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,, ^ 7 ? 3 ,i~ i NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1317 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 boundarylayer equation and the natural boundary conditions, nothing needs to be neglected or assumed, and not too much time is required for the calculation of a boundarylayer 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 1:4. 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 Verfahren zur Berechnung der laminaren Grenzschicht." Zentrale fur wissenschaftliches Berichtswesen der Luft fahrtforschung des Generalluftzeugmeisters (ZWB) Forschungsbericht Nr. 1741, BerlinAdlershof, February 25, 1943. 2 NACA TM 1317 I. INTRODUCTION The flow processes in the laminar boundary layer may be described by Prandtl's boundarylayer equation. If one limits oneself to the two dimensional steady case and introduces, in a suitable region around 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 vs + vn s + p'(s) = 1 2Vs (1) 6s on R an2 when vs, 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 calculation1. Equation (1) is complemented by the continuity equation 3V+ n = 0 (2) s 6n The transformation n = n R v = vn l( yields, instead of equations (1) and (2), the equation system Vs as+ v ? + p's) (3) )OT) v. + 1 0 (4) ds ar in which R no longer appears explicitly. 1A mathematically complete derivation of equations (1) 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 Ngherung der NavierStokesschen Differentialgleichungen bei unbegrenzt wachsender Reynoldsscher Kennzahl" (Prandtl's boundary layer equation as an asymptotic approximation of NavierStokes' differ ential equations for indefinitely increasing Reynolds number) Deutsche Mathematik, 6, Heft 4/5, pp. 307322. A survey of related literature is given in H. Schmidt's and K. Schro'der's report "Laminare Grenzschichten, I. Teil" (Laminar boundary layers, part I) Luftfahrtforschung 19, Lieferung 3, 1942. NACA TM 1317 3 The following boundary conditions for the integration of equa tions (3) and (4) are usually selected in boundarylayer theory as the natural ones from the physical point of view. For an initial value s = so an entrance profile vs = vs(so,) is prescribed as a function of I (entrance condition). Furthermore, in consequence of the adherence of the fluid to the contour, the rela tions fvs] =0 0 = 0 which are to be interpreted as limiting processes, are to be valid along C (adherence condition). Finally, for svalues larger than or equal to so the velocity component vs is to converge for Tr ) 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 9  m 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 rj  o denotes, on the contrary, the asymptotic transition to the boundary values, resulting along C of the outer potential flow obtained for R* 0. This can best be made clear by the example of the stagnationpoint flow at the flat plate, treated in the second report indicated in footnote 1 (by the author and H. Schmidt). Whereas the quantity 5 there specified as boundarylayer thickness tends like 1/VR toward zero, a quantity d tending toward zero, for instance, like l/ R, 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 2It is assumed, of course, that this limiting process is meaning ful. NACA TM 1317 flow alonL: C then there corresponds the limiting process lim n = lim d/iR = R) R;* So far, an appropriate existence and uniqueness theorem for this boundaryvalue 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 once3 that for mal power series developments of the function representing the solution with respect to n 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 boundaryvalue 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 boundarylayer 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 boundarylayer 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 profile4. 3Compare S. Goldstein "Concerning Some Solutions of the Boundary Layer Equations in Hydrodynamics," Proc. Cambridge Phil. Soc. 26, 1930, pp. 130, L. Prandtl, "Zur Berechnung der Grenzschichten" (Concerning Calculation of the Boundary Layers) ZAMM. 18, 1938, pp. 7782, (NACA TM 959) and H. Gdrtler, "Weiterentwicklung eines Grenzschichtprofils bei gegebenem Druckverlauf" (Further development of a boundarylayer profile for prescribed pressure variation) ZAM4. 19, 1939, pp. 129140. 