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f 00 3 : ' '"  NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORAiDUM NO. 1121 THEORETICAL INVESTIGATION OF DRAG REDUCTION BY MAINTAINING THE LAMINAR BOUNDARY LAYER BY SUCTIOI;' By A. Ulrich ABSTRACT Maintenance of a laminar boundary layer by suction was suggested recently to decrease the friction drag of an immersed body, in particular an airfoil section 111. The present treatise makes a theoretical contribution to this Question in which, 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 tnhreby obtained. OUTLINE I. Statement of the Problem II. Symbols III. Examnation of the Stability of Laminar Velocity Profiles (a) The flat plate with suction and blowing vo(x) k 1/ (b) The flat plate with uniform suction (c) The flat plate with an impinging jet IV. Stability Calculations V. ApplicaL.on of the Results to Drag Reduction by Maintaining the Laminar Boundary Layer *"Theoretische Untersuchungen Uber die Widerstandser sparnis durch Lami.narhaltung mit Absaugung." Aerodynamisches Institute der Technischen Hochschule Braunschweig, Bericht Nr. 44/8, March 20, 1944. 2 NACA TM No. 1121 VI. Measurements of the Velocity Distribution in the Laminar Boundary Layer with Suction VII. Summary VIII. Bibliography I. STATEMENT OF TIE PRO7'LE.1 Recent investigations established the fact that the drag on a wing may be reduced by maintaining a la..inar boundary layer. The first method to obtain a large region of laminar flow is to select a profile which has the minimum pressure position as far back as possible. Suction of the boundary layer as indicated by Betz [1] is another means to move the transition point from laminar to turbulent flow as far back as possible. The present investigation makes a theoretical contribution to the problem of transition laninar/turbulent in the bouniay layer under suction and blowing conditions. To this end an examination of the stability of the laminar boundary layer with suction was made. Suction always has a stabilizing effect on the laminar layer, that is, the transition point is moved downstream; whereas blowing has destabilizing effect. The stabilizing effect of the suction results from: first, a reduction of the boundary layer thickness (a thin boundary layer is less inclined to become turbulent than a thicker one, other conditions being equal); and second, changes in the shape of the laminar velocity distribution and therefore: an increase in the critical Reynolds number of the boundary layer (U6*/'U)ri (U = velocity outside crit the boundary layer, 6" = displacement thickness, see chapter II, v = kinematic viscosity). In both cases there is an analogy to the influence of a negative pressure gradient on the boundary layer. Thoretical calculation of the transition laminar/turbulent must be based on Lnowledge of the laminar velocity distribution and requires considerable accuracy. H. Schlichting and K. Bssima~ [5] and R. Iglisch [4] gave exact solutions of the boundary layer with suction and blowing and these solutions are suitable for an IAGC, TM Ho. 1121 exCLAin_.ti.n of stc.bility. The follo.'inc c:.ses were invei3 t i ' C. d : (1) Th'Ie boundqr?' larrer on th.3 fl Ct plate in long. r i.:tn filon with .suction rndri b, owing distributed accordin to v (x) i//: (. = distance alon, the plato.) (2) The :.undar;, layer on the f.s t .late in Icnl itudin'i flw wit u.nifor : 1 C t :, : = c:,nst., starting ; t the front e~e of tle .iat e. () The b.iuniar,l yie r for aflEt l:.t3 with an ImpTngfng jet uithiI unjifo'ri! suc i on or uni fon or* .owi.: Cnii tihe r.S iits Cf i.n't .1 ti io n Tf stobi iit 7 of a. a in... b t,,iA.nd :'r la)r "ni fl t _l ,tr in lonitudi inal flow and n th uni for, ,i Ln are at .resent avai label , For this ca:s a, the thici n':n s oc the cound,liy layer i.. cons' L nt i 9 l 1 ~ 4iist nce fri:i t.:. '_le .din. n : ,d r'o f the pPr. 6 = L ( 1) prThe v lisety deistriutio n yf this as. ,pt otic suction profile is de,n.nent un y onn ,, at,. is, ,rc.:odlng to IH. chlic'hting ['] aT follows: ;Z I.y iJTh =T 10 ; v(:.:,'r t s= const. ) p'rofil, i'. : c cor,di. to K. Bu.srsminn and H. M'.LCn.z [2] (U6i. )corit = :.00. Since for the ..lst in lonituirinal low th "C uct i t I.., tn T ),., flow *'ithcrT suction (U'L o) = 5'i., the suction in this case inc v. se3 s the critical Reynolds nuimber bj, a factor of &bot .t 100. Fror.: the known Reynolds numibe;r with suction, the minimumr suction quantity n'cLssary rfcr miri ntsining tie laminsr boundary layer can be determined immediately (since C"I 5 i t crit NACA TM No. 1121 Uo0 I Uo vo "' Uo6   U vo crit Then from eqution (1) vo6*/i = 1, the minimum suction quantity is: crt o > 1 0.14 x 104 (3) q crit Uo 70,000 Since the quantity is small the maintenance of a laminar boundary layer by suction seeiis rather promising; therefore further investigations of the stability for the boundary layer with suction were carried out in this report. The minimilu suction quantity for maintaining the l.'r.Ii.r boundary layer and the cdru reduction. shall be ststb'lihei. II. S'"TOLS x, y rectanlr coordinates parallel and perpendicular to the wvall; x = y = 0 at the leading edge