Skip to main content
SpringerLink
Log in

A new model of granular flows over general topography with erosion and deposition

  • Published:
Acta Mechanica Aims and scope Submit manuscript

Summary

A fundamental issue for describing gravity-driven flows over general topography is the search for an “optimal” coordinate. Bouchut and Westdickenberg [1] proposed an arbitrary coordinate system (BW) for general topography. The unified coordinate (UC) system (e.g., [2], [3]), which was developed for computational fluid dynamics, combines the advantages of both Eulerian and Lagrangian systems, so that the coordinates can instantaneously move with some singular surface within the flows. By utilizing the benefit of the BW coordinates and UC system, a new model of gravity-driven flows over general topography is derived, in which the erosion and deposition processes at the bed are considered. The depth-integrated mass and momentum equations are presented in the time-dependent and terrain-following coordinate system, which coincides with the interface distinguishing between the static and flowing layers. A shock-capturing numerical scheme is implemented to solve the derived equation system. Simulation results present the new features of this model and reveal a new physical insight of the erosion/deposition processes.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bouchut F. and Westdickenberg M. (2004). Gravity driven shallow water models for arbitrary topography. Commun. Math. Sci. 2(3): 359–389

    MATH  MathSciNet  Google Scholar 

  2. Hui W.H. (2004). A unified coordinates approach to computational fluid dynamics. J. Comput. Appl. Math. 163: 15–28

    Article  MATH  MathSciNet  Google Scholar 

  3. Hui W.H. (2007). The unified coordinate system in computational fluid dynamics. Commun. Comput. Phys. 2(4): 577–610

    MathSciNet  Google Scholar 

  4. Chang K.-J., Taboada T.A. and Chan Y.-C. (2005). Geological and morphological study of the Jiufengershan landslide triggered by the Chi-Chi Taiwan earthquake. Geomorphology 71: 291–309

    Article  Google Scholar 

  5. Chen R.-F., Chang K.-J., Angelier J. and Chan Y.-C. (2006). Benoît Deffontaines, Chyi-Tyi Lee, Ming-Lang Lin. Topographical changes revealed by high-resolution airborne lidar data: the 1999 tsaoling landslide induced by the Chi-Chi earthquake. Engng. Geol. 88: 160–172

    Article  Google Scholar 

  6. Savage S.B. and Hutter K. (1989). The motion of a finite mass of granular material down a rough incline. J. Fluid Mech. 199: 177–215

    Article  MATH  MathSciNet  Google Scholar 

  7. Koch T., Greve R. and Hutter K. (1994). Unconfined flow of granular avalanches along a partly curved surface. II. Experiments and numerical computations. Proc. R. Soc. Lond. A 445: 415–435

    MATH  Google Scholar 

  8. Wieland M., Gray J.M.N.T. and Hutter K. (1999). Channelized free-surface flow of cohesionless granular avalanches in a chute with shallow lateral curvature. J. Fluid Mech. 392: 73–100

    Article  MATH  Google Scholar 

  9. Gray J.M.N.T., Wieland M. and Hutter K. (1999). Gravity-driven free surface flow of granular avalanches over complex basal topography. Proc. R. Soc. A 455(1985): 1841–1874

    Article  MATH  MathSciNet  Google Scholar 

  10. Pudasaini S.P. and Hutter K. (2003). Rapid shear flows of dry granular masses down curved and twisted channels. J. Fluid Mech. 495: 193–208

    Article  MATH  MathSciNet  Google Scholar 

  11. Zwinger T., Kluwick A. and Sampl P. (2003). Numerical simulation of dry-snow avalanche flow over natural terrain. In: Hutter, K. and Kirchner, N. (eds) Dynamic response of granular and porous material under large and catastrophic deformations, pp 161–194. Springer, Berlin

