A nonlinear mathematical model of motions of a planning monohull in head seas

Date Author Address

September 2008 Deyzen, Alex van

Deift University of Technology Ship Hydromechanics Laboratory Mekelweg 2, 26282 CD Deift

A nonlinear mathematical model of motions of

a planning monohull ¡n head seas

by

Alex van Deyzen

Report No. 1595-P

₂₀₀₈

Published: 6th International Conference_{on High-Performance} Marin Vehicles, University of Napels, 18-19 _{September 2008,} Italy, ISBN: 88-901174-9-4

TU De Ift

Deift Ufliversity of Technology

6th

_{International Conference}

on

High-Performance Marine Vehicles

HIPER'08

Na pies, Italy, 18-19 Septe m be r 2008

University of Naples "Federico U" Department of Na vai Architecture

and Marine Engineering

Dr. Carlo Bertorello (Editor)

© Comitato Organizzatore RIPER 08

No part of the material protected by this copyright notice may be reproduced or

utilised in any form by any means, electronic

or mechanical including

photocopying, recording or by any information storage and retrieval system,

without prior written permission from the copyright owners.

ISBN 88-901174-9-4

Printed & Published at Comitato Organizzatore HIPER 08

Department of Naval Architecture and Marine Engineering

University of Naples Federico II,

Via Claudio 21, 80125 Naples, Italy

Table of Contents

Appropriate Tools For Flow Analyses For Fast Ships

I

Volker Bertram

Influence OfHeel On Yacht Sailpian Performance

II

Fabio Fossati, Sara Muggiasca

On An Oceangoing Fast SWATH Ship Without Pitching Resonace

27

Hajime Kihara, Motoki )'oshida, Hidelsugu Iwashita, Takeshi Kinosliita

A Potential Panel Method For The Prediction OfMidchord Face And Back Cavitation

33

S. Gaggero, S. Brizzolara

Nonlinear Seakeeping Analysis OfCatamarans With Central Bulb

47

Dario Bruzzone, Alessandro Grasso, Igor Zotti

Human Body Vibration Response Models In The Context OfHigh Speed Planing Craft And Seat Isolation

63

Systems

Thomas Coe, RA. Shenoi, J. T. Xing

Cure Optimization Of High Performance Resins For Marine Vehicles

71

Njarama/ala Rabearison, Christian Jochum, Jean-Claude Grandidier

Marine Propulsion System Dynamics Dùring Ship Manoeuvres

81

Michele Viviani, Marco A/toso/e, Marco Cerruti, Arcangelo Menna, Giulio Dubbioso

Hydrodynamic Lift In A Time-Domain Panel Method For The Seakeeping Of Fast Ships

95

Pepijnde Jong. Frans van J'a/ree

The Use OlA Vertical Bow Fin For TheCombined Roll And Yaw Stabilization Of A Fast Patrol Boat

107 J Alexander Keuning, Guido L Visch

Numerical And Experimental Study Of Wave Resistance For Trimaran Hull Forms

117 Thomas Maynard, Prasanta K Sahoo, Jon Mikke/sen, Dan McGreer

Optimisation Of Composite Boat Hull Structures As Part Of A Concurrent Engineering Environment

133 Adam Sobey, James Blake, Ajit Shenoi

Application Of The Orthotropic Plate Theory To Garage Deck Dimensioning

147 Antonio Campanile, Masino Mandarino, Vincenzo Piscopo

Adhesive Bonding In Marine Structures

163

Dirk Brügge. Karsten Fach, Wolfgang Franzelius

On The Saint-Venant Bending-Shear Stress In Thin-Walled Beams

.. 173

Vincenzo Piscopo

A Nonlinear Mathematical Model Of Motions Of A Planing Monohull In Head Seas

187

Alex van Deyzen

A Design Method For Contra Rotating Propellers Based On Exact Lifting SurfaceCorrection

. 201

Emilio Tincani, Davide Grassi, Stefano Brizzolara

Reliability Of Weight Prediction In The Small Craft Concept Design

215

lzvor Grubisic

An Optimization Procedure For The Preliminary Design Of High-Speed RoRo-Passenger Ships

227

Sotirios Skoupas; George Zaraphonitis

A Study On The Prediction Method Of Wave Loads Of A Multi-Hull Ship Taking Account Of The Side

Hull Arrangement

239

)'oshiiaka Ogawa

Trimaran Maneuvering Simulation, Based On A Three-Dimensional Viscous Free Surface Flow Solver

249

An Experimental And Numerical Study On Cavitation Of Hull Appendages

257

Luca Savio, Chiara Pittaluga, Michele Viviani, Marco Ferrando, Francesco Conti

Disposal And Recycling Of HSC Materials

271

Henning Gran2ann, Reinhard Krapp, Volker Bertram

Surf Hydromechanics

281

Carlo Bertorello, Luciano Oliviero

Numerical Simulation Of High Speed Ship Wash Waves

291

Kunihide Ohashi, Jun Hasegawa, Ryohei Fukasawa

Dynamics And Stability Of Racing Boats With Air Wings

301

Nikolai Kornev, Lutz Kleinsorge, Gunther Migeotte

Development Of Numerical Tool For Hydrodynamics Simulation Of High Speed Planing Crafts

311 Ebrahim Jahanbakhsh, Roozbeh Panahi, Mohammad SaeedSeif

Avoiding Common Errors In High-Speed Craft Powering Predictions

317

John A/meter

Optimization Of The Geometrical Parameters Of A Bonded Stiffener: Finite Element Analysis

