/vcAT^-a^w 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 Translation of *Ein Naherungsverfahren zur Berechnung der laminaren Grenzschicht mit Absaugung bei beliebiger Korperform" from Aerodynamisches Institut der Technischen Hoqhschule Braunschweig, Bericht 43/13, Jiine 1943 Washington March 1949 A' OF FLORIDA •ITS DEPARTMENT ..:TON SCIENCE )X 11 7011 ^VILLE.FL 32611-7011 LIBRAF Y U5A HATIONAL ADVISORY COMMITTEE FOR AEEONAUT'ICS TECHNICAL MEMORAOT)UM NO. 12X6 AN APPROXIMATE METHOD FOR CALCULAT'ION 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 -wizh. Suction rv. The Generiil Approximation Method for /arbitrary Pressure Distribution and Arbitrary Distribuoion of the Velocity' of Suction (a) The expression for the velDcity distribution (b) The differential equation for the momentuji 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 two-dimensional 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 two-dimensional stagnation point flow with homogeneouo suction VI. Examples (a) The circular cylinder with homogeneous suction Tor various suction quantities (b) Symmetrical Joukowsicy profile for c^ = with homogeneous suction VII. SuDEiary VIII. Bibliography IX. Appendix * Ein Naherungsverfahren zu.'- Berechniuifr der laminaren Grenzachicht mit Absaiigung bei beliebiger Kjrperform. Aerodynamisches InsLitut der Technlschen Hochschulf.- BraunRcnweig, Bericht ^3/l3, Jane 12, 19^+3. NACA TM No„ 12l6 INTRODUCTION Various ways were tried recently to decrease the friction drag of a bnd^' In a flow, they all employ influencing the houndary layer (reference l) • One of them consists in keeping the houndary layer laiiinar by suction; promising tests have been carried out by Holstein (references 2 and 3) aJ^d Ackeret (reference k) . 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 req,uired for suction taken into account. The boimdary 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 Guc^lon 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 boimdary layer (U5*/v)g^^^ 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 q_uantitles 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. Schllchting (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. Schllchting and K. Bussmann (reference 9) Investigated the two- dimensional stagnation point flow with homogeneous suction and the plate in longitudinal flow with Vq ^ l//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 ^he velocity distribution and the other boundary layer quantities. K- Bussmann, H. Miinz (reference 8), and A Ulrich (reference I6) calculated the transition fran 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 tvirbulent is exerted by the pressure gradient along the contour in the flow for impermeable wall. Both influences (pressure gradient and suction) will be pref'ent simultaneously for the intended maintenance of a laminar boundary layer for a wing. Both Influences have a stabilizing effect for the suction in the region of pressure drop; in the region of pressure rise, however, pressure gredient and suction have opposite influences. Whereas without suction, :'or prrrssure rise, mostly transition in the boundary NACA TM No. 12l6 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 q.uestions. 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 well-known Pohlhausen method for impermeable wall. It permits the calculation of the laminar boxmdary 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, re^^^v^ctively, to the wall wetted by the flow (x = y = 0: stagnation point and leading edge of the plate, respectively) 00 &* displacement thickness of the boundary layer ■a momentum tnickness of the boundary layer u^(l - uAj)dy ^y=0 8, measure of the boundary layer thickness 2 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 potential velocity outside of the friction layer free stream velocity given suction velocity at the wall Vq < 0: suction U(x) Uo vo(^) k NACA TM No. 12l6 (c) Other Quantities T o wall shearing stress T wall shearing stress for asymptotic solution of "boundary °° layer on plate in longitudinal flow with homogeneous suction Q total suction quantity lb/ Vodxl cq dimensionless mass coefficient of suction; Cq > 0: suction f ^ 1 c*Q reduced mass coefficient of suction I Cq u ~^ J T) dimensionless distance from wall (y/5i) Fi(t)> F2(ti) "basic functions for velocity distribution in boundary layer, eq.uatlon8 (8), (9) K form parameter of boundary layer profiles, equation (6) X,Xi dimensionless boundary layer thickness, equations (12), (13) K,itj^ dimensionless momentum thickness, equations (22), (23) i dimensionless length of boundary layer III. THE EQUATIONS OF THE BOUNDARY LAYER WITH SUCTION Following we shall consider the plane problem, thus the boundary layer on a cylindrical body In a flow (fig- l)- x, y are assumed to be the coordinates along the wall and perpendicular to the wall, respectively, Uq 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: Vq(x) > 0: blowing; Vo(x) < 0: suction Vq/Uq may be assumed to be very small (O.Ql to O-OOOl). 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, 12l6 ^u du dU ^ a^u al ^ ^ = ° (^"^ y = 0: u = 0; V = Y^ix) (2) y =00 : u = U(x) The system of eq.uatlons (l) , (2) differs from the ordinary boundary layer theory merely by the fact that one of the boundary conditions for y = is changed from v = to v = VqCx) ^ 0- Thereby the character of the solutions changes decisively: the solutions differ greatly according to whether it is a case of Vq > (blowing) or v© *^ (suction) A special solution of the eq.uations (l), (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) = Vq = const < and U(x) = Uq- For this case the boundary layer thickness becomes constant at some distance from the leading edge of the platej also, the velocity distribution becomes independent of x (reference 5)* From ^ = follows because s ox " . of the continuity 2X f and hence dy v(x, y) = Vq = Constant From equation (l) then follows for the velocity distribution Uo V^ - e y = Uo^l + e J u(x, y) = u(y) = Uo \i - e J = Uq^^I + e J (3) with NACA TM No. 12l6 signifying the displacement thickness of the asymptotic solution. The wall shearing stress for this solution is: ■^o = ^(^) = -pUo^o (^a) 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 eq.uatlons (l), (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 ll); 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 eq.uatlons 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 (l), (2) over y between the limits y = and y = « one obtains in the known manner (reference l8) : U^^ . (2^ . 