Calculation of Vortex-Shedding flow around oscillating circular and Lewis-form cylinder

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CALCULATION OF VORTEX-SHEDDING FLOW AROUND

OSCILLATING CIRCULAR AND LEWIS-FORM CYLINDER

YOSHIHO IKEDA

and

YOJI HIMENO

UNIVERSITY OF OSAKA PREFECTURE

DEPARTMENT OF NAVAL ARCHITECTURE

SAKAI ..

OSAKA

.

JAPAN

* will be read at the Third International

Conference

on Numerical Ship Hydrodynamics,held on the 16-19

June 1981 at Paris.

Abstract

Symmetric vortex-shedding flows around swaying circular and Lewis-form cylinders are calculated using a discrete vortex model.

The boundary layer on the body surface is replaced bya discrete vortex at the zero-shear

point, where the location is obtained by

Schli-chting's oscillatory boundary layer theory and the strength is determined using the ordinary

boundary layer assumption,. The paths of the

vortices downstream of the zero-shear point are determined by potential flow theory using a numerical time-step integration. The calculated

flow fields are compared with the flow

visuali-zation results to show fairly good qualitative agreements. The pressure and the drag force shows an agreement with the experiment in the region of K-C number between 5 and 8, if the wake effect on the determination of the

zero-shear point is empirically taken into account. 1. Introduction

For the prediction of the wave forces acting on ships and ocean structures, and for the esti-mation of their motions, it is important to

de-velope a prediction method for the viscous forces of oscillating bluff bodies with a

suf-ficient accuracy.

The experimental studies have been most common in this field, since the theoretical ap-proach is difficult due to the existence of the

flow separation. Keulegan and Carpenter £1] measured the viscous forces acting on a flat plate and circular cylinder submerged in the

standing waves, and found that the viscous forces acting on the bluff bodies in an oscil-lating flow depend mainly on the relative dis-placement. The relative displacement defined by them is usually called Keulegan-Carpenter number, namely K-C number. The K-C number is described

as

UmT/D (Ur,;

maximum speed, T; period, D; representative length), and becomes _2rrya/D

(j; amplitude of the motion) in the case of harmonically oscillating bodies with constant amplitude. Since then, many experimental studies [2,3,4] in this field have been carried out to confirm the fact. Nowadays, these experimental results for a circular cylinder and a flat plate are used on the practical estimations of the wave forces acting on pile [5] and the roll damping of the bilge keels of ships [6].

CALCULATION OF VORTEX-SHEDDING FLOW AROUND OSCILLATING CIRCULAR AND LEWIS-FORM CYLINDERS

Yoshiho Ikeda and Yoji Himeno University of Osaka Prefecture

Sakai , Osaka

Recently, the experiments for the oscillating bluff bodies with various shapes other than a

flat plate and a circular cylinder have been carried out by Bearman et al.[7] , Kudo et al.

[8] and Tanaka et al.[9]. Tanaka et al.[9] found from their experiments that the drag coefficient of oscillating bluff bodies varies significantly with the slight change of the body shape at low K-C number region. The characteristics of the

separation forces of oscillating bluff bodies are, therefore, much complicated.

The theoretical calculation is also one of powerful tools for clearifying the complicated

charactaristics of the separated flow associated. Some theoretical works using discrete inviscid vortex model have recently been proposed, most of which treated the oscillating bodies with

sharp edge like a flat plate set normal to the flow, or a rectangular cylinder [10,11,12]. There have been, however, a few theoretical works for the bodies with round corner like a

circular cylinder or Lewis-form cylinder [13,14]

mainly because of the difficulties in resonably determining the location of the vortex-shedding point and the vortex strength. Two ways to this problem can be considered; one is to apply the Kutta assumption or other method to determine

the vortex strength at the empirically deter-mined separation point, and another is to use a

theoretical prediction formula from viscous-flow

theory.

