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Introduction
In 2004, the author offered a training seminar on "Flood Channel
Design". A question has been raised in the class several times as to
how a permissible velocity recommended for drop structure designs was translated
into its limiting flow Froude number. The Urban Storm Drainage Criteria
Manual (USDCM), 2001, recommends that a drop height be designed with a
permissible velocity of 5 fps or flow Froude number of 0.6 for soil types A and
B and of 7 fps or flow Froude number of 0.8 for soil types C and D. This paper
presents an investigation on the energy dissipation for a flow overtopping a
vertical drop and then identifies the optimal flow Froude number to achieve the
highest dissipation. It was found that the maximal dissipation rate is related
to flow Froude number and channel roughness. Using Manning's coefficients of
0.04 and 0.045 for riprap and gabion linings, it is found that the optimal
dissipation rate can be 80 to 85 percent of normal depth when the flow Froude
number ranges between 0.6 and 0.8.
Drop Height
When a channel is too steep for the design condition, roll waves occur between
banks (Guo 1999a). Erosion and scour occur to the channel bed and embankments.
To mitigate the channel aggradation and degradation problems, a drop structure
is often installed across the channel bed to create a backwater pool upstream of
the structure and a dissipation pool downstream of the structure (ASCE and WEF,
1982). The site of a drop structure should be located in a reasonably straight
channel reach with neither upstream nor downstream curved within 100 to 200 feet
of the structure. The foundation material must provide required supporting
strength to resist sliding force and overturning moment. The width of a drop
structure has to be wide enough to pass the design discharge. The height of a
drop structure is governed by the design flow condition, available construction
material, required structural stability, and cost. From a hydraulic point of
view, the height of a drop illustrated in Figure 1 is calculated as (Guo 1997):
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in which H = drop height, Sn = proposed channel bed slope, So = existing channel bed slope, and L = length of reach.
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The proposed channel slope depends on the permissible flow velocity and can be determined by re-arranging Manning's equation as:
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in which N = Manning's roughness, U = cross sectional average velocity in fps, and R = hydraulic radius in ft. The relationship between the permissible velocity and its flow Froude number is defined as:
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in which F = Froude number, g = gravitational acceleration, D = hydraulic depth, Q = design discharge, and T = top width. Substituting Eq 3 into Eq 2 yields
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Aided by Eq 4, the drop height is calculated as:
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(7) |
The height of a drop structure represents the energy dissipation for the selected reach length, L. In current practice, a value of 0.8 is recommended for the limiting Froude number (City of Denver in 2001, and Clark County in 1999). Using the trapezoidal channel in Figure 2 as an example, Eq 3 becomes
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(8) |
in which B = bottom width, Z1 = left side slop, and Z2 = right side slope. For the selected variables: B, Z1, and Z2, Eq 8 is solved for flow depth, Y, for the specified F and Q. Having known the flow depth, Y, the proposed slope can then be calculated by Eq 2 or Eq 6 and the drop height is determined by Eq 1 or Eq 7.
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Maximal Dissipation Rate
In addition to Froude number and design discharge discussed above, it is also
important to properly choose the reach length, L. It is recommended that a
vertical drop be limited to seven feet (Editor's Note: The USDCM recommends a
maximum drop height of three feet in urban areas for safety reasons) because
of expensive construction costs on retaining walls and footings. Eq 7 represents
the criteria of erosion control by reducing the flow Froude number. From the
hydraulic point of view, the higher the value of H, the more efficient the drop
structure will be. As a result, the objective for the design of drop structure
is set to be
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(9) |
The solution for Eq 9 is subject to the selected design parameters.
Design Example and Sensitivity
To illustrate the design procedure, a trapezoidal channel is employed for the
design of drop height. The design parameters include: L=100 ft, B=10 feet,
Z1=Z2=4, So= 0.05, and Q=1000 cfs. As recommended by Chow (1959), Henderson
(1966), and Barnes (1967), Manning's coefficient ranges from 0.040 to 0.045 for
grouted rocks and riprap. In this study, Manning's coefficient is set to be
N=0.045. As indicated in Eq 9, the dissipation rate depends on the flow Froude
number. As an example, let us set the erosion control criteria with F= 0.8.
