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'72 2. ( 2 377 '7 W" NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS TECHNICAL MEMORANDUM 1315 DISPLACEMENT EFFECT OF THE LAMINAR BOUNDARY LAYER AND THE PRESSURE DRAG* By H. Gortler 1. INTRODUCTION The following considerations refer to twodimensional, not neces sarily steady laminar movements of an infinitely extended fluid without free surfaces, for very small friction, about a body immersed in this fluid; they deal with the mutual influence between the boundary layer developing in the proximity of the body and the outer flow which is practically free from friction. We denote by X and Y the coordinates of a Cartesian righthand system of the plane fixed with respect to the body, by x and y the customary boundary layer coordinates defined in a zone near the wall (that is, an orthogonal righthand system where x signifies the wallarc length measured from a fixed contour point, y the wall distance measured positively from the wall toward the fluid). Furthermore, R(x) is assumed to be the radius of curvature of the wall which we define in the manner that has become customary for boundary layer investigations, in contrast to the usual mathematical definition, as positive for walls which are convex with respect to the fluid.2 Finally, u and v are assumed to be the velocity components of the boundarylayer flow, U and V the corresponding components of the potential flow about the body in x or y direction, U and V the velocity components of the potential flow in X or Y direction. *"Verdringungswirkung der laminaren Grenzschichten und Druckwiderstand." IngenieurArchiv, vol. 14, 1944, pp. 286305. 'The contour of the cylinder cross section in the flow plane should, if no other requirements are made explicitly, be of continuous curvature and fre .of multiple point singularities; it may go to infinity. 2In order to have in the defined range of x, y a reversible onetoone relation between X, Y and x, y, the requirement must be made that there R + y # 0. This does not impose any further limitation, however, since according to presupposition in the boundary layer region y/R << 1. NACA TM 1315 If the wellknown boundary layer omissions are permissible, the general hydrodynamic equations of motion are reduced to the following boundarylayer equations: ou 6u 6u 1 aP 2u + u + v = +v (l.la) Ot ox oy P ox 6y2 L+ = 0, (l.lb) ox 6y where t signifies the time, p the constant density, and v the constant kinematic viscosity of the fluid. Furthermore, p = p(x,t) is the pressure impressed on the boundary layer by the outer flow. Insofar as the outer flow noticeably deviates from the regular potential flow about the body, p(x,t) must be determined on the basis of special considerations or be taken from pressure measurements. In the present report, we consider only cases where the potentialtheoretical pressure distribution in first approximation is sufficient for the boundarylayer calculation according to equations (1.1). In this boundarylayer theory approximation, the potential velocity prevailing at the "outer edge" of the boundary layer is replaced by the potential flow prevailing about the body itself, thus by U(x,0,t), and the pressure term in equation (l.la) is calculated from Euler's equation of motion with U(x,O,t) a Uo(x,t) as 1 p 6U0Uo Uo S = + Uo . (1.2) 7+ P Ox at 0 x The boundary condition required in addition to the adherence conditions u = v = 0 at the wall y = 0 is that U assume asymptotically for increasing y Re the value3 Uo(x,t). (Re is the Reynolds number formed with a characteristic length and a characteristic velocity.) The described omissions, correct in the limit for indefinitely increasing Reynolds numbers, are only an approximation for the large but still finite Revalues for which laminar flow is plainly still possible. Particularly the connection between the boundarylayer flow thus calculated and the outer potential flow, assumed to be undisturbed, The imaginary line, denoted above, in the customary manner of expression, as the "outer edge" of the boundary layer, runs at the distance from the wall y = 8(x,t)("boundarylayer thickness") at which u practically attains the value Uo(x,t) for the accuracy requirements in each case. NACA TM 1315 does not satisfy continuity. Thus the question arises: What actual course does the outer, practically frictionless flow which is modified by the boundary layer take? This problem and the related one regarding the calculation of the pressure drag will be discussed below. The considerations will be applied to an example in all details. 2. DISPLACEMENT EFFECT OF THE BOUNDARY LAYER ON THE OUTER FLOW We assume a body which, from a state of rest, is set in motion relative to the surrounding fluid. In this case, the boundary layer gradually developing from the start of the movement will be very thin at first, so that for sufficiently small times of movement the regular potential flow about the body itself, as outer flow, may be taken as a basis for the boundary layer calculation in good approximation. The wellknown investigation concerning the origin of the boundary layers (H. Blasius ), and later extensionsSo are based on this assumption. However, with increasing times t the boundary layer increases, for a finite Reynolds number, up to a noticeable thickness; in general, it will influence the outer flow by mass displacement away from the wall to a corresponding extent. For a corresponding improvement of the customary boundarylayer theory approximation sketched in section 1, one must therefore consider the mutual influence of the increasing boundary layer and the outerflow. It suggests itself to attain such an improvement in first approximation by using at first the displacement effect on the basis of the original boundarylayer calculation, and hence determining a correspondingly corrected course of the outer flow. In a second step one would then have to calculate the boundarylayer flow anew, taking the above new outer flow as a basis. However, it will generally not be possible to carry out this improved boundarylayer calculation, using the new outer pressure distribution, according to the old scheme (equations (1.1) with the boundary conditions formulated there in the text), since the curvature effects, so far neglected in the boundarylayer equations, generally contribute amounts which can no longer be neglected within the scope of such an improved boundarylayer calculation. We shall return to this later on (section 4). 