Nonlinear Processes in Geophysics www.nonlin-processes-geophys.net 1023-5809 1607-7946 16 3 2009 10.5194/npg-16-399-2009 http://www.nonlin-processes-geophys.net/16/399/2009/ http://www.nonlin-processes-geophys.net/16/399/2009/npg-16-399-2009.html http://www.nonlin-processes-geophys.net/16/399/2009/npg-16-399-2009.pdf 399 407 2009-05-20 Energy considerations in accelerating rapid shear granular flows S. P. Pudasaini B. Domnik domnik@geo.uni-bonn.de University of Bonn, Steinmann Institute, Department of Geodynamics & Geophysics, Nussallee 8, 53115 Bonn, Germany We present a complete expression for the total energy associated with a rapid frictional granular shear flow down an inclined surface. This expression reduces to the often used energy for a non-accelerating flow of an isotropic, ideal fluid in a horizontal channel, or to the energy for a vertically falling mass. We utilize thickness-averaged mass and momentum conservation laws written in a slope-defined coordinate system. Both the enhanced gravity and friction are taken into account in addition to the bulk motion and deformation. The total energy of the flow at a given spatial position and time is defined as the sum of four energy components: the kinetic energy, gravity, pressure and the friction energy. Total energy is conserved for stationary flow, but for non-stationary flow the non-conservative force induced by the free-surface gradient means that energy is not conserved. Simulations and experimental results are used to sketch the total energy of non-stationary flows. Comparison between the total energy and the sum of the kinetic and pressure energy shows that the contribution due to gravity acceleration and frictional resistance can be of the same order of magnitude, and that the geometric deformation plays an important role in the total energy budget of the cascading mass. Relative importance of the different constituents in the total energy expression is explored. We also introduce an extended Froude number that takes into account the apparent potential energy induced by gravity and pressure. % REFERENCE 1 Bartelt, P., Buser, O., and Kern, M.: Dissipated work, stability and the internal flow structure of granular snow avalanches, J. Glaciol., 51(172), 125–138, 2005. Bartelt, P., Buser, O., and Platzer, K.: Fluctuation-dissipation relations for granular snow avalanches, J. Glaciol., 52(179), 631–643, 2006. Bartelt, P., Buser, O., and Platzer, K.: Starving avalanches: Frictional mechanisms at the tails of finite-sized mass movements, Geophys. Res. Lett., 34, L20407, doi:10.1029/2007GL031352, 2007. Bouchut, F., Mangeney-Castelnau, A., Perthame, B., and Vilotte, J.-P.: A new model of Saint Venant and Savage-Hutter type for gravity driven shallow water flows, C. R. Math, 336(6), 531–536, doi:10.1016/S1631-073X(03)00117-1, 2003. Buser, O. and Bartelt, P.: Production and decay of random kinetic energy in granular snow avalanches, J. Glaciol., 55(189), 3–12, 2009. Castro, M., Gallardo, J. M., and Parés, C.: High order finite volume schemes based on reconstruction of states for solving hyperbolic systems with nonconservative products. Applications to shallow-water systems, Math. Comput., 75(255), 1103–1134, 2006. Dutykh, D. and Dias, F.: Energy of tsunami waves generated by bottom motion, P. R. Soc. A, 465, 725–744, 2009. Erismann, T. H. and Abele, G.: Dynamics of Rockslides and Rockfalls, Springer, Berlin, Germany, 2001. Fine, I. V., Rabinovich, A. B., Thomson, R. E., and Kulikov, E. A.: Numerical modeling of tsunami generation by submarine and subaerial landslides, in: Submarine Landslides and Tsunamis, edited by: Yalciner, A. C., Pelinovsky, E., Okal, E., and Synolakis, C. E., Kluwer Academic Publishers, Netherlands, 69–88, 2003. Gray, J. M. N. T., Wieland, M., and Hutter, K.: Gravity-driven free surface flow of granular