Numerische Hydrodynamik - Rankine Source Methods for Seakeeping Problems

Numerische Hydrodynamik

Rankine Source Methods for Seakeeping Problems

Rankine-Panelmethoden zur Berechnung des Seeverhaltens

Dr.-lng. Volker Bertram

Universitat Hamburg

Dr.-lng. H i r o n o r i Yasukawa

Mitsubishi Heavy Industries, Nagasaki

Summary. Ein Überblicl< über

Ranl<ine-Singula-ritaten-IVIettioden zur linearisierten Berechnung

des Seeverhaltens von Schiffen mit

Vorwartsge-schwindigkeit umfaBt Verfahren im Zeit- und

Fre-quenzbereich sowie Hybridmethoden mit Kopplung

an eine stark analytisch gepragte Lösung im

Fern-feld. Wenige Verfahren sind bisher bis zur

Anwen-dung auf richtige Schiffsformen oder die

Berech-nung des Zusatzwiderstands in Wellen entwickelt

worden. Keine der IVIethoden kann bislang als

prak-tisches Hilfsmittel für den Entwurf von

Handels-schiffen angesehen warden. Die Arbeiten des MIT

sind auf diesem Gebiet zur Zeit am weitesten

fort-geschritten.

1 Introduction

Seakeeping computations have made much progress

since Weinblum and St. Denis (1950) focussed

atten-tion of researchers on this topic in their landmark paper

"On the motions of ships at sea". The seakeeping

prop-erty of principal interest for ship designers is the added

resistance. Other aspects involve large amplitude

mo-tions of ships (capsizing, slamming or shipping of green

water) and require special tools based on time

simu-lations. We w i l l not discuss these aspects further and

limit this review to methods which might improve our

capabilities to predict added resistance for practical

pur-poses. We are aware that some of our statements are only

true within this limited scope.

A ship is still optimized most of the times f o r

perform-ance i n still water. However, most of its lifetime it w i l l

operate i n a seaway. This seaway is only known (at best)

as a statistical distribution of amplitudes, directions, and

frequencies of the waves. The general approach to

pre-dict added resistance uses a Fourier decomposition of

the seaway, calculates the ship motions and resulting

added resistance individually (frequency-domain

ap-proach) for all considered sinusoidal waves assuming

unit wave amplitude, and then adds all the contributions

weighted by actual amplitude (and Fourier weighting

factor). This superposition of waves and reactions is only

vahd f o r waves of small amplitude. The problem is thus

reduced to predicting the added resistance of a ship in a

harmonic wave of small amplitude. The added resistance

is always approximated as the second-order

longitudi-nal force. (The time-average of the harmonic first-order

force is zero.) Alternatively, the time-domain approach

can simulate the behaviour of a ship i n a superposition

of waves of various frequencies i n one calculation

(us-ing considerably more CPU time). Then the response

can be attributed to the individual waves (frequencies)

by Fourier decomposition. Both time-domain and

fre-quency-domain approaches assume generally an ideal

f l o w (free of viscosity and rotation). This is justified

because the problem is driven mainly by gravity (wave)

forces and only to a small degree by viscosity. The total

velocity potential is decomposed into the steady

poten-tial due to forward motion of the ship in calm water, the

incoming wave potential, the diffraction potential due

to the interaction of the motionless ship with incoming

waves and the radiation potentials due to forced motions

s h i p g e o m e t r y r a d i a t i o n p r o b l e m diffraction p r o b l e m added m a s s , d a m p i n g e.xcitmg forces

>. — . . ..

- restoring t e r m s , m a s s m a t r i x ( e m p i r i c a l c o r r e c t i o n s ) ship m o t i o n s a d d e d r e s i s t a n c e in r e g u l a r wave I ( e m p i r i c a l c o r r e c t i o n s ) wave s p e c t r u m a d d e d r e s i s t a n c e i n seaway

Fig. 1. Flow chart for added resistance computations Abb. 1. FluBdiagramm zur Berechnung des Zusatzwi-derstands

This eliminates time as a variable, but all unknown quan-tities are now complex, i.e. they require twice the com-puter space.

The boundary condition on the h u l l involves second derivatives o f the steady potential f o r the "/n-terms" (terms i n the boundary condition on the hull). The m-terms are difficult to evaluate. Some methods neglect all terms involving second derivatives (and more) to make the analysis easier, others have to refer to complicated numerical schemes to evaluate them.

3 Frequency-Domain Metliods

Nakos and Sclavounos (1990a,b), Nakos (1990) use double-body flow to approximate the steady flow i n their frequency-domain code S W A N (Ship Wave ANalysis). The method solves directly for the potential. Applica-tion of Stokes' theorem avoids the otherwise necessary evaluation o f the second derivatives o f the potential i n the /n-terms. H u l l and free surface are discretized with flat panels. The unknown velocity potential is approxi-mated by the linear superposition o f bi-quadratic spline base functions. They enforce radiation by imposing at the upstream end of the free-surface grid:

f A

ico^-U (p = 0 (3) d x j

? V

ico^-U- 0 = 0 (4)

where f is complex amplitude of the diffraction/radia-tion potential. See e.g. Nakos (1990) for a discussion of origin and physical interpretation o f these two upstream conditions. Both are necessary to ensure physically meaningful solutions. I n theoi-y, this method should be limited to T > 0.25. However, "for T < 0 . 2 5 and w i t h increasing Froude numbers, the amphtude of the waves upstream of the ship decreases relative to that of the trail-ing wave pattern and [the above radiation conditions] perform weU i f the truncation boundary is sufficiently removed f r o m the ship. No conditions are necessary on the transverse and downstream boundaries", Nakos and Sclavounos (1990b). No special condition is enforced on the transverse boundaries but the grid spacing in-creases towai'ds the transverse outer boundary. Increased grid spacing usually increases numerical damping. It could be that no problems w i t h wave reflections are observed because the increased grid spacing acts as an inherent numerical beach. A filtering algorithm is in-voked which eliminates waves 30 times smaller than the ship length. Fhst applications were limited to heave and pitch motions for a Wigley and a Series-60 i n head waves. The results for the Series-60 are compared with results f r o m experiments and a strip method, Fig.4. The strip method yields overall better prediction of the ship motions. Later publications compare only with experi-ments or time-domain versions o f SWAN.