4L. Prandtl and H. Gortler (reports quoted in footnote 3) arrive at the same conclusion, although on another basis. NACA TM 1317 II. GC~ERAL REPRESENTATION OF THE METHOD If one introduces into equation (3), instead of s, the new inde pendent variable S= J dt (6) so U(t) under the assumption that U(s) / 0 for s = so whereby dS 1 uU ds U(s) is valid, and if one uses the new designations U( ,iT) = vs(s,q), U(M) = U(s(0)), u*(C,TI) = u(,i) U(() there follows from equations (3) and (4) by way of with equation () taken into consideration dour initial equation w eu o (o ao with equation (5) taken into consideration, our initial equation bu* u 2u* 1 5u oj a2 U() ai " d  J' 6k u* u U(0) M According to the statement of the problem in the introduction, we have to find a solution u = u (,1l) of equation (7) for all points ( ,) the plane of the rectangular Cartesian approaching the straight lines in the right upper quadrant of ,T] coordinates which in S= 0 or T = 0 respectively 5If 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 prescribed functions: lim u*(,t) = vs(so,y) U(so) ( 0) (8) or lim u*(,t) = () ( > o0) (9) TIO 0 and which vanishes for '>m lim u*(,n) = 0 (t 0) (10) The fundamental 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 course, for the boundarylayer 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) aLu 62u + u*a = 0 6B 2 dB 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 NACA TM 1317 also appears on the left side of our initial equation (7). In integrating (under the boundary conditions (8), (9), and (10)) the differential equation u* 2 = f(S,) (11) at an2 into which had been introduced for abbreviation the function f ".UI16u _n U 6u (12) f(l f) ( ( U (12) U( 6y)J0o 6 u() OE 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. Lew 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. Collatz7. If one covers the right upper quadrant of the E.n plane by a net of lattice points with the coordinates Ep = pk Ta = aZ (p,a = 0, integers) (compare fig. 1) and introduces at the same time, with a view to later applications, the new designations up,o = u(tp,1r), u*p, = u*(Pp,7i) [Laul ru] Iul = EPIJ [ J p,4m o LO[jpJLl [ojitp,= H] p,o 6R. Courant, 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. 3274, particularly pp. 4752. 7L. 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. 8 NACA TM 1317 there corresponds to the differential equation (11) the difference equa tion of first approximation U p+l,C U*p,o k u*p,a+l 2u*,a If one selects the step magnitudes k and I not independent of each other but so that 2 2 in  and n direction (14) equation (13) is transformed into the simpler difference equation * U*p,a+1 + u*p,al u p+l,o = + kfp, (15) It can be shown that the solution of equation (15) for the corresponding boundaryvalue problem for 1*O and therewith also for kwO con verges toward the known solution of the boundary value problem of equa tion (11). Since the values u p,o (P ? o) u*0,a (a o) fp,o (P oa ro0) are known, one may, according to equation (15), successively calculate all values u p, (p 0,0 2 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 (11) 1[u* [^JP, a + u*p,al = fp,a (13) NACA TM 1317 was replaced by the difference quotient of first approximation u*p+l,a u*p,a k (Compare the following section.) One notes that due to the transitional condition (10) for the entrance profile u 0 necessarily must vanish for aao dnd that fp likewise vanishes for c, since even the approximate solution used for the formation of fp was supposed to satisfy the condition (10); hence one recognizes that the corrected solution (obtained with the aid of the difference calculation in the manner described above) also satis fies the transitional condition (10). 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 E,rnplane; 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 = so is best taken from the wellknown powerseries developments by BlasiusHiemenz, the coefficients of which for the first three terms were given in table form by Howarth8. For reasons of con vergence, these brokenoff series will represent a good approximation of the solution of the boundarylayer 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 svalues for which the "first boundary layer bond" =O p(s) = d (16) 8Compare 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, 1934. 