E L S E V I E R 0 1 4 1 - 1 1 8 7 ( 9 4 ) 0 0 0 2 5 - 5
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Transient fluid motion due to the forced horizontal
oscillations of a vertical cylinder
p. Mclver
Department of Mathematical Sciences, Loiigltborough University, Loiigliboroiigh, Leics, UK, LEll 3TU (Received 20 June 1994)
The problem o f the forced horizontal oscillations of a vertical cylinder extending throughout the fluid depth is considered on the basis o f the linearised theory of water waves. A new integral f o r m is given for the frequency-domain solution and the procedure is then used to obtain an explicit time-domain solution.
1 I N T R O D U C T I O N
W i t h i n the c o n t e x t o f the linearised t h e o r y o f water waves, there is a small n u m b e r o f geometries f o r w h i c h explicit a n a l y t i c a l solutions are possible. One such geometry is a v e r t i c a l cyhnder extending t h r o u g h o u t the d e p t h o f the fluid a n d this paper is concerned w i t h the wave field r e s u l t i n g f r o m the f o r c e d h o r i z o n t a l m o t i o n o f this type o f c y l i n d e r . T h e s o l u t i o n i n the f r e q u e n c y d o m a i n is usually expressed i n terms o f an i n f i n i t e series ( D e a n & D a l r y m p l e , ' Section 6.4) a n d solutions i n the t i m e d o m a i n m a y then be o b t a i n e d w i t h the a i d o f the F o u r i e r t r a n s f o r m . H o w e v e r , this series f o r m o f the f r e q u e n c y - d o m a i n s o l u t i o n contains the f r e q u e n c y o n l y i m p l i c i t l y t h r o u g h the solutions o f the dispersion r e l a t i o n a n d so the F o u r i e r t r a n s f o r m m u s t be carried o u t n u m e r i c a l l y . T h e m a i n result o f the present w o r k is an i n t e g r a l f o r m f o r the t i m e - d o m a i n s o l u t i o n c o r r e s p o n d i n g t o an i m p u l s i v e m o t i o n o f the cyhnder. T h i s m a y be used t o construct t i m e - d o m a i n s o l u t i o n s f o r m o r e general m o t i o n s o f the cyhnder w i t h o u t the need f o r n u m e r i c a l t r a n s f o r m s between the f r e q u e n c y a n d t i m e d o m a i n s .
T o illustrate the s o l u t i o n procedure, i n w h a t is perhaps a m o r e f a m i h a r context, a new i n t e g r a l f o r m f o r the f r e q u e n c y - d o m a i n s o l u t i o n is first o b t a i n e d . I n the c o n v e n t i o n a l f r e q u e n c y - d o m a i n s o l u t i o n procedure, e i g e n f u n c t i o n s are constructed t h a t satisfy the h o m o -geneous free-surface c o n d i t i o n a n d then these are c o m b i n e d t o satisfy the n o n - h o m o g e n e o u s b o d y b o u n d a r y c o n d i t i o n . H e r e , the i n f i n i t e - f r e q u e n c y l i m i t o f the s o l u t i o n is first subtracted o u t w h i c h leads t o a n e w b o u n d a r y - v a l u e p r o b l e m w i t h a n o n - h o m o g e n e o u s free-surface c o n d i t i o n a n d a homogeneous b o d y
© C r o w n copyright (1995).
c o n d i t i o n , the reverse s i t u a t i o n t o the c o n v e n t i o n a l a p p r o a c h . E i g e n f u n c t i o n s t h a t satisfy the b o d y c o n -d i t i o n are c o m b i n e -d t o satisfy the free-surface c o n -d i t i o n . T h i s a p p r o a c h t o the f r e q u e n c y - d o m a i n analysis was suggested b y the standard a p p r o a c h to t i m e - d o m a i n p r o b l e m s w h i c h employs w h a t is essentially the same c o n s t r u c t i o n . T h e result o f this f r e q u e n c y - d o m a i n analysis is a f o r m o f the s o l u t i o n i n w h i c h the f r e q u e n c y appears e x p l i c i t l y a l l o w i n g solutions i n the t i m e d o m a i n t o be o b t a i n e d a n a l y t i c a l l y b y the F o u r i e r t r a n s f o r m . F o r completeness, the same procedure is t h e n used i n the t i m e d o m a i n t o o b t a i n solutions d i r e c t l y w i t h o u t reference t o the F o u r i e r t r a n s f o r m o f the f r e q u e n c y d o m a i n s o l u t i o n . F i n a l l y , f o r the purposes o f i l l u s t r a -t i o n , a s o l u -t i o n is g i v e n f o r -the p a r -t i c u l a r case o f a cylinder m o v i n g under a sinusoidal f o r c e a f t e r the fluid is i n i t i a l l y at rest.
