kA-jy)-t 
 
 NATIONAL ADVISORY COMMITTEE 
 FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1315 
 
 DISPLACEMENT EFFECT OF THE LAMINAR BOUNDARY 
 
 LAYER AND THE PRESSURE DRAG 
 
 By H. Gbrtler 
 
 Translation of " Verdrangungswirkung der iaminaren Grenzschichten 
 und Druckwiderstand." Ingenieur-Archiv, vol. 14, 1944 
 
 Washington 
 October 1951UN,V^RS.1Y OF Pm^^^^^^ 
 
 y ,,.io:.ov..cNCE LIBRARY 
 
 Pw ,;w,^ 117011 
 
 r.A'M!^GViLLE,FL 32611-7011 US/= 
 
?^^^6</<i2^ 377?/y3 
 
 NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS 
 
 TECHNICAL MEMORANDUM 1315 
 
 DISPLACEMENT EFFECT OF THE LAMINAR BOUNDARY 
 LAYER AND THE PRESSURE DRAG* 
 By H. Gbrtler 
 
 1. INTRODUCTION 
 
 The following considerations refer to two-dimensional, 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. 
 
 V/e denote by X and Y the coordinates of a Cartesian right-hand 
 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 right-hand system where x signifies the 
 wall-arc 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. Finally, u and v are assumed to be the velocity 
 components of the boundary-layer 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. 
 
 "Verdrangungswirkung der laminaren Grenzschichten und 
 Druckwider stand. " Ingenieur-Archiv, vol. 1^, 19^^, pp. 286-305. 
 
 ■kche 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. 
 
 2 
 
 In order to have in the defined range of x, y a reversible 
 
 one-to-one 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 ™ 1315 
 
 If the well-known boundary layer omissions are permissible^ the 
 general hydrodynamic equations of motion are reduced to the following 
 boundary-layer equations: 
 
 Su Bu Su 1 ^p ,, S U fT T \ 
 
 — +u — +v — = - i-s.+ v — _ ( 1 . la ) 
 
 St 5x Sy P bx Sy^ 
 
 ^+^=0, (1.1b) 
 
 ox oy 
 
 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 potential-theoretical pressure 
 distribution in first approximation is sufficient for the boundary-layer 
 calculation according to equations (l.l). In this boundary-layer 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,0,t) B UQ(x,t) as 
 
 i — = — ^ + U^ —2-. (1.2 
 
 P Sx at ax 
 
 The boundary condition required in addition to the adherence conditions 
 u = V = at the wall y = 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 ommissions, correct in the limit for indefinitely 
 increasing Reynolds numbers, are only an approximation for the large 
 but still finite Re-values for which laminar flow is plainly still 
 possible. Partic\ilarly the connection between the boundary-layer flow 
 thus calculated and the outer potential flow, assumed to be undisturbed, 
 
 ~T]he 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 = 6(x,t) ("boundary-layer thickness") at 
 which u practically attains the value UQ(x,t) for the accuracy 
 
 requirements in each case. 
 
MCA TM 1315 
 
 does not satisfy continuity. Thus the question arises: What act\ial 
 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 press\ire 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 
 well-known investigation concerning the origin of the boundary layers 
 
 (H. Blasius ), and later extensions'^^ 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 boundary-layer theory approximation sketched in section 1, 
 one must therefore consider the mutual influence of the increasing 
 boundary layer and the outer-flow. It suggests itself to attain such 
 an improvement in first approximation by using at first the displacement 
 effect on the basis of the original boundary-layer calculation, and 
 hence determining a correspondingly corrected course of the outer flow. 
 In a second step one would then have to calculate the boundary-layer 
 flow anew, taking the above new outer flow as a basis. However, it will 
 generally not be possible to carry out this improved boundary-layer 
 calculation, using the new outer pressure distribution, according to the 
 old scheme (equations (l.l) with the boundary conditions formulated 
 there in the text), since the curvature effects, so far neglected in 
 the boundary-layer equations, generally contribute amounts which can no 
 longer be neglected within the scope of such an improved boundary-layer 
 calculation. We shall return to this later on (section h) . 
 
 TBlasius, H.: Grenzschichten in Flussigkeiten mit kleiner Reibung 
 (The Boundary Layers in Fluids with Little Friction) . Dissertation 
 Gottlngen, Zeitschrift fur Mathematik und Physik 56, I908, p. 1. 
 (Available as NACA TM I256.) 
 
 <Boltze, E. : Grenzschichten an Rotationskorpem in Fllissigkeiten 
 mit kleiner Reibung. (Boundary Layers on Bodies of Revolution in Fluids 
 with Little Friction). Diss. Gottingen, I908. 
 
 "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 mjmerical determination of the pressure drag'. This hecomes 
 particularly obvious, if one vis\ializes a body of finite dimension 
 transferred from a state of rest into a state of rectilinear -uniform 
 movement relative to the surrounding fluid of infinite extension. For 
 this terminal state the potential-theoretical pressure distribution, so 
 far, in first approximation, taken as a basis for the boundary-layer 
 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 potential-theoretical 
 pressure gradient, they alone constitute the pressixre 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 bovmdary-layer theo- 
 retical approximation. However, we should like to refer here to a 
 related report by M. SchwabeS where the pressure distribution after 
 completed boundary-layer separation is determined according to an 
 empirical formulation for the example of the circular cylinder set into 
 rectilinear-uniform 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 time-dependent source-sink 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. Pretsch-^ developed an 
 approximative method for theoretical determination of the pressure drag 
 
 '"pressure drag" = the component in direction of the movement of 
 the resiiltant pressure force on the entire body surface, also called 
 (less correctly) "form drag". Pressure drag + friction drag = total 
 drag (presupposing non-existence of free surfaces). 
 
 Sschwabe, M. : Uber Druckermlttlung in der nichtstationaren ebenen 
 Stromung (On the Determination of Pressure in a Nonsteady Two -Dimensional 
 Flow). Diss. Gottingen. Ing.-Archiv 6, 1935^ p. 3^. 
 
