In the previous chapters, we considered one-dimensional flows. However, the assumption of one-dimensional flow may not be valid in many situations — e.g., flow in a non-prismatic channel (i.e., channel with varying cross section and alignment), flow downstream of a partially breached dam, or lateral flow from a failed dyke. Although flow in these situations is three-dimensional, we may simplify their analysis by considering them as two-dimensional flows by using vertically averaged quantities. Such an assumption not only simplifies the analysis considerably but yields results of reasonable accuracy.
In this chapter, we discuss the analysis of two-dimensional flows. First, we derive the equations describing unsteady two-dimensional flows. Then, we present explicit and implicit finite difference methods for their solution.
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References
Abbott, M. B., 1979, Computational Hydraulics: Elements of the Theory of Free Surface Flows, Pitman, London.
Alcrudo, F., Garcia-Navarro, P., and Saviron, J. M., 1992, “Flux difference spht-ting for 1D open channel flow equatione,” Inter. Jour. for Numerical Methods in Fluids, vol. 14, pp. 1009-1018.
Anastasiou, K., and Chan, C. T. 1997, “Solution of the 2D Shallow Water Equa-tions Using the Finite Volume Method on Unstructured Triangular Meshes,” Inter. Jour. Numer. Methods Fluids, vol. 24, pp. 1225-1245.
Anderson, D. A., Tannehill J. D. and Pletcher, R.H., 1984, Computational Fluid Mechanics and Heat Transfer. McGraw-Hill, New York.
Anton, H., 1981, Elementary Linear Algebra. Wiley and Sons, New York.
Beam, R. M., and Warming, R. F., 1976, “An Implicit Finite-Difference Al-gorithm for Hyperbolic Systems in Conservation-Law Form.” Jour. Comp. Phys., Vol. 22, pp. 87-110.
Benning, R. M., Becker, T. M., and Delgado, A., 2001, “Initial Studies of Pre-dicting Flow Fields With an ANN Hybrid,” Adv. Eng. Software, vol. 32, pp. 895-901.
Benque, J. P., Hauguel, A., and Viollet, P. L., 1982, Engineering Applications of Computational Hydraulics, Pitman Advanced Publishing Program, London, England.
Chaudhry, M. H., 1987, Applied Hydraulic Transients. 2nd edition, Chapter 12, Van Nostrand Reinhold, New York, NY.
Chua, L. H. C., and Holz, K. P., 2005, “Hybrid Neural Network-Finite Element River Flow Model,” Jour. Hyd. Engineering, vol. 131, no. 1, pp. 52-59.
Cockburn, B., Karniadakis, G., Shu, C. W., and Griebel, M., (eds.), 2000, “Dis-continuous Galerkin Methods: Theory, Computation and Applications,” Lec-ture notes in computational science and engineering, Springer, Berlin.
Courant, R., 1936, Differential and Integral Calculus. vol. II, Interscience, New York, NY.
Cunge, J. A., Holly, Jr., F. M., and Verwey, A., l980, Practical Aspects of Com-putational River Hydraulics, Pitman, London.
Dibike, Y. B., and Abbott, M. B., 1999, “Application of Artificial Neural Net-works to the Simulation of a Two-Dimensional Flow,” Jour. Hyd. Research, vol. 37, no. 4, pp. 435-446.
Fagherazzi, S., Rasetarinera, P., Hussaini, M. Y., and Furbish, D. J., 2004, “Nu-merical Solution of the Dam-Break Problem With a Discontinuous Galerkin Method,” Jour. Hyd. Engineering, vol. 130, no. 6, pp. 532-539.
Fennema, R. J., 1985, “Numerical Solution of Two-Dimensional Transient Free-Surface Flows,” Ph. D. Disseration, Washington State University, Pullman, WA.
Fennema, R. J., and Chaudhry, M. H., 1986, “Second-Order Numerical Schemes for Unsteady Free-Surface Flows with Shocks,” Water Resources Research, vol. 22, no. 13, pp. 1923-1930.
