A Roe-type scheme for two-phase shallow granular flows over variable topography
ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 851-885.

We study a depth-averaged model of gravity-driven flows made of solid grains and fluid, moving over variable basal surface. In particular, we are interested in applications to geophysical flows such as avalanches and debris flows, which typically contain both solid material and interstitial fluid. The model system consists of mass and momentum balance equations for the solid and fluid components, coupled together by both conservative and non-conservative terms involving the derivatives of the unknowns, and by interphase drag source terms. The system is hyperbolic at least when the difference between solid and fluid velocities is sufficiently small. We solve numerically the one-dimensional model equations by a high-resolution finite volume scheme based on a Roe-type Riemann solver. Well-balancing of topography source terms is obtained via a technique that includes these contributions into the wave structure of the Riemann solution. We present and discuss several numerical experiments, including problems of perturbed steady flows over non-flat bottom surface that show the efficient modeling of disturbances of equilibrium conditions.

DOI : 10.1051/m2an:2008029
Classification : 65M99, 76T25
Mots-clés : granular flows, two-phase flows, thin layer approximation, non-conservative systems, numerical model, finite volume schemes, Riemann solvers, well-balanced schemes 
@article{M2AN_2008__42_5_851_0,
     author = {Pelanti, Marica and Bouchut, Fran\c{c}ois and Mangeney, Anne},
     title = {A {Roe-type} scheme for two-phase shallow granular flows over variable topography},
     journal = {ESAIM: Mod\'elisation math\'ematique et analyse num\'erique},
     pages = {851--885},
     publisher = {EDP-Sciences},
     volume = {42},
     number = {5},
     year = {2008},
     doi = {10.1051/m2an:2008029},
     mrnumber = {2454625},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1051/m2an:2008029/}
}
TY  - JOUR
AU  - Pelanti, Marica
AU  - Bouchut, François
AU  - Mangeney, Anne
TI  - A Roe-type scheme for two-phase shallow granular flows over variable topography
JO  - ESAIM: Modélisation mathématique et analyse numérique
PY  - 2008
SP  - 851
EP  - 885
VL  - 42
IS  - 5
PB  - EDP-Sciences
UR  - http://archive.numdam.org/articles/10.1051/m2an:2008029/
DO  - 10.1051/m2an:2008029
LA  - en
ID  - M2AN_2008__42_5_851_0
ER  - 
%0 Journal Article
%A Pelanti, Marica
%A Bouchut, François
%A Mangeney, Anne
%T A Roe-type scheme for two-phase shallow granular flows over variable topography
%J ESAIM: Modélisation mathématique et analyse numérique
%D 2008
%P 851-885
%V 42
%N 5
%I EDP-Sciences
%U http://archive.numdam.org/articles/10.1051/m2an:2008029/
%R 10.1051/m2an:2008029
%G en
%F M2AN_2008__42_5_851_0
Pelanti, Marica; Bouchut, François; Mangeney, Anne. A Roe-type scheme for two-phase shallow granular flows over variable topography. ESAIM: Modélisation mathématique et analyse numérique, Tome 42 (2008) no. 5, pp. 851-885. doi : 10.1051/m2an:2008029. http://archive.numdam.org/articles/10.1051/m2an:2008029/

[1] T.B. Anderson and R. Jackson, A fluid-dynamical description of fluidized beds: Equations of motion. Ind. Eng. Chem. Fundam. 6 (1967) 527-539.

[2] E. Audusse, F. Bouchut, M.-O. Bristeau, R. Klein and B. Perthame, A fast and stable well-balanced scheme with hydrostatic reconstruction for shallow water flows. SIAM J. Sci. Comput. 25 (2004) 2050-2065. | MR | Zbl

[3] D. Bale, R.J. Leveque, S. Mitran and J.A. Rossmanith, A wave-propagation method for conservation laws and balance laws with spatially varying flux functions. SIAM J. Sci. Comput. 24 (2002) 955-978. | MR | Zbl

[4] F. Bouchut, Nonlinear stability of finite volume methods for hyperbolic conservation laws, and well-balanced schemes for sources. Birkhäuser-Verlag (2004). | MR | Zbl

[5] F. Bouchut and M. Westdickenberg, Gravity driven shallow water models for arbitrary topography. Comm. Math. Sci. 2 (2004) 359-389. | MR | Zbl

[6] M.J. Castro, J. Macías and C. Parés, A Q-scheme for a class of systems of coupled conservation laws with source term. Application to a two-layer 1-D shallow water system. ESAIM: M2AN 35 (2001) 107-127. | Numdam | MR | Zbl

[7] M.J. Castro, J.A. García Rodríguez, J.M. González-Vida, J. Macías, C. Parés and M.E. Vázquez-Cendón, Numerical simulation of two layer shallow water flows through channels with irregular geometry. J. Comput. Phys. 195 (2004) 202-235. | MR | Zbl

[8] R.P. Denlinger and R.M. Iverson, Flow of variably fluidized granular masses across three-dimensional terrain: 2. Numerical predictions and experimental tests. J. Geophys. Res. 106 (2001) 553-566.