 9 
 Pretsch, J.: Zur theoretischen Berechn\ang des Prof ilwider standee 
 
 (Concerning Theoretical Calculation of the Profile Drag). Diss. Gottingen, 
 
 Jahrbuch 1938 der deutschen Luftfahrtforschung, p. I 60. (Available as 
 
 NACA TM 1009, ) 
 
NACA TM 1315 
 
 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 sliglit 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 boundary-layer 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 meas\ire for the displacement 
 effect of the boundary layer on the outer flow. For that purpose the 
 well-known boimdary-layer theoretical length presents itself which is 
 called "displacement thickness" and denoted by 6*. 
 
 The formulation of the boimdary condition for u(x,y,t) for 
 indefinitely increasing y VRs was based on the conception that the 
 potential theoretical velocity distribution for the large (though still 
 finite) values of the Reynolds nimbers 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 S* is likewise based on 
 the conception of this streamline approximation [u(x,y,t) = U_(x,t) 
 
 and thus for reasons of continuity V(x,y,t) = -y^UQ/Sx] . 
 
NACA ™ 1315 
 
 The q.uantity 5-^ is explained as follows: The fluid volume trans- 
 ported at a fixed point x at a fixed time t between the wall y = 
 and the outer edge y = 6(x,t) of the boundary layer in unit time 
 
 equals J u dy. The potential flow about the body would transport, if 
 
 the streamline approximation described above were taken for a basis, the 
 volume U (x,t)5 through the same cross-section in unit time. The 
 
 difference between these two quajitities, that is, the loss in rate of 
 flow per unit time caused by the viscosity effect, leads by virtue of 
 
 5 
 
 to the definition of the length 5* 
 
 6*(x,t) = ^ f (Uq- u)dy. (3.1) 
 
 
 
 [compare figure 1; the hatched areas are equal. Of course, any other 
 length y = yi > 6 may be selected instead of the upper limit 6 of 
 
 the integral in equation (3.I), as long as the velocity U(x,y,t) 
 in ^ y ^ yi is replaced by Uo(x,t)J According to definition, 
 
 6* 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 = 6*(x,t) a potential flow 
 with y = 6* as streamline is assumed to flow. Within ^ y - 5* 
 one assumes water at rest relative to the body. In this model flow the 
 same q\aantity 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 131::^ 
 
 Ag 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 = 6* 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 potential-flow velocity along the dividing line 
 y = 6*(x,t) should vanish every moment. Then y = 6* is a streamline, 
 
 A simple example which we shall investigate more thoroughly later 
 (section 5) will serve to clear up these conditions. VJe 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 
 
 r2\ 
 
 <1> = btlr + — Icos t3 
 
 where r and 13 signify polar coordinates in the flow plane, referred 
 to the center of the circle, r=R + y, -3=tc- x/R, and b denotes 
 a constant acceleration. This potential flow yields the pressure 
 variation on r = R required for the usual boimdary-layer calculation. 
 For very small times (small compared to the time t = t. of the start 
 
 of separation) this calculation results in a displacement thickness 
 b* increasing proportionally to V^- For these very small times after 
 the start of the motion the potential of our improved outer flow 
 therefore reads 
 
 * = bt 
 
 ^ (R + c fvt : 
 
 cos a (t > 0, r > R + cyvt, 
 
 &* = c ^vt, c = const.) 
 
8 ' NACA TM 1315 
 
 The line y = c -/vt is a streamline. V/hile it travels out into the 
 fluid starting from t = 0, the streeimline pattern of the entire outer 
 flow correspondingly varies continuously 10. 
 
 In order to clarify the conception we shall add a few remarks 
 regarding the mass displacement of the boundary layers. The streajnline 
 displacement thickness &* 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 = 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 = (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 6* does not present a measure for it. 
 
 10 '^ 
 
 We shall give here for comparison the velocity potential ^ of 
 
 the corresponding flow about an expanding impermeable circular cylinder. 
 
 If its contour is prescribed by r = R(t), one has 
 
 $ = bt 
 
 R^(t) 
 r + — ;: — - 
 
 „,^vdR(t) , r 
 
 cos -a + R(t) ^^ In - 
 
 dt R 
 
 The additively appearing soiirce yields that additional fluid in the 
 outer space r = R(t) which the impermeable circular cylinder itself 
 displaces while expanding 
 
 (dS\ ^ dR(t) 
 
NACA TM 1315 
 
 Looking for a measure for the mass displacement, one may start from 
 the relation following from equation (3.1) hy differentiation with respect 
 to X and consideration of the continuity equation 
 
 |- (U06*) = v(x,5,t) + 6 ^ (x,t) (3.2) 
 
 (Again any y = y-, > 5 may be substituted for 6) . The right side 
 
 indicates the excess of the v-velocity at the edge of the boundary 
 layer compared to the V-velocity - 5SUq/Bx of the corresponding 
 
 frictionless flow prevailing there. Thus the entire volume displaced 
 by friction effect between two points x^ and x, per unit time is 
 
 at any rate always prescribed by 
 
 r^i 
 
 / (v - V)dx = rUo6*1 - [Uo6*1 
 
 ^ 1- -J X=X-|_ ^ -1 X=Xq 
 
 We now visualize, as in the definition of &*, a model flow. We assume 
 a frictionless potential flow, with the quantity 6^ generally different 
 
 from 5*, to be prevailing for y = &^(x,t); the length S^ is assumed 
 
 to be fixed so that precisely the fluid which actually is displaced by 
 
 the friction effect would flow through = y = 5^ with the 
 
 velocity Uq. Inside = y = 6^ we assume for our model flow water 
 
 at rest relative to the body. Then there follows from equation (3.2) 
 
 |;("o5.;=^H^*). (3.3) 
 
 Hence there results by integration for the "mass displacement 
 thickness" 5^ the statement 
 
 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, 
 
10 
 
 NACA TM 1315 
 
 and therewith of the pressure field correction for large Re-values, 
 in such a manner that one would make not y = 5* but y = 5^ for all 
 
 times the streamline of the substitute flow, the result would remain 
 unchanged, according to equation (3.^). Both models differ at any 
 moment only in the numbering of the streamlines. 
 
