Hydrodynamics of Unconventional SWATH Vessels in Waves

Transcription

1 Hydrodynamics of Unconventional SWATH Vessels in Waves by Abiodun Timothy Olaoye MASSACHUSETTS INSTrTuTE OF TECHNOLOLGY JUL B.S., Petroleum & Gas Engineering University of Lagos (2010) LIBRARIES Submitted to the Department of Mechanical Engineering in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering at the MASSACHUSETTS INSTITUTE OF TECHNOLOGY June Massachusetts Institute of Technology All rights reserved. Signature redacted A u th o r., Abiodun Timothy Olaoye Department of Mechanical Engineering May 7, 2015 Signature redacted Certified by... Stefano Brizzolara Research Scientist and Lecturer Assista irector for Research at MIT Sea Grant 4 T0hesis Supervisor Signature redacted A ccepted by David E. Hardt Chairman, Committee on Graduate Students Department of Mechanical Engineering

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3 Hydrodynamics of Unconventional SWATH Vessels in Waves by Abiodun Timothy Olaoye Submitted to the Department of Mechanical Engineering on May 7, 2015, in partial fulfillment of the requirements for the degree of Master of Science in Mechanical Engineering Abstract The motion responses of unconventional Small Water-plane Area Twin Hull (SWATH) vessels are unique in the sense that viscosity has significant non-linear effects on their hydrodynamic parameters. The parametric optimization of the hull shape of these vessels to reduce the total resistance in waves yields an interesting hull form where viscous effects become significant and this kind of problem is generally more difficult to solve. This study aims to investigate the motion response of these special kind of ships in waves using both numerical and experimental approach with some theoretical simplifications to better understand the hydrodynamics of the ship. The two modes of motion of interest in this study are heave and pitch motions which were chosen in order to focus on the degrees of freedom which significantly contributes to the resistance of the ship in head waves. The vessel under investigation is an unmanned surface vessel (USV) proposed to be used to monitor a team of autonomous underwater vehicles. A scaled version of this model is built and some experiments were conducted at the MIT towing tank at zero speed. Additionally, the numerical methods are implemented using 2D and 3D potential flow solvers. As this is an ongoing project, the results obtained so far including the study of the effects of the inertial characteristics of the ship on the response amplitude operator (RAO) are presented. Thesis Supervisor: Stefano Brizzolara Title: Research Scientist and Lecturer Assistant Director for Research at MIT Sea Grant 2

4 Acknowledgement To almighty God be all the glory for the good health and sound mind that sustained me through this Masters program. With a feeling of accomplishment, I would like to express my sincere gratitude and appreciation to the Federal government of Nigeria and particularly to the President and Commander-in-Chief of the Nigerian Armed forces, President Goodluck Ebele Jonathan (GCFR) for initiating the Presidential Scholarship Scheme for Innovation and Development (PRESSID) under which I was fully funded to pursue a Master of Science degree in Mechanical Engineering at MIT. I would also like to thank the PRESSID implementation committee at the National Universities Commission, Abuja for every support that I have received through out this Masters program. Additionally, this thesis was actualized with the academic support provided by my Advisor, Professor Stefano -Brizzolara and other members of the MIT i-ship group including Luca Bonfigilo and Guliano Vernengo. The experimental part of the thesis carried out at the MIT towing tank was aided by the following research students: Jacob Israelevitz, Jeff Dusek, Amy Gao, and Stephanie Steele. Also, I would like to thank Giovani Diniz, and Heather Beem for their friendliness which really helped me through out the program. Finally, I would like to thank my family especially my parents, Mr. Joel Adebisi Olaoye and Mrs. Oluyemisi Beatrice Olaoye for their moral support in the course of this program. 3

5 Contents Title Page 1 Abstract 2 Acknowledgement 3 Table of Contents 4 List of Figures 6 List of Tables 7 List of Symbols 8 1 Introduction Literature Review Thesis Objectives Theoretical study of Motion of a Floating Body in Waves Theory of Ship Motions in Regular Waves Ship-Motion Strip Theory Equations of Motion of a SWATH Radiation and Diffraction Problems of SWATH Vessels in Regular W aves Wave Excitation Forces and Moments on SWATH Ships: Strip Theory M ethod Second-order Non-linear Forces Acting on Floating Bodies in Waves. 30 4

6 2.2.1 Added Resistance of Ships in Waves Salvesen's Method for Predicting Added resistance in Waves Other Methods of Predicting Added Resistance of a Ship in Waves Comparison between the Integrated Pressure Method and Salvesen's M ethod Results of Added Resistance Computation for a Series 60 hull Viscosity effects on the hydrodynamics of an unconventional SWATH Analytical Investigation of the effects of inertial characteristics on SWATH RAOs Sum m ary Computational Study of the Motion of a Floating Body in Waves D Strip Theory Method (PDSTRIP and CAT 5D) D Potential Flow Solver (WAMIT) Numerical Investigation of the Effects of Inertial Characteristics on m otion RAOs D iscussion Conclusions Experimental Study of the Responses of SWATH USV in Regular Head Waves The SWATH Model Assembling the Model Experimental Set-up Instrumentation and Procedure Laser Sensors Data Processing Results and Discussion D iscussion

