A STUDY OF METHODS TO PREDICT ADDED RESISTANCE IN WAVES Performed at Seaware AB

MARTIN ALEXANDERSSON [email protected] Master Thesis, KTH Centre for Naval Architecture STOCKHOLM JANUARY 2009

ABSTRACT This thesis is an overview of added resistance in waves, available methods and how a simplified method can be developed. Added resistance in waves is indeed a very complicated problem. The present methods, most of them developed during the 1970s, give reasonable results for head waves. But for quartering and following waves, the methods have low accuracy, due to uncertainties in strip calculated ship motions for these cases, and the ability to account for roll and yaw motions. In this thesis Gerristma and Beukelman’s method and Boese’s method to calculate added resistance is implemented and validated against published calculation results, from previous implementations. The implementations are also evaluated against published experimental results in head waves. These evaluations show that added resistance with good accuracy can be calculated for head waves using Gerritsma and Beukelman’s method. This thesis also shows how a simplified method to calculate added resistance in waves can be developed. This simplified method only uses the ships main particulars to describe the hull. The method is intended to be used when the entire hull geometry is not available, for instance in the pre-study of a new ship design. The idea with this method is to estimate transfer functions for added resistance without using strip calculations. The transfer functions are parameterized with three parameters, peak value, peak frequency and spreading. Expressions for these parameters are derived with regression analysis, based on analytical results from Gerritsma and Beukelman’s method. The simplified method has an accuracy of about 25%, which is about the same amount that usually can be expected of a method to calculate added resistance in waves.

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SAMMANFATTNING Denna rapport är en översikt av adderat vågmotstånd, tillgängliga metoder och en beskrivning av en förenklad metod. Adderat vågmotstånd är ett väldigt komplicerat problem. De metoder som används idag, utvecklades under 70-talet och ger rimliga resultat för mötande vågor. Metoderna ger dock sämre resultat då vågorna faller in snett, från sidan eller bakifrån. Detta beror på att de strip-beräknade fartygsrörelserna är osäkra för dessa vågriktningar och att metoderna inte tar hänsyn till fartygets rullning eller att det girar i vågorna. I denna rapport implementeras Gerritsma & Beukelmans metod och Boeses metod. Implementationerna jämförts med publicerade resultat från tidigare implementationer samt publicerade experiment-resultat, för mötande vågor. Dessa jämförelser bekräftar att Gerritsma & Beukelmans metod kan användas för att beräkna adderat vågmotstånd för mötande vågor. Detta examensarbete förslår en förenklad metod för att beräkna adderat vågmotstånd. Metoden använder endast enkla fartygsparametrar för att beskriva fartygets skrov. Den förenklade metoden är tänkt att användas då ett fartygs exakta skrovgeometri inte är känd, till exempel vid en förprojektering av en ny fartygsdesign. Idéen med metoden är att den ska generera transferfunktioner för adderat vågmotstånd, utan att använda stripberäkningar. Transferfunktionerna beskrivs med tre parametrar, pikvärde, pikfrekvens och spridning. Matematiska uttryck för dessa parameterar härleds med regressionsanalys, baserat på analytiska resultat från Gerritsma & Beukelmans metod. Den förenklade metoden har en exakthet på ungefär 25%, vilket är den exakthet som normalt kan förväntas av en metod för att beräkna adderat vågmotstånd.

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PREFACE Welcome to this master thesis! This is a Master Thesis at the Royal Institute of Technology, KTH, Centre for Naval Architecture. This master thesis project was performed at the company Seaware AB in Stockholm, Sweden. Seaware AB develops computer programs for weather routing and operational decision support. During my work with this master thesis project I have increased my knowledge about ship hydrodynamics and added resistance in particular. I have understood that this is a very complicated field where a lot of work still can be done. I also realize that I have a lot left to learn, and that there are a lot of aspects to the problem that I haven’t studied. I have tried to create this thesis as if it would be one that I myself had wanted to read, when I started out the literature study in this project. I would also like to thank: Mikael Palmquist, my supervisor at Seaware AB, and Karl Garme, my supervisor at KTH. I hope you will enjoy this master thesis! Stockholm, January 2009 Martin Alexandersson

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CONTENTS  Nomenclature ........................................................................................................................ 7  1  Introduction.................................................................................................................... 9  1.1  1.2  1.3  1.4  1.5  1.6  1.7 

2  2.1  3  4  5  5.1 

5.2  5.3  5.4  5.5  5.6  5.7  6  6.1  6.2 

6.3 

6.4  6.5  7  7.1  7.2  7.3  7.4  7.5  7.6 

8 

Added resistance in waves .............................................................................................................................. 9  The nature of added resistance ...................................................................................................................... 9  1.2.1  Motions are a first order problem......................................................................................................10  1.2.2  Added resistance is a second order problem ...................................................................................11  Added resistance in irregular waves ............................................................................................................11  Non dimensional added resistance .............................................................................................................12  Non dimensional wave frequency ...............................................................................................................12  Methods to calculate added resistance in waves .......................................................................................13  Relative velocity .............................................................................................................................................13  1.7.1  Coordinate systems ..............................................................................................................................14  1.7.2  Derivation of relative velocity ............................................................................................................15  Gerritsma & Beukelman’s method ........................................................................... 17  Physical interpretation...................................................................................................................................18  Boese’s method ............................................................................................................ 19  Faltinsen’s Asymptotic method ................................................................................. 21  Validation and Evaluations of the implemented methods .................................... 23  Gerritsma and Beukelman implementation...............................................................................................23  5.1.1  Compared to Salvesen .........................................................................................................................23  5.1.2  Head sea compared to SEAWAY......................................................................................................23  5.1.3  Head to Beam waves compared to SEAWAY ................................................................................25  Boese Implementation ..................................................................................................................................26  Following waves.............................................................................................................................................27  Following to beam waves .............................................................................................................................28  All implemented methods in all directions ................................................................................................29  Evaluation of Irregular sea calculations .....................................................................................................30  Evaluation summary......................................................................................................................................30  Simplified method to calculate added resistance ................................................... 31  Problem formulation .....................................................................................................................................31  Limitations ......................................................................................................................................................31  6.2.1  Only head waves ...................................................................................................................................31  6.2.2  Only radiation .......................................................................................................................................31  6.2.3  Even keel ...............................................................................................................................................32  Parameterization ............................................................................................................................................32  6.3.1  Non dimensional RAW , p , ω p and c ..................................................................................................33  6.3.2  Parameters of the model .....................................................................................................................34  Linear regression ............................................................................................................................................35  Parameter functions ......................................................................................................................................36  Application of the simplified model ......................................................................... 36  Linear regression of groups of ships ..........................................................................................................37  Evaluation of the regressions ......................................................................................................................37  Influence of the parameters .........................................................................................................................40  Linearized non linear regression .................................................................................................................40  Discussion .......................................................................................................................................................42  Further work...................................................................................................................................................43  7.6.1  Even keel assumption ..........................................................................................................................43  7.6.2  High frequencies ...................................................................................................................................44  7.6.3  Larger sample set ..................................................................................................................................44  7.6.4  Parameterization ...................................................................................................................................44  7.6.5  More advanced regression...................................................................................................................44  Wave direction ............................................................................................................. 44  5

8.1 

Approximation of RAW , p ( β , RAW , p , Head ) ......................................................................................................44 

8.2 

Approximation ω p , norm ( β , ω p , norm, Head ) ........................................................................................................45 

8.3 

Approximation of cnorm ( β , cnorm, Head ) ..........................................................................................................46 

8.4  9  9.1  10 

Conslusions.....................................................................................................................................................46  Summary and Discussion ........................................................................................... 47  Suggested reading ..........................................................................................................................................47  References ..................................................................................................................... 48  Appendix .............................................................................................................................. 49  1  Result of linear regression .......................................................................................... 49  2  Distortion of CP and LCG ............................................................................................ 49  CP distortion ..................................................................................................................................................50  2.1  2.2  LCG distortion ................................................................................................................................................50 

3  Parameter variations .................................................................................................... 51  3.1  3.2  3.3  3.4  3.5  3.6 

CP variation ....................................................................................................................................................51  LCG variation ..................................................................................................................................................52 

b variation ......................................................................................................................................................53  T variation.......................................................................................................................................................54 

ryy variation ....................................................................................................................................................55  Fn variation ....................................................................................................................................................56 

4  Check of non dimensional parameters ..................................................................... 56  5  The seven ships with their original parameters....................................................... 57  6  Validation of ship motions ......................................................................................... 58  6.1  6.2  7  7.1  7.2  7.3 

S.A. van der Stel in Head seas .....................................................................................................................58  S.A. van der Stel in Following waves..........................................................................................................60  Ship geometries ............................................................................................................ 63  S.A. Van der Steel ..........................................................................................................................................63  Series 60 hull model 4210 .............................................................................................................................64  S-175 ................................................................................................................................................................65 

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NOMENCLATURE Symbol

Quantity

Description

Wave model Φ

E( X ,Y , Z ) O( xo , yo , zo ) B ( xb , yb , zb ) xb

[ m]

η1 η2 η3 η4 η5 η6

[ m] [ m] [ m]

[rad ] [rad ] [rad ] −1 [m / s 2 ]

i g

ρ

[kg / m3 ] [rad / s ] [rad / m] [ m] [m / s ] [ m]

ω k

λ cw

λβ ωe ζa β

[rad / s ] [ m] [rad ]

Linear wave velocity potential. The earth fixed coordinate system. Mean position of ship coordinate system. Ship fixed coordinate system. x coordinate on the ship Surge Sway Heave Roll, port to starboard ( yo to zo ) Pitch, aft to forward ( zo to xo ) Yaw, xo to yo Imaginary unit. Acceleration of gravity. Density of water. Angular wave frequency, in E ( X , Y , Z ) . Wave number. Wave length. Wave celerity. Wave length in diagonal waves. Frequency of encounter, in O( xo , yo , zo ) . Wave amplitude. Heading angle, β = 0 following waves π β = waves from starboard side, β = π head sea. 2

V

[m / s ] [ m] [ m] [ m]

Lwl b T

Ship speed. Ship length in waterline. Ship beam. Draught.

Added resistance in waves RAW

[N ]

R AW

[N ]

Raw =

R AW

[ −]

ζ A ⋅ ρ ⋅ g ( b 2 / Lwl ) 2

R (ωe )

[ N / m2 ]

Sζ (ω )

[m ⋅ s / rad ] [ Pa ⋅ s ] [ Pa ] [ Pa ] [ m]

a33 b33 b' D Vzb

2

[m / s ]

Average added resistance in irregular waves. Added resistance in a regular wave. Non dimensional added resistance in a regular wave. Mean response curve. Wave energy spectrum. Added mass per meter. Damping per meter. Sectional damping. Mean depth. Vertical relative velocity, between strip and water.

Regression analysis RAW , p

[ −]

Peak of Raw . 7

ωp c ω p , norm

[rad / s ] [rad / s ] [ −]

cnorm

[ −]

RAW ,exp

[ −]

CP

[ −]

ryy

[ m]

LCG Fn X

[ m] [ −]

β ε Gn yest , Gn ( Gn ) yest , Gn ( Gall ships )

ε G ( Gn ) n

σ G ( Gn ) n

Peak frequency of Raw . Width of Raw peak. Non dimensional ω p . Non dimensional c . Raw approximated with exponential function. Prismatic coefficient. Radius of pitch gyration. Distance from aft peak to centre of gravity. Froude number Model parameter matrix Coefficient vector in linear regression. Error vector in linear regression. Group n where n = ( Containers , Reefers , RoRos , Tankers and All ships ). Estimation based on linear regression of the data in Gn evaluated for the same group Gn . Estimation based on linear regression of the data in Gn evaluated for all the ships Gall ships . Error vector for linear regression of the data in Gn evaluated for the same group Gn . Standard deviation of ε G ( Gn ) . n

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1 INTRODUCTION This Master Thesis handles three issues: Added resistance as phenomenon, implementation of three methods and development of a simplified method to calculate added resistance in waves. This thesis thereby contain: a summary of the literature study of this project, a thorough description of three conceptually important methods and a methodology to simplify a method, using regression analysis. 1.1 ADDED RESISTANCE IN WAVES Added resistance in waves is the part of a ship’s total resistance that is caused by encountering waves. Calculations of added resistance can be used as an addition to the calm water resistance to predict the total resistance of a ship in a seaway. There will always be waves on the sea, so there will always be added resistance. A ship can experience a 15-30% resistance increase in a seaway [12], where the added resistance is the main reason for this increase. Being able to predict added resistance due to waves is therefore a vital part of the prediction of a ships resistance. Prediction of added resistance can for instance be used in the following problems: •

Weather margin The so called Weather Margin for new ship designs can be decided, where the maximum resistance increase due to weather can be predicted, to decide engine install and so on.

•

Weather Routing Weather Routing is very important due to its economical effect on ship exploitation. It is for instance very important to make good estimations of the time it will take for a ship to travel a route, so the cargo owners know when the ship will arrive in port, minimizing the costs of storage and so on. It is also very important to be able to optimize routes in order to reduce the fuel consumption and emission. A good prediction of Added resistance in waves is important for both these tasks.

•

Performance analysis The previous two problems use the prediction of added resistance to get the total resistance, the reversed problem is however also of interest. Being able to get rid of the influence of the stochastic waves in a seaway, can be used to calculate a ship’s “real” calm water resistance. This “real” calm water resistance can be used as a measurement of the ships performance over time. The ship owners could use this information to determine the value of a ship, and how often it should be docked for antifouling and so on.

