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Tel.; 015-786873- Fax 015- 781838ON VISCOUS DRAG OF OSCILLATING BLUFF BODIES
YIIKEDA and NUTANAKA
Department of Naval Architecture
University of Osaka Prefecture
Japan
* This paper will be read at the 12th Scientific
and Methodological Seminar of Ship Hydrodynamics
(SMSSH'83), held on the 19-23. Sep.,1983 at yama
,Bulgaria.
* This paper will be read at the 12th Scientific and Methodological Seminer of Ship Hydrodynamics (SMSSH'83), held on the 19-23 Sep.,
1983 at yama, Bulgaria
ON VISCOUS DRAG OF OSCILLATING BLUFF BODIES
Y.IKEDA añd N.TANAKA
. INTRODUCTION
Time-dependent viscous förces take important roles in several problems in the ship and marine hydrodynamics, for example, the roll damping of ships, the wave forces acting on offshore structures and the nttions of floating structures. Since the time-dependent viscous forces become important when the flow separation occurs, it is
dif-ficult to treat them theoretically. Thus, many experimental works have been carried out in these twenty years,to reveal the characteristics of the viscous drag acting on oscillating bluff bodies (or bluff
bod-ies in oscillating flow) to some extent. The characteristics of the viscous hydrodynamic forces acting on oscillating bluff bodies depend on the wake pattern around the bodies. The wake pattern or flow pattern depends strongly onKc number
(Keulegan-Carpenter number), as Keule-gan and Carpenter pointed out already 30 years ago
CiJ.
Fig.i shows the wake patt-ern around a circular dylinder in oscilla-ting flow in various ranges of the Kc number C2J. At very low Xc number(sxnaller than Kc=4 for a circular cylinder), no separation of the flow occurs. In thisregion, a theoretical calculation by the time-dependent boundary layer theory can be applicable. At low Xc number (in the region of Kc=4 to 8 for a circular cylin-der), a thin separation bubble is formed
at the surface first, and gradually increases its thickness, and then forms a symmetrical wake. The hydrodynamic viscous force in this Kc-region pays an im-portant role in the roll motion of a ship in waves ,as will be described later.
At high Xc number (larger than Kc8 for a
circular cylinder), the wake behind a bluff body becomes asymmetric as shown in Fig.1. In this region, the vortices move complicatedly due to the indúced velocities by the other vortices. The asymmetric wake causes a transverse force (lift force), which takes an important role as well as the in-line force (drag force).
Im the present paper, the time-depen-dent viscous drag is classified into four
classes from the point of view of the wake pattern, that is, the drag without separa-tion, that due to the thin separation bubble,
that due to the syntrica1 and finally due to
theasymmetrical wake. Each drag has différent characteristics respectively, and we should do the different theoretical and experimental approaches to each of th. The characteristics of the viscous drags for each classes are discussed in detail respectively. Practical examples fran the field of the ship and marine hydrodynamics are presented and discussed from the point of view of the classification.
2. DRAG WITHOUT SEPARATION
At very low Kc number, no flow sepa-ration occurs around an oscillating bluff body, and the viscous drag acting on the body is mainly caused by the shear stress on the body surface. Usually, the time-dependent drag of this kind takes only
a minor part in the ship and marine
hydrodynamics
-The theoretical and experimental studies on the drag and flow of this type
Kc<4 (noparatiOfl) 4<KC<S(symgnetrical wake)
8<Kc<13' 13<Kc<20 (asymmetrical wake)
Kc>26
Fig. i Schematic view of the evolution of vortices around a circular cylinder in oscillating flow in various range of the Kc number C2J
give us many informations about the time-dependent viscous flow passing an oscill-ating bluff body. In fact, they provide
us with indispensable knowledge
about the treating of separated f low,for example, the location of the separation, the movement of the separation point, the strength of the shedding vortex and so on.
2.. 1. OSCILLATING BOUNDARY LAYER THEORY The boundary layer around an oscill-ating bodies of arbitrary shape was solved by Schlichting [3] on the assumption of low Kc number. A theoretical
calcula-tion has been used for the prediccalcula-tion of the zero-shear point (sometimes called the separation point, but generally speaking the zero-shear point does not coincides with the separation point in unsteady
flows) by Sawaragi C4J , Himeno [5] and Ikeda et al [6]
The calculation procedure is as follows.