7 5 Conclusion Summary of Results Recommendations Biblography 79 Appendices 81 List of Figures 2-1 Transverse Section of a Theoretical SWATH Vessel Added Drag Versus Encounter Frequency for Series60 with Cb=0.6 at F,=0.283 in Head Waves Added Drag Versus Wavelength for Series60 with Cb=0.7 at Zero Speed in Head W aves Comparison of WAMIT and PDSTRIP Heave RAOs for Series 60 ship with Cb=0.7 at Zero Speed Comparison of WAMIT and PDSTRIP Pitch RAOs for Series 60 ship with Cb=0.7 at Zero Speed Effect of Varying the pitch Gyradius on Heave RAO at Constant VCG Effect of Varying the pitch Gyradius on Pitch RAO at Constant VCG Effect of Varying the VCG on Heave RAO at Constant Pitch Gyradius Effect of Varying the VCG on Pitch RAO at Constant Pitch Gyradius Profile View of Model during Experiment Frontal view of model during experiment Plan view of Model during experiment Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No

8 4-6 Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Heave and Pitch Motion Time History for Wave Test No Cut Heave and Pitch Responses of SWATH USV Comparison of Experimental and Computational Predictions of Heave Motions of SWATH USV Comparison of Experimental and Computational Predictions of Pitch Motions of SWATH USV List of Tables 4.1 Main Model Parameters Tank Dimensions and Model Test Factors Wave Parameters for Different Cases of Model Test

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10 List of Symbols aij A Aij AW bij Bij (w) Cb cij Cij da df F(t) Fi F FP TFf" (Fji)* Fn Fs g GML hj (t) Hij Iij - Sectional added mass coefficient - Wave amplitude - Added mass coefficient - Water plane area - Sectional damping coefficient - Damping coefficient - Block coefficient - Sectional restoring coefficient - Restoring coefficient - Stern displacement - Bow displacement - Excitation force due to waves - Complex amplitude of excitation force - Froude-Krylov force on port side due to waves - Froude-Krylov force on starboard side due to waves - Hydromechanic force acting in ith direction - Complex conjugate of the Froude-Krylov part of the exciting force - Froude number - Second-order mean steady force - Acceleration of gravity - Metacentric height - Harmonic motion of ship in the jth mode of motion - Complex analytical function of added mass and damping coefficients - Ship's mass moment of inertial for rotational modes of motion 9

11 k - Wave number (Kv) - Pitch gyradius 11 - Distance between bow sensor and LCG 12 - Distance between stern sensor and LCG 13 - Longitudinal distance between stern sensor and aftmost point of the ship L - Length overall of ship LS M Paw Pi R RAOmax - Distance between sensors - Mass of ship - Energy dissipated by the ship during one encountering period - Linear hydrodynamic pressure obtained from Bernoulli's equation - Added resistance - RAO at resonance R, - Incident wave contribution to the added drag. Rf R 7 t UP3 g3 V Vz VCG w we Wn x Xb y Yb - Diffraction contribution to the added drag. - A function of the sectional damping coefficients in heave and sway. - time co-ordinate - Wave particle orbital velocity - Wave orbital acceleration - Velocity vector in two or three dimensions - Relative vertical velocity - Distance from waterline to center of gravity. Positive above the waterline - Circular wave frequency - Wave encountering frequency - Natural Frequency - Longitudinal axis of ship - X-coordinate of a given section - Transverse axis of ship - Y-coordinate of a given section 10

12 YO z Zb Zb Zh OP - Half the distance between center planes - Vertical axis of ship - Z-coordinate of a given section - Vertical distance between top of bow and CG in deflected position - Heave displacement - Pitch angle ZS - Vertical distance between top of stern and CG in deflected position 0 - Angle between direction of wave propagation and ship heading 'Ei - Phase difference r1 - Regular wave elevation 7 - Damping ratio A - Wavelength P V - Dynamic viscosity - Spatial derivative #j - Radiated wave potential #0 - Incident wave potential # 7 - Diffraction potential IT - 2D time domain linear potential p - Density of water 0 - Angular displacement of body 0 - Total fluid potential OB - Potential due to body disturbance 0* - conjugate of the incident wave potential (j - Response of ship(in the jth mode of motion) to sinusoidal waves (j (w) - Complex amplitude of ship motion in the jth degree of freedom - Ship acceleration - Ship velocity 11

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14 Chapter 1 Introduction The hydrodynamics of floating structures in waves is unique because of the frequency dependence of hydrodynamic parameters such as added mass, damping and the wave exciting force. For a SWATH, the effect of viscosity on the added mass and damping is non-negligible and must be taken into consideration when predicting the motion response of this kind vessels in waves. This paper presents the theoretical, numerical and experimental study of the motion responses of floating bodies in waves with focus on an unconventional SWATH USV proposed to be used to monitor a team of autonomous underwater vehicles (Brizzolara, 2011). The use of this watercraft is expected to reduce maintenance costs associated with manned ships and fixed platforms. Additionally, using remotely operated surface vessels such as this would guarantee safety and make very remote areas more readily accessible. The design operations of this vessel involves recharging batteries of AUVs and retrieving the AUVs back to onshore locations. A small version of this vessel can retrieve at least one autonomous vehicle per trip. A literature review of the advancement made in the optimization of this class of water crafts is presented as a basis on which future research is being carried out to study the hydrodynamics of this vessel. The objectives of this thesis are to compute the motion responses of the SWATH USV using 2D and 3D potential flow solvers, report experimental results obtained so far and attempt to validate the numerical results. The importance of this work 13