of the plate, or the stag nation point (figs. 1, 4, and 7) ]. len t of plate 'b width of plote U(x) potential flow outside of lthe boundary layer; 5 = ulx for the plane stagnation flow U = Uo .for the flat plate in longitudinal flc.' Uo freostrnam velocity u, v components of velocity in the boundary layer parallel and perpen dicular to the wall, respectively MACA TM No. 1121 V,(x) To 0 o* =/ (i 0  dy I CQ =  zb x U U'Tv o = 0 o\ 2 Uo : = 2 = KU " x "" V0 Co  cf = b L U o2 2 0 prescribed normal velocity at the wall; vo > 0 blowing, vo < 0 suction shesr stress at the rall displacement thickness of the boundary layer moiintuim thickness of the boundary layer total suction quantity and blowing quantity, respectively, for he late In longi tud.in.l flow; Q < 0 auction, Q > 0 blm~inPg nondimensional quantity rate of flow coefficient for th3 flat plate' f)r tle plate ..with uriforrm suction tiis coefficient becomes v c = (v = ccnst); 'I U o c 0 suction, c <: ' blowing reducE flow coefficient f.or the flat plate with the :;.tion distributed. as V (x) 1./, nondimensionol extent of laminar flow for the pl'ite flow ith uniform suctin reduced flow coeffici,=nt for the plane stagnation flov friction drag coefficient for the flat polte in longi tudinal flow (plate wetted on one side) NACA TN No. 1121 III. EXAMINATION OF THE STABILITY OF LAMINAR VELOCITY PROFILES (a) Flat plate with suction and blowing according to vo = 1/'. The first series of the investigated velocity pro files concerns the flat plate in longitudinal flow with continuous suction where the suction velocity is distribu ted according to vo 1/VI (fig. 1). Schlichting and Bussmann [5] gave exact solutions of the differential equations for the boundary layer with. suction and blowing for this case. It is a characteristic of this case that 6ach velocity profile along the plate is related to a prescribed mass flow coefficient. The reduced flow coefficient C = c i = 'e'~= 7, appears decisive. Herein c = O/Z b Uo stands for the ordinary flow coefficient for the plate of the length and the width b, and Re = Uot/u 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 profiles for the flow coefficients C = ; 1 and ,that i., on.eblowing roIfile with a nolnt of inflection and three suction profiles were selected for the investigations of stability. These profiles together with the )rofile for C = 0 (flat plate without suction) are iven in terms of y/'"' in figure 2. The second derivatives of these velocity distributions, which are essential for the calculation of stability, are HIACA TL. IO0. 1121 dres'n in figure Tsble 5 shows the connection between "6: and the r' mt:on of the lmIrrnar flow x. (b) Fl1t r,i3te with uniform suction: V = const. The' second osries of velocity rProfiles iaong LIe plate ;re i: o the c,,,se of uniL orL suctl.in (v = cons.: L ) these r: ';fil. s were csci.ulsatd ex;tly b'I I tlisch [ L . Figure lich '.va tr.e flow ?lon tI.h: flt lacx with uniform sue..t io, which wc irnv. t i gt Ili iaoh, and f iuAre 5 represents Lhe ei;:?::in': e!r cit. rrofiles. The vye' c ity p: f. 1les it c inc rea.r n, S'r x . , starting. at t.: oi f ror:,m :f the B3lsius o .file &at C0, graduially i ,.,prc. the a;,rrpCt tjo c suction ..,rofi l accord ingi to equs..:. in (2) [E5]. The second Jt. i;ti .ves of t.er"S velocity t r.,f ls ae dr:wn in fi .,ire D. A11 the velocity 7rcf.. les a&ve nzw'ative cIur',atre th.ouhout (62i' .'< 0). J1 'ih incrrazinp the .:soliut vs lae3 o 6 u/.7 ncro:.'c raidly. F'i.ur en t 7 r. t:~'~lr es'tnt the Iion:.li iional optP.net c f th, o n :;.r y lr i4er, na.miel t, di iisj cement rhici:nr Ts ' /, :c I t thickne.:s 0 shear str:. T, /iU and sh~:' E parnrteter F'/ .: functions o't ,' c ccrdin;. t Ili ch's c ld1.' I : .s L)] Six r file with' t he pa";.,::te. = ..'."5; C .02" .Od; O .1l" 0.32 5 :1 0.5 were ex:,rinedl v.'it h ce se .ct to state it The r es uts for 6 = 0 .nC. = n.ro !:noPn frm [2 . (c) Flat oa t e with an is rin Tin j "w :i h unifo rm, sacticn. As a third siies several s ta.nat L n int pr ofiles from those c.lc''lat.i cd ex ctly 'by Schlichtin S nd Eussmann [5] were a:. ULi x) = uI x; V(y) = v. y 8 NACA TM No. 1121 Figure 8 shows this plane stagnation flow. Figure 9 shows the velocity profiless which were investigated. Their shape changes with the flow coefficient v Co (5) Positive values of C correspond to suction and negative values of Co to blowing. The profile with Co = 0 is Hiemenz' profile of the boundary layer; this profile results from the first term in the power series starting at the stagnation noint for the boundary layer of the circular cylinder. The velocity profiles with the flow coefficients Co = 3,1905; 1.198; 0; 0.5; 1.095 and 1.9265, tht is, two blowing profiles, the profile with an impermeable wall Co = 0 and three suction profiles were selected for the stability investigations. Figure 10 shows the second derivatives of these velocity profiles which have negative values throughout. T'ble 7 shows the corresponding boundarylayer oaramieters. IV. STABILITY CALCULATIONS The examination of stability for these laminar velocity profiles was carried out according to the r!thod of small vibrations in the