Sawaragi et al.[14] presented a calculation method for a circular cylinder in an oscillating flow using none of experimental informations. They determined the location of the separation point using the oscillatory boundary layer theory of Schlichting [15] assuming the zero-shear point coincides with the separation point and the strength of shedding-vortex using ordi-nary boundary layer assumption in the same way as Sarpkaya [16]. They calculated the flow field, the pressure on the surface, the drag and the lift forces, though they treated only the first swing after the flow started.

Although the calculation method presented here is basically similar to that presented by Sawn-ragi et al.,several improvements are made, i.e., the determination of the separation point, the strength of the vortex and so on. The calcula-tions are carried out in several swings in order to take the wake effect into account.

The application of the method to ship-like sections is important

in

the field of ship hydrodynamics. The present method for a circular cylinder is easily applicable to the ship-like sections sInce ño empirical information is used. Using the Lewis

transformation

formula [17],

several typical outcomes of the calculation for Lewis-form cylinders are also shown.

2. Calculation procedure

We consider here the case of a two-dimensional

circular cylinder in a time-dependent flow

in-stead of the cylinder oscillating in still water.

The relative flow

patterns with the.fixed axis

on the cylinder are the same for, these two cases.

The pressure on the cylinder surface and the

force are, however, different due to the

exist-enceof the pressure gradient in the

tirne-depen-dent main flow. Ñote that the coefficient of the

hydrodynamic force proportional to the

accelèr-ation of the main-flow acting on a circular

cylinder in a time-dependent flow is the twice

of. that for the oscillating one. We also assume

that the flow field around the cylinder is

symmetry.

Th

complex velocity potential

w

of the flow

that there are a two-dimensional circular

cylin-der of radius R and a number N of vortices

located at. Zn wth strength

r

in a

time-depen-dent flow U, is given by

R2 j N.

W = U(z +--)

* .

{1r log(z-z,.)

N R2 N

-E1r log(z_) +1r log? }

(1)

where is assigned positive for clockwie

circulation.

2.1. Path of vortex

The vortices, move with their local velocities,

so that the equation of motion of k-th vortéx is

giveñ by

Uk iVk

d(w_rk log(z-zk))

(2)

where

(xk,yk) denotes the location of the k-th

vortex, and

(ukvk)

the velocity conponeñts

there.

As usually., the paths of the vortices, are

determined through equation (2) by a numerical

time-step integration. The Euler formúla is used

ïri the present calculation in order to save the

computation time.

2.2.

Determination of shedding-vortex

In the oscillation problem of a sharp-edged bòdies, the determination of the shedding point

and strength of the shedding vortex is not so

difficult, since we

can f iÍid

the shedding point easily

and

use Kutta assumption for

determining

the strength. The problem is, however, more difficult for bluff bodies with round corner like a circular cylinder and ship-like sections.

Several researchers have used the empirically determined shedding-paint and applied the

Kutta

assumption there.

Sawaràgi et 1.114] applied the Schlichting's oscillatory boundary layer theory for predicting shedding-point and an ordinary boundary-layer

assumption for the strength of shedding vortex.

We also adopt here asimilar method to Sawaragi et al. in order to easily apply the model to

other body shapes as well

as a circular cylinder without any experimental information.

By the Schlichting's oscillatory boundary

layer theory, we can obtain the location of the

zero-shear point where, the velocity gradient

du/dy at the cylinder surface becomes zero..

It

has beeñ known from recent studies [18] that the

zero-shear

point does not coincide with, the

sepa-ration point where the vortex sheds from the

surface into the outer potential flow in the

oscillating case, and that the boundary layer

assumption still holds at the zero-shear point.

At the zero-shear point, the following expréssion

is given by the

Schlichting's theory,

.dUms(OJ_

./.sin(wt#v/4)

w ¿ {(/-1)sin(2wt +

îr/4)-O.'SJ

where Ums(e) denotes the amplitude of fluid.

velocity at the outer edge of the boundary layer

at the zero-shear point, x the coordinate along

the cylinder surface the origin of which is

locate4 at the front stagnation point. Using.

equatioïì(3) we can get the location of the.

zero-shear point for any arbitary ylinder.. In the.,

case of a circular cylinder, equation(3) takes

the form,

K-c

sin(wt

+ ,T/4)

--CO8O

_-

_{2U/2-2)sin(2wt}

_ir/4)-O.5}

(4)

where

O

denótesthe

location anglé of the

zero-shear point from the rear stagnation point. 'The

location of the zero-shear paint at the moment

of a certain time-phase wt depends only oh K--C

number. The zero-shear point appeares .firstiyat

the rear stagnation point and then moves upstream.

