AcAT/ii'/3)-7 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1317 A SIMPLE NUMERICAL METHOD FOR THE CALCULATION OF THE LAMINAR BOUNDARY LAYER By K. Schroder Translation of ZWB Forschungsbericht Nr. 1741, February 25, 1943 Washington April 1952 UNIVERSITY OF FLORIDA . o';^; ooiujut udRAflY . . . ...X i17011 GAiNL:3VILLE, FL 32611-7011 LfcA 2H0 -vr /-T f ^-77 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 131? A SIMPLE NUMERICAL METHOD FOR THE CALCULATION OF THE LAMINAR BOUNDARY LAYER* By K. Schroder ABSTRACT A method is described which permits an arbitrarily accurate calcu- lation of the laminar boundary layer with the aid of a difference cal- culation. The advantage of this method is twofold. Starting from Prandtl's boundary- layer eq\mtion and the natural bo\indary conditions^ nothing needs to be neglected or assumed^ and not too much time is required for the calculation of a boundary- layer profile development. So far^ the method has been tested successfully in the continuation of the Blasius profile on the flat plate, on the circular cylinder Inves- tigated by Hiemenz, and on an elliptical cylinder of fineness ratio 1:^, Above all, this method offers for the first time a possibility of con- trol by comparison of methods known so far, all of which are burdened with more or less decisive presuppositions. OUTLINE I. INTRODUCTION II. GENERAL REPRESENTATION OF THE METHOD III. PRACTICAL EXECUTION IV. NUMERICAL EXAMPLES AND RESULTS V. REMARKS REGARDING THE CONVERGENCE OF THE ITERATION PROCESS *"Ein einfaches numerisches Verfaliren zur Berechnung der laminaren Grenzschicht." Zentrale fur wissenschaftliches Berichtswesen der Luft- fahrtfors Chung des Generalluftzeugmeisters (ZWB) Forschungsbericht Nr. 17^1, Berlin-Adlershof, Febr\iary 25, 19^3. NACA TM I3IT I. INTRODUCTION The flow processes in the laminar boundary layer may be described by Pramdtl's boundary-layer equation. If one limits oneself to the two- dimensional steady case and Introduces, in a suitable region aroiind a profile contour C situated in a flow, a curvilinear coordinate system s,n, the coordinate lines of which consist of parallel curves and nor- mals of C, that equation reads vg — £■ + vn -^ + p'(s) = i — ^ (1) ds on -^ dn"^ when Vg, Vn signify the velocity components in the s,n system, R the Reynolds number, and p = p(s) the pressure distribution along C taken from a measurement or calculation^. Equation (l) is complemented by the continuity equation The transformation T) = n I/r Vt^ = Vn ^^ yields, instead of equations (l) and (2), the equation system V in which R no longer appears explicitly. ^.v, -^I^^p'(s) =^ (3) Bs 'I Sri St|2 A mathematically complete derivation of equations (l) and (2) based on physically plausible assumptions may be found in H. Schmidt's and K, Schroder's report entitled "Die Prandtlsche Grenzschichtgleichung als asymptotische Naherung der Navier -Stoke sschen Differentialgleichungen bei unbegrenzt wachsender Reynoldsscher Kennzahl" (Prandtl's bovmdary- layer equation as an asymptotic approximation of Navier -Stokes' differ- ential equations for indefinitely increasing Reynolds number) Deutsche Mathematik, 6, Heft 4/5, pp. 307-322. A survey of related literature is given in H. Schmidt's and K. Schroder's report "Laminare Grenzschichten, I. Teil" (Laminar boundary layers, part l) Luftfahrtforschiong 19^ Lieferung 3, 19^2. NACA TM 1317 The following boiindary conditions for the integration of equa- tions (3) and (k-) are usually selected in boundary-layer theory as the natural ones from the physical point of view. For an initial value s = Sj- an entrance profile vs = Vs(So,Tl) is prescribed as a function of tj (entrance condition). Fiirthermore^ in consequence of the adherence of the fluid to the contour, the rela- tions Ho'° W = which are to be interpreted as limiting processes, are to be valid along C (adherence condition). Finally, for s-values larger than or equal to Sq the velocity component Vg is to converge for t\ > 00 toward the velocity U(s) which is connected with the prescribed pressure distribution p(s) by U(s) U'(s) = - p'(s) (5) (transitional condition). The general significance of these boundary conditions will be dis- cussed more thoroughly In the second part of the Luftfahrtforschung report quoted in footnote 1. Here we shall only point out that the tran- sitional condition formulated for t\ — ^ « must not be confused with a condition for n > since for the latter limiting process the veloc- ity components converge toward those of the basic flow^. The limiting process r\ > <» denotes, on the contrary, the asymptotic transition to the boundary values, resulting along C of the outer potential flow obtained for R' — > 00. This can best be made clear by the example of the stagnation-point flow at the flat plate, treated in the second report indicated in footnote 1 (by the author and H. Schmidt). Whereas the quantity 6 there specified as boundary- layer thickness tends like 1/ /R toward zero, a quantity d tending toward zero, for instance, like 1/ \|^, can be prescribed in such a manner that, the flow outside of a layer of the thickness d adhering to the contour for R — > "> converges toward the outer potential flow. However, to the asymptotic transition toward the boundary values of this assumed potential ful. '^It is assumed^ of course, that this limiting process is meaning- k NACA ™ 1317 flow along C then there corresponds the limiting process lim T] , = lim d\/R = » R — ><» R — *" So far, an appropriate existence and uniqueness theorem for this boundary-value problem does not exist. However, the results obtained with the new method described below show that the statement of the problem is perfectly sensible. In the literature it has been pointed out more than onceS that for- mal power series developments of the function representing the solution with respect to t) make the fact plausible that the entrance profile cannot be selected completely arbitrarily, but that it is dependent on the pressure distribution p = p(s). Our method for the