Three Dimensional Ship Hydrodynamic Coefficients Using …cradpdf.drdc-rddc.gc.ca/PDFS/unc01/p517953.pdf · Three Dimensional Ship Hydrodynamic Coefficients Using the Zero Forward

Defence R&D Canada

Three Dimensional Ship Hydrodynamic

Coefficients Using the Zero Forward Speed

Green Function

Kevin McTaggart

Technical Memorandum

DRDC Atlantic TM 2002-059

April 2002

Copy No.________

Defence Research andDevelopment Canada

Recherche et développementpour la défense Canada

Three Dimensional Ship HydrodynamicCoefficients Using the Zero Forward SpeedGreen FunctionKevin McTaggart

Defence R&D Canada - Atlantic



April 2002

Abstract

This report presents a method for predicting ship hydrodynamic coefficients andpressures in waves. Computations are performed in the frequency domain usingthe Green function for zero forward speed. Pressures and coefficients at non-zeroforward speed are computed using extensions to zero speed results. The approachhas been implemented using the Python programming language. The present im-plementation gives robust results over a wide range of wave frequencies. A newalgorithm facilitates efficient computation of hydrodynamic terms for a large num-ber of combinations of ship speed, heading, and incident wave frequency. Samplecomputations for the HALIFAX class demonstrate application of the method.

Resum e

Dans ce rapport, nous presentons une methode de prediction des coefficients hydro-dynamiques et de la pression exercee sur un navire battu par la houle. Les calculssont effectues dans le domaine des frequences,a l’aide de la fonction de Greenpour un navire dont la vitesse avant est nulle. Les pressions et les coefficients pourles vitesses avant non nulles sont calculesa partir d’extensions jusqu’aux resultatsa l’arret. Nous avons programme cette methode avec le langage Python. Le pro-gramme actuel donne des resultats robustes pour une vaste gamme de frequencesde houle. Un nouvel algorithme facilite le calcul efficace des termes hydrodyna-miques pour un grand nombre de combinaisons de vitesses de navire, de caps etde frequences de la houle incidente. Pour illustrer l’utilisation de la methode, nouspresentons des calculs types pour des bateaux de la classe Halifax.

DRDC Atlantic TM 2002-059 i

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ii DRDC Atlantic TM 2002-059

Executive summary

Introduction

Strip theory is the most commonly used approach for computing ship motions andsea loads in waves. While computationally efficient, strip theory cannot give ac-curate hull pressure predictions, and has deteriorating accuracy as ship slendernessdecreases. This report presents a three-dimensional method for computing hydro-dynamic forces and pressures using the frequency domain Green function for zeroship speed. Forward speed effects are included using suitable approximations.

The present work is part of a larger effort to develop an object oriented softwarelibrary for modelling and simulation of ship motions and sea loads in waves. Thislibrary will aim to incorporate capabilities that are currently spread among a numberof existing codes, including SHIPMO7, PRECAL, and FREDYN.

Principal Results

The presented computational method gives robust and efficient computations ofthree-dimensional hydrodynamic coefficients. Using an 800 MHz Pentium III pro-cessor, sample radiation computations for the HALIFAX class require approxi-mately ten seconds per encounter frequency when using a mesh with 225 panelson half of the wet hull. A new algorithm facilitates efficient computation of hy-drodynamic terms for a large number of combinations of ship speed, heading, andincident wave frequency.

Significance of Results

The efficiency and robustness of the present three-dimensional approach indicatesthat it is a suitable replacement for strip theory. The three-dimensional methodwill likely give significantly improved accuracy for sea loads, which can be verysensitive to three-dimensional effects. The influence of three-dimensional effectsbecomes more important as ship slenderness decreases, suggesting that the presentwork will be particularly relevant to the KINGSTON class. In comparison to striptheory, the three-dimensional method also has the advantage of giving hull pres-sures.

Future Plans

The three-dimensional method will be incorporated into a new version of DRDC’sSHIPMO program. The resulting program will build upon SHIPMO’s unique strengths

DRDC Atlantic TM 2002-059 iii

in the areas of lift and viscous forces, which have been recently demonstrated bycomparisons with full-scale trials and predictions from other ship motion codes.

Kevin McTaggart; 2002; Three Dimensional Ship HydrodynamicCoefficients Using the Zero Forward Speed Green Function;DRDC Atlantic TM 2002-059; Defence R&D Canada - Atlantic.

iv DRDC Atlantic TM 2002-059

Sommaire

Introduction

La methode la plus couramment utilisee pour le calcul des mouvements des naviresdans la houle et les forces qu’elle exerce sur eux est la theorie des bandes. Bienqu’efficace d’un point de vue numerique, elle ne permet toutefois pas de predireprecisement les pressions exercees sur la coque. Qui plus est, plus la coque est fine,moins les resultats de la methode sont precis. Nous presentons dans ce rapport unemethode tridimensionnelle de calcul des forces et des pressions hydrodynamiquespour un navire ayant une vitesse nulle, qui utilise la fonction de Green dans ledomaine des frequences. Des approximations adequates permettent de deduire leseffets dusa la vitesse avant.

Ce travail fait partie d’un projet plus important visanta mettre au point une bi-bliotheque logicielle orientee-objet pour modeliser et simuler les mouvements desnavires causes par la houle et les charges qu’elle exerce sur eux. Nous prevoyonsinclure dans cette bibliotheque les capacites de calcul actuellement dispersees dansdiff erents logiciels, notamment SHIPMO7, PRECAL et FREDYN.

Resultats principaux

La methode que nous presentons permet d’effectuer des calculs robustes et efficacesdes coefficients hydrodynamiques dans les trois dimensions. Sur un processeur Pen-tium III cadence a 800 MHz, il faut environ dix secondes par frequence incidenteapparente pour effectuer des calculs pour une fregate de la classe Halifax simuleepar une demi-coque immergee et modelisee par un reseau de 225 panneaux. Unnouvel algorithme facilite le calcul efficace des termes hydrodynamiques pour ungrand nombre de combinaisons de vitesses de navire, de caps et de frequences dehoule apparente incidente.

Importance des r esultats

Efficace et robuste, la methode tridimensionnelle pourra remplacer convenablementla theorie des bandes. Nous croyons que la methode tridimensionnelle amelioreragrandement la precision des calculs des charges de mer, lesquelles dependent for-tement des effets tridimensionnels. Puisque l’impact de ces effets s’accroıt avec lafinesse de la coque des navires, le present travail pourraitetre tres pertinent pourles navires de la classe Kingston. En outre, la methode tridimensionnelle presentel’avantage de calculer les pressions sur la coque, ce que ne permet pas la theoriedes bandes.

DRDC Atlantic TM 2002-059 v

Plans pour l’avenir

On incorporera la methode tridimensionnellea la nouvelle version du logiciel SHIPMOde RDDC. Le programme resultant exploitera les capacites particulieres du pro-gramme SHIPMO au chapitre des forces portantes et visqueuses, lesquelles onteterevelees lors de comparaisons avec les resultats d’essais en grandeur reelle et lespredictions d’autres programmes simulant les mouvements des navires.

Kevin McTaggart; 2002; Calcul des coefficients hydrodynamiquesdes navires dans les trois dimensions, a partir de la fonction deGreen pour une vitesse avant nulle; DRDC Atlantic TM 2002-059;Defence R&D Canada - Atlantic.

vi DRDC Atlantic TM 2002-059

Table of contents

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Resume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

Executive summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii

Sommaire . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v

Table of contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii

List of tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . x

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

2 Hydrodynamic Forces on a Ship in Waves . . . . . . . . . . . . . . . 2

3 Solution of Velocity Potentials Using Source Distribution Method . . 5

3.1 Velocity Potentials for Zero Forward Speed . . . . . . . . . . 5

3.2 Three-Dimensional Green Function for Zero Forward Speedin Deep Water . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3.3 Radiation Velocity Potentials for Non-Zero Forward Speed . . 8

3.4 Incident and Diffracted Wave Potentials . . . . . . . . . . . . 9

3.5 Evaluation of Influence Matrix Terms from Panel Sources . . . 10

4 Panelling of Ship Hull for Seakeeping Calculations . . . . . . . . . . 11

5 Numerical Implementation for Solution of Velocity Potentials andHydrodynamic Forces . . . . . . . . . . . . . . . . . . . . . . . . . . 15

5.1 Influence Matrices for Hull with Lateral Symmetry . . . . . . 16

5.2 Evaluation of Frequency Dependent Portion of Green Function 17

5.3 Variation of Green Function from Panel Source . . . . . . . . 18

6 Efficient Computation of Hydrodynamic Terms for Seakeeping andSea Loads Computations . . . . . . . . . . . . . . . . . . . . . . . . 18

DRDC Atlantic TM 2002-059 vii

7 Sample Computations for the HALIFAX Class . . . . . . . . . . . . . 20

7.1 Hydrostatic Properties . . . . . . . . . . . . . . . . . . . . . . 20

7.2 Added Mass and Damping Coefficients at Zero Forward Speed 20

7.3 Motions in Waves . . . . . . . . . . . . . . . . . . . . . . . . 36

7.4 Computational Requirements . . . . . . . . . . . . . . . . . . 43

8 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

Symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

Document Control Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

viii DRDC Atlantic TM 2002-059

List of tables

1 Main Particulars for HALIFAX Class Frigate . . . . . . . . . . . . . 20

2 Hydrostatic Properties for Different Panel Meshes . . . . . . . . . . . 22

DRDC Atlantic TM 2002-059 ix

List of figures

1 Coordinate System for Solution of Hydrodynamic Forces . . . . . . . 3

2 Sea Direction for Solution of Hydrodynamic Forces . . . . . . . . . . 3

3 Relative Error in Integration of Potential Term∫

Ss1/R dS from a

Rectangular Source Panel to Own Centroid, Simpson’s Rule with 11Points per Side . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

