Inviscid and Viscous 2D Unsteady Flow Solvers Applied to FPSO Hull Roll Motions

Transcription

1 O C E AN E NGINEERING G R O UP Inviscid and Viscous 2D Unsteady Flow Solvers Applied to FPSO Hull Roll Motions Bharani Kacham December 2004 Report No ENVIRONMENTAL AND WATER RESOURCES ENGINEERING DEPARTMENT OF CIVIL ENGINEERING THE UNIVERSITY OF TEXAS AT AUSTIN Austin, TX 78712

2 Copyright by Bharani Kacham 2004

3 Inviscid and Viscous 2D Unsteady Flow Solvers Applied to FPSO Hull Roll Motions by Bharani Kacham, B.Tech. Thesis Presented to the Faculty of the Graduate School of The University of Texas at Austin in Partial Fulfillment of the Requirements for the Degree of Master of Science in Engineering The University of Texas at Austin December 2004

4 Inviscid and Viscous 2D Unsteady Flow Solvers Applied to FPSO Hull Roll Motions APPROVED BY SUPERVISING COMMITTEE: Supervisor: Spyros A. Kinnas Reader: Kamy Sepehrnoori

5 To family and friends

6 Acknowledgements At the outset, I would like to express my gratitude to my advisor, Dr. Spyros A. Kinnas for his unending support, encouragement and valuable advice which kept me going for the entire duration of my masters program. His understanding nature and fatherly concern are worth mentioning and thanking. I would also like to thank Dr. Kamy Sepehrnoori for agreeing to be the reader of my thesis in spite of his busy schedule. His comments and suggestions were of immense help in giving this thesis a final shape. It is with great pleasure that I mention the names of my CHL buddies; Dr. Lee, Shreenaath, Hua, Vimal, Yi-Hsiang, Apurva, Bikash, Hong and Yumin, from whom I have learnt a great deal. They have always been more than willing to help and were fun to work with. Special thanks to Yi-Hsiang, without whose help the thesis progress would have been very slow. I am indebted to my parents and my brother Shravan for all the support and freedom they have given me. It is very difficult not to list my friends Shilpa, Swapna, Kranthi, Jeetain and Gopal who were of great support and strength during the inevitable tough times. I wish them all great health and prosperity. Finally, I would like to thank the Offshore Technology Research Center for providing financial support through their Cooperative Agreement with the Minerals v

7 Management Service (MMS) and its Industry Consortium, and also the faculty of UT for the superior quality of education they imparted. Disclaimer: The views and conclusions contained in this document are those of the authors and should not be interpreted as representing the opinions or policies of the U.S. Government. Mention of trade names or commercial products does not constitute their endorsement by the U.S. Government. vi

8 Inviscid and Viscous 2D Unsteady Flow Solvers Applied to FPSO Hull Roll Motions by Bharani Kacham, M.S.E. The University of Texas at Austin, 2004 SUPERVISOR: Spyros A. Kinnas The roll dynamics of a Floating, Production, Storage and Offloading (FPSO) hull are of special interest in the present offshore industry. The FPSOs, while on duty need to be stationary for long periods of time in order to enable smooth drilling and oil transfer to the shuttle tankers. The present research is aimed at providing insights into the effectiveness of using anti-roll appendages, like bilge keels, in mitigating roll motion of FPSOs operating in mid-seas. Numerical modeling is a tool that can be extensively used to simulate and investigate real ship motions. The present work details a 2D unsteady Boundary Element Method and Navier-Stokes solver based on Finite Volume Method and their application to modeling roll motions of an FPSO hull. The Navier Stokes solver is a viscous solver and is advantageous when compared to the traditional potential flow solvers due to its ability to capture the effects of viscosity and separation past the bilge keel on the motion of the hull. vii

9 The method could be applied to three dimensional hulls by using either strip theory or by including the third dimension in the formulation. viii

10 Table of Contents Acknowledgements Abstract List of Tables List of Figures Nomenclature v vii xii xiii xix Chapter 1. Introduction Background Motivation Objective Overview Chapter 2. Literature Review Hull Motion Prediction Vortex Tracking Method Chapter 3. 2D Boundary Element Method and Its Applications Background Numerical Formulation Green s Theorem Application of Green s Formula for a two-dimensional body Numerical Implementation Validations of the method Prismatic cylinder of circular cross-section Prismatic cylinder of elliptic cross-section Prismatic cylinder of square cross-section ix

11 3.3.4 Prismatic cylinder of cross shaped cross-section Roll motions of a submerged body Forces and added mass coefficient D submerged hull without bilge keels D submerged hull with bilge keels Oscillating hull at free surface Boundary Conditions Numerical Implementation Time-marching Forces and Hydrodynamic coefficients Tip Vortex Tracking Method Numerical Formulation and Implementation Application to flow over a foil Chapter 4. Numerical Formulation of 2D Viscous Solver Non-dimensional governing equation Finite Volume Method Upwind scheme Time Marching Pressure Correction Scheme Chapter 5. Applications of 2D Navier-Stokes solver D Channel Flow Numerical Wavemaker Boundary Conditions Heave and Roll Motions Assumptions Coordinate System and Grid details Froude number and Reynolds number Heave Motion Roll Motion Results Roll motion of a semi-circular hull x

