Bilge Keel Roll Damping. Mark Jan van Kampen. Combining CFD and local velocities. Literature Review

Transcription

1 Bilge Keel Roll Damping Combining CFD and local velocities Delft University of Technology Offshore and Dredging Engineering

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3 Bilge Keel Roll Damping Combining CFD and local velocities In partial fullfillment of the degree of Master of Science in Offshore and Dredging Engineering at the Delft University of Technology May 8, 2015 Faculty of Mechanical, Maritime and Materials Engineering (3mE) Delft University of Technology

4 The work in this thesis was supported by SBM Offshore. Their cooperation is hereby gratefully acknowledged. Copyright c All rights reserved.

5 Abstract The aim of this thesis is to develop a method to determine the roll damping and maximum roll of an FPSO with aberrant bilge keels in irregular waves in such a manner that the time for calculation is within reason and usable for the design stage. For this end roughly four stages were identified: validation of the hull-pressure part of the ITH method for aberrant bilge keels and if necessary adaption; application of ITH to regular waves using local kinematics if necessary; application of regular wave theory to irregular waves using linearized damping and potential flow theory; furthermore if necessary the implementation of memory effect. The tools to be used to accomplish the above goals will be a literature review, experiments by Schut, results from the 2006 Roll JIP and a CFD model by Pelerin verified using the results of Schut and the Roll JIP. This literature review has a focus on three components: roll damping itself, CFD and vortices. The roll damping is a subject that has been discussed extensively and where three relevant problems have been identified. First the memory effects are due to the flow history where the flow history has influence on the local flow velocity at present time. These memory effects are quantifiable through experiments and are significant. Research shows that memory effects can be incorporated in the time-domain and regular waves. For irregular waves this approach would introduce non-linearities, a more simplified approach might be to assume linearity and convolute the RAOs in such a manner to account for the memory effect. Second, local kinematics play a large role in irregular and regular waves and disregarding them can cause roll damping underestimation up to 75%. Research has been done successfully to incorporate the local kinematics in the bilge keel normal force damping, but not yet for the hull pressure. The third issue at hand is the linearization of roll damping for spectral analysis. Stochastic linearization is a suitable candidate but the assumed Rayleigh distribution for the MPM roll maxima should be further investigated and developed. It seems possible to use CFD modeling to generate accurate data to evaluate hull pressure damping. Various models are possible, but the most likely candidates are RANS and DVM modeling, where for the Reynolds-Averaged Navier-Stokes (RANS) model a Realizable κ ɛ turbulence model seems most suitable. As Jean-Luc Pelerin of SBM Offshore has already developed a SST RANS model it seems pragmatic to try and verify & validate this model for use in this research.

6 ii The vortices shed by the bilge keels are one of the most complicated subjects of this review and the thesis. Not many hard facts are available on the hull pressure created by vortices and the influence of the free-surface and especially irregular wave orbital motions. A reasonable assumption seems to be that the free-surface absorbs or deflects vortices nullifying their effect on the hull-pressure. Wave orbitals seem to alter the paths of the vortices and require further investigation. Conducting this literature review has given the author an insight on how to approach the problem of FPSO roll damping, resulting in the following goals and steps to be undertaken: Can the Computational Fluid Dynamics (CFD) model be validated using the results of Schut and be used as a basis for further research? Determine if the formulation by ITH for hull-pressure damping is valid for FPSOs with large bilge keels and abnormal geometries. Adapt if necessary. Validate the ITH method in a regular wave, if not determine if the use of local kinematics instead of global motions is applicable Implement the (stochastically) linearized roll damping in a (possibly iterative) potential flow scheme to enable spectral analysis Implement the memory effect in the scheme mentioned in item 4. Reference is made to Appendix B for a planning.

7 Table of Contents Preface Acknowledgements ix xi 1 Introduction Document Structure Document Purpose and Scope Conventions History and background Floating Production, Storage and Offloading (FPSO) Hydrodynamics Problem Statement, Research Question and Goal Methodology Methodology Search set-up Research area Search terms Databases and search engines Planning On Roll Damping Introduction Roll Hydrodynamics Roll Damping Results Ikeda et al Recent Developments Discussion

8 iv Table of Contents 4 On Vortices Introduction Vorticity and velocity Velocity and pressure variations Results General Results Near-surface influence Influence of (irregular) wave-induced orbital velocities Discussion On Computational Fluid Dynamics Introduction Navier-Stokes equations Potential Flow Theory Background results Eulerian Methods Lagrangian Methods Hybrids Discretization methods Solver Methods Existing Model Assumptions Verification and Validation Literature Results Discussion Conclusion 43 A Exploration Results 45 A-1 Mindmap A-2 Tabulated results B Planning 49 Bibliography 51 Glossary 59 List of Acronyms

9 List of Figures 1-1 Axis conventions according to Journée A representation of an FPSO including its exaggerated bilge keel Mindmap of search terms A simplified 2D representation of an FPSO including its exaggerated bilge keel and bilge-keel induced hull pressures Comparison of estimation methods and measured data Results from Orozco where Ikeda s formulation for roll damping was stochastically linearized and multiplied by four to account for irregular waves as became clear from experiments and using HYDROSTAR to solve Calculation scheme utilizing stochastic linearization by van t Veer Regular wave Response Amplitude Operator (RAO)s for potential diffraction-refraction based software, using the method by Hajiarab and from experiments by Hajiarab Regular wave RAOs for potential diffraction-refraction based software, using the method by Hajiarab and from independent experiments by Brown et Al Potential flow calculation scheme including possible options for incorporating roll damping and memory effects, excluding iterative schemes Example of a von Karman vortex street due to vortex shedding, source: Malikiaa Path of the highest vorticity concentrations of a fixed rectangle in a regular wave. The solid line and circle is positive vorticity, while the dotted line and circle are negative vorticity. The points a through e represent the time of the snapshots. For the seaward side this is an inverse cosine and for the leeward side this is an inverse sine for wave elevation 8 cm before the barge (seaward) and 8 cm behind the barge (leeward). From Jung A comparison of the simulation of the separation bubble caused by a turbulent flow past a blunt plate, it should be noted that the standard κ ɛ model under predicts the size, while the Realizable κ ɛ is in very good agreement with experiment. Courtesy of ANSYS

10 vi List of Figures 5-2 A comparison between the average pressures over ten periods simulated by the CFD model and recorded during the experiments by Schut using the 2-norm to determine discrepancy A-1 Mindmap of key topics B-1 The Gantt chart of the planning for the nine month graduation period

11 List of Tables A-1 Research area of bilge keels overview in order of decreasing record count (Web of Science (WoS)) A-2 Research area of roll damping overview in order of decreasing record count (WoS) 47 A-3 Research area of ship-related CFD in order of decreasing record count (WoS).. 48

12 viii List of Tables

13 Preface This literature review is part of my M.Sc. graduation thesis. After proposing a different subject to SBM Offshore interest was caught and dr.ir. R. van t Veer approached me to conduct my thesis on variety of other subjects. The subject of FPSO roll damping piqued my interest as it seemed a very relevant and potentially usable subject, depending on my performance. For me it seemed to provide sufficient depth for my thesis while keeping in touch with the needs and interests of the industry. Furthermore both prof.dr.ir. R.H.M. Huijsmans and dr.ir. R. van t Veer seemed very engaged with the subject which continues to be an asset and motivator for me. There was little original work done by me in this literature review as it was the aim to inventory the current and past work relevant to this research. For me it was a quite interesting read which broadened and deepened my knowledge on fluid flows and hydrodynamics.