4 Relative Error in Integration of∫

Ss1/R dS from 1× 1 Source Panel

to Normal Field Point . . . . . . . . . . . . . . . . . . . . . . . . . . 12

5 Relative Error in Integration of∫

Ss1/R dS from 1× 1 Source Panel

to Field Point in Panel Plane . . . . . . . . . . . . . . . . . . . . . . 12

6 Relative Error in Centroid Approximation for Integration of VelocityTerm

∫Ss

∂(1/R)/∂R dS from 1× 1 Source Panel to Field PointNormal to Panel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

7 Relative Error in Centroid Approximation for Integration of VelocityTerm

∫Ss

∂(1/R)/∂R dS from 1× 1 Source Panel to Field Point inPanel Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

8 Coarse Mesh for HALIFAX . . . . . . . . . . . . . . . . . . . . . . . 21

9 Medium Mesh for HALIFAX . . . . . . . . . . . . . . . . . . . . . . 21

10 Fine Mesh for HALIFAX . . . . . . . . . . . . . . . . . . . . . . . . 22

11 Surge Added Mass and Damping Coefficients for Coarse MeshHALIFAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

12 Sway Added Mass and Damping Coefficients for Coarse MeshHALIFAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

13 Heave Added Mass and Damping Coefficients for Coarse MeshHALIFAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

14 Roll Added Mass and Damping Coefficients for Coarse MeshHALIFAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

15 Pitch Added Mass and Damping Coefficients for Coarse MeshHALIFAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

x DRDC Atlantic TM 2002-059

16 Yaw Added Mass and Damping Coefficients for Coarse MeshHALIFAX . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

17 Surge Added Mass and Damping Coefficients for HALIFAX . . . . . 30

18 Sway Added Mass and Damping Coefficients for HALIFAX . . . . . 31

19 Heave Added Mass and Damping Coefficients for HALIFAX . . . . . 32

20 Roll Added Mass and Damping Coefficients for HALIFAX . . . . . . 33

21 Pitch Added Mass and Damping Coefficients for HALIFAX . . . . . 34

22 Yaw Added Mass and Damping Coefficients for HALIFAX . . . . . . 35

23 HALIFAX Surge Motion in Stern Quartering Seas, 18 knots, 30Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

24 HALIFAX Surge Motion in Bow Quartering Seas, 18 knots, 150Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

25 HALIFAX Sway Motion in Stern Quartering Seas, 18 knots, 30Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

26 HALIFAX Sway Motion in Bow Quartering Seas, 18 knots, 150Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

27 HALIFAX Heave Motion in Stern Quartering Seas, 18 knots, 30Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

28 HALIFAX Heave Motion in Bow Quartering Seas, 18 knots, 150Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

29 HALIFAX Roll Motion in Stern Quartering Seas, 18 knots, 30 Degrees 40

30 HALIFAX Roll Motion in Bow Quartering Seas, 18 knots, 150 Degrees 40

31 HALIFAX Pitch Motion in Stern Quartering Seas, 18 knots, 30Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

32 HALIFAX Pitch Motion in Bow Quartering Seas, 18 knots, 150Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

33 HALIFAX Yaw Motion in Stern Quartering Seas, 18 knots, 30 Degrees 42

DRDC Atlantic TM 2002-059 xi

34 HALIFAX Yaw Motion in Bow Quartering Seas, 18 knots, 150Degrees . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

xii DRDC Atlantic TM 2002-059

1 Introduction

When evaluating the motions and loads acting on a ship in waves, solution of theoscillatory flow in the vicinity of the hull is a key element. Strip theory, which uses2 dimensional approximations when solving the flow in the vicinity of a ship sec-tion, has been the most commonly used approach for solving motions and loads ofa ship in waves. DRDC’s SHIPMO7 [1] is an example of a strip theory code. Ad-vances in computational power have made three-dimensional approaches practicalfor solving the oscillatory flow around a ship in waves. The Cooperative ResearchShips code PRECAL [2] is an example of a three-dimensional ship motion code.

A recent comparison of measured motions with predictions for HMCS NIPIGON[3] indicates that SHIPMO7 gives very good motion predictions for a slender shipin moderately rough seas. In contrast, Version 4 of PRECAL gives good heaveand pitch predictions but poor roll predictions. The ability of SHIPMO7 to givegood lateral plane predictions is partially due to the work of Schmitke [4], who wasamong the first to consider the influence of lift forces on ship motion predictions.McTaggart [5] gives a more recent discussion of the influence of lift forces on shipmotions and sea loads.

The strong predictions by DRDC’s SHIPMO7 suggest it would be worthwhile toextend its capability to include three-dimensional hydrodynamic forces. This ca-pability will improve the accuracy of predictions for lowL/B vessels such asDND’s KINGSTON class. Sea load predictions are particularly sensitive to three-dimensional effects, and will benefit from the present work. Three-dimensionalhydrodynamic computations will also provide hydrodynamic pressures that can beused for structural analysis.

The present work is part of an effort to develop an object oriented library for shipmotions and sea loads in waves, replacing the present FORTRAN code for DRDC’sSHIPMO7. The move to object oriented software will contribute to code reuse,maintainability, and ease of integration to applications such as modelling and sim-ulation.

This report describes a method for computing three dimensional hydrodynamic co-efficients in the frequency domain using the Green function for zero forward speed.Section 2 describes the theory of hydrodynamic forces acting on a ship in waves.Section 3 describes the solution of hydrodynamic forces using a source distributionmethod. The solution procedure requires a panel representation of the ship hull,which is discussed in Section 4. Section 5 describes numerical implementation ofthe solution of the hydrodynamic forces. A new approach for efficient computationof hydrodynamic forces is presented in Section 6, followed by sample computationsfor the HALIFAX class in Section 7. Section 8 gives final conclusions.

DRDC Atlantic TM 2002-059 1

2 Hydrodynamic Forces on a Ship in Waves

The hydrodynamic forces acting on a ship in waves are solved using potential flowtheory. The present approach is similar to that used by many other investigators, in-cluding Beck and Loken [6], Papanikolaou and Schellin [7], and Hsiung and Huang[8].

The hydrodynamic forces are solved with respect to the ship centre of gravity; how-ever, the coordinate system for the present report has its origin in the still waterplanealigned vertically with the ship centre of gravity (see Figure 1). The coordinatesystem translates with the steady forward speed of the ship. The reason for usinga coordinate system not located at the CG is that many of the terms used for solv-ing hydrodynamic forces are dependent on elevation relative to the still waterplane;thus, it is most convenient to usez = 0 for the still waterplane. Figure 2 gives theconvention for sea direction.

The total oscillatory velocity potential in the vicinity of a floating body in waves isbroken into components as follows:

Φ(~x, t) = Real

{[φI(~x, β, ωI) + φD(~x, β, ωI) +

6∑j=1

ηjφj(~x, U, ωe)

]eiωet

}(1)

whereΦ is total velocity potential,~x is location,t is time, Real{. . .} denotes thereal part of the term in brackets,φI is incident wave velocity potential,β is relativesea direction,ωI is incident wave frequency,φD is diffracted wave velocity poten-tial, ηj is the complex motion amplitude for modej, φj is radiation potential formodej, andU is steady ship forward speed. The wave encounter frequencyωe isgiven by:

ωe = |ωI − kI U cos βs| (2)

Deep water conditions are assumed, giving the following relationship for incidentwavenumber:

kI =ω2

I

g(3)

Similarly, the wavenumber for radiated waves is given by:

ke =ω2

e

g(4)

The pressure at a point in the fluid is given by Bernoulli’s equation for unsteadyflow:

p = −ρ

(∂Φ

∂t+

1

2|∇Φ|2 + g z

)(5)

2 DRDC Atlantic TM 2002-059

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zη3

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...................η5....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

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yη2

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η1•

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η6

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y...............................................................................................................................

z

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Figure 1: Coordinate System for Solution of Hydrodynamic Forces

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.............................................................................................................................................................................................................................................................................................................................................................................................................................................................. ...............................Ship

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.Sea

•....................................................................

