s
i.
Introduction
In a remarkable paper on the mtion of pendulums
Stokes showed that the expression for t1e force
on a
sphere oscillating in ai unlimited viscous fluid
con-
sists of two ternis, one iiivolving the acceleration of
(lie sphere and t lie ot lier tue velocity [1] 2 Further-
iore, the inertia coeflicient involved in the accelera-
on tena is
mo(lifie(l because of viscosity and,
indeed, is augmented over the theoretical value valid
for irrotational flow. The drag coefficient associated
vitli the velíity term is modified because of the
acceTeration, a ad its vailue is greater t han it would
&thejiliie were inovilig with
a constant veIok.
Sulisequent'to Stokes' siulics, the forces ou
a spTliere
moving in a viscous fluid in
ari arbitrary manner
were investigated by Boussinesq a ¡1(1 also by Basset
[2, 3].
They found that the force experienced by
a
s here at a. riven time dc wiids in reneral, on the
entire history o its HcCC ciation as well as t
ie instan-
taneous_velocity and acceleration. As
an example,
iîa sphereis accelera teT,
say with a constant accel-
eration, from a position of iest t.o
a finite velocity
and is then kept at this velocity, the force (luring
the initial instants of uniform velocity differs from
the force occurring at a later time.
Rayleigh has
given the formula for the force for this
case [4]. The
force expression of Boussinesq-Bassct contains three
terms, one of which is in the form of
an integra!
involving the history of acceleration. If the integral
evaluated when the acceleration is represented by
sinusoidal function it then yields the modifications
of the inertia and drag coelhicients in Stokes' formula.
One expects quantitatively different results
when
the oscillating velocities
are large and the flow
turbulent.
As yet a theoretical analysis of the
problem is difficult and much of the desired mf
orma-
tion must be obtained experimentally.
In this
respect the experimental studies have been dealt
with variously.
One method is due to McNown
and Wolf [5], who considered the force
on a two-
Investigation sponsored by the Office of Naval Research.
Figures in brackets indicate the literature referentes at the end
of this paper.
dimensional object immersed in a flow as made up
of three parts:
F= Ap
d(kU)
+pzlS+ UDpU!UI,
(1)
where F is the force per unii, length in the direction
of flow. x; U tine velocity at points far removed from
the object; p, the x-comnponent of the ambient
pressure in the absence of the body; dS, au element
of flic surface area; C, the coefficient of drag; and
k, the virtual mass coefficient. nue dimension of the
body normal to the flow is D. and
l
is a circular
nua, i1o=rD2/4, to which (lie added mess is referred.
If A is the cross-sectional tuca of tue body, A=rA0,
r being a ratio, them
p2dS=prilo
LIt'
and finally
F A0p
[d(kU)
+r]
+ caDpUU.
In tItis approach tite variability of the mass co-
efficient, k,
is implie(l.
Cf}15
introducing a new
coefficient k' sudi that
k'=±(kU)
dt dt
and putting
o
G,,t= (k'+r),
there is obtained from eq (1),the expression
F=OmpAo GdDpUjU,
(4)
which in fact constitutes a second approach utilized
first by Morison and coinvestigators
[n,
7]. The
form of time expression is in agreement with the
Stokes formula for force on a sphere oscillating in
a
viscous medium.
In a general sense one may still
regard Cm as a kind of mass or inertia coefficient.
