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The surge motion of a circular cylinder containing a concentric cylindrical hole in finite depth
Ocean Engng, Vol. 13, No. 4, pp. 409--416, 1986. Printed in Great Britain.
002%8018/86 $3.00 + .00 Pergamon Journals Ltd.
TECHNICAL NOTE
THE SURGE MOTION OF A C I R C U L A R CYLINDER CONTAINING A CONCENTRIC CYLINDRICAL HOLE IN FINITE DEPTH M. ILKI~IKand K. KAFALI Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, Istanbul, Turkey Abstract--This paper deals with hydrodynamic forces of a single semisubmerged circular cylinder containing a concentric cylindrical hole constrained to move in a water domain of finite depth. The fluid domain is divided into inner and outer regions. The Laplace equations governing velocity potentials for the three regions are solved by separation of variables and expressed in terms of eigenfunctions of the resulting equations which satisfy appropriate boundary conditions, Continuity of pressure and velocity at the interface of the regions provides the necessary equations from which the velocity potentials, pressures and forces are obtained. Numerical results are plotted for added mass and damping coefficients for different draft-to-depth and radius-to-depth values and for various wave amplitudes. 1.
INTRODUCTION
ADDED mass and damping of bodies oscillating in a free surface has been a subject of considerable research interest. In recent years, hydrodynamic characteristics of vertical circular cylinders and elliptical cylinders oscillating in free surface have been studied by Sabuncu and Cali~al (1981). In computing hydrodynamical forces, numerical techniques must be resorted to. These include the traditional wave source distribution m e t h o d by Garrison (1975), finite element variational formulations by Chen and Mei (1974). Scattering and radiation problems for various obstacles including the vertical circular cylinder were also treated by the Schwinger variational technique by Black et al. (1971). The axisymmetry of the geometry, the problem can be treated simply by the use of eigen functions alone (Garrett, 1970), developed the expressions of the interior and exterior problems in terms of the potential at the c o m m o n boundary and match the normal derivatives together. In this work, the interior and exterior problems were treated as a Dirichlet and N e u m a n n type, respectively, as if the conditions on the c o m m o n boundary were known. But the radiation problem of a circular cylinder containing a concentric cylindrical hole, has not been solved. The objective of this p a p e r is to present a simple and efficient m e t h o d for computing the hydrodynamic characteristics of a circular cylinder containing a concentric cylindrical hole for a surge motion. 2.
FORMULATION
The theory is based upon the usual assumptions of classical hydrodynamics, i.e. that the fluid is inviscid and of uniform density, and that the motion starts from rest, and remains irrational. Nonlinear terms in the equations of motion are neglected. It is also assumed that the motion is three-dimensional and the body possesses both a vertical axis of symmetry and a horizontal plane of symmetry. The coordinate system of Oxyz and the geometry of the body are shown in Fig. 1. The origin is at the b o t t o m and z is positive upwards. The centre of the hole is in the origin. The radius of the hole is b and the radius of the cylinder is a, and draft is T = h - d where 409
410
M. ILKI~IK and K. KAI-AI.I
---3-~-
~ ~7
V __z~l~
Region I Region ! ~ -
I
f
i[I
'i
z:d
i
iI [ L . IJ
lll/llfllll//lO//i-7ill/lllllllll Y
FIG. 1.
d is the gap between the cylinder and the bottom. The undisturbed free surface is at z = h. The non-dimensional complex velocity potential is defined in three different regions (I, II, III) as follows:
d~(r,O,z,t) = Re {D6(r,z) cos0e -i~'} .
(2.1)
Region one O<_z<~h
r<~b.