327

A NONLINEAR MATHEMATICAL MODEL OF MOTIONS OF A PLANING MONOHULL

IN HEAD SEAS

Alex van Deyzen, Deift University of Technology, The 'Netherlands

SUMMARY

A nonlinear mathematical model for the simulation of motions and accelerations of planing monohulls, having a constant

deadrise angle, in head waves has been formulated. The model is based on 2-dimensional strip theory. The original

model, developed by Keuning [9], who based his model on Zarnick [25], is extended to three degrees of freedom: surge,

heave and pitch motion can be simulated The simulations can be carried out for a planing boat sailing in (ir)regular

head seas, using either a constant forward speed or constant thrùst. The hydromechanic coefficients in the equations of motion are determined by a combination of theoretical and empirical relationships. The sectional hydromechanic forces are determined by the theory of a wedge penetrating a water surface. The wave excitation in vertical direction is directly integrated in theexpressions for the hydromechanic forces and is caused by the vertical orbital velocity in the wave and the geometrical properties of the wave, altering the total wetted length and the sectional wetted breadth and immersion. An overall frictional resistance has been estimated. A constant thrust force can be set as input. When simulating with a constant thrust, the surgemotion is induced by the frictional resistance and the horizontal component of the hydrodynamic force(hull pressure resistance)

The total calm Water resistance has been validated. Existing experimental data of two models, a conventionaldouble chinedplaning monohull(DCH) and a modern axebow (Axehull),planing in calm water showeda fair agreement with the calculated drag. A large sensitivity of the hydromechanic coefficients on the computated results for the total resistance, vertical motionsandaccelerations was found as well.

187

NOMENCLATURE ffrn Sectional hydrodynamic lift associated with the

change of fluid momentum

Deadrise angle of cross section FN Froude number over displacement (,/gI/3)

V

t

Dynamic viscosity of waterPropulsor shaft's angle

F51,

F

Total hydrostatic force

Total hydromechanic force in x-direction

o,Ò,Ö Pitch angle, velocity and acceleration Body fixed coordinate system

Total hydromechanic force in x-direction minus terms associated with motion acceleration

A Cross sectional area _{TOtal hydromechanic force in z-direction}

a Reduction length for transomcorrection function Total hydromechanic force in z-direction minUs

Qbf Buoyancy force correction factor terms associated with motion acceleration Buoyancy moment correction factor h Immersion of cross section

anondim Dimensionless reduction length for transom

cor-rection fUnction 'a_4y

Pitch moment of inertia of the total added mass Pitch moment of inertia

A.

Total wetted area _L Wetted chine length

b Half breadth of cross section _{Lt Wetted keel length}

CD, Cross flow drag coefficient Lm Mean wetted length

Cj Friction coefficient M Mass of ship

C,n Addedmass coefficient _Ma Total added mass

Cpu Pile-tip factor _ma Sectional added mass

Cir Transomcorrection function _Qa Total added mass moment Froudenumber overbreadth ( y, Rp Frictional resistanceof bare hull

\

)

Total frictional resistance force along the hull

D R

Reynolds number ()

F0 Total hydromechanic pitch moment Rp Hull pressure resistance Total hydromechanic pitch moment minus terms RSR Total spray railsresistance

associated with motion acceleration Rs Total spray resistance

fa

Sectional buoyancy force

Rp

Viscous pressure resistance

fcfd Sectional viscous lift associated with the cross Ry Total viscous resistance

flow drag of a calm water penetrating wedge Rw Total wavemaking resistance

This publication wasnot found, but is mentioned for the completeness

188

range (FNV < 2) (semi-displacement), where the seakeep-ing characteristics are very similar to the displacement hull and the high speed range (FNV > 2), where the hydrody-namic lift forces are predominateand where high impact forcescan occur.

Fridsma [2, 3] executed systematic model tests with a serie of constant-deadrise models, varying in length.

His results, presented in the form of response characteris-tics, cover a wide range ofoperating conditions andshow, quantitatively, the importance ofdesign parameterson the rough water performance of planing htills. At this time it already became apparent that planing monohulls show a significant nonlinear behaviour in head waves;

The study of planing monohulls is closely related to the

study offlat and V-bottom prismaticplaning surfaces and to the study ofa 2-dimensional wedge penetratinga calm water surface. These studies were initially carried out in order to get a better understanding of the hydrody-namics involved with the landing of seaplanes (for over

nearly hundred years these topics have been studied, but the works of Von Karman [22] and Wagner [23, 24] can

still be seen as the most important contributions in this

field), bUt later were also used to get moreinsight of plan-ing of monohulls, see for exampleSavitsky's work [17]. Zarnick [25], [26] 1) developed a nonlinear mathematical

model of motions of a planing monohulls in head seas,

where the solution is solvedin the timedomain. l-Lis model is based on 2-dimensional strip theory and the forces

act-ing on a cross section are determined by the theory of a wedge penetrating a fluid surface. The instantaneous values of wetted length, trim and sinkage are taken into account using strip theory in the time domain. The co-efficients in the equations of motion are determined by

a combination of theoretical and empirical relationships. His model showed remarkably good agreement with ex-perimental data.

His work forms the theoretical basis for the

simula-tions models developedby Akers (Powersea) [1] and Ke-uning (Fastship)[9].

Garme[6, 4] developeda similar time domain simula-tion model forthe mosimula-tions of a planingmonohulls in head

seas, but his model distinguishes from Zarnick's model,

becausehe implemented pre-calculated cross section data, so that the hull geometry is better accounted for.

Later,Garme[5] improved his time domain simulation model by adding a near-transom pressure correction func-tion, which reduces the pressure near the stern gradually to zeroat the stern.

In the present research the original mathematical model

of motions of a planing monohull inheadseas, developed by Zamick and later extended by Keuning, is extended to three degrees of freedom: the surge, heave and pitch mo-tion in (ir)regular head seas can be simulated. The simu-lations can be carriedwith either a constant forward speed orconstant.thrust.

