﻿ 无限水深下波浪与二维水面物体作用的简单格林函数方法
 海洋与湖沼  2022, Vol. 53 Issue (4): 822-829 PDF
http://dx.doi.org/10.11693/hyhz20211200318

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#### 文章信息

TENG Bin, YU Mei. 2022.

A BEM WITH SIMPLE GREEN'S FUNCTION FOR WAVE INTERACTION WITH A 2D BODY AT THE SURFACE OF INFINITE WATER

Oceanologia et Limnologia Sinica, 53(4): 822-829.
http://dx.doi.org/10.11693/hyhz20211200318

### 文章历史

A BEM WITH SIMPLE GREEN'S FUNCTION FOR WAVE INTERACTION WITH A 2D BODY AT THE SURFACE OF INFINITE WATER
TENG Bin, YU Mei
Dalian University of Technology, State Key Laboratory of Coastal and Offshore Engineering, Dalian 116024, China
Abstract: To understand the interaction of waves with a two-dimensional surface body in infinite water depth, the traditional wave Green's function has complex form and slow calculation. In order to improve the calculation efficiency and accuracy, the watershed was divided into inner domain around the object and outer domain far away from the object. Simple Green's function method was adopted in the inner domain, and multi-pole expansion method was adopted in the outer domain. The velocity potential of any point in the watershed can be obtained through coupling solution by matching inner and outer domain boundaries. The wave excitation force, additional mass, radiation damping and transmission and reflection coefficients of the object under wave action can also be calculated. The method was applied to calculate two-dimensional water surface floating semicircle and water surface floating square box, and the numerical results show that the method can conveniently, accurately and quickly calculate the interaction between waves and arbitrary floating objects in infinite water depth.
Key words: infinite water depth    boundary element method (BEM)    simple Green's function    multipole expansion

1 数学模型和数值计算方法 1.1 问题的数学模型

 图 1 波浪与无限水深中浮体的作用及流域分区示意图 Fig. 1 Interaction between waves and floating bodies in infinite water depth and schematic diagram of watershed zoning 注: SF: 自由水面; SB: 物体表面; SJ: 内外域分界面; Ω: 内域; RJ: 半径; Oxz表示笛卡尔直角坐标系

(1)

(2)

(1) 自由水面条件

(3)

(2) 物面条件

(4)

(3) 深水条件

(5)

(4) 散射势ϕj(j = 1, 2, 3, 4)向左右两端传播的远场条件

(6)

(7)

1.2 积分方程和水面多极子展开

(8)

(9)

(10)

(11)

(12)

(13)

(14)

ψ1a为水面反对称偶极子, 其表达式为

(15)

(16)

(17)

ψ2msψ2m + 1a分别为对称和反对称的远场无波速度势, 即非传播项, 其表达式为

(18a)
(18b)

Ursell(1949, 1950)已证明对任意漂浮物体(Ka < 1.5)和淹没物体, 均存在式(10)的多极展开, 且对于某一特定的物面边界条件, 其未知待定系数cj, 2mcj, 2m + 1具有唯一解。

(19a)

(19b)
1.3 匹配边界元法的数值实现

(20)

(21)

(22a)
(22b)

(23)

1.4 波浪力、水动力系数和波高

(24)

(25)

(26)

(27)

(28)

(29)

(30)

(31)
2 数值算例

 图 2 简单格林函数边界元与波动格林函数边界元求解水面固定半圆激振力对比 Fig. 2 Comparison between simple Green's function boundary element and wave Green's function boundary element in solving the excitation force of fixed semicircle on water surface 注: RkGreen表示简单格林函数法, FsGreen表示波动格林函数法; 横坐标Ka是深水波数和半圆半径的乘积, 纵坐标fx、fz分别为纵荡激振力和升沉激振力, ρgAa为无因次参数

 图 3 简单格林函数边界元与波动格林函数边界元求解透射反射系数对比 Fig. 3 Comparison between simple Green's function boundary element and wave Green's function boundary element in solving transmission and reflection coefficient 注: RkGreen表示简单格林函数法, FsGreen表示波动格林函数法; 横坐标Ka是深水波数和半圆半径的乘积

 图 4 水面方箱附加质量及辐射阻尼随波数的变化曲线 Fig. 4 Variation curve of added mass and radiation damping of water surface square box with wave number 注: a、b分别为附加质量和辐射阻尼; 横坐标KB为深水波数和方箱宽度的乘积; 纵坐标a11、a33分别为纵荡和升沉方向的附加质量, b11、b33分别为纵荡和升沉方向的辐射阻尼, ρB2和为无因次参数

 图 5 水面方箱上波浪激振力随波数的变化曲线 Fig. 5 Variation curve of wave excitation force with wave number on water surface square box 注: KB为深水波数和方箱宽度的乘积; fx、fz分别为纵荡激振力和升沉激振力, ρgAB为无因次参数
3 结论

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