12 5.4 Submerged hull motions Fixed coordinate system and fixed grid Fixed coordinate system and moving grid Convergence Studies Hull with bilge keels Chapter 6. Conclusions and Recommendations Conclusions Recommendations Bibliography 136 Vita 143 xi

13 List of Tables 5.1 Comparison of roll added mass coefficients obtained from viscous and potential solvers for a submerged hull without bilge keels undergoing roll motion xii

14 List of Figures 1.1 Terra Nova FPSO (source: IcebergNet/gallery.htm) Description of motion under six degrees of freedom for a ship Volume confined by a surface Body B and a unit source P confined in a finite domain Figure showing the discretized body surface and corresponding index notation An infinitely long cylinder of circular cross-section subjected to a sinusoidal inflow Comparison of analytical and numerical values of perturbation potential on the circle Time history of the force on the circle in the x-direction An infinitely long cylinder of elliptic cross-section subjected to a sinusoidal inflow Time history of the force on the ellipse in the x-direction An infinitely long cylinder of square cross-section subjected to roll motion Time history of the moment on the square in the z-direction An infinitely long cylinder of cross shaped cross-section subjected to roll motion Time history of the moment on the cross in the z-direction Figure showing cross-section of submerged hull without bilge keels Figure showing cross-section of submerged hull with bilge keels Comparison between numerical (BEM) and analytical pressure on a heaving circle at Comparison between numerical (BEM) and analytical pressure on a heaving circle at Geometry details and boundary conditions for a submerged hull without bilge keels undergoing roll motion Time history of the moment on the hull without bilge keels undergoing roll motion xiii

15 3.19 Convergence of the roll added mass coefficient with respect to number of panels on the hull (without bilge keels) surface Error convergence plot for the roll added mass coefficient obtained for a submerged hull without bilge keels Geometry details and boundary conditions for a submerged hull with bilge keels undergoing roll motion Time history of the moment on the hull with bilge keels undergoing roll motion Geometry details and boundary conditions for a floating hull undergoing harmonic heave motion Force history for a hull undergoing heave motion for Comparison of heave added mass coefficients obtained from the BEM solver [Vinayan 2004] with those presented in [Newman 1977] and obtained from Euler solver [Kakar 2002] Comparison of heave damping coefficients obtained from the BEM solver [Vinayan 2004] with those presented in [Newman 1977] and obtained from Euler solver [Kakar 2002] A bilge keel with a trailing wake and a tip vortex A 2D foil subjected to a uniform inflow with a lateral sinusoidal gust Description of the initial wake and tip vortex geometry Figure showing trailing wake for a foil subject to a uniform inflow and a lateral sinusoidal gust Vorticity being shed tangentially into the shear layer Time history of the lift force on the foil Geometry details of the cell based scheme Description of the boundary conditions applied for a 2D channel flow The velocity and pressure contours for the fully developed flow in a 2D channel obtained from the viscous solver Comparison of horizontal velocity profile obtained from the viscous solver at the outflow boundary with analytical solution Description of the boundary conditions applied for a numerical wavemaker Pressure contours under a wave and the corresponding wave elevation at Pressure contours under a wave and the corresponding wave elevation at xiv

16 5.7 Depiction of coordinate system and domain for a floating body undergoing harmonic motions Grid details for a rectangular hull without bilge keels Grid details for a rectangular hull without bilge keels Comparison of hydrodynamic coefficients for a 2D hull undergoing heave motion obtained from the present solver with those measured by Vugts [1968] as given in [Newman 1977] and Euler solver [Kakar 2002] Force history for a heaving rectangular hull over one time period and for Pressure contours at different time steps for a 2D rectangular hull undergoing heave motion Wave profiles at different time steps for a 2D rectangular hull undergoing heave motion Bilge and keel geometry details Boundary conditions applied for a body undergoing forced harmonic roll motion at the free surface Figure explaining how to evaluate roll added mass and damping coefficients from the moment history plot itself Moment history of a hull without bilge keels undergoing harmonic roll motions for = Pressure contour plots at various time instants for a hull without bilge keels undergoing roll motion Comparison of roll added mass coefficients from the present solver with those obtained from the BEM solver, the Euler solver [Kakar 2002] and Vugts [1968] for a hull without bilge keels Comparison of roll damping coefficients from the present solver with those obtained from the BEM solver, the Euler solver [Kakar 2002] and Vugts [1968] for a hull without bilge keels Wave profiles at various time instants for a hull without bilge keels undergoing roll motion Moment history of a hull with 4 bilge keels undergoing harmonic roll motions for = Comparison of roll added mass coefficients from the present solver with those obtained from the BEM solver, the Euler solver [Kakar 2002] and Yeung et al. [2000] for a hull with 4 bilge keels Comparison of roll damping coefficients from the present solver with those obtained from the BEM solver, the Euler solver [Kakar 2002] and Yeung et al. [2000] for a hull with 4 bilge keels xv

