Calculation of viscous interference effect between two circular cylinders in tandem arrangement in harmonically oscillating flow

INSTITUT FUR SCHIFFS- UND MEERESTECHNIK

CALCULATION OF VISCOUS INTERFERENCE EFFECT BETWEEN TWO CIRCULAR CYLINDERS IN TANDEM ARRANGEMENT IN

HARMONICALLY OSCILLATING FLOW

by Dr. Yshiho Ikeda

University of Osaka Prefecture A. y. Humboldt Guest Scientist at. TUB

Cöntents

Page

Abstract 2

Introduction 3

Calculation Precedure 3

2.1. Complex Velocity Potential 3.

2.2. Path of Vortex 5

2.3. Zero-shéar Point 6

2.4. Replacement by Inviscid Vortex io

2.5. Vortex MödeI li

2.6. Separation Condition . 12

2.7. Neglect of Vortex too neàr Surf'àce 13 2.8. Calculation of Pressure Veriation 1.3

2.9. Hydrodynainic Force 15

Simulation Results and DiscusSioñ. 15

3.1. Effect of Time Step Size . 15

3.2. Simulation Results 17

Hydrodynarnic. Intcrference 21

4.1. Vortex Interférence 21

4.2. Interference Effect on Drag and 25

Inertia Coefficients

Conclusions 26

Acknowledgements 27

'References 28

Appendix 29.

Abstract

A calculatjon method for predicting the interference èfféct between two circular cylinders in tandem arrangement in harmonically oscillating flow is developed using a discrete vortex method. The flow is assumed to be symmetrical in order to reduce the. computer time. The zero-shear poiit and the

strength of the shedding-vortex are determined by an unsteady. boundary layer theOry.

The simulation results reveal sOme different kinds of viscous interference èffects according to différent bet aviourS Of shedding-vortex movemnts.

Hydrodynamic interference effect among multi-cylinders in oscillating flow is. one of the: important problems ïn the

assessment of the structural strength and motions of an ocean structure in waves.

The hydrodynamic interférence effects among multi-cylinders in wavès have béen calculated using a diffraction wave théory by

Ohkusu /1/, Isaacson /2/, Masumoto et àl. /3/ and Mïnematéu et al. /4/, and a lot of fruitful results have been obtained.

However, only a few investigations on the hydrodynamic

inter-ference effeOt ö. thé. vortex-shédding flow among multi-cylinders, which are significant in the case when cylinders have, Small

dimensions compared to the amplitude öf th5 öscillating flow or the wav length, have beén carried out by Sarpkaya /5/, Sawaragi et al. /6/ and Chakrabarti /7/. Therefore, the

inter-ference effects of vortex-shedding flows among multi-cylinders

have not, been clear yet.

In the present paper, as the first step of thè theoretical

approach to this problem, the interfereice betweén two circular-cylinders in tandem arrangement in haronically öscìllating flow

is çalculated by a.. discrete vortex method.

Calculation Procedure

2.1. COmplex Véiócity Potential

The complex velocity potential w of the flow around two circular cylinders with the same radius R in tandem arrangement, in the presence of a nimiber N of inviscid. vortices lodated at z with

strength rn in à timedependent. flow U, is given by:

W = U(z

(i)

where

=r04r

X and

r0'/R

( ro' : the distance betweer the centers of the two cylinders).

w*Z4CTJo:(z)

4(Z.)

Fj.1

rranernent cf \rrrtex: j.1s.. image. vorties

(2)

where is defined positive for cloçkwise circulation, z1, z2 denote the locations of the first image. vortices and z'. , z'

nl n2

the locations of the second. image vortices as shown ih Fig. 1, and

n the cönjugate complex of Z.

The potential component. i deduced by the f ist örder theory

Since the vortices move with their locaL velocities the equation

of motion. of the k-th. vortex is given by,

d-xk, id4

.

-d*

dt

k

d-flb,(Z-Zk)

d (3)

where (xkl denOtes the location of the k-th vortex, and Uk, Vk) the velocity component at (xk, which consists of the potential flow velocity and the velocity induced by the other vortices.

The paths of, the vortices can be calculated through Eq. (3) by

a numerical time-step ihtegration. The following two formulae of

Eqs. (4) and (5) have been trïed, and the simple Euler Equation of Eq. (5) is found to be.better than Eq. (4). This is because the formula of Eq. (4) increases a. random movement of a vortex rapidly

when it begins. .