Fennema, R. J., and Chaudhry, M. H., 1987, L“Simulation of One-Dimensional Dam-Break Flows.” Jour. Hydraulic Research, International Association for Hydraulic Research, vol. 25, no 1, pp. 41-51.
Fennema, R. J., and Chaudhry, M. H., 1989, “Implicit Methods for Two-dimensional Unsteady Free-Surface Flows,” Jour. Hyd. Research,” Inter. As-soc. for Hydraulic Research, vol. 27, no. 3, pp. 321-332.
Fennema, R. J., and Chaudhry, M. H., 1990, “Explicit Methods for Two-dimensional Unsteady Free-Surface Flows,” Jour. Hyd. Engineering,” Amer. Soc. of Civ. Engrs., vol. 116, no. 8, pp. 1013-1034.
Franke, C., and Schaback, R., 1997, “Convergence Orders of Meshless Collo-cation Methods and Radial Basis Functions,” Technical Report, Dept. of Mathematics, University of Gottingen, Gottingen, Germany.
Gabutti, B., 1983, “On Two Upwind Finite-Difference Schemes for Hyperbolic Equations in Non-Conservative Form,” Computers and Fluids. vol. 11, No. 3, pp. 207-230.
Garcia, R. and Kahawita, R. A., 1986, “Numerical Solution of the St. Venant Equations with MacCormack Finite-Difference Scheme,” International Jour. for Numerical Methods in Fluids, vol. 6, pp. 259-274.
Godunov, S. K., 1959, “A Finite Difference Method for the Computation of Discontinuous Solutions of the Equations of Fluid Dynamics,” Matematich-eski/u/i Sbornik. Novaya Seriya (Mathematics of the USSR-Sbornik),vol. 47, pp. 357-393.
Hardy, R. L., 1971, “Multiquadric Equations of Topography and Other Irregular Surfaces,” Jour. Geophys. Res., vol. 76, no. 26, pp. 1905-1915.
Hirsch, H., 1990, Numerical Computation of Internal and External Flows. Vol.2: Computational Methods for Inviscid and Viscous Flows, Wiley, New York, NY.
Hon, Y. C., Lu, M. W., Xue, W. M., and Zhu, Y. M., 1997, “Multiquadric Method for The Numerical Solution of a Biphasic Mixture Model,” Appl. Math. Comput., vol. 88, no. 2, pp. 153-175.
Hon, Y. C., Cheung, K. F., Mao, X. Z., and Kansa, E. J., 1999, “Multiquadric Solution for Shallow Water Equations,” Jour. Hydraulic Engineering, vol. 125, no. 5, pp. 524-533.
Jameson, A., Schmidt, W., and Turkel, E., (1981). “Numerical Solutions of the Euler equations by Finite Volume Methods Using Runge-Kutta Time-Stepping Schemes,” Proc., AIAA 14th Fluid And Plasma Dynamics Confer-ence, Palo Alto, CA, AIAA-81-1259.
Jimenez, O., 1987, Personal communications with M. H. Chaudhry.
Katopodes, N., 1984a, “Two-Dimensional Surges and Shocks in Open Chan-nels,” Jour. Hydraulic Engineering, Amer. Soc. Civil Engrs., vol. 110, no. 6, pp. 794-812.
Katopodes, N. D., 1984b, “A Dissipative Galerkin Scheme for Open-Channel Flow.” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., Vol. 110, No. HY6, pp. 450-466.
Katopodes, N. D., and Strelkoff, T., 1978, “Computing Two- Dimensional Dam-Break Flood Waves.” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., vol. 104, no. HY9, pp. 1269-1288.
Lax, P. D. and Wendroff, B., 1960, “Systems of Conservation Laws.” Com. Pure Appl. Math., vol. 13, pp. 217-237.
Lai, C., 1986, “Numerical Modeling of Unsteady Open-Channel Flows,” in Ad-vances in Hydroscience, vol. 14, Academic Press, New York, NY., pp. 161-333.
Lax, P. D. and Wendroff, B., 1960, “Systems of Conservation Laws.” Com. Pure Appl. Math., vol. 13, pp. 217-237.