 As to the relation between 6* 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 = y = y-, with y^ = & followed up 
 upstream finally blends completely in a region where the flow is 
 frictionless (case of approach flow), 6* = 5^. If, however, this 
 
 presupposition is not fulfilled, as for instance in the examples with 
 vanishing S^ 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 boiondary layer material 
 from the tody with subsequent readherence takes placed upstream 
 (case of longitudinal flow.) The value of S^ 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 = x^j for eqixation (3.^) then yields only the 
 difference statement 
 
 Uq^* 
 
 x 
 
 Uq6* 
 
 JO 
 
 "TEt is always presupposed that the flow in the x-interval 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. 
 
 -^^It must be noted, though, that cases exist where it is impossible 
 on principle to give such a value. Let us visualize for instance the 
 bo\andary-layer 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 
 
 rx 
 
 v(x,yi,t) + y^ — !=!■ (x,t) 
 ox 
 
 dx 
 
 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*. 
 
 k. 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 boundary-layer calculation according 
 to the equations (l.l) 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 bo\indary-layer calculation it would thus be necessary 
 to take corresponding further terms into consideration in the equations 
 of motion as well. 
 
 The general Navier-Stokes equations of motion and the continuity 
 equation read in ovr curvilinear coordinates x,y in full strictness -^ 
 
 3u 
 
 R Bu Su uv 
 u — + V — + 
 
 dt R + y bx 
 
 ^ R + y 
 
 R dp 
 
 p R + y Sx 
 
 Su 
 
 + V 
 
 R 
 
 "u 
 
 1^2. 
 
 idr 
 
 + 
 
 6^u 
 
 R + y Sy (R + y)2 Sx^ 
 
 + 
 
 2R Sv k 
 ^ '''^ 
 
 (R + yr (R + y) S 
 
 R 
 
 dR 
 
 (R + y)3 dx 
 
 V + 
 
 Ry dR du 
 
 (R + y)3 dx Sx 
 
 (l^.la) 
 
 13 
 
 Compare, for instance: Tollmien, W.: Grenzschichttheorie 
 
 (Boundary-Layer Theory). Handbuch der Experimentalphysik (Manual of 
 
 Experimental Physics) Vol. k, Part I, p. 2l+8, Leipzig 1931. 
 
12 
 
 NACA TM 1315 
 
 u 
 
 1 Sp 
 
 dt R + y Sx by R + y P by 
 
 bv B. bv bv 
 
 + u — + V 
 
 + "v 
 
 2R Su 
 
 Sfv 
 
 Sy^ (R + y)^ ^x 
 
 _i Sv R^ 
 
 R + y By (R + y)^ Sx^ 
 
 ^ +— £ .^u + 
 
 (R + y)^ (R + y)3 dx 
 
 (i+.2a) 
 
 Ry dR ^ 
 
 (R + y)-^ dx Sx_ 
 
 R Su Sv V 
 
 ^- + T— + 
 
 R + y dx dy R + y 
 
 = 0. 
 
 (i^.3a) 
 
 Therein R(x) denotes the radius of curvature of the wall contour y = 
 (contrary to the mathematical definition positive on walls convex with 
 respect to the flow). 
 
 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 boundary-layer equations (l.l)l^. If one considers in 
 improved approximation also the terms of the order 0(&/L) compared 
 with 0(1), there results 
 
 Su c)u Su 1 c3p 
 — +u — +v — + — - V 
 
 at 
 
 ax 
 
 By p hx 
 
 3fu 
 
 _ y Su uv y Sp V Su 
 R dx R pR Sx R Sy 
 
 (^.1) 
 
 1 Sp ^ uf 
 
 P Sy R ' 
 
 (i^.2) 
 
 1^, 
 
 Compare, for instance, V/. Tollmien, footnote I3. 
 
KACA TM 1315 
 
 13 
 
 Su I ^v _ y 5u V 
 Sx hy R hx R* 
 
 (J+.3) 
 
 On the left side are the terms from equation (l.l)^ on the right the 
 newly added terms. 
 
 The correction of the outer pressure we obtained above is, in 
 general, of this same order 0(&/L). Thus, if one wants to perform with 
 this improved outer flow an improved boundary-layer 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 y-direction through the boundary layer. 
 Equation (^.2) serves this purpose: It yields Sp/Sy sufficiently 
 accurately for this approximation, if we substitute in it on the right 
 side u from the first boundary-layer approximation. 
 
 If y = &(x,t) denotes the "outer edge" of the boundary layer, 
 and p[x,6(x,t),"ir] is the pressure distribution of the improved outer 
 
 potential flow along this line, one has for 
 approximation 
 
 = y = 6 in this 
 
 p(x,y,t) = p(x,&,t; 
 
 rW 
 
 u dy. 
 
 (h.k) 
 
 The pressure drag of the unit length of the cylinder in the flow becomes 
 
 W- 
 
 D 
 
 K 
 
 K 
 
 p(x,0,t) cos cp dx 
 
 p(x,S,t) - _P / u^dy 
 R(x) Jq 
 
 cos 9 dx, 
 
 (^^.5) 
 
Ik NACA TM 1315 
 
 Wherein cp 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. -^ 
 
 -'Additional 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 
 boundary-layer zone according to equation (k.h). 
 
 k. Again calculation of the velocity components u and v of the 
 boundary layer in second approximation from equations (^.1) and (^.3)^ 
 using th'? 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 15 
 
 5. example!^ 
 
 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 ciirrounding infinitely extended fluid, onward 
 from t = 0. The relative velocity of the tindisturbed approach flow 
 with respect to the body visualized as. being at rest is assumed to 
 
 ^he 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 Navier-Stokes equations is known, and 
 the flow near the wall calculated on the basis of the boundary-layer 
 theory is kno'^m to agree with the exact solution. If one replaces the 
 potential-theoretical 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 > & 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 boxmdary-layer 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 
 boundary-layer equations (l.l) therefore are valid except for terms of 
 the order 0(6^/R^) compared with 1) is the case of longitudinal flow 
 over a plate. According to Blasius, elsewhere, 6* = 1.73 yvx/Uoo 
 (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) = ^ 
 
 1 + 1.35 He(x) 
 
 (Re(x) = Uoox/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 . 
 
