ship manoeuvring in waves

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Ship Manoeuvring in Waves

A LITERATURE REVIEW

WL Rapporten00_096


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F-WL-PP10-1 Versie 04GELDIG VANAF: 12/11/2012

Ship Manoeuvring in Waves

A li terature review

Tello Ruiz, M. ; Candries. M., Vantorre, M. ; Delefortrie, G. ; Peeters, D.; Mostaert, F.

November2012

WL2012R00_096_18rev2_0


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This publication must be cited as follows:

Tello Ruiz, M. ; Candries. M., Vantorre, M. ; Delefortrie, G. ; Peeters, D.; Mostaert, F. (2012). Ship Manoeuvring in

Waves: A literature review. Version 2_0. WL Rapporten, 00_096. Flanders Hydraulics Research & Ghent

University: Antwerp, Belgium.

Waterbouwkundig Laboratorium

Flanders Hydraulics Research

B-2140 Antwerp

Tel. +32 (0)3 224 60 35

Fax +32 (0)3 224 60 36

E-mail: [email protected]

www.watlab.be

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Technologiepark Zwijnaarde 904

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Tel +32 (0)9 2645555

Fax +32 (0)9 2645843


www.maritiem.ugent.be

Nothing from this publication may be duplicated and/or published by means of print, photocopy, microfilm or

otherwise, without the written consent of the publisher.
http://www.maritiem.ugent.be/http://www.maritiem.ugent.be/http://www.maritiem.ugent.be/


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Document identification

Title: Ship Manoeuvring in Waves: A literature review

Customer: Flanders Hydraulics Research Ref.: WL2012R00_096_18rev2_0

Keywords (3-5): manoeuvring, seakeeping, 6 DOF, literature

Text (p.): 20 Appendices (p.): /

Confidentiality: Yes Exceptions: Customer

Internal

Flemish government

Released as from:

No Available online

Approval

Author

Tello Ruiz, M.Candries, M.

Reviser

Vantorre, M.

Project Leader

Delefortrie, G.

S&A Director

Peeters, P.

Division Head

Mostaert, F.

Revisions

Nr. Date Definition Author(s)

1_0 18/07/2012 Concept version Tello Ruiz, M. ; Candries, M.

1_1 18/07/2012 Revision Delefortrie, Guillaume

1_2 23/08/2012 Revision Vantorre, Marc

2_0 19/11/2012 Final version Delefortrie, Guillaume

Abstract

Manoeuvring in calm water and wave induced motion in open water have been major concerns over the past

years. Although the importance of such theories, developed to improve the safety at sea, their analysis is

constrained because of the assumptions and considerations employed. While the ship performs manoeuvres in

open sea or sheltered areas, she is subjected to a large number of phenomena thus impairing the overall

performance and increasing the probability of hazards and capsizes. Therefore, the conventional analysis of

manoeuvring, based only on the horizontal motions, cannot be applied if a more realistic study of the ship

dynamics is intended, thus it is necessary to incorporate the vertical motions. In this scope some light has beenshed in the recent years to address these problems and the present work aims to present a literature review of

the most promising methods up to date and the discussion of the relevant phenomena involved.


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Ship Manoeuvring in Waves: A literature review

Final version WL2012R00_096_18rev2_0 IF-WL-PP10-1 Version 04RELEASED AS FROM: 12/11/2012

Contents

1 Introduction ............................................................................................................................................... 1

2 Rigid body dynamics ................................................................................................................................ 3

3 Manoeuvring in Waves ............................................................................................................................. 7

3.1 General discussion ............................................................................................................................ 7

3.2 Additional consideration to manoeuvring in a seaway ...................................................................... 7

4 Methods based on two-time scale models ............................................................................................. 10

5 Methods based on hybrid approach ....................................................................................................... 14

6 Final observations .................................................................................................................................. 17

7 Conclusion .............................................................................................................................................. 18

8

References ............................................................................................................................................. 19


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Final version WL2012R00_096_18rev2_0 IIF-WL-PP10-1 Version 04RELEASED AS FROM: 12/11/2012

List of figures

Figure 1 Manoeuvring reference frame, (a) NED reference frame and (b) body fixed referenceframe,

. .................................................................................................................................................... 3

Figure 2 Euler angles and yaw-pitch-roll rotation sequence ............................................................................ 3

Figure 3 Variation of the wetted surface as function of the roll angle. ............................................................. 8

Figure 4 Description of the two-time scale method and coupled time matching and data exchange, (a)

sequential evaluation and (b) a parallel evaluation. ...................................................................................... 10

Figure 5 Reference frames used in: (a) Skejic and Faltinsen (2008), (b) Yasukawa and Nakayama (2009)

and (c) Seo and Kim (2011). ......................................................................................................................... 12

Figure 6 Reference frames proposed by Hamamoto and Kim (1993): (a) earth fixed axes, (b) general body

axes and (c) horizontal body axes. ................................................................................................................ 15


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Final version WL2012R00_096_18rev2_0 IIIF-WL-PP10-1 Version 04RELEASED AS FROM: 12/11/2012

