OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 90 Chapter  SPILLWAYS _________________________________________________________________________ 5.1. Introduction 5.2. General formula 5.3. Sharp-crested weir 5.4. The overflow spillway 5.5. Broad-crested weir _________________________________________________________________________ Summary Spillways are familiar hydraulic structures built across a stream to control the water level. This chapter emphasizes the classification of weirs and spillways as well as the application of hydraulic formulas for designing their shape and dimensions. Key words Spillway; weir; crest; design head ________________________________________________________________________ 5.1. INTRODUCTION Spillways are used at both large and small dams for letting flood flows pass, thereby preventing overtopping and failure of the dam. A spillway as sketched in Fig. 5.1 is the most common type. Three zones can be distinguished: the crest, the face and the toe – each with its separate problems. Fig. 5.1. General view of a spillway A weir is a notch of regular form through which water flows. The term is also applied to the structure containing such a notch. Thus a weir may be a depression in the side of a tank, a reservoir, or a channel, or it may be an overflow dam or other similar structure. Classified in accordance with the shape of the notch, there are rectangular weirs, triangular or V-notch weirs, trapezoidal weirs, and parabolic weirs. Weirs are usually designed to control water levels in rice fields or wetlands. They are commonly used as a means of flow measurement. Of primary importance for hydraulic structures considered in this chapter is the magnitude of backwater level they cause upstream of the structure for the given discharge; that is, the head-discharge relationship for the structure. Both gradually varied and rapidly varied flows are possible through these structures, but one-dimensional methods of analysis usually are sufficient and well-developed in this branch of hydraulics. Essential to the crest face toe OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 91 “hydraulic approach” is the specification of empirical discharge coefficients that have been well established by laboratory experiments and verified in the field. The determination of controls in hydraulic analysis also is important, and critical depth often is the control of interest. The energy equation and the specific-energy head diagram are useful tools in the hydraulic analyses of this chapter. 5.2. GENERAL FORMULA The equation for discharge over a weir cannot be derived exactly, because not only the flow pattern of one weir differs from that of another, but also the flow pattern for a given weir varies with the discharge. Furthermore, the number of variables involved is too large to warrant a rigorous analytical approach. Approximate derivations are presented in most texts. These derivations show effects of gravitational forces in an approximate manner, but do not include the effects of viscosity, surface tension, the ratios of the dimensions of the weir to the dimensions of the approach channel, the nature of the weir crest, and the velocity distribution in the approach channel. A simplified derivation will be made here to show the general character of the relationship between the discharge and the most important variables and to demonstrate the nature of the effect of some of the variables. The derivation will be made for sharp-crested weirs, but as will be shown later, a similar derivation would apply to weirs that are not sharp-crested. Now, we consider a rectangular weir, over which the water is flowing as shown in Fig. 5.2. Fig. 5.2. Rectangular weir Let, H = height of the water above the crest of the weir, L = length of the weir, and C d = coefficient of discharge (see Chapter 1). Let us consider a horizontal strip of water of thickness dh at a depth h from the water surface as shown in Fig. 5.2. Area of the strip = L.dh (i) Assuming that the flow does not contract when passing over the weir (i.e. neglecting streamline curvature), and that the pressure is atmospheric across the vertical section above the weir and the upstream velocities are small, the velocity of the water through