Inlet control occurs when the culvert barrel is capable of conveying more flow than the inlet will accept. Inlet control is possible when the culvert slope is hydraulically steep (d

c

> d

u

). The control section of a culvert operating under inlet control is located just inside the entrance. When the flow in the barrel is free surface flow, critical depth occurs at or near this location, and the flow regime immediately downstream is supercritical. Depending on conditions downstream of the culvert inlet, a hydraulic jump may occur in the culvert. Under inlet control, hydraulic characteristics downstream of the inlet control section do not affect the culvert capacity. Upstream water surface elevation and inlet geometry (barrel shape, cross-sectional area, and inlet edge) are the major flow controls.

A fifth-degree polynomial equation based on regression analysis is used to model the inlet control headwater for a given flow. Analytical equations based on minimum energy principles are matched to the regression equations to model flows that create inlet control heads outside of the regression data range. Equation 8-4 only applies when 0.5 ≤ HW

ic

/D ≤ 3.0.

Equation 8-4.

where:

Equation 8-5.

where:

For HW

i

/D > 3.0, Equation 8-6, an orifice equation, is used to estimate headwater:

Equation 8-6.

where:

where:

Generally for TxDOT designs, it is not considered efficient to design culverts for HW

i

/D < 0.5. However, if such a condition is likely, an open channel flow minimum energy equation (weir equation) should be used, with the addition of a velocity head loss coefficient. The minimum energy equation, with the velocity head loss adjusted by an entrance loss coefficient, generally describes the low flow portion of the inlet control headwater curve. However, numerical errors in the calculation of flow for very small depths tend to increase the velocity head as the flow approaches zero. This presents little or no problem in most single system cases because the flows that cause this are relatively small.

In many of the required calculations for the solution of multiple culverts, the inlet control curve must decrease continuously to zero for the iterative calculations to converge. Therefore, computer models modify this equation to force the velocity head to continually decrease to zero as the flow approaches zero.

The “Charts” in (FHWA, Hydraulic Design of Highway Culverts) provide guidance for graphical solution of headwater under inlet control.

Outlet control occurs when the culvert barrel is not capable of conveying as much flow as the inlet opening will accept. Outlet control is likely only when the hydraulic grade line inside the culvert at the entrance exceeds critical depth. (See Chapter 6 for Hydraulic Grade Line Analysis.) Therefore, outlet control is most likely when the culvert is on a mild slope (d

n

> d

c

). It is also possible to experience outlet control with a culvert on a steep slope (d

n

< d

c

) with a high tailwater such that subcritical flow or full flow exists in the culvert.

The headwater of a culvert in outlet control is a function of discharge, conduit section geometry, conduit roughness characteristics, length of the conduit, profile of the conduit, entrance geometry (to a minor extent), and (possibly) tailwater level.

The headwater of a culvert under outlet control can be adjusted, for practical purposes, by modifying culvert size, shape, and roughness. Both inlet control and outlet control need to be considered to determine the headwater. The following table provides a summary conditions likely to control the culvert headwater. Refer to Figure 8‑4 and Figure 8‑5 to identify the appropriate procedures to make the determination.

Table 8-2: Conditions Likely to Control Culvert Headwater

Description	Likely Condition
Hydraulically steep slope, backwater does not submerge critical depth at inside of inlet	Inlet control
Hydraulically steep slope, backwater submerges critical depth at inside of inlet	Outlet control
Hydraulically steep slope, backwater close to critical depth at inlet	Oscillate between inlet and outlet control.
Hydraulically mild slope	Outlet control

Outlet control headwater is determined by accounting for the total energy losses that occur from the culvert outlet to the culvert inlet. Figure 8‑7 and Figure 8‑8 and associated procedures in Section 4 should be used to analyze or design a culvert.

Outlet control headwater HW

oc

depth (measured from the flowline of the entrance) is expressed in terms of balancing energy between the culvert exit and the culvert entrance as indicated by Equation 8-8.

