Computations of second order wave drift forces acting on a slowly oscillating object in 2D

Computations of secoiid order wave drift forces

acting on a slowly oscillating object Ill 2D

B. M. Jagt

Student TWI

iiborato1um 'mor $cheepshydromcchaiica

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Mekeiweg 2, 2628 CD Ooft 'eL 015 - Fa 015 7I

Contents

3

Linearizing the model

14

3.1 Subdividing the velocity potential . . 14

:3.1.1 The free-surface condition . 15

3.1.2 The body-boundary condition 17

3.2 The double-body potential . 18

3.3 The forces 18

i

Introduction

2 The mathematical model for motions

2.1 Defining the actual model

2.2 The equations of motions 2.3 The boundary conditions

2.3.1 the free-surface boundary

2.3.2 The object boundary

2.3.3 The bottom boundary

2.4 The radiation conditions

2.5 The two-dimensional. body fixed coordinate system

2.6 Transformation to the new coordinate system 2.7 The forces ori the object

2.8 A summary of our model so far . . .

i

5 5 6 . 7 7 8 8 . 9 9 10 12 F3

CONTENTS ii

3.3.1 Approximating the force 19

3.4 The added mass and darping coefficients 21

3.5 The drift forces ., . . . 22

3.6 Summary of the linearized model . . 24

4 The numerical model

25

4.1 Iteration relationship for the potential . . 26

4.1.1 Splitting up the domain 26

4.2 Numerical model for the added mass and damping coefficients 31

4.2. 1 Numerical model for the drift forces . 32

5

Results

34

5.1 The computer programs 34

5.2 The added mass and damping coefficients 35

5.3 The drift forces 38

5.4 The drft dampiig coefficient 40 5.5 Conclusions

A The validity of the double-body approximation

45

A.1 More about the double-body potential

... 45

A.2 Comparison of the Green's functions ...46

B The Green's fuñction

50

BA Constructing the Green's function

B.1.1 A sub-model for the Green's function 51

B.1.2 The original Green's model 52

B.1.3 Fourier transformation 53

CONTENTS

C Discretising the free surface condition

D Some important theorems

E Summary

F Samenvatting

111 56 58 60 62

NOmenclature

Symbols denoting coordinates

Symbol

= (X,Y,Z)

2= (x,y,z)

=

(,()

Dimension Description

Earth bound coordinate system Object bound coordinate system

Help coordinates Radial coordinates

Properties of the object

Symbol

Dimension DescriptiQn

Î

[in] Displacement vector

[in] _{Total displacement (rotational and translation)}

[-J Rotational displacement

[s1] Frequency of the slow varying motion of the object

(i [in.s1J Velocity of the object caused by it' own propulsión

ü

[m s]

_{Velocity in the fluid domain in the earth-bound system} [in s_1] Velocity of the object in the earth-bound system

V [ni T'] Velocity of the object in the object-fixed system

fr [N] Force

(1) 2)

[N] First and second order forces

R [m] Radius of the object

d [m] Diameter of the object

Nomenclature (continuèd)

Properties of the domain

Symbol

_{Dimension Description}

h [in] _{Depth of the fluid domain}

û _{[in .s'] Velocity in the huid domain in the earth-bound}

system

[in s']

_{Velocity in the fluid domain in the object-fixed system}

m2s]

Velocity pötential in the earth-bound _system

m2s]

Velocity potential in the object-fixed system

[m2s1] Steady potential

[m21]

Slow oscillating potential

[m2 s_1] _{Unsteady potential}

inc, diff, [in2 s1] _{Potential caused by the incoming wave, the (liffracted wave}

and the movement of the object

p5db [in2 s _{Double-body potential}

p [Pa] _{Fluid pressure}

Po [Pa] _{Atmospheric pressure}

p [kg m_3] Fluid density

g [m .s2] _{Gravity constant}

[in] _{Height of the free surface}

wo [_1] _{Frequency of the incûming waves (sometimes w is used)}

w [s1] _{Frequency of encounter of the object}

w'

_[-]

_{Frequency of the incoming waves made non (hinlensional}

A _{[in] Wave length}

[rn']

Wave number

A [kg] _{Added mass coefficients matcix}

B

[kg .s1] _{Damping coefficients matrix}

M

[kg] Mass matrix

R

[kg.s2] _{Restoring coefficients matrix}

[kg] _{Drift added mass coefficients matrix}

Nomenclature (continued)

Symbols denoting an

area

Symbol

Description

N Ship hull (boundary)

No Average position of the hull fl05 Oscillatory position of the hull

S Interior domain

ÔS Total boundary of the interior domain

D Exterior domain

13 Artificial boundaries

f

Free-surface boundary

b _{Bottom boundary}

W1,..., W6 Part of integration contours

Other symbols

Symbol

Dimension Description

T fl,fl t A1,A2

c,O,ã

i

D,E

o-Time-scale Normal vector Time Time step

Constants. help variables Green's functions

Complex unit

Matrices used in the iteration scheme Real part of the parameter

Imaginary part of the parameter

Soilirce term

chapter i

Introduction

When considering ships at open sea, several factors influence the velocity and the

direction of the vessel. Random waves, wind, the power delivered by the engine and

the current are some of them. These kind of iìifluences are also present if the ship is not moving by itself, for example when it is moored to do _{some exploring of the}

sea bottom, or fo example when it is loading _{or unloadiùg goods (think of an oil}

tanker).

All these forces and moments which act on the ship can be separated into first order terms and second order terms. The first order forces and

moments are proportional to thewave height of

the incoming waves and contain the same fre-quencies as the incoming waves.

The second order forces, the so-called drift forces. are proportional to the square of the_{wave heights} and are the resi.ilt of the interaction between the different first order forces. Compared to the first order forces, the drift forces have small

frequen-cies.

Pitch * Sway Figure 1.1: 3-Dimensional

niove-Because the first order fórces have the largest t1iflts

influence on the motion of the vessel, _{they have been the subject of research for} some decades now and can be calculated fairly well. We, on the other hand, will

focus more on the second order forces.

Especially in mooripg systems at open sea, the drift forces can be of great impor-tance. Because of the fact that the drift forces _{can contain low frequent components,} the possibility exists that these _{frequencies comenear or even become equal to the}

eigenfrequencv of the total mooring _{system. In that case the system could he}

darn-aged sevefeiv. Because the low frequent components of the drift forces can induce large amplitude low frequency _{motions they are also indirect responsible for the}

Heave Surge

Yaw

t

Ro11

main part of the mooring forces.

As for the forces, the motions can also be separated into different orders of value.

We distinguish a _{mean excursion, a slowly varyìng motion and a higher frequency}

oscillation around the slowly varying position. _{The higher frequency oscillation} coincides with the wave motion while the slowly vatying motion coincides in period

with the slow oscillating part of the drift forces.

To calculate the motions of _{a vessel extensive use is made of Newton's second laws} the equation of motion:

82Î

_ôÎ

Mô2 +B--+CÎ=1

However in order to be able to calculate the slowly varying motion from the drift forces, we also need the value of the damping term B. This was already indicated

by Remery and Hermans[5} in 1971.. Since _{than there has been more research}

con-sidering this drift damping term.

