Hostname: page-component-78c5997874-94fs2 Total loading time: 0 Render date: 2024-11-15T03:21:45.024Z Has data issue: false hasContentIssue false
  • English
  • Français

Gas Kinetic Scheme for Anisotropic Savage-Hutter Model

Published online by Cambridge University Press:  03 June 2015

Wen-Chi Chen ,
Chih-Yu Kuo ,
Keh-Ming Shyue  and
Yih-Chin Tai
Wen-Chi Chen*
Affiliation:
Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan
Chih-Yu Kuo*
Affiliation:
Division of Mechanics, Research Center for Applied Sciences, Academia Sinica, Taipei 115, Taiwan
Keh-Ming Shyue*
Affiliation:
Department of Mathematics, National Taiwan University, Taipei 106, Taiwan
Yih-Chin Tai*
Affiliation:
Department of Hydraulic and Ocean Engineering, National Cheng Kung University, Tainan 701, Taiwan
*
Email:[email protected]
Corresponding author.Email:[email protected]
Email:[email protected]
Email:[email protected]
Get access

Abstract

The gas-kinetic scheme is applied to a depth-integrated continuum model for avalanche flows, namely the Savage-Hutter model. In this method, the continuum fluxes are calculated based on the pseudo particle motions which are relaxed from nonequilibrium to equilibrium states. The processes are described by the Bhatnagar-Gross-Krook (BGK) equation. The benefit of this scheme is its capability to resolve shock discontinuities sharply and to handle the vacuum state without special treatments. Because the Savage-Hutter equation bears an anisotropic stress on the tangential space of the topography, the equilibrium distribution function of the microscopic particles are shown to be bi-Maxwellian. These anisotropic stresses are the key to preserve the coordinate objectivity in the Savage-Hutter model. The effect of the anisotropic stress is illustrated by two examples: an axisymmetric dam break and a finite mass sliding on an inclined plane chute. It is found that the propagation of the flow fronts significantly depends on the orientation of the principal axes of the tangential stresses.

