Design of Naval Architecture Parameters for Wave Tank

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Design of Naval Architecture Parameters for Wave Tank

Andrés Pastor Sánchez

July 8, 2021

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List of Figures

1 Hydrodynamic experiences tank. Marin. . . 10

2.1 Scheme of engineering model of a wave tank. Source: own elaboration. . . 14

2.2 Model of offshore salmon fish farming. Source: [2]. . . 15

2.3 Scheme of general design of a structure. Source: own elaboration. . . 20

3.1 Scheme of simple flap. Source: [4]. . . 24

3.2 Scheme of double flap. Source: [1]. . . 24

3.3 Scheme of piston. Source: [4]. . . 24

3.4 Scheme of Vertical wedge. Source: [1]. . . 24

3.5 Scheme of pneumatic. Source: [1]. . . 25

3.6 Two-Dimensional Wave Flume Definition Sketch. Source: [5]. . . 25

3.7 Dispersion relationship for progressive waves. Source: [5]. . . 29

3.8 Dispersion relationship for standing waves. Source: [5]. . . 29

4.1 Scheme of Tank. Source: University. . . 36

4.2 Scheme of different zones in the tank. Source: Own elaboration. . . 36

4.3 Scheme of Flap Actuator. Source: [6]. . . 37

4.4 Scheme of Piston Actuator. Source: Own elaboration. . . 37

4.5 Interface of the program. Source: Own elaboration. . . 37

4.6 Flap stroke limit at 0.9 meters of water depth. . . 39

4.7 Piston stroke limit at 0.9 meters of water depth. . . 40

4.8 Flap Stroke limit in various water heights. . . 41

4.9 Piston Stroke limit in various water height. . . 41

4.10 Maximum wave height in flap at various water depths. . . 42

4.11 Maximum wave height in piston at various water depths. . . 42

4.12 Maximum wave height in flap at various water depths, 3D. . . 43

4.13 Maximum wave height in piston at various water depths, 3D. . . 43

4.14 Maximum Piston and Flap real stroke. . . 44

4.15 Maximum Piston and Flap wave height. . . 44

4.16 Wave Lenght. . . 45

4.17 Celerity. . . 45

4.18 Tank, Water on one side. Source: Own elaboration. . . 46

4.19 Scheme of Flap Actuator. Source: [6]. . . 46

4.20 Total force, Piston and Flap, water on one side. . . 47

4.21 Total force and Wave board motion, Piston and Flap. T = 0.5 s. . . 48

4.25 Separate force, Piston, water on one side. . . 49

4.26 Separate Force, Flap, T = 0.5 s. . . 49

4.27 Separate Force, Piston, T = 0.5 s. . . 49

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4.32 Tank, Water on both sides. Source: Own elaboration. . . 51

4.33 Scheme of piston actuator. Source: Own elaboration. . . 51

4.34 Total force, Piston and Flap, water on both sides. . . 51

4.45 Mean power, water on one side. . . 54

4.46 Maximum power, water on one side. . . 55

4.47 Separate Power, Flap, T = 0.5 s. . . 55

4.48 Separate Power, Piston, T = 0.5 s. . . 55

4.53 Mean power, water on both sides . . . 57

4.54 Maximum power, water on both sides . . . 58

4.61 Reference sketch flap. Source: [6]. . . 60

4.62 Actuator force, Flap, T = 0.5 s. . . 62

4.63 Actuator force, Flap, T = 1.5 s. . . 62

4.64 Flap Force at different water depths. . . 62

4.65 Reference sketch piston. Source: Own elaboration. . . 63

4.66 Actuator force, Piston, T = 0.5 s. . . 64

4.67 Actuator force, Piston, T = 1.5 s. . . 64

4.68 Piston Force at different water depths. . . 64

4.69 Flap actuator axis. Source: Own elaboration. . . 65

4.70 Piston actuator axis. Source: Own elaboration. . . 66

4.71 Actuators velocity. . . 66

4.72 Motions and velocity for T = 0.5 s. . . 67

4.73 Motions and velocity for T = 1.5 s. . . 67

4.74 Flap Actuators. Source: [6]. . . 68

4.75 Piston Actuators. Source: Own elaboration. . . 68

4.76 Actuators capacity. Source: [7]. . . 69

4.77 Actuators design. Source: [7]. . . 69

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4.78 Actuator extension. Source: [7]. . . 69

4.79 Constant slope. Source: [1]. . . 70

4.80 Variable slope. Source: [1]. . . 70

4.81 Progressive constant slope. Source: [1]. . . 71

4.82 Progressive variable slope. Source: [1]. . . 71

4.83 Progressive vertical wave absorber. Source: [8] . . . 71

4.84 SIPWA absorber. Source [9]. . . 72

4.85 Progressive vertical mesh. Source [9]. . . 72

4.86 Progressive porous parabola. Source [9]. . . 73

4.87 Constant slope with small stones. [9]. . . 73

4.88 Constant slope beach. Source: Own elaboration. . . 74

4.89 Progressive vertical mesh. Source: Own elaboration. . . 74

5.1 Scattered wave height in the mesh of the tank with the prism from Seafem. Source: Own elaboration. . . 76

5.2 Scattered wave height in the mesh of the tank with the cylinder from Seafem. Source: Own elaboration. . . 76

5.3 Measurement on the scattered wave in some nodes, cylinder. Source: Own elaboration. . . 77

5.4 Representation of wave scattering the front of the cylinder. Source: Own elaboration. . . 77

5.5 Representation of wave scattering on the side of the cylinder. Source: Own elaboration. . . 78

5.6 Measurement on the scattered wave in some nodes, box. Source: Own elaboration. 78 5.7 Representation of wave scattering the front of the box. Source: Own elaboration. 79 5.8 Representation of wave scattering on the side of the box. Source: Own elaboration. . . 79

