Calculation of lift force acting on circular cylinder in oscillating flow

TECHNISCHE UNIVERSITÎ BERLIN

INSTITUT FUR SCHIFFS- UND MEERESTECHNIK

CALCULATION OF LIFT FORCE ACTING ON CIRCULAR CYLINDER. IN OSCILLATING FLOW

by Dr. Yoshiho Ikeda

University of Osaka Prefeôture

A.. y. Humboldt Guest Scientist at TUB

Abstract

The asymmetrical flow pattern around a circular cylinder in harmonically oscillating flow is calculated by a discrete vor-tex method. The obtained vorvor-tex-shedding frequency shows a good

agreement with experimental results. The pressure arid lift force

acting on the cylinder are calculated. i

Introduction

The vOrtex shedding phenomenOn from a bluff body in oscillating flow causes. a transverse force as well as an in-line force

acting on it. This transverse fOrce plays an important role in the wave forces acting on piles, cables and elementsöf an ocean structure since it sometimes becomes of the same order

of magnitude as the. in-line force.

The characteristics of the transverse force are much more

com-plicated than those. of the in-line förce. For example, the f

re-quency variation with Kc number, some stochastic behaviöu±- and some beat appearing in Observed records, thé lock-in phenomenon, some peaks appearing :in the lift coefficient and so on /1/, /4/. In order to get a deep understanding of these characteristics it is necessary to know the behaviour of the. vortex-shedding

flo around a bluff body in detail.

For this purpose flow, visualization and velocity measurements are very usefül. However, it is sOmetimes not so easy to get detailed

informations

frOm these experimental results partic-ularly for the case of oscillating flow probléms.

A discrete. vortex method is one 'of möst powerful tools to reveal such a complicated flow characteristic, due t'o. vortex-shedding

from a bluff body, tt gives, us detaied knowledge of the

devel-opment. f the wake, thé. vortex-shédding phenomenon1 and the

relation between the vortices and the hydrodynamic fordes actsg on

the body.

CalculatiOn Procedure

2.1 Outline of' CalculatiOn Procedure

The calculation procedure iS almöst. the same as that presented in the previous papers /2/, /3/ by the authors.. The outline of

it is as follOws:

The zero shear point on a circular cylinder in oscillating flow is predicted by Schlichting's oscillating boundary layer theory /5/. The conditiön of tie zero-shear point is expressed as1

-

,(g+-)

co dz.

_-

_{(J_')in(2*_)_o.5}

where Udenotes thè fluid velocity on the body surface ánd

the cöordinate along the surface.

If the potentiai flow velocity ön the body

is used as

U

in

Eq. (1),

Eq.

(1) can be changed into the following expression:. (1)

COS _Os

I

ir s;n(wt+)

k

{(r2_I)Sir%(2iJt.)

5'J

(2)

where

Os

denotes the location angle of thé zero-shear point

and Kc the Keulegan-Carpenter number defined as Umax. T/D.

Shedding-vortices from the body affect the location of this

zero-shear point. The method to take this effect into account will be described in the next chapter.

At the predicted zero-shear point the réai. vortex in a boundary layer is replaced. by an invscid vortex with the same circulation strength asthat calculated by boundary layer theory. The invis-cid vortex generated at each time Step is assùmed to keep its strength constant after the substitution. The trace of each vortex is calculated using a potential flow theory by a time-step integration,

and.

these inviscid vortices sithulate _thß

behav-iour of the wake béhind the body n the time domain.

Separation is assuine

to occur when the ratio. of the transverse velocity to the horizontal: one of the vortex moving along the body surface after the generation at the zero-shear point bej-comes of the order of unity. In the calculation of the strength of the subStituted vortex: at thé zero-shear point and the

pres-cos O

=

4

sure variation on the body surface, only the separated vortices

are taken into account because the unseparated. vortices re-present the boundary layer near the. body surface..

In order to take the effect of the diffusion of a vortex into account, the circumferential velocity of the inviscid vortex

s modified as shown in Fig. 1. This vortex, model avoid the unrealistic large induced. velocity when a vortex approaches the other one or the body surface too closely.

