Time-domain numerical simulation of ocean cable structures

This paper describes the numerical features of WHOI Cable, a computer program for analyzing the statics and dynamics of oceanographic cable structures. The governing equations include the effects of geometric and material nonlinearities, bending stiffness for seamless modeling of slack cables, and a model for the interaction of cable segments with the sea floor. The program uses the generalized-α time integration algorithm, adaptive time stepping, and adaptive spatial gridding to produce accurate, stable solutions for dynamic problems. The nonlinear solver uses adaptive relaxation to improve robustness for both static and dynamic problems. The program solves surface and subsurface single-point mooring problems, multi-leg and branched array systems, and towing and drifting problems. User specified forcing can include waves, currents, wind, and ship speed.     

Modification of the damping function in the k–ε model to analyse oscillatory boundary layers

A simple relationship has been developed between the wall coordinate y⁺ and Kolmogorov's length scale using direct numerical simulation (DNS) data for a steady boundary layer. This relationship is then utilized to modify two popular versions of low Reynolds number k–ε model. The modified models are used to analyse a transitional oscillatory boundary layer. A detailed comparison has been made by virtue of velocity profile, turbulent kinetic energy, Reynolds stress and wall shear stress with the available DNS data. It is observed that the low Reynolds number models used in the present study can predict the boundary layer properties in an excellent manner.     

A finite element cable model and its applications based on the cubic spline curve

For accurate prediction of the deformation of cable in the towed system, a new finite element model is presented that provides a representation of both the bending and torsional effects. In this paper, the cubic spline interpolation function is applied as the trial solution. By using a weighted residual approach, the discretized motion equations for the new finite element model are developed. The model is calculated with the computation program complier by Matlab. Several numerical examples are presented to illustrate the numerical schemes. The results of numerical simulation are stable and valid, and consistent with the mechanical properties of the cable. The model can be applied to kinematics analysis and the design of ocean cable, such as mooring lines, towing, and ROV umbilical cables.     

An assessment model for the fate and environmental effects of offshore drilling mud discharges

The benthic boundary layer transport (bblt) model was developed to assess potential impact zones from drilling mud discharges from offshore oil and gas drilling. The model focuses on the drift, dispersion and concentration levels of the suspended fraction of the drilling mud fines in the benthic boundary layer with the assumption of a spatially homogeneous environment. The current version of the model includes a wave boundary layer, a breakup module for drilling mud flocs, a dose–response module for scallops, and a graphical user interface (GUI). The GUI was written in Java which makes the code largely platform independent. Simulations of suspended barite concentration near Sable Island on the Scotian Shelf during drilling in the fall of 1999 reproduce the very low concentrations (generally less than 1 μg L⁻¹) observed during the Environmental Effects Monitoring program. However, the simulations also exhibited concentrations in excess of the no-effects concentration for scallops (100 μg L⁻¹) prior to the sampling program. The model estimates that the potential impact on scallops in the vicinity of the drilling is a few days of lost growth over scales of a few kilometers.     

深海观测系统脐带缆形态分析及计算

通过对水下拖曳系统脐带缆索的受力情况、形态进行分析，建立了三维空间缆索的物理模型和数学模型，并在此基础上编制了计算程序，对6000m深海观测系统脐带缆的性状进行了模拟计算，其结果为深拖系统最佳缆长的确定提供了理论依据。     

Coupled dynamic analysis for wave interaction with a truss spar and its mooring line/riser system in time domain

A full time-domain analysis program is developed for the coupled dynamic analysis of offshore structures. For the hydrodynamic loads, a time domain second order method is developed. In this approach, Taylor series expansions are applied to the body surface and free-surface boundary conditions, and the Stokes perturbation procedure is then used to establish the corresponding boundary value problems with time-independent boundaries. A higher-order boundary element method (HOBEM) is developed to calculate the velocity potential of the resulting flow field at each time step. The free-surface boundary condition is satisfied to the second order by fourth order Adams–Bashforth–Moultn method. An artificial damping layer is adopted on the free surface to avoid the wave reflection. The mooring-line/tendon/riser dynamics are based on the rod theory and the finite element method (FEM), with the governing equations described in a global coordinate system. In the coupled dynamic analysis, the motion equation for the hull and dynamic equations for mooring-lines/tendons/risers are solved simultaneously using the Newmark method. The coupled analysis program is applied for a truss Spar motion response simulation. Numerical results including motions and tensions at the top of mooring-lines/risers are presented, and some significant conclusions are derived.     

