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W. M. SWANSON
A s s o c i o t e P r o f e s s o r o f M e c h a n i c a l E n g i n e e r i n g ,
W a s h i n g t o n U n i v e r s i t y , S t . L o u is , M o .
A s s o c . M e m . A S M E
T h e M a g n u s E f f e c t A S u m m a r y o f
I n v e s t i g a t i o n s t o D a t e
T h e M a g n u s f o r c e on a r o t a t i n g b o d y t r a v e l i n g t h r o u gh a f l u i d s p a r t l y r esp o n s i b l e f o r
b a l l i s t i c m i s si l e a n d r i f l e sh e l l i n a c cu r a c i e s a n d d i s p er s i o n a n d o r t h e s t r a n g e d e v i a -
t i o n a l b e h a v i o r o f su c h s p h e r i c a l m i s si l e s a s g o l f a l l s a n d b a s e b a l l s . A g r e a t d e a l o f
e f f o r t h a s b e en e x p en d e d n a t t e m p t s o p r e d i c t h e i f t a n d d r a g o r c es a s u n c t i o n s o f h e
p r i m a r y p a r a m et e r s, R e yn o l d s n u m b e r , r a t i o of p e r i p h e r a l t o r e e-s t r ea m v el o c i t y , a n d
g eo m e t r y . T h e f o r m u l a t i o n a n d s o l u t i o n o f t h e m a t h e m a t i c a l p r o b l e m i s of su f f i c i e n t
d i f f i c u l t y h a t e x p er i m e n t a l r e su l t s g i v e t h e o n l y r e l i a b l e n f o r m a t i o n o n t h e p h e n o m e n o n .
T h i s p a p er su m m a r i z es so m e o f t h e ex p er i m en t a l r e su l t s t o d a t e a n d t h e m a t h e m a t i c a l
a t t a c k st h a t h a v e b e en m a d e o n t h e p r o b l e m .
Introduction
A SPINNING missile or body traveling through the a ir
in such a way tha t the bod y axis of rotation is at an angle with the
flight path will experience a Magnus force component in a direc-
t ion perpendicular to the plane in which the f l ight path and rota-
t ional axis l ie . Th e magn itude of this force is a functi on of the
spin rate, the f l ight veloc ity , and the sha pe of the m issile.
This Magnus force has been the subject of much invest igation
in order t o determ ine its effect on the f l ight path of spinning shells
and as a possib le high lift device . M an y ballist ic missiles are g iven
some spin for stabilizat ion even though some may have f ins for
guid ance in the propelled and guided stages. For such missiles
the angle of yaw between the f l ight path and spin axis is usually
small so that a small perturb ation analysis is possib le . For large
yaw , or when the axis is at right angles to the f l ight direct io n, th e
flow separates ( in the case of moderate or large Reynolds num-
bers) and an y meth od of analysis, even appr oxim ate, is v irtually
impossib le at the present t ime.
A review of som e of the solutions is briefly presented along with
the experimental results for the part icular case of a two-dimen-
sional ( infinitely long) cylinder. Th e three-dimension al results
Contributed by the Fluid Mechanics Subcommittee of the Hy-
draulic Division and presented at the Winter Annual Meeting,
New York, N . Y. , November 27 -December 2 , 1960 , of THE AMERICAN
S O C IE T Y O F M E C H A N I C A L E N G I N EE R S . M a n u s c r i p t r e c e i v e d a t A S M E
Headquarters, August 1, 1960. Paper No. 60 WA -15 0.
cover so ma ny different geometr 'es that they are only briefly con
sidered here.
Historical Notes
Th e first record of the drift deviat io n of a spinning bo dy was
descr ib ed b y G . T . Wa lker in 1 6 71 . T he b ody wa s a s l i ce d
tennis b a l l . G . Ma gn us [ l ] ,
1
in faying to account for the drift of
spinning project iles, performed some crude experiments with
musk et balls and with a cylinder in an air jet . H e corr ect ly
a scr ib ed the dr i f t t o a n a erodyna mic fo rce produced b y the
interact ion between the rotation and fl ight velocity which gave
rise to an unsym metrica l pressure distribution pr odu ced by the
Bernou lli effect . His projec t ile drift experiments were perfo rme d
by laying musket balls with their center of mass not coincident
with the geometric center in a musket with their center of mass
in a known orientation. I f the center of mass were to the
right , an impulsive clockwise rotation about a vert ical axis would
be produced by the center of pressure act ing through the geo-
metric center and the inert ia l react ion act ing through the center
of mass. W hen fired, the ball should drift to the right it di d.