It reaches the mid beam of the circular cylinder

,where

the velocity gradient

dUms(0)/dz

is equal

to zero, at the nomeñt wt=3rr/4.

We can replace the boundary layerby a 'discrete vortex at the zero-shéar point, and assume thät the generated vortex keeps the strength constant

since the wall shear stress is 'small dOwnstream of the zero-shear point compared with that in

up-stream. The distance of the replaced vortex from

the surface at the zero-shear 'point is

assumed to

be half of the boundary layer thickness. In the

present calculation, the boundary layer thickness

is assumed as

ó4.61'Aii

, which is determined

as the velocity decrease in the boundary layeris 1% of the oùte± flow by the Stokes' solution for an oscillating flat plate r15J.

The strength r

of the vortex at the.zero-shear

point is a functiOn of both time t and locatión s. Then r can be expressed as

dF

=--

dt

at as (5)

Using the ordinary boundary layer assumption, equation(5) becomes to

r

=4uo62t

# u'O8R e

(6)

where U(O3) denotes the velocity outside of the boundary layer at the zero-shear point and O is

the displacement of the location angle of the

zero-shear point during the time increment 1st.

If the zero-shear point does not move, as in the steady case, the second term vanishes.

Note that the pressure is impressed on the boundary layer by the outer flow at the zero-shear point since the boundary layer assumption still holds there and even downstream to the

separation point. In other words, the pressure can be calculated by the potential flow theory excluding the vortices in the boundary layer.

2.3. Separation

In the present calculation, the vortex gener-ated at the zero shear point moves downstream along the surface like a boundary layer as seen

in the flow pattern at wt=rT/3 shown in Figure 2.

The separation takes place when the ratio of the vertical velocity to the horizontal one is of the

order unity. The numerical value of the ratio for the separation criterion is assumed to be 0.3 in the present calculation. According to the recent studies on the unsteady separation [19], the ratio increases rapidly to the infinity near

the separation point. Therefore, it can be safely said that the calculation results is not much affected by the slight change of the numerical

value of the ratio.

2.4. Assumption of vortex diffusion

In the real fluid, the vortex gradually

dif-fuses with time. Several authors took into account this effect on their inviscid vortex

models [20]. In the present calculation, the circumferential velocity V8 for the isolated viscous vortex described in the paper of J.W.Sch-aefer et al. [23] is also used.

y0 - exp(-r2/(4v(t-to))} (7)

The circumferential velocity V0 shown in Figure 1 has a maximum at r=r*, and the value r* increases

with tine. In the inner region of r*, the effect of viscosity is dominant. In the present calcu-lation, the initial time to in equation(7) is determined when it is created at the zero-shear point, with r* equal to 0.56.

Using the present vortex model, we can avoid the infinite induced velocity in case when vortices get too close each other or to the

sur-face. In the real fluid, a vortex close to the body surface loses its energy creating a boundary

layer on the surface. In order to take into account this effect, we neglect the vortex within the distance r* from the body surface.

2.4. Pressure

The pressure p(0) on the cylinder surface can be obtained by the following pressure equation.

o

potential vortex

vortex model using lore

(equation (7))

Fig.l Peripheral velocity of isolated vortex model.

(0) = al i U(0)2 (8)

3t 2 p at

the

surface

where is the velocity potential and U(0) the

velocity on the surface. The first term of equation(8) for a circular cylinder can be expressed as the form using equation(i),

au N

u..+iv

-p .-. = -p.real[2 R coSO -- z r ot at

2ir1

ZZ

R2 2rrn&i "fl(3 _-R2)

(u_iv)

+-- -'-{log(z-z)-log(z- )+logz}] 27rat (9) Since the vortex strength is assumed to be constant after the generation at the zero-shear point, the last term, which is proportional

to

arnJat , is zero except at the moment of the

generation. As mentioned above, the pressure is impressed on the boundary layer by the outer flow since the boundary layer assumption still holds at the zero-shear point. Then we may neglect the last term.