1.2 THE NATURE OF ADDED RESISTANCE When a ship is oscillating due to waves, it supplies energy to the surrounding water, energy that will increase the resistance. This energy is primarily transmitted with the waves radiating from the ship (Figure 1). The supplied energy is due to damping of the oscillatory motions. Hydrodynamic damping is dominating for heave- and pitch motions, which are the biggest contributors to added resistance. The viscous damping can therefore be neglected, which means that added resistance can be considered as a non viscous phenomenon [13]. This means that potential theory can be used. The radiation induced resistance is dominating when the ship motions are big. This happens in the region of the resonance frequency of heave and pitch motions (figure 2). The reflection of incident waves is also causing added resistance. The so called diffraction induced resistance is dominating for high wave frequencies (Figure 2), where the ship motions are small.

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Figure 1, Radiating waves due to oscillation.

Energy is also transmitted to the surrounding water by waves generated by the forward speed of the ship. But this is referred to as the calm water resistance, which is not handled in this thesis. The added resistance in a seaway is considered to be independent of the calm water resistance [13].

16

14

12

10

8

6

Radiation induced resistance Diffraction induced resistance

4

2

0 0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

L

λ Figure 2, Radiation induced resistance and diffraction induced resistance, for different wave frequencies.

1.2.1

Motions are a first order problem

Usually ship motions and forces are modeled as a so called LTI system (Linear Time Invariant system). This means that a ship is considered as a system which uses a linear sine-wave, representing the water wave, as input signal and delivers a linear sine-wave, representing for instance a motion or a force, as response to this signal. The LTI system is allowed to respond with a phase lag on the input signal and a linear change of the amplitude. These restrictions give a very advantageous property of the LTI system in that the superposition principle can be used. This means that if a signal x ( t ) can be expressed as the sum of sub signals xk ( t ) , the response to this signal y ( t ) can be expressed as the sum of the responses of the sub signals yk ( t ) :

x ( t ) = ∑ xk ( t ) k

→ y ( t ) = ∑ yk ( t ) k

10

(1.1)

This means that ship motions and forces in irregular waves can be expressed as the sum of the responses in regular waves, which is a very powerful property of a LTI system. In reality ships do not respond linearly to the waves. In order to model the responses as a LTI system, the responses have to be linearized. This linearization gives good accuracy according to [5], since the linear part is dominating the responses. Ship motions are therefore considered to be a first order problem. 1.2.2

Added resistance is a second order problem

The added resistance is the mean force in the heading direction of the ship. Calculating the mean force using a linear force from 1.2.1 will give a zero mean value. This is because the time mean value of an arbitrary sine wave with an arbitrary amplitude A and period time Te is zero: T

1 e ⋅ A ⋅ cos (ω ⋅ t + ε ) ⋅ ∂t = 0 Te ∫0

(1.2)

A second order sine wave however, will give a non zero time mean value: T

2 1 e A2 ⋅ ∫ ( A ⋅ cos (ω ⋅ t + ε ) ) ⋅ ∂t = Te 0 2

(1.3)

Therefore the quadratic term in the response has to be included in the problem. The quadratic term is small compared to the linear term but has to be included to obtain a mean value. Ström-Tejsen [13] has shown in experiments that the added resistance in regular waves varies linearly with the wave height squared at a constant wave length, added resistance is therefore considered to be a second order problem. It is unfortunately hard to get good predictions of added resistance, since it is a second order problem. If the motions are predicted with an accuracy of approximately 10-15%, the second order added resistance can not be expected to be of accuracy better than 20-30% [2]. The wave is usually expressed with a velocity potential function. The velocity potential function is derived from boundary conditions that can be linearized. This is referred to as linear wave theory, which will give a linear wave velocity potential. The linear theory is applicable until the wave steepness becomes sufficiently large, that non-linear effects become important. Although added resistance is a second order problem, the linear wave velocity potential is the only one needed. Higher order velocity potentials are not needed, to study the added resistance [5]. 1.3 ADDED RESISTANCE IN IRREGULAR WAVES Added resistance is the time mean value of a second order force. Consider a signal x ( t ) consisting of two signals x1 ( t ) and x2 ( t ) :

x1 ( t ) = A1 ⋅ cos (ω1 ⋅ t + ε1 ) x2 ( t ) = A2 ⋅ cos (ω2 ⋅ t + ε 2 )

(1.4)

x ( t ) = x1 ( t ) + x2 ( t ) The quadratic response to this signal:

A12 A2 2 A12 A2 + + ⋅ cos ( 2ω1 ⋅ t + 2ε1 ) + 2 ⋅ cos ( 2ω2 ⋅ t + 2ε 2 ) 2 2 2 2 + A1 ⋅ A2 ⋅ cos ( (ω1 − ω2 ) t + ε1 − ε 2 ) + A1 ⋅ A2 ⋅ cos ( (ω1 + ω2 ) t + ε1 + ε 2 ) x (t ) = 2

11

(1.5)

The second order force in an irregular wave can therefore not be expressed with superposition, because of the trigonometric cross terms A1 ⋅ A2 ⋅ cos(...) . But added resistance is the time mean value of this second order force, where the trigonometric terms from (1.5) disappears, so that the time mean value of (1.5) can be expressed as:

x (t )

2

A12 A2 2 = + 2 2

(1.6)

The added resistance in irregular waves can therefore be expressed with superposition of the regular wave responses. Ström-Tejsen [13] has shown this relation in experiments and that the average added resistance RAW in irregular waves with good accuracy can be expressed as: ∞

RAW = 2 ∫ R(ω ) ⋅ Sζ (ω ) ⋅ ∂ω 0

R (ω ) =

R AW (ω )

ζ a2

(1.7)

1 Sζ (ω ) = ⋅ ζ a (ω ) 2 2 R (ω ) is the mean response curve, and Sζ (ω ) is the wave energy spectrum. The evaluation of (1.7), made

by Ström-Tejsen, was done by inserting R(ω ) and Sζ (ω ) from regular wave experiments into (1.7), and compare that to the corresponding irregular wave experiment. The usual way to calculate added resistance in irregular waves, is therefore to first calculate the added resistance in regular waves for different wave frequencies and then use (1.7). This is why almost all available methods to calculate added resistance in waves, focus on regular waves. The added resistance for different wave frequencies can be presented in a transfer function like the schematic one in figure 2. It is also important to be aware that the choice of wave energy spectrum Sζ (ω ) , will have a big influence on the integrated mean added resistance RAW . The relation between the spectral peaks in the wave energy spectrum Sζ (ω ) and the mean response curve R (ω ) will have a big impact on the result. So it is reasonable to conclude that to find an accurate wave energy spectrum Sζ (ω ) , is as important as to find an accurate prediction of the added resistance in regular waves R(ω ) . 1.4 NON DIMENSIONAL ADDED RESISTANCE The full scale added resistance R AW in regular waves can be made non dimensional using the following expression:

Raw =

ζA

2

R AW ⋅ ρ ⋅ g ( b 2 / Lwl )

(1.8)

This relation has been confirmed by [13] in model tests, using models of the same ship with varying scale. In the rest of this thesis mostly the non dimensional added resistance Raw is considered, unless anything else is stated. 1.5 NON DIMENSIONAL WAVE FREQUENCY The peak of the added resistance transfer function (figure 2) usually occurs at a frequency where the wavelength is about the same size as the ships length. This is due to the big influence of pitch motion, which has its peak here, according to figure 3.

12

λ

Figure 3, wavelengths near the ship length will produce heavy pitching, and added resistance.

This means that the length of the ship will have a big influence on where the peak of the added resistance will be. To capture this relation it is usual to present the transfer functions with a non dimensional frequency, normalized with the ships length in some way. This can be done in a variety of ways, and different authors tend to invent their own way of normalizing the frequency. In the validation part of this thesis, where implementations made in this project are compared to previously published experiments and implementations, the frequency has been made non dimensional in the same way as the compared implementations. The non dimensional frequencies used are:

L

ωnorm =

(1.9)

λ

Which can be related to the wave frequency on deep water as:

ωnorm =

L

λ

=ω⋅

L 2 ⋅π ⋅ g

(1.10)

A non dimensional frequency of encounter is also used:

ωnorm = ωe ⋅

L 2 ⋅π ⋅ g

(1.11)

1.6 METHODS TO CALCULATE ADDED RESISTANCE IN WAVES Three methods to calculate added resistance in waves have been implemented in this project, Gerritsma and Beukelman’s method, Boese’s method and Faltinsen’s asymptotic method. Gerritsma and Beukelman’s method is a so called radiated energy method. This problem starts out by trying to describe the energy that the oscillating ship transmits to the surrounding water. It is assumed that to maintain a constant forward ship speed, this energy should be delivered by the ship’s propulsion plant. Boese’s method is a so called pressure integration method, which basically means that the linear pressure in the undisturbed wave is integrated over the ship hull, to obtain a mean force in the heading direction of the ship. It may seem strange that the linear pressure would give a mean force, but it does in this case since the ship hull, where the integration is performed, is moving. Both these methods primarily deal with radiation induced resistance. Faltinsen’s asymptotic method on the other hand, only deals with diffraction induced resistance, and neglects the ship motions. 1.7 RELATIVE VELOCITY Both Gerritsma and Beukelman’s method and Boeses method to calculate added resistance use Relative velocity. The relative velocity is the vertical velocity of the water related to a point on the ship (Figure 4).

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A point on the ship hull Vertical relative velocity

The velocity of a point on the ship hull

Relative velocity

Velocity of the water in the wave

Figure 4, definition of vertical relative velocity

1.7.1

Vzb .

Coordinate systems

Figure 5, the coordinate systems.

In the derivation of relative velocity three coordinate systems are used, E ( X , Y , Z ) , O( xo , yo , zo ) and B ( xb , yb , zb ) (Figure 5). The earth fixed coordinate system E ( X , Y , Z ) is fixed in space and has its X-axis aligned with the direction of the wave propagation, it also contains the other two coordinate systems. O( xo , yo , zo ) is positioned at the mean position of the ships centre of gravity. This means that it is traveling with the ship speed along a line with origin in E ( X , Y , Z ) and with an angle β to the X-axis. The coordinate system O( xo , yo , zo ) only moves in the X − Y − plane in E ( X , Y , Z ) with no rotations. B ( xb , yb , zb ) is fixed with the ship. The origin of B ( xb , yb , zb ) situated at CG is expressed by the coordinates η1 (surge), η2 (sway) and η3 (heave) (Figure 6). And if B( xb , yb , zb ) is rotating in O( xo , yo , zo ) around its origin, these rotations are expressed by η4 (roll, yo to zo ), η5 (pitch, zo to xo ) and η6 (yaw, xo to yo ). 14

zb η3

η5 xb

zo

B(xb,yb,zb) O(xo,yo,zo)

Figure 6, relation between

1.7.2

η1 xo O( xo , yo , zo )

and

B ( xb , yb , zb ) .

Derivation of relative velocity

The relative velocity is the velocity of the water particles in the ship fixed coordinate system B( xb , yb , zb ) . The velocities of the water particles in the earth fixed coordinate system E ( X , Y , Z ) can be obtained from the wave velocity potential [5]:

ΦE ( X ,Y , Z , t) =

i ⋅ g ⋅ζ a

ω

⋅ e k ⋅Z ⋅ ei⋅(ω ⋅t − k ⋅ X )

(1.12)

Particle velocities in E ( X , Y , Z ) :

∂Φ E = ω ⋅ ζ a ⋅ e k ⋅Z ⋅ ei⋅(ω ⋅t − k ⋅ X ) ∂X

(1.13)

∂Φ E = i ⋅ ω ⋅ ζ a ⋅ e k ⋅Z ⋅ ei⋅(ω ⋅t − k ⋅ X ) ∂Z

(1.14)

uE =

wE =

uE in X -direction, wE in Z -direction. These velocities can be transformed to the ship fixed system B ( xb , yb , zb ) to obtain the relative velocity. First (1.13) and (1.14) are expressed with xb using the following relations between X in E ( X , Y , Z ) and xb in B( xb , yb , zb ) :

X = (V ⋅ t + xO ) ⋅ cos ( β ) xO = η1 + xb ≈ xb

(1.15)

→ X = (V ⋅ t + xb ) ⋅ cos ( β ) Inserted in (1.13) and (1.14):

u E = ω ⋅ ζ a ⋅ e k ⋅Z ⋅ e

i ⋅(ω ⋅t − k ⋅⎡⎣(V ⋅t + xb )⋅cos ( β )⎤⎦ )

wE = i ⋅ ω ⋅ ζ a ⋅ e

k ⋅Z

= ω ⋅ ζ a ⋅ e k ⋅Z ⋅ e

⋅e

⎛⎛ ω2 ⎞ ⎞ i ⋅⎜ ⎜ ω − V ⋅cos( β ) ⎟⋅t − k ⋅ xb ⋅cos ( β ) ⎟ ⎟ ⎜⎜ ⎟ g ⎠ ⎝⎝ ⎠

⎛⎛ ω2 ⎞ ⎞ i ⋅⎜ ⎜ ω − V ⋅cos( β ) ⎟⋅t − k ⋅ xb ⋅cos( β ) ⎟ ⎟ ⎜⎜ ⎟ g ⎠ ⎝⎝ ⎠

15

(1.16)

(1.17)

And introducing the frequency of encounter:

⎛

ωe = ⎜ ω − ⎝

ω2

⎞ V ⋅ cos ( β ) ⎟ g ⎠

→ u E = ω ⋅ ζ a ⋅ e k ⋅Z ⋅ e

(1.18)

i ⋅(ωe ⋅t − k ⋅ xb ⋅cos( β ) )

i ⋅ ω ⋅t − k ⋅ xb ⋅cos ( β ) ) wE = i ⋅ ω ⋅ ζ a ⋅ e k ⋅Z ⋅ e ( e

(1.19)

This is still velocities in E ( X , Y , Z ) , but expressed with xb from B( xb , yb , zb ) . The velocities can be transformed to the moving system O( xo , yo , zo ) :

wO = wE

(1.20)

uO = −V + u E ⋅ cos ( β )

(1.21)

wb = ( ( uO − η1 ) ⋅η5 + wO − η3 + xb ⋅η5 )

(1.22)

And transformed to B( xb , yb , zb ) :

The surge velocity η1 can be neglected:

wb = ( uO ⋅η5 + wO − η3 + xb ⋅η5 )

(1.23)

Inserting the expressions for uO (1.21) and wO (1.20):

wb = ( −V + u E ⋅ cos ( β ) ) ⋅η5 + wE − η3 + xb ⋅η5

(1.24)

And finally inserting the expressions for uE (1.18) and wE (1.19), obtaining the relative velocity: i ⋅ ω ⋅t − k ⋅ xb ⋅cos ( β ) ) ⎤ wb = −V ⋅η5 − η3 + xb ⋅η5 + ( cos ( β ) ⋅η5 + i ) ⎡ω ⋅ ζ a ⋅ e k ⋅Z ⋅ e ( e ⎣ ⎦

(1.25)

cos ( β ) ⋅η5 is small and is neglected in [1]: i ⋅ ω ⋅t − k ⋅ xb ⋅cos ( β ) ) ⎤ Vzb ≈ wb = −V ⋅η5 − η3 + xb ⋅η5 + i ⋅ ⎡ω ⋅ ζ a ⋅ e k ⋅Z ⋅ e ( e ⎣ ⎦

(1.26)

Calculations have been carried out in this project using both (1.25) and (1.26) to verify that cos ( β ) ⋅η5 can be neglected from (1.25).

16

2 GERRITSMA & BEUKELMAN’S METHOD Gerritsma and Beukelman’s method [1] for calculation of added resistance is a so called radiated energy method. The added resistance is calculated with the following expression: 2 −k ⋅ cos( β ) ' ⋅ ⋅ ∂xb Raw = b V z b 2 ⋅ ωe ∫0 L

(2.1)

This method is very much related to the Strip theory, where (2.1) is an integration along the ships length, over the strips. b ' is the sectional damping coefficient for speed, for the different strips [1]:

b ' = b33 − V ⋅

∂a33 ∂xb

(2.2)

Vzb is the amplitude of the relative velocity, which is the water velocity related to the strip: i ⋅ ω ⋅t − k ⋅ xb ⋅cos( β ) ) Vzb = −V ⋅η5 − η3 + xb ⋅η5 + i ⋅ ω ⋅ ζ a ⋅ ek ⋅Z ⋅ e ( e

(2.3)

This is an equation for various strips (different xb ), but it is also an equation for various values of Z , representing the depth where the water velocity is evaluated. In [1] the water velocity is evaluated at a mean depth D for every strip:

D=

A' B'

(2.4)

Where A ' is the area of the “wet” part of the strip, and B ' is the beam of the strip in the waterline. The relative velocity can now be written: i ⋅ ω ⋅t − k ⋅ xb ⋅cos( β ) ) Vzb = −V ⋅η5 − η3 + xb ⋅η5 + i ⋅ ω ⋅ ζ a ⋅ e − k ⋅D ⋅ e ( e

(2.5)

The damping coefficient (2.2) and the relative velocity (2.5) only contain heave ( η3 ) and pitch motion ( η5 ), so Gerritsma and Beukelman’s method does not account for roll ( η4 ) or yaw motion ( η6 ). η3 , η5 can be expressed in a complex way:

η3 = ηˆ3 ⋅ ei⋅ω ⋅t

(2.6)

η5 = ηˆ5 ⋅ ei⋅ω ⋅t

(2.7)

e

e

ηˆ3 and ηˆ5 are complex amplitudes, which means that they contain both amplitude η3 , η5 and phase φ3 , φ5 :

ηˆ3 = η3 ⋅ ei⋅φ

3

ηˆ5 = η5 ⋅ e This gives the final expression for the relative velocity:

17

i⋅φ5

(2.8) (2.9)

Vzb = ⎡⎣ −V ⋅η5 + i ⋅ ωe ( xb ⋅η5 − η3 ) + i ⋅ ω ⋅ ζ a ⋅ e− k ⋅D ⋅ e− i⋅k ⋅ xb ⋅cos( β ) ⎤⎦ ⋅ ei⋅ωe ⋅t

(2.10)

…and the amplitude:

Vzb = −V ⋅η5 + i ⋅ ωe ( xb ⋅η5 − η3 ) + i ⋅ ω ⋅ ζ a ⋅ e− k ⋅D ⋅ e− i⋅k ⋅ xb ⋅cos( β )

(2.11)

Note: This expression contains ωe as well as ω . 2.1 PHYSICAL INTERPRETATION In [6] a derivation of Gerritsma and Beukelman’s method (2.1) is made. The basic idea with the method is to calculate the radiated wave energy during one period of oscillation, in regular waves. This would in other words be the energy required to create waves, when the ship is oscillating. And it is assumed that to maintain a constant forward ship speed, this energy should be delivered by the ship’s propulsion plant. According to [1] the radiated energy can be calculated with this equation: Te L

E = ∫ ∫ b '⋅ Vz b 2 ⋅ ∂xb ⋅ ∂t

(2.12)

0 0

Studying the expression for Vz in (2.10) enables the possibility to express Vz : b

b

Vzb = Vzb ⋅ cos(ωe ⋅ t + ε Vz )

(2.13)

b

εV is the phase lag of the relative velocity. The time integration in (2.12) can be performed: zb

Te L

E = ∫ ∫ b '⋅ Vz b 2 ⋅ ∂xb ⋅ ∂t = 0 0

2 2 Te π ⋅ ∫ b '⋅ Vz b ⋅ ∂xb = ⋅ ∫ b '⋅ Vz b ⋅ ∂xb ωe 0 2 0 L

L

(2.14)

The radiated energy during one period of oscillation can also be expressed in terms of added resistance Raw [6]:

⎛ ⎞ ⎛ ⎞ λ 2π E = Raw ⋅ λβ = Raw ⋅ ⎜ ⎟ = Raw ⋅ ⎜ ⎟ ⎝ − cos( β ) ⎠ ⎝ − k ⋅ cos( β ) ⎠

(2.15)

λβ (Figure 7) is the wave length that the ship experiences when it is heading diagonally through the waves. (2.15) together with (2.14) gives Gerritsma and Beukelman’s equation for added resistance (2.1).

18

β λ

λβ

Figure 7,

λβ

is the wave length experienced by the ship.

3 BOESE’S METHOD Boese’s method is a so called pressure integration method. This text is an interpretation of the derivation of Boese’s method made in [6]. The added resistance is calculated as the time average of the integrated pressure. The pressure in the undisturbed wave can be expressed with Bernoullis equation:

p+ρ⋅g⋅z+ρ

∂Φ ρ + ⋅∇Φ ⋅∇Φ = C ∂t 2

(3.1)

In Boese’s metod the linear part of the pressure is used:

p = −ρ ⋅ g ⋅ z − ρ

∂Φ ∂t

(3.2)

The pressure is integrated over the strip, from the bottom of the strip to the wave surface (Figure 8): ξ

f ( xb , t ) =

∫ p ⋅ ∂z

b

(3.3)

Zk

So this is a force per unit length. ξ is the wave surface and Z k is deepest point of the strip (Figure 8):

Z k = Ds − η3 + xb ⋅η5

19

(3.4)

Z

ξ

Zk

X Figure 8, pressure integration on the strip.

A Mean force per unit length is calculated: Te

f ( xb ) = ∫ f ( xb , t ) ⋅ ∂T

(3.5)

0

The horizontal part of this force per unit length can be obtained:

⎛ ∂y ⎞ f ( xb )η1 = f ( xb ) ⋅ ⎜ w ⎟ ⎝ ∂xb ⎠

(3.6)

This “mean force per unit length” is integrated over the ship length to obtain the added resistance: L ⎛ ∂y ⎞ Raw1 = 2 ⋅ ∫ f ( xb ) ⋅ ⎜ w ⎟ ⋅ ∂xb ⎝ ∂xb ⎠ 0

(3.7)

Which can be expressed, according to [6], as:

Raw1 =

ρ⋅g 2

∫V

zb

L

2

⎛ ∂y ⎞ ⋅ ⎜ w ⎟ ⋅ ∂xb ⎝ ∂xb ⎠

(3.8)

A second contribution is calculated as:

Raw 2

1 = Te

Te

∫ ρ ⋅∇ ⋅η ⋅η 3

0

20

5

⋅ ∂t

(3.9)

This is basically the vertical force on the ship ( ρ ⋅∇ ⋅η3 ), projected on the surge direction η1 with small angle approximation, sin (η5 ) ≈ η5 . And the total added resistance calculated with Boeses method is expressed as:

Raw = Raw1 + Raw 2

(3.10)

4 FALTINSEN’S ASYMPTOTIC METHOD Faltinsen’s asymptotic method [5] is a method to calculate diffraction induced added resistance (figure 2). Diffraction induced resistance is dominating the total added resistance for high wave frequencies, since the radiation induced resistance is small for high frequencies. The idea with Faltinsen’s asymptotic method is to calculate the force that a wave exerts on the ship hull, when it hits the hull and bounces off (diffraction). It is assumed that the hull sides are vertical and that the ship does not start to oscillate because of the contact with the wave. It is also assumed that the wave only affects the ship on the “upstream” side. This is more or less the same thing as if a wave would hit a vertical wall. The mean force for a wave hitting a vertical wall can be expressed as [5]:

F=

ρ⋅g 2

⋅ ζ a 2 ⋅ Lwall

(4.1)

Where Lwall is the length of the wall. This force can also be integrated over a curved wall L1 (representing a hull side):

F=

ρ⋅g 2

⋅ ζ a 2 ⋅ ∫ sin (θ + β ) ⋅ n ⋅ ∂l

(4.2)

L1

Where n is the normal vector of the hull and L1 is the non shadow part of the hull side shown in figure 9.

Z

10

L1

5

0 −50 0 50 100 150

−25

−15

−20

X

−10 Y

Figure 9,

21

L1

−5

0

5

10

F is called drift force, working in both direction xb and yb . The force in direction xb is the added resistance for a ship with no speed. F in (4.2) can be multiplied with a speed factor to be valid for speed as well. So that the added resistance with Faltinsen’s asymptotic method can be expressed as [5]:

Raw =

ρ⋅g

⎛ 2 ⋅ ω ⋅U ⎞ ⋅ ζ a 2 ⎜1 + ⎟ ⋅ sin (θ + β ) ⋅ nx ⋅ ∂l 2 g ⎠ L∫1 ⎝

(4.3)

Added resistance calculated for the hull design Model4210 (Appendix 7.2, page 64), using (4.3) is presented in figure 10.

β=0 Fn=0.150

β=16 Fn=0.150

−2 0

0.5

1

−2

−3

1.5

−1.5 RAW ρ · g · ζ 2 (b2 /L)

−1.5

−2.5

0

β=49 Fn=0.150

0

0.5

1

1.5

−2 −2.5

0

0.5

1

1.5

−0.6

1

1.5 0

β=147 Fn=0.150

0.5

1

1

1.5

0

0.5

1

1.5

1

0.5

1.5

0

0.5

1

1.5

0.5 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

1.5

1

β=180 Fn=0.150

1.5

2

0.5

2

β=164 Fn=0.150

2.5

1.5

3

2

1

1.5

1

β=131 Fn=0.150 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

0.5

0

β=115 Fn=0.150

1

0.5

−0.4

2.5

0

0

β=82 Fn=0.150

−1.5

1.5 RAW ρ · g · ζ 2 (b2 /L)

−3

−0.2

β=98 Fn=0.150

RAW ρ · g · ζ 2 (b2 /L)

−2.5

1.5

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

−2.5

1

1

−1

−2

0.5

0.5

−2

β=65 Fn=0.150

−1.5

−3

β=33 Fn=0.150

−1 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

−1

0

0.5

1

1.5

0.4 0.3 0.2

0

0.5




1

1.5

L λ

Figure 10, Added resistance calculated with Faltinsen’s Aymptotic for the hull design Model4210 (Appendix, page 64).

Figure 10 shows that the added resistance calculated with Faltinsen’s Asymptotic method is positive in head waves and negative in following waves. The heading angle β = 131° , where a big part of the ship hull is exposed to the waves, results in the highest added resistance.

22

5 VALIDATION AND EVALUATIONS OF THE IMPLEMENTED METHODS A lot of effort in this master thesis project has been made to validate the results of the implementation of the different methods. The implementations have been compared to available experimental results and results from earlier published implementations of the methods. These comparisons are therefore both validations of the implementations done in this project, and evaluations of how the methods themselves correspond to experimental data. 5.1 5.1.1

GERRITSMA AND BEUKELMAN IMPLEMENTATION Compared to Salvesen

The implementation of Gerritsma and Beukelman’s method is validated in figure 11 against an implementation of the method in head seas made by Salvesen [2]. The added resistance is calculated for a Series 60 hull (Appendix 7.2, page 64). The main particulars used in [2] where not known, instead main particulars where taken from [14], which are supposed to be the right ones. RAW 25 GoB Impl. GoB (Salvesen) Experiment

20

RAW ρ · g · ζ 2 (b2 /L)

15

10

5

0

0

1

2

3 ωe

Figure 11,

Raw

4



5

6

7

L g

of Series 60 hull model 4210.