the strip can theoretically be derived to be: H dh h water surface L OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 92 theoretical velocity = 2gh (ii)  The discharge per unit width, q, flowing over a weir is generally expressed as: d dq C area of strip theoretical velocity   d dq C .L.dh. 2gh (iii) The total discharge over the weir may be found by integrating the above equation within the limits 0 and H:  H 3 H 2 3 2 d d d o 0 h 2 Q C .L.dh. 2gh C .L. 2g. C .L. 2g.H 3 3 2               or the discharge per unit width, q, is: 3 2 d 2 q C 2g.H 3  (5-1) where H is to be conceived of as the total upstream specific-energy head on the weir crest supposing that the upstream velocities are negligibly small; C d is a discharge coefficient, which can be approximated by Rehbock’s experimental formula (1929): C d = 0.611 + 0.08 H P       (5-2) where P is measured from top of the crest of the weir to the bottom of the reservoir; P is called the weir height. Assuming P very large, C d becomes equal to 0.611. In this case, Eq. (5-1) can be written as: 3 2 q 1.80 H [1.80] = m ½ s -1 (5-3) Experiments show that the rise from the sharp weir crest to the highest point of the nappe (i.e. the “spillway crest”) is 0.11H (see Figs. 5.4 and 5.6). Using this fact we can express Eq. (5-3) in terms of H D , the head over the spillway crest. We obtain: 3 2 D q 2.14 H [2.14] = m ½ s -1 (5-4) where H D may be termed the design head. Example 5.1: A rectangular weir, 4.5 m long, has a head of water 30 cm. Determine the discharge over the weir, if the coefficient of discharge is 0.6. Solution: Given: length of weir: L = 4.5 m head of water: H = 30 cm = 0.3 m coefficient of discharge: C d = 0.6 Discharge over the weir Q? Using the relation: OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 93 3 2 d 2 Q C .L. 2g.H 3  = 1.31 m 3 /s Ans. Example 5.2: The daily record of rainfall over a catchment area is 0.2 million m 3 . It has been found that 80% of the rainfall reaches the storage reservoir and then passes over a rectangular weir. What should be the length of the weir, if the water is not to rise more than 1 m above the crest? Assume a suitable value of the coefficient of discharge for the weir. Solution: Given: rainfall = 0.2 x 10 6 m 3 per day discharge into the reservoir: Q = 80% of rainfall Q = 0.8 x 0.2 x 10 6 m 3 /day = 1.85 m 3 /s head of water: H = 1 m Let, L = length of the weir Take: coefficient of discharge: C d = 0.6 Using the relation: 3 2 d 2 Q C .L. 2g.H 3   1.85 = 1.77 L  L = 1.045 m Ans. 5.3. SHARP-CRESTED WEIR 5.3.1. Experiments on sharp-crested rectangular weirs All tests on weirs of this type were made with the nappe fully aerated. When the crest length L of a horizontal weir (see Fig. 5.3) is shorter than the width of the channel b, as well as in the case of V-notch weirs, aeration is automatic. However, for horizontal weirs extending over the full width of the channel, i.e. L/b = 1, air at atmospheric pressure must be provided by vents. Otherwise the air beneath the nappe will be exhausted, causing a reduction of pressure beneath the nappe, with a corresponding increase in the discharge for a given head. Fig. 5.3. Weirs, definition sketch L b crest V 2 V 2g  H P crest 2  2 V 2g  P H crest section through sharp-crested weir horizontal- crested weir L/b < 1 V-notch weir round-crested weir OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 94 The sharp-crested weir is of fundamental interest, because its associated theory forms a basis for the design of spillways. Because the edge is sharp, opportunities for boundary- layer development are limited to the vertical face of the weir, where velocities are low; we may therefore expect the flow to be substantially free from viscous effects and the resultant energy dissipation. Fig. 5.4. The sharp-crested weir Fig. 5.4 shows a longitudinal section of flow over such a weir. An elementary analysis can be made by assuming that the flow does not contract as it passes over the weir (i.e. neglecting streamline curvature), and that the pressure is atmospheric across the whole section AB. Under these assumptions the velocity at any point such as C is equal to 2gh (Henderson, 1966), and the discharge q per unit width accordingly equal to: 2 o 2 o 3 3 V 2 