Abbreviations

NED North-East-Down axes system

IRF Impulse Response Function

CFD Computational Fluid Dynamics

ODE Ordinary differential equation

ST Strip theory

BVP Boundary value problem

Subscripts

M Manoeuvring

W Wave

R Rudder

P Propulsion

H Hull

Nomenclature

,

,

Roll, pitch and yaw Euler angles of rotation [

]

/ Euler-rotation matrix form the NED frame to the internal frame 1 []/ Euler-rotation matrix form the internal frame 1 to the internal frame 2 []/ Euler-Rotation matrix from the internal frame 2 to the body frame []/, Euler-rotation matrix from the body frame to the NED frame [] Non orthogonal matrix []

Inertial axes system [

]

Body fixed axes system [],, Unit vectors of the Inertial axes system [],, Unit vector of the body axes system [] Linear momentum [/] Angular momentum [2/] Angular momentum with respect to the origin [2/] Total external force applied at the bodys centre of gravity []


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Final version WL2012R00_096_18rev2_0 IVF-WL-PP10-1 Version 04RELEASED AS FROM: 12/11/2012

,, Total external force components ,,or ,, direction respectively []

Total moment of external force applied at the bodys centre of gravity [

]

Total moment of external force applied at the origin [],, Total external moments of force about ,,or ,, direction

respectively[]

Linear velocity of the centre of gravity [/] Linear velocity of the origin [/], , Linear velocities in ,, direction [/]

,

,

Linear acceleration in

,

,

direction [

/

]

Angular velocity [/],, Angular velocity components about ,,direction [/], , Angular velocity components about ,, direction [/] Mass of the ship [],, Position of the centre of gravity in ,,direction respectively []

,

,

Angular momentum components in

,

,

direction respectively [

2/]

, , Time derivative of ,,in ,, direction respectively [2/2] Tensor of inertia [2] Principal moment of inertia about [2], Product of inertia about ,, direction respectively [2] Principal moment of inertia about [2]

,

Product of inertia about

,

,

direction respectively [

2]

Principal moment of inertia about [2], Product of inertia about , , direction respectively [2] Frequency of encounter [/] Wave length []

Incident wave angle relative to the ship []


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Final version WL2012R00_096_18rev2_0 1F-WL-PP10-1 Version 04RELEASED AS FROM: 12/11/2012

1 Introduction

The description of rigid body dynamics applied to a ship is often separated into manoeuvring in calm waterand seakeeping. The division in these two sub-cases is reasonable since most of the manoeuvring takes

place in calm water environments while during voyages overseas usually a constant speed and a straightcourse are set.

However, even if manoeuvring is mostly associated to calm water, other environmental forces such aswaves, wind and tidal currents may still be present. Hence the ship can be subjected to additional forceswhich might be of considerable magnitude. The incorporation of such forces, however, requires a detailedanalysis. Wave effects are perhaps the most important since moderate values of this phenomenon canresult in large magnitude of forces and moments.

These wave forces are generally divided in the seakeeping analysis in three components: the diffractionforces, the radiation forces and the Froude-Krylov forces. The diffraction forces can be understood as adisturbance of the flow pattern by the presence of the body, this component is found by applying the no flowboundary condition over the normal vector on the hull surface. The radiation forces are resultant of theexcitation given by the body to disturb a flow, e.g. when a rock located at certain high is dropped over acalm water then part of its potential energy is converted into a generation of waves. Finally the last force

component, the Froude-Krylov force is the wave force resultant of the pressure integration over the hullsurface considering no disturbance of the initial flow pattern.

The incorporation of such a wave forces will bring additional issues commonly observed in seakeepinganalysis such as: propeller and rudder emergence and a variation of the wetted surface, which are a resultof the heave and pitch motions in waves. This however must be studied in advance to foresee theirapplication or not.

The manoeuvring in a seagoing environment implies a more complex problem and requires some insightinto the fluid phenomena acting upon the ship. The fluid effects involving the viscous and the potentialcontribution and, the nonlinear behaviour as a result of the rigid body motions, increases the complexity ofthe problem. The assumptions taken into account for the manoeuvring in calm water and seakeepinganalysis might not be applicable, at least not directly. Hence, in order to solve the ship dynamics whilemanoeuvring in a seaway, a method must be sought that integrates manoeuvring and seakeeping aspects

and includes the hydrodynamic effects corresponding to both.

The manoeuvring motion in a seaway has been a topic of discussion over the recent years and the different

approaches found in the literature have been classified according to the ITTC (The Manoeuvring

Committee., 2011) as:- experimental methods;

- methods based on two-time scale models;

- methods based on unified theory;

- methods using CFD.

However, the classification established above might lead to misunderstanding while dealing with unified

methods, such as the work of Skejic and Faltinsen (2008) which is a unified method based on two-time

scale, as will be explained in4 . The following classification is therefore proposed:- experimental method;

- method based on two-time scale models;

- methods based on a hybrid approach;

- methods using CFD.

