Ocean Engineering 32 (2005) 1199–1215
www.elsevier.com/locate/oceaneng
Effect of shallow and narrow water on added mass
of cylinders with various cross-sectional shapes
Z.X. Zhou, Edmond Y.M. Lo*, S.K. Tan
MPA-NTU Maritime Research Center, School of Civil and Environmental Engineering,
Nanyang Technological University, Singapore 639798, Singapore
Received 7 September 2003; accepted 7 December 2004
Available online 25 February 2005
Abstract
The sway, heave and roll added masses of three uniform cylinders with semi-circular, rectangular
and triangular cross-sectional shapes in shallow and narrow water are numerically analysed. The
method is based on simulation of the potential flow induced by the cylinder’s mode of motion. The
effects of shallow and narrow water on added mass are analysed and presented. It is concluded that
the shallow and narrow water effects on added mass depend on the different cross-section shapes of
the cylinders. In particular, the water depth effect on sway added mass is stronger than that on heave
added mass while the narrow water effect on sway is weaker than that on heave. The shallow water
effect on added mass tends to weaken the narrow water effect. Lastly the effect of shallow and narrow
water on added mass on a rectangular cylinder is the strongest while that on a triangular cylinder is
the weakest.
q 2005 Elsevier Ltd. All rights reserved.
Keywords: Added mass; Shallow water; Narrow water; Potential flow
1. Introduction
Numerous factors affect the motions of a large floating body. The prediction of such
motions requires various hydrodynamic coefficients including added mass and damping.
When a seabed is close to the floating body, i.e. the water is shallow, the added mass and
damping change significantly due to the proximity to the seabed and the more intensive
* Corresponding author. Fax: C65 6792 1650.
E-mail address: cymlo@ntu.edu.sg (E. Lo).
0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.
doi:10.1016/j.oceaneng.2004.12.001
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Z.X. Zhou et al. / Ocean Engineering 32 (2005) 1199–1215
free surface fluctuation, e.g. Newman (1985) and Lamb (1932). Barr (1993) reviewed
various ship manoeuvring simulation methods, some of which apply to the shallow water
case. He ranked those methods in terms of the efficiency of the methods in addressing the
shallow water and free surface effects. De Tarso et al. (1996) investigated the added mass
and damping of rectangular cylinders mounted at the seabed and presented the shallow
water effect on added mass and damping. Aoki (1997) focused on the hydrodynamic
coefficients of very large floating structures in shallow water and quantified the depth
effect on the hydrodynamic coefficients. Beukelman (1998) deduced the manoeuvring
coefficients for a model wing in deep and shallow water from experiments and discussed
the changes caused by shallow water. Abul-Azm and Gesraha (2000) proposed an
approximation for the hydrodynamics of floating pontoons in shallow water under incident
oblique waves and obtained the depth effect on the pontoon hydrodynamics. Lopes and
Sarmento (2002) analysed the hydrodynamic coefficients of a submerged pulsating sphere
in finite depth using linear wave theory and presented the changes of added mass and
damping with water depth.
The added mass of a large submerged body is among the hydrodynamic characteristics
that play an important role in determining the body motion. Due to large seawater density,
the added mass force is often comparable to other terms in the equations of motion and
thus cannot be neglected. As such the evaluation of the added mass of a large floating body
is of considerable interest in applications such as prediction of seacraft manoeuvrability
and control. While results on frequency dependent added mass of floating cylinders with
simple shapes in deep-water are available in the literature, the literature on this same topic
in shallow water is considerably less.
For an object in translatory motion, it is difficult to obtain the zero-frequency added
mass based on the pressure integral over the body’s wetted surface. This is because the
pressure force as obtained from the potential theory calculation equals zero when the free
surface effect is ignored. The double body theory which is based on the assumption that the
free water surface fluctuation is weak when combined with conformal mapping
techniques, however, proves to be a useful tool to determine the zero-frequency added
mass associated with sway, e.g. Newman (1985). The semi-circular cylinder as a
simplification of the object’s cross-section is often adopted in the analysis of added mass.
The added masses of circular cylinders and cylinders with other cross-sectional shapes in
deep water are available, Newman (1985). Different semi-theoretical methods had also
been proposed to determine the added mass of circular cylinder in shallow water, e.g.