6*) u^ ■ ■ Uv =''-^ dx dx o (5) NACA TM No. 12l6 i3 Sjgnifles the Diomentijm thickness, 6* the displacement, thickness, and the wall shearing st.ress- The epproximation method :'or o calculation of the boundary la;y'er to be chosen here proceeds in such a nanner that e plausible expression is given for the velocity distribution in the boundat-y layer u(x, y) which is contained in eq.uation (5) in '^, 5* and Tq. Thus an ordinary differential eq.uation for ^x) results from equation (5)» after this different^ial equation has been solved one obtains the remaining characteristics of the bouiidary layer &*(x) , """oCx), and the velocity distribution u(x, y) in the boundary la^y'er- 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 boijndary layer with impermeable wall. The velocity profiles in the boundary layer were approximated by a one -parametric family and the approxl 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 tlie cne -parametric expression ^=Fi(ri) +KF2(ti); t, = -f- (6) Fj^(t^) and F2(il) are fixed prescribed functions which are immediately expressed explicitly^ K = K(x) is a form parameter of the boundary layer prof iles, the distribution of which along the length is dependent on the body shape and the suction law; 5l(x) is a measure for the local boundar-y layer thickness. The connection between 6i and 6* and ^ is given later. It proved useful to choose other expressions for the fujictions ?Q^(ii) and F2(ti) than Pohlhausen for the impermeable wall- For the velocity profile according to equation (6) the following five b-undary conditions are prescribed; they all follow from the differential equations of the boundary layer with suction, equation (l), (2): y = O: u = C; v^g^ = ug^ ^ v _ (7a,b) ^r y = oo: U = Uo 2 ou ^ o u u = Uoi r^ = O; ^ = (7c,d,e) o-v •y dy 8 NACA TM No. 12l6 The selection of ri{T]) and F2(t) is to be mede 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 setisfactorJly 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 Fi(n) = 1 - e-T (8) and correlates the values K = and 5i = &* tc the asymptotic suction profile. Furthermore, the expression (6) naturally should yield usable results also for the limiting case of disappearing suction. To this pui-pose a good presentation of a typical boundary layer profile without suction is required. One chooses as this profile the plate flow for impenneable 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 that the function -r = sin(aii) gives a very good approximation to the Blasius profile (a = Constant).^ Thus one puts < r, < 3: F2(ti) = Fi - sin^Ti) (9) r\>_ 3: F2(n) = Fj_ - 1 = -e"T and then obtains with K = -1 a good approximation for the plate flow without suction. The corresponding value of &i is given later- The functions Fj_(t]) and FgC^) are given in figure 2 and table 1. Thus one has for the velocity distribution in the boundary layer the expression: ^That the sine function is a good approximation for the velocity distribution at the plane plate without suction, resulted from an investigation of Mr. Iglisch about the asymptotic behavior of the plcr.e stepr.etlon point flow for large blowing quantity (reference 2o) . NACA TM No. 12l6 0 3- j}- = i-(K + i)e"'n ®] y (10) By selection of the functions Fi and F2 the boundary conditions (7a>c,d,e) ere per se satisfied. The last boundary condition, equation (Tb), results, because of ^^=v(|X = g[x.4-C)] (i:) in the following qualifying equation for K: ift*<^-s)] = '"'-^i^''*'^' and from it with X, = 61^' (12) ^1=- -^o^l (13) for K the equation K = \l - 1 ^1 (i-i) (lu) X and ^2. &^e ^^^'-^ dimenslonless boundary layer parameters. A quantity analogous to X was already used by Pohlhausen for the boundary layer without suction; X± is newly added by the suction. For the asymptotic suction profile with 82^ = 5, X^. = 1 according to equation (U). The form parameter K as a function of X and Xi is represented in figure 3- 10 NACA TM No. 12l6 (b) The Differential Equation for the Momentum Thickness In order to obtain by meana of the expressions {6), (fi; and (9) from equation (5) the differential equation for the momentum thickness one must first set up the relations between '^ ,^*, and 6-, . For the displacement thickness there results: « 00 n 00 1^ = (1 - Fi)dTi - K I FgdTi (15) The calculation of the integrals gives: I = 1 - k(2 - {) = g*(K) (16) For the momentiom thickness one obtains: g- = (Fi + KF2)(1 - Fi - KF2)dTi g- = Co + CiK + C2k2 = g(K) (17) The calculation of the integrals gives: Co = Fid - Fi)dTi = i (l8a) n 0^=1' (F2 - -'F^FgJdr, = -1 + | - S ^ = 0.066!)6 (l8b) Jo -i.(D NACA TM No. 12l6 11 1 ^ " C2 = -f F22dTi = -3 + ^ - 5 6e_ ^ .0.02358 (l8c) '0 1 + (I) Hence C-L - C2 = 2 - I = O.090IU Thus there Is ^1 o ^ = 3 + O.0665SC - 0.02358^=^ = g(K) (iTa) For the form parameter of the houndary layer profiles 6*/i3 used later one obtains therefore 1 - '^(^ - 1) S» V "-^ 1 - 0-0901^K A •| + Cj^K + C2K^ "I + O.O665SC - 0.0235ac^ Furthermore there results according to equations (ll) and (17): (19) :-°^«[-<-D]- (K) (20) The functions g(K) , 5*/3 and t^^-^Mv according to equation (ll) are represented in figiire k and table 2. In order to derive from equation (5) a differential equation for ^(x) one writes equation (5) in the form -E1"k^tJ-t- = -v (21) Furthermore one introduces according to Holstein and Bohlen (re^'eronce 19) 12 NACA TM No, 12l6 \J'5^ = K = Xg2 (22) and •Vot3 V = K = XjLg (23) With {2k) then ic = ZU'i K-L = -Vq^^ (25) is valid. With equations (22) to (25) as well as equation (2o) the differential equation (21) is transformed into 1 ,, dZ 1 - K 2 + C^ - 1) g(K) K + Kj^ = f(K) (26) If one finally puts for abbreviation: G(k , Ki) = 2f - 2k 1 - K 2 + c-s g(K) 2K. (27) the differential equation for Z(x) becomes: (28) NACA TM No. 12l6 13 If the function G(i^ , k.^) is known, the integral curve Z(x) can be calculated from this equation by moans of the isocline method. For carrying out the calculation in practice it is usefiol to introduce dimenaionless quantities- One forms them with the aid of the free stream velocity Uq and a length of reference Z (for instance, chord of the wing). Thus one puts -^o(^*) IVal Ur f (x*) • (29) Then equation (28) becomes: da* " uAJo ' ' - ^ Uo dx> -^l - ^l(^*) /Z* (30) The function G(k, kj) is calcvilated as follows: First, one obtains k and K]_ as fimctions of X and \]_ from equations (22) and (23), if one takes the connection between K and X, X]^ according to equation (l^^) into consideration: ~N K = g2(K)X = g2(X, \-^)\ Kj_ = g(K)X.jL = g(X, Xi)Xi From equations (27) and (3I) follows: ic = gb. + k(i - 1)1 - xg2 2 + 1 - K C-l) g >^is |g = g^ Wk(i -|)-2Xg - x[l -K^-l)] - Xi (31) G = 2gF(X., \i) (32) Ik NACA TM No. 12l6 with F(X, Xi) = 1 + k(i - "I)- 2Xg - X 1_1 - k(2 - |)J - \^ (33) Hence G can Tae calculated first as function of X, X^ and then, because of equation (3l)> also as function of k, k^- The functions k(X, X^) and k-j^(X, Xj^) are represented in figure 5 and table 3- The function thxis determined G( «, Kj) 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 = requires special considerations. In order that the initial inclination of the integral curve (dZ/dJc)o at this point be of finite value, G( k, k^) must equal zero. This gives the corresponding initial values Kq, Kiq. Since the function g(K) does not have a zero for the values of K considered (compare fig. k) the determination of the initial values k ©» Kj^q amoi-u^ts to the zeros of F(Xo, \io) = (3^^) The resulting initial values at the stagnation point Xq, Xj^q are given in table k, together with the initial values <^, k^q calc\ilated additionally according to equation (3l)' To each pair of values Kq* "^lo corresponds a mass coefficient of suction which results from V ^o' = i^oJ Z - "^lo as -Vo(o) _ '^10 = Co In figure 7 the initial values k.