Ship motions by a three dimensional Rankine Panel Method

ABSTRACT

A Rankine Panel Method is presented for the solution of

the complete three-dimensional steady and time-harmonic potential flows past ships advancing with a forward veloc-ity. A new free-surface condition is derived, based on lin-earization about the double-body flow and valid uniformly from low to high Froude numbers.

Computations of the steady ship wave patterns reveal

sig-nificant detail in the Kelvin wake a sigsig-nificant distance

downstream of the ship, permitted by the cubic order and

zero numerical damping of the panel method. The wave

pattern appears to be sensitive to the selection of the free-surface condition only for full ship forms.

The heave and pitch hydrodynamic coefficients, exciting

forces and motions of a Wigley and a Series-60 hull have

been evaluated in head waves over a wide range of fre-quencies and speeds. A robust treatment is proposed of

the rnterms which are found to be critical importance

for the accurate solution of the problem. Tri all cases the agreement with experiments is very satisfactory indicating a significant improvement over strip theory, particularly in the cross-coupling and diagonal pitch damping coefficients.

1. INTRODUCTION

Theoretical methods for the prediction of the seakeeping of ships have evolved in three phases over the past 40 years.

The first phase involved the development of strip theory,

and was followed by a series of developments in

slender-body theory which formulated rationally the ship motion

problem and produced several refinements of strip theory.

The advent of powerful computers in the early 80's al-lowed the transition into the third and current phase of

seakeeping research which aims at the numerical solution of the three-dimensional problem. This paper presents our progress in that direction.

The pioneering work of Korvin-Kroukovsky (1955) stimu-lated a number of studies on the strip method which led to the theory of Salvesen, Tuck and Faltinsen (1970). Its

pop-ularity to date arises from its satisfactory performance in

the prediction of the motions of conventional ships and its computational simplicity. Well documented are however

its limitations in the prediction of the derived responses,

TECHNISCHE UNIVERSITEIT

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Scheepehydromechanica

rchief

Mekeiweg 2, 2628 CD Deift

TeLi 015-7a6873. Fax 015 781838

Ship Motions by a Three-Dimensional

Rankine Panel Method

D. Nakos, P. Sclavounos (Massachusetts Institute of Technology, USA)

-structural wave loads and in general the seakeeping char-acteristics ofships advancing at high Fraude numbers [e.g. O'Dea and Jones (1983)[.

The 60's and 70's witnessed several analytical studies aim-ing to extend the slender-body theory of aerodynamics to the seakeeping of slender ships. The rational justification

of strip theory, as a method valid at high frequencies and

moderate Froude numbers, was presented by Ogilvie arid Tuck (1969). This theory was extended to the diffrac-tion problem by Faltinsen (1971) and was further refined

by Maruo and Sasaki (1974). The high-frequency

restric-tion in earlier slender-ship theories was removed by the unified theory framework presented by Newman (1978). Its extension to the diffraction problem was derived by

Sciavounos (1984) and applied to the seakeeping of ships by Newman and Sclavounos (1980) and Sclavounos (1984). Subsequent slender-ship studies by Kim and Yeung (1984)

and Nestegard (1986), accounted directly for convective forward-speed wave effects near the ship hull and repre-sented the transition to numerical studies aiming at the

solution of the three-dimensional ship-motion problem. By the mid-SO's, the performance of slender-body theory

for the seakeeping problem could only be validated from

experimental measurements. Moreover, it had become

ev-ident that end-effects at high Froude numbers cannot be

modelled accurately by slender-body approximations and

the need for a numerical solution of the complete three- -dimensional had emerged. Early efforts towards this goal by Chang (1977), Inglis and Price (1981) and Guevel and Bougis (1982) were not conclusive because the significant

computational effort necessary for the evaluation of the time-harmonic forward-speed Green function limited the total number of panels used on the ship surface. More

recently, King, Beck and Magee (1988) circumvented this difficulty by solving the same problem in the time domain,

therefore making use of the zero-speed transient Green

function which is easier to evaluate.

The last decade witnessed the growing popularity of

Rank-me Panel Methods for the solution of the steady poten-tial flow past ships. The success of the early work of Gadd (1976) and Dawson (1977) motivated several anal-ogous studies which concentrated upon the prediction of

the Kelvin wake and evaluation of the wave resistance. The principal advantages of the method are twofold- the

Rank-me singularity is simple to treat computationally and the

distribution of panels over the free surface allows the en-forcement of more general free-surface conditions with

vari-3$3itiC H31t33i

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1CC3)

rK,1fl13$mC,bÇßßta

able coefficients. A drawback of Rankine-panel methods is

that they require about twice as many panels as methods

based on the distribution of wave singularities over the ship surface alone. The resulting computational overhead is as-sociated with the solution of the resulting matrix equation, but may not be significant if an out-of-core iterative solu-tion method is available.

This paper outlines the solution of the three-dimensional time-harmonic ship motion problem by a Rankine Panel Method. For the steady problem, the theory for the

anal-ysis of the properties for such numerical schemes was

in-troduced by Piers (1983) and generalized by Sclavounos and Nakos (1988). The extension of this numerical anal-ysis to the time-harmonic problem is presented in Nakos

and Sclavounos (1990). In this reference the convergence

properties of a new quadratic-spline scheme are derived,

which has been found to be accurate and robust for the so-lution of both steady and time-harmonic free-surface flows

in three dimensions. This scheme is applied in this paper to the solution of the time-harmonic radiation/diffraction potential flows around realistic ship hulls and the evalu-ation of the hydrodynamic forces and motions in regular

head waves.

A new three-dimensional free-surface condition is derived, using the double-body flow as the base disturbance due to

the forward translation of the ship. This is shown to be

valid uniformly from low to high Froude numbers and over

the entire frequency range. Known low-Froude-number conditions for the steady problem, as well as the Neumann-Kelvin condition, are obtained as special cases. The

ship-hull condition includes the rnterms which are evaluated from the solution of the three-dimensional double-body

flow. An important property of the solution scheme is that the evaluation of the double gradients of the double-body flow is circumvented by an application of Stokes theorem.