X.k (±41)

= Zk(*)

Uf)

-(t

)

h(t)

3

(k) -

Tk

(*-t)}

(4) =

(t)

t

¿it. Uk()

(5)

k(-) =

+ ¿ii. (1JìCt)

oscillating böuidary layer theory as follows,

I

d (1 (8) 5m (c

t

Tr/4)

¿u

dX

_{{(T21)sin,(2c&J±Tr/)_o.J}

where (8) denotes the amplitude of fluid. velocity at the outer edge of the bouíidary iaer at the zero-shear point, X the

coordi-nate along the cylinder surface, the origin of which is located at the front stagnation pöint. When we ko the velocit.y gradient distribution on the surface of a òylinder, the löcation of the zero-shear point at the möinent of a certain time-phase t can be obtained by Eq. (6) In the present calculation the velocity distrbtition of the potential plow without vortices is used in the calculation of dOms (0)/dX in Eq. (6).

The function on the right-hand side öf Eq. (6) during one half cycle is shown ;fl Fig. 2.

on tifle rig'.-f-._}- side óf

J.J.d1L1

U RdO

Os b.

'o

-p,, s. '4

t

t

t t t

t

//

t

,,

_'F

t

t

/

t

'I..-'

t

i

.% t

,

,

-0.5-'a.

-- -r

-

a..->1'

Fi:r.3

'Teftcjj-j

rndient distribt.inn

irnc th

siirÑe

f (nlrder A

r0:Ioo

U ;U..,sn .,*

U(b):US.'w*

distance between cylinders is far, the curvè Of the velocity gradient approches to a cosine curve with negative sign as shown by the solid line in Fig. 3. The. shorter the distance between. two cylinders, the more signifiôantl.y the. velocity-gradient

distribütion is affected..y the other cylinder as shown in Fig. 3. Fig. 4 shows the löcation of minimum velocity-gradient,, where the zero-shear point occurs first during the first. half cycle..

two circular cylinders center, is 2.5 and, Kc numbér is 10 (Xc number is defined as

_UmT/D)

In real flow, the location of the zero-shear point is affected by generated rorI-ices. In the present calculation, however, this effect is nöt taken into

account.

&: Ioc..io.

aìil. cf

ZErO-SAEIY _po;at

r. i

g I

I

I

/

I

I

I zem-sJerv

pot

zem-sIe'

po..st on 4sey_A

1jn of

zero-gier noint

o

te to oírrii1pr

o11nders durjn

one hif c'c'r1e.

I /

I

/

/

2.4.. Replacement by Inyiscid Vortex

A real vortex in a boundary

layer

is replaced by an inviscid

vortex at the zero-shear point where the velocity gradient du/dy at the cylinder surface becomes zero. Nöte that the zero-shear point does not coincide with the separation point in oscillating

flow. Downstream of the zero-shear pöint, a very thin reverse flow region appears as obtained by Schlichting's theory. Sinöe the reverse flow is slow, we can safely assume that the. vortices are. not affected by body surface in this region., and that. thé substituted invisc'id vörtex keeps its strength cönstant after the replacement.

The strenth öf the. Vörtek at thé zero-shear point can be deter-mined by boundary layer theory aS follows,

4- U(es)! LJe)/4* ¿.j(&R40

(7)

where U(O5) denotes the. velocity. Qutsïde of the boundary layer

at the zero-shear pöint and O is the displacemént of the loöation angle of the zerö-sheár point during the time increment t. In the present calculatión; u(es) consists of a potential flow component and an induced. velòcity component. due to separated

vortices. This means that the induced, velocity due tO the vortices

in. the boundary layer is .not include. i. U(Os).

The inviscid vortex is set in the middle of the. bOundary layer as shown in Fig. 6. The boundary: layer thickness is assumed as

6

4..6L/2/

on the basis of the unsteady boundary layer

theory for an oscillating fiat plate. Then the place of the. replaced. vortex can be expreésed as

((R

/2)Co5

Os,

(p+/2)Sin

c as shown in Fig. 6. Note that this location depends on Reynolds

2.5. Vörtèx Mödei

In real fluid a. vortex gradually diffuses with time. In the present. calculation, in, order to take this effect into account the circumferential velocity u is modified as follows on the basis of theoretical esults for an isolated vortex:

= 27rr

-

exp(_r2/4p(*o.)}

(8)

On this. vortex model the circumférential. velocity u0. has

distribution like the sold line in Fig.. 7 am. has a maximum at r r* which increáses with. time. The initial time t0 in

Eq. (8) is determined as r* = 0.5 5 when the vortex is substituted at the zero-shear point. The vortex model avoids the unrealstic large induced velocity of. vortices getting too close to each other and to the body surface..

èo-her i'ojnt av

sertj,or.

o,i.nt.

Pip..7 Circifereroii veloci-br of t,he nre.sert vortéx

rno1 el.

2.6. Separatiön Conditïön

As stated before, the zero.-shear point does. not cöincide with the separation pöint i. unsteady flow problems. The inviscid vortices generated at the zero-shéar point move downstream along the surface like a boundary 'layer, thén shed into the outer flow automatically at separation point, make a. vortex spiral or a vOrtex lump as shown iii Fig. 8.

