Study on Characteristics of 3-D Translating-Pulsating Source Green Function of Deep-Water Havelock Form and Its Fast Integration Method

The singularities, oscillatory performances and the contributing factors to the 3-＇D translating-pulsating source Green function of deep-water Havelock form which consists of a local disturbance part and a far-field wave-like part, are analyzed systematically. Relative numerical integral methods about the two parts are presented in this paper. An improved method based on LOBATTO rule is used to eliminate singularities caused respectively by infinite discontinuity and jump discontinuous node from the local disturbance part function, which makes the improvement of calculation efficiency and accuracy possible. And variable substitution is applied to remove the singularity existing at the end of the integral interval of the far-field wave-like part function. Two auxiliary techniques such as valid interval calculation and local refinement of integral steps technique in narrow zones near false singularities are applied so as to avoid unnecessary integration of invalid interval and improve integral accordance. Numerical test results have proved the efficiency and accuracy in these integral methods that thus can be applied to calculate hydrodynamic performance of floating structures moving in waves.

Numerical study on wave loads and motions of two ships advancing in waves by using three-dimensional translating-pulsating source

A frequency domain analysis method based on the three-dimensional translating-pulsating （3DTP） source Green function is developed to investigate wave loads and free motions of two ships advancing on parallel course in waves. Two experiments are carried out respectively to mea- sure the wave loads and the free motions for a pair of side-by- side arranged ship models advancing with an identical speed in head regular waves. For comparison, each model is also tested alone. Predictions obtained by the present solution are found in favorable agreement with the model tests and are more accurate than the traditional method based on the three dimensional pulsating （3DP） source Green function. Numer- ical resonances and peak shift can be found in the 3DP pre- dictions, which result from the wave energy trapped in the gap between two ships and the extremely inhomogeneous wave load distribution on each hull. However, they can be eliminated by 3DTP, in which the speed affects the free sur- face and most of the wave energy can be escaped from the gap. Both the experiment and the present prediction show that hydrodynamic interaction effects on wave loads and free motions are significant. The present solver may serve as a validated tool to predict wave loads and motions of two ves- sels under replenishment at sea, and may help to evaluate the hydrodynamic interaction effects on the ships safety in replenishment operation.

Study on HOBEM Based on Analytical Panel Integrals Related to Translating-Pulsating Source for Hydrodynamic Responses of Vessels Sailing in Waves

A higher-order boundary element method(HOBEM)incorporated with analytical panel integrals related to translat-ing-pulsating source Green’s function is proposed for the hydrodynamic response prediction of ships advancing in waves.In this method,the 9-node bi-quadratic curvilinear elements employed to discretize the mixed-source/dipole boundary integral equation are mapped into the parametric plane through a coordinate transformation.Then in order to ease the numerical instability problem,a novel analytical quadrature is derived to calculate the influence coefficients by changing the integral order and using integration by parts.The singularity caused by infinite discontinuity is analyzed and eliminated by adopting some mathematical techniques.Through the calculations of panel integrals of Green’s function and its x-derivative,the analytical integral method is proved to be always accurate even for field points approaching the free surface,where numerical quadrature is impossible to give reasonable results.Based on this,a higher-order seakeeping program is developed and applied in the motion response prediction of two different types of ships(i.e.,a wall-sided ship Wigley III and a non-wall-sided ship S175).By comparing the computed results with the corresponding experimental data and numerical solutions of the translating-pulsating and higher-order Green’s function methods based on traditional Gauss quadrature,it is found that the HOBEM based on analytical quadrature is of better accuracy and stability.For the non-wall-sided ship,only the present method can produce reasonable pre-diction of motion responses,while obvious oscillatory phenomenon is observed in the results of the other two numerical methods based on Gauss quadrature.

Study on Far Field Wave Patterns and Their Characteristics of Havelock Form Green Function

A new mathematical integral representation including five integrals about the far field wave shape fimction of Havelock form translating-pulsating source is obtained by performing variable substitution. Constant-phase curves and propagation wave patterns are investigated by applying stationary phase analysis method to the new representation. Some findings are summarized as follows： （1） when 0〈r 〈0.25 （where r is the Strouhal number）, three types of stationary phase curves corresponding to three propagation wave patterns such as fan wave pattern, inner V and outer V wave patterns, are found in the integral representation. （2） When r 〉0.25, besides three types of wave patterns such as a ring-faning wave pattern, a fan wave pattern and an inner V wave pattern, a new one called parallel wave pattern is also found which not only exists in the integrals about the ring-fan wave and fan wave, but also in the integrals whose interval is [0, 7＇] ~ In addition, Characteristics about these parallel waves such as mathematical expressions, existence conditions, propagation directions and wave lengths are obtained, and cancellation relationships between these parallel waves are stated, which certificates the fact that there are no parallel waves existing in the far field.

Valid hydrodynamic interaction regions of multiple ships advancing in waves

Based on the 3-D surface panel method combined with the translating-pulsating source Green function, an approximate approach is developed to solve the hydrodynamic interacting problem of multiple ships advancing parallel in waves. Focus is on improving the calculating efficiency. In this approach, each ship is assumed to be in each other＇s far-field, and the near-field term in this Green function is neglected if the source point falls on one ship and the field point on others. Further, a matching relationship between the far-field waves and the interfered regions, which are defined as the overlapping areas between the mean wetted body surface of one ship and the propagating regions of the waves generated by another ship, is introduced to avoid the unnecessary computation of the relative terms of the Green function, if the field point is not in the overlapping areas. The approach is validated through studying the hydrodynamic terms and the free motions of two or three ships in side-by-side arrangement by comparing the obtained results with the model tests and the predictions of the exact method. The average calculating speed for the present approximate method is about 1.65-1.8 times of that for the exact method for solving the hydrodynamic interaction problem of two ships, and 2.56-2.73 times for that of three ships.