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NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1298 ON TBE PROBLEM OF GAS FLOW OVER AN INFINITE CASCADE USING CHAPLYGIN'S APPROXIMATION* By G. A. Bugaenko 1. Some wellknown results of Chaplygin's method (reference 1) are first presented. For the adiabatic law of the state of the gas when p = kP, the following relations hold: P = PC(1 a > (1.1) p = 00 2 where p is the gas pressure, p is the gas density, V is the modulus of velocity, p0 and P0 are the values of p and p at the critical point of the flowat which the velocity becomes zero, k is the coefficient of proportionality, ? is the ratio of specific heats, and a and g are constants. If the angle 6 between the velocity and the xaxis and the magnitude T equal to V2/2a are considered, then, as was shown by Chaplygin, the equations of gas motion assume the form 5" 2"T 0e Sw (lT) 3P I, (1.2) bT = 2(1T)+l e' 04p 1(20+1)T OV where 9p is the velocity potential and W is the stream function. *"K Voprosu o Struinom Obtekanii Beskonechnol Reshetki Gazom v Priblizhennoi Postanovke S. A. Chaplygina." Prikladnaya Matematika i Mekhanika, T. X7II, No. 4, 1949, pp. 449 456. . 1 NACA TM 1298 Chaplygin reduced these equations to the very simple form 60 6e (1.3) a2 =_ 1 6e cp K 6V by introducing the new variable 0 and the constant K defined by the formulas 2= f I dT 2 (1.4) K = 1(2P+1)  (lT)2+1 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 CauchyRiemann; hence, w = 0 + iT will be an analytical function of the complex variable f = c + i*. The equation for the elementary vector along a streamline in the approximate treatment has, as is known (reference 3), the form dz = (aei + bei)d (a = b = ( 2A 2 2 2 A 2 2 ) (1.5) The complex pressure is given by the equation (reference 3) Y +i = 02(ei e ei)d) + p0(l T2)B+ d1 (1.6) 2. The steady potential flow of a gas through an infinite cascade according to the wellknown scheme of Kirchhotf with separations of the jet is next considered. The vanes of the cas cade will be assumed to be plane (fig. 1). Velocity of the gas in the flow at infinity is denoted by V,1 and its angle with the xaxis by Ql; velocity of the gas in the jet at infinity is denoted by V62 and its angle with the NACA TM 1298 xaxis by Q,2. The angle e between the velocity vectors and the xaxis lies within the range 21 The velocity field of the flow repeats itself for each dis placement by the pitch of the cascade, that is, by the vector heik The condition of constancy of the mass flow for steady flow of the gas gives Q = pl V,1 h sin (X+ l) = Pg2 =V2 n (2.1) where pl is the density of the gas at infinity in the flow, p,2 is the density of the gas at infinity in the jet, and n is the width of the gas jet at infinity. .y making use cf expressions (1.1) for p, equation (2.1) can be represented in the form ( 2 P V h (1 2 sit (x+ Qe) = v2 n 1 2 2 (2.2) The behavior of the function f = O + iy in the zplane of the gas flow is now considered. For simplicity, the function f is assumed equal to zero at the critical point 0 of the flow (fig. i). From the relation d9 = ds = Vs ds it follows that on moving along the streamline W = 0, the function 9 varies monotonically from c at the point E (infinity in the flow) to +c at the point C (infinity in the jet) and passes through the zero value at the critical point 0 where the stream line branches. The value of the potential f = + iV at the critical point 0' displaced by the period he relative to the point 0 is found and (fig. 1) (0') V ds d = V, ds + Vs ds + Vs ds OMM'O' OM MM' M'O' where NI4' is a.cut parallel to the axis of the cascade. NACA TM 1298 The first and last integrals mutually