In the present calculatIon, it is required to distinguish between the vortex in boundary layer and the vortex separated from it because only separated. vortices should be taken into

account jn the calculation of the vortex strength and the pressure on the surface. The flow separation takes place whèn the ratiø of the vertical velocity to the horizontal one is of the order

unity. A nimìerical. value of the ratio for the separation. _criterion

is assumed to be 0.3. he effect of the iumerical value _{IÜay not} be so significant because the ratio increases repidly near the separation point.

2.7. Neglect f Vbrtex tôo near Surface

Some vortices approaching the body surface stay there for a long time because the. vortex model used in the present calculation reduces 1ts induced, velocity when it is clOse to the body _surface. In some cases, the trapped vOrtex affects the generation _and.

development of the new. vortices generated in the following cycles. In order to avoid this porblein, a vortex approaching the sürface more thana certain distance _{from the surface is usuälly}

omitted in a discrete. vortex method. Physically _{thIs corresponds} to the vortex extinction due to the energy loss by generating

OdaTlayer on the surface. There is, howevér, no definite method to determine this criterion _{In the. present calculation the} value of is assumed to be r*, which is shown in Fig. 7,

since the. viscous effect is dominant in the inner _{region of .r*.}

2.8. Calculatibn Of

P.vt1

The pressure P(G) ón the cylinder surface can be obtained by the following pressure equation.,

¿1(8)2/

,, th. cy(nder sur4lce

where denotes the. velocity potential and u (0) the velocity on the surface, both of which consists of the ptential flow

component and the induced velocity component by separated. vOrtices.

Then Eq.: (9) can b expressed as:

p(5)-f4

_f'_ffIue)+Uv(8)}/

(10)

4te

where suffix p donotes the potential flow component and suffix y the vortex cömponent.

The first term of Eq.. (10) is expressed as:

-f

=

When r0 is infinity Eq.. (11) becomes to:

2f k

_frU cos O ₍₁₂₎

which Is the. value for single. circular cylinder, jn: a harmoniòàlly

oscillating flow. The Second term of. Eq. (i can be expressed as:

r

=

(U:i M,.k

(LÇ

1f,)

174 k=i

k 1 ; a real. vortex

k. = 25.; the image. vortices in cylinuers

where (uk

I vflk) denotes the induced, velocity on the surface by

the n-th Vortex and (unk Vflj the velocity

due

to

the

movement of the n-th. vortex. In this calculation, the pressure term, which is propoftional to

Î' /'k ,

is omitted because it has a finite. value only at the zerb-shear point whfr the. vörtex is in the

In the figures of the present paper, the. pressure coefficient

C , which is obtained by dividing the. pressure p(0) in Eq. (9)

by

2

f

will be plotted.

2.9. Hydrödynamic Force

The hydrodynamic force acting on thé cylinder can be obtained by integrating the pressure (e) over thé surface. Nöte that a lift force dOes not act on the bodies bécause of the present assumption of the symmetry of the. flow.

3. SimulatiOn Results and Discussibn

In the preSent simulation only symmetrical flow is treated because

of the tine Iiitit of ±i cuter. In order to. calculate. asymmetrical

flow past two cylinders, it takes about four times the cömputer time of the syÌtretrical case. Note that no transverse fòrce acts on cylinders in the symmetrical case. The flow pattern in the upper half domain will be shown, since the flow is assumed to be

symmetri-cal

3. 1. Effect f Time Step Size

As a matter

f course a large time-step-size causes. unrealistic vortex movements like the entrance of a vortex into tie cylinder. The smaller the time-step-size, the closer thé flow model

approaches a reál. vortex sheét. small time-step-size,, hôwever, requires much computer time which is approximately proportional to the square of the number of the calculation steps. Since there

is no definite method to determine the best time-step-size in the numèridal calculation with a discrete vortex methôd, we can not help using a trial and. error method. to choose a suitable Step

size for the case considered. A suitäble time-step-size may

Reynolds number.

Figs. 3.1 - 3.3 show the. simuiation results for the same condition for differexit time-step-sizes. In these calculation a half period is divided into 30, 50and80 time-steps. Fig. 3.1 illustrates that the 30 time-step is tòo large to make a vortex Spiral in good

Vorder at the moment when the flow velocity is maximum ((a), (d) in Fig. 3.1). For 50 and 80 time-stepS, the vortex spiral rolls up in fai.1y goòd. order although some random mövements are seen in the vortex lump since no special techniques to keep the. vortex line in good order as used by Fink et al. /9/ and FaltinSen e.t al. /10/ are used in the present calculation. At w t = r in

Fig. 3.1 (b)randöm movements of the vortices, in the vortex spiral increase, but the flow pattern difference between for these two kinds of time-step is not So significant. The Iydro-dynamic force acting on Cylinder A shown in Fig. 3.2 shows that

the agreement between the cases for 50 and 80 time-stepS is fairly good up to w t = 4.5. From w t 4.7 to 6, the

hydro-dynamic forces show. some differeice'. However., such. small variations

of forces due to Slight differences of. vortex behaviour sometimes appear in simulation results with a discrete. vortex method since the motion o' each. vortex is .a little bit stochastic as shown

iii Fig. 3.1.

