experimental and numerical methods for the prediction of wave loads in extreme sea conditions

7/31/2019 Experimental And Numerical Methods For The Prediction Of Wave Loads In Extreme Sea Conditions

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D. I. N A V.Dipartimento di Ingegneria NAVale

Academic Year 2010/2011

Experimental and Numerical Methods for the

Evaluation of Wave Loads on Offshore Platforms in

Extreme Sea Conditions.

Supervisor

Prof. Ing. Stefano Brizzolara

Co-supervisors

Prof. Ing. Claire De Marco - University of Malta

Prof. Ing. Tonio Sant - University of Malta

Candidate: Umberto Ghisaura


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Ad Angelina, Giancarlo, Stefano e Riccardo


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Experimental and Numerical Methods for the Evaluation of Wave Loads on Offshore

Platforms in Extreme Sea Conditions

Umberto Ghisaura - Universita degli Studi di GenovaII

Conservative form of the governing equations of fluid flow 52

Differential and integral forms of the general transport equations 52

THE FINITE VOLUME METHOD 55

Step 1: Grid generation 56

Step 2: Discretization 56Step 3: Solution of equations 58

MAIN FEATURES OF THE MESH 59

Physic models 59

Control Volume and Mesh refinement 60

WAVES REPRESENTATION 62

Free Surface Definition and VoF Theory 62

Progressive Waves 63

COMPARISON AND ANALYSIS OF DATA 65Comparison between simulations and tank tests 65

Spline Interpolation of the height. 75

Analysis of experimental non dimensionalized data. 77

Vertical Force 77

Horizontal Force 81

Pitch Moment 87

CONCLUSIONS AND FUTURE DEVELOPMENTS 91

LEGEND 93

REFERENCES 94


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Umberto Ghisaura - Universita degli Studi di GenovaIII

TABLE OF FIGURES

Figure 1: Wave Theories Graph 25

Figure 2: Generated Wave Order 26

Figure 3: Wave Sampling - Wave Length vs. Period 27

Figure 4: Wave Sampling - Wave Height vs. Period 28

Figure 5: Wave Sampling - H/L vs. Period 29

Figure 6: Wave Sampling - Wave Height vs. Crank Length 30

Figure 10: KLC1 Calibration 39



Figure 13: Sawtooth test 41


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Umberto Ghisaura - Universita degli Studi di Genova1

ABSTRACT

Experimental and Numerical Analysis of the Wave

Loads on a Floating Wind Turbine Platform in

Extreme Sea Conditions

The aim of the research was to characterise the hydrodynamic surge, heave and pitch forces

of a floating wind turbine semi-submerged tension leg platform. A series of experimental

measurements on a 1/100th

scaled model were undertaken and the results were compared

with those obtained from a numerical RANSE model. Relevance and innovation consisted inusing a model strictly close to reality during experimental tests, and in using non-linear

waves both in the wave tank and with the state-of-the-art RANSE solver.

The geometry of the structure consisted of four cylindrical bodies joined to a central large

cylinder (5 meters diameter, 20 meters length) which supported the wind turbine tower by

means of eight tubular connections (3.3 meters diameter, 12.3 meters length). Each body was

composed of one cylinder (7.8 meters diameter, 20 meters length) with a further larger

cylindrical body (9 meters diameter, 7.8 height) to increase the stability.

The study concentrated on the prediction of non-linear hydrodynamic forces acting on the

complex structure. The scaled model was tested in a water wave maker in a fixed condition.Three separate load cells were used to measure the forces and moment components.

A preliminary study was dedicated to the verification of the second and third order Stokes

waves obtained with the wave maker by a sampling of the waves measured in the tank for

each of the reported test condition. The period of the water waves ranged between 6 seconds

and 1.15 seconds, while wave height varied between 3.37 to 10.7 meters.

The numerical RANSE model was based on a special non-structured (trimmed type) mesh

grid with anisotropic refinement in the three cardinal directions to ensure a good convergence

property with a limited expense in terms of total number of cells, while representing the

various geometrical details of the structure (such as the connections and the cylinders).

Experimental results were compared in a scientific and non-dimensionalized graphic form

with RANSE values versus the wave slope (k*H) and the Keulegan-Carpenter number (Kc).

Finally, a common engineering law for predicting these forces was identified.


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INTRODUCTION

Survey of word offshore wind energy plants

Europe is the world leader in offshore wind power, with the first offshore wind farm being

installed in Denmark in 1991. In 2008, offshore wind power contributed 0.8 GW of the total

28 GW of wind power capacity constructed that year. By October 2009, 26 offshore wind

farms had been constructed in Europe with an average rated capacity of 76 MW, and as of2010 the United Kingdom has by far the largest capacity of offshore wind farms with 1.3

GW, more than the rest of the world combined at 1.1 GW. The UK is followed by Denmark

(854 MW), The Netherlands (249 MW), Belgium (195 MW), Sweden (164 MW), Germany

(92 MW), Ireland (25 MW), Finland (26 MW) and Norway with 2.3 MW. Based on current

orders, BTM, an independent consultancy company specializing in services pertaining to

renewable energy, expects 15 GW more between 2010 and 2014.

Main features of the current offshore plants

Current offshore installations are made by

high-power units, which allow to exploit

better the best offer from the offshore

wind resource.

The location at sea has the advantage of

better wind resources and therefore an

higher production of energy. Furthermore

it yields less turbulences into the wind and

consequently a better durability of

mechanical parts. It is not tooverlook the higher availability of

sites, since the on-shore ones are

subjected to saturation, even for

the difficult acceptance by the

involved people in the installation

areas.

On the other hand, there is a

complex set of static and dynamic

forces on the structure and on the


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turbine which increase the difficulty of the challenge. Sea corrosion, higher wind resources,

ice on northern seas, far distances from the ashore which means using relevant electrical

equipment are the difficulties you find at the planning stage.

During the installation on site, there are procedures of transport, assembly and ready for use

settings much more difficult and different than onshore ones; thus times, expenses and

dimensions are in a different scale.

Studying floating structures, for the exploitation of wind energy, rise from the will of

reducing the environmental impact with the seabed and the needing to move the platform in

different places. The structure is done by two essential components: the floating platform

with its anchoring system and the wind turbine rigidly bound to the structure. Many kinds ofstructures have been studied:

-Semisub Dutch tri-floater

-Barge

-Spar-buoy with 2 tiers of guy-wire

-3-arm mono-hull

-Concrete TLP with gravity anchor

-Deep water spart

Recently a concept

wind farm made by a

unique huge floating

structure which brings

multiple wind turbines

has been developed;

this would improve the

stability of the whole

system and easier

accessibility for the

installation and the

maintenance.


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DEFINITION OF THE PROBLEM

The idea which gives rise to the joint work of the two Universities is to evaluate the wave

loads of an offshore platform in extreme non-linear conditions. In this project the University

of Malta offered the wave tank and its competence in the construction of models, while the

Universita` degli Studi di Genova shared its proficiency in solving such kind of

hydrodynamic problems both in an experimental and numerical way.