5.9 Wind turbine model. Source: Own elaboration. . . 80

6.1 Scheme of tank with floating wind turbine model. Source: Own elaboration. . . 86

6.2 Inclinometer. Source: [14]. . . 87

6.3 Inclinometer cover. Source: [14]. . . 87

6.4 Inclinometer position. Source: Own elaboration. . . 87

6.5 Accelerometers position. Source: Own elaboration. . . 88

6.6 Accelerometer [15]. . . 88

6.7 Cover of the Accelerometer [15]. . . 88

6.8 Laser sensor [16]. . . 89

6.9 Strain gauge. Source [18]. . . 90

6.10 Strain gauge sketch. Source [18]. . . 90

6.11 Doppler laser profiler [19]. . . 91

6.12 Doppler laser profiler position. Source: Own elaboration. . . 91

6.13 Calibre . . . 91

6.14 FBG-scan 804D [20]. . . 92

6.15 Connections. . . 92

6.16 PCi-6321. Source: [21]. . . 92

7.1 Program interface. . . 95

7.2 Program interface, input zone. . . 96

7.3 Program interface, Strokes and Waves heights. . . 97

7.4 Stroke limit comparative flap code. . . 97

7.5 Stroke limit flap at different water depths code. . . 98

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7.6 Waves height flap at different water depths code. . . 99

7.7 Program interface, Forces. . . 99

7.8 Dynamic resistive term for each period. . . 100

7.9 Dynamic inertia term for each period. . . 101

7.10 Forces and board motion for lap and piston. . . 102

7.11 Program interface, Power. . . 102

7.12 Mean power for flap and piston. . . 103

7.13 Max power for flap. . . 104

7.14 Power for a particular period flap. . . 105

7.15 Program interface, Actuator. . . 105

7.16 Different forces in the flap actuator over a cycle. . . 106

7.17 Different forces in the piston actuator over a cycle. . . 107

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Symbol Description Units Chapter 2

F_G Gravity force N

F_I Inertia force N

F_v Viscous force N

F_D Drag force N

Fp Pressure force N

F_e Elastic force N

M Mass Kg

g Gravity m/s²

˙u Velocity m/s

µ Viscosity Pa

A Area m²

C_d Drag coefficient -

ρ Density kg/m³

E Young modulus KPa

L Length m

λ Scale factor -

Fn Froude number -

Rn Reynolds number -

St Strouhal number -

KC Keulegan-Carpenter number -

Ur Ursell number -

Ch Cauchy number -

Chapter 3

φ Wave velocity potential

h Water depth m

h Radius m

v_n Velocity normal to the board m/s

X(z, t) Stroke at each time step m

θ Aperture angle of flap

k Wave number m⁻¹

S₀ Stroke at free surface m

H Wave amplitude m

n₁(x, t) Wave height at each time step m

w Angular velocity rad/s

p₀(z, t) wave board pressure, water on one side Pa/m p₀₀(z, t) wave board pressure, water on both sides Pa/m F_t₀(t) Wave board force, water on one side N/m F_t₀₀(t) Wave board force, water on both sides N/m P₀(t) Wave board power, water on one side W/m P₀₀(t) Wave board power, water on both sides W/m

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T Period s

a_max Ballscrew maximum acceleration m Sc Maximum physical stroke at 1.6 meters m

So_flap Maximum flap stroke m

S_o_piston Maximum piston stroke m

L Wave length m

C Celerity m/s

F_ATT Actuator force N

h_flap Height of the flap m/s

F_H₂_O Water force N

h_water_max Maximum height reached by water m

I_G Inertia of the board kg · m²

FW Force due to board weight N

F_B Buoyancy force N

h_water_min Minimum height reached by water m V_submerged Volume of the board submerged m³

V_flap Volume of the flap m³

m_flap Flap mass kg

m_steel Mass of steel kg

Chapter 5

h Water depth m

T Draft m

b Width of the tank m

L Length of the model m

B Beam m

∇_float Volume of the float m³

R_inn− R_out Inner and outer radius m

H_inn− H_out Inner and outer height m

∆float Mass of the float kg

∆_ballast Mass of the ballast kg

h_inn − h_out Inner and outer height of the nacelle m w_inn− w_out Inner and outer width of the nacelle m l_inn− lout Inner and outer length of the nacelle m w_inn− w_out Inner and outer width of the nacelle m l_inn− l_out Inner and outer length of the nacelle m

F_tendon Force supported by one tendon N

CS Security coefficient -

σmax Tension supported by one tendon N/m² D_section Section diameter of the tendon m²

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ε Deformation -

E_pvc Young’s modulus of the tendons N/m²

∆λ Length variation by the wave light refractive in the fiber - λ₀ Wave length at the beginning of the test -

k refractive index -

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Abstract

The objective of this thesis is the evaluation of the potentialities of the tank, recently built at the Naval Architecture and Ocean Engineering school, in terms of carrying out different research and teaching activities and design it. This objective is achieved by studying the linear water waves theory and then investigating a first suitable actuation system to cover our working area.

In this thesis a wave generator will be designed based on the capabilities of the tank. Related to that, a wave absorber will also be designed to minimize the possible reflection of the waves.

A wind turbine model will also be developed along with the necessary instrumentation to explore the different research and educational activities that can be carried out in the tank. All the previously mentioned designs will take into account that the main limitation of our tank is its small size.

Figure 1: Hydrodynamic experiences tank. Marin.

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Chapter 1 Introduction

1.1 Motivation and antecedents

From the 2019-2020 academic year, the Naval Architecture and Ocean Engineering school built a small-sized hydrostatic and hydrodynamic experience tank, in order to provide the school with new capabilities.

As a result of this the desire arose from me of knowing how is composed a seakeeping tank and the theory behind that. That is because most new constructions in the naval field must to be tested before the prototype can be built.

1.2 Objective

The objective of this thesis is to determine what are the capacities of the tank, recently built at the Naval Architecture and Ocean Engineering school, to carry out different research and teaching activities.

To do this, the first step is to stablish the state of the art of Naval Architecture experimentation.

With this in mind, the entire system in the tank must be designed to reproduce the environment necessary to test different structures. With this, it will be possible to know the capacities of the facility to carry out the tests.

Finally, the definition of parameters that can be measured in the tank and the instrumentation that is required for the measurement. The last part will be the design of a model with all the necessary instrumentation to measure all the parameters when different research and teaching activities are carried out.

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1.3 Structure of the thesis

This thesis has been structure and developed in 8 chapters.

Chapter 1. Introduction.

Brief introduction to the thesis with the motivation and objective together with the structure of the thesis.

Chapter 2. State of art in test modeling.

Description of why testing is still necessary today, the different models that can be developed, the laws that allow it and a planning scheme to carry out a test. A brief introduction to the wave generator and their types. To finish with some of the most important facilities in the world. The problems and limitations expected during the thesis are also shown.

Chapter 3. Wavemaker theory.

Study of the different types of wave generator and development of the first order theory that will be used to design the wave maker.

Chapter 4. Wavemaker design with first order water theory.