It is possible that Nakos and Sclavounos "get away" with no special treatment f o r T < 0.25 because they limit them-selves to head waves. K r i n g and Sclavounos (1991) apply the method to a Wigley catamaran. For ships with

1 1 1 >- ' 1 T 1 *

-0.5 1.0 1.5 2.0 X / L 0.5 1.0 1.5 2.0 X / L

Fig. 4. Heave and pitch motions in head waves for Series-60, = 0.2, Nakos and Sclavounos (1990b); • experiment, Strip theory, SWAN

Abb. 4. Tauch- und Stampfbewegung in See von vorn für Series-60, f ^ = 0.2, Nakos and Sclavounos (1990b); • experiment, Strip theory, SWAN

Rankine Source Metliods for Seakeeping Problems 415

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and

=ed

in-sed

. It

are

an

in-the

md

;ad

/ith

The

hip

sri-/ith

;m-31)

'ith

Fig. 5. Heave and pitch motions in head waves for S-175, F^= 0.275, Nakos and Sclavounos (1993); ' experiment, SWAN ('linear'), SWAN ('nonlinear')

Abb. 5. Tauch- und Stampfbewegung in See von vorn ftir S-175, 0.275, Nakos and Sclavounos (1993); • Experiment, SWAN ('linear'), SWAN ('nichtlinear')

flared sections, a quasi-nonlinear approach is necessary.

Sclavounos et al. (1993), Sclavounos and Nakos (1993)

compute the steady flow based on a double-body

hne-arisation. The waterline is adjusted for the squat thus

computed and a line integral over the new waterline

accounts f o r the steady wave profile. This "nonhnear"

approach greatly improves results for ships with flared

sections. Fig.5. Furthermore, a strip of panels forming

a 'wake' trailing the ship was found necessary f o r ships

with strong flare or transom sterns. Morino's Kutta-type

conditions enforce smooth detachment of the f l o w at the

stern. Heave and pitch motions i n head, beam and

quar-tering waves are shown f o r the SL7, S-175, and an

ge-neric l A C C yacht hull. Sclavounos and Nakos (1993)

show additional apphcations to yachts including

com-putations o f added resistance also for obhque waves.

However, leeway and heel angles are assumed to be zero,

so that geometrical symmetry can be exploited.

Conver-gence tests with up to 8000 panels confirm the solutions

for motion and added resistance. The authors claim that

"the accurate solution of the boundary-value problem

governing the steady and time-harmonic problems

re-quired the use of at least 4000 panels over the yacht hull

and the free surface." This statement requires a comment.

The number o f unknowns (source strengths) drives the

computational time and storage requirements. I f the

time-harmonic problem f o r oblique waves is

decom-posed into antisymmetric and symmetric subproblems,

it is sufficient to arrange panels only on the port half and

use mirror images, e.g. Bertram (1990b). This reduces

the number of unknowns by half. Even 2000 panels on

the port half would indicate inefficient arrangement o f

panels i n the grid for most cases. Nevertheless, the work

of the M I T group is impressive not only because

appli-cations have advanced to real ship forms, but also

be-cause o f the depth o f numerical sensitivity analyses.

Among other things, systematic convergence studies

were conducted aiming to establish the sensitivity o f the

numerical solution upon the number o f panels, their

aspect ratio and the truncation distance of the free

sur-face grid upstream, sideways and downstream o f the

ship. However, the approach uses a direct method

solv-ing for the potential (and not the strength o f the

sin-gularities). This makes it difficult to include derivatives

on a wavy surface. The method is i n essence limited to

double-body flow approximation of the steady flow with

some ad hoc approach to include dynamic squat. I n the

long run, this approach w i l l be outperformed by

meth-ods that linearize around the f u l l y nonlinear steady f l o w

solution or time-domain approaches. Iwashita (1994),

see also Takaki et al. (1995), Bertram and Iwashita

(1996), uses a similar approach incorporating most of

the techniques of Nakos and Sclavounos (1990a,b).

However, his application is original and interesting as

he investigates a fishing-boat at high speeds (F„=0.2 to

0.6). Results f o r Froude numbers 0.4 and 0.6 are not

presented as "they were not good". The case f o r Froude

number 0.2 shows problems due to the transom stem

which is not treated specially. Also the hull and

free-surface grids are much coarser than for the applications

of Nakos and Sclavounos. It is therefore d i f f i c u l t to

de-termine i f problems are due to the approach or the

dis-cretisation. Hong and Choi (1994a,b) propose a direct

higher-order boundary element method ( H O B E M ) to

solve steady and unsteady ship wave problems. 8-node

bi-quadratic elements describe the geometry o f ship and

free surface. The velocity potential varies also

biquadratically on the ship hull. On the free surface the b i

-quadratic spline scheme of Sclavounos and Nakos (1988)

is used. The elements are evaluated numerically by

Fig. 6. C B I E M near-field discretization; box-shaped (left) or cylindrical (right) Abb. 6. C B I E M Nahfeld-Diskretisierung; kastenförmig (links) oder zylindrisch (rechts)

Gauss quadrature. The steady f l o w potential is crudely approximated by parallel flow. For T > 0.25, the same radiation condition as i n Nakos (1990) is used at the upstream boundary. For T < 0.25, Hong and Choi pro-pose a Sommerfeld-type condition. This condition en-forces an oscillation of wave surface and potential cor-responding to the upstream wave travelhng i n x-direc-tion. (However, oblique waves are not exactly captured by this condition and therefore some questions concern-ing its vahdity remain.) On the lateral tmncation bound-ary, (pyy = 0 is enforced. Only the radiation problem i n head waves has been solved so far for a modified Wigley hull.

A research group o f t h e Hiroshima University proposes a hybrid method called C B I E M (Combined Boundary Integral Equation Method), Iwashita et al. (1992,1993), L i n et al. (1993a,b,c), Takaki et al. (1994). A near-field R S M solution is matched to a far-field G F M solution overcoming problems with the radiation condition. The near-field domain is shaped like a box or cylinder, Fig.6. Quadrilateral panels of constant strength are used. The total velocity potendal is expressed as superposition of double-body flow, steady Kelvin wave potential and an unsteady wave potenfial. However, a free-surface con-dition neglecting the Kelvin wave potential is enforced in the near field. The direct method requires a numeri-cal differentiation of the potential on the free surface for some of the terms i n the free-surface condition. Three-point finite difference operators are used for first and second derivatives. The derivatives with respect toxuse upstream operators, all other derivatives use central dif-ference operators. This scheme is o f quadratic order for numerical dispersion and cubic order for numerical damping for equidistant grids. However, it is not shown how large the numerical damping actually is f o r usual grid sizes. The numerical differentiation requires spe-cial care f o r the free-surface grid which is streamlined, orthogonal and wateriine fitted. The upstream differen-cing suppresses to some extent upstream waves i n the

near field regardless of the value of T. This is question-able f o r T < 0.25 and subject to current research. On the vertical (artificial) control surface the solutions of near-field and far-near-field are matched enforcing continuity of potential and normal derivative of potential. The G F M distributes singularities over the vertical control surfaces fulfilling a simple free-surface condition linearized about uniform flow. The line-integral term is neglected. The authors stress that the hne contour i n C B I E M would be over the intersection of vertical control surface and free surface (thus away f r o m the ship). Still, perhaps the neglection is rather driven by convenience than by phys-ics. Apphcations are hmited to an ellipsoid and a sphe-roid. Results (added mass, motions) for these simple test cases agree well w i t h experiments except f o r short waves. The method gives reasonable results for the po-tential (roughly coupled to the wave surface) for both