4Blasius, H.: Grenzschichten in FlUssigkeiten mit kleiner Reibung (The Boundary Layers in Fluids with Little Friction). Dissertation Gottingen, Zeitschrift fur Mathematik und Physik 56, 1908, p. 1. (Available as NACA TM 1256.) Boltze, E.: Grenzschichten an Rotationskbrpern in Flissigkeiten mit kleiner Reibung. (Boundary Layers on Bodies of Revolution in Fluids with Little Friction). Diss. G6ttingen, 1908. Goldstein, S., and Rosenhead, L.: Proc. Cambridge Phil. Soc. 32, 1936, p. 392. NACA TM 1315 The improved flow calculation aspired to is of decisive importance for the numerical determination of the pressure drag7. This becomes particularly obvious, if one visualizes a body of finite dimension transferred from a state of rest into a state of rectilinearuniform movement relative to the surrounding fluid of infinite extension. For this terminal state the potentialtheoretical pressure distribution, so far, in first approximation, taken as a basis for the boundarylayer calculation, does not yield any pressure drag (d'Alembert's paradox). Thus, small as the changes in pressure gradient at the body caused here by the boundary layer may be relative to the potentialtheoretical pressure gradient, they alone constitute the pressure drag. We want to emphasize here once more that, as we discussed before in the introduction, we do not include in our considerations flows where, for instance, by pronounced separation of the boundary layer, the outer frictionless flow is considerably transformed compared to the potential flow about the body. Of course, precisely such processes are of foremost importance for the originating pressure drag in many flow problems. Such processes cannot be included within the scope of the following considerations which are based on the boundarylayer theo retical approximation. However, we should like to refer here to a related report by M. Schwabe8 where the pressure distribution after completed boundarylayer separation is determined according to an empirical formulation for the example of the circular cylinder set into rectilinearuniform motion from a state of rest. The space taken up by the pair of vortices developing on the back of the cylinder after the separation is determined by observation and then simulated by calculation by superposition of a suitable timedependent sourcesink flow on the ordinary potential flow about the cylinder. Dhe then obtains outside of this space a streamline pattern which corresponds well to actual conditions, and a pressure drag caused by the nonsteady acceleration fields. Furthermore, we want to point out that J. Pretsch9 developed an approximative method for theoretical determination of the pressure drag pressure drag" = the component in direction of the movement of the resultant pressure force on the entire body surface, also called (less correctly) "form drag". Pressure drag + friction drag = total drag (presupposing nonexistence of free surfaces). 8Schwabe, M.: Uber Druckermittlung in der nichtstationaren ebenen Str6mung (On the Determination of Pressure in a Nonsteady TwoDimensional Flow). Diss. Gottingen. Ing.Archiv 6, 1935, p. 34. 9Pretsch, J.: Zur theoretischen Berechnung des Profilwiderstandes (Concerning Theoretical Calculation of the Profile Drag). Diss. G6ttingen, Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Available as NACA TM 1009.) NACA TM 1315 5 of profiles in a steady flow by taking the displacement effect of the boundary layer into account. In a procedure similar to ours he pre supposes that no separation of the boundary layer, or only a slight one, takes place; in the steady case this signifies, however, a limitation to slender profiles at a small angle of attack so that here the total profile drag is due in the greatest part to surface friction. Our consideration, related in the basic idea, refers at the outset to non steady flows as well and aims chiefly at the problem of the drag origin in movements of cylindrical bodies of arbitrary profile from a state of rest for small times after the start of motion, naturally without being limited to these movements. Also, it follows a different method. Whereas Pretsch makes the additional assumption that the friction losses in the wake behind the body may be neglected, probably satisfied in good approximation in view of his presupposition quoted above ("Bodies of Small Drag"), and then is able to determine, within this scope, the pressure drag from momentum considerations in a simple and general manner without actually having to calculate the corrected outer potential flow in every single case, we abstain from this or a similar simplifying assumption because we set ourselves a problem of a different type. In the following considerations we shall first deal with the first step of problem formulation, namely, the correction of the outer poten tial flow by consideration of the displacement effect (resulting from the boundarylayer flow calculated in first approximation), and shall discuss a few questions arising in case of nonsteady conditions. After that the second step which leads to the calculation of the pressure drag will be treated in general and carried out numerically on an example. 3. CORRECTION OF THE OUTER POTENTIAL FLOW First, one has to find an appropriate measure for the displacement effect of the boundary layer on the outer flow. For that purpose the wellknown boundarylayer theoretical length presents itself which is called "displacement thickness" and denoted by 5*. The formulation of the boundary condition for u(x,y,t) for indefinitely increasing y e was based on the conception that the potential theoretical velocity distribution for the large (though still finite) values of the Reynolds numbers of interest may be regarded practically as constant in first approximation for the very small thick ness of the boundary layer and may, therefore, be put equal to U(x,0,t). The definition of the displacement thickness 5* is likewise based on the conception of this streamline approximation [U(x,y,t) = U0(x,t) and thus for reasons of continuity V(x,y,t) = ybU0o/x]. NACA TM 1315 The quantity 5* is explained as follows: The fluid volume trans ported at a fixed point x at a fixed time t between the wall y = 0 and the outer edge y = &(x,t) of the boundary layer in unit time .8 equals u dy. The potential flow about the body would transport, if the streamline approximation described above were taken for a basis, the volume U0(x,t)6 through the same crosssection in unit time. The difference between these two quantities, that is, the loss in rate of flow per unit time caused by the viscosity effect, leads by virtue of U0 u dy = UO5* to the definition of the length 5* b*(x,t) = / (U u)dy. (3.1) [Compare figure 1; the hatched areas are equal. Of course, any other length y = yl > 5 may be selected instead of the upper limit 5 of the integral in equation (3.1), as long as the velocity U(x,y,t) in 0 y 5 yl is replaced by Uo(x,t).] According to definition, 5* is, therefore, a measure of the displacement of the streamlines of the outer potential flow away from the body on the basis of the reduction (caused by the friction layer near the body) in the quantity of fluid flowing by the point x at the time t. (Therein a streamline in its identity for all times is prescribed by the fact that