determination of the velocity profiles yields a numerical solution of the mentioned boundary-value problem with the aid of the difference calculation; it is superior to other methods because it requires no assumptions beyond equations (3) and (4) and the boundary conditions. In our method, the boundary-layer bonds of the entrance profile do not appear directly and thus do not cause any difficulties in the numerical calculation. A severe violation of these bonds causes, in our method, the variation of the successive boundary-layer profiles to become completely disordered. Small violations of these bonds, in contrast, do not exert any considerable effect on the further develop- ment of the profile^. ^Compare S. Goldstein "Concerning Some Solutions of the Boundary- Layer Equations in Hydrodynamics," Proc. Cambridge Phil. Soc. 26, 1930, pp. I-30, L. Prandtl, "Zur Berechnung der Grenzschichten" (Concerning Calculation of the Boundary Layers) ZAMM. I8, 1938, pp. 77-82, (NACA TM 959) and H. Gortler, "Weiterentwicklung eines Grenzschichtprof ils bei gegebenem Druckverlauf" (Further development of a boundary-layer profile for prescribed pressure variation) ZAMI4. I9, 1939, PP. 129-lifO. TL. Prandtl and H. Gortler (reports quoted in footnote 3) arrive at the same conclusion, although on another basis. NACA TM I3IT 5 II. GEKERAL REPRESEIWATIOW OF THE METHOD If one introduces into equation (3), instead of s, the new inde- pendent variable under the assumption that U(s) ^0 for s = Sq whereby ■ dl ds U(s) is valid, and if one uses the new designations U(|,T1) = Vs(s,Tl), U(0 = U(S(|)), U*(l,Tl) = U(I,T1) - 11(1) there follows from eqiiations (3) and (4) by way of , ^ dTi- U(0 Jq St with equation (5) taken into consideration, our initial equation ^ . 25i! - _1_ ^ r Su ,^ .JlL ^ (7) According to the statement of the problem in the introduction, we have to find a solution u = u (|,T1J of equation (7) for all points (| x\) in the right upper qiiadrant of the plane of the rectangular Cartesian |,ti coordinates5 which in approaching the straight lines 1=0 or T] = respectively 5lf separation phenomena appear, the solution will, in general, be of interest only up to the separation point or possibly a little way beyond it. NACA TM 1317 tends toward prescrilDed f\mctions: lim u*(|,Ti) = Vs(So,Tl) - U(so) (t) ^ 0) (8) or lim u*(|,Ti) = - U(fJ a ^ 0) (9) T)-*0 and which vanishes for i] — ><» lim u*(^,Ti) =0 (4^0) (10) T] >» The fymdamental formulation of our method consists in using the functional relation (7) - in the sense of the known method of successive approximations - for the calculation from a prescribed approximate solu- tion which already satisfies the indicated boundary conditions of a sequence of corrected functions which converges toward the actual solu- tion of the problem; one substitutes the last obtained approximate solu- tion every time on the left side of equation (7) and integrates the resulting partial differential equation of the type of the inhomogeneous heat conduction equation. The examples so far calculated numerically showed that the iteration process is obviously convergent. Nevertheless^ a general proof of this fact would be very desirable and we reserve returning, in a given case, to a mathematical examination of these problems. (Compare also Section V.) One may characterize the method by stating as the desired result a continual improvement of a given approximate solution in the sense of Oseen's method of linearization. Then this linearization of the hydro- dynamic equations of motion (which, of co\arse, for the botmdary-layer flow taken by itself is not permissible) consists in introducing the velocity loss u* and in neglecting all nonlinear terms in u*,v and their derivatives. From equation (3) one would thereby obtain U(s) ^-^+ u*^- as ^2 ds thus on the left side (aside from the term u*^ which, however, does ds not alter the character of the equation) precisely the expression which MCA TM 1317 also appears on the left side of o\rr Initial equation (7). In integrating (under the bo\jndary conditions (8), (9), and (lO)) the differential equation (11) into which had been introduced for abbreviation the function U(|) OT ^0 a5 ^ dTi - ^ au U(U ^^ (12) to be regarded as known in the sense of our approximations, one may now use successfully the difference calculation. For the homogeneous equa- tion this has been done, simultaneously with a proof of convergence, by R. Courant, K. Friedrichs, and H. Lewy°. For the inhomogeneous equation here dealt with, the proof of convergence together with a formula for error estimation may be found in a paper by L. CollatzT. If one covers the right upper quadrant of the ^,t\ plane by a net of lattice points with the coordinates ^r Pk T\0 r; crz > (PfO = 0, Integers) (compare fig. 1) and introduces at the same time, with a view to later applications, the new designations Up, a = ^Up,-^a), ^*p,a = ^*i^.p,na) SI i=^o,^^<. SI Su ]P,0 L^^j5 = lp,Tl=Tla Su p,o T^. Coixrant, K. Friedrichs, and H. Lewy: "Uber die partiellen Differenzengleichungen der mathematischen Physik" (On the partial differ- ence equations of mathematical physics), Math. Ann. 100, 1928, pp. 32-7^^ particularly pp. k-'~[-^2. 