same way as previously liv'en in detail by W. Tollmien [10] and H. Schlichting [ol . A plane disturbance motion in the form of a wave motion progressing in the direction of the flow is superimpniosed over the basic flow. It is essential that the basic flow U(y) be assumed dependent on the coordinate y only. Then the amplitude distribution of the basic flow also is a function of y only. The equation ( , v' (U V6 I x by 6x *(x,y) = c(y) eia(x ot) (6) NACA TM io. 1121 is valid for thi flow function j(x,y) of the snperirmo:. disturba.nc. motor (u', v' ). i real and ,ives t:.he wave length X of the disturbance :notion, X = 2n/u. c = cr + ici is complex; Cr represents the wave ve3ocO r of transmission and c, positie or negative, the excit ci.:n or damrping, respectively. '(y) = ,r() + ic () gives the complex amplitude function. The ordinary linear differenuial equation of the fourc.h order T i (Ti cJ (" a2 2a + a) (7 CLA with the boundar' conditions S= .: = .',,' = 'r : = .,: '= ,P' = o (2) is obtained i .,r that fincti..n fr.im ilaoierSt:ke=' eiqu3tiros. In equate. .cl i(7) all values arred rcnle rndimnension:.1 with as .uiabl. boundarylayer tiickne.c u and the potential velccity U. R =',/ .l/ stands for the Reynl..:"ds nurber; si:niflis differentiatiton ith respect to y/o. The boundar.layer cbnditlnns result from the disappearance of the normal rand: tangential ccmponent of the cdistr'iincc velocity at the w ll and outside c"f the boLundar.r.layeer (7 = c"). The ecamijnat.on of sta'ili]... of the prescribe.. tasic flow Ui (y) is a ci.racceristic value .robler! of that diffsrentii eiuaticn in the follo'.7in0 ser.se, since U(y) the tia, lenlsth = 2T//a and the R.eynclds number U/i, are prescribed. The corrle:. characteristic value c = C + icI is requir'..T' for every t..o cc:responding values a, t; from the real part of c results the velocity of transmission cf the supr i. llp .sed_ disturbance; ti!e irmaginar"y part of c is decisive for the stability. DiZturbance occurrir. for the condition of neutral st.':, i lity (ci = 0) are especially interesting, and lie ,on a curve (neutral stability curve) in the a, t plane. The neutral stability curve separates the stable disturbances from the unsctrlle ones. '.e figs. 1 to 1( )Tihe tangent to the neutral stability curve parallel to the aa;is gives the smallest Keynolds number at .ich a neutrally stable disturbance is still possible. This number is the critical Renubcr of the basic flow. JNCA TM 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, disarpearance of the normal and tangential disturbance velocities at the wall is required for the boundary layer with suction also, although the normal comronant of the basic flow at the wall is different from zero. The numerical solution of the characteristic value problem then takes exactly the same course as indicated by W. Tollien [10] and ,. Schlichting [6] and needs, therefore, no further explanation here. For the stability calculation, the velocity profiles are approximated by parabolas in the form: = 1 p (a y/)n with 6 = 56* The constants p, a, and n for tho three investi,?ted series of profiles are enuLmrated in table 1. The closest agreement with the exact velocity profiles near the .aell was important. (Figs. 2, 5, and 3.) The polar di. griis for the examined velocity profiles are given in fig.rs 11 to 1$ as an intermediate result of the stability calcu tation. The neutral stability curves were obtained from these dialcrams and a6 is plotted against Uo0 /, in figures 14 to 16 and corresponding values are tabul:t;d in tables 2 to 4. These curves show that the stability is greatly increased by suction while blowing decreases it. Figure 14 shows in detail that in the plate flow with continuous suction the neutral stability curves for positive flow coefficients C lie between those for the flat plate without suction and those for tho asymptotic suction profile, which were calculated previously by Bussmai and :.:hn [2] The case of blowing with C = demonstrates clearly the enlargement of the region of instability. NACA TM Nce. 1121 As to the flow along the plate, figure 15 shows that the neutral stability curves for the values of u used here also lie between the curves for = 0 (flat plate without suction) and r = a (asym).totic suction profile). For increasing E the region of instability diminishes and at : o = 0.5 the neutral stability curve a. ears to aporoacn the curve of the asymptotic suction profile. The stability in the stagnation flow also is rr'ttly increased by suction; with increasing flow coefficient Co the neutral stability curves (fig. lo) a,:.,roach the curve of the asymptotic sucti'n profile for the flat late. It is of interest that the neutral stability crves for the investigated cases of blowing still lie inside the curve for the irper;neable flat plate. The critical R:ynolds number (Uo:/ )crit is the tangent parallel to the ordinate of the neutral stability curves. Tna critical Reynolds numbers for the three series of stability examinations are enumerated in table 5. The following detailed result was obtained for the plate flow with continuous sucuion (Vo ~ 1/): (Uo1( /'Ujcrit = 24 for the blowing profile (C =\ and therefore is far below the critical Reynolds number 575 for the iimoermeable flat .'late. WVith suction the Peynolda tnitlber inirreases rapidly and reaches for C = the value of 19,100, thus evidently aonroach ing the critical Reynolds number 70p003 for the asynmptotic suction profile Ieternined by Bussinann ?'