 The Rehbock formula for discharge coefficient is:
where h= height of the weir.
 This equation holds up to H/h = 10.
 For H/h greater than 15, the weir becomes a sill, and the discharge is controlled by the critical section immediately upstream from the sill.
 For the sill, the discharge coefficient is:
C = 5.68[1 + (h/H)]^{1.5}

 Experiments have shown that the coefficient C remains approximately constant for sharpcrested weirs under varying heads
if the nappe is aerated.
14.3 CREST SHAPE OF THE OVERFLOW SPILLWAY

 From 1886 to 1888, Bazin made the first comprehensive laboratory investigation of nappe shapes.
 The use of Bazin's data in design will produce a crest shape that coincides with the lower surface of an aerated nappe over
a sharpcrested weir.
 In practice, there exists friction due to roughness on the surface of the spillway.
 Hence, negative pressures cannot be avoided.
 The presence of negative pressures will lead to danger of cavitation damage.
 In spillway design, avoidance of negative pressures is an objective.
Fig. 143 (Chow)


 The Army Corps pf Engineers has developed sharpcrested weir designs, called WES standard spillway shapes (See
spillway photos).
 They are represented by the following equation:
X and Y are the coordinates of the crest profile, with the origin at the highest point of the crest.
H_{d} is the design head excluding the velocity head of the approach flow.
K and n are parameters depending on the slope of the upstream face.
Slope on upstream face
 K
 n

Vertical
 2.000
 1.850

3V:1H
 1.936
 1.836

3V:2H
 1.939
 1.810

3V:3H
 1.873
 1.776

 For intermediate values, interpolation is possible.
14.4 DISCHARGE OF THE OVERFLOW SPILLWAY

 For WES shapes, the formula for discharge of an overflow spillway is:
where H_{e} is the total energy head of the crest, including the velocity head in the approach channel.
 Model tests have shown that the effect of the approach velocity is negligible when the height h of the spillway is greater than 1.33H_{d},
where H_{d} is the design head excluding the velocity head.
 When h/H_{d} > 1.33, then H_{e} = H_{d}, and the coefficient of discharge is C_{d} = 4.03 (U.S. customary units).
 For low spillways, with h/H_{d} < 1.33, the approach velocity will have an appreciable effect on the discharge and discharge coefficient.
Fig. 144 (Chow)


 Fig. 144 relates H_{e}/H_{d} with C/C_{d}, where C_{d} is 4.03, and C is the actual discharge coefficient.
 This is valid for verticalface WES spillways.
 For slopingface spillways, a correction factor is applied to C.
Example 141. Determine the crest elevation and the shape of an overflow spillway section having a vertical upstream face and crest length L = 250 ft.
The design discharge is 75,000 cfs. The upstream water surface at design discharge is at Elev. 1000, and the channel floor is at Elev. 880 ft.
Fig. 145 (Chow)


Solution.
 Assume a high overflow spillway, so that h/H_{d} > 1.33; then, the effect of the approach velocity is negligible: H_{e} = H_{d}, and C_{d} = 4.03.
 h + H_{d} = 1000  880 = 120 ft.
 The discharge equation is:
Q = C_{d}LH_{e}^{1.5}
75000 = 4.03 × 250 × H_{e}^{1.5}
H_{e} = 17.8 ft.
 The approach velocity is:
V_{a} = Q/[L × (h + H_{d})] = 75000/(250 × 120) = 2.5 fps.
 The velocity head is:
H_{a} = V_{a}^{2}/(2g) = 0.1 ft.
 The design head is:
H_{d} = H_{e}  H_{a} = 17.8  0.1 = 17.7 ft.
 The height of the dam is:
h = 120  17.7 = 102.3 ft.
 h/H_{d} = 102.3/17.7 = 5.77 > 1.33.
Assumption in Step 1 was correct.
 Crest elevation is at:
1000  17.7 = 982.3 ft.
 The crest shape:
X^{n} = K H_{d}^{n1} Y
Y = X^{n}/(K H_{d}^{n1})
Y = X^{1.85}/(2 H_{d}^{0.85})
Y = X^{1.85}/(2 × 17.7^{0.85})
Y = X^{1.85}/23.
 For the point of tangency:
Y' = (1.85/23) X^{0.85}
Y' = 0.08 X^{0.85}
Y' = ΔY /ΔX = 1.0/0.6
1.0/0.6 = 0.08 X^{0.85}
X = 35.6 ft.
Y = X^{1.85}/23 = 32.2 ft.
0.5 H_{d} = 8.8 ft.
0.2 H_{d} = 3.5 ft.
0.282 H_{d} = 5.0 ft.
0.175 H_{d} = 3.1 ft.
14.5 RATING OF OVERFLOW SPILLWAYS

 The profile is designed for one head, the design head.
 The spillway must operate under other heads.
 For lower heads, the pressure on the crest will be above atmospheric, but lower than hydrostatic.
 For higher heads, the pressure will be lower than atmospheric, and it may so low that separation may occur.
 Most experiments indicate that the design head may be safely exceeded by at least 50%.
 Beyond that, harmful cavitation may develop.
 For spillway shapes other than WES, Bradley has developed a universal curve shown in Fig. 146.
 This figure shows the relation between H_{e}/H_{D} and C/C_{D}.
 H_{D} is the design head including the approach velocity, and C_{D} is the corresponding coefficient of discharge.
 H_{e} is any head including the approach velocity, and C is the corresponding coefficient of discharge.
Fig. 146 (Chow)


 To find the coefficient of discharge for the design head for a given shape, use the Buehler method.
 The coefficient of discharge is computed by:
C = 3.97 (H_{e}/H_{D} )^{0.12}

H_{e} is the operating head.
H_{D} is the design head including the approach velocity for a verticalface spillway.
 The value of H_{D} is obtained from Fig. 147, matching a shape.
 If after matching, H_{D} differs on both sides, use the higher value.
 The dashed line indicates H_{D} = 45.
Fig. 147 (Chow)


RATING OF OVERFLOW SPILLWAYS II

 Pressure on the crest is close to atmospheric for the selected rate.
 The head above the spillway crest, excluding the approach velocity, is the design head H_{d}.
 The head above the spillway crest, including the approach velocity, is the design head H_{D}.
 The spillway rating is:
 Q = C (2g) ^{1/2} L H ^{3/2}
 C = discharge coefficient, dimensionless
 The theoretical value of C is based on the broadcrested weir equation:
 Q = (2/3) H L [g (2/3)H]^{1/2}
 C = 2/(3)^{3/2} = 0.3849
 H = total head, including approach velocity
 For ogeetype spillways:

Q = C_{D} (2g) ^{1/2} L H_{D}^{3/2}
 C_{D} is the value of C when H = H_{D}.
 Note that the actual C_{D} ranges from 0.3850.493, greater than the theoretical value C = 0.385.
 This means that the ogee spillway is very effective in passing the flow.
 Variation of C/C_{D} as a function of H/H_{D}
Check with ONLINE OGEE RATING.
Go to Chapter 15.
