UfVt:/VT^-(30^ 
 
 ^^^V^ 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1304 
 
 GENERALIZATION OF JOUKOWSKI FORMULA TO AN AIRFOIL 
 
 OF A CASCADE IN COMPRESSIBLE GAS STREAM 
 
 WITH SUBSONIC VELOCITIES 
 
 By L. G. Loitsianskii 
 
 Translation 
 
 "Obobshchenie Formuly Zhukovskogo na Sluchai Profilia v Reshetke 
 
 Obtekaemoi Szhimaemym Gazom pri Dozvukovykh Skorostiakh." 
 
 Prikladnaia Matematika i Mekhanika. Vol. XIII; No. 2, 1949. 
 
 Washington 
 
 Septembe]!'%r'i ' Y OF FLORIDA 
 
 OC,. , imTS DEPARTMENT 
 
 1 20 lv^y\h3T0N SCIENCE LIBRARY 
 
 F'.O. BOX 117011 
 
 GAINESVILLE. FL 32611-7011 USA 
 
^?i9ffc3 3^ ?7imi 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1304 
 
 GENERALIZATION OF JOUKOWSKI FORMULA TO AN AIRFOIL OF A CASCADE 
 IN COMPRESSIBLE GAS STRF'^M WITH SUBSONIC VELOCITIES* 
 By L. G. Loitsianskii 
 
 In the computation of the impellers of turbomachines, the Joukowski 
 formula is applied^ according to the formula, the lift force of the blade 
 is equal to the product of the density of the gas, the mean vector of the 
 velocities ahead of and behind the cascade, and the circulation about the 
 blade . 
 
 The presence of high oncoming relative velocities of the gas flow at 
 the blade make it necessary to account for the effect of the compressi- 
 bility of the air on the lift force of the blade. 
 
 In 1934, Keldysh and Frankl (reference l) showed that, in the case 
 of the isolated wing profile, the Joukowski formula retains its usual 
 form for both incompressible and compressible flow. As far as is known 
 to this author, no generalizations have yet been made to the case of the 
 airfoil in cascade. -'- 
 
 It is shown herein that, for subcritical values of the Mach number, 
 the Joukowski formula may, with an approximation entirely sufficient for 
 practical purposes, also be applied in its usual form to a cascade in a 
 compressible flow, provided that the density is taken as the arithmetical 
 mean of the densities of the gas at infinity ahead of and behind the 
 cascade. 
 
 This simple result may be useful for the computation of cascades, 
 especially in the solution of the problem of the resistance of a cascade 
 when it is necessary to separate the lift force from the total force deter- 
 mined by the momentum theorem (reference 2). 
 
 *"Obobshchenie Formuly Zhukovskogo na Sluchai Prof ilia v Reshetke 
 Obtekaemoi Szhimaemym Gazom pri Dozvukovykh Skorostiakh. " Prikladnaia 
 Matematika i MekJianika. Vol. XIII, No. 2, 1949, pp. 209-216. 
 
 Very recently, there had come to this author's attention the as yet 
 unpublished generalization of E. M. Berson, which differs from the one 
 given herein. 
 
NACA TM 1304 
 
 1. Vector equation of lift force of an airfoil of a tvo- dimensional 
 infinite cascade in an ideal incompressible fluid . - The flow about one 
 of the airfoils of a cascade situated in a tube of flow having at infin- 
 ity ahead of and behind the cascade the dimension equal to the pitch t 
 in the direction of the axis of the cascade is considered (fig. l) . The 
 density of the fluid is denoted by p, the velocities and pressures at 
 infinity ahead of and behind the cascade by W]_, wg and p-|_, pg, 
 respectively. When the momentum theorem is applied to the mass of fluid 
 enclosed in the flow tube between the infinitely removed sections a^ 
 and 02) the equation for the vector lift force may be written as 
 
 R = (P^-Pg) t + p (t • w-j^) w-,_ - p (t • Wg) Wg (l.l) 
 
 where the vector t is equal in magnitude to the pitch and is directed 
 at right angles to the axis of the cascade downstream of the flow. 
 
