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

Full Text 
NRC TnvM 1VIr
U 7Cc Le 740 f NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1256 THE BOUNDARY LAYERS INl FLUIDS WITH LITTLE FRICTION* By H. Blasius ITRODUCTION 1. The vortices forming in flowing water behind solid bodies are not represented correctly by the solution of the potential theory nor by Helmholtz's jets. Potential theory is unable to satisfy the condi tion that the water adheres at the wetted bodies, and its solutions of the fundamental hydrodynamic equations are at variance with the obser vation that the flow separates from the body at a certain point and sends forth a highly turbulent boundary layer into the free flow. Helmholtz's theory attempts to imitate the latter effect in such a wa that it joins two potential flows, jet and still water, nonanalytical alon: a stream curve. The admissibility of this method is based on the fact that, at zero pressure, which is to prevail at the cited stream curve, the connection of the fluid, and with it the effect of adjacent parts on each other, is canceled. In reality, however, the pressure at these boundaries is definitely not zero, but can even be varied arbitrarily. Besides, Helmholtz's theory with its potential flows does not satisfy the condition of adherence nor explain the origin of the vortices, for in all of these problems, the friction must be taken into account on principle, according to the vortex theorem. When a cylinder is dipped into flowing water, for example, the flow corresponds, qualitatively, to the komwn potential, but as the water adheres to the cylinder, a boundary layer forms on the cylinder wall in which the velocity rises from zero at the wall to the value given by the potential flow. In this boundary layer, the friction plays an essential part because of the marked velocity difference; on it also depends the extent of the velocitydecreasing wall effect, which must be conveyed by shearing forces into the fluid, that is, the *"Crenzschichten in Fludssigpeiten mit kleiner Reibung." Zeitschrift fur Mathematik urnd Physik, Band 56, Heft 1, 1903, pp. 1 37 2 IACA TM 1256 thickening of the boundary layer. That the outer flow separates at a certain place, and that the water, set in violent rotation at the boundary, leads into the open, must be explainable from the processes in the boundary layer. The exact treatment of this question was undertaken originally by Prandtl (Verhandlungen des intern. Math. Congress, 1904). This explanation of the separation is repeated below. Since the integration of the hydrodynamic equations with friction is a too difficult problem, he assumed the internal friction as being small, but retained the condition of adherence at the boundary surface. In the present report, several problems, based upon the simplified hydrod.rnamic equations resulting from Prandtl's article, are worked out. They refer to the formation of boundary layers on solid bodies and the origin of separation of jets from these boundary layers suggested b, Prandtl. The writer wishes to thank Prof. L. Prandtl for the sugges tion of this article. 2. The constant of the internal friction is assumed small as in Prandtl's report. The boundary layers then become correspondingly thin; the fluid maintains its normal (potential) velocity up to near the boundary surface. Nevertheless, the decrease in velocity to value zero, and, as the calculation will show, the separation in this boundary layer must, naturally, continue, and so the potential flow is not completely regained, even at arbitrarily little friction; rather the separation and the transformation of the flow effected through it behind the body must prevail even at arbitrarily small friction. Figure 1 The procedure is limited to twodimensional flow and coordinates parallel and at right angles to the boundary (arc length and normal distance). In spite of its curvature, the type of the basic equations in the narrow space of the boundary does not differ perceptibly from that for rectangular coordinates. With e as order of magnitude of the boundarylayer thickness iACA TM 12.6 1 c&u 1 e :. LI  as the velocity u over this distance is to increase from zeroto normal values; u, ,a O, and ot JX u have normal value; from the Ok equation of continuity follows then . 1, and by integration, v ~ e. oy The terms in the fundamental order of magnitude p equations obtain then the following Ou ru I +  + v ,x oy xop 1 ii E. (uV \Pt +u v Ox ",'I, E 1.iE E.l jou u + I + 7x2 + y2, 1 1 1 E2 +k + .2^ 2v) y_2 Oy E E +cv o ox 6y 1 1 The friction gains influence when it is put at k ~ E2; this gives the relationship between boundarylayer thickness and smallness of friction constant. In the first equation, the term 62u/2x2 cancels out; in the second equation, only op/5y e or, when allowing for the coordinate curvature, ~ 1 remains. In both cases, LAllowance for the curvature of the coordinates produces, as is apparent when reforming the differential quotients, only in the second equation a nottobeneglected term pu2/r if r is the radius of curvature. This term is of the order of magnitude, unity. NACA TM 1256 the effect of the pressure on y is to be disregarded since, in the narrow space of the boundary layer, the integration of 6p/6y can, at the most, produce pressure differences of the order of magnitude o2 or e, or, in other words, pressure and pressure difference 6p/Ix are independent of y, hence, are "impressed" by the outer flow on the boundary layer. The velocity of the outer flow next to the boundary layer is denoted by 9 and is to be regarded solely as function of x because the really existing dependence on y, when compared with the substantial variations in the boundary layer itself, can be ignored in the sense of the foregoing omissions; v is accord ingly e = i, hence becomes zero with k. The remaining fundamental equations for the boundary layers are then: adu 6u. du /du \,du pcu + u + v p + u + k t x yt ox o y2 ox 6y Boundary conditions are for y = O: u = 0 v =0 for y = m: u = a These equations establish, to a certain extent, a basis for a special mechanics of boundary layers, since the outer flow enters only in "impressed" manner. 