it, together with the wall y = 0, includes a stream tube of temporally constant through flow.) In order to obtain, instead of the potential flow about the pre scribed body, a corrected outer potential flow which takes the dis placement effect of the boundary layer into account, we visualize the following model flow: Outside of a line y = 5*(x,t) a potential flow with y = 5* as streamline is assumed to flow. Within 0 y < 6* one assumes water at rest relative to the body. In this model flow the same quantity of fluid is to flow past the body per unit time as in the viscous fl?. Equation (3.1) then indicates the value of 6*(x,t) in first approximation for sufficiently high Re values. NACA TM 1315 As to the line y = 5* which is to be a streamline of our substitute flow, it must be taken into consideration that for the general nonsteady case the course of this line varies with time. The line y = 5* then is a movable dividing line between potential flow and water at rest. In contrast to the dynamic dividing lines, it therefore consists in general not permanently of the same fluid particles, but represents, for reasons of continuity, a permeable line. In mathematical formulation the boundary condition to be stipulated expresses that the normal component of the potentialflow velocity along the dividing line y = 5*(x,t) should vanish every moment. Then y = 5* is a streamline. A simple example which we shall investigate more thoroughly later (section 5) will serve to clear up these conditions. We assume that a circular cylinder of the radius R is set at the time t = 0, from a state of rest, relative to the surrounding infinitely extended fluid, into a rectilinear and uniformly accelerated motion perpendicular to its axis. The frictionless flow relative to the body is then given by the velocity potential 0 = bt(r + )cos o 0 r where r and signify polar coordinates in the flow plane, referred to the center of the circle, r = R + y, 3 = n x/R, and b denotes a constant acceleration. This potential flow yields the pressure variation on r = R required for the usual boundarylayer calculation. For very small times (small compared to the time t = tA of the start of separation) this calculation results in a displacement thickness 5* increasing proportionally to vt. For these very small times after the start of the motion the potential of our improved outer flow therefore reads 1= bt r ++ cCt)2o s (t > 0, r > R + c Vt 6* = c vt, c = const.) NACA TM 1315 The line y = c 4~v is a streamline. While it travels out into the fluid starting from t = 0, the streamline pattern of the entire outer flow correspondingly varies continuouslyl. In order to clarify the conception we shall add a few remarks regarding the mass displacement of the boundary layers. The streamline displacement thickness 6* does by no means always represent at the same time a measure for the mass displacement. This becomes immediately clear from the following simple example. An unlimited plane wall which up to the time t = 0 is at rest relative to the surrounding infinitely extended fluid is assumed to be moved in itself, according to an arbitrary acceleration law, starting from t = 0. A certain velocity profile (which is known to be easily defined) develops, and one obtains a dis placement thickness 6* different from zero. However, a mass displace ment away from the wall does not take place due to v = 0 u = u(y,t) . We could have made clear the difference in the two displacement phenomena in a perfectly analogous manner on the case of the circular cylinder set rotating about its axis from a state of rest; thus we should have spared ourselves the fiction of an infinitely extended body. Here, just as in the above limiting case of the plane wall, a boundary layer develops; a mass displacement away from the cylinder in radial direction, however, cannot take place due to reasons of continuity and symmetry. If one relinquishes this symmetry by selecting for instance instead of the circular cylinder a cylinder slightly wavy in comparison, the mass displacement normal to the contour will be, in general, different from zero; however, the length b* does not present a measure for it. 10 A We shall give here for comparison the velocity potential 0 of the corresponding flow about an expanding impermeable circular cylinder. If its contour is prescribed by r = R(t), one has ^~ R2(t) dR(t) r 0= btr + R cos a + R(t) In r tdt R The additively appearing source yields that additional fluid in the outer space r = R(t) which the impermeable circular cylinder itself displaces while expanding dr\ r ) dR(t) kdr R(t) dt NACA TM 1315 Looking for a measure for the mass displacement, one may start from the relation following from equation (3.1) by differentiation with respect to x and consideration of the continuity equation Uo ( (Ub*) = v(x,b,t) + 8 ,x (x,t) (3.2) (Again any y = yl > may be substituted for 6). The right side indicates the excess of the vvelocity at the edge of the boundary layer compared to the Vvelocity 85UO/dx of the corresponding frictionless flow prevailing there. Thus the entire volume displaced by friction effect between two points x0 and x, per unit time is at any rate always prescribed by xx= X=XO We now visualize, as in the definition of 5*, a model flow. We assume a frictionless potential flow, with the quantity 86 generally different from 8*, to be prevailing for y 8&,(x,t); the length 6, is assumed to be fixed so that precisely the fluid which actually is displaced by the friction effect would flow through 0 = y = 5 with the velocity UO. Inside 0 y = 5 we assume for our model flow water at rest relative to the body. Then there follows from equation (3.2) U (UO0*,. (3.3) 6x ox( Hence there results by integration for the "mass displacement thickness" 68 the statement (5, 6") = f(t), (3.4) that is, this difference is only dependent on t. Hence one can recognize: If one were to proceed in the determination of the first correction of the outer potential flow described above, NACA TM 1315 and therewith of the pressure field correction for large Revalues, in such a manner that one would make not y = 8* but y = 6, for all times the streamline of the substitute flow, the result would remain unchanged, according to equation (3.4). Both models differ at any moment only in the numbering of the streamlines. As to the relation between 5* and 6 one may state: If one assumes that at the respective time t the stream tube which includes at the point x the region 0 y yl with yl = followed up upstream finally blends completely in a region where the flow is frictionless (case of approach flow), 5* 6b. If, however, this presupposition is not fulfilled, as for instance in the examples with vanishing 56 selected above, the conclusion drawn above is not valid, either. In every flow of this type the streamtube just considered is either bounded by the wall in its entire course upstream, or it leaves the wall in stretches as when a separation of the boundary layer material from the body with subsequent readherence takes placell upstream (case of longitudinal flow.) The value of 6b at the time t can be given for all x if one is able to give, in addition to the variation of 5*(x,t), known according to equation (3.1), the value of 6, at a single point x = x0;12 for equation (3.4) then yields only the difference statement It is always presupposed that the flow in the xinterval of interest of the body contour and outside of the boundary layer next to the body is frictionless. Thus for instance the case where the body gets into its own wake during the motion is left out of consideration. 