^ and k^^^, are plotted against the local mass coefficient at the stagnation point. The initial value Zq corresponding to Kq ^^ obtained by NACA TM Noo 12l6 " • 15 ^o = ^ (35) The following connection exists between the distribution function of the suction fi(x) = ^g ^ i/ -^ and Cq'. Uq' = KiUq/I, Kj being a profile constant. Thus there is /V^ = ^o/^ /S and V j^ fl(o) C. = -^= (36) 'o ^ 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 f2^(o) one first determines Cq according to equation (36). One obtains the corresponding initial values Kq and ^^.o f^'om figure 7, and according to equation (35) the initial value Zq of the Integral curve- If the suction does nob begin at the stagnation point but further downstream, Cq = Oj K-j^Q = and according to figure 7 0-0709 iCq = 0.0709; Zq = Uo' Separation point .- The separation point is defined by the fact that the wall shearing stress there equals zero. This gives for K, according to equation ( 11), the value K = ~ ■ " = -2.099- For the asymptotic suction 6 - n profile K = 0; this Is slmiJ.taneously the greatest possible value of K- To K = -2.099 corresponds the value \ = = -1.099 for all \i, and 16 NACA TM No. 12l6 ic = -0.0721 for all "^i- 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 \ is not unequivocal there (compare fig. k) . The function G(k, k±) 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 >^ - value of an exact separation profile according to Hartree (reference 13)- For this latter there is: separation: k^ = ( v A~ ) ~ "'^•'^^^ (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 ^ and 5* at the separation point for four different calculation methods: Case fv = K^ V u' = V 6* New method: (sine approximation ) Pohlhausen P^ (reference 15) Exact Hartree (reference I3) Exact Howarth (reference 1*+) (-0.0721) -0.1567 -0.0682 -o.o8i+i (-1.55) -1.92 -1.22 -1.25 (Ji.6ii) 3-50 k.03 3. 81+ 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/Ug partly assumes, shortly ahead of the separation point, values which are slightly larger than 1. NACA TM No. 12l6 17 (d) Performance of the Calculation for the General Case By means of the system of formioles given abo7e one may perform the calcxilatlon of the boundary layer for aii 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 -Vq(x) corresponds the total suc:.ion q,uantity and the reduced mass coefficient 0-^* = c^f-^ (36) and the reduced suction distribution function according to equation (29) -v„(x.) /U„l fl(x.) = ^— ^ Thus there is (39) If the suction begins at the stagnation point, one determines with Ki = tT~ ( r" 1 the mass coefficient C„ at the s^a^ynation point according to equation (36). Then one obtains »«o ^^^ "^lo -"^r-om figure 7 and Zq according to equation (31')- With these Initial value? the differential equation (30) can now be graphically integrated by means of the diagram in fig^ure 6- The calculation is carried out up to the point where k reaches the value tc^ = -0.0682. This Integration immediately yields Z*, K, tc^ as function of x*, with 18 NACA TM No. 12l6 F--\^ {ho) The remaining 'boundary layer parameters then result as follows: By means of figure 5 one obtains after k and k-^ the parameters X. and X.3_ and additionally from figure 3 the form parameter K- After K one obtains from figiire k the form parameter 6*/^ and thus I 1/ V ^ |/z* From equation (2o) one then also obtains the wall shearing stress t To Fn f(K) uUJfUo {^ (40a) Finally, the parameter &i is required for the velocity distribution in the boundary layer. According to equations (6), (T)> and {hO): n = g- _^-g(K) .'L.j^ — (ui) Examples of such boundary layer calculations are given in chapter VI. V. SPECIAL CASES A. WITHOUr 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 Vq(x) = Oj then NACA TM No. 12l6 19 X.3_ = 0; Kj = (without suction) and equation (ll4-) is transformed into K = X - 1 {k2) Therewith, according to equation (IT) a 1 gj = g(>^) = 2 + Ci(X - 1) + C2(X - 1) = = "1 + 0.06656 (x - 1) - 0.02358 (X - 1)^ (i;3) The differential equation (28) for the momentum thickness "becomes dZ _ G(t) dx ~ u ; K = ZU' ikk) G('«) is, according to equation (32) and (33): G = 2gF(x) (i^5) F(X) = 1 + (X - l)(l - I) - 2X I + Ci(X - 1) + C2(X - 1)^ [1 . (1 - X)(2 - I)] F(X) = -2C2X3 . (2C2 - 2 . |)x2 .(1 " | - |) ^■5 (hS) Furthermore, according to equation (31): K = g\ = X n 2 - + Ci(x - 1) + C2(X - 1) {hi) 20 NACA TM No. 12l6 The values of G and k calculated according to eqviations (^3)» C^'^)* (^6), (U7) are given in table 3- (a) The Plane Plate in Longitudinal Flow The boundary layer at the plate in longitudinal flow without suction (Blasius) wi .,h U = Uq is obtained for A, = i^ = 0. Then, according to equations (U2) to {k6)- "N K = -1 g(0, 0) = i - Ci + C2 =1 - I > r(o, 0) = I G(0, 0) = 2|(f - I = 2 - I = 0.ii29 y {k8) With the initial value Zq = the integration of eq.uation (^^4-) then gives: Z'Q-{)1 or /U - It /Vx ^ /vx ^ =i/-^r-:/- = 0.6^5 i/y- (^9) P'or the form parameter 6*/^ follows from eq.uation (19) ¥ = 2"— H = 2.66 {yo) NACA TM No. 12l6 21 and thus for the displacement thickness 6* = (it - 2) J-^—r^ = ^'^^^i\f (51^ Wr of the For the coefficient of the total friction drag Cf = - — tt plate of the width b and the length I, wetted on one side, one obtains because of Cj. = --— 'f^'IE'^-^°\^x Finally, the velocity distribution is, according to eq.uation (lO) , < n < 3: u = Uq sin (^^J y Therein is n = r- and ^1 (52) (^^?) In figure 8 the velocity distribution according to equations (53) and (54) is compared with the exact solution of Blasius; the agreement is very good. Furthermore the characteristics of the bovindary layer according to the present approximation calculation are compared with the values of the exact solution of Blaslus In the following table. For further Campari son the values according to the approximation method of Pohlhausen (reference 15) also have been given- The agreement of our new approxi- mation method with the exact solution is excellent for all boundary laj^er parameters; the drag coefficient, in pai'tlculai-, shows an error of only 2 cercent. 