Naval Architecture, Ocean and Civil Engineering

Modification of Velocity Formulations in a Two-Layer Boussinesq-Type Model for Water Waves

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  • 1. College of Transportation Engineering, Dalian Maritime University, Dalian 116026, Liaoning, China
    2. State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, Dalian 116024, Liaoning, China

Received date: 2021-09-06

  Revised date: 2021-12-10

  Accepted date: 2021-12-16

  Online published: 2022-11-25

Abstract

In order to improve the accuracy of velocity formulation in a Boussinesq-type wave model, with a two-layer Boussinesq-type model with the highest spatial derivative of 2 being chosen as the research object, a third-order term with constant coefficient is proposed to modify the velocity formulation. The coefficient is optimized by minimizing the error between the summation of the integration of horizontal and vertical velocities of the equation and that of the analytical linear Stokes wave velocity components in the range of 0<kh< 8 (where k is wave number, h is still water depth). At a 1% tolerance error, the applicable water depths of the modified formulations for horizontal and vertical velocities are up to kh=7.34 and kh=7.83, respectively, which are larger than those of the original formulations. The evolution of the steady-state wave and the focused wave is numerically simulated by using the numerical model. The horizontal velocity under the maximum surface elevation crest is in good agreements with the analytical solution of stream function and published experimental data, which verifies the effectiveness of the modified formulations. The studies show that the velocity accuracy of the improved equation is greatly improved. This method provides an important reference for the improvement of velocity field of other Boussinesq-type models.

Cite this article

LIU Zhongbo, HAN Qingliang, REN Shuangshuang, WANG Yan, FANG Kezhao . Modification of Velocity Formulations in a Two-Layer Boussinesq-Type Model for Water Waves[J]. Journal of Shanghai Jiaotong University, 2023 , 57(2) : 177 -182 . DOI: 10.16183/j.cnki.jsjtu.2021.337

References

[1] LIU Z B, FANG K Z. A new two-layer Boussinesq model for coastal waves from deep to shallow water: Derivation and analysis[J]. Wave Motion, 2016, 67: 1-14.
[2] LIU Z B, FANG K Z, CHENG Y Z. A new multi-layer irrotational Boussinesq-type model for highly nonlinear and dispersive surface waves over a mildly sloping seabed[J]. Journal of Fluid Mechanics, 2018, 842: 323-353.
[3] KIRBY J T. Boussinesq models and their application to coastal processes across a wide range of scales[J]. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2016, 142(6): 03116005.
[4] 张尧, 谢欣, 陶爱峰, 等. Boussinesq相位解析的海岸水动力学数学模型研究进展[J]. 海洋通报, 2018, 37(5): 481-493.
[4] ZHANG Yao, XIE Xin, TAO Aifeng, et al. Review of Boussinesq phase-resolving coastal hydrodynamic model[J]. Marine Science Bulletin, 2018, 37(5): 481-493.
[5] 孙家文, 房克照, 刘忠波, 等. 关于Boussinesq型水波方程理论和应用研究的综述[J]. 海洋学报, 2020, 42(5): 1-11.
[5] SUN Jiawen, FANG Kezhao, LIU Zhongbo, et al. A review on the theory and application of Boussinesq-type equations for water waves[J]. Acta Oceanologica Sinica, 2020, 42(5): 1-11.
[6] MADSEN P A, FUHRMAN D R. Trough instabilities in Boussinesq formulations for water waves[J]. Journal of Fluid Mechanics, 2020, 889: A38.
[7] LIU Z B, FANG K Z, SUN J W. A multi-layer Boussinesq-type model with second-order spatial derivatives: Theoretical analysis and numerical implementation[J]. Ocean Engineering, 2019, 191: 106545.
[8] 林鹏程, 刘忠波, 刘勇. 基于Boussinesq数值模型的波浪速度垂向分布模拟研究[J]. 海洋湖沼通报, 2021, 43(4): 7-15.
[8] LIN Pengcheng, LIU Zhongbo, LIU Yong. Simulation of vertical distribution of wave velocity field based on Boussinesq numerical model[J]. Transactions of Oceanology and Limnology, 2021, 43(4): 7-15.
[9] 刘必劲, 张振伟, 刘忠波, 等. 基于Boussinesq水波模型的聚焦波模拟[J]. 海洋学报, 2021, 43(3): 31-39.
[9] LIU Bijin, ZHANG Zhenwei, LIU Zhongbo, et al. Simulating the evolution of a focused wave group by a Boussinesq-type model[J]. Haiyang Xuebao, 2021, 43(3): 31-39.
[10] SUN J W, LIU Z B, WANG X G, et al. Effect of the coefficient on the performance of a two-layer boussinesq-type model[J]. China Ocean Engineering, 2021, 35(1): 36-47.
[11] LIU Z B, FANG K Z. Numerical verification of a two-layer Boussinesq-type model for surface gravity wave evolution[J]. Wave Motion, 2019, 85: 98-113.
[12] BALDOCK T E, SWAN P H, TAYLOR. A laboratory study of nonlinear surface waves on water[J]. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences, 1996, 354: 649-676.
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