14 x Preface

15 Acknowledgements I would like to thank my supervisor dr.ir. R. van t Veen for his support and his guidance during this review. Furthermore I would like to thank my colleagues and fellow students for providing with the necessary breaks between reading the papers and for allowing me to spar with them on certain subjects on which my knowledge was not yet solidified. I would like to thank my predecessor ir. X. Schut as well for his hard work during the making of his graduation thesis, as well as his continued, enthusiastic support during my thesis. Furthermore I would like to thank the Delft University of Technology and Prof. Dr. Ir. R.H.M. Huijsmans for making my education possible and the support given and for getting me out of my tunneled vision. Last but not least I would like to thank you, the reader, for showing your interest by reading this far. Delft, University of Technology May 8, 2015

16 xii Acknowledgements

17 Everything should be made as simple as possible, but not simpler Albert Einstein

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19 Chapter 1 Introduction The purpose of this Chapter is to provide the reader with a sense of the subject and relevancy as well as background information. This is done through a historical background of the Floating Production, Storage and Offloading (FPSO) unit, hydrodynamics and roll damping, furthermore the problem statement, the definition of the goals of the thesis, as well as the relevancy and the methodology will be discussed. 1-1 Document Structure This document s structure is along the general lines for a literature review and states a methodology, the introduction, results and discussion for each relevant subject and finally a conclusion. 1-2 Document Purpose and Scope This document is intended to gain an insight into the workings of roll damping, as well as the selection of an appropriate method to make an attempt to derive an useful formulation for the hull-pressure induced roll damping. The scope of this document is limited to the methodology to conduct a literature review, an overview of available literature and the selection of a method to proceed. Further research is not included, neither are extensive recommendations for further development. This is reserved for the main report [1]. 1-3 Conventions Following Journée [2] the axis are defined as in Figure 1-1 below.

20 2 Introduction Figure 1-1: Axis conventions according to Journée [2]. In Figure 1-1 three translations and three rotations all originating from the Center of Gravity (CoG) are depicted. With the translation in x, y and z being respectively the surge in the longitudinal direction positive towards the bow, the sway in the lateral direction positive towards port and heave in the vertical direction positive upwards. The rotations φ, θ and ψ being respectively the roll around the longitudinal axis positive right turning, the pitch around the lateral axis positive right turning and yaw around the vertical axis positive right turning. 1-4 History and background In this Section an overview is provided on the historical background of FPSOs and hydrodynamics. It provides an introduction into the relevant subjects for this review FPSO As hydrocarbon supplies dwindle, technology develops and hydrocarbon prices rise it is becoming more and more economical to develop hydrocarbon fields not just simply offshore but in ultra-deep waters as well. Traditional production platforms, such as a jack-up, a gravity based structure and others are not usable in these water depths. Furthermore a complication arises due to the distances from shore, resulting in long and expensive pipelines which are economically unattractive. Various concepts such as a Single Point Anchor Reservoir (SPAR) and a Tension Leg Platform (TLP) have been developed, but in general an FPSO is favored as an FPSO is flexible, quickly commissioned and cost-effective. An FPSO is a ship-shaped production platform that is usually connected to a subsea template from which a mix of water, gas and oil is produced. This mix is processed by the FPSO, after which the processed oil is stored on-board until a tanker is available for offloading. There are

21 1-4 History and background 3 Figure 1-2: A representation of an FPSO including its exaggerated bilge keel a few types of FPSOs which are identified by the presence and type of turret and the type of hull, either new-built or converted. For this literature review the vessels highest relevance are the vessels that are spread-moored and thus do not have an internal or external turret through which risers are fed, but rather a riser balcony on one of the sides. Furthermore the vessels that are converted from a Very Large Crude Carrier (VLCC) are of interest. These vessels are most suspicable to roll which is relevant for this research. A spread-moored FPSO uses a number of mooring lines at various locations to keep its position. This means it does not weather-vane and can encounter beam waves which can cause large roll motions. A converted FPSO is more suspicable to roll motions as for its original purpose as VLCC it is designed as a ship with forward speed which are less sensitive to roll. To counter these large roll motions, for which a maximum is maintained of 9 within Single Buoy Mooring (SBM) Offshore, bilge keels are employed. Figure 1-2 shows a hull with exaggerated bilge keels attached. Bilge keels on FPSOs are large in comparison to normal vessels as normal vessels move at a forward speed. As bilge keels size increases the wetted area and thus friction with the water increase. This makes it important for regular vessels that have a forward speed to minimize the bilge keel size. Furthermore having a forward speed reduces roll motions as lift is created, limiting the need for a large bilge keel. As FPSOs are generally stationary bilge keel size is not limited and thus FPSOs are fitted with larger bilge keels which usually have a deviating geometry, mostly due to the use of bracings to maintain sufficient structural integrity when loaded in multi-axial fatigue Hydrodynamics Hydrodynamics is the science of moving water and objects in moving water and was developed by well known names such as Blaise Pascal, Isaac Newton, Daniel Bernoulli, Jean le Rond d Alembert, Leonhard Euler, William Froude, George Gabriel Stokes, Osborne Reynolds and many others. On the subject of roll damping a great deal of work has been done by Ikeda and Himeno in , summarized in [3] by Himeno. In Himeno s work it is assumed that roll damping is composed of seven components of which three are relevant for bilge keel damping. The relevant components are: the normal-force damping, the hull-pressure damping and wave damping. The focus of this document is on hull-pressure damping, although the other components will be touched upon. Hull-pressure damping is caused by the vortices that are shed when the bilge keel encounters a flow. The vortices influence the pressure on

22 4 Introduction the hull, which results in a moment countering the roll motion in (almost) all cases. As this phenomenon depends on velocity it is considered damping. Empirical formulations are stated by Himeno [3] which are the result of extensive experiments. The predictions from the stated formula are quite accurate, but are limited for certain bilge keels and ship shapes and are not directly applicable to regular and irregular waves. Other limitations are the exclusion of the free surface in the experiments. The method that Ikeda and Himeno developed assumes a pressure distribution deduced from experiments. An analytical expression is formed to represent the pressure distribution as a two-dimensional curve, which then can be integrated over the length of the vessel. The pressure was identified to be in proportion with the square of the roll frequency for positive pressure and the negative pressure with the period of the roll. 1-5 Problem Statement, Research Question and Goal While the current methods by Ikeda, Himeno and as stated by International Towing Tank Conference (ITTC) can be quite accurate they lack flexibility as the empirical foundations are limited to certain bilge keel sizes and geometry. Furthermore they do not allow proper implementation in irregular waves. SBM Offshore and the industry in general requires a new, more accurate formulation of the hull-pressure and hull-pressure damping based on actual occurring phenomena and which would preferably be applicable in irregular waves as well. This would allow a reduction or elimination of experiments and optimization of bilge keel geometry and size. The research question is thus as follows: How can the roll damping and most probable maximum roll of an FPSO with aberrant bilge keel geometry in irregular waves be determined within a timeframe reasonable for a design stage? The goal of this literature review is to provide a theoretical background on the hull-pressure induced by bilge keels. As the goals of the literature review are directly linked to the goals of the thesis the goals are stated below. They are to: 1. Determine if the formulation by ITH for hull-pressure damping is still valid for FPSOs with large bilge keels and abnormal geometries and adapt if necessary. 2. Validate the ITH method in a regular wave, if not determine if the use of local kinematics instead of global motions is applicable 3. Implement the stochastically linearized roll damping in a (possibly iterative) potential flow scheme to enable spectral analysis 4. Implement the memory effect in the scheme mentioned in item 3. Using the goals and the original research question various subquestions can be formed that this literature will aim to answer:

23 1-6 Methodology 5 1. What is an appropriate CFD method that balances accuracy and computation time with the capability to accurately compute hull-pressures and vortex shedding? 2. What is the current state-of-the-art in the field of roll damping? 3. Which approaches have been taken to solve the roll damping of an FPSO and what angles seem promising? 4. What are the physical workings of vortices, dependency on velocity and their influence on hull pressure, focusing on the effects of the free-surface and wave orbital velocities? 1-6 Methodology The Delft Design Guide is used to select a methodology for generating keywords [4], while the Delft Literature Guide [5] is used as a general method and guideline for the survey. For further details reference is made to Chapter 2.