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......................................... β

Figure 2: Sea Direction for Solution of Hydrodynamic Forces


whereρ is water density. Neglecting second and higher order terms, the oscillatorypressure acting on the ship hull is given by:

p = −ρ

(i ωe − U

∂

∂x

) [φI + φD +

6∑j=1

ηjφj

]− ρgζ3 (6)

The last term of the above equation denotes the oscillatory pressure due to hydro-static pressure, withζ3 being the oscillatory vertical displacement given by:

ζ3 = η3 + yη4 − xη5 (7)

The oscillatory motions of the ship can be solved as follows:{−ω2

e ([M ] + [A]) + iωe[B] + [C]}{η} =

{F I + FD

}(8)

where[M ] is the ship mass matrix,[A] is the added mass matrix,[B] is the radiationdamping matrix,[C] is the hydrostatic stiffness matrix,

{F I

}is the incident wave

force vector, and{FD

}is the wave diffraction force vector. The added mass and

damping matrix terms are given by:

Ajk = − ρ

ωe

∫Sb

[Imag(φk) −

U

ωe

Real

(∂φk

∂x

)]nj dS (9)

Bjk = −ρ

∫Sb

[Real(φk) −

U

ωe

Imag

(∂φk

∂x

)]nj dS (10)

whereSb denotes the surface of the ship. The normals for each motion modek aregiven by:

n1 = nx (11)

n2 = ny (12)

n3 = nz (13)

n4 = ynz − (z − zCG)ny (14)

n5 = (z − zCG)nx − xnz (15)

n6 = xny − ynx (16)

wherenx, ny, andnz are the directional cosines for the unit normal pointing out-ward from the hull andzCG is the height of the ship centre of gravity above thewaterline. The wave excitation forces on the ship are given by:

F Ij = ρ ωe

∫Sb

(iφI −

U

ωe

∂φI

∂x

)nj dS (17)

FDj = ρ ωe

∫Sb

(iφD − U

ωe

∂φD

∂x

)nj dS (18)


3 Solution of Velocity Potentials UsingSource Distribution Method

A source distribution method is used to solve for the radiation and diffraction poten-tials presented in the previous section. The radiated or diffracted velocity potentialat a location in the fluid domain is expressed as follows:

φ(~x) =1

4π

∫Sb

G(~x, ~xs) σ(~xs) dS (19)

where~xs is source location on the surface of the ship,G(~x, ~xs) is the Green functiondescribing the flow at~x caused by a source of unit strength at~xs, andσ(~xs) is thestrength of the source at~xs. The Green function satisfies the continuity conditionand all boundary conditions, with the exception of the following normal velocityboundary condition on the hull surface:

∂φ(~x)

∂n= vn(~x) onSb (20)

wherevn(~x) is the flow normal velocity on the hull surface. The source strengthsare solved such that the following equation is satisfied:

− 1

2σ(~x) +

1

4π

∫Sb

∂G(~x, ~xs)

∂n~x

σ(~xs) dS = vn(~x) onSb (21)

For radiation potentials, the hull boundary condition is:

∂φk(~x)

∂n= i ωe nk onSb (22)

For the wave diffraction potentials, the hull boundary condition is:

∂φD(~x)

∂n= − ∂φI(~x)

∂nonSb (23)

3.1 Velocity Potentials for Zero Forward Speed

Solution of the three-dimensional radiation and diffraction potentials for zero for-ward speed is discussed in detail by Faltinsen and Michelsen [9], Hogben andStanding [10], and Garrison [11]. The ship hull is discretized intoNp panels, andthe source strengthσ(~xs) is taken as being constant over each panel. The normalvelocity boundary condition is satisfied at the centroid of each panel. Panel sourcestrengths for each radiation mode and for wave diffraction are solved by satisfyingthe following: {

∂φ

∂n

}= [D] {σ} (24)


where[D] is the normal velocity influence matrix with terms given by:

Djk = −δjk1

2+

1

4π

∫Sk

∂G(~xj, ~xs)

∂n~xj

dS (25)

whereδjk is the Kroenecker delta function andSk denotes the surface of panelkon the ship surface. Once the source strengths are solved, the potentials on the hullsurface are given by:

{φ} = [E] {σ} (26)

where[E] is the velocity potential influence matrix with terms given by:

Ejk =1

4π

∫Sk

G(~xj, ~xs) dS (27)

The flow velocities in thex direction, which are required for the forward speedcase, are given by: {

∂φ

∂x

}= [H] {σ} (28)

where[H] is thex velocity influence matrix with terms given by:

Hjk = −δjk1

2nx−j +

1

4π

∫Sk

∂G(~xj, ~xs)

∂x~xj

dS (29)

3.2 Three-Dimensional Green Function for ZeroForward Speed in Deep Water

When solving the radiation and diffraction potentials, evaluation of the Green func-tion is the most time consuming computational task. Many authors [9, 10, 11, 12,13, 14] have written about its solution. Telste and Noblesse [15] have developed anefficient method that is commonly used for evaluating the Green function. Usingthe sign conventions in the present report, the Telste and Noblesse formulation ofthe Green function is:

G(~x, ~xs, ke) =1

R+

1

R1

+ G0(~x, ~xs, ke) (30)

whereR is the distance from the field point~x to the source at~xs, R1 is the distancefrom the field point~x to the image of the source at~xs, andG0 is the frequencydependent term of the Green function. Garrison [11] discusses the Green function


at the zero and infinite frequency limits. The zero frequency (i.e.,ke = 0) limit isdenotedG0 and given by:

G0(~x, ~xs, ke) =1

R+

1

R1

for ke = 0 (ωe = 0) (31)

At infinite frequency (i.e.,ke = ∞), the Green function is denotedG∞ and givenby:

G∞(~x, ~xs, ke) =1

R− 1

R1

for ke = ∞ (ωe = ∞) (32)

Accordingly, the frequency dependent portionG0 has the following limits:

G0(~x, ~xs, ke) = 0 for ke = 0 (ωe = 0) (33)

G0(~x, ~xs, ke) = − 2

R1

for ke = ∞ (ωe = ∞) (34)

Based on Telste and Noblesse, the frequency dependent term is expressed as fol-lows:

G0(~x, ~xs, ke) = 2 ke

[R0(h, v)− i π J0(h) exp(v)

](35)

whereR0(h, v) is a function to be evaluated andJ0(h) is the Bessel function of thefirst kind of order zero. The function argumentsh andv are as follows:

Rxy =√

(x− xs)2 + (x− xs)2 (36)

h = ke Rxy (37)

v = ke (z + zs) (38)

Thus,Rxy is the horizontal distance from a field point to a source,h is the non-dimensional horizontal distance, andv is the non-dimensional elevation of the fieldpoint relative to the image of the source.

The derivatives of the Green function are required for computing flow velocitiesinduced by a source. The zero frequency portion of the Green function has thefollowing derivatives,

∂G0

∂x=

−(x− xs)

R3− (x− xs)

R31

(39)

∂G0

∂y=

−(y − ys)

R3− (y − ys)

R31

(40)

∂G0

∂z=

−(z − zs)

R3− (z + zs)

R31

(41)


Teltse and Noblesse give the following equations for the derivatives of the fre-quency dependent portion of the Green function

∂G0

∂Rxy

= −2k2[R1(h, v)− iπJ1(h) exp(v)

](42)

∂G0

∂x=

(x− xs)

Rxy

∂G

∂Rxy

(43)

∂G0

∂y=

(y − ys)

Rxy

∂G

∂Rxy

(44)

∂G0

∂z= 2k2

[1

ke R1

+ R0(h, v)− iπJ0(h) exp(v)

](45)

whereR1(h, v) is a function to be evaluated, andJ1(h) is the Bessel function of thefirst kind of order one.

The majority of Telste and Noblesse’s paper is devoted to the efficient solution ofthe termsR0(h, v) andR1(h, v), which will not be repeated here. The significanceof Telste and Noblesse’s work is demonstrated by its frequent citation in the litera-ture, including a recent paper by Chakrabarti [14] confirming the accuracy of theirapproach.

3.3 Radiation Velocity Potentials for Non-ZeroForward Speed

For computational efficiency, velocity potentials for non-zero forward speed aredetermined from velocity potentials for zero forward speed. Beck and Loken [6],Papanikolaou and Schellin [7] and Salvesen et al. [16] have also used this approach.Once the velocity potentials at zero forward speed have been solved, velocity po-tentials at non-zero forward speed are:

φ1(U, ωe) = φ1(0, ωe) (46)

φ2(U, ωe) = φ2(0, ωe) (47)

φ3(U, ωe) = φ3(0, ωe) (48)

φ4(U, ωe) = φ4(0, ωe) (49)

φ5(U, ωe) = φ5(0, ωe) +U

iωe

φ3(0, ωe) (50)

φ6(U, ωe) = φ6(0, ωe) −U

iωe

φ2(0, ωe) (51)

Thex derivatives of the forward speed potentials can be easily evaluated using thex derivatives of the zero speed potentials.


It should be emphasized that the speed correction terms given in this report are ap-proximations, and are based on the assumption ofU/ωe being small. When consid-ering the influence of ship speed using Equations (9), (10), (17), (18), (50) and (51),computations showed that motion predictions became unreliable for largeU/ωe. Inpractice, an upper limit should likely be imposed uponU/ωe when performing com-putations. Motion predictions presented in Section 7 suggest thatL/2 could be asuitable upper limit forU/ωe.

Many investigators have developed more sophisticated and accurate approaches forincorporating forward speed effects in the frequency domain. Huang and Hsiung[17] discuss the influence of the scattered steady flow potential induced by the pres-ence of the ship hull. Inglis and Price [18] present a frequency domain Green func-tion which incorporates forward speed effects and has been used extensively byothers. Unfortunately, computation of the forward speed Green function is verycomputationally intensive. Among the various algorithms for computing the for-ward speed Green function, Ba and Guilbard’s [19] is likely the most efficient andmost commonly used.