!unz [2]. Figure 17 shows how th? transition .oint on the 3.ate is shifted backward w ith increasing suction. The Re.nolds numbers formed by the disnlacemennt thickness are, for different flow coefficients C, plotted against the Re numbers formed front the distance (x) measured along the .late in this figure 17 ind also figure 13, and values are tabulated in table 5. The stability limit of the imnermeable wall (Uox/u)cn]it = 1.1 x 105; but for a flow coefficient C = 1 critical Re :xc'eds 107 12 NACA TM No. 1121 and therefore reaches the Renumber region of todayts large and fast airplanes.1 Figure 19 shows the dependency of the critical Reynolds number on the shaie parameter 5*/ . Flow along the plate with uniform suction (vo = const.), the critical Reynoldsnumbers for different /, compiled in table 5 lie between the critical Reynolds numbers 575 for the impermeable wall and 70p00 for the .":'. totic suction profile. Figure 20 shows the result of the stability calculation in which the critical R:;,'rolids numbers (Tor'/U)crit are represented as functions of the non.'iiE.1r.i.nonal. flow distance \/s. In figure 21 .he onset of instability is ascertained. Here both the stability limit (Uc /'/uI according to figure 20 and the nondimension'.l r.:, n..1J.i..,; I :e. c er'L. i ss_ Uo,: '/! plotted against \ e lw ,", c 'lc nrit 2 CQ = vo/ T are sh i,'wn. Th. l.' : .i ylA:;er ti'icknes is obtained from (lci) (o0 the values of v 6:'/ n function: :.f are given in figure 7. SIt3ce i 1r ': Ve":'."r = 1, the separate curves have t.e asymrtotes ___ ,_V,D t ,' n) The point of tr.nsrit io^nis iven bt thle :.rters.ction of a curve Uo / it" th' st */iUi t lrim t (;U,, '/ ) c.it* 1 1 means c, = thi ,'ith io" c, = 5.16 x 104. NACA TM No. 1121 The onset of instability occurs before V = 0.1 for Vo 1 1 1 the flow coefficients 70,000' 20,000' ,00 on the other hand, there is no intersection for Vo 1 U, 8500 Flow coefficients c1it = 1.18 x 0" (12) cWit 8500 are, therefore, sufficient for mPr.ntainin the laminar boundary layer for the entire preliminary laminar flow region. This value is to be compared with the value Qcrit = 7000 = 0 1 x 10 determined by Bussmann and CQcrit 70000 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 presumption. The minimum suction quantity2 found herewith is Q t = 1.18 x 104; 2One could consider the possibility of reducing the total suction quantity still further than CQcrit = 1/S500. One would have to select such a distribution of vo(x) as to make the curve Uo6/, in figure 20 remain ever,.here just underneath the stability liniit (Uo6:/u) rit. The necessary distri bution [vo(x)/Uo]c is given to a first approximation crit by the intersections of the curves Uo6 /v for different vo/Uo with the stability limit (fig. 21). One then obtains up to about 0/ = 0.1 an increasing local flow coefficient [Vo(x)/Uo]ci; the maximum is reached with 1/8500 at about V = 0.1; for higher Vi there is again a decrease down to the constant asymptote (vo/Uo) = 1/70000. The total suction quantity, however, would hardly be reduced under the constant value CQcrit = 1/8500 for practical purposes if such an "optimum" nonuniform distribution of suction were selected: For a plate with the Reynolds number Uol/u = 10' and NACA TM No. 1121 this quantity is still so small that the maintaining of the laminar boundary layer by suction appears quite promising. We mention, for comparison with experimental results, that the necessary suction quantities in Holstein's [7] measurements on the supporting wing are cQ = 1.1 x 10"4 to 2.8 x 104. This value is, however, not exactly comparable to the thoeretical one since the suction in the measurements. was produced through slots. Figure 19 represents the critical Renumber (Uo 6*/U)rit as a function of the shape parameter 6*s/& Simult3aneously, the results of the stability calc.lat:_cn for the velocity profiles of the flat plate with continuous suction vo 1/Vi are drawn into this diagram One ccn see that the critical Reynolds numbers of the two stability calculations.. lie on the same curve. Hence it is concluded that the critical Reynolds number (Uo6/:1)criit is dependent on the shape parameter 6:/ only. Plane stagnation flow. The critical Reynolds number (U o::U)rit for the impermeable wall (C. = 0) is 12,500; for the suction quantity (Co = 1.9265) this figure increases to 38,000 (fig. 22). .ith blowing the critical Rey;olds number decreases slowly; the value of 707 is reached at C, = 5.1905 (table 5). These critical Reynolds numbers also are given as functions of 6:/ in figure 19; one can s'e that they ta!:e a course similar co the flow along the 'late although they lie somewhat underneath this curve. o/U 10", 4 = 0.5.; therefore the region where oQ can be considerably smaller than 1/8500 is still far beyond the end of the'plate. NACA TM No. 