10 NACA TM 1317 which is a direct result of equations (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 svalues as well as possible by a polynomial of the form U(s) = uls + u3s3 + u S5 for the case of a profile symmetrical in freestream direction, or respectively, of the form U(s) = uls + us2 + u3 3 for the case of a profile unsymmetrical in freestream direction; one may sometimes get by with only two terms. After having determined, in the manner described above, the value so > 0 at which the continuation method may start, one first sets up the connection (given by equation (6)) = dt (s > so) SU(t) by evaluating the integral on the right side, for instance according to the trapezoidal rule. One graphically represents the functions t = E(s) and U = U(s) in a common diagram so that U = U(0) can be immediately taken from it. The step magnitudes k and 1, connected by equation (14), 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 t direction is possible. When profiles of not too pronounced Sshape (near the separation point) are to be calculated, eight to ten equidistant subdivision points generally will be sufficient to define the profile. In upward direction (that is, for large i 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 E and variable , the derivatives (obtained in first approximation by formation of 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(a,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 1, too, will necessarily be small. Since, however, the boundarylayer 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 5 direction without necessarily having to accept a step reduction in j direction. At the same time we shall then be able to indicate a criterion by which the necessity of a step reduction in t 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 fo 0 appearing in equation (15), in order to be able to execute the first step in t direction. It should be noted that together with the initial pro file at s = so also the values of u for values s < so may be taken from the series developments. Particularly the values u _C (that is, 1, a the profile one step ahead of the initial profile) are thus known. We then'put for a first approximation of the ul occurring in fo,a:i ul u0, UO1, [o0,a k Therewith jdl too can be evaluated numerically. Our 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 t points where the deriva ous u d? come to tives u become very large whereby the values = dqO come to 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 sufficient number of subdivision points are distributed over the profile to allow application of tie trapezoidal rule with sufficient accuracy. 12 NACA TM 1317 If one puts, furthermore, with good approximation u= ua+l uO',1 (17) u lo and ul, may be calculated in first approximation. The values thus obtained will be denoted by k* 11, and [ul 1,J It will be best to arrange the entire calculation procedure in the form of a table (compare table I on page 31). With the values obtained i]1a one will form corrected values of the derivatives in according to the scheme u ul [Ul] 1,c 1,0 W[ O0,o 2k whereupon one obtains (with the aid of table II on page 31) a second approximation [],o for the values Ul., with the values 70a,a, and Ca taken from the first table. Whereas the derivatives [I0jl formed in first approximation might show at a few points a an irregular course, this will generally no longer be the case for the corrected deriva tives I The columns for the quantities DCJ, 12j] ,a and 030 2 1,Co occurring further on in table II will be explained only later. This procedure is continued until the values obtained in the third fromlast 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 E0 the values up no longer varied except for an error of about 1/4 to 1/2 percent of the maximum velocity U(tp) 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 course. If the u = u(F,const) are concave in respect to the E 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 occurs 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 F axis as is the case for instance in the boundarylayer flow at the flat plate. NACA TM 1317 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 16L If one were to select instead the representation of higher approxi mation * uu P+l,a u pl,a p, a 2k one would obtain by maintaining equation (14) u u* 2ki() pu+lia = u pl.o + u p,+l + u pa1 2up, + p (18) 