Lockwood-Taylor (1930), and Kennard (1967). Clarke (2001a) used the techniques of
conformal mapping and calculated the added mass of circular cylinder in shallow water.
He demonstrated the effect of water depth through a comparison between results from
different methods based on conformal mapping techniques and concluded that the
approach of using a row of distributed dipoles gave the best accuracy. The added mass for
the more complex case of elliptical cylinder in shallow water was given by Clarke (2001b)
who used a more general mapping technique based on the Schwartz–Christoffel method.
Clarke (2003) further used a similar method to calculate the added mass of elliptical
cylinder with vertical fin stabiliser in shallow water.
Clarke and the others’ work on the zero-frequency added mass does not address the
narrow water effect that can produce changes in added mass values similar to those in
Z.X. Zhou et al. / Ocean Engineering 32 (2005) 1199–1215
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shallow water. Such effects would be important for example in manoeuvring predictions in
a narrow waterway such as that investigated by Sarioz and Narli (2003) for manoeuvres
of large ships through the Strait of Istanbul. This paper focuses on a general
numerical algorithm used to investigate the effects of shallow water and narrow water
on the zero-frequency added mass of a cylinder with various cross-sectional shapes.
The cross-sections of semi-circle, rectangle and triangle are considered. The solution
involves using the finite difference scheme on an unequal mesh to solve the 2D Laplace’s
equation that governs the induced flow-field. The cases for sway, heave and roll are
simulated.
2. Mathematical model and equations
Fig. 1(a)–(c) show the three cylinders: semi-circular, rectangular and triangular crosssections in shallow and narrow water, along with the Cartesian coordinate system used. A
flow-field is induced as the cylinder moves or rotates around its center. The flow boundary
comprises three flat planes: the seabed and two side-walls (i.e. seashores), two water levels
and the surface of the body. The seawater could be in motion as well. The co-ordinate
system with the origin set at the center of the object at the water-plane, the y sway axis and
z heave axis is used to describe the flow induced by the cylinder’s mode of motion. The
seawater is assumed to be inviscid and incompressible, and the induced water motion
irrotational. The induced flow velocity potential is governed by the Laplace’s equation as
Dfi Z 0
(1)
Ð i whose
in which fi is the induced velocity potential by the cylinder’s mode of motion U
component in the ith direction is unity and zero for the other five components. Here the
subscript i varies between 2, 3 and 4 corresponding to sway, heave and roll, respectively.
The water-side surfaces of two static side-walls are considered rigid and impermeable,
i.e.
vfi
j
Z0
vy jyjZb
(2)
in which b is the half of the distance between the two side-walls.
In general, there is a free surface effect at the water surface but the effect is ignored here
because the zero-frequency case is addressed using the twin-hull approximation. Then the
sway case is solved by using a zero vertical velocity condition at the free surface leading to
vf2
j Z0
vz zZ0
(3)
The heave case is different from the sway case in that it requires that the water level be
permeable only in the vertical direction. The following boundary condition at the water
level, based on the twin-hull approximation, is adopted:
f3 jzZ0 Z 0
(4)
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Fig. 1. (a) Semi-circular cylinder in shallow and narrow water. (b) Rectangular cylinder in shallow and narrow
water. (c) Triangular cylinder in shallow and narrow water.
For the roll case, the water level must be permeable to the vertical water flow in order to
satisfy the global mass conservation law. Thus similar to the heave case, the boundary
condition is imposed:
f4 jzZ0 Z 0
(5)
The cylinder body surface is assumed to be rigid and impermeable such that the following
wall condition is applied. Depending on the body mode of motion, this is given by
vfi
Ð i $Ð
j
ZU
nh
vn rÐZÐr h
(6)
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where nÐh is the normal unit vector to the cylinder hull surface pointing into the
body and rÐh is the position vector that defines the body surface. Three modes
Ð 2 Z ð0; 1 m=s; 0; 0; 0; 0Þ), heave
of motion are used corresponding to the sway (U
Ð
Ð
(U 3 Z 0; 0; 1 m=s; 0; 0; 0Þ) and roll (U 4 Z ð0; 0; 0; 1 rad=s; 0; 0Þ).
