Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-11T09:03:36.816Z Has data issue: false hasContentIssue false
  • English
  • Français

Hybrid Flux-Splitting Finite-Volume Scheme for the Shallow Water Flow Simulations with Source Terms

Published online by Cambridge University Press:  05 May 2011

W.-D. Guo ,
J.-S. Lai  and
G.-F. Lin
W.-D. Guo*
Affiliation:
Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
J.-S. Lai*
Affiliation:
Hydrotech Research Institute, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
G.-F. Lin*
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Postdoctoral Researcher
**Associate Research Fellow
***Professor, corresponding author
Get access

Abstract

The extension of the hybrid flux-splitting finite-volume (HFF) scheme to the shallow water equations with source terms is presented. Based on the monotonic upstream schemes for conservation laws (MUSCL) method, the scheme is second-order-accurate both in space and time. An accurate and efficient surface gradient method (SGM), in conjunction with the HFF scheme, is adopted for the discretization of source terms, including the bed slopes and friction slopes. The resulting scheme has several desirable properties: ease of implementation, satisfaction of entropy condition, sharp shock resolution and preservation of well-balancing. The HFF scheme with SGM is verified through the simulations of steady transcritical flow over a hump and steady flow over an irregular bed. Besides, the effects of the limiter functions, the grid sizes and the Manning roughness coefficients on the simulated results are investigated for the steady transcritical flow problems. Using the laboratory measurements, the scheme is also applied to the dam-break flows: with an adverse slope, with a triangular hump, and with a constriction. Furthermore, the HFF scheme is employed in the simulation of typhoon flood flow with natural-irregular river topography to demonstrate the practical engineering application. The results show good agreements compared with the exact solutions, the experimental data and the field measurements.