1121 V. APPLICATION OF THE RESULTS TO DRAG :ZDUCTION BY MAINITAIIIING THE LAKIITHAR BOUITD.IY LAYER The drag coefficient cf is plotted against the Reynolds number in figure 23 for the laminar and turbulent flow in the boundary layer of the flat plate with continuous suction vo ~ 1lx. The drag coefficient curves from reference [33 for different quantities of suction and blowing C also are shown in the figure []J in this diagram; the coefficients Cf increase with increasing suction quantities.3 Drag may be reduced by suction in the region between the curve for the laiiinar flow of the flat plate (C = 0) and the fully turbulent curve, if the laminar boundary layer car be maintained there by suction. The result of the stability calculation given in figure 18, that is, the critical Reynolds number (Uox/u)crit as a function of the mass coefficient C, was 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 lamiinar boundary layer at tne respective Reynolds number. The drag reduction by maintaining the laminar boundary layer for different Reynolds numbers can be specified immediately by neans of this diagram. The minimum suction quantity cQ crit = Ccrit/V necessary for maintaining the laminar boundary layer is determined for a given Reynolds number; then the drag coefficient Cf for the fully turbulent and laminar flow with suction is read off the ordinate. This calculation is carried out for the most interesting Reynolds numbers 6 Q from 2 x 10 to 10 in table 6. One can see that for instance for Re = 107 a drag reduction of more than 70 percent can be obtained. This statement, however, does not yet make allowance for the power required 3The 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 theparts sucked off spent their ximpulse fully in the boundary layer already. Compare Schlichting C8]. NACA TM No. 1121 for the suction blower. But this power is not excessive, since only very small suction quantities'are needed here. Figure 23 was concerned with the plate flow with continuous suction; in a similar way, in figure 24 the laminar frictional drag coefficient cy (determined according to Iglisch's C[] calculations) for the plate flow with uniform suction is plotted against the Reynolds.number Uol/u with the flow coefficient q = vo/Uo as parameter. For very small Reynolds numbers, all curves converge to the curve of the plate without suction. cf becomes constant with the value cfj = 2vo/Uo for high Reynolds numbers where the larger part of the plate lies within the region of the asymptotic solution with constant boundary layer. The curve c crit = = 1.18 x 1 0" named "stability limit" U0 was drawn into the diagran (fig. 21) as the result of the stability calculation. Figure 25 represents the same condition again; but, different from figure 21, cf is given on the crdinate in ordinary scale. Both represen tations show thit for instance at Re = 107 a drag reduction of 380 percent of the fully turbulent TF~itional drag can be achieved. Figure 25 compares the drag reductions by maintaining the laminar boundary layer and the critical suction quantities for the two cases uniform suction and vo ~ 1/VX. A comparison of the results obtained under the assumption of uniform suction vo = constant with the results based on the suction rule vo l/VT demon strates the following: The critical suction quantity cQ crit for continuous suction vo 3/x is variable with Re = UoZ/u and is for all Re< 7 x 107 larger than the suction quantity for uniform suction which is constant cq crit = 1.18 x 104. The drag reduction for uniform suction also is larger in the considered region of Reynolds numbers than for vo 1/VS; for instance, the drag reduction at Re = 107 is 80 percent against 75 percent. Therefore, the uniform suction is at any rate preferable to the suction with vo ~ 1/yx for the whole region 5 x 106 < Re < 108 that is, the main region of interest for practical purposes. IICha Tir. Ho. 1121 At h!ixh Renumbers (ever ,10') only the suction accordin t to the rule v,,(x'l 1/' shov:s sm allr critical ucticn quantities and higher drayg reduction than uniform suctirn. Table 6 and fi__ure 23 .'ive a corrm::.ario.n f tih3 critical su.ction. quanttities and th2 drag reductions for" the two vucti.n rules. VI. I'.EASLUFEIT TS OfIF T':E VELOCITY DIST'ITBUTI':D IF THE LAM.ITAP 30U1IDARY L .' .TTF S'TCTION Finally, 9 few results of e.l rimr=nts about thiG bound ary 15s,.7r with suction sall b ;*.':, n. In figl 27 two nmieas, r .d '.locit:i distribjtions v.ith tLe asymps:.tic suction ro.i:file [5] a;i ti lasiu: i profilel for thJ L Iie an lat. j'n lon'itui din::1 ilow .itlhout sucri'n are co.,iear, d. Th. first !ie .3u.erii;. nt f' ,*.*arri.d out by Holstein 1[] on a wing with the i.rofile ITACA 0u12o1; the tit ksur.ms nt wi:. tak'.r at x/7 = t0.9 ( 7 = wind chord) of the w.iri center sect:,.on on th.e si ct.:n side (a = O ), with six suction slots ,of th e suction side o,..ned. The valocit7 distribution which .:s cnv.crted il.to the dlis placeri, nt t:icknccs conforms .'..th. r we ll with the asyJin:t otic suction .rofiil of the flat lite .'