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f'( rIA\ 1303
1 (L/ ,c 7 // " NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1303 RESISTANCE OF CASCADE OF AIRFOILS IN GAS STREAM AT SUBSONIC VELOCITY* By L. G. Loitsianskii A method of computing the resistance of a cascade of airfoils in a viscous compressible gas flow is described. The case of an incompressible gas is considered in reference 1 and appears herein only as a simple particular case of the general theory of resistance of a cascade in a compressible gas. The investigation was restricted to subsonic velocities (that is, when the local velocity of sound is nowhere reached on the air foil surface) because the required assumption of isentropic flow, that is, the absence of shock waves in any region of the motion, is valid only under these conditions. The second reason for the restriction to relatively small values of Mach number is the possibility under this assumption of applying a lift formula analogous to the wellknown Joukowsky formula (reference 2) and of thus assigning a definite meaning to the term "cascade resistance" or, more accurately, the "resistance of an airfoil in cascade." The resistance formula can be derived for an isolated airfoil, as is known, by applying the momentum theorem between two parallel cross sections of the flow at an infinite distance upstream and down stream of the airfoil. In the problem of cascade resistance, dif ficulty is encountered, namely, the absence of an external potential flow downstream of the cascade where the boundary layers (wakes) from the individual airfoils merge. This essential difficulty, which is expressed quantitatively in the impossibility of employing the boundarylayer (wake) equation up to a plane at an infinite distance, can be circumvented by introducing the plane of merging of the boundary layers (wakes) and by establishing relations between the gas dynamic elements in this plane and in the plane at infinity downstream of the cascade. *Soprotivlenie reshetki profile v gazovom potoke s dokriti cheskimi skorostiami, Prikladnaia Matematika i Hekhanika, vol. XIII, no. 2, 1949 NACA TM 1303 An essential assumption of the present investigation is that a small degree of nonhomogeneity of the flow exists in the section of the aerodynamic wake of the cascade where the boundary layers from the individual airfoils, considered as layers of finite thick ness, merge; the larger powers of the small velocity differences may then be neglected. The same assumption was made in the investi gation of cascade resistance in an incompressible gas (reference 1) and was subsequently confirmed experimentally. The plane of merging of the boundary layers is then assumed to be the control surface required for the application of the momentum theorem and in the case of the isolated airfoil is taken to be the plane at an infinite distance downstream of the airfoil. It is evident that when the relative pitch of the cascade is increased, this plane will be farther and farther away from the axial plane of the cascade and in the limit, for a relative pitch equal to infinity, that is, in the case of an isolated airfoil, will go to infinity. This assumption may evi dently be made for cascades with moderate solidities, a case that corresponds in practice to turbine and compressor cascades. Any method of calculating the boundary layer in a compressible gas may be used to compute the characteristic thicknesses of the layer and to estimate the effect of the compressibility of the gas on the external flow. The solution of the proposed problem reduces to a straightforward and direct form that is independent of the method of computation. 1. Resistance of airfoil in cascade. Joukowsky force as com ponent of total force exerted by incompressible fluid on airfoil.  