f\lftr^T?w-QSb|^» . ^'^ ^ NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1256 THE BOUNDARY LAYERS IN FLUIDS WITH LITTLE FRICTION By H. Blasius Translation of * Grenzschichten in Fliissigkeiten mit kleiner Reibung" Zeitschrift fixr Mathematik und Physik, Band 56, Heft 1, 1908 Washington February 1950 DOCUMENTS DEPARTMENT NATIONAL ADVISOEY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM I256 TEE BOUKDARY LAYERS IN FLUIDS WITH LITTLE FRICTION* By H. Blasius INTRODUCTION I. The vortices forming in flowing vater "behind solid "bodies are not represented correctly "by the solution of the potential theory nor "by Helmholtz's jets. Potential theory is una"ble to satisfy the condi- tion that the water adheres at the wetted bodies, and its solutions of the fundamental hydrodynamic eqiuations 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 way that it Joins two potential flows, jet and still water, nonanalytical along 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, q_ualitatively, to the known 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 velocity— decreasing wall effect, which must be conveyed by shearing forces into the fluid, that is, the *"Grenz3chlchten in Friissigkelten mit kleiner Reibung." Zeltschrift fur Mathematik und Physik, Band 56, Heft 1, I9O8, pp. 1-37- WACA 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 boiondary, leads into the open, must he explainable from the processes in the hoiondary layer. The exact treatment of this question was undertaken originally by Prandtl (Verhandlungen des intern. Math. Kongress, 190^*-) . 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 hydrodynamic 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 by 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. y k Figure 1 The procedure ia limited to two-dimensional flow and coordinates parallel and at right angles to the boundary (arc length and normal distance). In spite of its ciurvature, 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 boundary— layer thickness WACA TM 1256 Su _ 1 ^u. J^ as the velocity u over this distance is to increase from, zero to normal values; u, —, 2^, and ^-^ have normal value; from the St Bx 5x2 equation of continuity follows then -21 ~ l, and "by integration, v ~ e The terms in the fundamental eq^uations ohtaln then the following order of magnitude-^ /bu. Su Su\ Sp /S^u S^uN p — - + u-— + V—- = + M + — - V^t Bx hjj Bx \ax2 Sy2y 1 1.1 .411^ J^ . u^ , .Sv\ _ ^ ^ ,/#v ^ #v "^^Bt Bx By; By VBx2 5y2 l-e e.l 1 ^ + ^ = Bx hj p The friction gains Influence when it is put at k ~ e ; this gives the relationship "between boundary— layer thickness and smallness of friction constant. In the first equation, the term S^u/Bx^ cancels out; in the second equation, only Bp/By ~ e or, when allowing for the coordinate curvature, ~ 1 remains-'-. In "both cases, -^Allowance for the curvature of the coordinates produces, as is apparent when reforming the differential quotients, only in the second equation a not— to— be— neglected term pu2/r if r is the radius of curvature. This term is of the order of magnitude, unity. MCA TM 1256 the effect of the pressure on y is to be disregarded since, in the narrow space of the "boiindary layer, the integration of Sp/Sy can, at the most, produce pressure differences of the order of magnitude €2 or g, or, in other words, pressure and pressure difference Bp/Sx are independent of y, hence, are "impressed" by the outer flow on the boimdary layer. The velocity of the outer flow next to the boundary layer is denoted by u 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 emissions; v is accord- ingly •»» 6 = tk, hence becomes zero with k. The remaining fundamental equations for the boundary layers are then: (^ ba ^\ /Su _Su \ , S^u :-— + U-;— + Vt^ = p -- + u^ + k at hx ciyj \bt Sx J ay2 ^ + ^ = Sx Sy Boundary conditions are for y=0: u=0 v=0 for y = 00: u = u These eq^uations 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 cLualitative explanation for the separation of flow according to Prandtl 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 MCA TM 1256 zero for presstire 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 du = for y = This explanation does not work like the- Helmholtz Jet theory with an ad hoc assumption, 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 ^t develops around the separation point [x] in the following manner: i = cjj^ + 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, ao that the assumption of potential flow is insufficient for quantitative accuracy of results and must be replaced by experimental recording of the pressure distribution. NACA TM 1256 I. BOUNDARY LAYEE FOE THE STEADY