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~fK A*M 1? 39
NATIONAL ADVISORY COMMITTEE FOR AERONrAUTICS TECHNICAL MIEMORAIIDUM 15i98 FORMLATIOIJ OF A VORTEX AT THE EDGE OF A PLATT* By Leo Anton 1. IN~TRODUCTIIOIN If a plate is moved from the state of rest tranlsversely to its width through a fluid at rest, a smrall vortex forms at every edge at the start of the motion; by continual added influx of' new parts of the fluid, that vortex becomes larger and grows into the flow. In case the flow is perpendicular to the plate, the position of the two vortices is syrmmetrical at the beginning of the motion of the fluid. In a later stage of development it is, as von Karman has proved, urrstable. Then new vortices originate alternately at the upper and.lower end of the plate which group themselves behind the plate in a certain mannier (von Karma~n's vortex streett. In what follows, we shall treat the initial flow Just described for a plate perpendicular to the approaching flow, where, therefore, flow direction and plate form a right angle. Since in this case the vortices are in the positions of reflected images, our investigation is limited to the fonrmtion and the growth of one vortex at o~ne plate edge. We visualize the plate as suddenly set in motion and then moved uiniformly at constant velocity. For a system of axes moved with the plate, the velocity at infinity is therefore to remrain inv;ariable throlgho~ut the flow duration. Moreover, we assume the plate to be laterally extended to infinite length so that we deal with a plane nonstationary flow phenomenon. Solution of the problem is considerably facilitated by disregard of the friction. This is permissible since the friction of most fluids coming into question is very small. Thus we take as a basis an ideal fluid where, therefore, the viscosity is zero, and require furthermore that it be homogeneous and incompressible. In this fluid the flow described is then to be interpreted as a potential flow with free vor tices distributed over a discontinuity surface, a socalled vortex sheet. The velocity field of such a flow is u~niquely determined byr the velocity at infinity and by the form and vortex distribution of' this surface. A solution for the motion of a fluid about a plate, starting from the state of rest, has, so far, been achieved only for infinitely small angles of attack. This borderline case has been treated by H. Wagne; Ausbildung eines Wirbels an der Kante einer Platte," Gr~ttinger Dissertation, IngenieurArchiv, vol. X, 1939, pp. 4ll427. MACA TM 1598 (ref.1). Hre the free vortices lie in a plane vortex sheet which extends from the trailing edge. The solution of the flow problem consists in the calculation of the distribution density of the discontinuity sur face. Wagner performed these calculations for accelerated and for uniform motion of the plate. For finite angles of ~attack, as in our case, the discontinuity surface forms a strongly curved vortex sheet which obviously is rolled up into a spiral vortex. Thus, besides the distribution density, the shape of the discontinuity surface also is unknown. We now make the following simplification. At the very beginning of the motion, of the fluid, the magnitude of a vortex at the~ plate edge in proportion to the width of the plate is still very small. Consequently the vortices at both edges still lie outside of their mutual :interference domain so that one deals practically with a flow about a plate of semi infinite width. This means a great advantage. Since no unit length exists which would influence the process, the flow conditions must be independent of the magnitude of the vortex at the time. The growth of the vortex therefore consists in a similar increase of it. In this connection the treatises of L. Prandt1 (ref. 2) and H. Kade~n (ref. 3) must be men tioned. Prandt1 performed a general investigation concerning the~ flow around a corner for certain laws of acceleration. No solution for the plate flow resulted from the Prandt1 fomuationr; however, one can derive from it the laws of similitude. In the~ treatise mentioned, Prandt1 sur mises that the rolledup, vortex sheet in its interior may be of the type of a spiral R = const./iPR (R and q, signify polar coordinates counted from the center of the spiral, m is a number). Kaden treats the rolling up of an unstable discontinuity surface, unilaterally of infinite length, into a spiral vortex. As we shall see later, Kaden's problem. becomes identical with ours for the interior of the vortex. The same laws of similitude are valid for both cases. The solution for the vortex core R = const. 