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FHWA > Engineering > Hydraulics > HEC 14 > Chapter 7 
Hydraulic Design of Energy Dissipators for Culverts and Channels

for rows 1, 3, and 5 
for rows 2 and 4 
The splash shield height and length are calculated by first calculating the jet height:
h_{1} = 1.25y_{c} = 1.25(0.822) = 1.03 m
h_{1} + h = 1.03 + 0.42 = 1.45 m
Since culvert rise is only 1.2 m, an enlarged section of culvert with a 1.45 m rise is required. Use 1.5 m based on constructibility and materials availability.
With culvert rise set to 1.5, use Equation 7.9 to calculate splash shield height:
h_{2} = 1.5(D(h_{1}+h)) = 1.5(1.51.45) = 0.075 m
Take h_{2} = 0.075 m since this is greater than the minimum (0.05 m).
Length of splash shield = L/2 = 3.57/2 = 1.78 m.
Design summary:
Design Example: Tumbling Flow in a Box Culvert (CU)
Design a concrete box culvert for tumbling flow (see Figure 7.2). Determine if the outlet velocity is less than 10 ft/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this example the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error with Manning's Equation.
y_{n} = 1.12 ft
V_{n} = 22.4 ft/s (Since this is greater than 10 ft/s, energy dissipation is required.)
Step 3. Compute critical depth and velocity. First compute the unit discharge,
q = Q/B = 100/4 = 25 ft^{2}/s
Using Equation 7.1,
V_{c} = Q/(y_{c}B) = 100/(2.69 (4)) = 9.3 ft/s (Since the outlet velocity will be critical velocity, a proper design will meet the design criterion of 10 ft/s.)
Step 4. Size the element height, element spacing, and splash shield. We will use the recommended procedure of 5 rows of equal height roughness elements. Roughness element height, h, longitudinal spacing, L, and transverse spacing, W_{1} and W_{2}, are as follows:
From Equation 7.2:
L = 8.5h = 8.5(1.36) = 11.56 ft
From Equations 7.6 and 7.7:
for rows 1, 3, and 5 
for rows 2 and 4 
The splash shield height and length are calculated by first calculating the jet height:
h_{1} = 1.25y_{c} = 1.25(2.69) = 3.36 ft
h_{1} + h = 3.36 + 1.36 = 4.72 ft
Since culvert rise is only 4 ft, an enlarged section of culvert with a 4.72 ft rise is required. Use 5.0 ft based on constructibility and materials availability.
With culvert rise set to 5.0, use Equation 7.9 to calculate splash shield height:
h_{2} = 1.5(D(h_{1}+h)) = 1.5(5.04.72) = 0.42 ft
Take h_{2} = 0.42 ft since this is greater than the minimum (0.2 ft).
Length of splash shield = L/2 = 11.56/2 = 5.78 ft.
Design summary:
Tumbling flow in circular culverts can be attained by inserting circular rings inside the barrel as shown in Figure 7.4. Geometrical considerations are more complex, but the phenomenon of tumbling flow is the same as for box culverts. For box culverts, only bottom roughness elements were considered, whereas in circular culverts the elements are complete rings. The culvert is treated as an open channel, which greatly simplifies the discussion, and the diameter is varied to obtain vertical clearance for free surface flow.
Figure 7.4. Definition Sketch for Tumbling Flow in Circular Culverts
Design procedures have been described by Wiggert and Erfle (1971). Their experiments for tumbling flow in circular culverts were run with a 152 mm (6 in) plexiglass model and a 457 mm (18 in) concrete prototype culvert. Slopes ranged from 0 to 25 percent, h/D_{1} ranged from 0.06 to 0.15 and L/D_{1} ranged from 0.3 to 3.0 (L/h from 5 to 20). The experimental variables are illustrated in Figure 7.4. The variables that determine whether or not tumbling flow will occur are: roughness height, h, spacing, L, slope, S_{o}, discharge, Q, and diameter, D_{1}.
A functional relationship for the roughness height can be described as:
h = f (L, S_{o}, Q, D_{ 1}, g)
Establishing dimensionless groupings yields:
h/D_{1}= f(L/D_{1}, S_{o}, Q/(gD_{1}^{5})^{1/2} )
Practical design limits can be assigned to h/D_{1} and L/D_{1} to simplify the functional relationship. Based on qualitative laboratory observations, tumbling flow is easiest to maintain when L/D_{1} is between 1.5 and 2.5 and when h/D_{1} is between 0.10 and 0.15. Assigning these limits for circular culverts is analogous to assigning values for L/h in the design procedure for box culverts. The previous functional relationship can be rewritten:
Constant = f(S_{o}, Q/(g D_{ 1}^{5})^{1/2} )
or
Q/(gD_{1}^{5})^{1/2} = f(S_{o})
Theoretically f(S_{o}) could be any function involving the slope term. Empirically f(S_{o}) was found to be approximately a constant. The slight observed dependence of f(S_{o}) on slope is considered to be much less significant than the inaccuracies associated with measuring flow characteristics over the large roughness elements. Based on model and prototype data, f(S_{o}) ranges from 0.21 to 0.32 if the slope is between 4 percent and 25 percent. For slopes less than four percent, the culvert should be designed for full flow rather than tumbling flow. (See section 7.2.)
With the observed limits on f(S_{o}), the following expression is developed:
0.21 < Q/(g D_{ 1}^{5})^{1/2} < 0.32
Rewriting for use in design:
(7.10)where,
D_{1} = Diameter of the enlarged culvert section, m (ft)
Equation 7.10 is the basic design equation for tumbling flow in steep circular culverts. If the diameter of the roughened section of the culvert is sized according to this equation, tumbling flow will occur and the outlet velocity will be approximately critical velocity. This design is limited to the following conditions:
Since tumbling flow is an open channel phenomenon, gravity forces prevail and the Froude number, V/(gy)^{1/2}, should be used as the basis for design (or interpretation of model results). Watts (1968) established, by reference to several publications, that h/y is an important scaling parameter for roughness elements in open channel flow. In both of these dimensionless terms, y is a characteristic flow depth. The validity of using D_{1} in lieu of a characteristic flow depth in Q/(gD_{1}^{5})^{1/2} must be carefully examined for culverts flowing less than full. The characteristic depth for tumbling flow, however, is critical depth, which is uniquely defined by Q and D_{1}; so D_{1} can be substituted for y in this special case of partially full culverts.
Furthermore, the higher coefficient in Equation 7.10 resulted from the 152 mm (6 in) model data rather than from the 457 mm (18 in) prototype. Differences in model and prototype data were attributed to experimental difficulties with the prototype; nevertheless, if there are scaling errors, they appear to be on the conservative side.
As with box culverts, a major concern is that silt may accumulate in front of the roughness elements and render them ineffective. This is perhaps unwarranted as the element enhances sediment transport capacity and tends to be selfcleansing. In their original list of possible applications, Peterson and Mohanty (1960) noted that by "using roughness elements to induce greater turbulence, the sedimentcarrying capacity of a channel may be increased."
Water trapped between elements may cause difficulties during dry periods due to freezing and thawing and insect breeding. Narrow slots in the roughness rings (less than 0.5h) can be used to allow complete drainage without changing the design criteria. Sarikelle and Simon (1980) performed field studies of internal rings on circular pipes and found that modifications to ease installation (effectively adding slots) did not impair energy dissipation performance.