Experimental methods are still the most reliable to investigate ship manoeuvring and course keeping in

waves. S. K. Lee et al. (2009) presented experimental results with a KVLCC model, manoeuvring in waves

for various wave-lengths and wave-amplitudes ratios. The results showed the dominant influence of the

second order wave forces on the trajectory for turning and zigzag manoeuvres.

Perhaps the most widely applied methods to deal with manoeuvring in a seaway are methods based on

two-time scale models or the hybrid approach. The two-time scale model separates the basic motion

equations into two groups, one is for the high frequency wave induced motion and the other is for the low

frequency manoeuvring motion.


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The hybrid approach, on the other hand, integrates the low frequency manoeuvring motion and the high

frequency wave induced motions into a generic set of equations to describe the manoeuvring in waves. To

achieve this, several reference frames are used, generally three. These methods will be discussed in

greater detail in Section4 and Section5 respectively.

CFD methods in principle provide an adequate description of all physics. But this approach is still highly

challenging from a computational point of view. Moreover, according to Sutulo and Guedes Soares (2006a)another problem is connected to difficulties in arranging an appropriate turbulence model which is especially

difficult in the highly challenging case of curvilinear motion of a surface displacement ship as the flow

around the ships hull is then rich in separations, re-attachments, vortex formation and substantial

interaction with the rudder and propeller.


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2 Rigid body dynamics

Two right handed Cartesian reference frames are generally used to describe the rigid body motion: an

inertial reference frame and a body fixed reference frame. A typical inertial reference frame, North-East-Down (NED), , with the unitary vectors, ,,, and a body fixed system, , with the unitaryvectors , , , ,are represented in Figure 1. Notice that the origin of the latter, ,does not correspond to thelocation of the centre of gravity,.

Figure 1 Manoeuvring reference frame, (a) NED reference frame and (b) body fixed reference frame,.The two reference frames are related using the Euler angles following the 321 rotation rule, first a yaw

rotation, followed by a pitch and finally a roll rotation, as indicated inFigure 2

Figure 2 Euler angles and yaw-pitch-roll rotation sequence

Each sequence leads to a matrix rotation relating the previous reference frame to the next fame. Thus the

resulting matrix for each transformation results in:

/ = (1)/ = (2)


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/ = (3)Where /2means the rotation from intermediate frame (2) to the body fixed frame (). Thus thecombination of the equations (1-3) results in the orthogonal matrix rotation from the body frame to the NED

frame as:

=/ = / / /= () (+ ) (+ ) () (4)

Thus, if any arbitrary vector

is written alternatively as:

=+ + = + + =

For the angular velocity, the following relation to the Euler angles is established:

=X+ Y+ Z = + + If we now consider the angular velocity in the body frame of reference () and employ the matrixtransformation presented above, then the angular velocity in the frame of reference expressed in theterms of the Euler angles is given by:

=0

0

+ /2 00

+ 2/1 00then the following relation between the angular velocity in the NED, and the Euler angles can bedrawn:

XYZ =

=

Where S is given by:

= 1 0 sin0 cos cos sin0 sin cos cos

Now considering the rigid body ship, as external forces are applied the linear momentum, , and theangular momentum, , will vary as a function of the external forces, this is given by:

= =

(5)


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= (6)As the origin differs from the centre of gravity, the velocity,, the angular momentum, , and themomentum of force,

, can be expressed with respect to the origin,

, by:

= + / => = + /Where /is the position of the centre of gravity respect to the origin O:

= / / = / => = + /

Considering the set of equations expressed above, the forces an moment of force can be rewritten in:

= = + / (7) = + / (8)

where:

=

=

= = + +

is the angular momentum, , is the inertia tensor and is the angular velocity expressed in the bodyfixed reference frame.

The resulting equation expressed above can be expressed into: surge, sway, heave, roll, pitch and yaw

components, as follows:

=[( +) + ( ) + (+ ) ( + )] (9) =[( + ) + ( ) + ( + ) ( + )] (10) =[(+ ) + ( ) + ( + ) ( + )] (11) = + +( ) +[() ()] (12)

= + +( ) +[() ()] (13) = + +( ) +[() ()] (14)

where X,Y, Z are the forces in surge, sway an heave respectively and the momentum of forces K,M,N are

roll, pitch and yaw, respectively. It is important to mention that the equations shown above are expressed in

the body rigid frame.


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So far the general description of the rigid body motion is given by equations (9-14), based only on the

assumption of the ship as a rigid body. The complex relations observed, such as high order expressions

and coupling between different degrees of freedom (DOF), makes their practical application almost

impossible even when the fluid, propeller and rudder forces have not been introduced yet.

In general, some level of simplification must be considered as in manoeuvring in calm water where asignificant simplification is achieved by considering only horizontal components and small motions, as in the

work of Bailey et al. (1998).


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3 Manoeuvring in Waves

3.1 General discussion

Manoeuvring in waves involves several nonlinearities associated with the rigid body motion, the propellerand rudder effects and with the complex fluid phenomena (fluid separation, vortex formation, viscous andpotential effects). When only the influence of the hull form is considered, neglecting the importantcontribution of the propeller and rudder effects, and the well established studies of seakeeping on one handand manoeuvring in calm water on the other, some insights into the fluid phenomena acting upon the shiphull can be addressed better.