For twodimensional flow of a real fluid, the resistance (or drag) of an isolated cylindrical wing of infinite span referred to unit length of the wing is the component of the total force exerted by the fluid on the wing in the direction of the velocity of the approach ing flow at infinity, or, in other words, of the velocity component of motion of the wing in an incompressible medium. This definition is invalid in the case of an airfoil in a twodimensional cascade, because in this more general case there is no unique velocity direction at infinity upstream and downstream of the cascade and there are no considerations by which preference is to be given to any particular direction for determining in this direction the resistance component of the total force acting on the wing. In this case, the problem is to determine what may be termed resistance. An isolated .in;! of finite span is now considered. In this case, as also in the case of an airfoil in cascade, for each section of the rln., in vi. of t:e presence of vortex systems (films) shed from the NACA TM 1303 v'n,. and poss.:ng. do;:strean to infin it ., two velocities :"ferent in *na_.nitude 3nd direction exist at in!"r..ty u.strcam rund downstream of the :;in;. For iiei flo.! about a ting of finite span in ccordance . lth the sche.ae of liftinE lines, the totl pressure force of the fio4 at ia cven section of The win is known to be ;.erendicular to the velocity of the flow at zhe correspond 'n point of he section under consideration on ..he 1ftin line. This velocity, which repre sents half the vector sum of the velocities :t 'nfinlt' urstream:! and do;.nstrea.a of the wing: is ass'.ied at the section considered as the effective velocity oi flow; the on le cetween the chord of the wing section and the direction o' tne effective velocity is considered as the effective angle of attack: a.nd so forth. For a twodimensional infinite cascade of airfoils, a similar assumption is made with the difference only that in the theory of the ;ing of finite span the effective velocity differs slightly from the velocity of the approaching floi; whereas in the case of the cascade the jump is of the same order c.s the geometrical angle of attack. In the aerodynamics of a wing of finite span, the profile dra,, is the difference between the he.d esistance, which is represented by the component of the total force exerted by the real (viscous) .a.s low' in the direction of the velocity at infinity upstream of the win., nid the induced dra, ;ihic' is the comronent in the same Jirect:on o' the effective lift force. For a small difference between the directions of the effective velocity and the velocity of the approaching flow, this Cefinition of the profile JraE of a win. section differs by small terms of higher order from the true profile 'ira,, strictly defined as the vector difference between the totl force exerted by the real flow on the win; section and the effective lift force for a real fluid. In the case of the twodinensional cascade, it is natural to assume for the profile dr.s R' the difference between the vector of the total force R (fig. 1) and the Joukoe:sky force Rj (in the terminology of reference i) which for an incompressible gas is given by R. = pV cn (1.1) NACA TM 1303 acting in the direction perpendicular to the fictitious velocity at infinity V determined as V = 1(V + V) (1.2) m 2 DO am where Vl. (u, v ) and V?. (ua, v,) are the vector velocities at infinity upstream and downstream of the cascade, p is the den sity of the fluid, and r is the circulation determined by the equation r = (v2 1 Vlt (1.3) where t is the pitch of the cascade. Introducing the concept of the vector pitch t, which is equal in length to the magnitude of the pitch t and directed at right angles to the axis of the cascade downstream of the flow, gives the Joikowsky force by the following vector equation (reference 2): Rj = PVm x (t x Vd) (1.4) where the following vector Vd = V2 V I (1.5) gives the vector change of velocity produced by the cascade. The preceding formulas are valid not only for the flow of an ideal incom pressible fluid but also for a viscous fluid. The profile drag R' is as follows (reference 1): R' = p't (1.6) where p' is the pressure loss in the cascade determined by the equation NACA TM 1303 P' =(Pl + pVi2) (P 2c + L V2O (1.7) P 2) =2) =Pl 2 1 PV) (" 2 