16 
 
 NACA ™ 1315 
 
 be U = C t^(c = const., n = 0, t = o). Then the potential 
 
 velocity U^(x,t) along the body boundaries can be represented in the 
 
 form 
 
 Un(x,t) = 
 
 for t = , 
 (l(x)t'^ for t ^ 
 
 (5.1) 
 
 We assume \^(x,y,t) to be the stream function of the boundary-layer 
 flow defined by u = b^/hj, v = - S\|f/Sx 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 (l.l), thus in the custofliary first approximation for 
 very large Re-values. Generalizing the series developments set up 
 by Blasius ' for the special cases n = (sudden transition from state 
 of rest to motion at constant velocity) and n = 1 (uniform acceleration) 
 one obtains the result 
 
 ^tr(x,y,t) = 2 fvt [^t^^j,^o(T) + <l^'t^^^^n,l(^) + •••]^ 
 
 (5.2) 
 
 thus a series development in powers of t (that is, in powers of 
 
 the distance covered by the body). The appearing coefficient functions 
 
 i Wi ^ (ti)j ••. with 7] = y/2 -Jvt are universal functions of T[. 
 n,0 n,l 
 
 More details may be found in the appendix to this report. 
 From equation (5*2) follows 
 
 .ni, , , , 2n+l^ , , 
 U = qt C' + qq't ^' + . 
 n,0 "-f-^ 
 
 V = -2 A/^t[q't^:^^Q + (q'2 + ^^''^^''^^l + ..]. 
 
 } (5.3) 
 
 17, 
 
 Compare footnote k on page 3. 
 
NACA TM 1315 
 
 17 
 
 The boundary conditions are satisfied by virtue of 
 
 ^n,o(°)=^n,l(°) = --=°^ 
 
 ^ ' ^(0) = r ,(0) = 
 
 n,0 n,l 
 
 = 0, 
 
 (5.^) 
 
 ^'n,0(= 
 
 ^n,l(= 
 
 0, 
 
 J 
 
 where there is always, here and later on 
 
 n,K dT) n^K 
 
 For the streamline displacement thickness 5* there results 
 
 &*(x,t) = 2 \fvi\cL^ - q'(x)t^+l^^^^(c«) - ..^ (5.5) 
 
 with 
 
 a = lim 
 n 
 
 T] > 
 
 T - ^- M 
 
 n,0 
 
 (5.6) 
 
 (a few numerical values <x^ are given in the appendix.) 
 
 V/e consider as an example the flow about a ■:;ircular cylinder, 
 the mode of notation of section i, the velocity potential of tiie 
 frictionler.c flow about the circular cylinder ic given by 
 
 In 
 
 Oq = U(t)(r + — ) cos i3 
 
 (5.7) 
 
13 
 
 NACA TM 1315 
 
 Hence the velocity at the periphery of the cylinder is calculated as 
 
 U (x,t) = q(x)t^ = 2U(t) sin i3. 
 
 (5.8) 
 
 and the resulting pressure distribution at the body surface is 
 
 p^(x^t) = p^ _ p 
 
 2R(cos {) + 1) ^ + 2U^(t) sin^d , 
 _ dt -I 
 
 (5.9) 
 
 where Pj('t) is the pressure at the forward stagnation point. The 
 
 potential-theoretical pressure drag of the length L of the cylinder 
 at rest is therefore 
 
 W 
 
 Dr 
 
 = - L 
 
 2it 
 
 ^ 
 
 Pq cos t3 Rdi3 
 
 2prf^ L — 
 dt 
 
 (5.10) 
 
 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 Vvt 
 
 °n + 2 ^ ^ 
 
 n,l 
 
 (c»)t 
 
 n+1 
 
 cos V + 
 
 (5.11) 
 
 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 = 6*(x,t) practically represents a 
 circle; that is to say, a circle with increasing radius r, the center 
 of which, for t>0, 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 3 + a 
 
 (1 - g sin^ 3)1 
 
NACA TM 1315 
 
 19 
 
 (a = radius, b = displacement of the center), and if b « a 
 
 r = a + b cos -3 
 
 ; of#i 
 
 We identify 
 
 L(t) =: R + 2an ^/vt^ 
 
 Mt) = it^ ^ (oo)Cot--^\ 
 
 R n,l ^ 
 
 (5.12) 
 
 If, temporarily, Y,"^ 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 + 6*, that is, r = a(t), is given by the potential 
 
 ^1 = (" - i)(^ * i) 
 
 2\_ 
 
 r cos t3 
 
 thus the potential flow here required in the polar coordinate system r,T3 
 fixed relative to the body by 
 
 $T = Ufl + 
 
 ? 2 
 
 r - 2br cos fl + b , 
 
 (r cos i3 - b) (5.13) 
 
 At first we introduce dimensionless quantities as follows: V/e 
 choose R as the characteristic length, and the time T the body requires 
 to cover the distance R, starting from the beginning of motion t = 0, 
 as the characteristic time, thus 
 
 T = 
 
 (n + 1) 
 
 _R_ 
 
 Co 
 
 _ 1 
 
 n+1 
 
 (5.li|) 
 
20 
 
 NACA TM 1315 
 
 Hence results the characteristic velocity 
 
 R _ ( ^0^ 
 
 1 
 
 n\ n+ 
 
 T \ n+1, 
 
 and the Reynolds number Re formed with this velocity and the length 
 R become slo 
 
 R^ /R\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 R /T^ and the 
 pressiore by dividing by pR2/t2 (we denote the dimensionless quantities 
 by adding a wavy line), we obtain 
 
 1 1 
 a = 1 + 2a^ Re 2'^ 2^ 
 
 b = 4(n + 1)^ (00) Re ^t ^j 
 n,l 
 
 (5.12a) 
 
 (n + Dt^l + 
 
 r^ + 2 b 
 
 '■Nrf z-^' 
 
 r cos X + b' 
 
 (? cos X + b) . 
 
 (5.13a) 
 
 ■'■°If one forms the Reynolds number Re^ with the length VvT 
 one obtains 
 
 Rej_ = (r/t) ffjr/v = Re^ 
 
NACA TM 1315 
 
 21 
 
 Furthermore, 
 
 y = = I Re^t"2 
 
 2 //vt 
 
 and 
 
 6* 
 R 
 
 ~T 
 
 Re 
 
 ^u. 
 
