Skip to main content Accessibility help

Shallow two-component gravity-driven flows with vertical variation

  • Julia Kowalski (a1) and Jim N. McElwaine (a2) (a1) (a3)


Gravity-driven geophysical mass flows often consist of a heterogeneous fluid–solid mixture. The complex interplay between the components leads to phenomena such as lateral levee formation in avalanches, or a granular front and an excess fluid pore pressure in debris flows. These effects are very important for predicting runout and the forces on structures, yet they are only partially represented in simplified shallow flow theories, since rearrangement of the mixture composition perpendicular to the main flow direction is neglected. In realistic flows, however, rheological properties and effective basal drag may depend strongly on the relative concentration of the components. We address this problem and present a depth-averaged model for shallow mixtures that explicitly allows for rearrangement in this direction. In particular we consider a fluid–solid mixture that experiences bulk horizontal motion, as well as internal sedimentation and resuspension of the particles, and therefore resembles the case of a debris flow. Starting from general mixture theory we derive bulk balance laws and an evolution equation for the particle concentration. Depth-integration yields a shallow mixture flow model in terms of bulk mass, depth-averaged particle concentration, the particle vertical centre of mass and the depth-averaged velocity. This new equation in this model for the particle vertical centre of mass is derived by taking the first moment, with respect to the vertical coordinate, of the particle mass conservation equation. Our approach does not make the Boussinesq approximation and results in additional terms coupling the momentum flux to the vertical centre of mass. The system is hyperbolic and reduces to the shallow-water equations in the homogeneous limit of a pure fluid or perfect mixing. We highlight the effects of sedimentation on resuspension and finally present a simple friction feedback which qualitatively resembles a large-scale experimental debris flow data set acquired at the Illgraben, Switzerland.


Corresponding author

Email address for correspondence:


Hide All
Bartelt, P., Salm, B. & Gruber, U. 1999 Calculating dense-snow avalanche runout using a voellmy-fluid model with active/passive longitudinal straining. J. Glaciol. 45 (150), 242254.
Benny, D. J. 1973 Some properties of long nonlinear waves. Stud. Appl. Maths 52, 4550.
Berres, S., Bürger, R. & Tory, E. M. 2005 On mathematical models and numerical simulation of the fluidization of polydisperse suspensions. Appl. Math. Model. 29, 159193.
Berzi, D. & Jenkins, J. T. 2009 Steady inclined flows of granular-fluid mixtures. J. Fluid Mech. 641, 359387.
Boyer, F., Guazzelli, É. & Pouliquen, O. 2011 Unifying suspension and granular rheology. Phys. Rev. Lett. 107, 188301.
Cassar, C., Nicolas, M. & Pouliquen, O. 2005 Submarine granular flows down inclined planes. Phys. Fluids 17, 103301.
Christen, M., Kowalski, J. & Bartelt, P. 2010 Ramms: numerical simulation of dense snow avalanches in three-dimensional terrain. Cold Reg. Sci. Technol. 63 (1–2), 114.
Denlinger, R. P. & Iverson, R. M. 2001 Flow of variably fluidized granular masses across three-dimensional terrain 2. Numerical predictions and experimental tests. J. Geophys. Res. 106 (B1), 553566.
Drew, D. A. & Passman, S. L. 1998 Theory of Multicomponent Fluids. Springer.
Gray, J. M. N. T. 2002 Rapid granular avalanches. In Dynamic Response of Granular and Porous Material Under Large and Catastrophic Deformations, Lecture Notes in Applied and Computational Mechanics , vol. 11. pp. 342. Springer.
Gray, J. M. N. T. & Chugunov, V. A. 2006 Particle-size segregation and diffusive remixing in shallow granular avalanches. J. Fluid Mech. 569, 365398.
Gray, J. M. N. T. & Kokelaar, B. P. 2010 Large particle segregation, transport and accumulation in granular free-surface flows. J. Fluid Mech. 652, 105137.
Gray, J. M. N. T. & Thornton, A. R. 2005 A theory for particle size segregation in shallow granular free-surface flows. Proc. R. Soc. Lond. A 461, 14471473.
Ishii, M. & Hibiki, T. 2006 Thermo-fluid Dynamics of Two-phase Flow. Springer.
Iverson, R. M. 1997 The physics of debris flows. Rev. Geophys. 35 (3), 245296.
Iverson, R. M. 2009 Elements of an improved model of debris-flow motion. In Invited Contribution to Powders and Grains 2009 Conference. American Physical Society.
Iverson, R. M. & Denlinger, R. P. 2001 Mechanics of debris flows and debris-laden flash floods. In Seventh Federal Interagency Sedimentation Conference, pp. IV–1–IV–8.
Iverson, R. M. & Major, J. J. 1999 Debris-flow deposition: effects of pore-fluid pressure and friction concentrated at flow margins. Geol. Soc. Am. Bull. 111 (10), 14241434.
Lavorel, G. & Le Bars, M. 2009 Sedimentation of particles in a vigorously convecting fluid. Phys. Rev. E 80 (4), 046324.
McArdell, B., Bartelt, P. & Kowalski, J. 2007 Field observations of basal forces and fluid pore pressure in a debris flow. Geophys. Res. Lett. 34, L07406.
McArdell, B. W., Zanuttigh, B., Lamberti, A. & Rickenmann, D. 2003 Systematic comparison of debris flows at the Illgraben torrent, Switzerland. In Debris-Flow Hazards Mitigation: Mechanics, Prediction and Assessment, pp. 647657. Millpress.
Pailha, M. & Pouliquen, O. 2008 Initiation of underwater granular avalanches: influence of the initial volume fraction. Phys. Fluids 20, 111701.
Pelanti, M., Bouchut, F. & Mangeney, A. 2008 A roe-type scheme for two-phase shallow granular flows over variable topography. ESAIM: Math. Model. Num. Anal. 42, 851885.
Pitman, E. B. & Le, L. 2005 A two-fluid model for avalanche and debris flow. Phil. Trans. R. Soc. Lond. 363, 15731601.
Pudasaini, S. P. & Hutter, K. 2007 Avalanche Dynamics–Dynamics of Rapid Flows of Dense Granular Avalanches. Springer.
Richardson, J. F. & Zaki, W. N. 1954 Sedimentation and fluidisation. Part 1. Trans. Inst. Chem. Engng 32, 3553.
Savage, S. B. & Hutter, K. 1989 The motion of a finite mass of a granular material down a rough incline. J. Fluid Mech. 199, 177215.
Savage, S. B. & Hutter, K. 1991 The dynamics of avalanches of granular materials from initiation to runout. part 1: Analysis. Acta Mechanica 86, 201223.
Scheid, B., Ruyer-Quil, C. & Manneville, P. 2006 Wave patterns in film flows: modelling and three-dimensional waves. J. Fluid Mech. 562, 183222.
Thornton, A. R., Gray, J. M. N. T. & Hogg, A. J. 2006 A three-phase mixture theory for particle size segregation in shallow granular free-surface flows. J. Fluid Mech. 550, 125.
Wieland, M., Gray, J. M. N. T. & Hutter, K. 1999 Channelized free-surface flow of cohesionless granular avalanches in a chute with shallow lateral curvature. J. Fluid Mech. 392, 73100.
MathJax is a JavaScript display engine for mathematics. For more information see

JFM classification

Related content

Powered by UNSILO

Shallow two-component gravity-driven flows with vertical variation

  • Julia Kowalski (a1) and Jim N. McElwaine (a2) (a1) (a3)


Full text views

Total number of HTML views: 0
Total number of PDF views: 0 *
Loading metrics...

Abstract views

Total abstract views: 0 *
Loading metrics...

* Views captured on Cambridge Core between <date>. This data will be updated every 24 hours.

Usage data cannot currently be displayed.