For example, in the standard manoeuvring in calm water, the fluid forces on the hull are strongly influencedby the viscosity and weakly by the potential effects. This suggests interaction between certain of these fluidactions, such as the potential Munk moments still contributing to the total yawing moment measured duringcaptive model test (Sutulo and Guedes Soares, 2006b).

The description of the fluid forces for a ship manoeuvring in calm water is often based on the Taylor

expansion of the forces with coefficients known as slow motion derivatives (Another approach is also thefunctional representation of the slow motion derivatives proposed by Bishop et al. (1984)). These slowmotion derivatives are quantities determined from experiments and evaluated at zero frequency (Bailey etal., 1998). Although the slow motion derivatives are strong dependent of the fluid viscosity one must bear inmind that they are still influenced by the potential flow contribution .

In seakeeping, on the other hand, viscous effects can be neglected in order to simplify the problem. Severaltheories have been proposed in this field and perhaps the most cited work is the strip theory presented bySalvesen et al. (1970). This method assumes the ship as a slender body in order to reduce the three-dimensional (3D) problem to a 2D case.

The 2D representation of the ship is an approximation which gives good results for the vertical responseswhile for horizontal motions, viscous effects may become important and empirical corrections can then beintroduced. Another problem with strip theory is that some discrepancies may be found at low frequencymotions. Several methods have been proposed in the mean time, e.g. the panel method developed by C.

Lee and Newman (2004). An extended discussion can be found in Vasquez (2011) and Skejic (2008).

However, when the ship manoeuvres in a seaway, the influence of wave effects must be included into theanalysis in addition to the viscous forces of the manoeuvring problem. Those wave effects are associatedto:

- the diffraction,- the radiation,- the Froude-Krylov and- the 2

ndorder wave forces.

As the study now includes the wave-induced motion, the description of the rigid body motion in wavesimplies modelling the 6DOF in contrast to the restricted horizontal motions analysis. Moreover, the differentwave force components requires a discussion on their importance, e.g. the diffraction force being mostlyimportant for wave scenarios where the wave length is small in relation to the vessel length and the Froude

Krylov for wave lengths which are large in relation to the vessel length.

3.2 Additional consideration to manoeuvring in a seaway

The incorporation of the wave effects while manoeuvring in a seaway is a straight forward decision;

however, one of the key issues is to discuss which wave phenomena should be considered.

For example, the first order wave forces are important while predicting the motion responses linked to

seakeeping criteria and to study the no less significant phenomena related to: harmonic and parametric

resonance, pure loss of stability, surf ridding and broaching to (Hamamoto et al., 1995). These indeed are

important factors to be considered since large motions responses could impair the propulsive force and

steering control. The 2nd

order wave forces, on the other hand, are needed to include due to the steady

force drifting the ship from its original path.


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An additional issue related to the variation of the wetted surface is important to include in some scenarios,

for example if the motions experienced by the ship (seeFigure 3)from 0 to roll to 7.5 the change of the

wetted surface is gentle and an assumption of constant form can be considered approximate, however if the

variation is considered from 0 degrees to 15, an important change can be observed and the constant

wetted surface is no longer valid. Large amplitude motions for example were found for a set of fishing

vessel at moderate to severe sea sates in Tello et al. (2011).

The variation of the wetted surface thus leads to a change of the associated wave forces such as: the

diffraction, radiation and Froude Krylov forces. However, an often practical solution is to consider as a

constant wetted surface for the diffraction and the radiation forces and to restrict the varying wetted surface

to the hydrostatic and Froude-Krylov forces (there are methods to include the nonlinear effects into the

diffraction and radiation forces but they are expensive in terms of computational time). By considering that

the variation of the wetted surface affects only the hydrostatic and the Froude-Krylov components is a

practical and simpler method for implementation (Fonseca and Guedes Soares, 1998).

Nonlinear phenomena were considered by Vasquez et al. (2011) and Vasquez (2011) and were found to be

only important when the ship waterplane area changes considerable from a lower draft to a higher draft

which may occur when the ship is subjected to large amplitude motions or in the case of hulls with high

flare.

Figure 3 Variation of the wetted surface as function of the roll angle.

The 2nd

order wave forces, on the other hand, are the result of two or more wave systems (McCreight,

1991). The resulting wave profile induces a mean wave-drift force and, a slowly-varying and rapidly- varying

wave-drift forces. The last two terms act at higher and slower frequencies than the incident waves and are

usually neglected due to their small magnitude compared to the linear forces. However, the mean wave-drift

force cannot be omitted from the analysis since it induces a steady force on the ship, which influences

turning manoeuvres and induce a speed reduction for zigzag manoeuvres (Artyszuk, 2003; Skejic and

Faltinsen, 2008).

Two alternatives are available to estimate the mean wave-drift forces: the direct integration of the pressure

on the body surface and the method which depends on momentum conservation. The pressure-integration

method is more general, but it generally requires a more accurate solution to avoid local numerical errors.