12It must be noted, though, that cases exist where it is impossible on principle to give such a value. Let us visualize for instance the boundarylayer flow originating when an unlimited wall, deviating from a mean plane by surface waviness, is moved in its mean plane out of a state of rest. If no special conditions prevail, the expression for the total volume displaced at the time t up to a point x in unit time x(xylt) + _I (x,t) dx V, ax t will not even be unique. NACA TM 1315 11 Therewith we shall close this little digression concerning mass displacement. Following we shall make use only of the streamline displacement thickness 6*. 4. CALCULATION OF THE PRESSURE DRAG IN FIRST APPROXIMATION With the improvement in the calculation of the outer potential flow a more accurate specification of the pressure distribution in the outer frictionless flow becomes possible. However, we must warn here against the following fallacy: In general, one cannot use the resulting pressure distribution at the edge of the boundary layer as impressed *pressure p(x,t) for an improved boundarylayer calculation according to the equations (1.1) and (1.2), for the obtained correction of the outer pressure gradient is, in general, of the same order of magnitude as the terms which have been neglected in the equations of motion. To obtain an improved boundarylayer calculation it would thus be necessary to take corresponding further terms into consideration in the equations of motion as well. The general NavierStokes equations of motion and the continuity equation read in our curvilinear coordinates x,y in full strictnessl3 u R bu bu uv 1 R p + u I 6t R+ y 6x 6y R + y pR + y ox y2 2 .2 1 du R o u  + R + y 0y (R + y)2 jx2 u + 2R V (.la) (R + y)2 (R + y) ox R dR (R + y)3 dx Ry dR du (R + y)3 dx x 13Compare, for instance: Tollmien, W.: Grenzschichttheorie (BoundaryLayer Theory). Handbuch der Experimentalphysik (Manual of Experimental Physics) Vol. 4, Part I, p. 248, Leipzig 1931. NACA TM 1315 R R+y oyv v + 2+ v=0. oy R +y Therein R(x) denotes the radius of curvature of (contrary to the mathematical definition positive respect to the flow). the wall contour y = 0 on walls convex with If one considers in these equations, on the basis of the customary estimates founded on the physical picture, only the terms of highest order for very large Reynolds numbers Re = UL/v (U is a characteristic velocity, L a characteristic length), one arrives, in the known manner, at the boundarylayer equations (1.1)14. If one considers in improved approximation also the terms of the order 0(5/L) compared with 0(1), there results S u 1 6p + v + oy p ax 8u 2y 6y y R uv R + y_ pP pR dx v du R by 1 dp u2 P dy R (4.1) (4.2) 14Compare, for,instance, W. Tollmien, footnote 13. dv U + bx v dy u2 R + y = 1 dp+ p yv v (R + y)2 Ry (R + y)3 . 62v 2R y2 (R + y)2 R2 2,v (R + y)2 6x2 R dR + u + (R + y)3 dx dR .vy dx Odx au + dx (4.2a) (4.3a) R R + y du 6t ,u u  dx NACA TM 1315 6du + v_= Y u v (4.3) 6x 6y R ox R On the left side are the terms from equation (1.1), on the right the newly added terms. The correction of the outer pressure we obtained above is, in general, of this same order O(b/L). Thus, if one wants to perform with this improved outer flow an improved boundarylayer calculation (second approximation for large Re), one can in general no longer neglect the right sides and can, therefore, no longer calculate with a pressure p = p(x,t) impressed on the boundary layer. (Compare addi tional remark at the end of this section.) For calculation of the pressure drag in first approximation for large Re, the problem that interests us here, the solution is simpler. We now know, according to the expositions of section 3, the outer pressure field, far remote from the body, sufficiently accurately; it only remains for us to continue this pressure field up to the body surface by deter mining the pressure gradient in ydirection through the boundary layer. Equation (4.2) serves this purpose: It yields op/by sufficiently accurately for this approximation, if we substitute in it on the right side u from the first boundarylayer approximation. If y = 6(x,t) denotes the "outer edge" of the boundary layer, and p[x,&(x,t),t1 is the pressure distribution of the improved outer potential flow along this line, one has for 0 = y = 6 in this approximation p(x,y,t) = p(x,8,t) R J 2u dy (4.4) The pressure drag of the unit length of the cylinder in the flow becomes WD = p(x,O,t) cos ( dx IKE J (4.5) = 5.9 px&t) P JOE) u2dy cos p dx, K nw0 NACA TM 1315 Wherein ) signifies the angle between surface normal and main flow direction, and the integral is to be formed over the entire contour K of the cylinder cross section.15 15Additional remark at the time of proof correction: Regarding the problem of an improved calculation of the boundary layer flow (second approximation for moderately large Reynolds numbers), not followed up further in the present report, the following calculation procedure seems to me to be promising: 1. Calculation of the boundary layer in the customary first approximation for very large Re. 2. Hence correction of the outer frictionless flow according to Ziff's method. 3. Improved calculation of the pressure field p(x,y,t) in the boundarylayer zone according to equation (4.4). 