MOTION ON A FLAT PLATE IMMERSED PARALLEL TO THE STREAM LINES The flow proceeds parallel to the x— axis. The plate starts in the origin of the coordinates and lies on the positive x— axis. In this very elementary case, there is no pressure difference; hence, no separation is expected. However, the calculation is carried out to illustrate the mode of calculation to he used later. The fundamental eq_uations read: , &u SuA , S^u P I'Ut— ■ + V—- = k- Bx Sy/ Sy2 ^ + ^ = Sx Sy The equation of continuity is integrated "by introducing the stream function \|/ : _ 'b^ _ _ ^ Sy Sx Boundary conditions are: for y=0: u=0, v=0 for y = °°: u = u, < — constant 1. According to the principle of mechanical similitude, the eq^uations can he simplified when a similitude transformation converting differential equations and boundary conditions are known: multi- plying X, y, u, V, \|r by the factors Xq, Jq, Uq, Vq, and ijfo results in P^o k ^o^o . XO y^2 xo as conditions that the problem and its solution are transformed, and that, through the transformation, p, k, u = 1 are created. The four MCA TM 1256 1 eq_uatlons still leave a degree of freedom in the choice of the factors Xq, Jq, Uq, Vq, and ^q. The last three eq^uations define the factors assumed by u, v, and \|r through the transformation; the first states that the desired solution of the problem, transforms in itself J provided only that pu ^o _ -, k Xq or in other words, with consideration of the factors which Uj v, and ^ assume, the condition can depend only on pu y2 k X By this argument, the number of Independent variables is reduced. Next I = 1/2 n/eS . JL /k ^ y p I are introduced; t is then sole function of | and u = 1/2 u^' ' Va ^i ( !?• - ^) Insertion in the differential eq_uation gives f f" = _ f"i Boundary conditions: for 1=0: ^'=0 ^=0 from u = 0; v = 0; for ^ = 00: ^' = 2 from u = u 8 WACA TM 1256 2. The Integration of these and suhsequent eq^uations is effected hy expansion in series: expansion in powers for i = 0, asymptotic approx-ijuations for I = 00. The "boundary conditions at "both points "being given, one and two integration constants, respectivelyj occur in the expansions. They are defined "by the fact that "both e^cpansions must agree, at an ar"bitrary 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 a"bove equation "by expansion in powers for 1 = with the "boundary conditions at this point ^« = C = is effected "by s=S n=0 which is so chosen that the coefficients Cn to "be defined are whole positive numbers, which simplifies calculation. The factor a "brings out the nature of entry of the integration constant; cq, which otherwise would occur as such, can then "be put as Co = 1. The recursion formula for c-^ reads n-1 An-l^ The first of the thus computed coefficients are: Co = 1 c-L = 1 Cg = 11 C3 = 375 cj^ = 27,897 C5 = 3,817,137 2cg = 865,871^,115 c^ = 298,013,289,795 On account of the convergence, the denominator (3n + 2)'. was used in the previous equations; t,' and ^" are easily formed. p The coefficients eg and Cj in the original thesis are incorrect. This error has no effect until the foixrth decimal. WACA TM 1256 9 h. There is an additive integration constant for ^ in the asymptotic approximation of | "because for ? = 00: ^' = 2 hence, ^ = 2| + const. = 2t\ so that Ti appears as new coordinate shifted toward | . To compute a first correction ^-, , put ^ = 2ti + ^^ which gives with the sq_uares of the corrections disregarded, hence by integration: t, = 7 / dT] / e-^l dT] = 7ri / e""! dri + ^""l 1 lioo '/co Joo />T1 2 2 5^' = 7 / e-^ dTi t,{ = /e-^l Uoo The general procedure for computing the other terms is such that further minor corrections ^^ ^^9 added and its squares dis- regarded. The result is a set of linear differential equations for ^n, 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 (,-. 2t1^i" = -^1 arises from the original equation itt 10 NACA TM 1256 when the roughest approximation t, = 2i] is inserted at the left for 5 Obviously, ^ has the least effect at this point, and the differential equation is then Integrated as if ^ were known at this point. i t = pi^ Tdrie-^ ^^-^ The three integration constants are contained in the arbitrary low limits. Putting t, = 2f] at the right gives 1^-, at the left, but putting ^ = 2t) + 5j_ at the right gives = r\. ra.a-i^-/'V'> 5 = 1 dri / dTie '' e or with consideration to the boundary conditions C = 2ti + 7 / dTi / dTie-'l^l 1 - / ?idTi (/CO t/00 \ Ifoo Hence, the second asymptotic approximation ^2 = -7^ /* dTi / dT) . e^^ / dTi / dTi / e""! dii By partial Integration 2 _T,2 /1I -,,2 2 I ,v, 2 I 2 ^ ,,.=^,e-Ve-^^-? /e-^^. . 2._„2 /"l._r,2.. r2 I /"'.-n2^ I .r2/'\^n2 ^2 -T^-v:-^ ^-f^ y:«"^^ ^?