3.4 Incident and Diffracted Wave Potentials

The complex velocity potential of a regular wave system is as follows:

φI =i g a

ωI

exp [−i kI (x cos β − y sin β)] exp(kI z) (52)

wherea is the incident wave amplitude. The above equation is based on the con-vention of the wave crest being at thex − y origin at timet = 0. Fluid velocitiesare given by the following derivatives:

∂φI

∂x= a ωI cos β exp [−i kI (x cos β − y sin β)] exp(kI z) (53)

∂φI

∂y= a ωI sin β exp [−i kI (x cos β − y sin β)] exp(kI z) (54)

∂φI

∂z= i a ωI exp [−i kI (x cos β − y sin β)] exp(kI z) (55)

The incident wave elevation is given by:

ζI = a exp [−i kI (x cos β − y sin β)] (56)

Knowing the incident wave potential and its derivatives, the diffracted wave poten-tial is solved in a manner similar to that for the radiation potentials. It is assumedthat the only influence ship forward speed has on incident and diffracted wave po-tentials is due to encounter frequency; however, ship forward speed influences re-sulting pressures and forces due to the termU ∂φ/∂x in Equations (6), (17) and(18).


3.5 Evaluation of Influence Matrix Terms from PanelSources

Garrison [11] gives a detailed discussion of evaluation of influence matrix termsfor determing velocity potentials using panel sources. If a field point is in closeproximity to a source panel, then the variation of the Green function term1/R overthe source panel must be considered. The variation of the term1/R1 over the imagesource panel can also be significant if the field point is close to the image source;however, this situation is less common. Fortunately, the frequency dependent termG0 can usually be considered constant with location over the source panel.

Garrison gives equations for integrating1/R and its derivatives from a source panel.Integration of1/R from a source panel requires numerical integration over thelength of each side of the source panel. When evaluating velocity potential at thecentroid of a ship panel, the source potential from itself has a dominant effect dueto 1/R being large. For a quadrilateral source panel, the integral

∫Ss

1/R dS to thefield point at its own centroid has the following analytical solution:∫

Ss

1/R dS = 2

√As

b

{ln

[b +

√b2 + 1

]+ b ln

[1 +

√b2 + 1

b

]}(57)

whereb is the aspect ratio of the panel. Figure 3 shows relative errors in∫

Ss1/R dS

versus panel aspect ratio for Simpson’s rule using 11 points per side of the panel.The relative error is less than 0.001 for panel aspect ratios less than 4.4.

The velocity potential at a field point will be largely influenced by the∫

Ss1/R dS

terms from nearby panels. Figures 4 and 5 show relative errors in computed valuesof

∫Ss

1/R dS as a function ofR(~x, ~xsc), the distance from the field point location~x to ~xsc, the centroid of a square source panel. The assumed correct value forevaluating relative error is from numerical integration using Simpson’s rule with1001 points per side of the source panel. Figure 4 shows the case for field pointsnormal to the centroid of the source panel, while Figure 5 shows the case for fieldpoints planar to the source panel. Figures 4 and 5 indicate that Simpson’s ruleusing 11 points per side of the source panel gives excellent results. If the integralis approximated based on the value ofR(~x, ~xsc), then the field point must be adistance of at least3

√As from the centroid of a square source panel for the relative

error to be less than one percent.

The derivatives of1/R integrated from a source panel can be evaluated analyti-cally, with no need for numerical integration. Figures 6 and 7 show the relativeerrors when using the source centroid approximation for evaluating the derivativesof 1/R integrated from a source panel. The derivatives are important because theyrepresent the velocity (when divided by4π) induced by the1/R term of the source


0 1 2 3 4 5

Panel Aspect Ratiolx/ly

10−6

10−5

10−4

0.001

0.01

RelativeError

in∫Ss

1/R dS

...............................................................................................................

....................................................................................................................................................................................................................

.................................................

......................................................

............................................................

.......................................................................

...................................................................................

...................................................................

Figure 3: Relative Error in Integration of Potential Term∫

Ss1/R dS from a

Rectangular Source Panel to Own Centroid, Simpson’s Rule with 11 Points perSide

panel. The field point must be a distance of at least5√

As from the square sourcepanel for the relative error to be less than one percent. It should be noted that∂(1/R)/∂R is−1/R2; thus, its magnitude decreases rapidly asR increases.

4 Panelling of Ship Hull for SeakeepingCalculations

The solution of the three-dimensional velocity potentials for a ship hull requiresthat the hull geometry be modelled using panels. Hess and Smith [20] is one ofthe earliest and most influential works on application of panel methods. Garrison[11] gives a comprehensive overview on application of panel methods to offshorestructures in waves. At DRDC, Hally [21, 22, 23, 24] has done much work on panelmethods for solving the steady flow around a ship hull.

For the present work, new software was developed for panelling a ship hull usinginput offsets. It was decided that developing new software would be easier thanintegrating existing software from Hally. Seakeeping computations require coarserhull descriptions than steady flow computations; thus, development of panellingsoftware for the current application was relatively easy.


0 1 2 3 4 5

Distance from Source CentroidR(~x, ~xsc)

10−6

10−5

10−4

0.001

0.01

0.1

1

10

RelativeError

in∫Ss

1/R dS

. . . . . Centroid approx.∫Ss

1/R dS ≈ As × 1/R(~x, ~xsc).........................................................Simpson’s rule, 11 points per side..........................................................................................................................................................................................................................................................................................................

............................................................................................................................

Figure 4: Relative Error in Integration of∫

Ss1/R dS from 1 × 1 Source Panel to

Normal Field Point

0 1 2 3 4 5


10−6

10−5

10−4

0.001

0.01

0.1

1

10

RelativeError

in∫Ss

1/R dS

. . . . . Centroid approx.∫Ss

1/R dS ≈ As × 1/R(~x, ~xsc).........................................................Simpson’s rule, 11 points per side....................................................................

......................................................... ..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 5: Relative Error in Integration of∫

Ss1/R dS from 1 × 1 Source Panel to

Field Point in Panel Plane


0 1 2 3 4 5


0.01

0.1

1

10

RelativeError

in∫Ss

∂(1/R)/∂R dS

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 6: Relative Error in Centroid Approximation for Integration of Velocity Term∫Ss

∂(1/R)/∂R dS from 1 × 1 Source Panel to Field Point Normal to Panel

0 1 2 3 4 5


0.01

0.1

1

10

RelativeError

in∫Ss

∂(1/R)/∂R dS

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 7: Relative Error in Centroid Approximation for Integration of Velocity Term∫Ss

∂(1/R)/∂R dS from 1 × 1 Source Panel to Field Point in Panel Plane


Garrison gives useful guidelines for panelling of a body for computation of wave-induced forces. There should be a sufficient number of panels to adequately modelthe geometry of the body. Adjacent panels should be of similar size so that thesource potential from a larger panel does not unduly influence adjacent smallerpanels. Panel geometries should not be too elongated, with the upper limit on aspectratio being approximately 5. In practice, the geometry of a ship hull can often beadequately described with 200 panels on each side of the hull.

Coordinates of panel vertices for a faired hull are easily generated from input shipoffsets. If offsets are given for two ship sections, then longitudinal panels are gen-erated girthwise from the keel to the waterline. The desired panel area and limit onaspect ratio will determine the number of panels in the longitudinal and girthwisedirections. When generating panels between two sections, triangular panels can beused if the girth of one of the sections is zero (e.g., at the fore perpendicular).

Hess and Smith [20] describe evaluation of panel normals and centroids. For atriangular panel, the normal components are evaluated by taking the cross productof two of its edges and then applying a normalization factor. The centroid coor-dinates are merely the average of the coordinates of vertices. For a quadrilateralpanel computed using four input vertices, the normal components are evaluated bytaking the cross product of the two diagonals, thus yielding the rotational orienta-tion of the panel plane. The location of the panel plane is fixed by specifying that itpasses through the mean of the four vertex coordinates. Vertex coordinates are thenadjusted to lie on the computed panel plane.

Hull hydrostatic properties can be computed from panel properties. For consistencybetween pressure and motion computations, the hydrostatic stiffness terms for apanelled hull should be computed using the panel properties. The volume of apanelled hull is determined by the following discretized equation:

5 =

Np∑j=1

Aj nz−j zj (58)

whereNp is the total number of panels,Aj is the area of panelj, nz−j is theznormal component of panelj, andzj is thez value (relative to the waterline) of thecentroid of panelj. The location of the centre of buoyancy relative to the waterlineis:

zCB 5 =

Np∑j=1

Aj nz−j1

2z2

j (59)

The longitudinal centre of buoyancy, which corresponds with the longitudinal cen-


tre of gravity, is given by:

xCB 5 =

Np∑j=1

Aj nz−j zj xj (60)

For ship motion computations,xCB is typically computed and thenx coordinatesare adjusted such thatxCB = 0. The hull waterplane area, which is used for com-puting heave stiffness, is given by:

Awp = −Np∑j=1

Aj nz−j (61)

The longitudinal centroid of floatation is given by:

xwp Awp = −Np∑j=1

Aj nz−j xj (62)

Waterplane moment terms are:

Iwp−xx = −Np∑j=1

Aj nz−j x2j (63)

Iwp−yy = −Np∑j=1

Aj nz−j y2j (64)

Using the above computed values, hydrostatic stiffness terms for motion equationsare:

C33 = ρ g Awp (65)

C35 = −ρ g Awp xwp (66)

C44 = ρ g [5 zCB − 5 zCG + Iwp−yy] (67)

C53 = C35 (68)

C55 = ρ g [5 zCB − 5 zCG + Iwp−xx] (69)

5 Numerical Implementation for Solution ofVelocity Potentials and HydrodynamicForces

The solution of velocity potentials and hydrodynamic forces has been implementedusing the Python programming language [25, 26] and the Numeric Python exten-sion library [27]. Python is a high level object oriented programming language. Its


syntax and capabilities are similar to C++ and Java; however, its high level natureleads to source code that is typically shorter and easier to read than C++ and Java.