In figure 11 GoB Impl. refers to this implementation of Gerritsma and Beukelman’s method, GoB (Salvesen) refers to Salvesens implementation of Gerritsma and Beukelman’s method [2]. Experiment refers to experimental data from [13]. The implementation of Gerritsma and Beukelmans’s method made in this project, seem to correspond well to the implementation made by Salvesen. The two implementations differ a bit for high wave frequencies, where the implementation from this project seems to correspond best to the experimental data. Generally Gerritsma and Beukelman’s method seems to over predict the peak of the response curve. 5.1.2

Head sea compared to SEAWAY

There is a report about validation of the computer program SEAWAY written by Journée [7] where a lot of interesting data is presented. For instance the calculated ship motions in head seas as well as the calculated added resistance for the ship S.A. van der Stel (Appendix, page 63). This means that all of the input to Gerritsma and Beukelman’s method, except the 2D-coefficients, and all of the output (resulting added resistance) is available. Figure 12 shows Raw calculated with SEAWAY [7] and Raw calculated with

this implementation, using ship motions from EnRoute [17], as well as the ship motions from SEAWAY [7]. The corresponding validation of the ship motions can be found in the section: [Appendix 6.1 S.A. van der Stel in Head seas, page: 58] β=180 Fn=0.150

β=180 Fn=0.200

20

20 GoB Impl.(EnRoute) SEAWAY KTH>0 GoB Impl. (KTH>0) Experiment

15

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

15

10

5

0 0.5

10

5

0.6

0.7

0.8




0.9 L λ

1

1.1

0 0.5

1.2

0.6

0.7

20

15

15

10

5

0 0.5




0.9 L λ

1

1.1

1.2

1

1.1

1.2

β=180 Fn=0.300

20

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

β=180 Fn=0.250

0.8

10

5

0.6

0.7

0.8




0.9 L λ

1

1.1

1.2

0 0.5

0.6

0.7

0.8




0.9 L λ

Figure 12, Added resistance, S.A. van der Stel.

• • • •

GoB Impl. (EnRoute) refers to this implementation of Gerritsma and Beukelman’s method, with ship motions from EnRoute [17]. SEAWAY KTH>0 refers to the implementation of Gerritsma and Beukelman’s method made in SEAWAY with ship motions calculated with a method referred to as “KTH>0” in [7]. GoB Impl. (KTH>0) refers to this implementation with the “KTH>0”-method for ship motions. Experiment refers to experimental data that has been taken from figures in [7], originally obtained at the Delft University of Technology with a 1:50 scale model of S.A. van der Stel.

In figure 12 GoB Impl. (EnRoute) is quite similar to SEAWAY KTH>0, and corresponds just as good to the experimental data. GoB Impl. (EnRoute) has a tendency to have its peak at a higher frequency than SEAWAY KTH>0, which is a bit worrying. To see if this was a result of differences in the implementation or differences in the ship motions, the implementation presented in this report was calculated with the same ship motions used for SEAWAY KTH>0, giving GoB Impl. (KTH>0). This curve has its peak at the same frequency as SEAWAY KTH>0, so it is probably because of differences in the peak of the ship motions [Appendix 6.1 S.A. van der Stel in Head seas, page: 58], especially pitch motion, that results in the different peak values for GoB Impl. (EnRoute) and SEAWAY KTH>0.

24

5.1.3

Head to Beam waves compared to SEAWAY

To evaluate the implementation for different heading angles β , it is compared (Figure 13) to an implementation made by Journée [7] for the ship hull: S-175 Containership Design (Appendix 7.3, page 65). β=180 Fn=0.150

β=150 Fn=0.150

0.6

0.8

1

1.2

1.4

1.6

10

0

1.8

20

RAW ρ · g · ζ 2 (b2 /L)

10

0

0.6

0.8

β=180 Fn=0.200

0.8

1

1.2

1.4

1.6

0

1.8

0.6

0.8

1

1.2

1.4

1.6

1

1.2

1.4

1.6

0.6

0.8

1

1.2

1.4

1.6

1

1.2 L λ

1.4

1.6

1.8

0.8

1

1.2

1.4

1.6

1.8

0.6

0.8

1

1.2

1.4

1.6

1.8

1.6

1.8

20

10

0

1.8

β=120 Fn=0.300

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L) 0.8

1.6

10

0

1.8

20

0.6

0.6

β=150 Fn=0.300

10

1.4

20

β=180 Fn=0.300 20

1.2

β=120 Fn=0.250

10

0

1.8

1

10

0

1.8

RAW ρ · g · ζ 2 (b2 /L)

10

0.8

0.8

β=150 Fn=0.250 20

0.6

0.6

β=120 Fn=0.200

10

0

1.8

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

1.6

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L) 0.6

20

RAW ρ · g · ζ 2 (b2 /L)

1.4

20

β=180 Fn=0.250

0

1.2

20

10

0

1

GoB Impl. SEAWAY KTH>0 10

β=150 Fn=0.200

20

0

β=120 Fn=0.150

20

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

20

0.6

0.8

1

1.2 L λ

1.4

1.6

1.8

10

0

0.6

0.8

1

1.2 L λ

1.4

Figure 13, Added resistance for different headings (S-175 Containership Design).

The two implementations in figure 13 are quite similar, “GoB Impl.” refers to the implementation of Gerritsma and Beukelman’s method made in this project. “SEAWAY KTH>0” refers to the implementation made by Journée [7].

25

5.2 BOESE IMPLEMENTATION The implementation of Boese’s method has been evaluated against calculations with Boese’s method made with SEAWAY for S-175, published by Journée [7]. β=180 Fn=0.150

β=150 Fn=0.150

0.6

0.8

1

1.2

1.4

10

0

1.6

20

RAW ρ · g · ζ 2 (b2 /L)

10

0

0.6

0.8

β=180 Fn=0.200

0.8

1

1.2

1.4

0.6

0.8

1

1.2

1.4

1

1.2

1.4

0.6

0.8

1

1.2

1.4

1

1.2

0.8

1.4

1.6

1.2

1.4

1.6

0.6

0.8

1

1.2

1.4

1.6

1.4

1.6

20

10

0

1

β=120 Fn=0.300

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L) 0.8

1.6

10

0

1.6

20

0.6

0.6

β=150 Fn=0.300

10

1.4

20

β=180 Fn=0.300 20

1.2

β=120 Fn=0.250

10

0

1.6

1

10

0

1.6

RAW ρ · g · ζ 2 (b2 /L)

10

0.8

0.8

β=150 Fn=0.250 20

0.6

0.6

β=120 Fn=0.200

10

0

1.6

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

0

1.6

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L) 0.6

20

RAW ρ · g · ζ 2 (b2 /L)

1.4

20

β=180 Fn=0.250

0

1.2

20

10

0

1

Boese Impl. SEAWAY KTH>0 10

β=150 Fn=0.200

20

0

β=120 Fn=0.150

20

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

20

0.6

0.8

1

L λ

1.2 L λ

1.4

1.6

10

0

0.6

0.8

1

1.2 L λ

Figure 14, Comparison between this Boese implementation and the Boese implementation in SEAWAY [7].

26

5.3 FOLLOWING WAVES Transfer functions for added resistance calculated with Gerritsma and Beukelman’s method, and Boese’s method are presented in figure 15, together with experimental results from [8]. β=0 Fn=0.150

β=0 Fn=0.200

1.5

1.5 GoB Impl. GoB SEAWAY Boese Impl. Experiment

1

1 0.5

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

0.5

0

0 −0.5

−0.5 −1 −1

−1.5

−1.5

0

0.2

0.4

0.6




0.8

1

1.2

−2

1.4

0

0.2

0.4

0.6

0.8
L λ

L λ

β=0 Fn=0.250

1

1.2

1.4

1.6

β=0 Fn=0.300

1.5

1.5

1

1

0.5

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

0.5 0 −0.5

0

−0.5 −1 −1

−1.5 −2

0

0.5




1

−1.5

1.5

L λ

0

0.2

0.4

0.6




0.8

1

1.2

1.4

L λ

Figure 15, Added resistance for S.A. van der Stel calculated with this implementation and SEAWAY [7].

None of the implementations in figure 15 correspond well to the experimental data [8]. These experimental results have unfortunately been the only one available in this project, and are therefore a bit uncertain. All the implementations and the experimental data in figure 15 indicate that the added resistance in following waves is quite small.

27

FOLLOWING TO BEAM WAVES β=23 Fn=0.150

1.2 1.4

0.6 0.8

β=0 Fn=0.200

0.6 0.8

−2 0.6 0.8

1

−2

1.2 1.4

0.6 0.8

1

RAW ρ · g · ζ 2 (b2 /L)

−2 0.6 0.8

1

0.6 0.8

0.6 0.8

β=0 Fn=0.300

1

0.6 0.8

RAW ρ · g · ζ 2 (b2 /L) 0.6 0.8

1 1.2 1.4
L λ

0.6 0.8

1

−2 0.6 0.8

1 1.2 1.4
L λ

1.2 1.4

0 −2

1.2 1.4

0.6 0.8

1

1.2 1.4

β=68 Fn=0.300

2

0

1

β=68 Fn=0.250

β=45 Fn=0.300

2

−2

−2

2

−2

1.2 1.4

1.2 1.4

0

1.2 1.4

0

β=23 Fn=0.300

2

1

2

−2

1

β=68 Fn=0.200

β=45 Fn=0.250

0

1.2 1.4

0

−2

1.2 1.4

2

0

0.6 0.8 2

0

β=23 Fn=0.250

2

−2

1.2 1.4

2

0

β=0 Fn=0.250

1

0

β=45 Fn=0.200

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

1.2 1.4

2

0

−2

β=23 Fn=0.200

2

RAW ρ · g · ζ 2 (b2 /L)

1

RAW ρ · g · ζ 2 (b2 /L)

−2

0

RAW ρ · g · ζ 2 (b2 /L)

1

0

2

RAW ρ · g · ζ 2 (b2 /L)

−2

β=68 Fn=0.150

2

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

0

0.6 0.8

RAW ρ · g · ζ 2 (b2 /L)

β=45 Fn=0.150

2

2

RAW ρ · g · ζ 2 (b2 /L)

β=0 Fn=0.150

2

RAW ρ · g · ζ 2 (b2 /L)

GoB Impl. Boese Impl.

RAW ρ · g · ζ 2 (b2 /L)

5.4

0 −2 0.6 0.8

1 1.2 1.4
L λ

0 −2 0.6 0.8

1 1.2 1.4
L λ

Figure 16, Gerritsma and Beukelman’s method and Boese calculated for various following seas.

In Figure 16 the point of zero frequency of encounter ωe is marked with a circle. Note that Gerritsma and Beukelman’s method will result in a singularity for that point, since ωe is in the denominator of (2.1). Boese’s method does not have the same problem.

28

ALL IMPLEMENTED METHODS IN ALL DIRECTIONS

β=0 Fn=0.150

β=16 Fn=0.150

5 0 −5

10

5 0 −5

0.5

1

1.5

2

2.5

1

β=49 Fn=0.150

2

2.5

0.5

0

2

5 0

2.5

β=98 Fn=0.150

1

1.5

2

0.5

1

1.5

2

5 0

2.5

β=147 Fn=0.150

1

1.5

2

0.5

1

1.5

2

2.5

1.5

2

2.5

β=180 Fn=0.150

5 0 −5

0.5

1

10 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

−5

2.5

0

β=164 Fn=0.150

0

2

5

2.5

10

5

2.5

−5 0.5

10

1.5 β=131 Fn=0.150

−5 0.5

1

10 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

−5

2

0

β=115 Fn=0.150

0

2.5

5

2.5

10

5

2

−5 0.5

10

1.5 β=82 Fn=0.150

−5 1.5

1

10 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

5

1

0

β=65 Fn=0.150

−5

RAW ρ · g · ζ 2 (b2 /L)

1.5

10

0.5

5

−5 0.5

10

RAW ρ · g · ζ 2 (b2 /L)

β=33 Fn=0.150

10 RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

GoB Impl. Boese Impl. 10 Asymptotic Faltinsen

RAW ρ · g · ζ 2 (b2 /L)

5.5

5 0 −5

0.5

1

1.5

2

2.5

0.5

1

1.5
L λ

Figure 17, All the implemented methods in all directions, for SA. Van der Stel.

Figure 17 is a summary of the three implemented methods in all wave angles. Gerritsma and Beukelman’s method and Boese’s method are quite similar for head to beam waves. Faltinsen’s Asymptotic method indicates that the added resistance is negative in following waves. Gerritsma and Beukelman’s method seems to be very sensitive to low encounter frequencies, which gives questionable results in following waves. Journée [6] claims that Boese’s method is better than Gerritsma and Beukelman’s method for following wave, because of this.

