Methodology for the Assessment of the Effect of Wave Wash on Moored Small Craft

Methodology for the Assessment of the Effect of Wave Wash on Moored Small Craft

MISAEL GOICOECHEA

KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES

Methodology for the Assessment of the Effect of Wave Wash on Moored Small Craft

Misael Goicoechea

Master Thesis in Naval Architecture TRITA-SCI_GRU 2019:313 KTH Royal Institute of Technology Stockholm, SWEDEN

Abstract

Wake wash problems are a rather recent issue within the maritime industry.

They involve the potential harmful effects that the waves generated by ships can produce. Erosion of the shore, smothering of local fauna and excessive motion of vessels are just examples of these effects. Having accurate infor- mation about the probability and severity of wake wash problems is funda- mental in the design stage of a vessel, a shipping route or a channel passage.

A detailed study of previous works and existing tools for simulating the poten- tial different harmful effects of wake wash is performed in order to understand the challenges with wake wash simulations. From the conclusions of the study, a procedure has been developed to evaluate different wake wash problems.

The procedure relies in an OpenFOAM based tool to obtain the generated wave. A boundary element method software: AQWA; is used to evaluate ex- cessive motion as a potential harmful effect. The potential of the procedure is tested with a validation study and a practical application. The practical case consists in the evaluation of wake wash effects on a moored vessel. From the validation study it is concluded that the wave generation prediction is accu- rate. On the other hand, the practical case presents the developed methodol- ogy as a fast, precise and inexpensive alternative for wake wash evaluation.

Sammanfattning på svenska

Problemen med fartygs våggenerering är en ny utmaning inom den

maritima industrin. Problemen innefattar de potentiellt skadliga effekterna av de vågor som genereras av fartyg. Exempel på dessa skadliga effekter är erosion av stranden, kvävning av lokal fauna och överdriven rörelse av fartyg. Att ha genomgående kunskap om sannolikheten och

konsekvenserna för problem med våggenerering är grundläggande för designen av ett fartyg, en transportväg eller en kanalpassage. En detaljerad studie av existerande verktyg för att simulera de potentiellt skadliga effekterna av våggenerering. Slutsatserna från studien skapar en procedur för att utvärdera effekterna av olika våggenereringar.

Proceduren grundar sig i ett OpenFOAM-baserat verktyg, för att simulera vågor för att se effekterna av fartygs våggenerering. Programvara för en gränselementsmetod: AQWA; används för att utvärdera överdriven rörelse som en potentiell skadlig effekt. Procedurens potential testas med en

valideringsstudie och en praktisk tillämpning. Det praktiska fallet består i utvärderingen av effekterna av fartygs våggenerering på ett förtöjt fartyg.

Från valideringsstudien så bekräftas det att proceduren är ackurat. Å andra sidan presenterar det praktiska fallet den utvecklade metodiken som ett snabbt, exakt och billigt alternativ för utvärdering av våggenerering.

Preface

This thesis is a detailed report of the research/development project carried by the author at the Spanish company SIPORT21. The project consisted in the development of a procedure to assess potential wake wash effects on moored vessels. It was part of an internal R&D program on wake wash which origi- nates from SIPORT21’s necessity of having reliable and efficient procedures and tools to address such problems.

The research program on wake wash was initiated around 20 years ago in or- der to satisfy different customers’ necessities. This type of projects normally involved participation of universities and other companies to resolve complex simulations. However, the late increase in the demand of wake wash evalua- tion, as well as, the increasing availability of powerful simulation tools; justify the necessity to develop an in-house procedure to tackle these problems.

The document is structured in 7 chapters. The first chapter provides the reader an introduction into wake wash and the general grounds of this thesis works.

The second chapter delves into the theoretical and mathematical background for the thesis. The third one is a deep study of the latest development of deter- ministic wake wash prediction. In the fourth chapter, the wake wash problem is studied in detail. The problem is classified into different cases and the avail- able methodologies to tackle them are discussed. As a result of this study, the developed procedure is presented. In the fifth chapter, the developed wave generation tool, which is the backbone of the procedure, is presented and val- idated. In the next chapter, the evaluation of wake wash effects on a moored vessel is taken as a practical application case. In this application case, the use of the procedure is demonstrated. Finally, the last chapter opens a discussion about the applicability of the procedure and for further improvement.

Both OpenFOAM and ANSYS AQWA were used for the development of this master thesis. These programs are the responsible of performing the sim- ulations in the procedure. Auxiliary programs have been developed in Python to assist in the pre and post processing of the two simulation tools.

The author would like to acknowledge the counselling provided by the two supervisors of this thesis: Raúl Atienza at Siport21 and Karl Garme at KTH. Also, to my colleagues at Siport21 which have been always ready to lend a hand.

Table of contents

METHODOLOGY FOR THE ASSESSMENT OF THE EFFECT OF WAVE WASH ON MOORED SMALL CRAFT ... I ABSTRACT ... III SAMMANFATTNING PÅ SVENSKA ... V PREFACE ... VII TABLE OF CONTENTS ... IX INTRODUCTION ... 1 THEORETICAL BACKGROUND ... 7 STATE OF THE ART OF DETERMINISTIC WAKE WASH

PREDICTIONS ... 25 CASE AND METHODOLOGY ANALYSIS AND PROCEDURE

PROPOSAL ... 29 WAVE GENERATION TOOL. ... 43 PRACTICAL CASE: WAKE-WASH EFFECTS ON A MOORED

VESSEL. ... 65 CONCLUSION AND FURTHER DEVELOPMENT ... 85 REFERENCES ... 89

Introduction

It is a well-known fact that any vessel, or for a matter of fact any object moving in the surface of water, will generate a wave system when sailing. This wave system is normally known by the name of wake. In 1887, it was mathematically described for the first time by Lord Kelvin [1] as the waves generated by a moving pressure point. Havelock [2] [3] extended Kelvin’s work to account for shallow-water wakes and hull geometry. Since then, a rather large quantity of research has been dedicated to the wake: Inui [4] [5], Peters [6], Saunders [7], Guilloton [8] [9] or Eggers [10].

Some of these works were developed from a natural science point of view, as in “knowledge for its own sake”. Yet, the predominant objective of wake re- search has been understanding vessel wave resistance and its relation with hull design [11]. This has led to a rather vessel-centric frame of reference re- garding ship generated waves. These waves, however, can travel long distances before they dissipate and, in shallow waters, close proximity to shore or in dense traffic regions, they can reach a sensitive area where they can have harmful effects.

When the wake causes severe damages or has the potential to do so it is then known as wake-wash or wave wash. Wake-wash related incidents are a rather recent issue within the maritime industry and some have even reached the general media. “In March 1999, 113 residents filed a class-action lawsuit claiming the fast ferries' waves were harming beaches” [12]. Critics in Venice denounce that the waves ships create are eroding the foundations of the his- toric lagoon city [13]. However, the most alarming one was caused by Stena Discovery. In 1999, the wake generated by Stena Discovery high speed ferry swamped a 10 metres angling boat with the loss of one life [14].

Like Titanic and the development of SOLAS, the Stena Discovery incident kin- dled an international urge to develop a set of criteria and tools to take wake wash issues into account. In 2003, PIANC workgroup 41 [15] produced a de- tailed report on the subject. The high complexity of the problem was made manifest: on one hand, the lack of reliable deterministic tools to solve the physical problems (wave generation and wave train propagation); on the other hand, the wide variety of impacts that need to be taken into account and the lack of criteria to evaluate their associated risk. A collection of the perceived risks can be seen in Table 1. The workgroup recommended resorting to risk or impact evaluation methods to overcome these obstacles.

