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**Chapter 2: Design of Overflow Structures**

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**2.1 Overflow Structures 2.1.1 Overflow Gates:**

The overflow structure has a hydraulic behavior that the discharge increases significantly with the head on the overflow crest. Remember that Q=CLH3/2. The height of the overflow is usually a small portion of the dam height. Further, gates may be positioned on the crest for “overflow regulation”. During the floods, if the reservoir is full, the gates are completely open to promote the overflow. A large number of reservoirs with a relatively small design discharges are ungated.

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**Bottom pressure profile**

Gated Ungated Bottom pressure profile

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**Currently most large dams are equipped with gates to allow for a flexible operation.**

The cost of the gates increases mainly the magnitude of the flood, i.e.: with the overflow area. Improper operation and malfunction of the gates is the major concern which may lead to serious overtopping of the dam. In order to inhibit floods in the tailwater, gates are to moved according to gate regulation. Gates should be checked against vibrations.

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**The Advantages and Disadvantages of Gates**

The advantages of gates at overflow structure are: Variation of reservoir level, Flood control, Benefit from higher storage level. The disadvantages are: Potential danger of malfunction, Additional cost, and maintenance. Depending on the size of the dam and its location, one would prefer the gates for: Large dams, Large floods, and Easy access for gate operation.

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**Flap Gate Vertical Gate Radial Gate Hinged flap gates,**

Three types of gates are currently favored: Hinged flap gates, Vertical lift gates, Radial gates. Flap Gate Vertical Gate Radial Gate

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**The flaps are used for a small head of some meters, and may span over a considerable length.**

The vertical gate can be very high but requires substantial slots, a heavy lifting device, and unappealing superstructure. The radial gates are most frequently used for medium or large overflow structures because of their simple construction, the modest force required for operation and absence of gate slots. They may be up to 20m X 20m, or also 12 m high and 40 m wide. The radial gate is limited by the strength of the trunnion bearings.

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The risk of gate jamming in seismic sites is relatively small, if setting the gate inside a stiff one-piece frame. For safety reasons, there should be a number of moderately sized gates rather than a few large gates. For the overflow design, it is customary to assume that the largest gate is out of operation. The regulation is ensured by hoist or by hydraulic jacks driven by electric motors. Stand-by diesel-electric generators should be provided if power failures are likely.

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Overflow Types Depending on the site conditions and hydraulic particularities an overflow structure can be of various designs: Frontal overflow, Side-channel overflow, and Shaft overflow. Other types of structures such s labyrinth spillway use a frontal overflow but with a crest consisting of successive triangles or trapezoids in plan view. Still another type is the orifice spillway in the arch dam.

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**Main Types of Overflow Structures**

Frontal Overflow Side Overflow Shaft Overflow

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The non-frontal overflow type of spillways are used for small and intermediate discharges, typically up to design floods of 1000 m3/s. The shaft type spillway was developed in 1930’s and has proved to be especially economical, provided the diversion tunnel can be used as a tailrace. The structure consists of three main elements: The intake, The vertical shaft with a 90o bend, and The almost horizontal spillway tunnel. Air by aeration conduits is provided in order to prevent cavitation damage at the transition between shaft and tunnel. Also, to account for flood safety, only non-submerged flow is allowed such that free surface flow occurs along the entire structure, from the intake to the dissipator. The hydraulic capacity of both the shaft and the tunnel is thus larger than that of the intake structure. The system intake-shaft is also referred to as “morning glory overflow” due to similarity with a flower having a cup shape.

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**Morning Glory Spillway**

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The Monticello Dam

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**Morning Glory Spillway**

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**The side channel overflow was successively used at**

the Hoover dam (USA) in the late 1930’s. The arrangement is advantageous at locations where a frontal overflow is not feasible, such as earth dams, or when a different location at the dam site yields a better and simpler connection to the stilling basin. Side channels consist of a frontal type of overflow structure and a spillway with axis parallel to the overflow crest. The specific discharge of overflow structure is normally limited to 10 m3/s/m, but for lengths of over 100 m. This type of overflow is restricted to small and medium discharges.

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**Side-Channel Spillway**

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The frontal type of overflow is a standard overflow structure, both due to simplicity and direct connection of reservoir to tailwater. It can normally be used in both arch and gravity dams. Also, earth dams and frontal overflows can be combined, with particular attention against overtopping. (eg: Hasan Ugurlu Dam) The frontal overflow can easily be extended with gates and piers to regulate the reservoir level, and to improve the approach flow to spillway. Gated overflows of 20 m gate height and more have been constructed, with a capacity of 200 m3/s per unit width. Such overflows are thus suited for medium and large dams, with large floods to be conveyedto the tailwater. Particular attention has to be paid to cavitation due to immense heads that may generate pressure below the vapor pressure in the crest domain. Also, the gate piers have to be carefully shaped in order to obtain a symmetric approach flow. The downstream of frontal overflow may have various shapes. Usually, a spillway is connected to the overfall crest as a transition between overflow and energy dissipator.

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Also, the crest may abruptly end in arch dams to include a falling nappe that impinges on the tailwater. Another design uses a cascade spillway to dissipate energy right away from the crest end to the tailwater, such that a reduced stilling basin is needed. The standard design involves a smooth spillway that convey flow with a high velocity either directly to the stilling basin, or to a trajectory bucket where it is ejected in the air to promote jet dispersion and reduce the impact action.