3. The qualitative explanation for the separation of flow according to Prandti is as follows: the pressure difference, and with it the acceleration, is, apart from the friction term, constant throughout the boundary layer, but the velocity near the wall is lower. As a result, the velocity here drops sooner below the value IACA TM 125r6 zero for pressure rise than outside, thus giving rise to return flow and jet formation, as indicated by the velocity profiles in the figure below. Figure 2 The region of separation itself is therefore characterized by = 0 o j/ for y = 0 This explanation does not work like the'Helmholtz jet theory with an ad hoc asumilMption, but only with the concepts forming the basis of the present hydrodynamic equations. The stream line, which bounds the separated part of the flow, departs at a certain angle from the area of separation since the stream function develops around the separation point [x] in the following manner: t = c y3 + c2(x [x] )y2 As a less important effect, it is to be foreseen that, as a consequence of the stagnation of water effected by adhesion, the flow is pushed away, from the body. Through this and the reformed flow aft of the body:, the flow upstream from the body is, of course, affected also, so that the assumption of potential flow is insufficient for quantitative accuracy of results and must be replaced by experimental recording of the pressure distribution. 6 HACA. TM 125': I. BOUNDARY LAYER FOR THE STEADY MOTION ON A FLAT PLATE IMMERSED PARALLEL TO THE STREAM LINES The flow proceeds parallel to the xaxis. The plate starts in the origin of the coordinates and lies on the positive raxis. In this very elementary case, hence, no separation is expected. carried out to illustrate the mode The fundamental equations read: there is no pressure difference; However, the calculation is of calculation to be used later. u cluf l 2u pu + v = k \o oy I ir ayx dy Clu CV 0 The equation of continuity is function : integrated by introducing the stream u =  V  ox Boundary conditions are: for y = 0: for y = : u = 0, v = 0 u = U, constant 1. According to the principle of mechanical similitude, the equations can be Eimplified when a similitude transformation converting differential equations and boundary conditions are known: multi plying x, y, u, v, 4i by the factors Xo, Jo, Uo, Vo, and *o results Puo Xo k y2 0c UoYo "o= *0 = uoyo; Lo = T4 as conditions that the problem and its solution are transformed, and that, through the transformation, p, k, U = 1 are created. The four IIACA TM 1256 7 equations still leave a degree of freedom in the choice of the factors xo, yo, uo, Vo, and Io. The last three equations define the factors assumed by u, v, and \ through the transformation; the first states that the desired solution of the problem transforms in itself, provided only that L k xo or in other words, with consideration of the factors which u, v, and 1 assume, the condition can depend only on pu y2 k x By this arg;unent, the number of independent variables is reduced. Next = 1/2 . are introduced; 5 is then sole function of ; and u = 1/2 ut' v =/2 1 (  Insertion in the differential equation gives r'" = c', Boundary conditions: for E = 0: 0' = 0 0 = 0 frca u = 0; v = 0; for = m: = 2 from u = u 8 NACA TM 1256 2. The integration of these and subsequent equations is effected by expansion in series: expansion in powers for a = 0, asymptotic approximations for a = o. The boundary conditions at both points being given, one and two integration constants, respectively, occur In the expansions. They are defined by the fact that both expansions must agree, at an arbitrary point in the function value to the first and second differential quotient. The agreement of all differen tial quotients is then assured by the differential equation. 3. Solution of the above equation by expansion in powers C = 5'" for t = 0 with the boundary conditions at this point =0 = is effected by S (i)n ncLn+l 3n+2 n=O (3n + 2)! which is so chosen that the coefficients cn to be defined are whole positive numbers, which simplifies calculation. The n+l factor a brings out the nature of entry of the integration constant; co, which otherwise would occur as such, can then be put as co = 1. The recursion formula for cn reads n1 cn = 3I nv1) CvCn1v v=0 \ 3V The first of the thus computed coefficients are: co = 1 c c2 = 11 = 375 c = 27,897 c5 = 3,317,137 2c6 = 865,874,115 C7 = 298,013,289,795 On account of the convergence, the denominator (3n + 2)! was used in the previous equations; and t" are easily formed. `The coefficients c6 and c7 in the original thesis are incorrect. This error has no effect until the fourth decimal. IIACA TM 1256. 4. There is an additive integration constant for asymptotic approximation of F because for 5 = m: ' = 2 hence, ( = 2 + const. = 2T so that T appears as new coordinate shifted toward To compute a first correction i1, put t = 2r + t1 which gives 2Tiwith = l"ar with the squares of the corrections disregarded, hence by integration: 1 = 7 d ef dri T1 =2erl di' = 7 e 2 dl Y 2 fq '2 2 = 7 e"r drl + 7e 2 SI :" I The general procedure for computing the other terms is such that further minor corrections tn are added and its squares dis regarded. The result is a set of linear differential equations for : the left, homogeneous side always the same; at the right, the error appears as "impressed force" which the sum of the preceding approxima tions, inserted in the differential equations, leaves. 