﻿ 多极子面元法近水面椭球体兴波时域研究
 舰船科学技术  2018, Vol. 40 Issue (8): 14-22 PDF

Multipole panel method for the time-domain wave making research for ellipsoid near free surface
SHEN Wang-gang, ZHENG Yao-kun, LIN Zhi-liang
State Key Laboratory of Ocean Engineering School of Naval Architecture Ocean and Civil Engineering, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: Using Rankine source panel method for solving wave making problems of submerged body has certain advantages, but with the expansion of the scale of the problem, the solving efficiency will decline rapidly, making it difficult to applied to large-scale problem and time-domain research efficiently. Combining the traditional panel method with the fast multipole method can overcome this limitation, and the computation efficiency and calculation scale will be greatly improved. In this paper, the multipole panel method will be used to calculate the wave making problem of submerged body in time domain, and compared with the traditional panel method to prove its efficiency; the results will also be compared with previous studies to prove its accuracy.
Key words: Rankine source     panel method     wave making     time domain     fast multipole method
0 引　言

1 潜体兴波问题描述
 图 1 计算流场示意图 Fig. 1 The schematic of computational flow field

 ${\nabla ^2}\varphi = 0\text{。}$ (1)

 $\frac{{\partial \varphi }}{{\partial n}} = {{U}} \cdot {{n}}\text{，}$ (2)

 $\frac{{\partial \eta }}{{\partial t}} = - \tilde \nabla \varphi \cdot \tilde \nabla \eta + \frac{{\partial \varphi }}{{\partial z}} , \; z = \eta (x, y, t)\text{，}$ (3)

 $\frac{{\partial \varphi }}{{\partial t}} = - g\eta - \frac{1}{2}{\left| {\nabla \varphi } \right|^2}, \; z = \eta (x, y, t)\text{。}$ (4)

 $\varphi = 0, \; \frac{{\partial \varphi }}{{\partial t}} = 0 , \; \eta = 0\text{。}$ (5)
2 传统面元法求解

 $\varphi ({{x}}) = \sum\nolimits_{i = 1}^N {\frac{{{\sigma _i}}}{{4\pi }}} \iint_{{S_i}} {\frac{1}{r}{\rm d}S} , \; {{x}} \in \varOmega \text{，}$ (6)

 $\nabla \varphi ({{x}}) = \sum\nolimits_{i = 1}^N {\frac{{{\sigma _i}}}{{4\pi }}} \iint_{{S_i}} {\nabla \frac{1}{r}{\rm d}S}, \; {{x}} \in \varOmega \text{，}$ (7)

 $\frac{{\partial {\varphi _i}}}{{\partial {n_i}}} \!=\! \sum\nolimits_{j = 1}^N {\frac{{{\sigma _j}}}{{4\pi }}} \iint_{{S_j}} {\frac{{\partial (\frac{1}{r})}}{{\partial {n_i}}}{\rm d}S} \!= \!{{U}} \cdot {{{n}}_{i}}, \; (i \!=\! 1, \cdots , {N_B})\text{，}$ (8)
 ${\varphi _i} \!=\! \sum\nolimits_{j = 1}^N {\frac{{{\sigma _j}}}{{4\pi }}} \iint_{{S_j}} {\frac{1}{r}{\rm d}S} \!=\! 0, \; (i \!=\! {N_B} \!+\! 1, \cdots , {N_B} \!+\! {N_F})\text{。}$ (9)

 ${ A}\sigma = B \text{，}$ (10)

 $p = - \rho \frac{{D\varphi }}{{Dt}} - \frac{1}{2}\rho {\left| {\nabla \varphi } \right|^2} + \rho \nabla \varphi \cdot {{U}} - \rho gz\text{，}$ (11)

 ${R_L} = - \sum\nolimits_{i = 1}^{{N_B}} {\iint_{{S_i}} {{p_i} \cdot {n_{iz}}{\rm d}S}} \text{，}$ (12)
 ${R_W} = - \sum\nolimits_{i = 1}^{{N_B}} {\iint_{{S_i}} {{p_i} \cdot {n_{ix}}{\rm d}S}} \text{。}$ (13)