 The difference in pressures P-i -Po ^^ equation (l.l) may be 
 expressed in terms of the velocity vectors at infinity by the equation 
 
 ^1 " ^2 " 2 ^^2^"^1^^ = 2 (^2+^i) ' (^2"^l) ^^-^^ 
 
 The mean vector velocity Wjjj and the velocity w. characterizing 
 
 the vector change of velocity of the flow in passing through the cascade 
 are then introduced: 
 
 Wm = - (^T+Wp) 
 
 (1.3) 
 
 ^d = ^2 " ^1 
 Equation (1.2) then becomes 
 
 p^ - P2 = pwj^ • w^ (1.4) 
 
 The condition of the conservation of mass along the tube of flow 
 
 Jives 
 
NACA TM 1304 
 
 Wn • t = Wg • t = Wj^ • t 
 
 (1.5) 
 w^ • t = 
 
 From equations (1.4) and (1.5), equation (l.l) takes the form 
 
 H = P (Wm • w^) t - D (Wj^ • t) w^ = pwjjj X (t X w^) ' (1.6) 
 
 As is seen from equation (1.6), the_vector R lies in the flow 
 plane and is directed perpendicular to v^ toward the side determined 
 by the vector product on the right side of equation (1.6). 
 
 The magnitude of the vector R is given by 
 
 R = pWj^Wjjt (1.7) 
 
 2. Possible transformation of Bernoulli formula for compressible 
 gas at subsonic velocities. - In the flow of an ideal compressible gas 
 about a two-dimensional cascade, the condition of conservation of mass 
 flow along a flow tube yields 
 
 p^ (t • w^) = p2 (t • Wg) (2.1) 
 
 and with the notation of equation (1.3), the equation for the vector 
 lift force may be given in the form 
 
 R=(P-^^-P2) t+Pi (t • w-^) w_^- Pg (t • Wg) Wg = (P1-P2) ^- Pi (^ ' \) \ 
 
 (2.2) 
 
 where p and pg are the densities of the gas far ahead of and behind 
 the cascade. 
 
 The first component of equation (2.2) is now considered. When it 
 is assumed that the motion is adiabatic, the known relations for the pres- 
 sure and the density are 
 
4 
 
 NACA TM 1304 
 
 P = 
 
 ^0 r 
 
 k+l J 
 
 k 
 
 k-1 
 
 = P. 
 
 i\2 
 
 k+l 
 
 2(k+l)' 
 
 \* - 
 
 (2.3) 
 
 P = 
 
 "o ^ 
 
 k-1 , 2^ 
 
 kTi^y 
 
 k-1 
 
 = PO 
 
 1 ,2 2-k 
 
 1 - kTi ^ + 77Z:7^2 
 
 2 (k+l)' 
 
 (2.4) 
 
 where Pq and p denote the pressure and the density, respectively, 
 
 of gas adiabatically brought to rest, k is the ratio of specific heats 
 of gas at constant pressure c and constant volume c , and \ is the 
 
 ratio of the velocity of flow w to the critical velocity of gas a^, 
 expressed in terms of the velocity of sound a^^ in the adiabatic stag- 
 nation of the gas : 
 
 \ = 
 
 w 
 
 (2.5) 
 
 2_ _2k Pq 
 k+l ^0 = J k+l p^ 
 
 From expansion (2.3), the difference of the pressures becomes 
 
 ^2 = ill Po^^2^A^) 
 
 ^ - 2(lriy (^A^2^) +ep 
 
 (2.6) 
 
 where 
 
 ''P (\22-A^2)k/(k+i) 
 
 1 - 
 
 k+l ^1 , 
 
 k 
 k-1 
 
 k-1 
 
 k+l 
 
 k ' 
 k-1 
 
 \2' 
 
 1 /^ 2, A 2. 2-k 
 
 2\ 2,-v 4> 
 
 1 ^ 2^1^ V-^2 ) = 7^7772 (V -V V+V) - • • • (2-v) 
 
 6(k+l) 
 
 and X^ and Xp are the values of the parameter X ahead of and behind 
 the cascade. 
 