/. -"^^^^ 2 NACA TM 1256 11 6. A general statement atout such integrations reads as follows: According to the formula 1^ e-^^TindTi = - Iri^-le-^^ + 2L^^ ^ e-n^r^^^d^r^ to "be gained "by partial integration, each Integral of this form can he reduced to the functions e ' and / e ' di] multiplied hy powers of T] . After several such integrals are ohtained, the innermost is o transformed, if necessary, in the indicated manner. The integral e~^ 9^1 _ 2 or / e ^ dT] multiplied by powers, appears then below the penultimate Jco integral sign. The former gives no new dlfficultyj the latter can he _ 2 /"^ _ 2 reduced by partial integration to the two functions e ^ and / e ^ di] T) p 2 fl ? 12 dTi / e-il dT] = T] / e-^^dT] + ie-^l to t/oo Joo 00 •^2 p^ r^ 2 1 r^ 2 1 2 1 2 / Ti2dri / e~^ dTi = iriS / e"^ dr] + r'n^e''^ + ^"^ and so forth. (jco Uoo Uoo 2 If, as above, the integral can be quadratic in e"^ , four types must be distinguished: multiplied by powers of t) . The first and fourth tjrpes give nothing new. Partial integration provides for the second the fonnula f "^e-^^dTi / "^e-^^dTi Joo t/00 11 2 2 /'^ 2 /'■n ? e-n^dTi^ - / e-^ dT) / e-^ dT] 12 NACA TM 1256 or and "^e-^^ /"'e-^^dT) = iJ /"'e-^^dri; "^Tie-^% / V^\ti = -ie-^^ fV^^ +i / e-^^^t) ASe-^^dn Te-^^dTi = - irie-^^ Te-^^dii + i e-^-dTi + f < / \-^^dTi S' - ^^^' and so forth. Likewiae for the third type e ^ di] ?■ dr] = T] e^ dTi e-^^ / "^e-^^dT) - / 'e^^^dTi ^y P _Ti2 i 2 p TT 2 -^^ / e-^ dTi - i, / e~^^dTi > + ie~2^ and 30 forth Since no new types for integrals are introduced by these formulas, 2 the indicated tables of formulas govern all integrals in which e M occurs no more than twice. Any number of successive integrations over such functions are possible; the powers of t\ involved are unrestricted. The formulas for ^2 ii^ section 5 were obtained by this method. These integrations 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 MCA TM 1256 13 amissions. It Is easily verified that 2t\, 2ti + Cj^, 2ti + ^2_ + ^2 remain "below tlie true value of ^. An upper limit can also "be found "by employing tlie previously given (see Section 5) form^ somewhat modified -/ (C^Tl)dTl "1 P"^ .2 ^ = 2ti + 7 / dii / drje ^ e of computing a finer from a rougher approximation: a rather arlDitrarily chosen upper limit, such as the first term of the semlconvergent expansion of ^^.j ^o^ instance, is entered for ^ — 2t\, thus 2ti < I £"'^ k .2 and an asymptotically finer upper limit for ^", ?', ^ is computed from this assumption. It is insured so long as the latter remains "below the assumed one. The calculation gives (according to the general formula) 2 \-^^^ = ± £^ _ V + 1 /' e-Tl2_iTL ri ^^ 2 ^v+1 2 J^ ,^v+2 2 ^v+1 ^-n^ (^-2n)dTi stemming from ^2j "the "bottom, nimierals from the upper limit defined in (y)- (The latter is, as stated before, rather rough.) The "temporary assumption" a"bout the upper limit of s gives: ^ <2 + O.O92 7. The upper limits are therefore guaranteed (reference 7) . Hence, the result a = 1.3266, X = 2.O49I1, 7 = 0.9227, O = 0.9508) It can he safely assumed that a ranges "between I.326 and I.327. 9- Frcam it, it can "be computed, for example, what drag a plate of width "b and length I is subjected to when dipped parallel to the flow lines into a flow moving at velocity u. The drag per unit of surface is y ^y 2 2 V k ^ - 1 f -' ^ Integration over the plate gives n h ■ / X dx - ^ ' /v^7ri3 % Jkplu- hence, when the water flows at "both sides of the plate Jkplu^ drag = 1.327 - "b KACA TM 1256 19 II. CAKTUIATIOW OF REGION OF SEPARATION BEHIKD A BODY DIPPED INTO A UNIFORM FLOW 1. The following problem is treated: In an othBrwiee parallel flow, a cylindrical "body is inmerBed symmetrically to the direction" of flow. The h oundary— layer 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 u^ + v^ = u-^ + ^ ^^ Sx hj 5x p ^y2 Su _j_ Bv bx 5y 5u ^ Bv _ Q u = 42 1=0 ^Z+1 the formula t =2lx^(y)x2^+l 1=0 is used, with due regards to the symmetrical conditions for the stream function ^^J u and v are obtained then "by differentiation. 