKNCM O 
 
 CM o ONCOCO irvdi^oj r-t 
 
 P^ [*-*J3 \0 \0 NO NO \0 sO ^O 
 
 r-co-diTvO 
 
 r-t P-C^O t^ 
 O (J-^O'O o 
 OOD P-r-»J3 
 
 NO LTNLP-lTNtTN 
 
 iho o knonOno r-iTNO 
 
 OJ ONt'-^^NO ONOJ C^-dKN 
 P-- (3NCM LTCO f-t IT-CO OJ NO 
 iTN-d-d KN fM CM •-' O O ON 
 GO CO CO CO CO CO CO CO CO c^ 
 
 ITM/NCM ITCO KNrHNXJ kn u^ 
 r— O OCDC^rH r-t t^rHfM 
 00 LfNCJNOO i/nO"CMnO On 
 CTnOnCOCO P-nO LOLP_d"KN 
 On O^ONONO^ONONO^ ON ON 
 
 NO LP rH OJ rHCO OJ »H On"^ 
 
 r-t tnoNfM i/NC^o ir\_d-d 
 
 CMCO_drH p-KNOCO lP-OJ 
 KNOJ CMCMrHrHrHOOCJ 
 ^OnC- O^O^O^O^O^O^ on 
 
 _d KNKNCM CM r»JNO 
 ONNO K-. O C^-d ON U^CM _d 
 OnOnOnOnODCO C-C— t--^^ 
 
 cococococococococoao 
 
 r-t ONt/Np^lTN 
 
 KN_da5 KNO 
 OldGO KNOO 
 OnCO P^P^-^ 
 
 F-c^p-p-p- 
 
 O _dGO rH CM lP«^0 CM LfNOJ 
 O OnnO O no O rH on OJ rH 
 
 OJ p-p-rH p-c^^-p-o^»^ 
 
 O O rH K\_:+ ufNsO f-CO O 
 OOO O O O O O O rH 
 
 COCMnD on*-*cm oknknp^ 
 
 _dCO CM r-^ rH OnnO rH NO 
 KN ONNO (M OnnO CM U~^ rH nO 
 rH rH CM KN KN_d ^^ "J^"^ ^-O 
 
 _d l^ CTn LP\ li^sO f-CM O O 
 CM CO -drH CO U^rHCOsO r-t 
 
 4 rH CM CM CM CM CM 
 
 sO rHsO O KNUNnO t/NCMC 
 
 iT^ U~> IfN 
 
 njipip— CM ir»»oco o 
 
 CI>COCO0O ^ONOnOnO^O 
 
 j_dNO00 O CM U~«00 O ITN 
 
 SOOOrHl-HrHMCMCM 
 
 O u-nO tPO iTnO mo U^ O itnO 1/nO 
 KNK>_d_dl^l'**^'^ P-P^ coco ONO^O 
 
46 
 
 NACA ARR No. L4I29 
 
 
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 CrCO CTNLnO ^ o a\<-H_d- o <-t c^Hco o\nj o o 
 
 
 cr\C~-trN«H t— 3-^-1co_ 
 
 so trxr-icO-O h-nO ONf-lT* 
 
 O 0^ 
 
 3D CTN-d-t 
 CTsCrsOCr^XKXjCO t^t^C^ vO^C-O LTMTMTMJ'NJd--:*-^ -ZT-lt »0 K\ K\ t\J ry rH rH O O <H CM t03*sOCO OOJ ir\ 00 KNONf-r-l 
 
