Absolute dimensions of Karman vortex motion

NMIONAL ADYS'RY COMMITTEE

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TECHNICAL NOTES
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.

No, 126.

ABSOLUTE DIMENSIONS OF KARMAN VORTEX MOTION.
By Werner Heisenberg.

From "Phyeikalische Seitacbrift ," September 15, 1932.

January, 19S3.

NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.

TECHNICAL NOTE NO. 136.

ABSOLUTE DIMENSIONS OF KARKAN VORTEX MOTION. *
By Werner Heisenberg.

Professor Karman succeeded in calculating the resistance
W of a plate, moving perpendicularly to -its surface through
water with the velocity U, from data obtained directly from
the phenomenon of the flew. * *

Professor Karman investigated the flow for some distance
behind the plate. The experiment showed a regular arrangement
of vortex lines, a "vortex street" (Fig. 2), which, remaining
behind the. plate, advanced mere slowly than the latter. The
relative dimensions of this arrangement, i.e. the ratio of the
distance I between the vortices to the width h of the vor-
tex street (Fig. 1) , were determined on the basis of a stabil-
ity investigation. The velocity u and a linear dimension
of the vortex system (for instance, the distance I between two
successive vortices rotating in the same direction) had to be
found experimentally, in order that the resistance W could
actually be deduced* The question as to the origin of the
vortex system remained unanswered.

| According to the law of Helmholta that no vortex can form

f

in a fricticnless or non-viscous fluid, the viscosity is obvi-

I *

J ousl y responsible for the formation of the vortices. Karman also ;

j * From Physikalische Zeitschrift, September 15, 1922, pp. 363-366.
I **Karman, G&ttinger Nachr. 1911, p. 509, and 1912, p. 5,4? •> Karman;
' and Rubach, Physikalische Zeitechrift, 1912, p. 49.

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Considered, that an investigation into the phenomena of the bound-
ary layer at the plate would be necessary* in order to calculate
the unknown u and I of the vortex tail. According to the cal-
culations of Oseen (Ann. d. Phya, 1915, pp. 231 and 646) , we must
nevertheless assune that the influence of viscosity in immediate
proximity to the plate is less than afc some distance behind it.
Furthermore, it was shown by Jaffe' (Phys. Zeitschr. 1920, p. 541),
that, even in a viscous fluid, in general, no vortex can originate
and that therefore the reason for the formation of vortices ap-
pearing everywhere in hydrodynamics cannot be found in the viscos-
ity. According to Jaffe, there is much more cause for the forma-
tion of vortices, when there are discontinuities in the external
forces or in the velocity of the fluid. In the case luider con-
sideration., there are certainly such discontinuities present in th
vicinity of the plate. We will therefore attempt to determine the'
quantities u and I, respectively § and ^' (d = width of plate
from a consideration of these discontinuities.

The only* hitherto known form of flow of water past flat

»
plates, when the flow velocity past the edge of the plate is riot

infinite, is the HeDjfcfcoltz-Kirohhcff unstable potential motion
(Fig. 1).**

The practical inpossibility of this flow, which, on account
of its surfac e? of -■discontinuity , must be regarded as a vortex

* Prof. Jaffe has kindlv reminded me that this is not exactly
accurate. A = R, fiichardaon (Phil, Uag. 1919, p, 433) offers an-
other very interesting Eiathematical solution, which, however, can
hardly have any physical significance.
** Lamb, Lehxbuch der Hydro dynamik , IV, Sec. 76, p. 113.

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flow, lies likewise in the lack of stability of the vortex shoe to.
There is always the conjecture that the -unstable potential motiou
must exist at least in the immediate vicinity of the plate.

In fact, photographs show that the actual flow in the imme-
diate vicinity of the plate is remarkably similar to the Xirchhoff
potential motion. In connection with the origin of the Kamaa
vortex system, we will therefore investigate the following hy-
pothesis (Fig. l)*.

In the immediate vicinity of the plate there is first -formed
the unstable potential motion, At some distance from the plate,
the surfaces of discontinuity, i.e. the vortex layers of this
potential motion, on account of their lack of stability, roll up
and form the Karman vortex arrangement at a still greater distance
from the plate.

