Propagation of Nonlinear Flood Waves in Rivers

Abstract

The propagation of flood waves in rivers is governed by the Saint Venant equations. Under certain simplifying assumptions, these nonlinear equations have been solved numerically via computationally intensive, specialized software. There is a cogent need for simple analytical solutions for preliminary analyses. In this paper, new approximate analytical solutions to the nonlinear kinematic wave equation and the nonlinear dynamic wave equations in rivers are presented. The solutions have been derived by combining Adomian’s decomposition method (ADM), the method of characteristics, the concept of double decomposition, and successive approximation. The new solutions compare favorably with independent simulations using the modified finite-element method and field data at the Schuylkill River near Philadelphia. The time to peak calculated by the analytical and numerical methods is in excellent agreement. There appears to be some minor differences in the peak magnitude and recession limb, possibly because of numerical dissipation. Including the momentum equation in the analysis causes a decrease in the magnitude of the flow rate at all times. Except for the flood peak, the nonlinear analytical solution exhibits lower flow rates than the numerical solution. The numerical solution also shows higher dispersion. The new analytical solutions are easy to apply, permit an efficient preliminary forecast under scarce data, and an analytic description of flow rates continuously over the spatial and temporal domains. They may also serve as a potential source of reference data for testing new numerical methods and algorithms proposed for the open channel flow equations. The ADM nonlinear kinematic and nonlinear dynamic wave solutions exhibit the usual features of nonlinear hydrographs, namely, their asymmetry with respect to the center of mass, with sharp rising limbs and flatter recession limbs. Linear approximations of the governing equations usually miss these important features of nonlinear waves. The greatest portion of the magnitude of discharge is given by the initial nonlinear kinematic wave component, which implies that in the lower Schuylkill River the translational components dominate the propagation of flood waves, in agreement with previous research. The nonlinear dynamic wave better predicts the flow rate during peak times and especially during recession and low-flow periods. Thus, while both the nonlinear kinematic and the nonlinear dynamic wave models are based on simple approximate analytical solutions that are easy to implement, the nonlinear kinematic wave equation model requires less data and less computational effort.