 o (x<o ir>c7st\jco ^-c^^K^ ca*^ irco k>ih oco o t-r 
 
 -,-_ . -. „. . 17\ir\t^K\^ Q\C\i CVJ iTNCM 
 
 ONr-LTNrH r-_:t»-*CO lArH t--irN(\l OCOirNN-\r\jOCO l/^K^H <T\sO_:J'OsO KNt- 
 
 ffstr^ONO^cococo C^C^^- vo^O'sO^ ifMr\irMrMr\_:t -zf-^_^icvrCMr\»ooj oj r-i 
 
 H O oj O rr\C^_:d-OMrs(7N 
 
 r-O H r-t*-(\J KNCTsCM rH CO fTsr-rH r- 
 
 
 QsCVJ^^-d-rOiKSKM^M/Sr- _ 
 
 O O O r-1 r\j tc\^ir*^ C^ CO O iH KN_d- 
 
 cr^rrxr— »r\CT\ 
 
 I rH iH rH rH 
 
 - -_ iTN rH o rH irvfvi K><7\i/\^o <H Tu r-r-KMr>_::t<o mo 
 
 ITnO U>C5 iTNONrH O C-rH KNK\rC«(\J OCO iTvvO f-CO OSCTvO>COCOvO_cJ-rHCO rH 
 
 o>co irvc\j r-_zfoj cTMr-ry co-o_:iTvj o ^-trv:j-(\j o co^o j-nj oco i/ncm o^-d" 
 
 K\0 ir^^OOD lTnOvOvO-tJ- 
 *r»vD CMTCO ON^-O ctnKN 
 *-• ovltncn'^o ir*^ OnOJCO 
 
 CO rH iT'O l^^t-tCO^-O K\ 
 rHrHOOOrHpjry KXJ^ 
 
 rH 0«H l/M/N 
 
 rH fxi ir»aN_:t 
 
 tr^K^c^J^g_3• 
 
 ltnlTssO C^hd 
 
 I I I 
 
 O J-0_d-t^ir>SD t/MTVlT mOvOtOCM-d-*-<irNrHcrs ONt^OOl/MJ-NO rH\0_^ 
 
 O oojnS iTNO nj OJ o t^ rr\_d-c7\Ovo r^^t-oj K\ fy ONf^X-d-fMCO K\njco^ 
 
 irsG ir\OsO rH K\rr\rHVO O iH .H r\J rH O OnO PCMJ-N t-OO O rH (\J fU KMTSOJ iH 
 
 o>co iT^fM c^ir-oj ctn^^ cm , . _ . __ - - _ _ _ . . . 
 
 C—OO O »-4 CM fvj KM<SfM "H 
 Oco ^-irNfTvrHCO l/NfCCO 
 
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 OsSO CM CO irMrNKNv£>_H'U> 
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 iTSrH t*-K^ON 
 
 I I 
 
 I 
 
 I I 
 
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 O r-t^<OCO ir\J"OJNO m CO C-*0 U-Nt*-fM OJ O rH cm CM_ct*^ rH O rH OvO O rH 
 
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 iTNO iTvrH t^CM U">0 IfNrTs CO O CM K~S_d- 1^ iT* C*- <-* _i± CO rH^iJ-t^ONCM iTvCTs^H U^ OD 'M-li'^ 0^'-H^2> _H-CM O 
 
 crco mcM t^lr^cM cr>^ ^f^ ONCOso_d-CM Oco o-^i3_3- cm /H crsr-uSlq-Hco r-<M r-KNX) KNC0_d-O ltncS if\ .„ , 
 
 ONCTNONcrcococo t^t^t^ vij^ONC^^ovo umtnu-mA u^lrtJJ■_:J_:^_3■-d■'<^K^K^ cMfMrHr^oaoor^rH r-iojrvj rf\fr\ 
 
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 CJnOvO onO O (TsCnnO O •-* C^-CO CM O 
 
 CTv-lKO K\C0 
 
 0^<J-'0'^Ch£OCOJX>CO Z—t^ C — sO VO *sO ^O sO vO iTilTm/n 
 
 K\CM CM fM r-tr^ rHOOO 00(Hr-4*H 
 
 rHOOCM^ r-rH^O t-CO_H- CM CO U^inONCT^CO OmTNC"- _d<D CO 00 iP. KN O rH li^OO rH fM C7n CT^O rH rCCO O^CO t-r-Q\^0 
 