This hypothesis can approach reality, only in so far as the
vortices, originating at some distance, influence the flow in
the immediate vicinity of the plate. Furthermore, it can hold
good only for the front side and edges of the plate, since, in
the "dead water," there is always a very indefinite vortex motion
(but not vortex formation).

However simple and plausible the thus formulated .statement
may at first seem, serious objections may be raised against it,
as Professor Prandtl lias very kindly informed me. (See his remarks

at the close of thi s article). Professor Karman put these objec-

* In Fig. 1 the velocities (w = ii - i v) are in part introduced
in -double fashion. The bracketed data refer to velocities with
relation to the plate, while the other data refer to velocities
with reference to the fluid at infinity.

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tions in the following form: "If the statement is applied literally
to the Bobileff case of a symmetrical wedge, it gives, for all the^
angles of the wedge, the same d:?ag coefficient, (namely, the same ■,
as for the flat plate)." The question remains open, therefore, ae J
to why, 3\ist in oxnr case, the statement corresponds so closely to *
the fact. In any case, further calculation will show that, for ;
the plate, the above hypothesis agrees remarkably well with exper- :
ience. . '

!• Condition for the Velocity of the Vortex System .

For calculating the unknown u and l, two equations are
readily found. It follows at once from the law of the conserva-
tion of the vortex moment, which we apply to both vortex sheets,
that the vornex moment originating per unit of time at each of
the two edges of the plate must equal the vortex moment, of either.,
direction of rotation, carried away per unit of time by the vortex;
syctemi This' vortex moment has the value ,

~ (U - u) k (1) '

when £ stands fox the moment of the individual vortex filament. ;'
On the other hand, in order to calculate the vortex moment origi- ,
nating on the edge of the plate, we note that, in the Helmholtz :
unstable potential irotion, there is in the "dead water" behind the
plate (considered stationary) the velocity 0, while in the "free 1
surfaoe proceeding from the plate it. is U, and hence the discon-
tinuity of the velocity is in the instability surface U (Fig. 1). I
If we separate the instability surface into its elements d f ,

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which we oonsider as vortex lines , and form the velocity integral
around a closed circuit enclosing such a vortex line df (Fig.
i), we find its moment to be U d f. Moreover, this element d f
is not at rest with reference to the plate, since we would then
have. the velocity -U/2 in the "dead water" and U/2 ouuside
of it. The element moreover moves farther from the plate on the
instability surface with the velocity U/3. Hence we conclude
that the vortex moment originating per unit of time, at the edge
of the plate is

u| (2) ;.

From Equations (l) and (3) we obtain

U| =| (U - u) (3)

2, Condition for the .Vortex Interval .

A second relation follows from the fact that there is always
a certain amount of water (Fig. 2), which is driven forward be-" - _
tween the two series of vortioes. While the fluid outside the
"vortex street" on the average is at rest (i.e. any fluid parti-
• ole does not depart in the course of time any great distance from
its original position), a stream flows continuously forward with-
in the "street. " The quantity of water carried along in this
stream must equal the quantity constantly pushed ahead by the
"plate in its motion, which is Ud.

In order to calculate the former, we write, following Karman. t
the complex potential of the vortex flow

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*J ,_ sin (z p - z)~Y

43 " sin (z + 3)2
z I hi

in which h denotes the distance between the two series of vor-
tices.

z = x + i y (The X-axis runs, ae shown in Fig. 1, parallel

i

to the vortex series and divides the space evenly between them).
In order to obtain the desired quantity of water, we integrate,
along an arbitrary- path of line elements d s (u = velocity
.vector; w = u - i v; J denotes "Imaginary portion of"),

b

b _ b b

u n ds = / J (wdz) = J /
a a a

/ u n ds = /~ J (wdz) = J /" H dz = J (X)

The limits a and b of the integral are the two streamlines
bounding the desired mass of water (dash lines in Pig. 3), in
which the particular position of the points a and b on the
stream lines is of no significance. In the case under consider-,
ation, the limiting stream lines are the only ones (in the Y-
direction) which extend to infinity, ''e -therefore extend the in-
tegral simply from z=+i» toz=-i<» , In Fig. 2, the in- .
tegration path from - i» extends along a dash line near the
vortex, then transversely aoross the stream bed and thence again
along the limiting stream line to + i<» . The quantity of fluid
passing through the stream per unit of time is acoo rdingly

J ^+ioo ~ X_io» ). .