5.1 Influence Matrices for Hull with LateralSymmetry

It is assumed that the ship hull has lateral symmetry. Radiation potentials are eval-uated only on the port side of the hull, while diffraction potentials are evaluated onboth sides of the hull. Hull potential influence matrices are partitioned, leading tothe following equations:{

∂φport/∂n

∂φstar/∂n

}=

[Dport,port Dport,star

Dstar,port Dstar,star

] {σport

σstar

}(70){

φport

φstar

}=

[Eport,port Eport,star

Estar,port Estar,star

] {σport

σstar

}(71){

∂φport/∂x

∂φstar/∂x

}=

[Hport,port Hport,star

Hstar,port Hstar,star

] {σport

σstar

}(72)

where the superscriptport denotes the port side of the hull and the superscriptstardenotes the starboard side of the hull. Due to symmetry, the following relationsexist: [

Dport,port]

=[Dstar,star

](73)[

Dport,star]

=[Dstar,port

](74)[

Eport,port]

=[Estar,star

](75)[

Eport,star]

=[Estar,port

](76)[

Hport,port]

=[Hstar,star

](77)[

Hport,star]

=[Hstar,port

](78){

σportj

}=

{σstar

j

}for j = 1, 3, 5 (79){

σportj

}=

{−σstar

j

}for j = 2, 4, 6 (80){

φportj

}=

{φstar

j

}for j = 1, 3, 5 (81){

φportj

}=

{−φstar

j

}for j = 2, 4, 6 (82)

where indexj denotes the motion mode in the above equations. In the Pythoncode, source strengths are determined by solving the following using the functionLinearAlgebra.solvelinear equations from the Numeric Python library:[

Dport,port + Dport,star] {

σportj

}=

{∂φport

j /∂n}

for j = 1, 3, 5 (83)


[Dport,port − Dport,star

] {σport

j

}=

{∂φport

j /∂n}

for j = 2, 4, 6 (84)[Dport,port Dport,star

Dport,star Dport,port

] {σport

D

σstarD

}=

{− ∂φport

I /∂n

∂φstarI /∂n

}(85)

Once the source strengths are solved, the velocity potentials are computed by:{φport

j

}=

[Eport,port + Eport,star

] {σport

j

}for j = 1, 3, 5 (86){

φportj

}=

[Eport,port − Eport,star

] {σport

j

}for j = 2, 4, 6 (87){

φportD

φstarD

}=

[Eport,port Eport,star

Eport,star Eport,port

] {σport

D

σstarD

}(88)

Thex derivatives are computed in a similar manner.

A Python class HullRadDif has been created for solution of radiation and diffrac-tion potentials of a symmetric ship hull. Upon class initialization, all zero andinfinite frequency influence matrices are computed. Class methods are available forevaluating radiation potentials and hydrodynamic forces at specified encounter fre-quencies and ship speeds. Similarly, methods are available for evaluating diffrac-tion potentials and incident wave forces at specified combinations of ship speed,relative wave heading, and wave frequency.

5.2 Evaluation of Frequency Dependent Portion ofGreen Function

When computing hull hydrodynamic forces in the frequency domain, most compu-tational time is used for computing the frequency dependent portion of the Greenfunction. Initial Python code for its evaluation was based on equations presentedby Telste and Noblesse [15]. Subsequent code profiling revealed that the ascendingseries and Haskind algorithms presented in the equations were very slow. Theseformulations were re-written based on Fortran source code presented by Telste andNoblesse, which uses rational approximations that are not presented in the equa-tions of their paper.

Python is an interpreted language, and is thus typically slower than compiled lan-guages such as C++ and Fortran. Many Python library components, including muchof Numeric Python, are written in C or C++ for improved performance. Lutz [25]gives details regarding calling C functions from Python. The frequency dependentversion of the Green function was re-written in C and compiled using the C++Builder 5 compiler. A basic version of this compiler is available from the Borlandweb site (http://www.borland.com/bcppbuilder/freecompiler/) at no cost.


5.3 Variation of Green Function from Panel Source

When evaluating the Green function and its derivatives from a panel source, thevariation of terms1/R and1/R1 with location over the panel source is considered.The frequency dependent portion of the Green functionG0 is assumed to be con-stant with location over the panel source. As mentioned previously,G0 approaches0 at low frequencies and approaches−2/R1 at high frequencies. To minimize theinfluence of the variation ofG0 with location on a source panel, the method forevaluating the integral of the Green function from a source panel depends on theencounter frequency:∫

Ss

G(~x, ~xs, ke) dS ≈∫

Ss

1

RdS +

∫Ss

1

R1

dS

+ As G0(~x, ~xsc, ke) for ωe ≤ ωte (89)∫

Ss

G(~x, ~xs, ke) dS ≈∫

Ss

1

RdS −

∫Ss

1

R1

dS

+ As

[2

R1(~x, ~xsc)+ G0(~x, ~xsc, ke)

]for ωe > ωt

e (90)

where~xsc is the location of the centroid of the source panel, andωte is a specified

transition frequency that determines which of the above two equations is used. Thederivatives of the Green function integrals are evaluated in a similar manner. Selec-tion of a suitable transition frequencyωt

e can be done empirically and will dependmainly on the size of the ship. The next section of this report shows that a transitionfrequency of 2.0 rad/s gives good results for the Canadian frigate HALIFAX.

When evaluating the1/R integral terms, Simpson’s rule with 11 points per side isused when the field point is close to the source. By default, the centroid approxi-mation is used whenR > 5

√As; however, this value can be changed. Similarly,

the centroid approximation is used for1/R1 integral terms whenR1 > 5√

As.

6 Efficient Computation of HydrodynamicTerms for Seakeeping and Sea LoadsComputations

Computation of three-dimensional ship hydrodynamic terms can be computation-ally intensive due to the large number of times that the Green function must beevaluated. The number of Green function evaluations for a symmetric hull will be:

NG = 2 Nportp Ne (91)


whereNportp is the number of panels on the port side of the hull andNe is the num-

ber of encounter frequencies. A common approach is to evaluate the hull Greenfunctions for the encounter frequency arising from each combination of ship speed,heading, and wave frequency. This approach can lead to a large number of en-counter frequencies. An alternative approach would be to store hull Green functionsfor a range of encounter frequencies (e.g., 0.1, 0.2,. . ., 6.0 rad/s) and to use inter-polation between stored Green functions; however, this approach would require alarge amount of memory.

A new approach has been developed for efficient evaluation of radiation and diffrac-tion potentials. The user provides an input range of encounter frequencies for ra-diation calculations. The user also provides ship speeds, headings, and wave fre-quencies for diffraction computations. Green functions and radiation potentials aresolved for the lowest two encounter frequencies. Each combination of ship speed,heading, and wave frequency is then checked to see if the associated encounterfrequency lies between the two encounter frequencies for which Green functionsare currently stored, and linear interpolation of Green functions is used where ap-propriate. This process is repeated for all input encounter frequencies. Computedradiation potentials can be stored for all input encounter frequencies and computeddiffraction potentials can be stored for all input combinations of ship speed, head-ing, and wave frequency. The accuracy of the interpolation procedure will dependon the intervals between input encounter frequencies. The input intervals should besufficiently small to describe variations in added mass and damping for all modesof motion.

For encounter frequencies below and above the input encounter frequency range,the zero and infinite frequency Green functions (Equations (31) and (32)) are usedfor solving velocity potentials. Normalized velocity potentialsφ/(iωe) are evalu-ated and stored at the zero and infinite frequency limits in a manner similar to thatused for 2D potentials in Reference 28.

The present computational procedure has been implemented into a Python classHullRadDiffDB, which is considered to be a database extension of the class Hull-RadDiff described earlier. Class HullRadDiffDB can take between several minutesand several hours to initialize, depending primarily of the number of seaways (i.e.,number of combinations of ship speed, heading, and wave frequency) for diffractioncomputations. Once initialized, class HullRadDiffDB can quickly provide requiredpotentials using interpolation of stored values.


7 Sample Computations for the HALIFAXClass

To verify implementation of the three-dimensional method, sample computationshave been performed using the HALIFAX class frigate. Table 1 gives the shipconfiguration for the present computations, which is the same as for the samplecase in the SHIPMO7 manual [1].

Table 1: Main Particulars for HALIFAX Class Frigate

Length,L 124.5 m

Beam,B 14.8 m

Midships draft,Tmid 5.025 m

Trim by stern,ts -0.135 m

Displacement,4 4732 tonnes

Vertical centre of gravity,KG 6.23 m

Dry roll radius of gyrationrxx 4.506 m

Dry pitch radius of gyrationryy 29.815 m

Dry yaw radius of gyrationrzz 29.815 m

7.1 Hydrostatic Properties

Hull panels were generated from input offsets by specifying ship draft, trim, nomi-nal panel area, and limiting panel aspect ratio. Three different panelled hulls weregenerated, with nominal panel areas of 5.0, 2.5, and 1.0 m2. A maximum panelaspect ratio of 4.0 was specified for each hull. Figures 8 to 10 show the resultingmeshes. The colour of each plotted panel is a function of its nominal elevation. Ta-ble 2 shows that hydrostatic properties have excellent agreement among the threedifferent meshes, with the greatest relative difference being 0.7 percent for volumesbetween the coarse and fine meshes.

