Diffusive wave model in a finite length channel with a concentrated lateral inflow subject to different types of boundary conditions

Abstract

The diffusive wave model is one of the simplified forms of Saint-Venant equations, and it is often used instead of the full model. In this paper, we present an analytical solution for the linearized diffusive wave model represented by a simultaneous system of two first-order partial differential equations focused on spatial variation of a lateral inflow in a finite channel. A concentrated lateral inflow from a small-width tributary is considered through the Dirac delta function. We use the Laplace transform method to solve these equations analytically. Two types of upstream boundaries are considered here in the form of a flow-discharge hydrograph and a flow-depth hydrograph, while keeping a flow-depth hydrograph as the downstream boundary. Using unit-step responses of the lateral inflow, the effect of different boundaries on the flow-depth responses and the flow-discharge responses is studied for different values of the Peclet number (Pe). The flow depth is observed to be more sensitive to the downstream boundary and other parameters used in this work. Consideration of the flow depth as the upstream boundary reflects the effect of all the parameters on the unit-step responses presented. These responses are compared with the available semi-infinite channel responses, which are found to be an inappropriate substitute for the finite channel responses for Pe<5 which implies that the downstream boundary cannot be ignored for these cases. However, for the case Pe>5, although the semi-infinite channel responses are found to satisfactorily estimate the discharge along the entire channel, they can approximate the flow depth at the locations closer to the upstream boundary only.