Computations are presented of the steady wave patterns trailing a fine Wigley model and a fuller Series-60 hull. The cubic order and zero numerical damping of the free-surface discretization allows the prediction of significant detail of the Kelvin wake at a large distance downstream of the ship. A comparison of the wave patterns obtained form the Neumman-Kelvin and the more general double body free-surface conditions reveals good agreement for the Wigley hull, while evident differences appear in the

respective Series-flU wakes.

Predictions of the heave and pitch added-mass and damp-ing coefficients and excitdamp-ing forces are found to be in very good agreement with experimental measurements both for

the Wigley and the Series-60 hull. The contribution of

the complete rnterms is found to be important,

partic-ularly in the cross-coupling coefficients. The validity of a more general set of Timman-Newman relations is observed and conjectured in connection with free-surface conditions based on the double-body flow.

The heave and pitch motion amplitudes and phases

pre-dicted by the present method are found in very good

agree-ment with experiagree-ments and present an improveagree-ment over

strip theory.

2.

TRE BOUNDARY VALUE PROBLEM

Define a Cartesian coordinate system = (z, y, z) fixed

on the ship which translates with a constant speed U.

The positive zdirection points upstream and the

posi-tive zaxis upwards. The boundary-value problem will be expressed relative to this translating coordinate system, therefore the flow at infinity is a uniform stream and the

ship hull velocity is due to its oscillatory displacement from its mean position.

The fluid is assumed incompressible and inviscid and the

flow irrotational, governed by a potential function (i, t)

which satisfies the Laplace equation in the fluid domain

V2'(,t) = O

. (2.1)

Over the wetted portion of the ship hull (B), the

compo-nent of the fluid velocity normal to (B) is equal to the

correspoading component of the ship velocity VB, or

= (.i)(i,t),

(2.2)

where the unit vector points out of the fluid domain.

The fluid domain is also bounded by the free surface, de-fined by its elevation z = ç(x,y, t) and subject to the

kine-matic boundary condition,

(

+ vw .

y)

[z - ç(z,y,t)] = O

on z =

(2.3)

The vanishing of the pressure on the free surface combined with Bernoulli's equation, leads to the dynamic free surface condition

ç(x,y,t) = -

(wt + .

-

u)

. (2.4)

The elimination of ç from (2.3) and (2.4) leads to

O onz = ç.

(2.5)

If the fluid domain is otherwise unbounded, the additional

condition must be imposed that at finite times the flow

velocity at infinity tends to that of the undisturbed stream.

-Linearization of the free surface condition

Physical intuition suggests that linearization of the pre-

-ceding boundary value problem is justified when the dis-t'irbance of the uniform incoming stream due to the ship is in some sense small. Small disturbances may be

justi-fied by geometrical slenderness, slow forward translation, or a combination of the above. Full-shaped ships typically advance at low speed and cause a small steady wave distur-bance. Fine-shaped ships, on the other hand, often advance at high Froude numbers. Yet the steady disturbances they

generate, is small if their geometry is sufficiently thin or slender. Linearization may therefore be justified both at

low and high Froude numbers F, as long as it is tied to the hull slenderness e. Linearization of the unsteady flow is

also supported by the assumption of a small ambientwave

amplitude.

The linearized free surface condition derived next is uni-formly valid between these two limits, and its validity is

heuristicallY justified if the parameter cF2 is sufficiently small. The details of the derivation outlined below are given in Nakos (1990). The total flow field W(î,t) is de-composed into a basis flow '(), assumed to be of 0(1), the

steady wave flow (ì), and the unsteady wave flow i/'(, t)

=

. _(2.6)

The double-body flow is chosen as the basis flow, a selection

primarily motivated by the body boundary condition as

well as the simplifications it allows in the ensuing analysis.

Thus, ' is subject to the rigid wall condition

= O

, on z = O . (2.7)

The wave disturbances and b are superposed upon the

double-body flow and are taken to be small relative to the

Linearization of (2.4-5), correct to leading order in

and , leads to the conditions

V V(V. V)+ V(V. V)

.

V) = V(V. V) .

(U2

- V.

_{onz = O}

ç(x,y) =

!

U2 + v.

v)

J

g \2

(2 8)

sb,, + 2V' V, + V' V (Va'. Vb)

+

V(VVV+gb--

_{(i,b + V. Vb) = 0, on z =}

_(2.9)

ç(z,y,t) =

( + V

for the steady and unsteady flows, respectively.

For slender/thin ships with e small, and for Froude num-bers of 0(1), the uniform incident stream Ur maybe used as the basis flow. In this case, (2.8-9) reduce to the well-known Neumann-Kelvin conditions. In the opposite limit

of bluff ships with e of 0(1) advancing at low Froude num-bers, (2.8-9) reduce to the conditions of slow-ship theory. The condition (2.8) contains all terms present in Dawson's

(1977) condition, and it is closest to the one proposed by Eggers (1981). This property may explain the fact that, even though Dawson's and Egger's conditions have been derived as low Foude number approximations, they have

been found to perform satisfactorily over a wider range of forward speeds.

Linearization of the body boundary condition

The linearization of the ship hull boundary condition may

also be derived from the decomposition (2.6). _By defi-nition, the velocity potential of the double-body _{flow is}

subject to

= O

, on (B) . (2.10)

Consequently, the steady wave flow also satisfies the homo-geneous condition

= o

,

on ()

, (2.11)

leaving the right-hand-side of (2.8) as the only forcing of

the steady wave problem.

The unsteady forcing due to the oscillatory motion of the

vessel is accounted for by the unsteady wave flow l. If

is the oscillatory displacement vector measured from the

mean position of the vessel (B), it follows by substituting

of (2.6) in (2.2) that

a'

₌

a.

. n

V( +

_{) .} , on (B) . (2.12)

an

at

Assuming that the magnitude of the displacement vector

d is small and comparable to the ambient wave amplitude,

the boundary condition (2.12) may be linearized about the mean position of the hull surface [Timman and

New-man(1962)],

±

=

_{,on ()}

(2.13)

The last term in (2.13) accounts for the interaction

be-tween the steady and unsteady disturbances in a manner

consistent with the assumptions underlying the derivation

of the free-surface conditions (2.8). An alternative form

of (2.13) may be derived in terms of the rigid-body global

displacements (eI,e2,eo) and rotations (e4,eo,ee), along

the axes (z, y, z) respectively,

=

an +

ern.)

