| ZHAO Hongjun,SONG Zhiyao,LI Ling,KONG Jun. 2014. On the Fourier approximation method for steady water waves. Acta Oceanologica Sinica, 33(5):37－47 |
| On the Fourier approximation method for steady water waves |
| 稳恒水波的Fourier近似解研究 |
| Received:May 10, 2013 Revised:October 18, 2013 |
| DOI：10.1007/s13131-014-0470-1 |
| Key words:steady water waves Fourier series Newton’s method relaxation technology wave properties |
| 中文关键词: 稳恒水波 Fourier近似 Newton迭代 松弛技术 波动特征 |
| 基金项目:The Jiangsu Province Natural Science Foundation for the Young Scholar under contract No. BK20130827; the Fundamental Research Funds for the Central Universities of China under contract No. 2010B02614; the National Natural Science Foundation of China under contract Nos 41076008 and 51009059; the Priority Academic Program Development of Jiangsu Higher Education Institutions. |
| Author Name | Affiliation | E-mail | | ZHAO Hongjun | State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China | oyhg@hhu.edu.cn | | SONG Zhiyao | Key Laboratory of Virtual Geographic Environment, Key Laboratory for Numerical Simulation of Largescale Complex System, Nanjing Normal University, Nanjing 210023, China | | | LI Ling | School of Engineering, The University of Queensland, St. Lucia, Qld 4072, Australia | | | KONG Jun | State Key Laboratory of Hydrology-Water Resources and Hydraulic Engineering, College of Harbor, Coastal and Offshore Engineering, Hohai University, Nanjing 210098, China | |
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| Abstract: |
| A computational method for steady water waves is presented on the basis of potential theory in the physical plane with spatial variables as independent quantities. The finite Fourier series are applied to approximating the free surface and potential function. A set of nonlinear algebraic equations for the Fourier coefficients are derived from the free surface kinetic and dynamic boundary conditions. These algebraic equations are numerically solved through Newton's iterative method, and the iterative stability is further improved by a relaxation technology. The integral properties of steady water waves are numerically analyzed, showing that (1) the set-up and the set-down are both non-monotonic quantities with the wave steepness, and (2) the Fourier spectrum of the free surface is broader than that of the potential function. The latter further leads us to explore a modification for the present method by approximating the free surface and potential function through different Fourier series, with the truncation of the former higher than that of the latter. Numerical tests show that this modification is effective, and can notably reduce the errors of the free surface boundary conditions. |
| 中文摘要: |
| 该文基于势函数理论，通过对波面和速度势的Fourier近似，由自由面运动学和动力学边界条件，构造了各项Fourier系数满足的非线性代数方程，进一步采用引入松弛系数的Newtion迭代法，对各项Fourier系数进行了数值求解，从而构造了物理空间的稳恒水波的Fourier近似数值解法。对不同水深和非线性的稳恒水波进行了数值计算，并研究了不同波动特征量随波陡的变化，结果显示稳恒水波的超高和跌落均是随波陡非单调变化的变量，且波面的Fourier谱明显宽于速度势。基于后一现象，通过对波面和速度势采用不同截断项数的Fourier近似（波面Fourier级数的截断项数高于速度势），进一步对本文数值解法进行了改进，数值实验表明，改进方法有效，可显著降低自由面边界条件的误差。 |
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