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 i NATIONAL ADVISORY COMMITTEE 
 
 FOR AERONAUTICS 
 
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 TECHNICAL MEMORANDUM 1303 
 
 RESISTANCE OF CASCADE OF AIRFOILS IN GAS 
 
 STREAM AT SUBSONIC VELOCITY 
 
 By L. G. I.oitsianskii 
 
 Translation 
 
 Soprotivlenie reshetki profile! v gazovonn potoke s dokriticheskimi 
 skorostianni, Prikladnaia Matematika i Mekhanika, vol. XIII, no. 2, 1949. 
 
 Washington 
 
 September 1951 
 
 UNIVERSITY OF FLORIDA 
 DOCUMENTS DEPARTMENT 
 120 MARSTON SCIENCE LIBRARY 
 RO. BOX 117011 
 GAINESVILLE. FL 32611-7011 USA 
 
y ry ^^7 <5 7 SI i^n^ 
 
 NATIONAL ADVISORY COM^HTTEE FOR AERONAUTICS 
 
 TECHNICAL MEf/IORANDUM 1303 
 
 RESISTANCE OF CASCADE OF AIRFOILS IN GAS 
 STREAfI 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 num.ber is the possibility under this assiimrjtion of 
 applying a lift formula analogous to the well-known 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 paireillel 
 cross sections of the flow at an infinite distance upstream and doT-m- 
 stream of the airfoil. In the problem of cascade resistance, dif- 
 ficulty is encoimtered, 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 boundary- layer (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 profilei v gazovom potoke s dokriti- 
 cheskimi skorostiami, Prikladnaia Matematika i Meldianika, vol. XIII, 
 no. 2, 1949 
 
NACA ™ 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 vake of the cascade where the hoiindary 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 l) 
 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 momentiom theorem ajid 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, ■jid.ll 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 ti^ro- dimensional 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 coraponent 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 
 t-tro- dimensional 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 
 v.-in^'. In this case, the problem is to determine what may be termed 
 resistance. 
 
 An isola.ted >n.nc of finite span is now considered. In this case, 
 as also in the case of an airfoil in cascade, for each section of the 
 wing, in viev; of the presence of vortex systems (films) shed from the 
 
NACA T?-l 1303 
 
 \:'-..nQ and pcsslna. dc-.TT-Strecn to Infinity, two velocities dirferent in 
 magnitude and direction exist at infinity upstream and dovnstream 
 of the vrin^^'. For Ideal flov.' about a vring of finite span in accordance 
 v.-ith the scheviie of lifting lines, the total pressure force of the 
 flov; at a given section of che '-rin,-: is I^iowti to be perpendi cu]_ar to 
 the velocity of the flow at the corresponding point of the section 
 under consideration on the iiftin';' line. This velocity, vmich repre- 
 sents half the vector sum of the velocities at Jnfinity upstreara and 
 do'imstreain of the win^, is assuined at the section considered as the 
 effective velocity of flow; the anrle between the chord of the wins 
 section and the direction of the effective velocity is considered 
 as the effective angle of attac';, and so forth. 
 
 For a two-dimensional infinite cascade of airfoils, a similar 
 assumption is made with the difference only that in the theory of 
 the wine oi' finite span the effective velocity differs slightly from 
 the velocity of the approaching flow; whereas in the case of the 
 cascade the jump is of the same order as the geometrical angle 
 'of attack. 
 
 In the aerodynamics of a wing of finite span, the profile draf, 
 is the difference between the head resistance, which is represented 
 by the component of the total force exerted by the real (viscous) 
 gas flow in the d'Jrection of the velocity at infinity upstream of 
 the wins, £'nd the induced drag, which is the component in the same 
 direct:' on of the effective lift force. 
 
 For a small difference between the directions of the effective 
 velocity and the velocity of the approaching flow, this definition 
 of the profile drag of a wing section differs by small terms of 
 higher order from the true profile drag, strictly defined as the 
 vector difference between the total force exerted by the real flov? 
 on the wing section and the effective lift force for a real fluid. 
 