20 NACA TM 1256 Figure 5 Consistent with the general houndary conditions, the functions ■^7(7) must then satisfy the houndary conditions X^' = ^7 = ° f or y = I = ^Z for y = hence r^ is the constant of integration. From insertion in the first fundamental equation, the differential equations for X are ohtained as: I I kv t" II^2X + 1)(\' Vx' - \Vx") = ZI(2^ + ^^\^l-X ^ ?l' x=o x=o which for 2=0 is quadratic, for Z > linear in the 'X^ function to Ise defined. This equation can, like the proceeding problem, he solved "by expanding y = in powers, for y = <» approximating asymptotically and Joining hoth. Subsequently, it is shown that the asymptotic approximation can he emitted, since the power series already identifies the asymptote and therefore the integration constant with sufficient accuracy. The calculation is restricted to x„ and x-.. NACA TM 1256 21 that is, the first and third powers of x. Because, since the corre- sponding coefficients q^ and q^-, in u already indicate a first increasing, then decreasing velocity — the case, in which presixmahly separation occiirs, is characterized ty q^ > 0^ q < — the type of pressure distrihution required in the introduction (3) is already supplied hy the first two powers; hence, it is to be ea^jected that Xq and X-j_, even though not quantitatively exact, alr-eady represent the effect of the separation. In one of the problems treated in similar manner later on the next approximation was also cam.puted; and it substantiated the admissibility of the llm.itation to the first two powers of x. 2. The eq^uationa for \ and Xt are \} ^3^3 ^o "*" P'^o ^o'^l' -\^- k. 1" -3(x,'x,'-x,x,")=^Vi-^fV" The manner of entry of q q.-, , k, P can be established by mechanical similarity. Here also, the first two terms indicate universal significance in some respects. Hence, writing u = q^x±qi3 ^ = x x ± X 'X? o 1 o 1 and introducing the following q^uantities '^ -&^ ^0=/!,^. ..= 5^, for X, y, Xq, Xi gives u = (I ± 1^) 22 NACA TM 1256 t = M^'oi ^ ^x^3, u = |M ^o' 5 ± V ?3) ,to. ^^ and ^-j_ satisfy, as functions of t\, the differential eq_uation8 ^o'^-U^o" =^ + ^0" Boundary conditions for Ti = 0: ?c = ^o'-° ^1^ = ?,• .0 for T) = 00: ?o'= = V=2 3. For ^ the power series ^ =11 — r^T^ n=2 ^^• is entered. Insertion in the differential eq^uation gives: ^2 arbitrarily = 1, since a already is integration constant. a^Po = —h; since, in the formula of the integration constant a, no allowance was made for the homogeneity of the equation for ^ , a appears again in this eq^uation. b]^ = Oj the curvature of the velocity profile does not change, at first, since the friction in its effect is two terms ahead of the MCA TM 1256 23 inertia; staarting fram m =- 5^ it is m ^=2 m - 3 II - 1 m- 3 h >[Dr-l-\i The coefficientB of these recurrence fonnuljas can, like all numbers ccEibined this vay from hinamial coefficients, he computed from, a diagram similar to Pascal's triangle, vhose start is the folloving: and in which each term is the sum of those ahove it . Only the framed— in portion, consistent with the foregoing limits of sums, is counted. The first I3 coefficients are \ = 1 ^ = 1 ^8 = -1 '11 = 27 - 161)33 '3=-J. \ = ^6 - '^^ b, - 2^3^ s = -*^ ^10 = -16^3^ h22 = iSlh^ ^ = SkOhJ^ k. Besides a, two more integration constants due to the asymptotic approximation are involved, vhich, as in the preceding problem, should Join the computed power series. For the present 2k KACA TM 1256 pirrposes (calculation of point of separation), it is, hovever, 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. 1 ^o Put Zq = — ^, H = ari and plot -rs^ as function of H from dH the power series. dZ^ dH Itself is still dependent on b^ = — — and J ai, shall, for the correct value of a, approach the asymptote k_ dZo dH a'- 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 = I.515 is obtained; the last cipher is no longer certain. dZ dH 1.0 - 2 >^ 0. 87 / <^ ^^^ / ^^^oT87— *— ' ^^ 0.88 /^:^^o.9o— ^__ 0.5 2\ =\1- 00 a2 \ 1.0 2.0 Figure 6 5. The calculation of ^2. ^y "t^© above linear equation and the boundary conditions is effected in similar manner: power formula ^^•^s-^^ MCA TM 1256 25 C2 C4 Ij since S already is Integration constant^ 5co = — 16; 0} and for m > 5: Cm = H=2 "4 -3^ *<-l] r.' '^^'^\l^hDr-l-^Ji Here also the coefficients in these formulas can he computed from a diagram whose first line (m = 3) consists of the numhers -1, +4, —3, while the others follow "by addition: The first coefficients are Cg = 1; 5Ct = — l6j cj, =0} Cr = ^ } cg = 6a^CT — 8; cy = -32c^; C8 = lTa°j c^ = 30a°C2 - 22lw3. c^Q = -576a^c - 256; c-jj_ = 20l^8c + 29ka/; « 9 6 6 3 C22 = 783a Co - 5092a J Cq^o = -17392a Co + 59648a ; 3 12 ^ c^^l^ = 221952a-'co - 315a - 136192; c,^ = -11025a'^c^ - 1024000c^ - 5U861w,^: 15 3 3 °l6 ^ lT^l68a^c - 221296a . 6. The asymptotic approach is again disregarded,, the integration constant 5 being defined hy the condition that ^-|.' nmst have the asymptote ^ =2. Figure 7 shows the terms of the power series 26 WAG A TM 1256 for ^ 2_ J those free from Co and those multiplied hy Co, as ciirveB A and B, that is ^-|_« = S • A - 16 ■ B after which ^^' is plotted for different values of B. J 1 1 I ■ I >■ 71 0.5 1.0 1.5 1.7 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 Ti = 1.6. In any event, the terms indicate, when identical powers of a are com"bined, a satisfactory variation so that the series are still practicahle. The correct value of 5 ranges "between 8o20 and 8. 