@) P E R G A M O N Appl ied M a t h e m a t i c s Let ters 13 (2000) 59-64

Applied Mathematics Letters

w w w . e l s e v i e r , n l / l oca t.e / a m l

Fluid Forces on a T r a n s v e r s e l y Osc i l l a t ing C y l i n d e r

S. I~[OCABIYIK Department of Mathematics and Statistics

Memorial University of Newlbundland St. John's, Newfoundland, Canada A1C 5S7

(Received February 1999; rvvised aTzd accepted ,June 1999)

A b s t r a c t - - T h e numer ica l solut ion of t he uns t eady two-dimens ional Navier-Stokes equa t ions is used to inves t iga te t he fluid forces exper ienced by a t r ans l a l i ng and t ransverse ly osci l lat ing cylinder. Ca lcu la t ions are first per formed in an oscillatory frequency range outs ide the synchron iza t iou when osc i l la tory- to- t rans la t ionM velocity rat io is 1.5 and at a fixed Reynolds n u m b e r R = 10 a. The objecl of th i s s t u d y is to exami ne t he effect of increase of forced oscillation f requency on the fluid iorces. T h e resu l t s of th i s s t u d y are in good agreement wi th previous exper imenta l predict ions. @ 2000 Elsevier Science Ltd. All r ights reserved.

K e y w o r d s - - U n s t e a d y , Incompressible , Viscous, qh'ansverse recti l inear oscillation, Cyl inder .

This paper deals with the unsteady flow generated ill a viscous, incompressible fluid of an infinite

extent by an infinitely long circular cylinder of radius a. At time t = 0, tile cylinder suddenly

starts to move with uniform translational speed U at right angles to its axis and at. the same

instant starts to oscillate with the velocity U,~ cos cot in the transverse direction to that of trans-

lation as shown in Figure 1. Here Um is the maximum oscillatory velocity and v., - 2rr.f is the

angular fl'equency of oscillation, where f is the fl'equency of oscillation. The cylinder axis coin- cides with the z-axis, and translational motion of the cylinder is in the negative x-direction while

oscillatory motion is about the y-axis. This class of flow provides a simplified two-dimensional

idealization of the more complicated problem of wave loading on offshore structm'es. The phe-

nomenon is controlled by three dimensionless numbers, ttie Rexmolds number R = 2aU/l/, where

u is the coefficient of the kinematic viscosity of the fluid, forcing Strouhal nmnber ~ = ,,co/U, and velocity ratio a = U,~/U.

The object of this numerical study is to investigate the effect of the oscillation frequetlcy on the

forces acting on tile cylinder at a fixed, high enough, lReynolds number when the velocity ratio

is c~ = 1.5. The influence of flow paranleters on the forces experienced by the cylinder has been extensively studied (see, for example, [1 4]). Earlier numerical studies have generally confined to velocity ratios c~ < 1 with emphasis on excitation at or near the vortex-shedding Dequency

(i.e., synchronization phenomenon, f i f o ~ 1, where f0 is the vortex-shedding frequency for the flow past the cylinder without oscillation). Here f0 is normalized with the constant speed of the cylinder translation U and the cylinder radius a; it remains practically constant (namely, at the

wdue of 0.1) when R = 10 3. It may be noted that f/Ji? - g~/(0.2rr). Previous numerical studies

0893-9659 /00 /$ - see front m a t t e r @ 2000 Elsevier Science Ltd. All r ights reserved. Typese t by A.~dS-'FI.)\ PlI : S0893-9659(99)00209-8

60 S. KOCABIYIK

u

r = e

Figure 1. The configuration of the cross-section.

investigated effect of the flow parameters on the near-wake structure as well as the hydrodynamic

forces on the cylinder either

(i) when the cylinder velocity has an oscillatory perturbat ion superimposed on its uniform velocity component (i.e., c~ < 1; see for example [4]), or

(ii) when oscillatory-to-translational velocity ratio is 1 (i.e., c~ = 1; see for example [1,3,4]).

To my knowledge, from a survey of existing literature, no purely numerical s tudy has been made to investigate the fluid forces on the cylinder when a > 1.

