ABSTRACT

A Rankine Panel Method is presented for the solution ofthe complete three-dimensional steady and time-harmonicpotential 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 uniformlyfrom low to high Froude numbers.

Computations of the steady ship wave patterns reveal sig-nificant detail in the Kelvin wake a significant distancedownstream of the ship, permitted by the cubic order andzero numerical damping of the panel method. The wavepattern appears to be sensitive to the selection of the free-surface condition only for full ship forms.

The heave and pitch hydrodynamic coefficients, excitingforces and motions of a Wigley and a Series-60 hull havebeen evaluated in head waves over a wide range of fre-quencies and speeds. A robust treatment is proposed ofthe rnterms which are found to be critical importancefor the accurate solution of the problem. Tri all cases theagreement with experiments is very satisfactory indicatinga significant improvement over strip theory, particularly inthe cross-coupling and diagonal pitch damping coefficients.

1. INTRODUCTION

Theoretical methods for the prediction of the seakeeping ofships 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 motionproblem 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 ofseakeeping research which aims at the numerical solutionof the three-dimensional problem. This paper presents ourprogress in that direction.

The pioneering work of Korvin-Kroukovsky (1955) stimu-lated a number of studies on the strip method which led tothe theory of Salvesen, Tuck and Faltinsen (1970). Its pop-ularity to date arises from its satisfactory performance inthe prediction of the motions of conventional ships and itscomputational simplicity. Well documented are howeverits limitations in the prediction of the derived responses,

21

TECHNISCHE UNIVERSITEIT..aboratorf um voor

Scheepehydromechanicarchief

Mekeiweg 2, 2628 CD DeiftTeLi 015-7a6873. Fax 015 781838

Ship Motions by a Three-DimensionalRankine Panel Method

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

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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 tothe seakeeping of slender ships. The rational justificationof strip theory, as a method valid at high frequencies andmoderate Froude numbers, was presented by Ogilvie aridTuck (1969). This theory was extended to the diffrac-tion problem by Faltinsen (1971) and was further refinedby Maruo and Sasaki (1974). The high-frequency restric-tion in earlier slender-ship theories was removed by theunified theory framework presented by Newman (1978).Its extension to the diffraction problem was derived bySciavounos (1984) and applied to the seakeeping of shipsby Newman and Sclavounos (1980) and Sclavounos (1984).Subsequent slender-ship studies by Kim and Yeung (1984)and Nestegard (1986), accounted directly for convectiveforward-speed wave effects near the ship hull and repre-sented the transition to numerical studies aiming at thesolution of the three-dimensional ship-motion problem.

By the mid-SO's, the performance of slender-body theoryfor the seakeeping problem could only be validated fromexperimental measurements. Moreover, it had become ev-ident that end-effects at high Froude numbers cannot bemodelled accurately by slender-body approximations andthe need for a numerical solution of the complete three-

-

dimensional had emerged. Early efforts towards this goalby Chang (1977), Inglis and Price (1981) and Guevel andBougis (1982) were not conclusive because the significantcomputational effort necessary for the evaluation of thetime-harmonic forward-speed Green function limited thetotal number of panels used on the ship surface. Morerecently, King, Beck and Magee (1988) circumvented thisdifficulty by solving the same problem in the time domain,therefore making use of the zero-speed transient Greenfunction 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 ofGadd (1976) and Dawson (1977) motivated several anal-ogous studies which concentrated upon the prediction ofthe Kelvin wake and evaluation of the wave resistance. Theprincipal advantages of the method are twofold - the Rank-me singularity is simple to treat computationally and thedistribution of panels over the free surface allows the en-forcement of more general free-surface conditions with vari-

3$3itiC H31t33i10cv 1CC3)

rK,1fl13$mC,bÇßßta

no

able coefficients. A drawback of Rankine-panel methods isthat they require about twice as many panels as methodsbased on the distribution of wave singularities over the shipsurface 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-dimensionaltime-harmonic ship motion problem by a Rankine PanelMethod. 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 Sclavounosand Nakos (1988). The extension of this numerical anal-ysis to the time-harmonic problem is presented in Nakosand Sclavounos (1990). In this reference the convergenceproperties 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 flowsin three dimensions. This scheme is applied in this paperto the solution of the time-harmonic radiation/diffractionpotential flows around realistic ship hulls and the evalu-ation of the hydrodynamic forces and motions in regularhead waves.