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**Significance of Overflow Structure**

According to ICOLD* (1967),the overflow structure and the design discharge have a strong impact on the dam safety. Scale models of overflow structures are currently needed in cases where: The valley is narrow and approach velocity is large such that asymmetric flow pattern develops. The overflow and pier geometry is not of standard shape, and Structures at either side of the overflow may disturb the spilling process. For all other cases, the design of the overflow is so much standardized that no model study is needed, except for details departing from the recommendations. * ICOLD= International Commisson for Large Dams, Paris

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**2.2 Design Discharge of Spillway**

Concept of Crest Height A spillway is a safety structure against overflow. It should inhibit the overflow of water at locations which were not considered. The spillway is the main element for overflow safety and especially the safety against dam overtopping. A structure is known to be safe against damage if the load is smaller than the resistance. What are the determining loads, and the resulting resistance with respect to the overflow safety? Are these Crest heights Discharges Water volumes ??

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**In concept of crest height, the load is composed of the sum of:**

Initial depth, h A depth increase, r, due to a flood A wave depth, b load=h+r+b …… (1) Accordingly, the resistance of the crest height, k, (i.e.: the lowest point) may be defined as: (h+r+b)-k ≤ ……………………………. (2) Overtopping occurs provided that (h+r+b)-k >0. The corresponding probability of overtopping is the complementary value of the security, and also coupled to the probabilities of the parameters h, r, b, and k.

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**Dam with reservoir Probability of overtopping, p**

Initial depth, h, before flood flow, Depth, r, of flood level Wave height, b Height of crest, k Maximum reservoir depth, hmax Maximum run-up height, bmax freeboard, f

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The probability of the initial depth h follows from the policy of reservoir operation. For an existing dam, it can be obtained from reservoir level records. For a future dam, it must be estimated from the planned policy. The probability of the flood depth increase, r, can be determined from flood routing. The result depends on the probabilities of the approach flood and reservoir outflow. The probability of the approach flood depends on general flood parameters such as: Maximum discharge, Time to peak, Time of flood, and Flood volume. Regarding the reservoir outflow, the degree of aperture of the outflow structure has to be accounted for.

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**The probability of wave depth b, depends mainly on:**

The character of the wind,its direction, and The run-up slope of the dam relative to the water surface. In particular cases, ship waves and surge waves due to shore instabilities ( including rock and snow avalanches, and land slides) are also included. normally, the parameter k is considered as a fixed number and not a stochastic value, which is an acceptable assumption for concrete dams. For high earth dams, the probability of settlement could be introduced, but this is often omitted and a maximum settlement is accounted for. The probability of combination of the parameters h, b, r, and k thus mainly results in a variation of six basic variables: Initial flow depth, h Peak flood discharge, Time to peak, Aperture degree velocity and direction of wind Assuming that these basic variables are stochastically independent, the result is straight forward.

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**s= number of combinations of (h+r+b-k) **

As an example, one could apply the Monte Carlo Method* to obtain a representative number of combinations, that is let s= number of combinations of (h+r+b-k) n= number of those sums with s≤0, Then the safety q against overflow is: q=n/s …………………………………(3) If m=s-n is the number of combinations where s>0, then the probability of overflow is: p=m/s……………………………………(4) The concept described is obvious, as it accounts for the relevant parameters: depth of reservoir, and depth of crest. It can be used for sensitivity analysis and answer questions such as: What is the change of overtopping safety for changes in reservoir operation? (relating to parameter h) What is the effect of time to peak for a given peak discharge? (relating to parameter r) (*Monte Carlo method is a risk and decision analysis tool)

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**A set of stochastically independent parameters can normally not be assumed:**

In certain regions, large floods are combined with thunderstorms and flood waves are thus related to wind waves. In other regions, the floods have their origin, far upstream from the reservoir and accompaniying winds may hardly reach the dam. Normally, dams are erected to store water during the rain period for the dry season. Accordingly, the reservoirs have usually reached a high level at the end of the rain period. The determining floods occur often at the end of the rain period, such that the initial depth is stochastically related to the flood depth.

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The design of spillways is often based on a fixed value of kfix for the crest height. Further, the wave run-up height is determined more or less independently from the season and a maximum value of bmax of undetermined probability is considered. Then the modified equation will become: h+r+bmax-kfix ≤ (5) Further, it is assumed that the initial reservoir level is equal to the maximum reservoir level, i.e.: at the maximum reservoir height, hmax. Therefore above equation will be: hmax +r+bmax-kfix ≤ (6) The only remaining stochastic parameter is thus r. It is impossible to determine the security against overflow along this model, because extreme values and stochastic values can not simply be superposed. Another popular approach uses instead of parameters kfix and hmax, the free board height f = kfix-hmax, and requires that: r ≤ (f-hmax)

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**2.2.2 Concept of Water Volumes**

The heights h, r, b, and k are related to particular volumes of reservoir such as Vh= resrvoir volume at initial reservoir level, Vr= flood storage volume, Vb= wave run-up volume, Vk= maximum reservoir volume up to the dam crest level. (h+r+b-k) ≤ 0 equation may be written analogously with regard to volume V as: (Vh+Vr+Vb-Vk) ≤ (9) And all concepts presented earlier can similtaneously be transposed. Again the result would be the same: the security against overtopping can not be predicted by using stochastic parameters which are not indepent to each other.

