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Simulation of Hydraulic Shock Waves by Hybrid Flux-Splitting Schemes in Finite Volume Method

Published online by Cambridge University Press:  05 May 2011

J.-S. Lai ,
G.-F. Lin  and
W.-D. Guo
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.
W.-D. Guo*
Affiliation:
Department of Civil Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C.
*
*Assistant Research Fellow
**Professor
***Postdoctoral Researcher
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Abstract

In the framework of the finite volume method, a robust and easily implemented hybrid flux-splitting finite-volume (HFF) scheme is proposed for simulating hydraulic shock waves in shallow water flows. The hybrid flux-splitting algorithm without Jacobian matrix operation is established by applying the advection upstream splitting method to estimate the cell-interface fluxes. The scheme is extended to be second-order accurate in space and time using the predictor-corrector approach with monotonic upstream scheme for conservation laws. The proposed HFF scheme and its second-order extension are verified through simulations of the 1D idealized dam-break problem, the 2D oblique hydraulic shock-wave problem, and the 2D dam-break experiments with channel contraction as well as wet/dry beds. Comparisons of the HFF and several well-known first-order upwind schemes are made to evaluate numerical performances. It is demonstrated that the HFF scheme captures the discontinuities accurately and produces no entropy-violating solutions. The HFF scheme and its second-order extension are proven to achieve the numerical benefits combining the efficiency of flux-vector splitting scheme and the accuracy of flux-difference splitting scheme for the simulation of hydraulic shock waves.

Keywords

Hybrid flux-splitting finite-volume schemeShallow water flowsFlux-vector splittingFlux-difference splitting
Type
Articles
Information
Journal of Mechanics , Volume 21 , Issue 2 , June 2005 , pp. 85 - 101
Copyright
Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2005

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References

1.Hirsch, C., Numerical Computation of Internal and External Flows, 2nd Edition, John Wiley & Sons, New York (1990).Google Scholar
2.Toro, E. F., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, Berlin (1997).CrossRefGoogle Scholar
3.Roe, P. L., “Approximate Riemann Solvers, Parameter Vectors, and Difference Schemes,” J. Comput. Phys., 43, pp. 357–372 (1981).CrossRefGoogle Scholar
4.Steger, J. L. and Warming, R. F., “Flux Vector Splitting of the Inviscid Gas Dynamic Equations with Application to Finite Difference Methods,” J. Comput. Phys., 40, pp. 263–293 (1981).CrossRefGoogle Scholar
5.Osher, S. and Solomone, F., “Upwind Difference Schemes for Hyperbolic Systems of Conservation Laws,” Mathematics and Computers in Simulation, 38, pp. 339–374 (1982).Google Scholar
6.Nujic, M., “Efficient Implementation of Non-Oscillatory Schemes for the Computation of Free-Surface Flows,” J. Hydraulic Res., 33(1), pp. 101–111 (1995).CrossRefGoogle Scholar
7.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
8.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
9.Tseng, M. H., “Explicit Finite Volume Non-Oscillatory Schemes for 2D Transient Free-Surface-Flows,” Int. J. Numer. Meth. Fluids, 30, pp. 831–843 (1999).3.0.CO;2-6>CrossRefGoogle Scholar
10.Tseng, M. H. and Chu, C. R., “Two-Dimensional Shallow Water Flows Simulation using TVDMacCormack Scheme,” J. Hydraulic Res., 38(2), pp. 123–131 (2000).Google Scholar
11.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(8), pp. 861–873 (2003).CrossRefGoogle Scholar
12.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
13.Liou, M. S. and Steffen, C. J., “A New Flux Splitting Scheme,” J. Comput. Phys., 107, pp. 23–39 (1993).CrossRefGoogle Scholar
14.Liou, M. S., “A Sequel to AUSM: AUSM+,” J. Comput. Phys., 129, pp. 364–382 (1996).CrossRefGoogle Scholar
15.Wada, Y. and Liou, M. S., “An Accurate and Robust Flux Splitting Scheme for Shock and Contact Discontinuities,” SIAM Journal of Scientific Computing, 18(3), pp. 633–657 (1997).CrossRefGoogle Scholar
16.Niu, Y. Y., “Advection Upstream Splitting Method to Solve a Compressible Two-Fluid Model,” Int. J. Numer. Meth. Fluids, 36, pp. 351–371 (2001).CrossRefGoogle Scholar
17.Evje, S. and Fjelde, K. K., “Hybrid Flux-Splitting Schemes for a Two-Phase Flow Model,” J. Comput. Phys., 175, pp. 674–701 (2002).CrossRefGoogle Scholar
18.Hus, U. K., Tai, C. H. and Tsa, C. H., “All Speed and High-Resolution Scheme Applied to Three-Dimensional Multi-Block Complex Flow Field System,” Journal of Mechanics, 20(1), pp. 13–25 (2004).Google Scholar
19.Tan, W. Y., Shallow Water Hydrodynamics, Elsevier, New York (1992).Google Scholar
20.Toro, E. F., Shock-Capturing Methods for Free-Surface Shallow Water Flows, John Wiley & Sons, New York (2001).Google Scholar
21.Bellos, C. V., Soulis, J. V. and Sakkas, J. G., “Experimental Investigation of Two-Dimensional Dam-Break Induced Flows,” J. Hydraulic Res., 30(1), pp. 47–63 (1992).CrossRefGoogle Scholar
22.Stoker, J. J., Water Waves: Mathematical Theory with Applications, Wiley-Interscience, Singapore (1958).Google Scholar
23.Harten, A. and Hyman, J. M., “Self Adjusting Grid Methods for 1D Hyperbolic Conservation Laws,” J. Comput. Phys., 50, pp. 235–269 (1983).CrossRefGoogle Scholar
24.LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge University Press, U.K. (2002).CrossRefGoogle Scholar
25.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
26.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
27.Hager, W. H., Schwalt, M., Jimenez, O. and Chaudry, M. H., “Supercritical Flow near an Abrupt Wall Deflection,” J. Hydraulic Res., 32(1), pp. 103–118 (1994).CrossRefGoogle Scholar