On the other hand, the momentum method is restricted to the horizontal components of the force and the

vertical component of the moment (C. Lee and Newman, 2004).


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When dealing with manoeuvring in a seaway both methods seem applicable. Skejic and Faltinsen (2008)

and Skejic (2008) applied the pressure integration and the momentum conservation, while Seo and Kim

(2011) used only the former method. However, according to C. Lee and Newman(2004), it is highly advised

that in case of the applicability of both methods they should be compared in order to evaluate the

convergence.

Additional issues regarding the variation of the incident wave angle for the computation of the wave forcesare important from the seakeeping point of view. The responses and the wave exciting forces are frequency

dependant of the encounter frequency (), which changes continuously as the ship manoeuvres, e.g. in aturning cycle.

Attempts to deal with these additional issues are the consideration of a quasi-steady behaviour during a

step of the simulation so that the incident wave angle will be defined by the actual relative heading. This will

permit to estimate the encounter frequency. This approach has been employed by . Skejic and Faltinsen

(2008) and Skejic (2008) and Yasukawa and Nakayama (2009).


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4 Methods based on two-time scale models

This method deals with manoeuvring in waves by a separate, but coupled, solution of two sub problems;

one associated with the manoeuvring motion and one with the wave induced motions. This separation is

based on the assumption of a rapidly varying time scale of the linear wave induced motions and a slowlyvarying time scale associated with the manoeuvring motion (Seo and Kim, 2011; Skejic, 2008; Yasukawa

and Nakayama, 2009).

The independent analysis of each sub problem is solved while including the effects of the counterpart set,

e.g. the manoeuvring problem or module uses the hull linear wave forces in its analysis and gives the new

kinematic parameters,,, , and the current relative incident wave angle () for the next seakeepingcalculations.

Two slightly different approaches are found within this methodology, a sequential (seeFigure 4a) evaluation

where the seakeeping part is evaluated after the manoeuvring part and repeating this process until the

simulation time has been reached; and the parallel (see Figure 4b) method where the seakeeping is

evaluated several times while the manoeuvring runs only one step time. The latter has been employed by

Seo and Kim (2011) while the former by Skejic and Faltinsen (2008), Skejic (2008) and Yasukawa andNakayama (2009).

Figure 4 Description of the two-time scale method and coupled time matching and data exchange,

(a) sequential evaluation and (b) a parallel evaluation.

A general agreement while dealing with the manoeuvring and the seakeeping problems in this

methodology, as adopted by the works mentioned above, is to treat the independent problems as:

-

a

4, surge, sway, yaw and roll problem for the manoeuvring and- a 6for the seakeeping counterpart.In the manoeuvring model the following considerations are also included:

- the modular approach in the mathematical model for the manoeuvring problem (see Eq. 15) and

- the incorporation of the mean second order wave forces.

Thus the total external forces and moments for the 4manoeuvring model, are given by: = + + + = + + +

=

+

+

+

= + + +(15)


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where,,are the corresponding surge and sway forces and the ,are the roll and yaw moments offorce respectively. The subscripts,,,,, indicates the contribution of the hull, rudder, propeller and themean wave-drift forces/moments.

When dealing with the seakeeping analysis a quasi-steady behaviour for the time scale of the seakeeping is

considered, then a constant heading angle is assumed till the new global position is given by the

manoeuvring model.The major difference between referred works are related to the chosen approaches for the seakeeping

analysis. While Skejic and Faltinsen (2008) and Yasukawa and Nakayama (2009) employed frequency

domain analysis, Seo and Kim (2011), on the other hand, solved directly the seakeeping problem in time

domain.

According to Skejic (2008) the seakeeping analysis, based on the assumption of a quasi-steady behaviour

for a differential of heading in a small period of time, will present advantages in comparison to the Cummins

equation since the solution of the convolution integral requires each time step the numerical computation of

the IRF. This due to the dependence of the wave forces of the incident wave angle,, and the frequency ofencounter, . Thus, neglecting the memory effects will decrease the expensive computational timerequired when dealing with the memory effects.

Although the direct solution of the boundary value problem (BVP) used by Seo and Kim (2011) seems abetter approach, to the authors best knowledge, no report regarding the advantages of the direct solution of

the BVP is in respect to the method used by Skejic and Faltinsen (2008) and Yasukawa and Nakayama

(2009) has been published .

In addition to the differences already pointed out, other distinctions between the works within the category

of two-time scale models can be found:- the method used for the estimation of the slow motion derivatives,

- the method used to obtain the linear wave induced motion,

- the method used to calculate the mean second order wave forces,

- the time scale for the manoeuvring and the seakeeping counterpart, and

- the number of reference frames employed.

The manoeuvring mathematical model employed by Skejic and Faltinsen (2008) is based on the 3

(surge, sway and yaw) model of Sding (1982) with modifications to include roll motion and the mean

second order wave forces. In Yasukawa and Nakayama (2009), the description of the hydrodynamic forces

for the low frequency problem is based on the work of Hamamoto and Kim (1993). Seo and Kim (2011), on

the other hand, use the model given in Yasukawa (2006) with modifications as to how the actual fluid

inertial terms are estimated by their proposed time domain potential code.