[9] R.P. Denlinger and R.M. Iverson, Granular avalanches across irregular three-dimensional terrain: 1. Theory and computation. J. Geophys. Res. 109 (2004) F01014, doi:10.1029/2003JF000085.

[10] T. Gallouët, J.-M Hérard and N. Seguin, Some approximate Godunov schemes to compute shallow-water equations with topography. Comput. Fluids 32 (2003) 479-513. | MR | Zbl

[11] D. Gidaspow, Multiphase Flow and Fluidization: Continuum and Kinetic Theory Descriptions. Academic Press, New York (1994). | MR | Zbl

[12] E. Godlewski and P.-A. Raviart, Numerical Approximation of Hyperbolic Systems of Conservation Laws. Springer-Verlag, New York (1996). | MR | Zbl

[13] L. Gosse, A well-balanced flux-vector splitting scheme designed for hyperbolic systems of conservation laws with source terms. Comput. Math. Appl. 39 (2000) 135-159. | MR | Zbl

[14] N. Goutal and F. Maurel, Proceedings of the 2nd Workshop on Dam-Break Wave Simulation. Technical report EDF-DER Report HE-43/97/016/B, Chatou, France (1997).

[15] J.M.N.T. Gray, M. Wieland and K. Hutter, Gravity driven free surface flow of granular avalanches over complex basal topography. Proc. R. Soc. London S. A 455 (1999) 1841-1874. | MR | Zbl

[16] J.M. Greenberg and A.Y. Leroux, A well-balanced scheme for the numerical processing of source terms in hyperbolic equations. SIAM J. Numer. Anal. 33 (1996) 1-16. | MR | Zbl

[17] A. Harten and J.M. Hyman, Self-adjusting grid methods for one-dimensional hyperbolic conservation laws. J. Comput. Phys. 50 (1983) 235-269. | MR | Zbl

[18] K. Hutter, M. Siegel, S.B. Savage and Y. Nohguchi, Two-dimensional spreading of a granular avalanche down an inclined plane, part I. Theory. Acta Mech. 100 (1993) 37-68. | MR | Zbl

[19] R.M. Iverson, The physics of debris flows. Rev. Geophys. 35 (1997) 245-296.

[20] R.M. Iverson and R.P. Denlinger, Flow of variably fluidized granular masses across three-dimensional terrain: 1, Coulomb mixture theory. J. Geophys. Res. 106 (2001) 537-552.

[21] R.M. Iverson, M. Logan and R.P. Denlinger, Granular avalanches across irregular three-dimensional terrain: 2, Experimental tests. J. Geophys. Res. 109 (2004) F01015, doi:10.1029/2003JF000084.

[22] F. Legros, The mobility of long-runout landslides. Eng. Geol. 63 (2002) 301-331.

[23] R.J. Leveque, clawpack. http://www.amath.washington.edu/ claw+.

[24] R.J. Leveque, Wave propagation algorithms for multi-dimensional hyperbolic systems. J. Comput. Phys. 131 (1997) 327-353. | Zbl

[25] R.J. Leveque, Balancing source terms and flux gradients in high-resolution Godunov methods: The quasi-steady wave-propagation algorithm. J. Comput. Phys. 146 (1998) 346-365. | MR | Zbl

[26] R.J. Leveque, Finite Volume Methods for Hyperbolic Problems. Cambridge University Press (2002). | MR | Zbl

[27] R.J. Leveque and D.L. George, High-resolution finite volume methods for the shallow water equations with bathymetry and dry states, in Proceedings of Long-Wave Workshop, Catalina, 2004, P.L.-F. Liu, H. Yeh and C. Synolakis Eds., Advances Numerical Models for Simulating Tsunami Waves and Runup, Advances in Coastal and Ocean Engineering 10, World Scientific (2008) 43-73.

[28] R.J. Leveque and M. Pelanti, A class of approximate Riemann solvers and their relation to relaxation schemes. J. Comput. Phys. 172 (2001) 572-591. | MR | Zbl

[29] A. Mangeney, F. Bouchut, N. Thomas, J.-P. Vilotte and M.-O. Bristeau, Numerical modeling of self-channeling granular flows and of their levee-channel deposits. J. Geophys. Res. 112 (2007) F02017, doi:10.1029/2006JF000469.

[30] A. Mangeney-Castelnau, J.-P. Vilotte, M.-O. Bristeau, B. Perthame, F. Bouchut, C. Simeoni and S. Yernini, Numerical modeling of avalanches based on Saint-Venant equations using a kinetic scheme. J. Geophys. Res. 108 (2003) 2527, doi:10.1029/2002JB002024.

[31] A. Mangeney-Castelnau, F. Bouchut, J.-P. Vilotte, E. Lajeunesse, A. Aubertin and M. Pirulli, On the use of Saint-Venant equations to simulate the spreading of a granular mass. J. Geophys. Res. 110 (2005) B09103, doi:10.1029/2004JB003161.