Five roughness rings at the outlet end of the culvert are sufficient to establish tumbling flow. The diameter computed from Equation 7.10 is for the roughened section only, and will not necessarily be the same as the rest of the culvert. The American Concrete Pipe Association (ACPA, 1972) introduced the telescoping concept in which the main section of the culvert is governed by the usual design parameters (presumably inlet control) and the roughened section is designed by Equation 7.10. They suggest telescoping the larger diameter pipe over the smaller "for at least the length of a normal joint and using normal sealing materials in the annular space." This concept is shown in Figure 7.4.
The design procedure requires computation of both the normal depth in the culvert based on the culvert diameter D and the critical depth based on the internal diameter of the roughened section, D_{i}. Using the definition sketch of Figure 7.5, the following geometric relationships are determined given a depth.
(7.11)where,
θ = internal angle, degrees
(7.12) (7.13)Figure 7.5. Definition Sketch for Flow in Circular Pipes
The outlet velocity for tumbling flow is approximately critical velocity. It can be computed by determining the critical depth, y_{c}, for the inside diameter of the roughness rings. Critical flow for an open channel of any shape will occur when:
(7.14)where,
T_{c} = water surface width at critical flow condition, m (ft)
A_{c} = flow area at critical flow condition, m (ft)
Once the critical depth is found, critical velocity is determined using the continuity equation.
Design Example: Tumbling Flow in a Circular Culvert (SI)
Design concrete pipe culvert for tumbling flow. Determine if the outlet velocity is less than 3 m/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this example the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error and the geometric relations of Equations 7.11, 7.12, and 7.13:
y_{n} = 0.445 m
V_{n} = 7.3 m/s (Since this is greater than 3 m/s, energy dissipation is required.)
Step 3. Compute critical depth and velocity. First we need to compute the diameter of the roughened section using Equation 7.10 and taking the range midpoint:
h = 0.125D_{1} = 0.125(1.67) = 0.21 m
D_{i} = D_{1}2h = 1.672(0.21) = 1.25 m
By trial and error, using Equation 7.14 and D = D_{i},
y_{c} = 0.913 m
A_{c} = 0.961 m^{2}
V_{c} = Q/A_{c} = 2.8/0.961 = 2.9 m/s (Meets design criteria of 3 m/s)
Step 4. Determine the remaining design component: element spacing.
L = 2D_{1} = 2(1.67) = 3.34 m
Design summary (see figure below; all dimensions not otherwise indicated are in meters):
Solution for Tumbling Flow Example for a Circular Culvert (SI)
Design Example: Tumbling Flow in a Circular Culvert (CU)
Design a concrete pipe culvert for tumbling flow. Determine if the outlet velocity is less than 10 ft/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this example the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error and the geometric relations of Equations 7.11, 7.12, and 7.13:
y_{n} = 1.46 ft
V_{n} = 24.2 ft/s (Since this is greater than 10 ft/s, energy dissipation is required.)
Step 3. Compute critical depth and velocity. First we need to compute the diameter of the roughened section using Equation 7.10 and taking the range midpoint:
h = 0.125D_{1} = 0.125(5.51) = 0.69 ft
D_{i} = D_{1}2h = 5.512(0.69) = 4.13 ft
By trial and error, using Equation 7.14 and D = D_{i},
y_{c} = 3.01 ft
A_{c} = 10.45 ft^{2}
V_{c} = Q/A_{c} = 100/10.45 = 9.6 ft/s (Meets design criteria of 10 ft/s)
Step 4. Determine the remaining design component: element spacing.
L = 2D_{1} = 2(5.51) = 11.0 ft
Design summary (see figure below; all dimensions not otherwise indicated are in feet):
Solution for Tumbling Flow Example for a Circular Culvert (CU)
The methodology described in this section involves using roughness elements to increase resistance and induce velocity reductions. Increasing resistance may cause a culvert to change from partial flow to full flow in the roughened zone. Velocity reduction is accomplished by increasing the wetted surfaces as well as by increasing drag and turbulence by the use of roughness elements.
Tumbling flow, as described in the previous section, is the limiting design condition for roughness elements on steep slopes. Tumbling flow essentially delivers the outlet flow at critical velocity. If the requirement is for outlet velocities between critical and the normal culvert velocity, designing increased resistance into the barrel is a viable alternative.
The most obvious situation for application of increased barrel resistance is a culvert flowing partially full with inlet control. The objective is to force full flow near the culvert outlet without creating additional headwater. Although based on the same principles, the design approaches for circular and box culverts differ.
Morris (1963) studied all pertinent rough pipe flow data available and concluded that there are three flow regimes and each has a different resistance relationship. Conceptually, the description of these regimes also applies to box culverts. The three regimes illustrated in Figure 7.6 are:
Figure 7.6. Flow Regimes in Rough Pipes
The design procedure for increased resistance in box and circular culverts may be summarized in the following steps:
Step 1. Verify the culvert is in inlet control.
Step 2. Compute normal flow conditions in the culvert to determine if the discharge conditions at the outlet require mitigation.
Step 3. Select initial design scale ratios. Determine Manning's n value for the roughened section of culvert.
Step 4. Compute mitigated depth and velocity. Check mitigated velocity against design criteria. One of three conditions will be observed:
Step 5. Complete sizing the element heights, element spacing, and other design features. Design details differ for box and circular culverts and are described in the following sections.
Wiggert and Erfle (1971) studied the effectiveness of roughness rings as energy dissipators in circular culverts. Although their study was primarily a tumbling flow study, they observed in many tests that they could get velocity reductions greater than 50 percent without reaching the roughness level necessary for tumbling flow. They did not derive resistance equations, but they did establish approximate design limits.
From these studies, good performance was observed when h/D was 0.06 to 0.09 using five rings. (See Figure 7.7.) Doubling the height, h_{1}, of the first ring was effective in triggering full flow in the roughened zone. Adequate performance was obtained with four identical rings, but with double spacing between the first two. However, the same pipe length is involved if a constant spacing is maintained and five rings used, with the first double the height of the other four. The additional ring should help establish the assumed full flow condition. In addition, the last (downstream) ring must be located no closer than onehalf the ring spacing from the end of the culvert.
Figure 7.7. Conceptual Sketch of Roughness Elements to Increase Resistance
Subsequent experience reported by the American Concrete Pipe Association (ACPA, 1972) indicated a need to consider lower values of h/D, and to establish approximate resistance curves for evaluating a design in order to avoid installations that will propagate full flow upstream to the culvert inlet.
Based on experience with large elements used to force tumbling flow (see Section 7.1) and the work of Wiggert and Erfle (1971), five rows of roughness elements with heights ranging from 5 to 10 percent of the culvert diameter are sufficient.
A key element in the design of increased roughness elements is determination of the roughness regime and, subsequently, the appropriate Manning's n value. Although much of the literature relative to large roughness elements in circular pipes expresses resistance in terms of the friction factor, "f", all resistance equations are converted to Manning's "n" expressions for this manual.
Isolated roughness flow was introduced in Section 7.2. The overall friction or resistance, f_{IR}, is made up of two parts:
(7.15)f_{IR} = f_{s} + f_{d}
where,
f_{s} = friction on the culvert surface.
f_{d} = friction due to form drag on the roughness elements.
The friction due to form drag is a function of the drag coefficient for the particular shape, the percentage of the wetted perimeter that is roughened, the roughness dimensions and spacing and the velocity impinging on the roughness elements. Morris (1963) related the velocity to surface drag and derived the following equation:
(7.16)where,
C_{D} = drag coefficient for the roughness shape
L_{r}/P = ratio of total peripheral length of roughness elements to total wetted perimeter
r_{i} = pipe radius based on the inside diameter of roughness rings measured from crest to crest
L_{r} may be less than P to facilitate constructibility of the rings or to permit a low flow opening at the bottom of the ring.
