PHYSICAL MODELING FOR
SIDECHANNEL WEIRS
by
KaLeung Lee and E. R. Holley
Center for Research in Water Resources
The University of Texas at Austin
Austin, TX 78712 USA
CRWR Online Report 022
http://www.crwr.utexas.edu/online.shtml
prepared for
Harris Flood Control District
Suite 200
9900 Northwest Freeway
Houston, TX 77092 USA
April 15, 2002
i
TABLE OF CONTENTS
1  Introduction.................................................................................................................................1
1.1  Regional Basins .................................................................................................................. 1
1.2  Objectives ........................................................................................................................... 2
2  Background.................................................................................................................................5
2.1  Side Weir Flow Conditions ................................................................................................ 5
2.2  No Weir Flow ..................................................................................................................... 6
2.3  Reverse Flow ...................................................................................................................... 6
2.4  Forward Flow ..................................................................................................................... 7
2.4.1  Water Surface Profiles................................................................................................. 7
2.4.2  Previous Work of Others ............................................................................................. 8
2.5  Previous Work at CRWR on Hydraulic Computations.................................................... 10
2.5.1  Purpose ...................................................................................................................... 10
2.5.2  Forward Weir Flow.................................................................................................... 11
2.5.2.1  Method of Analysis in the Previous Project ....................................................... 11
2.5.2.2  Flow Asymmetry ................................................................................................ 12
2.5.2.3  Downstream Depth ............................................................................................. 13
2.5.2.4  Kinetic Energy and Momentum Correction Factors........................................... 14
2.5.2.5  Weir Discharge ................................................................................................... 15
2.5.2.6  Upstream Depth.................................................................................................. 15
2.5.3  Reverse Weir Flow .................................................................................................... 16
2.5.3.1  General Approach............................................................................................... 16
2.5.3.2  Discharge Coefficients........................................................................................ 16
2.5.3.3  Submergence Correction Factor ......................................................................... 17
2.5.3.4  Channel Depths and Additional Head Change ................................................... 17
2.6  Previous Experimental Work at CRWR........................................................................... 18
2.6.1  Introduction................................................................................................................ 18
2.6.2  Results for Unsubmerged Forward Flow................................................................... 18
2.6.3  Results for Submerged Forward Flow....................................................................... 19
2.6.4  Unsubmerged Flow in Tapered Channels.................................................................. 20
2.6.5  Results for Weirs Downstream of Bends................................................................... 21
2.6.6  Reverse Flow ............................................................................................................. 21
2.7  Valves on Drainage Culverts............................................................................................ 22
2.7.1  Tideflex Valves.......................................................................................................... 22
2.7.1.1  General................................................................................................................ 22
2.7.1.2  Manufacturer's Information ................................................................................ 22
2.7.1.3  Calculation Method for Submerged Valves ....................................................... 26
2.7.1.4  Calculation Method for Unsubmerged Valves ................................................... 28
2.7.2  Flap Gates .................................................................................................................. 31
ii
3  Experimental Facilities.............................................................................................................33
3.1  Physical Model for Previous Project ................................................................................ 33
3.2  Modification of the Physical Model to 4H:1V Side Slopes ............................................. 36
3.3  Measurement of Discharges ............................................................................................. 38
3.3.1  Calibration of Venturi Meter ..................................................................................... 38
3.3.2  Calibration of VNotch Weir..................................................................................... 39
3.3.3  Calibration of Flow Sensor........................................................................................ 40
3.4  Measurement of Velocities............................................................................................... 41
3.5  Measurement of Water Surface Elevations ...................................................................... 43
4  ReEvaluation of Side Weir Discharge Coefficients ................................................................45
4.1  Derivation of Equation for Changes in Depth .................................................................. 45
4.1.1  Unsubmerged Flow in Prismatic Channels................................................................ 49
4.1.2  Submerged Flow in Prismatic Channels.................................................................... 51
4.1.3  Unsubmerged Flow in Tapered Channels.................................................................. 51
4.2  Optimization and Regression Analysis for Channel with 2.5H:1V Side Slopes.............. 51
4.2.1  Unsubmerged Flow in Prismatic Channels with 2.5H:1V Side Slopes..................... 51
4.2.1.1  Constant Discharge Coefficient.......................................................................... 51
4.2.1.2  Variable Discharge Coefficient .......................................................................... 52
4.2.2  Submerged Flow in Prismatic Channels with 2.5H:1V Side Slopes......................... 53
4.2.3  Unsubmerged Flow in Tapered Channels.................................................................. 54
4.3  Comparison between Measured and Calculated Values for 2.5H:1V Side Slopes .......... 55
4.4  Effects of Channel Slope and Roughness for 2.5H:1V Side Slopes ................................ 69
5  Discharge and Head Loss Experiments for 4:1 Side Slopes.....................................................73
5.1  Introduction ...................................................................................................................... 73
5.2  Model results .................................................................................................................... 73
5.2.1  Analysis of Data using Method A.............................................................................. 73
5.2.2  Analysis of Data using Method B.............................................................................. 76
6  Flow Asymmetry ......................................................................................................................85
6.1  Introduction ...................................................................................................................... 85
6.1.1  Background................................................................................................................85
6.1.2  Related Literature ...................................................................................................... 85
6.1.3  Objective.................................................................................................................... 87
6.2  Equations for the Channel Flow ....................................................................................... 87
6.3  Flow Conditions ............................................................................................................... 89
iii
6.4  Velocity Measurements .................................................................................................... 90
6.4.1  Measurement Procedures........................................................................................... 91
6.4.2  Integrations ................................................................................................................ 92
6.4.3  Case A........................................................................................................................ 92
6.4.4  Case B........................................................................................................................ 93
6.4.5  Case C........................................................................................................................ 94
6.4.6  Case D........................................................................................................................ 96
6.4.7  Case E ........................................................................................................................ 97
6.4.8  Case F ........................................................................................................................ 97
6.4.9  Case G........................................................................................................................ 98
6.4.10  Case H...................................................................................................................... 99
6.5  ? and ? Values at Downstream End of Weir.................................................................. 100
6.6  Components of ? and ?.................................................................................................. 102
6.6.1  Variation with Flow Distance.................................................................................. 102
6.6.2  Variation with Diversion ......................................................................................... 106
6.7  Length for Flow ReEstablishment................................................................................. 106
6.8  Momentum and Energy Balances................................................................................... 109
6.9  Application ..................................................................................................................... 112
7  Diversion Culverts..................................................................................................................115
7.1  Introduction .................................................................................................................... 115
7.2  The physical model......................................................................................................... 115
7.3  Model results .................................................................................................................. 117
7.3.1  Unsubmerged flow................................................................................................... 117
7.3.2  Submerged flow....................................................................................................... 118
7.4  Calculation Procedure..................................................................................................... 120
8  Conclusions ............................................................................................................................121
9  References...............................................................................................................................125
iv
10  Appendices ...........................................................................................................................127
Appendix 1  Data from previous project (Tynes, 1989).........................................................127
Appendix 1.1  Unsubmerged Flow Conditions ..................................................................127
Appendix 1.2  Submerged Flow Conditions.......................................................................133
Appendix 1.3  Tapered Channels........................................................................................134
Appendix 2  Weir and channel geometries investigated in previous project for
unsubmerged flow .............................................................................................137
Appendix 3  Results of simulation of side weir flow for different slopes and roughness......139
Appendix 3.1  Results of simulation using Method A .......................................................140
Appendix 3.2  Results of simulation using Method B........................................................151
Appendix 4  Summary of model data for 4H:1V side slopes.................................................163
Appendix 5  Components of ? and ? .....................................................................................165
Appendix 5.1  Variation of components of ? and ? with distance for 2.5H:1V side
slopes...........................................................................................................165
Appendix 5.2  Variation of components of ? and ? with distance for 4H:1V side
slopes...........................................................................................................166
Appendix 5.3  Components of ? and ? just downstream of weir for 2.5H:1V side
slopes...........................................................................................................166
Appendix 5.4  Components of ? and ? just downstream of weir for 4H:1V side
slopes...........................................................................................................167
Appendix 6  Momentum and energy balances .......................................................................169
Appendix 6.1  Momentum balance for Case A ..................................................................169
Appendix 6.2  Energy balance for Case A..........................................................................169
Appendix 6.3  Momentum balance for Case B...................................................................170
Appendix 6.4  Energy balance for Case B..........................................................................170
Appendix 6.5  Momentum balance for Case C...................................................................171
Appendix 6.6  Energy balance for Case C..........................................................................171
Appendix 6.7  Momentum balance for Case D ..................................................................172
Appendix 6.8  Energy balance for Case D..........................................................................172
Appendix 6.9  Momentum balance for Case F...................................................................173
Appendix 6.10  Energy balance for Case F ........................................................................173
Appendix 7  Data for diversion culverts.................................................................................175
Appendix 7.1  Results for diversion culverts with three barrels, unsubmerged flow.........175
Appendix 7.2  Results for diversion culverts with three barrels, submerged flow.............176
Appendix 7.3  Results for diversion culverts with two barrels, unsubmerged flow...........176
Appendix 7.4  Results for diversion culverts with two barrels, submerged flow...............177
v
LIST OF FIGURES
Fig. 2.1  Schematic diagram of sidechannel weir..........................................................................5
Fig. 2.2  Water surface profiles in a channel beside a sideweir ....................................................8
Fig. 2.3  Definition sketch for sidechannel weirs........................................................................12
Fig. 2.4  Assumed velocity distributions and effective flow area.................................................14
Fig. 2.5  Tideflex valves (from Red Valve Co., Inc. catalog).......................................................23
Fig. 2.6  Sample of curves obtained from Red Valve Co., Inc. for Tideflex valves.....................24
Fig. 2.7  Q
half
for Tideflex valves .................................................................................................25
Fig. 2.8  Scaled downstream areas for Tideflex valves ................................................................25
Fig. 2.9  Head loss coefficients for Tideflex valves .....................................................................26
Fig. 2.10  Height of Tideflex valves.............................................................................................29
Fig. 2.11  Assumed open shape of Tideflex valves ......................................................................29
Fig. 3.1  Plan and elevation views of the model weir...................................................................34
Fig. 3.2  A photograph of the model and weir..............................................................................35
Fig. 3.3  Longitudinal profiles of channel invert and weir crest (arbitrary datum).......................37
Fig. 3.4  Low flow calibration of Venturi meter...........................................................................39
Fig. 3.5  Calibration of Vnotch weir ...........................................................................................40
Fig. 3.6  Lowflow calibration of Vnotch weir ...........................................................................40
Fig. 3.7  Calibration of flow sensor ..............................................................................................41
Fig. 3.8  Effects of averaging time on longitudinal velocities......................................................42
Fig. 4.1  Definition sketch for channel with lateral flow..............................................................46
Fig. 4.2  Comparison of measured and calculated model values of sideweir discharge for
unsubmerged flow using Method A.......................................................................57
Fig. 4.3  Comparison of measured and calculated model values of upstream head on the
weir for unsubmerged flow using Method A .........................................................57
Fig. 4.4  Comparison of measured and calculated model values of sideweir discharge for
submerged flow using Method A...........................................................................58
Fig. 4.5  Comparison of measured and calculated model values of upstream head on the
weir for submerged flow using Method A .............................................................58
Fig. 4.6  Comparison of measured and calculated model values of side weir discharge for
tapered channels using Method A..........................................................................59
Fig. 4.7  Comparison of measured and calculated model values of upstream head on the
weir for tapered channels using Method A ............................................................59
Fig. 4.8  Comparison of measured and calculated model values of side weir discharge for
unsubmerged flow using Method B.......................................................................60
Fig. 4.9  Comparison of measured and calculated model values of upstream head on the
weir for unsubmerged flow using Method B .........................................................60
Fig. 4.10  Comparison of measured and calculated model values of side weir discharge
for submerged flow using Method B .....................................................................61
Fig. 4.11  Comparison of measured and calculated model values of upstream head on the
weir for submerged flow using Method B .............................................................61
Fig. 4.12  Comparison of measured and calculated model values of side weir discharge
for tapered channels using Method B ....................................................................62
vi
Fig. 4.13  Comparison of measured and calculated model values of upstream head on the
weir for tapered channels using Method B ............................................................62
Fig. 4.14  Comparison of measured and calculated model values of side weir discharge
for unsubmerged flow using Method C .................................................................63
Fig. 4.15  Comparison of measured and calculated model values of upstream head on the
weir for unsubmerged flow using Method C .........................................................63
Fig. 4.16  Comparison of measured and calculated model values of side weir discharge
for unsubmerged flow using Method D .................................................................64
Fig. 4.17  Comparison of measured and calculated model values of upstream head on the
weir for unsubmerged flow using Method D .........................................................64
Fig. 5.1  Observed and calculated C
e
values.................................................................................74
Fig. 5.2  Measured and calculated h
u
using Method A.................................................................75
Fig. 5.3  C
1
from regression equation and from optimization with C
2
= 0.85 .............................77
Fig. 5.4  Measured and numerically optimized h
u
with C
2
= 0.85 ...............................................78
Fig. 5.5  C
1
from regression equation and from optimization ......................................................79
Fig. 5.6  Values of h
u
from regression equation and from optimization ......................................80
Fig. 5.7  Measured and calculated Q
w
using Method B................................................................81
Fig. 5.8  Measured and calculated h
u
using Method B.................................................................82
Fig. 6.1  Typical locations of velocity measurements looking downstream.................................92
Fig. 6.2  Longitudinal distributions of velocity at downstream end of weir crest (Case A).........93
Fig. 6.3  Longitudinal distributions of velocity 4.3 ft downstream from end of weir crest
(Case A) .................................................................................................................93
Fig. 6.4  Longitudinal distributions of velocity 22.3 ft downstream from end of weir crest
(Case A) .................................................................................................................94
Fig. 6.5  ? and ? values for Cases A  C ......................................................................................94
Fig. 6.6  Longitudinal distributions of velocity 4.4 ft downstream from end of weir crest
(Case C) .................................................................................................................95
Fig. 6.7  Left boundaries of regions from which weir flow comes...............................................96
Fig. 6.8  ? and ? values for Case D..............................................................................................97
Fig. 6.9  Longitudinal distributions of velocity 2.5 ft from downstream end of weir crest
(Case F)..................................................................................................................98
Fig. 6.10  Variation of ? and ? (Case F) ......................................................................................99
Fig. 6.11  Variation of ? and ? (Case G)....................................................................................100
Fig. 6.12  ? and ? values at end of weir for 2.5H:1V side slopes ..............................................101
Fig. 6.13  ? and ? values at end of weir for 4H:1V side slopes .................................................102
Fig. 6.14  Assumed velocity distribution for calculating B
s
.......................................................103
Fig. 6.15  Components of ? and ? for 54% diversion (Cases A and B).....................................103
Fig. 6.16  Components of ? and ? for 25% diversion (Case C).................................................104
Fig. 6.17  Components of ? and ? for forced separation zone (Case D)....................................104
Fig. 6.18  Components of ? and ? for forced separation zone (Case F).....................................105
Fig. 6.19  Components of ? and ? for forced separation zone (Case G)....................................105
Fig. 6.20  Variation of components of ? and ? near end of weir with diversion for
2.5H:1V side slopes .............................................................................................107
vii
Fig. 6.21  Variation of components of ? and ? near end of weir with diversion for 4H:1V
side slopes ............................................................................................................107
Fig. 6.22  Exponential decay of excess ? for 2.5H:1V side slopes ............................................108
Fig. 6.23  Exponential decay of excess ? for 4H:1V side slopes ...............................................109
Fig. 6.24  Length of flow reestablishment region......................................................................110
Fig. 7.1  Schematic diagram of diversion culverts in model (not to scale) ................................115
Fig. 7.2  Diversion culverts.........................................................................................................116
Fig. 7.3  Adjusted loss coefficients for flow from point 0 to point 2 .........................................119
ix
LIST OF TABLES
Table 3.1  Effects of turbulent averaging time on velocities........................................................42
Table 4.1  tstatistics for coefficients in Eq. (4.23) ......................................................................52
Table 4.2  tstatistics for coefficients in Eq. (4.24) ......................................................................53
Table 4.3  tstatistics for coefficients in Eq. (4.25) ......................................................................54
Table 4.4  tstatistics for coefficients in Eq. (4.26) ......................................................................54
Table 4.5  tstatistics for coefficients in Eq. (4.27) ......................................................................54
Table 4.6  Methods of calculating Q
w
and h
u
...............................................................................55
Table 4.7  Statistics of differences between measured and calculated model values of Q
w
and h
u
(Method A) .................................................................................................65
Table 4.8  Statistics of differences between measured and calculated model values of Q
w
and h
u
(Method B)..................................................................................................66
Table 4.9  Statistics of differences between measured and calculated model values of Q
w
and h
u
(Method C) .................................................................................................67
Table 4.10  Statistics of differences between measured and calculated model values of
Q
w
and h
u
(Method D) ...........................................................................................67
Table 4.11  Comparison between Method A and Method B ........................................................68
Table 4.12  R
2
(Q
w
) and R
2
(h
u
) for comparison between measured and calculated values
of Q
w
and h
u
...........................................................................................................70
Table 4.13  Geometric conditions used in simulation ..................................................................70
Table 4.14  Largest differences between values of h
u
calculated from Methods A and B ...........71
Table 4.15  Largest ratios between values of Q
w
calculated from Methods A and B ..................71
Table 5.1  rms of ?Q
w
, ?Q
w
/Q
w(mea)
, ?h
u
and ?h
u
/h
u(mea)
...........................................................83
Table 6.1  Flow conditions for Type 1..........................................................................................90
Table 6.2  Flow conditions for Type 2..........................................................................................91
Table 6.3  Summary of errors in balancing momentum and energy equations ..........................111
xi
ACKNOWLEDGMENTS
This project was supported by the Harris County Flood Control District. Mr. Steve
Fitzgerald was very helpful with both the technical and administrative aspects of this project.
The project was conducted at the Center for Research in Water Resources of the Univer
sity of Texas at Austin. Kevin Wei assisted with some of the experimental work. Part of the
project work is reported by Burgin and Holley (2002). That report is a user's manual for the
computational scheme that has been developed for watershed hydrology, channel and sideweir
hydraulics, and filling and emptying of detention basins.
Red Valve Co., Inc., Pittsburgh, Pennsylvania, provided information on their Tideflex
check valves as presented in Section 2.7.1.
1
PHYSICAL MODELING FOR
SIDECHANNEL WEIRS
By KaLeung Lee and E. R. Holley
Center for Research in Water Resources
The University of Texas at Austin
Austin, TX 78712
1  INTRODUCTION
1.1  REGIONAL BASINS
As watersheds become urbanized, the additional impervious cover and land improve
ments produce an increase in the volume and speed of storm water runoff. In consequence,
downstream flooding becomes more recurrent and more severe, motivating the affected property
owners to demand that restrictions be placed on the further land development in the watershed.
The conflict between upstream and downstream interests has led many jurisdictions to adopt
regulations allowing new development only when it causes no increase in the maximum
discharge downstream. Developers can satisfy the regulations by using detention basins to
reduce peak flow rates. Onsite detention provides temporary storage for excess discharges near
their source, serving to redistribute the excess runoff from a single development.
A more comprehensive solution employs one or more regional detention basins to
consolidate the capacity of a number of separate, small detention facilities into fewer and larger
facilities. Storm runoff is allowed to enter a receiving channel. If flow in the channel
approaches that which will cause flooding, a portion of the flow is diverted into a regional deten
tion basin for temporary storage. When the flow in the channel has decreased sufficiently on the
falling limb of the hydrograph, the water stored in the regional detention basin is released back
into the stream. A sidechannel weir can be used as the structure that diverts excess discharges
from the main channel into the regional detention basin.
This report presents a method to assist in designing sidechannel weir and detention
systems. To model the performance of a trial design, the method connects a hydrologic model, a
channel hydraulics model, and a sidedischarge hydraulics model into a recursive system that
adjusts assumed diversions until they are matched by calculated diversions.
2
1.2  OBJECTIVES
In a previous project conducted at the Center for Research in Water Resources (CRWR)
and sponsored by the Harris County Flood Control District (HCFCD), experiments were
conducted (Tynes, 1989) to determine the hydraulic characteristics of embankmentshaped side
weirs, and a design and modeling method (Davis and Holley, 1988) was developed for side
weirs. The previous method used manual iteration between HEC1, HEC2, and a new program,
SIDEHYDR, which was developed specifically for the task of modeling flow beside and over
sidedischarge weirs.
The present project has built directly on the work done in the previous project. The
objectives of the present project were as follows:
1. Develop a computer program to automatically perform the iterations between the programs
HEC1, HEC2, and SIDEHYDR for the design of sidechannel diversion weirs;
2. Add "popup" screens for input and for graphical display of the results of the iterations on the
computer monitor;
3. Identify the source of computational oscillations in the computer program SIDEHYDR and
change the program to remove the oscillations;
4. Prepare a user?s manual for the entire computational package of programs, including an
improved treatment of the potential pitfalls and error messages in the SIDEHYDR program;
5. Expand the SIDEHYDR program to calculate culvert drainage of water stored in the deten
tion basin below the weir crest;
6. Modify the SIDEHYDR program to allow the choice of either side weirs or culverts for flow
diversion;
7. Conduct hydraulic model experiments to evaluate the effects of channel side slopes on side
weir hydraulics;
8. Modify the existing side weir physical model and conduct experiments to determine the size
and hydraulic effects of the separation zone created in the main channel by the side weir
diversion flow;
9. Reanalyze data from the previous project and use computations of water surface profiles
along side weirs to evaluate the potential effects of channel slope and roughness on weir
hydraulics;
10. Conduct experiments to evaluate the effects of channel flow on the hydraulics of culverts
used for diversion and basin drainage at detention facilities,
3
11. Change the method used in the computational program for flow from the channel into the
detention basin based on the results from Tasks 7 and 8,
12. Extend the work of Task 8 to include channels with 4H:1V side slopes.
Only subcritical channel flows are considered in the computational methods and experiments in
this report.
Tasks 7 ? 10 and 12 are all related to experimental work and are addressed in this report.
The other tasks are related to the computational scheme and are addressed in a companion report
(Burgin and Holley, 2002). The computational scheme presented in that report uses the empirical
results given in this report.
5
2  BACKGROUND
As the name implies, sidechannel weirs (Fig. 2.1) are placed along the side of a channel
parallel (or at a small angle relative to) the flow in the channel. The crest elevation, the crest
length, and the length of the weir can be designed to control the operating characteristics of the
weir.
weir
channel
flow
Fig. 2.1  Schematic diagram of sidechannel weir
2.1  SIDE WEIR FLOW CONDITIONS
There are three general types of flow conditions that can exist with side weirs:
(a) During the rising and falling parts of the hydrograph when the water level in the channel
is lower than the weir crest, gradually varied flow exists in the channel section where the
weir is located.
(b) When the water level in the channel is above the weir crest and above the water level in
the basin, forward flow takes place from the channel into the basin.
(c) If the basin fills to the point that the water level in the basin is above the weir crest,
reverse flow from the basin back into the channel will occur when the water level in the
channel fall below the water level in the basin during hydrograph recession.
Depending on the relative values of the heads on the weir from both the channel and the basin
sides, the weir flow in both directions may have either free or submerged flow conditions. A
brief summary is given first for no weir flow and for reverse flow from the basin into the river
channel, and then a more detailed treatment is given for forward flow from the channel into the
basin.
6
2.2  NO WEIR FLOW
When there is no flow in either direction over the weir, HEC2 could be used for compu
tations as if the weir were not present. However, the overall computational scheme is based on
using HEC2 in the sections of the channel with no weir for diversion and using the program
SIDEHYD, which is a revised version of SIDEHYDR from the previous project (Davis and
Holley, 1988), for computations in the channel where the weir is located. Thus, when there is no
flow over the weir, the depth at the downstream end of the weir is taken from the HEC2 com
putations. Then SIDEHYD computes the water surface profile in the part of the channel where
the weir is located. This computation is based on the differential momentum equation, which can
be written for gradually varied flow (Yen and Wenzel, 1970) as
?
?
?
y
x
=
?
?
SS
1Fr
of
2
(2.1)
where y = flow depth in the channel, x = longitudinal distance which is positive in the flow direc
tion, S
o
= bed slope, S
f
= friction slope, ? = momentum correction factor, and Fr = channel
Froude number which is defined as
T
A
g
U
Fr =
(2.2)
where U = average channel velocity (Q/A), A = channel flow area and T = top width of flow.
These computations in SIDEHYD give the water surface elevation at the upstream end of the
weir. This elevation is put into the HEC2 input file for restarting the HEC2 calculations for the
channel upstream of the weir.
2.3  REVERSE FLOW
For reverse flow from the basin back into the channel, the weir behaves as a normal weir
rather than as a side weir. The discharge equation for normal broadcrested weirs can be written
as
2/3
snw
h)x(g
3
2
3
2
CCQ ?= (2.3)
where Q
w
= weir discharge, C
n
= discharge coefficient for a broadcrested weir, C
s
= submer
gence correction factor, g = acceleration due to gravity, h = head on the weir, and ?x = increment
of length along the weir crest. Eq. (2.3)assumes that the approach velocity is small, as it should
be since the flow back over the weir is coming from the detention basin. The sign convention is
7
that reverse flow from the basin to the channel is positive (Eq. (2.3)) while forward flow from the
channel to the basin is negative (Eq. (2.4), Eq. (2.7)). If the weir crest is inclined (e.g., parallel to
the invert of an improved channel), then the head on the weir will decrease in the upstream direc
tion since the water level in the detention basin will normally be horizontal.
For reverse flow with subcritical channel flow, both the head loss in the channel due to
the disturbance caused by the flow coming over the weir and the increasing discharge in the
downstream direction mean that the depth in the channel decreases in the downstream direction.
As a result of this change of depth in the channel, C
s
can vary along the length of a weir when
submerged flow conditions exist.
2.4  FORWARD FLOW
Flow over side weirs depends on the head on the weir, among other factors. The head
depends on the water surface profile along the channel where the diversion is taking place.
While the primary factor affecting the water surface profile is the diversion itself, the channel
slope and roughness also have an effect on the water surface profile just as they do in a channel
without a side weir. Depending on the flow conditions and the channel geometry, the flow over
the side weir will cause the flow remaining in the channel to develop a lateral distribution of
velocity that is asymmetrical and may cause the flow to separate from the side of the channel
opposite the weir.
2.4.1  Water Surface Profiles
Some of the possible longitudinal water surface profiles in a channel along a side weir for
forward flow from the channel into the basin are illustrated in Fig. 2.2, which has been adapted
from Henderson (1966). There are several things that are illustrated or implied in this figure that
have a direct bearing on the flow diversion problem. One is that, for subcritical flow (Fig. 2.2a),
the water surface elevation usually increases in the downstream direction. The second thing is
that it is possible to have a hydraulic jump (Fig. 2.2c) in the channel because of the outflow. The
possible occurrence of the jump depends on the hydraulics of the outflow and does not require
supercritical flow in the channel upstream of the weir. Thus, in a channel with subcritical flow, it
is possible for the outflow itself (even on a horizontal or very mild slope channel) to cause the
flow in the channel to pass through critical depth at the upstream end of the weir giving super
critical flow, then a hydraulic jump, and finally subcritical flow again. Since the calculation of
water surface profiles for subcritical flows depends on knowing a downstream boundary condi
tion (depth), a third thing implied by the first two is that it is impossible to correctly calculate the
depths and water surface profile in the channel upstream of a weir without first considering the
8
details of the flow over the weir and the type of profile which exists at the weirs. Only subcriti
cal flow along the full length of the weir is considered in this design procedure.
(a) Subcritical flow throughout
(b) Jump due to side discharge with subcritical flow upstream and downstream
(c) Supercritical flow throughout
(d) Jump due to supercritical flow upstream and subcritical flow downstream
Fig. 2.2  Water surface profiles in a channel beside a sideweir
2.4.2  Previous Work of Others
The water that remains in the channel experiences the normal frictional head losses as the
channel flow occurs along the weir, and these losses tend to reduce the total head in the flow
direction. Hager (1987) discussed the fact that, when onedimensional analysis is used, the
hydraulic characteristics of side weir flow cause an additional head change that may be either
positive or negative, depending on the flow conditions. This condition is also discussed by Idel
chik (1986) in conjunction with flow bifurcations in ducts. However, it was found in this project
9
that using the kinetic energy correction factor (?) eliminated the need for including an additional
head change in the energy equation (Section 6.8).
Most of the previous work has considered only forward flow from the channel into the
basin. The earliest studies of the hydraulic characteristics of sidechannel weirs were concerned
primarily with the analytical prediction of the effects of the weirs on the longitudinal water
surface profile in the channel for the idealized case of a rectangular channel with a vertical weir
plate and a constant discharge coefficient (Forchheimer, 1930; de Marchi, 1934; Ackers, 1957;
Collinge, 1957; Frazer, 1957; Chow, 1959; Henderson, 1966; Bos, 1976). Some other studies on
evaluation of the discharge over the sidechannel weir are those of Mostafa and Chu (1974),
Subramanya and Awasthy (1972), and Hager (1987).
Part of Hager's (1987) analysis was based on the side weir discharge per unit length of the
weir (q
w
) written as
2/3
1w
hg
3
2
3
2
C
dx
dQ
q =?= (2.4)
where Q is the flow rate in the channel, C
1
is an empirical coefficient, and h is the head at any
point along the weir. C
1
may be constant or variable along the weir. Hager wrote C
1
as C
n
?
where C
n
is a discharge coefficient for a normal weir of the same geometry as the side weir and ?
is a lateral flow coefficient given by
2/1
3
2
3
2
)C1(2Fw
)C1()2Fw(
?
?
?
?
?
?
?
?
?+
?+
=? (2.5)
where Fw is defined as
gh
U
Fw =
(2.6)
and is called a weir Froude number since it is based on the head on the weir rather than the flow
depth. C
3
is a residual pressure coefficient that is related to the pressure distribution at the con
trol section for the weir flow and is less than unity. Hager used a value of 2/3 for C
3
in Eq. (2.5).
Apparently C
3
should depend on the particular type of weir under consideration. The effective
discharge coefficient C
n
? is variable along the weir.
10
2.5  PREVIOUS WORK AT CRWR ON HYDRAULIC COMPUTATIONS
2.5.1  Purpose
Engineers designing side weir and detention basin facilities have to determine the side
weir and basin dimensions necessary to reduce the channel flow depth and discharge to accept
able levels for given channel characteristics and a given storm. Although HEC1 and HEC2
contain some capabilities for modeling diversions, neither program is flexible enough to repre
sent some of the essential hydraulic features of side weir flows. For example, experimental work
indicates that side weir discharge coefficients vary with channel velocity and head on the weir as
they change during the passage of the hydrograph, but HEC1 and HEC2 cannot represent these
changes. Also, the programs cannot predict when submergence of side weirs occurs as the basin
fills, nor can they model the flow of water from the basin back to the channel as the channel
water level drops. Thus, a program originally called SIDEHYDR was developed in the previous
project to model side weir hydraulic characteristics. SIDEHYD used in the modeling presented
later in this report is a revision of SIDEHYDR.
SIDEHYDR represents side weir flow including the effects of channel flow character
istics, possible submergence as the basin fills, the discharge characteristics for an embankment
shaped weir, and possible reverse flow over the weir as the channel water level drops during the
recession limb of the hydrograph. The channel flow and flow over a side weir interact in such a
way that trial and error computations are normally required to determine the side discharge and
all of the depths in the channel (at the weir and both upstream and downstream of the weir and in
the basin). The side discharge depends on the depths in the channel, but the depths are controlled
from downstream for subcritical flow and these depths depend on the discharge, which cannot be
known until the side discharge is known. In addition, the depth at the downstream end of one
weir can depend on other weirs downstream of it, and the discharge at the upstream end of a weir
depends on other upstream weirs. Because of all of these interdependencies, it is necessary to
iterate between HEC1, HEC2, and SIDEHYDR. In the previous project, these iterations were
done manually. The procedure was to
(1) run HEC1 with an assumed diversion hydrograph at the weir to obtain hydrographs in the
channel,
(2) run HEC2 for times throughout the hydrograph to obtain stage hydrographs at the weirs,
(3) run SIDEHYDR using the discharge hydrographs from HEC1 and the stage hydrographs
from HEC2 and weir discharge characteristics from the experimental part of the project to
calculate the weir diversion hydrograph and the stage hydrograph in the basin,
(4) run HEC1 again using the calculated diversion hydrograph, and
11
(5) continue looping through these programs until the diversion hydrographs at the beginning and
end of an iteration loop agreed within a specified tolerance. The manual iterations were
extremely time consuming. Thus, part of the present project has been to automate the itera
tion process.
This section gives a summary of the general computational approach that is used for the
hydraulic parts of the problem for various flow conditions. Only subcritical flow along the entire
weir length (Fig. 2.2a) is considered. Thus, the computations to determine the weir discharge
and the water surface profile along the weir (or the depth change between the downstream (sub
d) and upstream (subu) ends of the weir) begin with the downstream water level and the down
stream head on the weir.
2.5.2  Forward Weir Flow
2.5.2.1  Method of Analysis in the Previous Project
In this section, the method of analysis used in the previous project is reviewed. All of
Tynes' (1989) test data are listed in Appendix 1 of this report. The tests were conducted in a
channel with a trapezoidal cross section with 2.5H:1V side slopes.
For sidechannel weirs, the head and the discharge coefficient vary along the length of the
weir crest. However, side weirs can be calibrated so that the total side discharge (Q
w
) can be
written in terms of a bulk discharge coefficient (C
e
). For broadcrested weirs, this expression is
QCC gAh
wes w
=?
2
3
2
3
12/
(2.7)
where A
w
= a representative flow area (e.g., Lh in Eq. (2.3) for normal weir flow) and C
e
= bulk
discharge coefficient. The sign convention is that flow into the channel is positive while flow
out of the channel is negative. Thus, Eq. (2.3) is positive while Eq. (2.4) and Eq. (2.7) have a
negative sign. In Eq. (2.7), some convention must also be established for defining h since the
head varies along the length of the weir. Likewise, a convention is needed for defining A
w
.In
the previous project, h was taken at the downstream (subd) end of the part of the weir crest
parallel to the channel invert and A
w
was taken as h
d
times the average length of the flow area
over the weir (Fig. 2.3). Thus, Eq. (2.7) can then be written as
[]
3/2
d
dsew
hEShLg
3
2
3
2
CCQ +?= (2.8)
where h = height of the water surface above the side weir crest, subd = downstream end of side
weir crest, L = length of the weir crest parallel to the bed slope (Fig. 2.3), and ES = slope of the
12
ends of the side weir (e.g., ES = 6 for a 6H:1V slope). The subscript u, which will appear later,
denotes the upstream end of the weir crest.
2.5.2.2  Flow Asymmetry
For forward flow from the channel into the basin, the flow in the channel develops an
asymmetrical velocity profile compared to the one that would exist with no diversion (Chapter
6). Frequently, as part of the flow goes toward and over the weir into the detention basin, a sepa
ration zone forms in the channel on the side opposite to the weir. The flow going over the weir
effectively pulls the flow that remains in the channel away from the opposite side of the channel.
When separation occurs, the flow in the channel at the downstream end of the weir may be con
centrated on the side of the channel next to the weir, as shown by the velocity profiles in Section
6.4. Thus, the true velocity head and true momentum flux cannot be obtained from the average
velocity given by Q
d
divided by the channel area. This condition needs to be taken into account
in determining the actual depth in the channel at the downstream end of the weir from the depth
indicated by HEC2 calculations.
Fig. 2.3  Definition sketch for sidechannel weirs
As pointed out in the HEC2 User?s Manual (US Army Corps Of Engineers, 1984) in
conjunction with the flow conditions downstream of bridges, some channel length is required for
flow expansion to take place downstream of a separation zone. Nevertheless, it was assumed in
the previous project that the channel length for this flow expansion is negligible. Thus, there
were two cross sections essentially adjacent to each other at the downstream end of the weir.
Cross section db corresponds to the conditions calculated by HEC2 for the downstream end of
the weir but is actually at the downstream end of the zone of flow expansion since HEC2 inher
ently assumes that the flow fills the entire cross section. Cross section da corresponds to the
13
actual conditions at the downstream end of the weir including the flow separation. The designa
tions da and db are used as subscripts.
2.5.2.3  Downstream Depth
The energy equation can be used to relate the depths at cross section da and db since
()
2g
U
y
2g
UU
K
2g
U
y
2
db
db
2
dbe
E
2
da
dada
+=
?
??+
(2.9)
where U
da
=Q
d
/A
da
= the apparent velocity at cross section da, ? = the kinetic energy correction
factor, U
e
= the effective velocity at cross section da, i.e., the average velocity in the part of the
cross section in which flow is actually taking place, and K
E
= expansion loss coefficient. This
form for head loss term (Henderson, 1966) is more appropriate for this type of flow expansion
than the form used in HEC2. From Eq. (2.9),
()
2g
UU
K
2g
U
2g
U
yy
2
dbe
E
2
da
da
2
db
dbda
?
+??+=
(2.10)
The velocity head at cross section da (the third term on the righthand side of Eq. (2.10)) is
always greater than at cross section db (the second term) for subcritical flow, and K
E
is less than
unity. The result is that y
da
is less than y
db
and that the flow asymmetry and resulting flow re
establishment at the downstream end of the weir suppress the head on the weir relative to y
db
.
Although Eq. (2.10) may be helpful toward understanding why y
da
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
+=
(2.39)
Since this equation is undefined at Q/Q
half
= 0, it was used only for Q/Q
half
> 0.036, which corre
sponds to the second smallest value read from the manufacturer's curves. For lower values, a
linear interpolations was used, giving
()
()
036.0
Q
Q
for
Q
Q
44.6
036.0
Q/Q
0.036
09059.0
0.036log9277.09127.0
A
A
halfhalf
half
5967.0
half
valve
?=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
++=
(2.40)
The curves for Eq. (2.39) and Eq. (2.40) are shown in Fig. 2.8.
Fig. 2.5  Tideflex valves (from Red Valve Co., Inc. catalog)
24
Fig. 2.6  Sample of curves obtained from Red Valve Co., Inc. for Tideflex valves
25
A
valve
=A
half
(ft
2
)
02468101214161820
Q
half
(c
f
s
)
0
50
100
150
200
250
D(in.)
847260483624
Fig. 2.7  Q
half
for Tideflex valves
As noted above, the manufacturer supplied both the head loss in the valve itself (H
L
)and
the "total head loss", which was defined as the head loss in the valve plus the velocity head in the
pipe. However, the actual loss associated with a submerged valve is the loss in the valve plus
exit loss or the velocity head of the jet leaving the valve (not the velocity head in the pipe), Thus,
only the head loss in the valve was used in calculating a loss coefficient. The exit loss was
included in the calculations separately. The head loss coefficient (K
L
) was defined by
Q/Q
half
0123456
A
va
l
v
e
/A
half
0.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
1.8
+
+
+
+
+
+
+
+
+
X
X
X
X
X
X
X
X
X
X
D(in.)
24
36
48
60
72
84
Eq.
Fig. 2.8  Scaled downstream areas for Tideflex valves
()
g2
VV
KH
2
pipevalve
L
valve
L
?
=
(2.41)
Eq. (2.40)
Eq. (2.39)
Eq. (2.38)
26
where V
valve
is the exit velocity from the valve and V
pipe
is the velocity in the culvert. The head
loss coefficients defined in this manner are shown in Fig. 2.9. The mean value of K
L
is 1.0 with
a standard deviation of 3%. There is a large amount of scatter for small values of Q/Q
half
, but the
velocities and therefore the head losses are small for these small flow rates.
Q/Q
half
0123456
K
L
0.90
0.95
1.00
1.05
1.10
1.15
+
+
+
+
+
+
+
+
+
X
X
X X
X
X
X
X
X
X
D(in.)
24 60
36 72
48 84
Fig. 2.9  Head loss coefficients for Tideflex valves
2.7.1.3  Calculation Method for Submerged Valves
This section summarizes the calculations used when the Tideflex valve is submerged.
The valve is considered to be submerged when the tailwater is above the downstream soffit of the
culvert pipe. In the calculations, the tailwater depth (TW) is measured from the downstream
invert of the pipe (not the valve). As described by Burgin and Holley (2002), the culvert flow is
calculated by trial and error based on assumed flows. The assumed flows are adjusted until the
head required for the assumed flow matches the head available in the detention pond. For the
assumed flow rate (Q), the calculation procedure is as follows, with all values being in feet and
cfs:
1) Calculate the culvert cross sectional area in ft
2
from A
pipe
= ?D
2
/4.
2) Calculate half of the culvert cross sectional area in ft
2
from A
half
=A
pipe
/2.
3) Calculate the flow rate in cfs corresponding to having an open area in the Tideflex valve
equal to A
half
from Eq. (2.38).
4) Calculate the open area (A
valve
) of the Tideflex valve corresponding to the specified Q from
Eq. (2.39) or Eq. (2.40).
5) Since the valve will flow full for submerged conditions, calculate the valve exit velocity
head from
27
g2
1
A
Q
g2
V
2
valve
2
valve
?
?
?
?
?
?
?
?
= (2.42)
where Q = assumed flow rate in the culvert.
6) Since the pipe will also flow full for submerged conditions, calculate the pipe velocity head
from
g2
1
A
Q
g2
V
2
pipe
2
pipe
?
?
?
?
?
?
?
?
= (2.43)
7) Since the valve loss coefficient based on the difference in the entrance and exit velocity
heads (Eq. (2.41)) is 1.0, the sum (H
L
) of the valve and exit head losses is
g2
V
g2
V
2
g2
V
g2
V
g2
V
H
2
pipe
2
valve
2
valve
2
pipe
2
valve
L
?=+
?
?
?
?
?
?
?
?
?
?
?=
(2.44)
8) It is now possible to calculate the head at the end of the culvert (i.e., at the upstream end of
the valve). Because of the way that the program for culvert flow does its calculations, the
equivalent tailwater (TW
equivalent
) at the downstream end of the culvert without the valve is
needed. That is, a culvert with TW
equivalent
and with no valve would have the same flow as
the culvert with the valve. TW
equivalent
to account for the valve and exit losses is
?
?
?
?
?
?
?
?
?
?
?+=
?+=
g2
V
g2
V
2TW
g2
V
HTWTW
2
pipe
2
valve
actual
2
pipe
Lactualequivalent
(2.45)
9) Calculate the headwater (HW) based on TW
equivalent
and the assumed flow rate (Q). HW is
relative to the upstream invert of the culvert.
10) Store TW and the final Q and HW for each time step in the calculations. Only the values for
the most recent time step are kept in storage. These values are needed for unsubmerged
conditions discussed in the next section.
11) At the end of the calculations for a given time step, check to determine whether TW is below
the downstream soffit. If so, the calculations then shift to those described in the next
section.
28
2.7.1.4  Calculation Method for Unsubmerged Valves
This section summarizes the calculations for unsubmerged conditions that are assumed to
exist when the tailwater is below the downstream soffit. An approximate method of calculation
is needed since information on the flow area of the valve and depth of flow in the valve outlet
could not be obtained for unsubmerged conditions. Even when the valve outlet is not
submerged, the head loss in the valve may cause the downstream end of the culvert to still be
flowing full. At the end for the calculations for the assumed unsubmerged conditions, a check is
made to determine if the downstream end of the culvert is full.
1) At the end of the first time step with the tailwater below the downstream soffit, a linear inter
polationisusedtodetermineQ
soffit
and HW
soffit
corresponding to having the TW at the
downstream soffit.
2) For subsequent time steps, it is assumed that the water depth (y
exit
) in the downstream valve
opening decreases linearly in proportion to the decreasing headwater in the detention pond
unless the tailwater is higher than the value indicated by this linear interpolation. Thus,
?
?
?
?
?
?
?
?
= TWD,
HW
HW
maxy
soffit
exit
(2.46)
where y
exit
is measured from the pipe invert, not from the bottom of the valve. The depth in
the valve opening (y
v
exit
) is measured from the bottom of the valve, so
2
DH
yy
exitv
exit
?
+= (2.47)
where H is the height of the downstream end of the valve (Fig. 2.10) and is given by H =
1.64D.
3) From Q
soffit
, the valve open area (A
soffit
) corresponding to having the tailwater at the soffit is
obtained from Eq. (2.39) or Eq. (2.40).
4) For each subsequent tailwater which is below the downstream soffit, it is assumed that the
valve open area decreases linearly, i.e.,
H
y
AA
exit
v
soffitvalve
= (2.48)
29
D = culvert diameter (ft)
01234567
H
=
va
lv
e
hei
ght
(
f
t)
0
2
4
6
8
10
12
H = 1.64D
Fig. 2.10  Height of Tideflex valves
5) It is further assumed that the open area of the valve is composed of two triangles with coinci
dent bases and with heights equal to half of the valve height, as shown in Fig. 2.11. The base
width of the two triangles is given by
Fig. 2.11  Assumed open shape of Tideflex valves
H
2A
b
valve
= (2.49)
6) The surface width (T
exit
) and the flow area in the valve opening (A
exit
) are calculated from
b
T
y
ve
xi
t
water
surface
30
2
H
yfor
2
y
TA
H/2
y
bT
2
H
yfor
2
yH
TAA
H/2
yH
bT
exit
exit
exit
exit
exit
exit
v
v
exitexit
v
exit
v
v
exitvalveexit
v
exit
?
?
?
?
?
?
?
?
=
=
>
?
?
?
?
?
?
?
=
=
(2.50)
7) The velocity head at the valve outlet is calculated from
()
g2
A/Q
g2
V
2
exit
2
exit
=
(2.51)
8) Since K
L
= 1, the head loss in the valve is
()
2g
Q/A
g2
V
H
2
flow
2
exit
L
valve
?=
(2.52)
where A
flow
is the flow area corresponding to the flow depth (y
pipe
)attheendoftheculvert
pipe.
9) TW
equivalent
is calculated from as follows: The energy equation between the end of the pipe
and the valve exit is
() ()
()
2g
V
2y
2g
Q/A
2y
2g
V
y
2g
Q/A

