﻿ 泰勒展开边界元法的船舶兴波阻力计算
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 哈尔滨工程大学学报  2019, Vol. 40 Issue (5): 872-877  DOI: 10.11990/jheu.201809025 0

### 引用本文

CHEN Jikang, DUAN Wenyang, LI Jiandong, et al. Numerical calculation on wave-making resistance based on Taylor expansion boundary element method[J]. Journal of Harbin Engineering University, 2019, 40(5), 872-877. DOI: 10.11990/jheu.201809025.

### 文章历史

Numerical calculation on wave-making resistance based on Taylor expansion boundary element method
CHEN Jikang , DUAN Wenyang , LI Jiandong , HUANG Shan
College of Shipbuilding Engineering, Harbin Engineering University, Harbin 150001
Abstract: For moving the collocation points forward to simulate the wave elevation due to steady potential by the source method, this paper propose the Taylor Expansion Boundary Element Method (TEBEM) to construct the boundary integral equation. The first-order derivatives of steady potential are solved directly, which overcomes some defects of the source method, such as difference method for computing the first and second order derivatives of the steady potential, and it is difficulty to construct the boundary integral equation. The wave-making resistance on the Wigley and S60 hull at different forward speed is calculated. Compared with numerical results and experimental measurements, good agreement can be obtained for the numerical results including wave patterns and wave profiles. It isn't necessary to shift the location of collocation points through the TEBEM method involved in this paper, which avoids the influence of shifting distance, and improves the robustness of numerical method.
Keywords: Taylor expansion boundary element method    boundary integral equation    wave-making resistance    steady wave profiles    Wigley ship    S60 ship    collocation points    numerical method

1 兴波阻力数值模型 1.1 定常兴波的定解问题

1) 拉普拉斯方程：

 $\nabla^{2} \phi=0$ (1)

2) 物面条件：

 $\nabla \phi \cdot \boldsymbol{n}=0$ (2)

3) 自由面条件：在自由面z=ζ(x, y)上，满足自由表面条件为：

 $\frac{1}{2} \phi_{x}(\nabla \phi)_{x}^{2}+\frac{1}{2} \phi_{y}(\nabla \phi)_{y}^{2}+g \phi_{z}=0$ (3)

4) 辐射条件：船体远前方无波。

 $\phi=-U x+\mathit{\Phi}+\varphi$ (4)

 $\left\{\begin{array}{l}{\nabla^{2} \mathit{\Phi}=0} \\ {\frac{\partial \mathit{\Phi}}{\partial n}=U n_{x}} \\ {\frac{\partial \mathit{\Phi}}{\partial z}=0, \quad z=0} \\ {\mathit{\Phi}=0, \sqrt{x^{2}+y^{2}+z^{2}} \rightarrow \infty}\end{array}\right.$ (5)

 $\begin{array}{l}{\frac{1}{2} \mathit{\Phi}_{x}\left(\mathit{\Phi}_{x}^{2}+\mathit{\Phi}_{y}^{2}\right)_{x}+\frac{1}{2} \mathit{\Phi}_{y}\left(\mathit{\Phi}_{x}^{2}+\mathit{\Phi}_{y}^{2}\right)_{y}+} \\ {\mathit{\Phi}_{x}\left(\mathit{\Phi}_{x} \varphi_{x}+\mathit{\Phi}_{y} \varphi_{y}\right)_{x}+\mathit{\Phi}_{y}\left(\mathit{\Phi}_{x} \varphi_{x}+\mathit{\Phi}_{y} \varphi_{y}\right)_{y}+} \\ {\frac{1}{2} \varphi_{x}\left(\mathit{\Phi}_{x}^{2}+\mathit{\Phi}_{y}^{2}\right)_{x}+\frac{1}{2} \varphi_{y}\left(\mathit{\Phi}_{x}^{2}+\mathit{\Phi}_{y}^{2}\right)_{y}+g \varphi_{z}=0}\end{array}$ (6)

 $\left\{\begin{array}{l}{\nabla^{2} \varphi=0} \\ {\frac{\partial \varphi}{\partial n}=0} \\ {\varphi=0, \sqrt{x^{2}+y^{2}+z^{2}} \rightarrow \infty}\end{array}\right.$ (7)

1.2 兴波阻力与波面升高

 $p+\rho g z+\frac{1}{2} \rho(\nabla \phi)^{2}=p_{\infty}+\frac{1}{2} \rho U^{2}$ (8)

 $C_{p}=\frac{p-p_{\infty}}{0.5 \rho U^{2}}$ (9)