Equation 8-8.

where:

For convenience energy balance at outlet, energy losses through barrel, and energy balance at inlet should be considered when determining outlet control headwater.

When the tailwater controls the outlet flow, Equation 8-10 represents the energy balance equation at the conduit outlet. Traditional practice has been to ignore exit losses. If exit losses are ignored, the hydraulic grade line inside the conduit at the outlet,outlet depth, H

o

, should be assumed to be the same as the hydraulic grade line outside the conduit at the outlet and Equation 8-10 should not be used.

Equation 8-10.

where:

The outlet depth, H

o

, is the depth of the hydraulic grade line inside the culvert at the outlet end. The outlet depth is established based on the conditions shown below.

Table 8-3: Outlet Depth Conditions

If...	And...	Then...
Tailwater depth (TW) exceeds critical depth (d c ) in the culvert at outlet	Slope is hydraulically mild	Set H o using Equation 8-10, using the tailwater as the basis.
Tailwater depth (TW) is lower than critical depth (d c ) in culvert at outlet	Slope is hydraulically mild	Set H o as critical depth.
Uniform depth is higher than top of the barrel	Slope is hydraulically steep	Set H o as the higher of the barrel depth (D) and depth using Equation 8-10.
Uniform depth is lower than top of barrel and tailwater exceeds critical depth	Slope is hydraulically steep	Set H o using Equation 8-10.
Uniform depth is lower than top of barrel and tailwater is below critical depth	Slope is hydraulically steep	Ignore, as outlet control is not likely.

NOTE: For hand computations and some computer programs, H

o

is assumed to be equal to the tailwater depth (TW). In such a case, computation of an exit head loss (h

o

) would be meaningless since the energy grade line in the culvert at the outlet would always be the sum of the tailwater depth and the velocity head inside the culvert at the outlet (h

vo

).

Department practice is to consider flow through the conduit occurring in one of four combinations:

If free surface flow is occurring in the culvert, the hydraulic parameters are changing with flow depth along the length of the culvert as seen in Figure 8‑9. It is necessary to calculate the backwater profile based on the outlet depth, H

o

.

By definition, a free-surface backwater from the outlet end of a culvert may only affect the headwater when subcritical flow conditions exist in the culvert. Subcritical, free-surface flow at the outlet will exist if the culvert is on a mild slope with an outlet depth (H

o

) lower than the outlet soffit or if the culvert is on a steep slope with a tailwater higher than critical depth at the culvert outlet and lower than the outlet soffit.

The is used to determine the water surface profile (and energy losses) though the conduit. The depth, H

o

, is used as the starting depth, d

1

. For subcritical flow, the calculations begin at the outlet and proceed in an upstream direction. Use the depth, H

o

, as the starting depth, d

1

, in the Direct Step calculations.

When using the direct step method, if the inlet end of the conduit is reached without the calculated depth exceeding the barrel depth (D), it verifies that the entire length of the conduit is undergoing free surface flow. Set the calculated depth (d

2

) at the inlet as H

i

and refer to Energy Balance at Inlet to determine the headwater.

When using the direct step method, if the calculated depth (d

2

) reaches or exceeds the barrel depth (D), the inside of the inlet is submerged. Refer to Type AB - Free surface at outlet and full flow at inlet for a description. This condition is possible if the theoretical value of uniform depth is higher than the barrel depth.

If full flow is occurring in the conduit, rate of energy losses through the barrel is constant (for steady flow) as seen in Figure 8‑10. The hydraulic grade line is calculated based on outlet depth, H

o

, at the outlet.

Full flow at the outlet occurs when the outlet depth (H

o

) equals or exceeds barrel depth D. Full flow is maintained throughout the conduit if friction slope is steeper than conduit slope, or if friction slope is flatter than conduit slope but conduit is not long enough for the hydraulic grade line to get lower than the top of the barrel.