2g
V
2g
Q/A
y
2
exit
exit
2
flow
pipe
2
exit
exit
2
flow
2
exit
2
flow
pipe
+=+
+=
?
?
?
?
?
?
?
?
?+
(2.53)
This equation is solved by trial and error to obtain y
pipe
,whichisTW
equivalent
if y
pipe
0.5is
()[]
97.3
d
50.4
d
t
1
s1
1Fwln62.55.0
h
h
5.1900.1
C
C
+?
?
?
?
?
?
?
?
?
??= (4.25)
For h
t
/h
d
? 0.5, C
1s
=1. The tstatistics for the coefficients in Eq. (4.25) are shown in Table 4.3.
R
2
for Eq. (4.25) is 0.965. All the coefficients are significant at the 5% level except 5.62. The
term involving Fw
d
is retained because of the improved correlation. However, Eq. (4.25) may
give negative values of C
1s
/C
1
. When this situation happens, Eq. (4.26) below is used instead of
Eq. (4.25) to determine C
1s
/C
1
.
The regression equation with h
t
/h
d
as the only parameter is
76.4
d
t
1
s1
5.0
h
h
6.22887.0
C
C
?
?
?
?
?
?
?
?
??= (4.26)
54
The tstatistics for the coefficients in Eq. (4.26) are shown in Table 4.4. R
2
for Eq. (4.26) is
0.799, which is substantially lower than that of Eq. (4.25). Although the coefficient 22.6 is not
significant at the 5% level, the term involving h
t
/h
d
is needed because the plot of C
1s
/C
1
vs. h
t
/h
d
shows a definite relationship between the two variables.
Table 4.3  tstatistics for coefficients in Eq. (4.25)
Coefficient
tstatistic
1.00
19.5
4.50
5.62
3.97
24.24
2.82
8.95
1.35
3.79
4.2.3  Unsubmerged Flow in Tapered Channels
The values of C
1
were obtained for the 65 tests for tapered channels using the same
procedure described in Section 4.2.1. C
2
wasadjustedto0.70tominimizetheerrorsinh
u
.The
best regression equation for C
1
is
Table 4.4  tstatistics for coefficients in Eq. (4.26)
Coefficient tstatistic
0.887
22.6
4.76
13.7
1.15
3.83
?
?
?
?
?
?
+=
P
h
ln105.0101.1C
d
1
(4.27)
The tstatistics for the coefficients in Eq. (4.27) are shown in Table 4.5. Both coefficients are
significant at the 5% level but R
2
for Eq. (4.27) is only 0.395. Again R
2
for Q
w
(calculated) vs.
Q
w
(measured) is much higher. Values are given in Section 4.3.
Table 4.5  tstatistics for coefficients in Eq. (4.27)
Coefficient tstatistic
1.101
0.105
34.0
6.41
Recall that
w
Ucos? =C
2
(?/P)U (Eq. (4.19)). A smaller value of C
2
for tapered channels
seems to suggest that the longitudinal component of the velocity of side weir discharge is smaller
55
in tapered channels than in prismatic channels. This speculation can be evaluated only by
detailed velocity measurements in the channel as well as in the region of the outflow.
4.3  COMPARISON BETWEEN MEASURED AND CALCULATED VALUES FOR
2.5H:1V SIDE SLOPES
Based on empirical coefficients from regression equations in Section 2.6 and on the equa
tions in Section 4.2, the various methods of calculating Q
w
and h
u
described in the previous sec
tions are summarized in Table 4.6. Method A is the method used in the previous project (Tynes,
1989). Method B uses a C
1
that is constant along the weir but varies with the hydraulic condi
tions at the downstream end of the weir and also uses a variable head along the length of the
weir; while Method C uses a C
1
that varies along the length of the weir based on ? and a variable
head. In Method D, the average of the values of
'
1
C obtained from the 238 tests was used with a
variable head along the weir.
For unsubmerged flow, the calculated flow depth at the upstream end of the weir using
Method A was supercritical for Test A3B19N (identification code used by Tynes, 1989) and no
solution could be obtained for h
u
for Tests A1C20W and A5C18N. This condition presumably
resulted from a critical or supercritical solution not being found. When Method B was used,
supercritical flow depths were obtained in the computed water surface profile for Test A5C18N.
The flow conditions calculated at the upstream end of the weir using Method C and Method D
were subcritical for all of the 238 tests.
Table 4.6  Methods of calculating Q
w
and h
u
Method Physical equations Empirical equations ?
Unsub
merged
Sub
merged
Tapered Unsub
merged
Sub
merged
Tapered
A Eq. (2.8),
Eq. (2.23)
Eq. (2.8),
Eq. (2.23)
Eq. (2.8),
Eq. (2.36)
Eq. (2.28),
Eq. (2.29)
Eq. (2.28),
Eq. (2.29),
Eq. (2.32),
Eq. (2.33)
Eq. (2.34),
Eq. (2.35)
not
used
B Eq. (2.4),
Eq. (4.18)
Eq. (2.4),
Eq. (4.18)
Eq. (2.4),
Eq. (4.22)
Eq. (4.19),
Eq. (4.23)
Eq. (4.19),
Eq. (4.23),
Eq. (4.25)
Eq. (4.19),
Eq. (4.27)
not
used
C Eq. (4.18),
Eq. (4.21)
Eq. (4.19),
Eq. (4.24)
Eq.
(4.20)
D Eq. (4.18),
Eq. (4.21)
Eq. (4.19),
'
1
C = 1.063
Eq.
(4.20)
Note: Also see Table 4.11.
56
The calculated values of Q
w
and h
u
are plotted against the measured values in Fig. 4.2 to
Fig. 4.17. The averages (avg), standard deviations (stdev) and rootmeansquare values (rms) of
the differences between measured and calculatedvaluesareshowninTable4.7toTable4.10,in
terms of both absolute values and relative values. (If the sum in the calculation of the standard
deviation is divided by N instead of N  1, then rms = (avg)
2
+ (stdev)
2
).
From Table 4.7 to Table 4.10, the differences between stdev and rms of either ?Q
w
,
where ?Q
w
=Q
w(mea)
Q
w(cal)
as shown in Table 4.7 through Table 4.10, or ?Q
w
/Q
w(mea)
were
at most 2%, except for tapered channels using Method A. This result is illustrated in the figures
by the fact that the points are scattered close to the 1:1 line. Similar observations were obtained
for the upstream head on the weir. The differences between stdev and rms were generally larger
for ?h
u
,where?h
u
=h
u(mea)
h
u(cal)
as shown in Table 4.7 through Table 4.10, and for
?h
u
/h
u(mea)
than for ?Q
w
and ?Q
w
/Q
w(mea)
. The largest difference was still only 8%, except for
submerged flow using Method A and for tapered channels.
The rootmeansquare values were used to compare the different methods of analysis.
For meaningful comparison, the statistics had to be based on the same number of tests. There
fore, when comparing Methods A and B, Tests A1C20W, A3B19N and A5C18N were excluded
and when comparing Methods B, C and D, Test A5C18N was excluded.
Method B gave smaller rms for the side weir discharge than Method A in terms of both
?Q
w
and ?Q
w
/Q
w(mea)
. For unsubmerged flow, the rms of ?Q
w
/Q
w(mea)
was reduced only by
2% when Method B was used instead of Method A. However the reduction was 57% for sub
merged flow and 44% for tapered channels.
For unsubmerged flow, Method B gave smaller rms for ?h
u
but larger rms for ?h
u
/h
u(mea)
than Method A. For submerged flow, Method B gave smaller rms for both ?h
u
and ?h
u
/h
u(mea)
than Method A whereas for tapered channels, Method B gave larger rms than Method A for both
?h
u
and ?h
u
/h
u(mea)
. Nevertheless, the differences between the rms for ?h
u
using either method
were only about 0.001 ft. Therefore, compared with Method A, Method B gave improved results
or results of comparable accuracy.
Method B takes into consideration more details of the side weir hydraulics, specifically
the water surface profile along the weir. Hence the head correction required in Method A (Eq.
(2.29), Eq. (2.33), and Eq. (2.35)) is eliminated in Method B. Moreover, Method B explicitly
accounts for the channel slope and roughness so that it is applicable for different slopes and
roughnesses. However, the improvement for submerged flow is partly due to the use of two
parameters in the regression equation instead of one (Eq. (4.25) vs. Eq. (2.33)). Table 4.11
summarizes the differences between Method A and Method B.
57
Q
w
(measured) (cfs)
0123456
Q
w
(
c
alcu
l
a
t
ed)
(c
f
s
)
0
1
2
3
4
5
6
Fig. 4.2  Comparison of measured and calculated model values of side
weir discharge for unsubmerged flow using Method A
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(c
a
l
c
u
l
a
t
e
d)
(f
t)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.3  Comparison of measured and calculated model values of
upstream head on the weir for unsubmerged flow using Method A
58
Q
w
(measured) (cfs)
0123456
Q
w
(
c
alc
u
lat
ed)
(
c
f
s
)
0
1
2
3
4
5
6
Fig. 4.4  Comparison of measured and calculated model values of side
weir discharge for submerged flow using Method A
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(
c
a
l
cul
a
te
d)
(f
t)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.5  Comparison of measured and calculated model values of
upstream head on the weir for submerged flow using Method A
59
Q
w
(measured) (cfs)
0123456
Q
w
(
c
al
cul
a
te
d)
(c
fs)
0
1
2
3
4
5
6
Fig. 4.6  Comparison of measured and calculated model values of side
weir discharge for tapered channels using Method A
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(calcula
ted)
(
ft)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.7  Comparison of measured and calculated model values of
upstream head on the weir for tapered channels using Method A
60
Q
w
(measured) (cfs)
0123456
Q
w
(
c
alculat
ed)
(cf
s
)
0
1
2
3
4
5
6
Fig. 4.8  Comparison of measured and calculated model values of side
weir discharge for unsubmerged flow using Method B
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(c
alculat
ed)
(f
t
)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.9  Comparison of measured and calculated model values of
upstream head on the weir for unsubmerged flow using Method B
61
Q
w
(measured) (cfs)
0123456
Q
w
(c
alcu
l
a
ted
)
(
c
f
s
)
0
1
2
3
4
5
6
Fig. 4.10  Comparison of measured and calculated model values of side
weir discharge for submerged flow using Method B
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(c
alculat
ed)
(f
t
)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.11  Comparison of measured and calculated model values of
upstream head on the weir for submerged flow using Method B
62
Q
w
(measured) (cfs)
0123456
Q
w
(
c
alc
u
l
a
te
d)
(
c
f
s
)
0
1
2
3
4
5
6
Fig. 4.12  Comparison of measured and calculated model values of side
weir discharge for tapered channels using Method B
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(c
alculat
ed)
(f
t)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.13  Comparison of measured and calculated model values of
upstream head on the weir for tapered channels using Method B
63
Q
w
(measured) (cfs)
0123456
Q
w
(
c
alcu
l
a
ted)
(c
fs
)
0
1
2
3
4
5
6
Fig. 4.14  Comparison of measured and calculated model values of side
weir discharge for unsubmerged flow using Method C
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(c
alculat
ed)
(f
t)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.15  Comparison of measured and calculated model values of
upstream head on the weir for unsubmerged flow using Method C
64
Q
w
(measured) (cfs)
0123456
Q
w
(
c
alc
u
late
d)
(c
fs
)
0
1
2
3
4
5
6
Fig. 4.16  Comparison of measured and calculated model values of side
weir discharge for unsubmerged flow using Method D
h
u
(measured) (ft)
0.00 0.05 0.10 0.15 0.20 0.25 0.30 0.35 0.40 0.45
h
u
(
c
al
cu
la
te
d
)
(
ft)
0.00
0.05
0.10
0.15
0.20
0.25
0.30
0.35
0.40
0.45
Fig. 4.17  Comparison of measured and calculated model values of
upstream head on the weir for unsubmerged flow using Method D
65
Table 4.7  Statistics of differences between measured and calculated model
values of Q
w
and h
u
(Method A)
?Q
w
(cfs) ?Q
w
/Q
w(mea)
?h
u
(ft) ?h
u
/h
u(mea)
Unsubmerged
N = 238
avg
stdev
rms
0.00474
0.124
0.123
0.0112
0.0680
0.0688
Unsubmerged
N = 235
(1)
avg
stdev
rms
0.000983
0.112
0.112
0.00819
0.0628
0.0632
0.00327
0.00877
0.00934
0.0123
0.0662
0.0672
Submerged
N=35
avg
stdev
rms
0.0737
0.328
0.332
0.0751
0.298
0.303
0.00280
0.00382
0.00469
0.0118
0.0177
0.0210
Tapered
N=65
avg
stdev
rms
0.114
0.161
0.197
0.0898
0.127
0.155
0.00280
0.00452
0.00529
0.0185
0.0372
0.0413
Notes: ?Q
w
=Q
w(mea)
Q
w(cal)
, ?h
u
=h
u(mea)
h
u(cal)
, mea = measured, cal = calculated,
N = number of tests.
(1)
Tests A1C20W, A3B19N and A5C18N excluded
In the tests for tapered channels, large values of Q
u
and small diversions were associated
with small values of both h
d
and h
u
that were two to three times those of h
d
. Under these condi
tions Fw
d
was high and C
e
was very large. As a result, C
e
was found to be proportional to
3
d
Fw .
The relationship between C
e
and Fw
d
in Eq. (2.34) is an artifact of calculating Q
w
in terms of h
d
in Method A. In Method B, the variation of the head along the weir is considered directly in the
computation. Fw
d
does not enter into the regression equation of C
1
for tapered channels.
When Method B was used, the errors in the estimation of the side weir discharge aver
aged about 6%, 13% and 9% for unsubmerged flow, submerged flow and flow in tapered chan
nels respectively. Even though the error is larger for submerged flows, the flows themselves are
normally small because of the small head difference across the weir. The errors in the estimation
of the upstream head on the weir at prototype scale were on average about 0.20 ft, 0.10 ft and
66
0.14 ft for unsubmerged flow, submerged flow and flow in tapered channels respectively for a
1:25 scale model.
Method B and Method C had comparable accuracy in predicting Q
w
while Method D was
less accurate. The rms values for ?Q
w
and ?Q
w
/Q
w(mea)
from Method D were about 50% higher
than those from Method B. Although Eq. (4.24) has a low R
2
, using this equation to estimate
'
1
C
is better than simply using the average value of
'
1
C . The rootmeansquare values of ?h
u
and
?h
u
/h
u(mea)
obtained from Methods B, C, and D were all about 0.008 ft and 0.1 respectively at
model scale. Therefore these three methods were comparable in accuracy in predicting h
u
.
Table 4.8  Statistics of differences between measured and calculated
model values of Q
w
and h
u
(Method B)
?Q
w
(cfs) ?Q
w
/Q
w(mea)
?h
u
(ft) ?h
u
/h
u(mea)
Unsubmerged
N = 237
(1)
avg
stdev
rms
0.00173
0.106
0.106
0.00361
0.0620
0.0619
0.00165
0.00807
0.00822
0.0338
0.101
0.107
Unsubmerged
N = 235
(2)
avg
stdev
rms
0.00306
0.105
0.105
0.00296
0.0618
0.0617
0.00167
0.00797
0.00813
0.0339
0.100
0.106
Submerged
N=35
avg
stdev
rms
0.00334
0.136
0.135
0.00181
0.132
0.130
0.00100
0.00403
0.00410
0.00554
0.0200
0.0205
Tapered
N=65
avg
stdev
rms
0.00017
0.120
0.119
0.00840
0.0865
0.0862
0.00154
0.00531
0.00549
0.0265
0.0541
0.0598
Notes: ?Q
w
=Q
w(mea)
Q
w(cal)
, ?h
u
=h
u(mea)
h
u(cal)
, mea = measured, cal = calculated,
N = number of tests.
(1)
Test A5C18N excluded.
(2)
Tests A1C20W, A3B19N and A5C18N excluded.
67
Table 4.9  Statistics of differences between measured and calculated
model values of Q
w
and h
u
(Method C)
?Q
w
(cfs) ?Q
w
/Q
w(mea)
?h
u
(ft) ?h
u
/h
u(mea)
Unsubmerged
N = 238
avg
stdev
rms
0.00250
0.120
0.120
0.00936
0.0639
0.0644
0.00233
0.00799
0.00831
0.0390
0.100
0.107
Unsubmerged
N = 237
(1)
avg
stdev
rms
0.00362
0.119
0.119
0.00875
0.0633
0.0638
0.00236
0.00799
0.00831
0.0393
0.100
0.107
Notes: ?Q
w
=Q
w(mea)
Q
w(cal)
, ?h
u
=h
u(mea)
h
u(cal)
, mea = measured, cal = calculated,
N = number of tests.
(1)
Test A5C18N excluded.
Table 4.10  Statistics of differences between measured and calculated
model values of Q
w
and h
u
(Method D)
?Q
w
(cfs) ?Q
w
/Q
w(mea)
?h
u
(ft) ?h
u
/h
u(mea)
Unsubmerged
N = 238
avg
stdev
rms
0.0305
0.153
0.156
0.0134
0.0913
0.0921
0.00251
0.00871
0.00904
0.0362
0.0959
0.102
Unsubmerged
N = 237
(1)
avg
stdev
rms
0.0311
0.154
0.156
0.0132
0.0915
0.0922
0.00278
0.00764
0.00812
0.0373
0.0946
0.102
Notes: ?Q
w
=Q
w(mea)
Q
w(cal)
, ?h
u
=h
u(mea)
h
u(cal)
, mea = measured, cal = calculated,
N = number of tests.
(1)
Test A5C18N excluded.
68
Table 4.11  Comparison between Method A and Method B
Method A Method B
Bulk discharge equation Discharge equation in terms of discharge
per unit length
Discharge coefficient related to Fw
d
and
L/B
Discharge coefficient related to Fw
d
and
h
d
/P
Q
w
and h
u
obtained from separate equa
tions, h
u
from energy equation but esti
mated Q
w
required to calculate h
u
Q
w
and h
u
obtained from the same analysis
based on momentum equation
? obtained from simplified assumption of
velocity distribution in separation zone
? obtained from interpolation (in some
cases extrapolation) of empirical results
and assumed to vary linearly between up
stream and downstream ends of weir
Details of hydraulics along weir not con
sidered
Water surface profile along weir computed
h
c
given by empirical equation h
c
not used
Assumed ()UP/CcosU
2w
?=?
Method C uses a simpler predictive equation for the discharge coefficient. However the
derivation of the lateral flow coefficient does not seem to have a sound theoretical basis. The use
of C
3
= 2/3 in the present calculation also lacks strong justification. Nevertheless, the lateral
flow coefficient does seem to account for at least part of the variation of the discharge coefficient
along the weir. The results should be regarded as preliminary and further study is needed.
The four methods of calculation were also compared using values of R
2
(Q
w
)andR
2
(h
u
)
defined as
()
()
()
?
?
?
?
?=
N
1
2
)mea(w)mea(w
N
1
2
)cal(w)mea(w
w
2
QQ
QQ
1QR
(4.28)
and
()
()
()
?
?
?
?
?=
N
1
2
)mea(u
)mea(u
N
1
2
)cal(u)mea(u
u
2
hh
hh
1hR
(4.29)
69
where Q
w(mea)
and h
u(mea)
are the averages of Q
w(mea)
and h
u(mea)
respectively. R
2
(Q
w
)and
R
2
(h
u
) are not coefficients of determination for regression equations; rather they are defined
using the concept of the coefficient of determination. A value of R
2
(Q
w
) close to unity indicates
that there is good agreement between the values of Q
w(mea)
and Q
w(cal)
and similarly for h
u
.
Although the values of R
2
for Eq. (4.23), Eq. (4.24), and Eq. (4.27) are low at 0.682,
0.337 and 0.395 respectively, the values of R
2
(Q
w
)andR
2
(h
u
) are all close to unity (Table 4.12).
Method B gave higher values of R
2
(Q
w
) than Method A for all flow conditions and higher values
of R
2
(h
u
) than Method A for both unsubmerged and submerged flows in prismatic channels.
Both Methods A and B gave the same R
2
(h
u
) for flow in tapered channels. Method B gave
higher R
2
(Q
w
) than both Method C and D and Methods B, C and D gave the same values of
R
2
(h
u
).
In view of the above comparison using rms, R
2
(Q
w
)andR
2
(h
u
), Method B is the best and
is recommended for the calculation of side weir discharge and upstream head on the weir.
4.4  EFFECTS OF CHANNEL SLOPE AND ROUGHNESS FOR 2.5H:1V SIDE SLOPES
Simulations for prototype conditions with unsubmerged flow were performed to evaluate
the effects of slope and roughness on Q
w
and h
u
. The geometric conditions were selected from
those used in the model study and are shown in Table 4.13 for a 1:25 model. From the model
test data, the maximum and minimum Q
u
for each geometric condition and the maximum and
minimum h
d
for each Q
u
were selected and scaled to the prototype values.
Four values of Manning's n (0.0125, 0.02, 0.03 and 0.04), and four values of channel
slope (0.000385, 0.0008, 0.0012 and 0.0016) were used in the simulation so that there were 16
combinations of different slopes and roughnesses. The model had a Manning's n of 0.0125 and a
slope of 0.000385. Manning's n of 0.04 and slope of 0.0016 are probably maximum limits of
prototype conditions in Harris County. Results of simulation of Q
w
and h
u
for different geomet
ric and flow conditions using Methods A and B are presented in Appendix 3.
In Method A, the calculation of Q
w
does not involve the slope and roughness. Therefore,
only one value of Q
w
was obtained for each particular geometry and flow. However, for calcula
tion of h
u
, Eq. (2.23) takes into account the roughness and slope. The largest differences
between values of h
u
calculated from Methods A and B are shown in Table 4.14 and the largest
ratios between values of Q
w
calculated from Methods A and B are shown in Table 4.15. The
worst cases give a 75% difference in Q
w
anda1.5ftdifferenceinh
u
. In general, larger slope and
smaller roughness give larger values of h
u
(B)h
u
(A) but smaller values of Q
w
(B)/Q
w
(A).
The following observations were obtained from the results of the simulation. For a
particular slope, h
u
increased as roughness was increased since higher head was required to over
come the increased frictional resistance. For a particular roughness, h
u
decreased as the slope
70
was increased. In most cases, a major contribution to the decrease in h
u
was the increase in the
elevation of the upstream end of the weir crest relative to the downstream end. For flows with
low velocities and thus negligible frictional loss, the decrease was entirely due to the change in
elevation.
Table 4.12  R
2
(Q
w
)andR
2
(h
u
) for comparison between measured and cal
culated values of Q
w
and h
u
Method
Flow
condition
Number
of tests
R
2
(Q
w
)R
2
(h
u
)
A Unsubmerged
Unsubmerged
Submerged
Tapered
238
235
(2)
35
65
0.988
0.990
0.868
0.969
0.988
0.985
0.992
B Unsubmerged
Unsubmerged
Submerged
Tapered
237
(1)
235
(2)
35
65
0.991
0.991
0.978
0.989
0.991
0.991
0.989
0.992
C Unsubmerged
Unsubmerged
238
237
(1)
0.989
0.989
0.991
0.991
D Unsubmerged
Unsubmerged
238
237
(1)
0.980
0.980
0.989
0.991
Notes:
(1)
Test A5C18N excluded.
(2)
Tests A1C20W, A3B19N and A5C18N excluded.
Table 4.13  Geometric conditions used in simulation
Weir length (ft) Invert width (ft) Weir height (ft)
model prototype model prototype model prototype
23.91 598 3.4 85 0.52 13.0
10.00 250 3.4 85 0.52 13.0
15.00 375 1.8 45 0.52 13.0
10.00 250 1.8 45 0.52 13.0
20.00 500 3.4 85 0.70 17.5
10.00 250 3.4 85 0.70 17.5
20.00 500 1.8 45 0.70 17.5
10.00 250 1.8 45 0.70 17.5
71
Table 4.14  Largest differences between values of h
u
calculated from
Methods A and B
Parameter Case 1 Case 2
L (ft) 500 250
B (ft) 45 85
P (ft) 17.5 13.0
Q
u
(cfs) 30012 30022
h
d
(ft) 3.00 3.55
n 0.04 0.02
S
o
0.000385 0.0012
Q
w
(A) (cfs) 3977 3280
Q
w
(B) (cfs) 6962 2856
Q
w
(B)/Q
w
(A) 1.75 0.87
h
u
(A) (ft) 4.48 0.73
h
u
(B) (ft) 3.92 2.20
h
u
(B)h
u
(A) (ft) 0.56 1.47
Table 4.15  Largest ratios between values of Q
w
calculated from Methods
AandB
Parameter Case 1 Case 3
L (ft) 500 598
B (ft) 45 85
P (ft) 17.5 13.0
Q
u
(cfs) 30012 10156
h
d
(ft) 3.00 0.97
n 0.04 0.0125
S
o
0.000385 0.0016
Q
w
(A) (cfs) 3977 1082
Q
w
(B) (cfs) 6962 430
Q
w
(B)/Q
w
(A) 1.75 0.40
h
u
(A) (ft) 4.48 0.36
h
u
(B) (ft) 3.92 0.10
h
u
(B)h
u
(A) (ft) 0.56 0.26
Since the side weir discharge is primarily a function of the head on the weir (Eq. (2.4)),
the same trends of variation with roughness and slope were observed for the side weir discharge.
That is, for a particular slope, Q
w
increased as roughness was increased and for a particular
roughness, Q
w
decreased as the slope was increased.
In addition to the effects of the method of calculation (Table 4.14 and Table 4.15), the
amount of variation of Q
w
and h
u
for different slopes and roughnesses depends on the geometric
and flow conditions. For example, for L = 598 ft, B = 85 ft, P = 13 ft, Q
u
= 30144 cfs and h
d
=
72
4.85 ft, the difference between the maximum and minimum Q
w
was about 7000 cfs and the
difference between the maximum and minimum h
u
was about 3 ft for the different S
o
andnfor
which calculations were done.
All of these results indicate that it is definitely beneficial to use Method B to account for
different channel roughnesses and slopes.
73
5  DISCHARGE AND HEAD LOSS EXPERIMENTS FOR 4:1
SIDE SLOPES
5.1  INTRODUCTION
The objective of the work reported in this section was to conduct hydraulic model experi
ments to evaluate the effects of channel side slope on weir hydraulics. The channel and weir
were modified to have 4H:1V side slopes (Section 3.2). Slopes of 4H:1V are the expected
extreme of flatter slopes, as contrasted to the previous experiments at the opposite extreme of
steeper slopes at 2.5H:1V.
5.2  MODEL RESULTS
Twentyfour tests were conducted for unsubmerged flow for the same general hydraulic
conditions as some of the previous experiments with 2.5H:1V side slopes. The model height of
the weir was 0.5 ft. Two weir lengths of 5 ft and 10 ft were investigated. There were 15 tests for
the 10 ft weir and 9 tests for the 5 ft weir. Six of the 15 tests for the 10 ft weir were duplicate
measurements that confirmed the reproducibility of the results. The test data are tabulated in
Appendix 4. The results of these tests were compared with the previous results to determine the
effect of side slope. The methods of analysis have been described in Sections 4.1  4.3.
5.2.1  Analysis of Data using Method A
In Fig. 5.1, values of C
e
obtained from experimental results (C
e
(observed)) are plotted
against values calculated from Eq. (2.28) (C
e
(regression)); the conditions for each test (A1, A2,
etc.) are given in Appendix 4. The data for the previous tests with 2.5H:1V side slopes, B = 1.8
ft and P = 0.52 ft are also shown in the figure. For this particular geometry, the figure shows a
positive bias in the coefficients calculated from the regression equation, i.e., the values of C
e
(regression) are all larger than the values of C
e
(observed). The bias is observed in the data for
the tests with 4H:1V side slopes as well as for the tests with 2.5H:1V side slopes. It is noted that
the data for unsubmerged tests in the previous project (Tynes, 1989) as a whole do not show a
bias.
The measured values of h
u
are plotted against the values calculated using Method A in
Fig. 5.2. The measured side weir discharges were used in this calculation. Due to the small
hump in the channel invert, Eq. (2.23) was modified to be
cdu
2
d
2
u
2
u
fuddu
h)ELEL(
A
A
1
g2
U
hPPhh +??
?
?
?
?
?
?
?
?
??+?+=
(5.1)
74
C
e
(r
egre
ssio
n
)
0
.
4
0
.5
0.
6
0
.7
0.8
0
.9
1.0
C
e
( o bs e r v e d)
0.40.50.60.70.80.91.0
X5
1
V
:
2
.5
H
5
ft
1V:
2
.5
H
1
0
f
t
1
V
:4
H
5
ft
1V:
4
H
1
0
f
t
A1
A2
A3
B1
B2
B3
C1
C2
C3
AA3
BB
1
BB3
CC1
X1
X2
X3
X4
Y1
Y2
Y3
Z1
CC2
CC3
Fi
g
.
5.1

