kcA^tn-ms^^ ^'^' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1293 APPROXIMATE METHOD OF INTEGRATION OF LAMINAR BOUNDARY LAYER IN INCOMPRESSIBLE FLUID By L. G. Loitsianskii Translation "Priblizhennyi Metod Integrirovania Uravnenii Laminarnogo Pogranichnogo Sloia v Neszhimaemonn Gaze." Prikladnaya Matematika i Mekhanika, USSR, Vol. 13, no. 5, Oct, 1949. Washington July '' ' '' ' UNIVERSrrY OF FLORIDA DOCUMENTS DEPARTViENT 120 MARSTON SCIIiNCE LIBRARY P.O. BOX 11 7011 GAINESVILLE, FL ocfui ' K^ z^ (^u(^ Ss"? i^p^^ NATIONAL ADVISORY COM^'a:TTEE FOR AERONAUTICS TECHNICAL I^M0RANI}U14 1293 APPROXIMATE METHOD OF INTEGRATION OF LAMINAR BOLiNDAxRY LAYER IN INCOMPRESSIBLE FLUID* By L. G. Loitsianskii Among all existing methods of the approximate integration of the differential eouations of the laminar boundary layer, the most vrLdely used is the method hased on the application of the momentum equation (reference l) . The accuracy of this method depends on the more or less successful choice of a one-parameter family of velocity profiles. Thus, for example, the polynomial of the fourth degree proposed by Pohlhausen (reference l) does not give velocity distributions closely agreeing vrith actual values in the neighbor- hood of the separation point, so that in the computations a strong retardation of the separation is obtained as compared with experi- mental results (reference 2) . The more-accurate methods employed in recent times (references 2 to 4) assume as a single -parameter fa'rdly of profiles the exact solutions of some special class of flows with given simple velocity distributions on the edge of the boundary layer (single term raised to a power, linear function). The transition to the more complicated two- and more -parameter famlies of profiles would require, in addition to the momentum equation, the emploiTnent of other possible equations (for example, the equations of energy (reference 5) and others (reference 6)). A greater accuracy might also then be expected for relatively siiiple velocity profiles that satisfy only the fundamental boundary con- ditions on the surface of the body and on the edge of the boundary layer. This second approach, however, as feir as is known, has not been considered except for very simple solution for the case of axial flow past a plate (reference 7). In the present paper, a solution is given of the problem of the plane lajninar boimdary layer in an incompressible gas; the method is based on the use of a system of equations of successive moments (including that of zero raoment, the momentum equation) of the equation of the boundary layer. Such statement of the problem "Priblishennyi Metod Integrirovania Uravnenii Laminarnogo Pogra- nichnogo Sloia v Keszhimaemom Gaze." Prikladnaya Matematika i Mek- hanika, USSR, Vol. 13, no. 5, Oct. 1949, p. 513-525. NACA TM 1293 leads to a complex system of equations, which, however, is easily solved for simple supplementary assximptions . The solution ohtained is given in closed form by very simple formulas and is no-less accurate than the previously mentioned complicated solutions that are based on the use of special classes of accurate solutions of the boundary-layer equations . 1. Derivation of Fundamental System of Successive Moments of Boundary-Layer Equation. The well-knovn equations of the stationary plane Isjrdnar bo^ondary layer in the absence of compressibility have the form > - ^2 ou ou ,„,, ,-, o u u ^— + V -— = UU' + u — - ox dy ^,/ (1.1) ox oy where u(x,y) and v(x,y) are the projections of the velocity at a section of the boundpa-y layer on the axial and transverse axes of coordinates x and y, U(x) is a given longitudinal velocity on the outer boundary U' = dU/dx, and u is the kinematic coeffi- cient of viscosity. VvTien the equation of continuity is applied, the first of equations (l.l) m.ay be given the miore convenient forjn L(u,v) = I- \u{\].n)\ + I- rv(U-u)1 + U'(U-u) - V ^J}tA = q ox •- -• oy •- -■ -^2. (1.2) The left side of equation (1.2) is multiplied by y and integrated wLtYi respect to y from, zero to infinity in the case of an asymptotically infinite layer or from zero to the outer limit of the layer y = 6(x) for the assumption of a layer of finite thickness. In either case, the following expression is obtained: >*,S /-f»,6 p»,5 L(u,v)y^ dy = ^ /Si(U-u) dy + j'^ ^ [v(U-u)] dy + U' y^(U-u) dy - U y^ MIMO ^^ ^ ^ ^ .