(cA-tt^^Um NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM No. 1121 THEORETICAL INVESTIGATION OF DRAG REDUCTION IN MAINTAINING THE LAMINAR BOUNDARY LAYER BY SUCTION By A. Ulrich Translation "Theoretische Untersuchungen liber die Widerstandsersparnis durch Laminarhaltung mit Absaugung" Aerod3mamisches Institut der Technischen Hochschule Braunschweig Bericht Nr. 44/8 '^QE^ Washington . UNIVERSITY OF FLORIDA J une iy4 1 poCUMENTS DEPARTMENT 120 MARSTON SCIENCE UBRARy RO. BOX 11 7011 GaMNFSVILLE. FL 32611-7011 U£ \00 173 377 5i^o-r-j_ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1121 THEORETICAL IDIVESTIGATION OF DRAG REDUCTION BY MAINTAINING THE LAMINAR BOUNDARY LAYER BY SUCTIOK"'-- By A. Ulrlch ABSTRACT Maintenance of a laminar 'boiindary layer by suction was suggested, recently to decrease the friction drag of an iiTiniersed "body^ in particular an airfoil section jl] . The present treatise makes a theoretical contribution to this question in whichj for several cases of suction and blowing^ the stability of the laminar velocity profile is investigated. Estimates of the minimum suction quantities for maintaining the laminar boundary layer and estimates of drag reduction are thereby obtained. OUTLINE I, Statem.ent of the Problem II . Symbols III . Examination of the Stability of Laminar Velocity Profiles (a) The flat plate with suction and blowing Vq(x) 7& 1//X (b) The flat plate with uniform suction (c) The flat plate with an impinging jet IV. Stability Calculations V. Application of the Results to Drag Reduction by Maintaini ng the Laminar Boxindary Layer ^^^^Theoretlsche Untersuchungen iiber die Wider standser- sparnis durch Laminarhaltung mit Absaugung." Aerodynamisches Institut der Technischen Hochschule Braionschweig, Berlcht Nr. hk/Q, March 20, 19^[4. NACA TM llo. 1121 VI. Measurements of the Velocity Distribution in the Larainar Boundary Layer v/ith Suction VII. Summary VIII. Bibliography I. STATEMEi\TT OP TKE PROBLEM Recent investigations established the fact that the drag on a v.'lng may be reduced by maintaining a larranar boundary lajer. The first method to obtain a large region of laminar flow is to select a profile which has the minimimi pressure position as far back as possible. Suction of the bomidar^/- layer as indicated by Betz [l] is another means to move the transition point from laminar to turbulent flow as far back as possible. The present investigation makes a theoretical contributloii to the problem of transition la:ninar /turbulent in the boundary layer under suction and blowing conditions. To this end an examination of the stability of the lai'iiinar bo^"iJldary layer with suction v/as made. Suction always has a stabilizing effect on the lam.inar layor^ that is, the transition point is m.oved downstream^ whereas blov/ing has destabilizing effect. The stabilizing effect of the suction results from; first, a reduction of the boundary lay^r thiclcness (a thin boundary layer is less inclined to become turbulent than a thicker one, other conditions being equal); and second, ch.anges in the shape of the laminar velocity distribution and therefore ? an increase in the critical Reynolds number of the boundary layer {l]d''''/v) . (U = velocity outside crib the boundary layer, 5"" = displacem.ent thickness, see chapter 11^ v - kinematic viscosity). In both cases there" is an analogy to the influence of a negative pressure gradient on the boundary layer*. Theoretical calculation of the transition laminar /turbulent must be based on l.nov;ledge of the la::iinar velocity distribution and requires considerable accuracy, H, Schlichting and K. Bussmaixa [j] and R, Iglisch CI}.] gave exact solutions of the boundary layer \'ith suction and blowing and these solutions are suitable for an NACa TM No, 1121 exa::!iin.2.tion of stability, , Th© rollov.nng ci.ses were investigated: (1) The boundary layer on the flat plate in longitudinal Tlow with suction and bl.owinp' distributed accordinf; to "^'■q(x) ~ l/\^~ (" = distance along the plate. ) (2) The boundary layer on the j?lat plate in longitudinal flow with unif om suction, v = const. , starting at the front edge of the plate. (3) The boundary-layer for a flat plats with an ImpTngjfhg jet v/ith uniform suction or unifor:-" blowing. Only the results of rai investigation of the stability of a laminar boundary layer on a flat plate in longitudinal flow and 7/ith aniforiii suction are at present available. For this case, the thickness of the boundary layer is constant at a large distance from the leading edge of the plate: 5''' = -^ (1) The velocity distribution of this asymptotic suction profile is dependent un y only, and is, according to H. Schlichtlng [5] > as follows: _ 1 - e ; v(x,y) = Vq = const. (2) 'o The critical Reynolds number for this asymptotic suction profile is according to K. Bussmann and H. Munz [2] (iT^eV-Ci)^,.^^ ~ yOpOO. Since for the plats in longitudinal flow without suction (U^5''"'/v)) ,, = 5755 the suction O ^ CPlt - 1 ^ ^ in this case increases the critical Reynolds number by a factor of about 100. Prom the known Reynolds number y/lth suction, the minimum suction quantity necessary for maintaining the laminar boundar7/' layer can be determined imiaediately (since S'"- <. 5'-"^^.^) NACA TM No. 1121 Then from equ'^ticn (1) -v ^5'''''/'o - 1, the minimum suction quantity is ; c ., = — > -^—7 - O.ll^- X 10"^- (3) q ci'lt Uq - 70000 ^ ^-^'' Since the quantity is smGl.1 the maintenance of a laminar bounda.ry l%er by suction seejus rather v>romislng; thez^efore fur-ther investigations of the stability for the boundary layer with suction were carried out in this repcrt. Th.e mininium suctlca quantity for maintaining the laminar bc^ondarv' layer and the dru^' reducti^jn shall be established. II. SYIvKGLS ■, y rectangular coordinates parallel and perpendicular to the wall; X = y = at the leading ede,e of the plate, or the stag- nation point (figs. 1, \f and 7) I lenp'th of plate b width cf olr^te U(x) potentir.l flov/ outside of the boundary layer; IJ =■ U]_x for the plane stsgn?