Keywords

76A0582B4086-0897M10Gas-kinetic schemeBGK equationextended Savage-Hutter equation
Type
Research Article
Information
Communications in Computational Physics , Volume 13 , Issue 5 , May 2013 , pp. 1432 - 1454
Copyright
Copyright © Global Science Press Limited 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1]Savage, S. B. and Hutter, K., The motion of a finite mass of granular material down a rough incline, J. Fluid Mech., Vol. 199, 177–215, 1989.Google Scholar
[2]Hutter, K.Siegel, .M, Savage, S.B. and Nohguchi, Y., Two-dimensional spreading of a granular avalanche down an inclined plane. part i: Theory. Acta Mech., Vol. 100, 37–68, 1993.Google Scholar
[3]Pudasaini, S. P. and Hutter, K., Rapid shear flows of dru granular massis down curved and twisted channels. J. Fluid Mech., Vol. 495, 193–208, 2003.Google Scholar
[4]Wang, Y., Hutter, K. and Pudasaini, S.P., The Savage-Hutter theory: A system of partial differential equations for avalanche flows of snow, debris and mud, J. App. Math. Mech., Vol. 84, 507–527, 2004.Google Scholar
[5]LeVeque, R. J., Finite Volume Methods for Hyperbolic Problems, Cambridge Univ. Press, 2003.Google Scholar
[6]Toro, E. L., Riemann Solvers and Numerical Methods for Fluid Dynamics, Springer-Verlag, New York, 1997.Google Scholar
[7]Jiang, G. and Tadmor, E., Non-oscillatory central schemes for multidimensional hyperbolic conservation laws, SIAM J. Sci. Comput., Vol. 19, 1892–1917, 1997.Google Scholar
[8]Tai, Y. C., Noelle, S., Gray, J. M. N. T. and Hutter, K., Shock-capturing and front tracking methods for granular avalanches, J. Comput. Phys., Vol. 175, 269–301, 2002.Google Scholar
[9]Xu, K., A well-balanced gas-kinetic scheme for the shallow-water equations with source terms, J. Comp. Phys., Vol. 178, 533–562, 2002.Google Scholar
[10]Gray, J. M. N. T., Wieland, M. and Hutter, K., Gravity-driven free surface flow of granular avalanches over complex basal topography. Proc. R. Soc. A, Vol. A455, 1841–1874, 1999.Google Scholar
[11]Wieland, M., Gray, J. M. N. T. and Hutter, K., Channelized free-surface flow of cohesionless granular avalanches in a chute with shallow lateral curvature, J. Fluid Mech., 73–100, 1999.Google Scholar
[12]Luca, I., Hutter, K., Tai, Y. C and Kuo, C. Y., A hierarchy of avalanche models on arbitrary topography, Acta Mech., Vol. 205, 121–149, 2009.CrossRefGoogle Scholar
[13]Hutter, K., Wang, Y and Pudasaini, S., The Savage-Hutter avalanche model: how far can it be pushed? Phil. Trans. R. Soc. A, Vol. 363, 1507–1528, 2005.Google Scholar
[14]Iverson, R. M. and Denlinger, R. P., Flow of variably fluidized granular masses across three-dimensional terrain. 1. coulomb mixture theory, J. Geophys. Res., Vol. 106, 537–552, 2001.Google Scholar
[15]De Toni, S. and Scotton, P., Two-dimensional mathematical and numerical model for the dynamics of granular avalanches, Cold Reg. Sci. Tech., Vol. 43, 36–48, 2005.Google Scholar
[16]Kelfoun, K. and Druitt, T. H., Numerical modeling of the emplacement of Socompa rock avalanche, Chile, J. Geophys. Res., Vol. 110, B12202, 2005.Google Scholar
[17]Einfeldt, B., Munz, C., Roe, P and Sjogreen, B., Spontaneously growing transverse waves in a plasma due to an anisotropic velocity distribution, Phys. Rev. Lett., Vol. 2, 83–84, 1959.Google Scholar
[18]Khazanov, G. V., Kinetic Theory of the Inner Magnetospheric Plasma, Springer, 2010.Google Scholar
[19]Pudasaini, S. P. and Hutter, K., Avalanche dynamics, Springer Verlag, Berlin/Heidelberg, 2007.Google Scholar
[20]Hungr, O., A model for the runout analysis of rapid flow slides, debris flows, and avalanches, Canadian Geotech. J., Vol. 32, 610V623, 1995.Google Scholar
[21]McDougall, S. and Hungr, O., A model for the analysis of rapid landslide motion across three-dimensional terrain, Can. Geotech. J., Vol. 41, 1084–1097, 2004.CrossRefGoogle Scholar
[22]Hungr, O., Simplified models of spreading flow of dry granular material, Canadian Geotech. J., Vol. 45, 1156–1168, 2008.Google Scholar
[23]Bhatnagar, P. L., Gross, E. P. and Krook, M., A model for collision processes in gases i: Small amplitude processes in charged and neutral one-component systems, Phys. Rev., Vol. 94, 1954.Google Scholar
[24]Prendergast, K. H. and Xu, K., Numerical hydrodynamics from gas-kinetic theory, J. Comp. Phys., Vol. 109, 53–66, 1993.Google Scholar
[25]Xu, K., Gas-kinetic scheme for unsteady compressible flow simulations, Lecture series, von Karman Institute for Fluid Dynamics, Caltech., 1998.Google Scholar
[26]Xu, K. and Guo, Z., Generalized gas dynamic equations, 47th AIAA Aerospace Sciences Meeting, AIAA 2009-672, 2009.Google Scholar
[27]Xu, K., A gas-kinetic bgk scheme for the Navier-Stokes equations and its connection with artificial dissipation and Godunov method, J. Comp. Phys., Vol. 171, 289V335, 2001.Google Scholar
[28]Bouchut, F., Nonlinear stability of finite volume methods for hyperbolic conservation laws and well-balanced schemes for sources, Birkhauser Verlag, 2004.Google Scholar
[29]Ghidaoui, M. S., Deng, J. Q., Gray, W. G. and Xu, K., A Boltzmann based model for open channel flows, Int. J. Numer. Methods Fluids, Vol. 35, 449–494, 2001.Google Scholar
[30]Tang, H., Tang, T and Xu, K., A gas-kinetic scheme for shallow-water equations with source terms, ZAMP, Vol. 55, 365V382, 2004.Google Scholar