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Concept of Discharges The effectively needed storage volume Vr can be determined from the mass balance as: Where Qa= reservoir inflow during the filling time T, Qz= reservoir outflow during the same time T, Qr= reservoir storage, The relation with the reservoir height h is: Where A=A(h) is the reservoir surface. The filling time ends when Qz=Qa, i.e.: when Qr=0

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**Therefore it can be written that:**

Vr ≤ VR Needed storage volume ≤ Available storage volume: or Vz-Va ≤ VR Where Vz= reservoir inflow volume during the filling time T, Va= corresponding reservoir outflow volume In fıgure below these relations are explained

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**Qz Qr Vz Vr Qa r Va Q V Qa Qz a) Hydrograph of inflow flood, Qz(t)**

b) Hydrograph of reservoir outflow, Qa(t) c) Hydrograph of superposition and required storage volume Vr until the end of time rise T, d) Hydrograph of reservoir storage, Qr(t) e) Hydrograph of increase of storage, r(t) Maximum of functions V Qa Qz

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**a) Hydrograph of inflow flood, Qz(t)**

b) Hydrograph of reservoir outflow, Qa(t) c) Hydrograph of superposition and required storage volume Vr until the end of time rise T, d) Hydrograph of reservoir storage, Qr(t) e) Hydrograph of increase of storage, r(t) Maximum of functions

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The probabilities of (Vz-Va), and VR may be used to determine the securities of overflow and relating probabilities. These computations become again useless if fixed values instead of stochastic values are admitted. A popular example involves the so-called (N-1) condition for reservoir outflow. Instead of relating the availability of a regulated outflow to probability, one is typically faced with a situation such as: out of the N outlets, there is one ( and the one with largest capacity ) not available.

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Other examples, such as fixing the maximum reservoir depth hmax as the initial depth for floods have already been mentioned. The most important parameter is the flood wave, such as shown in figure below, and characterized with the peak discharge Qzmax and time to peak tz. The temporal wave profile is given as an empirical function Qz(t). The determining flood for reservoir volume and spillway structure is called design flood.

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2.2.4 Design assumption Numerous dam failures due to overtopping point to the significance of the design flood. Its resulting probability of failure has to be minimum, at least infinitely small. The lifespan of a usual dam is of the order of 100 years. Therefore, the probability of failure has to be related also to this period. If a value of 1%, or 0.1 % for the entire lifespan is assumed, then the probability of failure per year is 10-4, and 10-5, respectively. For a large potential of damage, one would choose one or two orders of magnitude smaller . How can such small values of probability be guaranteed? It is not the purpose here to outline the corresponding philosophies of each country worldwide.

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**Computational or Intuitive Approach**

The probability of failure can be computed by combining the parameters mentioned with their probability of occurrence such that the probability of overtopping at the lowest point of the dam crest can be determined. The difficulty of the procedure is in the estimation of probabilities of some parameters. For dams, which have existed for several decades and whose safety against overflow is permanently checked, the wave run up has eventually been observed and the availability of the outflow structures is known. Also, information regarding the height of initial flow depth before floods is probably available.

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**For new dams, estimations have to be advanced, however**

For new dams, estimations have to be advanced, however. It was discussed that some parameters under consideration are stochastically dependent, which adds further complications. The probability of failure cannot be determined if some stochastic parameters are assumed to have a certain probability, and others are considered as fix values. The latter correspond normally to intuitively chosen maxima. Such a mixed approach is currently the common approach, however, and there is nothing to counter as long as no probability computation is performed. Normally, the mixed approach involves a rare design flood, for which considerations of probability are still appropriate.

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**As an example, a 1000-year flood is chosen**

As an example, a 1000-year flood is chosen. Then, intuitive security factor on parameters such as the initial reservoir outlets are introduced. As mentioned earlier, the maximum reservoir elevation is often set equal to maximum reservoir level and the (N-1) condition is added in relation to reservoir outlets Further, an immobile power plant is considered, that is the related power plant, the water supply station, or irrigation works are switched off, and the free board is chosen higher than the maximum wave run-up With these additional safety measures, a probability of overtopping well below 0.1 % within 100 years may be achieved, i.e. a value of practically zero.

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Design flood The design flood is a reservoir inflow of extremely small probability, of 1000 or even years of occurrence To estimate these rare values, a data series is evidently not available. There are conventions, however, by which extrapolations can be made based on a data series of several decades. These extrapolations of the reservoir inflow discharge, or the rainfalls are difficult to interpret. They include knowledge of local particularities, and a detailed hydrological approach The times when some flood discharge formula has been applied without particular reference to a catchments area have definitely passed. As an engineer would hardly transpose the geology of one dam site to the other, it is impossible to use hydrologic data to cases other than considered.