V Water entry velocity of penetrating wedge

vs Forward speed

w Weight of ship

w Vertical orbital velocity at the undisturbed water

level

X'y'z Earth fixed coordinate system

XI x-coordinate measured fromstern

x0 Moment armofhydrodynamic lift force

xb Moment arm:of hydrostatic lift force

xcG,xcG,.*cG Displacement, velocity and acceleration of

CG in x direction relative to earth fixed

axessys-tern

xd Moment arm of frictional resistance force x5,y5,z3 Steady translating coordinate system

x, Moment arm of thrust force

zcazcG,zcG Displacement, velocity and acceleration of

CG in z direction relative to earth fixed axes sys-tem

I INTRODUCTION

The behaviourof planing monohulls in waves has been a widely researchedtopic since moreand more semi-planing and planing monohulls appeared after the Second World War. Typical planing monohulls are: patrol vessels, pilot-boats, rescue vessels, coast guard vessels and small navy vessels.

The pressure acting on planing vessels running in calm water ischaracterized by a hydrostatic and hydrodynarnic

part. Due to the high forward speed and trim of the

ves-sel there is a relative velocity between the hull and water and ahydrodynamic pressure proportional to the square of this relative velocity is generated. At high forward speeda large part of the weight of the vessel is carried bythe dy-namic pressure. In wavesthe relative velocityand thus the

dynamic pressure gets additional contributions from the

vessel's motions andthemotions of thewaves. The result-ing nonlinear impact loads havea significant influence on the motions and accelerations in more or less every wave encounter and are crucial for the extreme responses.

For example, when such vessels are sailing in rough

head seas, violent motions and largevertical acceleration peaks occur. Thehullis subjectedto highimpact loads and the crew experiences high transient vertical accelerations and in most cases the crew needs to lower speed in order

to avoiddamageto the hull.

A good understanding of the behaviourof fast vessels in waves is necessary in order to be able to develop plan-ing vessels With large operability. Moreover, most of the

afore mentioned type of planing monohulls must beable

to operate in (extremely)rough weather.

In [18] Savitsky presented ai analysis is made of

Fridsma[3] discovered in his extended research onthebe-haviour of hard-chine planing monohulls in head seas that little or no surge motion was measured for models

sail-ing at high speeds. This would mean that if towed at constant forward speed, a model planing hull would

be-have exactly as if it would be tested at constant thrust. He proved that this hypothesis is also true for the lower speed

range (FNV 1.5). However, he only used one modeland

two seastates to proof his hypothesis With the present computational model Fridsma's hypothesis can be verified or rejected, using the calculated results of more than two scenarios.

In the industry there is an increasing need to predict the

motions and accelerations of a planing vessel inthe design state. The nonlinear mathematical model developed inthis research paper provides a computational design tool, with a rather simple input of the hull and little computer cal-culation time, for designers of fast planing monohulls to predict the operability in various sea states.

Moreover, in the near future, the effect of active

con-trol of the thrust (variation of the forward speed) when sailing in head seas on the vertical peak accelerations needs to be investigated. This mathematical model will

be a valuable simulation tool for this research.

2 THEORETICAL BACKGROUND

The nonlinear mathematical model presented in this sec-tionis an extension of the work of Zarnick [25] and Keun-ing [9]. The simulation model is termed Fastship.

2.1 APPLIED THEORY, ASSUMPTIONS AND

LIMI-TATIONS

Strip theory is used for the determination of the

mo-tions and acceleramo-tions of the system of a fast monohull

in waves. When strip theory is used the assumptions

are made that interaction effects within the 2-dimensional flowsof the cross sections:are negligible and thus that the hydromechanic forces, acting on the hull, can be

approx-imated by integrating forces on cross sections over the

ship's length.

Zarnick used the theory of a calm water penetrating

wedge for determiningthe forcesacting ona crosssectiOn When looking at one slice of water (no waves), a planing

monohull passing through it is like a wedge penetrating

the water surface with a constant velocity, see figure I.

189

rounded bilges; since the model simplifies cross sections to a knuckled wedge.

Zarnick used the time domain approach for the determi-natioñ of the behaviour of fast monohulls in head waves, because with the time domain approach the nonlinearities are seized better than with a frequency domain approach

[12].

Furthermore, he assumed that the flow around the hull

must be treated as quasi steady (every time instant the

equilibrium of the forces and moments are analysed and from there the accelerations-are determined) and that sur-face wave generation (wave resistance) and forces:associ-ated with unsteady circulatory flow can beneglected.

The wavelengths are assumed to be large in comparison with the ship's dimensions and the wave slope is small. Because of the large wavelengths diffraction forces can

be neglected (only the Froude-Krylov forces are of impor-tance).

The wave excitation in vertical direction is directly inte-grated in the expressions for hydromechanic forces and is caused by:

the geometrical properties of the wave, altering the total wetted length, the sectional wetted breadth and immersion and

the vertical orbital velocity.

Because the ships under consideration are generally shal-lowwith respect-to the height of the waves, theorbital ve-locity is takenat the undisturbed water surfacein the plane

z = O.

The influence of the horizontal orbital velocities on

both the horizontal and verticäl motions is neglected, be-cause these velocities are considered to be relatively small in comparison with the forward speed of the ship. The wave excitation in horizontal direction is very diffi-cult to model when applying strip theory. Together with the assumption that the wavelengths are large(small wave slopes), the present model is limitedto moderate surge mo-tions. Severe surge motions-when diving into a wave when sailing in head seas cannot be simulated. The most impor-tant forcein longitudinal directionat that specificmoment

is the Wave excitation in horizontal direction

(Froude-Krylov and diffraction).

