Investigation of the effect of reservior shape coefficient on dam-break waves using Leap-Frog and Lax methods in curvilinear coordinates

Abstract
Investigation of the effect of reservior shape coefficient on dam-break waves using Leap-Frog and Lax methods in curvilinear coordinates
Background and objectives
The prediction of hydraulic components of depth and speed has always been important for hydraulic engineers because of its impact on the severity of the disaster dam break. In the past, many studies have been carried out to investigate and predict the hydraulic properties of dam-break waves using numerical methods. The necessity of this research is the need to expand the scope of research in numerical solution of factors influencing the dam failure phenomenon. In this research, a comprehensive computer model has been developed in which using the explicit finite difference method and simultaneous use of Leap-Frog and Lax algorithms on the staggered mesh shallow water equations are solved to simulation dam break problem. This will increase the number of involved points in the computation and sharpen hydraulic gradients become smooth and the probability of oscillation and divergence will decrease without the use of artificial viscosity.

Materials and methods
In this study, the equations of interest are the governing shallow water equations. Due to the inability of the Cartesian coordinate system to reflect the irregular boundaries of the physical domain, in the curvilinear coordinate system on the staggered mesh are discretized. The method of discretization is a explicit method that simultaneously utilizes Leap-Frog and Lax algorithms.

Results:
In order to validate the present model, comparing its results with laboratory measures or with the results of other numerical models has been proposed by several researchers. One of these cases is the ideal failure in the canal with tail water, in which case the results of the discharge and the depth of water were simulated for the failure of the dam in a 100 meters horizontal channel with a high accuracy. The simulation of the dam failure with a trapezoidal reservoir with a maximum of 2.04 meters, a small 0.05 m, and a height of 2.02 meters in a canal with a dry bed was carried out, and the results of the model for the discharge and depth with the consistency well presented with laboratory results. The partial asymmetric dam failure in the wet bed is another case investigated in this study, which simulated an asymmetric failure in a reservoir with a length and width of one meter for three different shape coefficients of 1, 1.25, and 1.5. The flow and discharge hydrographs have been simulated for different situations, with the increase of the shape coefficient, discharge values and the level of the water surface increase due to increased reservoir volume.
Conclusion:
In this research, a computer model in the curvilinear coordinate system with the consideration of shallow-water equations and the use of Lax and Leap frog methods are presented simultaneously for the dam failure phenomenon. In the simulation of the ideal failure with tail water, the present model ability to approximate the analytical solution is highly accurate. In the simulation of failure in trapezoidal reservoirs on the dry bed, the results of the present numerical model are in agreement with experimental results, also in studying the main objective of the research, the results of the present numerical model for simulating partial asymmetric dam failure with different shape coefficients of reservoir in dry bed have been investigated. It was observed that water level increased with the change of shape coefficients following the deformation of the reservoir walls, compared with the simple reservoir. Also, the amount of discharge per unit width for a reservoir with a larger coefficient of shape than a reservoir with a lower coefficient is greater.