^tlon flow Ti = Uq for ""'the flat plate in lonr-iltudinal flow Uq froe-stroain velocity u, V components of velocity In the boundary layer pai-allel and perpen- dicular to the wall, respectively NACA TM No. 1121 Vq(x) prescribed normal velocity at the m'all; Vq > blowing, Vq < suction T shear stress at the v;all 5"' = / 1 1 - uj dy displaceuient thickness of the wO boundary layer ^ - I =j p. - Yf)dy moiaentuiii thickness of the *'0 ^ '"' boundary layer Q, total suction quantity and blowing quantity, respeotivelyj for the plate In longi- tudinal flow; ^ < " suction, Q > bloxidng Cq = nondimenslonal quantity rats ^ Ih X y^ of flow coefficient for the flat plate: for the plate Vu-lth uniform suction this coefficient becomes -v^ (Vq =: const) ; suction, c- < blowing Po^ C = Cq \/ reduced flow coefficient for "^ V " the flat plate with the =5 = U °Q> blowing suction distributed as V (x) - 1/v^ = r'.^':- ^-^0^ Q, ^ — n~ nondimenslonal extent of laminar flow for the plate flow Kith uniform suction Q z= — — reduced flow coefficient for '0 % the plane stagnation floY; friction drag coefflcieiit for ]3 7, P U 2 the flat plate in longi- 2 ° tudinal flow (plate wetted on oiie side) 6 MGA Tlvt No. 1121 III. EXAFINATTON OF THE STABILITY OF LAMIITAR VELOCITY PROFILES (a) Flat Plate with suction and blowing; according to Vq k l/v^- The first series of the investigated velocity pro- files concerns the flat plate in longlbudinal flow with continuous suction where the suction velocity is distribu- ted according to v^ ~ l/Vx (fig. 1). Schlichting and Bussmann [^] gave exact solutions of the differential equations for the boundary/ layer with suction and blowing for this case. It is a characteristic of tnis case that each velocity -profile along the plate is related to a prescribed mass flow coefficient. The reduced flow coefficient appears decisive. Herein c = - o/l b u stands for the ordinary flow coefficient for the olate of the length 'i and the width b, and Re = Uol/v for the Reynolds number of the plate. positive flow coefficients correspond to suction, negative ones to blowing. For this case all the blowing profiles have a point of inflection. The velocity 'orofiles for the flow coefficients 11 5 C = -j-; — ; 1 and ^,that 1j , oneblowingproXile with a point of inflection and three suction profiles were selected for the investigations of stability. These orofiles together v/ith the Tjx-.ofile for C = (flat plate v/ithout suction) are given in terns of y/6'"" in figure 2. The second derivatives of these velocity distributions, which are essential for the calculation of stability, are NAG A TM NO, 1121 Y drawn in fj.gure 3« Table 5 shows the connection between 5""' and tlie region of the laminar flow x. (b) Plat nlate with uniform suction: vo = const. The second series of velocity profiles along the plate are for the case of uniform suction (v^ = constant) these profiles virere calculated exactly by Igllsch [Lj.] . Figure li- shows the flow along the flat ..'late with uniform suction which was investigated by Iglisch, and figure 5 represents the examined velocity profiles. The velocity profiles with increasing ^ ~ KIT ~F" = °n- ~r ih) starting at the form of the Blasius oroflle at g - 0, gradually approach the asymptotic suction "orofile according to equation (2) C5J> The second dorivatives of these velocity profiles are drawn in figure 6. All the velocit7^ iDi'ofiles have negative curvature throughout (62iiyc)y2< 0) . V/ith increasing ^ the absolute values of 6 --ytj'- increase rapidly. Figure 7 ^-^d. table 7 represent the nondinunJional parameters of the boundary/ layer, namely the displacement thickness -v &''''/v> m.omentum -■■V •S^ - thickness — °_ shear stress T^5""'VU'Ur, and shar.e _,. ^ , O ' ' o parameter 6'''/^' as functions of ,-\/F according to Iglisch's calculations [J-L] . Six profiles with the parameters g^ = 0.005; 0.02; 0.08; 0.l3: O.52 and 0,5 were, examined with i-^espect to stability. The results for £^ = and f = " are known from [2], ( c ) Flat -plate v/ith an Impinging jet vfith uniform suction . As a third series several stagnation i^olnt profiles from_ those calculated exactly by Schllchting and Eussmann [5] were examined, with respect to stability. The potential flow is in this case (oompai'u fig". 7) lT{x) = u^ x; V(y) = -u^' y 8 NACA TM No. 1121 Figure 8 shows this plane stagnation flow. Figure 9 sho?/s the velocity nrofiles which were investigated. Their shape changes with the flow coefficient -V Co--zr~ (5) Positive values of C correspond to suction and negative values of Cq to blovifing. The profile with Cq = is tliemenz ' profile of the boundai^y layer; this profile results from the first term in the power series starting at the stagnation looint for the boundary layer of the circular cylinder. The velocity profiles v\fith the flow coefficients Cq = -3,1905; -1.1981 0; 0.5; 1.095 and 1.9265, that is, two blowing profiles, the profile with an impermeable wall Cq = and three suction profiles v/ere selected for the stability investigations. Figure 10 shovifs the second derivatives of these velocity profiles which have negative values throughout. Table 7 shows the corresponding boundary-layer 'jara-neters . IV. STABILITY CALCULATIONS The examination of stability for these laminar velocit77 profiles was carried out according to the m-ethod of small vibrations in the same v/ay as previously given in detail by /i/. Tollmien [10] and H. Schlichting [b] . A plane disturbance motion in the form of a wave motion progressing in the direction of the flew Is superimposed over the basic flow. It is essential that the basic flovir U(y) be assumed dependent on the coordinate y only. Then the amplitude distribution of the basic flow al; is a function of y only. The equation ; o (U' = -^, V' =r - -i 5y 5x ^t^(x,y) = ©(y) ei^(^ - ^^^ (6) NACA TM No. 1121 Is valid for the flo'v fixnotion \i/(x,y) of the supeririposed disturbance motion (u*, v'), a is real and gives the wave length ?