7.2 Added Mass and Damping Coefficients at ZeroForward Speed

Added mass and damping coefficients have been computed for HALIFAX at zeroforward speed using the 3 different panel meshes. When computing frequency inde-pendent terms of influence matrices, the user supplies input parameters indicating


Figure 8: Coarse Mesh for HALIFAX

Figure 9: Medium Mesh for HALIFAX


Figure 10: Fine Mesh for HALIFAX

Table 2: Hydrostatic Properties for Different Panel Meshes

Coarse Medium Fine

Nominal panel size (m2) 5.0 2.5 1.0

Number of panels on port side 225 428 1041

Volume (m3) 4537.8 4562.1 4571.9

LCB aft of FP (m) 64.542 64.562 64.563

CB wrt waterline (m) -1.906 -1.912 -1.916

Wetted surface area (m2) 1980.3 1984.4 1985.9


nondimensional threshold distances specifying when the centroid approximationcan be used for integrating potential from a source. For the present computations,the centroid approximation is used whenR2(~x, ~xsc) > 25As when considering the1/R term from a source. Similarly, the centroid approximation is used for the1/R1

integral from a source image whenR21(~x, ~xsc) > 25As.

Figures 11 to 16 show computed added mass and damping coefficients for HAL-IFAX using a coarse mesh with 225 panels on the port side of the ship. The co-efficients are nondimensionalized by the dry inertial force for each mode. Resultsare presented for transition encounter frequencies of zero and infinity, meaningthat all Green function integrals for each line are based on either Equation (89) orEquation (90). Forωt

e = ∞, the computations will be more accurate in the lowerfrequency range. Conversely, the computations will be more accurate in the higherfrequency range forωt

e = 0. The results indicate definite differences in results atlow and high frequency ranges. The most noticeable difference is that irregular fre-quency effects are less pronounced when using Equation (90) for evaluating Greenfunction integrals. This result is expected because irregular frequency effects occurat higher encounter frequencies. The two different evaluation methods give verysimilar results at a frequency of 2.0 rad/s, indicating that this is a suitable value forωt

e when performing computations for HALIFAX.

Figures 17 to 22 show computed hydrodynamic coefficients for HALIFAX using3 different mesh sizes. The transition frequencyωt

e has a value of 2.0 rad/s forcomputations with all 3 meshes. For wave frequencies up to 1.8 rad/s, the coeffi-cients show little sensitivity to mesh size. At higher frequencies, the coefficientsshow greater sensitivity to mesh size. This phenomenon is most likely due to thepresence of irregular frequencies. Ando’s method [29] suggests that the irregularfrequencies for HALIFAX will have a lower bound of 1.6 rad/s. Results for thefinest mesh size exhibit fewer irregularities, suggesting greater accuracy.

Hydrodynamic coefficients for HALIFAX approach their infinite frequency valuesin the upper range of presented frequencies. In this upper range, added mass varieslittle with encounter frequency and damping is effectively zero. It appears thatthe infinite frequency approximation can be used for frequencies above 6 rad/s forHALIFAX. In the lower frequency range, the zero frequency approximation issuitable for frequencies below 0.1 rad/s.

In comparison with the two dimensional coefficients results presented by McTag-gart [28], the most noticeable difference is that the present heave added mass coef-ficients approach finite values at lower frequencies. This difference could signifi-cantly influence computed motions and sea loads at lower encounter frequencies.


.........................................................ωte = ∞ (more accurate at lower frequencies)

............. ............. ....... ωte = 0 (more accurate at higher frequencies)

0 2 4 6

Encounter Frequencyωe (rad/s)

0.00

0.01

0.02

0.03

SurgeAddedMass

A11/4

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................

...........................................................................................................................................

................................................................................................................

........................................................................................................................................................................................................................

............................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ....................

...... .................................................... ............. ............. ............. ............. .......................

................ ....................................... ............. ............. ............. ............. .............

0 2 4 6


0.00

0.01

0.02

0.03

SurgeDamping

B11/(4ωe)

.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................

........................................................................................................................... .............................................................................. ............. .......................... ............. ............. ............. ............

.............. .......................... ............. ............. ............. ............. ............. .............................................................................. ............. ............. ............. ............. ............. .............

Figure 11: Surge Added Mass and Damping Coefficients for Coarse MeshHALIFAX




0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

SwayAddedMass

A22/4

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................

..............................................................................................................................................................

...........................................................................

.......................................................................................................................................................................................................................

............. ....................................................................................................................................................................................................................................................... ............. .......................................................

....................... .......................... .

............ ............. ............. ............. ............. ............. .................

...................... ............. ............. ............. ............. ............. ............. ...

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

SwayDamping

B22/(4ωe)

...........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..................................................................................................

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................... .............

............................................................................................................................................... ..................................................................................................................... ....................................... .................................................... ............. .......................... ............. ............. ............. .............

............. ............. ............. ............. ............. ............. ............. .......

Figure 12: Sway Added Mass and Damping Coefficients for Coarse MeshHALIFAX




0 2 4 6


0

2

4

HeaveAddedMass

A33/4

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................

...........................................................................

................................................................................................................................................................................................................ ............. ............. ............. ............. ..

........... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

0 2 4 6


0

2

4

HeaveDamping

B33/(4ωe)

......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............................

........... ............. ............. ............. ............. ............. ............. ........

Figure 13: Heave Added Mass and Damping Coefficients for Coarse MeshHALIFAX




0 2 4 6


0.00

0.05

0.10

0.15

0.20

RollAddedMass

A44/(4r2xx)

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................

..........................................................................................................................................................

.......................... ..........................

............. ........................................................................................................ ............. ............. ............. ............. ............. ............. ............. ............. ............. .......................... ............. ............. ..........................

............. ............. ............. ............. ............. ............. ............. .............

0 2 4 6


0.00

0.05

0.10

0.15

0.20

RollDamping

B44/(4r2xxωe)

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................ ............. ........................................................................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. ...........

Figure 14: Roll Added Mass and Damping Coefficients for Coarse Mesh HALIFAX




0 2 4 6


0.0

1.0

2.0

3.0

PitchAddedMass

A55/(4r2yy)

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................

.............................................................................................................................................................................................

..........................................................................................................................................................................

...........................................................................

........................................................................................................................

....................................................................................................................................................................................

............................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ............. ............. ..........................................

.......... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .........

.... ............. ............. ............. .

0 2 4 6


0.0

1.0

2.0

3.0

PitchDamping

B55/(4r2yyωe)

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................

............................................................................................................................................

............................................................................................................................................................................................................................................................................................ ............. ............. ............. .......................

................ ....................................... ............. ............. ............. ............. ........................................................................................... .......................... ............. ..............

............ ............. ............. ...

Figure 15: Pitch Added Mass and Damping Coefficients for Coarse Mesh HALIFAX




0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

YawAddedMass

A66/(4r2zz)

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................................................................................................................................................

.................................................................................................................................................................................................................................................

............. ............. ................................................................................................................................................................................................................ ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

............. ............. ............. ............. ............. ............. ............. .............

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

YawDamping

B66/(4r2zzωe)

....................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................. .............

..................................................................................................................... ........................................................................................... ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. .......

................... ............. ............. ............. ............. ............. .............

Figure 16: Yaw Added Mass and Damping Coefficients for Coarse Mesh HALIFAX


. . . . . Coarse mesh, 225 port panels

............. ............. ....... Medium mesh, 428 port panels

.........................................................Fine mesh, 1041 port panels

0 2 4 6


0.00

0.01

0.02

0.03

SurgeAddedMass

A11/4

..........................................................

............................

............................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ..........................

............. ................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........................

............. ..........

.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..............................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................

0 2 4 6


0.00

0.01

0.02

0.03

SurgeDamping

B11/(4ωe)

......................................................................

......................................

...................................................................................................................................................................................................................... ............. ............. ............. ............. ..........................

.......................... ............. .......................... .......................... ............. ............. ............. ............. .......................... ............. ............. ............. .......................... ...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 17: Surge Added Mass and Damping Coefficients for HALIFAX





0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

SwayAddedMass

A22/4

....................................................

.................

.....................

............. ....................................................................................................................................................................................................................................................... ............. ....................................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........................................................................................................................................................................................................................................................................................

.............................................................

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

SwayDamping

B22/(4ωe)

..........................................................................

........................... ............................................................................................................................................................ ............................................................................................................................................................ ............. .............

....................................... ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. .......................... ............. ...........................................................

..................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................................................................................................................................

.............................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 18: Sway Added Mass and Damping Coefficients for HALIFAX





0 2 4 6


0

2

4

HeaveAddedMass

A33/4

.................................................

................................

............................................................................................................................................................................................................................. ............. ............. ............. .............

............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

..........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.............................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................................................................................

0 2 4 6


0

2

4

HeaveDamping

B33/(4ωe)

....................................................................................

................................................................................................................................................................................................................................................................................. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .............

....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 19: Heave Added Mass and Damping Coefficients for HALIFAX





0 2 4 6


0.00

0.05

0.10

0.15

0.20

RollAddedMass

A44/(4r2xx)

........

................................................

................

............. ............. ..........................

..................................................................................................................... ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ....................................... .............

.......................... ............. ............. ............. ..........

........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.....................................................................................

0 2 4 6


0.00

0.05

0.10

0.15

0.20

RollDamping

B44/(4r2xxωe)

..............................................................