 2a - 4(n + l)f T (°°)t cos x 
 n ^ ^n,l 
 
 (5.11a) 
 
 We consider first the case n = 1 of uniform acceleration from a 
 state of rest. Following, F^ without more precise data will represent 
 
 ,2/m2, 
 
 those additive portions of the pressure p, = p, /(pR /T ) which do not 
 
 make any contributions to the pressure and are therefore not of interest 
 to us. Furthermore, terms of the order 0(62/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/r 
 
 ^ = ll (1 ^ 3a^Re ^^) cos x + P^ + ol^] 
 
 & - € 
 
 - U- r-^ 1 = 16 ^ t^ sin^ + Po + 
 
 2\y ax R 2 
 
 Thus one obtains 
 
 Pi 
 
 -. |. t) = ^(1 + 
 
 1 1\ 
 3a^Re"^^y co 
 
 s x + 16 ^ t^sln^ 
 R 
 
 (5.16) 
 
22 
 
 MCA TM 1315 
 
 According to equation (^.^) 
 
 p^(x, 0, ^) = p^(Sc, |, ^) 
 
 6/R 
 
 u^dy. 
 
 (i+.ita) 
 
 In the small times of Interest to us there is 
 
 u = U ^ sin x^ ' {t\) - 16 ^ sin x cos x^ ' (t)). 
 
 1,0 
 
 1.1 
 
 therefore 
 
 I u dy = - 16 t^sin^x / C r,^ + 
 
 ^0 ^ 
 
 -4o^^2~ 
 
 6/R 
 
 128 t^sin'^x cos x / r r dy _ 
 ^0 1,0 1,1 
 
 r6/R 
 
 256 t°sin^ cos^x / ^' ^dy. 
 Jo 1,1 
 
 ^(5.17) 
 
 Because of lim ^ (-q) = 1, lim ^ (ti) = the two last integrals 
 Ti-> 00 1,0 T)-> 00 1.1 
 
 on the right are. If S/R is chosen so large that these asymptotic 
 values are attained with the desirahle accuracy, numbers independent of 
 the variation of the boundary-layer edge 6(x,t) with x. The first 
 integral at the right, together with the corresponding second term at 
 the right in equation (5.I6), may also be combined into an expression 
 independent of 5(x,'£), namely the expression 
 
 16 
 
 t^sin^^ / (1 - ^1 Q^)df. 
 
NACA TM 1315 23 
 
 Thus the variation of 5(x,^) 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 ^t^di]. 
 
 all together one obtains, therefore. 
 
 ( 2~ 2 ) 
 
 \1 + Sa^Re t Jc 
 
 p(x, 0, *?) = Uyl + So-jRe "t /cos x 
 
 1 9 
 
 — p 00 
 
 + 256 sin% cos X Re'^t^ / tj^ qI;^ ^dr] + P]^ + 0(Re-^). 
 
 
 
 (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 
 Re -values results 
 
 1 a 
 
 2prf^^LCQ 
 
 1 + -^Re 2(3a +16-^/ C' ^ .drj). (5.19a) 
 1 Oq i^U 1,X 
 
 2pj^ LC„ = W_ is the potential-theoretical pressure drag. (Compare 
 U Uq 
 
 equation (5. 10).) According to Blasius 'calc\ilations 3C'± = 2 /•Jt(=1.128) , 
 
 furthermore, according to the author's numerical evaluation 
 
 / C q dTi = 0.09804 
 ' ' 
 
2k 
 
 NACA TO 1315 
 
 Thus the result in dimencional form is 
 
 ^•'d = ■■^D. 
 
 1 + J^(-^^+ 0.392% t^ 
 
 ^ w^ 
 
 R 
 
 (5.19) 
 
 As friction drag 
 
 '2n 
 
 W = pv / — (x, 0, t) sin - dx 
 
 R 
 
 JO Sy 
 
 there results from Blasius' calculation results 
 
 19 
 
 u _ V Vvt f 2 n n?Q ^ tM 
 
 (5.20) 
 
 For sufficiently small times t(t « l) thus V/ = W - W . that 
 
 is, the friction drag increases immediately after start of the motion 
 according to the same law as the contribution of the skin friction to 
 the pressijre drag. 
 
 The total drag becomes 
 
 '' - '''b ^ \ = V 
 
 1 + 
 
 Vvt h 
 
 Co .h\ 
 
 4x-. 0,363 ^t 
 
 vV^ R^ 
 
 (5.21) 
 
 In figure 2 we represented these results using the dimensionless 
 "t = t/T (with T = \f2R/CQ for the present case n = 1), thus the 
 relations 
 
 J!]L=i+ ^fe ^(1.128 + l.569?S, 
 
 (5.19-l) 
 
 ^n order to obtain the term with t , one must here include in 
 equation (5.2) also the third term of the series. The necessary data 
 may be found in Blasius' report, footnote h. 
 
NACA TM 1315 
 
 25 
 
 ^= J!pt2(l.l28.0.1l6t^), 
 
 Br 
 
 (5.20 j^) 
 
 
 (5.211) 
 
 Our formiLLas caxi be applied with good approxlniation 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 developement (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 setting- in of the separation. 
 
 The value t = tA( = Vo • 586 ) = 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 > one obtains correspondingly as 
 pressure drag with n(t) = CQt^ and Wj^ = 2prf^2L du/dt 
 
 
 = 1 + 
 
 Vvt 
 
 R 
 
 i^^i) 
 
 °n 
 
 he. 
 
 nR 
 
 2n+2 
 
 n,0 
 
 n,l 
 
 dT] 
 
 (5.22) 
 
 .2n+2 
 
 For the calculation of the coefficient of t'^^'^'^ sufficient numerical 
 
 data concerning f ' _(ti) are lacking so far. For this reason we limit 
 =n,l 
 
 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 
 
 Wt, = Wj) - W- 
 
 "Br 
 
 obtained above for n = 1, for times Immediately after 
 
 R JJ UQ 
 
 start of the motion, is valid also for arbitrary n > 0. The friction 
 drag has for these ptmall times the value 
 
 "r = "d. 
 