Table 1- Wake wash risk classification [15]

Where Subject Risk Cause

Safety of People

Beach People

Caught in water Long-Period Wave inunda- tion

Knocked down Plunging and Breaking waves

Sea-Wall People Trapped against Sea Wall Long-Period Wave inunda- tion

Safety of Vessel

Open Water

Small Craft Swamped, Broached or Capsized

Short-Period and High Am- plitude

Large Vessel Difficulty in keeping course Long-Period Waves

Confined Water- ways

Vessels with small

UKC Grounded Long-Period Waves

Small Craft Difficulty in keeping course Long-Period Waves

Vessel Excessive Motion Resonant-Period and High Amplitude

Shallow Water (3m)

Small Craft Swamped, Broached or

Capsized Very Steep and Breaking waves Passengers Washed Overboard

Harbour

Moored Vessel Mooring Line Failure

Long-Period Waves Cargo Handling Disruption of Operation

Tug Operation Tug and vessel Collision Differential Response to waves

Safety of Structures

Sheltered Waters Floating Structures Excessive Motion Waves outside design pa- rameters

Coastline

Coast Structure Undercutting of Support

Structures Waves outside design pa- rameters Monuments Damage and Erosion

Environ- mental

Coastline Coastline Erosion Long Period Waves

Sheltered Waters Fauna and Flora Smothering and Species Removal

Increased immersion due to long period waves

Only the British Maritime and Coastguard Agency has incorporated PIANC’s solution to their legislation. As part of the high-speed craft code [16], British lawmakers demand a risk analysis study on any planned route in order to ob- tain the permit of operation. Other countries, such as Denmark, have devel- oped their own set of criteria based on field studies and the comparison of

conventional and high-speed vessels [17]. However, the search for global and general criteria and methods has since faltered.

Field studies are the preferred method to evaluate wake wash impact. Wave height measurement at different depths and distances of the sailing line for both a fast ferry catamaran and a conventional vessel were taken by Kofoed- Hansen and Kirkegaard [18] [19]. Didenkulova [20] introduced time-fre- quency analysis to evaluate the wake characteristics of the vessels in the Tallin Bay. Benassai [21] uses the same technique in the Bay of Naples to suggest modifications to the routes the vessels use to approach port.

Both of these solutions, field studies and risk assessment, have important drawbacks which to the author’s opinion make them insufficient. Risk analysis and risks management are very successful tools to deal with uncertainty. Yet, they are susceptible to a biased perception of risk and prone to under and over estimation. Overall, these methods are highly subjective and its success de- pendent on the available data. Field studies are deeply rooted in science and provide a quite high degree of confidence on their outcomes. However, they are expensive, have a long duration in time and must be made posteriori. The results are hard to extrapolate to other regions and areas, as well.

All in all, these methods fail to address very important necessities during the design stages of a vessel, a maritime route or, even, a dredged waterway or shore infrastructure. These necessities are the following:

• Quantitative data to evaluate the risks. Ideally, this quantitative data should be checked with pass-or-fail criteria.

• A priori data, so that risks can be assessed before they take place. This necessity is essential in early design stages as avoiding these risks should always be a design factor.

• Fast and inexpensive to obtain data, so that decisions can be made on the basis of broad knowledge of the problem and its alternatives. An ex- ample of this necessity is the choice of a speed limit as a mitigation op- tion. A field study would normally just give information on a few speed trials, which is suboptimal.

• A general method to obtain data. Therefore, a method that does not rely on local or specific information such as vessel data bases. The method must be able to produce data of different and new vessel types or differ- ent types of bathymetry without hinderance.

Despite the “wish-list” appearance of these necessities, deterministic simula- tion tools are a potential candidate to satisfy many of them. These tools can be

used to perform the numerical computation of the two dominant physical problems: wave generation in the proximity of the hull and wave propagation from the hull outwards. There is a wide variety of numerical methods to achieve this which will be discussed in the corpus of the thesis. Yet, two trends can be identified among the available reports and scientific papers. One gives more importance to the numerical computation of the wake wash generated by the vessel. Finite volume method or boundary element method are used to address the complexities of the hull and free surface interaction. Simplifica- tions are normally used for the wave propagation in these cases. The other trend focuses in analysing the propagation mechanism over long distances.

Two-dimensional models, such as the shallow water equations or the Bous- sinesq equations, are frequently used. Field study data or simplifications are commonly employed as an input for the generated wake.

Additionally, other numerical tools can be used to evaluate the impact and de- rive case-specific criteria. Fluid-structure interaction simulations can be used to evaluate coastal structural damages on the basis of yield or fatigue criteria.

Likewise, seakeeping models can evaluate safety of vessel risks (see Table 1) based on excessive motion criteria.

A procedure has been developed to solve wake wash in an efficient and reliable manner. It consists on a set of steps that make use of different numerical methods to evaluate wake wash impacts. OpenFOAM, a finite volume method opensource code, is employed to obtain a detailed characterization of the gen- erated wave. A Boussinesq equation solver is suggested in order to compute the transformation of the wave during its propagation. However, for simpler cases, an analytical method of propagation is proposed as an alternative. Fi- nally, AQWA, a commercial seakeeping software, is employed to evaluate a vessel response to wake wash and determine the potential risks to the vessel and its crew safety.

The aforementioned necessities have been the capital design objectives for the procedure. The choice of methods intrinsically ensures the aprioristic solution of the problem, as only the hull shape and bathymetry, which are available during the design stages, are needed. In order to establish the quality of the procedure, so it can be used quantitatively, a validation study has been per- formed on the wake generated by the KCS benchmarking container vessel [22].

Regarding the other two necessities: the generality and the velocity -and ease of use- of the procedure, it should be stated beforehand that this project was carried out for the Spanish port and maritime consultancy firm Siport21.

Therefore, as main stakeholders of the project, commercial interests as well as resources and workflow have been taken into consideration.

The overall methodology is faster than performing a field study. Yet, the wave generation stage can take over a day to simulate and twice as long to prepare for an experienced user. For this reason, an assisting tool has been developed to automatically process and prepare large batches of simulations.

The procedure has been designed to cover a scope as broad as possible. How- ever, the focus of the procedure lays on large shipping vessels wake and safety of vessels in accordance with the stakeholder’s interest. A test case scenario has been analysed for a large container ship wake incidence on a 75m long moored yacht.

Theoretical Background

In this chapter, the theoretical and mathematical framework behind the main concepts of the thesis are condensed.

First, the mathematical background for gravity water waves is exposed. The general equations are derived from the conservational principles and the free surface boundary conditions. Two main approximations are then presented:

the shallow water theory and the linear wave theory. The latter is explored deeper to account for the difference between shallow water and deep-water wave behaviour. This section is based in the works by Stoker [23], Hermans [24], Salmon [25], Fransson [26] and Garme [27].

Afterwards, linear wave theory is used to obtain a mathematical and physical description of wave transformation concepts. The main concepts treated are shoaling, refraction, diffraction and breaking. This section is based in the works by McCormick [28] and the Coastal Engineering Manual by the U.S.

Army Corps of Engineers [29].

