Hostname: page-component-6b989bf9dc-g5k2d Total loading time: 0 Render date: 2024-04-14T23:50:43.900Z Has data issue: false hasContentIssue false

Modeling Shallow Over-Saturated Mixtures on Arbitrary Rigid Topography

Published online by Cambridge University Press:  09 August 2012

I. Luca*
Department of Mathematics II, University Politehnica of Bucharest, Splaiul Independentei 313, 060042 Bucharest, Romania
C. Y. Kuo
Division of Mechanics, Research Center for Applied Sciences Academia Sinica Taipei, Taiwan 11529, R.O.C.
K. Hutter
Bergstrasse 5, 8044 Zürich Switzerland
Y. C. Tai
Department of Hydraulic and Ocean Engineering, National Cheng Kung University Tainan, Taiwan 70101, R.O.C.
*Corresponding author (
Get access


In this paper a system of depth-integrated equations for over-saturated debris flows on three-dimensional topography is derived. The lower layer is a saturated mixture of density preserving solid and fluid constituents, where the pore fluid is in excess, so that an upper fluid layer develops above the mixture layer. At the layer interface fluid mass exchange may exist and for this a parameterization is needed. The emphasis is on the description of the influence on the flow by the curvature of the basal surface, and not on proposing rheological models of the avalanching mass. To this end, a coordinate system fitted to the topography has been used to properly account for the geometry of the basal surface. Thus, the modeling equations have been written in terms of these coordinates, and then simplified by using (1) the depth-averaging technique and (2) ordering approximations in terms of an aspect ratio ϵ which accounts for the scale of the flowing mass. The ensuing equations have been complemented by closure relations, but any other such relations can be postulated. For a shallow two-layer debris with clean water in the upper layer, flowing on a slightly curved surface, the equilibrium free surface is shown to be horizontal.

Copyright © The Society of Theoretical and Applied Mechanics, R.O.C. 2012

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.)