 R 2n 
 
 (5.23) 
 
26 
 
 NACA TM 1315 
 
 In the appendix the exact expressions for f (t]) are derived. 
 
 n,0 
 
 However, the relation 
 
 2(2n + Dan 
 
 ^;o(°) 
 
 ,2n (n.')2 
 
 (2n); ^ 
 
 (5.2i^) 
 
 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 k) . Thus, for very small times after 
 start of the motion, there applies indeed 
 
 % = Wc - Wp 
 
 VIZ : 
 
 = Wt 
 
 Vvt ,2n n.'(n - 1) .' 
 (2n): ^ 
 
 ^0 R 
 
 > 
 
 (n > 0) 
 
 (5.25) 
 
 (One has 2^l{n - l).'/(2n)J \/it = 1.128^ for n = 1; 0.7522 for n = 2; 
 0.6018 for n = 3; 0.5158 for n = k, etc.) 
 
 One may now consider more general laws of motion of the form 
 
 U(t) = 
 
 10 for t = 
 
 f(t) for t = 0(f(0) = 0) 
 
 and carry out corresponding calculations. If one presupposes that the 
 function f(t) defined in t :^ can be developed into a Taylor series 
 
 < 
 
 after 
 
 around t = which converges for the small times = t « t^ 
 
 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 = CQt^ 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 
 
 27 
 
 that for all these laws of motion the portion of the pressure dra^ 
 
 caused by friction W 
 
 D 
 
 w. 
 
 D. 
 
 '0 
 
 increases immediately after the start of 
 
 This is 
 
 motion according to the same law (5.25) as the friction drag 
 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 
 
 one oh tains 
 
 U = Cq = const, for 
 
 t a-o 
 
 Wj^ = 2pnRLCo 
 
 n 
 
 °D 
 
 r2 
 
 i' ^' dT) 
 
 ^0,0 0,1 
 
 (5.26) 
 
 and 
 
 W^ = 2prf?LC 
 
 ii 
 
 ^0,0 
 
 (0) 
 
 
 §t2 
 
 (5.27) 
 
 The function 
 
 f-0,2a(^) 
 
 is explained at the end of the appendix. We 
 
 did not numerically determine the coefficients at t^ in eq\mtions (5.26) 
 
 and (5.27). Because of Oq 
 
 above 
 
 W = W 
 R D 
 
 W 
 
 i ^^'^(0)(= 1/ /{^= 0.56i+2) the law stated 
 
 D„ 
 
 is valid also in this limiting case n = of 
 
 the sudden start of motion, for times immediately after the start of 
 
 motion. Here in particular W 
 
 '0 
 
 (d'Alembert ' s paradox). 
 
28 
 
 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 ^^ ±i'^) 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 wo\ild have 
 disrupted the connection there, and second, because the da to, and tables 
 attained are of interest in their own right. 
 
 As assumed in section 5^ let a velocity proportional to t (n = O) 
 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) = Uo(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 = and n = 1, the expression 
 
 with 
 
 >tr(x,Ti,t) = 2/Vt XI tn+^(n+l)x^ ^ (x,Ti) 
 X=0 
 
 n,X 
 
 (1) 
 
 Ti = y/2 ffjt 
 
 for the stream function ^ of the boundary-layer flow is obtained and 
 one obtains for the functions X -y by substituting equation (l) 
 
 into the boimdary layer differential equation (l.l) a system of 
 differential equations solved by recursion; we limit ourselves here to 
 the two first equations of this system which read 
 
 b3x 
 
 n,0 
 
 ^' 
 
 a3x 
 
 bn' 
 
 + 2T) 
 
 kn — = - 4nq(x), 
 
 4-+ 2ti 
 
 S^X 
 
 h,l 
 
 ^ 
 
 S^X 
 
 W 
 
 bx T /ax .... ^ 
 
 , , , n,l , f n.O n.O 
 i|(2n + 1) — = k I '- '— 
 
 K,0 ^\,0 
 
 M2) 
 
 bx ^^' 
 
 - qq' . 
 
NACA TM 1315 
 
 29 
 
 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 
 
 n,0 n,0 
 
 \,1 = q(x)ci'(x)^n,l(n) 
 
 (3) 
 
 one obtains from equation (2) the ordinary differential equations 
 
 P'> + 2ti^" 
 n,0 
 
 n^O 
 
 i^n^' 
 n,0 
 
 hn. 
 
 (^) 
 
 ^;.. . 2,^;.^ - M2n . l)r^^ = 4(^;/ - ^oCo- 1)^ 
 
 (5) 
 
 The boundary conditions to be satisfied by f ^ and t ^ are 
 •' '' ^n,0 ^n,l 
 
 formulated in equation (5-^) 
 
30 NACA TM 1315 
 
 In the case of a plane wall moving in its own plane, 
 
 ■'I'' = 2 Yvt q't'^^- 0^^) represents the complete solution (because 
 
 of q. = const. )> not only in the boundary-layer theory approximation 
 set up here, but in strict fulfillment of the complete Navier-Stokes 
 
 ' 20 
 equations. The calculation of t, may take place as follows. The 
 
 n,0 
 
 temporary eplitting-off 
 
 i' M = 1 - e-^\(Ti) (6) 
 
 n,0 ^^ 
 
 transforms equation (h) into 
 
 ^n - ^^n - 2(2n + l)qp^ = (?) 
 
 Because of U_ = U (t), u = u(y,t) and v = 0, the Navier- 
 Stokes equations are for these motions simplified to 
 
 Su S u BU^ 
 = V + '-'■ 
 
 ht by^ St 
 
 The analogy between U^ - 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 
 Navier-Stokes equations. For the rest, one can see for the present 
 problem that the first term qt^' of the series development 
 
 following from equation (l) for u as a solution of the above equation 
 approximately satisfies the boundary-layer 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 = and n = 1. 
 
MCA TM 1315 
 
 31 
 
 For every integral 2n ^ the general solution of this differential 
 
 eqviatlon is 
 
 '^ - ii;^ 
 
 dri; 
 
 "(' 
 
 e'' Ct + C 
 
 ^Tl 2 
 
 1 ^ ^2 
 
 -A) =( 
 
 
 (8) 
 
 (C,^ Cp are integration constants). As is veil known, the Hermite 
 polynomials Hjjj(x) are given by 
 
 ■>»(") = A- Irf^-^' = „5Z^(- x)«(/J (Mi (.x: 
 
 m-2ic 
 
 m' 
 
 (9) 
 
 (in the form originally given "by Hermite). For further use we also put 
 Hjj^(ix) = l"^(x), thus 
 
 ij.) - e-=^(|,)V - I- (,:)li^(.,--, (.0, 
 
 0<K^ 
 
 furthermore 
 
 Yrr 
 
 e"-^ dx. 
 