15 extends to other classes of multi-hull ships as there exists only few available data in this range of crafts relative to well established mono-hull ships. Additionally, the effect of the inertial characteristics of the model, vertical position of the center of gravity (VCG) and gyradius in pitch (K.) on the RAO is investigated both analytically and numerically. 1.1 Literature Review Strip theory originally developed by Korvin-Krokowsky and Jacobs (1957) formed the basis of most numerical methods for computing the motions of various kinds of ships in waves. This was further extended to account for wave-induced vertical shear forces and bending moments according to Jacobs (1958). The original strip theory gave very good agreement with experiments especially for regular cruiser stern ships operating at moderate speeds in head waves (Salvesen et. al., 1970). Gerritsma and Beukelman (1967) further extended the method by proposing a modified strip theory which predicts the motions of high speed destroyer hull in head seas very well with respect to experiments. The strip theory method for heave and pitch motions in head waves in which the forward speed terms satisfy the symmetry relationship proved by Timman and Newman (1962) was properly derived independently by Soding (1969), Netsvetayev (1969) and Tasai and Takaki (1969). Following the significant improvements made in the computation of the sectional added mass and damping coefficients using closed-fit methods, Smith and Salvesen (1970) showed that this method can be used to predict the motions of high speed hulls with large bulbous bows in head seas with satisfactory accuracy. Major advancements in the computation of ship hydrodynamics using the strip theory method were proposed in the famous work of Salvesen, Tuck and Faltinsen (1970). Their method can predict satisfactorily the heave, pitch, sway, roll and yaw motions as well as wave-induced vertical and horizontal shear forces and bending moments and torsional moments for a ship advancing at constant forward speed with 14 I II I I Pi"op"1111IT III P1,11II, I "'I"11 I I I -IIIII I III II

16 arbitrary heading in regular sinusoidal waves. The lateral symmetry of the ship guarantees that the equations of motion in the six degrees of freedom can be written into two sets of coupled equations of motion as follows: one set of three coupled equations for surge, heave and pitch as well as another set of three coupled equations for sway, roll and yaw. It was proved that in the first set of coupled equations, the surge motion may be neglected if the ship is a slender body in addition to having lateral symmetry. Hence, the equations of motion in the first set reduces to two coupled equations for heave and pitch. Of course, the strip method defined assumes that viscous effects are negligible and the assumption is credible because viscous damping is insignificant for vertical ship motions. Soding (1987) further improved the strip theory but the predictions were still limited to linear equations of motion, rigid-body responses to periodic exciting forces and moments due to free surface waves with small slopes. This method also assumes that the ship maintains mean forward speed in infinite water and a straight mean course. The strip theory method offers an immediate insight into the hydrodynamics of a ship saving computational costs at the preliminary design stage. The Numerical implementation of this method has been carried out using two-dimensional and threedimensional approaches. Both approaches give reasonably accurate predictions for the response amplitude operator of the ship motions; however, the latter is more efficient for predicting hydrodynamic pressures and forces. SWATHs are special types of watercrafts which are designed basically to reduce their motions in waves by reducing the water plane area of the sections interacting with surface waves. Hence, SWATHs are generally expected to have better seakeeping performance in waves than equivalent mono-hulls and catamarans. Earlier works on determining the motions and wave loads of SWATHs showed that analytical methods need to account for the effects of wave interaction between the two hulls. Soding (1988) developed a strip theory method that takes this interaction into account and the obtained results are more accurate than those obtained by solving merely the boundary value problem involving two-dimensional twin cylinders. This is because it 15

17 11FOOPm- considers the speed effects on the radiated and diffracted waves between the hulls. Dallinga et. al. (1989) developed a seakeeping program to gain a physical understanding of SWATH motions and described improved prediction methods. Their approach utilizes the 3D diffraction theory as well as empirical information on the associated loads and this was compared with traditional strip theory method. Hullfins interaction and free surface effects was found to be significant and are difficult to estimate using simple methods. In addition, they posited that accurate predictions of the seakeeping of SWATHs will require a more detailed description of the dynamics of the flow around the vessel. A frequency domain based strip theory method developed by Papanikolaou and Schellin (1991) accounts for the effects of forward speed on the hydrodynamic interaction between the hulls. This method gives motion predictions which agrees well the model test measurements and three-dimensional diffraction theory calculations. Brizzolara (2004) used an efficient differential evolution algorithm to implement a parametric design of a SWATH vessel which is optimized for total resistance. The optimized vessel is a variant of the original base vessel characterized by two humps at the bow and stern area and an intermediate hollow shape. This method automatically picks the optimized dimensions SWATH vessel with minimal total resistance. It was demonstrated that the operation profile of a watercraft should play a major role in shaping its hull during design. More advancements in design of SWATHs have focused on reducing wave pattern and added resistance. A major contribution to this development is achieved using the parametric optimization method as demonstrated by Brizzolara and Chryssostomidis (2012). This approach involves the modification of the submerged hull shape to maximize wave cancellation effects between both hulls. Such optimization of the SWATH ship hulls results in the reduction of total resistance by more than 25 percent at high speeds with respect to a conventional design. The computation of the hydrodynamics of multi-hull ships may or may not include the interaction between objects piercing through the free surface at different locations depending on the sophistication of the method employed. Although, it is important 16