The wave tank is made up of glass and its dimensions are L1* W1* H1=8m*0.75m*1m.The height of the water level from the seabed in calm water condition is D1=783 mm

Picture 1: Wave tank


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The wave tank consists of the following components:

-one wave generator made by a swinging panel hinged on the bottom of the tank as shown in

Picture 2: Swinging Panel

Picture 2: Swinging Panel


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-one frequency controlled electric motor which moves the panel by a crankshaft as shown in

Picture 3: Electric Motor

-two passive waves absorbers made of rolled up plastic nettings as shown in Picture 4 :

Wave Absorber

Picture 3: Electric Motor

Picture 4 : Wave Absorber


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The equipment for the measurement of

the forces is:

-3 load cells as shown in Picture 5:

Load Cell Setup

-2 data acquisition cards in order to

change the analogue value into digital

-1 analogue signal generator in order

to maintain a constant t during the

recording of the signal

-1 computer for data recording

The definition of the problem starts choosing:

-the typology of the structure

-the geometry of the platform

-the environmental conditions

Picture 5: Load Cell Setup

Picture 6 : Data Acquisition Equipment


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Typology of the structure

In order to exploit the traits

of the wave tank and theequipment at disposal we

decided to carry out fixed

body tests on a Semi-

submerged Quad-floater.

The model must be fixed to

the tank by an external

structure expressly built.

This kind of floating

system is anchored to the

seabed in reality with four

tension legs which mustremain in tension also when

the platform is in wave

trough, and preserve the

same distance as the

floating structure from the

bottom of the sea. Upper connections are semi-submerged in order to enhance the stability

with a larger inertia moment. Thus movements of heave, roll and pitch are blocked while

surge, sway and yaw are still present. The affinity between the degrees of freedom allowed

from the anchoring system and the nature of the measuring system permits us to assimilate

one to the other by approximation.

Tests will hive us only the wave load data, comprehensive of effects due to reflection on thewalls of the tank and the refraction on the body. They are the first step for a possible future

development as unfixed model tests.

Geometry of the platform

The main dimensions are determined by designing the semi-submerged platform which fits

the 3.6 MW wind turbine designed by General Electric and used for offshore plants. From

simple calculations about the GMT and the other Hydrodynamics requirements of theplatform, the following dimensions were decided taking into consideration the structural

requirements:

Overall length=L =36.25m

Overall width=W=36.25m

Main Cylinder Diameter = 7.5m

Main Cylinder Height= 20m

Stability Cylinder Diameter=9.0m

Stability Cylinder Height=7.6m

Connection Arm Diameter=3.4m

Connection Arm Length=12.9m=1150 t

Picture 7: Geometry of the Platform


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V=1122 m3

ZB=10.6 m

ZG=34.55 m

x=1.34 m

Aw=380 m2

Iy=47046 m4

RT=41.1 m

GMt=15.81 m

Environmental conditions

According with the need to design a structure which sustain bad weather and rough seas, we

decided to consult a document of the BMT Argoss - Wave Climate in order to decide the

main features of the waves approaching the structure. Table 1:Stochastic Mediterranean SeaWave Data refers to the distribution of significant wave heights versus mean periods,

measured with SAR technique in which the direction of propagation is not measured. It is

interesting to note that the highest height values (Hs= 5.2 m --- 8 m) are related with the

highest values of the mean period (P=8 s --- 11s).

lower 1 2 3 4 5 6 7 8 9 10 11

lower upper 2 3 4 5 6 7 8 9 10 11 12 total

0.1 0 0.2 0 0.008 0.871 1.547 0.338 0.06 0.006 0 0 0 0 2.829

0.3 0.2 0.4 0 0 1.599 4.414 2.214 0.314 0.055 0 0 0 0 8.596

0.5 0.4 0.6 0 0.002 2.17 4.968 3.664 0.942 0.082 0.021 0 0 0 11.849

0.7 0.6 0.8 0 0.002 1.793 5.708 4.051 1.365 0.141 0.017 0.002 0 0 13.079

0.9 0.8 1 0 0 0.541 5.649 4.111 1.58 0.187 0.008 0 0 0 12.076

1.1 1 1.2 0 0 0.023 4.178 3.902 1.608 0.227 0.004 0 0 0 9.942

1.3 1.2 1.4 0 0 0 2.537 3.424 1.481 0.282 0.021 0 0 0 7.7451.5 1.4 1.6 0 0 0 1.15 3.676 1.414 0.371 0 .021 0 0 0 6.632

1.7 1.6 1.8 0 0 0 0.291 3 .403 1.42 0.398 0.015 0 0 0 5.528

1.9 1.8 2 0 0 0 0.023 2.526 1.466 0.392 0.03 0.002 0 0 4.439

2.1 2 2.2 0 0 0 0 1.559 1.747 0.36 0.053 0 0 0 3.719

2.3 2.2 2.4 0 0 0 0 0.699 1.987 0.377 0.072 0.004 0 0 3.139

2.5 2.4 2.6 0 0 0 0 0.2 1.612 0.402 0.074 0.004 0 0 2.292

2.7 2.6 2.8 0 0 0 0 0.051 1.252 0.4 0.076 0.004 0 0 1.784

2.9 2.8 3 0 0 0 0 0.004 0.836 0.506 0.08 0.011 0 0 1.437

3.1 3 3.2 0 0 0 0 0.002 0.51 0.533 0.076 0 .021 0 0 1.142

3.3 3.2 3.4 0 0 0 0 0 0.232 0.52 0.072 0.011 0 0 0.836

3.5 3.4 3.6 0 0 0 0 0 0.072 0.48 0.099 0.01 0 0 0.66

3.7 3.6 3.8 0 0 0 0 0 0.015 0.402 0.118 0.01 0 0 0.544

3.9 3.8 4 0 0 0 0 0 0.008 0.324 0.112 0.008 0 0 0.451

4.1 4 4.2 0 0 0 0 0 0 0.168 0.141 0.002 0 0 0.31

4.3 4.2 4.4 0 0 0 0 0 0 0.118 0.166 0.008 0 0 0.291

4.5 4.4 4.6 0 0 0 0 0 0 0.04 0.12 0.015 0 0 0.175

4.7 4.6 4.8 0 0 0 0 0 0 0.013 0.124 0.01 0 0 0.147

4.9 4.8 5 0 0 0 0 0 0 0.004 0.108 0.019 0 0 0.131

5.1 5 5.2 0 0 0 0 0 0 0.004 0.027 0.017 0 0 0.048

5.3 5.2 5.4 0 0 0 0 0 0 0 0.046 0.013 0 0 0.059

5.5 5.4 5.6 0 0 0 0 0 0 0 0.013 0.013 0 0 0.027

5.7 5.6 5.8 0 0 0 0 0 0 0 0.015 0.008 0 0 0.023

5.9 5.8 6 0 0 0 0 0 0 0 0.008 0.015 0 0 0.023

6.1 6 6.2 0 0 0 0 0 0 0 0.004 0.01 0 0 0.013

6.3 6.2 6.4 0 0 0 0 0 0 0 0 0.013 0 0 0.013

6.5 6.4 6.6 0 0 0 0 0 0 0 0 0.008 0 0 0.008

6.7 6.6 6.8 0 0 0 0 0 0 0 0 0.006 0.004 0 0.01

6.9 6.8 7 0 0 0 0 0 0 0 0 0 0 0 0

7.1 7 7.2 0 0 0 0 0 0 0 0 0 0 0 0

7.3 7.2 7.4 0 0 0 0 0 0 0 0 0.002 0 0 0.002

7.5 7.4 7.6 0 0 0 0 0 0 0 0 0 0.002 0 0.002

7.7 7.6 7.8 0 0 0 0 0 0 0 0 0 0.002 0 0.002

7.9 7.8 8 0 0 0 0 0 0 0 0 0 0 0 0

0 0.011 6.996 30.466 33.823 19.922 6.788 1.742 0.244 0.008 0 100

Percentage of occurrence of wave height (m) in rows versus mean wave period (s) in columns

total

Copyright ARGOSS, October 2010Table 1:Stochastic Mediterranean Sea Wave Data


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The wave length, according with the linear wave theory, for deep water condition is strictly

related with the wave period by the dispersion relation:

So the range of wave lengths will be:

Summarizing we will take cue from this data as to generate our waves both in the tank and in

the simulation software

2

2gT

=

mT

T

mT

T

1882

)(*81.911

1002

)(*81.98

22

22

2

111

===

===

mm

sTs

mHm

W

S

188100

118

82.5


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MODEL CONSTRUCTION

Selection of Scale

From the main dimensions obtained and described in the previous chapter we need to define

a scale to represent the project we did with a model which has to fit into the wave tank. On

one hand we prefer to avoid by choosing a too big scale: it would mean small forces and an

accurate measurement would be very difficult; furthermore the representation of the physicalphenomena due to the fluid flow could be different from the reality. On the other hand

choosing a too small scale would invalidate our tests because of the small space around the

model, which would produce a great reflection phenomena of the waves on the tank walls.

Thus, according to the previous statements and considering the convenience of dealing with

an easy scale, we decide that the length scale will be:

Given the nature of the problem and the slow velocity of the water particles also when they

are on the wave crest, gravitational effects prevail over viscous ones. This predominance is

relevant enough for our model to be represented with the similitude of Froude and viscous

forces will be omitted.

According to the Similitude of Froude and considering the length scale stated we can write:

100

1==

R

M

L

L

where U is the velocity and a L a characteristic

length of the Model/Real objectM

M

M

R

RR

FrLg

U

Lg

UFr ===

**


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Finally we can determine the force scale trough equalising the Newton numbers of the

platform and of the model:

The wave field will also be influenced by the same scale , thus we will consider this aspect atthe time of generating waves.

Given the length scale, we can easily obtain the model dimensions:

L =36.25cm

W=36.25cm

Main Cylinder Diameter = 7.6cm

Main Cylinder Height= 20cm

Stability Cylinder Diameter=9.0cm

Stability Cylinder Height=7.8cm

Connection Arm Diameter=3.4cm

Connection Arm Length=12.9cmV=1122 cm

3

==

===

==

===

R

M

M

R

RM

RM

R

M

R

RR

M

MM

R

M

R

M

R

M

T

T

T

T

LT

TL

U

U

T

LU

T

LU

L

L

gL

gL

U

U

**

*

;

*

*

where is the time scale10

1==

3

22

1

424

24

24

2424

)(***

*

**

=

===

===

R

M

RR

MM

R

M

R

RR

R

MM

MM

F

F

TL

TL

F

F

NwTL

F

TL

FNw


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ZB=10.6 cm

Aw=380 m2

Iy=47046 m4

RT=41.1 m

Construction and junction materials

The material of construction are pipes of polypropylene (PP), easy to find in different

diameters and with good mechanical properties for machining. Its only problem is its smooth

surface, and because of this the PP has to be glued with a special bonder such as the

"LOCTITE 430 Super Bonder Instant Adhesive". It is a single component, general industrial

grade, cyanoacrylate adhesive with a low viscosity.

Machine working and assembly

Accurate workings can be done only by accurate machines and good technicians; in fact both

affect the precision of the work and represent the upper limit of the accuracy we can obtain.

The laboratory of the University of Malta is equipped with high precision machines, as

lathes, vertical drills and numerical control machine (NCM), but also with welding

equipment and all the necessary tools for working. The design of the model and all the other

components for the measurement of the forces has been discussed with the trained and

experienced workers of the University as to match our design with the needs of that working

procedure. Furthermore each machining had to be presented on its own drawing, approved bythe supervisor Eng. John Borg.

During the machining we found some difficulties with the PP. In fact the machines are made

to work with iron, steel and metal in general, so we had to reduce the velocity of machining

to decrease the heat due to friction and avoid melting the PP.

The assembly was initially made only with the bonder "LOCTITE 430 Super Bonder Instant

Adhesive" and using a sort of potting, a structure that was keeping the components blocked

and aligned on the right position. Then after a failed hardness test performed by the Prof.

Eng. Tonio Sant we decided to improve the model with few layers of fibreglass.

Construction tables and drawings are into the drawing appendix.


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WAVE TANK

The first time we worked on the wave tank we focused on the features of the waves we were

able to generate and on their quality. As the tank was built for a different kind of

experiments, many problems has to be solved.

Wave Period

The swinging panel which generates the

waves is moved by an electric motor

equipped with an inverter that allows us

to modulate the frequency, and

consequently also the round per minute

of the shaft and the period of the waves.

The frequency, before our modification,

was adjustable from 0 to a maximum of

50 Hz, that corresponds to a wave period

of 0.6 seconds. Working on the setup

menu of the inverter, shown in Picture 8:

Electric Inverter for the Regulation of

the Wave Period, we allowed the

frequency of the motor to reach a

maximum value of 60 Hz which

corresponds to a wave period of 0.5 s.

Even if the motor was not designed toexceed the maximum of 50 Hz, we could

do it because of the small energy

required; in fact to a small period

corresponds a small wave and small load

energy and heat for the motor.

Furthermore we checked that the

amperage did not exceed the level

recommended by the producing

company.

Picture 8: Electric Inverter for the Regulation of the

Wave Period


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Wave Height

According to the linear wave theory1, the flux of energy transmitted by a cylindrical wave

through a plan is

)(cos)2sinh(

21

2

1 22tkx

kd

kd

kga

t

E

+=

It is interesting to remember that the time derivate of the energy is the Power.

If we want to find out the transmitted energy in one period we have to integrate the flux of

energy in the time T at the point x=0:

=

+=

+=

=

dtt

kd

kdcgadttkx

kd

kd

kga

t

EE

TT TT

)(cos)2sinh(

21

2

1)(cos

)2sinh(

21

2

1 2222

+=

+=

)2sinh(

21

4

1*

)2sinh(

21

4

1 22

kd

kdgaT

kd

kdcga

+=

)2sinh(

21

4

1 2

kd

kdgaE

T

The mean energy transmitted during a period T, which represent a power, is:

+=

+==

)2sinh(

21

4

1

)2sinh(

21

4

1 22

kd

kdcga

kd

kdga

TT

EE T

Furthermore the dispersion equation says

)(*2 kdtghgk= )(* kdtghgk=

thus the velocity )(*)(*

kdtgh

k

g

k

kdtghgk

k

c ===

where

a is the amplitude of the linear wave

T

2= is the frequency

T is the wave period

2=k is the wave number

1

Fenton, J.D. (1985). "A Fifth -order Stokes Theory for Steady Waves". J. Waterway, Port,Coastal and Ocean Eng., Vol.111.


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is wave lengthd is depth of the bottom

In deep water d

1)( kdtgh and 2)sinh(

1 +kd

kd

Thus we can rewrite both equations

22ggk==

2

*2 2

2

Tgg ==

2

* g

k

gc == cgaE 2

2

1=

The dispersion equation shows that the wave length is proportional to the square of the

period and the velocity is proportional to the square root of the wave length. This yields that,

in the linear wave theory in deep water condition, if we increase the mean transmitted

energy, keeping constant the rpm of the motor and consequently the wave period, only the

amplitude of the waves will increase. In our case, increasing the power at constant rpm

means extending the arm of the crankshaft and generate different higher waves. Thus it will

be useful to deal with an adjustable crank.