5. The object is reached much quicker by the following argument: The differential equation for (t 2.q = 5l't arises from the original equation tt = tin in the NACA TM 1256 when the roughest approximation t Obviously, t has the least effect equation is then integrated as if = 2I is inserted at the left for t. at this point, and the differential t were known at this point. S= fIdTq fdTenfd The three integration constants are contained in the arbitrary low limits. Putting t = 2rj at the right gives (1 at the left, but putting C = 2T + 51 at the right gives C J f TI de  ei f t dri or with consideration to the boundary conditions = 2q7 + 7 drj dnel2 1 )d = 2T + y 7 / dTi I dTe2/TI dT Hence, the second asymptotic approximation 2 = y2 dTl d e 12 e Ld7 /dn /et 2dI By partial integration S(22 1)e 2 lTI 2d 2" = &(22 + l)e / 2d  2co 72 2 4 Se 2 d cco e272 4 O e d + 2 2"2 + e 7,2 2 i .r2 Cg2 =e o 72 .i 4 ,. T I 2 2 I e dr a/C + 2 /Tie22dn 2 tca NACA TM 1256 6. A general statement about such integration reads as follows: According to the formula eT1 = nle2 +  e2 d t' 2 2 (/,, to be gained by partial integration, each integral of this form can be 2 Tjl 2 reduced to the functions e and / er' dr multiplied by powers of q. After several such integrals are obtained, the innermost is transformed, if necessary, in the indicated manner. The integral eT2 or fIe2 dn multiplied by powers, appears then below the penultimate integral sign. The former gives no new difficulty; the latter can be reduced by partial integration to the two functions eT2 and eT .r2dT jI;b f er'i2dri = TI f eAdr + .e'1* T T1e2d i L le2d 1 /2 + 2 1ee2 Tdr er = d r 92d e d + r  T d 3 1 1 T12 122 1 2 T r2dr ej dT 2 3 =_2 / + 2 eTi + 2 eT + and so forth. J3 0 If, as above, the integral can be quadratic in e2 four types must be distinguished: e f eI2 (e / r e2 2 fTe2 e eTI2 ed d, e2d j 2e multiplied by powers of iT. The first and fourth types give nothing new. Partial integration provides for the second the formula 7 1 r fI '.2 /ITl r2 / 2e e21drl e1 dr, = e, d  j Te 2d eT 2dT (Jo I.000 (U CO NACA TM 1256 .Ie2dT e 2d = 2 e2d 2 '1.O 1. and T 2 re2d = n6_92 dT) L e11 = 2T .e72 j Jdo + e2id r ed = .e1 2 Co2 e 2dT + and so forth. Likewise for the third type 1 e 2dT 2 d =2 2 Z/o coc I" + eT2 e T2d  e2T2dr 0Ggo  2 2 e oT2 l 2 I 12 a'?44. + e2 eq2 + e212 and so 8 forth. Since no new types for integrals are introduced by these formulas, the indicated tables of formulas govern all integral in which eT2 occurs no more than twice. Any number of successive integration over such functions are possible; the powers of i involved are unrestricted. The formulas for 2 in section 5 were obtained by this method. These integration will be met again later. With the type of integration results thus known, the calculations can be made by utilizing a formula with indeterminant coefficients. 7. With this differential equation, it is possible also to define the error that afflicts the present solution as a result of the effected S 2e2  e2q2 go NACA TM 1256 omissions. It is easily verified that 2T9, 2r + Si, 2n] + +1 + 2 remain below the true value of t. An upper limit can also be found by employing the previously given (see Section 5) form, somewhat modified S= 2Tr + 7' dr 1 dTie 1e 1 u/OO of computing chosen upper expansion of a finer from a rougher approximation: a rather arbitrarily limit, such as the first term of the semiconvergent SJ, for instance, is entered for t 2n, thus  29 < e'2 4 2 and an asymptotically from this assumption. below the assumed one. formula) finer upper limit It is insured so The calculation for t", t', t is computed long as the latter remains gives (according to the general Seq12 . J100 e I . eT 2 + 00 v + 1 en2 d 2 1 v+2 1 e12 2 v+l Ti j I 2d ( i)d Ti NACA TM 1256 .0 a Figure 3 The upper limit for e' is then found according to the above figure as e') ,' 1 + ao e 0 So where Io is the highest existing value of 'J, hence corresponds to the value of the coordinates for which t is to be computed. It results in S< 2r + ? f dTI d el2 1( + f/T I/T, a8 e'2 / < l2 <2] + + 1 " IACA TM 1256 similarly for (' and. ": e'212 '> 2 + ay eo 32 A 4 2 + a2 eal 11 8 3 A more accurate execution of the integrals affords result. 8. The connection of the two developments and of the integration constants (a, y, and r i) is separate the integration constant a, z = , a more accurate the determination as follows: To X =/at is introduced in the power development (3), which results in S= ,Zf d _ ' dX2 a ( "1)n cn x3n+2 n=o (3n + 2)! = ( 1)n Cn 13n+1 n=O (3n + ) 1)n (3n) The displacement of I the integration constant 3 relative to E is expressed by introducing Qr = x 0 NACA TM 1256 The formulas are completed by inserting S= 2tq + 1 + 2 s' = 2 + t + 2 = I + t2" from the asymptotic approximation (4) and and its differential quotients fran which quoted: (5). A graph is made for the following values are X = 0 0.8 1.0 1.2 1.4 1.9 2.0 2.05 2.1 Z = 0 0.317 0.492 0.701 0.938 1.63 1.79 1.8561 1.94 d = 0 .784 .961 1.121 1.257 1.50 1.53 1.5479 1.56 dX d2Z = 1 .639 .34 .28 .2582 .23 c7X23 The terms of the power series are computed up to ; further terms 23' are extrapolated, in part, from the difference series of the logarithms of the coefficients. The location of the asymptotes is already quite apparent in figure 4. IJACA TM 1256 1.0 2.0 Figure 4 Since owing to 2 = 2r asymptotic, Z = 1, (X ), rough approximation a2/3 values can already be read for a and P: a = 1.30, P = 0.96, with X = 2.05 as connecting coordinate for = 1.00. The corresponding values of 0 and (" give for 7: 7 = 0.92. The calculation is more rigorous when a, 7, and the connecting coordinate related to T = 1 are varied by minor corrections and these then computed from linear equations. To judge the accuracy, it is stated that our calculation for X = 2.05 gave 2 = 1.8.61,, S= 1.5479, d1C d28 d2 = 0.2582, dX d3 = 0.479 dX3 where the fourth decimal is no longer certain. Asymptotic approximation for i = 1 gives (using Markoff's e t2dt) t 18 NACA TM 1256. S=2+ 0.04454 + J0.00012 2 0.OO010bJ 0.00076 2 S= 2 0.13940 0.00423 0" = 0.36788 7 + 1.0 } 2 the top numerals in the { stemming from %2, the bottom numerals from the upper limit defined in (7). (The latter is, as stated before, rather rough.) The "temporary assumption" about the upper limit of gives: < < 2 + 0.092 7. The upper limits are therefore guaranteed (reference 7). Hence, the result a = 1.3266, X = 2.0494, 7 = 0.9227, (B = 0.9508) It can be safely assumed that a ranges between 1.326 and 1.327. 9. From it, it can be computed, for example, what drag a plate of width b and length Z is subjected to when dipped parallel to the flow lines into a flow moving at velocity U. The drag per unit of surface is X = k = kkV',fl 1 Xy y 2 2Vk j Integration over the plate gives b Xy dx = b kpu o y 2 V hence, when the water flows at both sides of the plate drag = 1.327 b \kpu3 NACA TM 1256 19 II. CALCULATION OF REGION OF SEPARATION BEHIND A BODY DIPPED INTO A UNIIFORM FLOW 1. The following problem is treated: In an otherwise parallel flow, a cylindrical body is immersed symmetrically to the direction of flow. The boundarylayer coordinates are computed from the point of division of the flow. The quantity u is expanded as function of x in a power series. For the integration of the fundamental equations ,Ou i u Uu k 62u u + 7 = + ox oy ox p 6y2 u + v 0 ox oy u =jI_ q,x2 l=o the formula ,4 = > (., 2 + Z=o is used, with due regards to the symmetrical conditions for the stream function $i; u and v are obtained then by differentiation. NACA TM 1256 Figure 5 Consistent with the general boundary conditions, the functions 'X(y) must then satisfy the boundary conditions =0 = 0 for y = 0 2'. = q for y = m hence X = qy + r r, is the constant of integration. From insertion in the first fundamental equation, the differential equations for X are obtained as: (2, ('2%")= + (2% + l)q q 1 +k k=O \=0 which for 2 = 0 is quadratic, for I > 0 linear in the Y, function to be defined. This equation can, like the proceeding problem, be solved by expanding y = 0 in powers, for y = m approximating asymptotically and joining both. Subsequently, it is shown that the asymptotic approximation can be omitted, since the power series already identifies the asymptote and therefore the integration constant with sufficient accuracy. The calculation is restricted to y and X1, IACA TM 1256 that is, the first and third powers of x. Because, since the corre sponding coefficients qo and q1 in u already indicate a first increasing, then decreasing velocity the case, in which presumably separation occurs, is characterized by qo > O, ql < 0 the type of pressure distribution required in the introduction (3) is already supplied by the first two powers; hence, it is to be expected that Yo and Xl, even though not quantitatively exact, already represent the effect of the separation. In one of the problems treated in similar manner later on the next approximation was also computed; and it substantiated the admissibility of the limitation to the first two powers of x. 2. The equations for x and N, are '2 Xo = qo 2 + k~Xo, o1 ,oi" + 3(ki = 4qqI + k The manner of entry of qo' q1' k, P can be established by mechanical similarity. Here also, the first two terms indicate universal significance in same respects. Hence, writing i = q x3 f = x X X3 o 1 o 1 and introducing the following quantities 1 r i, _,, X1 = 1 n= k " Xo o for x, y, ., yI gives u qF 3) u V1 22 NACA TM 1256 = (to + ti3) V2pql u o l+ '3) etc. to and t1 satisfy, as functions of Tr, the differential equations o12 o o = + to" 4o i o 3C 1 = 16 + l1"' Boundary conditions for 0 = 0: =o = 0 = = 0 o 1 for 1 = : = 2 C' = 2 3. For 0 the power series o Sa9+1b p=2 9 ' is entered. Insertion in the differential equation gives: b2 arbitrarily = 1, since a already is integration constant. a4p = 4; since, in the formula of the integration constant a, no allowance was made for the homogeneity of the equation for (o, a appears again in this equation. b4 = 0; the curvature of the velocity profile does not change, at first, since the friction in its effect is two terms ahead of the NACA TM 1256 23 inertia; starting from m =.5, it is A bm =) (m 3] b'bm_ m =2 \[ ~ 9n The coefficients of these recurrence formulas can, like all numbers combined this way from binomial coefficients, be computed from a diagram similar to Pascal's triangle, whose start is the following: / 5 o 5 1// / // 6 /1,2/0 2 / / 1 7 .1 3/0/2/31 / / 8 /1,, /o0 / 54 X1 / and in which each term is the sum of those above it. Only the framedin portion, consistent with the foregoing limits of sums, is counted. The first 13 coefficients are 2 =1 b = b=0 b =1 b = 2b b = 2b 2 0 3 7 3 b9 =4b3 b10=ib3 b8 = 1 b = 4b bio = 1ob 2 bll= 27 16b33 b12 = 181b b13 = 840b 2 4. Besides a, two more integration constants due to the asymptotic approximation are involved, which, as in the preceding problem, should join the computed power series. For the present NACA TM 1256 purposes (calculation of point of separation), it is, however, sufficient