Heisenberg, Werner

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..
TECHNICAL NOTES
NATIONAL ADVISORY COMMITTEE FOR AERONAUTICS.
.
.
NO, 126.
ABSOLUTE DIMENSIONS OF KARMAR VORTEX MOTION.
By Werner Heisenberg.
*
From ‘PhysikaMsche Zeitschrift,n September 15, 1922.
January, 1923.
.——— -.— -.
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NATIONAL .4DV1S0R% CCMVI~TEE FOR AERO ‘M.UTI~%.
TEOHNICAL NOTE NO. 126.
ABSOLUTE DIWENSICINS OF KAFM_4NVORTEX MOTION. *
Professor
W of a plate,
water with the
the phenomenon
Professox
Ey Wexner Heisenberg.
.
Kazznan succeeded in calcuhting the resistance
~o~~ing -perpendicular’l,y to -it~ surface through
velocity U, from data obtained directly from
of the flew.**
Kamnzn investigated the flow for some distance
behind the plate. The ex~eriment 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 ~rr~~ge~e~~, i.e. the ratio of the
distance 1 ‘~~t~een the ~~~tice~ to the width h of the ‘vor-“
tex street (F~g. 1), were determined on the basis of a stabil-
ity investigation. The velocity u and a lihear dimension
of $he vortex system (fez instance, the distance Z 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
vort=x syste~mremaine’du.nanmered.
i According to the law of Helmholtz that no vortex can form
f in a fricticnl.ess or non-viscous fluid, the viscosity is obvi- ,
I
ously rs~ponsible for the formation of the vortices. Karman also +f
I * From Plrjjsikali~ckeZeitschrift, Se@ember 15, 1922, pF. 363-366.’
I
**@iw~n, Wttinger I?achr.1911, P. 509, and 1912, p. 5.47;Karman :
and Rubach, Physikalische Zeitechrift, 1912, p. 49.
-2-”
Considered that an investigation into the phenmena of the bnund-
ary layer at
the umknowm
cula.tions of
nevertheless
proximity to
Furthermore,
the plate WG=JIC1be necessary, in ozder ,to calculate
u and 2 of the vortex tail. Accmding to the cal-
the plate is less than at some distance behind It.
it was shown by Jaff’e’(Phys. Zeitschr. 1920, p. 541),
that, even in a viscouG fluid, in general , no vortex can o?:ginate
and that thcrefoce the reaaon for the formation of vortices ap
pearing ever-ywhere in ~:drad~-naxics cannot be fcund in the viscos-
ity. According tO Jaff6~ t-hereis murh more cause for the forna-
tion of vo:tices, when there aye di~contirmities in the external
forces or in the velocity of the fluicl. In the case cnder con-
sidera~iOn: thsre are cerhinh.~ SUCh discontinuitiss pr~~entj in t;l
vicinity of the Flate. “~?ewill thezefore attem~t to determine the’
from a cnns<de:ca-:ioaof the6e Ciiscontinuitiesa
The on2y* hither-to known form of flow of water Fast flat
plate~, when the flow -velo&ity past the edge of the Elate is tiot
infinitej is the lie~nb.oltz-fi:i~c~offunstable ~otential motion
(Fig. 1).**
The practi~l inpocsikility of this flow, which, on account
—
,.
.-3-
flow, lies likewise in the lack of ~tabtlity of the vortex sheetn.
There i.~always the conjecture t-hattl% unstable Fctential i’notio~l
must exist at least in the imfileciiatevlclnity of the plate.
In fact, photographs. s?lowthat the actual flew in the imme-
diate vicinity of the plate is remarkably simi~ar to the Xirckhoff
,
.-
potentiel motion. In connection with the origin of the Ka~-’uan .
l
vortex system, we will therefore inv~~~igate the following b~- .
pothesis (Fig. 1)*.
In the immediate vicinity Of the plate there is first-formed
the unstable pctential mction, At some distance frcm the plate,
the surfaces of ”dificon~inuity,i.e. the vortex layers of this
potential motions on account of their lack of stability, roll up
.
and form the Karman vortex a~zangement at a still greater distance
fxGm the plate.
