UFDC Home  Search all Groups  World Studies  Federal Depository Libraries of Florida & the Caribbean   Help 
Material Information
Subjects
Notes
Record Information

Full Text 
A icA m (i\
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM NO. 1216 AN APPROXIMATE METHOD FOR CALCULATION OF THE LAMINAR BOUNDARY LAYER WITH SUCTION FOR BODIES OF ARBITRARY SHAPE* By H. Schlichting Outline: I. Introduction: Statement of the Problem II. Symbols III. The Boundary Layer Equation with Suction IV. The Generail Approximation Method for Arbitrary Pressure Distribution and Arbitrary Distribution o:' he Velocitj of Suction (a) The expression for the velocity distribution (b) The differential equation for the momentum thickness (c) Stagnation point and separation point (d) Execution of the calculation for the general case V. Special Cases A. Without suction (a) The plane plate in longitudinal flow (b) The twodimensional stagnation.point flow B. With suction (a) The growth of the boundary layer for the plane plate in longitudinal flow with homogeneous suction (b) The twodimensional stagnation point flow with homogeneous suction VI. Examples (a) The circular cylinder with homogeneous suction for various suction quantities (b) Symmetrical Joukowsky profile for ca = 0 with homogeneous suction VII. Summary VIII. Bibliography IX. Appendix *"Ein N'lherungovtrahren aw^ Berechnunr der laminaren Grenzschich. mit Absa',ung be! belieblper Kjrper'oriam. Aerodynamisches Institut dr. Technischen Hochschule Brounsenweip, Bericht 43/13, June 12, 1943. NACA TM No. 1216 I. INTRODUCTION Various ways were tried recently to decrease the friction drag of a bndy in a flow; they all employ influencing the boundary layer (reference 1). One of them consists in keeping the boundary layer laminar by suction; promising tests have been carried out by Holstein (re!'erpnces 2 and 3) and Ackeret (reference 4). Since for large Reynolds n unbers the friction drag of the laminar boundary layer is much lower than that of the turbulent boundary layer, a considerable saving in drag results from keeping the boundary layer laminar, even with the blower power required for suction taken into account. The boundary layer is kept laminar by suction in two ways: first, by reduction of the thickness of the boundary layer and second, by the fact that the sjcion changes the form of the velocity distribution so that it becomes more stable, in a manner similar to the change by a pressure drop (reference 7). Thereby the critical Reynolds number of the boundary layer (US*/V)crit becomes considerably higher than for the case without suction. This latter circumstance takes full effect only if continuous suction is applied which one might visualize realized through a porous wall. Thus the suction quantities required for keeping the boundary layer laminar become so small that the suction must be regarded as a very promising auxiliary means for drag reduction. Various partial solutions exist at present concerning the theoretical investigation of this problem. Thus H. Schlichting (references 5 and 6) investigated the plane plate in longitudinal flow with homogeneous suction. At large distance from the leading edge of the plate a constant boundary layer thickness and an asymptotic suction profile result. Later H. Schlichting and K. Bussmann (reference 9) investigated the two dimensional stagnation point flow with homogeneous suction and the plate in longitudinal flow with vo 1//x (x = distance from the leading edge of the plate). In all cases a strong dependence upon the mass coefficient of the suction resulted for the velocity distribution and the other boundary layer quantities. K. Bussmann, H. Minz (reference 8), and A Ulrich (reference 16) calculated the transition from laminar to turbulent (stability) of the boundary layer with suction for several cases; in all of them the stability limit was found to have been raised considerably by the suction. As is known from earlier investigations (reference 7), the same amount of influence on the transition from laminar to turbulent is exerted by the pressure gradient along the contour in the flow for Impermeable wall. Both influences (pressure gradient and suction) will be present simultaneously for the intended maintenance of a laminar boundary layer for a wing. Both influences have a stabilizing effect for 'th suction n nL:c region of pressure drop; in the region of pressure rise, h,'wever, pressure ri'ed'cn'. and suction have opposite influences. Whereas w!.hut nu.ct 'on, "'r, rrrrA.re rise, mostly transition in the boundary NACA TM No. 1216 layer occurs; here the important problem arises whether this transition can be suppressed by moderate suction. The solutions for the laminar boundary layer with suction existing so far are not sufficient for answering these questions. An exact calculation of the boundary layer with suction encounters insuperable numerical difficulties just as in the case of the impermeable wall. Thus it is the more important to have an approximation method at disposal which permits one to check the calculation of the boundary layer with suction for an arbitrary body. Such a method will be developed in the present treatise. The method given here is an analogon to the wellknown Pohlhausen method for impermeable wall. It permits the calculation of the laminar boundary layer with suction for an arbitrarily prescribed shape of the body and an arbitrarily prescribed distribution of the suction velocity along the contour in the flow. II. SYMBOLS (a) Lengths x,y coordinates parallel and perpendicular, reL,,ctively, to the wall wetted by the flow (x = y = 0: stagnation point and leading edge of the plate, respectively) 6* displacement thickness of the boundary layer ( (l u/U)dy) 0 momentum thickness of the boundary layer f u/U(1 u/)ldy) 51 measure of the boundary layer thickness I plate length or wing chord, respectively b plate width (b) Velocities u,v velocity components in the friction layer, parallel and perpendicular to the wall U(x) potential velocity outside of the friction layer Uo free stream velocity given suction velocity at the wall vo< 0: suction NACA TM No. 1216 (c) Other Quantities T 0 wall shearing stress T wall shearing stress for asymptotic solution of boundary B> layer on plate in longitudinal flow with homogeneous suction total suction quantity b. o vodx CQ dimensionless mass coefficient of suction; c > : suction C*Q reduced mass coefficient of suction ( V) 7 dimensionless distance from wall (y/61) Fl(&), F2(j) basic functions for velocity distribution in boundary layer, equations (8), (9) K form parameter of boundary layer profiles, equation (6) X,IX dimensionless boundary layer thickness, equations (12), (13) Ii1 dimensionless momentum thickness, equations (22), (23) Sdimensionless length of boundary layer 5J 2 III. THE EQUATI O NS OF THE B UNDARY LAYER WITH SUCTION Following we shall consider the plane problem, thus the boundary layer on a cylindrical body in a flow (fig. 1). x, y are assumed to be the coordinates along the wall and perpendicular to the wall, respectively, Uo the free stream velocity, U(x) the potential flow outside of the friction layer, and u(x, y), v(x, y) the velocity distribution in the friction layer. Suction and blowing is introduced into the calculation by having along the wall a normal velocity Vo(x) prescribed which is different from zero and generally variable with x: vo(x) > 0: blowing; Vo(x) < 0: suction vo/Uo may be assumed to be very small (0.01 to 0.0001). Only the case of continuous suction will be considered, where, therefore, vo(x) is a continuous function of x. One may visualize this case as realized by a porous wall. The tangential velocity ar. the wall should, for every case, equal zero. The boundary layer differential equations with boundary conditions are for the steady flow case NACA TM No. 1216 5 au au dU v 2u u + dxv +V (1) au &y + = (la) y = 0: u = 0 v = vo(x) ( (2) y =0 : u = U(x) The system of equations (1), (2) differs from the ordinary boundary layer theory merely by the fact that one of the boundary conditions for y = 0 is changed from v = 0 to v = vo(x) J 0. Thereby the character of the solutions changes decisively: the solutions differ greatly according to whether it is a case of vo> 0 (blowing) or vo< 0 (suction). A special solution of the equations (1), (2) which forms the basis for the theory of the boundary layer with suction and is used again below is the solution for the plane plate in longitudinal flow with homogeneous suction, thus vo(x) = vo = const < 0 and U(x) = Uo. For this case the boundary layer thickness becomes constant at same distance from the leading edge of the plate; also, the velocity distribution becomes independent of x (reference 5). From = 0 follows because of the continuity y = 0 and hence v(x, y) = vo = Constant From equation (1) then follows for the velocity distribution u(x, y) = u(y) = Uo e = Uo y/+ e (3) with *1 =  (4) VO NACA TM No. 1216 signifying the displacement thickness of the asymptotic solution. The wall shearing stress for this solution is: S c = pUovo (4a) It is independent of the viscosity. This asymptotic solution is one of the very rare cases where the boundary layer differential equations can be integrated in closed form. For