NACA TM 1304 
 
 From equation (2.4), the arithmetic mean p of the densities p-, 
 and p2 can be expressed as 
 
 where 
 
 Pm= 2 (Pl+Ps) = Pq 
 
 1 - 2(iTiy ^I'-h^^ + 
 
 C„ = - 
 
 k-1 , 2 
 
 1 
 
 k-1 
 
 ^ - ¥. h' 
 
 1 ' 
 x-1 
 
 1 + 
 
 (2.8) 
 
 2-k 
 
 2lk+lT 1 ^ 4(k+l)2 ^1 2 ^ 
 
 (2.9) 
 
 When equations (2.6) and (2.8) are compared, 
 
 k ^0 , 2 2>/^Pm 
 
 Pi-P2 = ^;t(-2-i )(^^S-^p^ 
 
 = p w • W- 
 •^m m c 
 
 
 (2.10) 
 
 Use of equations (2.5) and (1.3) yield, respectively, 
 
 k ^0 
 
 1 ^0 
 
 k+1 „ „ 2 2 2 2 
 PQ^* PO^O 
 
 2 2 1 - - 
 
 V - Wn = — V • W = 
 
 2 1 2 HI a. 
 
 From equation (2.10) it follows that, with a certain error, the 
 order of which will subsequently be discussed, the differences in pres- 
 sures ahead of and behind the cascade may be determined by an equation 
 similar to equation (1.4) for the incompressible fluid if the density of 
 the gas is replaced by the arithmetic mean of the densities ahead of and 
 behind the cascade. 
 
NACA TM 1304 
 
 3. Generalization of Joukovski formula to flow of a compressible 
 gas at subsonic velocities about a cascade . - When equation (2.1) is 
 used, the second component of equation (2.2) becomes 
 
 p^(t . w^)w^ = j|p^(t . w^) + p^(t . w^) - p^(t . W^)J w^ 
 
 = Pj^ ■ ^m)^d + 
 
 PlPg ,- - 
 
 (* • ^m) - Pm(t • W_,) W, 
 
 m 
 
 P1P2 
 
 m' '-^m^ m' d 
 
 
 = p (t • w^)wj + , 
 
 r^Tn\ m' 0. \ p ' m 
 
 m^ 
 
 m 
 
 P ) (t • w^)w-| 
 ^m / ^ m' d 
 
 Pm -P1P2 , ,- - ,- 
 = 11 - "2" Pm^^- V^d 
 
 m 
 
 1 - 
 
 P1-P2 
 P1+P2. 
 
 P ("t • ■w„)w-, 
 ^m m' d 
 
 In order to estimate the second component in the brackets, note 
 that, according to equation (2.4), 
 
 Pi-P2 = ^(^'-V)[i-2(My(V-V)-- • -J 
 
 Pi ^ P2 = ^PQ L^ " 2(1^ Q-i^+h^) + . . J 
 
 (3.1) 
 
 Hence, 
 
 Therefore, 
 
 p^(t • w^)w^ = pjj^(t . wjv^ - ep'Pin(t ■ wjw^ 
 
 (3.2) 
 
 (3.3) 
 
NACA ™ 1304 
 
 where, from what has just been shown and when, for convenience, 
 K = l/(k-l), 
 
 ,2 
 'Pi - P2 
 
 I =4 
 
 2 1 (Pi-Pg)' 
 