If no disturbance s given in. the calculation, the. vortex flow

keeps its synunetrical pattern forever. However, experimental results show that the vortex-shedding flow around a circular cylinder in oscillating flow is asymmetrical for Kc niiber

above. 8. In order to get an asymmetrical, vortex pattern, the

strength of the generated. vortices at the zero-shear point of the lower side of the cylinder are halved artificially during the 1/20 times the period of the oscillating flow after the beginning of the. vortex generatiòn in the first half cycle.

2.2 Effect Of Shedding V n...erb-shear Point

Shedding vortices from .the body afféct the location of the zero-sheàr point

and

the separation point..In the prediction of the separatiOn point thjs effect is taken into account since the movement of each. vortex is calculated using the induced.

velocities due to all, vorticeS in the flow field and thei.r image. vortices in the body.

In

the condition of the zerO-shear point expressed by Eq. (2.), however, this effect is not taken into account.

In order to také this effect into account, the condition of the zero-shear point of Eq. (2) is modified as:

ir Sifl (wt .

-il

_{1W-') sin(2t.*#2E}

) -o. 5j

5

where denotes the magnificatibn factor öf the increase of the flow amplitude due to the shedding vortices and. the

phase advance due to the shedding vörtices. In our previous paper /2/ only was considered using experimental results. Fig. 2 shows an èxample of the velocity variation during one half cycle at

9

= 7T/2,.. where _O denotes the location angle from the rsar stagnation point. The potential flow, velocity due to the incident f.low is a sine curve as shown by the broken

line, in this figure, while the calculated flow: velocity

in-cluding- ices' presented by the solid line shows a bigger

axiplitude and a phase advance...

u (mis) 0.2 -0.2 6 / / /

/

calculated potential flow. 37r

Fig. 2: Velöcity variation at the side top' of, a circular'

cylinder in oscillating flow during one half òycle.

In the present calculation the value of /3 and E is pbtaiied. .at the beginning of the each half cycle by analysing the

velo-city variation at _G = Tr/2 during the former one cycle by Föurier Analysis. Up to the end of the third half cycle this

effect is not taken into account in the present simulation. Fig. 3 shows the values of and _E versus Kc number. The values of and E decrease with increasing IÇc number, and

at Kc.above 10 and E become almost unity and zero,

respec-tively.

i

o

Fig,. 3 Values of

and ¿.

o io _Kc

3. Calculated

Results

In this chapter sorné examples of calculated results of, the vortex pattern, the pressure distribution and the hydrodynamio force acting on a circular cylinder in oscillating flow will be shown.

3.1 The Case for Xc=8.

Fig. 4 shows the calculated. vörtex pattern around a circular cylinder for Kc=8. In the f irSt half cycle (

wt=o

TT )

asym-metrical. vortldeS gradually grow up behind the cylinder. At t=3 TI _{/2 in. the second half cycle the ne.ly generated. vortIces}

behind the cylinder are significant'ly affècted by the old. vor-tices, A1 and B1. The vortex pair formed by Vortex A1 and Vor-tex B1 moves toward. the left dircetion at an angle with respect to the x axis, and causes an asymmetrical flow pattern around

the cylinder. At aJt=27T the.vortices behind the cylinder are

almost symmetric. In. the third half; cycle a similar vortex behaviour can be seen, and thé new: vortex pair formed by Vor-tex B2 and VorVor-tex A2 moves toward. thé riht direction at an. angle with respedt to thé x axis. As shown in the. vortex, pat-tern

at

wt=771 /2, the'generated. vorticeé in. each half cycle make a vortex pair, and it leaveé gradually fröm the cylInder.

The preéSure distributions on. thé 'body Surfâce at t=3 and

7 Tr /2 are shown in Fig. '5. At t=:3.1T negative preésure

Is

creáted by. vortices ön the surface 'of thé: rear half side

(- w/2

< 8

. ff12), and. this pressure contributes to the decrease

of the. 'hydrodynainic inertia force. At Wt=7 ' /2 when the

inci-dent flow velocity is maximum, negative 'preésure ïs created on

the rear half. side (-lT< _G <-Tr/2 and. _{T72<8<7T )} by. vortices,

and it cuases thé pressure 'drag force.' The significant différ-ence of the pressure distribution betweén on thé. .upper and low-.er half sideé. cauSes a large 'lift' force.