17 5.25 Wave profiles at various time instants for a hull with 4 bilge keels undergoing roll motion; The vertical axis represents the wave elevation, scaled by the beam length, Pressure on the hull with 4 bilge keels for = 0.8 at = 0.8 (The discrepancies between the pressures from the current viscous solver and other solvers shown in the figure led to investigation and changes in the formulation of the solver, which are presented in the succeeding sections of the chapter.) A close-up view of the grid near the semi-circular hull Plot of pressure on the semi-circular hull curve length at various time instants Description of the main length parameters for a submerged hull undergoing roll motions a typical grid used for forced harmonic motions of a submerged hull Description of boundary conditions applied for the submerged roll problem Pressure on the submerged hull without bilge keels at for a non-moving grid Pressure on the submerged hull without bilge keels at for a non-moving grid Pressure contours around the submerged hull without bilge keels at for a non-moving grid Pressure contours around the submerged hull without bilge keels at for a non-moving grid Figure explaining the terms used in transformation of the unsteady term in the Navier-Stokes equations for a moving grid in a fixed inertial coordinate system Grid orientation for a submerged hull without bilge keels at and Pressure evaluated on the submerged hull without bilge keels at using a fixed coordinate system and a moving grid in the case of viscous solver Pressure evaluated on the submerged hull without bilge keels at using a fixed coordinate system and a moving grid in the case of viscous solver Pressure contours around the submerged hull without bilge keels at xvi

18 5.41 Pressure contours around the submerged hull without bilge keels at Comparison between hydrodynamic moment obtained from viscous and potential solvers for a submerged hull without bilge keels undergoing roll motion Flow field with respect to the submerged moving hull without bilge keels, shown around the hull at time instant Flow field with respect to the submerged moving hull without bilge keels, shown around the hull at time instant Flow field with respect to the submerged moving hull without bilge keels, shown around the hull at time instant Flow field with respect to the submerged moving hull without bilge keels, shown around the hull at time instant Comparison of the grid densities around the submerged hull without bilge keels used in the convergence study Comparison of the pressure on the submerged hull without bilge keels for increasing number of cells at Comparison of the pressure on the submerged hull without bilge keels for increasing number of cells at Comparison of the pressure on the submerged hull without bilge keels for increasing number of cells at Comparison of the hydrodynamic moment on the submerged hull without bilge keels between three different grids for the first time period a typical grid used for computation of forced harmonic motions of a submerged hull with bilge keels Comparison of the pressure on the submerged hull with bilge keels undergoing roll motion for varying Reynolds number at Comparison of the pressure on the submerged hull with bilge keels undergoing roll motion for varying Reynolds number at Comparison of the pressure on the submerged hull with bilge keels undergoing roll motion for varying Reynolds number at Comparison of the pressure on the submerged hull with bilge keels undergoing roll motion between viscous and potential solvers at Comparison of the pressure on the submerged hull with bilge keels undergoing roll motion between viscous and potential solvers at xvii

19 5.58 Comparison of the hydrodynamic moment on the submerged hull with bilge keels undergoing roll motion between viscous and potential solvers for the first time period xviii

20 Nomenclature Latin Symbols wave amplitude added-mass coefficient damping coefficient beam of the ship wave celerity, F &% water depth propeller diameter,, or (in deep water) draft of the ship, body force per unit mass column matrix for the derivative terms Froude number based on propeller diameter, "!$# Froude number based on beam, '% % )( Froude number based on draft, *( non-dimensionalized total - and + -direction force, gravitational acceleration G column matrix for the + derivative terms xix

21 + + wave number, Keulegan-Carpenter number non-dimensional bilge-keel depth reference length used in non-dimensionalization, or wavelength, or length of 2-D channel deep-water wavelength moment about the axis, normal vector pressure atmospheric pressure, body velocity Q U column matrix containing the source terms residual of the continuity equation Reynolds number based on reference length, area of cell in two-dimensional formulation non-dimensional time time period of motion column matrix for time derivative terms flow velocity at infinity amplitude of oscillating velocity function amplitude of heave velocity + and -direction velocities, total velocity vector!#" ship speed % $ computational cell volume coordinate system, location vector on the ship fixed downstream, upward and port side coordinates respectively xx