30. The curve rises, at first, very q^uickly and approaches its asymptote from ahoveo This marked influence on u near t] = compared to 11=00 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 (l) that, quantitatively, the results are not exact, since only the first and third powers of x were taken into consideration. The point of separation [ij is defined "by WACA TM 1256 27 « = l=iSi^'^""H^^^"f^]^' -' -° or by (3) and (5) cc3 ± 5 0^ = By (1|) and (6), respectively, a = I.515, B = 8.25. Thus, in the case of the lower prefix, the only one of interest, the coordinate of the point of sepaiTation is [1] = 0.65 hence with u = Q-o^ - li^^ The maximum of the velocity (minimum pressure) lies therefore at X = 0.5TT\[P K^i while zero velocity in the outside flow would not ha reached fio till X = 1 • \| Accordingly, the point of separation is 12 percent y I1 of the total "boundary— layer 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 1I'' = 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 't given in (2) reads 28 NACA TM 1256 Ho^ 1 ^= V^ ST^^^o'" fd - ^1'" B]')^3 -^ 3(^0" - 3^," n^)U - [I] h^) ^ = gives for the divergent stream line -a3 J — = 3:_L§J = 11.5 I -[I] 1603-1,0 36J_^_-^ or in the not— reduced coordinates X - K V PI0 These formulas are characterized by considerahle uncertainty hecause only two terms of the development of ^ were ccmputed and the hirjher differential quotients, which represent more subtle processes, are always less accurately computed than the fonner. III. FORMATION OF THE BOIMDABY 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 = 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 eq^uations 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 ^- + u^^ + v-^— = u-,— - + K — - dt dx dj dx ^y2 u = i! v=-^ By Sx K substitutes for — . The potential flow which is set up first gives the "boundary value u as function of x. Since the process for t = 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 Uq ^Ur^ S Ur o _ ,0^ ^o = K. ^^ Sy2 The integral of this ecLuation ^o Tl = 2-^ satisfies the conditions of supplying a vanishing "boundary layer for t = and of joining the outside flow Uq = u for y = «. The su"bseq[uent approximation is obtained by inserting Uq in the convective terms, while time and friction terms obtain u = Uq + u-]_. The resultant equation for u-^ reads Su-i S Ut _;^r, ^ = K — _i + u^ (function of ^) dt ^y2 di 30 KACA TM 1256 According to mechanical similarity, this equation is eatiafied hy the formula U^ = tU^ f(T)) -^ OX vhlch is also not contradictory to the "boundary condition u-, = for y = and y = 00. 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 cf r\, that is, still contain t. These functions are also still dependent en x, tut this time x enters the differential equations only as parameter. 3. The formula for if is accordingly 00 t = 2\/7t ZZtVx^(iTi) V=o n = 2\[Kt and hence the differential equations for X s3x aS( ax y=i/ ax ,_^ a^x,, ax,, a^x \ for |i = 1, the right— hand side contains — Uu^^li. di As "before, the calculation is limited to the first two terms, that is ''o - '"tc<1> ''l = ^i?!*"' WACA TM 1256 31 hence The eq.iiatiorL3 for ^ and ^ read then ^ '« + 2ti^ " = ,2 Boundary conditions for Ti = 0: C = t • = ' ^o ^o ^1 = ^^. = for Ti = «: ^o' = ^ ^1' = ° 4. The solutions of the ahove differential equations, which are to "be used in the subsequent prohlem, are ohtalned hy quadrature when the homogeneous equations are Integrated. The latter integrals were obtained "by the following consideration: The homogeneouB pajrbs of the equations stem from the time and friction term which together form the heat conduction equation ^ = k-^ 3u v^^n St Sy2 Of this equation integrals of the form 2ykt ! 32 NACA TM 1256 exist, according to similarity considerations, wherety fn satisfies the differential eqijation fn" + 2T,f^' - l^nf^ = which the above form possesses. Thus, for example (see above) ^ /^^ _.2 For T) = "° = ^° = i 1^^^^ u = 1 when t > Uq = when t < For T] = 0, hence, for y = 0, u^ is proportional to t^, hence must be representable by Buperpoeition of Bolutiona u^ in the following form (00 , \ n-1. ^-^ JoH , -^— '•'° ''^ ^(t - 1^) = n Tu, ( , ^ ] ■ to^^dt. °\2\jk{t - to) since for y = it is Pt = n / to°-X = ^" Vo For the evaluation of this integral, put t — t^ = t NACA TM 1256 33 insert herein. _2_ 2\l~kt = m 2 VkT 5k 2-* ^ .2' and finally o'btaln dT --^^■> 2k: ^3 ^-p'^-my^)'^'-!! e dfl Calculation of this integi^-l "by the "binomial theorem, and the previously cited method of partial integration finally gives ^ ^(2n - l)...3.l' '/ ' rn v=l 1 y~(-l)^^+^ iiii^^ ^ (2^ - l)...