A new three-dimensional free-surface condition is derived,using the double-body flow as the base disturbance due tothe forward translation of the ship. This is shown to bevalid uniformly from low to high Froude numbers and overthe entire frequency range. Known low-Froude-numberconditions 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 evaluatedfrom the solution of the three-dimensional double-bodyflow. An important property of the solution scheme is thatthe evaluation of the double gradients of the double-bodyflow is circumvented by an application of Stokes theorem.

Computations are presented of the steady wave patternstrailing 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 significantdetail of the Kelvin wake at a large distance downstreamof the ship. A comparison of the wave patterns obtainedform the Neumman-Kelvin and the more general doublebody free-surface conditions reveals good agreement forthe Wigley hull, while evident differences appear in therespective Series-flU wakes.

Predictions of the heave and pitch added-mass and damp-ing coefficients and exciting forces are found to be in verygood agreement with experimental measurements both forthe Wigley and the Series-60 hull. The contribution ofthe complete rnterms is found to be important, partic-ularly in the cross-coupling coefficients. The validity of amore general set of Timman-Newman relations is observedand conjectured in connection with free-surface conditionsbased 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 experiments and present an improvement overstrip theory.

22

2. TRE BOUNDARY VALUE PROBLEM

Define a Cartesian coordinate system = (z, y, z) fixedon 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 beexpressed relative to this translating coordinate system,therefore the flow at infinity is a uniform stream and theship hull velocity is due to its oscillatory displacement fromits mean position.

The fluid is assumed incompressible and inviscid and theflow 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 thecorrespoading 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 combinedwith Bernoulli's equation, leads to the dynamic free surfacecondition

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 additionalcondition must be imposed that at finite times the flowvelocity 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 shipis 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 typicallyadvance at low speed and cause a small steady wave distur-bance. Fine-shaped ships, on the other hand, often advanceat high Froude numbers. Yet the steady disturbances theygenerate, is small if their geometry is sufficiently thin orslender. Linearization may therefore be justified both atlow and high Froude numbers F, as long as it is tied to thehull slenderness e. Linearization of the unsteady flow isalso supported by the assumption of a small ambient waveamplitude.

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 sufficientlysmall. The details of the derivation outlined below aregiven in Nakos (1990). The total flow field W(î,t) is de-composed into a basis flow '(), assumed to be of 0(1), thesteady wave flow (ì), and the unsteady wave flow i/'(, t)

= . (2.6)

The double-body flow is chosen as the basis flow, a selectionprimarily motivated by the body boundary condition aswell 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 thedouble-body flow and are taken to be small relative to the

Linearization of (2.4-5), correct to leading order inand , 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)Jg \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 usedas the basis flow. In this case, (2.8-9) reduce to the well-known Neumann-Kelvin conditions. In the opposite limitof 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 byEggers (1981). This property may explain the fact that,even though Dawson's and Egger's conditions have beenderived as low Foude number approximations, they havebeen found to perform satisfactorily over a wider range offorward speeds.

Linearization of the body boundary condition

The linearization of the ship hull boundary condition mayalso be derived from the decomposition (2.6). By defi-nition, the velocity potential of the double-body flow is

23

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 ofthe steady wave problem.

The unsteady forcing due to the oscillatory motion of thevessel is accounted for by the unsteady wave flow l. Ifis the oscillatory displacement vector measured from themean position of the vessel (B), it follows by substitutingof (2.6) in (2.2) that

a' a.= . n V( + ) . , on (B) . (2.12)an at

Assuming that the magnitude of the displacement vectord is small and comparable to the ambient wave amplitude,the boundary condition (2.12) may be linearized aboutthe 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 mannerconsistent with the assumptions underlying the derivationof the free-surface conditions (2.8). An alternative formof (2.13) may be derived in terms of the rigid-body globaldisplacements (eI,e2,eo) and rotations (e4,eo,ee), alongthe axes (z, y, z) respectively,

=an + ern.) , on () , (2.14)

i='where tn, , j = 1, ...6, denote the so-called rn-terms fOgilvieand Tuck (1969)].