Observed
and
cal
cul
a
t
e
d
C
e
values
2.5H:1V
5
f
t
2.5H:1V
1
0
f
t
4H:1V
5
f
t
4H:1V
5
f
t
75
h
u
(
c
alculated
)
(
ft)
0.0
0
0.05
0.1
0
0.15
0
.
20
0
.
25
0.30
h
u
(m easure d )( ft)
0.0
0
0.0
5
0.1
0
0.1
5
0.2
0
0.2
5
0.3
0
X5
1
V
:2
.5
H
5
ft
1V
:2
.5H
1
0
f
t
1
V
:4
H
5
ft
1V
:
4
H
1
0
f
t
A1
A2
A3
B1
B2
B3
C1
C2
AA3
BB1
BB3
CC1
X1
X2
X3
X4
Y1
Y2
Y3
Z1
CC2
C3
CC3
Fi
g
.
5.2

Measured
and
cal
cul
a
t
e
d
h
u
using
M
ethod
A
2.5H:1V
5
f
t
2.5H:1V
1
0
f
t
4H:1V
5
f
t
4H:1V
5
f
t
76
where P
u
and P
d
are the weir heights at the upstream and downstream ends of the weir and EL
u
and EL
d
are the channel invert elevations at the upstream and downstream ends of the weir. The
figure shows that there are larger discrepancies between measured and calculated values of h
u
for
the tests with 4H:1V side slopes than for the tests with 2.5H:1V side slopes.
Since the calculation of the side weir discharge by Method A does not take into account
the longitudinal slope and roughness, it is more appropriate to analyze the data using Method B.
5.2.2  Analysis of Data using Method B
In the analysis using Method B, the local invert slope and a local weir height were used
for each of the computational step. Fig. 5.3 shows the comparison between values of C
1
obtained
from numerical optimization with C
2
= 0.85 and values of C
1
calculated from Eq. (4.23). Super
critical flow depths were obtained in the computed water surface profiles for Tests B3, BB3, C2,
CC2, C3 and CC3. Therefore, no results are presented for these tests. There is also a positive
bias in the coefficients calculated from the regression equation for the tests with 2.5H:1V side
slopes. However, the data points for the tests with 4H:1V side slopes exhibit a different pattern
with about half of them above the 1:1 line. The values of h
u
are shown in Fig. 5.4. While most
of the points for the tests with 2.5H:1V side slopes lie close to the 1:1 line, the points for the tests
with 4H:1V side slopes are all above the 1:1 line. In the analysis of the data of the previous pro
ject, C
2
was adjusted to minimize the discrepancies between the measured values of h
u
and the
values from the numerical optimization. Hence the larger discrepancies between the measured
values of h
u
and the values from the numerical optimization for the tests with 4H:1V side slopes
suggested that C
2
= 0.85 was inappropriate for this set of data.
C
2
was changed to 1.10 and the results are shown in Fig. 5.5 and Fig. 5.6. Supercritical
flow depths were obtained in the computed water surface profiles only for Tests C3 and CC3.
For C
2
= 1.10, the values of h
u
from the numerical optimization are in good agreement with the
measured values for the tests with 4H:1V side slopes. Moreover, most of the points in the plot of
C
1
(optimization) vs. C
1
(regression) are below the 1:1 line and follow a pattern similar to that
for the tests with 2.5H:1V side slopes. Nevertheless, data points for a few tests (A3, AA3, B3,
BB3 and X5) are still above the 1:1 line. These five tests had only about 10% diversion.
In Fig. 5.7 and Fig. 5.8, the measured values of Q
w
and h
u
are compared with the values
calculated using discharge coefficients calculated from Eq. (4.23). Fig. 5.7 illustrates that even
though the results for A3, AA3, B3, BB3 and X5 do not follow the general trend in Fig. 5.5, this
behavior should not be of concern in terms of the estimated side weir discharge because the
anomaly occurs only for low discharges. All the data points in Fig. 5.8 lie close to the 1:1 line
indicating good agreement between measured and calculated values of h
u
.
77
C
1
(regr
es
s
i
on)
0.
6
0
.
7
0.8
0
.
9
1.
0
1
.
1
1.
2
1
.3
C
1
(opti m iz ation)
0.60.70.80.91.01.11.21.3
X5
A1
A2
A3
B1
B2
C1
AA
3
BB1
CC
1
X1
X2
X3
X4
Y1
Y2
Y3
2.5H:
1
V
5
ft
2.5H
:1V
1
0
f
t
4H
:1V
5
f
t
4H:
1
V
1
0
f
t
Fi
g.
5
.
3

C
1
f
r
o
m
r
e
g
r
e
ssion
e
qua
tion
a
nd
f
r
o
m
optimiza
tion
w
ith
C
2
=0
.
8
5
78
h
u
(
o
pt
im
i
z
at
io
n)
0.
0
0
0.
05
0
.
1
0
0
.
1
5
0.
20
0.
2
5
0
.
3
0
h
u
(m eas ur ed)
0.
00
0.
05
0.
10
0.
15
0.
20
0.
25
0.
30
X5
2.
5
H
:
1
V
5
f
t
2.5
H
:1V
1
0
f
t
4H:1
V
5
ft
4H
:1V
1
0
f
t
A1
A2
A3
B1
B2
C1
AA3
BB
1
CC
1
X1
X2
X3
X4
Y1
Y2
Y3
Fi
g
.
5.4

Measured
and
num
eri
cal
l
y
opt
i
m
i
z
ed
h
u
with
C
2
=0
.
8
5
79
C
1
(regres
s
i
on)
0.6
0
.
7
0.
8
0
.
9
1.
0
1
.
1
1.
2
1
.3
C
1
( o p t im iz a t i o n)
0.
6
0.
7
0.
8
0.
9
1.
0
1.
1
1.
2
1.
3
X5
A1
A2
A3
B1
B2
C1
C2
AA3
BB1
CC1
X1
X2
X3
X4
Y1
Y2
Y3
Z1
CC2
2.5H:1V,
5
f
t
,
C
2
=
0
.85
2.5H:1V,
10
ft
,
C
2
=
0
.85
4H:1V,
5
ft,
C
2
=
1
.10
4H:1V,
10
ft,
C
2
=
1
.10
Fi
g.
5
.
5

C
1
f
r
o
m
r
e
g
r
e
ssion
e
qua
tion
a
nd
f
r
o
m
optimiza
tion
80
h
u
(optimization)
0.
00
0.
05
0.
10
0
.
15
0
.
2
0
0.
25
0
.
3
0
h
u
( m ea s u r ed)
0.
00
0.
05
0.
10
0.
15
0.
20
0.
25
0.
30
X5
A1
A2
A3
B1
B2
C1
C2
AA
3
BB
1
CC
1
X1
X2
X3
X4
Y1
Y2
Y3
Z1
CC2
2
.
5H:1
V
,
5
ft,
C
2
=0
.
8
5
2
.
5H:1
V
,
10
ft,
C
2
=0
.
8
5
4
H
:1V
,
5
ft,
C
2
=1
.
1
0
4
H
:1V
,
1
0
ft,
C
2
=1
.
1
0
F
i
g
.
5.6

Values
of
h
u
f
r
o
m
r
e
g
r
e
ssion
e
qua
tion
a
nd
f
r
o
m
optimiza
tion
81
Q
W
(
c
a
l
c
u
lat
ed)
0
.
0
0
.
5
1.
0
1
.
5
2.
0
2
.5
3.
0
3
.5
4
.
0
4
.5
Q
W
( m easur ed)
0.
0
0.
5
1.
0
1.
5
2.
0
2.
5
3.
0
3.
5
4.
0
4.
5
X5
1
V
:2
.5H
,
5
f
t,
C
2
=0
.
8
5
1
V
:2
.5H
,
10
f
t
,
C
2
=0
.
8
5
1V
:4H
,
5
f
t,
C
2
=1
.
1
0
1
V
:4
H
,
10
ft
,
C
2
=1
.
1
0
A1
A2
A3
B1
B2
B3
C1
C2
AA
3
BB1
BB3
CC
1
X1
X2
X3
X4
Y1
Y2
Y3
Z1
CC
2
Fi
g
.
5.7

Measured
and
cal
cul
a
t
e
d
Q
w
using
M
ethod
B
2.5H:1V
5
f
t
,
C
2
=0
.
8
5
2.5H:1V
1
0
f
t,
C
2
=0
.
8
5
4H:1V
5
f
t
,
C
2
=1
.
1
5
4H:1V
5
f
t
,
C
2
=1
.
1
5
82
h
u
(cal
cula
ted)
0.0
0
0.05
0.10
0
.
15
0.20
0.25
0.30
h
u
( m eas ur ed)
0.000.050.100.150.200.250.30
X5
A1
A2
A3
B1
B2
B3
C1
C2
AA3
BB1
BB3
CC
1
X1
X2
X3
X4
Y1
Y2
Y3
Z1
CC2
1
V
:
2
.5
H,
5
f
t,
C
2
=0
.
8
5
1
V
:2
.5
H
,
1
0
f
t
,
C
2
=0
.
8
5
1
V
:
4
H
,
5
ft,
C
2
=1
.
1
0
1
V
:4
H,
1
0
ft,
C
2
=1
.
1
0
Fi
g
.
5.8

