Solutions for Gradually Varied Flow Profiles in Prismatic and Wide Rectangular Channels by Perturbation Method
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RESEARCH PAPER
Solutions for Gradually Varied Flow Profiles in Prismatic and Wide Rectangular Channels by Perturbation Method Hamed Reza Zarif Sanayei1 · G. Reza Rakhshandehroo2 Received: 5 May 2020 / Accepted: 9 September 2020 © Shiraz University 2020
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. Keywords Gradually varied flow · Wide rectangular channel · Semi-analytical solution · Water surface profile · Ordinary rectangular channel · Perturbation method
1 Introduction Water depth estimation along a channel containing steady non-uniform flow is a fundamental step for the channel design. It is obtained by solving gradually varied flow (GVF) equation which is a nonlinear ordinary differential equation. Solution may be obtained numerically or analytically. However, due to the extreme nonlinearity in the equation, providing an analytical solution is impossible unless under highly simplified conditions. On the other hand, many numerical techniques have been presented in the last three decades to obtain GVF profiles in open channels. Researchers have investigated numerical solutions for the * Hamed Reza Zarif Sanayei [email protected]; [email protected] 1
Faculty of Engineering, Shahrekord University, Shahrekord, Iran
Department of Civil and Environmental Engineering, Shiraz University, Shiraz, Iran
2
GVF equation in open channels by finite difference method (FDM), Newton–Raphson method, differential quadrature method (DQM), Runge–Kutta method, genetic programming and other methods (Rhodes 1998; Dey 2000; Sen and G
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