Computation of Wave Ship Interactions

\FoT Advances in Marine Hydrodynamics, M. Olikusu, Editor] [Computational Meclianics Publications, 1995]

Computation of Wave Ship Interactions

Paul D. Sclavounos

Department of Ocean Engineering Massachusetts Institute of Technology

Cambridge MA 02139, USA

Abstract

This chapter surveys the modeling and computation of free-surface potential flows past ships in calm water and in waves by a Rankine Panel Method (RPM). Physical models for steady and unsteady flow are reviewed and the fundamental numerical analysis underlying the development of Rankine Panel Methods is outlined. Several apphcations o f the method are discussed, including the computation of ship Kelvin wave patterns and resistance, motion and wave induced loads in the frequency and time domains and wave-added resistance of sailing yachts.

1. INTRODUCTION

Computer based simulations of free surface flows past ships and sailing yachts have enjoyed rapid growth in use and popularity since the early 80's and in recent years have been firmly established as a versatile and inexpensive design tool at the disposal of the modem naval architect.

Viscous and nonlinear free surface effects often play an important role in certain flow regknes around the ship hull, yet their numerical treatment still presents a major challenge to the computational fluid dynamicist. Examples include the inflow to the propeller, the spray and wave breaking around a ship bow and the splash zone of a slamming event in steep ambient waves. Progress towards the treatment of such flows is being reviewed in other chapters of this volume.

Where viscosity and strong free surface nonlinearities are confined in a small flow region and do not affect appreciably the quantity being predicted, e.g. wave resistance or motions in waves, significant progress has been achieved by implementing the potential flow

model. The Laplace equation is enforced in the fluid domain, subject to the condition o f zero normal flux on the ship hull, the dynamic and kinematic conditions on the exact or linearized positions o f the free surface and the "radiation condition" that the flow is quiescent at infinity over finite times. The corresponding exact and linearized statements of the boundary value problem are made in Section 2.

The solution o f the linear or nonlinear set of equations modeling the free surface potential flow past a ship advancing with forward speed is far from simple. We lack a proof that even the linearized mathematical problem is well posed, except for the simplest cases o f a thin ship o f vanishingly small beam and for certain submerged bodies. Early efforts towards the solution of the linearized steady and unsteady ship wave problems were based on the distribution of fimdamental wave singularities, or Wave Green Functions (WGF), over the ship surface, its centerplane or its axis, depending upon the desired accuracy! Perhaps all the WGF's relevant to ship flows have been derived in closed from in the treatise o f Wehausen and Laitone (1960). In spite o f the approximate nature of the free surface flow they model, WGF's have been the source of invaluable physical insight into steady and unsteady free surface flows past ships. As a result they have been the foundation for a number of approximate wave-ship theories, which were the primary source of theoretical advice to the practicing naval architect in 60's and 70's when computers lacked the capacity they enjoy today. Examples include the strip theories o f Salvesen, Tuck and Faltinsen (1971) and Ogilvie and Tuck (1969), their extension by the ordinary slender body theory of Maruo and Tokura (1978) & Maruo (1982) and the unified slender body theory of Newman (1978) and its extension by Sclavounos (1984). For all the wealth of information extracted from WGF's, they remain hard to compute efficiently, their mathematical statement depends upon the time dependence of the underlying flow (steady, frequency or time domain) and with few exceptions their use is limited to the solution of one class of linearized free surface flow problems known as Neumann-Kelvin problems. Partly as a result of the prohibitive computational effort necessary in the late 70's and early 80's for the direct solution of ship wave flows by a distribution of WGF's over the ship hull, a new class of methods emerged. In the pioneering papers of Gadd (1976) and Dawson (1977) the use was suggested o f the elementary singularity for infinite-domain potential flows, the Rankine Source, as an altemative to WGF's for the solution of ideal free surface flows past ships. The principal attractiveness of this alternative Green fianction lies with its simplicity and versatility. Unlike WGF methods, in Rankine Panel Methods panels must be distributed over the free surface. This permits the enforcement of a wide range of linear, higher-order or fiilly nonlinear free surface conditions irrespective of the time dependence of the underlying flow. Moreover, the computation of the hydrodynamic influence coefficients between pairs of panels may be carried out in essentially the same manner as in potential flows in aerodynamics. Therefore, the stage was set for a new computational method which appeared to offer flexibility and simplicity in its computational set up.

In RPM methods an attempt to represent the wealth of ship wave systems by a distribution of Rankine sources and dipoles over a mesh of panels on the free surface is bound to mn

into discretization errors which, unless properly controlled, may prove detrimental to the accuracy of the numerical solution. Moreover, numerical radiation conditions are necessary for the effective absorption of the energy carried by the ship waves away from the computational domain. In many respects these are issues not unlike those encountered in other areas, of computational fluid dynamics and more generally in the numerical solution of partial differential equations.

The first effective treatment of such discretization errors was proposed by Dawson in his original RPM method by using an upwind finite difference scheme for the enforcement of the parabolic-like free surface condifion in the steady flow past a ship. Subsequent RPM's refined Dawson's scheme yet in most cases preserved its principal attributes, namely the solution of the steady flow, the representation of the velocity potential as a distribution of sources only over the ship hull and the free surface and the use of upwind differencing for the enforcement of the free surface and radiation conditions. A nonexhaustive list o f such studies for steady flow includes Raven (1992), Jensen and Soding (1989), Xia and Larsson (1986), Reed et al. (1990) and Rosen et al. (1993). Extensions to time domain forward speed ship flows have been presented by Maskev/ (1992) and Beck et al. (1993). Yet, a rational numerical analysis of the RPM method analogous to that routinely carried out in other areas of computational fluid dynamics was lacking. Extending loosely the Lax Equivalence Theorem [Ritchmeyer and Morton (1967)], for any free surface discretization by an RPM to be convergent it has to be shown to be consistent and stable. The consistency (order) and stability of RPM methods were addressed rationally in Sclavounos and Nakos (1988) for steady, Nakos and Sclavounos (1992) for time-harmonic and Nakos (1993) and Vada and Nakos (1993) for time-domain free surface flows with forward speed.

Several conclusions o f practical importance in the design o f RPM's were drawn from these studies. Upwind differencing introduces numerical damping into the discretization of steady free surface flows, with magnitude and sign which depend upon the choice o f the finite difference formula and the parameters of the free surface discretization. This damping is analogous to the Rayleigh viscosity, the device used to enforce the radiation condhion in the analytical derivation of WGF's. The drawback of upwind differencing is that the magnitude and sign of the numerical damping may not be possible to control with increasing numbers o f panels.

Rankine Panel Methods free of numerical damping and small numerical dispersion are possible to derive and they enjoy superior convergence properties relative to RPM methods using upwind differencing. With damping-free methods, a radiation condition must be enforced at the boundary of the free surface discretization and stability criteria must be derived and enforced restricting the choice of the panel aspect ratio relative to the grid Froude number and frequency of oscillation. In steady-state and time-harmonic problems such radiation and stability conditions are outhned in Section 3.

In time-domain Rankine Panel Methods, stability condhions must be derived restricting the choice of marching scheme, time step and grid Froude number. The corresponding stability analysis is discussed in Section 3. The enforcement of the radiation condhion in time domain ship flows entails the absorption of the energy of transient ship wave patterns in all directions. With RPM's this may be best accomplished by introducing a dissipative beach extending beyond the free surface discretization, designed to absorb most of the energy of the ship radiation and diffraction disturbances. The design o f such a beach and its performance are presented in Sections 3 and 4.

