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On the two-layer high-level Green-Naghdi model in a general form

  • Special Column on the 34th NCHD (Guest Editor Zheng Ma)
  • Published:
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Abstract

The traditional high-level Green-Naghdi (HLGN) model, which uses the polynomial as the shape function to approximate the variation of the horizontal- and vertical-velocity components along the vertical direction for each-fluid layer, can accurately describe the large-amplitude internal waves in a two-layer system for the shallow configuration (h2 / λ ≪ 1, h1 / λ ≪ 1). However, for the cases of the deep configuration (h2 / λ ≪ 1, h1 / λ = O(1)), higher-order polynomial is needed to approximate the variation of the velocity components along the vertical direction for the lower-fluid layer. This, however, introduces additional unknowns, leading to a significant increase in computational time. This paper, for the first time, derives a general form of the HLGN model for a two-layer fluid system, where the general form of the shape function is used during the derivation. After obtaining the general form of the two-layer HLGN equations, corresponding solutions can be obtained by determining the reasonable shape function. Large-amplitude internal solitary waves in a deep configuration are studied by use of two different HLGN models. Comparison of the two HLGN models shows that the polynomial as the shape function for the upper-fluid layer and the production of exponential and polynomial as the shape function for the lower-fluid layer is a good choice. By comparing with Euler’s solutions and the laboratory measurements, the accuracy of the two-layer HLGN model is verified.

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References

  1. Boyer T. P., Antonov J. I., Baranova O. K. et al. World Ocean Database 2013 [R]. NOAA Atlas NESDIS 72, 2013.

  2. Camassa R., Choi W., Michallet H. et al. On the realm of validity of strongly nonlinear asymptotic approximations for internal waves [J]. Journal of Fluid Mechanics, 2006, 549: 1–23.

    Article  MathSciNet  Google Scholar 

  3. Grue J., Rusas P. O., Sveen J. K. et al. Properties of large-amplitude internal waves [J]. Journal of Fluid Mechanics, 1999, 380: 257–278.

    Article  MathSciNet  Google Scholar 

  4. Kodaira T., Waseda T., Miyata M., et al. Internal solitary waves in a two-fluid system with a free surface [J]. Journal of Fluid Mechanics, 2016, 804: 201–223.

    Article  MathSciNet  Google Scholar 

  5. Du H., Wei G., Wang S. D. et al. Experimental study of elevation- and depression-type internal solitary waves generated by gravity collapse [J]. Physics of Fluids, 2019, 31(10): 102104.

    Article  Google Scholar 

  6. La Forgia G., Sciortino G. The role of the free surface on interfacial solitary waves [J]. Physics of Fluids, 2019, 31(10): 106601.

    Article  Google Scholar 

  7. Yao J. J., Zhen X. W., Huang Y. Experimental investigation of internal solitary wave loads on artificial seabed [J]. Ocean Engineering, 2023, 285: 115316.

    Article  Google Scholar 

  8. Zhang R. R., Wang H. W., Chen K. et al. Experimental investigation and prediction model of the loads exerted by oblique internal solitary waves on FPSO [J]. China Ocean Engineering, 2022, 32(2): 179–190.

    Article  Google Scholar 

  9. Wang X., Zhou J. F. Numerical and experimental study of internal solitary wave loads on tension leg platforms [J]. Journal of Hydrodynamics, 2021, 33(1): 93–103.

    Article  Google Scholar 

  10. Michallet H., Barthélemy E. Experimental study of interfacial solitary waves [J]. Journal of Fluid Mechanics, 1998, 366: 159–177.

    Article  Google Scholar 

  11. Zhao B. B., Wang Z., Duan W. Y. et al. Experimental and numerical studies on internal solitary waves with a free surface [J]. Journal of Fluid Mechanics, 2020, 899: A17.

    Article  MathSciNet  Google Scholar 

  12. Miyata M. An internal solitary wave of large amplitude [J]. La Mer, 1985, 23(2): 43–48.

    Google Scholar 

  13. Choi W., Camassa R. Weakly nonlinear internal waves in a two-fluid system [J]. Journal of Fluid Mechanics, 1996, 313: 83–103.

    Article  MathSciNet  Google Scholar 

  14. Choi W., Camassa R. Fully nonlinear internal waves in a two-fluid system [J]. Journal of Fluid Mechanics, 1999, 396: 1–36.

    Article  MathSciNet  Google Scholar 

  15. Jo T., Choi W. Dynamics of strongly nonlinear internal solitary waves in shallow water [J]. Studies in Applied Mathematics, 2002, 109(3): 205–227.

    Article  MathSciNet  Google Scholar 

  16. La Forgia G., Sciortino G. Interfacial solitons propagating through a background shear current [J]. Physics of Fluids, 2020, 32(10): 106603.