22 NACA TM No. 12l6 Coefficients of the Boimdary Layer at the Flat Plate in Longitudinal Flow Without Suction CalcuJ-Etion method rvx f vx 6* i3 <# To ^ New method (sine approxl mntion) 1.7^+0 0.655 2.66 1.310 0.215 Pohlhausen Tk (reference 15) 1.750 0.685 2.55 1.370 0.231+ Exact (Blasius) 1-721 0.66k 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. (h) The Plane Stagnation Point Flow For the plane stagnation point flow the velocity of the potential flow U(x) = Uq_x. All boundary layer characteristics are in this case Independent of the length x. The initial value of the mamentim thickness Zq is obtained from equation {kh) for G(ito) = 0- Since g(K) does not vanish in the range of the values of K considered, there must be F(Xq) = 0. From equation (^+6) one finds £is zero of F(x.) the value \q = 0.35^7 (stagnation point without suction) {%) The corresponding values of K and k according to equations (U2) and (U7) are K^ = -0.6^+53 and k^ = 0.0709; furthermore there is, according to equ&tion (43): g(X.o) = C.'+^7. Therewith the momentum thickness for the plane stagnation point flow becomes: = f^j-i = ^"^^i 0.266 i- (55) The form parameter 6*/'3 results from equa:iion (19) as b*/^ = 2.37; therewi th one has NACA TM No. 12l6 23 6* = 2.37/^ J~ = 0.630/^ (56) Furthermore, according to equation (U3)> ^t = 0.595/"' Thus there results from equation (ll) for the wall shearing stress: Lo K . 1.163 mU / ^1 (57) The velocity distribution results from equation (lo) as: 0^ T] < 3: ^ = o.357'+(l - e'T) + 0.6i+53 sin^-T]) T] > 3: ff = 1 - 0. 357^^6 -n (58) y with -.^ 1.6% /^ (58a) 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. 2k NACA TM No. 12l6 Coefficients of the Boundary Layer of the Plane Stagnation Point Flow without Suction Calculation method ^If -/? 6^ i3 New method (sine approximation) 0.266 0.630 2.37 1.163 0.310 Pohlhausen PU (reference 15) 0.278 0.6U1 2.31 1.19 0.331 Fxact (Hiemenz) 0.292 0.61+8 2.21 1.23U 0.360 The agreement of the new method with the exact solution is for this case somewhat less satisfactory than for the plane platej neither is it q.uite 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 for 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 xj hence all boundary layer parameters are constant. The corresponding asymptotic solution has been given already in equations (3), i^) > {^bl) • The applying values are: o 61 = 6*00; >^i = li "^1 =^ (59) NACA TM No. 12l6 25 One now calcvilates the growth of the 'bovLndary 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: X = Oi K = Therewith becomes according to equation (lU) with the abbreviation 1 - I = c = O.U76I1 (60) K = '-.LJi (61) 1 - c\i and according to eq.uation (17) 5- = g(0, Xi) = : (62) 1 (1 - cXi)"^ with Po = "i + 1= 0.110986 Pi = 5 ^ I - TT - K = °-°6^56 P2 = -2 ^ f * I (1 - D ^ -^1 = 0-05819 Furthermore there is according to equation (16) g^ = 3 - ;7-^i(:3 - i?-^J (62a) (63) 26 NACA TM No. 12l6 The val3 shearing stress becomes according to eq.uation8 (ll) and (I3): ^o = ^t ^ = -PVoUo-— -^ (6U) ^1 1 - cXi >.i(l - cXi) The differential equation (28) aesumee for the present case the form: dZ G(0, i^i) fz ,^. — = •, Ki = -Vo^fvi "^o = Constant < (65) The integration of this differential equation requires the explicit expression for G(o, k^). According to (32) and (33)- G(o, Ki) = 2g( — 1 Xi) = 2g(l - Xi) ^ i {6€) 1 - cX^ Thus G(o, k^) = for X^^ = 1- Therefore X^ = 1 is a solution of the mcmentuK equation; it corresponds to the asymptotic solution- The Init.ial value at the leading edge of the plate is X^ = 0. For the length of graving boundary layer X^. v'aries from to 1. If one Introduces as dlmensionless distance along the plate the different.ial equation (65) can be written in the form: ^^^ = G(k-l) (68) dl Initial value: I =^ 0; k^ = (68.a) NACA TM No. 12l6 2? The connection between k^ and X.^ is given by ^1 = Xig(Xi) (69) with g(Xi) 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: dKi dX-i Here all quantities can be best expressed by Xt so that a differential equation for Xi(= ) results. With iK-^/dX^^ according to equation (69) one obtains from equation (70) after division by 2g since the latter does not disappear in the range ^ X.^^ ^ 1: (j^^l^ .^!^ = n ->./L1!^ ^1 U - ^1 ^ H-r^ = (1 - ^i) ' — - (Ti) Wd.! cXj^ Initial value: | = O: Xj, = (Tla) Because of g(o, o) = Pq = — - ■^ one obtains from it in the nelghbovirhood of 1 = 0, X, = 0: d? 6 , 6/^ £\, ax^ = iT Po^i = n V« - 2^ ^1 (72) and ^ = 1 (|-i)^l' = 0.391Xi2 (73) 28 NACA TM No, 12l6 Hence follows for the neighborhood of the leading edge of the plate (I = 0) \l = 1.60 /r or -v.^5 o°l = 1 60 /F Because of — = ? - - follows hence: ^* = <"->/=^^ or 5* = (n - 2) \ - As the ccmparlson with equation (51) shows, the boundary layer thickness starts, therefore, at the leading edge of the plate with the value for the plate without suction. In order to integrate the equation (71), one has to insert the explicit values of g(Xi) and dg/dXi according to equation (62). After some intermediate calculating (compare appendix l) one obtains ^1 (1 - cXi)2(l - Xi)Q- cX^ (TU) with Po = Pi = P2 = P, = p^ = 0.^+0966 2p-]_ + PqC = -0.U66T 3p2 = 0.17^+57 -cp2 = -0.02772 > (75) NACA TM No. 12l6 29 The breeOcing up Into partial fractions yields: dl P^ Ki Kg Ko Ki^ ,, X , ^x d^l c3 X-L - 1 cXi - I (cXi - l)'^ cXi - 1 The integration vith the initial value X-^ = for 1=0 yields I = ^Xi ^ Ki In (1 - Xi) + f 2n (1 - I X^ \ XT p - Ko — ^^— + -r ^n (1 - cXi) = f(Xi) (77) -' cX2_ - 1 ^ The K-jL, • • • , Ki^ result from the breaking up into partial fractions as K^ = -6.9560; Kg = 3-'+70i^; K^ = -0.2281+; K^^ = -0.1569 (78) Thus the solution finally reads (79) 1 = -0.256^^X3^ - 6.956 In (1 - Xi) + 7-2846 In (l - 0.9099Xi) + 0.228U -7-7-7 - 0.3293 In (1 - 0.ii764Xi) O.476UX1 - 1 -^ 2 . For development of this solution in the neighborhood of 5=0, X-]^ = 0, the coefficient of X^ must, because of equation (73) > equal zero; the coefficient of X-, must equal -( ^) = O.39I. The resiat is: -^ - Ki - |C2 + K3 - K,^ = (77a) Wlch the numerical values of equation (78) one may verify that these equat.ions are satisfied. 30 NACA TM No. 12l6 The solution X^^C^) calculated accordingly is given in table 5' From Xx(^) a]] remaining boundary layer parameters can then he calcvilated Immediately according to equations (61), (62), (63), and (6U). They also are -VqS* 5« Tq5* given in table 5- In figure 9 — r; — , 7~ and are plotted against /I. ^ ^ mUq The displacement thickness of the boundary layer reaches 0-95 of its asymptotic value after an extent of the growing boundary layer of sj^ =f — j — — z h.'