The governing equation of the boundary layer around an oscillating cylinder can be written as,
au a U u
+ u - +
V = +
-y t x ay2
where, x is a coordinate along the body surface from the front stagnation point, y is measured normal to the wall, u and y denote the velocity components in x and y directions respectively, U denotes the 'ielocity at the outer edge of the boundary
layer and y is the kinematic viscosity.
Suppose that the velocity U is expressed by,
U(x,t)
= UA(x)et
(2)Expanding the velocity .0 in the boundary layer in series,
u=u0(x,y,t)+u1(x,y,t)+.. (3)
and putting Eq.3 into Eq.), and calculate on the assumption of the low Kc number, then we can obtain the following two equa-tions,
U at Vay7 - at
au1_ a2u1 au0
U
-uo
-v
E ax 3x ay = , U1e (wt+ll/4)LJ
(4) (5)The first order solution obtained from Eq.4 is as follows,
iwt1
-(I+i)r)
uo=UAe
-e
} (6)where, n =
yviJ
This solution coincides with the exact one for an oscillating flat plate obtained by Stokes [7]
For bluff bodies, it is difficult to determine the value of UA in Eq.6 , since
the potential flow velocity changes signi-ficantly along y . The composit expansion
technique gives us a solution which is valid in the entire flow field. It is as following
U0
= U+ Uo
-iwt -(t+i)rì
= u
+ Ue
ewhere u denotes the potential flow velocity and U the amplitude of the potential flow velocity on the body surface.
The shear stress on the body sur-face can be obtained by differentiating the velocity expressed by Eq.7 as lollows.
(7)
(8)
Bachelor ( 9] deduced a forni«1 a for
predicting the viscous drag acting on an oscillating cylinder includinq the pressure contribution using the same express)
on of
Eq.6. According to his theory, the viscous drag F for an unit length cylinder can be
expressed as,
F=
u0 t'where
=/7
and U0 the external stream velocity. For a circular cylinder, Eq.9 can be transformed to the follcMing expression:+
(10)
Note that the drag coefficient CD is a function of the Xc number as well as of the Reynolds number.
More interesting and important infor-nation about
the viscous
flow and the flow separation are given by the second ordersolution obtained from Eq.5. The second order solution for the velocity field can be expressed as follows,setting U=UAsinwt.
u =.u0 + u10 + u,.,.
u0= UA(sinwt - e"sin(wt-iì)) UAUA ( +
e1
+ e1 (cosn +25ml)
- cos1)}-e'
sin(2wt-/n) + e sin(2ot-r) - ne'cos(2wt-n) - ne1sin(2ut-n))where
Ç
= dUA/dx . In Eq.11, u, is afirst-harmonic(basic) term, u,.0 a steady term and u,.,. a second-harmonic term. As a matter of course, the second and the third terms make no contribution to the damping of the motion. Both terms , however, contribute significantly to the occtance
of
the
flow separation.The shear stress t,., on the body surface can be obtained by differentiating Eq.11 as follows,
= p/ UA[sin(wt+-)
+A{
-(I-1)sin(2wt+)}} (12)
Putting r,.,=0 , the following relation can
be obtained.
vi
sin(t + ii]4) ir CD=4ir
R0
KC _i = dx .3- -(/-1)sin(2ct + 5/4)This equation expresses the condition of the occurance of the zero-shear point where
(9)
(13)
t=O and au/ay=0 on the body surface. As
mentioned above, the zero-shear point does not coincide with the separation point in the case of the unsteady boundary layer, since no vortex sheds into the outer flow at the zero-shear point. In other words, the boundary layer assumption is held even in the downstream of the zero-shear point. Since many discussions on the unsteady flow separation have been done [io] , the
discu-ssion is not done in the present paper. Eq. 13 shows that the location of the zero-shear point at a given time t can be calcu-lated if we know the velocity gradient on the body surface dUA/dx. We can also
:1st
theàry
--: 2nd order theory
uO :measured 8 =95
Fig.2
Velocity distribution around an oscillating
circular cylinder.