4. Again calculation of the velocity components u and v of the boundary layer in second approximation from equations (4.1) and (4.3), using the pressure field calculated above and replacing the right sides of equations (4.1) and (4.3) by the known expressions of first approxi mation so that the newly added terms of the order 0(6/R) compared with 1 appear in the calculation as prescribed functions. NACA TM 1315 5. EXAMPLE1I Following we shall consider as an example a nonsteady flow which originates if a body is, from a state of rest, set into a rectilinear motion relative to the surrounding infinitely extended fluid, onward from t = 0. The relative velocity of the undisturbed approach flow with respect to the body visualized as.being at rest is assumed to 16The simplest, almost trivial example on which the method developed may be tested is the case of the plane steady stagnation point flow. Here the strict solution of the NavierStokes equations is known, and the flow near the wall calculated on the basis of the boundarylayer theory is known to agree with the exact solution. If one replaces the potentialtheoretical stagnation point flow at the wall by the stag nation point flow at the wall shifted by 6*(= const.), one obtains as the corrected outer flow for y > 6 also full agreement with the exact solution. Since the outer pressure gradient parallel to the wall has remained unchanged in this correction and the wall is plane, an improvement of the boundarylayer calculation proves to be impossible as it has to be. Thus the exact solution has already been attained with this one step. A further example with plane body boundaries (for which the boundarylayer equations (1.1) therefore are valid except for terms of the order 0(52/R2) compared with 1) is the case of longitudinal flow over a plate. According to Blasius, elsewhere, 6* = 1.73 ,vx7,U. (U. free stream velocity, x distance from the leading edge of the plate). With the improved outer pressure distribution calculated as suggested above, there results as the superimposed pressure for an improved boundary calculation p(x) = P El + 1.35 Re(x1 (Re(x) = Uox/v). One does obtain here a considerable pressure correc tion since the new outer flow has a stagnation point, but one recognizes that the pressure correction has already dropped at Re(x) = 75 to 1 percent of the stagnation pressure. However, in the proximity of the leading edge of the plate (small Re(x) values) Blasius' boundary layer equation cannot be used. An improved calculation of the flow which in this range does not use the boundary layer approximations is still lacking. Thus for the time being the method of improvement for larger Re(x) suggested in this report cannot be utilized for this example. NACA TM 1315 be = Cot C = const., n = 0, t ). Then the potential velocity U0(x,t) along the body boundaries can be represented in the form 0( for t o, , u0(x,t) = t f > (5.1) q(x)tn for t > 01 We assume *(x,y,t) to be the stream function of the boundarylayer flow defined by u = d'/oy, v = 6//dx which is obtained when the pressure gradient to be calculated from equation (5.1) according to equation (1.2) is taken as a basis, on the strength of the boundary layer equations (1.1), thus in the customary first approximation for very large Revalues. Generalizing the series developments set up by Blasius1' for the special cases n = 0 (sudden transition from state of rest to motion at constant velocity) and n = 1 (uniform acceleration) one obtains the result x(x,y,t) = 2 rt qtnno(n) + qq't2n+ .,l101 + ..] (5.2) thus a series development in powers of t n+ (that is, in powers of the distance covered by the body). The appearing coefficient functions S (,O l (n), ... with = y/2 fv4t are universal functions of i. More details may be found in the appendix to this report. From equation (5.2) follows qtn i it2n+l + u = qt n + qq't n ' n.0 nll (5.3) v = 2 't n,0 + (q2 + qq)t 2n+ n ,l + n,OJ 17Compare footnote 4 on page 3. NACA TM 1315 The boundary conditions are satisfied by virtue of n,o(O) = n,l(0) = *** =, (0) = (0) = .. = 0, n,O n,l n,O n,9  where there is always, here and later on '1 d .n n., K d n.K For the streamline displacement thickness 5* 5*(x,t) = 2 tvtai q'(x)tn+l 1(m) L .nn.,l there results  .. with an =lim I n,0 (A few numerical values an are given in the appendix.) We consider as an example the flow about a circular cylinder. the mode of notation of section 3, the velocity potential of the frictionlezs flow about the circular cylinder ij given by r2 0 = (t; (r + R) cos 0 0 r (5.4) 0, ' J (5.5) (5.6) (5.7) NACA TM 1315 Hence the velocity at the periphery of the cylinder is calculated as UO(x,t) = q(x)tn = 2U(t) sin (5.8) and the resulting pressure distribution at the body surface is pO(x,t) = pd p[2R(cos 3 + 1) + 2d+ t) sin20 (5.9) where Pd(t) is the pressure at the forward stagnation point. The potentialtheoretical pressure drag of the length L of the cylinder at rest is therefore W = L P cos .3 Rd. = 2pnR2L d. (5.10) DO J 0 dt Thus one obtains, as is known, an increase in inertia by double the amount of the inert mass of the fluid displaced by the cylinder itself. For the improved calculation of the outer potential flow there results 5*(x,t) = 2 vt a + 2 C (n)tn+l cos + .... (5.11) L R nl Following we limit ourselves to small times t for which the two first terms of the series development represent a sufficient approximation. In this approximation the line y = 5*(x,t) practically represents a circle; that is to say, a circle with increasing radius r, the center of which, for t>d, does not coincide with that of the cylinder cross section, but travels slowly downstream. The equation of such a circle reads formally r = b cos + r = b cos 3 + a(l b sin2 B)2 a/ NACA TM 1315 (a = radius, b = displacement of the center), and if r = a + b cos a [+ , We identify a(t) = R + 2an A TF 4( n,l b(t) R n,l ~) 0 t (5.12) If, temporarily, T7, denote polar coordinates about the center of this circle in motion relative to the cylinder, the potential flow appertaining to the latter, the streamline of which is the line r = R + 5*, that is, r = a(t), is given by the potential = (= )( + cos T thus the potential flow here required in the polar coordinate system r,1 fixed relative to the body by +l = Ul+ 2 (r cos a b) 2 c r 2br cos 1 + b2 cos b) (5.13) At first we introduce dimensionless quantities as choose R as the characteristic length, and the time to cover the distance R, starting from the beginning as the characteristic time, thus follows: We T the body requires of motion t = 0, (5.14) T = (n + 1 n+l 1)  b << a NACA TM 1315 Hence results the characteristic velocity 1 R CORn) p T T \ n+1 l and the Reynolds number Re formed with this velocity and the length R becomes8 Re = (R= R~ (5.15) By making dimensionless the variable lengths x, y, r, a, b by dividing by R, the time t by dividing by T, the velocity U(t) by dividing by R/T, the potential by dividing by R2/T, and the pressure by dividing by pR2/T2 (we denote the dimensionless quantities by adding a wavy line), we obtain 1 1 = 1 + 2an Re 22 1 (5.12a) b = 4(n + 1) (o) Re 2t 2; n,l S= (n + 1)n 1 + a (r cos x + b). (1 + 2 br cos x + bl (5.13a) 181f one forms the Reynolds number Rel with the length VvT one obtains Re= (RT) 1=Re Rel = (R/T)4 v = Re2 NACA TM 1315 Furthermore, 1 1 I = Y Re2 2 Jt 2 and  4(n + 1)t nn,1(m)n+l cos ]. co xJ. R= e n Re We consider first the case n = 1 of uniform acceleration from a state of rest. Following, Pi without more precise data will represent those additive portions of the pressure pl = p/(pR2/T2) which do not make any contributions to the pressure and are therefore not of interest to us. Furthermore, terms of the order 0(52/R2) compared with 1 are neglected. One obtains for the pressure calculation according to the Bernoulli equation at the distance from the wall Y = 6/ 4 1 + 3alRe t cos + P +(R. S01r 2  1)2 10l 2 sin2x + p2+ F2L = R 2R2 )2 ij Thus one obtains Pj(, , ) = 4(l + 3aI le~ ) cos I + 16 2s ~ P + o (5.16) (5.11a) 3a~1 g2\ = 0 '2 NACA TM 1315 According to equation (4.4) (4. 