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High-level Green–Naghdi model for large-amplitude internal waves in deep water

Published online by Cambridge University Press:  03 June 2024

Binbin Zhao Open the ORCID record for Binbin Zhao [Opens in a new window] ,
Tianyu Zhang ,
Zhan Wang Open the ORCID record for Zhan Wang [Opens in a new window] ,
Masoud Hayatdavoodi Open the ORCID record for Masoud Hayatdavoodi [Opens in a new window] ,
Ying Gou ,
R. Cengiz Ertekin Open the ORCID record for R. Cengiz Ertekin [Opens in a new window]  and
Wenyang Duan
Binbin Zhao
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin, PR China
Tianyu Zhang
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin, PR China
Zhan Wang*
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin, PR China Qingdao Innovation and Development Center of Harbin Engineering University, 266000 Qingdao, PR China
Masoud Hayatdavoodi
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin, PR China Civil Engineering Department, School of Science and Engineering, University of Dundee, Dundee DD1 4HN, UK
Ying Gou
Affiliation:
State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology, 116024 Dalian, PR China
R. Cengiz Ertekin
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin, PR China Department of Ocean & Resources Engineering, University of Hawai'i, Honolulu, HI 96822, USA
Wenyang Duan
Affiliation:
College of Shipbuilding Engineering, Harbin Engineering University, 150001 Harbin, PR China
*
†Email address for correspondence: zhan.wang@hrbeu.edu.cn
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Abstract

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

JFM classification

Geophysical and Geological Flows: Internal waves Geophysical and Geological Flows: Stratified flows Waves/Free-surface Flows: Solitary waves
Type
JFM Papers
Information
Journal of Fluid Mechanics , Volume 988 , 10 June 2024 , A32
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Copyright
© The Author(s), 2024. Published by Cambridge University Press

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