Memorandum presenting an investigation of boundary-layer suction and its favorable effect in deferring transition points as long as possible in order to avoid turbulence

Wuest, W.

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N 62 58336NATIONAL ADVISORY COMMITTEEFOR AERONAUTICSTECHNICAL MEMORANDUM 1336DEVELOPMENT OF A LAMINAR BOUNDARY LAYERBEHIND A SUCTION POINTBy W. WuestTranslation of "Entwicklung einer laminaren Grenzschicht hintereiner Absaugestelle." Ingenieur Archiv, Vol. 17,1949.JNASA-TM-79874) DEVELOPMENT OF A 1 A H I N A FBOON-DABY L A Y E B h E E E I N D A SOCTICH POINT(Nati'onal AdvisipEyAeronautics) ;19 pN .78-7 8 513Onelas00/02 32238WashingtonMarch 1952NACA TM 1336National Advisory Committee for Aeronautics.DEVELOPMENT OF A LAMINAR BOUNDARY LAYERLAYER BEHIND A SUCTION POINT. (Entwicklungeiner laminaren grenzschicht hinter einerabsaugestelle). W. Wuest. March 1952. 18p.diagrs. (NACA TM 1336. Trans, from Ingenieur-Archiv, v. 17, 1949, p.199-206).A theoretical investigation is made of the develop-ment of a laminar boundary layer behind a suctionslot that is assumed to cut off part of the boundarylayer without exerting any sink effect. The develop-ment, which is approximate, is based on the heatconduction equation. The heat conduction equationenters the analysis through a linearization of thePrandtl-Mises -form of the boundaryJayer equation.Copies obtainable from NACA, Washington1. Flow, Laminar(1.1.3.1)I. Wuest, WalterII. NACA TM 1336in. Ingenieur-Archiv, v. 17,1949, p.199-206NATIONAL ADVISORY COMMITTEE FOR AERONAUTICSTECHNICAL MEMORANDUM 1336DEVELOPMENT OF A LAMINAR BOUNDARY LAYERBEHIND A SUCTION POINT*By W. Wuest1. INTRODUCTIONBoundary-layer suction originally was applied to reduce theboundary-layer thickness and therewith the inclination to flow sepa-ration; however, since the properties of bodies with small drag havebeen improved more and more, attention was drawn to an increased extentto the reduction of surface friction. One now strived toward keepingthe boundary layer laminar as long as possible, thus to defer the tran-sition point to turbulence as far as possible. Boundary-layer suctionwas recognized to have a favorable effect in this sense, and therewiththe velocity distribution in a laminar boundary layer behind a suctionpoint acquired heightened interest. The stability of a laminar velocityprofile is very severely affected'by the shape of this profile..In a considerable number of theoretical reports (reference 1) thecase of continuous suction was treated for reasons of mathematicalsimplicity; permeability of the wall surface was assumed.. In furtherreports, the stability of laminar boundary-layer profiles in case ofcontinuous suction was treated and a considerable rise in the stabilitylimit was determined; however, a technical realization of such perme-able walls with sufficiently smooth surface and adequate materialstrength characteristics is difficult. For structural reasons, it issimpler to arrange single-suction slots. In addition to the suctioneffect proper, there appears here the sink effect first discussed indetail by L. Prandtl and 0. Schrenk (reference 2) and recently treatedby 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 practicalrealization of such a case. We assume that on a flat plate A, a laminarboundary layer ("Blasius boundary layer") develops at constant pressure.We assume a second plate B arranged beginning from a certain point XQ.at the distance yo parallel to the first plate so that a .suction slot"Entwicklung einer laminaren Grenzschicht hinter einerAbsaugestelle." Ingenieur Archiv, Vol. 17, 19̂ 9, pp. 199-206.NACA TM 1336is formed between the two plates. The magnitude of the power require-ment for suction is assumed to be precisely such that merely'the partof the boundary layer situated between the two plates is removed. Thus,there begins above the plate B a new laminar boundary layer which isdistinguished from the Blasius boundary layer by another initial condi-tion. The new boundary layer forms at its start the outer part of aBlasius boundary layer.2. BOUNDARY-LAYER EQUATION AND ASYMPTOTIC BEHAVIOR' By introduction of the stream function and the total pressure, theboundary- layer equation may be transformed by the well-known method(reference 4) into£ = vu L* (1)where g = p + §• u2 and — -- = u. We limit ourselves to the case that2 dythe flow takes place outside of the boundary layer at the velocityu-^ = const., thus to the flat plate and put furthermoreg = -| u-̂ l - q(x,i|r)) + Const. (2)or, respectivelyu = Vq (3)This statement has been chosen so that for large \jf-values, q assumesthe value 1. Equation (l) is thereby transformed intoQdq • i— ° q /, i-^ = vu-j^q — % (10ox x dl|2From the definition of the stream function and from equation (3)> onefurther obtains with r\ = i|r/ /vu, x* INACA TM 1336 3 ..u uxBy 2 x V(5)(6)y = •Mi- (7)In order to investigate the asymptotic behavior of the differentialequation (3).