Available online at ScienceDirect. Aquatic Procedia 4 (2015 )

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1 Available online at ScienceDirect Aquatic Procedia 4 (5 ) INTERNATIONAL CONFERENCE ON WATER RESOURCES, COASTAL AND OCEAN ENGINEERING (ICWRCOE 5) A 3D numerical wave tank study for offshore structures with linear and nonlinear mooring Shivaji GT a, Sen D b * a Indian Register of Shipping, 5-A, Adi shankaracharya Marg, Opp Powai Lake, Powai, Mumbai-47, India b Indian Institute of Technology, Kharagpur, WB 73, India. Abstract As offshore industry progressively moves towards deeper water, coupled dynamic analysis of such structures with mooring lines and risers becomes increasingly important because of the growing influence of mooring lines on the response of these structures. Experimental studies for such deepwater structures with mooring lines face serious problems due to the involved scaling and modeling issues. For such studies therefore a numerical tool capable of simulating the coupled dynamics of offshore structures is almost indispensable where the effect of mooring lines, loads arising from structural elements such as heave damping plates which are primarily due to fluid viscosity, external effects such as wind loads in case of floating wind turbines etc. are all coupled with the wave loads acting on the structure. It is widely acknowledged that for such a general and versatile numerical tool a time domain solution scheme is preferred over frequency domain solutions, particularly if nonlinearities in wave loads, mooring lines etc. are to be considered albeit in some simplified way. Based on such a premise, a solution scheme following a 3D numerical wave tank approach but with significant difference and novelty in its implementation have been under development. The developed method is capable of simulating long duration wave-structure interactions including important nonlinearities in wave loads, and in principle can also consider effects from many external sources such as linear and nonlinear mooring stiffness. While many aspects of the scheme including a validation and verification process is reported elsewhere, the purpose of the present paper is to study in detail the effect of linear and nonlinear mooring line stiffness on practical offshore configurations. As regards wave induced hydrodynamic loads, both linear loads as well as nonlinear loads arising from steep nonlinear incident wavers are considered. Results demonstrate that nonlinear mooring stiffness can have significant influence on the response of the structure. The influence of nonlinearities in mooring stiffness is relatively more pronounced when water depth is less while in deeper water a linear stiffness appears to be adequate in capturing the structure s response. The influence of hydrodynamic nonlinearity is however significant in all cases. These and other important results for practical offshore structures with different mooring configurations are presented and discussed in the paper. * Corresponding author. Tel.: ; fax: address: deb@naval.iitkgp.ernet.in 4-4X 5 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license ( Peer-review under responsibility of organizing committee of ICWRCOE 5 doi:.6/j.aqpro.5..64