solution of the boundary layer differential equations (1), (2) for the general case where the contour of the body and hence U(x) and also vo(x) are prescribed arbitrarily one could consider developing the velocity distribution from the stagnation point into a series in terms of x in the same way as for impermeable wall (reference 11); the coefficients of this series then are functions dependent on y for which ordinary differential equations result. K. Bussmann (reference 17) applied this method for the circular cylinder; with a very considerable expenditure of time for calculations the aim was attained there. However, for slenderer body shapes the difficulties of convergence increase so much that this method which works directly with the differential equations is useless for practical purposes. IV. THE GENERAL APPROXIMATION METHOD FOR ARBITRARY PRESSURE DISTRIBUTION AND ARBITRARY DISTRIBUTION OF THE SUCTION VELOCITY (a) The Expression for the Velocity Distribution For that reason one applies an approximation method which uses instead of the differential equations the momentum theorem which represents an integral of these differential equations. By integration of the equations (1), (2) over y between the limits y = 0 and y = one obtains in the known manner (reference 18): U2d + (2 + *) Ud Uv = ( dx dx o ) NACA TM No. 1216 7 0 sign!fies the momentum thickness, 5* the displacemeir: thickness,and To= the wall shearing stress. The approximation method :'or calculation of the boundary layer to be chosen here proceeds in such a manner that a pleasible expression is given for the velocity distribution in 'he boundary layer u(x, y) which is contained in equation (5) in 6, 5* and To. Thus an ordinary differential equation for 3(x) results from equation (5); after this differential equation has been solved one obtains the remaining characteristics of .he boundary layer 6*(x), To(x), and the velocity distribution u(x, y) in the boundary layer. The usefulness of this approximation method depends to a great extent on whether one succeeds In finding for u(x, y) an expression by appropriate functions. Pohlhausen (reference 15) first carried out this method for the boundary layer with impermeable wall. The velocity profiles in the boundary layer were approximated by a oneparametric family and the approxi mation function for the velocity distribution expressed as a polynome of the fourth degree. The coefficients of this polynome are determined by fulfilling for the velocity profiles a few boundary conditions which result from the differential equations of the boundary layer. This method proved to be satisfactory for the boundary layer without suction. Thus one proceeds in the same way for the boundary layer with suction. For the velocity distribution in the boundary layer one chooses the oneparametric expression *= Fl() + KF2(h) = (6) U 81(x) Fl(7) and F2(6) are fixed prescribed functions which are immediately expressed explicitly; K = K(x) is a form paramet.er of the boundary layer profiles,the distribution of which along the length is dependent on the body shape and the suction law; 51(x) is a measure for the local boundary layer thickness. The connection between 61 and 5* and 4 is given later. It proved useful to choose other expressions for the functions F1() and F2(j) than Pohlhausen for the impermeable wall. Fur the velocity profile according to equation (6) the following five b ndary conditions are prescribed; they all follow from the differential equations of the boundary layer with suction, equation (1), (2): JLA 6u 2 y = C: u = 0; v, Ut +V (7a,b) :oy oX (y2 Ou 22u y = m: u = Uo,; = 0 2 0; (7c,d,e) OY ,jv'2 NACA TM No. 1216 The selection of F1(n) and F2(1) is to be made from the view point that a few typical special cases of velocity profiles of the boundary layer with suction are represented by equation (6) as satisfactorily as possible. In particular we shall require the asymptotic suction profile according to equation (3) to be contained in the expression (6). This condition is satisfied if one puts Fl(n) = 1 e' (8) and correlates the values K = 0 and 81 = 8* to the asymptotic suction profile. Furthermore, the expression (6) naturally should yield usable results also for the limiting case of disappearing suction. To this purpose a good presentation of a typical boundary layer profile without suction is required. One chooses as this profile the plate flow for impermeable wall according to Blasius (reference 11). Since no convenient analytical formula exists for the exact solution of this case, a good approximation formula for Blasius' plate profile is needed. It is found u that the function o = sin(aq) gives a very good approximation to the Blasius profile (a = Constant).1 Thus one puts 0 < r 3: F2() = Fl sinn) (9) n > 3: F2(h) = FI 1 = e' and then obtains with K = 1 a good approximation for the plate flow without suction. The corresponding value of 51 is given later. The functions Fl(j) and F2(n) are given in figure 2 and table 1. Thus one has for the velocity distribution in the boundary layer the expression: IThat the sine function is a good approximation for the velocity distribution at the plane plate without suction, resulted from an invest iptie n of Mr. Iglisch about the asymptotic behavior of the plane stagnetion point flow for large blowing quantity (reference 20). NACA TM No. 1216 0 < T 3: = 1 + K [ i (10) > 3: = 1 (K + l)e By selection of the functions F1 and F2 the boundary conditions (7a,c,d,e) are per se satisfied. The last boundary condition,equation (Tb), results, because of I V [ K 1 (13) in the following qualifying equation for K: vU1 + K 1 =UU vU(1 + K) and from it with 1 U (12) Xl = v5 for K the equation K + X1  K : (1k) X and X1 are two dimensionless boundary layer parameters. A quantity analogous to X was already used by Pohlhausen for the boundary layer without suction; X1 is newly added by the suction. For the asymptotic suction profile with 51 = 6, X1 = 1 according to equation (4). Tne form parameter K as a function of X and X1 is represented in figure 3. NACA TM No. 1216 (b) The Differential Equation for the Momentum Thickness In order to obtain by means of the expressions (6), (8; from equation (5) the differential equation for the momentum one must first set up the relations between ,6*, and 51. displacement thickness there results: = 1 "=0 (1 Fl)dq K SF2d. 0 The calculation of the integral gives: = 1 K = g*(K) For the momentum thickness one obtains: 1 (FI + KF2)(1 F1 KF2)dT 1 ,0 =1 Co + ClK + C2K2 = g(K) The calculation of the integrals gives: Co = Fl(1 Fl)di = 10 2! C1= (F2 ,FIF2)dT, = 1 + = .6656 ^CT and (9) thickness For the (15) (16) (17) (18a) (18b) NACA TM No. 1216 F22d = 3* 12  Cl C2 = 2  I 1  6e = 0.02358 1 +\ 6 = 0.09014 Thus there Is i= 61 = For the form parameter one obtains therefore t + o.06656K 0.0235K2 = g(K) of the boundary layer profiles 5*/. (17a) used later 0, = I. ^  S+ C1K + C2K 1 0.09014K S+ o.o0656 o.02358K2 2 Furthermore there results according to equations (ll) and (17): 1 = g[ K (1 =f(K) (20) The functions g(K), 8*/3 and To01/,U according to equation (11) are represented in figure 4 and table 2. In order to derive from equation (5) a differential equation for 8(x) one writes equation (5) in the form US d0 U is 2 T 0 d + 2 + 9 dx v 0 U (21) Furthermore one introduces according to Holstein and Bohlen (reference 1?) C2 =  0l" Hence (18c) (19) NACA TM No. 1216 2 U1 = K = Xg2 Vo3 = El = %lg (22) (23) (24) S2 Z = v K = ZU'; KI = Vo Z (25) is valid. With equations (22) to (25) as well as equation (20) the differential equation (21) is transformed into 1 K(2 6 U + L = f(K) dx g(K)  If one finally puts for abbreviation: G(K iK) = 2f 2K 2   2K1 (K)the differential equation for () become: the differential equation for Z(x) becomes: (26) (27) (28) With then NACA TM No. 1216 If the function G(K, Il) is known, the integral curve Z(x) can be calculated from this equation by means of the isocline methAd. For carrying out the calculation in practice it is useful t,. introduce dimensionless quantities. One forms them with the aid ,of the free stream velocity Uo and a length of reference 1 (for instance, chord of the wing). Thus one puts ZUo x vo(*) JJ z* = ; x* =; Uo V) = (x*). (29) Then equation (28) becomes: Z C(K, K1) z z ( dU G( = u/uol) = Uo 6 K = fl(x*) f7 (*0) dx' U/U0 ZUdx' The function G(K, cl) is calculated as follows: First, one obtains K and Ki as functions of X and X1 from equations (22) and (23), if one takes the connection between K and X, XI according to equation (1I) into consideration: K = g(K)X = g2(X, X1)X (1) K = g(K)xl = g(x, xl)j1 From equations (27) and (31) follows: 21 =g[1+K( ] 2 K( )21 2Kj Xg G = g 1 + Kl ) 2 [ 2 ] x G = 2gF(X, il) (32) NACA TM No. 1216 with F(X, Xj) 1 + K( 2g X [1 K(2 J 1 (33) Hence G can be calculated first as function of ), X, and then, because of equation (31), also as function of K, Kl. The functions K(X, X1) and Kl(X, Xi) are represented in figure 5 and table 3. The function thus determined G(K, Kl) is given in figure 6 and table 3. (c) Stagnation Point and Separation Point The behavior of the differential equation (28) at the stagnation point where U = 0 requires special considerations. In order that the initial inclination of the integral curve (dZ/dx)o at