,

on () ,

(2.14)

i='

where tn, ,

j

= 1, ...6, denote the so-called rn-terms fOgilvie

and Tuck (1969)].

If the basis flow is approximated by the uniform stream the

only non-zero rn-terms are m5 = Un3 and m6

= Un3

which merely account for the 'angle of attack effect' due to yaw and pitch. This approximation of the rn-terms has

been employed in most previous studies of the ship motion

problem, consistently with the linearization steps leading to the Neumann-Kelvin free surface boundary condition. The performance of this linearization in practice will be

the subject of numerical experiments presented in section

7.

-Frequency domain formulation of the unsteady problem

The unsteady excitation is due to an incident monochro-

-matic wave train. The frequency of the incident wave, as viewed from the stationary frame S Wo, while in the

the frequency of encounter . 1f ß is the angle between

the phase velocity of the incident wave and the forward

velocity of the ship, c is given by

w=IoU-cosß

(2.15)

In the frame i, the velocity potential of the incident wave of unit amplitude, in deep water, is given by the real part

of the complex potential çO

çao(i, t) = i --

°' c1'' . _(2.16)

The linearity of the Boundary Value Problem that

gov-erns the physical system, along with the form of the body

boundary condition (2.14), suggest the decomposition of

the wave flow as follows,

Given the solution of the potential flow problem formu-lated in the preceding section, the hydrodynamicpressure

follows from Bernoulli's equation. Of particular interest, in practice, is the pressure distribution on the ship wetted

surface and resultant forces and moments necessary for the

determination of the ship motions.

The pressure on the hull is given by

P=

+ VW.V_

U2+gz]

. (3.1)

2

ZE(B)

The unsteady portion of (3.1), correct to leading order in

çt', may be expressed as follows

p = - p

+ V' V0((

-p (ä.V)(V.V+gz)

₂

E(B)

Under the assumption of small monochromatic motions at

the frequency of encounter , the components of the un-steady force F = (F1, F2, F3) and momentM= (F4, F5 , F6)

acting on the ship, accept the familiar decomposition

(3.2)

F(t) =

{ei

where,

X=

[Ax. +

(w2a,5ib

-

ci1)]

}

p

{ii i(çao + ça7) + V

(B)

V(ça + ça7)]d8

= -

_{ii

(iça5 + V Vça5) n da

=

_{ff (iça5 + V}

_{Vça5) n da} 6 (B) (B) (3.4)

= p

ff(d.V)(V.V+9z) n da

(B)

fori,j=1,...,6.

The exciting forces X and the added mass and damping co-efficients, a,, and b, are therefore functions of the forward

speed and the frequency of oscillation . The restoring

coefficients ci,, on the other hand, include the classical

hy-drostatic contribution augmented by a dynamic term due to the gradients of the double-body flow. The latter con-tribution depends linearly upon the deflection of the ship surface from its mean position and quadratically on the

ship speed. It is therefore expected to be substantial at

high Fraude numbers.

The equations governing the time-harmonic responses of the ship follow from Newton's law. Using the definitions

(3.5) of the forces acting on the hull, the familiar six-degree of freedom system of equations is obtained

j=1

(3.6)

where m5 is the ship inertia matrix, the complex ampli-tudes of the oscillatory ship displacements, and the restor-ing coefficients c5 are modified to include the moments in pitch and roll due to the corresponding displacement of the center of gravity.

4. THE INTEGRAL FORMULATION

Green's second identity is applied for the unknown

poten-tials, 'i',

or p j = 1, ..., 7, using the Rankine source

potential,

G(i)

_{= 2}

(4.1)

as the Green function. The fluid domain is bounded by the hull surface (B), the free surface (FS) and a cylindri-cal 'control' surface (S). The resulting integral equation

takes the form

=

e"'

A(çao + ça7) +

5=1

eis]

}

, (2.17)

where A is the amplitude of the incoming wave train, çO

is the complex diffraction potential, and ça5,

_{j =}

are the complex radiation potentials due to the harmonic

oscillation of the ship in each of the six rigid-body degrees of freedom, at frequency .ìand with unit amplitude.

Upon substitution of the linear decomposition into (2.9), the free surface conditions for ça,, i = 1, ...7, are derived. It is important to point out that the free surface condition

for the diffraction problem is inhomogeneous, the forcing arising from the interaction of the incoming wave train with

the double-body flow. In the limit of slender/thin ships,

where the uniform stream may be taken as the basis flow, this inhomogeneity vanishes.

4,()

_If

a4,(i') - -., -,

f!

4,(.\aG(z;z)d

-

G(x;x)dz +

_an (IS) (FS)u(B) (B)

where 4, stands for any of the potentials , 4', so, ,

j =

i, ...,7, introduced in the preceding sections. The surface integrals over the control surface (S,,) can be shown to vanish in the limit as (S) is removed to infinity with

kept finite.

The derivatives of c, 4, and ço normal to the ship surface (B) are known. The corresponding vertical derivative on the free surface (FS) is replaced by the appropriate

com-bination of the value and tangential convective derivatives, according to the corresponding free surface condition.

Of particular interest is the treatment of the integral over

the ship hull which accounts for the rnterms in the bound-ary condition (2.14). This is of the form

ffm G(i;i') di'

,

j

1,...,6 . (4.3) (3)

The evaluation of the rnterms in (4.3) requires the

com-putation of second order derivatives of the double-body po-tential 'I' on the ship hull. When it comes to the evaluation of gradients of the solution potential, low-order panel meth-ods are known to be sensitive to discretization error, unless their implementation and panel distribution is carefully

se-lected. The evaluation of double gradients of the solution

are known to introduce serious difficulties, as illustrated by Nestegard (1984) and Zhao and Faltinsen (1989). Here, an alternative expression for the evaluation of the in-tegral (4.3) is derived by an application of Stokes' theorem. Given that the basis flow satisfies a zero flux condition

on the ship hull and the z = O plane, it follows that, for

j=1,...,6,

ffm1 G(2;i')

di'

= -

ff

[V(i'). V5G(i;i')] n1

(_) (ll) (4.4)

The right-hand side of (4.4) involves only first derivatives

of '' on the hull, consequently it is clearly superior from the computational standpoint.