In this region the non-dimensional velocity potential is expressed as:
+l(r,z) =Ao-Jl(~m-gr-) Ro(z) + ~ AN ll(m~r) R~(z) J'i (mob ) n= l l'l ( m.b )
(2.2)
where J~ is the Bessel function of the first kind of order 1. The orthonormal R~(z) functions in the interval 0 ~< z ~< h are defined as:
Rn(z) =
N~71/2coshm~rz,
n=0
NfillZcosmnz,
n~ l
(2.3)
Technical Note
411
with normalizing factors given by 1/2 I 1 +
sinh 2moh 2moh
I
sin2mnh
I
n = 0
(2.4)
Nn = 1/21 1 +
n>t 1
2mnh where mo and mn are the roots of the dispersion equation. This potential must satisfy the free-surface boundary condition, kinematic boundary condition at the body surface, the bottom of the fluid r = b and the Laplace equation.
o)2
4z - - g
4 = 0
z = h at the free surfaces
(2.5)
r= b,d~
(2.6)
04 - Vscos0 Or
at the body surface where Vs is the velocity of cylinder due to surge motion
O4 Oz
-
0
z = 0 at the bottom.
(2.7)
Region two: O<<.z~d
b<~r<~a.
In this region the non-dimensional velocity potential is expressed as:
ll(m'rrr/d)
41I(r'z)- B°2 ar + .,=1~ Bm Ii(mrra/d)
m~z cos~
(2.8)
where 11 is the modified Bessel function of the first kind Bo and B,, are obtained as:
Bm
= ~ -2 I d 4u(a,z)
cos
m~z d
dz
m=0,1,2 . . . . . m.
(2.9)
This potential must satisfy: 0 1 0 02 ) O ~ + - r --Or + ~ 4(r,z) = O b <~ r ~ a, 0 <~ z <~ d
04
Oz
_0
z=Oatthebottom.
(2.10)
(2.11)
412
M. ILKI~IKand K. KAFALJ
Region three r>~a
O<~z~h.
In this region the non-dimensional velocity is expressed as: H,(rnor) cbni(r,z) = Co H'l(moa)
Ro(z) +
(2,12)
3~ Cq I-~m~r~ R~l(z ) ,t=1 K'l(mqa)
the orthonormal Rq(z) functions are the same in the region one. Where Ht,K~ denote Hankel and modified Bessel functions. This potential must satisfy the Laplace equation, the free-surface boundary condition, and the kinematic boundary condition at the bottom, and the radiation condition. 3.
S O L U T I O N OF THE PROBLEM
The appropriate solution of the velocity potential in each region is found and the solutions are matched at the boundary so as to have the continuity of velocity potential at r = b, and r = a for 0 ~< z ~< d and its normal velocities are satisfied: ¢bl (b,z) = ~bii(b,z)
r = b
(3.1)
O<~z<_d ~,h (h Or ~bn(a,z)
~v~.v,2z,= OChn(b,z) Or
r = b
(3.2)
+lli(a,z)
r = a
(3.3)
=
()<~z<~d
O~ll(a,z) Or
= Om,a,z I Or
r = a.
(3.4)
Thus, the potential coefficients are found as: • tf
1 fd O~bn(b'z) Rn(z)dz + I
A,, = rn,,h
Bm =
21
Or
.d
d
l
i
cbm(a,z)cos
d
=
mnh
2I
d-
V, R ~ ( z ) d z
(3.5) dh(b,z) cos
mTrz d
"dz
(3.6)
d
Cq
1 I o+.(a,z) Or
mqh .
o
n,q = 0, 1 . . . . .
n,q.
.h V, Rq(z) dz
(3.7)
Technical Note
413
Then four system of equations reduce into one system of infinite number of complex unknowns, hence oc B m OLsm =
-V
(3.8)
m
s=O
where
QooLos ( Ht(moa) 2 etMo H'1 (moa)
Ofo0 - -
Jl(mob) ) ~ QqoLqs ( Kl(mqa) J'l (mob) + q=l 2 MqoL K'~(rnqa)
ll(mqb) - i,l(mqb) ) - 500
(3.9)
and
_ O~sm +
m~r Qomtos ~M 0
(l'l(mrra/d) Hl(moa) _ l',(marb/d) Ii(m~ra/d) H'l(rnoa) ll(m~rb/d) Kl(m~a)) \ Jl(m~ra/cO K'l(mqa)
m'rr QqmZqs (l',(m~ra/d)
q=l 5Mq
J,(mob) ) J'l(mob)
l',(m,rrb/d) I,(mqb) t - 8,m. It(maxb/d) I~(mqb) ] " (3.10)
4.
