Two-layer shallow water formula with slope and uneven bottom solved by finite volume method

Iranian Journal of Numerical Analysis and Optimization

دوره 13، شماره 2 - شماره پیاپی 25، شهریور 2023، صفحه 317-335   

نوع مقاله: Research Article

شناسه دیجیتال (DOI): 10.22067/ijnao.2022.79205.1190

نویسندگان

U. Habibah^{* 1}؛ I.R. Lina²؛ W.M. Kusumawinahyu¹

¹Department of Mathematics, Faculty of Mathematics and Natural Sciences, University of Brawijaya, Malang, Indonesia.

²Department of Data Science, University of Insan Cita Indonesia, Jakarta, Indonesia.

چکیده

This paper proposes a numerical approach to solve a two-layer shal-low water formula with a slope and uneven bottom. The finite volume method (FVM) is applied to solve the shallow water model because the method is suitable for computational fluid dynamics problems. Rather than pointwise approximations at grid points, the FVM breaks the domain into grid cells and approximates the total integral over grid cells. The shallow water model is examined in two cases, the shallow water model in the steady state and the unsteady state. The quadratic upstream interpo-lation for convective kinetics (QUICK) is chosen to get the discretization of the space domain since it is a third-order scheme, which provides good accuracy, and this scheme proves its numerical stability. The advantage of the QUICK method is that the main coefficients are positive and satisfy the requirements for conservativeness, boundedness, and transportation. An explicit scheme is used to get the discretization of the time domain. Finally, the numerical solution of the steady state model shows that the flow remains unchanged. An unsteady-state numerical solution produces instability (wavy at the bottom layer). Moreover, the larger slope results in higher velocity and higher depth at the second layer.

کلیدواژه‌ها

Two-layer shallow water؛ Finite volume method؛ QUICK scheme؛ Explicit scheme

مراجع

[1] Abedian, R. A modified high-order symmetrical WENO scheme for hy-perbolic conservation laws, Int. J. Nonlinear Sci. Numer. Simul. 2022.
[2] Abedian, R. A finite difference Hermite RBF-WENO scheme for hyper-bolic conservation laws, Int. J. Numer. Methods Fluids, 94(6) (2022), 583–607.
[3] Abgrall, R. and Karni, S. Two-layer shallow water system: A relaxation approach, SIAM J. Sci. Comput. 31(3) (2009), 1603–1627.
[4] Arachchige, J.P. and Pettet. G.J. A finite volume method with lineari-sation in time for the solution of advection–reaction–diffusion systems, Appl. Math. Comput. 231 (2014), 445–462.
[5] Castro, M.J., Garcia-Rodriguez, J.A., Gonzalez-Vida, J.M., Macias, J., Pares, C. and Vazquez-Cendon, M.E. Numerical simulation of two-layer shallow water flows through channels with irregular geometry, J. Comput. Phys. 195 (2004), 2002–235.
[6] Chakir, M., Ouazar, D., and Taik, A. Roe scheme for two-layer shallow water equations: Application to the strait of Gibraltar, Math. Model. Nat. Phenom. 4(5) (2009), 114–127.

[7] Chen, S C., and Peng, S.H. Two-dimensional numerical model of two-layer shallow water equations for confluence simulation, Adv. Water Resour. 29 (2006), 1608–1617.
[8] Chiapolino, A. and dan Saurel, R. Models and methods for two-layer shallow water flows, J. Comput. Phys. 371 (2018), 1043–1066.
[9] Cristo, C.D., Greco, M., Iervolino, M., Martino, R. and Vacca, A. A remark of finite volume methods for 2D shallow water equations over irregular bottom topography, J. Hydraul. Res. 59 (2021), 337–344.
[10] Kurganov, A. and Petrova, G. Central-upwind schemes for two-layer shallow water equations, SIAM J. Sci. Comput. 31(3) (2009), 1742–1773.
[11] Leveque, R.A. Finite-volume methods for hyperbolic problems. Cam-bridge University Press, 2002.
[12] Lina, I.R., Habibah, U., and Kusumawinahyu, W.M. Mathematical mod-elling of two layer shallow water flow with incline and uneven bottom, J. Phys. Conf. Ser. 1563 (2020), 0122019.
[13] Muhammad, N. Finite volume method for simulation of flowing fluid via OpenFOAM, Eur. Phys. J. Plus, 136 (10) (2021) 1–22.
[14] Mungkasi, S. Finite volume methods for one-dimensional shallow water equations. Ph.D. Thesis, Australian National University, 2008.
[15] Nikan, O. and Avazzadeh, Z.Coupling of the Crank–Nicolson scheme and localized meshless technique for viscoelastic wave model in fluid flow, J. Comput. Appl. Math. 398 (2021), 113695.
[16] Nikan, O., Avazzadeh, Z. and Rasoulizadeh, M.N. Soliton solutions of the nonlinear sine-Gordon model with Neumann boundary conditions arising in crystal dislocation theory, Nonlinear Dyn. 106 (2021), 783–813.
[17] Pascal, J.P. A model for two-layer moderate Reynolds number flow down an incline, Int. J. Non-Linear Mech. 36(6) (2001), 977–985.
[18] Rasoulizadeh, M.N., Ebadi, M.J., Avazzadeh, Z., and Nikan, O. A high-order symmetrical weighted hybrid ENO-ux limiter scheme for hyperbolic conservation laws, Comput. Phys. Commun. 185 (2014), 106–127.
[19] Rasoulizadeh, M.N., Ebadi, M.J., Avazzadeh, Z. and Nikan, O. An ef-ficient local meshless method for the equal width equation in fluid me-chanics, Eng. Anal. Bound Elem. 131 (2021), 258–268.
[20] Rasoulizadeh, M.N., Nikan, O. and Avazzadeh, Z. The impact of LRBF‑FD on the solutions of the nonlinear regularized long wave equa-tion, Math. Sci. 15 (2021), 365–376.

[21] Remi, A. and Smadar, K. Two-layer shallow water system: A relaxation approach, SIAM J. Sci. Comput. 31(3) (2009), 1603–1627.
[22] Swartenbroekx, C., Zech, Y.V. and Soares-Frazao, S. Two-dimensional two-layer shallow water model for dam break flows with significant bed load transport, Int. J. Numeric. Met, Fluids, 73 (2013), 477–508.
[23] Vallis, G.K. Atmospheric and oceanic fluid dynamics. Cambridge Uni-versity Press, USA, 2006.
[24] Versteeg, H. K. and Malalasekera, W. An introduction to computational fluid dynamics: The finite volume method, Pearson Education Limited, 2007.
[25] Yadav, A., Chakraborty, S. and Usha, R. Steady solution of an inverse problem in gravity-driven shear-thinning film flow: Reconstruction of an uneven bottom substrate, J. Non-Newton. Fluid Mech. 219 (2015), 65–77.