3uIwy,
(
Kc=3.52 Y=Y0coswt\.l,/
Rn=1 .869x1O(PbO0
O
wt=Oo
wt=:
'tT°
'=:
o
wt180°
G a u aû Û
find that the zero-shear point occurs at
the location where the value of dUA/dx is
minimum at first, and moves to upstream. Note that the zero-shear point at a given time t for a certain cylinder obtaimed by Eq.13 depends only on the I<c number.
ti i o 2 Q o Q 2 Q
i
o 2 o 2i
Qi
ut=30° 6' 1011,
II wt=90°o
IO
'0
¡/'
lfl T Ot
'to
o o
/
/
'I
o
o
e
o
o
wt=I 20° wt=150° wt=i8O°00
o
o
Kc3.52 Rn=1 .869x10°o
A/
/
I
o
loll
11Fig.3 Velocity distribution around an oscillating circular cylinder.
8 =65°
- : ist ordèr theory ---:2nd order theory
Q :measued
t.
2.. 2 COMPARISON WITH EXPERIMENTS
The comparison between the theoretical solutions mentioned above and the experi-mental results may show not only the
vali-dity of the theory but also its limitations.
Figs.2 and 3 show the experimeñtal results of the velocity distribution around an oscillating circular cylinder in air. These data were obtained by a hot-wire ane-mometer. The Kc number and Reynolds number
in the experiment are 3.52 amd 1.869 x 10 respectively. Fig.2 shows the result at 8=95°, where the second order terms u10 and
u11 are negligibly small siñce the value of dUA/dx is almost zero. The experimental results are im fairly good agreement with the first and second order solutions
except at wt=0 and 1800. The disagreement im the region of n=3 to 6 seems to be caused by the vortices created during the swing. The experimental results at 8=65° are shown in Fig.3 with the first and second order solutions. The first order solution is in fairly good agreement with the experiment at the region of ut smaller than 60°, but at the larger ut reg-ion the velocity obtained by the first order theory
is much larger than the experixiEntal one near the
body. This may be caused by the developing vortices near the body surface. The second
order solution shows a different tendency
from the first order one. Far from the body, the second order solution shows a inich higher velocity than the first order
and the experimental one. The difference is caused by the steady flow u10 of the
90° es 45° 00
ri
predicted zero-shear point by Eq.13 -Q: measured separation lotFig.4 Location of the measured separation point and the calculated zero-shear point on an oscillating circular cylinder at
maximum swaying speed.
o 5 Kc io
ut =60
\_ ,
Fig. 5 Shear stress distribution on rolling ship hull surface.
b -roll amp. 80=100 roll period 1=1 .6sec half beam b=0. 11825m / ,
cted zero-shear point has a similar tendency
as the measured separation point as shown in this figure, but is located a little bit downstream of the measured separation point. This is a contradiction, because the zero-shear point always appears before the flow separation. Ikeda et.al. C6J pointed out that the wake effect due to the vortices created during the previous swings causes the discrepancy and showed that the zero-shear point moves to a little upstream of the separation point if the wake effect is taken into account. experimentally.
Fig.5 shows the comparison of the local shear stress distribution between the
experiment [8] and the first and the second order solutions for a rolling cylinder with
a ship-hull shape.. The first or4er solu-tion (solid line) is in fairly good agree-ment with the measured results except just downstream of the bilge where there may be
a small separation bubble. The second order solution (broken line) shows a
smaller value there, and it is in better agreement with the measured ones. This
figure also shows the validity of the time-dependent boundary layer theory at low Kc number. roll axis - roll motioñ -: tst order theory 2nd order theory Omeasured C) Cf=Tw/O.5 )bsi() o)
second order solution, which does not vanish at a large distance from the body and is independent of the viscosity as shown in Eq.11. Since iñ the experiments
the measurement of the velocity was done after several oscillations from the rest, the steady flow far from the body surface
seemed to have been not fully developed yet.
Al-though there is a large disagreement between the second order solution and the experiment at a long distance from the body, the second order solùtion shows a better result than the first order one near the
body. The second order solution shows almost zero-shear (or au/ayO on the
sur-face) at wt=90°, and the thin reverse flow at wt=120°. However, the thicicness of the reverse flow is too This may sugg-est that the boundary layer theory desc-ribed in the previous chapter is not valid any more in the aftbody of a circular cylinder even for Kc=3.52.