................... ............. ........................................................................................... ............. .................................................... ............. ............. .......................... ............. ............. ............. ............. ............. ............. ............. .......................... ..........................

............. .......................... ............. ............. ............. .............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 20: Roll Added Mass and Damping Coefficients for HALIFAX





0 2 4 6


0.0

1.0

2.0

3.0

PitchAddedMass

A55/(4r2yy)

........................................................

....................

.........

............................................................................................................................................................................................................................. ............. ............. ............. ............. ............. .............

............. ...........................................................

...... .......................... .......................... ............. ............. .......................... .......................... ............. ............. ............. ............. .............

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................

0 2 4 6


0.0

1.0

2.0

3.0

PitchDamping

B55/(4r2yyωe)

..........................................................................

.............

...................................................................................................................................................... ........................................................................................................ ............. ............. ............. ..............................

................................... ....................................... .......

............................................. ............. ............. ............. .......................... .......................... .......................... ............. .......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 21: Pitch Added Mass and Damping Coefficients for HALIFAX





0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

YawAddedMass

A66/(4r2zz)

...........................................................

....................

............. ............. ............................................................................................................................................................................................................................. ............. ............. ............. ............. ............. .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ..........

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

................................................................................................................................................................................................................................................................................................................

........

0 2 4 6


0.0

0.2

0.4

0.6

0.8

1.0

YawDamping

B66/(4r2zzωe)

................................................................

........................................ ............................................................................................................................................... ........................................................................................... ............. .......................... .......................... ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. ............. .................................................................................

................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

Figure 22: Yaw Added Mass and Damping Coefficients for HALIFAX


7.3 Motions in Waves

To verify the present computational methods, motions in waves have been com-puted for comparison with SHIPMO7 strip theory results. Three-dimensional pre-dictions are based on potentials evaluated directly using Python class HullRadDiffand interpolated potentials using class HullRadDiffDB. For speed corrections withthe three-dimensional predictions, the termU/ωe has been assigned an upper limitof L/2. Figures 23 to 34 show predicted motions in waves for the HALIFAX classtravelling at a speed of 18 knots in stern quartering (30 degrees) and bow quartering(150 degrees) waves. It has been found that 3D predictions from directly evaluatedand interpolated potentials give virtually identical results; thus, the motions frominterpolated potentials are not included in Figures 23 to 34.

The significant differences between 3D and SHIPMO7 strip theory predictions arisefrom several factors. Strip theory, and to a lesser extent the present 3D theory, isbased on the assumption of high encounter frequency; thus, differences betweenthe two approaches will be greater at low encounter frequencies. While strip theorycodes often do not include surge motions, SHIPMO7 evaluates surge motions byassuming that surge diffraction forces are negligible. The SHIPMO7 predictionsinclude viscous roll damping forces which aren’t included in the present three-dimensional predictions. The most noticeable differences between the strip theoryand three-dimensional predictions in Figures 23 to 34 are for surge and sway mo-tions in stern quartering seas at a wave frequency of approximately 1.2 rad/s. At thiscombination of speed, heading, and wave frequency, encounter frequency is nearzero, thus violating theoretical assumptions for both strip theory and the present3D approach. It should be noted that SHIPMO7 alters wave frequencies to avoidcomputations at low encounter frequencies. For surge motions in stern quarteringseas (Figure 23), diffraction forces could contribute to the differences between striptheory and 3D predictions. Significant differences in yaw motions occur at lowerwave frequencies in both stern and bow quartering seas (Figures 33 and 34), andare likely due to three-dimensional effects.

Due to assumptions regarding high encounter frequency, the present results at for-ward ship speed for bow quartering seas provide the best basis for consideringwhether the current 3D approach has been correctly implemented. The very goodagreement between strip theory and 3D theory for bow quartering seas (with the ex-ception of yaw, as previously noted) verifies the implementation of the 3D theory.SHIPMO7 typically gives good agreement with experimental results in head andbow quartering seas [30, 31]; thus, the present computational results suggest thatthe three-dimensional theory will likely give very good agreement with experimentsin bow quartering seas.


0.0 0.5 1.0 1.5 2.0

Wave FrequencyωI (rad/s)

0

1

2

3

SurgeAmplitude|η1|/a

.........................................................3D, coarse mesh, 225 port panels

. . . . . SHIPMO7 strip theory

.................................................

.....................................................

.......................................................................................................................................................................................................................................................................................................................................

...................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

......

......

............................................................

Figure 23: HALIFAX Surge Motion in Stern Quartering Seas, 18 knots, 30 Degrees

0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

0.8

1.0

SurgeAmplitude|η1|/a



.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................. . . . . ......

Figure 24: HALIFAX Surge Motion in Bow Quartering Seas, 18 knots, 150 Degrees


0.0 0.5 1.0 1.5 2.0


0

1

2

3

4

SwayAmplitude|η2|/a



....................................................................................................................................................................................................................................................................................................................................................

................

................

................

................

................

................

................

................

................

................

................

................

................

................

................

................

................

................

..............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................

..............................

Figure 25: HALIFAX Sway Motion in Stern Quartering Seas, 18 knots, 30 Degrees

0.0 0.5 1.0 1.5 2.0


0.0

0.1

0.2

0.3

0.4

0.5

SwayAmplitude|η2|/a


. . . . . SHIPMO7 strip theory.............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................. . ..... . . . . . . . . . .

Figure 26: HALIFAX Sway Motion in Bow Quartering Seas, 18 knots, 150 Degrees


0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

0.8

1.0

HeaveAmplitude|η3|/a


. . . . . SHIPMO7 strip theory.........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.........................................................................................................................................................................................................................................................................

...................................................................................................................................................................................

..........................................................................

Figure 27: HALIFAX Heave Motion in Stern Quartering Seas, 18 knots, 30 Degrees

0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

0.8

1.0

HeaveAmplitude|η3|/a


. . . . . SHIPMO7 strip theory.......................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.................................................................. . . . . ...

Figure 28: HALIFAX Heave Motion in Bow Quartering Seas, 18 knots, 150 Degrees


0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

RollAmplitude|η4|/(ka)



......................................................................................................................................................................................................................................................................................................................

.....................................................................................................................................................................................................................................................................................................................................................................................................

......................................................................................................................................................................................................................................................................................

.......

..................................................................

............

Figure 29: HALIFAX Roll Motion in Stern Quartering Seas, 18 knots, 30 Degrees

0.0 0.5 1.0 1.5 2.0


0

1

2

3

4

5

RollAmplitude|η4|/(ka)



........................................................

...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......

........................................................................................ . ........ .

Figure 30: HALIFAX Roll Motion in Bow Quartering Seas, 18 knots, 150 Degrees


0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

0.8

1.0

PitchAmplitude|η5|/(ka)


. . . . . SHIPMO7 strip theory........................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

....................................................................

Figure 31: HALIFAX Pitch Motion in Stern Quartering Seas, 18 knots, 30 Degrees

0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

0.8

1.0

PitchAmplitude|η5|/(ka)



...............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

.......................................................... . . . . . . . . . . . . .

Figure 32: HALIFAX Pitch Motion in Bow Quartering Seas, 18 knots, 150 Degrees


0.0 0.5 1.0 1.5 2.0


0.0

0.2

0.4

0.6

0.8

1.0

YawAmplitude|η6|/(ka)



............................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

........................................................................................................................................................................................................................................................................................................

.............

.................................................. . .

Figure 33: HALIFAX Yaw Motion in Stern Quartering Seas, 18 knots, 30 Degrees

0.0 0.5 1.0 1.5 2.0


0.0

0.1

0.2

0.3

0.4

0.5

YawAmplitude|η6|/(ka)



.................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................................

..........

............................................ . . . ..... . . . . . . . . . .

Figure 34: HALIFAX Yaw Motion in Bow Quartering Seas, 18 knots, 150 Degrees


7.4 Computational Requirements

Computational requirements are an important factor when considering the practi-cal application of scientific computations. For the present computations using theHALIFAX coarse mesh (225 panels on half of hull), the required computer memoryis approximately 60 MB. The amount of required memory will be approximatelyproportional toN2

p , whereNp is the number of panels on the hull surface.

When evaluating ship motions using the hull potential database class HullRad-DiffDB, almost all of the computational time is required for initialization. Afterthe database has been initialized, radiation and diffraction forces can be obtainedusing negligible amounts of computational time. The database can be generatedonce for a given hull loading condition, and can be stored and reused for futurecomputations using the Python “pickle” module (see Reference 26). The initializa-tion time for the database class HullRadDiffDB depends on the number of panels,the number of encounter frequencies, and the number of seaways (i.e., combina-tions of ship speed, heading, and wave frequency) for diffraction computations.For the HALIFAX coarse mesh running on an 800 MHz Pentium III processor, ap-proximately 45 seconds is required to generate a HullRadDiff object that is usedas input to the database class HullRadDiffDB. Subsequent initialization computa-tions require approximately 10 seconds per encounter frequency plus 5 seconds perseaway. A comprehensive database for a given ship loading condition can be devel-oped using 50 input encounter frequencies, 6 ship speeds, 19 wave headings, and20 wave frequencies (2280 seaway conditions), leading to a total initialization timeof approximately 3 hours.