to know a, and, as stated before, it will be seen that a can already be computed with sufficient accuracy by means of the power series. dZ Put Zo a, H = ar and plot as function of H from a ddZ the power series. Z0 itself is still dependent on b and dEH 3 a dZo 2 shall, for the correct value of a, approach the asymptote  dE O2 For other values of a, it approaches no asymptote at all, as a result of which as fig. 6 shows, the method for defining a is very sensitive. The value a = 1.515 is obtained; the last cipher is no longer certain. dZ o dH 1.0 0.87 Figure 6 5. The calculation of C 1 boundary conditions is effected by the above linear equation and the in similar manner: power formula c CC. NACA TM 1256 02 = 1, since 5 already is integration constant; 6c3 = 16; c4 = 0; and for m > 5: cm. An 3) + 4 3 3 )]3 "G+14 p=2 ^ 2) gf 1 t/ Here also the coefficients from a diagram whose first line +4, 3, while the others follow in these formulas can be computed (m = 3) consists of the numbers 1, by addition: 6 / 1/ Z6/ / Y Z3 8 / I /7 / / / / The first coefficients are c2 = 1; 5c3 = 16; c4 = 0; 0 = 4a3; C6 = 6a3c3 8; cy = 32c3; C8 = 17cLa6 010 = 576a3c3 256; 09 = 30a6,c3 2243; 1 = 2048c3 + 294x9; 7839c3 6 6 3 S= 783 c3 50922 13 = 17392a C3 + 59648a c14 = 221952a33 315a. 136192; 15 = 11025a12c3 1024000c 54864a9; 3 3 Cl6 = 174168a9c3 221296m6 6. The asymptotic approach is again disregarded, the integration constant 5 being defined by the condition that il' mist have the asymptote 11 = 2. Figure 7 shows the terms of the power series NACA TM 1256 for t1', those free from c3 and those multiplied by c3, as curves A and B, that is 1' = 5 A 16 B after which l' is plotted for different values of 5. Figure 7 This curve indicates that the convergence of the series is rather poor in spite of the great number of computed coefficients c, even at r = 1.6. In any event, the terms indicate, when identical powers of a are combined, a satisfactory variation so that the series are still practicable. The correct value of 5 ranges between 8,20 and 8.30. The curve rises, at first, very quickly and approaches its asymptote fron above. This marked influence on u near T = 0 compared to = oo permits u in the case of separation to change signs at the boundary before it does on the outside. 7. Proceeding to the calculation of the point of separation, it will be remembered from (1) that, quantitatively, the results are not exact, since only the first and third powers of x were taken into consideration. The point of separation [t] is defined by NACA TM 1256 27 0 0= u ( o~ l J3) for = 0 :OY 8kql or by (3) and (5) cL3 & = 0 By (4) and (6), respectively, a = 1.515, 5 = 8.25. Thus, in the case of the lower prefix, the only one of interest, the coordinate of the point of separation is hence [x] = 0.65 with S= qx qlx3 The maximum of the velocity (minimum pressure) lies therefore at S= 0.577 q while zero velocity in the outside flow would not be reached till x = 1 q* Accordingly, the point of separation is 12 percent of the total boundarylayer length behind the pressure maximum. The obtained figures are independent of friction constant, density, and a proportional increase of all velocities. According to Prandtl's diagram (section 3 of Introduction) the stream line 0 = 0 diverges from the boundary at a certain angle, which is computed as follows: In the vicinity of the point of separation, the development of the expression for given in (2) reads IACA TM 1256 t= vq,3i ^((st"B ~C11M [3)T3 +3 ( to" 3t 2~)(^ ] )n2) = 0 gives for the divergent stream line 2 3& l a S = 3I = 11.5 S B 16N3 400 or in the notreduced coordinates y = 1.5 These formulas are characterized by considerable uncertainty because only two terms of the development of were computed and the higher differential quotients, which represent more subtle processes, are always less accurately computed than the former. III. FORMATION OF THE BOUNDARY LAYER AND OF THE ZONE OF SEPARATION AT SUDDEN START OF MOTION FROM REST 1. The two preceding problems treated stationary flows. The problem of the growth of the boundary layer is now treated. Assume that a cylinder of arbitrary cross section is suddenly set in motion in a fluid at rest and from t = 0 is permanently maintained at constant velocity. At first, the state of potential flow is reached under the single action of the pressure distribution. The thickness of the boundary layer is zero to begin with, so far as the sudden velocity distribution can be obtained at all. The boundary layer develops in the first place under the effect of friction, then through the convective terms. The result is that, after a certain time, the separation starts at the rear of the body and, from there, progresses gradually. Since the fundamental equations refer only to thin boundary layers, they, naturally, represent only the start of the separation process, just as the previous problems dealt with the boundary layer only as far as the zone of separation. NACA TM 1256 29 2. The equations involved here are ou Au Mu 32u + u + v = u. + ot ox oy ox 6y2 U = V  oy ox K substitutes for  P The potential flow which is set up first gives the boundary value u as function of x. Since the process for t = 0 is singular, the type of development is, for the time being, still unknown; it must be established by successive approximation. The principal influence on the changes has (at small t) the friction, hence, for the first approximation uo uo a 2uo at oy2 The integral of this equation UO = f e7 2 Y  satisfies the conditions of supplying a vanishing boundary layer for t = 0 and of joining the outside flow uo = u for y = c. The subsequent approximation is obtained by inserting uo in the convective terms, while time and friction terms obtain u = uo + ul. The resultant equation for ul reads dU1 0 = + u (function of rI) Ot Jy2 ox NACA TM 1256 According to mechanical similarity, this equation is satisfied by the formula u f l = tu f() which is also not contradictory to the boundary condition ul = 0 for y = 0 and y = o. After further considerations, which in particular refer to the insertion of x, the quantity u is represented in an expansion in powers of t, the coefficients of which are functions if q, that is, still contain t. These functions are also still dependent on x, but this time x enters the differential equations only as parameter. 