 •* dx '^ 
 
 (11) 
 
 nTompare, for instance, E. Kamke "Differentialgleichungen: 
 Losungsmethoden und Lo'sungen, I Gewohnliche Differentialgleichungen" 
 (Differential eqiiations: Methods of solutions and solutions, I. 
 ordinary differential eauations) Leipzig 19^4-2, Part C, No. 2,lil, and 
 put there x = i ^5 t]. 
 
32 NACA TM 1315 
 
 (so that in particular 1 + $ = ft represents the error integral^ (j-, 
 
 the error function). By using these expressions, a simple recalculation 
 from equation (8) gives the general solution of equation (h) as 
 
 ^' (ti) = 1 - H (Ti)rc + 0*0 (ti; 
 n,0 2n |_1 2 
 
 2n (^' 
 
 2 
 
 '1^'^ ^ATY- ^^'^2n-J^\-l^^^ 
 
 with Cg* = -^C^. 
 
 C]^ = because of ^ '(<»)= 1 and Cg* = -n.'/(2n): because 
 
 of ^' (0) = and H2n(0) = (2n).'/n.'. [The polynomial sum in 
 
 equation (12) disappears for i] = and integral n term by term, 
 since either 2n - k or k - 1 is an odd number]. Thus the desired 
 
 particular integral of (4) reads 
 
 i' Jr\) = 1 + ""' 
 
 n,0 (2n).' 
 
 2n 
 
 K =1 ^ >^ ' 
 
 (13) 
 
 Because of ^ S^) = (- 1)*^" <I'-|(ti)H -.{r\) the solution may also be 
 
 written in the following form which is more elegant than equation (13) 
 but less serviceable for practical calculation 
 
 The direct and elementary derivation of the solution for 
 integral n r" 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 
 
 33 
 
 According to equation (13) one has in particular 
 
 ^0 0^"^^ " ^ ^ '^0^'^^ " ^^^^' 
 
 1,0 
 
 ^; Jti) = 1 + (2ti2 + 1)*q(ti) + t)$^(ti), 
 
 1 + - [(^Ti^ + 12ti2 + 3)0q(ti) + (2ti2 + ^)^<!>^{^)^., 
 1 + ^ [Bti^ + 60ti^ + 90ti2 + 15)$o(ti) + 
 (l|Ti^+ 28ti2+ 33)ti$^(ti)J , 
 
 16118+ 22I+T16+ 8l40ii^+ 8iK)Ti2+ 105) ^(ii) + 
 
 ^2,0(^ 
 
 4,0^^) 
 
 ^i,o(^) 
 
 1 + 
 
 105 
 
 (81)6+ 108ti'^+ 3T0ti2+ 279)t]*-l(ti)]. 
 
 (13b) 
 
 Thence one obtains by elementary quadratures ^ n^^^' likewise 
 
 expressed by $^ and with polynomial coefficients. The numerical 
 
 evaluation is reproduced in table 1 ■^. Because of the special impor- 
 tance of the solutions ^' (ti) as boundary-layer profiles u/qt^ 
 
 n,0 
 in the case of the plane wall (compare above) we represented them in 
 
 figure 3. [owing to 5* = 2^^ lim (t) - ^ Ar\)) = 2a^ Jvt the stream- 
 
 line displacement thickness &* can easily be taken from the numerical 
 cal table 1; compare -also table 2.j 
 
 Since it follows directly from equation (7), by single differ- 
 entiation, that the general integral cp , is the first derivative of 
 
 the general integral 9^ with respect to t], it is, with the 
 
 boundary conditions satisfied, easy to find as expressions for the basic 
 
 functions ^ {t\) with the integral n 
 
 
 n^O 
 
 ^n;0 - ^(^ - ^A,o)} 
 
 (iM 
 
 23 
 
 'All nimierical calculations were performed by Miss Ursula Ludewig. 
 
3^ 
 
 NACA TM 1315 
 
 Therein 
 
 (2n).' a/tt 
 
 n^O 
 
 (15) 
 
 In order to calculate the fundamental functions of the first order 
 t, -,(11)^ the total course of ^ n^^^ must be known (according to 
 
 equation (5)). Only the additional knowledge of t" (O) is required 
 
 n,l 
 
 for the problem which is of foremost interest, the question regarding 
 
 location and time of a possible separation of the boxmdary layer in 
 
 first approximation. 
 
 For because of 
 
 ^ ) = t^ 
 
 ^^y/y=o 2^/vt 
 
 ir (0) + qq't'^+^r (0) + . . ri 
 
 _ n,0 n,l J 
 
 one obtains in first approximation the connection 
 
 1 
 
 - C.o(°' 
 
 .-in+1 
 
 q'(x)t" (0) 
 ^n,l 
 
 (16) 
 
 for location x and time t of the separation. On the other hand, 
 
 we are interested in f (00) (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 
 
 well-known method as follows: ^(t^) is assumed to be a function of -t] 
 
 in = T] = 00 provided with the continuity properties required for the 
 
 following calculation. By partial integration one obtains the following 
 
 relations. If L and M are the differential operators 
 
 n 
 
 n 
 
 n 
 
 M = 
 n 
 
 dii"^ dT] 
 
 - l|(2n + 1), 
 
 d^ o„ ti 
 dTi^ dT] 
 
 - i'^ ^ i} 
 
NACA TM 1315 35 
 
 one has 
 
 /J^(^)^n[^A,l]^^ = p;,l - ^'^A,l - ^^^A,l]^ 'fo\l ^^'^'^' 
 
 (IT) 
 
 ^(t])L R' "]dT) = 1^^" - ^'?' + 2ti^^' + i M„M1" - 
 n|^n,lj 1^ n,l n,l n,l n,l Jq 
 
 (18) 
 
 We choose, therefore, -9 = t3 {r\) so that 
 
 a,0 
 
 M 
 n 
 
 with (17a) 
 
 ,3 (0) = - 1, 73 (00) = 0' 
 n,0 n,0 
 
 If then equation (5) and the boundary conditions valid for t (t^) 
 
 n,l 
 
 are taken Into consideration the result Is 
 
 On the other hand we choose ^ = i3„ (ti) so that 
 
 n^ CO* ' ' 
 
 S (0) .0, i, (J finite 
 
36 NACA TM 1315 
 
 Then one obtains, in analogy 
 
 ^n,l Jo n,co ^^n_,0 " ^n,0^n,0 " ^ 
 
 (18b) 
 dT). 
 