The preliminary test

we did during the

first attempt to the

wave tank showed

problems about the

oversized height of

the waves. This

because of the

excessive length of

the crank and its

scarse adjustability,

in fact there were

only 3 possible

regulation at 12, 24

and 36 centimetres

of length.

Picture 9: Initial Crank


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Thus we decided to improve

the wave generator with a

new adjustable rounded

shaped crank designed and

built by us in the University

of Malta. Now the generator

has 12 regulations of the

length of the crank at 2, 2.5,

3.2, 4, 5, 6, 7, 8, 9, 10, 11,

12 centimetres. The

accuracy of the distance of

the holes, which fits in the

connection rod, from the

rotation shaft is guaranteed

by the numerical control

machine.

Construction drawings are

attached to the thesis book.

Regularity and Turbulence of the waves

Since we are going to measure

the wave characteristics and the

acting forces on the model weneed to generate the more regular

and cylindrical waves than we

can. Considering the wave

height, if the wave was not

cylindrical for its whole width,

when we measured the height on

the tank wall we would find a

different value from the one in

the middle of the tank which runs

over the model. Furthermore it

would generate a transversaloscillatory movement and some

second order effects that we

cannot estimate.

Picture 10: New Circular Adjustable Crank

Picture 11: Initial Swin in Panel Setup


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The swinging panel was fixed in the

wave tank by an external support

with two coplanar upright. The

presence of these two elements on

the sides of the panel created two

vortices each half period which was

transported with the flow through the

whole tank. The presence of this

secondary flow self-generated by the

swinging panel would make us

miscalculate the forces, as we cannot

estimate the turbulence and its

effects on the floating structure. In

order to avoid this effect we decided

to change the geometry of the

support structure of the panelmoving backward the two uprights

and extending the swinging panel

close to the tank walls. The wave

generator is still fixed to the tank

through two horizontal rods placed

between the new uprights and the

base of the swinging panel.

Picture 12: New Swinging Panel Setup

Picture 13: Initial Waves with Turbolences


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Standing wave

One of the undesired disturbances of wave tanks which is most common and difficult toavoid is wave reflection from the tank end. It occurs when this generated wave on one side of

the tank is reflected on the opposite side and comes back. This undesired effect comes out by

the superposition of the reflected wave over the generated wave :

This formula describes a wave that oscillates in time, but has a spatial dependence that is

stationary: sin(kx). At locations x = 0, /2, , 3/2, ... called the nodes the amplitude is

always zero, whereas at locations x = /4, 3/4, 5/4, ... called the antinodes, the amplitude is

maximum. The distance between two conjugative nodes or anti-nodes is /2.

Open sea and ocean waves are generally represented with Stokes waves of order from 1 to 4,

so they are progressive, not stationary. It means that water particles change position at each

period T and the spatial dependence is not stationary. To avoid the reflection on the opposite

side of the tank, a wave absorber is commonly set up. It is possible to achieve good results

with one passive wave absorber whose length is the same length as the waves.

The wave absorber on the opposite side of the wave generator was built by cylinders made of

rolled-up plastic nettings. The waves flow through cylinders moving in the direction

)sin()cos(2

)sin()sin(

)sin(

)sin(

0

0021

02

01

kxtyy

tkxytkxyyyy

tkxyy

tkxyy

=

++=+=

=

+=

Picture 14: Example of Final Wave Generated


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perpendicular to the cylinder axis and the height goes on reducing both before and after the

reflection on the wall. The damping coefficient is related with the rate between the diameter

of the mesh of the netting and wave height; thus the smaller the wave height the thicker the

mesh has to be. To maintain this rate constant we have to reduce the diameter of the cylinder

and to thicken the netting as the waves flow through the absorber.

Even though the tank was equipped with a passive wave absorber some standing waves were

still present. The difference between the wave length which the absorber was designed for

and the length of the waves we decided to generate caused these disturbances. To avoid this

disturbances we implemented the absorber with 45 cylinders of a thicker mesh inserting them

into the sparse mesh cylinders.

Picture 16: Initial Wave Absorber

Picture 15: Final Wave Absorber


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WAVE SAMPLING

Generating Waves

First of all we need to prearrange all the equipment to measure, generate and video record the

waves. The frequency of the inverter is related with the frequency of the swinging panel so

we need a table in order to adjust the wave frequency working on the inverter.

Inverter

frequency

50 46 43 40 38 35.5 33.5 30 27.5 26.5

Wave

frequency

0.6 0.65 0.7 0.76 0.8 0.86 0.9 1 1.1 1.15

In order to measure the wave height and wave length we positioned on the side of the tank a

vertical rudder and a horizontal graduated stripe of paper adhesive tape, both of them

oriented with a plumb line. Thus, according to the range of periods and heights given by the

previous table, we started to generates waves. We will follow the same procedure for eachone of the generated waves: we set a length on the crank and a period on the inverter when

the water in the tank is calm. Each condition was recorded with a camera both the measuring

objects, and then we switched on the generator. The realized videos are essential in order to

measure the height of crest and trough and the length using a slow motion media player.

Unfortunately a wave probe was not available, so the wave measurements of the wave

characteristic was made visually during the test and on videos a posteriori.


29/101




These surveys allowed us to produce a complete sampling of the wave tank using the circular

crank we built at different settings. Our purpose is to generate non-linear waves, such as 2nd

and 3rd order Stokes waves, to represent extreme sea conditions. This is the reason why we

will leave aside waves above 3 centimetres and with a small slope.

Period [s] 1.15 Motor Freq 26.5 Hz

Crank amplitude [cm] Crest Height [cm] Trough Height [cm] H [cm] Mean Height [cm] Calm Water [cm] [cm] /H h/g*T^2

2

2.5

3.2

4

5

6

7

8 81.60 73.70 7.90 77.65 77.60 205.00 25.95 5.7E-02

9



2

2.5

3.2

4 80.20 76.50 3.70 78.35 78.25 188.00 50.81 0.06

5 81.00 76.00 5.00 78.50 78.25 189.00 37.80 0.06

6 81.45 75.60 5.85 78.53 78.25 190.00 32.48 0.067 82.25 75.05 7.20 78.65 78.25 190.00 26.39 0.06

8 82.90 74.30 8.60 78.60 78.25 191.00 22.21 6.3E-02

9

Picture 17: Sampling Waves


30/101




Period [s] 1 Motor Freq 30 Hz


2

2.5

3.2

4 80.95 75.90 5.05 78.43 78.25 157.00 31.09 0.08

5 81.80 75.30 6.50 78.55 78.25 160.00 24.62 0.08

6 82.45 75.05 7.40 78.75 78.25 162.00 21.89 0.087 83.10 74.40 8.60 78.75 78.25 164.00 19.07 0.08

