A NOVEL METHOD FOR THE M-TERMS OF SHIP WITH FORWARD SPEED

One of the major difficulties in linear wave-induced ship motion problem with forward speed is how to solve the m-terms accurately. This paper proposes a novel numerical method (Taylor Expansion Boundary Element Method, TEBEM) to compute the m-terms for arbitrary floating bodies. This method treats the m-terms as the Dirichlet type, uses the first-order derivatives terms on the right-handed side of boundary value problem, which is solved by TEBEM method. Numerical studies are performed for the hemisphere, mounted cylinder, and modified KVLCC2 ship models. Compared to the analytical solutions and other numerical results, a good agreement can be obtained by the TEBEM method.


Introduction
When the linear wave-induced ship motion problem with forward speed is considered, the interaction between the local steady flow and unsteady wave field should be considered in both the body surface and free surface boundary condition for the unsteady boundary value problem (BVP). The m-terms involved in the body surface condition is the second order derivatives of the velocity potential substantially. It is difficulty to calculate accurately for any ship body. With the increase in computer power, many researchers choose the CFD method to study the ship motion and added resistance, such as Lee et al [1], and avoid to calculate the m-terms involved in the BVP of ships with forward speed. Faltinsen (1974) [2] considered the influence of the m-terms in the calculation of the added mass and damping coefficient of the ships with forward speed. Many references have revealed the importance of m-terms, such as Inglis and Price (1981) [3], Iwashita and Bertram (1997) [4], Chen and Malenica (1998) [5], Duan and Price (2002) [6] etc. Iwashita and Bertram (1997) [4] pointed out the influence of m-terms on the wave pressure near the bow region. Chen and Malenica (1998) [5] found m-terms play an important role in the solutions of the added mass and damping coefficient by the numerical method inspired from Wu (1991) [7]. Duan and Price (2002) [5] even found that the local steady flow make a significant contribution to the slender body around the bow and stern.
The m-terms are usually neglected when taking the steady potential as the incoming uniform stream for the slender ship under the assumption of high speed. However, the impact cannot be ignored for the blunt ship with low forward speed. There are several attempts to deal with m-terms, which can be divided into the indirect method and direct method. For the indirect method, Bai (2000) [8], Teng et al. (2002) [9]and Kim et al. (2011) [10] all applied a modified Stokes formula to rewrite the effect of the second-order derivatives in m-terms in terms of the first-order derivatives and the corresponding waterline integral which involves additionally the first-order derivatives of the Green function. Shao and Faltinsen (2012) [11] calculated the added resistance of the ships under the body-fixed coordinate system without the second-order derivatives of the steady velocity potential on the body surface condition.
For the direct method, Zhao and Faltinsen (1989) [12] calculated the second-order derivatives at these points, which are offset in the fluid domain, because the singularity of Rankine source is weaken away from the boundary surface. The m-terms on the mean wetted body surface can be obtained by extrapolation. Raven (1996) [13] applied the desingularized panel method to calculate the wave-making resistance, whose kernel is making the singularities offset outside of the fluid domain to avoid the difficulty of dealing with the singularity of Green function, then the derivatives can be computed by differentiating the Green function. Nonetheless, the offset distance should be set carefully. Wu (1991) [7] handled the m-terms as Dirichlet type on the basis that the first-order derivatives could be accurately computed. Bingham and Maniar (1996) [14] represented the geometry and velocity potential by sixth-order B-splines method, which is solved by the higher-order boundary element method (HOBEM), then the first-order derivatives and m-terms could be evaluated by differentiating the shape functions. These methods could only offer satisfactory accuracy for the wave-body-current interactions problem without sharp corners.
How to calculate the m-terms for arbitrary floating bodies is the purpose of this paper. A novel BEM method is proposed to solve the BVP, named the Taylor Expansion Boundary Element Method (TEBEM), which can calculate the potential and velocity accurately for non-smoothed boundary, such as the ship structures. Then solve the second-order derivatives as solutions of a problem of the Dirichlet type. The mathematical formulations of m-terms are briefly reviewed in section 2. The scheme for calculating m-terms by the TEBEM is discussed in section 3. In section 4, the numerical issues of the hemisphere, the mounted cylinder, modified KVLCC2 ship models will show the superiority of the TEBEM.