The integral equation (4.2) will not accept unique solutions unless a radiation condition is imposed enforcing no waves

upstream. In practice the solution domain of (4.2) on the z = O plane will be truncated at a rectangular boundary located at some distance from the ship where appropriate 'end conditions' will be imposed enforcing the radiation condition. Due to the convective nature of the flow, the condition at the upstream boundary is the most critical and takes the form

(4.5)

where 4, stands for either the steady or the unsteady wave disturbance. The origin and physical interpretation of these

two upstream conditions are discussed in detail in Sciavouno and Nakc8 (1988) for a two-dimensional steady flow, and are extended to time-harmonic flows in Nakos (1990). It is shown that both are necessary in order to ensure physically meaningful numerical solutions of the steady and unsteady

problems. For r = wU/g > 1/4 no wave disturbance is

present upstream of the ship and the conditions (4.5) can be shown to enforce this property of the flow. For r < 1/4

and with increasing Froude numbers, the amplitude of the waves upstream of the ship decreases relative tothat of the trailing wave pattern and conditions (4.5) perform well if the truncation boundary is sufficiently removed from the ship. No conditions are necessary on the transverse and

downstream truncation boundaries.

5. THE NUMERICAL SOLUTION ALGORITHM

The solution of integral equation (4.2) for the steady and

unsteady flows is obtained using a Panel Method. The sys-tematic methodology for the study of the numerical proper-ties of Rankine Panel Methods for free surface flows devel-oped in Sclavounos and Nakos (1988) led to the design of a bi-quadratic spline-collocation scheme of cubic order, zero numerical dissipation and capable to enforce accurately the radiation condition (4.5).

The boundary domain - including the ship hull and the

free surface solution domain - is discretized by a collection

of plane quadrilateral panels [see Figure 1. The unknown

velocity potential is approximated by the linear superposi-tion of hi-quadratic spline basis funcsuperposi-tions B(i), as follows

4,(i) 0

B (i)

, (5.1)

where B1 is the basis function centered at the j'th panel

and 0 is the corresponding spline coefficient. By

collocat-ing the integral equation (4.2) at the panel centroids and enforcing the upstream condition (4.5), the discrete

for-mulation follows in the form of a system of simultaneous linear equations for the coefficients a1. The relation (5.1)

provides a C'-continuous representation of the velocity po- -tential and may be differentiated to give the velocity field on the domain boundaries. The free surface elevation and

hydrodynamic pressure are evaluated using the relations

(2.8_9) and (3.1-2), respectively.

The error and stability analysis of the bi-quadratic spline

scheme is presented in Nakc and Sclavounos (1990). It is based on the introduction of a discrete dispersion relation

governing the wave propagation over the discretized free surface. Comparison of the continuous and discrete dis-

-persion relations allows the rational definition of the

con-sistency, order and stability properties of the numerical

solution scheme. It is shown that the numerical dispersion

is of 0(h3) where h is the typical panel size and that no numerical dissipation is present. Both are valuable

prop-erties for the computation of ship wave patterns which are

not substantially distorted, damped or amplified by the

numerical algorithm.

Essential for the performance of the method is a stability

condition restricting the choice of the grid Froude number, Fh = U/.,/gh relative the panel aspect ratio, n = where h, h, are the panel dimensions in the streamwise

and transverse directions, respectively. This condition,

derived and discussed in detail in Nakos and Sciavounos (1990), establishes 'stable' domains on the (Ph, a) plane

with boundaries dependent on the frequency of oscillation. For a given a Froude number, a stable discretization for the highest frequency of oscillation is stable for all lowest fre-quencies. Therefore, no regridding of the ship hull and free

surface is necessary for the solution of the time harmonic

problem over a range of frequencies. The resulting complex linear system is solved by an accelerated block Gauss-Siedel iterative scheme which makes extensive use of out-of-core storage therefore permitting the use of discretizations with several thousand panels.

Experimental verification of the convergence of the solu-tion algorithm has been established by comparing com-putations of 'elementary' flows around singularities and

thin-struts with analytical solutions [Nakos and Sclavounos (1990) and Nakos (1990)]. The convergence of the hydro-dynamic added-mass and damping coefficients is discussed in Section 7.

6. STEADY AND UNSTEADY SHIP WAVE

PATTERNS

The forward-speed ship wave problems formulated in

Sec-tion 2 have been solved for two hull forms using the nu-merical algorithm outlined in the preceding section. This

section presents converged computations of the steady and time harmonic wave patterns around a Wigley and a Series-60 hull.

The Wigley model has parabolic sections and waterlines, a length-to-beam ratio L/B = 10 and beam-to-draft ratio

B/T

1.6. The grid used for the solution of the steady

problem consists of 40x10 panels on half the hull, providing

adequate resolution of the geometry, while the panels on

the free surface are aligned with those on the hull and have

a typical aspect ratio is a = h/h5 = 1. The grid Froude

number is Fh 6.3 . F, allowing an adequate resolution of the steady wave flow for Froude numbers as low as F =0.20

[see Nakos (1990)]. The free surface domain is truncated at a distance z,,=0.2L upstream of the bow and one ship length downstream of the stern.

The truncation in the

transverse direction is selected at y,,, zz0.75L, so that the

entire wave sector is included in the computational domain. The total number of panels in the grid is 2020.

Figure 1 : Discretjzation of the free surface and the hull for a modified Wigley model, using 1110 panels on half the configuration.

Figure 2 shows contour plots of the wave patterns resulting

from the steady forward translation of the Wigley model

at F = 0.25,0.35,0.40 .

