Abstract
The focus of the research presented here is the development of an efficient analytical model for the time-domain simulation of the evolution of a train of a three-dimensional steep random waves and its associated flow. Of particular interest here is the development of a tool that not only accurately predicts the surface elevation history of this wave train, but also the kinematics within the waves, particularly near the free surface. Fenton (Advances in coastal and ocean engineering, vol 5, World Scientific, Singapore, pp 241–324, 1999) reviews the rich literature of various computational models for water waves and is divided between analytical approaches that develop systems of equations to describe the evolution of waves in space and time, and computational approaches such as CFD (which will not be the focus here). What is required to analyze properly the practical ocean engineering problem described in the motivation below is a theory that predicts the three-dimensional evolution of waves that simultaneously has four essential characteristics: deep-water, random broadband seaways, steep waves (close to breaking), and demonstrated accuracy in both wave shape and near-surface wave kinematics. The number and variety of theories that satisfy some but not all of these characteristics is too voluminous to reference here; dynamic theories that exhibit the confluence of all four characteristics are apparently nonexistent. The intent of this paper is to make a step in the direction of filling this void by extending the Green–Naghdi theory of deep-water waves. It is shown that higher level GN models using distributed directors do have a bandwidth that is significantly larger than former GN models and have the same computational effort as using the traditional directors. The bandwidths achieved with the new approach are large enough to be useful in the context of many ocean engineering problems. Applications of these models to random wave situations will be reported in a subsequent article.
Similar content being viewed by others
Notes
The results communicated by Fenton were computed by use of his programs FOURIER, CNOIDAL, and STOKES, Fenton (2016b).
References
Chappelear JE (1961) Direct numerical calculation of wave properties. J Geophys Res 66:501–508
Cokelet ED (1977) Steep gravity waves in water of arbitrary uniform depth. Trans R Soc Lond A 286:183–230
De SC (1955) Contributions to the theory of Stokes waves. Proc Camb Philos Soc 51:713–736
Dean RG (1965) Stream function representation of nonlinear ocean waves. J Geophys Res 70:4561–4572
Demirbilek Z, Webster WC (1992) Application of the Green-Naghdi theory of fluid sheets to shallow-water wave problems. US Army Corps. of Eng. Waterways Experiment Station, Rep. No. CERC-92–11, Vicksburg, MS (date of publication, September 1992)
Fenton JD (1988) The numerical solution of steady water wave problems. Comput Geosci 14:357–368
Fenton JD (1999) Numerical methods for nonlinear waves. In: Liu PLF (ed) Advances in coastal and ocean engineering, vol 5. World Scientific, Singapore, pp 241–324
Fenton JD (2016a) Private communication (date of communication, April 2016)
Fenton JD (2016b) Technical Report, Steady Waves, URL. http://johndfenton.com/Steady-waves/Instructions.pdf
Fenton JD, Rienecker MM (1982) A Fourier method for solving nonlinear water wave problems: application to solitary wave interactions. J Fluid Mech 118:411–443
Green AE, Naghdi PM (1986) A nonlinear theory of water waves for finite and infinite depths. Philos Trans R Soc Lond Ser A 320:37–70
Green AE, Naghdi PM (1987) Further developments in a nonlinear theory of water waves for finite and infinite depths. Philos Trans Roy Soc Lond Ser A324:47–72
Kim JW, Bai KJ, Ertekin RC, Webster WC (2003) A strongly-nonlinear model for water waves in water of variable depth—the irrotational Green-Naghdi model. J Offshore Mech Arct Eng 125(1):25–32
Nwogu O, Demirbilek Z (2001) Bouss-2D: a Boussinesq wave model for coastal regions and harbors. ERDC Technical Report, US Army Engineer Research and Development Center, Vicksburg, MS, CHL TR-01-25
Schwartz LW (1972) Analytic continuation of Stokes’ expansion for gravity waves. Ph.D. Thesis, Stanford University
Webster WC (2009) Evolution and kinematics of steep, random seas, a comparison with usual engineering estimates. International Workshop on Applied Ocean Hydrodynamics, Rio de Janiero
Webster WC, Kim DY (1990) The dispersion of large amplitude gravity waves in deep water. 18th Symposium on Naval Hydrodynamics, Ann Arbor
Webster WC, Shields JJ (1989) Conservation of mechanical energy and circulation in the theory of in-viscid fluid sheets. J Eng Math 23:1–15
Williams JM (1981) Limiting gravity waves in water of finite depth. Philos Trans R Soc Lond Ser A 302(1466):139–188
Zhao BB, Duan WY, Webster WC (2011) Tsunami simulation with Green-Naghdi theory. Ocean Eng 38(2–3):389–396
Zhao BB, Duan WY, Demirbilek Z, Ertekin RC, Webster WC (2016) A comparative study between the IGN-2 equations and the fully nonlinear, weakly dispersive Boussinesq equations. Coast Eng 111:60–69
Zheng K, Zhao BB, Duan WY, Ertekin RC, Chen XB (2016) Simulation of evolution of gravity wave groups with moderate steepness. Ocean Model 98:1–11
Acknowledgements
The authors would like to gratefully acknowledge support for this research from the National Natural Science Foundation of China, (Numbers 11772099, 11572093, and 51490671) and the High-Tech Ship Research Projects Sponsored by the Ministry of Industry and Information Technology (MIIT) of China.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Webster, W.C., Zhao, B. The development of a high-accuracy, broadband, Green–Naghdi model for steep, deep-water ocean waves. J. Ocean Eng. Mar. Energy 4, 273–291 (2018). https://doi.org/10.1007/s40722-018-0122-1
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s40722-018-0122-1