Definition of the conditions for which the wavemaker must be design. Wave generator development using the first order theory previously shown. Development of the characteristics that the actuator system must have and its choice of catalogue. Introduction to the beach theory, which one is chosen and why.

Chapter 5. Models.

Study of the restriction that conditions the construction of the model and the subsequent design and development of the model and its mooring system.

Chapter 6. Instrumentation.

Description of all the measurement tools necessary to measure all the parameters that characterize the behaviour of the tested structure. It is also contemplated how the acquisition of all these information is collected from the instrumentation.

Chapter 7. Program developed.

Brief description of the program developed to perform the entire thesis calculation.

Chapter 8. Conclusion.

Presentation of the conclusions obtained during the development of the project together with the possible improvements that can be made and a general analysis of the project.

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Chapter 2 State of art in test modeling

2.1 Purpose of Model Testing

Models of a physic system are, in essence, a system from which the behaviour of the original physic system can be predicted. This type of models is particularly advantageous when the analysis of the prototype is complicated or uncertain, as well as when the construction of a prototype would not be economically viable and would carry a lot of risk without the prediction of the preliminary behaviour.

Test models offer great savings compared to full scale tests. However, this can also become an expensive business. The larger the model, the more accurate the data collected, but the higher the cost to run. Therefore, models must be carefully planned to reduce time and cost. Some advantages of the test models are:

• Investigate a situation that cannot be carried out analytically.

• Obtain the required empirical coefficients in analytical prediction equations.

• Justify an analytic technique by predicting the behaviour of the model and the relationship between predicted and actual behaviour.

• Evaluate the neglected high-order effects in the simplified analysis by correlating the differences between the predicted model behaviour and the actual model behaviour.

Therefore, the test model is a necessary experimental procedure when the analytical technique fails or is not capable of predicting the behaviour with the required accurate. Models can be classified into three groups:

1. First group consist of scale prototype replicas whose behaviour is equal in nature but not in magnitude.

2. The second group consist of distorted, in such a way that there is a general but not complete similarity. For example, a certain distance, such as the deep of the tank. In this case, some kind of prediction will have to be used to transfer the behaviour of the model to that of the prototype.

3. The third group is known as the analogue. The system consist of a different physical appearance but an analogue behaviour of the prototype.

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Today, computer modelling and analysis is a reality. Model tests are complementary to computer analysis to achieve the best possible design. In this thesis, only the test part of the model will be dealt with, in the future this thesis could be complemented with a computer analysis. It would be interesting to study a possible implementation of machine learning. A large amount of input data from previous models and their test results would be analysed by a RNN, recurrent neural network. Then, the RNN could give an output in which the design of a model would be executed and the RNN would give the behaviour of the model without the need to test it.

2.2 Models

Models in engineering projects can be classified in two categories:

• Display models

• Engineering models

Display models are commonly used to show both the product and its working or to sell the product. While the engineering models are used to recollect useful data that will help during the design. The engineering models are divided into two, constructibility models and measurement models. The last one is also known as testing model, Figure 2.1.

Constructibility models are built at scale to ensure that a design concept is feasible for construction. Measurement elements are not commonly used. A part of the system or a complete system is used to build a proof of a concept. An instance would be the habitability module inside of an offshore floating structure. Important improvements are usually developed thank to this type of model.

Measurement models are scale designs of the system with the objective of obtaining direct data for the prototype design and operation. To ensure the similarity between the prototype and the model a series of correlation laws must be followed. These correlation laws are often derived for the dimensional analysis. A dimensional analysis will be made in the following sections.

Figure 2.1: Scheme of engineering model of a wave tank. Source: own elaboration.

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2.3 Modeling criteria

The objective of an engineering model is to produce a model that can perform in a similar way to the prototype. In the naval industry often is not possible to build a full size model, thus it is necessary to build a scale model, Figure 2.2, that perform in a manner physically similar.

There are rules that guarantee the scalability of the model.

The rules for quantifying and scaling model responses are called the laws of similitude. There are two essential laws. The first is geometric similitude. In this law, the flow field and boundary geometry of the model and of the prototype must have the same shape. Therefore, the proportions of all model lengths to their corresponding prototype lengths are the same.

The second essential law of similitude is kinematic similitude, or motion similarity. According to this law, the proportions of corresponding velocities and accelerations must be equal between the model and the prototype. In wave modelling, there are three main factors that influence the scale choice [1]:

1. Model development.

2. Tank blockage.

3. Wave generation capacity.

For building models, commonly a larger scale results in easier construction and material selection. In terms of tank blockage, the smaller the scale is, the better will be the quality of waves in the tank. That means less contamination as a consequence of reflection from the tank to the model. Finally, the wave-maker must be capable of producing the scaled wave heights and periods desired for the tests. Each criterion can lead to different ”best scales”.

The scale is chosen as a compromise between technical requirements and the cost of the project for similitude. It should be noted that the widely held belief that the larger the scale model, the better it is, can not always be true. Several other considerations must be taken into account in the scale selection. Frequently, it is necessary more than one model to study the different phenomena experienced by a structure. For example, the entire structure can be at one scale, whereas a small section of it can be studied using a larger scale model.

Figure 2.2: Model of offshore salmon fish farming. Source: [2].

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While it is not possible to suggest an optimum scale factor for a structure without investigating all parameters of importance, a common scale factor used in wave effects in a tank is 1:50. A range of scales for a wave tank is normally between 1:10 and 1:100.

2.4 Dimensional analysis

Most structures can be investigated through small scale models whose behaviour is related with the prototype in a prescribed mode [3]. The problem in scaling is to obtain an appropriate scaling law that describes accurately this similarity. This requires an exhaustive understanding of the physical concepts involved in the system.

One method of relating the model properties to the prototype properties is the parametric approach in which the Buckingham Pi T heorem is applied to all applicable variables to obtain a group of relevant dimensionless quantities. In this method the most important variables on the dynamics of the system are identified first. This is the most important step in the similarity analysis by this method. If a important variable is omitted the result will be erroneous. On the other hand, if too many variables are included, even those least significant, then, the scaling law become too complicated and impossible to satisfy. It should be realized that a complete similitude cannot be achieved except at a one-to-one scale. The scale is chosen as a compromise between cost, complexity and technical requirements for similitude.

After the variables have been selected, it is possible to obtain an independent and convenient collection of dimensionless parameters, pi terms. The equality of the pi terms for the model and prototype produce the similitude requirements, or scaling laws, that must satisfied. If the corresponding pi terms are equal, the model and prototype structural systems are similar. This relationship provides the prototype values from the model test and is called the prediction equation of the system.