'L. Collatz: "Das Differenzenverfahren mit hoherer Approximation fur lineare Differenzengleichungen" (The difference method with higher approximation for linear difference equations), Schriften des Math. Sem. u.d. Inst. f. agewandte Math. d. Univ. Berlin, Bd. 3, Heft 1, 1935. NACA TM 1317 there corresponds to the differential equation (11) the difference equa- tion of first approximation k ^2 = ^P'^ ^^3) If one selects the step magnitudes k and Z in | and r\ direction not independent of each other but so that k = li (14) 2 equation (13) is transformed into the simpler difference equation * u*p,cT+l + u*p a-1 ,^ ,^, It can be shown that the solution of equation (15) for the corresponding boundary- value problem for I — ->0 and therewith also for k — *-0 con- verges toward the known solution of the boundary value problem of equa- tion (11). Since the values ^*p,0(P = 0) u*o,a(<^ = 0) fp^cj (p^ 0,0 ^0) are known, one may, according to equation (15)^ successively calculate all values ^p,a (P^O,a^O) progressing stepwise from lattice point column to lattice point column. Actually, however, we apply another correction at every step in order to compensate the systematic error originating by the fact that the derivative appearing on the left side of equation (ll) . P}^ NACA TM I3IT 9 was replaced by the difference quotient of first approximation u*p+l,a - u*p^cr k (Compare the following section. ) One notes that due to the transitional condition (lO) for the entrance profile u ^ necessarily must vanish for a — ^00 and that f^^ _ likewise vanishes for a — ^oo^ since even the approximate solution used for the formation of fp ^ was supposed to satisfy the condition (lO); hence one recognizes that the corrected solution (obtained with the aid of the difference calculation in the manner described above) also satis- fies the trauisitional condition (lO). III. PRACTICAL EXECUTION In practice one may vary the method in such a manner that one does not at all require an approximate solution prescribed at the outset in the first quadrant of the I^T]-plane; one rather determines this approx- imate solution for every step and then improves it to the desired accuracy before passing on to the next step. Thus one applies a combined system of continuation and correction. If one deals with the flow about a profile contour, the initial pro- file at the point s = Sq is best taken from the well-known power-series developments by Blasius-Hiemenz, the coefficients of which for the first three terms were given in table form by Howarth°. For reasons of con- vergence, these broken-off series will represent a good approximation of the solution of the boundary-layer equation only at a small distance from the forward stagnation point (s = 0) of the outer potential flow. In the permissible range they represent, as it were, an improved stagnation point flow. Our calculations so far have shown that the series are serviceable up to s-values for which the "first boundary layer bond" S^Vs W Jti=0 p.(s) = - MlI (16) dl Q Compare L. Howarth: On the calculation of steady flow in the boundary layer near the surface of a cylinder in a stream. R & M no. 1632, 193^. 10 NACA TM I3IT which is a direct result of eq\iations (3) and (6) is satisfied with sufficient accuracy. In practice^ one has therefore to start the calculation by approxi- mating the function U(s) for small s-values as well as possihle by a polynomial of the form U(s) = UlS + UoS-^ + UcS-' for the case of a profile symmetrical in free-stream direction, or respectively, of the form U(s) = UtS + u^s + u^s-' for the case of a profile unsymmetrical in free-stream direction; one may sometimes get by with only two terms. After having determined, in the manner described above, the value Sq > at which the continuation method may start, one first sets up the connection (given by eqviation (6)) faQ ' by evaluating the integral on the right side, for instance according to the trapezoidal rule. One graphically represents the functions I = ^(s) and U = U(s) in a common diagram so that U = U(|) can be Immediately taken from it. The step magnitudes k and Z, connected by equation (l^), must be selected so that, first, a sufficient number of subdivision points are distributed over the profiles to be calculated, and second, a sufficiently rapid continuation in | direction is possible. When profiles of not too pronounced S-shape (near the separation point) are to be calculated, eight to ten equldistajit subdivision points generally will be sufficient to define the profile. In upward direction (that is, for large T[ values) one will have to take so many subdivision points that the profile dies out sufficiently gradually toward the asymptotic value U. This provides a first indication for the selection of Z and therewith also of k. It should finally be remarked regarding the step magnitude k that it must be at least large enough to make, for fixed s and variable t], the derivatives ^ (obtained in first approximation by formation of o| difference quotients) take a reasonably regular course (compare the following discussion). Hence the lower limit is set for k and there- with also for Z . NACA TM 1317 11 On the other hand, one will be forced to choose the smallest possible step magnitude k at points t where the curves u = u(l, const) exhibit great curvatures (which occurs particularly directly ahead of the separa- tion point), in order to make a sufficiently exact calculation of the profiles possible. There Z, too, will necessarily be small. Since, however, the boundary-layer thickness has greatly increased at the sepa- ration point, one will have there a great many subdivision points dis- tributed over the profile. This is in one respect convenient - the posi- tion of the separation point is better defined. On the other hand, the expenditure of work increases at such points. However, at the end of this section we shall point out a possibility of reducing the steps in I direction without necessarily having to accept a step reduction in T) direction. At the same time we shall then be able to indicate a criterion by which the necessity of a step reduction in i direction may be recognized. Once a certain selection of step magnitudes has been decided upon, it is a question of obtaining a first approximation for the values f. 0,a appearing in equation (15)^ in order to be able to execute the first step in I direction. It should be noted that together with the initial pro- file at s also the values of u for values < So u from the series developments. Particularly the values the profile one step ahead of the initial profile) are thus known. -l,a may be taken (that is. We then "put for a first approximation of the in fo,a: occuring 0,a hn M U0,CT 0,0 U-1,CT Therewith '0 Su hi dri too can be evalioated niunerically. Oior calculation experience has shown that this integration may be very conveniently carried out with sufficient accuracy by use of the trapezoidal rule with the aid of the present subdivision; this can be done purely schematically by calculation according to tables. For at the | points where the derlva- tives — become very large - whereby the values / T^ ^'^ ^^"^^ ^° SI ^0 °^ be of great importance in the calculation of the profiles and must be determined relatively exactly as for instance in the neighborhood of the separation point - it will be necessary to select small k (and there- with also Z) values so that a s\ifficient number of subdivision points are distributed over the profile to allow application of the trapezoidal rule with s\ifficient accuracy. 