The resuh of this rational numerical analysis was the derivation o f an RPM free or numerical damping and with numerical dispersion of cubic order. It is based on the use of Green's theorem for the velocity potential which is equivalent to the distribution or sources and normal dipoles over the ship hull and the free surface. A bi-quadratic variation of the velocity potential is allowed over the panels, permitting the computation of the flow velocity as part of the solution. These attributes were implemented into an RPM for steady, time-harmonic and time-domain free surface flows past ships known as SWAN (ShipWaveANalysis).

Selected applications of SWAN for naval, commercial ships and sailing yachts in steady, time-harmonic and time-domain motion are presented in Section 4. Steady ship flows are discussed first using both the steady-state and time-domain versions o f SWAN. They include the prediction of Kelvin wave patterns and wave resistance o f cruiser and transom stems ships, twin-huU vessels and an Intemafional America's Cup Class (lACC) sailing yacht hull. In all cases the convergence of the computations is first established and is followed by a comparison whh experimental measurements.

The accurate solution of the steady flow lays the groundwork for the treatment of the seakeeping problem. Computations are presented in Section 4 of the wave induced motion of a broad range of ship hulls, using the linear frequency- and time-domain versions of SWAN. Derivative seakeeping quantities of importance in practice include the wave induced loads and added resistance. Computations of these quantities for naval, commercial ships and sahing yachts are also presented and compared to experimental measurements.

This survey article would be incomplete whhout a discussion of the various sources error arising in the modeling and computafion of ship wave interactions and their influence upon the model selection, code efficiency, code vahdation and use in ship design. These issues are addressed in Section 5, where the philosophy underlying the development of SWAN is outlined. The survey concludes with Section 6 where topics o f fiiture research are presented, including the prediction of the nonlinear wave induced loads, slamming and added resistance in steep ambient waves.

2. T H E O R Y

2.1 The Exact Ideal-Flow Equations

Consider a ship advancing whh a time dependent forward speed U(t) in ambient waves, as illustrated in Figure 1. The fluid flow equations of motion will be stated whh respect to a Cartesian coordinate system x=(x,y,z) translating whh velocity U(t) in the positive x-direction. The origin of the coordinate system is taken on the calm water surface which comcides whh the z=0 plane. Assuming potential flow, the disturbance fluid velocity v(x,t) is defmed as the gradient of the velocity potential 0(x,t), or v=VO. By virtue of continuity, O is subject to the Laplace equation in the fluid domain

(2.1) V^(D = 0.

The poshion of the free surface is defmed by the wave elevation C(x,y,t), which along whh the velocity potential 0(x,t) are the pair of unknown quantities, or state variables, to be determined by the Rankine Panel Method described in Section 3.

The state variables are related by two conditions on the free surface. The kinematic condhion requires that a fluid particle on the air-water interface at t=0 will stay there for all thnes. The corresponding mathematical statement relative to the translating reference fi-ame takes the form.

Figure 1: Coordinate System

The dynamic condition simply states that the fluid pressure on the firee surface must equal the atmospheric pressure, taken equal to zero. It follows fi-om Bemoulh's equation that

(D + lvO-V<D = -gC

The wave elevation C may be eliminated from (2.2) and (2.3) resulting into a more compact free surface condition which involves O explicitly and C implichly. Yet, there exist clear computational advantages in retaining both (<D,Q as unknowns and employing the pair of equations (2.2) and (2.3) as the free surface conditions.

On the ship huh, the fiow normal velocity equals the corresponding velocity of the rigid boundary. Denoting by n the unit vector normal to the instantaneous poshion of the ship hull, it follows that

(2.4) ^ ^ u - n + v-n,

where v is the oscillatory velocity of the ship hull due to the wave induced motions. Finally, at large distances from the ship the fiow velocity must vanish.

A special form o f (2.1)-(2.4) deserves mention. In the absence of ambient waves and in the limit U(t) -> U as t ^ o o , the state variables (<I),Q become time-independent. The resuhing equations govern the nonlinear steady free surface flow past a ship advancing with constant forward velocity in calm water, and have formed the basis of a number of computational methods developed for the prediction of the steady flow, Kelvin wave pattern and wave resistance of ships and sailing yachts.

The numerical solution of the exact set of equations (2. l)-(2.4) presents a very challenging task in steady or unsteady flow. Therefore, a number of linearizations have been suggested and are discussed next.

2.2 Linearization of Free Surface Conditions

Whh few exceptions, the linearization of (2.2)-(2.4) is justified i f both of the following condhions hold; a) the ambient wave slope is smah and b) the huh shape is sufficiently "streamlined", namely thin, slender or flat. At zero speed, the linearized equations follow trivially by dropping all quadratic terms in (2.2)-(2.3) and enforcing (2.2)-(2.4) over the mean positions of the free surface and ship hull.

(2.3)

With forward-speed, the linearization is less evident. The principal consequence of the assumption that the ship is streamlined is that the fluid disturbance velocity caused by the ship forward translation and its oscillatory motion in waves are both small compared to its speed U. The total velochy potential <D may therefore be broken dovm into two parts, the

basis-flow potential (po and the disturbance-flow potential cpi, defined as follows

(2.5) 0 = ^ 0 + ^ 3 ,

(2.6) | V < i ? , | « |V^0o.

A similar decomposition is adopted for the wave elevation C„

(2-7) C = ^o+C,

(2-8) C , « C o .

Evidently the decompositions (2.5)-(2.6) and (2.7)-(2.8) are not unique since there exist several choices for the basis velocity and wave elevation satisfying (2.6) and (2.8).

Two of the most popular linearizations of the free surface condition are discussed next.

Neumann-Kelvin Linearization

The simplest hnearization assumes that the basis fiow is the uniform stream. The corresponding basis wave elevation is zero and the resulting free surface condhion for the two state variables take the form

(2.9)

(2.10)

a ^

ac)

An advantage of the N-K hnearization is its simplichy and the possibihty to derive explicit elementary solutions knov^n as Wave Green Functions in steady flow, the frequency or thne domains. As discussed in the Introduction, the use of WGF's to solve wave ship mteractions requires the distribution of panels only over the ship hull. The resulting computational effort may however be substantial, particularly in unsteady forward speed flows.

The N-K model can be rationally justified only for ships of vanishingly small beam or draft and several analytical and numerical studies have confirmed its validity in these limiting cases. For conventional ships with finite beam and draft of comparable magnitude, a more accurate basis-flow model exists which accounts for the effects of the ship thickness. The resulting free surface linearization is discussed next.

Double-Body Linearization

The flow past the ship and hs positive image above the free surface may be selected as the basis flow and is hereafter referred to as the double-body flow. The advantage of this choice over the uniform stream is that the effects of the ship thickness are better modeled by the basis flow potential (po, at least over the ship huh.

The resuhing basis wave elevation Co fohows from Bernoulli's equation (2.3) in the form

(2.11)

8 ^ z = 0.

