'^//Vc/\'r<^>31^ 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1298 
 
 ON THE PROBLEM OF GAS FLOW OVER AN INFINITE CASCADE 
 
 USING CHAPLYGIN'S APPROXIMATION 
 
 By G. A. Bugaenko 
 
 Translation 
 
 'K Voprosu o Struinom Obtekanii Beskonechnoi Reshetki Gazom v 
 Priblizhennoi Postanovke S. A. Chaplygina." Prikladnaya 
 Matematika i Mekhanika, T. XIIL No. 4, 1949. 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 1 20 MARSTON SCIENCE LIBRAf Y 
 P.O. BOX 11 7011 
 iAlNESVILLE, FL 32611 -701 U 
 
 Washington 
 May 1951 
 
 SA 
 
jf(f f/^^/7 
 
 3^7y^33 
 
 NATIONAL ADVISOEY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEtvIORANDUM 1298 
 
 ON Tffli: PROBLEM OF GAS FLOW OVER AN INFINITE CASCADE 
 USING CHAPLYGIN'S APPROXIMATION* 
 By G. A. Bugaenko 
 
 1. Some well-laiown results of Chaplygin's method (reference l) 
 are first presented. For the adiabatic law of the state of the gas 
 
 when p = kP , the following relations hold; 
 
 PO 1 - 2^ J 
 
 P = PqU - 
 
 2\P 
 
 Y_ 
 2a,J 
 
 (1.1) 
 
 where p is the gas pressure, p is the gas density, V is the 
 modulus of velocity, Pq and Pq are the values of p and p at 
 
 the critical point of the flow- at which the velocity becomes zero, 
 k is the coefficient of proportionality, y is the ratio of specific 
 heats, and a and P are constants. 
 
 If the angle 9 between the velocity and the x-axis and the 
 magnitude T equal to V^/2a are considered, then, as was shown by 
 Chaplygin, the equations of gas motion assume the form 
 
 ar 
 
 2T 
 
 ae 
 
 b^ (i-T)P 3'-? 
 Si = - 2Th-T)P+l 06 
 
 a^ i-(2i3+i)T aij/ 
 
 where cp is the velocity potential and ^ is the stream function. 
 
 (1.2) 
 
 *"K Voprosu Struinom Obtekanii Beskonechnoi Reshetki Gazom v 
 Priblizhennoi Postanovke S. A. Chaplygina. " Prikladnaya Matematika i 
 Mekhanika, T. XIII, No. 4, 1949, pp. 449 - 456. 
 
KA.CA TM 1298 
 
 Chaplygin reduced these equations to the very simple form 
 
 So ^ Se 
 b^ s* 
 
 Sep 
 
 1 £0 
 
 K Sy 
 
 (1.3) 
 
 ty introducing the new variable and the constant K defined by 
 the formulas 
 
 _ J 2T 
 
 )P 
 
 dT 
 
 To.2 
 
 > 
 
 K = 1-(2P+1)T 
 (1-T)23+1 
 
 (1.4) 
 
 Chaplygin showed that for velocities far removed from the velocity of 
 sound, the magnitude K is approximately unity and equations (1.3) 
 can be integrated by assuming K equal to 1. For K = 1, these 
 equations go over into the conditions of Cauchy-Elemann; hence, 
 oj = + ia will be an analytical function of the complex variable 
 f = cp + i^. 
 
 The equation for the elementary vector along a streamline in 
 the approximate treatment has, as is known (reference 3), the form 
 
 dz 
 
 (ae^'^ + bei'^)dcp 
 
 1 + (1-^002^ 
 
 AT 
 
 2atj 
 
 1 - (l-'^2)" 
 2 ATi^g 
 
 The complex pressure is given by the equation (reference 3) 
 
 (1.5) 
 
 iX 
 
 — 2~J (® " ® >^^ + Po(^ -^o^2' J 
 
 dz 
 
 (1.6) 
 
 2. The steady potential flow of a gas through an infinite 
 cascade according to the well-known scheme of Kirchho:ff with 
 separations of the jet is next considered. The vanes of the cas- 
 cade will bs assumed to be plane (fig. l). 
 