(2^ -2v + 1) > Ti2V-le-^2 The other integral is algebraic ajid eq.ual to the above factor of / e ^ dri I 00 ^il .2tx ^ t^(2^i - i)...3.i 5. Quantity ^^ is determined as follows: With the boundary conditions taken into consideration 3^ NACA TM 125b f ' = 1 + -^ ( e~^^dTi whence "by Integration ^o=-^-^if^r-^^^-H while utilizing The second differential equation (of the second order for ^-1 ') assumes then the form 16 ^ " + 27it " - i+^ ' = — in _^2 8 -Ti2 16 e ' dri —!r\e ' + — Urr Jt ,-^ '"3-% I ' The integral of the homogeneous equation for T ' is "by (U) f = a(2Ti^ + 1) + P T^e ' + n ^2 (2t]2 + 1) e^ dT) The integral of the nonhomogeneous equation would then be ohtainahle "by quadratures. But it is also true that, hy twice differentiating, the differential equation finally 'becomes ^ ttni + 2ti^"" = function of r\ NACA TM 1256 35 which ia easier to Integrate as an equation of essentially first order. Hence t «" = e-' ■^ / e^ [function of T\jd.^ Since the Impressed force of the differential equation contains e ^ in each term after twice differentiating, e cancels oiit, and ^"" and then ^ can "be integrated, "because the functions behind the 2 Integrals contain, at the most, e twice, and in addition, powers of T], and must "be integrated several times, which can "be accam— plished "by the methods discussed previously (1,6). The result of the rather voluminous calculation reads ^1' ^Tie-^^ rv^\n + |(2ti2 _ 1) J rv^%v + 1 ,^^^ + —Tie ' - — , e ^"1 ~ ^ + a(2Ti2 + 1) + p |(2,2 _ i)e-^^ fy\ . 8^ gives a = p = _1 + ^ . 6. For computing the zone of separation, there is =^ = — 1- f u^ " + [t] n^^ "^ for Ti = Then f"=2_ r"=^+ 8 'o - /i '1 - ;« ~^ The condition for the time of separation [tj is hence, ^ must he negative. The separation occurs first where •^ ox ox has the greatest magnitude. The result applies to cylinders of any cross section; u is the correspondin^g potential flow. IV. DEVELOPMETIT OF ZONE OF SEPAEATION 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 WAG A TM 1256 37 "by an Intenruption of the fluid. Hence, let the solution of the prohlem assume that, stai-ting from the time t = 0, the Immersed "body is su"bjected to constant acceleration. In that case u = tw(x) 1 Sp ^ _Su , 2 ^ - ^ = TT— + Ut— = w + t^w-— p ox ot ox ox 2. From considerations similar to those made "before, the solution of the differential equation ^ ^ ^ 1 Sp „o u + Ut— + V— - = e^ + K- bt Sx Sy p Sx ^y2 is "based on the formula if = 2 \J^ . IZt^'''^"^^2v+l(xTi) v=o u = )~ t v=o Tl= ^ 2V+1 ^2v+l Sri 2^ Insertion in the "basic ecLuation gives a^x S^x Bx 2X+1 _ 2X+1 ,./^^ -.N 2X+1 — + 2ti k{2X + 1)^ Sti3 Sti2 ^n 38 NACA TM 1256 X^l = k H=o ^^2^+1 ^^2X-^u-l 2X-2M.— 1 2M.+1 Sri SxSy Sx ^rf for X. = Oj the right— hand side contains -kvj for X. = 1, -4w^. ox The calculation of the state is again limited to the first two terms, while it should he noted that through those two terms, the two terms of the pressure w + t^w^ are also taken into consideration. ox The impressed force of the next eq^uations 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 X-, and x^ the relationship of x can be introduced in the following manner: \ = w^-^(ti). X = 3 Sx 3 The differential eq^uations for ^ are then: + 2ti i4-^— Bti3 Sti2 S^ = -k i + 2ti ^ - 12^ = -1^ + 1+ St| w - ^ ''^ ' ^'} Boundary conditions: for T] = 0: for T] = od: Sti i = l. u = from V = from u = tw NACA TM 1256 39 3. According to the general solutions of the present type of differential eg_uation3 discussed in III {k) , r-^ can he written s^ 1 forthwith, since the nonhomogeneous term. -H- is disposed of hy - — = 1; L is ohtained hy integration hy the repeatedly cited method (I6) " fh 1 Sti2 "PL e~'n + 2ti / e-'H di] B^ Bn 1 = 14.^ \f^ T]e '' + (1 + 2ti^) e '' dT] 2^ rV^2 ^1 = •n + 1 ' 3{-- - 1 + (1 + Ti2)e-^^ + (3t1 + 2ti3) I e"'^ dTi These functions are q^uantitatively plotted in figure 8 and given in a tahle (see Section 6 following). ko NACA TM 1256 0.25 0.5 1.0 Figure 8 The Impressed force on the right— hand side of the second eq^uation is then 16 P''.-Ti2,_ . 16 e-n dri + iH f^ [L 3« k-T] e-T dTi + 2e-^ 16 3« Ucx, dT] + (3 + ^^) < e-^^dTi U. The integration of the second equation, in closed form, again succeeds by the same methods as in III (5)' For the part of the 2 impressed force quadratic in e ^ a formula with indeterminate coefficients is particularly advisable. NACA TM 1256 kl (a + "bTj^ + CT] )9 ''1 + (dT) + eTi3 + fTi5)e-Tl^ / e-^^dTi + (g + h.Ti'^ + ill + kT) ) This formula fails when the impressed force contains terms which exceed 2 6 _2ti2 5 _Ti2 n^ _.2 -^"^^ ^^ < /' "^-^^^^ V (compare III (5)). The T] e ■ J Ti'e ■ / ^ ' '^''Ij ^~^ "l / Q ^^1 coefficients are determined from linear equations. The other ^^ ^^ portions of ^— i- are easier to compute; ^^ and J- follow by integration and differentiation- So, when the integration constants are correctly computedj the final result is 3j«^ 3« (-T1 + 2Ti3)e^^' + (1 + 2ti2 + Sri^e-^^ / e-^^dri + (6ti + 8ti3 + 8ti5)< ^9-^% (16 + 36ti2 + 8Ti^)e-n^ (60ti + 80ti3 + l6rp) I e-^^dT] k2 NACA TM 1256 16 15Tt _Ti2 n^l 2 e ' + 2ti / e ' cLt] 9n r (8 + Ti2 + 2Ti^)e^'l' 1/ 00 2 /'^ 2 + (-9 + 18ti'^ + ^^^ + 8ti°) < / e-^ dTi U.