On the existence of flux as a function of the surface elevation for long wave solution of shallow water equations
Keywords:
Shallow water equations, turbulence, horizontal flux, surface elevationAbstract
The shallow water equations in mechanics of fluids, govern the motion of a shallow layer of water over a fixed impervious bed. In this paper, the bed form is assumed to be rough and horizontal, and the motion of water is assumed to be of the long wave type (Lamb [1], pp. 254-256) such that the free surface has a gradually varying propagating profile. Gravitation permits such motion but is resisted by the turbulence generated by the bed friction. A model of the governing equations based on the Reynolds averaged Navier-Stokes equations has recently been given by Bose [2], which is highly nonlinear. A heuristic approach of numerically solving the equations for the modified long waves is also presented in that article, by assuming that the horizontal flux across a section of flow is some function of the free surface elevation alone. This key assertion is analysed in this article and proved to hold provided some boundedness criteria are satisfied by the flux gradients. The theory is apparently applicable to find appropriate boundedness conditions on the flux of flow for numerically solving long wave equations in the case of other models for long wave propagation as well.
References
Lamb, H., Hydrodynamics, Dover, New York (2009).
Bose, S.K., Turbulent two-dimensional shallow water equations and their numerical solution, Archive of Applied Mechanics (2022), https://doi.org/10.1007/s00419-022-02243-w. DOI: https://doi.org/10.1007/s00419-022-02243-w
Bose, S.K., A numerical method for the solution of the nonlinear turbulent one-dimensional free surface flow equations, Comput. Geosci. 22, 81-86 (2018). DOI: https://doi.org/10.1007/s10596-017-9671-y
Garcia-Navarro, P., Murillo, J., Fern ´andez-Pato, P., Echeverribar, I., Morales-Hern ´andez, M., The shallow water equations and their application to realistic cases, Env. Fluid Mech. 19, 1235-1252 (2019), https://doi.org/10.1007/s10652-018-09657-7. DOI: https://doi.org/10.1007/s10652-018-09657-7
Li, M., Liu, X., Liu, Y., A highly nonlinear shallow water model arising from the full water waves with the Coriolis effect, J. Math. Fluid Mech. 25, 23 (2023), https://doi.org/10.1007/s00021-023-00785-9. DOI: https://doi.org/10.1007/s00021-023-00785-9
Geyer, A., Quirchmayr, R., Shallow water equations for equatorial tsunami waves, Phil. Trans. R. Soc. A 376 20170100 (2017), http://dx.doi.org/10.1098/rsta.2017.0100. DOI: https://doi.org/10.1098/rsta.2017.0100
Green, A., Naghdi, P., A derivation of equation for wave propagation in water of variable depth, J. Fluid Mech. 78, 237-246 (1976). DOI: https://doi.org/10.1017/S0022112076002425
Castro-Orgaz, O., Cantero-Chinchilla, F.N., Nonlinear shallow water flow modelling over topography with depth-averaged potential equations, Env. Fluid Mech. 22, 261-291 (2020), https://doi.org/10.1007/s10652-019-09691-z. DOI: https://doi.org/10.1007/s10652-019-09691-z
Chen, R.M., Gui, G., Liu, Y., On a shallow water approximation to the Green-Naghdi equations with the Coriolis effect, Adv. in Math. 340, 106-137 (2018), https://doi.org/10.1016/j.aim2018.10.003. DOI: https://doi.org/10.1016/j.aim.2018.10.003
Cienfuegos, R., Surfing waves from the ocean to the river with the Serre-Green-Naghdi equations, ASCE J. Hydraul. Eng. 149: 04023032 (2023). DOI: https://doi.org/10.1061/JHEND8.HYENG-13487
Liu, Y., Zhang, Y., Pang, J., Approximate solution to shallow water equations by the homotopy perturbation method coupled with Mohand transform, Frontiers in Phys. 10, 1-10 (2023), https://doi.org/10.3389/fphy 2022.1118898. DOI: https://doi.org/10.3389/fphy.2022.1118898
Yang, X., An, W., Li, W., Zhang S., Implementation of a local time stepping algorithm and its acceleration effect on two dimensional hydrodynamic models, Water. 12, 1148 (2020), https://doi.org/10.3390/w12041148. DOI: https://doi.org/10.3390/w12041148
Zhang, P-B, Zhang, Z-H, Zhang, H-S, Zhao, X-Yi., Numerical simulation of nonlinear wave propagation from deep to shallow water, J. Mar. Sci. Eng. 11, 1003 (2023), https://doi.org/10.3390/jmse11051003. DOI: https://doi.org/10.3390/jmse11051003
Lidyana, P., Ginting, B.M., Yudianto, D., Numerical solution for one-dimensional wave prpagation by solv-ing the shallow water equations using the Preissman implicit scheme, J. Civil. Eng. Forum, 8, 205-216 (2022), https://jurnal.ugm.ac.id/issue/archive. DOI: https://doi.org/10.22146/jcef.3872
Delis, A.I., Nikolas, I.K., Shallow water equations in hydraulics, Numerics and Applications. 13, 3598 (2021), https://doi.org/10.3390/w13243598. DOI: https://doi.org/10.3390/w13243598
Mader, C.L., Numerical modeling of water waves, CRC Press, Boca-Raton (2004). DOI: https://doi.org/10.1201/9780203492192
Hildebrand, F.B., Advanced calculus for applications, Prentice-Hall of India, New Delhi (1977).
Burkill, J.C., Burkill, N., A second course in mathematical analysis, Cambridge University Press, Cambridge (1970).
Garling, D.J.H., Inequalities, Cambridge University Press, Cambridge (2007).
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2024 Sujit K.Bose; Ganesh C. Gorain (Corresponding Author)
This work is licensed under a Creative Commons Attribution 4.0 International License.
Accepted 2024 Mar 21
Published 2024 Mar 23