桁架式Spar平台在波浪中的运动预报与实验测量.pdf

Numerical Prediction and Experimental Measurement on Truss Spar Motion and Mooring Tension in Regular Waves XU Guo-chun 1, MA Qing-wei1, 2, DUAN Wen-yang1, MA Shan1 (1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001, China; 2. School of Engineering and Mathematical Sciences, City University London, Northampton Square, London EC1V0HB, UK) Abstract: This paper presents a numerical and experimental study on the Truss Spar motions in reg鄄 ular waves. For the numerical calculation, a slender formulation is employed to evaluate wave loads on hard tank and truss structures of the Spar platform, and the Spar hull and its mooring lines are cou鄄 pled together by their interaction forces and motions. The Spar motions are solved by a set of non鄄 linear equations of a rigid body. A model test was carried out in the wave tank of HEU (Harbin En鄄 gineering University). During experiments, an optical measurement system is used to measure mo鄄 tions of the Truss Spar model in waves, and tension transducers are employed to record tension at the end of mooring lines. The numerical prediction results are compared with corresponding model test data. Satisfactory agreement is achieved for all the cases considered. Key words: Truss Spar; slender formulation; motions; model test; mooring loads; waves CLC number: U661.7Document code: Adoi: 10.3969/j.issn.1007-7294.2016.03.006 0 Introduction Truss Spar platforms have been utilized in deep water oil exploitation since about 20021. Generally, it is regarded as the second generation of Spar platforms, evolved from the Classic Spar. A typical Truss Spar is composed of three parts. The upper hard tank is a cylindrical hull similar to a Classic Spar. The middle truss section includes a number of slender braces and square heave plates. At the lower end of the truss section, there is a soft tank, containing solid ballast to keep stability and adjust draft of the whole platform. Now, some alternatives of the Truss Spar have been developed from the original Truss Spar, such as CT-Spar2, S- Spar3, and G-Spar4. Not only in the industry for oil and gas production, similar structures similar to Truss Spar are often adopted in the other offshore engineering field, for instance, McCabe Wave energy Pump5and SPAR-type floating wind turbine6. Compared to other floating structures, the Truss Spar has a number of distinctive advantages, such as less con鄄 structing costs, moderate heave motion due to heave plates, less susceptible to vortex-induced motion, and so on. Received date：2015-11-04 Biography：XU Guo-chun(1985-), male, Ph.D. candidate of Harbin Engineering University, xuguochun; MA Qing-wei(1955-), male, professor of Harbin Engineering University DUAN Wen-yang(1967-), male, professor/tutor of Harbin Engineering University. Article ID：1007-7294（2016）03-0288-18 第20卷第3期船舶力学Vol.20 No.3 2016年3月Journal of Ship MechanicsMar. 2016 A large body of research has been carried out on the loads on and response of Truss Spar platforms. The work may be summarized in several aspects as below. (1) Calculating the wave loads on Truss Spar platforms. The methods for estimating the wave loads on Truss Spar platforms may be split into two categories. One is based on slender-body theory using Morison equation and the other is based on diffraction/radiation theory for the Spar main hull while the Morison equation is used for truss components. Some examples for each category will be given below. Agarwal7used Mori鄄 son equation to compute the hydrodynamic pressure on the hull of a Spar to analyze its dy鄄 namic behaviors. Kurian8considered an axial divergence correction on the velocity of water particle to predict a truss Spar motions, which is different from the slender formulation con鄄 taining all nonlinear-terms employed by Ma and Patel9for a classic spar. The diffraction anal鄄 ysis based on potential theory to calculate the hydrodynamic force on the main hull of a truss Spar platform was adopted by many researchers, such as Kim et al10. They first solved hydro鄄 dynamic coefficients (the added mass and hydrodynamic damping, etc) in frequency domain, and then transferred them into time domain. Yang et al11calculated the velocity potential and then directly found hydrodynamic loads on Truss Spar in time domain by using higher-order boundary element method. Kim and Chen12made a comparison between slender body theory and diffraction analysis and found that the slender body theory