c2e21
¿/L
Q4i
F#8fIum voor'
(3)
f-
i
t
l. t
f
L eqt,v
Tournoi of Research of the National Bureau of Standards
Voi. 60, No. 5, May 1958
Research Paper 2857
Förces on Cylinders and Plates in
an Oscillating Fluid'
Garbis H. Keulegcin and Lloyd H. Carpenter
Thc inert ¡a nid drag coefficients of cylinders and plates in simple
sin isoichui eu rreti ts
are iìvetigated. i'he inidscf ion of a rectangular hui, with standing waves surging in it
s
scicctcd as the locale nf current s. The cylinders and plates are fixed horizont ally and bciuw
tite water surface. Tite average values of the inertia and drag coefficients
over a wave cycle
show variations wlicit
I he intensity of the current and the size of the cylinders or plates arc
changed. These variations, however, can be correlated with the period
par;imcier U,,, T/D,
2
X
where Um is the njaxitn,I,u intensity of the sinusoidal current, T is the period of tite
wave
and 1) is the diameter of the cylinder or the width of the plate. For tite cylinders U,,,T/D
eqi ahug 15 is a critical condii ion yielding the lowest valut of the inertia coefficient tind the
largest value of tite drag coefficient. For the plates tite higher values of the
drag coefficient
are associai ed with the smaller values of UmT/D and the higher values of tite mass coefficient,
with the larger values of Um T/1).
Tite variation of the coefficients with the phase of the
wave is examined and the bearing of this OIL the formula for the forces is discussed. The flow
patterns around the cylinders and plates are examined photographically, and
a suggestion is
advanced as to the physical meaning of the parameter UmT/D.
/
s
I
A thuud nppioarlì wns ploposed l)y Iveisen aìil
Baletit,
vhìo considered flic force on tin accelerated
(lisk iiovi1tg ill Oil(' direction [S].
Brielly,
F= CPDU2,
(5)
where
¡DU D (lU
p
1ehn lins (OlÌSi(I('1'C(j the case of accelera t cd cvlijidcrs
E 9] 1111(1 Buigi iu icflo t lia t of ac('c]cratod spheres [JO],
all notions liciiig iii one direction.
ITere tite resort
is to a single coeulicieiit G aiìd atten)1)ts to separate
the effects of accek'ratioii and viscosity ii:ve not
heeii sliowit to be successful.
Aecoi'dingly,
the
adoption of this method can have a meaning only
for inonoto flic Inotiolis su bj ecL to defini te linñ tations
as to initial and final conditions.
For oscilhutory motions, although tite forces are
more accurately described
either using eq (2) or
eq (4), the latter might he prefcrrc(.1 J)iovidcd the
coeíiicients G and C, could be predicted vit1i some
precision.
flie application of the expression to
vertical piling and large submerged objects by ileici
ftfl(t Bretscliiieider stresses tite necessity of having
these coefficients better cletcrmine(l [1 1].
On the basis of irrotational flow around the cyl-
ili(ICF, 0m should equal 2, and one may suppose that
tue value of Gd should be identical with that appli-
cattle to a constant velocity. \Ioiison and coinvesti-
gators have obtained the values of G flTl(1 Gm in
particular cases b
considering lite observed forces
iii the phases of tite wave cycle where U or ¿U/dl
vanishes.
Such determinations show considerable
variations of G,,. froiii the theoretical value and of
G, from the steady state value at the corresponding
Reynolds number.
Dealing with
field stlI(hcs at
Caplen, Texas, R. O. Reid found similar variations
in CÇ,. and G, [12].
The variations itì tite coefficients,
however, have not yet been
correlated with any
appropriate parameter.
The present investigation was undertaken with the
following two objectives in mind.
rflle first was in
regard to a supplementary function
that could
bC introduced in eq (4) for a truer representation of
force w-lien considering the coeíflcients Gm and G, as
being constant throughout a given wave cycle. The
necessity for the term
R is associated with tite
eventuality that the point values of
G,,. and
Gd
deviate from their average values.
The secoiid
objective was to examine the possibility of correlating
tite average values of Gm and Gd with a parameter
Um T/D, where U,,. is the amplitude of the harmoni-
cally varying velocity, T is the period of the oscilla-
tions, and D is time diameter of a cylinder or the
breadth of a rectangular plate.
Tite mid-cross
section of a large rectangular vessel with standing
waves surging in it was chosen as the field of harmnoni-
cally varying current. The cylinders and plates were
held fixed horizontally, totally submerged in water
and extending from one side of the vessel to the other
to approximate as closely as possible the condition
of infinite length.
2. Fluid Forces on an Immersed Body at
Rest in a Moving Liquid
it would be instructive to consider tire TltOInenl,ujn
e(ivaf ions discussed by Muritaghman for tite evaluation
of force ori objects immersed in a perfect liquid [i:J.
