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On the weak turbulent motions of an isothermal dry granular dense flow with incompressible grains: part II. Complete closure models and numerical simulations

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Abstract

The complete thermodynamically consistent turbulent closure models of isochoric and isothermal dry granular dense flows with incompressible grains and weak turbulent intensity are established on the basis of a linearized theory with respect to the granular coldness for the dynamic responses of the closure conditions. The models are applied to study a gravity-driven stationary turbulent flow down an inclined moving plane, and the numerical simulations are compared with the experimental outcomes. It shows that while the mean velocity decreases monotonically from its boundary value on the moving plane toward the free surface, the mean porosity and granular coldness display more “exponential-like” increasing/decreasing tendencies. Of particular interest is that the granular coldness evolves from its maximum value on the moving plane toward its minimum value on the free surface, leading to the turbulent dissipation evolving in a similar manner, while the turbulent kinetic energy demonstrate a reverse tendency. The obtained results show good agreements to the experimental outcomes and are similar to the characteristics of conventional Newtonian fluids in turbulent shear flows.

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Notes

  1. The turbulent viscosity models in [1, 2, 34] are based on Prandtl’s mixing length theory. However, they are proposed for rapid flows and are not appropriate for dense flows.

  2. \({\varvec{\varPhi }}\) should be homogeneous of the first degree of \({\varvec{D}}\) and is originally assigned to the Cauchy stress \({\varvec{t}}\). However, in our formulations, this concept is extended for the internal friction \({\varvec{Z}}\) and the mean production \(\bar{{\varvec{\varPhi }}}\) in turbulent flows.

  3. In the previous works, only the no-slip conditions for the velocity and the fixed value of the mean volume fraction on the moving plane are employed, and that the solid boundary behaves as an energy source/sink of the turbulent kinetic energy is not taken into account, even the flow is obviously turbulent, e.g., see [14, 15, 30, 43, 44]. This goes back to the laminar flow formulations. In the present study, the solid boundary as an energy source/sink is taken into account by prescribing the boundary values of \(\vartheta ^T\) on the moving plane.

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Acknowledgments

The authors are indebted to the National Science Council, Taiwan, for the financial support through the project NSC 101–2221–E–006–195–, and the reviewers and the editors for their comments and suggestions which led to improvements.

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Authors and Affiliations

  1. Department of Civil Engineering, National Cheng Kung University, No.1, University Road, Tainan City, 701, Taiwan

    Chung Fang

  2. Institute of Geotechnical Engineering, University of Natural Resources and Life Sciences, Vienna, Feistmantelstrasse 4, 1180, Vienna, Austria

    Wei Wu

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Correspondence to Chung Fang.

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Fang, C., Wu, W. On the weak turbulent motions of an isothermal dry granular dense flow with incompressible grains: part II. Complete closure models and numerical simulations. Acta Geotech. 9, 739–752 (2014). https://doi.org/10.1007/s11440-014-0314-3

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  • DOI: https://doi.org/10.1007/s11440-014-0314-3

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