could give the similar results to those of diffraction analysis as long as the diameter of Spar hull is less than 20% to the wave length. Although the diffraction analysis is potentially more accurate, it is theoretically not right for strong nonlinear problems and requires more computational costs. In contrast, the slender formulation is more computationally efficient and contains nonlinear terms. In this pa鄄 per, this formulation will be adopted for numerical analysis and all nonlinear terms are con鄄 sidered. (2) Estimating the interaction between Spar hull and mooring system. The mooring lines can be considered by static or coupled dynamic analysis. The former simplifies the mooring line as a weightless spring13or a catenary line, and the mooring force was included in the stiffness matrix or directly added into motion equations. The coupled dy鄄 namic method takes into account the wave loads on the mooring lines and the interaction be鄄 tween Spar and mooring system, and it models mooring lines as a number of hybrid beam ele鄄 ments14or elastic rods15when performing coupled analysis of the mooring lines and the Spar platform. The dynamic analysis method accounts for the wave loads but it needs more comput鄄 ing effort compared with the static methods. A simplified coupled method for calculating moor鄄 ing loads of the taut mooring system is adopted in this study. (3) Solving motions of a Spar platform. When motions of a Spar are simulated, the Spar is usually modeled as a rigid body with鄄 out considering its structural deformation. Two kinds of motion equations are usually adopted. One is the linear motion equation16assuming that the rigid bodys motion is small. The other one is the nonlinear equations of motions17. As for the solution of motion equations, the linear 第3期XU Guo-chun et al: Numerical Prediction and Experimental289 equation can be solved in frequency domain to evaluate the RAOs and in time domain. How鄄 ever, the nonlinear equation can only be solved in time domain. Although the linear equations can give very good results for small or moderate motions, the nonlinear ones are more suitable for the cases with large amplitudes of motions. In this paper, the numerical method and computer code described by Xu et al18are ex鄄 tended to predict the Truss Spar motions with six degrees of freedom. The hydrodynamic force on the hull and truss structures of a Truss Spar is estimated by a slender formulation and Spar motions are found by directly solving a set of nonlinear equations of motions. In addition, a Truss Spar model test was performed in the wave tank of HEU (Harbin Engineering Universi鄄 ty). The experimental and numerical results are compared to validate this numerical method. 1 Numerical model 1.1 The coordinate system Three coordinate systems are mainly adopted in this study as illustrated in Fig.1. One is the space-fixed coordinate system O-XYZ. The second one is the body-fixed coordinate sys鄄 tem Ob-XbYbZb. At the initial moment, the two coordinate origins coincide with each other. In order to compute the hydrodynamic force and its moment on the member of the truss section, one auxiliary coordinate system Oc-XcYcZcis introduced for each structure element, as seen in Fig.1(b). The origin of this coordinate system is fixed on the center of the axis of each mem鄄 ber and its zcaxis is along its centerline. 1.2 Equations of the Truss Spar In present study, the Truss Spar is assumed as a rigid body, and its motions are defined by a set of nonlinear equations given below: ? ? M dU 軑 0 dt =F 軋 (1) dX 軑 dt =U 軑 0 (2) (a) Space-fixed and body-fixed coordinate systems(b) Coordinate system of truss member Fig.1 Coordinate systems of Truss Spar 290船舶力学第20卷第3期 ?I d赘 軖 dt +赘 軖 ?I 赘 軖 軖軖=N 軑 (3) ? ?B d兹 軋 dt =赘 軖 (4) where X 軑 and U 軑 0 are translation displacement and velocity of the gravity center. 兹 軋 琢, 茁, 軖軖酌and 赘 軖 are Euler angles and angular velocity vectors.? ?Bis the transformation matrix between the time derivatives of the Euler angles and angular velocities.? ?Mand?Iare body mass and mo鄄 ment of inertia matrices. F 軋 and N 軑 are the total external force and moment subjected to the Truss Spar, and they can be expressed by: F 軋=F軋 G+F 軋 M+F 軋 H (5) N 軑=N軑 G+N 軑 M+N 軑 H (6) The subscript G denotes the gravity term; M is the term due to mooring loads; H-term comes from hydrodynamic and hydrostatic pressure. 