Modeling the gradually varied flow profile in circular and parabolic channels using the Adomian decomposition method
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ORIGINAL ARTICLE
Modeling the gradually varied flow profile in circular and parabolic channels using the Adomian decomposition method Nastaran Sheni Shahvand1 · Hamed Reza Zarif Sanayei1 · Reza Kamgar1 Received: 10 July 2020 / Accepted: 3 October 2020 © Springer Nature Switzerland AG 2020
Abstract To design dimensions of channels, examining the depth of water in a gradually varied flow is required. Therefore, it is essential to calculate the profile of gradually varied flow in the channels. The differential equation of the gradually varied flow needs to be solved to determine the flow’s depth along the channel to obtain this profile. Previous researches have been carried out to solve this equation by employing numerical methods. In this paper, however, a semi-analytical solution has been proposed to solve the gradually varied flow equation in the circular and parabolic prismatic channels using Adomian Decomposition Method (ADM). The ADM results have also been compared with the numerical Finite Difference Method (FDM) for some examples about the circular and parabolic channels. The results are then presented for three approximations of four sentences, five sentences, and six sentences in the ADM in each channel. Moreover, each approximation results have been compared with the results of the profiles obtained by the numerical finite difference method. The profiles of the ADM in this study are in good agreement with those obtained by the FDM. The ADM analytical solutions presented in this paper can validate other numerical methods in similar research. Keywords Gradually varied flow · Circular channel · Parabolic channel · Semi-analytic method · ADM
Introduction Open channel hydraulics has always been a significant area for the scientific and engineering activity because of the great importance of water for human life. The free surface flow that occurs in oceans, seas, and rivers is still one of the most complex processes in the environment (Szymkiewicz 2010). Moreover, in a prismatic channel, the steady nonuniform flow with step by step changes in free water surface elevation is referred to as a gradually varied flow (GVF). All the significant hydraulic engineering research involves the calculation of GVF profile length on the open channel flow (Zaghloul and Anwar 1991). Accurate computation of the water elevation in different channel sections can be done numerically, analytically, or semi-analytically. Numerical methods for solving the GVF equation in the channels are discontinuously solved, i.e. they only solve the problem at the node points of networking * Hamed Reza Zarif Sanayei [email protected] 1
Depratment of Civil Engineering, Shahrekord University, P.O. Box. 88186‑34141, Shahrekord, Iran
along the channel. For this reason, analytical and semi-analytical solutions are beneficial, because they provide a better view of the solution to the problem of discontinuous numerical solutions. Besides, analytical and semi-analytical solutions can be used to validate the answers of other methods. Many studies hav
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