9p2/3 found by Kaden applies, therefore, also to the interior of our vortex sheet; the abovementioned prediction of Prandt1 is thereby confirmed. ITe outer part of the discontinuity surface, thus the transitional region from the spiralshaped vortex core to the plate edge, we procure with the aid of graphical methods. The solution obtained is then used for finding the flow about the plate of finite width. 2. OUTLINE OF THE FIETHOD In this section we shall briefly describe the method by which one arrives at solution of the problem. We shall start from the plate of finite width, and shall then treat the plate of infinite width as a special case. In the case of the vortexfree potential flow about the plate placed transversely, the velocity at the edges is known to become infinitely large; however, under actual conditions this is never possible. We must therefore demand also in this case that the velocity there always has finite values. We attain this by assuming in the fluid socalled Helmholtz discontinuity surfaces which extend from the plate edges. By this expFression one~ understands vortex sheets consisting of free vortices and obeyilng B~elmholltz' laws. In contrast to this is the plate, which also is a discontinuity surface but no longer obeys Helmholtz' laws. It f'orms~ a rigid surface and exerts, therefore, pressures on the flow. The plane in which the flow takes place will henceforth be designated as complex zplane. The system of coordinates is selected so that the origin of the coordinates coincides with the center of the plate and the plate lies along an imaginary axis. The width of the plate is made equal to 2b. The movement of the plate is to take place from the left to the r~ight, or, which means thne same, the flow is to approach the plate' from the positive side. The constant velocity at infinity which, with oppo site sign, may also be interpreted as speed of the plate traveling in a fluid at rest is to be v, (fig. 1, left). We conformally ma~p the zplane onto a r;plane, by means of the transformation r= z2 + (1) with the plate, which represents, of course, a piece of the imaginary sxis in theF zplane, becoming a piece of the real axis in the new plane (fig 1,rigt).The edges z = tib shift into the zero point f = 0; the point, at z = O transforms to the points fb on the real Saxis. Fucrthelrmore, in this transformation the region at infinity in the zplane go~es over, without change, into that at infinity in the r;planet. There foire thie velocities at infinity in both planes are equal, thus v(Z = mo) = vt5 = m0) = vCO. As images of the discontinuity surfaces we obtain in the new plane again images which start from the point f = 0. Th~e new vertex sheets lie symmetrically to the real axis whereby the latter becomes a streamline so that the image of the plate also remains a st~reamrlinel. The flow in the 5plane corresponding to the plate flow is composed of the parallel flow and a flow caused by the free vortices of the discontinuity surfaces. We are now going to calculate its vielocity; field. I~or th.e case where the vortex sheets in the zplane no longer lie 3,mme~trical to the real axis, one must choose instead of (1) the tlranr;rrsflormto into a circle since the real axis in the fplane then is no! longer a streamlined. For our case the selected transformation is preferablel to that into the circle because~the graphical calculations Dicome co~nsiderably simpler. NACA TM 1598 4 NACA TM 1398 We denote by SF a variable point of the vortex sheet in the upper half plane and coordinate to it a circulation d f. The corresponding vortex point of the mirrored surface is then given by (Tp and has the same vortex strength, but with opposite sign, thus dr. Both ~vortex points produce at an arbitrary point 5; of the plane a velocity the~ cojnjugate complex value of which is given by If one puts dr= 7( F dF wherein 7(5F) represents the distribu ,tion density at a point SF, and d5F a line element of the surface, one obtains by integration over all points of the surfaces the velocity induced by them ((7' signifies the terminal point of the vortex sheet.) The parallel flow has the velocity v.For the total velocity of the superirmposed flow one may now write v(t;) = vm F 1~ 1" F 2 Between the velocities of both planes there exists the relationship v~z = ((dz from ~which one obtains the velocities for the zplane. The