Substitution in the nonlinear free-surface conditions and use of the linearization assumptions (2.6) and (2.8), leads to the following condhions for

(91,1^1)

(212)

a

_a

(2.13)

a

In the derivation of (2.12)-(2.13) it was assumed that the basis wave elevation Co is a sufficiently small quantity for the statement of (2.12) and (2.13) to be transferred on the z=0 plane whh small error. This may not be the case near the ship waterline and in particular near the ship bow. No attempt will be made here to defend (2.12)-(2.13) and the vahdity of linear theory in general in these flow regimes where strong nonlinear effects are often dominant.

Several variations of the double-body linearization (2.12)-(2.13) have been suggested in the literature for steady and unsteady ship flows. They were instigated by the low-speed wave resistance theories of Baba (1976), Maruo (1977), Newman (1976) and Keller (1979). A popular version of (2.12)-(2.13) is that of Dawson (1977), perhaps because he was the first to implement it in a Rankine Panel Method. An attempt to justify the validity and differences between ahernative double-body linearizations and their relationship to (2.12)-(2.13) whl steer this discussion into the field of perturbation theory and matched asymptotic expansions, which is not the objective of this survey.

Perhaps a better hnear model than (2.12)-(2.13) exists. Yet hs performance can be judged only after a convergent numerical solution has been compared to experimental measurements. Our experience has been that the model (2.12)-(2.13) performs consistently better than its N-K counterpart and has been adopted as the standard linear solution in the wave resistance and seakeeping problems in the frequency and time domains.

2.3 Linearization of the Body Boundary Condition

The linearization of the exact body boundary condhion entahs hs statement over the mean translating poshion of the ship hull, assuming that hs oscillatory displacement is small. The derivation o f the linearized body boundary condhions in the seakeeping problem dates back to Timman and Newman (1962) and Oghvie and Tuck (1969) and, unlike the free surface condhions, there exists little ambiguity as to their proper statement. The same however cannot be said about their numerical implementation.

In the presence of ambient waves, the ship undergoes an oscillatory motion with displacements ^j(t), j=l,..,6 in each of hs six degrees of freedom. Adopting the decomposhion of the total velocity potential O into basis and disturbance components, cpo and (pi, the former offsets the normal flux due to the forward translation of the ship, or

(2.14) ^ = Ü.n = Un,, onS,

ch

where n =(ni,n2,n3) is the unh vector normal to the ship hull pointing out of the fluid domain. The disturbance potential cpi consists of the incident, diffraction and radiation components. The body boundary condition satisfied by the diffraction potential on the ship hull is such that hs normal velocity offsets that of the ambient wave represented by the velochy potential cpi, or

(2.15)

The corresponding body boundary condition for the radiation disturbance potential is more involved and takes the form

(2.16) where (2.17) (2.18) _{{fn,m„m,) = {n-V)(U-V(p,)} on S (2.19) {m4,m„m,) = («• V ) [ f x (^7 - V ^ ^ ) ' .

It fohows from the (2.16)-(2.19) that the enforcement of the body boundary condition in the radiation problem requires the knowledge of the second gradients of the basis ilow potential, which except for the simplest possible choice for (po, are not easy to compute accurately.

Neumann-Kelvin Linearization

As in the derivation of the corresponding free surface condhion, the basis flow is approximated by the uniform stream, leading to the basis flow potential

(2.20) = 0.

Evidently hs second gradients vanish and the body boundary condhion (2.16)-(2.19) reduces to a quhe simple form. The accuracy of (2.20) is however questionable for ships with significant beam and drafl for which a more accurate model of the basis flow is required.

Double-Body Linearization

In this case the basis flow is modeled by the double-body flow potential which satisfies the exact boundary condhion over the ship hull due to its forward translation and a rigid lid condhion on the z=0 plane. This basis flow model leads to a more accurate set of body boundary conditions as confirmed by the ship fiow computations presented in Section 4.

2.4 Linear Time-Harmonic Ship Flows

An important special case of the time-domain free-surface and body boundary conditions arises when a ship advances with constant speed in regular monochromatic waves. By virtue of linearity, the time dependence of the unsteady component of the free surface flow becomes harmonic with frequency co, known as the encounter frequency and defined by

(2.21) o} = (Oo-U^cosP,

g

where 00 is the wave frequency relative to an inertial frame, P is the wave heading relative to the positive x-axis (P=180° for head waves) and U is the ship speed assumed constant. In fime harmonic flow, the disturbance potential (pi adopts the complex representation (2.22) (p,=Re[<^'^'

which allows the restatement of the linearized free-surface and body boundary conditions in terms of the complex potential ^ using the following interpretation of all time derivatives

(2.23) ^ = ia.

a

The frequency-domain linear seakeeping problem has attracted considerably more attention relative to hs time-domain counterpart, in sphe of their duality [cf Cummins (1962) and Ogilvie (1964)]. Perhaps, the primary reason for such a trend is the emergence of the length scale 27i:g/o)^ which allows the derivation of approximate formulations o f the seakeeping problem for slender ships, most notably strip theory.

In the limh of vanishing frequency of encounter, the free surface conditions reduce to those for the linearized steady flow past a ship which may thus be considered a special case of the frequency domain formulation.

2.5 Steep Ambient Waves - The Weak Scatterer Hypothesis

In ambient waves of large steepness the fially nonlinear seakeeping problem must be treated, a task which can only be accomplished in the time domain. The treatment o f the exact kinematic and dynamic free surface condhions (2.2) and (2.3) in the presence of a ship advancing with forward speed, represents a formidable computational task. Naval architecture practice requires that ship motions and wave induced loads be possible to simulate in a random environment over the period of up to a few hours in order to identify extreme events in a typical sea state. Computational algorithms must therefore be

developed with such applications in mind for the resuhing computational method to fmd use in ship design.

Two general approaches may be adopted for the treatment of nonlinear free surface flows in the time domain. The quasi-Lagrangian approach traces fluid particles on the free surface by solving a succession of potential flow problems bounded by the instantaneous poshion of the free surface. This method has been widely used in two dimensions and in some three dimensional flows. The tracing of free surface particles allows the accurate modeling of highly nonlinear and rapidly evolving localized flow regions arising in wave breaking [cf Longuet-Higgins and Cokelet (1976), Vinje and Brevig (1980), Dold and Perregrine (1984), Lin et al. (1984)], slamming and impact loads [cf Zhao and Faltinsen (1993)]. However, the implementation o f a quasi-Lagrangian approach to the solution of the three dimensional seakeeping problem is likely to incur prohibitively large computational costs.

The hydrodynamic forces, responses and wave induced loads on ships in steep waves are governed by large scale nonlinear effects which may be treated more efficiently by the Eulerian method. An important exception is slamming and the associated space- and time-localized strongly nonlinear free surface disturbance which may be treated by a different method and imbedded into the nonlinear seakeeping simulation.

Moreover, two types of wave disturbances of disparate origin are encountered in the seakeeping problem, the ambient and ship generated waves. The former are driven by the environment, their steepness depends upon the severity of the weather conditions and they are stochastic rather than deterministic processes. It is therefore natural for their modeling to be carried out by established techniques in oceanography [cf Kinsman (1965)] rather than attempt to incorporate their prediction into the numerical algorithm used for the modeling of the ship wave disturbance. Several numerical studies have successfully generated nonlinear wave records generated by an oscillating wavemaker. While such numerical wave records are very usefiil for the testing model nonlinear seakeeping problems, their value in naval architecture practice is limhed. The generation o f long records may not be possible or economical, the statistical properties of the resulting waves may not be easy to control and the extension to three dimensions may be prohibitively expensive.