 O rH (JnCO CO r- ON r-nj SO OCOirvO KMrNNO{M_;tU^ vi3\C\OvO-»OsO rHvOvO KN ^-:tQ^^'C' O O t— O OCD-zt- CO O O C\vO 
 
 IP-O UNCM "HCO KN^-O r^ ."^ sC .H\0 0_:+C0 CM QN^ HN O r-_j-rH CD ONONNO-J' OJ O ONaNONa^*H KMACn K\0N»^»-' CTv 
 
 OnOO UNCMCO ir«KNOa0 iTn C\J O OnC^^-^D-JCM cm O On CO t*-UV3'K\rH CnC^-*-0 KN O t^fCvO C^3'(M a^'^ '<^ rHCO^O J-rH 
 
 ONCTsCTNO^COCOCOCD C^C^ (-- C--,^ s© \0 sO \0 VO ^O iTx ITMJ^ iTn tTM/N ir\3"-d-=f^ ^KNKNKNCM CMCMrHrHrH rHOOOO 
 
 OJCOCO C— t-OJ U-v4C0_d- CNK\iH_d-_=f-:d'-4-»OCM-=J- 0^0'^K^K^O HNSOCO-d-O 
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 ^_, , O OcM rHCO irvKNrHONt^ CM^<^0^^-lf^ 
 
 -^ , ______ _ „ — -.-._.__,._„. .-._ ,_,_^.Ot-lr- O-C0-:jCO CTNONr-KN^CO CO iTNONrH O 
 
 iT-O^O^d-irvKNOvO rH^D On^O fCxO t^K\0 irw±lC\ KNCM 04 CM *-4 r-l^O i-i r-i^'^ PJ OnC— irvJ-_^U^t^ONOJ ^O rHvi> KNO 
 
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 ON^vONO^COCOCOCO r-C— C— t— C^C— M3vOv£)vi>vOvO VO^DVO irNirNl/MrM/M/Mrv _:t-^-^-d'f*NK\KMCvK\K\ fXJ OJ (M (M CM 
 
 O_:JC0 rH CM IT^^O CM ITNCM COOJ^O CJNrHfM O fOsKM^ .cf l/N CMTM/N^^ t^CMC_ ., ,^ 
 
 O CTNVO O'^ O rH crsCM H _j^ CM C^-zJ-H cr<^ rHsO C\JC0_^»HC0a>rHC0vOrH \0 >-*''£> O KMTMS U-\fM O 
 
 eg C^C--rHt^r-r^f-ON'-* KNOsnO cm CTn^^ cm irirH^O oj O-K^ovJ-O C^t^»<~<0 cm r-rH\0 O_;tC0 CMnO m pTinO CTsrH_: 
 
 2 2 C? 'T^""'**^ 1^-00 O rH rH CM fCi fCvJ- iTMTNnO vO t^ O-CO CO On O O ry CM H~\ IT^XI CO On rH CM KMr>0 F- OnO rH »r\_; 
 
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 LP« i/N iT* 
 
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 *-*OJ K\^lfNir«NO\0 t*-t- COaOCOCOONONONCNONO OOOOrHr^rHrHCMOJ HNKN_d"-d"irM/NvOvO r-t*- COCO CT^OO 
 
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 47 
 
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 C^O_^OJ frN>.0 O r<vj-_d- O tCVX) CM {7\r-ir\0 o 
 
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 . f-rH O OnKNOJO (M O KM/>0 O O-lTu^O 
 
 CNC— iTiO rr\CO c\|vO ctnoj u^rf\Oco l>-vO i/mtni/mtn ^ C'-co O c\i_zj^ ro^^o t*- rH CTsrH O c^'J^f^J rr\*H c^ 
 