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 (18) [2 *1 S+ 2k P. u + 2kfP (19) and forms the second derivative appearing in it according to the scheme 2 pp,+l21 from the first derivative T] already calculated in good approxi mation according to equation (17) by Jumping over. However, the case a = 1 14 NACA TM 1317 requires special consideration since I] is not known at first. But if one takes into consideration that according to equation (16) 1H,2 p,0 d one may put H ,u p1 p1 jcu Ip, LIpl $ pI O iu + = p + ) ) (21) 6Ip, dP 1 p,l 21 and hence calculate the value on the left from 1U already known LJp,i according to equation (17). We now use the relation (19) 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 6u become very large that the convergence relations here can be easily blurred (unless an especially small k value was selected); the values UPYU obtained by iteration do not remain quite fixed, but creep on continuously, although only by small amounts (compare also the remarks in section V). Rather we use equation (19) for making a correction in the values Ulge obtained after the last iteration in the manner described above. With the aid of the value (l (a 1), (already contained in the fourth column of table I) to which we add the value 1[ Just cal L1_0,0 culated according to equation (21) we determine (taking equations (20) and (14) into consideration) the values D u+ [2u * + (  Do = u 1, + 2k = u ._1 + l  IN 0O a+1 0[aI NACA TM 1317 We now assume, for instance, that the values [u2 ,a prescribed by table II were the final values even in the first procedure; we then insert the values Da in table II and calculate with the quantities A0 + B. appearing in them the values (corrected with respect to 1 21, = Da + 2(Au + Ba) these values, too, we note in the table. In the last column of this table we write the values [2i1,0 = +( *) 1,+ If the corrected values a deviate too much (that is, by more than 1/4 to 1/2 percent of U) from [u2]l, we calculate with the deri Svatives t 2lJ, l, 6t o,o 2k once more corrected values according to table III, p.31. The values [3U1 then represent the final values for the profile at the point 5 = 1.* For calculation of every step in 5 direction one must, therefore, calculate three to four 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 boundarylayer thickness, at the upper end no longer dies out gradually enough toward the asymptotic value U, one adds, NACA TM 1317 in calculating the following step, and T subdivision point in upward direction. The following condition should be mentioned which became 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 are 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 normal manner. According to our calculation experiences one can recognize that the step magnitude k must be reduced in t direction by the fact that the two i values obtained in the corrective calculation which pertain to the same i( (thus in the example considered above the values o2] and [3],i) deviate from each other by considerably more than 1/4 to 1/2 percent of the pertaining U value. If a new step magnitude in the E direction, kl, is selected kl< k (for instance, kl =) there appears as a result, because of equation (14), also a new step magnitude. 11 = 2k in I direction. If one wants to continue the calculation with the smaller steps kI for instance starting from S = r one needs as the initial values for further calculation the numbers u(Eroll) u(rklall) (a = 1,2, .) The first named numbers may be read off directly on the profile curve for t = tr already obtained. In order to obtain the latter, a double graphic interpolation must be made. One plots versus F the values NACA TM 1317 17 u tp zl) (a = 1,2 .) read off for the values of = 5p r) from the curves of the previously calculated profiles. Generally it will be sufficient to do this for the values u( I aoll) of three suc cessive profiles, thus for 0 = r 2, r 1, r. From the curves drawn through them u = u(t,all) (a = 1,2 .) one may then read off the values u. lrklal) (a = 1,2 .). If a boundary 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 boundarylayer thickness, the profiles are here very elongated, one would obtain, because of the small step magnitude I in q direc tion, a very great number of subdivision points over the profile; this would of course increase the time expenditure 