cancel and therefore as MM' approaches infinity in the flow, the following equation is obtained: V (0') = Vl h cos (X + e,,) (2.3) Furthermore, from the relation dQ P0 where dQ is the quantity of gas flowing in infinitely near streamlines, it follows that by the pitch of the cascade the function n' equal to Q/PO. Hence, (0o) = = (1  12 l unit time between on being displaced receives an increment hV sin (X + e6l) In this manner, the fplane with doublesided cuts along the halfstraight lines parallel to the axis of reals (fig. 2) corresponds to the region of the gas flow (fig. 1). All the cuts, because of the rule by which the cascade was constructed, are obtained from the initial one (the positive c@axis) by simultaneous displacement along verticals and horizontals at distances that are multiples of Q/PO and V i h cos (X + 8 i), respectively. * Because w = e + io is an analytic function of the problem may be solved by relating these functions of a parameter that varies in the upper semicircle of as in the LeviCivita method. f = p + i*, with the aid unit radius, By considering the rectilinearity of the cuts in the fplane, the analytic function f(t) is found, which brings about the con formal transformation of the fplane into the semicircle t. In the tplane, 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 t, and t.l that are symmetrical with respect to the circle and at the mirror reflection of these points in the diameter, that is, at the points ,s and t,1. At the origin of coordinates we place a sink of strength 2q (and a similar sink at infinity). (2.4) IACA TM 1298 In the constructed flow in the tplane, the upper semicircle of unit radius and the diameter of the semicircle are, of course, stream lines so that the stream function W maintains a constant value at the boundary of the upper semicircle t; in the fplane, this bcundary will correspond to straight cuts. The complex potential of the constructed flow will have the form f(t) = 1('Y + iq) log (t t) + (y + iq) log (t + ( + iq) log (t + ( + iq) log(t o) 2iq log t + constant or f(t) = 2 ( + iq) log t + 2 2M  (' iq) log (t + 2 + constant (2.5) where Y is the intensity of the vortex and q is the strength of the sources, and M = t+ t, The arbitrary constant in equation (2.5) is chosen so that f(t) becomes zero at a certain point t = eci, the position of which will be subsequently determined. Because the logarithm has multiple values, the upper semicircle of the tplane will correspond to an fplane with an infinite number of straight cuts v = kq(k = 0, 1, 2,...), where q changes from k? to +, as easily follows from equation (2.5) by substituting t = eie (the arc of the semicircle) and t = t1 where tl 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 tplane on the fplane with the doublesided 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 fplane on the half plane, the half plane on the unit circle, and finally the circle on the upper semicircle. NACA TM 1298 The point t = 0 is carried by the transformation (2.5) Into f = +m 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 = eic (the point eic corresponds to the origin of the doublesided cut in the folane). The condition df/dt = 0 for t = eic gives 7 + iq 7 i(2.6) = (2.6) cos CM cos cM In order to obtain the elements of the motion, an expression for the derivative df/dt is required that is represented in the form df q (teit)(teif)(tt1) dt (t (t)(t)(t 1)(2.7) The quantities 7 and q are next determined. When a point in the zplane of the gas flow is displaced by the pitch of the cascade he1 the point t, corresponding to the point in the zplane, goes over from one sheet of the Riemann surface to the