The trietimod, however, is now generalized to apply
lo inìperfeet liquids.
Consider the case of two-dimensional flow with
x
horizontal and z vertical.
The equation of motion
in the x-direction is
/òu òu
òu\
òr
òz
(6)
where u and w are the velocity components along
the axes x arid z, p the density of the liquid,
Pxx the
imormal stress on an elementary surface perpendicular
to r, and p
the tangential stress on an elementary
surface normal to z, the stress being in tite direction
of s.
Because of the incompressibility of the liquid,
òu òw
òx+òz
O
and eq (2) becomes
òu
pp ( U+-
uw)=Z?m+
òx
òx
Take tite immersed cylindrical body of surface S,
as in figure 1, arid draw a surface S' of arbitrary shape
winch encloses the cylinder.
Let w he the region
bounded liv Sand S' and latid'n the direction, cosines
of t lie normal drawn inward into the regioni.
Inte-
grating eq (S) throughout w, and in this making use of
Green's Theoremni, one finds
Pf/wPfU(lU+flW)dS_Pfu(lu+nw)dS'=
f(lPzx+flPzz)dS_f(1Pxx+flP)dS'.
I,
(ii
Fia tilE 1. Notation diagram for force analysis.
Over the surface S of (lie muueised body
iu+nw Vari-
ishes because flic body is at
rest.
Also
1', (hilt is, flic
x-componciìt of the force ex-
erted on the solid by the moving liquid.
It may be
assumed that if S' is i'eiiìoved
sufficiently from the
both- the tangential stress
p
on S' vanishes and the
normal stress p
reduces to the hydrostatic
pressure
p. Solving for F,
F= _Pf
(f''+ PfU(lU+nw)dS'+f lpdS'.
(10)
The later relation
may be given in another form,
suitable for the present
purpose.
Select the bound-
ing surface S' as the rectangular
strip shown in
figure 1.
The plane S to the left of
the cylinder
passes through ti e point x= x1 and the plane
S to
the right passes through
x=x1.
Denoting the hori-
zontal velocity components
at the points P1 nnd P2
with the common elevation
z by u1 and '2, and the
pressures by- Pi and P2, eq (10) now reduces
to
F= -f dw+f' (u?u)dz1 +f
p' p2)dz1,
(11)
which is the momentum equation
of familiar form.
This may he specialized
to evaluate the force on
a
circular cylinder when the
motion is irrotatiorial.
Letting U be the undisturbed
velocity and referring
to Lamb [14],
a2
cos
28]
sin 28
pdU/ a2\.
i
-=-
r+) sin 8 (u2+w2)
where a is the radius of the cylinder,
r is radial dis-
tance, and O is the angle between
a radius vector and
the vertical line x=0 passing
through the center of
Icylinder.
Clearly, u1=u2 and the
momentum
ation, eq (11), reduces to
F1= f
1i+f(pi_p2)dzi.
(13)
w
[utroducing the values of
u and p from eq (12), and
)Initting the straightforward but
somewhat lengthy
waluations, the result is
dU
F12ir
-- a2p,
r in terms of the diameter D of the cylinder
425
(12)
L'
pirT)2 1U
iCmTj
V]IeIC (7m2.
Next, suppose that flic undisturbed
velocity is con-
stant and (hut the body experiences
a drag. With the
liquid ctcnding to infinity and
ignoring the variation
of I)resSures from
the shedding eddies,
or, more prop-
erly, assuming that the surfaces
8 and S are far
removed froni flic cylinder,
PI=P2, and eq (ii) re-
(luces to
(15)
The velocity u1=U, and u2=mU,
where m is de-
pendent on z1/D and
on Reynolds number UD/y.
Thus,
(14)
Ca2f(1_m2)d
$.