18 to consider this hull interaction effects in SWATHs, its usually good practice to use simple and computationally cheap methods which gives reasonably accurate results at preliminary stages of design (Patrikalakis and Chryssostomidis, 1986). 1.2 Thesis Objectives The aim of this thesis is to predict the motion of an unconventional SWATH USV in waves using numerical and experimental approaches. There exist three dimensional fully viscous time domain simulations to account for non-linear free surface and viscosity effects but they can be computationally expensive. Hence, it's more efficient to develop quicker hybrid methods which account for these effects especially at early design stages. However, these new methods require sufficient experimental data for validation purposes and to boost the confidence in their applications. Only a few experimental data on the hydrodynamics of SWATHs are available in literature while some substantial numerical results can be found for existing SWATH designs. Moreover, in order to further progress with this work, the computationally predicted better performance of unconventional SWATHs in waves must be validated using experiments. Hence, the need for more experimental results such as those reported in this thesis. 17

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20 Chapter 2 Theoretical study of Motion of a Floating Body in Waves The motion of a body in random seas may be computed by the superposition of the responses of the body to each of the constituent elementary waves. Hence, it is very useful to study this problem using regular exciting waves which are easy to understand and apply the Wiener-Khinchin theorem to obtain the response of the ship in irregular waves. Another very important simplification is the assumption of potential flow and also the linearization of the free surface boundary condition in terms of the velocity potential. The solution to this problem is achieved by solving the diffraction and the radiation problems as would be discussed in following sections. 2.1 Theory of Ship Motions in Regular Waves This section discusses the theory of ship motions in the context of input and output into a linear time-invariant (LTI) system as assumed in studying small motions of a ship in the six degrees of freedom. Furthermore, this guarantees that if the input, that is- the exciting wave is sinusoidal, the response of the ship is also sinusoidal with the same frequency but in general, a different amplitude and phase. The magnification factor is otherwise known as the Response Amplitude Operator (RAO). Here, the RAO is a function of frequency but independent of the wave amplitude. This 19

21 concept as described is known as the linear theory of ship motions and it proves to be accurate for estimating the responses of ships in regular waves and irregular waves of considerable height. The regular wave elevation, q is given as follows: q(x, y, z, t) = Re{Aew t - kx} (2.1) The response of the ship to sinusoidal exciting waves, (j can be expressed as: (j(t) = Re{y (w)ewt} (2.2) The RAO which is the transfer function of the LTI system described above is the ratio of the complex amplitude of the ship motion to the wave amplitude, A. Hence, RAO = A (2.3) In model testing, the complex amplitude of the ship response is the measured amplitude of oscillation for a given frequency which can be measured using displacement sensors or derived from readings obtained using an accelerometer. Theoretically, the RAO is obtained from the solution of the equation of motion of the ship-water system using the strip theory method described in the next section. The equation of motion of a simple mass-spring system with a dashpot is a linear second-order differential equation and is given as: M + B +C( = F (t) (2.4) 20

22 2.1.1 Ship-Motion Strip Theory The strip theory is a relatively simple method of computing the forces acting on a submerged slender body by direct integration of the net pressure acting on a piece of length, dx over the entire length of the body. This tranche is generated by "tearing up" the body of the ship into sections also known as stations. Generally, viscous effects are neglected and the flow is assumed to be irrotational. Hence, the linear-ship motion strip theory is framed in the context of the potential flow theory. Also, the ship is considered to be very long compared to its beam, that is, the slender body theory is applied which allows surge motion to be ignored. The slender body theory applied in strip theory makes it possible to compute the exciting force and moment, hydrodynamic coefficients namely added mass and damping of the entire ship from two-dimensional potential flow solution for oscillating cylinders. Also, it is assumed that the disturbance of the free surface is due to the incident wave only. Hence, radiated waves can be ignored. For a ship with forward speed, the encounter frequency is assumed to be high. However, the strip theory gives good prediction for the heave and pitch motions even at low encounter frequencies because the Froude-Krylov force is dominated by the restoring force at low frequencies and is independent of the wave frequency. The strip theory calculations yield the hydrodynamic coefficients, that is, added mass and damping coefficient and the heave and pitch motion values with their corresponding phase relationship. These results are used as input to the formulation for predicting the added drag and must be accurate to ensure reasonable prediction of the added drag. The hydrodynamic coefficients may be determined using close-fit conformal mapping methods or by considering time harmonic motions of small amplitude for ship motions with the complex factor ei applied to the periodic quantities as described below. The total fluid potential around a ship hull using strip theory method is given as: 21