Predictions based on both the Neumann-Kelvin and the double-body linearizations are presented. Due to the slenderness of this Wigley model,

the two wave fields agree well even at high speeds. Small

differences are visible along the diverging portion of the wave system which originates from the stern, where the

Neumann-Kelvin solution tends to generate steeper waves,

particularly along the caustic. The opposite appears to be true in the 'bow wave system'. For all Froude numbers, the calculated wavelengths are not affected significantly by

0.50 0.00 -0.25 -0.75 0.50 0.00 -0.25 -0,75 0.00 -0.25 -0.75 -1.50

a

-1.00 -0.50

the selected linearization.

The second ship tested is the Series-60-05=0.6 hull which is significantly fuller than the Wigley model, with

length-to-beam and beam-to-draft ratios LfD = 7.5 and B/T = 2.5, respectively. The principal characteristics of the grid

used for the computations are the same to those employed for the Wigley model.

Figure 3 illustrates the wave patterns around the

Series-60 model for F = 0.20,0.25,0.35 , respectively. At low speeds (F < 0.30) the amplitude of the generated waves are comparable - if not smaller - than the ones computed

0.00 0.50 0.50 0.00 -0.25 -0.75 0.50 0.00 -0.25 -0.75 0.50 0 00 -0.25 -0.75

Fige 2 : Contour plots of the steady wave patterns due to the parabolic _{Wigley model advan, :.g}

at Froude numbers F= 0.25, 0.35, 0.40.

-1 50 -1.00 -0.50 0.00 0.50

around the Wigley model, despite the increase in the 'full-ness' of the hull shape. For the Wigley model the bow- and stern-wave systems are well formed while the

correspond-ing wave pattern around the Series-60 hull appears to be

more 'confused'.

Differences between the steady wave pattern computations from the Neumann-Kelvin and double-body linearizations are here clearly noticeable. Again, significant

discrepan-cies occur along the diverging portion of the stern-wave

system, where the Neumann-Kelvin solution shows larger amplitudes and shorter wavelengths. Moreover, the caustic

0.50 0.00 -0.25 -0.75 0.50 0.00 -0.25 -0.75

lines originating from the bow and stern appear at a larger

angle in the solution based on the double-body

lineariza-tion. The differences between the two solutions become

more pronounced as the speed increases, resulting in quite

different wave patterns at F =0.35 (see Figure 3c). Figure 4 is a snapshot of the time-harmonic wave pattern around a modified Wigley model translating at F = 0.2 and oscillating in heave at frequencies

_{i27 = 3 and}

/TL7=5. The grid used for this flow field has the same

density as that in Figure 1.

Both frequencies are

over-critical (r wU/g > 0.25), thus two wave systems appear

0.50 0.00 -0.25 -0.75 0.50 000 -0.25 0,75 0.50 000 -0.25 o s

Figure 3

Contour plots of the steady wave patterns due to the Series-60-cb =0.6 vessel advancing at Froude numbers F = 0.20,0.25,0.35.

-1 50 -1.00 -0.50 0.00 0.50

downstream. At F = 0.3, the

time-harmonic wave fields around the modified Wigleymodel are illustrated in

Fig-ure 5 and are obtained from the same grid as for F = 0.2.

For this larger Froude number, the wavelengths appearing

in Figure 5 are larger than their counterparts of Figure 4,

although the general structure of the wave field is similar.

Figure 6 illustrates the wave patternsaround the

Series-60-Cb = 0.7 hull advancing at F=0.2 and heaving at fre-quencies

/Z7=3 and

IL/gr4. Relative to the

cor-responding patterns generated by theWigley hull, the

di-verging wave system originating from the stern is more

pro-nounced and is attributed to the more three-dimensional

shape of the Series-60 geometry. In all cases the steady wave pattern has beenremoved.

Certain common features of these three-dimensional time harmonic wave patterns are worth emphasizing. The

short-est wavelength scales are associated with the transverse

wave system which appearsdownstream of the stern and

FIgure 4 : Snapshot8 of the time-harmonicwave patterns due to amodified Wigley model

ad-vancing at F=0.20 while oscillai.ing in heave at frequencies w/L/g=3.0,5.0.

propagates in the streamwise direction. Along the ship length, on the other hand, the wave field is dominated

by relatively long divergent waves which propagate in the

transverse direction and tend to be become more two

di-mensional as the frequency increases. This character of the time harmonic wave pattern therefore appears to

sup-port thi

" '.

-''dv

'rrv

Nes' he

ship hull the wave disturbance is convected primarily in the

transverse direction and becomes more focused as the fre- -quency increases. Its variation in the lengthwise direction is gradual since cancellation effects appear to significantly reduce the amplitude of the short transverse waves which

7. HYDRODYNAMIC FORCES AND MOTIONS

IN HEAD WAVES

The unsteady hydrodynamic pressure on the hull s

eval-uated from expression (3.2). The restoring component of the pressure which depends on the ship displacement and the gradients of the steady flow has been neglected since it been found to be small for the ship hulls and Fraude numbers considered below. The gradients of the steady

and time-harmonic potentials are obtained from the formal differentiation of the spline representation of the velocity

potential (5.1). Integration of the pressure over the hull

according to expressions (3.5), allows the determination of the added-mass, damping coefficients and exciting forces

from expressions (3.5), and Response Amplitude

Opera-tors from the solution of the linear system (3.6). Only the

coupled heave and pitch modes of motion in head waves

are considered in this paper.

In order to establish the convergence of the solution

algo-rithm, a systematic study of the effect of grid density on

the computations of the hydrodynamic coefficientswas

car-ried out for a modified Wigley model with L/B = 10 and

BIT = 1.6. The time-harmonic wave flow was solved ata

Fraude number F=0.3 for several frequencies of oscillation

in the range of practical interest /'Z7E [2.5,5.0] The free surface domain was truncated at a distance 0.25E

upstream of the bow, 0.5L downstream of the stern and L in the transverse direction. _{Four different grids were} considered, resulting in a systematic increase of the dis-cretization density on both the free surface and the hull.

These grids use 20, 30, 40 and 50 panels along the length

of the hull, respectively, while for all of them the aspect

ratio of the free surface panels is equal to 1.