condition that the velocity should assume finite values at the plate edges remains to be satisfied. As one can see very quickly, this requirement is ful fil~led when the expression (2) for 5 = 0 disappears, because the magni fication ratio d~/az becomes infinite for 5 = O or z = +ib. One obtains therewith the condition 2The part stemming from vl(() becomes real for 5 = 0. IHACA TM 1398 which must always be satisfied. It is, however, not sufficient for deter mir~ation of distribution density and shape of the discontinuity surface. As long as the vertices are still very small comrpared to the plate, the plate may be regarded as infinitelyr wide in comparison with the vor tices. For this case, the lack of a fixed comparative length is the reason that the form. of thne discontinuity surface is independent of the magnitude of the vortex configuration. The development of the discon tinuityl surface consists, therefore, initiallyr in a similar magnification. Thiis similar magnification can be fuilfilled only in the case of a certain shape of the discontinuity surface, and of a certain distribution of the vortices over it. On the basis of this condition, we can therefore cal culat~e the form and vortex distribution for the beginning of the motion. Starting from this initial condition we can then find, by further obser vation o~f the variation with time of the change in form, "also the shapes and vertex distributions for the later times when the vertices are no longer small compared to the width of the plate. Before performing the limiting process to the plate of infinite width, we choose a new system of coordinates z' which arises from the former coordinate system by parallel displacement, and the zero point of which is shifted to the upper edge (z = +ib). We write Z = Z + ib ) and replace, in addition, z' by z' = Z or Z '(6) b H whe re H represents a scale unit. Accordingly, equation (1) assumes the 'orm =H Z 2 2iHZ (7) Thie limiting process from the plate of finite width to that of infinite width (b  m) yields a new transformation =21HZ or Z = i (8) NACA TM 13198 This is the transformation of the plane with a slit along the negative imaginary axis onrto the upper ha~lf plane: of the Splane. ICf in the Splane v, continues denoting the freestreamw velocity, the interpre taction of v, a~s traveling or, respectively, freestream velocity, is lost in the Zplane since in this plans the velocity at infinity tends >0. The quantity v, appears in this plane as t~he velocity which prevails in the case of vortexfree flow a~t the point Z = in, All other considerations carried out so far concerning the plate of finite width may be directly transferred to the plate of infinite width. 3. THE INFINITELY WIDE PLAT (a) Laws of Similitude By the limiting process from a finitely wide to an infinitely wide plate which was just completed, a flow type was obtained for which. the successive flow patterns are similar and, accordingly, the~ circulation of the vortex and the distribution density on the vortex sheet are similarly enlarged for this growth. We shall briefly derive here the laws which characterize this behavior. (As was remarked in the Introduction, the laws of similitude could be immediately derived from the gaoted treatise of Prandtl.) We observe the distance 1 of two points remaining in similar position during the similar enlargement of the discontinuity surface and assume it to be at the time tl of the amount 1l and a~t the time t2 of the amount 22. We state as the time law for the enlargement 1 1 (9) where A is a number yet to be determined. The law for the circulation P about similarly situated regions may be obtained as follows. Since the freestream velocity in the 5plane is invariably va, the same veloc ity must prevail also for the similar enlargement at similarly located points. This velocity is composed of the freestream velocity v, and of the fields of the individual vortex elements. TIhe influence of a In the Zplane this is not the case, since here (for vortetxfree flow) the velocity is constant at the point Z = 1 i which is fixed during the similar enlargement, thus does not remain in similar position. N\ACA TMJ 1598 vortexe element nr, at the distance Ig from the point considered, on the velocity is AP/21t2 If during the growth AP and I5 are modified, AP/15, due to the constant velocity, also must be constant. For the c r~culastion C about similarly situated regions of the t;plane, therefore, 1, 25 r2 25 is valid. In the Zplane the