 In the case of the two-dimensional cascade, it is natural to 
 assume for the profile drag R' the difference between the vector 
 of the total force R (fig. l) and the Joukows'^ force R^ (in 
 
 the teimiinologi'' of reference l) which for an incompressible gas 
 is given by 
 
 R. = pV r (1.1) 
 
 .1 m 
 
NACA TM 1303 
 
 acting in the direction perpendicular to the fictitious velocity 
 at infinity V determined as 
 
 \ = i(V-^V (1-2) 
 
 where V, (u^ t v-, ) and V„ (u_ t v_ ) are the vector velocities 
 loo lao Jco 3oo * a» 2oo 
 
 at infinity upstream and downstream of the cascade, p is the den- 
 sity of the fluid, emd T is the circulation determined by the 
 equation 
 
 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 Joukowsky force by the following vector equation (reference 2): 
 
 R. = pV X (t X V-,) (1.4) 
 
 J *^ m ^ d' ^ ' 
 
 where the following vector 
 
 V, = V„ - Vt (1.5) 
 
 d 2oD loo ^ ' 
 
 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 l) : 
 
 R' = p't (1.6) 
 
 where p' is the pressure loss in the cascade determined by the 
 equation 
 
NACA TM 1503 
 
 P' = (Ploo + I P'^lJ) - (P2oo + I PV2» ) 
 
 (1.7) 
 
 The total force R is equal to the sum 
 
 R = R. + R' = pVjjj X (t X V^) + p't ' (1.8) 
 
 2. Resistance of tvo- dimensional cascade in real gas flov 3.t 
 subsonic velocities . - The expression for the total force R of 
 the interaction of the flow with a two-diraensional cascade at sub- 
 sonic velocities may be represented in the following form (reference 2) 
 
 R = (Pt„ - Ppjt + p, (V^ • t)V, - p^ (V„ • t)V„ (2.1) 
 
 1» -^200 loo ^ loo loo 2oo Sod 2oo 
 
 v/here p^ , p^ and p^ , p„ are the pressures and densities 
 loo' '^loD 2oD 2oo 
 
 upstream and do-vmstream of the cascade, respectively. The equivalent 
 
 expressions 
 
 Ploo^lcx- t = P2C.V2CO- 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 equal 
 in length to the pitch. 
 
 As was shoim (reference 2) also in the case of a compressible 
 gas for Mach nu^nbers not too near unity, the lift force of an airfoil 
 in cascade in an ideal gas flow may be represented in the for:n of 
 equation (l.4), provided that for the density p is taken the arith- 
 metical mean density p equal to 
 
 P,^ = o (Pt + Pq ) (2.3) 
 
 ' m 2 lo° '^2oo' ^ ' 
 
 The following o.pproximate expression of the Bernoiilli theorem 
 is employed: 
 
 P, - Vo = P V . V^ = i p (V„ 2 _ V 2) (2.4) 
 
 ■^loD -^200 "^m m d 2 ' m ^ 2oc loo 
 
NACA Wi 1305 
 
 This equation is valid with an accuracy to tenth parts of the square 
 of the difference of squares of the Mach ntiinbers at infinity upstream 
 and dovrnstream of the cascade. 
 
 In the case of the real (compressible and viscous) ^as, 
 
 p - p = 1 p (V 2 _ V 2) + p' (2.5) 
 
 lOD coc 2 HI '^* - 
 
 •.rhere p' characterizes the losses in the cascade due to the internal 
 friction in the gas; an equation may be obtained (reference 2) ana.lo- 
 gous to equation (1.8) 
 
 H = R. + R' = p V X (t X V^) + p't (2.6) 
 
 J ra m d 
 
 vhere p' is determined by the expression 
 
 P' = Pn - Vn - - P (V, 2 _ V 2) (2.7) 
 
 ^ -^loD ■" 2a) 2 rn ^ 2<x lex ' ^ ' 
 
 The prob]em of determining the profile drag force R' eiual to 
 
 R' = p't (2.8) 
 
 thus reduces to finding the losses p' \fhich depend on the shape 
 of the airfoil in the cascade and the character of the flow about 
 the airfoil. 
 
 3. Introduction of intermediate plane; relation betv/een gas 
 dynamic elements in this plane and corresponding values at infinite 
 
 distance from cascade . - In addition to the planes loc and 2 » 
 that were employed in the analysis of the incompressible fluid, 
 (reference l) 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 m.erge. The hydrodynamic and 
 therma], boundary layers in the wake downstream of the cascade are 
 hereinafter assumed to have the same pattern. 
 