One problem for calculating the drift forces _{which existed then was that there wasn't} enough calculating power to solve the complete _{problem in the time domaiù, which} forced the researchers to work in the frequency_{domain. This implies the assurnptiòn} of pure harmonical _{waves, which isn't very realistic at an open sea. The main data}

for these factors therefore still is obtained by conducting expensive model tests. These days the computer capacity has _{grown expÏosively which has made it possible} to make a start with creating models to solve the motions of the vessel in the time

domain. _{However there is still no complete solution, for theme is still no exact}

subscribed way to calculate the drift damping. _{A major problem which ar:ises when} constructing a model in the time-domain is the highly_{non-linearity of the equations}

in the model.

Prins and Hermans[4} quite recently made progress in this line of research by creating and implementing_{a linearized model for the time-domain into a computer program.}

They investigated the influence of the foiward _{speed of a ship on the forces and}

motions of the ship by constructing and implementing a time domain model in

a computer program which includes a vessel with a constant horizontal forward velocity. This was done for both _{a two-dimensional and the three-dimensional case.}

Sierevogel and Hermans[6] then improved the _{two-dimensional model by adjusting} the l)Oufldary conditions, which increased the _{speed of the progranì significant.}

In this report now we will try to calculate the drift damping coefficient by using a

computer program based on the earlier programs by Prins and Sierevogel. The idea behind the approach used in this report is illustrated by first looking at the way the drift forces and damping is measured in model _{tests There are in fact two methods} of doing so:

3

The extinction or motion decay tests These imply that in

a model system of a moored vessel and a wave emitter the vessel is given a small distortion whereafter the drift forces and dampiig can be measured with delicate

instru-ments;

The towing test This method uses a complete different approach. A vessel is

forced (towed) to carry out a prescribed low frequency motion, in the sway or

surge direction, while the forces, including the. drift forces, are measured. Be cause the drift forces will try to "correct" the forced motion, a slow oscillating part can be distinguished in the measured drift forces. For the slow oscillating

movement is already known, the drift clamping coefficient in sway or surge can

be calculated with the aid öf the equation of motion.

The last described test is not the exact way the tests are conducted in practice,

but it describes well the approach which is used in this report. While Prins and

Sierevogel use a ship with a constant velocity, we wilÏ use a slow oscillating vessel and

calculate with an adjusted program the drift forces and drift damping coefficients for thi situation.

Because the drift forces can also be written in the form of equation (1.1), by looking

at the components of the calculated drift fôrces which are oscillating in phase and out phase with the frequency of the velocity, we have a tool of calculating the wave

drift damping coefficients.

Another way of calculating the drift damping terms can also be derived from equatioí(1. I).

It shows that by calculating an approximation of the derivative of the averaged drift

forces with respect to the velocity (0>) we have an approximation of the wave

drift damping coefficient. One goal of this report therefore is to compare this last method with the method of using an oscillating speed and thereby verifying if these two different methods, which both should give the wave drift damping coefficient. really do give the saine outcome.

This is done for two models. The first model contains only minor adjustments to the model of Sierevogel and Prins. in fact the only difference is the implementation

of a slowly oscillating velocity in the model.

In the second model there are also extra terms taken into account, extra terms caused

by a now time-dependent velocity. This gives a different free surface condition and

the purpose of this model is to investigate the influence of these extra time-dependent

terms in the model.

In the next chapter, chapter 2. a mathematical potential model is constructed by

deriving a system of equations for a system which contains a slowly varying object in a fluid domain with an incoming current and incoming waves.

In chapter 3 the necessary linearisations are macle in order to he able to construct

the iteration schemes which are discussed in chapter 4. The model Which we discuss

4

would Iêave out these terms _{we would have the same model used by Sierevogel and}

Prins.

In chapter 5 finally the results of the two models are discussed.. This report contains also some appendices which _{go into more detail at some points than is done in the} precéding chapters.

Fiùally a summary of the model and the couc1üsioîs i given in both English and

Chapter 2

The mathematical model for

mot ions

In this chapter we will construct an exact mathematical model for an object which is floating at an open water. After we have done this for the common three dimensional

situation we will apply the so-called strip theory to our model and thereby restrict

ourselves to a two dimensional problem.

Also we will transform our model from an earth-fixed to an object-fixed coordinate system, which is needed for the numerical model in the chapters to _come.

2.1

_{Defining the actual model}

We consider an object which is floating horizontally at an open water, which could

for example be a sea.

z = o

'S

'SSS

.._c.__

SSS

Figure 2.1: The fixed coordinate system

The object is moving with a variable, time-dependent speed (1(t) in the negative X-direction. In fact is our object undergoing a slow oscillation. This is expressed

2.2. The equations óf motions

₆

by writing the velocity from _{now on as U(T). Hereby T = jit, which denotes a}

time-scale. The constant ji is a small scaling parameter in our model. By the Xdirection is meant. the X-direction in an earth-fixed coordinate system X = (X, Y, Z) as is shown in figure .2.1.

The Z = O plane is chosen to be eual

to the undisturbed free water surface

Furthermore the depth of the water is. represented by h.

We will make the following assumptions _{considering the fluid in which the object is} floating:

The fluid is incompressible. The fluid is inviscid.

The fluid is at all times irrotationaj.

In the fluid domain a current is absent, opposite to incoming waves which aré present

in the system. The object is further free to_{move in all directions or to rotate around}

each f its axis.

2.2

_{The equations of motions}

[f _{t) represents the velocity vector in the model as stated in the previous}

paragraph, then it has to satisfy the continuity equation. This equation in fact

represents the conservation of mass and looks like:

(2.1)

From. the assumption that the fluid is irrotational it follows that we caii introduce a velocity potential _{(X,t) now. This Potential is defined by}

(2.2)

Our completemathematical model will be based_{on this potential. Therefore we Use} equation (2.2) in equation (2.1) to get that the potential has to satisfy the Laplace

theory throughout the flûid dOmain. In equations:

= 0 (2.3)

2.3. The boundary conditions

₇

We can make a distinction between

three boundaries of the fluid:

the free-surface

the object boundary the bottom

On all three boundaries there has to be set conditions to the velocity potential.

P po

+

. + yZ + const (2.4)

Here p is the fluid density, g is the gravitational acceleration and po is the atinos-pheric pressure which is assumed constant. The term 'const' is a (yet) undefined

constant term, which arises in the equation l)eCaUSe the equation is in fact found by

integrating the Euler equation with respect to X.

With this equation we can later on find an expression for the wave elevation of the

free-surface, which will be discussed iii the next paragraph First we shall look at the various boundary conditions now.

2.3

_{The boundary conditions}

3

Figure 2.2: Side view of the boundaries

The conditions for the velocity potential at the three fluid boundaries will be worked

out in the following three paragraphs.

