Formation of a vortex at the edge of a plate

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Title:
Formation of a vortex at the edge of a plate
Series Title:
NACA TM
Physical Description:
36 p. : ill ; 27 cm.
Language:
English
Creator:
Anton, Leo
United States -- National Advisory Committee for Aeronautics
Publisher:
NACA
Place of Publication:
Washington, D.C
Publication Date:

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Subjects / Keywords:
Aerodynamics   ( lcsh )
Gas flow   ( lcsh )
Genre:
federal government publication   ( marcgt )
bibliography   ( marcgt )
technical report   ( marcgt )
non-fiction   ( marcgt )

Notes

Abstract:
The 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 vortex 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 form and vortex distribution of the surface. However, on the basis of a similitude 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 further development of the vortex distribution and shape of the surface are observed in the case of a plate of finite width.
Bibliography:
Includes bibliographic references (p. 24).
Funding:
Sponsored by National Advisory Committee for Aeronautics
Statement of Responsibility:
by Leo Anton.
General Note:
"Report date March 1956."
General Note:
"Translation of "Ausbildung eines wirbels an der kante einer platte," Göttinger Dissertation, Ingenieur-Archiv, vol. X, 1939."

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University of Florida
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All applicable rights reserved by the source institution and holding location.
Resource Identifier:
aleph - 003874042
oclc - 156913346
System ID:
AA00009196:00001


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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 wh-at 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 so-called 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, Ingenieur-Archiv, vol. X, 1939, pp. 4ll-427.







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 rolled-up, 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 above-mentioned prediction of Prandt1 is thereby
confirmed.

ITe outer part of the discontinuity surface, thus the transitional
region from the spiral-shaped 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 vortex-free 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 so-called
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 z-plane. 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~igh-t, 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
z-plane 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 z-plane, 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;
th-e point, at z = O transforms to the points fb on the real S-axis.
Fucr-thelrmor-e, in this transformation the region at infinity in the z-plane
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 5-plane 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 z-plane 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 f-plane 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 z-plane. 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 ver-tex 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



wh-e re H represents a scale unit. Accordingly, equation (1) assumes
the 'or-m



=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 S-plane. ICf in the
S-plane v, continues denoting the free-streamw velocity, the interpre-
taction of v, a~s traveling or, respectively, free-stream velocity, is
lost in the Z-plane 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 vortex-free 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 free-stream velocity in the 5-plane is invariably va, the same veloc-
ity must prevail also for the similar enlargement at similarly located
points. This velocity is composed of the free-stream velocity v, and
of the fields of the individual vortex elements. TIhe influence of a

In the Z-plane this is not the case, since here (for vortetx-free

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, du-e 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 Z-plane the corresponding distances are


11 1 2
2i~ 2/


Thuis -1 becomes in the Z-p~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 Z-pl~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 h-1
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 di~fference between the coordinates of the spiral center (a, h) and of the
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 S-plane.
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 Z-plane the velocities v' of
the f-plane are transformed into the velocities v of thet Z-plane,
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 5-plane to the Z-plane the field of this
vortex is deformed; hence an additional term appears in the conversion.
of the velocities. In the 5-plane the potential of the3 core vortex to be
omitted is







NACA TM 15S98


~,(5) 2ni~ i -0


In the Z-planre 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 Z-plane 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 5-plane or, respectively,



v dt (18)
additional .iI d d


in the Z-plane.

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 t-2/5


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 S-axis, 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 S-plane 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 S-axis (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 cross-ha~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 Z-plane a term ste-mming 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 Z-plane,


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 a-t
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







wh-e 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 abo-ve 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
tr-eated 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 =()/ bt-2/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 syg-plan onto a
S-plane 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 th-e 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 1-398


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


Plate


Figure 3.- Spiral according to the equation R = H/cp2/3























H /
Ktlrir


1


I 1 i I I I I I I


u,4




0. I




O 0.5 1.0 1.5 2.0 2.5


NACA M :1598


Figure 4.- Distribution density for
to the~ spiral


the region a/2 = 9p = 2.57T pertaining
R= H/ 2/3






NAcA TM 1598 27







P 9


s "d



rt=0
-~ t~t-----rH


Figure 5.- The spiral and its image in the p-plane.






































S
H
0.5 1.0 '1.5 2.0 2.5


NACA TM 1398


Ys H'/


0.2

0.1


O


Figure 6j.- Distribution density. 7(s ) of the spiral in the 5-plane.






HIACA TM 1598


d vp H3/
271T
ds' K


Figure 7.- Influence of the distribution on the velocity components
v5 and vf of the vortex center of gravity.







NACA TM 1398


Figure 8.- The quantities which are decisive for the variation of form
and distribution density for the initial spiral.






NACA TM 1398


Z= O)
X-r


-X


Figure 9.- First correction of the form.


b c


Figure 10. Distribution density and decisive quantities after the first
correction.







NACA TM 1598


-X C,1 C2 oSc itX




Plate






Figure 11.- Second correction of the forrn.


b c
Figure 12.- Distribution density and decisive quantities after the3
second correction.









































Ouer loop!
,G 1- Spiral R -H
4 : 2/3


0.5T 7


NACA TM4 1398


Figure 13. Final form of the spiral for the plate of infinite width.


1.5rr 2TT 2.577r 3-T 3.5TT 47~


Figure 14.- Relation between the radius R and the angle cp for the
final form and for the initial spiral























Y (s)

1.2





O.6





0.4
0.3
Q2

H


NACA TM 1598


O 05


1.5 210 2.5


Figure 15.-


Final distribution density in the
inint wridth.


case of the! plate of







NACA. TM 1398


Figure 16.- V~ariation of the form of the spiral with time in the case of
the plate of finite width (reduced to constant vortex magnitude).




z Plane


Figure 17. Vsriation of the form of the spiral with time in the case of
the plate of finite width (actual scale).













Y (s)




0.8
0.7
0.6

011

0.3

0.2

OSW
O .0 95 2. .


N;ACA TM 1398


Figure 18.-


Variation of the distribution density with time.






I~ae of infinite' widjth


,******Plate of finite width


b/v,


0 0.5 r.0 1.5

Figure 19.- Growth of the circulation with time.


NACA Langley Field, Va.







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