T> 0.25 and T < 0.25 with only some problems

attrib-uted to insufficient grid resolution. We deem hybrid methods promising and worthy of more intensive re-search. I t remains to be seen whether matching to G F M solutions is superior to absorbing boundary conditions. Takagi (1990a) proposes to f u l f i h the radiation condi-tion by applying a smah numerical damping ('Rayleigh viscosity' jS) to the free-surface condition. A n 'appro-priate' //-value for the damping would allow treatment Of all T-values. Takagi selects the most suitable value based on 2-D computations. This may appear question-able for real ships, but the pioneering work of Dawson (1977) for the steady wave problem was also based on selecting an 'appropriate' finite difference operator based on 2-D trial-and-error. A different issue is that the arti-ficial damping is applied in the whole domain, i.e. also directiy at the ship. The first author still has some doubts unphysical condition affects results. Takagi employs the double-body flow as steady base flow. Triangle constant source panels discretize hull and free-surface. Takagi (1990a) calculates hydrodynamic coefficients and d r i f t -ing forces (= added resistance f o r restrained ship) f o r a

Rankine Source Methods for Seakeeping Problems 417

0,6 0.8

X/L

Fig. 7. Added resistance acting on forepart of a restrained-blunt ship {F = 0.15); ' experiment, computation Abb. 7. Zusatzwiderstand auf das Vorschiff eines fixierten

völligen Schiffs (F =0.15); • Experiment, Rechnung

vertical circular cylinder and a hemisphere, Takagi

(1990b, 1991) the added resistance f o r a blunt ship

re-strained i n waves (diffraction only). The added

resist-ance curve shows remarkable humps and hollows

ver-sus wave length. Fig.7, not observed i n model

experi-ments. Takagi explains the discrepancy due to the

ef-fect of breaking waves near the ship bow in experiments.

To improve the accuracy, Takagi (1993) apphes a

higher-order panel method with spline interpolation function

on the free surface and second-order isoparametric

ele-ments for the hull. Calculated results fora half-immersed

ellipsoid show overall better agreement with the

experi-ments than those by strip method. However, our own

experience supports Hess (1990). Higher-order panels

generally improve accuracy fcr 'nice' bodies hke

spheres, ehipsoids, and Wigley huHs. For real ships,

higher-order panels - at least for steady flow problems

- have not resulted in consistent accuracy improvements.

Kashiwagi et al. (1994) present heave and pitch

ampli-tudes for a V L C C calculated byTakagi's (1993) method.

Fig.8. The results agree well with experiments.

Yasukawa (1990) derives a free-surface condidon

linearized for doublebody flow which corresponds d i

-rectly to Dawson's (1977) condition for steady flow.

Constant source panels are used for ship hull and free

surface. The radiation condition is enforced by

combin-ing 4-point upwind finite differenccombin-ing with a Rayleigh

viscosity term as in Takagi (1990a). The upwind finite

difference fulfills numerically the radiation condition for

the Kelvin waves distorted by oscillation. The Rayleigh

viscosity f u l f i l l s approximately the radiation condition

for the ring waves distorted by forward speed. The

method is apphcable for all rvalues. Apphcafion is

hm-ited to the diffraction problem for a half-immersed

el-lipsoid. The accuracy of results show room for

improve-ment. Maisonneuve et al. (1993) extend techniques of

the wave resistance code REVA, Delhommeau (1994),

to seakeeping problems i n their code A Q U A R E V A . The

steady flow is solved using double-body hnearisation.

The f l o w field and squat due to this solution is

consid-ered i n formulating the boundary conditions f o r the

seakeeping problem. However, the steady wave

eleva-tion is neglected. The use of an absorbing layer method

ahows the radiation condition to be f u l f i l l e d without

restriction. The free-surface grid covers a semi-circle

around the ship. The radius is 3 times the length of the

incident wave. For a semi-circle of 1 wave length, no

absorption is enforced. Outside this inner region, the

artificial beach starts with a damping that increases

quadratically towards the outer edge. The optimal value

of the damping constant is found i n numerical

experi-ments. We are not convinced that such a constant is

universal for all speeds and wave lengths, but probably

( » - 0 - g r

Strip Method

0

1

_{X / L}

0 3

0

-

Exp.

•

G a l . - —

'Strip Method

_ — Q = ö -1 •

-X / L

Fig. 8. Heave and pitch motions in head waves for V L C C {F^ = 0.131)

the approach works also for non-optimal values. The

damping zone grid is obtained by progressive

stretch-ing. This means that the grid density is progressively

reduced f r o m the interior to the exterior of the damping

zone. This arrangement was found to be optimum i n

terms of damping efficiency vs. number of additional

unknowns. No special technique is used to enforce the

radiation condition relying on the damping zone "to do

the j o b " . However, no figures of wave patterns are shown

to demonstrate the validity of this assumption. M a i

-sonneuve et al. solve the radiation problem and

diffrac-tion problem in head waves, but do not present results

for motions. Applications are limited to a submerged

elhpsoid and a modified Wigley hull. The work of Zhao

et al. (1988), Zhao and Faltinsen (1989) is included f o r

completeness, although they treat only the related

prob-lem of an offshore structure suject to waves and current.