Lord Rayleigh was the f irst to set up the ideal f low representa-
t ion. In 1877 he published a paper On the Irregular Fligh t of a
T ennis Ba l l [2 ] in which he presented the ma thema t ica l mod e l
of the f low as the classical potentia l f low around a cylinder with
circulation. A t the t ime he a lso stated that it was no t possib le
to g ive a comple te ma them a t ica l fo rmula t ion o f the a ctua l ph ys i -
1
Num bers in brackets designate References at end of paper.
.Nomenclature,
a = cylinder radius for case of unboun ded flow around a
cylinder
c = radial distance to external vortex ,
z = ee
i y
C
D
= drag coeff icient
C
D o
= stat ic drag coeff icient
C
L
= lift coeff icient
C
p
- pressure coeff icient =
jU = drag
i
= ima g ina ry q ua nt i ty
1
K
= a facto r relat ing velocity rat io and circu lation,
K
a
=
1
r
a a U
w
Journal of Basic Engineering
L
= lift
p = pressure
r = radius, radial co-ord inate
R = R eyno ld s numb e r
u = ra dia l ve loc i ty comp onent ,u ( r , 6 ) , general notation
U = ra dia l ve loc i ty com ponen t in po tent ia l , o r nea r ly i rro -
tational flow region
[ / = f ree s t rea m, uni form ve loc i ty o f a pproa ch a t
x =
v
= ta ngentia l ve loc i ty compon ent ,
v ( r , 6 ) ;
general notation
V
= ta ngent ia l ve loc i ty compon ent ,
V ( r , 6 ) ;
in potentia l or
nearly irrotational f low region
W
= com plex ve loc i ty ,
W = U + i V ,
a lso veloc ity magn itude
(Continued on next page)
S E P T E M B E R 1 9 6 1 / 46 1
Copyright 1961 by ASME
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cal process since no mathem atical meth ods were available to ex-
press the manner in which frict ion between the f luid and the
rotating cylinder would produce circulation.
The f irst quantitat ive data on record were determined by Lafay
in Paris between 1910 and 1912 [3 , 4 ] . He also obtaine d ne gative
lift forces at low rotational speeds.
Ma ny a t tempts ha ve b een ma de to use the h igh l i f t f o rces ob -
tainable on a spinning cylinder in the wind, or in an air stream,
but none has ever led to a f inancially successful venture. Th e
most nota b le a t temp t wa s ma de b y Anton F le t tner in Germa ny
during the period after the First World W ar. Flettner c onsulted
with Prandtl and the Gott ingen group on the idea of replacing a
ship's sails with rotors. In a cross wind, the Magn us effect would
prod uce a thrust man y t imes that for equivale nt sail area . I t
was, of course, necessary to drive th e rotors, bu t the pow er was
only a small fract ion of the power required for screw propulsion.
Acke ret and Buse mann [5 , 6 , 7 ] conduc ted a series of tests on
cylinders with and without end plates which indicated that the
meth od was feasib le . Tw o Flettner Roto r ships, the
B u c k a u
and
th e B a r b a r a [8, 9, 10, 11) were built an d on e sailed acros s th e
Atlan tic. Althou gh this meth od of propuls ion was quite inex-
pensive, the speed and reliability of screw propulsion was more
tha n compet i t ive .
T he Ma da ra s Rotor P ower P la nt , pa tented in 1 9 2 6 , wa s pro -
posed as a rather complicated method of generating power on a
large scale using large vert ical spinning rotors propelling them-
selves around a round or oval track by means of the Magnus
thrust . Generators attached to the wheels generated the pow er
output .
At tempts ha ve b een ma de to employ sp inning cy l inders a s
port ions of high lift wings, but the diff icult ies of complicated
drives and excessive weight have eliminated this applicat ion.
The f irst Magnus invest igation in this country was made by
E. G. Rei d in 1924, at the Langle y Field NA C A La bora tory [12],
using a single cylinder project ing th rough bo th sides of the f ive-
oo t diameter tunnel. Th e most interesting of these results were
never published.
T he most comple te exper imenta l work wa s done b y A . T hom
at the University of Glasgow and reported in his doctor 's dis-
sertation and in f ive Reports and Memoranda of the Brit ish Air-
craft Research Council covering a period of nine years from 1925
to 1 9 35 [1 3 -1 8 ] . T he e f fe cts o f Reyno lds numb er , sur fa ce con -
dit ion, aspect rat io , end condit ions, etc. , were invest igated.
Pressure, velocity , and circulation data were a lso obtained.
