12.1 Culvert Design and Inlet/Outlet Control

Culvert Design and Inlet/Outlet Control

Culvert Hydraulics Overview

Culverts are closed conduits used to convey water through highway embankments or other obstructions. Unlike open channels, culverts operate under complex hydraulic conditions that can transition between open-channel flow, gravity full pipe flow, and pressurized sewer flow. The design of a culvert is based on the concept of "control," which refers to the specific location that limits the hydraulic capacity of the structure. Culverts are classified as operating under either Inlet Control or Outlet Control.

Inlet Control Characteristics

Under inlet control, the flow capacity of the culvert is determined solely by the geometry of the inlet. The barrel of the culvert is capable of conveying more flow than the inlet can accept. Consequently, downstream factors such as culvert roughness, length, slope, and tailwater levels do not affect the headwater elevation, provided the tailwater does not submerge the inlet.

• Flow Regime: The flow inside the culvert barrel is typically supercritical, and the culvert behaves as an open channel.
• Governing Factors:
  • Inlet cross-sectional area ($A$).
  • Inlet shape (circular, box, arch).
  • Inlet edge configuration (e.g., beveled edges, square edges, projecting ends).
  • Headwater depth ($HW$).
• Inlet Edge Performance: The shape of the inlet edge significantly affects the flow contraction. A projecting barrel end causes the most flow contraction and has the lowest capacity, whereas a headwall with beveled edges or a tapered inlet minimizes contraction, resulting in a higher capacity for the same headwater depth.

For design, the Federal Highway Administration (FHWA) HDS-5 manual provides empirical equations to calculate the headwater depth ($HW$) required for a given discharge ($Q$). For unsubmerged conditions ($Q / A D^{0.5} < 3.5$): $\frac{HW_i}{D} = \frac{H_c}{D} + c \left( \frac{Q}{A D^{0.5}} \right)^M - 0.5 S_0$ For submerged conditions ($Q / A D^{0.5} \ge 4.0$): $\frac{HW_i}{D} = a \left( \frac{Q}{A D^{0.5}} \right)^2 + Y - 0.5 S_0$ Where:

• $HW_i$ = Headwater depth above inlet invert (ft)
• $D$ = Interior height of culvert barrel (ft)
• $H_c$ = Specific head at critical depth ($d_c + V_c^2/2g$) (ft)
• $Q$ = Discharge (cfs)
• $A$ = Full cross-sectional area of the barrel (sf)
• $S_0$ = Culvert barrel slope (ft/ft)
• $a, c, M, Y$ = Constants based on inlet type (from HDS-5 tables)

Outlet Control Characteristics

Under outlet control, the capacity of the culvert is limited by downstream conditions, barrel characteristics, and the inlet geometry. The flow is subcritical, or the barrel flows full, meaning the downstream water level or friction in the pipe exerts a backwater effect all the way to the inlet.

• Governing Factors:
  • Inlet geometry (cross-sectional area $A$ and entrance loss coefficient $k_e$).
  • Barrel characteristics (length $L$, roughness Manning's $n$, and slope $S_0$).
  • Tailwater depth ($TW$) at the outlet.
• Energy Equation: The headwater depth ($HW$) is calculated using the energy equation: $HW = H + h_o - L S_0$ Where:
  • $H$ = Total head loss (ft)
  • $h_o$ = Equivalent hydraulic depth at the outlet (ft)
  • $L$ = Length of the culvert barrel (ft)
  • $S_0$ = Slope of the culvert barrel (ft/ft)

The total head loss ($H$) is the sum of the entrance loss ($H_e$), friction loss ($H_f$), and exit loss ($H_o$): $H = H_e + H_f + H_o = \left( 1 + k_e + \frac{29 n^2 L}{R^{1.33}} \right) \frac{V^2}{2g}$ Where:

• $k_e$ = Entrance loss coefficient (dimensionless, see table below)
• $n$ = Manning's roughness coefficient
• $R$ = Hydraulic radius of the full barrel (ft)
• $V$ = Mean flow velocity inside the barrel (fps)
• $g$ = Acceleration due to gravity ($32.2\text{ ft/s}^2$ or $9.81\text{ m/s}^2$)

Determining the Governing Control

Since a culvert can operate under either inlet or outlet control, the design procedure requires calculating the headwater depth for both conditions. The actual operating headwater is the greater of the two calculated values: $HW = \max(HW_{inlet}, HW_{outlet})$ This is because the culvert will naturally operate under the control mechanism that requires a higher energy level (headwater depth) to pass the design discharge.

Tailwater and the $h_o$ Parameter

Tailwater ($TW$) is the depth of water in the outlet channel measured relative to the outlet invert. The value of $h_o$ used in the headwater equation depends on whether the outlet is submerged or unsubmerged:

1. Submerged Outlet ($TW \ge D$): The culvert outlet flows completely full. $h_o = TW$
2. Unsubmerged Outlet ($TW < D$): The flow profile at the outlet is transitional. The equivalent hydraulic depth $h_o$ is approximated as: $h_o = \max\left(TW, \frac{d_c + D}{2}\right)$ Where $d_c$ is the critical depth (ft) and $D$ is the culvert height (ft).

Energy Dissipation and Outlet Protection

When culverts flow under supercritical conditions or pressurized outlet control, discharge velocities at the outlet can be extremely high (often exceeding $10\text{ fps}$). These velocities can cause severe erosion and scour in the receiving channel, leading to structural failure of the culvert or embankment.

• Energy Dissipators: Structures such as riprap aprons, stilling basins (USBR Type III/IV), impact basins, and baffled outlets are designed to reduce velocity and transition the flow safely to the downstream channel.
• Riprap Aprons (HEC-14): A simple and widely used energy dissipator. The design parameters include:
  • Apron Length ($L_a$): Determined based on the discharge velocity ($V$), diameter ($D$), and tailwater conditions. For low tailwater ($TW < 0.5D$): $L_a = \frac{1.7 Q}{D^{1.5}} + 8D$ For high tailwater ($TW \ge 0.5D$): $L_a = \frac{3.0 Q}{D^{1.5}} + 4D$
  • Apron Width: Expands from the culvert width to a terminal width at the end of the apron.
  • Median Stone Size ($d_{50}$): Sized to withstand the shear stress of the discharge velocity. A common simplified formula for circular pipes is: $d_{50} = 0.02 D \left( \frac{Q}{D^{2.5}} \right)^{1.33} \cdot \left( \frac{D}{TW} \right)$ Riprap must be angular, graded rock placed over a geotextile filter fabric to prevent piping of fine soils underneath.