 Pi + P2 
 
 m 
 
 Fi - (k-i)/(k+i)\i2"|\ r . (k_i)/(k+i)\/T 
 
 [i - (k-i)/(k+i)\2j^- [i - {^.i)/{},+i)\^y 
 
 ^-^ (\2^-\l^)^ I 1 + M (>^l^+^2^) - • • 
 
 (3.4) 
 
 The equation of the order of the error e ' will be ignored for the 
 
 present; equation (2.2), on the basis of equations (2.10) and (3.3), may 
 then be represented in the form 
 
 R = Pm (^m* ^d) ^ - Pm (* • ^m^ ^d + «^R = Pm ^m ^ (^ ^^^) + € ^^ (3.5) 
 
 where c represents a certain small vector 
 R 
 
 PO 
 
 ^R = ^ (S"S^ Pm (^m- ^d) * + ^n'Pm (t • ^J w, 
 m 
 
 -p ^m 
 
 m' d 
 
 (3.6) 
 
 the order of magnitude of which will subsequently be discussed. 
 
 When this small vector C ^ is neglected, generalization of the 
 
 Joukowski formula to the flow of a compressible gas about an airfoil in 
 cascade yields 
 
 R= Pm^m^ (^^^d) 
 
 (3.7) 
 
NACA TM 1304 
 
 When equation (3.7) is coropaxed with the previously derived equa- 
 tion (1.6) for the flow of an incompressible fluid about a cascade, the 
 following result is obtained: At subsonic velocities, the lift force of 
 an airfoil in cascade can be approximately determined by the Joukowski 
 formula for the incompressible fluid if the density of this fluid is 
 equated to the arithmetic mean of the densities of the gas at a great 
 distance ahead of and beliind the cascade - 
 
 The absolute magnitude and the direction of the vector R with the 
 assumed approximation will now be determined and the difference from the 
 magnitude and the direction of the vector R determined for the incom- 
 pressible fluid by equation (1.6) will be shown. 
 
 The vector product t X v^ is perpendicular to the plane of flow of 
 the gas and therefore to the vector w . which yields 
 
 R = Pm^ml^'^ "^dl (3.8) 
 
 In contrast to the incompressible case, in the compressible case the 
 vector Wn is not perpendicular to the vector t, but is inclined by a 
 certain small angle to the axis of the cascade (fig. 2). Hence, in con- 
 trast to the condition of perpendicularity (equation (1.5)) in the case 
 of the incompressible gas, equation (2.1) gives 
 
 t . w^ = t . (w2-wi) = t • ^-^ PgV^ - ^ p3_wij = (t . p^w^) (^^ - j-J 
 t . w^ = I • (^1+V2) = I • (^ pA + ^ P2^2J = I (^ • Pi^i) (^ + 
 
 Hence, 
 
 ^•^d ^ 2 ^1 " ^2 Pi - P2 
 
 = 2^ (3.9) 
 
 t . wm Pi + P2 Pm 
 Whence, from equation (3.2), 
 
 ^ • ^d = 2 /^ t . w^ = YTI (^2^ V)(^ • ^m) [l -^ • • ] 
 
 (3.10) 
 
NACA TM 1304 
 
 Denoting the angles between the vector t and the vectors v^ and 
 w^ by and 9^ yields the following expression for the modulus 
 It X WH I : 
 
 t X w ■ 
 
 lf^6 
 
 tw^ sin e^ = /\/t'^w^2 _ t2^^2 ^^^2 ^^ _ ^ _^ 
 
 = V^S' - (^•^.)' 
 
 w. 
 
 ftV - ^Cp- (t . wj = tw^ /^/l - 4Cp' ( ^ ) cos- e^ 
 
 = tw- 
 
 (3.11) 
 
 The modulus of the vector R is thus given by 
 
 R = Pm^m^d^ 
 
 (3,12) 
 
 The direction of the vector R for compressible gas also remains 
 perpendicular to the vector of the mean velocity ^ (fig- 2). 
 