The hybrid approach uses RSM i n the near field, where

the f l o w is linearized about the double-body flow, and

G F M i n the far field. The solutions are matched by a

least square condition on a cylindrical control surface

some distance away f r o m the body. The cuiTent

veloc-ity is assumed to he small disregarding effects of the

steady wave system. Zhao and Faltinsen estimate the

method to be vahd for T < 0.15. Zhao et al. (1988) give

applications f o r a hemisphere, Zhao and Faltinsen

(1989) f o r floating vertical cylinder. The method per se

is not veiy interesting in the context of added resistance

of ships i n a seaway. However, the treatment of the m

-terms i n the body boundary condition deserves further

attention. Zhao and Fahinsen use flat panels of constant

source strengths. These popular first-order panels are not

capable of determining second-order derivatives of the

potential needed on the huh. To circumvent this

prob-lem, Zhao and Faltinsen compute the second derivative

teims away from the body and use extrapolation to

ob-tain the required values on the body surface. For the

sim-ple geometries investigated, this approach works well.

Bertram (1990b,c,d), Bertram and Söding (1990)

dis-tribute (desingularized) discrete point sources over the

free surface and panels of constant strength on the hull.

The f u l l y nonhnear steady flow is determined by

itera-tion first, Jensen et al. (1989), and the boundary

condi-tions f o r the unsteady f l o w are linearized around the

steady potential and wave elevation. Radiation

condi-tion and open-boundary condicondi-tion are enforced by

add-ing a row of collocation points on the upstream end and

a row of source points on the downstream end of the

free-surface grid, Bertram (1990a).This "shifting" technique

is only applicable for T > 0.25. For the corresponding

steady flow problem (wave resistance), Seto (1995)

shows that " s h i f t i n g " is equivalent to enforcing Nakos'

two upstream boundary conditions, see above. Then the

applicability to T < 0.25 should be the same as for the

SWAN code, but no such application has been tried yet.

The hull boundary condition and the pressure

integra-tion technique require for the radiaintegra-tion problem (and

thus for motions) second derivatives o f the steady

po-tential on the hull. These are very crudely estimated by

using a slender-body approximation, because the

em-ployed panels of constant strength can only predict first

derivatives of the potential. So i n essence this method

is hmited to diffraction problems for T > 0.25. W i t h

re-spect to a better capability of predicting added

resist-ance, this method is only of practical interestfor f u l l hull

forms (tankers). Apphcations are presented for a

sub-merged spheroid and the Series-60 (Cb= 0.7). Results

differ f r o m strip-method results only f o r short waves

where no experiments are available to proof that the

R S M approach really gives better accuracy. Hughes

(1996) modifies this approach by using a higher-order

panel method to compute the second derivatives of the

steady velocity potential on the hull, Hughes and

Bert-ram (1995). The computed second derivatives converge

for successive grid refinements for a Wigley hull. They

appear plausible and smooth f o r a Series-60, but no

comparison with experiments is possible. This approach

is the best physical model for the frequency domain but

is still hmited to T > 0.25 - at least formally, even though

the errors for -r< 0.25 may be small. Preliminary results

for a Series-60 ( Q = 0.7) give good results f o r exciting

forces. Differences i n computed motions are attributed

to possible eixors in input or programming. McCreight

(1991) outhnes a very similar approach. However, his

funding ran out before any publishable results could be

obtained (personal communication). The aspect of

computing first fhe steady flow offers a simple way to i m

-prove current methods including strip method approach.

The solution of the steady flow includes an accurate

prediction of squat. Most seakeeping predictions use the

still-water position of the ship. For ships with flared

sections, this leads to errors i n predicting the

hydrody-namic coefficients. These errors seem unnecessary

be-cause codes to solve the f u l l y nonhnear steady flow are

widely available and even linear wave resistance codes

can predict squat quite accurately The influence o f

dy-namic sinkage and trim on added resistance (or even

motions) has to our knowledge not been investigated i n

detail. Experimental and numerical studies are

recom-mended to clarify this point.

4 Time-Domain Metliods

Solutions i n the time-domain offer an alternative

avoid-ing the difficulty to enforce numerically a proper

radia-tion condiradia-tion. However, they require generally more

computational time and are even less mature than their

Rankine Source Metliods for Seakeeping Problems 419

frequency-domain counterparts. Beck et al. (1994) use

a mixed Eulerian-Lagrangian time-stepping procedure

to solve the radiation problem for heave and pitch. The

method is an extension of eariier work that presented

results only for the steady wave resistance problem, Cao

(1991), Beck et al. (1993). Desingularised single

Rank-ine sources are located above the free surface and

in-side the ship's hull. I n a first step, the steady flow prob->

lem is solved. The ship is moved f r o m rest in calm

wa-ter, gradually reaching the final speed. After letting the

ship move at constant speed for some time, a steady state

is reached. This solution gives the steady f l o w

includ-ing squat. I n the second step, the ship is forced to

oscil-late in either pitch or heave. The method solves for the

unknown source strengths. A fourth-order

Runge-Kutta-Fehlherg method is used i n the time stepping. The

com-putational domain on the free surface moves with the

body. A t each time step, a row of collocation points is

discarded downstream and a row of new collocation

points with specified potential and wave elevation is

added upstream. A t the upstream end, zero perturbation

potential and zero free-surface slope are prescribed.

Earlier works prescribed no special condition at the side

edge leading to few problems with wave reflection.

Ar-tificial damping overcomes this problem, Cao et al.

(1993). So far only applications to a Wigley hull have

been presented. Nakos etal. (1993), Kring (1994), Kring

et al. (1995), K r i n g and Sclavounos (1995) use a

simi-lar approach. Their method allows i n principle the ftiUy

nonlinear solution of the steady wave resistance

prob-lem including squat as the long-time limit of a

time-simulation. However, they use a double-body

lineari-zation for their seakeeping applications. The spatial

discretization is based on the approximation of all

un-knowns in terms of biquadratic B-spIine functions. A

semi-implicit backwards differentiation scheme is used

for time-stepping. This scheme is only first-order

accu-rate. A higher-order scheme with more favorable

stabil-ity properties is subject of current research. A n

artifi-cial wave-absorbing beach is located at the outer

por-tion of the free-surface computapor-tional domain. I n

addi-tion a low-pass filtering technique is necessary which

is designed to eliminate short waves. Waves longer than

4 to 5 panel sizes are not effected. Applications for free

heave and pitch motions in head seas are presented for

a Wigley hull. For a Series-60 hull, only the radiation

problem is solved.

Maskew (1991a,b) uses the commercial code USAERO

with the ESP (Free-Surface Program) addition. The FSP

module is based on a mixed Eulerian/Lagrangian

ap-proach to include a moving solid boundary. USAERO

covers a broad range o f applications, including

hehcop-ter rotor/body inhehcop-teractions, manoeuvering aircraft,

ma-rine propeller i n nonuniform flow, and high-speed

train-tunnel simulations. The same code is apphed to

wave-resistance problems (impulsive start and simulation until

a steady state is reached - o r allotted computer time runs

out), zero-speed seakeeping, Maskew (1992), and

for-ward-speed seakeeping. "The method has been applied

to an actual SWATH ship, but unfortunately the

meas-ured data are not available for pubhcation." However,

results are shown for a generic SWATH configuration

with propeher and foils included in the model. Fig.9.'