^ J ^0 Sy^ ITACA W 1293 It is assumed in this equation and in vrtiat follovs that^ in view of the very rapid approach of the velocity difference U - u to zero as y — ^cn, all integrals with the infinite upper liirdt have a finite value. For k = 0, ^00,6 _d_ dx p, JO v?here the magnitude ^oo,& i(U-u) dy + U' (U-u) dy = -2i \ -.(: ou \ (1.4) (1.5) ^^y/y=o represents the friction stress on the surface of the hody. Equation (1.4), the well-known impulse or momentum equation, is readily transform.ed into its usual form- whore d&** U'&** dx ^* = 8** = U ">oo,5 (2+H) w pU^ k - 1) dy H = &** (i.s) ■N / (1.7) J For k = 1, a new equation of the 'first moment' is obtained from equation (1.3) NACA TO 1295 OOD^S ''\»,& y(U-u) dy = UU (i.e) and, in general, for k > 2, the equations of successively increas- ing moments are obtained _d_ dx r>cx,& ^u(U-u) dy - k I y^-l\'(U-u) dy + U' I y^(U-u) dy p»,5 = k(k-l) u y^"^(U-u) dy (1.9) In all these equations, the transverse velocity v(x,y) is assuTiied expressed in terras of the axial u(x,y) from the equation of continuity. It is nov assuni.ed that the family of functions u = vf (x, y;>^i, ^2^ • • • ^^k) (1.10) satisfies the boundary conditions of the problem with k param- eters ^j_, . . . , ^jj, which are functions of x, such that the k successive moments of equation (1.3) na',8 :,-^L(u°, v°) dy (1.11) become zero. On the assumption that it is permissible to pass to the limit k '0==, it would then be possible to state that the function u(x,:) = lim u° [x, y> \(x), ^2(^')' • • • '^k^^J k -»°-- (1.12) with parameters ^j^(x), A.2(x), . . . , ^k'^) satisfying the infinite system of e'^uaticns NACA TM 1293 pa, 5 :H{^°, v°) dy = (k = 0, 1, 2, . . . ) or, what is equivalent, systems (1.4), (l.S), and (1.9) will "be an exact solution of the fundaifiental system (l.l) for the assumed boundary conditions . For this solution, it is merely necessary to recall the knoi^ii theorem that a continuous function, all successive derivatives of which are eoual to zero, is identically equal to zero (reference 3) The question of the proof of the validity of this theoreir: is not considered in the case of an infinite interval or of an inter- val the boundaries of which are functions of a certain variable ".■ri.th respect to which the differentiation is effected. A certain construction, not based it is true on a rigorous proof, of the solution of the problem will be employed with the aid of the suc- cessive equations of the moments of the basic boundary-la,; er equation. 2 . Choice of rara.T.eters of Family of Velocity Profiles at Soc- tions of Boundej'y La;:;er. Special Fonr. of Equations of Moments. As is seen from the previously discussed considerations,' the funda- mental difficulty lies in the choice of a familj- of velocity pro- files (l.lO) and the determination cf the parajneters k, of the family. One of the simplest methods of the solution of this prob- lem is indicated herein. In the converging part of the boundary layer, the velocity profiles at various sections of the layer are knoim to be almost similar; the velocity profile is deformed m.ainly in the diffuser part cf the boundary layer doivTistream of the point of minimum pres- sure. The deformation of the profile consists of the appearance of a point of flexu.ra that arises near the surface of the body and. then moves awav from it as the separation point is approached. The presence of this deformation of the profile near the sur- face should greatly affect the m.agnitude t ^^ proportional to the normal derivative of the velocitj' on the surface of the body; it will therefore dim.inish to zero as the point of separation is approached. The deform.ation of the profile will have a smaller effect or such Integral magnitudes as ^* "^nd ^** and very little effect en magnitudes that contain under the lnt,Ce^ral s^gn fur.ct^ms that rapidly decrease as the surface of the I ody i.s -/.pprcachei. NACA TM 1293 For the parameters characterizing the effect of the deforma- tion of the velocity profile, it is natural to assume those magni- tudes that depend relatively strongly on the