\, of the disturbance motion, X = 2Tr/a. c = o-^ + icj_ 'is complex; Cp represents the wave x'-elocity of transmission and Cf_, positive or negative, the excitation or daraping, respectively. cp(y) = cpp(y) + icr (y) gives the complex amplitude function. The ordinary linear differential equation of the fourth order (U - c) (-" - a2.T,) - Tj"cp - — (cp'"' - 2a2rp" + a'+co) (?) with the boundary conditions y = 0: cp := CO' = Oj y = cx^; (? - cpi = (8) is ob.talned lor that function frv?m Na^''ier-Stckes' equations In equation (7) all values are rendered nondimensional with a suitable boundary- layer thickness and the potential velccitjr U. R = I15/y stands for the Reynolds nurnberj ' signifies differentiation vrlth respect to y/e. The boundary'" -layer . conditions' x^e suit from the disappearance of the normal and tangential components of the dlstrtrbance velocity at the ?;all and outside of the boundary -layer (y = <») . 'The examination of stability of the prescribed basic flow ITly) is a characteristic value Tjroblem of that differential equation in the following sense, since U(y) the v/av.^ length X ~ Zrr/a and the Reynolds number US/i; are prescribed. The complex characteristic value c = Cr + ici_ is required for evsry two corresponding values a, R; from the real part of c results the velocity of transmission of the superimiDosed disturbance ° the imaginar;/ part of c is decisive for the stability. Disturbance occurring for the condition of neutral stabj.lity (cj_ = 0) are especially interesting, and lie on a curve (neutral stability curve) in the a, d plane. The neutral stability curve separates the stable disturbances from the unstable ones.Olee figs. 14 to l(ii )The tangent to the neutral stability curve parallel to the a-axis gives the smallest Reynolds number at which a neutrally stable disturbance is still possible. This number is the critical Re-nuinber of the basic flow. 10 NAGA Tlvi No. 1121 In examining the stability of the suction profiles, the boundary conditions of equation (8) were held the same as in the case of the Impermeable wall; that is, dlsaDpearance of the normal and tangential disturbaiice velocities at the wall Is required for the boundary layer vvlth suction also, although the normal component of the basic flow at the wall is different from zero. The nujiierlcal solution of the characteristic value problem then takes exactly the same course as indicated by W. Tolliiilen [10] and I:. Schlichting [6j and needs, therefore, no further explanation here. For the stability calculation, the velocity profiles are approxlmatsd by parabolas In the form.: -=l-p(a-y/5) (q) with 7. 5^ The constants p, a, and n for the three investigated series of profiles are enumerated in table 1. The closest agreem.ent v/ith the exact velocity profiles near the wall vi^as Importarit. (Pigs, 2, S, and 8.) The polar diagrams for the examined velocity profiles are given in figures 11 to 15 as an interm-ediate result of the stability calcu- tatlon. The neutral stability curves v;ere obtained from these diap-rams and aS""' Is plotted against UqC'"'/^ In figures lit to 16 and corresponding values are tabulated in tables 2 to I4.. These curves show that the stability is greatly increased by suction while blowing decreases it. Figure llf shows in detail that in the plate flow with continuous suction the neutral stability curves for positive flow coefficients C He between those for the flat plate ?jithout suction and those for the asymptotic suction profile, which were calculated previously by Bussm^ann and Tvliinz [2] . The case of blo¥/lng with C = - r- demonstrates clearly the enlargement of the region of instability. NACA TM Wc. 1121 11 As to th3 flow along the plate, figure I5 shov/s that the neutral stability curves for the values of g used here also lie between the curves for c, = (flat plate without suction) and £, = co (asymptotic suction profile). For Increasing g, the region of instability diminishes and at g = 0.5 the neutral stability curve appears to apr)roach"the curve of the asymptotic suction profile. The stability in the stagnation flov/ also is greatly Increased by suction; with increasing flovir coefficient Cq the neutral stability curves (fig. 16) approach the curve of the asymptotic suction profile for the flat ■nlate. It is of interest that the neutral stability curves for the investigated cases of blowing still lie inside the curve for the impermeable flat plate. The critical Reynolds nmnber (UoS'-'/u ) crit ^^ the tangent parallel to the ordinate of the netxtral stability curves. The critical Reynolds numbers for the three series of stability examinations are enumerated In t ab le 5 • The following detailed result was obtained for the plate flow with continuous suction (vo ~ 1/V3i): (Uo&V'o)crit - 20J4 for the blowing profile 'C =-7- \ ^^^'^ therefore is far below the critical Reynolds number 575 i"o^ "the impermeable flat olate. Vifith suction the Reynolds number increases rapidly and reaches for G ~ — the value of 19,100, thus evidently 2 a^oproaching the critical Reynolds number "JOpOO for the asymptotic suction ■•Drofile determined by Bussmann- Munz [2], Figure I7 shows how the transition point on the Dlate is shifted backward with increasing suction. The Reynolds n^'Uinbers formed by the displacement thickness are, for different flow coefficients' C, plotted against the Re numbers formed from the distance (x) measured along the relate in this figure I7 and also figure l3, and values are tabulated in table 5« The stability limit of the impermeable wall (Uox/u ) ^j-^it = 1.1 x 105; but for a flovif coefficient C = 1 critical Re exceeds 10? 