    Google Scholar 

  12. Chiou M.-C., Wang Y. and Hutter K. (2005). Influence of obstacles on rapid granular flows. Acta Mech. 175: 105–122

    Article  MATH  Google Scholar 

  13. Hutter K., Wang Y. and Pudasaini S. (2005). The Savage–Hutter model, how far can it be pushed?. Philos. Trans. R. Soc. A 363: 1507–41528

    Article  MathSciNet  MATH  Google Scholar 

  14. Bouchaud J.-P., Cates M.E., Ravi Prakash J. and Edwards S.F. (1994). A model for the dynamics of sandpile surfaces. J. Phys. I Fr. 4: 1383–1410

    Article  Google Scholar 

  15. Pitman E.B., Nichita C.C., Patra A.K., Bauer A.C., Bursik M. and Weber A. (2003). A model of granular flows over an erodible surface. Discret. Contin. Dynam. Sys. B 3: 589–599

    Article  MATH  Google Scholar 

  16. Naaim M., Faug T. and Naaim-Bouvet F. (2003). Dry granular flow modeling including erosion and deposition. Surv. Geophys. 24: 569–585

    Article  Google Scholar 

  17. Pouliquen O. (1999). Scaling laws in granular flows down rough inclined planes. Phys. Fluids 11(3): 542–548

    Article  MathSciNet  MATH  Google Scholar 

  18. Egashira, S., Miyamoto, K., Itoh, T.: Constitutive equations of debris flow and their applicability. In: Chen, C.-L. (ed.) First Int. Conf. on Debris-flow Hazards Mitigation: Mechanics, Prediction and Assessment, pp. 340–349, 7–9 August (1997)

  19. Egashira, S., Itoh, T., Miyamoto, K.: Debris flow simulations for san julian torrents in venezuela. In: Proc. 3rd IAHR Symp. on River, Coastal and Estuarine Morphodynamics, Barcelona, pp. 976–986 (2003)

  20. Bouchut, F., Fernándes-Nieto, E.D., Mangeney, A., Lagrée, P.-Y.: On new erosion models of Savage–Hutter type for avalanches. Acta Mech. (2007, in press)

  21. Hui W.H., Li P.Y. and Li Z.W. (1999). A unified coordinate system for solving the two-dimensional Euler equations. J. Comput. Phys. 153: 596–637

    Article  MATH  MathSciNet  Google Scholar 

  22. Hui W.H. and Koudriakov S. (2002). Computation of the shallow water equations using the unified coordinates. SIAM J. Sci. Comput. 23(5): 1615–1654

    Article  MATH  MathSciNet  Google Scholar 

  23. Leapage C.Y. and Hui W.H. (1995). A shock-adaptive godunov scheme based on the generalized Lagrangian formulation. J. Comput. Phys. 122: 291–299

    Article  Google Scholar 

  24. Nessyahu H. and Tadmor E. (1990). Non-oscillatory central differencing for hyperbolic conservation laws. J. Comput. Phys. 87: 408–463

    Article  MATH  MathSciNet  Google Scholar 

  25. Jiang G.-S. and Tadmor E. (1998). Non-oscillatory central schemes for multi-dimensional hyperbolic conservation laws. SIAM. J. Sci. Comput. 19(6): 1892–1917

    Article  MATH  MathSciNet  Google Scholar 

  26. Tai Y.C., Noelle S., Gray J.M.N.T. and Hutter K. (2002). Shock-capturing and front tracking methods for granular avalanches. J. Comput. Phys. 175: 269–301

    Article  MATH  Google Scholar 

  27. Pin, F.D., Idelsohn, S., Onate, E., Aubry, R.: The ale/lagrangian particle finite element method: a new approach to computation of free-surface flows and fluid–object interactions. Comput. Fluids 36, 27–38 (2007)

    Google Scholar 

  28. Boutreux T., Raphael E. and Gennes P.-G. (1998). Surface flows of granular materials: a modified picture for thick avalanches. Phys. Rev. E 58(4): 4692–4700

    Article  MathSciNet  Google Scholar 

  29. Greve R. and Hutter K. (1993). Motion of a granular avalanche in a convex and concave curved chute: experiments and theoretical predictions. Philos. Trans. R. Soc. A 342: 573–600