The principal forces encountered in a hydrodynamics model test [1] are:

• Gravity force:

F_G = Mg (2.1)

• Inertia force:

F_I= M ˙u (2.2)

• Viscous force:

F_V= µA(du/dy) (2.3)

• Drag force:

F_D= 1/2C_Dρ Au² (2.4)

• Pressure force:

F_p = pA (2.5)

• Elastic force:

F_e= EA (2.6)

Variables: M ≡ mass of the structure, u, ˙u ≡ velocity and acceleration of fluid, y ≡ vertical coordinate, A ≡ area, P ≡ fluid pressure.

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From this farces can be found the principal dimensionless number in modelling scale. These number are shown below, see Table 2.1.

Symbol Dimensionless number Definition Ratio

Fn Froude Number v²/gD Inertia/Gravity

Rn Reynolds Number pvD/µ Inertia/Viscous

St Strouhal Number f_eD/v Vortex Shedding Frequency KC Keulegan-Carpenter Number vT/D Period Parameter

Ur Ursell Number HL²/d³ Depth Parameter

Ch Cauchy Number ρ v²/E Elastic Parameter

Table 2.1: Common dimensionless number in modelling scale. Source: [1].

Hydrodynamics scaling laws are determined from the ratio of these forces. If all of these rules are satisfied the dynamic similitude will be achieved. But, most of cases only one of these is satisfies by the scale model. It is important to choose the right one. The most common in the water problem is the Froude’s law. While Reynolds number plays an important role too it is impossible to achieve the equality in froude number and reynolds number in scale model technology at the same time, it only occurs at scale 1:1, see Table 2.1.

In the case of water flow with a free surface, the predominant effect is gravitational. The effect of other factors, such as viscosity, surface tension, roughness, etc, is generally small and can be neglected [1]. That is why Froude’s law is usually chosen as the law to scale. Then, in this thesis the scale law to achieve a relationship between the behaviour of the prototype and the model will be Froude’s law. Therefore it is the law that will be following developed.

2.5 Froude’s law

The Froude number contemplate the effect of gravity on the system in question, which means that contains the gravitational acceleration term. The Froude number is defined as the ratio of the inertia force to the gravitational force developed on an fluid element in a medium.

Fn= u

√gL; (2.7)

Variables: Fn ≡ froude’s number, u ≡ velocity, g ≡ gravity, L ≡ corresponding length.

Scale, λ , is defined as the ratio between the length of the prototype and the length of the model.

Therefore, it is possible, matching the Froude term of the model and the prototype, to obtain the relation between velocities.

L_p= λ Lm, (2.8)

Fr= u²_p

gL_p = u²_m

gL_m; (2.9)

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Also once that the scale has been defined it is possible to obtain the relationship between all variables. The following tables, Table 2.2 and Table 2.3, show those relationships.

Variable Unit Scale

Principal dimension

Length L λ

Time T λ^1/2

Mass M λ³

Geometry

Area L² λ²

Volume L³ λ³

Angle - 1

Moment of inertia L⁴ λ⁴

M. of inertia mass ML² λ⁵ Kinematics, Dynamics

Velocity LT⁻¹ λ^1/2

Acceleration LT⁻² 1

Angular velocity T⁻¹ λ^1/2 Angular acceleration T⁻² λ⁻¹ Dumping coefficient MT⁻¹ λ^5/2

Momentum MLT⁻¹ λ^7/2

Angular momentum ML²T⁻¹ λ^9/2 Table 2.2: Principal dimensionless relationship

1.

Variable Unit Scale

Static

Stiffness ML³T⁻² λ⁵

Stress ML⁻¹T⁻² λ

Moment ML²T⁻² λ⁴

Shear MLT⁻² λ³

Section Modulus L³ λ³

Wave Mechanics

Wave Height L λ

Wave Period T λ^1/2

Wave Length L λ

Celerity LT⁻¹ λ^1/2

Wave Pressure ML⁻¹T⁻² λ

Keulegan-Carpenter - 1

Material Properties

Density ML⁻³ 1

Modulus Elasticity ML⁻¹T⁻² λ Modulus of rigidity ML⁻¹T⁻² λ Table 2.3: Principal dimensionless relationship 2.

According to Froude’s law the Keulegan-Carpenter number, Table 2.1, in the prototype and the model is the same,

(KC)_p= (KC)_m, (2.10)

whereas,

(Re)_p= λ³²(Re)_m; (2.11)

The Keulegan-Carpenter number follows Froude’s law. This allows the dependence on KC to guarantee the model values are applicable to the prototype. However, direct scaling is not possible if the quantities are strongly dependent on the Reynolds number. Also, for a small-scale model in a wave tank, the Reynolds number of the prototype cannot even be approximated due to the low fluid velocity in the model.

There are some problems with Strouhal number too. Fluid flow through a small member of a structure creates low pressure behind the member, and causes vortices to break away from the surface of the member. This vortex formation behind the limb is function of the Strouhal number. The eddy-shedding frequency, f e, in the Strouhal number depends on Re. Therefore, the Strouhal numbers in a model and a prototype are different and do not follow Froude’s law.

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Experiments have shown that flow characteristics in the boundary layer are presumably to be laminar at Re < 10⁵, while the boundary layer is turbulent for Re > 10⁶. Therefore, most small ratios model and test conditions, scaled by Froude number, will result in laminar flow conditions while full-scale conditions are certainly turbulent. Once the flow regime is turbulent, the drag coefficient depends only weakly on the Reynolds number while as long as it is laminar the drag coefficient depends strongly on the Reynolds number. The turbulent flow can be verified by visual examination and pressure transducer data. It can be confirmed that the resistance is only weakly a function of the Reynolds number within the turbulent region, by testing two models with identical shapes but with different scales [1]. A practical answer to this problem in model is to intentionally "trip" the laminar flow by some kind of roughness near the bow of the structure. In testing tanker models, different methods have been employed including struts placed upstream of the vessel, cables attached at a point just aft of the bow or sand-strips, studs or pins attached directly to the hull. Studs have been shown to appear to be the most effective method of stimulating turbulent flow at lower velocity and over a wider region of the wetted surface area.