4a) In the small times of interest to us there is n = 4 sin ( 16 sin (16 cos X'I (n), 1,0 1,1 therefore 8/R  I U2 = 16 3sn 8/R udy = 16 fsin2x ^0 io02d + 128 4sin2x cos x / R d1  JO 1,0 1,1 .3/R 256 6 sin2x cos dy. o 1,1 (5.17) Because of lim (I) = 1, lim (' (r ) = 0 the two last integral T ) o1,0 1.> i1, on the right are, if 6/R is chosen so large that these asymptotic values are attained with the desirable accuracy numbers independent of the variation of the boundarylayer edge 6(x,t) with x. The first integral at the right, together with the corresponding second term at the right in equation (5.16), may also be combined into an expression independent of (xE), namely the expression 16 2sin2 (1 02d. 0 P(Xr, 0, t = ) = 2dy. ~ \ / NACA TM 1315 Thus the variation of &5(,T) over x does not play any role in judging what terms make contributions to the pressure drag which is as it should be. If one finally inserts 1 1 dy = 2 Re 2t2d , all together one obtains, therefore, p( o, Y) = 4(1 +4 3 1e 3acRe t os 0 1 9 + 256 sina cos 1 Re2 22 J 0 io 1, P4 + P+0(Re1) (5.13) Thus the following pressure drag of a circular cylinder of the length L for small times after start of the motion and in first order for large Revalues results Wp 2pnR 2LCO 1 1 2e+, '*~'.9'11 ,`'0 IFn, (5.19a) 2pnR2LC0 = WD is the potentialtheoretical pressure drag. (Compare equation (5.10).) According to Blasius'calculations 3a1 = 2/rW(=1.128), furthermore, according to the author's numerical evaluation 0 q, q0 ,ldT = o.09o80 NACA TM 1315 Thus the result in dimensional form is tL 2 S 0.392 t(5.19) "D "o R R 2 As friction drag ;2n W / a(x, 0, t) sin dx R J y R there results from Blasius' calculation results19 WR / D 0.029 2 t (5.20) DO R (h R2 For sufficiently small times tt << 1) thus WR = WD WDo that is, the friction drag increases immediately after start of the motion .ccordinq to the same law as the contribution of the skin friction to the pressure drag. The total drag becomes S= JD+ ijR DO I + 0,363 R t4 (5.21) In figure 2 we represented these results using the dimensionless t = t/T (with T = f2R /Co for the present case n = 1), thus the re la 'tion T 1 = 1 + t (1.128 + 1.569 4), (5.191) IDO Tt, order to obtain the term with t 4 one must here include in .. (V.21 .lIso the third term of the series. The necessary data I, "' .. ir. Blasius' report, footnote 4. NACA TM 1315 1 W = (1.128 0.116), (5.201) WDo W = 1 + 17(2.257 + 1.4534). (5.211) WDO R Our formulas can be applied with good approximation only for very small times after start of the motion (solidly drawn parts of the curves), because we had, for the sake of simplicity, taken into consideration only a few terms of the development (5.2); but of course, with a little calculation expenditure they can be easily improved, at least so far that they are valid up to times shortly after settingin of the separation. The value t"= tA(= 0586) = 0.766 plotted in figure 2 indicates the time of the start of separation, in first approximation, at the rearward stagnation point according to Blasius. (Compare also appendix.) For an arbitrary integral n > 0 one obtains correspondingly as pressure drag with 7i(t) = COtn and WDO = 2pDn2L dU/dt WD = 1 + + n + t2n+2 n n,+ n (5.22) WDO For the calculation of the coefficient of t2n+2 sufficient numerical data concerning (_ ) are lacking so far. For this reason we limit n., ourselves in our numerical statements to the first term of the develop 1 ment with time (term with t2) and put the question whether the law WR = WD WDo obtained above for n = 1, for times immediately after start of the motion, is valid also for arbitrary n > 0. The friction drag has for these small times the value WR= WD rt (5.23) Do R 2n NACA TM 1315 In the appendix the exact expressions for (r) are derived. n,O However, the relation 2(2n + l)an = (0) = 22n (n!2 2 nO (2n) qr (5.24) is valid; we produce a very simple proof for it in the appendix (one may confirm it also with the aid of the tables of the appendix we calculated for n = 1, 2, 3, and 4). Thus, for very small times after start of the motion, there applies indeed WE = WD WD viz: = WD O R nJ'(n 1) (2n):! F (n > 0). (5.25) (One has 22nn'(n 1)!/(2n)! VfW = 1.1284 for n 0.6018 for n = 3; 0.5158 for n = 4, etc.) = 1; 0.7522 for n = 2; One may now consider more general laws of motion of the form (t) f for t 0 (t) for t 0(f(O) = 0) and carry out corresponding calculations. If one presupposes that the function f(t) defined in t 0 can be developed into a Taylor series around t = 0 which converges for the small times 0 = t << tA after start of the motion which are of interest to us, one may attain the result quite analogously with a series expression correspondingly generalized compared to equation (5.2). One can interpret the law U = Cotn which has been valid so far as the first term of the development with time of such a general law of motion. Hence it follows NACA TM 1315 that for all these laws of motion the portion of the pressure drag caused by friction WD WD increases immediately after the start of motion according to the same law (5.25) as the friction drag. This is a noteworthy quality of the circular cylinder. One may also include the case of a sudden start of motion in these considerations; it is true that one must then accept, corresponding to this degeneration of the form of motion at the time t = 0, infinite pressures and drags at the time t = 0. With U CO = const. for t 0 one obtains WD = 2pnRE C 4C2 2 + C WR = 2pdLCO 4 O i 0 t2 o Oo 0,1 2 R2 I The function (0t2a(q) is explained at the end of the appendix. We did not numerically determine the coefficients at t2 in equations (5.26) and (5.27). Because of aO = o'(0)(= 1/Ar = 0.5642) the law stated above W = W W is valid also in this limiting case n = 0 of R D DO the sudden start of motion, for times immediately after the start of motion. Here in particular WDO = 0 (d'Alembert's paradox). and (5.26) (5.27) NACA TM 1315 APPENDIX A FEW CALCULATIONS REGARDING THE DEVELOPMENT OF THE BOUNDARY LAYERS A few calculations will be given in this appendix which yield, among other data, those required for the preceding investigation (section 5) concerning the basic functions tn i(n) of the unsteady boundary layers; for the rest, they represent merely an extension of the related calculations by Blasius. We give these calculations apart from the previous considerations, first, because they would have disrupted the connection there, and second, because the datu and tables attained are of interest in their own right. As assumed in section 5, let a velocity proportional to tn(n > 0) be imparted to a body from a state of rest relative to the