> we put for large values of i|/q = 1 - q with a « 1Lw (8)In first approximation, one then obtains^2= VUj_ (9)This differential equation, however, is mathematically identical withthe differential equation of a nonsteady flow independent of x whichhas been treated before (reference 5)j the time t is now replaced bythe stipulated space coordinate x. It also corresponds to the well-known heat-conduction equation. The general solution is therefore given(10)NACA TM 1336Thereino Tz 2«=-!=/ e-y dyis the known error integral. W. Tollmien (reference 6) has investigatedthis solution for two special cases where the first integral disappears.For the boundary layer with suction, however-, this will no longer be thecase.3.. BLASIUS BOUNDARY LAYERAlthough we presupposed that the velocity u-[_ at the edge of theboundary layer is constant, the problem of the suction boundary layerto be treated here nevertheless differs from the flow on a simple flatplate ("Blasius boundary layer") by the fact that other initial condi-tions exist; rather, the Blasius boundary layer is contained as specialsolution among the suction boundary layers since there XQ = 0, thussuction point and beginning of the plate A (fig. l) coincide. Since weshall make use of this special solution for the later calculation, weshall first consider the Blasius boundary layer. It is distinguishedby the fact that q may be regarded as dependent merely on a quan-tity T} = \|rA/vu]_x. One then obtains from equation (k) the followingdifferential equation of the Blasius boundary layer(11)The solution may be written in the following form(12)NACA TM 1336 . 5The constants a-j_ therein have the following valuesa-, =--% a5 = 0.57627 x 10~6 aq = 3.7282 x 10"12•*• lp ^ "a2 =-^ a6 = 3.8907 x 10-9 aiQ = 3.2383 x 10'13a, = =1 a7 = -1.3986 x 10-9 a, , = -0.310̂  x lO"1^J (990 x 15) ra^ = 1.60333 X 10"6 ag = -3.9135 x 10"11 a12 = -l.oSl X 10'15Due to the boundary condition at the wall, one integration constant iszero. The second integration constant is determined from the asymptoticbehavior for large values of TJ. Because of ^(iJf^O) = 0 the firstintegral in equation (10) is eliminated. The second integral, however,yields by partial integration, with consideration of the asymptoticbehavior of the error integral, just as in the case treated before byW. Wuest the solution(13)The constants P in equation (12) and 7 in equation (13) are deter-mined by the fact that for large f\ values q and Sq/Sri accordingto equation (12) and equation (13) agree with each other. The recalcu-lation of the two constants yielded the following valuesP = 0.66̂ 2 7 = 0.828For comparison, L. Prandtl (reference 7) gives the following valuescalculated by Blasius and Toepfer which read, converted to the abovedesignationsP = 2 x 0.332 = 0.66*1 X7 = 2V5T0.231 = 0.819MCA TM 1336F. Riegels and J. A. Zaat give in a new report (reference 8) for ythe following value7 = " = 0.857The function q. with first and second derivative has been tabulatedand plotted in numerical table 1 and figure 2 .. ASYMPTOTIC BEHAVIOR OF THE SUCTION BOUNDARY LAYERFor calculation of the asymptotic behavior of the suction boundarylayer, we divide the function q^ defined by equation (8) into twopartsV +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. Fromequation (13) one then obtains= 7numerical table has been calculated with the valuesand 7 = 0-.819.= 0.66*4-NACA TM 1336Numerical Table 1. Blasius Boundary Layern0.1.2.3.4.5.6.7.8.91.0l.l1.21.31.41.51.61.71.81.92.02.53-03-54.04.55-0q(n)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• 9997q'Ol)o. 664o.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<I"(T])0.12751.17750.2114.2369.2561.2700.2796.2856. 