Another problem might be the material election of the model. Some quantities as cross-sectional area, section modulus, moment of inertia, etc, follow Froude scaling. However, if the same prototype material is used to build the model, the stiffness of the model, EI, where I is the moment of inertia, will not scale the prototype stiffness. Thus, the model behavior will not correctly predict the prototype performance. Therefore, to scale the stiffness of the structure, a suitable material should be chosen of the model so that the Young’s modulus scales linearly with the scale factor. Often, however, the same material as the prototype is used in model testing. In this case, the model is said to be distorted. In the model that will be developed in Chapter 5 a scale material will be chosen to reach the Froude’s law.

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2.6 Planing a model test

To make sure that the model test will be successful it is necessary to plan it before. An outline of the develop has been made to summarize the process of a model test, see Figure 2.3.

Figure 2.3: Scheme of general design of a structure. Source: own elaboration.

2.7 Hydrodynamic testing facilities in the world

In this section we are going to present some of the most important facilities in the world to have a general overview about the state of art of these tests and some references. As can be seen below, Table 2.4, most of them use piston as the way to generate waves. It is also important to note that it is rare the depth exceed 5 meters ans the usual wave height is about 0.5 meters.

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State and Institution Tank Tank dimensions [m] Wave maker type Regular Wave Height USA

Naval Surface Warfare Center, Carderock Division

Maneuvering and

Seakeeping Basin 106 x 76 x 6 Flap 0.6 m

Russian Federation Krylov State Research Centre

Maneuvering and

Seakeeping Basin 162 x 37 x 5 Flap 0.35 m

Public Republic of China

China Ship Scientific Research Center Seakeeping Basin 69 x 46 x 4 Flap 0.5 m

Japan

National Maritime Research Institute Actual Sea Model Basin 80 x 40 x 4.5 Flap 0.35 m South Korea

Korea Research Institute of Ships and Ocean Engineering

Towing Tank 221 x 16 x 7 Flap 0.5 m

Australia

Australian Maritime College Towing Tank 100 x 3.5 x 1.5 Flap 0.7 m

India Naval Science and Technological Laboratory

Seakeeping and

Maneuvering basin 135 x 37 x 5 Flap 0.5 m

Iran

National Iranian Marine Laboratory Towing Tank 400 x 6 x 4 Piston 0.5 m

Brazil

Brazilian Ocean Technology Laboratory Ocean Basin 40 x 30 x 15 Flap 0.5 m

Spain

University of Cantabria

Cantabria Coastal

and Ocean Basin 44 x 30 x 4 Flap 1 m

Spain

Hydrodynamic Experiences of the Pardo

Maneuvering and

Seakeeping Tank 150 x 30 x 5 Flap 0.9 m

Denmark

Aalborg University Wave and Current Basin 14.6 x 19.3 x 1.5 Piston 0.45 m

Table 2.4: Facilities in the world

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2.8 Conclusion

Taking into account everything seen so far and the type of tank built in the university department.

The expected limitations are:

• Tank blockage:

This occurs when due to the size ratio there is interference with the tank due to reflected waves. The waves can be reflected both on the bottom and on the walls.

• Wave development:

Development of the wave, due to the short length of the tank. One of the most important limiting factors in the design will be the wave generator and the distance required to generate a stable wave.

• Wave absorption:

Absorption of the wave, this occurs at the rear of the tank. The goal is to minimize it as much as possible so that it does not interfere with the test.

• Selection of material:

Choice of material, there may be problems with the material when choosing the model due to the large scale. Therefore, it is anticipated that large prototypes will not be able to be chosen to test your model.

The main limitation when choosing the types of tests that can be carried out is that they have to be static. That is, the tested structure will not move through the tank. The results will be measured with the static model in one place. The tests that are expected to be carried out on the tank are:

• Hydrostatic tests.

• Response tests of the structure against different types of regular waves.

• Mooring system tests, it will be possible to test the response of different mooring systems according to the type of waves.

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Chapter 3 Wavemaker theory

In this chapter, the develop of the first order water waves theory is developed. First, a brief description of wave generator a its types is presented. Later, the first order water waves theory is developed to find the wave potential of the board. With this wave potential it is possible to find the relation between the developed wave height and the stroke of the board. It is also essential to find the expression to obtain the pressure exerted by the board and therefore the force and power required.

3.1 Wave generator

An essential part when choosing the size of the models is the dimensions of the wave generator tank. According to documentation, the minimum size for this type of facility is usually 25 x 25 meters or more. With this size it is ensured that the test does not suffer any type of interference due to scale effects. Furthermore, the fact that they are square make it easier to simulate waves in any direction. Due to space problems at the university, the dimensions of the facility will be 8.4 x 2 x 1.6 meters.

The facility for model testing of an offshore structure generally consists of a wave generating basin, a towing tank, a wind generator and a current generating facility. It is advantageous to simulate the generation of waves, wind and current in a single basin so that their combined interaction with the structure model may be investigated. The wind generation in a wave basin is often carried out using a series of blowers located just above the water surface near the model.

The long period waves in the ocean can exhibit unidirectional behaviour. However, wind-generated ocean waves are generally multi-directional. In order to generate multi-directional waves, the wave basins must have widths comparable to their lengths. Many modern facilities are capable of producing multi-directional waves

This chapter will describe different types of wave generators, their design methods and the generation capabilities of many commercial wave testing facilities. Also describe the wave absorber theory and various types of waves absorber.

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3.1.1 Types of wave generators

There are many types of wave makers used in basins. The most important are summarized below. The wedge shaped wave generator is best suited for high frequency waves but has limited success in generating low frequency waves. On the other hand, the pneumatic wave maker can generate low frequency waves quite well but is limited to high frequencies of about 1 Hz due to the quick response time required to form these waves. On this thesis, considering the space of the basin, will be treated:

• Simple flap

• Piston

As can be shown below, Figures 3.1 and 3.2 show a sketch of a simple flap and double flap.

Figure 3.1: Scheme of simple flap.

Source: [4].

Figure 3.2: Scheme of double flap.

Source: [1].

The sketch of piston, Figures 3.3, vertical wedge, Figures 3.4, and pneumatic type, Figures 3.5, can also be shown below.

Figure 3.3: Scheme of piston.

Source: [4].

Figure 3.4: Scheme of Vertical wedge.

Source: [1].

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Figure 3.5: Scheme of pneumatic. Source: [1].

3.2 First order wavemaker theory

This is going to be the main reference sketch during the explanation, Figure 3.6.

Figure 3.6: Two-Dimensional Wave Flume Definition Sketch. Source: [5].