 ${C_L} = {R_L}/\frac{1}{2}\rho {U^2}{S_B},\;\;{C_w} = {R_w}/\frac{1}{2}\rho {U^2}{S_B}\text{。}$ (14)

 $\frac{{{\varphi ^{[n + 1]}} - {\varphi ^{[n]}}}}{{\Delta t}} = {{(\partial \varphi } / {\partial t}}{)^{[n]}}\text{，}$ (15)
 $\frac{{{\eta ^{[n + 1]}} - {\eta ^{[n]}}}}{{\Delta t}} = {{(\partial \eta } / {\partial t}}{)^{[n]}}\text{。}$ (16)

$t = (n + 1) \cdot \Delta t$ 时刻，将物体向运动方向移动 $U \cdot \Delta t$ 距离，更新物面和自由水面面元，将 ${\varphi ^{[n + 1]}}$ ${\eta ^{[n + 1]}}$ 作为已知变量，重复上述过程，最终可获得潜体在整个计算时间内的兴波波形，以及升力和阻力的变化情况。

3 多极子面元法

3.1 多极子法主要思想
 图 2 多极子计算示意图 Fig. 2 The schematic of multipole calculation

 图 3 多极子展开关键点 Fig. 3 Related points in multipole expansion

3.2 多极子法基本公式

 $\begin{split}G({{x}}, {{y}}) = & \frac{1}{{4\pi r}} = \frac{1}{{4\pi }}\sum\limits_{n = 0}^\infty \sum\limits_{m = - n}^n \overline {{S_{n, m}}} ({{x}} - {{{y}}_{c}}) \times \\& {R_{n, m}}({{y}} - {{{y}}_{c}} ) , \; \left| {{{x}} - {{{y}}_{c}}} \right| > \left| {{{y}} - {{{y}}_{c}}} \right|\text{，}\end{split}$ (17)

 ${R_{n, m}}({{x}}) = \frac{1}{{(n + m)!}}P_n^m(\cos \theta ){e^{im\varphi }}{r^n}\text{，}$ (18)
 ${S_{n, m}}({{x}}) = (n - m)!P_n^m(\cos \theta ){e^{im\varphi }}\frac{1}{{{r^{n + 1}}}}\text{。}$ (19)

 $P_n^m(x) = {(1 - {x^2})^{\textstyle\frac{m}{2}}}\frac{{{\rm d}m}}{{{\rm d}{x^m}}}{P_n}(x)\text{，}$ (20)

 $\begin{split} {F_i}({{x}}, {{y}}) =& \frac{{\partial G({{x}},{{y}})}}{{\partial {x_i}}} = \frac{1}{{4\pi }}\sum\limits_{n = 0}^\infty {\sum\limits_{m = - n}^n {\overline {{S_{n,m}}} ({{x}} - {{{y}}_c}) \times } }\\ & \frac{{\partial {R_{n,m}}({{y}} - {{{y}}_c})}}{{\partial {x_i}}},\left| {{{x}} - {{{y}}_c}} \right|\left. > {{{y}} - {{{y}}_c}} \right|\text{，}\end{split}$ (21)

 $\begin{split}\iint_{{S_i}} {G({{x}}, {{y}}){\sigma _i}{\rm d}S} =& \frac{1}{{4\pi }}\sum\limits_{n = 0}^\infty {\sum\limits_{m = - n}^n {\overline {{S_{n,m}}} ({{x}}, {{y}}) \times } } \\& {M_{n,m}}({y_c}),\left| {{{x}} - {{{y}}_c}} \right|>\left| {{y}} - {{{y}_c}} \right| \text{，}\end{split}$ (22)
 $\begin{split} \iint_{{S_i}} {{F_i}({{x}}, {{y}}){\sigma _i}{\rm d}S} =& \frac{1}{{4\pi }}\sum\limits_{n = 0}^\infty {\sum\limits_{m = - n}^n {\overline {{S_{n,m}}} ({{x}} - {{{y}}_c})} } \times \\&{{\tilde M}_{n,m}}({{{y}}_c}),\;\left| {{{x}} - {{{y}}_c}} \right|>\left| {{{y}} - {{{y}}_c}} \right| \text{。}\end{split}$ (23)

