Slow Drift Oscillations Research Articles

Experimental study and numerical simulation on the slow-drift oscillation of a semi-submersible in irregular waves

This paper presents an experimental investigation on the slow-drift oscillation of a semi-submersible. The soft-moored model was subjected to two sea states for studying the slow-drift response in irregular waves. The effect of wave direction was considered by setting the vessel in head, quartering and beam sea. Tests under a series of regular waves were carried out for measuring mean drift forces. Viscous damping was measured with decay tests in still water. In addition, a numerical simulation of slow-drift oscillation was also performed to complement the experimental work. In the numerical work, quadratic transfer functions were computed by using Pinkster's theory, and Aranha's approximation was employed to calculate drift damping. The comparison between experimental and numerical results showed generally good agreement. Both experimental and numerical work demonstrated that maximum amplitudes of slow-drift oscillation may be larger in modest seas than in rough seas.

A Study on Examination of an Estimation Method for Slow Drift Oscillation Amplitude Based on the Results of Open Sea Tests of Offshore Floating Type Wave Devise“Mighty Whale”

The coast in Japan has been often attacked by the typhoon. And then, structures that have been installed there have been hit high waves and strong winds. Especially, for floating structures, safety measures with these high waves and strong winds have been most important subject for their design. In the case of a kind of floating structure positioning by slack mooring system using mooring lines, it has been at risk for breaking of mooring lines by slow drift oscillations contributed non-linear long period fluctuating wave drift forces. Therefore, it has been necessary to evaluate the slow drift oscillation in design process of a floating structure. However, it seems that the evaluation has been not fully carried out, for paucity of the data with slow drift oscillations on open sea test site, and for complexity of calculation method due to non-linearity of slow drift oscillations.So, in this paper, a practical estimation method with slow drift oscillation amplitude has been presented. And this estimation method has been examined for its validity based on the results of open sea tests of an offshore floating wave device“Mighty Whale”. Additionally, coefficients being used in this presented method have been examined.

Physical Modeling of Wind Load on a Floating Offshore Structure

Wind is an important environmental parameter that influences the design of floating offshore structures, particularly in harsh environment. Because wind spectrum is broad-banded, computation of wind load on the floating structure is complicated. Moreover, the wind-induced slow-drift oscillation is an important design criterion. Simulated environment in a model test often includes wind effect. Accurate modeling of wind in a laboratory environment is, however, a difficult task. The wind tunnel provides a steady load on the superstructure quite accurately, but fails to show the effect of the changing free surface as well as dynamic effect. Therefore, simultaneous simulation of wind in the wave basin is desirable. A weight representing the steady wind load with a string and pulley arrangement at the center of the application of the superstructure is inadequate since it fails to simulate the variation of the wind spectrum. The generation and control of the design wind spectrum by an overhead bank of fans facing the model superstructure is an extremely difficult task due to large windage area. This paper presents an accurate and highly controllable method of the generation of variable wind simultaneously with waves and current in the wave basin that can be used with a variety of floating structure model. The concept was originally proposed by Kvaerner Oil & Gas International and implemented by the offshore model basin (OMB). In this method, a fan equipped with a constant-speed motor and blades with an adjustable pitch angle is directly mounted on the model deck above water. A digital signal generated from the specified wind spectrum is used to run the fan much like the wavemaker. A feedback system ensures the proper generation of the wind with the model motion. The method was successfully applied in several model tests of deepwater floating structures in which broad-banded wind spectra were generated. An example from an earlier such test is given here. The importance of the effect of the simulated wind spectrum on floating structures should be clear to a design engineer from this example.

Low frequency wave force spectrum influenced by wave-current interaction

The interaction between ocean current with sea waves plays an important role in the determination of the steady wave forces acting in a floating system and, in the limit when the velocity is small, an exact formula obtained [J Fluid Mech. 272 (1994) 147; J Fluid Mech. 313 (1996) 39], relates the steady wave forces with the standard drift forces in the seakeeping problems. This result is applied here, in conjunction to Newman's approximation, to express the first order influence of wave-current interaction in the second order low frequency wave force spectrum used in the study of the slow drift oscillation of a floating system.

