Effect of Friction on Spurious Oscillations in Open Channel Modeling with Variable Bathymetry or Roughness
Publication: Journal of Hydraulic Engineering
Volume 138, Issue 8
Abstract
Numerical issues for the friction-dominated steady-state Saint-Venant equations with a shock-capturing upwind finite-element scheme are studied using nonuniform-flow test cases and Fourier analysis. The friction-dominated case is a common phenomenon in open channel flow modeling when the depth becomes small compared with the discretization length. In the nonuniform-flow test cases, abrupt slope changes and abrupt roughness changes are introduced, and in the Fourier analysis, a periodic bed perturbation is used. Nondimensional parameter groups identified are as follows: the upwinding coefficient, the number of elements per wavelength, the average-flow Froude number, and the numerical friction number. The results show that errors in both depth and discharge variables are observed whenever there is any perturbation in the bed topography or bed roughness. These errors increase with increasing Froude number and increasing numerical friction number. A combined friction parameter is introduced for practical application. The combined friction parameter can be used to specify minimum depths or to guide mesh refinement. The analysis framework developed can also be used for other numerical schemes.
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Acknowledgments
This research was funded by the Natural Sciences and Engineering Research Council (NSERC) of Canada.
References
Bates, P. D., Aderson, M. G., Hervouet, J. M., and Hawkes, J. C. (1997). “Investigating the behavior of two-dimensional finite element models of compound channel flow.” Earth Surf. Processes LandformsESPLDB, 22(1), 3–17.
Berger, R. C., and Stockstill, R. L. (1995). “Finite-element model for high-velocity channels.” J. Hydraul. Eng.JHEND8, 121(10), 710–716.
Bermudez, A., and Vazquez, M. E. (1994). “Upwind methods for hyperbolic conservation laws with source terms.” Comput. FluidsCPFLBI, 23(8), 1049–1071.
Bradford, S. F., and Sanders, B. F. (2005). “Performance of high-resolution, nonlevel bed, shallow-water models.” J. Eng. Mech.JENMDT, 131(10), 1073–1081.
Burguete, J., Garcia-Navarro, P., and Murillo, J. (2008). “Friction term discretization and limitation to preserve stability and conservation in the 1D shallow-water model: Application to unsteady irrigation and river flow.” Int. J. Numer. Methods FluidsIJNFDW, 58(4), 403–425.
Chaudhry, M. H. (2008). Open-channel flow, Springer, New York.
Cunge, J. A., Holly, F. M. Jr., and Verway, A. (1980). Practical aspects of computational river hydraulics, Pitman, London.
Danish Hydraulic Institute (DHI). (2005). “MIKE 21 & MIKE 3 flow model FM.” Hydrodynamic and Transport Module, Scientific Documentation, Hørsholm, Denmark.
Dietrich, J. C., Kolar, R. L., and Westerink, J. J. (2006). “Refinements in continuous Galerkin wetting and drying algorithms.” Proc., 9th Int. Conf. on Estuarine and Coastal Modeling, ASCE, Reston, VA, 637–656.
Gallouët, T., Hérard, J., and Sequin, N. (2003). “Some approximate Godunov schemes to compute shallow-water equations with topography.” Comput. FluidsCPFLBI, 32(4), 479–513.
Godunov, S. K. (1959). “A difference method for numerical calculation of discontinuous solutions of the equations of hydrodynamics.” Mat. Sb., 47(3), 271–306.MATSAB
Goutal, N., and Maurel, F. (1997). “Proc. of the 2nd workshop on dam-break wave simulation.” EDF-DER Rep. HE-43/97/016/B, Département Laboratoire National d’Hydraulique, Groupe Hydraulique Fluviale, Electricité de France, France.
Heniche, M., Secretan, Y., Boudreau, P., and Leclerc, M. (2002). “Dynamic tracking of flow boundaries in rivers with respect to discharge.” J. Hydraul. Res.JHYRAF, 40(5), 589–602.
Hicks, F. E., and Steffler, P. M. (1992). “Characteristic dissipative Galerkin scheme for open channel flow.” J. Hydraul. Eng.JHEND8, 118(2), 337–352.