2 The total force R is equal to the sun R = R + R' = pVm x (t x Vd) + p't (1.8) 2. Resistance of twodimensional cascade in real as flo'. at subsonic velocities. The expression for the total force R of the interaction of the flow with a twodimensional cascade at sub sonic velocities may be represented in the follo..ingt for'r (reference 2): R = (Pl F F)t + 'iP (Vl r t( )V v p'= ( 2.1) where Pi', Pl and p2', p. are the pressures and densities upstream and downstream of the cascade, respectively The e:,uiva.l.ent expressions Pl, V t = V2, t (2.2) evidently express the rate of mass flow per second throu':h the section of the flow parallel to the axis of the cascade'and eDual in length to the pitch. As was shom (reference 2) also in the case of a co.ipressible ,gas for Mach nlubers not too near unity, the lift force of an sirfoil in cascade in :!i ideal g.as flow my,' be represented in the form of equation (1.4), provided that for the density p is taken the arith metical mean density pm equal to Om 1 ('b + P29) (2.3) The following opproximate expression of the Bernoulli theoreli is employed: S P PmV V = (V2 2) (2.4) lW 2. m d m 2m Iam NACA TM 1303 This equation is valid with an accuracy to tenth parts of the square of the difference of squares of the Mach numbers at infinity upstream and downstream of the cascade. In the case of the real (compressible and. viscous) Las, p1 P22 V2) + p' (2.5) PI P2o. m 2 10 ' where p' characterizes the losses in the cascade due to the internal friction in the gas; an equation may be obtained (reference 2) analo gous to equation (1.8) R = R. + R' = p V X (t x v) + p't (2.6) where p' is determined by the expression =l 2 (V 2 V2) (2.7) =P P2 2 Pm (V2o V10D The problem of determinin. the profile drag force R' ejual to R' = p't (2.8) thus reduces to finding the losses p' which depend on the shape of the airfoil in the cascade and the character of the flow about the airfoil. 5. Introduction of intermediate plane; relation between gas dynamic elements in this plane and corresponJin, values at infinite distance from cascle. In addition to the planes la and 2W that were empioy,ed in the analysis of the incompressible fluid, (reference 1) an intermediate plane 2 is introduced for the compress ible gas (fig. 2); plane 2 is located where the boundary layers (wakes) from the individual airfoils iner.e. The hydrodynnr.ic .nd thermal bou.nn.c.ary L],ve.'s in the wake downstream of the cascade are here:inafter assumed to have the same pattern. The followvi, assumptions with regard to the motion of the gas near ,i. ne 2 are necessary: By definition of the position of plane 2, no i'li'vidual boundary .;,'.e.;s exist in the flow downstream of this plane; the .eros.vni.lc and thermal u'.i.es of the airfoils are, however, TACA TM 1303 7 maintained and depressions in the velocity or totalpressure curves and also depressions or pea';s in the temperature curves result. A fundamental property of the boundary layer is that the pressure transverse to the wake is the same at all. points o'' a given normal section of the wakIe; that is, no pressure drop in the distribution curve occurs in this section. The I."ressure aJon' the wake changes sharply in the iruiediate neighborhood of Lhe tr'ling edge of the airfoil and is gradually e:{ullzed as the distance from the trailing. edge is increased. ';o sections of the 'ai:e are passed through the point of inter section of plane 2 ;ith the 3xis of the a.;a.e; one section lies in plane 2 and includes the yxis (f;7. 