A comparison of the Magnus lift force coeff icient as a function
of peripheral to free-stream velocity rat io as obtained by several
invest igators under a number of condit ions is shown in Fig . 1 .
One general trend indicated by these data is that due to aspect
ratio : A f inite and smaller aspect rat io prod uces a smaller l i ft
at a g iven velocity rat io and also g ives a peak lift which has not
been obtained with an infinite aspect-rat io cylinder.
7T
/ d
F i g . 1 S u m m a r y o f p r e v i o u s l i f t -c o e f f i c i e n t C
L
v e r s u s v e l o c i t y - r a t i o
a d a t a
C u r v e i n v e s t i g a t o r a n d
r e f .
A s p e c t
r a t i o
R e y n o l d s
n o .
R e m a r k s
a
( I d e a l f l u i d )
oo CO
b
T h o m [ 1 8 ]
1 2 . 5 5 . 3 - 8 . 8 X 3 X c y l - d i a e n d
h o m [ 1 8 ]
a n d 2 6 1 0
3
d i s k s
c R e i d [ 1 2 ] 1 3 . 3
3 . 9 - 1 . 6 X
e i d [ 1 2 ]
1 0
4
d
( G o t t i n g e n ) [ 2 1 ]
4 . 7
5 . 2 X 1 0
4
1 . 7 X c y l - d i a e n d
d i s k s
e
T h o m [ 1 4 ] 8
1 . 6 X 1 0
4
f
S w a n s o n
OO
3 . 5 X 1 0
4
- 3
X 1 0 5
9
T h o m [ 1 7 ] 5 . 7
3 - 9 X 1 0
4
R o u g h ( s a n d e d )
s u r f a c e
h
T h o m [ 1 7 ]
5 .7
3 - 9 X 1 0
4
S m o o t h s u r f a c e
i
( G o t t e n g e n ) [ 3 7 ] 4 . 7 5 . 2 X 1 0
4
i
S c h w a r t z e n b e r g 4 .5 5 . 4 , 1 8 . 6 X 1 0
4
U n p u b l i s h e d ( C a s e
I n s t . )
k
S w a n s o n
2
5 X 1 0
4
C o n t i n u o u s e n d
s e c t i o n s
-Nomenclature-
X =
X =
y =
Y =
a =
7 =
r =
e =
horizontal Cartesian co-ordinate
force compo nent in ^ -d irect ion
vert ical Cartesian co-ordinate
force component in indirect ion
complex variable, z = x + i y
velocity rat io ,
a = v
0
/U
argument of location of external vortex,z = c e
l y
circulation
general angular co-ordinate in z = r e
e
mea sured f rom
posit ive
x
axis whi ch is parallel to th e free stream flow
direct ion atx = < >
vortex strength, circulat ion
p = density
7/25/2019 (Swanson 1961) Swanson Magnus Effect Review Paper
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One of the first attempts at obtaining an analytical representa-
tion was made by W. G . Bickley [19] in 1928. This is prob ably
one of the most useful analyses with respect to providing an ex-
planation for some of the observed f low phenomena. Bickley
considered the potential flow resulting from a vortex in the neigh-
borhood of a cylinder with circulation in a free stream.
The most complicated mathematical model was formulated
and worked out by Torsten Gustafson [20] and published in 1933.
In a certain sense it may be considered as an extension of Bick-
ley's me tho d. Instead of a single vort ex used to represent the
vort icity shed in the wake, . . . a flow where vortices of constant
intensity are distributed on the logarithmically deflected stream-
lines in the wak e is considere d. Th is met hod is based on
Oseen 's methods of approximate analyses of viscous f lows as ex-
tended by Zei lon. Gustafson 's method is quite tedious and re-
quires numerical solution as well as determination of several
parameters from experimental data.
A recent work by E. Krahn [21] considers the very low
Reyn olds numbe r region in which no wake is formed . A laminar
boundary- layer analysis produces the c irculation as a function of
rotational speed. The m ethod is carried through only for the
case where the two stagnation points coincide at the bo ttom of
the cylinder.
The foregoing investigations have been carried out prima ri ly on
the case of a cyl inder normal to the flow direction. The Magn us
force and tumbling moment produced on spinning projecti les
and missiles at small yaw angles are also of interest and have been
the subject of much investigation [22, 23, 24, 25, 26, 27, 28, 29].
The Magnus ef fect is one of the primary factors contributing to
the drift and dispersion of spinning missiles.
More recently Glauert [30, 31, 32] and Moore [33] have made
some analyses for the small perturbation case of low peripheral to
free-stream velocity ratio.