2.3.1

_{the free-surface boundary}

We first introduce a new variable C( _{t) which will represent the height of the}

free-surface. At the free-surface we have two different conditions:

1. A fluid particle cannot leave the surface. This represents a kinematic boundary condition which is satisfied if the following equation is valid:

at

Z=

(2.5)

in which

D

=+uV=+VV

D Ô

2.3. The boundary conditions

This gives us

2. The dynamical condition at the free-surface

_{says that the pressure at the}

surface must equal the atmospheric pressure po. If we substitute p = Po in the

earlier given equation of Bernoulli (2.4) we get:

+ . + g + const = O (while Z

= (2.8)

We assume the disturbance, caused by the object, _{to be negligible at large} distance from the object. This implies that at infinity the wave height can be set to zero, which implies that the constant term in equation (2.8) _{equals zero} (there is no current,so O at infinity).

By rewriting the equation we _{now have a formula for the wave height} :

=

_,! (

_gôt

+ _(2.9)

2

₎

z=c

Because equation (2.9) is true for all time, we differentiate it and substitute the

found equation in equation (2.7) to get a partial derivation equation in which we will use from now on:

atZ=(

(2.10)

.3.2

The object boundary

We assume the floating object to be impermeable. _{This implies that the normal} velocity from the fluid on the hull must equal the normal velocity from the object:

atfl

(2.11)

In 'this equation _{is the hull of the object and t' is the velocity of the object in} our chosen coordinate system (notice by the way that because of the fact that we

assumed the fluid to be inviscid. slip along the hull _{could occur).}

2.3.3

The bottom botindary

As with the object we assume tile bottom to be also _{impermeable. Becai.ise. we also} assume the bottom to be horizontal and at a constant depth we can formulate the

following:

az

atZ=(

(2.7)

34

2.4. The radiation conditions

₉

24 The radiation

_conditions

As we can see with equation (2.10) the free-surface condition is a second order partial

differential equ3tion in time. This implies that we need two initial conditions. We get these by assuming that at t = O the fluid is ot disturbed. This gives us:

(,o)

O and

! (,g,o)

= o _{at t = 0}

(2.13)

To get a unique solution for the mathematical problem we also iieed to impose an extra condition, the so called radiation condition. Summarized this _{condition says}

that

waves can only be generated by the object, apart from the incoming waves

e the by the object generated _{waves move away from the object}

We will later on return to tl1is condition when we are dehning the numerical model.

2.5

_{The two-dimensional,}

_{body fixed coordinate}

system

Our earlier defined earth-fixed coordinate system has _{a big disadvantage as time}

goes on. _{lt's possible that, in time, the object will be at}

great distance from the origin, which won't improve further _{to come numerical calculations.}

A solution to this problem is choosing a body fixed coordinate system. But before we do this we restict our model to two dimensions.

We do this by making our object be a cylinder of infinite length and with radius R. The cylinder, which is made _{of homogene material, is floating}

in water of depth h. Because the mathematical model is the same for each cross-section of the cylinder we can (and will from now on) reduce our model to a two dimensional problem. In our svsteni there _{are regular incoming waves which are traveling in the positive} X-direction. Furthermore the cylinder is free to oscillate in both the

_{X- and the}

Z-direction but is restricted from _rolling.

The X-coordinate of the center of gravity of the cylinder is taken _{to be the}

time-dependent variable X0(t). We _{choose our cylinder to oscillate slowly in the}

X-direction, which explains the _{time-dependency of the l)lace-ver.tor.}

As mentioned before we now define an object-fixed coordinate _{system : with the}

following relationship _{to the earth-fixed coordinate system:}

2.6.

_{Transformation to the new coordinate system}

_io

This way the z-axis is chosen through the center of gravity of the cylinder _{and the} x-axis is again equal to the undisturbed _{free-surface plane.}

In this object-fixed coordinate _{system the (time dependent) movement of the ship}

can be translated to a time dependent current with velocity '(J(r)'. This implies

that in the earth-fixed coordinate system the ol)ject is moving with velocity '(J(r)'. This means in fact that the object is moving with velocity '(J(r)' to the left.

The parameter 'r' defines _{a time scale to express the slow oscillating movement of}

the cylinder (in the earth-fixed _{coordinate system).}

In equations we get:

Figure 2.3: The 2-dimensional model

TLt

(2.16)

in which _{¡ is a small (scaling) parameter.}

2.6

_{Transformation to the new coordinate system}

Now that we llave defined _{a new body-fixed coordinate system we must transform}

the formulas found in the previous chapter to this system.

We start by considering the Partial time-derivative. Because the object-fixed system

has a dependent shift with respect to the earth-fixed system, the partial time-derivative gets an extra term:

z=O

z=h

dXo(t) dt

- U(r)

with Heave j _t otj

Figure 2.4: 2-Dimensional

move-m en t s Sway (2.15) ô

ôtg

(ô

'ôt

(lXO(t))"\(lt

¡ô

dt

+ U(r)---

dx¿3 (2.17) X

and the second order time-derivative becomes ¿32

_(ô2

dU(r) it) 32 ₂ 32

=

À' (2.18)

(r)-2.6. Transformation to the

new coordinate system

_ii

= iZ+ (U(r))

Figure 2.5: Relationship between the earth-fixed and object-fixed system We now introduce the potential

_{(t) for the body fixed}

_{system. 1f we want to}

transform the equations found in the previous _{chapter to our new coordinatesystem,}

we have to find a relationship between _{(X, t) and t).}

Unfortunately the potential _{t) can't equal the potential t). This can be} seen by looking at the values of the potentials at infinity:

In our earth-fixed system there is _{no current so:}

hm I(,t) = 0

_(2.19)

X -<

because we assumed that the influence of the object is negligible far away.

In the object-fixed system however it _{seems that there is a time dependent} uniform current with velocity 'U(T)' present so:

um t) =

_Uer)i

_(2.20)

Notice by the way that if X_{- oc implies that x ; oc because the object oscillates} around the origin (so Xo(t) is finite).

To find a relationship between _{t) and t) we have to take a step back to} the definition of a potential. If we consider the relationship l)etween _{the velocity} û in the earth-fixed and _{in the object-fixed coordinate system, it's obvious that} the velocity in the earth-fixed system is equal to the velocity in the object-fixed system plus the velocity of the object itself

_{(= U(r)). If we use this relationship}

to construct the relationship between the potentials in the two systems, we get:

2.7. The forces on the object.

12

The term (J(7-)x could also be seen as a kind of córrection term for the difference between the two potentials at large distance from the object, at short distance from the object it will stay small.

By substituting the found relationships between and in the Laplace equation

(2.3) gives

o (2.22)

so the object-fixed potential also has to satisfy Laplace. Furtherïnore the equation

of Bernoulli (2.4) becomes

_P Po

= (

- U(r)x) +

(

-

(12(T)) ±gz (2.23)

Substitution of (2.21), (2.17). and (2.18), in the equations for thewave height (2.9.)

and the free surface condition (2.10) gives

ç .=.

-.

(

U(r)x) +

.

'(f2(r))

(2.24). and

(f U(r)x) + 2

(v

.

(I()dU(T).)