Coupled dynamics of tethered buoy systems

The three-dimensional coupled behavior during the interaction of buoys with their mooring systems is numerically analyzed. A time-domain model was developed to predict the response of a tethered buoy subject to hydrodynamic loadings. External loadings include hydrodynamic forces, tethers tensions, wind loadings and weight. System nonlinearities include large rotational and translational motions, and non-conservative fluid loadings. The mooring problem is formulated as a combined nonlinear initial-value and two-point-boundary-value problem which is directly integrated both in time and space. Buoy equations of motion are derived using small Eulerian angles. Coupling between rotational and translational degrees of freedom is included and coupling between the buoy and cable is effected by adopting the buoy equations of motion as boundary conditions at one end for the mooring problem. Numerical examples are provided to validate the formulation and solution technique; predicted responses of three types of buoy (sphere, spar, and disc) are compared with experimental results.     

On the motions of the underwater remotely operated vehicle with the umbilical cable effect

In this paper, a hydrodynamic model is developed to simulate the six degrees of freedom motions of the underwater remotely operated vehicle (ROV) including the umbilical cable effect. The corresponding hydrodynamic forces on the underwater vehicle are obtained by the planar motion mechanism test technique. With the relevant hydrodynamic coefficients, the 4th-order Runge–Kutta numerical method is then adopted to solve the equations of motions of the ROV and the configuration of the umbilical cable. The multi-step shooting method is also suggested to solve the two-end boundary-value problem on the umbilical cable with respect to a set of first-order ordinary differential equation system. All operation simulations for the ROV including forward moving, ascending, descending, sideward moving and turning motions can be analyzed, either with or without umbilical cable effect. The current effect is also taken into consideration. The present results reveal that the umbilical cable indeed significantly affects the motion of the ROV and should not be neglected in the simulation.     

Simulation of directional waves in a numerical basin by a desingularized integral equation approach

A numerical solution is developed to investigate the generation and propagation of small-amplitude water waves in a semi-infinite rectangular wave basin. The three-dimensional wave field is produced by the prescribed “snake-like” motion of an array of segmented wave generators located along the wall at one end of the tank. The solution technique is based on the boundary element approach and uses an appropriate three-dimensional Green function which explicitly satisfies the tank-wall boundary conditions. The Green function and its derivatives which appear in the integral equation formulation can be shown to be slowly convergent when the source and field points are in close proximity. Therefore, when computing the velocity potentials on the wave generators, the source points are chosen outside the fluid domain, thereby ensuring the rapid convergence of these functions and rendering the integral equations non-singular. Numerical results are shown which illustrate the influence of the various wavemaker and basin parameters on the generated wave field. Finally, the complete wave field produced by the diffraction of oblique waves by a vertical circular cylinder in a basin is presented.     

Earthquake response of sea-based storage tanks

An effective method for the linear analysis of dynamic response of submerged underwater oil storage tanks resting on a horizontal seabed to horizontal earthquake excitations is presented. The tank is axisymmetric in shape, has a flexible wall/roof, and is filled with oil and water. A general hybrid-finite element solution procedure has been formulated, wherein the tank structure, the interior fluids, as well as the near-field of the exterior water region are discretized into a toroidal finite-element network. The tank displacement is calculated as a superposition of the first few modes of the structure's free vibration. Contribution from the hydrodynamic interaction to the coupled motion is obtained by solving the Laplace equation with the appropriate boundary conditions, which includes a matching to the exterior far-field pressure (analytic) representation to simplify the computation process. The effects of fluids surrounding and inside the tank are studied. It is demonstrated that these effects have, in general, a significant impact on the tank earthquake response analysis.A comprehensive and predictive computer program for use in such tank response analysis has been developed for engineering applications.     

