Computing Inlet Control Headwater

For inlet control conditions, the capacity of the culvert is limited by the capacity of the culvert opening, rather than by conditions farther downstream. Extensive laboratory tests by the National Bureau of Standards, the Bureau of Public Roads, and other entities resulted in a series of equations, which describe the inlet control headwater under various conditions. These equations form the basis of the FHWA inlet control nomographs shown in the “Hydraulic Design of Highway Culverts” publication [FHWA, 1985]. The FHWA inlet control equations are used by the HEC‑RAS culvert routines in computing the upstream energy. The inlet control equations were developed for submerged and unsubmerged inlet conditions. These equations are:

Unsubmerged Inlet:

1)	$\begin{array}{l}\displaystyle \displaystyle \frac{HW_i}{D} = \frac{H_c}{D} +K \left[ \frac{Q}{AD^{0.5}} \right] ^M - 0.5S\end{array}$

2)	$\begin{array}{l}\displaystyle \displaystyle \frac{HW_i}{D} =K \left[ \frac{Q}{AD^{0.5}} \right] ^M\end{array}$

Submerged Inlet:

3)	$\begin{array}{l}\displaystyle \displaystyle \frac{HW_i}{D} =c \left[ \frac{Q}{AD^{0.5}} \right] ^2 +Y =0.5S\end{array}$

Symbol	Description	Units
$\begin{array}{l}HW_i\end{array}$	Headwater energy depth above the invert of the culvert inlet, feet
$\begin{array}{l}D\end{array}$	Interior height of the culvert barrel	feet
$\begin{array}{l}H_c\end{array}$	Specific head at critical depth (dc + Vc2/2g)	feet
$\begin{array}{l}Q\end{array}$	Discharge through the culvert	cfs
$\begin{array}{l}A\end{array}$	Full cross sectional area of the culvert barrel	feet²
$\begin{array}{l}S\end{array}$	Culvert barrel slope	feet/feet
$\begin{array}{l}K,M,c,Y\end{array}$	Equation constants, which vary depending on culvert shape and entrance conditions

Note that there are two forms of the unsubmerged inlet equation. The first form (1) is more correct from a theoretical standpoint, but form two (2) is easier to apply and is the only documented form of equation for some of the culvert types. Both forms of the equations are used in the HEC-RAS software, depending on the type of culvert.

The nomographs in the FHWA report are considered to be accurate to within about 10 percent in determining the required inlet control headwater [FHWA, 1985]. The nomographs were computed assuming a culvert slope of 0.02 feet per foot (2 percent). For different culvert slopes, the nomographs are less accurate because inlet control headwater changes with slope. However, the culvert routines in HEC‑RAS consider the slope in computing the inlet control energy. Therefore, the culvert routines in HEC-RAS should be more accurate than the nomographs, especially for slopes other than 0.02 feet per foot.