When dealing with the linear wave induced motion, most of the methods use mathematical approaches

based on the assumption of the slender body theory. Perhaps, the Rankine time domain panel method used

in Seo and Kim (2011) might present improvements with respect to the frequency method used by Skejic

and Faltinsen (2008) and Yasukawa and Nakayama (2009). This may be an important subject of further

analysis.

The mean second order wave forces, wave-drift forces, can either be estimated by the direct integration of

the pressure on the body surface or by a method which uses momentum conservation. Skejic and Faltinsen(2008) and Seo and Kim (2011) and apparently Yasukawa and Nakayama (2009) employed the former

method.

Although they used the same method, the only difference found then is the BVP used to estimate the mean

wave-drift force. The last two authors used the panel method and the former one employed four different

approaches based on the strip theory (ST). They are respectively, the direct integration pressure method by

Faltinsen et al., (1980); Salvesen, (1974) method; Loukakis and Sclavounos, (1978) method and the short-

wavelength asymptotic theory by Faltinsen et al., (1980). The four different theories used by Skejic and

Faltinsen (2008) are included in order to cover the whole wave-length to ship-length ratios (due to the

limitation of the method base on ST) where the ship might operate.

As the analysis is carried out in two sub problems and by using the standard codes developed, the data

exchange process between the methods has to be transformed to the respective problem before included


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for the next simulation step. This standard process can be achieved by the application of the Euler angles

as shown in Section 2.

Skejic and Faltinsen (2008) for example used three upright reference frames: one to describe the wave

system, and the next two frames, the inertial and the body axes (seeFigure 5b),used for both the seakeeping and the manoeuvring problem. The last two frames are based on the works of

Bailey et al. (1998) and Bishop and Price (1981). The axes system

, is introduced in order to simplify

the description of the incident wave angle relative to ship,, given by: = +, where is the incidentwave angle relative to and is the heading angle.These two last frames, and , the inertial and the body frames respectively, have alsobeen used by Seo and Kim (2011), here the inertial frames and the fixed body frame are the andaxes respectively. In that work no incorporation of a third frame for the wave system (seeFigure 5c) isintroduced.

Yasukawa and Nakayama (2009), on the other hand, used three axes systems: horizontal body

axes,, general body axes, , and earth fixed axes, , (seeFigure 5b). These axes were firstproposed by Hamamoto and Kim (1993) aiming to find a general equation integrating standard well

developed methods for the analysis of seakeeping and manoeuvring.

Figure 5 Reference frames used in: (a) Skejic and Faltinsen (2008),

(b) Yasukawa and Nakayama (2009) and (c) Seo and Kim (2011).

Even the works might differ in method while dealing with the axes systems, it seems a general rule to

consider the centre of gravity as the origin for the body fixed coordinates, this is pointed out in the works of

Skejic and Faltinsen (2008) (b) and Yasukawa and Nakayama (2009) and since there are no terms related

to the position of the centre of gravity in the work of Seo and Kim (2011) this is also understood.


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When dealing with the wave forces in the two-time scale method, the incorporation of the linear wave forces

and the mean second order wave forces are straight forward. Additionally, a general agreement to deal with

a 4DOF manoeuvring problem, based on the modular approach, instead of a full 6DOF is found. This

indeed will reduce the complexity of the problem resulting in a more practical implementation.

However, the simplification to a 4DOF manoeuvring part will be less accurate than a full 6DOF. According to

Skejic (2008) and Skejic and Faltinsen (2008) the results can be accepted as a practical solution for thesimulation of a ship manoeuvring in waves. A more exact solution will increase the computational time

required for the analysis.

When the mathematical model for the manoeuvring set is in discussion, the employed methods in the works

described within this category of two-time scale method are different from one author to another. The

resultant estimations and comparison of their respective values has not been found by the authors therefore

the convenience or relevance of one method respect to another could be an important aspect for further

study. This extent also to the seakeeping method for the linear wave induced motion and the mathematical

methods for the mean second order wave forces.

To the authors, it seems that the parallel approach of Seo and Kim (2011) would require more hardware

than the sequential approach. This as the evaluation of the seakeeping part is carried out several times in

one manoeuvring step-time. This however should be also a matter of further study.


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5 Methods based on hybrid approach

This method deals directly with the full 6DOF ship rigid body dynamics aiming to fuse the fluid effects into a

general equation describing the ship rigid body motions. Several works falling into the classification ofmethods based on hybrid approach can be found in the literature, such as the works of: Bailey et al. (1998),

Letki and Hudson (2005), Ayaz et al. (2006) and Sutulo and Guedes Soares (2006a, 2006b, 2008).

In this method the manoeuvring forces upon the ship are handled in the conventional modular approach

incorporating independently the hull (resistance), the rudder and the propulsive force. Additionally, taking

into account the assumption of a possible separation between the low frequency motion (calm water) and

the higher frequency motion (motion in waves), the contribution of the wave forces is included following up

the modular approach. Then the total force upon the ship can be written as:

where,,,are the corresponding surge, sway and heave forces and the ,,are the roll, pitch andyaw moments of force respectively. Here again the subscripts,,,,, indicates the contribution of thehull, rudder, propeller and wave-induced forces/moments.