1. Larcan, E., Mambretti, S. and Pulecchi, M., “A Procedure for the Evaluation of Debris Flow Stratification. Monitoring, Simulation, Prevention and Remediation of Dense and Debris Flows,” Transaction: Ecology and the Environement, 90, Eds. Lorenzini, G., Brebbis, C.A., Emmanouloudis, D., WIT Press (2006).Google Scholar
2. Takahashi, T., Debris Flow, Mechanics, Prediction and Countermeasures. Taylor & Francis, London, (2007).Google Scholar
3. Fraccarollo, L. and Capart, H., “Riemann Wave Description of Erosional Dam-Break Flows,” Journal of Fluid Mechanics, 461, pp. 183228 (2002).CrossRefGoogle Scholar
4. Morales de Luna, T., “A Saint Venant Model for Gravity Driven Shallow Water Flows with Variable Density and Compressibility Effects,” Mathematical and Computer Modeling, 47, pp. 436444 (2008).CrossRefGoogle Scholar
5. Luca, I., Hutter, K., Kuo, C. Y. and Tai, Y. C., “Two-Layer Models for Shallow Avalanche Flows Over Arbitrary Variable Topography,” International Journal of Advances in Engineering Sciences and Applied Mathematics, 1, pp. 99121 (2009).CrossRefGoogle Scholar
6. Fernández-Nieto, E. D., Bouchut, F., Bresch, D., Castro-Díaz, M. J. and Mangeney, A., “A New Savage-Hutter Type Model for Submarine Avalanches and Generated Tsunami,” Journal of Computational Physics, 227, pp. 77207754 (2008).CrossRefGoogle Scholar
7. Takahama, J., Fujita, Y., Hachiya, K. and Yoshino, K., “Application of Two Layer Simulation Model for Unifying Debris Flow and Sediment Sheet Flow and Its Improvement,” Debris-Flow Hazards Mitigation: Mechanics, Prediction, and Assessment, Eds. Rickenmann, & Chen, , Millpress, , Rotterdam, (2003).Google Scholar
8. Berzi, D. and Jenkins, J. T., “Approximate Analytical Solutions in a Model for Highly Concentrated Granular-Fluid Flows,” Physics Review E, 78, 011304, (2008).Google Scholar
9. Berzi, D. and Jenkins, J. T., “Steady, Inclined Flows of Granular-Fluid Mixtures,” Journal of Fluid Mechanics, 641, pp. 359387 (2009).CrossRefGoogle Scholar
10. Berzi, D., Jenkins, J. T. and Larcan, M., “Approximate Analytical Solutions in a Model for Highly Concentrated Granular-Fluid Flows,” Advances in Geophysics, 52, pp. 103138 (2010).CrossRefGoogle Scholar
11. Hutter, K., Jöhnk, K. and Svendsen, B., “On Interfacial Transition Conditions in Two Phase Gravity Flow,” ZAMP, 45, pp. 746762 (1994).Google Scholar
12. Svendsen, B., Wu, T., Jöhnk, K. and Hutter, K., “On the Role of Mechanical Interactions in the Steady-State Gravity Flow of a Two-Constituent Mixture Down an Inclined Plane,” Proceedings of the Royal Society of London, A, 452, pp. 11891205 (1996).Google Scholar
13. Wu, T., Hutter, K. and Svendsen, B., “On Shear Flow of a Saturated Ice-Sediment Mixture with Thermodynamic Equilibrium Pressure and Momentum Exchange,” Proceedings of the Royal Society of London, A, 454, pp. 7188 (1998).CrossRefGoogle Scholar
14. Sivakumaran, N. S., Tingsanchali, T. and Hosking, R. J., “Steady Shallow Flow Over Curved Beds,” Journal of Fluid Mechanics, 128, pp. 469487 (1983).CrossRefGoogle Scholar
15. Dewals, B. J., Erpicum, S., Archambau, P., Detrembleur, S. and Pirotton, M., “Depth-Integrated Flow Modeling Taking Into Account Bottom Curvature,” Journal of Hydraulic Research, 44, pp. 785795(2006).CrossRefGoogle Scholar
16. Iverson, R. M., “The Physics of Debris Flows,” Reviews of Geophysics, 35, pp. 245296 (1997).CrossRefGoogle Scholar
17. Pudasaini, S. P., Wang, Y. and Hutter, K., “Rapid Motions of Free Surface Avalanches Down Curved and Twisted Channels and Their Numerical Simulation,” Philosophical Transactions of the Royal Society, A, 363, pp. 15511571 (2005).Google ScholarPubMed
18. Pudasaini, S. P., Wang, Y. and Hutter, K., “Modeling Debris Flows Down General Channels,” Natural Hazards and Earth System Sciences, 5, pp. 799819 (2005).CrossRefGoogle Scholar
19. Bouchut, F. and Westdickenberg, M., “Gravity Driven Shallow Water Models for Arbitrary Topography,” Communications in Mathematical Sciences, 2, pp. 359389 (2004).CrossRefGoogle Scholar
