Large-amplitude Nonlinear Motion Characteristics Of A Submerged Taut-moored Buoy Under Wave Action

Tightly moored floating structures are widely used in offshore engineering,especially in the development and application of wave energy.The tightly moored offshore structures under wave action will have a large-amplitude half-wave frequency response phenomenon.It is of great significance to study the mechanism of this phenomenon,not for the safety of the structures,but also to transform the large-amplitude motion into available energy.In the potential flow theory,based on the body-exact condition in the time domain,a numerical model using the high-order boundary element method(HOBEM)is developed to study the largeamplitude motion response of the floating body under wave action.In this model,the linear free water surface and instantaneous wet body surface conditions are satisfied,considering the largeamplitude motion of the floating body,the generalized minimum residual iterative method is used to solve the linear equation with time-varying coefficient matrix and relevant physical quantities are updated by time stepping method.Based on the established numerical model based on the body-exact condition in the time domain,the calculation accuracy of the radiation wave force is studied.When the floating body moves in the large amplitude,the results of different time-domain wave force calculation methods will be significantly different.In order to obtain relatively accurate results of radiation wave force,it is necessary to study the effectiveness and applicable conditions of these methods.It is found that for the smooth bodies,the five wave force calculation methods can get accurate and consistent results.For the body with sharp edges,the results of the five methods are different.A series of submerged bodies that have sharp edges and rounded corners with small radii are considered to investigate the five methods’ reliabilities.In addition to the square term of spatial derivatives of velocity potential,new terms related to the spatial derivatives of velocity potential are introduced in different ways.Due to the singularity of the spatial derivatives of velocity potential,new terms lead to different effect on the wave force calculation results of the five methods.The all above methods are affected significantly by the sharp edge,especially the acceleration potential mrthod(AcP)and the auxiliary potential method(AuP).The rate of change of the fluid momentum(RCM)is not affected by the sharp edge.For the sharp edge with small fillet processing,the results of the five methods have a good agreement.After determining the wave force calculation method for large-amplitude motion of the floating body,the nonlinear time-domain numerical model based on a body-exact approach is applied to study the problem of the large-amplitude motion of a tightly moored submerged sphere.Firstly,numerical simulations are carried out for two wave steepness,each with various incident wave frequencies.It is found that for a smaller wave steepness,the oscillation frequency of the sphere is always the same as the wave frequency,while for a larger steepness,surge motion of the sphere may oscillate with a large amplitude at the frequency that is half of the incident waves as the incident wave frequency is close to twice the resonance frequency.The linear model satisfying both the linearized boundary conditions on the mean body surface and still water surface fails to capture this significant nonlinear phenomenon.Trough further study,it is found that as the incident wave steepness is larger,the scope of the incident wave frequency is larger for the half-wave frequency response.Then,different radius and different gravity and pretension ratio of two models,A and B,have experimented under regular waves.On the one hand,the experiments verify the large-amplitude half-wave frequency response phenomenon.On the other hand,it is found that the periodic sway oscillation occurs only in one of the experiments,and the motion characteristics are similar to surge motion.Based on verifying the applicability of the numerical model established by the time-domain body-exact condition,the viscous damping force is changed from linear damping to square damping,and it is found that the numerical results of the model with square damping are in better agreement with the experimental results.In order to explain the half-wave frequency response phenomenon in both the numerical and experimental results,the dynamic response of the tightly moored submerged sphere is analyzed.The dynamic equations of the mooring sphere system are established,by expanded by Taylor series of relevant parameters.the linear and nonlinear equations of motions are obtained.As the elongation of the mooring line is constant,the linear motion equation can be transformed into a typical Mathieu parametric equation with second-order excitation.When the parametric resonance condition is satisfied,double-period oscillation in the transverse direction can be simulated by linear time-domain model and nonlinear time-domain model with bodyexact condition.Surge motion is not affected by the parametric resonance condition.Even if the parametric resonance condition is not satisfied,as the frequency of the incident wave is close to twice the resonance frequency,due to second-order excitation,surge displacement shows a large-amplitude half-wave frequency response phenomenon.It is found that the nonlinear stiffness variation of the tightly moored sphere system creates the condition that the half-wave frequency response occurs in a larger range of the incident wave frequency.According to the difference frequency of the second-order excitation,the second-order exciting force formed by the interaction between the wave and the half-wave frequency component of the motion and the interaction between the motions of the body in different freedoms,is the cause of the half-wave frequency response in the surge direction.