29

5.6 EVALUATION OF IRREGULAR SEA CALCULATIONS In [13] added resistance has been calculated for full scale Series 60 hull 4210 (Appendix, page 64) in an irregular sea state. The sea state in [13] is defined by a Pierson-Moskowitz wave energy spectrum in SIunits:

Sζ (ω ) =

A

ω5

⋅e

−

B

ω4

A = 0.0872 ⋅ g 2 B=

(5.1)

3.1178 H1/32

The total added resistance in irregular waves calculated with (1.7) for a wave energy spectrum (5.1) with significant wave height H1/3 = 4.572[m] and analytical response curve calculated for ship speed Fn = 0.266 , is published in [13]. The method to calculate the added resistance in regular waves, used in [13], is unfortunately unknown and is referred to as “Ström-Tejsen Analytical” in table 1. Total added resistance in irregular waves for the situation described above is also calculated with response curves from the implementations made in this project. “Boese Impl.” and “GoB Impl.” in table 1, refer to total added resistance in irregular waves calculated with response curves from Boese’s method and Gerritsma and Beukelman’s method. Table 1, Total analytical added resistance for full scale Model4210 in irregular sea.

   Ström‐Tejsen Analytical.  Boese Impl.  GoB Impl. 

R AW [kN ] 372  295  441 

5.7 EVALUATION SUMMARY The implementations of Gerritsma and Beukelman’s method and Boese’s method in head to beam waves seem to correspond quite well to previous implementations of the methods according to 5.1.1, 5.1.2, 5.1.3 and 5.2. Reasons for the differences could be that different strip calculations have been used, and that there are some uncertainties concerning the hull geometries. The hull geometries used in this comparison, SA. Van der Stel and S175, have been digitalized from published body plans (Figure 18).

Figure 18, digitalizing of a hull.

30

It has been concluded that the implementations agree sufficiently well to be considered correct. The evaluations against experimental data in 5.1.1 and 5.1.2 indicate that Gerritsma and Beukelman’s method correspond quite well to experimental data, except for the peak of added resistance which probably is overestimated quite a bit. Added resistance in following waves is a very difficult problem. None of the implementations in (5.3 Following waves) correspond well to the experimental data [8]. These experimental results have unfortunately been the only one available in this project. Generally, experimental results in following waves seem to be very hard to find. It is however reasonable to conclude that the added resistance in following waves is quite small, even if the result from [8] is correct or not. Weather it is positive or negative, is not entirely clear, it probably depends a lot on the conditions. It is indeed hard to know if the radiated wave energy caused by ship motions in following waves should produce a positive or negative added resistance. It is nevertheless quite clear that the diffraction induced added resistance (Faltinsen’s Asymptotic) will be negative, giving the ship an easier ride in following waves. One reason for bad results in following waves is that the Strip theory that is used, is a high frequency theory. This means that it will have low accuracy for low encounter frequencies. This makes calculations of added resistance in following waves difficult.

6 SIMPLIFIED METHOD TO CALCULATE ADDED RESISTANCE In some applications of added resistance calculations it might be unnecessary complicated to use strip calculated ship motions, where the entire hull geometry has to be defined as input. For instance in the case of a pre study of a new ship design, where only the main particulars might have been decided. A simplified method to calculate added resistance in waves, that does not require strip calculations, have been developed in this project, to be used in the situation describes above. The idea is that the method should be derived with some kind of regression analysis. The ideal way to do this is of course to do a huge series of model experiments, varying different kind of parameters. In this case model experiments is not an option, instead systematic calculations are used. Gerritsma and Beukelman’s method is used for the systematic calculations. 6.1 PROBLEM FORMULATION The problem has been formulated as to develop a method that estimates transfer functions for added resistance in regular waves that fits the corresponding transfer functions, generated with Gerritsma and Beukelman’s method. It has been decided to use linear regression to derive this method. The method could therefore be described as semi empirical based on results from an analytical method. 6.2 LIMITATIONS For various reasons, discussed below, the following limitations have been put on the regression: • • • 6.2.1

Only head waves Only radiation Even keel Only head waves

The heading angle β has deliberately been left out of the linear regression, since its influence is hard to express as an addition to the result of the other parameters, which is a requirement for linear regression. How the heading could be included in the simplified method is discussed in section 8, Wave direction, on page 44. 6.2.2

Only radiation

This simplified method focuses on the frequencies of the transfer functions where the radiation of waves due to ship motions, dominates the added resistance (Radiation induced resistance, figure 2). For higher frequencies, the reflection of incident waves dominates (Diffraction induced resistance, figure 2). Gerritsma and Beukelman’s method has a “tail” for these high frequencies. It is however a bit uncertain, 31

according to experimental results from [13], whether this “tail” models the real situation in a satisfying way. It might be better to use a method specialized in dealing with diffraction, for instance Faltinsen’s asymptotic method [5], for these frequencies. The ship motions that require strip calculations are only needed to decide Radiation induced resistance. The Diffraction induced resistance will not suffer from the lack of strip calculations, which also is a reason that it has been left out of this study. 6.2.3

Even keel

This problem grows dramatically with the number of model parameters. The trim is not included in the regression because of this, which seems to be reasonable for small values of trim. This is discussed in 7.6.1 Even keel assumption, on page 43. 6.3 PARAMETERIZATION In order to estimate transfer functions for added resistance, it is needed to describe existing transfer functions, and to create new ones in some way. This is done in this simplified method by fitting a mathematical function, described by a few parameters, to the transfer function that should be described. Transfer functions created with Gerritsma and Beukelman’s method generally look very much like a normal distribution function, also known as the “bell curve” from statistics. Especially if only the Motion induced resistance is regarded. This simplified method therefore uses the normal distribution function as a curve fit of the transfer function, to express the transfer function in a parameterized way. A transfer function can now be expressed with three parameters: peak value RAW , p , peak frequency ω p , and “width” c , inserted in the normal distribution function:

RAW ,exp = RAWp ⋅ e

−

(ω −ω p ) 2 ⋅c

2

2

(6.1)

Figure 19 shows how well the normal distribution function (6.1) fits a transfer function created with Gerritsma and Beukelman’s method. Raw 25 Raw Raw

exp

RAW,p

RAW ρ · g · ζ 2 (b2 /L)

20

15

10

5

ω1
0

0.35

0.4

0.45

ω2


ωp 0.5

ω

0.55

0.6

0.65

Figure 19, Raw is approximated with an exponential function, similar to normal distribution function.

32

0.7

The curve fit of the expression (6.1) to a transfer function is done in the following way: RAW , p and ω p are achieved directly from the curve. c is calculated using ω '1 , which is defined as the frequencies where the value of (6.1) is half the peak value RAW , p .

c=

(ω

p

− ω1 ')

2 ⋅ ln(2)

(6.2)

Using the parameters RAW , p , ω p and c together with the approximation (6.1) provides fair agreement with the Radiation induced resistance, according to figure 19. Using these approximations also reduces the problem into a prediction of RAW , p , ω p and c , using regression analysis. 6.3.1

Non dimensional RAW , p , ω p and c

The added resistance can be made non dimensional using the expression (1.8):

Raw =

ζA

2

R AW ⋅ ρ ⋅ g ( b 2 / Lwl )

In this project this relation was tested using Gerritsma and Beukelman’s method with varying scale of ship hull Model4210. The study, summarized in (Appendix 4 Check of non dimensional parameters, page 56), proves that the added resistance calculated with Gerritsma and Beukelman’s method can be made perfectly non dimensional. This significantly simplifies the problem. It is the relation between the model parameters that should be studied, not their real magnitude. And therefore the length of the ship in the water line Lwl is used to make a non dimensional ship, (Table 2). The peak frequency ω p is normalized:

ω p ,norm = ω p ⋅

Lwl 2 ⋅π ⋅ g

(6.3)

The ”width” c of the peak is normalized in the same way:

cnorm = c ⋅

33

Lwl 2 ⋅π ⋅ g

(6.4)

6.3.2

Parameters of the model

An assumption about parameters that significantly affect the added resistance must be made. The chosen parameters should be accessible in the ship’s stability book and be able to describe the ship as good as possible. The selection of available parameters are however quite limited. Ideally if two ships have different added resistance, this would also mean that they have different values of their model parameters. This study has resulted in a selection of parameters, based on assumed relevance and availability: Table 2, parameters of the model.

Parameters

Non dimensional parameters 1

Prismatic coefficient

Lwl CP

Distance from aft peak to centre of gravity

LCG [m]

LCb , norm =

Beam

b [ m]

bnorm =

Draught

T [ m]

Tnorm

Radius of pitch gyration

ryy [m]

k yy =

Ship speed

V [m / s]

Fn

Length of ship in water line

CP LCG Lwl

b Lwl T = Lwl ryy

Lwl

Whether the ship speed V or the Froude number Fn should represent the ship speed is not obvious. But by trial and error it turns out that Fn should be used to make the added resistance non dimensional. So if a ship is scaled with constant values of the non dimensional parameters in table 2, the non dimensional added resistance (1.8) will not change.

34

6.4 LINEAR REGRESSION This part shows linear regression of RAW , p , but linear regression of ω p , norm and cnorm is done in the same way. Linear regression is basically an approximate solution to a linear equation system (6.5): ⎛ β const ⎞ ⎜ ⎟ ⎜ βC p ⎟ ⎜ ⎟ ⎛ Rawp ,1 ⎞ ⎡ 1 f C p ( CP ,1 ) f LCB ,norm ( LCB , norm ,1 ) f bnorm ( bnorm,1 ) fTnorm (Tnorm,1 ) f k yy ( k yy ,1 ) f Fn ( Fn1 ) ⎤ ⎜ β LCB ,norm ⎟ ⎛ ε1 ⎞ ⎥ ⎜ ⎟ ⎢ ⎜ ⎟ ... ... ... ... ... ... ⎥ ⋅ ⎜ β bnorm ⎟ + ⎜ ... ⎟ ⎜ ... ⎟ = ⎢... ⎜ ⎟ ⎜ ⎟ ⎢ ⎥ ⎜R ⎟ εn ⎠ awp , n ⎠ ⎝ ⎢⎣ 1 f C p ( CP , n ) f LCB ,norm ( LCB , norm , n ) f bnorm ( bnorm ,n ) fTnorm (Tnorm , n ) f k yy ( k yy , n ) f Fn ( Fnn ) ⎥⎦ ⎜ βTnorm ⎟ ⎝N  

  ⎜ ⎟ ε y X ⎜ β k yy ⎟ ⎜ ⎟ Fn ⎝β 

⎠ β

y is a vector containing either RAW , p , ω p , norm or cnorm . X is the model parameter matrix, with parameter

functions, f C ( CP ,1 ) and f b p

norm

(b

norm ,1

)

and so on. Assuming that the error vector ε is normal distributed

enables the possibility to find the vector β that best fits the data in y using the maximum-likelihoodmethod [15]:

β = ( XT ⋅ X −1 ) ⋅ XT ⋅ y

(6.6)

The estimation of y made with linear regression is calculated as:

yest = X ⋅ β

(6.7)

ε = y − yest

(6.8)

Error vector ε can be calculated as:

And the standard deviation of this error:

σ=

1 N

N

∑ε

2

(6.9)

i =1

The standard deviation of the error is a good measurement of the quality of the linear regression. It can be used to compare different linear regressions to each other, for instance comparison of linear regressions with different parameter functions.

35

6.5 PARAMETER FUNCTIONS In order to select reasonable parameter functions, model parameters have been varied one by one, starting off at a reference state. The shapes of the resulting curves have been used to choose the parameter functions. It is quite important that the parameters are varied one by one, exclusively. This is in many cases difficult, for instance when the draught T should be varied without changing the prismatic coefficient CP . The curves of the parameter variation for RAW , p can be found in (Appendix 3, Parameter variations, page 51). The chosen parameter functions are summarized in table 3. Table 3, Parameter functions.

f CP

f LCG ,norm

Rawp

CP

LCG , norm

ω p ,norm

CP

cnorm

CP

fbnorm

fTnorm

1 2

Tnorm

LCG , norm

bnorm

Tnorm

LCG , norm

bnorm

Tnorm

bnorm

f k yy

f Fn

1 k yy 2

Fn

1

1 Fn

k yy

2

1

1 Fn 2

k yy 2

7 APPLICATION OF THE SIMPLIFIED METHOD In order to derive an expression for added resistance using the method described above, a gathering of seven ships, one containership, two reefers, two RoRo-ships and two tankers, have been studied. Table 4, Original parameters of the seven ships.

  Ship 1  Ship 2  Ship 3  Ship 4  Ship 5  Ship 6  Ship 7 

Reefer  Reefer  Container  RoRo  RoRo  Tanker  Tanker 

CP

LCG , norm  

b [ m]  

T [ m]  

k yy  

Fn  

0.57  0.55  0.67  0.58  0.61  0.75  0.78 

0.47  0.46  0.46  0.44  0.47  0.53  0.52 

22.52  22.61  32.21  23.40  32.26  21.50  43.03 

8.00  7.50  10.50  5.00  8.50  6.00  10.00 

0.25  0.25  0.25  0.25  0.25  0.25  0.25 

0.21  0.21  0.14  0.24  0.16  0.15  0.10 

Fnmax 0.28  0.28  0.19  0.32  0.21  0.19  0.14 

L [ m] 142.5 140.5 279.7 128.4 195.1 140.7 242.5

In order to increase the number ships, the seven ships are distorted, to create simulated ships with different parameters (Appendix 2 Distortion of CP and LCG page 49). The hulls are distorted in the intervals:

LCG = ⎡⎣ LCG ,min , LCG ,max ⎤⎦ b = ⎡⎣ 0.85 ⋅ borig ,1.15 ⋅ borig ⎤⎦ T = ⎡⎣0.85 ⋅ Torig ,1.15 ⋅ Torig ⎤⎦ k yy = [ 0.200, 0.29] Fn = [ 0.5 ⋅ Fnmax , Fnmax ]

36

(7.1)

This way every ship has its own interval. LCG is a function of the distortion coefficient K L ( Appendix 2 CG

Distortion of CP and LCG page 49). The parameters with subscript orig are the original values of the ship. Subscript min and max are the smallest and biggest values allowed. For instance Fnmax (Table 4) is the maximum speed of the ship. The parameters are systematically varied, to cover all possible combinations of the parameters in the defined intervals (7.1). In this project the parameters have 3 points of variation in every interval. This will increase the number of ships, from the original 7 to 7 ⋅ 36 = 5103 ships. 7.1 LINEAR REGRESSION OF GROUPS OF SHIPS Linear regression is carried out for matrixes X consisting of various combinations of matrixes X ship , n

where X ship is a model parameter matrix consisting of distortions from shipn . The ships are thereby divided into the following groups: n

• • • • •

GContainer , containing ship3 . GReefers , containing ship1 and ship2 . GRoRos , containing ship3 and ship4 . GTankers , containing ship5 and ship6 . Gall ships , containing all ships ( ship1 , ship2 ,…, ship7 ).

