# Modelling and numerical simulation of two-phase debris flows

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## Abstract

Gravity-driven geophysical mass flows often consist of fluid–sediment mixtures. The contemporary presence of a fluid and a granular phase determines a complicated fluid-like and solid-like behaviour. The present paper adopts the mixture theory to incorporate the two phases and describe their respective movements. For the granular phase, a Mohr–Coulomb plasticity is employed to describe the relationship between normal and shear stresses, while for the fluid phase, the viscous Newtonian fluid is taken into account. At the basal topography, a Coulomb sliding condition for the solid phase and a Navier’s sliding condition for the fluid phase are satisfied, while the top free surface is traction-free for both the phases. For the interactive forces between the phases, the buoyancy force and viscous drag force are included. The established governing equations are expressed in a curvilinear coordinate system embedded in a curvilinear reference basal surface, above which an arbitrary shallow basal topography is permitted. Taking into account the typical length characteristics of such geophysical mass flows, the “thin-layer” approximation is assumed, so that a depth integration can be performed to simplify the governing equations. The resulting strongly nonlinear partial differential equations (PDEs) are first simplified and then analysed for a steady state in a travelling coordinate system. We find the current model can reproduce the characteristic shape of some flow fronts. Additionally, a stability analysis for steady uniform flows is performed to demonstrate the development of roll waves that means instabilities grow up and become clearly distinguishable waves. Furthermore, we numerically solve the resulting PDEs to investigate general unsteady flows down a curved surface by means of a high-resolution non-oscillatory central difference scheme with the total variation diminishing property. The dynamic behaviours of the granular and fluid phases, especially, the effects of the drag force and the fluid bed friction are discussed. These investigations can enhance the understanding of physics behind natural debris flows.

## Keywords

High-resolution scheme Mixture theory Roll waves Two phases## Notes

### Acknowledgments

The authors thank the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme FP7 for the financial support through Grant No. 289911.