deformation of the velocity profile. With regard to the magnitudes that vary little with the deformation of the velocity profile, however, it is natiiral to assume that they do not depend on the chosen parameters. For the fundamental parameters determining a change in the shape of the velocity profiles, which may be called form parameters, the nondimensional combination of the msignitudes t^^ 5* and 6** '.rtll be employed '^th the given functions U(x) and U'(x) and physical constants, namely, the parameters 2 A f = U'&^ Ku/u) .a(y/6**);„^, /y=o T^B*» tiU H • ** ) (2.1) J For the computation of the remaining magnitudes in the equa- tion of moments according to the assumption, the velocity profile will be assumed in a section of the boundary layer in a form that does not depend on the parameters f, ^, and H: u U i^)- 9 (t) (2.2) This assumption permits, as will be subsequently seen, obtain- ing on the basis of very simple computations a sufficiently accurate solution of the boundary- layer equations for arbitrary distribution of the velocity on the edge of the layer. The transformation of eqiiaticn (I.6) >n.ll now be considered. If the parajneter ^ is introduced, then by equation (2.1), d&** U'&** ,^ ,,^ V + —r— (^+H) = dx or U J_ h**^ ] 2 dx \ u / U 5** + (2+K) f = ^ MCA IM 1293 It is not difficult to obtain finally i^g^l-H.i^v.t (2.3) For the transformation of the left side of equation (1,8) the first integral can be written by eqimtion (2.2) (t^ = y/8 ) yu(U-u) dy = U^s**^ POC T)(p(l-'^) dTi = H^U-8**^ (2.4) where the magnitude H-, , equal to Ox Hi = Tl(?(l^) dT] (2.5) represents a constant computed by the given function '^(ti). In order to compute the following integral, the transverse %'elocity v is first expressed by the formula V = - ^ dy = - — )U 5*" Sx Sx U'5' *» 9dTi - U d&-» ^ dx (p dTi cpdT) - U 5**

: or, when it is noted that in _ J_ dx ~ dx y d5** 2 dx 5** = - r] 1 d5** 5** dx the following expression is obtained V = - U'S** 1 cp cLt) - U {t\) is the magnitude H5 + H-^ + Hg/2. Numerical inte- gration gives H5 + H7 + i Eg = 24.73 (3.8) after which there is immediately obtained H = 2.59 - 7.55 f (3.9) MCA TM 1293 13 Substituting this expression for H in equation (3.1) gives ^ = 0.22 + 1.85 f - 7.55 f^ (3.10) Finally, integrating the simple linear equation (2.14) gives X fXX u^-i(nde = o^ jj^'^m (3.11) u^-^ Uo Equations (3,9), (3.10) and (3.11) give the required solution. The simple, approximate solution Just obtained is now compared with the actual values. The almost complete agreement of the val- ues of f obtained with the first approximation (which is practic- ally the only one that is applied) of the preceding works (refer- ences 2 and 3) will be noted. The closed-form relation between ^ and f likewise differs little from the corresponding tabulated functions in the references cited. For comparison, the curves ^(f) and H(f) obtained accord- ing to the formulas of reference 2 and by the formulas (3.10) and (3,9) are shown in figure 1. The results obtained will also be compared with the formulas of Wright and Bailey (reference 9), An approximate method of computation of the laminar boundary layer is proposed therein in which the equation of momentum (1,6) is employed with T^ and 5** substituted by the formulas for the flow past a plate. By expressing the results of Wright and Bailey in the parameters of the present report, the analogs of equations (3,9), (3.10), and (3,11) are obtained, H = 2.59 t, = 0,22 + 4,09 f \ (3,12) f = 0.44 HiiE U It is easily sesn that this formula for f corresponds to equation (3,11) for b = 1, The straight lines for t, and H shown lotted in figure 1 indicate the considerable deviation of the formulas of Wright and Bailey from more accxirate formulas presented herein. 14 NACA TO 1293 For confirmation, the particular case of the laminar "boundary layer corresponding to the so-called single-slops velocity distri- bution at the outer boundary of the layer U = 1 - x will be con- sidered. This case has been theoretically solved and an exact solution in a tabulated form (reference lo) is available. The results of the recomputation of these accurate solutions in the form assumed by the parameters ■ are given in figure 2. Also shown for comparison are the corresponding curves obtained by the pro- posed approximate method and by the method of Wright and Bailey. 