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Actually, two different design cases are used in many countries, considering a smaller and a larger design flood. The smaller design flood has a return period of the order of 100 years. It must be received and diverted by the reservoir without damage. Often, a full reservoir level is assumed and all intakes for power plants etc. are blocked, and (N-1) spillway outlets are in operation. Whether the bottom outlet can be accounted for diversion is a question, but there is a tendency to include it in the approach. The freeboard is specified and must be observed. For the larger design flood, a return period of years is considered, for example As such an extra ordinary event can be extrapolated from the limited data available only as a rough estimation, other conventions are used.

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**One approach increases the 1000 year flood by 50% both in peak discharge and time to peak.**

Another approach is based on the concept of the possible maximum flood (PMF). Accordingly, a rainfall-runoff model with the most extreme combination of basic parameters is chosen, and no return period is specified. This design flood has to be diverted without a dam breaching. However, small damages at the dam and surroundings may occur. Therefore, the conditions for wave run-ups and the availability of (N-1) outlets, among others, are not entirely satisfied. In some countries, the definition of the design flood is also related to the potential damage due to a dambreak. As prediction of such a potential for the next century is difficult, hardly any figures are given. A usual compromise is to increase the design quantity if high dams and large reservoir volumes are involved.

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**Design Flood of Spillway Structure**

The design flood Qdmax of the spillway may be determined from Equation: that is from the inflow design flood, and includes the described effects of initial reservoir depth, reservoir freeboard, and availability of outflow structures. Where Qa= reservoir inflow during the filling time T, Qz= reservoir outflow during thesame time T, Va= available storage volume

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**2.3 Frontal Overflow 2.3.1 Crest Shapes Straight (standard) Curved**

Overflow structures of different shapes are: Straight (standard) Curved Polygonal Labyrinth The labyrinth structure has an increased overflow capacity with respect to the width of the structure. Plan view

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Labyrinth spillway

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2.3.2 Standard Crest Shape When the flow over a structure involves curved streamlines with the origin of curvature below the flow, the gravity component of a fluid element is reduced by the centrifugal force. If the curvature is sufficiently large, the internal pressure may drop below the atmospheric pressure and even attain values below the vapor pressure for large structures. Then cavitation may occur with a potential cavitation damage. As discussed, the overflow structure is very important for the dam safety. Therefore, such conditions are unacceptable. For medium and large overflow structures, the crest is shaped so as to conform the lower surface of the nappe from a sharp-crested weir.

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**The transverse section of an overflow structure may be rectangular, trapezoidal, or triangular.**

In order to have a symmetric downstream flow, and to accommodate gates, the rectangular cross section is used almost throughout. The longitudinal section of the overflow can be made either; Broad-crested. Circular crested, or Standard crest shape (ogee-type)

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**For heads larger than 3 m, the standard overflow shape should be used.**

Although its cost is higher than the other crest shapes, advantages result both in capacity and safety against cavitation damage.

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**The crest shape should be knife sharp, with a 2 mm horizontal crest, and 45o downstream bevelling.**

In order to inhibit the scale effects due to viscosity and surface tension, the head on the weir should be: H≥ 100 mm, and the height of the weir, W ≥ 2Hmax Then, the effects of approach velocity are insignificant. H W 45o 2 mm

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**Flow over a sharp-crested weir**

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**Flow over a sharp-crested weir**

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In order to inhibit the scale effects due to viscosity and surface tension, the head on the weir should be: H≥ 100 mm, and the height of the weir, W ≥ 2Hmax Then, the effects of approach velocity are insignificant. The shape of the crest is important regarding the bottom pressure distribution. Slight modifications have a significant effect on the bottom pressure, while the discharge characteristics remain practically the same. The geometry of the lower nappe cannot simply be expressed analytically. The best known approximation is due to US Corps of Engineers (USCE1970). They proposed a three arc profile for the upstream quadrant and a power function for the downstream quadrant, with the crest as origin of Cartesian coordinates (x,z).

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USCE Crest Shape

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The significant scaling length for standard overflow structure is the so-called design head, HD.. All other lengths may be nondimensionalized with HD. The radii of the upstream crest profile are: The origins of curvature O1,O2, and O3, as well as the transition points P1,P2, and P3, for the upstream quadrant are; Point O1 O2 O3 P1 P2 P3 x/HD 0.00 -0.105 -0.242 -0.175 -0.276 z/HD 0.500 0.219 0.136 0.032 0.115

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**The downstream quadrant crest shape was originally proposed by Craeger as:**

This shape is used up to so-called tangency point with a transition to the straight-crested spillway. The disadvantage of USCE crest shape is the abrupt change of curvature at locations P1 to P3 and at the origin. Such a crest geometry can not be used for computational approaches due to the curvature discontinuities. An alternative approach with a smooth curvature was provided by Hager:

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**Where (X*,Z*) are transformed coordinates based on USCE shape as:**

X*=1.3055(x’ ) Z*=2.7050(z’ ) with x’=x/HD, and z’=z/HD This equation has the property that the second derivative is The inverse curvature varies linearly with x*. For design purposes, the difference between the two crest geometries are usually negligible.