2883.2883.2857.2809.2743.2660.2563.2457.2340 '.2248.2088.1955.1303.0774.0381.0169.0066.0022\/S)0.2570.3620.4404.5046.5599.6079. 6502.6842.7212.7523.7798.8043.8265.8466.8645.8807.8954.9084.9200.9304.9671.9857.9940.9981• 9994.9998Therein i|r = 0 forms the nev wall streamline and Q̂ the .suctionquantity. The second part CL,O then must be chosen so that 'the bound-ary condition q^ = qw(0,x) is satisfied. If the asymptotic rela-tion q ~ 1 - a would rigorously apply in the entire domain of theboundary layer, there would have to be at the wall qw(0,x) = 1, becauseof q = 0; however, the asymptotic solution deviates from the rigoroussolution if it is continued up to the wall. Therefore q (0,x) is anunknown function regarding which we merely make the assumption that itdoes not become infinite. As initial condition for the part q one8 NACA TM 1336further has q_(\]r,XQ) = 0 since q,(i|r,XQ) already satisfies theinitial condition= 7 1 -which insures the connection with the Blasius solution. The contri-bution q to the solution also must obey the differential equa-tion (9). In the solution (equation (10)) the first integral is elimi-nated, because of 0.̂2 (̂ >xo) ~ °> whereas in the second integral onehas to put- qwl(0,x) = - 7 1 -so that the asymptotic solution reads-7 1 - $By partial integration one obtains with consideration of the asymptoticbehavior of the error integral (by W. Wuest, elsewhere)1 - 1-0- x0)/Because of the connection with the Blasius solution, however,q (0,XQ} = 7, if the asymptotic solution-is continued up to the wall,so that one finally obtains as 'the asymptotic solution for the suctionboundary layer+ 7$ IroW4vuixo/1 -- XQ)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 asymptoticbehavior of the latter according to equation (8) and equation (13) intoconsiderationq ~ q 1 - <b/Vu,(15)In this formula q-n represents the Blasius solution. The last form ofthe solution proves to be particularly expedient for the further con-siderations.5. APPROXIMATE SOLUTION FOR THE SUCTION BOUNDARY LAYERIt suggests itself to generalize the asymptotic solution which isvalid for. large values of \Jr in the following manner(16)Due to q = 0 for \|r = 0 and'because of equation (15) the function F(ijr,x) must satisfy the following conditions , "F(0,x) = q. (17)It was hoped at first that one could choose for F, as in the nonsteadyanalogue by W. Wuest, elsewhere correspondingly an exponential func-tion as the simplest formulation; besides equation (l?) the~~d"isappear-ance of the second derivative of q at the wall would be added as afurther condition; however, it was shown that such a formulation doesnot meet with success and even, in a certain domain, does not yield'anysolution at all. . . . > . .. • . ...10 NACA TM 1336For the further calculation ve introduce the following simplifiednotation ,.yvu-^x |/vu-j(x - x0) I/VU-LXso that the .solution (equation (l6)) reads<1 = q.B(T)) - FJjl - 1B(V))| (19):According to a suggestion by A. Betz, ve equate as first approxi-mation q. the function F to the value dependent only on xF0(x) = P(0,x).= %-F= = ̂O] (20)v̂ /at the vail. " Thus the first approximation reads(21)This formulation does not fulfill the boundary- layer equation (i|) exactly.In particular, the second derivative of q-, at the vail does not disap-pear; hovever, the dependency on the second derivative of the stabilityof the velocity profile is of a very sensitive nature so that one has tolook for a more accurate solution. By substitution of the approximatesolution '(equation (21).) into the boundary- layer equation (4), oneobtains• o / o ' -Hence there results . o" e-,/^1 as the error of this first approxi-mation. By subtraction of the exact solution in which F standsfor FQ and q for q , vhile e-j_ disappears, one then obtainsNA.CA TM 1336 11where>" = 2(x\is an unknovn function. The quantity Sp11 disappears for TJ' = 0and rif = °°. By integration of equation (22) one obtains(F - FQ)[l - qB(n')] = <?! + e2 (23)Therein e^ is to be determined graphically or numerically by repeatedquadratureOne may determine the asymptotic behavior of eg by substituting in theabove definition of 62" for ./qT and \/q. the asymptoticvalues ./qT ~ 1 - i- q, and ^q ~ 1 - i qw. Thereby one obtainsHence follows with use of equations (9)> (19)> (21), and repeated inte-(25)gration with respect to TJ' = i l r A v u (x -€ 2 - 7 ( ^ 0 0 = F012 MCA TM 1336As before, $ denotes the error integral. Generally we visualizeas represented in the following manner(26)By way of approximation we limit ourselves to the first two terms,with a-^ = 7(Fm - FQ) and a^ determined by the fact that q mustdisappear at the wall. We determine accordingly the function Fapproximately to beF = F0- qB(n')7(Foo - a Ji-a'OiMJJ(2?)