23 <D (x, y, z, t) =A(q5 0 +0~ 6 7 eiwt ) + Z Cjojeiwt(2) (2.5) The diffracted and radiated wave potentials must satisfy the following conditions: 1. Laplace equation in the fluid domain: a2<dj +y 2 1Z2= 0 (2.6) 1. Linearized free surface condition: ick~j + g 19) = 0 az (2.7) z=0, j=2,...,6 (2.8) 1. Ship hull no flux condition: = iwnae (j = 2,...,6) < radiated wave potential > (2.9) aopo an' (j = 7) < diffracted wave potential > (2.10) 1. Far field radiation potential: 22

24 an = 0, Z = 00 (2.11) According to the strip theory, the above complex amplitude of the wave potentials in 3D, D, may be replaced by 2D time domain body linear potentials, IV, using the relation below: O' (t, y, z) = <P (x, y, z)eiwt (2.12) The 2D time domain potentials, Oj (t, y, z)are obtained for each strip ordered from bow to stern. The linear hydrodynamic pressure is obtain from the Bernoulli's equation and the hydrodynamic force is obtained by integrating this pressure over the wetted section of the ship. The hydrodynamic forces acting on the ship can be expressed in terms of the 2D potentials as follows: Hij = P (x, y, z) ndds = -p ezit e-i ' ny ds (2.13) Hij is an analytic function which represents the radiation hydrodynamic coefficients and has the added mass, Aid and damping coefficients, Bid in its real and imaginary terms respectively. This is written as: Hij = w 2 Aij - iwbi (2.14) The strip theory method involves the integration of the force and moment per unit length (sectional force) acting on each strip over the entire length of the ship. This gives the total hydromechanics force and moments acting on the ship which is 23

25 added to the wave exciting force and moments respectively to obtain the equations of motion based on the Newton's second law. The hydromechanics force and moments consist of the hydrodynamic and hydrostatic components. For coupled heave and pitch motion, the equations of motion can be written as: In Heave: (M + A 33 ) h 3 + B 33 h 3 + C A 35 h 5 + B 35 h 5 + C 35 h 5 = F 3 eiwt (2.15) In Pitch: (155 + A 55 ) k 5 + B 55 h 5 + C 55 h 5 + A 53 h 3 + B 53 h 3 + C 53 h 3 = Fseiwt (2.16) The harmonic motion can be expressed as a complex amplitude of the motion in frequency domain: hj (t) = Re {(e't} (2.17) For harmonic motions with small amplitudes, the equations of motion described above can be expressed as a linear system in frequency domain given as: [C 33 - (M + A 33 ) w 2 + iwb 33 ] (3 + [C 35 - A 35 w 2 + iwb 3 s]( 5 = F 3 (2.18) 24

26 .M l [C55 - (M + A 55 ) w 2 + iwbs] (5 + [C53 - A 53 w 2 + ibs]( 3 = F 5 (2.19) For a ship with forward speed, the wave frequency, w which is the wave circular frequency is replaced with the wave encountering frequency, We. The added mass and damping are functions of the encountering frequency hence, are respectively calculated for each encountering frequency. The expressions which accounts for forward speed are given below (Salvesen, Tuck and Faltinsen, (1970): In Heave: F 3 = A eikxe-kds {pgb - w (wea33 - ib 33 )} dx - A eikxae-kdsw(wea A - ib ) (2.20) In Pitch: F 5 = - A eikxe-kds x [pgb - w (wea33 - ib 33 )] - Uw (wea33 - ib 3 3 ) dx S1We (2.21) + AU eikae-kdswxa(weaa - ib ) iwe The accuracy of the RAO especially for heave and pitch motions is very important since these motion results are the inputs to the added resistance formulation which is highly sensitive to their accuracy Equations of Motion of a SWATH The basic Newton's equation of motion as described in previous section is applicable to the SWATH vessel being investigated in much similar way as it is applicable to other floating bodies. However, the radiation and diffraction problem considers two hulls which are symmetric about the centerline from port to starboard and this yields the expressions below: 25

27 Zb Xb P S Yo Yo Figure 2-1: Transverse Section of a Theoretical SWATH Vessel In Heave: M - Fj = Fw (2.22) In Pitch: M -- F m= Fw (2.23) The response amplitude operator of the SWATH in heave and pitch is obtained from the above expressions. For a SWATH moving with forward speed, the Doppler shift must be taken into consideration and in fact, the interaction between the hulls becomes much more significant at low encountering frequencies. Also, because of the shape of the unconventional SWATH hulls, viscous effect plays an important role and cannot be neglected Radiation and Diffraction Problems of SWATH Vessels in Regular Waves The prediction of the motion response of a ship in waves based on the linear theory requires that the hydromechanics coefficients and exciting wave forces and moments 26

28 ... be determined. This is achieved by solving two main problems: * The Radiation Problem: This involves the excitation of the ship in calm water to create waves which carry energy away from the ship. The aim is to determine the added mass and damping coefficients which are expressed as the real and imaginary part of an analytic function derived from radiated wave potential, (Dr. hj = Re {( e}wt} (2.24) 41)r = (g i (2.25) As in the case of mono-hulls, the radiation force consists of the added mass and damping terms. It is important to note that these hydrodynamic coefficients are frequency dependent. While the added mass component is in phase with acceleration, the damping component is in phase with velocity. A more detailed proof can be found in Marine Hydrodynamics by Newman. The hydromechanics force acting on a twin hull swath oscillating in heave and pitch in calm water is given as: In heave: -Fh= (2a + 2b 31.f + 2c 31 x) + ( 2a 33 i + 2b 33 z + 2c 33 z) + (2a b c 35 9) (2.26) The first term accounts for the heave-surge coupling, the second term is the pure heave force contribution and the last term is the heave-pitch coupling term. In pitch: 27