The ships are described in table 4 and in (Appendix 5, The seven ships with their original parameters., page 57). Example: The equation system for GReefers is written:

⎛ yship1 ⎞ ⎡ X ship1 ⎤ ⎜⎜ ⎟=⎢ ⎥ ⋅ βGn + ε Gn yship2 ⎟⎠ ⎣⎢ X ship2 ⎦⎥ ⎝

  

(7.2)

XGn

yGn

Solving the system (7.2) using (6.6) will result in vector β G (in this case β G n

Reefer

), that can be used to

calculate yest , G , the estimation of yG : n

n

yest , Gn = XGn ⋅ βGn

(7.3)

So this is the estimation based on linear regression of the data in Gn evaluated for the same group. The resulting vectors β G from the linear regression can be found in (table 5 Appendix 1, Result of linear n

regression, page 49). 7.2 EVALUATION OF THE REGRESSIONS The error vector can be calculated as:

ε G = yG − yest , G n

n

And the standard deviation of this error:

37

n

(7.4)

σG = n

1 N

N

∑ε i =1

2

(7.5)

Gn

N is the number of ships in group Gn .

Figure 20 shows the standard deviation of the peak value σ G ( RAW , p ) for the groups in this linear n

regression. To get a feeling of the magnitude of this standard deviation, it is worth to mention that the mean value of RAW , p for all the ships is 8.8, the min value is 2.7 and the max value is 31.1.

Figure 20, standard deviation for the error of the peak values for the groups,

σ G ( RAW , p ) . n

In figure 21, added resistance transfer functions for the seven ships with their original parameters are compared to transfer functions calculated with linear regression, based on all the ships, β all ships . This is referred to as the Simplified method in figure 21.

38

Ship1

Ship2

20

Ship3

12

10

10

10

8

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

15

8 6 4

5

1

1.5

0 0.5

2

1

Ship4

0 0.5

2

4

4

2

2

1

1.5

2

0 0.5

2

6 4 2

1

1.5

2

0 0.5

1

1.5

Ship7 5

Gerritsma & Beukelman Simplified method

RAW ρ · g · ζ 2 (b2 /L)

4 3 2 1 0 0.5

1

1.5
L λ

2

2.5

8

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

6

0.5

1.5

10

6

0

1

Ship6

8

8

RAW ρ · g · ζ 2 (b2 /L)

1.5

Ship5

10

0

4 2

2 0 0.5

6

2.5

Figure 21, Added resistance transfer functions for the 7 ships with original parameters.

39

2

7.3 INFLUENCE OF THE PARAMETERS Linear regression can tell a lot about how much the parameters influence the final expression:

1 1 RAW , p = β const + βC p ⋅ CP + β LCB ,norm ⋅ LCb, norm + β bnorm ⋅ + βTnorm ⋅ Tnorm + β k yy ⋅ + β Fn ⋅ Fn (7.6) 2  



bnorm  

kyy 2    

   

Fn  

Tnorm LCb ,norm Cp kyy

bnorm

The influences of the parameters are presented in figure 22, for the seven ships with their original parameters.

Figure 22, Influence of the parameters.

Figure 22 shows that the prismatic coefficient CP has very little influence on the result. 7.4 LINEARIZED NON LINEAR REGRESSION Using linear regression has assumed that for instance the peak value can be expressed as:

RAW , p = β const + βC p ⋅ f C p ( CP ) + β LCB ,norm ⋅ f LCB ,norm ( LCB , norm ) + β bnorm ⋅ fbnorm ( bnorm ) + ...

(7.7)

Even if the parameter functions are perfect, this expression (7.7) does not have any connections between the parameters, for instance: fC ( CP ) ⋅ f L ( LCB , norm ) . It is quite likely that the parameters are in fact connected, so that for instance the influence of the draught T will differ for different speeds V . The peak value could be expressed as: p

CB , norm

RAW , p = β1 ⋅ CP β2 ⋅ LCG β3 ⋅ bnorm β4 ⋅ Tnorm β5 ⋅ kyy β6 ⋅ Fn β7

(7.8)

Expressions of this kind would require non linear regression analysis. (7.8) is however a bit special, since it can be treated with linear regression if it is expressed in a different way:

RAW , p = β1 ⋅ eln (CP )⋅β2 ⋅ e

ln ( LCG )⋅β3

⋅e

ln ( bnorm )⋅ β 4

Which is a non linear expression that can be linearized:

40

⋅e

ln (Tnorm )⋅ β5

⋅ eln ( kyy )⋅β6 ⋅ eln ( Fn )⋅β7

(7.9)

ln ( RAW , p ) = ln ( β1 ) + β 2 ⋅ ln ( CP ) + β 3 ⋅ ln ( LCG ) + β 4 ⋅ ln ( bnorm ) + β 5 ⋅ ln (Tnorm ) + β 6 ⋅ ln ( kyy ) + β 7 ⋅ ln ( Fn ) (7.10)

This is an expression that can be treated with linear regression and is therefore called linearized non linear regression. Using the model parameter matrix for all ships X all ships and vector y , that is used in the linear regression, gives the following estimation of the peak value, calculated with linearized non linear regression (7.10):

RAW , p = 206.5867 ⋅ CP 0.0440 ⋅ LCG 0.6376 ⋅ bnorm −1.2121 ⋅ Tnorm 0.6247 ⋅ kyy1.3611 ⋅ Fn 0.6377

(7.11)

The standard deviation of the error of this linearized non linear regression for all the ships is more or less the same as the standard deviation of the error of the linear regression. The two estimations seem to be very similar according to figure 23. Ship1

10

10

8

10

RAW ρ · g · ζ 2 (b2 /L)

15

12

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

Ship3

Ship2

20

8 6 4

5

1

1.5

0 0.5

2

1

Ship4

0 0.5

2

4

1

1.5

2

4

0 0.5

4 2

1

1.5

2

0 0.5

Gerritsma & Beukelman Simplified Method Linearized Non Linear

RAW ρ · g · ζ 2 (b2 /L)

4 3 2 1 0 0.5

1

1.5
L λ

2

2.5

6

Ship7 5

2

8

6

2

2

0.5

1.5

10

RAW ρ · g · ζ 2 (b2 /L)

RAW ρ · g · ζ 2 (b2 /L)

6

0

1

Ship6

8

8

RAW ρ · g · ζ 2 (b2 /L)

1.5

Ship5

10

0

4 2

2 0 0.5

6

2.5

Figure 23, Linearized non linear regression for the seven ships.

41

1

1.5

2

7.5 DISCUSSION The linear regression with the chosen model parameters and parameter functions for all the ships gives a standard deviation of the error of about 2.4 (Figure 20). If the regression is performed on a specific group of ships, the standard deviation of the error is a bit lower according to figure 20. So the method would be a bit more accurate for the corresponding group. The standard deviation for the peak value is about 25% of the mean peak value, which gives a feeling of the accuracy of the method. This may sound as a very high percentage, but it might be acceptable since the accuracy of added resistance calculations with strip theory is about 20-30% [2]. Error is introduced by the parameterization of the hulls. When the whole geometry is described with a few parameters, some information is inevitably lost, which introduces the parameterization error. Error is also introduces by the linear regression itself. When linear regression is performed on only one ship, with variation of the parameters, the standard deviation is about 50% of the standard deviation from the regression based on all the ships. In the case of linear regression based on variation of only one ship, it can be assured that it is only the selected model parameters that are varied. This means that the parameterization error is more or less zero, and the error is only introduced by the error due to the linear regression itself. In the linear regression with all the ships on the other hand, it can not be assumed that only the model parameters are varied. This means that the contribution of the parameterization error and the error due to the linear regression itself is of the same magnitude. It is quite clear that the chosen parameters do not capture all aspects of a hull geometry. An example of this is the two ships in figure 24. The two ships have the same values of the model parameters, but are in fact completely different ships. The peak value of the added resistance transfer function is twice as big for the Reefer, compared to the RoPax.

Reefer Cp(5.00)=0.60 Lcb=0.45 trim=0.00 b=0.15 T=0.040 kyy=0.25 beta=3.14 Fn=0.20Lpp=140.69

Z

10 5

Ap

0

Fp -60

-40

-20

0

20

10 0 -10

40

60 Y

X

RoPax Cp(0.00)=0.59 Lcb=0.45 trim=0.00 b=0.15 T=0.040 kyy=0.25 beta=3.14 Fn=0.20Lpp=128.88

Z

10 5

Ap

0

Fp -60

-40

-20

0

5 -50

20

40

60 Y

X

Figure 24, Top: Reefer, Bottom: RoPax

42

It is interesting to see that it is profitable to put the RoPax and the Reefer in different groups, avoiding the problem described above, when performing linear regression. Linear regression for the RoRo-group (containing the RoPax) has a significantly lower standard deviation than the total linear regression according to figure 20. From a physical point of view (7.8) seems more reasonable than (7.7), according to the discussion about the connections between the parameters from 7.4. So it is interesting that the linear regression and the linearized non linear regression (7.10) give very similar results. One advantage of the linearized non linear regression is that it does not require any parameter functions. 7.6 FURTHER WORK Further work and improvements, as well as weaknesses of this method are discussed and summarized in this section. 7.6.1

Even keel assumption

Generally in this project effort has been made to minimize the size of the problem. The current problem involves 5103 hulls, which means that 5103 strip calculations have to be performed, which is very timeconsuming. The parameter matrix grows dramatically with the number of parameters. Parameters that are assumed to only have a slight influence, are therefore not included. The Trim has because of this been left out off the regression. To see how much the method suffers from this, the trim has been varied for one of the Reefers, in a relevant interval. The resulting peak value Rawp is presented in figure 25, together with the prediction from the simplified method Rawpapprox. 13 Rawp Rawp

approx

12.5

Raw ρ · g · ζ 2 (b2 /L)

12

11.5

11

10.5

10

9.5 −1.5

−1

−0.5

0

0.5

1

1.5

Trim [m]

Figure 25, Variation for one of the Reefers.

Figure 25 shows that the simplified method responds to a change in trim. This is because a change in trim results in a change in LCG . But it does not respond properly to the change in geometry due to the change in trim. This results in the divergence in figure 25, which with reasonable accuracy has been neglected in the simplified method. 43

7.6.2

High frequencies

The diffraction induced resistance, dominating for high frequencies, has been left out of this regression. The method is therefore not valid for high frequencies. A way to improve this method for high wave frequencies would be to incorporate a method specialized on diffraction induced resistance. That method could be Faltinsen’s asymptotic method, described in this thesis. But there are also more advanced methods similar to Faltinsen’s asymptotic methods available, which for instance take into consideration that a ship is not infinitely deep. 7.6.3

Larger sample set

The linear regression performed in this project should be considered as an investigation of a methodology rather than being a finished method. A real method should be based on a linear regression on a larger set of different hull samples, to be more reliable in real conditions. 7.6.4

Parameterization

The parameterization can evidently be improved according to the discussions about the RoPax and the Reefer from figure 24. It seems as if the fullness in the ends of the ship will influence the added resistance. It was expected that the prismatic coefficient would capture this, but evidently it hasn’t. So some other parameter has to be used, for instance the longitudinal metacentric height, in order to improve this method. 7.6.5

More advanced regression

Linear regression and linearized non linear regression have been performed in this project. It would be interesting to perform non linear regression on this problem as well, where the peak value for instance could be expressed as:

(

)(

)(

)(

)(

)(

RAW , p = β1 ⋅ β 2 + CP β3 ⋅ β 4 + LCG β5 ⋅ β 6 + bnorm β7 ⋅ β8 + Tnorm β9 ⋅ β10 + kyy β11 ⋅ β12 + Fn β13

)

(7.12)

8 WAVE DIRECTION The simplified method is valid only for head waves. A brief investigation of how this method could be extended to be valid for other wave directions is presented in this part. In this thesis the wave direction is assumed to affect the total RAW , p  as:

RAW , p ( β , RAW , p , Head ) ≈ f β ( β ) ⋅ RAW , p , Head

(8.1)

RAW , p , Head is the peak value in head sea. This is the peak value that can be achieved with the simplified method. f β ( β ) is a parameter function. The same assumption, made for RAW , p (8.1), is made for ω p , norm and cnorm . 8.1

APPROXIMATION OF RAW , p ( β , RAW , p , Head )

The assumption has been tried by varying β and Fn for the seven ships. The resulting peaks as well as an approximation using (8.1) with f β ( β ) = cos (π − β ) is presented in figure 26.