3), and the second section lies in plane 2' normal to the axis of the wa,:e and includes the y 'axis. The following; minagnitudes re introduced: o+t "UU 2, = t +t 1 P CC, yr., v t v, 1 S P2 P Zp t P n Yo 'I 7o+t vt T y I O NACA TM 1303 which characterize the mean relative deviations of the hydrodynamic elements of the flow at the points of section 2 of the wake from the values of these elements at the boundaries of the wake at the points of intersection of the boundary layers. Section 2 will be assumed at such distance from the cascade that the differences u2 u, and also their mean relative values Au, may be considered small magnitudes, the higher powers and the products of which may be neglected. Moreover, the velocities at different points of section 2 are assumed parallel and in a general direction coinciding with that of the velocity at infinity behind the cascade. It follows at once that Au = vA (3.2) Comparison with analogous mean relative deviations in section 2' gives the magnitudes S u u' Au 1= tJ dy, (t' = t cos P2) (3.3) U.te T U21 2 O 2 In the subsequent discussion, it will be assumed that, for a sufficient distance of planes 2 and 2' from the axial plane of the cascade, all the magnitudes (3.3) and so forth are correspondingly equal to the L'agnitudes (3.1); that is, Au' =A ap' A Ap I (3.4) This additional assumption may be justified as a consequence of the assumption of a small degree of variation of the gas dynamic elements near plane 2 and behind it downstream of the flow. In accordance with the fundamental property of the wake A = 0, the following equation may be obtained: Ap = 0 (3.5) NACA TM 1303 Because of the smallness of the magnitudes u', Apo the gas dynamic magnitudes in the intermediate plane 2 are easily shown to be connected with the corresponding values of these magnitudes in the plane 2oo by relations that are analogous to the case of the incom pressible gas. For this purpose, a segment of a flow tube is assumed between sec tions 2 and 2m, where a length equal to the pitch t is taken for the transverse dimension of the tube in the direction parallel to the axis of the cascade. Application of the theorem of the conservation of mass then yields YO+t yO+t S pu dy =[ P 2 (P 2 P)] 22 u2 u)]dy= PU t Y0 YO Expanding the brackets and neglecting the product (p2 p)(u2 u) as a small quantity of higher order gives the following equation: yO+t S[P 2u2 u2(2 P) p2(u2 u) dy= p2o~u Y0 From this expression, the following relation is obtained in the notation of equation (3.1): 2u2(1 a \ u) = PZu2, or, with the same degree of accuracy, p2u2= p9,,u(l + Ap + A) (3.6) The momentum theorem in the projection on the xaxis applied to the same segment of the flow tube gives Y0+t y0+t Spdy +J pu2dy p2t P2ut = Yn YO NACA TM 1303 This equation may be written in the form yo+t  p]dy +r Yo [2 (P2 P] [u  (u2 u)]dy p t + P u 2t 200 z 2= t If the smallness of the differences p2 p, p2 p, and u2 u is taken into account, the following expression may be obtained: p2(l A) + p2u22(l A 2a) p + P2u2m (3.7) With the aid of equations (3.5) and (3.6) and the same approxima tion, the following equation may be written: P2 + P2ou2ou2(1 'u) = P2o + P2ou2O2 (3.8) The momentum theorem is now applied in the projection to the yaxis, which gives Y0+t f puv dy = p2,u2'v2Dt yo yo+t S 2 (P2 u2 (2 u YO [v2 (V2 vOldy = p2ou v2a, Rejection of small terms of higher order leads to the equation p2u2v2(l Ap A A) = pau2v2 YO NACA TM 1303 or, according to equation (3.6), v2(1 Av) = vZm v = v(1 + A) (3.9) The assumption of parallel directions of the velocity vectors in sections 2 and 2m, with the aid of equation (3.2), yields 2 = u2(1 + A) V2 = Va(l + ) = 2(1 + AU) (3.10) Equation (3.8) then gives (3.11) On the basis of equation (3.10), there tion (3.6) P2 = P2m,(1 + A ) Finally, from the Clapeyron equation, p P (P2 P) S 2 (2P) RT = RT2  p2 ( 23 p) also follows from equa (3.12)  (T2 ] or, when the smallness of the differences is accounted for, P2 P P2 2 RT2 T2 T + P2 = RT 1 T2 P 2 T2 From this equation, A A = tT is obtained by integration, or, (3.13) P2 = P2o P2 ( NACA TM 1303 Conversely, the same Clapeyron equation in planes 2 and 2cm yields, by equation (5.11) T2 P2 = P2 P P2a = P2 A R(T T2, __ 2 p2a P2 P22 P2 P2= or, by equation (3.13), T2 T= TB, = A T2 T that is, T2 = T2J1 + T) (3.14) 4. Relation of fictitious wake thicknesses to magnitudes A and A'. Expression of profile drag in terms of fictitious wake thick nesses. The momentum equation in the wake behind the airfoil of the cascade will now be employed; the equation contains the following fictitious wake thicknesses defined (reference 3) as integrals over section 