2.2 EQUATIONS OF MOTION

'z

CG

Xs

Zs

Figure 1: A planing monohull canbe seen as a wedge

pen-etrating the water surface Figure 2: Coordinatesystem

e an earth fixed coordinate system with x, y, zaxes,

with the xaxis lying in the undisturbed water

sur-face pointing in the direction of the forward velocity, e a steady translating coordinate system with x5, y3, z3 axes, with the xaxis lying in theundisturbed water surface pointing in the direction of the forward ve-locity and travelling with a given constant veve-locity

and

a body fixed coordinate system with Ei, x- and taxes, with the origin in the centre of gravity of the shipand of which the gaxis is the longitudinal

axis pointing forward.

The forces acting on a fast monohull are visualized in

fig-ure 3. D FL CG ¡ Fdyn

Figure 3: Definition of the forces actingon the ship The equationsof motion can be written as:

MCG

= Tcos(ø+t)Fdy,sinODcosO

MCG

= Tsin(O+t) -

FdcosO

F31+DsinO+ W

Iyy.Ö = TX,+FX0+Fs,0XbDXd

(1) The thrust force T is assumed to be constant. However, the efficiency loss of a (nearly) airborne propulsor has to be taken intoaccount. In the computational model the total thrust efficiency decreases linear to zero with the submer-gence of the aft section.

In order to be able to investigate the effect of the surge motion on the vertical peak accelerations when perform-ing simulations in head seas, a resistance dependent on the forward speed, wetted surface and pitch angle must be modelled. The estimate of the resistance will only be used to model a surge motion, not for resistance calcu-lations. Therefore, constant and/or negligible small resis-tance componentsareleft out of the equationof motion for surge. The inputted'thrust might not be the actual thrust of the vessel, but may be somewhat smaller, due to the un-derestimation of the resistance.

Air friction is not taken into account. The superstructure is not defined in the computational model and the depen-dence of this resistance force on variation of the forward speed is assumed to be minimal.

According to Müller-Graf [13] the total bare hull

resis-tance in calm water of a (semi-)planing monohull consist of the following components:

RIJ=RW+RP+RS-I-RSR+RV (2)

where:

190

Rw: wave resistance

o Rp: hull pressure resistance (horizontal component of thedynamic lift force3 here: Fd sinO)

Rs: spray resistance

RSR: spray rails resistance

Rv: viscousresistance

Zarnick assumed that wave resistance can be neglected. However the wave resistance can be significant, especially when semi-planing. For now, this resistance component has been left out of the equation, especially because no

direct formulation is available. The spray and spray

re-sistance are difficult to model, although recently a paper

has been published about this topic [19]. The results of

that study have not yet been incorporated in thesimulation model.

The viscous resistance of the bare hull consists of a fric-tional and a viscous pressure resistance component:

Ry=Rp+Rp

(3)

The viscous pressure resistance, caused by viscous effects of the hull shape and by flow separat ion and eddy making, can be neglected for

FN > 1.5.

This leaves only two time dependent resistance compo-nents (see also[l7]):

Rp

=

sinO

RF = D

The determination of these resistance components will be explained in section 2.4 and 2.7.

2.3 SECTIONAL HYDROMECHANIC FORCES

The forceacting on a cross section is visualized in figure 4 and consists of three components (forceper unit length):

o a hydrodynamic lift associated with the change of fluid momentum (fj,)

o a viscous lift force associated with the cross flow

drag of a water penetrating wedge(J.fd)

e a buoyancy force related to the momentaneous dis-placed volume (.fb)

-1

frm

fcfd

aft part, where the chines are immersed. The lift compo-nent associated with the cross flow drag of a penetrating wedge is small, but not negligible.

The sectional hydromechanic forces are determined

ac-cording to the theory of a calm water penetrating wedge.

The 2-dimensional penetrating wedge is replaced by a

flat lamina by the assumption that the fluid

accelera-tians are much larger than the gravitational acceleration

[22, 23,24]. The fiat laminaisexpanding at the same con-stant rate 'at which the intersection widthof an immersing wedge is increasing in the undisturbed water surface, see

figure5.This expanding rate is dependent on deadrise

an-gle:

db V

(4) Wagner included a term for water pile-up, which he gave

the value oft/2.

Figure5: Awedge penetrating a calm water surface and expanding lamina theory

Payne[15]presented an approximation of the added mass variation with chines immersed and a conventional cross flow drag hypothesis as an additional lift component l-le found, that the lift increment due to the chines immersed

added mass variation is the same as the one due to the

cross flow drag, so that adding the two togetherresults in a chines immersed dynamic force which is twice the cor-rect value.

In both Zarnick's as Keuning's computational model the additional lift component due to the cross flow drag

has been applied.

Payne [14] also suggested that using a pile-up factor of 7t/2 too high impact loads were found when compared

with experiments. Later, Payne [16] found that the results originally found by Pierson, in which he formulated that the pile-up is dependent on the deadrise, agreed very well

with results found by Zhao and Faltinsen [27]. The

ex-pression for a deadrise dependent pile-up factor is: Cpu =

-

(l

-

(5)

where a value for the pile-up factor of it/2 can be seen as the upper limit.

Hydrodynamic lift associated with the change of fluid momentum

191

Thehydrodynamic lift associated with the change of fluid momentum is given by the rate of change of momentum

of the oncoming fluid in terms of the added mass of the

particular cross section under consideration:

D dF

fím=j(ma.V)=ma.V+muV(maV)

(6) The difference with the ordinary strip theory methods is found in the time dependent added mass. Striptheory is2-dimensional, therefore a lengthwise variation of the added mass has to be included, which is represented in the last

term.

Change of the sectional added mass over the length

plays an important role. Since the added mass of the sec-tionsis relatedto the beamof that section at the momenta-neous waterline and since the beam of planing craft hulls

generally decreases in the aft body to minimize wetted

area, a negative lift could occur using these formulations. The formulation of the negative slope of the added mass

is neglected if it occurs and the hydrodynamic lift force arising from the fluid momentum is set to zero for these

sections[8].