Hubbard, M. E., and Garcia-Navarro, P. (2000). “Flux difference splitting and the balancing of source terms and flux gradients.” J. Comput. Phys.JCTPAH, 165(1), 89–125.
Hughes, T. J. R., and Mallet, M. (1986). “A new finite element formulation for computational fluid dynamics: III. The generalized streamline operator for multidimensional advective-diffusive systems.” Comput. Methods Appl. Mech. Eng.CMMECC, 58(3), 305–328.
Katopodes, N. D. (1984). “Fourier analysis of dissipative FEM channel flow model.” J. Hydraul. Eng.JHEND8, 110(7), 927–944.
Khan, A. A. (2000). “Modeling flow over an initially dry bed.” J. Hydraul. Res.JHYRAF, 38(5), 383–388.
Lax, P., and Wendroff, B. (1960). “Systems of conservation laws.” Commun. Pure Appl. Math.CPMAMV, 13(2), 217–237.
LeVeque, R. J. (1998). “Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm.” J. Comput. Phys.JCTPAH, 146(1), 346–365.
MacCormack, R. W. (1969). “The effect of viscosity in hypervelocity impact cratering.” AIAA Paper, American Institute of Aeronautics and Astronautics, Reston, VA, 66–354.
MacCormack, R. W. (1981). “A numerical method for solving the equations of compressible viscous flow.” AIAA Paper, American Institute of Aeronautics and Astronautics, Reston, VA, 81–110.
Miller, W. A., and Cunge, J. A. (1975). “Simplified equations of unsteady flow.” Unsteady flow in open channels, Mahmood, K. and Yevjevich, V., eds., Vol 1, Water Resources Publications, Fort Collins, CO, 485–508.
Petaccia, G., et al. (2009). “Flood wave propagation in a steep mountain river: Comparison of four simulation tools.” Proc., 33rd Int. Association for Hydraulic Engineering and Research (IAHR) Congress: Water Engineering for a Sustainable Environment, IAHR, Madrid, Spain, 5896–5903.
Ponce, V. M., and Simons, D. B. (1977). “Shallow wave propagation in open channel flow.” J. Hydraul. Div.JYCEAJ, 103(12), 1461–1476.
Roe, P. L. (1981). “Approximate Riemann solvers, parameter vectors, and difference schemes.” J. Comput. Phys.JRCPA3, 43(2), 357–372.
Schippa, L., and Pavan, S. (2008). “Analytical treatment of source terms for complex channel geometry.” J. Hydraul. Res.JHYRAF, 46(6), 753–763.
Steffler, P., and Blackburn, J. (2002). River2D: Two-dimensional depth averaged model of river hydrodynamics and fish habitat; Introduction to depth averaged modeling and user’s manual, Univ. of Alberta, Edmonton, Alberta, Canada.
Tchamen, G. W., and Kahawita, R. A. (1998). “Modeling wetting and drying effects over complex topography.” Hydrol. ProcessesHYPRE3, 12(8), 1151–1182.
Toro, E. F., and Garcia-Navarro, P. (2007). “Godunov-type methods for free-surface shallow flows: A review.” J. Hydraul. Res.JHYRAF, 45(6), 736–751.
Van Leer, B. (1979). “Towards the ultimate conservative difference scheme. V. A second-order sequel to Godunov’s method.” J. Comput. Phys., 32(1), 101–136.JRCPA3
Woolhiser, D. A. (1975). “Simulation of unsteady overland flow.” Unsteady flow in open channels, Mahmood, K. and Yevjevich, V., eds., Vol. 2, Water Resources Publications, Fort Collins, CO, 485–508.
Zhou, J. G., Causon, D. M., Mingham, C. G., and Ingram, D. M. (2001). “The surface gradient method for the treatment of source terms in the shallow-water equations.” J. Comput. Phys.JCTPAH, 168(1), 1–25.
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Published In
Journal of Hydraulic Engineering
Volume 138 • Issue 8 • August 2012
Pages: 695 - 706
Copyright
© 2012. American Society of Civil Engineers.
History
Received: Dec 23, 2010
Accepted: Feb 14, 2012
Published online: Feb 16, 2012
Published in print: Aug 1, 2012
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