2': displacement thickness 82 and lossofmomentum thick ness 2 ** 2' yO Yo'+t 82* = *2 S P yO' YO ' 1 2dy' P2V2 (4.1) )V 2V2 (1 y v2) When these thicknesses are connected with.the magnitudes A, NACA TM 1303 S2 YOI t P2 P2 CV2 (V2 V 2 Y ( 0y2V YO P2 p ' dy' y0'+t' YO' y0 V2 Vd V2 dy' = t'(A + Au') = (A + Au)t cos p2 y0 +t' 2 Y Y0 P2 (P2  p)] [ 2 v')] 2 v P2V2 V2 dy.3) (4.3) = t'Av' = tAu cos P02 The profile drag will now be determined; the magnitude p' must first be found. In equation (2.7), p' is expressed as a small dif ference between two large magnitudes and is therefore unsuitable either for experimental or for approximate theoretical determination of p'. In order to eliminate this defect, equation (2.'7) is rewrit ten in the form P' = Plm Pz= (PI + P2o) (v22 Vp2) (4.4) and the flow is considered between section la and the limits of the boundary layers that merge in plane 2. In this entire region, the flow is nonvortical so that the Bernoulli theorem may be applied without the additional term that accounts for such losses. The fol lowing expression similar to equation (2.4) may then be written: Pl P2 = (Pl+o P2) (V22 2) (4.5) YO' I+t = f y YO' S(4.2) dy' NACA TM 1303 Because by equation (3.11) p2 = P2,, the following expression is obtained when equation (4.5) is compared with equation (4.4): p' = ( + p (V2 2 1(p + p2 V) (V2 V2) p, (P 4 2 2 4 When V2 and p2 are replaced by their expressions in terms of V2o and p2w, then according to equations (3.10) and (3.12), p' = l (Po + p P + P+) [2(C + 2A ) ]  l (p + p32) (V2 2 V,) S(2_ v 2) A2 = (Pl2 + P2) V 2 + 1 a z PV(2) v or, by equations (4.2) and (4.3), 2 p*2i V S* p52* 2 P' = V 2 t c2 3+ 1 2 VL2) cs 52 (4.6) p m 2c t cos P0 4 P2V 20 t Cos 02. The following magnitude is now introduced: H2 = + A + which is for the case of the motion of an incompressible gas; the following simple equation is then obtained: P' = mV22 + (2 1) p2.(V2 Vl t cos p2 (4.8) The formula for profile drag is immediately obtained from equation (4.8), S= = 2 2 1) 52 (4.9) R' p't = [PmV2O2 + 14 (H2 1) p2Z(V2G v C (4.9) 2 ~21 cos BZp2+ (49 NACA TM 1303 From this expression, the profiledrag formulas for a cascade in an incompressible viscous fluid are obtained as particular cases and for the isolated airfoil in the general case. For the case of a cascade in an incompressible fluid (p = constant), P =P A =0 P H2= 1 and equations (4.8) or (4.9) are converted into , PVZo2c * P' = t cos Pg R, _pV225 2a * cos P02 (4.10) which are identical to equations (2.12) of reference 1. For an isolated airfoil in a viscous compressible fluid, Plo= 2P = Pm = V = V = V 02M = 0 Moreover, plane 2 extends to infinity, so that R' = PoV2 2,(* (4.11) NACA TM 1303 Equation (4.11) is the wellknown formula of the resistance theory for an isolated airfoil. The losses and the profile drag of the cascade are expressed by equations (4.8) and (4.9) in terms of known elements at infinity ahead of and behind the cascade and in terms of the elements H2 and 8 ** referred to plane 2, the position of which remains unknown, because up to the present no reliable theory of the turbulent wake exists. A formula will now be obtained for the profile resistance of the cascade; by the theory of the boundary layer at the airfoil, this expression makes possible the computation of the resistance of the cascade, and the dependence of the magnitudes H2 and 82** just mentioned on the elements of the boundary layer at the rear edge of the airfoil of the cascade can therefore be determined. 5. Establishment of relations between wake elements in sec tion 2 and boundarylayer elements in sections at trailing edge.  