NOTE: Refer to Type BA – Submerged Exit, Free flow at Inlet to determine whether the entire conduit flows full.

Equation 8-11determines the energy loss (friction loss) through the conduit:

Equation 8-11.

where:

Equation 8-12 is used to compute the depth of the hydraulic grade line at the inside of the inlet end of the conduit. Refer to Energy Balance at Inlet to determine the headwater.

Equation 8-12.

where:

Equation 8-13 is used to calculate friction slope. If friction slope is flatter than the conduit slope, the hydraulic grade line may drop below the top of the barrel. If this occurs, refer to Type BA - Full Flow at the outlet and free surface flow at the inlet.

Equation 8-13.

where:

S

f

= friction slope (ft./ft. or m/m)

If the friction slope is flatter than the conduit slope, it is possible that full flow may not occur along the entire length of the culvert (see the Table 8-5 on ). The following steps should be followed:

When the outlet is not submerged, full flow will begin within the conduit if the culvert is long enough and the flow high enough. Figure 8‑12 illustrates this condition. This condition is possible if the theoretical value of uniform depth is higher than the barrel depth. The following steps should be followed:

The outlet control headwater, HW

oc

, is computed by balancing the energy equation, depicted as Equation 8-15. The hydraulic grade at the inside face of the culvert at the entrance will need to be known. See . The velocity at the entrance (v

i

) is used to compute the velocity head at the entrance (h

vi

).

Equation 8-15.

where:

The entrance loss, h

e

, depends on the velocity of flow at the inlet, v

i

, and the entrance configuration, which is accommodated using an entrance loss coefficient, C

e

.

Equation 8-16.

where:

NOTE: The pipes of pipe runner SETs have been proven to be within the tolerance of the entrance loss equations. Therefore, the entrance should be evaluated solely for its shape and the effect of the pipes should be ignored.

Values of C

e

are shown on the following table (Entrance Loss Coefficients) based on culvert shape and entrance condition. (AASHTO Highway Drainage Manual Guidelines, 4th Ed, Table 4-1)

When the flow becomes unstable, a phenomenon termed slug flow may occur. In this condition the flow oscillates between inlet control and outlet control due to the following instances:

The methods recommended in this chapter accommodate the potential for slug flow to occur by assuming the higher of inlet and outlet control headwater.

The outlet velocity, v

o

, depends on the culvert discharge (Q) and the cross-sectional area of flow at the outlet (A

o

) as shown in Equation 8-17.

Equation 8-17.

For inlet control under steep slope conditions, estimate the depth at the outlet using one of the following approaches:

The Direct Step Backwater Method uses the same basic equations as the Standard Step Backwater Method but is simpler to use because no iteration is necessary. In the Direct Step Method, an increment (or decrement) of water depth (δd) is chosen and the distance over which the depth change occurs is computed. The accuracy depends on the size of δd. The method is appropriate for prismatic channel sections such as occur in most conduits. It is useful for estimating supercritical profiles and subcritical profiles.

The procedure for subcritical flow (d > d

c

) but steep slope (d

c

> d

u

) is similar with the following exceptions:

The procedure for supercritical flow (d < d

c

) and steep slope is similar with the following exceptions:

Figure 8-15 provides a sample plot of depth and momentum function and an associated specific energy plot. For a given discharge and given energy and momentum, there exist two possible depths, one less than critical depth (supercritical flow) a sequent (or conjugate) depth greater than critical depth (subcritical flow). With a proper configuration, the water flowing at the lower depth in supercritical flow can “jump” abruptly to its sequent depth in subcritical flow. This is called a hydraulic jump. With the abrupt change in flow depth comes a corresponding change in cross-sectional area of flow and a resulting decrease in average velocity.

By comparing the two curves at a supercritical depth and its sequent depth, it is apparent that the hydraulic jump involves a loss of energy. Also, the momentum function defines critical depth as the point at which minimum momentum is established.

