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5& 7f ' NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1336 DEVELOPMENT OF A LAMINAR BOUNDARY LAYER BEHIND A SUCTION POINT* By W. Wuest 1. INTRODUCTION Boundarylayer suction originally was applied to reduce the boundarylayer thickness and therewith the inclination to flow sepa ration; however, since the properties of bodies with small drag have been improved more and more, attention was drawn to an increased extent to the reduction of surface friction. One now strived toward keeping the boundary layer laminar as long as possible, thus to defer the tran sition point to turbulence as far as possible. Boundarylayer suction was recognized to have a favorable effect in this sense, and therewith the velocity distribution in a laminar boundary layer behind a suction point acquired heightened interest. The stability of a laminar velocity profile is very severely affected by the shape of this profile. In a considerable number of theoretical reports (reference 1) the case of continuous suction was treated for reasons of mathematical simplicity; permeability of the wall surface was assumed. In further reports, the stability of laminar boundarylayer profiles in case of continuous suction was treated and a considerable rise in the stability limit was determined; however, a technical realization of such perme able walls with sufficiently smooth surface and adequate material strength characteristics is difficult. For structural reasons, it is simpler to arrange singlesuction slots. In addition to the suction effect proper, there appears here the sink effect first discussed in detail by L. Prandtl and 0. Schrenk (reference 2) and recently treated by Pfenniger (reference 3) in an instructive experimental investigation. Below, the pressure variation along the wall as well as, in partic ular, the sink effect are disregarded. Figure 1 shows the practical realization of such a case. We assume that on a flat plate A, a laminar boundary layer ("Blasius boundary layer") develops at constant pressure. We assume a second plate B arranged beginning from a certain point xo at the distance yo parallel to the first plate so that a suction slot *"Entwicklung einer laminaren Grenzschicht hinter einer Absaugestelle." Ingenieur Archiv, Vol. 17, 1949, pp. 19920o. rz7/ f NACA TM 1336 is formed between the two plates. The magnitude of the power require ment for suction is assumed to be precisely such that merely the part of the boundary layer situated between the two plates is removed. Thus, there begins above the plate B a new laminar boundary layer which is distinguished from the Blasius boundary layer by another initial condi tion. The new boundary layer forms at its start the outer part of a Biasius boundary layer. 2. BOUNDARYLAYER EQUATION AND ASYMPTOTIC BEHAVIOR By introduction of the stream function and the total pressure, the boundarylayer equation may be transformed by the wellknown method (reference 4) into 3g = Vu (i)g ox (32 where g = p + u and u. We limit ourselves to the case that oy the flow takes place outside of the boundary layer at the velocity a1 = const., thus to the flat plate and put furthermore g = 2 u2(l q(x,*)) + Const. (2) or, respectively u = q (3) This statement has been chosen so that for large ivalues, q assumes the value 1. Equation (1) is thereby transformed into dq _q 7 =VUl q (4) Ox 6*2 From the definition of the stream function and from equation (3), one further obtains with q = '/vu1x I/v77 NACA TM 1336 ou _1 Uli uix oq By 2 xV V TI 2u Cy2 1 ul ulx 2 2 V) 2 y = xdq Y dqr  0^J rq In order to investigate the asymptotic behavior of the differential equation (3), we put for