next, passing once around the point E (t = t<), as a result of which the function f(t) receives an increment 7 + iq, as follows from equation (2.5). Because the corresponding increments of the functions P and J are equal to Va1 h cos (h + Q~1) and Q/P0, respectively, the following equations are obtained: 7 = 1l1 h cos (h + %,) ( T r (2.8) q = 1 2 V1 h sin (X + 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 w(t) is next determined. The function w = 9 + ic is regular within the semicircle t and has the following properties: 1. At the point 0(t = e1c), the real part of the function u has a discontinuity, equal to t, because of the branching of the streamline. On the arc AO the angle e is equal to it and on the arc OB it is equal to zero. NACA TM 1298 2. On the real diameter of the semicircle, the function u(t) = e + iT is real because its imaginary part J = 0 on the free jets where T =T2 3. At the origin of coordinates t = 0, the function W(t) is equal to A62 because at infinity in the jet 0 = 0 and C = Co2 From the preceding discussion, it follows that the function u(t) admits of analytical continuation in the lower semicircle and may be obtained by the Schwarz formula. Thus, i eip + t 1 eW +t t ei _ t W(t) a diP d = i log (3.1) 2) f ei T t 2 ei t 1 tei It = 1 E that branch of the logarithm being chosen that is equal to ic for t = 0. If the third property is used from equation (3.1) for t = 0, it is found that e = 9e2. The value of to is obtained from equation (3.1) by making use of the value of the velocity at infinity in the stream 1 to exp ioe2 w(t,) = ~l + iCl = i log exp i 1  exp i'662 to whence (reference 2) cho ,1 cos (0l 62) t cho cos ('e1 + e2) (3.2) shcOil sin 9 2 arg to = arc tg c cos e cos ch(col cos 02 cos 9 1 The pressure of the gas on a blade of the cascade is then com puted. Onthe forward side of the plate, which is a streamline, the pressure is obtained by equation (1.6) where Vg2 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 = p0 (1 T0 +1 NACA TM 1298 If the fact that along a streamline dtp = df is considered, a formula is obtained for the complex pressure in the form Po V02  Y + iX = 2 f (ei e" ) df or SiX = 2 I(e i e') df dt 2 f dt (3.3) where the integration is taken over the upper semicircle of the tplane in the clockwise direction. By considering that after analytical continuation in the lower semicircle the function W(t) assumes conjugate values at conjugate points, the following relation is obtained: Y ix =  Po V2 2 S 1 Iti = 1 ei((t) df dt dt 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 Y iX = 2 Itl = 1 The function under the integral sign in equation (3.5) has three poles, t = to t = t., and t = 0, all of which lie within the unit circle. The residues of the function at these points are, respectively, (t, ei )(t, ei) ( c Zo)(tGco e )1 (Eo ell)(E. erC) (3.4) (3.5) exp iWl1 exp iZ.l exp ie 2 eiu(t) (teic)(tei) (ttl)dt (ttw (ttol) (tt< (t% I) (I toc)(E, tj 1) NACA TM 129~ From equation (2.6), however, it follows that (t, eic)(to, eic) 7 + iq (to Z) ( 1) 2iq Hence, the residues may be represented in the form 7 + i exp(ie,, O l) 2iq 7 exp(ieO1 + ~c1) 2iq exp iG 2 From equation (3.5), applying the theorem on residues gives PoV,2 Y X = 2 [ 21q exp 1ie2 + (7 + iq) exp(i61 0Ga) ( iq) exp(ie01 +o ml) If the real and imaginary parts are separated, Y= [2q sin %2 + (7 cos e1 q sin Qcl) exp( O1)  (7 cos 1 + q sin Ql) expO 1] (3.6) S= 2 2q cos ^2 + (7 sin 06l + q cos 6,l) exp(o.l) (7 sin Oi q cos e8,) expOC,] (3.7) The velocity on the contour of the blade is assumed to remain finite; the totalpressure force on the blade will then be perpen dicular to the velocity and therefore X = 0, that