Preliminary results for forced pitch oscillations f o r a

f „ = 0.4 are not compared to experiments or other

com-putational results. Future applications " w i l l provide

added-mass and damping coefficients f o r the

configu-ration i n this motion." Exciting forces f o r a Series-60 at

F„ = 0.3 are presented i n head waves in a numerical

tow-ing tank. Despite a relatively coarse discretisation (640

panels on the h u l l including apparantly many inactive

panels above the waterplane and 885 panels on the free

surface) and low time resolution, the computation for

one case takes 20 minutes on a CRAY X M P or about

10 hours on a Silicon Graphics 4 D G T workstation.

Prins and Hermans (1994a,b), Prins (1995) use the

dou-ble-body flow to approximate the steady flow field. Their

method is intended to determine the drift forces on a

moored ship subject to current and waves. The problem

is identical to that of a ship in a seaway for potential flow.

But the speed of a current is usually much lower than

the speed of ship. So applications are limited to very low

Froude numbers, namely F„ < 0.08. For higher speeds,

problems occur due to the employed mesh generator:

"The current method used for the generation of the

free-surface mesh could not cope with these restrictions".

Prins (1995). Prins and Hermans derive an absorbing

boundary condition f r o m asymptotic expansion to treat

the radiation and openboundary condition. They i n

-crease the panel size towards the outer boundary. This

keeps the number of panels used quite low f o r the area

covered, but this approach may be unable to resolve the

Fig. 9. SWATH configuration investigated by Maskew (1992) Abb. 9. Von Maskew (1992) untersuchte SWATH-Konfigura-tion

waves properly. The method solves directly for the

po-tendal which works apparently well f o r simple

geo-metries such as a floating sphere, but problems with

numerical differentiation on the hull occur for a tanker.

"The spatial derivatives on both the hull and the free

surface have been discretized by second-order difference

schemes. Especially on the hull this has to be done very

c a r e f u l l y " . Prins and Hermans (1994). However, the

approach uses a more sophisticated time-stepping

scheme than Nakos et al. (1993) which is stable for every

time step and grid. The most sophisticated application

computes added resistance (and other second-order

forces) f o r a V L C C where problems at low frequencies

become apparent. Prins (1995) recommends an

exten-sion of the method from the cuiTently used flat panels

of constant strength to higher-order panels. Overall, this

approach seems well suited f o r the problem of d r i f t

forces on a moored ship, but an extension to compute

added resistance of ships seems not attractive i f at all

feasible.

5 Comparative Evaluation

There is no clear preference f o r the time/frequency or

indirect/direct method choice. Table I . However, time

domain methods involve fewer spatial derivatives of the

potential and thus lend themselves easier towards direct

methods approaches. The disadvantage of higher

com-putational effort w i l l decrease with time. I n the long run,

the simpler programming and versatihty (e.g. allowing

Table I . Reviewed seakeeping R S M

N o . Place Country Code Author Method Domain

1 M I T U S A SWAN Nakos, Sclavounos direct frequency 2 K R I / S N U Koreu H O B E M Hong, Chtii direct frequency 3 Hirosliima Japan C B I E M Iwashita el al. direct frequency 4 Osaka Japan Takagi indirect frequency 5 MHl Japan Yasukawa indirect frequency 6 Nantes France A Q U A R E V A Maissonncuvc ei al. indirect frequency 7 N T H Norway Zhao, Faltinsen indirect frequency 8 IIS Germany N E P T U N Bertram indirect frequency y irs Germany F R E D D Y Bertram, Hughes indirect frequency 10 Michigan U S A Cao el al. direct lime II M I T U S A SWAN Kring, Sclavounos direct lime 12 A M I U S A U S A E R O / F S P M a s k e w indirect lime 1.^ D c i n Holiand Prins direct lime

Table I I . Evaluation of methods

No. D R M A RS OW 3D TS WP U L F N 1 X X X X X X X ( X ) X 2 X X X 3 X X X X X X X 4 X X X ( X ) X X X X 5 X X X X 6 X X X X X 7 X X X X X 8 X (.X) ( X ) X X X X X ( X ) X 9 X X X X X X ( X ) X 10 X X X ( X ) X X X 1 1 X X X X X ( X ) X X X 12 X X X X ( X ) X X X 13 X X X X X X X

We considered the following items:

[D] Diffraction problem solved (exciting forces) [R] Radiation problem solved (added mass and damping) [M] Motions determined

[A] Added resistance determined

[RS] Application to real ship (incl. Series-60) vs. spheroids or Wigley hull

[OW] Applications also for obhque waves

[3D] 3-d steady flow components considered (double-body or fully nonlinear)

[TS] Automatically steady trim and sinkage considered' [WP] Steady wave profile considered

[UL] Not limited- to T > 0.25

[FN] Applications to realistic ship Froude numbers

direct simulations of motions in irregular seas) of

time-domain codes may lead to their dominance. A t present,

none of the methods has reached a f u l l y satisfactory

stage. Table I I .

6 Conclusion

No R S M is f u l l y satisfactoiy. The M I T group around

Sclavounos have the most mature method so far. Hybrid

methods matching a R S M near-field solution to G F M

far-field solutions seem attractive to overcome problems

in the frequency domain f o r T < 0.25, but questions

re-main concerning the R S M resolution o f various wave

systems i n the near field. Few real ship apphcations have

been presented world-wide. The state-of-the-art appears

not yet mature enough for standard apphcations to ship

flows. The best approach seems to couple selective R S M

' Some methods should allow in principle tlie automatic computation of steady trim and sinkage. However, in cases where this was not clearly stated we used (x). All methods can use a manual iterative approach to include trim and sinkage; this is not specially considered. Note that method 1 uses this manual approach in one case, though.

- We used (x) for methods which in theory do not allow T < 0.25, but may so in pracfice following the claim of the MIT group that the upstream wave is not important.

