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对称指数地形水槽内纵波和横波解析解研究

谢蓉 熊焱 童朝锋 王岗

谢蓉,熊焱,童朝锋,等. 对称指数地形水槽内纵波和横波解析解研究[J]. 海洋学报,2023,45(6):44–51 doi: 10.12284/hyxb2023081
引用本文: 谢蓉,熊焱,童朝锋,等. 对称指数地形水槽内纵波和横波解析解研究[J]. 海洋学报,2023,45(6):44–51 doi: 10.12284/hyxb2023081
Xie Rong,Xiong Yan,Tong Chaofeng, et al. Analytic solutions of longitudinal and cross waves in the wave flume with an exponential symmetric shoal[J]. Haiyang Xuebao,2023, 45(6):44–51 doi: 10.12284/hyxb2023081
Citation: Xie Rong,Xiong Yan,Tong Chaofeng, et al. Analytic solutions of longitudinal and cross waves in the wave flume with an exponential symmetric shoal[J]. Haiyang Xuebao,2023, 45(6):44–51 doi: 10.12284/hyxb2023081

对称指数地形水槽内纵波和横波解析解研究

doi: 10.12284/hyxb2023081
基金项目: 江苏省自然科学基金杰出青年基金(BK20220082);国家自然科学基金(52071128)
详细信息
    作者简介:

    谢蓉(1998—),女,四川省内江市人,主要从事水动力数值模拟研究。E-mail:395162775@qq.com

    通讯作者:

    熊焱(1992—),女,江苏省南京市人,讲师,主要从事极端水灾害数值模拟研究。E-mail:xiongyan@hhu.edu.cn

  • 中图分类号: TV139

Analytic solutions of longitudinal and cross waves in the wave flume with an exponential symmetric shoal

  • 摘要: 水槽实验通常用于波浪传播变形及防波堤护面块体稳定性等研究,涉及的波要素沿水槽纵向变化且在垂直于水槽的横向保持不变。然而实验中当波长与水槽宽度满足一定关系时,可能出现明显的横向波动现象。本文针对对称指数型隆起地形,基于线性长波方程分别推导了其内沿水槽方向的纵波与垂直于水槽方向的横波的解析表达。水槽内对称指数地形上的纵波可以表示为第一类和第二类一阶贝塞尔函数的形式,并结合自由水面及速度连续条件最终得到其完整解。对称指数地形上分别存在偶对称和奇对称模态的横波,可表示为第一类ν阶贝塞尔函数的形式。偶对称模态(n, m)沿水槽方向有n条波节线,在垂直于水槽方向存在2m条波节线;奇对称模态(n, m)沿水槽方向存在n条波节线而在垂直方向有2m − 1条波节线。
  • 图  1  波浪水槽示意图

    Fig.  1  Definition sketch of the wave flume

    图  2  归一化波幅沿波浪水槽的变化

    阴影部分标识隆起地形区域,其中L = 5 m,T = 4.6 s,λ = 0.46 m−1h1 = 0.5 m,h0 = 0.05 m

    Fig.  2  Variation of normalized amplitudes along the wave flume

    Shaded areas denotes the hump shoal, where L = 5 m, T = 4.6 s, λ = 0.46 m−1, h1 = 0.5 m and h0 = 0.05 m

    图  3  反射系数和透射系数随隆起地形顶部水深h0的变化

    T = 4.6 s,h1 = 0.5 m,L = 5.0 m,λ = ln(h1/h0)/Lω = 1.37 rad/s

    Fig.  3  Reflection and transmission parameters versus the depth at the top of hump shoal h0