12 NACA TM 1317 If one puts, furthermore, with good approximation ^ ^ ^0,0+1 - '-^,0-1 .0,0 21 (IT) u T ^ and l,a u-i „ may be calculated in first approximation. Kljl,a ^^^ Kjl,a- The values thus obtained will be denoted by |" i| i a It will be best to arrange the entire calciilation procediore in the form of a table (compare table I on page 31). With the values obtained Ml,. - ' ^ -'■- -' - ~ '"^"^ according to the scheme one will form corrected values of the derivatives ai _0,a dui h^ .Nm u .1,0 . 0,CT 2k whereupon one obtains (with the aid of table II on page 31) a second approximation P2 u a ^'~'^ ^^^ values u, with the values 7a, ^0, hi and C„ taken from the first table. Whereas the derivatives 0,0 formed in first approximation might show at a few points o an irregular coiirse, this will generally no longer be the case for the corrected deriva- tives N The columns for the quantities ^a, p2ji^, and ^2 1 a occiorring further on in table II will be explained only later. This procedure is continued until the values obtained in the third- from-last column of the table no longer vary in the desired decimal. In the examples we calculated the iteration was carried so far that for every step Ig the values Up ^ no longer varied except for an error of about l/k to 1/2 percent of the maximijm velocity U(|p) in each case. For the selected step magnitude k this was the case after two to three iterations . Due to the favorable position of the errors, the profiles calculated in the manner described generally show a very smooth coiirse. If the u = u(l, const) are concave in respect to the ? axis, as is the case for instance in the flow about the circular cylinder or the ellipse near the separation point (compare fig. 7 and fig. 12), the convergence occi^rs only on one side in the direction from larger to smaller values for u. The opposite behavior exists when the course of this curve is convex with respect to the I axis as is the case for instance in the boundary-layer flow at the flat plate. NACA TM 1317 13 If one wants to obtain with the described procedure a calculation of the u variation as accurate as possible without selecting too small a step magnitude k^ thereby increasing too much the expenditure in cal- culation^ one will find it necessary (as mentioned before) to make at every step a correction which takes the fact into account that in setting up the basic equation (15) the difference quotient of first approximation only was substituted for the derivative du* a^ . P^cr If one were to select instead the representation of higher approxi- mation ^ ai p,o * * ■^ p+l,a - ^ p-l^cr 2k one would obtain by maintaining equation (l4) u p+l_,a = ^Vl.a + ^%,a+l + ^*p,a-l " ^%,a + ^kfp^^ (I8) instead of equation (15). Since this relation^ however, (as can be seen Immediately) behaves considerably less favorably regarding propagation of errors than equa- tion (15)^ the profiles calculated with its aid will no longer show the smooth course mentioned before. Calculation practice has shown that one obtains very smooth curves if one writes instead of equation (I8) (H-l,a P-1,0 ?.2 * o u StI' + 2kf p,a (19) p,a and forms the second derivative appearing in it according to the scheme ;^2 * o u w bu br]_ p,a+l Su p,p-i p,a 21 (20) from the first derivative au L^^J already calculated in good approxi- p,a mation according to equation (17) by Jumping over. However, the case a = 1 Ik NACA TM 1317 reqiiires special consideration since bu ^ is not known at first. p,0 But if one takes into consideration that according to equation (I6) h^u ^^ p,0 di one may put P,0 = — -I + I h^u .^n Jp,0 3U Sn Jp,i , K^^l) - Kh-l) (21) and hence calculate the value on the left from according to equation (l?). N already known We now use the relation (I9) not as a substitute for (15) in the sense that the entire calculation is to be made with (19)^ for it became clear - particularly near the separation point where the derivatives ^ become very large - that the convergence relations here can be easily blurred (unless an especially small k value was selected); the values Up obtained by iteration do not remain quite fixed, but creep on continously, although only by small amounts (compare also the remarks in section V) . Rather we use equation (19) for making a correction in the values ^1 o^"ta-ined after the last iteration in the manner described above. With the aid of the value ari (a - 1), (already contained in the i_ -jO^a fourth column of table l) to which we add the value 3U Sri just cal- 0,0 culated according to equation (21) we determine (taking equations (20) and (ik) into consideration) the values u -1,0 + 2k ^u dT) = u 0,a -1 ,0-i Su ^njo,a+i [au 0,a- NACA TM 1317 15 We now assume^ for Instance, that the values P2 1 a prescribed hy tahle II were the final valiies even in the first procedure; we then Insert the values Dp in tahle II and calc\ilate with the quantities Ap + Bg appearing in them the values (corrected with respect to p pl these values, too, we note in the table. In the last column of this table we write the values If the corrected values P2 U deviate too much (that is, by more than 1/^ to 1/2 percent of U) from [^oli ^^ calculate with the deri- vatives '^U2 ^% hi. - H..., 0,a 2k once more corrected values according to table III, p. 31. The values then represent the final values for the profile at the point h]l,a i= ii- For