Use of the Gaussian hypergeometric function to solve the equation of gradually-varied flow

Gradually varied open-channel flow profiles normalized by critical depth and analytically solved by using Gaussian hypergeometric functions

“…The computation methods of GVF profiles in open channels have been discussed in many textbooks and journal papers (Chow, 1959;Subramanya, 2009;Jan and Chen, 2012;Vatankhah, 2012). The most widely used methods for computing GVF profiles could be classified into the step methods and the direct integration methods.…”

“…In the directintegration method, the one-dimensional (1-D) GVF equation is usually normalized to be a simpler expression in advance so as to allow the performance of direct integration. In most cases, the GVF equation is normalized by the normal depth y n (Chow, 1959;Subramanya, 2009;Jan and Chen, 2012;Venutelli, 2004;Vatankhah, 2012), while in some cases, it is normalized by the critical depth y c (Chen and Wang, 1969). Many attempts have been made by previous investigators on the direct-integration method.…”

“…The varied-flow function (VFF) needed in the direct-integration method conventionally used by Bakhmeteff (1932), Chow (1955Chow ( , 1957Chow ( , 1959, Kumar (1978), among others, has a drawback caused by the imprecise interpolation of the VFF values. To overcome the drawback, Jan and Chen (2012) already successfully used the Gaussian hypergeometric functions (GHFs)…”

“…Success to formulate the normal-depth(y n )-based nondimensional GVF profiles expressed in terms of GHFs for flow in sustaining channels, as reported by Jan and Chen (2012), does not necessarily warrant that it can likewise prevail to use y n in the normalization of the GVF equation for flow in adverse channels because y n for an assumed uniform flow in adverse channels is undefined. Even though one can use an imaginary flow resistance coefficient to evaluate y n for such an assumed uniform flow in adverse channels (Chow, 1959), as will be treated later in this paper, it is inappropriate to use such an y n in the normalization of the GVF equation for flow in adverse channels because y n so evaluated is fictitious and the two normalized variables in such y n -based dimensionless GVF equation collapse (in virtue of y n → ∞) as the channel-bottom slope approaches zero.…”

“…Before proceeding to formulate the two y c -based dimensionless GVF equations for sustaining channels and adverse channels, respectively, it is worthwhile to review all of what has been attained by previous investigators to compute the GVF profiles in both sustaining and adverse channels. We already reviewed quite comprehensively most of the previous work on the analytical computation of the GVF profiles in sustaining channels, as reported in our paper (Jan and Chen, 2012). Therefore, in this paper, we focus on the literature survey only of the computed GVF profiles in adverse channels.…”