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The crest shape given above for vertical spillways for which the velocity of approach is zero, i.e.; for HD/P→0, where P is the height of the spillway. In general, the shape of the crest depends on: The design head HD, The inclination of the upstream face, The height of the overflow section above the floor of the entrance channel (which influences the velocity of approach to the crest). The crest shapes have been studied extensively in the Bureau of Reclamation Hydraulic Laboratories. For most conditions, the data can be summarized as:

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**Elements of Nappe-Shaped Crest Profile**

xc yc x y HD h0 R2 R1 ha

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The portion upstream from the origin is defined as either a single curve and a tangent, or as a compound circular curve. The portion downstream defined by Where K and n are constants whose values depend on the upstream inclination and on the velocity approach head, ha.

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Values of K

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Values of n

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Values of xc

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Values of yc

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Values of R1 and R2

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**2.3.3 Discharge Characteristics**

The discharge over an ogee crest is given by the formula: Where: Q=discharge, C=discharge coefficient, L=effective length of crest, He=total head on the crest, including the velocity of approach head, ha. The discharge coefficient, C, is influenced by a number of factors: The depth of approach, Relation of actual crest shape to the ideal nappe shape, Upstream face slope, Downstream apron interface, Downstream submergence.

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**Pier and Abutment Effects**

Where crest piers and abutments are shaped to cause side contractions of the overflow, the effective length, L, will be less than the net length of the crest. The effect of end contractions may be taken into account by reducing the net crest length as follows: L=L’-2(NKP+Ka)He Where: L= effective length of crest, L’= net length of crest, N= number of piers, KP= pier contraction coefficient, Ka= abutment contraction coefficient, He= Existing total head on the crest.

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**2.3.4 Coefficient of Discharge for Ogee Crest**

i) The effect of depth of approach For a high sharp-crested weir placed in a channel, the velocity of approach is small and the underside of the nappe flowing over the weir attains the maximum contraction. As the approach depth (p+ho) decreaesed, the velocity of approach increases, and the vertical contraction diminishes. If the sharp-crested weir coefficients are related to the head measured from the point of maximum contraction instead of to the head above the sharp crest, coefficients applicable to ogee crests can be established. For an ideal nappe shape, i.e.:

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**In the text book, the discharge coefficient C is given as function of x=H/HD only:**

For x→0, the overflow is shallow and almost hydrostatic pressure distribution occurs. Then the overflow depth is equal to critical depth and the discharge coefficient C= 2/3√3= For design flow X=1, and Cd=0.495.

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The relationship of the ogee crest coefficient, C,, to various values of P/H, is shown on Fig. 1. These coefficients are valid only when the ogee is formed to the ideal nappe shape; that is, when He/H0 = 1.

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**Fig.1 Discharge Coefficients for vertical-faced Ogee Crest**

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**Effect of Heads Different from Design Head**

When the ogee crest shape is different from the ideal shape or when the crest has been shaped for a head larger or smaller than the one under consideration, the discharge coefficient will differ from that shown on Fig. 1. A wider shape will result in positive pressures along the crest contact surface, thereby reducing the discharge. With a narrower crest shape, negative pressures along the contact surface will occur, resulting in an increased discharge. Fig. 2 shows the variation of the coefficient as related to values of He/H0, where He, is the actual head being considered. An approximate discharge coefficient for an irregularly shaped crest whose profile has not been formed according to the undernappe of the overflow jet can be estimated by finding the ideal shape that most nearly matches it. The design head, HO, corresponding to the matching shape can then be used as a basis for determining the coefficients . The coefficients for partial heads on the crest, for preparing a discharge-head relationship, can be determined from Fig. 2.

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**Fig.2 Discharge Coefficients for other than the design head**

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**Effect of Upstream Face Slope**

For small ratios of the approach depth to the head on the crest, sloping the upstream face of the overflow results in an increase in the discharge coefficient. For large ratios the effect is a decrease in the coefficient. Within the range considered in this text, the discharge coefficient is reduced for large ratios of P/H, only for relatively flat upstream slopes. Fig. 3 shows the ratio for the coefficient for an overflow ogee crest with a sloping (inclined) face, Ci, to the coefficient for a crest with a vertical upstream face, Cv, as obtained from Fig. 1 (and as adjusted by Fig. 2 if appropriate), as related tovalues of P/H0,.

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**Fig.3 Discharge Coefficients for ogee-shaped crest with sloping upstream face**

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**Effect of Downstream Apron Interference and Downstream Submergence**

When the water level below an overflow weir is high enough to affect the discharge, the weir is said to be submerged. The vertical distance from the crest of the overflow to the downstream apron and the depth of flow in the downstream channel, as it relates to the head pool level, are factors that alter the discharge coefficient. Five distinct characteristic flows can occur below an overflow crest, depending on the relative positions of the apron and the downstream water surface: (1) flow can continue at supercritical stage; (2) a partial or incomplete hydraulic jump can occur immediately downstream from the crest;

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**(3) a true hydraulic jump can occur;**

(4) a drowned jump can occur in which the high-velocity jet will follow the face of the overflow and then continue in an erratic and fluctuating path for a considerable distance under and through the slower water; and (5) no jump may occur-the jet will break away from the face of the overflow and ride along the surface for a short distance and then erratically intermingle with the slow moving water underneath. Fig. 4 shows the relationship of the floor positions and downstream submergences that produce these distinctive flows. Where the downstream flow is at supercritical stage or where the hydraulic jump occurs, the decrease in the discharge coefficient is principally caused by the back-pressure effect of the downstream apron and is independent of any submergence effect from the tailwater.