+VV(V7)+g

= respectively.

at z = ((2.25)

The rest of the boundary conditions from the previous chapter. equation (2.11),

(2.12) and (2.13), become:

= i? . t -í (2.26)

ôn

where V will from now on denote the velocity of .the object in our object-fixed

coordinate system (V = Ç'

_{- (U(r),O)).}

= O

at z= h

(2.27)

and

O) =. O and -(Z. O) = 0 (2.28)

2.7

The. forces on the öbject

The common formula to calculate the forces which are working at the object, is

given by

2.8. A summary of

_{our model so far}

₁₃

To use this expression we need to know the pressure. For this purpose _{we can use} the earlier transformed to the object fixed system,

equation of Bernoulli (2.23):

p = p

((4

- (1(r)

+ (V

_{(12(r)) + g)}

_{at fl}

_(2.30)

In this equation we have substituted p for the term p Po from (2.23).

This equation is the base from which We will derive relationships for the different aspects of the force, like for example the drift force.

However, before we _{can do this, we first have to simplify our model in order to be}

able to derive an iteration scheme. This will be done in the next chapter throúgh

linearization of our model.

2.8

_{A stimmary of our model so far}

This is just a summary of the potential model _{which we have created, and which} will be the base for further calculations. We use the model for the body-fixed

coordination system. We have:

! (

(4)

(J(r)x) + V4)

. - U2(r)) (4) _{(J(r)x) ± 2 (v4).}

-

(f(r))

₊

+V4) . V (V4). V4)) + o on O) = O Radiation conditions

The Laplace equation

The wave height

at z = (

at N

at z = h

Iñitial values

For the forces We have the following

I F

_{= f pdS}

'ç

I. p p ((4)

U(r)x) + (V4) V4) - U2(r)) + yz)

at N

(2.31)

14

Chapter 3

Linearizing the model

As is simple to see, the boundatv conditions which were formulated in the previous chapter contain several non-linearities. To be able to formulate a numerical model

later on we have to linearize these conditions A condition for this procedure is that the amplitudes of both the movements as the incoming, wave field are restricted to

small values.

3.1

_{Subdividing the velocity potential}

The free-surface condition which we transformed in the previous section cofltains two

complications. First the non-linearity of the potential & For this we will assume

that the potential is of the following form:

= O(x) ± (r) + ç5(i?,.t)

(3.1)

Here is:

o the steady potential. Because in our model the velocity U(r)is time dependent this steady potential equals zero

(°

O)

(. r) the potential caused by the flow of the fluid along the object while the object is fixed in Ufl(listurbed situation. A very simple form of r)

would be q5(, r) (J(r). However we find this approximation too rough, it ignores the the object in the model completely. In a section further on We will

introduce a better, more accurate approximation for this potential.

(ii?. t) is the unsteady potential which contains all other time-dependent_parts

of the total potential. This potential consists of the potentials caused by the

incoming wave, the diffracted wave and the poteûtial caused l)y the movement

of'the object.. In formula:

3.1.

Subdividing the velocity potential

15

Notice bye the way that for this potential the normal time scale is used t

denote the difference between he slow and the fast oscillating parts of the

potential.

We now assume that both

(, T) and

(, t) are small. We also assume that for quadratic and for _{linear terms are sufficient, to kep our model less complex.}

Ef we look at the Laplace equation, which has to be satisfied by the total potential,

and we consider an undistúrbed situation, it follows that the potential has to

satisfy this condition alone to. As a result the potential th also has to satisfy the

Laplace condition, in equations they have to satisfy

8z2 Ox2

throughout the fluid.

As can be seen, no further linearization can be applied at these equ3tions, opposite to the boundary conditions, which will be discussed in the following paragraphs.

3.1.1

The free-surface condition.

If We apply the linearization on equation (2.25) we get:

+2

(vi.

V) +

V (.

v) +

+VV(VV)+g+g =0

at z=Ç (3.4)

A complication with formula (2.25) is that the wave height Ç is unknown. We

therefore need a formula for the wave height. First we rewrite (2.24) out in the

potentials and .

(= -

((

-

U(r)x) +

+

_{+ V.}

-

((T

())2)

_{at z o}

(3.5)

Although we still don't know the wave height Ç we can now expand equation (3.4) in a Taylor series around z =. O. This also implies that Ç has to be small which coincides with our assumptions that both q and stay small.

The result of this operation is split up in a slow and a fast oscillating I)art:

The slow oscillating part gives us:

(T(r)x) + 2

(vt.

4:_ U(r))

±

and

3.1.

_{Subdividing the velocity ptential}

₁₆

This equation holds for all values of x and t.

s For the fast oscillating part of C we have the following:

2(v. v)

V (vi.

) + v.v (v .v) +

+j

+ g-

(v. V

(f2 (r))

(o

+

---(--(J(r)x) {ô::z

ô2

₍

-ôz2kt±

-atz=O. ô

i (ôq

ô

\ Iô3

a2\

+g - -

-

-U(r)x)

+g-)

O

at z =0 (3.6)

As obvious the slowly oscillating part consists only of terms with

aid U(r).

1f we separate from this equation the terms of order O (i°) from the terms

with higher orders of (we defined í as a small parameter) we get:

ôçb

at z 0 (3.7)

In this last e uation we havè úsed (3.7) to apply further simplification.

The last equation can be written out to

Ç6tt + g

+ 2xrt +

_{+ xx±}

+t -

((J2()

_-

2) (

t)

+ +2

-

1 ( (

- U(r)x)) (+

g ± 2

±

-

-

X)) =

O for z 0 (3.9)

In the equation al)ove. subscripts denote the 1)artial derivatives. We will _{use these} from now on often to keep the equations small and hopefully simpler.

If we compare this free-surface condition to the. condition which, both Sierevogel and

rins used for their calculations, we notice that the' use a condition which contains only the first twO [ines of the equation above. This is a direct consequence of the fact

that they assumed the Velocity of the object (or the current) to he constant.Notice that if we would use that same assumptioi here, it Would cause the double body

potential t) to be time-independent as well and we would get the same

free-surface boundary condition as the' USe(l.

0 (3.8)

When we analyze the order of the 'extra' tetms (with respect to Prins' and Sierevo-gel's models) in the condition in the small variable and the velocity (J we get the

3.1.

Subdividing the vçloçity potential

17

following small table:

¿)x8t a ( (

- U(r)x))

( (

- U(r)x)) (Pa2at

a 'z

(J! \.

\\ (a

¿32 'at Y T1X11

_\aatawz

ax2at1

To see if the extra time-dependent terms (the current) in the model have a significant

effect on our calculations, it should be enough to inclùde only the largest terms in the numerical model. Therefore we shall omit the terms of order O (zU2) and of order O(u2U2). This reduces our free-surface condition to:

(3. 10)

for z = 0 (3.11)

Equation (3.11) will be used as the free-surface cohdition in the numerical model, which is discussed in the next chapter.

3.1.2

The bodyboundary condition

As is stated before, at the hull of the object we have the condition that the normal

velocity of the fluid equals the normal velocity of the Ol)jeCt:

¿N) -. -.