The added mass for a penetrating wedge can be approxi-mated by the high frequency solution:

m, = C,

p - b2

and its time derivative as:

ama db

=m=C,npb

(8)

where C, is the added mass coeffficient and b =h .cot13,

in whichh isthetime dependent immersion of the wedge. When the term for pile-up is included, the breadth is ex-pressed as: b = h cot 13. The determination of the

added mass coefficient C'a, will be explained insection 16.

Additional lift term duetocross flow drag

The additional lift term due the cross flow drag on the sur-faceof a water penetrâting wedgeis expressedias:

fcfd CD, . 00513 p b V2

where CD, is the cross flow drag coefficient. The

deter-mination of the cross flow drag coefficientCD, will be

explained in section 16 Buoyancy force

The buoyancy force on a segment is assumed to act verti-cally and to be equal to the equivalentstatic buoyancy of the section multiplied with a correction factor abf:

fh=abfpgA

(10)

where A is the immersedcrosssectional area of the wedge. The full amount of static buoyancy is never realized, be-cause at the high speeds under consideration the flow sepa-rates from the chines and thestern, reducing the pressures

at these locations to the atmospheric pressure. There-fore the total pressure distribution deviates considerably

(7)

(9) db/dt

from the hydrostatic pressure distribution when applying

Archimedes Law. Therefore, the coefficient a,j always has a value between O and I. When the moment of this

force is determined another correction factor, namely ab,,

isused.

The determination of the values of the buoyancy force and moment correction factors will be explained in section

2.6

2.4 TOTAL HYDROMECHANIC FORCES AND MO-MENTS

The total hydromechanic force on the ship in the vertical

plane is obtainedby thesummation of the three (ff1,f'fd

and f,,) force components for each segment and by inte-gration of these sectional forces over the length of the ship

,see figure 4.

The total hydroñechanicforce in each direction can be

ex-pressedas:

F =

FdSiflO =

= _fffmsin9d ffCfdsinOdr=

-f

{mav+mav

-

U(maV)}sinOd

_f

CØ,cCOS pb V2 Sf1

= FacosO-- sga =

= - ffjmcosOd - f fcjo'C°50d - f fbd =

=

-

f

{mar'+niav_ U(maV)}cosOd

(12)

cosOd

- f abf.pgAcff

and the total hydromechanic pitch moment around the cen-tre of gravity is expressed as:

F9 = FdynXa + F10xi, =

=

f Ifni

d+f Lfd.d+f 4cosO.d

=

=f{m(Ifr+thOv_U(m(Iv)}.rd

₍₁₃₎

+fCD,ccosJ.pbV2.?d

+fabn. pgA .{cosO+sinO}d

The velocities along U and normal V to the baseline of the ship can be expressed as:

U=XCGCosO(zCGw) sine

V_{=XCGSIflO+(ZCGW)C0SO--O}

(14)

192

And the acceleration normal to the baseline is expressed

as:

V=xcGsinO+zcGcos0O

+

cosO - ZCG sinO} (15)

w cosO

+

rw. sine

The hydromechanic forces can now furtherbe elaborated

into:

= {Ma..tCGsiflO MoCGCO50+QaÖ

M0 O(cG cosO - ZCG sinO) + f m0wcosOd

_fmaOwsinOd

_f

InuVd+fUV

- f Um0cosOdr f Um0Òd

-f C'

.cosI.pbV2d}

I={_MoCGSiflO_MoiCGCOS0+QaÖ

M0 (XCG

COSO -ZCG sinO) ± f m0wcosOd

-

f rn0Òwsin0d - f m,Vd + f UV

ainad

- f Um0cosOdE f Um0OdZ

-

f Ci,c cosf pbV2d}

cosO

- f abfpgAdT

Fo=Q0ïcGsinO+Q0cGCosO!aO

+Q0Ó(cGcos0*cGsin0)

fmotiìcosOE4

-

f Uy

+ f Um0

cosO

+fU?nUOdt+fCDccosI3.pbV2 d

i-f

abm pgA {cosO+sinO}dE

where

M0

= f m0

Ia=fma.d

(21)

2.5 NEAR-TRANSOM PRESSURE CORRECTION FUNCTION

Garme and Rosén [6, 7] studied the pressure distribution on the hull of planing craft incalmwater, head and oblique regular and irregular waves. Later, Garme[5] formulated a correction operating on both the hydrostatic andthe hydro-dynamicterms of the load distribution. He based his cor-rection on the assumption that the pressure is atmospheric at the dry transomstem. A strictly 2-dimensional analysis of the lift distribution on the planing hull over estimates the lift in near transom region [6], see figure 6.

Figure 6: The principal lift distribution on a hull with a

dry transom stern (the dotted line indicates the strictly 2-dimensional lift distribution)

Further, it is assumed that the difference between the 2-dimensional lift distribution and the actual pressure is largest at aft and decreasing afore. The correction ap-proach is to multiply the 2-dimensional load distribution

by a reduction function that is O at transom, that

ap-proaches I at a distance afore, and has a large gradient

at aft which decreases towards zero with with increasing distance from the stern.

The near-transom pressure correction function is

ex-pressed as:

Cir(Xi) =

tanh(.xt)

(22)

in whichaisa reduction length, sec figure 6.

Garme rewrote the reduction length into nondimensional

form:

anond,m =

a

(23)

Dm

in whichBmis the full breadthof the mainsection andC istheFroudenumber over breadth:

=

sJgHm

After a systematic research on several model experiments Garme chose a value of 0.34 foranondEm,which

isapplica-ble for medium and high speed configurations,C 2.

Thenear-transom pressure correction function can noW be rewritten into:

Cir =_{tanh (} 2.5

0.34 Bm. c,

tir)

(24)

193

in which ris the body fixed coordinate of the stem.

Garme validated the reduction function on basis of the model test measurements of the near-transom pressure, and on published model data on running attitude. This

correction improves the simulation in both calm water and in waves for a widerspeed range.