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 au* 62u* 1 du d + + 1 (22) 2 0 atN2 u(t)'70To d (I 2) 2 d as the initial equation, and then performing the integration as before. If one again denotes the step magnitudes in t direction by k, those in I direction by i, one obtains instead of equation (14) the relation z2 k  To the same I as in the first considered case, therefore, there corre sponds half the step magnitude in tdirection. In this manner the step reductions were carried out for the following examples of boundarylayer 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 .a. The numerical calculation showed further that a further step reduc tion in t direction, still for the same 2, for instance with the aid of the initial relation i u* 8lu* 3 u a 3 dU() 4as nt2 (sadvi abec e 4 was not advisable because the values assumed in the upper profile parts NACA 'T .. 7 1 TM 1317 on the right side are given as differences of two (approximately equal) large 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 k = ta (and therewith a = Ba) is found by graphic interpolation, or extrapolation, of the values L 0 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 1/2 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. NUMERICAL EXAMPLES AND RESULTS 1. Continuation of a Blasius Profile at the Flat Plate. The value s = 0 is to correspond to the leading edge of the plate. For the boundarylayer equation (3) which because of p'(s) = 0 is simplified to v a + v , = V23 together with the continuity equation hav av Vs + = 0 1 NACA TM 1317 then exists according to PrandtlBlasius, of the form Vs= :1 U'(0) 2 for which applies v 40 for for as is well known, a solution ^^ k with for S10 T4cO and all T and all s 0 The function 9 = c(p) satisfies the ordinary differential equa tion of the third order (p = Vp,, and the boundary conditions 9(0) = 0 p'(o) = 0 lim 9'() = 2. The values of ('(5) are C j q'(P ) 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 0 0.0664 0.1328 0.1989 0.2647 0.3298 0.3938 0.4563 0.5168 0.5748 0.6298 We choose U = 1 to be taken from the following table: 1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9 2.0 1 j'(C) 0.6813 0.7290 0.7725 0.8115 0.8460 0.8761 0.9018 0.9233 0.9411 0.9555 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 3.0 0.9670 0.9759 0.9827 0.9878 0.9915 0.9942 0.9962 0.9975 0.9984 0.9990 so that we may put S= S v,4U NACA TM 1317 and start our continuation procedure at s = 1. As step magnitude in q direction we take Z = 0.6 so that k becomes equal to 0.18. 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: TI 0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 u(0.82,ri) 0 0.225 0.430 0.626 0.782 0.894 0.955 0.982 0.995 1 T I u(1,l) 0 0.6 1.2 1.8 2.4 3.0 3.6 4.2 4.8 5.4 6.0 6.6 0 0.1989 0.3938 0.5748 0.7290 0.8460 0.9233 0.9670 0.9878 0.9962 0.9990 1 Six steps (that is, up to s = 2.08) were calculated 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 of diameter 2r = 9.75 the flow at a velocity the pressure distribution on a circular cylinder centimeters immersed in water and approached by of 19.2 centimeters per second. In order to make the quantities appearing in the basic equations dimensionless, one introduces the reference length 2 = 1 centimeter and the reference velocity Vo = 7.151 centimeters per second which corre sponds for v = 0.01 centimeter2 per second to a Reynolds number R =  = 715.1 9K. Hiemenz: Die Grenzschicht an einem in den gleichfdrmigen Flius igkeitsstrom eingetauchten geraden Kreiszylinder (The boundary layer on a rectilinear circular cylinder immersed in the uniform fluid flow). Dissertation G'3ttingen, 1911, published in Dingler's polytechn. J. Vol. 326, 1911, pp. 321342. NACA TM 1317 then the velocity distribution measured for 0 = s = 7, that is up to the separation point, observed shortly before s = 7 (corresponding to an angle a of 800 to 82 from the forward stagnation point) may be represented satisfactorily by the polynomial U(s) = s 0.006289 s3 0.000046 s5 On the basis of the previous indication, the solution of Blasius Hiemenz could be used up to the value s = 4.5 (a ~ 550) so that our calculation starts at s = 4.5 (as does GBrtler'slO). The value I = 3.4, and thus k = 0.08 were selected as step magnitudes for the first steps. The representation of S= 1 dt and U = U(s) againstt s may be seen from figure 3. against s may be seen from figure 3. The initial profiles at E = 0.08 and = tables, are compiled, together with the values of from the same tables, in the following table: 0 0.4 0.8 1.2 1.6 2.0 2.4 2.8 3.2 3.6 00 u(0,08,T) 0 1.282 2.229 2.871 3.269 3.488 3.596 3.649 3.667 3.672 3.674 0 1.289 2.2268 2.948 3.384 3.628 3.749 3.813 3.834 3.839 3.842 0, taken from Howarth's 4. resulting 6814.5 [ s ] ss s4.5j 0 0.008 0.104 0.249 0.367 0.448 0.497 0.529 0.538 0.540 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 t = 0.32) were calculated. 