Then, the case of ship generated waves is presented. Kelvin’s pressure point is used to explain the pattern of these waves. A qualitative description of the pat- terns for both shallow and deep water is then presented. This section is based in the works by Stoker [23] and Sclavounos [30].

After looking at the waves generated by sailing vessels, the opposite case, ves- sel motions as a response to external waves, is discussed. The general equation of motion for a vessel excited by waves is presented. Two different approaches, frequency domain and time domain, can be used to solve the equation. This section is based on the work by Bergdahl [31].

Finally, the motion induced interruption criteria developed by Graham [32] is introduced.

Gravity water waves

The waves that can be appreciated on the surface of lakes, seas and oceans are normally described as gravity waves. These oscillations are caused by the re- storing action of the gravitational force to any disturbance in the surface of water. Such disturbances can be originated from wind, currents, earthquakes, the pull from the moon and sun and, as will be discussed later, vessels.

The fluid flow of water waves is described by the incompressible Euler equa- tion (1). The assumption of incompressibility is consequential for water,

whereas, the assumption of inviscid flow is coherent for the general descrip- tion of the waves and their propagation.

{

𝜕𝒖

𝜕𝑡 + 𝒖 ⋅ 𝛁𝐮 = −𝛁p ρ − 𝒈

𝛁 ⋅ 𝒖 = 0

(1𝑎) (1𝑏)

Under the assumption of irrotationality, the velocity can be expressed by po- tential functions. This leads from the equation (1b) to the Laplace equation.

{

𝚫𝜙 = 0 𝑢_𝑖 = 𝜕𝜙

𝜕𝑥_𝑖

(2) In addition to this set of partial differential equations, the fluid flow must sat- isfy some boundary conditions. These are of application for the interface be- tween air and water, normally known as free surface, which will be denoted as 𝜁(𝒙, 𝑡) = 𝑧 − 𝜂(𝑥, 𝑦, 𝑡) , and for the bottom of the sea, which will be considered as slowly sloping surface: 𝐻 = 𝑧 + ℎ(𝑥, 𝑦).

𝜕𝜙

𝜕𝑥

𝜕𝜂

𝜕𝑥+𝜕𝜙

𝜕𝑦

𝜕𝜂

𝜕𝑦+𝜕𝜂

𝜕𝑡 =𝜕𝜙

𝜕𝑧 𝑜𝑛 ζ (3)

𝜕𝜙

𝜕𝑧 = 0 𝑜𝑛 𝐻 (4)

Both these equations (3) and (4) have the same physical meaning: fluid parti- cles in the surface are bounded to it. This principle is known as the kinematic condition. In the case of the free surface another principle must be satisfied:

the dynamic condition, which states that the free surface is in a pressure equi- librium.

𝑔𝜂 +𝜕𝜙

𝜕𝑡 +1 2(𝜕𝜙

𝜕𝑥_𝑗

𝜕𝜙

𝜕𝑥_𝑗) +𝑝₀

𝜌 = 0 𝑜𝑛 𝜁 (5)

The two boundary conditions affecting the free surface, (3) and (5), present non-linearities that complicate finding a solution. Not knowing the exact po- sition of the surface where the boundary conditions need to be imposed is also a difficulty that needs to be overcome. There are two main hypothesis that simplify the problem to linear approximations. The first considers that the amplitude of the waves can be considered small and leads to the linear wave theory. The second considers that the water depth can be considered small and leads to the shallow water non-linear models.

The linear wave model might seem insufficient if someone evokes a romantic visualization of waves: big plunging waves. However, this model can account for most cases of sea waves. For the north Atlantic wave statistics compiled by Hogben [33], only in 1% of the recorded waves does the ratio amplitude wave length surpass 0.05. Additionally, the simpler model allows to gain insight into properties and phenomena of wave propagation.

Under the small amplitude assumption, the free surface can be described by 𝜁(𝒙, 𝑡) = 𝑧 − (𝜂₀+ 𝜖𝜂₁+ 𝜖²𝜂₂+ ⋯ ) = 0, where 𝜖 is a small dimensionless pa- rameter. Likewise, a solution of the form 𝜙 = 𝜙₀+ 𝜖𝜙₁+ 𝜖²𝜙₂+ ⋯, where Δ𝜙_𝑖 = 0, is sought after. Both these conditions, when introduced in (3) and (5), lead to the following expressions, respectively:

{

𝜕𝜂₁

𝜕𝑡 =𝜕𝜙₁

𝜕𝑧

𝜕𝜂₂

𝜕𝑡 =𝜕𝜙₂

𝜕𝑧 −𝜕𝜙₁

𝜕𝑥

𝜕𝜂₁

𝜕𝑥 −𝜕𝜙₁

𝜕𝑦

𝜕𝜂₁

𝜕𝑦 + 𝜕𝜙₁

𝜕𝑧𝜕𝑧𝜂₁

(6)

{

𝑔𝜂₁+𝜕𝜙₁

𝜕𝑡 = 0 𝑔𝜂₂+𝜕𝜂₂

𝜕𝑡 +1 2

𝜕𝜙₁

𝜕𝑥_𝑖

𝜕𝜙₁

𝜕𝑥_𝑖 + 𝜂₁ 𝜕𝜙₁

𝜕𝑡𝜕𝑧= 0

(7)

These conditions need to be satisfied in 𝜁. Since 𝜂₀ = 0 if the amplitudes are small, this equates to 𝜁: 𝑧 = 0. In other words, the conditions are applied in the free surface at rest.

The basic problem, for which the linear solution is of application, is defined by the first equations in (6) and (7) and some initial conditions. The solution for the case of progressing waves in finite depth water can be obtained through separation of variables.

𝜙(𝒙, 𝑡) =𝐴𝜔 𝑘

cosh(𝑘(𝑧 + ℎ))

sinh(𝑘ℎ) sin(𝑘_𝑥𝑥 + 𝑘_𝑦𝑦 − 𝜔𝑡) (8)

𝑘² = 𝑘_𝑥²+ 𝑘_𝑦² (9)

𝜔² = 𝑔𝑘 tanh(𝑘ℎ) (10)

𝜂(𝑥, 𝑦, 𝑡) = 𝐴 cos(𝑘_𝑥𝑥 + 𝑘_𝑦𝑦 − 𝜔𝑡) (11) The linear solution for the wave elevation (11) corresponds to that of a planar wave travelling in the (𝑘_𝑥, 𝑘_𝑦, 0) direction at speed 𝑐 = 𝜔/𝑘. Superposition of the general solution (8) and (11) can be used to define more complex phenom- ena. Therefore, in principle, any initial value problem (12)(13) can be solved

through application of the Fourier transform. For the one-dimensional case, 𝑘_𝑦 = 0, the procedure would be the following.

