7.5 Outlet Control Computations

For outlet control, computations focus on the losses experienced through the culvert. The losses are added to the downstream water surface elevation to determine a headwater elevation. Assuming that the culvert flows full for at least a portion of its length, these losses are found as follows.


where	h_en	=	the head loss between sections 3 and BU (from Figure 7.1), due to the entrance geometry (ft, m)
	K_en	=	the entrance loss coefficient (ranging from 0.2 to 0.9)
	V²/2g	=	the velocity head of the culvert flowing full (ft, m)

Figure 7.10 shows four common entrance and exit conditions and the associated entrance loss coefficient for each. Table 7.3 lists entrance loss coefficients for a variety of culvert inlets.


where	h_ex	=	the head loss between sections BD and 2, due to the exit conditions (ft, m)
	K_ex	=	the exit loss coefficient (normally equal to 1.0)
	V_TW	=	the average velocity at Section 2 in the downstream channel (ft, m)

Table 7.3 Entrance loss coefficients for common entrance shapes under outlet control (FHWA).
Culvert Type	Entrance Type	Entrance Loss Coefficient, K_en
Pipe, Concrete	Projecting from fill, socket end (groove end)	0.2
	Projecting from fill, square-cut end	0.5
	Headwall or headwall with wingwalls Socket end of pipe (groove end) Square edge Rounded (radius = D/12)	0.2 0.5 0.2
	Mitered to conform to fill slope	0.7
	End section conforming to fill slope¹	0.5
	Beveled edges, 33.7° or 45° bevels	0.2
	Side- or slope-tapered inlet	0.2
Pipe or Pipe Arch, Corrugated Metal	Projecting from fill (no headwall)	0.9
	Headwall or headwall and wingwalls square edge	0.5
	Mitered to conform to fill slope, paved or unpaved slope	0.7
	End section conforming to fill slope^a	0.5
	Beveled edges, 33.7° or 45° bevels	0.2
	Side- or slope-tapered inlet	0.2
Box, Reinforced Concrete	Headwall parallel to embankment (no wingwalls) Square edged on three sides Rounded on three edges to radius of ¹/12 barrel dimension or beveled edges on three sides	0.5 0.2
	Wingwalls at 30° to 75° to barrel Square edged at crown Crown edge rounded to radius of ¹/12 barrel dimension or beveled top edge	0.4 0.2
	Wingwalls at 10° to 25° to barrel, square edged at crown	0.5
	Wingwalls parallel (extension of sides), square edged at crown	0.7
	Side- or slope-tapered inlet	0.2

¹"End Section conforming to fill slope," made of either metal or concrete, are the sections commonly available from manufacturers. From limited hydraulic tests they are equivalent in operation to a headwall in both inlet and outlet control. Some end sections incorporating a closed taper in their design have a superior hydraulic performance. These latter sections can be designed using the information given for the beveled inlet.

For hand computations using Equation 7.5, the exit loss coefficient (Kex) is assumed equal to 1 and the tailwater velocity head is often assumed to be negligible, resulting in a conservative estimate of the exit loss. Equation 7.5 then reduces to the exit loss equal to the culvert velocity head. In HEC-RAS however, the tailwater velocity head is computed and then subtracted from the culvert velocity head.


where	h_f	=	the head loss due to friction through the culvert barrel (ft, m)
	n	=	the Manning coefficient for the culvert material (dimensionless)
	L	=	the length of the culvert (ft, m)
	R	=	the hydraulic radius of the culvert (ft, m)
	k	=	1.486 for English units, 1.0 for SI

Combining the entrance, exit, and friction losses (Equation 7.4 through Equation 7.6) yields the following equation for loss:

Equation 7.7 is valid for English units. For SI units, replace the constant 29.1 with 19.6.

Normally, the entrance loss is the only coefficient (other than n) required to solve Equation 7.7. This equation is mainly used in hand computations for outlet control analysis with both the tailwater and headwater velocity heads considered negligible. HEC-RAS does not use Equation 7.7, but rather Equation 7.4 through Equation 7.6 to compute individual losses under outlet control conditions and includes the tailwater and headwater velocity heads in the analysis.

Example 7.2 Analysis of culvert under outlet control

A 6 ft wide by 4 ft high concrete box culvert is 100 ft long, with upstream and downstream invert elevations of 342 and 341.7 ft, respectively. The entrance consists of a headwall with 45° bevels. For a discharge of 200 ft³/s, compute the headwater elevation for a tailwater depth (a) two feet above the top of the downstream end of the culvert, (b) equal to the elevation of the top of the downstream end of the culvert, and (c) equal to 2 ft below the top of the downstream end of the culvert. Assume the headwater and tailwater velocity heads are negligible.