It,
:1 PP(a1S that in ordinary casos where the flow
departs from irrotatioiiljty
and becomes urist cacly
and eddviri, eq (i I ) is still flic basis
for evaluating
the force. SiIle( the first and third
integrals niay be
associai ('(h.
vitli acceleration and the
second
vith
(Irag.
l'huit
is, the coefficients C,,, and C
are de-
rive(l fiota eq (13) and (15)
providsl the velocities
nial l)I0sSI.u!es car-i be
giVen.
'flic fuNe of the state-
fieri t is only ilca(lCflIiC, since iii tIte flows involving
separat ion
t'ud intermittent eddy formation
the
pressures arid velocities arc not known
auch the
integrations itt eq (11), at present,
cannot be carried
out.
N everilueless,
experience
suggests
that eq
(4) remains useful at least for
sinusoidal motions, if
allowance can be made for the
variations in
C,,
and
Cd.
I-lad one carried out the integrations
in eq (11)
for an extended plate using the
known velocity ex-
pressions derived from the Kirchoff
solution for the
impact on a lamina, definite values for
Cm and C
would have resulted.
This would have shown in
principle the existence of
a relation between Gm
and Cd in the absence of eddy formation.
In the
Kirchofï solution the wake is of infinite
length and
this is entise for concern. McNown
overcomes this
difficulty by considering the
case of a closed wake
as between two plates and finds a relation between
k and Gd or between Gm and
0d [15]. This result is
very significant as it points to the path to be fol-
lowed in analytical approaches
taking into account
also the effect of the eddy
processes. With cylinders
the chuuiging separation seats
are a cause of added
difficulty.
T\Ieanwhule, the tasks of the experimental
investi-
gations become more
necessary.
Not only are the
needs of flic applied arts to be
fulfilled, but also
there must be clarification
as regards the flow pro-
cesses during unsteady flows.
ti
u i
u
Fi=CdpD
(16)
where
Ii
3. Cylinder in a Field of Sinusoidal Motion
Forces oli n evlitider admit a n easier representation
when the undisturbed portion of the flow, infimiit e
in extent, is varying hannomiically. Let the velocity
he given by
U=U,,,cosoi,
(17)
where Um is the SeflhiluÌiJ)litUdC of the current, T
hie period of the alternations, and o=27r/T.
The
force on the cylinder per unit length F is in general
l'=J(t, T, U,,,,D,p,v). (IS)
Grouping the variables on the basis of dimensional
reasoning
F
¡'t
LTmT UmD\
pUm2D=\T' D '
y
)'
or introducing
0=27rt/T,
()
(19)
F
/ UmT UmD
D '
y
(20)
where UD/v is a Reviiolds number and TjmT/D
will be termed the ''period parameter.''
Bearing
in mind that F is periodic, and that because of flow
symmetry
F(0)= F(8+ir),
ve have
PUDA1 sin 0+A3 sin 30+A5 sin 50+.
+B1 cos 0+B cos 30+115 (OS 50+ . . .
. (21)
Here the coeflicients
111,113 ...,
amI(l B, 113
. . . aie
independent of 0, and are at most functions of
U,7'/D and U,D/v. A simrt
method of approach
iii the analysis of the observed force curve is to re
sort to a Fourier analysis to determine the coefli-
cients A1 . . .
B1 .
ire/o
P
i "2FF sin nO
U,W
do
and
1 ' F cos nO
B=
UW
do.
7rj0
p
Once the coefficients are obtained, their dependence
On UmT/D and UmD/P may be established, provided
the observational data are of sufficient iiumher and of
large extent.
The above general and fundamental relation, eq
(21), may be reconciled with eq (4), which is the forni
which M.orison and coinvestigators Reid, Bret-
schneider and others, have adopted in their numerous
studies.
Introducing U from
eq (17) into eq (4)
irC
.O
tI
0 0 24
P1J,D
m
Sill
COS
COS
426
By time rule of Fourier
('2'r
I
cos
olcos 0 cos nOdO
./0
cos Ocos 0==
=ao+aj cos O+a2 cos 20+a3 cos 30+.
where
a,,=0 for n even,
!±
8
a=(-1) 2
fornodd,
n(n2-4)ir
8 8 8
a1=, a3=-- a5=-
37r loir lOöir
Introducing this in eq (21), and writing
B=B3 B1
a1
B=B5 B1
a1
one has
PUDAI sin 0-FA3 sin 30+A5 sin 50+
-I-Bn cos 0! cos 0+B cos 3O+B cos 50+
Now eq (24) and (27) may be compared.