𝜂(𝑥, 0) = 𝐹(𝑥) (12)

𝜕𝜂

𝜕𝑡 (𝑥, 0) = 𝐺(𝑥) (13)

𝜂(𝑥, 𝑡) = ∑𝐴_𝑛cos(𝑘_𝑛𝑥 − 𝜔_𝑛𝑡) + ∑𝐵_𝑛sin(𝑘_𝑛𝑥 − 𝜔_𝑛𝑡) (14) 𝐴_𝑛 = 1

2𝜋∫ 𝐹(𝑥) cos(𝑘_𝑛𝑥) 𝑑𝑥

∞

−∞

+ 1

2𝜋𝜔_𝑛∫ 𝐺(𝑥) sin(𝑘_𝑛𝑥) 𝑑𝑥

∞

−∞

(15) 𝐵_𝑛 = 1

2𝜋∫ 𝐹(𝑥) sin(𝑘_𝑛𝑥) 𝑑𝑥

∞

−∞

− 1

2𝜋𝜔_𝑛∫ 𝐺(𝑥) cos(𝑘_𝑛𝑥) 𝑑𝑥

∞

−∞

(16) So far, the obtained results are no different from the wave description ob- tained in other fields. However, under close inspection of equations (8) and (10), important differences on behaviour can be seen depending in the relative depth of the bottom.

Equation (10) is known as the dispersion relation and determines the velocity with which the waves travel. It is related to depth by the expression tanh (𝑘ℎ) for which the expansions tanh(𝑘ℎ) = 𝑘ℎ and tanh(𝑘ℎ) = 1 are accurate as 𝑘ℎ → 0 and 𝑘ℎ → ∞. For a given wavenumber, these expansions correspond to shallow water (𝑘ℎ < 0.3) and deep water (𝑘ℎ > 3). The effects can be appreci- ated from equations (17) and (18). In deep water, waves travel at a speed de- fined exclusively by the wavelength. The longer the wavelength, the faster the wave travels. The resulting effect at a point far from the initial perturbation is that waves will arrive sorted in decreasing wavelength. In shallow water, all waves travel exactly at the same speed and therefore no dispersion takes place.

However, the wave speed still depends on the depth of the water and therefore waves in shallower waters will travel slower. This phenomenon, known as re- fraction, will lead to local changes to the wave propagation direction and pro- duce an apparent bending of the waves, as can be appreciated in Figure 1.

𝐷𝑊: 𝜔² = 𝑔𝑘 → 𝑐 = 𝜔

𝑘 = √𝑔

𝑘 (17)

𝑆𝑊: 𝜔² = 𝑔𝑘²ℎ → 𝑐 =𝜔

𝑘 = √𝑔ℎ (18)

Equation (8) describes the flow of the entire fluid volume. As in equation (10) the solution depends on depth in the hyperbolic terms. The same expansions, 𝑘ℎ → 0 for shallow waters and 𝑘ℎ → ∞ for deep ones, can be applied to it. In the deep-water case, the hyperbolic term disappears except when 𝑧 → ∞ also.

In shallow water case, it necessarily depends on the depth. This can be

appreciated, for example, in the differences between the particle motions in both cases, see Figure 2.

𝐷𝑊: 𝜙 =𝐴𝜔

𝑘 sin(𝑘_𝑥𝑥 + 𝑘_𝑦𝑦 − 𝜔𝑡) (19) 𝑆𝑊: 𝜙 = 𝐴𝜔

𝑘 (1

𝑘ℎ+ 𝑘𝑧) sin(𝑘_𝑥𝑥 + 𝑘_𝑦𝑦 − 𝜔𝑡) (20)

Figure 1- Refraction of waves on the shoreline. In the far field waves travel parallel towards the viewer, as appreciated from the visible wave crests. In the shoreline against the waves, those break directly. However, in the shoreline tangential

to the propagation direction, waves refract towards the shoreline, turning against it as well.

Figure 2- Particle motion for deep water (A): circular motion that start to decrease in amplitude close to the bottom. Par- ticle motion for shallow water (B): elliptical motion with minor axis on the z-direction.

From the inspection of the equations, the main conclusion to be drawn is the importance depth has in the behaviour of waves. Bathymetry might then play an important role in the wave propagation process. However, the variations in

depth are not included in the propagation model derived before. The next sec- tion introduces the different phenomena that transform a wave during prop- agation and some methods to evaluate it.

Wave propagation and transformation phenomena

Mathematically, equation (11) describes planar waves at constant depth and with a small amplitude-wavelength ratio. These conditions are applicable for fully developed seas in offshore regions where the depth can be assumed to be constant or not influential. In these cases, the wave equation can be used to describe the sea state in a certain region. For example, measurements taken from an offshore wave buoy can be used to obtain a description of the sea around it. However, when using the same buoy to predict the waves at a dif- ferent location, shortcomings of the theory can be found, especially when propagating the wave inshore.

The effects that changes in bathymetry and obstacles, such as seawalls, pro- duce; are not modelled in the linear wave theory. Additionally, other phenom- ena are not modelled due to their non-linear behaviour. The objective of this section is to present those phenomena from a qualitative perspective in order to convey the complexities of propagation modelling.

Wave group and wave energy

According to linear wave theory, a complex sea state can be described by su- perposition of simpler sinusoidal waves with different phases and amplitudes.

This superposition leads to either constructive or destructive interferences. Of special interest is the interference resulting from addition of waves of equal amplitude and slightly different frequency.

𝜂₁ = 𝐴 𝑐𝑜𝑠(𝑘𝑥 − 𝜔𝑡)

𝜂₂ = 𝐴 cos[(𝑘 + 𝛥𝑘)𝑥 − (𝜔 + 𝛥𝜔)𝑡] (21) 𝜂 = 𝜂₁+ 𝜂₂ = 2𝐴 cos (𝛥𝑘

2 𝑥 −𝛥𝜔

2 𝑡) cos[(𝑘 + 𝛥𝑘)𝑥 − (𝜔 + 𝛥𝜔)𝑡] (22) The resulting wave is known as wave group, and after approximating equation (22) with Δ𝑘 → 0 and Δ𝜔 → 0, the wave can be seen as the initial wave with an amplitude modulation, equation (23). The amplitude modification is also a progressing wave that travels with velocity Δ𝜔/Δ𝑘 which is commonly known as group velocity. The wave group consists of a normal wave of a certain speed contained inside an enveloping wave that determines the amplitude and trav- elling at the group velocity, see Figure 3.

𝜂 = 2𝐴 cos (𝛥𝑘

2 𝑥 −𝛥𝜔

2 𝑡) 𝜂₂ (23)

𝑐_𝑔 =Δ𝜔

Δ𝑘 ; 𝑐_𝑔(Δ𝜔, Δ𝑘 → 0) = 𝑑𝜔 𝑑𝑘 = 𝑐

2+ (1 + 2𝑘ℎ

sinh(2𝑘ℎ)) (24) 𝐷𝑊: 𝑐_𝑔 =𝑐

2 𝑆𝑊: 𝑐_𝑔 = 𝑐

Figure 3 - Wave group: Envelope of amplitudes presented in dotted line and actual wave presented in continuous line.

Equation (24) presents the wave group speed for any given wavenumber. After deep water approximation, it can be seen that the wave group travel exactly at half the wave speed. On the other hand, in shallow water, the group travels exactly at the same speed. The resulting effect is that in deep water, waves are apparently generated at the node of the wave group, travel through it and van- ish in the following node. However, for shallow water wave groups, the same shape is always conserved as the wave propagates.

Wave energy can be obtained by considering a small wave element of length 𝑑𝑥. Potential energy per crest length can be obtained by considering the wave mass above the waterline and integrating over a wavelength.

