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A Hyperbolic Model of Strongly Nonlinear Waves in Two-Layer Flows of an Inhomogeneous Fluid

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Abstract

We propose a mathematical model for the propagation of nonlinear long waves in a two-layer shear flow of an inhomogeneous fluid with free boundary taking into account the dispersion and mixing effects. The equations of fluid motion are presented in the form of a hyperbolic system of first-order quasilinear equations. Solutions are constructed in the class of traveling waves that describe damped oscillations of the internal interface. The two-layer flow parameters for which large-amplitude waves can form are found. Unsteady flows that arise when flowing around a local obstacle are numerically modelled. It is shown that, depending on the oncoming flow velocity and the obstacle shape, disturbances propagate upstream in the form of a monotonous or undular bore.

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REFERENCES

  1. V. Yu. Lyapidevskii and V. M. Teshukov, Mathematical Models of Long Wave Propagation in an Inhomogeneous Fluid (Izd. Sib. Otd. Ross. Akad. Nauk, Novosibirsk, 2000) [in Russian].

    Google Scholar 

  2. O. Castro-Orgaz and W. H. Hager, “Non-hydrostatic free surface flows,” in Advances in Geophysical and Environmental Mechanics and Mathematics (Springer-Verlag, Berlin, 2017), p. 699.

  3. A. E. Green and P. M. Naghdi, “A derivation of equations for wave propagation in water of variable depth,” J. Fluid Mech. 78, 237–246 (1976).

    Article  MATH  Google Scholar 

  4. O. Le Metayer, S. Gavrilyuk, and S. Hank, “A numerical scheme for the Green–Naghdi model,” J. Comput. Phys. 229, 2034–2045 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  5. D. Lannes and F. Marche, “A new class of fully nonlinear and weakly dispersive Green–Naghdi models for efficient 2D simulations,” Comput. Phys. 282, 238–268 (2015).

    Article  MathSciNet  MATH  Google Scholar 

  6. D. Givoli and B. Neta, “High-order non-reflecting boundary conditions for the dispersive shallow water equations,” J. Comput. Appl. Math. 158, 49–60 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  7. C. Besse, P. Noble, and D. Sanchez, “Discrete transparent boundary conditions for the mixed KDV–BBM equation,” J. Comput. Phys. 345, 484–509 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  8. V. Yu. Liapidevskii and K. N. Gavrilova, “Dispersion and blockage effects in the flow over a sill,” J. Appl. Mech. Tech. Phys. 49 (1), 34–45 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  9. N. Favrie and S. Gavrilyuk, “A rapid numerical method for solving Serre–Green–Naghdi equations describing long free surface gravity waves,” Nonlinearity. 30, 2718–2736 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  10. A. A. Chesnokov and T. H. Nguyen, “Hyperbolic model for free surface shallow water flows with effects of dispersion, vorticity and topography,” Comput. Fluids. 189, 13–23 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  11. A. A. Chesnokov and V. Yu. Liapidevskii, “Hyperbolic model of internal solitary waves in a three-layer stratified fluid,” Eur. Phys. J. Plus 135 (5900), 1–19 (2020).

    Google Scholar 

  12. V. Y. Liapidevskii, A. A. Chesnokov, and V. E. Ermishina, “Quasi-linear equations of dynamics of internal solitary waves in multilayer shallow water,” J. Appl. Mech. Tech. Phys. 62 (4), 552–562 (2021).

    Article  MathSciNet  MATH  Google Scholar 

  13. S. Busto, M. Dumbser, C. Escalante, N. Favrie, and S. Gavrilyuk, “On high order ADER discontinuous Galerkin schemes for first order hyperbolic reformulations of nonlinear dispersive systems,” J. Sci. Comput. 87 (48), 1–47 (2021).

    MathSciNet  MATH  Google Scholar 

  14. S. L. Gavrilyuk, V. Yu. Liapidevskii, and A. A. Chesnokov, “Interaction of a subsurface bubble layer with long internal waves,” Eur. J. Mech. B/Fluids 73, 157–169 (2019).

