Effect of Gravity on the Vertical Force of an Oscillating Wedge at Free Surface

Transcription

1 Effect of Gravity on the Vertical Force of an Oscillating Wedge at Free Surface Master of Science Thesis For the degree of Master of Science in Offshore and Dredging Engineering at Delft University of Technology Wen Hu 25 th November, 2015 Committee Prof. Dr. Ir. R.H.M. Huijsmans TU Delft Dr. Ir. Ido Akkerman TU Delft Dr. Ir. S.A. Miedema TU Delft Ir. G.K. Kapsenberg MARIN

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3 Abstract As for a ship oscillating at free surface with frequency that ranges from those that are common for ship motion in waves to very high frequency associating with the impact problem, the vertical force acting on the ship for the lower limit of frequency range is dominated by gravity effect, which is usually ignored in the impact theory. As a consequence there must be a transition range in between those gravity-dominant and gravity-negligible conditions. Indeed these two limiting conditions have been studied deeply by many researchers for decades, but the transition range has not gain much attention. This thesis concerns the determination of the limit for each of these extreme conditions and the quantification of the gravity effect in this transition frequency range. The hydrodynamic load caused by ship motion in waves normally is estimated by linear potential BEM in which the free surface boundary condition is linearized on the undisturbed water level, while the assessment of impact induced load mainly involves momentum method and nonlinear potential BEM. To be able to deal with nonlinearity that the linear potential theory is lack of and gravity effect that the impact theory ignores, it is necessary to apply a CFD method to study the fluid domain. In this thesis, the CFD method is based on the Navier-Stokes equations for an incompressible, viscous fluid with a numerical discretization of the finite volume method. To simplify the simulation, wedge-shaped bodies with deadrise angle of 15º, 30ºand 45ºare used to represent the cross section of a ship. The thesis starts with defining the limit of gravity-dominant condition, in which both linear potential BEM and CFD method are applied. By comparing the hydrodynamic coefficients of added mass and damping, the application limit of linear potential theory is found as well and an estimation method is developed for determining the limits. In the slamming-frequency oscillations, in addition to the CFD simulations, momentum method and 2D-BEM proposed by Zhao, Faltinsen and Aarsnes are also used to calculate the impact loads. The vertical forces are compared with each other to verify the accuracy. By concerning the relative gravity force to the total, the limits of gravity-negligible condition are obtained. Finally, after determining the limits of those two extreme conditions, the quantification of the gravity effect is performed by applying the simulations with and without gravity acceleration in CFD program.

4 Acknowledgement Firstly, I would like to thank all the committee members at TU Delft. Especially, I would like to express my sincere gratitude to Prof. Rene Huijsmans for creating this precious opportunity for me to work on the subject I am interested in, and for his suggestion of always open the mind. A special gratitude goes to my supervisor at Marin, Geert Kapsenberg, who has always been supportive, guides me through my thesis and infects me with his gentleness and inspiration. Finally, I would like to thank my family: my parents and my sister. To my husband, Dai Li: thanks for supporting me financially and emotionally during my master study and for being in my life. Delft, 25 th November 2015 Wen Hu

5 Nomenclature Abbreviations BEM CFD CIP FEM FVM FPSO GABC SPH VOF WF Boundary Element Method Computational Fluid Dynamics Constrained Interpolation Profile Finite Element Method Finite Volume Method Floating Production Storage and Offloading vessel Generating and Absorbing Boundary Condition Smoothed Particle Hydrodynamics Volume of Fluid Wetting Factor List of Symbols a a l a t b b cr B B t cflmin cflmax, d dt max Added mass Froude length scaling factor Froude time scaling factor Damping Critical damping Buoyancy to the undisturbed water level Instantaneous buoyancy to the undisturbed water level Minimum CFL-number Maximum CFL-number Wedge immersion depth Maximum time step

6 vi Nomenclature Fz F Hys F Hyd h x h y h z k p T u v w W W l Y z z z z a β ω λ Φ ρ Vertical force Hydrostatic force Hydrodynamic force Mesh size in x direction Mesh size in x direction Mesh size in x direction Wave number Pressure Oscillation period Velocity component in x direction Velocity component in x direction Velocity component in x direction Width of wedge Width of wedge at water level Half width of wedge Heave motion Heave velocity Heave acceleration Oscillation amplitude Deadrise angle of wedge Oscillation frequency Wave length Velocity potential Fluid density µ Dynamic viscosity coefficient

7 Table of Contents ABSTRACT... III ACKNOWLEDGEMENT... IV NOMENCLATURE... V TABLE OF CONTENTS... VII LIST OF FIGURES... X LIST OF TABLES... XIII CHAPTER 1 INTRODUCTION Motivation and Objectives Research Background Oscillation Problem Slamming Problem Outlines... 4 CHAPTER 2 APPROACH AND THEORIES Approach CFD Governing Equations Boundary Conditions Calculation of forces Numerical Solution Radiation BEM Mathematical Model Numerical Solution Water Entry BEM Hydrodynamic Coefficients Potential Theory Impact Theory CHAPTER 3 MODEL AND CONVERGENCE STUDY Model Geometry and Coordinate Frequency Selection Mesh Size... 17

8 viii Table of Contents Frequency 6 rad/s Frequency 20 rad/s Frequency 100 rad/s Time step Boundary Condition on the Wedge Surface Non-reflecting Boundary Condition CHAPTER 4 WAVE-FREQUENCY OSCILLATION Vertical Force Amplitude Time Series Hydrodynamic Coefficients Fitting Calculation Added Mass Damping Discussion CHAPTER 5 SLAMMING-FREQUENCY OSCILLATION D BEM Water Entry with Constant Velocity Oscillation Comparison between ComFLOW, 2DBEM and Momentum Method Pressure Distribution Gravity Effect CHAPTER 6 OSCILLATION WITH FREQUENCY IN TRANSITION Vertical Force Pressure Free Surface and Velocity Field Gravity Effect Analysis Gravity Force Effect of Gravity Acceleration Ratio of Gravity Force to Total Force CHAPTER 7 STUDY ON DEADRISE ANGLE Introduction Application Limits of Linear Potential Theory Force Amplitude Hydrodynamic Coefficients Limits of Gravity-Negligible Condition Gravity Effect in between Limits... 72

9 Table of Contents ix 7.5 Deadrise Angle Effect Effect on Force Amplitude Effect on Application Limits Summary CHAPTER 8 CONCLUSION AND RECOMMENDATION Conclusion Recommendation REFERENCES... 78

10 List of Figures Figure 1-1 FPSO in ocean waves (left) and FPSO model in steep fronted waves by MARIN (right)... 1 Figure 2-1 Study approach scheme... 6 Figure 2-2 CFD simulation domain with boundary conditions... 8 Figure 2-3 Water entry coordination and free surface in approximation BEM [14] Figure 2-4 Scheme of added mass when wedge is moving into water Figure 2-5 Non-dimensional added mass of different estimation methods Figure 3-1 Wedge geometry Figure 3-2 Time series of vertical force for different grid sizes for ω = 6 rad/s Figure 3-3 Snapshots of velocity field in 6 rad/s for t=t and 1.5T for grid condition No Figure 3-4 Grids for condition No.1 and No.3 in Table Figure 3-5 Time series of vertical force for different grid sizes for ω = 20 rad/s Figure 3-6 Snapshots of velocity field in 20 rad/s for t=1/2t and 3/4T Figure 3-7 Time series of vertical force for different grid sizes for ω = 100 rad/s Figure 3-8 Time series of vertical force for different CFL ranges Figure 3-9 Vertical force for free slip and no slip boundary conditions at wedge surface Figure 3-10 Vertical force for different outflow boundary conditions right wall Figure 4-1 Maximum, minimum force for oscillating amplitude Za/D of 14%, 10%, 8% and 6% Figure 4-2 Maximum, minimum force for different frequencies Figure 4-3 Vertical force with non-dimensional time at different frequencies Figure 4-4 Time series of vertical force of ComFLOW, fitting curves, SHIPMO and Buoyancy to undisturbed water level for oscillation amplitude Za/D 6% and 10% Figure 4-5 Added mass coefficients at different frequencies with amplitude Za/D of 14 %, 10%, 6% and 4% Figure 4-6 Damping coefficients at different frequencies with amplitude Za/D of 14 %, 10%, 6% and 4%... 34

11 List of Figures xi Figure 4-7 Illustration of immersion depths Figure 4-8 Illustration of pile-up and jet generation Figure 4-9 Horizontal velocity vs wave speed at Za/D of 14%, 10%, 6% and 4% Figure 4-10 Free surface elevation for frequencies 10 rad/s, 16 rad/s, 20 rad/s and 22 rad/s for oscillating amplitude Za/D of 14% Figure 5-1 Elements on body and free surface for different times for condition No Figure 5-2 Time series of vertical force for different element sizes for ω = 100 rad/s Figure 5-3 Time series of vertical force for water entry wedge with different entering velocity Figure 5-4 Snaptshots of ComFLOW for different entering velocities Figure 5-5 Non-dimensional added mass of different estimation methods and SHIPMO Figure 5-6 Time series of vertical force for oscillation amplitude Za/D 2% Figure 5-7 Time series of vertical force for oscillation amplitude Za/D 4% Figure 5-8 Time series of vertical force for oscillation amplitude Za/D 6% Figure 5-9 Time series of vertical force for oscillation amplitude Za/D 8% Figure 5-10 Time series of vertical force for oscillation amplitude Za/D 10% Figure 5-11 Description of force amplitude of momentum method Fa.m and of ComFLOW Fa.c Figure 5-12 Ratios of Fa. mfa. c against oscillating amplitude Za/D Figure 5-13 Snapshots of pressure field for cω=3.1 and 6.3 at t=0t, ¼T, ½T and ¾T Figure 5-14 Description of gravity force Fg and force amplitude Fa Figure 5-15 Ratios of gravity force to force amplitude Fg/Fa for Za/D 2%, 4%, 6%, 8% and 10% Figure 5-16 Frequency coefficient limits of gravity negligible condition Figure 6-1 Force ratio Fz/B history for difference gravity contributions (with 0, ¼, ½, ¾ and 1 gravity acceleration) for Za/D=10% Figure 6-2 Force ratio Fz/B history for difference gravity contributions (with 0, ¼, ½, ¾ and 1 gravity acceleration) for Za/D=8% Figure 6-3 Force ratio Fz/B history for difference gravity contributions (with 0, ¼, ½, ¾ and 1 gravity acceleration) for Za/D=6%... 56

12 xii List of Figures Figure 6-4 Maximum and minimum forces for Za/D10% 8% and 6% and the fitting curves Figure 6-5 Pressure distribution along wedge surface Figure 6-6 Pressure distribution at moment of minimum vertical force for different frequencies Figure 6-7 Free surface profiles and velocity fields during the first period of frequencies with cω=1.3 and Figure 6-8 Time series of gravity force component at different oscillation amplitudes Figure 6-9 Illustration of gravity force Fg and instantaneous buoyancy Bt Figure 6-10 Ratio of gravity force to the buoyancy to the undisturbed water level Figure 6-11 Ratio of maximum gravity force to the buoyancy to the undisturbed water level.. 64 Figure 6-12 Gravity force against different gravity acceleration Figure 6-13 Percentage of gravity force Figure 7-1 Wedges with deadrise angle of 15ºand 30º Figure 7-2 Maximum, minimum force for deadrise angles of 15ºand 30º Figure 7-3 Added mass and damping for deadrise angles of 15 and 30º Figure 7-4 Fitting calculation for deadrise angle 15ºand 30º Figure 7-5 Horizontal velocity vs wave speed for β of 15º and 30 º at Za/D of 10%, 6% and 2% Figure 7-6 Threshold frequency coefficients of linear theory application conditions for β of 15º, 30 ºand 45º Figure 7-7 Ratios of gravity force to force amplitude for deadrise angle 15ºand 30º Figure 7-8 Frequency coefficient limits of gravity negligible conditions Figure 7-9 Gravity ratios for wedges with deadrise angle 15ºand 30ºof oscillation amplitude Za/D 2%, 6% and 10% Figure 7-10 Maximum and minimum ratios of vertical force to mean buoyancy for wedges with deadrise angle 15º, 30ºand 45º Figure 7-11 Frequency coefficient limit for wedges with deadrise angle 15º, 30ºand 45º Figure 7-12 Summary of low limits and high limits... 75