.with uniform suction while differing ,r.;tly from the l.:,aius profit~ aith imoermoable ws.ll. The second rmeqsuranent was te!:on by Ackeret [9] ; [ . suction channel with n.umn.rous narrow. slots :? short diRst:!nco behind thc: suction length ' S iLs us.d. This ve:locity .iistr:'.tution lso ts.:.es course similar tc the as:ymtoLic su.ct.I on rcfile. Thi r.fore, 1: fw existin g ricosurcrmnts show hoocd a .r.;ement :wir.h the theory as .to the fnrm of laminar velocity distrib utLon with End v:ithout zs.ction ,r3s'ectiv; 1.y. VII. 3SU',." ARY Stability calculations vrwre carried out on three series of exactly calculated velocity profiles for the laminanr boundary layer with suction: (a) on the'i flat plate ith continuous suction according. to tho rule vo 1/7, (b) on the flat plate with uniform suction NACA TM 11W. 1121 vo = const., (c) on the flat plate with an irmpin3 .ig jet with uniform suction. It became obvious that the stability of the boundary layer is greatly incr,aszd by suction, on the other hand greatly reduced by blowing. While the suction quantities are increased slightly, the critical Reynolds number increases greatly arid approaches the value found by Bussmannj%.Infi (UoS/u)crit = 70p00 for the asymptotic sucticn profile. Then the minimum suction quantities necessary for maintaining the laminar boundary layer were det=rriIined; they were c Q= 1.1 x 10 to 2.8 x 10' for ti plate with continuous suction and 1.18 x 104 for the 19te with uniform suction. The drag reduction obtained by maintaining the laminar boundary layer at Re 10i is 80 percent of the turbulent drag without suction. Translated by Miary L. Mahler National Advisory Committee for Aeronautics IIACA Tri1 N?. 1121 VIII. BIBLIOGRAPHY 1. A. Betz: Besinflussung. der Reituncsschicht und ihre r rakiis che Verwertunc. Sciriftsn der DeuLtchien Akademile ,der Luftfatnrtforschauni, Heft Lv9 (1i L); v1l. inch J~hrbuch der Deutschen a !kelemnit der Lftr'fi;thrtforscrLung 19'9/K0, 3. 246. 2. K. B.s':',nn u. H. iKrinz: Die Std'bilitit Jder laminaren e.ibi.:.rn Cschichu ilit Ab Uti.IT.ni. j,To.lrochl 19k2 der DeutLchen LJ'Ctfcjlhrtforecha .mc, 3. I 36. 5. !. Schlichting u. }:. BusEr.u nn EXAl:t Lsunre n fr'r die larr.intpne ( iGen S'.sc'.icht rnit AbsauLArung r Und Ausblacen. cohriftzn der Deutschien Akade.le der Tiftfahr,tf'or.schun BF:. T, 19' ie t 2. S. R. Iglisch: Exkte Losungein 'Lir lie 1mjinare Grenzschicht .n der l1i.n,.sane stiniten ebenen letter 'it lioriogener Absn: _iir,. Berir cht ',/2 de ? Aernodyn. Instiluts tder T. H. BrAunschweig. 5' H. SchIlciting7: Die Grenzschicht mit Absaul un, uLind Ausblasen. Luftf hrtforsciun' 1942, S. 179. 6. H. Schlichting: Ueber die throretische Ber'eciinunl der kritischen Reynolsschern 'Zahli inner ri3ibunrsschicht in beschlE.unigter und verzogernterr Str;mung. Jthtrbuch 194i' der deutsc hen Luritfahrtfor crsc hung, S. I '?7. 7. H. Holstein Hiessunpen zi.c Lsmi.nqrlhaltung der Grenzschicht duirc Absaugiun. 9nn einem Trawfl'.xuel mint 0r0ofil 1 ACA 0012/4 . Bericht der Aerodvnamischen Versuchicsanstalt Gotr.inren. F3 1S, (19:42). 8. H. Schlichtin:: Die Beeinflussun_ der iGrenzsc.hicht durch Absaugi~ng und Ausbiasen. Jahrbuch 19 j/441 der Deutschen Ak:.:ademiie der Luftfahrtforschiung. 9. J. Ackeret, li. Ras, W. Pfeningir: Verhinderung des Turbul'entwerdens einer Orenzschicht durch Absaugung. Nlaturwissenscheften Bd. 29 (191), S. u22. 20 NACA TM No. 1121 10, W. Tollmien: Ueber die Entstehung der Turbulenz. Nachr. Ges. ..Lss. G6ttingen, Math. 2hys. Klasse 1929. N..CA TM No. 1121 TABLE 1 A~' OXI'"ATIIO] OF THE VELOCIfTY ROFCILZS F'O: VArIOUS C, \C AND Cc, O TH'IE T'iREE IiVESTIGAT ED FLO.vS u Iyn ACCORDIiG T'O EQUATION (9): = 1 a  C p a n Flst plte with conti nuioi s.ect ion V0 I/: 0.25 10 1 1.5 Sp a n Flat .::.i te withh uniform suction vo = const, .005 .02 S13 .52 .5 W Plan3 stargn tiion flow with uniform suction .0707 L , .141 5 .707 o 5.190'5 1.19 ; 0 .50 i100 1. 0 1. ~C'u * 1Z)3 i. :0o i. 'o ', 1. OLh 2 1. 042 .13.5 Si '0254 P 1. 0550 .100 .1375 * i 0.95 2 1.015 2 i.6 2 L L~ 14 .0 15 i .oo0 . 5 3 .1JO 1.66 7 lc54 4 .05o .r4 1.642 1. 33 1.622 2 2 2 2 4 4 n 22 11CA TM to. 1121 TABLE 2 :;UI'E.ITCAL VALUES OF TIIE NEUTE..L STABILITY CURVES CF THE PLATE FLOW WITH CONTINUOUS SUCTiOi vo I/ C c/U0o y c c/Uo YK/6 a: i x 103 ] 0.20 0.587 0.092 3.5 4 .25 .483 .0 1.75 .568 .696 ., I .341 .Lo .756 .37 i. +98 Asymptote .6 .6 5 .45 64 5 .846 .27 .217 7 .879 '5 .567 0.05 0.084 .J.)3 857 .05 .084 .0. 1920 .10 .150 . 5 .10 .150 s 206 .15 228 .11 105 .15 i228 1 1.5.3 1 .20 .312 .1 26.2 2 .20 .512 .3 15.7 .25 .390 55 15.5 .25 .90 .1)1 6.14 .50 .774 .1 5.50 .50 .47 .. 5.53 .51 .9 70 5.6 _______ ~ ~ __ 49__ u__ ___ 0.05 .05 .10 .10 .15 .15 .20 .20 .25 0.069 .o69 .158 .158 .207 .207 .280 .280 .357 0.038 .021 .071 :or .050 0cJ0 .1531 500 2764 115 55.2 ) .92 9.9d 0.05 0.022 0.0:7 1150n .05 .022 .017 5770 _ .10 .044 .o77 571 2 .10 .oJ 1055 505 S .15 i .' .7 .., 102 .15i .067 .055 6.2 .20 .0 .ll. 32.1 .20 .090 .22.5 NACA TM io. 1121 2 ;.LE 3 .ERICL VALUES O'F THE NEUT''.AL STH:ILL.TY OTJU.rES OF THE O0Pil30IID PLATE FLOJ WITH UITIFpI J. SUCTIOII a o x. 10'3 U ' 0 0 3.10 0.056 1S  .20 .077 7.20 .20 .1!9 57.7 .25 .101 35.01 .25 .133 12.0 .129 1.53 .0 .225 ,.61 :1 5 5 ;. ; .l1 .55 .2 23. _ .35 .159 .89. .55 .252 2.0 .575 *181 .S .375 .26 1.42 .LO .2 05 .653 .1o .274 1.02 .L2 .259 .,105 .42 .275 .713 24 NACA TM No. 1121 TABLE 3 Continued NUTTERICAL VALUES OF THE NEUTRAL STABILITY CURVES OF THE OiCOi"iG PLATE FLO' 7ITH UTIiFOR,; SUCTION 'Continued i rc YK Zr Ui" 0 x103 o 6" D 0.005 0.10 0.055 0.076 630 .10 .055 .0365 131 .20 .114 .157 22.7 .20 .114 .085 7.9 .50 .175 .224 3. .50 .173 .145 1.86 .35 .205 .237 1.51 .55 .196 1.16 .02 .10 .053 .074 778 .10 .055 .0o5 198.5 .20 .108 .14 28.9 .20 .108 .074 12.8 .25 .138 .176 9.5 .25 .153 .108 5.06 .50 .167 .200 5.85 .30 .167 .14 2.59 .08 .10 .050 .075 543 .10 .050 .05LL 204 .20 .100 .15. 