From a different viewpoint, it is possible to consider

to

take into account the term ar,1Jat

at the separation point where the vortex sheds into the outer potential flow. In the present calculation, the two methods for the pressure calculation are used, one excludes the last term of equation(9), and the other includes the term at the separation point.

2.5. Drag force

The lift force does not yield because we treat only the symmetrical flow in the present

calcu-lation. The drag force acting on the cylinder can be obtained by integrating the pressure over

the surface. It can also be obtained more easily by using the time-dependent Blasius

theorem or from the time-derivative of the total impulse [21].

N N

P z

(y_yj)#2p7rR2

-'

n=i

n=it

(10) where (X,y) denotes the location of the real

vortex,

(Xjj)

the location of the image vortex, (u,v) the velocity of the real vortex

at (Xy), and

the velocity of the

image vortex at

(Xj,yj).

The third term is

derived from the potential flow theory without vortex, and is equal to

pirR2(3(J/t)

for a oscil-lating circular cylinder. The second term can be neglected with the same reason as the pressure

mentioned above. Note that the drag obtained by integrating the pressure over the surface is slightly different from the one obtained by equa-tion(lO) since the induced vortex velocity is determined as equation(7) in the present

calcu-lation.

3. Calculation results

The calculation results of flow pattern around a circular cylinder in periodic flow at K-C

num-ber 9 are shown in Figure 2. The circles in

the figure denote the vortices with clockwise circulation, and the crosses those of opposite

sign. The flow begins to move at t=O, with the velocity

U=U,,pin(wt).

Acàording to the

Schlichtings theory, the zero-shear point ap-pears at the rear

stagnation

point at wt=O.B5rad in this case, and then moves upstream. As seen

from the flow pattern of wt=7r/3 in Figure 2, the

vortices form a line along the surface like a boundary layer, gradually roll up, and finally they form a lump of the separation region. Since

the main flow velocity is small at the last stage of the f rst swing, the vortices gradually moves

toward upstream by the induced velocity of the

image vortices. At wt=5v/4 after the reverse turn, the vortex lump created during the first swing moves downstream rapidly due to the main-flow in addition to the induced velocity of the image vortices and the real ones in the other

side. Att31T/2, the vortex lump of the first swing lies far from the cylinder and the vortices with anticlockwise circulation create a new

vortex lump behind the cylinder. At Ut=511/2 of

the third swing, the first-swing vortex lump lies

in the left of the cylinder about three times of the diameter, the second-swing vortex lump is in the right of the cylinder, and the new vortex lump created by the third swing is formed near

the cylinder.

The strength of the vortices created in the second swing is greater than those created in the first swing as shown in Figure 3. This is caused by the wake effect of the vortices of the

previ-ous swing. The strength of the third-swing vertices is almost the same as those in the

second swing.

Figure 4 shows the calculated pressure distri-butions on the cylinder surface for the same

condition as the flow field calculations shown

in Figure 2. The pressure coefficient C is

defined as the amount of the pressure p(s) obtained by equation(8) devided by pU, /2 . The

black circles in the figure denote the pressure exclding the last term of equation(9) and the white circles denote that including the term

arwt at the separation point. The solid line shows the pressure distribution by the potential flow theory without vortex. Note that the pres-sure shown in Figure 4 is of the case for a

5

first swing

second swing

: third Swing

czsin (wt)

/ Sin

(it)

J

Na

wt51/2

KC number = LJT/'2R CC number 9 - 7111

vortex With clockwisc circulation vortex with anticlockwise circulation

Fig.2 Calculated flow field around circular cylinder.

wt1T/3 Wt7T/2 _{Wt37T/4 wt=11}

wt51T/4 t4t311/2

o

start of

lt

end of

a swing _{a swing}

Fig.3 Strength of vortex generated at

circular cylinder in a periodic flow, so that the pressure is different from that of an oscillating case by the amount

-.p(dTJ/dt)Rcos(8).