An alternative expression of the external forces is given Sutulo and Guedes Soares (2006a, 2006b, 2008),

In these studies, the subtraction of the potential forces for zero wave amplitude,

(0,

), is considered

and the resulting equation is then:

where (,)means the theoretical external forces while manoeuvring in waves, (0,)is the stillwater forces predicted by the theoretical model and () is the chosen experimental determinedmanoeuvring model.

According to Sutulo and Guedes Soares (2006a, 2006b, 2008) the subtraction of the zero wave amplitude

is made aiming to subtract the small contribution of the potential forces included into the manoeuvring

model () (mostly viscous and lift components). Thus, the expression given in Eq. 17 attempts toincorporate only the non-potential contribution measured in the experimental determined manoeuvring

model.

Although, the expression in Eq. 16 and Eq. 17 looks different, they include the same wave-induced forces

which are accounted for:- the Froude-Krylov forces/moments,

- the diffraction forces/moments,

- the radiation forces/moments and,

- the nonlinear Froude-Krylovforces/moments related to the variable wetted surface.

Additionally, the frequency domain analysis is expressed in the time domain by the Cummins equation

(Cummins, 1962). This includes the memory effects by the convolution integral equation based on the

Volterra functional forms. Then the solution requires the evaluation of the convolution integral and the

impulse response function (IRF) for every time step. This approach is found in: Bailey et al. (1998), Ayaz

and Vassalos (2003), Ayaz et al.(2006), Nishimura and Hirayama (2003) and S.-K. Lee (2000).

= + + +

=

+

+

+

= + + + = + + + = + + + = + + +

(16)

(,) =(,) (,) + () (17)


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However, with respect to the convolution integral, Sutulo and Guedes Soares (2006a, 2006b, 2008)

introduced simplifications while applying the inverse Fourier transform for the radiation problem. This

simplification is based upon the assumption that any complex added mass can be approximated by a

rational function employing auxiliary states variables and once this is introduce the Fourier Inverse

transform will derive in a simple representation of the radiation forces in the time domain by a set of ordinary

differential equations (ODEs) with constant coefficients. The discussion of this methodology can be found in

Sutulo and Guedes Soares (2006a, 2006b, 2008).

The use of the IRF in general is found to be time consuming as it has to be estimated for every incident

wave angle. For example, Ayaz et al. (2006) stored the values of the added mass and the damping

coefficients for every 10 heading angle between 0 to 360 and then interpolated for a particular wave

heading during simulation. Hence, the representation of the radiation problem in a form of ODEs seems of

practical application. This should in principle reduce the highly computational time required while evaluating

the time domain responses of the manoeuvring ship.

A particular mathematical model, which can also be considered to be using a hybrid approach, is the

approach of Bailey et al. (1998). This analysis is constrained to small amplitude motion and included only

the linear wave forces, which leads to a simplified mathematical expression for the manoeuvring and the

seakeeping equations. Hence, it is possible to build a fused generic equation to express the hydrodynamic

fluid forces based on the slow motion derivatives, oscillatory derivative, hydrodynamic coefficients and theIRF.

The simple representation of the hull-fluid forces based on the acceleration and velocity derivatives

(potential flow effects) will lead to an inaccurate prediction of the fluid effects. It has to incorporate the lift

effects, viscous cross flow and ship resistance (Eloot, 2006). Therefore, the manoeuvring increases its

complexity and it is impossible to find a fused relationship between the methods as in the work of Bailey et

al. (1998).

It is however, a common practice to employ well established manoeuvring models such as: the Japanese

manoeuvring model (MMG) together with the empirical formulation of Tasai (1961) as applied by Ayaz and

Vassalos (2003), Ayaz et al. (2006), Nishimura and Hirayama (2003); or the manoeuvring model of Inoue et

al. (1981) used by Sutulo and Guedes Soares ( 2006a, 2006b, 2008).

With respect to the reference frames, a general agreement between the authors is to use the proposedframes by Hamamoto and Kim (1993) (See Figure 6). These particular reference frames are defined in

order to incorporate the standard reference frames used in manoeuvrability, stability and seakeeping.

Figure 6 Reference frames proposed by Hamamoto and Kim (1993): (a) earth fixed axes,

(b) general body axes and (c) horizontal body axes.

The new set of reference frames incorporate three different frames: the first one an earth fixed system,

defined by 0 , the second defined by are general body axes which are fixed on the ship withthe origin G being located at the centre of gravity of the ship. The third frame uses horizontal body axes

fixed in the ship with the origin at G and defined by .The last system can rotate around vertical zaxis and can move vertically following to the movement of the centre of gravity , but the plane defined by

and

is kept as a horizontal surface.


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The origin of coordinates at the ships centre of gravity is chosen in order to simplify the resulting

mathematical expression of the application of the Euler angles (see section 2.) which lead to the

assumption of zero product of inertia. This is extensively explained in Ayaz et al. (2006).