12 NACA TM No. '1121 and therefore readies the Re-number region of today's large and fast airplanes. Figure I9 shov/s the dependency of the critical Reynolds n^amber on the shape parameter 5"-''/6 . Flow along the plate with uniform suction (v^^^ = const,)^ the critical Reynolds -numbers for different \^ compiled in table 5 lie betvreen the critical Re/ynolds numbers 575 for the im-permeable wall end 70pOO for the asymptotic suction profile. Figure 20 shovv's the result of the stab-illty calculation in whicli the critical Reynolds nunbers (Uo^'"'/''-') -r.-lt ''^■^'^ represented as functions of the nondlmensional . flow distance v'ci. In figure 21 the onset of instability is ascertained. Here both the stability limit (Up,5'"/u) ,^ according to figure 20 and the nondlmsnslonal boundary -la^^er thlctaess Uo^"'"'y"U plotted against a/c, for dlffei-ent flov/ coefficients Cq - - Vq/Tjq are shown. The boundary -layer thickness is obtained from -ir- = -V-3^ (10) the values of -v 5'""/''^ '"^s functions of \/^ are given in figure 7* Since for c,-^^^; -V(_^o'""/u - 1> the separate curves h9.ve the asymptotes C?, = 03 . 1 ^^^ I -vo The point of transition is given by the Intersection of a curve UqS'"'/'^ with the stability limit (U(-,6'"'/u ) crit ' 'O s Cn I 1 = 5.16 X 10"^ Iq = 1 means Cn i/~|- = 1^ thatis,with (1^1/1^ = lO''. NACA TM No. 1121 13 The onset of Instability occvirs before Vl = 0« 1 for V T n T the flow coefficients -^ = j^^^' W;^' iB^iS^' on the other hand, there is no intersection for "Uq > 8500' Plow coefficients Co .4. = -^ = 1.18 X 10"^ ^rit - 8500 (12) are, therefore, s-gfficient for mPintaining the laminar boundary layer for the entire preliminary laminar flow region. This value Is to be compared with the value °Qcrlt ~ r.Q QQo = O.lU X 10"^ determined by Bussmann and Munz (reference 2) for the asympototic suction profile. The minimum flow coefficient necessary for maintaining the laminar boundary layer is, therefore, increased by about the factor 10, if the preliminary laminar flow region is taken into consideration. The earlier investiga- tion had already led to this pres\Miption. The minimum • suction quantity^ found herewith is Cn = 1. I8 x 10"^; ^crit p " ' ' """ One could consider the possibility of reducing the total suction quantity still further than Cq . . = 1/8500, One would have to select such a distribution of -Vq(x) as to make the curve Uq5'--/u in figure 20 remain everwhere just underneath the stability limit (Uq5---/i3) .. . The necessary distri- bution [-v^(x)/Uol is given to a first approximation u '-'-'crit by the intersections of the curves 11^5-"-/^ for different -Vq/Uq with the stability limit (fig. 21). One then obtains up to about VI = 0« 1 ^^^ increasing local flow coefficient [-Vn(x)/Up,] .^; the maximum is' reached u / o-'crit _ with 1/8500 at about VI = 0.1; for higher y/g there is again a decrease down to the constant asymptote (-Vq/Uq)^ = 1/70000, The total suction quantity, however, would hardly be reduced under the constant value CQ j^ = 1/8500 for practical purposes if such an "optimtjm" nonuniform distribution of suction were selected: For a plate with the Reynolds number UqI/u =10' and ill NACA TM No. 1121 this quantity is still so small that the maintaining of the laj:;iinar boundary layer by suction appears quite promising. We I'liention, for comparison v/ith experimental results, that the necessary suction quantities in Holstein's [7] measurements on the supporting wing are cq = 1,1 X 10~^ to 2,8 X 10" ■'. This value is, hov/ever, not exactly comparable to the thoeretical one since the suction in the measiiremients v/as produced thorough slots. Figure I9 represents the critical Re-number (U 5-;;-/u )^^^^ as a function of the shape paraiueter 5-"-/^f Simultaneously, the results of the stability calc\:lat:.cn for the velocity profiles of the flat plate v/ith continuous suction v„ - I/a/x are di^'awn into this diagram,, One a^n see that the critical Reynolds numbers of phe tv;o stability calculations . lie on the sarae cui've. Hence it is concluded that the critical Reynolds number (UqO-:;-/-d )^^^^ is dependent on the shape parameter 5-::-/.^ only. Plane star;nation flov/ ,- The critical Reynolds nura^ber (U 5-;:-/u)^ for the im.permeable wall (C^ = 0) is 12,500; for the sxiction quaiitity (C^ = 1,9265) this figure increases to 58*000 (fig, 22), With blowing the critical Reyiaolds number decreases slowly; the value of 70? is reached at Cq = -5,1905 (table 5)» These critical Reynolds niAm.bers also are given as functions of 5---/^ in figure 19; one can sse that they take a course similar to the flow along the plate although they lie somewhat underneath this curve. -^o/^o = 10"^> \/^T= (—-^).\^^ = -O.5.; therefore the region where c--^ can be considerably smaller than 1/8500 Is still far beyond the end of the' plate. NACA TM No. 1121 15 V. APPLICATION OP THE RESULTS TO DRAG REDUCTION 3Y I/IAIITTAINING THE LAIIINAR BOUNDARY LAYER The drag coefficient c-f is plotted against the Reynolds number in figure 25 for the laminar and turbulent flow in the boiondary layer of the "flat plate v;ith continuous suction Vq ~ l^Sc. The drag- coefficient curves from reference [5] for different quantities of suction and blowing C also are shown in the figure [5] in this diagraxi; the coefficients Of increase with increasing suction quantities.