Rankine Souice Metliods for Seakeeping Problems 421

results to a good strip method. For slender ships with

flared sections, e.g. container ships, squat predicted by

a Rankine source method could easily improve

predic-tions of mopredic-tions and added resistance. For f u l l hulls, e.g.

tankers, solving the diffraction problem in head seas and

computing the resuUing added resistance contribution

should improve prediction while being relatively

sim-ple as relevant waves w i l l lead to nondimensional f r e - .

quency parameters T > 0.25. The most efficient approach

in the short mn w i l l be in the frequency domain, but in

the long run time-domain approaches may prevail.

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Erörterungen

Prof, Dr.-lng. GUnter GroBmann, Berlin: Zusatzwiderstand im Seegang

Die Autoren versuchen, aus 63 Veröffentlichungen eine Metho-de zur Ermittlung Metho-des Zusatz-WiMetho-derstanMetho-des filr Schiffe im Seegang als „praktisches Hilfsmittel fiir den Entwurf von Handelsschiffen" zu ermitteln. Sie behaupten, daB die Arbei-ten des M I T auf diesem Gebiete am weitesArbei-ten fortgeschritArbei-ten sind. Alle diskutierten Arbeiten scheinen rein theoretischer Natur zu sein, alle Ansatze benötigen offenbar einen erheblichen Rech-neraufwand, sie bauen alle auf irgenwelchen theoretischen An-satzen iiberWellenlange A und Wellenhöhe h und dem Strömungs-feld um das Schiff auf. Da noch niemand den Seegang so ein-deutig beschreiben konnte, das jeder genau weiB, was auf das Schiff fiir ein Zusatzwiderstand zukommt, wenn in der China-See die Briicke in den Logabstracts festhült, daB das Schiff ge-gen 9 Bft angedampft ist, kann keine Rede davon sein, daB auf diesem Wege überhaupt eine praktisch verwertbare Aussage gewonnen werden kann, Es tut mir leid, aber die ganzen

Unter-suchungen sind von der Praxis noch viel weiter entfernt, als die üblichen Schlepptank Untersuchungen.

Das Schiff, der Propeller und die Maschine überwinden den Seegang doch irgendwie, ohne daB es zu gröBeren Schaden führt, wenn die Besatzung keine groBen Fehler macht, was j a doch bisher nur ganz selten der Fall ist. Warum wird nicht untersucht, wie die Schiffe dieses, ohne finite Elemente zu kennen und ohne die Flow Chart aus Fig. 1 berUcksichtigt zu haben, vollbringen? Warum stellt man nicht fest, daB eine Lösung fUr dieses Problem schon 1992 in einer Arbeit von Yan [1] gegeben wurde? Der Widerstand des Schiffes wird nach Yan mit folgender Glei-chung berechnet

mit«„ = 1,2819 W t o - o d e r r t / = 4,8445 W f w A ) - , dieVerdriin-gung A„ wird mit

Rankine Source Metliods for Seakeeping Problems 423

angesetzl. Diese Gleichung wurde aus der Auswertung von Schönwetter-Etmalen aus Reisebericliten von 14 Schiffen ge-wonnen, das gröl5te war d i c T T "Lagena", das kleinste die "Me-teor".

DieVerbindung zwischen Schiff und Propeller iiber die Sogziffer

t und die Nachstromziffer w fiir das Original-Schiff geschieht

nach Kloster [2] und Wilhelm [3] aus Werten, die von [1] aus den Schönwetter-Etmalen gewonnen wurden.

w = 0.333-C^-^

f = 0.181'C

0.5

(3)' (4) Mit diesen Gleichungen der Annahme der Propellerflügelzahl Z, des Fliichenverhaltnisses AJA^, der Reynoldszahl Re und ei-niger möglicher Steigungsverhiiltnisse P/D lassen sich die K^-} Kurven dieses Propellers nach den Bestinimungsgleichungen der Wageninger B-Serie berechnen.

Fiir den Zusatzwiderstand bei schlechtem Wetter fanden Smolka und Yan aus den entsprechenden Aufschreibungen iiber die Ver-anderung des Verhaltnisses V/zi^, was bei konstant angesetzter Sogziffer w der Veründerung der Fortschrittszahl / entspricht, folgende einfache und praktische Gleichung.

0.5 F^AWS - ^AWO '^w'

(5)

mit C^,^„.= 0.3106 W f H ! " ' • Bft') oder C^„,„= 0.7058 kN • s W und L = L . pp

Hierbei sind die Windgeschwindigkeit V, bzw. die Windstarke in Bft kennzeichnend fiir den jeweiligen Zustand der See, wo-bei vorausgesetzt wird, das der Seegang sich voll ausgebildet hat. Überraschend an diesen Gleichungen ist, daB die Schiffs-geschwindigkeit V , der Tiefgang T und die „signifikante Wel-lenlange" X keinen am Schiff feststellbaren EinfluB haben. Weiter ist überraschend, daB ein Containerschiff mit L = 190 m und B = 28.41 m einen gröBeren Zusatzwiderstand bei gleicher Windstarke als ein viel gröBeres, aber reladv schlankeres Schiff mit L = 266 m und B = 32,3 m hat,

Nicht überraschend ist dann, daB bei kleinerer Schiffsgeschwin-digkeit der EinfluB des Zusatzwiderstandes erheblich gröBer wird, weil der Basiswiderstand R^ auf Grund der geringen Ge-schwindigkeit natürlich sinkt. Da die obigen Gleichungen aus dem Schiffsbetrieb bei realem Seegang, einer über 24 Stunden gemittelten Drehzahl und einem aus dem wirklichen Brennstoff-verbrauch gewonnenen Drehmoment - also einer Real Reality - mit dem Propeller als MeBinstrument gewonnen wurden, sind wir ganz sicher, daB sie für normale Handelsschiffe mit Bugwulst durchaus ein praktisches Hilfsmittel fiir den Entwurf sind, auch wenn die Werte nicht die geringsten Ahnlichkeiten mit den Kurven in Abb, 1 und 7 des Vorabdrucks haben. Sie haber aber den Vorteil, daB sie mit sehr wenig Aufwand an Hand der Bord-aufzeichnungen überprüft werden können, wie die Abb, 4-4 aus [1] zeigt, Sicherlich lassen sich für andere, stark abvveichende Schiffsformen, wie z.B. in Abb. 9 des Vorabdruckes gezeigt, aus dem Schiffsbetrieb genau so einfach verlaBUche Gleichungen für den Glattwasserwiderstand und den Zusatzwiderstand gewinnen, wenn die Genauigkeitsansprüche nicht über das hinaus ge-schraubt werden, was an Bord mit Bordmitteln gem essen und auch spüter überprüft werden kann.