 The following assumptions with regard to the motion of the gas 
 near plane 2 are necessary: By definition of the position of plane 2, 
 no individual boundary layers exist in the flow dowTistream of this 
 plane; the aerodynaiaic and thermal wakes of the airfoils are, hov.-ever, 
 
NACA TM 1303 
 
 maintained and depressions in the velocity or total-pressure curves 
 and also depressions or peaks in the temperature curves result. 
 A- fundamental property of the boundary lawyer is that the jjressure 
 transverse to the wake is the same at all points of a pjiven normal 
 section of the wake; that is, no pressure drop in the distribution 
 curve occurs in this section. The pressure alonr the wake chan^^es 
 sharply in the iiiimediate neighborhood of the trailing edce of the 
 airfoil and is gradually equalized as the distance from the trailing 
 edge is increased. 
 
 'P'wo sections of the wake are passed through the point of inter- 
 section of plane 2 with the axis of the Avakej one section lies in 
 plane 2 and includes the y-axls {iJ-'Z. o) , and the second section 
 lies in plane 2' normal to the axis of the v/ake and includes the 
 y '-axis. 
 
 The following magnitudes are introduced: 
 
 ^+t 
 
 A 
 
 u 
 
 A 
 
 \ 
 
 ^ 
 
 
 'z 
 
 -^J 
 
 Jq+^ 
 
 u.> 
 
 P2 - P 
 
 yr 
 
 V 
 
 A = 
 
 yo+t 
 
 = a 
 
 dy 
 
 
 Yo+t 
 
 
 
 1 
 
 J' 
 
 Vo - 
 
 V 
 
 t 
 
 Vo 
 
 
 
 '0 
 
 
 
 
 Yo+t 
 
 
 
 1 
 
 t 
 
 J" 
 
 yo 
 
 yo+t 
 
 P2 - 
 P2 
 
 P 
 
 J. 
 
 1 
 
 V2 - 
 
 V 
 
 dy 
 
 dy 
 
 ay 
 
 To - T 
 
 dv 
 
 "^ 
 
 > 
 
 (3.1) 
 
 J 
 
NACA TM 1303 
 
 which characterize the mean relative deviations of the hydro dynamic 
 elements of the flow at the points of section 2 of the wake from 
 the values of these elements at the hoxindaries of the wake at the 
 points of intersection of the boundary layers. 
 
 Section 2 idLll be assumed at such distance from the cascade 
 that the differences Ug - u, . . ., and also their mean relative 
 
 values A■^» • • • niay 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 
 
 ^u=^v=^V (3.2) 
 
 Comparison \n.th analogous mean relative deviations in section 2' 
 gives the magnitudes 
 
 .■4.1 
 
 u„' - u' 
 
 — — ; dy', . . . (f = t cos |3„J (3.3) 
 
 Jo 2 
 
 In the subsequent discussion, it >ri.ll 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 magnitudes (3.l)> that is, 
 
 Au' =Au Ap' =Ap . . . (3.4) 
 
 This additional assumption may be justified as a consequence 
 of the assumption of a small degree of variation of the gas dynamic 
 elernents near plane 2 and behind it downstrean of the flow. 
 
 In accordance \rith the fundamental property of the wake a ' = 0, 
 the following equation may be obtained: 
 
 Ap = (3,5) 
 
NACA TM 1303 
 
 Because of the smallness of the magnitudes A , A , . . . , 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 2od 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 2», 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 
 J pu dy = J^ [P2 - (P2 - P)] ^2 - (^2 " ^^^^^ = P2ooU2oo* 
 
 Expanding the brackets and neglecting the product (p„ - p) (ug - u) 
 as a small quantity of higher order gives the following equation: 
 
 J [P2^2 - ^2(P2 - P) - P2(^2 - ^fl^y = P2o=^a»* 
 ^0 
 
 From this expression, the following relation is obtained in the 
 notation of equation (3.l): 
 
 PoUo(l - A - A ) = po Uo 
 ^2 2^ p u' "^200 2oo 
 
 or, with the same degree of accuracy, 
 
 P2^2 = P2=o^2c»(l +^p + V (3.6) 
 
 The momentum theorem in the projection on the x-axis applied to 
 the same segment of the flow tube gives 
 