8

on the cylinder. As expected from the. vortex pattern and. the pressure distribution in Figs. 4 and 5, the lift force is large ca

_{the momentwe}

_{the. Incident flow velocity is large,}

-

-.

4 I

-T

Qt 3Tr4 I -

G:ȓ

----Tr -A1 o

'k

X

.-'

"ls_

Fig.4 (1) Vortex pattern around a circular cylinder in oscillating flow at Kc=8.

Fig.4(2) Vortex pattern around a circular cylinder in oscillating

Fig.4(3)

kc8

Fig.4(4)

-2

-4

2

cf

--11 . -. 0 o o -' o o O wt=3T 'oi 7ir/2 o O a

--o----.

e O o e

'--I O'

J.. I o d I

/

\\

,16

Co

/

J o 6

\

/

N

/

a e a o I I o

.

/

Io.,

I

/ /

I

/

_/

/

_/

/

Fig.5

Pressure distribution on the surface of

a circular cylinder in Oscillating flow at Kc=8. o

.

s-.

O e

1T9

e I I I O -S-I 2 C o .

R/O23I

Fig.6

- .1.6

3.2 The Case fbr Kc=18.

Fig. 7 shows the calculated vortex patterìi for Kc=18. At the

early stage of the first. half cycle, the upper. vortex develops

faster than, the lower vortex, and a part of the vortex sheds

into the flow. Thereafter, the lower, vortex develops as shown

in the figure at £u=4 iT /3. At ut= iT

we can see three

vor-tex-lumps behind the cylinder. In, thé second. half cycle, a part

of Vortex B1 and a part of Vortex A1, námed A

in the figure,

make a vortex pair and it passes around below the cylinder and

leaves in the left direction. This behaviour is similar to the

observed results /1/. As shown in thé figure at

u)t=51r/2,

even in the third half cycle a part of Vortex A1 which

gener-ated in the first half cycle .stàys near the cylinder yet, and

it has a significant effeòt on the pressure on the surface.

This fact may suggest it is necessary to take the decay of the

vortices into account. Although in the present calculation the

diffusion of the. vortices is taken into account as shown in

Fig.

i which was deduced theoretically from Navier Stokes

Equa-tion for isolated. vortex /2/, the strength of the circulaEqua-tion

of a vortex does flot. decay.

Fig. 8 shows pressure distribution on the cylinder. At the end

of one cycle

(

t2 ir

and 3Tr) the pressure created by

vor-tices on the rear side. öf the cylinder reduces the value fröm

the pressure due to the potential flow cöxnponent. At

t=5ff/2

negative 'preSsure acts on the 'font upper sjde of the cylinder

( 71/((7T )

as well as on.t'he 'rear side of it. This is

be-cause of the Vortex A1 which 'remains, near thê. cylinder yet as

shown in Fig. 3..4

,

and this pressure significantly redüce the

in-line drag force.

Fig. 9 shöws the calculated transverse ând in-line forces. The

beháviour of thé lift force is mOre 'cöinplicated than that for

Kc=8. During one half cycle, we can see 'aböut three. 'peaks, of

thé transverse force, and this number coincides with that of

generated vortex-lumps obtained fröin the calculation. of the

ki8

17 -.

Fiq.7(1) Vortex pattern around a circular cylindér in oscillating flow at Kc18.

. 18

-6)t

31'/2

*0 * o

f.

_" e xi ¿1 *

.1.

_t SI_I X

J

'

S

o,

.p%,.L ¡ ¿ -X b O . b

Fig.7(2) Vortex pattern around a circular cylinder in

kc1

n M * . * I I K s I X° e Fig.7(3)

Vortex pattern around a circular cylinder in osc±llating flow at Kc=18.

21r I A. a I G K K n X K X K K

e

.

e s K n K A o S X S s e -7 K e e e e .-( B1 X

.

l/'

p' K * N w 8N N S S o X K K o X X X o X -'C

X,

Fig.7(4)

Vortex pattern around a circu].ar cy).inder in oscillating

flow at Kc=18.

kc 18

B, w K 6 o ' /

K..