Measured
and
cal
cul
a
t
e
d
h
u
using
M
ethod
B
2.5H:1V
5
f
t
,
C
2
=0
.
8
5
2.5H:1V
1
0
f
t,
C
2
=0
.
8
5
4H:1V
5
f
t
,
C
2
=1
.
1
5
4H:1V
5
f
t
,
C
2
=1
.
1
5
83
Table 5.1 shows the rootmeansquare values of ?Q
w
, ?Q
w
/Q
w(mea)
, ?h
u
and ?h
u
/h
u(mea)
for the data presented in Fig. 5.7 and Fig. 5.8. The accuracy of the prediction of Q
w
for the tests
with 2.5H:1V side slopes was similar to that for the tests with 4H:1V side slopes. The rms of
?h
u
for the tests with 4H:1V side slopes was about half that of the tests with 2.5H:1V side
slopes.
Table 5.1  rms of ?Q
w
, ?Q
w
/Q
w(mea)
, ?h
u
and ?h
u
/h
u(mea)
?Q
w
(cfs) ?Q
w
/Q
w(mea)
?h
u
(ft) ?h
u
/h
u(mea)
2.5H:1V 0.155 0.119 0.00942 0.0590
4H:1V 0.172 0.130 0.00961 0.0839
The above discussion shows that the regression equation for discharge coefficient for
channels with 2.5H:1V side slopes is applicable for channels with 4H:1V side slopes. However,
C
2
should be increased to 1.1 for channels with 4H:1V side slopes. For slopes between 2.5H:1V
and 4H:1V, linear interpolation may be used to estimate values for C
2
.
85
6  FLOW ASYMMETRY
6.1  INTRODUCTION
6.1.1  Background
Flow diversion at sidechannel weirs causes an asymmetry in the velocity distribution for
the flow that remains in the channel. For higher diversions and/or flatter channel side slopes, a
separation zone is formed on the side of the channel opposite to the weir. Downstream of the
weir, the velocity distribution gradually reestablishes itself to the conditions that would exist for
the discharge downstream of the weir if there were no diversion. The asymmetry and possible
separation zones are important because they can cause the momentum and kinetic energy correc
tion factors (? and ?, respectively) for the channel flow at the downstream end of the weir to be
significantly greater than unity. For subcritical flows with downstream control, the result is that
the head at the downstream end of the weir can be significantly lower than would be calculated if
the flow were assumed to be symmetrical with ? and ? values near unity. When a separation
zone is formed, it is also important for the traditional reasons such as sediment deposition.
Although there have been several papers on the hydraulics of sidechannel weirs, most of
them have addressed discharge coefficients for side weirs and/or the water surface profile in the
channel along a side weir. Very few publications have addressed flow asymmetry and related
considerations. Related literature is summarized in Section 6.1.2 below. Because of the sparsity
of literature on flow asymmetry, the importance of this phenomenon was not recognized at the
beginning of the previous project and experiments for the previous project were planned without
making provisions for measurement of the effects of the separation zone. As a result, the effects
of the separation zone could be included in the previous project only by estimating the size of the
separation zone, not by directly measuring either its size or its effects on the channel and weir
hydraulics. For some situations, that approximate analysis indicated that the flow asymmetry can
cause a head decrease of one foot or more on the weir compared to the water level at the down
stream end of the region of flow asymmetry. An effect with such a magnitude should be based
on direct measurements, not on inferred or estimated characteristics of the separation zones, as
was done previously.
6.1.2  Related Literature
Subramanya and Awasthy (1972) conducted experiments in a rectangular channel with a
side weir. They stated simply that a separation zone was observed on the side of the channel
opposite the weir, but they did not give any quantitative information or further details. The
velocity profiles in their Fig. 4 show no evidence of separation zones, so they apparently were not
86
present for all of the test conditions. They said that ? =1.02and? = 1.04 at the upstream end of
the weir. They did not give any values for the downstream end and said that the values could be
taken as unity without appreciable error.
ElKhashab and Smith (1976) presented a figure with velocity contours showing no sepa
ration zone in a rectangular channel for Q
w
/Q
u
= 56%, where Q
w
= weir discharge and Q
u
=
channel discharge at the upstream end of the weir. They stated that a separation zone existed for
Q
w
/Q
u
? 70% for subcritical flow. They also stated that they used ? = 1. More of their results
are considered in Section 6.8 and Section 6.9.
Balmforth and Sarginson (1983) did experiments in a rectangular channel for the five
flow types given by Frazer (1957). The flow types were identified by Balmforth and Sarginson
as flows in a mild slope channel with (I) a low weir and no downstream throttle, (II) higher weirs
and a downstream throttle, and (III) low weirs and a downstream throttle. Type IV and V were
said to be similar to types I and III but for steep channels. The authors did not give their ranges
of flow conditions and diversions. They said that ? had only small deviations from 1.05, but they
evaluated ? for a tapered channel with a width that decreased in the flow direction beside the
weir. This type of tapering can keep separation zones from forming, even in trapezoidal channels
(Tynes, 1989).
For nearly prismatic, rectangular channels with small slopes, Hager (1981) gave ? as
? =1+
Q'B
Q
2
?
?
?
?
?
?
(6.1)
where Q' = dQ/dx, x = longitudinal coordinate, and Q = discharge in the channel. This equation
was for channels with side weirs and also with side and bottom orifices. It effectively gives not
only ? but also its variation along a side weir since both Q' and Q vary in a channel along a weir.
Hager stated that the effects of ? on the hydraulic calculations are limited to subcritical flows,
and later Hager and Volkart (1986) concluded that the effects of ? for rectangular channels are
negligible in comparison to other effects and assumed ? = 1. Hager and Volkart show velocity
profiles which indicate asymmetry but no separation zone for Q
w
/Q
u
= 50%.
Cheong (1992) used ? = 1 for calculations for comparison with his experimental results
for trapezoidal channels and for both subcritical and supercritical flows upstream of the side
weir. (Note that Cheong?s paper uses the symbol ? for something other than the momentum
correction factor.) Even though he had Q
w
/Q
u
values as high as almost 90%, he did not mention
anything about separation zones.
87
6.1.3  Objective
The objective of the experimental work and associated analysis presented in this chapter
was to determine the effects of the flow asymmetry and separation zones on the channel hydrau
lics and the weir flow. The primary focus was on the asymmetry, ?, and ?, but limited informa
tion was also obtained on the length of the flow reestablishment region downstream of side weir.
6.2  EQUATIONS FOR THE CHANNEL FLOW
Either the momentum equation or the energy equation can be used to calculate the change
in stage in the channel due to flow asymmetry. The momentum equation was written for the
prismatic channel used in the experiments described in this report as
()()()+??= ??????
?
A h h F QU QU
21 2
2 1
(6.2)
where ? = fluid density, Q = flow rate, U = x component of cross sectional average velocity, ? =
fluid specific weight, A = flow area, h = water surface elevation = piezometric head, F
?
=x
component of shear force on the channel bed, x is horizontal (not parallel to the flow, so there is
no weight component in the equation) and positive in the downstream direction, 1 and 2 are
respectively the upstream and downstream cross sections (e.g., at the downstream end of the weir
and the downstream end of the flow reestablishment region), and h
1
h
2
is small. The term
?A
2
(h
1
h
2
) in Eq. (6.2) accounts for the pressure forces at cross sections 1 and 2 plus the x
component of the pressure force on the channel bed and sides. In applying Eq. (6.2), it was
assumed that the channelslopeissmallsothatF
?
is essentially horizontal and so that it is not
necessary to distinguish between the direction normal to the bed and vertical. Dividing Eq. (6.2)
by ?A
2
gives
() ()
A
A
U
g
h
F
A
U
g
h
1
2
2
1
1
2
2
2
2
?
?
?
?
+? = +
(6.3)
where g = gravitational acceleration. The momentum correction factor (?) is defined as
2
A
2
U
dAu
A
1 ?
=?
(6.4)
where u is the x component of the total point velocity and U is the cross sectional average veloc
ity. The instantaneous value of u at a point can be written as
'uuu += (6.5)
88
where u = timeaveraged velocity and u' = turbulent fluctuation. An overbar on any quantity is
used to indicate time averaging. Substitution of Eq. (6.5) into Eq. (6.4) gives
321321
2
2
2
1
2
2
2
2
U
'u
U
u
U
u
?
+
?
==?
(6.6)
and the brackets indicate an average over the cross sectional area. Note that ? and therefore the
momentum flux include the turbulent flux. The components ?
1
and ?
2
in Eq. (6.6) are consid
ered in Section 6.4.8. F
?
was calculated from
Fx
?
??
=
(P)+(P)
12
2
? (6.7)
where ?x is the channel length for which F
?
is being calculated, P = wetted perimeter, and ? in
English units is (Henderson, 1966)
3/1
h
2
R21.2
UUn?
=? (6.8)
where R
h
= hydraulic radius. Since F
?
was frequently the smallest term in Eq. (6.2), it was not
necessary to include the effects of the flow asymmetry in calculating ?.
The energy equation can be written as
H
1
h
f
=H
2
(6.9)
where h
f
is the head loss due to boundary friction and H is the total head given by
H=h+?
U
g
2
2
(6.10)
where ? is the kinetic energy correction factor defined as
3
2
3
A
2
3
A
2
U
uV
U
dAuV
A
1
U
AdVV
A
1
==
?
=?
??
rr
(6.11)
where V is the magnitude of the point velocity vector (
r
V )and
r
A is the area vector pointing
downstream. Using Eq. (6.5), similar expressions for v and w, which are the y (transverse) and z
(vertical) components of velocity, and V
2
=u
2
+v
2
+w
2
, ? in terms of the components of the
velocity is
89
3
2
33
2
3
3
3
3
2
3
2
3
2
3
2
3
2
3
3
U
'w'u
U
'w'uw2
U
'v'u
U
'v'uv2
U
'u
U
'wu
U
wu
U
vu
U
'vu
U
'uu3
U
u
++++
++++++=?
(6.12)
As with ?, ? includes the turbulent flux of kinetic energy. The experimental results showed that
several of the terms in Eq. (6.12) contributed less than 1% to ? so that ? could be evaluated from
{ 32132143421
4
3
2
3
3
2
2
3
2
1
3
3
U
vu
U
'vu
U
'uu3
U
u
?
+
?
+
?
+
?
??
(6.13)
The various components of ? in Eq. (6.13) are considered in Section 6.4.8.
In English units, the head loss due to boundary friction was approximated as
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
+
?
?
?
?
?
?
?
?
?
=
2
3/4
h
2
2
1
3/4
h
2
22
f
RA
Q
RA
Q
42.4
xn
h
(6.14)
Note that the only head loss in Eq. (6.9) is that due to boundary friction, i.e., there is no expan
sion loss. As will be discussed in Section 6.9, Eq. (6.9) does not need to include any other head
loss terms for the region of flow reestablishment downstream of a side weir if appropriate values
of ? are used in defining the total head.
6.3  FLOW CONDITIONS
The experiments for studying the flow asymmetry were organized into two types. In
Type 1, velocities and water surface elevations were measured for at several cross sections in the
region of flow reestablishment for various flow conditions (Cases A  D for 2.5H:1V side slopes
and Case F and G for 4H:1V side slopes, Table 6.1). For Cases A, B, C, and F, there was diver
sion over the side weir. Since there was only about 27.2 ft of channel length downstream of the
weir, the channel was not long enough for laterally symmetrical velocity distributions to be re
established for Cases A, B, and F. For Cases D and G, the side weir was blocked and the flow
conditions at the downstream end of the weir for Cases A and F were recreated at the upstream
end of the channel so that about 60 ft of flow length could be used for measurements. As shown
in Fig. 6.22, this attempt to reproduce the separation zone at the upstream end of the channel was
not successful for Case D because of the way in which the separation zone was created; the
90
problem was corrected for Case G. For Cases E and H, there was no separation zone and the
weir was blocked in order to determine ? and ? for symmetrical velocity distributions.
Measurements were made near the downstream end of the channel. For Cases A, B, and C, the
first cross section (x = 0) was at the bottom of the downstream sloped access ramp. The analysis
showed significant transverse velocities at this cross section. Therefore, for Case F the first cross
section was 2 ft farther downstream; the transverse velocities were much smaller at this cross
section. Also, for the Type 2 measurements (Table 6.2), measurements were made at the down
stream end of the access ramp (x = 2 ft). Measurements showed that for a total discharge of 8.9
cfs with 54% and 26% diversion, there was only a 0.001 ft to 0.002 ft difference between the
water surface elevations at the cross sections at x = 0 and x = 2 ft. The purpose of the tests in
Type 2 was to get additional information on the flow asymmetry, ?,and? at the downstream end
of the weir.
Table 6.1  Flow conditions for Type 1
Case Channel
side slope
Total dis
charge (cfs)
Diver
sion (%)
Remarks
A
2.5H:1V
8.9 54 Side weir diversion
B 2.5H:1V 3.0 54 Side weir diversion
C 2.5H:1V 8.9 25 Side weir diversion
D 2.5H:1V 4.1 N/A Separation zone at
the upstream end of
the channel
E 2.5H:1V 4.1 N/A No flow asymmetry
F 4H:1V 6.1 54 Side weir diversion
G 4H:1V 4.5 N/A Separation zone at
the upstream end of
the channel
H 4H:1V 4.6 N/A No flow asymmetry
6.4  VELOCITY MEASUREMENTS
Velocities were measured using an acoustic Doppler velocimeter (Section 3.4). This
section discusses the measured velocities and the ? and ? values (Appendix 5) that were obtained
from those measurements. The downstream end of the weir is x = 0 for Cases A, B, C, and F
while zero distance is at the downstream end of the flow straighteners for Cases D, E, and G.
The components of the ? and ? values are discussed in Section 6.6.
91
Table 6.2  Flow conditions for Type 2
Channel
side slope
Total dis
charge (cfs)
Diver
sion (%)
2.5H:1V 8.9 54
2.5H:1V 8.9 40
2.5H:1V 8.9 25
2.5H:1V 6.0 55
2.5H:1V 6.1 39
2.5H:1V 3.0 54
2.5H:1V 3.0 40
2.5H:1V 3.0 25
4H:1V 9.0 54
4H:1V 9.0 40
4H:1V 9.0 25
4H:1V 6.0 54
4H:1V 6.0 40
4H:1V 6.0 25
4H:1V 3.0 54
4H:1V 3.0 40
4H:1V 3.0 25
6.4.1  Measurement Procedures
The measurements were taken on seven verticals spaced at approximately 1.1 ft intervals.
The middle vertical was at the centerline of the channel. The two outer verticals on each side
were above the side slope. For smaller flow depths, the outermost vertical on each side was less
than 3.3 ft from the centerline in order to maintain a sufficient distance between the sampling
volume and the boundary. The other verticals were not moved for the smaller depths.
Measurements were taken at three points on each of the middle three verticals. The top set of
measurements was about 2.5 in. below the water surface. The bottom set of measurements was
about 0.4 in. above the invert of the channel. The middle set of measurements was midway
between the top and bottom measurements. Only one measurement was made on the outermost
vertical on each side at about the same elevation as the top set of measurements on the middle
three verticals. Two measurements were taken on the nexttooutermost vertical on each side at
about the same elevations as the top and middle sets of measurements on the middle three
verticals. Hence 15 points were measured in a cross section. The flow depth was measured at
0.4 ft to the right of the centerline. Water surface elevations were measured at the seven verticals
for each cross section.
92
Transverse position (ft)
5 4 3 2 1 0 1 2 3 4 5
+++
+++++
+++++++
Left
Right
weir
Fig. 6.1  Typical locations of velocity measurements looking downstream
Due to the timeconsuming nature of the measurements, several days were required to
complete the measurements for a particular combination of total discharge and percentage diver
sion. The water surface elevations were measured in one day. The flow rate and percentage
diversion quoted for each case in Table 6.1 referred to those during the day of the measurement
of the water surface elevations. The flow rates on different days were within 2% of each other,
thereby indicating that there was also good reproduction of the flow depths from day to day.
6.4.2  Integrations
Numerical integrations of the measured velocities over the flow area had to be done to
calculate ? and ? values (Eq. (6.4) and Eq. (6.11)). The velocities were also integrated to obtain
the flow rate as a check against the flow rate from the flow meters. For all of the integration,
each measured velocity was assumed to represent an area defined laterally and vertically by the
midpoints between the measurements, the channel boundary, or the water surface. The area inte
grations were done first vertically then laterally using a trapezoidal rule in both directions. The
integrands were assumed to be the same at the water surface as for the top measurement, while
they were assumed to be zero at both the lateral boundary and the bottom boundary.
6.4.3  Case A
Case A had Q
u
= 8.9 cfs with 54% diversion for the channel with 2.5H:1V side slopes.
Velocities were measured at x = 0, 4.3 ft, 10.2 ft, 17.3 ft and 22.3 ft.. Fig. 6.2 to Fig. 6.4 show
the longitudinal velocity distributions for the cross sections at x = 0, 4.3 ft, and 22.3 ft. In Fig.
6.2, the presence of a separation zone is indicated by the upstream flow at the top on the left side
of the channel and by the higher velocities on the right for the middle and bottom measurements;
the weir is on the right side. At x = 4.3 ft, there was still a region of average upstream velocity
but it had diminished in size (Fig. 6.3). Along the length of the weir, the flow on the right side of
the channel next to the weir had a strong transverse velocity component due to the flow over the
weir. Immediately downstream of the weir, the residual transverse velocity continued to pull the
water to the right side of the channel. As a result, the velocity distribution was more skewed at x
93
= 4.3 ft than for x = 0. Fig. 6.4 shows that even at the end of the channel, the velocities were
higher on the right than the left.
Transverseposition(ft)
65432101234
u
(
ft/s)
0.5
0.0
0.5
1.0
1.5
top
middle
bottom
_
Fig. 6.2  Longitudinal distributions of velocity at downstream end of weir
crest (Case A)
Transverse position (ft)
65432101234
u
(
ft/s)
0.5
0.0
0.5
1.0
1.5
top
middle
bottom
_
Fig. 6.3  Longitudinal distributions of velocity 4.3 ft downstream from
end of weir crest (Case A)
The ? and ? values for Case A are shown in Fig. 6.5. From both Fig. 6.4 and Fig. 6.5 it is
evident that the channel was not long enough downstream of the weir for the velocities distribu
tions to return to symmetry and for ? and ? to reach their asymptotic values for Case A.
6.4.4  Case B
Case B had Q
u
= 3.0 cfs with 54% diversion for the channel with 2.5H:1V side slopes.
Velocities were measured at x = 0, 4.4 ft, 10.3 ft, 17.4 ft and 22.3 ft. The velocity distributions
resembled those at the corresponding cross sections in Case A. The velocity measurements indi
cate that the relative velocity distributions depend primarily on the percent diversion without a
strong dependence on the flow rate. This conclusion is also supported by the ? and ? values that
are shown in Fig. 6.5 in comparison with the values for Case A. There are small differences in
94
the ? and ? values immediately downstream of the weir, but for x ? 10 ft, the values are much
closer together.
Transverse position (ft)
65432101234
u
(
ft
/s
)
0.5
0.0
0.5
1.0
1.5
top
middle
bottom
_
Fig. 6.4  Longitudinal distributions of velocity 22.3 ft downstream from
end of weir crest (Case A)
?
1.0
1.2
1.4
1.6
1.8
2.0
X
X
X X X
X
x = Longitudinal distance (ft)
0 5 10 15 20 25
?
1.0
1.5
2.0
2.5
3.0
3.5
X
X
X X X
symmetrical
velocity
distribution
Case A
Case B
Case C
Fig. 6.5  ? and ? values for Cases A  C
6.4.5  Case C
Case C had Q
u
= 8.9 cfs with 25% diversion for the channel with 2.5H:1V side slopes.
Velocities were measured at x = 0, 4.4 ft, 10.2 ft, 17.4 ft and 22.2 ft. Fig. 6.6 shows the longitu
dinal velocity distributions at x = 4.4 ft. The velocities were higher on the left than on the right,
in contrast to Cases A and B. This characteristic continued further downstream. The measured
velocities were all positive, even at x = 0. The lateral flow over the weir was apparently not
95
strong enough to pull the bulk of the flow to the right side of the channel to create a separation
zone. Dye tests showed that there were no separation zones for any diversions on the order of
30% or less for 2.5H:1V side slopes. For trapezoidal channels, specification of the conditions for
initiation of a separation zone is somewhat subjective. Even very small diversions may cause a
region of nearly zero or upstream velocity for a width of one to two inches in the model at the
edge of the channel opposite to the weir. Also, upstream flow in a separation zone may be inter
mittent for low diversions. A separation zone was said to exist in the model when there was con
sistent upstream flow at the left most velocity measurement position (y = 3.3 ft).
Transverse position (ft)
65432101234
u
(
ft/
s
)
0.0
0.5
1.0
1.5
2.0
_
top
middle
bottom
Fig. 6.6  Longitudinal distributions of velocity 4.4 ft downstream
from end of weir crest (Case C)
While there is no absolutely conclusive explanation for the change in the velocity distri
butions for Case C relative to Cases A and B, the following comments are offered as speculation.
For several flow conditions, dye tests were done to determine the parts of the channel from which
the flow over the weir was coming. Dye was injected into the flow at the upstream end of the
weir. The injection tube was moved laterally until the dye streak at the downstream end of the
weir was split about evenly with half going over the weir and half going downstream in the chan
nel. This procedure was done for injections at the surface and at the bed. These dye tests indi
cated that the weir flow comes from farther away from the weir at the bottom of the channel than
at the top for most diversions (Fig. 6.7). This behavior is reasonable because the flow has higher
velocities near the surface than near the bed so the flow near the water surface has more down
stream momentum and a resulting stronger tendency to continue down the channel rather than go
over the weir. Applying this rationale to Case C, it is possible that the lower velocities on the
side of the channel near the weir are a result of water being drawn from the lower regions of the
approach flow to fill in the region vacated by water going over the weir. (Fig. 6.7 is for the chan
nel after it had been modified to have 4H:1V side slopes. Earlier qualitative tests with the
2.5H:1V side slopes showed the same trends.)
96
Left Right
Transverse Position (ft)
432101234
Diversion
54%
40
%
30
%
9%
weir
Fig. 6.7  Left boundaries of regions from which weir flow comes
For Case C with Q
u
= 8.9 cfs and 25% diversion, the ? and ? values are shown in Fig.
6.5. The values immediately downstream of the weir are only slightly greater than their asymp
totic values, and they rapidly reach their asymptotic values.
6.4.6  Case D
The channel downstream of the side weir was not long enough for the flow to completely
reestablish itself for diversions on the order of 50%. To allow more channel length to study the
reestablishment for a 54% diversion with a separation zone, the side weir was blocked and a
separation zone was created at the upstream end of the channel after measurements had been
made for separation zones created by outflow over the side weir. The discharge into the channel
was adjusted to be the same as the flow rate downstream of the side weir for Q
u
= 8.9 cfs with
54% diversion. For Case D, the left side of the channel cross section upstream of the flow
straighteners at the head box was blocked to create a separation zone. The flow straighteners are
thin vertical sheets 2 ft long and 2.4 in apart. Also, vertical wood strips were used to adjust the
velocity distribution at the upstream end of the channel to be similar to that downstream of the
side weir in the case with flow diversion. Even though the timeaveraged velocities for the
?forced? separation had reasonable agreement with the actual separation zone, analysis of the
measurements (Section 6.7, Fig. 6.22) showed that there were significant differences in the turbu
lence for the two cases so that the forced separation zone did not accomplish the desired
objective for Case D. The straighteners apparently had a significant effect on the turbulence
created by the flow separation. Since this difference was not discovered until after the channel
had been modified to have 4H:1V side slopes, it was not possible to repeat the measurements.
For Case G (Section 6.4.9), blocking of part of the channel was done downstream of the flow
straighteners. The agreement between the results for Cases F and G is much better than between
Cases A and D (Section 6.7, Fig. 6.23) .
Case D had Q = 4.1 cfs with a forced separation zone at the upstream end of the channel
with 2.5H:1V side slopes. This flow condition was similar to that downstream of the weir for Q
u
= 8.9 cfs with a 54% diversion; the flow of 4.1 cfs is 46% of 8.9 cfs. Velocity measurements
97
were made at the cross sections 7.4 ft, 12.5 ft, 25.5 ft, 37.2 ft, 48.9 ft and 60.6 ft from the down
stream end of the flow straighteners.
The values of ? and ? are shown in Fig. 6.8. Even with 60 ft of channel length, the ? and
? still do not reach their asymptotic values but they get much closer than for Cases A and B.
Longitudinal distance from headbox (ft)
0 10203040506070
?
1.0
1.5
2.0
2.5
3.0
3.5
?
1.0
1.2
1.4
1.6
1.8
2.0
symmetrical
velocity
distribution
Fig. 6.8  ? and ? values for Case D
6.4.7  Case E
Case E had Q = 4.1 cfs with no separation zone in the channel with 2.5H:1V side slopes.
With the side weir blocked with a thin metal sheet and with no flow modification at the headbox
(other than the packed bed and flow straighteners to remove the large scale eddies generated in
the headbox), velocity measurements were made at the cross section 60.6 ft from the downstream
end of the flow straighteners to determine ? and ? values for established flow. These velocity
distributions were essentially symmetrical about the channel centerline.
6.4.8  Case F
Case F had Q
u
= 6.1 cfs with 54% diversion for the channel with 4H:1V side slopes.
Velocities were measured at x = 2.5 ft, 8.1 ft, 13.6 ft, and 19.2 ft.. Fig. 6.9 shows the longitudi
nal velocity distributions at x = 2.5 ft for two sets of measurements made on different days. For
the top and middle measurements, the agreement is good. The bottom measurements are indica
tive of a problem in many of the measurements for the 4H:1V side slopes, namely that it was
difficult to obtain good reproducibility of the bottom measurements. Because of the steep gradi
98
ents of velocity near the bottom of the channel, small differences in the vertical position of the
velocity probe could make a significant difference in the velocities. Nevertheless, it is difficult to
believe that this potential problem is the source of the different velocities since the position of
the probe relative to the boundary was measured with the acoustic probe itself. Except for this
problem with the bottom set of measurements, the velocity distributions for the 4H:1V side
slopes were very similar to those for the 2.5H:1V side slopes. Fortunately, the problem with the
bottom set of measurements did not greatly affect the results for ? and ?. For the two sets of
measurements in Fig. 6.9 the two ? values were 1.99 and 1.91 (4% difference) and the two ?
values were 3.83 and 3.80 (less than 1% difference).
Transverse position (ft)
6 5 4 3 2 1 1 2 3 40
u
(
ft/
s
)
0.8
0.4
0.4
0.8
1.2
1.6
0.0
2.0
top1
middle1
bottom1
top2
middle2
bottom2
_
Fig. 6.9  Longitudinal distributions of velocity 2.5 ft from downstream
end of weir crest (Case F)
The ? and ? values are shown in Fig. 6.10. The values immediately downstream of the
weir are a little larger than for Cases A and B (Fig. 6.5), and the values decrease more rapidly
than for Cases A and B. The average values for Cases A and B are shown by the dashed lines in
Fig. 6.10. The faster decrease is presumably a result of a greater influence of boundary shear
with the flatter side slopes.
6.4.9  Case G
Case G for the channel with 4H:1V side slopes is similar in purpose to Case D for
2.5H:1V side slopes in that a separation zone was created artificially at the upstream end of the
channel to allow additional channel length for reestablishment of the flow. However, this time
the flow was blocked on the downstream side of the flow straighteners. The flow rate was 4.5
cfs, which corresponds to the flow downstream of a weir with Q
u
= 9.0 cfs and 50% diversion.
The agreement between the results for Cases F and G is much better than between Cases A and
D, as discussed in Section 6.7. Velocity measurements were made at 15 ft, 17.2 ft, 19.3 ft, 21.5
99
ft, 23.6 ft, 25.8 ft, 28 ft, 32.3 ft, 34.5 ft, 46.5 ft and 58.5 ft from the downstream end of the flow
straighteners. The large number of measurements was due to the variation of ? and ? (Fig. 6.11)
being somewhat irregular and the desire to try to determine the variation correctly.
0 5 10 15 20 25
?
1.0
1.5
2.0
2.5
3.0
3.5
x = Longitudinal distance (ft)
?
1.0
1.2
1.4
1.6
1.8
2.0
symmetrical
velocity
distribution
Case F
Cases A&B
Fig. 6.10  Variation of ? and ? (Case F)
Visual observation of the flow indicated that the flow immediately downstream of the
obstruction blocking part of the channel to create the separation zone was not similar to the flow
conditions at the downstream end of the weir. Thus, it was decided to start the measurements 15
ft downstream of the headbox. As the initial increases in ? and ? (Fig. 6.11) indicate, the asym
metry did not start to decrease until almost 20 ft downstream of the headbox.
6.4.10  Case H
Case H had Q = 4.6 cfs with no separation zone in the channel with 4H:1V side slopes.
With the side weir blocked with a thin metal sheet and with no flow modification at the headbox
(other than the packed bed and flow straighteners to remove the large scale eddies generated in
the headbox), velocity measurements were made at the cross section 62.5 ft from the downstream
end of the flow straighteners to determine ? and ? values for established flow. These velocity
distributions were essentially symmetrical about the channel centerline.
100
Longitudinal distance from headbox (ft)
0 10203040506070
?
1.0
1.5
2.0
2.5
3.0
3.5
?
1.0
1.2
1.4
1.6
1.8
2.0
symmetrical
velocity
distribution
Fig. 6.11  Variation of ? and ? (Case G)
6.5  ? AND ? VALUES AT DOWNSTREAM END OF WEIR
The parameters ? and ? at the downstream end of the weir are important since at least
one of them is needed to relate the head at the downstream end of the weir to the water level
downstream of the flow reestablishment region. The importance of having reasonable values for
? or ? increases as the flow velocity increases. Eq. (6.3) shows that there is a linear relationship
between ?U
2
/g and water surface elevation (or flow depth). If U is 6 ft/s at the downstream end
of the weir for prototype conditions, then assuming that ? = 1 when the actual value is 1.75 will
produce an error of 0.84 ft in the head on the weir while the error is only 0.05 ft when U = 1.5
ft/s. Eq. (6.10) shows a linear relationship between ?U
2
/2gandhforagivenHsoassuming? =
1 when the actual ? = 3 for U = 6 ft/s gives an error of 1.12 ft in h while the error is only 0.07 ft
when U = 1.5 ft/s.
In Fig. 6.12, measured ? and ? values at the end of the weir (Appendix 5.3) are plotted as
functions of Q
w
/Q
u
for 2.5H:1V side slopes As mentioned earlier, the primary dependence of
both ? or ? is on Q
w
/Q
u
, or equivalently Q
u
/Q
d
since Q
u
/Q
d
=1/(1Q
w
/Q
u
). The secondary
variation (scatter about the curve) comes from the fact that both ? and ? decrease slightly as Q
u
increases. Several attempts were made to find a suitable dimensionless parameter to represent
this variation, but none could be found. The best relationships that could be found for 2.5H:1V
side slopes using dimensionless parameters are
101
?=
?+
?
?
?
?
?
?
?
?
?
?
?
108
0 991 0 301 0 298
2
.
Q
Q
Q
Q
u
d
u
d
.. .
for 1
Q
Q
u
d
??125.
for 1.25
Q
Q
u
d
<<2.
(6.15)
?=
?+
?
?
?
?
?
?
?
?
?
?
?
123
166 170 108
2
.
..
Q
Q
Q
Q
u
d
u
d
.
for 1
Q
Q
u
d
??125.
for 1.25
Q
Q
u
d
<<2.
(6.16)
where Q
u
/Q
d
=1/(1Q
w
/Q
u
). To obtain these relationships, polynomials were first fitted to the
data points for flows with diversions. Then the intersections of the polynomials with the ? and ?
values for undisturbed flow (? =1.08and? = 1.23) were found. The values for undisturbed flow
were assumed to apply below the intersections, both of which occurred at Q
u
/Q
d
=1.25orQ
w
/Q
u
=0.2.
Q
w
/Q
u
=Diversion
0.0 0.1 0.2 0.3 0.4 0.5 0.6
?
and
?
1.5
2.0
2.5
3.0
3.5
1.0
?
?
Fig. 6.12  ? and ? values at end of weir for 2.5H:1V side slopes
The results for 4H:1V side slopes are given in Appendix 5.4. Curve fitting to the data
points gave
?
?
?
?
?
+?
=?
d
u
Q
Q
987.0073.0
12.1 20.1
Q
Q
1for
d
u
??
2.2<
Q
Q
<1.20for
d
u
(6.17)
102
?
?
?
?
?
+?
=?
d
u
Q
Q
93.228.2
17.1 18.1
Q
Q
1for
d
u
??
2.2<
Q
Q
<1.18for
d
u
(6.18)
as the best fit equations.
Q
w
/Q
u
= Diversion
0.0 0.1 0.2 0.3 0.4 0.5 0.6
?
and
?
2.0
3.0
4.0
5.0
1.0
?
?
Fig. 6.13  ? and ? values at end of weir for 4H:1V side slopes
In the absence of a detailed investigation, Tynes (1989) assumed that the velocity was
zero in the separation zone and uniform in the effective flow area. The ratio of the effective flow
area to the crosssectional area was taken to be Q
d
/Q
u
. Accordingly, immediately downstream of
the side weir,
? =
?
?
?
?
?
?
Q
u
Q
d
2
and ? =
Q
u
Q
d
(6.19)
The assumption by Tynes (1989) about the velocity distribution in the separation zone leads to
overestimation of ? and ?. For example, depending on the flow rates, for 54% diversion, ? is
overestimated by 40% to 70% and ? is overestimated by 20% to 30%.
6.6  COMPONENTS OF ? AND ?
6.6.1  Variation with Flow Distance
Appendix 5.1, Appendix 5.2, Fig. 6.15, Fig. 6.16, Fig. 6.17, Fig. 6.18 and Fig. 6.19 give
the components of ? and ? (Eq. (6.4) and Eq. (6.13)) for the tests for which velocities were
measured at different longitudinal distances in the channel. The components were evaluated to
determine the relative significance of the various terms in the momentum and kinetic energy
transport. The downstream end of the weir is x = 0 for Cases A, B, C, and F while zero distance
is at the downstream end of the flow straighteners for Cases D, E, and G. The results for some of
103
the components of ? and ? were inaccurate for Case F, so there are some missing values in
Appendix 5.2 and Fig. 6.18.
The longitudinal distances were normalized with respect to a transverse length scale (B
s
)
associated with the asymmetrical velocities at the end of the weir. To calculate B
s
,itwas
assumed that all of the flow at the downstream end of the weir is in an effective area (A
e
,Fig.
6.14) and that the velocity is zero in the remainder of the channel that has a width of B
s
.The
velocity (u
e
) in the effective area was calculated so that A
e
u
e
= Q. Since Q also is equal to AU,
u
e
/U = A/A
e
. Then, from Eq. (6.4) neglecting the turbulent transport of momentum,
e
2
2
e
2
e
A
2
e
2
A
2
A
A
A
Q
A
A
Q
A
1
U
dAu
A
1
U
dAu
A
1
=
?
?
?
?
?
?
?
?
?
?
?
?
?
?
===?
?
?
(6.20)
A
e
B
s
u
y
u=0
u
e
Fig. 6.14  Assumed velocity distribution for calculating B
s
x/B
s
= Dimensionless distance
012345678
Co
mpon
ents
o
f
?
and
?
3
6
3
6
3
6
0.001
0.01
0.1
1
3
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.15  Components of ? and ? for 54% diversion (Cases A and B)
104
B
s
was then calculated as the value needed to give A
e
so that ? in Eq. (6.20) is equal to the
empirical ? value at the downstream end of the weir. The results are shown in Appendix 5.1 and
Appendix 5.2. Since Case D with the forced separation zone at the upstream end of the channel
was supposed to represent the same flow conditions as Case A, B
s
was also taken to be the same
for Case D as for Case A. For the same reason, B
s
forCaseGisthesameasforCaseF. There
were no B
s
values for Cases E and H. The locations of the measurement cross sections for Cases
A  C were essentially the same; the larger dimensionless distances for Case C are the result of
the smaller B
s
value, not larger x values.
x/B
s
= Dimensionless distance
0 5 10 15 20 25 30
Com
pone
nts
o
f
?
and
?
3
6
3
6
3
6
0.001
0.01
0.1
1
3
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.16  Components of ? and ? for 25% diversion (Case C)
x/B
s
= Dimensionless distance
0 5 10 15 20 25
Compon
ents
o
f
?
an
d
?
6
3
6
3
6
3
6
0.001
0.01
0.1
1
3
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.17  Components of ? and ? for forced separation zone (Case D)
105
x/B
s
= Dimensionless distance
01234567
Comp
onen
ts
of
?
an
d
?
3
6
3
6
3
6
3
6
0.001
0.01
0.1
1
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.18  Components of ? and ? for forced separation zone (Case F)
x/B
s
= Dimensionless distance
0 5 10 15
Co
mponen
ts
of
?
and
?
6
3
6
3
6
3
6
3
6
0.001
0.01
0.1
1
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.19  Components of ? and ? for forced separation zone (Case G)
Several trends are apparent from the measurements:
(1) Most of the measurements at the end of the weir for the 2.5H:1V side slopes (Fig. 6.15 and
Fig. 6.16) do not fit the trends downstream from the weir. Thus, the trend lines are not
extended to x = 0. The apparent reason is the relatively strong transverse timeaveraged
velocity component toward the weir at this cross section. One of the clearest indications of
this behavior is that ?
4
, which includes the timeaveraged transverse velocity (
2
v ), is almost
two orders of magnitude larger at the end of the weir for Cases A and B than would be indi
cated by extrapolating the trend of the other points back to the weir. At the end of the weir,
most of the other components are a little smaller than would be indicated by extrapolation
from the downstream points. The same type of problem does not exist for Case F since the
first measurement cross section was at x = 2.5 ft, not x = 0.
106
(2) Downstream from the weir, ? and ? and each of their components decrease with increasing
longitudinal distance as the flow asymmetry decreases.
(3) For Case C with the smaller diversion, the relative magnitudes of the turbulence terms are
smaller than for Cases A, B, and F. There are no trend lines for Case C (Fig. 6.16) since the
components for the first cross section were affected by the transverse velocities and since
asymptotic conditions were reached upstream of the last two cross sections.
(4) The relative magnitudes of the turbulence terms are smaller for Case D with the forced sepa
ration zone for the channel with 2.5H:1V side slopes than for Cases A and B even though the
timeaveraged normalized longitudinal velocity distributions were nearly the same for all
three cases. This comparison indicates that the manner in which the separation zone was
created for Case D was not a good reproduction of the effects of the weir, as mentioned
earlier. The same problem does not exist for Case G where the area was blocked downstream
of the flow straighteners to create the separation zone.
(5) For Case E, the values of ? and ? were found to be 1.08 and 1.23. These values agree with
the downstream values for Case C, so they were adopted as the values for undisturbed flow in
the channel with 2.5H:1V side slopes. The corresponding values for the 4H:1V side slopes
were ? =1.12and? = 1.17.
6.6.2  Variation with Diversion
Appendix 5.3, Appendix 5.4, Fig. 6.20, and Fig. 6.21 give the components of ? and ? at
the downstream end of the weir. All components for both ? and ? increase as the diversion
increases, but ?
2
and ?
2
through ?
4
increase at approximately the same rate and more rapidly
than ?
1
and ?
1
. Similar rates of increase might be expected for ?
2
, ?
2
,and?
3
since all of these
terms have the mean of a squared turbulent velocity, but ?
4
has about the same rate of increase
with diversion even though it includes only timeaveraged velocities and is the smallest of the
terms. The relative importance of the turbulent transport increases as the diversion increases; ?
2
varies from 1% to 10% of ? while ?
2
supplies 2% to 15% of ?, ?
3
supplies 0.2% to 2.4%, and ?
4
supplies only 0.1% to 1.5%. All of the other terms that come from a complete expansion of
(Eq. (6.12)) are less that 1% of ?, even at the highest diversion rates.
6.7  LENGTH FOR FLOW REESTABLISHMENT
Downstream of a side weir, the velocity distribution gradually returns toward symmetry.
At the beginning of the measurements and analysis, it was not clear how to best quantify the
asymmetry of the velocity distributions. Thus, three parameters (in addition to ? and ?)were
used and were calculated separately for the top, middle and bottom sets of measurements at each
cross section. These parameters represented (a) the rootmean squared variation of differences
107
between the velocities on the right side of the channel and the velocities at the corresponding
points on the left side, (b) the skewness of the velocity distributions, and (c) the area under the
velocity distribution curve in the right half of the channel and that in the left half. None of these
parameters proved to be significantly more informative than ? and ?,so? and ? were used as the
primary parameters to represent the amount of asymmetry in the velocity distributions.
Q
w
/Q
u
= Diversion (%)
15 25 35 45 5520 30 40 50
Co
mpone
nts
o
f
?
an
d
?
3
6
3
6
3
6
0.001
0.01
0.1
1
3
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.20  Variation of components of ? and ? near end of weir with
diversion for 2.5H:1V side slopes
Q
w
/Q
u
= Diversion (%)
15 25 35 45 5520 30 40 50
Compon
ents
of
?
and
?
3
6
3
6
3
6
3
6
0.001
0.01
0.1
1
?
1
?
2
?
1
?
2
?
3
?
4
Fig. 6.21  Variation of components of ? and ? near end of weir with
diversion for 4H:1V side slopes
The variations of ?  ?
o
,where?
o
is the value for symmetrical velocity distributions, with
distance were used to determine the flow length (L
s
) required for the flow to return to symmetry.
108
For 2.5H:1V side slopes, ?
o
=1.08 while the value is 1.12 for 4H:1V side slopes. The variations
of ?  ?
o
were also studied, but they gave slightly different results. It was decided to use ? since
it is used in the computer program rather than ?. The variations of ?  ?
o
with distance for
2.5H:1V side slopes are given in Fig. 6.22 on a semilogarithmic plot since decay processes fre
quently have an exponential decay as they approach their asymptotic values. For this figure, the
results for Case D were treated as if the section at 12.5 ft from the flow straighteners were 4.3 ft
downstream from the end of the weir crest. This matching is based on the fact that the measured
velocities for these two cross sections were essentially the same.
x/B
s
= Dimensionless distance
024681012141618
?
1
.
0
8
0.01
0.1
1
Case
A
B
C
D
Fig. 6.22  Exponential decay of excess ? for 2.5H:1V side slopes
One of the prime objectives for Case D was to get a direct indication of the length
required for the flow asymmetry to disappear. Since the turbulence for Case D was less intense
than for Case A (Section 6.4.8), the asymmetry disappeared more slowly for Case D than for
Case A. Nevertheless, Case D has data over a larger distance and indicates that there is indeed an
exponential decay of ?  ?
o
. Based on this type of behavior, the best fit line through the points
for Cases A and B (except for x = 0, for reasons discussed earlier) is extrapolated to a value of
0.05 to represent the point at which ? decays to within 5% of its asymptotic value. This process
gives L
s
/B
s
= 12.5 for 54% diversion. For Case C, the last two points are not plotted since ? had
reached its asymptotic value upstream of these points. The bestfit line through the remaining
points gives L
s
/B
s
= 0.6 for 25 % diversion. However, it must be recognized that these values of
L
s
come from extrapolation of the measurements, that different values of L
s
would be obtained if
different parameters other than ? were used, and that the results for L
s
are very limited. Thus,
these values must be viewed as only an indication of the length of the flow reestablishment dis
109
tance. Fortunately, as is illustrated in Section 6.9, it is normally not necessary to know L
s
with
high accuracy.
Similar results for 4H:1V side slopes are given in Fig. 6.23. For these measurements, the
blocked area to create a separation zone was downstream of the flow straighteners. It can be seen
that the trends for Cases F (with actual diversion) and G (with a forced separation zone) are
essentially the same. For these measurements L
s
/B
s
=7.3.
x/B
s
= Dimensionless distance
024681012
?
1
.
1
2
0.01
0.1
1
Case
F
G
Fig. 6.23  Exponential decay of excess ? for 4H:1V side slopes
Assuming that L
s
/B
s
would be zero for no diversion and would remain small for diver
sions less than 30% since there is essentially no separation zone for those conditions, an esti
mated variation of L
s
/B
s
is given in Fig. 6.24. This figure, as well as a comparison of Fig. 6.22
and Fig. 6.23, shows that the flow conditions for 4H:1V side slopes return to symmetry more
rapidly than for 2.5H:1V side slopes. These figures give the results in terms of dimensionless
distances (x/B
s
and L
s
/B
s
), but the same conclusion applies for actual distances (x and L
s
).
Because of the very limited data, Fig. 6.24 needs to be used with caution.
6.8  MOMENTUM AND ENERGY BALANCES
The measurements allowed all terms in Eq. (6.3) except F
?
to be calculated, and F
?
could
be found from Eq. (6.7) and Eq. (6.8). Thus, to investigate the accuracy of the measurements, the
momentum equation (Eq. (6.3)) was written as
( ) ( )
2
2
2
M
2
1
1
2
2
1
h
g
U
h
A
F
h
g
U
A
A
+
?
=??
?
?+
?
?
(6.21)
110
where ?h
M
is a residual term to account for any inaccuracies in the measurements in balancing
the momentum equation. The term ?h
M
was evaluated from the measurements with cross
sections 1 and 2 in Eq. (6.21) being successive measurement cross sections. The measurements
also allowed determination of a residual term (?h
E
) for the energy equation written as
H
1
h
f
 ?h
E
=H
2
(6.22)
Q
w
/Q
u
=Diversion(%)
0 102030405060
L
s
/B
s
=
D
ime
n
si
onl
es
s
l
e
ngth
of
f
l
ow
r
e

