New formulation of the two-dimensional steep-slope shallow water equations. Part II: Numerical modeling, validation, and application

Numerical models based on the two-dimensional (2D) shallow water equations (SWE) are commonly used for flood hazard assessment, although the basic assumption of small bottom slopes is not always strictly satisfied, such as in mountain areas. When terrain slopes are large, the steep-slope shallow water equations (SSSWE) are theoretically more suitable because the restrictive hypothesis of small bottom slopes is not introduced in deriving these equations. A new formulation of the 2D SSSWE, in which the water depth is measured in the vertical direction, and the flow velocity is assumed parallel to the bottom surface, is proposed in the companion paper (Part I). The pressure distribution on the vertical is assumed linear (yet non-hydrostatic), and the effect of flow curvature is neglected. In this paper, the new SSSWE are solved with an explicit MUSCL-type second-order accurate finite volume scheme using the centered FORCE method for flux evaluation. The SSSWE model is validated against existing experimental data of one-dimensional (1D) dam-break flows on sloping channels with fixed slopes. The numerical results of the SSSWE and SWE models are compared both in this benchmark test case and in other numerical tests, including a 1D dam-break flow moving on an adverse slope, a 2D dam-break flow spreading on an inclined plane, and a 2D dam-break flow propagating in a sloping parabolic channel. Finally, the two models are applied to the real-field test case of the Cancano dam (Adda River, northern Italy), which is characterized by very steep and irregular topography, especially in the upper portion of the valley. The results show that, on the whole, the SSSWE are more accurate in describing dam-break flows over steep topographies than the conventional SWE and predict less severe flooding with slower wave propagation. The two models are practically equivalent when bottom slopes are relatively small.

Automated summary

Numerical models based on the two-dimensional shallow water equations (SWE) are commonly used for flood hazard assessment, but they may not be suitable for areas with large terrain slopes. Steep-slope shallow water equations (SSSWE) are theoretically more suitable as they do not introduce the restrictive assumption of small bottom slopes. In this study, a new formulation of the 2D SSSWE is proposed, and it is validated against experimental data. The results show that the SSSWE are more accurate in describing dam-break flows over steep topographies and predict less severe flooding with slower wave propagation compared to the conventional SWE when bottom slopes are relatively large.