«6J /^Vn2 1/5 16 'H6<^ 1.5P. (33T1 + 28ti3 + i+Ti5)e-^^ + (15 + 90ti2 + 60ti^ + 8ti^) / e-^^dri hencej "by integration ^ =--2- ^ 3^L 2 P^ 2 e"^ + 2ti / e-^ dTi 15ff 2 n1 2 T^e-^ + (1 + 2ti2) / e-T dt] 315« L ;^ ^.2. (1^9ti + llTl3 + 10Ti5)e^^^ + 768 y^ e-^^^dTi C/00 MCA TM 1256 43 (-537 + 198t|2 + 6J+Ti^ + U0Ti6)e-^^ f Q-^^dri + (-3I5TI + 210ti3 + 81iTi5 + UOt]'^) . r (21+ + 87ti2 + hQr\ + h^^)%~'^' (105T1 + 210ti3 + BliTi^ + 8ti^) / e-^% / 128 128 9 \l575\p3 105^ li^\[7 ) These three functions, plotted in figure 9j rigorously satisfy the differential eq.ufltions and the "boundary conditions for the coefficient X- 5- The condition for the zone of separation has the form = 2ii 1 + t2|2 i 51)2 S^ Sti2 T1=0 whence, hy the foregoing formul^as 3 _ St^S ,/lf Stj2 ^1 25L_ 15lfTr 225^ hk MCA TM 1256 The eq.uation for the separation time ItJ reads 1 60 225ny Sx The next term in the separation equation ^ = would oy read: -^=—t^ ^, and in order to "be ahle to allow for it, too, the coefficient in this separation equation, rather than the total variation of Xc* is computed. 0. 13 8 -» Figure 9 The development term Xj- satisfiee the equation S3x 5^Y S)c ^ + 2ti ^ - 20-^ = U sy a2 B2) ^x^ a-^Xi Sx-L S X3 Sx^ a'^x3 ax^ - -1 S-q SxSti ^x Sti^ ^^ ^:cdT] Sx ^2 The entry of x in X, and X is known, and calculation of the right— hand side confirms tliat X aBBumes the form NACA TM 1256 J^5 Since tw cancels out, the condition of separation reads h^L hr]'^ + t2^ _ T1=0 d^^. Sti^ + tM^^2 Jn=0 ^\ L ' Jti=0 + t\& Sx2 a^^ ^ M' = and 6. This leaves the calciilation of the coefficients T1=0 ^rf 1=0 ^ Sn^ Jti=0 For ^, the differential equation reads S + 2ti S - 20—^ = 8-J: -^ - 1;^ i SnS aTi2 Sti ^^ S11 1^ 2 and the boundary conditionB s^. C = 0, — ^ = for T] = Oj — ^ for T) = The inrpressed force fCt^) is given "by the previously written ~a2^r functions. The desired coefficients Green's method as follows: Sti^ are computed hy Ti=0 k6 NACA TM 1256 ^ ^ + 2t)— -^ - 20—2 dTi o Uti3 V ^^ .^ L V J o Jo ^1 ^2 3n / Then, if ,3 is made to satisfy the adjunct differential eq.uation ^ _ 2t^ - 22^ = and the hoiuidary conditions •6(0) = -1 t3(oo) = the result is t^ . f . dT| 1=0 f(Ti) is given previouBly; the influence coefficient -3 (Greenes function) is obtained hy integration of his differential equation ^(ri) = 9^5 vfn (2895T1 + 528071^ + 2352ti^ + 3527)'^+ 1611^) + (91^5 + 9^50x1^ + 12600ti^ + 50U0ti^ + T20ti^ + 32iil°)e^^ ' e-^^ri The curve of -a is shown in figure 10, along with the product t3 • f . The area of this last curve gives the desired c oaf fie lent o WACA TM 1256 47 0.5 - -0.5 - - 1.0 T* V Figure 10 For CQm.pu.tlng ^^ ^ StjS the equations Jti=0 B^C b%^ ^^ 22: + 2t\ — -22: _ 20-^22: = k Sti3 'dr\^ OTi S^l B^ ^- ^ ^\ 5t1 StI ^^ti2 = g(Tl) B2^ 5a Sti2 = / -g . g . cLti ,=0 ^o are available; -9 . g is plotted In figure 10 according to the values Indicated "below. 48 WACA TM 1256 The c cmputed values are the following: Tl r 1 0.25 0.50 1.00 1.50 00 ^1 .061 .211 .638 ^ Tl - 0.376 ^^1 .450 .720 .9^3 — 1 2.26 1.396 .799 .201 0.035 ^ .022 .060 .115 — .138 •137 .150 .020 — ^^3 5,2 .96k .231 -.092 -.156 -.05 f(Tl) .315 .750 .i^57 — ^(ti) -1 -.327 -.11? -.018 — ^ . f -.103 -.o&k -.008 — g(Tl) .12 It .2i|0 -.016 — ^ . g — Olil -.027 .0003 — The area of the two curves Is approximately #^ 1 Sti^ = -0.058 T1»=0 S2^ ^ ^' = -0.023 T1«0 7o The equation of separation therefore reads L + [tlS^f-Jl 2^\ _ [t]¥^^ . 0.058 - [t]^w^ . 0.023 = jt Sxyi5^ 225l«3y \ax/ Sx2 MAC A TM 1256 k9 or 1 + 0o42T . [t]2|^ - 0.026 . [t]Y|^f - 0.01 . [tj^v^ = Since the newly added correction term Is even negative, the existence of the zero position appears to he certain. The position and time of separation is according to the earlier approximation (without the term computed last) &]% = ^'3^ ax For the case of a cylinder symmetrical to the direction flow, \ f =0 at the rear point where the separation starts, the newly computed correction gives H% = -^-08 Bx 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. APPUCATION OF THE EESUITS OF THE SEPARATION PEOBLEM TO THE CIRCULAR CYLINDER 1. On the circular cylinder u = 2V sin § — is called the reduced coordinate X; V is the velocity at which E the parallel flow flows toward the right, and the cylinder moves 50 WACA TM 1256 toward the left, respectively. In the steady case, the aepajration starts according to Part II, Section 7 at XgQ-n, = O.65 J— ; the pi maximum velocity lies at where X = 0„57T, ^- max \ 1 1-1 u. = q X — q X-' ^o ^1 y-* Figure 11 Taking the ordinary development in powers of sine X = 1.59 . R, X =91^; X = l.J+1 . Rj X = 81*^ Sep. Sep. ^ 1^ ' max ' m«-«" max MCA TM 1256 51 But approximating the sine in the interval — n ty the method of least sq^uares, gives a = f: . 