calculation of every step in % direction one must, therefore, calculate three to fovoo of the calculation tables mentioned. The time expenditure may be estimated at approximately three to four hours per step. It should be stressed that all calculation operations are of purely schematic character and can therefore readily be performed by assistants. The values obtained are plotted on millimeter graph paper and the curve drawn through them. If slight scatter has resulted, after all, at one point or the other, one eliminates it with the aid of the drawing before starting on the next step. If the graph of the profile calculated just now shows that the curve, due to the increase in boundary-layer thickness, at the upper end no longer dies out gradually enough toward the asymptotic value U, one adds. l6 NACA TM 1317 In calculating the following step, and t) subdivision point in upward direction. The following condition should be mentioned which becajne evident in the practical calculation. If a step magnitude not sufficiently small is selected, two successive profiles may, due to accumulation of errors, show points where they axe somewhat too close, or else somewhat too dis- tant from each other, compared to their actual course. In the calcula- tion this can be recognized by the fact that the third profile following these two profiles shows a behavior, at these points, compared to the second profile opposite to the behavior of the first compared to the second profile. For t) values at which the first two profiles were too close one notices a gap somewhat too wide between the last two and vice versa. If one does not want to repeat the calculation with smaller steps, one may, as was found practically to be useful, once omit the corrective calculation mentioned before for the profile to be calculated next, thereby eliminating the fluctuating of the profiles, and may then continue calculating in the nonnal manner. According to our calculation experiences one can recognize that the step magnitude k must be reduced in i direction by the fact that the two u values obtained in the corrective calculation which pertain to the same t](j (thus in the example considered above the values | Tip \ L and U3 l,a deviate from each other by considerably more than l/k to 1/2 percent of the pertaining U value. If a new step magnitude in the i direction, k]_, is selected k.]_< k Ifor instance, ki = — j there appears as a result, because of equation (ik) , also a new step magnitude. '1 = ]pi in T] direction. If one wants to continue the calculation with the smaller steps k^ for instance starting from I = ^ one needs as the initial values for further calculation the numbers u(lr,aZi) u(f.r-ki,aZi) (a = 1,2, . . .) The first named numbers may be read off directly on the profile curve for I = Ij, already obtained. In order to obtain the latter, a double graphic interpolation must be made. One plots versus E, the values MCA TM 1317 17 uAp^aZi^ (cT = 1,2 . . .) read off for the values of ^ = lYp 5 t\ from the c\arves of the previously calculated profiles. Generally it will be sufficient to do this for the values u/^ln crZ]_^ of three suc- cessive profiles, thus for D = r - 2, r - 1, r. From the curves drawn through them u = u(|,aZl) (a = 1,2 . . .) one may then read off the values uAj.-ki,a2ij (a = 1,2 . . .). If a boimdary layer is to be calculated up to the separation point, it will in general be necessary to select, in the proximity of the sepa- ration point, rather small steps k. Since, however, due to the large increase in boundary- layer thickness, the profiles are here very elongated, one would obtain, because of the small step magnitude Z in t) direc- tion, a very great nimber of subdivision points over the profile; this would of coiirse increase the time expendit\ire for the calculation of a step. However, one may save a great deal of calculation expenditure by selecting, instead of equation (7)^ for instance 1 hu* S2u^ u(i) Vo ^^ Vua) y^^ ^ 2 SI ^^2 as the initial equation, and then performing the integration as before. If one again denotes the step magnitudes in I direction by k, those in T] direction by 2, one obtains instead of eqioation (ih) the relation -^ To the same Z as in the first considered case, therefore, there corre- sponds half the step magnitude in ^-direction. In this manner the step reductions were carried out for the following examples of boundary- layer flow on the circular and elliptic cylinder. The convergence of the iterations now occurred no longer only on one side toward the limit but alternately (except for the values assumed for small The numerical calculation showed further that a further step reduc- tion in I direction, still for the same Z, for instance with the aid of the initial relation 1 5ul ^ _ _]^ Bu pail dn f^L- + i\ au ^ 3 mi was not advisable because the values assumed in the upper profile parts 18 NACA TM 1317 on the right side axe given as differences of two (approximately equal) laxge numbers and therefore scatter widely; by this the convergence rela- tions may be concealed. Thus, if one is forced to reduce the step mag- nitude k still further, one will do so in the manner described above with the aid of the relation (22). It was found that one arrived in this manner, even for the extreme example of the circular cylinder, at a tolerable work expenditure even for the steps immediately ahead of the separation point. The separation point i = i^^ {and therewith s = sa) is found by graphic interpolation, or extrapolation, of the values — contained in the tables. The example of the Blasius flow at the flat plate shows very clearly the high degree of accuracy attained with this method. Here the profile obtained by continuation could be compared with the exact profile. After calculation of six steps, the calculated values deviated so little from the exact ones that they could hardly be distinguished within the scope of drawing accuracy. The differences amount to less than l/2 percent referred to U. In order to enable following the mode of calculation in