Therefore, the approach adopted in the development of the present method is that ambient waves are modeled independently of the ship generated disturbance and theh nonlinear interaction is accounted for in an iterative manner. The first step of this herative process is known as the weak scatterer hypothesis [cf Pawlowski (1992)]. Observations o f the interaction of ships with steep ambient waves in the towing tank or in the ocean often suggest that the magnitude of the ship wave disturbance is small compared to that o f the ambient waves. Again, an important exception are slamming events where the ship wave disturbance is highly nonhnear but localized near the ship hull. A somewhat loose justification in support of such observations is that a freely floating ship behaves as a

ship length. This invites the linearization of the ship generated wave disturbance about the nonlmear ambient wave profile coupled with the exact statement of the body boundary condhion over the ship hull.

The incident-wave potential and elevation, including a model for the basis fiow due to the ship forward translation are defined as the basis state variables (^3o,4'o)> here being time dependent quantifies. Denoting by ((pi,CO the remainder of the ship wave disturbance caused by its time dependent forward and oscillatory motions, the following linearization assumption is postulated

(2.24) |V^9,| « \W(p,

(2.25)

Since the ambient wave satisfies exactly the nonlinear free surface condhions, the substhution of the basis and perturbation state variables in (2.2) and (2.3) leads to a set o f linearized free surface condhions for {(p^X\) over a time-dependent basis surface defined by the incident wave profile

(2.26) ^ 1

a

a

a

_a

a

a

The body boundary condition for the basis and disturbance potentials are stated exactly over the instantaneous position of the ship hull.

The solution o f (2.26)-(2.27) may be carried out along lines similar to the linear set of equations (2.12)-(2.13) as discussed in Section 3. An important difference is that with (2.26)-(2.27) the basis surface is time dependent and must be rediscretized at each time step along with the instantaneous poshion of the ship surface. It may be easily verified that (2.26)-(2.27) reduce to (2.12)-(2.13) if the double-body flow is used as the basis flow and the statement of both condhions is transferred to the z=0 plane.

3. T H E R A N K I N E F R E E S U R F A C E P A N E L METHOD

3.1 The Continuous Formulation - Green's Integral Equation

Denote by cp the unkown basis or disturbance velocity potential to be determined over S, the mean translatmg position of the ship hull, and over F, the z=0 plane. The Laplace equation in the fluid domain bounded by S and F may be enforced by an application of Green's theorem for the velocity potential (p(x) and the Rankine source potential

(3.1) Gix;l) =

Green's identity leads to an integral relation betv^een the value and the normal derivative of (p over S and F which takes the form

(3.2) ^ ( x , / ) + J j ^ , ( ^ , / ) ^ ( x ; | ) ^ ^ -

JJMIlO

F+S '^i F+S

The contribution from a closing surface at infinity vanishes due to the decay of (p(x) and G(x,^), as X ->oo for fixed values of ^.

Over S, (pn is known and supphed by the body boundary condhions. Over F, the linearized free surface condhions (2.12) and (2.13) relate (pn=(pz to the value and tangential gradients of (p and C- Upon substitution of this relationship, (3.2) reduces to a pair of integro-differential equations for cp over S and ((p,Q over F which is solved by the panel method described below.

3.2 Spatial Discretization

The surfaces S and F are subdivided into a large number of plane quadrilateral panels as illustrated in Figure 2 which presents half of the panel mesh due to the ship symmetry. Over the j - t h panel, a bi-quadratic spline variation is assumed for ((p,Q of the form

(3.3) <p(x,t)=,j;^(^).(t)Bj(x)

where the basis function Bj(x,y) is centered at the j - t h panel and provides inter-panel continuity of ((p,Q and their first tangential gradients. It is noted that the principal unknowns are the spline coefficients ((p,Qj which are not equal to the values of the respective unknowns on the panel centroids but are linearly related to them.

In space, the discretized integral equation is enforced at the panel centroids, leading to systems of equations for the spline coefficients of the state variables. The state variables are also functions of time, therefore a temporal disretization is necessary and is discussed next.

3.3 Temporal Discretization

The evolution in time may be carried out analytically in steady or time-harmonic flow by making use of the complex notation (2.22) and the identity (2.23) for the time derivative. In time-domain flows a marching scheme must be selected for the approximation of the time derivatives of the state variables ((p,Q, or equivalently their spline coefficients introduced by (3.3) and (3.4). The lowest order scheme is the Euler marching, defined as fohows

Figure 2: Ship Hull and Free Surface Discretizafion

where the superscript n marks the time-step. Upon substitution of (3.5) in the free surface conditions (2.12)-(2.13) and Green's identity (3.2), the remaining terms in (2.12)-(2.13) not involving a time derivative may be enforced either at the n-th or at the (n+l)st time step. The former choice leads to an explicit and the second to an implicit method. A mixture of the,two is possible i f for example the explicit scheme is used in the kinematic [eq. (2.12)] and the implich scheme with the dynamic condhion [eq. (2.13)] respectively. The resulting method wih be referred to as emplicit. The choice is also possible among a wide variety of higher-order marching schemes, explicit, implicit or emplicit. Their relative merits are dictated by their stability properties discussed below.

3.4 Numerical Dispersion, Damping and Stability

The ship wave patterns propagating over the free surface panel mesh will be distorted due to discretization error, perhaps to the point that the solution becomes too inaccurate to be trusted. The discretization error is quantified by the numerical dispersion and damping introduced by the free surface discrefization. They measure the discrepancy in the phase and amplitude between the discrete and continuous wave disturbances, respectively. According to the Lax Equivalence Theorem, the condhion which must be met for the numerical error of a consistent discrefization to vanish in the limit as the panel size h-^0 is known as numerical stability criterion.

A rational study of the numerical dispersion, damping and stability of free surface Rankine Panel Methods in two dimensions was carried out in Sclavounos and Nakos (1988). One of the principal conclusions was that schemes which employ upwind differencing in the enforcement o f the free surface condition introduce significant numerical damping into the wave disturbance which may prevent their convergence in the limit as h^O. As a consequence, the bi-quadratic spline scheme outlined in Section 3.2 was studied and was found to enjoy excellent numerical properties which are summarized below for three dimensional flow.

The free surface discretization is characterized by two parameters, the panel aspect ratio

(3.6)

and the grid Froude number

(3.7)

where hx and hy are the panel dimensions in the streamwise and transverse directions, respectively. In time-harmonic flow, the reduced frequency parameter

T =

s

must also be introduced, where co is the frequency of encounter.

A Von-Neumann stabilhy analysis of the centroid collocation bi-quadratic spline scheme was carried out by Nakos and Sclavounos (1992). The method was found to be free o f numerical damping enjoying a numerical dispersion o f O(h^). This entails a very rapid convergence of the discrete to the continuous wave patterns, if the scheme is stable. The method was found to be convergent only i f a certain stability criterion is met which restricts the choice of the three parameters (a,Fh,T). Figure 3 identifies the stability regimes in the parameter space (a,Fh) parametrically with respect to x. It follows that for the bi-quadratic spline scheme to converge, Fh must be greater than a minimum critical value which depends on a and x . Alternatively, given Fh or the ship speed and panel size in the streamwise direction, the panel aspect ratio a must not exceed a certain critical value which depends on Fh and x.