0.856, It = ^ • 0-093 X = I.97.R, X = II30 Sep. ' ' Sep. -^ X = 1.75 . E, X = 101° max max , In any case, the point of separation lies, "by the present ' calculation, at frcaa 11 percent to 12 percent of the total toundary— layer length "behind the TnaTHTniTm 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 distrl"bution of the tur'buJ.ent "bodies "behind the cylinder. The sole purpose of the present calculations is to indicate that separation is actually obtained by the hydrodynamic eq.uations. Further development of the calculating methods, especially for the more important pro"blemB of solids of revolution, premises, therefore, success. 2. If the cylinder with constant velocity is suddenly set in motion u = 2T sin X ^ = il ^og x The time of separation [tj is, according to III (6), given "by (^ ^ h) m - - [t] = -0.35 ^ Y COB X 52 NACA TM 1256 The separation starts for X = n, cos X = — 1 at time t- « 0.35^ Up to that, the cylinder has travelled a distance S = Vto = 0.35 . E All this Is Independent of velocity, density, and friction coefficient (little friction assumed). 3. At constant acceleration u = tv(x) = 2Vt sin ^ R where V is then the acceleration of the cylinder In the flow. The separation time is (TV, 7) for the start of separation [t]^!^ = -2.31^ or =-2.08 respectively, or R ^-^ _ T nl, R ftr = -1.17 or = -1.04 V cos X V cos X respectively. The distance covered "by the cylinder is S = ivt2 2 at start of separation (X = «) S = 0.59 . R or = 0.52 . R respectively. MCA TM 1256 53 k. The resistance vhlch the cylinder experiences at constant acceleration is conrputed next. The stress components are Xy = k(^ + ^ py Owing to the smallness of the friction, -^ and ^ cancel with respect to —, leaving as force in direction of the outside flow 07 ■ ' ^C"!^ K = 2 . B _ / (p COB X + k^ sin X j. EdX B is the width of the layer (height of immerBed part of cylinder) The pressure portion is computed as follows; VesBure = ^BE / p cob XdX = -SBE^ J ^ sin XdX Then hp {^ _^u\ _ = p — + u — ) u = tw Si \^ Bx/ = p (w + t^w^ ) V = 2V sin X Sx. ^k NACA TM 1256 The Becond term, cancels out In the Integration; the first gives pressure = 2«pBR^ hence, an Increase in Inertia by twice the amoxint of displaced fluldo The friction portion is friction - ^ MtSh ^t^^illl sin Xdi friction 2^J^^ ^g ^^^^2 / ^^1 where k = k/p. Again, the second term disappeara "because ^2 and =i are merely constants, leaving St,2 friction = ^NJ^fPtt . 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 E, V, k are arhltrary; hence, necessitate the Introduction of reduced quantities for x, y, t, ijc, and u, so that R, V, k disappear. It is accomplished hy I— I 2 X = RX, t = JSt, y =\k/M_Y i = ^R^ K^v ^y u = x/rvtj I ^ NACA TM 1256 55 ■by -whlcb. tJie fcarmulaa (compare IV" (2) and V(3)) ' y^ dx. : 2^ w « 2T Bin i B t2^ . 2l£ cos i "become the follovdng reduced eq^uatioaas f = 1^t3/2 aln X . ( ^ + 2T^ cos X . L) U = 2T Bin X . --i + 21^=^ cos X . —^ \oTi ^^ j y 2yT The curve f. is then plotted against Y = 2^ . t] for a fixed time T for a numlaer of coordinate values X, and the position of the values ^ = constant read from these curves. In figure 12, the cylinder is shovn from X = n/2 to X ■ jr. The separation time is o given "by 2T cos X « -2.34, that is, the start of the separation 56 MCA TM 1256 "by T = 1.080 In figure 12 2T^ = 5, hence T = I.58, vas choseno For this chosen time, the separation point has already progressed up to "beyond 60° at the cylinder; nevertheless the "boundary layer still is fairly thin, the relative sizes correspond to the values E = 10 cm, 2 K = 0. 01.22^ (water), V = 0. 1 ^^ , that is, to a very small acceleration. sec sec' Accordingly, t = I5.8 sec. Figure 12 MCA TM 1256 57 Tlie picture obtained "by the previous reduction f omnilas for T = 10-2a_ after I.58 sec. is represented in figure 12. The sec thickening of the "boundary layer would he diminished in the ratio of 1 :JlO. Translated hy J. Vanler National Advisory Canmittee for Aeronautics NACA-Langley - 4-22-55 - 75 UNIVERSITY OF FLORIDA D0GU^.1£NTS ' ""■ '"^^ ^^, 17011 .L.E,FL 32611-7011 USA