K 6 e S s X _N., N N ox 4

18

cP2

e.-. O 6

/

_/

>0

/

\ .

O

/

_/

/

.

J

,

O e o =

4..

. o

,

e

/

e o O a e Fig.8(1).

Pressure distribution on the surface of a circular cylinder in oscillating flow at Kc=18.

/

/

s

/

¡<C '1 Cp f o

_-2

cf, D

)fi3W

II

e-

-- a-Is-. 0 LiJt e e

ô \

'e

\

4-o a Fig.8(2)

Pressure distribution on the surface of

a circular cylinder in oscillating flow at Kc=18. e u e -e a e e o a

,

.

e s.-5% e

f

o e o e e e

/

/

, e

/

/

_/

/

-e

/

/

e

/

e e

I

_,

e'

.5..

,

o s e e a a -a .-_. 11

s ._

r-o--..

CD 2 /1l D o Fiq.9

Transverse and in-line forces

acting on a circular cylinder in

24

-3.3 The Case for ,Kc=40.

In the calculation for this casé öne half cycle was divided

into 80 time-steps which is twide for previóus two cases...

Fig. 10 shows the calculated. vortex pattern. At lT , six vortex-lumps can be seen in. the flow field, four shedding

vor-tex-luinps which are arranged like Karman vortex and two

vortex-lumps just behind the cyliñder. Since these vortex-vortex-lumps

gener-ated in the first half cyc].e pass the cylinder with the reversed

incident flow during the second half çycle, the flow has a more complicated pattern as shown in the figure at wt=3 lT /2.

Fig. 11 shOws the pressuré distrïbution on the cylinder. At

t=3 /2 and. 5 lT /2, the asymmetry of the pressure

distri-bution betweèn on the uppe± and the lower sides is significant,

and at t=2Tr , the asymmetry is slight. As the Same .s the case for Kc=18 negative pressure due to thé. vortices generated

in the former half. cycle acts on thé front. side of the cylinder (- 1T/2(9<Q at t3 ir /2 and -ir< 0 < 71/2:

at wt=5

1T /2),

and it causes the décrease of the in-line drag force as pointed

in the previous. chapter.. In order to solve this problem,

de-tailed experimental and theoretical studies on thé decay of

shedding-vortices should be. carried out.

Fig. 12 shows the trànsverse and the in-line forces acting on

thé cylin.ér. Since the time step nuinbér increases, the

simu-lation could be carried on only up to 1.3 cycles, for which

it takes Cyber 170 computer at the 'cOmputer centre of the

k4O

,AJ.°

25

Fia.1O(1) Vortex pattern around a circular cylinder in oscillating

k=4o

26 £&)t=1T XXX .

.0 .

.

o o

Io

X X +Lrd vovt'e 54f

Çirst ecLLs yßyfQY

'L, X C

o f i

S e. X iv Y * e . o s o * wt' TTr/2 -V IO

g1r

I

Flg.1O('2)

_{Vortex pattern around a circular cylinder in}

oscillating flow at Kc=40.

X X _e g I g o X g X X

r

s o

0

_r X _o X I X s o . X

.

S I S e. I 0 s

I

I _s C o s it C 'X X

:

g

:

o.

_o.

X s o . e s V . o o s X . 0 e g . s

IX

0o

1 1'

kc:4O

Cf o

_-4

_-2 -ir

wt 31T/2

û o .

\

.

\

W*

2T

e

\

O o e.. O o .

/

_/

/

_/

/

_/

o . e . . e

---_____

---e e e-. . e e C s.'

.

,

/

_/

/

o

0/,

/

e

/

/ /

Fig.11 (1) Pressure distribution

on the surface of. a circular

cylinder in osci]].ating flow at Kc=40. / s s s o

/

e.

e

k

o _-2

_-L

o

wt

511/7 s o

/

.0/

/

Fig.11 (2)

Pressure distribution on the surface

of a circular cylinder in oscillating flow at Kc=40. ir . I o

0,

e,

/

/

/

I s

/

s

/

e

/

e

/

o . G o o e

- ---.