 Velocity of the gas in the flow at infinity is denoted by 
 Vo--|_ and its angle with the x-axis by ^3_; velocity of the gas in 
 
 the jet at infinity is denoted by '\^2 ^^^ ^"^^ angle with the 
 
MCA TM 1298 3 
 
 X-axis by %,2* '^^^ angle 6 between the velocity vectors and the 
 X-axis lies within the range -2jt<0^O. 
 
 The velocity field of the flow repeats itself for each dis- 
 placement by the pitch of the cascade, that is, by the vector he"-^ 
 
 The condition of constancy of the mass flow for steady flow of 
 the gas gives 
 
 Q = - Pcoi Vo.1 h sin (\+ Qooi) = i::to2 ^2 ^ (2.1) 
 
 where p^^-^ i^ the density of the gas at infinity in the flow, p^^z 
 
 is the density of the gas at infinity in the Jet, and n is the 
 width of the gas jet at infinity. 
 
 By .-naking use cf expressions (l.l) for p, eq^uation (2.1) can be 
 represented in the form 
 
 - V^-L h [l - -|i-J sin (\+ ^i) = V^2 ^ V^ - -|i-y (2.2) 
 
 The behavior of the function f = cp + i"i|/' In the z-plane of the 
 gas flow is now considered. For simplicity, the function f is 
 assumed equal to zero at the critical point of the flow (fig. l). 
 From the relation 
 
 Ss 
 
 dcp = -^ ds = Vg ds 
 
 it follows that on moving along the streamline ij/ = 0, the function 
 cp varies monotonically from -«» at the point E (infinity In the 
 flow) to +<» at the point C (infinity in the jet) and passes 
 through the zero value at the critical point where the stream- 
 line branches. The value of the potential f = 9 + iv at the 
 critical point 0' displaced by the period he"'^'^ relative to 
 the point is found and (fig. l) 
 
 CP(O') = J Vg ds = J Vg ds + J Vg ds + J 
 
 Vg ds 
 
 OMM'O' OM MM' M'O' 
 where MM' is a cut parallel to the axis of the cascade. 
 
KACA TM 1298 
 
 The first and last integrals mutually cancel and therefore 
 as MM' approaches infinity in the flow, the following equation is 
 obtained : 
 
 ;p (0') = 7„i h cos (\ + e^^) (2.3) 
 
 Furthermore, from the relation 
 
 dQ 
 
 d\j/ = — 
 
 where dQ is the quantity of gas flowing in unit time between 
 infinitely near streamlines, it follows that on being displaced 
 by the pitch of the cascade the function ^ receives an increment 
 equal to Q/pq* Hence, 
 
 nO') .^ = - (l - ^^J hV^i sin (\. e^i) (2.4) 
 
 In this manner, the f-plane with double-sided cuts along t?ie 
 half-straight lines parallel to the axis of reals (fig. 2) corresponds 
 to the region of the gas flow (fig. l). All the cuts, because of the 
 rule by which the cascade was constructed, are obtained from the 
 initial one (the positive cp-axis) by simultaneous displacement along 
 verticals and horizontals at distemces that are multiples of Q/Pq 
 
 and V -^ h cos (\ + 9 -^) , respectively. 
 
 • Because 'm - e + ±0 is an analytic function of f = cp + 1^ , 
 the problem ms.y be solved by relating these functions with t?ie aid 
 of a parameter that varies in the upper semicircle of unit radius, 
 as in the Levi-Civita method. 
 
 By considering the recti linearity of the cuts in the f-plane, 
 the analytic function f(t) is found, which brings about the con- 
 forme,! transformation of the f -plane into the semicircle t. In the 
 t-plane, the flow of an ideal fluid about the boundary of the semi- 
 circle is constructed. For this purpose, sources and vortices of 
 strengths and intensities are located, as shown in figure 3, at the 
 points tec, and t^~~ that are symmetrical with respect to the circle 
 and at the mirror reflection of these points in the diarseter, that 
 is, at the points toe and too" • At the origin of coordinates we 
 place a sink of strength 2q (and a similar sink: at infinity). 
 