22 + ( ( ( Greek Symbols angle of roll for FPSO hull, amplitude of roll motion pressure difference time step size cell size in and + direction vertical coordinate of free surface dynamic viscosity of water kinematic viscosity of water ( frequency of periodic heave and roll forcing function ( (, vorticity vector perturbation potential ( fluid density phase of the wave, Subscripts node numbers cell indices node or cell indices in each direction; is axial, is radial, and is circumferential. face (in two-dimensions) indices at north, west, south and east of a cell xxi

23 Superscripts intermediate velocity or pressure velocity or pressure correction time step indices Acronyms BEM CFD CPU FPSO FVM MIT RANS SIMPLE Boundary Element Method Computational Fluid Dynamics Central Processing Unit (time) Floating, Production, Storage and Offloading (vessels) Finite Volume Method Massachusetts Institute of Technology Reynolds Averaged Navier-Stokes (equations) Semi-Implicit Method for Pressure Linked Equations Computer Program Names FLUENT WAMIT commercial CFD software panel method based wave-structure interaction analyzer xxii

24 Chapter 1 Introduction 1.1 Background Oil in various forms has become an indispensable commodity in present day lives. The oil basins or reserves in shallow waters have long been dried up or reduced to non-profitable resources. The ever increasing demand for oil has forced explorers to look towards deeper seas for newer and better opportunities. Deeper seas were explored as early as 1950s. But, exploring, drilling and production in deeper seas are not without their share of problems. It is an arduous and expensive task to set up drilling and production units in deep seas mainly due to the depth of the sea bed and the prevailing harsh environmental conditions. The option of fixed structures is hence limited in deep waters. The best and obvious alternative for this purpose is the use of floating structures. Floating structures are used in all fields of marine technology, particularly in exploration work. Typical floating structures used for offshore operations are semi-submersibles and drill ships. Semi-submersibles are mainly floating drilling platforms which use pontoons and columns flooded with sea water to stay afloat. Floating, Production, Storage and Offloading (FPSO) vessels and Floating, Storage and Offloading (FSO) vessels are the common drilling vessels. FPSO vessels are nowadays extensively being used 1

25 for oil extraction. Both FPSOs and Semi-submersibles are usually anchored with mooring lines which in some cases can be assisted by dynamic positioning thrusters. A typical FPSO operating in mid-seas is shown in figure 1.1. Figure 1.1: Terra Nova FPSO (source: IcebergNet/gallery.htm) 1.2 Motivation Floating structures, together with moorings, risers and other equipment can be considered as a single system. Such systems usually have a low stiffness and hence have a low natural frequency. This in turn can cause the ship to move in all six degrees of freedom when the structure is subject to three-dimensional loads due to various factors like random waves, currents and winds. The translatory motions that occur along the three axes and the rotational motions that occur about the axes form the six degrees of motion. Translatory motions along the X-, Y- and Z-axes are called surge, sway and heave respectively. The three rotational motions in the same order 2

26 are roll, pitch and yaw respectively. The six degrees of freedom are depicted in figure 1.2. One can represent the energy carried by the waves as an area of spectral density in the frequency domain. From the wave energy spectrum one can observe that the energy intensive range of the spectrum is concentrated in the low frequencies. Hence, resonance phenomenon can be very problematical for floating structures as their natural frequency is low. Of the possible motions, pitch, roll and heave are significant and need to be studied carefully. The prime origin of roll motions of FP- SOs is the non-collinearity of wind, current, wind driven seas and swell. In storm conditions the wind driven seas are normally collinear with the wind and dominant over currents. Therefore, in extreme conditions FPSOs usually encounter seas and wind head on or at a small angle. It should be noted that the wind driven seas exhibit a directional spreading and that the FPSOs oscillate around the mean heading at the same time. Both the phenomena contribute to transverse wave loading, sway, yaw and roll motions. Swells originating from remote storms may arrive from the beam direction. In cases where the currents govern the heading of the vessel, vessels have to cope with the onslaught of beam seas. Other sources of transverse excitation are the variations in wind direction and current direction. During wind shifts or change of wind direction or tidal change of current, the FPSO may turn and experience bow quartering or beam waves for a certain period. The main focus of the present work is roll motion because the possibility of extremely large motions and even capsizing make roll one of the most critical aspects of ship motions and sea keeping. The FPSOs that are operating in deep waters need to be stationary for long periods of time in order to facilitate smooth drilling operations. Hence, the mitigation of roll motion acquires significant importance and needs to be studied extensively. The roll motion plays an important role in determining the loads on deck cargo of an offshore 3

27 Figure 1.2: Description of motion under six degrees of freedom for a ship vessel. The range of operability of the ship also can be predicted accurately if the roll motion near resonance can be estimated correctly. Field observations indicate that FPSOs roll more than expected based on their design and model tests. This can lead to riser fatigue, loads on mooring system, turntable and turret and operational difficulties (degraded process performance, operational limits on material transfer, helicopter operations, crew comfort and safety and effectiveness). FPSOs consist of both ballast tanks and cargo tanks which have a free surface at all times. Cargo tanks are basically used to store oil till the oil shuttle tankers make their trip. Sloshing in these tanks could pose a major problem due to roll motions at resonance. Hence, it becomes imperative that roll motions be avoided as much as possible for the FPSOs. Many types of devices and methods have been designed by sailors and naval architects to reduce the roll motions. The means to control the roll motion can be divided into the following groups: 4