2 2 2 H 2g o o V 2g V V2 2ghdh 2g H 3 2g 2g                                          (5-5) the depth h being measured downwards form the total energy-head line, and not from the upstream water surface. V o is the approach velocity to the weir. The effect of the flow contraction may be expressed by a contraction coefficient C c leading finally to the result: 3 3 2 2 2 2 3 o o 2 c d V V2 2 q C 2g H C 2g.H 3 2g 2g 3                         (5-6) where the discharge coefficient: 3 3 2 2 2 2 o o d c V V C C 1 2gH 2gH                        (5-7) We should expect both C c and the ratio (V o 2 /2gH) to be dependent on the boundary geometry alone, in particular on the ratio H/P; it follows that C d should be a function of H/P alone, which was indeed found by Rehbock (1929); see Eq. (5-2). H h V o 45 A B 2 o V 2g  2 V 2g  p  p  total energy-head line P C 2 V 2g  OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 95 In early experiments on weirs only small quantities of water were available. In most cases results are given in the form of Eq. (5-1), with a discharge coefficient C d .  Tests on weirs of this type were conducted by Kindsvater and Carter (1959). Their tests cover a range of values of H/P from approximately 0.1 to 2.5, a range of heads from 3 cm to 22 cm, and weir heights from 9 cm to 44 cm. They also varied the weir length and the channel width from 3 cm to 82 cm. In presenting their data they adopted the method used by Rehbock of including the effect of H in the main body of the equation. Kindsvater and Carter also introduced a method that includes the effect of the weir length L in the main body of the equation. Their method is shown in the following three equations: 3 2 e e Q C .L .H (5-8) e L L L k  (5-9) e H H H k  (5-10) In these equations k L and k H are factors representing the effects of viscosity and surface tension, and the subscript e indicates effective values, that is, H e is the effective energy- head. By treating the variables in this manner the authors were able to obtain a single linear relationship between C e and H/P for all values of H. The values of k L and k H were obtained by trying successive values of k L and k H until the values of C e were obtained that were the most independent of H and L. They did this not only for their own data, but for several other groups of experiments as well. Their equations for C e , with correcponding values of k H and k L are given hereafter.  The Kindsvater and Carter tests yielded e H C 1.78 0.22 P   (5-11) k H = 0.001 m; k L = - 0.001 m  The Bazin tests yielded e H C 1.80 0.25 P   (5-12) k H = 0.004 m; k L = 0 m  The Schoder and Turner tests yielded e H C 1.77 0.25 P   (5-13) k H = 0.001 m; k L = 0 m OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 96  The US Bureau of Reclamation (USBR) tests yielded e H C 1.78 0.24 P   (5-14) k H = 0.001 m; k L = 0 m 5.3.2. Other types of sharp-crested weirs used for flow measurement Of the many types of weirs developed in the last 100 years, only a few survived and find practical use today. Excluding the suppressed (i.e. without lateral contraction effects) rectangular weir, only the contracted rectangular weir and the triangular weir (see Fig. 5.5) are employed with any frequency. Other types, such as the parabolic, the circular and the compound-form weirs have been used from time to time for special applications.  The contracted rectangular weir This type of weir was subject to considerable experimentation in the USA in the past century. Most notable were the large scale tests by Francis, 1835, and Hamilton Smith, 1884. The Francis experiments were conducted with weirs between 2.44 m to 3.05 m in width, with a crest made of a cast iron plate, 6.3 mm in width and carefully planed and machined in the upstream corner. Francis suggested that the total discharge was diminished with respect to the suppressed weir due to the contractions occurring at the sides. An empirical correction was devised, that decreased the width of the weir by 5% of the head h (see Fig. 5.5) for each lateral contraction. The Francis discharge equation was: d 2 H Q C 1 0.1 bH 2gH 3 b            (5-15) with C d = 0.623 The more modern formulae of Hegly, 1921, and the Swiss Society of Engineers (SIA), 1924, for rectangular contracted weirs reproduce basically the same idea. These formulae are quoted below: Hegly: 2 d 2 b 0.004 H C 0.608 0.045 1 1 0.55 B H (H P)                         (5-16) SIA: 2 2 4 2 d 2 b 3.615 3 b b H B C 0.578 0.037 1 0.5 B 1000H 1.6 B (H P)                                                (5-17) In these relations, b is the width of the weir and B the width of the rectangular channel. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 97 Fig.5.5. The contracted rectangular thin-plate weir (top) and triangular thin-plate weir (bottom) and detail of crest and sides of notch (right) 1 cm to 2 cm  /4 radians upstream face of weir plate B H P b 4 to 5 H max. head measuring section B H P 4 to 5 H max. head measuring section  OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 98  Triangular or V-Notch weir Because the relative error in the measurement of the head becomes important for the smaller heads (say less than 0.05 m) in a rectangular weir, it has been found advantageous to replace this type of weir by triangular weirs or notches, when the discharges to be measured are smaller than about 0.020 m 3 /s. This type of weir was devised by James Thomson in England, 1858. Thomson experimented with 90 notches, but it was found later that other angles were needed for particular applications, in the range from 15 to 120. In this type of weir the crest should be finished as in the rectangular weir, with a sharp upstream corner and a crest width of the order of 1 mm to 3 mm. As the weir nappe does not span the whole width of the channel, it is not necessary to provide aeration ducts beyond the weir plate. For historical reasons (derived from Weisbach’s theory) it has become customary to define a flow equation for a triangular weir with a notch angle  as: 2 8 Q Cd tan H 2gH 15 2         (5-18) In many instances the product (8/15)C d is replaced by a more convenient non-dimensional coefficient m. This coefficient is expected to vary with the fluid properties and the weir geometry in much the same way as in the case of the rectangular weir. The velocity-of- approach effect is, however, much smaller than for rectangular weirs, as the cross-sectional area of the notch is small in comparison with the channel cross-section. This condition would ordinarily apply to weirs designed for accurate flow measurement. On the other hand the surface tension effects are of the same order as in the case of the rectangular weirs. 5.4. THE OVERFLOW SPILLWAY 5.4.1. The spillway crest Normally the crest is shaped so as to conform to the lower surface of the nappe from a sharp-crested weir, as shown in Fig. 5.6. The pressure on the crest will then be atmospheric, provided that the resistance of the solid surface to flow does not induce a material change in the pressure distribution. This could be happen only if the boundary layer over the crest were very thick; and it is known (see Henderson, 1966) that the boundary layer, which will grow effectively only from the neighbourhood of the point A, is in fact a very small fraction of the head over the crest. Therefore we may expect the pressure over the crest to be atmospheric, and in this fact lies the virtue of this crest shape; pressures above atmospheric will reduce the discharge, and pressures below atmospheric will increase the discharge, but at the risk of introducing instability and cavitation. OPEN CHANNEL HYDRAULICS FOR ENGINEERS ----------------------------------------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------- Chapter 5: SPILLWAYS 99 Fig. 5.6. The crest and the equivalent weir of an overflow spillway We consider the case of the high spillway. For the equivalent weir, H/P = 0 and the substitution of the Eq. (5-2) into Eq. (5-1) then leads to the result of Eq. (5-3): 3 2 q 1.80 H [1.80] = m ½ s -1 (5-19) Experiments show that the rise from the weir crest to the high point of the nappe (the spillway crest) is 0.11 H, as in Fig. 5.6. Using this fact we can express Eq. (5-19) in terms of H D , the head over the spillway crest. We obtain Eq. (5-4): 3 2 D q 2.14 H [2.14] = m ½ s -1 (5-20) where H D may be termed the design head; as we have seen, operation at this head will make the pressure over the crest atmospheric. However, the spillway will also have to operate at lower heads, and possibly higher heads as well. The former will evidently result in above-atmospheric pressures on the crest. As to the details of the crest shape, extensive experiments by the U.S. Bureau of Reclamation (USBR) have resulted in the development by the U.S. Army Corps of Engineers of curves which can be described by simple equations, yet approximately close to the nappe profiles measured in the USBR experiments. The profile for a vertical upstream face is shown in Fig. 5.7; others were also developed for various angles of the upstream face to the vertical. H H D 0.11H A C P V o [...]