Fig.4 shows a comparison between the experimental separation point and the predicted zero-shear point obtained by Eq.
13 for an oscillating circular cylinderC6J. The experimental result was obtained from the flow visualization test around an oscillating circular cylinder. The loca-tion of the separaloca-tion point was deter-mined by finding the slight difference of the curvatures between the body surface and the nearest stream line. The
predi-3. DRAG DUE TO SÉPARATION BUBBLE
As mentioned in the previous chapter, no separation occurs in very low Kc number. As Kc number increases more, a flow
sepa-ration will appear At low Kc number, the separated flow reattaches to the body sur-face, and forms a separation bubble as shown in Fig.6. The separation bubble is so small compared with the size of the body that the drag due to it can be usually meg-ligible. For example, in the case of the wave forces acting on a fixed cylinder, it
is sufficient to take into account only the mass force ( or inertia force ) for such
low Kc numbers.
/
separation bubble
mid-ship section of container ship
sway motion
at the moment of wt=90'
mid-ship section. of barap
separt ion bubble
sway motion Cdt90'
d... separation búbble
The viscous forces due to the small separation bubble, however, take a very important role in some cases of the ship and narine hydro-dynamics. The most important exanple is the roll damping of ships without bilge keel. Ikeda et al [B] confir-med by the measurements of the flow velocity around rolling cylinders that the small separation bubbles cause significant viscous effects. on the roll damping of ships. This is partly because the wave damping component is very small and partly because the separation bubbles are formed at the most effective location for generating the roll moment. Ikeda et al [ii] investigated the viscous effect due to the separation in detail, and found that roll damping moment of the eddy-making
component is proportional to the square of the roll angular velocity, in other words, the drag coefficient is independent of Kc number, or roll amplitude. Thèy proposed a prediction formula for the eddy component of the roll damping on the basis of many experimental results.
The separation bubble may affect the damping forces other than the roll damping, if the other damping components,
like the wave component, are very small,e.g. for a semi-submergible's motion in low
frequency. In such a case, the viscous
drag at low Kc number has a significant effect on the motions in resonance.
Recently, Ikeda et al [6J and Bearman et al C12J applied the discrete vortex model to this problem. Ikeda et al C6J calculated the flow field around swaying Lewis-form cylinders at low Kc number. From the calculation, it was found that the vortices staying near the body affect
sway mot
wt180°
:vortex with unticlock-wise circulation .:vortex with clock-wise circulation
w t = 2.70
t= 225'
14"
the old vortex lump has significant effect on the qenerat ion and deve-lopment of new vortices.
Fig. 7 Calculated flow field around swaying Lewis-form cylinder (H0=l.25 and o=0.97) at low Kc number by a discrete votex model [6].
roll damping with a discrete vortex model. However, it is questionable if the method gives good results for a ship without sharp bilge, which has thinner separation bubbles than for a barge with sharp bilge
Applying a theoretical method to this problem, the hydrodynainic characteri-stics of the separation bubble should be well known. The experimental stixlies, in
which the separation bubble in a steady flow was investigated, reported that the pressure is impressed into the bubble, in other words, the pressure on the body surfäce can be determined by that of the
outer flow like in the well-Icsn 1undary layer
assumption. This fact suggests that we should examine carefully whether the pressure on the surface and the drag can be calculated by the potential flow theory using inviscid vortices or not, when a dis-crete vortex model is applied to the problem.
Suppose that the pressure is impressed on the separation bubble, then a simpler theoreticäl model based on the concept of the displacement effect may be possible. The concept is sometimes used for the rough prediction of the pressure drag or the form drag using a boundary layer theory.
Usually, the body is assumed to fat up by the displacement thickness of the boundary layer, and the pressure is calculated by the potential flow theory for the slightly
thicker body than the original one. In
the same manner, the oscillating body is assumed to fat up by the thickness of the separation bubble.