T h e v e r t i c a l c i r c u l a r c y l i n d e r is o f radius a a n d extends t h r o u g h o u t w a t e r o f constant d e p t h /; . C y l i n d r i -cal p o l a r coordinates (/•, 9, z) are chosen w i t h the o r i g i n o f z i n the m e a n free surface, and z directed v e r t i c a l l y u p w a r d s , a n d the o r i g i n o f (r, 9) being the c y l i n d e r axis. T h e c y l i n d e r is f o r c e d t o p e r f o r m h o r i z o n t a l o s c ü l a -t i o n s , w i -t h a prescribed -time-dependen-t v e l o c i -t y , i n the d i r e c t i o n 0 = 0. U n d e r the usual assumptions o f the linearised t h e o r y o f w a t e r waves the m o t i o n m a y be described b y a v e l o c i t y p o t e n t i a l s a t i s f y i n g Laplace's e q u a t i o n t h r o u g h o u t the fluid d o m a i n .
2 F R E Q U E N C Y D O M A I N
F o r the case o f steady t i m e - h a r m o n i c oscillations o f angular f r e q u e n c y w a n d v e l o c i t y a m p l i t u d e U persisting f o r a l l t i m e , the v e l o c i t y p o t e n t i a l m a y be w r i t t e n
T h e time-dependent p o t e n t i a l (f) m u s t satisfy the free-surface b o u n d a r y c o n d i t i o n
dz K(}) o n
z =
0 (2)w h e r e the f r e q u e n c y p a r a m e t e r K = u? jg a n d g is the acceleration due t o g r a v i t y , the bed c o n d i t i o n
dz 0 o n the b o d y c o n d i t i o n
d4>
dr cos Q o n r = a a n d the r a d i a t i o n c o n d i t i o n ^ 0 asoo
(3) (4) (5)H e r e k is the r o o t o f the dispersion r e l a t i o n
K = ktanhkh (6) I n w h a t f o l l o w s , the l i m i t i n g p o t e n t i a l as the
f r e q u e n c y tends t o i n f i n i t y is r e q u i r e d . T h i s ' i n f i n i t e -f r e q u e n c y ' p o t e n t i a l w i U be denoted b y Q.{r, 6, z) a n d w i l l satisfy the above c o n d i t i o n s i n the l i m i t i<r ^ oo so t h a t the free-surface c o n d i t i o n (2) is replaced b y
= 0 o n
z =
0 a n d the r a d i a t i o n c o n d i t i o n (5) byoo
(7) 0 as ;• (8) S e p a r a t i o n o f variables a n d s a t i s f a c t i o n o f aU b o u n d a r y c o n d i t i o n s except t h a t o n the b o d y givesf2 = cos ö ^ A„Ki {p„r) c o s p „ { z + /;) n = \ where p„ = [In - l)7r/2/7 (9) (10) a n d Ki denotes the m o d i f i e d Bessel f u n c t i o n o f the second k i n d a n d order one. T h e b o d y b o u n d a r y c o n d i t i o n n o w requires
^ A„p„K'\ {.PnO) cosp„{z + h) = \ (11)
a n d the expansion coefficients f o l l o w f r o m the o r t h o -g o n a l i t y o f the set {cosp„{z + h); n= 1 , 2 , . . . } over the d e p t h so t h a t A., = 2 ( - l ) plhK\{Pnay 1 , 2 , . . . (12) T o o b t a i n the s o l u t i o n f o r a r b i t r a r y f r e q u e n c y consider the h a r m o n i c f u n c t i o n : i 3 ) s a t i s f y i n g the b o u n d a r y c o n d i t i o n s 9 0 az dz &tp
Ih
dxp dr a n d = 0 o nz =
—A = 0 o n /• = a as r —> DO (14) (15) (16) (17) T h e s o l u t i o n f o r Ü given b y (9) a n d (12) m a y be s u b s t i t u t e d i n t o the free-surface b o u n d a r y c o n d i t i o n (14) t o o b t a i n dtj^ dz ' where Kip = R{r) cos ö o nz =
0 (18) R{r) =-2Y:
Ki{p„r) '^.p„hK\{p„a) (19) 11=1 I n c o n t r a s t t o 0 , w h i c h m u s t satisfy a homogeneous free-surface c o n d i t i o n a n d a n o n - h o m o g e n e o u s b o d y b o u n d a r y c o n d i t i o n , the new p o t e n t i a l ip is r e q u i r e d to satisfy a n o n - h o m o g e n e o u s free-surface c o n d i t i o n a n d a h o m o g e n e o u s b o d y b o u n d a r y c o n d i t i o n .T h e s o l u t i o n proceeds by separation o f variables, first n o t i n g t h a t the b o d y b o u n d a r y c o n d i t i o n is satisfied i d e n t i c a l l y b y
C„{qr) = Uqr)Y!Ma) - Y„{qr)J!,{qa) (20) where / „ a n d ¥„ denote Bessel f u n c t i o n s a n d a p r i m e denotes d i f f e r e n t i a t i o n w i t h respect t o the f u n c t i o n a r g u m e n t . T h u s
d/ C„ {qr) = 0 o n r (21)
A s o l u t i o n f o r ip s a t i s f y i n g a l l except the free-surface a n d r a d i a t i o n c o n d i t i o n s is
Ip = cosd A{q)Ci (qr) cosh q{z + h)dq (22) T h e u n k n o w n f u n c t i o n A{q) is f o u n d b y a p p l y i n g a m o d i f i e d v e r s i o n o f Weber's i n t e g r a l t h e o r e m ( H u n t & B a d d o u r , ^ eqn (54)) w h i c h states t h a t C„{qr) ƒ ( ' • ) \H[iqa)\ •gdq C„{qr'yfir')dr' (23)
where Hi denotes a H a n k e l f u n c t i o n o f the first k i n d , p r o v i d e d r^^'^f{r) is i n t e g r a b l e over [a, oo). S u b s t i t u t i o n o f (22) i n t o the free-surface c o n d i t i o n (18) a n d a p p l i c a t i o n o f (23) gives
A{q){K cosh qh — qsinh qh)
\H[{qa)
where e has been i n t r o d u c e d t o a l l o w the interchange o f the i n t e g r a t i o n and s u m m a t i o n w h e n (19) is substi-t u substi-t e d i n substi-t o (24). N o w f r o m A b r a m o w i substi-t z & Ssubsti-tegun,^ eqn (11.3.29),
lp
l i m C,{qr)Ki{pr)rAr-Trq{p^ + q^) K[{pa) (25) so t h a t A{q){Kcosh qh — qsmhqh) 4 ^ 1 Itanhqh 'nh\H[{qatt'xPl+l' (26) •\Vi"l\ n = \ f n i y 'i^q\H[{qa)\where the series has been summed using eqn (1.421(2)) o f G r a d s h t e y n & Ryzhik.'* The s o l u t i o n f o r ij) is therefore
2 c o s é '
t a n h qh C j [qr) cosh cj{z + h) \H[{qa)\^q[K cosh qh - qsmhqh
where the p a t h o f i n t e g r a t i o n has been chosen t o r u n beneath the p o l e at the real r o o t of K = q tanh qh i n o r d e r t o satisfy the r a d i a t i o n c o n d i t i o n . T h i s can be c o n f i r m e d b y a s t a n d a r d residue calculus technique as described, f o r example, by M e i , ^ p . 379. I n d e e d this s o l u t i o n m a y be w r i t t e n i n series f o r m as
lp = COS0< ^ sin k„ h Ki {k„r) cos k„ (z + li) K{ik„a)klhNf,
+
+
2{-l)"Ki{p„r) cosp„iz + h) K[{p,.a)plh s i n h kh Hi (kr) cosh /c(z + h) (28) K N = i H[{ka)k^hNl j where {/c„; n= 1 , 2 , . . . } are the real, p o s i t i v e r o o t s o f- / c „ t a n / c „ / ; (29)
' - ^ ) ™
a n d /CQ = i / c . W h e n c o m b i n e d w i t h the i n f i n i t e-f r e q u e n c y s o l u t i o n Ü this gives the ' w e l l - k n o w n ' -f o r m o f the f r e q u e n c y - d o m a i n s o l u t i o n f o r r a d i a t i o n b y a c y l i n d e r as presented i n Section 6.4 o f D e a n & D a l r y m p l e . '
F o r the p u r p o s e o f o b t a i n i n g s o l u t i o n s i n the t i m e d o m a i n b y F o u r i e r t r a n s f o r m , the advantage o f (27) is t h a t the f r e q u e n c y appears e x p h c i t l y t h r o u g h K = w'^/g. I n the f o r m (28), the f r e q u e n c y appears o n l y i m p l i c i t l y t h r o u g h the solutions o f (29).