Effects of mooring line drag damping on response statistics of vessels excited by first- and second-order wave forces

The drag-induced damping in a mooring cable due to combined first- and second-order wave excited motion of a moored vessel has been determined by statistical linearisation. A dynamic stiffness approach developed elsewhere is used to deal with the dynamics of the mooring cables. The power spectral densities of low- and wave-frequency responses are obtained which clearly show the influence of mooring line damping. The non-Gaussian probability density functions (pdf) and expected crossing rates of vessel responses and dynamic cable tensions are determined using the Kac–Seigert technique, and the influence of drag damping is highlighted.     

Simulation studies of the response of deeply towed vehicle to various towing ship maneuvers

The behavior of a long cable towed at slow speeds through the ocean depends in a complex fashion on the path followed by the towing ship relative to the water. A cable simulation program was used to characterize the response of the cable by using idealized towing ship maneuvers as input to the program. The response of the cable was noted and it was found that the behavior of the cable was strongly dependent on the fundamental period of the towing vessel maneuvers. Sinusoidal deviations of the towing ship from a straight towing track resulted in delayed and reduced excursions of the towed vehicle from the tract; the estimated response ratio varied from 0·002 to 0·800, depending both on the period of the deviations (periods ranged from 5·5 to 4·0 hr) and on the towing depth (2 or 6 km). The ship's speed was 3 km/hr. The time lag between ship motion and vehicle response was approximately 0·5 hr for the shallow case and 1·3 hr for the deep case. Simulations runs of a low dragk (faired) cable showed that the behavior of the vehicle when towed at a depth of 6 km was similar to that obtained with a conventional cable at 2 km depth. The response of the towed vehicle to a right-angle turn of the towing ship was investigated and a generalized model of the response developed. The effects of a controllable side force on the towed vehicle were also simulated and it was noted that a deviation (2-hr period) of the towed vehicle from a straight-line track could be reduced from 40 to 2 m by impressing a side force on the vehicle with an average magnitude of 150 newtons (30 lb).     

Dynamic modeling of cable towed body using nodal position finite element method

This paper analyses nonlinear dynamics of cable towed body system. The cable has been modeled and analyzed using a new nodal position finite element method, which calculates the position of the cable directly instead of the displacement by the existing finite element method. The newly derived nodal position finite element method eliminates the need of decoupling the rigid body motion from the total motion, where numerical errors arise in the existing nonlinear finite element method, and the limitation of small rotation in each time step in the existing nonlinear finite element method. The towed body is modeled as a rigid body with six degrees of freedom while the tow ship motion is treated as a moving boundary to the system. A special procedure has been developed to couple the cable element with the towed body. The current approach can be used as design tool for achieving improved directional stability, maneuverability, safety and control characteristics with the cable towed body. The analysis results show the elegance and robustness of the proposed approach by comparing with the sea trial data.     

On the heave radiation of a rectangular structure

In this paper, an analytic solution to the heave radiation problem of a rectangular structure is presented. To solve the problem analytically, the nonhomogeneous boundary value problem is linearly decomposed into homogeneous ones, which can be readily solved. To provide further comparisons to the present analytic solution, a boundary element method is also presented to solve the problem. The present analytic solution is compared with the result by Black et al. [(1971)] Radiation and scattering of water waves by rigid bodies. J. Fluid Mech. 46, 151–164], and the boundary element solution, and the comparisons show very good agreements. Upon examination of the present analytic solution, it is shown that the solution satisfies the nonhomogeneous boundary condition in a sense of series convergence. Using the present analytic solution, the generated waves, the added mass and the radiation damping coefficients, as well as the hydrodynamic effects of the submergence and the width of the structure, are investigated.     

Simulation of unsteady oceanic cable deployment by direct integration with suppression

An improved algorithm is developed for predicting the transient response of a system of serially connected cables and bodies during unsteady deployment from a surface vessel. The governing equations of a cable-body system are derived with dependent variables of cable velocities, direction cosines and tension magnitude to form a nonlinear combined initial-value and boundary-value problem. The problem is then solved by introducing a stable Newmark-like implicit integration scheme in time and by a direct integration method with suppression of extraneous erroneous solutions. Special boundary conditions simulating actively controlled payout and slack-cable/ocean-bottom contact boundary conditions are included in the present model.     

