Two-dimensional numerical modeling of dam-break flow using a new TVD finite-element scheme

A new numerical scheme based on the finite-element method with a total-variation-diminishing property is developed with the aim of studying the shock-capturing capability of the combination. The proposed model is formulated within the framework of the one-step Taylor–Galerkin scheme in conjunction with the Van Leer’s limiter function. The approach is applied to the two-dimensional shallow water equations by different test cases, i.e., the partial, circular, and one-dimensional dam-break flow problems. For the one-dimensional case, the sub- and supercritical flow regimes are considered. The results are compared with the analytical, finite-difference, and smoothed particle hydrodynamics solutions in the literature. The findings show that the proposed model can effectively mask the sources of errors in the abrupt changes of the flow conditions and is able to resolve the shock and rarefaction waves where other numerical models produce spurious oscillations.

[1]  P. Lauritzen.  Numerical techniques for global atmospheric models , 2011 .

[2]  C. Klingenberg,et al.  Well-balanced central finite volume methods for the Ripa system , 2015 .

[3]  Hans-Joachim Bungartz,et al.  Dynamically adaptive Lattice Boltzmann simulation of shallow water flows with the Peano framework , 2015, Appl. Math. Comput..

[4]  Junqiang Gao,et al.  The numerical simulations based on the NND finite difference scheme for shallow water wave equations including sediment concentration , 2015 .

[5]  N. Z. Moghaddam,et al.  Simulating free surface problem using isogeometric analysis , 2016 .

[6]  Vittorio Selmin.  Finite element solution of hyperbolic equations II. Two dimensional case , 1987 .

[7]  E. Toro.  Shock-Capturing Methods for Free-Surface Shallow Flows , 2001 .

[8]  A. Boulahia,et al.  Simulation of Viscous and Reactive Hypersonic Flows Behaviour in a Shock Tube Facility: TVD Schemes and Flux Limiters Application , 2014 .

[9]  Ali Tavakoli,et al.  On numerical stabilization in the solution of Saint-Venant equations using the finite element method , 2011, Comput. Math. Appl..

[10]  A. Harten.  High Resolution Schemes for Hyperbolic Conservation Laws , 2017 .

[11]  J. S. Wang,et al.  FINITE-DIFFERENCE TVD SCHEME FOR COMPUTATION OF DAM-BREAK PROBLEMS , 2000 .

[12]  A. Hazzab,et al.  A Petrov–Galerkin scheme for modeling 1D channel flow with varying width and topography , 2013 .

[13]  Zhonghan. Zhang,et al.  Numerical simulation of flow and bed morphology in the case of dam break floods with vegetation effect , 2016 .

[14]  A TVD-WAF-based hybrid finite volume and finite difference scheme for nonlinearly dispersive wave equations , 2015 .

[15]  Xinya Ying,et al.  Numerical Simulation of Flood Inundation due to Dam and Levee Breach , 2003 .

[16]  Z. Chambers.  A Characteristics Finite Element Algorithm for Computational Open Channel Flow Analysis , 2000 .

[17]  Robert A. Dalrymple,et al.  Modeling Dam Break Behavior over a Wet Bed by a SPH Technique , 2008 .

[18]  Nicholas P. Chotiros,et al.  Finite element modeling of propagation and reverberation shallow water waveguide with a variable environment , 2015 .

[19]  MingLiang Zhang,et al.  Integrating 1D and 2D hydrodynamic, sediment transport model for dam-break flow using finite volume method , 2014 .

[20]  Ali Triki.  A Finite Element Solution of the Unidimensional Shallow-Water Equation , 2013 .

[21]  H. C. Yee,et al.  Implicit Total Variation Diminishing (TVD) schemes for steady-state calculations. [in gas dynamics , 1983 .

[22]  J. Z. Zhu,et al.  The finite element method , 1977 .

[23]  Pilar García-Navarro,et al.  A HIGH-RESOLUTION GODUNOV-TYPE SCHEME IN FINITE VOLUMES FOR THE 2D SHALLOW-WATER EQUATIONS , 1993 .

[24]  I. Akkerman,et al.  Isogeometric analysis of free-surface flow , 2011, J. Comput. Phys..

[25]  Yu Luo,et al.  A MacCormack-TVD finite difference method to simulate the mass flow in mountainous terrain with variable computational domain , 2013, Comput. Geosci..

[26]  S. Liang,et al.  Solution of shallow-water equations using least-squares finite-element method , 2008 .

[27]  Pablo Ortiz.  Shallow water flows over flooding areas by a flux-corrected finite element method , 2014 .

[28]  J. Sládek,et al.  Extrapolated local radial basis function collocation method for shallow water problems , 2015 .

[29]  Alessandro Dal Palù,et al.  GPU-enhanced Finite Volume Shallow Water solver for fast flood simulations , 2014, Environ. Model. Softw..

[30]  Bassam A. Younis,et al.  Selective application of a total variation diminishing term to an implicit method for two‐dimensional flow modelling , 2015 .

[31]  Qiang Xu,et al.  MacCormack-TVD Finite Difference Solution for Dam Break Hydraulics over Erodible Sediment Beds , 2015 .

[32]  J. Douglas,et al.  Numerical Solutions of the One-Dimensional Primitive Equations Using Galerkin Approximations With localized Basis Functions , 1972 .

[33]  Tony W. H. Sheu,et al.  High resolution finite-element analysis of shallow water equations in two dimensions , 2001 .

[34]  Hong-Ming Kao,et al.  Numerical simulation of shallow-water dam break flows in open channels using smoothed particle hydrodynamics , 2011 .

[35]  Ziyi Huang,et al.  Modeling the Spatial Evolution of Roll Waves with Diffusive Saint Venant Equations , 2015 .

[36]  Roger Alexander Falconer,et al.  Water Quality Simulation Study of a Natural Harbor , 1986 .

[37]  Ping Wang,et al.  A large-scale wave-current coupled module with wave diffraction effect on unstructured meshes , 2014 .