3 I M P U L S I V E M O T I O N
A t t e n t i o n is n o w t u r n e d t o the transient m o t i o n
generated by a c y l i n d e r started f r o m rest; the f o r m u l a -t i o n f o l l o w s -the d e s c r i p -t i o n o f -the s -t a n d a r d -t i m e - d o m a i n t h e o r y as given, f o r example, i n Section 7.11 o f M e i . ^ T h e fluid response t o a n i m p u l s i v e v e l o c i t y F ( T ) 5 ( / - r ) i m p o s e d at t i m e t = T \s described by the p o t e n t i a l
^r,e,z,t)^V{T)[n{r,e,z)8{t-T)
T{r,9,z,t^T)H{t-T) (31)
T h e first t e r m o n the r i g h t - h a n d side o f (31) describes the i n i t i a l pressure i m p u l s e t r a n s m i t t e d t h r o u g h o u t the fiuid a n d the second t e r m describes the subsequent fiuid m o t i o n . H e r e S{t) is the delta f u n c t i o n , H{t) is the Heaviside step f u n c t i o n a n d T{r,6,z,t) is a h a r m o n i c f u n c t i o n s a t i s f y i n g the i n i t i a l a n d b o u n d a r y c o n d i t i o n s dq (27) ^
r =
0 o n z =9r_ _ Ö0
dt~^ dz
d^Tdr_
W ^ ^ T z '
dr
dz
a n d 0 at ; = 0 o n z = 0 at / = 0 0 o n z = 0 f o r 0 < ? < oo = 0 o n z g;: = O o n r (32) (33) (34) (35) (36) T h e s o l u t i o n proceeds i n a s i m i l a r w a y t o the f r e q u e n c y -d o m a i n s o l u t i o n given i n the previous section. T h e b o -d y a n d bed c o n d i t i o n s ( 3 5 ) - ( 3 6 ) are satisfied b y t a k i n gcos( A{q, t)Ci {qr) cosh q{z + h) dq (37)
where C j is d e f i n e d i n eqn (20). T h e free-surface c o n d i t i o n (34) is satisfied p r o v i d e d
d'A
dt--+ W'A = 0
w h i c h has a general s o l u t i o n A = a{q) cos Wt + f3{q) sin Wt where W' = gq tanh qh T h e i n i t i a l c o n d i t i o n s ( 3 2 ) - ( 3 3 ) r e q u i r e t h a t a n d a{q)Ci{qr) coshqhdq = 0 p{q) W{q)Cl {qr) coshqh dq = -gR{r) (38) (39) (40) (41) (42) where R{r) is d e f i n e d i n eqn (19). I t f o l l o w s f r o m the m o d i f i e d v e r s i o n o f W e b e r ' s i n t e g r a l t h e o r e m , eqn (23), a n d f o l l o w i n g s i m i l a r m a n i p u l a t i o n s as c a r r i e d o u t i n eqns ( 2 4 ) - ( 2 6 ) t h a t
a n d
2g-tanh (7/7
TrqWcos)iqh\H[{qa)\' T h i s n o w has given
(44)
r
Ig cos ( ' sin Wt t a n h qh Ci (qr) cosh q{z + h) qW cosh qh\H[{qa)fdq (45) T h i s is related t o the F o u r i e r t r a n s f o r m o f the f r e q u e n c y - d o m a i n s o l u t i o n t h r o u g h ' i / i e -''"'dw (46) w h e r e tp is given by (27) a n d m a y be w r i t t e n i n the f o r m 2gcos9 t a n h qh {qr) cosh q{z + h) , 0 ^ ( w 2 _ W') coshqh\H[{qa)Y (47) T o recover (45) f r o m (46), UJ is replaced b y w + ie i n (47), so t h a t the p o l e o n the real q axis is m o v e d i n t o the u p p e r h a l f o f the c o m p l e x q plane, a n d then the resuh
l i m is used. 27r _ d ^ = - _ s i n f F / (48) 4 R E S P O N S E T O A S I N U S O I D A L F O R C I N G A c c o r d i n g t o M e i , ^ p . 374, the response t o a c o n t i n u o u s v e l o c i t y V{t) imposed o n the c y l i n d e r is given b y
$ = VLV{t) + r r ( / - r ) F ( r ) d r (49) J—00
Consider, f o r example, the s i n u s o i d a l f o r c i n g
V{t)=H{t)Vsmujt (50)
0) O u O
Fig. 1. Non-dimensional force F{t)/{p-Ka^ojV), due to the sinusoidal forcing (50), versus non-dimensional time T = tot for a/h = 0-1, Ka = 0-5: ( • • • • ) time-domain solution; ( )
frequency-domain solution.