_ 7 -

But according to Equation (4)

1

i£,~ e

= -5^lS

Hence, for (3->- « , a = (a does not eater into the imagi-
nary portion of X),.

x +i<*> ~ 3rr S e 4 31

= 2rr S V e 4 + 31-

■-loo

Therefore the quantity of fluid (no attention being paid
to the signs, since only the absolute value concerns us) is

|J (X +ico - x^i*,)!--^ (5)

from which is obtained the second relation

Ud = ^ (6)

3. Conclusion.

For the numerical evaluation of Equations (3) and (6), we
utilise Earman's formulas (i.e.)

I /8

v

= u

(7)

h

nd - = 0. 283 . (8)

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From Equation (3) then follows
£U S = u J 8 (U - u) ,

i.a,/T-a(i-j),

^ = 0.229. . (9)

The other. root u/V = 0.771 is excluded, for a reason about
to be given. Equation (6), on account of Equations (7) and (8),
is converted into

U* d = ui /IT- I - 0.283.

V

Whence the-re follows , according to Equation (9)

I

= 5.45; (10)

d / 8 • 0. 283 • 0. 229

| = 1. 54 (11)

The other root for u/V would give a much smaller value
for l /d and h/d, Since, however, the discontinuity surfaces
of the potential flow are directed outward, it is quite obvious
that this second possibility is unattainable,

Karman gives, as empirical values,

I

S = 0, 20; -J =■ 5.5.

From the theoretical values for u/V and l/d we have
ty n = 0.90 as the value for the specific coefficient of drag

V w , defined by

W = ty w - L- d • U s - P

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in which. L = length of plate at right angles to the plane of
the diagram and p = density of fluid.

This value seems to agree well with the latest determina-
tions.

Institute for Theoretical Physics,

Munich, July 18, 1923.

Remarks by Prof, L. Prandtl on the Foregoing Article .

My objections to- the statements of Mr. Heisenberg, already
referred to by him, are briefly formulated in the following par-
agraph.

1. The resistance of the plate is about twice as great as
given by tte Eirchhoff formula. Behind the plate there are on-
ly small velocities and hence a constant pressure clear -to the
edges. To the fall in 'pressure from the middle of the front
side to the edge, which is about twice as great, there also
corresponds a value of U s /3 about twice as great. Hence the
vortex production per second must be a- U s /3, in which a
is about 3 for the flat plate. (To be exact, the motion back /
of the plate also exerts a slight influence on the vortex pro&uc— .
tion).

3. It is not impossible that appreciable portions of the
positive and negative vortex moments mutually eliminate one an-
other by intermixing in the turbulent zone behind the plate and
hence are no longer present in the vortex system as it flows

/

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away. If the eliminated fraction is {3, we would nave, in
place of Equation (3)

(1 -p)a ^ =f (U- u)
3, The conclusion, that the quantity of fluid flowing be-

i

tween the vortices is exactly Ud, is not convincing. The lim-
it of the vortex zone advances with the plate into the fluid
(regarded as at rest) and this place must also toe filled toy the
flow. An opposite effect may toe introduced toy the contraction
of the vortex zone in forming the vortex series, so that Equa-
tion (6) also has an undetermined factor, which may toe greater
or less than unity.

In my opinion, ¥r ; Heisenberg's computation, though very
instructive, is only adapted to yield definite conclusions when
used in connection with experimental data concerning the correc-
tion factors referred to above.
Ggttingen, July 29, 1932.

Translated toy the National Advisory Committee for Aeronautics.

7.1'*. I

Fig. 2

x\

-ico