e
stal
i
s
hme
n
t
r
egi
on
0
2
4
6
8
10
12
14
Empirical
2.5H:1V
4H:1V
Estimated
Estimated
Fig. 6.24  Length of flow reestablishment region
From the measurements, ?h
M
(Eq. (6.21)) and ?h
E
(Eq. (6.22)) were calculated for each
pair of consecutive cross sections (Appendix 6.1 to Appendix 6.10). The rate at which momen
tum was transported across a section was calculated as ??Q
d
2
/A, where Q
d
is the measured flow
rate, for all cross sections except x = 0.
?
A
dAu wasusedinlieuofQ
d
for the cross section at
the downstream end of the weir crest because of the outflow over the downstream ramp. The
velocity head was calculated as ?Q
d
2
/(2gA
2
). In each case, Q
d
during the water surface
elevation measurements was used for the cross sections downstream from the side weir.
In the calculation of
?
A
dAu , a parabolic distribution was assumed for u below the
bottom measurement point. Between the water surface and the top measurement point, u was
assumed to be the same as that of the top measurement; u was assumed to vary linearly between
the top and middle measurement points and between the middle and bottom measurement points.
The vertical integration was done first using the assumed distribution of u and then laterally with
u = 0 at the sides of the channel. For the cross section at the downstream end of the weir crest
for Case A, the difference between
?
A
dAu and Q
d
was 8%. For all the other measurements, the
differences were less than 6%. The discrepancies were considered acceptable given that veloci
ties were measured only at 15 points in a cross section. There was lateral flow over the ramp at
the downstream end of the side weir. The higher discrepancy for the cross section at the down
stream end of the weir crest for Case A was probably due to the larger flow over the ramp.
111
As mentioned in Section 3.4, the distances from boundary reported by the data acquisition
program were inaccurate for measurements above the side slopes. It was estimated that the errors
in the calculation of
?
A
dAu , ? and ? due to the inaccuracy in the distance measurements were
less than 1%.
The water surface elevations were measured at the same cross sections and the same
seven transverse locations where velocities were measured. In the momentum and energy
balances, the water surface elevation for each cross section was taken as the average of the meas
urements for that cross section. For the cross sections other than that at the downstream end of
the weir crest, the differences between water surface elevations for the same cross section were at
most 0.003 ft. For the section at the downstream end of the weir crest, there were larger differ
ences because of the drawdown due to the flow over the weir crest. Therefore, for that section,
the measurements at y = 1.1 ft, 2.2 ft, and 3.3 ft (i.e., the measurements on the weir side of the
channel) were not included in the average for Case A, and the measurements at y = 3.3 ft were
not included in the averages for Cases B and C; the drawdown at the weir was much smaller for
Cases B and C than for Case A.
The largest values of ?h
M
and ?h
E
were between the cross section at the end of the weir
and the next cross section since there was still a significant transverse velocity at the end of the
weir. Excluding the values at the end of the weir, the residuals in balancing the equations are
smaller. The residuals in Table 6.3 are rather small given that the measurement accuracy for
water surface elevations was on the order of 0.001 ft to 0.002 ft and only 15 velocities were
measured in each cross section. It was essential to include the turbulent fluxes of momentum and
energy to obtain this good degree of closure for the momentum and energy equations. An indi
cation of the magnitude of the turbulence flux terms is given in Section 6.4.8. The momentum
and energy balances were not done for Case G; the excellent results for the other cases indicated
that accurate measurement techniques were being used.
Table 6.3  Summary of errors in balancing momentum and
energy equations
?h
M
?h
E
(ft) (ft)
All cross
sections
Exclude
end of
weir
All cross
sections
Exclude
end of
weir
maximum 0.0009 0.0009 0.0023 0.0023
average 0.0010 0.0004 0.0003 0.0001
minimum 0.0077 0.0016 0.0045 0.0015
standard deviation 0.0019 0.0006 0.0014 0.0009
112
The discussion above on the momentum and energy balances relates to the flow reestab
lishment region downstream of the weir. ElKhashab and Smith (1976) made detailed velocity
and depth measurements in a rectangular channel in the region beside a side weir. The velocity
and flow depth upstream of the weir were 3.9 ft/s and 0.85 ft. The diversion was 70%. (These
values are only approximate since they had to be read from a graph or calculated from values
read from a graph. Also, the 70% diversion is based on an estimated ? of 3 at the downstream
end of the weir.) They found a large imbalance in trying to close the energy equation for the
section of the channel along the weir. The imbalance was approximately 0.8 in. between the
upstream and downstream ends of the weir; this 0.8 in. was 16% of ?U
2
/2g at the upstream end
of the weir and 1.7 times ?U
2
/2g at the downstream end of the weir. ElKhashab and Smith
(1976) included the lateral and vertical velocities in their kinetic energy terms but not the turbu
lent transport of kinetic energy. If the estimated diversion is approximately correct, then there
was little or no separation zone for this flow condition in their rectangular channel. For their
channel, it is difficult to imagine that this 0.8 in. is head loss due only to flow asymmetry. Omit
ting the turbulence terms apparently accounted for at least part of the excess head loss.
6.9  APPLICATION
To illustrate the importance of flow asymmetry and ?, consider an improved trapezoidal
channel with a bed slope of 0.0008, a base width of 85 ft, side slopes of 2.5H:1V, a Manning?s n
of 0.035, Q
u
= 25,000 cfs, and a 50% diversion so that Q
d
= 12,500 cfs. The flow depth (d
2
)at
the downstream end of the flow reestablishment region is the normal depth of 16.1 ft. From Eq.
(6.15), ?
1
= 1.58 at the downstream end of the weir. Eq. (6.2) or Eq. (6.3) with ?
2
=1.08gives
the depth (d
1
) at the downstream end of the weir as 15.4 ft. The calculation of d
1
must be done
by iteration or by using a solver since B
s
(64.6 ft) depends on d
1
and ? as described previously.
From Fig. 6.24, L
s
/B
s
? 9 giving L
s
? 580 ft. For this situation, the flow asymmetry at the end of
the weir causes the flow depth and therefore the head on the downstream end of the weir to be
0.7 ft smaller than would be indicated by the downstream flow depth. Increasing or decreasing
L
s
by 50% gives essentially no change in d
1
. Even though the change in L
s
gives a significant
change in F
?
, the change in L
s
also gives a compensating change in the water surface elevations at
the two cross sections (h in Eq. (6.3)). Thus, it is not necessary to know L
s
with a high degree of
accuracy. If the downstream controls give d
2
= 12.1 ft (half way between the critical and normal
depths), then d
1
= 11.4 ft so that the depth at the downstream end of the weir is still 0.7 ft smaller
than farther downstream.
The previous examples assume that Q
w
is known. However, another level of iteration is
required in most calculations since Q
w
cannot be determined until the head on the weir is known.
For river channels, the flow (Q
u
) approaching a side weir is determined by the hydrology of the
113
watershed upstream of the weir but the flow depths are controlled from downstream for subcriti
cal flows. Thus, iterative calculations are required to determine the flow depth (d
1
) at down
stream end of a weir and therefore to determine the flow (Q
w
) over a weir since Q
w
depends on
the head which depends on the downstream depth which depends on the downstream flow (Q
d
)
which is equal to Q
u
Q
w
. A typical computational approach would be to assume Q
w
then use
onedimensional gradually varied flow calculations to obtain the water surface elevation at the
downstream end of the flow reestablishment zone for Q
d
=Q
u
Q
w
and thereby to obtain the
righthand side of Eq. (6.3) using ? for established flow. From this depth, the head on the weir
and then Q
w
can be calculated. This process can be continued until the assumed and calculated
values of Q
w
agree. This is the type of calculation that is done in SIDEHYD (Burgin and Holley,
2002).
If the energy equation is used rather than the momentum equation, the approach is basi
cally the same except that ?, Eq. (6.9) and Eq. (6.10) are used rather than ? and Eq. (6.3). It is
important to recognize that the only head loss which is needed in Eq. (6.9) applied to the flow re
establishment region is the head loss due to the boundary shear stress, provided that appropriate
? values are used. There is no additional head loss needed downstream of the weir to account for
the flow asymmetry or even the separation zones, for those flows with separation zones. This
condition is indicated by the excellent closure of the energy equation for the laboratory measure
ments when using only Eq. (6.9) with no additional head loss terms. If ? = 1 were assumed
throughout the flow reestablishment region, then the energy equation might be written as
h
U
g
hK
U
g
h
U
g
fL
+
?
?
?
?
?
?
?
?
?? =+
?
?
?
?
?
?
?
?
2
1
2
22
2
22
(6.23)
where K
L
would appear to be a head loss coefficient. However, the laboratory measurements
indicate that K
L
would need to be negative to balance the energy equation so K
L
could not actu
ally be a head loss coefficient. Rather, it would be due to the fact that U
1
2
/2g is too small to
account for the true velocity head, which is ? U
1
2
/2g. These observations are consistent with
those of Idelchik (1986) for other types of branching flows.
Some of the results presented here are dependent on the channel geometry. ElKhashab
and Smith (1976) did experiments in a rectangular channel 1.51 ft wide with heights of 0.33 ft to
0.82 ft for thin plate weirs. They found that separation zones formed only for diversions of 70%
or greater. The experimental work for this project was done in a trapezoidal channel with
2.5H:1V side slopes, and separation zones formed for diversions of about 30% and greater. Flow
visualization was done in a trapezoidal channel with 4H:1V side slopes. For this channel, sepa
ration zones formed for diversions of 20% and greater. This type of trend seems reasonable since
114
flatter side slopes give larger regions of low velocity on the side opposite to the weir, thereby
making it easier for a separation zone to form.
115
7  DIVERSION CULVERTS
7.1  INTRODUCTION
For some small diversions, it may be beneficial to use culverts for diversion rather than
side weirs. Just as the discharge coefficients for side weirs depend on the channel flow charac
teristics as well as the normal weir parameters, it is to be expected that the flow through diver
sion culverts will also depend on the channel flow. Therefore the analysis of diversion for
culverts needs to be modified to account for the effects of the channel flow.
The objective of these experiments was to evaluate the effects of channel flow on the
hydraulics of culverts at diversion facilities.
7.2  THE PHYSICAL MODEL
The diversion culvert model (Fig. 7.1) was built in the channel with 4H:1V side slopes.
Part of the side weir was blocked leaving an opening 1.271 ft long for the culverts. The culvert
model was made of 3/4 in. plywood with a base sitting on the weir crest of the side weir and with
vertical walls at the upstream and downstream ends of the culverts. The vertical walls had a
trapezoidal shape matching the 4H:1V side slopes of the embankment of the channel. Two verti
cal walls, 0.563 in. long, divided the culvert into three barrels 0.38 ft wide. The culvert model
did not have a top so that the flow in the culvert could be observed more clearly. The invert of
the culvert was about 0.55 ft from the invert of the channel. Manning?s n for the plywood culvert
model was assumed to be 0.0012 (Henderson, 1966). Some tests were done with flow in all three
barrels. Tests were also done with only two barrels. For these tests, a false wall was installed
parallel to the end wall and in line with one of the walls separating the barrels. An inclined cover
in line with the sloped side of the channel was then placed over the opening between the two ver
tical walls so that the geometry for the two operating barrels was similar to that for the three bar
rels.
Fig. 7.1  Schematic diagram of diversion culverts in model (not to scale)
support
model weir
4
1
0.563 ft
0.55 ft
1.625 ft
0.406 ft
channel
vertical end wall wall between culvert barrels
point 0 1 2 3 4 5
116
Photographs of the model are shown in Fig. 7.2. For all flow conditions, large eddies
developed upstream of the first and last barrel. Both eddies were caused by flow over the end
walls; there was flow in the upstream direction, back into the culverts, over the downstream wall.
The flow plunged over the walls, causing the eddies. The eddy on the most upstream barrel was
frequently more pronounced than the one at the most downstream barrel since gravity added to
the channel velocity as the flow plunged over the upstream wall while the gravityinduced flow
and the channel flow were in opposite directions at the downstream wall.
a) Higher channel velocity
b) Lower channel velocity
Fig. 7.2  Diversion culverts
channel flow
channel flow
117
7.3  MODEL RESULTS
Experiments were conducted for four different upstream discharges. For each of four
upstream discharge, different diversions were investigated. The total number of tests was 38.
The results are given in Appendix 7. For most of the tests, there was free or unsubmerged flow
at the downstream end of the culverts (cross section 4 in Fig. 7.1). A few tests were done with
submerged conditions.
7.3.1  Unsubmerged flow
Discharges and water levels were measured as described in Sections 3.3 and 3.5. In most
cases, the water surface elevations upstream and downstream of the culvert were the same; the
largest difference was only 0.002 ft.
Flow through the culvert was very complex. The simplified analysis of the flow was
done as follows. Although there were multiple barrels, an average flow depth both upstream and
downstream of the culvert barrels was used. Critical flow was assumed to exist at the down
stream end of the barrels for unsubmerged flow conditions. Gradually varied flow calculations
were performed from the downstream end of the barrels to the upstream end of the barrels. The
entrance loss at the upstream end of the barrels, h
entr
, was calculated as
( )
g2
VV
3.0h
2
2
2
3
entr
?
= (7.1)
where V
3
is the average velocity downstream of the entrance to the barrels and V
2
is the average
velocity upstream of the entrance; Fig. 7.1 shows the numbering of the various cross sections in
the flow. Hence
g2
V
h
g2
V
2
3
3entr
2
2
2
+?=?+? (7.2)
where ?
2
and ?
3
are the average flow depths immediately downstream and upstream of the
entrance. The total head immediately upstream of the barrels was related to the total head in the
channel by
g2
V
z
g2
V
K
g2
V
WS
2
2
22
2
2
E
2
o
o
++?=?+ (7.3)
where z
2
= invert elevation immediately upstream of the entrance of the barrels, WS
o
= water
surface elevation in the channel measured at the centerline, V
o
= average velocity in the channel,
and K
E
= head loss coefficient for flow from the channel to cross section 2 just before the
entrance into the barrels.
118
The velocity head in the channel was calculated using
? the average velocity in the channel upstream of the culvert
? the average velocity in the channel downstream of the culvert
? the mean of the average velocities in the channel upstream and downstream of the
culvert, and
? no velocity head for the channel (V
o
= 0 in Eq. (7.2)).
Different values of K
E
were obtained using different methods. Including the velocity head in the
channel gave better correlations than not including it. However the results for the three velocity
heads were very similar. The downstream velocity head was preferred because the calculations
proceed in the upstream direction for subcritical flows. K
E
obtained using the downstream
velocity for V
o
was correlated with the upstream channel Froude number (F
u
), downstream chan
nel Froude number (F
d
), upstream weir Froude number (Fw
u
), downstream weir Froude number
(Fw
d
), and Q
w
/NQ
u,
where Q
w
= discharge through the culverts, N = number of barrels, and Q
u
=
channel discharge upstream of the culverts. The correlations with the downstream Froude num
ber and with Q
w
/NQ
u
were slightly better than the other correlations, but the channel Froude
number does not include any parameters related to the flow through the culverts and the correla
tion with Q
w
/NQ
u
produced a more complicated predictive equation than the correlation with
Fw
d
. Thus, the correlation with the downstream weir Froude number is recommended. The
results are shown by the symbols that are capital letters in Fig. 7.3. The regression equation for
thedatainFig.7.3ais
d
Fw50.2
c
4
E
10248.0K
?
?
=
(7.4)
For unsubmerged flow, ?
4
/?
c
=1since?
4
, which is the depth at cross section 4, is equal to the
critical depth (?
c
) for flow in the culverts. The coefficient of determination (R
2
) for Eq. (7.4) is
0.965 (versus 0.992 using the channel Froude number and 0.972 with Q
w
/NQ
u
in Fig. 7.3b). As
stated above, the correlations with the weir Froude number is preferred since the correlation with
Q
w
/NQ
u
, requires a more complicated equation, namely
()[]
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
?
=
2
uw
NQ/Qlog504.0
675.0
u
w
c
4
E
10
NQ
Q
020.1K
(7.5)
7.3.2  Submerged flow
For submerged outflow from the culverts, the loss coefficients for the flow from the chan
nel to cross section 2 were larger than for unsubmerged flow. It was found that multiplying the
119
loss coefficients by ?
c
/?
4
caused them to follow the same general trend as for unsubmerged flow,
as shown in Fig. 7.3. Thus, Eq. (7.4) can be used for both unsubmerged and submerged flows.
For submerged conditions, the water level in the basin can be used to obtain ?
4
,whichis
assumedtobeequalto?
5
.
Fw
d
0.0 0.1 0.2 0.3 0.4 0.5 0.6
1
10
E
E
E
F
F
F
G
G
H
H
e
f
g
h
A
A
A
A
B
B
B
B
C
C
C
C
D
D
D
D
a
ab
b
c
c
d
d
0.5
K
E
(
?
c
/
?
4
)
=
Adju
st
e
d
he
ad
l
o
ss
co
eff
i
cie
n
t
Q
u
(cfs)
1.6
3.2
6.6
9.1
3 barrels
Free flow
A
B
C
D
3 barrels
Submerged
a
b
c
d
2 barrels
Free flow
E
F
G
H
2 barrels
Submerged
e
f
g
h
Q
w
/NQ
u
0.006 0.03 0.06 0.20.01 0.1
0.6
3
6
1
10
E
E
E
F
F
F
G
G
H
H
e
f
g
h
A
A
A
A
B
B
B
B
C
C
C
C
D
D
D
D
a
a b
b
c
c
d
d
K
E
(
?
c
/
?
4
)
=
Ad
justed
h
e
a
d
l
o
ss
coe
f
fici
e
n
t
Q
u
(cfs)
1.6
3.2
6.6
9.1
3barrels
Free flow
A
B
C
D
3 barrels
Submerged
a
b
c
d
2 barrels
Free flow
E
F
G
H
2 barrels
Submerged
e
f
g
h
Fig. 7.3  Adjusted loss coefficients for flow from point 0 to point 2
Eq. (7.4)
Eq. (7.5)
a)
d
4
c
E
Fwvs.K
?
?
b)
u
w
4
c
E
NQ
Q
vs.K
?
?
120
7.4  CALCULATION PROCEDURE
The experimental results and the resulting correlations are valid for 0.007 < Q
w
/NQ
u
<
0.12 for 2 or 3 barrels. For calculation of the flow through diversion culverts, it is assumed that
the known information includes the culvert geometry, the flow conditions in the channel at the
downstream end of the culverts, and the water level in the detention basin. For these conditions,
the major steps in the calculation procedure for diversion culverts are as follows:
1) Assume a value of Q
w
.
2) From Q
w
and N, calculate the critical depth (?
c
) in the culvert barrels.
3) Using Q
w
/N, use gradually varied flow computations to get the water surface profile through
the culvert barrel to obtain ?
3
, which is the depth in the barrels at the upstream end of the
barrels. These calculations start with ?
4
= ?
c
if the water level in the basin is below ?
c
.
Otherwise, ?
4
comes from the water level in the basin.
4) Use Eq. (7.1) and Eq. (7.2) to calculate V
2
and ?
2
just upstream of the culvert entrance.
5) Use Eq. (7.3) and Eq. (7.4) to calculate the water surface elevation (WS
o
) in the channel.
6) If WS
o
agrees with the know water level in the channel, then the assumed Q
w
is correct. If
not, assume another Q
w
and repeat the previous steps until agreement is obtained.
7) Because Q
w
/Q
u
is small for diversion culverts, the water surface elevation in the channel at
the upstream of the culverts can be assumed to be the same as at the downstream end.
When the stage in the channel first rises above the culvert invert, the headwater will be very
small giving a very large Fw
d
. The flow through the culverts will be very small giving a small
Q
w
/NQ
w
. Using either Eq. (7.4) or Eq. (7.5), the calculated value of K
E
can be extremely large.
In the computer program for diversion culverts (Burgin and Holley, 2002), the value of K
E
was
taken as 10 if the calculated value was larger than 10. (See Fig. 7.3.) For these very small flows,
any inaccuracy in the value of K
E
does not have any practical significance.
121
8  CONCLUSIONS
Sidechannel weirs provide a viable means of flood control by diverting flow from chan
nels into detention basins. The hydraulics of the weirs have been studied and improved methods
have been developed for calculating the flow over sidechannel weirs. Hydraulic information on
flap gates and Tideflex valves for drainage culverts has been obtained and analyzed to provide
relationships for calculating the culvert drainage from detention basins. The valves prevent flow
from the channels into the detention basins but, to varying degrees, they also slow the drainage
flow by restricting the flow area at the downstream end of the culverts.
This report addresses the experimental part of the project work. Another part of the pro
ject work is reported by Burgin and Holley (2002). That report is a user's manual for the com
putational scheme that has been developed for watershed hydrology, channel and sideweir
hydraulics, and filling and emptying of detention basins. The computational scheme uses the
results of the experimental work presented in this report.
An improved computational scheme was developed for the estimation of side weir
discharge and upstream head on the weir. Of the four calculation methods tested in Chapter 4,
Method B is recommended. In Method B, the side discharge per unit length of weir is calculated
from a differential equation for changes in water surface elevation in the channel along the weir.
This equation was derived from the momentum principle for spatially varied flow in prismatic
and tapered channels with trapezoidal cross sections. The computational scheme involves the
calculation of the water surface profile along the weir. The calculations explicitly account for
channel roughness and slope since the results of simulations showed that the roughness and slope
can have a significant effect on the side weir discharge and upstream head on the weir. The
physical equations to be used for calculating the side discharge and the changes in flow depth in
the channel for different situations are listed in Table 4.6 for the recommended Method B as well
as for the other methods that were tested.
Regression equations were obtained for the empirical coefficients required in the
computation; both previously published and new experimental results were used to develop the
equations. Accuracy of the estimated values of side weir discharge and upstream head on the
weir was generally good and was comparable to or better than that using the method of analysis
in the previous project. The regression equations for the various empirical coefficients to be used
in the calculations are also listed in Table 4.6.
The effects of channel side slope on weir hydraulics were investigated. Experiments were
conducted in a channel with 4H:1V side slopes for unsubmerged conditions. The results were
compared with those obtained in Chapter 4 for tests in a channel with 2.5H:1V side slopes.
Using discharge coefficients (C
1
) predicted by the regression equation obtained for tests in the
122
channel with 2.5H:1V side slopes, the agreement between measured and calculated values of the
side weir discharge and upstream head on the weir for tests with 4H:1V side slopes was
comparable to that for tests with 2.5H:1V side slopes. Therefore the same regression equation is
applicable for side slopes of both 2.5H:1V and 4H:1V. However, the coefficient (C
2
)intherela
tionship between the crosssectional average velocity in the channel and the velocity of the lateral
flow changes from 0.85 for 2.5H:1V side slopes to 1.10 for 4H:1V side slopes. For slopes
between 2.5H:1V and 4H:1V, linear interpolation may be used to estimate values for C
2
.
For channels with side weirs, flow over the weir creates an asymmetrical velocity
distribution in the channel. For diversions of 30% or more of the approach flow in a trapezoidal
channel with 2.5H:1V side slopes or 20% or more for 4H:1V side slopes, a separation zone is
created on the side of the channel opposite to the weir. Downstream of the weir, there is a region
of flow reestablishment as the asymmetry is eliminated. As a result of the asymmetry, the flow
depth at the end of the weir is usually less than at the downstream end of the reestablishment
region.
The flow depth at the end of the weir can be determined from the depth at the end of the
flow reestablishment region using the momentum or energy equation with a momentum correc
tion factor (?) or a kinetic energy correction factor (?) to account for the flow asymmetry. Sec
tion 6.5 gives the empirical ? and ? values for trapezoidal channels with 2.5H:1V and 4H:1V
side slopes. Linear interpolation may be used to estimate ? and ? values for other side slopes
between these two values.
The ? and ? values depend primarily on the ratio of the weir discharge (Q
w
) to discharge
(Q
u
) upstream of the weir and increase as Q
w
/Q
u
increases, where Q
u
/Q
d
=1/(1Q
w
/Q
u
). The ?
and ? values at the end of the weir are 1.6 and 2.6, respectively, for 50% diversion for a channel
with 2.5H:1V side slopes and 1.9 and 3.6 for 50% diversion for a channel with 4H:1V side
slopes. The ? and ? values in this report include the turbulent fluxes of momentum and kinetic
energy. Including the turbulent fluxes was important in obtaining good closure of the momentum
and energy balances for the measurements. The turbulent flux of momentum was as much as
10% of the total momentum flux, while the turbulent flux of kinetic energy was as much as 17%
of the total. The type of results presented in this report depends on channel geometry.
The limited data obtained for the length of the flow reestablishment region were used to
give an approximate relationship for this length. Fortunately, it is not necessary to know this
length with high accuracy since it does not have a strong influence of the depth calculated at the
downstream end of the weir.
Some experiments were done using culverts rather than weirs for diversion. The experi
ments were used as a basis for developing a method for calculating flow rates for diversion cul
123
verts. This method is included in the computer programs (Burgin and Holley, 2002). Since the
results are rather limited, this method for diversion culverts should be used with caution.
125
9  REFERENCES
Ackers, P. (1957) ?A Theoretical Consideration of Side Weirs as Stormwater Overflows,? Pro
ceedings, Institute of Civil Engineers, London, vol. 6, pp. 250, February.
Balmforth, D. J. and Sarginson, E. J. (1983) ?An experimental investigation of the discharge
capacity of side weirs,? Intern. Conf. on Hydraulic Aspects of Floods & Flood Control, Paper
No. F2, London, BHRA, Sept. 1315.
Bos, M. G. (ed.) (1985), Discharge Measurement Structures, International Institute for Land
Reclamation and Improvement, Wageningen, Netherlands, 464 pp.
Burgin, J. F. and Holley, E. R. (2002) SideWeir Analysis System, Online Rept. 022, Center for
Research in Water Resources, Univ. of Texas, Austin, TX
(http://www.crwr.utexas.edu/online.shtml)
Cheong, H.F. (1992) ?Discharge coefficient of lateral diversion from trapezoidal channel,? J.
Irr. and Drainage Engineering, vol. 117, no. 4, pp. 461475.
Chow, V. T. (1959) Open Channel Hydraulics, McGrawHill, New York, 680 p.
Collinge, V. K. (1957) ?The Discharge Capacity of Side Weirs,? Proceedings, Institute of Civil
Engineers, London, vol. 6, pp. 288304.
Davis, J. E. and Holley, E. R. (1988) ?Modeling sideweir diversions for flood control,? Hydrau
lic Engineering, Proc., National Conf. ASCE, pp. 979984.
de Marchi, G. (1934) ?Essay on the performance of lateral weirs,? (in Italian) L?Energia Ellec
trica, Milan, vol. 11, no. 11, pp. 849, November.
DOT (Department of Transportation) (1973) ?Hydraulics of Bridge Waterways,? Hydraulic
Design Series, No. 1, U.S. Dept. of Transportation/Federal Highway Administration.
ElKhashab, A. and Smith, K. V. H. (1976) ?Experimental investigation of flow over side weirs,?
J. Hydraulics Div., Proc. ASCE, vol. 102, no. HY9, pp. 12551268.
Forchheimer, P. (1930) Hydraulics, Teubner Verlagsgevellschaft, Berlin, 3rd ed., pp. 406409.
Frazer, W. (1957) ?The behaviour of side weirs in prismatic rectangular channels,? Proc., Inst. of
