|
Article information
2021 , Volume 26, ¹ 3, p.4-25
Khakimzyanov G.S., Fedotova Z.I., Dutykh D.
Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation. II. Fourth, sixth and eighth orders
In the numerical simulation of medium-length surface waves in the framework of nonlinear dispersive (NLD) models, an increased accuracy of reproducing the characteristics of the simulated processes is required. A number of works (Kirby (2016), e. g.) describe approaches to improve the known NLD-models. In particular, NLD-models of the fourth order of the long-wave approximation have been proposed and, based on a comparison of numerical results with experimental data, their high accuracy has been demonstrated (Ataie-Ashtiani and Najafi-Jilani (2007); Zhou and Teng (2010)). In these new models, the horizontal component of the velocity vector of the threedimensional (FNPF-) model of potential flows at a certain surface located between the bottom and the free boundary was chosen as the velocity vector. The result was a very cumbersome form of equations. In addition, the laws of conservation of mass and momentum do not hold for these models. The main result of this work is the derivation of a two-parameter fully nonlinear weakly dispersive (mSGN4) model of the fourth order of the long-wave approximation, which is a generalization of the well-known Serre – Green – Naghdi (SGN) second order model. In the derivation, the velocity averaged over the thickness of the liquid layer was used. The assumption about the potentiality of the three-dimensional flow was used only at the stage of closing the model. The movement of the bottom is taken into account. For the derived model, the law of conservation of mass is satisfied, and the law of conservation of total momentum is satisfied in the case of a horizontal stationary bottom. The equations of the mSGN4-model are invariant under the Galilean transformation and are presented in a compact form similar to the equations of gas dynamics. The dispersion relation of the mSGN4-model has the fourth order of accuracy in the long wave region and satisfactorily approximates the dispersion relation of the FNPF-model in the short wave region. Moreover, with a special choice of the values of the model parameters, an increased accuracy of approximating the dispersion relation of the FNPF-model at long waves (sixth or eighth order) is achieved. Analysis of the deviations of the values of the phase velocity of the mSGN4 model from the values of the “reference” speed of the FNPF model in the entire wavelength range showed that the most preferable is the mSGN4 model with the parameter values corresponding to the Pad’e approximant (2,4).
[link to elibrary.ru]
Keywords: long surface waves, nonlinear dispersive equations, dispersion relation, phase velocity
doi: 10.25743/ICT.2021.26.3.002
Author(s):
Khakimzyanov Gayaz Salimovich
Dr. , Professor
Position: Leading research officer
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Ac. Lavrentiev ave. 6
Phone Office: (383) 330 86 56
E-mail: khak@ict.nsc.ru
SPIN-code: 3144-0877Fedotova Zinaida Ivanovna
PhD.
Position: Senior Research Scientist
Office: Federal Research Center for Information and Computational Technologies
Address: 630090, Russia, Novosibirsk, Lavrentiev ave. 6
Phone Office: (383) 334-91-21
E-mail: zf@ict.nsc.ru
Dutykh Denys
Office: University Grenoble Alpes, University Savoie Mont Blanc, CNRS LAMA
Address: 73376, France, Chambery, Lavrentiev ave. 6
Phone Office: (330) 4 79 75 94 38
E-mail: Denys.Dutykh@univ-smb.fr
References:
1. Madsen P.A., Fuhrman D.R. High-order Boussinesq-type modelling of nonlinear wave phenomena in deep and shallow water. Advances in Coastal and Ocean Engineering. 2010; (11):245–285.
2. Brocchini M. A reasoned overview on Boussinesq-type models: The interplay between physics, mathematics and numerics. Proceedings of Royal Society of London. A. 2013; 469(2160):20130496.
3. Kirby J.T. Boussinesq models and their application to coastal processes across a wide range of scales. Journal of Waterway, Port, Coastal and Ocean Engineering. 2016; 142(6):03116005.
4. Khakimzyanov G., Dutykh D., Fedotova Z., Gusev O. Dispersive shallow water waves. Theory, modelling, and numerical methods. Lecture Notes in Geosystems Mathematics and Computing. Basel, Birkha¨user; 2020: 284.