A generalization is given herein of the known device of setting up relations between the elements of the boundary layer at the trailing edge of the airfoil and in the wake behind it at infinity, as proposed for the case of the isolated airfoil in the incompressible fluid by reference 4. In this generalization, for the case of the cascade the section at the trailing edge is connected not with the plane at an infinite distance downstream of the flow but with plane 2 of the merging of the boundary layers or, more accurately, with plane 2' inclined to it by the angle P2 Moreover, the generalization requires passing to the compressible case. The momentum equation for the wake behind a body may easily be derived from the general equations of the plane boundary layer in a compressible gas Vs yV' dp _r Vs + Vn = d + +s n 0ds a(pVs) ((PVn) +s n = 0 0 NACA TM 1303 where for the longitudinal (coordiante s) and transverse (coordinate n) projections of the velocities, the symbols Vs and Vn are used in contrast to the velocity projections u and v connected with the axes Ox and Oyj p is the local density, T the friction stress, and p the pressure on the outer boundary of the layer. By rewriting the system (5.1) in the following form, according to the second of equations (5.1) and the general Bernoulli equation, S(pVVs) + (pVsVn) = pVs ds an ds a(pVs) a(pVn) as an = 0 where p and Vs denote the density and the longitudinal velocity at the outer limit of the boundary layer. Both sides of the second equation are then multiplied by Vs to yield (5.2)  (pVV s as a ,dVs + 6 (OVsVn) 0V d O an as The first of equations (5.2) is then subtracted term by term from the equation just obtained; the resulting equation is then inte grated, which gives s pv(Vs Vs3 + n(s V s PVs) =  anan ds an along the normal to the section of the wake, which is considered either infinite in the usual sense of the theory of asymptotic boundary layer or finite, as is assumed in the theory of the finite thickness layer. In either case, the following relation holds: d+,5 dr dVs +W 5 pVs, Vs) dn + d (v pVs) dn = 0 s s sr The following expression is then obtained: OVs2 V + 7V V 1 s sVs Vs ds V dn = 0 TVs1 NACA TM 1303 By expanding the parentheses and introducing the notation of reference 4, (5.3) \, 6 the required momentum equation is finally obtained. 2 dV 1 dp\ 1 dV , differs from the corresponding equation for the incompressible gas 1 only in the term T1 d/ds (and, of course, in the definitions of the magnitudes 6* and 8**). If the momentum equation for the incompressible gas is considered for the case of axial symmetrical motion, the term Tld1/ds, which expresses the effect of the variable density of the gas, may be taken equivalent to the term that takes into account the transverse curvature. In addition to the momentum equation, the heat equation is con sidered; it can be easily set up by a method analogous to the preceding method from the known heat equation of the boundary layer. PVs C) + i+ =a (5.5) where a = c p/X is the Prandtl number. i = JCpT 3(n a 2 NACA TM 1303 The value of q is given in the case of the laminar boundary layer; equation (5.5) holds also for the turbulent layer, but in this case q would be expressed in a different form. The socalled temperature of adiabatic stagnation T* is now considered; it is given by s = T 2 (5.6) T =T+2Jc By means of the continuity equation, the following system of equations may be set up: s (pVsT') + n (pvT*) = 1 JCp (5.7) T (PVsTi) + 3 (pVT*) = 0 In the second equation of the system, the stagnation temperature T* at the outer limit of the boundary layer, which is constant (because the external flow is isentropic), is taken under the sign of the derivatives in the continuity equation. Subtracting one equation of the system (5.7) from the other and successively integrating over the cross section of the wake gives +40, 5 dJ pVs(( " T*)dn = 0 (5.8) +0, & S pVs(T* T*)dn = constant , &S The fictitious thickness of the wake is now introduced J+C pVs T v T., a.S NACA TM 1303 which may be termed the thickness of the energy loss; equation (5.8) may then be rewritten as pT*e = constant (5.10) Equation (5.4) is again considered. After each side is divided by 5**, the expression is integrated along the wake from section k at the trailing edge of the airfoil to plane 2', previously defined. The result is (k) In (IT(*in ds (5.11) The notation of equation (4.7) is used for the ratio of the ficti tious wake thicknesses, H (5.12) 8** and it is noted that equation (5.11) is integrated