Civil Engineers, Vol. 6, pp. 305327.
Hager, W. H. (1981) Die Hydraulik von Verteilkan&&alen (Hydraulics of distribution channels),
Doctoral Dissertation, parts I and II, Federal Institute of Technology (ETH), Zurich.
Hager, W. H. and P. U. Volkart (1986) ?Distributions channels,? J. Hydraulic Engineering, vol.
112, no. 10, pp. 935952.
126
Hager, W. H. (1987) "Lateral outflow over side weirs," J. Hydraulic Engineering, vol. 113, no. 4,
pp. 491504.
Hammons, M. A. and Holley, E. R. (1995) Hydraulics characteristics of flush depressed curb
inlets and bridge deck drains. Research Report #14091, Center for Transportation Research,
The University of Texas at Austin, Austin, Texas, 168 pp.
Henderson, F. M. (1966) Open Channel Flow, Macmillan Publishing Co., Inc., New York, 522 p.
Idelchik, I. E. (1986) Handbook of Hydraulic Resistance, 2nd Ed., Hemisphere Publ., Washing
ton, 640 p.
Lasdon, L. S. and Waren, A. D. (1994) GRG2 User's Guide. Dept. of Management and Infor
mation Sciences, Univ. of Texas, Austin, unpublished manuscript, 54 p.
Mostafa, M. G. and Chu, G. H. (1974) ?Experimental and Mathematical Investigation of Flow
Over Side Weirs in Open Channels,? Report No. ERC74012F, Engineering Research Cen
ter, Cal. State Univ., Long Beach Foundation, Report prepared for the Orange County Flood
Control District. June.
Nagler, F. A. (1923) ?Hydraulic tests of Calco automatic drainage gates,? The Transit, vol. 28,
no. 4, pp. 7172, 90.
Press, Wh. H., Flannery, B. P., Teukolshy, S. A. and Vetterling, W. T. (1989) Numerical Recipes.
The Art of Scientific Computing (FORTRAN Version). Cambridge Univ. Press, Cambridge,
128 p.
Subramanya, K. and Awasthy, S. C. (1972) ?Spatially varied flow over sideweirs,? J. Hydr.
Div., Proc. ASCE, vol. 98, no. HY1, pp. 110.
Tynes, K. A. (1989) Hydraulics of SideChannel Weirs for Regional Detention Basins,M.S.
Thesis, Dept. of Civil Engineering, University of Texas, Austin, 128 pp.
Yen, B. C. and Wenzel, H. G. (1970) "Dynamic equations for steady spatially varied flow," J.
Hydraulics Div., Proc. ASCE, vol. 96, no. HY3, pp. 801814.
U. S. Army Corps of Engineers (1984) HEC2 Water Surface Profiles Users Manual,Hydrologic
Engineering Center, U. S. Army Corps of Engineers, Davis, CA, 40 p. + appendices.
127
10  APPENDICES
APPENDIX 1  DATA FROM PREVIOUS PROJECT (TYNES, 1989)
Appendix 1.1  Unsubmerged Flow Conditions
Test L B P Q
u
Q
w
h
u
h
d
C
e
Fw
d
F
d
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
A1A10W 23.91 3.40 0.52 3.250 0.293 0.019 0.039 0.978 0.293 0.512 0.0156
A1A20W 23.91 3.40 0.52 3.243 0.813 0.040 0.061 0.611 0.225 0.724 0.0149
A1A40W 23.91 3.40 0.52 3.232 1.199 0.054 0.076 0.443 0.180 0.766 0.0150
A1A50W 23.91 3.40 0.52 3.237 1.713 0.072 0.092 0.292 0.129 0.820 0.0125
A1B10W 23.91 3.40 0.52 6.516 0.595 0.036 0.062 1.474 0.547 0.517 0.0313
A1B20W 23.91 3.40 0.52 6.493 1.199 0.046 0.089 1.037 0.453 0.603 0.0389
A1B30W 23.91 3.40 0.52 6.471 1.810 0.062 0.111 0.780 0.375 0.651 0.0409
A1B40W 23.91 3.40 0.52 6.527 2.554 0.082 0.130 0.591 0.304 0.723 0.0386
A1B50W 23.91 3.40 0.52 6.535 3.142 0.099 0.140 0.468 0.247 0.795 0.0302
A1C20W 23.91 3.40 0.52 9.566 2.374 0.080 0.161 0.904 0.507 0.485 0.0570
A1C40W 23.91 3.40 0.52 9.614 3.612 0.102 0.180 0.688 0.403 0.623 0.0539
A1C50W 23.91 3.40 0.52 9.646 4.823 0.126 0.194 0.519 0.313 0.742 0.0474
A4A10W 10.00 3.40 0.52 3.287 0.281 0.041 0.054 0.818 0.285 0.711 0.0102
A4A20W 10.00 3.40 0.52 3.205 0.666 0.075 0.087 0.502 0.216 0.814 0.0084
A4A30W 10.00 3.40 0.52 3.199 0.961 0.096 0.108 0.383 0.182 0.842 0.0087
A4A40W 10.00 3.40 0.52 3.188 1.307 0.117 0.129 0.282 0.144 0.871 0.0083
A4A50W 10.00 3.40 0.52 3.205 1.580 0.133 0.143 0.225 0.120 0.897 0.0064
A4B10W 10.00 3.40 0.52 6.497 0.669 0.073 0.100 1.054 0.485 0.660 0.0216
A4B20W 10.00 3.40 0.52 6.472 1.328 0.117 0.145 0.705 0.379 0.738 0.0220
A4B30W 10.00 3.40 0.52 6.428 1.984 0.152 0.176 0.520 0.302 0.815 0.0187
A4B40W 10.00 3.40 0.52 6.417 2.484 0.177 0.199 0.414 0.253 0.842 0.0169
A4B50W 10.00 3.40 0.52 6.414 3.170 0.203 0.230 0.300 0.193 0.855 0.0210
A4C10W 10.00 3.40 0.52 9.607 0.926 0.074 0.142 1.238 0.665 0.532 0.0427
A4C20W 10.00 3.40 0.52 9.586 1.984 0.145 0.200 0.797 0.487 0.667 0.0355
A4C30W 10.00 3.40 0.52 9.613 3.005 0.184 0.241 0.586 0.384 0.753 0.0386
A4C40W 10.00 3.40 0.52 9.573 4.005 0.228 0.274 0.437 0.300 0.818 0.0328
A4C50W 10.00 3.40 0.52 9.590 4.852 0.258 0.300 0.340 0.241 0.858 0.0305
A4D10W 10.00 3.40 0.52 1.560 0.154 0.028 0.036 0.488 0.141 0.720 0.0045
A4D20W 10.00 3.40 0.52 1.548 0.351 0.050 0.058 0.312 0.112 0.796 0.0044
A4D30W 10.00 3.40 0.52 1.555 0.500 0.062 0.072 0.239 0.095 0.816 0.0063
A4D40W 10.00 3.40 0.52 1.575 0.622 0.074 0.084 0.195 0.083 0.802 0.0063
A4D50W 10.00 3.40 0.52 1.583 0.779 0.086 0.093 0.153 0.068 0.859 0.0033
A5A10W 5.00 3.40 0.52 3.191 0.324 0.075 0.084 0.601 0.258 0.810 0.0080
A5A20W 5.00 3.40 0.52 3.194 0.636 0.115 0.127 0.398 0.204 0.829 0.0105
A5A30W 5.00 3.40 0.52 3.180 0.964 0.151 0.161 0.286 0.161 0.860 0.0085
A5A40W 5.00 3.40 0.52 3.195 1.256 0.177 0.189 0.219 0.132 0.865 0.0103
A5A50W 5.00 3.40 0.52 3.197 1.552 0.200 0.213 0.167 0.105 0.879 0.0112
A5B09W 5.00 3.40 0.52 6.446 0.557 0.104 0.122 0.929 0.466 0.774 0.0145
A5B10W 5.00 3.40 0.52 6.546 0.724 0.126 0.144 0.808 0.434 0.773 0.0177
A5B20W 5.00 3.40 0.52 6.534 1.240 0.174 0.193 0.586 0.356 0.825 0.0170
A5B30W 5.00 3.40 0.52 6.542 1.929 0.227 0.241 0.416 0.274 0.891 0.0122
128
Appendix 1.1  Unsubmerged Flow Conditions (continued)
Test L B P Q
u
Q
w
h
u
h
d
C
e
Fw
d
F
d
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
A5B31W 5.00 3.40 0.52 6.347 1.957 0.231 0.247 0.389 0.259 0.868 0.0145
A5B40W 5.00 3.40 0.52 6.527 2.652 0.276 0.288 0.296 0.208 0.910 0.0105
A5B50W 5.00 3.40 0.52 6.586 3.170 0.304 0.319 0.235 0.171 0.915 0.0131
A5B51W 5.00 3.40 0.52 6.431 3.199 0.305 0.322 0.220 0.161 0.909 0.0147
A5C10W 5.00 3.40 0.52 9.547 0.899 0.154 0.175 1.061 0.624 0.701 0.0232
A5C11W 5.00 3.40 0.52 9.639 1.011 0.148 0.185 0.968 0.574 0.721 0.0206
A5C20W 5.00 3.40 0.52 9.695 1.857 0.226 0.255 0.673 0.453 0.781 0.0235
A5C30W 5.00 3.40 0.52 9.679 3.055 0.298 0.327 0.444 0.326 0.846 0.0241
A5C40W 5.00 3.40 0.52 9.603 3.980 0.347 0.369 0.332 0.254 0.896 0.0188
A5C50W 5.00 3.40 0.52 9.599 4.987 0.395 0.406 0.245 0.193 0.952 0.0097
A6A10W 2.00 3.40 0.52 3.317 0.300 0.126 0.131 0.452 0.234 0.819 0.0042
A6A20W 2.00 3.40 0.52 3.322 0.614 0.188 0.190 0.300 0.180 0.879 0.0013
A6A30W 2.00 3.40 0.52 3.305 0.975 0.234 0.242 0.208 0.137 0.905 0.0071
A6A40W 2.00 3.40 0.52 3.287 1.244 0.274 0.279 0.159 0.110 0.890 0.0042
A6A50W 2.00 3.40 0.52 3.192 1.638 0.315 0.320 0.105 0.077 0.907 0.0042
A6B10W 2.00 3.40 0.52 6.575 0.622 0.189 0.200 0.630 0.386 0.813 0.0090
A6B20W 2.00 3.40 0.52 6.499 1.260 0.279 0.289 0.394 0.276 0.844 0.0085
A6B30W 2.00 3.40 0.52 6.355 1.919 0.342 0.351 0.273 0.205 0.892 0.0077
A6B40W 2.00 3.40 0.52 6.455 2.573 0.396 0.400 0.207 0.162 0.931 0.0031
A6C10W 2.00 3.40 0.52 9.552 0.940 0.247 0.254 0.733 0.491 0.799 0.0054
A6C20W 2.00 3.40 0.52 9.578 1.973 0.354 0.364 0.450 0.342 0.856 0.0080
A6C30W 2.00 3.40 0.52 9.611 2.826 0.416 0.430 0.334 0.267 0.889 0.0116
A6D20W 2.00 3.40 0.52 1.636 0.356 0.137 0.140 0.182 0.097 0.867 0.0022
A6D30W 2.00 3.40 0.52 1.625 0.495 0.161 0.164 0.142 0.080 0.918 0.0022
A6D40W 2.00 3.40 0.52 1.620 0.580 0.180 0.184 0.118 0.070 0.879 0.0032
A6D50W 2.00 3.40 0.52 1.741 0.810 0.218 0.222 0.090 0.057 0.879 0.0032
A3A22N 15.00 1.80 0.52 3.271 0.692 0.067 0.091 0.759 0.356 0.532 0.0261
A3A31N 15.00 1.80 0.52 3.281 0.996 0.081 0.108 0.593 0.300 0.590 0.0272
A3A39N 15.00 1.80 0.52 3.268 1.236 0.093 0.122 0.480 0.256 0.607 0.0280
A3A49N 15.00 1.80 0.52 3.247 1.571 0.107 0.137 0.361 0.202 0.646 0.0279
A3B19N 15.00 1.80 0.52 6.307 1.219 0.099 0.142 1.064 0.603 0.475 0.0491
A3B31N 15.00 1.80 0.52 6.314 1.990 0.115 0.177 0.750 0.464 0.552 0.0549
A3B40N 15.00 1.80 0.52 6.336 2.554 0.130 0.207 0.569 0.374 0.556 0.0623
A3B46N 15.00 1.80 0.52 6.328 3.005 0.141 0.210 0.493 0.326 0.640 0.0573
A3B51N 15.00 1.80 0.52 6.331 3.281 0.154 0.221 0.431 0.291 0.645 0.0560
A4A10N 10.00 1.80 0.52 3.199 0.312 0.060 0.068 1.008 0.412 0.555 0.0107
A4A13N 10.00 1.80 0.52 3.232 0.413 0.069 0.082 0.866 0.385 0.552 0.0142
A4A23N 10.00 1.80 0.52 3.206 0.739 0.094 0.111 0.608 0.308 0.621 0.0162
A4A31N 10.00 1.80 0.52 3.217 0.986 0.110 0.129 0.490 0.265 0.657 0.0174
A4A41N 10.00 1.80 0.52 3.227 1.298 0.130 0.148 0.379 0.217 0.699 0.0160
A4B12N 10.00 1.80 0.52 6.411 0.754 0.098 0.130 1.234 0.669 0.496 0.0322
A4B21N 10.00 1.80 0.52 6.391 1.350 0.124 0.174 0.863 0.527 0.565 0.0393
A4B31N 10.00 1.80 0.52 6.390 1.984 0.152 0.211 0.634 0.417 0.613 0.0448
A4B40N 10.00 1.80 0.52 6.404 2.535 0.194 0.228 0.517 0.350 0.693 0.0293
A4B50N 10.00 1.80 0.52 6.377 3.259 0.211 0.257 0.371 0.262 0.737 0.0371
A5A08N 5.00 1.80 0.52 3.185 0.256 0.077 0.083 0.910 0.409 0.652 0.0092
A5A20N 5.00 1.80 0.52 3.187 0.627 0.127 0.135 0.552 0.305 0.742 0.0093
129
Appendix 1.1  Unsubmerged Flow Conditions (continued)
Test L B P Q
u
Q
w
h
u
h
d
C
e
Fw
d
F
d
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
A5A31N 5.00 1.80 0.52 3.187 0.989 0.166 0.174 0.383 0.235 0.779 0.0085
A5A41N 5.00 1.80 0.52 3.205 1.320 0.196 0.204 0.284 0.186 0.802 0.0080
A5A51N 5.00 1.80 0.52 3.191 1.628 0.222 0.227 0.213 0.145 0.830 0.0050
A5B11N 5.00 1.80 0.52 6.457 0.709 0.133 0.172 1.011 0.617 0.569 0.0336
A5B23N 5.00 1.80 0.52 6.457 1.454 0.209 0.236 0.657 0.453 0.695 0.0262
A5B30N 5.00 1.80 0.52 6.439 1.935 0.250 0.268 0.521 0.376 0.749 0.0193
A5B35N 5.00 1.80 0.52 6.392 2.235 0.271 0.290 0.443 0.329 0.758 0.0196
A5C18N 5.00 1.80 0.52 9.568 1.723 0.269 0.295 0.822 0.613 0.568 0.0267
A6A09N 2.00 1.80 0.52 3.204 0.289 0.130 0.131 0.632 0.344 0.789 0.0023
A6A10N 2.00 1.80 0.52 3.263 0.364 0.141 0.148 0.578 0.332 0.806 0.0059
A6A18N 2.00 1.80 0.52 3.262 0.595 0.186 0.193 0.423 0.270 0.829 0.0059
A6A20N 2.00 1.80 0.52 3.197 0.644 0.201 0.205 0.377 0.245 0.806 0.0043
A6A30N 2.00 1.80 0.52 3.284 0.940 0.239 0.243 0.299 0.208 0.866 0.0032
A6A32N 2.00 1.80 0.52 3.189 1.007 0.254 0.258 0.258 0.183 0.832 0.0040
A6A39N 2.00 1.80 0.52 3.177 1.236 0.282 0.287 0.206 0.152 0.839 0.0048
A6A40N 2.00 1.80 0.52 3.287 1.294 0.280 0.289 0.213 0.158 0.867 0.0078
A6A49N 2.00 1.80 0.52 3.202 1.552 0.314 0.317 0.158 0.120 0.875 0.0028
A6A50N 2.00 1.80 0.52 3.283 1.633 0.316 0.321 0.158 0.121 0.899 0.0041
A6B10N 2.00 1.80 0.52 6.407 0.633 0.204 0.216 0.813 0.539 0.722 0.0125
A6B11N 2.00 1.80 0.52 6.576 0.706 0.210 0.230 0.789 0.538 0.719 0.0133
A6B19N 2.00 1.80 0.52 6.418 1.215 0.290 0.302 0.524 0.393 0.750 0.0119
A6B20N 2.00 1.80 0.52 6.498 1.320 0.291 0.301 0.530 0.398 0.820 0.0077
A6B30N 2.00 1.80 0.52 6.490 1.913 0.348 0.355 0.391 0.310 0.871 0.0055
A6B40N 2.00 1.80 0.52 6.482 2.560 0.395 0.412 0.282 0.235 0.875 0.0140
A6B50N 2.00 1.80 0.52 6.433 2.977 0.426 0.442 0.228 0.194 0.887 0.0134
A6C09N 2.00 1.80 0.52 9.615 0.902 0.243 0.271 0.982 0.708 0.681 0.0193
A6C10N 2.00 1.80 0.52 9.626 0.961 0.274 0.274 0.979 0.711 0.711 0.0020
A6C14N 2.00 1.80 0.52 9.639 1.294 0.291 0.332 0.758 0.587 0.669 0.0275
A6C20N 2.00 1.80 0.52 9.615 1.697 0.345 0.364 0.657 0.526 0.736 0.0118
C2A09W 20.00 3.40 0.70 3.201 0.296 0.028 0.038 0.683 0.180 0.639 0.0051
C2A21W 20.00 3.40 0.70 3.213 0.677 0.047 0.061 0.451 0.149 0.713 0.0084
C2A30W 20.00 3.40 0.70 3.189 0.961 0.059 0.073 0.355 0.127 0.770 0.0081
C2A39W 20.00 3.40 0.70 3.217 1.227 0.070 0.084 0.290 0.111 0.794 0.0079
C2A53W 20.00 3.40 0.70 3.192 1.693 0.087 0.101 0.193 0.080 0.826 0.0075
C2B07W 20.00 3.40 0.70 6.347 0.472 0.039 0.052 1.150 0.353 0.634 0.0154
C2B13W 20.00 3.40 0.70 6.343 0.794 0.052 0.069 0.915 0.320 0.694 0.0176
C2B17W 20.00 3.40 0.70 6.338 1.074 0.064 0.083 0.772 0.294 0.708 0.0184
C2B30W 20.00 3.40 0.70 6.325 1.935 0.094 0.116 0.515 0.228 0.764 0.0191
C2B41W 20.00 3.40 0.70 6.351 2.658 0.115 0.139 0.381 0.182 0.794 0.0199
C2B50W 20.00 3.40 0.70 6.327 3.199 0.132 0.153 0.300 0.150 0.824 0.0165
C2C11W 20.00 3.40 0.70 9.525 1.055 0.063 0.089 1.187 0.467 0.625 0.0334
C2C19W 20.00 3.40 0.70 9.537 1.836 0.088 0.123 0.866 0.394 0.662 0.0368
C2C30W 20.00 3.40 0.70 9.552 2.935 0.115 0.158 0.620 0.314 0.719 0.0403
C2C39W 20.00 3.40 0.70 9.569 3.798 0.141 0.180 0.489 0.261 0.760 0.0356
C2C50W 20.00 3.40 0.70 9.545 4.890 0.169 0.207 0.353 0.200 0.787 0.0334
C3B08W 15.00 3.40 0.70 6.498 0.522 0.051 0.068 0.994 0.346 0.617 0.0177
C3B18W 15.00 3.40 0.70 6.497 1.112 0.079 0.102 0.689 0.288 0.705 0.0214
130
Appendix 1.1  Unsubmerged Flow Conditions (continued)
Test L B P Q
u
Q
w
h
u
h
d
C
e
Fw
d
F
d
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
C3B25W 15.00 3.40 0.70 6.486 1.566 0.102 0.124 0.550 0.251 0.734 0.0197
C3B28W 15.00 3.40 0.70 6.463 1.779 0.109 0.131 0.504 0.235 0.766 0.0194
C3B42W 15.00 3.40 0.70 6.418 2.718 0.146 0.166 0.334 0.173 0.808 0.0165
C3B50W 15.00 3.40 0.70 6.338 3.266 0.164 0.183 0.257 0.138 0.833 0.0151
C3C10W 15.00 3.40 0.70 9.610 0.947 0.056 0.099 1.131 0.467 0.629 0.0423
C3C17W 15.00 3.40 0.70 9.629 1.643 0.096 0.140 0.819 0.393 0.638 0.0412
C3C21W 15.00 3.40 0.70 9.611 1.995 0.121 0.155 0.724 0.363 0.661 0.0326
C3C30W 15.00 3.40 0.70 9.637 2.921 0.141 0.184 0.559 0.302 0.739 0.0386
C3C40W 15.00 3.40 0.70 9.611 3.921 0.181 0.215 0.418 0.240 0.775 0.0301
C3C50W 15.00 3.40 0.70 9.611 4.948 0.209 0.232 0.321 0.190 0.867 0.0201
C4A09W 10.00 3.40 0.70 3.270 0.269 0.044 0.050 0.600 0.181 0.754 0.0350
C4A18W 10.00 3.40 0.70 3.254 0.580 0.072 0.081 0.398 0.150 0.773 0.0061
C4A31W 10.00 3.40 0.70 3.269 0.979 0.107 0.115 0.274 0.122 0.755 0.0052
C4A42W 10.00 3.40 0.70 3.262 1.328 0.128 0.136 0.202 0.096 0.786 0.0047
C4A51W 10.00 3.40 0.70 3.272 1.638 0.144 0.149 0.160 0.079 0.839 0.0017
C4B11W 10.00 3.40 0.70 6.368 0.663 0.080 0.092 0.781 0.312 0.725 0.0115
C4B20W 10.00 3.40 0.70 6.358 1.269 0.120 0.132 0.543 0.254 0.788 0.0106
C4B30W 10.00 3.40 0.70 6.369 1.924 0.154 0.167 0.399 0.206 0.822 0.0109
C4B39W 10.00 3.40 0.70 6.391 2.535 0.184 0.196 0.305 0.169 0.837 0.0096
C4B44W 10.00 3.40 0.70 6.377 2.826 0.197 0.208 0.267 0.152 0.848 0.0085
C4B49W 10.00 3.40 0.70 6.365 3.142 0.210 0.222 0.230 0.134 0.848 0.0092
C4C11W 10.00 3.40 0.70 9.591 1.011 0.102 0.124 0.958 0.437 0.693 0.0224
C4C12W 10.00 3.40 0.70 9.526 1.175 0.107 0.133 0.887 0.417 0.721 0.0251
C4C18W 10.00 3.40 0.70 9.593 1.723 0.150 0.166 0.709 0.366 0.743 0.0163
C4C32W 10.00 3.40 0.70 9.589 3.084 0.208 0.225 0.459 0.269 0.814 0.0155
C4C41W 10.00 3.40 0.70 9.587 4.048 0.241 0.258 0.347 0.214 0.854 0.0148
C4C46W 10.00 3.40 0.70 9.580 4.497 0.256 0.278 0.298 0.190 0.838 0.0189
C5A09W 5.00 3.40 0.70 3.204 0.281 0.072 0.076 0.459 0.169 0.789 0.0029
C5A19W 5.00 3.40 0.70 3.210 0.611 0.116 0.120 0.301 0.136 0.821 0.0027
C5A31W 5.00 3.40 0.70 3.209 0.979 0.153 0.158 0.211 0.107 0.834 0.0035
C5A39W 5.00 3.40 0.70 3.203 1.252 0.176 0.182 0.166 0.089 0.841 0.0045
C5A52W 5.00 3.40 0.70 3.213 1.653 0.209 0.213 0.117 0.067 0.848 0.0024
C5B10W 5.00 3.40 0.70 6.408 0.649 0.119 0.127 0.641 0.296 0.795 0.0083
C5B20W 5.00 3.40 0.70 6.377 1.265 0.176 0.184 0.430 0.233 0.834 0.0076
C5B31W 5.00 3.40 0.70 6.392 1.962 0.227 0.235 0.305 0.182 0.850 0.0072
C5B40W 5.00 3.40 0.70 6.382 2.580 0.264 0.273 0.229 0.145 0.859 0.0079
C5B50W 5.00 3.40 0.70 6.420 3.229 0.300 0.308 0.172 0.114 0.868 0.0068
C5C10W 5.00 3.40 0.70 9.599 0.940 0.149 0.162 0.805 0.413 0.768 0.0143
C5C17W 5.00 3.40 0.70 9.591 1.624 0.205 0.219 0.582 0.338 0.795 0.0142
C5C21W 5.00 3.40 0.70 9.601 1.995 0.229 0.245 0.505 0.307 0.803 0.0156
C5C29W 5.00 3.40 0.70 9.575 2.851 0.279 0.291 0.383 0.248 0.848 0.0117
C2A15N 20.00 1.80 0.70 3.258 0.472 0.038 0.054 0.765 0.252 0.598 0.0105
C2A19N 20.00 1.80 0.70 3.246 0.595 0.042 0.060 0.683 0.236 0.643 0.0122
C2A22N 20.00 1.80 0.70 3.262 0.709 0.052 0.070 0.597 0.222 0.606 0.0120
C2A28N 20.00 1.80 0.70 3.239 0.882 0.058 0.075 0.527 0.202 0.679 0.0108
C2A41N 20.00 1.80 0.70 3.248 1.294 0.073 0.092 0.382 0.161 0.729 0.0123
C2A46N 20.00 1.80 0.70 3.230 1.463 0.080 0.099 0.328 0.143 0.736 0.0122
131
Appendix 1.1  Unsubmerged Flow Conditions (continued)
Test L B P Q
u
Q
w
h
u
h
d
C
e
Fw
d
F
d
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
C2A50N 20.00 1.80 0.70 3.239 1.604 0.084 0.103 0.295 0.131 0.760 0.0121
C2B22N 20.00 1.80 0.70 6.327 1.376 0.070 0.104 0.889 0.396 0.642 0.0275
C2B30N 20.00 1.80 0.70 6.340 1.935 0.091 0.129 0.677 0.332 0.649 0.0297
C2B39N 20.00 1.80 0.70 6.386 2.516 0.106 0.144 0.548 0.281 0.712 0.0292
C2B52N 20.00 1.80 0.70 6.333 3.372 0.128 0.172 0.365 0.202 0.724 0.0335
C2C13N 20.00 1.80 0.70 9.604 1.231 0.082 0.120 1.357 0.644 0.461 0.0321
C2C20N 20.00 1.80 0.70 9.593 1.899 0.087 0.146 1.078 0.556 0.526 0.0404
C2C31N 20.00 1.80 0.70 9.470 3.027 0.123 0.187 0.741 0.424 0.571 0.0435
C2C34N 20.00 1.80 0.70 9.613 3.334 0.133 0.190 0.713 0.411 0.613 0.0392
C2C41N 20.00 1.80 0.70 9.594 4.048 0.132 0.211 0.576 0.347 0.632 0.0519
C2C51N 20.00 1.80 0.70 9.570 5.046 0.161 0.228 0.439 0.272 0.698 0.0455
C3A10N 15.00 1.80 0.70 3.256 0.300 0.037 0.049 0.861 0.271 0.585 0.0081
C3A16N 15.00 1.80 0.70 3.243 0.517 0.051 0.065 0.668 0.240 0.655 0.0096
C3A18N 15.00 1.80 0.70 3.257 0.580 0.055 0.070 0.626 0.233 0.656 0.0105
C3A28N 15.00 1.80 0.70 3.245 0.875 0.071 0.086 0.484 0.198 0.722 0.0102
C3A42N 15.00 1.80 0.70 3.219 1.345 0.092 0.110 0.323 0.148 0.759 0.0127
C3A51N 15.00 1.80 0.70 3.245 1.643 0.107 0.122 0.257 0.123 0.790 0.0097
C3B09N 15.00 1.80 0.70 6.402 0.546 0.053 0.074 1.321 0.504 0.567 0.0190
C3B18N 15.00 1.80 0.70 6.373 1.112 0.076 0.110 0.908 0.414 0.628 0.0272
C3B32N 15.00 1.80 0.70 6.386 2.039 0.118 0.149 0.599 0.312 0.718 0.0243
C3B41N 15.00 1.80 0.70 6.365 2.645 0.138 0.177 0.448 0.251 0.711 0.0304
C3B48N 15.00 1.80 0.70 6.383 3.099 0.153 0.187 0.378 0.216 0.764 0.0263
C3C11N 15.00 1.80 0.70 9.641 1.093 0.087 0.125 1.345 0.650 0.506 0.0286
C3C20N 15.00 1.80 0.70 9.668 1.968 0.101 0.152 1.046 0.549 0.672 0.0339
C3C30N 15.00 1.80 0.70 9.659 2.949 0.149 0.209 0.703 0.421 0.610 0.0406
C3C38N 15.00 1.80 0.70 9.659 3.684 0.166 0.231 0.573 0.358 0.650 0.0443
C3C45N 15.00 1.80 0.70 9.680 4.407 0.183 0.239 0.491 0.310 0.736 0.0387
C4A09N 10.00 1.80 0.70 3.168 0.271 0.045 0.054 0.794 0.262 0.675 0.0062
C4A18N 10.00 1.80 0.70 3.174 0.580 0.075 0.086 0.529 0.216 0.705 0.0078
C4A32N 10.00 1.80 0.70 3.174 1.018 0.103 0.113 0.364 0.168 0.807 0.0066
C4A42N 10.00 1.80 0.70 3.181 1.328 0.126 0.137 0.272 0.137 0.777 0.0074
C4A51N 10.00 1.80 0.70 3.182 1.633 0.143 0.155 0.207 0.109 0.786 0.0083
C4B11N 10.00 1.80 0.70 6.406 0.706 0.080 0.100 1.049 0.459 0.678 0.0165
C4B18N 10.00 1.80 0.70 6.388 1.139 0.106 0.131 0.796 0.392 0.716 0.0199
C4B30N 10.00 1.80 0.70 6.418 1.919 0.151 0.176 0.543 0.303 0.754 0.0197
C4B35N 10.00 1.80 0.70 6.310 2.265 0.168 0.193 0.452 0.262 0.767 0.0197
C4B41N 10.00 1.80 0.70 6.367 2.665 0.185 0.217 0.375 0.228 0.746 0.0256
C4B46N 10.00 1.80 0.70 6.387 2.956 0.202 0.225 0.336 0.208 0.780 0.0180
C4B50N 10.00 1.80 0.70 6.393 3.221 0.212 0.232 0.303 0.189 0.809 0.0154
C4C10N 10.00 1.80 0.70 9.597 1.015 0.094 0.137 1.259 0.632 0.594 0.0272
C4C20N 10.00 1.80 0.70 9.592 1.889 0.135 0.193 0.861 0.500 0.639 0.0373
C4C30N 10.00 1.80 0.70 9.594 2.963 0.201 0.230 0.638 0.397 0.754 0.0211
C4C37N 10.00 1.80 0.70 9.590 3.636 0.230 0.258 0.516 0.336 0.767 0.0206
C5A09N 5.00 1.80 0.70 3.250 0.291 0.076 0.080 0.643 0.255 0.753 0.0033
C5A21N 5.00 1.80 0.70 3.269 0.663 0.125 0.128 0.408 0.200 0.801 0.0020
C5A31N 5.00 1.80 0.70 3.243 0.982 0.156 0.161 0.297 0.161 0.811 0.0037
C5A40N 5.00 1.80 0.70 3.256 1.265 0.180 0.184 0.235 0.134 0.834 0.0027
132
Appendix 1.1  Unsubmerged Flow Conditions (continued)
Test L B P Q
u
Q
w
h
u
h
d
C
e
Fw
d
F
d
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
C5A51N 5.00 1.80 0.70 3.245 1.624 0.207 0.211 0.170 0.103 0.847 0.0026
C5B10N 5.00 1.80 0.70 6.421 0.644 0.116 0.130 0.895 0.441 0.759 0.0134
C5B21N 5.00 1.80 0.70 6.421 1.328 0.181 0.196 0.570 0.334 0.786 0.0137
C5B30N 5.00 1.80 0.70 6.411 1.935 0.227 0.238 0.423 0.268 0.820 0.0100
C5B39N 5.00 1.80 0.70 6.406 2.516 0.265 0.274 0.323 0.216 0.832 0.0081
C5B47N 5.00 1.80 0.70 6.438 3.055 0.294 0.304 0.254 0.177 0.840 0.0088
C5C10N 5.00 1.80 0.70 9.634 0.961 0.153 0.177 1.057 0.594 0.676 0.0200
C5C20N 5.00 1.80 0.70 9.614 1.924 0.224 0.247 0.703 0.451 0.764 0.0196
C5C25N 5.00 1.80 0.70 9.624 2.386 0.253 0.280 0.588 0.397 0.760 0.0225
133
Appendix 1.2  Submerged Flow Conditions
Compared L B P Q
u
Q
d
h
u
h
d
h
b
Fw
d
C
es
h
cs
Test to Test (ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft) (ft)
C3C21WS C3C40W 15.0 3.40 0.70 7.152 1.486 0.192 0.212 0.202 0.421 0.300 0.016
C3C26WS C3C40W 15.0 3.40 0.70 7.686 2.006 0.197 0.217 0.204 0.414 0.391 0.016
C3C32WS C3C40W 15.0 3.40 0.70 8.310 2.665 0.191 0.216 0.192 0.413 0.523 0.020
C3C35WS C3C40W 15.0 3.40 0.70 8.696 3.106 0.183 0.211 0.176 0.417 0.633 0.023
C3C37WS C3C40W 15.0 3.40 0.70 9.016 3.387 0.184 0.214 0.173 0.415 0.675 0.025
C3C38WS C3C40W 15.0 3.40 0.70 9.193 3.548 0.180 0.211 0.155 0.421 0.723 0.025
C3C40WS C3C40W 15.0 3.40 0.70 9.430 3.848 0.187 0.215 0.152 0.410 0.761 0.023
C3B32WS C3B50W 15.0 3.40 0.70 4.566 1.473 0.163 0.178 0.170 0.264 0.393 0.010
C3B45WS C3B50W 15.0 3.40 0.70 5.437 2.434 0.168 0.185 0.165 0.249 0.610 0.012
C3B30NS C3B48N 15.0 1.80 0.70 4.724 1.414 0.166 0.185 0.174 0.384 0.355 0.015
C3B38NS C3B48N 15.0 1.80 0.70 5.290 2.017 0.163 0.185 0.165 0.380 0.506 0.018
C3B46NS C3B48N 15.0 1.80 0.70 6.073 2.812 0.161 0.188 0.145 0.373 0.688 0.022
C3C10NS C3C38N 15.0 1.80 0.70 6.641 0.652 0.212 0.230 0.223 0.577 0.116 0.016
C3C21NS C3C38N 15.0 1.80 0.70 7.683 1.643 0.203 0.231 0.207 0.580 0.290 0.025
C3C31NS C3C38N 15.0 1.80 0.70 8.703 2.745 0.188 0.229 0.182 0.576 0.491 0.034
C4B28WS C4B49W 10.0 3.40 0.70 4.499 1.248 0.216 0.224 0.216 0.230 0.332 0.005
C4B36WS C4B49W 10.0 3.40 0.70 5.169 1.873 0.211 0.222 0.207 0.235 0.505 0.008
C4B44WS C4B49W 10.0 3.40 0.70 5.828 2.586 0.210 0.222 0.191 0.231 0.698 0.009
C4C19WS C4C41W 10.0 3.40 0.70 6.833 1.320 0.252 0.261 0.252 0.342 0.273 0.006
C4C36WS C4C41W 10.0 3.40 0.70 9.164 3.221 0.240 0.256 0.218 0.375 0.688 0.013
C4B33NS C4B50N 10.0 1.80 0.70 4.746 1.542 0.219 0.233 0.217 0.305 0.384 0.011
C4B42NS C4B50N 10.0 1.80 0.70 5.420 2.295 0.217 0.230 0.202 0.300 0.584 0.010
C4B49NS C4B50N 10.0 1.80 0.70 6.148 3.005 0.214 0.233 0.162 0.299 0.749 0.015
C4C07NS C4C30N 10.0 1.80 0.70 7.128 0.524 0.222 0.233 0.221 0.628 0.131 0.010
C4C19NS C4C30N 10.0 1.80 0.70 8.220 1.552 0.193 0.232 0.190 0.636 0.390 0.032
C5B15WS C5B40W 5.0 3.40 0.70 4.430 0.636 0.264 0.268 0.263 0.233 0.219 0.003
C5B25WS C5B40W 5.0 3.40 0.70 5.050 1.244 0.265 0.271 0.258 0.231 0.420 0.005
C5B34WS C5B40W 5.0 3.40 0.70 5.806 2.006 0.268 0.275 0.245 0.228 0.659 0.006
C5C12WS C5C29W 5.0 3.40 0.70 7.704 0.923 0.286 0.292 0.280 0.385 0.273 0.005
C5C23WS C5C29W 5.0 3.40 0.70 8.685 1.973 0.281 0.290 0.259 0.383 0.590 0.008
C5B12NS C5B39N 5.0 1.80 0.70 4.471 0.537 0.269 0.273 0.266 0.328 0.179 0.003
C5B23NS C5B39N 5.0 1.80 0.70 5.084 1.175 0.271 0.278 0.261 0.320 0.379 0.006
C5B32NS C5B39N 5.0 1.80 0.70 5.712 1.830 0.262 0.272 0.235 0.324 0.613 0.009
C5C05NS C5C25N 5.0 1.80 0.70 7.676 0.415 0.275 0.278 0.270 0.594 0.134 0.005
C5C16NS C5C25N 5.0 1.80 0.70 8.610 1.350 0.262 0.275 0.236 0.600 0.444 0.013
134
Appendix 1.3  Tapered Channels
Test L ?B' PQ
u
Q
w
h
u
h
d
F
d
Fw
d
C
e
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
C2A08T 20.0 0.080 0.70 3.205 0.263 0.024 0.024 0.288 1.289 1.137 0.005
C2A20T 20.0 0.080 0.70 3.189 0.647 0.044 0.051 0.232 0.723 0.895 0.005
C2A24T 20.0 0.080 0.70 3.194 0.779 0.050 0.058 0.216 0.635 0.886 0.004
C2A32T 20.0 0.080 0.70 3.189 1.044 0.060 0.071 0.186 0.497 0.873 0.004
C2A41T 20.0 0.080 0.70 3.172 1.324 0.070 0.084 0.155 0.384 0.857 0.005
C2A51T 20.0 0.080 0.70 3.179 1.619 0.081 0.095 0.127 0.298 0.868 0.003
C2B11T 20.0 0.080 0.70 6.339 0.683 0.058 0.032 0.542 2.111 1.912 0.012
C2B13T 20.0 0.080 0.70 6.321 0.836 0.062 0.042 0.512 1.751 1.551 0.011
C2B21T 20.0 0.080 0.70 6.445 1.324 0.077 0.073 0.442 1.165 1.061 0.010
C2B28T 20.0 0.080 0.70 6.347 1.758 0.090 0.096 0.374 0.871 0.928 0.008
C2B40T 20.0 0.080 0.70 6.327 2.612 0.112 0.132 0.278 0.562 0.845 0.007
C2B51T 20.0 0.080 0.70 6.351 3.341 0.131 0.153 0.214 0.407 0.860 0.001
C2C15T 20.0 0.080 0.70 9.694 1.473 0.103 0.035 0.782 2.916 3.602 0.022
C2C16T 20.0 0.080 0.70 9.663 1.491 0.104 0.041 0.765 2.645 2.870 0.021
C2C22T 20.0 0.080 0.70 9.601 2.096 0.116 0.090 0.621 1.488 1.221 0.012
C2C29T 20.0 0.080 0.70 9.595 2.785 0.126 0.130 0.511 1.041 0.922 0.011
C2C43T 20.0 0.080 0.70 9.573 4.195 0.154 0.182 0.358 0.633 0.825 0.003
C2C53T 20.0 0.080 0.70 9.552 5.085 0.170 0.203 0.284 0.480 0.843 0.004
C3A10T 15.0 0.107 0.70 3.274 0.298 0.045 0.046 0.275 0.900 0.639 0.006
C3A21T 15.0 0.107 0.70 3.285 0.661 0.067 0.074 0.226 0.592 0.687 0.006
C3A30T 15.0 0.107 0.70 3.279 0.947 0.081 0.091 0.192 0.459 0.716 0.006
C3A40T 15.0 0.107 0.70 3.271 1.281 0.096 0.110 0.157 0.343 0.723 0.007
C3A52T 15.0 0.107 0.70 3.273 1.643 0.112 0.126 0.124 0.255 0.751 0.005
C3B11T 15.0 0.107 0.70 6.411 0.663 0.079 0.053 0.522 1.597 1.146 0.008
C3B19T 15.0 0.107 0.70 6.358 1.195 0.100 0.094 0.423 0.994 0.860 0.007
C3B29T 15.0 0.107 0.70 6.316 1.830 0.123 0.131 0.336 0.682 0.788 0.007
C3B41T 15.0 0.107 0.70 6.298 2.612 0.149 0.165 0.255 0.469 0.784 0.004
C3B51T 15.0 0.107 0.70 6.298 3.311 0.170 0.191 0.195 0.338 0.790 0.003
C3C12T 15.0 0.107 0.70 9.503 1.215 0.118 0.036 0.786 2.892 3.781 0.011
C3C13T 15.0 0.107 0.70 9.496 1.223 0.117 0.044 0.769 2.569 2.807 0.014
C3C14T 15.0 0.107 0.70 9.601 1.328 0.119 0.067 0.725 1.988 1.606 0.020
C3C15T 15.0 0.107 0.70 9.485 1.298 0.119 0.053 0.743 2.275 2.245 0.013
C3C21T 15.0 0.107 0.70 9.426 2.051 0.138 0.123 0.563 1.174 0.974 0.011
C3C30T 15.0 0.107 0.70 9.567 2.928 0.164 0.175 0.449 0.806 0.801 0.009
C3C39T 15.0 0.107 0.70 9.561 3.848 0.188 0.213 0.355 0.589 0.772 0.006
C3C51T 15.0 0.107 0.70 9.559 5.006 0.215 0.248 0.263 0.410 0.788 0.000
C4A11T 10.0 0.160 0.70 3.180 0.337 0.057 0.056 0.255 0.762 0.794 0.003
C4A20T 10.0 0.160 0.70 3.212 0.619 0.082 0.084 0.217 0.537 0.780 0.002
C4A32T 10.0 0.160 0.70 3.197 1.022 0.108 0.114 0.169 0.365 0.799 0.002
C4A39T 10.0 0.160 0.70 3.191 1.240 0.120 0.127 0.147 0.303 0.818 0.002
C4A49T 10.0 0.160 0.70 3.191 1.556 0.139 0.146 0.118 0.229 0.823 0.000
C4B11T 10.0 0.160 0.70 6.430 0.698 0.096 0.074 0.492 1.290 1.070 0.006
C4B20T 10.0 0.160 0.70 6.427 1.236 0.127 0.121 0.397 0.835 0.880 0.005
C4B30T 10.0 0.160 0.70 6.427 1.919 0.159 0.164 0.312 0.576 0.844 0.003
C4B41T 10.0 0.160 0.70 6.423 2.652 0.190 0.201 0.240 0.408 0.841 0.000
C4B49T 10.0 0.160 0.70 6.390 3.177 0.211 0.223 0.195 0.317 0.851 0.003
C4C10T 10.0 0.160 0.70 9.508 0.961 0.133 0.054 0.772 2.343 2.395 0.010
135
Appendix 1.3  Tapered Channels (continued)
Test L ?B' PQ
u
Q
w
h
u
h
d
F
d
Fw
d
C
e
h
c
(ft) (ft) (ft) (cfs) (cfs) (ft) (ft) (ft)
C4C12T 10.0 0.160 0.70 9.512 1.183 0.140 0.080 0.704 1.782 1.608 0.008
C4C20T 10.0 0.160 0.70 9.539 1.946 0.172 0.151 0.542 1.034 0.976 0.003
C4C29T 10.0 0.160 0.70 9.541 2.851 0.207 0.210 0.418 0.696 0.842 0.001
C4C39T 10.0 0.160 0.70 9.547 3.831 0.238 0.254 0.325 0.503 0.829 0.001
C5A10T 5.0 0.320 0.70 3.149 0.324 0.079 0.075 0.247 0.640 0.929 0.001
C5A13T 5.0 0.320 0.70 3.149 0.409 0.092 0.090 0.231 0.551 0.876 0.002
C5A22T 5.0 0.320 0.70 3.142 0.689 0.127 0.128 0.188 0.384 0.833 0.001
C5A31T 5.0 0.320 0.70 3.147 0.996 0.154 0.156 0.154 0.290 0.867 0.000
C5A39T 5.0 0.320 0.70 3.192 1.236 0.174 0.178 0.134 0.237 0.862 0.001
C5A51T 5.0 0.320 0.70 3.196 1.628 0.205 0.209 0.100 0.166 0.863 0.001
C5B10T 5.0 0.320 0.70 6.442 0.658 0.129 0.109 0.464 1.018 1.034 0.006
C5B20T 5.0 0.320 0.70 6.462 1.244 0.179 0.174 0.360 0.645 0.901 0.004
C5B31T 5.0 0.320 0.70 6.433 1.968 0.229 0.232 0.271 0.432 0.871 0.001
C5B41T 5.0 0.320 0.70 6.453 2.619 0.268 0.274 0.213 0.318 0.866 0.002
C5B50T 5.0 0.320 0.70 6.439 3.214 0.298 0.306 0.168 0.241 0.874 0.003
C5C12T 5.0 0.320 0.70 9.531 1.135 0.188 0.146 0.617 1.191 1.103 0.007
C5C20T 5.0 0.320 0.70 9.496 1.878 0.234 0.219 0.475 0.775 0.919 0.003
C5C27T 5.0 0.320 0.70 9.516 2.619 0.276 0.275 0.382 0.570 0.861 0.000
137
APPENDIX 2  WEIR AND CHANNEL GEOMETRIES INVESTIGATED IN PREVIOUS
PROJECT FOR UNSUBMERGED FLOW
Weir
height
Weir
length
Channel
invert
width
Number
of
tests
PLB
(ft) (ft) (ft)
0.52 23.91 3.4 12
0.52 15.00 1.8 9
0.52 10.00 3.4 20
0.52 10.00 1.8 10
0.52 5.00 3.4 19
0.52 5.00 1.8 10
0.52 2.00 3.4 16
0.52 2.00 1.8 21
0.70 20.00 3.4 16
0.70 20.00 1.8 17
0.70 15.00 3.4 12
0.70 15.00 1.8 16
0.70 10.00 3.4 17
0.70 10.00 1.8 16
0.70 5.00 3.4 14
0.70 5.00 1.8 13
139
APPENDIX 3  RESULTS OF SIMULATION OF SIDE WEIR FLOW FOR DIFFERENT
SLOPES AND ROUGHNESS
The results for Method A begin on page 140 while the results for Method B begin on p.
151. For Method A, there is no way to specify different channel slopes when calculating Q
w
.
Therefore, all calculated values of Q
w
are the same for each case.
Notes for the tables:
All results are for prismatic channels.
(1) in the table means supercritical condition at upstream end.
(2) in the table means
uw
QQ > and iteration stopped.
(3) in the table means negative flow depth and iteration stopped.
(4) in the table means
uw
Q6.0Q > in final solution.
?Max. diff.? is the largest difference between the values in the row or column.
140
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10156
c
f
s
h
d
=0
.
9
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
1082
1082
1082
1082
0
0
.0125
0.47
0.19