Solution

The culvert slope is 0.003 ft/ft. Paved slopes less than about 0.005 normally result in subcritical flow and outlet control is the expected flow condition through the culvert. Solving for both y_n and y_c in the box culvert (using the procedures in Chapter 2) for a flow of 200 ft³/s results in a critical depth of 3.26 ft and a normal depth of 3.78 ft for a Manning's n = 013. Because the normal depth exceeds the critical depth, the flow will be subcritical, and outlet control will govern. For the entrance conditions specified, Table 7.3 lists the entrance loss coefficient (K_en) as 0.2.

(a) TW elevation = 341.7 + 4 + 2 = 347.7 ft. For this tailwater, the culvert exit is submerged and the culvert will flow full (condition A for outlet control from Figure 7.7). The headwater elevation is computed and is compared to the elevation of the top of the culvert on the upstream end (346 ft). If the HW depth exceeds the vertical height of the culvert by at least 20 percent, the culvert will flow full and Condition A is confirmed. For full culvert flow, the flow area is 24 ft² and the culvert velocity is 200/24 = 8.33 ft/s. The n value for concrete is assumed to be 0.013 and the hydraulic radius for full culvert flow is A/P = 24/20 = 1.2 ft. Applying Equation 7.7 for outlet conditions gives

The headwater elevation is found by adding the tailwater elevation (347.7 ft) and the head losses through the culvert (1.71 ft), yielding a headwater elevation of 349.41 ft and a headwater depth of 7.41 ft (HW elevation minus culvert invert elevation). The headwater depth exceeds 120 percent of the height of the culvert by 2.41 ft, confirming that full culvert flow is occurring and Condition A is the outlet flow situation.

(b) TW elevation = 341.7 + 4 = 345.7 ft. Because the tailwater depth equals the height of the culvert, Condition B or C of Figure 7.7 could be appropriate. As for part (a), the headwater depth is computed and compared to 120 percent of the culvert height. For this tailwater condition, all the values computed in part (a) are the same except for the tailwater elevation. Therefore, the new headwater elevation is 345.7 + 1.71 ft = 347.41 ft. Thus, the headwater depth is 5.41 ft, which exceeds 120 percent of the vertical height of the culvert (4 ft), confirming that Condition C displays the correct profile.

(c) TW elevation = 341.7 + 2 = 343.7 ft. The tailwater elevation is one-half of the culvert height, thus the culvert may not flow full. Also, the tailwater elevation is less than the elevation of critical depth at the outlet (341.7 + 3.26 = 344.96 ft), so critical depth at the culvert exit becomes the tailwater elevation and Condition D or Condition E from Figure 7.7 will occur. The water surface profile through the culvert must be computed starting at critical depth at or near the culvert exit. Because the culvert is prismatic, either the direct step or standard step backwater solution (presented in Chapter 2) may be applied to determine the flow depth at the culvert entrance. Performing a backwater computation similar to that in Example 2.12 gives a depth of 3.64 ft at the culvert entrance. Because this depth is less than the vertical height of the culvert, the culvert does not flow full, and Condition E from Figure 7.7 appears appropriate. The friction loss through the culvert (needed for the direct step and standard step methods) is computed by determining the average friction slope from the s_f values at the culvert entrance and exit. These values are computed as 0.0044 at the exit and 0.00329 at the entrance, yielding an average friction slope value of 0.00383. Velocity at the entrance is 9.16 ft/s at a depth of 3.64 ft, and the critical velocity at the exit is 10.22 ft/s. Thus, the head losses through the culvert are

The headwater elevation is found from adding the tailwater elevation for critical depth (344.96 ft) plus the head loss (2.26 ft) to obtain a headwater elevation of 347.25 ft at the culvert entrance. The headwater depth is therefore 5.22 ft, which exceeds 120 percent of D by 0.42 ft. Therefore, Condition D would initially appear to be appropriate for the culvert. This would be the end of the example for hand computations. However, although the entrance is computed as submerged, the water surface is below the top of the culvert immediately inside the upstream end of the structure, as determined by direct step backwater computations. This condition indicates that Condition E is correct. The initial simplification of assuming the headwater and tailwater velocity heads are negligible causes this conflict.

In reality, these two velocity heads are not negligible for part (c) and are probably 0.3 to 0.6 ft (corresponding to 4-6 ft/s) if the geometry outside the culvert were known. These values for velocity head are significant to these computations. It is further noted that the exit loss for this computation (1.62 ft) is over 70 percent of the total loss through the culvert, again significantly affected by the negligible tailwater velocity assumption. If the tailwater velocity is 5 ft/s, the exit loss drops to 1.23 ft, using the full form of Equation 7.5. If the actual headwater and tailwater velocities and velocity heads were incorporated, as they are in a HEC-RAS computation, smaller entrance and exit losses would result, giving a smaller total loss and a headwater depth significantly lower than 347.22 ft. In addition, because velocity is neglected in this example, the computed HW depth is actually to the energy grade line. Subtracting the velocity head would yield a lower water surface elevation. The headwater elevation is less than 347 and Condition E from Figure 7.7 is applicable for this example.