﻿ 基于方向谱的短峰畸形波数值模拟研究
 舰船科学技术  2016, Vol. 38 Issue (11): 75-79 PDF

Numerical simulation of freak wave based on directional spectrum
XUE Ya-dong, SHI Ai-guo, ZHANG Ben-hui, LI Dong
Abstract: Freak wave is a kind of accidental wave in the sea with short duration but great harm, so that the research on the generation and evolution of freak wave has important practical significance. In this paper, based on the direction of the spectrum and improved phase modulation method, a numerical simulation method of short-peak freak waves was proposed, which increased the probability of freak wave and realized to generate freak waves in predetermined time and place, and in the Matbab language environment to achieve the numerical simulation. Simulation results show that the result of spectral analysis had good agreement with target spectrum though the maximum entropy method.
Key words: freak wave     directional spectrum     phase modulation     Matlal
0 引言

1 波浪数学模型

1.1 规则波

 $\eta {\rm{ = }}A\cos \left( {\omega t} \right),$ (1)
 $\left\{ {\begin{array}{*{20}{l}} \begin{array}{l} u = A\omega {e^{kz}}\cos (kx - \omega t){\rm{,}}\\ v = 0{\rm{,}}\\ w = A\omega {e^{kz}}\sin (kx - \omega t) 。 \end{array} \end{array}} \right.$ (2)

1.2 长峰不规则波

 $\eta = \sum\limits_{i = 1}^n {{A_i}\cos ({k_i}x + {\omega _i}t + {\varepsilon _i})} ,$ (3)
 $\left\{ {\begin{array}{*{20}{l}} \begin{array}{l} u = \sum\limits_{i = 1}^n {{A_i}{\omega _i}{e^{{k_i}z}}\cos ({k_i}x + {\omega _i}t + {\varepsilon _i})} {\rm{,}}\\ v = 0{\rm{,}}\\ w = \sum\limits_{i = 1}^n {{A_i}{\omega _i}{e^{{k_i}z}}\sin ({k_i}x + {\omega _i}t + {\varepsilon _i})} 。 \end{array} \end{array}} \right.$ (4)
 图 1 长峰不规则波的谱密度分布 Fig. 1 Spectral density distribution of nagamine irregular waves

 $\begin{array}{*{20}{l}} \begin{array}{l} \hat \omega = ({\omega _{i - 1}} - {\omega _i})/2,\\ {A_i} = \sqrt {2{S_\eta }(\hat \omega )\Delta \omega } 。 \end{array} \end{array}$
1.3 短峰不规则波

 图 2 短峰不规则波子波叠加示意图 Fig. 2 Superposition of short peak irregular wave wavelet

 $\eta = \sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {{a_{ij}}\cos ({k_i}x\cos {\theta _j} + {k_i}y\sin {\theta _j} - {\omega _i}t + {\varepsilon _{ij}})} } 。$ (5)

 $\sum\limits_{i = 1}^I {\sum\limits_{j = 1}^J {\frac{1}{2}} } a_{ij}^2 = \int_{ - {\rm{ \mathsf{ π} }}}^{\rm{ \mathsf{ π} }} {\int_0^\infty {S(\omega ,\theta )} } {\rm{d}}\omega {\rm{d}}\theta 。$ (6)

 $S(\omega ,\theta ) = S(\omega )D(\omega ,\theta ) 。$ (7)

 $D(\omega ,\theta ) = {k_n}{\cos ^n}\theta (\left| \theta \right| \le \frac{{\rm{ \mathsf{ π} }}}{2}) 。$ (8)

 图 3 方向谱密度分布 Fig. 3 Directional spectrum density distribution

 $\begin{array}{*{20}{l}} \begin{array}{l} n = 2,\quad {k_2} = 2/{\rm{ \mathsf{ π} ,}}\\ n = 4,\quad {k_2} = 8/\left( {3{\rm{ \mathsf{ π} }}} \right) 。 \end{array} \end{array}$

2 基于方向谱的相位调制技术

 $\begin{array}{*{20}{l}} \begin{array}{l} \eta (x,y,t) = \sum\limits_{i = 1}^{{M_1}} {\sum\limits_{j = 1}^J {a_{ij}}\cos ({k_i}x\cos {\theta _j} + {k_i}y\sin {\theta _j} - {\omega _i}t + {\varepsilon _{ij}})} + \\ \;\;\;\;\;\;\;\sum\limits_{i = {M_1} + 1}^M \sum\limits_{j = 1}^J {{a_{ij}}\cos ({k_i}x\cos {\theta _j} + {k_i}y\sin {\theta _j} - {\omega _i}t + {\varepsilon _{ij}}),} \end{array} \end{array}$ (9)
 $\begin{array}{l} {\eta _1}(x,y,t) = \sum\limits_{i = 1}^{{M_1}} {\sum\limits_{j = 1}^J {{a_{ij}}} } \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\cos ({k_i}x\cos {\theta _j} + {k_i}y\sin {\theta _j} - {\omega _i}t + {\varepsilon _{ij}}) 。 \end{array}$ (10)
 $\begin{array}{l} {\eta _2}(x,y,t) = \sum\limits_{i = {M_1} + 1}^M {\sum\limits_{j = 1}^J {{a_{ij}}} } \times \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\cos ({k_i}x\cos {\theta _j} + {k_i}y\sin {\theta _j} - {\omega _i}t + {\varepsilon _{ij}}) 。 \end{array}$ (11)

1）当${k_i}({x_c}\cos {\theta _j} + {y_c}\sin {\theta _j}) - {\omega _i}{t_c} < 0$时，令整数$N = {\rm{int}}\left[{\left( {{k_i}({x_c}\cos {\theta _j} + {y_c}\sin {\theta _j}) - {\omega _i}{t_c}} \right)/2{\rm{ \mathsf{ π} }}} \right]$，此时N < 0，式（7）可以写为：

 $\begin{array}{l} {\eta _2}({x_c},{y_c},{t_c}) = \sum\limits_{i = {M_1} + 1}^M \sum\limits_{j = 1}^J {{a_{ij}}\cos \left( {{k_i}({x_c}\cos {\theta _j} + {y_c}\sin {\theta _j})} \right.} - \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\left. {{\omega _i}{t_c} - 2N{\rm{\pi }} + {\varepsilon _{ij}}} \right) 。 \end{array}$ (12)

3 数值模拟及验证 3.1 数值模拟方案

3.2 数值模拟结果

 图 4 tc=10 s瞬时波面图 Fig. 4 Instantaneous skiodrome when tc=10 s

 图 5 x=4 m，y=0 m位置波高时历 Fig. 5 Wave height time series at x=4 m, y=0 m
3.3 结果验证

 图 6 浪高监测阵点布设 Fig. 6 The Arrangement of wave monitor

 图 7 数值模拟得到的方向谱 Fig. 7 Numerical simulation of the spectrum direction

4 结语

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