0
.09

0.36
0.83
0.02
1082
1082
1082
1082
0
0
.02
0
.58
0
.31
0
.04

0.23
0.81
0.03
1082
1082
1082
1082
0
0
.03
0
.80
0
.53
0
.28
0
.03
0
.77
0.04
1082
1082
1082
1082
0
0
.04
1
.08
0
.83
0
.59
0
.35
0
.73
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
6
1
0
.
6
4
0
.
6
8
0
.
7
1
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10116
c
f
s
h
d
=2
.
3
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4905
4905
4905
4905
0
0
.0125
1.67
1.40
1.14
0.87
0.80
0.02
4905
4905
4905
4905
0
0
.02
1
.73
1
.46
1
.20
0
.94
0
.79
0.03
4905
4905
4905
4905
0
0
.03
1
.84
1
.58
1
.33
1
.08
0
.76
0.04
4905
4905
4905
4905
0
0
.04
1
.99
1
.74
1
.50
1
.26
0
.73
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
3
2
0
.
3
4
0
.
3
6
0
.
3
9
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29894
c
f
s
h
d
=4
.
0
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
9829
9829
9829
9829
0
0
.0125
(
1
)
(
1)
(
1
)
(
1)
0.02
9829
9829
9829
9829
0
0
.02
1
.06
(
1)
(
1
)
(
1)
0.03
9829
9829
9829
9829
0
0
.03
2
.92
2
.54
2
.15
1
.76
1
.16
0.04
9829
9829
9829
9829
0
0
.04
4
.20
3
.91
3
.63
3
.35
0
.85
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
3
.
1
4
1
.
3
7
1
.
4
8
1
.
5
9
(
S
ee
notes
on
p.
139.)
141
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
30144
c
f
s
h
d
=4
.
8
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
14305
14305
14305
14305
0
0
.0125
0.97
(
1
)
(
1)
(
1
)
0.02
14305
14305
14305
14305
0
0
.02
2
.36
1
.78
1
.08
(
1)
1.28
0.03
14305
14305
14305
14305
0
0
.03
3
.54
3
.18
2
.82
2
.44
1
.10
0.04
14305
14305
14305
14305
0
0
.04
4
.55
4
.26
3
.98
3
.70
0
.85
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
3
.
5
8
2
.
4
8
2
.
9
0
1
.
2
6
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
30022
c
f
s
h
d
=3
.
5
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
3280
3280
3280
3280
0
0
.0125
(
1
)
(
1)
(
1
)
(
1)
0.02
3280
3280
3280
3280
0
0
.02
1
.71
1
.31
0
.73
(
1)
0.98
0.03
3280
3280
3280
3280
0
0
.03
2
.82
2
.61
2
.40
2
.18
0
.64
0.04
3280
3280
3280
3280
0
0
.04
3
.76
3
.60
3
.45
3
.29
0
.47
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
2
.
0
5
2
.
2
9
2
.
7
2
1
.
1
1
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29969
c
f
s
h
d
=7
.
5
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
14482
14482
14482
14482
0
0
.0125
6.09
5.94
5.79
5.65
0.44
0.02
14482
14482
14482
14482
0
0
.02
6
.19
6
.04
5
.90
5
.76
0
.43
0.03
14482
14482
14482
14482
0
0
.03
6
.39
6
.25
6
.11
5
.97
0
.42
0.04
14482
14482
14482
14482
0
0
.04
6
.64
6
.50
6
.38
6
.25
0
.39
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
5
5
0
.
5
6
0
.
5
9
0
.
6
0
(
S
ee
notes
on
p.
139.)
142
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
4875
c
f
s
h
d
=0
.
9
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
523
523
523
523
0
0
.0125
0.73
0.63
0.52
0.42
0.31
0.02
523
523
523
523
0
0
.02
0
.74
0
.64
0
.53
0
.43
0
.31
0.03
523
523
523
523
0
0
.03
0
.76
0
.66
0
.55
0
.45
0
.31
0.04
523
523
523
523
0
0
.04
0
.79
0
.68
0
.58
0
.48
0
.31
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
0
6
0
.
0
5
0
.
0
6
0
.
0
6
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
4947
c
f
s
h
d
=2
.
3
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2431
2431
2431
2431
0
0
.0125
2.12
2.02
1.92
1.81
0.31
0.02
2431
2431
2431
2431
0
0
.02
2
.13
2
.02
1
.92
1
.82
0
.31
0.03
2431
2431
2431
2431
0
0
.03
2
.14
2
.03
1
.93
1
.83
0
.31
0.04
2431
2431
2431
2431
0
0
.04
2
.15
2
.05
1
.95
1
.84
0
.31
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
0
3
0
.
0
3
0
.
0
3
0
.
0
3
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10222
c
f
s
h
d
=2
.
2
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2657
2657
2657
2657
0
0
.0125
1.54
1.34
1.15
0.95
0.59
0.02
2657
2657
2657
2657
0
0
.02
1
.68
1
.49
1
.30
1
.11
0
.57
0.03
2657
2657
2657
2657
0
0
.03
1
.94
1
.75
1
.58
1
.40
0
.54
0.04
2657
2657
2657
2657
0
0
.04
2
.26
2
.09
1
.92
1
.76
0
.50
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
7
2
0
.
7
5
0
.
7
7
0
.
8
1
(
S
ee
notes
on
p.
139.)
143
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10147
c
f
s
h
d
=3
.
4
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5595
5595
5595
5595
0
0
.0125
2.77
2.58
2.40
2.22
0.55
0.02
5595
5595
5595
5595
0
0
.02
2
.84
2
.66
2
.48
2
.31
0
.53
0.03
5595
5595
5595
5595
0
0
.03
2
.99
2
.81
2
.64
2
.47
0
.52
0.04
5595
5595
5595
5595
0
0
.04
3
.17
3
.01
2
.84
2
.68
0
.49
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
4
0
0
.
4
3
0
.
4
4
0
.
4
6
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
19709
c
f
s
h
d
=3
.
5
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4720
4720
4720
4720
0
0
.0125
(
1
)
(
1)
(
1
)
(
1)
0.02
4720
4720
4720
4720
0
0
.02
1
.19
(
1)
(
1
)
(
1)
0.03
4720
4720
4720
4720
0
0
.03
2
.92
2
.64
2
.36
2
.06
0
.86
0.04
4720
4720
4720
4720
0
0
.04
4
.05
3
.85
3
.65
3
.44
0
.61
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
2
.
8
6
1
.
2
1
1
.
2
9
1
.
3
8
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
19784
c
f
s
h
d
=5
.
5
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
11534
11534
11534
11534
0
0
.0125
3.80
3.49
3.17
2.82
0.98
0.02
11534
11534
11534
11534
0
0
.02
4
.11
3
.84
3
.57
3
.28
0
.83
0.03
11534
11534
11534
11534
0
0
.03
4
.61
4
.38
4
.16
3
.93
0
.68
0.04
11534
11534
11534
11534
0
0
.04
5
.15
4
.96
4
.77
4
.57
0
.58
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
1
.
3
5
1
.
4
7
1
.
6
0
1
.
7
5
(
S
ee
notes
on
p.
139.)
144
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9997
c
f
s
h
d
=1
.
7
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
1080
1080
1080
1080
0
0
.0125
1.23
1.09
0.96
0.83
0.40
0.02
1080
1080
1080
1080
0
0
.02
1
.34
1
.21
1
.08
0
.95
0
.39
0.03
1080
1080
1080
1080
0
0
.03
1
.56
1
.43
1
.31
1
.19
0
.37
0.04
1080
1080
1080
1080
0
0
.04
1
.84
1
.72
1
.60
1
.49
0
.35
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
6
1
0
.
6
3
0
.
6
4
0
.
6
6
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10084
c
f
s
h
d
=3
.
7
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4465
4465
4465
4465
0
0
.0125
3.15
3.03
2.91
2.79
0.36
0.02
4465
4465
4465
4465
0
0
.02
3
.20
3
.07
2
.96
2
.84
0
.36
0.03
4465
4465
4465
4465
0
0
.03
3
.29
3
.17
3
.06
2
.94
0
.35
0.04
4465
4465
4465
4465
0
0
.04
3
.42
3
.30
3
.19
3
.08
0
.34
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
2
7
0
.
2
7
0
.
2
8
0
.
2
9
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
20034
c
f
s
h
d
=3
.
2
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2666
2666
2666
2666
0
0
.0125
(
1
)
(
1)
(
1
)
(
1)
0.02
2666
2666
2666
2666
0
0
.02
(
1)
(
1
)
(
1)
(
1
)
0.03
2666
2666
2666
2666
0
0
.03
2
.82
2
.60
2
.37
2
.13
0
.69
0.04
2666
2666
2666
2666
0
0
.04
3
.88
3
.72
3
.57
3
.41
0
.47
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
1
.
0
6
1
.
1
2
1
.
2
0
1
.
2
8
(
S
ee
notes
on
p.
139.)
145
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
19928
c
f
s
h
d
=6
.
4
2f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
10657
10657
10657
10657
0
0
.0125
5.12
4.95
4.79
4.62
0.50
0.02
10657
10657
10657
10657
0
0
.02
5
.26
5
.10
4
.94
4
.79
0
.47
0.03
10657
10657
10657
10657
0
0
.03
5
.52
5
.37
5
.23
5
.08
0
.44
0.04
10657
10657
10657
10657
0
0
.04
5
.84
5
.70
5
.57
5
.44
0
.40
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
7
2
0
.
7
5
0
.
7
8
0
.
8
2
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10003
c
f
s
h
d
=0
.
9
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
998
998
998
998
0
0
.0125
0.64
0.42
0.21
0.00
0.64
0.02
998
998
998
998
0
0
.02
0
.66
0
.45
0
.24
0
.03
0
.63
0.03
998
998
998
998
0
0
.03
0
.72
0
.51
0
.30
0
.10
0
.62
0.04
998
998
998
998
0
0
.04
0
.81
0
.60
0
.39
0
.19
0
.62
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
1
7
0
.
1
8
0
.
1
8
0
.
1
9
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9975
c
f
s
h
d
=2
.
5
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5068
5068
5068
5068
0
0
.0125
2.15
1.93
1.73
1.52
0.63
0.02
5068
5068
5068
5068
0
0
.02
2
.16
1
.95
1
.74
1
.54
0
.62
0.03
5068
5068
5068
5068
0
0
.03
2
.19
1
.98
1
.77
1
.57
0
.62
0.04
5068
5068
5068
5068
0
0
.04
2
.23
2
.02
1
.82
1
.62
0
.61
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
0
8
0
.
0
9
0
.
0
9
0
.
1
0
(
S
ee
notes
on
p.
139.)
146
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29766
c
f
s
h
d
=2
.
2
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2964
2964
2964
2964
0
0
.0125
1.21
0.90
0.60
0.29
0.92
0.02
2964
2964
2964
2964
0
0
.02
1
.50
1
.21
0
.93
0
.64
0
.86
0.03
2964
2964
2964
2964
0
0
.03
2
.03
1
.77
1
.52
1
.26
0
.77
0.04
2964
2964
2964
2964
0
0
.04
2
.67
2
.43
2
.20
1
.97
0
.70
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
1
.
4
6
1
.
5
3
1
.
6
0
1
.
6
8
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29828
c
f
s
h
d
=5
.
1
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
14738
14738
14738
14738
0
0
.0125
4.05
3.80
3.54
3.29
0.76
0.02
14738
14738
14738
14738
0
0
.02
4
.16
3
.91
3
.66
3
.42
0
.74
0.03
14738
14738
14738
14738
0
0
.03
4
.37
4
.13
3
.90
3
.66
0
.71
0.04
14738
14738
14738
14738
0
0
.04
4
.65
4
.42
4
.19
3
.97
0
.68
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
6
0
0
.
6
2
0
.
6
5
0
.
6
8
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10219
c
f
s
h
d
=1
.
2
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
842
842
842
842
0
0
.0125
1.06
0.95
0.85
0.74
0.32
0.02
842
842
842
842
0
0
.02
1
.07
0
.97
0
.86
0
.76
0
.31
0.03
842
842
842
842
0
0
.03
1
.10
1
.00
0
.89
0
.79
0
.31
0.04
842
842
842
842
0
0
.04
1
.14
1
.04
0
.93
0
.83
0
.31
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
0
8
0
.
0
9
0
.
0
8
0
.
0
9
(
S
ee
notes
on
p.
139.)
147
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10225
c
f
s
h
d
=3
.
7
2f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5215
5215
5215
5215
0
0
.0125
3.48
3.37
3.27
3.17
0.31
0.02
5215
5215
5215
5215
0
0
.02
3
.49
3
.38
3
.28
3
.18
0
.31
0.03
5215
5215
5215
5215
0
0
.03
3
.50
3
.39
3
.29
3
.19
0
.31
0.04
5215
5215
5215
5215
0
0
.04
3
.52
3
.41
3
.31
3
.21
0
.31
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
0
4
0
.
0
4
0
.
0
4
0
.
0
4
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29972
c
f
s
h
d
=3
.
1
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
3023
3023
3023
3023
0
0
.0125
2.39
2.25
2.11
1.97
0.42
0.02
3023
3023
3023
3023
0
0
.02
2
.50
2
.37
2
.23
2
.10
0
.40
0.03
3023
3023
3023
3023
0
0
.03
2
.73
2
.60
2
.47
2
.34
0
.39
0.04
3023
3023
3023
3023
0
0
.04
3
.03
2
.90
2
.78
2
.65
0
.38
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
6
4
0
.
6
5
0
.
6
7
0
.
6
8
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29938
c
f
s
h
d
=6
.
9
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
13816
13816
13816
13816
0
0
.0125
6.21
6.09
5.97
5.85
0.36
0.02
13816
13816
13816
13816
0
0
.02
6
.25
6
.13
6
.01
5
.89
0
.36
0.03
13816
13816
13816
13816
0
0
.03
6
.32
6
.20
6
.09
5
.97
0
.35
0.04
13816
13816
13816
13816
0
0
.04
6
.43
6
.31
6
.20
6
.08
0
.35
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
2
2
0
.
2
2
0
.
2
3
0
.
2
3
(
S
ee
notes
on
p.
139.)
148
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10181
c
f
s
h
d
=1
.
3
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
1499
1499
1499
1499
0
0
.0125
0.92
0.69
0.47
0.25
0.67
0.02
1499
1499
1499
1499
0
0
.02
0
.99
0
.76
0
.55
0
.33
0
.66
0.03
1499
1499
1499
1499
0
0
.03
1
.13
0
.91
0
.69
0
.48
0
.65
0.04
1499
1499
1499
1499
0
0
.04
1
.31
1
.10
0
.89
0
.68
0
.63
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
3
9
0
.
4
1
0
.
4
2
0
.
4
3
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10122
c
f
s
h
d
=2
.
5
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4604
4604
4604
4604
0
0
.0125
2.09
1.86
1.65
1.43
0.66
0.02
4604
4604
4604
4604
0
0
.02
2
.13
1
.90
1
.69
1
.48
0
.65
0.03
4604
4604
4604
4604
0
0
.03
2
.21
1
.99
1
.78
1
.57
0
.64
0.04
4604
4604
4604
4604
0
0
.04
2
.31
2
.10
1
.89
1
.69
0
.62
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
2
2
0
.
2
4
0
.
2
4
0
.
2
6
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
30012
c
f
s
h
d
=3
.
0
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
3977
3977
3977
3977
0
0
.0125
(
1
)
(
1)
(
1
)
(
1)
0.02
3977
3977
3977
3977
0
0
.02
1
.46
0
.85
0
.00
(
1)
1.46
0.03
3977
3977
3977
3977
0
0
.03
3
.10
2
.76
2
.43
2
.08
1
.02
0.04
3977
3977
3977
3977
0
0
.04
4
.48
4
.22
3
.96
3
.70
0
.78
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
3
.
0
2
3
.
3
7
3
.
9
6
1
.
6
2
(
S
ee
notes
on
p.
139.)
149
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29906
c
f
s
h
d
=5
.
7
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
15172
15172
15172
15172
0
0
.0125
3.78
3.40
3.01
2.59
1.19
0.02
15172
15172
15172
15172
0
0
.02
4
.13
3
.79
3
.45
3
.09
1
.04
0.03
15172
15172
15172
15172
0
0
.03
4
.72
4
.42
4
.14
3
.84
0
.88
0.04
15172
15172
15172
15172
0
0
.04
5
.37
5
.11
4
.86
4
.61
0
.76
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
1
.
5
9
1
.
7
1
1
.
8
5
2
.
0
2
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9900
c
f
s
h
d
=1
.
3
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
835
835
835
835
0
0
.0125
1.11
1.00
0.89
0.78
0.33
0.02
835
835
835
835
0
0
.02
1
.15
1
.03
0
.93
0
.82
0
.33
0.03
835
835
835
835
0
0
.03
1
.22
1
.10
1
.00
0
.89
0
.33
0.04
835
835
835
835
0
0
.04
1
.31
1
.20
1
.09
0
.99
0
.32
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
2
0
0
.
2
0
0
.
2
0
0
.
2
1
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9944
c
f
s
h
d
=3
.
8
8f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5175
5175
5175
5175
0
0
.0125
3.57
3.46
3.36
3.25
0.32
0.02
5175
5175
5175
5175
0
0
.02
3
.59
3
.48
3
.37
3
.27
0
.32
0.03
5175
5175
5175
5175
0
0
.03
3
.61
3
.51
3
.40
3
.30
0
.31
0.04
5175
5175
5175
5175
0
0
.04
3
.65
3
.54
3
.44
3
.34
0
.31
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
0
8
0
.
0
8
0
.
0
8
0
.
0
9
(
S
ee
notes
on
p.
139.)
150
A
ppe
ndix
3.1

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
A
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29991
c
f
s
h
d
=3
.
4
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2925
2925
2925
2925
0
0
.0125
1.76
1.43
1.05
0.52
1.24
0.02
2925
2925
2925
2925
0
0
.02
2
.31
2
.06
1
.80
1
.52
0
.79
0.03
2925
2925
2925
2925
0
0
.03
3
.11
2
.92
2
.74
2
.54
0
.57
0.04
2925
2925
2925
2925
0
0
.04
3
.94
3
.79
3
.63
3
.48
0
.46
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
2
.
1
8
2
.
3
6
2
.
5
8
2
.
9
6
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29969
c
f
s
h
d
=6
.
4
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
10778
10778
10778
10778
0
0
.0125
5.14
4.98
4.82
4.66
0.48
0.02
10778
10778
10778
10778
0
0
.02
5
.28
5
.13
4
.97
4
.82
0
.46
0.03
10778
10778
10778
10778
0
0
.03
5
.55
5
.40
5
.26
5
.11
0
.44
0.04
10778
10778
10778
10778
0
0
.04
5
.89
5
.75
5
.62
5
.48
0
.41
M
a
x
.
d
i
f
f
.
0000
M
a
x
.
d
i
f
f
.
0
.
7
5
0
.
7
7
0
.
8
0
0
.
8
2
(
S
ee
notes
on
p.
139.)
151
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10156
c
f
s
h
d
=0
.
9
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
951
739
564
430
521
0.0125
0.66
0.41
0.16

0
.10
0
.76
0.02
1039
821
637
485
554
0.02
0.76
0.51
0.28
0.03
0.73
0.03
1216
991
795
622
594
0.03
0.95
0.72
0.49
0.27
0.68
0.04
1461
1226
1020
832
629
0.04
1.18
0.97
0.76
0.55
0.63
Max
.
dif
f
.
510
487
456
402
Max
.
dif
f
.
0.52
0.56
0.60
0.65
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10116
c
f
s
h
d
=2
.
3
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4801
4348
3924
3523
1278
0.0125
1.81
1.55
1.29
1.03
0.78
0.02
4875
4429
4010
3612
1263
0.02
1.87
1.61
1.35
1.10
0.77
0.03
5019
4587
4179
3787
1232
0.03
1.97
1.72
1.48
1.24
0.73
0.04
5203
4789
4399
4018
1185
0.04
2.10
1.87
1.64
1.41
0.69
Max
.
dif
f
.
402
441
475
495
Max
.
dif
f
.
0.29
0.32
0.35
0.38
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29894
c
f
s
h
d
=4
.
0
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
6407
(
2
)
(
3)
(
3
)
0
.0125
1.39
(
2
)
(
3)
(
3
)
0.02
8104
7287
6478
5642
2462
0.02
2.29
1.97
1.65
1.32
0.97
0.03
10257
9598
8987
8382
1875
0.03
3.28
3.05
2.84
2.63
0.65
0.04
12380
11819
11266
10715
1665
0.04
4.20
4.02
3.83
3.64
0.56
Max
.
dif
f
.
5973
4532
4788
5073
Max
.
dif
f
.
2.81
2.05
2.18
2.32
(
S
ee
notes
on
p.
139.)
152
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
598
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
30144
c
f
s
h
d
=4
.
8
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
10338
9104
(
3
)
(
2)
1234
0.0125
1.95
1.45
(
3
)
(
2)
0.50
0.02
11947
10916
9875
8950
2997
0.02
2.79
2.39
1.96
1.66
1.13
0.03
14069
13247
12401
11675
2394
0.03
3.85
3.52
3.16
2.93
0.92
0.04
15895
15268
14656
14034
1861
0.04
4.69
4.48
4.27
4.04
0.65
Max
.
dif
f
.
5557
6164
4781
5084
Max
.
dif
f
.
2.74
3.03
2.31
2.38
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
30022
c
f
s
h
d
=3
.
5
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2845
2661
2469
2243
602
0.0125
2.13
1.83
1.46
0.85
1.28
0.02
3174
3014
2856
2692
482
0.02
2.63
2.42
2.20
1.96
0.67
0.03
3721
3583
3449
3314
407
0.03
3.33
3.18
3.03
2.87
0.46
0.04
4368
4238
4112
3986
382
0.04
4.05
3.92
3.79
3.66
0.39
Max
.
dif
f
.
1523
1577
1643
1743
Max
.
dif
f
.
1.92
2.09
2.33
2.81
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29969
c
f
s
h
d
=7
.
5
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
15596
15380
15170
14960
636
0.0125
6.51
6.37
6.23
6.09
0.42
0.02
15699
15487
15280
15073
626
0.02
6.59
6.46
6.32
6.19
0.40
0.03
15903
15696
15496
15295
608
0.03
6.75
6.63
6.50
6.37
0.38
0.04
16170
15971
15778
15586
584
0.04
6.96
6.84
6.73
6.61
0.35
Max
.
dif
f
.
574
591
608
626
Max
.
dif
f
.
0.45
0.47
0.50
0.52
(
S
ee
notes
on
p.
139.)
153
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
4875
c
f
s
h
d
=0
.
9
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
481
437
397
358
123
0.0125
0.78
0.67
0.57
0.47
0.31
0.02
485
441
400
362
123
0.02
0.79
0.68
0.58
0.48
0.31
0.03
493
449
408
370
123
0.03
0.81
0.70
0.60
0.51
0.30
0.04
504
460
419
380
124
0.04
0.83
0.73
0.63
0.53
0.30
Max
.
dif
f
.
23
23
22
22
Max
.
dif
f
.
0.05
0.06
0.06
0.06
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
4947
c
f
s
h
d
=2
.
3
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2451
2367
2287
2208
243
0.0125
2.17
2.07
1.96
1.86
0.31
0.02
2454
2370
2290
2211
243
0.02
2.18
2.07
1.97
1.87
0.31
0.03
2460
2376
2296
2218
242
0.03
2.19
2.08
1.98
1.88
0.31
0.04
2469
2385
2305
2227
242
0.04
2.20
2.09
1.99
1.89
0.31
Max
.
dif
f
.
18
18
18
19
Max
.
dif
f
.
0.03
0.02
0.03
0.03
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10222
c
f
s
h
d
=2
.
2
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2604
2422
2251
2085
519
0.0125
1.64
1.47
1.30
1.13
0.51
0.02
2715
2535
2367
2203
512
0.02
1.75
1.59
1.43
1.27
0.48
0.03
2927
2753
2590
2430
497
0.03
1.97
1.82
1.67
1.52
0.45
0.04
3201
3029
2871
2717
484
0.04
2.23
2.09
1.96
1.83
0.40
Max
.
dif
f
.
597
607
620
632
Max
.
dif
f
.
0.59
0.62
0.66
0.70
(
S
ee
notes
on
p.
139.)
154
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10147
c
f
s
h
d
=3
.
4
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5708
5479
5261
5042
666
0.0125
2.77
2.62
2.45
2.28
0.49
0.02
5776
5550
5336
5122
654
0.02
2.82
2.69
2.53
2.36
0.46
0.03
5910
5688
5481
5276
634
0.03
2.93
2.81
2.67
2.52
0.41
0.04
6085
5869
5667
5471
614
0.04
3.07
2.95
2.84
2.71
0.36
Max
.
dif
f
.
377
390
406
429
Max
.
dif
f
.
0.30
0.33
0.39
0.43
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
19709
c
f
s
h
d
=3
.
5
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
(
2
)
(
3)
(
3
)
(
3)
0.0125
(
2
)
(
3)
(
3
)
(
3)
0.02
3969
3578
3167
(
3
)
802
0.02
1.99
1.71
1.43
(
3
)
0
.56
0.03
5199
4925
4661
4394
805
0.03
2.97
2.80
2.64
2.47
0.50
0.04
6342
6083
5863
5646
696
0.04
3.79
3.63
3.50
3.38
0.41
Max
.
dif
f
.
2373
2505
2696
1252
Max
.
dif
f
.
1.80
1.92
2.07
0.91
Input
geom
etr
y
data:
L
=
375
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
19784
c
f
s
h
d
=5
.
5
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
10427
10025
9610
9156
1271
0.0125
3.91
3.66
3.39
3.08
0.83
0.02
10785
10424
10061
9680
1105
0.02
4.17
3.97
3.75
3.51
0.66
0.03
11387
11069
10766
10453
934
0.03
4.49
4.39
4.27
4.10
0.39
0.04
(
4
)
11796
11513
11237
559
0.04
(
4
)
4
.74
4
.65
4
.57
0
.17
Max
.
dif
f
.
960
1771
1903
2081
Max
.
dif
f
.
0.58
1.08
1.26
1.49
(
S
ee
notes
on
p.
139.)
155
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9997
c
f
s
h
d
=1
.
7
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
1064
999
939
880
184
0.0125
1.34
1.22
1.11
0.99
0.35
0.02
1116
1051
990
931
185
0.02
1.44
1.33
1.21
1.10
0.34
0.03
1219
1155
1094
1035
184
0.03
1.63
1.52
1.42
1.31
0.32
0.04
1360
1294
1233
1174
186
0.04
1.87
1.77
1.67
1.57
0.30
Max
.
dif
f
.
296
295
294
294
Max
.
dif
f
.
0.53
0.55
0.56
0.58
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10084
c
f
s
h
d
=3
.
7
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4728
4608
4492
4376
352
0.0125
3.21
3.09
2.97
2.84
0.37
0.02
4764
4645
4530
4416
348
0.02
3.25
3.13
3.02
2.90
0.35
0.03
4835
4718
4606
4494
341
0.03
3.34
3.22
3.11
3.00
0.34
0.04
4929
4816
4707
4598
331
0.04
3.46
3.34
3.24
3.13
0.33
Max
.
dif
f
.
201
208
215
222
Max
.
dif
f
.
0.25
0.25
0.27
0.29
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
20034
c
f
s
h
d
=3
.
2
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
(
3
)
(
3)
(
3
)
(
3)
0.0125
(
3
)
(
3)
(
3
)
(
3)
0.02
2343
2155
1940
(
2
)
403
0.02
1.93
1.66
1.19
(
2
)
0
.74
0.03
3087
2942
2800
2654
433
0.03
2.96
2.80
2.64
2.46
0.50
0.04
3797
3678
3563
3448
349
0.04
3.77
3.65
3.54
3.42
0.35
Max
.
dif
f
.
1454
1523
1623
794
Max
.
dif
f
.
1.84
1.99
2.35
0.96
(
S
ee
notes
on
p.
139.)
156
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
13.0
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
19928
c
f
s
h
d
=6
.
4
2f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
10959
10768
10582
10394
565
0.0125
5.31
5.17
5.04
4.89
0.42
0.02
11082
10898
10718
10536
546
0.02
5.41
5.30
5.17
5.03
0.38
0.03
11319
11143
10974
10803
516
0.03
5.58
5.49
5.40
5.29
0.29
0.04
11624
11455
11293
11133
491
0.04
5.81
5.72
5.64
5.56
0.25
Max
.
dif
f
.
665
687
711
739
Max
.
dif
f
.
0.50
0.55
0.60
0.67
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10003
c
f
s
h
d
=0
.
9
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
887
729
593
476
411
0.0125
0.72
0.51
0.31
0.10
0.62
0.02
909
750
612
492
417
0.02
0.74
0.54
0.34
0.14
0.60
0.03
954
793
652
527
427
0.03
0.80
0.60
0.40
0.20
0.60
0.04
1017
853
708
578
439
0.04
0.88
0.68
0.49
0.29
0.59
Max
.
dif
f
.
130
124
115
102
Max
.
dif
f
.
0.16
0.17
0.18
0.19
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9975
c
f
s
h
d
=2
.
5
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5099
4768
4456
4152
947
0.0125
2.21
2.00
1.79
1.59
0.62
0.02
5117
4786
4475
4172
945
0.02
2.22
2.01
1.81
1.60
0.62
0.03
5152
4823
4515
4213
939
0.03
2.25
2.04
1.84
1.64
0.61
0.04
5199
4875
4569
4270
929
0.04
2.29
2.09
1.89
1.69
0.60
Max
.
dif
f
.
100
107
113
118
Max
.
dif
f
.
0.08
0.09
0.10
0.10
(
S
ee
notes
on
p.
139.)
157
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29766
c
f
s
h
d
=2
.
2
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2785
2503
2243
1995
790
0.0125
1.59
1.34
1.09
0.84
0.75
0.02
3050
2768
2505
2253
797
0.02
1.83
1.59
1.36
1.12
0.71
0.03
3567
3286
3023
2768
799
0.03
2.25
2.04
1.83
1.62
0.63
0.04
4254
3966
3699
3445
809
0.04
2.76
2.57
2.38
2.19
0.57
Max
.
dif
f
.
1469
1463
1456
1450
Max
.
dif
f
.
1.17
1.23
1.29
1.35
Input
geom
etr
y
data:
L
=
500
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29828
c
f
s
h
d
=5
.
1
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
14698
14130
13581
13032
1666
0.0125
4.08
3.83
3.59
3.33
0.75
0.02
14885
14327
13789
13251
1634
0.02
4.18
3.94
3.70
3.46
0.72
0.03
15246
14708
14191
13674
1572
0.03
4.37
4.15
3.93
3.70
0.67
0.04
15712
15198
14705
14213
1499
0.04
4.62
4.41
4.21
4.00
0.62
Max
.
dif
f
.
1014
1068
1124
1181
Max
.
dif
f
.
0.54
0.58
0.62
0.67
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10219
c
f
s
h
d
=1
.
2
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
787
736
687
641
146
0.0125
1.11
1.01
0.91
0.80
0.31
0.02
794
742
694
647
147
0.02
1.13
1.02
0.92
0.82
0.31
0.03
808
756
708
661
147
0.03
1.16
1.05
0.95
0.85
0.31
0.04
828
776
727
679
149
0.04
1.20
1.09
0.99
0.89
0.31
Max
.
dif
f
.
41
40
40
38
Max
.
dif
f
.
0.09
0.08
0.08
0.09
(
S
ee
notes
on
p.
139.)
158
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10225
c
f
s
h
d
=3
.
7
2f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5243
5133
5028
4923
320
0.0125
3.53
3.43
3.33
3.22
0.31
0.02
5248
5138
5033
4928
320
0.02
3.54
3.43
3.33
3.23
0.31
0.03
5257
5148
5043
4939
318
0.03
3.55
3.45
3.34
3.24
0.31
0.04
5271
5161
5057
4953
318
0.04
3.57
3.46
3.36
3.26
0.31
Max
.
dif
f
.
28
28
29
30
Max
.
dif
f
.
0.04
0.03
0.03
0.04
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29972
c
f
s
h
d
=3
.
1
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2921
2823
2729
2637
284
0.0125
2.60
2.47
2.35
2.22
0.38
0.02
2998
2900
2807
2715
283
0.02
2.70
2.58
2.46
2.33
0.37
0.03
3154
3056
2963
2872
282
0.03
2.91
2.79
2.67
2.55
0.36
0.04
3364
3268
3175
3084
280
0.04
3.17
3.06
2.95
2.84
0.33
Max
.
dif
f
.
443
445
446
447
Max
.
dif
f
.
0.57
0.59
0.60
0.62
Input
geom
etr
y
data:
L
=
250
f
t
B
=
85
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29938
c
f
s
h
d
=6
.
9
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
14199
14019
13846
13673
526
0.0125
6.29
6.17
6.06
5.94
0.35
0.02
14243
14064
13892
13720
523
0.02
6.33
6.21
6.10
5.98
0.35
0.03
14331
14154
13984
13813
518
0.03
6.40
6.29
6.17
6.06
0.34
0.04
14453
14278
14109
13941
512
0.04
6.50
6.39
6.28
6.16
0.34
Max
.
dif
f
.
254
259
263
268
Max
.
dif
f
.
0.21
0.22
0.22
0.22
(
S
ee
notes
on
p.
139.)
159
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10181
c
f
s
h
d
=1
.
3
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
1493
1291
1116
955
538
0.0125
1.01
0.80
0.60
0.39
0.62
0.02
1552
1349
1171
1007
545
0.02
1.07
0.86
0.67
0.47
0.60
0.03
1672
1468
1283
1115
557
0.03
1.20
1.00
0.80
0.61
0.59
0.04
1835
1629
1442
1266
569
0.04
1.36
1.17
0.98
0.80
0.56
Max
.
dif
f
.
342
338
326
311
Max
.
dif
f
.
0.35
0.37
0.38
0.41
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
10122
c
f
s
h
d
=2
.
5
7f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
4907
4573
4255
3948
959
0.0125
2.13
1.91
1.69
1.48
0.65
0.02
4952
4622
4307
4002
950
0.02
2.16
1.95
1.73
1.52
0.64
0.03
5042
4717
4409
4108
934
0.03
2.24
2.03
1.82
1.61
0.63
0.04
5160
4844
4545
4249
911
0.04
2.33
2.13
1.94
1.73
0.60
Max
.
dif
f
.
253
271
290
301
Max
.
dif
f
.
0.20
0.22
0.25
0.25
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
30012
c
f
s
h
d
=3
.
0
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2902
2386
(
2
)
(
3)
516
0.0125
1.14
0.66
(
2
)
(
3)
0.48
0.02
3866
3446
3043
2635
1231
0.02
1.96
1.67
1.38
1.07
0.89
0.03
5412
5033
4670
4309
1103
0.03
3.02
2.80
2.58
2.37
0.65
0.04
6962
6631
6315
6003
959
0.04
3.92
3.75
3.59
3.43
0.49
Max
.
dif
f
.
4060
4245
3272
3368
Max
.
dif
f
.
2.78
3.09
2.21
2.36
160
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
500
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29906
c
f
s
h
d
=5
.
7
0f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
13613
12799
12011
11239
2374
0.0125
3.38
2.94
2.53
2.14
1.24
0.02
14273
13566
12841
12106
2167
0.02
3.78
3.44
3.06
2.68
1.10
0.03
15337
14746
14163
13559
1778
0.03
4.38
4.12
3.86
3.58
0.80
0.04
16472
15965
15473
14974
1498
0.04
4.95
4.77
4.58
4.38
0.57
Max
.
dif
f
.
2859
3166
3462
3735
Max
.
dif
f
.
1.57
1.83
2.05
2.24
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9900
c
f
s
h
d
=1
.
3
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
825
772
723
675
150
0.0125
1.17
1.06
0.96
0.85
0.32
0.02
840
787
738
690
150
0.02
1.20
1.09
0.99
0.89
0.31
0.03
872
818
769
720
152
0.03
1.27
1.16
1.06
0.96
0.31
0.04
916
862
812
763
153
0.04
1.35
1.25
1.15
1.05
0.30
Max
.
dif
f
.
91
90
89
88
Max
.
dif
f
.
0.18
0.19
0.19
0.20
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
9944
c
f
s
h
d
=3
.
8
8f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
5427
5315
5208
5102
325
0.0125
3.59
3.49
3.38
3.28
0.31
0.02
5437
5326
5219
5113
324
0.02
3.61
3.50
3.40
3.29
0.32
0.03
5457
5346
5240
5134
323
0.03
3.63
3.53
3.42
3.32
0.31
0.04
5485
5375
5270
5165
320
0.04
3.66
3.56
3.46
3.36
0.30
Max
.
dif
f
.
58
60
62
63
Max
.
dif
f
.
0.07
0.07
0.08
0.08
(
S
ee
notes
on
p.
139.)
161
A
ppe
ndix
3.2