As seen from the result at wt=1r/2, the pressure

is nearly uniform over a large part of the rear of the cylinder like the case of steady flow. However, there is a slight negative peak near

e=3o degree, and the peak becomes more recogniz-able at the moment ofmaximum flow velocity in the second and the third swing, wt=3v/2 and

wt=51T/2. The pressure included the last term of

equation(9) at the separation point has an abrupt

cp

lIt 37/4

e o

o...

discontinuity at the separation point as

mentioned by several authors [14,21]. This dis-continuity is due to the discrete vortex model.

Sarpkaya [21] avoided the discrepancy by replacing the shear layers by a combination of

Ideal vortices and an infinite number of vortex

sheets.

C1

P(8)/(pU)

KC NO. 9

7111

potential flow calculation present method including the last

term of eq. (9)

o _{present method excluding the last}

term of eq. (9) at the separation point

Fig.4 _{Calculated pressure distribution on circular cylinder in oscillating flow.}

7/2 A

o

o"

.

NIt = 71/2 NIt = 311/2

li

"

-

V

potential flow

&dt 11/2 11/2 K-C No.= 9 No.= 7 K-C No. 5

potential flow

K-C No.= 5 K-C No.= 7 K-C No.= 9

Fig.6 K-C number effect on surface pressure distribution for circular cylinder.

11/2 lOt 311/2 Or. 311/2 lot 371/4 Rn 7111

-K-C NIl.-K-1 No. 1

- I K( No.

7 K-C No. S

Fig.5 K-C number effect on calculated flow field around oscillating circular cylinder. The flow field at different Keulegan-Carpenter number is shown in Figure 5. The scale of the vortex lump behind the circular cylinder is greater with the increase of K-C number, and the vortex lump created during the first swing

separates far away with K-C number at wt=371/2.

The pressure of the rear of the cylinder de-creases with the increase of K-C number as seen

from Figure 6.

Figure 7 shows the Reynolds number effect on

the flow field. In the present calculation, the Reynolds number effect is caused by the differ-ence of the initial location of the vortex at the zero-shear point and by the induced velocity decrement in the vortex core as described in chapter 2.3.. From Figure 7, it is found that the Reynolds number effect is not so large compared with the K-C number effect, and the tendency coincides with the experimental results.

n example of calculated drag forces is shown

in Figure 8. The drag coefficient CD and the inertia coefficient CM can be obtained using following definitions with the calculated forces

as in Figure 8.

21T _{F cos(wt) d(wt)}

CDTf

D

Ot =

Fig.7 _{Reynolds number effect on calculated flow field around oscillating circular cylinder.}

I NIl 3307 Re 7111 Re 1]500 lOt 311/4

jt

fr

'J

11 N 11/2

Fig.8 Calculated time history of drag force. cc ci. S pkuy (2) o _/ '105 5290 x.. 7000 CM = lo q. (13)) s - - l0 ,..cio.

Fig.9 Drag coefficient of oscillating circular

cylinder.

LImT 27r

FD Sin(()t).d(Wt)

ub

(12)

m

In the calculation of drag coefficient, the

second term related with is neglected.

The calculated drag coefficient is shown by a

solid line in Figure 9 Though there is a

similar tendency to the expe±imént, the cal-culated value is much lower thab the experiment. The broken line in the figure represents the non-separation viscOus drag force calculated by the Bachelor's method [22] which is described as

follows,

= 4rr( 2 (13)

As seen from the figure, the non-separation drag force is much lower than the separation drag in

the K-C number region above 4, a±id can be

negligible.at present discussïon.

4. Improvement of the method

Although the calculated flow field is similar

to the experimental, ones, the calculated drag

coefficient shown in Figure 9 has qualitative and quantitative disagreement with the

experi-ments. _{For example, the calculated value seems} to be have a maximum at lower K-C number regioni

and is lower than the experimebtal one in the whole range of K-C number.