w h i c h is t u r n e d o n at Z = 0; here H{t) is the Heaviside step f u n c t i o n . F r o m (49) a n d (50) the p o t e n t i a l f o r the r e s u l t i n g disturbance is
f
$ = ÜV{t) + V V{t - r ) s i n w r d r (51) where f2 is given b y (9) a n d (12) a n d F is given b y (45). T h e result is
^ =V[{Q. + Rt1|})smüJt + ^ , t>0 (52) where K&ip is the p r i n c i p a l value equivalent o f (27) a n d
* = -
2ci;cos( °° sin Wt0 W
t a n h qh C j {qr) cosh q{z + h) \H[{qa)\'q{K coshqh - q sinh qh
•dq (53)
I t is expected t h a t p u r e l y t i m e - h a r m o n i c m o t i o n w i l l be achieved a f t e r a l o n g t i m e , t h a t is
$ - R e { i F , ^ e - ' " ' }
= V[{Q, + Reip) sin cot - Imi/j cos uit] as ; ^ 00 (54) where (p is d e f i n e d i n eqn (1) a n d Im.ip is the c o n t r i b u t i o n t o (27) f r o m the i n t e g r a t i o n a r o u n d the pole a n d is given by , sinh/c/2Ci(/c/-)cosh/c(z-f/!) \H[{ka)\'k'hNl (55) T h i s l i m i t i n g b e h a v i o u r (54) can be c o n f i r m e d b y w r i t i n g ^ , w h i c h appears i n (52), as w c o s ö ^ f' * = I m F AWt TT J-00 W t a n h qh {qr) cosh q{z + h) \H[{qa)fq{K cosh qh - qsinh qh)
dq (56)
a n d s h i f t i n g the p a t h o f i n t e g r a t i o n i n t o the u p p e r h a l f o f the c o m p l e x q plane b y a l l o w i n g q to have a smaU positive i m a g i n a r y p a r t . T h e l i m i t i n g b e h a v i o u r arises f r o m the poles o n the real axis w h i l e the r e m a i n i n g p a r t o f the i n t e g r a l decays t o zero as —> 00. T h e details are s t r a i g h t f o r w a r d b u t messy a n d so are o m i t t e d here.
T h e s o l u t i o n f o u n d above m a y be used, f o r example, t o calculate the free-surface elevation at a n a r b i t r a r y p o i n t o r the h y d r o d y n a m i c f o r c e o n the c y l i n d e r . T h e latter is used f o r purposes o f i l l u s t r a t i o n . T h e t i m e -dependent f o r c e i n the d i r e c t i o n ö = 0 is
— cos9dS (57) where S denotes the w e t t e d surface o f the c y l i n d e r .
C a r r y i n g o u t the i n t e g r a t i o n s gives F{t)
Ki{p„a) la
h fx{Pna)'K[{p„a) cos Lot •
X
(cos Wt - cos u)t) t a n h qh {qa) dq{58) 10 \H{{qa)\^q^{Kcothqh-q) T h i s expression is r e a d i l y evaluated b y s t a n d a r d
n u m e r i c a l techniques and a sample c a l c u l a t i o n is given i n F i g . 1 s h o w i n g the r a p i d a p p r o a c h t o the frequency-d o m a i n s o l u t i o n .
R E F E R E N C E S
1. Dean, R. G. & Dalrymple, R. A . , Water Wave Meclianics for Engineers and Scientists. W o r l d Scientific, 1991.
2. Hunt, J. N . & Baddour, R. E., Nonlinear standing waves bounded by cylinders. Quarterly Journal of Meclianics and Applied Mathematics, 33, 357-371.
3. Abramowitz, M . & Stegun, I . A . , Handboolc of Mathe-matical Functions. Nadonal Bureau of Standards, Washington, D C , 1964.
4. Gradshteyn, I . S. & Ryzhik, I . M . , Tables of Integrals. Series and Products. Academic Press, New Y o r k , 1980. 5. M e i , C. C , The Apphed Dynamics of Ocean Surface Waves.
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