Open channel hydraulics for engineers. Chapter 5 spillways
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Open channel hydraulics for engineers. Chapter 5 spillways
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.
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Nội dung Text: Open channel hydraulics for engineers. Chapter 5 spillways
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  Chapter SPILLWAYS _________________________________________________________________________ 5.1. Introduction 5.2. General formula 5.3. Sharpcrested weir 5.4. The overflow spillway 5.5. Broadcrested 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. crest face toe 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 Vnotch 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 headdischarge relationship for the structure. Both gradually varied and rapidly varied flows are possible through these structures, but onedimensional methods of analysis usually are sufficient and welldeveloped in this branch of hydraulics. Essential to the  Chapter 5: SPILLWAYS 90
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  “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 specificenergy 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 sharpcrested weirs, but as will be shown later, a similar derivation would apply to weirs that are not sharpcrested. Now, we consider a rectangular weir, over which the water is flowing as shown in Fig. 5.2. water surface dh h H L Fig. 5.2. Rectangular weir Let, H = height of the water above the crest of the weir, L = length of the weir, and Cd = 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:  Chapter 5: SPILLWAYS 91
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  theoretical velocity = 2gh (ii) The discharge per unit width, q, flowing over a weir is generally expressed as: dq Cd area of strip theoretical velocity dq Cd .L.dh. 2gh (iii) The total discharge over the weir may be found by integrating the above equation within the limits 0 and H: 3 H h 2 Q Cd .L.dh. 2gh Cd .L. 2g. H Cd .L. 2g.H 2 3 2 3 3 2 0 o or the discharge per unit width, q, is: q C d 2g.H 2 2 3 (51) 3 where H is to be conceived of as the total upstream specificenergy head on the weir crest supposing that the upstream velocities are negligibly small; Cd is a discharge coefficient, which can be approximated by Rehbock’s experimental formula (1929): H Cd = 0.611 + 0.08 (52) P 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, Cd becomes equal to 0.611. In this case, Eq. (51) can be written as: q 1.80 H [1.80] = m½s1 3 2 (53) 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. (53) in terms of HD, the head over the spillway crest. We obtain: q 2.14 H D [2.14] = m½s1 3 2 (54) where HD 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: Cd = 0.6 Discharge over the weir Q? Using the relation:  Chapter 5: SPILLWAYS 92
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  Q 2 3 Cd .L. 2g.H 2 = 1.31 m3/s Ans. 3 Example 5.2: The daily record of rainfall over a catchment area is 0.2 million m3. 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 106 m3 per day discharge into the reservoir: Q = 80% of rainfall Q = 0.8 x 0.2 x 10 6 m3/day = 1.85 m3/s head of water: H = 1 m Let, L = length of the weir Take: coefficient of discharge: Cd = 0.6 Using the relation: Q Cd .L. 2g.H 2 2 3 3 1.85 = 1.77 L L = 1.045 m Ans. 5.3. SHARPCRESTED WEIR 5.3.1. Experiments on sharpcrested 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 Vnotch 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. b V2 L 2 V2 2g 2g crest H H V P crest crest P section through horizontal Vnotch weir roundcrested sharpcrested weir crested weir weir L/b < 1 Fig. 5.3. Weirs, definition sketch  Chapter 5: SPILLWAYS 93
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  The sharpcrested 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. Vo 2 2g B total energyhead line h H C p V2 Vo 2g A P p V2 2g 45 Fig. 5.4. The sharpcrested 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: H o V2 V 2 2 Vo2 2 3 3 2g 2ghdh H 2g 2 o (55) Vo 2g 2g 2 2g 3 the depth h being measured downwards form the total energyhead line, and not from the upstream water surface. Vo is the approach velocity to the weir. The effect of the flow contraction may be expressed by a contraction coefficient Cc leading finally to the result: V 2 2 3 Vo2 2 2 3 q C c 2g H Cd 2g.H 2 o 3 2 (56) 2g 2g 3 3 Vo2 2 Vo2 2 3 3 where the discharge coefficient: C d Cc 1 (57) 2gH 2gH We should expect both Cc and the ratio (Vo2/2gH) to be dependent on the boundary geometry alone, in particular on the ratio H/P; it follows that Cd should be a function of H/P alone, which was indeed found by Rehbock (1929); see Eq. (52).  Chapter 5: SPILLWAYS 94