The computer time limit is also. very important factor to determine the time-Step-Size as mentioned above.. It is necessary to calculate for 1.5 cycle at leaSt. .It.tkes the 'Cybe'r 170 computer of the

computer center of Technical University of Berlin about 3Ó minutes for the Simulation duriflg 1.5 cycle in the 50 time-stepS case.

3.2 Slmu.iatiòn Results

Simulations ha'e béen carried out for the following conditions and the results are showriin Fiqs. 3.4 to 3.22.

Table. i. ...

where Kc = UmT/(2R)i R U (2R)/'J an N1. denotes the number of

the time-steps in,.e:half cycle.

Fig1 3.4 shows the results fOr r = 10. In the first and. econd half cycles ((a) -. (d) and (e) - (g) i Fig. 3. 4) the. develOpment

and behav±our of vortices around Cylinder A are similar to those for a single cylinder in oscillating flow: The; generated. vortices

form a. vortex line along the body surface like a boundary layer the front end of the. vortex line reverses along the surface and makea vortex spiral (c) and then the vortex spiral rolls

up and,moves toward. upstream (to the left directión) due to the

i-duced. velocity of the image vortices (d). At the. end of the

conditions . Figure

Kc Rn. r0 N1 . Flow. Pattern

. Pressure Force

8 6820 1000 ' 50 . Fig. 3.10 Fig. 3.17

8 682.0 10 50 Fig. 3.4; F.ïg. 3.11; Fig. 3.17

8 6820 8 .50 Fig. 3..5 Fig. 3.12 Fig. 3.17

8 6820 6 50 Fig. 3.6 Fig. 3.13;

Fig. 317

8 6820 3 50 Fi.g. 3.7 Fig. 3.14. Fig. 3.18

8 6820 2.5 50 Fig. 3.8 Fig. 3.15; Fig.o 3.18

8 6820 2.1 50 Fig. 3.9 Fig. 3.16: Fig.. 3.18

12 6820 2..5 50 Fig. 3.19 Èig. 3.22

16 6820 2.5 50 Fïg. 3.20 Fig. 3.22

second. half cycle (g), the. vortex lump B1 which was created

at Cylinder B during the first. half cycle approaches Cylinder A. Vortex B1 with clockwisè circulation and Vortex A2 with

anti-clockwise circulatiôn niake a. vortex pair (h), and it moves upward

due to the induced, velocity of each other (i). The effects of Vortex B1 and the. vortex pair of A2 and B1 on the pressure

distribution seem tö be slight as shown in Fig. 3.11 ((f)'' (i)).

The slight difference of hydrodynamic force between for r0 = 100

and for r, 10 in the early stage of the 3rd half cycle shown

in Fig. 3.17 (b) is due to the interference effèct of these two vortices, toô.

Fig. 3.5 shows the results for r0 = 8. The flow patterns are

similar to thôsé for r0 = 10 shôwn in Fig. 3.4. However Vortex B1 reaches Cylinder A at the late stage of the second half cycle (g) and affects the pressure distributions on Cylinder A. as shown in

Fig. 3.12 ((g) and (h)) and the. hydrödynamic force acting on it as

shown in Fig. 3.11. An interesting fact is that the vortex pair is always formed near Cylinder A in this situation. If we assume U =-U sin ( wt), the vortex pair iS always formed near Cylinder B. In other words, it depends on the initial: condition. Then,

Cylinder A and B have different. hydrodynamic characteristicS from each other. In real fluid like in waves some disturbances

seems to change the situation, and probably a cylinder ometimes has two different hydrodynainic characteristics.

Fig. 3.6 shöws the results for r0 = 6. As r0 decreases, Vortex B1 reaches Cylinder A in the earlier stage of the second. half cycle. The deviation of the surface pressure from the potential one

at wt = 3 /2 as showh in FIg. 3.13 Cd) is due to the effect

of Vortex B1. Then we can expect an interference effect appears on the drag coefficient as well as on the inertia coefficient.

At wt

2 r, Vortex gets. over Vortex A2 as shown in Fig. 3.6

(e), and then they make a vortex pair (g), Note that the relative locations of Vortex A2 and B1 are opposite to that in the case for r0 8 and 10. TherefOre the. vortex pair moves downward

and stays betweén the two cylinders (h). The pressure distribution in Fig. 3.13 illustrates significant differences from those. for

r0 = 100 after wt = '3Tr/2. The peak appearing in the. hydrodynamic force acting on Cylinder A at wt. = 5.8 shown in Fig. 3.1 (d) may be due to the. effect of Vortex B1. The. vortex pair formed by Vortex A2 and B1 seems to. survive between two cylinders bécause any energy loss due to the interference effèct between vortiOes is nat taken into account.