If the basis flow is approximated by the uniform stream theonly non-zero rn-terms are m5 = Un3 and m6 = Un3which merely account for the 'angle of attack effect' dueto yaw and pitch. This approximation of the rn-terms hasbeen employed in most previous studies of the ship motionproblem, consistently with the linearization steps leadingto the Neumann-Kelvin free surface boundary condition.The performance of this linearization in practice will bethe subject of numerical experiments presented in section7.

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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, asviewed from the stationary frame S Wo, while in the trans-lating frame of reference î, the incident wave arrives at

the frequency of encounter . 1f ß is the angle betweenthe phase velocity of the incident wave and the forwardvelocity of the ship, c is given by

w=IoU-cosß (2.15)

In the frame i, the velocity potential of the incident waveof unit amplitude, in deep water, is given by the real partof 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 bodyboundary condition (2.14), suggest the decomposition ofthe wave flow as follows,

Given the solution of the potential flow problem formu-lated in the preceding section, the hydrodynamic pressurefollows from Bernoulli's equation. Of particular interest,in practice, is the pressure distribution on the ship wettedsurface and resultant forces and moments necessary for thedetermination 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 inçt', 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 atthe 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)

24

F(t) = {ei

where,

X=

[Ax. + (w2a,5ib - ci1)]

}

p{ii i(çao + ça7) + V V(ça + ça7)]d8

(B)

= - {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 forwardspeed and the frequency of oscillation . The restoringcoefficients ci,, on the other hand, include the classical hy-drostatic contribution augmented by a dynamic term dueto the gradients of the double-body flow. The latter con-tribution depends linearly upon the deflection of the shipsurface from its mean position and quadratically on theship speed. It is therefore expected to be substantial athigh Fraude numbers.

The equations governing the time-harmonic responses ofthe ship follow from Newton's law. Using the definitions(3.5) of the forces acting on the hull, the familiar six-degreeof 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 inpitch and roll due to the corresponding displacement of thecenter 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 sourcepotential,

G(i) = 2 (4.1)

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

= e"' A(çao + ça7) +5= 1

eis]}

, (2.17)

where A is the amplitude of the incoming wave train, çOis the complex diffraction potential, and ça5, j =are the complex radiation potentials due to the harmonicoscillation of the ship in each of the six rigid-body degreesof freedom, at frequency .ì and 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 conditionfor the diffraction problem is inhomogeneous, the forcingarising from the interaction of the incoming wave train withthe double-body flow. In the limit of slender/thin ships,where the uniform stream may be taken as the basis flow,this inhomogeneity vanishes.

3. THE HYDRODYNAMIC FORCES

4,() Ifa4,(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 surfaceintegrals over the control surface (S,,) can be shown tovanish in the limit as (S) is removed to infinity withkept finite.

The derivatives of c, 4, and ço normal to the ship surface(B) are known. The corresponding vertical derivative onthe 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 overthe 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 evaluationof gradients of the solution potential, low-order panel meth-ods are known to be sensitive to discretization error, unlesstheir implementation and panel distribution is carefully se-lected. The evaluation of double gradients of the solutionare known to introduce serious difficulties, as illustrated byNestegard (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 conditionon 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 derivativesof '' on the hull, consequently it is clearly superior fromthe computational standpoint.

The integral equation (4.2) will not accept unique solutionsunless a radiation condition is imposed enforcing no wavesupstream. In practice the solution domain of (4.2) on thez = O plane will be truncated at a rectangular boundarylocated at some distance from the ship where appropriate'end conditions' will be imposed enforcing the radiationcondition. Due to the convective nature of the flow, thecondition at the upstream boundary is the most criticaland takes the form

(4.5)

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

25

two upstream conditions are discussed in detail in Sciavounoand Nakc8 (1988) for a two-dimensional steady flow, andare extended to time-harmonic flows in Nakos (1990). It isshown that both are necessary in order to ensure physicallymeaningful numerical solutions of the steady and unsteadyproblems. For r = wU/g > 1/4 no wave disturbance ispresent upstream of the ship and the conditions (4.5) canbe shown to enforce this property of the flow. For r < 1/4and with increasing Froude numbers, the amplitude of thewaves upstream of the ship decreases relative to that of thetrailing wave pattern and conditions (4.5) perform well ifthe truncation boundary is sufficiently removed from theship. No conditions are necessary on the transverse anddownstream truncation boundaries.

