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 哈尔滨工程大学学报  2019, Vol. 40 Issue (2): 234-239  DOI: 10.11990/jheu.201803089 0

### 引用本文

NING Dezhi, QIU Shihui, ZHANG Chongwei. Nonlinear characteristics of wave resonance in the gap between twin barges[J]. Journal of Harbin Engineering University, 2019, 40(2), 234-239. DOI: 10.11990/jheu.201803089.

### 文章历史

1. 大连理工大学 海岸和近海工程国家重点实验室, 辽宁 大连 116024;
2. 大连理工大学 海洋可再生能源研究中心, 辽宁 大连 116024

Nonlinear characteristics of wave resonance in the gap between twin barges
NING Dezhi 1,2, QIU Shihui 1,2, ZHANG Chongwei 1,2
1. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, China;
2. Offshore Renewable Energy Research Center, Dalian University of Technology, Dalian 116024, China
Abstract: Based on high-order boundary element method, a fully-nonlinear potential flow numerical wave flume is developed to study the nonlinear characteristics of wave resonance between two adjacent barges. Incidental waves are generated by an inner source method. Damping zones are used at two ends of the flume to dissipate outgoing waves. An artificial damping term is introduced to free surface boundary resonance conditions in the gap to approximate the energy dissipation. By comparing the numerical results with the experimental data, we determine the damping parameters in the narrow gap corresponding to different incident wave heights. Our findings show that, with the increase of incident wave heights, the damping strength that represents the viscous dissipation at resonance increases and the nonlinear characteristics of wave run-ups along the upstream side of the barge are enhanced, whereas the resonance frequency of the liquid in the gap is not evidently affected and the dimensionless wave height at resonance decreases.
Keywords: high-order boundary element    source wave formation    narrow gap    resonance    nonlinear waves    artificial viscosity    twin barges

1 基本理论与数值方法

 Download: 图 1 水槽示意 Fig. 1 Definition sketch of the wave flume

 ${\nabla ^2}\phi = {q^*}\left( {{x_{\text{s}}}, z, t} \right)$ (1)
 $\left\{ \begin{gathered} \frac{{{\text{d}}X\left( {x, z} \right)}}{{{\text{d}}t}} = \nabla \phi-{\mu _1}\left( x \right)\left( {X-{X_0}} \right) + k{\mu _2}^2\phi \hfill \\ \frac{{{\text{d}}\phi }}{{{\text{d}}t}} =-g\eta + \frac{1}{2}\left( {{N_s} - 1} \right){\left| {\nabla \phi } \right|^2} - \hfill \\ \;\;\;\;\;\;\;\;{\mu _1}\left( x \right)\phi - 2{\mu _2}{\left( {gk} \right)^{0.5}}\phi \hfill \\ \end{gathered} \right.$ (2)
 ${\left. {\frac{{\partial \phi }}{{\partial \mathit{\boldsymbol{n}}}}} \right|_{{S_B}}} = 0$ (3)
 ${\left. {\frac{{\partial \phi }}{{\partial \mathit{\boldsymbol{n}}}}} \right|_{{S_D}}} = 0$ (4)
 $\phi {|_{t = 0}} = \eta {|_{t = 0}} = 0$ (5)

 $\begin{gathered} V =- \frac{{\partial {\phi _i}}}{{\partial x}} =- \frac{{g{A_i}k\cosh \left[{k\left( {z + h} \right)} \right]}}{{\omega \cosh \left( {kh} \right)}}\cos \left( {kx - \omega t} \right) - \hfill \\ \frac{3}{8}\left( {{N_s} - 1} \right)A_i^22k\omega \frac{{\cosh \left[{2k\left( {z + h} \right)} \right]}}{{{{\sinh }^4}\left( {kh} \right)}}\cos \left[{2\left( {kx-\omega t} \right)} \right] \hfill \\ \end{gathered}$ (6)

 ${\omega ^2} = gk\tanh \left( {kh} \right)$ (7)

 $\alpha \phi \left( \xi \right) = \int\limits_S {\left[{\phi \left( \xi \right)\frac{{\partial G\left( {\xi, \eta } \right)}}{{\partial \mathit{\boldsymbol{n}}}}-G\frac{{\partial \phi }}{{\partial \mathit{\boldsymbol{n}}}}} \right]{\text{d}}S} + \int\limits_\varOmega {{q^*}G{\text{d}}\varOmega }$ (8)

 $G\left( {\xi, \eta } \right) = \frac{1}{{2\pi }}\left( {\ln {r_1} + \ln {r_2}} \right)$ (9)