H∞ Controller Design for Dynamic Positioning of a Turret Moored FPSO

This paper presents the controller design of a dynamic positioning system for a FPSO-turret moored in deep water, using a H∞ methodology. The controller must counteract slow-drift oscillations due to waves and static offsets induced by current, wind and waves. The controller must not respond to zero mean high frequency motions caused by first order wave forces. To the controller synthesis, a linear model is used, but tests are carried out with a complete non-linear FPSO model. Simulations considering several environmental conditions and two distinct turret positions confirm the robustness of the control system.

Analysis of wave-drift damping of a VLFS with shallow draft

Numerical analysis of the wave-drift damping of VLFS as a floating elastic plate is presented. The source distribution method is used to analyze the drift force of the floating plate advancing with low-forward speed in waves. In the analysis, the shallow-draft assumption of the floating body is utilized; this assumption leads to the neglect of the steady disturbance field, which simplifies the analysis to a great extent. The consideration of the elastic deformation is made by modal expansion of the response, and the unit-amplitude radiation potentials are computed for each mode. The numerical results have confirmed that the wave-drift damping will have appreciable magnitude at the frequency region where the slow-drift oscillations are dominant. The formula for the wave-drift damping proposed by Aranha (J Fluid Mech 1996;313:39–54) has been compared with the present numerical results, but fundamental disagreement has been observed as reported by Finne and Grue (J Fluid Mech 1998;357:289–320).

A comparison of methods for the statistics of slow-drift oscillations

A probabilistic model for the slowly varying horizontal motion of a moored floating structure, excited by nonlinear, second order wave forces is derived. The model consists of a coupled system of three white-noise excited stochastic differential equations that govern a continuous vector Markov process. Different methods are applied for the investigation of the first passage time probabilities of the displacement of the marine structure. The methods include time history simulation estimation of the first four moments, time history simulation of the first passage problem, and finite element solution of the backward Kolmogorov equation governing the first passage time distribution. The impact of non-Gaussian aspects of the response on extreme value statistics is demonstrated.

Slow drift damping force acting on a horizontal cylinder

The hydrodynamic forces acting on a circular cylinder and a rectangular cylinder undergoing slow drift oscillation in regular waves were investigated experimentally and numerically. Forced oscillation tests with low frequency and large amplitude in regular waves and forced two-harmonic oscillation with combined low and high frequencies were carried out in the experimental study. In the numerical study a finite-difference method was used to simulate viscous flow around a two-dimensional oscillating cylinder. The results of experiments showed that a horizontal rectangular cylinder oscillating slowly in waves has much higher damping coefficients than one oscillating in two-harmonic mode, while the numerical study indicated that such differences in the damping coefficient arise partly from the different effects of one-direction harmonic flow and rotating flow on the vortex shedding caused by the slow drift oscillation.

Design and Model Experiments on Thruster Assisted Mooring System

The paper is concerned with a dynamic positioning control of a moored floating platform model. Moorings are used to maintain the position of the platform against quasi-constant external forces induced by current and wind. The control purpose is to suppress the slow drift oscillation due to drifting force in irregular waves using thrusters so that they do not respond to linear exciting force in wave frequency range. Control of motions in wave frequency region has to be ignored because the amplitude of those motions is smaller than those of the slow drift oscillation and enormous thruster power is needed to control them. The problem is formulated in the framework of H∞ control, and is solved using LMI (Linear Matrix Inequalities) Control Toolbox on Matlab ; some successful results are shown.

Statistical Analysis of Nonlinear Response of Moored Vessels in Random Seas

This paper presents statistical analyses for second-order responses of tension-leg platform.Second-order responses of moored vessels in random seas include two kinds of elements, difference-frequency response and sum-frequency response. It is well known that difference-frequency response leads to slow drift oscillation. Sum-frequency response is generally eliminated for an analysis of slack mooring.In the case of heaving motion of tension-leg platform, sum-frequency response is large because of its high natural frequency, and it leads to springing response. Non-normality of two kinds of responses are investigated, and compared these two responses for surging, heaving and pitching modes.Main results are as follows :1) Sum-frequency response of exciting force is a little larger than difference frequency response of it. Both of them have large non-normality.2) Sum-frequency response of heaving and pitching motions are much larger than difference, and surging shows opposite result. These three large responses have behavior similar to normal distribution.3) Gamma-distribution approximation and DVI method can be applied to sum-frequency response.4) Non-normality increases and standard deviation becomes small as damping increases.