1.3 The hydrodynamic loads on the Truss Spar The wave loads on the Spar hull is divided into two parts, e.g. the inviscid loads and drag force. The inviscid part on the hull including hard tank and truss structures is calculated by a slender formulation19, which is composed of three parts: the transverse forces on per unit length, the immersed end loads and wave surface part. For completeness, the formulations are summarized below. The transverse force on unit immersed length is given by: d f 軆 1=籽Sa 軆-g軆軆軆 n + M a ? ?a軆+ l 軆? ? V l 軆軖軖v 軆 r 軆軆 - M a ? ?U 觶 軆軆 -2 M a ? ?赘軖v軆 ra 軖軖(7) where 籽, S, g 軆 are the water density, the cross-section area and gravitational acceleration, re鄄 spectively; a 軆 and v 軆 stand for the acceleration of the fluid particle and its velocity; U 軑 represents the velocity of the structure at a corresponding point; v 軆 r=v 軆-U軑 is the relative velocity of the flu鄄 id particle and v 軆 ra is its axial component;Ma ? ?, 赘 軖 and? ?Vare the added mass, angular ve鄄 locity and velocity gradient matrices, respectively.軆 軆 n denotes the component normal to the centerline of the member, and l 軆 is the unit axial vector. The immersed end loads are expressed as: f2=pSl 軆- 1 2 v 軆 r Ma? ?v 軆 r軖軖 l 軆+ l 軆v軆 r 軖軖Ma ? ?v 軆 r (8) where p includes static water pressure and the incident wave hydrodynamic pressure. The stat鄄 ic water part and the integration -g 軆軆 軆 n along the centerline of the member is the buoyancy of the body in water. 第3期XU Guo-chun et al: Numerical Prediction and Experimental291 Wave surface hydrodynamic pressure is given by: f 軆 3= 1 2 tan渍 子 軋 v 軆 r 軋軋軋軋Ma 軋 軋v 軆 r - 子 軋 l軆 M a 軋 軋v 軆 r 軋軋軋軋l 軆v軆 r 軋軋(9) where 渍 is the acute angle between the axis of the slender member and the normal vector of the undisturbed wave surface, and 子 軋 is the unit vector in intersection point. The drag force d f 軆 4 on the hull of the Spar, it is estimated by a squared velocity formula of Morison equation. As for the hydrodynamic loads on hard tank, F 軋 H1 is composed of integration d f 軆 1 and d f 軆 4 from immersed end to top point of wetted length, f 軆 2 and f 軆 3. But for truss member, there is a little difference due to excluding wave surface force and relatively small immersed end loads. Thus its hydrodynamic force and moment are expressed by: F 軋 H2= m1 i=1 移 Li 乙d f 軆 1+ Li 乙g 軆 n 乙 乙 乙 乙 乙 乙 乙 乙 + m1 i=1 移-籽g 軆荦 i 軋軋+ m1 i=1 移 Li 乙d f 軆 4 乙 乙 乙 乙 乙 乙 乙 乙 (10) N 軑 H2= m1 i=1 移 Li 乙r 軆 bc i +r 軆 c i 軋軋d f 軆 1+ Li 乙r 軆 bc i +r 軆 c i 軋軋g 軆 n 軋軋 乙 乙 乙 乙 乙 乙 乙 乙 + m1 i=1 移r 軆 bc i -籽g 軆荦 i 軋軋軋軋軋軋 + m1 i=1 移 Li 乙r 軆 bc i +r 軆 c i 軋軋d f 軆 4 乙 乙 乙 乙 乙 乙 乙 乙 where m1is the total number of members in the truss structures; 荦iand Listand for the wet鄄 ted volume and the length of the i th truss member, respectively; r 軆 bc i is the vector pointing from the origin of the body-fixed coordinate system to the origin of the auxiliary coordinate system of the i th truss structure member; r 軆 c i represents a vector from the origin of the i th truss mem鄄 ber to a point concerned. The hydrodynamic force on heave plates and soft tank is calculated by the equation em鄄 ployed by Prislin et al20. The total hydrodynamic force F 軋 H can be solved by summing of the cor鄄 responding wave loads of different parts of Truss Spar, including hard tank, truss structures, heave plates and soft tank. 1.4 The mooring loads on the Truss Spar In present study, the taut mooring line is modeled as an elastic rope that withstands spring force, damping forces and hydrodynamic loads. The hull of the Truss Spar and each mooring line are coupled at fairleads. The mooring force and its moment of the i th mooring line are ex鄄 pressed by: F 軋 M i =F 軋 s i+F 軋 d i +F 軋 h i (12) N 軑 M i =N 軑 s i+N 軑 d i +N 軑 h i (13) (11) 292船舶力学第20卷第3期 In Eqs.(12) and (13), F 軋 s i and N 軑 s i are the spring force and its moment of the i th mooring line on the hull of Spar. They are estimated by the following equations: F 軋 s i= F 軋 M0 i +KAiPi 軑軑 - AiPi0 軑軑 軑軑軑軑E 軑 i (14) N 軑 s i= r 軆 i fE 軑 i 軑軑F 軋 M0 i +KAiPi 軑軑 - AiPi0 軑軑 軑軑軑軑(15) where Aiis the i th anchor point in the seabed; Pi0is the initial position of the ith fairlead and Piis its position at an instant after movement. F 軋 M0 i is the pre-tension of the ith mooring line; K is the stiffness of the mooring line; AiPi0 軑軑 and AiPi 軑軑 represent the mooring lines lengths at the initial moment and at any time instant, respectively; r 軆 i f stands for the vector pointing from the origin of the body-fixed coordinate system to the ith fairlead, and E 軑 i is the unit vector a鄄 