Comparative study on ship motions in waves based on two time domain boundary element methods
Introduction
An accurate prediction of ship motions in waves is an important issue in ship early design. Not only the wave-induced motion has an effect on powering performance, but also the wave-induced drift forces may lead to poor manoeuvrability and even cause safety problems.
Early studies were mostly based on two-dimensional (2D) strip theory due to its practical applicability and high efficiency [1], [2], [3] within the assumptions of high encounter wave frequencies and low Froude numbers, therefore were not reliable for the complex hull with high forward speed. Buoyant requirements on more advanced methods along with the increased computing power thus promote the rapid development of three-dimensional (3D) panel methods based on potential flow theory.
Within the linear potential flow theory, the 3D panel methods for seakeeping problems can be classified into either Green's function method or Rankine source method. In the Green's function method, the integral only needs to be implemented on the body surface. Moreover, compared to solving Green's function in the frequency domain, the Transient Free surface Green's Function (TFGF) method is adopted in the time domain, which is less difficult to compute and can be more easily expanded when considering the forward speed and the nonlinearities. In terms of the TFGF method for linearized forward speed problems, the studies can be found in Liapis and Beck [4], [5], Magee and Beck [6], Bingham [7], Korsmeyer and Bingham [8], Zhu et al. [9], Sun et al. [10]. Efforts have also been made to account for the nonlinearities in time domain analysis, such as Lin and Yue [11], Singh and Sen [12]. Using the Rankine source method, Nakos [13] conducted the pioneering research on ship motion problems in the frequency domain. Variants of this method were then widely adopted for the seakeeping computations. Main changes were embodied in the progress from linear analysis to consideration of nonlinear factors, e.g., Kring [14], Huang [15], Kim and Kim [16], Kim et al. [17], Shao and Faltinsen [18], Söding et al. [19], Chen et al. [20]; and from frequency domain to time domain, e.g., Zhang et al. [21], Zhang and Zou [22], He et al. [23].
Despite the successful developments with the application of the Green's function method and the Rankine source method, the respective drawbacks of these two methods should also be recognized. One of the main concerns of the TFGF method is that the contributions of the steady flow components cannot be considered in the body boundary condition because of the uniform flow assumption. Moreover, the singularities near the free surface are also a tricky problem in terms of stability of solutions, especially for a ship with large flare; while the artificial damping zone on the truncated free surface is an important consideration when applying the Rankine source method. Though these drawbacks cannot get in the way of their respective applications for seakeeping analysis, only limited publications can be found that contain a concurrent comparative study of the two different methods, especially in terms of the accuracy of solutions.
The main objective of this study is to carry out a systematic comparison of the two time domain methods, so as to identify the dominant influences that lead to the differences in the computational accuracy of the numerical results, especially the hydrodynamic coefficients in low frequency ranges and the wave-induced motions near the resonant wavelengths. Another motivation is to develop a robust and efficient analysis tool for the accurate prediction of ship motions, so as to lay a foundation for further research into the accurate prediction of second-order wave forces, which have been shown to be of critical importance when dealing with the problem of ship manoeuvring in waves [24]. In this context, a TFGF method and a Rankine Higher Order Boundary Element Method (HOBEM) based on B-spline function are applied to study the motion responses of a ship advancing in waves. In the TFGF method, the Precise Integration Method (PIM) [25] is employed to evaluate the wave part of the transient Green's function. In the Rankine HOBEM, three different models based on different levels of linearization approximations are adopted for comparisons.
Section snippets
Mathematical formulation
It is assumed that the fluid is inviscid and incompressible, and the flow is irrotational. For a ship advancing with a forward speed U0 in a regular wave, a right-handed coordinate system o-xyz moving together with the ship is defined in the earth-fixed coordinate system O-XYZ. The o-xy plane is on the undisturbed free surface, with the x-axis positive pointing to the bow and the y-axis to the port side. The total velocity potential Ψ(x, y, z, t) can be written as:
Evaluation of BIE with the TFGF method
The BIE can be derived from Green's second identity:where IW is the waterline integral defined as:
The Rankine part G0 and the wave part of the transient Green's function are expressed as:where J0 is the zeroth order Bessel
Study objects
In order to verify the codes developed, the Wigley I [29] and Series 60 with a block coefficient CB = =0.7 [30] are chosen as study objects. Their main particulars are listed in Table 1, where L, B and T are the ship length, breadth and draught, respectively; ∇ is the displacement volume, xCoG and zCoG are the longitudinal and vertical coordinates of ship's gravity center, respectively; kyy is the radius of gyration.
In this paper, the hull surface and the free surface are discretized with
Conclusions
In this study, two time domain panel methods, a transient free surface Green's function (TFGF) method and a Rankine higher order boundary element method (HOBEM), are applied to comparatively study the motions of two ship models, the Wigley I and the Series 60 (CB = =0.7), advancing in head waves. The numerical results of wave exciting forces, hydrodynamic coefficients and ship motions are obtained by using the developed codes based on the two methods. From the analysis of the numerical results,
Acknowledgments
The first author gratefully acknowledges the financial support from China Scholarship Council (CSC), and the Lloyd's Register Foundation (LRF) through the joint center involving University College London, Shanghai Jiao Tong University, and Harbin Engineering University. LRF help protect life and property by supporting engineering-related education, public engagement, and the application of research.
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