RAW , p ( β , RAW , p , Head ) ≈ cos (π − β ) ⋅ RAW , p , Head ( Fn )

44

(8.2)

Figure 26,

β

and

Fn

variation together with cos-approximation

The Red surface represent “the real peak values” calculated with Gerritsma and Beukelman’s method and the Green surface represent the approximation according to (8.2). The approximation seems to correspond reasonably well to the data. 8.2

APPROXIMATION ω p , norm ( β , ω p , norm, Head )

How the non dimensional peak frequency ω p , norm is affected by the heading angle β is investigated in this part. It is assumed that the peak frequency will occur at a certain frequency of encounter, for all wave angles. This means that the non dimensional peak frequency of encounter will be the same for a ship in head sea ωe, p , Head and for a ship with a wave angle ωe, p , β :

ωe , p , β = ω p ,norm ( β , ω p ,norm, Head ) −

ω p ,norm ( β , ω p ,norm, Head ) ⋅V ⋅ cos ( β ) 2

g

ωe , p , Head = ω p ,norm, Head +

ω p ,norm , Head 2 ⋅ V

ωe , p , Head = ωe , p , β

g

(8.3)

(8.4) (8.5)

This means that the non dimensional peak frequency for a wave angle β can be expressed as a function of ωe, p , Head , using (8.3) and (8.5):

45

ω p ,norm ( β , ω p ,norm , Head ) = g ⋅

Figure 27,

β

and

Fn

1 − ωe, p , Head (ω p ,norm, Head ) ⋅ g 2 − 4 ⋅ V ⋅ cos ( β ) 2 ⋅V ⋅ cos ( β )

variation together with approximation of

(8.6)

ω p , norm ( β , ω p , norm , Head ) .

The Red surface represents “the real peak frequencies” calculated with Gerritsma and Beukelman’s method and the Green surface represents the approximation according to (8.6). This approximation seems to be remarkably good. 8.3

APPROXIMATION OF cnorm ( β , cnorm , Head )

The following expression for cnorm has turned out to be a good approximation:

cnorm ( β , Fn ) ≈

1

4

cos (π − β )

⋅ cnorm, Head ( Fn )

(8.7)

8.4 CONSLUSIONS The approximations (8.2), (8.6) and (8.7) are promising. They are examples of how the simplified method could be made valid for head to beam waves. The approximations have been performed for a variety of ships, with very good results for the peak frequency ω p , norm ( β ,ω p , norm, Head ) and quite good results for the peak value RAW , p ( β , RAW , p , Head ) and the width cnorm ( β , cnorm, Head ) . The peak values RAW , p ( β , RAW , p , Head ) (Figure 26) for ship3 differs from the other ships in that head sea does not produce the biggest peak, which is very unusual. This results in poor predictions with the peak approximation (8.2) for ship3 . The peak approximation (8.2) can thereby definitely be improved, to deal with this. It can also be improved for beam waves, where it states that the peak value should be zero, which is not true. 46

9 SUMMARY AND DISCUSSION This master thesis project has consisted of a literature study of the work that has been done previously in the field of added resistance. It has also consisted of implementations of three methods to calculate added resistance. A simplified method based on linear regression of systematic calculations with one of the implemented methods has also been investigated. It seems as if most of the fundamental work in this field was conducted in the 1970’s. Back then, the primary objective with added resistance calculations was to determine the Weather margin. Calculation of added resistance in other than head sea is very difficult. You can however avoid this difficulty when the Weather margin is concerned, since head sea normally produces the highest added resistance [13]. This has been the general approach to this problem over the years. Today there is a growing interest for two other problems: Weather routing and Performance prediction, mentioned in the introduction to this thesis. These problems are more complex than the Weather margin problem, since other wave directions than head sea has to be considered. It seems as if both Gerritsma and Beukelman’s method and Boese’s method give satisfactory predictions of the added resistance in head sea. They both seem to over predict the resistance peak though. None of the methods account for roll and yaw motions, which probably gives poor predictions for beam sea. The implementations in this project have shown poor agreement with experimental data in following waves. But it has been concluded that the added resistance in following waves is quite small. More work has to be done in this field, in order to improve the precision of the added resistance calculations in beam, quartering and following waves. Added resistance is indeed a very difficult problem; there is a reason that little progress has been made since the fundamental work was conducted. But the growing interest for performance predictions will hopefully provide the means that are needed for experiments and research. This master thesis project has also consisted of a study of how a method to predict added resistance transfer functions can be derived, using linear regression analysis. The evaluation of this method has shown that the error of the method is about 25%, but that this is about the same accuracy that generally can be expected of a theoretical method for added resistance in waves. The method presented in this thesis is limited to radiation induced resistance, even keel and head sea. It will thereby require further work to develop an unlimited method. The methodology described in this thesis together with some further work, mentioned above, has however good potential. 9.1 SUGGESTED READING In the literature study of this project, I have come across a lot of books and technical reports about added resistance and ships hydrodynamics. I would like to suggest some of them that have been particularly helpful. • • •

“Added Resistance in Waves” [13], probably the most complete paper about added resistance that I have found. It contains a lot of experimental data, and I have used it a lot. “Sea loads on ships and offshore structures” [5] is a very educational book about the basics in ship hydrodynamics. “Theoretical manual of SEAWAY” [6] explains Gerritsma and Beukelman’s method, and Boese’s method in a very detailed way. Free to download on the web!

47

10 REFERENCES 1. Gerritsma, J and Beukelman, W: Anlysis of the resistance increase in waves of a fast cargo ship. International Shipbuilding Progress, 19:285-293, 1972. 2. Salvesen, N: Added resistance of ships in waves. Journal of Hydronautics, 12(1):24-34, 1978. 3. Salvesen, N: Ship motions and sea loads. Naval ship research and development center, Washington D. C., November 1970. 4. De Jong, B: Computation of the hydrodynamic coefficients of oscillating cylinders. Report No.145S, Netherlands Ship Research Centre TNC. 1973 5. Faltinsen, O.M. : Sea loads on ships and offshore structures. Department of marine technology, Norwegian institute of technology, Cambridge University Press 1990. 6. Journée, J.M.J: Theoretical manual of SEAWAY. Delft University of Technology Shiphydromechanics Laboratory, (Release 4.19, 12-02-2001). 7. Journée, J.M.J: Verification and Validation of Ship Motions Program SEAWAY. Delft University of Technology Shiphydromechanics Laboratory, Report1213a, February 2001 8. Journée, J.M.J: Motion and resistance of a Ship in Regular Following Waves. Delft University of Technology Shiphydromechanics Laboratory, Report0404, September 1976 9. Garme, K. : Marin Hydromekanik. Centre for Naval Architechture, KTH Royal Institute of Technology, Stockholm October 2007. 10. Journée, J.M.J: Motions, Resistance and Propulsion of a Ship in Regular Head Waves. Delft University of Technology. Report 428, May 1976. 11. Pjedsted Pedersen, B: A comparative study of methods for predicting added resistance in Waves. Section of coastal, maritime and structural engineering, Technical University of Denmark, 1st February 2006. 12. Péres Arribas, F: Some methods to obtain the added resistance of a ship advancing in waves. Naval architecture school of Madrid, 27 September 2006. 13. Ström-Tejsen, J., Hugh Y.H. Yeh and Moran D.D. :”Added Resistance in Waves”, Society of Naval Architects and Marine Engineers, Transactions, Vol. 81, 1973, pp. 109-143. 14. Ming-Chung, Fang: “Second-order steady forces on a ship advancing in waves”, Dept. of Naval Architecture and Marine Engineering, National Cheng Kung Univ., Tainan, Taiwan, R.O.C, Int. Shipbuild. Progr., 38, no. 413 (1991) pp. 73-93. 15. Blom, G. : ”Statistikteori med tillämpningar”, Studentlitteratur, Lund, Andra upplagan 1984. 16. Rosén, A. :”Quicklines”, Centre for Naval Architechture, KTH Royal Institute of Technology, Stockholm. 17. EnRoute, Seaware AB, Stockholm, Sweden, “http://www.seaware.se”

48

APPENDIX 1 RESULT OF LINEAR REGRESSION Table 5,

βG

βL

 

n

β const  

βC

βContainer  

13.6 

3.09 

‐30.4 

0.06 

100.0 

‐0.205

21.7

β Reefers  

26.5 

‐1.05 

‐60.9 

0.21 

54.8 

‐0.529

58.2

β RoRos  

4.6 

‐3.02 

‐10.7 

0.15 

95.7 

‐0.254

22.8

βTankers  

17.5 

‐1.13 

‐36.7 

0.11 

111.0 

‐0.285

44.8

‐13.9   

‐0.27   

21.3   

0.10   

165.7 

‐0.334  

β all ships    

βC

p

CB , norm

βb

norm

βL

βT

norm

 

 

βk

37.5  

β Fn

β const  

βContainer  

‐0.64 

‐0.16 

2.74 

‐0.76 

‐8.38 

0.017

0.30

β Reefers  

0.31 

0.05 

0.88 

‐0.60 

‐3.02 

0.008

0.15

β RoRos  

‐0.02 

0.02 

1.42 

‐0.34 

‐5.87 

0.012

0.22

βTankers  

0.55 

‐0.01 

0.57 

‐0.20 

‐10.35 

0.011

0.18

‐0.15   

‐0.53   

‐1.37   

‐6.18 

 

1.21   

0.011  

cnorm  

β const  

βC

βContainer  

‐0.85 

0.00 

1.89 

0.44 

‐2.760 

0.008 0.00152

β Reefers  

‐0.04 

0.02 

0.15 

0.05 

‐0.165 

0.002 0.00078

β RoRos  

‐0.41 

0.10 

0.80 

0.32 

‐2.051 

0.005 0.00155

βTankers  

‐0.31 

0.09 

0.74 

0.45 

‐5.624 

0.006 0.00073

β all ships  

0.34 

‐0.01 

‐0.25 

‐0.44 

‐3.124 

0.005 0.00093

β all ships  

p

  CB , norm

βL

CB , norm

p

CB , norm

βb

 

RAW , p = β const + βC p ⋅ CP + β LCB ,norm ⋅ LCb, norm + β bnorm ⋅

ω p ,norm = β const + β C ⋅ CP + β L

norm

norm

1 bnorm

2

 

 

βT

yy

ω p ,norm  

p

βb

 

βk

β Fn

RAW , p  

norm

 

 

βT

norm

+ βTnorm ⋅ Tnorm + β k yy ⋅

⋅ LCb ,norm + β bnorm ⋅ bnorm + βTnorm ⋅ Tnorm + β k yy ⋅

cnorm = β const + βC p ⋅ CP + β LCB ,norm ⋅ LCb ,norm + β bnorm ⋅ bnorm + βTnorm ⋅ Tnorm + β k yy ⋅

2 DISTORTION OF

βk

 

yy

yy

0.20  

β Fn

1 + β Fn ⋅ Fn (9.1) kyy 2

1 1 + β Fn ⋅ (9.2) 2 kyy Fn

1 1 + β Fn ⋅ 2 (9.3) 2 kyy Fn

CP AND LCG

The hulls have been distorted using code from the program QuickLines [16]. The code distorts the hulls by changing the position of the strips in the hull geometry definition. The change of position of the strips is found by using a distortion function that sets a translation of the sections:

49

Distortion Form Function 5 4.5 4

section translation [m]

3.5 3

KA

2.5 2 1.5

KF

1 0.5

WA 0

0

0.1

0.2

0.3

WF

0.4

0.5 x/L

0.6

0.7

0.8

0.9

1

Figure 28, Distortion function.

In this figure K A is the maximum translation for the strips in the aft half of the hull. K F is the maximum translation for the strips in the fore half of the hull. K A , K F < 0 is translation backwards and K A , K F > 0 is translation forwards. WA and WF decide the location of the maximum translations K A and K F should be. WA = 0 is mid ship, WA = 1 is AP, WF = 0 is mid ship, WF = 1 is FP. 2.1

CP DISTORTION

CP is distorted by:

K A = − KC p K F = KC p

(9.4)

WA = WF = 0.5 K C p is the distortion coefficient for CP .

2.2

LCG DISTORTION

K A = 1.6 ⋅ K LCG K F = K LCG WA = 0 WF = 1 K LCG is the distortion coefficient for LCG .

50

(9.5)

3 PARAMETER VARIATIONS CP VARIATION 13 Rawp

12.5

12

11.5

11

10.5

10

9.5

0.48

0.5

0.52

0.54

0.56

0.58

0.6

C

p

Figure 29

CP

variation of one of the Reefers.

→ f CP = CP

Fp

60 40

10 Z

9 0.46

20

5

0

Ap

-20

0

-40 -60

10

0

-10

X

Y

Fp

60 40

10 Z

Raw ρ · g · ζ 2 (b2 /L)

3.1

20

5

0

Ap

-20

0

-40

10

0

-60 -10

X

Y

Figure 30, Visualisation of extreme values of

51

CP .

0.62

0.64

LCG VARIATION

11.5 Rawp

11

10.5

10

9.5

0.49

0.495

0.5

0.505

0.51

0.515

0.52

LCG,norm Figure 31,

LCG

variation of Model4210.

→ f LCG ,norm = LCG , norm

Fp

80 60 40 20

10 Z

9 0.485

0

5

-20

Ap

0

-40 -60

10

0

-10

-80

X

Y

Fp

80 60 40 20

10 0 Z

Raw ρ · g · ζ 2 (b2 /L)

3.2

5

-20 Ap

0

-40 -60

10

0

-80 -10

X

Y

Figure 32, Visualisation of extreme values of LCG .