surrounding indefinitely extended fluid, beginning at t = 0. The potential theoretical circumferential velocity U(x,0,t) = U0(x,t) then has the form (5.1). For calculation of the boundary layer development from t = 0, if a generalization is made of the series set up by Blasius for n = 0 and n = 1, the expression *(x,q,t) = 2 Yt L tn+X(n+l)X (x,rl) O rn, with (1) T = y/2 4t for the stream function of the boundarylayer flow is obtained and one obtains for the functions Xn, by substituting equation (1) into the boundary layer differential equation (1.1) a system of differential equations solved by recursion; we limit ourselves here to the two first equations of this system which read a3 n,0 K0 n,O n + 2T 4n = 4nq(x), N3 n,J o ,1 l (2n n,O nO + 2 4(2n + 1) = 4(2) q3 2~ \ an aox 10 a2 ",o _6 c qq 2 NACA TM 1315 We assume first n (later 2n) to be an integer. Then the solutions may be represented with the aid of Hermite polynomials. With the statements SO= q(x)t (T), n,O n,O ,l = n one obtains from equation (2) the ordinary differential equations ' + 2I'' 4n' = 4n, n,O n,0 n,0 '" + 2" 4(2n+ l)C' nl nl nil The boundary conditions to be satisfied by n,0 formulated in equation (5.4) S O2 n  = 4 nO n,O~n,O i . and tn,l are NACA TM 1315 In the case of a plane wall moving in its own plane, S= 2 V%7 qtnfn, () represents the complete solution (because of q = const.), not only in the boundarylayer theory approximation set up here, but in strict fulfillment of the complete NavierStokes 20 equations. The calculation of tn may take place as follows. The prary plttn,ngoff temporary splittingoff 2, n,O() = 1 e" n() transforms equation (4) into n 2)pn 2(2n + )(n = 0 20Because of U0 = UO(t), u = u(y,t) and v = 0, the Navier Stokes equations are for these motions simplified to bu a2u oU ) = V + +a t oy ~t Cy l. The analogy between U0 u and the corresponding solutions of the problem of heat conduction has been known for a long time; it offered one of the few possibilities of attaining exact solutions of the NavierStokes equations. For the rest, one can see for the present problem that the first term qt',0 of the series development following from equation (1) for u as a solution of the above equation approximately satisfies the boundarylayer equation in the sense that only the terms of highest order are taken into consideration for small times after start of the motion, whereas the quadratic inertia terms are neglected. The iterative improvement of this first approximation for small times then yields step by step the ascending terms of the series we set up formally at the outset. This consideration led Blasius, (elsewhere), at the time to his special series formulations for n = 0 and n = 1. NACA TM 1315 For every integral equation21 is 2n t 0 the general solution of this differential Ph= n (2Cl + C2 ed 21 (d )2n dw) (C1, C2 are integration constants). As is well known, the .Hermite polynomials Hm(x) are given by Hm(x) = e( = m(K) (2x)m2" 0< K<_ (in the form originally Lm(ix) = iftm(x), thus given by Hermite). For further use we also put m() = e e) 2 .) m 2 ) (2x)m2 K 0< K. furthermore S() ( =o x exdx, 0 A . (11) . d(x) = dxK 21Compare, for instance, E. Kamke "Differentialgleichungen: Losungsmethoden und Losungen, I Gewohnliche Differentialgleichungen" (Differential equations: Methods of solutions and solutions, I. ordinary differential equations) Leipzig 1942, Part C, No. 2,41, and put there x = i VI. (10) dadeX1 NACA TM 1315 (so that in particular 1 + 00 = 0 represents the error integral, e0 the error function). By using these expressions, a simple recalculation from equation (8) gives the general solution of equation (4) as '() = 1 H() + C2 (*)1 + 2n (12) Ic (7) 7 2n ) , (TI ) Hr K=1 2n_(TK1 with C C. 2 2 2 C1 = 0 because of 'n0 (w) = 1 and C2* = n'/(2n)! because of S' (0) = 0 and B2n(0) = (2n)./ni. The polynomial sum in equation (12) disappears for n = 0 and integral n term by term, since either 2n n or K 1 is an odd number]. Thus the desired particular integral of (4)22 reads 2n Sn,0() = 1 + (2n()0 i() 2n ( ()2n()HI( ( n,(O + ( 2n l' ( n) * 0 = (13) Because of 0 (7) = ( I)1 ( )H K (n) the solution may also be written in the following form which is more elegant than equation (13) but less serviceable for practical calculation 2n (n= + n! 2 D)(T). (13a) 22The direct and elementary derivation of the solution for integral n > 0, and therewith for integral 2n 0, compare below, fails to work if n does not have this property. But in that case, too, the solutions are easily found if one makes use of the analogy to the corresponding solutions of linear heat conduction (compare footnote 20 to this report) and represents the solution according to the singularity method. NACA TM 1315 According to equation (13) one has in particular ' O}C) = 1 + C(TI) = t(n), ' (1) = 1 + (2n2 + 1)$0(n) + nl() 1,0 g,0o() = 1 + 1(4n4 + 12n2 + 3)00 () + (2C2 + 5)i1(TI[, S(n) = 1 + 1 [8q6 + 6094 + 9012 + 15)00(C) + 3,015 (Ci4+ 28T2 + 33)1(C~], S,0(n) = 1 + 1 6i8 + 224n6+ 84oT4 + 840T2 + 105)o(Ti) + (8,6 + 1084 + 370n2 + 279)7nl (i)1 ( (13b) Thence one obtains by elementary quadratures n,0(n), likewise expressed by t0 and T0 with polynomial coefficients. The numerical evaluation is reproduced in table 123. Because of the special impor tance of the solutions ', (n) as boundarylayer profiles u/qtn n,0 in the case of the plane wall (compare above) we represented them in figure 3. Owing to 6* = 2\ l (T)) = 2an ;t the stream lie dslaoo e (I 0 ) 2 line displacement thickness 5 can easily be taken from the numerical cal table 1; comparealso table 2.3 Since it follows directly from equation (7), by single differ entiation, that the general integral 1 is the first derivative of n 2 the general integral %, with respect to n, it is, with the boundary conditions satisfied, easy to find as expressions for the basic functions + 1 (q) with the integral n o =1 1 Lno + 2m(1 ,0) (14) numerical calculations were performed by Miss Ursula Ludewig 23All numerical calculations were performed by Miss Ursula Ludewig. 34 NACA TM 1315 Therein v" (0) = 22 (n1)2 2 (15) n,0 (2n)' in In order to calculate the fundamental functions of the first order (n,1(), the total course of n,0(r) must be known (according to equation (5)). Only the additional knowledge of (" (0) is required n,l for the problem which is of foremost interest, the question regarding location and time of a possible separation of the boundary layer in first approximation. For because of (U t :n t n+ltt S = n 2 .+ ( + qq'tn" .