^ Drag raay be reduced by suction in the region betv/een the curve for the larriinar flow of the flat plate (C = 0) and the fully turbulent curve, if the laminar boundary layer can be maintained there by suction. The result of the stability calculation given in figure I8, that is, the critical Reynolds number (Uox/u)^^^. ^ as a function of the mass coefficient C, v/as transferred to this diagram and yields the curve denoted "stability limit." This curve signifies that for conditions (C, Re) above this limit the suction quantity is sufficient to maintain the laminar boiondary layer at the respective Reynolds niimber. The drag reduction by maintaining the laminar boundary layer for different Re^Tiolds numbers can -be specified immediately by means of this diagrain. The minimum suction quantity Cq Q-^lt ~ ^crit/'^'^^ necessary for maintaining the laminar boundary la^^er is determined for a given Reynolds number; then the drag coefficient Cf for the full^?- turbulent and laminar flow vvlth suction is read off the ordinate. This calculation is carried out for the m.ost interesting Re^niolds niAmbers 6 8 from 2 X 10 to 10 in table 6. One can see that for instance for Re = 10' a drag reduction of more than 70 percent can be obtained. This statement, however, does not yet make allowance for the power required The frictional drag coefficients represent in the present case, of continuous suction the total drag, because there is no additional sink drag of the suction quantity since the 'parts sucked off spent their x-impulse fully in the boundary layer already. Compare Schlichting CS]. l6 NACA TM No. 1121 for the suction blower. But this power is not excessive, since only very small suction quantities 'are needed here. Pigiire 25 was concerned v/ith the plate f lov/ with continuous suction; in a similar way, in figi.ire 2l(. the larainar frictional drag coefricient c^ (determined according to Iglisch's [I].] calculations) for the plate flow with imif orra suction is plotted against the Reynolds number UqI/u with the flow coefficient cq = -Vq/Uq as parameter. For very small Reynolds numbers, all c^oi'-ves converge to the curve of the plate v;ithout suction, c-f becomes contant with the value C£>^ = -2vq/Uo ^°^'' high Reynolds nuxibers where the larger part of the plate lies within the region of the asymptotic solution virlth constant boundary layer. The curve 'Q, crit ~ ~ 1.18 X 10"^, naraeo. ■■s-caoij.i-cy Co p.r.T-f- = ~^ - 1.18 X 10"^, named "stability limit" -^o v\;as drawn into the diagram (fig. 2i|) as the result of the stability calculation. Figui''e 25 represents the sarae condition again; but, different i'rom figiire 2l\., Cf is given on the ordinate in ordinary scale. Both represen- tations shoY/ that for instance at Re = 10' a drag reduction of 80 percent of the fully turbulent frictional drag can be achieved. Figure 26 compares the drag reductions by maintaining the laminar boiondary layer and the critical suction quantities for the two cases uniform suction and v^ ~ l/v^« A comparison of the results obtained under the assumption of uniform suction Vq = constant with the results based on the suction rule Vq - l/v^ demon- strates the following: The critical suction quantity Cq crit ^°^ continuous suction Vq ~ 3/vx is variable with Re = UqI/u and is for all Re < 7 x lo''' larger than the suction quantity for uniform suction which is constant Cq crit = I.I8 x 10"4-. jhe drag reduction for uniform suction also is larger in the considered region of Reynolds nijmbers than for Vq ~ l/v^T; for instance, the drag reduction at Re =10' is 80 percent against 73 percent. Therefore, the uniform suction is • at any rate preferable to the suction with Vq ~ l/V^" for the v^-hole region 5 x 10^ < Re < 108 that is, the main region of interest for practical purposes. i^i^Cii TM No, 1121 17 At high Re-numbers (ever 10°) only the suction accordlni;- to the rule Vq{x) ^ l/v^ shov.'s smaller critical suction quantities and. higher d.i?ag reduction than uniform suction. Table 6 and figure 26 give a comparison of the critical suction quantities and the drag reductions for the two suction rules. . VI. ]'-iiEA3UREIGi;iTTS OF THE VELOCITY DISTRIBUTION IW THE LAMINAR BOUNDARY LAYER ."ITH -SUCTION Finally, a few results of experiments about the boundary layer with suction shall be siven. In figure 27 tv/o measured velocity distributions v/ith the asymptotic suction or of lie [5] and the Blasius profile for the plane plate in longitudinal flow without suction are compared. The first measurement vvas' carried out by Holstein [7] on a wing with the profile NACA 0012-6I|l| the measurement was taken at n/l = 0.9 ( I- = v^ing chord) of the wing center section on the suction side (a - 0'-'), with six suction slots of the suction side opened. The velocity distribution which 'was converted into the dis- placement thickness conforms rather well v/ith the asymptotic suction profile of the flat plate with uniform suction while differing greatly from the Blasius profile with imp erm.e able wall. The second measurement was taken by Ackeret [9] ; a suction channel with numerous narrow slots a short distance behind the suction length vi/as used. This velocity distribution also takes =^- course similar to the asymptotic suction nroflle. Therefore, the fev/ existing moasurem.ents show good agreement V'/ith the theory as Tto the form of laminar velocity distri?Jution vn'.th and without suction ^resoectivoly. VII. SUMKARY Stability calculations were carried out on thi'^ee series of exactly calculated velocity profiles for the