is, NACA TM 1298 (7 sin e,, + q cos 6l1) exp( a, )  (7 sin Ql q cos ql) exp Ogl = 2q cos (2 (3.8) Thus the pressure force of the gas on a blade of the cascade is by equations (3.6) and (2.8) equal to POV iV V i V2\P, Y = =2 h 2(l sin( + ,,i) sin eQ2 + / 2a) [cos( X+ 01) cos Gl + 1 I sin (0 + 10l) sin ol ]s exp( ,)  [cos(X+ e1) cos 6 1 2! ) sin (x+ 0e1) sin 0a1 exp Gol (3.9) If in equation (3.9) 3 is set equal to 0 and the magnitude 0,,l, determined by equations (1.4), is correspondingly replaced by Tol dTr i og 109. = log log log  1 2 'og2 2 W^2 the formula for the pressure is obtained for the case of an ideal fluid (reference 2). In order to determine the angle X, entering equation (3.9), between the axis of the cascade and the xaxis, the ratio (2.6) and the values of Y' and q from equations (2.8) are used. Thus ce i o 2 cos (M + M) (3.0) ctg(\ + eml) = i 1 M .10) 2c 2 M / In order to compute the length of a cascade blade, equation (1.5) is used. Replacing dcp by df gives dz = a tei b t ei+ (t eiC)(t ei)(t t^)dt = eic t teiC i/ (t t.)(t %) (t to )t .1) NACA TM 1298 This expression may be put in the form dz = a gl(t)dt + 2b g2(t)dt where SeiC(teiE)2(ttl) 1g(t 92(t) (tt) (ttw1) (tZ,) (ttz1) = ei(tei)2 (tt1) (tt) (tt (tt)(t 1 expansions of gl(t) are of the form AV g,U(t) = +  tt= I and g2(t) into the sum of simple C, ttti D, + 1 EV + t t (V = 1,2) Al = C2 = Y exp(iEl1 al) 2iq B1 = D2 = isC exp(iew1 + o0,l) 2iq E1 = ei C = A2 = 7 exp(aol iol) 2iq D1 = B2 = 1 exp( 161, Oml) 2iq E2 = ei These expressions for the coefficients are obtained if the following relations are used: (3.11) The fractions where I NACA TM 1298 ltceic exp iul = eita (teif teL i +1 (toU) (L 1I) 2iq exp iei = l (te i( e ) i (Et (bt 1) 21q If equation the tplane in a z(1)z(l) = Z a cascade blade 1 = (3.11) is integrated over the upper semicircle in counterclockwise direction and if relation is used, the following expression for the length of is obtained: 1 A t  lt1 [S Allg  + Bllg + C0lg + S1t 11 lt 1 !t, Dllg + 1I1 qb A21 D21g  E1lg (1) + 1t + B2g + !~t0 1 1to1 E2lg (1) NACA TM 1298 t +1 (A1 + Cl)lg o + too1 1+1 (B1 + D)ig 1 0to 1 to,1+1 E91l(+lB] + ELA2 2g _ 1 (B2 + D2)lg = q (A, + B1 Ti I t11 to 1 + (A2 + B2 + Cl + D1)lg + E2)lg (1)] R R, + ia (Al + Cl B1 D1) + (A, + B1 + E1) i I + R (A, + B1 + C1 + Dl)lg R + R2 ia (A1 + C, B1 D1) + (Cl + D1 + E2) Iti The magnitudes R1, R2, and a that enter this equation are shcwn in figure 4. Substituting the values of the coefficients and making use of equation (3.8) gies the following expression for the length of a blade: + = + 2 (a + b) cos G2 Ig R + It R2 a (a + b)[(7 cos 8 q sin cxI) exp( Ol) + (y cos 9,1 + q sin e.1) expol]+ ab [(7 cos 0oS q sin ei) exp( a01) 2j (7 cos 8,,o + q sin e~1) expO,1]+ qa sin eg2 = qa 7 14 NACA 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 = 0 if a = V.2 b =0 q = Vl h sin (X + e0) 7 = V,1 h cos (A + 0l) icol = Ig V FERENCES REFERENCES 1. Chaplygin, S. A.: Gas Jets. NACA TM 1063. 1944. 2. Akhiezer, N. I.: On the PlaneParallel Flow through an Infinite Cascade. No. 2, Nauchnie Zapi 3: Kharkovskogo aviainstituta, 1934. 3. Slezkin, N. A.: On the Problem of the Plane Motion of a Gas. No. 7, Uchenie Zapiski Moskovskogo Uniersiteta, 1957. Translated by S. Reiss National Advisory Committee for Aeronautics NACA TM 1298 Figure 1.  I  0 ~1 0' We Figure 2. /R NACA TM 1298 I orc1 (J9 tc' Figure 3. Figure 4. NACALangley 51151 1000 0 0 ScN re 0 q a z U +3 0u .0 4, 01 < I 0) ', 40 +) rd 4 0 4 mO0 a, fl +d 0 a) o a4 Co o 0 a, ro 4 ct *H 4 EC rI CD 0 p. 'dO Is a o u 2 H r P0 :3  *4 ro > 0) 0) Qa) +a, )M 0 a W o +3 A T 3 rj a a 0 + 4 0) al a, C, O4 0 "' vc) UNIVERSITY OF FLORIDA 3 1262 08106 639 0 