long the axis of the ith mooring line at any moment. F 軋 d i and N 軑 d i are structural damping force and its moment of ith mooring line, the detailed computation process can be found in publica鄄 tion of Raman-Nair and Baddour21. F 軋 h i and N 軑 h i are hydrodynamic loads on the ith mooring line. They are written as F 軋 h i =F 軋 h1 i +F 軋 h2 i (16) N 軑 h i =r i fF 軋 h i (17) where F 軋 h1 i is inviscid part of hydrodynamic loads and F 軋 h2 i is viscous drag force. They are es鄄 timated by using the similar method as in Chen et al22. F 軋 h1 i = LM i 乙籽Ai 1+Ca n a 軆 n-u 觶 軑 L n 軑軑軑軑dL+ LM i 乙籽AiCa t a 軆 t-u 觶 軑 L t 軑軑dL(18) F 軋 h2 i = LM i 乙1 2 籽DiCd n v 軆 n-u 軋 L n 軑軑v 軆 n-u 軋 L n dL+ LM i 乙1 2 籽DiCd t v 軆 t-u 軋 L t 軑軑v 軆 t-u 軋 L t dL(19) where Distands for the diameter of mooring line. Ca n and Ca t are normal added mass coefficient and tangential added mass coefficient; Cd n and Cd t are normal drag coefficient and tangential drag coefficient; a 軆 and v 軆, u觶 L 軑 and u 軋 L are the acceleration and velocity of fluid and of the mooring line at concerned point, respectively; superscripts n and t represent their normal and tangen鄄 tial components. In hydrodynamic calculation of mooring line, one of key tasks is to solve velocity and ac鄄 celeration of points of mooring line. Several methods have been proposed. Huang 23 treated mooring cable to be lump-mass-and-spring model and velocity and acceleration of point in cable are evaluated by solving motion equation of discrete lump mass, whereas, in Ref.15, 第3期XU Guo-chun et al: Numerical Prediction and Experimental293 they solved a linear momentum conservation equation from slender rod theory to decide moor鄄 ing lines velocity and acceleration vectors. An equilibrium equation is employed by Jeon et al24to compute kinematic quantities of catenary mooring cable. In above research, they have to solve ordinary or differential equation by different numerical methods such as the finite difference or finite element scheme, to estimate velocity, acceleration and position vectors of mooring line, whereas in this study the mooring lines velocities u 軋 L n and u 軋 L t are estimated by: u 軋 n L= u 軋 i f-u 軋 i ft LM i L 軌 M i (20) u 軋 t L= u 軋 i ft LM i L 軌 M i (21) where u 軋 i f is the velocity of ith fairlead and u 軋 i ft is its tangent component. L 軌 M i is the distance from anchor point to the concerned point of the mooring line. Comparison with above studied methods, solving extra differential equation of mooring line is not needed. Thus the computa鄄 tion time will be saved by this velocity solving scheme. As for the hydrodynamic force due to u 觶 L 軌 term, it is neglected in the cases discussed in this paper but it may be worth to be taken into account in other cases in future. 1.5 Numerical implement These Eqs, from (1) to (4), are a set of nonlinear equations. To successfully find the solu鄄 tion for Spar motions e.g. X 軑, 兹軋, U軑 0, 赘 軖 軖軖, the following tasks need to be performed: (1) Calculating the wetted length of the hard tank This is equivalent to find the common point between the instantaneous wave surface and the centerline of the truss Spar. For this purpose, an auxiliary function is defined, which de鄄 pends on the wave surface and the position/attitude of the Spar centerline. The detailed descrip鄄 tion can be found in Xu et al18. (2) Efficiently computing hydrodynamic loads on the hull of the Truss Spar After the instantaneous wetted length is found, the hydrodynamic loads on the Spar hull are evaluated by the adaptive Simpson integral method. The characteristic of this integral method, as well known, is to double the number of the integral intervals for each iteration cycle before the error tolerance is met. If this integral method is used directly along the long wetted length of the hull element of the Spar, the computational time will be unnecessarily increased since it does not reflect the fact that the hydrodynamic pressure on the hull decays exponentially a鄄 long the water depth. To avoid it, the wetted length is divided into several parts and the adap鄄 tive Simpson integral method is applied for each part separately. (3) Dealing with the coupling between the Spar and mooring lines The vector AiPi 軖軌 in Eqs.(14) and (15) is used to determine the force and moment on the 294船舶力学第20卷第3期 Spar platform but depends on its motions. As a result, Eqs.(14) and (15) are coupled with the Eqs.(1)-(4). In the procedure to solve the problem of the whole system,