Experimental Measurements of the Two-Dimensional
Magnus Forces
As previously mentioned, Lafa y was the f irst to m ake q uantita-
tive measurements of Mag nus forces. He also made measure-
ments of the pressure distribution around the rotating cylinder.
The extensive data gathered by Thom for several end conditions
including various end shapes and combinations of end disks give
a variety of data for many basic shapes for the case of 90-deg
yaw . The se results along with those of other investigators are of
primary interest in indicating the effect of finite aspcct ratio.
The smaller the aspect ratio, the smaller the maximum l i f t ob-
tained and the smaller is the velocity ratio at which this maximum
is reached. Leak age flow and consequen t pressure equaliza tion
around the ends of the cylinder is responsible for this aspect-ratio
ef fect. The best approximation to two-dimensional f low ( i .e. ,
indepen dent of end effects) is obta ined by exte nding the cylinder
through the tunnel walls with a ver y small clearance. Th e only
investigation using such an apparatus seems to have been made by
Reid [12] in 1924. M any attemp ts have been made to approach
two-dimensional or inf inite cyl inder end conditions b y adding end
disks. End disks, how ever, give entirely different flow condition s
from those of an infinite aspect-ratio cylinder so that the flow
pattern will be one due to a finite cylinder plus that due to the end
disks resulting in secondary axial flows and end flows around
the disks. Und er such circumstan ces, no combinat ion of disks
on a f inite cyl inder woidd b e expected to produce cond itions
similar to those for ail infinite cylinder.
The closest approach to infinite cylinder conditions is believed
to have been obtained with a three-section apparatus used by the
author [34] . A l iv e cyl inder section mounte d on a long shaft
supported by canti lever strain-gage beams was f lanked by dum my
cylinders running on shafts conce ntric with the main shaft. All
three sections were spun simultaneously using couplings that
transmitted torque, but negl igible transverse thrust. A ve ry
close clearance (0.010 to 0.015 in.) was maintained between the
6- in . diameter cyl inder sections. The du mm y cyl inders were also
extended through the wind-tunnel walls with a c lose c learance to
obtain minimum end ef fects . Dat a were obtained for as broad a
range of Reynolds numbers (based on free-stream velocity and
cyl inder diameter) and velocity ratios as was possible with the
appara tus. Th e force coefficient data are presented in Figs. 2, 3, 4,
and 5.
One of the primary objectives of this investigation was to de-
termine whether or not a maximum (peak) l i f t coef f ic ient indi-
cated by Prandtl [35] would be obtained for an inf inite aspect-
ratio cyl inder. No ne was obtained and it can be seen that the
Magnus lift was still increasing uniformly at a velocity ratio of 17.
Contrary to this behavior , Prand tl had predicted that a max imum
in l i f t coef f ic ient would exist at a velocity ratio of about 4. His
prediction was based on the belief that the stagnation points
would coincide at the bottom of the cyl inder at this velocity
ratio, beyo nd w hich no more vortic ity could be shed; the l i ft
would then remain constant at a value near that corresponding
to an ideal velocity ratio of 2, i .e., C
L u m x
= 4ir. His predictio n
was based on his wel l -known observation and photograph s [36,
37] of flow patterns which indicated that the real fluid flow pat-
tern for a velo city ra tio of 4 correspond ed t o the ideal fluid flow
at 2. Th e apparatus with which these observations were made
had two restrictions which have subsequently proved them to
produce incorrect deductions regarding the actual f low patterns:
First, the patterns are of the flow on a free surface which is at a
uniform pressure and, second, the Reynolds number ( free-stream
velocity, cyl inder diameter) was in the neighborhood of 4000.
A comparison of Prandtl ' s f low patterns with those obtained by
Prof . F. N. M. Brown [33] at Notre Dame University shows a
striking difference. In the actua l flow with a wake behind the
cylinder, the flow patterns will be much different from those
obtained by Prand tl on which he made his hypothesis . One in-
teresting fact shown by the smoke tunnel patterns obtained by
Brown is the lack of rotation of the forward stagnation point pre-
dicted by the ideal flow representation of the flow by a doublet
and single lifting vor tex in a free stream. This pheno men on w ill
be discussed further.