I I e

/

_/

/

o

\

e e e s I

cp

2 o 2

-i

Tir t&iI Fig.12

- 30

Vortec Shedding Frequency

Measurements. of the transverse force acting on the circular

cylinder in oscillating flow showed it to be intermittent /1/, and Ikeda et al. /4/ cofimed experimentally that the frequèn

of, the transverse förce coincides with the vortex-shedding frequency.

Fig. 13 shows the comparison. of the. vortex shedding frequency

between the present calculation and the experimental results by Ikeda at al. /4/. The agreement between them is good except

at Kc=12. At Kc=12 two, vortex-lumps are created behind the

cylinder during the first half cycle, and three vortex-lumps are created during second and third hàlf cycles. This may sug-gest that at Kc=12 the vortex-shedding frequency is sensitive

to the disturbance in the flow.

Hydrodynafliic Lift Fbrce

In the present calculation thé sïmulation could have been car-ried on about up to wt=5 for relatively low Kc number (time-step numbers = 40 in. one hàlf cyclé), and only up to wt=5 iT /2 for high Kc number (time-step.ntunbers = 80) because of the time limit of the computer. Since, the: flow, has not. reàched the steady condition, the quantitative discussions on the. hydrodynainiö

force seem to be difficult.

Fig. 14 shows a plot of: _{versus Kc núinbér, where Cax is} the maximum. value of the transverse force devided by

f

UD/2... The experimental. values.were obtaiñed by Ikeda et al.. /4/ by averging the maximum. values in any cycle during 50 cycleS. The value obtained by the. present calculation in this figure shows

the maximum value during the simulatiOn. Although CLmax by the present calculation is much lower thàn the experimental one,

4

2

31

-fv: frequency of vortex generation f : frequency of oscillating flow

Q: present cal.

experinntal results by S.Ikeda et aJ.../4/

IO 20 30 40 60

k

Fia.13 Frequency of shedding-vortex from a cicu1ar

o

o

o

-. 32

-O

present cl. experimental results by S. Ikeda et al./4/

Fig.14 Lift force acting on a circular cylinder in oscillating flow.

Io 20. 30

Ô

3

2

- 33 -,

6. Conclusions

The computer program to simulate the vortex-shedding flow

around.a circular cylinder in oscillating flow has been

devel-oped using a discrete vortex method, and the following cOn-clusions have been obtained.

) The obtained vortex pattern is similar to the observed

one. The obtained vortex-shedding frequency is in good

agreement with the measured one..

The calculated lift coéfficient has a similar tendency

to the experimental, one with Kc number although the calculated. value is lower than the experimental _one.

More detailed studie.s on the decay of shedding vortices are necessary in order to get acc.urate hydrodynamic forces.

7. . Acknowledgements

This work has been done during the stay of the author _{at the} Technical University of Berlin as a Research Fellow _{of the}

Alexander. von Humb.ldt Foundatin.

The author should like to express his. sincere _{thanks to}

Professor Hörst Nowacki. of the Technical University of Berlin for his kind support and valuable Suggestion's.

34

-8. References

[1.] Turgut Sarpkaya, Michael Isaacson: Mechanics öf Wave

Forces on Off shOre Structures, Van Nostrand Reinhold Cömpany, 1981

[2..] Yoshïho Ikeda, Yoji Hiineno:. Calculation of.

Vortex-Shedding Flow around Oscillating Circular and Lewis-Form Cylinder, Pròc. of the Third International Confer-ence on Numerical Ship Hydrodynamics, 1981

Yöshiho Ikeda: Calculation of Viscous-Interference Effect between Two Circular Cylinders in Tandem

Arrangement in Harmonically Oscillating Flow, Report of ISM, Technical University of Berlin, No 84/11,

Jan. 1984

Syunsuke Ikeda, Yoshimichi Yamamoto: Lift Force on Cylinders in. Sinusoidally Oscillating Flows, Nagare Vol. 2, No. 1. 1983 (in Japanese)

H. Schlichting:. Boundary Layer Theôry, 6th edition,