Although, a constant correction length is questionable if the ship motions are large and the wetted hull length is small as for sequences when the hull leaves or is close to leaving the water.

The transom reduction function reduces the sectional

forces in the aft ship and has to be inserted within the in-tegrals for the sectional hydrodynamic forces as follows:

= -

_f

C1, ffmSiflOd JC'ir fcfd51flOd (25)

F2 = _fClrffmCOSOdfCtrfcfdCOSOd

_fCirfbdE

F9 =

f

C:, Jim d+f Cir leid

+fClrfbcosOd

2.6 DETERMINATION OF HYDROMECHANIC CO-EFFICIENTS

The integrals for the total hydromechanic forces and mo-merits can be evaluated when the four hydromechanic co-efficients(CD,,Cm, abfand abni) are known.

The lilI force dùe to the cross flow drag is of minor

im-portance, when compared with mass flux and buoyancy, so fixing the value of the cross flow drag coefficient has

only a marginal effect on the total lift. Both Zamick as Keuning fixed the value of CDC, according to the approach of Shuford [20]. Zarnick assumed thatCD, =1.0 and and

Keuning assumed thatCD, = 1.33. The latter one is used in the present computational model.

Originally, Zarnick used constant values forCm, aj,j and

a07. He assumed that the added mass coefficient C,,, was

equal to 1 and that the buoyancy correctionaòj wasequal to 1/2 and that ahm, the correction for the longitudinal dis-tribution of the hydrostatic lift, was equal to 1/2 .

a.

He

used apile-up factor independent of deadrise:

=

Keuning showed that Zarnick's method is only

appli-cable to very high speeds, because of the constant

val-ues he used for the hydrodmechanic coefficients. Keun-ing. together with Kant [8], approximated the trim angle

and sinkage of the craft under consideration using

poly-nomial expressions derived from the results of systematic model tests, the DeIft Systematic Deadrise Series (DSDS)

The solution of the equations of motion, describing the

steady state planing in calm water, is known, because of these polynomial expressions. Substituting these values

for sinkage and trim in the equations of motion results in

a system of two equations and three unknowns.

Keun-ingand Kant assumedthat there is noadditional factor for the correction of the longitudinal distribution of thehydro-static lift: ahm = 1. The values of Cm and abf can now be determined.

By determining the hydromechanic coefficients in this way, the hydrodynamic lift is brought into the

computa-tional model With a higher level of accuracy than in the original Zarnick model and the model can be used for a

broader speed range. The present model is applicable for speeds (FN > 1.5), but it also restricted to hull forms sim-ilar to the models used in the DSDS.

Determination of hydromechanic coefficients C, and

a,f

A planing vessel, sailing in calm water with a constant

speed, is sailing in stationary condition. Sinkage and trim are constant in time. The sinkage and trim are determined

by three components of the hydromechanic force in the

vertical force and moment equilibrium. 1f only steady state planing isconsideredthe following simplifications may be introduced in theequations:

OXCG 2CG

= O

U=ICGCOSO (28)

V=XCG'SiflO

The equations of motion in the stationary condition in

calm water are reduced tO(XCG = constant):

Heave:

F2+w

= O Pitch:

F=0

where:

I =fCir()UV.CO5Od

-

jC,r(f)CD,ccosI3pbV2cosOd

-

fCsr(e)abfpgAd

F_fCir(,)UV

+fC:r().CD,ccOs3.pbV2 .Tdr;

(31)

+fCir()

pgA. {cosO+ sinO}d

in which F and F are the total hydromechanic forces mi-nus terms associated with motion accelerations.

2.7 FRICTIONAL RESISTANCE FORCE The total frictional resistance can be determined by:

1 ₂

D=CF--p-U -A

194

The velocity along the baseline U can be expressed as:

U

=

XcG cosO - zcG sinO The influence of the orbital

velocities is negligible and can therefore be omitted. At each timestep the mean wetted length, the Reynolds number and the friction coefficient are calculated. The mean wetted length is the average between the wetted keel length and wetted chine length and is formulated as:

Lk + L

Lm

-andthe Reynolds number as: U Lm

R

-V

The friction coefficient is determined using the lTl'C

for-mula.

0.075

C1

(logR,1 2)2

The total wetted surface minus the di-y stern is estimated by adding the surfaces of the wetted sections. The moment arm of this force is estimated by assuming that the centre of effort lies halfway the average immersion of the cross sections.

2.8 SOLUTION OF THE EQUATIONS OF MOTION

The equations of motion form a set of three coupled

second order nonlinear differential equations, which are solved in the time domain using standard numerical tech-niques. The equations of motions can be written in matrix

form:

M-i=F=.

/ M+Msin20

MasinOcosO

\

Q,sinO

M0 sin OcosO M+ M0cos2O Q0 COSO Q0 sinO Q0 CO50

(I+I)

/

Tcos(O+t)+PDcosO

= (

Tsiñ(O+t)+F+DsinO+W

Tx1 +FDxa

in which I, F2' and F1 are the total hydromechanic forces minus terms associated with motion accelerations. The solution of these sets may be found by:

(35)

The procedure used to solve these equations isthe

Runge-Merson method Knowing the vessel's orientation in the

earth fixed coordinate system and the velocities at t = to the equations are simultaneously solved for thesmall time (32) _{increment dt to yield the solution on t + dt.}

).(

3 VALIDATION OF CALM WATER RESISTANCE Data of existing model tests were used to validate the to-tal calm water resistance [21]. Two different hull shapes were used for these tests: a double chined planing mono-hull (DCH, 17° deadrise angle in aft ship) and a modern axebow (Axehull). The main dimensions are given in table I and a sketch of the hull geometriesare given in figures 7

and 8.

Table 1: Main dimensions DCH and Axehull

U.U..;..]