The balance of forces is represented using a momentum function (Equation 8-22):

Equation 8-22.

where:

The term A $\stackrel{\_}{d}$
represents the first moment of area about the water surface. Assuming no drag forces or frictional forces at the jump, conservation of momentum maintains that the momentum function at the approach depth, M

1

, is equal to the momentum function at the sequent depth, M

s

.

The potential occurrence of the hydraulic jump within the culvert is determined by comparing the outfall conditions with the sequent depth of the supercritical flow depth in the culvert. The conditions under which the hydraulic jump is likely to occur depend on the slope of the conduit.

Under mild slope conditions (d

c

< d

u

) with supercritical flow in the upstream part of the culvert, the following two typical conditions could result in a hydraulic jump:

Under steep slope conditions, the hydraulic jump is likely only when the tailwater is higher than the sequent depth.

A direct solution for sequent depth, d

s

is possible for free surface flow in a rectangular conduit on a flat slope using Equation 8-23. If the slope is greater than about 10 percent, a more complex solution is required to account for the weight component of the water. FHWA provides more detail for such conditions.

Equation 8-23.

where:

A direct solution for sequent depth in a circular conduit is not feasible. However, an iterative solution is possible by following these equations:

where:

Equation 8-25.

where:

Equation 8-26.

Equation 8-27.

Equation 8-24 applies to other conduit shapes having slopes of about 10 percent or less. The first moment of area about the surface, A $\stackrel{\_}{d}$
, is dependent on the shape of the conduit and depth of flow. A relationship between flow depth and first moment of area must be acquired or derived.

Where water flows both over the roadway and through a culvert (see Figure 8‑17), a flow distribution analysis is required to define the hydraulic characteristics. This is a common occurrence where a discharge of low design AEP (low probability of occurrence) is applied to a facility designed for a lower design frequency.

For example, a complete design involves the application and analysis of a 1% AEP discharge to a hydraulic facility designed for a much smaller flood. In such a case, the headwater may exceed the low elevation of the roadway, causing part of the water to flow over the roadway embankment while the remainder flows through the structure. The headwater components of flow form a common headwater level. An iterative process is used to establish this common headwater.

The following procedure is an iterative approach that is reasonable for hand computations and computer programs:

For any given culvert, the control (outlet or inlet) might vary with the discharge. Figure 8‑18 shows sample plots of headwater versus discharge for inlet and outlet control. The envelope (shown as the bold line) represents the highest value of inlet and outlet headwater for any discharge in the range. This envelope is termed a performance curve.

In this example, inlet control prevails at lower discharges and flow transitions to outlet control as the discharge increases. The flatter portion represents the effect of roadway overflow. The performance curve can be generated by calculating the culvert headwater for increasing values of discharge. Such information is particularly useful for performing risk assessments and for hydrograph routing through detention ponds and reservoirs.

The traditional assumption in the design of typical highway culverts is continuity of the hydraulic grade line. At the outlet, this implies that when the tailwater is higher than critical depth and subcritical flow exists, the hydraulic grade line immediately inside the barrel is equal to the tailwater level. This is reasonable for most normal culvert designs for TxDOT application. However, by inference there can be no accommodation of exit losses because the energy grade line immediately inside the culvert can only be the hydraulic grade line plus the velocity head, no matter what the velocity is in the outfall.

Occasionally, an explicit exit loss may need to be accommodated. Some examples are as follows:

If such a need arises, base the starting hydraulic grade level (H

o

) is based on balancing between the outside and inside of the culvert face at the outlet. A common expression for exit loss appears in Equation 8-30. This assumes that the tailwater velocity (v

TW

) is lower than the culvert outlet velocity (v

o

) and the tailwater is open to the atmosphere. If the above approach is used, it is most likely that the outlet depth (H

o

) will be lower than the tailwater. This conforms to basic one-dimensional hydrostatic principles.

Equation 8-29.

where:

Equation 8-30.

where: