mfirr/v\'i3%\ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1336 DEVELOPMENT OF A LAMINAR BOUNDARY LAYER BEHIND A SUCTION POINT By W. Wuest Translation of "Entwicklung einer laminaren Grenzschicht hinter einer Absaugestelle. " Ingenieur Archiv, Vol. 17, 1949. Washington March 1^^2 ci npiDA )N SCIENCE LIBRARY l^AINESViLL^a 32611-7011 USA 7(^( !>' ^ ' 3 vv NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM I336 DEVELOPMENT OF A LAMINAR BOUNDARY LAYER BEHIND A SUCTION POINT* By W, Wuest 1. INTRODUCTION Boundary- layer suction originally was applied to reduce the boundary- layer 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. Boundary- layer 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 boundary- layer 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 single-suction 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 x^ at the distance y^ parallel to the first plate so that a suction slot "Entwicklung einer laminaren Grenzschicht hinter einer Absauges telle." Ingenieur Archiv, Vol. 17, 19^9, pp. 199-206. 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 Blasius boundary layer. 2. BOUNDARY-LAYER EQUATION AND ASYMPTOTIC BEHAVIOR By introduction of the stream function and the total pressure, the boundary- layer equation may be transformed by the well-known method (reference k) into ^ = vu ^ (1) op ^ at where g = p + 75- u and -— = u. We limit ourselves to the case that 2 Sy the flow takes place outside of the boundary layer at the velocity u-]_ = const., thus to the flat plate and put furthermore g = -| u^2(i _ q(x^^)) + Const. (2) or, respectively u = u^V^ (3) This statement has been chosen so that for large t-"^3-lLies, q assumes the value 1. Equation (l) is thereby transformed into From the definition of the stream function and from equation (3), one further obtains with t] = f/ /vu, x NACA TM 1336 ay 2 X V V ^^ (5) (6) y = X Ui X i^Jo \^ d-q (7) In order to investigate the asymptotic behavior of the differential equation (3)^ we put for large values of ilr q = 1 - q^ with q^ « 1 (8) In first approximation, one then obtains Sqv ax hf (9) 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 heat-conduction equation. The general solution is therefore given fey do at i + ^lr' i - t' wl^i'^ - -o)y w^^^(^ - ^0), q^(0,x')-^ f ^lr ^x' \\Jkvu^{x - x')^ dx' d\|f' + (10) NACA TM 1336 Therein $ = A r e-y^ dy is the known error integral. W. Tollraien (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.. BLASIUS BOUNDARY LAYER Although we presupposed that the velocity u-^ 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 Xq = 0, thus suction point and beginning of the plate A (fig. l) 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 T^ = ^/Jvul-^x. One' then obtains from equation (h) the following differential equation of the Blasius boundary layer .2 ^.^=0 (11) The solution may be written in the following form ^B = PMi + ^iir -^^2-T +^3hr + • • •/ (12) NACA TM 1336 The constants a-j_ therein have the folloving values an = a„ = - a-j = a, = 2 15 a^ = 0.57627 X 10 1 90 % = 3.8907 X 10" -7 a^ = -1.3986 X 10 (990 X 15) 1.60333 X 10' -6 ag = -3.9135 X 10 ■11 aq = 3.7282 X 10 a^Q = 2.2383 X 10' -12 •13 ^11 = -0.3104 X 10-1^ a = -1.081 X 10"15 Due to the boundary condition at thg wall^ one integration constant is zero. The second integration constant is determined from the asymptotic behavior for large values of ri. " Because of q^(\j/,0) = the first integral in equation (lO) 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 \ %B'~'^ * \\Jhvuj^x (13) The constants P in equation (12) and 7 in equation (13) are deter- mined by the fact that for large rj values q and c3q/(3ri according to equation (12) and equation (13) agree with each other. The recalcu- lation of the two constants yielded the following values 3 =;..0.66i+2 0.828 For comparison, L. Prandtl (reference 7) gives^Vthf following values calculated by Blasius and Toe^'er whic?i read, converted to the above desipriations - Xl, ' p = 2 X 0.332 = o.66i+ 7 = 2\Jr{ 0.231 = 0.819 6 NACA TM 1336 F. Riegels and J. A. Zaat give in a new report (reference 8) for 7 the following value 7 = 0.3i|2\^ = 0.857 The function q with first and second derivative has been tabulated and plotted in numerical table 1 and figure 2 . k. ASYMPTOTIC BEHAVIOR OF THE SUCTION BOUTTDARY LAYER For calculation of the asymptotic behavior of the suction boundary layer, we divide the function q^ defined by equation (8) into two parts \= ^^l"- %r2 The first part is to be selected so that it satisfies the initial condi- tion at the suction point x = Xq; this is done by extending the asymp- totic solution of the Blasius boundary layer to x > Xq as well. From equation (I3) one then obtains %1 1 - <5 /^ + t [\f^^J "T]he numerical table has been calculated with the values (3 = 0.66^1 and 7 = O-.819. NACA IM 1336 Numerical Table 1. Blasius Boundary Layer n ^o) ~ *-*' whereas in the second integral one has to put \^{^,^) = %(0,x) - q^i(0,x) = q^(0,x) - 7 1-0 ^r lkvu.-.x so that the asymptotic solution reads S. = 7 /ilr + \lf 1 - * y^ifVUjX^ + / <|q^(0,x') - 7 1 - $ [JkVUj Sx' [^kvn^{x - x-o)) dx' By partial integration one obtains with consideration of the asymptotic behavior of the error integral (by W. Wuest, elsewhere) q ~ 7 ^w ' 1-0 '^ + t \]Jk\Mjy + iSr(°^^) - ^ 1 - $ ^kVn l^j 1-0/ /4Va-L(x - xq)^ Because of the connection with the Blasius solution, however, q^^O, XQ^ = 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 %~ ^ 1-0 y\/i+Vu^ + 701 ^, \Jkvuj_i 1-0/ ^^IIVU^^X - Xq) (li+) NACA TM 1336 Instead of the error integrals $ one may for large values of ^ again go back to the Blasius solution if one takes the asymptotic behavior of the latter according to equation (8) and equation (I3) into consideration Wl^Vu i^oy ^B »/vui(x - Xq) (15) In this formula q-n 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 tIt in the following manner U/vI^ X ^^ \/vUi(x - Xq)^ (16) Due to q = for \|f = and because of equation (15) the func- tion F(ilr, x) must satisfy the following conditions F(0,x) = q. B t fV^^x F(-,x) i v\r^^, (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 t + t, = n t = n' ^fr so that the solution (equation (16)) reads ^0 ^ = ^0 + X - ^ .. V (18) ci = y,) -F[(i-q^(v))] According to a suggestion by A. Betz, we equate as first approxi- mation q the function F to the value dependent only on x Fq(x) = F(0,x) = qgj % rv = 1^. ■b{\) at the wall. ' Thus the first approximation reads d-L = %i^) •0 (i-d^CV)) (19) (20) (21) This formulation does not fulfill the boundary- layer equation (k) exactly. In particular, the second derivative of q-, 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 boundary- layer equation (U), one obtains ^,"M^^ ^^foE-%(v3} a^e. dx V -L ^^,^ 2/2 Hence there results S 6-|/5t]' as the error of this first approxi- mation. By subtraction of the exact solution in which F stands for Fq and q for q , while e-, disappears, one then obtains MCA TM 1336 11 ^ {(^ - ^o)E - %(V1 (22) where a^6. Sti 7|=^2"=2(x-Xo)^ V\ - \/^" is an unknown function. The quantity e^** disappears for t]' and Tj' = °°. By integration of equation (22) one obtains = (F - Fo)[l - qB(v[| = ^1+ 62 (23) Therein €j_ is to be determined graphically or numerically by repeated quadrature T* p-no ^2 00 "J 00 2 d 6-1 — ^ dri^ drig dT]^ (24) One may detenaine the asymptotic behavior of eg by substituting in the above definition of eg'' for ./qT and yq the asymptotic values ./q, ~ 1 - ^ q^ and yq ~ 1 - ^ q . Thereby one obtains 62" ~ (^ - '^)^N - (21), and repeated inte- gration with respect to t)' = "if/Jvu (x - y:^\ 62 ~ 7(f,o - Fq) 1 - .fl^) (25) 12 NACA TM 1336 As before, denotes the error integral. Generally we visualize go as represented in the following manner K'l 21 ^(^) 1 - ^{^t) (26) By way of approximation we limit ourselves to the first two terms, with a-^ = 7(F^ - Fq') and a^ determined by the fact that q must disappear at the wall. We determine accordingly the function F approximately to be F = Fq + Tzi^ l^i(nSx) . 7(F„ - Fo)[i - o(^j] ..^\i- $'(vl]| (27) I2 = -^j^iO,^) - 7(F^ - Fq) (28) Calculation example.- \|r„ //Vu, x = 0.125 was selected as numerical example;. F was calculated for the values x/^ = 1-23^, I.562, h.3^, and 9- 78 and plotted in figure 3- For x/xq = I.562 the error was determined by substitution of the approximated solution into the boundary- layer 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. k) . 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 tj^* = ^q/JvoTx^ will be supplemented. By the degree of suction 8 we here understand 8=1 Si* (29) 5-[_* being the displacement thickness immediately ahead of the suction point and Sg"*^ immediately behind it. Therewith 8 is given by NACA TM 1336 13 _Jo \\f^' J dijf 3 Jo \\/^ (30) The resulting values are tabulated in table 2 and plotted in figure "J. Table 2. Degree of Suction ^0* 0.1 0.2 o.k 0.6 0.8 1.0 2.0 G 0.392 0.520 0.671 0.762 0.82i+ 0.870 0.97^ The suction quantity ^^ is, furthermore, given by the following relation ^0 = \/^Vo^o' 6. SUMMARY (31) 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 boundary- layer equation in the form indicated by Prandtl-Mises 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 Ik NACA TM 1336 REFERENCES 1. Tollmien, W., and Mangier, W.: Stationare laminare Grenzschichten. Monogr. Fortschr. Luftfahrtforsch. Aerodyn. Vers.-Anst. Gottlngen (AVA-Monogr.), Vol. 1, 19^6, CF. also H. Schlichting, Ing.-Arch., Vol. 16, 19ij-8_, p. 201. 2. Schrenk, 0.: Z. angew. Math. Mech., Vol. 13, 1933, P- I80. 3. Pfenniger, W.: Untersuchungen iiber Reibungsverminderungen an Tragflugeln, insbesondere mit Hilfe von Grenzschlchtabsaugimg. Mitt. Inst. Aerodyn. Tech. Hochschule Zurich Nr. 13, 1946. (Available as MCA TM II8I.) k. Prandtl, L.: Note on the Calculation of Boundary Layers. Z. ange-w. Math. Mech., Vol. I8, 1938, p. 77-82. (Available as NACA TM 959-) 5. Wuest, W.: Beitrag zur instationaren laminaren Grenzschicht an ebenen Wanden. Ing.-Arch., Vol. I7, 19^9, PP- 193-198. 6. Tollmien, W. : Uber das Verhalten einer Stromung langs einer Wand am Susseren Rand ihrer Reibvmgsschicht . Bet z -Festschrift AVA-Gottingen, 19^5, 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 ungestorte stromung. Nachr. Akad. Wiss. Gottingen. Math.-Phys. Kl. 19^+7, pp. ii2-45. NACA TM 1336 15 A / T Xo B W^ Suction Figure 1.- Boundary -layer suction at the flat plate without sink effect. q,q,2q 0.5 1.0 1.5 2.0 2.5 3.0 3.5 Figure 2.- The function q(Ti) of the Blasius boundary layer with first and second derivative. 16 NACA TM 1336 F(^:x) 0.05 0.10 Figures.- Aiixiliary function F(ti',x) for calculation of the suction boundary layer for +q/\^vu2Xq = 0.125 0.02 -0.02 -0.04 -0.06 Second / approximation ' l/l/U, (X-Xq) Q5 1,0 1.5 2.0 2,5 Figure 4.- Error of the first and second approximation for x/x^ = 1.562 NACA TM 1336 17 2 0,4 0.6 0-8 IJ Figure 5.- Velocity profiles of the suction boundary layer for iI^q/J vuj^Xq = 0.125 and various distances from the suction point. 0.050 0.100 Figure 6.- Second derivative of the velocity profiles of the suction boundary layer for + Alvu^x = 0.125. 18 NACA TM 1336 Deg ree of suction e 1 n 0.8 0J6 04 y ^ "" / / D'' / '0 -^ )* ^ n- , r<^« - 0.2 0.4 0.6 0.8 1.0 1.2 1.4 Figure 7.- Degree of suction (ratio of the cross-hatched and the total shaded area in fig. 1). 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