Computational Geosciences

, Volume 22, Issue 1, pp 87–106 | Cite as

Refinement on non-hydrostatic shallow granular flow model in a global Cartesian coordinate system

  • L. Yuan
  • W. Liu
  • J. Zhai
  • S. F. Wu
  • A. K. Patra
  • E. B. Pitman
Original Paper


Current shallow granular flow models suited to arbitrary topography can be divided into two types, those formulated in bed-fitted curvilinear coordinates and those formulated in global Cartesian coordinates. The shallow granular flow model of Denlinger and Iverson (J. Geophys. Res. 109, F01014, 2004) and the Boussinesq-type shallow granular flow theory of Castro-Orgaz et al. (2014) are formulated in a Cartesian coordinate system (with z vertical), and both account for the effect of nonzero vertical acceleration on depth-averaged momentum fluxes and stress states. In this paper, we first reformulate the vertical normal stress of Castro-Orgaz et al. (2014) in a quadratic polynomial in the relative elevation η. This form allows for analytical depth integration of the vertical normal stress. We then calculate the basal normal stress based on the basal friction law and scaling analysis. These calculations, plus certain constitutive relations, lead to a refined full non-hydrostatic shallow granular flow model, which is further rewritten in a form of Boussinesq-type water wave equations for future numerical studies. In the present numerical study, we apply the open-source code TITAN2D for numerical solution of a low-order version of the full model involving only a mean vertical acceleration correction term. To cure the numerical instability related with discretization of the enhanced gravity, we propose an approximate formula for the enhanced gravity by utilizing the hydrostatic pressure assumption in the bed normal direction. Numerical calculations are conducted for several test cases involving steep slopes. Comparison with a bed-fitted model shows that even the simplified non-hydrostatic Cartesian model can be used to simulate shallow granular flows over arbitrary topography.


Granular flow Depth average Cartesian coordinate Arbitrary topography Non-hydrostatic pressure Basal normal stress 


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L. Yuan, W. Liu, J. Zhai thank the support of state key program for developing basic sciences (2010CB731505, MJ-F-2012-04) and Natural Science Foundation of China (11321061, 11261160486, 91641107). S. Wu thanks the support of the Department of Education of Guangdong Province (2014KQNCX175). A. Patra and E. Pitman acknowledge the support of NSF grants 0620991, 0757367, 0711497, 1228217, and NSF/OAC-1339765.

The modified TITAN2D code can be downloaded from∼lyuan/code.html.


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Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.LSEC and Institute of Computational Mathematics and Scientific/Engineering ComputingAcademy of Mathematics and Systems Science, University of Chinese Academy of Sciences, Chinese Academy of SciencesBeijingChina
  2. 2.Department of Computer ScienceGuangdong Polytechnic Normal UniversityGuangzhouChina
  3. 3.Department of Mechanical and Aerospace EngineeringUniversity at Buffalo SUNYBuffaloUSA
  4. 4.Department of MathematicsUniversity at Buffalo SUNYBuffaloUSA

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