Mathematica formulas for m-terms
in the whole fluid domain.
2) The kinematics condition on the mean wetted body surface For unsteady boundary value problem, the condition on the body surface is: Where  and  are both first-order infinitesimal quantities, but the steady velocity   Then wn  can be expressed on the static water plane as following: Finally, the body surface boundary condition for unsteady boundary value problem can be written as: The well-known m-terms in details are given as While based on Neumann-Kelvin (NK) assumption, the m-terms can be obtained as follows:

The scheme for calculating m-terms by the TEBEM
Here, the steady potential and its first-order derivatives have been solved accurately for arbitrary floating body by the TEBEM, which can be obtained from reference [15]. Then the m-terms can be solved based on the accurate first-order derivatives. The scheme is shown as following.
Firstly, it introduces a new potential function For this Dirichlet type BVP, the other 2N equations are needed to form the closed equations. Hence, the tangential first-order derivatives of the field point p at x and y direction for Eq. (7) are adopted and the Taylor expansion procedure as Eq. (14) is applied to the dipole strength. The supplementary equations are ( , ) Where all the influence coefficients can be evaluated analytically on each element. The procedure to compute the influence coefficient is shown in the reference [15]. The gradients of can be obtained by solving the combining equation set (15), (16) and (17). Then we also

Hemisphere model
To examine the validity of TEBEM method, the case of a hemisphere of radius 1.0 Rm = floating in infinite depth is considered. The analytical solution of the double-body flow velocity potential  for hemisphere can be expressed as: Where ( ) The present TEBEM has only j=1 convergence rate and the slope is 0.46 for m3 term. We could find the TEBEM method could retain the contribution of first and second order derivatives of the dipole strength, which could improve the accuracy and convergence rate of the velocity potential, the corresponding information can be referred to Duan (2015) [15].
However, the m-terms involve the second-order derivatives of the velocity potential.
Compared with the velocity potential, the accuracy of second-order derivatives would decrease, as expected. . The detailed information is illustrated in Fig. 7. Here, there are 500 and 2000 quadrilateral discretization panels on the body and free surface, respectively. The analytical solution of the double-body flow velocity potential  for the mounted cylinder can be expressed as, Where ( )

KVLCC2 ship model
The influence of the m-terms for ship motion is discussed in this section. Here the modified KVLCC2 ship is taken as an example. Yasukawa (2019) [17] developed the seakeeping and manoeuvring research in the physical tank. The main dimensions of the modified KVLCC2 ship are shown in Table 1 as following:  Fig.11. Comparison of ship pitch motion amplitudes between DB and NK scheme by TEBEM method for different forward speed.

Conclusions
This paper introduces a new method, named the TEBEM method, to solve the m-terms precisely, which involves in the wave-induced motion problem of the floating body with A novel method for the J.K. Chen m-terms of ship with forward speed W.Y. Duan 107 forward speed. It is validated by the numerical results of the hemisphere, the mounted cylinder, and the modified KVLCC2 ship models. The main conclusions can be obtained are as follows: (1) Compared with the analytical solutions of the hemisphere and the mounted cylinder, it is demonstrated that the proposed method could compute precisely m-terms for any boundaries with sharp corners.
(2) For DB assumption, a good agreement can be obtained for the unsteady ship motion.
Large difference is shown in the resonant frequency domain notably. The greater the forward speed, the more important role m-terms play in ship motions.

Acknowledgement
The authors acknowledge financial support from the National Natural Science Foundation of China (Grant No. 51709064，51679043，51779050) and the Numerical Tank Project sponsored by the Ministry of Industry and Information Technology (MIIT) of P.R. China.