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  In early experiments on weirs only small quantities of water were available. In most cases results are given in the form of Eq. (51), with a discharge coefficient Cd. 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: Q Ce .L e .H 3 2 (58) Le L k L (59) He H k H (510) In these equations kL and kH are factors representing the effects of viscosity and surface tension, and the subscript e indicates effective values, that is, He is the effective energy head. By treating the variables in this manner the authors were able to obtain a single linear relationship between Ce and H/P for all values of H. The values of kL and kH were obtained by trying successive values of kL and kH until the values of Ce 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 Ce, with correcponding values of kH and kL are given hereafter. The Kindsvater and Carter tests yielded C e 1.78 0.22 H (511) P kH = 0.001 m; kL =  0.001 m The Bazin tests yielded C e 1.80 0.25 H (512) P kH = 0.004 m; kL = 0 m The Schoder and Turner tests yielded C e 1.77 0.25 H (513) P kH = 0.001 m; kL = 0 m  Chapter 5: SPILLWAYS 95
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  The US Bureau of Reclamation (USBR) tests yielded C e 1.78 0.24 H (514) P kH = 0.001 m; kL = 0 m 5.3.2. Other types of sharpcrested 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 compoundform 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: 2 H Q Cd 1 0.1 bH 2gH (515) 3 b with Cd = 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: b 0.004 H2 Cd 0.608 0.045 1 1 0.55 H (516) B (H P) 2 SIA: b 2 3.615 3 b B 1 0.5 b C d 0.578 0.037 2 4 2 2 H 1000H 1.6 B (H P) (517) B In these relations, b is the width of the weir and B the width of the rectangular channel.  Chapter 5: SPILLWAYS 96
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  B b head measuring H section 4 to 5 H max. P 1 cm to 2 cm /4 radians upstream face of weir plate B head measuring H section 4 to 5 H max. P Fig.5.5. The contracted rectangular thinplate weir (top) and triangular thinplate weir (bottom) and detail of crest and sides of notch (right)  Chapter 5: SPILLWAYS 97
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  Triangular or VNotch 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 m3/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: 8 Q Cd tan H 2 2gH (518) 15 2 In many instances the product (8/15)Cd is replaced by a more convenient nondimensional 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 velocityof approach effect is, however, much smaller than for rectangular weirs, as the crosssectional area of the notch is small in comparison with the channel crosssection. 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 sharpcrested 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.  Chapter 5: SPILLWAYS 98
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  Vo H HD C A 0.11H P 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. (52) into Eq. (51) then leads to the result of Eq. (53): q 1.80 H [1.80] = m½s1 3 2 (519) 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. (519) in terms of HD, the head over the spillway crest. We obtain Eq. (54): q 2.14 H D2 [2.14] = m½s1 3 (520) where HD 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 aboveatmospheric 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.  Chapter 5: SPILLWAYS 99
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  0.282HD origin of coordinates 0.175HD X R= 0.2HD R= 0.5HD X 1.85 2.0 Y HD 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. In this case acceleration and boundary layer development are both taking place during much of the journey down the spillway face, as shown in Fig. 5.8. Turbulence does not become fully developed until the boundary layer fills the whole cross section of the flow, at the point marked C. Downstream of this point the flow might be expected to conform to the S2profile (see Chapter 4) but the extreme steepness of the slope introduces more complications, chiefly the phenomenon of air entrainment, or “insufflation”. H 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  Chapter 5: SPILLWAYS 100
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  increasing proportion of air, continues to accelerate until uniform flow occurs, or the base of the spillway is reached. Clearly the designer will wish to know the velocity reached at the base, or toe, of the spillway, but the above remarks make it clear that the computation of this velocity would be tedious and difficult, even if one were certain of the correct assumptions to adopt concerning the nature of the flow. Considerable work have been done on this problem at the U.S. Bureau of Reclamation. The mechanism of air entrainment is not yet completely understood, and reliable field data on the concentration of entrained air are extremely sparse. 