Adopting the form of the dynamic μ(Fr,h) basal friction law and introducing a second-order viscous term into the Saint Venant-type equations, we simulated two-dimensional granular roll waves generated in a rectangular chute applying the second-order-accurate total-variation-diminishing MacCormack scheme. Consistent with previous findings, we found that the amplitude and wavelength of the wave increased with transit distance. Although the μ(Fr,h) dynamic basal friction law is the more influential factor in shaping the waves, introducing a viscous term slightly reduces the wave peaks. A sensitive analysis indicates a quadratic relationship between wave peak height and the coefficient in the depth-averaged viscosity with ν≤2.4 × 10⁻² m^1.5/s. In the numerical simulation of roll waves subject to initial perturbations of different magnitudes, three stages in their evolution (namely, emerging, ripening, and attenuating) are evident from their difference in transit behavior. In the emerging stage, the normalized maximum height of the roll waves increases exponentially during propagation. In the ripening stage, this height grows linearly with transit distance whereas the growth rate decreases with the strength of the initial perturbation. During the attenuating stage, the maximum roll-wave height forms a hump in the upstream region where the waves attenuate. Roll-wave peaks and troughs for all three stages tend to approach equilibrium heights further down the chute. Subcritical flow roll-wave simulations show that as flow speed decreases, a reverse transformation pattern is found; specifically, the amplitude and wavelength of the wave decrease further down the chute.

通过采用动态μ（Fr，h）基础摩擦定律的形式并将二阶粘性项引入Saint Venant型方程，我们模拟了在矩形斜槽中生成的二维粒状滚动波，应用了二阶精确的减少总变化的MacCormack方案。与以前的发现一致，我们发现波的振幅和波长随着传输距离的增加而增加。尽管μ（Fr，h）动态基础摩擦定律是影响波浪成形的更重要因素，但引入粘性项会稍微降低波浪峰值。敏感的分析表明，波峰高度与深度平均粘度系数为ν之间存在二次关系≤2.4×10 ^-2 m ^1.5/ s。在对具有不同大小的初始扰动的卷波进行数值模拟时，从传播行为的差异可以看出，卷波的发展分为三个阶段（即出现，成熟和衰减）。在新兴阶段，滚动波的归一化最大高度在传播过程中呈指数增长。在成熟阶段，该高度随运输距离线性增长，而生长速率则随着初始扰动强度的降低而降低。在衰减阶段，最大侧倾波高度在波衰减的上游区域形成一个驼峰。所有这三个阶段的波峰和波谷趋于在滑道的下方逐渐接近平衡高度。亚临界流辊波模拟表明，随着流速的降低，找到了相反的转换模式。