Fig. 8 shows the separation bubble on the bottom of a rolling cylinder with sharp bilge. The shape of the bubble was
obtained from the results of the velocity measurements C8J . Suppose for th? sake of
simplicity that the separation bubble keeps
the same shape during a swing , the
velocity, the pressure and the roll damping due to the separation bubble can be cal-culated as shown in Figs. 9 thru. 11
Fig.9 shows the calcultated velocity at the edge of the assumed bubble. At the fore part of the bubble, the velocity keeps a constant value and then gradually decreases. The calculated velocity is in fairly good agreement with the experimental one. Fig.10 shows the calculated pressure vari-ation on the bottom. In this calculation, the pressure is assumed to be impressed on
separat ion bubble shape
Fig. 8 Separation bubble shape obtained from the measured velocity around a rolling rectangular cylinder. u, e O :measured velocity : calculated velocity edge of the assumed bubble.
Fig. 9 Velocity distribution on the assumed separation bubble calculated by a source distribution method.
o
O :measured pressure - : predicted assumed bubble shapes at the separation 0.02 =BEx(V/BT,) where V:c]ispLacemeflt B B:bam LIenqrh Eo
0.01 0 Cp ACAp/0.Sc (de0 2 where d: draft, 8 roll amp. 10Fig. 10 Comparison of the pressure on the bottom of a rolling rectangular cylinder between measured and calculated ones.
predicted Q :measured
Fig. 11 Comparison of the eddy roll damping for a rolling rectangular cylinder between experiment and prediction by separation bubble model. a C b/2 bilge bottom o IO 80(deq.> 20 b/2 bilge bottom
the separation bubble from the outer flow
like in the bounda±y layer assumption. The
agreement between the calculated and the measured pressure variations is fairly
good. The calculated roll damping is shown in Fig.11 with the experimental results. In the calculation, a
separa-tion bubble of the same shape and size is also assumed on the side wall, since there is no experimental data a1ut the separation bubble on the side wall. As seen from Fig.11, the calculated roll damping coef-f icient is proportional to the roll amp-litude as the saine tendency as the experi-mental results. The agreement between the calculated and the experimental results is fairly good. At present, however, we can not use the method to predict the roll damping of a ship in practice, since we have no means to determine the shape of the separation bubble. It may be possible to determine the shape of it by a discrete vortex model.
4. DRAG DUE TO SYMMETRICAL WAKE
In this chapter, the time-dependent viscous drag due to a symmetrical wake behind a bluff body will discussed. As the Kc number increases, the thin separation bubble which was discussed previously, develops. The vortices shed into the outer flow, and then form a symmetri-cal wake behind the body. Generally speaking, the Kc number range where a symmetrical wake exists is very narrow, since a symmetrical wake formêd by a pair of vortices is essentially unstable. For instance., a symmetrical wake appears in the Kc-range from 4 to 8 for an oscillating circular cylinder as shown in Fig.1. Sarpkaya has reported, however, that there is a 90 percent chance that the asymmetry will appear at even Kc=5, and pointed Out that the minimum value of the Xc at which the asymmetry in the vortices appears, is extremely sensitive to the experimental conditions C2J
If the axis of symmetry is a boundary, the wake is kept symnietical up to
larger Xc number. In many cases in
ship and marine hydrodynamics, the free surface and the body surface work as this boundary, and contribute to keep the metrical wake (exactly speaking, the
sym-metry between a real vortex and an imaged vortex). Let us refer to some examples.
The first example is the sway motion of a ship. The viscous effect on the sway damping coefficient can be neglected in
moderate ampl1tisic of sway motion. In
large amplitude, however, the viscous effect appears as shown in Fig.12, and reaches about 36 percent of the total
damping at YA/d0.4 (where A denotes the
amplitude of sway motion and d the draft of a ship) in this case. The flow pattern around the swaying cylinder in large ampl-itude is shown in Fig.13. We can see that
a large wake behind the body covers over
the side wall of the ship and causes the significant viscous effect on the sway damping.
22 with bilge keels SR 108 container Ship
YA:sway amp.
0.5 61=0.55
(=sÌi7)
d :draft O :measured---e iscos effect
(wt'.0') (1) Beginning o
(2) at the moment of the maximum sway speed
(t=90°)
wave-making damping(cal.)
0.5 YA/d
Fig. 12 VIscous effect on the sway damping in large amplitude. faswing sway ;.:., motion
J
JI(3) Ending stage of a awing
_0
:-.-'
-
'
(t"1 800)
\k
ì_;,mid-ship section of SR-lOS container Ship with bilge keels.