detail, we add the complete calculation of the first step in the continuation of a Blasius profile at the flat plate. IV. NIMERICAL EXAMPLES AND RESULTS 1. Continuation of a Blasius Profile at the Flat Plate. The value s = is to correspond to the leading edge of the plate. For the boundary- layer equation (3) which because of p'(s) =0 is simplified to SVq SVc! ^Vo Vg — a + v_ — S. = s Ss ' St) 3t]2 together with the continuity equation ^+ ^= ^s Stj NACA TM 1317 19 then exists according to Prandtl-Blasius, as is well known^ a solution of the form vs = i «p'(U with 4^^ for which applies Vg — >0 for Tl— )0 ^U for s-^0 and all tj and >U for and all s = The function q) = cp(0 satisfies the ordinary differential equa- tion of the third order q) cp' ' = _ cp' ' ' and the boundary conditions 9(0) = cp'(o) =0 lim 9'(0 = 2. The values of cp'CO are to he taken from the following table: ^ |cp'(0 0.1 0.0664 0.2 0.1328 0.3 0.1989 Q.k 0.261+7 0.5 0.3298 0.6 0.3938 o.T a.4563 0.8 0.5168 0.9 0.57^8 1.0 0.6298 ^ |9'(0 1.1 0.6813 1.2 0.7290 1.3 0.7725 l.k 0.8115 1.5 0.8460 1.6 0.8761 1.7 0.9018 1.8 0.9233 1.9 0.9411 2.0 0.9555 ? |cp'(U 2.1 0.9670 2.2 0.9759 2.3 0.9827 2.4 0.9878 2.5 0.9915 2.6 0.9942 2.7 0.9962 2.8 0.9975 2.9 0.9984 3.0 0.9990 We choose U = 1 so that we may put ^ = 3 20 NACA TM 1317 and start our continuation procedure at s = 1. As step magnitude in Ti direction we take I = 0.6 so that k becomes equal to O.I8. The initial profile then may be taken directly from the above table, whereas the profile one step farther back, thus the profile at s = 0.82, is to be obtained from this table by graphic Interpolation. The values are contained in the following table: r\ u(0.82,Ti) 0.6 0.225 1.2 0.i^30 1.8 0.626 2.4 0.782 3.0 O.89J+ 3.6 0.955 k.2 0.982 k.Q 0.995 ^.h 1 Tl U(1,T)) 0.6 0.1989 1.2 0.3938 1.8 0.57^8 2.k 0.7290 3.0 0.8I+60 3.6 0.9233 k.2 0.9670 h.8 0.9878 ^A 0.9962 6.0 0.9990 6.6 1 Six steps (that is, up to s = 2.08) were calc\ilated by the method described. The calculation of the first step is contained completely In the table added at the end of the report. The results are represented in figure 2. 2. Circular Cylinder According to Hiemenz. Hiemenz9 measured the pressure distribution on a circular cylinder of diameter 2r = 9.75 centimeters Immersed in water and approached by the flow at a velocity of 19.2 centimeters per second. In order to make the quantities appearing in the basic equations dimensionless, one introduces the reference length 1=1 centimeter and the reference velocity Vq = 7-151 centimeters per second which corre- sponds for V = 0.01 centimeter^ per second to a Reynolds n\mber R = ZV(- V 715.1 -TC. Hiemenz: Die Grenzschicht an einem in den gleichformigen Fliiss- igkeitsstrom eingetauchten geraden Kreiszylinder (The boundary layer on a rectilinear circular cylinder Immersed in the uniform fluid flow). Dissertation Gottingen, I9II, published in Dingier 's polytechn. J. Vol. 326, 1911, PP. 321-3^2. NACA TM 1317 21 < < then the velocity distribution measured for = s = 7^ that is up to the separation pointy observed shortly before s = 7 (corresponding to an angle a of 80° to 82° from the forward stagnation point) may be represented satisfactorily by the polynomial u(s) = s - 0.006289 s3 - O.OOOOi+6 s5 On the basis of the previous indication, the solution of Blasius- Hiemenz could be used up to the value s = k.'^ (a ~ 55°) so that our calculation starts at s = k,^ (as does G<5rtler' s^O) . The value I = OA, and thus k = O.O8 were selected as step magnitudes for the first steps. The representation of ^=/J5urtT^'""''" = "^^^ against s may be seen from figure 3. The initial profiles at I = - O.O8 and ^ = 0, taken from Howarth's tables, are compiled, together with the values of from the sajne tables, in the following table: A. 5 resulting n u(-0,08,Ti) u(0,Tl) [5s s=l+.5 o.k 1.282 1.289 0.008 0.8 2.229 2.2268 O.IOI+ 1.2 2.871 2.91+8 O.2I+9 1.6 3.269 3.38i+ 0.367 2.0 3.^88 3.628 0.1+1+8 2.k 3.596 3.7^9 0.1+97 2.8 3.61+9 3.813 0.529 3.2 3.667 3.831+ 0.538 3.6 3.672 3.839 0.5^ 00 3.6lh 3.81+2 When the latter values are used, the calculation of the first step requires only one worksheet of the type described before. With the step magnitudes indicated, first four steps (up to f. = 0.32) were calculated. 10, See footnote 3. 22 NACA TM I317 The profiles obtained are represented together with the Initial profiles in figure k (partly displaced with respect to each other). With the initial relations (22) as a basis, five further steps (up to I = 0.52) were calculated for the same I = O.k and the required k = 0.04. Like wise with the use of equation (22), one step (I = 0.5^) with I = VO.OS = 0.283 and k = 0.02 and finally two more steps with Z = 0.2, k = 0.01 (up to I =0.56) were calculated. The pro- files are also represented in figure 5. By plotting of the values — i— ^^ — - the sepsiration point was found dr] to be Ig = 0.5697, that is Sggp = 6.87 (compare figure 6). Thus all together twelve steps were to be calculated. Figure 7 shows the curves u = u(| , const). Their steep decline in the neighborhood of the separation point is remarkable. Figure 8 shows a comparison of a few of the profiles obtained by us (-S) with those of Blasius-Hiemenz ( — --B-H), Pohlhausen ( — - — ^P), and Gbrtler ( G) which were obtained for the same pressure distribution!!. The comparison shows, first of all, that the Blasius-Hiemenz solution becomes insufficient in the neighborhood of the separatiDn point; the reason obviously lies in the fact that the series developments used con- verge for large s only slowly, if at all, and that, therefore, with merely the first three terms the actual course is not satisfactorily represented there. Our values agree best and most systematically with those obtained by Gortler. The differences are increasingly noticeable toward the sepa- ration point. The deviations from the values obtained by Pohlhausen, considered as a whole, remain for this example within tolerable limits although a systematic variation of the differences cannot be determined. It