1rhe résults for r0 = 3 in Fig. 3.7 illustrates anothe.r kind of interference phenomenOn. Even in the first half cycle, we can see the difference of vortex shâpe betweén VOrtex A1 and B1. This is bedaüSe a bOdy and the. vortices generated neàr it begin to affect vortices created behind another body directly. Moreóver the

inter-ference betweèn Vortex A2 and B1 changes into a differént mode from those mentioned above: As shown in. Fig. 3.7 (f) Vortex B1 gets over Vortex A2, and goes away to the left without forming vortex pair. During this process Vortex B1 affects. significantly the generation and developmentof new vortices A2 and B2. The

pressure distributions on Cylinder A are much. different fran.thoë for r0 = 100 even at the early stage of the first half cycle as shown

in Fig. 3.14.

Fig. 3.8 shows the results for r0 = 2.5. At wt = 2.199 in the first half cycle the. vortex sheet generated frOm Cylinder A attaches to the surface of Cylinder B as shown in Fïg. 3.8 (c). This is the other mode of the interference effect. Strange to sày, Vortex B2 generated during the second half cycle iS not similar to Vortex ((b) and (f)). This may be mainly because of. the

effect of Vortex B1 on the generation and development of Vortex B2. It is not sure. whether such a. phenomenon appears in real fluid, and. experimental investigation of this point is required.

Fig. 3.9 shows the results för r0 = 2.1. The attachment of Vortex A1 to the suface of Cylinder B appears earlier than for the case

for r0. =2.5, and at wt = a slight mixture between. Vortex and B1 can be seen.

In Fig. 3.19 to 3.12, the. effect of Kc number on the. vortex patterns for r0 = 2.5 is shown. As Kc. number increases, the vortex formed behind a cylinder becomes largér, and the interference efféct is expected to be. more significant.

Fig. 3.19 showS the result for Kc. 12. The attachment of

Vortex A1 to Cylinder appears much earlier than the case for Kc = 8' (Fig. 3.5). At wt =2.199 Vortex A js elongated and mixed with Vortex B. The randöm movement of vortices becomes Siqnificant.

Fig. 3.20 shows the results for Kc.= 16. At wt.= 2.199, Vortex.A1 is elòngated and lumped togethér wth Vortex B1, and they ioók like one vortex wake.

Fig. 3.21 shows the resultS for Kc. 214. Since the randbm movements of vortices are 'too severe to get a wke

shpe,

a smaller

4. Hydrbdynainhc Interference

4. 1. Vortex Interference

We can obtain a lot of physical insights into the interference

effects between two circular cylinders from the present simulation.

Fig. 9 shows a schematic. View Of the behaviour of vortices for relatively large r. In this figure A dénotes a vortex lump created during the i-tb half cycle nea:r Cylinder A, and B1 that

reated near Cylinder B. Thé incident flow is expressed U

UX

sin (w t), and the. simulation. starts frOm t = O.

When the distaflce betweén the two cylinders is far enough, A1

goes away toward -x direction and B2 goes away toward the opposite direction. A2 and B1 make a vortex pair and go upwards as shown

in Fig. 9 (a) near the middle between the two cylinders.

As decreases, B1 approaches Cylinder A and makes a vortex pair with A2 near cylinder A (Fïg. 9 (b)). The interference, effect.

due to B1 is expected to appeàr first on added mass since B1 approaches Cylinder A at the. end stage of a half cycle.

As r0 decreases more, B1 approaches more clo.séiy to Cylinder A, and reaches it (FIg. 9 (c.)) B1 meéts A2. over Cylinder A, and make

a vortex pair with ït, and then the. vortex pair goes upward. as

shown in Fig. 9 (c) . The inteference effeät increases. Note.

that the Vortex pairs are always made near Cylinder A in this situation. Then we can expect that the. hyd.rodynamic force acting Cylinder A is different from that acting on Cylinder B. If the incident flow starts in the opposite direction, thê. vortex pairs are always made near Cylinder B. In real oscillating flow, the disturbance in the flow may cause. the transition f-rom one

situation to another. The two different hydrodynain.i.c forces

obtained by thé experïments /11/ seems to be caused by this phenomenon.

At r0 = 6 shôwn in Fïg. 9 (d), we see another type of vortex

pair.. At thi.s situation, B1 reaches Cylinder A before A2 has

fully developed, and runs across A2. However since the incident

flow reverses before B1 passes Cylinder A, B1 moves back toward

thè right and makes a vorte

pair with A2. Since their relative

locations are opposité in the case shown in Fig. 9

(a)

- (c),

the pair moves dOwnward and stays between the two cylinders..