HYDRODYNAMIC FORCES
Forces will be calculated by integrals taken over the body surface as follows: F = - p ~-
d~.n.ds .
(4.1)
s
Thus, non-dimensionalized added mass and damping coefficients are calculated in the following manner
a2
pA
+i
b22
opA
d
A
dpn(a'z)c°s2OadOdz-
h
+I(b'z)bc°s2OdOdz (4.2)
a22: added mass coefficients; b22: damping coefficients; A: is the volume of the cylinder. 5.
NUMERIC SOLUTION AND RESULTS
The numerical results are achieved by IBM 4331 in the computer centre of Istanbul Technical University. The solution of the boundary value problem is written as the sum of the series where the coefficients of infinite equations are truncated after 11 equations;
M. [I.KI~IK and K. K..XFALI
414
such an approximation is found to be appropriate for a sufficient convergence, computations for different values are plotted for various wave amplitudes, added mass coefficients and damping coefficients. The results obtained by this formulation are compared with the results given by Sabuncu and Cali~al (1981). The trends of the curves of added mass coefficients of a concentric cylindrical hole show similar characteristics observed in the case of simple vertical cylinder. Added mass takes greater values with respect to the simple vertical cylinder. Added mass values are observed to increase as radius of hole-to-depth is increased (Figs 2 and 4); damping coefficients for surge arc observed to increase as the draft increases, The curves for damping coefficients are given in Figs 3 and 5.
a~
,/~/~ a / h = l.O b / h = 05
2O d /h -- 025
ZO
+
05
,{
I
I0
15
FIo. 2.
b22/,~r
L.)
alh =I0 blh = 05 20
8 k3
tO.
dlh:025
~-O~2 a/g - - 0 ' 5
Z'O
Fro. 3.
I,'5
Technical Note
415
a22/ A a / h = 1,0
t~
b / h = 03
~
2.0
L~
1.0
d/h=O,25
"< 0:5
I 10
I 1.5
I 2.0
= to2a/g
FIG. 4.
b 2 2 /o,.f z~
alh -- 1.0 b / h =0.3 d/h=0.25 0.5
/ I
05
I
I
I
10
1.5
2.0
• t~2a/g
FIG. 5. FIGS. 2 - 5 . ADDED MASS (a22) AND DAMPING (b22) FOR SURGE MOTION OF A CIRCULAR CYLINDER CONTAINING A CONCENTRIC CYLINDRICAL HOLE.
A w e a k p o i n t in this p r o c e d u r e is the o m i s s i o n of the vortex f o r m a t i o n a r o u n d the p e r i p h e r y of the cylinder. H o w e v e r , this s i n g u l a r b e h a v i o u r m a y be e l i m i n a t e d by a d d i n g n e w c o r r e c t i n g vortex terms. I n o u r next study we shall i n c l u d e this terms. Acknowledgements--The first author would like to thank Prof. Sabuncu and the staff of the Faculty of Naval Architecture and Ocean Engineering ITO, Istanbul.
REFERENCES ABRAMOWITZ, M. and STEGUN, I. 1967.'Handbook of Mathematical Functions. Dover Publication, New York. BLACK, J . L . , MEI, C.C. and BRAY, M.C.G. 1971. Radiation and scattering of water waves by rigid bodies. J.
Fluid Mech. 46, 151. CHEN, H.S. and MEI, C.C. 1974. Oscillationsand wave forces in a manmade harbor in the open sea. lOth Syrup. Naval Hydrodynamics, Cambridge, Mass.
416
M. h,Kl~lK and K. K;,F,.~LI
GARRET'r, C.J.R. 1970. Wave forces on a circular dock. ,1. Fluid Mech. 46(1), 129-139. GARRISON, C.J. 1975. Hydrodynamics of large objects in the sca--ll. Motion of free fl~)ating bodies, ,1 Hydronautics 9(2), 58. SABUNCU, T. and CALI~;AL, S. 1981. Hydrodynamic coefficients for vertical, circular cylinders at finitc d e p t h . / Ocean Engng 8, 25-63. YEUNG, R.W. 1981. Added mass and damping of a vertical cylinder in linitc depth waters. Apl?l. Ocean Res 3(3), 119.