𝐸_𝑝 = ∫ (𝜌𝜂(𝑥))𝑔𝜂(𝑥)

2 𝑑𝑥 = 𝜌𝑔𝐴²

2 ∫ cos(𝑘𝑥 − 𝜔𝑡) 𝑑𝑥

𝜆 0 𝜆

0

=𝜌𝑔𝐴²𝜆

4 (25)

𝐸_𝑘 = ∫ ∫ 1

2𝜌(𝑢²+ 𝑤²)𝑑𝑥𝑑𝑧

𝜆 0

=𝜌𝑔𝐴²𝜆 4

0

−ℎ

(26) 𝐸 = 𝐸_𝑘 + 𝐸_𝑝 =𝜌𝑔𝐴²𝜆

2 (27)

Of more interest, is the energy flux or wave power. Wave flux represents the amount of energy that travels in the propagation direction per unit of time.

The only energy term that varies with time is the time-rate of change of energy per unit of area normal to the propagation direction. The average power or energy flux per period is obtained through integration.

𝜌𝜕𝜙

𝜕𝑡 ∇𝜙 (28)

𝑃 = 𝜌

𝑇∫ ∫ 𝜕𝜙

𝜕𝑡 ∇𝜙𝑑𝑧𝑑𝑡

0

−ℎ 𝑇 0

=𝜌𝑔𝐴²𝑐

4 (1 + 2𝑘ℎ

sinh(2𝑘ℎ)) (29) From equation (29) it can be interpreted that energy is propagated at the wave group velocity.

Wave refraction and shoaling

As has been discussed and demonstrated by equations (18), (21) and (24), waves parameters such as wavelength, phase and group speed depend on depth. A wave propagating through a varying bathymetry, therefore, is sub- jected to changes in its main parameters during propagation. The transfor- mation undergone by the wave due to variations in depth is known as shoaling.

One of the most elemental cases of shoaling is that of a plane wave propagating from deep water towards a slowly sloping coastline, such as in Figure 4. In deep-water, the planar wave has a certain wavelength 𝜆₀ and a certain velocity 𝑐₀ which correspond to a 𝜔₀. As the waves progresses and because wave fre- quency is conserved, the wavelength starts shortening and the wave slows down, as per equations.

𝜆

𝜆₀ = tanh (2𝜋

𝜆 ℎ) (30)

𝑐 = 𝑐₀ 𝜆

𝜆₀ (31)

Figure 4 -Waves propagating against a sloping shore.

Despite the shortening, energy flux between two points along the wave prop- agation must be conserved. Therefore, as the wave propagates it also increases in amplitude.

𝑃₀ = 𝑃 (32)

𝐴 = 𝐴₀√ 𝑐_𝑔0

𝑐_𝑔 → 𝐾_𝑆 = 𝐴

𝐴₀ = √ cosh²𝑘ℎ

𝑠𝑖𝑛ℎ 𝑘ℎ cosh 𝑘ℎ + 𝑘ℎ (33)

Refraction is a specific case of shoaling that occurs when the propagation di- rection and the bathymetry gradient are not aligned. The result of refraction is twofold: in first place, propagation direction is locally modified along a wave crest producing a bending of the waves; secondly, the wave increases in am- plitude. The simpler example is to consider a deep-water wave travelling oblique to a sloping coast, see Figure 5. As in the previous example, wave flux must be conserved throughout propagation.

𝐴 = 𝐴₀√𝑐_𝑔0 𝑐_𝑔 √𝑏₀

𝑏 = 𝐾_𝑠𝐾_𝑟 (34)

Due to the bending caused by the difference in propagation speed between both sides of the depth contour, the wave experiences a smaller shoaling. The difference is represented by the refraction coefficient 𝐾_𝑟 and is proportional to the square root of the change in the length of the wave crest. If the water depth is considered to be discrete, Snell’s law of optics can be applied to obtain 𝐾_𝑟. As can be seen by equation (31), the refraction coefficient is determined by the incidence angle 𝛼₀ and by depth according to (32) and (28).

𝐾_𝑟 = √cos(𝛼₀)

cos(𝛼) (35)

𝜆₀

𝜆 =sin(𝛼₀)

sin(𝛼) (36)

Figure 5- Wave refraction in a sloped beach

Wave diffraction

The waves that have been considered are planar waves of infinite wave crest length. Additionally, the wave has the same amplitude along the crest direc- tion. This description does not account for waves of finite crest, such as the ones generated by ships or other perturbation. Another common case is the restriction of wave crest when the plane wave meets a breakwater or other ob- stacle. In these cases, energy is transferred laterally from points of greater am- plitude to those of smaller one. This phenomenon is known as wave diffrac- tion.

Diffraction is a common phenomenon in all physical waves. A general descrip- tion is obtained from the Huygens principle: every point pertaining to a wave front acts as a secondary source of spherical waves. Therefore, the following wave front can be predicted geometrically as the interference of the numerous wavelets, see Figure 6.

Figure 6- Huygens principle applied to the propagation of planar waves and cylindrical waves.

One important characteristic of spherical or cylindrical waves is that the en- ergy flux must be conserved along the different wave fronts. Therefore, the energy per length of wave crest must decrease the further away from the source.

Despite not presenting the mathematical theory behind this phenomenon, due to its complexity, two main qualitative concepts can be described. First, is that as finite-crested waves propagate the crest extends and the amplitude de- creases as energy is spread. Second, is that the shadow zone of a breakwater or other structures are subjected to diffracting waves that propagate from the main waves.

Wave breaking

Due to the aforementioned effects, specially shoaling, waves can transform into high amplitude waves. These waves have a great energy content that might make the wave break if a certain critical energy level is exceeded. This

breaking process is characterized by a sudden change in the wave structure or wave profile, with the crest of the wave arching over the through.

Wave breaking is mathematically described to occur when the velocity of the crest particles equals the propagation velocity of the wave. Under this condi- tion, crest particles move faster than through particles ensuing in the produc- tion of multiple vortices. During wave breaking, a great part of the wave kinetic energy is transformed into turbulent energy and is therefore dissipated.

From this description is evident that the mathematical model described at the beginning of this chapter is of no use to predict dissipation due to breaking.

For one, viscosity and turbulence play an important role and were not in- cluded in the equations. Additionally, run up towards breaking is character- ized by high amplitude waves. This requirement breaks the main hypothesis for the linear theories. Higher order models are normally used to describe waves before break up. Other computational models that either resolve the Navier-Stokes equations or other non-linear approximations are used for the breaking process.

Ship generated waves (Kelvin Pattern)

A vessel moving along the surface of water produces a strong perturbation on the water surface. In a basic approximation, the disturbance produced by the vessel is similar to that of a moving pressure point. This moving pressure point is the source of small cylindrical wavelets along the vessel track. Integration of the wave elevation contribution of the wavelets renders the final wave ele- vation.

𝜂(𝑥, 𝑦, 𝑧, 𝑡) = −1

2𝜋𝑔𝜌∫ 𝜔

∞ 0

sin(√𝑔𝑘𝑡) 𝑒^𝑘𝑧𝑘𝑑𝑘 ∫ cos (𝑘√𝑥²+ 𝑧²cos 𝛽) 𝑑𝛽

𝜋 2 0

(37) Through the use of the stationary phase principle, the wave surface elevation, 𝜂, is obtained from the impulse (37) contribution. This principle states that, when measuring the contribution of different sources far from them, most of the contributions interfere destructively with each other as the wave phase 𝛼 = 𝑘𝑥 − 𝜔𝑡 varies very quickly. The only contribution that can be seen is that for which the phase does not vary, 𝑑𝛼/𝑑𝑘 = 0, which is known as the stationary phase.