The variation of the l i f t and drag at low Reynolds numbers and
velo city ratios is also interesting. Th e variation in beha vior with
Reyn olds numbe r is attr ibuted to the nonsym metric bou ndary -
layer separation from the top and bottom surfaces of the cyl inder
as explained by Krahn [21] and Swanson [34] . As one example,
consider curve g of Fig. 3 for which the Reyno lds number is sub-
critical (with respect to laminar-turbulent boundary- layer transi -
tion) at 1.52 X 10
5
. At low cyl inder to free-stream velocity
ratios , a, between 0 and 0.1, both the upper and lower boundary
layers remain fully laminar up to their separation points. Th e
upper separation point moves rearward (in the direction of the free
stream) and the lower separation point moves forward. The ad di-
tional length of boundary attachment on the top surface gives a
greater region of negative pressure coefficient here than over the
bottom where the length of attached boundary layer has de-
creased. As a conse que nce of the pressure distributio n, a posit ive
lift is produced.
The boundary layer on the stationary cyl inder at this Reynolds
number is already slightly transitional as is indicated on Fig. 6.
2
As a is considered to further increase beyond 0.1, the relative
2
This
CD
versus R plot for the static cylinder is of interest in that
it indicated the degree of turbulence in the tunnel. Also, the transi-
tion region of what is usually referred to as that where the drag co-
efficient su d d e n l y decreases actually covers about three quarters of
the Reynolds number range up to the point of minimum CD -
Journal of Basic Engineering
S E P T E M B E R 1 9 6 1 / 46 3
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/
F
/
a
Y
4
R
/
/
\
\
p
/
/
/
A
/
\
0
\
A
/
\
/
/
/
/
s
V
Y
/
\
/
N .
/
\
/
1
\
7
j
\
V
/
/
\
i
/
/
\
\
/
/
\
\
I //
V
N
1 /
/
N
/
s
N
i i
f ,
I
A
/
/
/
H
1
/
\
I
E x p e r i me n t a l
( E q u a t i o n ( 7 ) .
l K
a
= 2 . 8 9 , c / a - 3 . 5 1 , Y = 2 4 2 . 1
( E o u a t i o n ( 7 )
\
i
E x p e r i me n t a l
( E q u a t i o n ( 7 ) .
l K
a
= 2 . 8 9 , c / a - 3 . 5 1 , Y = 2 4 2 . 1
( E o u a t i o n ( 7 )
i
/ J
t
E x p e r i me n t a l
( E q u a t i o n ( 7 ) .
l K
a
= 2 . 8 9 , c / a - 3 . 5 1 , Y = 2 4 2 . 1
( E o u a t i o n ( 7 )
w
/
U
0
- 2 . 8 9 , c / a
4 , r = 224
. /
1 2
3
4 5
6 7
8
9
10 11 12
1 3
14
a
F i g .
5
D r a g c o e f f i c i e n t
C o
v e r s u s v e l o c i t y r a t i o
a .
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7/25/2019 (Swanson 1961) Swanson Magnus Effect Review Paper
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1 2 5 4 5
B x
1 0
5
F i g . 6 D r a g c o e f f i c i e n t C o v e r s u s R e y n o l d s n u m b e r R p l o t t e d o n a l i n e a r s c a l e . T h e l e t t e r s
a c e e t c ., c o r r e s p o n d t o t h e f r e e - s t r e a m R e y n o l d s n u m b e r s f o r t h e c u r v e s s h o w n i n F i g . 2 .
velocity over the top of the cylinder is decreased, but it is in-
creased over the lower half . I t is useful to introduce here the con-
cept o f a re la t ive Reyno lds num b er . Where the sur fa ce is
traveling with the free-stream velocity , the transit ion can be ex-
pecte d to be delayed , and convers ely . Th e relat ive veloc ity
R
r o
i
= R ( l a ) ( + s ign fo r b o t to m a nd s ign for t op) . T he
boundary-layer behavior can then be correlated with the relat ive
Reynolds number and a qualitat ive est imation of the relat ive por-
t ions of laminar and turbulent boundary layer and total length
of boundary layer can be obtaine d. Referrin g back to curve g
of Fig . 3 , for a increasing from 0 .1 , the upper relat ive Reynolds
number decreases so that condit ions in the upper boundary layer
can be considered to be similar to those for Reynolds numbers
less tha n 1.5 X 10
E
on F ig . 6 . T he lower b ounda ry - la yer cond i -
t ions correspond to Reyn olds num bers greater than 1 .5 X 10
5
.
For example, for a = 0 .2 , the upper relat ive Reynolds number
would be 1.5 X 10
s
(0.8) = 1.2 X 10
6
and the lower would be 1 .8 X
10
6
. Fig . 6 then indicates a longer attached boun dary layer on the
bot tom than on the top. This effect counte racts that previous ly
mention ed and the lift begins to decrease. A t higher rotational
speeds, a greater port ion of the lower boundary layer becomes
turbulen t and therefore mor e reattached . Th e greater reattac h-
ment on the lower surface produces the negative lift effect shown
and. also results in a lower drag as show n in Fig . 3. Th e lowe r
boundary layer will f inally reach a fully developed turbulent state
near the point of maximum negative C
L
.
