Basic Equations for the Gradually-Varied Flow
The one-dimensional gradually-varied flow (GVF) is a steady non-uniform flow in a prismatic channel with gradual changes in its water surface elevation.
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Basic Equations for the Gradually-Varied Flow
1.1 Introduction The one-dimensional gradually-varied flow (GVF) is a steady non-uniform flow in a prismatic channel with gradual changes in its water surface elevation. Many hydraulic engineering works involve the computation of one-dimensional GVF surface profiles such as the drawdown produced at a sudden drop in a channel and the backwater produced by a dam or weir across a river, as indicated by Chaudhry (2006), Chow (1959), Subramanya (2009), Jan and Chen (2012), Vatankhah (2012), among others. The evaluation of steady one-dimensional gradually-varied flow profiles under a specific flow discharge is very important in open-channel hydraulic engineering. Two basic assumptions are involved in the analysis of GVF. One is the pressure distribution at any section assumed to be hydrostatic. The other is the resistance to flow at any depth assumed to be given by the corresponding uniform flow equation, such as Mannings equation. Almost all major hydraulic-engineering activites in free surface flow involve the computation of GVF profiles. The various available procedures for computing GVF profiles can be classified as: the graphical-integration method, the direct integration, and the numerical method, as shown in Chow (1959) and Subramanya (2009). The development of the basic GVF dynamic equation and the classification of flow profiles in a prismatic channel is reviewed and discussed in this chapter. The direct integration method for analytically solving the GVF equation by using the Gaussian hypergeometric function (GHF) will be presented in subsequent chapters.
1.2 The GVF Equation for Flow in Open Channels The GVF equation for flow in open channels with various cross-sectional shapes can be readily derived from the energy or momentum conservation law. Therefore, two types of the GVF equation are obtainable: One is based on the energy conservation law and the other on the momentum conservation law. The difference between them simply lies in the convective acceleration term expressed using the different C.-D. Jan, Gradually-varied Flow Profiles in Open Channels, Advances in Geophysical and Environmental Mechanics and Mathematics, DOI: 10.1007/978-3-642-35242-3_1, © Springer-Verlag Berlin Heidelberg 2014
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1 Basic Equations for the Gradually-Varied Flow
Fig. 1.1 Derivation of the gradually-varied flow equation
velocity-distribution correction factor (or coefficient), namely the energy coefficient, α, and the momentum coefficient, β, based on the energy and momentum conservation laws, respectively. The theoretical expressions of α and β for turbulent shear flow in wide channels can be derived from the power law, and the exponent (m) of the power-law velocity distribution is the sole parameter that determines the values of α and β, as indicated by Chen (1992). Many investigators, such as Chow (1959), have adopted α for its wide recognition and comprehension. Thus, following in most researchers footsteps, we opt to use α rather than β. Nevertheless, before elaboratin
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