Throughout Morris' work, he used measurements from crest to crest of a roughness element ring as the effective diameter, D_{i}. To convert the expression for roughness to Manning's n, the following expressions are needed:
f_{s} = α (n/D^{1/6})^{2}
f_{IR} = α (n_{IR} /D_{i}^{1/6})^{2}
α represents a unit conversion constant equal to 124 in SI and 184 in CU. Equation 7.16 can then be converted to Manning's n:
(7.17)where,
n_{IR} = overall Manning's "n" for isolated roughness flow
n = Manning's "n" for the culvert surface without roughness rings
D = nominal diameter of the culvert, m (ft)
D_{i} = inside diameter of roughness rings, m (ft) (D_{i} = D2h)
For sharp edge rectangular roughness shapes, a constant value of 1.9 can be used for C_{D}. It is noteworthy that the overall resistance, n_{IR}, decreases as the relative spacing, L/D_{i}, increases for this regime.
The friction in this regime is independent of friction on the culvert surface:
(7.18)where,
f_{HT} = overall friction for hyperturbulent flow
φ = function of Reynolds number, element shape, and relative spacing
By restricting application of Equation 7.18 to sharp edged roughness rings and to a spacing greater than the pipe radius, φ can be neglected.
Substituting the following expression:
f_{HT} = α (n_{HT} /D_{i}^{1/6})^{2}
Equation 7.18 can then be converted to Manning's n:
(7.19)where,
n_{HT} = Manning's n for hyperturbulent flow
α = unit conversion constant, 0.0898 (SI) and 0.0737 (CU)
The effect of the roughness height, h, is included inherently in D_{i}. From Equation 7.19 it can be seen that n_{HT} increases as the spacing increases for this regime.
Since resistance increases when the spacing increases for the hyperturbulent regime and when the spacing decreases for the isolated roughness regime, the boundary between the regimes occurs when the resistance equations are the same. The boundary is determined by equating n_{IR} in Equation 7.17 to n_{HT} in Equation 7.19. For design, both are calculated; the lowest n value is used and indicates which regime is applicable.
The recommended design is limited to the following conditions:
Once these ranges are selected, the roughness element height is computed as follows:
(7.20)where,
h= roughness element height, m (ft)
c= ratio of h/D_{i}
Once h is calculated, values of D_{i} and L follow directly and the roughness values are calculated.
Design Example: Increased Resistance in a Circular Culvert (SI)
Design a concrete pipe culvert for increased roughness. Determine if the outlet velocity is less than 3 m/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this case, the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error and the geometric relations of Equations 7.11, 7.12, and 7.13:
y_{n} = 0.445 m
V_{n} = 7.3 m/s (Since this is greater than 3 m/s, energy dissipation is required.)
Step 3. Select initial design scale ratios and determine Manning's n value for the roughened section of culvert.
Try L/D_{i} = 1.1 and h/D_{i} = 0.06
Calculate h from Equation 7.20:
(round to 0.06 m) 
D_{i} = D  2h = 1.2  2(0.06) = 1.08 m
L = 1.1D_{i} = 1.1(1.08) = 1.19 m
For this design, we will not have gaps in the rings, therefore, L_{r}/P = 1.
Now, we calculate Manning's n for the isolated roughness (Equation 7.17) and hyperturbulent flow (Equation 7.19) to determine flow regime and Manning's n.
Since n_{IR} < n_{HT} the roughness is characterized as isolated roughness and n = 0.035
Step 4. Compute mitigated depth and velocity. Check mitigated velocity against design criteria.
With the internal diameter, D_{i}, and the Manning's n values calculated in step 3, the normal depth for the roughened condition is calculated to be (by trial and error);
y_{n} = 0.932 m
V_{n} = 3.3 m/s
Compared with the design goal of 3.0 m/s, this velocity is unacceptable even though it has been reduced significantly from the unmitigated velocity. Since the depth is less than D_{i}, we can increase h to further slow the velocity. (We also need to increase the culvert size because the full flow velocity of 3.1 m/s still exceeds our design criteria.) Steps 3 and 4 must be repeated.
Step 3 (2nd iteration). Select trial design scale ratios and determine Manning's n value for the roughened section of culvert.
Maintain L/D_{i} = 1.1, increase D = 1.50 m (next available size) and try h = 0.1 m
D_{i} = D  2h = 1.50  2(0.10) = 1.30 m (h/D_{i} = 0.077)
L = 1.1D_{i} = 1.1(1.3) = 1.43 m
For this design, we will not have gaps in the rings, therefore, L_{r}/P = 1.
Now, we calculate Manning's n for the isolated roughness (Equation 7.17) and hyperturbulent flow (Equation 7.19) to determine flow regime and Manning's n.
Since n_{IR} < n_{HT} the roughness is characterized as isolated roughness and n = 0.040.
Step 4 (2nd iteration). Compute trial of the mitigated depth and velocity. Check mitigated velocity against design criteria.
With the internal diameter, D_{i}, and the Manning's n values calculated in step 3, the normal depth for the roughened condition is calculated to be (by trial and error);
y_{n} = 1.025 m
V_{n} = 2.5 m/s
Compared with the design goal of 3.0 m/s, this velocity is acceptable. However, we must check the culvert capacity using Manning's Equation.
We assume that the culvert is flowing full and estimate the wetted perimeter.
A = πD_{i}^{2}/4 = π(1.30)^{2}/4 = 1.33 m^{2}
P = πD_{i} = π(1.30) = 4.08 m
R = A/P = 1.33/4.08 = 0.326 m
Using Manning's Equation,
Since this flow is greater than the design flow of 2.8 m^{3}/s we know that the design is acceptable. If this had not been the case, a larger culvert barrel could be evaluated going back to step 3 or tumbling flow or another type of dissipator could be considered.
Step 5. Complete sizing the element heights, element spacing, and other design features.
Roughness height and spacing have been established as well as an oversized culvert section. For 5 rows of roughness elements, the length of the oversized section with increased roughness is 5.06 m.
Design summary (see figure below):
Sketch for Increased Resistance In a Circular Culvert Design Example (SI)
Design Example: Increased Resistance in a Circular Culvert (CU)
Design a concrete pipe culvert for increased roughness. Determine if the outlet velocity is less than 10 ft/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this case the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error and the geometric relations of Equations 7.11, 7.12, and 7.13:
y_{n} = 1.45 ft
V_{n} = 24.1 ft/s (Since this is greater than 10 ft/s, energy dissipation is required.)
Step 3. Select initial design scale ratios and determine Manning's n value for the roughened section of culvert.
Try L/D_{i} = 1.1 and h/D_{i} = 0.06
Calculate h from Equation 7.20:
(round to 0.21 ft) 
D_{i} = D  2h = 4.0  2(0.21) = 3.58 ft
L = 1.1D_{i} = 1.1(3.58) = 3.94 ft
For this design, we will not have gaps in the rings, therefore, L_{r}/P = 1.
Now, we calculate Manning's n for the isolated roughness (Equation 7.17) and hyperturbulent flow (Equation 7.19) to determine flow regime and Manning's n.
Since n_{IR} < n_{HT} the roughness is characterized as isolated roughness and n = 0.036
Step 4. Compute mitigated depth and velocity. Check mitigated velocity against design criteria.
