Abstract
Abstract. The equation of one-dimensional gradually varied flow (GVF) in sustaining and non-sustaining open channels is normalized using the critical depth, yc, and then analytically solved by the direct integration method with the use of the Gaussian hypergeometric function (GHF). The GHF-based solution so obtained from the yc-based dimensionless GVF equation is more useful and versatile than its counterpart from the GVF equation normalized by the normal depth, yn, because the GHF-based solutions of the yc-based dimensionless GVF equation for the mild (M) and adverse (A) profiles can asymptotically reduce to the yc-based dimensionless horizontal (H) profiles as yc/yn → 0. An in-depth analysis of the yc-based dimensionless profiles expressed in terms of the GHF for GVF in sustaining and adverse wide channels has been conducted to discuss the effects of yc/yn and the hydraulic exponent N on the profiles. This paper has laid the foundation to compute at one sweep the yc-based dimensionless GVF profiles in a series of sustaining and adverse channels, which have horizontal slopes sandwiched in between them, by using the GHF-based solutions.
Highlights
Many hydraulic engineering works involve the computation of surface profiles of gradually varied flow (GVF) that is a steady non-uniform flow in an open channel with gradual changes in its water surface elevation
This study focuses on the direct integration method that used to analytically compute the GVF profiles in sustaining and non-sustaining channels, in which the GVF equation is normalized by using yc and analytically solved by using the Gaussian hypergeometric function (GHF)
Success to formulate the normal-depth(yn)-based GVF profiles expressed in terms of GHFs for flow in sustaining channels, as reported by Jan and Chen (2012), does not warrant that it can likewise prevail to use yn in the normalization of the GVF equation for flow in horizontal and adverse channels because yn for an assumed uniform flow in horizontal and adverse channels is undefined
Summary
Many hydraulic engineering works involve the computation of surface profiles of gradually varied flow (GVF) that is a steady non-uniform flow in an open channel with gradual changes in its water surface elevation. They evaluated the two indefinite integrals using the partial-fraction expansion and got the elementary-transcendental-function (ETF)-based solution for each of the cross-sectional shapes under study, such as rectangular, triangular and parabolic cross-sectional shapes Another approach taken by Zaghloul (1990, 1992) to integrate the GVF equation for the profiles in circular pipes was the same as that used by Allen and Enever (1968) by substituting the geometric elements of a circular conduit section into Z and K in the GVF equation before integrating it by use of Simpson’s rule (Zaghloul, 1990) or both the direct step and integration methods (Zaghloul, 1992). For computing GVF profiles in sustaining channels, one can solve the yn-based dimensionless GVF equation by using the direct integration method and the Gaussian hypergeometric function (GHF), as presented by Jan and Chen (2012).
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