29 -Fa"m = (2an1; + 2b 5 l + 2c 51 x) + ( 2a 53 i + 2b c 5 3 z) + (2a b c 55 0) (2.27) The first term accounts for the pitch-surge coupling, the second term is the pitchheave force contribution and the last term is the pure pitch coupling term. In both cases described above, the surge terms are neglected for this problem because only the heave and pitch motion responses have significant contributions to the added resistance of the vessel in head waves which is the end objective of this project. * The Diffraction Problem: The goal here is to determine the wave exciting forces and moments by fixing the ship and sending harmonic waves towards the ship. The excitation wave forces and moments are computed starting from the diffracted wave potential, FDd. In the special case of a swath, the interaction between the parallel hulls must be considered. The hydrodynamic interaction between the hulls comprises of two wave systems: * The radiated waves produced by the motion of one hull striking the other hull * The incident waves acting on one of the hulls which are modified due to presence of the other hulls (reflection and transmission). However, the effects of the interaction between the hulls is more significant at small encountering frequencies for ships with forward speed Wave Excitation Forces and Moments on SWATH Ships: Strip Theory Method The diffraction problem in which the floating body is constrained in the presence of incident waves yields the exciting wave forces comprising of the Froude-Krylov and 28

30 diffraction force components. In the case of a Swath with zero speed in regular head waves, the excitation forces can be expressed in terms of the 2D sectional potential mass and damping components, the wave particle orbital velocities and accelerations and the sectional Froude-Krylov force. The first order wave potential for deepwater case is given as: 4D = Re ei(wt-kbcos,6-kybsin8)+kzb (2.28) For the case of head waves, / is 180 degrees. In the vertical direction, the following expressions for the wave particle orbital velocity and acceleration for the portside (p) and starboard side (s) can be written as: = Re gka et(wt+kxb)+kzb } = 8 (2.29) nt = Re {-igkae(t+kxb)+kzb} its (2.30) The exciting wave force in heave and pitch is obtained by integrating the 2D sectional components listed above along the length of the ship. This is given as follows: In heave: F3'= j [M 3.(iP + its) + N3 3. (UP + u3) + (FPFK + F(2FK) (2.31) In pitch: 29

31 F3= [M' 3.(h+hi)+ Ni.(u +ui)+(ffk3+ffk)bb (2.32) The exciting forces in the remaining degrees of freedom namely yaw, roll, surge and sway can be trivially obtained from the same concept described above. 2.2 Second-order Non-linear Forces Acting on Floating Bodies in Waves The strip theory described so far is based on the linearized seakeeping theory. Using this concept, the integration of the first order pressure force on a floating boding over one wave period is effectively zero. However, any floating body in waves experience a steady force which does not go to zero over the wave period. This steady force is a second-order nonlinear force known as the drift force. The contribution to the drift force may result from the non-zero component of the time integral of the non-linear total wave potential but can be expressed in terms of known first order terms (Salvesen, 1978). Also, second order forces may result from first order pressure terms integrated over the actual wetted surface and in irregular seas, second order terms may be due to wave coupling. The first-order terms which contribute to the second-order drift force on a floating body in head waves are the heave and pitch motions already discussed for general floating bodies and specifically for the SWATH-type of vessel under investigation in this project. Hence, it is very important to accurately predict the motion response of the vessel in order to obtain a reasonable value for the second-order drift forces. The drift force could act transversely, longitudinally or vertically with respect to the ship fixed coordinate system. The expression below gives the theoretical representation of the second-order longitudinal drift force according to Salvesen (Salvesen, 1978): 30

32 Fs = p ]] VC* + -V() ds (2.33) 2 ff ( an an )( 0 2 B Added Resistance of Ships in Waves The added resistance also referred to as the added drag is that part of a ship's total drag which is due to its encountering of waves. The other part is known as the calm water resistance resulting from its forward speed in still water. The added resistance is typically in the range of 15% to 30% of the calm water resistance for a conventional merchant ship. Some of the factors that increase the added drag of a ship in realistic sea/weather conditions are wave reflection, ship motion in waves and wind effect on ship's superstructure. The added drag acts along the longitudinal axis of the ship hence, it creates additional resistance to the motion of the ship. It can be deduced from Havelock's formulation that the added resistance results from the phase relationship between the ship motion and the exciting wave forces. Also, the energy loss associated with the added resistance is dissipated mainly by the radiated waves generated by ship motions and negligibly by fluid friction. The added resistance consists of three components namely the drift force, diffraction effects and viscous effects. The drift force component results from the interference between the incident wave and the radiated waves generated due to ship motions specifically in heave and pitch. The diffraction effect is the component which results from the interaction between the diffracted wave and the radiated wave also due to ship motion in heave and pitch. The viscous effect is due to the damping of the ship's vertical motion. It is important to note that all components of the added drag are additive and each of them is proportional to the square of the wave amplitude which makes the added drag a non-linear problem. Also, the viscous damping is very small compared to the hydrodynamic damping of the entire ship. Hence, energy lost due to friction is negligible. Therefore, in practical terms, added drag can be considered as a non- 31