Although the use of the three reference frames discussed above, is found in most of the works, a slight

variation is employed by Sutulo and Guedes Soares ( 2006a, 2006b, 2008). They do not consider the

centre of gravity as the origin of coordinates they chose an arbitrary position on the mean water level. Thesame arbitrary position was chosen by Bailey et al.(1998) however this time an upright coordinate system

was employed.

The discussion of the coordinate systems however is not more important as the wave effects included into

the analysis. Most of the works, beside the linear wave forces, given by the standard seakeeping tools,

include the estimation of the nonlinear wave forces due to the variation of the wetted surface. However the

importance of these force contributors might not be relevant as discussed in the previous section.

Additionally the incorporation of the convolution integral and its evaluation might be an important subject of

further analysis as well the radiation forces expressed in a set of ODEs and proposed by Sutulo and

Guedes Soares ( 2006a, 2006b, 2008).

An important fact to criticize about all the methods within this classification is that none of them incorporated

the second order wave effects. The omission of the steady fluid components will lead to a wrong prediction

for manoeuvring studies capabilities such as turning circles and zigzag manoeuvres (Araki et al., 2011; S.K. Lee et al., 2009; Skejic and Faltinsen, 2008).

If the incorporation of the second order components is sought, according to Skejic and Faltinsen (2008) this

will turns the problem more complex since the second order wave forces will imply a second order

expansion of the Volterra series, and as the IRF is dependant of the incident wave angle and the frequency

of encounter the estimation of each IRF for the first order and the second order wave forces will turning a

high task to accomplish.


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6 Final observations

The analysis of the ship dynamics while manoeuvring in waves is a challenging problem and some level of

simplification is ought to be accounted for in order to present a realistic and practical approach.The incorporation of the wave forces is a straightforward decision however, the remaining question to

answer is which components should be considered for, e.g. the first order wave forces are important while

predicting the emergence of the propeller and the rudder, which will degrade the steering control; or the

incorporation of the mean second order wave forces which subject the ship to a steady force component

being crucial for ship capabilities such as turning circles, zigzag and spiral manoeuvres.

Several phenomena are involved while the ship is manoeuvring in waves and the incorporation of them are

fundamental and must be evaluated for some scenarios, types of ships, and zones of operability. On the

other hand, identifying phenomena which are less or insignificant to the problem will indeed simplify the

mathematical model and its implementation.

The two methods, the two-time scale model and the hybrid approach, discussed above, although both

looking very different, are based on the general assumption of a possible separation of fluid effects into acontribution of the conventional manoeuvring in calm water and the wave-induce motions related to the low

frequency and high frequency problems, respectively. However, the evaluations of those systems are

completely different.

What is not a general consensus between the works discussed above, are the models implemented,

although they are based on the classical methods; in respect to seakeeping and manoeuvring. Apparently,

a determinant factor while choosing one or another methodology is based on the accessibility to the codes.

For example, Skejic and Faltinsen (2008) employed ST for the wave induced motion while Seo and Kim

(2011) used the Rankin time domain panel method. This leads to a straight forward question; which of the

methods are for better implementation in simulation tool for manoeuvring in waves?

Moreover, it is also important to mention that all the works found in the literature are developed for a ship

manoeuvring in waves in deep water and, to the authors best knowledge none mathematical model for a

shallow water environment have been developed yet. The introduction of the bottom constraint will indeed

introduce another excitation forces. These, however, as the other observation mentioned above, requires

an extend discussion and further study.


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7 Conclusion

The analysis of a manoeuvring ship in waves is found to be based on the assumption of the possible

separation of fluid effects related to the low frequency and high frequency motion. Within this considerationtwo possible methods are found to deal with manoeuvring in waves: (a) a two-time scale model and (b) a

hybrid mathematical theory.

Both approaches employ the modular approach for the mathematical model already implemented in the

standardized manoeuvring in calm water and the incorporation of wave effects. This later estimated by the

seakeeping codes available to date. On the other hand, the main differences are that the former method

solves the manoeuvring and the seakeeping independently but with data exchange while the latter method

fuses the manoeuvring equations with the seakeeping equations and solves one general set of equations.

When the incorporation of the wave forces is under discussion, the importance of the linear and mean

second order wave forces is found. Other effects such as the nonlinear contribution of the wetted surface

under ship motions as well as the inclusion of the convolution integral (if the Cummins equation is used)

needs further evaluation.

Based on this literature review, the following recommendations or suggestions for further research are

made:

- Investigate the suitability of the methods to calculate the second order mean drift forces,

such as: the far-field method based on the momentum-conservation theory, and the near-

field method by integrating the pressure on the body surface.

- Determine the relevance of the second order wave forces resulting from the non-linear ship

motions and its simplified approach based on the pressure integration of the Froude-Krylov

forces.

- Study of the effects of the ship under keel clearance on the manoeuvrability of the ship in

waves.

- Since the study is carried out for a simulation of a ship manoeuvring in waves, the evaluation

of the time response of the methods should also be accounted for.


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