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SC ? 37 7i1 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1304 GENERALIZATION OF JOUKOWSKI FORMULA TO AN AIRFOIL OF A CASCADE IN COMPRESSIBLE GAS STRFAM 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 1) 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.1 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 Profilia v Reshetke Obtekaemoi Szhimaemym Gazom pri Dozvukovykh Skorostiakh." Prikladnaia Matematika i Mekhanika. Vol. XIII, No. 2, 1949, pp. 209216. Very recently, there had come to this author's attention the as yet unpublished generalization of E. M. Berson, which differs from the one given herein. 2 NACA TM 1504 1. Vector equation of lift force of an airfoil of a twodimensional 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. 1). The density of the fluid is denoted by p, the velocities and pressures at infinity ahead of and behind the cascade by wl, w2 and p1, P2, respectively. When the momentum theorem is applied to the mass of fluid enclosed in the flow tube between the infinitely removed sections al and 02, the equation for the vector lift force may be written as R = (plp2) T + p (F i) i P ( ." 2) W (1.1) 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 plp2 in equation (1.1) may be expressed in terms of the velocity vectors at infinity by the equation p p = (w22w2) P (W2+7wj) (_l) (1.2) 1 2 2 '1 2 w1 2 1) (1.2) The mean vector velocity 1m and the velocity wd characterizing the vector change of velocity of the flow in passing through the cascade are then introduced: S (1.3) wd = v2 1Wj Equation (1.2) then becomes pl 2 = m wd (1.4) The condition of the conservation of mass along the tube of flow gives NACA TM 1304 wl t w2 t = wm t From equations (1.4) and (1.5), equation (1.1) takes the form R = p (w wd) t ( ) w m ( ~) Px ( d) (1.6) As is seen from equation (1.6), the vector R lies in the flow plane and is directed perpendicular to wm 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 = pwmwdt (1.7) 2. Possible transformation of Bernoulli formula for compressible gas at subsonic velocities. In the flow of an ideal compressible gas about a twodimensional cascade, the condition of conservation of mass flow along a flow tube yields 01 (t W) = P2 (t w2) (2.1) and with the notation of equation (1.3), the equation for the vector lift force may be given in the form = (Plp2) t+ P (t w) v P2 ( 2 = 2 = 1( P2 ( ;1) wd (2.2) where p1 and p2 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 NACA TM 1304 k P ki k1 1 i k X2 k \4 SfO k+l 2(k+l)2 1 E2 2k 4 0 k+l 2(k+1)2 J where po and pO 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 cp and constant volume c and X is the ratio of the velocity of flow w to the critical velocity of gas a,, expressed in terms of the velocity of sound aO in the adiabatic stag nation of the gas: a, (2.5) F2 4 2k PO a* ai 0 = k+ PO From expansion (2.3), the difference of the pressures becomes 2 i p) 1 (112+22) + p] (2.6 p1 P2 = k1 P0(%2ro ) 2 %1 ( 1 where 1 p (X22_, 2)k/(k+1) k kl k1 1 ki Xl 1 + 2(1 (X12+2) 2k (2 14 + 122+X24) (2.7) 6(k+1)2 and X1 and X2 are the values of the parameter X ahead of and behind the cascade. k" k1 1 ( k+1 ) 2 NACA TM 1304 5 From equation (2.4), the arithmetic mean pm of the densities pi and P2 can be expressed as (2.8) Pm = (PliP2) = PO 2(k+l) b21 + 1 k 2 k1 k+l I 1 ( 2 2 2(k+71 1 2 ki+ 1 1 k 1" k+l 2 1 2k 4(k+l) 2 (Xl4+X4) (2.9) When equations (2.6) and (2.8) are compared, k PO 2 2 P P2 = k+l 2 (2 1 a* Pm Wd (+m VO p P 0P I ( I (2.10) Use of equations (2.5) and (1.5) yield, respectively, k P0 k+l POa*2 kpo 1 2 2 POO 2 2 1  w2 1 W 2 m Wd 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. where 1p NACA TM 1304 3. Generalization of Joukowski 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(" .)d = [Pm m) + P+ = Pm( md + ) = Pm ^m P d + / m 12 ( 2Pm 1T   ( m. w) m( m)] id Lp (T" m) Pm(i m) Wd  m J /Pfl2 \ Pm ) % W1)Wdj Pmc' Wm)Wd PM 'w' Wd In order to estimate the second component in the brackets, note that, according to equation (2.4), PI 2 2 2 12 )2 2(2 12 + 2) (3.1) P + P2 = 2p0 (1 = 1 (N _1 )2 4(k+1) S1+ (12+X 2) + ki ( 2 ) + .] Therefore, pl1(t. &)wd = pnm(' w)d Ep'Pm(t Wm)d Hence, ( P1Pp2 pl 2 (3.2) (3.3) NACA TM 1304 where, from what has just been shown and when, for convenience, K = l/(kl), P(l + P2 ? 1 2ii4 p 1 (P1P2 2 4 p 2 Pm S (k,)/(k+l)12 1 (k)/(k+1X22 1[ 1)(kh+i1)l2]K 1(k1)/(k+1)A2 K[ 1 (2 2_1) 4(k+l)2 1 + ki +1+H (hl 2 +2) . (3.4) The equation of the order of the present; equation (2.2), on the basis then be represented in the form error c will be ignored for the of equations (2.10) and (3.3), may R = pm (wm'. ) Pm m) d + R Pjm (t xd) + R (3.5) where ,R represents a certain small vector PO R p (P) m (m" d) m t+ p'pm (t Wm) Wd the order of magnitude of which will subsequently be discussed. When this small vector ER is neglected, generalization of the Joukowski formula to the flow of a compressible gas about an airfoil in cascade yields R = Pm m (x~ d) (3.6) (3.7) 8 NACA TM 1304 When equation (3.7) is compared 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 behind 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 wd is perpendicular to the plane of flow of the gas and therefore to the vector wm' which yields R = p m T X d (3.8) In contrast to the incompressible case, in the compressible case the vector wd 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 T P 1 (1 1 1 tiwd = t *(w2) = 2 PA + >(T P^) ( 2 p2 P2 2 1 2 Hence, t Pi P2 Pi P2 = 2 = = 2 (3.9) t w PI + P2 Pm p Whence, from equation (3.2), t d 2 7' m 1 2 2))(E m) [1+ j (3.10) "Wd= 'm k + 1"" NACA TM 1304 Denoting the angles between the vector t and the vectors wm and wd by em and Gd yields the following expression for the modulus [tX dl= td sin 9d = t2 d2 t2wd2 cos2 9= t2w (2 d) = t2 d2  4p' (~t )2 Stwd 1l 4c \ cos2 em (WM m = tW 2c' ) cos2 9 The modulus of the vector R is thus given by R = PmWmwdt 1 2c~p' ( The direction of the vector R for compressible gas also remains perpendicular to the vector of the mean velocity wm (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 CR. The simplest method of estimating is applied in writing the evident inequality PO R : p C PmWmWdt cos emd + (p PmWmWdt cos Gm or, when the cosines are replaced by their maximum values, R< C I + C m0 dt (3.11) cos2 em C o s ) M W PmWmWdt (3.12) (4.1) NACA TM 1304 The ratio p /p is readily estimated from the known equations: 0OM k1 I + 2 M, =P2 1 k 2 ) 2 Setting M1 = M2 = 1 yields PO

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