24 6 Introduction

25 Chapter 2 Methodology This Chapter will discuss the methodology used for this literature review. This literature review aims to provide a summary of current knowledge and understanding of the relevant subjects, identify less-defined areas in the literature and thus formulate additional questions that need answering, as well as selecting an appropriate method that will serve as a basis for the Computational Fluid Dynamics (CFD). The literature review will executed using the following steps, reference is made to the Delft Literature Guide [5] for a more comprehensive overview: Extract search terms Set-up a search plan Determine sources, i.e. databases and engines used Perform search for important papers Refine search terms Perform exhaustive search, storing terms, engines and number of hits Systematically answers stated questions 2-1 Search set-up In this Section the prerequisites for the search are defined, i.e. the search terms, the search engines to be used and the base papers.

26 8 Methodology Research area To start the literature review an inventory is made of key topics, journals, conferences, authors and institutions to define a research area. For this the database Web of Science (WoS) is used as it is a large database with advanced analytic options that has better historical papers than for example SCOPUS. This enables the user to select key journals, authors etc. To determine the impact factor of each journal Journal Citation Reports (JCR) is used. The following keywords are used for the initial, explorative search: Bilge keel Maritime CFD Ship roll damping Vortex shedding For the results reference is made to Appendix A. The mind map in Figure A-1 provides an overview of key topics. These will be used to refine the search terms and identify possible information of interest. The results shown in Tables A-1, A-2, A-3 provide an overview of journals, conferences, authors and institutions that seem to play a role in these areas Search terms This section deals with how the search terms are generated from the research questions. Below the sub-questions are restated: 1. What is an appropriate CFD method that balances accuracy and computation time with the capability to accurately compute hull-pressures and vortex shedding? 2. What is the current state-of-the-art in the field of roll damping? 3. Which approaches have been taken to solve the hull-pressure component of roll damping? 4. What are the physical workings of vortices and their influence on hull pressure? To ensure that all possible subjects are covered the search terms are categorized in four domains: bilge keels, vortex shedding, CFD and roll damping. This ensures full coverage. For the resulting mindmap reference is made to Figure Databases and search engines Various databases and search engines are available to find scientific articles, proceedings and other materials. A variety of sources are used in the literature review, including but not limited to:

27 2-2 Planning 9 ScienceDirect Scopus WoS OnePetro (Society of Petroleum Engineers (SPE)) Delft University of Technology (TU Delft) Library ASME Local Single Buoy Mooring (SBM) and Maritime Research Institute Netherlands (MARIN) sources Google Scholar The various arguments for and against the sources are discussed in length in a multitude of papers and are left outside the scope of this document as all sources are well known, including their limits. 2-2 Planning The planning is as shown in Figure B-1 found in Appendix B. It should be noted that as this planning is very global as specifics will consistently change depending on the results obtained.

28 10 Methodology Bilge keel structural Bilge keel geometry Bilge keel fatigue Bilge keel pressure Bilge Keel Vortex Shedding Wallbounded plate Vortex wake pressure Vortex shedding pressure Hull-pressure Induced Roll Damping CFD Roll Damping Ship damping moment Vortex method Ship roll motion Ship roll CFD Vortex shedding CFD Rollsupressing Ship roll damping Ship roll stability Figure 2-1: Mindmap of search terms

29 Chapter 3 On Roll Damping This Chapter will focus on the roll damping aspect of this literature review and will give an introduction to roll damping and hydrodynamics, discuss the current methods and provide an overview of the recent developments. 3-1 Introduction This Section will provide a short introduction of roll hydrodynamics and damping which will serve as a basis for the results of the literature review Roll Hydrodynamics To understand roll damping knowledge of the basic equations that describe the motions of a vessel are necessary. The most basic equation applicable to regular roll, but also easily modified to other motions is as follows: A φ φ + Bφ φ + Cφ φ = M φ cos (ωt) (3-1) with φ, φ and φ being the roll acceleration, velocity and angle respectively. With A φ, B φ and C φ being the roll moment of inertia coefficient, the roll damping coefficient and the restoring (or spring) coefficient respectively and M φ being the magnitude of the external roll moment and ω the frequency. The focus of this study is the roll damping coefficient B φ. For more information reference is made to Journeé [2]. Bernoulli s equation In 1738 Daniel Bernoulli finished his book Hydrodynamica and with it the so called Bernoulli s Equation: 1 2 ρ U 2 + ρgz + p = C (3-2)

30 12 On Roll Damping where p is the pressure, g the gravitational constant, z the depth and C a constant. The Bernoulli equation describes a direct relation between pressure and flow velocity if the depth is constant and can thus be rewritten to: p = 1 2 ρ U U C p (3-3) with C p being the pressure coefficient, which will be discussed in Section This equation is the basis for the Ikeda et al. formulation of hull pressure damping [6] Roll Damping In 1981 Himeno wrote a report on the state of the art of the prediction of roll damping [3]. Even now the fundamentals used have not been changed as for example International Towing Tank Conference (ITTC) still follows the same basic practice as recommended by Himeno. This practice is based on the notion that roll damping is caused by a variety of effects and that these effects can be described separately. The following components are assumed, neglecting interactions between them, reference is made to Himeno [3] and Chakrabarti [7]: 1. Friction damping 2. Eddy damping 3. Lift damping 4. Wave damping 5. Normal-force bilge keel damping 6. Hull-pressure bilge keel damping 7. Wave bilge keel damping As the naked hull damping is not the focus of this study points one through four are not discussed. The focus is on the bilge keel damping. As seen above three components are identified which are discussed below. Normal-force Bilge Keel Damping The normal-force on the bilge keel is due to the pressure difference between the front and back of the bilge keel. With the front of the bilge keel defined as the face of the bilge keel that is facing the flow and the back the face that points in the direction where the flow goes. This force can be described using the well-known Morison equation which is the result of a research on which a 1950 paper by Morison, Johnson and Schaaf has been published [8]. The basic Morison equation is: F = 1 2 ρc DA U U + ρv C I Ü (3-4) with ρ being the density of the water, A being the projected surface which in the case of a bilge keel is the bilge keel height multiplied with the length, V the reference volume and U

31 3-1 Introduction 13 being the displacement and thus U and Ü being the flow velocity and acceleration respectively. C D and C I are the most important and at the same time most uncertain factors. They are respectively the drag and mass or inertia coefficients. These coefficients are dependent on a multitude of factors and are usually found through experiments. The results are incorporated in comprehensive tabulations and graphs from which appropriate values can be chosen or formulated in an empirical manner. The Morison equation was modified by Ikeda et al. [9] for the purpose of predicting roll damping. The formula was linearized and the inertia component neglected as it was assumed to play no role in roll damping. It was shown that the C D and C I values are highly dependent on the so called KC number named after Keulegan and Carpenter [10]. The non-dimensional number is, in the case of bilge keels, dependent on the bilge keel height and amplitude of the motion and independent of the frequency. A correction factor f was introduced into the KC number to accommodate flow velocity increases due to hull geometry, with f being f = exp( 160(1 σ A )) (3-5) where σ A is the sectional area coefficient. This leads to an expression for C D : C D,Ikeda = 45 KCf (3-6) Although the Morison equation is usually regarded as quite accurate with the proper C D and C I values, for zero forward speed Floating Production, Storage and Offloading (FPSO) ships with large bilge keels and complex bilge keel geometries they are not readily available. Up to recently these values have been estimated and used until they were proven inaccurate and now are generated experimentally. In recent research and experiments by Schut [11] it was shown if the coefficients are chosen properly there is good agreement between experiments and the Morison equations. For a 303 FPSO the KC number was identified to have the following (empirical) relation with the C D coefficient: C D,303F P SO = { KC for KC < KC for KC 6 (3-7) which yields lower damping than the C D,Ikeda and agrees with experimental results. Further explanation of the normal-force damping is not provided as this is not the direct subject of this review. Reference is made to Ikeda [9] and Schut [11] for more information. Hull-pressure Roll Damping Hull-pressure roll damping results from pressure on the hull due to vortices created by the motion of the bilge keel. It results in positive pressure in front of the bilge keel and negative pressure behind the bilge keel. For a simplified distribution reference is made to Figure 3-1. From this Figure it can be noted that if the pressures and their arms are integrated over the hull a moment countering the motion is generated which is,as the pressure depends largely on velocity [6], a form of positive damping. Depending on the roll center, which is dependent on draft and weight distribution, reference is made to Yuck [12] and Park [13], as well as the pressure distribution the damping might become smaller or even negative. Reference is made to Section 3-2 for more in-depth information.