= -6(0,x) - - F (28)Calculation example.- T|r_//Vu"7xI = 0.125 was selected as numericalexample; F was calculated for the values X/XQ = 1.23̂ , 1.562, 4.3̂ ,and 9-78 and plotted in figure 3- For X/XQ = 1-562 the error wasdetermined by substitution of the approximated solution into theboundary- layer equation, and compared with the first approximationaccording 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 ofthe magnitude t| * = in/\/VulxO wi-L-'- l3e supplercented. By the degree ofsuction © we- here understand©.=5o*_2_.61*(29)8-]_* being the displacement thickness immediately ahead «of the suctionpoint and 82* immediately behind it. Therewith, © is given byNACA TM 1336 13*r dijr(30)oThe resulting values are tabulated in table 2 and plotted in figure 7-Table 2. Degree of SuctionV8.000.10.392'0.2•̂0.520o.k' 0.6710.60.7620.80.8241.00.8702.0• 0.97^The suction quantity ijr,. is, furthermore, given by the followingrelation '(3D6.' SUMMARYThe developmen%^of d laminar boundary layer behind a suction pointis investigated if̂ bĵ the suction merely the part of the boundary layernear the wall is "cut. off", with&it the slot exerting a sink effect.As basis of the calculation, we used the boundary-layer equation in theform indicated by Prandtl-Mises which is.closely related to the heatconduction equation oi\, respectively, to the differential equation ofthe nonsteady flowCSdiich i/iL independent of the coordinate x along thewall. With consideration of"̂ the*asymptotic behavior of the solution,'an approximate solution is developed which is similar in structure tothe solution of the nonsteady analogue which has been treated ,in anearlier report by W. Wuest,^elsewhere.\ Translated by Mary -L.MahlerNational Advisory Cesmmit^eefor Aeronautics14 NACA TM 1336REFERENCES1. Tollmien, W., and Mangier, W.: Stationare laminare Grenzschichten.Monogr. Fortschr. Luftfahrtforsch. Aerodyn. Vers.-Anst. Gottingen(AVA-Monogr.), 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- l80.3- Pfenniger, W.: Untersuchungen uber Reibungsverminderungen anTragflugeln, insbesondere mit Hilfe von Grenzschichtabsaugung.Mitt. Inst. Aerodyn. Tech. Hochschule Zurich Nr. 13,.1946.(Available as WACA TM Il8l.)4. Prandtl, L.: Note on the Calculation of Boundary Layers: Z. angew.Math. Mech., Vol. 18, 1938, p. 77-82. (Available as NACA TM 959-)5- Wuest, W.: Beitrag zur instationaren laminaren Grenzschicht anebenen Wa'nden. Ing.-Arch., Vol. 17, 19̂ 9, pp. 193-198.6. Tollmien, W.: Uber das Verhalten einer Stroraung langs einer.Wand amSusseren Rand ihrer Reibungs'schicht. Betz-Festschrift AVA-Gottingen,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 dieungestorte stromung. Nachr. Akad. Wiss. Gottingen. Math.-Phys.Kl. 19̂ 7, pp. 42-45.NACA TM 1336 15B• SuctionFigure 1.- Boundary-layer suction at the flat plate without sink effect.0.20 0.5 1.0 U5 2.0 2.5 3.0 3.5Figure 2.- The function q(i\) of the Blasius boundary layer with first andsecond derivative.16 NACA TM 13360.05 0.10Figure 3.- Auxiliary function F(n ' ,x) for calculation of the suctionboundary layer for W\|VU1X,Q = 0.125.0.02 h'Second 'approximationFirstapproximation-0.060 0.5 1,0 1.5 2.0 2.5Figure 4.- Error of the first and second approximation for x/x« = 1.562.NACA TM 1336 17Figure 5.- Velocity profiles of the suction boundary layer for*0/\l VU1X0 = 0-125 and various distances from the suctionpoint.0.050 0.100 0.150Figure 6. - Second derivative of the velocity profiles of the suctionboundary layer for t /vlvujx = 0.125.18 NACA TM 1336in0.8OJ60.40.2Degree of suction ®//fXX ^^-^, — • —*i«— ̂^M^̂ ™* \^0°"^TO ^0 0.2 0.4 0.6 0.8 1.0 1.2 1.4Figure 7.- Degree of suction (ratio of the cross-hatched and the totalshaded area in fig. 1).NACA-Langley - 3-19-52 - 1000

Entwicklung einer laminaren Grenzschicht hinter einer Absaugestelle

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Entwicklung einer laminaren Grenzschicht hinter einer Absaugestelle

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