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A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows

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Abstract

A non-hydrostatic depth-averaged model for dry granular flows is proposed, taking into account vertical acceleration. A variable friction coefficient based on the \(\mu (I)\) rheology is considered. The model is obtained from an asymptotic analysis in a local reference system, where the non-hydrostatic contribution is supposed to be small compared to the hydrostatic one. The non-hydrostatic counterpart of the pressure may be written as the sum of two terms: one corresponding to the stress tensor and the other to the vertical acceleration. The model introduced here is weakly non-hydrostatic, in the sense that the non-hydrostatic contribution related to the stress tensor is not taken into account due to its complex implementation. The motivation is to propose simple models including non-hydrostatic effects. In order to approximate the resulting model, a simple and efficient numerical scheme is proposed. It consists of a three-step splitting procedure and the resulting scheme is well-balanced for granular material at rest with slope smaller than the fixed repose angle. The model and numerical scheme are validated by means of several numerical tests, including a convergence test, a well-balanced test, and comparisons with laboratory experiments of granular collapse. The influence of non-hydrostatic terms and of the choice of the coordinate system (Cartesian or local) is also analyzed. We show that non-hydrostatic models are less sensitive to the choice of the coordinate system. In addition, the non-hydrostatic Cartesian model produces deposits similar to the hydrostatic local model as suggested by Denlinger and Iverson (J Geophys Res Earth Surf, 2004. https://doi.org/10.1029/2003jf000085), the flow dynamics being however different. Moreover, the proposed model, when written in Cartesian coordinates, can be seen as an improvement of their model, since the vertical velocity is computed and not estimated from the boundary conditions. In general, the non-hydrostatic model introduced here much better reproduces granular collapse experiments compared to hydrostatic models, especially at the beginning of the flow.

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Acknowledgements

This research has been partially supported by the Spanish Government and FEDER through the research projects MTM2015-70490-C2-2-R and RTI2018-096064-B-C22, and by the ERC contract ERC-CG-2013-PE10-617472 SLIDEQUAKES. The authors would like to thank Cipriano Escalante for the interesting discussions related to this work.

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Authors and Affiliations

  1. Dpto. Matemáticas, Edificio Einstein - U. Córdoba, Campus de Rabanales, 14014, Córdoba, Spain

    J. Garres-Díaz & T. Morales de Luna

  2. Dpto. Matemática Aplicada I, ETS Arquitectura - U. Sevilla, Avda. Reina Mercedes S/N, 41012, Seville, Spain

    E. D. Fernández-Nieto

  3. Institut de Physique du Globe de Paris, Seismology Team, U. Paris-Diderot, Sorbonne Paris Cité, 75238, Paris, France

    A. Mangeney

  4. ANGE Team, CEREMA, INRIA, Lab. J. Louis Lions, 75252, Paris, France

    A. Mangeney

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  1. J. Garres-Díaz
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Garres-Díaz, J., Fernández-Nieto, E.D., Mangeney, A. et al. A Weakly Non-hydrostatic Shallow Model for Dry Granular Flows. J Sci Comput 86, 25 (2021). https://doi.org/10.1007/s10915-020-01377-9

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