Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation

Abstract

Application of nonlinear dispersion wave hydrodynamics (NLD-) models for solving practical problems constantly stimulates the search for ways to expand their field of applicability and achieve a more accurate reproduction of the characteristics of the simulated processes. A productive step in this direction turned out to be the method proposed by Madsen & Sørensen (1992), which made it possible to increase the approximation order of the dispersion relation of the Peregrine model while preserving the third order of derivatives included in the original equations and the second order of long-wave approximation. Later, other approaches were proposed to achieve this goal, which had a noticeable effect on expanding the field of applicability of NLD-models (for example, Nwogu (1993), Beji & Nadaoka (1996)). In the present work, we set a similar goal — to improve the properties of the dispersion relation of the model (and, therefore, the phase velocity), providing the Pade approximation (2,2) of the dispersion relation of the 3D model of potential flows. In contrast to earlier works on this subject, where weakly non-linear models were considered, we proceed from the fully nonlinear weakly dispersive two-dimensional Serre — Green — Naghdi (SGN-) model. The novelty of the proposed method consists in modifying the formula for the non-hydrostatic part of the pressure, while the accuracy of the long-wave approximation is preserved. It is shown that in some special cases the obtained fully nonlinear model is close to the known models (for example, after appropriate simplification it coincides with the model from Beji & Nadaoka (1996)). A dispersion analysis was performed one of the results of which was the conclusion that for sufficiently long waves the approximation order of the dispersion relation of the 3D model increases from the second to the fourth and an improvement was also achieved for more short waves. The proposed modification of the SGN-model is invariant with respect to the Galilean transformation; the law of conservation of mass and the law of balance of the total momentum are satisfied. However, the law of conservation of total energy is not satisfied. Apparently all NLD-models with improved dispersion characteristics possess this negative quality.

Keywords

Bibliographic record

BibTeX Citation

@article{Khakimzyanov2020twodimensionalmodel,
  author = {Khakimzyanov, G.S. and Fedotova, Z.I. and Dutykh, D.},
  title = {Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation},
  journal = {Computational Technologies},
  year = {2020},
  volume = {25},
  number = {5},
  pages = {17--41},
  doi = {10.25743/ICT.2020.25.5.003},
  abstract = {Application of nonlinear dispersion wave hydrodynamics (NLD-) models for solving practical problems constantly stimulates the search for ways to expand their field of applicability and achieve a more accurate reproduction of the characteristics of the simulated processes. A productive step in this direction turned out to be the method proposed by Madsen & Sørensen (1992), which made it possible to increase the approximation order of the dispersion relation of the Peregrine model while preserving the third order of derivatives included in the original equations and the second order of long-wave approximation. Later, other approaches were proposed to achieve this goal, which had a noticeable effect on expanding the field of applicability of NLD-models (for example, Nwogu (1993), Beji & Nadaoka (1996)). In the present work, we set a similar goal — to improve the properties of the dispersion relation of the model (and, therefore, the phase velocity), providing the Pade approximation (2,2) of the dispersion relation of the 3D model of potential flows. In contrast to earlier works on this subject, where weakly non-linear models were considered, we proceed from the fully nonlinear weakly dispersive two-dimensional Serre — Green — Naghdi (SGN-) model. The novelty of the proposed method consists in modifying the formula for the non-hydrostatic part of the pressure, while the accuracy of the long-wave approximation is preserved. It is shown that in some special cases the obtained fully nonlinear model is close to the known models (for example, after appropriate simplification it coincides with the model from Beji & Nadaoka (1996)). A dispersion analysis was performed one of the results of which was the conclusion that for sufficiently long waves the approximation order of the dispersion relation of the 3D model increases from the second to the fourth and an improvement was also achieved for more short waves. The proposed modification of the SGN-model is invariant with respect to the Galilean transformation; the law of conservation of mass and the law of balance of the total momentum are satisfied. However, the law of conservation of total energy is not satisfied. Apparently all NLD-models with improved dispersion characteristics possess this negative quality.},
  keywords = {dispersion relation, phase velocity, long surface waves, nonlinear dispersive equations}
}