The main objective of this chapter is to describe an equation that describes the motion of the board in relation to the generated wave. The way to find out is from the boundary conditions of the wave potential [5] and a new boundary condition of the board. Therefore, the boundary conditions of the generated wave potential will be the starting point to find the equation that

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describes the characteristics of the wave generator. The boundary conditions of the wave potential [5] are:

• Laplace equation

∂²φ

∂ x² +∂²φ

∂ z² = 0; (3.1)

• Bottom Boundary condition for horizontal bottom

∂ φ

∂ z = 0 at z= −h; (3.2)

• Kinematic free surface condition

∂ η

∂ t +∂ φ

∂ x

∂ η

∂ x −∂ φ

∂ z = 0 at z= η; (3.3)

• Dynamic free surface boundary condition

∂ φ

∂ t +1 2

"

∂ φ

∂ x

2

+

∂ φ

∂ z

2#

+ gη = 0 at z= η. (3.4)

Variables: φ = φ (x, y, z) ≡ wave velocity potential, z ≡ distance from free surface to the bottom, t ≡ time, η ≡ wave height.

Now, the boundary condition of the wave board is formulated assuming that the wave board is flat, solid and waterproof. Therefore, the velocity of the fluid normal to the wave board has the same velocity as the wave board. Then, the boundary condition for this side becomes [5],

r∂ θ

∂ t = v_n, (3.5)

Variables: θ ≡ Aperture angle of flap in each time step, r ≡ Board length, vn ≡ Velocity normal to the board.

From the geometry of the previous reference sketch, Figure 3.1, the stroke can be developed at each time and water depth,

r= q

(X (z,t))²+ (l + h + z)²= (l + h + z) q

1 + (tan θ )², (3.6)

where normal velocity v_ncan be writen as function of linear and angular velocity,

v_n= u cos θ − ω sin θ . (3.7)

The horizontal position of the stroke in each time step may be writen as,

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X(z,t) =

1 + z h+ l

· X_o(t); (3.8)

Variables: r ≡ board length; X(z, t) ≡ Stroke at each time step; l ≡ Distance between hinge point and bottom; h ≡ Water deep; θ ≡ Aperture angle of flap in each time step; v_n ≡ Velocity normal to the board; u, ω ≡ Horizontal and vertical velocity.

Substituting these relations within the boundary condition,

(l + h + z) q

1 + (tan θ )²∂ θ

∂ t = u cos θ − ω sin θ , (3.9)

assuming θ to be small,

ucos θ ≈ u; ω sin θ ≈ 0; (tan θ )²≪ 1, (3.10)

then,

θ ≈ X(z,t)

(l + h + z); (3.11)

Substituting, it is possible to obtain the general wave board boundary condition [5]:

∂ φ

∂ x =

1 + z

h+ l

∂ X_o(t)

∂ t at x= X (z,t) (3.12)

• l = 0, boundary condition for a flap type.

• l = ∞, boundary condition for a piston type.

Once the boundary condition has been developed, it is time to obtain an equation that defines the wave generator. This solution for the wave generator will be obtained using the boundary conditions indicated above. To find the solution to the wavemaker problem, it is assumed to write φ₁in the following manner,

φ1(x, z,t) = X (x)Y (y)T (t); (3.13)

Making this assumption, it is possible to divide Laplace’s equation into ordinary differential equations with known solutions. Since of all them have to be contained in the potential function, it is written as follows [5],

φ₁(x, z,t) = φk1+ φk2+ φk3 (3.14)

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The kind of solutions are when the separation constant is respectively real, zero and imaginary.

• k²₁> 1

φ_k₁= [A₁cos (k₁x+ α1)]

C₁e^k¹²+ D₁e^−k¹²

T₁(t) (3.15)

• k₂= 0

φ_k₂ = (A₂x+ B₂) (C₂z+ D₂) T₂(t) (3.16)

• k²₃> 1

φ_k₃=

A₃e^|k³^|x+ B₃e^−|k³^|x

(C₃cos |k3| z + D₃sin |k3| z) T₃(t) (3.17)

Evaluating the potentials under boundary conditions and making simplifications it is possible to obtain,

φ_k₁ = A cosh [k₁(h + z)] [sin (k₁x− ω1t+ γ1) + sin (k₁x+ ω1t+ ε1)] ; (3.18)

φ_k₂ = (Bx + D); (3.19)

φ_k₃ = Ce^−|k³^|xcos [|k₃| (z + h)] cos (ω3t+ α3) ; (3.20)

where the dispersion relationships are,

ω₁²= gk₁tanh(k₁h); (3.21)

ω₃²= −g|k₃| tan(|k₃|h); (3.22)

Variables: ω ≡ Angular velocity; g ≡ Gravity; k ≡ Wave number; h ≡ Water deep.

φ_k₂ will give, after the application of the vector differential operator ∇ , a constant horizontal velocity component. Since it is not possible to have flow through the paddle, we can set B equal to zero. The remaining constant D is arbitrary and so it is possible to choose it again equal to zero.

As shown in Figure 3.7 and Figure 3.8, setting a value for ω and h, there is only one solution k₁ for the first dispersion relationship and infinite solutions k3for second dispersion relationship.

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Figure 3.7: Dispersion relationship for progressive waves.

Source: [5].

Figure 3.8: Dispersion relationship for standing waves.

Source: [5].

As already seen previously, k₁ is the wave number of a progressive wave, while the infinite values of k₃, that satisfy the last equation for fixed ω and h, are wave numbers of standing waves. These latter quantities result in many values of φ_k₃ that must be considered as a summation. The origin of these standing waves is due to the fact that the solid wave board does not exactly follow the velocity motion of a progressive first order wave.

Summing up the last results,

φ₁(x, z,t) = φk₁+ φk₂+ φk₃, (3.23)

φ₁(x, z,t) = A cosh [k₁(h + z)] sin (k₁x− ωt) + cos(ωt)

∞

∑

n=1

C_ne^−k³ⁿ^xcos [k₃_n(z + h)] , (3.24)

Variables: k₁≡ Wave number of progressive waves; k₃_n ≡ Wave number of standing waves;

h ≡ Water deep; z ≡ Distance from the free surface to the bottom; ω ≡ Angular velocity.

The first term of the potential φ₁is a progressive wave propagating in the positive x direction with its wave number k₁. The second term is a series of standing waves, each one of them with its wave number k3_n. As can be seen, these standing waves decay exponentially with the distance x from the paddle. According to Hughes [5], the amplitude of the first standing wave, the most important one, will decrease by 99% at x = 3h. This means that at three times the depth of the water the effect of standing waves can be neglected.