to completion if the magnitude H is replaced by some average value; for example, the following relation may be set up: S= Hp = (H2 + k) (5.13) By this simplification, the following expression is immediately obtained: \6kn + Hk) nV or finally, 52 POk /k 2 + (H2+Hk) k V2 (5.14) This equation connects 62** and 5k**, but does not explicitly contain 52*; the exponent on the right contains the magnitude H2, which is equal to the ratio 62* /2**. From equations (4.8) and (4.9) previously derived, equation (5.14) serves as one of the equations NACA TM 1303 for expressing the two unknowns 52*" and H2 entering in the equations for the losses in the cascade and the resistance in terms of the elements of the boundary layer at the trailing edge of the airfoil. The second equation is obtained by use of equation (5.10), which may be rewritten as follows: P2V2T2 *2 = PkVkTk k or, because of the isentropic character of the motion outside the boundary layer, T2 = Tkj the expression then becomes P2V2z2 = PkVKdk (5.15) In this equation a new unknown quantity e2 appears to enter; because of the small degree of nonhomogeneity of the fields of hydro dynamic elements in planes 2 or 2', however, this term can actually be expressed in terms of the previous unknowns. When the small degree of nonhomogeneity and the formulas relating the elements in planes 2 and 2m (derived in section 3) are accounted for, equation (5.14) and then equation (5.15) are transformed. By equation (5.14), SP( 2(H2+Hk v 2 +Hk) P2I V 2 k) ((1 +[2 + (H(5.16) Pk 2 + 1 (H2 + H Because in this section everything will be expressed in terms of the unknowns 2"** and H2, equations (4.2) and (4.3) are applied; the following expression is then obtained: NACA TM 1503 pk v 22 k2 (H2 ) 2  S2 2 +) j or 82 k ** Pk V 2 k2 1 1+Hk PzpYM(Cz~ 2) ] For a first approximation, the subtrahend in the brackets on the right may be neglected in comparison to unity to obtain 2**= S Pk2 /VL 2 2 P2c M20) (5.18) The second fundamental equation (5.15) is similarly transformed. With the chosen degree of accuracy, +c, 8 e2 ~f Pv5 p2V2 p2V2(T2* 2 P T2* dn G PVs(T2* T*) dn SP2V2T2  W158 dn = ^= *Vs/fl T2* T* 2 dn T2 * *4 2** (5.17) NACA TM 1305 Therefore, e2 (T2 T)dn 2 p~8 T2 + V22/2Jc~ + V22/2JC, Vs2/2Jcp dn w,6 T2 + V22/2Jcp 2 f T2 n + 1 + V22/2JCpT2 T2 2 C72jc roJ 'V22/2Jcp [v22 2V2(V2 V]l/2Jcp dn i T2 + V22/2Jc 1 + v22/2Jc T2 oDb r V22/2JecP t S2 T22 + V2/J t (I Jc T2 2 T20J+ 20. ST2 J T2 2m A T20 a 2W f or, by equations (4.2) and (4.3), 2TV (H2 1) 2** SJCpT2w T2 82 (5.19) The term e2/t', which by equation (5.19) is proportional to 52**/t', is a small quantity of the first order; with the assumed order of approximation, equation (5.15) may be transformed into [pv2 9J V 2 (2 1) pT2= T 82 = Pk k T2oJ [ 2 9 p2M (1 I(2 a T2 PkVkT2 k P (5.20) NACA TM 1303 The system of equations (5.18) and (5.20) gives the required system of equations for determining the two unknown magnitudes 52* and H2 as functions of the parameters of the boundary layer, of the external flow near the trailing edge of the airfoil, and of the density, velocity, and temperature at infinity behind the cascade. For the solution of this system of equations, it is noted that the unknowns are readily separated if equations (5.18) and (5.20) are divided one by the other. This procedure yields h Va 2 V2nm/Jp (H2 1)T3, the Mach number M2 is introduce S=V2 2 Ca2 V2~ 1 2* t;; S22 (k 1)M2=2 (H2 1 \V, J(k 1)M2 + 1 142H _l+ f ^ ed at infinity behind the V2aw 4 (k l)J5pT2 l) 6k /V2 1 8k Vk This transcendental equation in H2 may be solved by one of the approximate methods in any concrete case. According to equation (5.20), S**= 1+ 1)M 22 PVk 8k 1 + (k 1)M2 2 e pV2 (5.22) The terms 52** and H2 referring to section 2, the location of which is unknown, are thus eliminated and expressed in terms of the magnitudes k** and ek either measurable or computable by any method of boundarylayer theory and in terms of the velocity, density, and temperature at the outer limit of the boundary layer near the trailing edge of the air foil and at infinity behind the cascade. The