Hostname: page-component-cd9895bd7-mkpzs Total loading time: 0 Render date: 2024-12-26T17:11:38.322Z Has data issue: false hasContentIssue false
  • English
  • Français

A Dispersive Numerical Model for the Formation of Undular Bores Generated by Tsunami Wave Fission

Part of: Basic methods in fluid mechanics Hyperbolic equations and systems Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  31 January 2018

I. Magdalena
I. Magdalena*
Affiliation:
Industrial and Financial Mathematics Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung, Jalan Ganesha 10, Bandung 40132, Indonesia
*
*Corresponding author. Email address:[email protected] (I. Magdalena)
Get access

Abstract

A two-layer non-hydrostatic numerical model is proposed to simulate the formation of undular bores by tsunami wave fission. These phenomena could not be produced by a hydrostatic model. Here, we derived a modified Shallow Water Equations with involving hydrodynamic pressure using two layer approach. Staggered finite volume method with predictor corrector step is applied to solve the equation numerically. Numerical dispersion relation is derived from our model to confirm the exact linear dispersion relation for dispersive waves. To illustrate the performance of our non-hydrostatic scheme in case of linear wave dispersion and non-linearity, four test cases of free surface flows are provided. The first test case is standing wave in a closed basin, which test the ability of the numerical scheme in simulating dispersive wave motion with the correct frequency. The second test case is the solitary wave propagation as the examination of owing balance between dispersion and nonlinearity. Regular wave propagation over a submerged bar test by Beji is simulated to show that our non-hydrostatic scheme described well the shoaling process as well as the linear dispersion compared with the experimental data. The last test case is the undular bore propagation.

Keywords

Undular boresdepth integrated Euler equationsfinite volume methodpredictor correctortsunami fission

MSC classification

Secondary: 65M08: Finite volume methods 76M12: Finite volume methods 35L65: Conservation laws
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 7 , Issue 4 , November 2017 , pp. 767 - 784
Copyright
Copyright © Global-Science Press 2017 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Beji, S. and Battjes, J.A., Experimental investigation of wave propagation over a bar, Coast. Engng. 19, 151–162 (1993).Google Scholar
[2] Bai, Y. and Cheung, K.F., Depth-intergrated free surface flow with a two-layer on-hydrostatic formulation, Int. J. Numer. Meth. Fluids 69, 411–429 (2011).Google Scholar
[3] Grue, J., Pelinovsky, E.N., Fructus, D., Talipova, T. and Kharif, C., Formation of undular bores and solitary waves in the Strait of Malacca caused by the 26 December 2004 Indian Ocean tsunami, J. Geophys. Res. 113, C05008 (2008).Google Scholar
[4] Ioualalen, M., Pelinovsky, E., Asavanant, J., Lipikorn, R. and Deschamps, A., On the weak impact of the 26 December Indian Ocean tsunami on the Bangladesh coast, Nat. Hazards Earth Syst. Sci. 7, 141–147 (2007).Google Scholar
[5] Kuang, D.Y. and Lee, L., A conservative formulation and a numerical algorithmfor the double-gyre nonlinear shallow-water model, Numer. Math. Theor. Meth. Appl. 8, 634–650 (2015).Google Scholar
[6] Lee, W.T., Tridiagonalmatrices: Thomas algorithm, MS6021, Scientific Computation, University of Limerick, 2011.Google Scholar
[7] Magdalena, I., Erwina, N. and Pudjaprasetya, S.R., Staggered momentum conservative scheme for radial dam break simulation, J. Sci. Comput. 65, 867–874 (2015).Google Scholar
[8] Matsuyama, M., Ikeno, M., Sakakiyama, T. and Takeda, T., A study of tsunami wave fission in an undistorted experiment, Pure Appl. Geophys. 164, 617–631 (2007).Google Scholar
[9] Magdalena, I., Pudjaprasetya, S.R. and Wiryanto, L.H., Wave interaction with an emerged porous media, Adv Appl. Math. Mech. 6, 680–692 (2004).Google Scholar
[10] Nwogu, O., Alternative form of boussinesq equations for nearshore wave propagation, J. Waterw. Port. Coast. Ocean Eng. 119, 618–638 (1993).Google Scholar
[11] Peregrine, D.H., Calculations of the development of an undular bore, J. Fluid Mech. 25, 321–330 (1966).Google Scholar
[12] Pudjaprasetya, S.R. and Groesen, E.V., Unidirectional waves over slowly varying bottom Part II. Quasi-homogeneous approximation of distorting waves, Wave Motion 23, 23–38 (1996).Google Scholar
[13] Pudjaprasetya, S.R. and Magdalena, I., Momentum conservative scheme for dam break and wave run up simulations, East Asian J. Appl. Math. 4, 152–165 (2014).Google Scholar
[14] Stelling, G.S. and Duinmeije, S.P.A., A staggered conservative scheme for every Froude number in rapidly varied shallow water flows, Int. J. for Numer. Meth. Fluids 43, 1329–1354 (2003).Google Scholar
[15] Soares-Frazão, S. and Guinot, V., A second-order semi-implicit hybrid scheme for one-dimensional Boussinesq-type waves in rectangular channels, Int. J. Numer. Methods Fluids 58, 237–261 (2008).Google Scholar
[16] Soares-Frazão, S. and Zech, Y., Undular bores and secondary waves-experiments and hybrid finite-volume modeling, J. Hydr. Research 40, 33–43 (2002).Google Scholar
[17] Stelling, G.S. and Zijlema, M., An accurate and efficient finite-difference algorithm for non-hydrostatic free-surface flow woth application to wave propagation, Int. J. for Numer. Meth. Fluids 43, 1–23 (2003).Google Scholar
[18] Zijlema, M. and Stelling, G.S., Efficient computation of surf zone waves using the nonlinear shallow water equations with non-hydrostatic pressure, Coast. Engng. 55, 780–790 (2008).Google Scholar
[19] Zhang, W., Tong, L. and Chung, E., Efficient simulation of wave propagation with implicit finite difference schemes, Numer. Math. TMA 5, 205–228 (2012).Google Scholar