26.6 .20 .100 .o8o 15.5 .25 .127 .160 9.67 .25 .127 .il 6.65 .275 .105 .163 6.17 .275 .1405 .152 4.67 .295 .151 .165 5.95 NACATiM Loc. 1121 2Z TIBLE3 Concluded IHJ1'" ~IC' ,.'VL'M' S 07' i" iIFUTRAL ,0' 1"i P IR'/l S 0'n. THI C .(, : IlU1 .A.TL FLO\. ..Il U.H' I AL. Li Ol Concl i c U 8"w > b:: O" o" i c I 10* 1 i 0.1i 0.10 0.047 '.07 '0?. .10 .017 S,.u 89.7 2 .0 o 1.2"5 20. ' .10 .L .0. lu. .2 1 I1 i . 2 C.05 0.02:. C.017 12250 .03 ,0'2"I 5 .010 4217 1 i5 .00 0 'o .'i .15 .06.2 .O:', 0o.2 0., .0,2 .0 i '2.0 .5 0.0 O. 1064 5e. .0o2,, .. ,N7 O .r4 1, .066 .*0"'4 i017.0* .15 .Oco6 .'056 o .20 .0.0;' .111 51.' .20 .O8;:; .0.1i 2n..9 0.0 0' .0 2' .09 . 09 .15. .15 .175 .051, . I0 7 * 1 ,, . I: ,3 0.0176 n. O .000 .0526 .00 5 .OYQ5 L 1200 104i00 1500 70C 147 115 70 26 NACA TM ie. 11u1 TABLE 4 TUCTERICAL VALTUS OF THE .iEUT~RAL STABILITY CUV.'E ,.' THE pLT.A STA.ITATION FLOW WITH UNIFORM S'CTIT iT C a,6 xo6* 10 o UO .15 .20 .50 .50 .35 .10 .10 .15 .15 .20 .20 .25 .25 .29 .05 .05 .10 .10 .15 .15 .20 .20 .23 .235 0.109 .054 .079 .207 .'15 .2357 .200 .071 .07, ,106 .0 5 .039 .oi6 .125 .016 .0534 .098 .055 .119 .082 .125 .102 77.6 25.1 20.7 8.6 5.70 2.02 1.58 1.12 655 218 107 27.4 14.5 9.65 0. 14 7 0 lG7o 711 292 114 51 18.1 14.8 13.8 4 ________________  5.1905 1.198 0 INACA TM No. 1121 7 7 TA.LE 14 Corc.luded liUTzr' IC, L V, LiES n' 0 'FE !IEUTR'L ra2T LITY CURVES *'~' THE ?L..,A. T ".GtL.TT ICli FLOR/ Yi"'H UiiIFORi'l &UCTTOII C,.c,.o 'ded c U,,'u _ ," a6'  x 1i0 .o 1'D0 .^, .CO ii7 D : .10* *'13 '' ^0 1* C. ;,1 .1 F L 15 1  .2:' 1 p, 7 ? 7l ' .2 .1 . , _ ' *.7 '* .* .1 1 .. I ,, .01 '2' 10 *.', 1* L 1i .' 1 0 .io .1 ? .0 1.'\ 5 ..' .i 1'; 21 '1 .0 .014 L1 0 .10 *(4 71* 15 1 124 175 lI.l 5Q 5 .175 *''T7 49.7 S. 19 *lr1 7^ ~ NACA TM No. 1121 TABLE 5 z0 CRITICAL REYNOLDS NUMBERS AS FPrUiCTIONS OF C, , AYD Co FOR THE T7ERT INVESTIGATED FLOWS oC U =  \/U cr * /c rit oU i a . c it Flat late 0.25 2.77 204 1.03 x 10 with con 0 2.59 575 1.10 x 10? tinuous .5 2.1 2986 525 x 10 suction 1 2.29 9550 8.1 x 10A vo ~ 1/ 1.5 2.22 19100 4.90 x 100 Asymptotic 2 70000  suction profile Flat plate with uni form suc tion vo = const. v lUx ; 0 .0707 .141 .285 .566 .707 "; 5 2.59 2.553 2.47 2.39 2.51 2.25 2.21 2 /U 6 U c rit I i 575 1122 1820 3935 7590 13500 21900 70000 v 4iul y \" rit ,lane stag 3.1905 2.538 707 nation flow 1.198 2.552 460 with uni 0 2.21 12300 form .5 2.172 17360 suction 1.095 2.126 27700 1.9265 2.088 38000 thCA TM :Io. 1121 C'4 C, 4' r r A ri .j+ r. '4 L iJ + * 'r "<  "I r o l r 0 )k j  I1~ 4 i Lrc Lr', r"I c; " l' ~J. ) c. i  0 iJ C3 ]. J O  L ,' .L '  LC. CO C' 0 ,0 ,0 [ [ D ' .X ,  r r rl  r4 X X XX lry L CJIL1 , I C., 7 c E 0 C > '0 i o O \0 * 3 [ ( .0 CO Cjj 3) 0 UL. l. L.\ L, 3 *t r ^~~~ r 4^ "j ' 3 0' E' C 0 0 C7 0000 rI r 4 r 4 r e ,I ri x x x ,\I LP. CJ LP, A! D H OD .4; 4  c, ? > *r I 44: i SII i 2 rO +  a c~ :   t    CmI NACA TM No, 1121 TABLE 7 THE CHARACTERISTIC BOUNDARYLAYER PARAMETERS OF THE ITVESTIGI.TED LAMINAR VELOCITY PROFILES :.ITH SUCTION C = C *"o 1xp v0)x ToO lUo Flat plate 0.25 2.010 0.7L0 2.77 0.500 with con 1.721 .66 2.59 .53 tinuous .5 1.5 2.41 .682 suction 1 1.047 .458 2.29 .765 o ~ 1/V7 1.5 .863 .590' 2.22 .818 Flat plate with uniform suction v = const. 0 Plane s; ag nation flow wi t~h uni for suct i r.il 77 = /U0x .0707 .141 .212 .28d .554 .L24 .495 .656 .707 5.1905 1.193 .5 1.095 1.9265 v u  v v 4 4 + .211 .5''1 .51 *'. v J l.C'1J .6. . :. 'J ;'' .^.*F) .125 .192 .221 .27j3 .2'95 .515 .5 u 2.59 2:4, 2.59 2.3'5 2.51 2.23 2.25 2.21 2 oT 0 0.571 .607 .631 .671 .699 .726 .773 94 .8o50 1 T 0 0 0.772 .292 .250 .209 .167 2.54 2.55 2.22 2.17 2.15 2.09 0.608 .917 .917 NACA TM No. 1121 o  > "^ hi c ^H6 y jO Xy^ Figure 1. Explanatory chart: Boundary layer at the plate in longitudinal flow with continuous suction according to the rule vo (x) =  y Uo x =3 L 2 Figure 2. Velocity distributions u/Uo against y/S for various flow coefficients C of the flow from Fig.l; WP = point of inflection. Comparison with the approximation of equation (9). Figs. 1,2 NACA TM No. 1121 Figure 3. The second derivatives of the velocity distributions from Fig. 2. (C= flow coefficient). Fig. 3 NACA TM No. 1121 Fig. 4 3 0 .0 o 0 ' 3 /S4 i 0I / S ,"/ // " ', ',r'),y 4 / / / / . CM 0 rc. Fig. 5 NACA TM No. 1121 < ..i_ _.,,___I U 45 05 vii ""y' Figure 5. Velocity distributions u/Uo against y/& for different according to Iglisch [41 ; comparisons with the approximation of equation (9). S Uo_x Uo ( )2 0 qo NACA TM No. 1121 Figure 6. The second derivatives of the velocity distributions of Fig. 5, according to Iglisch [4 . Fig. 6 NACA TM No. 1121 __ VI o fs uI Figure 7. The boundary layer parameters for the plate with uniform suction: Vo o J*and yv 7 , f against n V5 according to Iglisch [4]. (Flow Fig.4). i suction V0 = onst i suction VQ=const. Figure 8. Explanatory chart: Boundary layer of the suction vo= const. plane stagnation flow with uniform Figs. 7,8 NACA TM No. 1121 Figure 9. Velocity distribution u/Uo against y/j* for different Co, according to SchlichtingBussrann [3] ; comparison with the approximation of equation (9). Fig. 9 NACA TM No. 1121 0 / 2 3 5 J 6 Figure 10. The second derivatives of the velocity distributions from Fig.8 according to SchlichtingBussmann [3] . Fig. 10 NACA TM No. 1121 Fig. 11 _,, /, I /. <., . '.' y,, , ^ /7. ,I .' 