 ${M_{n, m}}({{{y}}_{c}}) = \iint_{{S_i}} {{R_{n, m}}({{y}} - {{{y}}_{c}}){\sigma _i}{\rm d}S}\text{，}$ (24)
 ${\tilde M_{n, m}}({{{y}}_{c}}) = \iint_{{S_i}} {\frac{{\partial {R_{n, m}}({{y}} - {{{y}}_{c}})}}{{\partial {x_i}}}{\sigma _i}{\rm d}S}\text{。}$ (25)

 $\begin{split} {M_{n,m}}({{{y}}_{c}}) \!=\! & \iint_{{S_i}} {{R_{n,m}}({{{y}}_{c}}){\sigma _i}{\rm d}S} \!=\!\! \\& \sum\limits_{n' = 0}^n {\sum\limits_{m' \!=\! - n'}^{n'} {{R_{n',m'}}({{{y}}_{c}} - {{{y}}_{c}}) \cdot {M_{n \!-\! n',m \!-\! m'}}({{{y}}_{c}}} } )\text{，}\end{split}$ (26)

 $\!\!\!\!\!\!\!\!\iint_{{S_i}} {G({{x}}, {{y}}){\sigma _i}{\rm d}S} = \frac{1}{{4\pi }}\sum\limits_{n = 0}^\infty {\sum\limits_{m = - n}^n {{R_{n, m}}({{x}} - {{{x}}_{{L}}}) \cdot {L_{n, m}}({{{x}}_{{L}}}} } )\text{，}$ (27)

 $\begin{split}{L_{n,m}}({{{x}}_{L}}) = & {( - 1)^n}\sum\limits_{n' = 0}^n {\sum\limits_{m' = - n'}^{n'} {\overline {{S_{n + n',m + m'}}} ({{{x}}_{L}} - {{{y}}_c}) \times } } \\& {M_{n',m'}}({{{y}}_{c}}), \left| {{{{y}}_{c}} - {{{x}}_{L}}} \right| > \left| {{{x}} - {{{x}}_{L}}} \right| \text{。}\end{split}$ (28)

 ${L_{n, m}}({{{x}}_{{{L'}}}}) = \sum\limits_{n' = n}^\infty {\sum\limits_{m' = - n'}^{n'} {{R_{n' - n, m' - m}}({{{x}}_{{{L'}}}} - {{{x}}_{{L}}}) \cdot {L_{n', m'}}({{{x}}_{{L}}}} } )\text{，}$ (29)

3.3 多极子面元法计算步骤

 图 4 正方形单元相互关系 Fig. 4 Relations between square cells

 图 5 向上和向下传递过程 Fig. 5 Upward and downward passes

4 计算结果与分析 4.1 近水面回转椭球体匀速直航兴波问题

 图 6 椭球网格划分 Fig. 6 The mesh generation of elipsoid

 ${(\Delta t)^2} \leqslant \frac{8}{\pi }\frac{{\Delta x}}{g} , \; \Delta x \leqslant \frac{8}{\pi }\frac{{U_x^2}}{g}\text{。}$ (30)

4.1.1 自由面静网格

 图 7 t=8 s时波面图 Fig. 7 The wave surface at t=8 s

 图 8 t=8 s时自由面二维波形图（y=0） Fig. 8 The 2D waveform of free surface at t=8 s (y=0)

 图 9 兴波水动力参数随时间变化情况 Fig. 9 The change of wave making hydrodynamic parameters over time

4.1.2 自由面动网格与网格密度

 图 10 不同物面网格下兴波水动力参数随时间变化情况 Fig. 10 The change of wave making hydrodynamic parameters over time in different object surface mesh

4.1.3 计算效率对比

 图 11 多极子面元法和传统面元法单步计算效率比较 Fig. 11 Comparison of single-step computational efficiency between FMBEM&BEM
4.2 不同航速及潜深下椭球定常兴波
 图 12 兴波水动力参数 Fig. 12 The wave making hydrodynamic parameters

5 结　语

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