8 83.60 73.90 9.70 78.75 78.25 165.00 17.01 7.5E-02

9



2

2.5

3.2

4 81.05 75.45 5.60 78.25 78.25 130.00 23.21 0.09

5 81.85 75.10 6.75 78.48 78.25 132.00 19.56 0.09

6 82.75 74.80 7.95 78.78 78.25 134.00 16.86 0.09

7 83.30 74.50 8.80 78.90 78.25 135.00 15.34 0.09

8 84.50 74.40 10.10 79.45 78.25 137.00 13.56 9.4E-02

9



2

2.5

3.2 79.65 75.90 3.75 77.78 77.60 113.00 30.13 0.10

4

5

6

7

8 84.70 74.00 10.70 79.35 78.25 125.00 11.68 1.0E-01

9

Period [s] 0.8 Motor Freq 38 Hz


2

2.5 79.37 76.00 3.37 77.69 77.60 98.00 29.08 0.12

3.2 79.95 75.60 4.35 77.78 77.60 99.00 22.76 0.12

4 81.50 75.50 6.00 78.50 78.25 100.00 16.67 0.12

5 82.60 75.20 7.40 78.90 78.25 102.00 13.78 0.12

6 82.90 74.30 8.60 78.60 78.25 105.00 12.21 0.12

7 83.80 74.30 9.50 79.05 78.25 106.00 11.16 0.12

8

9



2

2.5 79.50 76.00 3.50 77.75 77.60 88.00 25.14 0.13

3.2 80.20 75.40 4.80 77.80 77.60 90.00 18.75 0.13

4

5 83.00 75.15 7.85 79.08 78.25 94.00 11.97 0.136 83.35 74.45 8.90 78.90 78.25 97.00 10.90 0.13

7

8

9



2

2.5 79.60 75.95 3.65 77.69 77.60 76.00 20.82 0.16

3.2 80.30 75.40 4.90 77.85 77.60 79.00 16.12 0.16

4 81.65 75.20 6.45 78.43 78.25 82.00 12.71 0.16

5

6

7

8

9


31/101






2

2.5 79.65 75.85 3.80 77.75 77.60 68.00 17.89 0.18

3.2 80.80 75.50 5.30 78.15 77.60 70.00 13.21 0.18

4

5

67

8

9



2

2.5 79.80 75.70 4.10 77.75 77.60 60.00 14.63 0.21

3.2

4

5

6

7

8

9

The surveys were carried out at least twice in order to avoid mistakes due to stochastic

effects. From these values we can obtain features such as the mean height and the values

2gT

hand

2gT

Hthat are essential to identify the effective order of the waves as it will be

shown later. Furthermore we added some further graphs to better understand how waves

parameters change with the wave period and to have a useful table for future experiments.


32/101




Figure 1: Wave Theories Graph


33/101




Figure 2: Generated Wave Order


34/101




Figure 3: Wave Sampling - Wave Length vs. Period

WaveLength

40.0

0

60.0

0

80.0

0

100.0

0

120.0

0

140.0

0

160.0

0

180.0

0

200.0

0

220.0

0

240.0

00.5

0.6

0.7

0

.8

0.9

1

1.1

1.2

1.3

Period(s)

L(cm)

2.5cm

3.2cm

4cm

5cm

6cm

7cm

8cm

L

=g*T^2/6.28


35/101




Figure 4: Wave Sampling - Wave Height vs. Period

WaveHeight

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Period(s)

H(cm)

2.5cm

3.2cm

4cm

5cm

6cm

7cm

8cm


36/101




Figure 5: Wave Sampling - H/L vs. Period

H/L

0.0

0

10.0

0

20.0

0

30.0

0

40.0

0

50.0

0

60.0

00.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Period(s)

H/L

2.5cm

3.2cm

4cm

5cm

6cm

7cm

8cm


37/101




Figure 6: Wave Sampling - Wave Height vs. Crank Length

Figure 1: Wave Theories Gh summarizes linear and non-linear wave theories dividing the

area of the chart in different zones through some lines defined by equations. The independent

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

11.00

2

3

4

5

6

7

8

9

Crank(cm)

Waveheight(cm)

0.65

0.7

0.76

0.8

0.9

1

1.1


38/101




variable2

gT

h, which relates the depth h with the period T, is relevant to identify if the

seabed influences the wave shape, while the dependent variable2

gT

H, which relates the

wave height H with the period P, is related to the shape and to the slope of the wave.

Figure 2: Generated Wave Order

shows that our waves are both of the second and third order of Stokes and that points for

small periods are in the initial "transitional water" zone. This is a marginal problem, because

height and length are only slightly affected by the deepness by a mean 1.5%. Anyway, we

decide to do experimental tests for most of the points even though they are in the transitional

water area, while for the numerical testing we will take only those points marked with a

circle.

Figure 3: Wave Sampling - Wave Length vs. Period verifies the good quality of the generated

waves comparing them with the dispersion relation

2

* 2Tg= . We can ascertain that the

obtained wave length is coherent with theoretical values except for small a reasonable gap

due to measuring errors.

Figure 4: Wave Sampling - Wave Height vs. Period shows the little decreasing slope of the

value of the height at the change of the period, using the crank length as parameter.

Figure 5: Wave Sampling - H/L vs. Period is meant to be as advice while generating waves.It is important to monitor the slope in order to avoid the breaking waves and evaluate how

much more we can decrease the period while generating at constant crank length.

Figure 6: Wave Sampling - Wave Height vs. Crank Length confirms the theory we discussed

in the previous paragraph about the Wave Height. At equal period, there is an almost

proportional relation between the two variables.

Reading the exact wave order and the seabed deepness on the graph we can decide which

waves to use for experimental test and which for numerical ones.

Crank 0.6 s 0.65 s 0.7 s 0.76 s 0.8 s 0.86 s 0.9 s 1.0 s 1.1 s 1.15 s

2.5 cm exper not tested exper not tested exper and num --- --- --- --- ---

3.2 cm --- exper not tested exper exper and num exper --- --- --- ---

4.0 cm --- --- exper --- exper and num --- exper and num exper exper ---

5.0 cm --- --- --- exper exper and num --- exper and num exper exper ---

6.0 cm --- --- --- exper exper and num --- exper and num exper exper ---

7.0 cm --- --- --- --- exper and num --- exper and num exper and num exper ---

8.0 cm --- --- --- --- --- exper exper and num exper exper exper

Period


39/101




Experimental Wave Profile Control

This verification wants to carry out a deeper analysis comparing the generated wave profile

with the theoretical profile developed by Fenton2

. From one frame of the video for themeasurement of the wave length we digitize the wave profile referring some points to a

couple of axis. For theoretical profiles, we reproduce the same wave with linear, second

order third order and fifth order Stokes waves, using the deepness, the wave height and the

wave length data already sampled .

Picture 18: Period 0.9 s Crank 4 cm Wave Profile Control Digitization - Second Order Stokes Wave

2

Fenton, J.D. (1990) "Non linear Waves Theories". The Sea,Vol.9, Wiley & Son Inc., NewYork.


40/101




Picture 19: Period 1.0 s Crank 5 cm Wave Profile Control Digitization - Second Order Stokes Wave

Picture 20: Period 1.0 Crank 7 cm Wave Profile Control Digitaztion - Third Order Stokes Wave


41/101




The theoretical model developed by Fenton3

for the second order waves uses the formula

below to define the profile :

)2cos(***cos** 22

1 bb fakfa +=

where

1bf =1

2bf =

h)*(ksinh*4

h))*cosh(2k(2*h)*cosh(k3

+

Fenton4

developed also a model for the third order Stokes waves:

)3cos(***)2cos(***cos** 332

2

2

1 bbb fakfakfa ++=

where

1bf and 2bf are those parameter explained above

3bf =

)*(sinh*64

)3)*(cosh*24(6

6

hk

hk +

For the fifth order wave, we decided to use the Skjelbreia-Hendrickson5

model.