 J pdy +J pu2dy - pgj: - Pg^u^^t = 
 
 yo yo 
 
10 NACA TM 1303 
 
 This equation may be written in the form 
 Yo+t yo+t 
 
 J [P2 - ^P2 - PQ^y +J [P2 - ^P2 " pO &2 - K - ^3<^y 
 
 yo yo 
 
 2 
 = Po t + p u t 
 
 If the smallness of the differences Po - P? Po " P; ^^^ u„ - u 
 is taken into account, the following expression may be obtained: 
 
 P2(l - ^) + P2^2'tl - ^p - 2AJ - Pa, + P2^U2^2 (3.7) 
 
 With the aid of equations (3.5) and (3.6) and the same approxima- 
 tion, the following equation may be written: 
 
 P2 + P2o."2o=^2(l - ^i) = Pax + P2<x^2oo^ (3-8) 
 
 The mementum theorem is now applied in the projection to the 
 y-axis, which gives 
 
 yo^t 
 
 J puv dy = P2eo^2»^2oo^ 
 
 yo 
 
 or 
 
 yo+t 
 
 I J [^2 - (P2 - P)] [^2 - (^2 - ^0 [^2 - (^2 - ^O'iy = P2a^ax''2oo 
 
 yo 
 
 Rejection of small terms of higher order leads to the equation 
 
 P2^2^2(l - \ - ^ - \) = P2»^2cx.^2a» 
 
WACA TM 1303 11 
 
 or, according to equation (3.6), 
 
 ^2 = ^200(1 + A^) J 
 
 (3.9) 
 
 The assumption of parallel directions of the velocity vectors in 
 sections 2 and 2od, with the aid of equation (3.2), yields 
 
 (3.10) 
 
 Equation (3.8) then gives 
 
 Pg = Pg^ (3.11) 
 
 On the basis of equation (3,10), there also follows from equa- 
 tion (3.6) 
 
 p2 = P2Jl+Ap) (3-12) 
 
 Finally, from the Clapeyron equation, 
 X3 Po - (Pp - P) 
 
 = RT = r[T2 - (Tg - T)] 
 
 P Pg - (pg - p) 
 
 or, when the smallness of the differences is accounted for. 
 
 P2 ( P2 - P P2 
 
 1 - ~T + 
 
 p\ / T2 - T \ 
 
 P2 \ P2 P2 
 From this equation, A - A = Ap is obtained by integration, or, 
 
 A = - Ap (3.13) 
 
12 
 
 NACA TM 1303 
 
 Conversely, the same Clapeyron equation in planes 2 and 2ao 
 yields, by equation (3.11) 
 
 R(T2^ - T^) 
 
 =: hs-- El= P2c» P2 - PZu. _ P2» 
 
 P2» P2 P2oc 
 
 P2 
 
 P2c 
 
 A. 
 
 or, by equation (3.13), 
 
 ^20. - ^2 
 
 T„ A = 
 2oD p 
 
 ^^2^ 
 
 that is, 
 
 ^2 = ^2 Jl -*■ V 
 
 (3.14) 
 
 A' 
 
 4. Relation of fictitious vake thicknesses to magnitudes A and 
 Expression of profile drag in terms of fictitious vake thlck- 
 
 nesses. - The momentiim 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 6p and loss -of -momentum thick- 
 
 ness 
 
 y.'+t' 
 
 ^2* 
 
 
 ♦ ♦ 
 
 =1 
 
 yn'+t 
 
 
 
 yo' 
 
 pv 
 
 P2V2 
 
 ^y 
 
 \ 
 
 J 
 
 (4.1) 
 
 When these thicknesses are connected with. the magnitudes A, 
 
NACA TM 1303 
 
 -X 
 
 P2-P n*-' Vg-V 
 
 P2 
 
 dy 
 
 -; 
 
 V, 
 
 dy' 
 
 yo' 
 
 Yr 
 
 = t'(A ' +A^') = (A- +A^)t cos P2c 
 
 13 
 
 5^*. P^^'^' [l . [P2- (P2-P)] [VS- (V2- V)] ^.,.^ 
 
 > (4.2) 
 
 y 
 
 •=x 
 
 yn'+t- 
 
 [pg - (P2 - p)] [Vg - (Vg - V)] Vg - V >> 
 
 yo' 
 
 ^2^2 
 
 dy' 
 
 > 
 
 = t'Ay' = tA^ cos P2c 
 
 (4.3) 
 
 ^^ 
 
 The profile drag will now be determlnedj 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' = Pio,- P2oo 
 