this point be of finite value, G( i, KI) must equal zero. This gives the corresponding initial values Ko, %lo. Since the function g(K) does not have a zero for the values of K considered (compare fig. 4) the determination of the initial values no, 'lo amounts to the zeros of F(ko, Xlo) = 0 (34) The resulting initial values at the stagnation point \o, Xlo are given in table 4, together with the initial values Ko, Klo calculated additionally according to equation (31). To each pair of values Ko, Ilo corresponds a mass coefficient of suction which results from ,o2 o(o0)o Vy = *o v *10 as vo(o) K10 Co In figure 7 the initial values Ko and lo are plotted against the local mass coefficient at the stagnation point. The initial value Zo corresponding to Ko is obtained by NACA TM No. 1216 Zo = _) Uo ' The following connection exists between the distribution function of the suction fl(x) = vx) and Co: Uo' = KlUo/, K, being Uo a profile constant. Thus there is and So(o) vo(o) / 1 UV Uo0 fl(o) Co (36) The determination of the initial values of the integral curve proceeds, therefore, as follows: With the given initial value of the suction velocity at the stagnation point fl(o) one first determines Co according to equation (36). One obtains the corresponding initial values Ko and Klo from figure 7, and according to equation (35) the initial value Zo of the integral curve. If the suction does not begin at the stagnation point but further downstream, Co = 0; Klo = 0 and according to figure 7 0.0709 Ko = 0.0709; Zo =  Uo ' Separation point. The separation point is defined by the fact that the wall shearing stress there equals 7pro. This gives for K, according 6 to equation (11), the value K = = 2.099. For the asymptotic suction profile K = 0; this is simultaneously the greatest possible value of K. 11 To K = 2.099 corresponds the value = = 1.099 for all X1, and 6  NACA TM No. 1216 K = 0.0721 for all Ml. However, if one would want to carry the boundary layer calculation up to this point, certain difficulties result in the last part shortly ahead of this point, since the correlation between K and X is not unequivocal there (compare fig. 4). The function G(K, 1i) against K also is not unequivocal shortly ahead of this point. Thus it is useful to select a point situated somewhat further upstream as separation point where the boundary layer calculation has to stop. Such a point results if one chooses the K value of an exact separation profile according to Hartree (reference 13). For this latter there is: separation: KA = ( A = 0.0682 (37) One defines this point as separation point of the present boundary layer calculation for all mass coefficients of suction. The following table gives a survey of the values of 4 and 5* at the separation point for four different calculation methods: The selection of the separation point thus made is somewhat arbitrary; however, it may be accepted unhesitatingly since, as is well known, the approximation methods for the boundary layer calculation in the region of the pressure rise are always somewhat uncertain and only a rough estimate but no exact calculation of the boundary layer parameters is possible here. For the same reason one may also accept the fact that for the present case the velocity distribution u/Uo partly assumes, shortly ahead of the separation point, values which are slightly larger than 1. .2 5*2 Case 2U' = A U' = New method: (sine (New method: in 0.0721) (1.55) approximation) Pohlhausen P4 (reference 15) 192 Exact Hartree 682 (reference 13) Exact Howarth 1 (reference 14) NACA TM No. 1216 (d) Performance of the Calculation for the General Case By means of the system of formulas given above one may perform the calculation of the boundary layer for an arbitrarily prescribed body shape and an arbitrary distribution of the suction velocity along the wall in the flow. It takes the following course: To the distribution of the suction velocity the total suction quantity Q1= x=0 v,(x) corresponds Vo(x) dx = cqUobl and the reduced mass coefficient C* = cQ^ Rv and the reduced suction distribution function according to equation (29): v (x*) U fi(x*) = Thus there is cQ* = *=o fl(x*) dx* (39) If the suction begins at the stagnation point, one determines with KI = ( the mass coefficient C, at the saar.na;on pint according to equation (36). Then one obtains Ko and K from figure 7 and Zo according to equation (3'.) Witn these initial value! the differential equation (30) can now be graphically inLegrated by means of the diagrem in figure 6. The calculation is carried ot up to the point where i reaches the value KA = 0.0682. This intehre.:n immediately yields Z*, i, ~i as function of x*, wi h (3') NACA TM No. 1216 r= . (40) The remaining boundary layer parameters then result as follows: By means of figure 5 one obtains after i and Kl the parameters X and X1 and additionally from figure 3 the form parameter K. After K one obtains from figure 4 the form parameter 6*/0 and thus From equation (20) one then also obtains the wall shearing stress To To Vi f(K) I = (40a) U Uo U Finally, the parameter 61 is required for the velocity distribution in the boundary layer. According to equations (6), (7), and (40): Y Y (K)gl g (41) Examples of such boundary layer calculations are given in chapter VI. V. SPECIAL CASES A. WITHOUT SUCTION Our general system of formulas is to be specialized in this section for a few typical special cases for which one can partly give solutions in closed form. First, the case without suction in particular shall be treated for which, naturally, our equations also must give satisfactory results. This case one obtains for vo(x) 0; then NACA TM No. 1216 X1 O l1 5 and equation (14) is transformed K Therewith, according to equation S1 s = g(x) = + 0 (without suction) into = X 1 (17): C1(x 1) + C2(X 1)2 (42) = + 0.06656 (x 1) 0.02358 (x 1) (43) The differential equation (28) for the momentum thickness becomes dZ G9I) zu dx u G(K) is, according to equation (32) and (33): c = 2gF(X) F() = 1 + (X 1)( ) 2x + Cl( 1) + C2( 1)2] 1( + (i )2 ,6)] F(X) = 2C23 + (2C2 2 + )2 + )2 + (44) (46) (46) Furthermore, according i = g2A to equation (31): = X + Cl(x 1) + C2(X 1) 2 (47) NACA TM No. 1216 The values of G and K calculated according to equations (43), (45), (46), (47) are given in table 3. (a) The Plane Plate in Longitudinal Flow The boundary (Blasius) which U to equations (42) layer at the plate in longitudinal flow without suction = Uo is obtained for X = K = 0. Then, according to (46): K = 1 1 6= g(O, 0) =2 C C2 2 F(0, O) = 6 G(O, 0) = 2 ( = 2 = 0.429 6 n ; 2 (1486) With the Initial value Zo = 0 the integration of equation (44) then gives: Z = 2 ) or 4 / = , = 0o.655 Cv For the form parameter 8*/a follows from equation (19): S 2 = 2 = 2.66 4  (49) (o0) NACA TM No. 1216 and thus for the displacement thickness w = ( 2) = 1.740 (l) wH For the coefficient of the total friction drag cf = WUo of the QUo2b plate of the width b and the length 1, wetted on one side, one 26 obtains because of cf =  cf = = 1.308 (52) Finally, the velocity distribution is, according to equation (10), 0 < <_ 3: u = Uo sin ( ) 3: u = U y Therein Is = g and 1 = = 1. 60i (54) In figure 8 the velocity distribution according to equations (53) and (CA) is compared with the exact solution of Blasius; the agreement is very good. Furthermore the characteristics of the boundary layer according to the present approximation calculation are compared with the values of the exact solution of Blasius in the following table. For further comparison the values according to the approximation method of Pohlhausen (reference li) also have been given. The agreement of our new epjroxl ma:,in nmehod with the exact solution is excellent for all boundary layer paerametLers; the drag coefficient, in particular, shows an error of onl: 2 percent. NACA TM No. 1216 Coefficients of the Boundary Layer at the Flat Plate in Longitudinal Flow Without Suction Calculation method V U  * r b / uo Pohlhausen Ph (reference P ) 1.750 0.685 2.55 1370 0.234 (reference 15) Exact (Blasius) 1.721 0.664 2.59 1.328 0.220 The deviations of our sine approximation from the exact solution are, for most characteristics, even somewhat smaller than in the Pohlhausen method. (b) The Plane Stagnation Point Flow For the plane stagnation point flow the velocity of the potential flow U(x) = ulX. All boundary layer characteristics are in this case independent of the length x. The initial value of the momentum thickness Zo is obtained from equation (44) for G(Ko) = 0. Since g(K) does not vanish in the range of the values of K considered, there must be F(Xo) = 0. From equation (46) one finds as zero of F(X) the value Xo = 0.3547 (stagnation point without suction) (54) The corresponding values of K and K according to equations (42) and (47) are Ko = 0.6453 and Ko = 0.0709; furthermore there is, according to equation (43): g(Xo) = C.447. Therewith the momentum thickness for the plane stagnation point flow becomes: = = orii = 0.66/ (55) The form parameter 6*/l results from equation (19) as 5*/8 = 2.37; therewith one has NACA TM No. 1216 6* = 2.37 = 0.630 Furthermore, according to equation (43): 61 = 0.595 r . results from equation (11) for the wall shearing stress: T v  = 1.163 The velocity distribution results from equation (10) as: o0 T Thus there (7)  = 0.3574(1 e') + 0.6453 sinCfi 1 21 3: *= 1 0.3574en Uo j (58) (58a) TI = = 1.68y b 1 Figure 8 gives a comparison between velocity distribution according to equation (58) and the exact solution by Hiemenz (reference 12); here also the agreement is satisfactory. Furthermore the characteristics of the boundary layer according to the present calculation are again compared with the exact solution by Hiemenz and with the approximate calculation by Pohlhausen in the following table. (56) with NACA TM No. 1216 Coefficients of the Boundary Layer of the Plane Stagnation Point Flow without Suction Calculation method 13 [ 8* U 1 i 0' New method (sine approximation) 0.266 0.630 2.37 1.163 0.310 Pohlhausen P4 (reference 15) 0.278 0.661 2.31 1.19 0.331 Exact (Hiemenz) 0.292 0.648 2.21 1.234 0.360 The agreement of the new method with the exact solution is for this case somewhat less satisfactory than for the plane plate; neither is it quite as good as the approximation of Pohlhausen. But even here the new method yields still very useful values. B. WITH SUCTION In this section a few cases with suction will be treated far which the solutions can be given in closed form. First we shall treat the boundary layer at the plate in longitudinal flow with homogeneous suction, already investigated formerly (reference 6). The following results are considerably more accurate than those former ones. (a) Growth of the Boundary Layer for the Plate in Longitudinal Flow with Homogeneous Suction For this case the boundary layer is at large distance from the leading edge of the plate independent of x; hence all boundary layer parameters are constant. The corresponding asymptotic solution has been given already in equations (3), (4), (4a). The applying values are: SV g5 * 5' rg~ 81 = 5* E Tom PUovo 1 ; = i l = i 1 K = 0 (59) NACA TM No. 1216 One now calculates the growth of the boundary layer from the value zero at the leading edge of the plate to the given asymptotic value. In our system of formulas one has to put for it: Therewith bees 0;to equation () with the abbreviation Therewith becomes according to equation (1i) with the abbreviation S1 = c = 0.764 6 K = 1 1 1 c01 (6c) (61) and according to equation (17) = g(O, X1) 1 Po + Pl1 + P2)kl2 (1 cXl)2 Po = = 0.30986 6= 1 = 0.06656 p2 = 2 + + 2 + C = 0.05819 7C 2 \ V Furthermore there is according to equation (16) 8* = 6 + 51  with (62) (62a) (63) NACA TM No. 1216 The wall shearing stress becomes according to equations (11) and (13): Uo 6 To = S 1] 1 c11 ii pvoUo j(1 cX1) The differenLial equation (28) assumes for the present case the form: dZ G( (, C1). dx Uo i = "VoV vo = Constant < 0 (65) The integration of this differential equation requires the explicit expression for G(o, "l). According to (32) and (33): G(o, AI) = 2g(  ScX = 2g(l X1) 1 cX1 Thus G(o, Ki) = O for X1 = 1. Therefore X1 = 1 is a solution of the momentum equation; it corresponds to the asymptotic solution. The initial value at the leading edge of the plate is X1 = 0. For the length of growing boundary layer X1 varies from 0 to 1. If one introduces as dimensionless distance along the plate 2 (67) the differential equation (65) can be written in the form: d(=i2) = CG(K1) dS Initial value: 9 = O0 K1 = 0 (64) (66) (68) =T V3 V7) (68a) NACA TM No. 1216 The connection between l1 and. I is given by 1r = klg(Xl) (69) with g(Xk) according to equation (62). The differential equation (68) can be solved according to the isocline method. For the present case, however, an analytical solution, too, is possible which is preferable. From equation (69) first follows: 2 !K1 Gd 1 dKl dXl d Here all quantities can be best expressed by equation for xl() results. With dal/d1l one obtains from equation (70) after division does not disappear in the range 0 I X, 1: = (1 x1) XI so that a differential according to equation (69) by 2g since the latter  ckI 1 cX1 Initial value: I = 0: X1 = O Because of of t = 0, (71) (71a) 6 ? g(o, o) Po = " one obtains from it in the neighbourhood X1 = 0: dS 6 66 0 1 "= n p = 2) l t = 2 12 = .391Xi2 it O (70) and (72) (7?) X1 8( dg dX dXl dC1I 28 NACA TM No. 1216 Hence follows for the neighborhood of the leading edge of the plate (P = 0) Because of A1 = 1.60o or 6  follows hence: 11 8* 81 v 8*  = ( 2) As the ccmIparison with equation (51) shows, Lhe boundary starts, therefore, at the leading edge of the plate with the plate without suction. layer Lhickness the value for In order to integrate the equation (71), one has to insert the explicit values of g(X1) and dg/dXi according to equation (62). After some intermediate calculating (compare appendix I) one obtains 1pP0 + Pll P212 + PI3) (1 cX)2(l Xi) cX Po = Po = 0.409e6 Pl = 2p + Poc = 0.4667 P2 = 3p2 = 0.17157 P, = cp2 = 0.02772 (74) (75) Y 1= 1.60V V 14I' =2 _/ It with 8 (t 2) T_ U d @71 ' NACA TM No. 1216 The breaking up into partial fractions yields: d_ P; S + dX c'3 Kl K2 K. >1 1 C11 (cl 1)2 The integration with the initial value X1 = 0 = c3 K4 + = f'(Xl) cA1 1 for t = 0 yields + KI In (1 X1) + 2 In   K X +  In (1 ca1) = f(k1) 3 C lA 1 The Kl, ..., K4 result from the breaking up into partial fractions as K1 = 6.9560; K2 = 3.4704; K3 = 0.2284; K4 = 0.1569 Thus the solution finally reads I = 0.2564i 6.956 In (1 x1) + 7.2846 In (1 0.9099X1) 4 0.2284 l + 0.22846 x 0.3293 In (1 0.476411) 0.4764x1 1 (79) 2 For development of this solution in the neighborhood of t = 0, I1 = 0, the coefficient of X1 must, because of equation (73), equal zero; the coefficient of 2 J. must equal ( ) = 0.391. n d The result is: 3 K1 + X K = 3 3 K4 0 C K1 c 2 K2 + CKIKC 4it \V/ (77a) (77b) WiTh the numerical values of equation (78) one may verify that these equations are satisfied. (76) (77) (78) NACA TM No. 1216 The solution Xl(S) calculated accordingly is given in table 5. From X1(t) all remaining boundary layer parameters can then be calculated immediately according to equations (61), (62), (63), and (64). They also are "vo8* 705* given in table 5. In figure 9 5 and. are plotted against (. The displacement thickness of the boundary layer reaches 0.95 of its asymptotic value after an extent of the growing boundary layer of tA )2 UV2 4.5. The velocity profiles in the growing boundary .voy layer in the plotting u/Uo against  = ~11 are represented in figure 10. For the wall shearing stress one obtains from equations (64) and (0a) TO 6 = 6 (80) o. X1(1 chl) The wall shearing stress is plotted in figure 11 as a function of / Drag. In view of the reduction of the drag by maintenance of a laminar boundary layer the friction drag in the extent of growing boundary layer is of particular interest for this solution. For the asymptotic solution the local friction drag along the wall is constant with Tom = PVoUo; thus the coefficient of the total friction drag also equals this value W TOG v S  (81) Uo2 b o20 For small suction quantities vo/Uo the extent of growing boundary layer is sometimes so large that the growth is not finished by far at the end of the plate. According to former investigations (references 8 and 10) it is to be expected that for homogeneous suction at the plate the maintenance of a laminar boundary laier is possible even for Reynolds U. numbersof the order of magnitude j = 107 to 10 with a very small suction quantity of the order of magnitude Q = = 10 For 10 Uo0 7 U Uo and = 107 or 108 one has at the end of the plate I = 2 U! = 0.1 or 1, that is, the growth of the boundary layer is not finished by far. NACA TM No. 1216 Since the friction layer over the extent of growing boundary layer is thinner, the friction drag there is considerably larger than for the asymptotic solution. For this reason the calculation of the drag over the extent of growing boundary layer will be given completely. The total friction drag for the plate wetted on one side is: W = b To dx (82) o(82) and with the value of o7 according to equation (80) and with To, according to equation (4a): W = oUobl l) 'Oo6 L0 Xl(0 cxl) With dx = dt according to equation (67) this equation becomes: 2 v W = pbUo2 V  S (1 c)dt 6 X1(1 ci) t=0 with E, signifying the value of t at the end of the plate, thus: (L2 St lD U0 = f(lo) V= r(X0) Therein Xlo signifies the value of X1 at the end of the plate which is obtained from equation (77) for t = ~1i therefore f(klo) 3 lo + Kl n (1 lo) + In 1 c33 3xio 4 i ( (%.itl0) K3 lo + In (1 cXlo) (85) cXlo 1 (83) (84) )g2 2L NACA TI No. 1216 Introducing in equation (83) for = f'(x) dX1 to equation (76) one obtains: the expression according W = pbUo2 F(klo) To and W cf =  Uo2 bl (86) = 2 IF(olo) vO1 with F()io) signifying =lo 1Xl=0 F(0lo) = "I 6 fl'(X) d x Xl(1 cxi) Finally introducing V . r2 f(Xlo) according to equation (84) into equation (86) one obtains =v F(Xlo) To c = 2 = 2 G(Xlo) S Uo F(Xlo) Uo cf = cfW G(Alo) (88) Because of the connection between Xlo and EZ according to equation (84), the total drag coefficient for the extent of growing boundary layer is thereby given as a function of the dimensionless distance along the plate = ) On the other hand, equations (84) (87) NACA TM No. 1216 and (88) give for prescribed mass coefficient of the suction vo/Uo the drag law cf also against Uol/V in the form of a parameter representation. The parameter Xlo is the dimensionless boundary layer thickness at the end of the plate: lo =1v The values Sv x= of Xlo lie between 0 and 1, the first value being valid at the leading edge of the plate, the latter for the asymptotic solution, after the growth of the boundary layer has ended. The calculation of the integral F(Xlo) according to equation (87) gives (appendix II) F(Xio) = KlC 1 c K2n( clo) c4 K + In(l cklo) + 1 Zn(1 lo) + AK2 In ( lo) cXlo  (K3 + K _) 1 cXlo K2 2+ cX1o(2 c l) (1 cXlo) (89) c is, according to equation (60), c = 1 , and Kl) ..., K4 are given by equations (75) and (78). After insertion of the numerical values according to equations (60) and (78) follows: F(Xlo) = 0.3288 In (1 0.4764X1o) 6,956 In (1 10o) + 7,2846 In (1 0.9099X1o) 0. 