Computations of the heave and pitch added-mass and damp-ing coefficients obtained from these grids, are illustrated in Figure 7. _{The convergence rate is very satisfactory and}

Figure 5

_{: Snapshots of the time-harmonic wave patterns due to a modified Wigley model} ad-vancing at Fr0.30 while oscillating

_{in heave at frequencies w.,/L/g30,5.0.}

appears not to depend strongly on the frequency. Having established the convergence of the numerical algo-rithm, the hydrodynamic coefficients and ship motions ai-e

next compared to experimental measurements and strip theory. A systematic set of experiments for a modified

Wigley hull were recently conducted by Gerritsma(1986).

The diagonal heave and pitch added-mass and damping

coefficients at F = 0.3 are illustrated in Figure 8.

The

experimental measurements are compared to strip theory

and the present method. The solid line, hereafter denoting results from SWAN (Ship WaveANalysis), is based on the double-body free-surface condition (2.9) and the complete

treatment of the rnterms. The Neumman-Kelvin curve is

obtained from the solution of the linearized problem using

the present Rankine panel method and is obtained by ap-proxirnating the steady flow by the uniform stream -Ux

both in the free-surface and body boundary conditions.

The agreement between SWAN and experiments is quite

satisfactory and represents an improvement over strip the-ory. For the diagonal coefficients, SWAN and the Neumman

Kelvin problem are in good qualitative and quantitative

agreement.

Significant differences between the three theoretical pre-dictions occur in the heave and pitch cross-coupling co-efficients illustrated in Figure 9. These coefficients are

known to be sensitive to end-effects, therefore their ac-curate prediction requires the complete treatment of the

rnterms which attain large values near the ship ends.

This is confirmed by the very good agreement between

SWAN and the experimental measurements. In spite of its three-dimensional character, the departure of the Neumman Kelvin solution from the experiments is mainly attributed

to the incomplete treatment of the rnterms.

-Figure 6 :

Snapshots of the time-harmonic wave patterns due to the Series-60-c6 = 0.7 vessel

advancing at F=020 while oscillating in heave at frequencies ./Z7=3.0,4.0.

g, o N o w o e e N q. + + e e +

Of interest is also the observed symmetry of the experimen-tal measurements and the SWAN predictions of the cross-coupling coefficients. The modified Wigley hull is sym-metric fore and aft and a generalization of the Timman-Newman symmetry relations appears to hold. The

origi-nal Timman-Newman relations were shown to be exact for submerged vessels and the Neumman-Kelvin free-surface

condition. It is here conjectured that they are also exactly

valid for surface piercing vessels when the free-surface

con-dition is based on the double-body flow. No proof has yet been attempted using the condition (2.9).

o L ? q. N 4 4 I

Figure 10 compares experimental measurements with the

strip..theory and SWAN and predictions for the heave and

pitch exciting-force and motion modulus and phase. The

pitch radius of gyration of the modified Wigley hull is k5=

0.25L, and the center of gravity is taken at z = y = z = O. The agreement of SWAN with the experiments is in all cases very satisfactory. The strip-theory predictions have been obtained from the MIT 5-D Ship Motion program which is regarded a standard strip-theory code. The

dis-crepancy between the strip-theory and experimental heave

and pitch resonant frequencies, is attributed to the poor

prediction of the b and the cross-coupling coefficients by

strip theory (Figures 8 and 9).

r- e o + x Discretization (A) Discretization (B)

+ Discretization (A)

F. Discretization (B) A Discretization (C) Discretization (C)

\DiscrezaretiZa;on

(D) -

-o- Discretization (D)

0o 3.00 1.00 5.00 8.00 00 3.00 4 00 5 00 E 00

Figure 7 :

Numerical convergence study for the heave and pitch hydrodynamic coefficients of a modified Wigley model advancing at F = 0.3.

C o

o

r,

o

o nl

o

o

'-3 e.

e

o

-e

Experiments

-. - Strip Theory

- SWAN

- -- - Noimann-}Celvin 4 4 e e

o

o

'2.00 6.00

FIgure 8

: Diagonal hydrodynamic coefficients in heave and pitch for a modified Wigley model

advancing at Froude number F =0.3.

Figures 11 and 12 compare experiments with the strip the-ory and SWAN predictions of the heave and pitch

added-mass and damping coefficients of the Series-60-Cb = 0.7

model, advancing at Froude number F = 0.2. The experi-mental data are due to Gerritsma, Beukelman and

Glans-dorp (1974). The performance of SWAN is in all cases very satisfactory, offereing a significant improvement over strip

theory.

Due to the fore-aft asymmetry of the Series-60 model, the Timman-Newman relations for the cross-coupling

coeffi-cients do not hold. It is interesting, however, to notice

that the curves corresponding to035 and b35 are very close

o L

e

o

u., r-Q

o

o

u.,

e

o

e

200

o

Experiments

s

_.- Strip Theory

_{- SWAN}

5

--

Neumann-Kelvin

e

-o

to being mirror images of the those corresponding to 053

and C53 respectively about a non-zero value. In strip the-

-ory, for example, it may be shown easily that 035

-and b35 - b53 are symmetric about the corresponding

co-efficients at zero forward speed (F=0), but no such proof

is yet available in three dimensions.

The Series-60 heave and pitch motion amplitude and phase are shown in Figure 13. The agreement between theory and experiments is again satisfactory for both strip-theory and

SWAN, with a slight detuning of the strip-theory

predic-tions again attributed to its discrepancies with experiments in the cross-coupling coefficients andb55.

3.00 4.00 5.00 6.00

o o w o o

I

o e N 0 o o e e r', o O a

o

Experiments

-. - Strip Theory

SWAN Neumann-Kelvin a e '2.00

Figure 9

_{Cross-coupling bydrodynarnic coefficients between heave and pitch for a modified}

Wigley model advancing at Froude number Fr0.3.

8. CONCLUSIONS AND FUTURE WORK

A new three-dimensional Rankine Panel _{Method method,} referred to as SWAN, has been developed for the solution of the complete three-dimensional steady and time-harmonic

ship-motion problem. Its principal attributes are:

P' L r' 2.0 Experiments

.Strip Theory

- SWAN

Neumann-Kelvin s a 3.0 4.0 5.0 6.0

The use of a new free-surface condition based on the double-body flow and valid uniformly from low to high

Froude numbers.