R
e
s
ult
s
of
sim
u
lat
i
on
using
M
e
t
hod
B
(
c
ont
i
nue
d)
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29991
c
f
s
h
d
=3
.
4
3f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
2596
2448
2302
2152
444
0.0125
2.13
1.90
1.65
1.37
0.76
0.02
2859
2722
2588
2453
406
0.02
2.54
2.35
2.16
1.95
0.59
0.03
3325
3201
3081
2961
364
0.03
3.18
3.03
2.88
2.73
0.45
0.04
3881
3763
3653
3544
337
0.04
3.84
3.71
3.59
3.47
0.37
Max
.
dif
f
.
1285
1315
1351
1392
Max
.
dif
f
.
1.71
1.81
1.94
2.10
Input
geom
etr
y
data:
L
=
250
f
t
B
=
45
f
t
P
=
17.5
f
t
I
nput
f
l
ow
c
onditions
:
Q
u
=
29969
c
f
s
h
d
=6
.
4
5f
t
Calc
ulated
Q
w
(cf
s
)
C
a
l
cu
l
a
te
d
h
u
(f
t)
Slope
Slope
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
Manning'
s
n
0.000385
0.0008
0.0012
0.0016
Max
.
dif
f
.
0.0125
10689
10481
10280
10077
612
0.0125
4.92
4.76
4.60
4.43
0.49
0.02
10846
10643
10447
10250
596
0.02
5.06
4.91
4.75
4.60
0.46
0.03
11150
10956
10768
10580
570
0.03
5.33
5.19
5.05
4.91
0.42
0.04
11541
11356
11178
11000
541
0.04
5.67
5.54
5.41
5.28
0.39
Max
.
dif
f
.
852
875
898
923
Max
.
dif
f
.
0.75
0.78
0.81
0.85
(
S
ee
notes
on
p.
139.)
163
AP
P
E
NDIX
4

S
UMMARY
OF
MODEL
DATA
F
OR
4H:1V
S
IDE
S
LOP
E
S
Te
s
t
L
B
P
u
P
d
Q
u
Q
w
h
u
h
d
Fw
d
F
d
C
e
(f
t)
(f
t)
(f
t)
(f
t)
(cf
s
)
(
cf
s)
(f
t)
(f
t)
A1
10
1.8
0
.513
0.506
3.208
1.685
0.138
0.168
0.216
0.137
0.717
A2
10
1.8
0
.513
0.506
3.208
0.992
0.092
0.126
0.403
0.226
0.666
A3
10
1.8
0
.513
0.506
3.208
0.246
0.027
0.060
0.928
0.377
0.522
B1
10
1.8
0
.513
0.506
6.608
3.172
0.201
0.237
0.351
0.253
0.775
B2
10
1.8
0
.513
0.506
6.608
1.915
0.140
0.201
0.566
0.383
0.614
B3
10
1.8
0
.513
0.506
6.608
0.783
0.077
0.133
1.016
0.583
0.485
C1
10
1.8
0
.513
0.506
9.165
3.714
0.222
0.268
0.490
0.369
0.742
C2
10
1.8
0
.513
0.506
9.165
2.379
0.164
0.237
0.694
0.499
0.581
C3
10
1.8
0
.513
0.506
9.165
0.983
0.125
0.162
1.201
0.748
0.443
AA3
1
0
1
.8
0.513
0.506
3.200
0.256
0.026
0.060
0.922
0.375
0.543
BB1
1
0
1
.8
0.513
0.506
6.570
3.196
0.202
0.237
0.345
0.248
0.781
BB3
1
0
1
.8
0.513
0.506
6.570
0.794
0.078
0.132
1.010
0.579
0.495
CC1
10
1.8
0
.513
0.506
9.094
3.742
0.218
0.278
0.462
0.352
0.697
CC2
10
1.8
0
.513
0.506
9.094
2.463
0.167
0.237
0.678
0.488
0.602
CC3
10
1.8
0
.513
0.506
9.094
0.980
0.125
0.160
1.204
0.746
0.451
X
1
5
1
.8
0.513
0.490
3.184
1.627
0.215
0.236
0.165
0.120
0.717
X
2
5
1
.8
0.513
0.490
3.184
1.339
0.190
0.213
0.217
0.152
0.703
X
3
5
1
.8
0.513
0.490
3.184
0.943
0.151
0.174
0.320
0.207
0.696
X
4
5
1
.8
0.513
0.490
3.184
0.591
0.108
0.132
0.472
0.273
0.689
X
5
5
1
.8
0.513
0.490
3.184
0.286
0.064
0.085
0.743
0.357
0.678
Y
1
5
1
.8
0.513
0.490
6.428
1.491
0.207
0.239
0.518
0.377
0.643
Y
2
5
1
.8
0.513
0.490
6.428
1.021
0.161
0.196
0.690
0.467
0.617
Y
3
5
1
.8
0.513
0.490
6.428
0.546
0.110
0.138
1.030
0.608
0.592
Z1
5
1
.8
0.513
0.490
8.989
0.886
0.157
0.190
1.066
0.713
0.565
165
APPENDIX 5  COMPONENTS OF ? AND ?
Appendix 5.1  Variation of components of ? and ? with distance for 2.5H:1V
side slopes
Case
Q
u
(cfs)
Diver
sion (%)
B
s
(ft)
s
B
x
?
?
1
=
2
2
U
u
?
2
=
2
2
U
'u
?
?
1
=
3
3
U
u
?
2
=
3
2
3
U
'uu
?
3
=
3
2
U
'vu
?
4
=
3
2
U
vu
A 0.00 1.689 1.554 0.135 3.180 2.466 0.365 0.0483 0.2782
8.9 1.16 1.636 1.494 0.142 2.859 2.376 0.395 0.0612 0.0135
54 2.75 1.520 1.420 0.100 2.541 2.165 0.311 0.0477 0.0040
3.72 4.66 1.326 1.250 0.076 1.986 1.711 0.230 0.0346 0.0034
5.98 1.287 1.225 0.062 1.876 1.640 0.198 0.0305 0.0041
B 0.00 1.794 1.630 0.165 3.483 2.669 0.461 0.0689 0.2593
3.0 1.18 1.813 1.681 0.132 3.309 2.779 0.426 0.0750 0.0048
54 2.77 1.545 1.437 0.108 2.588 2.171 0.338 0.0605 0.0026
3.72 4.67 1.388 1.304 0.084 2.166 1.855 0.262 0.0438 0.0012
6.00 1.324 1.246 0.078 2.006 1.694 0.260 0.0385 0.0027
C 0.00 1.142 1.133 0.010 1.396 1.341 0.0273 0.0026 0.0241
8.9 2.60 1.107 1.098 0.009 1.295 1.258 0.0281 0.0029 0.0050
25 6.13 1.091 1.083 0.008 1.245 1.213 0.0249 0.0027 0.0035
1.67 10.41 1.082 1.075 0.006 1.219 1.191 0.0202 0.0024 0.0040
13.30 1.084 1.077 0.007 1.227 1.199 0.0210 0.0024 0.0041
D 1.98 1.516 1.500 0.016 2.428 2.361 0.0481 0.0067 0.0081
4.1 3.35 1.437 1.420 0.017 2.225 2.166 0.0448 0.0070 0.0045
N/A 6.84 1.273 1.261 0.012 1.802 1.754 0.0394 0.0056 0.0009
3.72 9.99 1.221 1.206 0.015 1.639 1.579 0.0482 0.0070 0.0018
13.13 1.186 1.174 0.013 1.531 1.481 0.0417 0.0057 0.0012
16.28 1.149 1.137 0.013 1.421 1.372 0.0409 0.0059 0.0011
E N/A 1.083 1.077 0.006 1.229 1.203 0.0200 0.0034 0.0014
166
Appendix 5.2  Variation of components of ? and ? with distance for 4H:1V side
slopes
Case
Q
u
(cfs)
Diver
sion (%)
B
s
(ft)
s
B
x
?
?
1
=
2
2
U
u
?
2
=
2
2
U
'u
?
?
1
=
3
3
U
u
?
2
=
3
2
3
U
'uu
?
3
=
3
2
U
'vu
?
4
=
3
2
U
vu
F 0.51 1.989 1.833 0.156 3.828 3.164 0.532 0.168 0.0080
6.1 0.51 1.908 1.763 0.145 3.800 3.211 0.481 0.083 0.0108
54 2.02 1.695 1.584 0.112 3.283 2.755 0.431 0.118 0.0105
3.99 2.02 1.728 1.617 0.112 3.226 2.784 0.354 0.0669 0.103
3.41 1.398 1.377 2.166 2.063
4.80 1.215 1.198 1.630 1.554
G 0.00 1.885 1.832 0.053 3.276 3.094 0.177 0.0253 0.0024
4.5 0.54 1.960 1.910 0.050 3.497 3.346 0.130 0.0169 0.0019
N/A 1.08 1.934 1.876 0.059 3.450 3.236 0.197 0.0232 0.0044
3.99 1.63 1.818 1.765 0.053 3.240 3.065 0.147 0.0206 0.0037
2.16 1.733 1.672 0.062 3.000 2.785 0.198 0.0236 0.0042
2.71 1.612 1.555 0.057 2.724 2.521 0.185 0.0243 0.0037
3.26 1.427 1.390 0.037 2.128 1.995 0.115 0.0186 0.0046
4.34 1.305 1.280 0.026 1.874 1.783 0.0785 0.0079 0.0017
4.89 1.236 1.209 0.028 1.671 1.565 0.0953 0.0107 0.0031
6.52 1.185 1.167 0.018 1.516 1.444 0.0571 0.0107 0.0044
7.89 1.201 1.184 0.017 1.536 1.495 0.0505 0.0052 0.0031
10.90 1.132 1.118 0.014 1.371 1.317 0.0443 0.0112 0.0018
H N/A 1.123 1.115 0.008 1.172 1.153 0.0133 0.0022 0.0035
Appendix 5.3  Components of ? and ? just downstream of weir for 2.5H:1V side slopes
Case
Q
u
(cfs)
Diver
sion (%)
B
s
(ft)
s
B
x
?
?
1
=
2
2
U
u
?
2
=
2
2
U
'u
?
?
1
=
3
3
U
u
?
2
=
3
2
3
U
'uu
?
3
=
3
2
U
'vu
?
4
=
3
2
U
vu
25 8.91 1.122 1.111 0.011 1.330 1.292 0.030 0.0029 0.0041
25 3.01 1.119 1.106 0.012 1.310 1.265 0.037 0.0055 0.0016
39 6.13 1.336 1.281 0.056 1.871 1.710 0.129 0.0216 0.0067
40 8.92 1.268 1.213 0.055 1.732 1.554 0.145 0.0204 0.0144
40 3.01 1.341 1.291 0.050 1.874 1.717 0.124 0.0215 0.0055
54 8.91 1.617 1.455 0.162 2.737 2.195 0.421 0.0633 0.0403
54 3.01 1.839 1.692 0.147 3.333 2.740 0.475 0.0803 0.0222
55 5.97 1.785 1.620 0.165 3.184 2.657 0.413 0.0615 0.0376
167
Appendix 5.4  Components of ? and ? just downstream of weir for 4H:1V side slopes
Diver
sion
(%)
Q
u
(cfs)
?
?
1
=
2
2
U
u
?
2
=
2
2
U
'u
?
?
1
=
3
3
U
u
?
2
=
3
2
3
U
'uu
?
3
=
3
2
U
'vu
?
4
=
3
2
U
vu
25 3.0 1.262 1.238 0.024 1.692 1.618 0.0652 0.0077 0.0017
25 6.0 1.140 1.104 0.036 1.394 1.267 0.122 0.0355 0.0011
25 9.0 1.297 1.208 0.089 1.823 1.538 0.275 0.0562 0.0057
40 3.0 1.704 1.631 0.074 2.837 2.581 0.211 0.0304 0.0040
40 6.0 1.504 1.301 0.203 2.393 1.794 0.576 0.1278 0.0080
40 9.1 1.561 1.448 0.113 2.501 2.131 0.327 0.0739 0.0083
53 3.0 2.513 2.322 0.191 6.016 4.995 0.817 0.1347 0.0234
53 6.1 1.989 1.833 0.156 3.828 3.164 0.531 0.1683 0.0080
53 9.1 2.137 1.882 0.255 4.366 3.356 0.944 0.2023 0.0137
53 2.9 2.524 2.309 0.215 5.844 4.749 0.880 0.1793 0.0051
55 6.0 1.908 1.763 0.145 3.800 3.211 0.481 0.0831 0.0108
169
AP
P
E
NDIX
6

M
OMENTUM
AND
ENERG
Y
BALANCES
Appendix
6.1

M
om
entum
b
alance
f
o
r
C
ase
A
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
A
?
2
?
?
QU
h
()
2
1
2
h
h
gA
?
?
?
F
M
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
4
/s
2
)
(f
t
4
/s
2
)
(
ft
)
(
ft
4
/s
2
)(
f
t
4
/s
2
)(
f
t
)
0.0
0
.986
4.284
5.361
1.69
5.361
1.2449

1
.158
0.051

0
.003
4.3
0
.991
4.040
4.584
1.64
4.730
1.2511

0
.585
0.066

0
.002
10.2
0
.997
4.040
4.222
1.52
4.356
1.2542

0
.364
0.078
0.001
17.3
1
.007
4.068
3.681
1.33
3.747
1.2561

0
.255
0.052

0
.001
22.3
1
.022
4.017
3.414
1.29
3.564
1.2574
Appendix
6.2

E
nergy
b
alance
f
o
r
C
ase
A
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
V
A
?
2
?
g
U
2
2
?
hH
f
h
E
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
5
/s
3
)
(f
t)
(f
t)
(f
t)
(f
t)
(f
t)
0.0
0
.986
4.284
7.476
3.18
0.0271
1.2449
1.2720
0.0003

0
.001
4.3
0
.991
4.040
5.555
2.86
0.0220
1.2511
1.2731
0.0004

0
.001
10.2
0
.997
4.040
4.854
2.54
0.0192
1.2542
1.2734
0.0004
0.002
17.3
1
.007
4.068
3.764
1.99
0.0146
1.2561
1.2707
0.0003

0
.000
22.3
1
.022
4.017
3.282
1.88
0.0132
1.2574
1.2706
170
Appendix
6.3

M
om
entum
b
alance
f
o
r
C
ase
B
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
A
?
2
?
?
QU
h
()
2
1
2
h
h
gA
?
?
?
F
M
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
4
/s
2
)
(f
t
4
/s
2
)
(
ft
)
(
ft
4
/s
2
)(
f
t
4
/s
2
)(
f
t
)
0.0
0
.853
1.418
0.764
1.79
0.764
1.1205

0
.092
0.008

0
.000
4.4
0
.860
1.345
0.686
1.81
0.721
1.1211

0
.062
0.011
0.000
10.3
0
.865
1.367
0.600
1.54
0.610
1.1215

0
.109
0.012

0
.000
17.4
0
.871
1.353
0.523
1.39
0.543
1.1222

0
.129
0.008

0
.001
22.3
0
.893
1.342
0.474
1.32
0.500
1.1230
Appendix
6.4

E
nergy
b
alance
f
o
r
C
ase
B
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
V
A
?
2
?
g
U
2
2
?
hH
f
h
E
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
5
/s
3
)
(f
t)
(f
t)
(f
t)
(f
t)
(f
t)
0.0
0
.853
1.418
0.445
3.48
0.0049
1.1205
1.1254
0.0001

0
.000
4.4
0
.860
1.345
0.353
3.31
0.0043
1.1211
1.1254
0.0001
0.001
10.3
0
.865
1.367
0.285
2.59
0.0033
1.1215
1.1248
0.0001

0
.000
17.4
0
.871
1.353
0.228
2.17
0.0027
1.1222
1.1249
0.0001

0
.000
22.3
0
.893
1.342
0.192
2.01
0.0023
1.1230
1.1253
171
Appendix
6.5

M
om
entum
b
alance
f
o
r
C
ase
C
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
A
?
2
?
?
QU
h
()
2
1
2
h
h
gA
?
?
?
F
M
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
4
/s
2
)
(f
t
4
/s
2
)
(
ft
)
(
ft
4
/s
2
)(
f
t
4
/s
2
)(
f
t
)
0.0
0
.899
6.403
9.224
1.14
9.224
1.1603

0
.593
0.158

0
.008
4.4
0
.906
6.278
8.503
1.11
9.736
1.1639

0
.050
0.220

0
.000
10.2
0
.912
6.368
8.537
1.09
9.500
1.1642
0.135
0.263
0.000
17.4
0
.917
6.458
8.642
1.08
9.352
1.1634

0
.310
0.173

0
.002
22.2
0
.934
6.369
8.210
1.08
9.136
1.1652
Appendix
6.6

E
nergy
b
alance
f
o
r
C
ase
C
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
V
A
?
2
?
g
U
2
2
?
hH
f
h
E
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
5
/s
3
)
(f
t)
(f
t)
(f
t)
(f
t)
(f
t)
0.0
0
.899
6.403
14.218
1.40
0.0345
1.1603
1.1948
0.0010

0
.005
4.4
0
.906
6.278
12.170
1.30
0.0345
1.1639
1.1984
0.0013
0.000
10.2
0
.912
6.368
11.975
1.24
0.0325
1.1642
1.1967
0.0016
0.000
17.4
0
.917
6.458
12.047
1.22
0.0313
1.1634
1.1947
0.0010

0
.001
22.2
0
.934
6.369
11.050
1.23
0.0300
1.1652
1.1952
172
Appendix
6.7

M
om
entum
b
alance
f
o
r
C
ase
D
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
A
?
2
?
?
QU
h
()
2
1
2
h
h
gA
?
?
?
F
M
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
4
/s
2
)
(f
t
4
/s
2
)
(
ft
)
(
ft
4
/s
2
)(
f
t
4
/s
2
)(
f
t
)
7.4
0
.993
3.962
4.075
1.52
4.448
1.2537

0
.244
0.058

0
.000
12.5
0
.992
4.033
4.008
1.44
4.223
1.2550

0
.446
0.144
0.000
25.5
1
.015
4.195
3.719
1.27
3.622
1.2573
0.000
0.127

0
.000
37.2
1
.007
4.138
3.509
1.22
3.512
1.2573

0
.039
0.128

0
.000
48.9
1
.011
4.094
3.317
1.19
3.392
1.2575

0
.261
0.123

0
.001
60.6
1
.039
4.121
3.133
1.15
3.161
1.2588
Appendix
6.8

E
nergy
b
alance
f
o
r
C
ase
D
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
V
A
?
2
?
g
U
2
2
?
hH
f
h
E
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
5
/s
3
)
(f
t)
(f
t)
(f
t)
(f
t)
(f
t)
7.4
0
.993
3.962
4.428
2.43
0.0189
1.2537
1.2726
0.0003

0
.000
12.5
0
.992
4.033
4.291
2.22
0.0174
1.2550
1.2724
0.0008
0.001
25.5
1
.015
4.195
3.664
1.80
0.0132
1.2573
1.2705
0.0007
0.000
37.2
1
.007
4.138
3.270
1.64
0.0123
1.2573
1.2696
0.0007
0.000
48.9
1
.011
4.094
2.924
1.53
0.0113
1.2575
1.2688
0.0006

0
.000
60.6
1
.039
4.121
2.562
1.42
0.0097
1.2588
1.2685
173
Appendix
6.9

M
om
entum
b
alance
f
o
r
C
ase
F
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
A
?
2
?
?
QU
h
()
2
1
2
h
h
gA
?
?
?
F
M
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
4
/s
2
)
(f
t
4
/s
2
)
(
ft
)
(
ft
4
/s
2
)(
f
t
4
/s
2
)(
f
t
)
2.5
0
.776
2.765
3.995
1.99
4.304
1.1920

0
.6095
0.0642
0.000
8.1
0
.776
2.546
2.887
1.70
3.670
1.1970

0
.7260
0.0646

0
.001
13.6
0
.773
2.618
2.534
1.40
3.045
1.2030

0
.6237
0.0632

0
.002
19.2
0
.787
2.791
2.430
1.22
2.570
1.2080
Appendix
6.10

E
nergy
b
alance
f
o
r
C
ase
F
Dis
t
anc
e
D
epth
dA
u
A
?
dA
u
V
A
?
2
?
g
U
2
2
?
hH
f
h
E
h
?
(f
t)
(f
t)
(cf
s
)
(
f
t
5
/s
3
)
(f
t)
(f
t)
(f
t)
(f
t)
(f
t)
2.5
0
.776
2.765
5.588
3.83
0.0338
1.1920
1.2258
0.0005

0
.001
8.1
0
.776
2.546
3.740
3.28
0.0290
1.1970
1.2260
0.0005
0.003
13.6
0
.773
2.618
2.718
2.17
0.0194
1.2030
1.2224
0.0005
0.000
19.2
0
.787
2.791
2.336
1.63
0.0137
1.2080
1.2217
175
APPENDIX 7  DATA FOR DIVERSION CULVERTS
Appendix 7.1  Results for diversion culverts with three barrels, unsubmerged
flow
Upstream
discharge
Diversion Upstream
head on
weir
Down
stream
head
on weir
Down
stream
Froude
number
Down
stream
weir
Froude
number
Ratio of
critical
depth to
depth at
culvert
outlet
Loss coeffi
cient from
0to2
Q
u
Q
w
h
u
h
d
F
d
Fw
d
?
c
/?
4
K
E
(cfs) (cfs) (ft) (ft)
1.6 0.560 0.324 0.330 0.113 0.143 1 0.642
1.6 0.438 0.277 0.283 0.128 0.171 1 0.751
1.6 0.303 0.215 0.221 0.149 0.218 1 0.804
1.6 0.173 0.148 0.154 0.174 0.294 1 1.050
3.2 0.574 0.336 0.342 0.178 0.223 1 0.876
3.2 0.459 0.291 0.297 0.194 0.255 1 1.006
3.2 0.320 0.229 0.235 0.218 0.312 1 1.213
3.2 0.185 0.162 0.168 0.249 0.406 1 1.823
6.6 0.482 0.317 0.323 0.278 0.356 1 1.943
6.6 0.376 0.274 0.280 0.299 0.401 1 2.449
6.6 0.273 0.230 0.236 0.323 0.461 1 3.360
6.6 0.185 0.185 0.191 0.350 0.542 1 4.758
9.1 0.412 0.306 0.312 0.338 0.437 1 3.476
9.1 0.355 0.282 0.288 0.351 0.467 1 4.023
9.1 0.257 0.241 0.247 0.376 0.528 1 5.597
9.1 0.180 0.207 0.213 0.398 0.592 1 7.950
176
Appendix 7.2  Results for diversion culverts with three barrels, submerged flow
Upstream
discharge
Diversion Upstream
head on
weir
Down
stream
head
on weir
Down
stream
Froude
number
Down
stream
weir
Froude
number
Ratio of
critical
depth to
depth at
culvert
outlet
Loss coeffi
cient from
0to2
Q
u
Q
w
h
u
h
d
F
d
Fw
d
?
c
/?
4
K
E
(cfs) (cfs) (ft) (ft)
1.6 0.584 0.341 0.347 0.109 0.135 0.887 0.816
1.6 0.187 0.187 0.193 0.162 0.251 0.583 1.838
3.2 0.546 0.333 0.339 0.179 0.225 0.983 1.080
3.2 0.188 0.192 0.198 0.236 0.360 0.645 3.145
6.5 0.457 0.317 0.323 0.277 0.355 0.946 2.257
6.6 0.190 0.212 0.218 0.334 0.492 0.703 6.273
9.1 0.408 0.315 0.321 0.334 0.427 0.923 3.734
9.2 0.188 0.224 0.230 0.388 0.560 0.760 8.731
Appendix 7.3  Results for diversion culverts with two barrels, unsubmerged flow
Upstream
discharge
Diversion Upstream
head on
weir
Down
stream
head
on weir
Down
stream
Froude
number
Down
stream
weir
Froude
number
Ratio of
critical
depth to
depth at
culvert
outlet
Loss coeffi
cient from
0to2
Q
u
Q
w
h
u
h
d
F
d
Fw
d
?
c
/?
4
K
E
(cfs) (cfs) (ft) (ft)
1.6 0.381 0.334 0.337 0.121 0.153 1 0.672
1.6 0.273 0.267 0.270 0.139 0.190 1 0.722
1.6 0.194 0.214 0.217 0.156 0.230 1 0.850
3.2 0.372 0.329 0.332 0.186 0.236 1 0.776
3.2 0.261 0.263 0.266 0.209 0.287 1 0.997
3.2 0.190 0.217 0.220 0.228 0.335 1 1.300
6.6 0.314 0.317 0.320 0.283 0.363 1 1.894
6.6 0.169 0.229 0.232 0.326 0.470 1 3.503
9.1 0.284 0.312 0.315 0.337 0.435 1 3.065
9.1 0.178 0.252 0.255 0.371 0.516 1 5.044
177
Appendix 7.4  Results for diversion culverts with two barrels, submerged flow
Upstream
discharge
Diversion Upstream
head on
weir
Down
stream
head
on weir
Down
stream
Froude
number
Down
stream
weir
Froude
number
Ratio of
critical
depth to
depth at
culvert
outlet
Loss coeffi
cient from
0to2
Q
u
Q
w
h
u
h
d
F
d
Fw
d
?
c
/?
4
K
E
(cfs) (cfs) (ft) (ft)
1.6 0.202 0.296 0.299 0.137 0.180 0.473 0.662
3.2 0.178 0.284 0.287 0.205 0.274 0.454 1.109
6.6 0.182 0.298 0.301 0.294 0.386 0.461 2.692
9.1 0.198 0.312 0.315 0.338 0.437 0.482 3.764