5. THE NUMERICAL SOLUTION ALGORITHM

The solution of integral equation (4.2) for the steady andunsteady 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 abi-quadratic spline-collocation scheme of cubic order, zeronumerical dissipation and capable to enforce accurately theradiation condition (4.5).

The boundary domain - including the ship hull and thefree surface solution domain - is discretized by a collectionof plane quadrilateral panels [see Figure 1. The unknownvelocity potential is approximated by the linear superposi-tion of hi-quadratic spline basis functions B(i), as follows

4,(i) 0 B (i) , (5.1)

where B1 is the basis function centered at the j'th paneland 0 is the corresponding spline coefficient. By collocat-ing the integral equation (4.2) at the panel centroids andenforcing the upstream condition (4.5), the discrete for-mulation follows in the form of a system of simultaneouslinear 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 fieldon the domain boundaries. The free surface elevation andhydrodynamic pressure are evaluated using the relations(2.8_9) and (3.1-2), respectively.

The error and stability analysis of the bi-quadratic splinescheme is presented in Nakc and Sclavounos (1990). It isbased on the introduction of a discrete dispersion relationgoverning the wave propagation over the discretized freesurface. Comparison of the continuous and discrete dis- -

persion relations allows the rational definition of the con-sistency, order and stability properties of the numericalsolution scheme. It is shown that the numerical dispersionis of 0(h3) where h is the typical panel size and that nonumerical dissipation is present. Both are valuable prop-erties for the computation of ship wave patterns which arenot substantially distorted, damped or amplified by thenumerical algorithm.

Essential for the performance of the method is a stabilitycondition 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

=1!iE (FS)ur) . (4.2)

and transverse directions, respectively. This condition,derived and discussed in detail in Nakos and Sciavounos(1990), establishes 'stable' domains on the (Ph, a) planewith boundaries dependent on the frequency of oscillation.

For a given a Froude number, a stable discretization for thehighest frequency of oscillation is stable for all lowest fre-quencies. Therefore, no regridding of the ship hull and freesurface is necessary for the solution of the time harmonicproblem over a range of frequencies. The resulting complexlinear system is solved by an accelerated block Gauss-Siedeliterative scheme which makes extensive use of out-of-corestorage therefore permitting the use of discretizations withseveral thousand panels.

Experimental verification of the convergence of the solu-tion algorithm has been established by comparing com-putations of 'elementary' flows around singularities andthin-struts with analytical solutions [Nakos and Sclavounos(1990) and Nakos (1990)]. The convergence of the hydro-dynamic added-mass and damping coefficients is discussedin Section 7.

6. STEADY AND UNSTEADY SHIP WAVEPATTERNS

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. Thissection presents converged computations of the steady andtime 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 ratioB/T 1.6. The grid used for the solution of the steadyproblem consists of 40x10 panels on half the hull, providingadequate resolution of the geometry, while the panels onthe free surface are aligned with those on the hull and havea typical aspect ratio is a = h/h5 = 1. The grid Froudenumber is Fh 6.3 . F, allowing an adequate resolution ofthe steady wave flow for Froude numbers as low as F =0.20[see Nakos (1990)]. The free surface domain is truncatedat a distance z,,=0.2L upstream of the bow and one shiplength downstream of the stern. The truncation in thetransverse direction is selected at y,,, zz0.75L, so that theentire 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 1110panels on half the configuration.