海底电缆在埕岛油田的应用及发展趋势

随着埕岛油田原油产量逐年递增,电力需求不断增加,海底电缆不断发展。文中主要介绍了埕岛油田采用的海底电缆的结构类型及特点,并对在海底电缆的维修、铺设中应注意的问题进行了分析,较为系统地对整个海上油田海底电缆进行了描述,并对今后埕岛油田海底电缆的发展方向进行了分析。     

Fully nonlinear numerical wave tank (NWT) simulations and wave run-up prediction around 3-D structures

A finite-difference scheme and a modified marker-and-cell (MAC) algorithm have been developed to investigate the interactions of fully nonlinear waves with two- or three-dimensional structures of arbitrary shape. The Navier–Stokes (NS) and continuity equations are solved in the computational domain and the boundary values are updated at each time step by the finite-difference time-marching scheme in the framework of a rectangular coordinate system. The fully nonlinear kinematic free-surface condition is implemented by the marker-density function (MDF) technique developed for two fluid layers.To demonstrate the capability and accuracy of the present method, the numerical simulation of backstep flows with free-surface, and the numerical tests of the MDF technique with limit functions are conducted. The 3D program was then applied to nonlinear wave interactions with conical gravity platforms of circular and octagonal cross-sections. The numerical prediction of maximum wave run-up on arctic structures is compared with the prediction of the Shore Protection Manual (SPM) method and those of linear and second-order diffraction analyses based on potential theory and boundary element method (BEM). Through this comparison, the effects of non-linearity and viscosity on wave loading and run-up are discussed.     

A practical surface panel method to predict velocity distribution around a three-dimensional hydrofoil including boundary layer effects

A practical, low order and potential-based surface panel method is presented to predict the flow around a three-dimensional rectangular foil section including the effect of boundary layer. The method is based on a boundary-integral formulation, known as the “Morino formulation” and the boundary layer effect is taken into account through a complementary thin boundary layer model. The numerical approach used in the method presents a strongly convergent solution based on the iterative wake roll-up and contraction model including the boundary layer effect. The method is applied to a three-dimensional foil section for which the velocity distribution around the foil was measured using a 2D Laser Doppler Velocimetry system in a large cavitation tunnel. Comparison of the predicted velocity distributions both inside and outside of the boundary layer of the foil as well as the boundary layer shapes obtained from the numerical model show fairly good correlation with the measurements, indicating the robustness and practical worthiness of the proposed method.     

The discrete methods for free vibration analyses of an immersed beam carrying an eccentric tip mass with rotary inertia

In this paper, a beam without contact with water is called the “dry” beam and the one in contact with water is called the “wet” beam. For a partially (or completely) immersed uniform beam carrying an eccentric tip mass possessing rotary inertia, the conventional analytical (closed-form) solution is achieved by considering the inertial forces and moments of the tip mass and rotary inertia as the boundary conditions at the tip end of the beam. However, it has been found that the approximate solution for the last problem may be achieved by two techniques: Method 1 and Method 2. In Method 1, the basic concept is the same as the conventional analytical method; but in Method 2, the tip end of the beam is considered as a free end, while the inertial forces and moments induced by the tip mass and rotary inertia are considered as the external loads applied at the tip end of the beam. The main differences between the formulation of Method 1 and that of Method 2 are: In Method 1, the “normal” shapes of the “dry” beam are functions of the frequency-dependent boundary conditions but the external loads at the tip end are equal to zero; On the contrary, in Method 2, the “normal” mode shapes of the “dry” beam are determined based on the zero boundary conditions at the tip end of the beam but the external loads at the tip end due to the inertial effects of the tip mass and rotary inertia must be taken into consideration for the free vibration analysis of the “wet” beam. Numerical results reveal that the approximate solution obtained from Method 2 are very close to that from Method 1 if the tip mass moment of inertia is negligible. Besides, the two approximate solutions are also very close to the associated analytical (closed-form) solution or the finite element solution. In general, it is hoped that there exist several methods for tackling the same problem so that one may have more choices to incorporate with the specified cases. It is believed that the two approximate methods presented in this paper will be significant from this point of view.