4. Possible Methods of Rendering the Foregoing Solution More Accurate. The method described in the preceding sections was based on the assumption of a slight dependence of E^ on the form param- eters f, ^, and H. This assujnption may be eliminated and the method rendered more accurate, although it thereby becomes con- siderably more complicated. In order to discuss the possible generalizations of the method, the complete system of equations, for example, for the three- parameter case is written out; that is, a three-parameter family of velocity profiles is assumed In place of equation (2.2). ^ = q)h; f, C, H) (ri = ^) (4.1) By substituting this velocity profile in the system of the three equations of successive moments (1.6), (1.8) and (2.15), there is obtained a system of three ordinary nonlinear differential equations that determine the magnitudes of the parameters f, ^, and H: = if [l - (2Hi+H3+H4)f] + §; (Hi-i Eg) f (4.3) Hi - - Ho + f M dx MCA TM 1293 15 I ^ (3H5-2H6) f (4.4) in which, in addition to the previous notations, the following definitions are chosen: Kl = 1^ dr]} (l-U.5) ^ 16 NA.CA TM 1293 It is noted that, In the system of equations (4.2), (4.3), and (4.4), Hj^ and K^ are not constant magnitudes , as pre- viously, but known fxinctions of the form parameters f , C, and H; the form of these functions depends on the chosen family of profiles (4.1). The equations (2.3), (2.14), and (2.22) earlier employed evi- dently represent a particular case of the system (4.2), (4.3), and (4,4) on the assumption that the family of velocity profiles at the different sections of the houndary layer has the form of equa- tion (2.2); in other words, these profiles are similar to one another. All values of K^ are of course then equal to zero and E^ is constant. The proposed method may be rendered considerably more accurate by assuming, for example, the single -parameter family of velocity profiles u § = cp(ri; f) (4.6) Then 3h, Sh^ Sh^ Sh^ 2 Sc ^ bE ^ ac ^ Sh and the system of equations (4.2), (4.3) and (4.4) is transformed as follows: f + Hf = ^ 2 U' dx "*"y ' 2 ^,2 J 1 - I H2 .(ki. |i)fJM = ^ [1 - (2Hi.H3.H4)f] . i(3H5-2H6) ^^K^.i^jf ILL u H - H5+H 7-1 H3): df dx + i F (^%-2S6)f (4.7) (4.8) (4.9) MCA TM 1293 17 Equation (4.8) can be given the form af _ u- 1 - (2Hi+H3+H^)f U" % - 2 52 dx ~ U - "^ %" 9 ^2 + (Ki+^H-L/Sf)f "' H^- i Hg + (K3^+SH3^/Bf)f (4.10) vhich represents a generalization of equation (2.12) where equa- tion (4.10) approximates equation (2.12) because of the small change in Hj^ with change in the parajneter f and the amallness of the magnitude (Kj+SH-i^/Sf )f in comparison with Hj - Hg/S. This gen- eralization permits obtaining the integral of equation (4.10) by introducing a correction to the solution of eqtiation (2.12). By dividing both aides of equation (4.9) by the corresponding sides of eqiiation (4.3) and thus eliminating df/dx, there is obtained H = (H5+H7+iH8)f + i(3H5-2H6) ^(K4^1^)f I 1 W^ t - (2Hl+H3-.H4)f] .+ Hj^ - 5 H - ,,^ f (3=5-3H6)^(K4^|^ )f (^-1^) 7 (3%-2H6) (4.11) By similar considerations on the smallness of the magnitudes (K4+I/2 SKs/Sf )f in comparison with (3H5-2Hg)/4 and of (K^+^H^ySf )f in comparison with H]^ - H2/2 and on the slight variability of H^, it may be concluded that the value of H determined by equation (4.11) is an Improvement in the accuracy of the approximate value of H according to equation (3.6). It may be remarked that in this more accvirate approximation there is no longer that universal relation between the parameters H and f, independent of the form of the function U(x), character- izing the given particular problem. The presence in equation (4.11) 18 MCA TM 1293 of a second term with the factor UU''/U' shows that in the moro accurate approximation the magnitude H in a given section of the layer depends not only on the value of the form parajneter f In this section, as was the case in the rougher approximation of equa- tion (3.5) or (3.9), but also on the value of the magnitude UU' '/U' in the section considered, that is, on the values of the func- tion U(x) and its first two derivatives. It is readily observed that the second term on the right side of equation (4.11) will give a small correction to the solution (3.6) for relatively small values of the magnitude UU"/U'^. The same considerations hold for the expression for 5, which may be obtained by substituting df/dx from equation (4.10) and H from equation (4.11) into equation (4.7): 1 - (2Hi+H3+H4)f 1^3 bE. {'^^} Iz 4 (3H.