At this poin t it is helpful to introdu ce the con cep t of a bo un d-
ary-layer orig in. ' ' Th e posit ions of the boun dary- layer instability
and separation points are a function of a boundary-layer length
Rey nol ds num ber. For a stat ionary surface such as a f lat plate,
a irfoil , or nonrotatin g cylinder, the boun dary -layer length is
measured from the front stagnation point . Th e shear (or rota-
t ion) in the bound ary layer has oppo site direct ion (or is of o pp o-
site sign) on opposite sides of the body from this point (Fig . 7 ) .
I t seems logical to define the beginning of the boundary layer, or
a bound ary-laye r orig in, with respect to such a condit ion if we
accep t the definit ion and idea of a boundar y layer as a shear layer.
For a stat ionary surface, this boundary-layer orig in is coincident
with the stagnation poin t ; howe ver, for a mo ving surface this
boundary-layer orig in will , in general, not coincide with the stag-
nation poin t . On the rotating cylinder, the forward stagnation
poin t is translated in a direct ion oppos ite to the direct ion of rota-
t ion. (In addit ion, it is interest ing to note that the stagnation
point as a point of zero velocity can no longer lie on the surface
F i g . 7 R e p r e s e n t a t i o n o f b o u n d a r y l a y e r s a r o u n d r o t a t i n g c y l i n d e r f o r
a = 0 . 2 a n d R X I 0
4
of the bod y. ) I f the boun dary or shear layers are no w inve st i-
gated, we f ind that the upper and lower layers conforming to the
required condit ions for a boundary layer begin at a point where
the cylinder surface velocity is equal to the f luid velocity ( i .e . ,
where the relat ive veloc ity is zero) . Th e bounda ry layers are
then as shown in Fig . 7 . I t is seen that the b ound ary-lay er
origin is translated in the same direct ion as the direct ion of m o-
t ion of the surface (direct ion of rotation) but opposite to that of
the stagnation poin t . Boun dary-lay er profiles obtained for the
cylinder rotating at two different veloci ty rat ios are presented in
Figs. 8 and 9 . Th e outlines of the bound ary layers are show n.
It is interest ing to note that , for a = 1 , both b oun dary layers are
abou t the same length. In order to calculate characterist ics of the
boundary layer, it is necessary to have a set of init ia l condit ions.
The previously described boundary-layer orig in is the most logical
posit ion at which to attempt to start a boundary-layer solution
for a geom etry where the wall is in a nonu niform motio n with
respect to the external flow.
As the rotational speed is further increased (beyond a value
correspond ing to a ~ 0 .5) the boun dary-l ayer orig in mov es
further back on the upper surface with a consequent increased
length of f low attachm ent produ cing more Magn us lift . I t is
believed that both boundary layers are in the fully developed
turbulen t state at a veloc ity rat io of 1 .0 . An idea of the f low
pattern may be obtained by referring to Fig . 8 . Altho ugh the
relat ive velocity between the surface and free stream is much
466 / S E P T E M B E R 1 9 6 1
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F i g . 8 B o u n d a r y - l a y e r p r o f i l e s o b t a i n e d a t p o s i t i o n s e v e r y 3 0 d e g a r o u n d
t h e c y l i n d e r f o r a = 1 . T h e d o t t e d l i n e r e p r e s e n t s t h e a p p r o x i m a t e l i m i t
f o r t h e t h r e e b o u n d a r y l a y e r s . T h e v e r t i c a l s c a l e o n t h e p r o f i l es is t e n
t i m e s t h a t f o r t h e c y l i n d e r . S c a l e s a r e v /t o r o v e r s u s y / r o .
F i g . 9 S a m e a s F i g . 8 ; a = 1
smaller in the upper boundary layer than in the lower boundary
layer, the velocit y gradients appear to be abo ut the same at equal
distances fro m the boundary- layer origin. Th e main indication
that both the layers are ful ly developed turbulent at and beyond
this point lies in the fact that the drag in this region of a is a
minimum corresponding to the ful ly developed turbulent bound-
ary layer over the nonrotating cyl inder at R ~ 3.5 X 10
s
; in
addition, no further dependence of either C
L
or C
D
on R is indi-
cated beyon d a = 1. Th e foregoing is only a very qual itative
description of the assumed flow beh avior in the light of the e x-
perimental results avai lable. Since no hot-wire data have been
taken within the boun dary layer, the foregoing assumptions have
not been veri f ied.