The complete and accurate treatment of the

rnterms.

A high-order non-dissipative numerical algorithm for the enforcement of the free-surface and radiation conditions.

3.00 4.00 5.00 6.00 J o o e o Q a

a e o o C o e e o o s 4 e - .-. Experiments

-. - Strip Theory

-SWAN

s e s-I i i

Í.

-.

o

-

ExperimentsStrip Theory

S WAN

¡e

- «

t.0

t.5

2.0 2.5 .5 t.0 LS 2.0 2.5

A/L A/L

Figure 10 : Heave and pitch exciting forces and motions of a modified Wigley model advancing at

Froude number F=O.3 through regular head waves.

o e o o C e

-o N 'C o o o o o w o ç', o o o

500

Experimente

- - - Strip Theory

- SWAN

3.00 4.00 5.00 6.00

Computations of steady and time-harmonic ship wave pat-tenis illustrate the capability of the method to resolve con-siderable detail in the wave disturbance and at a significant downstream of the ship.

Predictions of the heave and pitch added-mass, damping coefficients, exciting forces and motions of a Wigley and

the Series-60 hull are found to be in very good agreement

with experiments and present a significant improvement

over strip theory. A complete treatment of the mternis

o o, 's o o o

Figure 11 : Diagonal hydrodynarnic

coefficients in heave and pitch for the Series-60-c6=O.7 _vessel

advancing at Froude number F=0.2.

has been developed and found to be essential for the accu-rate prediction of the cross_coupling coefficients_{and ship} motions.

In summary, all important features of the three-dimensional time-harmonic flow around the ship_{appear to be well pre-}

-dicted by the present method. This will_{permit the}

accu-rate prediction of the hydrodynamic pressure distribution, wave loads, derived responses and added-resistance_by

di-rect use of the velocity potential and its gradients on the

ship hull and the free surface.

V Q o

o

o

e

a

r-Q

o

o

b_o

o

o

o

o

a

('j Q Experiments Strip Theory

- SWAN

I F F

o

'2.00 .00 4 . 00 5.00 6.00

Figure 12

Cross-coupling hydrodynarnic coefficients between heave and pitch for the Series-60-Cb=O.7 vessel advancing at Froude number F=O.2.

Future research towards the further development of the present rankine panel method in the steady problem,will concentrate upon thedetermination of the ship wave

spec-trum from the available numerical data over the discretized

portion of the free surface. This information is useful for the characterization of ships from their Kelvin wake and the accurate and robust evaluation of the wave resis-tance. The properimplementation of the present

numeri-cal scheme to hull forms with significant flare willalso be

studied in both the steadyand time-harmonic problems.

The application is also planned of the samemethod to the

prediction of the seakeeping properties ofunconventional

ship forms (e.g. SWATH ships and SES's) the

hydrody-narnic analysis of which is particularly amenable by the

"o

'J _o 't,

vi

Q Esperimenta Strip Theory SWAN

-i

i

¡

I

Th.5 1.0 1.5 2.0 2.5

Figure 13 : Heave and pitch

_{motions of the Series-60.cb=O.7 vessel advancing at Froude number} F= 0.2 through regular head waves.

9. ACKNOWLEDGEMENTS

This research has been supported by the Applied

Hydrorne-chanics Research Program administered by the Office of

Naval Research and the David Taylor Research Center (Con-tract: N00167-86-K-OOl0) and by A. S. Ventas Research of

Norway. The majority of the computations reported in this

paper were carried out on the National Science Founda-tion Pittsburgh YMP Cray underthe Grant 0CE880003P. This award is greatly appreciated. We are also indebted to the Computer AidedDesign Laboratory of the

Depart-ment of Ocean Engineering at MIT for their assistance in

the preparation of the time-harmonic ship wave patterns

on their IRIS Workstation.

REFERENCES

Chang, M.-S., 1977, 'Computations of three-dimensional ship motions with forward speed', 2nd International

Con-ference on Numerical Ship Hydrodynamics, USA.

Dawson, C. W., 1977, 'A practical computer method for

solving ship-wave problems', 2nd International Conference on Numerical Ship Hydrodynamics, USA.

Eggers, K., 1981, 'Non-Kelvin Dispersive Waves around

Non-Slender Ships', Schiffstechnik, Bd. 28.

Faltinsen, O., 1971, 'Wave Forces on a Restrained Ship in

Head-Sea Waves', Ph.D. Thesis, University of Michigan,

USA.

Gadd, G. E., 1976, ' A method of computing the flow and surface wave pattern around full forms', Trans. Roy. Asst. Nay. Archit., Vol. 113, pg. 207.

Gerritsma, J., 1986, 'Measurments of Hydrodynamic Force and Motions for a modified Wigley Model', (unpublished). Gerritsma, J., Beukelman, W., and Glansdorp, C. C., 1974,

'The effects of beam on the hydrodynamic characteristics

of ship hulls', 10th Symposium on Naval Hydrodynamics,

USA.

Guevel, P., and Bougis, J., 1982, 'Ship Motions with

For-ward Speed in Infinite Depth', International Shipbuilding

Progress, No. 29, pp. 103-117.

Inglis, R. B., and Price, W. G., 1981, 'A Three-Dimensional

Ship Motion Theory - Comparison between Theoretical

Predictions and Experimental Data of Hydrodynamic Co-efficients with Forward Speed', Transactions of the Royal In8titution on Naval Architects, Vol.124, pp. 141-157. King, B. K., Beck, R. F., and Magee, A. R., 1988, 'Seakeep-ing Calculations with Forward Speed Us'Seakeep-ing Time-Domain Analysis', 17th Symposium on Naval Hydrodynamics, The Netherlands.

Korvin-Kroukovsky, B. V., 1955, 'Investigation of ship

mo-tions in regular waves', Soc. Nov. Archit. Mar. Eng., Trans. 63, pp. 386-435.