32 14 On Roll Damping Figure 3-1: A simplified 2D representation of an FPSO including its exaggerated bilge keel and bilge-keel induced hull pressures Wave bilge keel damping Radiation waves are waves that are radiated outwards from the ship. This wave radiation presents an energy loss and thus causes damping. This component is not discussed in this literature review as for bilge keels both Himeno [3] and Schut [11] as well as others concluded that wave induced bilge keel damping is negligible compared to the normal force and hull pressure damping. 3-2 Results In the subsequent Sections the results of the literature review are detailed and afterwards discussed in Section Ikeda et al. Ikeda et al. developed an empirical formula in 1977 [6] to describe hull-pressure damping. At the time there were limited measurements of the hull-pressure caused by a bilge keel. A vertical set-up using three models, two 2-Dimensional (2D) models and one ellipsoid model, was implemented to perform additional experiments. To limit the influence of hydrostatic pressure and free-surface effects the 2D models were mounted perpendicular to the freesurface. The models were excited using forced oscillations and the pressure was measured at various spots around the bilge keel. From these experiments a non-dimensional C p was determined at the various sensor locations at the moment when the flow velocity was maximal, thus at a roll angle of zero degrees. The expression is as follows: C p = p 1 2 ρ(rωφ 0) 2 (3-8) where p is the pressure at the maximum flow velocity, r the distance from the roll center to the bilge keel and φ 0 the roll amplitude. Plotting C p for the different measurement points

33 3-2 Results 15 shows a pressure distribution. When results for various experiments are put together certain dependencies become clear. It seems that the pressure is proportional with the roll frequency. The pressure from the back face of the bilge keel seems to be dependent of the KC number which for a bilge keel is defined as: KC = πrφ 0 h BK (3-9) where h BK is the height of the bilge keels. Furthermore the length of the pressure distribution seems dependent on the KC number, as well as the roll amplitude. The pressure distribution from the front face of the bilge keel seems independent of the KC number. Using these results an empirical formulation for the hull-pressure can be formulated. Using the Bernoulli Equation, reference is made to Equation 3-3 and modifying it to the problem at hand the hull pressure created by a bilge keel can be expressed as: p = 1 2 ρr2 f 2 φ φ Cp (3-10) where f is the velocity modification factor as defined in Section Multiplying the pressure by the moment arm and integrating over the surface yields a roll damping moment which can be rewritten as a non-linear damping coefficient: B BK ( φ) = 1 2 ρr2 f 2 φ C p lds (3-11) S where B BK ( φ) is the non-linear damping coefficient, l the moment arm and S the hull surface. Using a Fourier series expansion of a general form of B with harmonic excitation and the assumption that the energy loss due to damping during half a period is that same for linear and non-linear damping the following equation can be obtained: B e = B π ωφ 0B ω2 φ 0 2 B 3 (3-12) with B e being the equivalent linear damping coefficient and where B 1, B 2 and B 3 are the linear damping coefficients proportional to respectively φ, φ φ and φ2. Rewriting equation 3-11 into equation 3-12 and making ω and B e non-dimensional the resulting damping is: ˆB e = 4 r 2 f 2 ˆωφ 0 3π B 2 C p lds (3-13) S with B and being the beam and water displacement of the ship respectively. In this equation all variables are known from the exciting frequency and amplitude and the geometry and mass distribution of the vessel. The parameter that remains to be solved is C p. C p is divided into two regions: the negative pressure behind the face of the bilge keel and the positive pressure in front of the bilge keep, respectively C p + and C p. These pressure coefficients are assumed empirically based on the results of the experiments as detailed by Ikeda et al. [6].

34 16 On Roll Damping C p + is assumed to have a maximum at the face of the bilge keel of: C p + = 1.2 (3-14) The distribution is assumed to decrease linearly with distance from the bilge keel and reaches zero at the free surface or the center line of the bottom of the hull. C p is assumed to be equal to the positive pressure coefficient minus C D,Ikeda as in equation 3-6: C p = 1.2 C D,Ikeda (3-15) The distribution is assumed to be a trapezoid which is constant to S BK /2 and linearly decreases to zero at S BK which is determined empirically as: S BK h BK = 0.4 πfrφ 0 h BK (3-16) where S BK is the distribution length. This means equation 3-13 can now be solved. There are various methods to simplify the calculation and make it more practical for which reference is made to the paper by Ikeda et al. [6]. The method described here is still recommended by ITTC [14] in Recent Developments This Section is aimed at recent developments that are not yet generally practiced, but look promising and are relevant to this literature review. On a general note Oliveira suggests that quadratic damping is not applicable to FPSO roll damping and proposes bilinear or hyperbolic damping to get better results as roll damping becomes linear at larger angles [15]. It should be noted that Oliveira focuses on roll-decay test and subsequent damping coefficients and does not separate damping into the ITH components. Van Dijk showed that with a proper (tuned) damping coefficient good agreement can be reached between theory and full-scale measurements if wave-spreading is taken into account. He further notes good agreement between model tests and full-scale measurements which shows adequate handling of scaling effects [16]. Korpus used Computational Fluid Dynamics (CFD) and potential theory to investigate the difference between potential theory and Reynolds-Averaged Navier-Stokes (RANS) [17]. By subtracting the potential theory roll moment from the RANS roll moment the shear and vortex effects could be captured. The shear roll moment was negligible, while the vortex effect was identified to have significant difference in phase and magnitude compared to the potential theory. Furthermore for tests including bilge keels higher harmonics (third and fifth) were identified as significant.

35 3-2 Results 17 Memory effect Flow memory effects are due to flow velocities and thus vortices from previous cycles. This means that if the system is not in a steady state the velocity field depends on one or more previous cycles, as well as the roll motion itself. Van t Veer [18], Katayama [19] and earlier Ikeda [20] confirmed that memory effects are present and Schut [11] showed that the results from regular oscillations are not directly applicable on irregular oscillations. Katayama et al. proposed a manner relying on global roll velocity to cope with memory effects and irregular oscillations [21]. In a 2010 paper of Katayama it was shown that the drag coefficient changes when a test device undergoes a forced oscillation from rest [19]. It is found that after the fourth swing the drag coefficient stabilizes. Various formulations were made to incorporate the memory effect and changing drag but a definite conclusion is note made. Three estimation methods are tested against measured data: 1. Method one uses a C D number that is based on a position dependent KC number and is updated with each time step. It excludes memory effects. 2. Method two expands on method one by including memory effects 3. Method three utilizes constant KC and C D numbers but does include a factor f for flow velocity at the bilge. It seems that the results from method two are in best agreement with the measurements as seen in Figure 3-2. Ibrahim outlines a method to incorporate the pressure and thus force created by waves generated in previous cycles [22]. He utilizes a convolution integral in the following form: F = α V t K(t τ)v (τ)dτ (3-17) with α being the added mass, V the ship velocity and K the retardation or memory function. δφ(t τ) K(t τ) = ρ sdσ (3-18) δτ with s being the normal vector of the surface element dσ. A similar method may be useful for the memory effect of flow velocity and damping as well, but needs to be evaluated more properly for a conclusion. (Ir)Regular waves Roll damping in irregular waves is observed to be four times as high as in regular waves by Orozco [23], which is largely attributed to the local kinematics. Orozco applied the method of Ikeda et al. to irregular waves. For this he linearized the non-linear damping coefficients using stochastic linearization which takes into account the spectra of the incoming waves. This yielded better results than the regular wave linearization where the quadratic damping is assumed to dissipate the same energy as the linear damping in one oscillation cycle. Using