Some singular points exist with at least two normal directions in the flow-field models
shown in Fig. 1(a)–(c). For example, at any of the four right-angle corners of the flow
domain, the following wall conditions apply:
vfi
vf
Z i Z0
vy
vz
(7)
The corners between the water level and cylinder are assumed to have the normal vectors
determined by the cylinder body to avoid the singularity. In addition, the normal vector at
any bottom corner of the rectangular and triangular cylinders is considered to be the
algebraic average of the vectors at the two points adjacent to the corner, thus avoiding the
singularity.
The seabed is considered rigid and the application of the impermeable wall condition
generates
vfi
j
Z0
vz zZKH
(8)
in which H is the water depth.
Based on the spatial distribution of the induced velocity potential derived from the
governing equation, Eq. (1), and the associated boundary conditions, Eqs. (2)–(8), the
determination of the sway added mass, heave added mass and roll added mass as given by
Newman (1985) is
maij Z rw
#f n
i hj
ds
(9)
S
where rw is the seawater density and S is the wetted body surface.
3. Numerical method and computational parameters
Numerical simulation using finite difference is adopted to generate the velocity
potentials induced by the different cylinder modes of motion. A central difference scheme
on an unequal mesh is adopted to discretize Eq. (1). The implicit discrete form is given by
cdef
1
1
fðj K 1; kÞ
C fðj C 1; kÞ
fðj; kÞ Z
cd C ef
cðc C dÞ
dðc C dÞ
1
1
C fðj; k K 1Þ
C fðj; k C 1Þ
(10)
eðe C f Þ
f ðe C f Þ
where the symbols j and k refer to the y-coordinate and z-coordinate of the node (j, k),
respectively; c, d, e and f represent the corresponding distances from node (jK1, k) to node
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(j, k), node (jC1, k) to node (j, k), node (j, kK1) to node (j, k), and node (j, kC1) to node (j,
k), respectively.
An imaginary layer of nodes is added at every impermeable wall, such as the side-walls
and seabed, to define the discrete wall conditions. The discretization of cylinder hull
condition is done by means of various first-order difference schemes, to ensure that the
cylinder mode of motion is imposed on the hull to induce the flow-field. The backward
difference scheme is adopted at the points with negative y-coordinate while the
forward difference scheme is applied to the points with positive y-coordinate. These
two first-order difference schemes capture the outward disturbance by the rigid cylinder
and have enough accuracy to generate added mass values, as is shown below.
Eq. (10) constrained by the discrete boundary conditions is solved by iteration. Iteration
is carried out until the following convergence criterion is satisfied:
M X
N
X
jfc ðj; kÞ K fp ðj; kÞj% 10K8
(11)
jZ1 kZ1
where M and N refer to the number of nodes in the y-direction and the z-direction,
respectively. In addition, the subscripts c and p mean the current and previous iteration
steps, respectively.
The cross-section planes of the cylinders (Fig. 1(a)–(c)) are subdivided evenly into
sufficient segments to ensure the convergence. More elements are needed for the shallow
and narrow water (typically 40). The water depth changes from deep water depth (up to
20r0) to the smallest water depth of 1.1r0 used. The distance between the two side-walls
varies over the range of 20r0 to 1.2r0 to capture the narrow water effect.
The second-order approximation is adopted in the evaluation of the integral in Eq. (9),
leading to the following discrete form:
maij Z rw
M
1X
½f ðk K 1Þnhj ðk K 1Þ C fi ðkÞnhj ðkÞDlk
2 kZ1 i
(12)
in which the variable k is the kth segment on the cylinder surface and Dlk represents the
linear length of the kth segment.
4. Numerical results and discussion
The numerical results on the added masses of the three basic cylinders and the induced
flow patterns by modes of motion are presented in Tables 1–3 and Figs. 2(a)–10(c).
Table 1
Added mass of semi-circular cylinder in deep water
Added mass
Sway (ma22/rwpr20)
Heave (ma33/rwpr20)
Sources
Newman’s analytical
value (1985)
Present simulation
(bZ10r0, HZ10r0)
Difference (%)
0.5
0.5
0.5026
0.5026
0.52
0.52
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Table 2
Added mass of rectangular cylinder in deep water
Added mass
Sway (ma22/rwr20)
Heave (ma33/rwr20)
Roll (ma44/rwr40)
Sources
Newman’s analytical
value (1985)
Present simulation
(bZ10r0, HZ10r0)
Difference (%)
2.377
2.377
0.3625
2.392
2.392
0.3653
0.63
0.63
0.77
Figs. 3(a)–4(c) show the flow patterns induced by the rectangular cylinder at various
modes of motion in the form of velocity magnitude contours normalized with 1 m/s.