In the present study, numerical calculations first performed at a fixed Reynolds number outside the synchronization range when a > 1 to examine the effects of the Strouhal number on the forces experienced by the cylinder. In this sense, the result of this research differs from the

previous similar numerical studies, and it adds to the knowledge of hydrodynamic forces acting on the cylinder. The velocity ratio and the Reynolds number are maintained at c~ = 1.5 and

R = 103, respectively, and the forcing Strouhal number takes low and moderate values: ft =

re/4, rr/2, and rr, or f / f o = 1.25, 2.5, and 5. The unsteady two-dimensional Navier-Stokes are solved in their vort ici ty-stream function formulation using an existing spectral finite-difference method. Detailed features of numerical method and systematic validations have been outlined in [3,4]; therefore, Section 2 only briefly summarizes the equations and the numerics. Several experimental ly observed phenomena on drag and lift coefficients at small and moderate oscillation

amplitudes, A = Ur~/Oa, are numerically predicted. It may be noted tha t A/a = c~/fL

M E T H O D O F S O L U T I O N S U M M A R Y

The same basic formulation of the problem described in [3] is adopted. Modified polar coor- dinates (~, 0) are used, where ~ = log(r/a), with the origin at the centre of the cylinder. The governing equations are given in the form

= ,,,ae + oe ] + g oe oe b7 ( la)

and

o~--~ + ~ =

F l u i d F o r c e s ~il

where '~) is the s t ream function and ( is the (negative) scalar vorticity. These quanti t ies are

all dimensionless and are defined in [1]• The boundary conditions are based on the no-slip and

impermeabi l i ty conditions on the cylinder surface and fi 'ee-stream conditions away from it.,

0'¢ '~/~ -- -- 0, when ~ O. (2a) a(

0(., i) ~,., c - " -+ sin 0 - a'cos(~Qr) cos 0, e - - ' ~ - ~ cos 0 + (~ (:os(~r) sin 0, as ( ~ e,.:. (21))

O~ 00

One of the problems using the vortici ty equation is tha t its boundary conditions are unknown

which could introduce errors if not t reated correctly. ~Ib ow;rcome this difficulty, one of Green 's

identities is applied to the domain of the field of flow to t ransform the boun(lary condit ions on

the s t ream flmction into a set of global conditions, termed integral conditions on the vorticity,

natnely,

for tdl integers 't~ >_ 0, where 51,1 = 1 and d,,.,t = 0 if 7, ;~ 1. These are employed in the solution

procedure to ensure tha t all necessary conditions of the problem are satisfied. Expansions fl)r (.,

and ( are assumed in the respective forms

1 >o '~/,({, 0, T) = ~F0(~, r) + E ( F , , ( ~ , r ) c o s nO + f,~(~, T) sin n0). (4a)

n 1

1 ~o ((~, O, T) = ~aO(~, ~-) + y ] ( a , ( ~ . 7-) c, ,s ,~0 + ,~,,(~, r) sin ,,.0). (41,)

rz 1

The equat ions and bounda ry conditions satisfied by the various fimctions appear ing in series (4)

and the solution procedure using boundary- layer coordinates have been described bv Nguyen and

Kocabiyik in [3]. The grid size A~. in the coordinate a.:, defined by { 2(2r/'/~)~/ezr. is more

or less independent of R. The value A z = 0.05 was taken tbr the Reynokts number I? 10 a

considered here. The max imum value of the computa t ional field length was :~:,~l = 8 and the

max imum number of terms used in series (4) corresponds to replacing the infinite upper limit in each sum by the finite integer N = 20. The discussion of the choice of these various parameters is

in principle the same as tha t given by Nguyen and Kocabiyik in [3], and the t ime steps A r used

for the evolution of the solution follow exactly those set out in tha t discussion, since these are

found to be sat isfactory and are checked carefully. The under-relaxat ion parameter fin' the surface

vort ic i ty is chosen by trial-and-error, and the values 0.5, 0.4, and 0.25 are used in obta in ing the

results for 9~ = re/4, re/2, and re, respectivel> Convergence is reached when the difference between two successive i terations of the surface vortici ty fails below the specified tolerance ~ - [!} r,.