36 18 On Roll Damping Figure 3-2: Comparison of estimation methods and measured data [21]. measurement data method 1 method 2 method 3 the method of applying a Rayleigh distribution to the maximum values of roll a estimate can be found for the roll motion maxima, although it should be noted that this is only valid for linear damping. It is used in Orozco s research as it provides an indication which should be sufficient for determining the influences of various different sizes bilge keels. After applying a factor of four to the roll damping the numerical model, which utilizes HYDROSTAR, showed good agreement with the experimental results. Reference is made to Figure 3-3. Jung identified in an experiment that local kinematics dominates the generation of the vortices and not ship motions, when the wave period is longer than the natural roll period. I.e. the flow velocities are higher than the body s roll velocities [24]. Jung continued his research for a larger range of period for a rolling barge fixed in other Degree-of-freedom (DOF)s in regular waves [25]. He concluded that for waves at the roll natural period wave the generated vortices were behind the body s motion, resulting in viscous damping, with the same results for waves with shorter periods than the roll natural period. It was confirmed that for longer periods damping became negative and thus viscous effects added to the body s motions. Van t Veer [26] uses a similar approach as Orozco, but instead of relying on global motions use is made of local kinematics as an input for the Morison/Ikeda formula resulting in a calculation scheme as in Figure 3-4. Instead of using the velocity increment factor as proposed by Ikeda, reference is made to Equation 3-5, velocities are obtained directly from potential flow theory. This allows van t Veer to incorporate the local kinematics missing in the research done by Orozco. Van t Veer remarks that there still is quite some work to be done regarding the complex flows around the appendage: It is for seen that such a development [an heuristic damping model] will utilize a (local) KC-dependent drag coefficient in combination with a local flow velocity obtained from potential flow. Among others one difficulty lies in finding a proper relationship between the complex flow behaviour seen around the appendage and the potential flow velocities in wave conditions. Another advantage of using local velocities is the ability to differentiate between the bilge

37 3-2 Results 19 Figure 3-3: Results from Orozco where Ikeda s formulation for roll damping was stochastically linearized and multiplied by four to account for irregular waves as became clear from experiments and using HYDROSTAR to solve [23]. keels on both sides which is relevant in for example beam seas where one keel will experience much higher local velocities than the other. Van t Veer continued research on bilge keels leading to a 2012 paper on bilge keel normal forces [27]. The goal of this research was to provide insight into the forces on the bilge keel for structural calculation, not roll damping. Nonetheless these results are relevant for this research. The bilge keel forces are calculated using local fluid velocity Response Amplitude Operator (RAO)s, the drag equation with a Keulegan-Carpenter number [-] (KC)-dependent drag coefficient and a correction factor to account for unknowns such as the memory effect and the free surface. Especially the free surface seems to make an impact at larger roll amplitudes and is well observed during experiments. The local relative velocity is dependent on vessel motions, radiation velocities, diffraction velocities and wave orbital velocities. Another point that was raised is the difference between exposed and leeward bilge keel velocities and thus loads in mostly beam seas. In 2013 van t Veer published another paper [28]. The focus was on irregular waves and bilge keel forces (not pressures on the hull). In this research van t Veer used the Bureau Veritas (BV) software Hydrostar to calculate the total local velocity RAOs around the bilge keel composed of wave orbital velocities, diffraction velocities and radiation velocities. The peak of this relative velocity is used to calculate the maximum force on the bilge keel. Further research is mostly aimed at determining bilge keel forces in irregular waves, including the inertia term and higher harmonics, showing their relevance when predicting correct loads. Brown and Patel [29] developed a theory using the Discreet Vortex Method (DVM). This method entails the use of potential flow in combination with discreet vortices to model viscous effects within an inviscid model. Reference is made to Section for more details. The results of the model were captured in the following formula which can be used in the frequency domain:

38 20 On Roll Damping Figure 3-4: Calculation scheme utilizing stochastic linearization by van t Veer [26]. M vs = f 1 (φ 0 )f 2 (r/d)e i(ωt+α) (3-19) Where M vs is the vortex shedding induced moment, φ 0 the roll amplitude, r the roll center measured upwards from the keel, d the draft and α the phase. Brown and Patel thus state that the roll moment and thus damping is dependent on the roll frequency, amplitude, roll center and draft. Their results seem to yield a fair estimate but differ at resonant frequency with large amplitude motion. A disadvantage of this method is that the roll center needs to be estimated as the roll center is determined from the motions of the vessel. Downie and Graham developed a method using the DVM to perform an one-off calculation to determine a vortex shedding moment coefficient to be used in potential flow calculations to estimate roll damping [30] which is based on the work of Brown and Patel [29]. Hajiarab continued this work with Downie and Graham as documented in [31] and [32] and finished his PhD thesis recently which involved a black box model compatible with most potential flow-based hydrodynamics software [33]. The focus was on rectangular cylinders with sharp corners and not round corners with bilge keels. The results are promising when compared to experiments, reference is made to Figure 3-5 and Figure 3-6. There seems to be an overestimation of the RAOs at wave periods higher than the natural period and underestimation at wave periods lower than the natural period for the Hajiarab model tests, while comparison to the Brown et Al. data shows a reversed error. Spectral analysis One of the goals is to develop a tool to determine roll damping and more specifically to predict the Most Probable Maximum (MPM) roll amplitude, for this purpose spectral analysis is employed. To determine these roll maxima it is common to assume a Gaussian distribution for the incoming waves, assume a narrow-band spectrum and a Rayleigh distribution for the wave height and through a linear system thus the MPM of the motions, reference is made to

39 3-2 Results 21 Figure 3-5: Regular wave RAOs for potential diffraction-refraction based software, using the method by Hajiarab and from experiments by Hajiarab [33]. Figure 3-6: Regular wave RAOs for potential diffraction-refraction based software, using the method by Hajiarab [33] and from independent experiments by Brown et Al. [34].

