(swanson 1961) swanson magnus effect review paper

7/25/2019 (Swanson 1961) Swanson Magnus Effect Review Paper

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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


3/10

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 -


S E P T E M B E R 1 9 6 1 / 46 3



4/10

/

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 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 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 .



6/10

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

Transactions of the

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7/10

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

.


S E P T E M B E R 1 9 6 1 / 46 7



8/10

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


9/10

(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 -


10/10

2

Lord

Rayleigh, On the Irregu1ar Flight. of a Tennis Ball,

a) "Scientific Papers, vol. 1, 1B57, pp. 344-6. " Messenger oC

Mathematics,

vol. 7,

1B77, p.

14.

3 LaCay, Sur l'Inversion du

Ph6nomene

de Magnus, Comptes

Rend1l8, vol. 161, HHO.

4 Lafay,

Contribution Experiment.ale n l'

Acrodynamique

du

Cylindrc

,"

Rermea Mechaniqlle,

vo l. 30, 1912.

5 A. Bet.z, Del' Magnuscffckt., die Grundlage del' F lettner

Walze, VOl, 1925, p. 9. English translation, NACA TM 310,

1925.

6 L. PrandU, Mngnuseffekt uud Windkrnftsehiff, Die

Natur

wissenschaften, 1925, p. 93. English translation, NACA TM 367,

1926.

7 A.

Buseman,

Mcssungen an

Rolicrcnden Zylindern, Ergib

nisse del' Aerodynamik Versuehsanstalt zu Gottingen,

IV

Lieferung,

1932,

p.

101.

B R. Rizzo, The Flettner

Rotor

Ship

in

the

Light of

the

Kutta

Joukowski Theory and oC

Experimental

Results, NACA TN 22B

1925.

9 H. Ziekelldraht,

Magnuseffekt, Flettner-Rot.or

und Dllsen

FIUgel," Schweiz Tech7i. Zcitschr. , vol. I, 1926,

pp.

B37-843, B55-857.

10 F. Ahlborn, Del' MagnuseO'ekt in

Theorie und

Wirklichkeit,

Zeilachr. /. Flllotechm k

laid

Motorlll/lschiJJahrt,

Dec. 28, 1929,

pp.

642- 653. English

translation,

NACA TM 567.

11 A. Flettner. The Flettncr Rotor Ship,"

l -'1I0illce,.i,10,

vo l. 19,

January 23, 1925, pp. 117- 120.

12 E. G. Reid, Tests of Rotating Cy linders, NACA

TN

209,

1924.

13 A. Thom, Th e Aerodynamics of a Rotating Cy linder, thesis,

University

of Glasgow, 1926.

14 A. Thorn, Experiments ou

the

Air Forces on Rotating

Cy

Hllders,"

ARC

Rand M lOIS, 1925.

15

A.

Thorn,

The

Pr e

ssure s

Round

a

Cylinder Rotating

in

au

Ail' Current, ARC

Rand

M lOS2, 1926.

16 A. Thorn, Experiments on

the

Flow Pnst a Rotating Cylin

der.

ARC

Rand M 14

10,1

931.

17 A.

Thorn aud

S. R. Sengupta, AirTorquc

on

a

Cylinder

Rotat

ing in an Air Stream, ARC Rand M 1520, 1932.

18 A. Thorn, Effects of Discs

011

the Air Forces on a Rotatillg

Cylinder, ARC Rand M 1623. 1934.

19 W. G. Bickley, The InBuence ofVorticcs

Upo

n thc Hesistallce

Experienced

by

Solids Moving Through a Liquid, Proc

eccii1

lgs,

Roya

l

Society (London),

vol. 119, scries A, 102S,

pp.

146-156.

20 T. Gustafson, On

the

Magnus Effect According to the Asymp

totic Hydrodynamic Theory," Hakan Ollissons Buchdruckel'ei, Lund

(Sweden), 1933.

NACA

trans . N-2M121, 1954.