^. The velocity profiles in the growing boundary layer in the plotting u/Uq against — — = ^X-^ are represented in figure 10- For the wall shearing stress one obtains from equations (6U) and (1+a) : To 6 "^o^ Xi(l - cXi) (80) The wall shearing stress Is plotted in figure 11 as a function of /T- 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 Tq^ = -qVqUq; thus the coefficient of the total friction drag also equals this value W ^o„ y c = -2,70 (81) For Rmall suction quantities "VqA^o '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 lO) it is to be expected that for homogeneous suction at the plate the maintenance of a laminar boundary layer is possible even for Reynolds Uot 7 8 numbers of the order of magnitude -^ = 10 ' to 10 with a very small suction quantity of the order of magnitude cq = fr-^ = lO" . For ' — = lO" UqI 7 ft ^o Uo and -^ = 10 ' or 10*-^ one has at the end of the plate . _ /"Vo\2 UqZ ^I "lUoy V = 0.1 or 1, that is, the growth of the boundary layer Is not finished by far. NACA TM No. 12l6 31 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 ccanpletely. The total friction drag for the plate wetted on one side is: W = b I Tq dx , (82) and with the value of t^ according to equation (8o) and with t^^ according to equation (^a): « I dx W = -pVoUob^ 6 J^ Xi(l - cXi) o With dx =f ~j Tj" dl according to equation (67) this equation becomes: W = pbU,2^| I ' , ^^ , (83) with ^2 signifying the value of ^ at the end of the plate, thus: Vo\^ UqI h -(f) ^ = f(^io) (8^) Therein ^^.o signifies the value of X-i at the end of the plate which is obtained from equation (77) for I = ^l> therefore f(Xio) = -^ ^lo + Ki In (1 - Xio) -H ^ In (T - f Xi^ K^ —^ -^ ^ In (1 - cXio) (85) cX^o - 1 32 NACA TM No. 12l6 Introducing in equation (83) for r— = f '(Xi) the expression according to equation (76) one obtains: ^o and Cf = ,T-V- = 2-i^F(Xio) (86) 2"o with F(Xio) signifying F(Xio) = f (1 1 Ho f'(Xi) dXi Xj^d - cXi) Xl=0 (87) Finally introducing i^ ^ = vj-^ f(X.j_o) according to equation (8U) into equation (86) one obtains ^ Uq F(\io) U or Cf = Cf G(X.1o) (88) Because of the connection between X-io and 1 2 according to equation (8U), the total drag coefficient for the extent of growing boundary layer is thereby given as a function of the dlmensionless (VqN^ UqX fj~ ) ~v~" '-'^ ^^® other hand, equations {6k) NACA TM No. 12l6 33 and (88) give for prescribed mass coefficient of the suction -Vq/Uq the drag law Cf also against Uo^A i-ti the form of a parameter representation. The parameter X^q is the dlmensionless boundary layer thickness at the end of the plate: X-^o = ( ) ■ The values ^ ^ x=Z of X]^Q lie between 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(X.j_o) according to equation (87) gives (appendix II) zr /'p? Kic Ko N ^(Xlo) = { -^ - ^-7-^ - - - K3 + KiJ ln(l - cXio) + |ci ln(l - X.10) ^ ;|k2 In (l - ^Xj_^ - (-K^ + Ki,) i^ ^ ^ -i2 12- (89) 1 - cXio '^ (1 - cXio) c is, according to eq.uation (60), c = 1 - ?-, and K^-, ..., Ki,. are given by eq.uations (75) and (78). After Insertion of the numerical values according to equations (60) and (78) follows: F(>.io) = -0.3288 In (1 - O.k76k\io) - 6,956 In (l - ^lo) + 7,28U6 In (1 - 0.9099X10) o.k-jekXir, o.U76i+Mo (2 - 0.476UX10) - 0.037^+ -^7 0.05980 1 - 0.ii76iiXio (1 . o.U764Xio)^ The values of F(Xio) and G(Xiq) are given in table 6. For X2.0 — ^1 that Is I, — >■» (growth of boundary layer ended) one has, as can immediately be seen from equations (89) and (77): 3U NACA TM No. 12l6 F()tio) Xio-^l: -J- r = G(Xio) = 1 (89a) and thus Cf — > cr for I 7 --> " . On the other hand one has in the neighborhood of the leading edge of the plate, that is, for X]_q — > according to equation (73) and thus according to equation (87) Ho->0= F(Xio) =C|-£)^ic and therefore 2 o Xio— >0: G(Xio) = f ,— (89b) - ^lo If one substitutes this value into equation (88) and takes into consideration that ^10 -{h mFD is valid for small X^q according to equation (73) > one obtains Cf = 2 "t/V//^ NACA TM No. 12l6 35 or = /^T^ (V)"'^ thus the drag law of the plate without suction according to equation (52). The drag law of the length of growing houndary layer ia therefore for very small lengths of growing boundary ii asymptotically transformed into the drag law of the plate without auction. The drag law according to equation (88) is represented in figure 12, where Cf/cf^ is plotted against Ij. Fiirthermore figure 13 gives the drag law in the form Cf against Ugl/v for various values of the mass coefficient -Vq/Uq. 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 -2Vq certain transition region, into the asymptotic curve c^^ = -ri — • The o 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 lO) 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 moment'jm 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) = u^x and the suction velocity Vq(x) = Vq(o) = Vqo < 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, icl) = 0; this in turn requires, as was discussed in detail in chapter IV c, F()\., X,i) = 0. F(X, \x) is given by equation (33)- The values of Xq, X-^q 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 36 NACA TM No. 12l6 immediately give the complete solution since the boimdary layer thickness and all other parameters are independent of the length of growing boundary layer x. Besides Xj^, \ one further obtains K according to equation (l^t^) , g according to equation (17), 6*/i3 according to equation (19), and Kq aj^d. k-^^ according to equation (31)- The mass coefficient Cq = r^" is obtained according to equation (3'+)' From k.q finally follows t3 and therewith 5*. The results are compiled in table h. Naturally, momentiim and displacement thickness decrease with increasing suction quantity. With Cq — ^ " the form parameter 6*/^ approaches the value 2 of the asymptotic suction profile. In figure Ik ■d /ui/v and 6*y'ui/v are plotted against Cq 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. 12l6 37 n -l Sl"^ vo m o • ■ . CM cvj C\J ^ -ct (£ o 1^ VO ^-^ II H j- ITN cy Ji J- CM o ■HI CM ^1 H * • ■ »»^^ ■^luD O o O Cm ON CO On ^ ^ 00 lTA CO CS" ♦ 1 H o o H o H (Ol(0 • • H l-{ H TT ^^_^ ^_, H cy cy k; O o o H|CM s OJ Si f- t~- vo i; O o 6 o 6 1 o cp o H O o "^ 'o' H <>e o o H ^ t— ,_^ ^ On H H ^ o 0^1 ON H o d << O • X^ o ^_^ ro H m ON ^ H 5 On O W ' o •^ • • ^ o 1 CVJ 1 tM ;h O -P o 1 G g O -p w O :3 +J -H c 4J -H m C +^ T-{ C O +J •H O o-^ T-1 +J o © +3 o d ft-d o 3 m ;i P< a) a> p< tt) GO ed O C -P O ^ G -p o OJ C -H o -^ Xi CO o -P O O (m 03 -p fe^ oJ o P4 P o -P P< o ^^ H -P w 0) m P^ " en < n w P4 s <; >«« w M to ID 8 4) I 0] o (D -P (D m •d O ■p s 4) X! ■P g •H -P O O CO j3 -p tH O ^1 ■H Jh O G 0) ,a > Xi a o •H -p o a> CO -p Cm O I'd I'd (D Xi Eh 38 ' NACA TM Noo 12l6 are given In figure 15- For the case without auction qp = 101.7° results as separation polntj this is slightly further to the front than for the customary Pohlhausen method (q) = 108-9°) -for which the calculation was performed elsewhere (reference 7)- With Increasing suction q.uantity results a reduction of the boundary layer thickness and a shifting of the separation polrit 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- Such 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 ll) • Except for the neighborhood of the separation point the agreement is quite satisfactory- (b) Syrametrical Joukowsky Profiles for c^ = As second example a symmetrical Joukowsky profile of 15 percent thickness has been calculated for c^ = 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 Cq = 0-^17, that is fi(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 well-known 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 syrametrical Joukowsky profile with homogeneous suction were calculated as examples. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM No. 12l6 39 VIII. APPENDIXES APPENDIX I Concerning the Length of Growing Boundary Layer for the Plane Plate with Homogeneous Suction According to equation (71) is From equation (6?) one finds: dg _ (1 - c>Li)(pi -t- 2p2>^i) + 2c(l - cXi)(po + Pl^l -^ P2^1^) ^^" (1 - cXi)^ Pi + 2cp + (p.c + 2p2)X-, -i 2 1- ^^ (1^1) (1 - cXi)- Substitution of equations ( I, l) and (62) into equation (71) gives: ko NACA TM No. 12l6 cvi H H o 1^-^ kIV£) H H 1 - H 1 ^ OJ p< CM + on o ^— ^ H H P< ^ <^ O + 1 o H ft o OJ 5? I + OJ H + o o o o •H -p o •H •H -P -^ (0 K|VO H H OJ o H 3 CM H + o Pi o CVJ p< o CM H H Pi O P( CM O I H I << I H H o I CM H CO H p» o Pi CM O P) CM o CM H o I CM H Pi ^ + 1 H H ^ H CM m H + ^ o O Vs^ o p< o + rvi Pi H o Pi 1 n a CO H (^ (ii .** o S/ Pi CO II fl o OJ ^ m ^1^ i -P NACA TM No„ 12l6 kl APPENDIX II Concerning the Calculation of the Drag of the Plane Plate with Homogeneoxis Suction The calculation of (87) '\-^=0 gives vith f'(Xi) according to eq.uation (T6), if Xj. ^^ replaced "by z, p, ri° dz lz=0 ^lo dz ^^^l°^=3i . z(l - cz) " ^1 ,1 ^ (z-l)z(l-cz) 'z=0 >^lc + K2 dz !>^lc + K, dz ^^Q (cz - |)z(l - cz) ^ (j^^Q (cz - l)^z(l - cz) ^lo K dz ^ .1 . (cz - lY'^z z=0 ^ = I + II + III + IV + V (11,1) The integrals are solved by hreaking up into partial fractions. One finds: = ^ l^ln z - m (^z - £) '^l. -■ z=0 II = -K, In z + In ( z - 1) + — ^ In (z - 7) c-1 1-c V13/ - z=0 1^2 NACA TM No. 12l6 III = -K2 I m z . i 2n (z - i) - ;| Zn (z - ^ z=0 IV = K- In z - Zn(cz - l) cz - 1 2 (cz - D' ^lo -^ [- V = Ki^ I -In z + + Zn(cz - l) cz - 1 ^lo -I When summed up, all terms with ['" i'° cancel each other, hecause of equation (TTa). After insertion of the limits the remaining terms give: |ln(cz - 1)1 ° = Zn(l - cXio)j Rn(z - 1)1 ° = Zn(l - X^^) ^lo -cX [--t]T-^4-%>'[^r-. lo cX lo 1 n^lo ^^lo(2 - cXio) r 1 n ^io ;:::i (1 - cXio)' NACA TM No. 12l6 i+3 Thus there results by simplification from eq.uation (ll,l): + K ln(l - Xio) + |k2 ln(l - ^Xio) ^ ^ 1 - cX-io '^ (1 - cXio)^ (89) hk NACA TM No. 12l6 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 longitudinsil flow with homogeneous suction the exact solution from the differential eq.uation8 also was given in an unpublished report by Iglisch. A comparison of the approximate solution above with that exact solution is given in figures l8 and 19' Figure l8 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 I = 0.5» since the exact solution does not yet completely exist.) NACA TM No. 12l6 1+5 REFERENCES 1. Betz, A.: Beeinflussung der Grenzschicht und ihre praktlache Verwertung. Jahrb. Dtsch. Akad. Luftfahrtforschuxig 1939Ao, p- 246 and Schriften d. Dtsch. Akad. LuftfahrtforschuDg Heft U9 (19^2). 2. Holstein, H. : Messungen zur Lamlnarhaltung der Grenzschicht durch Absaugung an einem Tragflugel- Berlcht S. 10 der Lilienthal- Gesellschaft fiir Luftfahrtforschung 1914-2. 3- Holstein, H. : Messvmgen zur Lamlnarhaltiong der Relbungsschicht durch Absaugung an einem Tragflugel mit Profll NACA 0012-6iv, FB 1651+, 19h2. k. Ackeret, J., Ras, M. , and Pfennlnger, W- : Verhlnderung des Turbulent- werdens einer Relbungsschicht durch Absaugung. Die Naturvissen- schaften 19^1, p. 622. 5- Schllchting, H. : Die Grenzschicht mlt Absaugung und Ausblasen. Luftfahrtforschung Bd. 19, p- 179 (19^2). 6- Schllchting, H-: Die Grenzschicht an der ebenen Platte mlt Absaugung und Ausblasen. Luftfahrtforschung Bd- 19, p- 293 (19I+2). 7- Schllchting, H. , and Ulrich, A. : Zur Berechnung des Umschlages laminar /turbulent. Berlcht S. 10 der Lllienthal-Gesellschaft fiir Luftfahrtforschung, p. 75, 19^+2 and Jahrb. 19^+2 der Dtsch. Luftfahrt- forschung, p. I 8. 8. Bussmann, K. , and Miinz, H. : Uber die Stabllltat der lamlnaren Relbungsschicht. jahrb. 19h2 der dtsch. Luftfahrtforschung, p. I 36- 9- Schllchting, H. , and Bussmann, K- : Exakte L'dsungen fiir die laminare Grenzschicht mlt Absaugen und Ausblasen. Schriften der Dtsch. Akad. d. Luftfahrtforschung, 19^3- 10. Schllchting, H. : Die Beeinflussung der Grenzschicht durch Absaugen und Ausblasen. Lecture to the Deutschen Akademle der Luftfahrt- forschung, May 7, 19^3> to be published soon. 11- Blaslus, H- : Grenzschichten in Fliissigkelten mlt klelner Reibung. Zschr. Math. u. Phys. , Bd. 56, p. 1 (1908) . 12. Hiemenz, K. : Die Grenzschicht an einem in den gleichmassigen Fliissigkeitsstrom eingetauchten Kreiszylinder. Dingl. Polytechn. Journal. Bd. 326, p. 321 (1911)- k6 NACA TM No. 12l6 13. Ear tree, D. R.: On an Eq.uatlon Occurring in Fallcner and Skan's Appr^"imete Treatment of the Equations of the Boundery Layer. Camtrldge Phil. Soc Vol. 33, p. 223 (1937)- Ik. Howarth, L. : On the Solution of the Laminar Boundary Layer Equations. Proc. Roy. Soc London A No. 919, Vol. 16^+ (1938), P- 5^7- 15. Pohlhausen, K. : Zur naherungswelsen Integration der Differential- gleichung der laminaren Grenzschicht. Zechr- angew. Math. u. Mech. Bd. 1, p. 252 (1921). 16. Ulrich, A. : Die Stabilitat der laminaren Reibungsschicht an der langsangestronten Platte mit Absaugung und Ausblasen. Bericht I4.3/9 des Aerodynamischen Instituts der T. H. Braunschweig; to be published soon. 17. Bussmann, K. : Exakte Losungen fiir die Grenzschicht am Kreiszylinder mit Absaugen und Ausblasen. Not published. 18. Prandtl, L. : The Mechanics of Viscous Fluids. Durand, Aerodynamic Theory vol. Ill, Berlin 1935- 19. Holstein, H. , and Bohlen, T. : Ein vereinfachtes Verfahren zur Berechnung laminarer Reibungsschichten, die dem Ansatz von K. Pohlhausen geniigen. Bericht S. 10 der Lilienthal-Gesellschaft fiir Lxiftfahrtforschung 19^^- 20. Igllsch, R.: Uber das asymptotlsche Verhalten der Losungen einer nichtlinearen gewohnlichen Different ialgleichung 3- Ordnung. Bericht ^3/l'^- des Aerodynamischen Instituts der T. H. Braunschweig; to be published soon. NACA TM Noo 12l6 hi TABLE 1 THE BASIC FUNCTIONS F^ AND F2 FOR THE VELOCITY DISTRIBUTION IN THE BOUNDARY LAYER WITH SUCTION ^1 ^2 .2 .181^ .0768 .1+ .3297 .1221 .6 4512 .IU23 .8 5507 .IU52 1.0 6^21 .1322 1.2 6988 .1112 l.U 753^ .O8U3 1.6 7981 •0551 1.8 83U7 .0263 2.0 86ii7 - . 