-2HR)f + {^^^m m U' 2f + (H5+H7+ ^ Ha)f^- - (^1-1^) 'ii - ^'^^-'^^^ f]-("^ 4 1^) (^^-i ^^) 1 2^2- FW (4.12) As is seen, in this new approximation, In contrast to the pre- ceding one, there is no universal relation between presence of a term with the factor UU''/U' ( and f. The makes the magnitude t depend not only on the value of the parameter f but also on the form of the function U(x) and its first two derivatives in the given section of the boxmdary layer. It is of Interest to remark that in this approximation the position of the point of separation of the boundary layer, that is, the value of x = Xg for which ^ is equal to zero, will no longer be determined by some universal value of the form param- eter f „ , but in each Individual case the value of X = x^ must be determined for which the right side of equation (4.12) becaaes zero. NACA Tl'4 1293 19 By assuming a particular form of a family of velocity profiles (4.0), employing^ for example^ the sets of velocity profiles applied in the previous investigations (references 2 to 4), the values of the functions Hj_ and K-j^ are determined; the form parameters f, ^, and H, that is, the thickness of the momentum loss S**, the friction stress T^ and the displacement thick- ness 5* may then be found without difficulty. The solution of equation (4.10) and the determination of H and C hy equa- tions (4.11) and (4.12) offers no particular difficulty. Further improvement in the accuracy requiring the solution of a system of the type of equations (4.2), (4.3) and (4.4) is hardly of practical interest. In the previous discussion, the scheme of the asymptotically infinite boiindary layer was used, but similar equations may be obtained also for the case where the boimdary layer is assumed to be of finite thickness. The method here proposed may evidently also be applied to the case of the thermal boundary layer. The characteristic feature of the method for the cases of both the dynamic and the thermal bound- ary layer lies in the fact that the friction stress and the quantity of heat given off by a unit area of the body are expressed in inte- gral form and not in terms of the derivatives of functions that represent the approximate velocity and temperature distributions in the sections of the boundary layer. Trajislated by S. Reiss National Advisory Committee for Aeronautics . REFERENCES 1. Loitsianskii, L. G.: Aerodynamics of the Boimdary Layer. GTTI, 1941, pp. 170, 187. 2. Loitsianskii, L. G.: Approximate Method for Calculating the Laminar Boundary Layer on the Airfoil. Comptes Rendus (Doklady) de I'Acad. des Sci. de L'URSS, vol. XXXV, no. 8, 1942, pp. 227-232. 3. Kochin, N. E., and Loitsianskii, L. G.: An Approximate Method of Computation of the Boundary Layer. Doklady AN SSSR, T. XXXVI, No. 9, 1942. 20 MCA TM 1293 4. Melnikov, A. P.: On Certain Problems in the Theory of a Wing in a Nonideal Medium. Doctoral dissertation, L. Voenno- vozdushnaia inzhenernaia akademia, 1942. 5. Leibenson, L. S.: Energetic Form of the Integral Condition in the Theory of the Boundary Layer. Rep. No. 240, CAHI, 1935. 6. Kochin, N. E., Kibel, I. A., and Roze, N. V.: Theoretical Hydrodynamics, pt. II. GITI, 3d ed., 1948, p. 450. 7. Sutton, W. G. L.: An Approximate Solution of the Boundary Layer Equations for a Flat Plate. Phil. Mag. and Jour. Sci., vol. 23, ser. 7, 1937, pp. 1146-1152. 8. Carslaw, H., and Jaeger, J.: Operational Methods in Applied Mathematics. 1948. 9. Wright, E. A., and Bailey, G. W. : Laninar Frictional Resistance with Pressure Gradient. Jour. Aero. Sci., vol. 6, no. 12, Oct. 1939, pp. 485-488. 10. Howarth, L.: On the Solution of the Laminar Bounaary Layer Equations. Proc. Roy. 3oc. (London), vol. 164, no. A919, Feb. 1938, pp. 547-579. NACA TM 1293 21 0.35 H P-,- (reference 9)7"^^ ^ H Figure 1. 0.08 Figure 2. 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