    Article  Google Scholar 

  30. Kreyszig E. (1999). Advanced engineering mathematics. Wiley, New York

    Google Scholar 

  31. Macquorn Rankine W.J. (1857). On the stability of loose earth. Philos. Trans. R. Soc. Lond. 147: 9–27

    Article  Google Scholar 

  32. Hutter K., Siegel M., Savage S.B. and Nohguchi Y. (1993). Two-dimensional spreading of a granular avalanche down an inclined plane. Part I: Theory. Acta Mech. 100: 37–68

    Article  MATH  MathSciNet  Google Scholar 

  33. Rericha E.C., Bizon C., Shattuck M.D. and Swinney H.L. (2002). Shocks in supersonic sand. Phys. Rev. Lett. 88(1): 014302

    Article  Google Scholar 

  34. Gray J.M.N.T., Tai Y.C. and Noelle S. (2003). Shock waves, dead-zones and particle-free regions in rapid granular free surface flows. J. Fluid Mech. 491: 161–181

    Article  MATH  MathSciNet  Google Scholar 

  35. Bouchut F. (2004). Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources. Birkhäuer, Basel

    MATH  Google Scholar 

  36. Noelle S., Pankratz N., Puppo G. and Natvig J. (2006). Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213(2): 474–499

    Article  MATH  MathSciNet  Google Scholar 

  37. LeVeque R.J. and Shyue K.M. (1995). One-dimensional front tracking based on high resolution wave propagation methods. SIAM J. Sci. Comput. 16(2): 348–377

    Article  MATH  MathSciNet  Google Scholar 

  38. Hutter K. and Koch T. (1991). Motion of a granular avalanche in an exxponentially curved chute: experiments and theoretical predditions. Philos. Trans. R. Soc. A 334: 93–138

    Article  Google Scholar 

  39. Chen H. and Lee C.F. (2000). Numerical simulation of debris flows. Can. Geotech. J. 37: 146–160

    Article  Google Scholar 

  40. Tai, Y.C., Hsieh, M.J., Liu, I.C.: An application of unified coordinate to 1D Savage–Hutter model for granular flows. In: 2006 Cross-Strait Workshop on Engineering Mechanics, Taipei, Taiwan, p. 10 (2006)

  41. Wang Y., Pudasaini S.P. and Hutter K. (2004). The Savage–Hutter theory: a system of partial differential equations for avalanche flows of snow, debris and mud. ZAMM – J. Appl. Math. Mech. 84(8): 507–527

    Article  MATH  MathSciNet  Google Scholar 

  42. Pudasaini S.P., Wang Y. and Hutter K. (2005). Rapid motions of free-surface avalanches down curved and twisted channels and their numerical simulations. Philos. Trans. R. Soc. A 363: 1551–1571

    Article  MathSciNet  MATH  Google Scholar 

  43. Pudasaini, S.P., Hsiau, S.-S., Wang, Y., Hutter, K.: Velocity measurements in dry granular avalanches using particle image velocimetry technique and comparison with theoretical predictions. Phys. Fluids 17(9), 093301 SEP (2005)

    Google Scholar 

  44. Pudasaini S.P. and Hutter K. (2007). Avalanche dynamics. Springer, New York

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Civil Engineering, National Chi Nan University, No.1, University Road, Puli, Nantou, 545, Taiwan

    Y. C. Tai

  2. Division of Mechanics, Research Center of Applied Sciences, Academia Sinica, Taipei, Taiwan

    C. Y. Kuo

Authors
  1. Y. C. Tai
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. C. Y. Kuo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Y. C. Tai.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Tai, Y.C., Kuo, C.Y. A new model of granular flows over general topography with erosion and deposition. Acta Mech 199, 71–96 (2008). https://doi.org/10.1007/s00707-007-0560-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00707-007-0560-7

Keywords

Search

Navigation