On considering the reasons for these disagee-ments, it can be noted that two effects have not been taken into account in the present

calcul-ation. _{Firstly, the flow field is assumed to be} F =F3/)PSt)

K-C NO. 10.5

R9 7115

o exclude the second ters of eq. (10)

O include the second ter,n of eq. (10)

p,esoot seti)ooixq sq. (14))

(0,7111)

'

-prososO ,thod(ooiflq Oq.(4)(

(R,.7111)

symmetry, though the real flow is asymmetry at the K-C number above 8. For solving the former disagreement, it is neóessary tO càlulate asym-metric vortex flow at the K-C number above 8.

Secondly, no effect of the vortex wake of the former swing is taken into account on determi-nation of the zero-shear point. As seen from

Figure 2, 'there ïs à vortex lthnp created durïng thé first swing near, the cylinder on- the early

stage of the next swing tò afféct the location of

the zero-shear point. It is, however, dïfficult

tocônsTider this, effect exactly on

the'determi-nation of the location of the zero-shear point. Instead, we try to take into accOunt the effect of the wake of the former-swing in following simple method of the phase

imdxfica-tion. Thé vélocity measurement around an oscillating circular cylinder suggests that the wake effect due to the vortex lump created

during the former swing can be considered as' if

the Outer flow had the phase advance against the main sinusoidal flow. Figure 10 shows the

results of the velocity measurements at the 'side top (e=Tr/2) of the oscillating circular cylinder

From this figure, the phase advance Just outer edge of the boundary layer is about ¶/8 due

to

the wake effect In Figure 11, the separation

point obtained from the resülts of flôw

visuali-zation for an osòïllaing ci±culü cylinder are shown together with the zerò-shea± points by Schlichting's theory. it is a contradiction that the zero-shear point is located downstream of the separation point, because the zero'-hear point occures earlier than the flow sèparation.

-3.3 ((/2 05 "/4 0.6 _{bostiol flow} 0 5 K-C number lo

Fig.11 Location of separation point and -zero-shear point.

/ pOx,, di

10 _y(,( 20 0 '((/2' ut

ve100ity at the edqo

of dory loyer

p0000tlol flow nol000ty (y.20ow)

SV,

Fig.lO Experimental results of velocity distri-bution at the side top of an oscillating

circUlar cylinder. O ,mrasurcd Separation pain t - calculated zero-shear toint by cq.(4( calculated zero-sI-ar - point by eq.114)

Qn the basis of the experimental results shown

in Figure 11, we assume that the phase advance

of the outer flow is 11/8.

Thén the equation for

the zero-shear point takes the form,

i dUras(0)

/ 8ifl(Wt + Sir/8)

w d. - {(/-i)sin(2wt +iî/2)-O.5

(14)

The zero-shear point ôbtàined by equation(14) is

also shown in Figure 11, by a broken line and.

located a little upstream of the measured

sepa-ration point.

The calculation results of the drag coefficient

using equation(14) on the

determination

of the

zero-shear

point

shown in Figure

9 show bettèr

agreement with

the experimental ones in the

region of K'-C number between 5 and 8.

The

cal-culation results tend to zero at nearly K-C=4,

and the d.iscrete vortex model may not be Suitable

in. the: low K-C number, region under 4.

Iñ the

region, the separated f 1w reattachs on the body

surface to form a thin separation bubble as many

flow visualization results show.

If the

sepa-ration bubble should be replaced by discrete

vortices like the present method, the

calculated surface pressure would

rIse because of the.

velocity reduction

on the surface due to the

vortex induced velocity and then the

negative

drag force would act on the body.

It is one of

the remaining problem how to theo±etically obtain

the drag fOrcé in such low K-C number region.

Figure 12 shows the comparison between the

calculated and the measured Reynolds number

effect on the drag. force of a oscillating

circu-lar cylinder.

The.ágreernent is fairly good in

the region of Reynolds number above 5000.

At

low Reynolds number under

5000, the experimental

results are higher than the calculation.

The

disagreement may be resonable because the

discrete ïnvisid vortex model is meaningful at

the high Reynolds number where the viscous effect

is restricted in thé thin region.