2 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) The Authors. Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license 5 The Authors. Published by Elsevier B.V. ( Peer-review under responsibility of organizing committee of ICWRCOE 5. Peer-review under responsibility of organizing committee of ICWRCOE 5 Keywords: 3D numerical wave tank, wave-structure interactions, offshore structures, mooring analysis, nonlinear wave loads. Introduction Offshore structures are required to operate in a hostile ocean environment. Wave induced motions and loads are the major factor affecting the normal operation of floating offshore units. Wind and current loads contribute to the drifting of these structures, adding to the drifting caused by mean second order wave loads. Restoring loads from the mooring line generally follows nonlinear force displacement trend. To study the combined effect of wave, wind, and current loads on a moored floating offshore structure, a time-domain solution is generally preferred over frequency domain solution as some of these loads and effects are nonlinear in nature. A brief survey of literature reveals that there have been many studies on moored structure response in waves in the past. Van Oortmerssen (976) and Wichers (979) have studied motion of moored ships exposed to ocean environment. Coupled analysis of moored SPAR platforms have been reported in Chen et al. (), Kim and Sclavounos (), Tahar et al. (6), Zhang et al. (8), Chen and Zhu (), and Yang et al. (). For a turret moored FPSO, a coupled dynamics analysis have been reported by Ormberg and Larsen (998), Heurtier et al. () and Kim et al. (3) have discussed the importance of mooring line dynamics in deeper water. Column stabilised spread moored semisubmersible coupled analysis have been presented in Garrett (5) and Kim et al. (3). Almost in all of the above time domain studies except Chen and Zhu () and Yang et al. () have followed a similar approach in which the frequency-dependent hydrodynamic coefficients (added mass, and damping) are usually computed by Morison equations, strip theory, or by linear radiation-diffraction theory, and frequency domain equations of motion are then transformed into time domain through impulse response function as suggested by Cummins (96). Mooring system dynamics is formulated using either FEM or lumped mass method and the combined equation of motion of the floating body and the mooring lines are then integrated in time domain. These studies have also shown that mooring line dynamics has substantial influence on floating body motions for water depth greater than m. In the present work, a direct time domain coupled analysis of moored floating ocean structures exposed to steep nonlinear waves have been performed following a method based upon a form of a three dimensional numerical wave tank (3D NWT) approach. The developed scheme has certain simplifying approximation to make it suitable for routine industry use without sacrificing on the accuracy. The developed numerical method can consider steep and nonlinear incident waves. The basic solution algorithm developed here is the extension of previous work detailed in Sen (), and Sen and Srinivasan (7). Rankine panel method is employed to determine the unknown flow field variables like potential and velocity, combining this with appropriate time integration schemes for updating the flow field variables as well as equations of body motions. Nonlinearities associated with incident steep waves are captured completely while the interaction hydrodynamics are linearized. Elastic catenary theory has been utilized to determine the horizontal restoring forces from the mooring lines. The coupled analysis method adopted here can be termed a coupled quasi-static method because the dynamics of mooring lines are neglected in the adopted static catenary theory. The results of developed numerical scheme in its fully linearized version are validated with other available numerical and experimental results. The necessity of nonlinear Froude Krylov (FK) and hydrostatic restoring formulation over a linear formulation, and the importance of taking a fully nonlinear incident wave are discussed. Finally, the influence of nonlinear mooring stiffness on the structure response is discussed.. Mathematical background At the outset, two right handed Cartesian coordinate systems are introduced: an earth fixed coordinated system Oxyz with origin at the undisturbed water surface z = with +z directed vertically upwards, and a body fixed system Ox y z with origin coinciding with body centre of mass. Fig. shows the brief sketch of the coordinate systems. Based on the assumption of potential flow theory, fluid velocity vector can be described by gradient of velocity potential (φ). Consider a circular fluid domain Ω with the interacting body placed at the centre as shown in Fig..

3 494 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) The fluid domain is bounded by a time variant free surface on top, by the instantaneous wetted hull surface, by a time invariant flat bottom surface and by an exterior control surface. Fig. Coordinate system, Computational Domain The fluid flow problem is defined by the following governing equations and the corresponding boundary conditions: (, xyzt,;) fluid domain Ω () n VB. n n t x x y y z g. t on Ω D () on, (3) on z= η, (4) In addition, a suitable radiation condition on exterior control surface Ω C needs to be imposed to close the boundary value problem specification. In the above, is the free surface elevation, g is gravitational constant, t is time, h is the water depth, represent the velocity vector at any point on body surface, and is the exterior normal on the body surface. Eq. () given above is the Laplace equation, Eqs. () & (3) are impermeability conditions on rigid bottom surface Ω D and on instantaneous wetted body surface Ω B (t), Eq. (4) are the well known kinematic and dynamic free surface boundary conditions applicable on instantaneous free surface Ω F (t). Rankine panel method is adopted to solve the initial boundary value problem given above. At first, the Laplace equation given in Eq. is transformed into a well known boundary integral relation Eq.5 below which can be derived by application of Green s second identity to the velocity potential function and the basic Rankine source G = /4r in the computational fluid domain Ω, ( Q) ( P) ( P) ( Q) d( ) n (, ) (, ) B F D C Q r P Q r P Q n (5) Q In the above, P and Q are field and source points respectively, σ(p) is the solid angle, and is the radial distance. Constant panel boundary element method (BEM) is applied to solve the above boundary integral equation Eq. (5). Sea bed surface Ω D is discarded from the computational domain by making use of Rankine image sources. Exterior surface Ω C is discarded by enforcing a radiation condition in form of a wave absorbing beach on an outer annular part of the free surface (see eg. Cointe et al., 99; Ferrant, 996) and explained in Sen and Srinivasan (7). These approximations allow the discretized computational domain consisting of only body and free surface panels for the solution of the boundary value problem.