With the internal diameter, D_{i}, and the Manning's n values calculated in step 3, the normal depth for the roughened condition is calculated using Manning's Equation to be (by trial and error);
y_{n} = 3.10 ft
V_{n} = 10.7 ft/s
Compared with the design goal of 10 ft/s, this velocity is unacceptable even though it has been reduced significantly from the unmitigated velocity. Since the depth is less than D_{i}, we can increase h to further slow the velocity. (We also need to increase the culvert size because the full flow velocity exceeds our design criteria.) Steps 3 and 4 must be repeated.
Step 3 (2nd iteration). Select trial design scale ratios and determine Manning's n value for the roughened section of culvert.
Maintain L/D_{i} = 1.1, increase D = 4.5 ft and try h = 0.32 ft
D_{i} = D  2h = 4.5  2(0.32) = 3.86 ft (h/D_{i} = 0.083)
L = 1.1D_{i} = 1.1(3.86) = 4.25 ft
For this design, we will not have gaps in the rings, therefore, L_{r}/P = 1.
Now, we calculate Manning's n for the isolated roughness (Equation 7.17) and hyperturbulent flow (Equation 7.19) to determine flow regime and Manning's n.
Since n_{IR} < n_{HT} the roughness is characterized as isolated roughness and n = 0.041
Step 4 (2nd iteration). Compute trial of the mitigated depth and velocity. Check mitigated velocity against design criteria.
With the internal diameter, D_{i}, and the Manning's n values calculated in step 3, the normal depth for the roughened condition is calculated to be (by trial and error);
y_{n} = 3.07 ft
V_{n} = 9.9 ft/s
Compared with the design goal of 10 ft/s, this velocity is acceptable. However, we must check the culvert capacity using Manning's Equation.
We assume that the culvert is flowing full and estimate the wetted perimeter.
A = πD_{i}^{2}/4 = π(3.86)^{2}/4 = 11.70 ft^{2}
P = πD_{i} = π(3.86) = 12.13 ft
R = A/P = 11.70/12.13 = 0.964 ft
Using Manning's Equation,
Since this flow is greater than the design flow of 100 ft^{3}/s we know that the design is acceptable. If this had not been the case, a larger culvert barrel could be evaluated going back to step 3 or tumbling flow or another type of dissipator could be considered.
Step 5. Complete sizing the element heights, element spacing, and other design features.
Roughness height and spacing have been established as well as an oversized culvert section. For 5 rows of roughness elements, the length of the oversized section with increased roughness is 17.0 ft.
Design summary (see figure below):
Sketch for Increased Resistance In a Circular Culvert Design Example (CU)
Material for this section was drawn primarily from a preliminary FHWA report on fish baffles in box culverts (Normann, 1974). This report used Morris' categorization of flow regimes and basic friction equations (Morris, 1963), but a more representative approach velocity, V_{A}, in one of the regimes. Experimental data by Shoemaker (1956) were also utilized to define the transition curves. For several reasons, modifications to the fish baffle development were necessary to adapt to energy dissipator design. In fish baffle design, the interest is in a conservative estimate of resistance in order to size a culvert; whereas, in this manual, a conservative estimate of the outlet velocity is also important. Also, fish baffle design curves involve bottom roughness only.
As before, both the hyperturbulent and isolated roughness flow regimes are considered. For box culverts in hyperturbulent flow, Manning's roughness may be estimated by:
(7.21)where,
n_{HT} = Manning's n for the hyperturbulent flow regime
n = Manning's roughness coefficient for the culvert without the roughness elements (maximum n value is 0.015 for this equation)
L_{r}/P = ratio of total peripheral length of roughness elements to toal wetted perimeter
For the isolated roughness regime, a high and low Manning's range are considered as shown by the following equations:
(7.22a) (7.22b)where,
n_{IR,LOW} = Manning's n for the isolated roughness flow regime, low design range for estimating velocity.
n_{IR,HIGH} = Manning's n for the isolated roughness flow regime, high design range for estimating depth.
n = Manning's roughness coefficient for the culvert without the roughness elements (maximum n value is 0.015 for these equations)
L_{r} will equal the bottom width, B, for installations with bottom roughness only and will equal the wetted perimeter, P, when roughness elements are attached to all sides or when the roughness elements extend through the flow. The presence of drainage slots in the roughness elements is ignored in estimating L_{r}.
The equations are based on C_{D} = 1.9, f = 0.14 (where f is the Darcy friction factor for the culvert surface without roughness elements), and V_{A}/V = 0.60 or 0.85. The lower value of V_{A}/V is implicitly included in Equation 7.22a and the higher value in Equation 7.22b. The use of a representative approach velocity, V_{A}, allows an opportunity to input culvert parameters that will lean towards either an overprediction or an underprediction of resistance. It is assumed that (R/R_{i})^{1/3} is approximately one to simplify the analysis. R is the hydraulic radius of the culvert proper and R_{i} is the hydraulic radius taken inside the crests of the roughness elements. For designs considered in this section, the approximation is reasonable.
For this manual, it is appropriate to develop high as well as low resistance curves. Rather than attempt to define the transition between these curves, an abrupt transition is used as the worst condition for the high curves, and a straightline transition is assumed as the mildest condition for the low curves. This is illustrated in Figure 7.8. Observations by Powell (1946) are the basis for assuming the 6 to 12 range of L/h for the transition curve between isolated roughness and hyperturbulent flow. An L/h=10 is chosen for design because it yields the largest n value.
For estimating velocities, it is appropriate to estimate resistance based on the straightline (low) relationship shown in Figure 7.8. Therefore, Equation 7.21 (hyperturbulent) is evaluated at L/h=6 and Equation 7.22a (isolated roughness) is evaluated at L/h=12. A linear interpolation between the two for L/h=10 results in the relationship provided as follows:
(7.23)Figure 7.8. Transition Curves between Flow and Regimes
For determining the lower value of Manning's n, n_{LOW}, for the purpose of estimating outlet velocity, Equation 7.23 applies to values of h/R_{i} less than or equal to 0.3. For ratios above 0.3, n_{LOW} is calculated directly from Equation 7.22a evaluated for L/h = 10.
For determining the upper value of Manning's n, n_{HIGH}, for the purpose of estimating flow depths, the abrupt (high) value indicated in Figure 7.8 is desired. For h/R_{i} greater than 0.2, Equation 7.22b is evaluated at L/h = 10 and used for n_{HIGH}. For ratios less than or equal to 0.2, Equations 7.22b and 7.21 are both evaluated at L/h = 10 and the lower value is taken for n_{HIGH}. Both are compared to avoid unrealistic values from Equation 7.22b.
Since the above equations are normal flow equations and roughness elements may be relatively small using this method, it is necessary to compute the length of the culvert to be roughened. The momentum equation, written for the roughened section of culvert, is used to compute the number of rows of roughness element needed.
(7.24)where,
N = number of roughness element rows
B = culvert bottom width, m (ft)
y_{n} = normal depth in the culvert approaching the roughened section, m (ft)
y_{i} = normal depth in the roughened section of the culvert, m (ft)
V_{n} = normal velocity in the culvert approaching the roughened section, m/s (ft/s)
V_{i} = normal velocity in the roughened section of the culvert, m/s (ft/s)
C_{D} = coefficient of drag (taken as 1.9)
A_{f} = wetted frontal area of a roughness row, m^{2} (ft^{2}), equal to B(h) for bottom roughness
V_{w} = average wall velocity action on the roughness elements, m/s (ft/s), equal to (V_{n}+V_{i})/6
Regardless of the result of Equation 7.24, the number of rows should never be less than five. Furthermore, it is recommended that one large element be used at the beginning of the roughened zone to accelerate the asymptotic approach to normal flow. The recommended height of the larger element is twice the height of the regular elements. The spacing is the same for all rows of elements.