... weir (Fig 5. 11.b) According to Hubert Chanson (1999), model studies suggest that undular weir flow occurs for: For low discharge (i.e q gh 3 min  1 .5 (5- 34) where hmin is the minimum flow depth upstream of the first wave crest Another criterion is: HT  P  0.1 (5- 35) L crest -1 05 Chapter 5: SPILLWAYS OPEN CHANNEL HYDRAULICS FOR ENGINEERS.. . -103 Chapter 5: SPILLWAYS OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 5. 5 BROAD-CRESTED WEIR 5. 5.1 Introduction Sharp-crested weirs have proved to be too fragile to be considered reliable gauging structures in open channels, especially in irrigation channels Thus, other types of structures with a... Theoretical and experimental results for the latter case are shown in Fig 5. 10, which displays the behavior of po, the pressure at O, for angles  (in Fig 5. 9.a) of 45 and 90 It is seen that the free vortex method gives results approximating closely to those of the -102 Chapter 5: SPILLWAYS OPEN CHANNEL HYDRAULICS FOR ENGINEERS ... recently presented an authoritative work on the topic -104 Chapter 5: SPILLWAYS OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 5. 5.2 Broad-crested weir discharge formula Consider the section on the weir crest in case of a critical flow condition At this 2 section, Vc  gh c , where.. .OPEN CHANNEL HYDRAULICS FOR ENGINEERS - 0.282HD origin of coordinates 0.175HD X R= 0.2HD R= 0.5HD X    HD  1. 85 Y  2.0 Y HD Fig 5. 7 Standard spillway crest (US Army Engineers Waterways Experiment Station) 5. 4.2 The spillway face Flow down the steep face of the spillway is normally at about 45 to the horizontal... than V1, and the depth ho greater than h1, so that Eq (5- 21) will not be correct po  V = V1 R1 h1 A R h0 O velocity distribution (a) Spillway toe (b) Flip bucket Fig 5. 9 Flow at the spillway toe, after Henderson and Tierney (1962, 1963) -101 Chapter 5: SPILLWAYS OPEN CHANNEL HYDRAULICS FOR ENGINEERS ... result of Eq (5- 21) However, the latter fails to predict any thickening of the jet and therefore seriously underestimates the total thrust and bending moment on the sidewalls 1.0 R   e  2.718  h1   po 1  1  2  0.8 65 1 e   V12 2 0.9 0.8 complete theory free vortex approximation 0.7 po 1  V12 2 Eq (5- 21) 0.6 0 .5 0.4 experiment ( = 45 )  = 90  = 45 0.3 0.2 0.1 0.0 1 2 3 4 5 6 7 8 9 10... limit of boundary layer C Z  Fig 5. 8 Boundary layer development on spillway face It is now generally agreed that insufflation begins at this very point C, where the boundary layer meets the water surface The resulting mixture of air and water, containing an ever -100 Chapter 5: SPILLWAYS OPEN CHANNEL HYDRAULICS FOR ENGINEERS ... with respect to the weir crest as indicated in Fig 5. 11 The discharge above the weir, equals: 2 q  Vc h c  Vc (H T  P) 3 q or 2 2 3 g  HT  P  3 3 (5- 28) The above equation may be conveniently rewritten as: q  1.7 05  H T  P  3 2 [1.7 05] = m½s-1 (5- 29) These equations are used for ideal fluid flow calculation Notes:  In a horizontal rectangular channel, assuming a hydrostatic pressure distribution,... P  1 .5 L crest HP  0. 05 Lcrest H  P  0.06 m Remarks (4) deduced from laboratory experiments based upon laboratory and field tests HP  1 .5 P P  0. 15 m H P 0. 15 . Eq. (5- 21) experiment ( = 45 )  = 90  = 45 1.0 0.9 0.8 0.7 0.6 0 .5 0.4 0.3 0.2 0.1 0.0 1 2 3 4 5 6 7 8 9 10 11 OPEN CHANNEL HYDRAULICS FOR ENGINEERS. C  H 0.282H D 0.175H D R= 0.2H D R= 0.5H D origin of coordinates X Y 1. 85 D D X Y 2.0 H H        OPEN CHANNEL HYDRAULICS FOR ENGINEERS -----------------------------------------------------------------------------------------------------------------------------------