is remarkable that the differences assume higher values precisely in the proximity of the velocity maximum (I ~ O.36, s ~ 6, a~ 71°) (com- pare the curve for | =0.32 represented in figure 8) while again sijb- siding to some extent toward the separation point. The separation point was found according to Gortler in good agree- ment with our value Sggp = 6.8, according to Hiemenz at Sggp = 6.98, and according to Pohlhausen at Sggp = 6.9^. An approximately correct position of the separation point is, therefore, by itself not yet deci- sive for the usefulness of a method. llCompare K. Hiemenz, paper quoted in footnote 9, H. Gortler, paper quoted in footnote 3, and K. Pohlhausen, "Zur naherungsweisen Integration der Different ialgleichung der laminaren Grenzschicht" (On the approximate ■Integration of the differential equation of the laminar boundary layer), Z.A.M.M. Bd. 1, 1921, pp. 252-268. NACA TM 1317 23 3. Elliptic Cylinder of the Aspect Ratio l:k. As a further example, we calcvilated. the boundary layer for an ellip- tic cylinder of the aspect ratio 1:^, taking as a basis the pressure distribution resulting from the potential theory. ^ and Vo 10 ^.3 u. vere chosen as reference quantities for the introduction of .dimensionless quantities, with Iq being half the circumference of the ellipse and Uq the free stream velocity. The (dimensionless) velocity at the edge of the boundary layer coiild be taken directly from a table by Schlichting and Ulxich.-'-^ It is represented in figure 9 together with the function In the interval = s = 0.2 satisfactorily by the polynomial -vS I = Us) = dt ^0.2 U(t) it was possible to represent U U(s) U(s) = s - 5.116 s3 The initial profile, however, was chosen at s = O.I63 (^ = - 0.25) for the reasons mentioned before. For the first six steps Z =0.5 and k = 0.125 were taken as step magnitudes. The two initial profiles are represented in the following table, together with the values \by£} s = 0.163 r\ u(-0.375,T]) u(-0.25,Ti) Ss , s = 0.163 0.5 0.0573 0.0589 0.09^9 1.0 0.0951 0.0990 0.2672 1.5 0.1166 0.1226 o.i+235 2.0 0.1268 0.i3to 0.5197 2.5 0.1310 0.1389 0.5703 3.0 0.1323 0.llK)3 0.5858 0.1327 0.llK)8 12h. Schlichting und A. Ulrich, "Zur Berechnung des Umschlages laminar - turbulent" (On the calculation of the transition from laminar to turbu- lent) Bericht S 10 der Lilienthal-Gesellschaft (19^), PP. 75-135- 2k NACA TM 1317 From I =0.5 onward the steps could, first, be Increased. The following further steps were calculated: Seven steps with 2 = \lo3 = O.TO7I and k = 0.25 (up to ^ = 2.25), three steps with Z = 1, k = 0.5 (up to I = 3.75), four steps with I = \/2 = l.klh and k = 1 (up to | = 7.75), and nine steps with 1=2 and k = 2 (up to I = 25.75). With the aid of the starting equation (22) another step reduction was made. The further steps were: Three steps with Z = 2 and k = 1 (up to i = 28.75) and finally one step with 1=^/2 = 1.4li<- and k = O.5 (up to I = 29.25). A complete calciilation was thus made of 33 steps altogether. It became clear that selection of larger steps is not advisable, particularly at the point where the curve U = U(s) turns from its steep ascent to the flatter course (compare figure 12). On the "high plateau" of velocity distribution itself one could have chosen steps somewhat larger but they would have had to be reduced again when approaching the separation point. A large nimiber of the pro- files we calculated can be seen in figure 10. The separation point was determined from the variation of 1^ figure 11), . as ssep = 8.iv75 (compare LotiJti=0 Since Schlichting and Ulrich completely calculated-^3 the same example once according to the ordinary Pohlhausen method (P^-method), and then according to a Pohlhausen method modified by taking a polynomial of the sixth degree as a basis (P5-method), the comparison co\ild be made for a n\mber of profiles. The results are compiled in figures I3 and Ik. As far as the pressure minimum the deviations between our curves and the Pi^.- and Pg-corves are not too large. However, larger deviations appear in the proximity of the separation point. There the profiles of the original Pohlhausen method agree with ours better than the profiles of the P^-method, especially for small t^ values. The resulting separa- tion point was, according to the Pi^ -method, at Sggp = 8.38, according to the Pg-method, at Sgg = 8.26; thus these values (especially that of the Pj^ -method) do not deviate too widely from our value. ■'■^Quoted in footnote 12. NACA TM 1317 25 V. REMARKS REGARDING THE CONVERGENCE OF THE ITERATION PROCESS Regarding the conditions of convergence of the iteration process described in section III, we can prove the following theorem which will probably be sufficient for the requirements of practical calculation. If there applies for the profile at the point I = |p for all r\ '^ t^CTq °- f^^4p,ri< "(^P) and \bu\ W|d.^ U(^p) U(lp) the sequence of the velocity values obtained by the iteration process [^nlpfl^a (^^ ^o) converges with increasing n. Thus one is always able to predict, when calculating a new step, whether the iteration process will converge. The presuppositions of the theorem are satisfied with certainty when for all t| % jjao 0^ [u] .fl,a = Max T=l,2,. . . jO < - ^n P+1,T Un-1 P+1,T - (a ^ oo) with the presuppositions taken into consideration, obviously Uufl P+1,1 uu P+1,1 < 1 2U(lp) ^ Su ^P'^1 Wp.l,a 2U(lp) •- -IP+^Z NACA TM 1317 27 with < a < 1 and a beind, independent of n. Furthermore, one can see for oneself that a (0 < a < 1, independent of n) can be chosen so that slm\iltaneously the estimations Uu+1 F^l,a Ui ■u 0+1 ,^^=--[^^f>.l,. '■Lim-1 p+l,a u, (H-l,a < uTc du L . P+l,a exist so that d-u+l Pfl,a^^ \^^ fx-l^a must be true as well. Hence there exists the limit un PH-1,CJ ^1 P+1,0 ^ U2 _ r. 0+1,0 I ^1 PH-l^CJ + . . . + U. U 0+1,0 I ' ■n-1 PH-l,cr < + ... (cr = CTq) since the series at right may be majorized by a convergent geometrical series. One recognizes further that - if equations (I8) or (19) are used instead of equation (15) - if convergence of the iteration process can be proved under the assumptions that for | = ^p and all t) '^ t](j < = M. < u(lp) 28 NACA TM 1317 and since one then finds the estimations UU4-1 0+1,1 - \^\x f>+l,l < U(l, U(^p) - [u] ^0.