40 22 On Roll Damping Journée [2]. While this is valid for most motions it is not for roll, as the Rayleigh distribution is only valid for a linear process and roll is non-linear due to viscous effects. To solve this problem various solutions have been proposed which can broadly be divided in two areas: the linearization, a different distribution that can cope with non-linearities or (time-consuming) time domain simulations. The linearizations for irregular waves are stochastic linearizations which are based on maintaining the statistic properties of the non-linear damping in the linear damping. This means that the wave state is incorporated in the damping and iteration has to be performed, solving and converging the Equation of Motion (EoM) to obtain the proper equivalent linear damping coefficient, reference is made to Orozco [23] and Droby [35]. This iteration can be avoided by using an approximation as proposed by Drobyshevski [35], but is mostly useful as an initial approximation. After linearization usually the Rayleigh distribution is applied. Leloux compared a two-parameter Weibull and the Rayleigh distribution but are both found lacking for the MPM roll [36]. Leloux makes reference to a third method, which is the so called Linearize & Match & Iterate (LMI) method which approximates a non-linear system with a non-linearity in the form of u u by another non-linear system which is based around a cubic polynomial, reference is made to Prevosto [37] and Minko [38]. The system is then supplemented with a variety of linear systems to ensure statistical equality. Gachet and Kherian assessed the impact of stochastic linearization on ship operability [39]. It is compared to constant damping and is found to be more favorable as it yields higher operability. Unfortunately Gachet does not evaluate the accuracy of stochastic linearization. Choi attained good agreement when using regular wave linearization of a damping coefficient obtained from experiments [40]. Leloux concluded that spectral linearization is more applicable than the harmonic, regular wave approach. It seemed to yield reasonable results compared to the experiments, but it seemed to underestimate the roll damping when the wave peak frequency was not located near the roll natural frequency. An alternative is selecting a distribution that is more fitting for the response and the MPM values. It should furthermore be noted that the non-linearities are also introduced due to changing underwater geometry and non-linearities in the waves. The spectrum of the waves of furthermore often assumed to be narrow-band while it often is more medium-band. This implies that just linearization of the system is not sufficient to justify the use of the Rayleigh distribution. Nonetheless the combination of linearization and a Rayleigh distribution is often used for roll as it yields reasonable results as long as roll angles are not too large [41]. 3-3 Discussion From the results as detailed in Section 3-2 it seems to become apparent that there are a few issues that need further investigation to develop a robust application of the hull-pressure roll damping in irregular waves: 1. The influence of memory effects in irregular waves and how to incorporate them

41 3-3 Discussion The implementation and influence of local kinematics 3. The linearization of the damping 4. The validity of the Rayleigh distribution when damping is linearized Memory effects Memory effects are quantifiable [21], [11] and are significant (up to 15%). As for irregular waves true memory effect can probably only be generated through the use of a wavetrain and introduces non-linearity into the problem as the superposed wavetrains start interacting with each other. It seems more likely to define some convolution for the RAO to account for the memory effects, such is done for a Dynamic Positioning (DP) observer design [42]. Local kinematics Local kinematics seem to play a large role in irregular waves, Orozco [23] even attributed a 400% increase in the damping to it. From the results of van t Veer [26] it becomes apparent that this might be true as the relative local normal velocity, including wave orbital velocity, can be up to three or more times as high as the velocities induced due to ship rolling. It might be necessary to adjust Ikeda s formulation for the hull-pressure damping to incorporate local kinematics. Linearization of the damping Most sources agree that stochastic linearization is the most applicable when linearizing for use in irregular waves and a spectrum analysis. Rayleigh distribution Although the Rayleigh distribution is considered to be not entirely accurate it is still widely used and seems to be applicable to give an indication of the MPM roll angle, as well as sensitivity to certain seastates. Further development of a more suitable distribution would be welcome but most likely outside the scope of this report. A candidate seems the LMI method. Summary Summarizing the most immediate problems at hand are the reformulation of Ikeda s hull-pressure theory to local velocities and irregular waves, as well as the incorporation of the memory effects if necessary. These can then be incorporated into the potential flow calculation as shown in for example Figure 3-7.

42 24 On Roll Damping Geometry and other BC Local velocities Method 1 Hull pressures Calculate pressure Loads and moments Method 2 Mass and damping matrices RAOs Method 3 Roll damping and memory RAO Calculate damping RAO incl. damping Figure 3-7: Potential flow calculation scheme including possible options for incorporating roll damping and memory effects, excluding iterative schemes

43 Chapter 4 On Vortices This Chapter will focus on the creation, the movements and the pressure of the vortices generated by the bilge keels. The surface-effects as well as wave-induced water velocities are considered. 4-1 Introduction Vortices are not defined in a very exact manner, but can be seen as rotating elements in a flow, such as hurricanes, in the wake of a ship or airplane or as shed by a body in a flow. An example is the von Karman vortex street visible in Figure 4-1 where vortices are initiated due to a small asymmetry after which vortices are created and start shedding. Usually vortices are found in a viscous flow where a no-slip condition on a wall introduces viscous stresses which in turn introduce a velocity curl and thus vorticity. From this it immediately becomes apparent why the potential flow theory fails to properly describe the roll damping, as it assumes an inviscid, irrotational flow, reference is made to White [43]. Vortices, when formed, are almost always relevant when looking at fluid flows. First due to the velocity and pressure variations they introduce, secondly due to the force caused when a vortex is shed and third as they induce memory effects, as during a next pass of a body a previously shed vortex might still be present. These subjects will be introduced one by one, starting with the velocity and pressure variations as those are the cause of the other effects. But first some key concepts surrounding vortices are discussed. Vorticity is defined as the curl of the velocity field in a flow: i j k δ δ δ curl U = δx δy δz u v z (4-1)

44 26 On Vortices Figure 4-1: Example of a von Karman vortex street due to vortex shedding, source: Malikiaa To clarify, vorticity is the local angular rotation, best visualized as the rotation of a particle around its own axis. This vorticity is relevant to a vortex as it partly describes the character of the vortex. A vortex usually starts as a rotational or rigid-body vortex. This means that the angular velocity around the center of the vortex stays the same for every point in the vortex. This also means that the vorticity is the same in any spot in the vortex. As a vortex moves away from a surface, i.e. if no external forces are present, it decays to an irrotational vortex, which has no vorticity and an angular velocity around the center that decreases the further from the center a point is. This is due to viscous effects. In some calculation schemes the rate of vorticity shed from the separation point depends on the local relative velocity at the separation point, reference is made to Braathen [44] δcurl U δt = 1 2 U r U r (4-2) Kolmogorov microscales are the scale at which the smallest vortices exist. At this scale the viscosity of the fluid is dominant and vortices are dissipated into heat. This means that no vortices exist that are smaller than: η = ( ) ν 3 1/4 (4-3) where η is the Kolmogorov microscale in meters, ν the kinematic viscosity and ɛ the rate of dissipation of turbulent kinetic energy. ɛ Vorticity and velocity There is interaction between vorticity and the velocity gradient which is essential to turbulence. Two elements are important here: vortex stretching and vortex tilting. Vortex stretching is the stretching and thinning of a vortex tube along its rotational axis when its rotation is

45 4-2 Results 27 accelerated. Vortex tilting is when a velocity gradient normal to the rotation axis of a vortex exists which causes rotation of the vortex tube and thus vorticity on a second axis. As these effects are essential to turbulence, it becomes apparent that a 3-Dimensional (3D) simulation has to be done to accurately model turbulence as in a 2-Dimensional (2D) simulation these effects are not present Velocity and pressure variations Observing Bernoulli s equation, reference is made to Section 3-1-1, it is found that local pressure and velocity are inversely dependent on each other. This means as the vortex influences the flow velocity a pressure increase or decrease can be found, depending on the rotation direction and location relative to the vortex. 4-2 Results The results of the literature review are presented and discussed in the following Sections General Results Sarpkaya and O Keefe [45] performed experiments based on a flat plate attached to a wall in an oscillating flow. Three vortex shedding regimes depending on the Keulegan Carpenter number were identified: 1. KC < < KC < 8 3. KC > 8 For KC < 3 when a new vortex is created it sheds away with the vortex created in the previous motion cycle, creating a counter-rotating pair that moves away at a 45 upwards angle to the left or right of the tip of the plate. The direction is random and dependent on starting conditions, once a direction is established it is continued for an indefinite period of time. For 3 < KC < 8 the vortices of the previous cycles have started decaying and thus will start orbiting around the newly shed, stronger vortices. This results in a more complex flow pattern that does not reset itself each cycle such as is the case with KC < 3. This means that on each side of the plate a new vortex is generated each half cycle around which the older vortex starts orbiting. The time for shedding a fully developed vortex and the decay time are identified to be crucial to determine flow and pressure characteristics. For KC > 8 the vortex shedding approaches a steady state in which one large vortex is shed each half cycle in addition to various smaller ones which develop if the longer duration of a cycle allows them to. Increasing the KC number leads to more vortices being shed.

