A two-dimensional depth-averaged (I)-rheology for dense granular avalanches
A two-dimensional depth-averaged (I)-rheology for dense granular avalanches
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发表时间：
2015
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通讯作者：
J. L. Baker;T. Barker;J. Gray
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作者：
J. L. Baker;T. Barker;J. Gray
稳定的均匀颗粒溜槽fl在工业上是常见的，为新的理论模型提供了一个重要的测试案例。本文在动量平衡方程中引入了深度积分粘性项，利用物质框架无关性或客观性原理，将最新的稠密颗粒flOWS的深度平均µ(I)-流变学推广到两个空间维度。通过对流道宽度上的横坡坐标和一维定常均匀解上的速度进行标度，我们证明了定常二维下坡速度file与尺度无关。唯一的控制参数是渠道纵横比、坡度倾角以及溜槽和侧壁的摩擦特性。对于无滑移条件和在壁面上的恒定库仑摩擦，都构造了解。对于狭窄的滑道，会产生一个明显的抛物线状的深度平均下游速度file。然而，对于很宽的通道，flow几乎是均匀的，靠近侧壁的边界层很窄。这两种情况都与传统的无粘性雪崩模型形成了直接的对比，传统的无粘性雪崩模型没有发展出横坡PROfiLE。用fiNite单元法计算了全三维µ(I)-流变学的稳态数值解。结果表明，这些解也与规模无关。对于SUFfi浅水航道，由全解计算的水深平均流速分布与水深平均理论的结果符合得很好。通过假定Bagnold-fi速度与深度成正比，从深度平均理论可以重建整个下游速度。对于宽滑道，这与完全三维计算的结果非常接近。为了进行实验验证，利用激光fi测速仪和天平在一定的倾斜角和流道宽度范围内确定了溜槽内总质量flUX与fl厚度之间的关系，并使用粒子图像测速仪记录了相应的表面速度fiLes。实测值与新深度平均理论的重建解有很好的定量一致性。
Steady uniform granular chute flows are common in industry and provide an important test case for new theoretical models. This paper introduces depth-integrated viscous terms into the momentum-balance equations by extending the recent depth-averaged µ( I ) -rheology for dense granular flows to two spatial dimensions, using the principle of material frame indifference or objectivity. Scaling the cross-slope coordinate on the width of the channel and the velocity on the one-dimensional steady uniform solution, we show that the steady two-dimensional downslope velocity profile is independent of scale. The only controlling parameters are the channel aspect ratio, the slope inclination angle and the frictional properties of the chute and the sidewalls. Solutions are constructed for both no-slip conditions and for a constant Coulomb friction at the walls. For narrow chutes, a pronounced parabolic-like depth-averaged downstream velocity profile develops. However, for very wide channels, the flow is almost uniform with narrow boundary layers close to the sidewalls. Both of these cases are in direct contrast to conventional inviscid avalanche models, which do not develop a cross-slope profile. Steady-state numerical solutions to the full three-dimensional µ( I ) -rheology are computed using the finite element method. It is shown that these solutions are also independent of scale. For sufficiently shallow channels, the depth-averaged velocity profile computed from the full solution is in excellent agreement with the results of the depth-averaged theory. The full downstream velocity can be reconstructed from the depth-averaged theory by assuming a Bagnold-like velocity profile with depth. For wide chutes, this is very close to the results of the full three-dimensional calculation. For experimental validation, a laser profilometer and balance are used to determine the relationship between the total mass flux in the chute and the flow thickness for a range of slope angles and channel widths, and particle image velocimetry (PIV) is used to record the corresponding surface velocity profiles. The measured values are in good quantitative agreement with reconstructed solutions to the new depth-averaged theory.