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**Fig.4 Effect of downstream influences on the flow over the weir crest**

Subcritical flow Supercritical flow Downstream depths where jump occur Downstream depth insufficient to form a good jump depths sufficient to form a good jump depths excessive to form a good jump Drown jump with diving jet No jump jet on surface

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Fig.5 shows the effect of downstream apron conditions on the discharge coefficient. It should be noted that this curve plots, in a slightly different form, the same data represented by the vertical dashed lines on Fig.4. As the downstream apron level nears the crest of the overflow, (hd + d)/H, approaches 1.0, and the discharge coefficient is about 77 percent of the coefficient for unretarded flow. On the basis of a coefficient of 4.0 for unretarded flow over a high weir, the coefficient when the weir is submerged will be about 3.08, which is virtually the coefficient for a broad-crested weir. From Fig.4, it can be seen that when (hd + d)/H, exceeds about 1.7, the downstream floor position has little effect on the coefficient, but there is a decrease in the coefficient caused by tailwater submergence.

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**Fig.5 Ratio of discharge coefficients resulting from apron effects**

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Fig.6 shows the ratio of the discharge coefficient where affected by tailwater conditions to the coefficient for free flow conditions. This curve plots, in a slightly different form, the data represented by the horizontal dashed lines on Fig.4. Where the dashed lines on Fig.4 are curved, the decrease in the coefficient is the result of a combination of tailwater effects and downstream apron position.

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**Fig.6 Ratio of discharge coefficients caused by tailwater effects**

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**2.3.5 Uncontrolled Ogee Crest Design Example on Design of a Spillway Crest**

Design an uncontrolled overflow spillway crest, to discharge 56 m3/s at 1.5-meter head. The upstream face of the crest is sloped 1:1, and the entrance channel is 30 m. long. A bridge is to span the crest, and 50 cm- wide bridge piers with rounded noses are to be provided. The bridge spans are not to exceed 6m. The abutment walls are rounded to a 1.5 m radius, and approach walls are to be placed at 30o with the centerline of the spillway entrance.

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Procedure 1: First assume the position of the approacah anad downstream apron level with respect to crest level, say 0.60 m below the crest level, i.e: Let P=0.60 m Then He+P≈ 2.1 m approximately To evaluate the approach channel losses, assume a value of C to obtain an approximate approach velocity:

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**In order to compute the frictional losses, we can use the Manning formula:**

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**Assuming an entrance loss into the approach channel equal to =0**

Assuming an entrance loss into the approach channel equal to =0.1ha, the total head loss in the approach channel is approximately: The effective head H0 is equal to: H0= =1.46 m. From Fig.1, C0=2.08 Fig.3 is used to correct the discharge coefficient for the inclined upstream slope:

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Next the relationships of (hd+d)/He and hd/He are evaluated to determine the downstream effects. The value of (hd+d)≈H0+P= =2.06 If supercritical flow prevails, hd should be equal to:

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**With the indicated unit discharge q=3**

With the indicated unit discharge q=3.803 m3/s/m, the downstream velocity will be approximately: The closeness of the values of hd and hv, verifies that flow is supercritical . From Fig.252 it can be seen that the downstream effect is due to apron influences only and the corrections shown in Fig. 5 will apply.

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**To evaluate the effecet of submergence: C=2**

To evaluate the effecet of submergence: C=2.05 This coefficient has now been corrected for all influencing effects. The next step is to determine the required crest length. For the design head, H0, of 1.46 m, the required effective crest length, L, is equal to:

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To correct for the pier and abutment effects, the net length is: If the bridge spans are not to exceed 6 m, two piers will be required for the approximate 16 m total span and N will be equal to 2. Then: This procedure establishes a coefficient of discharge for the design head. For computing a rating curve , Q v.s H, coefficient for lesser heads must be obtained. Since variations of the different corrections are not consistent, the procedure for correcting the coefficients must be repeated for each lesser head.

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Procedure 2: First assume an overall coefficient of discharge, say The discharge per unit length, q, is then equal to: Then the required effective length of crest ,L, is equal to: Next, the approach depth is approximated by use of Fig.1

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**Thus the approach depth P can not be less than 0. 39 m**

Thus the approach depth P can not be less than 0.39 m. To allow for other factors which may reduce the coefficient, an approach depth of about 0.60 m might be reasonably assumed. With P=0.60 m the computation for approach losseswill be the same as in procedure 1 solution, and the effective head H0 will become 1.46 m. Similarly, the value of Cinc=2.12. Since, the overall coefficient of 2.00 was assumed for 1.5 m gross head, the corresponding coefficient for 1.46 m effective head is:

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**The submergence ratio:**

Therefore the downstream apron should be placed =0.71 m below the crest level. Since it was demonstrated previously that the pier and contraction effects are small, they can be neglected in this example, and the crest length is therefore m. This crest length and the downstream apron position can be varied by altering the assumptions of overall coefficient, and approach depth.

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Crest Shape:

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**Therefore ha=0.198 m. ( h+P) (m) ha (m) 2.06 0.162 1.898 0.19 1.87**

0.196 1.864 0.1976 1.862 0.198 1.862 0.198 Therefore ha=0.198 m.