10See footnote 3. 22 NACA TM 1317 The profiles obtained are represented together with the initial profiles in figure 4 (partly displaced with respect to each other). With the initial relations (22) as a basis, five further steps (up to E = 0.52) were calculated for the same 2 = 0.4 and the required k = 0.04. Likewise with the use of equation (22), one step (E = 0.54) with I = 0.08 = 0.283 and k = 0.02 and finally two more steps with I = 0.2, k = 0.01 (up to 5 = 0.56) were calculated. The pro files are also represented in figure 5. By plotting of the values the separation point was found to be ksep = 0.5697, that is Ssep = 6.87 (compare figure 6). Thus all together twelve steps were to be calculated. Figure 7 shows the curves u = u(t ,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 BlasiusHiemenz ( BH), Pohlhausen ( P), and Gortler ( G) which were obtained for the same pressure distribution. The comparison shows, first of all, that the BlasiusHiemenz solution becomes insufficient in the neighborhood of the separation 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 GCrtler. 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 (t ~ 0.36, s ~ 6, a ~ 710) (com pare the curve for E = 0.32 represented in figure 8) while again sub siding to some extent toward the separation point. The separation point was found according to Girtler in good agree ment with our value Ssep = 6.8, according to Hiemenz at Ssep = 6.98, and according to Pohlhausen at Ssep = 6.94. 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 Differentialgleichung der laminaren Grenzschicht" (On the approximate integration of the differential equation of the laminar boundary layer), Z.A.M.M. Bd. 1, 1921, pp. 252268. ' NACA TM 1317 23 3. Elliptic Cylinder of the Aspect Ratio 1:4. As a further example, we calculated the boundary layer for an ellip tic cylinder of the aspect ratio 1:4, taking as a basis the pressure distribution resulting from the potential theory. S= zn and Vo = 4.3 Uo 10 were chosen as reference quantities for the introduction of.dimensionless quantities, with lo being half the circumference of the ellipse and Uo the free stream velocity. The dimensionlesss) velocity at the edge of the boundary layer could be taken directly from a table by Schlichting and Ulrich.12 It is represented in figure 9 together with the function dt = .(s) =4.2 RY 0.2 UV t) In the interval 0 < a 0.2 it was possible to represent U = U(s) satisfactorily by the polynomial U(s) = s 5.116 s3 The initial profile, however, was chosen at s = 0.163 (E = 0.25) for the reasons mentioned before. For the first six steps 1 = 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 as = 0.163 T u(0.375,i) u(0.25,i) s =0.163 a s j= 0.163 0 0 0 0 0.5 0.0573 0.0589 0.0949 1.0 0.0951 0.0990 0.2672 1.5 0.1166 0.1226 0.4235 2.0 0.1268 0.1340 0.5197 2.5 0.1310 0.1389 0.5703 3.0 0.1323 0.1403 0.5858 0.1327 0.1408 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 LilienthalGesellschaft (1940), pp. 75135 24 NACA TM 1317 From 9 = 0.5 onward the steps could, first, be increased. The following further steps were calculated: Seven steps with Z = 0.5 = 0.7071 and k = 0.25 (up to t = 2.25), three steps with Z = 1, k = 0.5 (up to t = 3.75), four steps with I = 2 = 1.414 and k = 1 (up to t = 7.75), and nine steps with Z = 2 and k = 2 (up to 5 = 25.75). With the aid of the starting equation (22) another step reduction was made. The further steps were: Three steps with I = 2 and k = 1 (up to t = 28.75) and finally one step with Z = 42 = 1.414 and k = 0.5 (up to E = 29.25). A complete calculation 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 number of the pro files we calculated can be seen in figure 10. The separation point was determined from the variation of [ as ssep = 8.475 (compare figure 11). Since Schlichting and Ulrich completely calculated13 