YA/d=O. 67
Fig. 13 Flow visualization results of the vortex flow around the swaying cylinder in large amplitude.
10
Another important example is the roll damping created by bilge keels. In this case, the hull surface prevents an asymmetrical shedding and movements of the vortices. The drag coefficient of the bilge
keel was measured by Ikeda et al C16J as shown in Fig.14, and a prediction formula was proposed:
CD
2;5
+ 2.4 (14)The experimental results show a fairly good agreement with those for oscillating flat plates in unbounded fluid ( Keulegan et al. [1) , Kudo et al.C13J and Yuasa et al.[14J)
despite the difference of flow pattern beten
the synuetrical abd asynuetrical type of ke.Only the
experimental results by Bearman et al.[12J and Tanaka et al.[15J show lower values for the drag coefficient of an oscillating flat
plate iñ unbounded fluid than that for a
bilge keel. It is not clear whether
the difference of the drag coefficients is due to the difference between the synuetrical and the asymmetrical flow patterns.
The pressure variation on the hull surface created by bilge keels also plays an important role in the roll damping as well-known. Although a prediction formula of the roll damping due to it on the basis of the experimental results was proposed by Ikeda et al. C17J , a theoretical analysis and
some more detailed experimental studies
are necessary to clear up this damping ccxçonent.
The discrete vortex model may be a powerful tool to reveal the complicated characteristics of the viscous drag in this class. In this type of flow, there may be
no problem to replace the real vortex by an
Inviscid vortex and to calculate the pressure
5
o
experlmeñtal data for oscillating %Jfjat plate by Keulegan et al.
O:measured CD of bilge
O
keelprediction formula for 0bilge keel by Ikeda et al.
o
experimental data for oscillating flat plate
by Bearman et al. and Tanaka et. ai.
Fig. 14 Drag coefficient of the bilge keel.
on the surface using the invisid vortices
since the vortices will roll up and form
vortex lumps relatively far from the body. An example of the calculated results for a swaying circular cylinder at Kc=9 by a discrete vortex method is shown in Fig.15. This figure shows the f loti field during the third swing after the start of the motion. The vortex lump on the left side is that created in the first swing, and that on the right side is that created in the second swing.
We can also s a nJ
vortex lump developing during the third swing just behind the cylinder. The
vorti-ces created behind the cylinder ncve up to the
opposite side due to the induced velocities
of the imaged vortices, and move away from the cylinder. The fairly good agree-ment between the measured drag and the
calculated one was obtained for Kc6 to 8 by Ikeda et al. [6J
The drag of this type acting on an
oscillating flat plate has been
calcu-lated by Graham C2OJ , Kudo C19J and BimenoC 5J using inviscid vortex model.
The drag coefficient calculated by Kudo is in good agreement with experiments. Graham and Himeno obtained the following formulae for predicting the drag coeffi-cient:
CD = 8.0
Kc4
(Graham) (15) CD = 9.555// (Himeno) (16)These formulae also give fairly good results.
5. DRAG DUE TO ASYMMETRICAL WAKE
At high Kc number, the wake pattern behind an oscillating cylinder becomes asymmetrical. As noted earlier, the value of Ke number at which the flow field becomes asymmetrical depends not only on
the shape of the cylinder but also on the environment like the turbulence in a flow. The asymmetrical wake causes a
trans-verse force as well as sa fluctuations of
the drag. The magnitude and the frequency of the transverse force also play an
important role in some cases of marine
hydrodynamics.
Although the vortices form a Karman
vortex array behind the body in regular way in steady flow, in an oscillating flow
.
S S S0
oj
e.1
S oPOS
vortex lump created during the first swing
Si
vortices shedding from the body surface move in complicated manner due tothe interraction of the vortices as
des-cribed in detail by Sarpkaya [ 2] . The
detailed characteristics of the
asymmetr-ical wake of an oscillating cylinder have not been cleared yet, and it is necessary
to perform both systematic experimental
and theoretical stüdies of this problem.
As a practical example in the field of the naval architecture, the
hydrodynam-ic viscous forces of this type play an
important role in the prediction of the wave forces acting on the small elements of an offshore structure. Since it is difficult im the present situation to cal-culate them theoretically, we can mot help but using experimental data. Let us refer to some experimental data which can be used in practice.