Vn

dn

g = + X (

-'-at 7-t (3.12)

The slowly-oscillating part of the potential () was defined as the potential which is caused by the flow of the fluid along the object while the object is fixed in an

undisturbed situation If this is the case then it is obvious that the velocit of the object V is zero so for the Potential we have:

ôn at 7-L (3.13)

Furthermore in Newman [2] it is showed that the condition (3.12) can be linearized

to:

Dn = V)

v]

.g at (3.14)

In the last equation is the total displacement vector, given by definition by:

(3. 15)

ct ± Y& + ± + +

((12(T)

)

+

3.2. The dotible-body potential

18

1f we would restrain the object from moving, this becOmes with (3.2) ()diff inc

ôn -

¿in

32 The double-body potential

In the previous sections we discussed the boundary conditions. For the object bound-ary we saw that the normal derivatives of the slowly-oscillating potential equal zero.

This implies that, according to our linearization no waves are created by the object With thi cve can. approximate the slowly-oscillating potential with the double-body

potential This is the potential which we get in the situation where we take the com putational domain the normal domain extended with its mirror image in the plane

z O. In other words we look at the case in Which the free surface is approximated

by a rigid wall, i.e;, (x, O, T) = O, and in which there is no normal velocity at the

cylinder.

In our problem the potential then becomes:

/

R2

= db(,t) =

(J()x

+ _{2 + 2}

Ê

= f pdS

(3.17)

(3. 18)

In this equation R still denotes the radius of the cylinder. li fact this potential can be split up in the steady oscillating potential due to the current plus a disturbance potential caused by the object in the fluid (o U(r)x +

In appendix A it is proven that at small distance from the object, this approximation

is acceptable.

3.3

The forces

In the previous chapter we gave the common formula to calculate the forces. which are working oh the object:

(3. 19)

in which f is the translation and & the rotational movement of the object relative

to the center of gravity of the object ().

However, because we restricted the object from rolling, the second term in (3.15) disappears. We therefore get as linearized boundary condition

3.3. The forces

₁₉

Where the pressure was given by

p = p

((

- U(i-)x) + (V .

-

(12(T)) +

gz)

The formula above is exact. However, we can't calculate it exact beëause we don't

know the exact position of the surface 7-t. A second difficulty with the integral (2.29)

is that the length of the wetted surface, _{over which there lias to be integrated, is}

also unknown.

A third difficulty lies in the fact that _{we have used linearizations in the previous}

chapter, which has as a consequence that the quantities are only known at an average position.

3.3.1

_{Approximating the force}

To solve these problems _{we have to use an approximation for the integral. In this} approximation we follow the method which was used by Prins [4], who extended and

improved a method outlitied by Pinkster [3].

Because we assumed the fast oscillating _{part to be small and because the slow}

oscillating only varies slowly in time, _{we assume the movements of the object to be}

small. With this assumption we now expand the pressure at the actual position of the object into a Taylor series around the. average position fl0:

= p10 + 5

p0 +

(s.

) p0 + o (ii)

Furthermore we estimate the size of the _{actual wetted surface by the average surface} plus an oscillatory disturbance of the average surface. This gives us for any function

f(.) the following equation:

ff(1S=ff()d.c+ f f(2)dS

(3.22)

7-io líosc

We approximate the integral over the oscillatory wetted surface

at 7-t (3.20)

now by: (3.21)

In the right hand term above, 'wI' is an abbreviation of 'water line'.

To evaluate the integrals resulting from the _{estimates that we made, we assume that}

all the quantities can be written in a perturbation series:

f

)dS

f]

f()dzdl

_(3.23)

3.3. The forces

20

= p

+ ± O (E3)

Î

=

Î1

+

E2XH + o (E3)

=

+ E22 + O

(3)

= ₊ X +

E22)

x

+ o

(f3)

F = P+P(1)+E2F(2)+o(E3)

where the term with the overbar denotes the static value l-Ïowever in our case that would mean that all first terths equal zero because all the variables in our model are time dependent. Because we assumed the slow oscillation to be of small frequency ₍ is chosen small), we join these terms of order¡t in the first terms of the perturbations

above.

The perturbation series in are written in the common form. For our model the

object is restricted from rotating, which implies that all terms containing Il can be

removed. This leaves for the normal for example oniy the term ii.

Notice also the difference between

_{and Î:}

denotes the coordinates of a point in the earth-bound coordinatesystem

X is the displacement vector relative to the gravitational center of our object

If we substitute the perturbation series in the Taylor linearization which we

per-formed on the pressure (equation (3.21)), and collect the equal powers of E, we

get:

pl.' = 1

p(l)

= ₊

Î'

. = (2)

₊

Î2

_{. V73 +}

Î1 Vp

₊

(îw v)2

5 (3.24) (3.25) (3.26)

In order to he able to calculate the terms p, (') and (2) _{we first have to linearize} the exact equation for the pressure (2.30) in the same way as we did: in the previous chapter. This gives Us:

p = p (

- U(r)x) +

+

! (vi.

(J2( _at

(3.27)

As with the wave height, p represents the slowly varying pressure which is caused by the slow oscillating velocity. So we gt for the components of the pressure. se ries

on the average wetted surface:

p

(:

(

-

U(r)x) + (V.

(J2(r)) + (:3.28)

1)

= _ (

(1)

+

3.4. The. added

mass and damping coefficients

₂₁ where (2) = Fhydro =

f jidl =

(

O

\ pg5

Now that we have found expressions _{for the various orders of the pressure term we}

can substitute these in the integral equation for the _{forces, equation (2.29). This}

way we get a power series for the force in . The first term in the series consists _of

two parts, a hydrostatic part whicjh loo! s like

and a slow oscillating, hydrodynainic_{part which is caused by the first term in(3.28).}

Although equation (3.28) looks fine, it's incorrect. The double body approximation which we use, cannot include wave occurrences. This is caused by the condition

= 0 at z

O which we used. _{Therefore we will leave this force term, which}

contains these corrupted terms, out.

3.4

_{The added mass and damping coefflcients}

For the first order term of the force _{we get the following expression:}

F'

=

f (2) dS

=

-f

(vi.

v1

Ô)

+ .

y)

v+

71

+Î

(

(

-

(J.(r)x)))

(1S (3.30) (3.31) (3.32)

[f looking closer at the order of the terms in (3.32) it _{can be seen that in fact the} first order term of the force can be written in a form ₌

_F1

_{+ where the}

"n-term" is caused by the factor _{( - U(-)x). In our model we will leave this}

term of order jt out for the first order _{term of the force.This gives}

(1)

=

-f (v

V(1) +

idS

(3.33)

From Newtons _{second law it follows that the system of forces can be written as}

A - added mass matrix

B- - damping matrix

M -

mass matrix

R

- matrix of restoring coefficients

The matrix R contains the restoring coefficients, and is therefOre build up by the zero-speed restoring coefficients. The term v (x .

y) V

in (3.32) is proportional to the movement of the object and can therefore ajso be seen as a restoring coefficient.

Therefore R is also partly built up of this term.