 It is easily confirmed that 
 
 furthermore by comparison with equation (7) 
 
 \0^^^ = ^' P2n.l,0(^) - ^]^ (^) 
 
 Therewith the desired functions i3 ^ and 3 are traced back, to 
 
 n^O n, 00 
 
 the known basic functions of zero order ^,-. n ^ 
 
 2ii+l,0. 
 
 Numerical evaluation yielded for n = 0, 1, 2, 3, and k all 
 together the data here of interest given in table 2. It also shows the 
 numerical values of a^. For the n-values 1, 2, 3, and k one finds 
 the law (5.2^) confirmed. A general proof of this law may be produced 
 with a few calculations on the basis of the known expressions 
 for ^j^ 0^^^' ^ much simpler proof of the relation (5.24) will be 
 presented below. As mentioned above, the expression 
 
 u = <lt-?;^^(ri) 
 
 is the strict solution for the boundary- layer profile on a plane wall 
 with U = qt^ (q = const.) outside of the boimdary layer. The wall 
 shearing stress according to the momentum theorem of the boundary 
 layers is generally 
 
 ^0 = U f r Pudy - ± r pu^dy r' p Su ^ _ ^ Sp ^^1) 
 ^x ->0 Sx ^0 ^0 at Bx 
 
NACA TM 1315 37 
 
 thus for the above flow^ due to the velocity distribution u(Ti,t) being 
 
 independent of x as well as due to -i- = _ p -— and with 
 
 ox dt 
 
 "^ - ,^-^o[^ - ^n,0^^] 
 
 T = p :^ / (U - u)dy = p A(U6*) = pq(2n + l)a Jv t~^. (22) 
 
 ° at Jo at n» . 
 
 On the other hand 
 
 1 
 au(0,t) ^„ , , a/V ^"" ? , > 
 
 ^ Sy n,0 2 
 
 and thus in combination with equation (15)^ as asserted, 
 
 22n+l( ,n2 
 
 In the case of the flow about a circular cylinder moved rectilinear ly 
 out of a state of rest, investigated in section 5 [compare equation (5.8)n 
 one has q(x) = 2Cq sin x/R and therefore -Q.'(x)iiiax " ^Cq/R 
 for x/R = n. Thus the separation starts according to equation (I6) 
 at the rearward stagnation point at the time 
 
 •A = p;;,o(°>/^o?;i«'^ 
 
 i_ 
 
 in+l 
 
 (compare table 3). The distance covered by the cylinder during that 
 time is 
 
 ^A 
 
 C-t,W(n + 1) = R^" (0)/2(n + 1)^" (O) 
 ^ ^ ' n,0 n,l 
 
38 NACA TM 1315 
 
 Finally we want to give a few indications where to find further 
 data regarding the "basic fimctions of the plane nonsteady boundary-layer 
 flow. The series development (5.2) written down up to the third term 
 reads 
 
 f-2^ ,t-[^^„ . t-1..?^^, . t''---^\,'\,, . q,"?„,2,) . ...]. 
 
 Blasius gives, in addition to the ftmctions ^ o^^^ ^^^ ^1 n^^^ 
 
 calculated above, the rigorous solutions t (t\) and C (ti). 
 } & ^0,1 ' "1,1 ' 
 
 Beyond that, he calculates the numerical values ^'' p (o) and ^'' py^(o) 
 
 which are of interest for the determination of the separation. 
 
 S. Goldstein and L. Rosenhead^^ give the exact expressions for ^q 2a(^) 
 
 and L, p, (t)). These integrals were, by the way, n\imerically determined 
 
 before by Boltze^^ on the occasion of treatment of the corresponding 
 problem n = of rotationally symmetrical flows which seems to have 
 escaped the attention of the authors. 
 
 Translated by Mary L. Mahler 
 National Advisory Committee 
 for Aeronautics 
 
 Compare footnote 6 of this report. 
 
 2"^ 
 ''Compare footnote 5 of this report. 
 
NACA TM 1315 
 
 39 
 
 
 
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ko 
 
 NACA TM 1315 
 
 TABLE 2 
 
 n 
 
 °n 
 
 ^;;,o(o) 
 
 ^n,l(") 
 
 ^n,l(0) 
 
 
 
 0.5642 
 
 1.128i| 
 
 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 
 
 k 
 
 0.2293 
 
 4.1266 
 
 0.033 
 
 0.552 
 
 TABLE 3 
 START OF SEPARATION t^^ AND DISTANCE TRAVELLED S;^ IN THE 
 CASE OF A CIRCULAR CYLINDER MOVED RECTILINEARLY 
 OUT OF A STATE OF REST 
 
 n 
 
 1 
 (c l^^+l 
 
 a[rJ 
 
 ^A 
 R 
 
 
 1 
 2 
 
 3 
 4 
 
 0.351 
 1.082 
 1.258 
 1.300 
 1.302 
 
 0.351 
 0.586 
 
 0.663 
 0.714 
 0.748 
 
MCA TM 1315 
 
 1+1 
 
 ^^^^^^^^^^^^^ 
 
 Figure 1.- Regarding definition of b"* 
 
I^2 
 
 NACA TM 1315 
 
 0.2 
 
 0.4 
 
 
 
 
 
 («j^ 
 
 1 (^D 
 
 
 y^ 
 
 
 ^ 
 
 Reh 
 
 1^ 
 
 
 ^D^ 
 
 1— 
 
 0.6 
 
 \0.6t 
 
 tA 
 
 Figure 2.- Pressure drag Wj-,, friction drag Wj^, and total drag 
 
 W = W-p) + Wp 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 
 
 h3 
 
 1.5 
 
 1.25 
 
 1.0 
 
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 0.5 
 
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 ^ 
 
 ^ 
 
 r y^ 
 
 ^ 
 
 
 0.25 0.5 0.75 1.0 Tn.O 
 
 Figure 3.- Course of the functions t q(t)) for n = 0, 1, 2, 3, and 4 
 
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