For a circular cylinder, Sarpkaya's experimental works [2.J are most extensive and detailed. He investigated the effects of the Kc number, the Reynolds number and the roughness on the hydrodynamic viscous forces acting on a circular cylinder in oscillating flow in detail over a wide range of these parameters. Fig.16 show an example of the measured drag, mass and lift coefficients for a smooth circular cylinder.
For cylinders other than the circular cylinder, the experimental works by Tanaka et al. C151[181 and Kudo et al.C13J may be most detailed and systematic. Tanaka et al. investigated the Kc number effect, the edge-roundness effect and the attack angle effect on the drag and the added-mass coefficients acting on cylinders with
new developing vortex lump ,.
5.,
..
N'
n. S .S S
SIss
S CL N S.S.n
#
S...
vortex lump created during the second swing
Fig. 15 Symmetrical vortex pattern aroùnd an oscillating circular cylinder calculated by the discrete vortex method C6J.
o' 3.0. Drag coefficient 2.0. CD 1.0 0.5 0.325 3.0 2.0 CM 1.0 0.5 0.43 55 4.0 3 0 8=497 1107 , 2.0 1.0 0.5 448Q... 5260 8370 5 10 20 30 50 100 200 Kc 5260 smooth circular cylinder 8= 7 784 1107 1985 3123 8=5260 ..4480 1985 107 84 497 10 20 30 50 100 200 Kc B=Rn/Kc
Fig. 16 Drag. Mass and Lift coefficients versus Kc for various values of the
frequency parameter 8 (=Re/Kc)C2 J
various cross-section shapes. Unfortunately they did not measure the transverse force.
The discrete vortex model may be also useful to reveal the characteristics of the viscous forces of this type. Usually an artificial disturbance is given in the early stage of the calculation to obtain the asymmetrical flow. This disturbance decreases with time at low Kc numbei, and
lo
I3 10 20 30 50 100 200
developes gradually at higher Kc numbers. Several calculations using the discrete vortex model have been made, and Some of
them showed fairly good results. We can anticipate the applicability of the method by oncoming research works.
6. CONCLUSIONS
In this paper, the viscous drag acting on an oscillating bluff body (or a bluff body in an oscillating flow) is classified to four classes from the point of view of the flow pattern difference around it.
The characteristics of it in each classes
are discussed. The following conclusions can be obtained.
At very low Kc number, the theoretical calculations on the basis of the time-depen-dent boundary layer theory are useful not only for prediction of the frictional force but also for that of the zero-shear point. The reverse flow obtained by the theory, however, is much thinner than in the experiments.
The viscous drag due to a thin
separa-tion bubble plays an important
role in some problems in the field of the ship and marine hydrodynamics, and the prediction methods for each practical problems have been developed on the basis
of experimental results. When the
discrete vortex method is applied to the problem, we should examine carefully whether the pressure in the separation bubble can be evaluated by the potential flow theory or not. It may depend on the thickness of the separation bubble. The possibility of another simple model on the basis of the concept of the displa-cement effect of the bübble is also shown in this paper.
The time-dependent viscous drag due to a symmetrical and asymmetrical wake plays an important role in several problems of naval architecture. The discrete vortex
method may be the riost promising theoretical
approach to reveal the complicated charac-teristics of the viscous flow in these classes. It should be emphasized that the theoretical results by this method should be compared with experimental data over a wide range as possible.
7. ACKNOWLEDGEMENT
The authors wish to express their a appreciation to Prof. Y.Himeno of University of Osaka Prefecture for continuous encou-ragement and valuable advice.
A part of the work has been done during the stay of the first authors in the Technical University of Berlin as a research fellow of the Alexander von Hum-kldt Foundation. Prof. H.Nowacki and Dr. A.Papanikolaou of the Technical University of Berlin are sincerely acknowledged for their kind help and warm encouragements.
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IKEDA Y., Research Associate, Dr.Eng. Tanaka N., Prof., Dr.Eng.
Department of Naval Architecture University of Osaka Prefecture 4-804 Mozu-Umemachi, Sakai-shi Osaka, 591, JAPAN