13 List of Tables Table 3-1 Main particulars Table 3-2 Froude scaling factors Table 3-3 Grid sizes for ω = 6 rad/s Table 3-4 Computational Domain Table 3-5 Grid sizes for ω = 20 rad/s Table 3-6 Computational Domain Table 3-7 Grid sizes for ω = 20 rad/s Table 4-1 Threshold frequency coefficients from method of comparison between SHIPMO and ComFLOW and method of Vh = c Table 5-1 Element and domain definition for ω = 100 rad/s Table 5-2 Force amplitude of ComFLOW and momentum method Table 5-3 Threshold frequency coefficients for gravity negligible conditions Table 7-1 Main dimensions of wedges Table 7-2 Threshold frequency coefficients from method of comparison between SHIPMO and ComFlow and method of Vh = c Table 7-3 Threshold frequencies for gravity negligible conditions... 71

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15 Chapter 1 Introduction 1.1 Motivation and Objectives For a ship floating or sailing at water surface, estimating its hydrodynamic load caused by ship motion is a common and important problem in marine and offshore industry and has been studied in many ways. For instance, when a ship is excited by the ocean waves and oscillates at the wave frequencies, it is normally considered as a diffraction and radiation problem and studied by potential theory. Particularly the linear potential BEM in frequency domain has been applied extensively in engineering projects. Sometimes when a ship sails in rough seas, the high relative velocity between ship and wave surface makes it to impact on the water heavily, leading to extreme high load on the ship hull. Since the time scale during the impact is extremely short, the motion of ship can be recognized as an oscillation with a very high frequency. The assessment of the impact induced load on structures has been researched by analytical, numerical and experimental methods and many methods have been verified to be able to give good estimations. The photographs in Figure 1-1 show a FPSO in ocean waves and an experiment of FPSO model in steep fronted waves. Figure 1-1 FPSO in ocean waves (left) and FPSO model in steep fronted waves by MARIN (right) In terms of gravity effect on the vertical loads acting on the oscillating ship, it changes significantly with the different oscillating frequencies. The vertical force in the lower frequency relating to common ship heaving is dominated by gravity effects, but for impact theory it is usually ignored. As a consequence, there must be a transition in between those gravity-dominant and gravity-negligible conditions. Although these two limiting conditions have been studied deeply by many researchers for decades, but the range of transition has not gain much attention. Thus, to fill in the gap in frequency of oscillation is a meaningful work. Besides, the linear potential theory has its validity in frequency due to the linearized boundary condition to the undisturbed free surface without consideration of water elevation, and most impact methods ignoring the gravity effect require the lower applicable limit of frequency. Therefore, this thesis concerns determination of limit for both of the gravity-dominant and

16 2 Chapter 1 Introduction gravity-negligible conditions, determination of application limits of the widely applied linear radiation method and impact method, and also quantification of the gravity effect in this transition frequency range. 1.2 Research Background The previous researches relating to this thesis majorly involve the wave-frequency oscillation and the high frequency slamming problems. In this section, the previous effort associating with these problems is introduced in a general and feasible way rather than describing the details of theories Oscillation Problem A variety of analytical and numerical methods has been developed over the past decades for the treatment of a body oscillating in periodic motion at the water surface. The solution can provide information of hydrodynamic load as the basis to study the problems of wave and structure interaction, strength and stability of floating structures. Early studies on radiation loads were usually based on linear analytical and numerical methods for the regular geometries, such as circular cylinder, hemisphere, rectangular and wedge. The first relevant investigation on this subject was given by Ursell (1949) [1] who considered an infinitely long horizontal cylinder of circular cross-section half-immersed in a fluid, in which an analytical solution is derived based on the construction of polynomial set of stream functions. Havelock (1955) [2] studied a floating hemisphere with periodic heaving oscillations, using the theory similar to Ursell's. After that, many researchers had expanded the application to other geometries, to other motion degrees and had modified the solutions. For instance, Kim (1965) [3] studied harmonic oscillation of a rigid body with the form of either an elliptical cylinder or an ellipsoid by using a Fredholm integral equation to obtain numerical results and Hulme (1982) [4] provided a number of modifications to Havelock s treatment of the heaving-hemisphere proble.. Besides, Lebreton et al. (1966) and Hsu and Wu (1997) [5] gave their study on linear heave radiation problem of a rectangular geometry. Later after that, more efforts were spent on generalizing the loads from regular geometries to arbitrary geometries to make the results more applicable. One method achieving this purpose is by conformal mapping. In two-dimensional problem, the added mass and damping are firstly determined for a simple shape like a cylinder for which there is an analytical solution available for potential flow and subsequently mapped to the intended arbitrary shape. Ghadimi et al. (2011) [6] mapped a wedge section onto a line in solving the water entry problem. Many types of mapping technique were developed for the analysis of loads on and flow around marine and offshore structures. Another valid method for computing the arbitrary geometry is the numerical BEM based on potential flow theory, in which potential sources are assumed to be distributed in the boundary elements over the wet surface and free surface. Generally, the wet surface is linearized to its calm water position, while the free surface condition is linearized to the undisturbed water surface, enabling the application of Green functions that automatically satisfy the linearized free surface condition. Inspired by the growing application of offshore structures which are of similar dimension in all directions, the two-dimensional radiation theory was extended to three dimensions, using the same assumption of linearity.

17 1.2 Research Background 3 During last three decades, the non-linear effect started to draw a great deal of attention. Initially, extension of the linear theoretical models to second-order is based on perturbation expansion schemes with respect to wave steepness, and the Taylor approximation of boundary conditions on the mean water and body surfaces. Hatchmann (1991) formulated an efficient method extrapolating hydrodynamic pressure above the calm water level. The above second-order methods were in the frequency domain, besides the perturbation expansion methods have also been applied in time-domain calculations on the basis of the retardation function or memory integral formulated by Cummins (1962). Although the time domain method enables the consideration of exact boundary condition, the considerable computational effort limits its applicability in engineering projects Slamming Problem Since von Kármán published his paper in 1929 about a two-dimensional wedge impacting with a calm water surface, the slamming problem has been invested by many researchers by theoretical and experimental ways. Generally speaking, the theories applied for estimating slamming loads includes momentum theory, potential theory and Computational Fluid Dynamic (CFD) method on basis of Navier-Stokes equations. The investigation on impact between a body and water began with von Kármàn (1929) [7], who used momentum theory to estimate the load on the floats of landing seaplanes. Kapsenberg (2010) [8] proposed an approximation method based on momentum theory accounting for pile-up effects caused by the immersion depth and bow waves, in which the added mass of the ship is calculated through radiation BEM using panels on the hull. A good mathematical basis founded on potential flow theory was laid by Wagner (1932) [9], who paid attention to the pile-up which can arrive at about 50% of the immersion. In his paper, he proposed an asymptotic solution for water entry of two-dimensional wedges with small deadrise angles. Much work has been done by other researchers to further develop the application of potential flow on slamming problem. Dobrovol skaya (1969) [10] improved the approximation solution for the free surface by transferring the potential flow problem into a self-similar flow problem in complex plane, known as the similarity solution. It is valid for a wedge with any deadrise angle, but cannot be applied to arbitrary geometries. A numerical method of BEM being applicable for arbitrary shaped bodies has been adapted for tackling this problem in recent thirty years. Greenhow & Lin [11] and Greenhow (1987) [12] initially presented a non-linear numerical solution for the wedge water entry problem followed by Zhao and Faltinsen (1993) [13]. To further develop the non-linear numerical solution, Zhao et al. (1996) [14] presented a fully non-linear numerical solution with flow separation and an approximation solution with linearized free surface boundary condition at the horizontal line at the height of intersection between free surface and body surface. With rapid development of computational techniques, CFD method based on directly solving Navier-Stokes equations becomes a potential way in studying slamming problems. But due to its high request on computational cost, CFD effort on the slamming problems is still not as rich as developing other methods and the research mainly restricts to 2D problems. Different CFD techniques have been applied for solving 2D water entry problems. The free-surface-capture method Volume of Fluid (VOF) was used by Fekken [15] and Kleefsman (2004) [16] in Finite Volume Method (FVM) based program ComFLOW to calculate the water entry problem. Oger G., Doring M., Ferrant P (2005), Koukouvinis (2013) implemented Smoothed Particle Hydrodynamics (SPH) method for a wedge water entry. Zhu et al. (2006) [17] studied the Constrained Interpolation Profile (CIP) method. All the above solutions consider the body as

18 4 Chapter 1 Introduction rigid. Recently, codes based on explicit FEM which can analyze hydro-elastic problem began to be applied to predict local slamming loads. Luo et al. (2012) and Wang et al. (2012) studies the hydroelastic response a wedge in a drop test Apart from the theoretical studies on the slamming problem, a number of experimental researches were carried out, which majorly included drop test and constant velocity test representing as vessel slamming and planing problem respectively. For instance, Bisplinghoff and Doherty (1952), Ochi and Motter (1973), Zhao et al. (1996), Aarsnes (1996), Wu et al. (2004) and Tveitnes, Fairlie-Clarke et al. (2008) [18] carried out the experimental research on water entry problems. 1.3 Outlines In this thesis simulation of a wedge oscillating at free surface is presented. To describe the method to accomplish this task the thesis is outlined as follows. The first chapter is an introduction to the problem and the objectives of this thesis. Also the research background relating to this subject is introduced. In chapter 2, the approach to study the problem is described and the theories involved in the selected approach are introduced. First, the CFD theory in program ComFLOW is described in Section 2.2, including the governing equations, boundary conditions and the numerical methods. In Section 2.3, the linear potential BEM applied in program SHIPMO is introduced. Also the non-linear potential theory for calculating the slamming loads in program 2DBEM is given in Section 2.4. Finally, the hydrodynamic coefficients in the potential theory are discussed, and it gives some estimation methods provided by previous researchers. Chapter 3 describes the model for simulation. After that the convergence study in ComFLOW is performed. First studies on mesh size for different frequencies is presented. Section 3.3 shows the analysis on influence of time steps on the vertical force. Furthermore, the conditions on the body surface and outflow boundary are studied as well. Chapters 4, 5 and 6 give the simulations and discussion of an oscillating wedge with deadrise angle of 45º. The study of oscillation at wave frequencies is provided in chapter 4, first presenting the vertical forces from ComFLOW and comparing the hydrodynamic coefficients of added mass and damping extracted from CFD results to the linear potential method of SHIPMO. In chapter 5, the slamming-frequency oscillation is performed. An introduction of program 2DBEM and the added mass estimation of momentum method is given at first. The gravity effect on the total vertical force indicates the high frequency limits of gravity negligible conditions. After obtaining the limits of the two extreme conditions, the study on gravity effect in between the limits is carried out in chapter 6. Details such as vertical force, pressure distribution, free surface elevation and fluid velocity field are presented. Furthermore, the gravity effect is discussed by simulations with and without gravity acceleration in ComFLOW. To verify the results and conclusions in chapter 4,5 and 6, wedges with deadrise angle of 15º and 30º are involved in chapter 7. The application limits of liner potential theory for radiation and non-linear potential theory for slamming are considered first. Then the simulations of oscillation at frequencies in the transition range is performed to look into the gravity force acting on the wedge. In the end, there are some conclusions and recommendation about this thesis given in chapter 8.

19 Chapter 2 Approach and Theories This chapter describes the study approach applied to achieve the objectives with the present investigation. Also the computational methods involved in this approach are explained. Three kinds of numerical methods for fluid flow are applied and they are stated in the order of CFD, radiation BEM and water entry BEM, which are simulated in programs ComFLOW, SHIPMO and 2DBEM respectively. For each method, the mathematical model, boundary conditions and numerical solutions are discussed. 2.1 Approach A wedge oscillating at the same amplitude at different frequencies varying from those that are common for ship motions in waves to very high frequencies with a limiting case of a drop test (impact theory) is studied in this thesis. The floating wedge oscillating at the water surface in periodic heaving motion generates radiation waves, which are propagating away. According to the dispersion relation, the relatively low oscillating frequencies generate waves with longer wave length so that the gentle wave steepness enables this situation to be recognized as a linear problem. However, as the oscillating frequency increases, the propagating speed is declining, resulting in that the energy cannot be delivered away immediately. The collected energy may cause water pile up or even a jet flow. The large free surface elevation leads to significant non-linear effect. Thus, a method which is capable of non-linear effect is required to simulate this situation. Besides, to quantify the gravity force specially, it should be able to estimate the interaction load with and without gravity effect. Based on the above considerations, a numerical CFD method directly solving Navier-Stokes equations in the fluid domain being able to satisfy the requirements is selected for this subject. The available CFD program ComFLOW is determined to be used for the simulations, which has been proven by Fekken (2004) in [15] and Kleefsman (2004) in [16] individually to be suitable for wedge water entry and rectangular oscillation cases, by comparing the numerical results with experiment results and numerical results from other researchers in other theories. The FVM based ComFLOW is developed by the offshore industry, MARIN and the Universities of Groningen and Delft together, and details of it will be further introduced in the next Section. For verification of the results, the general methods used for the wave-frequency oscillation and slamming problem are applied as well, to be compared with the CFD method. The linear potential based 2D-Strip program named SHIPMO will be used for the vertical force for wave-frequencies oscillation. A program known as 2DBEM developed for the prediction of slamming loads on the basis of non-linear potential theory in time domain in two-dimensional is capable for the verification of slamming-frequency oscillation in ComFLOW.