5. Witting J.M. A unified model for the evolution of nonlinear water waves. Journal of Computational Physics. 1984; 56(2):203–236.
6. Madsen P.A., Murray R., Sørensen O.R. A new form of the Boussinesq equations with improved linear dispersion characteristics. Coastal Engineering. 1991; (15):371–388.
7. Madsen P.A., Banijamali B., Sch¨affer H.A., Sørensen O.R. Boussinesq type equations with high order accuracy in dispersion and nonlinearity. Coastal Engineering. 1996; (25):95–108.
8. Madsen P.A., Sch¨affer H.A. Higher order Boussinesq-type equations for surface gravity waves: Derivation and analysis. Philosophical Transactions of the Royal Society of London. A. 1998; (356):3123–3181.
9. Gobbi M.F., Kirby J.T., Wei G. A fully nonlinear Boussinesq model for surface waves. Part 2. Extension to O(kh)4. Journal of Fluid Mechanics. 2000; (405):181–210.
10. Gobbi M.F., Kirby J.T. Wave evolution over submerged sills: Tests of a high-order Boussinesq model. Coastal Engineering. 1999; (37):57–96.
11. Wei G., Kirby J.T., Grilli S.T., Subramanya R. A fully nonlinear Boussinesq model for surface waves. Part 1. Highly nonlinear unsteady waves. Journal of Fluid Mechanics. 1995; (294):71–92.
12. Ataie-Ashtiani B., Najafi-Jilani A. A higher-order Boussinesq-type model with moving bottom boundary: Applications to submarine landslide tsunami waves. International Journal for Numerical Methods in Fluids. 2007; (53):1019–1048.
13. Lynett P., Liu PL-F. A multi-layer approach to wave modeling. Proceedings of Royal Society of London. A. 2004; (460):2637–2669.
14. Zhou H., Teng M. Extended fourth-order depth-integrated model for water waves and currents generated by submarine landslides. Journal of Engineering Mechanics. 2010; 136(4):506–516.
15. Khakimzyanov G.S., Fedotova Z.I., Dutykh D. Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation. Computational Technologies. 2020; 25(5):17–41. DOI:10.25743/ICT.2020.25.5.003. (In Russ.)
16. Shokin Yu.I., Fedotova Z.I., Khakimzyanov G.S. Hierarchy of nonlinear models of the hydrodynamics of long surface waves. Doklady Physics. 2015; 60(5):224–228.
17. Fedotova Z.I., Khakimzyanov G.S. The basic nonlinear-dispersive hydrodynamic model of long surface waves. Computational Technologies. 2014; 19(6):77–93. (In Russ.)
18. Benjamin T.B., Bona J.L., Mahony J.J. Model equations for long waves in nonlinear dispersive systems. Philosophical Transactions of the Royal Society of London. A. 1972; (272):47–78.
19. Peregrine D.H. Long waves on a beach. Journal of Fluid Mechanics. 1967; (27):815–827.
20. Whitham G.B. Linear and Nonlinear Waves. New York: John Wiley & Sons Inc.; 1974: 636.
21. Fedotova Z.I., Khakimzyanov G.S. Characteristics of finite-difference methods for dispersive shallow water equations. Russian Journal of Numerical Analysis and Mathematical Modelling. 2016; 31(3):149–158.
22. Fedotova Z.I., Khakimzyanov G.S., Gusev O.I., Shokina N.Yu. History of the development and analysis of numerical methods for solving nonlinear dispersive equations of hydrodynamics. II. Two-dimensional models. Computational Technologies. 2017; 22(5):73–109. (In Russ.)
23. Khakimzyanov G., Dutykh D., Gusev O., Shokina N. Dispersive shallow water wave modelling. Part II: Numerical simulation on a globally flat space. Communications in Computational Physics. 2018; 23(1):30–92.
Bibliography link:
Khakimzyanov G.S., Fedotova Z.I., Dutykh D. Two-dimensional model of wave hydrodynamics with high accuracy dispersion relation. II. Fourth, sixth and eighth orders // Computational technologies. 2021. V. 26. ¹ 3. P. 4-25
|