^1 U(iD) f>+l,a ^u+1 (>t-l,CT u, ■u P+l,a U(lc) U(|p) - [u] ^P'% uCSd) thus ^u+l p+l,a < = a i-u p+l,a again with < a < 1 and a independent of n. These presuppositions are satisfied for instance for the profiles before the pressure minimum. However, in the proximity of the wall, if the profiles there show an approximately rectilinear course, the convergence will take place only very slowly. Beyond the pressure minimum one can, therefore, not arrive at a general statement on the convergence. As mentioned before, our calcu- lations in the proximity of the separation point showed that the case of divergence may actually occur. Therewith the procedure we selected, using the relation (I9) merely for the correction calculation, proves to be perfectly reasonable also from the general point of view now considered. Correspondingly, one recognizes that the iteration process performed with the aid of relation (22) certainly is convergent at the point I = Ip for a "^ Oq if there for all n "^ r\aQ 0^ [u], - i If one takes generally the Initial relation 1 Su* aV 1 Su / ■' Su._ / u (-#. + i^-i) dU(l) dl (m^l) convergence at the point | = | for a = Oq would be assured under the following sufficient conditions: For all t] = r\aQ there shall be valid o'^ H^P,n< u(lp) and ^ ^ ^D,11 < Udo) l.tk^.ork] — + m ^p^^< 1 U(lp) u(l. in m \ HI U(lp) u(lp) 30 NACA TM 1317 Thus^ if one wants on the right side a positive limit also for m must be not greater than 3. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM 1317 31 H ° + , — ^ * II « ♦ D t> + H D 1 1 W ' ' ^ 3 * ♦ ■• 13 II rH D D + II • * ... .?' — ' II * * ... 1 1 * * • ... ^ ex S]/ry • « to II D p , — ° » • . . u c D ° ~ s: ^ + r-i * ♦ ... D + H b '* 'i 11 ,^ • * • • • S * * • ■ ■ id" •s + OJ 1 1 II .* * ■ • • D M D + M 1 1 ca 3 *^ * 13 * * • • • + D < * » ... II * • ... < ca. II * * ... ca /o /t> f II » » ... D II ♦ » ... D ~ s: • • • 13 O * * 1-^ <° H ID ♦ « • • II D pq + * * . . M M M D < t. a 1 I II « * • • D t> * * • • II D , 9 cx /p \ ^ D II * ♦ ... D II ♦ » . . • ~ ~ . . . 32 NACA TM 1317 rH + D -* 0\ Q cy C\J ro ^ On H ir\ •• r-i -1 J- cy [:: § s^ ^ f= 51 % ?\ H m m VO CO ON CJN ON ' H ' « 3 II d d d d d H D D -f Q ^£> -* r^ CO CO OJ ^ ^ c^ lA •1 ^D m u^ r^ Ch C7\ on 81 CO H 8 H D OJ -:t t^ ^ a\ ir\ 'V' m CO ^ J- r-i r^ + H 1 ■?■ 9 9 9 9 9 9 9 9 9 9 t 1 11 ^ D 0" *:j H OJ MD VO c^ in in ^ \o On ITN m CO cv^ m 3 CO v£) H & + CM s r^ m t^ 0\ rH 5^ vo -:* OJ rH ?■ 9 9 9 9 9 9 9 9 9 9 4 ^ LTN OJ lA -^ ON on f- 5 \r\ FP m ^JD on ir\ VX) ro i-t cy on ro on OJ ^ + 4' d d 1 1 9 d 1 9 d 1 9 9 D C?N a\ CO % -4- J- m ^ H rH <-{ c- cy D f cy cy ^~^ m II =?■ "?■ 9 9 d 9 9 9 D 1 1 * 8 2( ^ CO cy H an -4- On g S ti 1 1 :S 3 on H LTN 8 w S_ f-( .-< =? 9 9 9 d 9 d 1 9 9 d 11 \o en t— S ir\ On OJ ON ITN sy CO -d- CO m CO on H *& 8 H H 8 8 8 < ca II 9 d d t d d d 1 d 1 9 9 D CO CO f- s -:t -^ t^ t^ t~- b on g 8 on 8 8 ?^ 3 d d d d d d d d d II D 4 S OJ SI, m s cy\ CO t— m on CO CT\ nH § m -:t On |gU' fH m ~ 8i H °. 8 d d d d d d d d ^ t3 ir\DO m t^ on ro vo m O\00 fig lAO lA on CD Q II ON §^ lA m m m r^ ir\ 2? -* CM on nn lAF- cy CO a? /5|/D SS ^^ 3 CO &8 0\ H OS 99 d d 1 1 d d 1 1 99 99 9 9 9 9 9 9 9 9 9 9 9" d 1 C* 11 t3 ^5 ^IR 3s 3^:3 t--CO M3 OJ ^S. ro on en on 8d ?^,s- \R\R

Q II H d 9 9 9 9 9 9 9 u^ t^ PJ OJ \D lA m CO CO On a ro m f= rH CO t- CM t^^ s ITN -^ ^ r-i Q t- ir\ W f-t 9 d 1 d 9 9 9 d 9 9" 9 rH + h" to t-- ro C\J ro in CM I— NO c^ H ir\ iH ^ C\J I— R 3 s= r-l CJN 00 CJN §^ ;?' 1 ^~l * H m in VD . CO On CjN ON CJn CJN ) '^ II d d d d d d d d d d H D + m r— CO t^ m CO m -* M 0\ lA -* MD m J- cu 8^ CJn ro CM ^TN r-t r-l t3 CM 5 t— CM lA CM CD -. « CO -d- (^^ W r-l '* n ,^ + r-^ -i ro CVJ rH + • • \ 9 ' — ~- ^\\ Xv\ 00 6 1 /^ \ \;^ 1 ) \^ L \ u> CVJ ro OJ ^ / - 00 to ^^""^^ ::^:^. CVJ ^^ ^. II X 1 t-H >-l Oj 00*0 u. QJ II -H J-"' ni hD fa k2 NACA TM 1317 0.27 0.23 U(s) 0.19 0.15 B Utsi- _>" 1 7*1 6 -X /" /lis) y^ 2 / / / ^ ^ / 0.8 1.6 2.4 Figure 9(a).- The functions | = 5(s) and U = U(s) for the elliptic cylinder. 32 0.29i— 28 0.28 U(s) 0.27 0.26 24 e 20 16 0.25 1— 8 U(s) / x ^ y / \ / ^.) \ / / y / 8 % Figure 9(b).- The functions 5 = l(s) and U = U(s) for the elliptic cylinder. NACA TM 1317 ^+3 Figure 10.- Velocity profiles for the elliptic cylinder. kk NACA TM 1317 m r)-0 0.015 QOI2' 0009 ^ \ \, 0.006 0.003 \ \ \ ., \ 26.75 27.75 28.75 2925 29.43 Figure 11.- Determination of the separatior point for the elliptic cylinder. NACA TM 1317 i^5 8 c to I// /^^ -^ y^ ^ y^ y/^ / / — / — / / /' 0/ CO / ^-J / / oj/ CVJ CO ll 7/ / / / \ 1 ^ / 11/ / / / // / / \ / / ^ \ ^ r-^ ""^ OJ ID CO 8 ro d CM d o o •r-t . — t a % u o to G o to o Q) Eh I I— ( > o o -rH . — I 0) Jh O •4-1 w O o w •rH cd s o o I CO u bo NACA TM 1317 1^7 Figure 14.- Comparison of the profiles for the elliptic cylinder. NACA-Langley - 4-22-52 - 1000 s < < z. 2 -c 51 c- v >» ^-* o CO fi ° 'w ^ s o - < < z t« 0} B S *j C- ■[- - . > m5 (1.Q S ™ i TO *- ^ o a CD cj ^«aB ^ -^ rt CO ■3 - u>? <-■ - ii S5 " 3 <^ T3 ti 1 -g J3 *-" 5 fr. a <^ « r/l EC ^ •■= S T3 3 _^ -n c OJ a; en ) H — < B .0 ° « s 'a "5 .S P 01 ■" Ceo 3 1-3 " >.& a o ■< u < z Q> = T3 -^ -F- O "tS — < ■ii'.£'i'-i!c5°3 j2 « 65 a E O 0) u , > = 2. c . o 3 w '3 « :3 Qi ■s .-:0 < 2 •= < b CO ^ N (0 t/] n|S-g=-3 .sg .5 -a — o ..s ^ C m c ^ o c ^ - o - ^ == ■ •y S -o s: ^ ts m ■§5 T3 U) n flj « S -a ^2 c o g ii w « O O 'S 0) .^ 3 dj tjo C !t3 ^ •5 E •o g o o S -^ a 0) E 5 iH c ■0.2 *2 CD ji: si S I* ^ 50£ <:£ - ^ ^- fl) ' o* c a S p Oj V- a c o 3 t. S ^ « o o 3 r o i =! 3 CO I ID H ■a ^ -:0 < 1 M Z m -S « = 3 S^ « 5 s C 3 11 tie W 3 £ fa Q u u (h ^:3 ^ "en :S J fi Ih c --) >. -b n a> OJ U] m < - ^n a nap , g s < z z < u •a o & 2- s < (J < z ^^ e §s P S Q. 5 >^ a "S a.S R *« c S o -S btiS < C C 5^ a t. 5-i Oj c- CJ <«-( 2 CO 3 ° c 1-H m 5 o . o H < w o Q. O o < z CQ V u B S v3 o 5^ r- a> ^ s o - H < Sii < z a