## References

- 1.Anderson TB, Jackson R (1967) Fluid mechanical description of fluidized beds. Equations of motion. Ind Eng Chem Fundam 6:527–539CrossRefGoogle Scholar
- 2.Arai M, Huebl J, Kaitna R (2013) Occurrence conditions of roll waves for three grain-fluid models and comparison with results from experiments and field observation. Geophys J Int 195:1464–1480CrossRefGoogle Scholar
- 3.Bagnold RA (1954) Experiments on a gravity-free dispersion of large solid spheres in a Newtonian fluid under shear. Proc R Soc A 225:49–63CrossRefGoogle Scholar
- 4.Bouchut F, Fernandez-Nieto ED, Mangeney A, Narbona-Reina G (2015) A two-phase shallow debris flow model with energy balance. Math Model Numer Anal 49:101–140MathSciNetCrossRefzbMATHGoogle Scholar
- 5.Chen CL (1988) Generalized viscoplastic modeling of debris flow. J Hydraul Res 114:237–258CrossRefGoogle Scholar
- 6.Chiou MC, Wang Y, Hutter K (2005) Influence of obstacles on rapid granular flows. Acta Mech. 175:105–122CrossRefzbMATHGoogle Scholar
- 7.Chugunov V, Gray JMNT, Hutter K (2003) Group theoretic methods and similarity solutions of the Savage–Hutter equations. In: Hutter K, Kirchner NP (eds) Dynamic response of granular and porous materials under large and catastrophic deformations. Springer, Berlin, pp 251–261Google Scholar
- 8.Drew DA (1983) Mathematical modelling of two-phase flow. Ann Rev Fluid Mech 15:261–291CrossRefGoogle Scholar
- 9.George DL, Iverson R (2011) A two-phase debris-flow model that includes coupled evolution of volume fractions, granular dilatancy, and pore-fluid pressure. In: Genevois R, Hamilton D, Prestininzi A (eds) The 5th international conference on debris-flow hazards, Padova, Italy. Ital J Eng Geol Environ, 415–424Google Scholar
- 10.Gray JMNT, Wieland M, Hutter K (1999) Gravity-driven free surface flow of granular avalanches over complex basal topography. Proc R Soc A 445:1841–1874MathSciNetCrossRefzbMATHGoogle Scholar
- 11.Gray JMNT, Edwards AN (2014) A depth-averaged \(\mu (\varvec {I})-\)rheology for shallow granular free-surface flows. J Fluid Mech 755:503–534MathSciNetCrossRefzbMATHGoogle Scholar
- 12.Gray JMNT, Tai CY (1998) On the inclusion of a veloity-dependent basal drag in avalanche models. Ann Glaciol 26:277–280Google Scholar
- 13.Greve R, Koch T, Hutter K (1994) Unconfined flow of granular avalanches along a partly curved surface I. Theory. Proc R Soc A 445:399–413CrossRefzbMATHGoogle Scholar
- 14.Hungr O (2000) Analysis of debris flow surges using the theory of uniformly progressive flow. Earth Surf Proc Land 25:483–495CrossRefGoogle Scholar
- 15.Hutter K, Siegel M, Savage SB, Nohguchi Z (1993) Two-dimensional spreading of a granular avalanche down an inclined plane Part I. Theory. Acta Mech 100:37–68MathSciNetCrossRefzbMATHGoogle Scholar
- 16.Hutter K, Jöhnk K, Svendsen B (1994) On interfacial transition conditions in two phase gravity flow. J App Math Phys 45:746–762MathSciNetCrossRefzbMATHGoogle Scholar
- 17.Ishii M, Zuber N (1979) Drag coefficient and relative velocity in bubbly, droplet or particulate flows. AIChE J 25:843–855CrossRefGoogle Scholar
- 18.Iverson RM (1997) The physics of debris flows. Rev Geophys 35:245–296CrossRefGoogle Scholar
- 19.Iverson RM, Denlinger RP (2001) Flow of variable fluidized granular material across three dimensional terrain 1: Coulomb mixture theory. J Geophys Res 106(B1):537–552CrossRefGoogle Scholar
- 20.Iverson RM, George DL (2014) A depth-averaged debris-flow model that includes the effects of evolving dilatancy. I. Physical basis. Proc R Soc A 470:20130819MathSciNetCrossRefGoogle Scholar
- 21.Jiang G, Tadmor E (1998) Non-oscillatory central schemes for multidimensional hyerbolic conservation laws. SIAM J Sci Comput 19:1892–1917MathSciNetCrossRefzbMATHGoogle Scholar
- 22.Kowalski J, McElwaine J (2013) Shallow two-component gravity-driven flows with vertical variation. J Fluid Mech 714:434–462MathSciNetCrossRefzbMATHGoogle Scholar
- 23.Needham DJ, Merkin JH (1984) On roll waves down an open inclined channel. Proc R Soc A 394:259–278CrossRefzbMATHGoogle Scholar
- 24.Nessyahu H, Tadmor E (1990) Non-oscillatory central differencing for hyperbolic conservation laws. J Comput Phys 87:408–463MathSciNetCrossRefzbMATHGoogle Scholar
- 25.Ouriemi M, Aussillous P, Guazzelli È (2009) Sediment dynamics. Part 1. Bed-load transport by laminar shearing flows. J Fluid Mech 636:295–319MathSciNetCrossRefzbMATHGoogle Scholar
- 26.Pelanti M, Bouchut F, Mangeney A (2008) A Roe-type scheme for two-phase shallow granular flows over variable topography. Math Model Numer Anal 42:851–885MathSciNetCrossRefzbMATHGoogle Scholar
- 27.Pitman EB, Patra AK, Kumar D, Nishimura K, Komori J (2013) Two phase simulations of glacier lake outburst flows. J Comput Sci 4:71–79CrossRefGoogle Scholar
- 28.Pitman EB, Le L (2005) A two-fluid model for avalanche and debris flows. Philos Trans R Soc A 363:1573–1601MathSciNetCrossRefzbMATHGoogle Scholar
- 29.Pouliquen OP, Forterre Y (2002) Friction law for dense granular flows: application to the motion of a mass down a rough inclined plane. J Fluid Mech 453:133–151CrossRefzbMATHGoogle Scholar
- 30.Prochnow M, Chevoir F, Albertelli M (2000) Dense granular flows down a rough inclined plane. In: Proceedings of XIIIth international congress on rheology, Cambridge, UKGoogle Scholar
- 31.Pudasaini SP, Wang Y, Hutter K (2005) Modelling debris flows down general channels. Nat Hazards Earth Syst 5:799–819CrossRefGoogle Scholar
- 32.Pudasaini SP (2012) A general two-phase debris flow model. J Geophys Res 117(F3):1–28CrossRefGoogle Scholar
- 33.Pudasaini SP (2014) Dynamics of submarine debris flow and tsunami. Acta Mech 225:2423–2434CrossRefGoogle Scholar
- 34.Pudasaini SP, Miller SA (2012) Buoyancy induced mobility in two-phase debris flow. Am Inst Phys Proc 1479:149–152Google Scholar
- 35.Que Y-T, Xu K (2006) The numerical study of roll-waves in inclined open channels and solitary wave run-up. Int J Numer Mech Fluids 50:1003–1027MathSciNetCrossRefzbMATHGoogle Scholar
- 36.Rao IJ, Rajagopal KR (1999) The effect of the slip boundary condition on the flow of fluids in a channel. Acta Mech 135:113–126MathSciNetCrossRefzbMATHGoogle Scholar
- 37.Richardson JF, Zaki WN (1954) Sedimentation and fluidisation: Part I. Trans Inst Chem Eng 32:82–100Google Scholar
- 38.Savage SB, Hutter K (1989) The motion of a finite mass of granular material down a rough incline. J Fluid Mech 199:177–215MathSciNetCrossRefzbMATHGoogle Scholar
- 39.Svendsen B, Wu T, Jöhnk K, Hutter K (1996) On the role of mechanical interactions in the steady-state gravity flow of a two-constituent mixture down an inclined plane. Proc R Soc A 452:1189–1205CrossRefzbMATHGoogle Scholar
- 40.Tai YC, Noelle S, Gray J, Hutter K (2001) Shock-capturing and front-tracking methods for granular avalanches. J Comput Phys 175:269–301CrossRefzbMATHGoogle Scholar
- 41.Truesdell C (1984) Rational thermodynamics. Springer, BerlinCrossRefzbMATHGoogle Scholar
- 42.Wang Y, Hutter K (1999) A constitutive model for multi-phase mixtures and its application in shearing flows of saturated soild–fluid mixtures. Granul Matter 1:163–181CrossRefGoogle Scholar
- 43.Wang Y, Hutter K, Pudasaini SP (2004) The Savage–Hutter theory: a system of partial differential equations for avalanche flows of snow, debris, and mud. J App Math Mech 84(8):507–527MathSciNetzbMATHGoogle Scholar
- 44.Wang Y, Hutter K (1999) A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol Acta 38:214–223CrossRefGoogle Scholar
- 45.Wang Y, Hutter K (2001) Comparisons of numerical methods with respect to convectively dominated problems. Int J Numer Methods Fluids 37:721–745CrossRefzbMATHGoogle Scholar