28 Hull design (main dimensions, distribution of displacement, cross section shape); Passive devices such as bilge keels, skegs and fins; Active systems based on moving weights and stabilizer fins; Active and passive anti-rolling tanks; Rudder-roll control and heading control; In hull design, distribution of displacement mainly involves distributing the weight on to either side of the vessel away from the centreline. Making the topside of the vessel lighter also helps in increased stability by lowering the center of gravity, minimizing the moment of inertia and thus reducing the roll moments. But this approach might not be feasible for an FPSO which houses a drilling unit on its deck. According to [Kasten 2002], active stabilizers can cause up to 90 roll reduction but they are most effective only when the ship is moving at its maximum speed. Stabilizer fins and rudder-roll control are based on lift force generation with forward speed and therefore not applicable for stationary vessels such as FPSOs. They are also relatively expensive and complex to install. Using anti-roll tanks, roll reductions in both amplitude and acceleration to the order of 50 to 60 have been possible. Vessel speed is not an issue in this case. The main disadvantage is the added displacement required to carry the extra deadweight of the tank contents. Space provision for antiroll tanks can lead to compromises in spaces for interior and storage. Another major disadvantage is the possible effects on stability of the vessel due to the large free surface effect in the tank. Orienting the ship into the wave direction using thrusters 5

29 is one desirable way of reducing ship motions, but, in rough weather the waves are random both in direction and individual characteristics. Bilge keels are appendages that form an obstruction to roll motion. A bilge keel generally runs over the midship portion of the hull, perpendicular to the turn of the bilge. According to [Kasten 2002], for sailing vessels, long, low aspect ratio bilge keels offer roll reduction of the order of 35 to 55 and their efficacy is independent of the vessel speed. There is some added frictional resistance due to increased wetted surface area. Bilge keels are relatively inexpensive and also simple to build and are hence widely preferred. They also have the advantage of having no moving parts and require no more maintenance than that devoted to the hull surface. Properly designed bilge keels create minimal drag and increase roll period while reducing roll amplitude. The effectiveness of using bilge keels in FPSO hull roll mitigations needs to be studied and hence, is the main focus of this thesis. Accuracy in the prediction of ship motions under extreme conditions and the resulting hydrodynamic loads is of great importance to the ship design process and is a challenging task. Accurate predictions of the motion are also necessary for the development of control methods. Early predictions of the ship motions were based on scale model tests in given wave conditions in wave basins. Though these tests yield fairly good results, it is cumbersome and expensive to model these tests. These model tests are still being used, but, are limited by the time and efforts taken in conducting them. The scale effects also pose a substantial problem in the experiments. It is usually difficult for researchers to get a good correlation between Reynolds number and Froude number for models of practical hull forms. In contrast, theoretical and numerical methods offer greater ease of use and are relatively 6

30 very inexpensive. A number of techniques were developed for the estimation of the roll damping moment. Among these are empirical and semi-empirical formulae that are derived from experimental data. Two types of experiments are generally used for the estimation of roll damping; free decay and forced roll tests. Development of numerical methods made ship motion prediction easier and faster. Continuous advancements in technology has only helped increase the usage and effectiveness of these numerical methods. Most of the numerical methods till now have been potential based methods due to the simplicity in modeling the problem and cost effectiveness in terms of computer time and storage. The lack of proper computational methods for prediction of ship motions also arose mainly due to the complexity of the problem and limited knowledge of the actual governing physics. The main cause of roll damping in the case of a hull, fitted with bilge keels, is the vortex created due to the separated flow past the bilge keels in addition to the outwardly radiating free surface waves. Accurately modeling the complex flow around the bilge keels in the presence of the hull geometry and a free surface acquires great importance in the process of determining the effectiveness of the bilge keels in roll attenuation. There are existing potential solvers such as WAMIT (Waves at MIT) which are used to determine the hydrodynamic coefficients; added mass and damping coefficients for ships undergoing motions under six degrees of freedom. Potential solvers have been proved fairly accurate in predictions of heave and sway motions. But, potential solvers fail to predict the roll motion accurately because the flow around the hull can neither be assumed to be inviscid nor irrotational in the roll case. Viscous effects dominate and separation of flow past sharp edges play a major role when a ship WAMIT is a registered trademark of WAMIT, Inc. ( 7

