| Unsteady flow in open channels can be treated as propagation of shallow water waves in channel flow. It can be mathematically approximated by the Saint-Venant equations or by simplified wave approximations, including the kinematic wave, noninertia wave, gravity wave and quasi-steady dynamic wave models. These approximations differ not only in their computational complexities, but also in the basics of the wave propagation mechanisms. Issues related to the hydrodynamics of this phenomenon are studied: the role of downstream backwater in wave propagation processes; the difference between the noninertia wave and diffusion wave; and the applicability of different wave approximations to unsteady flow problems. Three mathematical approaches are employed. In the linear wave analysis, the Laplace transform method is implemented to obtain analytical spatio-temporal expressions of upstream and downstream channel responses, facilitating a critical comparison of the physical characteristics among different waves. In addition, linear stability analysis is introduced to discuss the free surface stability mechanism. The wave celerity and attenuation characteristics over the wave number spectrum are derived for each wave approximation in the M1, M2 and S1 type flow regimes. In the nonlinear wave analysis, a uniformly valid asymptotic nonlinear description of the wave in a rectangular channel is derived, at and away from the wave shock, as well as near the downstream boundary where the backwater effect is significant. Results of this study show that each wave approximation has its distinct wave celerity and attenuation. Results also demonstrate that a generalized diffusion wave equation can be formulated from all levels of the wave approximations, whereas the noninertia wave approximation is a special case of the diffusion wave equation. This study suggests that the downstream backwater effect comes from two mechanisms: that of the negative characteristic wave, and that of the pressure instantly transmitted upstream. Derived theoretical results are further examined to obtain applicability criteria in terms of dimensionless physical parameters representing the unsteadiness of the wave disturbance, channel characteristics, base flow properties, downstream backwater, and variations in space and time. For the various wave approximations, the proposed criteria can be applied to a specified point and time in unsteady flow simulation. |