corresponding distances are 11 1 2 2i~ 2/ Thuis 1 becomes in the Zp~lane A~/2 =\ = (10) If we designste by V similitude, we obtain portionali to r/1 the velocities which correspond to the law of for the velocity V(Z) in the Zpl~ane which is pro VVI 12/r /1 _2 t 1 1/2 (11) If we replace in the ratios above the time tl by the time 1 and the time t2 y th~e time t, the laws of similitude read I= t = 11 1 A/2 1 h/2 (12) RACA TM 1596l wherein the number A remains to be determined. For calculation of the number A we form the velocity of growth of the distance t dZ h1 at Since this velocity, like all velocities, must vary proportionally thus Therevith the laws of similitude read 16v~1/5 p l1/5 (15) I = 11 2/5 (b) Solution for the Vortex Core Dur task now consists in finding a shape for the spiral which satisfies these laws. They are the same laws Kaden (ref. 9) obtained as a result. Thus one deals here, too, with a simnilar problem. In particular, both prob lems become identical for the inner part of the vortex where in both cases a discontinuity surface consisting of free vortices is to be rolled up into a spiral vortex. It is therefore sufficient if we refer to Kaden's paper a2nd here only briefly mention the results. Let R and tp be polar coordinates of a system of coordinates the zero point of which coincide~s with the center of the spiral. For form and circulation r of the vortex core! bounded by a circle of the radius R, Kaden obtained R = X2/3, 1. = 2X Ri (14) ElACA TM 1398 The circumferential velocity, for radius R is r _x vu (15) 2R35IAli The re in t is the time, counted from the beginning of the motion. The qua~nt ity X is a constant, still unknown for the time being. It has the dimension velocity times square root of a length. Thus we may also intro duce for it the dimensionless constant k (16) (c) Solution for the Outer Loops of the S~piral As was shown before, the behavior of the spiral is characterized in th~e interior, that is, for large angles cP, by R .For small ang les cp, in contrast, especially where the discontinuity surface adjojins the edge, considerable deviations from this form occur. This tiransitional region, on be found as follows. We visualize the plate as liIng parallel to the straight line rP = O; this assumption is insignifi cant for practical purposes since the spiral windings approach, in the Iinwir~d direction, a circular form. We isolate, furthermore, at one point t~he inner part of the spiral, in which the form is prescribed with suffi cien~t accuracy by R I. For calculation of the velocity field ou~ltsid~e of the spiral core, we visualize the latter as replaced by an Is:islt~ed vortex at the center of gravity of the circulation. This is dirrectly permissible since almost circular symmetry prevails in the interior of the spiral. The dissyrmmetry caused by the separation point orings it about that the center of gravity of the circulation does not coincide with the center of the spiral. For the calculations indicated ait~er on, the separation point is placed at the point cP = 2.5 ce~nter of gravity (ai, hi) may be calculated, similarly as by Kaden, (ref. 3) to be approximately NACA TM 1598 ai a = 0.006 Ri and hi h = 0.051 Ri (17) if Ri is the radius of the core of the3 spiral. The outer .loop which we now want to calculate, extends from the separation point (cp = 2.5xr) to the plate edge (P := ci4). Its shape and vortex distribution 7 as well as the position and circulation Piof the vortex core we assume, at first, arbitrarily, with a factor of simili tude~ X commn to both quantities and at present not ylet. determined, (equations (114) and (ly))1 and make therevith the transition into the plne.The rein 7 is transformed into S_ dz d5 whereas the circulation ri remains unchanged. In this mapping plane ( plane) the velocities stemming from the vortex mayr be calculated as the field of the vortex core and the distribution on the outer winding, likewise their mirrored images. ~By superimposition of the undisturbed ve locity v, there results the velocity field v' in the Splane. First, one ascertains the velocity a~t the point II = 0. From the condi tion that this velocity must be zero (equation (4)) results factor X, still, undetermined at first, for the circulation ri and the distribu tion density 3r. In the conformal mapping onto the Zplane the velocities v' of the fplane are transformed into the velocities v of thet Zplane, according to the relationship v = v  dZ where v and v' signify the conjugate values of