31 is undergoing roll motion, but, potential solvers can solve only for attached flows. Hence, there arises a need for solvers which can take into account viscous and separation effects and predict the flow accurately. A Reynolds Averaged Navier-Stokes (RANS) solver provides a good alternative to potential solvers. The whole motion problem can be split up into the sum of harmonic oscillations of the ship in still water and waves coming in on the restrained ship and the two fields can be investigated entirely separately [Vugts, 1968]. In this thesis, the aim is to deal with the harmonic oscillations of the ship hull. For all practical purposes the problem can be assumed to be two-dimensional and solved for a general rectangular cross-section of a hull. Strip theory then can be used to integrate the solutions of all the cross-sections along the ship s length and an approximate 3D solution can be obtained. 1.3 Objective The objective of the work presented in this thesis is to develop and validate a twodimensional Navier-Stokes solver to solve the problem of radiation due to the roll motions of a 2D FPSO hull at the free surface. Hulls with and without bilge keels are considered. The free surface effects and the viscous effects are decoupled and studied separately to get a better understanding of both phenomena. The present method aims at capturing the separated flow and the radiated wave profile and at predicting the roll hydrodynamic coefficients. The ultimate objective of the present work is to develop a 3D solver which can simulate roll motions of 3D ship hulls. In the present work, investigations have shown that the solution is not affected much by a change in the Reynolds number and hence, the present solver is based on laminar flow alone. 8

32 1.4 Overview Chapter 2 presents a literature review of the past work done on the prediction of hull motions using various schemes. Also, a brief review of vortex tracking methods is given. Chapter 3 discusses the detailed formulation of the boundary element method. The chapter then presents results for various applications of the method. Comparisons with existing solutions and convergence studies are provided. It also provides a brief outline of the vortex tracking method. Chapter 4 describes the numerical formulation of 2D unsteady Navier-Stokes solver. The Crank-Nicolson scheme and SIMPLE pressure correction method are explained. Chapter 5 presents the application of the 2D finite volume method to the problem of a floating hull undergoing forced harmonic motions. It presents the comparisons between the present solver results and results from theory or experiments. It also presents the formulation and results for the problem of submerged body undergoing forced harmonic motions. Chapter 6 includes conclusions on the present work done and recommendations for the work to be done in the future. A copy of this thesis may also be downloaded from the following website: 9

33 Chapter 2 Literature Review This chapter reviews literature related to different methods and approaches to study the roll motion of hull forms. The first section discusses previous literature which deals with the problem of 2D and 3D hull forms undergoing roll motions in the presence as well as the absence of a free surface. The second section discusses literature that deals with vortex tracking methods based on potential theory. 2.1 Hull Motion Prediction Prediction of roll motion is very important in ship dynamics and has interested researchers for long. Most of the currently available techniques for the analysis of ship motions and sea loads are based on potential flow assumptions. These potential solvers have proven adequate in the analysis of sway, pitch and heave motions. But, these solvers fail to predict roll motion accurately due to their fundamental assumption of irrotationality and absence of viscous effects. Vugts [1968] was probably one of the first to do a comprehensive study of ship motions and observe the importance of viscous effects in the case of rolling bodies. Yeung et al. [1996] states that viscous effects are known to have significant influence on hydrodynamic forces on bluff- 10

34 shape bodies. Ocean structures in long waves and roll damping arising from bilges of a ship hull are important example. In the potential methods viscous effects may be accounted for by empirical, semi-empirical formulations (Tanaka [1960], Ikeda et al. [1977], Himeno [1981]). The empirical and semi-empirical formulations depend mostly on various model tests and are hence used only on a trial and error basis. They are also incapable of dealing with motions of bodies with complex geometries. Another component of damping is the free surface waves which are well predicted by potential theory. There have been efforts by Fink and Soh [1974], Brown and Patel [1985], Braathen and Faltinsen [1988a], Cozens [1987] and Downie et al. [1974] to predict viscous damping without relying on empiricism. But, none of them were able to model accurately the interaction of hull geometry, vorticity generation and free surface simultaneously until Yeung and Vaidhyanathan [1994] who developed a Free-Surface Random Vortex Method. On the other hand, a RANS equations based technique, naturally incorporates the effect of viscosity and hence, produces better results in cases where viscosity plays an important role [Sarkar and Vassalos 2000]. They can easily be extended to 3D practical ship forms and the creation of vorticity in the boundary layer and vortex shedding during separation can be readily tackled. Among the available techniques to predict vessel motions, the strip theory based Seakeeper, or the panel method diffraction codes such as WAMIT (Waves at MIT) assume inviscid flow and operate in the frequency domain. Klaka [2001] observes that viscous forces are important and the non-linear nature of roll response requires time domain modeling. According to Gentaz et al. [1997], viscous effects are important for rectangular bodies in sway or roll motion. Therefore numerical developed by Formation Design Systems Pvt. Ltd 11