It may sûrvive thére, and affect thé behaviour, of the following

vortices. Since in a reál fluid 'it loses its energy by mixing Of

vortices with opposite 'circulatiòn, the 'thxing effect Should be

taken into account.

Below r0

3, B1. runs over A2 and goes away tOward. -x direction.

B1 affects thé generation and development of A2 as well as B2.

in addition to the interference, effect due to old. vortex, the

effect of the. existence of another body and another nèw vortex.

on the development of vortices Occurs in

this situation.

Òf course the critical, value of r0 for transition from one

inter-ference mode tO another significantly depends on Kc niainber..

Fig. 10 shows the schematic view Of the vortex, interference for

r0

2.5 for variOus Kc numbers. At Kc = 8 A1 and B1 affect each

other and the shape

f them differs. Since two cylinders are

located, very near, the effect of A. on the pressurei Cylinder B

and makes a closed region betweén two cylinders as

shown in

Fig. 10 (b). in. this situation, the flow pattern' around the two.

cylinders looks like 'that' around one cylinder with different

shape from a circular cylidèr. As Kc nunther increases, A1 is

elongated and covers B1 as shown in Fig. 10

(c:). At Kc.= 16 and

A.

k=8

t'

r q _{Sc'1e!.. tjc View. of vorey i"terfere'ce te1weer ew1r}

CkC/L.

F'i.1O Schemtic view of vortex interference be-twee the vort±ces er.erated.. in te seme. half cycle..

4.2. Interferene Effect of Drag and InertïaCöefficient

Hydrodynamic force F acti.ng on'cylinder in oscillating flow can be expressed as.

The drag coefficient CD and thf inertia coefficient CM obtained from the simulation results shown in Figs. 3.17 and 3.18 using Fourier analysis. In Fig. 11 obtained CD and CM values àre shown. The drag coefficient decreases with decréàsing spacing and shows a negative value at r0 = 2. 1 In the contrary the inèrtia

coef.fiòient increases with decreasing spacing.

o =

fDCDtIUI 1

FjrCo0

alas d.tu- 0Ç cyI-lov

da

dt

IO --ioo r0

'ç

-/0 IbO _r0 (14)

Fi11 T)ra

nd inertia coefficients of hydrodmamic forces acting on Cylinder A.

CM

o o

5. Cbnclusion

Simulâtion program of. hydrodynamic interference of two circular cylinders in tandem arrangeme in harmonically oscillating flow

has been developed using. a discretè vortex methòd, and the following conclusions have been obtained.

The discrete vortex method used in this paper has proven useful for providing a better physical understanding of complex vortex-shed4ing flows around multi-cylinders in oscillating flow.

Hydrodynamic interference can be divided into two clässes. The one is the interference betweén the former generated vortex and the new generated. vortex. Anöthér oe iS that

betê

two. vortices generated in the same half cycÏe.. The latter occurs only when thé distance between two cylinders is near, but the íoriner remains even for relatively large distance between them.

The behaivöur of. vorticeE can be classified into some types. The critical, value of thé distance of twò cylinders whén the transition bétweén one type to the other one mainly depends

on Kc nu±nber.

6. Acknowledgements

This work has been dnè during thé stay of the author at the Technical University of Berlin as a Research Fellow of the Alexander von Humboldt Foundation.

The author wishes to express his sincee thanks to Professor Hörst Nowacki of the Technical University of Berlin fOr his kind support and. valuable iscusions. The author should like to express his gratitude to PrOfessor Nono Tanaka, Associate Professor Yoji Himeno of University of Osaka Prefécture and Dr. Apostolos Papanikoiaou of TU Berlin for their valuable suggestions and discussions.

7. Referencês

1.] Ohkusu, M.: Hydrodynamic Forces on Maltiple Cylinders in Waves, Symp. ön the Dynamics of Marine Vehiclés and

Structures in Wave, 1974

[ 2.] Isaacson, M.: Interférence Effects Between Large Cylinders

in Wave, OTC paper, No. OTC 3069, 1978

Masumoto, A., Yarnagüchi, Y., Sakata,. R.: WaVê Exciting Forces on Groups of Floating Bodies, Jour, of the Society of Naval Architects of Japan, Vol. 145, 1979

Minematsu, H., Noun, N., Mita, S.: An Experimental Study of Wave Forces on a Structure Supported by Vertical Cylinders, Jour. of the Kansai Society of Naval.Architects,

Japan, No. 178, 1980

Sarpkaya, T., Omar, N.: Hydrodynamic Interference of. Two Cylinders in Harmonic Flow, OTC Paper No. 3775,. 1980

[ 6.] Sawaragi, T., Nakarnura, T.: In Interference Efféct of Wave

Force Acting on Multi Cirular Cylinders (translation from Japanese title), Proc. of the 27th Coastal Engeneering Conf., Japan, 1980