 CO tC\\o lr^a^K^c^^-'-^co 
 
 KVd'ON^-O 
 
 q^o^^jit-it^ o_d-a\(\J 
 
 0_:J-^ LTCO 0JvO_d-'-« K^ 
 
 
 
 f\jvO qn'*^ u^hr\ru c— »-« o c--co_^-^o ir\ 
 J-*o O t^o^cJ-iTsNO oco O ryco k>vo 
 H iroo r-co_3^ru rH r-ON \o<5r-ii' ' 
 
 ITMT. 
 
 a\a\c\CT<6co c^r-t— ^b vo^ovbvovi) r-c^^-r-o- f^-c6coco d\o\<D r^ rH ro Tf^'r^^ eg ry o^co^ocb r^^^csi i/> 
 
 conj ONt-co r-»Hvo irsfH r-t c\i,:d<o uMr>covo_:i<o c^cy-drvjco o iTNrH ir\K> t'-co Orj^^ rrs_^«H onC^ (>u%iho h 
 
 Of-_d"0 mrr\0 CTNt-- ^~ C--COCO c^O iH rcvjut^^^- 00 O pr\uAt^O J-qnOJ <H OJ^C^.AJ ct^X) O LT^^f- CO^O^O NA^O 
 
 ONONCrNCT^COCOaO t^t— C-- t^t^C^C— coco coco coco go OMJNCTsOnO ( 
 
 rrJ3- LT^ C"-CO O KMTCQ «-i 
 i rH na OJ C\i OJ KA 
 
 
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 0^0^ CNZOroCO t^C^t — C~- 
 
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 o t^i/vz+u-Nt^rH K^^-f\J 
 
 CMrHrHHi-HrHC\ifMCMf<A 
 
 t^ [^ f- r- r— c^ t-- 1>- C-- c^ 
 
 (XlCOvO K^CO r^ iHCO CM cr\ 
 
 CO LPiNAfVI t-t CM ONC^rH t- 
 KV^ ITNVO t^-OO ON rH rCvVO 
 
 o- 1^ c^ e^ r-- c- f-<xj tx) CO 
 
 KAOMA-^t^-d-rH o ir\os c*-irNvoao lo 
 
 onco ir»^ca av>o--ovo_d- co uacm c—co 
 
 00 K^cM_z^a^^-co rn^OMA rHHojrrvirN 
 
 O ir^C5 U^O^O CM O^UAOJ CTN'^ nao f- 
 
 OSOnO O rH rH cm OJ KN_ZT -J-irw^ t^ t^ 
 
 iH mcyvO OGO rHCO C^K\ 
 *HvO CsKMTNrH rH UAt~-CO 
 ir\rH(M KNC^OJ <T<X) O Lf^ 
 0>CO^O_:JOJ CM »-H rH CM CM 
 C^ CTn CTs On 0^ CT^ On On cr^ ON 
 
 rH O O OJ J■lr^KM/Na^C^■ 
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 rfNKN_:3-Lr\Lr«0 r--o-co on 
 Cr> ON C7^ (7n ON CJN On ON ON Qn 
 
 ITNNO CM_rfOC0 t-f-t..^^^ 
 
 knO H irNrr\KNO_d^ C^ 
 
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 C3 M OJ CM NAJ-vO CO CTNCM 
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 o o o c 
 
 O rvj ctnno nj irNi/NK\o o oncmcono c*- 
 
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 CM KNsD ONfM U^OnOJ LfNf- OnO CTnCOnO 
 
 irco «H_::tCD r-i^:i<X> t-i^ C-r-i K\nO on 
 
 rH rH CM CM CM KNKAMA^rt^" _:t UN U^ tTN LfN 
 
 rHr^r^^^rHrHr^rH»-^rH rHiHrHrHrHrHrHrHrHrH rHrHrHrHf 
 
 lA^CM rCvCM <M ONNOCO CM _d-Ol^^O_:tONDOONO_d■ K^H OCO^ KMTNCO OnHA O K\_rt KNnO cm n£> Cw^-O CO CM O O O 
 