For velocity ratios greater than one, the force behavior is indi-
cated to be essential ly independent of Rey nolds num ber for 2 X
10' 1 is laminar. How ever, boundary - layer
instabi l i ty and transition, and the transition from laminar to
turbulent flow, in general , are functions of Reyn olds number based
on absolute velocity (considering the Reynolds number and its
consequences to be representative of the ratio of inertial to viscous
forces) . Ove r the upper surface the absolute velocit y is quite
high so that, for free-stream Reynolds numbers corresponding to
laminar separation for the static cylinder, it is possible that the
upper boundary layer is turbulent for the case of the rotating
cyl inder. One of Brow n's photog raphs for 1 < a < 2 (Fig. 14,
ref . [36] ) favors the second argum ent: Th e smoke f i lament on
the upper surface indicates a turbulent boun dary layer. Similar
arguments can be presented for the behavior of the lower bound-
ary layer.
Th e knee of the li f t curve near a 3 is produce d when the
bounda ry- layer origin reaches the top of the cyl inder. At any a
greater than this value, the cyl inder is everywhere travel ing at a
velocity greater than that attainable by the external f low; conse-
quently the boun dary- laye r origin as defined previously no longer
exists . Prand tl considered that the flow pattern produced by the
stagnation point coincidence at the bottom of the cyl inder would
produce a f low pattern and force system that would not change
by any further increase in rotational speed. He reasoned that no
further vortic it y could be shed; therefore the
C
L
would level off
at a value of 4ir . Th e foregoing condition where the boundary -
layer origin reaches the top may be considered to pose a similar
l imitation, except that the C
L
is seen to continuous ly increase, b ut
at a s lower rate than before. As the rotational speed is further in-
creased beyo nd a value corresponding to a = 3, the vortic ity shed
in the upper boundary layer goes from positive (c lockwise) to
negative (counterclockwise) while that shed in the lower bound-
ary layer is alw ays negat ive. In a real fluid, the lift and shed
vortic ity wil l always have opposite s igns ( for the co-ordinate
system orientation shown) and there wil l be a correspondence be-
tween the magnitude of the shed vortic ity and the li f t . As a
is further increased, the vortic ity shed in both boundary layers
is nega tive and increases in mag nitud e. Th e field induce d by this
shed vortic ity has the ef fect of rotating the f low pattern around
the cyl inder in a counterclockwise direction. (An approxim ate
calculatio n of this effec t will be given la ter.) Th e pressure dis-
tributio n produc ed b y this patter n is seen to increase the lift. If
this is the primary ef fect producing the l i f t rise beyo nd the knee,
i t appears as i f some l imit should exist as the wake form ed b y the
separating boundary layers rotates around to a position near the
front of the cyl inder. N o such limiting condition was reached
within the limits of data taken.
The drag behavior is also indicated as a consequence of the
foregoing f low behavior . As a increases beyo nd 1, the drag, sur-
prisingly, increases to a value much greater than the drag on the
nonrotating cyl inder, even though the wake area is decreasing.
Th is large dra g increase with increasing a is prod uced b y flow re-
attachment over the rear of the cyl inder accompanied by a move-
ment of the rear stagnation point and the wake in a counterclock-
wise direction into the region near the bottom of the cyl inder.
The drag peaks in the region where the l i f t knee occurs . Th e
bounda ry- layer origin is at the top of the cyl inder and the separa-
tion points and wake are near the bot tom of the cyl inder. An in-
crease of a as described produces a further rotation of the
wake toward the front of the cyl inder. Th e resultant flow pattern
and pressure distr ibution p roduc e a decrease in
C
D
.
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A Proposed Ideal Flow Analysis
The potentia l f low pattern of a free stream, doublet , and bound
vortex combined to y ield the f low on a cylinder with lift is quite
familiar. This simple potentia l function and its solution yield a
lift,C
L
= 2 i r a ( a = v
0
/U , but no drag is predic ted.
As one s tep b eyond the s imple Kut ta -J oukowski theorem o f
lift , consider the force on the cylinder produced by the circulation
from a singularity (potentia l vortex T) within the boundary as in
the Kutta-Joukowski theory and an addit ional vortex downstream
from the cylinder representing the shed vort icity in the wake that
is continuo usly being carried down stream . This movin g vortex
sets up an induced velocity f ie ld which acts to produce a force in
addit ion to that arising from the interact ion of the free-stream
velocity and the circulation within the body contour.