20 6 Chapter 2 Approach and Theories The following steps are designed and a schematic diagram indicating the steps is shown in Figure First step is to find the limit of gravity-dominant condition and the application limit of linear potential theory. In the wave frequency oscillation, perform radiation analysis in SHIPMO and search for an as high as possible frequency. Then, simulations with same frequency, oscillation amplitude and mean draft are carried out in CFD program ComFLOW. By comparing their results, study the non-linearity and gravity effect, and determine the limit of gravity-dominant condition and the application limit of linear potential theory. 2. Step two is to study the slamming-frequency oscillation to find the high frequency limits. The widely applied slamming methods including momentum method and non-linear 2DBEM are used to study the periodic oscillation at high frequency. Then, study the same cases in ComFLOW with and without the consideration of gravity. Then by comparing the results and analyzing the difference, determine the high frequency limits. 3. After determining the transition range, perform CFD simulations with and without gravity to quantify the gravity effect in between the limits. Figure 2-1 Study approach scheme

21 2.2 CFD CFD The offshore industry, MARIN and the Universities of Groningen and Delft have worked together successfully in the development of the FVM based program ComFLOW since 1995 to study complex free-surface flows around offshore structures. The simulation of fluid flow in ComFLOW is based on the Navier-Stokes equations for an incompressible, viscous fluid. The governing equations are discretized using FVM. ComFLOW is capable of dealing with one-phase flow and two-phase flow, capturing the free surface by VOF technique. From [19] it is known that only when the relative angle between the body and free surface is below 5º, the compressibility effects of air is important in the pressure underneath the body because the air is being compressed below the body in the phase just before the impact. In this thesis, no deadrise angle lower than 5º will be discussed. Therefore, the compressibility of the air can be ignored and no need is required to include the air flow in the calculation, so that one-phase flow model can meet the requirement. Thus, the following computational models are introduced for one-phase flow Governing Equations The flow of a homogeneous incompressible viscous fluid is described by the continuity equation (2-1) and the Navier-Stokes equation (2-2), which represent the conservation of mass and the conservation of momentum respectively [20]. u = 0 (2-1) u t + (uut ) = 1 ρ p + μ ρ 2 u+f (2-2) Here, u is velocity vector with u = (u, v, w), ρ is fluid density, p is pressure, µ is dynamic viscosity coefficient, and F is external force vector Boundary Conditions To solve the governing equations, boundary conditions are needed at the solid walls, in- and out-flow boundaries and free surface. Symmetry First, no roll angle of the wedge is considered, resulting in geometrical symmetry about the centerline, which allows the flow to be simulated in half of the domain. Therefore to reduce the computational cost, the left side wall is defined as symmetry. Body surface It is possible to define the no-slip or free-slip condition at the wedge domain boundaries, indicating that no fluid can go through the boundary. Comparison of the effect of both the boundary conditions will be studied in this report. This condition is described by u = 0 for fixed boundaries and u = u b for moving objects, where u b is the object moving velocity. For free-slip, the above velocity is applied to the velocity normal to the body surface u n, while the velocity tangential to the body surface is free u t.

22 8 Chapter 2 Approach and Theories Out flow boundary In order to prevent wave reflection from the right boundary, it is defined as out flow boundary. In ComFLOW, two kinds of methods can be applied to simulate the out flow boundary, which are Sommerfeld and a generating and absorbing boundary condition (GABC). At the outflow boundary, a Sommerfeld condition is very appropriate in cases where a regular wave is used. In the case of an irregular wave or a much deformed regular wave (e.g. due to the presence of an object in the flow) a damping zone is added at the end of the domain [16]. The proper outflow boundary condition setting for this wedge oscillation will be studied. Bottom At the bottom boundary, it is demanded that no fluid can go through the boundary and that the fluid sticks to the wall because of viscosity. Thus the boundary condition at bottom is defined as no-slip condition with fluid horizontal and vertical velocities equal to zero. Free surface To describe the displacement of the free surface, Equation (2-3) is used in which the position of free surface is given by an equation s(x, t) = 0. Ds = s Dt t + (u )s = 0 (2-3) At the free surface, boundary conditions are required for the pressure and the velocities to satisfy the balancing force. Continuity of normal and tangential stresses lead to the equations p + 2μ u n n = p 0 + 2σH (2-4) μ ( u n t + u t n ) = 0 (2-5) Here, p is the pressure, μ is the dynamic viscosity, n is the normal of free surface, t is a direction tangential to the free surface, u n is the normal component of the velocity, u t is the tangential component of the velocity, p 0 is the atmospheric pressure, σ is the surface tension and 2H denotes the total curvature. Figure 2-2 CFD simulation domain with boundary conditions

23 2.3 Radiation BEM Calculation of forces The solution of governing equations gives the pressure distribution over the entire fluid. The force acting on the object in the fluid flow normally consists of two parts, namely the pressure force and the shear force. In ComFLOW, the shear force is neglected, since it is typically much smaller than the pressure force. The force is calculated as the integral of the pressure along the boundary of the object S by formula: F p = s p n ds (2-6) Numerical Solution In ComFLOW, fixed rectangular (Cartesian) grids are used for discretization of the governing equations, which may be uniform or stretched. Different from geometry-aligned grid, the Cartesian grids do not require regeneration of new grid every time step, saving a plenty of time-consuming. But in this case, geometry will cut through the grid cells. To be able to simulate fluid flow in arbitrary shaped bodies, the cut-cell technique is employed in which volume and edge apertures are introduced to define which part of the cell volume and cell edge respectively are open to fluid. Another function known as VOF function is introduced to track the free surface by identifying the fraction of a cell that is filled with fluid. To prevent flotsam and jetsam, the VOF function is combined with a local height function The variable are staggered, which means that the velocities are defined at cell faces, while the pressure is defined in cell centers. The continuity and Navier-Stokes equations are discretized spatially using FVM associated with first order upwind method. In time discretization, forward Euler method is used. This first-order method is accurate enough because the order of the overall accuracy is already determined by the first order accuracy of the free surface displacement algorithm. 2.3 Radiation BEM The program SHIPMO is applied for the simulation of cases of wave oscillation. SHIPMO calculates the behavior of a ship in a seaway in frequency domain. The program is based on the well-known 'strip theory', which is 2D linear diffraction theory. It is a program for estimating the ship behavior, but this thesis only concerns its function of estimating the hydrodynamic load, therefore only the radiation theories is to be introduced. In SHIPMO, the 2D radiation problem and the 2D diffraction problem are solved for a cross section of a ship. It makes use of the Green function module and symmetry relations in case the section is symmetrical Mathematical Model In potential theory, the fluid is assumed to be inviscid and incompressible and the flow is irrotational. Thus, the fluid motion can be described by a velocity potential function, which must satisfy the Laplace equation in the fluid domain:

24 10 Chapter 2 Approach and Theories 2 Φ = 0 (2-7) With assumption: Φ = R(ϕe iωt ) (2-8) The boundary conditions include the followings: At free surface: ϕ z = ω2 g ϕ at z = 0 (2-9) On body: ϕ n = v n on S Far away: ϕ 0 as y (2-10) (2-11) Through the linearized Bernoulli s equation, the dynamic pressure can be obtained from the velocity potential and expressed as: P = ρ Φ t = iωρφ = iωρϕe iωt (2-12) The vertical wave force acting on the wedge can be calculated by integrating the wave pressure along the submerged surface: F = P da = iωρe iωt S ϕ da (2-13) 0 S Numerical Solution This mathematical model can be solved numerically by BEM, which performs computations only on the boundary of the domain. Avoiding detailed computations inside the domain makes the BEM more efficient than the domain type methods [5]. A potential source is assumed at each element. To utilize the BEM, the boundary value problems must be first converted into an integral equation representation. Using Green s second identity: ( φ q φ q Γ )dγ = n n Ω ( φ 2 q q 2 φ )dω (2-14) In which q means fundamental solution of the governing equation, Γ is the boundary of the solution domain, Ω is the solution domain, φ is the velocity potential at a selected element of the boundary. The fundamental solution for Laplace equation is : q = 1 2π ln (1 r ) (2-15) where, r represents the distance from the source point to the field point. Therefore, from equation (2-14) any velocity potential φ j at the boundary is given by:

25 2.4 Water Entry BEM 11 β 2π φ j = ( φ q φ q Γ )dγ (2-16) n n where j is the source point, β is the internal angle of the source point j. On wet surface of the body, the velocity potential ϕ is the unknown, the velocity ϕ known. At each panel, an equation about ϕ can be listed. Rearranging the equations at all the panels in such a way that all unknowns are taken to the left hand side and all the knowns are move to the right hand side, then the velocity potential can be solved linearized. In linear diffraction and radiation problems, according to Lamb (1932), the potential φ at a point (x, y, z) on the mean wetted body surface S owning to body motion and diffraction potential can be represented by a continuous distribution of single sources on the body surface: n is φ (x, y, z) = 1 4π s σ(x, y, z ) G(x, y, z, x, y, z ) ds (2-17) in which, φ (x, y, z) is the potential function in a point (x, y, z) on the mean wetted body surface S, σ(x, y, z ) is the complex source strength in a point (x, y, z ) on the mean wetted body surface S due to body motion, G(x, y, z, x, y, z ) is the Green s function or influence function of the pulsating source σ(x, y, z ) in a point located at (x, y, z ) on the potential φ (x, y, z) at point (x, y, z), singular for (x, y, z ) = (x, y, z). This Green s function satisfies the Laplace equation, the linearized boundary conditions on free surface and the radiation condition at infinity. 2.4 Water Entry BEM At MARIN, program 2DBEM on basis of 2D non-linear potential theory is available for simulating the impact problem. It is developed for the prediction of slamming forces and pressure on a body that penetrates an initially calm free water surface. The theories behind it is based on the approximation method developed by Zhao, Faltinsen and Aarsnes [14]. Important inputs to the computer program include the body geometry, initial displacement, initial inclination and time history of the vertical velocity. Being different from radiation BEM, 2DBEM uses a non-linear method, considering the free surface elevation in time domain. The effect of gravity is neglected compared with the large fluid pile-up acceleration. Figure 2-3 Water entry coordination and free surface in approximation BEM [14]

26 12 Chapter 2 Approach and Theories The problem as an initial value problem is solved in time domain. The velocity potential satisfies two-dimensional Laplace equation in the fluid domain. A boundary value is solved by using Equation (2-16). The linearized dynamic free surface condition is simplified to be applied on horizontal L lines that starts at the intersection points between the body and the free surface, shown in Figure 2-3. The boundary conditions are as following: At free surface: φ = 0 at line L (2-18) On body: ϕ n = v n (2-19) The kinematic free surface condition that a fluid particle remains on the free surface, is used to determine free surface elevation at next time step. At the free surface, the velocity potential at free surface far from the body can be expressed as vertical dipole (symmetrical part) and a multipole (asymmetrical part) in infinite fluid from the boundary condition (2-18). In program 2DBEM, the horizontal computational domain is defined as a factor of the instantaneous immersion depth. Scaling factors between the neighboring elements are used to define the element size on the wet surface and free surface. The smallest elements are near the intersection point between wet surface and free surface. 2.5 Hydrodynamic Coefficients Potential Theory In radiation theory, the hydrodynamic coefficients can be derived from radiation velocity potential of the fluid. Hydrodynamic reaction force referred to as radiation force depends on the body velocity and acceleration, added mass and damping: F Hyd.3 = a 3 z b 3 z (2-20) The radiation velocity potential Φ r which is associated with the body oscillation in still water, can be written in terms Φ j for 6 degrees of freedom as: Φ r (x, y, z, t) = Φ j (x, y, z, t) 6 j=1 (2-21) To derive added mass a 33 and damping b 33, radiation potential is written as function of velocity of the motion: Φ 3 (s, t) = R{ 3 (s)v 3 (t)} = R{ 3 v a3 e iωt } = R{ 3 iω s a3 e iωt } (2-22) 3 (s) is only space dependent and v 3 (t) is only time dependent. v a3 is complex amplitude of heave velocity. s a3 is complex amplitude of heave displacement. Hydrodynamic reaction force can be obtained by integration of radiation pressure. According to Bernouli equation, the radiation force can be expressed as:

27 2.5 Hydrodynamic Coefficients 13 F 33 = ρ Φ 3 S t n 3 ds = R { (ρ ( 3 iω s 3 e iωt ) n 3 ) ds} S t (2-23) = R { ρ ω 2 s a3 3 n 3 ds e iωt } S Let equation (2-23) equals to equation (2-20), the hydrodynamic coefficients resulting from heave motions are: a 33 = R {ρ S b 33 = I {ρω S 3 n 3 ds} (2-24) 3 n 3 ds} (2-25) Impact Theory Apart from the numerical solution for impact problem introduced in section 2.4, there is another kind of method on the basis of momentum theory which is the oldest and most common theory to tackle the slamming problem. In this theory, the impact force relates to the change rate of momentum of added mass. F = (M a v) t = M a t v + M a v t (2-26) Therefore, the key point of momentum theory is to estimate the added mass M a. Many researchers have derived added mass values directly for a range of cross sectional shapes by studying the two-dimensional water entry problem. Tveitnes (2001) gave a detailed review about the researches on added mass of a wedge in his Ph.D thesis [21]. Normally, the researchers express the added mass as the product of the mass of the fluid volume obtained by taking the projected area of the cylinder in that direction and evaluating on a half-cylinder and a factor relating to deadrise angle as shown in Figure 2-4 and the expression is given in following equation. m zz = 1 2 ρπy2 C m (2-27) where, ρ means water density, y is the wetted width of wedge, C m is the deadrise angle relating factor. Figure 2-4 Scheme of added mass when wedge is moving into water

28 14 Chapter 2 Approach and Theories Various two-dimensional added mass formulas have been derived by many researchers for the case of a wedge water entry, and a list of the few is shown as follows. von Kármán m zz = π 2 ρy2 (2-28) Wagner m zz = π 2 ρy2 ( π 2 )2 (2-29) Wagner-Sydow m zz = π π ρ( 2 2β 1)2 z 2 (2-30) Lewis, Taylow m zz = π 2 ρ y2 (1 β 2π )2 (2-31) here, ρ is water density, y means half width at the undisturbed water level, z means the immersion depth of wedge, β is deadrise angle, l means the length of the float. The corresponding non-dimensional added mass m zz ρy2 from the above estimation methods for different deadrise angles are plotted in Figure 2-5. Only Wagner-Sydow, Lewis and Taylow concern the contribution of deadrise angle von Kármán Wagner Wagner-Sydow Lewis, Taylow Deadrise angle [º] Figure 2-5 Non-dimensional added mass of different estimation methods

29 Chapter 3 Model and Convergence Study In this chapter, first the model selected for simulation is introduced, then there is a description of the study on convergence. In the simulations, the wedge oscillates with frequency varying from ship heaving frequency in waves to ship slamming duration, so that the features of excited waves in the fluid domain are considerably different. For convergence study, different numerical parameters are required for different oscillating frequencies. For instance, the low frequency oscillation generates long waves, which means mesh length can be larger in horizontal direction rather than in vertical direction, while the short and higher waves derived from high frequency oscillation requires smaller mesh to capture the features of the wave and smaller time step as well. In order to make sure the convergence of simulation in ComFLOW, the results are studied by varying the following parameters: Mesh size CFL number Boundary condition on the wedge surface Computational domain 3.1 Model Geometry and Coordinate The geometry is chosen as an experimental scaling and a wedge with deadrise angle of 45º and with half width of 0.3 m is selected for the simulation of the whole range of frequency first, later a study on the other deadrise angle will be carried out to consider the effect of deadrise angle and to verify the obtained conclusions from 45º. The dimension of the wedge with 45º is shown in Table 3-1 and Figure 3-1. The origin of the coordination is set at the free surface with x-axis positive to right and z-axis upward positive and the local coordination of the wedge is placed at the apex.

30 16 Chapter 3 Model and Convergence Study Figure 3-1 Wedge geometry Table 3-1 Main particulars Item Unit Value Width W m 0.6 Half-width Y m 0.3 Chine height H m 0.3 Deadrise angle β º 45 Mean draft D m 0.19 Water depth m 1 The periodic heave motion is considered with locating at the highest location at initial instant. The draft of the wedge can be expressed by formula: d = D + z a cos (ω t) (3-1) here, D is mean draft z a is the oscillating amplitude and ω is oscillating frequency Frequency Selection The range of frequency should be determined initially for simulation. Since the range from the common ship motion to the ship slamming is interested, the lowest frequency and high frequency can therefore be chosen based on the wave frequencies and slamming duration. The wave data in different ocean regions provided in [22] indicates the spectral peak periods are around 7 or 15 s for ocean waves and swell waves. The Froude scaling is needed to convert the real period to the model period and a Froude scaling table for different physical parameters are given as Table 3-2. Assuming the 0.6 m width wedge reflects a ship with 30 m width, length scaling factor therefore is a l = 50. Thus, the period scaling factor and frequency scaling factors are a t = a l = 7.07 and a f = 1/ a l = 0.14 respectively. In consequence, the corresponding period of 15s is 2.12 s and its scaled frequency is around 3 rad/s which can be chosen as the lower limit of the frequency range.

31 3.2 Mesh Size 17 The time scale for bottom slamming entering water is extremely short, having a duration of the order of 0.02 s from the recording graph in Dutch destroyers in [23] and having durations for laboratory test data that varies from a small fraction of a millisecond (0.05 ms) to 20 or 30 milliseconds based on many studies on laboratory and full-scale test data by many investigators [24]. In terms of bowflare slamming, the spontaneous recorded data in [23] represents the duration relatively long in seconds. If we take the impact duration from first contact between water and body to the moment of peak vertical force as a quarter period of a sinusoidal oscillation and assume it as 0.02 s, then the corresponding oscillating frequency is around 500 rad/s. In conclusion, frequencies initially selected for study cover the range from ones to hundreds in rad/s, but these values can be adjusted based on the analysis of the simulation results. Table 3-2 Froude scaling factors Physical Parameter Unit Multiplication factor Length m a l Time a t s a l Frequency a f rad/s 1/ a l 3.2 Mesh Size With respect to grid sizes, their requirements are much different based on the oscillating frequencies. For instance, a lower frequency such as 6 rad/s leads to stable and sinusoidal waves, while for a higher frequency like 30 rad/s, pile-up even wave breaking may occur and the local pressure varies largely, therefore, much finer meshes are required near the wet surface to capture the feature of the waves. Three frequencies: 6 rad/s, 20 rad/s and 100 rad/s with oscillation amplitude 0.019m (10% of draft) are illustrated here representing low, intermediate and high frequency oscillations respectively as examples showing the convergence study on grids. In ComFLOW, both the flow domain and the bodies are covered with a Cartesian grid, thus cells of different characters appear. This difference in character is incorporated in the numerical method by introducing edge and volume apertures. These edge and volume apertures are a measure for which part of the cell face or cell volume is open to flow Frequency 6 rad/s When the wedge oscillates at frequency 6 rad/s, stable and sinusoidal radiation waves are generated with wave length of 1.7 m according to wave dispersion relationship in deep water. An initial simulation with coarse grids in ComFLOW implies that the corresponding generated wave height is around 0.036m. Three levels of grid size studied are shown in Table 3-3. The non-dimensional parameters of grid sizes are used, such as grid number per wave length and grid number per wave height. The refinement ratio between the neighboring conditions is. 2. Figure 3-2 shows that the coarse grid No.1 gives underestimated force results compared to grid conditions No. 2 and No.3, results of which are highly close. Thus, the simulating results are grid-independent when grid size is smaller than condition No.2. Figure 3-3 illustrates the snapshots of velocity field for frequency 6 rad/s of grid condition No.2. The top one represents the moment of the end of the first period, when the wedge is at the top position with maximum acceleration downward. As we can see, the maximum water particle velocity is away from the body surface, representing the wave is propagating away.

32 18 Chapter 3 Model and Convergence Study Condition No. Table 3-3 Grid sizes for ω = 6 rad/s Cells per Cells per dx wave length wave height [mm] dz [mm] Figure 3-2 Time series of vertical force for different grid sizes for ω = 6 rad/s Figure 3-3 Snapshots of velocity field in 6 rad/s for t=t and 1.5T for grid condition No. 2

33 3.2 Mesh Size Frequency 20 rad/s Four levels of grid sizes with local refinement are applied for frequency 20 rad/s as shown in Table 3-5. The region near the wedge wet surface is refined with scaling factor of 2. The dimensions of computational domain and refined domain are shown in Table 3-4. The refinement ratio between the neighboring conditions is 2. Because for frequency 20 rad/s the waves tend to break and it is difficult to define the wave length and height, the total cells numbers are illustrated here, which denotes basic cell number excluding the refined grids. dx and dz means the grid size, where the numbers before and after slash are basic size and refined size respectively. The grids of condition No.1 and No.3 are shown in Figure 3-4. Figure 3-5 illustrates the time series of vertical force for different grid sizes given in Table 3-5. For the initial phase (1-0.25s), the different grid sizes give negligible differences on the numerical results of vertical force. At the second crest ( s), apparent differences show up with about 15% gap between condition No.1 and No. 2, but the most refined conditions No.3 and No.4 provide very close results. Therefore, it can be conclude that the results convergence for condition No.3 and No.4. In Figure 3-6, there are snapshots of velocity at half period and four third period. The left picture representing the moment of maximum acceleration illustrates that the velocity is very small. In the right one, the water particles near the body surface move upward with the maximum moving velocity at the water surface. Table 3-4 Computational Domain Limits Domain Refined domain x [m] z [m] x [m] z [m] Min Max Condition No. Table 3-5 Grid sizes for ω = 20 rad/s Grid number Grid number dx in x in z [mm] dz [mm] / / / / / / / /2.3

34 20 Chapter 3 Model and Convergence Study Figure 3-4 Grids for condition No.1 and No.3 in Table 3-5 Figure 3-5 Time series of vertical force for different grid sizes for ω = 20 rad/s

35 3.2 Mesh Size 21 Figure 3-6 Snapshots of velocity field in 20 rad/s for t=1/2t and 3/4T Frequency 100 rad/s Three levels of grid sizes with local refinement are applied for frequency 100 rad/s as given in Table 3-7. In order to reduce computational effort, the region near the wedge wet surface is refined twice with scaling factor of 2. The dimensions of computational domain and refined domain are shown in Table 3-6. The corresponding time series of vertical force for the three given grid sizes are plotted in Figure 3-7. The differences at force crest and trough are relatively obvious but still can be ignored especially for difference between grid condition No.2 and No. 3. Table 3-6 Computational Domain Limits Domain Refined domain 1 Refined domain 2 x [m] z [m] x [m] z [m] x [m] z [m] Min Max Condition No. Table 3-7 Grid sizes for ω = 20 rad/s Grid number Grid number dx in x in z [mm] dz [mm] /16.7/ /18.9/ /8.3/ /9.4/ /4.2/ /4.7/2.4

36 22 Chapter 3 Model and Convergence Study Figure 3-7 Time series of vertical force for different grid sizes for ω = 100 rad/s 3.3 Time step The selection of time steps during the simulation influences the accuracy and stability of the results significantly. A proper time step should be selected to balance the accuracy against the computational cost. Furthermore, the selection of time step also depends on the method of time discretization. Implicit methods require an extra computation, being much harder to implement, but they are used because many problems arising in practice are stiff, for which the use of an explicit method requires impractically small time steps to keep the error in the results bounded. For such problems, to achieve given accuracy, it takes much less computational time to use an implicit method with larger time steps. In ComFLOW, the time discretization uses the Forward Euler method, which is an explicit method. In ComFLOW, time step during the simulation is adjusted using the CFL-condition. The CFL-number is calculated as: CFL = max ( u ijk δt i,j,k h x.i + v ijk δt h y.j + w ijk δt h z.k ) (3-2) Where, u, v, w are velocity components and h x, h y, h z denote grid size in the corresponding directions. The fundamental stability condition of most explicit schemes for wave and convection equations expresses that the distance covered during the time interval δt, by the disturbances propagating with speed v, should be lower than the minimum distance between two mesh points. The CFL stability condition CFL<1 expresses that the mesh ratio δt/h has to be chosen in such a way that the domain of dependence of the differential equation should be entirely contained in the numerical domain of dependence of the discretized equations. In other words, the numerical scheme defining the approximation u in mesh point i must be able to include all the physical information which influences the behavior of the system in this point. If this is not the case, then a change in physical conditions would not be seen by the numerical scheme and therefore the difference between the exact solution and the numerical solution could be made arbitrary large. Hence, the scheme will not be converged.