3. The formula for is accordingly t = 2 tt Z X (XI) TI=0 u.= .tV v=o u tvV V = OTI and hence the differential equations for X +. 2  = 0 I 12 4 for g = 1, the righthand side contains 4u . ox As before, the calculation is limited to the first two terms, that is xo = o(n) x = NACA TM 1256 hence The equations for u = u + tu:i to and t read then to'" + 2Tt = 0 t" + 2T1 4 = (2 t " 1) Bodary conditions Boundary conditions for q = 0: o=0 0o =0 (1l= (o' = 0 0o 0 = 0 for n = o: o' = 1 y1 = 0 4. The solutions of the above differential equations, which are to be used in the subsequent problem, are obtained by quadrature when the homogeneous equations are integrated. The latter integral were obtained by the following consideration: The homogeneous parts of the equations stem from the time and friction term which together form the heat conduction equation au = u at y2 Of this equation integrals of the form Un = tnfn() 2 = P.r NACA TM 1256 exist, according to similarity considerations, whereby fn the differential equation fn" + 2Tfn' 4nfn = 0 which the above form possesses. Thus, for example (see above) o 2 UO = f o =  Tit; T e 2d fl^j LIe uo = 1 when t > 0 Uo = 0 when t < 0 For n = 0, hence, for y = 0, un is proportional be representable by superposition of solutions uo form un = n uo tt T S2 (t to) to tn, hence must in the following  t dto =n Uq ( y ) to0nldto Uo 2 \(t to) since for y = 0 it is = n tndto = t For the evaluation of this integral, put t to = T un = .n uo ) (tT T "t l2 Y satisfies For Tj = 0 NACA TM 1256 insert herein T = T 2^ 2 t1^ Wk 1 2 T =1= y; 1 2' T=^4k d 2T 2k 3 and finally obtain u = r 2n F3 1 1 nI PeS2d ^ / Uco Calculation of this integral by the binomial theorem and the previously cited method of partial integration finally gives n n + 12 2Vle7 12 v=1 = (2p )...(2 2 + 1) The other integral is algebraic and equal to the above factor of r e 2dl n 2(n n =o(21 1)...3.1 5. Quantity to is determined as follows: With the boundary conditions taken into consideration TACA TM 1256 S 1 + ee d a +ar e S d whence by integration 1 0 f it4 1 (iI eT11 +2 d T 1+ce 1 1 while utilizing o = 50 T The second differential equation (of the second order for assumes then the form 8 _2 \mTI L" + 2j"1 LO' 16 1 1 f 7 I' ) 16  .e4T2 'eI2d 2e22 The integral of the homogeneous equation for ' fl (22 + 1) + 2 + (22 + 1) The integral of the nonhomogeneous equation would then be obtainable by quadratures. But it is also true that, by twice differentiating, the differential equation finally becomes t "H' + 2Ti "" = function of 1T is by (4) e 2dj  em dT IIACA TM 12,6 which is easier to integrate as an equation of essentially first order. Hence C "" = eT2 feq2[function of r] d Since the impressed force of the differential equation contains e 2 in each term after twice differentiating, eI2 cancels out, and s"" and then r can be integrated, because the functions behind the 2 integrals contain, at the most, e twice, and in addition, powers of r, and must be integrated several times, which can be accom plished by the methods discussed previously (1,6). The result of the rather voluminous calculation reads 6 I = e2 dle2d 2(2 1) 1 x Soo V 2 2 + 22r2 it + 12 _2 4 + ca(2T2 + 1) + " =  (2n2 1)e  (2r12 + 3)e12 fn Seq2dn __2 + 32 P [neq2 + (212 + 1) I e62dn] TI 2 d +8 ) 2 { nI 2 _2 2T2 0 df + dS 7e +8 e 2 32 + 4aq + 2e21 + 4 e d The reason that the equation here could be integrated in closed form, despite its affinity with the previously stationary problem, is 3b NACA TM 1256 due to the fact that ou is simpler than uS, although both have, )t dx according to the order of the differential quotient, "heat conduction character." The determination of a and 0 fran the limiting conditions = 0 for r = 0 and q = = gives a=0 P = + . 6. For computing the zone of separation, there is 0 = U 1 ut + 1t]u C" 2y 2 \Jt \ o ax 1 for q = 0 " =22 o f t=2 an2 + 8 1 l 7# The condition for the time of separation [t] is 1 + 1 +L) [t] u= hence, must be negative. The separation occurs first where ax ox has the greatest magnitude. The result applies to cylinders of any cross section; f is the corresponding potential flow. IV. DEVELOPMENT OF ZONE OF SEPARATION FROM REST AT UNIFORMLY ACCELERATED MOTION 1. Against the physical principles of the foregoing problem, the objection may be raised that the sudden shock might be accompanied Then NACA TM 1256 37 by an interruption of the fluid. Hence, let the solution of the problem assume that, starting from the time t = 0, the immersed body is subjected to constant acceleration. In that case = tw(x) 1 op u 67u ,2 aw + u+ = w + t w p 8x at 6x x 2. From considerations similar to those made before, the solution of the differential equation 6u bu 2u 1 Sp ?u + + v 1 + K it 3x dy p x .y2 is based on the formula S= 2 st It2V+lX2v+1(xT) V=o v=o ^T 2 t Insertion in the basic equation gives d3X 2X oX 21 + 2 2X 4(2X + 1)x+ a3+3 N2 0YT NACA TM 1256 xl Tax 8~2x = 2p+l 2X 2p13 =o bZ xy Px c2+1 ox6T2 for X = 0, the righthand side contains 4w, for X = 1, 4w. ox The calculation of the state is again limited to the first two terms, while it should be noted that through those two terms, the two terms of the pressure w + t2wv are also taken into consideration. ox The impressed force of the nexr equations contains only earlier development coefficients. For the final equation, however, which supplies the zone of separation, the coefficient of the next term is computed also. For X1 and the relationship of x can be introduced in the following manner: 1 = w (T), 3 Ox ' The differential equations for ( are then: 3 1 62 1 =1  + 2TrI   4 273 oT2 lj a3 ;3 23 + 2T. ST,2  123 4 + 4 oil Boundary conditions: for I = 0: for t = oc. tl = 0, t3 = 0, 3 = 1, 61 0 0^ ? 