Figure 7: Sketch DCH

...irnri

1UU1UjW'

Figure 8: Sketch Axehull

The values of the hydromechanic coefficients aj and Cm

were difficult to determine, because of the thct that the geometry of two models deviate significantly from the

DSDS. Therefore the coefficients were estimated by using a parent hull from the DSDS with a comparable deadrise in theaftship(model 363, 19° deadriseangie)asa reference. This resulted in the following values:

195 E w 0 -ii C us

Table 2: Used values of hydromechanic coefficients

For the validation the total resistance has been calculated at a constant forward speed. The working line of the thrust force T and the frictional resistance force D act

through CG(no additional moments) andthe vertical com-ponents of T and D arenegligiblesmall with respect to the other hydromechanic forces involved (no additional verti-cal force cornponents)[25, 9].

Figures 9 to 13 show respectively the resuitsof measured and calculatedsinkage, trim, wetted surface, friction coef-ficient and resistance of the DCH.

The general trend of the sinkage has been captured, al-though a srnall'deviation can-be seen in the lower to middle speed range. At higher speeds the trim is underpredicted. Perhaps water spray onthe two sprayrails of the hull cause a larger trim angle than calculated. In the mathematical model no spray rails have been modelled The wetted sur-face is underpredicted over the whole speed range. This results in a small underprediction of the total frictional re-sistance force, although a small overprediction of the

fric-tion coefficient compensates this effect slightly.

The-residuary resistance is clearly underpredicted. The underprediction of the trim at higher speeds might yield a underestimation of the hull pressure resistance. At lower speeds the magnitude of the wave resistance might be sig-nificant. This resistance component should decrease to-wards higher speeds. At lower speeds, the calculated

residuary resistance is about 25 percent of the measured residuary resistance, while at higher speeds it is 50 per-cent. The magnitude of thespray and spray rails resistance

is-still unknown. SlnkageDCH 20 10 0 10 .20 -30 .40 50 -60 00 S' 05 10 15 2.0 25 30 35

Froude number over dlsp!acement (.J DCII meas ---s--- DCII cale

Figure 9: Measured and calculatedsinkage DCH

40

Designation Unit DCH Axehull Length over all m 19.34 20.00

Beam over all m 6.3 5.65

E o 00 100 90 C ₈₀ 70 60 50 40 30 20 10 60 50 40 01 30 20 w 10

J

00 0.0023 0.0022 T 0.0021 b 0.0020 0.0019 0.0018 0.0017 00 05 10 15 20 25 30 35

Fraude number over dlsplacement(-J

J-.-- OCHn,eas --.-- OCHah

Figure 10: Measured and calculated trim DCH

Wettedsurface DCH

's

05 10 15 20 25 30 35

Fraude number over displacement I-i

--- DCII,eeas --.--' OCHcaIc j

05 10 15 20 25 30 35

Fraude number over displacemen (-j DCH aseas --.-- OCH cak

J

40

40

Figure 12: Measured and calculated friction coefficient

DCH

Resistance DCII

---- 001-fit meas ----DcH-Rf meas a-- DcH-Rrest meas --U- -. OCH-St calc - - -. ocH-Fit cdc -- *- -. OCH-Orest edit

6

os

si

E

o

-0

Axehull meas --e -. Axehull cale

Figure 14: Measured and calculatedsinkage Axehull

Thm Axehuil

00 05 10 15 20 25 30 35

Froude number over displacement ('j

Wetted surface Axehuii

05 10 15 20 25 30 35 40

Fraude number over displacemen [-j Aaehull meas ---U--- Asehisli caic

Friction coefficient Axehuii

0.0023 0.002 2 'r 0.0021 b 0.0020 0.0019 0.0018 -L. 0.00 17 00 05 10 35 20 25 30 35 40

Fraude number over displacement I-j

40

Figure 16: Measured and calculated wetted surface Axe-hull

-U--- AxehalI mea. - - U--- Asehisll

Figure 11: Measured and calculated wetted surface DCH

Figure 15: Measured and calculated trim Axehull

Friction coefficient DCII

U Aiseiwil mea. ---a- -. Axehull cale

Figure 13: Measured and calculated resistance DCH

Figure 17: Measured and calculated friction coefficient

Figures 14 to I 8show respectively the resultsof measured Axehull

196

Thm DCH _{and calculatedsinkage, trim, wetted surface, friction}

coef-ficient and resistance of the Axehull.

Sinkage Axehuil 20 10 E o s

.

U. U u s., 20 3° _.5. -4° 5° 40 00 05 1.0 15 20 25 30 35 40

Fraude number over displacement [-j

00 05 10 15 20 25 30 35

Froude number over displacement I-)

Resistance Axehuli

05 10 55 20 2.5 30

Froude number over displacement (-J -U-- AaehuIt-ft meas e--- DCet'UIJ-RI meas a--- OCaFLII-arest meas - - e- -. DCeItst.Rt caic --4--- OCa44I-Rtcalc --a--- DCeI$ill-Rrestcalc

Figure 18: Measuredandcalculated resistance Axehull The measured sinkage of the Axehüll is nearly zero over the whole speed range. The mathematical model does not predict this constant valúe. Thé trim is overpredicted. The wetted surfaceis underpredicted over the wholespeed range, probably because of the overprediction of the rise of the vessel. This results in an underprediction of the total frictional resistance force.

The erroneous calculated results for the sinkage and trimof the Axehùll might also be caused by the estimated values for the hydromechanic coefficients. Perhaps the

used values are not applicable for axebow hull shapes. At this moment it is not known what more appropriate values should be for these kindof hull shapes.

Generally, it canbe concluded that the frictional resistance forceispredicted accurate eriough But the residuary resis-tance andtherefore the total resisresis-tance is clearly underpre-dicted using only the hull pressure resistance. Theremain-ing resistancecomponents shouldibe incorporated into the model.