46 28 On Vortices Yeung utilized a different approach, using a vertical, partially submerged plate with an angular forced oscillation at the emerged end, which thus includes the free surface and an angular movement [46]. Yeung s results are not directly comparable due to differences between the setup and KC number. What can be compared are the identified flow regimes. Yeung observed two flow regimes, the so-called symmetrical and asymmetrical regimes. The asymmetrical regime is similar to the area identified by Sarpkaya for KC < 3 with a shedding of vortex pairs in one 45 direction. The symmetrical flow regime is similar to the 3 < KC < 8 regime, which is described by Yeung as a vortex pair which is not strong enough to move away from the plate and where the older vortex is absorbed by the newer, stronger one. While there are discrepancies between the work of Sarpkaya and the work of Yeung, the identified regimes are similar. Various experiments very similar to the work of Yeung have been done by Klaka et al. [47]. A distinct difference is that Klaka et al. compares 2D and 3D effects. Unfortunately the flow is not visualized and only forces, moments and damping coefficient are considered. One important note is made. Klaka et al. found a transition in the 2D model at a certain frequency that caused a (relatively) large shift in the roll moment generated, which was not encountered in the 3D model. They theorized that the transition observed from symmetric to asymmetric vortex shedding by Yeung is the most likely source of the transitional phenomenon. This brings into question the results of Yeung for a 3D case. Aloisio performed a Particle Image Velocimetry (PIV) analysis of a ship model with a bilge keel during a free roll decay test [48]. At a Froude number of zero (no forward speed) Aloisio identifies the formation of the Kelvin-Helmholtz instability (mostly found at the separation between two fluids). This instability is characterized by acceleration instead of velocity. The intensity of the vortex is found to be dependent on the roll amplitude. The vortex behavior seems similar to that identified by Sarpkaya and O Keefe where in the beginning the flow regime is similar to the 3 < KC < 8-regime as identified by Sarpkaya, although from the data provided by Aloisio it is impossible to determine to relevant KC number. Oliveira studied the effect of vortex shedding due to large bilge keels on the roll damping on an FPSO using numerical methods and experiments with PIV [49]. During a decay test the behavior of the vortices is analyzed. A 45 shedding angle is observed, which gives the impression of the KC < 3 area identified by Sarpkaya. This hold for smaller roll angles, but at larger angles the interaction between the vortices becomes too strong, resulting in a split of the pair into two pairs. One pair will move away from the hull while the other hugs the wall. This separation can explain the limit on roll damping at large angles where it seems to reach saturation. After some periods a vortex street can be observed. It is furthermore determined through regular wave experiments that the size of the bilge keel has a influence of the natural roll frequency, i.e. larger bilge keels lead to lower natural frequency. This is most likely due to additional added mass. Avalos performed experiments and numerical calculations to study vortex shedding and roll damping around bilge keels on sharp and rounded bilges [50]. After the first one-and-ahalf oscillation the last vortex interacts with the previous vortex through which both are dissipated. From this point on the flow field seems to reach a steady state similar to the regime identified by Sarpkaya for KC < 3 with pairs shedding at a 45 angle, most likely as the outer vortex is less strong. It should be noted that from simple calculations the KC number belonging to Avalon s data seems higher that three. It should also be noted that the

47 4-2 Results 29 KC-number is hard to determine as the roll center position is unclear. It was identified that as the bilge keel becomes smaller the flow becomes more complex and the vortices shed start hugging the hull, similar to other experiments Near-surface influence From sources such as Bernal [51] it is stated that as a vortex approaches the surface, the vortex lines open, resulting in vortex lines that run from and to the surface. Ohring numerically shows that a vortex can connect and be absorbed by the surface, connect and create a secondary vortex or bounce from the surface while creating multiple secondary vortices [52]. This depends on the vortex velocity and the amount of surface tension present. At larger angles damping is Rood performed similar research in 1994, with similar results [53]. He shows with a thought experiment that vorticity is not conserved, while not ignoring any physical laws such as conservation of momentum and mass. Imagine an infinite horizontal plate with a fluid on top and bounded by another plate on top of the fluid. If the lower plate starts moving a velocity gradient will be created which will results in a steady-state of constant vorticity. If the upper plate is replaced with a free surface the flow velocity will become equal to that of the moving plate resulting in zero vorticity as no gradient is present. In reality the free surface is an interface between two fluids where vorticity is transferred from one fluid to another. In his paper Rood reviews other literature (including Bernal s work) that confirm this hypothesis Influence of (irregular) wave-induced orbital velocities While studies have been done towards vortices generated by a fully submerged cylinder under wave action, less work has been done on bluff bodies, bodies with appendages and partially submerged bodies. Jung performed an experiment in 2002 with a fixed rectangular structure in waves, reference is made to [54] and [55]. PIV recordings were made and analyzed. Vortex paths were analyzed and presented as in Figure 4-2. The paths seem to venture quite far from the hull, but the vorticity extends quite far from the center of intensity. Significant differences were found between the leeward and seaward sides. It is observed that water surface level and velocities around the barge have a large impact on the shedding and path of the vortices. The absolute vorticity on the leeward side is observed to be roughly half of that on the seaward side. Jung does not discuss pressure, so no conclusions can be made on the impact on this thesis. In 2005 Jung continued work on the rectangular body (without keels) in waves, but now instead of fixed it was free on the roll axis using a hinge through the Center of Gravity (CoG) [24]. In waves with a period longer than the natural period of the structure is was identified that the rolling body moved in the same direction as the fluid flow, but the fluid flow had a larger magnitude. This means that the relative velocity was lower leading to less strong vortices, as opposed to what was described in Section 3-2. For a wave with the natural period this is not valid and the vortices are generated in the wake of the body roll motion and cause positive damping. The vortex shedding patterns at the natural period were not discussed while these are of interest for this review. Other patterns were found to be similar to the fixed case, reference is made to the previous Paragraph.

48 30 On Vortices Figure 4-2: Path of the highest vorticity concentrations of a fixed rectangle in a regular wave. The solid line and circle is positive vorticity, while the dotted line and circle are negative vorticity. The points a through e represent the time of the snapshots. For the seaward side this is an inverse cosine and for the leeward side this is an inverse sine for wave elevation 8 cm before the barge (seaward) and 8 cm behind the barge (leeward). From Jung [55]. In 2006 Jung continued his research but with a larger spread of wave periods and aimed at viscous damping [25]. The research contains PIV images, from which it is identified that the vortices at the leeward and seaward side are the same size and magnitude for the natural roll period. At shorter periods the leeward side shows a decrease of size and magnitude of the vortices compared to the seaward side. In 1999 Oshkai and Rockwell applied PIV to a submerged cylinder at various depths, subjected to wave action [56]. It is noted that decreasing the depth of the cylinder resulted in the retardation of the orbital motion and variations in the shedding point of the vortices. Chen performed Reynolds-Averaged Navier-Stokes (RANS) simulations and used PIV measurements by Jung [54] on a fixed and rolling rectangular barge subjected to wave motions. The validated RANS code was used to simulate a barge in regular waves that capsized due to extreme roll motions [57] and one Degree-of-freedom (DOF) large amplitude roll motions of a barge in a regular wave [58]. Unfortunately the discussion about the large amplitude roll motions in a regular wave is limited to the influence of the wave period, where it was identified that waves with the same period as the free-decay period were causing resonance. As for the capsizing simulation more results were visualized. It is surmised that the wave-induced velocities are strong enough to generate vortices due to flow separations at the barge corners, with a strong positive vortex being created when the surface elevation rises. As the surface level drops the positive vortex decays and an elongated negative vortex is created which is subsequently shed as the flow velocity forces it downward. 4-3 Discussion While the research done into vortices is extensive, this is less so in the case where the freesurface or waves are involved. Regarding the free-surface it seems safe to assume that vortices

49 4-3 Discussion 31 are not reflected and mostly absorbed. This seems especially true for larger scales where surface tension plays a smaller role. Furthermore reflected vortices and secondary vortices seem to move away from the shedding point and thus have little impact on the hull-pressure. A more complicated topic is the generation and path of vortices in (ir)regular waves. The consensus is that there is an influence of the path of the vortices. One of the issues at this point is that it is hard to translate the deviating path to a different hull-pressure. A possibility is to use potential theory to determine fluid velocities and assume that the vortices will follow these vectors to determine possible influence on the hull-pressure.

