Green-Naghdi Theory, Part B: Green-Naghdi Equations for Deep Water Waves

Abstract

“Green — Naghdi Theory, Part A: Green — Naghdi (GN) Equations for Shallow Water Waves” have investigated the linear dispersion relations of high-level GN equations in shallow water. In this study, the GN equations for deep water waves are investigated. In the traditional GN equations for deep water waves, the velocity distribution assumption involves only one representative wave number. Herein, a new velocity distribution shape function with multiple representative wave numbers is employed. Further, we have derived the three-dimensional GN equations and analyzed the linear dispersion relations of the GN-3 and GN-5 equations. In this study, the finite difference method is used to simulate focus waves in the time domain. Additionally, the GN-5 equations are used to validate the wave profile and horizontal velocity distribution along water depth for different focused waves.

Keywords

Green—Naghdi equations / Finite difference / Water waves / Deep water / Focused waves

References

[1] Baldock TE, Swan C, Taylor PH. A laboratory study of nonlinear surface waves on water. Philosophical Transactions of the Royal Society of London. Series A: Mathematical, Physical and Engineering Sciences, 1996, 354(1707): 649-676

[2] Demirbilek Z, Webster WC. Application of the Green-Naghdi theory of fluid sheets to shallow-water wave problems. Report 1. Model Development, 1992, Vicksburg: Coastal Engineering Research Center

[3] Ertekin RC, Webster WC, Wehausen JV. Waves caused by a moving disturbance in a shallow channel of finite width. J Fluid Mech, 1986, 169: 275-292

[4] Green AE, Laws N, Naghdi PM. On the theory of water waves. Proc R Soc Lond A Math Phys Sci, 1974, 338: 43-55

[5] Green AE, Naghdi PM. Directed fluid sheets. Proc R Soc Lond A Math Phys Sci, 1976, 347: 447-473

[6] Hayatdavoodi M, Treichel K, Ertekin RC. Parametric study of nonlinear wave loads on submerged decks in shallow water. Journal of Fluids and Structures, 2019, 86: 266-289

[7] Hayatdavoodi M, Liu J, Ertekin RC. Bore impact on decks of coastal structures. Journal of Waterway, Port, Coastal, and Ocean Engineering, 2022, 148(2): 04021051

[8] Hayatdavoodi M, Ertekin RC. On some nonlinear wave diffraction and refraction solutions in shallow waters. International Conference on Offshore Mechanics and Arctic Engineering, 2022, 85895: V05AT06A004

[9] Kostikov VK, Hayatdavoodi M, Ertekin RC. Drift of elastic floating ice sheets by waves and current: Multiple sheets. Physics of Fluids, 2022, 34(5): 057113

[10] Liu J, Hayatdavoodi M, Ertekin RC. Bore pressure on horizontal and vertical surfaces. International Conference on Offshore Mechanics and Arctic Engineering, 2019, 58844: V07AT06A008

[11] Liu J, Hayatdavoodi M, Ertekin RC. On bore dynamics and pressure: RANS, Green-Naghdi, and Saint-Venant equations. Journal of Offshore Mechanics and Arctic Engineering, 2020, 142(2): 021902

[12] Wang Z, Zhao BB, Duan WY, Ertekin RC, Hayatdavoodi M, Zhang TY. On solitary wave in nonuniform shear currents. Journal of Hydrodynamics, 2020, 32(4): 800-805

[13] Webster WC, Kim DY (1991) The dispersion of large-amplitude gravity waves in deep water. Water Waves, 397–416. https://trid.trb.org/view/439579

[14] Webster WC, Duan WY, Zhao BB. Green-Naghdi theory, part A: Green-Naghdi (GN) equations for shallow water waves. Journal of Marine Science and Application, 2011, 10(3): 253-258

[15] Webster WC, Zhao BB. The development of a high-accuracy, broadband, Green — Naghdi model for steep, deep-water ocean waves. Journal of Ocean Engineering and Marine Energy, 2018, 4(4): 273-291

[16] Zhao BB, Duan WY, Ertekin RC. Application of higher-level GN theory to some wave transformation problems. Coastal Engineering, 2014, 83: 177-189

[17] Zhao BB, Zheng K, Duan WY, Ertekin RC, Shao YL. Time domain simulation of focused waves by High-Level Irrotational Green—Naghdi equations and Harmonic Polynomial Cell method. European Journal of Mechanics-B/Fluids, 2020, 82: 83-92

[18] Zheng K, Zhao BB, Duan WY, Ertekin RC, Chen XB. Simulation of evolution of gravity wave groups with moderate steepness. Ocean Modelling, 2016, 98: 1-11