The wave board is assumed to follow the following sinusoidal motion,

X_o₁ =S_o

2 sin(ωt); (3.25)

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Operating, it is possible to obtain the coefficient A and C_n,

A=ω S_o 2k₁

R₀

−hf(z) cosh [k₁(h + z)] dz R₀

−h(cosh [k₁(h + z)])²dz , (3.26)

C_n= −ω S_o 2k₃_n

R₀

−hf(z) cos [k₃_n(z + h)] dz R₀

−h(cos [k₃_n(z + h)])²dz , (3.27) Variables: S₀≡ Stroke at z = 0 (Free surface); f(z) ≡ Board geometry description.

To complete the analysis, the expression of the surface elevation η₁ and the general transfer function between wave board stroke, S₀, and wave height H has to be found.

By substituting the expression for φ₁ within the first order Dynamic Free Sur f ace Boundary Condition and evaluating it at z, it is possible to obtain the surface elevation in the wave tank.

η₁(x,t) =ω A

g cosh (k₁h) cos (k₁x− ωt) + sin(wt)

∞ n=1

∑

ωC_n

g e^−k³ⁿ^xcos (k_3nh) , [m] (3.28)

Variables: η1≡ Wave height in each step.

As previously stated, from a distance of x = 3h from the wavemaker, there are not disturbances. For this reason, far from the wavemaker, it is possible to neglect the summation terms that mathematically describe this phenomenon. Then, the surface elevation can be described as the progressive wave solution,

η₁(x,t) =H

2 cos (k₁x− ωt) , [m] (3.29)

Variables: H ≡ Wave amplitude.

Equating the last two expressions for η₁ without considering the terms of the series, the general transfer function between wave board stroke S₀and wave height H is obtained,

H= 2ωA

g cosh (k₁h) ; [m] (3.30)

This is a general expression from which the relationship for different types of wavemaker can be found. It is possible to substitute in f (z) the function that describes the particular geometry of the wave board .

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3.3 Piston and Flap type

With reference to the main sketch, the wave board geometry is described by,

f(z) =

1 + z

h+ l

; (3.31)

As already mentioned, this expression can describe the geometry of both piston-type and flap-type wave generators:

• l = 0 , boundary condition for a flap type.

• l = ∞ , boundary condition for a piston type.

Developing the expressions for A and C_n, it is obtained,

A= 2ωS_o

k(sinh 2kh + 2kh)

sinh kh +(1 − cosh kh) k(h + l)

, [m²/s] (3.32)

C_n= − 2ωS_o

k_n(sinh 2k_nh+ 2k_nh)

sin k_nh+(cos k_nh− 1) k_n(h + l)

, [m²/s] (3.33)

Variables: ω ≡ Angular velocity; S₀ ≡ Stroke at free surface; k ≡ Wave number of progressive waves; k_n ≡ Wave number of standing waves; h ≡ Water deep; l ≡ Distance between hinge point and bottom; z ≡ Distance from the free surface to the bottom.

Substituting the expression A in the expression H and using the dispersion relationship to explicit ω, the General First-Order Wavemaker Solution is obtained,

H

S_o = 4 sinh kh sinh 2kh + 2kh

; (3.34)

From this relation, the transfer function can be found for both piston and flap type wavemakers:

• l = 0 , boundary condition for a flap type H

sinh kh +(1 − cosh kh) kh

; (3.35)

• l = ∞ , boundary condition for a piston type H

S_o = 4 sinh kh

sinh 2kh + 2kh[sinh kh]; (3.36)

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3.3.1 Wave Board Pressure

Assuming that there is only water on one side of the wave board, a first order expression is had for the pressure distribution in the paddle from the linearized Bernoulli equation [5],

p(x, z,t) = −ρ∂ φ1

∂ t − ρgz; (3.37)

Variables: p(x, z, t) ≡ Pressure; ρ ≡ Density; φ1≡ Generated wave potential; g ≡ Gravity;

z ≡ Distance from free surface to bottom.

The first term is called the dynamic pressure term and second term is known as the hydrostatic term. Substituting the expression for φ₁ and evaluating the result at x = 0, the general relationship for the instantaneous pressure acting on the wave board is obtained. The particular expressions for the piston and flap type wavemaker can be derived using l = ∞ or l= 0 respectively.

p_o(z,t) = [ρωA cosh k(h + z)] cos ωt +

"

ρ ω

∞

∑

n=1

C_ncos k_n(h + z)

#

sin ωt − ρgz; (3.38)

The first element is the resistive term of the dynamic pressure fluctuation and is in phase with the velocity of the paddle. The second element is the inertia term of the dynamic pressure fluctuation, this term is in phase with the acceleration of the paddle. The third element is the hydrostatic pressure acting on the paddle and it is usually the most relevant of the three terms.

If there is water of the same depth on both sides of the board, it is necessary to add the contribution that comes from the waves that propagates in the negative x-direction. This results in the doubling of the dynamic pressure terms and also the elimination of the hydrostatic term.

p_oo(z,t) = 2[ρωA cosh k(h + z)] cos ωt + 2

"

ρ ω

∞

∑

n=1

C_ncos k_n(h + z)

#

sin ωt; (3.39)

3.3.2 Wave Board Force

Assuming again water on only one side, the total instantaneous force acting per unit width of the paddle is,

F_T_o(t) = Z 0

−h

p_o(z,t)dz, (3.40)

and this yields to,

F_T_o(t) =

ρ ω A sinh kh k

cos ωt + ρ ω

∞

∑

n=1

C_n

k_n sin k_nh

!

sin ωt +ρ gh²

2 ; (3.41)

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If there is water of equal depth on both sides of the board, it can be written,

F_T_oo(t) = Z ₀

−h

p_oo(z,t)dz, (3.42)

and then,

F_T_oo(t) = 2

ρ ω A sinh kh k

cos ωt + 2 ρ ω

∞ n=1

∑

C_n

k_n sin k_nh

!

sin ωt; (3.43)

The single sub 0 meaning water on one side and the double sub 0 meaning water on both sides.