Keywords

Hybrid flux-splitting finite-volume schemeSource termsMonotonic upstream schemes for conservation lawsSurface gradient methodDam-break flows.
Type
Articles
Information
Journal of Mechanics , Volume 23 , Issue 4 , December 2007 , pp. 399 - 414
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Alcrudo, F. and Garcia-Navarro, P., “A High-Resolution Godunove-Type Scheme in Finite Volumes for the 2D Shallow Water Equations,” Int. J. Numer. Meth. Fluids, 16, pp. 489–505(1993).CrossRefGoogle Scholar
2.Mingham, C. G. and Causon, D. M., “High-Resolution Finite-Volume Method for Shallow Water Flows,” J. Hydraulic Eng., 124(6), pp. 605–614 (1998).CrossRefGoogle Scholar
3.Brufau, P. and Garcia-Navarro, P., “Two-Dimensional Dam Break Flow Simulation,” Int. J. Numer. Meth. Fluids, 33, pp. 35–57 (2000).3.0.CO;2-D>CrossRefGoogle Scholar
4.Erduran, K. S., Kutija, V. and Hewett, C. J. M., “Performance of Finite Volume Solutions to the Shallow Water Equations with Shock-Capturing Schemes,” Int. J. Numer. Meth. Fluids, 40, pp. 1237–1273 (2002).CrossRefGoogle Scholar
5.Lin, G. F., Lai, J. S. and Guo, W. D., “Finite-Volume Component-Wise TVD Schemes for 2D Shallow Water Equations,” Adv. Water Resour., 26, pp. 861–873 (2003).CrossRefGoogle Scholar
6.Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin (1997).CrossRefGoogle Scholar
7.Toro, E. F., Shock-capturing Methods for Free-surface Shallow Water Flows, John Wiley & Sons, New York (2001).Google Scholar
8.Bermudez, A. and Vazquez, M. E., “Upwind Methods for Hyperbolic Conservation Laws with Source Terms,” Computers & Fluids, 23, pp. 1049–1071 (1994).CrossRefGoogle Scholar
9.Nujic, M., “Efficient Implementation of Non- Oscillatory Schemes for the Computation of Free- Surface Flows,” J. Hydraulic Res., 33, pp. 101–111 (1995).CrossRefGoogle Scholar
10.Leveque, R. J., “Balancing Source Terms and Flux Gradients in High-Resolution Godunov Methods: the Quasi-Steady Wave-Propagation Algorithm,” J. Comput. Phys., 146, pp. 346–365 (1998).CrossRefGoogle Scholar
11. Vazquez-Cendon, M. E., “Improved Treatment of Source Terms in Upwind Schemes for Shallow Water Equations in Channels with Irregular Geometry,” J. Comput. Phys., 148, p. 497 (1999).CrossRefGoogle Scholar
12.Garcia-Navarro, P. and Vazquez-Cendon, M. E., “On Numerical Treatment of the Source Terms in the Shallow Water Equation,” Computers & Fluids, 29, pp. 951–979(2000).CrossRefGoogle Scholar
13.Zhou, J. G., Causon, D. M., Mingham, C. G. and Ingrams, D. M, “The Surface Gradient Method for the Treatment of Source Terms in the Shallow Water Equations,” J. Comput. Phys., 168, pp. 1–25 (2001).CrossRefGoogle Scholar
14.Burguete, J. and Garcia-Navarro, P., “Efficient Construction of High-Resolution TVD Conservative Schemes for Equations with Source Terms: Application to Shallow Water Flows,” Int. J. Numer. Meth. Fluids, 37, pp. 209–248(2001).CrossRefGoogle Scholar
15.Brufau, P., Vazquez-Cendon, M. E. and Garcia-Navarro, P., “ANumerical Model for the Flooding and Drying of Irregular Domains,” Int. J. Numer. Meth. Fluids, 39, pp. 247–275 (2002).CrossRefGoogle Scholar
16.Goutal, N. and Maurel, F., “A Finite Volume Solver for 1D Shallow-Water Equations Applied to an Actual River,” Int. J. Numer. Meth. Fluids, 39, pp. 1–19 (2002).CrossRefGoogle Scholar
17.Zhou, J. G., Causon, D. M., Ingrams, D. M. and Mingham, C. G., “Numerical Solutions of the Shallow Water Equations with Discontinuous Bed Topography,” Int. J. Numer. Meth. Fluids, 38, pp. 769–788 (2002).CrossRefGoogle Scholar
18.Tseng, M. H., “The Improved Surface Gradient Method for Flows Simulation in Variable Bed Topography Channel Using TVD-Maccormack Scheme,” Int. J. Numer. Meth. Fluids, 43, pp. 71–91 (2003).CrossRefGoogle Scholar
19.Rogers, B. D., Borthwick, A. G. L. and Taylor, P. H., “Mathematical Balancing of Flux Gradient and Source Terms Prior to Using Roe S Approximate Riemann Solver,” J. Comput. Phys., 192, pp. 422–451 (2003).CrossRefGoogle Scholar
20.Rebollo, T. C., Delgado, A. D. and Nieto, E. D. F., “A Family of Stable Numerical Solvers for the Shallow Water Equations with Source Terms,” Comput. Methods Appl. Mech. Engrg., 192, pp. 203–225 (2003).CrossRefGoogle Scholar
21.Brufau, P. and Garcia-Navarro, P., “Unsteady Free Surface Flow Simulation Over Complex Topography with a Multidimensional Upwind Technique,” J. Comput. Phys., 186, pp. 503–526 (2003).CrossRefGoogle Scholar
22.Nelida, C. Z., Senka, V. and Luka, S., “Balanced Finite Volume WENO and Central WENO Schemes for the Shallow Water and the Open-Channel Flow Equations,” J. Comput. Phys., 200, pp. 512–548 (2004).Google Scholar
23.Lai, J. S., Lin, G. F. and Guo, W. D., “Simulation of Hydraulic Shock Waves by Hybrid Flux-Splitting Schemes in Finite Volume Method,” Journal of Mechanics, 21, pp. 85–101 (2005).CrossRefGoogle Scholar
24.Tan, W. Y., Shallow Water Hydrodynamics, Elsevier, New York (1992).Google Scholar
25.Hirsch, C., Numerical Computation of Internal and External Flows, John Wiley & Sons, New York (1990).Google Scholar
26.Liou, M. S. and Steffen, C. J., “A New Flux Splitting Scheme,” J. Comput. Phys., 107, pp. 23–39 (1993).CrossRefGoogle Scholar
27.Liou, M. S., “A Sequel to AUSM: AUSM+,” J. Comput. Phys., 129, pp. 364–382 (1996).CrossRefGoogle Scholar
28.Wada, Y. and Liou, M. S., “An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities,” SIAMJournal of Scientific Computing, 18, pp. 633–657 (1997).CrossRefGoogle Scholar
29.Hu, K., Mingham, C. G. and Causon, D. M., “A Bore- Capturing Finite Volume Method for Open-Channel Flows,” Int. J. Numer. Meth. Fluids, 28, pp. 1241–1261 (1998).3.0.CO;2-2>CrossRefGoogle Scholar
30.Valiani, A., Caleffi, V. and Zanni, A., “Case Study: Malpasset Dam-Break Simulation Using a Two- Dimensional Finite Volume Method,” J. Hydraulic Eng., 128, pp. 460–472 (2002).CrossRefGoogle Scholar
31.Zhao, D. H., Shen, H. W., Tabios, G. Q., Lai, J. S. and Tan, W. Y., “Finite-Volume Two-Dimensional Unsteady-Flow Model for River Basins,” J. Hydraulic Eng., 120, pp. 863–883 (1994).CrossRefGoogle Scholar
32.Alcrudo, F. and Benkhaldoun, F., “Exact Solutions to the Riemann Problem of the Shallow Water Equations with a Bottom Step,” Computers & Fluids 30 643–671 (2001).CrossRefGoogle Scholar
33.LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge, UK (2002).CrossRefGoogle Scholar
34.Zhou, G. J. and Stansby, P. K., “2D Shallow Water Flow Model for the Hydraulic Jump,” Int. J. Numer. Meth. Fluids, 29, pp. 375–387 (1999).3.0.CO;2-3>CrossRefGoogle Scholar
35.Aureli, F., Mignosa, P. and Tomirotti, M., “Numerical Simulation and Experimental Verification of Dam- Break Flows with Shocks,” J. Hydraulic Res., 38, pp. 197–206 (2000).CrossRefGoogle Scholar
36.Zhou, J. G., Causon, D. M., Mingham, C. G. and Ingrams, D. M., “Numerical Prediction of Dam-Break Flows in General Geometries with Complex Bed Topography,” J. Hydraulic Eng., 130, pp. 332–340 (2004).CrossRefGoogle Scholar
37.Rebollo, T. C., Nieto, E. D. F. and Marmol, M. G., “A Flux-Splitting Solver for Shallow Water Equations with Source Terms,” Int. J. Numer. Meth. Fluids, 42, pp. 23–55 (2003).CrossRefGoogle Scholar
38.Macchione, F. and Morelli, M. A., “Practical Aspects in Comparing Shock-Capturing Schemes for Dam Break Problems,” J. Hydraulic Eng., 129, pp. 187–195 (2003).CrossRefGoogle Scholar
39.Lai, J. S., “Feasibility Study on the Raise Elevation of the Zhongshan Bridge in the Keelung River—Based on Hydraulic Data Obtained from the Physical Modeling,” Research Report No. 441, Hydrotech Research Institute, National Taiwan University (2002) (in Chinese).Google Scholar
40.Environmental Modeling Research Laboratory, “SMS 8.0 Tutorials,” Brigham Young University, Copyright (2002).Google Scholar