0017 2.2 8892 -.O2I1O 2.I1 9093 -.0416 2.6 9257 -.052U 2.8 9392 -.0552 3-0 9502 -.oi^98 3-5 9698 -.0302 k.o 9817 -.0183 i^.5 9889 -.0111 5-0 9933 -.0067 6.0 9975 -.0025 7-0 9991 -.0009 1 1^8 NACA TM No, 12l6 TABLE 2 PARAMETER OF BOUNDARY LAYER WITH SUCTION 5* b* Vi ^0^ -K e = 6l 8*-q 4 ^xu t+u a 0.5 1 2 1 0.5 • 1 .1^931 1.009 2.05 .9521+ .1+696 .2 .1+856 1.018 2.10 .901+7 .1+393 •3 .i+779 1.027 2.15 .8571 .1+096 .k .1+696 1.036 2.21 .8091+ .3801 •5 .1+608 I.0U5 2.27 .7^8 .3510 ., -^ .1+516 1.051+ 2.33 .711+2 .3225 ^ .61^53 .1+1+72 1.058 2.37 .6926 .3097 -7 .1+1+19 1.063 2.1+1 .6665 .291+5 .8 .1+317 1.072 2.1+8 .a89 .2672 •9 .1+210 1.081 2.57 .5712 .21+05 <5l.O .1+099 1.090 2.66 .5236 .211+6 1.1 .3982 1.099 2.76 .1+760 .1895 1.2 .3862 1.108 2.87 .1+283 .1651+ 1-3 •3736 1.117 2.99 .3807 .ii+a l.k .3606 1.126 3.12 .3330 .1201 1.5 .3»+71 1.135 3.27 .2851+ .0991 1.6 •3331 1.11+1+ 3.1+3 .2378 .0792 1.7 .3186 1.153 3.62 .1901 .0606 1.8 .3037 1.162 3.82 .11+25 .01+33 1.9 .2883 1.171 1+.09 .091+8 .0273 2.0 .2721+ 1.180 1+.33 .01+72 .0129 ^) 0.5 0.590 0.1292 -0.5^^2 O.k 0.935 0.113 -0.1+75 .h .600 .100 -.1+00 •3 •950 .0825 -.350 •3 .ai .0722 -.265 .2 .965 .0538 -.225 .2 .622 .0U63 -.136 .1 .982 .0220 -.100 .1 .637 .0222 -.016 1.0 .0192 -.2 1.01+5 -.01+60 .195 .652 .099 -.1+ 1.115 -.0810 .320 -.2 .688 -.0380 .291+ -.1+ -.6 -.8 -1.099 .738 .808 .892 1.172 -.0660 -.0830 -.0910 -.0721 .U38 .520 .555 .358 -1.099 -.0721 -.01+2 (c^ = 0.6 X >^1 IC G(ic,Ki) Kl = o.k 0-3 .2 1.116 0.088 -0.327 -.206 \ Xi K GC*,**!) 1.122 .056 .1 1-135 .020 -.072 o.k 0.770 0.1068 -0.1+60 1.15^ •3 .782 • 0777 -.321+ -.2 1.200 -.01+8 .190 .2 .800 .0500 -.200 -.1+ 1.251+ -.086 .^1+0 ,1 .815 . 0202 -.n8n .008 ici = 0.7 .835 .035 -.2 .875 .935 -.01+15 -•0730 .232 .375 X ^1 K G(k,Hi) 0.3 1.291+ 0.089 -0.125 -1.099 -.0721 .158 .2 1.292 1.296 .059 -.082 • 1 .030 -.005 I.30I+ .089 -.2 1.338 -.051+ .262 -.1+ 1.1+01 -.096 .1+02 NACA TM No. 12l6 51 EH o M B CQ W >1 CO o . p^ VO Ov CO I^ CO vO vo t- tD • oo t>- O • * CM 8 a H ^^ H CM OJ CO A [^xl On CO H O -4- CO vo H CO vS CJ vo o CM H • • 60 O o L4^ H m U3 o -ll- O CO CTn CM OO CM O H OJ CO o o [5;i> OJ OJ oo o\ H CO OO J- H ITN CM H CJn "^ o o t— CO CO li^ CM CO CM ;^i'^ no CO • CM OJ CM H • H OJ OJ CM oJ oJ CM oJ CM o OO ^r^ o On m -4- o JtX 5^ o CO CJs co 0\ VO CO o • • • 8 o H rH CM CO -d- o o OS ^ H CO ^ CM OO ITN CO o o H o H CM OJ CO 0^1 liA it o " • • • • • ■ On vo ^ ON C7N UA CO O CO C3\ H ir\ CTn O t>- ITN CT) CM CM H O Si! O 6 o O O O O o o r- OJ o\ .•D S -4- ;^- o j- IfN UA VO C— CO o t30 6 -=f J- J- • -d- -4- LTN IfN o o Cri O MS vo vS «s 0-) t~- OJ -4- ir\ UA -d- ^ oo OJ i>4 6 1 •' '■ 1 1 ■* o l/A ^ r— ON o ® OJ ir\ t— CO o J- ^ 6 OJ H H H o o o H CM -* IfN VO tr- CO o ^ o iH 52 NACA TM. No. 12l6 TABLE 5 THE BOUJNDARY -LAYER PARAMETERS AT THE PLANE PLATE IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION; LENGTH OF GROWING BOUNDARY LAYER 1 ^1 -K 5* 5l 6* V txUo ^0 fi -PVqUo 0.000171 . 000662 - 0015^ . 00291 .02 .01+ .06 .08 1 -989 •979 -968 -956 1.090 1.089 1.088 1.087 1.086 2.66 2.65 2.61+ 2.63 2.62 -0218 -01+35 -0652 .0869 0.572 -576 .581 -586 -591 CO 26.1+3 13-3^ 8.98 6.80 .0131 .0257 -0392 .051+0 -001+56 -01139 -02037 -031+1 • 0517 .10 -15 -20 -25 .30 .91+5 -916 .&8k -851 .817 1.085 1.083 1.080 1-077 l-07i+ 2. a 2.58 2.55 2-53 2.50 .1085 .1621+ • 2159 -2692 .3221 -597 -610 .625 -61+0 -656 5-50 3-76 2.89 2.38 2.6i+ .0676 .1068 .11+26 .1^5 .227 .078^ .1121+ •1551 .2127 .2879 .35 -t+o .i+5 -50 -55 .780 .7i+l -700 -656 .610 1.070 I-O67 1-063 1-059 1-055 2.1+7 2.1+1+ 2.1+1 2-37 2. 31+ .371+6 .1+267 .5296 -5802 -673 -690 .707 .728 'lh9 1.80 1-617 1.1+81 1.375 1.290 .280 .335 .391+ .1+61 .536 .388^ -5209 -7091 -9756 1-373 .60 -65 -70 •75 .80 .560 -507 .1+50 -389 •323 1.050 1-01+6 l.oi+l 1-035 1.029 2.31 2.27 2.21+ 2.20 2.16 -630^ -6797 • 7281+ •7763 •8233 • 110 -793 .817 .81+? .871 1.222 1.167 1.122 1.086 1.058 .621+ .722 .81+2 .988 1.172 2.007 3-163 5-81+0 10.556 IU.733 00 •85 -90 -95 -98 -99 1.00 .252 •175 • 091 • 037 .019 1.023 1.016 1.008 1-003 1-002 1 2.12 2.08 2.0U 2.02 2.01 2 .869-^ .911+2 -9578 -9833 -9917 1 .900 .931 .961+ .985 .993 1 1-035 1.018 1.007 1.002 1.001 1 1.1+16 1.780 2.1+15 3.250 3.835 00 MCA TM No. 1216 53 TABLE 6 DRAG LAW OF THE PLANE PLATE IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION H 1 = f(Xi) F(Xi) G(Xi) 00 .01 . ooooJ+06 • ooii665 115.02 .02 • 0001706 .008883 52.08 • 03 • 0003682 . 012732 3^^.56 .ok . 0006612 .01662? 25.11+ .06 •001535 .026i+6 17.29 .08 . 002909 .03608 12.2+0 .10 • 001+56 • 01+719 10.35 .15 •01139 .07U27 6.528 .20 . 02037 .10506 5.159 • 25 .03^+05 .1391^9 1+.096 .30 .05172 .18066 3.1+93 •35 •07833 .23006 2.937 .ko .112U .2872 2.556 ■ h3 •1551 .3539 2.281 .50 .2127 • 1^357 2.01+8 • 55 .2879 .5357 1.861 .60 .3883 .66lk 1.701+ •65 .5209 .8199 1-571^ •70 .7091 1^ 031+7 1.1^59 • 75 .9756 I.328I+ 1^362 .80 1.3731 1.7538 1.277 .85 2.0075 2.1+165 I.20U .90 3^1630 3.6018 1.139 .95 5^8!+03 6. 3098 1.080 .98 10.556 II.01+3 I.0I+6 •99 lk.l3 15^23 1.031+ • 992 16.15 16.61+ 1.031 • 99i+ 18.01 18.51 1.0275 • 996 20.69 21.19 1.021+1 .997 22.62 23.12 1.0220 .998 25.37 25.87 1-0196 • 999 30.11 30. a 1.0166 • 9995 3i+.90 35.1+0 1.011+3 1 00 00 1 = G(Xi) "V, "'- • T„ ^k Y^kCk TM No. 1216 TABLE 7 PARAMETERS FOE THE VELOCITY DISTEIBUTION OVER THE LENGTH OF GROWING BOUNDARY LAYER FOR THE PLANE PLATE IN LONGITUDINAL FLOW WITH HOMOGENEOUS SUCTION (TO FIG- 12) 1^ H -K (i H -K 1 1.0 0.1% O.38I+ .1 .1U3 .919 i.k .8U8 .255 .2 .267 .8i+0 1.8 .902 .171 .k A53 • 696 3.0 .973 •053 .6 .590 .573 00 1 .8 .685 .i^67 NACA TM No. 12l6 55 TABLE 8 RESULTS OF THE BOUNDARY LAYER CALCULATIONS FOR THE CIRCULAR CYLINEiER WITH SUCTION (a) Co = «p° s R R If V 6* /UqR R if V K \ \ Xl -K 0.1883 0.1+1+2 0.0709 0.355 0.653 k .0698 .1881+ .1+1+^ .0708 .355 .653 8 • 1396 .1888 .1+1+6 .0705 .353 .657 12 .209^+ .1892 .1+1+7 .0700 .350 .660 16 .2793 .1897 .1+1+8 .0696 .31+8 .662 20 ■3^9 .1913 .1+52 .0668 .31+5 .661+ 25 .i+36 .191+1+ .1+59 .0685 .3^+3 .665 30 .^2k .1985 .1+69 .0682 .31+2 .667 35 .611 .2030 .1+79 .0675 .3^0 .669 ko .698 .2071 .1+88 .0657 .?