Slots in the roughness elements are provided for low flow drainage. The slot opening should not exceed h/2.
The procedure is limited to solid strip roughness elements with sharp upstream edges. Rectangular cross section roughness elements will best fit the assumptions made.
Due to the assumed velocity distribution, application of the procedure must be limited to small roughness heights and to relatively flat slopes. The roughness height should not exceed ten percent of the flow depth. This restriction is inherently included in the suggested range of h/R_{i} in the design procedure.
The recommended design is limited to the following conditions:
Design Example: Increased Resistance for a Box Culvert (SI)
Design a concrete box culvert for increased roughness using bottom roughness elements. Determine if the outlet velocity is less than 3 m/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this case the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error:
y_{n} = 0.34 m
V_{n} = 6.8 m/s (Since this is greater than 3 m/s, energy dissipation is required.)
Step 3. Select initial design scale ratios. Determine Manning's n value for the roughened section of culvert for estimating the mitigated depth and velocity.
Try L/h = 10 and h/R_{i} = 0.3
We will assume that the culvert is not flowing full for computation of the wetted perimeter. (If the culvert did flow full for this computation, it will be surcharged when we compute the high Manning's n for the capacity check. Using the sides and bottom of the culvert will provide the lowest (short of full flow), and therefore, conservative value of n for estimating velocity in the roughened section. This assumption may be revised in subsequent iterations.
P = 2D+B = 2(1.2)+1.2 = 3.6 m
L_{r} = B = 1.2 m (bottom roughness only)
Using Equation 7.23 and n = 0.013 (maximum value is 0.015):
Step 4. Compute mitigated depth and velocity and roughness height, h. Check mitigated velocity against design criteria. Using trial and error:
y_{i} = 0.783 m (note that culvert is not flowing full)
A_{i} = 0.783(1.2) = 0.940 m^{2}
P_{i} = 2(0.783)+1.2 = 2.77 m
R_{i} = 0.940/2.77 = 0.340 m
h = (h/R_{i})(R_{i}) = 0.3(0.340) = 0.102 m (round to 0.10 m)
V_{i} = 2.98 m/s
Since this velocity is less than or equal to 3 m/s, the dissipation design is satisfactory. However, we must check the culvert capacity using the high estimate of Manning's n from Equation 7.22b. Two possible flow limiting scenarios exist.
First, assume the culvert is flowing nearly full. Using Equation 7.22b and n = 0.013 (maximum value is 0.015):
Area and hydraulic radius are calculated as:
y = D  h = 1.2  0.10 = 1.10 m
A = yB = 1.10(1.2) = 1.32 m^{2}
P = 2y+B = 2(1.10)+1.2 = 3.40 m
R = A/P = 1.32/3.40 = 0.388 m
Using Manning's Equation,
Q is greater than the design flow using this scenario.
Second, assume the culvert is flowing full. Using Equation 7.22b and n = 0.013 (maximum value is 0.015):
Area and hydraulic radius are calculated as:
A = yB = 1.10(1.2) = 1.32 m^{2}
P = 2(y+B) = 2(1.10+1.2) = 4.6 m
R = A/P = 1.32/4.6 = 0.287 m
Using Manning's Equation,
Since both flows are greater than the design flow of 2.8 m^{3}/s we know that the design is acceptable. If this had not been the case, a larger culvert barrel could be evaluated going back to step 3 or tumbling flow or another type of dissipator could be considered.
Step 5. Complete sizing the element heights, element spacing, and other design features.
Spacing between roughness elements is calculated to be:
L = (L/h)h = (10)0.10 = 1.0 m
Number of rows of roughness elements is estimated using Equation 7.24. First calculate:
A_{f} = B(h) = 1.2(0.10) = 0.12 m^{2}
V_{w} = (V_{n}+V_{i})/6 = (6.8+2.98)/6 = 1.63 m/s
Round up to the nearest whole number, N = 26
Design summary:
Design Example: Increased Resistance for a Box Culvert (CU)
Design a concrete box culvert for increased roughness using bottom roughness elements. Determine if the outlet velocity is less than 10 ft/s. Given:
Solution
Step 1. Verify the culvert is in inlet control. In this case the culvert is in inlet control.
Step 2. Compute normal flow conditions. Using trial and error:
y_{n} = 1.11 ft
V_{n} = 22.5 ft/s (Since this is greater than 10 ft/s, energy dissipation is required.)
Step 3. Select initial design scale ratios. Determine Manning's n value for the roughened section of culvert for estimating the mitigated depth and velocity.
Try L/h = 10 and h/R_{i} = 0.3
We will assume that the culvert is not flowing full for computation of the wetted perimeter. (If the culvert did flow full for this computation, it will be surcharged when we compute the high Manning's n for the capacity check. Using the sides and bottom of the culvert will provide the lowest (short of full flow), and therefore, conservative value of n for estimating velocity in the roughened section. This assumption may be revised in subsequent iterations.
P = 2D+B = 2(4.0)+4.0 = 12.0 ft
L_{r} = B = 4.0 ft (bottom roughness only)
Using Equation 7.23 and n = 0.013 (maximum value is 0.015):
Step 4. Compute mitigated depth and velocity and roughness height, h. Check mitigated velocity against design criteria. Using trial and error:
y_{i} = 2.54 ft (note that culvert is not flowing full)
A_{i} = 2.54(4.0) = 10.17 ft^{2}
P_{i} = 2(2.54)+4.0 = 9.083 ft
R_{i} = 10.17/9.083 = 1.119 ft
h = (h/R_{i})(R_{i}) = 0.3(1.119) = 0.336 ft (round to 0.34 ft)
V_{i} = 9.84 ft/s
Since this velocity is less than or equal to 10 ft/s, the dissipation design is satisfactory. However, we must check the culvert capacity using the high estimate of Manning's n from Equation 7.22b. Two possible flow limiting scenarios exist.
First, assume the culvert is flowing nearly full. Using Equation 7.22b and n = 0.013 (maximum value is 0.015):
Area and hydraulic radius are calculated as:
y = D  h = 4.0  0.34 = 3.66 ft
A = yB = 3.66(4.0) = 14.64 ft^{2}
P = 2y+B = 2(3.66)+4.0 = 11.32 ft
R = A/P = 14.64/11.32 = 1.293 ft
Using Manning's Equation,
Q is greater than the design flow using this scenario.
Second, assume the culvert is flowing full. Using Equation 7.22b and n = 0.013 (maximum value is 0.015):
Area and hydraulic radius are calculated as:
A = yB = 3.66(4.0) = 14.64 ft^{2}
P = 2(y+B) = 2(3.66+4.0) = 15.32 ft
R = A/P = 14.64/15.32 = 0.956
Using Manning's Equation,
Since both flows are greater than the design flow of 100 ft^{3}/s we know that the design is acceptable. If this had not been the case, a larger culvert barrel could be evaluated going back to step 3 or tumbling flow or another type of dissipator could be considered.
Step 5. Complete sizing the element heights, element spacing, and other design features.