0 1 2 3

Figure 3; Stability Diagram for Steady and Time-Harmonic Flow

Knowledge o f the stability criterion of an RPM method is essential for hs implementation to the study of flows past realistic ship forms. For example, an attempt to establish convergence by increasing the panel density only in the transverse direction is bound to lead to high aspect ratio panels near the wateriine and the transition into unstable territory, as would be the case with the present scheme. Such attempts have been reported in the literature of RPM's leading to wateriine wave elevations which lacked physical meaning.

3.5 Stability of Temporal Discretization

The stabihty analysis o f marching schemes in time-domain flows entails the derivation of criterion which'restricts the choice of the non-dimensional time step

(3.9) P = ^ .

At

The stability properties of a broad range of time marching schemes were studied by Nakos (1993) and Vada and Nakos (1993). The emplich Euler scheme discussed in Section 3.3 was found to enjoy very good properties and was thus analyzed in detail. Figure 4 illustrates hs stability diagram which plots the critical value of P as a fimction of Fh and a. The curves plotted in Figure 4 specify the upper bound for the time step At which at low speeds scales with h'^ and at higher speeds with h ^ . Therefore, high ship speeds require smaller time steps with the emplicit Euler scheme. More general methods allowing the selection of large time steps at high speeds were studied in Vada and Nakos (1993).

20.0 17.5 15.0 12.5 10.0 7.5 5.0 2.5 0.1 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 F, = u/(gh.)"'

Figure 4; Stabilty Diagram for Emplich Euler Time Marching Scheme

The stability analysis o f time-domain flows is of critical importance for the efficiency of a time-domain method simulating free surface flows past realistic ship forms where large numbers of panels and time steps are necessary. The larger the maximum time step allowed by the stabihty analysis the smaller the total number of steps for a simulation record of given length. In linear and eventually nonlinear ship motion and wave load analyses in random waves simulation records a few hours long may be necessary in order to obtain the necessary statistics of the extreme responses.

3.6 Spatial Filtering

Rankine Panel Methods for free surface flows are known to develop spurious grid-scale oscillations with wavelength equal to 2h, where h is the panel size. This numerical error arises from the aliasing of the energy of the sub-grid scales onto the maximum Nyquist wavenumber iz/h the free surface panel mesh can resolve. Therefore, the energy of physical wavelengths of the order 2h or smaller, which is often quite significant in free surface flows past ships, may be severely distorted by the free surface discretization and must therefore be removed.

The occurrence of the grid scale spurious oscihafions was first detected by Longuet-Higgins and Cokelet (1976) in their simulation of surface wave breaking in two dimensions. The remedy they suggested removes the grid-scale energy by low-pass filtering which acts to smooth out the simulated surface wave disturbance. They proposed a filter shape based on a 7-point spatial averaging scheme which is designed to have no effect on length scales longer than approximately 4-5 panel sizes h.

Several ahernative fihers are plotted in Figure 5 as fianctions o f the wavenumber. By design, all filters taper o f f to zero at the Nyquist wavenumber 7t/h at different rates, depending upon the application. In the present RPM the 7-point modified filter is used which may be seen to taper off to zero at the Nyquist wavenumber n/h less rapidly than the 7-point filter of Longuet-Higgins and Cokelet. Numerical experiments revealed that its frequent application tended to smooth excessively the simulated ship wave disturbance. An optimal filtering rate was thus determined which was found to depend on the ship speed and frequency of oscillation.

3.7 Radiation Conditions

The proper enforcement of the radiation condition is essential for the success o f Rankine Panel Methods in simulating ship flows. Several statements exist for the radiation conditions of steady, time-harmonic and transient wave patterns generated by a ship advancing with forward speed. Their mathematical derivation is detailed in Wehausen and Lahone (1960) and their numerical enforcement by the present RPM is discussed below in several special cases.

3.7.1 Steady Flow

The simplest statement of the radiation condition satisfied by a ship Kelvin wave pattern requires that the wave elevation vanishes far upstream. In the derivation of the corresponding Wave Green Function, the Kelvin source potential, this condition is enforced by the introduction of an artificial damping term known as Rayeigh viscosity [cf Lighthih (I960)].

In the original RPM of Dawson, the use o f an upwind differencing scheme introduces numerical damping or a Rayleigh viscosity which acts to enforce the radiation condition. The drawback of this approach is that this numerical damping is often excessive whh detrimental effects upon the accuracy of the numerical solution.

The present RPM is free of numerical damping, therefore the radiation condition must be enforced explichly. It was shown in Sclavounos and Nakos (1988) that enforcing a zero wave elevation and slope at the upstream boundary of the free surface grid is equivalent to the presence of a rigid lid upstream of the free surface panel mesh. Moreover, by virtue of the convective nature of the Kelvin wave pattern, no radiation condition is necessary at the transverse and downstream boundaries of the free surface mesh, as long as they are sufficiently removed from the ship. This radiation condition was found to perform very well as will be illustrated in the applications discussed in Section 4.

3.7.2 Time-Harmonic Flow

For values of the reduced frequency x=öU/g > 1/4, the radiation condhion is similar to that for steady flow, namely that the wave disturbance vanishes upstream. Therefore, enforcing a zero wave elevation and slope at the upstream boundary was found to work as effectively for time-harmonic as for steady flow, for T>1/4.

For T < l / 4 , one wave system of the ship unsteady wave pattern propagates upstream and the radiation condhion used for T>1/4 is strictly speaking invalid. Yet, the amplitude of this system is generally small and decreases with increasing Froude number. Therefore, the enforcement of the condition of vanishing wave amphtude and slope was found to perform well for Froude numbers greater than about 0.25. At lower Froude numbers, the

effectiveness o f this radiation condition was found to deteriorate, a condhion which may be remedied m part by increasing the size of the free surface mesh. A more effective treatment of fi-ee surface flows for x < 1/4 is possible whh the time-domain version o f the method discussed below.

3.7.3 Time-Domain Flow

When the simulation of the free surface flow past a ship is treated as an initial value problem, wave systems propagate in all directions and are subject to the radiation condition that theh amplitude vanishes at infinity. With the present RPM, this condition may be enforced very effectively by introducing a wave absorbing beach, consisting of a layer of panels which surrounds the free surface mesh. The free surface condifion enforced on the beach panels is dissipative and designed to absorb the wave energy with minimal reflection back towards the ship. The design and use of such absorbing layers in wave propagafion problems is reviewed by Israeli and Orszag (1981).

In the present method, the following free surface condition is enforced over the beach, stated here at zero speed for the sake of simplicity,

(3.10) ^^=-g^

(3.11) C = ^ . - 2 v C + - ^ . ,

g

where v > 0 is the strength of the damping parameter which must be properly tuned to minimize wave reflection. The dispersion relation corresponding to (3.10)-(3.11) may be readily derived in the form

(3.12) (o = i v ± ^ .