37 3.4 Boundary Condition on the Wedge Surface 23 In ComFLOW, time step is restricted by three parameters: dt max, cflmin and cflmax, which represent maximum time step, minimum CFL-number and maximum CFL-number respectively. During the simulation, if the computed CFL-number is greater that cflmax, the time step will be decreased. If the computed CFL-number is smaller than cflmin during 10 successive time steps, the time step will be doubled. For the Forward Euler method, it is suggested that in ComFLOW dt max is set up as T/250, cflmin and cflmax are defined as 0.2 and 0.5 respectively, in which T means wave period. Still in order to study the influence of the time step on the simulation results, other values of CFL are considered as well. Two allowable CFL range are used in the calculation, including and The corresponding vertical forces for frequencies 20 rad/s and 100 rad/s are shown in Figure 3-8, from which it can be seen that the given CFL ranges have negligible influence on the numerical results of the total vertical force both in low frequency and high frequency. Figure 3-8 Time series of vertical force for different CFL ranges 3.4 Boundary Condition on the Wedge Surface There have been no-slip and free-slip boundary conditions applied at the body surface in CFD problems. The no-slip condition for viscous fluids states that at a solid boundary the fluid will have zero velocity relative to the boundary. Free-slip boundary condition implies the fluid can flow along body surface freely without friction. Comparison of the effect of both the boundary conditions has been studied in order to select the proper boundary conditions at the body surface. Theoretically, the no slip boundary condition satisfy the reality, but it requires smaller mesh size near the body surface. It is indicated that in ComFLOW the partial slip boundary condition has not been tested properly, no slip boundary condition is suggested. Anyhow, both the two boundary conditions have been studied for the selection of the proper boundary conditions. The cases of frequencies 20 rad/s and 100 rad/s with coarse mesh and fine mesh have been studied and plotted in Figure 3-9. The dashed line indicating the no-slip boundary condition has good match with the solid line representing free-slip boundary condition in both cases of coarse mesh and fine mesh. The difference between these two boundary conditions is relatively obvious in frequency 20 rad/s compared to frequency 100 rad/s, but it still can be ignored. Therefore, based on the comparison between free-slip and no-slip boundary conditions, the

38 24 Chapter 3 Model and Convergence Study no-slip condition at the body surface which is suggested in ComFLOW is applied in the simulation. Figure 3-9 Vertical force for free slip and no slip boundary conditions at wedge surface 3.5 Non-reflecting Boundary Condition The right wall of the fluid domain represents the truncation of the infinite fluid domain for computational purpose. Therefore, non-reflecting boundaries have to be considered, which must allow the waves to leave the truncated domain avoiding spurious reflections that may pollute the solution in the interior of the computational domain of interest. When the wedge oscillates at different frequency, the corresponding wave lengths vary considerably, for example the wave length generated by oscillation of frequency 2 rad/d is around 15 m while the oscillation of frequency 8 rad/s creates waves of 1 m length. With respect to the cases of high frequency, the reflections are less important when a short simulation time is used therefore the waves have not propagated till the right wall yet. However, for the cases of lower frequency, the truncation of fluid domain at the right boundary is necessary, otherwise an extreme long computational domain is required. In ComFLOW, the two kinds of boundaries including Sommerfeld conditions and GABC can be applied to perform the wave outflow boundary. The Sommerfeld boundary so called radiation boundary is a classic example of no-reflecting boundary, that takes its name from the German theoretical physicist Arnold Sommerfeld. It relates the temporal derivative and the

39 3.5 Non-reflecting Boundary Condition 25 normal derivative of the unknown in the case of boundaries far away from sources and normal to the propagating wave. Thus, it is applicable when the sources are concentrated in a region of the space and the exterior boundary is a sphere surrounding with its center at the source region. Additionally, the spherical surface has to be far away from the source, so that one can assume that the impinging waves only have radial component when they reach the artificial boundary. Besides, a generating and absorbing boundary condition (GABC) has been implemented in ComFLOW in order to prevent wave reflection from the domain boundaries. The GABC can be used at the inflow ends of the domain, where waves are generated (incoming waves), and it can be used at outflow ends of the domain, where waves leave the domain (outgoing waves). The GABC can be defined as long as the waves is propagating with angle of incidence of range (0º, 90º). In case of parallel propagation, the GABC cannot be specified on the side walls that are parallel to the propagation direction of the waves. The Sommerfeld boundary and GABC are considered. The computational domain is one wave length long, which means the waves arrive at the right wall at t = T. Besides the enough long computational domain is defined to eradicate the reflection for reference. From Figure 3-10, GABC and Sommerfeld non-reflecting boundary conditions give exactly same results for the first period. Compared to the case of no reflection, GABC and Sommerfeld give different vertical force after the first period of oscillation, which is caused by the reflections of waves from the right. From the results, it can be concluded that the boundary conditions GABC and Sommerfeld do not perform the outflow condition eligibly. Therefore, the best way to deal with the simulation is to try to use a short simulation time with enough long computational domain. Figure 3-10 Vertical force for different outflow boundary conditions right wall

40 Chapter 4 Wave-Frequency Oscillation This chapter represents study on the wedge oscillation with wave frequencies. Here, theories of CFD and radiation BEM are applied to the simulations. The vertical forces acting on the oscillating wedge are analyzed. Also the hydrodynamic coefficients containing added mass and damping from CFD and radiation BEM are compared and the differences are discussed. 4.1 Vertical Force In ComFLOW, the variation of total force with time is obtained by integration of the vertical component of pressure along the wet surface of wedge at each time step. Instead of vertical force, the ratio of vertical force Fz to Archimedes buoyancy B is analyzed and discussed, in which B means the buoyancy regarding to the mean draft D (here 0.19 m) as follow. B = ρg D2 tan (β) (4-1) Here, ρ means water density, g is gravity acceleration, D means mean draft and β represents deadrise angle of wedge. To remove the effect caused by the dimension of wedge on the frequency, frequency coefficient c ω is presented, and the frequency is non-dimensionalized by the following equation: c ω = ω π g (4-2) B l where, ω is frequency in rad/s, g is gravity acceleration, and B l is width of wedge at water level (here, 0.38 m) Amplitude In Figure 4-1, there are maximum and minimum forces extracted from the time series of vertical force for different frequencies and amplitudes. Based on the force amplitudes, we can divide it into two phases. The first phase is for frequency coefficient c ω lower than around 0.7 (there is a slight change of the threshold frequency for different oscillating amplitudes, but the small difference is ignored here), in which the maximum and minimum forces keep almost constant and the force ratio is close to one.

41 4.1 Vertical Force 27 This means in this phase gravity plays a dominant role through the form of buoyancy and the system is dominated by stiffness. In other words, the hydrostatic force acting on the wedge is much more important than hydrodynamic force. There is a very slight and negligible declining in force amplitude due to the added mass force, because in cosine periodic oscillation the directions of motion and acceleration are always opposite so that the increasing added mass force is offsetting the hydrostatic force. In conclusion, the first phase denotes the gravity dominant region, in which the total vertical force is approximately equal to the buoyancy. The second phase is at where frequency coefficient is beyond 0.7. In this phase, the amplitudes of vertical force gradually increase, and actually the increment tends to be proportional to frequency squared. As the frequency increases, the rapidly increasing acceleration makes the added mass force start to become the dominant component. Rather than being recognized as constant all the time in potential theory, added mass is better to be considered as time-dependent with increases in oscillating frequency or amplitude. As we can image, when the wedge is immersed deeper in the water, it can move larger volume of fluid because it has larger project area in horizontal plane. Consequently, the negative force amplitude should be larger than the positive force amplitude, which can be observed from Figure 4-1. To make a conclusion, the rapidly increasing acceleration makes the added mass force increase significantly being proportional to frequency squared, while the relatively larger added mass for wedge immersed deeper results in larger amplitude in negative force than in positive force. To investigate the relationship between vertical force and oscillating amplitude, the force amplitudes are plotted against oscillation amplitude in Figure 4-2, which illustrates clearly that the vertical force is proportional to the oscillation amplitude. Figure 4-1 Maximum, minimum force for oscillating amplitude Za/D of 14%, 10%, 8% and 6% Figure 4-2 Maximum, minimum force for different frequencies

42 28 Chapter 4 Wave-Frequency Oscillation Time Series In this section, the details of time series of force ratios are presented. Figure 4-3 gives the time series of force ratio F Z /B for frequency coefficients from 0.3 to 1.9 with the same oscillation amplitude (Za/D=10%). To make it clearly, a horizontal line with value of 1 is shown in each picture, representing the buoyance to mean draft. Besides, a picture illustrating z, z and z is placed on the top for reference since the force can be expressed as F Z = a z b z c z + B, in which z, z and z are wedge heaving displacement, velocity and acceleration respectively. The wedge starts cosine oscillation from the top position moving downward with the displacement z changing as follows: z = z a cos (ωt) (4-3) here, z a is oscillating amplitude, ω is oscillating frequency and t means time. The second picture shows the time series of vertical force ratio in the first phase or gravity dominant phase (c ω <0.7, mentioned in previous one section). At frequency coefficient of 0.3, the vertical force appears the sinusoidal shape and it is synchronized with z, which suggests one more time that the force is dominant by the restoring force. But as frequency increases, the peak of force ratio moves from half period (t/t=0.5) where hydrostatic force is maximum to around a quarter period (t/t=0.25) where the wedge velocity is maximum with decrease in the value of peak force. It is inferred that the damping of system gradually becomes important, which is proportional to heave velocity. However, the trough of vertical force does not have obvious change. In the third picture, there are time series of vertical force ratios for frequency coefficient from 0.6 to 1.0, in which range the features of the vertical force change considerably. It is clear that the shape of the time series changes significantly. In terms of the peak, the amplitude increases with a small change in phase. An obvious change happens at the trough. First, it switches from t/t=1 to t/t=0.5, which denotes it switches from displacement synchronized to acceleration synchronized owing to the significant increase in acceleration as frequency becomes higher. Therefore, the range can be considered as the transition range in which the force is transferring from displacement synchronized to acceleration synchronized, or in other words from gravity dominant range to added mass dominant range. Time series of vertical force ratios for frequency coefficient from 0.9 to 1.9 are plotted in the bottom picture. The phases for the force histories are not changing anymore. The only change is that the force amplitude increases dramatically. As described in section 4.1.1, the amplitude increases in frequency squared. The reason is that in this range the system the force is purely dominated by added mass. With respect to the shape of the time series of vertical force, it is similar to sinusoidal but with a wide crest and a narrow and deep trough. The asymmetry along horizontal line Fz/B=1 is caused by the change of the body surface along height.

