Numerical simulation of two-layer shallow water flows through channels with irregular geometry.

*(English)*Zbl 1087.76077Summary: This paper deals with the numerical simulation of flows of stratified fluids through channels with irregular geometry. Channel cross-sections are supposed to be symmetric but not necessarily rectangular. The fluid is supposed to be composed of two shallow layers of immiscible fluids of constant densities, and the flow is assumed to be one-dimensional. Therefore, the equations to be solved are a coupled system composed of two shallow Water models with source terms involving depth and breadth functions. Extensions of the \(Q\)-schemes of van Leer and Roe are proposed where a suitable treatment of the coupling and source terms is performed by adapting the techniques developed in [M. E. Vázquez-Cendón, J. Comput. Phys. 148, No. 2, 497–526 (1999; Zbl 0931.76055); Comput. Fluids 29, No. 8, 17 ff (2000); M. Castro et al., M2AN, Math. Model. Numer. Anal. 35, No. 1, 107–127 (2001; Zbl 1094.76046)]. An enhanced consistency condition, the so-called C-property, introduced in [Comput. Fluids 23, No. 8, 1049–1071 (1994; Zbl 0816.76052)] is extended to this case ,and a general result providing sufficient conditions to ensure this property is shown. Then, some numerical tests to validate the resulting schemes are presented. First, we verify that, in practice, the numerical schemes satisfy the \(C\)-property, even for extremely irregular channels. Then, in order to validate the schemes, we compare some approximate steady solutions obtained with the generalized \(Q\)-scheme of Van Leer with those obtained by using the asymptotic techniques developed by L. Armi and D. Farmer [Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. 455, No. 1989, 3221–3258 (1999; Zbl 0951.76021)] for channels with simplified geometries. Finally, we apply the numerical scheme to the simulation of the flow through the Strait of Gibraltar. Real bathymetric and coast-line data are considered to include in the model the main features of the abrupt geometry of this natural strait connecting the Atlantic Ocean and the Mediterranean Sea. A steady-state solution is obtained from lock-exchange initial conditions. This solution is then used as initial condition to simulate the main semidiurnal and diurnal tidal waves in the Strait of Gibraltar through the imposition of suitable boundary conditions obtained from observed tidal data. Comparisons between numerical results and observed data are also presented.

##### MSC:

76M20 | Finite difference methods applied to problems in fluid mechanics |

76B70 | Stratification effects in inviscid fluids |

76U05 | General theory of rotating fluids |

86A05 | Hydrology, hydrography, oceanography |

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\textit{M. J. Castro} et al., J. Comput. Phys. 195, No. 1, 202--235 (2004; Zbl 1087.76077)

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##### References:

[1] | Armi, L., The hydraulics of two flowing layers with different densities, J. fluid mech., 163, 27-58, (1986) |

[2] | Armi, L.; Farmer, D., Maximal two-layer exchange through a contraction with barotropic net flow, J. fluid mech., 164, 27-51, (1986) · Zbl 0587.76168 |

[3] | Armi, L.; Farmer, D., The flow of the Mediterranean water through the strait of gibraltar. the flow of the atlantic water through the strait of gibraltar, Prog. oceanogr., 21, 1-105, (1988) |

[4] | Bermúdez, A.; Vázquez, M.E., Upwind methods for hyperbolic conservation laws with source terms, Comput. fluids, 23, 8, 1049-1071, (1994) · Zbl 0816.76052 |

[5] | Bermúdez, A.; Dervieux, A.; Désidéri, J.A.; Vázquez, M.E., Upwind schemes for the two-dimensional shallow water equations with variable depth using unstructured meshes, Comput. methods appl. mech. engrg., 155, 1-2, 49-72, (1998) · Zbl 0961.76047 |

[6] | Bryden, H.; Candela, J.; Kinder, T.H., Exchange through the strait of gibraltar, Prog. oceanogr., 33, 201-248, (1994) |

[7] | Castro, M.J.; Macı́as, J.; Parés, C., A Q-scheme for a class of systems of coupled conservation laws with source term. application to a two-layer 1-D shallow water system, Math. model. numer. anal., 35, 1, 107-127, (2001) · Zbl 1094.76046 |

