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Unformatted text preview: Mild & Steep Surface Water Profiles By Mohamed S. Ghidaoui 2002 Introduction 1. Uniform flow theory is often used to size artificial channels (e.g., sewers). 2. Uniform flow serves as the standard reference for experimental and theoretical work (e.g. studies aimed at understanding dissipation and turbulent flow behavior in channels; stability to gravitational waves etc). However, it needs to be emphasized that uniform flow (UF) in open channels is the exception rather than the rule. Factors that cause the flow to depart from being uniform include (i) irregularities in cross-section, alignment, roughness and slope of natural channels; (ii) man-made obstructions such as dams, bridge piers; (iii) control devices such as gates; and (iii) unsteadiness of flow caused by dynamic control structures and/or by time and spatial varying inputs and outputs such as runoff and infiltration. Even in the laboratory it is difficult, if not impossible, to produce a truly uniform flow because the length of the flume is often not sufficient to establish this flow regime. The departure from uniform flow means that the flow velocity and depth vary from location to location along the channel. If flow non-uniformity occur over a relatively short distance, then the flow velocity and depth vary rapidly. That is, the rate of change of depth and velocity with distance is very large. This type of non-uniform flow is referred to as rapidly varying flow (RVF). Examples of RVF include hydraulic jumps and hydraulic bores. When the flow is a RVF, skin friction effects can frequently be neglected, and in many instances, solutions may be obtained by simultaneously considering the mass conservation law and the momentum equation. It must be stressed that momentum and energy are not equivalent concepts across hydraulic jumps and bores. Indeed, across hydraulic jumps and bores, momentum is conserved but energy is not. On the other hand, if the flow varies gradually, skin friction resistance gives rise to surface profiles of considerable extent (sometimes called ``Backwater Curves'') whose quantitative description requires the integration of either the equations of continuity and momentum or continuity and energy. The fact that we can use the momentum or the energy implies equivalency of these two concepts for Gradually Varied Flow (GVF) problems. To be sure, most cases of non-uniform flow in open channels represent a combination of both RVF and GVF. Types of Surface Water Profiles Recalling that the differential form of the energy equation is: f f f S S gA Q y dx d S S g V y dx d S S dx dE- = + ⇒- = + ⇒- = 2 2 2 2 2 Or +- = ⇒- = + ⇒- = + 2 2 2 2 2 2 1 2 1 1 2 1 2 A dy d g Q S S dx dy S S dx dy A dy d g Q dx dy S S A dx d g Q dx dy f f f Or 2 2 2 2 3 2 4 2 2 2 1 1 1 2 2 1 1 2 1 c V S S gA T A Q S S gA T Q S S A dy dA A g Q S S A dy d g Q S S dx dy...
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- Spring '10
- Fluid Dynamics, Manning, uniform flow, control point