large values of * q = 1 q. with q << 1 In first approximation, one then obtains aq, a qw vul 2 OX This differential equation, however, is mathematically identical with the differential equation of a nonsteady flow independent of x which has been treated before (reference 5); the time t is now replaced by the stipulated space coordinate x. It also corresponds to the well known heatconduction equation. The general solution is therefore given by q( *,x) = L J q''( 2 x A0 _, O )  dq *1 ^ (xl x 0vul(x x Cd'( x 4vul(x x') (10) qw(ojx')1o NACA TM 1336 Therein 0 = 2 ey dy is the known error integral. W. Tollaien (reference 6) has investigated this solution for two special cases where the first integral disappears. For the boundary layer with suction, however, this will no longer be the case. 3. BLASrUS BOUNDARY LAYER Although we presupposed that the velocity ul at the edge of the boundary layer is constant, the problem of the suction boundary layer to be treated here nevertheless differs from the flow on a simple flat plate ("Blasius boundary layer") by the fact that other initial condi tions exist; rather, the Blasius boundary layer is contained as special solution among the suction boundary layers since there x0 = 0, thus suction point and beginning of the plate A (fig. 1) coincide. Since we shall make use of this special solution for the later calculation, we shall first consider the Blasius boundary layer. It is distinguished by the fact that q may be regarded as dependent merely on a quan tity 11 = /fvuix. One then obtains from equation (4) the following differential equation of the Blasius boundary layer TI + 0 (11) The solution may be written in the following form qB = 3r(l + a+ j) + 3 )2 + a3 ( + (12) NACA TM 1336 The constants ai therein have the following values a_ 2 a2 S"90 7 a3 (990 x 15) a4 = 1.60333 x 100 a, = 0.57627 x 10 a, = 3.890( x 109 a = 1.398o .< 109 a8 = 3.9135 x 1011 a9 = 3.7282 x 1012 a10 = 2.2383 x 1013 all = 0.3104 ;., 014 a12 = 1.081 x 1015 Due to the boundary condition at th wall, one integration constant is zero. The second integration constant is determined from the asymptotic behavior for large values of r. 'Because of q.w(,0) = 0 the first integral in equation (10) is eliminated. The second integral, however, yields by partial integration, with consideration of the asymptotic behavior of the error integral, just as in the case treated before by W. Wuest the solution (13) The constants 3 in equation (12) and 7 in equation (13) are deter mined by the fact that for large I values q and oq/f.r according to equation (12) and equation (1l) agree with each other. The recalcu lation of the two constants yielded the following values 0 = .0.66.42 7 ='0.828 For comparison, L. Prandtl (reference 7) givesc\thy f6flowing values calculated by Blasius and ToAer which read, converted to the above designations 0 = 2 / 0.332 = o.664 S= 2 F 0.231 = 0.819 qwB'.~) 1 'Y1 [ _ V4=Vulx] NACA TM 1336 F. Riegels and J. A. Zaat give in a new report (reference 8) for 7 the following value y = 0.342 24 = 0.857 The function q with first and second derivative has been tabulated and plotted in numerical table 1 and figure 21. 1. ASYMPTOTIC BEHAVIOR OF THE SUCTION BOUNDARY LAYER For calculation of the asymptotic behavior of the suction boundary layer, we divide the function qw defined by equation (8) into two parts w = v1 + w2 The first part is to be selected so that it satisfies the initial condi tion at the suction point x = xO; this is done by extending the asymp totic solution of the Blasius boundary layer to x > xO as well. From equation (13) one then obtains q wl = 7 1 0 4vulx] The numerical table has been calculated with the values o = 0.664 and 7 = 0.819. NACA TM 1336 Numerical Table 1. Blasius Boundary Layer q'(i) I "(T) 4 4 ______ 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 2.5 3.0 3.5 4.0 4.5 5.0 .06606 .13106 .1939 .2546 .3135 .3695 .4228 .4681 .5211 .5659 .6081 .6469 .6831 .7167 .7474 .7758 .8017 .8252 .8465 .8657 .9352 .9715 .9881 .9962 .9988 .9997 o.6640o .6555 .6427 .6206 .5981 .5734 .5471 .5195 .4912 .4625 .4337 .4062 .3766 .3487 .3217 .2955 .2705 .2466 .2238 .2022 .1820 .1013 .0509 .0217 .0085 .0029 .0009 .12751 .17750 .2114 .2369 .2561 .2700 .2796 .2856 .2883 .2883 .2857 .2809 .2743 .2660 .2563 .2457 .2340 .2248 .2088 .1955 .1303 .0774 .0381 .0169 .0066 .0022 0 .2570 .3620 .4404 .5046 .5599 .6079 .6502 .6842 .7212 .7523 .7798 .8o4 3 .8265 .8466 .8645 .8807 .8954 .9084 .9200 .9304 .9671 .9857 .9940 ..9981 .9994 .9998 Therein 4 = 0 quantity. The ary condition forms the new wall streamline and r0 the suction second part qw2 then must be chosen so that the bound qw = qw(O,x) is satisfied. If the asymptotic rela tion q ~ 1 q would rigorously apply in the entire domain of the boundary layer, there would have to be at the wall qw,(0,x) = 1, because of q = 0; however, the asymptotic solution deviates from the rigorous solution if it is continued up to the wall. Therefore qw(O,x) is an unknown function regarding which we merely make the assumption that it does not become infinite. As initial condition for the part qw2 one 1 qq) NACA TM 1336 further has qw2(*,x0) = 0 since qwl(*,xo) already satisfies the initial condition q w = 7 1 0 which insures the connection wtth the Blasius solution. The contri bution qw2 to the solution al'b must obey the differential equa tion (9). In the solution (equation (10)) the first integral is elimi nated, because of qw2(,x0o) = 0, whereas in the second integral one has to put qw%2(0,x) = q%(0,x) ql(O,x) = q(0,x) 7 1  so that the asymptotic solution reads { + \ /[ c / 2O)) , = 7 1 +f q (O,x') 7 1 x' By partial integration one obtains with consideration of the asymptotic behavior of the error integral (by W. Wuest, elsewhere) 7[ (D**o + (oxO) 7 K i 4 1 ( \ vulx ul Iul(x x0)^I Because of the connection with the Blasius solution, however, q(O, xo) = 7, if the asymptotic solution is continued up to the wall, so that one finally obtains as the asymptotic solution for the suction boundary layer qv Y + y( ( o)[ (14) NACA TM 1336 Instead of the error integrals 4 one may for large values of 4 again go back to the Blasius solution if one takes the asymptotic behavior of the latter according to equation (8) and equation (13) into consideration q + *0\ 0 q q ~ qB 1 qB X (1) Vul /ulx0 Vul x x0 In this formula qB represents the Blasius solution. The last form of the solution proves to be particularly expedient for the further con siderations. 5. APPROXIMATE SOLUTION FOR THE SUCTION BOUNDARY LAYER It suggests itself to generalize the asymptotic solution which is valid for large values of 4 in the following manner q = qB* F(,x) 1 B (16) Vu /SVul x x0) Due to q = 0 for 4 = 0 and because of equation (15) the func tion F(i,x) must satisfy the following conditions F(O,x) = qB \ F(),x) = u 0 (17) It was hoped at first that one could choose for F, as in the nonsteady analogue by W. Wuest, elsewhere correspondingly an exponential func tion as the simplest formulation; besides equation (17) the disappear ance of the second derivative of q at the wall would be added as a further condition; however, it was shown that such a formulation does not meet with success and even, in a certain domain, does not yield any solution at all. 10 NACA TM 1336 For the further calculation we introduce the following simplified notation + t0 + 0 x Xo = = = + (18) _ulx VulX x) vu x so that the solution (equation (16)) reads q = q(nl) F 1 qB') (19) According to a suggestion by A. Betz, we equate as first approxi mation q1 the function F to the value dependent only on x FO(X) = F(O,x) = QB o) (20 at the wall. Thus the first approximation reads l= qB() Fo 1 qB(')) (21) This formulation does not fulfill the boundarylayer equation (4) exactly. In particular, the second derivative of ql at the wall does not disap pear; however, the dependency on the second derivative of the stability of the velocity profile is of a very sensitive nature so that one has to look for a more accurate solution. By substitution of the approximate solution .(equation (21)) into the boundarylayer equation (4), one obtains qB' )x x 2 {[o } 2 (x xO) 2I Hence there results 2 61 in2 as the error of this first approxi mation. By subtraction of the exact solution in which F stands for FO and q for q1, while el disappears, one then obtains NACA TM 1336 r2 =) 21 22E. 2 F FFO) [1 ( + (22) where S2 =( t 2(x xo)~2L is an unknown function. The quantity e2'' disappears for q' = 0 and ij' = m. By integration of equation (22) one obtains (F Fo)E 1 qB(T')] = i + '2 (23) Therein eI is to be determined graphically or numerically by repeated quadrature El j fr2 2 l2 (24) I = o 61 d11 dT2 One may determine the asymptotic behavior of e2 by substituting in the above definition of e2'' for and q the asymptotic values q 1 qwl and yq 1 qw. Thereby one obtains 2"x x0) (qv qwl) ox Hence follows with use of equations (9), (19), (21), and repeated inte gration with respect to q' = i/vux x0) E2 F(F Fo) [ V (Io ) (25) NACA TM 1336 As before, 0 denotes the error integral. Generally we visualize g2 as represented in the following manner o I 2 Z a(x) Q ) (26) X.1 By way of approximation we limit ourselves to the first two terms, with aI = 7(F. FO) and a2 determined by the fact that q must disappear at the wall. We determine accordingly the function F approximately to be F = FO + q ') 1(',x) + Y(F. FO) + a2  (27) a2 = 1(0,x) 7(F% FO) (28) Calculation example. 0/F vu,x0 = 0.125 was selected as numerical example;. F was calculated for the values x/xO = 1.234, 1.562, 4.34, and 9.78 and plotted in figure 3. For x/xO = 1.562 the error was determined by substitution of the approximated solution into the boundarylayer equation, and compared with the first approximation according to equation (21). Compared to the first approximation, a con siderable improvement results particularly in the region near the wall (fig. 4). In figure 5 the results are converted to the velocity pro file, in figure 6 the second derivative is represented. As a supplement, the connection between the degree of suction and the suction quantity of the magnitude T0* = 01F/Vulx will be supplemented. By the degree of suction e we here understand 8 * S= 1 2 (29) 81" 81* being the displacement thickness immediately ahead of the suction point and 62* immediately behind it. Therewith E is given by NACA TM 1336 r of 1) d e = Jo (30) f 1 d* The resulting values are tabulated in table 2 and plotted in figure 7. Table 2. Degree of Suction 0q* O 0.1 0.2 0.4 0.6 0.8 1.0 2.0 8 0 0.392 0.520 0.671 0.762 0.824 0.870 0.974 The suction quantity 40 is, furthermore, given by the following relation :0 = tlxo0o* (31) 6. SUMMARY The development of a laminar boundary layer behind a suction point is investigated if by the suction merely the part of the boundary layer near the wall is "cut off", without the slot exerting a sink effect. As basis of the calculation, we used the boundarylayer equation in the form indicated by PrandtlMises which is closely related to the heat conduction equation or, respectively, to the differential equation of the nonsteady flow which is independent of the coordinate x along the wall. With consideration of the asymptotic behavior of the solution, an approximate solution is developed which is similar in structure to the solution of the nonsteady analogue which has been treated in an earlier report by W. Wuest, elsewhere. