Debris flows and general steep slope shallow water flows numerical simulation
Authors:   , Medina Iglesias Vicente César de 
Published: August 2023

The main target of this research is to develop a numerical model for debris flow simulations. As it is known, in general, this kind of flows occur in steeped mountain slopes. When dealing with the complete 3D physic system of equations that model the phenomenon this singular characteristic has not special effect.
To simulate large events (typical scales in real world) is not possible to use the complete equations (three dimensional, variable density, non-hydrotactic…) so a spacial dimension reduction is necessary combined with several simplifications to reduce the complexity of the system, along the present text previous works references are introduced to justify the selected simplifying hypothesis.
During the mathematical manipulation of the equations, performed in order to reduce the spacial dimension (3D & 2D), the real complexity of the problem emerges, important consequences on the coordinate system appear. This means that, to obtain a simpler version of the physical model of the phenomenon, complex mathematical operations are needed.
In the approach presented in this work the complexity of the problem is reduced
in to manners:
• Applying direct physical hypothesis on the flow characteristics.
• Applying mathematical hypothesis.
An example of these physical simplifications could be the monophasic fluid hypothesis.An example of the mathematical simplifications could be the fact of the curvature terms neglecting.
In this work, the coordinate system selected for the model is named Proposed Coordinates System (PCS), this coordinates system is based on the work of Bouchut and Westdickenberg (2004) and Berger and Carey (1998a). The metric characteristics of the system are calculated and important conclusions are extracted from the analysis of the curvature terms. A link exist between the curvatures and the Christoffel symbols, if the curvatures are discarded also the Christoffel symbols should be discarded for model consistency. In the model governing differential equations, the use of curvilinear coordinates (PCS) provokes the existence of metric source terms defined as a Christoffel symbols functions. So, neglecting these terms the equations become simplified.
Strictly talking the model is valid for steep but slowly varying slopes, where the curvatures are very small, although in the real test cases used for validating the model, the curvatures does not fit this condition. The rough approximation to the debris flow process through simplified physics used in the model probably hides the low accuracy of the model in strongly curved areas. Many other different coordinates system choices exist in the scientific literature, in this work some of them are commented and analyzed.
Along the development of the model, different problems appear and different strategies and methodologies are proposed in order to overcome them. The first important problem is related to the physics of the process, the debris flows tend to naturally develop flow pulses, the flowing mixture stops and later is remobilized. The model includes what is called ”stop and go” mechanism to capture this behavior.
The second problem is related with the boundary conditions, standard debris flow hydrograph includes sharp gradients which can become flow shocks, in this work a new methodology is proposed to introduce the boundary conditions in a way that shocks are correctly solved also in the boundary of the computational domain, contrary to the standard method of characteristics (Henderson, 1966; Abbott, 1966).
The general family of the numerical methods selected to solve the resulting system is the Finite Volume Method (FVM) using the Riemann solver. This approach is extensively used in the debris flow simulation framework, so its selection is justified. In order to fit the special requirements of the PCS coordinates system the methodology developed by Rossmanith et al. (2004) and Rossmanith (2006) is used. This approach solves the problem of the non-orthogonal coordinates in the FVM framework.
To validate the code analytical, experimental and real test cases are selected. The model validation process is partial because the lack in analytical solutions, especially for complex geometries.