Analytical solution to one-dimensional mathematical model of flow and morphological evolution in open channels

The evolution of open-channel flow and morphology can be simulated by one-dimensional(1D) mathematical models. These models are typically solved by numerical or analytical methods. Because the behavior of variables can be explained by explicit mathematical determinations,compared to numerical solutions,analytical solutions provide fundamental and physical insights into flow and sediment transport mechanisms. The singular perturbation technique derives a hierarchical equation of waves and describes the evolutionary nature of disturbances in hyperbolic systems. The wave hierarchy consists of dynamic,diffusion,and kinetic waves. These three types of waves interact with each other in the process of propagation. Moreover,the Laplace transform is implemented to transform partial differential equations into ordinary differential equations. Analytical expressions in the wave front are subtracted by the approximation of kinetic and diffusion models. Moreover,an analytical solution consists of a linear combination of the kinetic wave front and the diffusion wave front expressions,pursuing to describe the physical mechanism of motion in open channels as completely as possible. Analytical solutions are presented as a combination of exponential functions,hyperbolic functions,and infinite series. The obtained analytical solution was further applied to the simulation of flood path and morphological evolution in the Lower Yellow River. The phenomenon of increased peak discharge in the downstream reach was successfully simulated. It was encouraging that the results were in good agreement with the observed data.