E 7.] Chakrabarti, K: Wave Forces on Vertical Aayof Tubes1

Proc, of. Civil ngineèring in the Ocean IV, 1979

1 8.] 1mai, I.: HydrOdynamics, Shôkabo K. K., 1973

[ .9,j Fink, P. T., Soh, W. K.: Calculation. of. Vortex Sheets in

Unsteady 'iow and Applications in Ship Hydrodynamics, Proc. of the 10th Symp. Naval HydrodynamIcs, 1974

(10.] Faltinsen, O. M., Pettersen, B.: Separated Flow around

Marine Structures, Proc. of the Second International Symposium on Ocean Engineering and Ship Handling, 1983

[11.] HOne,. H.,: MeaSurement of Hydrodynaniic Force Acting on

Appendix

The complex. velocity potential for the flow passing two circular cylinders has been. obtained, by Imai /8/ as follows:

Two circulars cylinders with the radius o a and b are located .at z = O and z = c rèspectiveÏy. Whén the distance between them

is far, each of cylinders can be assumed to be in a uiiform flow. Therefore each cylinders can be represented by a doublet located at its center. The strength of the doub..ets are represented by

1LL and ,z' respectively.

Then the. cylinder K is approximately assumed to generatê the flow whose complex. velocity potential is given as:

JL

(a.

The complex. velocity potential can be expanded as follow near

z = O,

fO)

2

The effect of cylinder K can be öbtained.by the circle theorn of Mime-Thomson:

f

=

j:

= f0'O .1-

f0

a2/z

. _{(a. 3)}

In other words, the. efect is equal to the existence of a doublet with the strength of

= - f0'co

a2

-

(u

' E2

(a. 4)

In the same way, the effect of cylinder K, can be represented by a doublet with the strength of ,N locatéd at thé center,

u#4)

i42

Ta2b2 ¡ cl4 a2b

ici4

From Eq. ('a.. 4) and Eq. (a. 5.), ,AAand /A'jg obtained as,

Va2

Ob2

(a. 5)

(a. 6)

For the. case of two. circular cylinders with thé same radius in tañdem arrangement. in a flöw, Eq. (1) in Chapter 2.1 can be obtained Eq. (a. 6').

--a---.-.-. tJ'=8'O

(a)

,.)i:1.r7f («1T/2)

it

f.371 #:1Ç71

-i.-,

'

N.:30

Ni.O

r0

lo

r(fo a

Fig. 3.1(1) Effect of tine-step size

on

calculated vortex pattern.

9

(d)

4i)f:4. 712

Wii

an1i.cIodii' Ci rcuIatio.i:

Fig. 3.1(3) Effect of time step size on calculated vortex pattern.

(e)

(t/,- V)

CF

.3

F

CF7PU2R)

---.----N1 -o

F: kyJ'c.

UYCQ

c1idA.

_._.-_ N0

N,:

e nuIrn

vÇ+;n,e seps

dw'1,

kat cycle..

r'

poteti'o I

/\..

/

/

\

\'

.. /

t

S . / F /

,

/

S' 'S 'S s

Rs- ¿aio

yo

lo

Fig. '3.2

Effect of time-step size on h'drodynainic force acting

Cylinder A.

-3

¿,41,Ç

ycIe

Secod

---h

.

cp 3 c

U (=-Usf)

FJrO

---+--;

7)2 p

2 r -

P- ydd1r'a.' pr4sQfe

f

p-, '-b, b F 1 *

,

-.% r

_t.

Fig. 3.3 Effect of time-step size 'on _{pressure distribution on}

1257

(a)

wt- 377O

(e)

_=

L

r

r1oR

LA.)f

6.23

2Tr)₍ i -a -I A j j ,1 s

.

s a o

-wt=4.712

_.

a .0 a

Fïg. 3.Lf(2) Flow pattern around Cylinder A (Kc=8,, Rn=6820, r0=ì

I +h clock piîSe c,rcuk1i(', 1th QMti-clockwiSe a s a I a

I

s

Ia

a o o I g o g g. s $ s s

5$

(ti)

(t)t f ti .. s _e a . I s s s a s O

(4)

M M * 11* K o X o lo 7o

Fig. 3.Lf(3)

Flow pattern around Cylinder A (Kc=8, Rn=6820,, r0=1O).

o

$

X

K

j(a)

CA.)

(.25?

(d)

(.e)

3.170

(C)

vt: 1442

=Tr A o

(b).

iU*= 1.571 Tr2 A A t

Fig.. 3.5(1) Flow pattern around Cylinder A (Kc=8,. Pn=6820, rO=& )e

V

X

(4)

_{() jfr f,.2}

2IT

(vj-. 4:'7 2

3i4

A a a a s a $ a X s X s e o _o' . X a

Fig. 3.5(2)

Flow pattern around Cylinder A (Kc8, Pn=6820,, r0=8):.