This hypothesis can
vertices, origimting at
approa~h reality, only in so far as the
some distance, influence the flow in
the immediate vicinity of the plate. Furthermore, it can hold
good only for the fzont side and edges of the plate, since, in
.
the “dead wat.er,fithere is always a very indefinite vortex motion
(but not vortex formation).
However simple and plausible the thus formulated ,statement
may at fizst seem, serious objections may be raised against it,
as Professor Prandtl has very kindly informed me. (See his remarks
at the close of ~hiq article)- Profes&orKarman put these objec-
—. .
* In Fig. 1 the -velo~ities (w = u - i v) are in part introduced
in .Qmble fas-hicn.. Tke.bracketed data refer to velocities with
relation %0 tM plate, while the other data refer to velocities .
with reference to the fluid at infinity.
-4-
-.
tions in the following form: ‘*Ifthe statement is applied 1~tczallX
:
, to the Bo’oileff case of a syznmstrioal weti.ge,it gives, fOr all i@~
angles of the wedge, the same
as for the flat plate). “ The
to why, Sust in Gur case, the
dnag coefficient, (riardely,the same ~
question remains open, therefore, as”
.
statez!er.tcorre~ponds so closely to :4
the fact, In any case, further c-al~ulationwill show that, for ~
.
the Plate, the above hypothesis agvees rema~kably well with exper-~
ience. I
1, Condition for the Velocitv of the Vortex SyQtem.
For calculating the qnknown u and I, two equations are .
readily fov.nd, It foll~~ws at once from the law of the conser~a- “.
“.
tion of the vortex moment, which we apply to bcth vortex sheets, .
>
tkt the vorhexmoaent originating pm unit of time at each of -
-,
the two edges of the plate must equal the vortex mcment, of either,.
directicn of rotation, carried away per unit of tine by the vortsxj
9p tern; Thi~’vortex moment has the value j
,, .
On the other hand.,
naiing on the edge
unstable potential
+(U-u)
the moment of the individual vortex filament. ~
in order to calculate the vortex moment origi- .
of the pSate, we note that, in the ?delmholtz y.
motion, there is in the “deaclwater” behind the
p~ate (consid.ered’~tationary) the velocity 0, while in the ‘~freei
surface proceed.lng f:romthe plate it,is U, and hence the diacon-r
.
tinuity of the velocity is in the instability surface U (Fig,l). ~
i,
, If we separate the instability surface into its elements d f, ;!
“,
-5-
which we oonsider as vortex lines, and form the velocity integral
around a closed cirait enclcsing such a vortex Qne d f (Fig.
i), we find its mment to be U d f. l~oreover, this element d f
is not at rest with reference to the plate, since we wcmld then
have.the velocity _u/CJ in the “dead water” and U/2 ouzside
of it. The element moreover moves farther from the plate on the
instability surface with the velocity u/2. Hence we conclude
that the vo~tex moment originating per unit of time,at the @@e _
of the ~late is
From Equations (1] and (2) we obtain
T.3:=+ (u-u)
(2) :
(3)
.
Z. Condition for the ,Vortex Interval.
A second relation follows from the fact that there is always
a oertain amount of wate~ (Fig. 2), which is driven forward be- . .
tween the two series of vortioesi While the fluid outside the
‘fvortexstreet” on the’average is at rest “(i.e. any-fluid parti-
a ole does not depart in t-he course of time any great distance from
its original position), a stream flows continuously forvard with-
in the ‘lstreek‘~ The quantity
stream must equal the quantity
‘plate in its mction, whioh is
In order to c.Qculate the
of water oarried along in this
constantly pushed ahead by the
Ud.
former, we write, following Karmang
the complex potential of the vortex flow ,
—
r-6-
,
(4)
..
in which h denotes the distanm between the two ssrtes o’fvor-
tices. .
z = x + i y (The X-axis runs, as shown in Fig. 1, ~rallel
to the vertex series and divides the space evenly between them).’