Spacing between roughness elements is calculated to be:
L = (L/h)h = (10)0.34 = 3.4 ft
Number of rows of roughness elements is estimated using Equation 7.24. First calculate:
A_{f} = B(h) = 4.0(0.34) = 1.36 ft^{2}
V_{w} = (V_{n}+V_{i})/6 = (22.5+9.84)/6 = 5.39 ft/s
Round up to the nearest whole number, N = 25
Design summary:
Peterka (1978) has described the design process for a baffled apron that makes use of roughness elements on the floor of a box culvert or chute as shown in Figure 7.9. The roughness elements, referred to as baffles or blocks, perturb the flow pattern such that flow slows as it approaches each block and then accelerates as it passes each block and approaches the next row. By placing the baffles from the top of the culvert or chute to the bottom, the baffles prevent excessive acceleration of flows regardless of the total drop height. Based on model studies reported by Peterka, the baffled apron design produces velocities at the bottom of the apron equal to no more than onethird of the critical velocity if the design guidance is followed. This approach works satisfactorily with or without downstream tailwater and is generally not susceptible to trash or debris accumulation.
Figure 7.9. USBR Type IX Baffled Apron (Peterka, 1978)
The USBR Type IX baffled apron is limited to the following site conditions and design limits:
The baffled apron is not a device intended to slow excessive approach velocity, but to prevent excessive acceleration during the vertical drop. According to Peterka (1978), the recommended approach velocity is 1.5 m/s (5 ft/s) less than critical velocity. Velocities near or above critical velocity tend to cause the flow to be thrown into the air after striking the first row of baffles and jumping past the first two or three baffle rows. This is of particular concern for relatively short aprons. One strategy for reducing the approach velocity is providing a recessed approach prior to the entrance of the apron as shown in Figure 7.9.
Another key design element is the selection of the baffle dimensions (height, width, and spacing). Based on model testing and prototype observations by Peterka, baffle height, H, should be about 0.8 times the critical depth, y_{c}. The height may be increased as high as 0.9y_{c}, but should not be less than 0.8y_{c}.
As shown in Figure 7.9, the baffle widths and horizontal baffle spacing should be equal to 1.5H, but not less than H. Each row should be alternating and partial blocks will likely be necessary at the culvert/chute walls.
Longitudinal spacing between rows is based on an assumed apron slope at the maximum slope of 50%. Under these conditions, successive rows of baffles are placed 2H apart as measured along the slope. For baffles less than 0.9 m (3 ft) in height, the row spacing may be greater, than 2H, but not exceeding 1.8 m (6 ft). For apron slopes less than 50 percent, the spacing along the apron slope may be increased such that the vertical drop between baffle rows is 0.89H.
Four rows of baffles are required to establish full control of the flow, although fewer rows have been successful. As shown in Figure 7.9, the chute is generally extended below the bed level of the downstream channel with the lower row of baffles buried to control scour. Riprap consisting of 6 to 12inch rock should be placed at the downstream ends of the sidewalls to prevent turbulence from undermining the walls, but should not extend appreciably into the channel.
The design procedure for the USBR Type IX baffled apron may be summarized in the following steps:
Step 1. Compute normal flow conditions in the culvert/chute to determine if the discharge conditions at the outlet require mitigation.
Step 2. Verify that the approach flow conditions are acceptable.
Step 3. Compute discharge velocity. If this velocity meets criteria, the USBR Type IX may be an appropriate energy dissipation approach.
Step 4. Size the baffle height, spacing, and other design features.
Design Example: USBR Type IX Baffled Apron (SI)
Design a USBR Type IX baffled apron for energy dissipation in a concrete box culvert with an overall vertical drop of 8 m. Determine if the outlet velocity is less than 3 m/s. Given:
Approach channel:
Box/Chute:
Solution
Step 1. Compute normal flow conditions in the box/chute to determine if a baffled apron is needed. Using trial and error and the geometric relations of Equations 7.11, 7.12, and 7.13:
y_{n} = 0.19 m
V_{n} = 12.2 m/s (Since this is greater than 3 m/s, energy dissipation is required.)
Step 2. Verify that the approach flow conditions are acceptable. Estimate the approach flow conditions using trial and error and Manning's Equation:
y_{n} = 0.92 m
V_{n} = 2.54 m/s
This approach velocity must be compared with critical velocity computed from:
q = Q/B = 2.8/1.2 = 2.33 m^{3}/s/m
V_{c} = (qg)^{1/3} = (2.33(9.8))^{1/3} = 2.84 m/s
Since V_{c} > V_{n} (approach velocity) and the unit discharge is less than 5.6 m^{3}/s/m, the baffled apron is applicable.
Step 3. Compute discharge velocity. Discharge velocity is no more than onethird of critical velocity; in this case discharge velocity = (2.84 m/s)/3 = 0.95 m/s. This is well below the design requirement of 3 m/s, therefore, the USBR Type IX may be an appropriate energy dissipation approach.
Step 4. Size the baffle height, spacing, and other design features. Critical depth must be computed first.
y_{c} = q/V_{c} = 2.33/2.84 = 0.82 m
Baffle height, H = 0.8y_{c} = 0.8 (0.82) = 0.66 m
Baffle width = 1.5H = 1.5 (0.66) = 0.99 m
Baffle spacing (horizontal) = 1.5H = 0.99 m
Vertical drop between baffle rows, Δ h = 0.89H = 0.89 (0.66) = 0.59 m
Spacing (measured along apron) between baffle rows, L = 1.87 m
Minimum sidewall height = 3H = 3(0.66) = 1.98 m (Since this is a closed box with a rise = 1.2 m, splash is not a concern.)
Design summary:
Design Example: USBR Type IX Baffled Apron (CU)
Design a USBR Type IX baffled apron for energy dissipation in a concrete box culvert with an overall vertical drop of 26.2 ft. Determine if the outlet velocity is less than 10 ft/s. Given:
Approach channel:
Box/Chute:
Solution
Step 1. Compute normal flow conditions in the box/chute to determine if a baffled apron is needed. Using trial and error and the geometric relations of Equations 7.11, 7.12, and 7.13:
y_{n} = 0.62 ft
V_{n} = 40.2 ft/s (Since this is greater than 10 ft/s, energy dissipation is required.)
Step 2. Verify that the approach flow conditions are acceptable. Estimate the approach flow conditions using trial and error and Manning's Equation:
y_{n} = 2.98 ft
V_{n} = 8.4 ft/s
This approach velocity must be compared with critical velocity computed from:
q = Q/B = 100/4 = 25 ft^{3}/s/ft
V_{c} = (qg)^{1/3} = (25.0(32.2))^{1/3} = 9.31 ft/s
Since V_{c} > V_{n} (approach velocity) and the unit discharge is less than 60 ft^{3}/s/ft, the baffled apron is applicable.
Step 3. Compute discharge velocity. Discharge velocity is no more than onethird of critical velocity; in this case discharge velocity = (9.3 ft/s)/3 = 3.1 ft/s. This is well below the design requirement of 10 ft/s, therefore, the USBR Type IX may be an appropriate energy dissipation approach.
Step 4. Size the baffle height, spacing, and other design features.
Critical depth must be computed first.
y_{c} = q/V_{c} = 25/9.3 = 2.69 ft
Baffle height, H = 0.8y_{c} = 0.8 (2.69) = 2.15 ft
Baffle width = 1.5H = 1.5 (2.15) = 3.23 ft
Baffle spacing (horizontal) = 1.5H = 3.23 ft
Vertical drop between baffle rows, Δ h = 0.89H = 0.89 (2.15) = 1.91 ft
Spacing (measured along apron) between baffle rows, L = 6.08 ft
Minimum sidewall height = 3H = 3(2.15) = 6.45 ft (Since this is a closed box with a rise = 4.0 ft, splash is not a concern.)