50 32 On Vortices

51 Chapter 5 On Computational Fluid Dynamics This Chapter will elaborate on the various Computational Fluid Dynamics (CFD) methods available and give a general introduction to CFD. 5-1 Introduction Computational Fluid Dynamics (CFD) is a term that encompasses all techniques that utilize numerical techniques to approximate and predict the motion of fluids. The early development of CFD started of with the following quote of Lewis Fry Richardson after discovering complex geometries that would require very complex analytical solutions: Further than this, the method of solution must be easier to become skilled in than the usual methods (i.e. analytical solutions). Few have time to spend in learning their mysteries. And the results must be easy to verify, much easier than is the case with a complicated piece of algebra. Moreover, the time required to arrive at the desired result by analytical methods cannot be foreseen with any certainty. It may come out in a morning, it may be unfinished at the end of a month. It is no wonder that the practical engineer is shy of anything so risky. From this perspective Richardson published a paper on using finite differences [59], although at that time computations had to be done by hand, taking weeks to complete even simple problems. In the late 1960s the use of CFD for marine applications increased with the advent of the panel method by Smith and Hess [60]. The main focus of CFD as we know it today are methods to solve and approximate the Navier-Stokes equations, which can yield impressive results.

52 34 On Computational Fluid Dynamics Navier-Stokes equations The Navier-Stokes equations are the basis for a large part of the modern CFD techniques. The Navier-Stokes equations are comprised of four parts that act on a element [43]: 1. Gravity forces 2. Viscous forces 3. Pressure forces 4. Inertia This results in the following equations: ρg x δp ( ) δx + u µ(δ2 δx 2 + δ2 u δy 2 + δ2 u δu δz 2 ) = ρ δt + uδu δx + uδu δy + uδu δz ρg y δp ( δy + µ( δ2 v δx 2 + δ2 v δy 2 + δ2 v δv δz 2 ) = ρ δt + v δv δx + v δv δy + v δv ) δz ρg z δp ( δz + w µ(δ2 δx 2 + δ2 w δy 2 + δ2 w δw δz 2 ) = ρ δt + w δw δx + w δw δy + w δw ) δz Navier-Stokes equations (5-1) with g i being the gravity in i = x, y, z direction, µ the viscosity of the medium and u, v and w velocity in x, y and z direction. Combined with the continuity equation for incompressible flow, closure is obtained with for equations and four unknowns: δu δx + δv δy + δw δz = 0 (5-2) Potential Flow Theory Another theory used to supplement CFD is potential flow theory. The basis of the potential flow theory is that there is an expression, the velocity potential, for each point in the fluid. The derivative of this function in a direction is equal to the velocity in that direction. In an expression with Φ being the potential function: u = δφ δx v = δφ δy w = δφ δz Potential function derivatives (5-3) From this potential function it becomes apparent that the flow must be assumed rotation free as the vorticity is defined as the curl of the velocity and the curl of a gradient of a vector,

53 5-2 Background results 35 i.e. the gradient of the potential flow or velocity, is zero. Another assumption is inviscid and continuous flow. These assumptions mean that potential flow is invalid for flows where rotation is present such as boundary layers and turbulent wakes. For these flows it is necessary to switch to another theory such as Reynolds-Averaged Navier-Stokes (RANS) or add a vortex model such as Discreet Vortex Method (DVM) if vortices are involved, such as is the case with roll damping and bilge keel which is shown from experiments, reference is made to Brown [34]. For further information on potential flow theory, reference is made to Journée [2] and Richardson[59]. 5-2 Background results The methods available for solving CFD can be divided in two segments: Eulerian methods and Lagrangian methods. Eulerian methods are based around a grid where the properties are set for a specific time and place. Lagrangian methods are based around a different principle and depend on particles. For example for tracking flow velocity an Eulerian method is equivalent to placing flow meters everywhere, while Lagrangian methods would track a specific particle or group of particles, i.e. releasing floating balls to visualize the flow and determine velocity. For more detailed and recent information on the discussed CFD methods and additional methods, reference is made to Davidson [61] Eulerian Methods The Eulerian-based mathematical models described in this section are based around the Navier-Stokes equation, but involve some simplifications to keep computing time to acceptable levels. They rely on discretization and thus grids. DNS Direct Navier-Stokes (DNS) is simply put a process of directly solving the Navier- Stokes equations [62]. This approach is the most accurate and only contains discretization errors. It computes the flow velocities at every time and length scale, all vortices are calculated. Its disadvantage is the high computational costs due to very small grid size and time steps, limiting it to simpler flows or extreme computing times. The information it yields is very extensive, but not always more useful in engineering applications than other simplified methods. LES Large Eddy Simulation (LES) is a method that puts emphasis on the large scale motions as they have the largest transport capacity of conserved properties. Its computational costs are much lower compared with DNS. In a LES model the smaller motions are filtered out and modeled as dissipated using viscosity instead of calculated. RANS Reynolds-Averaged Navier-Stokes (RANS) is a method where unsteady flow properties of a statistically steady flow are averaged out and unsteady flow properties are regarded as turbulence. This turbulence is modeled using turbulence models, which are discussed in the next Section. RANS is also applicable to unsteady flows, although time averaging is not

54 36 On Computational Fluid Dynamics Figure 5-1: A comparison of the simulation of the separation bubble caused by a turbulent flow past a blunt plate, it should be noted that the standard κ ɛ model under predicts the size, while the Realizable κ ɛ is in very good agreement with experiment. Courtesy of ANSYS. usable. RANS is usually employed when there is interest in certain properties which can be modeled sufficiently accurate using RANS, such as average forces. Turbulence Models As the RANS equations cannot be closed a turbulence model is introduced. There are various models available, such as the κ ɛ model, the κ ɛ v 2 f model, the Realizable κ ɛ model, the Wilcox s κ ω model, the Reynold s Stress Model (RSM) and the SST κ ω model. The choice of turbulence model has a large impact on the results of a simulation as for example can be witnessed in Figure 5-1. The κ ɛ model is based around two transported variables, turbulent kinetic energy κ and turbulent dissipation ɛ. The two values present the two parameters that can be used to characterize a turbulent flow: the energy and the length scale. Turbulence is introduced into the equations as increased viscosity in the Reynolds stress. The Realizable κ ɛ contains a new formulation for turbulent viscosity and a new equation for ɛ has been derived. It seems to provide more accuracy in flows involving rotation, as well as (steady) flows with a square cylinder bounded to a plate, reference is made to Davis [63]. The Wilcox s κ ω model which utilizes the specific dissipation rate ω = ɛ κ and provides a better estimate of the near-wall region. It is mostly used in the aerospace industry. The SST model combines the near-wall qualities of the κ ω models with the κ ɛ for far field, which results in a pure κ ω model near the wall and a pure κ ɛ model further from the wall. The v 2 f model is not only based around κ and ɛ, but also v 2 which is the velocity variance and f which is an elliptic relaxation function. This means two additional transport equations are necessary. It is more accurate in capturing near-wall turbulence. The RSM do not employ turbulent viscosity, but directly calculates the Reynolds stresses. This means an additional six equations and a length scale equation need to be solved. This means it requires a great deal more computational power, has a potential for convergence