the same example once according to the ordinary Pohlhausen method (P4method), and then according to a Pohlhausen method modified by taking a polynomial of the sixth degree as a basis (P6method), the comparison could be made for a number of profiles. The results are compiled in figures 13 and 14. As far as the pressure minimum the deviations between our curves and the P4 and P6curves 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 P6method, especially for small q values. The resulting separa tion point was, according to the P4method, at sep = 8.38, according to the P6method, at Ssep = 8.26; thus these values (especially that of the P4method) do not deviate too widely from our value. 13Quoted in footnote 12. NACA TM 1317 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 t = Ep for all I 1oo 0 [u] < U(yp) and f( p) [<] t p,T < I + the sequence of the velocity values obtained by the iteration process ["nup+l,a (a$ ao) 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 9 s1 oo 0 o [u: < U(Cp) and [u 0 pp, apply as is the case for instance for the profiles before the pressure minimum; for then is 5a s cothr I tp, q TJ p,1 is valid if n* is'selected so that uETPI, = = [U]pn < u U(p) (TI T oo) T, 1 l6 p, 26 NACA TM 1317 The presuppositions are satisfied also for the Sshaped profiles beyond the pressure minimum when u does not become too large. For the examples we calculated, boundarylayer flow for circular and elliptic cylinder, the presuppositions are satisfied, up to the separation point. Proof of the theorem: If one takes into consideration that according to the procedure of section III one has to put [u n+1lp+l, p,o+1 4 u*p,al = + k [oLu 1, p+ T up1,T 22k k iUu]p+l,g Upl,a U( uI U(5p) 2k there follows with = Max T=1,2,... , u IlP1,T un.lp+1T I I (a o) with the presuppositions taken into consideration, obviously uul+1 P+1. u ]p+1,l 1 2U( 1p) 2 (IU(p) upl) 2U(p) dN +1,0a a [du] , dn]+1,0 Slu1 dp]+1a NACA TM 1317 with 0 < a < 1 and a behind independent of n. Furthermore, one can see for oneself that a (0 < a < 1, independent of n) can be chosen so that simultaneously the estimations uLu p+l, a p+I12 L c1 uuljp+la Iu p+la a [dup+i,a exist so that [du+1]pl+ a [dn pl, must be true as well. Hence there exists the limit up+ ,T = im [in]p+l, I\^ L + 1 = [uIp+la + ([u2+1,0 [u p+ ,a + up+1 uni p ) + ..' (a co) since the series at right may be majorized by a convergent geometrical series. One recognizes further that if equations (18) or (19) are used instead of equation (15) if convergence of the iteration process can be proved under the assumptions that for t = Fp and all q 4 io 0 Is (U]tp,, < U() NACA TM 1317 and since one then minds the estimations[u since one then finds the estimations (u 1 p,~ U(p) [ult [u ,+1, [uu+ p+,a u]P1,o < U(p ]p) Id~ + ^e )1^1" thus [u+1 p+ 4C [ iduop+,a again with 0 < 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 (19) 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 S= Sp for o S ao if there for all T) T oo o u]pn < U(p) ulp.+1 1 up 1,I< NACA TM 1317 and (1) < U(p) + u for l 1 + for >1 2 U( P) f (Ep ) 2 If one takes generally the initial relation 1 u* 62u* 1 3u 11u u_ I aU (1i i) d convergence at the point t = tp for a = ao the following sufficient conditions: For all valid would be assured under Y= nuo there shall be )+ for 1 m' ( ) U() * UC1 P) U(t) M 6( (m> 1) and CU(p) 0 < []p,< ( Ep) 30 NACA TM 1317 Thus, if one wants on the right side a positive limit also for [u],1 (YO) U(o m must be not greater than 3. Translated by Mary L. Mahler National Advisory Committee for Aeronautics NACA TM 1317 .4 + II 0 . +  r4 5 o0 * 4 + i4 92 *1 4 B?5 0 o 5I I l 0 4+ 0 MI o SO I 0 * 4 0 .4 r 0 I rI ,^ r 11 I, 0 fI 0a o + S0 III C r1 . 0 2ii i o .5o 03 ^ 5  0 * 5 .s a9 m" In 0 4 .2 0 0 C o c[ *^ 0 0 OK D '' 0 . o * 0  01 a . 0 . 0 . S . a a . o  01 01 0 + S* rd, 0 u 4f, ft F1 Is ' 0 0 C II 01 4  ^ 4 * * 11 II o o o, a o o 8 0 0 a 9 9 0, 9 7 o o 9 9 9 0 0a a 1' % 0' ID m W L N 0  * I 0% C 0 00 0C 9_o* 9 9 9 9 9 9 0 S 8 o , 9 C 9 o! 0 0C8 8 o N 0r0 0 0 0 0 C3 C 0 0 C t C 0? 0 0 0 0 0 0 0 C Y\ r C 0 0 0 0 0 0 0 0 H 0 0 0 0 0 0 0 S0' 0" 0 0 00 o  S i< . I .0 .^fl co O ? l i?' r r 9 a 0' ?r H S05 0 0* 0 0 0 8 S6 0o o o 0 o o, 9u CU! .4 1' (41 d 9 0 '.V0 7 C .0 Cs 0 a C ( . 0 I 0 a a'4 01 m rr4 c 0 0 0 0 r0 * n t d0' d d d (b^d si1 ~ 00co ^0 Co C 0 a < :0 '0 *' < 0 ^ o? f ' 0lROh~ 11 r 0 a1 Q" ~ F* 1*' T ^ P0 r m O'lN i 3 0 pp o u^^o 3^1 'toid 'iiQFI.