002%8018/86 $3.00 + .00 Pergamon Journals Ltd.
TECHNICAL NOTE
THE SURGE MOTION OF A C I R C U L A R CYLINDER CONTAINING A CONCENTRIC CYLINDRICAL HOLE IN FINITE DEPTH M. ILKI~IKand K. KAFALI Faculty of Naval Architecture and Ocean Engineering, Istanbul Technical University, Istanbul, Turkey Abstract--This paper deals with hydrodynamic forces of a single semisubmerged circular cylinder containing a concentric cylindrical hole constrained to move in a water domain of finite depth. The fluid domain is divided into inner and outer regions. The Laplace equations governing velocity potentials for the three regions are solved by separation of variables and expressed in terms of eigenfunctions of the resulting equations which satisfy appropriate boundary conditions, Continuity of pressure and velocity at the interface of the regions provides the necessary equations from which the velocity potentials, pressures and forces are obtained. Numerical results are plotted for added mass and damping coefficients for different draft-to-depth and radius-to-depth values and for various wave amplitudes. 1.
INTRODUCTION
ADDED mass and damping of bodies oscillating in a free surface has been a subject of considerable research interest. In recent years, hydrodynamic characteristics of vertical circular cylinders and elliptical cylinders oscillating in free surface have been studied by Sabuncu and Cali~al (1981). In computing hydrodynamical forces, numerical techniques must be resorted to. These include the traditional wave source distribution m e t h o d by Garrison (1975), finite element variational formulations by Chen and Mei (1974). Scattering and radiation problems for various obstacles including the vertical circular cylinder were also treated by the Schwinger variational technique by Black et al. (1971). The axisymmetry of the geometry, the problem can be treated simply by the use of eigen functions alone (Garrett, 1970), developed the expressions of the interior and exterior problems in terms of the potential at the c o m m o n boundary and match the normal derivatives together. In this work, the interior and exterior problems were treated as a Dirichlet and N e u m a n n type, respectively, as if the conditions on the c o m m o n boundary were known. But the radiation problem of a circular cylinder containing a concentric cylindrical hole, has not been solved. The objective of this p a p e r is to present a simple and efficient m e t h o d for computing the hydrodynamic characteristics of a circular cylinder containing a concentric cylindrical hole for a surge motion. 2.
FORMULATION
The theory is based upon the usual assumptions of classical hydrodynamics, i.e. that the fluid is inviscid and of uniform density, and that the motion starts from rest, and remains irrational. Nonlinear terms in the equations of motion are neglected. It is also assumed that the motion is three-dimensional and the body possesses both a vertical axis of symmetry and a horizontal plane of symmetry. The coordinate system of Oxyz and the geometry of the body are shown in Fig. 1. The origin is at the b o t t o m and z is positive upwards. The centre of the hole is in the origin. The radius of the hole is b and the radius of the cylinder is a, and draft is T = h - d where 409
410
M. ILKI~IK and K. KAI-AI.I
---3-~-
~ ~7
V __z~l~
Region I Region ! ~ -
I
f
i[I
'i
z:d
i
iI [ L . IJ
lll/llfllll//lO//i-7ill/lllllllll Y
FIG. 1.
d is the gap between the cylinder and the bottom. The undisturbed free surface is at z = h. The non-dimensional complex velocity potential is defined in three different regions (I, II, III) as follows:
d~(r,O,z,t) = Re {D6(r,z) cos0e -i~'} .
(2.1)
Region one O<_z<~h
r<~b.
In this region the non-dimensional velocity potential is expressed as:
+l(r,z) =Ao-Jl(~m-gr-) Ro(z) + ~ AN ll(m~r) R~(z) J'i (mob ) n= l l'l ( m.b )
(2.2)
where J~ is the Bessel function of the first kind of order 1. The orthonormal R~(z) functions in the interval 0 ~< z ~< h are defined as:
Rn(z) =
N~71/2coshm~rz,
n=0
NfillZcosmnz,
n~ l
(2.3)
Technical Note
411
with normalizing factors given by 1/2 I 1 +
sinh 2moh 2moh
I
sin2mnh
I
n = 0
(2.4)
Nn = 1/21 1 +
n>t 1
2mnh where mo and mn are the roots of the dispersion equation. This potential must satisfy the free-surface boundary condition, kinematic boundary condition at the body surface, the bottom of the fluid r = b and the Laplace equation.
o)2
4z - - g
4 = 0
z = h at the free surfaces
(2.5)
r= b,d~
(2.6)
04 - Vscos0 Or
at the body surface where Vs is the velocity of cylinder due to surge motion
O4 Oz
-
0
z = 0 at the bottom.
