Open Boundary Condition For Unsteady Open-channel Flow

Abstract

Initial and boundary conditions are needed to solve the unsteady flow equations for flood routing in rivers. As the boundary conditions, inflow discharge hydrograph at the upstream boundary and a rating curve (stage discharge relationship) at the downstream boundary are employed. But in the computation of long wave propagation the open boundary condition is applied at the boundary and a rating curve is not necessary as in the computation of flood routing. Since the governing equations of the unsteady flow and the long wave propagation are essentially the same, the open boundary condition in the case of the computation of the long wave propagation is applicable to the case of the computation of flood routing. Introduction The computation of the flood routing is the most important subject in flood control of rivers. It is very evident from that EEC 2 model were developed recently in the United States. The usual methods to solve the governing equations for the unsteady flow are the finite difference method or the method of characteristics. During the numerical computation, the initial and boundary conditions are necessary to solve differential equations. This study applies the open boundary condition to the method of characteristics and proposes the method of the flood routing without a rating curve. The first application of the open boundary condition to the computation of the long wave propagation was done by Hino (Hino, 1987; Hini and Nakaza, 1989) for the case of the finite difference scheme. In this study, for given discharge hydrograph at the upstream boundary the rating curve at the downstream boundary is computed during the flood Transactions on Ecology and the Environment vol 12, © 1996 WIT Press, www.witpress.com, ISSN 1743-3541 224 Hydraulic Engineering Software Up st re am B ou nd ar y => Computational Zone D o w n s t r e a m B ou nd ar y Figure 1: Coordinate system routing and the characteristics of the rating curve is investigated. The stationary water depth at the downstream boundary for the flood routing is numerically determined. Problem Formulation The continuity equation of unsteady flow in the uniform channel of rectangular cross section is given by dA dx (1) in which A = cross-sectional area; / = time; Q = discharge; and x = the coordinate system along the flow direction. The dynamic equation of unsteady flow is also written by dh dx <, o dx gAdt ° * (2) in which g = the gravitational acceleration; B = top width of cross section; h = water depth; So = the bottom slope; and Sf = the energy slope. The coordinate system is represented in Fig.l. By using the Manning's formula, the energy slope is expressed as