5.4.3. The spillway toe When the flow reaches the end of the inclined face of the spillway it is deflected through a vertical curve into the horizontal or into an upward direction (Fig. 5.9). In the latter case we have the skijump and the buckettype energy dissipators, to be discussed in Chapter 6. In either case, centrifugal pressures will be developed which can set up a severe thrust on the spillway sidewalls. These pressures cannot be accurately calculated by elementary means, but certain approximations suggest themselves; e.g., if one assumed that the depth ho at the center of the curve (Fig. 5.8.a) is equal to the depth h1 of the approaching flow, then the centrifugal pressure at the point O will be equal to: V12 h1 po (521) R where V1 and R are also defined in Fig. 5.8.a. This result can only be an approximation, for a pressure rise along AO must, by the Bernoulli equation, be accompanied by a fall in velocity, so that the velocity profile will be somewhat as shown in Fig. 5.8.a. The average velocity will then be less than V1, and the depth ho greater than h1, so that Eq. (521) will not be correct. V = V1 R1 h1 R A h0 O velocity distribution (a) Spillway toe (b) Flip bucket Fig. 5.9. Flow at the spillway toe, after Henderson and Tierney (1962, 1963)  Chapter 5: SPILLWAYS 101
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  A better approximation can be made by assuming that the streamlines crossing OA form parts of concentric circles, and that the velocity distribution along this line is accordingly the same as that in the free, or irrotational, vortex, i.e., V C (522) r where C is a constant and r is the radius of any streamline. Since the streamlines are concentric circles, r is also a measure of distance along AO from A to O. If R1 is the radius of the streamline at A, then C = V1R1. The discharge q across AO is given by: q V1h1 Vdr V1R 1 r 1dr V1R 1 ln R R R (523) R1 R1 R1 ln h1 R h1 R 1 R i.e., and ln (524) R1 R1 R R R1 Since h1 and R are known in advance, R1 can be obtained by trial from this equation. Given R1/R, we can obtain p o, the pressure at O, from the condition: p o Vo2 V12 1 1 (525) 2 2 V R 2 2 1 o 1 1 po i.e., (526) 1 V12 V1 R 2 assuming no energy dissipation between A and O; this assumption appears to be justified by experiment. The “freevortex” method leads to results that are quite accurate within a certain range, but it suffers from a curious limitation, arising from the fact that the function (lnx)/x has a maximum value of 1/e, which occurs when x = e, the base of the natural logarithms. Applying this result to Eq. (524), we see that the ratio R/h1 has a minimum value of e when R/R1 = e, even though R/h1 is by the nature of the problem an independent variable, which may in practice assume any value at all. The effect of this curious result is that the theory cannot be applied when R/h1 < e, and a curve displaying the results of the theory must, as in Fig. 5.10, terminate at the point where R/h1 = e, although lower values of R/h1 are quite possible. The corresponding terminal value of p o will, from Eq. (526), be equal to 1 2 . 1 1 e 2 V1 2 A complete solution of the problem requires the use of the mathematical theory of irrotational flow. This has been done by Herderson and Tierney (1962, 1963) for the case where there is an open sluice through the spillway, as shown by broken lines in Fig. 5.9.a and for the more usual case discussed above, where the toe is a curved solid surface. 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  Chapter 5: SPILLWAYS 102
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  complete theory when R/h1 > 6, as does the elementary result of Eq. (521). 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 e 2.718 R h1 0.9 po 1 2 0.865 1 1 V12 e 0.8 2 complete theory 0.7 free vortex approximation Eq. (521) 0.6 po experiment ( = 45) V12 1 0.5 2 = 90 0.4 = 45 0.3 0.2 0.1 R 0.0 h1 1 2 3 4 5 6 7 8 9 10 11 Fig.5.10: Flow at spillway toe: Maximum pressure – theory and experiment (after Henderson and Tierney, 1962, 1963) All the above discussions imply the assumption that the flow is irrotational. This assumption is a reasonable one, since losses must be small over the short length of spillway involved, and the highly turbulent approaching flow must have a transverse velocity distribution very close to the uniform distribution which is characteristic of irrotational flow of a perfect fluid. Also, there is little risk of separation, when a solid boundary is continually curving into the flow, as in this case. Pressure distributions should therefore be close to those in the irrotational flow of a perfect fluid, and this conclusion is confirmed by the good agreement between theory and experiment shown in Fig. 5.10. Further, the effect of gravity has been ignored, so that the pressures derived are purely those due to centrifugal action. We take gravity into account simply by adding hydrostatic pressure to the pressure obtained above. This additional pressure may be substantial in the case of the buckettype energy dissipator, in which a structure like that of Fig. 5.9.b. is deeply drowned under a turbulent but stationary eddy.  Chapter 5: SPILLWAYS 103