 4. Estimate of accuracy of generalized Joukowski formula . - In order 
 to answer the question of the possibility of application of the previously 
 derived equations, it is necessary to estimate the order of the neglected 
 terms and also the interval of Mach numbers in which, with a preassigned 
 accuracy, the proposed approximate formula may be applied. 
 
 For this purpose, consider the estimate of the order of magnitude of 
 the vector £j^. The simplest method of estimating is applied in writing 
 the evident inequality 
 
 ^0 
 "^m 
 
 s - s 
 
 Pm^rr^WH*- ^OS fl ^ + C^'p W W,t COS 9, 
 
 ^m"m"d^ 
 
 p ""^m m d 
 
 m 
 
 or, when the cosines are replaced by their maximum values. 
 
 ^R< 
 
 ■"m 
 
 + £. 
 
 p w w^t 
 m m d 
 
 (4.1) 
 
10 NACA TM 1304 
 
 The ratio p /p is readily estimated from the known equations; 
 m 
 
 1 
 
 k-1 
 
 Assiuning, however, that M-, = Mp = yields 
 
 Pq > Pi Pq > P2 
 
 The result immediately follows: 
 
 r:<(-; 
 
 1<T^<(^) K=kfT (^-2) 
 
 The estimate of the remaining magnitudes entering the parentheses of 
 equation (4.1) are now considered. For this purpose, the following 
 simple device is used: The values k t= 4/3 = 1.33 and k = 3/2 = 1.5, 
 "between which the region of most practical values of k is found (for 
 air and certain other gases), are considered and for these values of k, 
 the values of € , € ^ and e^' are computed. Similar computations may 
 
 easily be made if desired for the interval 5/4 < k -^ 4/3, and so forth. 
 
 For the chosen values of k, the exponents of the terms in equa- 
 tions (2.7), (2.9), and (3.4) become integers and the entire computation 
 is easily carried out. Then, for k = 4/3, 
 
NACA TM 1304 
 
 11 
 
 2x4 
 
 e = 
 
 (1 - i K')' - (1 - i x/) 
 
 7 (^2 
 
 K') 
 
 1 ^ K (^l' ^ ^2') 
 
 . i (X^^ + \^2X22 ^x/) - 1 (^^6 ^ X/ X/ + >^l' ^2^ + ^2^) 
 
 4-7 
 
 S = i 
 
 .1-7^0 ^ U-^2^ 
 
 - ^ ^ n ^' ^ ^') 
 
 -^(\/+\/ ) --^(>^i'+^2') 
 
 2-7' 
 
 , 1 (Pl-P2^^ PQ^ 
 
 ^P ~ 4 ^2 ~ , 2 
 
 •^ D.„ 4n 
 
 m 
 
 Pm 4:p. 
 
 2-7 
 
 i.iV - i-iv 
 
 I^ ^) <^^^ -^^'^ 
 
 my 
 
 -K^^^^ 
 
 (x-iv)(l-ix3^).(x-i^,^J 
 
12 
 
 NACA TM 1304 
 
 and for k = 3/2, 
 
 (1 - i\/)2 - (1 -^^/)^ 
 
 0-5 
 
 
 1 + I (>^2 ^ X^^) 
 
 1 
 2 
 
 '^ - 1 \f ^ (- I ^^f 
 
 l + i(\/.X3^) 
 
 2-5 
 
 2 (V + ^2") 
 
 ' _ _ 
 
 1 (P1-P2) Pc 
 
 ' P.' 
 
 m 
 
 4p. 
 
 m 
 
 -^x= 
 
 1 - 3 ^2^ 
 
 n2 
 
 4-5 \^m/ 
 
 Therefore, 
 
 tp - «^P 
 
 = -4(x/ -X/)' [l-ll(^i'+ ^2')] for 
 
 4 
 
 k = - 
 
 3 
 
 k = l 
 
 (4.3) 
 (4.4) 
 