33 All viscous fluid flow phenomenon which makes it possible to apply potential flow theory to this problem. Added drag is important for choosing an accurate and reasonable value of weather margin during ship propulsion design and also important for mapping out the routes of an ocean going vessel to reduce fuel consumption, stress resulting from resistance and maximize the efficiency of the ship. This involves using the calm water resistance obtained from subtracting the predicted added resistance from the current ship's total resistance Salvesen's Method for Predicting Added resistance in Waves The total potential of the fluid around the ship hull is given as: # = 0 + #B (2.34) 0 is the incoming wave potential, #B is the potential due to the body disturbance and diffraction effects. The added drag is the negative of the mean second-order wave force acting in the longitudinal direction. There are three main contributing terms to the added drag predicted by Salvesen as shown below: R = E {R+Rf } + R 7 (2.35) j=3,5 The first term, R is the incident wave contribution to the added drag. This term is a function of the sum of the Froude-Krylov part of the excitation force in heave and pitch. The second term, Rf is the diffraction contribution to the added drag. It is a function of the sum of the diffraction part of the excitation force in heave and pitch. 32

34 The third term, R 7 is a function of the sectional damping coefficients in heave and sway. The added drag can be further expressed in terms of sectional quantities which are obtained from solving a two-dimensional problem of a cylinder oscillating in free surface as follows: R =- k cos ) (3 (Fj)* + Pf I + R 7 (2.36) j=3,5 (F/)* is the complex conjugate of the Froude-Krylov part of the exciting force. F 9 is the same as the diffraction part of the exciting force except that the incident potential, 5 0 is replaced by its conjugate, <b*. The conjugate of the Froude-Krylov force and moment is given as: (FI)* = ip f N 3 J ds < heave > (2.37) (FJ)* = -ip j xn 3 *ds < pitch > (2.38) The term F 9 is closely related to the diffraction part of the exciting force and moment and is expressed as: P3 = 3 (x) dx < heave > (2.39) FPD=- X+- i)h 3 (x)dx pitch> (2.40) 5 L W 33

35 h3 (x) = -pk j4 3 (N 3 + in 2 sin /)<djdl (2.41) JC0 Salvesen's formula treats the ship as a slender body and a 'weak scatterer' justifying the assumption that the body potential due to body-disturbance and diffraction effects (damping), ib's small compared to incident-wave potential, #0. Hence, quadratic terms involving the body potential is neglected. This assumption is justified when the wavelength is considerably longer than at least two main dimensions of the ship as assumed in the derivation. Typically, for navy ships, the maximum added drag is reached when the wavelength is about seven times larger than the beam of the ship when operating with normal speed in head waves while the wavelength is only about three and half times larger for zero speed ship. Immediately, it is expected that Salvesen's formulation predicts the added drag for a ship with normal operating speeds in head and bow waves more accurately than for a ship with zero speed in beam and following waves Other Methods of Predicting Added Resistance of a Ship in Waves (1) The Momentum and Energy Method: This method basically involves defining a control volume around the ship hull and deriving the momentum balance in the control volume. The added resistance is related to the linear momentum flow through the control volume. The fluid velocity potential is divided into the incident wave potential, diffracted wave potential and the radiated wave potential. The solution to the added resistance problem involves finding the harmonic potential which satisfies three basic conditions namely: The linearized free surface potential, the far field radiation potential and the ship hull no flux condition. These conditions 34

36 are simplified by considering a slender-body ship which its lengths much greater than its beam and draft. Joosen (1966) using this approximations expanded Maruo's results in an asymptotic series of power of a slenderness parameter and considered only first order terms. His formulation is given as: 1 W 3 Raw = - [B 3 h2 + B 5 h2-2b 3 5 h 3 h 5 cos (E3-65)] (2.42) 2 g 3 5 The encounter frequency, We accounts for speed effects on the ship in waves. (2) The Radiated Energy Method: This method equates the energy loss by the ship during one period of oscillation in regular waves to the added resistance of the ship. The most important parameter in this approach is the average relative vertical velocity. The average relative vertical velocity, V is the vertical velocity of the water particles relative to a point on the ship. It can also be described as the velocity of the water particles expressed in the ship's coordinate system. Vz = W h3 - Xbh5 + Vh 5 - * (2.43) We =[1 Yb exp (kzb) dzbi (2.44) yw -T The radiated wave energy of the ship is equal to the work absorbed by the added resistance. For one encountering period, the energy dissipated by the ship is given as: 7r Paw = jb 33 V 2~ dxb (2.45) We I The radiated wave energy during one period of oscillation is related to the added 35

