Finite Volume Model for Two-Dimensional Shallow Environmental Flow
Publication: Journal of Hydraulic Engineering
Volume 137, Issue 2
Abstract
This paper presents the development of a two-dimensional, depth integrated, unsteady, free-surface model based on the shallow water equations. The development was motivated by the desire of balancing computational efficiency and accuracy by selective and conjunctive use of different numerical techniques. The base framework of the discrete model uses Godunov methods on unstructured triangular grids, but the solution technique emphasizes the use of a high-resolution Riemann solver where needed, switching to a simpler and computationally more efficient upwind finite volume technique in the smooth regions of the flow. Explicit time marching is accomplished with strong stability preserving Runge-Kutta methods, with additional acceleration techniques for steady-state computations. A simplified mass-preserving algorithm is used to deal with wet/dry fronts. Application of the model is made to several benchmark cases that show the interplay of the diverse solution techniques.
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References
Alcrudo, F., and Garcia-Navarro, P. (1993). “A high-resolution Godunov-type scheme in finite volumes in 2D shallow water equations.” Int. J. Numer. Methods Fluids, 16, 489–505.
Anastasiou, K., and Chan, C. T. (1997). “Solution of the 2D shallow water equations using the finite volume method on unstructured triangular meshes.” Int. J. Numer. Methods Fluids, 24, 1225–1245.
Balzano, A. (1998). “Evaluation of methods for numerical simulation of wetting and drying of shallow water flow models.” Coastal. Eng., 34, 83–107.
Barth, T. J. (1993). “Recent developments in high order k-exact reconstruction on unstructured meshes.” 31st AIAA Aerospace Sciences Meeting and Exhibit, Reno, NV, January 11–14, 1993, 15 pages AIAA Paper 93–0668.
Batten, P., Lambert, C., and Causon, D. M. (1996). “Positively conservative high-resolution convective schemes for unstructured elements.” Int. J. Numer. Methods Eng., 39, 1821–1838.
Beffa, C., and Connell, R. J. (2001). “Two-dimensional flood plain flow. I: Model description.” J. Hydrol. Eng., 6(5), 397–405.
Begnudelli, L., and Sanders, B. F. (2006). “Unstructured grid finite-volume algorithm for shallow-water flow and scalar transport with wetting and drying.” J. Hydraul. Eng., 132(4), 371–384.
Begnudelli, L., Sanders, B. F., and Bradford, S. F. (2008). “Adaptive Godunov-based model for flood simulation.” J. Hydraul. Eng., 134(6), 714–725.
Bradford, S. F., and Sanders, B. F. (2002). “Finite-volume method for shallow water flooding of arbitrary topography.” J. Hydraul. Eng., 128(3), 289–298.
Brufau, P., Garcia-Navarro, P., and Vázquez-Cendón, M. E. (2004). “Zero mass error using unsteady wetting-drying conditions in shallow flows over dry irregular topography.” Int. J. Numer. Methods Fluids, 45, 1047–1082.
Brufau, P., Vázquez-Cendón, M. E., and Garcia-Navarro, P. (2002). “A numerical model for the flooding and drying of irregular domains.” Int. J. Numer. Methods Fluids, 39, 247–275.
Crossley, A. J., Wright, N. G., and Witlow, C. D. (2003). “Local time stepping for modeling open channel flows.” J. Hydraul. Eng. 129(6), 455–462.
Gottlieb, S., Shu, C. -W., and Tadmor, E. (2001). “Strong stability-preserving high-order time discretization methods.” SIAM Rev., 43(1), 89–112.
Harten, A., and Hyman, J. M. (1983). “Self adjusting grid methods for one-dimensional hyperbolic conservation laws.” J. Comput. Phys., 50, 235–269.
Horritt, M. S. (2002). “Evaluating wetting and drying algorithms for finite element models of shallow water flow.” Int. J. Numer. Methods Eng., 55, 835–851.
International Association of Hydraulic Research (IAHR). (2007). “Harmonizing the demands of art and nature in hydraulics.” Proceedings of the 32nd IAHR Congress, Venice, Italy, 2007.
Jameson, A., and Mavriplis, D. (1986). “Finite volume solution of the two-dimensional Euler equations on a regular triangular mesh.” AIAA J., 24(4), 611–618.
Krivodonova, L., Xin, J., Remacle, J. -N., Chevaugeon, N., and Flaherty, J. E. (2004). “Shock detection and limiting with discontinuous Galerkin methods for hyperbolic conservation laws.” Appl. Numer. Math., 48, 323–338.
Murillo, J., Garcia-Navarro, P., Burguete, J., and Brufau, P. (2007). “The influence of source terms on stability, accuracy, and conservation in two-dimensional shallow flow simulation using triangular finite volumes.” Int. J. Numer. Methods Fluids, 54, 543–590.
Rausch, R., Batina, J., and Yang, H. (1991). “Spatial adaptation procedures on unstructured meshes for accurate unsteady aerodynamic flow computation.” AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, 32nd, Baltimore, MD, Apr. 8–10, 1991, Technical Papers. Pt. 3 (A91–31826 12–39). Washington, DC, American Institute of Aeronautics and Astronautics, 1991, 1904–1918. AIAA Paper 1991–1106
Roe, P. L. (1981). “Approximate Riemann solvers, parameter vectors, and difference schemes.” J. Comput. Phys., 43, 357–372.
Rusanov, V. V. (1961). “Calculation of intersection of non-steady shock waves with obstacles.” J. Comput. Math. Phys. USSR, 1, 267–279.
Sanders, B. F. (2008). “Integration of a shallow water model with a local time step.” J. Hydraul. Res., 46(4), 466–475.
Swanson, R. C., and Turkel, E. (1997). “Multistage schemes with multigrid for Euler and Navier-Stokes equations, components and analysis.” NASA Technical Paper 3631, Langley Research Center, Hampton, Va.
Synolakis, C. E. (1987). “The runup of solitary waves.” J. Fluid Mech., 185, 523–545.
Titov, V. V., and Synolakis, C. E. (1995). “Modeling of breaking and nonbreaking long-wave evolution and runup using VTCS-2.” J. Waterway, Port, Coastal, Ocean Eng., 121(6), 308–316.
Toro, E. F. (2001). Shock-capturing methods for free-surface shallow flows, John Wiley & Sons, Ltd., Chichester, West Sussex, England.
van Leer, B. (1979). “Towards the ultimate conservative difference scheme, V: A second order sequel to Godunov's method.” J. Comput. Phys., 32, 101–136.
Venkatakrishnan, V. (1995). “Convergence to steady state solutions of the Euler equations on unstructured grids with limiters.” J. Comput. Phys., 118, 120–130.
Yoon, T. H., and Kang, S. -K. (2004). “Finite volume model for two-dimensional shallow water flows on unstructured grids.” J. Hydraul. Eng., 130(7), 678–688.
Information & Authors
Information
Published In
Journal of Hydraulic Engineering
Volume 137 • Issue 2 • February 2011
Pages: 173 - 182
Copyright
© 2011 ASCE.
History
Received: Dec 10, 2008
Accepted: Jun 10, 2010
Published online: Jan 14, 2011
Published in print: Feb 2011
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