 $R_{w}=\iint\limits_{S_{\mathrm{B}}} p n_{x} \mathrm{d} S$ (10)

 $C_{\mathrm{w}}=\frac{R_{\mathrm{w}}}{(1 / 2) \rho U^{2} S_{\mathrm{w}}}=\frac{1}{S_{\mathrm{w}}} \sum\limits_{i=1}^{N_{\mathrm{B}}} C_{p}(i) n_{x i} \Delta S_{i}$ (11)

 $\zeta(x, y)=\frac{1}{2 g}\left(U^{2}-\mathit{\Phi}_{x}^{2}-\mathit{\Phi}_{y}^{2}-2 \mathit{\Phi}_{x} \varphi_{x}-2 \mathit{\Phi}_{y} \varphi_{y}\right)$ (12)
2 泰勒展开边界元方法

 $\varphi_{z}=f_{1}(\mathit{\Phi}) \varphi_{x}+f_{2}(\mathit{\Phi}) \varphi_{y}+f_{3}(\mathit{\Phi})$ (13)

 $2 \pi \varphi+\iint\limits_{S_{\mathrm{H}}+S_{\mathrm{F}}} \varphi \frac{\partial}{\partial n}\left(\frac{1}{r}\right) \mathrm{d} s=\iint\limits_{S_{\mathrm{F}}} \frac{1}{r} \frac{\partial \varphi}{\partial n} \mathrm{d} s$ (14)

 $\varphi(q)=\varphi\left(q_{0}\right)+\overline{\xi}\left.\varphi_{, \overline{\xi}}\right|_{q_{0}}+\overline{\eta}\left.\varphi_{, \overline{\eta}}\right|_{q_{0}}$ (15)

 $\begin{array}{l}{\iint\limits_{S_{\mathrm{F}}} \frac{1}{r} \frac{\partial \varphi}{\partial n} \mathrm{d} s=2 \pi \varphi+\iint\limits_{S_{\mathrm{H}}+S_{\mathrm{F}}} \overline{\eta} \frac{\partial \varphi}{\partial \overline{\eta}} \frac{\partial}{\partial n}\left(\frac{1}{r}\right) \mathrm{d} s+} \\ {\iint\limits_{S_{\mathrm{H}}+S_{\mathrm{F}}} \varphi \frac{\partial}{\partial n}\left(\frac{1}{r}\right) \mathrm{d} s+\iint\limits_{S_{\mathrm{H}}+S_{\mathrm{F}}} \overline{\xi} \frac{\partial \varphi}{\partial \xi} \frac{\partial}{\partial n}\left(\frac{1}{r}\right) \mathrm{d} s}\end{array}$ (16)

3 数值计算与分析

3.1 Wigley 4船数值结果

Wigley 4的船型方程为：

 $y=\frac{B}{2}\left(1-\frac{4 x^{2}}{L^{2}}\right)\left(1-\frac{z^{2}}{T^{2}}\right)$ (17)

 Download: 图 3 TEBEM方法自由面区域收敛性验证 Fig. 3 Free surface convergence verification of TEBEM

 Download: 图 5 Wigley船型兴波阻力系数曲线 Fig. 5 The curves of wave-making resistance coefficient of Wigley ship

 Download: 图 6 不同航速下Wigley船型的波面等高线图 Fig. 6 Wave contour map of Wigley ship at different Fr

 Download: 图 7 不同航速下wigley船型的船侧波面升高 Fig. 7 Wave height of ship side of Wigley ship at different Fr
3.2 S60船数值结果

S60船型(CB=0.6)是被ITTC组织认可的标准船型。学者基于该船型进了大量兴波阻力预报研究。S60半船划分793个四边形网格，自由面划分30×70个面元，并在船舶艏艉处适当加密。

 Download: 图 8 S60船型兴波阻力系数曲线 Fig. 8 The curve of wave-making resistance coefficient of S60 ship

 Download: 图 9 不同航速下S60船型的波面等高线图 Fig. 9 Wave contour map of S60 ship at different Fr
 Download: 图 10 不同航速下S60船型的船侧波面升高 Fig. 10 Wave height of ship side of S60 ship at different Fr
4 结论

1) 通过与试验值的比较可以看出，本文方法得到的计算结果与试验结果基本一致。

2) 泰勒展开边界元法可以有效地预报船舶兴波阻力与流场波形；且避免Dawson方法中需要将自由面配置点前移的缺陷。叠模势、兴波扰动势的一阶导数项求解无需差分，大大降低求解兴波扰动势的边界积分方程组难度。

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