the velocities. The velocity of the individual vortex concentrated at the core is obtained by omission of the field of this vortex. It must, however, be noted that in the transition from the 5plane to the Zplane the field of this vortex is deformed; hence an additional term appears in the conversion. of the velocities. In the 5plane the potential of the3 core vortex to be omitted is NACA TM 15S98 ~,(5) 2ni~ i 0 In the Zplanre the potential. of the vortex to be emitted is 4KZ) 2i In Z ZO) ri In(12 _2 2ni~ Oj 25ri 2n1i In order to calculate the velocity for the Zplane we must, therefore, sub~tract, besides the potential QK((), also the additional potential i_ In ( + 5( to which corresponds a velocity ri Additional lj , in the 5plane or, respectively, v dt (18) additional .iI d d in the Zplane. Onee one has calculated the velocities for individual points of the diiiontinuity surface one resolves them, most advantageously, into normal and tangential velocities vn and vt, and plots the latter as functions of' the arc length s. B~y combination of the function values one obtains the velocities pertaining to each point of the vortex sheet; from them results the motion of the assumed vortex sheet. A fluid particle at the point P of the spiral having the velocity v (fig. 2) moves by the distance PP1 = v~ in the unit time. Due to the 12 NACA TM 1598 similar enlargement, the point P is tran~formed during this time into the similarly located point P2' The velocity aut which the~ simlarly situated points are displaced is assumed to be V so that the distance is P)P2 = V. To have P1 come to lie on the similarly enlarged spiral, the normal components vn and V, of the~ two velocities v and V must be equal or (19) avn n v, n = o We? observe the point P on its path to the similarly situated point P2 and find that this path does not represent the motion. of a fluid particle. Rather, the fluid flows, with the ~velocity v V, through the point moving from P to P2* If VT atnd VT signify the tangential components of the velocities v and V, the circulation in flowing with the fluid into the core inside of P or P2, respectively, during the3 time at is aF = (vT V )7at Hencee there results for the region inside of P, which is being similarly enlarged (vT VT)T According to the laws of similitude (15i), one mrust have for a region. being similarly enlarged ar ~1 t2/5 The two values of dP/dt must agree in order to have the similitude satisfied, therefore the equation Thle duplication of the equation numbers 20 and 21 follows that of the original Germain document. N~ACA TMJ 1598, ar r S (vT VT)T  O at 3t muTLst aFpply. Since the distribution. density is as we may write instead of (20) also (20) (21) With the aid of the equations (19) and (21) it is now possible to find the hshae and vortex distribution of the outer loop. We shall perform the calculation of these quantities for a time t t x95/2 X (20)4 for which the equation (114) of the spiral core assumes the simplified fo rm (21)4 R  We select as the initial forn for the entire spiral the form R = H 9/ and place it as a first try so that it joins the plate at the point gj = *~l = lr/2 (fig. 3). The break originated at the edge is merely a flow which, as one readily understands, disappears at the next moment. Except for the factor X, the circulation r = 2x S (22) a, (s vT , VT) O NACA TM 1598 is known to us. For the core which we shall. assume~ to begin at we have cP = 2.51(, H 1 215 Thus r. becomes ri = 2x (2.rjn)2/35 The distribution density is (23) as _ ar aR aR as SX R 9g/Lil + R5 X (24) In figure 4 this distribution density 7 is plotted as a function of the arc length s. The conformal mapping (8), in combination with the reflection at the Saxis, leads to the double spiral. indicated in figure 5. The~ distribution density in this plane (image plane) is represented in figure 6. When we calculate the velocity at the zero point, we obtain, on the basis of equation (4), a condition for the still undetermined value X. There results xl = 2.356 Hv, or kl= 2.36 As an example for the determination of the velocity field we shall here briefly reproduce the calculation of the velocity at the center of gravity. In the Splane an element of the~ discontinuity surface as' induces at the center of gravity the velocity components avS sin \I as , 1 2xRs' av = cos r as' I 2ns RACA TM 1798 when Rs' denotes the distance of the element from the center of gravity and JI the angle of the radius v~etor from the center of gravity to