(0) + (Yy=/ 2 vt L 0 nPl one obtains in first approximation the connection 1 (X),T(0) n+l  for location x and time t of the separation. On the other hand, we are interested in Cn (no) (compare section 5), with a view to the n,l calculation of the displacement thickness. These two data can be determined without solution of the differential equation (5) by a wellknown method as follows: (qT) is assumed to be a function of in 0 m provided with the continuity properties required for the following calculation. By partial integration one obtains the following relations. If Ln and Mn are the differential operators Ln = 2 + 2n 4(2n + 1), n d= 2 d 2)' NACA TM 1315 one has (B)LI din [ 'n, n li L 1n f0 n Mn0,] dI, (17) f(1)L [' ]di = [" OJ nll n nsl n,1 .O/ n, J h .di We choose, therefore, I = UO (r) so that Mn[3n0,O] = 0 with with n (0) 1, nO a (o) = n,0 If then equation (5) and the boundary conditions valid for t (T) are taken n,to consideration the result is are taken into consideration the result is n,1 () = no O n.02 n,0 n,0 9 ". On the other hand we choose 8 = 6~m ( s) so that MOn] = 1 with n, (0) = 0, n'sD n, (.) finite n,"D + 213t' + Mn ];  n,l l 0 (18) (17a) (17b) (18a) NACA TM 1315 Then one obtains, in analogy (18b) n = 00 0% O)4(n,02 n0n,0 1)d. It is easily confirmed that 9n,)(() = 2( 4n + 3) n furthermore by comparison with equation (7) n,0() = e2 2 1( 1 ] (19) Therewith the desired functions and 3 n,0 n, ao the known basic functions of zero order + 2q+1,0. are traced back to Numerical evaluation yielded for n = 0, 1, 2, 3, and 4 all together the data here of interest given in table 2. It also shows the numerical values of an. For the nvalues 1, 2, 3, and 4 one finds the law (5.24) confirmed. A general proof of this law may be produced with a few calculations on the basis of the known expressions for no(O(). A much simpler proof of the relation (5.24) will be presented below. As mentioned above, the expression u = qtn 10 is the strict solution for the boundarylayer profile on a plane wall with U = qtn (q = const.) outside of the boundary layer. The wall shearing stress according to the momentum theorem of the boundary layers is generally d /5 _ pudy J pu2dy  dx .0 SP dy b P J0 it ox (20) 7 = u x O 0 x J0O (21) NACA TM 1315 thus for the above flow, due to the velocity distribution u(n,t) being independent of x as well as due to p and with ox dt n= l 0 ,0()] 6 1 T = p (U u)dy = p (Ub) = pq(2n + l)a tn 2. (22) 6 Ito 60 t n On the other hand 1 Tu(o,t) 2n T0 == pqIn (0) 4 t n (23) 6y n,O 2 and thus in combination with equation (15), as asserted, 22n+z(n,)2 2(2n + l)% = I, o(O) = (5.24) In the case of the flow about a circular cylinder moved rectilinearly out of a state of rest, investigated in section 5 [compare equation (5.8)] one has q(x) = 2C0 sin x/R and therefore q'(x)max = 2C/R for x/R = i. Thus the separation starts according to equation (16) at the rearward stagnation point at the time 1 tA = [n,o(O)/2CO 1(O n+l (compare table 3). The distance covered by the cylinder during that time is SA = COtAn+/(n + 1) = Rt" (O)/2(n + 1)t" (0) Sn,O n,l NACA TM 1315 Finally we want to give a few indications where to find further data regarding the basic factions of the plane nonsteady boundarylayer flow. The series development (5.2) written down up to the third term reads 4 = 2 t qtn [,0 + tn+lqtn,l + t2(n+1)(qi2 n,2a + q"n,2b) + a..]. Blasius gives, in addition to the functions 0,0(1) and 51,0() calculated above, the rigorous solutions 0,1(C) and t, (17). Beyond that, he calculates the numerical values ," 2a(0) and 2b(0) which are of interest for the determination of the separation. S. Goldstein and L. Rosenhead24 give the exact expressions for S0,2a(1) and 0,2b(T). These integrals were, by the way, numerically determined before by Boltze25 on the occasion of treatment of the corresponding problem n = 0 of rotationally symmetrical flows which seems to have escaped the attention of the authors. Translated by Mary L. Mahler National Advisory Committee for Aeronautics 2%Compare footnote 6 of this report. 2Compare footnote 5 of this report. __ NACA TM 1315 %Dcoco %4 uc% r ni0rC U o\%0 m cu o\I: a\ t nc c Ii m  Mh M8~AMRc4oom8oo8888ooooo 888Il I0 I l H 00000000 0000000000 IIII OOOO OOOOOOOOOOOO O Oooo I III I I in o n I \ O M o t1 %D o C\Y( otm c;c o\ o o o r4 i r MM doto ', CC 'm 'M & ', ,0 0 \ Q\ '1 .0 oIr.cuCC)...r.t.coc MC ..o.00ooo 111111111 o .000000 0000000000000o o i i i i i i i 1 i 0 r %DI I I I I I I t oo (M c M n u \ Q to a o r"Q cy m r\Co t, 6 1 , ooJU oooIHooooo_^*C~\ ^oooooooooo 0n o'ori.IUL r\%r' Ch CN M M tF H 0 1 0 o A O r\o m %D M[O \o \ o ln0cu mI O OO 1oI I S00000000 0000r% 0000000000 1 .. ......................... I I I I I I I .C Oco oiroo Oodo ".ooooo. oooo I I I I I I SOOWH Bmoo O oII,,,,,,,o , Sr O D .0 hA  CoJ ,I 0 0 0 00 1 I I I I 'CO .111 l=l OJ CU ,fr 0 0 0 0 0 0 0 0 0 0 0 0. 0. 0 0 lo 1 1 1 1 1 S0 OOOOOOOOOOOOOOOOOOOO 0 0 I I I I  o o oI 1 oo\ o: oG o r\ o 8 o\ o,, a ,,, n0 *.q ( m t'0 C9 0 * 0 no I I I I I r0000 1WJ%'.O.NQJHH01111111 0 Ra S 1 5\ c R Ch mo rioo & i i i 0t0 flC000000000000000000oo ua a 0 yC t on Cr O' (o o oo a a a M i 0 O 0% Oh COOI I OO N OO OOOOOOOOOOOmOOHOOHHHIIIII 0 c ; rip o C (co oi c io 0 6c; I I I 8 8 6 r r A o a 0 4M U CY) NC 0 10 (0M 0 0H 0 0A a' 'ma'%a a''OO I I I I I I I I S0000 00000 000 MH 0 ,H 00 1 I I I I I I Ia 0q OOOrillIII o o0 o o0 o EO to o C O 60 i CT O \ 0\ O0\ \ 0\ r a I I 0 C OOrO O g\ 1 g0\ 0\ 0\ 0\ 0\ 0\ 0\ Og O~~~h~~9 MmO I C4. 0 0 _dd dh dddPddddd C iMll40 8 as o\ s\ ooooooo o o oo 01 0 t MM 000000000000000 0 000 0 00 ICcI C C 0 0*1 il COf Ch H< \D %Q L zt 1 :t f tgcI 0\0 CU M % o 00 01 0% 0% o 0 00 0 0% 0 0% 0 0 0 0 1 0 0 C\ o o a'0 ' Ch )000r0000000 000 H 0 0000 111 SR* m\ ilQ MCO aQt \ r 4 0 & MO S\ Os 6 5k r4 at 00 11 1, oooooooooTooooooooooooo ,,, oooo O wb iOrnQQ O (1 b^M3comi1 4000000 \B O IRO O R O h MhOOi OJOO OOMOMM a o o roo o ooooooo oooooor ooooooooooo 0 C\C [O t M"OOO ( 4 OOOO\O O Or l OO 0' l OI O O O OO IO,, O OOoM 60 o o o od C cy66(66Cs6 C;8o 1oo1o ,ooooooooooooooo ooo\B adddddd g m7 o> 666666(o30o 0orro o C; o 00880 6 (3 OfO roD c oo O ai y m k c o 1 cn mm co C mn CM ..MCC) 8 'D "" r mCU 0 0000%0 0%00 000 0 0 o %D m m \ r r h 00 0 0% 0 % o 4 a CU 0\V AC\oC) O Vo\ Om oM&MM ooooo U\ N t U\ q A 9 %D N ri :tcc) H 00 % Lr n t Lr .9 C! C! C! 9. ! I C% r? U\, 1: c ,, 0 0. 0u m0 Lr 0 O cO 0 0 0 0 4 U0O 0%D .! 0! %). l" ,O P,.cO 0)0 4 + u MI Lr\% r ,CO.0 % i CU Lt. U\D t ,O C ,"l. , NACA TM 1315 TABLE 2 TABLE 3 START OF SEPARATION tA AND DISTANCE TRAVELLED SA IN THE CASE OF A CIRCULAR CYLINDER MOVED RECTILINEARLY OUT OF A STATE OF REST n Can On,0(0) 5n,9() Hn,1(0) 0 0.5642 1.1284 0.418 1.607 1 0.3761 2.2568 0.138 0.963 2 0.3009 3.0090 0.072 0.756 3 0.2579 3.6108 0.046 0.632 4 0.2293 4.1266 0.033 0.552 SSA tAt_} ___ R 0 0.351 0.351 1 1.082 0.586 2 1.258 0.663 3 1.300 0.714 4 1.302 0.748 NACA TM 1315 Figure 1. Regarding definition of 5*. NACA TM 1315 0 0.2 0.4 0.6 I0.d6 tA Figure 2. Pressure drag WD, friction drag WR, and total drag W = WD + WR of the circular cylinder for very small times after start of the motion for uniform acceleration out of a state of rest. NACA TM 1315  I o 0.25 05 0.75 .O r n, Figure 3. Course of the functions Cn,0(n) for n = 0, 1, 2, 3, and 4. 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