avalanches over complex basal topography, P. R. Soc. A, 455, 1841–1874, 1999. Gwiazda, P.: On measure-valued solutions to a two-dimensional gravity-driven avalanche flow model, Math. Method Appl. Sci., 28, 2201–2223, 2005. Heim, A.: Bergsturz und Menschenleben, Fretz & Wasmuth, Zürich, 1932. Hsü, K.: On sturzstroms-catastrophic debris streams generated by rockfalls, Geol. Soc. Am. Bull., 86, 129–140, 1975. Jin, S. and Wen, X.: An efficient method for computing hyperbolic systems with geometrical source terms having concentrations, J. Comput. Math., 22, 230–249, 2004. Le Roux, A. Y.: Riemann solvers for some hyperbolic problems with a source term, in: ESIAM Proceedings/Actes du 30EME Congres d'Analyse Numerique: CANum'98, 75–90, 1998. Mangeney, A., Heinrich, P., and Roche, R.: Analytical solution for testing debris avalanche numerical models, Pure Appl. Geophys., 157, 1081–1096, 2000. Noelle, S., Pankratz, N., Puppo, G., and Natvig, J.: Well-balanced finite volume schemes of arbitrary order of accuracy for shallow water flows, J. Comput. Phys., 213, 474–499, 2006. Noelle, S., Xing, Y., and Shu, C.-W.: High Order Well-balanced Finite Volume WENO Schemes for Shallow Water Equation with Moving Water, J. Comput. Phys., 226, 29–58, 2007. Pudasaini, S. P. , Hsiau, S.-S., Wang, Y., and Hutter, K.: Velocity measurements in dry granular avalanches using Particle Image Velocimetry-Technique and comparison with theoretical predictions, Phys. Fluids, 17(9), 93301, doi:10.1063/1.2007487, 2005. Pudasaini, S. P. and Hutter, K.: Rapid Shear Flows of Dry Granular Masses Down Curved and Twisted Channels, J. Fluid Mech., 495, 193–208, 2003. Pudasaini, S. P. and Hutter, K.: Avalanche Dynamics: Dynamics of Rapid Flows of Dense Granular Avalanches, Springer, Berlin, Germany, 2007. Pudasaini, S. P., Hutter, K., Hsiau, S.-S., Tai, S.-C., Wang, Y., and Katzenbach, R.: Rapid Flow of Dry Granular Materials down Inclined Chutes Impinging on Rigid Walls, Phys. Fluids, 19(5), 053302, doi:10.1063/1.2726885, 2007. Pudasaini, S. P. and Kröner, C.: Shock waves in rapid flows of dense granular materials: Theoretical predictions and experimental results, Phys. Rev. E, 78(4), 041308, doi:10.1103/PhysRevE.78.041308, 2008. Pudasaini, S. P., Wang, Y., and Hutter, K.: Modelling Debris Flows Down General Channels, Nat. Hazard Earth Sys., 5, 799–819, 2005. Pudasaini, S. P., Wang, Y., and Hutter, K.: Rapid motions of free-surface avalanches down curved and twisted channels and their numerical simulation, Philos. T. R. Soc. A, 363(1832), 1551–1571, 2005. Pudasaini, S. P. , Wang, Y., Sheng, L.-T., Hsiau, S.-S., Hutter, K., and Katzenbach, R.: Avalanching granular flows down curved and twisted channels: Theoretical and experimental results, Phys. Fluids, 20(7), 073302, doi:10.1063/1.2945304, 2008. Rao, N. M.: Avalanche Protection and Control in the Himalayas, Defence. Sci. J., 35(2), 255–266, 1985. Rudenko, O. V., Sobisevich, A. L., and Sobisevich, L. E.: Nonlinear dynamics of slope flows: simple models and exact solutions, Dokl. Earth Sci. 416(7), 1109–1113, 2007. Saint-Venant, A. J. C.: Theorie du mouvement non-permanent des eaux, avec application aux crues des rivieres et a l'introduction des marees dans leur lit, Comptes rendus des seances de l'Academie des Sciences, 36, 174–154, 1871. Savage, S. B. and Hutter, K.: The motion of a finite mass of granular material down a rough incline, J. Fluid Mech., 199, 177–215, 1989. Ui, T.: Volcanic dry avalanche deposits-identification and comparison with nonvolcanic debris stream deposits, J. Volcanol. Geoth. Res., 18, 135–150, doi:10.1016/0377-0273(83)90006-9, 1983. Ward, S. N. and Day, S.: Particulate kinematic simulations of debris avalanches: interpretation of deposits and landslide seismic signals of Mount Saint Helens, 1980 May 18., Geophys. J. Int., 167, 991–1004, 2006.