Figs. 5(a)–10(c) show the added mass elements in sway, heave and roll. Various
corresponding results from Newman (1985), Reddy and Arockiasamy (1991) and Clarke
(2001a) are also shown in Tables 1–3 and Fig. 2(a) and (b) for comparison.
4.1. Verification of model and algorithm
The comparison of the added masses of the three cylinders in deep and wide water (i.e.
semi-infinite domain) with the analytical results given by Newman (1985) and
computational results listed by Reddy and Arockiasamy (1991) is shown in Tables 1–3.
The writers’ calculations are based on a depth HZ10r0 and width bZ10r0. As is shown
later, these depth and width values effectively simulate the semi-infinite domain. The
comparison indicates that the present numerical method for the semi-infinite domain
generates reliable values of added mass for semi-circular, rectangular and triangular
cylinders.
A further comparison is shown in Fig. 2(a) and (b) for the sway added masses of semicircular and rectangular cylinders as functions of water depth in wide water i.e. side-walls
far from the cylinders and width bZ10r0 from the origin. Results from the semitheoretical analysis by Clarke (2001a) and calculations by Reddy and Arockiasamy (1991)
are also shown in Fig. 2(a) and (b). All the results show large increase in the added mass
with decreasing depth. The plots show that there are only small differences between the
writers’ results and those of the references for very shallow water depth. At depth H
approaching r0, the added mass values become infinite, a consequence of the assumption
that seawater is incompressible and seabed is rigid: cylinders can no longer move in the
sway direction.
Table 3
Added mass of triangular cylinder in deep water
Added mass
Sway (ma22/rwr20)
Heave (ma33/rwr20)
Roll (ma44/rwr40)
Sources
Reddy and Arockiasamy
(1991)
Present simulation
(bZ10r0, HZ10r0)
Difference (%)
1.1938
1.1938
0.10053
1.1981
1.1981
0.10113
0.36
0.36
0.597
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Fig. 2. (a) Sway added mass of semi-circular cylinder in shallow and wide water (CaZma22/rwpr20 and the
subscript d denotes the deep water value). (b) Sway added mass of rectangular cylinder in shallow and wide water
(CaZma22/rwr20 and the subscript d denotes the deep water value).
4.2. Flow patterns
Fig. 3(a)–(c) present the flow patterns induced by the modes of motion for the
rectangular cylinder when the size of flow-domain is 8r0 by 8r0. It can be seen from the
contours that the side and bottom walls increase the added mass through the generation of
weak wall flows. It is also noted that these effects are weakest for the roll motion. The
additional flow generated at the far boundaries is small resulting in only negligible
increases in added mass. As such the added mass values approach those of the semiinfinite case.
Fig. 4(a)–(c) show the corresponding flow patterns at shallow (HZ1.5r0) and narrow
(bZ1.5r0) water. The density of the contour lines increases significantly indicating a
significant increase in the induced flow velocity and thus the corresponding increase in
added mass. Similar observations are seen for the circular and triangular cylinders, and are
not shown.
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Fig. 3. (a) Velocity magnitude contours caused by rectangle sway mode of motion (bZ8r0, HZ8r0 and contour
increment is 0.02). (b) Velocity magnitude contours caused by rectangle heave mode of motion (bZ8r0, HZ8r0
and contour increment is 0.0285). (c) Velocity magnitude contours caused by rectangle roll mode of motion
(bZ8r0, HZ8r0).