35 simulations based on inviscid flow theory cannot give satisfactory results. It has been shown in Yeung and Ananthakrishnan [1992] that for strongly separated flow, the shear stress is of secondary importance. This is illustrated in Kakar [2002] and Kinnas et al. [2003], where the flow past a flat plate is determined using both an Euler and a Navier-Stokes solver. The values of the drag and inertia coefficients from both the solvers compare very well with each other as well as with experiments (experimental data presented in Sarpkaya and O Keefe [1995]). Some of the past work done on the subject of roll motions includes an investigation into the eddy-making damping in slow-drift motions performed by Faltinsen and Sortland [1987]. The authors showed the importance of bilge-keel depth, especially for low Keulegan-Carpenter numbers. [Sarpkaya, 1995] presented experimental results for two- and three-dimensional bilge keels subject to an oscillating flow. The authors conclude that bilge keel damping is affected by the vortex shedding from the edge of the bilge keel and the use of damping coefficients from flat plates in a free stream are not necessarily accurate for wall bounded bilge keels. Korpus and Falzarano [1997] were the one of the first researchers to use a RANS solver to tackle the problem of ship roll motion. Their work aimed at studying the viscous and vortical flows around the hull corners and appendages in the absence of a free surface. They performed a series of parametric studies in order to identify the individual contributions of viscosity, vorticity, and pressure. Yeung et al. [1998] applied the Free-Surface Random Vortex Method (FSRVM) to a rectangular ship-like section oscillating in roll motion and compared the hydrodynamic coefficients obtained from the method with those obtained from their experiments. Their study shows that the added mass coefficients are not affected by 12

36 further increase in the amplitude of roll beyond five degrees. A composite model representing the effect of flow separation on the hydrodynamic moment is also developed. The moment is expressed as the sum of the added mass inertia, a linear damping associated with surface wave generation and a quadratic damping associated with vortex generation. In Yeung et al. [2000], the authors extended the work to include modeling of the complex flow around the bilge keels. In the FSRVM, the flow-field is solved by decomposing it into irrotational and vortical parts. The irrotational part is solved using a complex-valued boundary-integral method, utilizing Cauchy s integral theorem for a region bounded by the body, the free surface and the open boundary. The rotational part is solved by solving the vorticity equation using the fractional step method. Results obtained using the solver are compared to experimental data as well as results obtained by Alessandrini and Delhommeau [1995] for various bilge keel depths and forcing function amplitudes. The increase in size of the keels increased the added inertia and the damping coefficients. Miller et al. [2002] was one of the first to use three-dimensional RANS calculations to simulate roll motions of a circular cylinder with bilge keels. The numerical results are compared with experiments performed at the Circulating Water Channel at the Naval Surface Warfare Center, Carderock Division. The results compared well for immersed body computations but emerged body results need to be improved further. These calculations demonstrate that RANS can play an important role in variety of hull motions in the near future. At the same time Wilson and Stern [2002] presented results for unsteady simulation of a surface combatant under roll motion. Though the authors did not have experimental data to validate their results, their efforts showed the efficacy of a RANS solver in naval architecture applications. Other works in this area include Sturova and Motygin [2002], where the authors solve, using a multi pole 13

37 expansion method, a system of boundary integral equations describing the linear two-dimensional water-wave problem, for a horizontal cylinder undergoing small oscillations at the interface of two layers of different densities. Most recently, Felli et al. [2004] conducted free decay roll experiments on a DDG551 ship model with forward sped at the INSEAN facility in order to study the 3D flow field around the hull. The flow field is resolved in phase with the roll motion using Laser Doppler Velocimetry (LDV). The study is performed for a bare hull as well as a fully appended hull (rudder, brackets and bilge keels). Bilge keels are found to be the major contributors towards roll damping. The authors observe that LDV results could be improved significantly by using Particle Image Velocimetry (PIV). Bishop et al. [2004] conducted experiments at the Naval Surface Warfare Center, Carderock Division to explore the viscous flow field in the region of the bilge keels while the ship is undergoing roll motions.the model used in the experiments is DTMB model Irvine et al. [2004] also conducted towing tank experiments for an advancing surface combatant (DTMB model 5512) in free roll decay. For free roll decay experiments, results are presented for all motions under all the six degrees of freedom. All the studies conclude that with increasing forward speed, the roll damping increases. This is attributed to the lift effect caused by the bilge keels. These studies could be useful for validation tests when the present solver is made capable of handling 3D flows. 2.2 Vortex Tracking Method This section does a review of some of the past work done in the field of prediction and tracking of vortices that are shed from edges using inviscid flow theory. Rott 14