The resulting wave elevation equation is obtained after a complex mathemat- ical derivation. However, under the ansatz that the resulting waves consist on a system of planar waves, the stationary phase equation can be obtained easily.

For a ship travelling in deep water at a speed U, the flow potential can be ex- pressed in a vessel frame of reference, equation (38). Because waves must

travel at the same velocity as the vessel, the dispersion relation is constrained by this speed. From this relation between dispersion and vessel speed, equa- tion (39), it can be inferred that there is a fixed propagation direction for each wave length.

𝜙 =𝐴𝜔

𝑘 sin(𝑘(𝑥 𝑐𝑜𝑠(𝜃) − 𝑦 𝑐𝑜𝑠(𝜃)) + (𝑘𝑈𝑐𝑜𝑠(𝜃) − 𝜔)𝑡)

𝑋 = 𝑥 + 𝑈𝑡; 𝑌 = 𝑦; 𝑍 = 𝑧 (38)

𝑐 =𝜔

𝑘 = 𝑈𝑐𝑜𝑠(𝜃) → 𝜔² = 𝑈²𝑘²cos²𝜃 (39) 𝜔² = 𝑔𝑘 → 𝜆 =2𝜋𝑈²cos²𝜃

𝑔 (40)

The principle of the stationary phase is applied to the potential function (38).

The stationary phase equation (39) defines the phase of the contributing waves. On the other hand, the crests are curves of constant phase (43). By combining both equations (41) and (42), the resulting wave crests can be ob- tained. A graphical representation of the intersecting solution is presented in Figure 7.

𝑑𝛼

𝑑𝑘 =𝑑(𝑘(𝑥𝑐𝑜𝑠𝜃 − 𝑦𝑐𝑜𝑠𝜃) + (𝑘𝑈𝑐𝑜𝑠𝜃 − 𝜔)𝑡)

𝑑𝑘 =𝑑(𝑘(𝑥𝑐𝑜𝑠𝜃 + 𝑦𝑠𝑖𝑛𝜃))

𝑑𝜃

𝑘(𝜃) = 𝑔

𝑈²cos²𝜃 → 𝑑𝛼 𝑑𝑘 = 𝑑

𝑑𝜃( 𝑔

𝑈²cos²𝜃(𝑥 𝑐𝑜𝑠𝜃 + 𝑦 sin 𝜃)) = 0

(41)

𝑥𝑐𝑜𝑠(𝜃) + 𝑦𝑠𝑖𝑛(𝜃)

cos²𝜃 = 𝐶 (42)

This set of curves represents the typical wave pattern after a ship. Despite the fact that all the wave crest is part of the same wave system, they are normally divided into two. In Figure 7 the curve in blue, is the divergent wave system that ranges in propagation directions from 90 to 35.16 degrees. However, most of the wave crest is propagating in approximately a 34 degrees angle from the sailing line. In orange, the transverse wave system that ranges from 35.16 to 0 degrees. As was the case with the diverging system, most of the wave crest propagates parallel to the vessel sailing course. Another important fact is that the cusp where the two wave systems meet is always in a relative position of 19º28’ degrees. This indicates that all wave crests are contained in a 20º wedge, commonly known as Kelvin Wedge.

Figure 7 - Two main groups of vessel generated waves. In blue, the divergent system, with waves propagating from 35.16 degrees to 90 degrees. In orange the transversal system, with waves propagating from 34,16 to 0 degrees.

In equation (40), the deep-water dispersion relation was used and certainly it can be seen that all vessels in deep-water generate this pattern. However, for shallow waters the behaviour can be quite different. This behaviour is indi- cated by the depth Froude Number, which is the relation between the vessel speed and the shallow water wave speed.

𝐹_𝑛ℎ = 𝑈

√𝑔ℎ (44)

For values below one, the deep-water pattern occurs and it is normally known as subcritical regime. For values above one, the vessel is sailing faster than the shallow water wave speed. This means that equation (45) is not satisfied for all values of the propagation direction 𝜃. Therefore, the waves components propagating in the vessel sailing direction do not occur. The resulting wave system can be seen in Figure 8. This regime is known as super critical.

Figure 8 - Waves generated by a vessel in supercritical regime. Only a divergent wave train can be seen. The propagation direction of the waves is limited by the Froude number.

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8

-6 -5 -4 -3 -2 -1 0

Y

X

The critical regime occurs for depth Froude numbers around 1. In this case, the transversal wave system travels exactly at the shallow water speed. How- ever, as equation (24) points out, so does the wave energy. Therefore, the en- ergy that the vessel transmits to the water is accumulated exactly at the vessel.

The resulting pattern is a big transversal wave travelling alongside the ship.

During this stage, resistance is greatly increased due to the big energy of the waves. Therefore, the vessel naturally transitions either to the supercritical re- gime by overcoming the wave or back to sub-critical by the reduction in power.

Ship motion due to wave excitation

In a first approximation, a vessel subjected to external waves can be consid- ered to respond as a solid rigid. Therefore, the vessel can be described by a ship-fixed frame of reference, which is represented in the absolute frame of reference by 𝜼 = (𝜂₁, 𝜂₂, 𝜂₃, 𝜂₄, 𝜂₅, 𝜂₆). The first three terms represent the ab- solute coordinates of the vessel centre of gravity. The last three, under the small angle hypothesis, are respectively the rotation around the absolute axes.

The six components of 𝜂 are commonly known as surge, sway, heave, roll, pitch and yaw. The position of any point of the ship can be quickly referred to the absolute coordinates through the rotation matrix. A small angle approxi- mation (45) is normally used for wave-motion analysis.

( 𝑋 𝑌 𝑍

) = ( 𝜂₁ 𝜂₂ 𝜂₃) + (

𝜂₄ 𝜂₅ 𝜂₆) × (

𝑥 𝑦 𝑧

) (45)

The motions of the vessel are determined by Newton’s Law: 𝑀̅̅𝜼̈ = 𝑭_𝒆𝒙𝒕. The external forces can be divided into hydrodynamic reaction forces, wave-ex- cited forces and other forces, such as mooring, wind or other couplings. The former, being reactive forces, depend on the position derivatives. Under the assumption that the motion consists of small oscillations around an equilib- rium position, the dependency can be expressed as proportional (46).

𝑭_𝒉(𝜼, 𝜼̇, 𝜼̈) = 𝐴̅̅𝜼̈ + 𝐵̅̅𝜼̇ + 𝐶̅̅𝜼 (46) (𝑀 + 𝐴)

̅̅̅̅̅̅̅̅̅̅̅

̅̅̅̅̅̅̅̅̅̅̅𝜼̈ + 𝐵̅̅𝜼̇ + 𝐶̅̅𝜼 = 𝑭_𝒘+ 𝑭_{𝒐𝒕𝒉𝒆𝒓} (47) 𝑀 + 𝐴

̅̅̅̅̅̅̅̅

̅̅̅̅̅̅̅̅ =

(

𝑚 + 𝐴₁₁ 0 0 0 𝑚𝑧_𝐺 + 𝐴₁₅ 0

0 𝑚 + 𝐴₂₂ 0 −𝑚𝑧_𝐺+ 𝐴₂₄ 0 0

0 0 𝑚 + 𝐴₃₃ 0 0 0

0 −𝑚𝑧_𝐺+ 𝐴₄₂ 0 𝐼₄+ 𝐴₄₄ 0 0

𝑚𝑧_𝐺+ 𝐴₅₁ 0 0 0 𝐼₅+ 𝐴₅₅ 0

0 0 0 0 0 𝐼₆+ 𝐴₆₆)