764Xlo 0.4764X1o (2 0.476411o) 0.0374 0.05980 1 0.4764Xlo (1 o.4764xlo)2 The values of F(Xlo) and G(Xlo) are given in table 6. For Xlo 1 that is S>o (growth of boundary layer ended) one has, as can immediately be seen from equations (89) and (77): NACA TM No. 1216 F(Xlo) lo >:1 o(klo) = 1 f(klo) (89a) and thus cf cr for s )4 On the other hand one has in the neighborhood of the leading edge of the plate, that is, for Xlo> 0 according to equation (73) \lo >0: f(klo) = ( C lo2 It 2] 0 and thus according to equation (87) lo >0: F(o10) = i )l and therefore o 0: c(Xlo) = 1 = lo (89b) If one substitutes this value into equation (88) and takes into consideration that i tI 2) is valid for small Xlo according to equation (73), one obtains Cf = 2 /IF Uo 4 /Ir" rU0 2"/ NACA TM No. 1216 or c = 2 U y1/2 thus the drag law of the plate without suction according to equation (52). The drag law of the length of growing boundary layer is therefore for very small lengths of growing boundary tj asymptotically transformed into the drag law of the plate without suction. The drag law according to equation (88) is represented in figure 12, where cf/cf, is plotted against Et. Furthermore figure 13 gives the drag law in the form cf against Uol/V for various values of the mass coefficient vo/Uo. The larger the suction quantity the smaller the Reynolds number at which the respective cf curve separates from the drag curve of the plate without suction and is transformed, after a 2vo certain transition region, into the asymptotic curve cf = Uo The Reynolds number at which the latter is reached is the larger, the smaller the suction quantity. The drag coefficients given here represent the total drag of the plate with suction. No special sink drag is added (compare reference 10) since for continuous suction, as in the present case, the sucked particles of fluid have already given up their entire x momentum in the boundary layer so that this momentum is contained in the friction drag. In order to obtain the total drag power of the plate with suction, however, one must, aside from the drag given here, take into account the blower power of the suction. (b) The Plate Stagnation Point Flow with Homogeneous Suction Another special case which can be solved in closed form is the plane stagnation point flow with homogeneous suction. Since for this case the exact solution from the differential equations of the boundary layer has been given elsewhere (reference 9) it shall also briefly be treated here. The potential flow is U(x) = ulx and the suction velocity Vo(x) = vo(o) = voo <0. If the integral curve of equation (28) is to have a finite value at the stagnation point x = 0, there has to be G(K i) = 0; this in turn requires, as was discussed in detail in chapter IV c, F(X, Xl) = 0. F(x, 1i) Is given by equation (33). The values of Xo, Xlo which belong together follow from it; they give for the general case the initial values of the boundary layer calculation at the stagnation point; for the present case of stagnation point flow they NACA TM No. 1216 immediately give the complete solution since the boundary layer thickness and all other parameters are independent of the length of growing boundary layer x. Besides Mi, X one further obtains K according to equation (14), g according to equation (17), 8*/C according to equation (19), and so and Klo according to equation (31). The mass coefficient Co = _oo is obtained according to equation (34). From Ko finally follows o and therewith 8*. The results are compiled in table 4. Naturally, momentum and displacement thickness decrease with increasing suction quantity. With Co> the form parameter 8*/C approaches the value 2 of the asymptotic suction profile. In figure 14 9 fuj/V and 8* ul/v are plotted against Co and compared with the exact solution. The agreement is quite satisfactory. As conclusion of these considerations of the special cases the characteristic boundary layer parameters for these special cases are compiled in the following table. NACA TM No. 1216 I i OJ 0  o0 t 0 O CJ 0 r o lo o 00 00 Q0 r4 0 0 0 oo 0t rI 03r OP H 0 0 c0 U H1  C U o 0 O I I*  C\ S0 0 0 0 o I SI V C .) 0 ,C 0 :: pq 0 0 m 0 0 P4 4) a  0 M +) C 0 4  S 43 4, 0 0 1 0) 40 0 0l 0 pq I I3 < tBI 0v ; +J C~ I4t 0 a, 3 , C: 0 + Sa fo m*r o 0 a,, ,C 0 a OH e 0 u CO T' Q 5 a oC a 0 . *rI 04' 0 0 0 *r4 NACA TM No. 1216 are given in figure 15. For the case without suction P = 101.70 results as separation point; this is slightly further to the front than for the customary Pohlhausen method (cp = 108.9) for which the calculation was performed elsewhere (reference 7). With increasing suction quantity results a reduction of the boundary layer thickness and a shifting of the separation point toward the rear. In order to completely avoid the separation for the circular cylinder, it is probably useful to select not a homogeneous suction along the contour, as in the present case, but a distribution of vo(x) which has considerably larger values on the rear than on the foreside. Euch calculations may also be carried out according to the present method without additional expenditure of time. A comparison of the present approximate calculation with an exact calculation by K. Bussmann (reference 17) for the displacement and momentum thickness is given in figure 16. The latter calculation is a development in power series starting from the stagnation point, as first indicated by Blasius (reference 11). Except for the neighborhood of the separation point the agreement is quite satisfactory. (b) Symmetrical Joukowsky Profiles for ca = 0 As second example a symmetrical Joukowsky profile of 15 percent thickness has been calculated for ca = 0, also with homogeneous suction. The suction extends over the entire contour. The same profile without suction has been calculated elsewhere (reference 7), also according to the Pohlhausen method. Here, too, a reduction of the boundary layer thickness and a shifting of the separation point toward the rear results with increasing suction quantity. For the suction quantity Co = 0.417, that is fl(0) = 3, a separation does no longer occur. VII. SUMMARY A method of approximation for calculation of the laminar boundary layer with suction for arbitrary body contour and arbitrary distribution of the suction quantity along the contour of the body in the flow is developed. The method is related to the wellknown Pohlhausen method for calculation of the laminar boundary layer without suction. The calculation requires the integration of a differential equation of the first order according to the Isocline method. The method is applied to several special cases for which there also exist, in part, exact solutions: Plate in longi tudinal flow and plane stagnation point flow with homogeneous suction. Furthermore the circular cylinder and symmetrical Joukowsky profile with homogeneous suction were calculated as examples. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM No. 1216 V I I. A P P E ND I X ES APPENDIX I Concerning the Length of Growing Boundary Layer for the Plane Plate with Homogeneous Suction According to equation (71) is ~ 6 dXE d1 1 cX1 l + (1 (1 cl) (71) a(d) dt 1 cx1 From equation (62) one finrs: dg (1 cX)(pl + 2p2X1) + 2c(1 cxl)(po + pi*X p2X2) dX1 (1 cx 1) S+ 2cpo + (Plc + 2p2)X1 = (I,1) (1 c )3 .Substitution of equations(I,l) and (62) into equation (71) gives: 40 40 RACA Th No. 1216 * H whl rA ^i cu Hm H O 0 H pH u so ,e4 I i II H _r_ +___ _ H + r 4 m +o++ 1 o R Hu H P a U H 11 H4 C H + A 2 i 0 A H H + r S)' + +  II . .0 + H H .4 1 ) 0 .4 0 c 35; NACA TM No. 1216 APPENDIX II Concerning the Calculation of the Drag of the Plane Plate with Homogeneous Suction The calculation of f'(Xl) dX1 X1(1 cxl) (87) F llo F(Xlo) =  x1=0 gives with f'(xl) according to equation (76), if 1I is replaced by 03(1 !) c l dz c3 z(1 cz) 1z=0 + K1 l z=0 dz (z l)z(l cz) + K2 J  K Z The integrals are finds: 03 dz cz 4)z(l cz) + 1 1 + K3 Iz0 Vz=O dz (cz 1) 2z( cz) diz z = = I + II + III + IV + V (cz l) z (II,1) solved by breaking up into partial fractions. One in z In z ~~1 z=0 II = K In z + n (z 1) + c 1  2In z=0 NACA TM No. 1216 II3 K2 1 n z + I n z ) n ( . z=0 IV = K3 [n z In(cz 1) z 2] cz 1 (cz 1)2 V = Kg In z +  + In(cz 1 I cz 1 When summed up, all terms with [n z lo cancel each other, because 0 of equation (77a). After insertion of the limits the remaining terms give: n(cz 1) Il = In(1 cxlo); [n(z 1 lo 0 0 = In(l Xlo) [n z o = In x cr 11lo L" ^ F 1 XAlo (cz 1) 2O cXlo(2 c1lo) (1 cXlo)2 S clo 1 'Xlo NACA TM No. 1216 Thus there results by simplification from equation (II,1): (Xloc) = 1  ^c3 1 c  K3 K c 3 K4) In(l clo) + 6 In(l Alo) + k2 In(l ilo ScXlo K cklo(2 cklo)  (K3 + K) + 2 1 c0o (1 cAXo) (89) NACA TM No. 1216 APPENDIX III To page 30. Table 10 gives the numerical table concerning the velocity distribution of figure 10. To page 35. For the boundary on the plate in longitudinal flow with homogeneous suction the exact solution from the differential equations also was given in an unpublished report by Iglisch. A comparison of the approximate solution above with that exact solution is given in figures 18 and 19. Figure 18 gives the comparison of the displacement and momentum thickness; particularly for the displacement thickness the agreement is good. Figure 19 gives the comparison for the wall shearing stress; here also the agreement is satisfactory. (So far, these comparisons can be carried out only for the front part of the length of growing boundary layer, up to 6 = 0.5, since the exact solution does not yet completely exist.) NACA TM No. 1216 REFERENCES 1. Betz, A.: Beeinflussung der Grenzschicht und ihre praktische Verwertung. Jahrb. Dtsch. Akad. Luftfahrtforschung 1939/40, p. 246 and Schriften d. Dtsch. Akad. Luftfahrtforschung Heft 49 (1942). 