38 [1956] was one of the first among to consider the effects of viscous separation and include it into calculations of fluid flow past sharp edges. In the problem of diffraction of shock waves he modeled the separation of flow by replacing the vortex region by a single concentrated vortex. Assuming that the flow was irrotational, he argued that neglecting viscosity would cause only a small deviation from real flow pattern and solved the problem using dimensional analysis. Researchers later tried to study problems involving unsteady motion of 2D vortex sheets past wedges. According to Pullin [1978], the appropriate similarity law for the wedge starting flow appears to have been originally discovered by Prandtl. Fink and Soh [1974] later made an attempt to model impulsive flat-plate (zero wedge angle) flow by a finite number of point vortices whose initial strengths and positions represent a discretized model of the disturbed sheet circulation. Pullin [1978] applied a model consisting of a vortex sheet, a cut and an isolated vortex developed by Smith [1968] to the impulsive starting flow past an infinite wedge. In Pullin [1978] a similarity solution is used to transform the time-dependent problem for the sheet motion into an integro-differential equation and finite difference solutions to the same are obtained. Two-dimensional methods based on a discrete vortex approach were used by Clements and Maull [1975] and Bearman and Graham [1980] to model vortex sheets. These methods were later applied to the problem of prediction of ship roll damping by Standing et al. [1988] based on the method developed by Bearman et al. [1982] and Cozens [1987]. Later this method was extended to three-dimensions and applied to ship roll damping problem by Downie et al. [1991]. Graham and Cozens [1988] adapted the Cloud-in-cell method (Christiansen [1973]) to model the vortex sheet 15

39 which is shed and rolls up from a single sharp edge. The method is a mesh method in which a discrete moving point vortex representation of the vorticity field is transferred to a fixed mesh. Numerical approximation to the velocity field is carried out on the mesh and transferred back to the moving points as a convection velocity. Another approach used in modeling vortex sheets was developed by Faltinsen and Pettersen [1987]. It was based on distributing sources and dipoles over boundaries and free shear layers. It was applied for oscillating flow over bodies with either curved surfaces or sharp edges. It was later extended to include the free surface effects and applied to a 2D floating body with sharp as well as round corners undergoing forced harmonic roll motion by Braathen and Faltinsen [1988b]. 16

40 Chapter 3 2D Boundary Element Method and Its Applications In this chapter, the first section presents the Boundary Element Method (BEM) or the Panel Method and its detailed formulation in two dimensions. Next, its application to a few standard problems are presented for validation purposes. A problem of a submerged hull undergoing forced harmonic roll motions is solved. The method is extended to include trailing vortex prediction for flow past a bilge keel (wedge). The ultimate objective of the present work with 2D BEM is to be able to predict the vortex shedding past a bilge keel for a 2D hull section thats rolling at a free surface. The motivation for using BEM is that it requires less computational time and storage to solve a problem when compared to viscous solvers. 3.1 Background Boundary Element Method is based on integral equations. Boundary value problems can be represented mathematically in terms of integral equations by transforming the governing partial differential equations into integral equations relating only boundary values. The integral representation of a problem relates the main variables (velocity potential in fluid flows, temperature in heat transfer problems, etc) with 17

41 functions of their derivatives (velocities and heat flux respectively). The advantages of using BEM are: 1) Only boundaries need to be discretized, hence, minimal computational storage and time are used 2) Problems involving infinite or semi-infinite domains can be easily solved since the boundaries at infinity need not be created 3) Problems involving some kind of singularity or discontinuity can be dealt with effectively 4) One need not perform any discretization in the plane of symmetry in case of problems involving symmetry 3.2 Numerical Formulation This section presents the numerical formulation and implementation of 2D Boundary Element Method Green s Theorem Consider a volume surrounded by a surface as shown in Figure 3.1. Suppose and are two functions that satisfy the Laplace equation inside, i.e, and inside the volume, then, according to Green s second identity the following equation holds: The formulation and numerical implementation of the method is based on the course work offered by Dr. Spyros Kinnas in CE 380 P.4, Boundary Element Methods,

42 n ν S Figure 3.1: Volume confined by a surface (3.1) where, is the unit vector normal to the surface pointing out of the domain as shown in the Figure Application of Green s Formula for a two-dimensional body Consider a body surrounded by a surface % in the two-dimensional space, as shown in Figure 3.2. Consider a unit source at a point outside. The potential associated with the unit source is given by:, where, is the position vector of the point P. Assume a potential (3.2) which satisfies the Laplace equation outside B. 19

43 (3.3) Consider a circle of radius surrounding the point P and a surface surrounding the body and the source. Applying the Green s theorem inside the volume surrounded by %, and, and considering the limits 0 and 0, we obtain the following equation:,, (3.4) The above equation shows that value of the potential at any point depends only on the values of and on the body boundary. It can also be seen that the potential! can be expressed as a superposition of the potentials due to distributions of sources and normal dipoles. The integral equals for a point outside the body, on the body and 0 inside the body. Following the same approach for a function that is harmonic inside the body, we obtain an integral that equals 0 outside the body, on the body and inside the body. Adding the two integrals provides an integral equation for the value of on a general 2D body which forms the governing equation of 2D Boundary Element Method. The governing equations is as follows:, Consider a body subject to an inflow of a velocity equal to, (3.5)!. If! is the velocity potential of the inflow and is the total potential of the resultant flow, then, 20