43 4.1 Vertical Force 29 Figure 4-3 Vertical force with non-dimensional time at different frequencies

44 30 Chapter 4 Wave-Frequency Oscillation 4.2 Hydrodynamic Coefficients Regarding to wave-frequency oscillation, potential flow theory with BEM numerical solution nowadays is the most common solution to estimate the excited wave force and radiation force, because of its small computation effort and enough accuracy for small ship motion. However, the CFD method being able to count the viscous effect and influence of free surface elevation is considered to be a more precise method for wedge oscillation problems, especially for high frequency oscillation. The hydrodynamic coefficients of CFD method calculated in ComFLOW and of linear theory obtained by SHIPMO are compared with each other. The details are described and discussed in this section. The dimensionless added mass coefficient is the added mass divided by the displaced fluid mass, while the dimensionless damping coefficient is the damping divided by critical damping. Thus, the non-dimensional hydrodynamic coefficients are calculated as following: Added mass coefficient C a = a M (4-4) Damping coefficient C b = b b cr (4-5) in which, b cr = 2 cm (4-6) here, a is added mass, M is the mass of displaced fluid, b is damping, b cr is the critical D damping, c is stiffness coefficient and equals to 2ρg for heaving of the wedge. tan (β) Fitting Calculation Apart from hydrostatic force, fluid loads acting on the wedge oscillating at free surface also include added mass force and damping force, which are proportional to wedge acceleration and velocity respectively. Therefore, the total vertical force can be expressed as: F Z = a z b z + F hys = a z b z c z + B (4-7) Here, Fz is total vertical force, F hys is hydrostatic force, representing buoyancy on the body, z, z and z are wedge heaving displacement, velocity and acceleration respectively, a is added mass, b is damping and c is stiffness and B is buoyancy to mean dra. The equivalent added mass and damping of ComFLOW results are obtained by fitting equation (4-7) to the time series results of vertical force. In Figure 4-4, the details of the fitting calculation for some frequencies are shown, including the time series of vertical force of ComFLOW, SHIPMO, buoyancy to the undisturbed water level and the fitting curves to the ComFLOW forces. There are four frequencies containing 4 rad/s, 12 rad/s, 20 rad/s and 30 rad/s (c ω are 0.3, 0.8, 1.3 and 1.9 respectively) for the oscillation amplitudes Za/D of 6% and 10%. In the low frequency coefficient of 0.3, ComFLOW forces and SHIPMO forces are close to each other and the total vertical force is synchronous to buoyancy, which represent the gravity plays dominant role. For the frequency coefficient of 0.8, the ComFLOW forces are not in the shape of sinusoid. With respect to the higher frequency coefficients of 1.3 and 1.9, SHIPMO gives underestimated force amplitude than ComFLOW, especially in the force trough.

45 4.2 Hydrodynamic Coefficients 31 Figure 4-4 Time series of vertical force of ComFLOW, fitting curves, SHIPMO and Buoyancy to undisturbed water level for oscillation amplitude Za/D 6% and 10%

46 32 Chapter 4 Wave-Frequency Oscillation Added Mass The physical meaning of the phenomenon of added mass may be explained as that it determines the necessary work done to change the kinetic energy associated with the motion of fluid. In a simple and visual way, it can be modeled as some volume of fluid moving with the body in the fluid. The constant mean added mass with respect to frequency can be related to the mass of the fluid entrapped by the wedge. Any motion of a fluid such as that which occurs when a body moves through fluid implies a certain positive, non-zero amount of kinetic energy associated with the fluid motion. Thus, these inertial force can be simply derived from energy conservation equation (4-8) and transformed to equation (4-9) [25]: F a v = d(1 2 m av 2 ) dt (4-8) F a dv ma (4-9) dt Here, m a is added mass, F a is inertial force, v is velocity of the added mass. The added mass coefficients extracted from ComFLOW and obtained from SHIPMO are shown in Figure 4-5. Frequency coefficients range up to 3 with four levels of oscillating amplitude Za/D of 14%, 10%, 6% and 4% has been studied. The added mass of SHIPMO declines steeply at frequency coefficient lower than 0.5, and after that increases gradually and tends to remain relatively constant with an increase in frequency. In the case of lower frequencies (c ω < 0.5), ComFLOW gives lower added mass than SHIPMO, due to the viscous effect. In SHIPMO, the fluid is assumed to be free of viscous effect i.e., both from solid to fluid friction and fluid to fluid friction as well. The ignorance of friction enables the fluid move with solid body without dissipating any kinetic energy, and hence the added mass at low frequency region shows an infinite growth as frequency approaching to zero. As a consequence, the CFD method based on the viscous fluid gives lower added mass than the potential fluid in lower frequencies. When c ω >= 0.5, ComFLOW starts to give higher added mass than SHIPMO. The larger the frequency or oscillating amplitude is, the larger the difference. In particular, it can be observed that the difference is more sensitive on oscillating amplitude than frequency, since the difference remains approximately constant when frequency coefficient is beyond a certain value. The reason of difference here is the non-linear effect caused by the free surface elevation. In SHPMO, the wet surface on the body and the free surface are linearized to the undisturbed water level. The pile-up of water and the increasing width with height of a wedge included in ComFLOW enlarge the projected area on the horizontal plane, therefore leading to the increment in added mass Damping Figure 4-6 illustrates damping coefficients obtained by ComFLOW and SHIPMO. A frequency coefficient range up to 3 with four levels of oscillating amplitude Za/D of 14%, 10%, 6% and 4% have been considered. It is needed to point out that the damping values of ComFLOW for frequency coefficients higher than around 0.6 are highly sensitive to the duration of simulation, since the reacting force is not strictly periodic because of the memory effect. Here the ComFLOW hydrodynamic coefficients are extracted from time series of one period long.

47 4.2 Hydrodynamic Coefficients 33 From Figure 4-6, it is observed that in SHIPMO damping increases first and starts to decline at around c ω = 0.3. The results of ComFLOW and linear theory show a slight difference at the lower frequencies (c ω ). In the following phase, ComFLOW dampings have a good agreement with SHIPMO damping in a certain range with frequency coefficient less than a threshold value, but after which ComFLOW damping and SHIPMO damping appear different trend, that ComFLOW damping is increasing quickly while SHIPMO damping is declining to zero as frequency increases. The threshold value of frequency coefficient is different for different oscillating amplitude and a discussion on the determination of this threshold value can be found in next section An explanation is given here to illustrate the difference in the high frequency range. Damping will lead to the dissipation of energy, representing the energy transmitted into water by the oscillating wedge. It is inferred that the causes of damping may mainly contain wave-making, viscous effect and wave-breaking. Viscous component only counts in very low frequency and can be ignored in this case. In linear potential theory, the radiation waves generated by the oscillation at high frequency are short, with small height and with less energy, which may explain why the linear theory gives zero-approaching damping as frequency increases, but this is not the real case. In reality, the waves not being able to be transported away will lead to water pile-up or even a jet. Consequently, wave-breaking damping becomes an important component in high oscillation frequencies. Figure 4-5 Added mass coefficients at different frequencies with amplitude Za/D of 14 %, 10%, 6% and 4%

48 34 Chapter 4 Wave-Frequency Oscillation Figure 4-6 Damping coefficients at different frequencies with amplitude Za/D of 14 %, 10%, 6% and 4% Discussion The differences between hydrodynamic coefficients of ComFLOW and SHIPMO mainly come from the difference of basic principles in these two numerical methods. In the linear potential theory, the boundary conditions of wet surface on the body and free surface are linearized to the undisturbed water level. As demonstrated in Figure 4-7, a wedge is moving downward. At instant t 0, the immersion depth of wedge equals to d 0, and the immersion depth d 1 at instant t 1 equals to the sum of the initial immersion depth d 0, the depth caused by water elevation d water and the depth by the displacement of the wedge d disp. Thus, to adapt the validity of linear theory, immersion depth of d water and d disp should be kept small enough. First, smaller motions or oscillating amplitude should be guaranteed to limit the immersion depth d disp. However, even for a small oscillating amplitude, the non-linear effect tends to be important as oscillating frequency increases due to the increasing immersion depth d water. In particular, when the wedge is moving downward, the wedge surface is pushing the water away. As the oscillating frequency increases, the propagating speed is reducing according to the dispersion relationship in deep water shown in Equation (4-10), but the moving velocity of wedge is increasing as shown in Equation (4-11). Therefore, there must be a threshold value, when oscillating frequency is higher than which, the radiation wave propagating speed will become lower than horizontal moving velocity of the wedge surface. This means that the volume of the water in where has been occupied by the wedge is not able to transmit away timely, and thus this volume of water has nowhere to go except flowing into the air, generating pile-up or a jet as shown in Figure 4-8. This occurred high free surface elevation makes non-linear effect to become significant. Therefore, second step is to restrict the oscillating frequency to make the linear method applicable.

49 4.2 Hydrodynamic Coefficients 35 Figure 4-7 Illustration of immersion depths According to dispersion relationship, wave propagating speed in deep water can be expressed as: c = g k = g ω (4-10) Here, c is wave propagating speed, k is wave number. For a cosine oscillation of motion, the horizontal velocity is: V h = z a ω sin (ω t)/tan (β) (4-11) Here, V h is horizontal velocity of the wedge wet surface, z a is oscillating amplitude, ω is oscillating frequency, t is time, β is deadrise angle of the wedge. Figure 4-8 Illustration of pile-up and jet generation Base on the above analysis, the horizontal velocity and wave speed for different oscillating amplitudes for a wedge with deadrise angle of 45º are plotted in Figure 4-9. Besides, if the maximum horizontal moving velocity of the wedge is assumed to equal to the wave propagating speed, the threshold frequency can also be obtained through: g tan (β) ω = Za (4-12) here, g is gravity acceleration, β is deadrise angle of the wedge, and Za is oscillating

50 36 Chapter 4 Wave-Frequency Oscillation amplitude. The threshold frequency coefficients from this method are given in Table 4-1, associated with the frequencies obtained from the comparison of dampings from ComFLOW and SHIPMO. This assumption is justified by the good agreement between these two methods. Concerning the free surface profile, take Za/D of 14% for example, when frequency coefficient is 1.2, the maximum horizontal velocity of wedge equals to wave speed, which means jets are formed for c ω > 1.2. As shown in Figure 4-10, the free surface profiles at moment t=t/2 when wedge is moving downward reaching the lowest position are given for frequency 10 rad/s, 16 rad/s, 20 rad/s and 22 rad/s (c ω 0.6, 1.0, 1.3 and 1.9 respectively) and the free surface elevation increases as increase in frequency. In picture c ω =0.6, the wave is propagating away without generation of jet. The jets are generated at c ω =1.0 and 1.3 and become quite apparent at c ω =1.9. This conclusion basically matches the outcomes from the comparison of added mass and damping. Figure 4-9 Horizontal velocity vs wave speed at Za/D of 14%, 10%, 6% and 4% Table 4-1 Threshold frequency coefficients from method of comparison between SHIPMO and ComFLOW and method of V h = c Frequency coefficient Difference Za/D SHIPMO vs ComFLOW V h = c [%] 4% % % %

51 4.2 Hydrodynamic Coefficients 37 ω=10 rad/s c ω =0.6 t=1/2t ω=16rad/s c ω =1.0 t=1/2t ω=20 rad/s c ω =1.3 t=1/2t ω=30 rad/s c ω =1.9 t=1/2t Figure 4-10 Free surface elevation for frequencies 10 rad/s, 16 rad/s, 20 rad/s and 22 rad/s for oscillating amplitude Za/D of 14%

52 Chapter 5 Slamming-Frequency Oscillation In this chapter, the high frequency oscillation relating to slamming problem is studied. ComFLOW, 2DBEM and momentum method are applied to study this problem and their results are compared to each other. The limits of the application of each method are given based on the study of gravity effect D BEM In 2D BEM, elements on the wet surface of body and on the free surface are applied to calculate the velocity potential numerically. Convergence study on element size is carried out, in which three levels of element size in Table 5-1 are considered and their results are shown in Figure 5-2. Mesh numbers are given separately on the body and on the free surface. The horizontal size of computational domain is defined as a factor of the instantaneous draft. The scaling factor denotes the ratio between sizes of neighboring elements on the body or on the free surface. The smallest element locates closest to the free-surface-body intersection. The elements for condition No.2 at different time are shown in Figure 5-1. In order to show the change of the free surface more clearly, the body is plotted fixed in the pictures. In the wedge-moving-downward phase, pile-up phenomenon occurs and the free surface profile goes up. In the wedge-moving-upward phase, the free surface is going down but with the intersection point still moving up, which obviously is not correct. This implies that the internal-defined estimation method of intersection point at next time step is not suitable for the body-moving-upward case. In Figure 5-2, the coarse-element condition No.1 gives a rough result especially in the initial phase, when the wedge velocity is zero. In 2DBEM, the time increment is automatically determined by internal-defined CFL number near the intersection point. Consequently, the extremely low velocity at beginning of cosine oscillation leads to a big time increment. With respect to the fine-element condition No.3, the resulting force curve at the initial phase is quite smooth, but has a big deviation in the later phase at time t=0.055s, owning to the error accumulation for the very small time increments. Nevertheless, they all lead to quite close curves of vertical forces.