R E S U L T S A N D D I S C U S S I O N

Dimensionless drag and lift coefficients are defined by CL) = D/f)U2a and CL L/I)U'a, where O and L are the total lift and drag on the cylinder. They are computed using tile formulae

CO = CFD -[- CpD and Cc = CpL + CpL +,'TAc, (5)

where

2 f 2~ CFO = ~ (()~=o sin 0 dO,

2 j l 2rr C..,~ = - ~ (()~=o cos 0 c/O,

62 S. KOCABIYIK

8.0-

6 . 0

C= 4.0-

(;~,~, Cro

2.0

0.0-

. 2 . 0

~.0

4.0

4 8 lJ2 16 2JO 2'4 28 ,32

C, 2.0

Cp~ C~

O. 0

- 2 . 0

4 . (

-6 .0 0 10 20 25 50

4.0- T

(b)

55 ~0

5.0

- . 1 . 0 | . ~ . , t , , , ~ ' ~ ' , , , 0 5 10 15 20 25 30 .$5 z-O

Figure 2. Var ia t ion of d rag coefficients C D : - - ; C p D : • "" ; C F D : . . . . with T at R = 10 3 and ~ = 1.5: (a) ~ = 7r/4, (b) ~ = Tr/2, (c) ~ = 7r.

H e r e CFD a n d CFL a r e t h e c o e f f i c i e n t s d u e t o t h e f r i c t i o n , a n d CpD a n d CpL a r e t h e c o e f f i c i e n t s

d u e t o t h e p r e s s u r e . T h e t h i r d t e r m o f CL i s t h e i n v i s c i d l i f t c o e f f i c i e n t d u e t o c y l i n d e r a c c e l e r a t i o n

A c = - c ~ s i n ~2~-. T h e c a l c u l a t e d v a l u e s o f C o a n d CL a r e b a s e d o n e x p r e s s i o n s (5) a n d a r e

p l o t t e d i n F i g u r e s 2 a n d 3 fo r t h e c a s e s o f R = 10 3 a n d a = 1 .5 w h e n ~ = ~ / 4 , 7r /2 , a n d 7r. I n

Fluid Forces (i3

2 0 ,

" 5 .

• 0 -

C, b

C<, ,2~ 0

5

"0

"19

2O

/ t / ', ..: ., / 't ~ <', .-.7 i

i I \ ,.. / ' ,,..,, ',

/

/

( at )

I ~ 2 '0 ."-:? 4 8 ! .~ " .'S >' a 2' 8

,10 /'1

~ | £' /'/ /~" A & .' / i .., /'.

i / !: /

i

. s o ~ ( b )

0 ~" I 0 15 20 PD .!,c~ ~!: 4 ' j

! T

i ;! ~l '1 ,I il I '! ' , u ~ ,:,

' ; I I i i ! ! " - i ' i i i i . ' ' i" ;' i ' [I , I / : Jt ' ~ i I '

c, f ' 1 . , i 0 : 1 .

- " 3 1 I ~i I ! ! ! I : , ~ I ' '

~ o l ! i j il i ! i i; ~ : l l ;! i - ~ J! it i 1 t i ~1 ', i I'~ i

I ! l i ! i J ! . 5 0 r ,,t LI I I i t ! ! t !

~ I I I !. il ' ~' : ! h !i Ili ~J lh i , il !

i ~ i ,' ii

ii I i i~ ii

i I !i :, ; I! I I I

i l i ! '.I i~ ! , i i~ j i I~ ;1 i

.!.~- !4 L t.~ k;--

~i ,! i ;

' : I i 'i It 'i ; ! i t I! : :i (c,

i

25 }, } .~ b 4 }

Figure 3. Var ia t ion of lift coefficients C L : - - ; C p L : ' " : ( ; F L : - - " • wilh r al R = 10 a and a = 1.5: (a) fl = rr/4, (b) f~ = vr/2, (c) ~'t = ~r.