It follows from (3.12) that v plays the role of Rayleigh's viscosity used to enforce the radiafion condifion in the frequency domain WGF's. Whh forward speed, the time derivatives in (3.10)-(3.11) are simply replaced by the convective operator {didi - U dld\). The design o f the absorbing beach entails the proper choice of its distance from the ship, its size and the selection of the magnitude and variation of the Rayleigh viscosity v to minimize wave reflection. The beach location and size have been determined from numerical experiments. The damping coefficient v is taken to start from a zero value over the inner beach boundary and increase quadraticahy in the radial direction towards hs outer boundary. More details are presented in Nakos, Kring and Sclavounos (1992). The beach has been found to perform very well over a broad range of wave frequencies and ship speeds, including T < 1/4 and zero speed. Selected ship flow simulations in the fime domain are presented and discussed in Section 4, illustrating its performance.

4. APPLICATIONS

The Rankine Panel Method described in Section 3 was implemented into a computer program named SWAN (for ShipWaveANalysis). During the early stages of development, SWAN was ejTercised for simple ship-like wave disturbances for which exact solutions exist, in order to test the validity and performance of the fimdamental numerical properties of the method described in Sections 2 and 3. Lherally countless numerical experiments were carried out for applied pressure distributions, thin ships and submerged bodies which confirmed the qualities of the basic panel method and led to the development of rational criteria for the parameters of the algorithm. They include the location of the boundaries of the free-surface panel mesh, the panel aspect ratio and typical number of panels for SWAN-1, the steady and frequency-domain version o f the code. The additional parameters for the time-domain version of the code, SWAN-2, include the parameters for the absorbing beach and the time step of the marching algorithm. Resuhs from these early studies are not reported in this Section in lieu of resuhs for reahstic ship forms.

In recent years, SWAN has been extended to flows past naval and commercial ships whh transom stems, twin huh vessels and lACC sahing yachts. The present section highlights the performance of SWAN-1 and SWAN-2 comparing computations with experiments where possible. The presentation starts whh a discussion of SWAN computations of Kelvin ship wave pattems and wave resistance and continues with studies of the seakeeping problem in the frequency and time domains, including the predictions of unsteady wave patterns, motions, wave induced loads and added-wave resistance.

4.1 Kelvin Wave Patterns. Wave Resistance

Series 60 - CB=0.6

The discretization o f the hull and free surface around the Series 60 hull by SWAN-1 is illustrated in Figure 6. The total number of panels shown in the figure is 5,400 on half of the computational domain due to the ship symmetry.

The rate of convergence of the Kelvin pattern is illustrated in Figure 7 where the number of panels along the ship length, NBX is progressively increased from 50 to 100, keeping the panel aspect ratio constant and equal to a = l . The densest grid corresponds to 5,400 panels over half the domain or 10,800 panels over the entire domain.

The contour plots shown in Figure 7 offer but qualitative evidence that the Kelvin wave pattern computations have converged for a Froude number 0.3. A more rational measure of convergence entails the computation of the spectrum H(u) which may be defined ehher by a longitudinal or transverse cut analysis as discussed in Eggers, Sharma and Ward (1967). Such an analysis is possible whh SWAN since the scheme is free of numerical damping and the wave disturbance far from the ship is not excessively distorted due to numerical error. In SWAN-1, transverse wave cut analysis is used as described in Nakos and Sclavounos (1992) to obtain the Kelvin wake spectrum plotted in Figure 7 for three successively finer dicretizations. The spectrum H(u) converges over a broad range o f wavenumbers, confirming the consistency and stability of the underlying numerical m_ethod.

Derivative quantities like the ship sinkage, trim and wave resistance may be evaluated either by pressure integration or by enforcing the momentum conservation principle. The proper evaluation of the wave resistance by computational methods based on the linearization of the free surface condhion has been the source of controversy for quhe sometime. There exist at least three altemative definitions of the wave resistance two o f which are based on pressure integration and the third on the use of Havelock's far field formula. As shown in Nakos and Sclavounos (1992), Havelock's definition is the most consistent and should be used when possible. In SWAN-1 the implementation o f Havelock's formula entails the integration of the energy carried by the Kelvin wave spectmm H(u), according to the formula

Computafions of the Series 60 wave resistance by SWAN-1 using pressure integration and the far field method based on (4.1) are illustrated in Figure 8 where the ITTC 1984 range of experimental measurements is also shown. The same figure also plots the wave resistance predictions obtained from pressure integration but keeping only the linear terms in Bemoulh's equation. It may be seen that the far field method performs best, particularly at higher speeds.

,v/ — — • • • —

0.15 0.20 0.25 0.30 FrNo

Figure 8: SWAN Predictions vs ITTC Measurements of Series 60 Wave Resistance

Naval Combatant - Model 5415

An extensive experimental program aiming to measure the Kelvin wave pattern of the naval combatant shown discretized in Figure 9 was carried out at the David Taylor Model Basin and reported in the Wake Off Survey of Lindenmuth, RatcliflFe and Reed (1991). One of the essential differences between the Series 60 and the Model 5415, is that the latter has a transom stem which must be handled with care by the panel method for the Kelvin wave pattem to be convergent. A second difference is the presence of the sonar dome at the bow of Model 5415, which is handled in SWAN-1 as a non-lifting appendage.

Ships whh transom stems are handled by SWAN-1 by the introduction of a strip of "wake" panels which trail the transom. Assuming that the transom draft is small and the ship speed sufficently high for the transom section to remain dry, a set of smooth detachement conditions of the flow must be enforced at the upstream end of the wake panels. These conditions require that the free surface elavation and slope match the same values of the ship hull at the transom and are hereafter referred to as transom "Kutta" conditions, in light of their similarity to the corresponding condhions in lifting flows. Equipped whh this model for the treatment of transom stem ships, SWAN-1 computations of the Kelvin wave pattem o f the Model 5415 advancing at a Froude number 0.41 were found to converge, as illustrated in Figure 10a, where N B X is again progressively increased from 25 to 100.

Convergence is confirmed by the evaluation of the Kelvin wake spectrum which is also seen in Figure 10b to be in very good agreement with the experimental measurements carried out by Lindenmuth et al. (1991) over a broad range of wavenumbers.

Figure 10b: Predictions and Measurement of Kelvin Wake Spectrum of Model 5415

The quahtative features of the contour plots of the Kelvin wake for the model 5415 appear to change appreciably in a sector trailing the transom with increasing number o f panels. As the density o f the free surface panel mesh increases, SWAN-1 attempts to resolve the short wavelength surface wave pattems which are often evident in the wake o f transom stem ships. A similar trend is evident in the divergent wave system near the bow o f the 5415. A ftirther increase of the panel density would lead to a better resolution of the short scales. Yet, beyond a certain panel density the physical meaning of the SWAN-1 Kelvin wake predictions become questionable due to the influence of viscosity and flow separation at the tip of the transom upon the trailing free surface wake. Evidence o f such effects is perhaps the reason for the discrepancy between the SWAN-1 predictions o f the Kelvin wake spectmm at high wavenumbers and experiments.

International America's Cup Class (lACC) Sailing Yachts

Distinct features of the hull shape of an lACC yacht are the bow and stem flare. An lACC huh is treated by SWAN as a vessel with a round transom stern of zero draft. Therefore a strip of wake panels must be included in the free surface discretization as illustrated in the top of Figure 11. Unlike hull forms with clearly defined transom stations, the width of the wake strip trailing an lACC yacht must be selected optimally for the Kelvin pattern and wave resistance predictions to be convergent. An alternative mesh topology is the polar grid shown in the bottom half of the figure which allows naturally for a higher panel density near the yacht wateriine in the radial and azimuthal directions where more resolufion is necessary for the modehng of the yacht wave disturbance.