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**Now we can determine the crest shape:**

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**Discharge rating curve**

He/H0 He C/C0 Ci Hd+d Hd+d/He Cs/C Cs q He+P Va ha Sf hm hl HG Q 0.1 0.146 0.82 1.74 0.746 5.11 1.00 0.097 0.13 0.001 0.00 0.0 0.15 1.50 0.2 0.292 0.85 1.802 0.892 3.05 0.284 0.32 0.0052 0.0001 0.29 4.41 0.4 0.584 0.90 1.908 1.184 2.03 0.852 0.72 0.0264 0.0002 0.01 0.59 13.20 0.6 0.876 0.94 1.993 1.476 1.68 1.634 1.11 0.0625 0.0004 0.89 25.33 0.8 1.168 0.97 2.06 1.768 1.51 0.982 2.023 2.554 1.44 0.1063 0.0005 1.18 39.58 1.0 1.46 2.12 1.41 0.966 2.05 3.613 1.75 0.1568 0.0006 0.02 1.48 56.00 1.2 1.752 1.03 2.184 2.352 1.34 0.95 2.07 4.811 0.2133 0.0007 1.77 74.58

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**2.3.6 Bottom pressure characteristics**

The bottom pressure distribution Pb(x) is important, because it yields: an index for the potential danger of cavitation damage, and the location where piers can end without inducing separation of flow. The bottom pressure head Pb/g nondimensionalized by the design head HD as a function of location X=x/HD for various c=H/HD is shown in figure below. The minimum pressure Pm occurs on the upstream quadrant. The most severe pressure minima along the piers due to significant streamline curvature effects.

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**Bottom pressure distribution**

Free surface profile Plane flow Between piers Pb/g Axial between piers along piers H/HD

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**A generalized analysis is given by Hager as**

Accordingly the minimum bottom pressure is positive compared to the atmospheric pressure when c<1. Also, the minimum pressure head Pm/g The location of the minimum pressure is: Xm = for c<1.5 Xm = for c>1.5 i.e.; just at the transition of the crest to the vertical abutment.

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**The crest bottom pressure index as a function of c is significantly**

above the minimum pressure. Approximately: The figure below refers the location of zero bottom pressure, X0=x0/HD, i.e.; where atmospheric pressure occurs. Hager gives this relation as

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**Minimum bottom pressure index**

Discharge coefficient in relation to relative head c=H/HD Crest pressure Location of atmospheric bottom pressure c0( c )

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**2.3.7 Cavitation Design Standart overflow with c<1 underdesigned**

c>1 over design and thus subatmospheric bottom pressures. Initially over design of dam overflows was associated with advantages in capacity. However the increase in discharge coefficients C for c>1 is relatively small, but the decrease of minimum pressure, Pm, is significant.

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**Overdesigning, thus adds to the cavitation potential.**

Incipient cavitation is a statistical process depending greatly on the water quality and the local turbulence pattern. Generally, one assumes an incipient pressure head: The limit head HL for incipient cavitation to occur is The constant b was introduced to account for additional effects, such as the variability of Pvi with c.

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Spillway Face The spillway face is a straight line between the point of tangency, P.T.(xt,yt) and the point of curvature P.C. It has a length of L1, and a slope of a, which is determined from the stabilty analysis. P.T P.C a L1

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Point of tangency At the point of tangency:

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Flow down the steep face of spillway, normally at about 45o to the horizontal, has a rather special charactecter which makes the methods of Gradually-Varied Flow unsuitable for its treatment. In this case acceleration and boundary layer development are both taking place along the spillway face, as shown in figure below. 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 S2 profile, but the extreme steepness of the slope introduces more complications, chiefly the phenomenon of air entrainment, or “insufflation”.

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Spillway Face Recommended radius of toe: R=H0+0.25P R P x

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**Air entrainment on the face of a spillway**

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**Flow on a spillway face and air entrainment Oldman river dam, Alberta**

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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-increasing proportion of air, continues to accelerate until uniform flow occurs, or the base of spillway is reached. Clearly, the designer will wish to know the velocity reached at the base, or toe, of the spillway. The computation of this velocity can be obtained by using boundary layer development over the spillway face. The boundary layer will start to develop from point A where spillway crest starts. The thickness of the boundary layer, d, the displacement thickness, d1, and the energy thickness d3, at the point of curvature is given by:

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**Spillway crest and boundary layer**

The boundary layer will start to develop from point A where spillway crest starts.

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Where: d= the boundary layer thickness at P.C. L=Lc+L1= total length of crest Lc= length of curved crest, L1= Length of face, k= roughness height of concrete= cm d1= the displacement thickness at P.C. d3= the energy thickness at P.C. The length of curved crest, Lc,can be obtained from Fig.7 as a function of xt/H0. The depth of flow at the point of curvature can be obtained from energy equation by assuming that the head loss is zero. This depth dp is called as potential-flow depth, because head loss is neglected.

