Solutions for Gradually Varied Flow Profiles in Prismatic and Wide Rectangular Channels by Perturbation Method

Abstract

Gradually varied flow (GVF) in open channel hydraulics occurs when free water surface level gradually varies in a steady flow due to geometry alteration in the channel. Investigation of GVF profile is important because water depth along the channel should be determined when designing the channel. Mathematically, GVF profile is obtained by solution of a nonlinear ordinary differential equation. However, due to extreme nonlinearity of the equation, providing an analytical solution is practically unattainable unless under highly simplified conditions. The current study presents semi-analytical solutions for the equation in prismatic ordinary and wide rectangular channels by perturbation method (PM). At the first step, a perturbation solution is derived for wide rectangular channels using a constant Chezy coefficient. At the second step, the wide rectangular channel is investigated using Manning equation as the resistance equation. Finally, the perturbation solution is presented for an ordinary rectangular channel using Manning equation. Obtained GVF profiles using the perturbation solutions are compared with finite difference method (FDM) profiles. Results show that PM profiles are in excellent agreements with FDM ones, and therefore, PM solutions may be used to obtain GVF profiles in mentioned cases with high levels of accuracy. It was concluded that developed semi-analytical solutions may be used as reference solutions for efficiency assessment in various numerical techniques that provide solution for GVF profile in open channels.