=

=

5

1

* )cos(1

i

i ik

ak**1 =

24

4

22

2*

2

*)*(*)*( BakBak +=

35

5

33

3*

3 *)*(*)*( BakBak +=

3 - 4 Fenton, J.D. (1990) "Non linear Waves Theories". The Sea,Vol.9, Wiley & Son Inc., New

York.

Scarsi, G. (1998). "Onde di gravita` regolari". Collana di Idraulica, Marina Edizioni

Litograph, Genova.

5Skjelbreia, L. and Hendrickson, J.A. (1960). "Fifth order Gravity Wave Theory". Proc. 7th ICCE, Vol.1.

Scarsi, G. (1998). "Onde di gravita` regolari". Collana di Idraulica, Marina EdizioniLitograph, Genova.


42/101




44

4*

4 *)*( Bak=

55

5*

5 *)*( Bak=

S=sinh(k*h)

C=cosh(k*h)

3

2

22S*4

1)C*C(2B

+=

9

2468

24S*384

21)C*322C*192-*504C*C(272B

++=

C

6

6

33S*64

1)C*(8*3B

+=

)1C*6(S*12288

81)-C*54C*6264C*21816C*54000*70848C*208224-C*(88128B

212

2468101214

35

+++=

C

)1C*6(S*384

21)-C*106C*48*48C*448-C*(768*CB

29

246810

44

++=

C

)3*11*8(*)1C*6(S*12288

225)C*1050C*1800C*7160C*7280*20160(

)3*11*8(*)1C*6(S*12288

)C*83680C*262720-C*(192000B

24210

246810

24210

121416

35

+

+++

+

+=

CC

C

CC


43/101




Figure 7: Period 0.9 s Crank 4 cm Wave Profile Comparison

Figure 8: Period 1.0 s Crank 5 cm Wave Profile Comparison


44/101




We choose these three waves in order to do three controls of the profile, referring to the

Figure 1: Wave Theories Graph, in points with a value of2*Tg

Hlargely different.

Except for few points, we can appreciate the good quality of the generated waves comparingthem to relative theoretical profile: the real crest is 1.16% lower than the wave amplitude

compared to the theoretical whereas the trough is coincident. Other minimal gaps are on the

body of the waves.

Even though our tank waves are only of second and third order we decided to add to the

graph the first and the fifth order profile to better evidence the order of magnitude of an

approximation we are doing. The RANSE solver we will use for the numerical simulations

can reproduce only linear and fifth order waves, so we have to choose one of these to

approximate our tank waves. The comparisons show the great difference between linear and

tank profile, and the close resemblance between the fifth and the other non-linear waves.

Figure 9: Period 0.9 Crank 7 cm Wave Profile Comparison


45/101




LOAD CELLS AND DATA ACQUISITION

Load Cells

Each load cell we used, manufactured by the Laboratory of

the University of Malta, is made of a circular steel ring

with two screws placed on the external side, diametrically

opposed to each other. The measurement is done by fourstrain gauges linked with a full bridge connection and

glued with a special bonder to the ring. The strain gauge is

just a resistor which modifies its resistivity coefficient

with its deformation. The full bridge connection plans to

put the four sensors on the diameter perpendicular to the

direction of strain, two on each side, one internal and one

external and doing the mean of the values surveyed.

Placing the gauges in this way may help us to obtain better

data because of the mean error effect. So, the load applied

to the ring deforms the shape and consequently also the

strain gauges which register the phenomena and send the

analogical data to the acquisition system. Notice that the

load cells are sensible to every kind of strains they are

subjected to, so it is

important to build an

external structure with appropriate constrains in order to

make them work only in the axial direction. Load cells must

be calibrated imposing a known load and noting down the

value of the voltage, then you can draw the calibration

equation and evaluate the dispersion of the points from the

line. Furthermore, it is compulsory that the deformationremains into the

elastic range in

order to maintain

the most

obtainable linear

relation between

the strain and

voltage during the

calibration.

Picture 22: Load cells setup

Picture 21: Strain gauge

Picture 23: Full Bridge


46/101




Calibration On-Site

The sensitivity of the load cells

does not allow us to do the

calibration and then mount the

equipment on the model, as there

would be pre-load effects and the

slope could change. The best way

to avoid data mistakes is doing our

calibration on-site: we mount all

the equipments and then we start

taking measurements. Furthermore,it is compulsory to do a calibration

each day we obtain the data and

wait some time when we turn on

the pc in order to warm up the

electric circuit: electric devices are

sensitive to temperature and they

do different measurement of the

same load if they are not working

in isothermal condition, reached

only after some half hours later than when they are just turned on.

The calibration consists in drawing a graphic by points, where the independent variable is theweight we impose and dependent variable is the voltage we read on the calibration software.

Then we find the trend line that relates the points and we check that points are on a straight

line by calculating the dispersion R. So, entering the equation of the calibration for each load

cell, when we will take the measurement we will read the force expressed in kilograms.

KLC1

y = 0.0011x - 0.0055

R2

= 0.9994

-0.006

-0.0055

-0.005

-0.0045

-0.004

-0.0035

-0.003

-0.5 0.0 0.5 1.0 1.5 2.0 2.5

Weight [Kg]

Voltage[V]

KLC1 Calibration Linear (KLC1 Calibration)

Figure 10: KLC1 Calibration

Picture 24: Calibration On-Site


47/101




KLC2

y = 0.0012x + 0.014

R2

= 1

0.0135

0.014

0.0145

0.015

0.0155

0.016

0.0165

0.017

-0.5 0 0.5 1 1.5 2 2.5

Weight [Kg]

Voltage[V]



KLC3

y = 0.0017x - 0.0194

R2

= 0.9997

-0.02

-0.019

-0.018

-0.017

-0.016

-0.015

-0.014

-0.500000 0.000000 0.500000 1.000000 1.500000 2.000000 2.500000 3.000000

Weight [Kg]

Voltage[V]




48/101




Data Acquisition System

The first electronic device where the signal of the strain gauges arrives is the data acquisition

card. Each card can process 2 signals with an amplifier and an ADC (Analog to DigitalConverter). The first one amplifies in order to avoid the noise due to the electric net of the

lab (such as neon lights in the room and the standard noise transmitted by other electrical).

The second one discretizes the signal at a constant time step, usually referring to the clock

rate of the pc. As we are dealing with waves with a minimum period of 0.6 seconds (1.67 Hz)

and we need to work with a minimum step of 10 points per half-period, our time step will be

t =snumberofpo

period

intmin

min= 0.6/20=0.03 second (33.3 Hz)

So we rounded down to t =0.025 seconds (40 Hz).

After few test we realized that the time step we were dealing with was too small for the pc

clock, the t is inconstant, so we had to enhance the current setup. The solution we found

was to add an analog signal generator which sends the input for the discretization in place of

the pc. Since the device can generate digital signals with different shapes till a maximum

frequency of 1000 Hz we verified the precision of the 40 Hz signal with "sawtooth test".

This test consists in emitting a sawtooth shaped signal with a frequency higher than the 40

Hz we need, such as 50 Hz like as we did. Then a plot of the points recorded with the data

acquisition system is prepared and the right position of the points and the shape of the

sawtooth is checked. The points must lay on the straight lines as the Figure 13: Sawtooth test

shows an exemple of this comparison.