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**However, a boundary layer is developing along the spillway face**

However, a boundary layer is developing along the spillway face. Hence a head loss will occur. Therefore, the actual depth, d, and the head loss,hl, can be computed by using the displacement and energy thicknesses as follows:

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**Example on Spillway Energy Loss**

P=107 m x Given: H0=10 m P=107 m k=6.1x10-4 m Face slope: 1:0.78 For a high spillway, The crest shape is: H0= 10 m x yT (P.T) xT Y2 (P.C) y

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**COMPUTE THE ENERGY HEAD ENTERING THE STILLING BASIN**

1. Boundary Geometry Length of curved crest, Lc

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b) Length of tangent, LT Total crest length, L L=Lc+LT= =144.3 m

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**2. Hydraulic Computation:**

Boundary-layer thickness, d: b) Energy thickness, d3: d3=0.22d=0.142 m. c) Unit discharge, q:

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**d) Potential flow depth dp and velocity U at PC of toe curve**

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e) Spillway energy loss, hl: f) Energy head entering to the stilling basin: Hb= = m g) Depth of flow, d, at PC of toe curve: d=dp+d1 d1=0.18 d=0.116 m d= =1.56 m Since d<d, no bulking of flow from air entrainment

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Spillway Toe When the flow reaches the end of inclined face of spillway it is deflected through a vertical curve into the horizontal or into an upward direction. In the latter case we have the ski-jump and the bucket-type energy dissipators, to be discussed later. In either case, centrifugal pressures will be developed which can set up a severe thrust on the spillway side walls. These pressures cannot be accurately calculated by elemantary means, but there are certain approximations. One of them is: Assume that the depth yO at the center of curve is equal to the depth y1 of approaching flow. Then the centrifugal pressure at point O will be equal to: Where V1= velocity of approaching flow, and R= radius of curvature of toe

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**Spillway toe and flip bucket**

Spillway toe Flip bucket

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**Spillway cross section**

A typical cross section of a spillway with a ski-jump

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If pressure is increasing, velocity must be decreasing by the Bernoulli Equation. Then the average velocity must be smaller than V1, and the depth must be greater than yO, so that this equation is not correct. A better approximation can be made by assuming that the streamlines crossing OA form parts of concentric circles, and the velocity distribution along this line is accordingly the same as that in free, or irrotational vortex, i.e.: 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 streamline at A, then C=V1R1. The discharge q across AO is given by

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Since y1 and R are known in advance, R1 can be obtained by trial from the last equation. Given R1/R, we can obtain PO, the pressure at O, from the condition :

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The “free vortex” method leads to results that are quite accurate within a certain range, but it has a 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 natural logarithms. Applying this result to the equation for y1/R ,we see that R/y1 has a minimum value of e, when R/R1 =e, even though R/y1 is by nature of the problem an independent variable, which may in practice assume any value at all. Therefore, the theory cannot be applied when R/y1 <e. 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, P=gy, to the pressures obtained above.

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**Gate controlled ogee crest**

Releases for partial gate openings for gated crests occur as orifice flow. With full head on a gate that is opened a small amount, a free discharging trajectory will follow the path of a jet issuing from an orifice. For a vertical orifice the path of the jet can be expressed by the parabolic equation where H is the head on the center of the opening. For an orifice inclined an angle q from the vertical, the equation is: If subatmospheric pressures are to be avoided along the crest contact, the shape of the ogee downstream from the gate sill must conform to the trajectory profile.

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Gates operated with small openings under high heads produce negative pressures along the crest in the region immediately below the gate if the ogee profile drops below the trajectory profile. Tests showed the subatmospheric pressures would be equal to about one-tenth of the design head when the gate is operated at small openings and the ogee is shaped to the ideal nappe profile: For maximum head Ho. The force diagram for this condition is shown on figure 8.

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**Subatmospheric crest pressures for undershot gate flow**

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The adoption of a trajectory profile rather than a nappe profile downstream from the gate sill will result in a wider ogee, and reduced discharge efficiency for full gate opening. Where the discharge efficiency is unimportant and where a wider ogee shape is needed for structural stability, the trajectory profile may be adopted to avoid subatmospheric pressure zones along the crest. Where the ogee is shaped to the ideal nappe profile for maximum head, the subatmospheric pressure area can be minimized by placing the gate sill downstream from the crest of the ogee. This will provide an orifice that is inclined downstream for small gate openings and will result in a steeper trajectory closer to the nappe-shaped profile.

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**Discharge Over Gate-Controlled Ogee Crests**

The discharge for a gated ogee crest at partial gate openings will be similar to flow through an orifice and may be computed by the equation: where: H = head to the center of the gate opening (including the velocity head of approach), D = shortest distance from the gate lip to the crest curve, and L = crest width.

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The coefficient, C, is primarily dependent upon the characteristics of the flow lines approaching and leaving the orifice. In turn, these flow lines are dependent on the shape of the crest and the type of gate. Figure 9, which shows coefficients of discharge for orifice’ flows for different q angles, can be used for leaf gates or radial gates located at the crest or downstream of the crest. The q angle for a particular opening is that angle formed by the tangent to the gate’lip and the tangent to the crest curve at the nearest-point of the crest curve for radial gates. This angle is affected by the gate radius and the location of the trunnion pin.

133
**Definition sketch for gated crests**

134
**Discharge coefficient for flow under gates**

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Basic Hydraulics: Open Channel Flow – II

Basic Hydraulics: Open Channel Flow – II

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