 ${r_1} = \sqrt {{{\left( {x-{x_0}} \right)}^2} + {{\left( {z-{z_0}} \right)}^2}}$ (10)

r2ηξ关于水底的镜像点的距离：

 ${r_2} = \sqrt {{{\left( {x-{x_0}} \right)}^2} + {{\left( {z + {z_0} + 2h} \right)}^2}}$ (11)

 $\begin{gathered} \alpha \phi \left( \xi \right)-\sum\limits_{j = 1}^{{N_B}} {\int\limits_{-1}^1 {\sum\limits_{i = 1}^3 {{k_i}\left( \zeta \right){\phi _{ji}}\frac{{\partial G}}{{\partial \mathit{\boldsymbol{n}}}}\left| {J\left( \zeta \right)} \right|{\text{d}}\left( \zeta \right)} } } + \hfill \\ \;\;\;\;\;\;\sum\limits_{j = 1}^{{N_F}} {\int\limits_{-1}^1 {\sum\limits_{i = 1}^3 {{k_i}\left( \zeta \right)G\frac{{\partial {\phi _{ji}}}}{{\partial \mathit{\boldsymbol{n}}}}\left| {J\left( \zeta \right)} \right|{\text{d}}\left( \zeta \right)} } } = \hfill \\ \;\;\;\;\; - \sum\limits_{j = 1}^{{N_B}} {\int\limits_{ - 1}^1 {\sum\limits_{i = 1}^3 {{k_i}\left( \zeta \right)G\frac{{\partial {\phi _{ji}}}}{{\partial \mathit{\boldsymbol{n}}}}\left| {J\left( \zeta \right)} \right|{\text{d}}\left( \zeta \right)} } } + \hfill \\ \;\;\;\;\; - \sum\limits_{j = 1}^{{N_F}} {\int\limits_{ - 1}^1 {\sum\limits_{i = 1}^3 {{k_i}\left( \zeta \right){\phi _{ji}}\frac{{\partial G}}{{\partial \mathit{\boldsymbol{n}}}}\left| {J\left( \zeta \right)} \right|{\text{d}}\left( \zeta \right)} } } + \hfill \\ \;\;\;\;\;\;\sum\limits_{j = 1}^{{N_Q}} {\int\limits_{ - 1}^1 {\sum\limits_{i = 1}^3 {{k_i}\left( \zeta \right){q^*}\left( \zeta \right)G\left( \zeta \right)} } } \left| {J\left( \zeta \right)} \right|{\text{d}}\left( \zeta \right) \hfill \\ \end{gathered}$ (12)

 $J\left( \zeta \right) = \left( {\frac{{\partial x}}{{\partial \zeta }}, \frac{{\partial z}}{{\partial \zeta }}} \right)$ (13)

 $\mathit{\boldsymbol{X}} = {\mathit{\boldsymbol{A}}^{-1}}\mathit{\boldsymbol{H}}$ (14)

2 数值计算及讨论

 Download: 图 2 在t=36T和t=40T两个时刻的波面分布对比 Fig. 2 Contrast of wave surface distribution at two moments at t=36T and t=40T

 Download: 图 3 窄缝中心无因次波高与入射波频率间的关系 Fig. 3 Comparison between experimental and numerical results

 Download: 图 4 窄缝中无因次波高与入射波波高间的关系 Fig. 4 The relationship between the dimensionless wave height and the incident wave height in the narrow gap
 Download: 图 5 阻尼系数随入射波波高的变化规律 Fig. 5 The variation of damping coefficient with the height of incident wave

 Download: 图 6 不同入射波波高作用下窄缝内波高随波数kh变化 Fig. 6 Variation of dimensionless wave height at the gap with kh for different incident wave height

 Download: 图 7 共振时不同位置的无因次波高随入射波高分布 Fig. 7 Variation of dimensionless wave height at different positions at resonance with incident wave height
 $f\left( {{x_0}} \right) =-1.254{x_0}-0.899\;7\sin {x_0} + 8.339$ (15)

 $f\left( {{x_0}} \right) =-0.045\;98{x_0} + 0.503\;4$ (16)

 $f\left( {{x_0}} \right) = 0.073\;54{x_0} + 1.72$ (17)

 Download: 图 8 非线性和线性无因次共振波高对比 Fig. 8 Comparison of nonlinear and linear non-dimensional resonance wave height
3 结论

1) 在共振频率处表征粘性耗散的阻尼项随入射波高的增大而增大；

2) 入射波波高的变化仅仅会使共振波高发生变化，而对共振频率影响有限；

3) 随着非线性的增强，窄缝处的无因次共振波高减小，双箱体迎浪侧非线性作用越明显，而背浪侧非线性并不显著，并得到了3个位置处的波高拟合曲线。

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