KACA TM 1298 5 
 
 In t.?ie constructed flow in the t-plane, the upper semicircle of 
 unit radius and the diameter of the semicircle are, of course, stream- 
 lines so that the stream function f maintains a constant value at 
 the boundary of the upper semicircle t; in the f -plane, this 
 boundary vrill correspond to straight cuts. The complex potential 
 of the constructed flow will have the form 
 
 f (t) = ^y Uy + iq.) log (t - ttt) + (-Y + iq) log (t - ^J+ 
 
 (-■> + iq) log (t - t^) + CY + iq) log ( t - i^ j - 2ig log t + constant 
 
 or 
 
 f (t) = ^ [(7 + iq) log ft + i - 2MJ - 
 
 (■> - iq) log f t + i - 2MJj + constant (2.5) 
 
 where 7 is the intensity of the vortex and q is the strength of 
 the sources, and 
 
 The arbitrary constant in equation (2.5) is chosen so that 
 f(t) becomes zero at a certain point t = e^-^, the position of 
 which will be subsequently determined. 
 
 Because the logarithm has multiple values, the upper semicircle 
 of the t-plane will correspond to an f-plane with an infinite number 
 of straight cuts '^ = kq(k = 0, ±1, ±2,...), where cp changes from 
 k7 to +CC, as easily follows from equation (2.5) by substituting 
 t = ei® (the arc of the semicircle) and t = t^^ where t]_ is the 
 real amount of the interval (-1, +l), the diameter of the semicircle. 
 
 In this manner, the function (2.5) establishes a conformal 
 mapping of the upper semicircle of the t-plane on the f-plane with 
 the double-sided cuts represented in figure 2. 
 
 The function (2.5) is used in the work of N. I. Akhiezer 
 (reference 2) where it is obtained by successive conformal mappings: 
 the f-plane on the half plane, the half plane on the unit circle, 
 and finally the circle on the upper semicircle. 
 
MCA TM 1298 
 
 The point t = is carried by the transformation (2.5) Into 
 f = +«» so that the radii AC and BC go over into the infinite 
 segments (figs. 2 and 3). The conformal property of the transfor- 
 mation breaks down at the points t = -1, t = +1, and t = e-'- ^ 
 (the point e^^ corresponds to the origin of the double-sided cut 
 in the f-olane). The condition df/dt = for t = ei^ gives 
 
 -^^-^-^ = -2L_:_ia (2.6) 
 
 cos £-M cos £-M 
 
 In order to obtain the elements of the motion, an expression for 
 the derivative df/dt is req.uired that is represented in the form 
 
 df q (t-e^g)(t-e-i^)(t-t-^) 
 dt n (t-t^) (t-i^-^) (t-t^) (t-t^-1) 
 
 The quantities y and q. are next determined. When a point in 
 the z-plane of the gas flow is displaced by the pitch of the 
 cascade he"-^'^, the point t, corresponding to the point in the z-plane, 
 goes over from one sheet of the Eiemann surface to the next, passing 
 once around the point E (t =1^1,), as a result of which the function 
 f(t) receives an increment y + iq, as follows from equation (2.5). 
 Because the corresponding increments of the functions cp and 'ii 
 are equal to Vcc^^ h cos (\ + Qk^) and Q/Pq^ respectively, the 
 
 following equations are obtained: 
 
 7 = "^1 h cos (\ + da^i) 
 
 (. 
 
 q = - I 1 - -^—j V„i h sin (\ + 0001 ) 
 
 (2.8) 
 
 From the expression for 7, it is evident, among other things, 
 that 7=0 corresponds to the case where the approaching flow has a 
 velocity at infinity perpendicular to the axis of the cascade. 
 
 3. The function ^{t) Is next determined. The function 
 uj = 9 + io is regular within the semicircle t and has the following 
 properties: 
 
 1. At the point 0(t = e^^), the real part of the function ^ 
 has a discontinuity, equal to n, because of the branching of the 
 streamline. On the arc AO the angle 6 is equal to -n and on 
 the arc OB it is equal to zero. 
 