3.3.3 Wave Board Power

Neglecting friction and mechanical losses, starting again with water on only one side of the paddle, the instantaneous power per unit width of board is,

P_o(t) = Z ₀

−h

p_o(z,t)u_o(z,t)dz, (3.44)

and this yields to,

P_o(t) =

ρ ω²S_oA 2k

(cos ωt)²+ (3.45)

ρ ω²S_o 2

_∞

∑

n=1

C_n k_n

sin ωt cos ωt +

ρ gω S_o 2

h²

2 − h³ 3(h + l)

cos ωt;

If there is water on both sides of the waves board, it can be written,

P_oo(t) = Z 0

−h

p_oo(z,t)u_o(z,t)dz, (3.46)

solving the integral,

P_o(t) = 2

ρ ω²S_oA 2k

(cos ωt)²+ (3.47)

2

ρ ω²S_o 2

_∞

∑

n=1

C_n k_n

sin ωt cos ωt;

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It is also possible to obtain the expression for the mean wave board power over a wave cycle writing,

P_o= 1 T

Z +T /2

−T /2

P_o(t)dt, (3.48)

then, substituting the dispersion relationship,

P_o=π ρ gS²_o kT

tanh kh sinh 2kh + 2kh

2

; (3.49)

For water on both sides of the paddle it is had,

P_oo = 2π ρ gS²_o kT

tanh kh sinh 2kh + 2kh

2

; (3.50)

The single sub 0 meaning water on one side and the double sub 0 meaning water on both sides.

According to Hughes [5], piston type wave generator are more efficient in shallow water because their motion approximates better the nearly uniform vertical distribution of the horizontal fluid particles. In contrast, flap type wavemakers are more efficient for generating deep water waves.

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Chapter 4 Wavemaker design with first order water waves theory

In this chapter, the design method of the wavemaker is presented. It is based on the linear water waves theory. Using this theory, the wave generation potentialities of the tank will be analysed.

For this, the maximum stroke reachable for every period and the corresponding maximum wave height will be evaluated for a period range of interest. Several other values for strokes and wave heights will be obtained to have a complete overview of the characteristics of the system. The graphs of the wave length and celerity with respect to the period, complete this first analysis.

Then, the forces and power necessary to generate the desired waves will be obtained. These results will be used later on the actuator system as a performance requirement.

Once the forces and powers are known, it is necessary to design the paddle and the actuation system. For that reasons, the velocities and forces acting on the paddle, to generate the desired waves, are calculated. The results of these analysis are used to make a first dimensioning of the actuation system, composed by a electric cylinder driven by a brush-less motor.

The last part will be the study of the possible types of waves absorbers, also called beaches, that can minimize the wave reflection in the tank according with the space problem.

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4.1 Problem definition

The water tank that will be equipped with the wavemaker is 8.4 m x 2 m x 1.5 m. A scheme is shown below in Figure 4.1.

Figure 4.1: Scheme of Tank. Source: University.

The range of interest of the wave period is up to 1.5 s. This period range is due to space problem. As mentioned in the previous chapter, the minimum distance to perform the test is three times the depth of the water plus the stroke necessary to develop the wave, which is around 3.2 meters. On the other hand, as will be seen later, the distance of the wave absorber is around 1 meter. Adding these distances and subtracting the total length of the tank, 8.4 meters, the testing space is around 4.2 meters. Wavelength must be shorter than testing place to avoid the reflecting of the same wave while the models are been testing.

Figure 4.2: Scheme of different zones in the tank. Source: Own elaboration.

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Due to there is a space problem, the flap and piston type wavemaker have been considered.

From now on, the text will be referred to the following sketches:

Figure 4.3: Scheme of Flap Actuator.

Source: [6].

Figure 4.4: Scheme of Piston Actuator.

Source: Own elaboration.

Both configurations will have an actuation system comprised of an electric cylinder driven by a brushless motor. In this way, the rotational movement of the motor is converted into a translation that drives the flap and piston motion. Both configurations, water on one side and water on both sides, will be discussed in the forces evaluation. Further on, it will be focused only on the configuration with water on both sides. The following results were calculate using a program develop in Python by me. The interface of the program can be shown below in Figure 4.5. The code and the program are attach in the pdf.

Figure 4.5: Interface of the program. Source: Own elaboration.

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4.2 Waves ideally generable in the tank

Summing up, the input data are:

• Tank geometry: 8.4 m x 2 m x 1.5 m

• Wave period interest range: T = 0.5 − 1.5 s

• Water depth: h = 0.9 m

• Actuator height: h_c= 1.6 m

First, the wave generation capability of the tank is evaluated. For each wave period of the above mentioned range, there will be a maximum wave height expected according to the linear water waves theory. To find this information, the following limits that constrain the system are introduced:

• Wave break limit, if this ratio is exceeded, it will not have regular shaped wave and the wave probably breaks. That limit is only relevant in high order theories. In this thesis only long wave theory will be developed, so this limit will no be taken into account.

H λ <1

7; (4.1)

• Maximum wave height in the tank, maximum wave height to prevent the water spill.

H_max = h_max − h = 1.5 − 0.9 = 0.6 m; (4.2)

• Ballscrew maximum acceleration, this value was assumed in first instance according to typical electric cylinder accelerations.

a_max= 1g, m/s², (4.3)

(t) = S_c

2 sin ωt Deriving twice respect to the time

−→ a(t) = −S_c

2ω²sin ωt, (4.4) S_c

2ω²≤ 9.81 → S_c= g

2π²T² m; (4.5)

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• Maximum actuator stroke, this limit arises from another limit on the control system.

If the control the position of the flap is wanted, the maximum recommended angle excursion, θ , is set to +/− 12^◦ [7]. Controlling the force, it is possible to reach a maximum angle of +/− 18^◦ [7]. Since the main interest is to control the position, it is considered,

θ = 12^◦, h_c= 1.6 m, S_c

2 = h_c· tan θ = 0.34 m ⇒ S_c= 0.68 m, (4.6)

S_c≈ 0.7 m; (4.7)

Therefore, at h = 0.9 meters of water depth, the maximum available stroke for flap and piston will be,

S_o_{f lap}= 2 · h · tan θ = 0.38 m, (4.8)

S_o_piston = S_c= 0.7 m; (4.9)

Then, using General First-Order Wavemaker solution and applying the previously mentioned limitations, the maximum stroke, S_o_max, will be calculated for piston and flap types.

H

, (4.10)

• l = 0 , boundary condition for a flap type, H

sinh kh +(1 − cosh kh) kh

; (4.11)

Figure 4.6: Flap stroke limit at 0.9 meters of water depth.

Design of Naval Architecture Parameters for Wave Tank