terms p' and R' may then be obtained with little difficulty by equations (4.8) and (4.9). or, when cascade, (5.21) + k NACA TM 1303 6. Approximate formulas for computing losses and resistances in cascade. At these relatively small subsonic Mach numbers considered herein, the nonisothermal character of the flow in the wake behind the airfoil of the cascade can occur mainly through the heating or the cooling of the surface of the airfoil and not through the internal transformations of kinetic energy into heat. In order to verify this fact, equations (5.21) is employed. The following notation is introduced for briefness: k 1 2 = 2 cOB H21=  2 3 + Hk= 2 (6.1) Equation (5.21), assumes the form which is transcendental relative to C, then m1 + m k V2 \~ 1+ m 2k k The unknown magnitude of the small parameter m. does not exceed 0.1) C is now expanded into a series in powers (For air the value of m at M2 < 0.7 c = C0 + Glm + C2m2 + ... (6.3) Substituting this series in equation (6.2) gives  + ( l)m ...]( m ...) = '~zpcO (6.4) or+ 1)m ...= 2 +C..  CO+ (I+COCj)m +C1 a M k Y (6.2) NACA TM 1303 By equating coefficients, the following equation is obtained: 2. = I 1, v' 0 k 2 ,vk/ (6.5) for determining C0. Because of the assumption previously made on the small heat transfer from the surface of the airfoil in the cascade, the quantity CO is considered small for M2, = 0. The following equation, accurate to small quantities of the second order, is then obtained:  Ak [1 + C 1 k (6.6) From equation (6.6), Ak S 1 + Ak in (V2JVk) The ratio VTJ/Vk generally differs little from the natural logarithm of this ratio is small so that may be written without great error. By equating the coefficients of m to the first equation (6.4), the expression C1 = 1 + O = 1 Ak with the same degree of accuracy. The approximate equation is then (6.7) unity; C =  (6.8) hence, Ak power, in is obtained c = Ak + (1 + Ak)m rm Ak where (6.9)  C = A CkO Vk ) NACA TM 1303 Equation (5.22) is now employed and in the new notation has the form 2, 1 + m PkVk k 2 (m. ) pan (. According to equations (6.11) and (6.7), 2 PkVk k P V2 2A 2M2a k S(5+Vk) k P V2 (6.10) (6.11) If it is assumed that at the trailing edge (6.11) and (6.7) assume the form Hk = 1.4, equations & * k V k 3.2 2 7 .2 Ak I V f \ 2.2 k 2 ** k vk ) (6.12) From equation (4.8), an approximate formula for the losses is readily obtained: P LPi22 + p V22 2)1 * pV= Y2 + p i E 2V2 V 2.2* [Pm 2 W 2 2w 2c 100 t cos 02w P= mV22 P C ) P2w V 20/ 3.2 6 r 2 s + (m Ak) 2~ 2 t cos P2 LPm V; and therefore a corresponding approximate formula for the resistance differing from the right side of the previous equation only in the factor t. 2 NACA TM 1303 The further possible simplifications of equation (6.13) are connected with the choice of devices for computing the characteristics of the boundary layer at the surface of the airfoil in the cascade and for taking into account the effect of the compressibility on the external flow. Translation by S. Reiss National Advisory Committee for Aeronautics REFERENCES 1. Loitsianskii, L. G.: Resistance of a Cascade of Airfoils in a Viscous Incompressible Fluid. Prik. Mat. i Mek., T. XI, No. 4, 1947. (Translation available on loan from NACA Headquarters.) 2. Loitsianskii, L. G.: Generalization of the Joukowski Formula to the Case of an Airfoil of a Cascade in a Compressible Gas Stream with Subsonic Velocities. NACA TM 1304, 1950. 3. Loitsianskii, L. G.: Inverse Effect of the Boundary Layer on the Pressure Distribution on the Surface of a Body in a Real Fluid. Prik. Mat. i Mek., T. XI, No. 2, 1947. 4. Squire and Young: Computation of the Profile Drag of a Wing. Collection of articles on the problem of Maximum Speed of an Airplane. N. Oborongiz, 1941, pp. 100126. 5. Loitsianskii, L. G.: Resistance of a Cascade of Airfoils in a Gas Flow with Subsonic Velocities. Prik. Mat. i Mek., T. XIII, No. 2, 1949, pp 171186. NACA TM 1303 29 Figure 1. Figure 2. /o. O X NACA TM 1303 Figure 3. 3 P. cu m vf ? tr, L. * c in 0 ~ Cu Cu cA 0 w I 2 . 2 's 2 a. 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