As mentioned above, a HullRadDiffDB object can be stored in a file for subse-quent computations. When a HullRadDiffDB object is created, there is an optionfor retaining panel potentials, which are required for pressure and sea load compu-tations. If hull potentials are not retained, then only the global ship hydrodynamiccoefficients and forces need to be stored. The size of the file for storing a Hull-RadDiffDB object depends upon the number of encounter frequencies, number ofseaways, number of hull panels, and whether panels potentials are to be retained.If hull potentials are not retained, then the storage file size is approximately 100kB plus 4 kB per encounter frequency plus 0.5 kB per seaway. For our examplecase above of 50 encounter frequencies and 2280 seaway conditions, the size of thestored database file would be approximately 1.5 MB. If hull potentials are retained,the storage file size will increase significantly and be approximately proportionalto the number of hull panels. For the HALIFAX coarse mesh, the storage file sizeis approximately 230 kB plus 120 kB per encounter frequency plus 80 kB per sea-way. For our example case above of 50 encounter frequencies and 2280 seawayconditions, the size of the stored database file including panel potentials would beapproximately 200 MB.


8 Conclusions

An efficient method has been developed for computing three-dimensional hydro-dynamic coefficients in the frequency domain. This approach is based on the zerospeed Green function, and includes extensions to include forward speed effects.Robustness and accuracy at both low and high frequency ranges are obtained byreferencing the frequency dependent portion of the Green function to either thezero or infinite frequency Green function, depending on the encounter frequency.A new algorithm based on interpolated Green functions for computing radiationand diffraction potentials reduces the number of Green function evaluations whenconsidering a variety of ship speeds, headings, and wave frequencies.

Sample computations for HALIFAX suggest that the approach gives reliable re-sults for a coarse mesh with 225 panels on half of the ship. Computations with afine mesh of 1041 panels appear to have somewhat better accuracy but require sig-nificantly more computational time. Comparisons of predicted motions with striptheory suggest that the new three-dimensional method is giving reliable results.


References

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2. SeaTech, BMT (1995). User’s Manual for Motion, Load and Pressure ProgramSuite - PRECAL. Technical Report. prepared for NSMB CRS PRECALWorking Group. Distribution limited to CRS members.

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13. Newman, J.N. (1985). The Evaluation of Free-Surface Green Functions. InFourth International Conference on Numerical Ship Hydrodynamics,Washington.

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15. Telste, J.G. and Noblesse, F. (1986). Numerical Evaluation of the GreenFunction of Water-Wave Radiation and Diffraction.Journal of Ship Research,30(2), 69–84.

16. Salvesen, N., Tuck, E.O., and Faltinsen, O. (1970). Ship Motions and SeaLoads.Transactions, Society of Naval Architects and Marine Engineers, 78,250–287.

17. Huang, Z.J. and Hsiung, C.C. (1993). An Improved 3-D Panel Method toCompute mj-Terms for Ship Motion. InProceedings of the Second CanadianMarine Dynamics Conference, Vancouver.

18. Inglis, R.B. and Price, W.G. (1981). Calculation of the Velocity Potential of aTranslating, Pulsating Source.Transactions, Royal Institution of NavalArchitects, 123, 163–175.

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20. Hess, J.L. and Smith, A.M.O. (1964). Calculation of Nonlifting Potential FlowAbout Arbitrary Three-Dimensional Bodies.Journal of Ship Research, 8(3),22–44.

21. Hally, D. (1989). POTFLO: A Suite of Programs for Calculating PotentialFlow about Ship Hulls. (DREA TM 89/210). Defence Research EstablishmentAtlantic.

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24. Hally, D. (1994). Integration of the Wave Resistance Green Function OverPlanar Panels. (DREA TM 94/206). Defence Research Establishment Atlantic.

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Symbols

[A] added mass matrixAj area of paneljAs source areaAwp waterplane areaa wave amplitudeB ship beamb aspect ratio of rectangular panel[C] hydrodynamic stiffness matrix[D] normal flow velocity influence matrix[E] velocity potential influence matrix{FD} diffracted wave force vector{FI} incident wave force vectorG Green functionG0 zero frequency Green functionG0 frequency dependent term of Green function relative to zero

frequency Green functionG∞ frequency dependent term of Green function relative to infinite

frequency Green functiong gravitational acceleration[H] x velocity influence matrixh non-dimensional horizontal distance from field point to sourceIwp−xx x moment of waterplane areaIwp−yy y moment of waterplane areaJ0(h) Bessel function of the first kind of order zeroJ1(h) Bessel function of the first kind of order oneKG centre of gravity above baselineke encounter wavenumberkI incident wavenumberL ship length between perpendiculars[M ] inertial matrixnk outward normal componentkNe number of encounter frequenciesNG number of evaluations of Green functionNp number of panelsp pressureport superscript denoting port side of hullR distance from field point to sourceRxy horizontal distance from field point to source


R0(h, v) term for evaluation of frequency dependent Green functionR1 distance from field point to source imageR1(h, v) term for evaluation of frequency dependent Green functionrxx roll radius of gyrationryy pitch radius of gyrationrzz yaw radius of gyrationSb surface of floating bodySk surface of hull panelkSs surface of sourcestar superscript denoting starboard side of hullTmid draft at midshipsts trim by sternU ship speedv non-dimensional vertical distance from field point to source imagevn normal flow velocity~x location in space~xs source location~xsc source centroid locationxwp x coordinate of centre of floatationxCB x coordinate of centre of gravityzCB z coordinate of center of buoyancyzCG z coordinate of center of gravityβ wave direction relative to shipδjk Kroenecker delta functionζ3 oscillatory vertical displacementηj motion amplitude in modejρ water densityσ source strengthΦ total oscillatory velocity potentialφD diffracted wave potentialφI incident wave potentialφj radiated wave potential due to motion modejωI incident wave frequencyωe wave encounter frequencyωt

e transition encounter frequency for Green function computation5 ship volume4 ship mass displacement


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DOCUMENT CONTROL DATA

(Security classification of title, body of abstract and indexing annotation must be entered when document is classified)

1. ORIGINATOR (the name and address of the organizationpreparing the document. Organizations for whom the documentwas prepared, e.g. Centre sponsoring a contractor’s report, ortasking agency, are entered in section 8.)

Defence R&D Canada - Atlantic

2. SECURITY CLASSIFICATION (overallsecurity classification of the document includingspecial warning terms if applicable).

UNCLASSIFIED

3. TITLE (the complete document title as indicated on the title page. Its classification should be indicated by theappropriate abbreviation (S,C,R or U) in parentheses after the title).

Three Dimensional Ship Hydrodynamic Coefficients Using the Zero ForwardSpeed Green Function

4. AUTHORS (Last name, first name, middle initial. If military, show rank, e.g. Doe, Maj. John E.)

McTaggart, Kevin A.

5. DATE OF PUBLICATION (month and year of publication ofdocument)

April 2002

6a. NO. OF PAGES(total includingAnnexes, Appendices,etc).

66

6b. NO. OF REFS (totalcited in document)

31

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Defence R&D Canada - Atlantic, PO Box 1012, Dartmouth, NS, Canada B2Y 3Z7

9a. PROJECT OR GRANT NO. (if appropriate, the applicableresearch and development project or grant number under whichthe document was written. Please specify whether project orgrant).

11GK

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10a. ORIGINATOR’S DOCUMENT NUMBER (the officialdocument number by which the document is identified by theoriginating activity. This number must be unique.)


10b. OTHER DOCUMENT NOs. (Any othernumbers which may be assigned this documenteither by the originator or by the sponsor.)

11. DOCUMENT AVAILABILITY (any limitations on further dissemination of the document, other than thoseimposed by security classification)

(X) Unlimited distribution( ) Defence departments and defence contractors; further distribution only as approved( ) Defence departments and Canadian defence contractors; further distribution only as approved( ) Government departments and agencies; further distribution only as approved( ) Defence departments; further distribution only as approved( ) Other (please specify):

12. DOCUMENT ANNOUNCEMENT (any limitation to the bibliographic announcement of this document. This willnormally correspond to the Document Availability (11). However, where further distribution (beyond the audiencespecified in (11) is possible, a wider announcement audience may be selected).


13. ABSTRACT (a brief and factual summary of the document. It may also appear elsewhere in the body of the documentitself. It is highly desirable that the abstract of classified documents be unclassified. Each paragraph of the abstract shallbegin with an indication of the security classification of the information in the paragraph (unless the document itself isunclassified) represented as (S), (C), (R), or (U). It is not necessary to include here abstracts in both official languagesunless the text is bilingual).

This report presents a method for predicting ship hydrodynamic coefficients andpressures in waves. Computations are performed in the frequency domain usingthe Green function for zero forward speed. Pressures and coefficients at non-zero forward speed are computed using extensions to zero speed results. Theapproach has been implemented using the Python programming language. Thepresent implementation gives robust results over a wide range of wave frequen-cies. A new algorithm facilitates efficient computation of hydrodynamic terms fora large number of combinations of ship speed, heading, and incident wave fre-quency. Sample computations for the HALIFAX class demonstrate application ofthe method.

14. KEYWORDS, DESCRIPTORS or IDENTIFIERS (technically meaningful terms or short phrases that characterizea document and could be helpful in cataloguing the document. They should be selected so that no security classificationis required. Identifiers, such as equipment model designation, trade name, military project code name, geographiclocation may also be included. If possible keywords should be selected from a published thesaurus. e.g. Thesaurus ofEngineering and Scientific Terms (TEST) and that thesaurus-identified. If it not possible to select indexing terms whichare Unclassified, the classification of each should be indicated as with the title).

heavepanel methodspitchrollsea loadsseakeepingship motionssurgeswayyaw


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