The added mass and damping matrices, which represent the dependency of the force

on the motion of the object, can now be calculated by considering a model ii which incoming waves are omitted and in which the object undergoes a forced oscillation.

The added mass and damping matrices _{can then be calculated by fitting the first} two terms of the force equatioì (3.32) to the acceleration (the other terms were put in R) and the velocity:

_f

₍

+ v.

v)

udS =

+

By-'

(3.35)

Here however, we encounter another complication in. our model. Because we modeled

the potential _{to be slow-oscillating, this causes a slow-oscillating factor- in the}

"constant" terms A and B (in fact these matrices do depend on the used incoming frequency w). In fact this would result in coefficients which are not only dependent on the freqüencies w and i but also on combinations of those like w +

_{j and w - ji.}

Therefore we will use a velocity U(r) of zero value to calculate added mass and

damping coefficients.

When discussing the results, we will corné back to this issue.

3.5

The drift forces

After we have calculted the movement of the object and the potential due to this

motion (and from (3.2)) our total fast oscillation potential is known, _{so we can}

calculate the drift forces. We have to look at second-order forces to calculate the

([rift .forces acting on the object.

1f we continue now with the Perturbation series for the forces, we find the second

order forces to be

-F

=

_p/ (vm.

+ 2))

udS+

3.5. The drift forces

23

_pf{v.

v

+ î. y

()

+ v. ve( +

(

_U(T)x))}dS

+

-f

(1)

y

{v (î

.

y) v

+ î

. y (

(

-

U(r)x))

} (3.36)

The first integral term in (3.36) can be further simplified by making lise of a pettur-.

bation series for the wave height ((2, t). The second order term in this perturbation

(2) ₍₂ . r

series namely equals

+ Vq V

(derived witb the aid of equation, (.L)).. As

a last simplification we will omit the last term in the integral, which is of order o (x), also because the displacement of the object stays small.

This gives us the following drift force: 2 C=R,z0

=

pg{}'

-

p/ {î V

(

+ V. V)

(:3.7)

where (r is the linearized relative wave height, which can be calculated by Bernoulli's

equation and the movement of the hull.

When we have computed these forces we will examine if the drift forces also contain

a slow-oscillating compoûent. This is done by fitting the oùtcome of the drift forces over two slow-oscillation periods in time at the function A1 sin(it) + A2 cos(jit) As with the actual forces we can distinguish an added mass and a damping compo-nent in the drift forces. This can also be done for the slow-oscillating terii in the drift forces which leads to au equation of the form

ô2î

th2 +

B

= Aisiu(pt) + A2cos(t)

here B denotes the drift-damping term, X, Is the slow-oscillating movement and A. denotes tile drift added mass term.

Because we already know the actual slow-oscillating movement of the object, we can

nOW simply cälculate the drift added WSS and drift damping terms. In the iinoclel we chose tile velocity U() = (J0 cos(at) which then gives

(2)

A = --A1

(3.38)

3.6. Summary of the linearized model

24

Summary óf the linearized model

For the forces we have the; following re1atonships:

F1

= pf(V.Vç±)iZdS

= M

+ A +

+ RÎ

The. double body potential

Incoming, diffracted and

movement potential Laplace

Ê2

pg{}RO

pf{î.v(+v.v)

±VVth+ (î .v) (v(î .v)v)}.is

Equation of motion (3.40) = O Total driftforce (3.41)

With the aid of these linearized relationships we will now create a numerical model in order to calculate some drift forces. This will be discussed in the next chapter. As in the previous chapter, we end this chapter with a summary of the model which

we have obtained so far.

For the potential we have the following relationships:

,t) = db(,t)

=

(J(r)(1

± ,t) t)+ djff(X, t) + q10(vx,t) O2 _a2

-

r3x2

-

x2

-

_z=0

+ gçb + + + +

+rxt

(U2()

-

_{) (&}

+

+2rtr -

( (

-

U(r)x)) (&

- gth) = O

z=O

Th

at fl

n -Î . (n . V)

at 7-1 Radiation conditions...

-Pf

d.5 = A frced.

-- Forced oscillation with

Chapter 4

The

nun.i

crica! model

Now that we have applied the necessary linearizations at our model. we will create

an iteration scheme, in order to write a computer program to do the extensive

calculating. The iteration schemes for both models which we discuss in this report. are almost the same. Because the formulas fr the model without an adjusted free-surface condition are equal to the formulas derived by Prins{4] and Sierevogel[6] we will only discuss the derivation of the iteration formulas for the model with the adjusted iteration scheme.

T1e main idea is to split up the total iteration time into small intervals and solve on such an interval the models for the potentials and the forces like

Give potentials initiation values

Solve the potentials at time level ì with the aid of the value of the potential on previous time levels

Solve the forces at time level, t with the found values of the potential increase the time level to t + t and return to step 2

For solving the potentials 'at a certain time step, we yill adept the method clevel oped by 1-lermans 'and Prins[4], but with the improved artificial boundar condition developed by Hermans and Siereogel[6] Because of the importance of tius method in oUr numerical model, it will be discussed in the next sections. After this we will discuss the calculation of the' forces. We conclude this chapter with the nïote detailed versions of the iteration schemes which are used in the computer programs.

4.1.

Iteration relationship for the potential

26

4.1

Iteration relationship for the potential

The method to solve the potential at a time step t + t is mainly based on solving thè formula given by Green's second theorem:

8«,t)

₌

j

(

t)vxi) Gt))

dS (4.1) as with I i

ES

I O elsewhere (4.2)

where ÔS denotes the boundary of the fluid domain S.

Because all the values we want to calculate are determined by the values at the

boundaries, we will only consider values of

at the boundaries. We divide the

boundary into intervals and assume that oli suth a interval the quantities are

constant. The collocation points are given by the midpoints of the intervals. This way we can write (4.1) as a set of linear equations:

D(t) = E(t)

(4.3)

However Prins [4] discovered that this couldn't be solved' in a straight time-iteratiOn,

fòr every size of the time step the scheme was unstable. Therefore he developed a

different iteration scheme which is based on sub'stituting th given boundary condii

tions for in (4.3).

Sierevogel improved Pçins' method by using a different boundary condition which speeded up the calculations. In this report we will follow Sierevogel's approach,

which consists of dividing lip the fluid domain into an interior and an exterior

domain.

4,1.1

Splitting up the domain

The dividing of the infinite fluid domain in an interior and an exterior, is shown in the next figure (4.1).

In the following sectioiis the different parts of the fluid domain will be treated.

Numerical model for the interior

For the interior S we lise the model which was constructed iii the previous chapter,

so we look for a function t) which satisfies the Laplace equation

4.1.

Iteration relationship for the potential

27

Figure 4.1: The geometry for the numerical model

In figure 4.1 we used the following symbols and characters:

7-1 - the hull boundary

S - the interior domain

V - the exterior domain

13 - the artificial boundaries

f

- the free-surface boundary b - the bottom boundary

and which also satisfies the free-surface condition which we constructed:

tt +g + _{+ + +} 1

-

(U2(r)

-

₎

_(xx

-

+

2

_!