(2.7)
Region two: O<<.z~d
b<~r<~a.
In this region the non-dimensional velocity potential is expressed as:
ll(m'rrr/d)
41I(r'z)- B°2 ar + .,=1~ Bm Ii(mrra/d)
m~z cos~
(2.8)
where 11 is the modified Bessel function of the first kind Bo and B,, are obtained as:
Bm
= ~ -2 I d 4u(a,z)
cos
m~z d
dz
m=0,1,2 . . . . . m.
(2.9)
This potential must satisfy: 0 1 0 02 ) O ~ + - r --Or + ~ 4(r,z) = O b <~ r ~ a, 0 <~ z <~ d
04
Oz
_0
z=Oatthebottom.
(2.10)
(2.11)
412
M. ILKI~IKand K. KAFALJ
Region three r>~a
O<~z~h.
In this region the non-dimensional velocity is expressed as: H,(rnor) cbni(r,z) = Co H'l(moa)
Ro(z) +
(2,12)
3~ Cq I-~m~r~ R~l(z ) ,t=1 K'l(mqa)
the orthonormal Rq(z) functions are the same in the region one. Where Ht,K~ denote Hankel and modified Bessel functions. This potential must satisfy the Laplace equation, the free-surface boundary condition, and the kinematic boundary condition at the bottom, and the radiation condition. 3.
S O L U T I O N OF THE PROBLEM
The appropriate solution of the velocity potential in each region is found and the solutions are matched at the boundary so as to have the continuity of velocity potential at r = b, and r = a for 0 ~< z ~< d and its normal velocities are satisfied: ¢bl (b,z) = ~bii(b,z)
r = b
(3.1)
O<~z<_d ~,h (h Or ~bn(a,z)
~v~.v,2z,= OChn(b,z) Or
r = b
(3.2)
+lli(a,z)
r = a
(3.3)
=
()<~z<~d
O~ll(a,z) Or
= Om,a,z I Or
r = a.
(3.4)
Thus, the potential coefficients are found as: • tf
1 fd O~bn(b'z) Rn(z)dz + I
A,, = rn,,h
Bm =
21
Or
.d
d
l
i
cbm(a,z)cos
d
=
mnh
2I
d-
V, R ~ ( z ) d z
(3.5) dh(b,z) cos
mTrz d
"dz
(3.6)
d
Cq
1 I o+.(a,z) Or
mqh .
o
n,q = 0, 1 . . . . .
n,q.
.h V, Rq(z) dz
(3.7)
Technical Note
413
Then four system of equations reduce into one system of infinite number of complex unknowns, hence oc B m OLsm =
-V
(3.8)
m
s=O
where
QooLos ( Ht(moa) 2 etMo H'1 (moa)
Ofo0 - -
Jl(mob) ) ~ QqoLqs ( Kl(mqa) J'l (mob) + q=l 2 MqoL K'~(rnqa)
ll(mqb) - i,l(mqb) ) - 500
(3.9)
and
_ O~sm +
m~r Qomtos ~M 0
(l'l(mrra/d) Hl(moa) _ l',(marb/d) Ii(m~ra/d) H'l(rnoa) ll(m~rb/d) Kl(m~a)) \ Jl(m~ra/cO K'l(mqa)
m'rr QqmZqs (l',(m~ra/d)
q=l 5Mq
J,(mob) ) J'l(mob)
l',(m,rrb/d) I,(mqb) t - 8,m. It(maxb/d) I~(mqb) ] " (3.10)
4.