^o i o rf ?^,8 ^fi ,^ ^? g8 c^ V C f O < O ^ O ^ A NACA TM 1317 _I 1 1 r I i .I I i 1 I 1 ;I 'I 1;1 b 0 Sg U 0 S.I NACA TM 1317 AL a 0 H 0% 0  Nr 'D inn 0 NQ in 0C 4 in if\ 0 in % 0 II 9 N H 0% C! 9 C o r 1. 0 0. 0 0  0 0 0 0 0 0 0 0 c o no o 0N Sio a 0 0 0 0 0 9I 9 9 9 9 9o 9 o9 o o o H0 0 0 0 0 0 0 0 0 0 D 0 0 H II 04 0 o i N C cu in H 7 HO C N 4 Nm0 0 0 9 0 9 0 0 9 0 9 0 in + 0 9 9 0 0 0 h 0 0 I 0 ts 09 m 9P 9 9 0 0 0 0 U i m 9 0 0 0 0M r 01 n N N in r 0 0 0 r i O % 0% 0% + 's i NH C C' C Ci 0' 1i in 0 l 0 Oj 0 H0 0w 0 0 0 0 00 0  0 N 0 0 d, N n 0 0a cu0 0 H 0 H N0 0 0 0 0  CQ 0 0 0 0 0 0 0 0 ,, o oy oo '" 0 of 9 o o 0 iifS ~ ~~ ^o (\ co r o do fi w ^ o ^ c 34 NACA TM 1317 11 1 SII S rI b 0 ( o o1 i M m c co 0 k Fl N h u rn 0 S0 0 C 0 0 0 0 H II 0 + H 0 r 0 t l n . 0 0 0 0 0 CU 0 H 0 0 C H 0 S0 0 0 0 0 0 0 0 0 0 0 a n 00 N00 S p0 ,0 r 0 0 0 4 4 4 4 4 0 a O n 0 0 0 C C.) 0I ? I a I I a, " 0 I 0 0U0 \ 0 0\ L0 o to 0 0o o 9o o o H 0 c H 4 C 0D 0 C CU C O O 0 0 0 Sa o o a o o o o o 0 O O 4 l CO 9 O O O b 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 Lr, 4 N HO m m m fl UN E O\ ONO\ II 9 I I I C 0 r o 4^ 0 n 4 w ( U 44 O0 f HCUa H NO __ M j a U\ 44 0 0A 400 0 00 0 0 0D HH; 0. 44% 44 q C?0 rlO ? C CUCO o o n N4 Cr CP CUC4 HOJ HH 00 00 00 V.0 CU 90 4 0 '1 CU _O 4 0 '\.\0 CUL it MACA TM 13.7 35 71 ____________ 2_.3) A Figure 1. Tre net of the lattice points. 36 NACA TM 1317 8 7 6 5 Calculated according to 1/ the difference method  Exact solution 4 3 3 2.08 1.90 1.72 1.54 2  w 1.36  1.18 0 0.2 0.4 0.6 0.8 1.0 u Figure 2. Continuation of a Blasius profile at the flat plate. NACA TM 1317 37 "C r4 r4t 0 ODO I0 1 S\ II ,UI  __ _ __ q 0\ S0D o 4. 4. I. I D <> o 0 0. i d j i w' o 38 NACA TM 1317 ro  U % CD 00 0 , _o / I i\ i 0 7I co0 o ~ O 4  j O a Yt; to cu NACA TM 1317 39 Figure 5. Further profiles for the circular cylinder. .NACA TM 1317 .52 .53 .54 .55 .56 .5697 Figure 6. Determination of the separation point for the circular cylinder &,/3.6 3. ,4.0 4 S  2 0 '2.0 3 U((,const) 0 0.2 0.4 0.6 Figure 7. The curves u = u(S const.) for the circular cylinder. NACA TM 1317 41 0 lo LO 0 .0 .4  Sdt 0 0' Li CL NL 0 0. I II \ C _'c d s^!~ l NACA TM 1317 2i 2. * J Figure 9(a). The functions t = E(s) and cylinder. 32 U(s) 0.29r 28 0.28 U (s) 0.27 U = U(s) for the elliptic Figure 9(b). The functions 5 = E(s) and U = U(s) for the elliptic cylinder. 0.27 0.23 U(s) 0.19 0.15 0.26  0.25 NACA TM 1317 43 0.1 0.2 U Figure 10. Velocity profiles for the elliptic cylinder. NACA TM 1317 0.015 0012 0.009 [dul [d1J 710 0.003 0 Figure 11. Determination of the separation point for the elliptic cylinder. NACA TM 1317 45 8 f N / o / N)/ ///S C, O o ::J 0 0 II 8 0 0 0 1I 46 NACA TM 1317 S~s \ L  o, m a I 5 1co \b _15 \ MID oew i s ^\\\ __ / \ u u Q  N v cn NACA TM 1317 0 0.1 0.2 Figure 14. Comparison of the profiles for the elliptic cylinder. Figure 14. Comparison of the profiles for the elliptic cylinder. NACALangley 42252 1000 I ii I 1 I I r j , L ,C El ', P El 0;: L. 0L.  In6c fSt2 at' 0.q0. Z 22a Q I' !S sa^ ^l2 Go L. . .0< L.fl inr' In?* ..ZJ2i6 UbM 0.0 i> Gi b m C2 0 0 4 Go G Ww E cu' =,a" 02" ^. C M5 : :gz0 r tuC Z cd '0 >0&. LJMC IOnDS 01 0 o ,I '. " cc ,. [.,3 o ,' ,.U ", M .0 U E BP.5 L:E u M c." a 0 Zt .S an ,." 6 l 2 'Li maS  Gina A. L u Of .0ll >bU I.U.dO w w q0 u g mas Z S.1 0 C .  ; = r . r^s 1 3 ; saS." / u063 G"i'0 [ :1. n *0 = u AR .5 . CcUp 6 w Gi" IMmZN In In ...I OInW < E n 1 z Z I 80 0J , C< . O*L. S c S ca 8o , t.2 UlG  c L O, 2 6 S 5 . .0 C Olu ra w5 Z z k cl Cal ZE S S So 0In *..LCU *0li^ 2ia~ ts .si a' 0 at *& *0 0u Ciu I 4 a IS s SBu o.0C Cu 1u1 .2' U *a  "* **o 0a a M 0 Cue I V E . ad i. uwI m ( W B c n w u'C Ii cu 10 A M Cc) w g0 . 'i g 2 t um t,.S &, V A= aj ~ E 0 .< ~ u~iao >E 'c S g QU .r BlB 0"'J ma ' L" 2 5 0 UU O c E"Sm b ^ A cu m j m m E w~i> L. E u Mr3^ * CU ILCU > L1 11 3pili ^OS'" ~ e r^0 Cy c 3 * u uir.iC 1C 6 u U A Aua g , u<^inQ5 < u i 4Su< z U1 .1 a wE) s!a S, e s. iI go W k 0 u B"g .d ra 00 1: .z  A 0d P Go z >" M cCT$c i c ;r^ i r w B ii.S3 2sei  : UC A. t gt~~rISgS'3^ !l^ .r  cu2. ... E sS^ Si *S ft E )oOet fz~aaZ^.s 0 U4 0 r oE U . ,.1" ZZ 2z C j' 1~ H Cu 'a0 0. R& cs i'S 2  * 0i 'E I! Cua * . 0  a 3g P U U fU V *0. Sa  0 S. .w CU0 CL 40) as SIf aU2 na o Cu "'2 0. a 682 itl tis ;1 & " ^g UNIVERSITY OF FLORIDA 262 0801 09 0 