    Article  MathSciNet  MATH  Google Scholar 

  15. E. L. Shroyer, J. N. Moum, and J. D. Nash, “Energy transformations and dissipation of nonlinear internal waves over New Jersey’s continental shelf,” Nonlinear Process. Geophys. 17, 345–360 (2010).

    Article  Google Scholar 

  16. J. C. Harris and L. Decker, “Intermittent large amplitude internal waves observed in Port Susan, Puget Sound,” Estuar. Coast. Shelf Sci. 194, 143–149 (2017).

    Article  Google Scholar 

  17. M. Brocchini and D. H. Peregrine, “The dynamics of strong turbulence at free surfaces. Part 1. Description,” J. Fluid Mech. 449, 225–254 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  18. M. Brocchini and D. H. Peregrine, “The dynamics of strong turbulence at free surfaces. Part 2. Free-surface boundary conditions,” J. Fluid Mech. 449, 255–290 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  19. R. H. J. Grimshaw, K. R. Khusnutdinova, L. A. Ostrovsky, and A. S. Topolnikov, “Structure formation in the oceanic subsurface bubble layer by an internal wave field,” Phys. Fluids 22, 106603 (2010).

    Article  MATH  Google Scholar 

  20. P. Lubin and S. Glockner, “Numerical simulations of three-dimensional plunging breaking waves: generation and evolution of aerated vortex filaments,” J. Fluid Mech. 767, 364–393 (2015).

    Article  MathSciNet  Google Scholar 

  21. V. Yu. Liapidevskii, A. K. Khe, and A. A. Chesnokov, “Flow regimes in a flat elastic channel in presence of a local change of wall stiffness,” J. Appl. Ind. Math. 13 (2), 270–279 (2019).

    Article  MathSciNet  Google Scholar 

  22. A. A. Chesnokov, S. L. Gavrilyuk, and V. Y. Liapidevskii, “Mixing and nonlinear internal waves in a shallow flow of a three-layer stratified fluid,” Phys. Fluids 34 (70), 075104 (2022).

    Article  Google Scholar 

  23. L. V. Ovsyannikov, “Two-layer “shallow water” models,” Prikl. Mekh. Tekh. Fiz. (2), 3–14 (1979).

  24. A. A. Chesnokov, G. A. El, S. L. Gavrilyuk, and M. V. Pavlov, “Stability of shear shallow water flows with free surface,” SIAM J. Appl. Math. 77, 1068–1087 (2017).

    Article  MathSciNet  MATH  Google Scholar 

  25. H. Nessyahu and E. Tadmor, “Non-oscillatory central differencing schemes for hyperbolic conservation laws,” J. Comput. Phys. 87, 408–463 (1990).

    Article  MathSciNet  MATH  Google Scholar 

  26. A. Treske, “Undular bores (Favre waves) in open channels: experimental studies,” J. Hydraul. Res. 32, 355–370 (1994).

    Article  Google Scholar 

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ACKNOWLEDGMENTS

The author expresses her gratitude to A.A. Chesnokov for his interest in this work.

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Authors and Affiliations

  1. Lavrentyev Institute of Hydrodynamics, Siberian Branch, Russian Academy of Sciences, Novosibirsk, 630090, Russia

    V. E. Ermishina

  2. Novosibirsk State University, Novosibirsk, 630090, Russia

    V. E. Ermishina

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  1. V. E. Ermishina
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Correspondence to V. E. Ermishina.

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Translated by V. Potapchouck

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Ermishina, V.E. A Hyperbolic Model of Strongly Nonlinear Waves in Two-Layer Flows of an Inhomogeneous Fluid. J. Appl. Ind. Math. 16, 659–671 (2022). https://doi.org/10.1134/S199047892204007X

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  • DOI: https://doi.org/10.1134/S199047892204007X

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