2. Holstein, H.: Messungen zur Lamlnarhaltung der Grenzschicht durch Absaugung an einem Tragflrgel. Bericht S. 10 der Lilienthal Gesellschaft fur Luftfahrtforschung 1942. 3. Holstein, H.: Messungen zur Laminarhaltung der Reibungsschicht durch Absaugung an einem Tragfliigel mit Prof l NACA 001264, FB 1654, 1942. 4. Ackeret, J., Ras, M., and Pfenninger, W.: Verhlnderung des Turbulent werdens einer Reibungeachicht durch Absaugung. Die Naturwissen schaften 1941, p. 622. 5. Schlichting, H.: Die Grenzschicht mit Absaugung und Ausblasen. Luftfahrtforechung Bd. 19, p. 179 (1942). 6. Schlichting, H.: Die Grenzschicht an der ebenen Platte mit Absaugung und Ausblasen. Luftfahrtforschung Bd. 19,p. 293 (1942). 7. Schlichting, H., and Ulrich, A.: Zur Berechnung des Umschlages laminar/turbulent. Bericht E. 10 der LilienthalGesellschaft fir Luftfahrtforschung, p. 75, 1942 and Jahrb. 1942 der Dtsch. Luftfahrt forschung, p. I 8. 8. Busamann, K., and Minz, H.: Uber die Stabilitft derlaminaren Reibungsschicht. Jahrb. 1942 der dtsch. Luftfahrtforschung, p. I 36. 9. Schlichting, H., and Bussmann, K.: Exakte Lisungen fur die laminare Grenzschicht mit Absaugen und Ausblasen. Schriften der Dtsch. Akad. d. Luftfahrtforschung, 1943. 10. Schlichting, H.: Die Beeinflussung der Grenzschicht durch Absaugen und Ausblasen. Lecture to the Deutschen Akademie der Luftfahrt forschung,May 7, 1943; to be published soon. 11. Blasius, H.: Grenzschichten in FlUssigkeiten mit kleiner Relbung. Zschr. Math. u. Phys., Bd. '6, p. 1 (1908). 12. Hiemenz, K.: Die Grenzschicht an einem in den gleichmissigen Flissigkeitestram eingetauchten Kreiszyllnder. Dingl. Polytechn. Journal. Bd. 326, p. 321 (1911). NACA TM No. 1216 13. Hartree, D. R.: On an Equation Occurring in Falkner and Skan's Appr";imate Treatment of the Equations of the Boundary Layer. Cambridge Phil. Soc. Vol. 33, p. 223 (1937). 14. Howarth, L.: On the Solution of the Laminar Boundary Layer Equations. Proc. Roy. Soc. London A No. 919, Vol. 164 (1938), p. 547. 15. Pohlhausen, K.: Zur nKherungsweisen Integration der Differential gleichung der laminaren Grenzschicht. Zechr. angew. Math. u. Mech. Bd. 1, p. 252 (1921). 16. Ulrich, A.: Die Stabilitit der laminaren Reibungsschicht an der langsangestramten Platte mit Absaugung und Ausblasen. Bericht 43/9 des Aerodynamischen Institute der T. H. Braunschweig; to be published soon. 17. Bussmann, K.: Exakte Lbsungen fUr die Grenzschicht am Kreiszylinder mit Absaugen und Ausblasen. Not published. 18. Prandtl, L.: The Mechanics of Viscous Fluids. Durand, Aerodynamic Theory vol. III, Berlin 1935. 19. Holstein, H., and Bohlen, T.: Ein vereinfachtes Verfahren zur Berechnung laminarer Relbungsschichten, die dam Ansatz von K. Pohlhausen geniigen. Bericht S. 10 der LilienthalGesellschaft fir Luftfahrtforschung 1942. 20. Igllsch, R.: Uber das asymptotische Verhalten der Ldsungen einer nichtlinearen gewbhnlichen Differentialgleichung 3. Ordnung. Bericht 43/14 des Aerodynamischen Institute der T. H. Braunschweig; to be published soon. NACA TM No. 1216 TABLE 1 THE BASIC FUNCTIONS FI AND F2 FOR THE VELOCITY DISTRIBUTION IN THE BOUNDARY LAYER WITH SUCTION SFI F2 0 0 0 .2 .1813 .0768 .4 .3297 .1221 .6 .4512 .1423 .8 .5507 .1452 1.0 .6321 .1322 1.2 .6988 .1112 1.4 .7534 .0843 1.6 .7981 .0551 1.8 .8347 .0263 2.0 .8647 .0017 2.2 .8892 .0240 2.4 .9093 .0416 2.6 .9257 .0524 2.8 .9392 .0552 3.0 .9502 .0498 3.5 .9698 .0302 4.0 .9817 .0183 4.5 .9889 .0111 5.0 .9933 .0067 6.0 .9975 .0025 7.0 .9991 .0009 m 1 0 NACA TM No. 1216 TABLE 2 PARAMETER OF BOUNDARY LAYER WITH SUCTION K 6 6* ToS0 Tro K g _ g s E T rU u a0 0.5 1 2 1 0.5 .1 .4931 1.009 2.05 .9524 .4696 .2 .4856 1.018 2.10 .9047 .4393 .3 .4779 1.027 2.15 .8571 .4096 .4 .4696 1.036 2.21 .8094 .3801 .5 .4608 1.045 2.27 .7618 .3510 .6 .4516 1.054 2.33 .7142 .3225 S.6453 .4472 1.058 2.37 .6926 .3097 7 .4419 1.063 2.41 .6665 .2945 .8 .4317 1.072 2.48 .6189 .2672 .9 .4210 1.081 2.57 .5712 .2405 C 10. .4099 1.090 2.66 .5236 .2146 1.1 .3982 1.099 2.76 .4760 .1895 1.2 .3862 1.108 2.87 .4283 .1654 13 .3736 1.117 2.99 .3807 .1461 1.4 .3606 1.126 3.12 3330 .1201 1.5 .3471 1.135 3.27 .2854 .0991 1.6 .3331 1.144 3.43 .2378 .0792 1.7 .3186 1.153 3.62 .1901 .0606 1.8 .3037 1.162 3.82 .1425 .0433 1.9 .2883 1.171 4.09 .0948 .0273 2.0 .2724 1.180 4.33 .0472 .0129 d2.099 .2562 1.189 4. 6 0 0 Asymptotic suction profile. bStagnation point without suction. cPlane plate without suction. dSeparation point. NACA TM No. 1216 TABLE 3 THE FUNCTION G(i, 1) FOR THE INTEGRATION OF THE DIFFERENTIAL EQUATION OF THE MOMENTUM THICKNESS mI = 0 without suction 0. = 0.1 IX 1 ,( K,K ) x x 11I G(K,rl) 0.1062 .0937 .0816 .0709 .0586 .0477 .0373 .0273 .0177 .0086 o .0159 .0298 .0419 .0528 .0602 .0666 .0711 .0738 .0748 .0742 .0721 .0682 1 1 , 0.2042 .1323 .0621 0 .0729 .1375 .2001 .2607 .3191 3753 .4292 .5301 .6213 .7023 .7730 8330 .8820 .9198 .9470 .9617 .9656 .9584 0.5 .4 .3 0 .2 .1 0 .2 .4 .6 .8 1.099 0.206 .210 .215 .221 .227 .235 .250 .270 .290 .320 0.1132 .0875 .0627 .0520 .0401 .0190 0 .0330 .0550 .0710 .0775 .0721 0.333 .192 .057 0 .072 .190 .300 .490 .635 .742 .792 .758 S 0.2 x IX I .% G 0.5 .4 .3 .2 0.50 .45 .40 .3547 .30 .25 .20 .15 .10 .05 0 .1 .2 .3 .4 .5 .6 .7 .8 .9 1.0 1.099 0.405 .415 .424 .432 .443 .454 .480 .515 .557 .620 0.1213 .0936 .0674 .0431 .0350 .0205 0 o .0355 .0590 .0770 .0835 .0721 0.450 .310 .172 .095 o .077 .188 .370 .516 .620 .657 .558 .1 0 .2 .4 .6 .8 1.099 I I NACA TM No. 1216 TABLE 3 Concluded THE FUNCTION G( K, i) FOR THE INTEGRATION OF THE DIFFERENTIAL EQUATION OF THE MOMENTUM THICKNESS Concluded 'K = 0.3 Kg = 0.5 X 1 K G(t,c1) x X1 1 G( C, IL) 0.5 .4 .3 .2 0 .2 .4 .6 .8 1.099 0.4 .3 .2 0 .2 .4 1.099 0.590 S.611 .622 .637 .652 .688 .738 .808 .892 1.172 0.770 .782 .800 .815 .835 .875 .935 0.1292 .100 .0722 .0463 .0222 .0192 10 .0380 .0660 .0830 .0910 .0721 0.1o68 .0777 .0500 .0202 .oo8 0 .0415 .0730 .0721 0.542 .400 .265 .136 .016 0 .099 .294 .438 .520 .555 .358 0.460 .324 .200 .080 o 0 .035 .232 *375 .158 ' ' 0.4 3 .2 .1 .4 1.099 0.3 .2 .1 0 .2 .4 0.935 .950 .965 .982 1.0 1.045 1.115 0.113 .0825 .0538 .0220 0 .0460 .0810 .0721 0.475 .350 .225 .100 0 .195 .320 .042 K = 0.6 x Xl I G( K, ) 1.116 1.122 1.135 1.154 1.200 1.254 0.088 .056 .020 0 .048 .086 KI = 0.7 X I X1 K G(l,1) 0.3 .2 .1 0 .2 .4 1.294 1.292 1.296 1.304 1.338 1.401 0.089 .059 .030 o .054 .096 o.327 .206 .072 0 .190 .340 0.125 .082 .005 .089 .262 .402 _ _I I ?1 = 0.4 X X1 1 G(K, l) NACA TM No. 1216 on I t 0 r H o o. a Ol H Cu 4 m 0 coT 00 O S0 0 0 v a m t ru E* * o( 0 1n %0 cu fn fn CC) a% S Do (n C O CU O Co o Cu r l 0 0. 0 t rr) 00 tt CU N O Cu  n c Cu cu cu H H 0o m 0 0 S .\ . 0 ; 0 o0 .Q \ C O O 0o H Ch cn om T0 01 C 0\ 0\ f\ C0 "o i 0o O i n H C O S0 0 0 0 0 0 SCU o U n (Y ) ot k C cu H H H t 0 0 0 0 0 1H1 NACA 4M No. 1216 TABLE 5 THE BOUNDARYLAYER PARAMETERS AT THE PLANE PLATE IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION; LENGTH OF GROWING BOUNDARY LAYER 5* 5* ro5* TO5* To l 1 v gUo pvoUo 0 0 1 1.090 2.66 0 0.572 o 0 0.000171 .02 .989 1.089 2.65 .0218 .576 26.43 0131 .ooo662 .04 .979 1.088 2.64 .0435 .581 13.34 .0257 .00154 .06 .968 1.087 2.63 .0652 .586 8.98 .0392 .00291 .08 .956 1.086 2.62 .0869 .591 6.80 .0540 .00456 .10 .945 1.085 2.61 .1085 .597 5.50 .0676 .01139 .15 .916 1.083 2.58 .1624 .610 3.76 .1068 .02037 .20 .884 .o080 2.55 .2159 .625 2.89 .1426 .0341 .25 .851 1.077 2.53 .2692 .640 2.38 .1845 .0517 .30 .817 1.074 2.50 .3221 .656 2.04 .227 .0783 .35 .780 1.070 2.47 3746 .673 1.80 .280 .1124 .40 .741 1.067 2.44 .4267 .690 1.617 335 .1551 .45 .700 1.063 2.41 .4784 .707 1.481 .394 .2127 .50 .656 1.059 2.37 .5296 .728 1.375 .461 .2879 .55 .610 1.055 2.34 .5802 .749 1.290 .536 .3883 .60 .560 1.050 2.31 .6303 770 1.222 .624 .5209 .65 .507 1.046 2.27 .6797 .793 1.167 .722 .7091 .70 .450 1.041 2.24 .7284 .817 1.122 .842 .9756 .75 .389 1.035 2.20 .7763 .843 1.086 .988 1.373 .80 .323 1.029 2.16 .8233 .871 1.058 1.172 2.007 .85 .252 1.023 2.12 .8693 .900 1.035 1.416 3.163 .90 .175 1.016 2.08 .9142 .931 1.018 1.780 5.840 .95 .091 1.oo8 2.o4 .9578 .964 1.007 2.415 10.556 .98 .037 1.003 2.02 .9833 .985 1.002 3.250 