26

Figure 2 shows contour plots of the wave patterns resultingfrom the steady forward translation of the Wigley model

at F = 0.25,0.35,0.40 . Predictions based on both theNeumann-Kelvin and the double-body linearizations arepresented. Due to the slenderness of this Wigley model,the two wave fields agree well even at high speeds. Smalldifferences are visible along the diverging portion of thewave system which originates from the stern, where theNeumann-Kelvin solution tends to generate steeper waves,particularly along the caustic. The opposite appears to betrue 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

27

the selected linearization.

The second ship tested is the Series-60-05=0.6 hull whichis 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 gridused for the computations are the same to those employedfor the Wigley model.

Figure 3 illustrates the wave patterns around the Series-60 model for F = 0.20,0.25,0.35 , respectively. At lowspeeds (F < 0.30) the amplitude of the generated wavesare 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, :.gat Froude numbers F= 0.25, 0.35, 0.40.

-1 50 -1.00 -0.50 0.00 0.50

-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- andstern-wave systems are well formed while the correspond-ing wave pattern around the Series-60 hull appears to bemore 'confused'.

Differences between the steady wave pattern computationsfrom the Neumann-Kelvin and double-body linearizationsare here clearly noticeable. Again, significant discrepan-cies occur along the diverging portion of the stern-wavesystem, where the Neumann-Kelvin solution shows largeramplitudes and shorter wavelengths. Moreover, the caustic

0.50

0.00

-0.25

-0.75

0.50

0.00

-0.25

-0.75

28

lines originating from the bow and stern appear at a largerangle in the solution based on the double-body lineariza-tion. The differences between the two solutions becomemore pronounced as the speed increases, resulting in quitedifferent wave patterns at F =0.35 (see Figure 3c).

Figure 4 is a snapshot of the time-harmonic wave patternaround a modified Wigley model translating at F = 0.2and oscillating in heave at frequencies i27 = 3 and

/TL7=5. The grid used for this flow field has the samedensity 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 advancingat Froude numbers F = 0.20,0.25,0.35.

-1 50 -1.00 -0.50 0.00 0.50

-1 50 -1.00 -0.50 0.00 0.50

downstream. At F = 0.3, the time-harmonic wave fields

around the modified Wigley model 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 appearingin 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 patterns around 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 the Wigley 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 steadywave pattern has been removed.

Certain common features of these three-dimensional timeharmonic wave patterns are worth emphasizing. The short-

est wavelength scales are associated with the transversewave system which appears downstream of the stern and

FIgure 4 : Snapshot8 of the time-harmonic wave patterns due to a modified Wigley model ad-

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

29

propagates in the streamwise direction. Along the shiplength, on the other hand, the wave field is dominatedby relatively long divergent waves which propagate in thetransverse direction and tend to be become more two di-mensional as the frequency increases. This character ofthe time harmonic wave pattern therefore appears to sup-port thi " '. -''dv 'rrv Nes' heship hull the wave disturbance is convected primarily in thetransverse direction and becomes more focused as the fre- -

quency increases. Its variation in the lengthwise directionis gradual since cancellation effects appear to significantlyreduce the amplitude of the short transverse waves whichare clearly visible downstream of the stern.

7. HYDRODYNAMIC FORCES AND MOTIONSIN HEAD WAVES

The unsteady hydrodynamic pressure on the hull s eval-uated from expression (3.2). The restoring component ofthe pressure which depends on the ship displacement andthe gradients of the steady flow has been neglected sinceit been found to be small for the ship hulls and Fraudenumbers considered below. The gradients of the steadyand time-harmonic potentials are obtained from the formaldifferentiation of the spline representation of the velocitypotential (5.1). Integration of the pressure over the hullaccording to expressions (3.5), allows the determination ofthe added-mass, damping coefficients and exciting forcesfrom expressions (3.5), and Response Amplitude Opera-tors from the solution of the linear system (3.6). Only thecoupled heave and pitch modes of motion in head wavesare considered in this paper.

In order to establish the convergence of the solution algo-rithm, a systematic study of the effect of grid density onthe computations of the hydrodynamic coefficients was car-ried out for a modified Wigley model with L/B = 10 andBIT = 1.6. The time-harmonic wave flow was solved at aFraude number F=0.3 for several frequencies of oscillationin the range of practical interest /'Z7E [2.5,5.0]The free surface domain was truncated at a distance 0.25Eupstream of the bow, 0.5L downstream of the stern andL in the transverse direction. Four different grids wereconsidered, 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 lengthof the hull, respectively, while for all of them the aspectratio 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 inFigure 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.

