﻿ 基于涡环栅格法的三体船斜拖水动力数值分析
 舰船科学技术  2018, Vol. 40 Issue (4): 22-26 PDF

Numerical analysis of trimaran oblique towing hydrodynamic derivatives based on vortex lattice method
WANG Hong-dong, YI Hong, YU Ping
State Key Laboratory of Ocean Engineering, Shanghai Jiaotong University, Shanghai 200240, China
Abstract: The maneuverability of trimaran is set as the background of this paper. Based on the 3D panel method of potential flow theory, the oblique towing motion of trimaran is stimulated, and the hydrodynamic derivatives is calculated. Vortex lattice method (VLM), which is traditionally used to calculate the lift force of wings, is used for the numerical stimulation of trimaran oblique towing motion. Ship hull is derived into many quadrilateral panels, and vortex lattice is placed in every panels and trailing vortex plane. By unpenetrated condition of the hull and Kutta Condition in the trailing edge, the vorticity of every panel could be calculated, and the displacement of pressure on the hull surface could be obtained. Then the lateral force and moment around Z direction could be obtained. By the obtained result, the hydrodynamic derivatives which is related with the drift angle could be calculated, and be used for comparison with the hydrodynamic derivatives which is calculated by software.
Key words: trimaran     dirigibility     vortex lattice method     horizontal force     yaw moment
0 引　言

1 数学模型 1.1 坐标系

 图 1 船体坐标系 Fig. 1 Hull coordinate system
1.2 控制方程以及边界条件

 $\varPhi = x \cdot {V_{0{{x}}}} + y \cdot {V_{0y}} + z \cdot {V_{0z}} + \varphi \text{，}$ (1)

 $\left \{\begin{array}{l}{\nabla ^2}\varphi = 0\text{，}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{\text{流场中}}} \right)\text{；}\\{{\partial \varphi } / {\partial n}} = - {{{V}}_0} \cdot {{n}}\text{，}\;\;\;\;\;\;\left( {{\text{物面上}}} \right)\text{；}\\{{\partial \varphi } / {\partial n}} = 0\text{，}\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{\text{自由面上}}} \right)\text{；}\\\varphi \to 0\text{，}\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left( {{\text{远方辐射条件}}} \right)\text{。}\end{array}\right.$ (2)

 ${p_{TE}}^ + = {p_{TE}}^ - \text{，}\;\;\;\;\;\left( {{\text{船体尾缘库塔条件}}} \right)\text{。}$ (3)

1.3 方程的离散与求解

 $\varphi (p) = - \frac{1}{{4\pi }}\int_s {m(q)\frac{\partial }{{\partial n}}} \frac{1}{{r(p,q)}}{\rm d}S\text{，}$ (4)

 $\begin{split}\varphi (p) = & - \frac{1}{{4\pi }}\sum\limits_{j = 1}^N {{m_j}(q)\int_{{s_j}} {\frac{\partial }{{\partial n}}\frac{1}{{r(p,q)}}{\rm d}S} } - \\&\frac{1}{{4\pi }}\sum\limits_{k = 1}^L {{m_k}(q)\int_{{s_k}} {\frac{\partial }{{\partial n}}\frac{1}{{r(p,q)}}{\rm d}S} } \text{。}\end{split}$ (5)

 ${{v}}(p) = - \frac{1}{{4\pi }}\sum\limits_{j = 1}^N {{m_j}} \oint_{{l_j}} {\frac{{{{r}} \times {\rm d}l}}{{{r^3}}}} - \frac{1}{{4\pi }}\sum\limits_{k = 1}^L {{m_k}} \oint_{{l_k}} {\frac{{{{r}} \times {\rm d}l}}{{{r^3}}}} \text{。}$ (6)

 图 2 涡格模型示意图 Fig. 2 Schematic diagram of voterx model

 ${m_w} = {m_{{\text{上}}}} - {{\rm{m}}_{{\text{下}}}}\text{，}$ (7)

 ${{v}}(p) = \sum\limits_{j = 1}^N {{m_j}} {{{C}}_j}(p)\text{。}$ (8)

 ${{{n}}_{{p_i}}} \cdot \sum\limits_{j = 1}^N {{m_j}} {{{C}}_j}({p_i}) = - {{{n}}_{{p_i}}} \cdot {{{v}}_\infty }\text{。}$ (9)

 $\sum\limits_{j = 1}^N {{K_{ij}}} {m_j} = {B_i} , i = 1,2......,N\text{。}$ (10)

 ${{{C}}_j}({p_i}) \!\!=\!\! \left\{\!\! \begin{array}{l}{{C}}_j'({p_i}) \text{，}\!\!\!\!\! {\text{不与马蹄涡相连单元}}\text{；}\\{{C}}_j'({p_i}) \!\!+\!\! {{{H}}_j}({p_i}) \text{，}\!\!\!\!\!{\text{与马蹄涡相连的单元}}\text{。}\end{array} \right.$ (11)

 $\begin{split}{{v}}({p_i}) = & - \frac{1}{2}gradm({p_i}) - \frac{1}{{4\pi }}\int_S {{n}} \times gradm(q) \times \\ &\frac{{{r}}}{{{r^3}}}{\rm d}S - \frac{1}{{4\pi }}\oint_l {m(q)\frac{{{r}}}{{{r^3}}}} \times {\rm d}l\text{，} \end{split}$ (12)

 $\frac{{\partial m}}{{\partial x}} = \frac{{{m_2} - {m_1}}}{{{l_1}}};\;\frac{{\partial m}}{{\partial y}} = \frac{{{m_3} - {m_1}}}{{{l_2}}}\text{。}$ (13)

 \begin{align}\frac{{\partial m}}{{\partial x}} \!=\! gradm \cdot {{{e}}_1} \!=\! {h_1} \!+\! {h_2}{{{e}}_1} \cdot {{{e}}_2};\\\frac{{\partial m}}{{\partial y}} = gradm \cdot {{{e}}_2} \!=\! {h_2} \!+\! {h_1}{{{e}}_1} \cdot {{{e}}_2}\text{。}\end{align} (14)

${{{e}}_1} \cdot {{{e}}_2} = {g_{12}}$ ，于是联立上式可以解得h1h2，进而得到式（14）第1项。解得各分布点速度后，可得由伯努利方程到物面的压力分布 ${C_{{p_i}}}$ ，再由可控制点的向径pi，由式（15）和式（16）求得船体受到的横向水动力和转首水动力矩：

 ${{{Y}}_h} = \sum\limits_{i = 1}^N {{p_i}{S_i}{{{n}}_{yi}}} \text{，}$ (15)
 ${{{N}}_h} = \sum\limits_{i = 1}^N {{S_i}} {{{p}}_i} \times ({p_i}{{{n}}_{xi}} + {p_i}{{{n}}_{yi}})\text{。}$ (16)
2 三体船斜航数值模拟及计算

 ${\rm{y}} = \frac{B}{2}\left[ {1 - {{\left( {\frac{x}{{L/2}}} \right)}^2}} \right]\left[ {1 - {{\left( {\frac{z}{T}} \right)}^2}} \right]\text{。}$ (17)

 图 3 三体船模网格 Fig. 3 Grid of trimaran model

 图 4 计算结果对比图 Fig. 4 Contrast figure of calculation results

3 结　语

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