𝐵̅̅ = (

𝐵₁₁ 0 0 0 𝐵₁₅ 0

0 𝐵₂₂ 0 𝐵₂₄ 0 0

0 0 𝐵₃₃ 0 0 0

0 𝐵₄₂ 0 𝐵₄₄ 0 0

𝐵₅₁ 0 0 0 𝐵₅₅ 0

0 0 0 0 0 𝐵₆₆)

𝐶̅̅ = (

0 0 0 0 0 0

0 0 0 0 0 0

0 0 𝐶₃₃ 0 𝐶₃₅ 0

0 0 0 𝐶₄₄ 0 0

0 0 𝐶₅₃ 0 𝐶₅₅ 0

0 0 0 0 0 0)

The coefficients of M are the mass and the inertia of the vessel. The A coeffi- cient is named added mass due to its physical interpretation. It represents the weight of the amount of water that is pushed by the vessel due to the acceler- ations. The coefficient C is related to the restoring force that keeps the vessel at the equilibrium position. Finally, the B coefficient is related to the damping of the motion, or in other words, to the amount of energy that is dissipated during the motion. For a floating body, all these components are dependant of the frequency.

Frequency-domain solution

When any physical system is subjected to a harmonic oscillation, the resulting motion stabilizes around a harmonic oscillation of the same frequency. If an- satzes are taken at both sides of the equation of motion, the equation is trans- formed into the frequency domain (48).

𝐹_𝑤 = 𝐹₀𝑐𝑜𝑠(𝜔𝑡); 𝜂 = 𝜂₀cos (𝜔𝑡 + 𝜖)

(𝑐 − (𝑚 + 𝑎)𝜔²)𝜂₀cos(𝜔𝑡 + 𝜖) − 𝑏𝜔𝜂₀sin(𝜔𝑡 − 𝜖) = 𝐹₀cos(𝜔𝑡) (48) If the terms are rearranged it can be seen that a relation between the incident wave amplitude and the vessel motions can be obtained. This relation (49) is normally known as the response amplitude operator or RAO.

𝑇(𝜔) =𝜂₀

𝐹₀ = 1

√(𝑐 − (𝑚 + 𝑎)𝜔²)²+ 𝑏²𝜔² (49) The great advantage of solving the problem in the frequency-domain is that complex sea states are frequently described with help of the Fourier transform (14). This transformation converts the wave information into the frequency domain so that there is a function 𝐴(𝜔) relating the amplitude of each wave frequency. The equation can then be solved and through linear superposition

the complex motion response can be obtained. The only complexity left is ob- taining the values 𝑎, 𝑏, 𝑐 and 𝐹₀ which depend on the vessel hydrodynamic re- sponse and the excited force by waves.

Time-domain response

The frequency domain is a very useful domain to solve the equation. In one hand, the mathematical complexity of the problem is greatly diminished and in the other, long-period statistical analysis can be performed over the solu- tion. However, the method can’t be applied to short-time motion prediction.

This is precisely the case that might be of interest in the case of finite wave train or the response to mooring equipment.

For quick responses a transient formulation needs to be employed. In this case, the motion can be supposed to behave according to an impulse function which defines the motion of the vessel to a unitary impulse (50). The homoge- neous solution to the second order system gives an expression for this impulse (51). The solution can be obtained as the sum of the contribution of all im- pulses that occurred until the time that is being analysed. The result is thus a convolution function of the unitary impulse response and the excitation force impulse (52).

𝜂(𝑡) = 𝑤(𝑡)𝐹Δ𝑡 (50)

𝜔_𝑟 = √ 𝑐

𝑚 + 𝑎− 𝑏² 4(𝑚 + 𝑎)²

𝑤(𝑡) = 1

(𝑚 + 𝑎)𝜔_𝑟𝑒⁻

𝑏𝑡

2(𝑚+𝑎)sin(𝜔_𝑟𝑡) (51)

𝜂(𝑡) = ∫ 𝑤(𝑡 − 𝜏)𝐹(𝜏)𝑑𝜏

𝑡 0

(52) The convolution function can be solved easily for constant coefficients. How- ever, as was mentioned before, the main coefficients of the equation of motion in the case of a floating body depend strongly in the frequency. Therefore, spe- cial convolution techniques must be employed to be able to transition between frequency-domain coefficients and time-domain solution.

Motion induced interruption

A wide set of criteria has been developed to evaluate the sea-worthiness of a vessel regarding the induced motion by the waves. The criteria are normally used as a pass or fail criteria to assess the ship operability. Because ship oper- ability is assessed over long period statistics and probabilistic theory, it is com- mon that these criteria are related to the vessel response in the frequency

domain. For example, criteria regarding excessive roll motions involve the root main square value of the roll motion in a certain sea state.

These criteria have no meaning for finite wave trains and deterministic wave predictions. The motion induced interruption is probably one of the few ex- ceptions. This criterion analyses the risk of any person to slide or trip over the deck. The theory behind it as a seakeeping criterion is that if any member of the crew suffers from a slide or tripping, their task would be interrupted for a while. However, it can also be seen as a comfort criterion as for passengers it would produce sensation of insecurity and force them to hold to banisters.

The criterion is derived from considering the person as a rigid solid. A sliding occurrence takes place when the inertial and gravitational forces overcome an experimentally modelled friction force (53) (54). A tipping occurrence takes place when the inertial and gravitational forces overcome the normal force (55) (56) (57) (58).

𝑆𝑙𝑖𝑑𝑖𝑛𝑔_{𝑝𝑜𝑟𝑡} → −𝜂_2𝑃̈ − 𝑔𝜂₄− 𝜇_𝑠𝜂̈_3𝑃 > 𝜇_𝑠𝑔 (53) 𝑆𝑙𝑖𝑑𝑖𝑛𝑔_𝑠𝑡𝑏 → +𝜂_2𝑃̈ + 𝑔𝜂₄− 𝜇_𝑠𝜂̈_3𝑃 > 𝜇_𝑠𝑔 (54)

𝑇𝑖𝑝𝑝𝑖𝑛𝑔_{𝑓𝑜𝑟𝑒} → −1

3ℎ𝜂̈₅−𝑑

ℎ𝜂̈_3𝑃 >𝑑

ℎ𝑔 (55)

𝑇𝑖𝑝𝑝𝑖𝑛𝑔_𝑎𝑓𝑡 → +1

3ℎ𝜂̈₅−𝑑

ℎ𝜂̈_3𝑃 >𝑑

ℎ𝑔 (56)

𝑇𝑖𝑝𝑝𝑖𝑛𝑔_{𝑝𝑜𝑟𝑡} → 1

3ℎ𝜂̈₄− 𝜂̈_2𝑃 − 𝑔𝜂₄ −𝑑

ℎ𝜂̈_3𝑃 >𝑑

ℎ𝑔 (57)

𝑇𝑖𝑝𝑝𝑖𝑛𝑔_{𝑝𝑜𝑟𝑡} → −1

3ℎ𝜂̈₄+ 𝜂̈_2𝑃 + 𝑔𝜂₄ −𝑑

ℎ𝜂̈_3𝑃 >𝑑

ℎ𝑔 (58)

State of the Art of Deterministic Wake Wash Predictions

In order to develop a deterministic procedure to evaluate wake wash, previous studies on the same topic have been examined. The objective in doing so is two-fold. First, to be able to compare the different methods used and point out the main advantages and disadvantages of each one. Second, to categorize and study the different cases of wake wash analysis. This information has been used to design the procedure the most reliable and broad manner.