30

appears not to depend strongly on the frequency.

Having established the convergence of the numerical algo-rithm, the hydrodynamic coefficients and ship motions ai-enext compared to experimental measurements and striptheory. A systematic set of experiments for a modifiedWigley hull were recently conducted by Gerritsma(1986).The diagonal heave and pitch added-mass and dampingcoefficients at F = 0.3 are illustrated in Figure 8. Theexperimental measurements are compared to strip theoryand the present method. The solid line, hereafter denotingresults from SWAN (Ship WaveANalysis), is based on thedouble-body free-surface condition (2.9) and the completetreatment of the rnterms. The Neumman-Kelvin curve isobtained from the solution of the linearized problem usingthe present Rankine panel method and is obtained by ap-proxirnating the steady flow by the uniform stream -Uxboth in the free-surface and body boundary conditions.

31

The agreement between SWAN and experiments is quitesatisfactory and represents an improvement over strip the-ory. For the diagonal coefficients, SWAN and the NeummanKelvin problem are in good qualitative and quantitativeagreement.

Significant differences between the three theoretical pre-dictions occur in the heave and pitch cross-coupling co-efficients illustrated in Figure 9. These coefficients areknown to be sensitive to end-effects, therefore their ac-curate prediction requires the complete treatment of thernterms which attain large values near the ship ends.This is confirmed by the very good agreement betweenSWAN and the experimental measurements. In spite of itsthree-dimensional character, the departure of the NeummanKelvin solution from the experiments is mainly attributedto the incomplete treatment of the rnterms.

-

-

Figure 6 : Snapshots of the time-harmonic wave patterns due to the Series-60-c6 = 0.7 vesseladvancing at F=020 while oscillating in heave at frequencies ./Z7=3.0,4.0.

g,o

No

w

oe

e

N

q.

+ +

ee

+

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 forsubmerged vessels and the Neumman-Kelvin free-surfacecondition. It is here conjectured that they are also exactlyvalid for surface piercing vessels when the free-surface con-dition is based on the double-body flow. No proof has yetbeen attempted using the condition (2.9).

32

o

L

?q.

N

4 4 I

Figure 10 compares experimental measurements with thestrip..theory and SWAN and predictions for the heave andpitch exciting-force and motion modulus and phase. Thepitch 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 allcases very satisfactory. The strip-theory predictions havebeen obtained from the MIT 5-D Ship Motion programwhich is regarded a standard strip-theory code. The dis-crepancy between the strip-theory and experimental heaveand pitch resonant frequencies, is attributed to the poorprediction of the b and the cross-coupling coefficients bystrip theory (Figures 8 and 9).

r- eo

+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 amodified Wigley model advancing at F = 0.3.

C

oo

r,oo

nloo

'-3

e.eo

-e

Experiments-. - Strip Theory- SWAN- -- - Noimann-}Celvin

4 4

ee

oo'2.00 6.00

FIgure 8 : Diagonal hydrodynamic coefficients in heave and pitch for a modified Wigley modeladvancing 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 = 0.7model, 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 verysatisfactory, offereing a significant improvement over striptheory.

Due to the fore-aft asymmetry of the Series-60 model, theTimman-Newman relations for the cross-coupling coeffi-cients do not hold. It is interesting, however, to noticethat the curves corresponding to 035 and b35 are very close

o

L

e

o

u.,r-Qo

ou.,e

33

oe200

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 proofis yet available in three dimensions.

The Series-60 heave and pitch motion amplitude and phaseare shown in Figure 13. The agreement between theory andexperiments is again satisfactory for both strip-theory andSWAN, with a slight detuning of the strip-theory predic-tions again attributed to its discrepancies with experimentsin the cross-coupling coefficients and b55.