Design summary:
An alternative to installing a steeply sloped culvert is to break the slope into a steeper portion near the inlet followed by a horizontal runout section. This configuration is referred to as a brokenback culvert and may be considered another internal (integrated) energy dissipator strategy if it is designed so that a hydraulic jump occurs in the runout section to dissipate energy. Figure 7.10 illustrates two cases: a double brokenback culvert, and a single brokenback culvert. In both cases, the exit or runout section is assumed to be horizontal. Under certain conditions of culvert properties and tailwater levels, a hydraulic jump will form in the runout section and reduce the outlet velocity from that associated with a supercritical depth to that associated with a subcritical depth. Modifications to the runout section may be used to induce a hydraulic jump within the culvert.
A hydraulic jump will form in a channel if either of the following two conditions occurs: (1) the momentum in the tailwater downstream from the culvert exceeds that in the barrel, or (2) the supercritical Froude number in the barrel is reduced to approximately 1.7 in a decelerating flow environment (Chow, 1959).
To solve for the hydraulics of a brokenback culvert, a gradually varied water surface profile is calculated within the culvert from the entrance down to the flat runout section. This supercritical profile is compared to the tailwater elevation and the sequent depth to determine whether or not a hydraulic jump will occur in the runout section.
Figure 7.10. Elevation view of (a) Double and (b) Single Brokenback Culvert
If a jump does occur, the design should ensure that the jump is confined to the runout section. First, the location of the jump referenced from the beginning of the runout section must be determined. This is accomplished by computing the profiles upstream and downstream of the jump to find where the momenta are the same, that is, where the jump is located.
Second, the length of the jump is estimated. The sum of these two quantities must be less than the runout section length. The jump length for a rectangular culvert or channel is given by:
(7.25)where,
L = jump length, m (ft)
y_{1} = supercritical flow depth, m (ft)
Fr = supercritical Froude number
For a circular barrel, the jump length is equal to six times the subcritical sequent depth, where the sequent depth is computed using an empirical formulation (French, 1985).
The hydraulic analysis of brokenback culverts has been simplified by the computer application entitled Brokenback Computer Analysis Program, or BCAP (Hotchkiss et al. 2004).
The recommended design is limited to the following conditions:
For situations where the runout section is too short and/or there is insufficient tailwater for a jump to be completed (or initiated) within the barrel, modifications may be made to the outlet that will induce a jump. Two modification alternatives are presented in the following sections.
Placing a weir near the outlet of a culvert will induce a hydraulic jump under certain flow conditions (see Figure 7.11). The weir spans the width of a box culvert and is located approximately 3 m (10 ft) upstream from the culvert outlet. This location will facilitate debris removal from the upstream side of the weir. Drain holes in the weir prevent water from standing upstream. The distance L_{w} is referenced to the break in slope from the more steeply sloped section of the culvert. The rise of the culvert must be greater than y_{2}.
Figure 7.11. Weir Placed near Outlet of Box Culvert
Weirs of this nature are intended for use in conjunction with brokenback culverts, but may be used for chutes. They are placed in the horizontal runout section downstream from the change in slope exiting the steep section a distance to be determined during the design process. The weir is best used when there will be no standing water or design tailwater downstream from the culvert. Because flow will pass over the weir without the mitigating effect of tailwater, the flow will pass through critical depth and become supercritical as it approaches the culvert outlet. The need for downstream channel protection will be decreased due to the presence of the weir.
Hotchkiss, et al. (2005) tested conditions similar to those investigated by Forster and Skrinde (1950). Weirs near culvert outlets will induce hydraulic jumps for approach Froude numbers between 2 and 7. Designers interested in this dissipator may also wish to compare with the stilling basin designs found in Chapter 8.
The recommended design is limited to the following conditions:
The approach hydraulic conditions may be determined for brokenback culverts (see Section 7.4) or for chutes or any other steep approach to a horizontal runout section. However, the design procedure that follows has only been developed for rectangular shapes. Future extensions of the methodology will need to be supported by additional experimental testing.
The procedure makes use of the critical depth and the sequent depth for the hydraulic jump. The critical depth for a rectangular culvert was given earlier by Equation 7.1. The sequent depth is as follows:
(7.26)where,
y_{2} = sequent depth, m (ft)
Design of the weir primarily involves selecting its location and height. The relationship between weir height, approach depth, and Froude number is given by:
(7.27)where,
h_{w} = weir height, m (ft)
y_{1} = depth at the beginning of the runout section, m (ft)
The distance from the break in slope to the weir, approximately equal to the length of the hydraulic jump, is calculated as follows:
(7.28)Equation 7.28 is empirically based on the experimental data. For this reason, and because of its simplicity, it is used in this design procedure rather than Equation 7.25.
To calculate conditions downstream of the weir, near the culvert outlet, it is necessary to solve the energy equation iteratively for the depth downstream from the weir assuming no losses:
(7.29)where,
y_{3} = theoretical depth leaving the culvert, m (ft)
B = culvert width, m (ft)
Equation 7.29 has two solutions: subcritical and supercritical. The supercritical solution is taken because after passing through critical depth going over the weir the flow will be supercritical. The theoretical depth is adjusted for energy losses from the experimental data of Hotchkiss, et al. (2005):
(7.30)where,
y_{o} = outlet depth, m (ft)
α = constant equal to 0.015 m in SI and 0.05 ft in CU
The corresponding outlet velocity is computed from:
(7.31)The following design procedure may be used:
Step 1. Find the depth of flow, y_{1}, velocity, and Froude number at the beginning of the horizontal runout section. (This can be calculated using BCAP (Hotchkiss et al. 2004), HY8 (FHWA culvert analysis software), or other calculation tool to determine the depth entering the runout section.)
Step 2. Find critical depth, y_{c}, using Equation 7.1.
Step 3. Find the weir height (Equation 7.27) and location (Equation 7.28).
Step 4. Solve the energy equation (Equation 7.29) iteratively for the depth downstream from the weir.
Step 5. Compute the outlet depth (Equation 7.30) and velocity (Equation 7.31). Evaluate if energy dissipation is sufficient. Check culvert height for sufficient clearance.
Design Example: Outlet Weir in a Box Culvert (SI)
Design an outlet weir in the runout section of a RCB and determine the outlet conditions. The approach depth to the runout section is y_{1} = 0.375 m. Given:
Solution
Step 1. Find the depth of flow, y_{1}, velocity, and Froude number at the beginning of the horizontal runout section. y_{1} was given.
V_{1} = Q/(By_{1}) = 14.2/((4.3)(0.375)) = 8.81 m/s
Step 2. Find critical depth, y_{c}, using Equation 7.1. Unit discharge, q = Q/B =14.2/4.3 = 3.302 m^{2}/s.
Step 3. Find the weir height (Equation 7.27) and location (Equations 7.27 and 7.28).
(round to 11.3 m) 
Step 4. Solve the energy equation (Equation 7.29) iteratively for the depth downstream from the weir. From this trial and error process, y_{3} = 0.561 m
Step 5. Compute the outlet depth (Equation 7.30) and velocity (Equation 7.31). Evaluate if energy dissipation is sufficient.
If this velocity is acceptable, then the weir design is appropriate. Also, verify that the depth inside the culvert does not touch the top of the culvert. In this case, the rise of the culvert (2.44 m) is higher than the jump height (2.25 m) and the design is acceptable.
Design Example: Outlet Weir in a Box Culvert (CU)
Design an outlet weir in the runout section of a RCB and determine the outlet conditions. The approach depth to the runout section is y_{1} = 1.23 ft. Given:
Solution
Step 1. Find the depth of flow, y_{1}, velocity, and Froude number at the beginning of the horizontal runout section. y_{1} was given.