However, the mathematical isnot developed for accu-rate calculations of the resistance; we are only interested in the time varying resultant force in longitudinal direction. For now, it is assumed that an accurate surge motion can be simulated, using only the hull pressure and frictional

resistance.

4 SIMULATIONS ADDRESSING THE

DIFFER-ENCE BETWEEN CONSTANT SPEED AND CON STA NT THRUST

Fridsma [3] observed little to no surge motion during his measurements at high speeds. Simulations with constant speed and thrust at high speeds show this trend as well. m'the lower speed range he obsérved some surgeinotion.

He tested a 100 deadrise model (LIB = 5, B = 22.9 cm)

at FNV = 1.5 in two sea states (a Pierson-Moskowitz

spec-trum with HS/B = t1444 and (H3/B = 0667) both with a

constant thrust and constant speed. The average total re-sistance agreed very well. The distribution ofthe crest and throughs of the heave andpitch motions were nearly equal, as well as the distributions of the vertical accelerations at the boW andthecentre of gravity.

Simulations carried out for the DCH and the Axehull in

197

moderate sea states in order to address a possible differ-ence between while simulating with a constant forward speed and a constant thrust, neither showed a remark-able difference. A wave realisation has been made ac-cording to the Jonswap spectrum. Three forward speeds (V5 = 20,30,40 kn), three significant wave heights (I1 = 1.0, 1.5,2.0 m) and three peak periods (T = 7, 10,13 s)

have been chosen. The total run length was 1100 seconds.

Whilesimulating withconstantthrust,,the averageforward

speed over the total ninlength has been used as a measure

for the thrust.

The computational model has been validated for the mo-tions in headseasby both Zarnick as Keuningusing the re-sultsof model tests. They carried oUt modeltests with con-stant forwards speed. Because of the fact that theresults of simulations carried out with constant thrust show no re-markable difference with the resultsof simulations carried out with constant forward speed, it can be assumed that the motions and accelerations are predicted with the same level of accuracy as the original computational model. However, theaccuracy of the calculated results for

themo-tions and accelerathemo-tions for the Axehull is still

question-able. The results are very sensitive to the used values of the hydromechanic coefficients.

5 CONCLUSIONS AND FUTURE WORK

A nonlinear mathematical model of a monohúll having a constant deadrise angle, planing in head waves, has been

formulated using strip theory. Keuning's [9] nonlinear

mathematical model, based on Zarnick's model [25], has been extended to the possibilty to simulate with either con-stant forward speed or concon-stant thrust.

The time domain approach is used for the determination of the motions. Each time step the sectional forcesareelab-oratedandthe total vertical and horizontal hydromechanic force and the total hydromechanic pitch moment are found by integrating the sectional forces and moments over the length of the vessel.

The surge motion is induced by a speed dependent

fric-tional forceand thehorizontal component of the hydrody-namic force (hull pressure resistance), which varies With speed, trim angle and wetted surface. The thrust force is assumed to be constant. In order to find a more accurate surge motion, the wave, spray and spray rails resistance [19] still need to be incorporated.

Diffraction forces are neglected, only Froude-Krylov forces are of importance. Therefore the assumption is made that the wave lengths are long in comparison to the vessel's length andthat wave slopes are small.

The coefficients inthe equations of motion are determined

by a combinatiOn of theoretical and empirical relatiOn

the trim angle and sinkage for the situation of steady state planingin calm water. The values of the twomost relevant coefficients aredeterminedby substituting these values for sinkage and trim in the equations of motion.

However, this approach is very sensitive to errors. If

the huligeometry deviatessignificantly from the DSDS,a

value for the hydromechanic coefficients cannot be found

and has to be estimated using a model of the serie with

similar deadrise in the aftship.

In order to increase the level of accUracy of the com-putational model for modem hullshapes a thorough inves-tigationinto thehydromechanic coefficientsshould be car-ned out. Instead of applying a constant buoyancy correc-tion factor and the added mass coefficient over the whole hull, a solution can be foUnd in sectional hydromechanic coefficients, dependent on forward speed, deadrise, trim and sectional width. Research of the pressure distribution of planing V-bottom prismatic surfaces might give some insight in a finding a more accurate approximation of the (sectional) buoyancy correction factor and theadded mass coefficient.

The computational model has been validated for the

mo-tions in head seas by both Zarnick as Keuning using the results of model tests. They carried out model tests with

constant forwards speed.

Simulations with constant forward speed and constant thrust, carried out for a double chined hull (DCH) and an axebow hull shape (Axehull) showed no remarkable differ-ences in motions and vertical accelerations. lt can there-fore be assumed that the motions and accelerations, cal-culated with constant thrust, are predicted with the same

level of accuracy as the original computational model

(constant forward speed). Fridma's hypothesisthat model

tests and thus simulations with constant forward speed

generate the same results for the motions and vertical

ac-celerations as model tests or simulations with constant

thrust has been verified for moderate sea states

However, it is still recommended to carry oUt model tests with free sailing self propelled models in head seas.

The relation between thrust, resistance, motions

accel-erations and wave profile needs to be studied more thor-oughly It is also expected that the number of (very) large vertical peak accelerations in higher sea states decreases when executing model tests with a constañt thrust. Next, the influence of active control of the thrust can be studied using this simulation modeL

The present nonlinear mathematical modUl of motions of

a planing monohull in head seas provides designers of planing vessels a computational tool, with little

calcula-tion time, that is able to predict the surge, heave and pitch motion and the vertical accelerationsin various seas states.

The model isapplicable for speeds largerthanFN 1.5.

The geometry of cross sections of the hull are

approxi-mated by the shape of a hard chined wedge. The designer isable to analyse magnitude and probability of exceedence of large vertical peak accelerations when sailing in head

seas.

198

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