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Gradually-varied Flow Profiles in Open Channels: Analytical by Chyan-Deng Jan

By Chyan-Deng Jan

Gradually-varied circulate (GVF) is a gentle non-uniform circulation in an open channel with slow alterations in its water floor elevation. The review of GVF profiles lower than a particular move discharge is essential in hydraulic engineering. This publication proposes a unique method of analytically resolve the GVF profiles by utilizing the direct integration and Gaussian hypergeometric functionality. either normal-depth- and critical-depth-based dimensionless GVF profiles are provided. the unconventional procedure has laid the root to compute at one sweep the GVF profiles in a chain of maintaining and antagonistic channels, which could have horizontal slopes sandwiched in among them.

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Extra info for Gradually-varied Flow Profiles in Open Channels: Analytical Solutions by Using Gaussian Hypergeometric Function

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As for the second way to express the solutions of the two integrals using an infinite series, one can also readily obtain the GHF-based solutions of the two integrals from the Mathematica software, as will be elaborated later in Chap. 3. However, it is unlikely that one can relate the two infinite series expressed in terms of the hypergeometric series (see Appendix A for detail) with those adopted by Bakhmeteff (1932),Chow (1955, 1957, 1959) in their computation of the VFF or derived by Ramamurthy et al.

Thus, if we want to evaluate these two integrals more precisely by some means other than the VFF table, we should first examine how Bakhmeteff and Chow computed the VFFvalues, which enabled them to construct the VFF table. We undertake this scrutiny next. 8) to form a sole integral, which he called the VFF. Four methods were proposed by him in the computation of the VFF-values, as explained in Appendix II of his book. In all the four methods except for the third one in which he used the approximate integration formula, he expanded the integrand of the VFF into an infinite series, thereby numerically integrating and calculating a sum of the first few terms of the expanded infinite series for each fixed value of N .

The definition of GHF can be written as 2 F1 (a, b; c; z) = Γ (c) Γ (a)Γ (b) → k=0 Γ (a + k)Γ (b + k) k z , Γ (c + k)k! 20) in which Γ (a), Γ (b) and Γ (c) are Gamma functions; a, b, and c are the function parameters and z is the variable. The infinite series in Eq. 3 GVF Solutions by Using Gaussian Hypergeometric Functions 43 if c > a + b (Olde Daalhuis 2010). 20) has the symmetry property 2 F1 (a, b; c; z) = 2 F1 (b, a; c; z). As presented in the next section, the general solutions of the infinite integrals related with the GVF profiles in this book, if expressed in terms of GHF, are all in following form φ+1 φ+1 u φ+1 uφ ; + 1; u N du = 2 F1 1, 1 − uN φ+1 N N + Const.

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