MCA TM 1298 
 
 2. On the real diameter of the semicircle, the function 
 oo(t) = + icJ is real "because its imaginary part a = on the 
 
 free jets where 
 
 <»2' 
 
 3. At the origin of coordinates t = 0, the f-onction ^{t) 
 is equal to ^g 'because at infinity in the jet 0=0 and 
 
 e = e, 
 
 'o=2' 
 
 From the preceding discussion, it follows that the function 
 uj(t) admits of analytical continuation in the lower semicircle and 
 may be obtained by the Schwarz formula. Thus, 
 
 r 
 
 lt| 
 
 J 
 
 e^^ + t 
 «i* - t 
 
 ^^ = ^ 
 
 >iP 
 
 >i<P 
 
 dcp 
 
 i lo£ 
 
 
 e^^ - t 
 
 1 - te 
 
 i€ 
 
 (3.1) 
 
 that branch of the logarithm being chosen that is equal to ie for 
 t = 0. If the third property is used from equation (3.1) for t = 0, 
 
 it is found that e = -a 
 
 'oo2- 
 
 The value of too is obtained from 
 
 equation (3.1) by making use of the value of the velocity at infinity 
 in the stream 
 
 (A)(0 = ^1 + iq»i = i log 
 
 1 - to3 exp iSoo2 
 exp i0t»2 " "^sc 
 
 whence (reference 2) 
 
 arg tQ = arc t^ 
 
 jchOo^i - cos (^2^ - ©002) 
 cho^^ - cos (e^^ + e^g) 
 
 sha^3_ sin e^^g 
 
 chO^-^ cos %=2 " '^°^ ^oox 
 
 (3.2) 
 
 The pressure of the gas on a blade of the cascade is then com- 
 puted. On the forward side of the plate, which is a streamline, the 
 pressure is obtained by equation (I.6) where V^^ is the velocity 
 
 in the stream.. 
 
 The back side of the plate is in the gas at rest where the 
 pressure is constant and equal to 
 
 p = p^ (1 - T ^)P+1 
 
MCA TM 1298 
 
 If the fact that along a streamline d^p = df is considered, a 
 formula is obtained for the complex pressure in the form 
 
 Y + iX = ^°32 r(e-i^ - e-i^) df 
 
 J 
 
 or 
 
 Y - iX = ^2-^ [{e^ - e^) 1£ dt (3.3) 
 
 ^ J dt 
 
 where the integration is taken over the upper semicircle of the 
 t-plane in the clockwise direction. By considering that after 
 analytical continuation in the lower semicircle the function u)(t) 
 assumes conjugate values at conjugate points, the following relation 
 is obtained: 
 
 Y - IX = - ^2jb r el^(t) il dt (3.4) 
 
 ' Itl-il '' 
 
 where the integration is taken over the entire arc of the unit 
 circle in the counterclockwise direction. 
 
 Substituting the value of df/dt from equation (2.7) in 
 equation (3.4) gives 
 
 
 f ei^(*) . (t-e^^)(t-e-i^)(t-t-l)dt , . 
 ^ . (t-tj(t-tc,-l)(t-tj(t-t»-l) 
 
 2" 1^1 -^_^ (t-tj(t-t«-l)(t-tj(t-t^-l) 
 
 The function under the integral sign in equation (3.5) has 
 three poles, t = to:, t = tec, and t = 0, all of which lie within the 
 
 unit circle. The residues of the function at these points are, 
 respectively, 
 
 (t»- e^M(tcx. - e-^S 
 
 yt oc - tocj (tct - t oc J 
 
 exp iuj^^ 
 
 (W- eie)(t^- e-ie) 
 
 Itoc - tocj (tct - t(x. ) 
 
 -exp ie^2 
 
 — exp i^cxi 
 
NACA TM 1298 
 
 From equation (2.6), however, it follows that 
 (to.,- e^^){t^- e-i£) 
 
 = 2_±_ii 
 
 (too- U(tc.- V-) 2iq 
 Hence, the residues may be represented in the form 
 
 2_±_iaexp(ie„.i-a^i) 
 
 2iq 
 
 - exp iScxg 
 From equation (3.5), applying the theorem on residues gives 
 
 Y - iX = ^ [- 2iq exp iS^g + 
 
 (7 + iq) exp(iQ5^;L " ^^c^i) - (7 - i<l) exp(i0^;L +O001)] 
 If the real and imaginary parts are separated, 
 