In a very interest ing paper, W. G. Bickley [19] has set up an
a pprox ima te ma thema t ica l mode l fo r th is prob lem in which the
net vort icity moving in the wake is considered to be concentrated
in a single vortex of opposite direct ion to the lift -circulat ion
around the cylinder (Fig . 10). Express ions for the transportation
ve loc i ty components o f the she d vortex a re g iven a nd f rom
these and the velocity potentia l for this f low model, the lift and
drag are determined for an arbitrarily prescribed location of the
shed vortex.
v o r t e x
Consider the f low around a circular cylinder of radius a w ith
posit ive circulation and superimposed on this, a vortex of strength
k
a t
2
=
a
A
= ce '
7
, and the necessary vortexes +
k
at the inverse
a
2
point ( image point in the circle)
z = z
B
=
e
17
,
an d
K at 2 = 0
c
to maintain the circle
z
= a as a streamline (Fig . 10). Th e vortex
a t A is a free vort ex; those atz = 0 are bound, and the vortex atB
is restricted to move such that it is a lways at the inverse point of
A . With th is a rra ngementz = a is preserved as a stream line of
the flow as
K
a
moves a ccording t o the ve loc i ty induced a t
A
by the
rest of the f ield. This poten tia l f low representation is an ap -
proximation to the actual f low where the net vort icity continu-
ously shed by the cylinder and washed downstream is considered
to be instantaneously concentrated with strength
a t
A ,
and
with an instantaneous velocity as determined from the rest of the
field. Th e net vort i city within the cylinde r conto ur is the li ft
circulation.
Th e com plex potentia l fun ction for the f low of Fig . 10 is
,
T r
( . a
2
\ i ( T + k ) i n 2 - Z
A
1)
a
2
.
where z . , .= ce
,y
an dz
B
= e
l y
. T he complex ve loc i ty i s g iven
c
2
d F t a
2
\
W > - U > - i V > - - - - U
m
( l - )
_ t r +
K) _
j k / I i _ \
2
TZ 2 i r \ 2 z
A
z z
B
)
For ideal motion of an incompressib le f low the Blasius theorem
yie lds the fo rce on the cy l inder b ound a ry :
d< f> -
~ d T
dz
^
The first contour integral on the right-hand side of (3 ) g ives the
convect ive or
W
2
contribution to the force and the second term
g ives the unstea dy or t ime-dependent contr ib ut ion . T he contour
of integration is the cylinder surface,
z = a .
After performing the necessary calculations, the lift and drag
coefficients can be determined as:
+
S T . ( ' 0 ^ 0 ^ f H
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(7)
cos y / a \ ,
- - ^ T ( 7 ) ^
Even considering that this model presents a good approxima-
tion to the actual f low condition, the parameters c / a , 7 and K
a
would be expected to be functions of or ; consequently, i f these
relationships could be determined, a better f i t to the experi -
mental results would result . Unfortun ately, there is no way by
which c / a ( a ) , 7 ( a ) , a n d K
a
( a ) can be determined.
It is interesting to note that the field induced by the trailing
vorticity has the effect of rotating the entire flow pattern in a
counterclockwise direction about the center of the cyl inder ( in
the same direction as the l i f t c irculation T ) . The forwa rd stag-
nation poin t is then locate d closer to the fron t (6 = 0) of the cy lin-
der and the rear stagnation poin t is c loser to the botto m (6 = 90
deg) . For example, for the values of
c / a = 4 , 7
= 2 24 deg, and
K
a
a = 2.89 c ited in the foregoing ex ample, the stagnation points
for a = 1 are locat ed at 9 deg and at 180 + 21 deg ; for a = 2,
at 19 deg and 180 + 50 deg. Th is pred icted effect is in agree -
ment with visual observations by Brown [36] and Gustafson
[20] and meas ureme nts by Swan son [34] (Figs. 8, 9). In reality,
the rear stagnation point d oes not exist because of the wake region
formed . It is of s ignif icance that for the range of Reyno lds nu m-
bers investig ated (3 X 10
4
to 5 X 10
6
) the wake is always present
at all values of a. The refor e, for a real fluid flow at high Re yno lds
numbers, the condition where the two stagnation points can
coincide will not exist and a separated wake will always be present.
Indeed, some smoke studies by Gustafson [20] at very large a (5
\
\
\
\
\
\
/ >
/
A
//
f
/
E x p e r i m e n t a l
( Y = 2 1 0
/ c / a -
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2
Lord
Rayleigh, On the Irregu1ar Flight. of a Tennis Ball,
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En g lish translation by J. P. den Hartog, McGraw-Hili Book
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