In the present

calculation, the initial location of the replaced

vortex becomes. too far from the surface in such

a low Reynolds number.

The calculated inertia coefficient CN

shown

in Figure 12 is also in good agreement with the

experiment at K-C number under 7.

The

disagree-ment at K-C ni.uber above 8 may be caused by .the

difference of flow pattern between the

calcula-tion

and

the experiment as well as the case of

drag coefficient.

co

- BaOLOrS method

5000 10000 R 15000

O measured

..._. _pre.rnt method uaieg eq.(14)

..

o

a

O

₀₀

-- o

Fig.12 Reynolds

number

effect on drag coef

fi-cient for oscillating

ciicular cylinder.

pmeaeet thed(ueieq eq.(14((

òxpertntot bySosiegan mt aiS

SorpkeyC (23

0 10 K-C lee.

Fig.13

Inertia coefficient of circular cylinder

in oscillating flow.

5.

Application to Lewis-form cyli der

The present calculation model is easily

appli-cable

to ship-like sections

using Lewis

transfor-mation.

In this chapter, several examples of

calculation of flow field

around

the oscillating

Lewis-form cylinders

are

presented.

The mapping function of Lewis form

can

be

re-presented as follows,

a1

a3

(15)

th

_x

iy

ill

The coefficient a1

and

a3 are the functions of

the section shâpe,

and M a

magnification factor,.

This function transforms a unit circle in c-plane

to the shIp-like section in z-plane.

The equation of

vortex motion at

-p1ane

can

be

obtained from

equation(l) and (15".

drlk d{w

irk

log(

ç-dt' dt dç dz.

(l6).

where

r) denotes -thé coordinate of -the k-th

vortex in

-p1ane.

We can get the instantaneous

location of each vortex át the real plane using

the mapping function of equation(15).

In the

sane manner as thé circular cylinder case, the

location of the zero-shear point

can

be given by

Schlichtings theory using the velocity

distri-bution over the real section, and the strength

of the vortex can be 'also' be determined.

FOr

sake of simplicity, the vortex is here assuméd

to be a potential vortex, although a more

sophisticated vortex has been used in the case of a circular cylinder.

Figure 14 is an example of the calculated flow

field for a 'swaying bow-section f H0(=half beam!

draft)=0.33

and

c1(=sectional area/beamxdraft)=

0.75} at K-C number 1.67.

The zero-shear pOint

fOr such a shape located at

the

side edge and

hardly moves.

In the first swing, the vortices

form a large vortex lwñp b hind thé cylinder.

At wt=311/2, the newly created vOrtex lump does

not roll up round in good order like those in

the first swing because the vortex lump created

during the first swing is still locatad near the

cylinder. As seen from thé figure at

wt2ir, the

vortices geñerated during the second swing are

K-C NO. I

(5)

Fig.14 Calculated flow field around swaying bow-section devided into two parts, and one of them moves

downstream with the vortex lump created in the fist swing düe to its strong Induced velocity.

The flow fiêld for a swaying midship section at K-C number=2.l is shown in Figure 15. For

this section, there are two zero-shear points, and the downstream one appears firstly because the flow acceleration is greater there than that

at the upstream one At utr/2, the vortices are

generated only at the downstream bilge. At

wt=31r/4, the downstream vortices fòrm a vortex

lump, while the upstream vortices do not grow

up. At ut=5ir/4 after the reverse turn, the

vortex lump nves towards downstream gradually. The vortex lump remains at the bottom even at

wts3v/2 and 771/4, affecting the new vortex

generatioi and the growth. It is one of the problem how to take into account the damping of the strength of the vortex which remains near

Wt = 17/2 71 uit = 371/2 -a

:.

cylinder(H0=O.33, cY=O.75, K-C No.=l.67 the body surfãôe.

The flow field around the oscillating flat plate at K-C number=2 is shown in Figure 16.

In the first swing, the vortices form a spiral vortex lump behind the plate, while in the

second swing, the vortices do not form such a spiral lump In the same tay as the case of the

bow section shoi.in in Figure 14.