the element with the Saxis (fig. 5). By integration over s' from the zero point to the core boundary (cP = 2.55%, s' = 2.62 II) and o~ver the corre ;spnding curve (S') of the Leage one obtains vil' and vill'. In fig uire 7 t b values dvi1'as and s', re spe actively. The quantities dvll'asI ae plotted against s or vSl' and vbl a~re obtained by cir c~umscribing, with the aid of a planim~eter, the crossha~tched areas bounded by the curves and the abscissa axis = vSl' = O.27v, vI Thezse values represent the influence of the outer loop of the spiral. The imag~e of the vortex core causes the velocities vS2 =; O.22 Vb = 0 q~2 if' h' signifies the distance of the center of gravity of the core from the (axis. The vortex core itself does not contribute to the velocity of its own center of gravity in the t;plane. However, it must be noted that, according to the explanations on pp. 10 and 11, in the conversion to the Zplane a term stemming from the vortex core = .11 vm O .11 v, aIppear~s Fojr the resultant velocity one obtains v vm "51 + $2 5 0.3v vyl' = Vrl1 + 2 5 q = 0.11v, RACA TM 1598 whence there results, by conversion to the Zplane, Cgyql + qvg' vx = H VY = H 52 12 = .29v, = O.17v, According to 'the .Law of similitude, vx = 0.37v, VY = 0 vould hazve to be valid. The center of gravity thus shows for the assumptions made, relatively to its required motion, a wrong motion obliquelyr downward, toward the plate, with the velocity components A`Vx, = x = Vx = 0.08V, Avy = vy Vy = 0.17v, manner the velocities for the points of the outer winding are Of course, one has to add a term which represents the influ core; on the other hand, the additional term v5 does not In the same calcu lated. ence of the appear here. We find the calculated normal, and tangential comrpone~nts plotted in figurres 8a and Sc. As can be seen from figures 8b, and 8&d, equations (19) and (21) are not satisfied. In other words, the shape and vortex distribution assumed deviate from the actual solution. We now modify the position of the core c.g. in the sense of Clvx and bly, an~d the shape and vortex distribution, of the outer loop in the sense of the dii'ferences Avn and ad Z. The form of the spiral core remains n at unchangedly R  We now repeat the~ calculation procedure NACA TMJ 1598 carried out so far. After several steps, the first two of which are represented in figures 9 to 12, one arrives finally at a solution that satisfies the laws of similitude; it is plotted in figure 13. 14e is the radius of the spiral; in figure 15 its distri is plotted as functions of the center angle and of the respectively. In figure bution density are length s, For more convenient further use, the most important constants necessary for characterization of the spiral vortex have been briefly compiled below. For the solution indicated in figures 15 to lyj, the calculation yields the time t = 1.42 1 Vm the co:nstant X =2.22H1/27 k = 2.22 the coordinates of the spiral center a =0.55K h = 0.22H the total circulation of the spiral vortex r = 4.20BHv, For the assumed core boundary a~t cp = 2.5%r the radius of the core iS Ri = 0.267Ei the circulation of the core Pi = 2.3Iyvx MACA TM 1598 For an arbitrary time t one obtains a = 0.4H~t2/3 h = 0.165Ht2/7 r =5.7rv,t1/3 R_0.8 Ht2/3 = 2 a Ri = 0.2Ht2/j = o.5a 'i = 2Hytl/?= = .15 a 4. THE PLATE OF FINITE WIDTH With the solution found for the plate of infinite width, we know the initial flow conditions about a plate of finite width, where the vortices are still small compared to the plate width. We know that for this initial state the successive vortex images originate from one another by similar enlargement. However, as soon as the vortices assume a magnitude which is no longer negligible compared to the plate width, the presupposition for a similar growth no longer holds true. Yet, starting from the form and distribution density we found for the still very small vortices, we can calculate the variation of this form and distribution density, using the values of avn ~19~ and A 2rfT1r which now, with growing vortices, more and more deviate from 0. We only must make use of the transformation by means of (1_) instead of the tranls formation by means of (8). If we start from a known form, and distribu tion density at a time t1, we obtain the deviation of the form and dis tribution density from those which would result for similar enlargement according to the time laws (15S), at the time t2 as being an = 12 Indt and byI = 2a t 1; 1_ IIACA TM 1598g 19 The graphical