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566 Chapter 10 ■ Open-Channel Flow (b) or Similarly, for the triangular weir, Eq. 10.32 gives where H and Q are in meters and m 3 s and C wt is obtained from Fig. 10.21. For example, with H 0.20 m, we find C wt 0.60, or Q 2.36 10.60210.202 52 0.0253 m 3 s. The triangular weir results are also plotted in Fig. E10.8. (c) For the broad-crested weir, Eqs. 10.28 and 10.29 give Thus, with P w 1 m or Q C wt 8 15 tan au 2 b 12g H5 2 C wt 8 15 tan145°2 2219.81 m s 2 2 H 5 2 Q 2.36C wt H 5 2 Q C wb b1g a 2 3 b 32 H 3 2 1.125 a 1 H 12 P w b b1g a 2 32 2 HP w 3 b H 3 2 Q 1.125 a 1 H 2 H b 12 12 m2 29.81 ms 2 a 2 3 b 32 H 3 2 Q 3.84 a 1 H 2 H b 1/2 H 3/2 (3) (2) where, again, H and Q are in meters and m 3 s. This result is also plotted in Fig. E10.8. COMMENTS Although it appears as though any of the three weirs would work well for the upper portion of the flowrate range, neither the rectangular nor the broad-crested weir would be very accurate for small flowrates near Q Q min because of the small head, H, at these conditions. The triangular weir, however, would allow reasonably large values of H at the lowest flowrates. The corresponding heads with Q Q min 0.02 m 3 s for rectangular, triangular, and broad-crested weirs are 0.0312, 0.182, and 0.0375 m, respectively. In addition, as discussed in this section, for proper operation the broad-crested weir geometry is restricted to 0.08 HL w 0.50, where L w is the weir block length. From Eq. 3 with Q max 0.60 m 3 s, we obtain H max 0.349. Thus, we must have L w H max 0.5 0.698 m to maintain proper critical flow conditions at the largest flowrate in the channel. However, with Q Q min 0.02 m 3 s, we obtain H min 0.0375 m. Thus, we must have L w H min 0.08 0.469 m to ensure that frictional effects are not important. Clearly, these two constraints on the geometry of the weir block, L w , are incompatible. A broad-crested weir will not function properly under the wide range of flowrates considered in this example. The sharpcrested triangular weir would be the best of the three types considered, provided the channel can handle the H max 0.719-m head. (Photograph courtesy of Pend Oreille Public Utility District.) 10.6.4 Underflow Gates A variety of underflow gate structures is available for flowrate control at the crest of an overflow spillway (as shown by the figure in the margin), or at the entrance of an irrigation canal or river from a lake. Three types are illustrated in Fig. 10.24. Each has certain advantages and disadvantages in terms of costs of construction, ease of use, and the like, although the basic fluid mechanics involved are the same in all instances. The flow under a gate is said to be free outflow when the fluid issues as a jet of supercritical flow with a free surface open to the atmosphere as shown in Fig. 10.24. In such cases it is customary to write this flowrate as the product of the distance, a, between the channel bottom and the bottom of the gate times the convenient reference velocity 12gy 1 2 12 . That is, q C d a12gy 1 (10.35) where q is the flowrate per unit width. The discharge coefficient, C d , is a function of the contraction coefficient, C c y 2a, and the depth ratio y 1a. Typical values of the discharge coefficient for free V10.15 Spillway gate y 1 y 2 a (a) (b) (c) F I G U R E 10.24 Three variations of underflow gates: (a) vertical gate, (b) radial gate, (c) drum gate.
The flowrate from an underflow gate depends on whether the outlet is free or drowned. V10.16 Unsteady under and over 10.6 Rapidly Varied Flow 567 0.6 Free outflow 0.5 0.4 C d 0.3 Drowned outflow 0.2 0.1 __ y 3 a = 2 3 4 5 6 7 8 0 2 4 6 8 10 12 14 16 __ y 1 a F I G U R E 10.25 Typical discharge coefficients for underflow gates (Ref. 3). y 1 q y 3 a F I G U R E 10.26 Drowned outflow from a sluice gate. outflow 1or free discharge2 from a vertical sluice gate are on the order of 0.55 to 0.60 as indicated by the top line in Fig. 10.25 1Ref. 32. As indicated in Fig. 10.26, in certain situations the depth downstream of the gate is controlled by some downstream obstacle and the jet of water issuing from under the gate is overlaid by a mass of water that is quite turbulent. The flowrate for a submerged 1or drowned2 gate can be obtained from the same equation that is used for free outflow 1Eq. 10.352, provided the discharge coefficient is modified appropriately. Typical values of C d for drowned outflow cases are indicated as the series of lower curves in Fig. 10.25. Consider flow for a given gate and upstream conditions 1i.e., given y 1a2 corresponding to a vertical line in the figure. With y 3a y 1a 1i.e., y 3 y 1 2 there is no head to drive the flow so that C d 0 and the fluid is stationary. For a given upstream depth 1y 1a fixed2, the value of C d increases with decreasing y 3a until the maximum value of C d is reached. This maximum corresponds to the free discharge conditions and is represented by the free outflow line so labeled in Fig. 10.25. For values of y 3a that give C d values between zero and its maximum, the jet from the gate is overlaid 1drowned2 by the downstream water and the flowrate is therefore reduced when compared with a free discharge situation. Similar results are obtained for the radial gate and drum gate. E XAMPLE 10.9 Sluice Gate GIVEN Water flows under the sluice gate shown in Fig. E10.9. The channel width is b 20 ft, the upstream depth is y 1 6 ft, and the gate is a 1.0 ft off the channel bottom. FIND Plot a graph of flowrate, Q, as a function of y 3 .
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Achieve Positive Learning Outcomes
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Sixth Edition F undamentals of Flui
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About the Authors vii Bruce R. Muns
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P reface This book is intended for
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Preface xi Life Long Learning Probl
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Preface xiii WileyPLUS offers today
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Featured in this Book xv LEARNING O
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C ontents 1 INTRODUCTION 1 Learning
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Contents xix 6.4.3 Irrotational Flo
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12.6 Axial-Flow and Mixed-Flow Pump
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10 8 Jupiter red spot diameter 2 Ch
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4 Chapter 1 ■ Introduction and do
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12 Chapter 1 ■ Introduction TABLE
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16 Chapter 1 ■ Introduction Crude
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18 Chapter 1 ■ Introduction Visco
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20 Chapter 1 ■ Introduction Kinem
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22 Chapter 1 ■ Introduction As th
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24 Chapter 1 ■ Introduction Boili
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26 Chapter 1 ■ Introduction E XAM
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28 Chapter 1 ■ Introduction The r
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30 Chapter 1 ■ Introduction fluid
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32 Chapter 1 ■ Introduction Secti
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2 Fluid Statics CHAPTER OPENING PHO
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40 Chapter 2 ■ Fluid Statics in a
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58 Chapter 2 ■ Fluid Statics Free
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66 Chapter 2 ■ Fluid Statics SOLU
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94 Chapter 3 ■ Elementary Fluid D
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100 Chapter 3 ■ Elementary Fluid
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148 Chapter 4 ■ Fluid Kinematics
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264 Chapter 6 ■ Differential Anal
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WARNING Stand clear of Hazard areas
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462 Chapter 9 ■ Flow over Immerse
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616 Chapter 11 ■ Compressible Flo
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646 Chapter 12 ■ Turbomachines Ex
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I ndex Absolute pressure, 13, 48 Ab
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■ TABLE 1.4 Conversion Factors fr
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■ TABLE 1.7 Approximate Physical
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