0 =OT = 0 OTf K :1 ,2 Oil _1 U =0 from v = from u = tw \OT, NACA TM 125r6 3. According to the general solutions of the present type of differential equations discussed in III (4), can be written ZOTl a1 forthwith, since the nonhomogeneous term 4 is disposed of by = 1; C is obtained by integration by the repeatedly cited method (16) 1 .+" + 2q eIq2 d9 6T2 2 2 S1 + e + (1 + 22)e n= + 2 1 + (1 + T2)e7,2 + (31 + 213) S eq2dT] These functions are quantitatively plotted in figure 8 and given in a table (see Section 6 following). NACA TM 1256 Figure 8 The impressed force on is then the righthand side of the second equation Se92dd + 16 3A [4i ei 2d + 2e92 ( + 2)e22 ++ (49 + 43)e 4. The succeeds by + (3 + 4T4) e 2dl 2 integration of the second equation, in closed form, again the same methods as in III (5). For the part of the 2 impressed force quadratic in e 2 a formula with indeterminate coefficients is particularly advisable. e 2di J o NACA TM 1256 (a + bi12 + cT4)e22 + (drT + en3 + fi5)eT12 J eT2d 00 + (g + h2 + iT4 + kTj6) e2dl 2 This formula fails when the impressed force contains terms which exceed 6 2T2 5e 2 de2 Ti4 {s eT d 2 (compare III (5)). The oo co coefficients are determined from linear equations. The other portions of : are easier to compute; 3 and  follow by as t3 o2 integration and differentiation. So, when the integration constants are correctly computed, the final result is 2t 3 4 _.2 32 eqe2drl Oa2 3 j 15x + (2 + 203)e2.2 31f + (1 + 2Ti2 + 8T4)e2 fne2 i + (6ri + 83 + 8Ti5) fi e 2 }T + 1 6 45'1 ) (16 + 36n2 + 8 )e12 + (60o + 80o 3 + 165) fe2d1L S.o T NACA TM 1256 4. eq2drl 3/1 a, 1 e 2 + 2q 151 L + (8 9i / e T d Li a, + 12 + 24 )e22 + (24n + 8r3+ 85)e2 Seledy + (9 + 18n2 + 12n4 + 86) 1 56 15 6 f? (33n + 28,N3 + 45 + (15 + 90n2 + 60o4 + 8r6) l e 2 hence, by integration 3 =2 e2 + 2T reL2d 3 Cif L u< I 8 e2 157 + (1 + 2Y2) 9I 21 a, J +  (49 + l113 + 10o5)e'22 + 768 315 L L1n(,il~ a~e22 6 f e2n2dn fm "T 6TI T 2 4T5)e2 NACA TM 1256 + (537 + 198n2 + 64T4 + 40T6)ea2 .eS2d + (315I + 21o13 + 84n5 + 40o17)  2 dTI dTI}2 00"" + .L (o 105 V6F h16 ) 24 + 87T2 45f/3 ) + 404 + 416)e 2 + (105sl + 210rT3 + 84T5 + 8T17) ( 128 + f10a.73 U1575 ,3 128 105 1/2 9) These three functions, plotted in figure 9, rigorously satisfy the differential equations and the boundary conditions for the coefficient X. 5. The condition for the zone of separation has the form = 1 L  jv w _2 ox6 J2 x6,2 Trj = whence, by the foregoing formulas 2 t 3 31 256 O3n2 '15if 225V3 0 by a2ir h NACA TM 1256 The equation for the separation time [t] reads 64 225n [t 2 = 0 ox The next term in the separation equation T = 0 dy would 2X read: t52, and in order coefficient in this separation variation of X5, is computed. 0.138 to be able to allow for it, too, the equation, rather than the total 0.25 Figure 9 The development term X5 + 2I2 20 6TI satii sfied t 6x = 4 3 __a2X Z) ax6N the equation 32,3 ",xr 62 aTI + ..l TT O1 Yx23 ax l 8x 1 The entry of x righthand side in X1 and X3 is known, and calculation Df the confirms that X assumes the form o;3 o^3~ 1 + 3 ( 60 NiACA TM 1256 X5 Ta X + w2Z 6X2 5m Since tw cancels out, the condition of separation reads + t23 ox 2  + t42 J w 2 TI=0 + _A_ + t l 2 t q0 6. This leaves the calculation of the coefficients P2 and  I and For the differential equation read For 5 the differential equation reads + 21__2 N2 =l 8~,  t 12 l ''  32 and the boundary conditions = 0, =0 5 =0 for i\ = 0  for Ni The impressed force f(i) is given by the previously written functions. The desired coefficients  are computed by Green's method as follows:   20 OTI 32 _I Tj=0o NACA TM 1256 '8 2adr 2O) A2r dTI = i  T 12 + 2^. 0 M+ L 0 2L 20)d sJo 6q F12 6t q Then, if 3 is made to satisfy the adjunct differential equation 2 r 22, = 0 and the boundary conditions (0) = 1 ,5( ) = 0 the result is .12 1=0 = f dr o f(n) is given previously; the influence coefficient a (Green's function) is obtained by integration of his differential equation T(Oq) = ( 28959 + 5280"3 + 2352i5 + 35217+ 106T9) 945 t L + (945 + 9450nr2 + 126,00i + 50O4n60 + 720T18 + 321310)el 2 "se2d] The curve of a is shown in figure 10, along with the product 9 f. The area of this last curve gives the desired coefficient. +2 . an;? C)1 "" IJACA TIM 12 6 0.5 0 0.5 1.0 For computing 6n13 25 0.50 Figure 10 '2 = 6_ 5m )= the equations 2 = g d T L_72 ]_ =0 are available; ? g is plotted in figure 10 according to the values indicated below. = g(l) ,2 (I r + 2Tr~~4 6T12 HACA TM 1256 The computed values are the following: all 821 3 323 ( N12 *C f f(T) 6(Tl) 3 f g(T) 3 g 0 0 2.26 0 0 .964 0 1 0 0 0 0.25 .061 .450 1.396 .022 .137 .231 .315 .327 .103 .124 .041 0.50 .211 .720 1 .799 .060 .150 .092 .750 .12 .084 .240 .027 1.00 .638 .943 .201 .115 .020 .156 .457 .018 .008 .016 .0003 1.50 0.035 .05 S o.376 1 0 .138 0 0 o 0 0 0 0 0 The area of the two curves is approximately 1h2 Ti0O = 0.058 LI [2 m0 7o The equation of separation therefore reads 4L + [t]2.~ _ 256\_ [t]Y ) 0.058 [t]4w 0.023 = 0 TIC L x 15{ii 225TAf3 \ 2 = 0. 