laminar boundary layer with suction; (a) on the' flat plate with continuous suction according to the rule Vq ^ l/v^5 (b) on the flat plate vvith uniform suction 18 NAG A TM No, 1121 Vq - const*, (c) on the flat plato with an impinging jet v;ith uniform suction. It became obvious that the stability of the boundary layer is greatly increased by suction, on the other hand greatly reduced by blowing. Yl/hile the suction quantities are Increased slightly, the critical Reynolds number increases greatly and ap-proaches the value found by Eussmann-Mtlnz (UQ5'""/u)gpj_-(^ = 70pOO for the asymptotic suction oroflle. Then the mlnimuin suction quantities necessary for maintaining the laminar boundary Isiyer were determined; they were Cq = 1.1 x 10"^ to 2.3 x 10,"^- for the plate with continuous suction and l.lc! x 10"^ for the olate with uniform suction. The drag r^eduction obtained by maintaining the laminar bo'ondary layer at Re = lOi is 80 percent of the turbulent drag without suction. Translated toy Iviary L. Mahler Ns-tlonal Advisory Committee for Aeronautics NACA TM No. 1121 I9 VIII. BIBLIOGRAPHY 1, A. Betz; Besinf lussung der Reibungsschlcht und Hire ppakbische Ver-wertung, Schriften der Deutschen Akademie der Luf tf ahrtf orschung, Heft kS ( 191+2), • vgl, anch Jahrtauch der Deutschen Akedemit der Luftfahrtfors Chung 1959/i^.O, S. P.k^ . 2. K. Bussrnann u. H. Ivliinz : Die Stabilitat der lamlnaren Reibungsschlcht ruit Absaugung. Jahrbuch 19^4-2 der Deutschen Luftfahrtfors chung, 3. I 56. 5. ¥', Schlichting u. K. Bussrnann: Exakte Losungen fur die lamlnare Grenzschicht rilt Absaugung und Ausblasen. Schriften der Deutschen Akademie der Luftfahrtfors Chung, Bd. 7B, 19i+5, Heft 2. ij-* R. Igllsch: Sxakte Lbsungen fiir die lamlnare Grenzschicht an der langsangestrbmten ebenen Platte i:iit homogener Absaugung. Bericht 43/22 des Aerodyn. Ins tl tuts der T. H. Braunschweig. UM 2061. 5* H. Schlichting: Die Grenzschicht mit Absaugung und Ausblasen. Luftfahrtfors chung 19lt-2, S. 179. 6. H. Schlichting; Ueber die theoretlsche Berechnung der kritischen Reynoldsschen Zahl einer Reibungsschlcht in beschleunigter und verzogerter Strbmung, Jahrbuch 1914-0 der deutschen Luftfahrtfors chung, s. I 197. 7" H. Holstein: Messungen zur Laminarhaltung der Grenzschicht durch Absaugung an einem Tragflugel mit Profil NACA OOI2/6I4.. Bericht der Aerodynamlschen Versuchsanstalt Gottingen. P3 I65IJ., (191+2). 8, H. Schlichting: Die Beelnf lussung der Grenzschicht durch Absaugung und Ausblasen. Jahrbuch 19l4-5/l|i4- der Deutschen Akademie der Luf tf ahrtf orschung. .9. J. Ackeret, M. Ras , W. Pfenninger; Verhinderung des Turbulentwerdens einer Grenzschicht durch Absaugung. Naturwissenschaften Bd. 29 (I9I4.I), S. o22. 20 NACA TM Ne» 1121 10 , W. Tollmien: IJebsr die i;ntstehung der Turbulenz . Nachr. Ges. Wiss. Gottlngen, Math. ?hys . Klasse 1929. Wx.GA TM No. 1121 2.1 TABLE 1 a:°^roximtion of the velocity profiles for various c, # and go of the three investigated floi^s ACCORDIIIG TO EQUATION (9) u = 1 - f i-i) n c P a n Flat plate with -0.25 1.23 8 0.95 2 continuous suction - 1.000 1.015 2 Vq - l/v/x .5 1.000 1.00 ■ 2 1 .133 i.6|36 1.614.2 1, 4 1-5 .137 k Af p a n Plat plate with 1.000 1.015 2 uniform suction .005 .0707 1.068 .968 2 Vq = const. .02 .lu 1.0Il2 .q8o 2 .08 .2B,5 1.072 .96b 2 •i? .^2i| .1296 1,667 k .566 .1316 1.660 k •5 .707 .1335 1.651^ k CO CO .00251+ U.ip k plane stagnation flow ^O p a. n -3.1905 1.0530 0.96^9 .954 1.6112 c with uniform suction -1.198 .100 .1375 2 .5 .1375 .ill 06 1.6J4.2 4 1.095 1.633 h # 1.9265 .i-'i)!6 1.622 h 22 NAG A TM No. 1121 TABLE 2 rrOlTERIGAL VALUES OP THE NSUTRr.L STABILITY CURVES OP THE PLATE FLO.V WITH GONTIIMUOUS SUCTION v^ ~ l/y^c c -Ao ^Yye-"" a 5'"" X 10^ V "1 J- 1 0.20 0.337 0.092 3.350 k .25 .1^.83 .120 1.373 .563 .696 .20k .3ifi WP .308 c:i .ko .756 .337 l.li98 Asymptote .ko .1x5 Ms .23k .361^ .21^9 .^95 .l-;-5 .Sl|.6 .267 .217 1 ^ .Ll( .0 ,9 .365 .367 0.05 0.03.1^ .058 3837 .05 .oSLi .0165 1920 .10 .150 .075 550 .10 .150 J 032, 206 .15 223 .101' 105 .15 J 223 .051 k3.8 1 o c — — ' .312 .132 26.2 2 .20 .312 .075 13.7 .25 .390 .155 13.5, .25 .pS'p .101 6.1k .30 .[i-7'+ .171^ 5.30 .30 ,7 , 'k9o .If 1.2 5.35 .31 .170 3.06 0.05 0.069 C.O58 13900 .05 .065 .021 276k .10 .13'^ .071 709 .10 .13s .03^ .094 ?)iS 1 .15 .207 115 .15 .207 .05^ 53.2 .20 .2o0 .127 29.0 .20 .2S0 .OoO 18.92 .25 .357 .131 9.98 0.05 0.022 0.057 11300 .05 .022 .017 3770 3 .10 .Oldi .077 571 2 .10 .0IJ4. .035 305 .15 .067 .100 102 .15 .067 .05s 63.2 .20 .090 .Ilk 32.1 .20 .090 .036 22.5 NAGA TM No. 1121 23 TABLE 5 nuiierigal values of the neutx^al stability curves of the 0ng0min5 plate plov/ vvtth uniporl? suction r ^ ae'"^ U 5'"" -2~ X 10-5 '0 ■ 0.10 0.03b 118. ■ .20 .077 7.20 .20 'Iks 57.7 .25 .101 3.01 .25 .138 12.0 .?o .129 1.53 .30 .223 i^.6i .3?^5 .114-5 1.15 .52 5 .233 5.29 .55 .159 .893 .35 .252 2.07 . -375 .181 .736 .575 ,26k 1.42 .Uo .633 .L:.o .274 1.02 s .42. .239 .605 .i|.2. .273 .713 2k NAG A TM No, 1121 TABLE 3 - Continued NTOIERICAL VALUES OF THE NEUTRAL STABILITY CURVES OF THE ONCOMING PLATS FLO;'. YvITH UNIFORM SUCTION -'Continued 5? 1 ; c ^^0 6""" a5''- 0.005 0.10 0.055 0.076 650 .10 .055 .0365 151 .20 ,11'! .157 22.7 .20 .11I4. .085 7.9 .30 .175 .22):. 5.5 .50 .175 .ll-:-5 1.86 .55 .203 .257 1.51 .55 .196 1.16 .02 .10 .055 . 07k 778 .10 .055 193.5 .20 .103 .11:-^ 28.9 .20 .108 .074 12.8 .25 .158 .176 9.3^ .25 .l:;o .108 S.06 .50 .167 .200 5.85 .50 .167 .II1-5 2.59 ,08 .10 .050 .075 5'4-5 .10 .050 .03/-!- 20J+ .20 .100 .Ink 26.6 .20 .100 ,080 15.5 .25 .127 ,160 9. §7 .25 .127 .111 6.65 .275 .i1l05 .160 6.17 .275 • .iir05 .132 1^.67 .295 .151 .165 .1 5.95 NAG ATM No. 1121 25 TABL3 3 - Concluded NUM3RIC^.L V*\LTri]S O:^ Tli^I] NEUTRAL STABlLir: nURV i]S OF THE OU®riN'> PLATE' PLOV. ^'.Illi UNIKRft SIIGTIOK - CorjcladGCi £ C ^K 5-- ao'-^' /. 