14.733 .99 .019 1.002 2.01 .9917 993 1.001 3835 1.00 0 1 2 1 1 1 " NACA TM No. 1216 TABLE 6 DRAG LAW OF THE PLANE PLATE IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION x1 I = f(xL1) F(X5) G(X1) 0 .01 .02 .03 .04 .06 .08 .10 .15 .20 .25 30 .35 .4o .45 .50 .55 .60 .65 .70 .75 .80 .85 .90 .95 .98 .99 .992 .994 .996 .997 .998 .999 .9995 .0000406 .oooowo6 .0001706 .0003682 .0006612 .001535 .002909 .00456 .01139 .02037 .03405 .05172 .07833 .1124 .1551 .2127 .2879 .3883 .5209 .7091 .9756 13731 2.0075 3.1630 5.8403 10.556 14.73 16.15 18.01 20.69 22.62 25.37 30.11 34.90 m I. J .004665 .oo8883 .012732 .016623 .02646 .03608 .04719 .07427 .10506 .13949 .18066 .23006 .2872 *3539 .4357 .5357 .6614 .8199 1.0347 1.3284 1.7538 2.4165 3.6018 6.3098 11.043 15.23 16.64 18.51 21.19 23.12 25.87 30.61 35.40 Go Cf C f S Uo cfeo =  11".02 52.08 34.56 25.14 17.29 12.40 10.35 6.528 5.159 4.096 3.493 2.937 2.556 2.281 2.048 1.861 1.704 1.574 1.459 1.362 1.277 1.204 1.139 1.080 1.046 1.034 1.031 1.0275 1.0241 1.0220 1.0196 1.0166 1.0143 1 = G(X1) NACA TM No. 1216 TABLE 7 PARAMETERS FOR THE VELOCITY DISTRIBUTION OVER THE LENGTH OF GROWING BOUNDARY LAYER FOR THE PLANE PLATE IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION (TO FIG. 12) SK T K 0 0 1 1.0 0.754 0.384 .1 .143 .919 1.4 .848 .255 .2 .267 .840 1.8 .902 .171 .4 .453 .696 30 .973 .053 .6 .590 .573 1 o .8 .685 .467 NACA TM No. 1216 TABLE 8 RESULTS OF THE BOUNDARY LAYER CALCULATIONS FOR THE CIRCULAR CYLINDER WITH SUCTION (a) Co = 0 Po 2 Ic ). K 1j K 0 0 0.1883 0.442 0.0709 0 0.355 o 0.653 4 .0698 .1884 .443 .0708 .355 .653 8 .1396 .1888 .446 .0705 .353 .657 12 .2094 .1892 .447 .0700 .350 .660 16 .2793 .1897 .448 .0696 .348 .662 20 .349 .1913 .452 .0668 .345 .664 25 .436 .1944 .459 .0685 .343 .665 30 .524 .1985 .469 .0682 .342 .667 35 .611 .2030 .479 .0675 .340 .669 40 .698 .2071 .488 .0657 .333 .673 45 .785 .2128 .502 .0641 .325 .680 50 .873 .2129 .522 .0612 .312 .690 55 .959 .2259 .540 .0585 .300 .703 60 1.047 .2326 .563 .0541 .280 .720 65 1.134 .2438 .591 .0502 .260 .740 70 1.222 .2552 .623 .0446 .232 .770 75 1.309 .2698 .669 .0377 .202 .805 80 1.396 .2881 .726 .0288 .156 .850 85 1.484 .3087 .790 .0166 .094 .905 90 1.571 .3332 .886 o 0 1.000 95 1.658 3583 1.010 .0223 .141 1.15 100 1.746 .3937 1.240 .0538 .420 1.34 S 101.7 1.776 .4062 1.290 .0682 .682 1.44 NACA TM No. 1216 TABLE 8 Continued RESULTS OF THE BOUNDARYLAYER CALCULATIONS Continued (b) Co = 0.5 UoO R */Uo 9 1 K .0698 .1396 .2094 .2793 .349 .436 .524 .611 .698 .785 .873 .959 1.047 1.134 1.222 1.309 1.396 1.484 1.571 1.658 1.746 1.833 1.854 0.1573 .1575 .1587 .1598 .1605 .1619 .1634 .1658 .1695 .1726 .1766 .1821 .1871 .1937 .2012 .2097 .2181 .2289 .2470 .2683 .2881 .3123 .3421 .3522 0.3681 .3686 *3714 .3739 3756 3788 3824 .3880 .3966 .404 .413 .426 .440 .455 .473 .495 .515 .543 .593 .645 .709 .835 .978 1.004 0.0495 .0495 .0494 .0493 .0492 .0490 .0487 .0482 .0471 .0456 .0441 .0430 .0403 .0375 .0331 .0287 .0230 .0170 .0109 0 .0144 .0338 .0606 .0682 0.1112 .1114 .1122 .1130 .1137 .1145 .1155 .1172 .1199 .1220 .1249 .1288 1323 1370 .1422 .1481 .1542 .1619 .1747 .1897 .2037 .2208 .2419 .2490 0.240 .240 .240 .240 .240 .240 .238 .234 .225 .220 .215 .208 .195 .185 .164 .141 .115 .089 .057 0 .075 .185 .415 .452 0.250 .250 .250 .250 .250 .250 .252 .255 .260 .267 .275 .290 .297 .305 .315 329 .350 .360 375 .410 .462 .525 .610 .625 0.580 .580 .580 .581 .582 .584 .585 .587 .589 .592 .595 .601 .608 .615 .622 .630 .640 .665 .695 .735 .781 .884 1.06 1.18 2.34 2.342 2.343 2.35 237 2.40 2.67 0 8 12 16 20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95 100 105 Slo 6 s lo6. 4 NACA TM No. 1216 TABLE 8 Continued RESULTS OF THE BOUNDARY LAYER CALCULATIONS Continued (c) Co = 1 UoR qp R v R* k I K 0 0 0.1350 0.3071 0.0365 0.1909 0.170 0.415 0.525 4 .0698 .1350 .3071 .0364 .1909 .170 .415 .525 8 .1396 .1351 .3074 .0362 .1911 .170 .415 .525 12 .2094 .1360 .3094 .0362 .1923 .170 .415 .525 16 .2793 1367 3109 .0361 1933 .169 .417 .525 20 .349 .1373 .3124 .0354 .1942 .166 .420 .525 25 436 .1383 .3146 .0347 .1956 .162 .425 .524 30 .524 .1400 .3178 .0339 .1966 .156 .430 .523 35 .611 .1419 .3221 .0330 .2007 .153 .440 .522 40 .698 .1442 .3273 .0319 .2039 .150 .445 .521 45 .785 .1466 .3328 .0304 .2073 .142 .455 .520 50 .873 .1491 3385 .0289 .2106 .135 .470 .519 55 .959 .1532 .3478 .0269 .2167 .127 .485 .517 60 1.047 .1595 .3613 .0254 .2256 .122 .495 .515 65 1.134 .1665 .3762 .0234 .2355 .107 .512 .509 70 1.222 .1729 .380 .0205 .2445 .098 531 .500 75 1.309 .1795 .394 .0167 .2538 .077 .554 .500 80 1.396 .1869 .410 .0121 .2643 .055 .580 .508 85 1.484 .1942 .435 .0066 .2746 .031 .600 .515 90 1.571 .2017 .450 0 .2852 0 .630 .520 95 1.658 .2090 .476 .0076 .2956 .040 .668 .535 100 1.746 .2207 .507 .0169 .3121 .088 .710 .560 105 1.833 .2291 .539 .0272 .3240 .140 .775 .592 110 1.920 .2500 .598 .0438 .3436 .225 .825 .680 115 2.008 .2775 .700 .0651 .3642 .357 .850 .835 s 115.5 2.016 .2814 .715 .0682 .3697 .375 .870 .870 NACA TM No. 1216 TABLE 8 Continued RESULTS OF THE BOUNDARY LAMER CALCULATIONS Continued (d) Co = 2 PP IR "O1 X I K 0 4 8 12 16 20 25 30 35 40 45 50 55 60 6.5 70 75 80 85 90 95 100 105 110 115 120 125 S 127.5  .0698 .1396 .2094 .2793 .349 .436 .524 .6n1 .590 .785 .873 959 1.047 1.134 1.222 1.309 1.396 1.484 1.571 1.658 1.746 1.833 1.920 2.008 2.095 2.183 2.227 0.1030 .1030 .1031 .1034 .1039 .1044 .1050 .1055 .1068 .1078 .1091 .1106 .1123 .1153 .1187 .1217 .1252 .1289 .1334 .1393 .1460 .1543 .1631 .1721 .1860 .2015 .2205 .2371 0.2281 .2281 .2284 .2290 .2301 .2312 .2326 .2340 .2360 .2382 .2400 .2437 .2472 .2524 .2580 .2647 .2712 .2795 .2881 .3002 *3132 .3290 .3464 .3660 *3891 .415 .452 .472 0.0212 .0211 .0210 .0208 .0205 .0198 .0194 .0190 .0185 .0178 .0168 .0157 .0145 .0133 .0119 .0101 .0081 .0058 .0031 0 .0040 .0083 .0138 .0202 .0306 .0406 .0557 .0682 0.2913 .2913 .2918 .2925 .2939 .2953 .2970 .2984 .3021 .3049 .3086 .3128 .3176 3261 .3357 .3442 .3541 .3646 3773 .3940 .4129 .4364 .4613 .4868 .526 .570 .635 .671 0.090. .090 .090 .087 .083 .079 .075 .073 .071 .069 .068 .066 .064 .060 .052 .045 .035 .025 .012 0 .020 .040 .068 .100 .140 .185 .220 .305 I I 0.620 .620 .620 .621 .623 .625 .628 .632 .638 .645 .654 .665 .677 .690 .710 .735 .755 .775 .795 .815 .845 .890 .925 .980 1.055 1.190 1.260 1.305 0.420 .420 .420 .419 .418 .416 .414 .412 .410 .405 399 .394 .388 .380 .370 .360 350 .340 .325 310 .295 .278 .255 .225 .190 .120 .038 .020 I I  i NACA TM No. 1216 TABLE 8 Concluded RESULTS OF THE BOUNDARYLAYER CALCULATIONS Concluded (e) Velocity Distribution NACA TM No. 1216 TABLE 9 RESULTS OF THE BOUNDARYLAYER CALCULATION FOR THE SYMMETRICAL JOUKOWSKY PROFILE J 015 FOR ca = 0 (a) fi(o) = 0; o = 0 ao L K 1 X X1 K 180 0 0.0370 0.0878 0.0708 0 0.355 0 0.650 177.5 .0493 .0377 .0895 .0697 .349 .653 175 .00986 .0405 .0964 .0674 .335 .661 172.5 .0148 .0438 .1048 .0630 319 .682 170 .0197 .0485 .1167 .0581 .298 .703 165 .0308 .0605 .1480 .0464 .247 .753 160 .o444 .0755 .1885 .0351 .197 .802 150 .0764 .1109 .284 .0193 .io8 .892 140 .1233 .1572 .411 .0053 030 .964 136 .1445 .1772 .472 0 0 1.000 130 .1775 .2062 .559 .0089 .052 1.062 120 .2416 .2608 .742 .0260 .172 1.182 110 .3132 .3217 .960 .0484 362 1.367 S 101.9 377 .3748 1.310 .o68o 0 .628 1.637 NACA TM No. 1216 TABLE 9 Continued RESULTS OF THE BOUNDARYLAYER CALCULATION Continued (b) fi(o) = 0.5; Co = 0.0695 B 0t 0T C X1 K t t rv t v j1 180 o 0.0359 0.0846 0.0667 0.0180 0.335 0.044 0.640 177.5 .00493 .0366 .0867 .0658 .0183 331 .045 .645 175 .00986 .0391 .0927 .0634 .0196 .322 .046 .654 172.5 .0148 .0430 .1025 .0603 .0215 307 .047 .674 170 .0197 .0474 .1140 .0556 .0237 .283 .051 .690 165 .0308 .0580 .1416 .0430 .0299 .227 .060 .734 160 .0444 .0737 .1830 .0334 .0368 .180 .074 .773 150 .0764 .1072 .268 .0169 .0540 .100 .110 .833 140 .1233 .1446 .367 .0045 .0720 .024 .154 .867 136 .1445 .1613 .415 0 .0790 0 .173 .880 130 .1775 .1850 .483 .0071 .0913 .035 .203 .906 120 .2416 .2274 .620 .0200 .1137 .112 .255 .976 110 .3132 .2793 .764 .0356 .1397 .225 .316 1.075 100 .391 .3339 .961 .0530 .1669 .360 .384 1.200 s 92.4 .4575 .3742 1.125 .0682 .1871 .505 .440 1.310 NACA TM No. 1216 TABLE 9 Continued RESULTS OF THE BOUNDARYLAYER CALCULATION Continued (c) f1(o) 1.0; Co = 0.139 u 0i tx *t KKX K 180 o 0.0348 0.0819 0.0626 0.0348 0.314 0.077 0.631 177.5 .00493 0358 .0842 .0626 .0358 .313 .079 .631 175 .00986 .0382 .0900 .0605 .0388 .308 .085 .640 172.5 .0148 .0416 .0985 .0564 .0403 .293 .095 .653 170 .0197 .0460 .1095 .0516 .0450 .265 .104 .667 165 .0308 .0560 .1345 .0376 .0560 .205 .128 .703 160 .0444 .0686 .1678 .0292 .0678 .154 .156 .748 150 .0764 .0985 .243 .0152 .0985 .080 .223 .776 140 .1233 .1342 332 .0038 .1340 .020 .300 .788 136 .1445 .1484 .368 0 .1478 0 335 .795 130 .1775 .1682 .421 .0059 .1682 .032 .380 .8oo 120 .2416 .2030 .505 .0162 .2025 .084 .465 .806 110 .3132 .2385 .595 .0270 .2400 .142 .556 .809 100 .391 .2790 .696 .0380 .28oo .200 .650 .812 90 .475 .3200 .800 .0499 .3203 .267 .750 .818 80 .561 .362 .916 .0636 .3622 .343 .858 .830 S 76.4 .593 .376 .952 .0682 .3766 .380 .892 .860 NACA TM No. 1216 TABLE 9 Continued RESULTS OF THE BOUNDARYLAYER CALCULATION Continued (d) fl(0) = 1.5; Co = 0.2085 SPt 'B 1 x K 180 0 0.0336 0.0792 0.0584 0.0504 0.295 0.110 0.635 177.5 .00493 0351 .0824 .0592 .0526 .293 .116 .635 175 .00986 .0372 .0878 .0580 .0557 .281 .125 .639 172.5 .0148 .0404 .0958 .0531 .o6o6 .265 .136 .648 170 .0197 .0442 .1052 .0482 .0662 .202 .150 .658 165 .0308 .0544 .1301 .0376 .0816 .192 .181 .690 16o .0444 .0675 .1596 .0280 .0980 .145 .220 .705 150 .0764 .0927 .224 .0135 .1391 .070 .310 .722 140 .1233 .1249 .297 .0027 .1852 .016 .415 .718 136 .1445 .1368 .325 0 .205 0 .454 .702 130 .1775 .1543 365 .0046 .230 .01 .512 .669 120 .2416 .1825 .430 .0127 .272 .060 .605 .630 110 .3132 .2107 .490 .0202 .315 .097 .698 .596 100 .391 .2388 .546 .0276 .359 .132 .785 .566 90 .475 .2683 .610 .0356 .401 .165 .870 .529 80 .561 .2981 .668 .0432 .443 .198 .950 .486 70 .645 .3255 .726 .0o04 .487 .230 1.034 .430 60 .734 3555 .770 .0578 .531 .260 1.114 .350 50 .807 .3789 .808 .0637 .568 .286 1.190 .250 s 4o .885 .4027 .835 .0682 .604 .305 1.265 .150 NACA TM No. 1216 TABLE 9 Continued RESULTS OF THE BOUNDAEYLAYER CALCULATION Continued (e) f1(0) = 3.0; Co = 0.417 P a tl X X K 180 0 0.0307 0.0720 0.0504 0.0921 0.242 0.210 0.602 177.5 .00493 .0323 .0755 .0511 .0940 .241 .213 .603 175 .00986 .0342 .0795 .0494 .1026 .229 .225 .604 172.5 .0148 .0365 .0852 .0433 .1095 .212 .240 .609 170 .0197 .0398 .0900 .0391 .1194 .195 .260 .612 165 .0308 .0486 .1150 .0300 .1398 .157 .310 .620 160 .0444 .0587 .136 .0216 .1705 .107 .376 .610 150 .0764 .0794 .184 .0099 .238 .051 .530 .570 140 .1233 .1015 .231 .0022 .309 .011 .666 .502 136 .1445 .1118 .248 0 335 0 .714 .471 130 .1775 .1240 .270 .0032 .370 .016 .781 .422 120 .2416 .1435 .303 .0080 .425 .033 .880 .306 110 .3132 .1594 333 .0119 .476 .049 .970 .153 100 .391 .1745 .358 .0148 .522 .060 1.045 .022 90 .475 .1881 .374 .0172 .564 .068 1.110 .090 80 .561 .2006 .396 .0197 .600 .072 1.172 .180 70 .645 .2109 .410 .0212 .633 .075 1.230 .230 60 .734 .2218 .424 .0226 .663 .078 1.280 .270 50 .807 .2302 .439 .0234 .691 .079 1.326 .300 40 ..885 .2383 .452 .0238 .715 .080 1.336 .330 30 .945 .2440 .464 .0248 ..732 .080 1.340 .350 20 .990 .2486 .470 .0256 .746 .080 1.344 .360 10 1.018 .2512 .476 .0259 .754 .080 1.348 .360 0 1.028 .2522 .479 .0261 .757 .080 1.350 .360 no separation NACA TM No. 1216 TABLE 9 Concluded RESULTS OF THE BOUNDARYLAYER CALCULATION Continued (f) Velocity Distribution NACA TM No. 1216 TABLE 10 VELOCITY DISTRIBUTION OVER THE REGION OF GROWING BOUNDARY LAYER FOR THE PLATE WITH HOMOGENEOUS SUCTION it = 0.1 (T o02 Ft = 0.' / = 0.6 f = o.8 vo? u "vo u "voy u v3y .vY . v Uo v Uo v Uo v Uo v Uo 0 0 0 0 0 0 0 0 0 0 .0286 .1108 .0534 .1168 .0910 .1279 .1180 .1373 137 1455 .0572 .2176 .1068 .2271 .182o .2448 .236 .2598 .274 .2727 .0858 .3205 .1602 .3317 .273 .?522 .354 .3697 .411 .3848 .1144 .4173 .2136 .4287 .'64 .4497 .472 .4676 .548 .4830 .143 .5107 .267 .5211 .455 .5302 .590 .5506 .685 .5704 .1716 .5967 .2204 .6054 .546 .6215 .708 .6351 .822 .6469 .2002 .6760 .3738 .6826 .637 .6948 .826 .7051 959 .7141 .2288 .7475 .272 .7518 .728 .7598 .944 .7666 1.096 .7724 .2574 .8105 .4806 .8126 .819 .8164 1.062 .8196 1.233 .8224 .286 .8663 .5340 .8661 .910 .8659 1.180 .8657 1.370 .8655 .3146 .9112 .5874 .9094 1.001 .9059 1.298 .9029 1.507 .900o .3432 .9475 .64o8 .9442 1.092 9382 1.416 .9331 1.644 .9287 .3718 .9738 .6942 .9697 1.183 .9621 1.534 .9557 1.781 .9501 .4004 .9899 .7476 .9856 1.274 .9776 1.652 .9708 1.918 .9650 .4290 .9959 .8010 .9920 1.365 .9848 1.770 .9787 2.055 .9734 .5005 .9975 .934. .9952 1.5925 .9908 2.065 .9871 2.3975 9839 .5720 .9985 1.0680 .9971 1.820 .9944 2.360 .9922 2.740 .9902 .6435 .9991 1.2015 .9982 2.0475 .9966 2.655 9953 3.0825 .9941 .715 .9995 1.3350 .9989 2.275 .9980 2.950 .9971 3425 .9964 .858 .9998 1.6020 .9996 2.730 .9992 3.540 .9989 4.11 .9987 1.001 .9999 1.8690 .9999 3.185 .9997 4.130 .9996 4.795 .9995 m 1.0000 1.0000 1.0000 m 1.0000 1.0000 /1t = 1.0 =1. = 1.8 3.0 =3 =0 0 0 0 0 0 0 0 0 0 0 .1508 .1518 .1696 .1617 .1804 .1682 .1946 .1772 .2 .1813 .3016 .2829 3392 .2986 .3608 .088 .1892 .3232 .4 .3297 .4524 .3966 .5088 .4149 .5412 .4269 .5838 .437 .6 .4512 .6032 .4950 .6784 .5137 .7216 .5259 .7784 .5430 .8 .5507 .75401 .5814 .848 .5984 .902 .6095 .973 6251 1.0 .6321 .9048 .6562 1.0176 .6704 1.0824 .6798 1.1676 .6929 1.2 .6988 1.0556 .7211 1.1872 .7119 1.2628 .7390 1.3622 .7489 1.4 .7534 1.2064 .7770 1.3568 .7840 1.4432 .7887 1.5568 .7952 1.6 .7981 1.3572 .8246 1.5264 .8280 1.6236 .8302 1.7514 .8333 1.8 .8347 1.5080 .8654 1.696 .8651 1.804 .8650 1.946 .8648 2.0 .8647 1.6588 .8984 1.8656 .8953 1.9e44 .8933 2.1406 .8905 2.2 .8892 1.8o96 .9253 2.0352 .9199 2.1648 .9164 2.3352 .9115 2.4 .9093 1.9604 .9458 2.2048 .9391 2.3452 .9347 2.5292 .9285 2.6 .9257 2.1112 .9614 2.3744 .9533 2.5256 .9486 2.7244 .9421 2.8 .9392 2.262 .9693 2.5440 .9629 2.7060 .9587 2.9190 .9528 3.0 .9502 2.639 .9814 2.968 .9775 3.1570 .9750 .34055 .9714 35 .9698 3.016 .9887 3.392 .9864 3.608 .9848 '.8920 .9827 4.0 .9817 3393 .9932 3.816 .9917 4.059 .9908 4.785 .9895 4.5 .9889 3.770 .9959 4.240 .9950 4510 .9944 4.865 .9931 5.0 .9933 4.524 .9985 5.088 .9981 5.412 .9979 5.838 .9976 6.0 .9975 5.278 .9994 5.936 .9993 6314 .9993 6.811 .9992 7.0 .9991 1.0000 1.0000 cc 1.0000 1.0000 a 1.0000 The parameter K and l, (See table 7.) NACA TM No. 1216 Figure 1. Explanatory sketch for the boundary layer with suction for arbitrary body shape. BI Figure 2. The functions F1 and F2 for the velocity distribution in boundary layer, see equation (9). NACA TM No. 1216 20   ITI S^ A K  I _^ \^ ^  I _,\N X4$, r: _ll I\s^^ $ ^S  *;sl\,, ,,',,',,', \,,\ \\\\\ ,, _S S .5 S S S ^ 0.5 a '. 70 \ .A A,= 1 08 06 04 02 0 02 Figure 3. The form parameter K of the velocity profile as a function of S, A 1, according to equation (14). NACA TM No. 1216 / 4.5 rts ft/U 0.5 0 Figure 4. Itj2L I I I _J I IL Oas 10 s. 2.0 6* To 1 The auxiliary functions G(K), and as functions of K, according to equations (17a), (19), and (20). 70 NACA TM No. 1216 Ij * __  ,II 4  a:3 : '.j ca d S t %  \ o  r r 8 1 4C _ I I J t_ I cn c0  I, IA zt c1 Ss ' p 0 __ 1 r a y "* '? ~ ~~ ^ i    I  rfS^W ^ ^  \   , r ^~ ll < % Ji ~ ~ i 1  QsE NACA TM No. 1216 Figure 6. Diagram for solution of the differential equation for the momentum thickness: G(s, x1). NACA TM No. 1216 10  0.0682, For all K! ,/ i  Separation point" _ SHortree 71r I r.v S^ _ _    O 0aoog' Stagnation point t (D withoutj suction _D mZ __ 0~~t NACA TM No. 1216 73 I"o ^"f 0 000 r08 0.0709 006  004 04  002 02 0 / 2 3 Vo K= o Figure 7. The initial values of the boundary layer at the stagnation point for various suction quantities. NACA TM No. 1216 LI U0o . Approxliate  i / i I  yf~ 1 1 i i i _ Figure 8. Comparison of the velocity distributions according to the approximate with the exact calculation. (a) Plane plate in longitudinal flow, exact calculation according to Blasius, approximate calculation according to equations (53) and (54). (b) Exact calculation according to Hiemenz, approximate calculation according to equations (58) and (58a). NACA TM No. 1216 Figure 9. The extent of growing boundary layer for the plane plate with V1T* 50 T05* homogeneous suction: against j v Uc i NACA TM No. 1216 Figure 10. Plane plate with homogeneous suction; region of growing boundary layer; velocity distribution. NACA TM No. 1216 16 1,2 ___ ^ .O I d00 04       04 0 B 1.2 1.6 20 ?.4 Figure 11. Plane plate with homogeneous suction; region of growing boundary T  layer; the local friction coefficient  against t. Oo 78 NACA TM No. 1216 41 a, 0 U a II k o *  u 8 h0o .I ILrbO 6;: S.4 a, 0 0o 4 Cd o h 0 CO ___g Cd cc U4 Z V a, ~ iv, V hCd c'J . NACA TM No. 1216 79 0 4'   "00 Sio/ I / :  I O  .:    f W / )f   1.4 0 0 S f i ' z \ 4 Ul 0 ^ f ^ ,.::I / 0 f Cd 0 0 0 S.DI.l Eo 0 A E: , V 0 Cd 14 4 Q) ., " " rri ~a~cl) ~ ~~ 8 ci c NACA TM No. 1216 03 o.0.8 \ 02 0,6 104  Exac t SApproximate   0.400 Figure 14. Plane stagnation point flow: comparison of the boundarylayer thickness and momentum thickness of approximate and exact calculation for various suction quantities, exact calculation according to reference 9. NACA TM No. 1216 Figure 15. The boundary layer on the circular cylinder with homogeneous suction for various suction quantities Co = o U Form parameter K = 2 U With increasing suction quantity the v ds separation point shifts rearward. NACA TM No. 1216 Ei 41 o bf vl  / el'1, XO dddX S o .,C. sO cco 00 ^ \ \  4 0 I I _ 1.0 c\ \ h0 co i1 a ,0 C (D d C L 0 0 ,~1 o Z l^ i  UiL!44 *saf q3 h ci *30' *ok ^ ^ 2 NACA TM No. 1216 008 Figure 17. The boundary layer on a symmetrical Joukowsky profile J 015 with ca = 0 for homogeneous suction with various suction quantities Co =fl according to equation (36): Co x fl(o) a dU (K1 = 51.7). = ds. With increasing suction quantity the separation point shifts rearward. 84 NACA TM No. 1216 O "4 o no I b la \g s \  \ I o  v \ I ' cdu $I c 0 \I C __ \ __ __a _._ 0 0) %\ \ / / S~ca" cd ,, 1 \CV / I' J I "4 r a NACA TM No. 1216 85 I  S n 9 0 o) F o . Q 0 ci SI C  0 S h ' S00 c I O o I x L2 r = a ~cua ob~ if C, C Cd I CC 00. a o .5 ci S4 I O) bi Scd oa _____ __ ____ ___ ___ ____ __ a N~~O + ,  ~r siia~ 2* UNIVERSITY OF FLORIDA 3 1262 08106 319 9 
Full Text 
xml version 1.0 encoding UTF8
REPORT xmlns http:www.fcla.edudlsmddaitss xmlns:xsi http:www.w3.org2001XMLSchemainstance xsi:schemaLocation http:www.fcla.edudlsmddaitssdaitssReport.xsd INGEST IEID ES1EG1JPP_7A48OZ INGEST_TIME 20120229T19:05:44Z PACKAGE AA00009241_00001 AGREEMENT_INFO ACCOUNT UF PROJECT UFDC FILES 