    Article  Google Scholar 

  17. Zhi C. H., Xu S. D., Han P. P. et al. Applicability of high-order unidirectional internal solitary wave theoretical model [J]. Acta Physica Sinica, 2022, 71(17): 174701.

    Article  Google Scholar 

  18. Zhao B. B., Zhang T. Y., Duan W. Y. et al. Internal solitary waves generated by a moving bottom disturbance [J]. Journal of Fluid Mechanics, 2023, 963: A32.

    Article  MathSciNet  Google Scholar 

  19. Xu C.G., Wang Z., Hayatdavoodi M. Error calculation of large-amplitude internal solitary waves within the pycnocline introduced by the strong stratification approximation [J]. Journal of Marine Science and Application, 2023, 22(1): 146–152.

    Article  Google Scholar 

  20. Cheng L., Wang C., Guo B. et al. Numerical investigation on the interaction between large-scale continuously stratified internal solitary wave and moving submersible [J]. Applied Ocean Research, 2024, 145: 103938.

    Article  Google Scholar 

  21. Choi W. High-order strongly nonlinear long wave approximation and solitary wave solution [J]. Journal of Fluid Mechanics, 2022, 945: A15.

    Article  MathSciNet  Google Scholar 

  22. Choi W. High-order strongly nonlinear long wave approximation and solitary wave solution. Part 2. Internal waves [J]. Journal of Fluid Mechanics, 2022, 952: A41.

    Article  MathSciNet  Google Scholar 

  23. Debsarma S., Das K. P., Kirby J. T. Fully nonlinear higher-order model equations for long internal waves in a two-fluid system [J]. Journal of Fluid Mechanics, 2010, 654: 281–303.

    Article  MathSciNet  Google Scholar 

  24. Zhao B. B., Ertekin R. C., Duan W. Y. et al. New internal-wave model in a two-layer fluid [J]. Journal of Waterway, Port, Coastal and Ocean Engineering, 2016, 142(3): 04015022.

    Article  Google Scholar 

  25. Zhao B. B., Zhang T. Y., Duan W. Y. et al. Internal solitary waves in the presence of linear and nonlinear shear-currents [J]. Wave Motion, 2023, 123: 103217.

    Article  MathSciNet  Google Scholar 

  26. Zhao B. B., Duan W. Y. Green-Naghdi wave theory: GN wave model [M]. Beijing, China: Tsinghua University Press, 2014(in Chinese).

    Google Scholar 

  27. Webster W. C., Zhao B. B. The development of a high-accuracy, broadband, Green-Naghdi model for steep, deep-water ocean waves [J]. Journal of Ocean Engineering and Marine Energy, 2018, 4(4): 273–291.

    Article  Google Scholar 

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Acknowledgements

This work was supported by the Fundamental Research Funds for the Central Universities (Grant No. 3072022FSC0101), the China Postdoctoral Science Foundation (Grant No.2022M710932), the Ph. D. Student Research and Innovation Fund (Grant No. BCJJ2023103) and the High-End Foreign Expert Recruitment Program and the Heilongjiang Touyan Innovation Team Program. The authors are grateful to Prof. J. Grue, Dr. T. J. Ellevold of Department of Mathematics, University of Oslo of Norway for providing the numerical code, IW2 (Two layer Internal Waves), to help us to obtain Euler’s solution.

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Authors and Affiliations

  1. College of Shipbuilding Engineering, Harbin Engineering University, Harbin, 150001, China

    Bin-bin Zhao, Tian-yu Zhang, Zhan Wang & Wen-yang Duan

  2. Qingdao Innovation and Development Center of Harbin Engineering University, Qingdao, 266000, China

    Zhan Wang

  3. Lavrentyev Institute of Hydrodynamics, Novosibirsk, Russia

    Alexander Chesnokov & Natalia Shmakova

Authors
  1. Bin-bin Zhao
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  2. Tian-yu Zhang
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  3. Zhan Wang
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  4. Wen-yang Duan
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  5. Alexander Chesnokov
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  6. Natalia Shmakova
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Corresponding author

Correspondence to Zhan Wang.

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Conflict of interest: The authors declare that they have no conflict of interest. Bin-bin Zhao, Wen-yang Duan are editorial board members for the Journal of Hydrodynamics and was not involved in the editorial review, or the decision to publish this article. All authors declare that there are no other competing interests.

Ethical approval: This article does not contain any studies with human participants or animals performed by any of the authors.

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Additional information

Project supported by the National Natural Science Foundation of China (Grant Nos. 12202114, 52261135547).

Biography: Bin-bin Zhao (1984-), Male, Ph. D., Professor

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Zhao, Bb., Zhang, Ty., Wang, Z. et al. On the two-layer high-level Green-Naghdi model in a general form. J Hydrodyn 36, 78–86 (2024). https://doi.org/10.1007/s42241-024-0012-z

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  • DOI: https://doi.org/10.1007/s42241-024-0012-z

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