 On the basis of the obtained formulas, an estimate of the expression 
 
 3 ^ 
 
 p w w,t P^ P 
 ^m m d m ' 
 
 ^S' 
 
 (4.5) 
 
NACA TM 1304 
 
 13 
 
 for any values of k in the Interval 4/3 S k < 3/2 is given: From 
 equations (5.2), (5.3) [Ed. note: equations (4.2) and (4.3)j, and so 
 forth, for k = 4/3, 
 
 98 7 + 4.72 ^ \7 
 
 >-'m. 
 
 s - S + S' 
 
 < 
 
 7\3 J_ 
 6>/ 98 
 
 4-7' 
 
 if^ 
 
 <^i' 
 
 K^f 
 
 or 
 
 0.033 (\ 2 _ >^ 2)2 _0 [e e„ + € ' < 0.133 {K 
 
 1 ^ — pP P p 1 
 
 ^2^)' 
 
 (4.6) 
 
 and for k = 3/2, 
 
 ■Tfe^i^(l)%i^V)^<r:K-'phV 
 
 < 
 
 V4; 150 4.52 \i/ 
 
 2l2 
 
 (\^ - \^) 
 
 or 
 
 0.032 (A^2 .X^^f<: ^ £p -cp +£p'< 0.108 (\^2 . X^^f (4.7) 
 
 Thus, for the magnitude e^,, the estimate is 
 
 K 
 
 0.033 i\^2.X^2f^ _^__ ^0^133 (^^^ _ ^^^^^ ^^^g^ 
 
 m m d 
 
 From the parameters K. and Xg^ "^^^ Mach numbers M-, = ^-i/^i ^^^ 
 Mg = w„/ap are obtained, where a and ag are the velocities ahead of 
 
 and behind the cascade. By the known equations of gas dynamics. 
 
14 
 
 NACA TM 1304 
 
 \' 
 
 k+1 
 2 
 
 k+1 
 2 
 
 M-, 
 
 1 + i (k-1) M^2 
 2 
 
 ^ 
 
 1 + i (k-1) Mg^ 
 
 VThence, 
 
 , 2 k+1 
 ^ = ~2" 
 
 2 ? 
 
 M - M 
 1 2 
 
 [l + I (k-1) M,2j [^1 ^ 1 (k-1) M^^" 
 
 Hence, 
 
 k+1 
 
 (m/ - M22)'< (\,2 . X22)2<(^)' (M,2 . M^^)' 
 
 (4.9) 
 
 Thus, in the interval 4/3 < k < 3/2, the following estimate may be 
 used in place of equation (4.8): 
 
 0.03 (M^^ _ ^^2f^_^__ ^Q_2 (M^S _ j^2)2 
 
 m m m 
 
 (4.10) 
 
 Equation (4.10) is a very rough estimate; in fact, the modulus of 
 the vector e^ vlll be much less. From this simple estimate, however, 
 
 the possibility of practical application of the approximate equation (3.7) 
 is evident. Thus, for example, if at the entrance to a turbine cascade 
 the Mach number is equal to M-, = 0.2 and at the exit Mo = 0.7, the 
 
 relative error will not exceed 4 percent, even for the very rough estimate 
 assumed. 
 
 Translated by S. Reiss, 
 National Advisory Committee 
 for Aeronautics. 
 
RACA TM 1304 
 
 15 
 
 REFERENCES 
 
 Keldysh^ M. V., and Frankl, F. I.: External Problem of Neumann for 
 the Nonlinear Elliptic Equations with Application to the Theory of 
 the Wing in a Compressible Gas. Izvestia AN SSSR, Otd. mat, i. 
 est. naiik, 1934. 
 
 Loitsianskii, L. G.: Resistance of Cascade of Airfoils in Viscous 
 Incompressible Fluid. PMM, vol. XI^ no. 4, 1947. (Translation 
 available from NACA Headquarters . ) 
 
16 
 
 NACA TM 1304 
 
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