(

-

(J(T)x))

-

xx) =

Furthermore the potential lias to fulfill:

= O

for z = h

(4.6) and

=i

n.(n.V)V

V

z=O

h

for z = 0 (4.5) at 7i (4.7)

Instead of the radiation conditions which we mentioned when the problem vas for-rnulated, we construct a different boundary condition at B in a later paragraph.

To solve the interior problem,as already stated. ve make use of Green's second theorem. Because we restrict ourselves to calculating oniy values of the potential on the boundaries this, if we have a Green's function which satisfies equations (4.4)

and (4.3), becomes

o(t)

=

4.1.

Iteration relationship for the potential

28

where ÔS is the boundary of the domain.

The Green's function G(,() we use now is giien by

-.-.. i

-

i

G(x,) = lnr+--Inr2

2ir 2ir

in which F

₌

-

and F2 =

-

with the image of with respect, to th,e bottom.

We cliscretise the integral equation (4.8) by dividing the boundary into intervals and

assuming the quantities to be constant on such an interval. The collocation pOints are the. midpoints of these ittervals. This allows us to rewrite the equation into a

form like

D&(t) = E&(t)

(4.10)

which represents a set of linear equations atid where T

is a vector (I).

Equation (4.10) stilI contains two unknown variables at each boundary, the free surface, the hull and the outer boundary 8. namely and . The method Prins

developed now consists of substituting tbe boundary conditions in (4.10). We follow

his method for all boundaries except for the boundary 8. This is worked out for

the free-surface condition in appendix C, for the other boundaries the method is analogue and gives us a matrix system of the form

D141

= D2

+ D_1 ± f+i

(4.11)

.l.stead of giving a condition on the boundary 8 (like the Sommerfeld condition) we take an exterior domain into account to complete our system of eqiations. This exterior domain is the subject of our next section.

Numerical model 'for the exterior

For the exterior we construct a different model than we used for the interior.. OJ)-posite to the interior we take the exterior domain fixed to earth. Thi implies that the interior (together With the object) moves with a velocity ¿Y(T) in the negative X-direction. This gives us a geometry which is showed in the next figure.

In the exterior ve neglect the influence of the ol)ject (* çj O) on the potential.

Because there also exists no current in this model, the slow oscillating potential

reduces to zero ( O).

This leaves our linearized free surface condition to. the following:

(4.9)

4.1.

Iteration relationship for the potential

29 U(r)

U(r)Lt

av

ul__ ÖS at t

ÔS at t + ¿t

Figure 4.2: The geometry for the exterior domain

To solve this exterior problem, we introduce t-lie Green's function, which satisfies the next model

at z =0

V2G=0

G=0

atz=h

= G = O limr -* oo

D

(4.13)

The factor is a result of the discretisation of the free surface condition (4.12)

to

Z,+1 +

_g(t)2

_{= g(t)2(2}

-

at z = 0 (4.14)

The last equatIon in the model (4.13)is the physical radiation condition. Again by using Green's theorem, we get the following potential in the exterior

=

¡

ôG()

dz

-where I i

xEVUÖDj

8=

(4.16) I O elsewhere

and V1 is the free surface of the exterior. Sierevogel and Hermans derived the

following Green's function which satisfies (4.13)

G(if)

= lnr+lnr2-

_2-i I 27r i 7e

((kd)coshk(z+h)coshk((+h)cosk(x)

+1

dk 7rJ k ksinh.kh+[lc'oslìkh

₎

(-J

g(zt)2

f G(,) (2() -

dx Df (4.15)

4.1.

Iteration relationship for the potential

30

(4.17)

= -

_{rn hm+}

/3 mk(z + h) cos Plk(Ç +

h)en

(4.18) with ¿ ₌

(,Ç)

and ß ₌ and mk imaginary parts of the purely imaginary

poles. The sum equation (4.18) was derived bye a transformation in the complex plane, and taking the summation of the residues. The imaginary poles in the time domain bave to satisfy

imktanh(imkh)

_-ß

- _mktan(mkh) = 7? E R (4.19)

The integral formulatjon must be used for small values of

-

_¿I because the sum

equation does not converge.

By using Green's theorem We have to integrate the Green's function with respect to

or . This way we are able to write the potential in the exterior in the following

form:

= (4.20)

where is a vector and where is the last term of equation(4.15).

Combining the interior and exteriOr model

Because we use two different coordinate systems for the interior and the, exterior domaiñ, the tWo boundaries of the domains differ. 1f there was no current in our

original model, we could simply combine equations (4.10) and (4.20), to get an

overall matrix equation, like

Dl1

D2i4' + D3&_1 +

f+1 +

E (4.21) with ,T a. vector

(1I

_).

However to combine the to found matrix equations two one model, we have to

find a relationship between the potentials on the different side boundaries of each

domain. Because of the current 11(r) we get a differetce of

Ut

between tbe 'interior

and exterior boundary for each step in time(At). This 'is also shown in figure 4.3. For the figure above, the same variables are used as in figure 4.2. Notice by the way that the width of ÔSt+L

equal is to (ist, the distance that is traveled after one

time-step.

If we consider the domain between the boundaries ß and B5 after a time-step t,

we can apply Green's secònd' theorem in the same way as before. We consider only the left boundary of the interior domain for now, the right l)oundary' goes analogue.

Applying Green's theorem, with the same Green's function as was used for the

exterior domain, gives us

=

f

(aG

_c±)

_{d+/3 f Gd}

+

f

_

G4)

thi d (4.22)

4.2.

Numerical model for the added mass and damping coefficients

31

If we take the potential q at the

non-cliscretised version of equation

[(aG

= I

(---¿

J \Ofl

13$ Ss

ast+t

4 I Df I

S at t

ÖS at t + t ÔD

boundary 8 (.

= ), and extract now the

(4.15) from (4.22), we get

_c±)

d+/

f

Gd

(4.23)

This gives us the last needed relationship between and For the right

bound-ary of the interior domain, we can construct a similar relationship . However for

the left boundary, in (4.22) must be taken equal to the potential of the exterior domain, while for the right boundary it must be taken equal to the potential of the interior. This can also be seen from figure 4.3. if you look at where the free surface of the exterior domain (Df) stops.

So now the potential in our model are completely defined: after computing the potential in the interior and the artificial boundary 5, we can compute with the aid of equation (4.15) the potential at the free-surface of the exterior.

Now that we have constructed a relationship to calculate the potential at a time

step t we will look in more detail at the way to obtain the forces. For this purpose however, first the added mass and damping coefficients must he calculated.

4.2

Numerical model for the added mass and

damp-ing coefficients

For our final goal, calculating the drift forces. ve need to have the added mass

and damping coefficients. As was mentioned in the previous chapter, these can

(a) left (b) right

Figure 4.3: The shifted boundaries after a time-step t

4.2. Numerical model for the added mass and damping coefficients

32

he obtained by giving the object a forced pure sinusoidal movement with a known

frequency.

