Open channel hydraulics for engineers. Chapter 2 uniform flow

The chapter on uniform flow in open channels is basic knowledge required for all hydraulics students. In this chapter, we shall assume the flow to be uniform, unless specified otherwise. This chapter guides students how to determine the rate of discharge,the depth of flow, and the velocity. The slope of the bed and the cross-sectional area remain constant over the given length of the channel under the uniform-flow conditions. The same holds for the computation of the most economical cross section when designing the channel. The concept of permissible velocity against erosion and sedimentation is introduced.......
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Open channel hydraulics for engineers. Chapter 2 uniform flow OPEN CHANNEL HYDRAULICS FOR ENGINEERS-----------------------------------------------------------------------------------------------------------------------------------Chapter UNIFORM FLOW 2.1. Introduction 2.2. Basic equations in uniform open-channel flow 2.3. Most economical cross-section 2.4. Channel with compound cross-section 2.5. Permissible velocity against erosion and sedimentationSummaryThe chapter on uniform flow in open channels is basic knowledge required for allhydraulics students. In this chapter, we shall assume the flow to be uniform, unlessspecified otherwise. This chapter guides students how to determine the rate of discharge,the depth of flow, and the velocity. The slope of the bed and the cross-sectional arearemain constant over the given length of the channel under the uniform-flow conditions.The same holds for the computation of the most economical cross section when designingthe channel. The concept of permissible velocity against erosion and sedimentation isintroduced.Key wordsUniform flow; most economical cross-section; discharge; velocity; erosion; sedimentation2.1. INTRODUCTION2.1.1. Definition Uniform flow relates to a flow condition over a certain length or reach of a streamand can occur only during steady flow conditions. Uniform flow may be also defined as theflow occurring in a channel in which equilibrium has been reached between gravitationalforce and shear force. Many irrigation and drainage canals and other artificial channels aredesigned to carry water at uniform depth and cross section all along their lengths. Naturalchannels as rivers and creeks are seldom of uniform shape. The design discharge is set byconsiderations of acceptable risk and frequency analysis, whereas the channel slope andthe cross-sectional shape are determined by topography, and soil and land conditions.Uniform equilibrium open-channel flows are characterized by a constant depth and aconstant mean flow velocity: h V 0 and 0 (2-1) s swhere s is the coordinate in the flow direction, h the flow depth and V the flow velocity.Uniform equilibrium open-channel flows are commonly called “uniform flows” or “normalflows”.Note: The velocity distribution in fully-developed turbulent open channel flows is givenapproximately by Prandtl’s power law (Fig. 2.1):  y N 1   V (2-2) Vmax  h where the exponent 1/N varies from ¼ down to ½ depending on the boundary friction andthe cross-section shape. The most commonly-used power law formulae are the one-sixth-----------------------------------------------------------------------------------------------------------------------------------Chapter 2: UNIFORM FLOW 25OPEN CHANNEL HYDRAULICS FOR ENGINEERS-----------------------------------------------------------------------------------------------------------------------------------power (1/6) and the one-seventh power (1/7) formulas. It should be noted that the velocityin open-channel flow is assumed constant over the entire cross-section. Vmax V velocity h distribution v y Fig. 2.1. Velocity distribution profile in turbulent flowSuch flow conditions are represented schematically in Fig. 2.2. Considering Bernoulli’stheorem of the conservation of energy, between cross-sections 1 and 2, leads to theexpression:  V2   V2  E1   z1  1  1 1   E 2  h L   z 2  2   2 2   h L p p (2-3)   2g    2g where 1 and 2 are the Corriolis-coefficients corresponding to the velocities V1 and V2,respectively. They are also called the kinetic-energy correction coefficients.  is equal toor larger than 1 but rarely exceeds 1.1. (Li and Hager, 1991). For a uniform velocitydistribution,  = 1.The slope of the energy gradient line S is equal to the bed slope i of the channel, or: S= i L h (2-4) L  energy-gradient line  V12 hL 2g V 22 S 2g h1  1 p hydraulic-gradient line E1 g h2  ...