53 Veritcal Force [kn] 5.1 2D BEM 39 Condition No. Table 5-1 Element and domain definition for ω = 100 rad/s Mesh number Domain Scaling factor body free surface factor body free surface Figure 5-1 Elements on body and free surface for different times for condition No Frequency 100 rad/s Time [s] No. 1 No. 2 No. 3 Figure 5-2 Time series of vertical force for different element sizes for ω = 100 rad/s

54 40 Chapter 5 Slamming-Frequency Oscillation Base on the convergence study, some tips observed are mentioned here when modelling the sinusoidal oscillation or water entry case in 2DBEM. - 2DBEM is not suitable for the body-moving-up cases or water-existing cases. - In the case of high frequency oscillation, some forces have the negative value. The default solution limit of minimum pressure is zero and should be extended to a negative limit, for instance, default solution limit is adjusted from [0, 10e99] to [ 10e99, 10e99]. - The results of oscillation case are sensitive to the element size, especially the element on the free surface near the free-surface-body intersection. - Zero velocity at initial time should be given as a very small value, otherwise 2DBEM is not able to compute a CFL number. - For water entry problem, zero immersion depth at initial time should be given as a very small value as well, since the horizontal computational domain of free surface is defined based on the instantaneous draft. 5.2 Water Entry with Constant Velocity The water entry of wedge-shaped object has been extensively investigated for the purpose of studying the impact or planing problem of a ship. To compare the results of 2DBEM and ComFLOW, the water entry cases with constant velocity are simulated. A wedge entering water with constant velocity (0.2 m/s, 0.5m/s, 1m/s and 2m/s) have been studied and their results are shown in Figure 5-3, with x-axis of non-dimensional immersion depth d = vt, where v is entering velocity and D means the height of chine (0.3m). The D simulation starts from the first touch of the free surface to the moment of d = 1. The simulation in 2DBEM stops when the chine gets wet. It can be seen that in all the pictures 2DBEM has the time series of vertical force proportional to the immersion depth, whereas the slope of ComFLOW increases with the increase in immersion depth, which is more apparent for smaller entering velocity. This is due to the hydrostatic force being proportional to the square of immersion depth, but the hydrostatic force is ignored in 2DBEM. Denoting the sum of 2DBEM and hydrostatic force computed based on the immersion depth to undisturbed water level, the item 2DBEM+Hys has a negligible discrepancy with force history of ComFLOW in velocities of 0.2 m/s, 0.5m/s. However, as entering velocity increases, hydrostatic force becomes relatively small and ignorable and 2DBEM tends to give higher estimation than ComFLOW. This difference is caused by the approximation of free surface boundary condition in 2DBEM, in which the velocity potential is assumed to be zero at the horizontal line across the intersection between free surface and body, therefore the difference between real free surface and the assumed free surface increases as increase in entering velocity. Some snapshots of free surface profile and pressure distribution in ComFLOW for these four entering velocities can be found in Figure 5-4. The left column illustrates the moment of non-dimensional draft d = vt = 1. In the right column, there are snapshots at the moment D 2 when wedge chine gets wet. From the snapshots for entering velocity of 0.2 m/s, it is hard to see wave generated. Water pile-up happens when entering velocity is 0.5 m/s. As for entering velocities of 1 m/s and 2 m/s, there is occurrence of jet and wave-breaking. Water reaches

55 5.2 Water Entry with Constant Velocity 41 chines of the wedge at 0.198s and 0.09s respectively, which represents the wetting factors WF (defined as the ratio of the wetted width of the section to its width at the undisturbed water surface) are about 1.52 and 1.66 for entering velocities of 1 m/s and 2 m/s respectively. Figure 5-3 Time series of vertical force for water entry wedge with different entering velocity

56 42 Chapter 5 Slamming-Frequency Oscillation d = 1/2 Chine get wet V= 0.2m/s V= 0.5m/s V= 1m/s V= 2m/s Figure 5-4 Snaptshots of ComFLOW for different entering velocities

57 5.3 Oscillation Oscillation Comparison between ComFLOW, 2DBEM and Momentum Method In this section, the cosine oscillation of first period in which a wedge starts from the top position with zero velocity is considered. They are studied through ComFLOW, 2DBEM and momentum method. In ComFLOW, the simulations with and without the consideration of gravity are studied in order to consider the gravity effect. From section 4.2.2, the added masse coefficient calculated in both SHIPMO and M ComFLOW tend to be around 1.1 for high frequencies, which equals to the so called non-dimensional added mass m zz ρy2 in momentum method for deadrise angle of 45º. As shown in Figure 5-5, this non-dimensional added mass is close to the value given by the equation obtained by Lewis and Taylow separately, Therefore, in this section Equation (2-31) is applied for estimation of added mass: a Deadrise angle [º] SHIPMO von Kármán Wagner Wagner-Sydow Lewis, Taylow Figure 5-5 Non-dimensional added mass of different estimation methods and SHIPMO Five oscillation amplitudes with Za/D of 2%, 4%, 6%, 8% and 10% are studied and the corresponding vertical force histories for some frequencies are plotted in Figures from Figure 5-6 to Figure The ComFLOW force without gravity gives low results than force with gravity, and as frequency increases the difference which represents the gravity force tends to become smaller. It can be observed that the difference has its maximum value at the trough of the vertical force, which denotes the moment when the wedge arrives at the lowest position. Due to the ignorance of gravity effect, the forces of 2DBEM are close to ComFLOW force without gravity. It is clear that the force amplitude of 2DBEM is slightly higher than ComFLOW results, but this overestimation becomes small as oscillation amplitude decreases. In 2DBEM, the time increment is automatically determined by internal-defined CFL number near the intersection point. Consequently, the extremely low velocity near time t=0 and t=t/2 leads to a relative big time increment and this results in the relative sick agreement at these

58 44 Chapter 5 Slamming-Frequency Oscillation moments. For each frequency, the moving downward phase has a better match between 2DBEM and ComFLOW than the moving upward phase, which is more obvious for c ω =2.5 and 3.1 for Za/D=10% in Figure It may be concluded that 2DBEM is more suitable for the case of wedge moving downward (or water entry problem) than the case of wedge moving upward (or water exit problem). The reason for this is inferred to be the improper estimation method of the intersection point for the body-moving-upward phase, which has been described in section 5.1. Regarding to momentum method, it has a good match with ComFLOW force with consideration of gravity, especially during the first force crest. But this agreement becomes slightly weak in the following force trough and crest because of the memory effect which is ignored in momentum method but is included in the other two numerical methods. The memory effect here means the effect caused by the disturbed free surface. The minimum forces of ComFLOW with gravity and momentum method are summarized in Table 5-2. The force amplitude is calculated by subtracting the buoyancy to the mean water level from the minimum force, as shown in Figure It is observed that the ratios of the force amplitude of momentum method to the force amplitude of ComFLOW Fa.m keep almost fixed for the same Fa.c oscillation amplitude for different frequencies and it increases as decrease in oscillation amplitude as plotted in Figure 5-12, which means a better match in a smaller oscillation amplitude. Besides, extending the curve to Za/D of zero, the ratio Fa.m tends to be 1. From the Fa.c comparison, it can be suggested that the momentum method with added mass equation given by Lewis and Taylow is able to give proper initial estimation of the vertical force for a wedge with deadrise angle of 45º oscillating at free surface.

59 5.3 Oscillation 45 Figure 5-6 Time series of vertical force for oscillation amplitude Za/D 2% Figure 5-7 Time series of vertical force for oscillation amplitude Za/D 4%

60 46 Chapter 5 Slamming-Frequency Oscillation Figure 5-8 Time series of vertical force for oscillation amplitude Za/D 6% Figure 5-9 Time series of vertical force for oscillation amplitude Za/D 8%

61 5.3 Oscillation 47 Figure 5-10 Time series of vertical force for oscillation amplitude Za/D 10% Figure 5-11 Description of force amplitude of momentum method Fa.m and of ComFLOW Fa.c

62 48 Chapter 5 Slamming-Frequency Oscillation Table 5-2 Force amplitude of ComFLOW and momentum method Frequency Fmin Force Amplitude Fa Ratio [rad/s] ComFLOW Momentum ComFLOW Momentum Za/D 2% Fa.m/Fa.c Za/D 4% Za/D 6% Za/D 8% Za/D 10% Figure 5-12 Ratios of Fa.m Fa.c against oscillating amplitude Za/D

63 5.3 Oscillation Pressure Distribution Some snapshots of free surface profile and pressure in ComFLOW for oscillating frequency coefficients c ω of 3.1 and 6.3 (frequencies of 50 rad/s and 100 rad/s) for Za/D 10% are shown in Figure The four pictures in each frequency illustrate the moments of 0, ¼, ½ and ¾ of one oscillating period separately. The pressure of c ω 3.1 is much lower than the pressure of c ω 6.3, in which the hydrostatic pressure is negligible. At the moment of half period when the wedge is at the lowest position with zero velocity and with maximum acceleration upward, it is seen that the pressure is negative and has it maximum value near the apex of wedge. For both of the frequencies, the wave-breaking appears in the last picture Gravity Effect From the previous section, it can be observed that the 2DBEM gives a good match with ComFLOW. In order to determine the limits of which the gravity effect can be ignored, the gravity forces at the moment of the half period t/t=0.5 when the gravity force has its maximum value are extracted. The difference between force with gravity and force without gravity represents gravity force F g, and the force amplitude Fa is obtained by subtracting the buoyancy to the mean water level from the minimum ComFLOW force. The ratios F g /F a of the maximum gravity force to the vertical force amplitude are calculated for each oscillating amplitude and are plotted in Figure If the criterion is assumed to be 0.1, which means when the gravity force F g is less than 10% of total force amplitude Fa then the gravity effect can be considered as negligible, the threshold frequency coefficients c ω are summarized in Table 5-3 and Figure Moreover, criterion of 20% is also considered.

64 50 Chapter 5 Slamming-Frequency Oscillation c ω =3.1 c ω =6.3 0 ¼T ½T ¾T Figure 5-13 Snapshots of pressure field for c ω =3.1 and 6.3 at t=0t, ¼T, ½T and ¾T

65 5.3 Oscillation 51 Figure 5-14 Description of gravity force Fg and force amplitude Fa Figure 5-15 Ratios of gravity force to force amplitude Fg/Fa for Za/D 2%, 4%, 6%, 8% and 10% Table 5-3 Threshold frequency coefficients for gravity negligible conditions Frequency coefficient c ω Za/D Criteria 10% Criteria 20% 2% % % % %

66 52 Chapter 5 Slamming-Frequency Oscillation Figure 5-16 Frequency coefficient limits of gravity negligible condition

67 Chapter 6 Oscillation with Frequency in Transition The previous two chapters studied the vertical force on the wedge oscillating at the wave frequencies and slamming frequencies, and found the limits of the two extreme conditions in which the gravity effect are dominant and negligible respectively and the application limits of linear radiation theory and impact theory. This chapter concerns the oscillation at the frequencies in between those two limits, aiming to specify the gravity effect affecting on the wedge. 6.1 Vertical Force The vertical forces acting on the oscillating wedge vary significantly with frequency in the transition range. In order to consider the gravity contribution to the total force, the cases with full gravity acceleration (g=9.81m/s), with partial g and without g have been simulated in ComFLOW. The frequencies with c ω from 1.3 to 6.3 are studied and there are three levels of oscillating amplitudes applied involving Za/D 10%, 8% and 6%. The corresponding time series of vertical force obtained are plotted in Figure 6-1 to Figure 6-3. At each picture, it can be observed that as the gravity acceleration increases, the vertical force increases, showing more clearly in lower c ω. This trend represents that the gravity gives a positive contribution to the total vertical force acting on the wedge, mainly owing to the component of buoyancy. A higher c ω shows a smaller relative difference between different gravity accelerations. In particular, significant difference can be observed in the pictures of lower frequencies (c ω =1.3 and 1.9), but when c ω >3.8 the forces with different ratios of gravity acceleration are quite close to each other. Moreover, for the same c ω more obvious effect exists in smaller oscillating amplitude. These variations of gravity effect are mainly due to the change of total force. The gravity effect on the vertical force will be discussed specifically later in Section 6.4. The maximum and minimum forces of the full gravity simulation are extracted from the time series data shown in Figure 6-1 to Figure 6-3 and are plotted in Figure 6-4. By fitting the data for each case, the black curves are obtained, and clearly the maximum and minimum forces increase as a second power of frequency. This is due to the dominance of added mass force, which is controlled by the acceleration and it is proportional to frequency squared in a cosine oscillation z = z a cos (ωt) as represented in Equation (6-1). But the fitting curves do not fit the data quite well during the lower frequencies, e.g. from c ω 1.3 to 2.5, since the inertia force is not absolutely dominant yet while the damping force and restoring force still have certain influence. Moreover, the amplitude of negative force is higher than the positive force. z = z a ω 2 cos (ωt) (6-1)