20. De Toni, S. and Scotton, P., “Two-Dimensional Mathematical and Numerical Model for the Dynamics of Granular Avalanches,” Cold Regions Science and Technology, 43, pp. 3648 (2005).CrossRefGoogle Scholar
21. Bouchut, F., Fernández-Nieto, E. D., Mangeney, A. and Lagrée, P.-Y., “On New Erosion Models of Savage-Hutter Type for Avalanches,” Acta Mechanica, 199, pp. 181208 (2008).CrossRefGoogle Scholar
22. Tai, Y. C. and Kuo, C. Y., “A New Model of Granular Flows Over General Topography with Erosion and Deposition,” Acta Mechanica, 199, pp. 7196 (2008).CrossRefGoogle Scholar
23. Tai, Y. C. and Lin, Y. C., “A Focused View of the Behaviour of Granular Flows Down a Confined Inclined Chute Into the Horizontal Run-Out Zone,” Physics of Fluids, 20, 123302 (2008).CrossRefGoogle Scholar
24. Pelanti, M., Bouchut, F. and Mangeney, A., “A Roe-Type Scheme for Two-Phase Shallow Granular Flows Over Variable Topography,” ESAIM: M2AN, 42, pp. 851885 (2008).CrossRefGoogle Scholar
25. Luca, I., Tai, Y. C. and Kuo, C. Y., “Non-Cartesian Topography Based Avalanche Equations and Approximations of Gravity Driven Flows of Ideal and Viscous Fluids,” Mathematical Models and Methods in Applied Sciences, 19, pp. 127171 (2009).CrossRefGoogle Scholar
26. Luca, I., Hutter, K., Tai, Y. C. and Kuo, C. Y., “A Hierarchy of Avalanche Models on Arbitrary Topography,” Acta Mechanica, 205, pp. 121149 (2009).CrossRefGoogle Scholar
27. Luca, I., Tai, Y. C. and Kuo, C. Y., “Modeling Shallow Gravity-Driven Solid-Fluid Mixtures Over Arbitrary Topography,” Communications in Mathematical Sciences, 7, pp. 136 (2009).Google Scholar
28. Dressler, R. F., “New Nonlinear Shallow Flow Equations with Curvature,” Journal of Hydraulic Research, 16, pp. 205222 (1978).CrossRefGoogle Scholar
29. Sivakumaran, N. S., Hosking, R. J. and Tingsanchali, T., “Steady Shallow Flow Over a Spillway,” Journal of Fluid Mechanics, 111, pp. 411420 (1981).CrossRefGoogle Scholar
30. Egashira, S., Myamoto, K. and Itoh, T., “Constitutive Equations of Debris Flows and Their Applicability,” Debris-Flow Hazards Mitigation: Mechanics, Prediction, and Assessment, Proceedings 1st International DFHM Conference, San Francisco, CA, USA, Ed. Chen, C.L., ASCE, pp. 340349 (1997).Google Scholar
31. Iverson, R. M. and Denlinger, R. G., “Flow of Variably Fluidized Granular Masses Across Three-Dimensional Terrain, 1. Coulomb Mixture Theory,” Journal of Geophysical Research, 106, pp. 537552 (2001).CrossRefGoogle Scholar
32. Egashira, S., Itoh, T. and Takeuchi, H., “Transition Mechanism of Debris Flows Over Rigid Bed to Over Erodible Bed,” Physics and Chemistry of the Earth, B, 26, pp. 169174 (2001).CrossRefGoogle Scholar
33. Schneider, L. and Hutter, K., “Solid-Fluid Mixtures of Frictional Materials in Geophysical and Geotechnical Context,” Advances in Geophysical and Environmental Mechanics and Mathematics, Springer Verlag, Berlin, Heidelberg, New York, 247p (2009).Google Scholar
34. Lien, H. P. and Tsai, F. W., “Sediment Concentration Distribution of Debris Flow,” Journal of Hydraulic Engineering, DOI: 10.1061(ASCE)0733-9429(2003) 129:12(995) (2003).Google Scholar
35. Kuo, C. Y., Tai, Y. C., Bouchut, F., Mangeney, A., Pelanti, M., Chen, R. F. and Chang, K. J., “Simulation of Tsaoling Landslide, Taiwan, Based on Saint Venant Equations Over General Topography,” Engineering Geology, 104, pp. 181189 (2009).CrossRefGoogle Scholar
36. Pitman, E. B. and Le, L., “A Two-Fluid Model for Avalanche and Debris Flows,” Philosophical Transactions of the Royal Society, A, 363, pp. 15731601 (2005).Google ScholarPubMed
37. Pudasaini, S. P. and Hutter, K., Avalanche Dynamics, Berlin Heidelberg, New York (2007).Google Scholar
38. Truesdell, V. and Toupin, R., Handbuch der Physik, III/1, Ed. Flügge, S., Springer Verlag, Berlin, Göttingen, Heidelberg (1960).Google Scholar
39. Truesdell, C., Rational Thermodynamics, McGraw-Hill Series in Modern Applied Mathematics, New York 1969, 2nd Ed., Springer, New York (1984).Google Scholar