The studies analysed come from two sources: projects carried by Siport21 and papers published in scientific literature. At the same time, these studies can be divided in those that have a practical case scenario and those that have a theoretical scenario. The latter are generally focused in testing a new tool.

However, not being connected to a real case can make hard to assess the pro- cedures holistically. The former is much preferred by this reason, yet, there is a clear scarcity of this type of publications.

A short summary of these studies is presented hereon.

Siport21: Dredged-channel feasibility [34]

This project consisted in analysing the nautical feasibility of the design of a new channel access. The new design consisted on a dredged bottom to allow the access of bigger vessels. There was a great concern about whether the wake wash of the new ships and the new modified bathymetry would have detri- mental effects. The main cause of concern was the presence of two old coastal structures on the margins of the channel access that were subjected to wake wash erosion.

The types of ship analysed were large and slow-steaming vessels, such as Su- ezmax oil tankers and New-Panamax container. The cruising speed of these vessels is well below the values for which critical waves are shed. Instead, the importance of wake wash comes from the blocking effect the depth and width restriction the channel imposes.

The procedure to analyse the scenario was divided in two stages: wave gener- ation and wave propagation. Each of the stages was outsourced to a third com- pany and different numerical methods were used to solve them. Wave gener- ation was studied with potential panel-method. Wave propagation was solved through the use of Danish Hydraulic Institute’s MIKE21 PMS software. A lon- gitudinal wave cut, which is a reflection of the time series measurement of a buoy, was used to transfer information between the two methods.

The output result was the maximum wave height that reached the coastal structure. This height was compared with the measured values of the wake of the usual passing vessels. The acceptance criteria being that it should not be greater.

Siport21: Wake wash study in Sevilla’s river [35]

In this report, wake wash coastal erosion of Guadalquivir river is studied. The possibility of adding underwater breakwaters to absorb part of the incident wave energy is evaluated.

The vessels analysed are like in the previous study slow-steaming ships, mainly tankers, bulk carriers and container ships. Once again, the main con- cern is the effect the restricted water has on wave generation.

The procedure employed is very similar to the one explained in the previous study. The main difference lays in the use of MIKE21 NSW, instead of the PMS module. The PMS module is based on the parabolic approximation to the el- liptic mild-slope equation. This module accounts for the refraction, diffraction and reflection of linear waves on a sloping bottom. [36] The NSW module, on the other hand, considers the effects of refraction, shoaling, energy dissipation and wave breaking. [37]

Samir Gharbi: Study of ship induced waves in St. Lawrence Water- way [38]

The effect of the introduction of a larger size container ship in the St. Lawrence Waterway is studied. Visible shore erosion on the margins of the channel pas- sage is the main motive for the study. The channel has a dredged bottom of 11.3 metres, which can be the cause of the increased wake wash.

Wake measurement was taken for two representative vessels. The OOCL Mon- treal with a length overall of 294m is a representative of the new larger con- tainer vessels. The Canmar Honour with 245m is a representative of the clas- sical traffic of the waterway. An estimation of the wave height in the waterway was obtained by approximating a decay function for the distance propagated.

𝐻_𝑚 = 𝛼 𝑥^𝑛 (59)

Where H is the wave height, alpha and n are experimental coefficients and x the distance to the sailing line.

The most interesting aspect of this study is that the results from the field study were used to validate the predictions made by a numerical method. The method, which was developed by N.J. MacDonald [39], consists in solving the shallow water equations for both the wave generation and wave propagation stages. This is achieved by the substitution of the vessel for a moving pressure

field equivalent to the hydrostatic pressure field. The predictions calculated to a great degree of confidence the drawdown wave. This wave is characteristic of sheltered water problems and is a suction wave that forms to the sides of the vessel. However, the divergent wave trains were not captured successfully.

One last aspect to comment about the study is the connection the author makes between the propagated wave and the actual impact. It is argued that coast erosion should be linked proportionally to the amount of energy carried in the wave. However, no actual criterium is used to determine what is an ac- ceptable amount of energy.

Anil Kumar: Ferry wake wash analysis in San Francisco Bay [40]

This study analyses a possible future fast ferry route between to passenger ter- minals in San Francisco Bay. The fast ferry is a high-speed catamaran. In this case, the vessel travels through open water. However, due to the high velocity of the vessel a large quantity of energy is transferred to the waves. Addition- ally, the vessel can sail indistinctly across deep and shallow water, transition- ing on the process from sub critical to super critical wave generation.

The wave generation phase in this case is solved through a hybrid CFD soft- ware: SHIPFLOW. This software combines at the same time a panel method for the regions of the domain can be considered as free flow and a finite vol- ume method for the regions near the hull and in the wake. The propagation stage is done analytically following the finite-depth spectral definition of the far field developed by Scragg [41].

Finally, the possibility of doing an in-depth study of the sediment transport for evaluating the damages is mentioned. However, no indications are given regarding the most adequate procedure to do so.

A. Robbins: A tool for the prediction of wave wake in deep water [42]

In this paper a tool for early design prediction of wake wash is presented. The main idea behind this tool is to be able to evaluate how different ship hulls are more or less prone to wake wash problems. Abstraction from concrete impact analysis is sought after. This is achieved through a parameter study of how quickly the propagated wave decays is used as a criterium for ponderation.

The wave generation stage is done with SHIPFLOW software in order to be able to define the wave system that follows the ship. In deep water regime, the maximum wave height takes place at the first crest of the divergent wave sys- tem. According to Havelock [3], height decay of these waves is proportional to the cube of the sailing distance.

𝐻_𝑤 = 𝛾 ⋅ 𝑦¹³ (60) The decay coefficient 𝛾 can be obtained from the wave decay measured in the CFD simulation of the wave generation stage. Robbins suggests using the de- cay factor obtained from different vessels as a measuring tool for comparing the wake wash risk of different hull shapes.

A great outcome of this method is that it is not necessary to link wake wash directly to a localization. Therefore, the results can be directly linked to a hull shape wake wash performance.

Zhou Li-Lan: Wash waves generated by high speed displacement ships [43]

In this paper a method similar to Siport21 procedure is used to obtain the wave generation and wave propagation. Afterwards the propagated wave results are employed to analyse how they might affect a moored vessel.

The wave generation stage is simulated with a NURBS-based panel method.

This method is an improvement over the common panel method as the distri- bution of singularities along the hull is greatly improved. For the propagation stage, a non-linear propagation method based on the Boussinesq equations is employed. The Boussinesq approach is probably the most adequate for the evaluation of propagation albeit its computational expense. This method ac- counts for most of wave propagation phenomena: diffraction, refraction, shoaling and non-linearities; it also extends over a wider validity region than other wave theories. The two stages are coupled through the wave information obtained from a longitudinal wave cut.

Finally, the induced motions by these waves are studied for a moored vessel.

A diffraction-refraction problem is solved in order to obtain the response am- plitude operator of the moored vessel. Through spectral decomposition of the propagated wave, the actual motions of the vessel can be estimated.