The polar grid was implemented with SWAN-2, the time-domain version of the method. The yacht huh is at rest for t<0 and at t=0 its speed increases to a constant value U in a step like manner. The resulting evolution of the free surface flow around the yacht hull is shnulated by enforcing the linearized condhions (2.12)-(2.13) on the z=0 plane and using the time domam extension of the RPM reviewed in Section 3. The radiation condition is again satisfied by the introduction of the absorbing beach which coincides with an outer annulus of panels on the polar grid of Figure 11. Otherwise, numerical experiments confirmed that the principal numerical properties of rectangular meshes discussed in Secfion 3 hold for polar grids as well.

The time record o f the yacht resistance is plotted in Figure 12 as a fiinction of time. Following an initial time interval when the force record is dominated by transient free surface effects, the resistance converges to a nearly steady state value. Two properties of the force record deserve ftirther discussion. The spikes or "heart beats" in the record indicate the time steps when the fihering discussed in Section 3.6 is apphed. Lack of fihering would resuh into a force record with a growing oscillatory instability due to the accumulation o f numerical error at the Nyquist wavenumber. Otherwise, the force record is quhe insenshive on the filtering rate.

9 knots

The low frequency oscillation in the force record arises from the exchation of a free surface disturbance at the critical frequency co = g/4U, corresponding to the value of

T=coU/g =1/4. It may be confirmed that the period of this oscihation is indeed T = 2 7 I / C Ö .

This permits the removal of this oscillatory behavior from the force record by filtering yielding the yaeht wave resistance.

The origin of this oscillatory behavior of the force record is not numerical. Unlike time-harmonic ship flow simulations which by their nature study the evolution of the ship wave pattem at a particular excitation frequency, time-domain flows starting from rest exche the entire frequency spectmm, including ca=g/4U, the resonant frequency. It was shown by Liu and Yue (1994) that forward speed wave disturbances oschlating at the resonant frequency become localized around the ship hull and radiate no energy at infinity. Therefore, the associated damping is zero which explains their persistence in the force record of Figure 12.

Converged wave resistance computations over a broad range of Froude numbers of interest in the design of lACC yachts are compared in Figure 13 with experimental measurements. Sinkage and trim are included in both computafions and experiments, but are found not to affect the wave resistance appreciably for yacht speeds exceeding 8 knots. The agreement is seen to be very good for speeds below 9 knots. At higher speeds nonlinear effects are important and are not modeled in the present version of SWAN-2.

Cw

10 11

Vessel Speed (knots)

4.2 Seakeeping

The Wigley Hull

The lack of exact solutions of the seakeeping problem for realistic ship hulls prompted the Society of Naval Architects and Marine Engineers (SNAME) to commision a set of carefully conducted seakeeping experiments at the University of Delft for a modified Wigley Hull. These measurements have since served as the benchmark for the calibration and validation of several seakeeping computational methods, including the present RPM. Converegent computations of the heave and pitch added-mass and damping coefficients have been obtained in the frequency domain by SWAN-1. The panel layout over the Wigley hull and the free surface around it is similar to that for the Series 60 case studied earher in this Secfion. The time-harmonic version of the linear free surface conditions (2.12)-(2.13) was enforced over the free surface, along with the corresponding set o f body boundary conditions (2.15)-(2.19) including the m-terms.

The computation o f the double gradients of the double body flow was circumvented via the application o f Stokes' theorem in Green's integral theorem, as detailed in Nakos and Sclavounos (1990). Even for a streamlined ship like the Wigley huh, the account of the double-body flow in the ship hull boundary condhion was found to be essential for the accuracy o f the SWAN-1 predictions. Figure 14 compares the SWAN-1 predictions for the heave phch cross-coupling coefficients with the Delft forced oscillation experiments for the Wigley hull at a Froude number 0.3. SWAN-1 computations are also shown based on the Neumann-Kelvin model which employs the uniform stream as the basis flow. The agreement with the experiments is clearly better with the double-body flow linearization, underscoring the importance of properly accounting for the double-body terms on the body boundary and free surface condhions. The close correlation between the SWAN-1 predictions and experiments for the heave and pitch exciting forces and motion amplitudes and phases is illustrated in Figure 15.

•41» • ExperinMnU " — S t r i p Theory SWAN ——Neumann- Kelvin 5 ^ l.O l.ii x/i. 2.0 _2.3 . —-1

\

i

L

J /

Kyushu Twin-Hull Vessel

A set of forced oscülation experiments on a twin-hull vessel advancing at a Froude number 0.3, were carried out at the towing tank of Kyushu University (private communication). Each huh ha(J parabolic waterlines, uniform draft and Lewis form sections with length/beam ratio of 6. Their separation/beam was selected equal to 2, allowing for significant three-dimensional wave interactions between the hulls.

Figure 16: Panel Mesh for Twin-Hull Vessel

The panel mesh used in SWAN-1 over the twin huhs, the free surface and the trailing wakes is illustrated in Figure 16. The two sheets of vorticity (or velocity potential disconfinuity) trailing the sharp sterns of the twin hulls have been introduced in order to prevent infinite cross flow velocity at the stern sections where a smooth detachment condhion is enforced. More detahs are presented in Kring and Sclavounos (1991).

Computations of the heave and phch damping coefficients whh SWAN-1 are compared in Figure 17 to the Kyushu experiments and strip theory. The agreement of the SWAN predictions whh experiments is very good, confirming the accurate modeling of the three-dimensional wave interactions between the hulls. Strip theory, on the other hand, is seen to predict a resonant response at the non-dimensional frequency (o(L/g)"^=2.3, which corresponds to a two-dimensional standing wave between the two hulls.

B 3 3 / p V V ( g / L ) -O E x p e r i m e n t 5 . 0 1 1 ( 1 S w a n - 1 . 2 S t r i p theory

Twin hull, lewis form ship Frno= 0.3 Sep/B=2.0 L/B=6.0 2 . 5 O 1 0 0 . 0 1 < • • ' • ' • l i l 0 . 5 0

The Series 60 Cb=0.7 Hull

An extensive set of seakeeping experiments were conducted for the Series 60 Cb=0.7 huh in the 60's at the University of Delft. They include measurements of the heave & phch added-mass and damping coefficients and motions in regular waves at a Froude number 0.2.

Seakeeping computations for this Series 60 hull were carried out in the frequency and time domams, aiming to establish the consistency between the two linear versions o f SWAN and validate their performance relative to the experimental measurements. The hull and free surface discretization around the Series 60 in SWAN-1 is similar to that for the Wigley and the steady-flow computations discussed eariier in this section. The corresponding panel discretization in SWAN-2 is illustrated in Figure 2, where the border of the numerical beach is also shown.

The SWAN-2 seakeeping computations proceed along the following hnes. The frequency dependent added-mass and damping coefficients are derived from forced heave and pitch motion oscihations at a prescribed frequency, starting from rest with zero heave and pitch displacement. The resulting force records converge quickly to a harmonic signal which upon Fourier analysis leads to the heave and pitch hydrodynamic coefficients. Figure 18 compares the SWAN-1 and SWAN-2 hydrodynamic coefficient predictions whh the experimental measurements. The agreement between SWAN-1, SWAN-2 and experiments is generally very good, confirming the consistency of the SWAN predictions and their performance relative to experiments. The small differences between the frequency and thne-domain predictions are due to discretization errors in the respective numerical implementations of the linearized seakeeping problem.