The results of flow visualizations performed to check the càlculations for Lewis-form cylinders are also shown in Figs. 17 thru 19. The cylinders are swaying in a small water tank by means of a forced oscillating mechanism. The oil particles with unit density in the tank

re illuminated by a lamp through a lit.

Particle tracks are taken by camera fixèd to the cylinder with ¡ relative long éxposure of 1/15

sec.. bit = 311/4 (2 (4) 4-uit 511/4 bit = 711/4

Fig.15 _{Calculated flow field around swaying nidship-section cylinder(H0=1.25, a=O.97, K-C-No.=2.l).'}

wt = 1) 4. t e.. 't

..'.

j q...

. ..

51T/4 4) 4.

;V ..

M

.

'e.

'

e.%

.

_{. e}

r.

7w/4 . s

:

The ship-like section shown in Figure 17 is the SS 9 section of a container ship, and has

the same H0 and values as the Lewis-form

cylinder shown in Figure 14. The cylinder shown in Figure 18 is also the midship section of the

same ship corresponding to the Lewis-form

cylin-der shown in Figure 15. As seen from these figures, the vortices at the moment of maximum

3V4

(2)1

4.

t

...

V

Fig.16 Calculated flow field around swaying flat plate(H00.l, 0=0.85, K-C No.=2).

= 91T/4 (s) e : M - r s r s S

s

X s se

s'

.,e Wt 311/2 5) = 2,T 7)

cylinder speed spread like a thin bubble on the cylinder surface. In the decreasing stage of the motion, the vortices roll up into a large-scale vortex lump. These experimental results agree qualitatively with the calculation results shown in Figures 14 and 15.

pictures area

1r

_w-section

cylinder

Fig.17 Flow visualization res1ts around swaying bow-sectiôn cylinder(K-C nuiñber=l.67 ).

ut - 5/2 )+2nhT( (eazimwn swaying speed)

- rPiCtUxeS area

:V

midship section cylinder

Fig. 18 Flow visualization results around swaying midship-section cylinder (K-C numbex=2.l) ut 7m/6 (+2ntr) ut - lt (+2n5)

(zero swaying speed)

CPt05 area

Fig.19 _{Flow visualization resuitsaround swaying flat plate(i(-C nuxnber=O.62).} at 1T/2 (+2ntT) ut - 77r/6 (+2nT) hit T (2n7T)

(iUníin sa ing speed) _{(zero iwaying speed)}

htart of swing the esment of maximtim swaying spied last stage of swing

ugh t

plate. A black shade in the right side of the photographs is a shadow of the plate because of using single slit lamp. The photographs show that the former vortex lump forms a vortex pair with the new vortex, and moves downstream

gradually. The behavior of the vortices is similar to the calculated one shown in Figure

16.

6. Conclusions

In this paper, computation of the symmetrical vortex-shedding flows around swaying two-dimen-sional cylinders are made using a discrete

vortex model. The results of the study can be summarized as the following items.

The calculated flow field by the present method is in fairly good guaritative agree-ment with the experiagree-ment

The ca1cu1atd forces acing on a circular cylinder show a similar tendency to the experiment though the values are lower than

the experiment. When taking account the wake effect on the

detérm.thation

of the zero-shear point, the calculated forces are improved to

show. fairly. good agreement with the

experi-ment

in the region of K-C number between 5

and 8.

It is necessary to treat asymmetric flow at

K-C number above 8.

The vortices created during the previous swing significantly affect on the generation, the strength and the behavior of the new

vortices.

The present method can easily be applicable to the ship-like sections using Lewis trans-form méthod, so that it would be useful for the prediction of the viscous effect on the ship motions and maneuvering.

The authors would like to thank Professor Nono Tanaka and Professor Toshio Hishida for their encouragements, and wish to thank Mr.Kenji Higashida and Mr.Yasuhiro Kashiwa, students of University of Osaka Prefecture, for their help on the numerical calculation and experiments. The computers FACOM M-200 and ACOS-600 at the computer center of Kyoto University and

University of Osaka Prefecture were used for the

numerica], calculation.

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