[8] | M.J. Castro, J. Macı́as, C. Parés, J.A. Rubal, M.E. Vázquez-Cendón, Two-layer numerical model for solving exchange flows through channels with irregular geometry, in: Proceedings of “ECCOMAS 2001”, Swansea, 2001 |

[9] | Farmer, D.; Armi, L., Maximal two-layer exchange over a sill and through a combination of a sill and contraction with barotropic flow, J. fluid mech., 164, 53-76, (1986) · Zbl 0587.76169 |

[10] | Garcı́a Lafuente, J.; Almazán, J.L.; Castillejo, F.; Khribeche, A.; Hakimi, A., Sea level in the strait of gibraltar: tides, Int. hydrogr. rev., LXVII, 1, 111-130, (1990) |

[11] | Garcı́a-Navarro, P.; Vázquez-Cendón, M.E., On numerical treatment of the source terms in the shallow water equations, Comput. fluids, 29, 8, 17-45, (2000) · Zbl 0986.76051 |

[12] | Harten, A., On a class of high resolution total-variation-stable finite-difference schemes, SIAM J. numer. anal., 21, 1, 1-23, (1984) · Zbl 0547.65062 |

[13] | Helfrich, K.R., Time-dependent two-layer hydraulic exchange flows, J. phys. oceanogr., 25, 3, 359-373, (1995) |

[14] | Izquierdo, A.; Tejedor, L.; Sein, D.V.; Backhaus, J.O.; Brandt, P.; Rubino, A.; Kagan, B.A., Control variability and internal bore evolution in the strait of gibraltar, Estuarine coastal shelf sci., 53, 637-651, (2001) |

[15] | Lawrence, G.A., The hydraulics of steady 2-layer flow over a fixed obstacle, J. fluid mech., 254, 605-633, (1993) |

[16] | J. Macı́as, Approximate Armi and Farmer solutions for flows through arbitrary channels, Internal Journal 0115 group on “Differential Equations Numerical Analysis and Applications”, University of Málaga, Spain, 2001 |

[17] | Roe, P.L., Approximate Riemann solvers parameter vectors and difference schemes, J. comput. phys., 43, 357-371, (1981) · Zbl 0474.65066 |

[18] | Roe, P.L., Upwinding differenced schemes for hyperbolic conservation laws with source terms, (), 41-51 |

[19] | J.B. Schijf, J.C. Schonfeld, Theoretical considerations on the motion of salt and fresh water, in: Proceedings of the Minn. Int. Hydraulics Conv., Joint meeting IAHR and Hyd. Div. ASCE., September 1953, pp. 321-333 |

[20] | Tejedor, L.; Izquierdo, A.; Sein, D.V.; Kagan, B.A., Tides and tidal energetics of the strait of gibraltar: a modeling approach, Tectonophysics, 294, 333-347, (1998) |

[21] | Toro, E.F., Riemann solvers and numerical methods for fluid dynamics. A practical introduction, (1997), Springer · Zbl 0888.76001 |

[22] | E.F. Toro, M.E. Vázquez-Cendón, Model hyperbolic systems with source terms: exact and numerical solutions, in: Proceedings of “Godunov methods: Theory and Applications”, 2000 |

[23] | van Leer, B., Towards the ultimate conservative difference scheme III. upstream-centered difference schemes for ideal compressible flow, J. comput. phys., 23, 263-275, (1977) · Zbl 0339.76039 |

[24] | B. van Leer, Progress in multi-dimensional upwind differencing. ICASE Report 92/43, NASA Langley Research Center, Hampton, VA, 1984, in: Proceedings of the 13th International Conference on Numerical Methods in Fluid Dynamics, Rome, 1992 |

[25] | M.E. Vázquez-Cendón, Estudio de Esquemas Descentrados para su Aplicación a las leyes de Conservación Hiperbólicas con Términos Fuente, PhD thesis, Universidad de Santiago de Compostela, 1994 |

[26] | Vázquez-Cendón, M.E., Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry, J. comput. phys., 148, 497-526, (1999) · Zbl 0931.76055 |

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