One can
write
r
Di
A1+A
sin 30
sin 59
+A5
Um
sitiO
sinO
and
B
B'
cos 30
B cos 50
cos O! cos O
! cos o! cos
0+
or
UmT
[A1+A3+A5+2(A3
ait)
D
+A5) cos 20+2A5 cos 40+
.
. .]
(28)
and
2
[2(BB)+4(BB) cos 20
G(0)-2B+1
cosO!
4B cos 40+
.
.
.1.
(29)
Thus if A3, A5, and B, B vanish, the coefficients
of mass and drag remain constant for all the phases
n=O
so
2,T
cos2 nodo
o
in the wave cycle and
Itì tile event that these coefficients vary with the
phase O of the wave cycle, the values given by eq
(30) and (31) are in a sense the weighted averages
C1
I
Cm(0) alU2 Odo
1rJ0
and'
P2r
Cd=+
I
Ce(o)
cos
ej
cos2 ode.
(33)
4Jo
With the above possibilities in mind, it is prefer-
able to adopt the expressions
F
sinO+13 cos O cos 0j+R (34)
or
F
ir Lhr.
C,,
C'mTr sin 0--- cos Ojcos 0+R, (34a)
where A1, B, Cm, and
cd
are constant, and R lias
the value
R=A3 Sill 30+115 Sin 5O+J3 cos 30+13 COS 50.
(:35)
The function 1? will be referred lo as the remamder
function, end thieii
i Iii
remainder function is ob-
tained h
slll)t.raeting the computed values of A1
sinO and B Icos O cos O horn the observed F/pU,D.
The, remainder thus obtained niiiy be examined in
regard to its Fourier structure alill also as to its
magnitude.
4. Characteristics
of the Experimental
Waves
The region under the nodal area of a si ending wave
that may be realized iñ a rectangular vessel furnishes
a velocity field of simple harmonic motion in the
velocity component U.
This circumstance is not
seriously modified even when the surges aie moder-
ately high.
Taking the x-axis in the plane surface of the un-
disturbed water, the z-axis vertical and upwards and
the origin at one end of the basin, (sec fig. 2), the
surf ace elevation as reckoned from the undisturbed
level, according to the second-approximation theory,
from Miche 16], is
ak
h=acos kx slnct+aTNlcos 2kx-
ak
a--N2 cos 2kx cos 2at, (36)
where
and
COSI) 2kH
N1.
smb 2k11
cosli2 k]? (coshì 2k11+2)
N2==
2
smnh kllsuihkH
Here k=ir/L, L being the length of the basin; o=
2ir/T, T being the period of oscillation; II the (lepih
of water; and a the semiwave height, that is, the
mean value of the extreme end deflections in a cycle.
The expression for the periodE is the same as iii the
first-approxunation theory, that is,
o'2=gk tanh ¡cli.
(37)
Focusing attention on the basin end x=0, tire surface
displacement is
/=asint+aNiaN2 cos 2ut;
x=0. (38)
Tlìus, the maximum elevation, occurring at t=ir/2, is
h1= a+a'[Ni+N2],
and the maximum depression, at t=3ir/2o, is
h2=
a+a[Ni+
N2].
l'ue ratio of the elevai ion to the depression is
Hic
<1
+llc[Ni+N21)/(1
-
[N1+N2]),
(40)
amid accordingly its value increases with wave height.
rflìe surface configuration for t=0 is
X
L
FIGURE 2. Notation diagram for wave profile.
(39)
2 UmT
_2 Um7'12'F5111 0(10
(30)
Cm
D
pUD
and
: P27 1' ,..
(31)
(Yd2B1=J
t=0.
(41)
li=a[N1N2 cos 2k.rJ,