Nonlinear mooring line induced slow drift motion of an ALP-tanker

A tanker moored to an articulated loading platform (ALP) is modeled by a 2-DOF system. The mooring line between the ALP and tanker is represented by an unsymmetric, piecewise-nonlinear restoring function. Forced vibrations of this system are investigated using direct numerical integration. The highly nonlinear characteristics, including multiple solution, subharmonics, and bifurcations are discussed. Slow drift oscillation could occur under harmonic excitations due to nonlinearities of the systems. The sensitivity of the dynamic response of the strong nonlinear system with respect to system parameters and initial conditions is examined.

Hydrodynamic Interaction among Multiple Floating Cylinders in Both Waves and Slow Current

The problem of interaction among fixed multiple floating cylinders in waves and currents is considered. The current velocity is assumed to be small. The problem is linearized based on a small parameter tau= U/g. Problems of different orders are solved for a single cylinder to determine the diffraction property of the cylinder, which gives the relationship between the incident wave amplitude and the diffracted wave amplitude. The results are then applied to the problems of multiple cylinders. The interaction among cylinders is represented by some additional incident waves emitted from the other cylinders towards the one under consideration. After solving the velocity potential near each cylinder, the hydrodynamic forces can be calculated by integrating the pressure along the wetted surface of each cylinder. It is found that current and steady potential have a strong effect on the wave-drift damping force. Also given is a procedure to evaluate the wave-drifting damping which plays an important role in the slow drift oscillation of moored ocean structures.

Hydrodynamic Coefficients of a Moored Semisubmersible in Waves

The hydrodynamic coefficients of a semisubmersible undergoing slow-drift oscillation were determined through a model test. The semisubmersible model was moored in head seas, fore and aft, with linear springs which were pretensioned and never became slack during any test run. At the beginning of each test run, the vessel was held at an initial displacement from its equilibrium position and then released, and the resulting line loads were recorded. The semisubmersible was tested in still water and in regular waves. The amplitude of the waves at a given period was varied. The added mass and damping of the semisubmersible were determined from the decayed oscillation of the loads. The semisubmersible experienced both linear and nonlinear damping. The hydrodynamic coefficients obtained from the semisubmersible as functions of wave height and period are compared with those found previously on a tanker.

A Numerical Study on Hydrodynamic Forces of an Oscillating Circular Cylinder in Harmonic Flow

The hydrodynamic forces acting on an oscillating circular cylinder in an uniform and harmonic flow are investigated through the use of direct solution of the incompressible Navier-Stokes equations in order to gain a deeper understanding of the nature of the slow-drift damping force of an ocean structure. In the calculations the time-dependent generalized coordinate system has been employed to obtain an accurate numerical representation of boundary conditions for a moving boundary, which is important for determining the hydrodynamic forces by integrating the pressure distribution on the cylinder surface. The computed results are compared with other experimental works and show good agreement. Several important conclusion regarding slow-drift damping and the characteristics of a two harmonic flow are obtained : in the range of small reduced velocity the high frequency oscillation of flow field obstructs the formation of an alternate vortex caused by steady flow or slow-drift oscillation, but in the range of large reduced velocity this oscillation excites the alternate vortex shedding. The values of the reduced velocity of the turning point can be determined from the Strouhal number of the cylinder in steady flow.An attempt to explain the mechanism of the viscous slow-drift damping force is also offered.

Drift Motions of a Floating Barge in Random Multi-Directional Waves

Abstract This paper presents results of a numerical and laboratory investigation into the mooring line forces and slow drift oscillations of large floating structures in multidirectional waves. A procedure for computing the spectral density of the second-order forces in random multi-directional waves based on the concept of a bidirectional, bifrequency quadratic transfer function is presented. Laboratory tests were carried out with a floating barge model, restrained horizontally by soft linear springs. The barge was subjected to random multi-directional waves with different degrees of directional spreading. The influence of wave directionality on the mooring line forces and low frequency motions is investigated by comparing results in unidirectional and multi-directional sea states with an identical frequency spectrum. The results indicate a significant reduction of the mean and standard deviation of the surge response, and an increased sway and yaw response. The mooring line forces were affected by wave directionality in a similar manner as the surge response.