Sawtooth

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

30 30.2 30.4 30.6 30.8 31 31.2 31.4 31.6 31.8 32

Time [s]

Volts[V]

Sawtooth

Figure 13: Sawtooth test

After the ADC the digital signal is recorded with the software LabVIEW in a text file.


49/101




System of axis

The system of axis which Forces will refer to is the one on the centre of buoyancy, where the

x-axis is opposite to the wave advancing direction, the z-axis is parallel to the direction of theforce of gravity with the negative side facing the seabed, and y-axis is perpendicular to the

previous two axis.

Geometry of the External Structure

The external structure is meant to rigidly sustain the model while satisfying the cinematic

conditions of blocked mode tests. As we decided to deal with symmetric body test problem

and carrying out this test for the first time we decided to evaluate only the forces on the x-z

plan: the force on the direction of the Surge, FX, the force on the direction of the Heave, FZ,and the moment on the direction of the Pitch, FY. Error! Reference source not found.

shows how we decided to arrange the layout of the load cells in order to measurement the

Forces. A hollowed aluminium rod is stacked inside the central cylinder and transfers the

forces to the three load cells. In order to measure only the normal force we decided to use

simply supported restraint. The load cell on the top of the model has its axis parallel to the

vertical axis and from now on it will be called KLC3. KLC3 measures only the vertical force

and we need to change only the sign because of the axis orientation:

Fz=-KLC3*g

The two load cells on the upper left side have the axis parallel with the X-axis and from now

on they will be called KLC2 (the upper) and KLC1 (the lower). The horizontal force and the

moment are strictly related with these two load cells by these relations:

Fx=(KLC2+KLC1)*g

My=(KLC1*Z1+KLC2*Z2)*g

where Z1 and Z2 are the distance of KLC1 and KLC2 from the centre of buoyancy.


50/101




CONSERVATION LAWS OF FLUID FLOW

Governing equations of fluid flow

The governing equations of fluid flow represent two mathematical statements of the

conservation laws of physics.

The mass of a fluid is conserved.

The rate of change of momentum equals the sum of the forces on a fluid particle (Newton's

second law).

The fluid will be regarded as a continuum. For the analysis of fluid flows at macroscopic

length scales (say 1 m and larger) the molecular structure of matter and molecular motionsmay be ignored. We describe the behavior of the fluid in terms of macroscopic properties, suchas velocity, pressure, density and temperature, and their space and time derivatives. These may

be thought of as averages over suitably large numbers of molecules. A fluid particle or point

in a fluid is then the smallest possible element of fluid whose macroscopic properties are not

influenced by individual molecules.

We consider such a small element of fluid with sides x, y and z.

The six faces are labeledN, S, E, W, T, B which stands for North, South, East, West, Top and

Bottom. The positive directions along the co-ordinate axes are also given. The centre of the

element is located at position (x,y,z). A systematic account of changes in the mass, momentum

and energy of the fluid element due to fluid flow across its boundaries and, where appropriate,

due to the action of sources inside the element, leads to the fluid flow equations.

All fluid properties are functions of space and time so we would strictly need to writep(x,y,z,t),

p(x,y,z,t), T(x,y,z,t) and u(x,y,z,t) for the density, pressure, temperature and the velocity vector

respectively. To avoid unduly cumbersome notation we will not explicitly state the

dependence on space co-ordinates and time. For instance, the density at the centre (x,y,z) of a

fluid element at time tis denoted by and the x-derivative of, say, pressurep at (x,y,z) andtime tby p/x. This practice will also be followed for all other fluid properties.


51/101




The element under consideration is so small that fluid properties at the faces can be

expressed accurately enough by means of the first two terms of a Taylor series expansion.

So, for example, the pressure at theEand Wfaces, which are both at a distance of1/2x fromthe element centre, can be expressed as

Mass conservation in three dimensions

The first step in the derivation of the mass conservation equation is to write down a mass

balance for the fluid element.

The rate of increase of mass in the fluid element is

Next we need to account for the mass flow rate across a face of the element which is given by

the product of density, area and the velocity component normal to the face. From the figure

beyond it can be seen that the net rate of flow of mass into the element across its boundaries

is given by

Flows which are directed into the element produce an increase of mass in the element and get

a positive sign and those flows that are leaving the element are given a negative sign.


52/101




The rate of increase of mass inside the element is now equated to the net rate of flow of mass

into the element across its faces. All terms of the resulting mass balance are arranged on the

left hand side of the equals sign and the expression is divided by the element volume xyz.

This yields

or in more compact vector notation

For an incompressible fluid(i.e. a liquid) the densityis constant and equation becomes


53/101




Rates of change following a fluid particle and for a

fluid element

The momentum and energy conservation laws make statements regarding the changes of

properties of a fluid particle. Each property of such a particle is a function of the position

(x,y,z) of the particle and time t. Let the value of a property per unit mass be denoted by . Thetotal or substantive derivative ofwith respect to time following a fluid particle, written as

D/Dt, is

A fluid particle follows the flow, so x/t = u, dy/dt= v and dz/dt = w. Hence the substantivederivative ofis given by

D/Dtdefines the rate of change of property per unit mass. As in the case of the massconservation equation we are interested in developing equations for rates of change per unit

volume. The rate of change of property per unit volume for a fluid particle is given by the

product ofD/Dtand density, hence

The mass conservation equation contains the mass per unit volume (i.e. the density ) as theconserved quantity. The sum of the rate of change of density and the convective term in the

mass conservation equation (2.4) for a fluid element is

The generalization of these terms for an arbitrary conserved property is


54/101




Formula (2.9) expresses the rate of change ofper unit volume plus the net flow ofout ofthe fluid element per unit volume. It is now re-written to illustrate its relationship with the

substantive derivative of:

The term [/t + div(u)] is equal to zero by virtue of mass conservation (2.4). In words,relationship (2.10) states

To construct the three components of the momentum equation and the energy equation the

relevant entries for and their rates of change per unit volume as defined in (2.8) and (2.10)are given below:

Momentum equation in three dimensions

Newton's second lawstates that the rate of change of momentum of a fluid particle equals the

sum of the forces on the particle.

The rates of increase ofx-, y- andz- momentum per unit volume of a fluid particle are given

by


55/101




We distinguish two types of forceson fluid particles:

surface forces

-pressure forces

-viscous forces

body forces

-gravity force

-centrifugal force

-Coriolis force

-electromagnetic force

It is common practice to highlight the contributions due to the surface forces as separate

terms in the momentum equation and to include the effects of body forces as source terms.

The state of stress of a fluid element is defined in terms of the pressure and the nine viscous

stress components. The pressure, a normal stress, is denoted by p. Viscous stresses are

denoted by . The usual suffix notationij , is applied to indicate the direction of the viscousstresses. The suffices i andj in ij indicate that the stress component acts in the j-direction on asurface normal to the i-direction.

First we consider the x-components of the forces due to pressurep and stress components xx,yx and zx shown in the figure above. The magnitude of a force resulting from a surface stressis the product of stress and area. Forces aligned with the direction of a co-ordinate axis get a

positive sign and those in the opposite direction a negative sign. The net force in the x-

direction is the sum of the force components acting in that direction on the fluid element.