4 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) The motion equation of rigid floating body is given in Newtonian form, [ M]{ v } { F}; { F} { FHS FHD FMoor Fother} {} v {};{} x v {} x (6) In Eq.6 [M] is the generalized mass matrix, is the generalized instantaneous acceleration vector, is the generalized velocity vector, and is the generalized total external force/moment vector acting on the body. The external forces consist of,,, and representing the generalized hydrostatic restoring force, total wave hydrodynamic force, mooring restoring force, and other forces such as viscous damping forces respectively. Instantaneous total forces and moments acting on the body are computed by following Eq.7. F pnd( ) ; M p( rn) d( ); p. t B() t B() t (7) In the above equation is the unit exterior normal to the body surface, is the radial distance between panel centroid and centre of mass, and p is the hydrodynamic pressure given by Bernoulli's equation. For linear computation (FK-Lin), the incident Froude Krylov forces and hydrodynamic interaction forces are computed on time invariant mean wetted surface i.e. on Ω B (), and hydrostatic restoring forces are expressed by means of wellknown hydrostatic restoring coefficients. In the approximate nonlinear formulation (termed FK-Nonlin here), incident Froude Krylov and hydrostatic restoring are computed on exact wetted surface at the displaced location Ω B (t) but the hydrodynamic interaction (diffraction + radiation) are linearized around mean wetted surface Ω B (). Details of numerical computation of total wave hydrodynamics loads are given in Sen (). In the developed numerical scheme, ambient incident waves can be prescribed by linear Airy wave theory, Stokes second order theory or even full nonlinear numerical wave theory. In the present work, we use a fully nonlinear incident wave defined by the Fourier approximation method of Rienecker and Fenton (98). Mooring line can be model as either linear or nonlinear springs. Nonlinear load excursion characteristics of the mooring lines are determined by static catenary theory. Assuming restraining forces and moment in the vertical direction (heave, roll and pitch) are comparatively much smaller than hydrostatic restoring forces/moments, in this paper only horizontal restoring forces from mooring lines are modeled. The horizontal stiffness of any elastic catenary mooring line as it translates in its own plane can be determined by the scope of the line for any given horizontal force. Fig. show a schematic view of single elastic mooring line with tension T at the fairlead connection. Horizontal load excursion plot can be established by the following relations (Barltrop 998): Fig.. Mooring line configuration for catenary solution

5 496 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) H wl H wl L S S T H H LL sinh sinh w H w H AE T wh H AE AE; L T H AE AE w T wh ; H AE AE L T H AE AE w (8) where, w is the mooring line submerged weight per unit length, h is water depth or vertical span of the line, A and E are the average cross sectional area and the effective modulus of elasticity of the line, S is the horizontal scope, L is the length of the line in water (distance between touchdown to fairlead), and L T is the total un-stretched length. As indicated in Fig.. H, V, and T are the horizontal, vertical and net tension force at the fairlead. Quantities with subscripts are corresponding quantities related to the initial working pretension, thus H o is horizontal pretension force. Usually floating offshore structures are moored with multiple mooring lines; the net horizontal load exerted on the floating body can be obtained by combining the contribution from each line given by Faltinsen (99). In the present study, nonlinear load excursion plot if fitted with a nd order polynomial and the restoring horizontal loads are computed for exact displacement of the connection point at every time step. The time evolving six degree of freedom (DOF) motion of floating body can be obtained by integrating the motion equations Eq.6 and establishing the total forces and moments acting on the body at every instant. Estimation of total force/moments due to wave loading are detailed in Sen () are not repeated here. Fourth order Runge- Kutta method and Adams-Bashforth-Moulton methods are used to update the flow field and kinetics of floating body. 3. Numerical results and discussion Numerical results based on linear formulation (i.e. when all the external forces are computed on mean wetted surface at mean location) for a rectangular barge and a six-column semisubmersible are compared with well known experimental results of (98). In the present numerical simulation, 5% of critical roll damping is introduced as a roll damping value. Body panels of both geometries are displayed in Fig 3. The length, breadth and draft of the barge are 5m., 5m., and m. The semisubmersible pontoons are m long, total breadth and draft are 76m and m respectively. For the purpose of comparison, time series motion results of each individual incident wave are converted to frequency domain using Fast Fourier transform. The translational are non-dimensionalized by wave amplitude (ξ a ) and rotational motions by kξ a where k is /λ is wave number, and λ is wave length. Fig. 4 shows the comparison of motions RAOs for the rectangular barge and semisubmersible in a head sea (wave headings β=8 ). Observed good agreement validates the present method. Further detailed verification and validation of the developed scheme for the linear solution has been reported elsewhere by the authors. Fig. 3. Barge Six leg semisubmersible.