 T (Pio. + P2<x) (VgJ - \J) 
 
 (4.4) 
 
 and the flow is considered between section Ice 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: 
 
 Plo 
 
 P2 = T (P1C.+ P2) (V - \J) 
 
 (4.5) 
 
14 NACA TM 1303 
 
 Because by equation (3.1l) pg = Pgoo^ ^^^ following expression 
 is obtained when equation (4.5) is compared with equation (4.4): 
 
 When V2 and pg are replaced by their expressions in terms of 
 ^2 00 ^^^ P2oo' "then according to equations (3.10) and (3.12) ^ 
 
 = I Cplx, + P2J V2„'a^ + \ Pa»(v2» - Vi^') ^ 
 
 or, by equations (4.2) and (4.3), 
 
 p' = p Vp 2 ^ + i: p (V 2 . V, 2 _2 2_ (^^g) 
 
 m 2» t cos p^ 4 "^^0=^ 2oD l^" ' t cos Pg^ 
 
 The following magnitude is now introduced- 
 
 5p* A,, + A^ A_ 
 
 which is for the case of the motion of an incompressible gas; the 
 following simple equation is then obtained: 
 
 C~\ So* * 
 
 PmV2„' + I (H2 - 1) P2JV2„' - ViJ)J ^ cos p,^ (^-S) 
 
 The formula for profile drag is immediately obtained from 
 equation (4.8), 
 
 R. = P't = [p^V^J 4 (Hg - 1) P2JV2J - V^2)] _^ (,.9) 
 
NACA TM 1303 
 
 15 
 
 From this expression, the profile-drag 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 
 
 "^m 
 
 A = 
 P 
 
 ^2 = 1 
 
 and equations (4.8) or (4.9) are converted into 
 
 P' = 
 
 t cos Pg^ 
 
 ** 
 
 •^ 
 
 cos p2» 
 
 J 
 
 (4.10) 
 
 which are identical to equations (2.12) of reference 1. 
 
 For an isolated airfoil in a viscous compressible fluid. 
 
 Ploo = P200 = Pm = Pc 
 
 
 P2co=0 
 
 Moreover, plane 2 extends to infinity, so that 
 
 R' = p V ^6 ** 
 
 (4.11) 
 
16 NACA TM 1303 
 
 Eqiiation (4.1l) is the well-known 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 E and 
 
 6„** 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 Hg and ^2** J^^'^ 
 
 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 boimdary- layer 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 Pgoo" 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 
 
 SV„ SVo dp Bt ^ 
 
 S , ^TT S 
 
 pv« ^^ + PV, ^ = - — + 
 
 s 
 
 Ss ^ bn ds bn 
 
 — ^ — + — s — = 
 
 OS on 
 
 > (5.1) 
 
NACA TM 1303 
 
 17 
 
 where for the longitudinal (coordiante s) and transverse (coordinate n) 
 projections of the velocities, the symbols Vg and V^^ are used in 
 
 contrast to the velocity projections u and v connected with the 
 axes Ox and Oy; 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.l) in the following form, according 
 to the second of equations (S.l) and the general Bernoulli equation. 
 
 OS on 
 
 ds 
 
 St 
 
 ■N 
 
 > 
 
 ^(pVs) 5(pVn) 
 
 — s — + — s — = 
 
 OS on 
 
 -/ 
 
 (5.2) 
 
 where p and Vg 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 Vg to yield 
 
 S , _ , S . _ , dVg 
 5J (PV,V3) . - (PV.VJ - PV3 jf . 
 
 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 
 
 dV, 
 
 St 
 
 ^KCTs - V3)] . |^K(V3 - V,)] . (PV, - PVJ _. = - ^ 
 
 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: 
 
 dsj 
 
 +0:,S 
 
 pVg(V3 - V^) dn + ^ 
 
 _ -K»^& 
 
 -00, & 
 
 r (pVg - pV^) dn 
 
 -oo)& 
 
 The following expression is then obtained: 
 
 -hx,8 
 
 dV„ / V„\ 
 
 dn 
 
 s ^ 's 
 
 d 
 
 -MX,0 
 
 sv3^r ^(1-^) 
 
 -Oo;'- 
 
 _ dV, 
 
 +OD^5 
 
 + pVp 
 
 ds 
 
 
 dn = 
 
18 
 
 MCA TM 1303 
 
 By expanding the parentheses and introducing the notation of 
 reference 4, 
 
 +CC,5 
 
 --S (-t) 
 