68 54 Chapter 6 Oscillation with Frequency in Transition Figure 6-1 Force ratio Fz/B history for difference gravity contributions (with 0, ¼, ½, ¾ and 1 gravity acceleration) for Za/D=10%

69 6.1 Vertical Force 55 Figure 6-2 Force ratio Fz/B history for difference gravity contributions (with 0, ¼, ½, ¾ and 1 gravity acceleration) for Za/D=8%

70 56 Chapter 6 Oscillation with Frequency in Transition Figure 6-3 Force ratio Fz/B history for difference gravity contributions (with 0, ¼, ½, ¾ and 1 gravity acceleration) for Za/D=6%

71 6.2 Pressure 57 Figure 6-4 Maximum and minimum forces for Za/D10% 8% and 6% and the fitting curves 6.2 Pressure Pressure distributions along the wedge surface at the instants when the maximum or minimum force happens are studied. In Figure 6-5, pressure distributions obtained with full-gravity simulation for oscillating amplitude Za/D 10% are plotted against non-dimensional horizontal position on the wedge surface y/y, here y is horizontal coordination and Y is the half width of wedge. To excluding the influence of frequency and the dimension of the wedge, the pressures p are made non-dimensional on 1 π 2 ρω2 Y 2, which is shown as follow: c p = p 1 π 2ρω2 Y 2 (6-2) In Figure 6-5, the zero pressure near y/y= 1 represents the part the wedge surface out of water or with a jet. About the series denoting the instant when the maximum force occures, the hydrostatic pressure plays an dominant role at the lower frequencies such as c ω =1.3 and 1.5, therefore the maximum pressure locates at the apex no mater at instant t = 1 2 T or at t = T. However as frequency increases, the dynamic pressure caused by the motion and acceleration of the water particals becomes important and lead to an approximately constant distribution of pressure from the apex to the around y/y=0.4. With respect to the pressure distribution at moment of minimum force, for the lower frequency the pressure near the apex is positive due to the hydrostatic pressure, then as y/y increases the reducing water depth results in negative pressure. In high frequencies with c ω =3.8, 5.0 and 6.3, their pressure coeffienct distribtuions are close to each other and tend to be invariable. Besides, contours of pressure in the fluid field at moments when minimum force and maximum force occur respectively for frequencies with c ω =1.9, 2.5, 3.8 and 5.0 can be found in Figure 6-6.

72 58 Chapter 6 Oscillation with Frequency in Transition Figure 6-5 Pressure distribution along wedge surface 6.3 Free Surface and Velocity Field In the intermediate frequencies, the profile of free surface has a great change with variation in frequency. The large free surface elevation leads the non-linear effect to become significant, and the breaking waves make the problem more complicated. In Figure 6-7 the profile of free surface and the corresponding velocity field of frequencies with c ω =1.3 and 6.3 are plotted at every quarter in one period for oscillating amplitude Za/D of 10%. During the first half period when the wedge is moving downward and pushing the water away, the arrows representing the velocity of water particle show that the water moves from near the wedge up into the air, generating pile-up and jet. Due to the gravity effect, the pile-up or jet falls down, causing flow separation from wedge surface which can be observed at the instant of t = 1 2 T. During the latter half period, the wedge is forced to move from the lowest position up to the highest position. With respect to lower frequency, gravity and the volume of space in fluid released by the up-moving wedge enable the water particles to move down. Inertia acquired during the falling movement causes the water to penetrate below its level of equilibrium, resulting in a sink of free surface near the wedge surface volume which can be observed in picture of c ω =1.3 at t=3/4 T and T. But for the high frequency with c ω =6.3, the high momentum in the jet enables it to continue to flow up into the air. When the frequency is high enough, the wedge-moving-up phase is too short that the water particle does not have enough time to fall down beneath the mean water level before the wedge starts to move downward again. From the view of wave propagation, it can also be understood as that the wave crest is not able to be transferred away timely. Therefore, in these cases, pile-up occurs all the time which is observed in picture of c ω =6.3.

73 6.3 Free Surface and Velocity Field 59 t = 1/2 T t = T c ω 1.9 c ω 2.5 c ω 3.8 c ω 5.0 Figure 6-6 Pressure distribution at moment of minimum vertical force for different frequencies

74 60 Chapter 6 Oscillation with Frequency in Transition c ω =1.3 c ω =6.3 t=1/4 T t=1/2 T t=3/4 T t=t Figure 6-7 Free surface profiles and velocity fields during the first period of frequencies with c ω =1.3 and 6.3

75 6.4 Gravity Effect Analysis Gravity Effect Analysis Gravity Force The effect of gravity on the wedge oscillating with frequency in the transition range has been established based on the data from ComFLOW simulations shown in Figure 6-1 to Figure 6-3. The full effect of gravity on total fluid force, referred to as gravity force component, for each case has been obtained by subtracting the corresponding zero gravity simulation force from the gravity simulation force. Based on the simulation results presented in Section 6.1, the time series of gravity force are plotted in Figure 6-8 for different frequencies against a non-dimensional time t/t. Besides, the instantaneous buoyancy B t on the wedge to the undisturbed water level is also plotted as the black solid line with name static. Figures from Figure 6-1 to Figure 6-3 show some spikes on vertical force history during the beginning phase, which leads to the high scattered data in gravity forces as plotted in Figure 6-8. In ComFLOW when the solid grid is moving into a fluid grid, the change of geometry apertures and labels will result in numerical pressure spikes and the incompressibility of the fluid forces a pressure pulse to travel over the whole domain in infinitely small time, consequently reflecting on the force history. A smaller size of grid will lead to severer spike problem. The higher spikes in cases of higher frequency are because of the smaller grids which are defined to capture the complex free surface in high-frequency oscillations. These details of spikes in ComFLOW are discussed by Kleefsman in [26]. On the other side, since the total vertical force in high frequency is extremely high, small numerical errors will result in large differences in the gravity force, although the results are convergence as shown in Figure 6-1 to Figure 6-3. Nevertheless it can be found that the highly scattered data still has the same trend as the instantaneous buoyancy B t, but they are higher than the buoyancy in the beginning phase and lower than it in the latter phase. The maximum and minimum gravity forces during the first oscillating period are considered specially obtained by filtering the high frequency noises. It is obvious that the maximum gravity force happens near t/t=0.5 denoting the moment when wedge arrives at the lowest position, while the minimum gravity force occurs at 1 when wedge locates at the highest position. By dividing the gravity forces by the corresponding buoyancy B t, the ratios F g /B t referred to as gravity ratios are obtained and Figure 6-9 is given to illustrate the gravity force F g and instantaneous buoyancy B t. The gravity ratios F g /B t at two instants involving t/t=0.5 and 1 are considered and are plotted in Figure Clearly, all the ratios at t/t=0.5 are higher than one while all the ratios at t/t=1 are less than one, but as oscillation amplitude decreases all the ratios are approaching to one. For the cases of t/t=0.5, the ratios decreases quickly with increasing frequency coefficient c ω less than 2, but beyond that the ratios seem to keep a constant value (close to 1). With respect to the ratios of t/t=1, they are not as regular as ratios of t/t=0.5 and highly scattered values appear especially at high frequencies, but they have the trend of diminishing with increasing frequency.

76 62 Chapter 6 Oscillation with Frequency in Transition Figure 6-8 Time series of gravity force component at different oscillation amplitudes Figure 6-9 Illustration of gravity force F g and instantaneous buoyancy B t

77 6.4 Gravity Effect Analysis 63 Figure 6-10 Ratio of gravity force to the buoyancy to the undisturbed water level A discussion is given that tries to explain the nature of the trends that are observed. In addition to causing a hydrostatic force which consists of the buoyancy to undisturbed water level B and the buoyancy caused by the free surface elevation B disturb, gravity also affects the flow and causes the hydrodynamic force contribution in gravity environment to differ from that in a non-gravity environment, and this effect is referred to as the dynamic gravity effect. The dynamic gravity effect can be explained physically as the force necessary to create the potential energy in the water pile up. Therefore, the reasons making the ratios unequal to 1 involve the hydrostatic force regarding to the free surface elevation B disturb and the dynamic gravity force. At instant of t/t=0.5, the wedge moves downward reaching at the lowest position with maximum acceleration upward. It is suggested that the pile-up of water that occurs during a water entry causes the hydrostatic pressure head to be greater than in corresponding static condition, and that the increased pressure head gives a gravity force that is greater than the buoyancy to the undisturbed water level B. In a higher frequency, a jet may happen, leading to a much high free surface elevation, but even so the results show a lower ratio compared to ratios in lower frequency. This phenomenon may be explained from three terms. Firstly, according to the boundary condition in which pressure at free surface equals to atmospheric pressure, the pressure in the quite thin layer of jet flow is suggested to be close to atmospheric pressure, and hence it is so small that it does not have noticeable contribution to the gravity force component. Furthermore, the flow separation happens and the water flow flows off the wedge surface leaving the wedge section sides dry, which implies that hydrostatic pressure head down to the atmospheric pressure as well. Lastly, in terms of dynamic gravity force, to pile up the water the upward force acting on the fluid is required, therefore, according to Newton's third law the dynamic gravity force acting on the wedge points downward which has a negative contribution to the total gravity force. Or in other words, the kinetic energy of the moving wedge is transferred to the potential energy in the water. Compared to lower pile-up, pushing a large amount of fluid up into high location requires more dynamic gravity force acting on the fluid to create the potential energy in the water pile up and jet, which means more negative contribution to total force in higher frequency oscillation.

78 64 Chapter 6 Oscillation with Frequency in Transition With respect to the instant of t/t=1, the sink of free surface in lower frequency explained in section 6.3 leads the buoyancy ratio to be less than one. In higher frequencies, pile up still happens when the wedge is moving up (explained in section 6.3), giving a higher hydrostatic gravity force. But the significant dynamic gravity force that gives negative contribution to the total gravity force makes the buoyancy ratio to be less than one. Actually to investigate the gravity effect, only the maximum gravity force during the transition range is of interest, which are extracted and plotted in Figure Clearly, for all oscillating amplitudes, first gravity ratios decrease gradually and remain approximately constant for c ω > 2. Figure 6-11 Ratio of maximum gravity force to the buoyancy to the undisturbed water level Effect of Gravity Acceleration Gravity affects the fluid force acting on the wedge in the form of gravity acceleration. Normally, the gravity acceleration g is 9.81 m/s 2, but in order to consider the relationship between gravity force and gravity acceleration, partial gravity accelerations of 0 g, ¼ g, ½ g, ¾ g and g are applied for the simulation and the results of vertical force are plotted in Figure 6-1 to Figure 6-3. In this section, the corresponding gravity force at instants of t/t=0.5 and 1 are plotted against the ratios of applied gravity acceleration g to the full gravity acceleration g in Figure According to Newton s Law F = ma, the gradient of each curve can be considered as representation of added mass. Obviously, the gravity force tends to be proportional to gravity acceleration, and have a higher gradient at t/t=0.5. With respect to the oscillation frequency, the low frequency gives a relatively high gradient. For instance, for both maximum and minimum cases frequency with c ω =1.3 gives the largest gradient while frequency with c ω =6.3 gives smallest gradient.

79 6.4 Gravity Effect Analysis 65 Figure 6-12 Gravity force against different gravity acceleration Ratio of Gravity Force to Total Force In addition to the absolute effect of gravity, the relative effect of gravity is of interest as well, which is studied by the ratios of gravity force to the total vertical force amplitude Fg/Fa as described in Section To present the results clearly, the corresponding ratios are plotted in logarithmic against frequency coefficient as shown in Figure It is already known that gravity force has a decline for c ω <2 and keeps constant for c ω >2, while the amplitude of total vertical force increases increase as a second power of frequency. Therefore, it can be inferred that ratio Fg/Fa declines as a -3 rd power of c ω in the range of c ω <2 and as a -2 nd power of c ω for c ω >2. Figure 6-13 Percentage of gravity force