HYDRODYNAMIC FORCES
Forces will be calculated by integrals taken over the body surface as follows: F = - p ~-
d~.n.ds .
(4.1)
s
Thus, non-dimensionalized added mass and damping coefficients are calculated in the following manner
a2
pA
+i
b22
opA
d
A
dpn(a'z)c°s2OadOdz-
h
+I(b'z)bc°s2OdOdz (4.2)
a22: added mass coefficients; b22: damping coefficients; A: is the volume of the cylinder. 5.
NUMERIC SOLUTION AND RESULTS
The numerical results are achieved by IBM 4331 in the computer centre of Istanbul Technical University. The solution of the boundary value problem is written as the sum of the series where the coefficients of infinite equations are truncated after 11 equations;
M. [I.KI~IK and K. K..XFALI
414
such an approximation is found to be appropriate for a sufficient convergence, computations for different values are plotted for various wave amplitudes, added mass coefficients and damping coefficients. The results obtained by this formulation are compared with the results given by Sabuncu and Cali~al (1981). The trends of the curves of added mass coefficients of a concentric cylindrical hole show similar characteristics observed in the case of simple vertical cylinder. Added mass takes greater values with respect to the simple vertical cylinder. Added mass values are observed to increase as radius of hole-to-depth is increased (Figs 2 and 4); damping coefficients for surge arc observed to increase as the draft increases, The curves for damping coefficients are given in Figs 3 and 5.
a~
,/~/~ a / h = l.O b / h = 05
2O d /h -- 025
ZO
+
05
,{
I
I0
15
FIo. 2.
b22/,~r
L.)
alh =I0 blh = 05 20
8 k3
tO.
dlh:025
~-O~2 a/g - - 0 ' 5
Z'O
Fro. 3.
I,'5
Technical Note
415
a22/ A a / h = 1,0
t~
b / h = 03
~
2.0
L~
1.0
d/h=O,25
"< 0:5
I 10
I 1.5
I 2.0
= to2a/g
FIG. 4.
b 2 2 /o,.f z~
alh -- 1.0 b / h =0.3 d/h=0.25 0.5
/ I
05
I
I
I
10
1.5
2.0
• t~2a/g
FIG. 5. FIGS. 2 - 5 . ADDED MASS (a22) AND DAMPING (b22) FOR SURGE MOTION OF A CIRCULAR CYLINDER CONTAINING A CONCENTRIC CYLINDRICAL HOLE.
A w e a k p o i n t in this p r o c e d u r e is the o m i s s i o n of the vortex f o r m a t i o n a r o u n d the p e r i p h e r y of the cylinder. H o w e v e r , this s i n g u l a r b e h a v i o u r m a y be e l i m i n a t e d by a d d i n g n e w c o r r e c t i n g vortex terms. I n o u r next study we shall i n c l u d e this terms. Acknowledgements--The first author would like to thank Prof. Sabuncu and the staff of the Faculty of Naval Architecture and Ocean Engineering ITO, Istanbul.
REFERENCES ABRAMOWITZ, M. and STEGUN, I. 1967.'Handbook of Mathematical Functions. Dover Publication, New York. BLACK, J . L . , MEI, C.C. and BRAY, M.C.G. 1971. Radiation and scattering of water waves by rigid bodies. J.
Fluid Mech. 46, 151. CHEN, H.S. and MEI, C.C. 1974. Oscillationsand wave forces in a manmade harbor in the open sea. lOth Syrup. Naval Hydrodynamics, Cambridge, Mass.
416
M. h,Kl~lK and K. K;,F,.~LI
GARRET'r, C.J.R. 1970. Wave forces on a circular dock. ,1. Fluid Mech. 46(1), 129-139. GARRISON, C.J. 1975. Hydrodynamics of large objects in the sca--ll. Motion of free fl~)ating bodies, ,1 Hydronautics 9(2), 58. SABUNCU, T. and CALI~;AL, S. 1981. Hydrodynamic coefficients for vertical, circular cylinders at finitc d e p t h . / Ocean Engng 8, 25-63. YEUNG, R.W. 1981. Added mass and damping of a vertical cylinder in linitc depth waters. Apl?l. Ocean Res 3(3), 119.