 Y = ^ — [2q sin ^3^3 + (^ cos 6^^ - q sin Q^-^) exp(- O^^) - 
 
 (7 cos Qa^i + q sin ^^^.^ expa^^^^J (3.6) 
 
 X = ^ [- 2q cos 00,2 + ^^ ^^^ ®ocl + ^ cos e^{) exp(- <3^^) - 
 
 (7 sin e^-^ - q cos 6^:^J expa^^l (3.7) 
 
 The velocity on the contour of the blade is assumed to remain 
 finite; the total -pressure force on the blade will then be perpen- 
 dicular to the velocity and therefore X = 0, that is, 
 
10 MCA TM 1298 
 
 (7 sin Oo^-^ + g cos e^^.^ exp(- 0^,^^ ) - 
 
 (7 sin ^2. - ^ ^°^ ^1) SXP ^0=1 = 2q cos %,2 (3-8) 
 
 Thus the pressure force of the gas on a blade of the cascade is 
 hy equations (3.6) euad (2.8) equal to 
 
 Y = - -1^^ h 1^- 2(1 - ^j sin(\ + e^^) sin S^g + 
 
 [cos(\+ e^^i) cos 60,3^ "^ (■'■ " ~2^) ^^^ ^^ ^ ^°"1^ ^^^ ^"'l] ®^P^" ''^l) 
 
 [C0S(\+ 0o;^) cos 80=1 - U - ^^J ^^'^ (^+ ^*l) ^^^ ®ool] ©Xp ao,;^} 
 
 (3.9) 
 
 If in equation (3.9) (3 is set equal to and the 
 magnitude O^-^^ determined by equations (1.4), is correspondingly 
 
 replaced by 
 
 rocl 
 
 ^ oc.]_ 
 
 ii- = ± log r. = log ± 
 
 "^"^ 2 Toag ^<»2 
 
 .J 
 
 T 0,0 
 
 the formula for the pressure is obtained for the case of an ideal 
 fluid (reference 2). 
 
 In order to determine the angle \; entering equation (3.9), 
 betv/een the axis of the cascade and the x-axis, the ratio (2.6) 
 and the values of ■> and q from equations (2.8) are used. Thus 
 
 ctg(X . e^J = i fi - "^Y ^ ^°^ ^ - (^ ^ ^) (3.10) 
 
 In order to compute the length of a cascade blade, equation (1.5) 
 is used. Replacing dtp by df gives 
 
 q ( 1 - te^^ ^ t - e^^ \ (t - e^^)(t - e-^^)(t - t-^^)dt 
 dz = J a — 7- + b — : ^ — -^-^ — t- 
 
 " V e'^^- t te^^ - 1/ (t - t,„)(t - €J(t - tcr~^)(t - V-1) 
 
NACA TM 1298 11 
 
 This expression may be put in the form 
 
 dz = aa g (t)dt + 21 gp(t)dt (3.11) 
 
 n -^ n 
 
 where 
 
 g (t) = eiqt-e-l£)2(t-t-l) 
 
 , (t-to,)(t-too-l)(t-t^)(t-t„-^) 
 
 ggCt) = e-iMt-e^^)^t-t-^) 
 
 (t-tj(t-t -l)(t-tj(t-t -1) 
 
 The expansions of g]_(t) and §2^^^ into the sum of simple 
 fractions are of the form 
 
 gy(t) = + + -r + — 7-3^ + — ^^ - 1^2J 
 
 "t — "toe ^"""^OD ^~"^03 '^■*"'^0D 
 
 where 
 
 H = ^2 = ^ exp(i^i -O^i) 
 
 B-, = Dp = ii:!! exp(iea>i + '^*l) 
 ^ ^ 2iq -^ ^ 
 
 E-|_ = -e' 
 
 -ie 
 
 ^1 = ^2 = ^^ exp(ao.i - i^<xl) 
 2iq -^ -^ 
 
 1^1 = ^2 = ^^ ®^(- ^®°-l -'^»l) 
 
 These expressions for the coefficients are obtained if the 
 following relations are used: 
 
12 
 
 NACA TM 1298 
 
 exp ii4x-]_ 
 
 l-to=? 
 