02 3 NACA TM 1256 1 + 0.427 [t]2w 0.026 [t] 2 0.01 t] 2w = 0 Since the newly added correction term is even negative, the existence of the zero position appears to be certain. The position and time of separation is according to the earlier approximation (without the term computed last) 2 t = 2.34 For the case of a cylinder symmetrical to the direction flow, = 0 at the rear point where the separation starts, the 0x2 newly computed correction gives [t] 2 = 2.08 equivalent to an error of about 10 percent. From this the quality of the approximation made in the other problems, where only the first powers were taken, can probably be also appraised. V. APPLICATION OF THE RESULTS OF THE SEPARATION PROBLEM TO THE CIRCULAR CYLINDER 1. On the circular cylinder S= 2V sin R x is called the reduced coordinate X; V is the velocity at which R the parallel flow flows toward the right, and the cylinder moves 50 NACA TM 1256 toward the left, respectively. In the steady case, the separation starts according to Part II, Section 7 at xeep, = 0.65 1; the maximum velocity lies at mx = 0577 I1 where l3 u= q x qx ol Figure 11 Taking the ordinary development in powers of sine S2V x = 1.59 R, X = 911 sep. sep. 4 2V 6R3 x = 1.41 R; X = 810 max max IACA TM 12 5u But approximating the sine in the interval 0 x by the method of least squares, gives 2v= o.856, lo R " ep. = 1.97.R, sep. x = 1.75 R, max a = L 0.093 R3 X = 113 sep. X = 1010 max In any case, the point of separation lies, by the present calculation, at from 11 percent to 12 percent of the total boundary layer length behind the maximum of the velocity. This statement makes, of course, no claim to accuracy, since only the first two powers of x are taken into consideration. Besides, test records of the pressure difference indicate that the state near the separation is difficult to attain by development from starting point of the boundary layer, because it is too strongly affected by the pressure distribution of the turbulent bodies behind the cylinder. The sole purpose of the present calculations is to indicate that separation is actually obtained by the hydrodynamic equations. Further development of the calculating methods, especially for the more important problems of solids of revolution, promises, therefore, success. 2. If the cylinder with constant velocity is suddenly set in motion u = 2V sin X The time of separation = cos X 8x R [t] is, according to III (6), given by + A Bz t] = 0.35 R V cos X NACA TM 1256 The separation starts for X = i, cos X = 1 at time to 0.395 Up to that, the cylinder has travelled a distance S = Vto = 0.35 R All this is independent of velocity, density, and friction coefficient (little friction assumed). 3. At constant acceleration u = tw(x) = 2Vt sin x R where V is then the acceleration of the cylinder in the flow. The separation time is (IV,7) for the start of separation or = 2.08 respectively, or t]2 = 117 R V cos X or = R V cos X respectively. The distance covered by the cylinder is S = lt2 2 at start of separation (X = t) S = 0.99 R or = 0.52 R respectively.  = 2.34 NACA TM 1256 4. The resistance which the cylinder experiences at constant acceleration is computed next. The stress components are Y  + 2k1 y 6a 6v Owing to the smallness of the friction, 2y and respect to leaving as force in direction of C7 6v cancel with the outside flow the outside flow K = 2 .B p co X + k sin X RdX B is the width of the layer (height of immersed part of cylinder). The pressure portion is computed as follows; Pressure = 2BR fp cos XdX $~uo = 2BR2 x sin XdX "so opx Then u = tw (= p +2 w = 2V sin X NACA TM 1256 The second term cancels out in the integration; the first gives Kpressure = 23pBR2 hence, an increase in inertia by twice the amount of displaced fluid. The friction portion is Kfriction 2kBR 2 Vi . 2 c 2 o2+ \ + t3 ) sin XAX 6x 2 / where K = k/p. Again, the second term disappears because  S2^3 and 2 are merely constants, leaving Friction = 4"pkt BRV 5. To give a picture of the flow conditions corresponding to these formulas, the flow curves for a specific state of motion of the uniformly accelerated cylinder are represented in a diagram. The parameters R, V, r are arbitrary; hence, necessitate the introduction of reduced quantities for x, y, t, *, and u, so that R, V, K disappear. It is accomplished by x = RX, t = , x=B , y = YjR u = U NACA TM 1256 by which the formulas (compare IV (2) and V(3)) * = 2t3/12w( 1 + TI = tvw S2V Kt w 2V Bin + t2 x _3\ t2 2Vt2 cos ex R R become the following reduced equations 3 = 4T3/2 sin X ( + 2T2 U = 2T sin X ' + 2T2 cos X Y 2 The curve I is then plotted against for a number of coordinate values X, values T = constant read from these Y = 2/T 1 for a fixed time T and the position of the curves. In figure 12, the cylinder is shown from X = w/2 to X = x. The separation time is given by 2T2 cos X = 2.34, that is, the start of the separation cos X 3) UACA TM 1256 by T = 1.08. In figure 12 2T2 = 5, hence T = 1.58, was chosen. For this chosen time, the separation point has already progressed up to beyond 600 at the cylinder; nevertheless the boundary layer still is fairly thin, the relative sizes correspond to the values R = 10 cm, 2 S= 0.01c2 (water), V = 0.1, that is, to a very small acceleration. sec sec2 Accordingly, t = 15.8 sec. Figure 12 NACA TM 1256 The picture obtained by the previous reduction fa orn as far V 10SL after 1.58 sec. is represented in figure 12. The 2 sec thickening of the boundary layer would be diminished in the ratio of 1 : . Translated by J. Vanier National Advisory CaUmittee for Aeronautics NACALangley 42255 75 i UNIVR1SIIT OF 0LOUIUA II1111111 11111 1 3 1262 08105 034 5 