The SWAN-2 predictions of the phch motion record of the Series 60 huh in regular head waves are illustrated in Figure 19. In the time-domain, seakeeping simulations in a regular or random stationary wave record are carried out by assuming that the hull is kept fixed at hs mean poshion for t<0 and is released at t=0. The resuhing heave and phch motion records indicate that convergence to a harmonic signal of constant amplitude and oscHlation frequency occurs very rapidly. The convergence properties of the heave and pitch motion calculations with respect to the panel density and time step are very good and are discussed in more detail in Kring (1994). Moreover, the simulation of the free motions of ships advancing in waves by SWAN-2 or any other time-domain method, require a separate stability analysis which serves to ensure that the equations of motions are integrated in a stable manner. This stability analysis is an essential complement to the numerical analysis of the wave propagation over the free surface panel mesh discussed in Section 2, and is presented in Kring and Sclavounos (1995).

The SWAN-1 and SWAN-2 predictions of the phch motion amplitudes and phase o f the Series 60 hull advancing at a Froude number 0.2, are compared to experiments in Figure 20 whh very good correlation.

0.02S 0.000 •nmo n" > 0.025 « « l i 0.75 0.60 0.25 0.076 0.025 0.000 O O O O Experiments SWAN1 SWAN2 0.400 > 0.200 ,c 0.000 -0.200 • ' • ' • • • • ' I I 2.0 2.5 3.0 3.5 4.0 4.6 6.0 6.5 ü)(L/g)" 2.0 2.6 3.0 3.6 4.0 4.5 5.0 5.5

The SL-7 Hull

Proceeding with hull shapes of increasing complexity, the SL-7 hull was originally designed as a high-speed containership and was later converted for naval use. Its distinctive feature relative to the classical cruiser-stem Series 60 hull, is its counter stem which remains dry with the vessel at rest. Yet, when cmising at a Froude number 0.3, the steady wave profile wets the counter stem and alters appreciably the ship waterplane area and therefore its restoring and seakeeping characteristics, as illustrated below.

Two sets of seakeeping computations were carried out for the SL-7 with the fi-equency domain SWAN-1 code. In the "linear" case, the SL-7 huh is discretized below the calm water, or static, wateriine. In the "nonlinear" case the steady flow is solved first and the mnning wateriine is determined, found to wet a significant portion of the counter stem at a Froude number of 0.3. The SWAN-1 hull discrefization is then carried out for the wet portion of the ship surface which is found to differ appreciably from hs static shape. The free surface discretization is then carried out only after the hull is stretched vertically for the mnning wateriine to coincide with the z=0 plane. This last step is an approximafion aiming to accomodate the requirement in SWAN-1 that the free surface panel mesh hes on the z=0 plane.

Figure 21 co^ipares tlie SWAN-1 "linear" and "nonlinear" heave and pitch motion predictions with experiments and the latter are found to agree much better with the measurements. The same conclusion is drawn for the wave induced midships vertical bending moment and shear force potted in Figure 22, which are seen to agree very well with experiments when the running wateriine stretching technique is used. More details are presented in Sclavounos, Nakos and Huang (1993).

•EO p g L B A 2.S0 0 0 0 0 0 S W A N ( U n « " » S W A N ( " n . « U m . « « ' ) 3 0 0 2.001-1 n i l . 0 0 0 l O O M . I 0 2.0 — 3.0 to s o 0 0 _{' 0 2.0 ^ 3.0 * 0 SO}

Figure 22: Midships Vertical Bending Moment & Shear Force on SL-7 Hull at Fr=0.3 It may be concluded from SL-7 and other related studies that the account of the runnmg wateriine, when appreciably different from its static counterpart, alters appreciably the hydrostatic restoring, hydrodynamic properties of the ship to the point that the motions and wave induced loads differ significantly from their purely linear values.

L V C C Yacht Hull

The hull shape of a typical International America's Cup Class (lACC) sailing yacht is characterized by the significant flare at the bow and stern, a consequence of the rating rule. The huh profile on the centerplane intersects the free surface at an angle as smah as

10 degrees at the stern (90 degrees corresponds to a wallsided hull). Therefore, particular care is necessary for the modelling of the steady and unsteady flows around such hull shapes.

The steady and unsteady flow past lACC hulls was extensively studied by SWAN-1 since 1989, and resuhs fi-om these studies contributed to the design o f ihe America^ and Young

America entries in the 1992 and 1995 America's Cup compethions, respectively. Figure

11 illustrates various discretizations of the hull and free surface around an lACC hull considered durmg the design of Young America. The rectangular mesh was used for the unsteady computations presented here, where the stem is treated as a curved transom station equipped with a strip of wake panels trailing the hull. The optimal width of the wake which ensures numerical convergence was found to depend on the Froude number. Moreover, the enforcement of smooth detachment conditions of the steady and unsteady flow around the stem was found necessary for convergence.

The quantity o f primary interest in the evaluation of the hull performance in waves is the added resistance. Its prediction involves the following general steps. A panel mesh is selected and the convergence of the steady flow is first established by selecting the optimal width o f the panel wakes for the specific Froude number under study. The mnning wateriine is determined as in the SL-7 study and a stretched hull is derived for used in the seakeeping study. Convergence is next established for the time-harmonic flow, leading to the computafion of the motions and added resistance using the theory described in

Sclavounos and Nakos (1993).

Figure 23 plots the SWAN-1 predictions of the added resistance o f an lACC hull shape which is a narrower version of Young America. Comparisons v^dth experimental measurements in head waves is seen to be very satisfactory. SWAN-1 computations of the added resistance are also presented in oblique waves which correspond to the realistic windward sailing conditions. In a sea state, the mean value o f the added resistance has been computed by combining the added resistance operator shown in Figure 23 with a representative San Diego wave spectrum. Further details on the SWAN-1 use in the design of lACC yachts, including the effect of appendages, will be presented in a separate article.

5. C O D E E F F I C I E N C Y Vs A C C U R A C Y

Efficiency is of critical importance i f a code for ship wave interactions is to find its way into ship design. For the sake of the present discussion, a computafional method is considered efficient i f a complete hydrodynamic analysis of a single ship design consumes at most a few hours on a powerfiil engineering workstation.

During the early stages of development of SWAN, concems about computational efficiency led to the selection of the Rankine source as the fimdamental singularity in the solution of the boundary value problems arising in potential flow wave ship interaction. RPM's benefit by the inherent efficiency in the set up of the panel mesh, computation of the influence coefficients and iterative solution of the resulting matrix equations. Furthermore, RPM's may be readily generalized to the treatment o f nonlinear ship wave interactions with minor changes to the mechanics of the underlying panel method.

However, in order to achieve the measure of efficiency indicated in the opening paragraph some accuracy must be sacrificed! In order of importance, the three principal sources of inaccuracy in computation are:

Human Error

Denotes primarily coding errors arising from the translation of a numerical algorithm into computer code. The process o f removal of human error is commonly known as code

debugging.

Modelling Error

Denotes the error arising from the discrepancy between the physical quantity under study and its mathematical model treated by the computational method. It is assumed that the result of a carefully conducted controlled experiment are error free.