^^ .67^ hy •785 .2128 .502 .061+1 .325 .680 50 .87^ .2129 .522 .0612 .312 .690 55 .959 .2259 .51+0 .0585 .300 .703 60 1.01+7 .2326 •563 .051+1 .280 .720 65 1.131+ .21+38 .591 .0502 .260 .71+0 70 1.222 .2552 .623 .01+1+6 .232 • 770 75 1-309 .2698 .669 .0377 .202 .805 8o 1.396 .2881 .726 .0288 .156 .850 85 1.1+81+ .3087 .790 .0166 .091+ .905 90 1.571 .3332 .886 1.000 95 1.658 .3583 1.010 -.022^ -.11+1 1.15 100 1.71+6 .3937 1.21+0 -.0538 -.1+20 1.3^ S 101.7 1.776 .1+062 1.290 -.0682 V -.682 ' i/ 1.1+1+ 56 TABLE 8 - Continued NACA TM No. 12l6 RESULTS OF THE BOUNDARY -LAYER CALCULATIONS - Continued (b) Co = 0.5 CP° 8 R ^ /UoR R/ V R|f V K •^1 X ' ^1 -K 0.1573 0.3681 0.0495 0.1112 0.240 0.250 0.580 2.34 h .0698 .1575 .3686 .0495 .1114 .240 .250 .580 8 .1396 .1587 •3714 .0494 .1122 .240 .250 .580 12 .209*^ .1598 .3739 •0493 .1130 .240 .250 .581 16 .2793 .1605 .3756 .0492 .1137 .240 .250 .582 20 .3U9 .1^9 .3788 .0490 .1145 .240 .250 .584 2.342 25 .k^6 .1634 • 3824 .0487 .1155 .238 .252 .585 30 .52U .1658 .3880 .0482 .1172 .234 .255 •587 35 .611 .1695 .3966 .0471 •1199 .225 .260 .589 i|0 .698 .1726 .404 .0456 .1220 .220 .267 .592 2.343 i^5 • 785 .1766 .1^13 .0441 .1249 .215 .275 .595 50 .873 .1821 .426 .0430 .1288 .208 .290 .601 55 .959 .1871 .440 .0403 •1323 .195 .297 .608 60 l.oJ+7 .1937 .455 .0375 .1370 .185 .305 .a5 65 1.13^^ .2012 .473 •0331 .1422 .164 .315 .622 2.35 70 1.222 .2097 .495 .0287 .1481 .141 .329 .630 75 1.309 .2181 .515 .0230 .1542 .115 .350 .640 80 1.396 .2289 .543 .0170 .16L9 .089 .360 .665 2.37 85 l.i^84 .21+70 .593 .0109 .1747 .057 .375 .695 90 1.571 .2683 .645 .1897 .410 .735 2.40 95 1.658 .2881 .709 -.0144 .2037 -•075 .462 .781 100 1.7^6 .3123 .835 -.0338 .2208 -.185 .525 .884 2.67 105 1.833 .3421 .978 -.0606 .2419 -.415 .ao 1.06 s 106.4 1.854 .3522 1.004 -.0682 .2490 -.452 .625 1.18 2.84 NACA TM Noo 12l6 57 TABLE 8 - Continued RESULTS OF THE BOUNDARY LAYER CALCULATIONS - Continued (c) Co « 1 >> ^ v>; ^ 1.5 \^ \^ f ^ ^ \ ^^ Vv Vv V\ >^ \ \\ >^ Vi ^ \ \ \\ N\ \\ ^ ^ ^ ^ \ ^' .V x\ N^ !^ N^ V v\i n\ \\ ^> s\ ^ N. \ nN \ N \\ \^ ^< ^ N n\ \ s\ \ sV N^ X ^ X \ \ .\ \ N> \ N> X n:] N N V^ \ \^ \ 0.5 \' ^ \^ \^ s^ s \ \ \^ V N^ ^ X \^ N^ s H \; \ \ \ \^ N^ N \ V \^ N T' \ \, \ \, N^ \: nN .N N \ \\^ \ II 1 ■^1 \ \ \ \ \ ,>\ \ \ K \ V \ V \ \ \ \ \ \ \ \ K\ -1.0 -0.5 O.S Ta A,= r Q8 O.B 0.'» 0.2 -0.2 Figure 3.- The form parameter K of the velocity profile as a function of A , A according to equation (14). NACA TM No. 12l6 69 ^.6^ 5* "^o^ 1 Figure 4. The auxiliary functions G(K), — , and ^ ' uU of K, according to equations (17a) , (19), and (20) as functions 70 NACA TM No. 12l6 00 ■^^ a o , v> s& (H ^ 3 ca rr (U ;:* o Cii -4-J C3 hn G ^ ca o o o CTi -a a W O T3 c; m s q; 0) NACA TM No. 12l6 71 -ao8 Figure 6.- Diagram for solution of the differential equation for the momentum thickness: G(k,k^). 72 NACA TM No. 12l6 ^ \ 1 CD J- -\^^- \ l'/' ' \ r\ \ y / J / /' / / f / / / v7 / -O.Oeez For all K * f *7 / / / f > / i ft > / f Separatl i — 1 — i-.^ on point 7 7 "7~ r 7 — - 7~ '~ r 7 --- 1 P cn Hortree _/ [// / / / / / / f / 1 / / 1 f /> 7 / / J / / J f / J / / t II 7 / / / / / / / r / 1 / 7 7 / f. J / / / f /> / / L / f /L t 1 M V"* 5 y J / '-/ / -11 / 'll i 1 / / / / / 1 Jl / / A 1 / / r JO o — 1 1 1 ' t > f 1 / / / 6 / / / / f / / V r f 1 / J / A 7 / / / / r / Hi /(5 y ' i / r f / I , CD y n M/ Ac Y / / .o :3 c o o 4 ^ v /> 7 / / f 1 / l_ 1 11 1 1 / KJ / / / J f 1 11 If 7 1 J J 1 1 II 1 / 1 1 / / fj 1 / / / / i f V 1 L / 1 1 / li // 7 / / / J f 111 / / 1 / / / Boundary layer with suction Work sheet (For figure 6 ) ill 1} V / / / / II 1 / 7 / 1 1 '1 1 / / I / / jj 7/ ' 1 7 f ', / 1 f 7 / / / / O.07O9 stagnation point h yi 1 ', i— ' without Riir.tion / If 7 \ / / I CD vl '1 / 7 / V // h / / / 1/ / 7 r // ^ ^ A NACA TM No. 12l6 73 ^o ^to 0.08 0.0709^ -U.8 — 0.06 rti\ 0.o\ 0.04. rt / V. uA \ \ ^^^^"^^ i 'Cio -^ ' 0.02 n o ^ < ^ U,2 ^ & ; f / ^ > 5 k " W^ Vicl Figure 7.- The initial values of the boundary layer at the stagnation point for various suction quantities. Ik NACA IM No. 1216 u_ Uo 0.8 0.6 0.^ 0.2 JSi ^® 1 1 ..-^ ^^ n — - ^ ^ " .y y .^ ^ i • / Exact (Hlemenz) // / Approximate / / / A 1 1 / 1 V/n* <— •v< 3 0.d 1.6 2V Z.1 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 calciilation according to Hiemenz, approximate calculation according to equations (58) and (58a). NACA TM No. 12l6 75 W ae Q6 04 az ,_ ==- t=^ ^ P= — =.,==. r,^' ^^ y •> t \' / \ \ 1 / k K 5; / \ •^ f- ? / % y^ / ^ 6 1 ~ ■ ■-— - ■ 1 a*^ 0.8 1.2 16 20 2.* 2.8 ^ Ml -VF-^r^ .52 3.6 Figure 9.- The extent of growing boundary layer for the plane plate with homogeneous suction: against fT. 16 NACA TM Noo 12l6 W U. 0.8 06 0.4 0.2 / /" /-^ -^ ^ ^= ^ = ■ — — F= — n f /o 7 /0.4 /W <^ / / 4^ ^ ^ 1/y j / / ^ [75 ) / ^ ^ # / 1 /4 f /A f yr-^f¥ r — V Q* fl« /2 /ff 20 2.k 2.B 55 Figure 10.- Plane plate with homogeneous suction; region of growing boundary layer; velocity distribution. NACA TM No„ 12l6 77 Figure 11.- Plane plate with homogeneous suction; region of growing boundary Tq 1— layer; the local friction coefficient against \^ I. O 00 78 NACA TM No. 12l6 0) o o a; o -4-> B w o (U ho o B o J3 t— I I O o hD nJ -O . — I ctJ o 0) u P4 NACA TM No. 12l6 79 <0 5 a 0) V o o >> a; ^ o 3 a o o o o ^^ §^ a) > ^1 o O ' Si CD ho o ■ a o o CD I — I TO I CO Pn 80 NACA TM Noo 12l6 yU^v Figure 14.- Plane stagnation point flow: comparison of the boundary-layer thickness and momentum thickness of approximate and exact calculation for various suction quantities, exact calculation according to reference 9. NACA TM No. 12l6 81 l/f U Vo- /T mSLo ^ ^f^ --T ^ / 1 1 1 . 0.8 ir N- ^ U / ^ f: ^- ^ i,o • r /? m / 1 1 1 1 A N ni. fta * ^ X H -^ ^ 1 1 ,--1 U.H iZfljl f -. 2n ' — -- 1 ' 1 / o" N 1 -*i -' 1 '1 ' / y* 1 20 40 6 80 ^o m \ 120 1 — ■ ^ 1 K ^, 0.04 ---^ 45 \ N, 1 ^--x^ \ 1 o -~^ ^. \ ■ )::Z^^^ -___ ^\ n\ ^ -^ 1 20 40 6 80 ■v. i20 1 ^l^ \ N, -OM w i\ \ 1 ^ IV J \ 1 \l \ \1 \| K=-€£68 vV \_ ^1 -nna Figiire 15.- The boundary layer on the circular cylinder with homogeneous suction for various suction quantities Cq = tt^x/ v — • Form parameter «; = — 4^ • With increasing suction quantity the V ds separation point shifts rearward. 82 NACA TM No. 12l6 w w crt s f^ CD O X! -t-> o -t-> d 3^ o . — 1 o Cli +-> ^>^ O o X o 0) ;3 n1 J^ o •f-H c:^ a ■^ QJ G ho o o 9, • rH o Si C) c^ •r-l . — 1 O CD ^ oi > u o a CT) ^ CO rt o 0) .a -t-> ■ rH «t-t o o 0) c ^ o c:) -t-> SJ- o fn u, (D H . — 1 o o >> t:^ LO '\j T3 o § II O JD o ^^ CU C) ir- ^ ,-H H -(-> fH a f ) ) 1 CO 0) •t-H o • rH I— 1 J*-* CD tn tf-H 0) CD CD u U ;h ;=! o M Ph NACA TM No. 12l6 83 ^ 1.0 ^/t to -0.08 Figure 17.- The boundary layer on a symmetrical Joukowsky profile J 015 with c„ = for homogeneous suction with various suction quantities Cq = - according to equation (36): C^ = -p — x f^Co) \[k{ / dU (K^ = 51.7). K = "TT "dT • With increasing suction quantity the separation point shifts rearward. Bk NACA TM No, 12l6 CO X "" r" «M ^- lO c o tti ex o w ^S 1=1 o o o T3 -a O nj CTi d 0) CD u xi o -t-> MH o o cs ^ CD "S cS S x: •* 1^ CD gS ••-1 Q) 5 -^ o 5 OSJ Q) ° 2 C G. o a •'-' ni hjD ™ 0) tf O 1 • 00 H w o 0) c CD bJ3 O -JH S nj O , — I • f-\ > — 1 > CTJ ^5 C <^-^ a o 0) ^•§ •r< O 5 J-' O Q. ^1 CX ho ct! c o •i-i ho 03 .—1 g, UNIVERSITY OF FLORIDA 3 1262 08106 319 9 UNIVERSITY OF FLORIDA DOCUMENTS DEPARTMENT 120 MARSTON SCIENCE LIBRARY P.O. BOX 117011 CAINFCVILI E PL 32611-7011 USA