 U 
 
 e -bCjD 
 
 (V,-e^'-)(t,-e-U) ^^^^ 
 
 exp ioCgoi = 
 
 1-W 
 
 .ie 
 
 e^^-t. 
 
 (tare-iS(Ve'-M 
 
 2iq, 
 
 If equation (3.11) is integrated over the upper semicircle in 
 the t-plane in a counterclockwise direction and if relation 
 z(-l)-z(l) = Z is used, the following expression for the length of 
 a cascade blade is obtained: 
 
 n 
 
 Ailg 
 
 1-t, 
 
 Bnlg 
 
 ■1-1 
 
 1-^. 
 
 + C^li 
 
 -1-t, 
 
 -1 
 
 1-t, 
 
 -1 
 
 -1-t -1 
 
 1-t "^ 
 
 7t 
 
 Aglg 
 
 •l-t„ 
 
 1-t. 
 
 h^s - 
 
 -l-to. 
 
 -1 
 
 1-t, 
 
 + -gig 
 
 l-t„ 
 
 -1 
 
 Dglg 
 
 1-^. 
 
 + Splg (-1) 
 
NACA TM 1298 
 
 13 
 
 
 t -1+1 
 
 V^^l 
 
 (Ai + Ci)lg -2L_^ + (Bi + Di)lg -^_- + 
 *^"1 tocT -1 
 
 (A^ + B^ + Ei^)lg(-l) 
 
 CLb 
 
 ■a 
 
 t^"^+l 
 
 (Ag + C2)lg — - + 
 
 too -1 
 
 (B2 + E2)lg "^ 
 
 toT^-i 
 
 + (Ag + Bg + E2)lg (-1) 
 
 3S 
 
 TC 
 
 Ri 
 
 (A3_ + B^ + C;l + ^l)lS ^ + 
 
 R-z 
 
 ia (A3_ + C]_ - B]^ - D-^) + {A-^ + B]_ + E]^) n: 
 
 qb 
 
 Ri 
 
 (A^ - B^ + C3_ + D3^)lg _i + 
 
 ^2 
 
 ia (A^^ + C3_ - Bt_ - D2_) + {C-^ + D-^ + Eg) it: 
 
 The magnitudes H-^, Rg^ ^^^ "■ that enter this equation are 
 
 sho^m in figure 4. Substituting the values of the coefficients and 
 malclng use of equation (3.S) gl-^es the following expression for 
 the length of a blade: 
 
 2q 
 
 El 
 
 I = + tl (a + b) cos 9^2 Ig -^ + 
 
 It Eg 
 
 - (a + b)[(7 cos Ga^i - q sin 0oc]_) exp(- O^-^) + 
 iy CCS 00=1 + 1 sin e^i) expao,J + 
 
 a-b 
 
 [(7 cos 0odi_ - q 3in ec»j_) exp(- aoj,-j_) - 
 
 (7 cos 0odj_ + q sin do:,j) expOoo]_"l+ qa sin ^2 
 
14 MCA TM 1298 
 
 The formula for the length of a blade in the case of an Ideal fluid 
 (reference 2) is obtained from the preceding equation for P = 
 if 
 
 1 
 a = 
 
 Va.2 
 
 b = 
 
 q = -X^l h sin {k + e^^ 
 
 7 ••= y^si h cos {K ■+ e^j_) 
 
 ^o=2 
 
 REFEEEWCES 
 
 1. Chaplygin, S. A.: Gas Jets. MCA TM 1063^ 1944, 
 
 2. Akhiezer, W. I.: On the Plane -Parallel Flow through an 
 
 Infinite Cascade. No. 2, Nauchnle Zapiski Kharkovskogo 
 aviainstituta, 1934. 
 
 3. Slezkin, W. A.: On the Problem of the Plane Motion of a 
 
 Gas. Wo. 7, Uchenie Zapiski Moskovskogo Universiteta, 1937. 
 
 Translated by S. Eeiss 
 National Advisory Committee 
 for Aeronautics 
 
NACA TM 1298 
 
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16 
 
 KACA TM 1298 
 
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UNIVERSITY OF FLORIDA 
 
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