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM 1336 REFERENCES 1. Tolimien, W., and Mangler, W.: StationKre laminare Grenzschichten. Monogr. Fortschr. Luftfahrtforsch. Aerodyn. Vers.Anst. GOttingen (AVAMonogr.), Vol. 1, 1946, CF. also H. Schlichting, Ing.Arch., Vol. 16, 1948, p. 201. 2. Schrenk, 0.: Z. angew. Math. Mech., Vol. 13, 1933, p. 180. 3. Pfenniger, W.: Untersuchungen Uber Reibungsverminderungen an Tragfl'Ugeln, insbesondere mit Hilfe von Grenzschichtabsaugung. Mitt. Inst. Aerodyn. Tech. Hochschule Zurich Nr. 13, 1946. (Available as NACA TM 1181.) 4. Prandtl, L.: Note on the Calculation of Boundary Layers. Z. angew. Math. Mech., Vol. 18, 1938, p. 7782. (Available as NACA TM 959.) 5. Wuest, W.: Beitrag zur instationaren laminaren Grenzschicht an ebenen Wanden. Ing.Arch., Vol. 17, 1949, pp. 193198. 6. Tollmien, W.: Uber das Verhalten einer Str5mung langs einer Wand am lusseren Rand ihrer Reibungsschicht. BetzFestschrift AVAG5ttingen, 1945, p. 218. 7. Prandtl, L.: F. W. Durand Aerodynamic Theory, Vol. 3, 1935, p. 88. 8. Riegels, F., and Zaat, J. A.: Zum Ubergang von grenzschichten in die ungestb'rte str5mung. Nachr. Akad. Wiss. OGttingen. Math.Phys. Kl. 1947, pp. 4245. NACA TM 1336 I I I Suction Figure 1. Boundarylayer suction at the flat plate without sink effect. i q,q,2q" . JB    ^_ (_r__ N 2_q11?) 0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 2. The function q(71) of the Blasius boundary layer with first and second derivative. ,4' (11) 4 / 1ffItI*^*1 NACA TM 1336 Figure . Auxiliary function F(q',x) for calculation of the suction boundary layer for ,0/\VUlXo = 0.125. Figure 4. Error of the first and second approximation for x/x0 = 1.562. NACA TM 1336 Figure 5. Velocity profiles of the suction boundary layer for ,0/J /vUlX = 0.125 and various distances from the suction point. 0 0.050 0.100 0.150 Figure 6. Second derivative of the velocity profiles of the suction boundary layer for / vux = 0.125. 13 i' 1 u NACA 9I 1336 Degree of suction G Figure 7. Degree of suction (ratio of the crosshatched and the total shaded area in fig. 1). NACALangley 31951 1000 . , U I Id1 t E  J . zE " E; 2% a; o, w Hu.r > < i w < 2,Lo 0 0 L '1 * C .a,  02 Cn e : . a3,* V, '4 "L rr E  * a D. 5~ mr 3a 3rf " C ^r Qj n ^ 'sCrI < C "zl. 4 2, 0 a.3'2 L.: 0 .03 5 CU~n  I  CD cB C3 C.)~.j~ , , C. '4SC ,n L ci '^ w  i I CJ d 'o )' 3 314 2 a a, u _ E U0 *nJ R :_ : = C: u~ CL ~ s 0Eh eg ^2'E roj a , z,_ < WC 5LI ,'O 0 u > a', t E a6 z J z m i Ei i .5 E3 ^ cn F. 4 .ScD r44 bg ,3 0 v Cd r 0 2 0dlC k $4 .. 'S z o~ I4 <0 E 9 .... co w'Z. 8 01 A : . 14 t c S N 24 E<0s u ta 0 awD 1 a U to 8 w w1. cz$44 k zzc1w .1s. 0u0 oiE Z Q* oM W0 M  . C, 0 < I 4 H ii m ItCJ I. I S l Ii 3 <.S Cd 3 u"N L Ln' < im F w uo 0 *Z cn 3 W ii SCO 2 g Sg LJ^ > < 0V < I cd s5 S " aj I r:> 0 C S E >, u 0 iC ^*~ Cuf g 2 z6 ac =c r . E^  L n n m rj C E> u3 n T i C ii >.: i<3' , CL ^ ." 5 a Ca i3 a Sw nZ ra3 l'r fr c rn Si  IDO ^t 0 0"i. E o C >  0 E Z Cu % 224 'i 3 ..E 3a z. I 0jL10 < o 6 f n mI Lj 0~ ZCEO u5 2 m WLU2iii j pj ^ ca L.C  > m 0 1..a  C) >5 0, Cu 0L ' .1 01 0 C co =5 0, 5ctv4 < 0Cu C a. _: 0 ^r^ 0J" 0ij 0 ZS?^ go ^ GO  u r r a in5 a, tn w .^ VU^j.. Q, a0 Cur La i n ci, Cu 5>a Cu c' C L. , a, C E .E WE fc .c 0 ef . * C C0' C. CI C, CU n a ^ o .EB3 e g o UNIVERSITY OF FLORIDA I I I1 1 111111 I IIIII 3 1262 08105 783 7 