1 U - 1) 0.18 0.10 .10 .20 .20 .25 .25 o.ok? .096 .096 . 120 . 120 0.073 . 0i|-6 .125 .082 .158 . 105 b05.3 89.7 26.9 lo.^ 10/13 8.07 .52 G.05 .05 .15 .3.5 .20 on 0.0225 ,0225 .068 .063 .092 .092 C.037 .01[|.. .099 .056 .122 .089 12230 i|277 96.7 60.2 28.9 22.6 .5 0.05 .05 .15 .15 .20 .20 0.0. ":2 .022 .066 .066 ■ .088 .088 0.058 .017 .09k .056 .111 .08i|. IO6J4.O 3727 107.0 67.5 31.5 2)1.9 ■ CO 0.025 .025 .09 .09 .15 .15 .175 0.02S5 .02S^ .O9I4-3 .091^3 .16 5 .165 .192 0.0176 .0080 .0618 .032.6 .0935 .O0O5 .088 ^7,200 IO6I1OO ISOO 7,55 ll^7 113 70 26 NAG A TM No. 1121 TABLE k FCJMERICAL ".^ALTJSS 0? TKE. xIEUTRAL STABILITY CUP.VES OP THE PLANE STAGNATION FLOW WITK UNIFORM SUCTION ■' 1 c — a5" •'in-' Uq V -3.1905 C . 1") . 109 77.6 .1-3 05I'- 25.1 . 20 M 20.7 .20 079 8.6 .50 207 3.70 .50 l'-'4- 2.02 ♦ 53 -37 I.5S .3'9 200 1.12 -I.I9S .10 071 653 .10 .03!^ 213 .15 iC6 107 .13 .20 OS 3 ll^2 27. k .20 .079 1^.5 .25 .171 0.63 • 25 .111, .loS .05 "' 7" 7 i|].;7o .05 .016 Mio .10 .00^ 711 .10 . 0;U- 292 •15 .0-8 11!^ . . It 0^. c . 20 • --: ^1 .20 .0:32 18.1 .25 .123 ii+.8 .2^ .10 2 13.8 MCA TM No. 1121 27 TABLE i^. " Concluded HTOIERICAL VALUES 0? THE NEUTRAL STABILITY CURVES 0? THE PL.:HL ST_'.GIL.TICN FLOY/ WITH UNIFORM SUCTIOIT - Concluded ^0 c Uo a5'"' 0.5 , COS .05 0.0^2 .007 15500 il7i^-0 .10 .065 890 .10 .051 557 . .15 .09lj- 158.0 .15 .OS5 79.5 .26 .120 .20 .078 . -26.5 .22 .Ilk 22.7 .22 .101 , 19.1 1.095 .05 .029 25^!..95 .05 .015 7512 .10 .067 ,951 .10 .OS 2 ■ ]|18 .IS .093 157.6 .IS .OSS ■ 85.7 .26 .112 35.6 .20 .085 29.2 .215 . 106 2k. 1.9265 .OS .05 .10 .051 .oik 21190 3156 .Ob 5 J'-^k .10 .0S2 [^8.5 .15 .091 12)i.8 .15 .055 85.5 .175 .175 . lok .073 59.5 fo.7 .19 . 093 38.0 28 NAG A TM No, 1121 TABLE 5 TIU CRI^'IGAL REYNOLDS HUMBERS AS FUIvTCTIONS OP C, A^'D Go FOR THE TFR.^E INVESTICtATED PLOWS v^. c = ^^ JY 5- - V /crlt • u .-^crit Plat niate -0.25 2.77 20]i 1.0^ X 10^ io5 10° io7 10° with con- tinuous .5 i:^i 2986 1.10 X 5.25 X suction 1 2.29 9550 8.31 X Vq~ 1/-^ 1.5 2.22 19100 -^.90 X Asymptotic 2 76000 - suction profile -v^ /U X 5'"' /U 5'-^ Plat plate . 1 ' u 'exit 2.59 575 with uni- .0707 2.^3 ■ 1122 form suc- .ikl 2.1i7 1320 tion .28^ .ii2ll 2.39 5935 Vq - const. 2.51 7590 .5t>6 2.25 15500 .707 2.21 2 21900 70000 c - '"^ 5-- 1^0^ ''lane stag- Vuiu ^ V^ irlt -3.1905 2.533 ,79'' nation flov^ -1.198 2.352 10160 v'ith uni- 2. 218 12300 form .5 2.172 17560 suction 1.095 2.126 27700 1.9265 2.088 38000 NACA TM No. 1121 29 o EH P4 vO b CO O W o o M Eh O o -P rH LC\i-H Ln® hTN-O - t^t© CC. nO rH \0 KMOl"^ . <-',.l --.0 C--- t-- CO CO CO lO -P • •»••• • •••••• t— 1 <1h g hT- (--! PrH iH Ln LJ\LP\ 1=> X sO rf> t-' -^ 0"'-. ON f-U-\_r}rH ^_^:X> w • • * • « • • • • e * « WH 01 ',:\j oj C'i oj <-! CVI a; OJ OJ OJ rH rH p^l CI Ph ; fr. _.r^ .. - 1 M- Cu rH P~^ r! -H Q-N ^ LO IT: Eh -C^ f TN CO MJ .J K'., L.^^ rH ijj nO _:r |s->, n"-. i-H rH rH W i-:i isffl 1 1 ' , ro> n • ■■0 "3 r-i,a tH "a-; rH t^ ■ hTx ON ^- -d' CM -Ct 0;.) t^. ^D N-N rH P 13 X *•*••■ • ••*•*• M-+^ -L-'i t<\ ai cvi c\j 0.1 _:H 1^-,, ,,-^ r<-\ OJ C\J 'AJ -'-v Q-' 1? U ■< r-i H ) X !iA LfN ^_ CD ( -P CO ai '^0 KN rH C\J OJ C\t rH rH ^ ■-^ r-H - /■ > •H f-1 - - a o> w (T-) i— ^ -P L' N L-A p ■j-l -^ ir-, --o CO 0^ 1 1 1 1 i i 1 u 1 1 1 1 ; 1 1 h-i * * rH 1 1 1 i 1 1 1 Eh ^ sO vO f- t— C^CO sO vO sO {>- l>- C~~ CO 0000 0000000 r^ r-< r-i r-^ i-< r-i r-i r-t ,-{ r-* rH r-{ rM r-{ ■ P tu XX XX XX XX OJ LfN OJ LPv OJ in OJ Lf^ •D . C rH «1 ri^^ rH ^ r; f-L, ;!< rH" D ?-! .H .r' +■> ^ 3 -P ) -p ,e; f|-: +3 II oJ -P d U.i -P .H r-1 .H .H P rH .H fl p ti, fS -P ro t> c^, ;« p w i> •H 4J ;3 to (V ai -P -H •H ■P e crt u-rt I-H H ^ — %. ^M •H -P- ,_^ C 3 1 w • >. ,ij -p ^ •H 3 +5 -P C cri i-H P rH CT' P Sh ^ ^^ — ^ P Ih a c •rl V ^ iH •H II ■■■ V) 30 NAG A TM No. 1121 TABLE 7 THE CHARACTERISTIC BOUNDARY-LAYER PARAMETERS OP THE INVESTIGATED LAI.IINAR VELOCITY PROFILES WITH SUCTION ,.. K' ." /^'° - Z'^- 5'"' T^O-"- C - o^\J ^ "■'\l -.. 'V.. „^ ^^0 Plat plate -0.25 2.010 0.7!+0 2.77 0.500 with con- 1.721 .661). 2.59 2. 41 'W tinuous .5 ^^'j'jy :l^^. .682 suction 1 i.okl 2.29 .618 Vq ~ iM 1.5 .865 .590 2.22 ^i. Plat -olate 1 y -v^5-- -^^0- Q'"' ^^0 u V C 2.59 0.571 : with .0707 .Ilk 0.01^5 2.53 2^7 .607 uniform .1^1 .211 086 .d;>1 suction .212 .303 .125 2.1+5 .671 V = const. .283 .:;-'! .160 2.39 .699 .55if Vr50 .192 2.35 .726 .k2l^ .511 .221 2.31 .750 .i;95 .706 .21.8 2.28 .775 .566 .61k .273 2.25 •in .656 .65S .295 2.23 .707 .605 .515 2.21 .850 cc 1 .5 2 1 Plane stag- y/u-,lJ •if v^ laU -3.1905 1.95^ 1.018 0.772 2.5^4- 0.608 nation flO'vV -1.198 .4-55 2.55 .693 v; i th .61l8 .292 2.22 .796 unif orFi .5 .250 2.17 .856 suction 1.095 ,209 2.15 .368 i 1.9265 .5i|.9 .167 2.09 : .917 NACA TM No. 1121 Figs. 1,2 Vo(X]^-^f^ FigTire 1. Explanatory chart: Boundary layer at the plate in longitudinal flow with continuous suction according to the rule v^ (x) = - —2-^/ T^o Figure 2. Velocity distributions u/U^ against y/fj' for various flow coefficients C of the flow from Fig.l; WP = point of inflection. Conparison with the approximation of equation ( 9) . Fig. 3 NACA TM No. 1121 1 " Ua dy' Figure 3. The second derivatives of the velocity distributions from Fig. 2. (C= flow coefficient ) . NACA TM No. 1121 Fig. 4 a i? § s ■vi :3 bo C o C! ■H 0) P. 0) tM O n3 >5 ' a m •a c o PQ o o o it o > O CQ e O Cd 3 Fig. 5 NACA TM No. 1121 *' /A- ^ Figure 5. Velocity distributions u/U^ against y/^* for different 5' according to Iglisch [4j ; comparisons with the approximation of equation (9). I = ( -^o )2 UqX or NACA TM No. 1121 Fig. 6 Figure 6. The second derivatives of the velocity distributions of Fig. 5, according to Iglisch [4] . Figs. 7,8 NACA TM No. 1121 iO 0,8 OA 0.2 uUn \ ^ — — 1 Jpj^ ^ ~~~^ ^ ^' V Q= 1,927 Ni r '<>■■ Co' ■?