 N 
 
 
 > 
 
 6** = 
 
 = 1 
 
 
 
 dn 
 
 (5.3) 
 
 the required momentum equation is finally obtained. 
 
 ds 
 
 ^Vvds^pds/^ ""yds 
 
 b"" = 
 
 (5.4) 
 
 In such form^ the momentum equation for the compressible gas 
 differs from the corresponding equation for the incompressible gas 
 only in the term p" dp/ds (and, of course, in the definitions of 
 the magnitudes S* and 5* *) . If the moraentiim equation for the 
 incompressible gas is considered for the case of axial symmetrical 
 
 motion, the term p~ dp'/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. 
 
 (pV3 I . PV„ I;) (. . ^) . ^ 
 
 (5.5) 
 
 where a = \xq. JX is the Prandtl number. 
 
 i = Jc T 
 P 
 
 on \a 2 ' 
 
MCA TM 1303 
 
 19 
 
 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 so-called temperature of adiabatic stagnation T is now 
 consideredj it is given by 
 
 rp* = T' 4- ^ 
 
 2Jc, 
 
 (5.6) 
 
 By means of the conti.nuity equation, the following system of 
 equations may be set up: 
 
 I- (PV3T*) + ^ (pVj^T*) = -^ $a 
 ds on Jc_ on 
 
 ^ (pVsT*) + ^ (pV^T*) = 
 
 ■> 
 
 (5.7) 
 
 J 
 
 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 
 
 +00,5 
 
 r 
 
 -oc,6 
 
 "N 
 
 1 v*.,^ 
 
 4 J" PV3(T*- T*)dn = 
 
 +00,5 
 
 
 pV (T* - T*)dn = constant 
 
 J 
 
 (5.8) 
 
 The fictitious thickness of the wake is now introduced 
 
 +00^6 
 
 J , PVs \ TV 
 
 -00,0 
 
 (5.9) 
 
20 NACA TM 1303 
 
 which may be termed the thickness of the energy loss; equation (5,8) 
 may then be rewritten as 
 
 pVT*0 = constant (S.IO) 
 
 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 
 
 /^\ _ ^^ /PkVk!\ _ r^'^5^ 
 
 d In Vg 
 
 (^0 
 
 The notation of equation (4.7) is used for the ratio of the ficti- 
 tious wake thicknesses, 
 
 H = 1^* (5.12) 
 
 and it is noted that equation (S.ll) is integrated to completion 
 
 if the magnitude H is replaced by some average value; for example, 
 
 the following relation may be set up: 
 
 H = H^p = I (Eg + E^) (5.13) 
 
 By this simplification, the following expression is immediately 
 obtained: 
 
 \6kW VpzVgV 2 
 
 ^M:^y= i"Mr^;-i (^2 ^ h^,) m 
 
 Z2 
 
 Vk 
 
 or finally. 
 
 V* Pk KV + I (H2+H^) 
 
 This eq\iation connects ^2** ^^^ ^k**' ^^^ does not explicitly 
 contain Bg*'" "^^^ exponent on the right contains the magnitude Hg^ 
 which is equal to the ratio 52*7^2* ** From equations (4.8) and (4.9) 
 previously derived, equation (5.14) serves as one of the equations 
 
NACA TM 1303 
 
 21 
 
 * ♦ 
 
 for expressing the two unknowns Sg and Hg 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*02 = PkVkTk*^k 
 
 or, because of the isentropic character of the motion outside the 
 boundary layer, T„ = T, j the expression then becomes 
 
 PgYgSg = Pk^k^k 
 
 (5.15) 
 
 In this equation a new unknown quantity 0^ 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 2or, (derived in section 3) are accounted for, equation (5.14) and 
 then equation (5.15) are transformed. By equation (5.14), 
 
 ^ 
 
 iL. 
 
 P2c.^l + %) 
 
 Lv^Ji^V^J 
 
 24(H2+Hj^) 
 
 ^(fe)^^'"^^"^'a-Vf-[-|("e^%3^} 
 
 >(5.16) 
 
 Pk 
 P2a- 
 
 V> \24(Hp+Hj 
 
 ^20./ 
 
 A 
 
 - [2 + 1 (H2 + \)\ ^,] 
 
 Because in this section everything will be expressed in terms of 
 the unknowns Sg** ^^^ ^2' equations (4.2) and (4.3) are applied; 
 the following expression is then obtained: 
 
22 
 
 6£* 
 
 2-4-iHv+iH. 
 
