Headwater under Inlet Control

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.
EquationObject236261
Equation 8-4.
where:
  • HW
    ic
    = inlet control headwater (ft. or m)
  • D
     = rise of the culvert barrel (ft. or m)
  • a
     to
    f
    = regression coefficients for each type of culvert (see the following
    table
    )
  • S
    0
    = culvert slope (ft./ft. or m/m)
  • F
     = function of average outflow discharge routed through a culvert; culvert barrel rise; and for box and pipe-arch culverts, width of the barrel, B, shown in Equation 8-5.
EquationObject237262
Equation 8-5.
where:
  • W
     = width or span of culvert (ft. or m).
Table 8-1: Regression Coefficients for Inlet Control Equations
Shape and Material
Entrance Type
a
b
c
d
e
f
RCP
Square edge w/headwall
0.087483
0.706578
-0.2533
0.0667
-0.00662
0.000251
-
Groove end w/headwall
0.114099
0.653562
-0.2336
0.059772
-0.00616
0.000243
-
Groove end projecting
0.108786
0.662381
-0.2338
0.057959
-0.00558
0.000205
-
Beveled ring
0.063343
0.766512
-0.316097
0.08767
-0.00984
0.000417
-
Improved (flared) inlet
0.2115
0.3927
-0.0414
0.0042
-0.0003
-0.00003
CMP
Headwall
0.167433
0.53859
-0.14937
0.039154
-0.00344
0.000116
-
Mitered
0.107137
0.757789
-0.3615
0.123393
-0.01606
0.000767
-
Projecting
0.187321
0.567719
-0.15654
0.044505
-0.00344
0.00009
-
Improved (flared) inlet
0.2252
0.3471
-0.0252
0.0011
-0.0005
-0.00003
Box
30-70º flared wingwall
0.072493
0.507087
-0.11747
0.02217
-0.00149
0.000038
-
Parallel to 15º wingwall
0.122117
0.505435
-0.10856
0.020781
-0.00137
0.0000346
-
Straight wingwall
0.144138
0.461363
-0.09215
0.020003
-0.00136
0.000036
-
45º wingwall w/top bevel
0.156609
0.398935
-0.06404
0.011201
-0.00064
0.000015
-
Parallel headwall w/bevel
0.156609
0.398935
-0.06404
0.011201
-0.00064
0.000015
-
30º skew w/chamfer edges
0.122117
0.505435
-0.10856
0.020781
-0.00137
0.000034
-
10-45º skew w/bevel edges
0.089963
0.441247
-0.07435
0.012732
-0.00076
0.000018
Oval
B>D
Square edge w/headwall
0.13432
0.55951
-0.1578
0.03967
-0.0034
0.00011
-
Groove end w/headwall
0.15067
0.50311
-0.12068
0.02566
-0.00189
0.00005
-
Groove end projecting
-0.03817
0.84684
-0.32139
0.0755
-0.00729
0.00027
Oval
D>B
Square edge w/headwall
0.13432
0.55951
-0.1578
0.03967
-0.0034
0.00011
-
Groove end w/headwall
0.15067
0.50311
-0.12068
0.02566
-0.00189
0.00005
-
Groove end projecting
-0.03817
0.84684
-0.32139
0.0755
-0.00729
0.00027
CM
Pipe arch
Headwall
0.111261
0.610579
-0.194937
0.051289
-0.00481
0.000169
-
Mitered
0.083301
0.795145
-0.43408
0.163774
-0.02491
0.001411
-
Projecting
0.089053
0.712545
-0.27092
0.792502
-0.00798
0.000293
Struct plate
Pipe arch
Projecting—corner plate (17.7 in. or 450 mm)
0.089053
0.712545
-0.27092
0.792502
-0.00798
0.000293
-
Projecting—corner plate (30.7 in. or 780 mm)
0.12263
0.4825
-0.00002
-0.04287
0.01454
-0.00117
CM arch
(flat bottom)
Parallel headwall
0.111281
0.610579
-0.1949
0.051289
-0.00481
0.000169
-
Mitered
0.083301
0.795145
-0.43408
0.163774
-0.02491
0.001411
-
Thin wall projecting
0.089053
0.712545
-0.27092
0.792502
-0.00798
0.000293
For HW
i
/D > 3.0, Equation 8-6, an orifice equation, is used to estimate headwater:
  • Determine the potential head from the centroid of the culvert opening, which is approximated as the sum of the invert elevation and one half the rise of the culvert. The effective area, A, and orifice coefficient, C, are implicit.
  • Determine the coefficient, k, by rearranging Equation 8-6 using the discharge that creates a HW/D ratio of 3 in the regression equation, Equation 8-7 (i.e., the upper limit of the
    Equation 8-1
    ):
EquationObject238263
Equation 8-6.
where:
  • HW
    i
    = inlet control headwater depth (ft. or m)
  • Q
    = design discharge (cfs or m
    3
    /s)
  • k
     = orifice equation constant
  • D
     = rise of culvert (ft. or m).
  • EquationObject239264
    Equation 8-7.
where:
  • Q
    3.0
     = discharge (cfs or m
    3
    /s) at which HW/D = 3.
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
HDS-5
(FHWA, Hydraulic Design of Highway Culverts) provide guidance for graphical solution of headwater under inlet control.