Leendertse, J. J., 1967, “Aspects of a Computational Model for Long Period Water-Wave Propagation,” Memo RM-5294-PR, Rand Corporation, Santa Monica, CA, May.
MacCormacK, R. W., 1969, “The Effect of Viscosity in Hypervelocity Impact Cratering.” Amer. Inst. Aero. Astro., Paper 69-354, Cincinnati, Ohio.
Matsutomi, H., 1983, “Numerical Computations of Two-Dimensional Inundation of Rapidly Varied Flows due to Breaking of Dams.” Proc., XX Congress, Inter. Assoc. Hyd. Research, Moscow, USSR, Subject A, vol. II, Sept. pp. 479-488.
Mingham, C. G., and Causon, D. M., 1998, “High-Resolution Finite-Volume Method for Shallow Water Flows,” Jour. Hydraulic Engineering, vol. 124, no. 6, pp. 605-614.
Morreti, G., 1979, “The λ-Scheme,” Computer and Fluids, vol. 7, pp. 191-205.
Richtmyer, R. D., and Morton, K. W., 1967, Difference Methods for Initial-Value Problems, John Wiley and Sons, New York, 2nd Edition.
Sakkas, J. G., and Strelkoff, T., l973, “Dam-Break Flood in a Prismatic Dry Channel.” Jour. Hyd. Div., Amer. Soc. Civ. Engrs., vol. 99, no. HY12, pp. 2195-2216.
Schwanenberg, D., and Harms, M., 2004, “Discontinuous Galerkin Finite-Element Method for Transcritical Two-Dimensional Shallow Water Flows,” Jour. Hyd. Engineering, vol. 130, no. 5, pp. 412-421.
Singh, V. 1996, “Computation of shallow water flow over a porous medium,” Ph.D. thesis, Indian Institute of Technology, Kanpur, India.
Sleigh, P. A., Gaskell, P. H., Berzins, M., and Wright, N. G., 1998, “An Unstruc-tured Finite-Volume Algorithm for Predicting Flow in Rivers and Estuaries,” Comput. Fluids, vol. 27, no. 4, 479-508.
Tamamidis, P., and Assanis, D. N. 1993, “Evaluation of Various High-Order-Accuracy Schemes With and Without Flux Limiters,” Int. Jour. Numer. Methods Fluids, vol. 16, pp. 931-948.
Toro, E. F. 1992, “Riemann Problems and the WAF Method for Solving the Two-Dimensional Shallow Water Equations,” Philos. Trans. Royal Soc., Lon-don, 338, 43-68.
Toro, E. F., 1999, Riemann Solvers and Numerical Methods for Fluid Dynamics, 2nd Ed., Springer, Berlin.
Yee, H. C., 1989, “A class of high-resolution explicit and implicit shockcapturmg methods,” NASA Technical Memorandum 101088, NASA Ames Research Center, CA.
Yoon, T. H., and Kang, S. K., 2004, “Finite Volume Model for Two-Dimensional Shallow Water Flows on Unstructured Grids,” Jour. Hyd. Engineering, vol. 130, no. 7, pp. 678-688.
Warming, R. F., and Beam, R. M., 1978, “On the Construction and Application of Implicit Factored Schemes for Conservation Laws.” Proc., Symposium on Computational Fluid Dynamics,SIAM-AMS, vol. 11, NY, pp. 85-129.
Wong, S. M., Hon, Y. C., Li, T. S., Chung, S. L., and Kansa, E. J., 1999, “Multi-Zone Decomposition for Simulation of Time-Dependent Problems Using the Multiquadric Scheme,” Comput. Math. Appl.
Zhao, D. H., Shen, H. W., Tabios, G. Q., Lai, J. S., and Tan, W. Y., 1994, “Finite-Volume Two-Dimensional Unsteady-Flow Model for River Basins,” Jour. Hyd. Engineering, vol. 120, no. 7, pp. 863-883.
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(2008). Two-Dimensional Flow. In: Open-Channel Flow. Springer, Boston, MA. https://doi.org/10.1007/978-0-387-68648-6_15
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