 
 NACA TM 1303 
 
 -1^1)^] 
 
 (5.17) 
 
 or 
 
 '"''^"U^) 
 
 2+^^432 
 
 [-(-i%4Ha)2f;*] 
 
 For a first approximation, the subtrahend in the "brackets on the 
 right may he neglected in comparison to unity to obtain 
 
 52**= 6^**£k. 
 P2c 
 
 
 24^1,+|h2 
 
 (5.18) 
 
 The second fundamental equation (5.15) is similarly transformed. 
 With the chosen degree of accuracy. 
 
 +00, 5 +00, & 
 
 "00^ o -05, 5 
 
 pVs(T2* - T*) 
 P2V2T2* 
 
 dn 
 
 +00,0 +ex>,8 
 
 _P P2V2(V - ^) .. - P ^2* - T* 
 
 dn 
 
MCA TM 1303 
 
 23 
 
 Therefore, 
 
 +OD^6 
 
 +a»S 
 
 
 (Tp; - T)dii ^ p V272Jc^ - Vs^/2JCp 
 
 P V2 /^J^T 
 v) _ To + 
 
 2 + Vg^/SJCp ^^^5 Tg + V22/2JC 
 
 dn 
 
 = 1 r T2 - T 
 
 1 + V„2/2Jc Tp J T2 
 
 2 ' P 2 ^,6 
 
 dn + 
 
 +CD,5 
 
 P V2V2Jc^ -[V22 - 2V2(V2 - V)] A Jc^ ^ _ 
 -00^ & 
 
 T, + V22/2JC 
 
 1 + V22/2JCpT2 
 
 V 
 
 V22/2JC 
 
 T^^+V2^/2Jc 
 
 1 _ /^^2te 
 
 '200 
 
 ^— V=(^*^^T:'^-*)V 
 
 '2cx> 
 
 p 2oo 
 
 -l2^-A -Z2» A\t' 
 
 .Jc„T 
 
 ■* u 
 
 p-'2cx 
 
 ^2» 
 
 T"p 
 
 or, by equations (4.2) and (4.3), 
 
 V2<x2 
 
 (Ho - 
 
 1)^ 
 
 ♦ ♦ 
 
 (5.19) 
 
 The term QgA ' ; which by equation (5.19) is proportional to 
 &2**/t', is a small quantity of the first order j with the assumed order 
 of approximation, equation (5.15) may be transformed into 
 
 Pax^2oo 
 
 V. 
 
 2cx. 
 
 p ^00 
 
 (Hp - 1) 
 
 '■2a. 
 
 *♦ 
 
 ^2 =PkVk 
 
 or, 
 
 2oD 2oci 
 
 X222_ - (H - l)T, 
 
 L-^^P 
 
 200 
 
 S**=nVT *8 
 °2 Pk^k^2» ^k 
 
 (5.20) 
 
24 NACA TO 1303 
 
 The system of equations (5.18) and (5.20) gives the required 
 system of equations for determining the tvo unknown magnitudes 5_** 
 
 and Hp as functions of the parameters of the bovmdary 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 
 
 fy^\ ^^ V.c^Jc, - (H, - l)Tao ^Jx.fY2^) ^"^^^ 
 
 
 Vv; 
 
 or, when the Mach number Mp is introduced at infinity behind the 
 cascade, 
 
 M - ^2cc_ V2» _ V2<x -^ 
 
 ^" ^200 V^RT^ >^/(k - l)JCpT2 
 
 :oo 
 
 \yJ |(k-l)M2 2-.l \*\\J 
 
 1„ } (5.21) 
 
 This transcendental equation in H^ may be solved by one of the 
 approximate methods in any concrete case. According to equation (5.20), 
 
 . . . ^ 4fc - ^'"^.^ ^^ e^ (5. 33) 
 
 The terms bg** and Hg referring to section 2, the location 
 of which is unknown, are thus eliminated and expressed in terms of 
 the magnitudes &k** and 9-^ either measurable or computable by 
 any method of boundary- layer 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). 
 