Schrifttum

1 Yan, X.: Beitrag zur Ermittlung des Zusatzwiderstandes von Schiffen im Seegang, Dissertation. D 83, T U Berlin, 1992 2 Kloster, Y.: Lay-out of Diesel Engine and Propeller for All Weatlier Service in Ships, Diplomarbeit, T U Berlin, 1993 3 Wilhelm, T : Motor - Propeller Auslegung schneller

Con-tainerschiffe, Diplomarbeit, T U Berlin, F G Schiffskraft-anlagen. Sept, 1995

Dr-Ing, Carsten Óstergaard und Dipt,-Ing. Helge Rathje, Ger-manischer Lloyd, Hamburg; Verschiedene Methoden zur Ermitt-lung dcs Seegangsverhaltens

Die Autoren haben eine interessante und nach unserer Kennt-nis auch vollstandige Ubersicht über die zum Thema gehören-den Entwicklungen der Rankine Quell-Senken-Methode zu-sammengetragen und diskutiert. Die Arbeit enthalt gleicherma-Ben Anregungen für den Fachmann wie Anleitungen fUr den Lemenden, Hierzu mochten wir die Autoren beglückwünschen. Als Beitrag zu einer Vortragsveranstaltung der S T G vermissen wir allerdings die Darstellung siibstantiellerEigenleistungen, die sich aus unserer Sicht allzu bescheiden hinter der angezogenen Literatur verbergen, Mit Blick auf Abb, 4 des Vortragstextes ware es z,B. für die Belange der Entwurfspraxis auf deutschen Werf-ten wünschenswert, wenn die Gegenüberstellung einerseits von BewegungsgroBen, die nach einer Streifen- und einer Rankine-Methode berechnet wurden, und andererseits auf Modellversu-che zurück gehen, durch Berechnungsergebnisse nach eigener Vorgehensweise der Autoren erganzt werden könnten, Ebenfalls im Hinblick auf den praktischen Schiffsentwurf auf deutschen Werften vermissen wir im Zusammenhang mit dem im schriftlichen Text gegebenen Ausblick auf die alternative Vorgehensweise der sog. Greenfunktions-Methode, die im Ver-gleich zur Rankine-Methode möglicherweise weniger Rechen-zeit benötigt, Hinweise auf Entwicklungen in unserem Lande, die dem deutschen Koautoren sicher bekannt sein dürften. Wir möchten hier einmal auf Entwicklungen beim Germani-schen Lloyd hinweisen und nehmen dazu Bezug auf die Abb. 4 des schriftlichen Vortragstextes, d.h,, wir analysieren das Be-wegungsverhalten des Series 60 Schiffes mit der Greenfunk-tions-Methode nach dem Programm GL-Panel, über das vor ei-nem Jahr vor der S T G berichtet wurde (C, Östergaard und T. E . Schellin, 1995),

Abb. E l . Modellierung des Series 60-Schiffes für C^ = 0.7 mit 2656 Panelen

GLPANEL SWAN (N.,S.,I9») -STRIP N .S.,1990 • EXPERIMENT (G..el aJ.,1974)

VVeüen'aenge^Scfill'clacngo

Abb. E 2 . Tauch-Übertragungsfunktion des Series 60-Schiffes fiir C = 0.7 bei F = 0.2

In unserer Abb. E l haben wir zunachst die notwendige Model-lierung der Schiffsoberflache durch ebene Fanele dargestellt. Fiir den Fachmann ist es in diesem Zusammenhang wahrscheinlich interessant zu erfahren, daB mit diesem Modell fiir alle Über-tragungsfunktionen der 6 Bewegungskomponenten des Schif-fes, dargestellt mit 20 Werten A/L, eine Rechenzeit von etwa 50 Minuten je Wellenrichtung benötigt wird. Vielleicht können die Autoren zum Rechenzeitbedarf nach der Rankine-Methode ei-nen entsprechenden Vergleichswert beistellen?

Die Ergebnisse für die Tauchbewegung finden sich in unserer Abb. E2. Auf den ersten BHck scheint die Streifenmethode am besten geeignet, die am Modell gemessenen Werte realistisch vviederzugeben. Allerdings basieren alle Berechnungsergebnisse auf linearen Betrachtungen, wahrend die Messungen nichtline-are Effekte enthalten, was man bei Gerritsma et al. (1956) nach-lesen kann. D.h., bei allen drei Berechnungsmethoden mUssen noch nichtlineare Korrekturen angebracht werden, wobei erwar-tet werden kann, daB die maximalen BewegungsgroBen entspre-chend kleiner ausfallen werden. Die Streifenmethode wird dann zu kleine Werte liefern, wahrend sich die beiden Panelmethoden (SWAN für Rankine und G L - P A N E L fur Greenfunktionen) in die richtige Richtung (auf die MeBergebnisse zu) bewegen wer-den. Wo also nichtlineare Effekte von Bedeutung sind, wie bei der Bestimmung des Seegangswiderstandsanteils, sind Panel¬ methoden der Streifenmethode vorzuziehen.

An der für einen Seegang notwendigen nichthnearen Korrektur linearer Berechnungsergebnisse nach G L - P A N E L arbeiten wir seit einiger Zeit intensiv. Erste Schritte dazu sind erledigt, ent-sprechende Werte für diese Programmversion werden wir bei passender Gelegenheit vorstellen.

Schrifttum

- Östergaard, C ; Schellin, T, E . : Entwicklung eines hydro-dynamischen Panelverfahrens für dieAnalyse des Seegangs-verhaltens von Schiffen in der Praxis. Jahrbuch der Schiff-bautechnischen Gesellschaft, 89. Band. 1995

Dr-Ing. Volker Bertram, Hamburg (SchluBwort)

Wir danken Herrn Östergaard und Herm Rathje für den interes-santen Diskussionsbeitrag, Dem Wunsch nach mehr Darstellung der eigenen Leistungen kommen wir gerne mit einer Abbildung nach. Ansonsten verweisen wir auf unsere zehn Veröffentlichun-gen in der Literatudiste, die vielleicht doch bei wohlwollender Betrachtung eine 'substantielle Eigenleistung' darstellen, Abbil-dung E3, aus Bertram (1996), zeigt für ein Series-60, = 0,6, die Übertragungsfunküonen für Tauchen und Stampfen mit ei-ner RSM, der Streifenmethode der HSVA und Modellversuchen von Gerritsma (1960). Die Vortelle der R S M sind zwar erkenn-bar, aber noch kein starkes Argument für den gröBeren Auftvand. Das Series-60 ist eben ein sehr schlankes Schiff, das bei einer Froude-Zahl von 0,2 sehr der Streifenmethode entgegenkommt, veiligere Schiffe und höhere Froude-Zahlen sind Anwendun-gen, die wir im nachsten Jahr zu untersuchen hoffen, um Vor-telle des Erfassens der Dreidimensionalitat der Strömung zu de-monstrieren.