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By trial and error, the flow condition includes: Y= 4.62 ft, A=131.6 ft2, R=2.74 ft, T=46.9 ft, and U=7.6 fps. Substituting the above variables into Eq 2 yields
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Eq 6 provides the same result. With a new channel slope, use Eq 1 to calculate the drop height as:
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(12) |
Of course, Eq 7 provides the same drop height as Eq 1. Eq 12 indicates that for this case, Fr=0.8 produces energy dissipation at 78% of the normal flow depth.
A good hydraulic design is the one that optimizes the integrated performance among design variables. At this point, a question is raised as to whether Eq 12 provides the maximal dissipation. For a given design condition, the flow Froude number in Eq 7 is the only design variable. Therefore, a sensitivity study was conducted to investigate the drop heights for Froude numbers ranging from 0.4 to 1.4. The results are plotted in Figure 3. For this example, the above flow condition can be further improved to increase the dissipation rate to 80% of normal depth or H/Y= 0.8 when reducing the flow Froude number to 0.6.
Furthermore, a similar analysis was performed to investigate the impact of Manning's coefficient. Consider N=0.04. As shown in Figure 3, the maximal dissipation can be H/Y= 0.85 at F=0.8. The trend in Figure 3 shows that the smoother the lining material is, the higher the flow Froude number can be. Of course, a stable channel does not sustain flows with a high Froude number. Therefore, the analysis in Figure 3 with N=0.03 is not recommended for designs.
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Conclusions
The vertical drop height represents the improvement on the channel bed slope.
Drop structures can be built with sheet piling, concrete walls with footings,
and riprap. In urban drainage design, it is more desirable to use a series of
low head drops than a high drop (Little and Murphy, 1982). Sediment erosion
control applied to an erodable channel can be achieved by reducing flow velocity
or flow Froude number. Design parameters include flow rate, water depth, flow
velocity, headwater difference, storage capacity upstream of the submerged dam,
seepage and uplift forces applied to the footings of the drop structure, and
channel flow regime, i.e subcritical flow or supercritical flow. Among these
variables, the design target dictates the type and height of the drop structure.
This study presents an approach to determine the drop height by maximizing the
dissipation to flow depth ratio. For a selected channel cross sectional
geometry, Eq 9 indicates that the maximal dissipation rate over a drop depends
on the flow Froude number and lining roughness.
Based on the sensitivity tests conducted in this study, the maximal dissipation rate over a drop structure ranges from 80 to 85 percent of the normal depth. The corresponding flow Froude number for the maximal dissipation rate ranges from 0.6 to 0.8. These findings agree with the current practice and recommendation. In practice, Eq 9 can be customized for the design condition. With a recommended drop height by Eq 9, the engineer needs to further examine flow hydraulics, channel morphologic, soil particles on the channel bed for the selection of building materials and design of underdrains.
Appendix I: References
American Society of Civil Engineers and Water Environment Federation (ASCE
and WEF) (1992). "Design and Construction of Urban Stormwater
Management Systems," ASCE Manuals and Reports of Engineering
Practice No. 77 and WEF Manual of Practice FD-20, New York, ASCE.
Barnes, H.H. (1967) "Roughness Characteristics of Natural Channels," Water Supply Paper No. 1849. Washington, DC, US Department of the Interior, Geologic Survey.
Chow, Ven T. (1959) "Open Channel Hydraulics," McGraw-Hill Book Company, New York.
Guo, James C.Y. (1997), "Open Channel Design and Flow Analysis," Water Resources Publication Company, Litleton, Colorado.
Guo, James C.Y. (1999a). "Roll Waves in High Gradient Channels," IWRA International J. of Water, Vol 3, No.1, March.
Little, W.C. and Murphey, J.B. (1982) "Model Study of Low Drop Grade Control Structures," J. of Hydraulic Division, 108 (HY10).
"Urban Storm Drainage Criteria Manual" (USWDCM) (2001), Ch 5, Vols 1 and 2, Urban Drainage and Flood Control District, Denver, Colorado.
Appendix II
B = bottom width
D = hydraulic depth
F = Froude number
g = gravitational acceleration
H = drop height
L = length of reach
N = Manning's roughness
k = unit conversion coefficient, equal to 1.48 for English units and one for
metric units
Q = design discharge
R = hydraulic radius
Sn = proposed channel bed slope
So = existing channel bed slope
T = top width
U = cross sectional average velocity
Z1 = left side slope
Z2 = right side slope
