Research Articles

A Finite Element Solution of the Unidimensional Shallow-Water Equation

[+] Author and Article Information
Ali Triki

Research Laboratory: Applied Fluid Mechanics,
Process Engineering and Environment,
National Engineering School of Sfax,
University of SFAX B.P.,
1173, 3038 Sfax, Tunisia
e-mail: ali.triki@enis.rnu.tn

Manuscript received July 20, 2010; final manuscript received July 25, 2012; accepted manuscript posted August 17, 2012; published online January 15, 2013. Assoc. Editor: Arif Masud.

J. Appl. Mech 80(2), 021001 (Jan 15, 2013) (8 pages) Paper No: JAM-10-1249; doi: 10.1115/1.4007424 History: Received July 20, 2010; Revised July 25, 2012; Accepted August 17, 2012

Based on the finite element method, the numerical solution of the shallow-water equation for one-dimensional (1D) unsteady flows was established. To respect the stability criteria, the time step of the method was dependent on the space step and flow velocity. This method was used to avoid the restriction due to the wave celerity variation in the computational analysis when using the method of characteristics. Furthermore, boundary conditions are deduced directly from the scheme without using characteristics equations. For the numerical solution, a general-purpose computer program, based on the finite element method (FEM), is coded in fortran to analyze the dynamic response of the open channel flow. This program is able to handle rectangular, triangular, or trapezoidal sections. Some examples solved with the finite element method are reported herein. The first involves routing a discharge hydrograph down a rectangular channel. The second example consists of routing a sudden shutoff of all flow at the downstream end of a rectangular channel. The third one deals with routing a discharge hydrograph down a trapezoidal channel. These examples are taken from the quoted literature text book. Numerical results agree well with those obtained by these authors and show that the proposed method is consistent, accurate, and highly stable in capturing discontinuities propagation in free surface flows.

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Yost, S. A., and Rao, P., 2000, “A Moving Boundary Approach for One-Dimensional Free Surface Flows,” Adv. Water Resour., 23, pp. 373–382. [CrossRef]
RaoP., 2005, “Numerical Modelling of Open Channel Flows Using a Multiple Grid ENO Scheme,” Appl. Math. Comput. Sci. Direct, 161, pp. 599–610. [CrossRef]
Liang, D., Falconer Ingram, R. A., and Lin, B., 2000, “Comparison Between TVD-McCormack and ADI-Type Solvers of the Shallow Water Equation,” Adv. Water Resour., 23, pp. 545–562. [CrossRef]
Streeter, V. L., and Wylie, E. B., 1967, Hydraulic Transients, McGraw-Hill, New York.
Wylie, E. B., Streeter, V. L., and Suo, L., 1993, Fluid Transients in System, Prentice Hall, Englewood Cliffs, NJ.
Chanson, H., 2009, “Application of the Method of Characteristics to the Dam Break Wave Problem,” J. Hydraul. Res., 47(1), pp. 41–49. [CrossRef]
Mohammadian, A., Le Roux, D. Y., and Tajrishi, M., 2007, “A Conservative Extension of the Method of Characteristics for 1-D Shallow Flows,” Appl. Math. Model., 31, 332–348. [CrossRef]
Roe, P. L., 1981, “Approximate Riemann Solvers Parameter Vectors and Difference Schemes,” J. Comput. Phys., 43, pp. 357–372. [CrossRef]
Toro, E. F., 2000, Shock Capturing Methods for Free Surface Shallow Flows, John Wiley, New York.
Fennema, R. J., and Chaudhry, M. H., 1987, “Simulation of One-Dimensional Dam-Break Flows,” J. Hydraul. Res., 25(1), pp. 41–51. [CrossRef]
Hicks, E. F., and Steffer, P. M., 1992, “Characteristics Dissipative Galerkin Scheme for Open Channel Flow,” J. Hydraul. Eng., 118(2), pp. 337–352. [CrossRef]
Szymkiewicz, R., 1991, Finite-Element Method for the Solution of the Saint-Venant Equations in an Open Channel Network,” J. Hydrol., 122, pp. 275–287. [CrossRef]
Masud, A., and Khurram, R., 2004, “A Multiscale/Stabilized Finite Element Method for the Advection-Diffusion Equation,” Comput. Methods Appl. Mech. Eng., 193, pp. 1997–2018. [CrossRef]
Masud, A., and Calderer, R., 2009, “A Variational Multiscale Stabilized Formulation for the Incompressible Navier–Stokes Equations,” Comput. Mech., 44, pp. 145–160. [CrossRef]
Calderer, R., and Masud, A., 2010, “A Multiscale Stabilized ALE Formulation for Incompressible Flows With Moving Boundaries,” Comput. Mech., 46, pp. 185–197. [CrossRef]
Buchanan, G. R., 2004, Finite Element Analysis, Tata McGraw-Hill, New Delhi.
Dhatt, G., and Touzot, G., 1984, Une Présentation de la Méthode des Eléments Finis, Edition Maloine SA, Paris.
Richtmeyer, R. D., and Morton, K. W., 1967, Difference Methods for Initial Value Problems, Intersciences, New York.


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Fig. 1

Schematic cross section of the channel

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Fig. 3

Depth velocity and discharge evolution

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Fig. 5

Comparison between the finite element method and the method of characteristics (Streeter and Wylie [4])

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Fig. 6

Comparison between the FEM method and Streeter and Wylie [4]

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Fig. 7

Depths and velocities evolution

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Fig. 8

Profiles of free surface evolution

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Fig. 9

Comparison between the FEM method and Streeter and Wylie [4]

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Fig. 10

Depth velocity and discharge evolution




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