'a / I; E. . / , 1 ,J 'I/ ,L I 1' : / '0 ' I / . ,, E lE.I i o ,,I, l I. I . .F /* .In . 't ,'" ,' . _ .n, Ci Si.I '.1 If L4 0 U I . > 0 .4. 1%. .o Cx. Fig. 12 NACA TM No. 1121 S' ' I, I awl p I a C.) Qu I. " S, I 77  ./ / J '. I' I S' / / C N..4. II_, 0 m I C4 i i ". I , o cd Z lc / / 0 5 ^ ii ^4$ 0) r I I I . ,, i t I 1 I I*ll s.; ;. C.4 I I, v I / _t.I If __I_,,,_____ >'. d S .' I /I I I1 . / I 0  1 I l " C0  ,,, a I V j I I\I Y ^/R O'O S.~I IL~e NACA TM No. 1121 Fig. 13 ,,1 /q own :, : L4 I I I _I,  '" 0" 1 4 , ^ /  Uz 7 /C: l{"7,,' I:,a " I ,'9 1 / i y I / i 'I C / i c 4 I : I .) . /(\/ 8/ 1 / I 0 110 n / cd Q, 11 .. I I I /r1 0 (1 0. / 11 0. 1 0 c Lu c1 cn 4 / . / , l I s I I l i i % Ll ,  *  ^^ 1 $ s 85 a o Fig. 14 NACA TM No. 1121 SStable (0 plane plate C= o 7 without suction asymptotic suction profile o I \ ".... o 0 1 01 10 I I 5 Figure 14. Result of the stability calculation for the flat plate in longitudinal flow with vo1/\. The neutral stability curves Os* against US* /rfor various flow coefficients C. NACA TM No. 1121 0,3 0,2 0,. O l 02 0o 4 /5 i6 6W* t 7 IV Figure 15. Result of the stability calculation for the plate in longitudinal flow with uniform suction: The neutral stability curveso6&* against U */V' for various ; V= 'o 2 Uox = 0: Plate without suction (Blasius) =oo: Asymptotic suction profile. Fig. 15 NACA TM No. 1121 0,2 \ Stable II I I Iuctio v f02 5 6 7 Figure 16. Result of the stability calculation for the plane stagnation flow with uniform suction: The neutral stability curvescxS*against US* /ij' for various flow coefficients c = Ul r Fig. 16 NACA TM No. 1121 S01 '1 I k _O l ___ _______  4 Shill I [aVI Fig. 17 a, e 8 0r E1 4 1i *4 bo No z 0 1 o o o 0 NS s0 t o f 'S* Li , 0, (1o cdi i o , ,i I .I Fig. 18 NACA TM No. 1121 0 41 o EO _ I o a 0 0 0 It 0 C .0 V  uL to C* 4) S 4 0, 4 rx.  II cc \ ^t c < \ 0 0 l"^ "" ^ "" t i ^ ^ ^ ^1 p * I '^1 I C Q) NACA TM No. 1121 5 2 5 2 3 5 2 O8 2 2,4 I? Figure 19. The critical Renumber (Uo S/y )crit as a function of the shape parameter S */1 for the plate in longitudinal flow with vo l/ /Y and vo const. and the plane stagnation flow. 760 0 asymptotic suction profile 1 \\ , 5751(plate without suction) PlaTeu u ut T o Plate with uniform suction o=co nst .r = continuous v ? ~ a Plane stagnation flow I I I Fig. 19 NACA TM No. 1121 f5 O 5 2 52 2 2 10 /,4 0,6S VII @r, Figure 20. The critical Renumber function of 0;o 0 u 0 ; V (UoS'/i,)crit as a for the flat plate in longitudinal flow with uniform suction. 70000 (asymptotic suction profile)   /_ e___t u___t_ 5775 (plate without suction) Fig. 20 r NACA TM No. 1121 f05 5 2 10" 5 2 0 o,2 0,9 6 08 o Figure 21. Ascertaining of the critical flow coefficient c9 for main taining the laminar flow for the plate in longitudinal flow with uniform suction. Uo r Asymptotes: ( "T Uo 1 oc vo Ce Fig. 21 NACA TM No. 1121 3 2 f 0 1 2 Slowing S uetion Uo0 Figure 22. The critical Reynolds number (crit as a function of the flow coefficient Co for the plane stagnation flow with uni form suction. Fig. 22 NACA TM No. 1121 Fig. 23 / 0 / ik I  I H17.n / H 1 / /// I _ N c& .I  l I I Y + I i t / I/ / * _: ,, l /Z 0 ii i t / / /" / / I .r L4 0; ii > c C1 NACA TM No. 1121 Z, '4 a <^ I I O C bo 0> 04 0 to I I I 43 443 c CD C c i=C4 o I Ca L o ,  00 ( +i 0 U 4 *a La 3 I 3 C. +3 3 C Iftlr tic t rl3 U Fig. 24 NACA TM No. 1121 Fig. 25 er I. 4 6 .. / C  0 U 2 a l e C > S .I a 0 0 O o. ., c1 L ,O +3 0 z" .9 =I ed 'i IA / ,. / / = _i Io n jC4L4 r  II ioiC W 8L G 7 . C o b.. o '4.' 3 3 *"~~~~" /^O' t U 3 .' ~ ~ ~ ~ ~ r ^* I,RC. i> ^ ~ ~ ~ ~ ~ ~ ~ ~ ~ *^ '' __ __ __ __3 g J>^. ^^:' ~~ ~ q   S8 3 y      ^ S 5  __ __ __ __ ___^ o ) i L ~ ~ ~ ~ ~~  r 3 L? mo c S     __ __ s < 10 ) M^.Q NACA TM No. 1121 FiL. 26 +cct.1o, 32 2,+ f6 48 I I I I I I I I 10 10' 2 3 5 7 VO7 2 3 5 7 fo U = Re VJ A Cf Figure 26. The relative drag reduction ) f t and the minimum (cf) fully turb. suction quantity necessary for maintaining the laminar boundary layer CQ crit as a function of Re = Uol/!r for the plate in longitudinal flow with uniform suction (vo0const) and with Vo 1/ fX. Cf= (Cf) (cf) fully turb. laminar with suction NACA TM No. 1121 Fig. 27 \ e a d e a C i A s V  0  o .o ac o0 I I a "_ a) a 1 . m \ . u o *o . o o  .aooa 1 ii I I0 \ .N I.. \ \ ,, O ,_ 0 0 0 \ 0 1 o CU S o S ( I I To ai C; ( k7' caW d i UNIVERSITY OF FLORIDA 3 1262 08106 301 7 
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