Die nichthnearen Effekte spielen bei den hier betrachteten E x -perimenten zum Series-60 nach Gerritsma (1960) für die Mo-delle mit = 0,6 und 0,7 kaum eine Rolle: „In general, the differences in [pitch] and [heave motion RAO] for wave heights

1/48 L and 1/40 L are very small: they approach the measuring error." Die angegebenen MeBwerte sind die für 1/48 L . Gerritsma erwahnt nichtlineare Einfiüsse beim Modell C , = 0.8 und bei höheren Froude-Zahlen. Wir stimmen allerdings zu, daB man die Vergleiche zwischen verschiedenen

Rechenverfahren und Messungen mit skeptischer Distanz

be-2.0 1.5 1.0 0.5 + 0 + + ° . o

XJL

0.5 1.0 1.5 2.0

6

4^

\ns\Llh

X/L

0.5 1.0 1.5 2.0

Abb. E 3 . Übertragungsfunktionen für Series 60, = 0.6 n = 180°, = 0.2; ' 3-d RSM, • Experiment, + Streifen-methode

Fig. E 3 . Response amplitude operators for Series 60, = 0.6 H = 180°, F = 0.2; - 3-d R S M , • experiment, + strip method

Rankine Source Metliods for Seakeeping Problems 425

trachten muB. Auch Messungen haben eineFehlerbandbreile, die normalerweise nicht in den Diagrammen wieder auftaucht. Alle veröffentlichten Reclienergebnisse für das Series-60, einschlieB-lich der Streifenmethoden-Ergebnisse, lassen allerdings auch einen Nachweis vermissen, daB die Ergebnisse netzunabhangig sind. Der Grund liegt nicht zuletzt darin, daB die Netzgenerie-rung und Rechnung für Schiffe recht aufwendig sind. Wir vermu-ten aber, daB die meisvermu-ten Ergebnisse, auch unsere eigenen, mit signifikanten Diskretisierungsfehlern behaftet sind.

Wir können leider nicht der Aussage zustimmen: „Wo nichtline-are Effekte von Bedeutung sind, wie bei der Bestimmung des Seegangswiderstandsanteils, sind Panelmethoden der Streifen-methode vorzuziehen." Starke nichtlineare Effekte müssen im Zeitbereich erfaBt werden. Dies kann auch von nichdineaien Strei-fenmethoden wie S I M B E L geleistet werden, die haufig durch Zeitsi mulation mehr nichtlineare Effekte erfassen als „nicht-lineare" Korrekturen bei dreidimensionalen Verfahren im Fre-quenzbereich. Der Zusatzwiderstand erfordert aber doch gera-de ein weitgehend lineares Vorgehen, da gera-der zugrungera-de gelegte Seegang nur mit wenigen statistischen Angaben (Spektrum) vor-liegt. Wie will man da vorgehen, wenn nicht durch Aufspalten in Einzel wellen, Berechnung der Antwort auf die einzeliie Wel-le und Überlagerung zu einer Gesamtantwort? Dieses Vorgehen setzt Linearitat voraus; ansonsten ist das Überlagerungsprinzip nicht zulassig.

Die eigentüche Rechenzeit der Verfahren, zudem abhangig von Anzahl der benutzten Elemente und Rechenleistung der benutz-ten Maschine, ist für die Praxis in der Regel sekundar, s. z.B. Berttam und Jensen (1993). Wichtig sind vielmehr Kosten (Bear-beitungsaufwand) und Gesamtantwortszeit. Die Kosten sind bei RSM genauso hoch wie bei G F M . Die Gesamtantwortszeit wird wohl bei einer für eine Werft (Stand 1996) üblichen Rechner-ausstattung bei etwa 2 bis 3 Tagen liegen, abhangig davon,-ob eine elektronische Beschreibung des Schiffes (etwa in NAPA) vorliegt oder noch erstellt v/erden muB. Dabei gehen wir davon aus, daB nur die Nachtstunden zum automatischen Berechnen des Seeverhaltens genutzt v/erden und dieWorkstations tagsüber

mit interakliven Arbeiten belegt sind. Sollte diese Antwortszeit für die Praxis noch zu lang sein, könnte man die Berechnungen durch Mehrgitter- und Cluster-Techniken beschleunigen, wie Söding (1996) sie für R S M zur Berechnung stationarer Strömun-gen vorgestellt hat. Vordringlich ist aber zunachst, daB die Ver-fahren auch für alle Begegnungswinkel mit praktischer Genau-igkeit Ergebnisse liefern.

Herr GroBmann schlagt empirisch basierte Entwurfsformeln vor Neben dem Modellversuch und der numerischen Simulation gibt es natürlich für eine Reihe von Fragestellungen einfache Erfah-rungs-Formeln oder -Diagramme, die schnell urid billig anzu-wenden sind. Diese sollte man auch soweit möglich nutzen. Die Formeln von Herrn GroBmann sind bestenfalls für eine frühe Abschützung im Vorentwurf nützlich, aber zu ungenau für den eigentlichen Entwurf. Zudem paBt die Dimension von a„ nicht zum Ansatz in seiner Gleichung (1).

Wir haben hoffentlich deutlich gemacht, daB auch R S M für die Vorhersage des Zusatzwiderstands noch nicht praxisreif sind. Wir halten sie aber für vielversprecliend und glauben, daB in ein bis zwei Jahrzehnten die Anwendung numerischer Codes zur Vorhersage des Zusatzwiderstands ahnliche Routine wie heute die Anwendung von Wellenwiderstand-Codes sein könnte. Da-bei wird man sich wohl auf langere Zeit mit quahtativen Aus-sagen (der Art „Form A ist deutlich besser als Form B") zufrie-den geben müssen.

Schrifttum

- Bertram, V. (1996), A 3-d Rankine panel method to compute added resistance of ships, IfS-Bericht 566

Bertram, v.; Jensen, G . (1993), Wirtschaftlichkeit des C F D --Einsatzes, 27. IfS-Kontaktstudium

- Gerritsma, J. (1960), Shipmotions in longitudinal waves, Int. Shipb. Progress 7/66

- Söding, H. (1996), Advances in panel methods, 21. Symp. Naval Hydrodyn., Trondheim