21 E. Krahn, Tho Laminar Boundary Layer

on

a Rotat.ing

Cylinder in CrossBow," NAVORD Rep. 4022, Aerobal. Res. Rep. 28B

U. S. NOL, Maryland, June, 1955.

22 J. C . Martin, On Magnus Effects Caused by the Boundary

Layer Displacement

Thickn

ess on Bodics of Revolution at Small

Angles

of

Attack,"

BRL

Rep.

870, D

ept. of

the

Army Prou.

5B03-03-

~ O l Ord.

Res. and Dev. Proj. TB3-0108,

Aberdeen

Proving Groun

d,

Maryland,

June, 1955.

23 Karl

H.

Stefan ,

Magnu

s Effect on a

Bod

y

of

Revolution at.

Various Ang les of Att.ack, Master's t.hesis, Dept. of Aero.

Engr

.

Univers

ity of

Minn.,

1949.

2 1 William E. Buford, Magnus Effect

in the

Case of Rotating

Cyli nders

and

Shell," BRL Memo

Report No.

821, July, 1954.

25

Wind

Tunnel Group,

CO

il vail'

Wind

Tunnel Handbook, vo l. L,

COil vail' Repl. 2T-043, April, 1955.

26

John W. MaeeoD,

Aerodynamics

of a Spinning Sphere,"

Jour1lal a the Royal Aeronautical Society,

vol. 32, 1928, p. 777.

27 J. M. Davies, The Aerodynamics of Golf Balls, Journal

0

Applied

Physica, vol. 20, 1949, p. 821.

28 R. H.

Heald,

J. G. Logan,

Jr

., H.

Spivak.

and W.

Squire,

"Aerodynamic Characteristics of a Hot.ating Model of the 5.0-Inch

47

I

SEPTEMBER 1961

Spin-Stabilized Rocket Model

32,

U. S.

Navy

BuOrd Tech. Note

No. 37, June, 1957.

29 M. J. Queijo Rnd H. S. Fletcher, Low Speed

Experimental

Inve

st

igation

of the

Magnus Effect on

VarioW

Sections

of Body of

R evo luLi

on

With

and

Without a

Propeller,

NACA

TN

4013,

August,

1957.

30 M. B. G lauert, A Boundary Layer Theory

With

Application

to

Rotating Cylinders,

Joumal

0 Flltid .

Me

c

ha1l'lca, yo l. II,

Part

I,

1957, p. 89.

31 M B. Glauer t , The Laminar Boundary Layer Oil Oscillating

Pla tes and Cylinders," JOllnlal 0/ Fluid M

ec

hallic8, \ 01. I, Part I,

May,

1956, p. 97.

32

M.

B.

Gl

auert,

The

~ l o w

Past

a

Rapidly Rotating Cylinder,

Proceedi1l01J

0/ ths Royal Society, vol. 242, series A, 1957, pp. 108-115.

33

D.

W. Moore,

The

Flow

Past

a

Rapidly

Rotating Circular

Cy

linder

in a Uniform Stream, Joltr1lal

0/

Ffltid e c h a l ~ i c s vol.

II,

Part 6, 1957, p. 541.

34 W. M. Swanson, An

Experimental

I

nvestigation

of tbe Mag

nus Effect, Final Report,

OOR

Proj.

No.

1082, Case Inst. of

Tech.,

Deecmber, 1956.

35 S. Goldstein , Modern Development.s in

Fluid

Dynamics,

vols. 1 and 2, C l

arendon

Press, Oxford,

En g

land, 1938.

36 R W.

Van

Aken a nd H. R. Kelly, The

Magnus

Effect on

Spinning

Cy

linders, " Reprint No. 712, paper prese

nted

at the 25th

Annual Meeting, LA.S., 1957.

37

L.

Prandtl and O.

Tietjens, Applied

Hydro and Aerody.

namics,

En g lish translation by J. P. den Hartog, McGraw-Hili Book

Co

mpany,

Inc., New York, N. Y., 1934.

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