«?7^ ^ -^ \ "S^/" ' ^— -ms^ ^ \^ X ^- 05^ \ ^ \^ 4J \ ^ ■^ 1 ^ ^< i198 / \, i N ^^^ J.f9f 5 jy_ 6 Figure 10. The second derivatives of the velocity distributions from Fig. 8 according to Schlichting-Bussmann [3] . NACA TM No. 1121 Fig. 11 . 1 C3- -4—^1 /'i ^7 // t cnj ^^ T^' lo CM C3- G *3 O ■H Lj +J o «3 tw rH 3 • — . O C) iH A (d ti O Ed 11 s >s ^ \ o "^ > :% x: +j ^ ■H 3 ra 3 o r fH o 5 ,-H U +J O Q-H a. iH >3-H . •h3 X3 1— ( •H nj 1 — 1 o ^J o w <1> <-\ u 0) ti-J a > o •H b^ Fig. 12 NACA TM No. 1121 In i Q T' 0) u o o g •H ■(J ■H 3 O rH cd o g II U II 3 S CO .^ e I u •^ o — J u fJ * (X •H CJ •-i 1— 1 >) •H ♦J XJ 0) •H crt u C) ♦J s o M •-H 0) W-l Ee. > O NACA TM No. 1121 Fig. 13 eg '■Jin 0) u o Cd II 3 f- \ .— .. 3 ^ Q C o ■H -P •^ rr (1) . ^ (U o X! .H ■fJ ii-i t^j c o o •H n ■fJ o n3 ■H +3 s> 3 01 M +j o CO m 0) d) c x: crt +3 M CX U o (m (1) x: ca ■H> m J • +J fO ■M l-l c; o a> rH ^ §! bo Fig. 14 NACA TM No. 1121 OA occT' 0.5 0.2 0.1 0^ 10' \ \ \ \ ^r* Sta ble "^ --^ ^ ^^ r \ y Jlate t suctl on 1 s wlthou 1 \ \ \ \ \ C=ir > \ 1 \ r X \ \ A \ \ /> N \ \\. \ \ /-- N \ N, ^. \ \ 1 >~^ \ \ asymptotic suction profile % \ N \ X S N, 1 1 ^ 1 ^ ^ ^ \ ■> ^ N \ II 575 1 c:> _ 19100/ :^ \ V N, ''^^ \ ^ V 1 \"*! ^ ^ 1 1 1 _ fO' 10^ 10' u6^ 10' V 10' Figure 14. Result of the stability calculation for the flat plate in longitudinal flow with Vp — l/y x. The neutral stability curves 0^6 against uS'/lrfor various flow coefficients C. NACA TM No. 1121 Fig. 15 Figure 15. Result of the stability calculation for the plate in longitudinal flow with uniform suction: The neutral stability curves c^S' a gains t U A /!/' for vari ous ! = 0: Plate without suction (Blasius) = co: Asymptotic suction profile. ^■■n^j Fig. 16 NACA TM No. 1121 10^ U6* fO^ >if Figure 16. Result of the stability calculation for the plaj^e stagnation flow with uniform suction: The neutral stability curvesotO against uS /V for various flow coefficients Co = 'V~^ lir NACA TM No. 1121 Fig. 17 0) Si 4J R g o r-l u J +J (U •H ■p r-1 cd •H rH 43 P- tri +J 0) CO ^ c ■tJ ■H tj Vj o O tM +J ■p c ■H •H U o o a. (U i^ Xi ^ ■p » 1>- rH f— 1 3 t) ^ (1) rH tJ l-l n5 -H :3 o 5 bo Fig. 18 NACA TM No. 1121 3 o x: c o U tM cd c •H T3 ♦J bo C o to u J2 I n3 o ■H +J •H Li O 0) x; o o c ■a c p. Q 00 0) Li -1-3 Cd 0) x; Li O 0) 11 c . 575 (plate wlthoi it suet Ion) 1 1 ;^ Plate with uniform suction k, = c o " " continuous " "o™ • , ex 1 V ^ Pla ne stag nation flow ^ X 22 Zh- 26 2,8 Figure 19. The critical Re-number (Uq3 ^^ 'crit ^^ ^ function of the shape parameter 6 / rj' for the plate in longitudinal flow with v^-^1 'Sp^ and Vq= const, and the plane stagnation flow. Fig. 20 NACA TM No. 1121 10- 5 2 10" 5 2 0' 70 000 (asyBiptotlc suction profile) y crit. 575 (plate without suction) 0.2 D.^t 0,6 0.8 Figure 20. The crit ical Re -number (U^o ^TA'crit ^^ ^ function of — ^^—V °^ -]l^ ^°^ ^^^ ^-"-^^ plate in longitudinal flow with uniform suction. NACA TM No. 1121 Fig. 21 10 ID 10' 1 1 1 1 1 1 1 -■ 70 000 (asraptotle suction proflla) 70001 J— I\f1 -^ ■^ ^-'' "^ ^^fl f / / / /^ 7 ^ 1 / / / jcnr^^ 1 "M L/ 20 000 / / /' — 1 1 / ''^>^ 1 10000'^^^ .X 1 1 / .^'/ -- ^-'- — "^S t--i^. "i 1 1 1 1 ■■■ / / / ^ 80L W 9000 \ 1 1 / V 1 / 1 / Vi / '^^crit" 8500 1.^^10'" f S- -75 (P i«t« irl thou t sue blon) 'i 0,2 0,¥ 0.6 0.8 1.0 Figure 21. Ascertaining of the critical flow coefficient Cq for main- taining the laminar flow for the plate in longitudinal flow with unifonn suction. Asymptotes: ( — —^ — — ' -V, CQ Fig. 22 NACA TM No. 1121 -J fU 6 1 — tU' -I ' ■'cnt I V ■—^ Tijti K=trt5? Jfj?^ ^ o ^ ^-^ //7^ ^^^ ^ F 5 <> «' Bl ovlng -O- 8ue tlon c= -Wb (/,« Figure 22. The critical Reynolds number ( Uo^ S / ji / 1 / ' / 1 c? / ^/// f 1 1 / / ' "5 1 //// Y fh ^ •o :§ CVJ K K> ^ lO og C3- CO 0) x; •!-> x; • +j 0) •H o 5 T1 J-1 3 CO c T3 o OJ ■H +J +J -tJ o OJ 3 s to II g o _• n CO Wa CJ 3 ■H > •H § CO cd .-H x: «X. (M CIJ -p w •H o s o •fj II 01 ^^ bo +J 3 c o m •H > r-t II -o 1 P. Ll <^-l O V> o O • . 1— 1 O ccl 8 ^ V-1 • 1 Q) i-H .-IICJ x; CO 1 ■tj CJ CO *"* r— 1 ^0 o (►J (C5 3 • •H (.-I bo CO r— 1 p QJ O CJ O rH +:> II 3 CJ 3 CO C O 5 CJ +J •H -H 0) XI O ce O -tJ CO bo w II Cm O 8 t o ^3 J-i O Cm g ■H ■(J o 3 CO x; •H E Fis. 26 NACA TM No. 1121 ^ ■<1 «? flff 0,7 0.6 OJS ■ Acf ft) fully £ *-rj tnrhul a-t rr. -= "~-^ ~^^ 1 ] rr^ _,^ !:>- ""-r-- — "^ * ^^^ ■^ __^ -— f '~"~^- ^ / ^A«^- ^.^-T^ -— •"' rx (ff / \ t^- CO lat X nlfora suction 10" 10' CQ -10 crit- Up I Re A Figure 26. The relative drag reduction and the minimum (Cf) fully turb. suction quantity necessary for maintaining the laminar boundary layer cq Qj.it ^^ ^ function of Re = UqI/it for the plate in longitudinal flow with uniform suction (v. 1/l/T Cf = (Cf ) fully turb. -(Cf) const ) and with laminar with suction NACA TM No. 1121 Fig. 27 1 ^ « ffl syaptotle suction profile theoretical easureaent on suction wing according to Holsteln easureaent on suction ehan according to Ackeret laslus without suction theoretlca \ l» * >5^ M • V 1 \^ t 1 ! 1 ! 1 1 a \ \ \ \ ^ \ \ V a « ■ m \\ ** o 8 BO tt ^ i m a « c m p. m 3 » ft a o ft \ \^ O o » /> • t^. < d h o o t • » «» ^■> k\^ \, B H M O h« a. a K ■ a — according to - Ackeret . >>: x: \ >j ^ \ ^ Ox ^1:^ ^^ ^ ^ ^ ><1- ^0 CM Q) u x: <\i -tj >. n1 c x: rH •H +j 0) ■H >1 *^ S u CO a --( DJ T-l o c X u 3 • •H o o g +J 42 +j M 3 XI l^ bo . •H g c O •H ♦J ■H ^'1 CO fc •H T3 ^ >, a) *J -U 1 ( ■H +j CO O CQ o c 0) +J ■-i •H C 0) 9^ ^ Ij > OT V ■H 3 ■P o ■tJ -Q C -tJ 0) ■H 0) t-l Lj E bo o +J CJ --1 0) ro ^ ■ H n3 -O ■!-> ■X3 p-H J-i a o 0) >1 CO CJ x: ■iJ • H CJ ■fj ■ H O cd • H CJ> o