effect of water depth on added mass

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

Ocean Engineering 32 (2005) 1199–1215

www.elsevier.com/locate/oceaneng

0029-8018/$ - see front matter q 2005 Elsevier Ltd. All rights reserved.

doi:10.1016/j.oceaneng.2004.12.001

* Corresponding author. Fax: C65 6792 1650.

E-mail address: [email protected] (E. Lo).

http://www.elsevier.com/locate/oceaneng

Z.X. Zhou et al. / Ocean Engineering 32 (2005) 1199–12151200

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 1201

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 cross-

sections 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)

in which fi is the induced velocity potential by the cylinder’s mode of motion ÐUi whose

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

vyjjyjZb Z 0 (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

vzjzZ0 Z 0 (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:

f3jzZ0 Z 0 (4)

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:

f4jzZ0 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

vnjÐrZÐrh

Z ÐU i$Ðnh (6)


where Ðnh is the normal unit vector to the cylinder hull surface pointing into the

body and Ðrh is the position vector that defines the body surface. Three modes

of motion are used corresponding to the sway ( ÐU 2Z ð0; 1 m=s; 0; 0; 0; 0Þ), heave

( ÐU3Z0; 0; 1 m=s; 0; 0; 0Þ) and roll ( ÐU4Z 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

vyZ

vfi

vzZ 0 (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

vzjzZKH Z 0 (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 #S

finhj ds (9)

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

fðj; kÞ Zcdef

cd Ceffðj K1; kÞ

1

cðc CdÞCfðj C1; kÞ

1

dðc CdÞ

�

Cfðj; k K1Þ1

eðe C f ÞCfðj; k C1Þ

1

f ðe C f Þ

�(10)

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


(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:

XM

jZ1

XN

kZ1

jfcðj; kÞKfpðj; kÞj%10K8 (11)

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

1

2

XM

kZ1

½fiðk K1Þnhjðk K1ÞCfiðkÞnhjðkÞ�Dlk (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 Sources

Newman’s analytical

value (1985)

Present simulation

(bZ10r0, HZ10r0)

Difference (%)

Sway (ma22/rwpr02) 0.5 0.5026 0.52

Heave (ma33/rwpr02) 0.5 0.5026 0.52

Table 2

Added mass of rectangular cylinder in deep water

Added mass Sources

Newman’s analytical

value (1985)

Present simulation

(bZ10r0, HZ10r0)

Difference (%)

Sway (ma22/rwr02) 2.377 2.392 0.63

Heave (ma33/rwr02) 2.377 2.392 0.63

Roll (ma44/rwr04) 0.3625 0.3653 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 semi-

circular 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 semi-

theoretical 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 Sources

Reddy and Arockiasamy

(1991)

Present simulation

(bZ10r0, HZ10r0)

Difference (%)

Sway (ma22/rwr02) 1.1938 1.1981 0.36

Heave (ma33/rwr02) 1.1938 1.1981 0.36

Roll (ma44/rwr04) 0.10053 0.10113 0.597

Fig. 2. (a) Sway added mass of semi-circular cylinder in shallow and wide water (CaZma22/rwpr02 and the

subscript d denotes the deep water value). (b) Sway added mass of rectangular cylinder in shallow and wide water

(CaZma22/rwr02 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 semi-

infinite 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.

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

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.

Fig. 5. (a) Added mass of semi-circular cylinder in shallow and wide water (bZ10r0). (b) Added mass of semi-

circular 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

Fig. 6. (a) Sway added mass of semi-circular cylinder in shallow and narrow water (CaZma22/rwpr02). (b) Heave

added mass of semi-circular cylinder in shallow and narrow water (CaZma33/rwpr02).


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

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.

Fig. 8. (a) Sway added mass of rectangular cylinder in shallow and narrow water (CaZma22/rwr02). (b) Heave

added mass of rectangular cylinder in shallow and narrow water (CaZma33/rwr02). (c) Roll added mass of

rectangular cylinder in shallow and narrow water (CaZma44/rwr04).


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.

Fig. 10. (a) Sway added mass of triangular cylinder in shallow and narrow water (CaZma22/rwr02). (b) Heave

added mass of triangular cylinder in shallow and narrow water (CaZma33/rwr02). (c) Roll added mass of triangular

cylinder in shallow and narrow water (CaZma44/rwr04).



Acknowledgements

Support by MPA-NTU Maritime Research Center, Nanyang Technological University,

Singapore is gratefully acknowledged.

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