3.00 4.00 5.00 6.003.00 4.00 5.00

oo

woo

Ioe

N0o

oe

e

r',o

Oa

o Experiments-. - Strip Theory

SWANNeumann-Kelvin

a

e'2.00

Figure 9 Cross-coupling bydrodynarnic coefficients between heave and pitch for a modifiedWigley 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 ofthe complete three-dimensional steady and time-harmonicship-motion problem. Its principal attributes are:

34

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 thedouble-body flow and valid uniformly from low to highFroude 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

Jo

o e

oQ

a

35

ae

o

oC

o

ee

oo

s

4

e

- .-.

Experiments-. - Strip Theory-SWAN

s es-

I

i i

Í.-.o- Experiments

Strip TheoryS 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 atFroude number F=O.3 through regular head waves.

oe

o

oC

e

-

o

N

'C

oo

ooo

wo

ç',oo

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 significantdownstream of the ship.

Predictions of the heave and pitch added-mass, dampingcoefficients, exciting forces and motions of a Wigley andthe Series-60 hull are found to be in very good agreementwith experiments and present a significant improvementover strip theory. A complete treatment of the mternis

36

o

o,

'soo

o

Figure 11 : Diagonal hydrodynarnic coefficients in heave and pitch for the Series-60-c6=O.7 vesseladvancing 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 shipmotions.

In summary, all important features of the three-dimensionaltime-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 theship hull and the free surface.

3.00 4.00 5.00 6.00

VQo

oo

ea

r-Qo

o

boo

oo

oa

('jQ

ExperimentsStrip 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 the determination 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 proper implementation of the present numeri-

cal scheme to hull forms with significant flare will also be

studied in both the steady and time-harmonic problems.

37

The application is also planned of the same method to theprediction of the seakeeping properties of unconventionalship forms (e.g. SWATH ships and SES's) the hydrody-

narnic analysis of which is particularly amenable by thepresent three-dimensional' panel method.

"o'J o

't,

38

vi

Q EsperimentaStrip TheorySWAN-

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 numberF= 0.2 through regular head waves.

9. ACKNOWLEDGEMENTS

This research has been supported by the Applied Hydrorne-

chanics Research Program administered by the Office ofNaval Research and the David Taylor Research Center (Con-

tract: N00167-86-K-OOl0) and by A. S. Ventas Research ofNorway. The majority of the computations reported in this

paper were carried out on the National Science Founda-tion Pittsburgh YMP Cray under the Grant 0CE880003P.This award is greatly appreciated. We are also indebted

to the Computer Aided Design Laboratory of the Depart-

ment of Ocean Engineering at MIT for their assistance inthe preparation of the time-harmonic ship wave patterns

on their IRIS Workstation.

REFERENCESChang, M.-S., 1977, 'Computations of three-dimensionalship motions with forward speed', 2nd International Con-ference on Numerical Ship Hydrodynamics, USA.

Dawson, C. W., 1977, 'A practical computer method forsolving ship-wave problems', 2nd International Conferenceon Numerical Ship Hydrodynamics, USA.

Eggers, K., 1981, 'Non-Kelvin Dispersive Waves aroundNon-Slender Ships', Schiffstechnik, Bd. 28.

Faltinsen, O., 1971, 'Wave Forces on a Restrained Ship inHead-Sea Waves', Ph.D. Thesis, University of Michigan,USA.

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

Gerritsma, J., 1986, 'Measurments of Hydrodynamic Forceand Motions for a modified Wigley Model', (unpublished).

Gerritsma, J., Beukelman, W., and Glansdorp, C. C., 1974,'The effects of beam on the hydrodynamic characteristicsof ship hulls', 10th Symposium on Naval Hydrodynamics,USA.

Guevel, P., and Bougis, J., 1982, 'Ship Motions with For-ward Speed in Infinite Depth', International ShipbuildingProgress, No. 29, pp. 103-117.

Inglis, R. B., and Price, W. G., 1981, 'A Three-DimensionalShip Motion Theory - Comparison between TheoreticalPredictions and Experimental Data of Hydrodynamic Co-efficients with Forward Speed', Transactions of the RoyalIn8titution 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 Using Time-DomainAnalysis', 17th Symposium on Naval Hydrodynamics, TheNetherlands.