4.3. Added mass in shallow and narrow water
Fig. 5(a) shows the depth variation of the sway and heave added masses of the circular
cylinder when the side-walls are far at bZ10r0. The variations of the added mass of the
circular cylinder in deep water (HZ10r0) caused by varying the distance to side-walls are
shown in Fig. 5(b). Both the sway and heave added masses increase as the water depth
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Fig. 4. (a) Velocity magnitude contours caused by rectangle sway mode of motion (bZ1.5r0, HZ1.5r0
and contour increment is 0.056). (b) Velocity magnitude contours caused by rectangle heave mode of
motion (bZ1.5r0, HZ1.5r0 and contour increment is 0.08). (c) Velocity magnitude contours caused by rectangle
mode of motion (bZ1.5r0, HZ1.5r0 and contour increment is 0.035).
decreases. The sway case depicts a larger increase. The water depth effect is small until a
depth H/r0 of about 2.5 or less. The deep water result is obtained when the depth H/r0
increases to eight or the larger. Similar effects are seen in the case of narrow and deep
water (Fig. 5(b)). The two added mass elements increase with decreasing width b/r0.
However, the increase in heave is more pronounced. The narrow water effect is small until
the width b/r0 is about 2.5 or less. The narrow water effect becomes negligible only at
distances larger than about 8r0.
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Fig. 5. (a) Added mass of semi-circular cylinder in shallow and wide water (bZ10r0). (b) Added mass of semicircular cylinder in narrow and deep water (HZ10r0).
Fig. 6(a) and (b) show the added mass values for the circular cylinder versus water
depth H/r0 at different values of side-wall distance b/r0. When the water depth H/r0 is
smaller, the added mass increase caused by decreasing side-wall distance b/r0 is
smaller, indicating that the shallow water effect weakens the narrow water effect. The
roll added mass is, of course, zero for the circular cylinder as the effect of viscosity is
neglected.
The changes of the sway, heave and roll added masses of the rectangular cylinder in
wide water (bZ10r0) at various water depths are shown in Fig. 7(a). Fig. 7(b) shows the
added mass of the rectangular cylinder in narrow and deep water (HZ10r0). Fig. 8(a)–(c)
present the added mass of the rectangular cylinder as a function of water depth at various
values of the side-wall distance. Figs. 7(a)–8(c) depict that the roll added mass increases
with decreasing water depth and/or side-wall distance but its relative increase is by far less
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Fig. 6. (a) Sway added mass of semi-circular cylinder in shallow and narrow water (CaZma22/rwpr20). (b) Heave
added mass of semi-circular cylinder in shallow and narrow water (CaZma33/rwpr20).
than those of the sway and heave added masses. This indicates that the roll added mass is
little changed by the shallow and narrow water. The trends of the shallow water effect and
narrow water effect on the rectangular cylinder (Fig. 8(a)–(c)) are similar to those on
the circular cylinder. In particular, the added mass increase due to decreasing side-wall
distance b/r0 is smaller at smaller water depth H/r0.
Figs. 9(a)–10(c) present the added mass values of the triangular cylinder in different
depth and width of the fluid domain such as deep and wide water (HZ10r0, bZ10r0),
shallow and wide water (bZ10r0), narrow and deep water (HZ10r0) and shallow and
narrow water. It is seen from these figures that the added mass trends on the shallow and
narrow water effects on the rectangular cylinder also hold for the triangular cylinder.
It is also deduced from Figs. 5(a)–10(c) that the added mass of the rectangular
cylinder has the largest relative increase in response to the change of water depth and/or
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Fig. 7. (a) Added mass of rectangular cylinder in shallow and wide water (bZ10r0). (b) Added mass of
rectangular cylinder in narrow and deep water (HZ10r0).
side-wall distance. The triangular cylinder shows the smallest relative increase. This
observation implies that the shallow and narrow water effect on the added mass of the
rectangular cylinder is the strongest and that on the triangular cylinder the weakest. This
is the direct consequence of the fact that the cylinders with the same breadth at water
level and draft have different areas: the rectangle area is the largest and the triangle area
is the smallest.
5. Conclusion
An algorithm for evaluating the added mass of the semi-circular, rectangular and
triangular uniform cylinders in shallow and narrow water has been developed.
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Fig. 8. (a) Sway added mass of rectangular cylinder in shallow and narrow water (CaZma22/rwr20). (b) Heave
added mass of rectangular cylinder in shallow and narrow water (CaZma33/rwr20). (c) Roll added mass of
rectangular cylinder in shallow and narrow water (CaZma44/rwr40).
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Fig. 9. (a) Added mass of triangular cylinder in shallow and wide water (bZ10r0). (b) Added mass of triangular
cylinder in narrow and deep water (HZ10r0).