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  5.5. BROADCRESTED WEIR 5.5.1. Introduction Sharpcrested 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 wider crest were developed. The intention was to prevent damage to the crest section from floating debris, and to shape the structure such that deposition of sediment on the upstream side would not seriously alter the coefficient of discharge of the weir. The structure that resulted from these considerations was termed a broadcrested weir. A broadcrested weir is a flatcrested structure with a crest length large compared with the flow thickness (see Fig. 5.11). total head line HT H hc P Lcrest (a) energyhead line HT H hmin P Lcrest (b) Fig. 5.11: Flow pattern above a broadcrested weir (a) Broadcrested weir flow. (b) Undular weir flow The ratio of crest length to upstream head over the crest must typically be greater than 1.5 to 3 (e.g. Chow, 1973; Henderson, 1966): 1.5 to 3 L crest (527) HT P where Lcrest is the crest length in the flow direction; HT is the upstream total head; and P is the weir height above the channel bed. When the crest is “broad” enough for the flow streamlines to be parallel to the crest, the pressure distribution above the crest is hydrostatic and the critical flow depth is observed on the weir crest. Broadcrested weirs are sometimes used as critical depth meters (i.e. to measure stream discharges). The hydraulic characteristics of broadcrested weirs were studied during the 19th and 20th century. Hager and Schwalt (1994) recently presented an authoritative work on the topic.  Chapter 5: SPILLWAYS 104
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  5.5.2. Broadcrested weir discharge formula Consider the section on the weir crest in case of a critical flow condition At this section, Vc gh c , where hc = (HT – P) is the critical depth and (HT –P) the specific 2 3 energy head at the section with respect to the weir crest as indicated in Fig. 5.11. The discharge above the weir, equals: q Vc .h c Vc (H T P) 2 3 q g HT P 2 2 3 or (528) 3 3 The above equation may be conveniently rewritten as: q 1.705 H T P [1.705] = m½s1 3 2 (529) These equations are used for ideal fluid flow calculation. Notes: In a horizontal rectangular channel, assuming a hydrostatic pressure distribution, the critical flow depth equals: hc E 2 (530) 3 where E is the specificenergy head. The critical depth and the discharge per unit width are related by: q2 hc 3 (531) g q gh 3 c (532) At the crest of a broadcrested weir, the continuity equation and the Bernoulli equation yield: HT P h c 2 (533) 3 5.5.3. Undular weir flow and discharge coefficients Undular weir flow HP For low discharge (i.e. 1 ), several researchers observed free surface P undulations above the crest of a broadcrested weir (Fig. 5.11.b). According to Hubert Chanson (1999), model studies suggest that undular weir flow occurs for: 1.5 q (534) gh 3 min where hmin is the minimum flow depth upstream of the first wave crest. Another criterion is: HT P 0.1 (535) L crest  Chapter 5: SPILLWAYS 105
 OPEN CHANNEL HYDRAULICS FOR ENGINEERS  This equation is a practical criterion based on the ratio of energy head on crest to weir length. In practice, design engineers should avoid flow conditions leading to undular weir flow. In the presence of free surface undulations above the crest, the weir cannot be used as a discharge meter, and waves may propagate in the downstream channel. Discharge coefficients Experiment measurements indicate that the discharge versus the total head relationship departs slightly from the equation for ideal fluid flow: 2 3 q g HT P 2 (536) 3 For a real flow rate it holds: 2 3 q Cd g H T P 2 (537) 3 The above equation is usually rearranged as: q Cd g HT P 2 2 3 (538) 3 3 where the discharge coefficient Cd is a function of the weir height, the crest length, the crest width, the upstream corner shape and the upstream total head, as given Table 5.1 below: Table 5.1. Discharge coefficient for broadcrested weir Reference Discharge coefficient Cd Range Remarks (1) (2) (3) (4) HT P 0.1 1.5 Sharp corner deduced from weir 2 laboratory 9 L crest Hager and 0.85 1 9 4 experiments 1 HT P Schwalt 7 (1984) L crest HP 0.05 Rounde corner based upon weir L crest r L crest r laboratory and 1 0.01 1 0.01 Lcrest P HP field tests H P 0.06 m Bos (1976) HP 1.5 P Ackers et al. 0.95 * P 0.15 m (1978) H P 0.15< T 0.6 P Notes: * Reanalysis of experimental data presented by Ackers et al. (1978); r = curvature radius of upstream corner.  Chapter 5: SPILLWAYS 106
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