52

0.525

b VARIATION 16 Rawp

15

14

13

12

11

10

0.85

0.9

0.95

1

1.05

1.1

1.15

b boriginal Figure 33,

b

Variation,

→ f bnorm =

1

bnorm 2

Fp

60 40

10 Z

9 0.8

20

5

0

Ap

-20

0

-40 5

-60 0

-5

X

Y

Fp

60 40

10 Z

Raw ρ · g · ζ 2 (b2 /L)

3.3

20

5

0

Ap

-20

0

-40 10

0

-60 -10

X

Y

Figure 34, Visualisation of extreme values of

53

b.

1.2

1.25

3.4

T VARIATION 14.5 Rawp 14

13.5

Raw ρ · g · ζ 2 (b2 /L)

13

12.5

12

11.5

11

10.5

10

9.5 0.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

T Toriginal Figure 35,

T

variation of Model4210.

→ fTnorm = Tnorm

Ship hull Model4210 has been used for these variations of T (Figure 36), since CP does not change that much with T , for this hull.

54

ryy VARIATION 14 Rawp

13

12

Raw ρ · g · ζ 2 (b2 /L)

3.5

11

10

9

8

7

6 0.2

0.21

0.22

Figure 36,

0.23

0.24

0.25

ryy Lwl

0.26

0.27

ryy variation of one of the Reefers. → f k yy =

55

0.28

1 k yy 2

0.29

0.3

3.6

Fn VARIATION

16 Rawp

15

14

Raw ρ · g · ζ 2 (b2 /L)

13

12

11

10

9

8

7 0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

Fn Figure 37,

Fn

variation for one of the Reefers.

→ f Fn = Fn

4 CHECK OF NON DIMENSIONAL PARAMETERS In order to verify that the approach of using non dimensional parameters in 6.3.2, added resistance transfer functions have been calculated for Model4210 at original size Rawp , X 1 and for Model4210 at twice the size Rawp , X 2 .

56

16 Rawp

X1

Rawp

X2

14

Raw ρ · g · ζ 2 (b2 /L)

12

10

8

6

4 0.1

0.12

0.14

0.16

0.18

0.2

0.22

0.24

0.26

0.28

0.3

Fn Figure 38, Speed variation of Model4210 at original size and at twice the size.

5 THE SEVEN SHIPS WITH THEIR ORIGINAL PARAMETERS.

Ap

Fp

-50

0

50

10 5 0

-50

Y

X

Ap

Z

10 5 0 -5

Z

Z

Cp(0.00)=0.57 Lcb=0.47 trim=0.00 b=0.16 Cp(0.00)=0.55 Lcb=0.46 trim=0.00 b=0.16 Cp(0.00)=0.67 Lcb=0.46 trim=0.00 b=0.12 T=0.056 kyy=0.25 beta=3.14 Fn=0.21Lpp=142.50 T=0.053 kyy=0.25 beta=3.14 Fn=0.21Lpp=140.55T=0.038 kyy=0.25 beta=3.14 Fn=0.14Lpp=279.65 Fp 0

15 10 5 0 -5

Ap -100

50

Fp 0

100 X

Y

X

Y

Ap

Fp

-50

0

50

Y

X

20 10 0

Ap

Z

10 5 0

Z

Z

Cp(0.00)=0.61 Lcb=0.47 trim=0.00 b=0.17 Cp(0.00)=0.58 Lcb=0.44 trim=0.00 b=0.18 Cp(0.00)=0.75 Lcb=0.53 trim=0.00 b=0.15 T=0.039 kyy=0.25 beta=3.14 Fn=0.24Lpp=128.39 T=0.044 kyy=0.25 beta=3.14 Fn=0.16Lpp=195.09T=0.043 kyy=0.25 beta=3.14 Fn=0.15Lpp=140.66 Fp

-50

0

50 X

100

10 5 0

-50 Y

Z

Cp(0.00)=0.78 Lcb=0.52 trim=0.00 b=0.18 T=0.041 kyy=0.25 beta=3.14 Fn=0.10Lpp=242.48 15 10 5 0 -5

Ap -100

Fp -50

0

50 X

100

Y

Figure 39, The seven ships with their original parameters.

57

Ap

Fp 0

50 X

Y

6 VALIDATION OF SHIP MOTIONS 6.1 S.A. VAN DER STEL IN HEAD SEAS In the following figures “EnRoute” and “SEAWAY KTH>0” refers to ship motions calculated respectively with EnRoute [17] and SEAWAY [7], “Experiment” refers to experimental data, obtained at the Delft University of Technology with a 1:50 scale model of the ship. The data has been taken from figures in the reports [1] and [10]. β=180 Fn=0.150

β=180 Fn=0.200

2

2 EnRoute SEAWAY KTH>0 Journée Experiment

1.5

η3 ζa

η3 ζa

1.5

1

0.5

0 0.5

1

0.5

0.6

0.7

0.8




0.9 L λ

1

1.1

0 0.5

1.2

0.6

0.7

2

1.5

1.5

1

0.5

0 0.5




0.9 L λ

1

1.1

1.2

1

1.1

1.2

β=180 Fn=0.300

2

η3 ζa

η3 ζa

β=180 Fn=0.250

0.8

1

0.5

0.6

0.7

0.8




0.9 L λ

1

1.1

0 0.5

1.2

0.6

0.7

0.8




0.9 L λ

Figure 40, Amplitude of heave, calculated in SEAWAY [7] and EnRoute [17].

58

β=180 Fn=0.150

β=180 Fn=0.200

1

1

0.5

0 −1

φ3 [rad]

φ3 [rad]

0 −0.5 −1

EnRoute SEAWAY KTH>0 Journée Experiment

−1.5 −2

0

0.2

0.4

0.6




−2 −3 −4 −5

0.8

1

1.2

−6

1.4

0

0.2

0.4

0.6

L λ

β=180 Fn=0.250




0.8

1

1.2

1.4

1

1.2

1.4

L λ

β=180 Fn=0.300

0

1 0

−1

−1 −2

φ3 [rad]

φ3 [rad]

−2 −3

−3 −4

−4

−5 −5 −6

−6 0

0.2

0.4

0.6




0.8

1

1.2

−7

1.4

0

0.2

0.4

0.6

L λ




0.8 L λ

Figure 41, Phase of heave, calculated in SEAWAY [7] and EnRoute [17].

Note that the phase lag has been forced to stay in the interval [−π , π ] , which explains the strange “jump”. β=180 Fn=0.150

β=180 Fn=0.200

0.04

0.04 EnRoute SEAWAY KTH>0 Journée Experiment

0.035 0.03

0.035 0.03 0.025

η5 ζa

η5 ζa

0.025 0.02

0.02

0.015

0.015

0.01

0.01

0.005

0.005

0

0

0.2

0.4

0.6




0.8

1

1.2

0

1.4

0

0.2

0.4

0.6

L λ

β=180 Fn=0.250




0.8

1

1.2

1.4

1

1.2

1.4

L λ

β=180 Fn=0.300

0.04

0.05

0.035 0.04

0.03

0.03

η5 ζa

η5 ζa

0.025 0.02

0.02

0.015 0.01

0.01

0.005 0

0

0.2

0.4

0.6




0.8

1

1.2

0

1.4

L λ

0

0.2

0.4

0.6




0.8 L λ

Figure 42, Amplitude of pitch, calculated in SEAWAY [7] and EnRoute [17].

59

β=180 Fn=0.150

β=180 Fn=0.200

−1.5

−1.5 −2

−2

−2.5

−2.5

φ5 [rad]

φ5 [rad]

−3 −3 −3.5

−4.5 −5

−4

EnRoute SEAWAY KTH>0 Journée Experiment

−4

0

0.2

0.4

0.6




−3.5

−4.5 −5 0.8

1

1.2

−5.5

1.4

0

0.2

0.4

0.6

L λ

β=180 Fn=0.250




0.8

1

1.2

1.4

1

1.2

1.4

L λ

β=180 Fn=0.300

−1

−1 −2

−2

φ5 [rad]

φ5 [rad]

−3 −3

−4

−4 −5

−5

−6

−6

0

0.2

0.4

0.6




0.8

1

1.2

−7

1.4

0

0.2

0.4

L λ

0.6




0.8 L λ

Figure 43, Phase of pitch, calculated in SEAWAY [7] and EnRoute [17].

S.A. VAN DER STEL IN FOLLOWING WAVES β=0 Fn=0.150

β=0 Fn=0.200

2

2 EnRoute SEAWAY KTH>0 Experiment

1.5

η /ξ

1

3

η3/ξ

1.5

0.5

0 0.5

1

0.5

0.6

0.7

0.8 0.9 sqrt(L/λ)

1

1.1

0 0.5

1.2

0.6

0.7

β=0 Fn=0.250 2

1.5

1.5

3

1

0.5

0 0.5

0.8 0.9 sqrt(L/λ)

1

1.1

1.2

1

1.1

1.2

β=0 Fn=0.300

2

η /ξ

η3/ξ

6.2

1

0.5

0.6

0.7

0.8 0.9 sqrt(L/λ)

1

1.1

0 0.5

1.2

0.6

0.7

0.8 0.9 sqrt(L/λ)

Figure 44, Amplitude of heave for S.A. van der Stel, calculated in SEAWAY [7] and EnRoute [17].

60

β=0 Fn=0.150

β=0 Fn=0.200

3

3 EnRoute SEAWAY KTH>0 Experiment

2

2 1

φ [rad]

0

0

3

φ3 [rad]

1

−1

−1

−2

−2

−3 0.5

0.6

0.7

0.8 0.9 sqrt(L/λ)

1

1.1

−3 0.5

1.2

0.6

0.7

0.8 0.9 sqrt(L/λ)

1.1

1.2

1

1.1

1.2

β=0 Fn=0.300

3

3

2

2

1

1

φ [rad]

0

0

3

φ3 [rad]

β=0 Fn=0.250

1

−1

−1

−2

−2

−3 0.5

0.6

0.7

0.8 0.9 sqrt(L/λ)

1

1.1

−3 0.5

1.2

0.6

0.7

0.8 0.9 sqrt(L/λ)

Figure 45, Phase of heave for S.A. van der Stel Calculated in SEAWAY [7] and EnRoute [17].

β=0 Fn=0.150

β=0 Fn=0.200

0.025

0.025 EnRoute SEAWAY KTH>0 Experiment

0.02

η /ξ [rad/m]

0.015

0.01

5

η5/ξ [rad/m]

0.02

0.005

0

0.015

0.01

0.005

0

0.2

0.4

0.6 0.8 sqrt(L/λ)

1

1.2

0

1.4

0

0.5

β=0 Fn=0.250

1.5

β=0 Fn=0.300

0.02

0.02

0.015

0.015

η /ξ [rad/m]

0.01

0.01

5

η5/ξ [rad/m]

1 sqrt(L/λ)

0.005

0

0.005

0

0.5

1

0

1.5

sqrt(L/λ)

0

0.2

0.4

0.6 0.8 sqrt(L/λ)

1

1.2

Figure 46, Pitch amplitude for S.A. van der Stel Calculated in SEAWAY [7] and EnRoute [17].

61

1.4

β=0 Fn=0.200 3

2

2

1

1

φ [rad]

0

0

5

φ5 [rad]

β=0 Fn=0.150 3

−1

−1 EnRoute SEAWAY KTH>0 Experiment

−2 −3 0.5

0.6

0.7

0.8 0.9 sqrt(L/λ)

−2

1

1.1

−3 0.5

1.2

0.6

0.7

1

1.1

1.2

1

1.1

1.2

β=0 Fn=0.300

3

3

2

2

1

1

φ [rad]

0

0

5

φ5 [rad]

β=0 Fn=0.250

0.8 0.9 sqrt(L/λ)

−1

−1

−2

−2

−3 0.5

0.6

0.7

0.8 0.9 sqrt(L/λ)

1

1.1

−3 0.5

1.2

0.6

0.7

0.8 0.9 sqrt(L/λ)

Figure 47, Pitch phase for S.A. van der Stel Calculated in SEAWAY [7] and EnRoute [17].

62

7 SHIP GEOMETRIES S.A. VAN DER STEL S175

AP

FP

20

15

[m]

7.1

10

5

0

-10

-5

0 [m]

5

10

Figure 48, Body plan S.A. van der Stel Table 6, S.A. van der Stel, main particulars [7]

Block coefficient  Lpp  Breadth, B  Draft (level trim)  Lcg  Vcg  Pitch gyradius/Lpp 

0.563 152.5 22.8 9.14 73.7 9.14 0.22

63

  [m]  [m]  [m]  [m]  [m]   

SERIES 60 HULL MODEL 4210 S175

AP

FP

20

15

[m]

7.2

10

5

0

-10

-5

0 [m]

5

10

Figure 49, Body plan of model4210 Table 7, Series 60 hull model4210, main particulars [13]

Block coefficient  Lpp  Breadth, B  Draft (level trim)  Lcg  Vcg  Pitch gyradius/Lpp 

0.6 190.5 25.412 10.17 95.25 7.9 0.25

64

  [m]  [m]  [m]  [m]  [m]   

S-175

S175 20

AP

FP

18

16

14

12

[m]

7.3

10

8

6

4

2

0 -10

-5

0 [m]

5

10

Figure 50, Body plan of S175.

Table 8, Containership S175, main particulars [7].

Block coefficient  Lpp  Breadth, B  Draft (level trim)  Trim by stern  Lcg  Vcg  Pitch gyradius/Lpp 

0.559 175 25.4 8.5 1 87.5 7.78 0.24

65

  [m]  [m]  [m]  [m]  [m]  [m]