In order to obtain the desired quantity of water, we integrate,
along an arb:trary path cf line elements d s (fi= velooity - ‘
1
!veptor; w = u- i v; J denotes ‘tImaginaryportion of!’),
The limits a and b of the integral are the two streamlines
bounding the desiued mass of water (dash lines in Fig. 2), in ~
which the particular position of the points a and b on the
stieam lines is of no significance. In the ease under consider-.
ation, the limiting stzeam lines are the only ones (in the Y-
direction) which extend to infinity. We therefore extend the in..
tegral simply from z = + i= to z = - i= .
tegration path from - im extends along a
vortex, then t~ansversely aoross the stream
along the limiting stream line to +im.
@ss~ng thraugh the st~eam per unit of time
In Fig. 2, the in-.
dash line near the
bed and thence again .
The quantity of fluid
is,acmrdingly
J m~m - x_ius’).
,
-7-.
But acccrding to Fquat~on (4)
Hence, for ~+~, a= O (a does not cuter into the
nary portion of X), ,
Therefore the quantity of fluid (no attention being
to the signe , since only ‘theabsolute value concerns us)
paid
is
(5)
(6)
For the numerical evaluation of Equations (3) and (6), we
utilize’ Karnan]s foxmuias (z.c.)
,-8-
From Equation (3) then follows
*U2 =Uy-(u -u),
1 = 2;/~:(1–$)J
*
$ = o.229* .(9)
The other root u/V = 0.771 is excluded, for a reason about
to be g~ven. Equation (6), on account of Equations (7)’and (8),
is converted into
.
/-U* d=u~ 8= to 0.283.N
Whence th&efollows, amcording to Equation (9)
i 1
= 5.45;
z=
r
8“ 0.2839 0.229
(lo)
(11)
The other root for u/V would give a much smaller value
for ~/d and h/d, Since, however, the discontinuity surfaces
of the Fotential fiow are directed outt~ard, it is quite obvious
this second possibility is
Karman gives, as empirioal
g
~=o.20; ;
unattainable.
values ,
= 5.5.
From the tbazetical values for u/V and Z/d we have
0.90 as the
defined by
value for the speoifie coefficient of drag
w= w~ “ L*d. U2”P
.-9- -
in i~h~c~ L = Zenith of p~lateat rigb.tangles to the plar.e of ~
the diagram and p = density of fluid.”
This value seems to agree well with the latest determina-
tions.
Institiite for Theoretical Physics,
Munich, July 18, 1922.
Vy objections to’the statements of ?[r.Helsenberg, already
referred to by him, are briefly formulated in the following par-
agraph.
1. The res~-stance of the plate is about twine as great as
given by the lli~chhoff form’ula. Behind the plate there are on-
ly small velocities and hence a constant pressure cl”~r to the
edges. To the fall in’pressure from the middle of the front
side to the edge} wtiich is about twice as great, there also
corresponds a value of U2/2 about twice as great. Hence the
vortex production per second ,must be ~ u2/2, in which a
is a’bo-ii2 for the flat plate. (To be exact, the motion back i
of the plate also exe~ts a slight influence on the vortex prod&-’\
tion).
2. It is not impossible
poiZtive and negative vortex
,
that appreciable portions of the
moments mutually eliminate one an-
other by intezmzxing in the turbulent zone bekind the F~ate and
hence aze no longer present in the vortex system as it flows i
L-m-
away, If the eliminated fraction is P>
place of Equation (3)
(1- P) ’+=:. (U-U)
3. The c~nclusion, that the quantity
.-
we would ‘have,in
of fluid flowing be-
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 be filled by the
flow. An opposite effect may be introduced by the contraction
of the vortex zone in forming the vortex series, so that Equa-
tion (6) also has an undetermined factor, which may be greater
or less than unity.
..
In my opinion, Wr, Hei6enbergTs computation; though very
instructive, is only adapted to yield definite conclusions when
used in connection with exp~rimental data concerning the correc-
tion factors referred to above.
G3ttingen, July 29, 1922. ~~
Translated.by the National Advisory Committee for Aeronautics.
I \
.-
I I
\.
<’
Fig.2 A
-ico


Absolute dimensions of Karman vortex motion

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