37 MI I I li II q i I III ill~ iln P n i s I T lli 1" 1111 M 1 91 lm lliphp slle I iff 111 "j '' 1, In resistance as follows: Raw = 7e b 3 3 V 2 dxb (2.46) This method does not give accurate results for following waves with phase speed approaching ship speed because the encountering frequency in the denominator tends to zero. Hence, the added resistance results tends to infinity in this case which is not practical. However, it gives good results in head to beam seas Comparison between the Integrated Pressure Method and Salvesen's Method Both methods compute the added drag using the products of first-order terms obtained from the linear ship-motion theory. In fact, all computational methods for predicting added drag uses the linear ship responses as inputs although the added drag is a highly nonlinear quantity (Salvesen, 1978). Both methods do not consider the interaction between neighboring stations as they employ transverse strip theory to compute the pressure on the submerged sections of the ship. The Salvesen's method considers diffraction and viscous effects contribution to added resistance whereas the integrated pressure method neglects these effects. Hence, for a bulky ship with high Cb, the Salvesen's method is expected to give better results than the PDSTRIP method. The integrated pressure method utilized in PDSTRIP gives more accurate results for oblique waves compared to other methods in theory including the Salvesen's method. The PDSTRIP approach over estimates the peak of the added resistance for high Froude numbers and under estimates this value at lower Froude number compared to other methods in theory (Arribas, 2006). Also, it is more reliable than the Salvesen's method for predicting added resistance in short waves where diffraction contribution 36

38 is dominant (Matulja, 2011). Generally, the PDSTRIP approach has more agreement with experiments than the Salvesen's method especially for ship forms with low Cb Results of Added Resistance Computation for a Series 60 hull The theoretical methods discussed above were implemented in different computer programs. Results were obtained for both zero speed and forward speed case and a brief discussion of the results are given based on theory. The drift force RAOs presented are for Normand Ranger and Series 60 mono-hull ships CI 18 "co 16 o 14 Y 12 Cn * PDSTRIP Ger&Beu - Salvesen c Experiment - -h C] In- op- 2~ Encounter Frequency (We(/g) 1/2) 7 Figure 2-2: Added Drag Versus Encounter Frequency for Series60 with Cb=0.6 at F,=0.283 in Head Waves 37

39 Cs * WAMIT Pressure - * WAMIT Momentum * PDSTRIP 0 c 41) Waveleng th (A/L) Figure 2-3: Added Drag Versus Wavelength in Head Waves for Series60 with Cb=0.7 at Zero Speed 38

40 2.2.6 Viscosity effects on the hydrodynamics of an unconventional SWATH Radiation forces are the most influenced by viscosity effects especially for SWATHtype of vessels (Bonfigilo and Brizzolara, 2014). The significance of the viscosity effects on the radiation hydrodynamic force acting on a multi-hull ship is dependent on the body and wave motions. Palaniswamy showed that the effect of non-linear free surface and viscosity can be neglected for a multi-hull when body and wave motions are small. Also, the kinematic boundary condition is unaffected by the effect of viscosity (Ananthakrishnan, 2012). However, for the kind of hull shape being investigated in this project, viscosity cannot be neglected even with somewhat small body and wave motions (Bonfigilo et. al., 2013a). The struts of conventional SWATH ships is continuous from bow to stern which allows for trapping of waves on the mean surface between the hulls. The modes of motion undergone by these waves can be divided into two: e Piston Mode of Motion: This mode of motion corresponds to the vertical oscillation of the mean surface between the hulls. This mode of motion is the most affected by the viscous effect especially at resonance (Ananthakrishnan, 2012). * Sloshing Mode of motion: This mode of wave motion is characterized by the back and forth transverse oscillation of the standing waves between the hulls. For the unconventional SWATH ship being investigated in this project, the struts are not continuous from bow to stern. The water plane of the vessel comprise of four struts piercing through the free surface. Since the radiation problem is conducted in head waves, only the vertical modes of motion are deemed important. Hence, the piston mode of motion also known as Helmholtz oscillation is expected to be the dominant mode of wave motion in this case. The fully viscous Navier-Stokes governing equation that accounts for viscosity of the fluid is given as: 39

41 p ( +V.VV V ( + gz)+ LV2V (2.47) ( t The Navier-Stokes equation must be solved for this kind of problem but this approach is usually computationally expensive and hybrid methods which gives reasonably accurate results may be used at preliminary stages of design. Since, the frequency domain analysis of ship seakeeping problems is formulated in the context of linear theory, a hybrid method must be developed to account for this non-linear effect using a wholly linear analysis method (Bonfigilio et. al., 2013a). The comparison between the potential flow based heave and pitch motion RAOs and experiments at resonance frequency in the later part of this work shows clearly that the viscous effects are non-negligible for this kind of hull topology Analytical Investigation of the effects of inertial characteristics on SWATH RAOs The RAO of a floating body in the context of linear theory of seakeeping is independent of the wave amplitude. The RAO is characterized by its peak amplitude and the frequency at which this peak occurs. The inertial characteristics under study are the vertical center of gravity relative to the waterline, VCG and the pitch gyradius, KY of the SWATH ship. This discussion is shaped to illustrate the effect of varying the VCG and KY on the peak RAO and the wavelength at which it occurs. The wavelength is derived from the dispersion relation for deepwater as follows: W2 = gk (2.48) Hence, 40