6 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) ξ /ξ a ξ 3 /ξ a ξ 5 /kξ a ξ /ξ a ξ 3 /ξ a Fig. 4. RAOs Comparison Barge Six-column semisubmersible. ξ 5 /ξ a.5.5 Next, a brief comparative study between linear (FK-Lin) and nonlinear (FK-Nonlin) formulation for moored floating structures are presented. Fig. 5 displays a schematic view of barge spread moored with six lines operating at water depth of m, and the semisubmersible operating at water depth of m moored with four lines. Fig. 5. Spread mooring system Barge 6 lines Semisubmersible 4 lines. Mooring line parameter to determine load excursion plot are taken from Kim BW et al. (3). Load excursion plot for a single mooring line operating at these two different water depth are determined based on (8). Note that slope of this load-excursion plot about origin gives linear mooring stiffness. When nonlinearities in mooring stiffness are considered, then the horizontal tension at the fairlead is determined from the exact displacement of the connection point. Fig. 6 shows the resulting six DOF motions of moored barge and semisubmersible in bow quartering sea. For clarification only a part of motion results are shown for vertical motions (heave, roll, and pitch). Complete motion results are shown for horizontal motions (surge, sway, and yaw) to demonstrate the importance of low frequency motion along with wave frequency motions. For both structures, it is observed that the motions in horizontal direction show significant differences between FK-Lin and FK-Nonlin formulation. There seems negligible difference in the first order motion between both formulations. The nonlinear formulation is able to capture the low frequency motions along with mean offset position. Minor variations are observed in the vertical motions (heave, roll, and pitch) between both formulations. Even for semisubmersible which is a small water plane area structure,

7 498 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) interesting results in horizontal motions are observed where in particular mean equilibrium position in yaw direction are captured by the nonlinear formulation. These results reveal the importance of nonlinear formulation over linear formulation for accurate prediction of six DOF motions of moored floating body. Fig. 6. Six DOF Motions for wave heading β =35 Barge ξ a =5, T= sec Semisubmersible ξ a =5, T= sec Next we present results to demonstrate the effect of considering linear and nonlinear mooring stiffness in modelling mooring line forces. Fig. 7 shows the motions of moored barge and semisubmersible for wave heading of 35, i.e. for the bow quartering waves. For both structures, nonlinear mooring stiffness provides greater restoring effect in surge and sway direction to restrict large mean offset, but also provides sufficient damping to arrest the low frequency motion to attain the equilibrium state. In case of the semisubmersible the effectiveness of nonlinear stiffness is realized only in yaw motion as compared to surge and sway motions. As expected, differences in vertical motions (heave, roll, and pitch) are absent. Fig. 7. Six DOF Motions for wave heading β =35 Barge in ξ a =5, T= sec Semisubmersible ξ a =5, T= sec

8 G.T. Shivaji and D. Sen / Aquatic Procedia 4 ( 5 ) Conclusion In this paper, a coupled time domain solution is developed based on a 3D NWT approach to determine six DOF motions of moored floating body. Rankine panel method is utilized to solve the initial boundary value hydrodynamic problem using lower order boundary element method. Two levels of numerical simulation are performed, a fully linear and an approximately nonlinear where FK and hydrostatic restoring forces are exact but the interaction hydrodynamic forces of radiation and diffraction are linearized. The developed method can consider the nonlinearity associated with mooring line stiffness which is determined based on static catenary equations. The results from the linear version of the present method have been compared with experimental results for a rectangular barge and twin pontoon six column semisubmersible and the computed motions results showed good correlation with experimental results. A comparative study between linear and nonlinear formulation showed significant variation in computed horizontal motion results between the two computations, revealing the importance of nonlinear formulation and in particular its ability to capture nonlinearity arising from the time variant wetted surface. Finally, motion results are compared when mooring line restoring forces are determined based on linear and nonlinear mooring stiffness. It is observed that the influences of nonlinear stiffness on horizontal motions are relatively more significant in shallow water compared to when water depth is large. References Barltrop NDP (Ed). (998) Floating structures: A guide for design and analysis, Volume II, Chapter 9. The Centre for Marine and Petroleum Technology Publication. Chen J and Zhu D. () A direct time domain simulation of floating structures with mooring lines. 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