determination of the required quantities is greatly hampered by the fact that the vortices are at the beginning very small and later very large. We can avoid this difficulty by visualizing the figur~el at every instant, enlarged or reduced in such a manner that the vortexr would always remain the samne if it would keep on growing according to the laws of similitude that are valid for the beginning. Th~e figure in actual size we denote as figure I and the quantities valid for it by the subscript I, the enlarged figure as figure II and the corresponding quantities by the subscript II. At the initial stage, the enlarged vor tex is to agree precisely with the one calculated for the plate of infinite width in the preceding section. L~et us call the ratio of mag nification in every case E. If we reduce in the magnified figure all velocities si~multaneously at suitable points in the proportion1/C the circulations r are magnified in the proportion E.The distribu tion densities 7 are reduced in the proportion 1/ / For the potential flow about the plate of the width 2b in the plane I there results in the neighborhood of the edge the velocity VI = (25) For the plate of infinite length treated in section 3 the corresponding velocitr, was whe rei n y' = b y signifies the distance from the edge of the plate. In order to obtain for the beginning of the motion the same conditions as fo~r thie flow about the plate of infinite width treated before, we select the width of the plate 2b in such a manner that b =H FoJr the above stipulation, exactly the velocity v, prevails in the mag~nified figure, while E is still very large, at a point at a distance of b/2 from the edge. In the case of the plate of infinite length treated before, this point lay at a distance H/2 from the edge. Sinee we equated b = H, we obtaba in the magnified figure precisely the flow treated in section 3. N~ACA TM 1598 According to equation (l1C), for the~ growing vortex core of the plate of infinite viath RXt2/ Since at the initial stage the flow about the edge of the~ plate of the finite width 2b = 2H coincides with the flow about the edge of the plate of infinite width, this formua applies also to the initial stage in the case of a plate of finite width. B3y means of the magnification by the factor E, this initial vortex is to be transformedl into the flow treated for which R = H b .Thus necessarily (p2Xt r2/5_ H~ene there results E =()/ bt2/l (27) Since in the plane I, at the beginning of the motion, the circulations ry grow, according to (15), with tl/5, we obtain in th~e plane II, where they appear magnified in the proportion \fe constant circulations and constant distribution densities. At the beginning of the motion C = m3. The plate edges lie in the plane II at an infinite distance from one another so that we actually have the case of the plate of infinlite width. In time, however, E attains smaller finite values, and we obtain in the plane II also a plate of finite width. Therewith the velocities become different, and we~ obtain deviations from the similar magnification. In order to calculate the velocities in the plane II, we transform this sygplan onto a Splane which we shall characterize by the subscript III, by means of the function S+ (Eb)2 (777= I (28) NACA TM 1398 For very large a this mapping.is transformed into the infinite plate according to (8) in the neighborhood of the edge. For finite E, however, other shapes result for the mapped vortices, and that is the vezry reason which causes the modifications of the velocities and there wihthe deviations from similar vortex growth. As soon as, due to this deviation, the form and distribution density of the vortices have changed with respect to the similar growth, this also contributes to the variation of the velocities. Since, however, 'o~rm and distribution density change, at first, only very slowly compared to the similar growth, one may in the plane II assume the form and distri butiojn density as constant in turn through a large time? interval, and need consider in this time interval only the modification of the trans formation in the plane III, due to the modification of the value E. For such a time interval tl to 't2 (the first starts with t = 0), one calculates for several intermediate times the normal and tangential components vnr and vtI of the velocities of the plane II with the sid of the transformation onto the plane III in the same manner as in the case of the plate of infinite length. One forms