MCA TM 1303 
 
 25 
 
 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 (5.2l) is employed. 
 The following notation is introduced for briefness: 
 
 9 ^00 
 
 >» 
 
 m 
 
 He 
 
 = € 
 
 > 
 
 (6.1) 
 
 3 + Hv 
 
 = P- 
 
 2 ^ 
 
 Equation (5.2l), which is transcendental relative to £, then 
 assumes the form 
 
 m - £ ^ 1 \ /^2o°'\ 
 1 + m 2 sT' VVk / 
 
 ^l+£ 
 
 (6.2) 
 
 The unknown magnitude £ is now expanded into a series in powers 
 of the small parameter m. (For air the value of m at Mg^^^ < 0.7 
 does not exceed O.l) 
 
 e = £q + £-|_m + figm + 
 
 (6.3) 
 
 Substituting this series in equation (6.2) gives 
 
 [-.o.(i-,,»-...](i-.-...) = i^(|f)^"°(|ff 
 
 £nm+. . . 
 (6.4) 
 
 or 
 
 ^0 + i^ + ^0 - ^i)"i - 
 
 ■•-^(^)^^^'^-(^)"-] 
 
26 NACA TM 1303 
 
 By equating coefficients, the following equation is obtained: 
 
 - ^0 
 
 
 for determining e . Because of the assiomption previously made on 
 the small heat transfer from the surface of the airfoil in the cascade, 
 the quantity £ is considered small for M^ =0- The following 
 equation, accurate to small quantities of the second order, is then 
 obtained: 
 
 '^ = \ (^)'° = \ [l ^ ^0 1- ^] (6.6) 
 
 where 
 
 A, = 1 3^ /l2«\ ^^'^""^^ (6.7) 
 
 ^^k 
 
 From equation (6.6), 
 
 ^ 
 
 ■0 1 + Aj^ m iV2j^^) 
 
 (6.8) 
 
 The ratio ^2ioJ^'k generally differs little from unity; hence, 
 
 the natural logarithm of this ratio is small so that e^ = - A, 
 
 Ok 
 
 may be written without great error. 
 
 By equating the coefficients of m to the first power, in 
 equation (6.4), the expression c, = 1 + C^ = 1 - A, is obtained 
 
 with the same degree of accuracy. 
 
 The approximate equation is then 
 
 C = - A, + (l + A, )m « m - A, (6.9) 
 
NACA TM 1303 
 
 27 
 
 Equation (5.22) is now employed and in the new notation has 
 the form 
 
 * * _ 1 + m Pk^k 
 
 2(m -e) p. V, 
 
 (6.10) 
 
 2oo 2oc 
 
 According to equations (6.1l) and (6.7), 
 
 5 ** ^ Pk^k % ** Pk_/Z]^\ 
 
 iC^-Hl.) 
 
 (6.11) 
 
 If it is assumed that at the trailing edge Hi^ = 1.4, equations 
 (6.11) and (6.7) assume the form 
 
 2 1^ P2»\V2oc^ 
 
 
 3.2 "^ 
 
 ^ 2 5,-^vJ 
 
 2.2 
 
 (6.12) 
 
 y 
 
 From equation (4.8), an approximate formula for the losses is readily 
 obtained: 
 
 P' = [Pm^2j ^ k ^^2.(^2^ - \Jll 
 
 ^^2 
 
 >v 
 
 t cos Bo 
 
 = P^V, 
 
 m 2oD 
 
 ^ P2*\^ 
 
 Vo J 
 2oo' 
 
 *♦ 
 
 t cos Pp 
 
 OD l_ 
 
 1 + ^ (m - Aj^) £2» ^2.^ - ^lo, 
 Pr. V^ 2 
 
 m 
 
 '2» jl* 
 
 >(6.13) 
 
 and therefore a corresponding approximate formula for the resistance 
 differing from the right side of the previous equation only in the 
 factor t . 
 
28 NACA TM 1303 
 
 The further possible simplifications of equation (6.13) are 
 connected with the choice of devices for computing the characteristics 
 of the bovmdary 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 . ) i 
 
 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. 100-126. 
 
 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 171-186. 
 
MCA TM 1303 
 
 29 
 
 Figure 1. 
 
 Figure 2. 
 
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UNIVERSITY OF FLORIDA 
 
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 UNIVEBSnY OF FLORIDA 
 
 DOCUMEI^S DEPARTMENT 
 
 120 MARSTON SCIENCE UBRARY 
 
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