Maruo, H., and Sasaki, N., 1974, 'On the Wave Pressure

Acting on the Surfa.ce of an Elongated Body Fixed in Head Seas', Journal of the Society of Naval Architects of Japan, Vol. 136, pp. 34-42.

Nak, D. E., 1990, 'Ship Wave Patterns and Motions by a

Three-Dimensional Rankine Panel Method', Ph.D. Thesis, Mass. Inst. of Technology, USA.

Nak, D. E., and Sciavounos, P. D., 1990, 'Steady and

Un-steady Ship Wave Patterns', Journal of Fluid Mechanics,

Vol 215, pp. 265-288.

Nestegard, A., 1984, 'End effects in the forward speed

ra-diation problem for ships', Ph.D. Thesis, Mass. Inst. of

Technology, USA.

Newman, J. N., 1978, 'The theory of ship motions',

Ad-vances in Applied Mechanics, Vol. 18, pp. 221-283.

Newman, J. N., and Sciavounos, P. D., 1980, 'The Uni-fied Theory of Ship Motions', 13th Symposium on Naval

Hydrodynamics, Japan.

O'Dea, J. F., and Jones, H. D., 1983, 'Absolute and relative motion measurment.s on a model of a high-speed contain-ership', Proceedings of the 20th ATTC, USA.

Ogilvie, T. F., and Tuck, E. O., 1969, 'A rational Strip Theory for Ship Motions - Part 1', Report No. 013, Dept. of Naval Architecture and Marine Engineering, Univ. of

Michigan, USA.

Piers, W. J., 1983, 'Discretization schemes for the

mod-elling of water surface effects in first-order panel methods for hydrodynamic applications', NLR report TR-83-093L, The Netherlands.

Salvesen, N., Tuck, E. O., and Faltinsen, 0., 1970, 'Ship motions and wave loads', Soc. Nao. Archit. Mar. Eng.,

Trans 78, pp. 250-287.

Sclavounos, P. D., 1984a, 'The Diffraction of Free-Surface Waves by a Slender Ship', Journal of Ship Research, Vol. 28, No. 1, pp. 29-47.

Sciavounos, P. D., 1984b, 'The unified slender-body the-ory : Ship motions in waves' 15th Symposium on Naval

Hydrodynamics, Germany.

Sclavounos, P. D., and Nakos, D. E., 1988, 'Stability anal-ysis of panel methods for free surface flows with forward

speed', 17th Symposium on Naval Hydrodynamics, The

Netherlands.

-Timman, R., and Newman, J. N., 1962, 'The coupled damp-,

ing coefficients of symmetric ships', Journal of Ship Re.

search, Vol. 5, No. 4, pp. 34-55.

Yeung, R. W., and Kim, S. H., 1984, 'A New Development in the Theory of Oscillating and Translating Slender Ships',

15th Symposium on Naval Hydrodynamics, Germany.

Zhao, R., and Faltinsen, 0., 1989, 'A discussion of the

rn-terms in the wave-current-body interaction problem', 3rd International Workshop on Water Waves and

DISCUSSION William R.

McCreight David Taylor

Research Center, USA Your predictions_{of added}

mass and damping

for the Series_{60 hull}

are better than

those for the_{Wigley hull,}

yet the motion

predictions

are not as good.

Could you_{describe the}

accuracy on the_{Series 60}

exciting-force

computations, which are_{not shown.}

If this does not

account for the

discrepancy, what do you

believe is the_{cause of this?}

AUTHORS' REPLY

In response _{to Dr.}

McCreight's_{question we}

want to state_{that the}

calculation of_{the heave/pitch}

exciting forces_typically

compare very well with

corresponding

experimental data.

Discrepancies between

the numerical _and

experimental

results for the motions _{of the}

Series-60 may be partly

attributed to the speed

dependent portion_of

the restoring

force, which _{was not}

included in the presented

calculations.

Additional

differences may also arise_{due to}

ambiguities about the

appropriate values for_{the pitch}

moment of inertia_{and the} vertical position of the center

of gravity,_{as well as about}

the location of the point

about which_{the heave/pitch}

motions are referenced. DISCUSSION Hoyte Raven Maritime Research Institute Netherlands, The Netherlands

This paper is_{very interesting}

for me,

in particular, as it addresses

some points_{studied in}

my paper. _{I have a}

question on _{the steady}

wave resistance.

You found

differences in_{the remote}

wave pattern

between the _{Kelvin and}

the show-ship

condition. _{These may,}

however, be due to subtle

changes in

interference between wave components. _{Did you}

find any

substantial

difference in wave

resistance? Secondly,_{as you noticed}

your free surface

condition is

intermediate in

form between_{those of}

Dawson and_{Eggers, 1979.} 1

have implemented

your FSC in our code _{to make}

the same

comparisons _{as in my}

paper, and

found that the result_{was also}

intermediate for the Series_{60 C5=0.60}

model: the

predicted Rw_is

6-8% lower than with

Dawson's

condition, while

Eggers is 20%

lower. _{For a full}

hull form, again the

resistance is lower than Dawson. but

better behaved _{than Egger's}

condition. _{Ref. Raven,}

H.C., 'Adequacy

of Free Surface

Conditions for the Wave

Resistance

Problem,' this_volume. AUTHORS' REPLY

We would like to thank

Dr. Raven

for implementing

and testing the free surface

condition

proposed in this paper.

The differences of the

wave patterns,

as predicted _{by different}

free surface

linearization models are indeed _reflected

on the

corresponding_{wave resistance} calculations. _{We strongly}

believe,

however, that _numerical'

evaluation of the relative

performance of _different

linearization

models is still

clouded due to the delicate

nature of the

underlying calculations.

The robustness_{of each}

scheme ought to be established individually before

comparison

argumenta _{can be stated.}

We are

currently working _towards this direction by employing the conservation of momentum as the self-consistency criterion ([1]).

[I] _{Nakos, DE.,}

1991,

'Transverse Wave Cut

Analysis by a

Rankine Panel

Method,' 6th _{lin. Workshop}

on Water

Waves and Floating Bodies,

Woods Hole,_{MA, USA.}