furthennore the differ ences with respect to the velocities of the similar magnification VnI and Vty and obtains then by transfer to the plane I the values (19) ny p n nn no (29) and corresponding to (21) =E (v V 6 If we, finally, replace in the last term t by (b/E)J/ n/X (equation (27)), we obtain S =E S I 13b2(30) H~ACA TM 1398 By graphical integration then results the displacement Lln of the points of the vortex sheet at right angles to it, and the modification of the distribution density compared to the similar magnification in the plane I as Any = 2 nngdt (71) C1 67= 12 y (32) and the modification of form and distribution density in the plane~ II as Llnll = ea Any (33) Therein E2 is the value of the magnification ratiO E at the time t2* Due to the finite magnitude of the vortices compared with the plate aIII width, avnI fO and a at 0; also, the condition (4), that in the t;plane at the zero point the velocity must be zero, will no longer be satisfied. In the ascertainment of the vortex for thet plate of infinite width we have been able to fulfill this condition by suitable definition of an as yet undetermined factor for the circulation. Due to this condition we found the quantity X or k, respectively. For the further development of the vortex, form and vortex strength and their variation with time are fixed. Only the strength of the vrortices shed at the plate edge is still undetermined since we can here not form. the differential quotient a/as occurring in (30). We must select it in such a manner that equation (4) is satisfied. We obtain therefore an additional modification of the distribution density starting fromt the plate edge which gradually is carried into the vortex by the flow. It NACA TMr 1398 is true that it was shown in the quantitative calculation that the devia tions from the condition (4) are extremely small, because the variation of the conformal representation C(1)] vith E TOsults in a positive velocity, whereas the modification of the form and distribution density according to (55) and (34) results in a negative velocity at the point (1 = O and the two almost cancel one another. The calculation was carried out for the intervalS E = m, to E = 3, E = 3 tO E = 2, and E = 2 tO E = 1. The results are compiled in the figures 16 to 193. True to expectation, the circulation r increases more slowly with time~ than it does according to the solution for the plate of infinite width (initial state, fig. 19). In the final state it would perhaps approach a constant value which corresponds to a steady state of flow. However, according to experience the symmetrical vertex configuration becomes unstable from a certain magnitude onward, so that this steady state is not attained. 5. STIWARY Th~e flow about the plate of infinite width may be represented as a potential flow with discontinuity surfaces which extend from the plate edges. For prescribed form and vertex distribution of the discontinuity surfaces, the velocity field may be calculated by means of a conformal representation. One condition is that the velocity at the plate edges must be finite. However, it is not sufficient for determination of the form1 anld vortex distribution of the surface. However, on the basis of a sim~ilitude requirement one succeeds in finding a solution of this problem for the plate of infinite width which is correct for the very beginning of the motion of the fluid. Starting from this solution, the fuirt~her development of the vortex distribution and shape of the surface are observed in the case of a plate of finite width. Finally, I should like to express my special gratitude to Professor Betz for his suggestion of this investigation and his active support in carrying it out. Translated by Mary L. M~abler National Advisory Cormmittee for Aeronautics 24 NACA TM 13~98 REERNES 1. Wagnet~r, H. : Z. angev. Math. Mech. 5, 1925, p. 17  2. "Prandtl, L. : Uer die Etntatehun von Vireln in, der idealen Fliissigkreit, mit Anwe~ndung auf die Tragfliigeltheorie und andere Aufgaben. Vortr~ige aus dem. Gebiete detr Hyd~ro und Aerodynamik, Inunsbruck 1922, p. 18. 3. K~aden, H.: In~g.Arch. ;2, 1951, p. 140O. NACA TM~ 1598 5 Plane voo z Plane va ~x b b Plate with vortex sheets starting from the edges (left), and conformal representation of the flow (right). Figure 1. Figuref 2. Actual velocity v of a fluid particle and displacement velo city V corresponding to the similar magnification. P jZ= X+iY
X /' TX P' 7 
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