[32] M. Massoudi, Constitutive relations for the interaction force in multicomponent particulate flows. Int. J. Non-Linear Mech. 38 (2003) 313-336.

[33] S. Noelle, N. Pankratz, G. Puppo and J.R. Natvig, Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows. J. Comput. Phys. 213 (2006) 474-499. | MR | Zbl

[34] C. Parés and M.J. Castro, On the well-balance property of Roe's method for nonconservative hyperbolic systems. Applications to shallow-water systems. ESAIM: M2AN 38 (2004) 821-852. | Numdam | MR | Zbl

[35] A.K. Patra, A.C. Bauer, C.C. Nichita, E.B. Pitman, M.F. Sheridan, M. Bursik, B. Rupp, A. Webber, A.J. Stinton, L.M. Namikawa and C.S. Renschler, Parallel adaptive numerical simulation of dry avalanches over natural terrain. J. Volcanology Geotherm. Res. 139 (2005) 1-21.

[36] M. Pelanti, Wave Propagation Algorithms for Multicomponent Compressible Flows with Applications to Volcanic Jets. Ph.D. thesis, University of Washington, USA (2005).

[37] M. Pelanti and R.J. Leveque, High-resolution finite volume methods for dusty gas jets and plumes. SIAM J. Sci. Comput. 28 (2006) 1335-1360. | MR | Zbl

[38] E.B. Pitman and L. Le, A two-fluid model for avalanche and debris flows. Phil. Trans. R. Soc. A 363 (2005) 1573-1601. | MR | Zbl

[39] E.B. Pitman, C.C. Nichita, A.K. Patra, A.C. Bauer, M.F. Sheridan and M. Bursik, Computing granular avalanches and landslides. Phys. Fluids 15 (2003) 3638-3646. | MR

[40] S.P. Pudasaini and K. Hutter, Rapid shear flows of dry granular masses down curved and twisted channels. J. Fluid Mech. 495 (2003) 193-208. | MR | Zbl

[41] S.P. Pudasaini, Y. Wang and K. Hutter, Modelling debris flows down general channels. Natural Hazards and Earth System Sciences 5 (2005) 799-819.

[42] S.P. Pudasaini, Y. Wang and K. Hutter, Rapid motions of free-surface avalanches down curved and twisted channels and their numerical simulations. Phil. Trans. R. Soc. A 363 (2005) 1551-1571. | MR | Zbl

[43] W.J.M. Rankine, On the stability of loose earth. Phil. Trans. R. Soc. 147 (1857) 9-27.

[44] P.L. Roe, Approximate Riemann solvers, parameter vectors, and difference schemes. J. Comput. Phys. 43 (1981) 357-372. | MR | Zbl

[45] S.B. Savage and K. Hutter, The motion of a finite mass of granular material down a rough incline. J. Fluid. Mech. 199 (1989) 177-215. | MR | Zbl

[46] S.B. Savage and K. Hutter, The dynamics of avalanches of granular materials from initiation to runout, part I. Analysis. Acta Mech. 86 (1991) 201-223. | MR | Zbl

[47] I. Suliciu, On modelling phase transitions by means of rate-type constitutive equations, shock wave structure. Internat. J. Engrg. Sci. 28 (1990) 829-841. | MR | Zbl

[48] I. Suliciu, Some stability-instability problems in phase transitions modelled by piecewise linear elastic or viscoelastic constitutive equations. Internat. J. Engrg. Sci. 30 (1992) 483-494. | MR | Zbl

[49] Y.C. Tai, S. Noelle, J.M.N.T. Gray and K. Hutter, Shock-capturing and front-tacking methods for dry granular avalanches. J. Comput. Phys. 175 (2002) 269-301.

[50] E.F. Toro, Riemann Solvers and Numerical Methods for Fluid Dynamics. Springer-Verlag, Berlin, Heidelberg (1997). | MR | Zbl

[51] B.G.M. Van Wachem and A.E. Almstedt, Methods for multiphase computational fluid dynamics. Chem. Eng. J. 96 (2003) 81-98.

[52] M.E. Vázquez-Cendón, Improved treatment of source terms in upwind schemes for the shallow water equations in channels with irregular geometry. J. Comput. Phys. 148 (1999) 497-526. | MR | Zbl

[53] P. Vollmöller, A shock-capturing wave-propagation method for dry and saturated granular flows. J. Comput. Phys. 199 (2004) 150-174. | Zbl

[54] C.B. Vreugdenhil, Two-layer shallow-water flow in two dimensions, a numerical study. J. Comput. Phys. 33 (1979) 169-184. | MR | Zbl

[55] Y. Wang and K. Hutter, A constitutive model of multiphase mixtures and its application in shearing flows of saturated solid-fluid mixtures. Granul. Matter 1 (1999) 163-181.

[56] Y. Wang and K. Hutter, A constitutive theory of fluid-saturated granular materials and its application in gravitational flows. Rheol. Acta 38 (1999) 214-223.

Cité par Sources :