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

39

Nak, D. E., 1990, 'Ship Wave Patterns and Motions by aThree-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 NavalHydrodynamics, Japan.

O'Dea, J. F., and Jones, H. D., 1983, 'Absolute and relativemotion 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 StripTheory for Ship Motions - Part 1', Report No. 013, Dept.of Naval Architecture and Marine Engineering, Univ. ofMichigan, USA.

Piers, W. J., 1983, 'Discretization schemes for the mod-elling of water surface effects in first-order panel methodsfor hydrodynamic applications', NLR report TR-83-093L,The Netherlands.

Salvesen, N., Tuck, E. O., and Faltinsen, 0., 1970, 'Shipmotions and wave loads', Soc. Nao. Archit. Mar. Eng.,Trans 78, pp. 250-287.

Sclavounos, P. D., 1984a, 'The Diffraction of Free-SurfaceWaves 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 NavalHydrodynamics, Germany.

Sclavounos, P. D., and Nakos, D. E., 1988, 'Stability anal-ysis of panel methods for free surface flows with forwardspeed', 17th Symposium on Naval Hydrodynamics, TheNetherlands.

-

-

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 Developmentin the Theory of Oscillating and Translating Slender Ships',15th Symposium on Naval Hydrodynamics, Germany.

Zhao, R., and Faltinsen, 0., 1989, 'A discussion of thern-terms in the wave-current-body interaction problem',3rd International Workshop on Water Waves and Float-ing Bodies, Norway.

DISCUSSION

William R. McCreightDavid Taylor

Research Center, USAYour predictions of added mass and damping

for the Series 60 hullare better than

those for the Wigley hull, yet the motionpredictions

are not as good.Could you describe the accuracy on the Series 60

exciting-force computations, which are not shown. If this does notaccount 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/pitchexciting forces typically compare very

well withcorresponding

experimental data.Discrepancies between

the numerical andexperimental results for the motions of the

Series-60 may be partlyattributed to the speed

dependent portion ofthe restoring

force, which was notincluded in the presented

calculations. Additional differences may also arise due to ambiguitiesabout the appropriate values for the pitch

moment of inertia and thevertical position of the center

of gravity, as well as aboutthe location

of the pointabout which the heave/pitch

motions are referenced.

DISCUSSION

Hoyte RavenMaritime Research Institute

Netherlands, The NetherlandsThis paper is very interesting for me, in particular, as it addressessome points studied in my paper. I have a question on the steady

wave resistance.You found

differences in the remote wave patternbetween the Kelvin and the show-ship

condition. These may,however, be due to subtle

changes ininterference between wave

components. Did you find anysubstantial difference in wave

resistance? Secondly, as you noticedyour free surface

condition isintermediate in form between those of Dawson and Eggers, 1979. 1

have implementedyour FSC in our code to make the same

comparisons as in mypaper, and found that the result was also

intermediate for the Series 60 C5=0.60 model: thepredicted Rw is

6-8% lower than with Dawson'scondition, while

Eggers is 20%lower. For a full hull form, again the

resistance is lower thanDawson. but better behaved than Egger's

condition. Ref. Raven,H.C., 'Adequacy of Free Surface Conditions forthe Wave

ResistanceProblem,' this volume.

AUTHORS' REPLY

We would like to thankDr. Raven

for implementingand testing the

free surfacecondition proposed in this paper.

The differences of thewave patterns, as predicted by different free surface

linearizationmodels are indeed reflected on the

corresponding wave resistancecalculations. We strongly

believe, however, that numerical'evaluation of the relative

performance of differentlinearization

models is stillclouded due to the delicate

nature of theunderlying

calculations. The robustness of eachscheme ought to be established

individually beforecomparison argumenta can be stated.

We arecurrently working towards this direction by

employing theconservation of

momentum asthe self-consistency

criterion ([1]).[I] Nakos, DE., 1991,'Transverse Wave Cut

Analysis by aRankine Panel

Method,' 6th lin. Workshop on WaterWaves and

Floating Bodies,Woods Hole, MA, USA.

40