The cylinder cross-sections have the same breadth at the waterline and draft. The modes of
motion include sway, heave and roll. The writers’ results compared well with the
corresponding results by Clarke (2001a), Newman (1985) and Reddy and Arockiasamy
(1991) at various limiting cases. The presence of bottom and side-walls induced stronger
flow patterns that produce significant increase in the added mass values from those of
the semi-infinite domain. Significant increase in the added mass is obtained for both
shallow and narrow water. It is also shown that the shallow water effect on sway added
mass of any cylinder is stronger than that on heave. However, the narrow water effect on
sway is weaker than that on heave. The shallow water has the same effect in changing the
roll added mass of the rectangular and triangular cylinder as the narrow water but both
effects are weaker. The shallow water effect weakens the narrow water effect. The shallow
and narrow water effect on the rectangular cylinder is the strongest, that on the circular
cylinder moderate and weakest on the triangular cylinder.
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Fig. 10. (a) Sway added mass of triangular cylinder in shallow and narrow water (CaZma22/rwr20). (b) Heave
added mass of triangular cylinder in shallow and narrow water (CaZma33/rwr20). (c) Roll added mass of triangular
cylinder in shallow and narrow water (CaZma44/rwr40).
Z.X. Zhou et al. / Ocean Engineering 32 (2005) 1199–1215
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Acknowledgements
Support by MPA-NTU Maritime Research Center, Nanyang Technological University,
Singapore is gratefully acknowledged.
References
Abul-Azm, A.G., Gesraha, M.R., 2000. Approximation to the hydrodynamics of floating pontoons under oblique
waves. Ocean Engineering 27 (4), 365–384.
Aoki, Shin-ichi, 1997. Shallow water effect on hydrodynamic coefficients of very large floating structures.
Proceedings of the 1997 7th International Offshore and Polar Engineering Conference, May 25–30 1997,
Honolulu, HI, USA 4 (1), 253–260.
Barr, R.A., Review and comparison of ship manoeuvring simulation methods, Transactions–society of naval
architects and marine engineers, 101, 1993 SNAME Annual Meeting, (Sep 14–19), New York, NY, USA,
609–635
Beukelman, W., 1998. Manoeuvering coefficients for a wing-model in deep and shallow water. Part
I—experiments and test results. International Shipbuilding Progress 45 (441), 5–50.
Clarke, D., 2001a. Calculation of the added mass of circular cylinders in shallow water. Ocean Engineering. 28
(9), 1265–1294.
Clarke, D., 2001b. Calculation of the added mass of elliptical cylinders in shallow water. Ocean Engineering 28
(10), 1361–1381.
Clarke, D., 2003. Calculation of the added mass of elliptical cylinders with vertical fins in shallow water. Ocean
Engineering 30 (1), 1–22.
De Tarso, P., Esperanca, T., Hamilton Sphaier, S., 1996. Added mass and damping on bottom-mounted
rectangular cylinders. Journal of the Brazilian Society of Mechanical Sciences 18 (4), 355–373.
Kennard, E.H., 1967. Irrotational flow of frictionless fluids, mostly of invariable density, David Taylor Model
Basin, USA Report. 2299, 108–112
Lamb, H., 1932. Hydrodynamics, sixth ed. Cambridge University Press, Cambridge, England (Reprinted 1945,
Dover, New York).
Lockwood-Taylor, J., 1930. Some hydrodynamic inertia coefficients. Philosophical Magazine Series 79 (55),
161–183.
Lopes, D.B.S., Sarmento, A.J.N.A., 2002. Hydrodynamic coefficients of a submerged pulsating sphere in finite
depth. Ocean Engineering 29 (11), 1391–1398.
Newman, J.N., 1985. Marine Hydrodynamics, third ed. MIT Press, Cambridge, Massachusetts and London,
England.
Reddy, D.V., Arockiasamy, M., 1991. Offshore Structures, vol. 1. Krieger Publishing Company, Malabar, Florida
p. 1991.
Sarioz, K., Narli, E., 2003. Assessment of manoeuvring performance of large tankers in restricted water
waterways: a real-time simulation approach. Ocean Engineering 30 (12), 1535–1551.