    T = 4.6 s, h1 = 0.5 m, L = 5.0 m, λ = ln (h1/h0)/L and ω =1.37 rad/s

    图  4  反射系数和透射系数随着入射波角频率的变化

    h1 = 0.5 m,h0 = 0.05 m,L = 5.0 m,λ = 0.46 m−1

    Fig.  4  Reflection and transmission parameters versus the angular frequency ω for the hump shoal

    h1 = 0.5 m, h0 = 0.05 m, L = 5.0 m and λ = 0.46 m−1

    图  5  反射系数和透射系数随着λ的变化

    h1 = 0.5 m,h0 = 0.05 m,L = 5.0 m,ω = 1.37 rad/s,λ = ln(h1/h0)/L

    Fig.  5  Reflection and transmission parameters versus the topography parameter λ for the hump shoal

    h1 = 0.5 m, h0 = 0.05 m, L = 5.0 m, ω = 1.37 rad/s and λ = ln(h1/h0)/L

    图  6  角频率ω随波数κ的变化

    2b = 1 m,h1 = 0.5 m,h0 = 0.05 m和λ = 0.46 m−1,实心圆是偶对称,空心圆是奇对称

    Fig.  6  Angular frequency ω versus wave number κ

    2b = 1 m, h1 = 0.5 m, h0 = 0.05 m and λ = 0.46 m−1, where the solid circles indicate the symmetrical mode and the hollow circles the anti-symmetrical mode

    图  7  角频率ω随着顶部水深h0的变化

    λ = 0.46 m−1κ1 = π m−1,实线是偶对称,虚线是奇对称

    Fig.  7  Angular frequency ω versus the top depth h0

    λ = 0.46 m−1 and κ1 = π m−1, where the solid lines indicate the symmetrical mode and the dash lines indicate the anti-symmetrical mode

    图  8  角频率ω随着地形参数λ的变化

    h0 = 0.05 m,κ1 = π m−1,实线是偶对称,虚线是奇对称

    Fig.  8  Angular frequency ω versus depth profile parameter λ

    h0 = 0.05 m and κ1 = π m−1, where the solid lines indicate the symmetrical mode and the dash lines indicate the anti-symmetrical mode

    图  9  波数κ1 = π m−1κ2 = 2π m−1κ3 = 3π m−1时偶对称模态(左侧)和奇对称模态(右侧)横波沿水槽方向的波幅分布

    h0 = 0.05 m,h1 = 0.5 m,λ = 0.46 m−1L = 5.0 m

    Fig.  9  Amplitude profiles of cross waves along the wave flume for symmetrical patterns (left column) and anti-symmetrical patterns (right column) with κ1 = π m−1, κ2 = 2π m−1 and κ3 = 3π m−1 respectively

    h0 = 0.05 m, h1 = 0.5 m, λ = 0.46 m−1 and L = 5.0 m

    图  10  偶对称模态的横波的空间分布

    宽2b = 1 m,长L = 5.0 m,顶部水深h0 = 0.05 m,地形参数λ = 0.46 m−1,隆起地形上波数κ2 = 2π m−1m = 0(a),m = 1(b),m = 2(c)

    Fig.  10  Spatial structure of the cross-wave amplitudes for symmetrical patterns over the exponential symmetric shoal

    2b = 1 m, L = 5.0 m, h0 = 0.05 m, h1 = 0.5 m and λ = 0.46 m−1 corresponding κ2 = 2π m−1, where m = 0(a), m = 1(b), m = 2(c)

    图  11  奇对称模态的横波的空间分布

    宽2b = 1 m、长L = 5.0 m、顶部水深h0 = 0.05 m、地形参数λ = 0.46 m−1隆起地形上波数κ2 = 2π m−1m = 1(a),m = 2(b),m = 3(c)

    Fig.  11  Spatial structure of the cross-wave amplitudes for anti-symmetrical patterns over the exponential symmetric shoal

    2b = 1m, L = 5.0 m, h0 = 0.05 m, h1 = 0.5 m and λ = 0.46 m−1 corresponding κ2 = 2π m−1, where m = 1(a), m = 2(b), m = 3(c)

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出版历程
  • 收稿日期:  2022-11-11
  • 修回日期:  2023-01-12
  • 网络出版日期:  2023-05-16
  • 刊出日期:  2023-06-30

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