 O C;-*^ QNOJ irNC^X) KAC— O rH_j-rH O rOCO r-t CTnCJn H iTNrHCOCO OnDNO ITKO t^t--^i3 KNt-C0_d-U~vrH O CM C-_d"£M H 
 
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 O^OO lTNKNOCOnO LTNKNCM rH O O CJN CT'^O CO CO C^ C^ C— "^nOsOnOnO LTNirNifNl/N UN l/N LT- LTMrNLTNO nO r" O- COCOonOnO 
 
 owoNCTNONONCococococo cococo c—^-r^r-c^c^c^ r-c-c^t^t^c^c^r-c^t^ t^t-c^c^c^r-c^r^c--f- r-c-i>-f-co 
 
 rNKV:HNO O CM kaC~-J-^ 
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 _ _ _ _ sOnO (/NU^irsLrNUArH_::tNO 
 
 OnCD ir>OJ ONf^U^CM oco NO ITN^KACVJ rH O O O^O 
 
 ononono^cococococo c^ r-r-c^r-c^c^r-c-'^NO 
 
 C^HACM KNCOnOCO rH_:3-0 I/nonCM rH ITM-H J-KNCM O^ 0^<J\lI^H\0\ 
 CM OnC^-^O^OCOO.' CI ONCTN r-3-rHNO CT\rH C3 C—rH t-i CO rH 0_dTy 
 ■" Ot<AK\_^r-CM CO ir^ KNrH O rH CM KAnO CTn OJ C^Cy t^«N 
 
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 O O CM rHCO iTNKArH On^- CM K^ON^-U^ 
 
 C^X3_3CO OnOnC^KNnOCO G0lrNC^r^O 
 
 KN CM CM CM rH"r^ NO .H rH NO CM ONC^iri_:t-J"i^C^CrNCM nO rHvO KAO 
 
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 (M C^NACJN-ZtO QnC^KN30 
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 rH (M Kv^i/NiT^ONO C^C^ COCOCOCOC7n(J\<JnC?nOnO <5000rHrHrHrHCMCM K\KXj-_:Ji/Nir*O^0 C^t- OOGO CT^CT^O 
 
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48 
 
 NACA ARR No. L4I29 
 
 
 O iH C^HCO CTNtNJ O O 
 
 co^ Ococo ry LP'lr^o^O 
 
 3" ON KVzJ- CTM/MTN CTN-zJ- O 
 OnC^iTnH iTNojGO KNOT ry 
 0^0^ OnO^COCO C^C—^^O 
 
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 Lrsu"vd-rOvtr\rvj m r-t o 
 
 iTNCy ONO_;JaD r-^ooo o^ 
 
 COCO_^CO C— ifNOJ ONU^KN 
 O OJ KV^tvDOO KNON^^ r\j 
 
 U-nOnOnO O KNi-i_:t'-< On 
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 ON»<Nr-»-i irNt-co-d--3'f\J 
 
 rH_da3 OJ 0\ 
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 ry r-ry kncq 
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 00 c\j «HNO ir.c^ifNr-c— KN no_iKo oj iH kni/nno cm 
 
 OnvO O^fi OJ LfNON'-H to ON*-' t^ry t~l\i lTn^O c^ 
 
 -ztONKNNNCo K^^y_:d■a^M^ _^o OnO t/xx) ry oncd 
 
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 _=J<0 roO-zj-iTNOvirco ITS 
 
 GO_J-0 OJ irNC^_^*^ LP- On 
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 coco ONOJ C*-_d'irNt*-H-\0 
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 ds-ifi-t o o 
 
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NACA ARR No. L4I29 
 
 49 
 
 a\0 _:trvj K^^J^ o tfvj+^j- 
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50 
 
 NACA ARR No. L4I29 
 
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NACA ARR No. L4I29 
 
 53 
 
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NACA ARR No. L4I29 
 
 54 
 
 
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NACA ARR No. L4I29 
 
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NACA ARR No. L4I29 
 
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