Slow drift damping due to drag forces acting on mooring lines

Hydrodynamic forces acting on mooring lines are essential in the case of a slow drift oscillation of a floating structure, because drag forces acting on mooring lines act as the damping force and have a great effect on the slow drift oscillation. In this paper a time domain Modified Approximate Method for calculating the dynamic tension of mooring chain is presented. The calculated results of the slow drift oscillation of floating structure and dynamic tensions of mooring lines obtained by using this method are compared with the experimental ones. The calculated results are in good agreement with experimental ones.

Statistical theory of total second order responses of moored vessels in random seas

A statistical estimation of total second order responses of moored floating structures subjected to stationary Gaussian random waves is presented which includes slow drift oscillations. Interference effects of first and second order responses are also analysed. Under the assumption that the total second order responses are represented in the form of a two term Volterra functional series, a theory of probability density functions (p.d.f.'s) is developed for an instantaneous response and its maxima, in order to predict the 1/nth highest expected amplitude. New formulas for the total second order p.d.f.'s which include not only quadratic but also linear responses are derived. These new p.d.f.'s can be represented by the generalized Laguerre polynomials of which the first term is a Gamma p.d.f. consisting of three parameters. Assuming that the response and its time derivative processes are mutually independent, the 1/nth highest expected amplitude can be evaluated numerically from the derivative of the instantaneous response p.d.f. This method is first applied to the sway motions of moored floating semi-circular and rectangular two dimensional structures, and the applicability of the method is studied by comparisons with the Naess'exact solution. The variation of the 1/nth highest expected amplitude of the total second order response is then investigated following increases in damping and restoring forces.

Time Domain Simulation of Slow Drift Motion of a Floating Structure in Waves

Time domain simulations of slow drift motion are compared with model experiments in wave basin. The simulations include various slow motion hydrodynamic forces such as drag coefficient depending on Kc number, time varying wave drift damping, added mass change due to waves etc. as well as second order wave excitation. Most of them are given by slow motion forced oscillation tests in regular waves. The simulation and the model experiments are carried out for two types of floating structures moored in bi-chromatic or irregular waves.They show excellent agreement with the experiments. It is clarified that importance of each slow motion hydrodynamic forces on the slow drift oscillation in irregular waves of a moored floating vessel depends on a type of vessel. Wave drift damping affects significantly the tuned motion amplitude of not only a barge but also a semi-submersible. Added mass change due to waves affects the tuned frequency only for a semi-submersible.

On the Motion Responses of Offshore Floating Structures in Directional Spectra Waves

In the previous paper, the 1st-order motion responses of semi-submersibles in short-crested directional spectra waves were investigated. This paper presents results of an investigation into the 2nd-order responses such as slow drift oscillations in directional waves.Model tests in various kinds of directional waves which have specified power spectra and directional spreading, were carried out with spring moored and catenary moored conditions. Test results in directional waves are compared with those in uni-directional (long-crested) waves, and the influence of wave directionality on drift forces, low-frequency oscillations etc. are investigated.In order to estimate the 2nd-order responses in directional waves, new practical methods are proposed. These are based on the Pinkster's expression' of the low-frequency drift force spectrum in directional waves. By approximating the estimation of quadratic transfer functions of the mean drift force in two-directional cross regular waves, mean and low-frequency drift forces in directional spectra waves can be estimated practically. The applicability of these methods are investigated by comparing estimated 2nd-order sway responses with experimental measurements.

Mooring Line Damping for Slow Drift Oscillation of Semi-Submersibles

Dynamic tensions of mooring lines may be ignored in the estimation of wave frequency motions of a floating structure, but for slow drift oscillations hydrodynamic forces acting on mooring lines are important, because drag forces acting on mooring lines act as the damping force and have a great effect on the slow drift oscillation. When dynamic tension of mooring lines are calculated in full consideration of added mass forces and drag forces acting on mooring lines, a numerical calculation such as Lumped Mass Method is usually used. The numerical calculation needs prohibitively long computer time and it is not practical in the early stage of the design of a floating structure. The authors have developed an Approximate Method for calculating the dynamic tension taking the added mass force and drag force into consideration. The method has a short computation time and the accuracy has been confirmed by comparing calculated results with experiments. In this paper a Modified Approximate Method extended to the time domain problem is presented. The calculated results are compared with the experimental ones and they are found to be in good agreement.