'w

(Ji)

i*

'7.

4

o s a IC .

. s

(0)

ci-f-=1,'?1

c

(u

YA '?

Fig. 3.6(1)

Flow pattern around Cylinder A & B (Kc=8, Rn=6820, r0=6).

(17)

wt 24qc

s

(e)

s

,*r6.2S3 (2Tr)

e e o e e o

Fig. 3.6(2)

Flow pattern around

Cylinder .A (Kc=8, Rn=682O, r0=6).

-Cd)

ûit4.7f2 (:.j

I o C e C e e e o o o o

('s)

Lr (= Tr) p s o

SB'

I s C S X u a a

F±g. 3.6(3)

Flow pattern around Cylinder A (Kc=8, Rn=6820,, r0=6).

o

I.

X a

w*- &i2

C s M . a e

ih

(.JJ-= 3T

s Q s o

i

o s * s w X J X o 3D

Fig.

3.6(Lf)

Flow pat:tern aroud

Cylinder A :(Kc=8, Pn6820, r0=6).

s

e

(Q)

ca.t=1.7Ç7

(C)

iif

2.1q

.(

).t

6.283

(71r)

s

(Li) »

?.-4

()

M o M I M M

Fig. 3.7(2.) Flow pattern around Cylinder J

(C)

t,

()

_¿,t-=1.57f

-F±g. 3.8(1) Flow pattei'n around Cy1nder

A & B (f(c=8, Pn=6820,

r02.5).

(e) wt

_.

t 4.712

()

wt

6.fl3

... .'

/>

D.

/

B3

(h) c*î.gç4

I

()

C4)t

1.251

()

t.5'7f

(C)

f-=qq

irlr

(e)

ijt3.?O

:A.

.

(f)tD

4.112

(

.28'3

21T . o

Fig. 3.9(2) Flöw pattern around Cylinder A & B (Kc=8, Pn=682Q, r02.l).

A *

2TT

C

2

wf'. f.2t?

c.it1.71

(C)

2. 1?

cd)

L+

3.142

CPI

-3

3 Cb -4

(f) t:4?f2

(V

t.2g3

71r

o.

-5

()

_{wt t. 27}

(C)

ql

(d)

wt=3442

-11

Cf'

-5

(h)

LA)

?.4

¿o+- 1.27

(b)

Lj

_2.1

(C)

t=3.f42

Tr

(t)wt= '1.gS-4

2'

CF

Fig. 3.13(2) Pressure distribution

_{on Cylinder A (Kc=8, Pn=6820, r0=6).}

(d) t

4.712

(C) c1*=2.1

()

ù'tì42

e)

f=3.77O

cP

(b)

wttS'?'1

o Ir 3 r0-IoO

Fig. 3.lk(1,) Pressure distribution on Cylinder A (Kc=8, Rn=682Ö,

_r0=3).

cP

(f)

t47j'

2TT

Fig. 3.lk(2) Pressure distribution on Cylinder A (Kc=8, Pn=682D

_r=3).

.2

cP

bL

r)

Cr

(3Tr).

b

(a)

wt= i 257

6t1.'7t

(c)

bi

2.19?

(d)

wt' 3. 44-Z

7T CF! o 3

I

N

(9)

.2 83 2Tt

(C)

(d)

W*3

₁₄₂

7r

(h)

LJ*

1.t4-C

wt

Fig. 3.17

Hydrodynamic force acting on C1inder A.

CF:2

(d)

r0 6

o

=3

(b)

2

Yo7.

o 2..J 4I cycle 3r

..----; (-4Ob

Fig. 3.18

Hydrodynamic force acting Ön Cylinder A.

(C)

2

r0

2.1

(b)

wt1qi

(Ç) D d

wt3.442

ir 2 7

(+) (Arts 2T

&f:.

Ç1T4

A

X

Fig. 3.19(2) Flow

aìtern aröund Cylinder A & E (Kc=12, Rn682O, r=2.5)0

s

X

(a)

j27

U)

6JtI.'?I

(C)

¿il-2.t

o G 9. 9 o G G o G u  G 9 I I o

Fig. 3.20(2) Flow pattern

around Cylinder. A & B (Kc=16, Rn682Ò,, r0=2.5).

(d)

GJf:3142

o o TÍ o o o

V * A s A s S I, V o o . V V V

d)

iit293

7Tr X o 70.

Fig,

(d)

e

o

o

o

Fig'. 321(2) Flow pattern around Cylinder A &B (Kc=2i, Pn682O,, r0=2.5).

j. 142

e s

:11

o o

t't

.341

y__J g p. -70 M -OE 1 k

I

wf7.24-

(V

Fig. 3.21(3) Flow pattern around Cylinder A

(Q)

kci12

r0'.

2

o o