56/101




On the pair of faces (E, W) we have

The net force in the x-direction on the pair of faces (N,S) is

Finally the net force in the x-direction on faces TandB is given by

The total force per unit volume on the fluid due to these surface stresses is equal to the sum

of the above terms divided by the volume xyz:

Without considering the body forces in further detail their overall effect can be included by

denning a source Smx of x-momentum per unit volume per unit time.

The x-component of the momentum equation is found by setting the rate of change of x-

momentum of the fluid particle equal to the total force in the x-direction on the element due to

surface stresses plus the rate of increase of x-momentum due to sources:

It is not too difficult to verify that the y-component of the momentum equationis given by


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and the z-component of the momentum equation by

The effects of surface stresses are accounted for explicitly; the source terms SMX,SMyand SMZ

in (2.14a-c) include contributions due to body forces only. For example, the body force due to

gravity would be modeled by SMX= 0, S\fy = 0 and Smz = -pg-.

Navier-Stokes equations for a Newtonian fluid

The governing equations contain as further unknowns the viscous stress components ij. The

most useful forms of the conservation equations for fluid flows are obtained by introducing asuitable model for the viscous stresses ij. In many fluid flows the viscous stresses can beexpressed as functions of the local deformation rate (or strain rate). In three-dimensional

flows the local rate of deformation is composed of the linear deformation rate and the

volumetric deformation rate.

The rate of linear deformation of a fluid element has nine components in three dimensions,

six of which are independent in isotropic fluids (Schlichting, 1979). They are denoted by the

symbol ey. The suffix system is identical to that for stress components (see section 2.4).

There are three linear elongating deformation components:

There are also six shearing linear deformation components:

The volumetric deformation is given by

In a Newtonian fluid the viscous stresses are proportional to the rates of deformation. The

three-dimensional form of Newton's law of viscosity for compressible flows involves two

constants of proportionality: the (first) dynamic viscosity, , to relate stresses to linear

deformations, and the second viscosity, , to relate stresses to the volumetric deformation. Thenine viscous stress components, of which six are independent, are


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Not much is known about the second viscosity , because its effect is small in practice. For

gases a good working approximation can be obtained by taking the value =-2/3 (Schlichting, 1979). Liquids are incompressible so the mass conservation equation is div u=

0 and the viscous stresses are just twice the local rate of linear deformation times the

dynamic viscosity.Substitution of the above shear stresses (2.31) into (2.14a-c) yields the so-called Navier-

Stokes equations:

Often it is useful to re-arrange the viscous stress terms as follows:

The viscous stresses in they- and z-component equations can be re-cast in a similar manner.

We clearly intend to simplify the momentum equations by 'hiding' the two smallercontributions to the viscous stress terms in the momentum source. Denning a new source by


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the Navier-Stokes equations can be written in the most useful form for the development of

the finite volume method:

Conservative form of the governing equations of

fluid flow

To summarize the findings thus far we quote in the table beyond the conservative or

divergence form of the system of equations which governs the time-dependent three-

dimensional fluid flow

Differential and integral forms of the general

transport equations

It is clear from the previous table that there are significant commonalities between the

various equations. If we introduce a general variable the conservative form of all fluid flowequations, including equations for scalar quantities such as temperature and pollutant

concentration etc., can usefully be written in the following form


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In words

This equation is the so-called differential general transport equation for property . It clearly

highlights the various transport processes: the rate of change term and the convective term onthe left hand side and the diffusive term (T =diffusion coefficient) and the source term

respectively on the right hand side. In order to bring out the common features we have, of

course, had to hide the terms that are not shared between the equations in the source terms. Note

that equations can be made to work for the internal energy equation by changing i into Tby

means of an equation of state.

Furthermore this equation is used as the starting point for computational procedures in the

finite volume method. By setting equal to 1, u, v, w and i (or T or h0) and selectingappropriate values for the diffusion coefficient and source terms we obtain special forms ofthe summarizing table for each of the five partial differential equations for mass, momentum

and energy conservation.

The key step of the finite volume method is the integration of the differential general transport

equation over a three-dimensional control volume CVyielding:

The volume integrals in the second term on the left hand side, the convective term, and in the

first term on the right hand side, the diffusive term, are re-written as integrals over the entire

bounding surface of the control volume by using Gauss' divergence theorem. For a vector a

this theorem states


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Applying Gauss' divergence theorem, the integral equation can be written as follows:

A diffusive flux is positive in the direction of a negative gradient of the fluid property , i.e.

along direction -grad . For instance, heat is conducted in the direction of negative

temperature gradients. Thus, the product n (- grad) is the component of diffusion fluxalong the outward normal vector, and so out of the fluid element. Similarly, the product n (grad), which is also equal to (-n (- grad), can be interpreted as a positive diffusion fluxin the direction of the inward normal vector -n, i.e. into the fluid element.

The first term on the right hand side , the diffusive term, is thus associated with a flux into

the element and represents the net rate of increase of fluid property of the fluid element dueto diffusion. The final term on the right hand side of this equation gives the rate of increase ofproperty as a result of sources inside the fluid element.In words, this relationship for the fluid in the control volume can be expressed as follows:


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THE FINITE VOLUME METHOD

Now we develop the numerical method based on this integration, the finite volume (or control

volume) method, by considering the simplest transport process of all: pure diffusion in the

steady state. The governing equation of steady diffusion can easily be derived from the

general transport equation for property by deleting the transient and convective terms. Thisgives

The control volume integration, which forms the key step of the finite volume method that

distinguishes it from all other CFD techniques, yields the following form:

By working with the one-dimensional steady state diffusion equation the approximation

techniques that are needed to obtain the so-called discretized equations are introduced.

Consider the steady state diffusion of a property in a one-dimensional domain defined inthe figure beyond. The process is governed by

where T is the diffusion coefficient and Sis the source term. Boundary values ofat pointsA and B are prescribed. An example of this type of process, one-dimensional heat conduction

in a rod, is studied in detail in this section:


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Step 1: Grid generation

The first step in the finite volume method is to divide the domain into discrete control

volumes. Let us place a number of nodal points in the space between A and B. The boundaries(or faces) of control volumes are positioned mid-way between adjacent nodes. Thus each node

is surrounded by a control volume or cell. It is common practice to set up control volumes

near the edge of the domain in such a way that the physical boundaries coincide with the

control volume boundaries.

At this point it is appropriate to establish a system of notation that can be used in future

developments. The usual convention of CFD methods is shown in the figure above.

A general nodal point is identified by P and its neighbors in a one-dimensional geometry, the

nodes to the west and east, are identified by Wand Erespectively. The west side face of the

control volume is referred to by 'w' and the east side control volume face by V. The distances

between the nodes W and P, and between nodes P and E, are identified by xWP and xPE

respectively. Similarly the distances between face w and point P and between P and face e aredenoted by xwP and xPe respectively. Furthermore the figure above shows that the controlvolume width isx = xpe.

Step 2: Discretization

The key step of the finite volume method is the integration of the governing equation (or

equations) over a control volume to yield a discretized equation at its nodal point P. For the

control volume defined above this gives

HereA is the cross-sectional area of the control volume face, Vis the volume and is theaverage value of source Sover the control volume. It is a very attractive feature of the finite

volume method that the discretized equation has a clear physical interpretation. This equation

states that the diffusive flux ofleaving the east face minus the diffusive flux ofentering thewest face is equal to the generation of, i.e. it constitutes a balance equ

experimental and numerical methods for the prediction of wave loads in extreme sea conditions

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