t h e p r e s e n t w o r k , t h e l o w a n d m o d e r a t e f o r c i n g S t r o u h a l n m n b e r s ( o u t s i d e l o c k - i n ) a r e c h o s e n

fo r t h e s t u d y s i n c e t h e f low s t r u c t u r e i n s u c h c a s e s is c h a r a c t e r i z e d b y f o r m a t i o n o f v o r t e x p a i r s

w h i c h c o n v e c t a w a y f r o m t h e b o d y , f o r m i n g w a k e s . T h e s e f i g u r e s s h o w t h a t t h e c o n t r i b u t i o n

o f f l ' i c t i o n a l f o r c e s t o t h e t o t a l i n - l i n e a n d t r a n s v e r s e f o r c e s is r e l a t i v e l y s m a l l , a n d t h e m a i n

c o n t r i b u t i o n c o m e s f r o m t h e p r e s s u r e f o r c e s s i n c e in t h i s t y p e o f f low a t m o d e r a t e a n d h i g h

64 S. KOCABIYIK

Reynolds numbers, the viscous flow effect is limited to the thin boundary layer and subsequent narrow near-wake region. The figures also indicate the periodic variation of the flow field in the near-wake region associated with vortex shedding. It appears that the lift and drag are affected

by the forced oscillations of the wake with frequencies f~ and 2fL respectively, as expected. Figure 2 shows tha t drag forces change greatly with the increase of oscillation frequency.

CD values tend to be smaller and distributed as 0.25 < CD < 0.75 when f~ increases from 7r/4 to 7r/2; however, CD values tend to be larger and distributed as - 4 < CD < 2.5 when f~ increases from :r/2 to :r. This tendency agrees with the experimental results of Kato et al. In

their work [5], the variation of CD as a function of Keulegan-Carpenter number K C = :rA/a for

different frequencies was investigated under practical offshore conditions in two typical directions:

the in-line oscillation and the transverse oscillation to a uniform flow. In the transverse oscillation case, drag reduction was reported with the increase of oscillation frequency when K C takes values above about 3; however, especially below K C -~ 3 (or above f~ ~ :r/2), CD values again tend to

be larger. In the case of c~ = 1.5 and D = :r, maximum oscillatory velocity dominates and flow oscillations exhibit high frequency and relatively low amplitude; this produces a higher pressure

field than the other two cases. Present results on CD are considerably different h'om the Cz) variations of the case of R = 103, D < rr when c~ = 0.5 (see for example [5], almost a s teady

drag coefficient CD ~ 0.5 with a small oscillation was observed in this case). For all frequencies, CL is basically periodic at the cylinder frequency and reaches approximately the same maximum

and minimum values in every cycle. The maximum values are at tained near zero position of the

cylinder during the upward and downward motion of the cylinder.

Figures 3a and 3b show a phase difference between CpL and CL at the two values of the forcing

Strouhal frequency :r/4 and 1r/2, and the phase difference decreases rapidly with the increase of fl. This is probably due to the fact that the small amplitude and high frequency case simply increases inertia effect and decreases the size of the separated flow region. It is also observed tha t as the forcing oscillation frequency increases from f~ = re/4 to f~ = :r/2, CL values increase by about 50%. Amplification in oscillating lift coefficient at moderate oscillation frequencies was

also reported in the numerical s tudy of Chilukuri [6].

R E F E R E N C E S

1. S. Kocabiyik and P. Nguyen, A finite difference calculation for a transverse superimposed oscillation, Can. Appl. Math. Quart. 4, 381-420 (1996).

2. B.M. Sfimer and J. Fredsce, Hydrodynamics Around Cylindrical Structures, World Scientific, (1997). 3. P. Nguyen and S. Kocabiyik, On a translating and transversely oscillating cylinder: Part 1--The effect of

the Strouhal number on hydrodynamic forces and near-wake structure, Ocean Engng. 24, 677-693 (1997). 4. S. Kocabiyik and P. Nguyen, On a translating and transversely oscillating cylinder: Part 2 The effect of

the velocity ratio on hydrodynamic forces and near-wake structure, Ocean Engng. 26, 21-45 (1999). 5. M. Kato, T. Abe, M. Tamiya and T. Kumakiri, Drag forces on oscillating cylinders in a uniform flow, Trans.

A.S.M.E. 107, 12-17 (1985). 6. R. Chilukuri, Incompressible laminar flow past a transversely vibrating cylinder, A.S.M.E. Journal of Fluids

Engineering 109, 166-171 (1987).