V_{1} = Q/(By_{1}) = 500/((14.0)(1.23)) = 29.0 ft/s
Step 2. Find critical depth, y_{c}, using Equation 7.1. Unit discharge, q = Q/B =500/14.0 = 35.71 ft^{2}/s.
Step 3. Find the weir height (Equation 7.27) and location (Equations 7.27 and 7.28).
(round to 37.2 ft) 
Step 4. Solve the energy equation (Equation 7.29) iteratively for the depth downstream from the weir. From this trial and error process, y_{3} = 1.847 ft
Step 5. Compute the outlet depth (Equation 7.30) and velocity (Equation 7.31). Evaluate if energy dissipation is sufficient.
If this velocity is acceptable, then the weir design is appropriate. Also, verify that the depth inside the culvert does not touch the top of the culvert. In this case, the rise of the culvert (8.0 ft) is higher than the jump height (7.43 ft) and the design is acceptable.
A drop in the culvert invert followed by a weir is shown in Figure 7.12. As with the weir near the culvert outlet (Section 7.5), this installation is intended to be used in conjunction with a brokenback culvert or chute where a steeply sloped section terminates with a horizontal runout section. The location of the drop beyond the break in slope from the steep barrel, L_{d}, is about 1.5 m (5 ft). The drop effectively decreases the slope of the steep culvert section, while the weir induces a hydraulic jump between the drop and weir. The drop may also be used if the height of the hydraulic jump for the design in Section 7.5 reaches the top of the culvert.
Figure 7.12. Drop followed by Weir
The design procedure is based upon Hotchkiss and Larson (2004) and Hotchkiss, et al. (2005). An extensive set of experiments was performed to define reductions in energy, momentum, and velocity at the culvert outlet due to the presence of a drop followed by a weir. Empirical results relate the drop height, h_{d}, to approach Froude number and weir height. Designers interested in this design may wish to compare the results with a straight drop structure (Section 11.1) at the end of the culvert or chute.
The approach hydraulic conditions, at location 1, may be determined for brokenback culverts (See Section 7.4) or for chutes or any other steep approach to a horizontal runout section. However, the design procedure that follows has only been developed for rectangular shapes. Future extensions of the methodology will need to be supported by additional experimental testing.
The procedure makes use of the critical depth and the sequent depth for the hydraulic jump. The critical depth for a rectangular culvert was given earlier by Equation 7.1. The sequent depth was also provided earlier as Equation 7.26.
Design of the weir primarily involves selecting its location and height. The relationship between weir height, approach depth, and Froude number is given by Equation 7.32. The weir height is also related to the drop height.
(7.32)where,
h_{w} = weir height, m (ft)
h_{d} = drop height, m (ft)
y_{1} = depth at the beginning of the runout section, m (ft)
To solve Equation 7.32, the ratio of h_{d}/h_{w} is selected to fall within the range of 0.60 and 0.65. The distance from the drop to the weir is calculated as follows:
(7.33)The quantity (y_{c} + h_{w}) approximates the sequent depth downstream from a classic hydraulic jump.
To calculate conditions downstream of the weir, near the culvert outlet, it is necessary to solve the energy equation iteratively for the depth downstream from the weir assuming no losses. Equation 7.29, presented earlier is used for this purpose. The theoretical depth calculated from Equation 7.29 is adjusted for energy losses from the experimental data of Hotchkiss, et al. (2005) using previously presented Equation 7.30.
The following design procedure may be used:
Step 1. Find the depth of flow, y_{1}, velocity, and Froude number at the beginning of the horizontal runout section. (This can be calculated using BCAP (Hotchkiss et al. 2004), HY8 (FHWA culvert analysis software), or other calculation tool to determine the depth entering the runout section.)
Step 2. Find critical depth, y_{c}, using Equation 7.1.
Step 3. Find the weir height (Equation 7.32), weir location (Equation 7.33), and drop height.
Step 4. Solve the energy equation (Equation 7.29) iteratively for the depth downstream from the weir.
Step 5. Compute the outlet depth (Equation 7.30) and velocity (Equation 7.31). Evaluate if energy dissipation is sufficient. Check culvert height for sufficient clearance.
Design Example: Drop and Outlet Weir in a Box Culvert (SI)
Design an outlet weir in the runout section of a RCB and determine the outlet conditions. The approach depth to the runout section is y_{1} = 0.375 m. Given:
Solution
Step 1. Find the depth of flow, y_{1}, velocity, and Froude number at the beginning of the horizontal runout section. y_{1} was given.
V_{1} = Q/(By_{1}) = 14.2/((4.3)(0.375)) = 8.81 m/s
Step 2. Find critical depth, y_{c}, using Equation 7.1. Unit discharge, q = Q/B =14.2/4.3 = 3.302 m^{2}/s.
Step 3. Find the weir height (Equation 7.32), weir location (Equations 7.27 and 7.33), and drop height. Select the ratio h_{d}/h_{w} = 0.64.
L_{W} = 6(y_{c} +h_{w}) = 6(1.04 + 0.771) = 10.87 (round to 10.9 m)
h_{d} = 0.64(h_{w}) = 0.64(0.771) = 0.49 m
Step 4. Solve the energy equation (Equation 7.29) iteratively for the depth downstream from the weir. From this trial and error process, y_{3} = 0.568 m
Step 5. Compute the outlet depth (Equation 7.30) and velocity (Equation 7.31). Evaluate if energy dissipation is sufficient.
y_{o} = 1.23y_{3} + α = 1.23(0.568) + 0.015 = 0.714 m
If this velocity is acceptable, then the weir design is appropriate. Also, verify that the depth inside the culvert does not touch the top of the culvert. In this case, the rise of the culvert (2.44 m) is higher than the jump height less the drop (2.25 0.49 m) and the design is acceptable.
Design Example: Drop and Outlet Weir in a Box Culvert (CU)
Design an outlet weir in the runout section of a RCB and determine the outlet conditions. The approach depth to the runout section is y_{1} = 1.23 ft. Given:
Solution
Step 1. Find the depth of flow, y_{1}, velocity, and Froude number at the beginning of the horizontal runout section. y_{1} was given.
V_{1} = Q/(By_{1}) = 500/((14.0)(1.23)) = 29.0 ft/s
Step 2. Find critical depth, y_{c}, using Equation 7.1. Unit discharge, q = Q/B =500/14.0 = 35.71 ft^{2}/s.
Step 3. Find the weir height (Equation 7.32), weir location (Equations 7.27 and 7.33), and drop height. Select the ratio h_{d}/h_{w} = 0.64.
L_{W} = 6(y_{c} +h_{w}) = 6(3.41 + 2.53) = 35.64 ft (round to 35.6 ft)
h_{d} = 0.64(h_{w}) = 0.64(2.53) = 1.62 ft
Step 4. Solve the energy equation (Equation 7.29) iteratively for the depth downstream from the weir. From this trial and error process, y_{3} = 1.86 ft
Step 5. Compute the outlet depth (Equation 7.30) and velocity (Equation 7.31). Evaluate if energy dissipation is sufficient.
y_{o} = 1.23y_{3} + α = 1.23(1.86) + 0.05 = 2.34 ft
If this velocity is acceptable, then the weir design is appropriate. Also, verify that the depth inside the culvert does not touch the top of the culvert. In this case, the rise of the culvert (8.0 ft) is higher than the jump height less the drop (7.43  1.62 ft) and the design is acceptable.
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