If we solve the potential at a time level t with the aid of the found matrix system

(4.21). we can than calculate the added mass and damping forçes by fitting the

forces to the acceleration and velocity of the object:

!9q.

ô2Î

_¿it

JW7id5=Aa2 ±B--

(4.24)

Because the fitting is doue over a period of time, the calculation of the added mass and clamping coefficients is done in a different program than where the drift fôrces

are calculated. In fact there are two programs, Heave and Sway, which calculate the added mass and damping coefficients for the horizontal and vertical movement

respectively.

They both use the following iteration scheme:

Sèt the initiations values for t = Create a mesh

Calculate the potèntial by solving matrix system (4.21) Calculate the forces

5 Inctease the time level with t and return to step 3 until the maximal time

level is reached (self chosen)

Fit the forces at each time level to the acceleration and velocity of the object at each time level

End.

After the added rnass and damping coefficients are known, we have all the necessary

data to calculate the drift forces.

4.2.1

Numerical model for the drift forces

When calculating the drift forces, we consider a model in which the object is free to

move in all directions. So instead of the case where we already know the movenent of the object, like when calculating the added mass and damping coefficients, ve now have to calculate the position of the ol)ject at each time level.

4.2. Numerical model for the added mass and damping coefficients

33

Set initiation values, t 0,

₌

= O

Create the mesh

Update the speed U(T), the double-body potential for the current time level Solve the diffracted potential by solving the matrix system (4.21)

Calculate the linear forces with the incoming and the diffracted potential Calculate the movement due to the linear forces which act on the object, e.g.

solve

ô2Î

aî

M

_ôt2 + A

ôt2 +

B_- +

= F

Calculate the potential due to the movement of the object the total

potential is now known

Calculate the drift forces at this time level

1ftmax is not reached yet, increase the time level with t aid return to step 3

Calcuiate the drift clamping and drift added mass coefficients

End.

Notice that in the scheme we make use of the separation of the potential into

+ ifl + Conseqùebtly, when we solve the diffractioñ potential in step 4,

we make use of a slightly adjusted model. This is due to the fact that at that point we restrain the object from moving 0).

The boundary conditions in this situation then become:

22diff

-f)j

--o + g

+ + + 5 + th (U2(r) _{) (c} + (th ( -

U(r)x))

-'- g(1;)' = O with qS ₌

_{+ diff}

{

at N

(4.25)

Of course there's also a boundary condition at the bottom (-- = O at z = h),

but l)ecause we choose the bottom relatively far aiay from the object, this boundary

Chapter 5

Results

In this chapter we will discuss the outcome of the various computer cakulations

which were done according to the models discussed previously First however we will give the relationship between the different program modules which we use.

51 The computer programs

With the aid of the iteration schemes, which were discussed in the previous

chapter, we created a software pro

gram. In fact I (only) had to adjust the programs' which were already written

by Prins and adjusted l)y Sierevogel.

A can be seen in the figùre aside, the

total program consists of three

subpro-grams:

Heave , which calculates the

added nass and clamping coeffi-cients in the vertical direction,

Sway, which calculates the added mass and damping coefficients in

the horizontal direction, and

Drift

, which calculates the

ac-tual drift forces with making use

of the outcome of Sway and Heave Figure 5.1: Relationship between

Héave Sway and Drift

'Which strengthened my preference for writing my own prograrris from scratch

5.2. The added mass and daniping coefficients

35

In the next paragraphs we'll discuss the input and outcome of the three progiams.

The programs5 by the Way, were written in the software language FORTRAN.

5.2

The added mass arid damping coefficients

Because the base structure of the programs Heave and Sway are the same (in fact they share a lot of files which perform the main calculations), we discuss them in one paragraph. Therefore if iiì the text in this paragraph. is spoken about "the

program", both programs are meant.

As already mentioned we calculate the added mass and damping coefficients for a. circular object which undergoes an oscillatory movemeùt with a frequency wo. This is done by giving the object at t = O an impulse so the object gets a motion

in the horizontal plane X which behaves as (a sin(wot)( i - e)3, a fast clamping

oscillation.

As input values in the programs we used the following

As

an be seen the depth of the domain does not equal oc. Implementing an

infinite depth can't be clone (yet), however Sierevogel showed in her report[6] that the depth taken here is large enough to avoid any influence from reflected waves.

The length of the time integration was chosen four periods in the frecuency wo which

were divided into 200 time steps. Only over the last two periods the actual fitting of the added mass and clamping coefficients was done, to evade any distortion caused

by the initialization of the system. For the interior a length of one wavelength \

was taken where K IS given by 2 wo

Ktan(th) =

-g (5.1) (5.2)

Symbol Meaning Value

w0 _{Frequency of the incoming waves}

3,..., 10

w Frequency of encounter at the object w0 + tJJ

- -.

-Wave number . Wz

U Velocity of the incoming field 0, 0.025, 0M5, 0.075 and 0.1

g Gravity constant 9,81

h Depth of the domain 6H R Radius of the object - d/2 = O 0 d Diameter of the object 0.1

Î

Displacement of the obje t ((a sin(wot) (1

_ct) 0)

5.2. The added mass and damping coefficients

36

For the exterior we chose a length of waves to return to the boundary within

This gave the plots as showed in figures by using w' ₌

_woVf.

0.8 0.7 0.3 0.2 0.1 O O 0.5 10.4 0.3 02 0.I 0.2

(a) Added mass

IS

Figure 5.2: Added mass and damping in heave

0.5

(a) Added mass

15

wavelengths Which is long enough to avoid

the total integration time..

5.2 to 5;5, where w was made dimensionless

04 0.35 25 0.15 0.l 0.05 (b) Damping 0.5 (b) Damping 15

Figure 5.3: Açldecl mas and damping iù sway

As cän be seen , the choice of value fo U does not inflüence the added mass an

damping coefficients very much for (J 0.1. However if we look at the coupled

acldel mass and damping coefficients. the value of U has more influence on the

outcome.

Although the value of these coefficients are a ordeç 10 smaller than the added mass and damping coefficients, this makes our decision to use the calculated coefficients

0.01 -0.01 1_002 N -0.04 -0.05. -0.06. -0.07 ..0.08 0.5 0.2 0I

-

U = O

U=0.025

---(1=0.05

(1=0.075 (1 = 0.1 0.5 w,

(a) Added mass (b) Damping

Figure 5.5: Coupled added mass and clamping in sway

for U = O in the program Drift discutable. This couic! be altered in a future version of Drift by updating the added mass and clamping coefficients as the velocity U changes in time. However these are optional future improvements3, in our current

version of the program ve use constant values for the added mass and clamping

coefficients .N otice by the way that especially for smaller values of w' (w' 1) the values of the coupled coefficients differ from the one for U = 0.

I5

i have already prepared the program for this updating, but due to lack of time I haven't been able to run and test these modifications

(a) Added mass (b) Damping

Figure 5.4: Coupled added mass and damping in heave

5.2. The added mass and damping coefficients

37

0.5

wii.

'5 0-s w, IS 0.07 0.06 :.:

o

0.02 0.01 o -.0.01 -0.02 0.5 w,